Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations 9783110641851, 9783110641240

Table of contents :
Contents
Preface
Notation
Introduction
1. Preliminaries
2. Almost periodic type solutions of abstract Volterra integro-differential equations
3. Almost automorphic type solutions of abstract Volterra integro-differential equations
Bibliography
Index

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Marko Kostić Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations

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Marko Kostić

Almost Periodic and Almost Automorphic Solutions to Integro-Differential Equations |

Author Prof. Dr. Marko Kostić University of Novi Sad Faculty of Technical Sciences Trg D. Obradovića 6 21125 Novi Sad Serbia [email protected]

ISBN 978-3-11-064124-0 e-ISBN (PDF) 978-3-11-064185-1 e-ISBN (EPUB) 978-3-11-064125-7 Library of Congress Control Number: 2019935657 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Cover image: awstok / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Contents Preface | IX Notation | XI Introduction | XVII 1 1.1 1.2 1.2.1 1.3 1.4 1.5 2 2.1 2.2 2.3 2.3.1 2.4 2.5 2.6

2.6.1 2.6.2 2.7

Preliminaries | 1 Fundamentals of the theory of operators and integration theory | 1 Multivalued linear operators | 6 Fractional powers | 8 Laplace transform of functions with values in Banach spaces | 9 Operators of fractional differentiation, Mittag-Leffler and Wright functions | 13 Degenerate (a, k)-regularized C-resolvent families | 17 Almost periodic type solutions of abstract Volterra integro-differential equations | 25 Almost periodic functions and asymptotically almost periodic functions | 25 Stepanov almost periodic functions and asymptotically Stepanov almost periodic functions | 30 Weyl almost periodic functions and asymptotically Weyl almost periodic functions | 36 Asymptotically Weyl almost periodic functions | 39 Almost periodic solutions of abstract degenerate first and second order Cauchy problems | 46 Almost periodic solutions of abstract Volterra integro-differential equations | 52 Asymptotically almost periodic solutions and Stepanov asymptotically almost periodic solutions of abstract Volterra integro-differential equations | 60 Stepanov (asymptotically) almost periodic properties of convolution products | 66 Generalized (asymptotically) almost periodic properties of degenerate C-semigroups and degenerate C-cosine functions | 72 Stepanov (asymptotically) almost periodic solutions of abstract fractional semilinear inclusions | 79

VI | Contents 2.8 2.9 2.9.1 2.9.2 2.10 2.10.1 2.10.2 2.11 2.11.1 2.11.2 2.12 2.13 2.13.1 2.14

2.14.1 2.14.2 2.14.3 2.15 2.15.1 2.16 3 3.1 3.2

Subspace (asymptotical) almost periodicity of C-distribution semigroups and C-distribution cosine functions | 86 Asymptotically almost periodic solutions of fractional relaxation inclusions with Caputo derivatives | 99 Subordinated fractional resolvent families with removable singularities at zero | 100 The nonanalyticity of semigroup (T (t))t>0 | 107 The use of fractional powers of multivalued linear operators | 111 Almost periodic solutions of abstract semilinear Cauchy inclusion (78) | 112 Almost periodic solutions of abstract semilinear Cauchy inclusion (103) | 117 Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations | 120 Weyl C (n) -almost periodic properties of convolution products | 130 Weyl-almost periodic properties and asymptotically Weyl-almost periodic properties of degenerate solution operator families | 136 Pseudo-almost periodic solutions of abstract semilinear Cauchy inclusions of first order | 138 On Besicovitch–Doss almost periodic solutions of abstract Volterra integro-differential equations | 141 Besicovitch–Doss C (n) -almost periodic solutions of abstract inhomogeneous Cauchy inclusions | 156 Generalized almost periodic solutions and generalized asymptotically almost periodic solutions of inhomogeneous evolution equations | 158 Generalized almost periodic solutions of inhomogeneous evolution equations | 160 Generalized asymptotically almost periodic solutions of inhomogeneous evolution equations | 165 Almost periodic and asymptotically almost periodic solutions of semilinear evolution equations with Stepanov coefficients | 168 Vector-valued almost periodic ultradistributions | 171 Generalizations of vector-valued almost periodic ultradistributions | 178 Notes and appendices | 182 Almost automorphic type solutions of abstract Volterra integro-differential equations | 193 Almost automorphic functions, asymptotically almost automorphic functions and their generalizations | 193 Two-parameter generalized almost automorphic functions | 200

Contents | VII

3.3 3.4

3.4.1 3.4.2 3.4.3

3.5

3.5.1 3.5.2 3.6

3.6.1 3.6.2 3.7 3.8 3.8.1 3.8.2 3.9 3.9.1 3.9.2 3.9.3 3.10 3.11

Weighted pseudo-almost periodic functions, weighted pseudo-almost automorphic functions and their generalizations | 203 Generalized asymptotically almost periodic and generalized asymptotically almost automorphic solutions of abstract multiterm fractional differential inclusions | 209 k-regularized C-propagation families for (173) | 210 Asymptotical behavior of ki -regularized C-propagation families for (173) | 216 Generalized asymptotically almost periodic and generalized asymptotically almost automorphic solutions of abstract multiterm fractional differential inclusions with Riemann–Liouville derivatives | 223 Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions | 232 Generalized (asymptotically) almost automorphic properties of convolution products | 232 Semilinear Cauchy inclusions | 238 Generalized weighted pseudo-almost periodic solutions and generalized weighted pseudo-almost automorphic solutions of abstract Volterra integro-differential inclusions | 240 Generalized weighted almost periodic (automorphic) properties of convolution products | 241 Weighted pseudo-almost automorphic solutions of semilinear (fractional) Cauchy inclusions | 245 Besicovitch almost automorphic solutions of nonautonomous differential equations of first order | 247 Vector-valued almost automorphic distributions and vector-valued almost automorphic ultradistributions | 256 Almost automorphy of vector-valued distributions | 256 Almost automorphy of vector-valued ultradistributions | 258 Asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized functions | 261 Asymptotical almost periodicity and asymptotical almost automorphy of vector-valued distributions | 262 Asymptotical almost periodicity and asymptotical almost automorphy of vector-valued ultradistributions | 266 An application to systems of ordinary differential equations in distribution and ultradistribution spaces | 271 Examples and applications | 274 Notes and appendices | 289

VIII | Contents Bibliography | 305 Index | 325

Preface The theory of abstract Volterra integro-differential equations is an active field of research of many mathematicians. A great number of papers in the existing literature deals with the almost periodic properties and asymptotically almost periodic properties of abstract non-degenerate Volterra integro-differential equations in Banach spaces. The main aim of this book is to focus on the almost periodic (automorphic) solutions and asymptotically almost periodic (automorphic) solutions of abstract Volterra integro-differential equations that are degenerate in time, in particular those equations whose solutions are governed by (degenerate) solution operator families with removable singularities at zero. At the very beginning, we would like to stress that some of our results are new even for the abstract non-degenerate differential equations of first order whose solutions are governed by strongly continuous semigroups of operators, and also for the abstract non-degenerate differential equations with almost sectorial operators. Fractional calculus and fractional differential equations have recently received great attention in many disciplines, primarily for their invaluable importance in modeling of various phenomena appearing in the fundamental and applied sciences. Concerning the abstract fractional differential equations in Banach spaces, we can freely say that, in the present situation, it would be very difficult to summarize so many important results obtained recently within the framework of only one research monograph. In this one, special emphasis is paid to the analysis of abstract fractional differential equations and inclusions with multivalued linear operators, as well as abstract semilinear Cauchy problems with integer and fractional order time-derivatives. Throughout the book, we use only Caputo fractional derivatives, Weyl–Liouville fractional derivatives and Riemann–Liouville fractional derivatives. This monograph consists of three individual chapters, which are further broken down into many sections and subsections. As in my previous monographs [245–247], the numbering of theorems, propositions, lemmas, corollaries, definitions, etc., is done by chapter and section, and the bibliography is by author in alphabetic order. We would like to note that the rigorous mathematical knowledge is not a prerequisite to reading this book. The text is accessible to the readers familiar with the theory of functions of one complex variable, fundamentals of functional analysis and integration theory, and the basic theory of abstract differential equations in Banach spaces. Concerning the target audience and the groups of people the content of the book would interest, we would like to isolate PhD students in mathematics, researchers in abstract partial differential equations and experts from all areas of functional analysis. Our attempt has not been to cite all references and research papers valuable for the development of this theory, but rather we have tested the reference list to avoid any form of plagiarism. The book is not intended to be exhaustively complete. https://doi.org/10.1515/9783110641851-201

X | Preface I would like to say a big thanks to my family, friends, colleagues and advisor Prof. S. Pilipović (Novi Sad, Serbia) for unstinting support of my work. My sincere appreciation also goes to V. Fedorov (Chelyabinsk, Russia), C.-C. Chen (Taichung, Taiwan), M. Li (Chengdu, China), B. Chaouchi (Khemis Miliana, Algeria), D. Velinov, P. Dimovski, B. Prangoski (Skopje, Macedonia), R. Ponce (Talca, Chile), C. Lizama (Santiago, Chile), P. J. Miana, L. Abadias, J. E. Galé (Zaragoza, Spain), M. Murillo-Arcila, J. A. Conejero, A. Peris, J. Bonet (Valencia, Spain), C. Bianca (Paris, France), E. M. A. ElSayed (Alexandria, Egypt), M. S. Moslehian (Mashhad, Iran), A. Arbi (Tunis, Tunisia), C.-C. Kuo (New Taipei City, Taiwan), V. Valmorin (Pointe-à-Pitre, Guadeloupe), D. N. Cheban (Chisinau, Moldova), V. Keyantuo (Rio Piedras Campus, Puerto Rico, USA), T. Diagana (Huntsville, USA) and G. M. N’Guérékata (Baltimore, USA). Loznica/Novi Sad January, 2018

Marko Kostić

Notation ℕ, ℤ, ℚ, ℝ, ℂ: the natural, integer, rational, real, and complex numbers, respectively.

For any s ∈ ℝ, we denote ⌊s⌋ = sup{l ∈ ℤ : s ⩾ l} and ⌈s⌉ = inf{l ∈ ℤ : s ⩽ l}.

Re z, Im z: the real and imaginary parts of a complex number z ∈ ℂ; |z|: the modulus of z, arg(z): the argument of a complex number z ∈ ℂ ∖ {0}.

ℂ+ = {z ∈ ℂ : Re z > 0}.

B(z0 , r) = {z ∈ ℂ : |z − z0 | ⩽ r} (z0 ∈ ℂ, r > 0). Σα = {z ∈ ℂ ∖ {0} : | arg(z)| < α}, α ∈ (0, π].

card(G): the cardinality of G. ℕ0 = ℕ ∪ {0}.

ℕn = {1, . . . , n}.

ℕ0n = {0, 1, . . . , n}.

ℝn : the real Euclidean space, n ⩾ 2.

If α = (α1 , . . . , αn ) ∈ ℕn0 is a multiindex, then we denote |α| = α1 + ⋅ ⋅ ⋅ + αn . α

α

xα = x1 1 ⋅ ⋅ ⋅ xn n for x = (x1 , . . . , xn ) ∈ ℝn and α = (α1 , . . . , αn ) ∈ ℕn0 . α

α

f (α) := 𝜕|α| f /𝜕x1 1 ⋅ ⋅ ⋅ 𝜕xn n ; Dα f := (−i)|α| f (α) .

If (X, τ) is a topological space and F ⊆ X, then the interior, closure, boundary, and complement of F with respect to X are denoted by int(F) (or F ∘ ), F, 𝜕F and F c , respectively. If Z is a vector space over the field 𝔽 ∈ {ℝ, ℂ}, then for each nonempty subset F of Z by span(F) we denote the smallest linear subspace of Z which contains F. X: complex Banach space. L(E, X): the space of all continuous linear mappings between Banach spaces E and X, L(X) = L(X, X).

X ∗ : the dual space of X.

A: a linear operator on X. 𝒜: a multivalued linear operator on X (MLO).

C: an injective continuous linear operator on X, if not stated otherwise. If F is a subspace of X, then we denote by 𝒜|F the part of 𝒜 in F. 𝒜∗ : the adjoint operator of 𝒜.

D(𝒜), R(𝒜), ρ(𝒜), σ(𝒜): the domain, range, resolvent set and spectrum of 𝒜.

N(𝒜) or Kern(𝒜): the null space of 𝒜. 𝒜: the closure of 𝒜.

σp (𝒜): the point spectrum of 𝒜. https://doi.org/10.1515/9783110641851-202

XII | Notation ρC (𝒜): the C-resolvent set of 𝒜.

χΩ (⋅): the characteristic function, defined to be identically one on Ω and zero elsewhere.

Γ(⋅): the Gamma function.

If α > 0, then gα (t) = t α−1 /Γ(α), t > 0; g0 (t) ≡ the Dirac delta distribution.

If 1 ⩽ p < ∞, (X, ‖ ⋅ ‖) is a complex Banach space and (Ω, ℛ, μ) is a measure space, then Lp (Ω, X, μ) denotes the space consisting of those strongly μ-measurable functions f : Ω → X such that ‖f ‖p := (∫Ω ‖f (⋅)‖p dμ)1/p is finite; Lp (Ω, μ) ≡ Lp (Ω, ℂ, μ).

L∞ (Ω, X, μ): the space consisting of all strongly μ-measurable, essentially bounded functions. ‖f ‖∞ = ess supt∈Ω ‖f (t)‖, the norm of a function f ∈ L∞ (Ω, X, μ).

Lp (Ω : X) ≡ Lp (Ω, X) ≡ Lp (Ω, X, μ), if p ∈ [1, ∞] and μ = m is the Lebesgue measure; Lp (Ω) ≡ Lp (Ω : ℂ).

Lploc (Ω : X): the space consisting of those Lebesgue measurable functions u(⋅) such that, for every bounded open subset Ω󸀠 of Ω, one has u|Ω󸀠 ∈ Lp (Ω󸀠 : X); Lploc (Ω) ≡ Lploc (Ω : ℂ) (1 ⩽ p ⩽ ∞).

Assume that I = ℝ or I = [0, ∞). By Cb (I : X) we denote the space consisting of all bounded continuous functions from I into X; C0 ([0, ∞) : X) denotes the closed subspace of Cb (I : X) consisting of functions vanishing at infinity. By BUC(I : X) we denote the space consisting of all bounded uniformly continuous functions from I to X. The sup-norm turns these spaces into Banach spaces. The abbreviation ACloc ([0, ∞) : X) (BVloc ([0, ∞) : X)) stands for the space of all X-valued functions that are absolutely continuous (of bounded variation) on any closed subinterval of [0, ∞). ACloc ([0, ∞)) ≡ ACloc ([0, ∞) : ℂ), BVloc ([0, ∞)) ≡ BVloc ([0, ∞) : ℂ).

BV[0, T], BVloc ([0, τ)), BVloc ([0, τ) : X): the spaces of functions of bounded variation.

C k (Ω : X): the space of k-times continuously differentiable functions (k ∈ ℕ0 ) from a nonempty subset Ω ⊆ ℂ into X; C(Ω : X) ≡ C 0 (Ω : X). 𝒟 = C0∞ (ℝ), ℰ = C ∞ (ℝ) and 𝒮 = 𝒮 (ℝ): the Schwartz spaces of test functions.

If 0 ≠ Ω ⊆ ℝ, then by 𝒟Ω we denote the subspace of 𝒟 consisting of those functions φ ∈ 𝒟 for which supp(φ) ⊆ Ω; 𝒟0 ≡ 𝒟[0,∞) .

󸀠 𝒟󸀠 (E) := L(𝒟, E): the space consisting of all continuous linear functions 𝒟 → E; 𝒟Ω (E) 󸀠 denotes the subspace of 𝒟 (E) containing E-valued distributions whose supports 󸀠 are contained in Ω; 𝒟0󸀠 (E) := 𝒟[0,∞) (E).

(Mp ): a sequence of positive real numbers satisfying M0 = 1 and the following conditions: Mp2 ⩽ Mp+1 Mp−1 ,

p ∈ ℕ,

(M.1)

Notation

Mp ⩽ AH p sup Mi Mp−i , 0⩽i⩽p

∞

∑

p ∈ ℕ, for some A, H > 1,

Mp−1

p=1

Mp

| XIII

(M.2) (M.3󸀠 )

< ∞.

Condition (M.3) will be used occasionally: ∞

sup ∑

p∈ℕ q=p+1

Mq−1 Mp+1 pMp Mq

< ∞.

(M.3)

𝒟(Mp ) : the space of Beurling ultradifferentiable functions; 𝒟{Mp } : the space of Roumieu

ultradifferentiable functions.

𝒟 (X) := L(𝒟∗ : X): the space of all X-valued ultradistributions of ∗-class. 󸀠∗

If k ∈ ℕ, p ∈ [1, ∞] and Ω is an open nonempty subset of ℝn , then we denote by W k,p (Ω : X) the Sobolev space which consists of those X-valued distributions u ∈ 𝒟󸀠 (Ω : X) such that, for every i ∈ ℕ0k and for every α ∈ ℕn0 with |α| ⩽ k, one has Dα u ∈ Lp (Ω : X); H k (ℝn : X) ≡ W k,2 (ℝn : X).

k,p (Ω : X): the space of those X-valued distributions u ∈ 𝒟󸀠 (Ω : X) such that, for Wloc every bounded open subset Ω󸀠 of Ω, one has u|Ω󸀠 ∈ W k,p (Ω󸀠 : X).

ℱ , ℱ −1 : the Fourier transform and its inverse.

If a function K(t) satisfies condition (P1) stated in Section 1.3, then we denote abs(K) = ̃ inf{Re λ : K(λ) exists}.

L1loc ([0, ∞)), resp. L1loc ([0, τ)): the space of scalar-valued locally integrable functions on [0, ∞), resp. [0, τ).

Jtα : Riemann–Liouville fractional integral of order α > 0.

Dαt : Riemann–Liouville fractional derivative of order α > 0. Dαt : Caputo fractional derivative of order α > 0.

W+α : Weyl fractional derivative of order α. γ

Dt,+ : Weyl–Liouville fractional derivative.

Eα,β (z): Mittag-Leffler function (α > 0, β ∈ ℝ); Eα (z) ≡ Eα,1 (z). Ψγ (t): Wright function (0 < γ < 1).

℘(R): the set consisting of all subgenerators of an (a, k)-regularized C-resolvent family (R(t))t∈[0,τ) .

a∗,n (t): the nth convolution power of function a(t). δj,l : Kronecker’s delta.

supp(f ): the support of function f (t).

Let I = ℝ or I = [0, ∞), and let 1 ⩽ p < ∞.

XIV | Notation AP(I : X): the Banach space consisting of all almost periodic functions from the interval I into X, equipped with the sup-norm. AAP([0, ∞) : X): the Banach space consisting of all asymptotically almost periodic functions from the interval [0, ∞) into X, equipped with the sup-norm. AP(I × Y : X): the vector space consisting of all almost periodic functions f : I × Y → X. AAP([0, ∞) × Y : X): the vector space consisting of all asymptotically almost periodic functions f : I × Y → X.

LpS (I : X): the Banach space of all Stepanov p-bounded functions, equipped with the Stepanov norm. APSp (I : X): the Banach space of all Stepanov p-almost periodic functions I → X, equipped with the Stepanov norm.

AAPSp ([0, ∞) : X): the Banach space of all asymptotically Stepanov p-almost periodic functions I → X, equipped with the Stepanov norm.

AAPSp ([0, ∞) × Y : X): the vector space consisting of all Stepanov p-almost periodic functions f : I × Y → X. C (n) − AP(I : X): the vector space of all C (n) -almost periodic functions I → X (n ∈ ℕ).

C (n) − APSp (I : X): the vector space of all Stepanov C (n) -p-almost periodic functions I → X (n ∈ ℕ).

C (n) − AAPSp ([0, ∞) : X): the vector space of all asymptotically Stepanov C (n) -almost periodic functions (n ∈ ℕ).

p e − Wap (I : X): the vector space of all equi-Weyl-p-almost periodic functions I → X. p Wap (I : X): the vector space of all Weyl-p-almost periodic functions I → X.

W0p ([0, ∞) : X) and e − W0p ([0, ∞) : X): the vector spaces consisting of all Weyl-p-vanishing functions and equi-Weyl-p-vanishing functions, respectively. p e − C (n) − Wap (I : X): the vector space consisted of all equi-Weyl-p-C (n) -almost periodic functions I → X (n ∈ ℕ). p C (n) −Wap (I : X): the vector space consisted of all Weyl-p-C (n) -almost periodic functions I → X (n ∈ ℕ).

PAP(ℝ × Y : X): the vector space consisting of all pseudo-almost periodic functions f : ℝ × Y → X.

Bp (I : X) and Bp (I : X): the vector spaces consisting of all Besicovitch–Doss-p-almost periodic functions I → X and all Besicovitch-p-almost periodic functions I → X, respectively.

Dp (I : X): the class consisting of all Doss-p-almost periodic functions I → X.

Bp0 ([0, ∞) : X): the vector space consisting of all Besicovitch-p-vanishing functions.

C (n) − Bp (I : X): the vector space consisting of all Besicovitch-p-C (n) -almost periodic functions I → X (n ∈ ℕ).

Notation | XV

C (n) − Bp (I : X): the vector space consisting of all Besicovitch–Doss-p-C (n) -almost periodic functions I → X (n ∈ ℕ). ANP0 (I : X): the linear span of almost anti-periodic functions I 󳨃→ X; ANP(I : X): the linear closure of ANP0 (I : X) in AP(I : X). AA(ℝ : X) and AAc (ℝ : X): the Banach spaces consisting of all almost automorphic functions and compactly almost automorphic functions, respectively, equipped with the sup-norm. C (n) − AA(ℝ : X): the vector space consisting of all C (n) -almost automorphic functions. C (n) − AAA([0, ∞) : X): the vector space consisting of all asymptotically C (n) -almost automorphic functions. C (n) − AASp (ℝ : X): the vector space consisting of all Stepanov-p-C (n) -almost automorphic functions. C (n) − AAASp ([0, ∞) : X): the vector space consisting of all asymptotically Stepanov-pC (n) -almost automorphic functions. W p AA(ℝ : X): the vector space consisting of all Weyl-p-almost automorphic functions.

C (n) − W p AA(ℝ : X): the vector space consisting of all Weyl-p-C (n) -almost automorphic functions.

Bp AA(ℝ : X): the vector space consisting of all Besicovitch p-almost automorphic functions. WPAP(ℝ, X, ρ1 , ρ2 ): the vector space consisting of all double-weighted pseudo-almost periodic functions. WPAA(ℝ, X, ρ1 , ρ2 ): the vector space consisting of all double-weighted pseudo-almost automorphic functions. Sp WPAP(ℝ, X, ρ1 , ρ2 ) (Sp WPAA(ℝ, X, ρ1 , ρ2 )): the vector space of all double-weighted Stepanov p-pseudo almost periodic (automorphic) functions. Sp WPAP(ℝ × Y, X, ρ1 , ρ2 ) (Sp WPAA(ℝ × Y, X, ρ1 , ρ2 )): the vector space of all doubleweighted Sp -pseudo almost periodic (automorphic) functions. 𝒫p,k (I : X): the vector space consisted of all Bloch (p, k)-periodic functions. 𝒜P p,k ([0, ∞) : X): the vector space of all asymptotically (p, k)-Bloch-periodic X-valued

functions.

B󸀠ap (X)

(B󸀠∗ ap (X)): the vector space consisting of all almost periodic X-valued distributions (ultradistributions of ∗-class).

B󸀠aa (X) (B󸀠∗ aa (X)): the vector space consisting of all almost automorphic X-valued distributions (ultradistributions of ∗-class). B󸀠aap (X) (B󸀠∗ aap (X)): the vector space consisting of all asymptotically almost periodic X-valued distributions (ultradistributions of ∗-class).

XVI | Notation B󸀠aaa (X) (B󸀠∗ aaa (X)): the vector space consisting of all asymptotically almost automorphic X-valued distributions (ultradistributions of ∗-class).

B󸀠+,0 (X) (B󸀠∗ +,0 (X)): the vector space consisting of all bounded vector-valued distributions (ultradistributions of ∗-class) tending to zero at plus infinity.

Introduction Nearly everyone who works professionally in different fields of mathematical analysis is familiar with the concepts of almost periodic functions and almost automorphic functions. There is an enormous literature devoted to the study of various almost periodic (automorphic) properties and asymptotically almost periodic (automorphic) properties of abstract Volterra integro-differential equations in Banach spaces. In the present state of our knowledge, besides the author’s recent papers, we can quote only two research papers concerning almost periodic (automorphic) and asymptotically almost periodic (automorphic) properties of abstract degenerate integro-differential equations: the paper [393] by Q.-P. Vu (pertaining to the study of asymptotical almost periodicity) and the paper [293] by N. T. Lan; in both papers, the authors have dealt with abstract degenerate differential equations of first order. We would like to point out that a great number of our results seems to be new even for abstract nondegenerate differential equations with almost sectorial operators, which naturally appear in the studies of the heat equation and elliptic higher-order differential equations in Hölder spaces (see [341, 342, 395]). The study of almost periodic strongly continuous semigroups in Banach spaces has been initiated by H. Bart and S. Goldberg in [43]. Concerning Volterra integrodifferential equations, it should be mentioned that, in [349, Section 11.4], J. Prüss has analyzed the almost periodic solutions, Stepanov almost periodic solutions and asymptotically almost periodic solutions of the following abstract nondegenerate Cauchy problem ∞ 󸀠

∞ 󸀠

u (t) = ∫ A0 (s)u (t − s) ds + ∫ dA1 (s)u(t − s) + f (t), 0

t ∈ ℝ,

0

where A0 ∈ L1 ([0, ∞) : L(Y, X)), A1 ∈ BV([0, ∞) : L(Y, X)), X and Y are Banach spaces such that Y is densely and continuously embedded into X. To the best knowledge of the author, this was the first work where the existence and uniqueness of various almost periodic solutions of abstract nondegenerate Volterra integro-differential equations have been deliberated. Only a year after the appearing the monograph [349], Q.-P. Vu [394] enquired into the almost periodicity of the abstract Cauchy problems like ∞

u󸀠 (t) = Au(t) + ∫ dBu(τ)u(t − τ) + f (t),

t ∈ ℝ,

0

where A is a closed linear operator acting on a Banach space X, (B(t))t⩾0 is a family of closed linear operators on X and f : ℝ → X is continuous. We should mention also the paper [11] by R. Agarwal, B. de Andrade and C. Cuevas, where the authors https://doi.org/10.1515/9783110641851-203

XVIII | Introduction have considered various types of periodicity for solutions of the following fractional differential equation: Dαt u(t) = Au(t) + Dα−1 t f (t, u(t)),

t ∈ ℝ,

(1)

where 1 < α < 2, Dαt u(t) is a Riemann–Liouville fractional type derivative of order α, A : D(A) ⊆ X → X is a linear, densely defined, sectorial operator on a complex Banach space X, and f : ℝ × X → X is a pseudo-almost periodic function satisfying suitable conditions in the space variable x (cf. also [18, 116] and [127]). Regarding the nondegenerate case, coextending investigations of almost automorphic solutions and generalized almost automorphic solutions of such equations have been carried out by numerous authors so far. Now we would like to explain, with a lot of technical details and maybe a little bit lengthily, the organization of material and results presented in this monograph. In Chapter 1, which consists of five individual sections, we collect some preliminary results that we will need throughout the remaining part of the book. Section 1.1 is devoted to the recapitulation of some rudimentary functional analytical methods and fixing notations. We recall the basic things about vector-valued functions, closed linear operators and integration in Banach spaces, fixed point theorems and Sobolev spaces. In Section 1.2, we look upon the multivalued linear operators in Banach spaces; for more details about this topic, we refer the reader to the monographs by R. Cross [112] and A. Favini, A. Yagi [188]. The main aim of Section 1.3 is to remind readers of the most intriguing properties of Laplace transform of functions with values in Banach spaces. Operators of fractional differentiation, Mittag-Leffler and Wright functions are investigated in Section 1.4. Without any doubt, the most demanding and unfamiliar to our readers will be Section 1.5, where we give a brief recollection of definitions and results about degenerate (a, k)-regularized C-resolvent families subgenerated by multivalued linear operators. Before we go any further, we would like to recall the basic facts about this general class of operator families. For the beginning, just a few historical remarks about the class of nondegenerate (a, k)-regularized C-resolvent families are in order. The notion of a resolvent family, which is an (a, k)-regularized C-resolvent family with k(t) ≡ 1 and C ≡ I, where I stands for the identity operator, was introduced by G. Da Prato and M. Iannelli [125] in 1980. A fairly complete theory of resolvent families and their applications to real word phenomena was presented in the monograph [349] by J. Prüss. Integrated solution operator families (k(t) ≡ t n /n!, C ≡ I) were introduced for the first time by W. Arendt and H. Kellermann [30] in 1987, C-regularized resolvent families (k(t) ≡ 1, C ∈ L(X) injective) were introduced for the first time by M. Li, Q. Zheng and J. Zhang [300] in 2007, while k-convoluted resolvent families (k(t) kernel, C ≡ I) were introduced by M. Kim [240] in 1995 and further studied by C. Lizama [313] in 2000; cf. also [316]. The notion of a nondegenerate (a, k)-regularized C-resolvent family in the Banach and locally convex space setting was introduced by the author [271, 272]

Introduction

| XIX

in 2009 and 2012, respectively. The class of degenerate (a, k)-regularized C-resolvent families subgenerated by multivalued linear operators has been recently introduced by the author in [275] and thoroughly analyzed in the third chapter of monograph [247]. A strong motivational factor for introduction of a general class of (a, k)-regularized C-resolvent families comes from its universality, as well as numerous applications in mathematical physics and the theory of fractional differential equations. We will mention only a few of them: (i) (see [246, Example 2.1.9]) Let Ap be the realization of the Laplacian with Dirichlet or Neumann boundary conditions on Lp ([0, π]n ), 1 ⩽ p < ∞. We know that Ap generates an exponentially bounded α-times integrated cosine function for every α ⩾ (n − 1)| 21 − p1 |. Assume c ∈ BVloc ([0, ∞)) and m(t) is a bounded creep function with m0 = m(0+) > 0, as well as that there exists a completely positive function c1 (t) such that a(t) = (c1 ∗ c1 )(t), t ⩾ 0 (see [349] for the notion and notation used). Then there exists a completely positive function b(t) such that dm ∗ b = 1. The problem [349, (5.34)] describing heat conduction in materials with memory can be mathematically modeled by the equation u(t) = (a ∗ Ap )(t) + f (t),

t ⩾ 0.

(2)

Applying subordination principles for (a, k)-regularized C-resolvent families, we may conclude that Ap is the integral generator of an exponentially bounded (n−1)| 1 − 1 |

2 p )(t))-regularized resolvent family, where ℒ−1 denotes the (a, 1 ∗ ℒ−1 ( λ1 c1̃ (λ) inverse Laplace transform, as well as that there exists ω > 0 such that Ap is the in-

−⌈ 1 (n−1)| 1 − 1 |⌉

2 p )-regularized tegral generator of an exponentially bounded (a, (ω−Ap ) 2 resolvent family, which has some obvious applications in the study of wellposedness of (2). Subordination principles for (a, k)-regularized C-resolvent families can be also applied in the analysis of the Rayleigh problem of viscoelasticity, which can be generally described by the initial-value problem (cf. [349, (5.45), p. 136]):

t

{ { {ut (t, x) = ∫ da(s)uxx (t − s, x) + f (t, x), t, x > 0, { { 0 { {u(t, 0) = g(t), t > 0; u(0, x) = u0 (x), x > 0,

(RP)

about which we assume that a(t) is a creep function with a1 (t) being log-convex. (ii) (see [246, Section 2.5]) Let k ∈ ℕ, aα ∈ ℂ, 0 ⩽ |α| ⩽ k, aα ≠ 0 for some α with |α| = k, and let 0 < γ < 2. Then (gγ , C)-regularized resolvent families can be essentially applied in the investigation of the well-posedness of the following abstract fractional differential equation: γ

in Lp -spaces.

(α) {Dt u(t, x) = ∑ aα f (t, x) + f (t, x), |α|⩽k { {u(0, x) = u0 (x),

t ⩾ 0, x ∈ ℝn , x ∈ ℝn

XX | Introduction (iii) Perturbation results for degenerate (a, k)-regularized C-resolvent families have been examined in [247, Section 3.7]. The obtained results are applied in the analysis of the following time-dependent perturbation of the Poisson heat equation in the space X = Lp (Ω):

(P)t−d

𝜕 { { { 𝜕t [m(x)v(t, x)] = Δv(t, x) + bv(t, x) + m(x)B(t)v(t, x), { (t, x) ∈ [0, ∞) × 𝜕Ω, {v(t, x) = 0, { m(x)v(0, x) = u (x), x ∈ Ω, 0 {

t ⩾ 0, x ∈ Ω;

where Ω is a bounded domain in ℝn with smooth boundary, b > 0, m(x) ⩾ 0 a. e. x ∈ Ω, m ∈ L∞ (Ω), 1 < p < ∞ and B ∈ C([0, ∞) : L(X)). Subordination principles for degenerate (a, k)-regularized C-resolvent families established in [247] enable one to consider the well-posedness results for the following fractional Poisson heat equation: γ

D [m(x)v(t, x)] = Δv(t, x) + bv(t, x), { { t v(t, x) = 0, { { m(x)v(0, x) = u0 (x), {

t ⩾ 0, x ∈ Ω; (t, x) ∈ [0, ∞) × 𝜕Ω, x ∈ Ω,

with the same suppositions being satisfied and the number γ belonging to the interval (0, 1). (iv) Following the ideas of C. Lizama and his coauthors, we can use degenerate (a, k)-regularized C-resolvent families, with appropriately chosen functions a(t) and k(t), in the analysis of a wide class of abstract multi-term fractional differential inclusions (see, e. g., [127, 236] and references cited therein). For example, using almost the same approach as in [236] (see also Example 3.10.7 below), we can consider the abstract two-term fractional differential inclusion β󸀠

Dαt +1 u(t) + c1 Dt u(t) ∈ 𝒜u(t) + f (t), { (k) u (0) = uk , 󸀠

t ⩾ 0, k = 0, 1,

where c1 > 0, 𝒜 is a closed multivalued linear operator satisfying certain conditions, c < 0, 0 < α󸀠 ⩽ β󸀠 ⩽ 1, and f (t) is a given X-valued function. (v) The class of (degenerate) (a, k)-regularized C-resolvent families is not most general and, in [247], we have also considered the class of (degenerate) (a, k)-regularized (C1 , C2 )-existence and uniqueness resolvent families. The notion of a (g1 , g1 )-regularized (C1 , C2 )-existence and uniqueness resolvent family has a crucial role in the study of well-posedness of the following backward Poisson heat equation in the space Lp (Ω):

(P)b :

𝜕 { { { 𝜕t [m(x)v(t, x)] = −Δv + bv, { {v(t, x) = 0, { {m(x)v(0, x) = u0 (x),

t ⩾ 0, x ∈ Ω;

(t, x) ∈ [0, ∞) × 𝜕Ω, x ∈ Ω,

Introduction

| XXI

where Ω is a bounded domain in ℝn , b > 0, m(x) ⩾ 0 a. e. x ∈ Ω, m ∈ L∞ (Ω) and 1 < p < ∞ (see [247, Example 3.2.23] for more details). For further information about degenerate solution operator families subgenerated by multivalued linear operators, or by pairs of closed linear operators, we refer the reader to the forthcoming monograph [247]. In Definition 1.5.6 and Proposition 1.5.7 (Definition 1.5.8 and Proposition 1.5.9), we introduce for the first time the class of degenerate K-convoluted C-groups (degenerate (a, k)-regularized C-resolvent group families) and prove its composition property [251]; these are the only original contributions of ours given in the first chapter. Chapter 2 is organized as described below. We start by listing some indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces (Section 2.1); special emphasis is paid to Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions (Section 2.2). Concerning original contributions in these two sections, it is only worth noting that, in Proposition 2.1.6, we reconsider the notion of an asymptotically almost periodic function depending on two parameters, while in Definition 2.2.5 we introduce the class of asymptotically Stepanov almost periodic twoparameter functions. A useful characterization of this class is proved in Lemma 2.2.7 following the ideas of W. M. Ruess, W. H. Summers [356] and H. R. Henríquez [220]. In Section 2.3, we recollect some basic definitions and results regarding equi-Weylalmost periodic functions and Weyl-almost periodic functions. The class of asymptotically Weyl-almost periodic functions, introduced in this section, seems to be not considered elsewhere even in the scalar-valued case. Concerning the introduced class of asymptotically Weyl-p-almost periodic functions, we would like to note that the class of equi-Weyl-p-almost periodic functions is a subclass of the class of asymptotically Weyl-p-almost periodic functions, whereas any of this classes extend the well-known class of asymptotically Stepanov p-almost periodic functions (1 ⩽ p < ∞). We actually introduce some new classes of asymptotically almost periodic functions and analyze relations between them. The reader may consult review paper [20] by J. Andres, A. M. Bersani and R. F. Grande (cf. also J. Andres, A. M. Bersani, K. Leśniak [21]) for an excellent survey of results about various classes of Stepanov and Weyl almost periodic functions. The main aim of Section 2.4, the material for which is taken from [251], is to introduce the notion of an almost periodicity for (a, k)-regularized C-resolvent families in Banach spaces, which can be degenerate or nondegenerate in time. In this section, we first observe that many structural results proved by Q. Zheng, L. Liu [421] and T.-J. Xiao, J. Liang [402, Section 7.1.1] continue to hold in the degenerate case; the corresponding part of this section is almost completely written in expository manner and the proofs are given only for a few results. Our main contributions are presented in the remaining part of Section 2.4, where we investigate almost periodic properties of (a, k)-regularized C-resolvent families subgenerated by multivalued linear operators

XXII | Introduction and almost periodic properties of (a, k)-regularized C-resolvent families generated by a pair of closed linear operators A, B with domains and ranges contained in X. The established abstract results seem to be completely new even in the nondegenerate case and they can be simply incorporated in the study of existence and uniqueness of almost periodic solutions of the following abstract degenerate Volterra equation: t

Bu(t) = ∫ a(t − s)Au(s) ds + f (t),

t ⩾ 0,

0

and the following abstract Volterra inclusion: t

u(t) ∈ 𝒜 ∫ a(t − s)u(s) ds + f (t),

t ⩾ 0,

(3)

0

where a ∈ L1loc ([0, ∞)), a ≠ 0, f : [0, ∞) → X is continuous and 𝒜 is a closed multivalued linear operator on X. The notion of an asymptotically almost periodic strongly continuous semigroup was introduced by W. M. Ruess and W. H. Summers [356] in 1986, while the notion of an (asymptotically) Stepanov almost periodic strongly continuous semigroup was introduced by H. R. Henríquez [220] in 1990. The main aim of Section 2.5 is to enquire into the most important asymptotically almost periodic properties, Stepanov almost periodic properties and asymptotically Stepanov almost periodic properties of abstract (degenerate) Volterra integro-differential equations in Banach spaces (this material has been published in our recent paper [259]). Regarding (asymptotically) Stepanov almost periodic solutions of integro-differential equations, our results seem to be completely new for degenerate equations (cf. [82, 220] and [353, 354] for some results established so far in the nondegenerate case). Especially, we prove that the asymptotical almost periodicity is preserved under the action of the subordination principle discovered by E. Bazhlekova [53, Theorem 3.1] (see Theorem 2.6.6(iii)) and that for each number α ∈ (0, 2) ∖ {1} the only Stepanov almost periodic non-degenerate (gα , C)-resolvent family (C ∈ L(X) injective, with dense range) is that generated by the zero operator (see Theorem 2.6.10). As we will see later, the asymptotical almost periodicity is a very exceptional property of considered vector-valued functions that remains unchanged after the action of this subordination principle. In Example 2.6.4, we will see that the notion of asymptotical almost periodicity is much more appropriate for dealing with the abstract fractional differential equations than that of almost periodicity. From examinations in this example, it readily follows that the proofs of the main results of [220], established for semigroups and cosine operator functions, are not applicable for fractional resolvent families of Caputo order α ∈ (0, 2) ∖ {1} (cf. Proposition 2.6.1, Proposition 2.6.3 and Proposition 2.6.5 for some other results given in this part). In Subsection 2.6.1, we explore the Stepanov (asymptotically) almost periodic properties of convolution products appearing in the variation of parameters formulae, while in

Introduction

| XXIII

Subsection 2.6.2 we extend the results of H. R. Henríquez [220] and A. S. Rao [353], as well as some results of V. Casarino [83], I. Cioranescu, P. Ubilla [106] and E. Vesentini [392], to (degenerate) C-semigroups and (degenerate) C-cosine functions in Banach spaces. Many of our results on generalized almost periodic properties and generalized almost automorphic properties of infinite convolution product applies also to the following abstract integral inclusion: t

u(t) ∈ 𝒜 ∫ a(t − s)u(s) ds + f (t),

t∈ℝ

(4)

−∞

where a ∈ L1loc ([0, ∞)), a ≠ 0, f : ℝ → X satisfies certain assumptions and 𝒜 is a closed multivalued linear operator on X, as well as to semilinear analogues of (4), like the most simplest one, namely t

u(t) ∈ 𝒜 ∫ a(t − s)u(s) ds + f (t, u(t)),

t ∈ ℝ;

−∞

for more details about the single-valued linear case, cf. C. Cuevas, C. Lizama [118] and H. R. Henríquez, C. Lizama [221]. The main goal of Section 2.7 is to analyze the existence of a unique almost periodic solution or a unique asymptotically almost periodic solution for a class of abstract semilinear Cauchy inclusions of first order with (asymptotically) Stepanov almost periodic coefficients (cf. M. Kamenskii, V. Obukhovskii, P. Zecca [232] for a different approach to semilinear Cauchy inclusions). For this purpose, we introduce the class of asymptotically Stepanov almost periodic functions depending on two parameters and prove a few composition principles in this direction (see, e. g., [62, 317] and references therein; the material of Section 2.7 will appear in our recent research [265]). In Theorem 2.7.1, we clarify that the composition principle [317, Theorem 2.2], proved by W. Long and H.-S. Ding, continues to hold for the functions defined on the real semi-axis I = [0, ∞) (see also [22] and [120] for some other references in this direction). The use of standard Lipschitz assumption has some advantages compared to the condition f ∈ ℒr (ℝ × X : X) used in the formulation of the above-mentioned theorem since, in this case, we can include the order of (asymptotic) Stepanov almost periodicity p = 1 in our analyses (cf. Theorem 2.7.2 for more details). In Propositions 2.7.3–2.7.4, we analyze composition principles for asymptotically Stepanov almost periodic twoparameter functions. The main aim of Lemma 2.7.5 is to prove that the function defined through the infinite convolution product t

t 󳨃→ ∫ R(t − s)f (s) ds,

t∈ℝ

−∞

is asymptotically almost periodic provided that (R(t))t>0 is exponentially decaying at infinity and has a removable singularity at zero, as well as that the coefficient f (⋅) is

XXIV | Introduction asymptotically Stepanov almost periodic. In the remaining part of book, an important role is played by the class of multivalued linear operators 𝒜 satisfying the condition [188, (P), p. 47] introduced by A. Favini and A. Yagi: (P) There exist finite constants c, M > 0 and β ∈ (0, 1] such that Ψ := Ψc := {λ ∈ ℂ : Re λ ⩾ −c(| Im λ| + 1)} ⊆ ρ(𝒜) and −β 󵄩󵄩 󵄩 󵄩󵄩R(λ : 𝒜)󵄩󵄩󵄩 ⩽ M(1 + |λ|) ,

λ ∈ Ψ.

In Theorems 2.7.6–2.7.7, we prove the existence of a unique almost periodic mild solution of the following semilinear differential inclusion of first order: u󸀠 (t) ∈ 𝒜u(t) + f (t, u(t)),

t ∈ ℝ,

(5)

where f : ℝ × X → X is Stepanov almost periodic and some extra conditions are satisfied. Also, of concern is the following semilinear Cauchy inclusion of first order: (DFP)f ,s

u󸀠 (t) ∈ 𝒜u(t) + f (t, u(t)), { u(0) = u0 .

t ⩾ 0,

In Theorems 2.7.8–2.7.9, we analyze the existence of a unique asymptotically almost periodic solution of semilinear differential inclusion (DFP)f ,s provided that the coefficient f (⋅, ⋅) behaves asymptotically in time as a Stepanov almost periodic function. Some simple consequences of Theorem 2.7.9 are stated in Corollaries 2.7.10 and 2.7.11. The main purpose of Remark 2.7.12(i) is to explain how we can use our established results with a view to prove a slight extension of [127, Theorem 4.4], one of the main results of an investigation conducted by B. de Andrade and C. Lizama [127]. Distribution semigroups of operators with densely defined generators were introduced by J. L. Lions in his celebrated and landmark paper [306] (1960). In all probability, this was the first attempt in seeking for solutions of abstract first-order differential equations that are not governed by strongly continuous semigroups of linear operators. Concerning the abstract ill-posed Cauchy problems, we need to mention the notions of C-regularized semigroups of operators, introduced by G. Da Prato [124] (1966), integrated semigroups of operators, introduced by W. Arendt [26] (1987), and convoluted semigroups of operators, introduced by I. Cioranescu and G. Lumer [105] (1994). Combinations of integrated or convoluted semigroups with C-regularized semigroups have received much attention recently as well (for instance, the notion of a C-distribution semigroup of operators has been introduced by the author [249] in 2008). A creation of parallel theory of abstract second-order differential equations starts presumably with the pioneering paper by M. Sova [371] (1966), who defined the class of cosine operator functions. The notion of an integrated cosine operator function was introduced by W. Arendt and H. Kellermann [30] (1987), while the notions of

Introduction

| XXV

C-regularized cosine operator function and integrated C-regularized cosine operator function seem to be defined for the first time in the doctoral dissertation of Y.-C. Li [302] (1991); the notion of a C-distribution cosine function was introduced by the author [250] (2012). For an account of the theory of abstract ill-posed Cauchy problems of first and second order, we refer the reader to the monographs [29, 128] and [245–247]. In Section 2.8, we take up the study of subspace almost periodic and subspace weak almost periodic properties of C-distribution semigroups and C-distribution cosine functions in Banach spaces (cf. [277, 290, 397]). The introduction of notions of subspace almost periodicity and subspace weak almost periodicity is motivated by the fact that the integral generators of almost periodic strongly continuous semigroups and almost periodic integrated C-semigroups need to satisfy rather restrictive spectral conditions (since in this section we primarily deal with semigroups and cosine operator functions consisting of unbounded linear operators, the notion of subspace uniform almost periodicity will not attract our attention here). On the other hand, many chaotic or subspace chaotic C-distribution semigroups and C-distribution cosine functions satisfying certain variants of the famous Desch–Schappacher–Webb criterion [131] are subspace almost periodic as well, with the subspace of almost periodicity being generally dense in the initial Banach space (see [246, Chapter III] and [247, Sections 2.10–2.12] for a more or less updated survey of results on hypercyclic and topologically mixing solutions of abstract fractional PDEs). Because of this, we can freely say that the notion of subspace almost periodicity is incredibly important considered from the application’s point of view. The classes of (bounded) almost periodic distribution groups and cosine distributions were considered for the first time by I. Cioranescu [101] in 1990. Six years later, in 1996, Q. Zheng and L. Liu [421] investigated the class of almost periodic tempered distribution semigroups. Almost periodic distribution (semi-)groups and cosine distributions examined in [101] and [421] are exponential and have densely defined integral generators, which is not the case with C-distribution semigroups and C-distribution cosine functions considered in Section 2.8. The reader with a little experience will easily observe that our approach is completely different from those employed in [101] and [421]: speaking only in terms of global n-times integrated semigroups and cosine functions, here we are interested in whether the nth derivatives of such semigroups and cosine functions exist and are almost periodic on a certain subspace X̃ of the pivot space X (the main aim in [101] and [421] is to analyze the almost periodicity of induced n-times integrated semigroups and cosine functions, see, e. g., [101, Theorem 1.1(ii)], which is very difficult to be satisfied in the case when n ∈ ℕ). In [405], L. Xie, M. Li and F. Huang have analyzed the asymptotically almost periodic C-regularized semigroups by assuming that their integral generators have no positive eigenvalues; the notion of Hille–Yosida space of a closed linear operator (see S. Kantorovitz [234]) has been essentially employed in [405], which is no longer the case in our investigations. Let γ ∈ (0, 1), and let 𝒜 be a multivalued linear operator on a Banach space X. Of importance is the following fractional relaxation inclusion:

XXVI | Introduction γ

(DFP)f ,γ

{

Dt u(t) ∈ 𝒜u(t) + f (t), u(0) = x0 ,

t > 0,

and its semilinear analogue (DFP)f ,γ,s γ

γ

{

Dt u(t) ∈ 𝒜u(t) + f (t, u(t)), u(0) = x0 ,

t > 0,

where Dt denotes the Caputo fractional derivative of order γ, x0 ∈ X and f : [0, ∞) → X, resp. f : [0, ∞) × X → X, is Stepanov almost periodic. In Section 2.9, we analyze the asymptotically almost periodic solutions of fractional Cauchy inclusions (DFP)f ,γ and (DFP)f ,γ,s . We would like to note that the existence and uniqueness of (asymptotically) quasi-periodic solutions of fractional relaxation equations with Caputo derivatives have not received much attention so far: in the existing literature, we have been able to locate only one research paper (F. Li, J. Liang, H. Wang [298]) concerning a similar problem; this paper is devoted to the study of S-asymptotically ω-periodic solutions to fractional relaxation equations with finite delay. Section 2.9 is broken down into two separate subsections: in Subsection 2.9.1, we examine the growth orders at zero and infinity of subordinated fractional resolvent families generated by multivalued linear operators satisfying condition (P), while in Subsection 2.9.2 we examine the case in which the initial semigroup of operators is not analytic in a sector around the positive real axis (here we essentially apply the results from the theory of C-regularized semigroups). The results of Section 2.9 are taken from our recent paper [266]. In [340, Section 6.3], A. Pazy has analyzed semilinear Cauchy problems with generators of analytic strongly continuous semigroups by using the results from the theory of fractional powers of operators. Concerning the study of existence and uniqueness of almost periodic solutions of nondegenerate semilinear Cauchy problems, it seems that the fractional powers of operators have been employed for the first time by M. Bahaj and O. Sidki in [37] (cf. also T. Diagana, C. M. Mahop, G. M. N’Guérékata [153] and M. M. El-Borai, A. Debbouche [181]). Suppose that γ ∈ (0, 1) and 𝒜 is a multivalued linear operator on a Banach space X satisfying condition (P). In Section 2.10, we examine the existence and uniqueness of almost periodic solutions of the semilinear Cauchy inclusion of first order (5) and its fractional relaxation analogue with Weyl– Liouville fractional derivatives of order γ ∈ (0, 1). We essentially employ some results about fractional powers of sectorial multivalued linear operators in the interest of establishing our structural results. It is worth noting that the conditions on function f (⋅, ⋅) used in this section are weaker than those employed in [37] for equation (78), resp. [181] for equation (9), where the authors have looked into the following condition (known already from [340]): (F) The function f : ℝ × [D((−𝒜)θ )] → X is such that there are finite numbers L > 0 and η ∈ [0, 1] such that ‖f (t1 , x1 ) − f (t2 , x2 )‖ ⩽ L(|t1 − t2 |η + ‖x1 − x2 ‖θ ) for all (t1 , x1 ), (t2 , x2 ) in ℝ × [D((−𝒜)θ )]; −𝒜 is the single-valued generator of an exponentially decaying C0 -semigroup.

Introduction

| XXVII

In this case, the constructed mild solutions are Hölder continuous, which need not be the case in our approach (without going into further details, we want only to observe here that the structural results from [37] and [181] can be reconsidered for degenerate semigroups of operators). Condition (F) implies the validity of usually considered Lipschitz type condition (74) used in our approach, and the validity of (74) or its generalization (73) does not imply the validity of condition (F). Furthermore, we consider the case in which the function f (⋅, ⋅) is not uniformly almost periodic but only Stepanov almost periodic. The material of Section 2.10 has appeared in [257]. The main purpose of Section 2.11 is to investigate Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integrodifferential equations and inclusions. The Weyl-almost periodicity and asymptotical Weyl-almost periodicity are very unexplored fields in the theory of almost periodic abstract differential equations in Banach spaces: with the exception of our recent papers [189] (joint work with V. Fedorov) and [273, 274], where both the equi-Weyl-almost periodic solutions and Weyl-almost periodic solutions of abstract fractional differential inclusions have been investigated, and the paper [55] by F. Bedouhene, Y. Ibaouene, O. Mellah and P. Raynaud de Fitte, where the authors have examined equi-Weyl-almost periodic solutions of abstract linear and semilinear equations with equi-Weyl almost periodic coefficients, almost nothing has been said about these topics even for abstract nondegenerate differential equations whose solutions are governed by strongly continuous semigroups of operators. In this section, we go a step further by investigating the Weyl-almost periodic and asymptotically Weyl-almost periodic properties of various classes of degenerate solution operator families subgenerated by multivalued linear operators. A slight generalization of Theorem 2.6.10 has been established as well, stating that for each number α ∈ (0, 2) ∖ {1} the only Weyl almost periodic non-degenerate (gα , C)-resolvent family, where C ∈ L(X) is injective and has dense range, is that generated by the zero operator. The class of C (n) -almost periodic functions was introduced by M. Adamczak [9] in 1997. From then on, the analysis of various generalizations of C (n) -almost periodic functions and their applications to the qualitative analysis of nondegenerate integrodifferential equations in Banach spaces has attracted the attention of many authors working in this field of functional analysis (see, e. g., [38, 154, 155, 180, 183] and [303]). In a separate part of Section 2.11, we introduce the class of (equi-)Weyl C (n) -almost periodic functions and the class of asymptotically (equi-)Weyl C (n) -almost periodic functions, as well as examine the existence and uniqueness of solutions of integrodifferential inclusions belonging these classes of functions (the Besicovitch–Doss C (n) -almost periodic functions and solutions are considered in Subsection 2.13.1; see also [255]). Besides this article, results presented in Section 2.11 are taken from [253] and [189]. The class of Banach space valued pseudo-almost periodic functions was introduced in the doctoral dissertation of C. Zhang [412] (1992), while the class of weighted pseudo-almost periodic functions was introduced by T. Diagana [134] (2006). From

XXVIII | Introduction 1992 onwards, many mathematicians have investigated pseudo-almost periodic solutions and weighted pseudo-almost periodic solutions for various classes of abstract Cauchy problems with integer or fractional time-derivatives (cf. [11–13, 17, 133, 299] and [413, 414]). For example, C. Zhang [416] has analyzed the existence and uniqueness of (pseudo-)almost periodic solutions of the following abstract nonlinear equation of first order: u󸀠 (t) = A(t, u(t)) + f (t),

t ∈ ℝ,

where A : ℝ × X → X and f : ℝ → X are given functions. In the finite-dimensional case X = ℝn , the author has imposed the following conditions: (K1 ) A(⋅, ⋅) is a continuous function; (K2 ) f (⋅) is a continuous function and there exists a finite constant N > 0 such that ‖f (t) − A(t, 0)‖ ⩽ N for all t ∈ ℝ; (H3 ) p(⋅) is a bounded continuous function with real values such that there exist positive real constants δ, δ1 , T0 , T1 satisfying p(t) ⩽ −δ, t ⩽ T0

and p(t) ⩽ −δ1 , t ⩾ T1 ;

(K4 ) for all (t, x, y) ∈ ℝ × ℝn × ℝn , we have [x − y, A(t, x) − A(t, y)] ⩽ p(t)‖x − y‖, where [x, y] := lim

h→0+

1 (‖x + hy‖ − ‖x‖), h

x, y ∈ ℝn , and so on and so forth.

In Section 2.12, we analyze pseudo-almost periodic solutions of abstract semilinear Cauchy inclusions of first order and abstract semilinear relaxation inclusions with Weyl–Liouville fractional derivatives (cf. [252]). The main aim of Theorem 2.12.4 is to prove the existence of a unique pseudo-almost periodic mild solution of the semilinear differential inclusion of first order (5), with f : ℝ × X → X being pseudo-almost periodic. In Corollary 2.12.5, we state a simple consequence of Theorem 2.12.4 provided that the usually considered Lipschitz type condition is satisfied. If condition (P) holds, then there exists a degenerate strongly continuous semigroup (T(t))t>0 ⊆ L(X) generated by 𝒜 and there exists a finite constant M > 0 such that ‖T(t)‖ ⩽ Me−ct t β−1 , t > 0. Regarding the Poisson heat equation in the Sobolev space X = H −1 (Ω), it is worth noting that condition (P) holds with the number β = 1 and the region Ψc = ℂ ∖ (−∞, c), so that (T(t))t>0 is bounded at zero and strongly continuous on D(𝒜) (cf. [188, Example 3.3, pp. 74–75, and Remarks, p. 52]). If this is the case, then we are in a position to investigate pseudo-almost periodic mild solutions of the semilinear fractional differential inclusion (1) of order α ∈ (1, 2), with f : ℝ × X → X being

Introduction

| XXIX

pseudo-almost periodic (cf. the notes and appendices to the second chapter as well as [11, Section 3] for more details; the Leray–Schauder alternative can be employed here). The analysis of Besicovitch-p-almost periodic solutions, Doss-p-almost periodic solutions and Besicovitch–Doss-p-almost periodic solutions of abstract Volterra integro-differential equations has not received much attention of the authors so far (see, e. g., [36] and [338] for some known results in this direction). The main purpose of Section 2.13 is to introduce the class of Besicovitch–Doss-p-almost periodic functions with values in Banach spaces, as well as to consider Besicovitch–Doss-p-almost periodic properties of both infinite and finite convolution products. In this section, we raise the issue whether the classes of Besicovitch-p-almost periodic functions and Besicovitch–Doss-p-almost periodic functions coincide in the vector-valued case (due to the important results of R. Doss [175, 176], we know that the answer is affirmative in the scalar-valued case); see also the paper [119] by A. N. Dabbouchy and H. W. Davies, where a slightly different characterization of scalar-valued Besicovitch-p-almost periodic functions has been shown. The results of Section 2.13 come from our recent research papers [254] and [283]. Starting presumably with the paper [85] by Y.-H. Chang and J.-S. Chen (1995), many authors have investigated the existence and uniqueness of various (asymptotically) almost periodic solutions of nonautonomous abstract differential equations in Banach spaces. For more details about the subject, we refer the reader to J.-B. Baillon, J. Blot, G. M. N’Guérékata, D. Pennequin [38], T. Diagana [132–138], H.-S. Ding, W. Long, G. M. N’Guérékata [169], H. Lee, H. Alkahby [294], L. Maniar, R. Schnaubelt [321] and the references cited therein. The main aim of Section 2.14, which is divided into three separate subsections, is to continue the research study of L. Maniar and R. Schnaubelt [321] by investigating the existence and uniqueness of (asymptotically) Weyl-almost periodic solutions and asymptotically Stepanov almost periodic solutions of inhomogeneous (semilinear) evolution equations in Banach spaces, as well as to reconsider some results of H.-S. Ding, W. Long and G. M. N’Guérékata [169] concerning the Stepanov almost periodic solutions of such equations. It is worth noting that our results from Section 2.14 cannot be formulated successfully for the class of Besicovitch– Doss almost periodic functions (cf. Section 3.7 for the corresponding results for the class of Besicovitch almost automorphic functions). The notions of bounded and almost periodic distributions have been introduced already in the foundational papers by L. Schwartz (see, e. g., [365]), who analyzed only the scalar-valued case. Bounded and almost periodic distributions with values in general Banach spaces have been investigated by I. Cioranescu in [101]: Let (X, ‖ ⋅ ‖) be a complex Banach space, and let 𝒟L1 denote the vector space consisting of all infinitely differentiable functions f : ℝ → ℂ such that for each number j ∈ ℕ0 we have f (j) ∈ L1 (ℝ). The Fréchet topology on 𝒟L1 is induced by the following system of semi-

XXX | Introduction norms: k

󵄩 󵄩 ‖f ‖k := ∑ 󵄩󵄩󵄩f (j) 󵄩󵄩󵄩L1 (ℝ) ,

f ∈ 𝒟L1

j=0

(k ∈ ℕ).

A continuous linear mapping f : 𝒟L1 → X is said to be a bounded X-valued distribution. The space of such distributions is usually denoted by B󸀠 (X); equipped with the strong topology, B󸀠 (X) becomes a complete locally convex space. We would like to observe the following: If f : 𝒟L1 → X is a bounded X-valued distribution, then there exist c > 0 and k ∈ ℕ such that ∞

∞

k k 1 󵄨 (j) 󵄨 󵄩 󵄩󵄩 󵄨 󵄨 [(x2 + 1)󵄨󵄨󵄨φ(j) (x)󵄨󵄨󵄨] dx 󵄩󵄩f (φ)󵄩󵄩󵄩 ⩽ c ∑ ∫ 󵄨󵄨󵄨φ (x)󵄨󵄨󵄨 dx = c ∑ ∫ 2 x + 1 j=0 j=0 −∞

−∞

∞

⩽c ∫ −∞

k dx 󵄩 󵄩 ⋅ ∑ 󵄩󵄩󵄩(⋅2 + 1)φ(j) (⋅)󵄩󵄩󵄩∞ , + 1 j=0

x2

φ ∈ 𝒮,

so that f|𝒮 : 𝒮 → X is a tempered X-valued distribution (here, the space of rapidly decreasing functions 𝒮 is endowed with the usual Fréchet topology). Following L. Schwartz [365] and I. Cioranescu [101], we say that a bounded X-valued distribution f ∈ B󸀠 (X) is almost periodic iff there exists a sequence of X-valued trigonometric polynomials converging to f (⋅) in B󸀠 (X). If B󸀠ap (X) denotes the vector space consisting of all almost periodic X-valued distributions, then AP(ℝ : X) is dense in B󸀠ap (X) by definition. Furthermore, by [101, Theorem 1.1], we have the following equivalence relations for an element f ∈ 𝒟󸀠 (X): (i) f ∈ B󸀠 (X), resp. f ∈ B󸀠ap (X). (ii) There is an integer k ∈ ℕ0 such that f = ∑kj=0 fj , where fj ∈ Cb (ℝ : X), resp. fj ∈ AP(ℝ : X), for 0 ⩽ j ⩽ k. (iii) For any φ ∈ 𝒟, we have f ∗ φ ∈ Cb (ℝ : X), resp. f ∗ φ ∈ AP(ℝ : X). (iv) The set of all translations of f (⋅), defined as usually, is bounded in 𝒟󸀠 (X), resp. relatively compact in B󸀠 (X). (j)

Here and hereafter, the symbol 𝒟 denotes the Schwartz space of test functions and 𝒟󸀠 (X) denotes the space of all continuous linear mappings 𝒟 → X, equipped with the strong topology. The spaces of vector-valued almost periodic distributions are systematically analyzed in a series of research papers by B. Basit and H. Güenzler (see, e. g., [46–48]). Here we would like to mention that they have proved [47] that any regular ∞ vector-valued distribution φ 󳨃→ ∫−∞ f (t)φ(t) dt, φ ∈ 𝒟, where f : ℝ → X is a Stepanov p-almost periodic function for some p ∈ [1, ∞), is almost periodic, as well as that, for every p ∈ [1, ∞), there exists a scalar-valued infinitely differentiable Weyl-p-almost periodic function f (⋅) such that the regular distribution given above is not almost periodic.

Introduction

| XXXI

Within the Komatsu theory of ultradistributions, the notion of a scalar-valued almost periodic ultradistribution has been introduced by I. Cioranescu [104]. In her approach, the corresponding sequence (Mp ) satisfies conditions (M.1), (M.2) and (M.3). The results from [104] have been reconsidered and slightly generalized by M. C. Gómez-Collado [201] and C. Fernández, A. Galbis, M. C. Gómez-Collado [190], within the theory of ω-ultradistributions (cf. R. W. Braun, R. Meise, B. A. Taylor [75]). Regarding scalar-valued almost periodic ultradistributions as boundary values of harmonic almost periodic functions, we would like to mention that I. Cioranescu [104] has proved that, for any such ultradistribution, we can find a harmonic function u(x, y) in the right-half plane such that, for every x > 0, the mapping y 󳨃→ u(x, y), y ∈ ℝ is almost periodic and limx→0+ u(x, y) = f in the ultradistributional sense. The notion of an almost periodic Colombeau generalized function has been introduced by C. Bouzar and M. T. Khalladi [72] (the Bloch-periodicity of Colombeau generalized functions has been investigated by M. F. Hasler [217]; cf. also [73] for the notion of an almost automorphic Colombeau generalized function and [74] for the notion of an almost automorphic distribution). We want also to draw the attention of our readers to the excellent survey of results [386] by V. Valmorin, concerning periodic generalized functions and their applications. The main aim of Section 2.15 is to introduce the notion of a vector-valued almost periodic ultradistribution and to investigate some generalizations of this concept (cf. Subsection 2.15.1). Albeit containing some original contributions, this section is primarily intended to review and slightly generalize some known results concerning scalar-valued almost periodic ultradistributions and their generalizations [104, 47, 201]. We contemplate the work of many authors, and transfer several known results on vector-valued almost periodic distributions to vector-valued almost periodic ultradistributions of Beurling and of Roumieu type. We basically follow Komatsu’s approach here, with the sequence (Mp ) satisfying conditions (M.1), (M.2) and (M.3󸀠 ); any use of condition (M.3) is explicitly emphasized. To the best knowledge of the author, the notion of a vector-valued almost periodic ultradistribution has not been yet introduced in the existing literature, even in the case that the sequence (Mp ) satisfies condition (M.3). And, more to the point, the notion of a scalar-valued almost periodic ultradistribution, introduced in this section, seems to be new in the case that (Mp ) does not satisfy condition (M.3). The material of Section 2.15 is taken from [269]. Chapter 3 pertains to the study of almost automorphic and asymptotically almost automorphic type solutions of integro-differential equations in Banach spaces. Let us recall that S. Bochner introduced the notion of a scalar-valued almost automorphic function in [66] (1962), as well as that the first extensive study of such a class of functions on topological groups was conducted by W. A. Veech [387, 388] (1965–1967). A continuous function f : ℝ → X is said to be almost automorphic iff for every real sequence (bn ) there exist a subsequence (an ) of (bn ) and a mapping g : ℝ → X such

XXXII | Introduction that lim f (t + an ) = g(t) and

n→∞

lim g(t − an ) = f (t),

n→∞

pointwise for t ∈ ℝ. Then f ∈ Cb (ℝ : X) and the limit function g(⋅) is bounded on ℝ but not necessarily continuous on ℝ. Similarly, a continuous function f : ℝ → X is said to be asymptotically almost automorphic iff there is a function h ∈ C0 ([0, ∞) : X) and an almost automorphic function q : ℝ → X such that f (t) = h(t) + q(t), t ⩾ 0. Let us recall that any almost periodic function ℝ 󳨃→ X is almost automorphic and any asymptotically almost periodic function [0, ∞) 󳨃→ X is asymptotically almost automorphic. We say just a few words about the progress made within the theory of abstract differential equations. Almost automorphic solutions to a class of semilinear fractional differential equation (1) have been examined by C. Cuevas and C. Lizama in [117] (for almost automorphic solutions of semilinear Cauchy problems, we also refer to T. Diagana, G. M. N’Guérékata [148] and J. A. Goldstein, G. M. N’Guérékata [200]; the nonautonomous case has been analyzed by H.-S. Ding, J. Liang and T.-J. Xiao [167]). Concerning Stepanov class of almost automorphic functions, we should mention the papers [166] and [145]. T. Diagana, V. Nelson and G. M. N’Guérékata have introduced the notion of an Sp(n) -almost automorphic function in [155], furnishing also some results

about C (m+N) -pseudo almost automorphic solutions to the higher-order abstract differential equation n−1

u(n) (t) + ∑ ai (t)u(i) (t) = f (t), i=1

t ∈ ℝ,

where ai : ℝ → ℝ satisfy certain conditions (i ∈ ℕ0n−1 ) and the function f : ℝ → ℝ is Stepanov-like C (m) -pseudo almost automorphic. Their method is based on converting of above equation into an equivalent first order matrix system and therefore is not applicable to abstract multiterm fractional differential equations. In [4], S. Abbas, V. Kavitha and R. Murugesu have examined Stepanov-like (weighted) pseudo almost automorphic solutions to the following fractional order abstract integro-differential equation: Dαt u(t) = Au(t) + Dα−1 t f (t, u(t), Ku(t)),

t ∈ ℝ,

where t

Ku(t) = ∫ k(t − s)h(s, u(s)) ds,

t ∈ ℝ,

−∞

1 < α < 2, A is a sectorial operator with domain and range in X, of negative sectorial type ω < 0, the function k(t) is exponentially decaying, the functions f : ℝ×X ×X → X and h : ℝ × X → X are Stepanov-like weighted pseudo almost automorphic in time for

Introduction

| XXXIII

each fixed elements of X × X and X, respectively, satisfying some extra conditions. It is noteworthy that almost automorphic solutions of certain classes of abstract Volterra integro-differential equations have been also examined in [64, 185, 195, 294, 319, 410]. In [84], V. Casarino has introduced the notions of (Stepanov) almost automorphic C0 -group and (Stepanov) asymptotically almost automorphic C0 -group on Banach space, where some equivalence relations between almost periodicity and almost automorphy for orbits of a C0 -group have been proved. We would like to observe that the extensions of her results to (degenerate) C-regularized groups of operators can be proved almost immediately. The assertion of Proposition 2.5.1 can be also straightforwardly formulated for various classes of (asymptotically) almost automorphic (a, k)-regularized C-resolvent families in Banach spaces. On the other hand, numerous very nontrivial and unpleasant problems occur if we try to reconsider some known assertions on the (asymptotical) almost periodicity of (a, k)-regularized C-resolvent families in Banach spaces, provided that the results from the Bohr–Fourier analysis of almost periodic functions are needed for their proofs. In the first three sections of Chapter 3, we pay special attention to the recapitulation of some known results on almost automorphic functions, asymptotically almost automorphic functions and their generalizations, like weighted pseudo-almost automorphic functions (in this part, essentially, the only new results are Propositions 3.1.6 and 3.2.5–3.2.6). The main aim of Section 3.4 is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives (this is a part of our joint research with Prof. G. M. N’Guérékata [210]). Of concern is the following abstract multiterm fractional differential inclusion: α

n−1

α

Dt n u(t) + ∑ Ai Dt i u(t) ∈ 𝒜Dαt u(t) + f (t), i=1

u(k) (0) = uk ,

k = 0, . . . , ⌈αn ⌉ − 1,

t ⩾ 0, (6)

where n ∈ ℕ ∖ {1}, A1 , . . . , An−1 are bounded linear operators on a Banach space X, 𝒜 is a closed multivalued linear operator on X, 0 ⩽ α1 < ⋅ ⋅ ⋅ < αn , 0 ⩽ α < αn , f (⋅) is an X-valued function, and Dαt denotes the Caputo fractional derivative of order α [53, 246]. Since we essentially follow the method proposed by C.-G. Li, M. Li and me [276] (see also [246, Subsection 2.10.1]), the boundedness of linear operators A1 , . . . , An−1 is crucial for applications of the vector-valued Laplace transform, and therefore, will be the starting point in our work. Subsection 3.4.1, which is written almost in an expository manner, is devoted to the study of k-regularized C-propagation families for (6). The main result of Section 3.4 is Theorem 3.4.10, where we investigate the asymptotic behavior of ki -regularized C-propagation families for (173). In the proof of this theorem, we use the well-known results on analytical properties of the vector-valued Laplace transform established by M. Sova in [370] (see, e. g., [29, Theorem 2.6.1]) in place of

XXXIV | Introduction E. Cuesta’s method established in the proof of [113, Theorem 2.1]. The proof of Theorem 3.4.10 is vastly easier and transparent than that of [113, Theorem 2.1] because of the simplicity of contour Γ in our approach. We will essentially use this fact for refinements of some known results on the asymptotic behavior of operator families governing solutions of abstract two-term fractional differential equations, established recently by V. Keyantuo, C. Lizama, M. Warma [236] and V. T. Luong [318]. Contrary to many papers from the existing literature, Theorem 3.4.10 is applicable to the almost sectorial operators, generators of integrated or C-regularized semigroups, and multivalued linear operators appearing in the analysis of (fractional) Poisson heat equation in Lp -spaces [188, 247]. In a separate subsection, we explain how one can similarly analyze the following multiterm fractional Cauchy problem: n−1

α

α

Dt n u(t) + ∑ Aj Dt j u(t) ∈ 𝒜Dαt u(t) + f (t), j=1

t ∈ (0, τ),

(7)

where n ∈ ℕ ∖ {1}, 𝒜 is a closed multivalued linear operator on a complex Banach space X, Aj ∈ L(X) for 1 ⩽ j ⩽ n − 1, 0 ⩽ α1 < ⋅ ⋅ ⋅ < αn , 0 ⩽ α < αn , 0 < τ ⩽ ∞, f (t) is an X-valued function, and Dαt denotes the Riemann–Liouville fractional derivative of order α. We analyze (asymptotically) almost automorphic solutions of integro-differential inclusions in Section 3.5. Briefly speaking, this section is organized as follows. Our main results are stated in Subsection 3.5.1, where we investigate the generalized (asymptotically) almost automorphic properties of various types of convolution products. After that, we analyze the following abstract Cauchy fractional relaxation inclusion γ

Dt,+ u(t) ∈ −𝒜u(t) + f (t),

t ∈ ℝ,

(8)

γ

where Dt,+ denotes the Weyl–Liouville fractional derivative of order γ ∈ (0, 1), and its semilinear analogue γ

Dt,+ u(t) ∈ −𝒜u(t) + f (t, u(t)),

t ∈ ℝ,

(9)

where f : ℝ × X → X is a generalized almost automorphic function. The main goal of Subsection 3.5.2 is to prove several assertions on the existence and uniqueness of generalized almost automorphic solutions of the semilinear Cauchy inclusions (5), (9) and (DFP)f ,γ,s . This section is written without giving the proofs of our results. The main reason for this lies in the fact that our results established in Subsection 3.5.1 and composition theorems for generalized almost automorphic functions established in Subsection 3.2 enable one to simply deduce the proofs of our results clarified in Subsection 3.5.2 by using an almost verbatim repeating of the arguments. The material is basically taken from [267], where the author has analyzed just one pivot Banach space.

Introduction

| XXXV

The class of weighted pseudo almost-periodic functions was introduced by T. Diagana in [134] (2006). This class extends the previously considered class of pseudoalmost periodic functions and has numerous applications in the study of qualitative properties of abstract Volterra integro-differential equations (see, e. g., [13, 61, 64, 111, 135, 230, 304, 418] and references cited therein for further information on the subject). The general theory of weighted pseudo almost-periodic functions is full of open problems and, as such, it is a potential field for further investigations. In contrast to the well-explored class of pseudo almost-periodic functions, several intriguing questions are still open for weighted pseudo almost-periodic functions (the situation is even worse for the class of weighted pseudo-almost automorphic functions, which was introduced by J. Blot, G. M. Mophou, G. M. N’Guérékata and D. Pennequin in [64] (2009)). In Section 3.6, we analyze the existence and uniqueness of generalized weighted pseudo-almost periodic solutions and generalized weighted pseudo-almost automorphic solutions of abstract Volterra integro-differential inclusions in Banach spaces (see [268]). Concerning already made applications to abstract Volterra integro-differential equations, we will mention here only a few important results obtained so far. In [132, Chapter 10], T. Diagana has examined the existence and uniqueness of weighted pseudo almost-periodic solutions of autonomous partial differential equation d 󸀠󸀠 [u (t) + f (t, Bu(t))] = w(t)Au(t) + g(t, Cu(t)), dt

t ∈ ℝ,

(10)

where A is a sectorial operator on X, B and C are closed linear operators acting on X, and f : ℝ × X → X, g : ℝ × X → X are pseudo almost-periodic functions in t ∈ ℝ uniformly in x ∈ X. The results on well-posedness of (10) clearly apply in the study of qualitative properties of solutions of the abstract nonautonomous third-order differential equation u󸀠󸀠󸀠 + B(t)u󸀠 + A(t)u = h(t, u),

t ∈ ℝ;

see [132, Chapter 10] for more details. Motivated by the ideas of T. Diagana, many authors have considered the qualitative properties of solutions for various classes of the Sobolev-type differential equations. For example, in [86], Y.-K. Chang and Y.-T. Bian have investigated the weighted asymptotic behavior of the Sobolev-type differential equation d [u(t) + f (t, u(t))] = A(t)u(t) + g(t, u(t)), dt

t ∈ ℝ,

where A(t) : D ⊆ X → X is a family of densely defined closed linear operators on a common domain D, independent of time t ∈ ℝ, f : ℝ × X → X is a weighted pseudo-almost automorphic function and g : ℝ × X → X is a Stepanov-like weighted pseudo-almost automorphic function satisfying some extra conditions. Here, it is worth noting that

XXXVI | Introduction the class of weighted Stepanov-like pseudo-almost automorphic functions has been introduced by Z. Xia and M. Fan in [400], where the authors have analyzed the existence and uniqueness of such solutions for the following abstract semilinear integrodifferential equation: t

u(t) = g(t) + ∫ a(t − s)f (s, u(s)) ds,

t ∈ ℝ,

−∞

under certain conditions. Weighted pseudo-almost automorphic solutions to functional differential equations with infinite delay have been recently considered by Y.-K. Chang and S. Zheng in [88]. They have looked upon the abstract nonautonomous differential equation d [u(t) + f (t, ut )] = A(t)u(t) + g(t, ut ), dt

t ∈ ℝ,

where A(t) : D ⊆ X → X is a family of densely defined closed linear operators on a common domain D, independent of time t ∈ ℝ, the history ut : (−∞, 0] → X defined by ut (⋅) := u(t + ⋅) belongs to some abstract phase space ℬ defined axiomatically, and f : ℝ × ℬ → X, g : ℝ × ℬ → X fulfill certain conditions. Denote by PAP(ℝ, X, ρ) and PAP(ℝ × Y, X, ρ) the spaces consisting of all weighted pseudo almost-periodic functions depending on one or two variables, respectively; let PAP0 (ℝ, X, ρ) be the space of so-called weighted ergodic components (see Section 3.6). The first problem we would like to address is that the unique decomposition of a weighted pseudo almost-periodic function into its almost-periodic and ergodic component is not unique, in general [304]. The second problem appearing is that the sum of AP(ℝ : X) and PAP0 (ℝ, X, ρ) need not be a closed subspace of Cb (ℝ : X) albeit both AP(ℝ : X) and PAP0 (ℝ, X, ρ) considered separately are closed subspaces of Cb (ℝ : X). To overcome this problem, and to analyze weighted pseudo almost-periodic properties of certain classes of semilinear first-order Cauchy problems, J. Zhang, T.-J. Xiao and J. Liang [418] have introduced the following (sometimes called modular) norm on the space PAP(ℝ, X, ρ): ‖f ‖ρ := inf[sup ‖gi (t)‖ + sup ‖qi (t)‖], i∈I

t∈ℝ

t∈ℝ

where I denotes the family of all possible decompositions of f (⋅) into almost-periodic and ergodic component. This norm turns PAP(ℝ, X, ρ) into a Banach space. After that, the authors of [418] proved a composition theorem for weighted pseudo almostperiodic functions and applied this result in the analysis of existence and uniqueness of semilinear Cauchy problems involving the generators of exponentially decaying strongly continuous semigroups. We would like to note that the results established in [418] can be also formulated for semilinear Cauchy inclusions with multivalued

Introduction

| XXXVII

linear operators satisfying condition (P), especially for almost sectorial operators, or for semilinear fractional relaxation inclusions with Weyl–Liouville derivatives. The existence of a weighted mean for almost periodic functions has been investigated by T. Diagana in [139], while translation invariance of weighted pseudo almostperiodic functions and some other problems for this class have been investigated by D. Ji and Ch. Zhang [230]. In what follows, we will present the main results of recent research study [111] by A. Coronel, M. Pinto and D. Sepúlveda. It is well-known that the space AP(ℝ : X) (PAP(ℝ : X)) is a convolution invariant space in the sense that for each function f ∈ AP(ℝ : X) (f ∈ PAP(ℝ : X)) and for each scalar-valued function k ∈ L1 (ℝ) we have k ∗ f ∈ AP(ℝ : X) (k ∗ f ∈ PAP(ℝ : X)), where ∞

(k ∗ f )(t) = ∫ k(t − s)f (s) ds,

t ∈ ℝ.

−∞

This is no longer true for the space PAP(ℝ, X, ρ), which is not convolution invariant, in general. It can be easily checked that the convolution invariance of space PAP(ℝ, X, ρ) is equivalent to that of PAP0 (ℝ, X, ρ) (the corresponding problem for the class of socalled double-weighted pseudo almost-periodic functions has been examined by T. Diagana in [141, 142]). One of the main results of [111] asserts that if a weighted measure ρ ∈ 𝕌∞ satisfies the condition sup sup

r>0 t∈Ωr,s

ρ(t + s) < ∞, ρ(t)

s ∈ ℝ,

(11)

where Ωr,s := {t ∈ ℝ : |t| ⩽ |s| + r} (s ∈ ℝ, r > 0), then the space PAP(ℝ, X, ρ) is convolution invariant. The convolution invariance of PAP(ℝ, X, ρ) also holds in the case when the following conditions are fulfilled: r

sup

r∈ℝ,|s|⩽r

s

1 ∫ |k(t − s)|ρ(t) dt < ∞, ρ(s)

sup

r∈ℝ,|s|⩽r

s

lim

l→+∞

−t

l

−∞

−l

∞

l

l

−l

1 ∫ |k(t − s)|ρ(t) dt < ∞, ρ(s) −r

1 ∫ ds ∫ |k(t − s)|ρ(t) dt = 0, ν(l, ρ)

and 1 lim ∫ ds ∫ |k(t − s)|ρ(t) dt = 0; l→+∞ ν(l, ρ) the translation invariance of PAP(ℝ, X, ρ) is ensured by the validity of condition (11) (cf. [111, Theorem 3.7 (b)]). Applications to certain classes of semilinear abstract integral equations are given in [111, Section 4].

XXXVIII | Introduction Almost periodic and almost automorphic type solutions of nonautonomous parabolic equations have been examined, among many other research papers, in [42, 56, 64, 132, 137, 143, 167, 169, 170, 185, 264, 311] and [419]. The main purpose of Section 3.7 is to analyze the existence and uniqueness of Besicovitch almost automorphic type solutions of nonautonomous differential equations of first order (see [263]). In Section 3.8, we first verify that all structural results proved by C. Bouzar and Z. Tchouar [74] holds for vector-valued almost automorphic distributions. The main aim of this section is to introduce the notion of a vector-valued almost automorphic ultradistribution in Banach space, as well as to provide several structural profilations for the introduced class [278]. The notion of a scalar-valued asymptotically almost periodic distribution was introduced by I. Cioranescu in [103] (1989), while the notion of a vector-valued asymptotically almost periodic distribution was investigated in the important research monograph [90] by D. N. Cheban (2009) following a different approach (see also I. K. Dontvi [173] and A. Halanay, D. Wexler [213]). The main purpose of Section 3.9, in which we follow the approach presented in [103], is to introduce the notions of an asymptotically almost periodic ultradistribution and an asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. The notion of an asymptotically almost periodic ultradistribution seems to be not considered elsewhere even in the scalar-valued case; the notion of a vector-valued almost automorphic distribution seems to be completely new as well. We provide several applications of the introduced concepts in the analysis of systems of ordinary differential equations (see [270]). The analysis of almost periodic and almost automorphic solutions to ordinary differential equations and their systems is almost completely without scope of this book; for more details on the subject, one may refer, e. g., to [80, 91, 109, 216, 291, 303, 337, 391] and references cited therein. We have made a choice to single out a special section for principal examples and applications of our abstract theoretical results, deeply believing that this decision will not impoverish the scope of the applicability of the methods developed and described in the book (cf. Section 3.10; many other, much more illustrative and elementary examples are presented in preceding sections). Here, we would like to note that this monograph does not intend to consider almost periodic type and almost automorphic type solutions of a substantially large class of (nonlinear) Volterra integrodifferential equations in finite-dimensional spaces. It is well known that Volterra integro-differential equations and fractional differential equations have a lot of important applications to real world problems and, before proceeding further, we would like to present a few relevant examples regarding this (cf. the monographs [204] by G. Gripenberg, S. O. Londen, O. J. Staffans, [349] by J. Prüss, and [246, 247] by the author, for further information in this direction):

Introduction

| XXXIX

(i) The transport of charged particles in a turbulent plasma is described by the system of equations 𝜕I { + α∇ ⋅ Φ = 0, { { { { 𝜕τ t { { 𝜕Φ + εΩ ⋅ Φ = −(εν)2 ∫ 𝒦(ε, λ) ⋅ Φ(τ − λ) dλ − 1 α∇I, { { { 𝜕τ 3 0 {

where I is the omnidimensional intensity, Φ is the flux and the coefficients (εν)2 and α > 0 are sufficiently small. (ii) Suppose that A(⋅) is a nonnegative locally integrable function defined on [0, ∞), S0 > 0, g(x) = 1 − e−x , x ⩾ 0 and the function f : [0, ∞) → [0, ∞) is defined as in [204, Example 2.1, p. 7]. The nonlinear Volterra integral equation t

x(t) = S0 ∫ A(t − s)g(x(s)) ds + f (t),

t⩾0

0

is a mathematical model for the spread of an epidemic in a population of fixed size. (iii) The following boundary value problem is commonly used for modeling some problems in viscoelasticity, like simple shearing motions and torsion of a rod: t

{ { {ut (t, x) = ∫ da(s)uxx (t − s, x) + h(t, x), t ⩾ 0, x ∈ [0, 1], { { 0 { u(t, 0) = u(t, 1) = 0, t ⩾ 0; u(0, x) = u0 (x), x ∈ [0, 1], { about which we assume that a : [0, ∞) → ℝ is a function of bounded variation on each compact interval J = [0, T] and a(0) = 0 (T > 0). (iv) The Basset–Boussinesq–Oseen equation u󸀠 (t) − cDαt u(t) + u(t) = f (t),

t ⩾ 0;

u(0) = 0,

(12)

where c ∈ ℝ and 0 < α < 1, describes the unsteady motion of a particle accelerating in a viscous fluid under the action of the gravity, while the two-term fractional differential equation u󸀠󸀠 (t) + cΔDαt u(t) + u(t) = f (t),

t ⩾ 0;

u(0) = x,

u󸀠 (0) = y,

where c ∈ ℝ and 1 < α < 2, describes the propagation of plane electromagnetic waves in an isotropic and homogeneous material, lossy dielectric. (v) Suppose 1 ⩽ p < ∞, 0 ≠ Ω ⊆ ℝn is an open bounded domain with smooth boundary, and X := Lp (Ω). The degenerate Volterra integral equation t

(α − Δ)utt = βΔut + Δu + ∫ g(t − s)Δu(s, x) ds, 0

t > 0, x ∈ Ω;

XL | Introduction u(0, x) = ϕ(x),

ut (0, x) = ψ(x),

where g ∈ L1loc ([0, ∞)), α ∈ ℝ and β ∈ ℝ ∖ {0}, appears in the theory of nonlinear viscoelasticity if n = 3. The main aim of Example 3.10.1 is to continue the analysis of R. Ponce and M. Warma [348] concerning diffusion Volterra integro-differential equations with memory, proving the existence of some specific classes of exponentially decaying degenerate (a, k)-regularized resolvent families. In the final part of this example, we consider asymptotically almost periodic solutions and Stepanov asymptotically almost periodic solutions of the related abstract Cauchy inclusion (3). Suppose that 0 ≠ Ω ⊆ ℝn is a bounded domain with smooth boundary 𝜕Ω. Denote by {λk } [= σ(Δ)] the eigenvalues of the Dirichlet Laplacian Δ in X := L2 (Ω) (recall that 0 < −λ1 ⩽ −λ2 ⋅ ⋅ ⋅ ⩽ −λk ⩽ ⋅ ⋅ ⋅ → +∞ as k → ∞; see [247] for further information), numbered in nonascending order with regard to multiplicities. By {ϕk } ⊆ C ∞ (Ω) we denote the corresponding set of mutually orthogonal (in the sense of L2 (Ω)) eigenfunctions. In Example 3.10.2, we analyze the almost periodic solutions of the following modification of the Barenblatt–Zheltov–Kochina equation (λ − Δ)ut (t, x) = iζ Δu(t, x),

t ∈ ℝ, x ∈ Ω,

equipped with the following initial conditions: u(0, x) = u0 (x),

x ∈ Ω;

u(t, x) = 0,

(t, x) ∈ ℝ × 𝜕Ω,

where ζ ∈ ℝ ∖ {0} and λ = λk0 ∈ σ(Δ), as well as the related abstract degenerate Volterra integral equations whose solutions are governed by exponentially bounded (a, k)-regularized resolvent families generated by pairs of closed linear operators. In Example 3.10.3, we investigate the existence of a unique (Stepanov, asymptotically) almost periodic solutions of the abstract linearized Boussinesq–Love equation (λ − Δ)utt (t, x) − α(Δ − λ󸀠 )ut (t, x) = β(Δ − λ󸀠󸀠 )u(t, x) + f (t, x),

u(0, x) = u0 (x),

ut (0, x) = u1 (x),

(t, x) ∈ ℝ × Ω;

u(t, x) = 0,

t ∈ ℝ, x ∈ Ω,

(t, x) ∈ ℝ × 𝜕Ω,

where λ, λ󸀠 , λ󸀠󸀠 ∈ ℝ, α, β ∈ ℝ and α, β ≠ 0. The existence and uniqueness of asymptotically almost periodic solutions of the following fractional semilinear equation with higher order differential operators in the Hölder space X = C α (Ω): γ

β {Dt u(t, x) = − ∑ aβ (t, x)D u(t, x) − σu(t, x) + f (t, u(t, x)), |β|⩽2m { {u(0, x) = u0 (x),

t ⩾ 0, x ∈ Ω; x ∈ Ω,

with some assumptions introduced by W. von Wahl [395] being satisfied, have been studied in Example 3.10.4.

Introduction

| XLI

The important contributions are given in the analysis of existence and uniqueness of almost periodic (automorphic) solutions of semilinear Poisson heat equation { 𝜕 [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)), { 𝜕t {v(t, x) = 0,

t ∈ ℝ, x ∈ Ω;

(t, x) ∈ [0, ∞) × 𝜕Ω,

asymptotically almost periodic (automorphic) solutions of semilinear Poisson heat equation 𝜕 { { { 𝜕t [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)), { v(t, x) = 0, { { m(x)v(0, x) = u0 (x), {

t ⩾ 0, x ∈ Ω;

(t, x) ∈ [0, ∞) × 𝜕Ω, x ∈ Ω,

asymptotically almost periodic (automorphic) solutions of the following fractional Poisson semilinear heat equation with Caputo fractional derivatives γ

D [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)), { { t v(t, x) = 0, { { {m(x)v(0, x) = u0 (x),

t ⩾ 0, x ∈ Ω; (t, x) ∈ [0, ∞) × 𝜕Ω, x ∈ Ω,

and almost periodic (automorphic) solutions of the following fractional Poisson semilinear heat equation with Weyl–Liouville fractional derivatives: γ

Dt,+ [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)),

t ∈ ℝ, x ∈ Ω

in the space Lp (Ω), where 1 < p < ∞; see Example 3.10.5 for more details. Let Δ denote the Dirichlet Laplacian in X := L2 [0, π], acting with the domain 2 H [0, π] ∩ H01 [0, π]. In Example 3.10.11, we consider the existence and uniqueness of Besicovitch almost automorphic solutions of the following nonautonomous onedimensional heat equation with Dirichlet (sometimes also called clamped) boundary conditions: ut (t, x) = uxx (t, x) + q(t, x)u(t, x) + f (t, x), u(0) = u(π) = 0,

t ⩾ 0, x ∈ [0, π];

u(0, x) = u0 (x) ∈ X,

where q : ℝ × [0, π] → ℝ is a jointly continuous function satisfying that q(t, x) ⩽ −γ0 , (t, x) ∈ ℝ×[0, π], for some number γ0 > 0. In order to do that, we employ the approach of T. Diagana [137, Section 4]. Concerning possible applications of C-regularized resolvent operator families, it is worth noting that, in Example 3.10.6, we investigate the existence and uniqueness of asymptotically almost periodic solutions of the following semilinear Cauchy problem: γ

(α) {Dt u(t, x) = ∑ aα f (t, x) + f (t, u(t, x)), |α|⩽k { {u(0, x) = u0 (x),

t ⩾ 0, x ∈ ℝn , x ∈ ℝn .

in the space Lp (ℝn ), where 1 ⩽ p < ∞, with some extra assumptions being satisfied. In addition to the above, we propose several open problems to our researchers.

1 Preliminaries 1.1 Fundamentals of the theory of operators and integration theory In this section, we recall the basic things about vector-valued functions, closed operators, integration in Banach spaces, as well as Sobolev spaces and fixed point theorems. Vector-valued functions, closed operators. Unless specified otherwise, by X we denote a Banach space over the field of complex numbers; the norm of an element x ∈ X is denoted by ‖x‖. Assuming that Y is another complex Banach space, by L(X, Y) we denote the space consisting of all continuous linear mappings from X into Y; L(X) ≡ L(X, X). The topology on L(X, Y) and X ∗ , the dual space of X, are introduced as usual. If not stated otherwise, by I we denote the identity operator on X. A linear operator A : D(A) → X is said to be closed iff the graph of the operator A, defined by GA := {(x, Ax) : x ∈ D(A)}, is a closed subset of X × X. Since no confusion seems likely, we will identify A and its graph. The null space and range of A are denoted by N(A) and R(A), respectively. It is well known that a linear operator A : D(A) → X is closed iff, for every sequence (xn ) in D(A) such that limn→∞ xn = x and limn→∞ Axn = y, the following holds: x ∈ D(A) and Ax = y. A linear operator A is called closable iff there exists a closed linear operator B such that A ⊆ B. Let us remind ourselves that the closability of the operator A is equivalent to saying that, for every sequence (xn ) in D(A) such that limn→∞ xn = 0 and limn→∞ Axn = y, we have y = 0. If A is closed, then we introduce the graph norm on D(A) by ‖x‖A =: ‖x‖ + ‖Ax‖, x ∈ D(A). The obtained Banach space is simply labeled by [D(A)]; a subspace Y ⊆ D(A) is called a core for A iff Y is dense in D(A) with respect to the graph norm. Let C ∈ L(X). Then the C-resolvent set of A, ρC (A) for short, is defined by ρC (A) := {λ ∈ ℂ : λ − A is injective and (λ − A)−1 C ∈ L(X)}; the resolvent set and spectrum of A are defined by ρ(A) := ρI (A) and σ(A) := ℂ ∖ ρ(A), respectively. If F is a linear submanifold of X, then the part of A in F, denoted by A|F , is a linear operator defined by D(A|F ) := {x ∈ D(A) ∩ F : Ax ∈ F} and A|F x := Ax, x ∈ D(A|F ). The power An of A is defined inductively (n ∈ ℕ0 ). Set D∞ (A) := ⋂n⩾1 D(An ). For a closed linear operator A acting on X, we introduce the subset A∗ of X ∗ × X ∗ by A∗ := {(x∗ , y∗ ) ∈ X ∗ × X ∗ : x∗ (Ax) = y∗ (x) for all x ∈ D(A)}. If A is densely defined, then A∗ is also known as the adjoint operator of A; it is a closed linear operator on X ∗ . Assuming α ∈ ℂ ∖ {0}, A and B are linear operators, we define the operators αA, A + B and AB in the usual way. The Gamma function will be denoted https://doi.org/10.1515/9783110641851-001

2 | 1 Preliminaries by Γ(⋅) and the principal branch will be always used to take the powers. Set, for every α > 0, gα (t) := t α−1 /Γ(α),

t > 0,

g0 (t) ≡ the Dirac delta distribution and 0ζ := 0. The nth convolution power of a locally integrable function a(t) ∈ L1loc ([0, ∞)) is denoted by a∗n (t). Then a∗n ∈ L1loc ([0, ∞)). For any two numbers s ∈ ℝ and n ∈ ℕ, we define ⌊s⌋ := sup{l ∈ ℤ : s ⩾ l}, ⌈s⌉ := inf{l ∈ ℤ : s ⩽ l}, ℕn := {1, . . . , n} and ℕ0n := {0, 1, . . . , n}. If X, Y ≠ 0, then we set YX := {f | f : X → Y}. Let I = ℝ or I = [0, ∞). By Cb (I : X) we denote the space consisting of all bounded continuous functions from I into X; the symbol C0 ([0, ∞) : X) denotes the closed subspace of Cb ([0, ∞) : X) consisting of functions vanishing at infinity. By BUC(I : X) we denote the space consisting of all bounded uniformly continuous functions from I to X; Cb (I) ≡ Cb (I : ℂ), C0 ([0, ∞)) ≡ C0 ([0, ∞) : ℂ) and BUC(I) ≡ BUC(I : ℂ). The sup-norm turns these spaces into Banach spaces. We refer the reader to [29] and [247] for the fundamental properties of analytical functions with values in Banach and locally convex spaces. Integration in Banach spaces. First of all, we recall the following elementary definition. Definition 1.1.1. (i) A function f : I → X is said to be simple iff there exist k ∈ ℕ, elements zi ∈ X, 1 ⩽ i ⩽ k and Lebesgue measurable subsets Ωk , 1 ⩽ i ⩽ k of I, such that m(Ωi ) < ∞, 1 ⩽ i ⩽ k and k

f (t) = ∑ zi χΩi (t), i=1

t ∈ I.

(13)

(ii) A function f : I → X is said to be measurable iff there exists a sequence (fn ) in X I such that, for every n ∈ ℕ, fn (⋅) is a simple function and limn→∞ fn (t) = f (t) for a. e. t ∈ I. (iii) A function f : I → X is said to be weakly measurable iff for every x∗ ∈ X ∗ , the function t 󳨃→ x∗ (f (t)), t ∈ I is measurable. (iv) Let −∞ < a < b < ∞ and a < τ ⩽ ∞. A function f : [a, b] → X is said to be absolutely continuous iff for every ε > 0 there exists a number δ > 0 such that for any finite collection of open subintervals (ai , bi ), 1 ⩽ i ⩽ k of [a, b] with ∑ki=1 (bi − ai ) < δ, we have ∑ki=1 ‖f (bi ) − f (ai )‖ < ε; a function f : [a, τ) → X is said to be absolutely continuous iff for every τ0 ∈ (a, τ), the function f|[a,τ0 ] : [a, τ0 ] → X is absolutely continuous. Assume that f : I → X and (fn ) is a sequence of measurable functions satisfying limn→∞ fn (t) = f (t) for a. e. t ∈ I. Then f (⋅) is measurable, as well. The Bochner integral

1.1 Fundamentals of the theory of operators and integration theory | 3

of a simple function f : I → X, f (t) = ∑ki=1 zi χΩi (t), t ∈ I is defined by k

∫ f (t) dt := ∑ zi m(Ωi ). i=1

I

It is easily seen that the definition of the Bochner integral does not depend on the representation (13). A measurable function f : I → X is said to be Bochner integrable iff there exists a sequence of simple functions (fn ) in X I such that limn→∞ fn (t) = f (t) for a. e. t ∈ I and 󵄩 󵄩 lim ∫󵄩󵄩󵄩fn (t) − f (t)󵄩󵄩󵄩 dt = 0;

n→∞

(14)

I

if this is the case, the Bochner integral of f (⋅) is defined by ∫ f (t) dt := lim ∫ fn (t) dt. I

n→∞

I

The definition of Bochner integrability of a measurable function is meaningful and is independent of the choice of a sequence of simple functions (fn ) in X I satisfying limn→∞ fn (t) = f (t) for a. e. t ∈ I and (14). It is well known that f : I → X is Bochner integrable iff f (⋅) is measurable and the function t 󳨃→ ‖f (t)‖, t ∈ I is integrable as well ∞ as that, for every Bochner integrable function f : [0, ∞) → X, one has ∫0 f (t) dt = τ

limτ→+∞ ∫0 f|[0,τ] (t) dt. The space of all Bochner integrable functions from I into X is designated by L1 (I : X); equipped with the norm ‖f ‖1 := ∫I ‖f (t)‖ dt, L1 (I : X) becomes a Banach space. It is said that a function f : [0, ∞) → X is locally (Bochner) integrable iff f (⋅)|[0,τ] is Bochner integrable for every τ > 0. The space of all locally integrable functions from [0, ∞) into X is denoted by L1loc ([0, ∞) : X). If no confusion seems likely, we will not distinguish a function and its restriction to any subinterval of its domain. The dominated convergence theorem and Fubini theorem are stated below. Theorem 1.1.2. (i) Suppose that (fn ) is a sequence of Bochner integrable functions from X I and that there exists an integrable function g : I → ℝ such that ‖fn (t)‖ ⩽ g(t) for a. e. t ∈ I and n ∈ ℕ. If f : I → X and limn→∞ fn (t) = f (t) for a. e. t ∈ I, then f (⋅) is Bochner integrable, ∫I f (t) dt = limn→∞ ∫I fn (t) dt and limn→∞ ∫I ‖fn (t) − f (t)‖ dt = 0. (ii) Let I1 and I2 be segments in ℝ and let I = I1 × I2 . Suppose that F : I → X is measurable and ∫I ∫I ‖f (s, t)‖ dt ds < ∞. Then f (⋅, ⋅) is Bochner integrable, the re1

2

1

2

peated integrals ∫I ∫I f (s, t) dt ds and ∫I ∫I f (s, t) ds dt exist and equal to the inte-

gral ∫I f (s, t) ds dt.

2

1

Let 1 ⩽ p < ∞ and let (Ω, ℛ, μ) be a measure space. By Lp (Ω : X) we denote the space consisting of all strongly μ-measurable functions f : Ω → X such that ‖f ‖p :=

4 | 1 Preliminaries (∫Ω ‖f (⋅)‖p dμ)1/p is finite. The space L∞ (Ω : X) consists of all strongly μ-measurable, essentially bounded functions; as it is well known, this space becomes a Banach space when equipped with the norm ‖f ‖∞ := ess supt∈Ω ‖f (t)‖, f ∈ L∞ (Ω : X). At this point, we identify functions that are equal μ-almost everywhere on Ω. By the Riesz–Fischer theorem, (Lp (Ω : X), ‖⋅‖p ) is a Banach space for all p ∈ [1, ∞]; moreover, (L2 (Ω : X), ‖⋅‖2 ) is a Hilbert space. If limn→∞ fn = f in Lp (Ω : X), then there exists a subsequence (fnk ) of (fn ) such that limk→∞ fnk (t) = f (t) μ-almost everywhere. In the case when the Banach space X is reflexive, Lp (Ω : X) is reflexive for all p ∈ (1, ∞) and its dual is isometrically p isomorphic to L p−1 (Ω : X). The fundamental properties of vector-valued absolutely continuous functions are collected in the following proposition. Proposition 1.1.3. t (i) Suppose −∞ < a < b < ∞, f ∈ L1 ([a, b] : X) and F(t) := ∫0 f (s) ds, t ∈ [a, b]. Then t+h

F(⋅) is absolutely continuous, F 󸀠 (t) = f (t) for a. e. t ∈ [a, b] and limh→0 h1 ∫t ‖f (s) − f (t)‖ ds = 0 for a. e. t ∈ [a, b], i. e., almost every point of [a, b] is a Lebesgue point of the function f (⋅). Furthermore, if f ∈ C([a, b] : X), then the above equality holds for all t ∈ [a, b]. (ii) Suppose −∞ < a < b < ∞, F : [a, b] → X is absolutely continuous and F 󸀠 (t) exists for a. e. t ∈ [a, b]. Then F 󸀠 (⋅) is Bochner integrable on [a, b] and F(t) = F(a) + t ∫a F 󸀠 (s) ds, t ∈ [a, b].

The space consisting of all X-valued functions that are absolutely continuous (of bounded variation) on any closed subinterval of [0, ∞) will be denoted by ACloc ([0, ∞) : X) (BVloc ([0, ∞) : X)). By C k (Ω : X) we denote the space of k-times continuously differentiable functions (k ∈ ℕ0 ) from a nonempty subset Ω ⊆ ℂ into X, C(Ω : X) ≡ C 0 (Ω : X). In the case that X = ℂ, then we also write ACloc ([0, ∞)) (BVloc ([0, ∞))) in place of ACloc ([0, ∞) : X) (BVloc ([0, ∞) : X)); the spaces BV[0, T], BVloc ([0, τ)), BVloc ([0, τ) : X) and the space Lploc (Ω : X) for 1 ⩽ p ⩽ ∞ are defined in a very similar way (T, τ > 0); Lploc (Ω) ≡ Lploc (Ω : ℂ). Sobolev spaces. Suppose that k ∈ ℕ, p ∈ [1, ∞] and Ω is an open nonempty subset of ℝn . Then the Sobolev space W k,p (Ω : X), sometimes also denoted by H k,p (Ω : X), consists of those X-valued distributions u ∈ 𝒟󸀠 (Ω : X) such that, for every i ∈ ℕ0k and for every multiindex α ∈ ℕn0 with |α| ⩽ k, we have Dα u ∈ Lp (Ω, X). Of course, the derivative Dα is taken in the sense of distributions. The subspace of 𝒟󸀠 (Ω : X) consisting of all X-valued distributions of the form u = ∑ u(α) α , |α|⩽k

(15)

1.1 Fundamentals of the theory of operators and integration theory | 5

where uα ∈ Lp (Ω : X), is denoted by W −k,p (Ω : X) (H −k,p (Ω : X)). In this space, we introduce the norm 1/p

‖u‖−k,p,X := inf{( ∑ ‖uα ‖pLp (Ω,X) )

},

|α|⩽k

where the infimum is taken over all representations of distribution u of form (15); W −k (Ω : X) ≡ W −k,2 (Ω : X) (H −k (Ω : X) ≡ H −k,2 (Ω : X)). Let us recall that W −k,p (Ω : X) k,p k,p (Ω : X)) we (Ω : X) (Hloc is a Banach space and W −k (Ω : X) is a Hilbert space. By Wloc 󸀠 denote the space of those X-valued distributions u ∈ 𝒟 (Ω : X) such that, for every bounded open subset Ω󸀠 of Ω, one has u|Ω󸀠 ∈ W k,p (Ω󸀠 : X). Fixed point theorems. Throughout the book, we use a few basic results from the fixed point theory. We will first remind ourselves of the Banach contraction principle, stated for the first time by Stefan Banach in 1922. Assume that (E, d) is a metric space. Then a mapping T : E → E is called a contraction mapping on E iff there exists a constant q ∈ [0, 1) such that d(T(x), T(y)) ⩽ qd(x, y) for all x, y ∈ E. Theorem 1.1.4 (Banach Fixed Point Theorem, 1922). Let (E, d) be a complete metric space, and let T : E → E be a contraction mapping. Then T admits a unique fixed point x in X (i. e., T(x) = x). There exists a vast amount of mathematical literature concerning various generalizations of the Banach contraction principle. We will use the following: Theorem 1.1.5 (Bryant Fixed Point Theorem, 1968). Let (E, d) be a complete metric space, and let T : E → E be such that there is an integer n ∈ ℕ such that T n : E → E is a contraction mapping. Then T has a unique fixed point x in E. Theorem 1.1.6 (Weissinger Fixed Point Theorem, 1952). Let (E, d) be a complete metric space, and let T : E → E be a continuous mapping. Assume that for each integer n ∈ ℕ we have the existence of a positive number θn so that d(T n x, T n y) ⩽ θn d(x, y),

x, y ∈ E, n ∈ ℕ

and ∞

∑ θn < ∞.

n=1

Then T has a unique fixed point x in E. In the existing literature, there are some controversies about Theorem 1.1.6 because somewhere it is also attributed to the Italian mathematician R. Caccioppoli, due to its paper published in Rend. Acad. Naz. Linzei as early as 1930. For further information about the fixed point theory, the reader may consult the monographs [12] and [202, 203].

6 | 1 Preliminaries

1.2 Multivalued linear operators The main aim of this section is to present a brief overview of the necessary definitions and results from the theory of multivalued linear operators that will be necessary for our further work. Assume that X and Y are two Banach spaces. A multivalued map (multimap) 𝒜 : X → P(Y) is said to be a multivalued linear operator (MLO) iff the following holds: (i) D(𝒜) := {x ∈ X : 𝒜x ≠ 0} is a linear subspace of X; (ii) 𝒜x + 𝒜y ⊆ 𝒜(x + y), x, y ∈ D(𝒜) and λ𝒜x ⊆ 𝒜(λx), λ ∈ ℂ, x ∈ D(𝒜). In the case that X = Y, then we say that 𝒜 is an MLO in X. It is well known that for any x, y ∈ D(𝒜) and λ, η ∈ ℂ with |λ| + |η| ≠ 0, we have λ𝒜x + η𝒜y = 𝒜(λx + ηy). If 𝒜 is an MLO, then 𝒜0 is a linear manifold in Y and 𝒜x = f + 𝒜0 for any x ∈ D(𝒜) and f ∈ 𝒜x. Define R(𝒜) := {𝒜x : x ∈ D(𝒜)}. The set 𝒜−1 0 := {x ∈ D(𝒜) : 0 ∈ 𝒜x} is called the kernel of 𝒜 and it is denoted henceforth by N(𝒜). The inverse 𝒜−1 of an MLO is defined by D(𝒜−1 ) := R(𝒜) and 𝒜−1 y := {x ∈ D(𝒜) : y ∈ 𝒜x}. It is easily seen that 𝒜−1 is an MLO in X, as well as that N(𝒜−1 ) = 𝒜0 and (𝒜−1 )−1 = 𝒜. If N(𝒜) = {0}, i. e., if 𝒜−1 is single-valued, then 𝒜 is said to be injective. Let us recall that 𝒜x = 𝒜y for some two elements x and y ∈ D(𝒜), iff 𝒜x ∩ 𝒜y ≠ 0; moreover, if 𝒜 is injective, then the equality 𝒜x = 𝒜y holds iff x = y. For any mapping 𝒜 : X → P(Y) we set 𝒜̌ := {(x, y) : x ∈ D(𝒜), y ∈ 𝒜x}. Then it can be simply shown that 𝒜 is an MLO iff 𝒜̌ is a linear relation in X × Y, i. e., iff 𝒜̌ is a linear subspace of X × Y. Assuming that 𝒜, ℬ : X → P(Y) are two MLOs, we define their sum 𝒜 + ℬ by D(𝒜 + ℬ) := D(𝒜) ∩ D(ℬ) and (𝒜 + ℬ)x := 𝒜x + ℬx, x ∈ D(𝒜 + ℬ). It is evident that 𝒜 + ℬ is likewise an MLO. Let 𝒜 : X → P(Y) and ℬ : Y → P(Z) be two MLOs, where Z is a complex Banach space. The product of 𝒜 and ℬ is defined by D(ℬ𝒜) := {x ∈ D(𝒜) : D(ℬ) ∩ 𝒜x ≠ 0} and ℬ𝒜x := ℬ(D(ℬ) ∩ 𝒜x). It can be simply checked that ℬ𝒜 : X → P(Z) is an MLO and (ℬ𝒜)−1 = 𝒜−1 ℬ−1 . The scalar multiplication of an MLO 𝒜 : X → P(Y) with the number z ∈ ℂ, z 𝒜 for short, is defined by D(z 𝒜) := D(𝒜) and (z 𝒜)(x) := z 𝒜x, x ∈ D(𝒜). Assume that X 󸀠 is a linear subspace of X, and 𝒜 : X → P(Y) is an MLO. Then we define the restriction of operator 𝒜 to the subspace X 󸀠 , 𝒜|X 󸀠 for short, by D(𝒜|X 󸀠 ) := D(𝒜) ∩ X 󸀠 and 𝒜|X 󸀠 x := 𝒜x, x ∈ D(𝒜|X 󸀠 ). It is clear that 𝒜|X 󸀠 : X → P(Y) is an MLO. The integer powers of an MLO 𝒜 : X → P(X) are defined inductively as follows: 𝒜0 =: I; if 𝒜n−1 is defined, set D(𝒜n ) := {x ∈ D(𝒜n−1 ) : D(𝒜) ∩ 𝒜n−1 x ≠ 0}, and n

𝒜 x := (𝒜𝒜

n−1

)x =

⋃

y∈D(𝒜)∩𝒜n−1 x

𝒜y,

x ∈ D(𝒜n ).

1.2 Multivalued linear operators | 7

It is easily proved that (𝒜n )−1 = (𝒜n−1 )−1 𝒜−1 = (𝒜−1 )n =: 𝒜−n , n ∈ ℕ and D((λ − 𝒜)n ) = D(𝒜n ), n ∈ ℕ0 , λ ∈ ℂ. Moreover, if 𝒜 is single-valued, then the above definitions are consistent with the usual definition of powers of 𝒜. Assume that 𝒜 : X → P(Y) and ℬ : X → P(Y) are two MLOs. Then the inclusion 𝒜 ⊆ ℬ is equivalent to saying that D(𝒜) ⊆ D(ℬ) and 𝒜x ⊆ ℬx for all x ∈ D(𝒜). If a linear single-valued operator S : D(S) ⊆ X → Y has domain D(S) = D(𝒜) and S ⊆ 𝒜, where 𝒜 : X → P(Y) is an MLO, S is called a section of 𝒜; in this case, we have 𝒜x = Sx + 𝒜0, x ∈ D(𝒜) and R(𝒜) = R(S) + 𝒜0. We say that an MLO operator 𝒜 : X → P(Y) is closed iff for any nets (xτ ) in D(𝒜) and (yτ ) in Y such that yτ ∈ 𝒜xτ for all τ ∈ I we have that the suppositions limτ→∞ xτ = x and limτ→∞ yτ = y imply x ∈ D(𝒜) and y ∈ 𝒜x. Because we are working only with Banach spaces, the nets can be replaced by sequences here. Let 𝒜 : X → P(Y) be an MLO. Then 𝒜 : X → P(Y) is an MLO, as well, so that any MLO has a closed linear extension, in contrast to the usually considered single-valued linear operators. We define the adjoint 𝒜∗ : Y ∗ → P(X ∗ ) of 𝒜 by ∗

∗

∗

∗

∗

∗

∗

𝒜 := {(y , x ) ∈ Y × X : ⟨y , y⟩ = ⟨x , x⟩ for all pairs (x, y) ∈ 𝒜}.

It can be easily checked that 𝒜∗ is a closed MLO, as well as that ⟨y∗ , y⟩ = 0 whenever y∗ ∈ D(𝒜∗ ) and y ∈ 𝒜0. The following lemma can be deduced by means of the Hahn–Banach theorem. Lemma 1.2.1. Suppose that 𝒜 : X → P(Y) is a closed MLO. Assume, further, that x0 ∈ X, y0 ∈ Y and ⟨x∗ , x0 ⟩ = ⟨y∗ , y0 ⟩ for all pairs (x∗ , y∗ ) ∈ X ∗ × Y ∗ satisfying that ⟨x ∗ , x⟩ = ⟨y∗ , y⟩, whenever y ∈ 𝒜x. Then y0 ∈ 𝒜x0 . Let Ω denote a locally compact and separable metric space, and let μ denote a locally finite Borel measure defined on Ω. We need the following important result from [247]: Theorem 1.2.2. Suppose that 𝒜 : X → P(Y) is a closed MLO. Let f : Ω → X and g : Ω → Y be μ-integrable, and let g(x) ∈ 𝒜f (x), x ∈ Ω. Then ∫Ω f dμ ∈ D(𝒜) and ∫Ω g dμ ∈ 𝒜 ∫Ω f dμ. Now we will examine the C-resolvent sets of MLOs in Banach spaces. Our standing assumptions here will be that 𝒜 is an MLO in X, as well as that C ∈ L(X) is injective and C 𝒜 ⊆ 𝒜C (the injectivity of C will be not assumed a priori in future). The C-resolvent set of 𝒜, ρC (𝒜) for short, is defined as the union of those complex numbers λ ∈ ℂ for which (i) R(C) ⊆ R(λ − 𝒜); (ii) (λ − 𝒜)−1 C is a single-valued linear continuous operator on X. The operator λ 󳨃→ (λ − 𝒜)−1 C is called the C-resolvent of 𝒜 (λ ∈ ρC (𝒜)); the resolvent set of 𝒜 is defined by ρ(𝒜) := ρI (𝒜), R(λ : 𝒜) ≡ (λ − 𝒜)−1 (λ ∈ ρ(𝒜)). If ρC (𝒜) ≠ 0, then

8 | 1 Preliminaries for each λ ∈ ρC (𝒜) we have 𝒜0 = N((λI − 𝒜)−1 C), as well as λ ∈ ρC (𝒜), 𝒜 ⊆ C −1 𝒜C and ((λ − 𝒜)−1 C)k (D(𝒜l )) ⊆ D(𝒜k+l ), k, l ∈ ℕ0 ; the equality 𝒜 = C −1 𝒜C holds provided, in addition, that ρ(𝒜) ≠ 0. The basic properties of C-resolvent sets of single-valued linear operators (see [245, 246]) continue to hold in our framework (observe, however, that there exist certain differences that we will not discuss here). For example, if ρ(𝒜) ≠ 0, then 𝒜 is closed. We have the following (cf. [247]): Theorem 1.2.3. (i) The next inclusions hold: (λ − 𝒜)−1 C 𝒜 ⊆ λ(λ − 𝒜)−1 C − C ⊆ 𝒜(λ − 𝒜)−1 C,

λ ∈ ρC (𝒜).

The operator (λ − 𝒜)−1 C 𝒜 is single-valued on D(𝒜) and (λ − 𝒜)−1 C 𝒜x = (λ − 𝒜)−1 Cy, whenever y ∈ 𝒜x and λ ∈ ρC (𝒜). (ii) Suppose that λ, μ ∈ ρC (𝒜). Then the resolvent equation (λ − 𝒜)−1 C 2 x − (μ − 𝒜)−1 C 2 x = (μ − λ)(λ − 𝒜)−1 C(μ − 𝒜)−1 Cx,

x∈X

holds. In particular, [(λ − 𝒜)−1 C][(μ − 𝒜)−1 C] = [(μ − 𝒜)−1 C][(λ − 𝒜)−1 C]. It is well known that ρC (A) need not be an open subset of ℂ if C ≠ I and A is a single-valued linear operator in X, as well as that ρ(𝒜) is always an open subset of ℂ, if 𝒜 is an MLO in X (cf. [188, Theorem 1.6]). The generalized resolvent equations hold for C-resolvents of multivalued linear operators as well; the results treating the analytical properties of C-resolvents of multivalued linear operators can be found in [247].

1.2.1 Fractional powers In this subsection, we assume that (−∞, 0] ⊆ ρ(𝒜) and that there exist finite numbers M ⩾ 1 and β ∈ (0, 1] such that −β 󵄩 󵄩󵄩 󵄩󵄩R(λ : 𝒜)󵄩󵄩󵄩 ⩽ M(1 + |λ|) ,

λ ⩽ 0.

Then there are two positive numbers c > 0 and M1 > 0 such that the resolvent set of 𝒜 contains an open region Ω = {λ ∈ ℂ : | Im λ| ⩽ (2M1 )−1 (c − Re λ)β , Re λ ⩽ c} of complex plane around the half-line (−∞, 0], where we have the estimate ‖R(λ : 𝒜)‖ = O((1 + |λ|)−β ), λ ∈ Ω. Let Γ󸀠 be the upwards oriented curve {ξ ± i(2M1 )−1 (c − ξ )β : −∞ < ξ ⩽ c}. In [188], A. Favini and A. Yagi define the fractional power 𝒜

−θ

:=

1 ∫ λ−θ (λ − 𝒜)−1 dλ ∈ L(X), 2πi Γ󸀠

1.3 Laplace transform of functions with values in Banach spaces | 9

for θ > 1 − β. Set 𝒜θ := (𝒜−θ )−1 (θ > 1 − β). Then the semigroup properties 𝒜−θ1 𝒜−θ2 = 𝒜−(θ1 +θ2 ) and 𝒜θ1 𝒜θ2 = 𝒜θ1 +θ2 hold for θ1 , θ2 > 1 − β (let us recall that the fractional power 𝒜θ need not be injective and that the meaning of 𝒜θ is understood in the MLO sense for θ > 1 − β). We equip the vector space D(𝒜) with the norm ‖ ⋅ ‖[D(𝒜)] := inf ‖y‖. y∈𝒜⋅

As it is well known, (D(𝒜), ‖ ⋅ ‖[D(𝒜)] ) is a Banach space and, since 0 ∈ ρ(𝒜), the norm ‖ ⋅ ‖[D(𝒜)] is equivalent to ‖ ⋅ ‖ + ‖ ⋅ ‖[D(𝒜)] . Since 0 ∈ ρ(𝒜θ ), (D(𝒜θ ), ‖ ⋅ ‖[D(𝒜θ )] ) is likewise a Banach space, and we have the equivalence of norms ‖ ⋅ ‖[D(𝒜θ )] and ‖ ⋅ ‖ + ‖ ⋅ ‖[D(𝒜θ )] for θ > 1 − β. For any θ ∈ (0, 1), the vector space 󵄩 󵄩 θ X𝒜 := {x ∈ X : sup ξ θ 󵄩󵄩󵄩ξ (ξ + 𝒜)−1 x − x󵄩󵄩󵄩 < ∞}, ξ >0

endowed with the norm 󵄩 󵄩 ‖ ⋅ ‖X θ := ‖ ⋅ ‖ + sup ξ θ 󵄩󵄩󵄩ξ (ξ + 𝒜)−1 ⋅ −⋅󵄩󵄩󵄩, 𝒜

ξ >0

θ becomes a Banach space. It is clear that X𝒜 is continuously embedded in X. We refer the reader to [187], [339] and [247, Section 2.12] for further information concerning interpolation spaces and fractional powers of multivalued linear operators. Because of its importance, we recall condition (P) from the introductory part: (P) There exist finite constants c, M > 0 and β ∈ (0, 1] such that

Ψ := Ψc := {λ ∈ ℂ : Re λ ⩾ −c(| Im λ| + 1)} ⊆ ρ(𝒜) and −β 󵄩 󵄩󵄩 󵄩󵄩R(λ : 𝒜)󵄩󵄩󵄩 ⩽ M(1 + |λ|) ,

λ ∈ Ψ.

1.3 Laplace transform of functions with values in Banach spaces Unquestionably, the most important monograph regarding Laplace transform of functions with values in Banach spaces is written by W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander [29]. We warmly recommend reading this monograph for any researcher endeavoring to acquire the basic skills and knowledge about Laplace transform and its applications.

10 | 1 Preliminaries Let f : [0, ∞) → X. We examine the existence of Laplace integral τ

∞

(ℒf )(λ) := f ̃(λ) := ∫ e−λt f (t) dt := lim ∫ e−λt f (t) dt, 0

τ→∞

0

for λ ∈ ℂ. If f ̃(λ0 ) exists for some λ0 ∈ ℂ, then we define the abscissa of convergence of f ̃(⋅) by absX (f ) := inf{Re λ : f ̃(λ) exists}; otherwise, absX (f ) := +∞. We say that f (⋅) is Laplace transformable, or equivalently, that f (⋅) belongs to the class (P1)-X, iff absX (f ) < ∞. Set (P1)≡ (P1)-ℂ. If there exist finite numbers ω ∈ ℝ and M > 0 such that ‖f (t)‖ ⩽ Meωt , t ⩾ 0, then we define ωX (f ) ∈ [−∞, ∞) as the infimum of all numbers ω ∈ ℝ with the above property; if such a number ω ∈ ℝ does not exist, then we define ωX (f ) := +∞. Further on, we abbreviate ωX (f ) (absX (f )) to ω(f ) (abs(f )), if no confusion seems likely. Set 󵄨󵄨 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 −λs ∗ 󵄨 ⟨x , f (s)⟩ ds󵄨󵄨󵄨 < ∞ for all x∗ ∈ X ∗ }, w abs(f ) := inf{λ ∈ ℝ : sup󵄨󵄨󵄨∫ e 󵄨󵄨 t>0 󵄨󵄨 󵄨 0 F∞ := limτ→∞ F(τ), if the limit exists in X, and F∞ := 0, otherwise. The following fundamental results hold true: Theorem 1.3.1. Let f ∈ L1loc ([0, ∞) : X). Then the following holds: (i) The Laplace integral f ̃(λ) converges if Re λ > abs(f ) and diverges if Re λ < abs(f ). If Re λ = abs(f ), then the Laplace integral may or may not be convergent. (ii) w abs(f ) = abs(f ). t (iii) Suppose that λ ∈ ℂ and the limit limt→+∞ ∫0 e−λs ‖f (s)‖ ds exists. Then f ̃(λ) exists as well. (iv) We have abs(f ) ⩽ abs(‖f ‖) ⩽ ω(f ). In general, any of these two inequalities can be strict. (v) We have abs(f ) = ω(F − F∞ ), ∞

f ̃(λ) = F∞ + λ ∫ e−λt (F(t) − F∞ ) dt,

Re λ > ω(F − F∞ ),

0

̃ f ̃(λ) = λF(λ),

Re λ > max(abs(f ), 0)

and abs(f ) ⩽ ω ⇐⇒ ω(F) ⩽ ω

(if ω ⩾ 0).

In particular, f (⋅) is Laplace transformable iff ω(F) < ∞.

1.3 Laplace transform of functions with values in Banach spaces | 11

(vi) The mapping λ 󳨃→ f ̃(λ), Re λ > abs(f ) is analytic, provided that f ∈ (P1)-X. If this is the case, the following formula holds: ∞

dn ̃ f (λ) = (−1)n ∫ e−λt t n f (t) dt, dλn

n ∈ ℕ, λ ∈ ℂ, Re λ > abs(f ).

0

The various operational properties of vector-valued Laplace transform are collected in the following theorem. Theorem 1.3.2. Let f ∈ (P1)-X, z ∈ ℂ and s ⩾ 0. (i) Put g(t) := e−zt f (t), t ⩾ 0. Then g(⋅) is Laplace transformable, abs(g) = abs(f )−Re z ̃ and g(λ) = f ̃(λ + z), λ ∈ ℂ, Re λ > abs(f ) − Re z. (ii) Put fs (t) := f (t + s), t ⩾ 0, hs (t) := f (t − s), t ⩾ s and hs (t) := 0, s ∈ [0, t]. Then s ̃(λ) = e−λs f ̃(λ) abs(fs ) = abs(hs ) = abs(f ), f̃s (λ) = eλs (f ̃(λ) − ∫0 e−λt f (t) dt) and h s (λ ∈ ℂ, Re λ > a). ̃ (iii) Let T ∈ L(X, Y). Then T ∘ f ∈ (P1)-Y and T f ̃(λ) = (T ∘ f )(λ) for λ ∈ ℂ, Re λ > abs(f ). (iv) Suppose that 𝒜 : X → P(Y) is a closed MLO, as well as f ∈ (P1)-X, l ∈ (P1)-Y ̃ and (f (t), l(t)) ∈ 𝒜 for a. e. t ⩾ 0. Then (f ̃(λ), l(λ)) ∈ 𝒜, λ ∈ ℂ for Re λ > max(abs(f ), abs(l)). (v) Suppose, in addition, ω(f ) < ∞. Put ∞

j(t) := ∫ 0

τ

2

2

e−s /4t e−s /4t f (s) ds := lim ∫ f (s) ds, τ→∞ √πt √πt

t>0

0

and ∞

k(t) := ∫ 0

τ

2

se−s /4t 3

2√πt 2

f (s) ds := lim ∫ τ→∞

2

se−s /4t 3

2√πt 2

0

f (s) ds,

t > 0.

Then j(⋅) and k(⋅) are Laplace transformable, 2

max(abs(j), abs(k)) ⩽ (max(ω(f ), 0)) ,

̃√ ̃ = f ( λ) j(λ) √λ

̃ and k(λ) = f ̃(√λ)

for all λ ∈ ℂ with Re λ > (max(ω(f ), 0))2 . (vi) Let f ∈ (P1)-X, a ∈ L1loc ([0, ∞)) and abs(|a|) < ∞. Then the mapping t 󳨃→ (a∗f )(t) = t

∫0 a(t − s)f (s) ds, t ⩾ 0 is Laplace transformable and ̃ ̃ f ̃(λ), a ∗ f (λ) = a(λ)

λ ∈ ℂ,

Re λ > max(abs(|a|), abs(f )).

(vii) Suppose f ∈ (P1)-X and t > 0 is a Lebesgue point of f (⋅). Then f (t) = lim (−1)n n→∞

n+1

1 n ( ) n! t

n f ̃(n) ( ). t

12 | 1 Preliminaries (viii) Suppose f ∈ (P1)-X, λ0 > abs(f ) and f ̃(λ) = 0 for all λ > λ0 . Then f (t) = 0 for a. e. t ⩾ 0. (ix) Suppose that 𝒜 : X → P(Y) is a closed MLO, as well as f ∈ (P1)-X, l ∈ (P1)-Y and ̃ (f ̃(λ), l(λ)) ∈ 𝒜, λ ∈ ℂ for Re λ > max(abs(f ), abs(l)). Then l(t) ∈ 𝒜f (t) for any t > 0 which is a point of continuity for both functions f (t) and l(t). The complex inversion theorem for the vector-valued Laplace transform reads as follows. Theorem 1.3.3. Assume a ⩾ 0, q : {λ ∈ ℂ : Re λ > a} → X is analytic and there exist M M > 0 and r > 1 such that ‖q(λ)‖ ⩽ |λ| r , λ ∈ ℂ, Re λ > a. Then there exist a continuous

function f : [0, ∞) → X and a number M 󸀠 > 0 such that ‖f (t)‖ ⩽ M 󸀠 t r−1 eat , t ⩾ 0 and q(λ) = f ̃(λ) for all λ ∈ ℂ with Re λ > a. The following extension of celebrated Arendt–Widder theorem has been proved by T.-J. Xiao and J. Liang (see, e. g., [402]). Theorem 1.3.4. Let a ⩾ 0, α ∈ (0, 1], ω ∈ (−∞, a], M > 0 and let q : (a, ∞) → X be an infinitely differentiable function. Then (i) ⇐⇒ (ii), where:

(i) ‖(λ − ω)k+1 q k!(λ) ‖ ⩽ M, λ > a, k ∈ ℕ0 . (ii) There exists a function F ∈ C([0, ∞) : X) satisfying F(0) = 0, (k)

α

∞

q(λ) = λ ∫ e−λt F(t) dt,

λ > a,

0 t 󵄩󵄩 󵄩󵄩 t+h (t + h − s)−α (t − s)−α 󵄩 󵄩󵄩 F(s) ds − ∫ F(s) ds󵄩󵄩󵄩 ⩽ Mheωt max(eωh , 1), 󵄩󵄩 ∫ 󵄩󵄩 󵄩󵄩 Γ(1 − α) Γ(1 − α) 0

0

for any t ⩾ 0 and h ⩾ 0, if α ∈ (0, 1), and 󵄩 󵄩󵄩 ωt ωh 󵄩󵄩F(t + h) − F(t)󵄩󵄩󵄩 ⩽ Mhe max(e , 1),

t ⩾ 0, h ⩾ 0

if α = 1. Moreover, in this case, F(⋅) is uniquely determined and for each α ∈ (0, 1] we have 2M α 󵄩 󵄩󵄩 h max(eω(t+h) , 1), 󵄩󵄩F(t + h) − F(t)󵄩󵄩󵄩 ⩽ αΓ(α)

t ⩾ 0, h ⩾ 0.

Define Σα := {reiθ : r > 0, θ ∈ (−α, α)}, α ∈ (0, π] and ℂ+ := {z ∈ ℂ : Re z > 0}. The following important characterization of analytical properties of vector-valued Laplace transform has been proved by the Czech mathematician M. Sova [370]. Theorem 1.3.5. Let α ∈ (0, π2 ], ω ∈ ℝ and q : (ω, ∞) → X. Then the following assertions are equivalent:

1.4 Operators of fractional differentiation, Mittag-Leffler and Wright functions | 13

(i) There exists an analytic function f : Σα → X such that supz∈Σβ ‖e−ωz f (z)‖ < ∞ for all β ∈ (0, α) and q(λ) = f ̃(λ) for all λ ∈ (ω, ∞).

(ii) The function q(⋅) has an analytic extension q̃ : ω + Σ π +α → X such that 2 ̃ supλ∈ω+Σ π +γ ‖(λ − ω)q(λ)‖ < ∞ for all γ ∈ (0, α). 2

Let abs(k) = 0. Following C. J. K. Batty, J. van Neerven and F. Räbiger [52], we say that a point λ = iν ∈ iℝ is a regular point for k(t) iff there are an open neighborhood ̃ U of λ in ℂ and a holomorphic function g : U → X such that g(z) = k(z) whenever z ∈ U ∩ ℂ+ . The singular set 𝔼 of k(t) is the set consisting of all points of iℝ which are not regular points. For further information concerning the vector-valued Laplace transform, the reader may consult [402, Chapter 1] and [247].

1.4 Operators of fractional differentiation, Mittag-Leffler and Wright functions Fractional calculus and fractional differential equations are rapidly growing fields of research due to their numerous applications in engineering, physics, chemistry, biology and other sciences. For the basic information about fractional calculus and nondegenerate fractional differential equations, we refer the reader to the monographs by K. Diethelm [158], A. A. Kilbas, H. M. Srivastava, J. J. Trujillo [239], V. Kiryakova [241], F. Mainardi [320], I. Podlubny [347], S. G. Samko, A. A. Kilbas, O. I. Marichev [360] and the author [245–247], as well as to the doctoral dissertation of E. Bazhlekova [53]. Assume that α > 0, m = ⌈α⌉ and I = (0, T) for some T ∈ (0, ∞]. Then the Riemann– Liouville fractional integral Jtα of order α is defined by Jtα f (t) := (gα ∗ f )(t),

f ∈ L1 (I : X), t ∈ I.

The Caputo fractional derivative Dαt u(t) is defined for those functions u ∈ C m−1 ([0, ∞) : m X) for which gm−α ∗ (u − ∑m−1 k=0 uk gk+1 ) ∈ C ([0, ∞) : X), by Dαt u(t) :=

m−1 dm [g ∗ (u − ∑ uk gk+1 )]. dt m m−α k=0

The existence of the Caputo fractional derivative Dαt u for t ⩾ 0 implies the existence of ζ the Caputo fractional derivative Dt u for t ⩾ 0 and any ζ ∈ (0, α). The Sobolev space W m,1 (I : X) can be introduced in the following way (see, e. g., [53, p. 7]):

14 | 1 Preliminaries

W m,1 (I : X) := {f | ∃φ ∈ L1 (I : X) ∃ck ∈ ℂ (0 ⩽ k ⩽ m − 1) m−1

f (t) = ∑ ck gk+1 (t) + (gm ∗ φ)(t) for a. e. t ∈ (0, τ)}. k=0

If so, then we have φ(t) = f (m) (t) in the distributional sense, and ck = f (k) (0) (0 ⩽ k ⩽ m − 1). The Riemann–Liouville fractional derivative Dαt of order α is defined for those functions f ∈ L1 (I : X) satisfying gm−α ∗ f ∈ W m,1 (I : X) by dm m−α J f (t), dt m t

Dαt f (t) :=

t ∈ I.

It is well known that the Riemann–Liouville fractional integrals and derivatives fulfill the following equalities: β

α+β

Jtα Jt f (t) = Jt

f (t),

Dαt Jtα f (t) = f (t),

for f ∈ L1 (I : X) and m−1

Jtα Dαt f (t) = f (t) − ∑ (gm−α ∗ f )(k) (0)gα+k+1−m (t) k=0

for any f ∈ L1 (I : X) with gm−α ∗ f ∈ W m,1 (I : X). γ The Weyl–Liouville fractional derivative Dt,+ u(t) of order γ ∈ (0, 1) is defined for t

those continuous functions u : ℝ → X such that t 󳨃→ ∫−∞ g1−γ (t − s)u(s) ds, t ∈ ℝ is a well-defined continuously differentiable mapping by γ Dt,+ u(t)

t

d := ∫ g1−γ (t − s)u(s) ds, dt

t ∈ ℝ.

−∞

Set D1t,+ u(t) := −(d/dt)u(t). For more details about the Weyl–Liouville fractional derivatives, we refer the reader to the paper [329] by J. Mu, Y. Zhoa and L. Peng. β+γ β+γ Suppose now that β > 0, γ > 0 and Dt u(t) is defined. Then the equality Dt u = β

γ

Dt Dt u does not hold in general (cf. [246, Preliminaries, Section 1.3, p. 14]). Assuming m u ∈ C([0, ∞) : X), resp. u ∈ C m−1 ([0, ∞) : X) and gm−α ∗ (u − ∑m−1 k=0 uk gk+1 ) ∈ C ([0, ∞) : X), we have the following: Dαt Jtα u(t) = u(t),

t ⩾ 0,

resp.

m−1

Jtα Dαt u(t) = u(t) − ∑ u(k) (0)gk+1 (t), k=0

The Laplace transform of function Dαt u(t) can be evaluated by ∞

m−1

0

k=0

̃ − ∑ u(k) (0)λα−1−k ; ∫ e−λt Dαt u(t) dt = λα u(λ)

t ⩾ 0.

1.4 Operators of fractional differentiation, Mittag-Leffler and Wright functions | 15

cf. the identity [246, (16)] for precise formulation. At some places (for example, in Subsection 2.9.1), we will use a slightly weakened notion of Caputo fractional derivatives; any such a place will be particularly emphasized. The Mittag-Leffler and Wright functions naturally occur as solutions of fractional integro-differential equations. Assume that α > 0 and β ∈ ℝ. Then the Mittag-Leffler function Eα,β (z) is defined by zn , Γ(αn + β) n=0 ∞

Eα,β (z) := ∑

z ∈ ℂ.

Set, for short, Eα (z) := Eα,1 (z),

z ∈ ℂ.

Perhaps the most intriguing feature of function Eα (z) is that Dαt Eα (ωt α ) = ωEα (ωt α ). For α = 1/2, E1/2 (z) is the error function: E1/2 (z) = exp(z 2 )erfc(−z), and for α = 2, E2 (z) is the hyperbolic cosine: E2 (z) = cosh(√z). The asymptotic behavior of entire function Eα,β (z) is described in the following important theorem (see, e. g., [399, Theorem 1.1]): Theorem 1.4.1. Let 0 < σ < 21 π. Then, for every z ∈ ℂ ∖ {0} and m ∈ ℕ ∖ {1}, Eα,β (z) =

m−1 z −j 1 + O(|z|−m ), ∑ Zs1−β eZs − ∑ α s Γ(β − αj) j=1

where Zs is defined by Zs := z 1/α e2πis/α and the first summation is taken over all those integers s satisfying | arg(z) + 2πs| < α( π2 + σ). In the case that α ∈ (0, 2) ∖ {1}, β > 0 and N ∈ ℕ∖ {1}, we have the following special cases of Theorem 1.4.1: Eα,β (z) =

1 (1−β)/α z 1/α z e + εα,β (z), α

󵄨 󵄨󵄨 󵄨󵄨arg(z)󵄨󵄨󵄨 < απ/2,

(16)

and Eα,β (z) = εα,β (z),

󵄨 󵄨󵄨 󵄨󵄨arg(−z)󵄨󵄨󵄨 < π − απ/2,

(17)

where N−1

εα,β (z) = ∑

n=1

z −n + O(|z|−N ), Γ(β − αn)

|z| → ∞.

(18)

16 | 1 Preliminaries We will use the following Laplace transform identity: ∞

∫ e−λt t β−1 Eα,β (ωt α ) dt = 0

λα−β , λα − ω

Re λ > ω1/α ,

ω > 0.

(19)

The function t 󳨃→ Eα,β (−t), t ⩾ 0 is completely monotonic (i. e., we have that (−1)n (dn /dt n )Eα,β (−t) ⩾ 0, t ⩾ 0, n ∈ ℕ0 ) provided α ∈ (0, 1] or β ⩾ α [320]. There are many identities for the Mittag-Leffler functions which will be not necessary for our examinations. It is only worth noting that the Mittag-Leffler function Eα,β (z) can be integrally represented in the form Eα,β (z) =

λα−β eλ 1 dλ, ∫ α 2πi λ − z

z ∈ ℂ,

G

where G is a contour (the Hankel path) which starts and ends at −∞ and encircles the disc |λ| ⩽ |z|1/α counter-clockwise. Let γ ∈ (0, 1). The Wright function Φγ (⋅) is defined by Φγ (t) := ℒ−1 (Eγ (−λ))(t),

t ⩾ 0.

The Wright function Φγ (⋅) is an entire function which can be equivalently introduced by the formula (−z)n , n!Γ(1 − γ − γn) n=0 ∞

Φγ (z) = ∑ Let us recall that: (i) Φγ (t) ⩾ 0, t ⩾ 0;

z ∈ ℂ.

γ

∞

(ii) ∫0 e−λt γst −1−γ Φγ (t −γ s) dt = e−λ s , λ ∈ ℂ+ , s > 0; and (iii)

∞ ∫0 t r Φγ (t) dt

=

Γ(1+r) , Γ(1+γr)

r > −1.

The asymptotic expansion of the Wright function Φγ (⋅), as |z| → ∞ in the sector | arg(z)| ⩽ min((1 − γ)3π/2, π) − ε, is given by M−1

Φγ (z) = Y γ−1/2 e−Y ( ∑ Am Y −M + O(|Y|−M )), m=0

where Y = (1 − γ)(γ γ z)1/(1−γ) , M ∈ ℕ and Am are certain real numbers (see, e. g., [53]). The Wright function Φγ (z) can be integrally represented by the formula Φγ (z) =

1 ∫ λγ−1 exp(λ − zλγ ) dλ, 2πi G

where G is the Hankel path used above.

z ∈ ℂ,

1.5 Degenerate (a, k)-regularized C-resolvent families | 17

1.5 Degenerate (a, k)-regularized C-resolvent families In this section, we will present a brief overview of definitions and results from the theory of degenerate (a, k)-regularized C-resolvent families [247], and remind ourselves of the basic facts about well-posedness of fractional differential inclusions and abstract degenerate Volterra integral inclusions. Let X and Y be two complex Banach spaces. Let 0 < τ ⩽ ∞, α > 0, a ∈ L1loc ([0, τ)), a ≠ 0, ℱ : [0, τ) → P(Y), and let 𝒜 : X → P(Y), ℬ : X → P(Y) be two given mappings (possibly non-linear). Suppose that B is a single-valued linear operator. Of concern is the following abstract degenerate Volterra inclusion: t

ℬu(t) ⊆ 𝒜 ∫ a(t − s)u(s) ds + ℱ (t),

t ∈ [0, τ),

(20)

0

as well as the following fractional Sobolev inclusions: (DFP)R

Dα Bu(t) ∈ 𝒜u(t) + ℱ (t), { t (j) (Bu) (0) = Bxj ,

t ⩾ 0, 0 ⩽ j ⩽ ⌈α⌉ − 1,

(DFP)L

ℬDα u(t) ⊆ 𝒜u(t) + ℱ (t), { (j) t u (0) = xj ,

t ⩾ 0, 0 ⩽ j ⩽ ⌈α⌉ − 1.

and

The following general definition of various types of solutions to the abstract degenerate inclusions (20), (DFP)R and (DFP)L has been recently introduced in [247]. Definition 1.5.1. (i) We say that a function u ∈ C([0, τ) : X) is a pre-solution of (20) iff (a ∗ u)(t) ∈ D(𝒜) and u(t) ∈ D(ℬ) for t ∈ [0, τ), as well as (20) holds. By a solution of (20) we mean any pre-solution u(⋅) of (20) such that additionally there exist functions uℬ ∈ C([0, τ) : Y) and ua,𝒜 ∈ C([0, τ) : Y) satisfying uℬ (t) ∈ ℬu(t) and ua,𝒜 (t) ∈ t

𝒜 ∫0 a(t − s)u(s) ds for t ∈ [0, τ), as well as

uℬ (t) ∈ ua,𝒜 (t) + ℱ (t),

t ∈ [0, τ).

A strong solution of (20) is any function u ∈ C([0, τ) : X) such that there exist two continuous functions uℬ ∈ C([0, τ) : Y) and u𝒜 ∈ C([0, τ) : Y) satisfying uℬ (t) ∈ ℬu(t), u𝒜 (t) ∈ 𝒜u(t) for all t ∈ [0, τ), and uℬ (t) ∈ (a ∗ u𝒜 )(t) + ℱ (t),

t ∈ [0, τ).

(ii) Assume that B = ℬ is single-valued. By a p-solution of (DFP)R we mean any X-valued function t 󳨃→ u(t), t ⩾ 0 such that t 󳨃→ Dαt Bu(t), t ⩾ 0 is well-defined,

18 | 1 Preliminaries u(t) ∈ D(𝒜) for t ⩾ 0, and the requirements of (DFP)R hold; a pre-solution of (DFP)R is any p-solution of (DFP)R that is continuous for t ⩾ 0. Finally, a solution of (DFP)R is any pre-solution u(⋅) of (DFP)R such that additionally there exists a function u𝒜 ∈ C([0, ∞) : Y) satisfying u𝒜 (t) ∈ 𝒜u(t) for t ⩾ 0, and Dαt Bu(t) ∈ u𝒜 (t) + ℱ (t), t ⩾ 0. (iii) A pre-solution of (DFP)L is any continuous X-valued function t 󳨃→ u(t), t ⩾ 0 such that the term t 󳨃→ Dαt u(t), t ⩾ 0 is well defined and continuous, as well as that Dαt u(t) ∈ D(ℬ) and u(t) ∈ D(𝒜) for t ⩾ 0, and that the requirements of (DFP)L hold; a solution of (DFP)L is any pre-solution u(⋅) of (DFP)L such that additionally there exist functions uα,ℬ ∈ C([0, ∞) : Y) and u𝒜 ∈ C([0, ∞) : Y) satisfying uα,ℬ (t) ∈ ℬDαt u(t) and u𝒜 (t) ∈ 𝒜u(t) for t ⩾ 0, as well as uα,ℬ (t) ∈ u𝒜 (t) + ℱ (t), t ⩾ 0. Henceforward, it will be assumed that 𝒜 and ℬ are multivalued linear operators. It is clear that Theorem 1.2.2 can be used to prove that any strong solution of (20) is already a solution of (20), provided that 𝒜 is closed. The notion of a (pre-)solution of problems (DFP)R and (DFP)L can be similarly defined on any finite interval [0, τ) or [0, τ], where 0 < τ < ∞, thus extending the notion of a strict solution of problem (E) given on pp. 33–34 of [188] (ℬ = I, α = 1, ℱ (t) = f (t) is continuous single-valued). In [247], we have recently analyzed the following notions of solution operator families for the abstract Cauchy problem (20). Definition 1.5.2. Suppose 0 < τ ⩽ ∞, k ∈ C([0, τ)), k ≠ 0, a ∈ L1loc ([0, τ)), a ≠ 0, 𝒜 : X → P(X) is an MLO, C1 ∈ L(Y, X), and C2 ∈ L(X). (i) It is said that 𝒜 is a subgenerator of a (local, if τ < ∞) mild (a, k)-regularized (C1 , C2 )-existence and uniqueness family (R1 (t), R2 (t))t∈[0,τ) ⊆ L(Y, X) × L(X) iff the mappings t 󳨃→ R1 (t)y, t ∈ [0, τ) and t 󳨃→ R2 (t)x, t ∈ [0, τ) are continuous for every fixed x ∈ X and y ∈ Y, as well as the following conditions hold: t

(∫ a(t − s)R1 (s)y ds, R1 (t)y − k(t)C1 y) ∈ 𝒜, t

t ∈ [0, τ), y ∈ Y

and

(21)

0

∫ a(t − s)R2 (s)y ds = R2 (t)x − k(t)C2 x,

whenever t ∈ [0, τ) and (x, y) ∈ 𝒜. (22)

0

(ii) Let (R1 (t))t∈[0,τ) ⊆ L(Y, X) be strongly continuous. Then it is said that 𝒜 is a subgenerator of a (local, if τ < ∞) mild (a, k)-regularized C1 -existence family (R1 (t))t∈[0,τ) iff (21) holds. (iii) Let (R2 (t))t∈[0,τ) ⊆ L(X) be strongly continuous. Then it is said that 𝒜 is a subgenerator of a (local, if τ < ∞) mild (a, k)-regularized C2 -uniqueness family (R2 (t))t∈[0,τ) iff (22) holds. Suppose that (R1 (t), R2 (t))t∈[0,τ) is a mild (a, k)-regularized (C1 , C2 )-existence and uniqueness family with a subgenerator 𝒜. Arguing as in the nondegenerate case (cf.

1.5 Degenerate (a, k)-regularized C-resolvent families | 19

the paragraph directly preceding [246, Definition 2.8.3]), we may prove that (a ∗ R2 )(s)R1 (t)y − R2 (s)(a ∗ R1 )(t)y

= k(t)(a ∗ R2 )(s)C1 y − k(s)C2 (a ∗ R1 )(t)y,

t ∈ [0, τ), y ∈ Y.

The notion of an (a, k)-regularized C-resolvent family is introduced as follows. Definition 1.5.3. Suppose that 0 < τ ⩽ ∞, k ∈ C([0, τ)), k ≠ 0, a ∈ L1loc ([0, τ)), a ≠ 0, 𝒜 : X → P(X) is an MLO, C ∈ L(X) and C 𝒜 ⊆ 𝒜C. Then it is said that a strongly continuous operator family (R(t))t∈[0,τ) ⊆ L(X) is an (a, k)-regularized C-resolvent family with a subgenerator 𝒜 iff (R(t))t∈[0,τ) is a mild (a, k)-regularized C-uniqueness family having 𝒜 as subgenerator, R(t)C = CR(t) and R(t)𝒜 ⊆ 𝒜R(t) (t ∈ [0, τ)). Any (a, k)-regularized C-resolvent family under our consideration will be also a mild (a, k)-regularized C-existence family and the condition 0 ∈ supp(a) will be assumed henceforth. We say that an (a, k)-regularized C-resolvent family (R(t))t⩾0 is exponentially bounded (bounded) iff there exists ω ∈ ℝ (ω = 0) such that the family {e−ωt R(t) : t ⩾ 0} ⊆ L(X) is bounded. If k(t) = gα+1 (t), for α ⩾ 0, then it is also said that (R(t))t∈[0,τ) is an α-times integrated (a, C)-resolvent family; 0-times integrated (a, C)-resolvent family is further abbreviated to (a, C)-resolvent family. We pay special attention to the case a(t) ≡ 1, resp. a(t) ≡ t, when we say that (R(t))t⩾0 is an α-times integrated C-semigroup (C-regularizing semigroup or C-semigroup, if α = 0), resp. an α-times integrated C-cosine function (C-regularizing cosine function or C-cosine function, if t α = 0); if K ∈ L1loc ([0, τ)), K ≠ 0 and k(t) = ∫0 K(s) ds, t ∈ [0, τ), then it is said that (R(t))t∈[0,τ) is a K-convoluted C-semigroup with subgenerator 𝒜. By χ(R) we denote the set consisting of all subgenerators of (R(t))t∈[0,τ) . It is clear that for each subgenerator 𝒜 ∈ χ(R) we have 𝒜 ∈ χ(R). The set χ(R) can have infinitely many elements; furthermore, if 𝒜 ∈ χ(R), then 𝒜 ⊆ 𝒜int , where the integral generator of (R(t))t∈[0,τ) is defined by t

𝒜int := {(x, y) ∈ X × X : R(t)x − k(t)Cx = ∫ a(t − s)R(s)y ds for all t ∈ [0, τ)}. 0

The integral generator 𝒜int of (R(t))t∈[0,τ) is always a closed subgenerator of (R(t))t∈[0,τ) , provided that τ = ∞. If 𝒜 and ℬ are two subgenerators of (R(t))t∈[0,τ) and α, β ∈ ℂ with α + β = 1, then C(D(𝒜)) ⊆ D(ℬ), 𝒜int ⊆ C −1 𝒜C and α𝒜 + βℬ is also a subgenerator of (R(t))t∈[0,τ) ; furthermore, if C is injective, then 𝒜int = C −1 𝒜C. We similarly define the notion of integral generator of a mild (a, k)-regularized C2 -uniqueness family (R2 (t))t∈[0,τ) . Applying the Laplace transform, we can deduce the following extensions of [246, Theorems 2.8.5 and 2.1.5].

20 | 1 Preliminaries Theorem 1.5.4. Suppose that 𝒜 is a closed MLO in X, C1 ∈ L(Y, X), C2 ∈ L(X), C2 is injective, ω0 ⩾ 0 and ω ⩾ max(ω0 , abs(|a|), abs(k)). (i) Assume that (R1 (t), R2 (t))t⩾0 ⊆ L(Y, X) × L(X) is strongly continuous, and the family {e−ωt Ri (t) : t ⩾ 0} is bounded for i = 1, 2. (a) Suppose (R1 (t), R2 (t))t⩾0 is a mild (a, k)-regularized (C1 , C2 )-existence and uniqueness family with a subgenerator 𝒜. Then, for every λ ∈ ℂ with Re λ > ω ̃ ̃ k(λ) ̃ 𝒜 is injective, R(C1 ) ⊆ R(I − a(λ) ̃ 𝒜), and a(λ) ≠ 0, the operator I − a(λ) ∞

−1 ̃ ̃ 𝒜) C1 y = ∫ e−λt R1 (t)y dt, k(λ)(I − a(λ)

y ∈ Y,

(23)

0

1 ̃ a(z) ̃ { : Re z > ω, k(z) ≠ 0} ⊆ ρC1 (𝒜) ̃ a(z)

(24)

and ∞

−λt ̃ k(λ)C [R2 (t)x − (a ∗ R2 )(t)y] dt, 2x = ∫ e

whenever (x, y) ∈ 𝒜.

(25)

0

Here, ρC1 (𝒜) is defined in the obvious way. (b) Let (24) hold, and let (23) and (25) hold for any λ ∈ ℂ with Re λ > ω and ̃ ̃ k(λ) a(λ) ≠ 0. Then (R1 (t), R2 (t))t⩾0 is a mild (a, k)-regularized (C1 , C2 )-existence and uniqueness family with a subgenerator 𝒜. (ii) Assume that (R1 (t))t⩾0 is strongly continuous, and the family {e−ωt R1 (t) : t ⩾ 0} is bounded. Then (R1 (t))t⩾0 is a mild (a, k)-regularized C1 -existence family with a ̃ ̃ k(λ) subgenerator 𝒜 iff for every λ ∈ ℂ with Re λ > ω and a(λ) ≠ 0, one has R(C1 ) ⊆ ̃ 𝒜) and R(I − a(λ) ∞

̃ ̃ 𝒜) ∫ e−λt R1 (t)y dt, k(λ)C 1 y ∈ (I − a(λ)

y ∈ Y.

0

(iii) Let (R2 (t))t⩾0 be strongly continuous, and let the family {e−ωt R2 (t) : t ⩾ 0} ⊆ L(X) be bounded. Then (R2 (t))t⩾0 is a mild (a, k)-regularized C2 -uniqueness family with ̃ ̃ k(λ) a subgenerator 𝒜 iff for every λ ∈ ℂ with Re λ > ω and a(λ) ≠ 0, the operator ̃ 𝒜 is injective and (25) holds. I − a(λ) Theorem 1.5.5. Let (R(t))t⩾0 ⊆ L(X) be a strongly continuous operator family such that there exists ω ⩾ 0 satisfying that the family {e−ωt R(t) : t ⩾ 0} is bounded, and let ω0 > max(ω, abs(|a|), abs(k)). Suppose that 𝒜 is a closed MLO in X and C 𝒜 ⊆ 𝒜C. (i) Assume that 𝒜 is a subgenerator of the global (a, k)-regularized C-resolvent family (R(t))t⩾0 satisfying (21) for all x = y ∈ X. Then, for every λ ∈ ℂ with Re λ > ω0 and ̃ ̃ k(λ) ̃ 𝒜 is injective, R(C) ⊆ R(I − a(λ) ̃ 𝒜), as well as a(λ) ≠ 0, the operator I − a(λ) ∞

−1 ̃ ̃ 𝒜) Cx = ∫ e−λt R(t)x dt, k(λ)(I − a(λ) 0

x ∈ X,

Re λ > ω0 ,

̃ ̃ k(λ) a(λ) ≠ 0, (26)

1.5 Degenerate (a, k)-regularized C-resolvent families | 21

{

1 ̃ a(λ) ̃ : Re λ > ω0 , k(λ) ≠ 0} ⊆ ρC (𝒜) ̃ a(λ)

(27)

and R(s)R(t) = R(t)R(s), t, s ⩾ 0. (ii) Assume (26)–(27). Then 𝒜 is a subgenerator of the global (a, k)-regularized Cresolvent family (R(t))t⩾0 satisfying (21) for all x = y ∈ X and R(s)R(t) = R(t)R(s), t, s ⩾ 0. We will use the following definition of a (local) K-convoluted C-group; cf. [245, Section 2.6] for more details about the nondegenerate case. Definition 1.5.6. Let C ∈ L(X) and K ∈ L1loc ([0, τ)), K ≠ 0. Suppose that τ ∈ (0, ∞] and ±𝒜 are the integral generators of K-convoluted C-semigroups (SK,± (t))t∈[0,τ) . Put SK (t) := SK,+ (t), t ∈ [0, τ) and SK (t) := SK,− (−t), t ∈ (−τ, 0). Then we say that (SK (t))t∈(−τ,τ) is a K-convoluted C-group with the integral generator 𝒜. Any (local, degenerate or nondegenerate in time) K-convoluted C-semigroup (SK (t))t∈[0,τ) , resp. K-convoluted C-cosine function (CK (t))t∈[0,τ) , where C is not necessarily injective, satisfies the well-known composition properties stated in [245, Proposition 2.1.5, resp. Theorem 2.1.13]. Similar composition properties hold for (local) C-semigroups and (local) C-cosine functions. Although we do not intend to analyze the class of degenerate K-convoluted C-groups in more detail, we will use hereafter some special cases of the general composition property of degenerate K-convoluted C-groups. This composition property is stated in the following proposition, which is of independent interest and may become useful some day (the continuity of mapping t 󳨃→ SK (t)x, t ∈ (−τ, τ) for x ∈ D(𝒜) is irrelevant here; cf. also the short discussion after Definition 1.5.8). Proposition 1.5.7. Suppose that (SK (t))t∈(−τ,τ) is a K-convoluted C-group with the integral generator 𝒜. Then, for every t, s ∈ (−τ, τ) with t < 0 < s and x ∈ X, one has: SK (t)SK (s)x = SK (s)SK (t)x

0

s

{∫ K(r − t − s)SK (r)Cx dr + ∫t K(t + s − r)SK (r)Cx dr, = { t+s s t+s {∫t K(t + s − r)SK (r)Cx dr + ∫0 K(r − t − s)SK (r)Cx dr,

t + s ⩾ 0, t + s < 0.

Proof. Let −τ < t < 0 < s < τ and t + s ⩾ 0. Proceeding as in the proof of [245, Theorem 2.6.7], we obtain similarly that, for every x ∈ X, s

SK (t) ∫ SK (σ)x dσ 0

s

s

r

= ∫ Θ(r)SK (t + s − r)Cx dr + ∫ K(r − s − t)C ∫ SK (σ)x dσ dr t+s

t+s

0

22 | 1 Preliminaries r

s

0

= ∫ Θ(t + s − r)SK (r)Cx dr + ∫ K(r − s − t)C ∫ SK (σ)x dσ dr, t+s

t

0

t

where Θ(t) = ∫0 K(s) ds, t ∈ [0, τ). Applying partial integration in the above equality and using a straightforward computation, we get that, for every x ∈ X, SK (t)SK (s)x =

s

d [S (t) ∫ SK (σ)x dσ] ds K 0

0

= ∫ K(t + s − r)SK (r)Cx dr t

+

s

s

d [Θ(−t) ∫ SK (r)Cx dr − ∫ Θ(r − s − t)SK (r)Cx dr] ds 0

0

t+s

t+s

= ∫ K(t + s − r)SK (r)Cx dr + ∫ K(r − s − t)SK (r)Cx dr. s

t

We can analogously prove that, for t + s < 0 and x ∈ X, t+s

s

SK (t)SK (s)x = ∫ K(t + s − r)SK (r)Cx dr + ∫ K(r − t − s)SK (r)Cx dr. t

0

Since (SK (−t))t∈(−τ,τ) is a K-convoluted C-group with the integral generator −𝒜, the obtained composition properties imply that SK (t)SK (s) = SK (s)SK (t) for all t, s ∈ (−τ, τ). The proof of the theorem is thereby complete. In the case that ±𝒜 are the integral generators of C-semigroups (T± (t))t∈[0,τ) , the C-group (T(t))t∈(−τ,τ) , defined similarly as above, satisfies the much simpler group property T(t + s)C = T(t)T(s) for all t, s ∈ (−τ, τ) with t + s ∈ (−τ, τ). We will use this functional equality for transferring [421, Theorem 2.1] to degenerate C-groups (see Theorem 2.4.1 below). Now we will introduce the general class of (a, k)-regularized C-resolvent groups. Definition 1.5.8. Suppose 0 < τ ⩽ ∞, k ∈ C([0, τ)), k ≠ 0, a ∈ L1loc ([0, τ)), a ≠ 0, C ∈ L(X), C 𝒜 ⊆ 𝒜C and ±𝒜 are the integral generators of (a, k)-regularized C-resolvent families (R± (t))t∈[0,τ) . Put R(t) := R+ (t), t ∈ [0, τ) and R(t) := R(−t), t ∈ (−τ, 0). Then we say that (R(t))t∈(−τ,τ) is an (a, k)-regularized C-resolvent group family with the integral generator 𝒜. Observe that the mapping t 󳨃→ R(t)x, t ∈ (−τ, τ) is continuous for all x ∈ D(𝒜) and that the strong continuity of (R(t))t∈(−τ,τ) in degenerate case (at zero) is not automatically guaranteed by Definition 1.5.8. Nevertheless, a composition property of

1.5 Degenerate (a, k)-regularized C-resolvent families | 23

(R(t))t∈(−τ,τ) can be deduced even in the case when the mapping t 󳨃→ R(t)x, t ∈ (−τ, τ) is not continuous for some x ∈ X ∖ D(𝒜) (we can similarly introduce the class of (a, k)-regularized (C1 , C2 )-existence and uniqueness group families and prove an analogous composition property; cf. [246, Section 2.8] for more details): Proposition 1.5.9. Let (R(t))t∈(−τ,τ) be an (a, k)-regularized C-resolvent group family with the integral generator 𝒜. Set kg (t) := k(t) for t ∈ [0, τ) and

kg (t) := k(−t) for t ∈ (−τ, 0]

and, for every x ∈ X, t

(a ∗g R)(t)x := ∫ a(t − s)R(s)x ds

for t ∈ [0, τ),

0

t

(a ∗g R)(t)x := ∫ a(s − t)R(s)x ds

for t ∈ (−τ, 0].

0

Then we have, for −τ < t, s < τ and x ∈ X, (a ∗g R)(s)R(t)x − R(s)(a ∗g R)(t)x

= kg (t)(a ∗g R)(s)Cx − kg (s)C(a ∗g R)(t)x.

(28)

Proof. Let x ∈ X be fixed. We will prove the composition property only in the case when −τ < s ⩽ 0 and 0 ⩽ t < τ; the proof in all other cases is similar. Performing a straightforward computation, we obtain that (a ∗g R)(s)y = R(s)x − k(−s)Cx,

whenever (x, y) ∈ 𝒜.

By employing this equality and elementary definitions, we get that (a∗g R)(s)R(t)x ∋ (a ∗g R)(s)[𝒜(a ∗g R)(t)x + k(t)Cx] = k(t)(a ∗g R)(s)Cx + (a ∗g R)(s)𝒜(a ∗ R)(t)x

= k(t)(a ∗g R)(s)Cx + R(s)(a ∗g R)(t)x − k(−s)C(a ∗g R)(t)x. This immediately implies (28). Remark 1.5.10. The composition property established for K-convoluted C-groups in Proposition 1.5.7 and the corresponding composition property presented in (28) are not the same. We will not draw a parallel between these composition properties here. For the sequel, we need to remind ourselves of the notion of an exponentially bounded (a, k)-regularized C-resolvent families generated by a pair of closed linear operators A, B acting on X; cf. [247, Subsection 2.3.3] for more details.

24 | 1 Preliminaries Definition 1.5.11. Suppose that the functions a(t) and k(t) satisfy (P1), as well as that R(t) ∈ L(X, [D(B)]) for all t ⩾ 0. Let C ∈ L(X) be injective, and let CA ⊆ AC and CB ⊆ BC. Then the operator family (R(t))t⩾0 is said to be an exponentially bounded (a, k)-regularized C-resolvent family generated by A, B iff there exists ω ⩾ max(0, abs(a), abs(k)) such that the following holds: (i) The mappings t 󳨃→ R(t)x, t ⩾ 0 and t 󳨃→ BR(t)x, t ⩾ 0 are continuous for every fixed element x ∈ X. (ii) The family {e−ωt R(t) : t ⩾ 0} ⊆ L(X, [D(B)]) is bounded. ̃ ̃ (iii) For every λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0, the operator B − a(λ)A is injective, ̃ R(C) ⊆ R(B − a(λ)A) and ∞

−1 ̃ ̃ k(λ)(B − a(λ)A) Cx = ∫ e−λt R(t)x dt,

x ∈ X.

0

Before proceeding further, we want to mention that the class of exponentially bounded (a, k)-regularized C-resolvent families generated by A, B, where the operator C ∈ L(X) is not injective, has not deserved the attention of authors so far. Principal properties of exponentially bounded (a, k)-regularized C-resolvent families generated by A, B are clarified in [247, Theorem 2.3.15]. For our purposes, it will be sufficient to recall the following facts: (T1) Let x ∈ X. Then the function t 󳨃→ u(t), t ⩾ 0, defined by u(t) := R(t)x, t ⩾ 0 is a solution of problem (20) with 𝒜 = A, ℬ = B and ℱ (t) = k(t)Cx, t ⩾ 0. (T2) Let x ∈ D(A) ∩ D(B). Then the function t 󳨃→ u(t), t ⩾ 0, defined by u(t) := R(t)Bx, t ⩾ 0 is a strong solution of problem (20) with 𝒜 = A, ℬ = B and ℱ (t) = k(t)CBx, t ⩾ 0. However, we do have t

u(t) = k(t)Cx + ∫ a(t − s)R(s)Ax ds, 0

t ⩾ 0.

(29)

2 Almost periodic type solutions of abstract Volterra integro-differential equations In this chapter, we investigate abstract degenerate Volterra integro-differential equations and abstract fractional differential equations in Banach spaces which do have almost periodic solutions and asymptotically almost periodic solutions. We pay special attention to the analysis of various classes of semilinear Cauchy problems and inclusions, which can be fractional or nonfractional in time, with almost periodic solutions and asymptotically almost periodic type solutions. Unless stated otherwise, throughout this chapter we assume that X is an infinitedimensional complex Banach space, and that A and B are two closed linear operators acting on X. The symbol I denotes the identity operator on X, and C ∈ L(X) denotes a possibly noninjective operator, if not stated otherwise. In some places, we need to have two different pivot spaces, so that we sometimes use the symbols Y, Z, . . . , in place of X. We start our work by recalling the fundamental properties of almost periodic functions and asymptotically almost periodic functions in Banach spaces, giving also some original definitions, observations and results.

2.1 Almost periodic functions and asymptotically almost periodic functions The concept of almost periodicity was introduced by Danish mathematician H. Bohr around 1924–1926 [68] and later generalized by many other authors (cf. [45, 97, 98, 108, 110, 132, 191, 206, 207, 224, 296] and [409] for more details on the subject). Let I = ℝ or I = [0, ∞), and let f : I → X be continuous. Given ε > 0, we call τ > 0 an ε-period for f (⋅) iff 󵄩 󵄩󵄩 󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 ⩽ ε,

t ∈ I.

(30)

The set constituted of all ε-periods for f (⋅) is denoted by ϑ(f , ε). It is said that f (⋅) is almost periodic, a. p. for short, iff for each ε > 0 the set ϑ(f , ε) is relatively dense in I, which means that there exists l > 0 such that any subinterval of I of length l meets ϑ(f , ε). Since for each ε > 0 we have ϑ(f , ε) ⊆ ϑ(‖f ‖, ε), which is a consequence of the inequality |‖x‖ − ‖y‖| ⩽ ‖x − y‖, x, y ∈ X, it immediately follows from definition that the almost periodicity of function f : I → X implies the almost periodicity of the scalarvalued function ‖f ‖ : I → X. As a matter of fact, a much more general result can be proved [296]: For any positive real number r > 0, we have that the function ‖f ‖r : I → X is almost periodic. Furthermore, it can be easily seen that the almost periodicity of https://doi.org/10.1515/9783110641851-002

26 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations function f : I → X implies the almost periodicity of vector-valued functions f (⋅ + a) and f (a⋅), where a ∈ I. We call f (⋅) weakly almost periodic, w. a. p. for short, iff for each x∗ ∈ X ∗ the function x∗ (f (⋅)) is almost periodic (it is well known that any function f ∈ BUC(I : X), which has a relatively compact range in X and which is w. a. p., needs to be a. p.; cf. [29, Proposition 4.5.12]). A family of functions ℱ ⊆ X I is said to be uniformly almost periodic iff for each ε > 0 there exists l > 0 such that any subinterval of I of length l contains a number τ > 0 such that (30) holds for all f ∈ ℱ . The space consisting of all almost periodic functions from the interval I into X will be denoted by AP(I : X). Equipped with the sup-norm, AP(I : X) is a Banach space. For the sequel, we need some preliminaries from the pioneering paper [43] by H. Bart and S. Goldberg. The translation semigroup (W(t))t⩾0 on AP([0, ∞) : X), determined by [W(t)f ](s) := f (t + s), t ⩾ 0, s ⩾ 0, f ∈ AP([0, ∞) : X) consists solely of surjective isometries W(t) (t ⩾ 0) and can be extended to a C0 -group (W(t))t∈ℝ of isometries on AP([0, ∞) : X), where W(−t) := W(t)−1 for t > 0. Furthermore, the mapping E : AP([0, ∞) : X) → AP(ℝ : X), defined by [Ef ](t) := [W(t)f ](0),

t ∈ ℝ, f ∈ AP([0, ∞) : X),

is a linear surjective isometry and Ef (⋅) is the unique continuous almost periodic extension of a function f (⋅) from AP([0, ∞) : X) to the whole real line. We have that [E(Bf )] = B(Ef ) for all B ∈ L(X) and f ∈ AP([0, ∞) : X). The most intriguing properties of almost periodic vector-valued functions are collected in the following two theorems (in the case I = ℝ, these assertions are wellknown in the existing literature; in the case I = [0, ∞), these assertions can be deduced by using the case I = ℝ and the properties of extension mapping E(⋅)). By c0 we denote the Banach space of all numerical sequences tending to zero, equipped with the sup-norm. Theorem 2.1.1. Let f ∈ AP(I : X). Then the following holds: (i) f ∈ BUC(I : X); (ii) If g ∈ AP(I : X), h ∈ AP(I : ℂ), α, β ∈ ℂ, then αf + βg and hf ∈ AP(I : X); (iii) Bohr’s transform of f (⋅), t

1 ∫ e−irs f (s) ds, t→∞ t

Pr (f ) := lim

0

exists for all r ∈ ℝ and t+α

1 Pr (f ) := lim ∫ e−irs f (s) ds t→∞ t α

for all α ∈ I, r ∈ ℝ. The element Pr (f ) is called the Bohr coefficient or Bohr–Fourier coefficient of f (⋅);

2.1 Almost periodic functions and asymptotically almost periodic functions | 27

(iv) If Pr (f ) = 0 for all r ∈ ℝ, then f (t) = 0 for all t ∈ I; (v) Bohr’s spectrum σ(f ) := {r ∈ ℝ : Pr (f ) ≠ 0} is at most countable; (vi) If c0 ⊈ X, which means that X does not contain an isomorphic copy of c0 , I = ℝ t and g(t) = ∫0 f (s) ds (t ∈ ℝ) is bounded, then g ∈ AP(ℝ : X); (vii) If (gn )n∈ℕ is a sequence in AP(I : X) and (gn )n∈ℕ converges uniformly to g, then g ∈ AP(I : X); (viii) If I = ℝ and f 󸀠 ∈ BUC(ℝ : X), then f 󸀠 ∈ AP(ℝ : X); (ix) (Spectral synthesis) f ∈ span{eiμ⋅ x : μ ∈ σ(f ), x ∈ R(f )}; (x) R(f ) is relatively compact in X; (xi) (Supremum formula) we have 󵄩 󵄩 ‖f ‖∞ = sup󵄩󵄩󵄩f (t)󵄩󵄩󵄩, t⩾t0

t0 ∈ I;

(31)

(xii) (Convolution invariance) if I = ℝ and g ∈ L1 (ℝ), then g ∗ f ∈ AP(ℝ : X), where ∞

(g ∗ f )(t) = ∫ g(t − s)f (s) ds,

t ∈ ℝ;

−∞

(xiii) If n ∈ ℕ and f1 ∈ AP(I : X1 ), . . . , fn ∈ AP(I : Xn ), then (f1 , . . . , fn ) ∈ AP(I : X1 × ⋅ ⋅ ⋅ × Xn ). Here, Xi is a complex Banach space for all i = 1, . . . , n; (xiv) If f1 ∈ AP(I : X1 ), . . . , fn ∈ AP(I : Xn ), then for each ε > 0 there exists a common relatively dense set ϑ(f1 , . . . , fn , ε) of ε-periods for any of these functions. Here, Xi is a complex Banach space for all i = 1, . . . , n. Theorem 2.1.2 (Bochner’s criterion). Let f ∈ BUC(ℝ : X). Then f (⋅) is almost periodic iff for any sequence (bn ) of numbers from ℝ there exists a subsequence (an ) of (bn ) such that (f (⋅ + an )) converges in BUC(ℝ : X). Remark 2.1.3. It is worth noting that (viii) holds in the case I = [0, ∞) as well. More precisely, let f ∈ AP([0, ∞) : X) and f 󸀠 ∈ BUC([0, ∞) : X). Then f 󸀠 ∈ AP([0, ∞) : X). To see this, it suffices to apply (vii) by noticing that the sequence fn (t) := n[f (t+1/n)−f (t)], t ⩾ 0 of almost periodic functions converges uniformly to f (t) for t ⩾ 0, because 󵄩󵄩 t+1/n 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩󵄩 󵄩󵄩 󸀠 󸀠 󵄩 󵄩󵄩fn (t) − f (t)󵄩󵄩 = 󵄩󵄩n ∫ [f (s) − f (t)] ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 t 󵄩 t+1/n

󵄩 󵄩 ⩽ n ∫ 󵄩󵄩󵄩f 󸀠 (s) − f 󸀠 (t)󵄩󵄩󵄩 ds, t

t⩾0

and f 󸀠 (⋅) is bounded and uniformly continuous on [0, ∞). Before proceeding any further, we would like to mention that the necessary and sufficient condition for X to contain c0 is given in [29, Theorem 4.6.14]: c0 ⊆ X iff there

28 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations exists a divergent series ∑∞ n=1 xn in X which is unconditionally bounded, i. e., there exists M ⩾ 0 such that 󵄩󵄩 m 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩∑ xn 󵄩󵄩󵄩 ⩽ M, 󵄩 j󵄩 󵄩󵄩󵄩 󵄩j=1 󵄩󵄩 whenever nj ∈ ℕ (j = 1, 2, . . . , m) such that n1 < n2 < ⋅ ⋅ ⋅ < nm . The importance of condition c0 ⊈ X has been recognized already by H. Bohr and later employed by many others (see, e. g., Kadet’s theorem [29, Theorem 4.6.11]). By APΛ (I : X), where Λ is a nonempty subset of I, we denote the vector subspace of AP(I : X) consisting of all functions f ∈ AP(I : X) for which the inclusion σ(f ) ⊆ Λ holds true. It can be easily seen that APΛ (I : X) is a closed subspace of AP(I : X) and therefore Banach space itself. For some other criteria stating the necessary and sufficient conditions for the almost periodicity of a given function, like Haraux’s and Maak’s criteria, we refer the reader to [97]. Arithmetic properties of almost periods of a given function f ∈ AP(I : X) have been investigated in [296, Chapter 3]. The conditional compactness of subsets in AP(I : X) has been examined by C. Corduneanu [109] (see also [132, Theorem 3.11] and the formulation of Lusternik’s theorem [97]). A function f ∈ BUC(I : X) is said to be weakly almost periodic in the sense of Eberlein iff {f (⋅ + s) : s ∈ I} is relatively weakly compact in X. This important class of functions will not be pursued in the rest of the book (for further information concerning this intriguing topic and connections between almost periodicity and Carleman spectrum of functions, one may refer to the monograph [29] and references cited therein). To the best knowledge of the author, the notion of asymptotical almost periodicity was introduced by A. S. Kovanko [286] in 1929 and later rediscovered, in a slightly different form we are using now, by M. Fréchet [193] in 1941 (for more details concerning the vector-valued asymptotically almost periodic functions and asymptotically almost periodic differential equations, see, e. g., [52, 90, 130, 132, 206, 207, 357, 358, 361, 405] and [415]). A function f ∈ Cb ([0, ∞) : X) is said to be asymptotically almost periodic iff for every ε > 0 we can find numbers l > 0 and M > 0 such that every subinterval of [0, ∞) of length l contains at least one number τ such that ‖f (t + τ) − f (t)‖ ⩽ ε for all t ⩾ M. The space consisting of all asymptotically almost periodic functions from [0, ∞) into X will be denoted by AAP([0, ∞) : X). It is well known that (see W. M. Ruess, W. H. Summers [356–359]) for any function f ∈ C([0, ∞) : X), the following statements are equivalent: (i) f ∈ AAP([0, ∞) : X). (ii) There exist uniquely determined functions g ∈ AP([0, ∞) : X) and ϕ ∈ C0 ([0, ∞) : X) such that f = g + ϕ. (iii) The set H(f ) := {f (⋅+s) : s ⩾ 0} is relatively compact in Cb ([0, ∞) : X), which means that for any sequence (bn ) of nonnegative real numbers there exists a subsequence (an ) of (bn ) such that (f (⋅ + an )) converges in Cb ([0, ∞) : X).

2.1 Almost periodic functions and asymptotically almost periodic functions | 29

The functions g and ϕ from (ii) are called the principal and corrective terms of the function f , respectively. We know that R(g) ⊆ R(f ) (see, e. g., [132, Lemma 3.43]). By C0 ([0, ∞) × Y : X) we denote the space of all continuous functions h : [0, ∞) × Y → X such that limt→∞ h(t, y) = 0 uniformly for y in any compact subset of Y. A continuous function f : I × Y → X is called uniformly continuous on bounded sets, uniformly for t ∈ I iff for every ε > 0 and every bounded subset K of Y there exists a number δε,K > 0 such that ‖f (t, x) − f (t, y)‖ ⩽ ε for all t ∈ I and all x, y ∈ K satisfying that ‖x − y‖ ⩽ δε,K . If f : I × Y → X, then we define f ̂ : I × Y → Lp ([0, 1] : X) by f ̂(t, y) := f (t + ⋅, y), t ⩾ 0, y ∈ Y. For the purpose of research of (asymptotically) almost periodic solutions of semilinear Cauchy inclusions, we need to recall the following well-known definitions and results (see, e. g., C. Zhang [416], W. Long, H.-S. Ding [317], and Proposition 2.1.6 below): Definition 2.1.4. Let 1 ⩽ p < ∞. (i) A function f : I × Y → X is called almost periodic iff f (⋅, ⋅) is bounded, continuous, as well as for every ε > 0 and every compact K ⊆ Y there exists l(ε, K) > 0 such that every subinterval J ⊆ I of length l(ε, K) contains a number τ with the property that ‖f (t + τ, y) − f (t, y)‖ ⩽ ε for all t ∈ I, y ∈ K. The collection of such functions will be denoted by AP(I × Y : X). (ii) A function f : [0, ∞) × Y → X is said to be asymptotically almost periodic iff it is bounded continuous and admits a decomposition f = g +q, where g ∈ AP([0, ∞)× Y : X) and q ∈ C0 ([0, ∞) × Y : X). Denote by AAP([0, ∞) × Y : X) the vector space consisting of all such functions. Observe that we automatically assume the boundedness of function f (⋅, ⋅), following the approach obeyed in [416]. The composition principles clarified below are well known in the existing literature (see, e. g., [416]). Theorem 2.1.5. (i) Let f ∈ AP(I × Y : X) and h ∈ AP(I : Y). Then the mapping t 󳨃→ f (t, h(t)), t ∈ I belongs to the space AP(I : X). (ii) Let f ∈ AAP([0, ∞) × Y : X) and h ∈ AAP([0, ∞) : Y). Then the mapping t 󳨃→ f (t, h(t)), t ⩾ 0 belongs to the space AAP([0, ∞) : X). In Definition 2.1.4(ii), many authors assume a priori that g ∈ AP(ℝ × Y : X). This is slightly redundant on account of the following proposition: Proposition 2.1.6. Let f : [0, ∞) × Y → X, and let S ⊆ Y. Suppose that, for every ε > 0, there exists l(ε, S) > 0 such that every subinterval J ⊆ [0, ∞) of length l(ε, S) contains a number τ with the property that ‖f (t + τ, y) − f (t, y)‖ ⩽ ε for all t ⩾ 0, y ∈ S (this, in particular, holds provided that f ∈ AP(I × Y : X)). Denote by F(t, y) the unique almost

30 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations periodic extension of function f (t, y) from the interval [0, ∞) to the whole real line, for fixed y ∈ S. Then, for every ε > 0, with the same l(ε, S) > 0 chosen as above, we have that every subinterval J ⊆ ℝ of length l(ε, S) contains a number τ with the property that ‖F(t + τ, y) − F(t, y)‖ ⩽ ε for all t ∈ ℝ, y ∈ S. Proof. Let ε > 0 be given in advance, let l(ε, S) > 0 be as above, and let J = [a, b] ⊆ ℝ. The assertion is clear provided that a ⩾ 0. Suppose now that a < 0; then we choose a number τ0 > 0 arbitrarily. It is clear that there exists τ󸀠 ∈ J = [τ0 , τ0 + b − a] ⊆ [0, ∞) such that ‖f (t +τ󸀠 , y)−f (t, y)‖ ⩽ ε for all t ⩾ 0, y ∈ S. Since τ := τ󸀠 −τ0 −|a| ∈ J, it suffices to show that ‖F(t + τ, y) − F(t, y)‖ ⩽ ε for all t ∈ ℝ, y ∈ S. Towards this end, fix a number t ∈ ℝ and an element y ∈ S. Since the mapping s 󳨃→ F(s+τ󸀠 −τ0 −|a|, y)−F(s−τ0 −|a|, y), s ∈ ℝ is almost periodic, equation (31) shows that 󵄩 󵄩󵄩 󸀠 󵄩󵄩F(t + τ − τ0 − |a|, y) − F(t − τ0 − |a|, y)󵄩󵄩󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩F(⋅ + τ󸀠 − τ0 − |a|, y) − F(⋅ − τ0 − |a|, y)󵄩󵄩󵄩∞ 󵄩 󵄩 = sup 󵄩󵄩󵄩F(s + τ󸀠 − τ0 − |a|, y) − F(s − τ0 − |a|, y)󵄩󵄩󵄩 s⩾τ0 +|a|

󵄩 󵄩 = sup 󵄩󵄩󵄩f (s + τ󸀠 − τ0 − |a|, y) − f (s − τ0 − |a|, y)󵄩󵄩󵄩 s⩾τ0 +|a|

󵄩 󵄩 = sup󵄩󵄩󵄩f (s + τ󸀠 , y) − f (s, y)󵄩󵄩󵄩 ⩽ ε. s⩾0

This ends the proof of proposition.

2.2 Stepanov almost periodic functions and asymptotically Stepanov almost periodic functions Let 1 ⩽ p < ∞, let l > 0, and let f , g ∈ Lploc (I : X), where I = ℝ or I = [0, ∞). We define the Stepanov “metric” by DpS [f (⋅), g(⋅)] l

x+l

1 󵄩 󵄩p := sup[ ∫ 󵄩󵄩󵄩f (t) − g(t)󵄩󵄩󵄩 dt] x∈I l x

1/p

.

(32)

Then we know that, for every two numbers l1 , l2 > 0, there exist two positive real constants k1 , k2 > 0 independent of f , g, such that k1 DpS [f (⋅), g(⋅)] ⩽ DpS [f (⋅), g(⋅)] ⩽ k2 DpS [f (⋅), g(⋅)], l1

l1

l2

(33)

as well as that (see, e. g., [57, pp. 72–73] for the scalar-valued case, and [21]) there exists DpW [f (⋅), g(⋅)] := lim DpS [f (⋅), g(⋅)] l→∞

l

(34)

2.2 Stepanov (and asymptotically Stepanov) almost periodic functions | 31

in [0, ∞]. The distance appearing in (34) is called the Weyl distance of f (⋅) and g(⋅). The Stepanov and Weyl “norms” of f (⋅) are defined by ‖f ‖Sp := DpS [f (⋅), 0] and ‖f ‖W p := DpW [f (⋅), 0], l

l

respectively. Taking into account (33), in the sequel of this section it will be appropriate to assume that l1 = l2 = 1. We say that a function f ∈ Lploc (I : X) is Stepanov p-bounded, or Sp -bounded in short, iff t+1

‖f ‖Sp

1/p

󵄩p 󵄩 := sup( ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds) t∈I

t

< ∞.

Equipped with the above norm, the space LpS (I : X) consisting of all Sp -bounded functions is a Banach space. A function f ∈ LpS (I : X) is said to be Stepanov p-almost periodic, or Sp -almost periodic in short, iff the function f ̂ : I → Lp ([0, 1] : X), defined by f ̂(t)(s) := f (t + s),

t ∈ I, s ∈ [0, 1]

is almost periodic (cf. [19] for more details). It is said that f ∈ LpS ([0, ∞) : X) is asymptotically Stepanov p-almost periodic, or asymptotically Sp -almost periodic in short, iff f ̂ : [0, ∞) → Lp ([0, 1] : X) is asymptotically almost periodic. It is a well-known fact that if f (⋅) is an almost periodic (resp. a. a. p.) function then f (⋅) is also Sp -almost periodic (resp. asymptotically Sp -a. a. p.) for 1 ⩽ p < ∞. The converse statement is false, however, as the following two examples from the book of M. Levitan [295] show: Example 2.2.1. Assume that α, β ∈ ℝ and αβ−1 is a well-defined irrational number. Then the functions f (t) = sin(

1 ), 2 + cos αt + cos βt

t∈ℝ

g(t) = cos(

1 ), 2 + cos αt + cos βt

t∈ℝ

and

are Stepanov p-almost periodic but not almost periodic (1 ⩽ p < ∞). The case α = 1 and β = √2 has been further considered by A. Nawrocki in [334], who proved with the help of Liouville’s theorem and some results from the theory of continuous fractions [334, Theorems 1 and 2] that t −2−ε = 0, t→+∞ 2 + cos t + cos √2t lim

ε > 0,

32 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations but t −2 t→+∞ 2 + cos t + cos √2t lim

does not exist. Of course, it was known before that the function t 󳨃→ 1/(2 + cos t + cos √2t), t ∈ ℝ is well-defined, continuous and unbounded. Example 2.2.2. Define sign(0) := 0. Then, for every almost periodic function f : ℝ → ℝ, we have that the function sign(f (⋅)) is Stepanov 1-almost periodic. It is worth noting that, under the same assumptions, the function sign(f (⋅)) is Stepanov p-almost periodic for any exponent p ∈ [1, ∞): Example 2.2.3. (i) Let ε > 0 be given and let ε0 := εp /2p−1 . By the conclusion from Example 2.2.2, we know that there exists l0 > 0 such that any subinterval I of ℝ, having length l0 , contains a point τ ∈ I such that t+1

󵄨 󵄨 ∫ 󵄨󵄨󵄨sign(f (x + τ)) − sign(f (x))󵄨󵄨󵄨 dx < ε0 , t

t ∈ ℝ.

(35)

For every t, τ ∈ ℝ, define Bt,τ,1 := {x ∈ [t, t + 1] : f (x + τ)f (x) < 0}

and Bt,τ,2 := {x ∈ [t, t + 1] : f (x + τ)f (x) = 0}. All that we need to prove is that (35) implies 󵄨p 󵄨 ( ∫ + ∫ )󵄨󵄨󵄨sign(f (x + τ)) − sign(f (x))󵄨󵄨󵄨 dx < εp , Bt,τ,1

t ∈ ℝ.

Towards this end, observe that we already know from (35) (∫B + ∫B )|sign(f (x + τ)) − sign(f (x))| dx < ε0 for all t ∈ ℝ as well as that t,τ,1

t,τ,2

󵄨p 󵄨 ( ∫ + ∫ )󵄨󵄨󵄨sign(f (x + τ)) − sign(f (x))󵄨󵄨󵄨 dx Bt,τ,1

(36)

Bt,τ,2

Bt,τ,2

󵄨p 󵄨 = 2p m(Bt,τ,1 ) + ∫ 󵄨󵄨󵄨sign(f (x + τ)) − sign(f (x))󵄨󵄨󵄨 dx Bt,τ,2

󵄨 󵄨 = 2p m(Bt,τ,1 ) + ∫ 󵄨󵄨󵄨sign(f (x + τ)) − sign(f (x))󵄨󵄨󵄨 dx Bt,τ,2

󵄨 󵄨 ⩽ 2p−1 [2m(Bt,τ,1 ) + ∫ 󵄨󵄨󵄨sign(f (x + τ)) − sign(f (x))󵄨󵄨󵄨 dx] Bt,τ,2

that

2.2 Stepanov (and asymptotically Stepanov) almost periodic functions | 33

⩽ 2p−1 ε0 = εp , as claimed; here, m(Bt,τ,1 ) denotes the Lebesgue measure of set Bt,τ,1 . (ii) For every almost periodic function f : [0, ∞) → ℝ, we have that the function sign(f (⋅)) is Stepanov p-almost periodic. This can be simply deduced with the help of (i) and the fact that sign(f (t)) = sign([Ef ](t)) for all t ⩾ 0. Denote by APSp (I : X) and AAPSp ([0, ∞) : X) the spaces consisting of all Sp -almost periodic functions I 󳨃→ X and of all asymptotically Sp -almost periodic functions [0, ∞) 󳨃→ X, respectively. For any Sp -almost periodic function f (⋅) and for any real number δ ∈ (0, 1), we define the function t+δ

fδ (t) :=

1 ∫ f (s) ds, δ t

t ∈ I.

Arguing as in the scalar-valued case [57], we can prove that the function fδ (⋅) is almost periodic (0 < δ < 1) as well as that ‖fδ − f ‖Sp → 0 as δ → 0+. Hereafter we will also use the Bochner theorem, which asserts that any BUC function which is Stepanov p-almost periodic has to be almost periodic (1 ⩽ p < ∞). The notion of a scalarly Sp -almost periodic function, slightly different from the notion of usually considered weakly Sp -almost periodic function, is given as follows: A function f ∈ LpS (I : X) is said to be scalarly Stepanov p-almost periodic iff for each x∗ ∈ X ∗ we have that the function x∗ (f ) : [0, ∞) → ℂ defined by x∗ (f )(t) := x∗ (f (t)), t ⩾ 0 is Stepanov p-almost periodic. Definition 2.2.4. A function f : I × Y → X is called Stepanov p-almost periodic, or Sp -almost periodic in short, iff f ̂ : I × Y → Lp ([0, 1] : X) is almost periodic. By [416, Theorem 2.6], we have that a bounded continuous function f : [0, ∞) × Y → X is asymptotically almost periodic iff for every ε > 0 and every compact K ⊆ Y there exist l(ε, K) > 0 and M(ε, K) > 0 such that every subinterval J ⊆ [0, ∞) of length l(ε, K) contains a number τ with the property that ‖f (t + τ, y) − f (t, y)‖ ⩽ ε for all t ⩾ M(ε, K), y ∈ K. We introduce the notion of an asymptotically Stepanov p-almost periodic function f (⋅, ⋅) as follows: Definition 2.2.5. Let 1 ⩽ p < ∞. A function f : [0, ∞) × Y → X is said to be asymptotically Sp -almost periodic iff f ̂ : [0, ∞) × Y → Lp ([0, 1] : X) is asymptotically almost periodic. The collection of such functions will be denoted by AAPSp ([0, ∞) × Y : X). It is very elementary to prove that any asymptotically almost periodic function is also asymptotically Stepanov p-almost periodic (1 ⩽ p < ∞). We need the assertion of [220, Lemma 1]:

34 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Lemma 2.2.6. Suppose that f : [0, ∞) → X is an asymptotically Sp -almost periodic function. Then there are two locally p-integrable functions g : ℝ → X and q : [0, ∞) → X satisfying the following conditions: (i) g is Sp -almost periodic; (ii) q̂ belongs to the class C0 ([0, 1] : Lp ([0, 1] : X)); (iii) f (t) = g(t) + q(t) for all t ⩾ 0. Moreover, there exists an increasing sequence (tn )n∈ℕ of positive reals such that limn→∞ tn = ∞ and g(t) = limn→∞ f (t + tn ) a. e. t ⩾ 0. Now we state the following two-variable analogue of Lemma 2.2.6: Lemma 2.2.7. Suppose that f : [0, ∞) × Y → X is an asymptotically Sp -almost periodic function. Then there are two functions g : ℝ × Y → X and q : [0, ∞) × Y → X satisfying that for each y ∈ Y the functions g(⋅, y) and q(⋅, y) are locally p-integrable, as well as that the following holds: (i) ĝ : ℝ × Y → Lp ([0, 1] : X) is almost periodic; (ii) q̂ ∈ C0 ([0, ∞) × Y : Lp ([0, 1] : X)); (iii) f (t, y) = g(t, y) + q(t, y) for all t ⩾ 0 and y ∈ Y. Moreover, for every compact set K ⊆ Y, there exists an increasing sequence (tn )n∈ℕ of positive reals such that limn→∞ tn = ∞ and g(t, y) = limn→∞ f (t + tn , y) for all y ∈ Y and a. e. t ⩾ 0. Proof. By the foregoing, we have that f ̂ : [0, ∞) × Y → X is bounded continuous and admits a decomposition f ̂ = G + Q, where G ∈ AP([0, ∞) × Y : Lp ([0, 1] : X)) and Q ∈ C0 ([0, ∞)×Y : Lp ([0, 1] : X)). Moreover, the proof of [416, Theorem 2.6] shows that, for every compact set K ⊆ Y, there exists an increasing sequence (tn )n∈ℕ of positive reals such that limn→∞ tn = ∞ and G(t, y) = limn→∞ f ̂(t + tn , y) for all y ∈ Y and t ⩾ 0. The remainder of the proof follows by bringing into effect Lemma 2.2.6 to the function f ̂(⋅, y), for a fixed element y ∈ Y, and the uniqueness of decomposition g(⋅) + q(⋅) in this lemma. It is noteworthy that Bochner’s criterion for Stepanov p-almost periodic functions has been proved by Z. Hu and A. B. Mingarelli in [225, Theorem 1]. In the case that p = 1, then we simply say that the function under our consideration is (asymptotically, scalarly) Stepanov almost periodic. Definition 2.2.8. ([9]) Let n ∈ ℕ. It is said that a function f (⋅) ∈ X I is C (n) -almost periodic, f ∈ C (n) − AP(I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AP(I : X).

2.2 Stepanov (and asymptotically Stepanov) almost periodic functions | 35

For example, the function sin nt , 4 n=1 n ∞

f (t) := ∑

t∈ℝ

is C (2) -almost periodic but not C (3) -almost periodic (see, e. g., [132, Example 3.39]). Various classes of Stepanov-like C (n) -pseudo almost automorphic functions have been considered by T. Diagana, V. Nelson and G. M. N’Guérékata in [155] (see, e. g., [155, Definition 2.18]). For our work, the classes of Stepanov C (n) -almost periodic functions and asymptotically Stepanov C (n) -almost periodic functions will be general enough (the derivatives appearing below will be taken in distributional sense; see also Notes and Appendices to Chapter 3). Definition 2.2.9. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (I : X). (i) It is said that a function f (⋅) is Stepanov-p-C (n) -almost periodic, f ∈ C (n) − APSp (I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ APSp (I : X). (ii) It is said that a function f ∈ Lploc ([0, ∞) : X) is asymptotically Stepanov-p-C (n) almost periodic, f ∈ C (n) − AAPSp ([0, ∞) : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AAPSp ([0, ∞) : X). Before proceeding further, we would like to point out that the inclusion f ∈ C (n) − APS (I : X) does not imply that f ∈ C n (I : X), in general. As a matter of fact, the inclusion f ∈ C (n) − APSp (I : X) (f ∈ C (n) − AAPSp ([0, ∞) : X)) implies that f ∈ C n−1 (I : X) (f ∈ C n−1 ([0, ∞) : X)) as well as that f (n) ∈ Lploc (I : X) (f (n) ∈ Lploc ([0, ∞) : X)) and f (n−1) (⋅) is locally absolutely continuous; see, e. g., [29, Proposition 1.2.2] and [39, Chapter I, Section 2.2]. This observation also holds for any (asymptotically) generalized C (n) -almost periodic function space considered below. p

Example 2.2.10. Let 1 ⩽ p < ∞, and let X be any nontrivial complex Banach space. Then, for any Sp -almost periodic function f (⋅) and for any real number δ ∈ (0, 1), we define the function fδ (⋅) as before. Assume that f (⋅) is Sp -almost periodic, but not almost periodic. Then, for 0 < δ < 1, one has: fδ󸀠 (t) :=

1 [f (t + δ) − f (t)] for a. e. t ∈ I. δ

Hence, fδ (⋅) ∈ C (1) − APSp (I : X) ∖ C (1) − AP(I : X) for 0 < δ < 1, so that C (1) − APSp (I : X) is a strict extension of C (1) − AP(I : X). By APSbp (I : X), resp. AAPSbp ([0, ∞) : X), we denote the space consisting of all essentially bounded functions from APSp (I : X), resp. AAPSp ([0, ∞) : X). We also introduce the following function spaces: Definition 2.2.11. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (I : X). (i) It is said that a function f (⋅) is Stepanov-p-Cb(n) -almost periodic, f ∈ Cb(n) − APSp (I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ APSbp (I : X).

36 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations (ii) It is said that a function f (⋅) is asymptotically Stepanov-p-Cb(n) -almost periodic, f ∈ Cb(n) − AAPSp ([0, ∞) : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AAPSbp ([0, ∞) : X). Arguing similarly as in the proof of [132, Proposition 3.4, p. 81] and Theorem 2.11.23 below, we can deduce the following result on the convolution invariance of introduced function spaces belonging to the Stepanov class: Proposition 2.2.12. Let n ∈ ℕ, and let f ∈ Cb(n) − APS1 (ℝ : X), resp. f ∈ APSb1 (ℝ : X). Then, for every g ∈ L1 (ℝ), we have that (g ∗f )(⋅) := ∫ℝ f (⋅−y)g(y) dy ∈ Cb(n) −APS1 (ℝ : X), resp. (g ∗ f )(⋅) ∈ APSb1 (ℝ : X). Hereafter we will use the following lemma (see, e. g., [57, p. 70] for the scalarvalued case): Lemma 2.2.13. Let −∞ < a < b < ∞, let 1 ⩽ p󸀠 < p󸀠󸀠 < ∞, and let f ∈ Lp ([a, b] : X). 󸀠 Then f ∈ Lp ([a, b] : X) and 󸀠󸀠

b

1/p󸀠

1 󵄩p󸀠 󵄩 [ ∫󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds] b−a a

b

1 󵄩p󸀠󸀠 󵄩 ⩽[ ∫󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds] b−a a

1/p󸀠󸀠

.

2.3 Weyl almost periodic functions and asymptotically Weyl almost periodic functions Unless specified otherwise, in this section it will be always assumed that I = ℝ or I = [0, ∞). The pivot Banach space will be denoted by X, as before. It is well known that the concept introduced by H. Weyl [253] suggests a very general way of approaching almost periodicity. The notion of an (equi-)Weyl-p-almost periodic function is given as follows (cf. also (32)). Definition 2.3.1. Let 1 ⩽ p < ∞ and f ∈ Lploc (I : X). p (i) We say that a function f (⋅) is equi-Weyl-p-almost periodic, f ∈ e − Wap (I : X) for short, iff for each ε > 0 we can find two real numbers l > 0 and L > 0 such that any interval I 󸀠 ⊆ I of length L contains a point τ ∈ I 󸀠 such that x+l

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 dt] l x∈I x

1/p

⩽ ε,

i. e.,

DpS [f (⋅ + τ), f (⋅)] ⩽ ε. l

p (ii) We say that the function f (⋅) is Weyl-p-almost periodic, f ∈ Wap (I : X) for short, iff for each ε > 0 we can find a real number L > 0 such that any interval I 󸀠 ⊆ I of

2.3 Weyl almost periodic functions and asymptotically Weyl almost periodic functions | 37

length L contains a point τ ∈ I 󸀠 such that x+l

lim sup[

l→∞ x∈I

1 󵄩󵄩 󵄩p ∫ 󵄩f (t + τ) − f (t)󵄩󵄩󵄩 dt] l 󵄩

1/p

x

⩽ ε,

i. e.,

lim DpS [f (⋅ + τ), f (⋅)] ⩽ ε.

l→∞

l

The spaces of (equi-)Weyl-p-almost periodic functions are not complete w. r. t. the Weyl norm; cf. [20] for further information in this direction. Let us recall that p p APSp (I : X) ⊆ e − Wap (I : X) ⊆ Wap (I : X)

in the set theoretical sense and that any of these two inclusions can be strict [20]. For example, the scalar-valued function f : ℝ → ℂ defined by f (x) := χ(0,1/2) , x ∈ ℝ is not Stepanov 1-almost periodic but it is equi-Weyl-almost-1-periodic (see, e. g., [20, Example 4.27]) and the scalar-valued function f : ℝ → ℂ defined by f (x) := χ(0,∞) , x ∈ ℝ is not equi-Weyl-almost-1-periodic but is Weyl-almost-1-periodic (see, e. g., [21, Example 1]); here, χ(⋅) denotes the characteristic function. We want also to point out p that the space of scalar-valued functions Wap (ℝ : ℝ) seems to be defined and analyzed firstly by A. S. Kovanko [285] in 1944 (according to the information given in the survey paper [20]); the compactness in the spaces of (equi-)Weyl-p-almost periodic functions has been analyzed by the same author [287, 288] in 1951/1953 with the help of Lusternik type theorems. It is well known that for any function f ∈ Lploc (I : X) its Stepanov boundedness is equivalent to its Weyl boundedness, i. e., ‖f ‖Sp < ∞

iff

‖f ‖W p < ∞.

As of now, we use abbreviations e − Wap (I : X) and Wap (I : X) to denote the spaces 1 1 e − Wap (I : X) and Wap (I : X), respectively (the case p = 1 will be the most important in our further analyses). Similarly, we say that a function is (equi-)Weyl-almost periodic iff it is (equi-)Weyl-1-almost periodic. p p (I : X), the space consisting of all essentially (I : X), resp. e − Wap,b Denote by Wap,b p p bounded functions from Wap (I : X), resp. e − Wap (I : X). p It is very instructive to state the following characterization of the space e − Wap (I : X); see, e. g., [20] for the scalar-valued case: p Theorem 2.3.2. Let 1 ⩽ p < ∞ and f ∈ Lploc (I : X). Then f ∈ e − Wap (I : X) iff for every ε > 0 there exists a trigonometric X-valued polynomial Pε (⋅) such that

DpW [Pε (⋅), f (⋅)] < ε. A Bochner type theorem holds for Weyl-p-almost periodic functions, as well [97]: p Theorem 2.3.3. Let 1 ⩽ p < ∞ and let f ∈ Wap (I : X) be uniformly continuous. Then f ∈ AP(I : X).

38 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations p It is well known that the functions from e − Wap (I : X) need to be Weyl uniformly continuous in the following sense (see [57, p. 84]): p Theorem 2.3.4. Let 1 ⩽ p < ∞ and f ∈ Wap (I : X). Then, for every ε > 0, there exist two finite numbers L > 0 and δ0 > 0 such that

DpS [f (⋅ + δ), f (⋅)] < ε L

for |δ| < δ0 . For some other notions of Weyl-p-almost periodicity, like equi-W p -normality and W p -normality, we refer the reader to [20, Section 4]. It can be simply proved that the limit of any uniformly convergent sequence of bounded continuous functions that are (asymptotically) almost periodic, resp. (asymptotically) Stepanov p-almost periodic, has again this property. The following result holds for the (equi-)Weyl-p-almost periodic functions: Proposition 2.3.5. Let (fn ) be a uniformly convergent sequence of functions from e − W p (I : X) ∩ Cb (I : X), resp. W p (I : X) ∩ Cb (I : X), where 1 ⩽ p < ∞. If f (⋅) is the corresponding limit function, then f ∈ e − W p (I : X) ∩ Cb (I : X), resp. f ∈ W p (I : X) ∩ Cb (I : X). Proof. We will prove the statement of proposition only for the equi-Weyl-p-almost periodic functions. It is clear that f ∈ Cb (I : X). Let ε > 0 be given in advance. Then there exists an integer n0 (ε) such that for each n ⩾ n0 (ε) we have that 󵄩 󵄩󵄩 󵄩󵄩fn (t) − f (t)󵄩󵄩󵄩 ⩽ ε,

t ∈ I.

(37)

By definition, we know that there exist two real numbers ln0 > 0 and Ln0 > 0 such that any interval I 󸀠 ⊆ I of length Ln0 contains a point τn0 ∈ I 󸀠 such that sup[ x∈I

1

ln0

x+ln0

󵄩p 󵄩 ∫ 󵄩󵄩󵄩fn0 (t + τn0 ) − fn0 (t)󵄩󵄩󵄩 dt] x

1/p

⩽ ε.

(38)

Then, for the proof of equi-Weyl-p-almost periodicity of function f (⋅), we can choose the same l := ln0 > 0 and L := Ln0 > 0, and the same τ := τn0 from any subinterval I 󸀠 ⊆ I; strictly speaking, we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩f (t + τ) − fn0 (t + τ)󵄩󵄩󵄩 + 󵄩󵄩󵄩fn0 (t + τ) − fn0 (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩f (t) − fn0 (t)󵄩󵄩󵄩 for all t ∈ I, so that a simple calculation involving (37) gives the existence of a finite constant cp > 0 such that x+l

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 dt] x∈I l x

1/p

2.3 Weyl almost periodic functions and asymptotically Weyl almost periodic functions | 39

⩽cp [ε + sup[ x∈I

1

ln0

x+ln0

󵄩p 󵄩 ∫ 󵄩󵄩󵄩fn0 (t + τ) − fn0 (t)󵄩󵄩󵄩 dt]

1/p

x

] ⩽ 2cp ε.

Then the final result simply follows from (38). Before we switch to the next subsection, we would like to note that L. I. Danilov [121] and H. D. Ursell [385] have established two interesting characterizations of equi-Weyl-p-almost periodic functions. Equi-Weyl-p-almost periodic functions have been also investigated by A. Iwanik [228] within the field of topological dynamics.

2.3.1 Asymptotically Weyl almost periodic functions For the beginning, we need to introduce the following notion. If q ∈ Lploc ([0, ∞) : X), then we define the function q(⋅, ⋅) : [0, ∞) × [0, ∞) → X by q(t, s) := q(t + s),

t, s ⩾ 0.

Definition 2.3.6. It is said that q ∈ Lploc ([0, ∞) : X) is Weyl-p-vanishing iff x+l

1 󵄩 󵄩 󵄩 󵄩p lim 󵄩󵄩q(t, ⋅)󵄩󵄩󵄩W p = 0, i. e., lim lim sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] t→∞󵄩 t→∞ l→∞ x⩾0 l x

1/p

= 0.

(39)

It is clear that for any function q ∈ Lploc ([0, ∞) : X) we can replace the limits in (39). We say that q ∈ Lploc ([0, ∞) : X) is equi-Weyl-p-vanishing iff x+l

1/p

1 󵄩 󵄩p lim lim sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] l→∞ t→∞ x⩾0 l

= 0.

x

(40)

Since the second limit in (39) always exists in [0, ∞] (on account of (34)) and the second limit in (40) always exists in [0, ∞] (on account of the fact that the mapping x+l t 󳨃→ supx⩾0 [ 1l ∫x ‖q(t + s)p ‖ ds]1/p , t ⩾ 0 is monotonically decreasing), condition (39) is equivalent to x+l

∀ε > 0 ∃t0 (ε) > 0 ∀t ⩾ t0 (ε) ∃lt > 0 ∀l > lt :

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] x⩾0 l

1/p

x

⩽ ε, (41)

while condition (40) is equivalent to x+l

∀ε > 0 ∃l0 (ε) > 0 ∀l ⩾ l0 (ε) ∃tl > 0 ∀t > tl :

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] l x⩾0 x

1/p

⩽ ε. (42)

40 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Before proceeding further, we would like to observe that there is a great number of very simple examples showing that for a function q ∈ Lploc ([0, ∞) : X) the situation in which ‖q(t, ⋅)‖W p ≠ ‖q(t 󸀠 , ⋅)‖W p for all t ≠ t 󸀠 can occur: Consider, for instance, the function q(t) := 2−1 (t + 1)(−1)/2 , t ⩾ 0 and the case in which p = 1; then a direct computation yields that ‖q(t, ⋅)‖W p = (t + 1)(−1)/2 , t ⩾ 0. The situation is completely different in the case of Besicovitch-p-vanishing functions, as we will see in the appendix section to this chapter. 1. Assume that q ∈ Lp ([0, ∞) : X). Then for each ε > 0 there exists t0 (ε) > 0 such t+1 ∞ that ∫t ‖q(s)‖p ds ⩽ εp , t ⩾ t0 (ε). A fortiori, ∫t ‖q(s)‖p ds ⩽ εp , t ⩾ t0 (ε) and the p function q̂ : [0, ∞) → L ([0, 1] : X) belongs to the class C0 ([0, ∞) : Lp ([0, 1] : X)). The converse statement is not true, however, since the scalar-valued function q(t) = t (−1)/2p , t > 0 satisfies q̂ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)) and q ∉ Lp ([0, ∞) : X). 2. If q ∈ Lploc ([0, ∞) : X) and q̂ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)), then the computation 1/p

x+l

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] x⩾0 l x

x+t+⌈l⌉

x+t+1

1 ⩽ sup[ ( ∫ + ⋅ ⋅ ⋅ + x⩾0 l x+t

∫ x+t+⌈l⌉−1

1/p

󵄩p 󵄩 )󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds]

1/p

⩽ (ε

⌈l⌉ ) l

⩽ 2p ε,

holding for any t ⩾ 0, shows that the function q(⋅) is equi-Weyl-p-vanishing, with t+1 l0 (ε) = 1 and tl = t0 (ε) chosen so that ∫t ‖q(s)‖p ds ⩽ εp , t ⩾ t0 (ε) (l > l0 (ε)). As the following simple counterexample shows, the converse statement does not hold in general: Example 2.3.7. Define ∞

q(t) := ∑ χ[n2 ,n2 +1] (t), n=0

t ⩾ 0.

n2 +1

Since ∫n2 ‖q(s)‖p ds = 1, n ∈ ℕ, it is clear that q̂ ∉ C0 ([0, ∞) : Lp ([0, 1] : X)). On the other hand, the interval [t, t + l] contains at most √t + l − √t + 2 squares of nonnegative integers, so that x+l

t+l

1 󵄩 1 󵄩󵄩 󵄩p 󵄩p ∫ 󵄩q(t + s)󵄩󵄩󵄩 ds ⩽ sup ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds l 󵄩 x⩾t l x

t

1 1 l ), ⩽ (√t + l − √t + 2) ⩽ (2 + l l √t + √l

x ⩾ 0, t ⩾ 0,

so that (42) holds with l0 (ε) > 0 sufficiently large and tl = l (l ⩾ l0 (ε)).

2.3 Weyl almost periodic functions and asymptotically Weyl almost periodic functions | 41

3. If q ∈ Lploc ([0, ∞) : X) and q(⋅) is equi-Weyl-p-vanishing, then q(⋅) is Weyl-p-vanishing. To see this, assume that (42) holds with l0 (ε) > 0 and put t0 (ε) := tl0 (ε) . Therefore, 1 sup[ l (ε) x⩾0 0

x+l0 (ε)

󵄩p 󵄩 ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds]

1/p

⩽ ε,

x

t ⩾ t0 (ε).

(43)

For any fixed t ⩾ t0 (ε), we set lt := l0 (ε). Then it suffices to show that for any l > lt we have x+l

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] x⩾0 l

1/p

x

⩽ 2ε.

This follows from (43) and a simple analysis involving the second inequality in part (i) of [21, Proposition 1]: x+l

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] x⩾0 l

1/p

1/p

⩽2

x

1 sup[ x⩾0 l0 (ε)

x+l0 (ε)

󵄩p 󵄩 ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds]

1/p

x

,

which is valid for l > lt = l0 (ε). Again, the converse statement does not hold in general, and a Weyl-p-vanishing function need not be equi-Weyl-p-vanishing: Example 2.3.8. Define ∞

q(t) := ∑ √nχ[n2 ,n2 +1] (t), n=0

t ⩾ 0.

Then it is clear that x+l

t+l

t+l 1 󵄩 1 󵄩󵄩 󵄩p 󵄩p ∫ 󵄩q(t + s)󵄩󵄩󵄩 ds ⩽ sup ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds ⩽ √ 2 , l 󵄩 l x⩾t l x

t

x ⩾ 0, t ⩾ 0,

so that (41) holds with t0 (ε) > 0 chosen such that √1/(t + 1) ⩽ ε for t ⩾ t0 (ε) and lt = (t + 1)2 . Hence, q(⋅) is Weyl-p-vanishing. On the other hand, q(⋅) cannot be equi-Weyl-p-vanishing because for each number l > 1 there does not exist a finite limit x+l

1/p

1 󵄩 󵄩p lim sup[ ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds] t→∞ x⩾0 l x

.

To see this, it suffices to observe that for each t > 0 and n ∈ ℕ such that n2 > t we have x+l

√n t 1 󵄩 󵄩p > . sup ∫ 󵄩󵄩󵄩q(t + s)󵄩󵄩󵄩 ds ⩾ l l x⩾0 l x

42 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations We would like to note that an equi-Weyl-p-vanishing function q(⋅) need not be bounded as t → +∞: Example 2.3.9. Define ∞

q(t) := ∑ n1/4p χ[n4 ,n4 +1] (t), n=0

t ⩾ 0.

Then, similarly as in Example 2.3.7, we can prove that x+l

t+l

1 󵄩 1 󵄩󵄩 󵄩p 󵄩p ∫ 󵄩q(t + s)󵄩󵄩󵄩 ds ⩽ sup ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds l 󵄩 x⩾t l x

l 1 ), ⩽ (2 + l √t + √l

t

x ⩾ 0, t ⩾ 0,

which implies the required conclusions. Denote by W0p ([0, ∞) : X) and e − W0p ([0, ∞) : X) the sets consisting of all Weyl-p-vanishing functions and equi-Weyl-p-vanishing functions, respectively. The symbol S0p ([0, ∞) : X) will be used to denote the set of all functions q ∈ Lploc ([0, ∞) : X) such that q̂ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)). By our considerations in points 1.–3., Examples 2.3.7 and 2.3.8, we have: Theorem 2.3.10. The following inclusions hold: Lp ([0, ∞) : X) ⊆ S0p ([0, ∞) : X) ⊆ e − W0p ([0, ∞) : X) ⊆ W0p ([0, ∞) : X) and any of them can be strict. We introduce the following function spaces: AAPW p ([0, ∞) : X) := AP([0, ∞) : X) + W0p ([0, ∞) : X),

e − AAPW p ([0, ∞) : X) := AP([0, ∞) : X) + e − W0p ([0, ∞) : X), AAPSW p ([0, ∞) : X) := APSp ([0, ∞) : X) + W0p ([0, ∞) : X),

e − AAPSW p ([0, ∞) : X) := APSp ([0, ∞) : X) + e − W0p ([0, ∞) : X), p p e − Waap ([0, ∞) : X) := e − Wap ([0, ∞) : X) + W0p ([0, ∞) : X),

p p ee − Waap ([0, ∞) : X) := e − Wap ([0, ∞) : X) + e − W0p ([0, ∞) : X), p p Waap ([0, ∞) : X) := Wap ([0, ∞) : X) + W0p ([0, ∞) : X),

p p e − Waap ([0, ∞) : X) := Wap ([0, ∞) : X) + e − W0p ([0, ∞) : X).

Then it is clear that AAPW p ([0, ∞) : X) ⊆ AAPSW p ([0, ∞) : X)

2.3 Weyl almost periodic functions and asymptotically Weyl almost periodic functions | 43

p p ⊆ e − Waap ([0, ∞) : X) ⊆ Waap ([0, ∞) : X),

e − AAPW p ([0, ∞) : X) ⊆ e − AAPSW p ([0, ∞) : X)

p p ⊆ ee − Waap ([0, ∞) : X) ⊆ e − Waap ([0, ∞) : X),

and that any of these inclusions can be strict. As already mentioned, the function f : [0, ∞) → ℂ defined by f (t) := χ(0,1/2) (t), t ⩾ 0 is equi-Weyl-almost periodic; since this function is also in the class e−W01 ([0, ∞) : X), p p p we have that the sums defining e−Waap ([0, ∞) : X), ee−Waap ([0, ∞) : X), Waap ([0, ∞) : p p X) and e − Waap ([0, ∞) : X) are not direct. For the first four spaces AAPW ([0, ∞) : X), e − AAPW p ([0, ∞) : X), AAPSW p ([0, ∞) : X) and e − AAPSW p ([0, ∞) : X), the sums in their definitions are direct, which follows from the following proposition: Proposition 2.3.11. Let 1 ⩽ p < ∞. Then W0p ([0, ∞) : X) ∩ APSp ([0, ∞) : X) = {0}. Proof. Assume q ∈ W0p ([0, ∞) : X) ∩ APSp ([0, ∞) : X). In order to see that q(t) = 0 for ̂ = 0, t ⩾ 0 in Lp ([0, 1] : X). Since q(⋅) ̂ is almost a. e. t ⩾ 0, it suffices to show that q(t) ̂ is equal to periodic, we only need to prove that any Bohr–Fourier coefficient of q(⋅) zero, i. e., that 1/p 󵄩󵄩p 󵄩󵄩 1 t 󵄩󵄩 󵄩󵄩 −irs lim (∫󵄩󵄩 ∫ e q(s + v) ds󵄩󵄩󵄩 dv) = 0, 󵄩󵄩 t→∞ 󵄩󵄩 t 󵄩 0󵄩 0 1󵄩

r ∈ ℝ.

(44)

To see that (44) holds true, observe first that p

󵄩󵄩 󵄩󵄩 1 t 󵄩󵄩 󵄩 (∫󵄩󵄩󵄩 ∫ e−irs q(s + v) ds󵄩󵄩󵄩 dv) 󵄩󵄩 󵄩󵄩 t 󵄩 0󵄩 0 1󵄩

1/p

1

t

0

0

p

1/p

1 󵄩 󵄩 ⩽ (∫[∫󵄩󵄩󵄩q(s + v)󵄩󵄩󵄩 ds] dv) t

,

which can be further majorized by using Lemma 2.2.13: 1

1/p

t

1 󵄩p 󵄩 ⩽ (∫ t p−1 ∫󵄩󵄩󵄩q(s + v)󵄩󵄩󵄩 ds dv) t 0

0

1/p

1 t

󵄩p 󵄩 = t (−1)/p (∫ ∫󵄩󵄩󵄩q(s + v)󵄩󵄩󵄩 ds dv)

.

0 0

Hence, we need to prove that 1 t

1 s+t

1 1 󵄩p 󵄩 󵄩p 󵄩 lim ∫ ∫󵄩󵄩󵄩q(s + v)󵄩󵄩󵄩 ds dv = lim ∫ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr dv = 0. t→∞ t t→∞ t

(45)

0 s

0 0

Let ε > 0 be given in advance. Since q ∈ W0p ([0, ∞) : X), we know that there exist two finite numbers t0 (ε) > 0 and l0 (ε) > 0 such that, for every l > l0 (ε), we have x+l

1 󵄩 󵄩p sup[ ∫ 󵄩󵄩󵄩q(t0 (ε) + s)󵄩󵄩󵄩 ds] x⩾0 l x

1/p

⩽ ε.

(46)

44 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Let T0 (ε) > 0 be such that for each t > T0 (ε) we have t ⩾ t0 (ε)2

and t − √t ⩾ l0 (ε).

(47)

The validity of (47) clearly implies by (46) that s+t

1 󵄩p 󵄩 ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds ⩽ ε, t − √t

s ∈ [0, 1].

(48)

s+√t

Since s+t

s+1

s

s

1 + ( t ⩽

s+⌈√t⌉

s+2

1 󵄩󵄩 1 󵄩p ∫ 󵄩󵄩q(r)󵄩󵄩󵄩 dr = ( ∫ + ∫ + ⋅ ⋅ ⋅ + t t s+1

s+⌈√t⌉+1

∫ s+⌈√t⌉

∫ s+⌈√t⌉−1

󵄩p 󵄩 )󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr

s+t

󵄩p 󵄩 + ⋅ ⋅ ⋅ + ∫ )󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr s+⌊t⌋

⌈√t⌉ t − ⌊t⌋ ‖q‖Sp + ε t t

by Sp -boundedness of q(⋅) and (48), equation (45) holds true. The proof of the proposition is thereby complete. It is very simple to prove that e − W0p ([0, ∞) : X) and W0p ([0, ∞) : X) are vector spaces, so that the introduced eight function spaces have a linear vector structure. Disregarding the term ([0, ∞) : X), and taking into consideration the previously defined spaces AAP and AAPSp , we have the following inclusion diagram of asymptotically almost periodic function spaces (see Theorem 2.3.10): AAP

⊆

e − AAPSW p

AAPW p

⊆

AAPSW p

⊆

p e − Waap

⊆

p Waap .

⊆

AAPSp

⊆

⊆

e − AAPW p

⊆

⊆

⊆

⊆ p ee − Waap

⊆

⊆ p e − Waap

By the foregoing, any inclusion of this diagram can be strict. Furthermore, for any two function spaces A and B belonging this diagram and satisfying additionally that there is no transitive sequence of inclusions connecting either A and B, or B and A, we have that A ∖ B ≠ 0 and B ∖ A ≠ 0 (the diagram can be expanded by constructing the sums of

2.3 Weyl almost periodic functions and asymptotically Weyl almost periodic functions | 45

spaces of (equi-)Weyl-p-almost periodic functions with S0p ([0, ∞) : X), which will not be examined here). In [1], S. Abbas has introduced the notions of a Weyl p-pseudo almost automorphic function and a Weyl p-pseudo ergodic component. Definition 2.3.12. Let p ⩾ 1. Then we say that a function q ∈ Lploc (ℝ : X) is a Weyl p-pseudo ergodic component iff it satisfies T

x+l

1 1 󵄩󵄩 󵄩p ∫ [ lim ∫ 󵄩󵄩q(t)󵄩󵄩󵄩 dt] T→+∞ 2T l→+∞ 2l

1/p

lim

dx = 0.

(49)

x−l

−T

The set of all such functions is denoted by Wp PAA0 (ℝ : X). Now we will prove that the class of Weyl-p-vanishing functions (extended by zero outside [0, ∞)) is contained in Wp PAA0 (ℝ : X): Proposition 2.3.13. Let 1 ⩽ p < ∞, and let q ∈ Lploc ([0, ∞) : X) be a Weyl-p-vanishing function. Let qe ∈ Lploc (ℝ : X) be given by qe (t) := q(t), t ⩾ 0 and qe (t) := 0, t < 0. Then qe ∈ Wp PAA0 (ℝ : X). Proof. We only need to prove that (49) holds with q(⋅) replaced therein with qe (⋅), i. e., that T

x+l

0

0

1 󵄩󵄩 1 󵄩p lim ∫[ lim ∫ 󵄩󵄩q(t)󵄩󵄩󵄩 dt] T→+∞ 2T l→+∞ 2l

1/p

dx = 0.

Let x ∈ [0, T] be fixed. It suffices to show that x+l

1 󵄩󵄩 󵄩p lim ∫ 󵄩󵄩q(t)󵄩󵄩󵄩 dt = 0. l→+∞ 2l

(50)

0

Towards this end, fix a number ε > 0. Owing to the fact that q(⋅) is Weyl-p-vanishing, i. e., that x+l

lim lim sup[

t→∞ l→∞

x⩾t

1 󵄩󵄩 󵄩p ∫ 󵄩q(s)󵄩󵄩󵄩 ds] l 󵄩 x

1/p

= 0,

we have the existence of numbers l0 (ε) > 0 and t0 (ε) > 0 such that x+l

1 󵄩󵄩 󵄩p ∫ 󵄩q(s)󵄩󵄩󵄩 ds ⩽ ε, 2l 󵄩 x

x ⩾ t0 (ε), l ⩾ l0 (ε).

(51)

Then we have x+l

x

x+l

0

0

x

1 󵄩󵄩 1 󵄩p 󵄩p 󵄩 󵄩p 󵄩 ∫ 󵄩q(t)󵄩󵄩󵄩 dt ⩽ [∫󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds + ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds], 2l 󵄩 2l

l > 0.

46 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations x+l

If x ⩾ t0 (ε), then the term (1/2l) ∫x ‖q(s)‖p ds is less or equal than ε by (51), which clearly implies the existence of a number l1 (ε) > 0 such that for each l ⩾ l1 (ε) we have x+l

1 󵄩󵄩 󵄩p ∫ 󵄩q(t)󵄩󵄩󵄩 dt ⩽ 2ε. 2l 󵄩 0

If x < t0 (ε), then we have x+l

t0 (ε)

t0 (ε)+l

x

x

t0 (ε)

1 󵄩󵄩 1 󵄩p 󵄩 󵄩p 󵄩 󵄩p ∫ 󵄩󵄩q(t)󵄩󵄩󵄩 dt ⩽ [ ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds + ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds] 2l 2l t0 (ε)

⩽

1 󵄩p 󵄩 ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds + ε, 2l x

l ⩾ l1 (ε),

which clearly implies the existence of a number l2 (ε) > l1 (ε) such that for each number l ⩾ l2 (ε) we have x+l

1 󵄩󵄩 󵄩p ∫ 󵄩q(t)󵄩󵄩󵄩 dt ⩽ 2ε. 2l 󵄩 0

This yields (50) and completes the proof of proposition. We round off the section by introducing the following definition. Definition 2.3.14. Let I = ℝ or I = [0, ∞), let (R(t))t∈I ⊆ L(X) be a strongly continuous operator family, and let ⊕ denote any of (asymptotical) almost periodic properties considered above. Then we say that (R(t))t∈I is ⊕ (asymptotically) almost periodic iff the mapping t 󳨃→ R(t)x, t ∈ I is ⊕ (asymptotically) almost periodic for all x ∈ X. It is said that (R(t))t∈I is uniformly almost periodic iff the family {R(⋅)x : ‖x‖ ⩽ 1} is uniformly almost periodic.

2.4 Almost periodic solutions of abstract degenerate first and second order Cauchy problems We start this section by reconsidering the structural results proved by Q. Zheng and L. Liu in [421]. The assumption on denseness of R(C) is crucial in this paper, but careful inspections of proofs show that the injectivity of regularizing operator C is superfluous in almost all places (if D(𝒜) is dense in X, 𝒜 generates a C-semigroup (T(t))t⩾0 (C-cosine function (C(t))t⩾0 ) and C is injective, then (T(t))t⩾0 is nondegenerate since T(0) = C (C(0) = C), which automatically implies that 𝒜 is single-valued). Unless specified otherwise, in this section, we assume that (T(t))t∈ℝ is a global C-group with the integral generator 𝒜; this means that 𝒜 generates a C-semigroup

2.4 Almost periodic solutions of abstract degenerate first and second order

| 47

(T(t))t⩾0 and −𝒜 generates a C-semigroup (T(−t))t⩾0 (a class of very simple counterexamples shows that the equality T+ (t)T− (t) = C 2 stated in the proof of implication (b) ⇒ (c) of [421, Theorem 3.1] does not hold for degenerate C-groups, even in the case that C = I), C can be possibly noninjective. Then it is very simple to prove that the assumption irx ∈ 𝒜x for some r ∈ ℝ implies T(t)x = eirt Cx, t ∈ ℝ. Keeping in mind this fact, as well as Theorem 2.1.1, parts (i)–(ii), (iv) and (vii), it is straightforward to extend the assertion of [421, Theorem 2.1] to degenerate C-groups; cf. also Theorems 2.4.3–2.4.4, Proposition 2.5.1 and Theorem 2.5.5 below. Theorem 2.4.1. Suppose that R(C) = D(𝒜) = X. Then (T(t))t∈ℝ is almost periodic iff (T(t))t∈ℝ is bounded and the set D consisting of all eigenvectors of operator 𝒜 corresponding to purely imaginary eigenvalues of operator 𝒜 is total in X (i. e., the linear span of D is dense in X). Before going any further, we would like to point out that Theorem 2.4.1 does not hold without assuming the denseness of 𝒜. For example, the a. p. degenerate group (T(t) ≡ 0)t∈ℝ has the integral generator {0} × X, but the set D is vacuous. Furthermore, almost periodic C-groups (T(t))t∈ℝ for which R(C) = D(𝒜) = X and 𝒜 is not single-valued really exist. To see this, observe that for each C ∈ L(X) we have that (T(t) ≡ C)t∈ℝ is a global C-group whose integral generator 𝒜 is given by 𝒜 = X × N(C); furthermore, in this case, we have that ρC (𝒜) = 0 provided that the operator C is not injective. Arguing as in [421], we can prove the following: 1. If c0 ⊈ X, ±𝒜 are the integral generators of global bounded C-uniqueness families (T(±t))t⩾0 and the mapping t 󳨃→ T(t)y, t ∈ ℝ is almost periodic for all y ∈ R(𝒜), then the mapping t 󳨃→ T(t)x, t ∈ ℝ is almost periodic for all x ∈ D(𝒜) (cf. [421, Proposition 2.4]). Here we use parts (vi) and (vii) of Theorem 2.1.1. 2. Let us recall that X is weakly sequentially complete iff every weak Cauchy sequence in X converges weakly. If this is the case, then the denseness of R(C) and D(𝒜) in X implies that the concepts weak almost periodicity and almost periodicity of a global C-group (T(t))t∈ℝ are mutually equivalent; furthermore, the almost periodicity of (T(t))t∈ℝ implies its uniform periodicity provided, in addition to the above, that {eλt : λ ∈ σp (𝒜)} is uniformly almost periodic. Keeping in mind Theorem 2.1.1, the proofs of these statements are almost the same as those of parts (a) and (b) in [421, Theorem 2.5]; cf. also the proof of [43, Theorem 3] and observe that for each r ∈ ℝ the graph of ir − 𝒜 is a closed convex subset of X × X, as well as that the partial integration and Theorem 1.2.2 together imply that for each r ∈ ℝ we have t

1 1 [Cx − e−irt T(t)x] ∈ lim (ir − 𝒜) ∫ e−irs T(s)x ds, t→∞ t t

x ∈ X.

0

The following is an extension of [421, Theorem 3.3] to degenerate C-semigroups.

48 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Theorem 2.4.2. Suppose that (T(t))t⩾0 is an almost periodic C-semigroup with the integral generator 𝒜. Then there exists a bounded, strongly continuous, almost periodic operator family (S(t))t∈ℝ ⊆ L(X) commuting with C and satisfying S(t) = T(t),

t⩾0

and

S(t)S(s) = S(t + s)C,

t, s ∈ ℝ.

(52)

Furthermore, if (T(t))t⩾0 is uniformly almost periodic, then (S(t))t∈ℝ is likewise uniformly almost periodic. In the case that C is injective, we have that (S(t))t∈ℝ is a global C-group with the integral generator 𝒜 (i. e., (S(−t))t⩾0 is a global C-semigroup with the integral generator −𝒜). Proof. Since (T(t))t⩾0 is almost periodic, it must be uniformly bounded. Let M := supt⩾0 ‖T(t)‖. Define S(t)x := [E(Tx (⋅))](t), t ∈ ℝ, x ∈ X, where Tx (t) := T(t)x, x ∈ X, t ⩾ 0. Since E is a linear surjective isometry between the spaces AP([0, ∞) : X) and AP(ℝ : X), we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩S(t)x󵄩󵄩󵄩 ⩽ sup󵄩󵄩󵄩S(s)x󵄩󵄩󵄩 = sup󵄩󵄩󵄩S(s)x󵄩󵄩󵄩 = sup󵄩󵄩󵄩T(s)x󵄩󵄩󵄩 ⩽ M‖x‖, s∈ℝ

s⩾0

s⩾0

x ∈ X, t < 0,

so that S(t) ∈ L(X) for all t ∈ ℝ, and supt∈ℝ ‖S(t)‖ = M < ∞. It can be easily seen that S(⋅) commutes with C. On the other hand, we have S(t)S(s)x = S(t)[E(Tx (⋅))](s) = [E(S(t)Tx (⋅))](s) = [E(CW(t)Tx (⋅))](s)

= C[E(W(t)Tx (⋅))](s) = C[W(s)W(t)Tx (⋅)](0) = C[W(t + s)Tx (⋅)](0) = C[E(Tx (⋅))](t + s) = CS(t + s)x,

t ⩾ 0, s ⩽ 0.

Since S(t) and S(s) commute, the proof of (52) is completed, which simply implies that (S(t))t∈ℝ is uniformly almost periodic provided that (T(t))t⩾0 is. If the operator C is injective, a simple computation shows that the integral generator ℬ of a global C-semigroup (S(−t))t⩾0 equals −𝒜 (in general case, we can only prove that ℬ ⊆ C −1 [−𝒜]C and −𝒜 ⊆ C −1 ℬC). If C is injective, then R(C) furnished with the norm ‖ ⋅ ‖R(C) := ‖C −1 ⋅ ‖ becomes a Banach space; we will denote this space simply by [R(C)]. Now we are ready to prove the following extension of [421, Theorem 3.4]: Theorem 2.4.3. Suppose that (T(t))t⩾0 is a global C-semigroup with the integral generator 𝒜, as well as R(C) = D(𝒜) = X. Then (T(t))t⩾0 is almost periodic iff (T(t))t⩾0 is bounded and the set D consisting of all eigenvectors of operator 𝒜 corresponding to purely imaginary eigenvalues of operator 𝒜 is total in X. Furthermore, in the case that the operator C is injective, then we have that 𝒜 = A is single-valued, as well as that: (i) ℂ ∖ iℝ ⊆ ρC (A); if the number ir is a pole of ρC (A), then ir is its simple pole (in particular, ir ∈ σp (A)), and its residue is Pr defined by

2.4 Almost periodic solutions of abstract degenerate first and second order

| 49

t

1 Pr x = lim ∫ e−irs T(s)x ds, t→+∞ t

x ∈ X.

0

(ii) If σp (A) is bounded, then A ∈ L([R(C)], X). Proof. Suppose that (T(t))t⩾0 is almost periodic. Then it is clear that (T(t))t⩾0 is bounded. Let (S(t))t∈ℝ be given by Theorem 2.4.2. Repeating literally the proof of Necessity in [421, Theorem 3.1], with T(⋅) replaced by S(⋅) therein, we obtain that the set D is total in X. For the converse, we can use the arguments contained in the proof of Sufficiency in the above-mentioned theorem since Theorem 2.1.1(vii) holds for the functions defined on the semiaxis [0, ∞) (recall that the mapping E defined above is a linear surjective isometry) and the assumption ir ∈ 𝒜x for some r ∈ ℝ and x ∈ X implies T(t)x = eirt Cx, t ⩾ 0. The remnant is a part of [421, Theorem 3.3]. 3.

The periodicity of abstract (degenerate) Volterra integro-differential equations is not our focus here (cf. the paper [40] by V. Barbu and A. Favini for some interesting applications given in this direction). We only want to observe the following: Suppose that (T(t))t∈ℝ is a global C-group with the integral generator 𝒜. Then we say that a number p > 0 is a period of (T(t))t∈ℝ iff T(t + p) = T(t) for all t ∈ ℝ. If this is the case, then it can be easily seen that p

(Cx,

∫0 e−λs T(s)x ds 1 − e−λp

) ∈ λ − 𝒜,

x ∈ X, λ ∈ ℂ ∖ 2πip−1 ℤ;

furthermore, if the set Dp consisting of all eigenvectors of operator 𝒜 corresponding to eigenvalues λ ∈ ℂ ∖ 2πip−1 ℤ of operator 𝒜 is total in X, then p is a period of (T(t))t∈ℝ (see [421, Theorem 5.1]). Similar statements can be proved for global C-cosine functions (see [421, Theorems 5.2–5.3]). Now we will analyze almost periodicity of abstract degenerate second order Cauchy problems. Let 𝒜 be the integral generator of a C-cosine function (C(t))t⩾0 . Set C(−t) := t

C(t), S(t) := ∫0 C(s) ds and S(−t) := −S(t) (t ⩾ 0). As it is well known, S(⋅) is said to be a sine function associated with C(⋅). It can be easily seen that d’Alambert functional equation 2C(t)C(s) = C(t + s)C + C(t − s)C holds for all t, s ∈ ℝ. The assertion of [421, Theorem 4.1] can be reformulated for degenerate cosine functions in the following way; the proof is standard and omitted therefore (the only thing worth noting is that the assumption −r 2 x ∈ 𝒜x for some r ∈ ℝ and x ∈ X implies C(t)x = cos(rt)Cx, t ∈ ℝ): Theorem 2.4.4. Let R(C) = D(𝒜) = X. Then (C(t))t∈ℝ ((S(t))t∈ℝ ) is almost periodic iff (C(t))t∈ℝ ((S(t))t∈ℝ ) is bounded and the set D consisting of all eigenvectors of 𝒜 corresponding to the real non-positive eigenvalues of 𝒜 is total in X.

50 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Remark 2.4.5. Observe that the injectivity of C (we do not need the condition R(C) = D(𝒜) = X here) and the almost periodicity of (S(t))t∈ℝ imply that there exists a > 0 such that σp (𝒜) ⊆ (−∞, −a2 ]; cf. the final part of proof of [421, Theorem 4.1] and the proof of Sufficiency in [402, Theorem 1.2, pp. 242–243] for the nondegenerate case. It is well known that the almost periodicity of (C(t))t∈ℝ does not imply the almost periodicity of (S(t))t∈ℝ , even in the case that C = I (see [186]); the most simplest counterexample refuting this assertion is: C(t) = I, t ∈ ℝ and S(t) = tI, t ∈ ℝ. Using the argumentation already employed in the nondegenerate case [421], we can deduce the following: 4. If c0 ⊈ X, then the almost periodicity of (C(t))t∈ℝ , taken together with the boundedness of (S(t))t∈ℝ , implies the almost periodicity of (S(t))t∈ℝ ; see [421, Theorem 4.3(a)–(b)]. 5. If c0 ⊈ X, the integral generator of a global bounded C-cosine function (C(t))t∈ℝ is densely defined and the mapping t 󳨃→ S(t)y, t ∈ ℝ is almost periodic for all y ∈ R(𝒜), then (C(t))t∈ℝ is almost periodic (cf. the proof of [421, Proposition 2.4]). 6. In the case when the state space X is weakly sequentially complete, as well as that R(C) = D(𝒜) = X, any w. a. p. C-sine (or C-cosine) function is automatically a. p. 7. The almost periodicity of (C(t))t∈ℝ ((S(t))t∈ℝ ) implies its uniform periodicity provided, in addition to the weak sequential completeness of X, that the family {eλt : λ2 ∈ σp (𝒜)} is uniformly almost periodic. The proofs of statements 6.–7. follows similarly as in that of [421, Theorem 4.4]. 8. Any degenerate semigroup (T(t))t⩾0 is exponentially bounded. By the proof of [29, Lemma 3.14.3], the above is also true for degenerate cosine functions. Keeping in mind this fact, we can repeat almost literally the proof of [402, Theorem 1.6, pp. 247–249] in order to see that the (weak) almost periodicity of a sine function (S(t))t∈ℝ implies that for each u ∈ D(𝒜) the mapping t 󳨃→ C(t)u, t ∈ ℝ is (weakly) almost periodic. Strictly speaking, a more general result holds true: Suppose that 𝒜 is the integral generator of an exponentially bounded C-cosine function (C(t))t⩾0 (we do not need the denseness of 𝒜 or the range of C in X, C can be noninjective). Then the (weak) almost periodicity of a C-sine function (S(t))t∈ℝ implies that for each u ∈ D(𝒜) the mapping t 󳨃→ C(t)u, t ∈ ℝ is (weakly) almost periodic. This can be seen by considering the function g(t, x) := S(t)y − ω20 S(t)x, t ⩾ 0, where y ∈ 𝒜x is arbitrarily chosen and ω0 is strictly greater than the exponential type of (C(t))t⩾0 . Then the computation given on pp. 247–248 of [402] shows that ∞

−

1 ∫ e−ω0 |t−s| g(s, x) ds = S(t)x, 2ω0

t ∈ ℝ,

−∞

and the final conclusion follows as in the nondegenerate case.

2.4 Almost periodic solutions of abstract degenerate first and second order

| 51

Suppose now that k ∈ ℕ and A is a closed, single-valued linear operator with nonempty resolvent set, say λ0 ∈ ρ(A). Then it is well known that: (i) A generates a global exponentially bounded k-times integrated semigroup iff A generates a global exponentially bounded (λ0 − A)−k -semigroup. (ii) A generates a global exponentially bounded (2k)-times ((2k + 1)-times) integrated cosine function iff A generates a global exponentially bounded (λ0 − A)−k -cosine function ((λ0 − A)−k−1 -cosine function). Slight extensions of the above statements are clarified in [245, Propositions 2.3.12 and 2.3.13]. The formulae obtained in these propositions show that one will be in a tight corner if wants to expect the boundedness of induced (λ0 − A)−k -semigroups, resp. (λ0 − A)−k -cosine functions ((λ0 − A)−k−1 -cosine functions), in the case when A does not generate a strongly continuous semigroup (cosine operator function). Nevertheless, Q. Zheng and L. Liu have investigated in [421, Section 6] various questions about almost periodicity of induced regularized semigroups and cosine functions, giving also a necessary and sufficient condition for the generation of almost periodic tempered distribution semigroups (we will not analyze here a similar problematic for tempered ultradistribution semigroups; see [245] for the notion). 9. Let us recall that X. Gu, M. Li and F. Huang have investigated the almost periodicity of C-semigroups, integrated semigroups and C-cosine groups in [205], by assuming that the range of C is not necessarily dense in X. Their results lean heavily on the use of Hille–Yosida’s spaces for closed single-valued linear operators which do not have eigenvalues in (0, ∞), and we would like to underline that these results cannot be so easily reworded in degenerate case. It is also worth observing the following: 10. Let 𝒜 be an MLO. Denote by D (F) the set consisting of all eigenvectors of operator 𝒜 corresponding to purely imaginary eigenvalues of operator 𝒜 (to nonpositive real eigenvalues of operator 𝒜). By D0 (F0 ) we denote the set consisting of all eigenvectors of operator 𝒜 corresponding to purely imaginary nonzero eigenvalues of operator 𝒜 (to negative real eigenvalues of operator 𝒜). Suppose that n ∈ ℕ0 , 𝒜 is a subgenerator of an n-times integrated C2 -uniqueness t family (Sn (t))t⩾0 , r ∈ ℝ and irx ∈ 𝒜x. Then Sn (t)x − gn+1 (t)C2 x = ir ∫0 Sn (s)x ds, t ⩾ 0, which simply implies that the mapping t 󳨃→ Sn (t)x, t ⩾ 0 is infinitely differentiable with all derivatives at zero of order less than or equal to n − 1 being zeroes. Hence, the mapping t 󳨃→ (dn /dt n )Sn (t)x = eirt C2 x, t ⩾ 0 is almost periodic for all y ∈ span(D) and the mapping t 󳨃→ (dn−1 /dt n−1 )Sn (t)x, t ⩾ 0 is almost periodic for all y ∈ span(D0 ). Similarly, if 𝒜 is a subgenerator of an n-times integrated C2 -cosine uniqueness family (Cn (t))t⩾0 , r ∈ ℝ and −r 2 x ∈ 𝒜x, then t

Cn (t)x − gn+1 (t)C2 x = ir ∫0 (t − s)Cn (s)x ds, t ⩾ 0, which simply implies that the mapping t 󳨃→ Cn (t)x, t ⩾ 0 is infinitely differentiable with all derivatives at zero of order

52 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations less than or equal to n − 1 being zeroes. Hence, the mapping t 󳨃→ (dn /dt n )Cn (t)x = cos(rt)C2 x, t ⩾ 0 is almost periodic for all y ∈ span(F), as well as the mappings t 󳨃→ (dn−1 /dt n−1 )Cn (t)x, t ⩾ 0 and t 󳨃→ (dn−2 /dt n−2 )Cn (t)x, t ⩾ 0 are almost periodic for all y ∈ span(F0 ). Here, dl /dt l ⋅ = g−l ∗ ⋅ for l ∈ −ℕ.

2.5 Almost periodic solutions of abstract Volterra integro-differential equations We start our work in this section by stating the following simple, but important result (disappointingly, we will be forced to reproduce Proposition 2.5.1 a few times henceforth, for different classes of functions and for different solution operator families). Proposition 2.5.1. Suppose that abs(|a|) < ∞, abs(k) < ∞ and 𝒜 is a subgenerator of a mild, strongly Laplace transformable, (a, k)-regularized C2 -uniqueness family (R2 (t))t⩾0 . Denote by D the set consisting of all eigenvectors x of operator 𝒜 corresponding to eigenvalues λ ∈ ℂ of operator 𝒜 for which the mapping fλ,x (t) := ℒ−1 (

̃ k(z) )(t)C2 x, ̃ 1 − λa(z)

t⩾0

(53)

is almost periodic. Then the mapping t 󳨃→ R2 (t)x, t ⩾ 0 is almost periodic for all x ∈ span(D); furthermore, the mapping t 󳨃→ R2 (t)x, t ⩾ 0 is almost periodic for all x ∈ span(D) provided additionally that (R2 (t))t⩾0 is bounded. Proof. Let x ∈ D be an eigenvector of operator 𝒜 corresponding to an eigenvalue λ ∈ ℂ of operator 𝒜. Then t

λ ∫ a(t − s)R2 (s)x ds = R2 (t)x − k(t)C2 x,

t ⩾ 0.

0

Performing the Laplace transform, we get that R2 (t)x = fλ,x (t), t ⩾ 0. This immediately implies the final conclusions by parts (ii) and (vii) of Theorem 2.1.1. Remark 2.5.2. Suppose that R2 (t)𝒜 ⊆ 𝒜R2 (t), t ⩾ 0 and x ∈ span(D). Then the mapping t 󳨃→ u(t) := R2 (t)x, t ⩾ 0 is a strong solution of the abstract Cauchy inclusion (20) with ℬ = I and ℱ (t) = k(t)C2 x, t ⩾ 0. Remark 2.5.3. Suppose that x ∈ D, λx ∈ 𝒜x and C2 x ≠ 0. Then the scalar-valued function ϑ(t) := t 󳨃→ ℒ−1 (

̃ k(z) )(t), ̃ 1 − λa(z)

t⩾0

(54)

is almost periodic. Since scalarly-valued almost periodic functions are uniform limits of trigonometric polynomials in BUC(ℝ), the principal case in which the above holds

2.5 Almost periodic solutions of abstract Volterra integro-differential equations | 53

is that there exist an integer n ∈ ℕ, real numbers r1 (λ), . . . , rn (λ), a positive real number ω(λ), and complex numbers α1 (λ), . . . , αn (λ), such that ̃ αn (λ) α1 (λ) k(z) = + ⋅⋅⋅ + , ̃ 1 − λa(z) z − ir1 (λ) z − irn (λ)

Re z > ω(λ).

(55)

It is worth noting that (55) holds for substantially large classes of kernels a(t) and regularizing functions k(t); for example, (55) holds in the case a(t) = k(t) = sin t, −1 λ ∈ (−∞, 1) ⊆ σp (𝒜), n = 2, ir1,2 (λ) = ±√λ − 1, α1,2 (λ) = ±2−1 √λ − 1 . If a(t) = gn (t) + n−1 n−1 ∑j=0 aj gj (t), k(t) = gn (t) + ∑j=0 bj gj (t), where aj , bj ∈ ℂ for 1 ⩽ j ⩽ n − 1 and there exists a nonempty subset Ω of σp (𝒜) such that for each λ ∈ Ω the polynomial n−1

Pλ (z) = z n − λ ∑ an−j z j − λ, j=1

z∈ℂ

has purely imaginary, pairwise disjoint, roots ir1 (λ), . . . , irn (λ), then (55) holds with appropriately chosen complex numbers α1 (λ), . . . , αn (λ). The situation is a little bit delicate in purely fractional case. For example, if a(t) = gα (t), where α ∈ (0, ∞) ∖ ℕ, and (55) holds for pairwise disjoint numbers rj (λ) and nonzero complex numbers αj (λ) (1 ⩽ j ⩽ n), then it is necessary to have λ ∈ (iℝ)α , λ = (irj (λ))α (1 ⩽ j ⩽ n), and the function n

k(t) = ∑ αj (λ)[eirj (λ)t − λt α E1,1+α (irj (λ)t)], j=1

t⩾0

should be independent of parameter λ; cf. the Laplace transform identity (19). From an application point of view, this is very difficult to be satisfied; see also Theorem 2.6.10 below. The following useful lemma is probably known in the existing literature; we will include the proof for the sake of clarity (cf. also [350, Theorem 1.3(iii)] for a similar result). Lemma 2.5.4. Let (R(t))t⩾0 ⊆ L(X) be a bounded strongly continuous operator family, ̃ and let x ∈ X. If a ∈ L1loc ([0, τ)), a ≠ 0, a(ir) exists for some r ∈ ℝ and PrR x

t

1 := lim ∫ e−irs R(s)x ds t→∞ t 0

exists, then Pra∗R x

t

1 = lim ∫ e−irs (a ∗ R)(s)x ds t→∞ t 0

exists as well, and the following holds: R ̃ Pra∗R x = a(ir)P r x.

(56)

54 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Proof. Let ε > 0 be given, and let ‖R(t)‖ ⩽ M 󸀠 , t ⩾ 0. Then there exists M > 0 such that 󵄨󵄨 󵄨󵄨 ∞ 󵄨 󵄨󵄨 −irs 󵄨󵄨 ∫ e a(s) ds󵄨󵄨󵄨 < ε. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 M Furthermore, t

t s

0

0 0

1 1 ∫ e−irs (a ∗ R)(s)x ds = ∫ ∫ e−ir(s−v) e−irv R(v)x dv ds t t t v

=

1 ∫ ∫ e−ir(s−v) e−irv R(v)x ds dv t 0 t t

t−v

0

0

1 = ∫[ ∫ e−irs a(s) ds]e−irv R(v)x dv. t Hence, for any t > M, we have 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 1 󵄩󵄩 R 󵄩 󵄩󵄩 ∫ e−irs (a ∗ R)(s)x ds − a(ir)P ̃ x r 󵄩 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 0

󵄩󵄩 t ∞ 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 −irs −irv 󵄩 = 󵄩󵄩 ∫[ ∫ e a(s) ds]e R(v)x dv󵄩󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩 0 t−v 󵄩

t ∞ 󵄩󵄩 󵄩󵄩 t−M 󵄩󵄩 󵄩󵄩 1 = 󵄩󵄩󵄩 [ ∫ + ∫ ][ ∫ e−irs a(s) ds]e−irv R(v)x dv󵄩󵄩󵄩. 󵄩󵄩 󵄩󵄩 t 󵄩 󵄩 t−v 0 t−M

Now equality (56) follows from (57) and the next estimates: 󵄩󵄩 t−M ∞ 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 ∫ [ ∫ e−irs a(s) ds]e−irv R(v)x dv󵄩󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 t−v t−M

1 t−M ⩽ ∫ εM 󸀠 dv = εM 󸀠 , t t 0

󵄩󵄩 󵄩󵄩 t M 󵄩 󵄩󵄩 1 󵄩󵄩 ∫ [ ∫ e−irs a(s) ds]e−irv R(v)x dv󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 t−M t−v

M

t−M M

⩽

M 1 󵄨 󵄨 󵄨 󵄨 ∫ [∫󵄨󵄨󵄨a(s)󵄨󵄨󵄨 ds] M 󸀠 dv = [∫󵄨󵄨󵄨a(s)󵄨󵄨󵄨 ds] M 󸀠 , t t t

0

0

󵄩󵄩 t ∞ 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩󵄩 ∫ [ ∫ e−irs a(s) ds]e−irv R(v)x dv󵄩󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 t−M M

(57)

2.5 Almost periodic solutions of abstract Volterra integro-differential equations | 55 t−M

1 M ⩽ ∫ εM 󸀠 dv = εM 󸀠 . t t t

Now we are ready to state the following necessary conditions for an (a, k)-regularized C-resolvent family (R(t))t⩾0 to be almost periodic. In some sense, this is a converse to Proposition 2.5.1. Theorem 2.5.5. Let 𝒜 be the integral generator of an almost periodic (a, k)-regularized C-resolvent family (R(t))t⩾0 , let R(C) = D(𝒜) = X, and let k(0) ≠ 0. Denote ̃ ℛ := {r ∈ ℝ : a(ir) exists}.

(58)

̃ Suppose that k(t) and |a|(t) satisfy (P1), limRe z→∞ a(z) = 0, as well as that Prk

t

1 = lim ∫ e−irs k(s) ds = 0, t→∞ t

r ∈ ℛ.

(59)

0

Then (R(t))t⩾0 is bounded and (Q) holds, where R ̃ (Q) PrR x ∈ 𝒜[a(ir)P r x], r ∈ ℛ, x ∈ X and the mapping R(t)PrR x = ℒ−1 (

̃ a(ir) ̃ k(z) )(t)CPrR x, ̃ ̃ a(ir) − a(z)

t ⩾ 0, x ∈ X,

is almost periodic for all r ∈ ℛ and x ∈ X. Suppose, in addition, that R(t)PrR x = k(t)CPrR x,

t ⩾ 0, r ∈ ℝ ∖ ℛ, x ∈ X.

(60)

Then the set D consisting of all eigenvectors of operator 𝒜 corresponding to eigenvalues ̃ −1 : r ∈ ℛ, a(ir) ̃ λ ∈ {0} ∪ {a(ir) ≠ 0} of operator 𝒜 is total in X. Proof. The boundedness of (R(t))t⩾0 follows from Theorem 2.1.1(i) and the uniform boundedness principle. Let x ∈ X. Since R(C) = D(𝒜) = X, we have that R(0) = k(0)C. By [247, (271)], (R(t))t⩾0 satisfies the following functional equality: R(t)(a ∗ R)(s)x − k(t)C(a ∗ R)(s)x = (a ∗ R)(t)R(s)x − k(s)(a ∗ R)(t)Cx,

(61)

for all t, s ⩾ 0; see also the identity preceding Definition 1.5.3. On the other hand, Lemma 2.5.4 implies that, for every r ∈ ℛ, we have that Pra∗R x exists and (56) holds. By employing this equation, as well as (59) and (61), we get that, for every r ∈ ℛ, R R ̃ ̃ R(t)a(ir)P r x − k(t)C a(ir)P rx

= R(t)Pra∗R x − k(t)CPra∗R x

56 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations σ

1 ∫ e−irs [R(t)(a ∗ R)(s)x − k(t)C(a ∗ R)(s)x] ds σ→∞ σ

= lim

0 σ

= lim

σ→∞

1 ∫ e−irs [(a ∗ R)(t)R(s)x − k(s)C(a ∗ R)(t)x] ds σ 0

σ

1 ∫ e−irs [R(s)x − k(s)Cx] ds σ→∞ σ

= (a ∗ R)(t) lim = (a ∗

R)(t)PrR x,

0

t ⩾ 0.

R ̃ Hence, PrR x ∈ 𝒜[a(ir)P r x], r ∈ ℛ and, performing the Laplace transform, we get that ̃ condition (Q) holds; notice only that the condition limRe z→∞ a(z) = 0 implies that for ̃ ̃ ̃ on some right half-plane. Assume now each r ∈ ℛ with a(ir) ≠ 0 we have a(ir) ≠ a(z) ̃ that a(ir) = 0 for some r ∈ ℛ. Then the previous computation gives (a ∗ R)(t)PrR x = 0, t ⩾ 0 so that R(t)PrR x = 0, t ⩾ 0, as well as R(0)PrR x = k(0)CPrR x and CPrR x = 0 due to condition k(0) ≠ 0. This simply implies 0 = R(t)PrR x = k(t)CPrR x, t ⩾ 0 and therefore 0 ∈ 𝒜PrR x. The validity of (60) implies that 0 ∈ 𝒜PrR x for all r ∈ ℝ∖ ℛ as well. Therefore, if PrR x ≠ 0 for some r ∈ ℝ, then PrR x ∈ D. Suppose that x∗ ∈ X ∗ and ⟨x ∗ , y⟩ = 0 for all y ∈ D. Then σ

1 lim ∫ e−irs ⟨x∗ , R(s)x⟩ ds = ⟨x ∗ , PrR x⟩ = 0, σ→∞ σ

r ∈ ℝ,

0

so that the almost periodicity of mapping t 󳨃→ R(t)x, t ⩾ 0 yields by Theorem 2.1.1(iv) that ⟨x∗ , R(t)x⟩ = 0 for all t ⩾ 0. Specifically, ⟨x∗ , k(0)Cx⟩ = ⟨x ∗ , Cx⟩ = 0, whence we may conclude by our assumption R(C) = X that x∗ = 0. This completes the proof of theorem. Remark 2.5.6. (i) Suppose that 𝒜 = A is single-valued and generates an almost periodic (a, k)-regularized C-resolvent family (R(t))t⩾0 . Since the equation [246, (22), Proposition 2.1.3] holds in our framework, we have that R(0) = k(0)C. Therefore, if we disregard the condition R(C) = D(𝒜) = X and accept all other conditions from the first part of formulation of Theorem 2.5.5, then (R(t))t⩾0 is still bounded and (Q) still holds. Let it be the case, and let (60) be fulfilled. Then the final part of proof of Theorem 2.5.5 shows that span(D)∘ ⊆ R(C)∘ , which simply implies by the bipolar theorem that R(C) ⊆ span(D). (ii) We feel duty bound to say that condition (60) from the formulation of Theorem 2.5.5 seems to be slightly redundant. This condition is satisfied in the usual considerations of almost periodicity of various types of semigroups and cosine operator functions.

2.5 Almost periodic solutions of abstract Volterra integro-differential equations | 57

In the following two propositions, we will reconsider our conclusions from the points 2., 6.–7. of previous section for (a, k)-regularized C-resolvent families. Proposition 2.5.7. Let 𝒜 be the integral generator of a weak almost periodic (a, k)-regularized C-resolvent family (R(t))t⩾0 , let R(C) = D(𝒜) = X, and let k(0) ≠ 0. Suppose that ̃ = 0, (59) holds (see (58)), and k(t) is almost periodic, |a|(t) satisfies (P1), limRe z→∞ a(z) σ

1 ∫ e−irs ⟨x∗ , R(s)x⟩ ds = 0, σ→∞ σ lim

r ∈ ℝ ∖ ℛ, x∗ ∈ R(𝒜∗ ), x ∈ X.

(62)

0

Let X be weakly sequentially complete. Then (R(t))t⩾0 is almost periodic. Proof. Since (R(t))t⩾0 is weakly almost periodic, the uniform boundedness principle and Mackey’s theorem together imply that (R(t))t⩾0 is bounded. Fix an element x ∈ X. Then the weak sequential completeness of X, in combination with the weak almost periodicity of (R(t))t⩾0 , yields that for each number r ∈ ℝ there exists an element Mr x ∈ X such that σ

1 ∫ e−irs ⟨x ∗ , R(s)x⟩ ds, σ→∞ σ

⟨x ∗ , Mr x⟩ = lim

x∗ ∈ X ∗ .

0

Let a pair (y∗ , x∗ ) ∈ 𝒜∗ and a number r ∈ ℛ be fixed. Then ⟨y∗ , R(t)x − k(t)Cx⟩ = ⟨x ∗ , (a ∗ R)(t)x⟩,

t ⩾ 0.

Making use of this equality, as well as Lemma 2.5.4 and (59), it readily follows that ̃ ⟨y∗ , Mr x⟩ = ⟨x∗ , a(ir)M r x⟩,

r ∈ ℛ.

Owing to Lemma 1.2.1, the above implies ̃ Mr x ∈ 𝒜[a(ir)M r x],

r ∈ ℛ.

̃ The proof of Theorem 2.5.5 shows that 0 ∈ 𝒜Mr x for all r ∈ ℛ with a(ir) = 0, as well as ̃ a(ir) ̃ R −1 k(z) that R(t)Mr x = ℒ ( a(ir)− ̃ ̃ )(t)CPr x, t ⩾ 0, r ∈ ℛ. By means of Lemma 1.2.1 and (62), a(z) we infer that 0 ∈ 𝒜Mr x

and R(t)Mr x = k(t)CMr x,

r ∈ ℝ ∖ ℛ, t ⩾ 0.

Repeating to the letter the final part of the proof of Theorem 2.5.5, we get that the set D consisting of all eigenvectors of operator 𝒜 corresponding to eigenvalues λ ∈ {0} ∪ ̃ −1 : r ∈ ℛ, a(ir) ̃ {a(ir) ≠ 0} of operator 𝒜 is total in X. Now the final conclusion follows by applying Proposition 2.5.1.

58 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations After proving Theorem 2.5.5, Proposition 2.5.7 and giving some useful observations in Remark 2.5.6, we want to blow the cobwebs away by stating a sufficient condition for the uniform almost periodicity of degenerate (a, k)-regularized C-resolvent families. For this, we first need to remind ourselves that a nonempty subset Λ of ℝ is called harmonious iff for each ε > 0 the set 󵄨 󵄨 ⋂ {τ ∈ ℝ : 󵄨󵄨󵄨eiλτ − 1󵄨󵄨󵄨 ⩽ ε}

λ∈Λ

is relatively dense in ℝ. It is well known that a subset of a harmonious set Λ is harmonious, as well as that for any finite set F the set Λ+F is also harmonious. Any nonempty finite set and certain lacunary infinite sequences are harmonious, too. Proposition 2.5.8. Let 𝒜 be the integral generator of an almost periodic (a, k)-regularized C-resolvent family (R(t))t⩾0 . Suppose that k(t) and |a|(t) satisfy (P1), (59) holds, and the set ̃ ̃ −1 ∈ σp (𝒜)} ∪ {r ∈ ℛ : a(ir) ̃ Λ := {r ∈ ℛ : a(ir) ≠ 0, a(ir) = 0} ∪ (ℝ ∖ ℛ) is harmonious. Then (R(t))t⩾0 is uniformly almost periodic. Proof. Without loss of generality, we may assume that R(⋅) is defined on the whole real line ℝ and strongly almost periodic there. Suppose that PrR x ≠ 0 for some r ∈ ℝ and x ∈ X with ‖x‖ ⩽ 1. Then the proof of Theorem 2.5.5 shows that r ∈ Λ. Since (R(t))t∈ℝ is almost periodic, this inclusion, in combination with the uniform boundedness principle, yields that {R(⋅)x : x ∈ X, ‖x‖ ⩽ 1} is a bounded subset of APΛ (I : X). Since Λ is harmonious, the claimed assertion follows from [43, Theorem 13]. Remark 2.5.9. If 𝒜 = A is single-valued, then it is sufficient to assume that the set ̃ ̃ −1 ∈ σp (𝒜)} ∪ (ℝ ∖ ℛ) Λ󸀠 := {r ∈ ℛ : a(ir) ≠ 0, a(ir) is harmonious. As a matter of fact, due to the proof of Theorem 2.5.5, we have that the ̃ assumption PrR x ≠ 0 for some r ∈ ℛ with a(ir) = 0 and some x ∈ X with ‖x‖ ⩽ 1 implies R ̃ PrR x ∈ A[a(ir)P x] = A0 = {0}, which is a contradiction. r Now we would like to point out a few facts concerning the possibilities for transferring the assertion of Theorem 2.4.2 to (a, k)-regularized C-resolvent families. Suppose that 𝒜 is the integral generator of an almost periodic (a, k)-regularized C-resolvent family (R(t))t⩾0 satisfying R(t)R(s) = R(s)R(t) for all t, s ⩾ 0. Set S(t)x := [E(Rx (⋅))](t), t ∈ ℝ, x ∈ X, where Rx (t) := R(t)x, x ∈ X, t ⩾ 0. Arguing as in the proof of Theorem 2.4.2, we get that (S(t))t∈ℝ ⊆ L(X) is a bounded, strongly continuous, almost periodic operator family such that S(t) = R(t) for all t ⩾ 0 and S(t)S(s) = S(s)S(t) for all t, s ∈ ℝ. Furthermore, the uniform almost periodicity of (R(t))t⩾0 is equivalent to that of (S(t))t∈ℝ . Equation (28) cannot be expected with (R(t))t∈ℝ replaced by (S(t))t∈ℝ . To explain this in more detail, suppose that the functions t 󳨃→ k(t), t ⩾ 0 and t 󳨃→ (a ∗ R)(t)x, t ⩾ 0 are

2.5 Almost periodic solutions of abstract Volterra integro-differential equations | 59

almost periodic as well. Let t ⩾ 0 and s ⩽ 0. Then (28) and the properties of extension mapping E : AP([0, ∞) : X) → AP(ℝ : X) show that (a ∗g S)(t)S(s)x = [W(s)(a ∗g S)(t)Rx (⋅)](0) = [W(s){S(t)(a ∗g S)(⋅)x

+ k(⋅)(a ∗g S)(t)Cx − k(t)C(a ∗g S)(⋅)x}](0)

= S(t)[E((a ∗g S)(⋅)x)](s)

+ [E(k)](s)(a ∗g S)(t)x − k(t)C[E((a ∗g S)(⋅)x)](s).

In the general case, [E(k)](s) ≠ k(−s) for s < 0 and the obtained composition property is clearly different from (28). An additional problem is how to compute [E((a ∗g S)(⋅)x)](s), so that we have reached the end of the road with this therapy. Taking the Laplace transform of both sides of (29), it is straightforward to prove the following analogue of Proposition 2.5.1 for exponentially bounded (a, k)-regularized C-resolvent families generated by a pair of closed linear operators. Proposition 2.5.10. Suppose that abs(|a|) < ∞, abs(k) < ∞ and (R(t))t⩾0 is an exponentially bounded (a, k)-regularized C-resolvent family generated by A, B. Denote by D the set consisting of all nonzero vectors x ∈ D(A) ∩ D(B) such that there exists λ ∈ ℂ, satisfying λBx = Ax, and the mapping ϑ(t), defined through (54), is almost periodic. Then the mapping t 󳨃→ R(t)Bx, t ⩾ 0 is almost periodic for all x ∈ span(D); furthermore, the mapping t 󳨃→ R(t)y, t ⩾ 0 is almost periodic for all y ∈ B(span(D)) provided additionally that (R(t))t⩾0 is bounded. Concerning the statement of Theorem 2.5.5, it is very difficult to say what will be the consequences of almost periodicity of an exponentially bounded (a, k)-regularized C-resolvent family (R(t))t⩾0 generated by A, B (with the exception of its boundedness, which is a trivial thing that must be satisfied). But, as an unambiguous consequence of Theorem 2.5.5, we can clarify some necessary conditions for the operator family (BR(t))t⩾0 ⊆ L(X) to be almost periodic. More precisely, we always have that (BR(t))t⩾0 is an exponentially bounded (a, k)-regularized C-resolvent family generated by the multivalued linear operator AB−1 (recall that AB−1 is closed provided that C = I, as well as that any multivalued linear operator is closable); cf. [247, Remark 3.2.6(iv)]. Since BR(0) = k(0)C, in the corresponding reformulation we do not need to employ the condition on the denseness of domain of the multivalued linear operator AB−1 . The interested reader may try to rephrase the assertions of Propositions 2.5.7–2.5.8 for the operator family (BR(t))t⩾0 . Before we move ourselves to the next section, we would like to observe that B. Basit and V. V. Zhikov [49] have also analyzed almost periodic solutions of abstract Volterra integro-differential equations in Banach spaces; cf. also [25, 129] and [346].

60 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations

2.6 Asymptotically almost periodic solutions and Stepanov asymptotically almost periodic solutions of abstract Volterra integro-differential equations We start this section by stating the following simple consequence of [52, Theorem 4.1], which provides proper extensions of [405, Theorems 3.7 and 3.9] (cf. also [27, Theorem 4.1]). ̃ ̃ k(λ) Theorem 2.6.1. Suppose that abs(k) = abs(|a|) = 0, a(λ) ≠ 0 for Re λ > 0, C1 ∈ L(Y, X), and 𝒜 is a closed subgenerator of a mild (a, k)-regularized C1 -existence family (R1 (t))t⩾0 . Let y ∈ Y be such that the mapping t 󳨃→ R1 (t)y, t ⩾ 0 is bounded and uniformly ̃ 𝒜 is injective for Re λ > 0, then continuous. If the operator I − a(λ) ∞

−1 ̃ k(λ) 1 H(λ) := ( − 𝒜) C1 y = ∫ e−λt R1 (t)y dt, ̃ ̃ a(λ) a(λ)

Re λ > 0.

(63)

0

Suppose that the singular set iS of mapping λ 󳨃→ H(λ), Re λ > 0, where S ⊆ ℝ, is at most countable. If for every μ ∈ S, we have that ∞

lim λ ∫ e−(λ+iμ)t R1 (t + s)y dt

λ→0+

0

exists, uniformly in s ⩾ 0, then the mapping t 󳨃→ R1 (t)y, t ⩾ 0 is asymptotically almost periodic. Proof. Applying Theorem 1.2.2 and the Laplace transform, we get that ∞

̃ ̃ 𝒜) ∫ e−λt R1 (t)y dt, k(λ)C 1 y ∈ (I − a(λ)

Re λ > 0,

0

i. e., ∞

−1 ̃ 1 k(λ) −1 ̃ ̃ 𝒜) C1 y ∋ ∫ e−λt R1 (t)y dt, ( − 𝒜) C1 y = k(λ)(I − a(λ) ̃ ̃ a(λ) a(λ)

Re λ > 0.

0

̃ 𝒜 is injective for Re λ > 0, we have that the set ((1/a(λ)) ̃ Since the operator I − a(λ) − 𝒜)−1 C1 x is a singleton, so that the last equality immediately implies (63). Now the proof

follows from a simple application of [52, Theorem 4.1].

Remark 2.6.2. Suppose α > 0, β ⩾ 0, a(t) = gα (t), k(t) = gβ+1 (t), X = Y, C1 = I and the set of all λ ∈ iℝ ∖ {0} such that λα ∈ σ(𝒜) is at most countable. Then the singular set iS of mapping λ 󳨃→ H(λ), Re λ > 0 is at most countable as well.

2.6 Solutions of abstract Volterra integro-differential equations | 61

Further information on connections between countable spectrum of operators and asymptotical almost periodicity can be obtained by consulting the monograph [29]. In order to proceed, we state the following proposition, which is very similar to Propositions 2.5.1 and 2.5.10. Proposition 2.6.3. Suppose that abs(|a|) < ∞, abs(k) < ∞ and 𝒜 is a subgenerator of a mild, strongly Laplace transformable, (a, k)-regularized C2 -uniqueness family (R2 (t))t⩾0 . Denote by D the set consisting of all eigenvectors x of operator 𝒜 corresponding to eigenvalues λ ∈ ℂ of operator 𝒜 for which the mapping fλ,x (⋅), given by (53), is asymptotically almost periodic. Then the mapping t 󳨃→ R2 (t)x, t ⩾ 0 is asymptotically almost periodic for all x ∈ span(D); furthermore, the mapping t 󳨃→ R2 (t)x, t ⩾ 0 is asymptotically almost periodic for all x ∈ span(D) provided additionally that (R2 (t))t⩾0 is bounded. Proof. Suppose that x ∈ D is an eigenvector of operator 𝒜 corresponding to an eigenvalue λ ∈ σp (𝒜). Exploiting the identity t

λ ∫ a(t − s)R2 (s)x ds = R2 (t)x − k(t)C2 x,

t ⩾ 0,

0

and performing the Laplace transform, we obtain that R2 (t)x = fλ,x (t), t ⩾ 0. This immediately implies the result since AAP([0, ∞) : X) is a closed subspace of BUC([0, ∞) : X). As explained in the previous section, one is in chancery if one wants to apply Proposition 2.5.1 with a(t) = gα (t), where α ∈ (0, ∞) ∖ ℕ. The situation is completely different if we consider the asymptotical almost periodicity, when Proposition 2.6.3 can be essentially applied. Example 2.6.4. Suppose that α ∈ (0, 2) and θ = π − πα/2. Let us consider the fractional Cauchy problem Dαt u(t, x) = eiθ uxx (t, x),

0 < x < 1, t ⩾ 0,

proposed already by E. Bazhlekova in her doctoral dissertation [53, Example 2.20] and equipped with initial boundary conditions like for the general problem of form (DFP)L with B = I. Let X := L2 [0, 1] and A := eiθ Δ, where Δ denotes the Dirichlet Laplacian. It is well known that A is the integral generator of a bounded (gα , I)-resolvent family (R(t))t⩾0 . Since A has eigenvalues λn = eiαπ/2 n2 π 2 and eigenfunctions xn = sin nπx, n ∈ ℕ, the Laplace transform identity (19) shows that fλn ,xn (t) := Eα (eiαπ/2 n2 π 2 t α )xn ,

t ⩾ 0, n ∈ ℕ.

In the case that α = 1, the above simply implies that (R(t))t⩾0 is almost periodic. The situation is not the same in the case α ∈ (0, 2)∖{1}: Then the asymptotic expansion formulae for the Mittag-Leffler functions (16), (18) imply that the mapping t 󳨃→ fλn ,xn (t), t ⩾ 0

62 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations 2/α 2/α

is asymptotically almost periodic for all n ∈ ℕ, because the mapping t 󳨃→ α−1 eitn π , 2/α 2/α t ⩾ 0 is almost periodic and the mapping t 󳨃→ Eα (eiαπ/2 n2 π 2 t α ) − α−1 eitn π , t ⩾ 0 is continuous, tending to zero as t goes to infinity. Hence, (R(t))t⩾0 is asymptotically almost periodic (since AP([0, ∞) : X) ∩ C0 ([0, ∞) : X) = {0}, the mapping t 󳨃→ R(t)x, t ⩾ 0 cannot be almost periodic for x ∈ span({xn : n ∈ ℕ}); this is a very intriguing fact for fractional resolvent families of order α close to 2− because H. R. Henríquez has proved [220, Theorem 3] that a strongly continuous cosine operator function (C(t))t⩾0 is almost periodic iff (C(t))t⩾0 is asymptotically almost periodic iff (C(t))t⩾0 is Stepanov asymptotically almost periodic). Finally, we would like to observe that we can similarly examine asymptotically almost periodic solutions of certain classes of abstract multiterm fractional Cauchy problems with Caputo derivatives, see also [246, Example 2.10.32], Section 3.4 and V. Keyantuo, C. Lizama, M. Warma [236]. Let us return to Proposition 2.6.3 once more. If x ∈ D, λx ∈ 𝒜x and C2 x ≠ 0, then the function ϑ(t) := t 󳨃→ ℒ−1 (

̃ k(z) )(t), ̃ 1 − λa(z)

t⩾0

needs to be asymptotically almost periodic. By our former examinations, it is very purposeful to analyze the case in which there exist an integer n ∈ ℕ, real numbers r1 (λ), . . . , rn (λ), a positive real number ω(λ), complex numbers α1 (λ), . . . , αn (λ), and a function f ∈ C0 ([0, ∞)) such that ̃ αn (λ) α1 (λ) k(z) = + ⋅⋅⋅ + + f ̃(z), ̃ 1 − λa(z) z − ir1 (λ) z − irn (λ)

Re z > ω(λ).

(64)

It is clear that (64) holds for substantially large classes of kernels a(t) and regularizing functions k(t). Proposition 2.6.3 and Proposition 2.6.5 below, whose proof is omitted, can be also formulated for the class of exponentially bounded (a, k)-regularized C-resolvent families generated by a pair of closed linear operators: Proposition 2.6.5. Suppose that abs(|a|) < ∞, abs(k) < ∞ and 𝒜 is a subgenerator of a mild, strongly Laplace transformable, (a, k)-regularized C2 -uniqueness family (R2 (t))t⩾0 . Denote by D the set consisting of all eigenvectors x of operator 𝒜 corresponding to eigenvalues λ ∈ ℂ of operator 𝒜 for which the mapping fλ,x (⋅), given by (53), is (asymptotically) Stepanov almost periodic. Then the mapping t 󳨃→ R2 (t)x, t ⩾ 0 is (asymptotically) Stepanov almost periodic for all x ∈ span(D); furthermore, the mapping t 󳨃→ R2 (t)x, t ⩾ 0 is (asymptotically) Stepanov almost periodic for all x ∈ span(D) provided additionally that (R2 (t))t⩾0 is bounded. Now we would like to inscribe some basic facts about asymptotical almost periodicity of subordinated fractional solution operator families (see, e. g., [53, Theorem 3.1] for a foundational result in this direction). Assume that 0 < α < β, γ = α/β and

2.6 Solutions of abstract Volterra integro-differential equations | 63

(Sβ (t))t⩾0 ⊆ L(X) is a strongly continuous operator family satisfying ‖Sβ (t)‖ ⩽ Meωt , t ⩾ 0 for some constants M ⩾ 1 and ω ∈ ℝ. Many subordination principles appearing in the theory of abstract (degenerate) Volterra integro-differential equations are closely connected with the following formula: ∞

Sα (t)x := ∫ Φγ (s)Sβ (st γ )x ds,

x ∈ X, t > 0

and Sα (0) := Sβ (0),

(65)

0

where Φγ (⋅) denotes the Wright function. Concerning the inheritance of asymptotical almost periodicity under the action of this subordination principle, we have the following result: Theorem 2.6.6. (i) Suppose that ω < 0. Then ‖Sα (t)‖ = O(t −γ ), t ⩾ 1. (ii) Suppose that x ∈ X and the mapping t 󳨃→ Sβ (t)x, t ⩾ 0 belongs to the space C0 ([0, ∞) : X). Then the mapping t 󳨃→ Sα (t)x, t ⩾ 0 belongs to the space C0 ([0, ∞) : X), too. (iii) Suppose that x ∈ X and the mapping t 󳨃→ Sβ (t)x, t ⩾ 0 belongs to the space AAP([0, ∞) : X). Then the mapping t 󳨃→ Sα (t)x, t ⩾ 0 also belongs to the space AAP([0, ∞) : X). Proof. By definition of Wright functions, we have that ∞

∫ e−zs Φγ (s) ds = Eγ (−z),

z ∈ ℂ.

(66)

0

Keeping in mind this identity, (65) and the asymptotic expansion formulae for MittagLeffler functions (16), (18), it readily follows that the assumption ω < 0 yields that, for every x ∈ X, t −γ 󵄨 󵄩 󵄩󵄩 γ 󵄨 + O(t −2γ )], 󵄩󵄩Sα (t)x󵄩󵄩󵄩 ⩽ M‖x‖󵄨󵄨󵄨Eγ (ωt )󵄨󵄨󵄨 = M‖x‖[− ωΓ(γ − 1)

t → +∞.

This proves (i). To prove (ii), choose a number ε > 0 arbitrarily. Then there exists M > 0 such that ‖g(v)‖ < ε, v ⩾ M. Suppose that ‖g(v)‖ < M 󸀠 , v ⩾ 0 for some finite constant M 󸀠 > 0. Then Mt −γ

∞

󵄩 󵄩 󵄩󵄩 󵄩 γ 󵄩 γ 󵄩 󵄩󵄩Sα (t)x󵄩󵄩󵄩 ⩽ ∫ Φγ (s)󵄩󵄩󵄩g(st )󵄩󵄩󵄩 ds + ∫ Φγ (s)󵄩󵄩󵄩g(st )󵄩󵄩󵄩 ds 0

Mt

Mt −γ

−γ

∞

⩽ ∫ Φγ (s)M ds + ∫ Φγ (s)ε ds 󸀠

0

Mt −γ

64 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Mt −γ

∞

⩽ ∫ Φγ (s)M ds + ∫ Φγ (s)ε ds 󸀠

0

0

Mt −γ

= ∫ Φγ (s)M 󸀠 ds + ε < 2ε,

t → +∞.

0

It remains only to prove (iii). By (ii), it suffices to show that, for every function f ∈ AP([0, ∞) : X), the function ∞

F(t) := ∫ Φγ (s)f (st γ ) ds,

t > 0;

F(0) := f (0)

0

belongs to the space AAP([0, ∞) : X). For every n ∈ ℕ, we can find a trigonometric polynomial fn (⋅) such that the sequence (fn )n∈ℕ converges to f in BUC([0, ∞) : X), as n → ∞. Define ∞

Fn (t) := ∫ Φγ (s)fn (st γ ) ds,

t > 0;

Fn (0) := fn (0).

0

By the proof of [53, Theorem 3.1], the function F(⋅) and functions Fn (⋅) are continuous (n ∈ ℕ). However, we do have ∞

󵄩 󵄩 ‖Fn − F‖∞ ⩽ sup ∫ Φγ (s)󵄩󵄩󵄩fn (st γ ) − f (st γ )󵄩󵄩󵄩 ds t⩾0

∞

0

⩽ ∫ Φγ (s)‖fn − f ‖∞ ds = ‖fn − f ‖∞ → 0,

n → ∞.

0

Keeping in mind that AAP([0, ∞) : X) is closed in the space BUC([0, ∞) : X), it remains to be proved that Fn ∈ AAP([0, ∞) : X) for all n ∈ ℕ. Here, an application of (66) strikes the right note, showing that Fn (⋅) is a linear combination of functions like Eγ (iθ⋅γ ) ⊗ x (θ ∈ ℝ, x ∈ X). The final conclusion is a consequence of the fact that Eγ (iθ⋅γ ) = 1 for θ = 0 and Eγ (iθ⋅γ ) ∈ C0 ([0, ∞) : X) for θ ≠ 0, which follows again from the asymptotic expansion formulae for the Mittag-Leffler functions. Remark 2.6.7. It is very nontrivial and difficult to say anything relevant about the invariance of asymptotical Stepanov p-almost periodicity under the action of this subordination principle. Let 1 ⩽ p < ∞ and let f : [0, ∞) → X be Stepanov p-almost periodic. Then the t+α Bohr–Fourier coefficients Pr (f ) := limt→∞ 1t ∫α e−irs f (s) ds exists for all r ∈ ℝ, independently of α ∈ ℝ, and the assumption Pr (f ) = 0 for all r ∈ ℝ implies that f (t) = 0 for a. e. t ∈ ℝ. Especially, with r = α = 0, we obtain that the first antiderivative of f (⋅)

2.6 Solutions of abstract Volterra integro-differential equations | 65

is exponentially bounded so that the function f (⋅) satisfies (P1). Keeping these facts in mind, we can repeat literally the proof of Theorem 2.5.5 in order to see that the following result holds true: Theorem 2.6.8. Let 𝒜 be the integral generator of a Stepanov p-almost periodic (a, k)-regularized C-resolvent family (R(t))t⩾0 , let R(C) = D(𝒜) = X, and let k(0) ≠ 0. ̃ = 0, as Define ℛ through (58). Suppose that k(t) and |a|(t) satisfy (P1), limRe z→∞ a(z) well as that (59) holds. Then we have R ̃ (Q)󸀠 PrR x ∈ 𝒜[a(ir)P r x], r ∈ ℛ, x ∈ X and the mapping R(t)PrR x = ℒ−1 (

̃ a(ir) ̃ k(z) )(t)CPrR x, ̃ ̃ a(ir) − a(z)

t ⩾ 0, x ∈ X,

is Stepanov p-almost periodic for all r ∈ ℛ and x ∈ X. Suppose, in addition, that (60) holds. Then the set D consisting of all eigenvectors of ̃ −1 : r ∈ ℛ, a(ir) ̃ operator 𝒜 corresponding to eigenvalues λ ∈ {0} ∪ {a(ir) ≠ 0} of operator 𝒜 is total in X. For the proof of Theorem 2.6.10 stated below, we need the following auxiliary lemma. Lemma 2.6.9. Suppose that α ∈ (0, 2) ∖ {1} and r ∈ ℝ ∖ {0}. Then the function t 󳨃→ Eα ((ir)α t α ), t ⩾ 0 is not Stepanov almost periodic. Proof. Suppose to the contrary that the function t 󳨃→ Eα ((ir)α t α ), t ⩾ 0 is Stepanov almost periodic. By the asymptotic expansion formula for the Mittag-Leffler functions, we have that Eα ((ir)α t α ) = α−1 (ir)1−β eirt + εα ((ir)α t α ),

t ⩾ 1,

where 󵄨󵄨 −α α α 󵄨 󵄨󵄨εα ((ir) t )󵄨󵄨󵄨 = O(t ), t ⩾ 1.

(67)

Furthermore, the function t 󳨃→ εα ((ir)α t α ), t ⩾ 0 needs to be Stepanov almost periodic since the function t 󳨃→ α−1 (ir)1−β eirt , t ⩾ 0 is almost periodic. By (67), we get that there exist two finite constants c1 , c2 > 0 such that 󵄨 󵄨 c1 t −α ⩽ 󵄨󵄨󵄨εα ((ir)α t α )󵄨󵄨󵄨 ⩽ c2 t −α ,

t ⩾ 1.

By employing these estimates, we obtain that t+1

c1 ∫ s t

−αp

t+1

t+1

t

t

󵄨p 󵄨 ds ⩽ ∫ 󵄨󵄨󵄨εα ((ir)α sα )󵄨󵄨󵄨 ds ⩽ c2 ∫ s−αp ds,

t ⩾ 1.

This simply implies that εα̂ ((ir)α ⋅α ) ∈ C0 ([0, ∞) : Lp ([0, 1] : X)), which is a contradicα α tion since εα̂ ((ir) ⋅ ) ∈ AP([0, ∞) : Lp ([0, 1] : X)).

66 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Theorem 2.6.10. Let C ∈ L(X) be injective, let A be a closed single-valued linear operator, and let R(C) = X. Suppose that α ∈ (0, 2) ∖ {1} and A generates a Stepanov almost periodic (gα , C)-resolvent family (R(t))t⩾0 . Then A = 0 ∈ L(X) and R(t) = C, t ⩾ 0. Proof. Suppose that r ∈ ℝ ∖ {0} and x ∈ X satisfies that PrR x ≠ 0. By Theorem 2.6.8 and injectiveness of C, we get that the function Eα ((ir)α t α )CPrR x = ℒ−1 (

zα

z α−1 )(t)CPrR x, − (ir)α

t⩾0

is Stepanov almost periodic. This is false due to Lemma 2.6.9, and therefore, PrR x = 0, r ∈ ℝ ∖ {0}, x ∈ X. Using dominated convergence and simple arguments, this implies t

1 lim ∫ e−irs R(s + ⋅)x ds = 0 t→∞ t

(in Lp ([0, 1] : X)),

r ∈ ℝ ∖ {0}, x ∈ X.

0

By spectral synthesis, R(t + ⋅)x = Const. in Lp ([0, 1] : X), t ⩾ 0, which simply implies by the continuity of mapping t 󳨃→ R(t)x, t ⩾ 0 and R(0) = C that R(t) = C, t ⩾ 0. Therefore, the integral generator A of (R(t))t⩾0 is the zero operator. 2.6.1 Stepanov (asymptotically) almost periodic properties of convolution products In this subsection, we will investigate the Stepanov (asymptotically) almost periodic properties of various types of convolution products (for almost periodicity and asymptotical almost periodicity, see [11, Lemmas 2.12 and 2.13] and [127, Lemma 4.1]). Our first result reads as follows (by Y we denote another Banach space over the field of complex numbers). Proposition 2.6.11. Let 1 ⩽ p < ∞, 1/p + 1/q = 1 and (R(t))t>0 ⊆ L(X, Y) be a strongly continuous operator family such that M := ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞. If g : ℝ → X is Sp -almost periodic, then the function G : ℝ → Y, given by t

G(t) := ∫ R(t − s)g(s) ds,

t ∈ ℝ,

(68)

−∞

is well-defined and almost periodic. Proof. Without loss of generality, we may assume that X = Y. It can be easily seen that, ∞ for every t ∈ ℝ, we have G(t) = ∫0 R(s)g(t − s) ds, and the last integral is absolutely convergent by Hölder inequality and Sp -boundedness of function g(⋅): ∞

∞ k+1

󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩g(t − s)󵄩󵄩󵄩 ds = ∑ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩g(t − s)󵄩󵄩󵄩 ds 0

k=0 k

2.6 Solutions of abstract Volterra integro-differential equations | 67

∞

t ∈ ℝ.

󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ‖g‖Sp = M‖g‖Sp , k=0

Let a number ε > 0 be given in advance. Then we can find a finite number l > 0 such t+1 that any subinterval I of ℝ of length l contains a number τ ∈ I such that ∫t ‖g(s + τ) − g(s)‖p ds ⩽ εp , t ∈ ℝ. Applying Hölder inequality and this estimate, we get that 󵄩 󵄩󵄩 󵄩󵄩G(t + τ) − G(t)󵄩󵄩󵄩 ∞

󵄩 󵄩 󵄩 󵄩 ⩽ ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩g(t + τ − r) − g(t − r)󵄩󵄩󵄩 dr 0

∞ k+1

󵄩 󵄩 󵄩 󵄩 = ∑ ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩g(t + τ − r) − g(t − r)󵄩󵄩󵄩 dr k=0 k

k+1

∞

󵄩p 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩g(t + τ − r) − g(t − r)󵄩󵄩󵄩 dr) k=0

k

1/p

t−k

∞

1/p

󵄩 󵄩 󵄩p 󵄩 = ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds) k=0

t−k−1

∞

󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ε = Mε, k=0

t ⩾ 0,

which clearly implies that the set of all ε-periods of G(⋅) is relatively dense in ℝ. Only ̂ is uniformly continuous, we have the continuity of G(⋅) remains to be proved. Since g(⋅) the existence of a number δ ∈ (0, 1) such that 1

󵄩p 󵄩 ∫󵄩󵄩󵄩g(t + s) − g(t 󸀠 + s)󵄩󵄩󵄩 ds ⩽ εp , 0

󵄨 󵄨 provided t, t 󸀠 ∈ ℝ and 󵄨󵄨󵄨t − t 󸀠 󵄨󵄨󵄨 < δ.

(69)

For any δ󸀠 ∈ (0, δ), the foregoing arguments yield: 󵄩 󵄩󵄩 󸀠 󵄩󵄩G(t + δ ) − G(t)󵄩󵄩󵄩 ∞

k+1

󵄩p 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩g(t + δ󸀠 − s) − g(t − s)󵄩󵄩󵄩 ds) k=0

δ

1/p

k

󸀠

󵄩󵄩 󵄩 󵄩 + ∫󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩g(t − s)󵄩󵄩󵄩 ds 0

∞

1

1/p

󵄩 󵄩 󵄩p 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] (∫󵄩󵄩󵄩g(t + δ󸀠 + s − k − 1) − g(t + s − k − 1)󵄩󵄩󵄩 ds) k=0

0

68 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations 1/p

t+δ󸀠

󵄩p 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [0,1] ( ∫ 󵄩󵄩󵄩g(s)󵄩󵄩󵄩 ds) t

,

t ∈ ℝ,

so that the final conclusion follows from (69) and the well-known fact that t+δ󸀠

󵄩p 󵄩 lim ∫ 󵄩󵄩󵄩g(s)󵄩󵄩󵄩 ds = 0,

δ󸀠 →0

t

t ∈ ℝ.

Remark 2.6.12. Let t 󳨃→ ‖R(t)‖, t ∈ (0, 1] be an element of the space Lq [0, 1]. Then the condition ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞ holds provided (R(t))t>0 is exponentially decaying at infinity or there exists a finite number ζ < 0 such that ‖R(t)‖ = O(t ζ ), t → +∞ and (i) p = 1 and ζ < −1; or (ii) p > 1 and ζ < (1/p) − 1. In this way, we have extended the assertion of [11, Lemma 2.12], where the authors have considered the case in which ζ < −1 and g : ℝ → X is almost periodic. Concerning asymptotical Stepanov almost periodicity, we can deduce the following proposition. Proposition 2.6.13. Suppose that 1 ⩽ p < ∞, 1/p + 1/q = 1 and (R(t))t>0 ⊆ L(X) is a strongly continuous operator family such that, for every s ⩾ 0, we have ∞

󵄩 󵄩 ms := ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [s+k,s+k+1] < ∞. k=0

Suppose, further, that f : [0, ∞) → X is asymptotically Sp -almost periodic and that the locally p-integrable functions g : ℝ → X, q : [0, ∞) → X satisfy the conditions from Lemma 2.2.6 (the use of symbol q will be clear from the context). Let there exist a finite number M > 0 such that the following holds: t+1 s (i) limt→+∞ ∫t [∫M ‖R(r)‖‖q(s − r)‖ dr]p ds = 0. t+1

(ii) limt→+∞ ∫t

mps ds = 0.

Then the function H(⋅), given by t

H(t) := ∫ R(t − s)f (s) ds,

t ⩾ 0,

0

is well-defined, bounded, continuous and asymptotically Sp -almost periodic. Proof. It is obvious that the function H(⋅) is well-defined and bounded because f (⋅) is Sp -bounded and m0 < ∞; cf. the proof of Proposition 2.6.11. To prove that H(⋅) is continuous on [0, ∞), we can use [29, Proposition 1.3.2] and a simple trick from the last paragraph of [29, p. 22]. Let the function G(⋅) be given by (68). Define

2.6 Solutions of abstract Volterra integro-differential equations | 69 t

∞

F(t) := ∫ R(t − s)q(s) ds − ∫ R(s)g(t − s) ds, t

0

t ⩾ 0.

(70)

The measurability of integrand functions and the local integrability of convolution products in (70) follow from the proofs of [29, Propositions 1.3.4 and 1.3.5] (henceforward, this will be the case with all other examinations of function F(⋅) given by (70)). Then it is clear that H(t) = G(t) + F(t) for all t ⩾ 0 and by Proposition 2.6.11 it suffices to show that the mapping F̂ : [0, ∞) → Lp ([0, 1] : X) belongs to the class C0 ([0, 1] : Lp ([0, 1] : X)). This mapping is clearly continuous and we need to prove that t+1

󵄩p 󵄩 lim ∫ 󵄩󵄩󵄩F(s)󵄩󵄩󵄩 ds = 0. t→+∞

(71)

t

Let M > 0 be such that (i) holds. Then it is clear that there exist finite constants cp > 0 and cp󸀠 > 0 such that, by Hölder inequality, p

s−M

s

󵄩 󵄩p 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩F(s)󵄩󵄩󵄩 ⩽ cp [( ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr) + ( ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr) 0

s−M

p

∞

p

󵄩 󵄩󵄩 󵄩 + ( ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩󵄩󵄩󵄩g(s − r)󵄩󵄩󵄩 dr) ] s

⩽

cp󸀠 [(

p

s−M

󵄩p 󵄩 󵄩 󵄩󵄩 󵄩 ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr) + 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [0,M] ‖q‖pLp [s−M,s] 0

p

∞

󵄩 󵄩 + ( ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [s+k,s+k+1] ) ‖g‖pSp ].

(72)

k=0

t+1

Let ε > 0 be given. Then there exists t0 (ε) ⩾ 1 such that ∫t ‖q(s)‖p ds ⩽ ε, t ⩾ t0 (ε). This implies that for each t ⩾ t0 (ε) + M we have ‖q‖pLp [s−M,s] ⩽ ⌈M⌉ε. Invoking this estimate and integrating (72) along the interval [t, t + 1], we obtain with the help of (i)–(ii) that (71) holds, as claimed. Remark 2.6.14. The proof of Proposition 2.6.13 is similar to those of [11, Lemma 2.13] and [127, Lemma 4.1]. Below some special situations are listed, in which the asymptotical Sp -almost periodicity of function H(⋅) is proved by applying Proposition 2.6.13 directly or by combining Proposition 2.6.11 and the proofs of the above-mentioned lemmas: (i) Suppose that (R(t))t⩾0 is strongly continuous, exponentially decaying, g : ℝ → X is Sp -almost periodic and q ∈ C0 ([0, ∞) : X). Then we can use Proposition 2.6.11, the proof of [127, Lemma 4.1], decomposition t

t/2

t

∫ R(t − s)q(s) ds = ∫ R(t − s)q(s) ds + ∫ R(t − s)q(s) ds, 0

0

t/2

t⩾0

70 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations and the estimates for the term ∫t R(s)g(t − s) ds given in the proof of Proposition 2.6.13, in order to see that function H(⋅) is asymptotically almost periodic. The case in which the function H(⋅) is asymptotically Sp -almost periodic but not asymptotically almost periodic can also occur: if we accept all above assumptions with the exception of q ∈ C0 ([0, ∞) : X), and suppose in place of this condition that ∞

t+1

p

s

󵄩 󵄩 lim ∫ ( ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr) ds = 0, t→+∞ t

s/2

then the same arguments as above show that function H(⋅) is only asymptotically Sp -almost periodic. In such a way, we have proved a proper extension of [127, Lemma 4.1], which can be further applied for stating a proper extension of [127, Theorem 4.2] and new results about inhomogeneous abstract Cauchy problems of third order: αu󸀠󸀠󸀠 (t) + u󸀠󸀠 (t) − βAu(t) − γAu󸀠 (t) = f (t),

α, β, γ > 0, t ⩾ 0,

appearing in the theory of dynamics of elastic vibrations of flexible structures [127]. (ii) We can prove a proper extension of [11, Lemma 2.13] as explained below. Suppose that (R(t))t⩾0 is strongly continuous, ζ < −1, ‖R(t)‖ = O(1 + t ζ ), t ⩾ 0, g : ℝ → X is Sp -almost periodic and q ∈ C0 ([0, ∞) : X); then we can use the same arguments as above, appealing to [11, Lemma 2.13] in place of [127, Lemma 4.1], to show that function H(⋅) is asymptotically almost periodic. The only thing worth noting here is that mt → 0 as t → ∞; for this, observe that there exists a finite number M 󸀠 ⩾ 1 such that (α = −ζ ): t+k+1

1/q

∞ dr 󵄩 󵄩 ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [t+k,t+k+1] ⩽ M 󸀠 ∑ ( ∫ αq ) r k=0 k=0 ∞

t+k

∞

󵄨1/q 󵄨 = M 󸀠 (αζ − 1)−1 ∑ 󵄨󵄨󵄨(t + k + 1)1−αq − (t + k)1−αq 󵄨󵄨󵄨 ⩽ M 󸀠 (αζ − 1)

k=0 ∞ 1/q−1

1 (t + k)α k=0 ∑

1 ⩽ Const. t −να , να k (1−ν)α t k=0 ∞

⩽ M 󸀠 (αζ − 1)1/q−1 ∑

t > 0,

provided (1 − ν)α > 1. If we assume, as in the first part of this remark, that t+1 s limt→+∞ ∫t (∫s/2 ‖q(r)‖ dr)p ds = 0, then function H(⋅) will be only asymptotically Sp -almost periodic.

2.6 Solutions of abstract Volterra integro-differential equations | 71

(iii) Proposition 2.6.13 can be applied provided (R(t))t>0 is exponentially decaying at infinity, (R(t))t>0 has a certain growth order at zero, and q : [0, ∞) → X has a certain growth order. For the sake of illustration, we will examine only the case in which the multivalued linear operator 𝒜 satisfies condition (P), see Section 1.3: Then it is well known that there exists a degenerate strongly continuous semigroup (T(t))t>0 ⊆ L(X) generated by 𝒜 such that ‖T(t)‖ = O(t β−1 ), t > 0. Furthermore, the proof of [188, Theorem 3.1] combined with the integral computation given in the proof of [29, Theorem 2.6.1] shows that ‖T(t)‖ = O(e−ct t β−1 ), t > 0. This estimate enables one to see that condition (ii) from the formulation of Proposition 2.6.13 holds. Therefore, if t+1

s

t

M

p

󵄩 󵄩 lim ∫ [∫ e−cr r β−1 󵄩󵄩󵄩q(s − r)󵄩󵄩󵄩 dr] ds = 0,

t→+∞

then we can apply Proposition 2.6.13 to conclude that function H(⋅) is asymptotically Sp -almost periodic (cf. [188, Theorem 3.7] and [188, Examples 3.3 and 3.6] for some applications in the study of inhomogeneous Poisson heat equation in the spaces H −1 (Ω) and Lr (Ω), where Ω is a bounded domain with smooth boundary and 1 < r < ∞). It is clear that Proposition 2.6.11 can be also applied here, which can be simply incorporated in the study of existence and uniqueness of almost periodic solutions of the following differential inclusion of first order u󸀠 (t) ∈ 𝒜u(t) + g(t),

t ∈ ℝ,

where g : ℝ → X is Sp -almost periodic (see, e. g., [94, Definition 2.1, Lemma 2.9], the extensions for multivalued linear operators are immediate even if the corresponding solution operator families have integrable singularities at zero). Remark 2.6.15. As many results established so far show, the key for the existence and uniqueness of certain asymptotically almost periodic solutions of abstract inhomogeneous Volterra integro-differential equations is exponential or, at least, polynomial decaying rate at infinity of a corresponding (a, k)-resolvent C-resolvent family (R(t))t⩾0 . Here we would like to present a simple, almost trivial idea, which applies also to many other solution operator families as well as to multiterm Volterra integral equations (see, e. g., [247, Definition 2.3.36, Theorem 2.3.37]). Let z ∈ ℂ, az (t) = e−zt a(t), t ⩾ 0, and kz (t) = e−zt k(t), t ⩾ 0. Then (Rz (t) ≡ e−zt R(t))t⩾0 is an (az , kz )-resolvent C-resolvent family having the same set of subgenerators (the integral generator) as (R(t))t⩾0 does.

72 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations 2.6.2 Generalized (asymptotically) almost periodic properties of degenerate C-semigroups and degenerate C-cosine functions Concerning Stepanov almost periodicity of degenerate C-semigroups, we will first state the following simple result (as announced earlier, the operator C is allowed to be possibly noninjective): Proposition 2.6.16. Suppose (S(t))t⩾0 is a bounded C-regularized semigroup with the integral generator 𝒜. If x ∈ X is such that the mapping t 󳨃→ S(t)x, t ⩾ 0 is Stepanov almost periodic, then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is almost periodic. Proof. Let us recall that (S(t))t⩾0 ⊆ L(X) is a strongly continuous operator family commuting with C, and that S(t)S(s) = S(t + s)C, t, s ∈ ℝ. Since the mapping t 󳨃→ S(t)x, t ⩾ 0 is Stepanov almost periodic and (S(t))t⩾0 is uniformly bounded, we have that the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is Stepanov almost periodic and bounded, so it only remains to prove that the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is uniformly continuous (any Stepanov almost periodic function f ∈ BUC([0, ∞) : X) has to be almost periodic by the Bochner theorem). But this follows from the uniform boundedness of (S(t))t⩾0 and the estimate 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩S(t)Cx − S(s)Cx󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩S(s)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩S(t − s)x − Cx 󵄩󵄩󵄩,

t, s ⩾ 0,

t ⩾ s.

If R(C) is dense in X and the mapping t 󳨃→ S(t)x, t ⩾ 0 is Stepanov almost periodic for all x ∈ X, then Proposition 2.6.16 and the fact that AP([0, ∞) : X) is a closed subspace of BUC([0, ∞) : X) together imply that the mapping t 󳨃→ S(t)x, t ⩾ 0 is almost periodic for all x ∈ X, as long as (S(t))t⩾0 is bounded. On the other hand, the strong Stepanov almost periodicity of mapping t 󳨃→ S(t), t ⩾ 0 does not imply a priori the boundedness of (S(t))t⩾0 ; in the present situation, the best we can do concerning this question is to prove the following slight extension of well-known H. R. Henríquez’s result [220, Theorem 1]: Theorem 2.6.17. Suppose that 1 ⩽ p < ∞ and (S(t))t⩾0 is a C-regularized semigroup with the integral generator 𝒜. Then the following holds: (i) Let x ∈ X be such that the mapping t 󳨃→ S(t)x, t ⩾ 0 is Sp -bounded. Then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is bounded. Suppose that the mapping t 󳨃→ S(t)x, t ⩾ 0 is Sp -bounded for all x ∈ X. Then we have the following: (ii) The mapping t 󳨃→ S(t)C 2 x, t ⩾ 0 is bounded and uniformly continuous for all x ∈ X, and there exists a finite constant M ⩾ 0 such that ||S(t)C‖ ⩽ M, t ⩾ 0. Therefore, if x ∈ X is such that the mapping t 󳨃→ S(t)C 2 x, t ⩾ 0 is Stepanov almost periodic, then it is almost periodic.

2.6 Solutions of abstract Volterra integro-differential equations | 73

(iii) If R(C) is dense in X, then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is bounded and uniformly continuous for all x ∈ X. Therefore, if x ∈ X is such that the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is Stepanov almost periodic, then it is almost periodic. Proof. Assume that (i) does not hold. Then there exists a strictly increasing sequence (tn )n∈ℕ in [1, ∞) such that limn→∞ tn = ∞ and limn→∞ ‖S(tn )Cx‖ = ∞. Let N := sups∈[0,1] ‖S(s)‖ < ∞. Then S(tn )Cx = S(s)S(tn − s)x, 0 ⩽ s ⩽ 1, n ∈ ℕ and therefore ‖S(tn − s)x‖ ⩾ ‖S(tn )Cx‖/N, n ∈ ℕ. Integrating this estimate over the interval [0, 1], we get tn

‖S(tn )Cx‖p 󵄩p 󵄩 , ∫ 󵄩󵄩󵄩S(s)x󵄩󵄩󵄩 ds ⩾ Np

tn −1

n ∈ ℕ,

contradicting the Sp -boundedness of mapping t 󳨃→ S(t)x, t ⩾ 0. This completes the proof of (i). For the rest of the proof, let us assume that the mapping t 󳨃→ S(t)x, t ⩾ 0 is Sp -bounded for all x ∈ X. By the uniform boundedness principle and (i), we have that there exists a finite constant M ⩾ 0 such that ||S(t)C‖ ⩽ M, t ⩾ 0. Now the uniform continuity of mapping t 󳨃→ S(t)C 2 x, t ⩾ 0 for any x ∈ X follows from the estimate 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 2 2 󵄩 󵄩󵄩S(t)C x − S(s)C x󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩S(s)C 󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩S(t − s)x − Cx 󵄩󵄩󵄩,

t, s ⩾ 0, t ⩾ s,

which simply completes the proof of (ii). If R(C) is dense in X, then for each x ∈ X we can find a sequence (xn )n∈ℕ in X such that limn→∞ Cxn = x. Hence, for every number ε > 0 given in advance, we can find an integer n0 ∈ ℕ and a positive real number δ > 0 such that 󵄩 󵄩 󵄩 󵄩󵄩 2 2 󵄩󵄩S(t)Cx − S(s)Cx󵄩󵄩󵄩 ⩽ 2M‖Cxn0 − x‖ + 󵄩󵄩󵄩S(t)C xn0 − S(s)C xn0 󵄩󵄩󵄩 ⩽ 2ε/3 + ε/3 = ε, provided t, s ⩾ 0 and |t − s| < δ, so that the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is bounded and uniformly continuous for all x ∈ X. This simply yields (iii). Before proceeding further, let us only recall that it is still an open problem in the theory of nondegenerate C-regularized semigroups (C ∈ L(X) injective) whether there exists a bounded C-regularized semigroup (S(t))t⩾0 that is not strongly uniformly continuous; cf. [128, Remark 5.19] for more details. Concerning the asymptotical Stepanov almost periodicity of degenerate C-regularized semigroups, we can clarify the following result. Theorem 2.6.18. Suppose that 1 ⩽ p < ∞, (S(t))t⩾0 is a C-regularized semigroup with the integral generator 𝒜, and the mapping t 󳨃→ S(t)x, t ⩾ 0 is Sp -bounded for all x ∈ X. Then we have the following:

74 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations (i) The asymptotical Sp -almost periodicity of mapping t 󳨃→ S(t)x, t ⩾ 0 for some x ∈ X implies that the mapping t 󳨃→ S(t)C 4 x, t ⩾ 0 is asymptotically almost periodic. (ii) If R(C) is dense in X and (S(t))t⩾0 is strongly asymptotically Sp -almost periodic, then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is asymptotically almost periodic for all x ∈ X. Proof. We shall content ourselves with sketching the proof, which is much like that of [220, Theorem 2]. By the foregoing, we have that there exists a finite constant M ⩾ 0 such that ‖S(t)C‖ ⩽ M, t ⩾ 0. Let x ∈ X be such that the mapping t 󳨃→ S(t)x, t ⩾ 0 is asymptotically Sp -almost periodic. Then for any sequence (tn )n∈ℕ of positive rê als there exists a subsequence (sn )n∈ℕ such that limn→∞ S(s n + ⋅)x exists in the space p Cb ([0, ∞) : L ([0, 1] : X)). Lemma 2.2.6 and the proof of the aforementioned theorem together imply that there exist two functions gx (⋅) and qx (⋅) such that the conclusions of this lemma hold with the function f (⋅) replaced therein by S(⋅)x, and that there exists a subsequence (rn )n∈ℕ of (sn )n∈ℕ such that gx (t) = limn→∞ S(rn + t)x a. e. t ⩾ 0. Arguing as in [220], we get that the limit limn→∞ S(rn + t)C 2 x exists in X for all t ⩾ 0. This implies that the sequence (S(rn + ⋅)C 4 x)n∈ℕ is Cauchy in the space Cb ([0, ∞) : X) and therefore convergent (observe here that 󵄩 󵄩󵄩 2 2 󵄩 4 4 󵄩 󵄩󵄩S(rn + t)C x − S(rm + t)C x󵄩󵄩󵄩 ⩽ M 󵄩󵄩󵄩S(rn )C x − S(rm )C x󵄩󵄩󵄩, for all m, n ∈ ℕ, t ⩾ 0), finishing the proof of (i). The proof (ii) is simple so we refrain from giving it here. By Example 2.6.4, for each number α ∈ (0, 2)∖{1} there exist examples of bounded, nondegenerate, asymptotically Stepanov almost periodic (gα , I)-resolvent families that are not (Stepanov) almost periodic. Now we specialize to the case α = 2. To the best knowledge of the author, the assertion of [220, Theorem 3] has not yet been reconsidered for C-regularized cosine operator functions. In order to do that, we need first to extend the well-known result of I. Cioranescu and P. Ubilla [106, Theorem 1] concerning the generation of uniformly bounded cosine operator functions in terms of boundedness and analyticity of subordinated strongly continuous semigroups as well as denseness of subspace consisting of exponential vectors (see Y. V. Radyno [351] and [106, Lemma, pp. 2–3]): Let C ∈ L(X) be injective, and let A be a closed single-valued linear operator comμ muting with C. Denote by DA the vector space consisting of all vectors x ∈ D∞ (A) such that there exists c > 0 satisfying ‖Ak x‖ ⩽ cμk , k ∈ ℕ0 (μ > 0). Equipped with the norm ‖x‖μ := supk⩾0 μ−k ‖Ak x‖, this space becomes a Banach space. The space of exponenμ tial vectors of A, ExpA for short, is defined by ExpA := ⋃μ>0 DA ; then it is clear that ExpA can be viewed as a subspace the linear span of all eigenfunctions corresponding μ to some eigenvalue of A. Let Aμ := A|Dμ and Cμ := C|Dμ (μ > 0). Then Aμ , Cμ ∈ L(DA ) A

A

mutually commute, Cμ is injective and (λ − Aμ )−1 Cμ = ((λ − A)−1 C)|Dμ for any λ ∈ ρC (A), A

with the estimate ‖(λ − Aμ )−1 Cμ ‖μ ⩽ ‖(λ − A)−1 C‖, λ ∈ ρC (A), μ > 0; see, e. g., the proof of

2.6 Solutions of abstract Volterra integro-differential equations | 75

[351, Theorem 11]. Moreover, if the mapping λ 󳨃→ (λ − A)−1 C, λ ∈ Ω is analytic on some open domain Ω ⊆ ℂ, then we know that R(C) ⊆ R((λ − A)n ), n ∈ ℕ and dn−1 (λ − A)−1 C = (−1)n−1 (n − 1)!(λ − A)−n C ∈ L(X), dλn−1

n ∈ ℕ. μ

In this case, we can inductively prove that (λ − Aμ )−n Cμ ∈ L(DA ), n ∈ ℕ, μ > 0 and ‖(λ − Aμ )−n Cμ ‖μ ⩽ ‖(λ − A)−n C‖, n ∈ ℕ, λ ∈ Ω, μ > 0; furthermore, the estimate 󵄩󵄩 󵄩󵄩 (λ − Aμ )−1 Cμ x − (z − Aμ )−1 Cμ x 󵄩 󵄩󵄩 + (λ − Aμ )−2 Cμ x󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩μ 󵄩󵄩 λ−z 󵄩󵄩 (λ − A)−1 Cx − (z − A)−1 Cx 󵄩󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩 + (λ − A)−2 Cx󵄩󵄩󵄩‖x‖μ , 󵄩󵄩 󵄩󵄩 λ−z

μ

λ ∈ Ω, μ > 0, x ∈ DA μ

enables one to see that the mapping λ 󳨃→ (λ − Aμ )−1 Cμ ∈ L(DA ), λ ∈ Ω is analytic as well (μ > 0). Keeping in mind these facts and the well known structural results from the theory of C-regularized semigroups and C-regularized cosine functions (see [245, 246]), a careful inspection of the proof of [106, Theorem 1] enables one to deduce the following result of independent interest (for (ii), define u(t) in the proof of abovementioned theorem by t 2n n Aμ Cμ xμ , μ=1 (2n)! ∞

u(t) := ∑

t ⩾ 0,

μ

for x = ∑∞ μ=1 xμ ∈ D(A), xμ ∈ DA for 1 ⩽ μ < ∞): Theorem 2.6.19. Let C ∈ L(X) be injective, and let A be a closed single-valued linear operator commuting with C. Then the following holds: (i) If A generates a bounded C-regularized cosine function, then A generates a bounded analytic C-regularized semigroup of angle π/2 and R(C) ⊆ ExpA . (ii) If A generates a bounded analytic C-regularized semigroup of angle π/2 and ExpA = X, then A generates a bounded C-regularized cosine function. (iii) Suppose that R(C) = X. Then A generates a bounded C-regularized cosine function iff A generates a bounded analytic C-regularized semigroup of angle π/2 and ExpA = X. Now we are ready to prove the following slight extension of [220, Theorem 3]: Theorem 2.6.20. Let C ∈ L(X) be injective, let A be a closed single-valued linear operator, and let R(C) = X. Suppose that A generates an asymptotically Stepanov almost periodic C-regularized cosine function (C(t))t⩾0 . Then (C(t))t⩾0 is almost periodic. Proof. The proof of theorem is much the same as that of [220, Theorem 3] and, because of this, we will only outline the main differences. It is well known that R(C) ⊆ D(A), so A is densely defined. Moreover, the mapping B : X → LpS ([0, ∞) : X) given by Bx :=

76 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations C(⋅)x (x ∈ X) is linear and closed, therefore continuous. Using the abstract Weierstrass formula [246, Theorem 2.4.18] and the proof of Theorem 3 in [220], we can simply verify that A generates a bounded analytic C-regularized semigroup of angle π/2. Denote by S(⋅) the induced C-regularized sine function generated by A. Due to Theorem 2.6.19(iii) and [421, Theorem 4.1], it suffices to show that the set D consisting of all eigenvectors of operator A corresponding to the real nonpositive eigenvalues of A is total in X. For this, define Pλ x and Qλ x, as well as the functions gx (⋅) and qx (⋅), as in the proof of the aforementioned theorem (λ ∈ ℝ, x ∈ X). Then the arguments contained in the proofs of [220, Theorem 3] and [421, Theorem 4.1] enable one to see that C(t)Pλ x = cos(λt)CPλ x, C(t)Qλ x = cos(λt)CQλ x, as well as {Pλ x, Qλ x} ⊆ N(λ2 + A), for all λ ∈ ℝ, x ∈ X. Let x∗ ∈ X ∗ , and let x∗ annulate D. By d’Alambert’s formula, for every a ∈ [0, 1], t ⩾ 0 and x ∈ X, we have 1

1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 2󵄨󵄨󵄨⟨x∗ , C(t)S(a)x⟩󵄨󵄨󵄨 ⩽ ∫󵄨󵄨󵄨⟨x∗ , qCx (t + s)⟩󵄨󵄨󵄨 ds + ∫󵄨󵄨󵄨⟨x∗ , qCx (t − s)⟩󵄨󵄨󵄨 ds. 0

0

Set ya∗ := S(a)∗ x∗ , a ∈ [0, 1]. Then the above yields limt→∞ ⟨C(t)∗ ya∗ , x⟩ = 0, x ∈ X, a ∈ [0, 1] and the boundedness of the set {C(t)∗ ya∗ : t ⩾ 0} in X ∗ , for every fixed a ∈ [0, 1]. If we define the sequences (un )n∈ℕ and (vn )n∈ℕ as in the proof of H. R. Henríquez, then the d’Alambert functional equation C(tn )Cx + C(t)Cx = 2C(un )C(vn )x, x ∈ X, n ∈ ℕ and the arguments contained on page 431 of the proof show that ⟨ya∗ , C(t)x⟩ = 0, t ⩾ 0, a ∈ [0, 1]. Hence, ⟨x∗ , S(a)C(t)x⟩ = 0, t ⩾ 0, a ∈ [0, 1], x ∈ X and therefore ⟨x∗ , S(a)Cx⟩ = 0, a ∈ [0, 1], x ∈ X. Since R(C 2 ) is dense in X and C 2 x = lima→0+ a−1 S(a)Cx, x ∈ X, we get that the set {S(a)Cx : 0 ⩽ a ⩽ 1, x ∈ X} is total in X, so that x∗ = 0. The conclusion in the theorem follows from this. We can similarly prove an analogue of [220, Proposition 1] for C-regularized cosine functions in weakly sequentially complete Banach spaces: Proposition 2.6.21. Let C ∈ L(X) be injective, let A be a closed single-valued linear operator, and let R(C) = X. Suppose that A generates a scalarly Stepanov almost periodic C-regularized cosine function (C(t))t⩾0 . Then (C(t))t⩾0 is almost periodic, provided X is weakly sequentially complete. The following proposition is motivated by V. Casarino’s result [83, Proposition 3.1; 2)] (we must say that, in the formulation of this statement, one has to impose the condition on uniform boundedness of considered cosine operator function); in our approach, the operator C ≠ I need not be injective and (C(t))t⩾0 can be degenerate in time: Proposition 2.6.22. Let 𝒜 be the integral generator of a uniformly bounded C-cosine function (C(t))t⩾0 . Suppose that x ∈ X is such that the mapping t 󳨃→ C(t)x, t ⩾ 0 is asymptotically almost periodic. Then the mapping t 󳨃→ C(t)Cx, t ⩾ 0 is almost periodic.

2.6 Solutions of abstract Volterra integro-differential equations | 77

Proof. The prescribed assumption implies that there exists a finite constant M > 0 such that ‖C(t)‖ ⩽ M, t ⩾ 0. Let ε > 0 be given in advance. Then we can find numbers l = l(ε) > 0 and K = K(ε) > 0 such that every subinterval I of [0, ∞) of length l contains at least one number τ such that ‖C(t + τ)x − C(t)x‖ ⩽ ε/(2M + ‖C‖) for all t ⩾ K. Let s ⩾ K. Using the d’Alambert functional equality C(t)Cx = 2C(s)C(t + s)x − C(t + 2s)Cx, t ⩾ 0, we infer that 󵄩 󵄩󵄩 󵄩󵄩C(t + τ)Cx − C(t)Cx 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ 2M 󵄩󵄩󵄩C(t + s + τ)x − C(t + s)x󵄩󵄩󵄩 + ‖C‖󵄩󵄩󵄩C(t + 2s + τ)x − C(t + 2s)x󵄩󵄩󵄩 ⩽ (2M + ‖C‖)ε/(2M + ‖C‖) = ε,

t ⩾ 0,

where τ ∈ I is chosen as above. This concludes the proof. Now we would like to raise the following problem concerning Theorem 2.6.20 and Proposition 2.6.22: Problem. Let 𝒜 be the integral generator of a bounded C-cosine function (C(t))t⩾0 . Suppose that x ∈ X is such that the mapping t 󳨃→ C(t)x, t ⩾ 0 is asymptotically Stepanov almost periodic. Is it true that the mapping t 󳨃→ C(t)Cx, t ⩾ 0 is almost periodic? The following extension of E. Vesentini’s result [392, Proposition 4] for degenerate C-groups is deduced similarly (cf. [247] for more details on the subject):

Proposition 2.6.23. Suppose that (S(t))t∈ℝ ⊆ L(X) is a bounded, strongly continuous operator family commuting with C, and S(t)S(s) = S(t + s)C, t, s ∈ ℝ. If x ∈ X is such that the mapping t 󳨃→ S(t)x, t ⩾ 0 is asymptotically almost periodic, then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is almost periodic. In a series of his research papers, A. S. Rao has investigated the conditions under which the Stepanov almost periodic (bounded) solutions of certain abstract differential equations are almost periodic (see, e. g., [353] and [354]). We close this subsection by explaining how we can prove a slight extension of the main theorem of [353] for infinitesimal generators of almost periodic C-regularized groups. Before doing that, let us agree on the following notion: Suppose that A and B are two closed, not necessarily densely defined, single-valued linear operators in X and a function f : ℝ → X is continuous. By a solution of the second-order differential equation u󸀠󸀠 (t) = Au󸀠 (t) + Bu(t) + f (t) a. e. on ℝ we mean any twice differentiable function u(t) with u󸀠 (t) ∈ D(A), u(t) ∈ D(B) for all t ∈ ℝ and satisfying the above equation a. e. on ℝ. Theorem 2.6.24. Suppose that X is a reflexive Banach space, f : ℝ → X is an S1 -almost periodic continuous function, and A is the infinitesimal generator of an almost periodic

78 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations nondegenerate C-regularized group (T(t))t∈ℝ , where C ∈ L(X) is injective. Let u : ℝ → X, with its derivative u󸀠 (t) ∈ D(A) for all t ∈ ℝ, be a solution of the differential equation u󸀠󸀠 (t) = Au󸀠 (t) + B(t)u(t) + f (t) a. e. on ℝ, where B : ℝ → L(X) is almost periodic. If u(⋅) is S1 -almost periodic and u(⋅) is S1 -bounded on ℝ, then C 2 u󸀠 (⋅) and C 2 u(⋅) are both almost periodic from ℝ to X. Proof. The proof of this result can be obtained by an insignificant modification of the proof of Theorem in [353]. First of all, note that for each x ∈ D(A) we have that the mapping t 󳨃→ T(t)x, t ∈ ℝ is continuously differentiable with (d/dt)T(t)x = T(t)Ax, t ∈ ℝ. Then the computation given in the proof of [353, Lemma 1] shows that t

Cu󸀠 (t) = T(t)u󸀠 (0) + ∫ T(t − s)[B(s)u(s) + f (s)] ds,

t ∈ ℝ,

0

which implies t

T(−t)u󸀠 (t) = Cu󸀠 (0) + ∫ T(−s)[B(s)u(s) + f (s)] ds,

t ∈ ℝ.

0

Furthermore, we can simply prove by definition that (T(−t))t∈ℝ is an almost periodic C-regularized group with the infinitesimal generator −A. The only thing that should otherwise be noted is that the assertion of [353, Lemma 2] holds for C-regularized groups. As a matter of fact, suppose that h(⋅) is an almost periodic function from ℝ to X. Then there exists a sequence of X-valued trigonometric polynomials hn (⋅) converging uniformly to h(⋅) as n → ∞. Using the fact that, if g ∈ AP(ℝ : X) and p ∈ AP(ℝ : ℂ), then gp ∈ AP(ℝ : X), we get that the mapping t 󳨃→ T(−t)hn (t), t ∈ ℝ is almost periodic. Since (T(−t))t∈ℝ is almost periodic, it is uniformly bounded and therefore T(−⋅)hn (⋅) converges uniformly to T(−⋅)(⋅) as n → ∞, so T(−⋅)(⋅) is almost periodic, too. Arguing similarly, we can prove some results on almost periodic solutions of the first-order nonlinear differential equation u󸀠 (t) ∈ [A + B(t)]u(t) + f (t) a. e. on ℝ, in reflexive Banach spaces. If A is the infinitesimal generator of an almost periodic nondegenerate C-regularized group (T(t))t∈ℝ , where C ∈ L(X) is injective, u : ℝ → D(A) is an S1 -almost periodic solution of the above differential equation (for the topology of X), f : ℝ → X is an S1 -almost periodic continuous function and B : ℝ → L(X) is almost periodic, then the mapping C 2 u(⋅) is almost periodic as well.

2.7 Solutions of abstract fractional semilinear inclusions | 79

2.7 Stepanov (asymptotically) almost periodic solutions of abstract fractional semilinear inclusions Composition theorems for two-parameter Stepanov p-almost periodic functions have been considered in [317, Theorem 2.2]. We start this section by investigating composition theorems for Stepanov two-parameter almost periodic and asymptotically Stepanov two-parameter almost periodic functions. The following result states that the assertion of [317, Theorem 2.2] continues to hold for the functions defined on the real semiaxis I = [0, ∞), with possibly two different pivot spaces X and Y. Theorem 2.7.1. Let I = ℝ or I = [0, ∞). Suppose that the following conditions hold: (i) f ∈ APSp (I × Y : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lf ∈ LrS (I) such that 󵄩 󵄩󵄩 󵄩󵄩f (t, x) − f (t, y)󵄩󵄩󵄩 ⩽ Lf (t)‖x − y‖Y ,

t ∈ I, x, y ∈ Y;

(73)

(ii) x ∈ APSp (I : Y), and there exists a set E ⊆ I with m(E) = 0 such that K := {x(t) : t ∈ I ∖ E} is relatively compact in Y; here, m(⋅) denotes the Lebesgue measure. Then q := pr/p + r ∈ [1, p) and f (⋅, x(⋅)) ∈ APSq (I : X). As observed in [169, Remark 2.5], condition (73) seems to be more conventional for dealing with than the usual Lipschitz assumption. But then we cannot consider the value p = 1 in Theorem 2.7.1: this is not the case if we accept the existence of a Lipschitz constant L > 0 such that 󵄩 󵄩󵄩 󵄩󵄩f (t, x) − f (t, y)󵄩󵄩󵄩 ⩽ L‖x − y‖Y ,

t ∈ I, x, y ∈ Y.

(74)

Joking aside, an insignificant modification of the proof of [317, Theorem 2.2] shows that the following result holds true: Theorem 2.7.2. Let I = ℝ or I = [0, ∞). Suppose that the following conditions hold: (i) f ∈ APSp (I × Y : X) with p ⩾ 1, L > 0 and (74) holds. (ii) x ∈ APSp (I : Y), and there exists a set E ⊆ I with m(E) = 0 such that K = {x(t) : t ∈ I ∖ E} is relatively compact in Y. Then f (⋅, x(⋅)) ∈ APSp (I : X). In the sequel of this section, it will be assumed that X = Y. Concerning asymptotically two-parameter Stepanov p-almost periodic functions, we can prove the following composition principle (cf. Lemmas 2.2.6 and 2.2.7; the use of symbol q is clear from the context): Proposition 2.7.3. Let I = [0, ∞). Suppose that the following conditions hold:

80 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations (i) g ∈ APSp (I × X : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lg ∈ LrS (I) such that (73) holds with the functions f (⋅, ⋅) and Lf (⋅) replaced by the functions g(⋅, ⋅) and Lg (⋅) therein. (ii) y ∈ APSp (I : X), and there exists a set E ⊆ I with m(E) = 0 such that K = {y(t) : t ∈ I ∖ E} is relatively compact in X. (iii) f (t, x) = g(t, x)+q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 ([0, ∞)×X : Lq ([0, 1] : X)) and q := pr/p + r. (iv) x(t) = y(t) + z(t) for all t ⩾ 0, where ẑ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)). (v) There exists a set E 󸀠 ⊆ I with m(E 󸀠 ) = 0 such that K 󸀠 = {x(t) : t ∈ I ∖ E 󸀠 } is relatively compact in X. Then q ∈ [1, p) and f (⋅, x(⋅)) ∈ AAPSq (I : X). Proof. By Theorem 2.7.1, we have that the function t 󳨃→ g(t, y(t)), t ⩾ 0 is Stepanov q-almost periodic. Since f (t, x(t)) = [g(t, x(t)) − g(t, y(t))] + g(t, y(t)) + q(t, x(t)),

t ⩾ 0,

it suffices to show that 1/q

t+1

󵄩q 󵄩 lim ( ∫ 󵄩󵄩󵄩g(s, x(s)) − g(s, y(s))󵄩󵄩󵄩 ds)

t→+∞

t

=0

(75)

and t+1

󵄩q 󵄩 lim ( ∫ 󵄩󵄩󵄩q(s, x(s))󵄩󵄩󵄩 ds) t→+∞ t

1/q

= 0.

(76)

To see that (75) holds, we can argue as in the proof of estimate [317, (2.12)]. More precisely, by (74) and Hölder inequality, we have 1/q

t+1

󵄩q 󵄩 ( ∫ 󵄩󵄩󵄩g(s, x(s)) − g(s, y(s))󵄩󵄩󵄩 ds) t

t+1

1/q

󵄩q ⩽ ( ∫ Lg (s) 󵄩󵄩x(s) − y(s)󵄩󵄩󵄩 ds) q󵄩 󵄩

t

t+1

1/r

t+1

1/r

t+1

1/p

󵄩p 󵄩 ⩽ ( ∫ Lg (s) ds) ( ∫ 󵄩󵄩󵄩x(s) − y(s)󵄩󵄩󵄩 ds) r

t

t+1

t

󵄩p 󵄩 = ( ∫ Lg (s) ds) ( ∫ 󵄩󵄩󵄩z(s)󵄩󵄩󵄩 ds) t

r

t

1/p

,

t ⩾ 0.

2.7 Solutions of abstract fractional semilinear inclusions | 81

Hence, (75) is proved by applications of Sr -boundedness of function Lg (⋅) and inclusion ẑ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)). The proof of (76) follows immediately from the facts that q̂ ∈ C0 ([0, ∞) × X : Lq ([0, 1] : X)) and K 󸀠 = {x(t) : t ∈ I ∖ E 󸀠 } is relatively compact in X. If we accept the Lipschitz assumption (74), then the following result holds true: Proposition 2.7.4. Let I = [0, ∞). Suppose that the following conditions hold: (i) g ∈ APSp (I × X : X) with p ⩾ 1, and there exists a constant L > 0 such that (74) holds with the function f (⋅, ⋅) replaced by the function g(⋅, ⋅) therein. (ii) y ∈ APSp (I : X), and there exists a set E ⊆ I with m(E) = 0 such that K = {y(t) : t ∈ I ∖ E} is compact in X. (iii) f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 ([0, ∞) × X : Lp ([0, 1] : X)). (iv) x(t) = y(t) + z(t) for all t ⩾ 0, where ẑ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)). (v) There exists a set E 󸀠 ⊆ I with m(E 󸀠 ) = 0 such that K 󸀠 = {x(t) : t ∈ I ∖ E 󸀠 } is relatively compact in X. Then f (⋅, x(⋅)) ∈ AAPSp (I : X). The following result is very similar to Proposition 2.6.11. Proposition 2.7.5. Suppose that (R(t))t>0 ⊆ L(X) is strongly continuous and ‖R(t)‖ = O(e−ωt t β−1 ), t > 0 for some numbers ω > 0 and β > 0. Let f ∈ AAPSq ([0, ∞) : X) with some q ∈ [1, ∞), let 1/q + 1/q󸀠 = 1, and let the following hold: q󸀠 (β − 1) > −1, provided q > 1 and

β = 1, provided q = 1.

(77)

Define t

H(t) := ∫ R(t − s)f (s) ds,

t ⩾ 0.

0

Then H ∈ AAP([0, ∞) : X). Proof. Suppose that the locally p-integrable functions g : ℝ → X, q : [0, ∞) → X satisfy the conditions from Lemma 2.2.6. Let the function G(⋅) be given by (68), with R(⋅) replaced therein by T(⋅); then we know from Proposition 2.6.11 that G(⋅) is almost periodic. Set t

∞

F(t) := ∫ T(t − s)q(s) ds − ∫ T(s)g(t − s) ds, 0

t

t ⩾ 0.

With Hölder’s inequality in mind, we can simply prove that H(⋅) is well-defined. Since H(t) = G(t) + F(t) for all t ⩾ 0, it suffices to show that F ∈ C0 ([0, ∞) : X). It is clear that

82 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations 󵄩󵄩 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩 󵄩󵄩 ∫ T(s)g(t − s) ds󵄩󵄩󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩 q󸀠 󵄩󵄩 󵄩󵄩 󵄩L [t+k,t+k+1] ‖g‖Sq 󵄩 󵄩󵄩 󵄩󵄩 k=0 t ∞

󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩L∞ [t+k,t+k+1] ‖g‖Sq ⩽ Const. e−ct ‖g‖Sq , k=0

t > 1,

so that limt→∞ ∫t T(s)g(t − s) ds = 0. Arguing as above, we get ∞

󵄩󵄩 󵄩󵄩 t/2 ⌈t/2⌉ 󵄩 󵄩󵄩 󵄩󵄩 ∫ T(t − s)q(s) ds󵄩󵄩󵄩 ⩽ ‖g‖Sq ∑ 󵄩󵄩󵄩R(t − ⋅)󵄩󵄩󵄩 q󸀠 󵄩󵄩 󵄩L [k,k+1] 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k=0 0 ⩽ M(1 + ⌈t/2⌉)e−c(t−⌈t/2⌉−1) ‖g‖Sq ,

t ⩾ 2,

t/2

so that limt→∞ ∫0 T(t − s)q(s) ds = 0. Therefore, it remains only to prove that t

limt→∞ ∫t/2 T(t − s)q(s) ds = 0 (observe that the integral in this limit expression converges by (77) and the Sq -boundedness of function q(⋅)). For that, fix a number t+1 ε > 0. Then there exists t0 > 0 such that ∫t ‖q(s)‖q ds < εq , t ⩾ t0 . Let t > 2t0 + 6. Then Hölder inequality implies the existence of a finite constant c > 0 such that 󵄩󵄩 t 󵄩󵄩 ⌊t/2⌋−2 󵄩󵄩 󵄩 󵄩󵄩 β−1 󵄩󵄩 󵄩󵄩 ∫ T(t − s)q(s) ds󵄩󵄩󵄩 ⩽ c ∑ 󵄩󵄩󵄩R(t − ⋅)󵄩󵄩󵄩 q󸀠 󵄩L [t/2+k,t/2+k+1] ε + ε󵄩󵄩⋅ 󵄩󵄩Lq󸀠 [0,2] 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 k=0 t/2 ⌊t/2⌋−2

󵄩 󵄩 󵄩 󵄩 ⩽ c ∑ 󵄩󵄩󵄩R(t − ⋅)󵄩󵄩󵄩L∞ [t/2+k,t/2+k+1] ε + ε󵄩󵄩󵄩⋅β−1 󵄩󵄩󵄩Lq󸀠 [0,2] k=0

⌊t/2⌋−2

󵄩 󵄩 ⩽ cεM ∑ e−c(t/2+k) + ε󵄩󵄩󵄩⋅β−1 󵄩󵄩󵄩Lq󸀠 [0,2] k=0

⩽ cεMe

−ct/2

∞

󵄩 󵄩 ∑ e−ck + ε󵄩󵄩󵄩⋅β−1 󵄩󵄩󵄩Lq󸀠 [0,2] .

k=0

This yields the final conclusion. Suppose now that condition (P) holds. Then there exists a degenerate strongly continuous semigroup (T(t))t>0 ⊆ L(X) generated by 𝒜 and we have the estimate ‖T(t)‖ ⩽ Me−ct t β−1 , t > 0 for some constant M > 0. Let us examine the following semilinear Cauchy inclusion of first order: u󸀠 (t) ∈ 𝒜u(t) + f (t, u(t)),

t ∈ ℝ.

(78)

By a mild solution of (78), we mean any continuous function u(⋅) such that u(t) = (Λu)(t), t ∈ ℝ, where t

t 󳨃→ (Λu)(t) := ∫ T(t − s)f (s, u(s)) ds, −∞

t ∈ ℝ.

2.7 Solutions of abstract fractional semilinear inclusions | 83

Theorem 2.7.6. Suppose that f ∈ APSp (ℝ × X : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lf ∈ LrS (ℝ) such that (73) holds with I = ℝ. Let the following condition hold: β = 1, provided r = p/p − 1,

and

pr 1 < , provided r > p/p − 1. pr − p − r 1 − β

Set q󸀠 := ∞, provided r = p/p − 1,

and

q󸀠 :=

(79)

pr , provided r > p/p − 1. pr − p − r

Assume that M = ∑∞ k=0 ‖T(⋅)‖Lq󸀠 [k,k+1] < ∞ and M‖Lf ‖Sr < 1. Then there exists a unique almost periodic mild solution of (78). Proof. Since the range of any function u ∈ AP(ℝ : X) is relatively compact in X, Theorem 2.7.1 yields that f (⋅, u(⋅)) ∈ APSq (ℝ : X), where q = pr/p + r. Since (T(t))t>0 is exponentially decaying at infinity, condition (79) holds and 1/q󸀠 + 1/q = 1, we can apply Proposition 2.7.5 in order to see that the mapping Λ : AP(ℝ : X) → AP(ℝ : X) is well-defined. Furthermore, for every t ∈ ℝ, we have by Hölder inequality: 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩(Λu)(t) − (Λv)(t)󵄩󵄩 = 󵄩󵄩 ∫ T(s)[f (t − s, u(t − s)) − f (t − s, v(t − s))] ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩0 ∞ k+1

󵄩 󵄩󵄩 󵄩 ⩽ ∑ ∫ 󵄩󵄩󵄩T(s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s, u(t − s)) − f (t − s, v(t − s))󵄩󵄩󵄩 ds k=0 k ∞

󵄩 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩T(⋅)󵄩󵄩󵄩Lq󸀠 [k,k+1] 󵄩󵄩󵄩f (t − ⋅, u(t − ⋅)) − f (t − ⋅, v(t − ⋅))󵄩󵄩󵄩Lq [k,k+1] k=0 ∞

󵄩 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩T(⋅)󵄩󵄩󵄩Lq󸀠 [k,k+1] 󵄩󵄩󵄩Lf (t − ⋅)[u(t − ⋅) − v(t − ⋅)]󵄩󵄩󵄩Lq [k,k+1] k=0 ∞

󵄩 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩T(⋅)󵄩󵄩󵄩Lq󸀠 [k,k+1] ‖Lf ‖Sr 󵄩󵄩󵄩u(t − ⋅) − v(t − ⋅)󵄩󵄩󵄩Lp [k,k+1] k=0 ∞

󵄩 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩T(⋅)󵄩󵄩󵄩Lq󸀠 [k,k+1] ‖Lf ‖Sr 󵄩󵄩󵄩u(⋅) − v(⋅)󵄩󵄩󵄩L∞ (ℝ) . k=0

Since M‖Lf ‖Sr < 1, we can apply the Banach contraction principle to complete the proof of theorem. We can similarly deduce the following result provided the Lipschitz type condition (74) holds: Theorem 2.7.7. Suppose that f ∈ APSp (ℝ × X : X) with p ⩾ 1, L > 0 and (74) holds with I = ℝ. Let the following condition hold: β = 1, provided p = 1,

and

p 1 < , provided p > 1. p−1 1−β

84 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Set q󸀠 := ∞, provided p = 1,

and

q󸀠 :=

p , provided p > 1. p−1

Assume that M = ∑∞ k=0 ‖T(⋅)‖Lq󸀠 [k,k+1] < ∞ and ML < 1. Then there exists a unique almost periodic mild solution of (78). Let the initial value u0 be a point of the continuity of semigroup (T(t))t>0 ; see, e. g., [188, Theorems 3.3 and 3.5] for more details regarding this question. Of importance is the following abstract semilinear Cauchy inclusion of first order: (DFP)f ,s

{

u󸀠 (t) ∈ 𝒜u(t) + f (t, u(t)), u(0) = u0 .

t ⩾ 0,

By a mild solution u(⋅) = u(⋅; u0 ) of problem (DFP)f ,s we mean any function u ∈ C([0, ∞) : X) such that t

u(t) = (ϒu)(t) := T(t)u0 + ∫ T(t − s)f (s, u(s)) ds,

t ⩾ 0.

0

Suppose that (73) holds for a. e. t > 0 (I = [0, ∞)), with locally integrable positive function Lf (⋅). Set, for every n ∈ ℕ, t xn

x2

Mn := M n sup e−ct ∫ ∫ ⋅ ⋅ ⋅ ∫ ecx1 (t − xn )β−1 t⩾0

n

0 0

0

n

× ∏(xi − xi−1 )β−1 ∏ Lf (xi ) dx1 dx2 ⋅ ⋅ ⋅ dxn . i=2

i=1

Then a simple calculation shows that 󵄩󵄩 n n 󵄩 󵄩󵄩(ϒ u) − (ϒ v)󵄩󵄩󵄩∞ ⩽ Mn ‖u − v‖∞ ,

u, v ∈ BUC([0, ∞) : X), n ∈ ℕ.

(80)

Now we are able to state the main result of this section: Theorem 2.7.8. Suppose that I = [0, ∞) and the following conditions hold: (i) g ∈ APSp (I × X : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lg ∈ LrS (I) such that (73) holds with the functions f (⋅, ⋅) and Lf (⋅) replaced by the functions g(⋅, ⋅) and Lg (⋅) therein. (ii) f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 (I × X : Lq ([0, 1] : X)) and q = pr/p + r. pr 1 < 1−β , provided r > p/p − 1. (iii) β = 1, provided r = p/p − 1, and pr−p−r (iv) (73) holds for a. e. t > 0, with locally bounded positive function Lf (⋅) satisfying Mn < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,s .

2.7 Solutions of abstract fractional semilinear inclusions | 85

Proof. Define the number q󸀠 as in the formulation of Theorem 2.7.6. By (i)–(ii) and Proposition 2.7.3, we have that f (⋅, x(⋅)) ∈ AAPSq (I : X) for any x ∈ AAP(I : X), where q = pr/p + r; here, it is only worth observing that the range of an X-valued asymptotically almost periodic function is relatively compact in X. Due to (iii), condition (77) holds. By means of Lemma 2.7.5 and the obvious equality limt→+∞ T(t)u0 = 0, we get that the mapping ϒ : AAP(I : X) → AAP(I : X) is well-defined. Making use of (80), (iv) and a well-known extension of the Banach contraction principle (see Theorem 1.1.5), we obtain the existence of a unique asymptotically almost periodic solution of inclusion (DFP)f ,s . The uniqueness of solutions can be also proved in the following way: let u(⋅) and v(⋅) be two mild solutions of inclusion (DFP)f ,s . Then we have t

󵄩 󵄩󵄩 󵄩 󵄩 −c(t−s) (t − s)β−1 Lf (s)󵄩󵄩󵄩u(s) − v(s)󵄩󵄩󵄩 ds, 󵄩󵄩u(t) − v(t)󵄩󵄩󵄩 ⩽ M ∫ e

t ⩾ 0.

0

This implies by the boundedness of function s 󳨃→ e−c(t−s) L(s), s ∈ (0, t] and [158, Lemma 6.19, p. 111] that u(s) = v(s) for all s ∈ [0, t] (t > 0 fixed). The proof of the theorem is thereby complete. Using Proposition 2.7.4 in place of Proposition 2.7.3, we can simply formulate and prove the following analogue of Theorem 2.7.8 in the case of consideration of classical Lipschitz condition (74): Theorem 2.7.9. Let I = [0, ∞). Suppose that the following conditions hold: (i) g ∈ APSp (I × X : X) with p ⩾ 1, and there exists a constant L > 0 such that (74) holds with the function f (⋅, ⋅) replaced by the function g(⋅, ⋅) therein. (ii) f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 (I × X : Lp ([0, 1] : X)). p 1 (iii) β = 1, provided p = 1, and p−1 < 1−β , provided p > 1. (iv) (73) holds for a. e. t > 0, with locally bounded positive function Lf (⋅) satisfying Mn < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,s . Now we would like to formulate the following important consequence of Theorem 2.7.8: Corollary 2.7.10. Suppose that I = [0, ∞), the function f (⋅, ⋅) is asymptotically almost periodic and (73) holds for a. e. t > 0, with locally bounded positive function Lf (⋅) satisfying Mn < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,s . Specifically, in the case that M1 < 1 in Theorem 2.7.9, we obtain the following corollary:

86 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Corollary 2.7.11. Suppose that I = [0, ∞), the function f (⋅, ⋅) is asymptotically almost periodic and (74) holds for some L ∈ [0, cβ M −1 Γ(β)−1 ). Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,s . Remark 2.7.12. (i) In the case of β = 1 and Lf ∈ L∞ ([0, ∞)) ∩ L1 ([0, ∞)), the proof of [127, Theorem 4.4] shows that ∑∞ n=1 Mn < ∞, so that the uniqueness of solutions follows immediately by applying the Weissinger’s fixed point theorem (see Theorem 1.1.6). If the above conditions are satisfied, then the proof of Theorem 2.7.8 can be used to state a proper extension of [127, Theorem 4.4]; strictly speaking, in our approach the term f (⋅, u(⋅)) need not be asymptotically almost periodic and it can be of the form (iii) from the formulation of Theorem 2.7.8, or asymptotically Stepanov almost periodic if we consider Theorem 2.7.9. Applications in the study of abstract semilinear Cauchy problems of third order: αu󸀠󸀠󸀠 (t) + u󸀠󸀠 (t) − βAu(t) − γAu󸀠 (t) = f (t, u(t)),

α, β, γ > 0, t ⩾ 0.

(81)

(ii) If 0 < β < 1, then it is not trivial to state a satisfactory criterion which would enable one to see that the inequality Mn < 1 holds for some integer n ∈ ℕ.

2.8 Subspace (asymptotical) almost periodicity of C-distribution semigroups and C-distribution cosine functions First, we recall the basic facts about vector-valued distribution spaces used henceforth. The Schwartz spaces of test functions 𝒟 = C0∞ (ℝ) and ℰ = C ∞ (ℝ) are equipped with the usual inductive limit topologies; the topology of rapidly decreasing function space 𝒮 is defined by the following system of seminorms pm,n (ψ) := supx∈ℝ |xm ψ(n) (x)|, ψ ∈ 𝒮 , m, n ∈ ℕ0 . If 0 ≠ Ω ⊆ ℝ, then by 𝒟Ω we denote the subspace of 𝒟 consisting of those functions φ ∈ 𝒟 for which supp(φ) ⊆ Ω; 𝒟0 ≡ 𝒟[0,∞) . If φ, ψ : ℝ → ℂ are measurable functions, the finite convolution product will be also denoted by φ ∗0 ψ; the ∞ infinite convolution product φ ∗ ψ is defined by φ ∗ ψ(t) := ∫−∞ φ(t − s)ψ(s) ds, t ∈ ℝ; here, some obvious abuse of notation appears. If φ ∈ 𝒟 and f ∈ 𝒟󸀠 , or φ ∈ ℰ and f ∈ ℰ 󸀠 , then we define the convolution f ∗ φ by (f ∗ φ)(t) := f (φ(t − ⋅)), t ∈ ℝ. For f ∈ 𝒟󸀠 , or for f ∈ ℰ 󸀠 , define f ̌ by f ̌(φ) := f (φ(−⋅)), φ ∈ 𝒟 (φ ∈ ℰ ). In general, the convolution of two distributions f , g ∈ 𝒟󸀠 , denoted by f ∗ g, is defined by (f ∗ g)(φ) := g(f ̌ ∗ φ), φ ∈ 𝒟. It is well-known that f ∗g ∈ 𝒟󸀠 and supp(f ∗g) ⊆ supp(f )+supp(g). For every t ∈ ℝ, we define the Dirac distribution centered at point t, δt for short, by δt (φ) := φ(t), φ ∈ 𝒟. The space 𝒟󸀠 (X) := L(𝒟, X) consists of all continuous linear functions 𝒟 → X; 󸀠 𝒟Ω (X) denotes the subspace of 𝒟󸀠 (X) containing X-valued distributions whose sup󸀠 ports are contained in Ω. Set 𝒟0󸀠 (X) := 𝒟[0,∞) (X). If X = ℂ, then the above spaces are 󸀠 󸀠 󸀠 also denoted by 𝒟 , 𝒟Ω and 𝒟0 . For more details about vector-valued distributions,

2.8 Subspace (asymptotical) almost periodicity | 87

we refer the reader to [363, 364]; the convolution of vector-valued distributions will be taken in the sense of [290, Proposition 1.1]. In this section, we assume that the operator C ∈ L(X) is injective and that any operator family under our consideration is nondegenerate. We first recall the definition of a C-distribution semigroup (see [245]): Definition 2.8.1. Let 𝒢 ∈ 𝒟0󸀠 (L(X)) satisfy C 𝒢 = 𝒢 C. Then it is said that 𝒢 is a C-distribution semigroup, shortly (C-DS), iff 𝒢 satisfies the following conditions: (i) 𝒢 (φ ∗0 ψ)C = 𝒢 (φ)𝒢 (ψ), for any φ, ψ ∈ 𝒟. (ii) 𝒩 (𝒢 ) := ⋂φ∈𝒟0 N(𝒢 (φ)) = {0}. A (C-DS) 𝒢 is called dense iff, in addition to the above, (iii) ℛ(𝒢 ) := ⋃φ∈𝒟0 R(𝒢 (φ)) is dense in X. Let 𝒢 ∈ 𝒟0󸀠 (L(X)) be a (C-DS) and let T ∈ ℰ0󸀠 , i. e., T is a scalar-valued distribution with compact support contained in [0, ∞). Define G(T) := {(x, y) ∈ X × X : 𝒢 (T ∗ φ)x = 𝒢 (φ)y for all φ ∈ 𝒟0 }. Then it can be easily seen that G(T) is a closed linear operator commuting with C. We define the (infinitesimal) generator A of a pre-(C-DS) 𝒢 by A := G(−δ󸀠 ). We know that C −1 AC = A as well as that the following holds: Let S, T ∈ ℰ0󸀠 , φ ∈ 𝒟0 , ψ ∈ 𝒟 and x ∈ X. Then we have: A1. G(S)G(T) ⊆ G(S ∗ T) with D(G(S)G(T)) = D(G(S ∗ T)) ∩ D(G(T)), and G(S) + G(T) ⊆ G(S + T). A2. (𝒢 (ψ)x, 𝒢 (−ψ󸀠 )x − ψ(0)Cx) ∈ A. We denote by D(𝒢 ) the set consisting of those elements x ∈ X for which x ∈ D(G(δt )), t ⩾ 0 and the mapping t 󳨃→ G(δt )x, t ⩾ 0 is continuous. By property A1, we have that D(G(δs )G(δt )) = D(G(δs ∗ δt )) ∩ D(G(δt )) = D(G(δt+s )) ∩ D(G(δt )),

t, s ⩾ 0,

which clearly implies G(δt )(D(𝒢 )) ⊆ D(𝒢 ), t ⩾ 0. We refer the reader to [29, 128] and [245, 246] for further information concerning fractionally integrated C-semigroups and fractionally integrated C-cosine functions in Banach spaces, properties of their subgenerators and integral generators. Lemma 2.8.2 ([245]). A closed linear operator A is the generator of a (C-DS) 𝒢 iff for every τ > 0 there exist an integer nτ ∈ ℕ and a local nτ -times integrated C-semigroup (Sn (t))t∈[0,τ) with the integral generator A. If this is the case, then the following equality holds: n

τ

𝒢 (φ)x = (−1) ∫ φ

(n)

0

(t)Sn (t)x dt, x ∈ X, φ ∈ 𝒟(−∞,τ) .

88 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Let us recall that the solution space for a closed linear operator A, denoted by Z(A), is defined as the set of all x ∈ X for which there exists a continuous mapping t t u(⋅, x) ∈ C([0, ∞) : X) satisfying ∫0 u(s, x) ds ∈ D(A) and A ∫0 u(s, x) ds = u(t, x) − x, t ⩾ 0. Hereafter, we will use the following characterization of space Z(A). Assume A generates a (C-DS) 𝒢 . Then Z(A) = D(𝒢 ). If x ∈ Z(A), then u(t, x) = G(δt )x, t ⩾ 0 and ∞ 𝒢 (ψ)x = ∫0 ψ(t)Cu(t, x) dt, ψ ∈ 𝒟0 . We need to recall the assertion of [245, Proposition 3.1.28(ii)] for later purposes. Lemma 2.8.3. Assume that, for every τ > 0, there exists nτ ∈ ℕ such that A is a subgenerator of a local nτ -times integrated C-semigroup (Snτ (t))t∈[0,τ) . Then the solution space Z(A) is the space which consists exactly of those elements x ∈ X such that, for every τ > 0, Snτ (t)x ∈ R(C) and the mapping t 󳨃→ C −1 Snτ (t)x, t ∈ [0, τ) is nτ -times continuously differentiable; if this is the case, then we have G(δt )x = (dnτ /dt nτ )C −1 Snτ (t)x, t ∈ [0, τ). For any x ∈ Z(A), the function u(⋅, x) ∈ C([0, ∞) : X) satisfying the above requirements is said to be a mild solution of the abstract Cauchy problem u ∈ C([0, ∞) : [D(A)]) ∩ C 1 ([0, ∞) : X), { { 󸀠 (ACP)1 : {u (t) = Au(t), t ⩾ 0, { {u(0) = x. Let ζ ∈ 𝒟[−2,−1] be a fixed test function satisfying ∫−∞ ζ (t) dt = 1. Then, with ζ chosen in this way, we define I(φ) (φ ∈ 𝒟) as follows: ∞

⋅

∞

−∞

−∞

I(φ)(⋅) := ∫ [φ(t) − ζ (t) ∫ φ(u) du] dt.

(82)

d Then I(φ) ∈ 𝒟, I(φ󸀠 ) = φ, dt I(φ)(t) = φ(t) − ζ (t) ∫−∞ φ(u) du, t ∈ ℝ and, for every 󸀠 G ∈ 𝒟 (L(X)), the primitive G−1 of G is defined by setting G−1 (φ) := −G(I(φ)), φ ∈ 𝒟. It is clear that G−1 ∈ 𝒟󸀠 (L(X)), (G−1 )󸀠 = G, i. e., −G−1 (φ󸀠 ) = G(I(φ󸀠 )) = G(φ), φ ∈ 𝒟 and that supp(G) ⊆ [0, ∞) implies supp(G−1 ) ⊆ [0, ∞). Now we recall definition of a C-distribution cosine function (cf. [246, 247] for more details): ∞

Definition 2.8.4. An element G ∈ 𝒟0󸀠 (L(X)) is called a pre-(C −DCF) iff G(φ)C = CG(φ), φ ∈ 𝒟 and (C − DCF1 ) : G−1 (φ ∗0 ψ)C = G−1 (φ)G(ψ) + G(φ)G−1 (ψ),

φ, ψ ∈ 𝒟;

if, additionally, (C − DCF2 ) : x = y = 0 iff G(φ)x + G−1 (φ)y = 0,

φ ∈ 𝒟0 ,

then G is called a C-distribution cosine function, in short (C − DCF). A pre-(C − DCF) G is called dense iff the set ℛ(G) := ⋃φ∈𝒟0 R(G(φ)) is dense in X.

2.8 Subspace (asymptotical) almost periodicity | 89

Notice that (DCF2 ) implies ⋂φ∈𝒟0 N(G(φ)) = {0} and ⋂φ∈𝒟0 N(G−1 (φ)) = {0}, and the assumption G ∈ 𝒟0󸀠 (L(X)) implies G(φ) = 0, φ ∈ 𝒟(−∞,0] . The (integral) generator A of G is defined by A := {(x, y) ∈ X × X : G−1 (φ󸀠󸀠 )x = G−1 (φ)y for all φ ∈ 𝒟0 }. It is well known that (G(ψ)x, G(ψ󸀠󸀠 )x + ψ󸀠 (0)Cx) ∈ A, ψ ∈ 𝒟, x ∈ X and (G−1 (ψ)x, −G(ψ󸀠 )x − ψ(0)Cx) ∈ A, ψ ∈ 𝒟, x ∈ X. We also need the following results [246]: Lemma 2.8.5. (i) Let G ∈ 𝒟0󸀠 (L(X)) and G(φ)C = CG(φ), φ ∈ 𝒟. Then G is a (C-DCF) in X generated by A iff G is a (𝒞 -DS) in X ⊕ X generated by 𝒜, where 0 A

𝒜≡(

I ), 0

C 0

𝒞≡(

0 ) C

and G ≡ (

G G −δ⊗C 󸀠

G−1 ). G

(ii) A closed linear operator A is the generator of a (C-DCF) G iff for every τ > 0 there exist an integer nτ ∈ ℕ and a local nτ -times integrated C-cosine function (Cnτ (t))t∈[0,τ) with the integral generator A. If this is the case, then the following equality holds: n

τ

G(φ)x = (−1) ∫ φ(n) (t)Cnτ (t)x dt, 0

x ∈ X, φ ∈ 𝒟(−∞,τ) .

A (C-DS) 𝒢 is said to be an exponential C-distribution semigroup, (E-CDS) in short, iff there exists ω ∈ ℝ such that e−ωt 𝒢 ∈ 𝒮 󸀠 (L(X)). A (C − DCF) G is said to be an exponential C-distribution cosine function, (E −CDCF) in short, iff G is an (E − 𝒞 DS) in X ⊕X. The above is equivalent to saying that there exists ω ∈ ℝ such that e−ωt G−1 ∈ 𝒮 󸀠 (L(X)). It is well known that A is the generator of an (E − CDCF) in X (an (E-CDS) in X) iff there exists n ∈ ℕ such that A is the integral generator of an exponentially bounded n-times integrated C-cosine function in X (an exponentially bounded n-times integrated C-semigroup in X). A function u(⋅; x, y) is said to be a mild solution of the abstract Cauchy problem

(ACP)2 :

u ∈ C([0, ∞) : [D(A)]) ∩ C 2 ([0, ∞) : X), { { 󸀠󸀠 u (t) = Au(t), t ⩾ 0, { { u(0) = x, u󸀠 (0) = y { t

iff the mapping t 󳨃→ u(t; x, y), t ⩾ 0 is continuous, ∫0 (t − s)u(s; x, y)ds ∈ D(A) and t

A ∫0 (t − s)u(s; x, y)ds = u(t; x, y) − x − ty, t ⩾ 0; in the sequel, we primarily consider the mild solutions of (ACP)2 with y = 0. Denote by Z2 (A) the set which consists of all x ∈ X for which there exists such a solution. Let π1 : X × X → X and π2 : X × X → X be the projections and let G be a (C − DCF) generated by A. Then G is a (𝒞 -DS) generated by

90 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations 𝒜 and we define G(δt )x := π2 (G(δt )(0x )), t ⩾ 0, x ∈ Z2 (A). Let us recall that, for every

x ∈ Z2 (A), one has G(δt )(Z2 (A)) ⊆ Z2 (A), t ⩾ 0, 2G(δs )G(δt )x = G(δt+s )x + G(δ|t−s| )x, ∞ t, s ⩾ 0 and G(φ)x = ∫0 φ(t)CG(δt )x dt, φ ∈ 𝒟0 .

Lemma 2.8.6 ([246]). Assume that, for every τ > 0, there exists nτ ∈ ℕ such that A is a subgenerator of a local nτ -times integrated C-cosine function (Cnτ (t))t∈[0,τ) . Then the solution space Z2 (A) consists exactly of those vectors x ∈ X such that, for every τ > 0, Cnτ (t)x ∈ R(C) and the mapping t 󳨃→ C −1 Cnτ (t)x, t ∈ [0, τ) is nτ -times continuously differentiable. If x ∈ Z2 (A) and t ∈ [0, τ), then G(δt )x = (dnτ /dt nτ )C −1 Cnτ (t)x. In [101], I. Cioranescu has investigated the conditions under which the abstract nondegenerate inhomogeneous Cauchy problem (DFP)L of first and second order (α ∈ {1, 2}) has a unique strong almost periodic solution for all initial values x ∈ D∞ (A). The notions of (bounded, almost periodic) distribution group and (bounded, almost periodic) cosine distribution have been introduced there (cf. [245] for further information concerning these subjects), and the spectral characterizations of generators of such distribution groups and cosine distributions have been given. Now we would like to introduce the following notion. Definition 2.8.7. Let G be a (C − DCF) generated by A, resp. let 𝒢 be a (C-DS) generated by A. Suppose that X̃ is a linear subspace of Z2 (A), resp. x ∈ Z(A). Then it is said that G ̃ ̃ is X-(weakly) almost periodic, resp. 𝒢 is X-(weakly) almost periodic, iff for each x ∈ X̃ the mapping t 󳨃→ G(δt )x, t ⩾ 0 is (weakly) almost periodic. Remark 2.8.8. (i) It is clear that the above notions can be introduced for arbitrary operator family (F(t))t⩾0 consisted of possibly nonlinear and possibly discontinuous singlevalued operators. (ii) Suppose that G is a (C−DCF) generated by A, resp. 𝒢 is a (C-DS) generated by A. Let ̃ X̃ be a linear subspace of Z2 (A), resp. x ∈ Z(A), such that G, resp. 𝒢 , is X-(weakly) almost periodic. Let G1 be a (C1 − DCF) generated by A, resp. let 𝒢1 be a (C1 -DS) ̃ generated by A. Then G1 , resp. 𝒢1 , is X-(weakly) almost periodic. Let A be a closed linear operator. Designate by D (H) the set consisting of all eigenvectors of operator A corresponding to purely imaginary eigenvalues of operator A (to nonpositive real eigenvalues of operator A); we assume henceforth that the set D0 (H0 ) consists of all eigenvectors of operator A corresponding to purely imaginary nonzero eigenvalues of operator A (to negative real eigenvalues of operator A). Proposition 2.8.9. (i) Suppose that 𝒢 is a (C-DS) generated by A. Then X̃ := span(D) ⊆ Z(A) and 𝒢 is t ̃ X-almost periodic. Furthermore, the mapping t 󳨃→ ∫0 G(δs )x ds, t ⩾ 0 is almost periodic for all x ∈ span(D0 ).

2.8 Subspace (asymptotical) almost periodicity | 91

(ii) Suppose that G is a (C − DCF) generated by A. Then X̃ := span(H) ⊆ Z2 (A) and t ̃ G is X-almost periodic. Furthermore, the mappings t 󳨃→ ∫0 G(δs )x ds, t ⩾ 0 and t

t 󳨃→ ∫0 (t − s)G(δs )x ds, t ⩾ 0 are almost periodic for all x ∈ span(H0 ).

Proof. We will prove only (i) and outline some basic facts needed for the proof of (ii). By Lemma 2.8.2, we know that, for every τ > 0, there exist an integer n = nτ ∈ ℕ and a local n-times integrated C-semigroup (Sn (t))t∈[0,τ) with the integral generator A. t

Suppose that r ∈ ℝ and irx = Ax. Then Sn (t)x − gn+1 (t)Cx = ir ∫0 Sn (s)x ds, t ∈ [0, τ), which simply implies that the mapping t 󳨃→ Sn (t)x, t ∈ [0, τ) is infinitely differentiable with all derivatives at zero of order less than or equal to n − 1 being zeroes and that (dn /dt n )Sn (t)x = eirt Cx, t ∈ [0, τ). Hence, Sn (t)x = (gn ∗ eir⋅ )Cx, t ∈ [0, τ), G(δt )x = eirt x, t ∈ [0, τ) and the last equality clearly continues to hold for all nonnegative reals t. Now the final conclusions follow from Lemma 2.8.3. The proof of part (ii) is quite similar, and can be deduced by using Lemma 2.8.5(ii), Lemma 2.8.6 and the equality G(δt )x = cos(rt)x, t ⩾ 0, provided that −r 2 x = Ax for some r ∈ ℝ.

It is also worthwhile mentioning that Proposition 2.8.9 continues to hold for C-distribution semigroups and C-distribution cosine functions in locally convex spaces, which follows from the fact that the equalities G(δt )x = eirt x, t ⩾ 0 and G(δt )x = cos(rt)x, t ⩾ 0 used above can be proved without assuming that the vector-valued distributions 𝒢 and G are of finite order (cf. [247] for the notion and [246] for semigroup case; for cosine operator case, combine the above result with Lemma 2.8.5(i), and [246, Lemma 3.2.33] with λ = ±ir). ̃ Suppose that 𝒢 is a (C-DS) generated by A, and 𝒢 is X-almost periodic. Define T(t)x := Tx (t) := G(δt )x, t ⩾ 0, x ∈ Z(A) and S(t)x := [E(Tx (⋅))](t), t ∈ ℝ, x ∈ X.̃ As before, we have 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩S(t)x󵄩󵄩󵄩 ⩽ sup󵄩󵄩󵄩S(s)x󵄩󵄩󵄩 = sup󵄩󵄩󵄩S(s)x 󵄩󵄩󵄩 = sup󵄩󵄩󵄩T(s)x 󵄩󵄩󵄩, s∈ℝ

s⩾0

s⩾0

x ∈ X,̃ t < 0,

and therefore, 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩S(t)x󵄩󵄩󵄩 = sup󵄩󵄩󵄩S(t)x󵄩󵄩󵄩, t∈ℝ

t⩾0

x ∈ X.̃

Furthermore, S(⋅) commutes with C and the following holds: Proposition 2.8.10. We have the following: (i) T(t)T(s)x = T(t + s)x for all t ⩾ 0, s ⩾ 0 and x ∈ Z(A). (ii) Suppose that t ⩾ 0, s ⩽ 0, x ∈ X̃ and G(δt )x ∈ X.̃ Then S(s)S(t)x = S(t + s)x. (iii) Suppose that t ⩾ 0, s ⩽ 0 and x ∈ X.̃ Then S(s)x ∈ Z(A) and T(t)S(s)x = S(t + s)x. (iv) Suppose that t ⩾ 0, s ⩽ 0, x ∈ X,̃ G(δr )x ∈ X̃ for all r ⩾ 0 and X̃ is closed. Then S(s)x ∈ X̃ and S(t)S(s)x = S(t + s)x. (v) Suppose that t ⩽ 0, s ⩽ 0, x ∈ X,̃ G(δr )x ∈ X̃ for all r ⩾ 0 and X̃ is closed. Then S(t)S(s)x = S(t + s)x.

92 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Proof. The part (i) follows immediately from property A1, while the proof of (ii) can be carried through by using definition and properties of extension mapping E, property A1 and the inclusion G(δt )x ∈ X:̃ S(s)S(t)x = S(s)G(δt )x = [E(TG(δt )x (⋅))](s) = [W(s)TG(δt )x (⋅)](0) = [W(s)W(t)Tx (⋅)](0) = [W(t + s)Tx (⋅)](0) = [E(Tx (⋅))](t + s) = S(t + s)x.

For (iii), we need only to prove that t

A ∫ u(r; S(s)x) dr = u(t; S(s)x) − S(s)x, 0

where u(r; S(s)x) := S(r + s)x, r ⩾ 0. It is checked at once that the mapping r 󳨃→ u(r; S(s)x), r ⩾ 0 is continuous and u(0; S(s)x) = S(s)x. Let us suppose first that t ⩾ −s. By definition of integral generator A, we need to prove that t

𝒢 (−φ ) ∫ u(r; S(s)x) dr = 𝒢 (φ)[u(t; S(s)x) − S(s)x], 󸀠

φ ∈ 𝒟0 .

(83)

0

We have t

−s

t

0

−s

𝒢 (−φ ) ∫ u(r; S(s)x) dr = 𝒢 (−φ )( ∫ + ∫ )u(r; S(s)x) dr 󸀠

󸀠

0

t

−s

= 𝒢 (−φ󸀠 ) ∫ [E(Tx (⋅))](r + s) dr + 𝒢 (−φ󸀠 ) ∫ G(δr+s )x dr 0

0

t+s

−s

= 𝒢 (−φ󸀠 ) ∫[E(Tx (⋅))](r) dr + 𝒢 (−φ󸀠 ) ∫ G(δr )x dr s

0

0

= 𝒢 (−φ󸀠 ) ∫ S(r)x dr + 𝒢 (φ)[u(t + s; x) − x], s

φ ∈ 𝒟0 ,

where the last equality follows from definitions of integral generator A and mild solution u(⋅; x). Hence, (83) is equivalent to 0

𝒢 (φ)S(s)x − 𝒢 (φ)x = 𝒢 (φ ) ∫ S(r)x dr, 󸀠

s

φ ∈ 𝒟0 ,

(84)

because u(t; S(s)x) = G(δt+s )x = u(t + s; x). For this, fix a test function φ ∈ 𝒟0 and put f (v) := 𝒢 (φ)S(v)x, v ⩽ 0. By the Newton–Leibniz formula, it suffices to show that f 󸀠 (v) = 𝒢 (−φ󸀠 )S(v)x,

v ⩽ 0.

(85)

2.8 Subspace (asymptotical) almost periodicity |

93

Observe first that properties A1–A2 along with definition of mild solution u(⋅; x) imply t

𝒢 (φ)Tx (⋅ + σ) − 𝒢 (φ)Tx (⋅) = ∫ 𝒢 (−φ )Tx (⋅ + σ) dσ, 󸀠

σ ⩾ 0,

0

i. e., σ

W(σ)[𝒢 (φ)Tx (⋅)] − 𝒢 (φ)Tx (⋅) = ∫ W(s󸀠 )[𝒢 (−φ󸀠 )Tx (⋅)] ds󸀠 ,

σ ⩾ 0.

0

Because of this, the pair (𝒢 (φ)Tx (⋅), 𝒢 (−φ󸀠 )Tx (⋅)) belongs to the graph of the integral generator of (W(t))t⩾0 on AP([0, ∞) : X), which is well known to be equal to the infinitesimal generator of (W(t))t⩾0 ((W(t))t∈ℝ ). Hence, lim

h→0

W(h) − I 󸀠 𝒢 (φ)Tx (⋅) = 𝒢 (−φ )Tx (⋅), h

(86)

for the topology of AP([0, ∞) : X). By virtue of (86), the group property of (W(t))t∈ℝ , the fact that E is a linear surjective isometry, and elementary definitions, we get that f 󸀠 (v) = lim [W(v) h→0

W(h) − I 𝒢 (φ)Tx (⋅)](0) h

= [W(v)𝒢 (−φ󸀠 )Tx (⋅)](0) = [E(𝒢 (−φ󸀠 )Tx (⋅))](v) = 𝒢 (−φ󸀠 )[E(Tx (⋅))](v) = 𝒢 (−φ󸀠 )S(v)x,

v ⩽ 0,

completing the proof of (85) and (iii) in the case that t ⩾ −s. The case t ⩽ −s is much easier and follows by applying (84) and elementary substitutions in integrals. In order to prove (iv), it suffices to show, by definition of an X-valued almost periodic function, that for each n ∈ ℕ there exists a positive number ln such that any interval I of length ln contains a number τn such that ‖S(v + τn )x − S(v)x‖ ⩽ 1/n, v ∈ ℝ and, in particular, ‖S(s + τn )x − S(s)x‖ ⩽ 1/n. Taking I ⊆ [−s + 1, ∞) we obtain that S(s)x ∈ {S(r)x : r ⩾ 0}. Since S(r)x = G(δr )x ∈ X̃ for all r ⩾ 0 and X̃ is closed, (iv) immediately follows from the above inclusion. To prove (v), observe that (iv) implies S(v)x ∈ X̃ for v ⩽ 0. Plugging y = S(s)x, the equality in (v) is equivalent with S(t)y = S(t + s)S(−s)y, which follows from an application of (ii), where we use the equality T(−s)S(s)y = y, which is true due to (iii). Remark 2.8.11. Parts (iv) and (v) hold in the case X̃ = Z(A), whether this subspace is closed or not in X. In the general case, it is not clear whether parts (iv) and (v) hold, provided that X̃ ≠ Z(A) and X̃ is not closed in X. Remark 2.8.12. Suppose that X̃ = Z(A) and Z(A) is dense in X. Due to Lemma 2.8.2, for every τ > 0, there exist an integer nτ ∈ ℕ and a local nτ -times integrated C-semigroup

94 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations (Sn+τ (t))t∈[0,τ) with the integral generator A. Define t

Sn−τ (t)x := ∫ 0

(t − s)nτ −1 CS(−s)x ds, (nτ − 1)!

t ∈ [0, τ), x ∈ Z(A).

Then it is not clear how one can prove that there exists a finite constant Mτ such that ‖Sn−τ (t)x‖ ⩽ Mτ ‖x‖, t ∈ [0, τ), x ∈ Z(A). Because of this, we are not able to extend Sn−τ (t) to a bounded linear operator S−nτ (t) acting on X, despite of our assumption made on the density of Z(A) in X. It is clear that, owing to Proposition 2.8.10, we can expect from (S−nτ (t))t∈[0,τ) to be a local nτ -times integrated C-semigroup generated by −A. Of course, the validity of the last statement for all numbers τ > 0 would imply by Lemma 2.8.3 ̃ that Z(A) ⊆ Z(−A) and −A generates a X-almost periodic C-distribution semigroup in X. 1.

2.

3.

In connection with Remark 2.8.12, we have the following comments to make: Suppose that A generates a global C-regularized semigroup (Q+ (t))t⩾0 . Denote by 𝒢+ the induced C-distribution semigroup generated by A. Then it can be easily seen that (Q+ (t))t⩾0 is almost periodic (let us recall that this means that for any element x ∈ X the mapping t 󳨃→ Q+ (t)x, t ⩾ 0 is almost periodic) iff 𝒢+ is R(C)-almost periodic. If this is the case, the operator −A generates an almost periodic C-regularized semigroup (Q− (t))t⩾0 , and therefore, −A generates a C-distribution semigroup 𝒢− that is R(C)-almost periodic. The situation is quite similar if we consider integrated C-semigroups. To explain this, suppose that A generates an exponentially bounded n-times integrated C-semigroup (Qn,+ (t))t⩾0 for some n ∈ ℕ. Then [245, Proposition 2.3.13] shows that there exists a positive real number λ such that A generates an exponentially bounded ((λ − A)−n C)-regularized semigroup (Rn,+ (t))t⩾0 , so that R((λ − A)−n C) ⊆ Z(A). Denote by 𝒢+ the induced exponential C-distribution semigroup generated by A, and suppose that 𝒢+ is R((λ − A)−n C)-almost periodic. Making use of the analysis from the above paragraph, the afore-mentioned proposition and Remark 2.8.7(ii), it is very simple to prove that −A generates an exponential R((λ − A)−n C)-almost periodic C-distribution semigroup. The same conclusion can be formulated for weak almost periodicity. Here we would like to propose an interesting question with regard to almost periodicity of C-distribution cosine functions. Suppose that G is a (C − DCF) generated ̃ by A, and G is X-almost periodic. Define C(t)x := Cx (t) := G(δt )x, t ⩾ 0, x ∈ Z2 (A) and C(t)x := [E(Cx (⋅))](t), t ∈ ℝ, x ∈ X.̃ As in the semigroup case, we have that C(⋅) commutes with C as well as that ‖C(t)x‖ ⩽ sups⩾0 ‖C(s)x‖, x ∈ X,̃ t < 0, and supt∈ℝ ‖C(t)x‖ = supt⩾0 ‖C(t)x‖, x ∈ X.̃ In the present situation, we do not know what conditions ensure the validity of expected equality C(−t)x = C(t)x, for t ⩾ 0 and x ∈ X.̃

2.8 Subspace (asymptotical) almost periodicity | 95

We continue by stating the following theorem. Theorem 2.8.13. ̃ (i) Suppose that 𝒢 is a (C-DS) generated by A, and 𝒢 is X-almost periodic for some linear subspace X̃ of Z(A). Put t

1 ∫ e−irs G(δs )x ds, t→∞ t

Pr x := lim

r ∈ ℝ, x ∈ X.̃

(87)

0

Then APr x = irPr x, r ∈ ℝ, x ∈ X.̃ Furthermore, if X̃ is dense in X, then the set D consisting of all eigenvectors of operator A corresponding to purely imaginary eigenvalues of operator A is total in X. ̃ (ii) Suppose that G is a (C − DCF) generated by A, and G is X-almost periodic for some linear subspace X̃ of Z2 (A). Define, for every r ∈ ℝ and x ∈ X,̃ the element Pr x through (87). Then APr x = −r 2 Pr x, r ∈ ℝ, x ∈ X.̃ Furthermore, if X̃ is dense in X, then the set H consisted of all eigenvectors of A corresponding to the real nonpositive eigenvalues of A is total in X. Proof. Let x ∈ X̃ and r ∈ ℝ be fixed. We will only prove that APr x = irPr x in the semigroup case and that APr x = −r 2 Pr x, in the cosine operator function case; then the remaining part of proof of (i)–(ii) follows by paraphrasing the final part of proof of [421, Theorem 2.1]. Consider first the semigroup case. Then we know that t

A ∫ u(s; x) ds = u(t; x) − x,

t ⩾ 0, x ∈ X,̃

0

i. e., t

𝒢 (−φ ) ∫ G(δs )x ds = 𝒢 (φ)[G(δt )x − x], 󸀠

t ⩾ 0, x ∈ X,̃ φ ∈ 𝒟0 .

0

Applying partial integration, the above yields t

t

s

0

0

1 −irt −irs 𝒢 (−φ )Pr x = 𝒢 (−φ ) lim [e ∫ G(δs )x ds + ir ∫ e ∫ G(δv )x dv ds] t→∞ t 󸀠

󸀠

0

= lim

t→∞

t

1 −irt [e 𝒢 (φ){G(δt )x − x} + ir ∫ e−irs 𝒢 (φ){G(δs )x − x} ds], t 0

t

which can be simply shown to equal ir ∫0 e−irs 𝒢 (φ)G(δs )x ds by the boundedness of function G(δ⋅ )x. This implies by definition of integral generator A that APr x = irPr x, as

96 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations claimed. Consider now the cosine operator function case. Then we have that [246]: t

A ∫(t − s)G(δs )x ds = G(δt )x − x,

t ⩾ 0, x ∈ X,̃

0

i. e., t

G(φ ) ∫(t − s)G(δs )x ds = G(I(φ))[G(δt )x − x],

t ⩾ 0, x ∈ X,̃ φ ∈ 𝒟0 .

󸀠

(88)

0

Fix an element x ∈ X̃ and a test function φ ∈ 𝒟0 , and define the function H(t) := t G(φ󸀠 ) ∫0 (t − s)G(δs )x ds, t ⩾ 0. Then (88), in combination with the almost periodicity of function G(δ⋅ )x, implies that H(t) is twice continuously differentiable, with H(t) and H 󸀠󸀠 (t) being bounded for t ⩾ 0. Then the Landau inequality 󵄨 󵄨 󵄨 ∗ 󸀠󸀠 󵄨 ∗ 󵄨2 󵄨󵄨 ∗ 󸀠 󵄨󵄨⟨x , H (t)⟩󵄨󵄨󵄨 ⩽ 4󵄨󵄨󵄨⟨x , H(t)⟩󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨⟨x , H (t)⟩󵄨󵄨󵄨,

t ⩾ 0, x∗ ∈ X ∗ ,

shows that the function t 󳨃→ H 󸀠 (t), t ⩾ 0 is weakly bounded and therefore bounded. Hence, there exists a finite constant M > 0 such that t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󸀠 󵄩󵄩 󵄩󵄩󵄩 󸀠 󵄩󵄩 ⩽ M, G(φ ) G(δ )x ds = H (t) ∫ 󵄩󵄩 󵄩󵄩 󵄩󵄩 s 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 0

t ⩾ 0.

(89)

On the other hand, applying partial integration twice, we get t

∫e

t −irs

G(δs )x ds = e

−irt

0

t

∫ G(δs )x ds + ire 0

t

s

0

0

−irt

∫(t − s)G(δs )x ds 0

− r 2 ∫ e−irs ∫(s − v)G(δv )x dv ds.

(90)

By means of (88) and (90), we obtain t

1 G(φ󸀠 )Pr x = lim e−irt [G(φ󸀠 ) ∫ G(δs )x ds] t→∞ t 0

+ G(I(φ)) lim ire t→∞ t

−irt 1

t

[G(δt )x − x]

1 − r lim ∫ e−irs G(I(φ))[G(δs )x − x] ds. t→∞ t 2

0

This equality, in combination with (89) and the boundedness of function G(δ⋅ )x, shows that G(φ󸀠 )Pr x = −r 2 G(I(φ))Pr x. Since φ ∈ 𝒟0 was arbitrary, we get that APr x = −r 2 Pr x, finishing the proof.

2.8 Subspace (asymptotical) almost periodicity | 97

Concerning the subspace weak almost periodicity of C-distribution semigroups, we have the following proposition. ̃ Proposition 2.8.14. Suppose that 𝒢 is a (C-DS) generated by A, and 𝒢 is X-weakly almost periodic for some linear subspace X̃ of Z(A). If X̃ is dense in X and X is weakly sequentially complete, then the set D defined above is total in X and 𝒢 is span(D)-almost periodic. Proof. Applying partial integration, we easily get t

1 1 (ir − A) ∫ e−irs G(δs )x ds = [x − e−irt G(δt )x], t t

t > 0, r ∈ ℝ, x ∈ X.̃

0

Then we can proceed as in the proofs of [43, Theorem 3] and [421, Theorem 2.1] in order to see that the set D is total in X. Hence, the statement of proposition is a simple consequence of Proposition 2.8.9(i). Concerning the subspace weak almost periodicity of C-distribution cosine functions, we would like to propose the following problem (cf. also [402, Theorem 1.5, ̃ pp. 246–247]): Suppose that G is a (C − DCF) generated by A, and G is X-almost perĩ ̃ odic for some linear subspace X of Z2 (A). If X is dense in X and X is weakly sequentially complete, is it true that the set H is total in X and G is span(H)-almost periodic? In [251], we have also investigated almost periodic (degenerate) strongly continuous semigroups, cosine operator functions and the associated sine operator functions acting on Banach spaces which do not contain an isomorphic copy of the space c0 (cf. Theorem 2.1.1(vi)). The interested reader may toy with the idea of formulating an analogue of Theorem 2.8.13(ii) in the case that G is a (C − DCF) generated by A and t ̃ periodic for all elements x belonging to the mapping t 󳨃→ ∫0 G(δs )x, t ⩾ 0 is X-almost some linear subspace X̃ of Z2 (A). It would be worthwhile to mention that the assertions of Propositions 2.8.9 and 2.8.10, Theorem 2.8.13 and Proposition 2.8.14 hold for C-ultradistribution semigroups of ∗-class and C-ultradistribution cosine functions of ∗-class in Banach spaces (cf. [247] for the notion). In the remaining part of this section, we examine the subspace asymptotical almost periodicity of C-distribution semigroups and C-distribution cosine functions. Our main aim here is to reconsider some structural results of L. Xie, M. Li and F. Huang [405]. Definition 2.8.15. Let G be a (C−DCF) generated by A, resp. let 𝒢 be a (C-DS) generated by A. Suppose that X̃ is a linear subspace of Z2 (A), resp. Z(A). Then it is said that G is ̃ ̃ X-asymptotically almost periodic, resp. 𝒢 is X-asymptotically almost periodic, iff for ̃ each x ∈ X the mapping t 󳨃→ G(δt )x, t ⩾ 0 is asymptotically almost periodic.

98 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Let G be a (C−DCF) generated by A, resp. let 𝒢 be a (C-DS) generated by A, and let X̃ ̃ be a linear subspace of Z2 (A), resp. Z(A). Assume that G, resp. 𝒢 , is X-asymptotically almost periodic. Let G1 be another (C1 -DCF) generated by A, resp. let 𝒢1 be another ̃ (C1 -DS) generated by A. Then G1 , resp. 𝒢1 , is X-asymptotically almost periodic, as well. The following characterization of subspace asymptotical almost periodicity of C-distribution semigroups is motivated by [405, Lemma 2.1, Theorem 2.2, Corollary 2.3]. Theorem 2.8.16. Let 𝒢 be a (C-DS) generated by A, and let X̃ be a linear subspace of Z(A). Then the following assertions are equivalent: ̃ (i) 𝒢 is X-asymptotically almost periodic. ̃ (ii) For every x ∈ Z(A) ∩ X, there exist elements y, z ∈ Z(A) such that y + z ∈ X,̃ x = y + z and the following two conditions hold: (a) The mapping t 󳨃→ G(δt )y, t ⩾ 0 belongs to the space AP([0, ∞) : X). (b) The mapping t 󳨃→ G(δt )z, t ⩾ 0 belongs to the space C0 ([0, ∞) : X). (iii) For every x ∈ Z(A) ∩ X,̃ there exist elements y, z ∈ Z(A) such that y + z ∈ X,̃ x = y + z and the following two conditions hold: (c) The mapping F : [0, ∞) → Cb ([0, ∞) : X), defined by F(t) := G(δ⋅+t )y, t ⩾ 0, belongs to the space AP([0, ∞) : Cb ([0, ∞) : X)). (d) The mapping H : [0, ∞) → Cb ([0, ∞) : X), defined by H(t) := G(δ⋅+t )z, t ⩾ 0, belongs to the space C0 ([0, ∞) : Cb ([0, ∞) : X)). Proof. Suppose first that x ∈ Z(A) ∩ X.̃ Arguing as in the proof of [405, Lemma 2.1], we get the existence of a strictly increasing sequence (tn )n∈ℕ of positive reals and a mapping φ ∈ C0 ([0, ∞) : X) such that limn→∞ tn = ∞, the mapping h : [0, ∞) → X defined by h(t) := limn→∞ G(δt+tn )x, t ⩾ 0 is almost periodic, and G(δt )x = h(t) + φ(t),

t ⩾ 0.

(91)

Set y := limn→∞ G(δtn )x, z := x − y and yn := G(δtn )x, n ∈ ℕ. Let t ⩾ 0 be temporarily fixed. Then limn→∞ yn = y and limn→∞ G(δt )yn = limn→∞ G(δt+tn )x = h(t). By the closedness of G(δt ), we get y ∈ D(G(δt )) and G(δt )y = h(t). Therefore, we have y, z ∈ Z(A), y + z = x ∈ X̃ and, due to (91), G(δt )x = G(δt )y + G(δt )z,

t ⩾ 0.

(92)

This implies (a)–(b) in (ii). To prove that (ii) implies (iii), choose a number ε > 0 arbitrarily. Then the almost periodicity of mapping h(⋅), defined as in the proof of implication (i) ⇒ (ii), yields that there exists l > 0 such that any interval I ⊆ [0, ∞) of length l contains an ε-period τ for h(⋅), so that supt⩾0 ‖h(t + τ) − h(t)‖ = supt⩾0 ‖G(δt+τ )y − G(δt )y‖ ⩽ ε. This is equivalent to saying that supt⩾0 ‖F(t + τ) − F(t)‖Cb ([0,∞):X) = supt,s⩾0 ‖G(δt+s+τ )y − G(δt+s )y‖ = supt⩾0 ‖G(δt+τ )y − G(δt )y‖ < ε, which immediately

2.9 Asymptotically almost periodic solutions of fractional relaxation inclusions | 99

implies (c). On the other hand, it is clear that the mapping H(⋅) is well defined because for each t ⩾ 0 we have that lims→+∞ G(δs+t )z = lims→+∞ φ(s + t) = 0, where we define φ(⋅) as before. Moreover, limt→+∞ ‖G(δ⋅+t )z‖Cb ([0,∞):X) = 0 is equivalent to saying that limt→+∞ [sups⩾0 ‖G(δs+t )z‖] = 0, which can be easily verified to be true since for any ε > 0 we have the existence of a sufficiently large number M > 0 such that ‖φ(v)‖ = ‖G(δv )z‖ < ε, v > M. The converse statement (iii) ⇒ (ii) is much easier because (c) and (d) in turn imply that the mapping t 󳨃→ G(δt )y ∈ X, t ⩾ 0 is almost periodic and the mapping t 󳨃→ G(δt )z, t ⩾ 0 belongs to the space C0 ([0, ∞) : X). The implication (ii) ⇒ (i) is trivial, finishing the proof. Remark 2.8.17. We have already mentioned that the authors of [405] have considered the asymptotical almost periodicity of C-regularized semigroups by assuming that their integral generators have no eigenvalues in (0, ∞). It is worth noting that this is not the case in our analysis, where we allow that the point spectrum of integral geñ erator of a X-asymptotically almost periodic C-distribution semigroup could have a nonempty intersection with any ray (ω, ∞), where ω > 0. This is very important in the case that X̃ ≠ X, because then we can construct a great number of nonexponential ̃ C-distribution semigroups (C-distribution cosine functions) that are X-almost periodic ̃ [277], with the subspace X being dense in X, by using the Desch–Schappacher–Webb criterion for chaos of strongly continuous semigroups [131, 179]; the point spectrum of integral generator of such a C-distribution semigroup (C-distribution cosine function) may contain ray (ω, ∞), for some ω > 0 (see, e. g., Proposition 2.8.9, [246, Theorem 2.2.10, Example 3.2.37(iii)] and [245, Theorem 3.1.36]). As mentioned in the introductory part, our recent research studies of hypercyclic and topologically mixing properties of strongly continuous semigroups and cosine functions (see [246, Chapter III] for a comprehensive survey of results) enable one to construct many other examples of subspace almost periodic strongly continuous semigroups and cosine functions that are not almost periodic in the usual sense. It is clear that Propositions 2.6.22–2.6.23 cannot be reconsidered for C-distribution semigroups and C-distribution cosine functions because the operators G(δt ) are generally unbounded in this framework (t ⩾ 0).

2.9 Asymptotically almost periodic solutions of fractional relaxation inclusions with Caputo derivatives Let γ ∈ (0, 1), and let 𝒜 be a multivalued linear operator on a Banach space X. Of importance is the following fractional relaxation inclusion: (DFP)f ,γ

γ

{

Dt u(t) ∈ 𝒜u(t) + f (t), u(0) = x0 ,

t > 0,

100 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations and its semilinear analogue γ

(DFP)f ,γ,s

{

Dt u(t) ∈ 𝒜u(t) + f (t, u(t)), u(0) = x0 ,

t > 0,

γ

where Dt denotes the Caputo fractional derivative of order γ, x0 ∈ X and f : [0, ∞) → X, resp. f : [0, ∞) × X → X, is Stepanov almost periodic. The main aim of this section is to investigate asymptotically almost periodic solutions of fractional Cauchy inclusions (DFP)f ,γ and (DFP)f ,γ,s . The established results of ours are brand new in the degenerate case and they seem to be new even for fractional relaxation equations with almost sectorial operators [342]. 2.9.1 Subordinated fractional resolvent families with removable singularities at zero In this subsection, we analyze the class of multivalued linear operators 𝒜 satisfying condition (P). Then degenerate strongly continuous semigroup (T(t))t>0 ⊆ L(X) generated by 𝒜 satisfies estimate ‖T(t)‖ ⩽ M0 e−ct t β−1 , t > 0 for some finite constant M0 > 0. Furthermore, (T(t))t>0 is given by the formula T(t)x =

1 ∫ eλt (λ − 𝒜)−1 x dλ, 2πi

t > 0, x ∈ X,

Γ

where Γ is the upwards oriented curve λ = −c(|η| + 1) + iη (η ∈ ℝ). Let 0 < γ < 1, and let ν > −β. Set Tγ,ν (t)x := t

γν

∞

∫ sν Φγ (s)T(st γ )x ds,

t > 0, x ∈ X.

0

Following E. Bazhlekova [53] and R.-N. Wang, D.-H. Chen, T.-J. Xiao [396], we set Sγ (t) := Tγ,0 (t) and Pγ (t) := γTγ,1 (t)/t γ ,

t > 0.

Recall that (Sγ (t))t>0 is a subordinated (gγ , I)-regularized resolvent family generated by 𝒜, which is not necessarily strongly continuous at zero. In [247], we have proved that there exists a finite constant M1 > 0 such that 󵄩 󵄩 󵄩 󵄩󵄩 γ(β−1) , 󵄩󵄩Sγ (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩Pγ (t)󵄩󵄩󵄩 ⩽ M1 t

t > 0.

(93)

For our later purposes, we need to refine the growth order of these operator families at infinity: Lemma 2.9.1. There exists a finite constant M2 > 0 such that 󵄩 󵄩󵄩 −γ 󵄩󵄩Sγ (t)󵄩󵄩󵄩 ⩽ M2 t ,

t⩾1

and

󵄩 󵄩󵄩 −2γ 󵄩󵄩Pγ (t)󵄩󵄩󵄩 ⩽ M2 t ,

t ⩾ 1.

(94)

2.9 Asymptotically almost periodic solutions of fractional relaxation inclusions | 101

Proof. By definition of (Sγ (t))t>0 and growth order of (T(t))t>0 , we have ∞

∞

0

0

γ 󵄩 󵄩󵄩 γ β−1 γ(β−1) cst γ ∫ ecst Φγ (s)sβ−1 ds, 󵄩󵄩Sγ (t)󵄩󵄩󵄩 ⩽ M0 ∫ Φγ (s)e (st ) ds = M0 t

for any t > 0. Since Φγ (s) ∼ (Γ(1 − γ))−1 , s → 0+, a Tauberian type theorem [29, Proposition 4.1.4; b)] immediately implies the first estimate in (94). We can prove the second estimate in (94) by using the same result, since ∞

γ 󵄩 󵄩󵄩 γ(β−1) ∫ ecst Φγ (s)sβ ds, 󵄩󵄩Pγ (t)󵄩󵄩󵄩 ⩽ M0 t

t > 0.

0

Suppose that x0 ∈ X belongs to the domain of continuity of (Sγ (t))t>0 , that is, limt→0+ Sγ (t)x0 = x0 (this holds provided that x0 ∈ X belongs to the space of continuity θ of (T(t))t>0 ; for example, in the case that x ∈ D((−𝒜)θ ) with 1 ⩾ θ > 1 − β or that x ∈ X𝒜 with 1 > θ > 1 − β). In this subsection, we employ the following definition of Caputo fractional derivaγ tives of order γ ∈ (0, 1). The Caputo fractional derivative Dt u(t) is defined for those functions u : [0, T] → X, for which u|(0,T] (⋅) ∈ C((0, T] : X), u(⋅) − u(0) ∈ L1 ((0, T) : X) and g1−γ ∗ (u(⋅) − u(0)) ∈ W 1,1 ((0, T) : X), by γ

Dt u(t) =

d [g ∗ (u(⋅) − u(0))](t), dt 1−γ

t ∈ (0, T].

Moreover, we will use the following definition (cf. [247, Section 3.5] for more details on the subject): Definition 2.9.2. By a classical solution of (DFP)f ,γ , we mean any function u ∈ γ C([0, ∞) : X) satisfying that the function Dt u(t) is well-defined on any finite interval (0, T] and belongs to the space C((0, T] : X), as well as that u(0) = u0 and γ Dt u(t) − f (t) ∈ 𝒜u(t) for t > 0. Set Rγ (t) := t γ−1 Pγ (t), t > 0. Then we need the following lemma (cf. also Propositions 2.6.11 and 2.7.5): Lemma 2.9.3. Let f ∈ AAPSq ([0, ∞) : X) with some q ∈ (1, ∞), let 1/q + 1/q󸀠 = 1, and let q󸀠 (γβ − 1) > −1. Define t

H(t) := ∫ Rγ (t − s)f (s) ds, 0

Then H ∈ AAP([0, ∞) : X).

t ⩾ 0.

102 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Proof. Let the locally p-integrable functions g : ℝ → X, q : [0, ∞) → X satisfy the conditions from Lemma 2.2.6, with the number p replaced with q therein, and let the function G(⋅) be given by (68). Since q󸀠 (γβ − 1) > −1 and, due to (93), ‖Rγ (t)‖ ⩽ M1 t γβ−1 , t ∈ (0, 1], we can apply Proposition 2.6.11 (see also Remark 2.6.12) in order to see that G(⋅) is almost periodic. On the other hand, it is clear that there exists a number η ∈ (0, 1) such that (1 − η)(1 + γ) > 1. Put t

∞

F(t) := ∫ Rγ (t − s)q(s) ds − ∫ Rγ (s)g(t − s) ds, t

0

t ⩾ 0.

Owing to Hölder inequality, we have that H(⋅) is well-defined. Since H(t) = G(t) + F(t) for all t ⩾ 0, we need to prove that F ∈ C0 ([0, ∞) : X). Evidently, ‖Rγ (t)‖ ⩽ M2 t −γ−1 , t ⩾ 1 and 󵄩󵄩 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩 󵄩󵄩 ∫ Rγ (s)g(t − s) ds󵄩󵄩󵄩 ⩽ ∑ 󵄩󵄩󵄩Rγ (⋅)󵄩󵄩󵄩 q󸀠 󵄩󵄩 󵄩󵄩 󵄩L [t+k,t+k+1] ‖g‖Sq 󵄩 󵄩󵄩 󵄩󵄩 k=0 t ∞

∞

‖g‖Sq 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩Rγ (⋅)󵄩󵄩󵄩L∞ [t+k,t+k+1] ‖g‖Sq ⩽ ∑ (t + k)γ+1 k=0 k=0 ∞

‖g‖Sq ⩽ Const. t −η(1+γ) ‖g‖Sq , η(1+γ) k (1−η)(γ+1) t k=0

⩽ Const. ∑

t > 1.

On account of this, we have that limt→∞ ∫t Rγ (s)g(t − s) ds = 0. Then a similar line of reasoning shows that ∞

󵄩󵄩 t/2 ⌈t/2⌉ 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 ∫ Rγ (t − s)q(s) ds󵄩󵄩󵄩 ⩽ ‖g‖Sq ∑ 󵄩󵄩󵄩Rγ (t − ⋅)󵄩󵄩󵄩 q󸀠 󵄩L [k,k+1] 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k=0 0 ⌈t/2⌉

󵄩 󵄩 ⩽ ‖g‖Sq ∑ 󵄩󵄩󵄩Rγ (t − ⋅)󵄩󵄩󵄩L∞ [k,k+1] k=0

⩽ M2 (1 + ⌈t/2⌉)(t − ⌈t/2⌉)

−γ−1

‖g‖Sq ,

t ⩾ 2;

t/2

t

hence, limt→∞ ∫0 Rγ (t − s)q(s) ds = 0. It remains to be proved that limt→∞ ∫t/2 Rγ (t − s)q(s) ds = 0 (observe that the integral in this limit expression converges by Hölder inequality, the estimate q󸀠 (γβ − 1) > −1 and the Sq -boundedness of function q(⋅)). Let t+1 a number ε > 0 be fixed. Then there exists t0 > 0 such that ∫t ‖q(s)‖q ds < εq , t ⩾ t0 . Suppose that t > 2t0 + 6. Then Hölder inequality implies the existence of a finite constant c > 0 such that 󵄩󵄩 t 󵄩󵄩 ⌊t/2⌋−2 󵄩󵄩 󵄩 󵄩󵄩 β−1 󵄩󵄩 󵄩󵄩 ∫ Rγ (t − s)q(s) ds󵄩󵄩󵄩 ⩽ c ∑ 󵄩󵄩󵄩Rγ (t − ⋅)󵄩󵄩󵄩 q󸀠 󵄩󵄩 󵄩󵄩 󵄩L [t/2+k,t/2+k+1] ε + ε󵄩󵄩⋅ 󵄩󵄩Lq󸀠 [0,2] 󵄩 󵄩󵄩 󵄩󵄩 k=0 t/2

2.9 Asymptotically almost periodic solutions of fractional relaxation inclusions | 103

⌊t/2⌋−2

󵄩 󵄩 󵄩 󵄩 ⩽ c ∑ 󵄩󵄩󵄩Rγ (t − ⋅)󵄩󵄩󵄩L∞ [t/2+k,t/2+k+1] ε + ε󵄩󵄩󵄩⋅β−1 󵄩󵄩󵄩Lq󸀠 [0,2] k=0

⌊t/2⌋−2

󵄩 󵄩 ⩽ cεM ∑ (t/2 + k)−γ−1 + ε󵄩󵄩󵄩⋅β−1 󵄩󵄩󵄩Lq󸀠 [0,2] k=0

∞

󵄩 󵄩 ⩽ cεM(t/2)−η(1+γ) ∑ k (1−η)(1+γ) + ε󵄩󵄩󵄩⋅β−1 󵄩󵄩󵄩Lq󸀠 [0,2] . k=0

This estimate simply completes the proof of lemma. Remark 2.9.4. Let f ∈ AAPSq ([0, ∞) : X) with some q ∈ (1, ∞), let 1/q + 1/q󸀠 = 1, and let q󸀠 (γβ − 1) > −1. Let (Rγ (t))t>0 ⊆ L(X, Y) be a strongly continuous operator family, and let H : [0, ∞) → Y be defined as above. Then H ∈ AAP([0, ∞) : Y), provided that (Rγ (t))t>0 has the same growth rate at zero and infinity as the operator family (Rγ (t))t>0 considered above. Keeping in mind Lemmas 2.9.1 and 2.9.3 and [247, Theorem 3.5.3], the following result is easily attainable: Theorem 2.9.5. Suppose that 1 ⩾ θ > 1 − β and x0 ∈ D((−𝒜)θ ), resp. 1 > θ > 1 − β and θ x0 ∈ X𝒜 , as well as there exists a constant σ > γ(1 − β) such that, for every T > 0, there exists a finite constant MT > 0 such that f : [0, ∞) → X satisfies 󵄩 󵄩󵄩 σ 󵄩󵄩f (t) − f (s)󵄩󵄩󵄩 ⩽ MT |t − s| ,

0 ⩽ t, s ⩽ T.

Let 1 ⩾ θ > 1 − β, resp. 1 > θ > 1 − β, and let θ f ∈ L∞ loc ((0, ∞) : [D((−𝒜) )]),

resp.

θ f ∈ L∞ loc ((0, ∞) : X𝒜 ).

Then there exists a unique classical solution u(⋅) of problem (DFP)f ,γ . If, additionally, f ∈ AAPSq ([0, ∞) : X) with some q ∈ (1, ∞), 1/q + 1/q󸀠 = 1 and q󸀠 (γβ − 1) > −1, then u ∈ AAP([0, ∞) : X). Proof. The first part of theorem is a simple consequence of [247, Theorem 3.5.3], which also shows that the classical solution of (DFP)f ,γ is given by the formula t

u(t) = Sγ (t)x0 + ∫(t − s)γ−1 Pγ (t − s)f (s) ds,

t ⩾ 0.

0

The second part of theorem immediately follows from this representation, Lemmas 2.9.1 and 2.9.3. θ Suppose now that 1 ⩾ θ > 1 − β and x0 ∈ D((−𝒜)θ ), resp. 1 > θ > 1 − β and x0 ∈ X𝒜 . In [247, Section 3.5], we have proved that there exists a finite constant M1,θ > 0 such that

󵄩 󵄩 󵄩 󵄩󵄩 γ(β−θ−1) , 󵄩󵄩Sγ (t)󵄩󵄩󵄩L(X,[D((−𝒜)θ )]) + 󵄩󵄩󵄩Pγ (t)󵄩󵄩󵄩L(X,[D((−𝒜)θ )]) ⩽ M1,θ t

t > 0,

(95)

104 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations resp. 󵄩 󵄩 󵄩 󵄩󵄩 γ(β−θ−1) , 󵄩󵄩Sγ (t)󵄩󵄩󵄩L(X,X θ ) + 󵄩󵄩󵄩Pγ (t)󵄩󵄩󵄩L(X,X θ ) ⩽ M1,θ t 𝒜 𝒜

t > 0.

(96)

We need to improve the estimates (95)–(96) for long time behavior: Lemma 2.9.6. There exists a finite constant M2,θ > 0 such that 󵄩 󵄩 󵄩 󵄩󵄩 −γ(θ+1) , 󵄩󵄩Sγ (t)󵄩󵄩󵄩L(X,[D((−𝒜)θ )]) + 󵄩󵄩󵄩Sγ (t)󵄩󵄩󵄩L(X,X θ ) ⩽ M2,θ t 𝒜

t > 0,

resp. 󵄩 󵄩 󵄩 󵄩󵄩 −γ(θ+1)−1 , 󵄩󵄩Pγ (t)󵄩󵄩󵄩L(X,[D((−𝒜)θ )]) + 󵄩󵄩󵄩Pγ (t)󵄩󵄩󵄩L(X,X θ ) ⩽ M2,θ t

t > 0.

𝒜

Proof. Let t > 2/|c|. By the proof of [188, Proposition 3.2] and the well known integral computation similar to that appearing in the proof of [187, Lemma 7.1], we have that (T(t)x,

1 ∫(−λ)θ eλt (λ − 𝒜)−1 x dλ) ∈ (−𝒜)θ , 2πi

x ∈ X,

Γ

as well as ξ θ [ξ (ξ − 𝒜)−1 − I]T(t)x =

ξ θ λ λt 1 e (λ − 𝒜)−1 x dλ, ∫ 2πi ξ − λ

ξ > 0, x ∈ X.

Γ

Let a(t) > 0 satisfy c2 (a(t)2 + 1) + a(t)2 = t −2 . Using Cauchy theorem, we can deform the path of integration Γ to the upwards oriented curve Γ󸀠 , obtained by replacing the union of segments [c, c(a(t) + 1) + ia(t)] ∪ [c(−a(t) + 1) − ia(t), c] of the curve Γ with the part of circle with radius 1/t and center at point c. Putting into action the computation contained in the proof of [29, Theorem 2.6.1], we get 󵄩 󵄩 󵄩 󵄩󵄩 󸀠 ct γ(β−θ−1) , 󵄩󵄩T(t)󵄩󵄩󵄩L(X,[D((−𝒜)θ )]) + 󵄩󵄩󵄩T(t)󵄩󵄩󵄩L(X,[D((−𝒜)θ )]) ⩽ M2,θ e t resp. 󵄩 󵄩󵄩 󵄩 󵄩 󸀠 ct γ(β−θ−1) . 󵄩󵄩T(t)󵄩󵄩󵄩L(X,X θ ) + 󵄩󵄩󵄩T(t)󵄩󵄩󵄩L(X,X θ ) ⩽ M2,θ e t 𝒜 𝒜 Now the final conclusions can be obtained by following the line of argument given in the proof of Lemma 2.9.1. Using Lemma 2.9.6 and the conclusion from Remark 2.9.4, we can simply prove that the following result holds true: Theorem 2.9.7. Let the requirements of the first part of Theorem 2.9.5 hold. Then there exists a unique classical solution u(⋅) of problem (DFP)f ,γ . Assume, further, that θ the initial value x0 satisfies limt→0+ Sγ (t)x0 = x0 in [D((−𝒜)θ )], resp. X𝒜 , as well as q q f ∈ AAPS ([0, ∞) : X) and 1 > θ > 1 − β, resp. f ∈ AAPS ([0, ∞) : X), with some q ∈ (1, ∞). Let 1/q+1/q󸀠 = 1 and q󸀠 (γ(β−θ)−1) > −1. Then u ∈ AAP([0, ∞) : [D((−𝒜)θ )]), θ resp. u ∈ AAP([0, ∞) : X𝒜 ).

2.9 Asymptotically almost periodic solutions of fractional relaxation inclusions | 105

Remark 2.9.8. (i) The condition q󸀠 (γ(β − θ) − 1) > −1 immediately forces that β − γ > 0 and therefore β > 1/2, which can be slightly restrictive in applications (see [188, Section 3.7] for more details). θ (ii) It is very simple to see that limt→0+ Sγ (t)x0 = x0 in [D((−𝒜)θ )], resp. X𝒜 , holds θ θ provided that limt→0+ T(t)x0 = x0 in [D((−𝒜) )], resp. X𝒜 . Concerning the space [D((−𝒜)θ )], we have that the last equality holds for any x0 ∈ (−𝒜)−θ (Ωc ), where Ωc denotes the domain of continuity of semigroup (T(t))t>0 ; strictly speaking, if y0 ∈ Ωc and x0 = (−𝒜)−θ y0 , then T(t)y0 − y0 ∈ (−𝒜)θ (T(t)x0 − x0 ), t > 0 and therefore ‖T(t)x0 − x0 ‖[D((−𝒜)θ )] ⩽ ‖T(t)y0 − y0 ‖, t > 0, which implies the claimed assertion. θ Concerning the space X𝒜 , the situation is a bit more subtle and, undoubtedly, not well explored in the existing literature.

For semilinear problems, we will use the following notion. Definition 2.9.9. By a mild solution of (DFP)f ,γ,s , we mean any function u ∈ C([0, ∞) : X) satisfying that t

u(t) = Sγ (t)x0 + ∫(t − s)γ−1 Pγ (t − s)f (s, u(s)) ds,

t ⩾ 0.

0

Set, for every x ∈ Cb ([0, ∞) : X), t

(ϒx)(t) := Sγ (t)x0 + ∫(t − s)γ−1 Pγ (t − s)f (s, x(s)) ds,

t ⩾ 0.

0

Suppose that (73) holds for a. e. t > 0 (I = [0, ∞)), with locally integrable positive function Lf (⋅). Set, for every n ∈ ℕ, t xn

x2

󵄩 󵄩 An := sup ∫ ∫ ⋅ ⋅ ⋅ ∫󵄩󵄩󵄩Rγ (t − xn )󵄩󵄩󵄩 t⩾0

n

0 0

0

n

󵄩 󵄩 × ∏󵄩󵄩󵄩Rγ (xi − xi−1 )󵄩󵄩󵄩 ∏ Lf (xi ) dx1 dx2 ⋅ ⋅ ⋅ dxn . i=2

i=1

Then a simple calculation shows that 󵄩󵄩 n n 󵄩 󵄩󵄩(ϒ u) − (ϒ v)󵄩󵄩󵄩∞ ⩽ An ‖u − v‖∞ ,

u, v ∈ BUC([0, ∞) : X), n ∈ ℕ.

(97)

The following result holds true: Theorem 2.9.10. Suppose that I = [0, ∞) and the following conditions hold: (i) g ∈ APSp (I × X : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lg ∈ LrS (I) such that (73) holds with f = g and Lf = Lg .

106 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations (ii) f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 (I × X : Lq ([0, 1] : X)) and q = pr/p + r. Set q󸀠 := ∞, provided r = p/p − 1 and

q󸀠 :=

pr , provided r > p/p − 1. pr − p − r

Assume also that: (iii) q󸀠 (γβ − 1) > −1, (iv) (73) holds for a. e. t > 0, with locally bounded positive function Lf (⋅) satisfying An < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . Proof. Due to (i)–(ii) and Lemma 2.7.3, we get that for each x ∈ AAP(I : X) one has f (⋅, x(⋅)) ∈ AAPSq (I : X), where q = pr/p + r; here we would like to recall only that the range of an X-valued asymptotically almost periodic function is relatively compact in X. Owing to the assumption (iii), Lemma 2.9.3 and the obvious equality limt→+∞ Sγ (t)x0 = 0, we get that the mapping ϒ : AAP(I : X) → AAP(I : X) is well-defined. Using now (97), (iv) and a well-known extension of the Banach contraction principle, we obtain the existence and uniqueness of an asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . This completes the proof of theorem. If we employ Lemma 2.7.4 in place of Lemma 2.7.3, then we are in a position to formulate and prove the following analogue of Theorem 2.9.10 in the case of consideration of classical Lipschitz condition (74): Theorem 2.9.11. Let I = [0, ∞), and let p > 1. Suppose that the following conditions hold: (i) g ∈ APSp (I × X : X) and there exists a constant L > 0 such that (74) holds with the function f (⋅, ⋅) replaced by the function g(⋅, ⋅) therein. (ii) f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 (I × X : Lp ([0, 1] : X)). p (γβ − 1) > −1. (iii) p−1 (iv) (73) holds for a. e. t > 0, with locally bounded positive function Lf (⋅) satisfying An < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . Since any function g ∈ AAP([0, ∞) × X : X) belongs to the class AAPSp ([0, ∞) × X : X) for all p > 1, and lim

p→+∞

p (γβ − 1) = γβ − 1 > −1, p−1

Theorem 2.9.10 immediately implies the following corollary:

2.9 Asymptotically almost periodic solutions of fractional relaxation inclusions | 107

Corollary 2.9.12. Suppose that I = [0, ∞), the function f (⋅, ⋅) is asymptotically almost periodic and (73) holds for a. e. t > 0, with locally bounded positive function Lf (⋅) satisfying An < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . It is not trivial to state a satisfactory criterion which would enable one to see that there exists an integer n ∈ ℕ such that An < 1. On the other hand, a very simple calculation involving the estimates (93) and (94) shows that A1 ⩽ L[

M1 M2 + ]. γβ γ

Hence, Theorem 2.9.11 implies the following: Corollary 2.9.13. Suppose that I = [0, ∞), the function f (⋅, ⋅) is asymptotically almost 1 + Mγ2 )−1 ). Then there exists a unique asympperiodic and (74) holds for some L ∈ [0, ( M γβ totically almost periodic solution of inclusion (DFP)f ,γ,s . 2.9.2 The nonanalyticity of semigroup (T (t))t>0 It is not difficult to observe that the analyticity of degenerate semigroup (T(t))t>0 examined in the previous subsection is a slightly redundant assumption. In this subsection, we investigate the case in which the operator C ∈ L(X) is injective and (T(t))t⩾0 ⊆ L(X) is a C-regularized semigroup, i. e., the mapping t 󳨃→ T(t)x, t ⩾ 0 is continuous for every fixed element x ∈ X, T(0) = C and T(t + s)C = T(t)T(s) for all t, s ⩾ 0; we will employ the classical definition of Caputo fractional derivatives of order γ ∈ (0, 1). The integral generator of (T(t))t⩾0 is defined as before (see [245–247] for more details on the subject). Let ‖T(t)‖ = O(ect ), t ⩾ 0 for some negative constant c < 0. Then we know ∞ that {λ ∈ ℂ : Re λ > c} ⊆ ρC (𝒜) as well as ∫0 e−λt T(t)x dt = (λ − 𝒜)−1 Cx, x ∈ X, Re λ > c (see Theorem 1.5.5 and [247]). Define the operator families Sγ (⋅), Pγ (⋅) and Rγ (⋅), as well as the sequence (An )n∈ℕ and the operator ϒ(⋅) as in the previous subsection. Then the operator 𝒜 = C −1 𝒜C is the integral generator of an exponentially bounded (gγ , C)-regularized resolvent family (Sγ (t))t⩾0 (cf. [247, Definition 3.2.2] for the notion), so that 𝒜 is closed, (Sγ (t))t⩾0 t

is strongly continuous for t ⩾ 0, Sγ (t)x − Cx ∈ 𝒜 ∫0 gγ (t − s)Sγ (s)x ds, t ⩾ 0, x ∈ X; t

Sγ (t)x − Cx = ∫0 gγ (t − s)Sγ (s)y ds, t ⩾ 0, whenever y ∈ 𝒜x; Sγ (t)𝒜 ⊆ 𝒜Sγ (t), t ⩾ 0 and Sγ (t)C = CSγ (t), t ⩾ 0. We continue by observing that the arguments in the proof of Lemma 2.9.1 show that there exist two finite constants M3 > 0 and M4 > 0 such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩Sγ (t)󵄩󵄩󵄩 + 󵄩󵄩󵄩Pγ (t)󵄩󵄩󵄩 ⩽ M3 ,

t > 0,

(98)

108 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations as well as 󵄩 󵄩 󵄩 󵄩󵄩 −2γ −γ 󵄩󵄩Sγ (t)󵄩󵄩󵄩 ⩽ M4 t , t ⩾ 1 and 󵄩󵄩󵄩Pγ (t)󵄩󵄩󵄩 ⩽ M4 t , t ⩾ 1.

(99)

Definition 2.9.14. Let x0 ∈ R(C), and let f : [0, ∞) → X be continuous. Then we say γ that a continuous function u : [0, ∞) → X is a classical solution of (DFP)f ,γ iff Dt u(t) is well-defined and continuous for t ⩾ 0 as well as (DFP)f ,γ holds identically for t ⩾ 0. Now we are in a position to state the following result: 1,1 Theorem 2.9.15. Suppose that C −1 x0 ∈ D(𝒜) and C −1 f ∈ Wloc ((0, ∞) : X). Then there exists a unique classical solution u(⋅) of problem (DFP)f ,γ . If, additionally, f ∈ AAPSq ([0, ∞) : X) with some q ∈ (1, ∞), 1/q + 1/q󸀠 = 1 and q󸀠 (γ − 1) > −1, then u ∈ AAP([0, ∞) : X).

Proof. The uniqueness of classical solutions is a simple consequence of the fact that 𝒜 generates a global (gγ , C)-regularized resolvent family [247]. To prove the existence of classical solutions, we first observe that the argumentation contained in the proof of [235, Theorem 5] (see also [248]) shows that ∞

∫ e−λt Rγ (t)x dt = (λγ − 𝒜) Cx,

x ∈ X, λ > 0.

−1

0

By the uniqueness theorem for Laplace transform, we get that (g1−γ ∗ Rγ (⋅)x)(t) = Sγ (t)x,

t ⩾ 0, x ∈ X.

(100)

t

Let C −1 x0 = y0 and z0 ∈ 𝒜y0 . Then Sγ (t)y0 − Cy0 = ∫0 gγ (t − s)Sγ (s)z0 ds, t ⩾ 0 and, γ on account of this, it is almost trivial to verify that Dt Sγ (t)C −1 x0 = Sγ (t)z0 , t ⩾ 0. Set t

Fγ (t) := ∫0 Rγ (t − s)C −1 f (s) ds, t ⩾ 0 and u(t) := Sγ (t)C −1 x0 + Fγ (t), t ⩾ 0. Using the proof of [29, Proposition 1.3.6], we infer that Fγ󸀠 (t) = (Rγ ∗ (C −1 f ) )(t) + Rγ (t)(C −1 f )(0), 󸀠

t > 0, x ∈ X,

γ

(101)

which simply implies that Dt Fγ (t) = (g1−γ ∗ Fγ󸀠 )(t), t ⩾ 0. By the foregoing, it suffices to show that (g1−γ ∗ Fγ󸀠 )(t) − f (t) ∈ 𝒜[Rγ ∗ C −1 f ](t),

t ⩾ 0,

i. e., by (100)–(101), (Sγ ∗ (C −1 f ) )(t) + Sγ (t)C −1 f (0) − f (t) 󸀠

= (Sγ ∗ C −1 f ) (t) − f (t) ∈ 𝒜[Rγ ∗ C −1 f ](t), 󸀠

t ⩾ 0.

2.9 Asymptotically almost periodic solutions of fractional relaxation inclusions | 109

Due to the closedness of 𝒜, the only thing that remained to be proved is (g1 ∗ (Sγ ∗ C −1 f ) )(t) − (g1 ∗ f )(t) ∈ 𝒜[g1 ∗ Rγ ∗ C −1 f ](t), 󸀠

t ⩾ 0,

i. e., due to (100), (Sγ ∗ C −1 f )(t) − (g1 ∗ f )(t) ∈ 𝒜[gγ ∗ Sγ ∗ C −1 f ](t),

t ⩾ 0.

This follows from Theorem 1.2.2 and the inclusion Sγ (t)x − Cx ∈ 𝒜(gγ ∗ Sγ (⋅)x)(t), t ⩾ 0, x ∈ X. The remainder of the theorem is a consequence of Lemma 2.9.3. Following our analyses from [247, Subsection 2.2.5], it will be convenient to introduce the following definition: Definition 2.9.16. We say that a continuous function t 󳨃→ u(t), t ⩾ 0 is a mild solution of the semilinear fractional Cauchy inclusion (DFP)f ,γ,s iff t

u(t) = Sγ (t)C x0 + ∫(t − s)γ−1 Pγ (t − s)C −1 f (s, u(s)) ds, −1

t ⩾ 0.

0

Keeping in mind the estimates (98)–(99) and the foregoing arguments, it is almost straightforward to formulate and prove the following analogues of Theorems 2.9.10–2.9.11 and Corollaries 2.9.12–2.9.13: Theorem 2.9.17. Suppose that I = [0, ∞), p > 1 and the following conditions hold: (i) g ∈ APSp (I×X : X), there exist a number r ⩾ max(p, p/p−1) and a function Lg ∈ LrS (I) such that (73) holds with f = g and Lf = Lg . (ii) C −1 f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 (I × X : Lq ([0, 1] : X)) and q = pr/p + r. Set q󸀠 := ∞, provided r = p/p − 1 and

q󸀠 :=

pr , provided r > p/p − 1. pr − p − r

Assume also that: (iii) q󸀠 (γ − 1) > −1; (iv) (73) holds with the functions f (⋅, ⋅) and Lf (⋅) replaced therein by the functions C −1 f (⋅, ⋅) and LC−1 f (⋅), respectively, for a. e. t > 0, where LC−1 f (⋅) is a locally bounded positive function satisfying An < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . Theorem 2.9.18. Let I = [0, ∞), and let p > 1. Suppose that the following conditions hold: (i) g ∈ APSp (I × X : X) and there exists a constant L > 0 such that (74) holds with the function f (⋅, ⋅) replaced by the function g(⋅, ⋅) therein.

110 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations (ii) C −1 f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 (I × X : Lp ([0, 1] : X)). p (iii) p−1 (γ − 1) > −1.

(iv) (73) holds with the functions f (⋅, ⋅) and Lf (⋅) replaced therein by the functions C −1 f (⋅, ⋅) and LC−1 f (⋅), respectively, for a. e. t > 0, where LC−1 f (⋅) is a locally bounded positive function satisfying An < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . Corollary 2.9.19. Suppose that I = [0, ∞), the function C −1 f (⋅, ⋅) is asymptotically almost periodic and (73) holds with the functions f (⋅, ⋅) and Lf (⋅) replaced therein by the functions C −1 f (⋅, ⋅) and LC−1 f (⋅), respectively, for a. e. t > 0, where LC−1 f (⋅) is a locally bounded positive function satisfying An < 1 for some n ∈ ℕ. Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . Corollary 2.9.20. Suppose that I = [0, ∞), the function C −1 f (⋅, ⋅) is asymptotically almost periodic and (74) holds with f (⋅, ⋅) replaced therein by C −1 f (⋅, ⋅) and some constant M M L ∈ [0, ( γ3 + γ4 )−1 ). Then there exists a unique asymptotically almost periodic solution of inclusion (DFP)f ,γ,s . Theoretical results established here can be applied to a class of abstract degenerate fractional differential equations, see, e. g., [188, Example 2.1] and [247, Example 3.2.14] (C = I). Instructive applications with C ≠ I can be given to abstract nondegenerate fractional differential equations. We close the section by providing basic information on asymptotically almost periodic solutions of abstract fractional inclusion Dγ+ u(t) ∈ −𝒜u(t) + f (t),

t ∈ ℝ,

(102)

γ

where D+ u(t) denotes the Weyl–Liouville fractional derivative of order γ and f : ℝ → X (see [329]). In this paper, J. Mu, Y. Zhoa and L. Peng have investigated various types of (asymptotically) generalized almost periodic and generalized almost automorphic solutions of (102) provided that the operator 𝒜 is single-valued and generates an exponentially decaying C0 -semigroup. By applying the Fourier transform in [329, Lemma 6], the authors have proposed the following definition of mild solution of (102): A continuous function u : ℝ → X is said to be a mild solution of (102) iff it has the following form t

u(t) = ∫ (t − s)γ−1 Pγ (t − s)f (s) ds,

t ∈ ℝ;

−∞

a semilinear analogue is t

u(t) = ∫ (t − s)γ−1 Pγ (t − s)f (s, u(s)) ds, −∞

t ∈ ℝ.

2.10 The use of fractional powers of multivalued linear operators | 111

We would like to observe that the proof of [329, Lemma 6] works in the case 𝒜 is an MLO generating degenerate semigroup with removable singularity at zero or 𝒜 is an MLO generating an exponentially decaying C-regularized semigroup. Therefore, we are in a position to analyze the existence and uniqueness of asymptotically almost periodic solutions ℝ 󳨃→ X of (semilinear) fractional Cauchy inclusion (102) (cf. C. Zhang [416] for the notion) by using similar arguments to those employed in the proofs of [329, Theorems 8, 9, 17 and 19]. We leave this for the reader to make explicit.

2.10 The use of fractional powers of multivalued linear operators Suppose that γ ∈ (0, 1) and 𝒜 is a multivalued linear operator on a Banach space X satisfying condition (P). Of concern is the abstract Cauchy inclusion of first order (78), the abstract fractional relaxation inclusion (102) and its semilinear analogue γ

Dt,+ u(t) ∈ −𝒜u(t) + f (t, u(t)),

t ∈ ℝ,

(103)

γ

where Dt,+ denotes the Weyl–Liouville fractional derivative of order γ, x0 ∈ X and f : ℝ × X → X is Stepanov almost periodic. In the present section, we use fractional powers of sectorial multivalued linear operators with a view of establishing new structural results on the existence and uniqueness of almost periodic solutions of semilinear Cauchy inclusions (78) and (103); it should be said, again, that our results are new even for fractional relaxation equations with almost sectorial operators. The aim of section is also to correct some small typographical errors and inconsistencies from our recent research paper [257]. Let us recall that the conditions on function f (⋅, ⋅) used in this section are relaxed compared to those employed in [37] for equation (78), resp. [181] for equation (103), where the authors have investigated the following condition: (F) The function f : ℝ×[D((−𝒜)θ )] → X is such that there are finite numbers L > 0 and η ∈ [0, 1] satisfying ‖f (t1 , x1 ) − f (t2 , x2 )‖ ⩽ L(|t1 − t2 |η + ‖x1 − x2 ‖θ ) for all (t1 , x1 ), (t2 , x2 ) in ℝ × [D((−𝒜)θ )]; 𝒜 is the single-valued generator of an exponentially decaying C0 -semigroup. Let us recall that degenerate strongly continuous semigroup (T(t))t>0 ⊆ L(X) generated by 𝒜 satisfies estimate ‖T(t)‖ ⩽ M0 e−ct t β−1 , t > 0 for some finite constant M0 > 0. Suppose now that β > θ > 1 − β. Set ∞

Tγ,ν (t)x := t γν ∫ sν Φγ (s)T(st γ )x ds,

t > 0, x ∈ X

0

and, following M. M. El-Borai, A. Debbouche [181], Pγ (t) := γTγ,1 (t)/t γ ,

t > 0.

112 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations In the sequel of this section, by M > 0 we denote the finite generic constant whose numerical value may change from line to line.

2.10.1 Almost periodic solutions of abstract semilinear Cauchy inclusion (78) Put Y := [D((−𝒜)θ )] and ‖ ⋅ ‖Y := ‖ ⋅ ‖[D((−𝒜)θ )] . We already know that (T(t)x,

1 ∫(−λ)θ eλt (λ − 𝒜)−1 x dλ) ∈ (−𝒜)θ , 2πi

x ∈ X, t > 0.

(104)

Γ

Set Tν (t)x :=

1 ∫(−λ)ν eλt (λ − 𝒜)−1 x dλ, 2πi

x ∈ X, t > 0 (ν > 0).

Γ

Moreover, we have that (A) ‖Tν (t)‖ ⩽ Me−ct t β−ν−1 , t > 0, ν > 0. Applying the mean value theorem, (A) and the obvious equality Tν󸀠 (t) = −Tν+1 (t), t > 0, ν > 0, we obtain that: (B) ‖Tν (t + h) − Tν (t)‖ ⩽ Mhe−ct t β−ν−2 , t > 0, ν > 0, h > 0. The following notion of a mild solution of (78) will be sufficiently good for the purposes of this subsection: Definition 2.10.1. Let f : I × Y → X, and let Y be continuously embedded in X. By a mild solution of (78), we mean any Y-continuous function u(⋅) such that u(t) = (Λu)(t), t ∈ ℝ, where t

t 󳨃→ (Λu)(t) := ∫ T(t − s)f (s, u(s)) ds,

t ∈ ℝ.

−∞

In order to proceed, we next state the following important lemma. Lemma 2.10.2. Let 1 < q, q󸀠 < ∞, 1/q + 1/q󸀠 = 1 and q󸀠 (β − θ − 1) > −1. Assume that f ̂ : I × Y → Lq ([0, 1] : X) is bounded continuous, and u ∈ C(ℝ : Y). Define t

(Ψu)(t) := ∫ Tθ (t − s)f (s, u(s)) ds, −∞

Then Ψu ∈ Cb (ℝ : X).

t ∈ ℝ.

2.10 The use of fractional powers of multivalued linear operators | 113

t+1

Proof. Let ∫t ‖f (s, y)‖q ds ⩽ M q for all t ∈ ℝ and y ∈ Y. We will first prove the right continuity of (Ψu)(⋅). For this, fix a number h ∈ (0, 1]. Then a straightforward computation involving Hölder inequality shows that 󵄩 󵄩󵄩 󵄩󵄩(Ψu)(t + h) − (Ψu)(t)󵄩󵄩󵄩 t

󵄩 󵄩󵄩 󵄩 ⩽ ∫ 󵄩󵄩󵄩Tθ (t + h − s) − Tθ (t − s)󵄩󵄩󵄩󵄩󵄩󵄩f (s, u(s))󵄩󵄩󵄩 ds −∞

t+h

󵄩 󵄩 󵄩󵄩 + ∫ 󵄩󵄩󵄩Tθ (t + h − s)󵄩󵄩󵄩󵄩󵄩󵄩f (s, u(s))󵄩󵄩󵄩 ds t

∞

󵄩󵄩 󵄩 󵄩 = ∫ 󵄩󵄩󵄩Tθ (s + h) − Tθ (s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s, u(t − s))󵄩󵄩󵄩 ds 0

h

󵄩 󵄩󵄩 󵄩 + ∫󵄩󵄩󵄩Tθ (s + h)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s, u(t − s))󵄩󵄩󵄩 ds 1

0

󵄩 󵄩󵄩 󵄩 ⩽ ∫󵄩󵄩󵄩Tθ (s + h) − Tθ (s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s, u(t − s))󵄩󵄩󵄩 ds 0

∞ k+1

󵄩 󵄩󵄩 󵄩 + ∑ ∫ 󵄩󵄩󵄩Tθ (s + h) − Tθ (s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s, u(t − s))󵄩󵄩󵄩 ds k=1 k h

󵄩 󵄩󵄩 󵄩 + ∫󵄩󵄩󵄩Tθ (s + h)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s, u(t − s))󵄩󵄩󵄩 ds 0

1

1/q󸀠

󵄩q󸀠

󵄩 ⩽ M(∫󵄩󵄩󵄩Tθ (s + h) − Tθ (s)󵄩󵄩󵄩 ds) 0

∞ k+1

󵄩 󵄩 + M ∑ ∫ 󵄩󵄩󵄩Tθ (⋅ + h) − Tθ (⋅)󵄩󵄩󵄩Lq󸀠 [k,k+1] k=1 k h

󵄩q󸀠

1/q󸀠

󵄩 + M(∫󵄩󵄩󵄩Tθ (s + h)󵄩󵄩󵄩 ds)

,

t ∈ ℝ,

0

which clearly implies that 󵄩 󵄩󵄩 󵄩󵄩(Ψu)(t + h) − (Ψu)(t)󵄩󵄩󵄩 1

1/q󸀠

󵄩q󸀠 󵄩 ⩽ M(∫󵄩󵄩󵄩Tθ (s + h) − Tθ (s)󵄩󵄩󵄩 ds) 0

(105)

114 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations ∞ k+1

󵄩 󵄩 + M ∑ ∫ 󵄩󵄩󵄩Tθ (⋅ + h) − Tθ (⋅)󵄩󵄩󵄩L∞ [k,k+1] k=1 k 2h

󵄩q 󸀠

1/q󸀠

󵄩 + M(∫󵄩󵄩󵄩Tθ (s)󵄩󵄩󵄩 ds) 0

for any t ∈ ℝ. It is clear that the third term in the last expression tends to zero as h 󸀠 tends to zero because of (A) and q󸀠 (β − θ − 1) > −1, which gives that ‖Tθ (⋅)‖q ∈ L1 [0, 1]. For the second term, we can use the estimate ∞ k+1

∞

k=1 k

k=1

󵄩 󵄩 ∑ ∫ 󵄩󵄩󵄩Tθ (⋅ + h) − Tθ (⋅)󵄩󵄩󵄩L∞ [k,k+1] ⩽ Mh ∑ e−ck ,

which simply follows from (B). For the first term, we can employ the dominated convergence theorem and (A); to sum up, we have proved the right continuity of (Ψu)(⋅). The left continuity of (Ψu)(⋅) can be proved similarly, and we only still need to show the boundedness of (Ψu)(⋅). This is a consequence of the following calculus obtained with the help of Hölder inequality: ∞

∞

0

k=0

k+1

󵄩q󸀠

1/q󸀠

󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(Ψu)(t)󵄩󵄩󵄩 ⩽ ∫ 󵄩󵄩󵄩Tθ (s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s, u(t − s))󵄩󵄩󵄩 ds ⩽ M ∑ ( ∫ 󵄩󵄩󵄩Tθ (s)󵄩󵄩󵄩 ds)

,

k

which is valid for any t ∈ ℝ, and the estimate (A). Define, for every X-valued bounded continuous function u(⋅), t

(Φu)(t) := ∫ Tθ (t − s)f (s, (−𝒜)−θ u(s)) ds, t ∈ ℝ. −∞

Since (−𝒜)−θ ∈ L(X, [D((−𝒜)θ )]), we know from Lemma 2.10.2 that Φ : Cb (ℝ : X) → Cb (ℝ : X) is well-defined as long as f (⋅, ⋅) satisfies the conditions from Lemma 2.10.2. Let Lf (⋅) be a locally bounded nonnegative function, and let M denote the constant from (A), with ν = θ. Set, for every n ∈ ℕ, xn

x2

−∞ −∞ n

−∞

t

n

Mn,θ := M sup ∫ ∫ ⋅ ⋅ ⋅ ∫ e−c(t−xn ) (t − xn )β−θ−1 t∈ℝ

n

× ∏ e−c(xi −xi−1 ) (xi − xi−1 )β−θ−1 ∏ Lf (xi ) dx1 dx2 ⋅ ⋅ ⋅ dxn . i=2

i=1

(106)

Since the norm of mapping (−𝒜)−θ ∈ L(X, [D((−𝒜)θ )]) is less or equal to 1, a simple calculation yields that 󵄩󵄩 n n 󵄩 󵄩󵄩(Φ u) − (Φ v)󵄩󵄩󵄩∞ ⩽ Mn,θ ‖u − v‖∞ ,

u, v ∈ Cb (ℝ : X), n ∈ ℕ.

Now we are ready to formulate the following result:

(107)

2.10 The use of fractional powers of multivalued linear operators | 115

Theorem 2.10.3. Suppose that (P) holds, β > θ > 1 − β and the following conditions hold: (i) f ∈ APSp (ℝ × Y : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1), as well as a locally bounded nonnegative function Lf ∈ LrS (ℝ) such that r > p/p − 1 and (73) holds. pr Set q := pr/p + r and q󸀠 := pr−p−r . Assume also that: (ii) q󸀠 (β − θ − 1) > −1. (iii) Mn,θ < 1 for some n ∈ ℕ. Then there exists an almost periodic mild solution of inclusion (78). The uniqueness of mild solutions holds in the case that 𝒜 = A is single-valued. Proof. Let us recall that the range of a Y-valued almost periodic function is relatively compact in Y. Clearly, u ∈ AP(ℝ : X) implies (−𝒜)−θ u ∈ AP(ℝ : Y); after that, using condition (i) and Theorem 2.7.1, we get that f (⋅, (−𝒜)θ u(⋅)) ∈ APSq (ℝ : X). Due to assumption (ii), inequality (A) and Proposition 2.6.11 (see also Remark 2.6.12), we get that the mapping Φ|AP(ℝ:X) : AP(ℝ : X) → AP(ℝ : X) is well-defined. Since (107) and (iii) hold, we can apply Theorem 1.1.5 in order to see that the mapping Φ|AP(ℝ:X) (⋅) has a unique fixed point φ(⋅). Observing that q󸀠 (β − 1) > −1 by (ii), we can employ Propot sition 2.6.11 again to conclude that the mapping t 󳨃→ ∫−∞ T(t − s)f (s, (−𝒜)−θ φ(s)) ds, t ∈ ℝ is well-defined. Due to Theorem 1.2.2 and (104), we obtain that t

φ(t) = ∫ Tθ (t − s)f (s, (−𝒜)−θ φ(s)) ds −∞

t

∈ (−𝒜)θ ∫ T(t − s)f (s, (−𝒜)−θ φ(s)) ds, t ∈ ℝ. −∞

This implies t

(−𝒜) φ(t) = ∫ T(t − s)f (s, (−𝒜)−θ φ(s)) ds, t ∈ ℝ. −θ

−∞

Taking into account that 󵄩 󵄩 ‖(−𝒜)−θ φ(t) − (−𝒜)−θ φ(s)‖Y ⩽ 󵄩󵄩󵄩φ(t) − φ(s)󵄩󵄩󵄩,

t, s ∈ ℝ,

the mapping (−𝒜)−θ φ(⋅) is Y-continuous so that, actually, (−𝒜)−θ φ(⋅) is an almost periodic mild solution of (78) (by almost periodicity, we mean the X-almost periodicity). To complete the proof of the theorem, we need to show the uniqueness of mild solutions in the case that 𝒜 = A is single-valued and (iv) holds. Then the fractional powers constructed in Subsection 1.2.1 coincide with those constructed in [342], so that

116 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations the fractional powers (−A)±θ are injective. Assume that u(⋅) is a mild solution of (78); hence, t

u(t) = ∫ T(t − s)f (s, u(s)) ds,

t ∈ ℝ.

−∞

Then

t

t

u(t) = ∫ T(t − s)f (s, u(s)) ds = ∫ (−A)−θ Tθ (t − s)f (s, u(s)) ds −∞

−∞

t

= (−A)−θ ∫ Tθ (t − s)f (s, u(s)) ds := (−A)−θ ζ (t),

t ∈ ℝ;

−∞

observe here that the mapping ζ (⋅) is well-defined, X-continuous and X-bounded by Lemma 2.10.2, because 1/q + 1/q󸀠 = 1 and q ∈ (1, p). This implies t

(−A) ζ (t) = u(t) = (−A) −θ

−θ

∫ Tθ (t − s)f (s, u(s)) ds −∞ t

= (−A)−θ ∫ Tθ (t − s)f (s, (−A)−θ ζ (s)) ds,

t ∈ ℝ.

−∞

By the injectivity of power (−A)−θ , we get that ζ (⋅) is a fixed point of mapping Φ : Cb (ℝ : X) → Cb (ℝ : X). Hence, ζ (⋅) is uniquely determined and, because of that, u(⋅) is uniquely determined, as well. Assuming the Lipschitz condition (74) in place of (73), we can deduce the following result. Theorem 2.10.4. Suppose that (P) holds, β > θ > 1 − β and the following conditions hold: (i) f ∈ APSp (ℝ × Y : X) with p > 1, and there exists a constant L > 0 such that (74) holds. p (ii) p−1 (β − θ − 1) > −1. (iii) Mn,θ < 1 for some n ∈ ℕ. Then there exists an almost periodic mild solution of inclusion (78). The uniqueness of mild solutions holds provided that, in addition to (i)–(iii), 𝒜 is single-valued. Before proceeding further, it would be worthwhile to mention that a direct computation shows that, in the concrete situation of Theorem 2.10.4, one has M1,θ ⩽ MLΓ(β − θ)cθ−β , where M denotes the constant from (A), with ν = θ.

2.10 The use of fractional powers of multivalued linear operators | 117

2.10.2 Almost periodic solutions of abstract semilinear Cauchy inclusion (103) In this subsection, we examine the existence and uniqueness of almost periodic solutions of abstract fractional inclusion (103); cf. also [329] and [181]. In our approach, we need to have two different pivot spaces for a reasonable definition of a mild solution of (103): Definition 2.10.5. Let f : I × Y → X, and let Y be continuously embedded in X. By a mild solution of (103), we mean any Y-continuous function u(⋅) such that u(t) = (Λγ u)(t), t ∈ ℝ, where t

t 󳨃→ (Λγ u)(t) := ∫ (t − s)γ−1 Pγ (t − s)f (s, u(s)) ds, t ∈ ℝ. −∞

Set Rγ (t) := t γ−1 Pγ (t),

t>0

and

∞

Rθγ (t) := γt γ−1 ∫ sΦγ (s)Tθ (st γ )x ds,

t > 0, x ∈ X.

0

We need to prove a few auxiliary lemmas. Lemma 2.10.6. There exists a finite constant Mγθ > 0 such that 󵄩󵄩 θ 󵄩󵄩 θ γ(β−θ)−1 , 󵄩󵄩Rγ (t)󵄩󵄩 ⩽ Mγ t

t ∈ (0, 1]

and

󵄩󵄩 θ 󵄩󵄩 θ −1−γ , 󵄩󵄩Rγ (t)󵄩󵄩 ⩽ Mγ t

t ⩾ 1.

(108)

Furthermore, Rθγ (t)x ∈ (−𝒜)θ Rγ (t)x,

t > 0, x ∈ X.

(109)

Proof. By introduced definitions, we have ∞

(−𝒜)θ Rγ (t)x = γt γ−1 (−𝒜)θ [ ∫ sΦγ (s)T(st γ )x ds],

t > 0, x ∈ X.

0 θ

Since the power (−𝒜) is closed, we can apply Theorem 1.2.2 and (104) to see that (109) holds. On the other hand, straightforward computation yields ∞

γ 󵄩󵄩 θ 󵄩󵄩 θ γ(β−θ)−1 ∫ e−cst Φγ (s)sβ−θ ds, 󵄩󵄩Rγ (t)󵄩󵄩 ⩽ Mγ t

0

for any t > 0. Since Φγ (s) ∼ (Γ(1 − γ)) , s → 0+, a Tauberian type theorem [29, Proposition 4.1.4; b)] immediately implies the second estimate in (108). The first esγ ∞ timate in (108) is clear, since e−cst ⩽ 1 for t, s > 0 and the integral ∫0 Φγ (s)sβ−θ ds converges. −1

118 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Lemma 2.10.7. There exists a finite constant Mγ,θ > 0 such that 󵄩 󵄩󵄩 θ −2−γ θ , 󵄩󵄩Rγ (t + h) − Rγ (t)󵄩󵄩󵄩 ⩽ Mγ,θ ht

t ⩾ 1, h > 0.

Proof. By the mean value theorem, it suffices to show that ‖(d/dt)Rθγ (t)‖ = O(t −2−γ ), t ⩾ 1. Towards this end, observe that ∞

(d/dt)Rθγ (t)x = γ(γ − 1)t γ−2 ∫ sΦγ (s)Tθ (st γ )x ds 0

∞

− γ 2 t 2γ−2 ∫ sΦγ (s)Tθ+1 (st γ )x ds, 0

for any t ⩾ 1 and x ∈ X. Therefore, making use of estimate (A), we get ∞

β−θ−1 󵄩 󵄩󵄩 γ−2 θ −cst γ (st γ ) ds 󵄩󵄩(d/dt)Rγ (t)󵄩󵄩󵄩 ⩽ γ(1 − γ)t ∫ sΦγ (s)e 0

∞

γ

β−θ−2

+ γ 2 t 2γ−2 ∫ sΦγ (s)e−cst (st γ )

ds,

0

for any t ⩾ 1. Now the final conclusion follows from an application of [29, Proposition 4.1.4; b)]. Lemma 2.10.8. Let 1 < q, q󸀠 < ∞, 1/q + 1/q󸀠 = 1 and q󸀠 (γ(β − θ) − 1) > −1. Assume that f ̂ : I × Y → Lq ([0, 1] : X) is bounded continuous, and u ∈ C(ℝ : Y). Define t

(Ψθγ u)(t) := ∫ Rθγ (t − s)f (s, u(s)) ds,

t ∈ ℝ.

−∞

Then Ψθγ u ∈ Cb (ℝ : X). Proof. The proof of lemma is almost the same as that of Lemma 2.10.2, and the only thing that should be explained in more detail is the fact that the second term in the last inequality of (105) tends to zero as h tends to zero. This is a consequence of the following computation involving the mean value theorem and estimate (108): ∞ k+1

󵄩 󵄩 ∑ ∫ 󵄩󵄩󵄩Tθ (⋅ + h) − Tθ (⋅)󵄩󵄩󵄩Lq󸀠 [k,k+1]

k=1 k

∞

k+1

q󸀠 (−2−γ)

⩽ hMγ,θ ∑ ( ∫ s k=1

k

1/q󸀠

ds)

2.10 The use of fractional powers of multivalued linear operators | 119

∞

󸀠 󸀠 󵄨1/q󸀠 󵄨−1 󵄨 󵄨 = hMγ,θ 󵄨󵄨󵄨q󸀠 (−2 − γ) + 1󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨(k + 1)q (−2−γ)+1 − k q (−2−γ)+1 󵄨󵄨󵄨

k=1

∞

⩽ hMγ,θ ∑ (k q (−2−γ) ) 󸀠

k=1

1/q󸀠

∞

= hMγ,θ ∑ k −2−γ . k=1

Suppose that (73) holds for a. e. t > 0, with a locally bounded nonnegative function Lf (⋅). Set, for every n ∈ ℕ, xn

t

x2

󵄩 󵄩 Bn := sup ∫ ∫ ⋅ ⋅ ⋅ ∫ 󵄩󵄩󵄩Rθγ (t − xn )󵄩󵄩󵄩 t⩾0

×

−∞ −∞ n 󵄩 ∏󵄩󵄩󵄩Rθγ (xi i=2

−∞

n

󵄩 − xi−1 )󵄩󵄩󵄩 ∏ Lf (xi ) dx1 dx2 ⋅ ⋅ ⋅ dxn . i=1

(110)

Let f (⋅, ⋅) satisfy the conditions of Lemma 2.10.8. Define, for every X-valued bounded continuous function u(⋅), (Φθγ u)(t)

t

:= ∫ Rθγ (t − s)f (s, (−𝒜)−θ u(s)) ds, t ∈ ℝ. −∞

Then we know from Lemma 2.10.8 that Φθγ : Cb (ℝ : X) → Cb (ℝ : X) is well-defined. As a matter of routine, one obtains that n

n

‖(Φθγ ) u − (Φθγ ) v‖∞ ⩽ Bn ‖u − v‖∞ ,

u, v ∈ Cb (ℝ : X), n ∈ ℕ.

(111)

Keeping in mind (111) and the last three lemmas, we can repeat almost verbatim the proof of Theorem 2.10.3 (with the mapping Φ(⋅), as well as the operator families T(⋅) and Tθ (⋅), replaced by the mapping Φθγ (⋅), as well as the operator families Rγ (⋅) and Rθγ (⋅), respectively) in order to see that the following results hold true:

Theorem 2.10.9. Suppose that (P) holds, β > θ > 1 − β and the following conditions hold: (i) f ∈ APSp (ℝ × Y : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1), as well as a locally bounded nonnegative function Lf ∈ LrS (ℝ), such that r > p/p − 1 and (73) holds. pr Set q := pr/p + r and q󸀠 := pr−p−r . Assume also that: (ii) q󸀠 (γ(β − θ) − 1) > −1. (iii) Bn < 1 for some n ∈ ℕ. Then there exists an almost periodic mild solution of inclusion (103). The uniqueness of mild solutions holds provided that, in addition to (i)–(iii), 𝒜 is single-valued.

120 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Theorem 2.10.10. Suppose that (P) holds, β > θ > 1 − β and the following conditions hold: (i) f ∈ APSp (ℝ × Y : X) with p > 1, and there exists a constant L > 0 such that (74) holds. p (ii) p−1 (γ(β − θ) − 1) > −1. (iii) Bn < 1 for some n ∈ ℕ. Then there exists an almost periodic mild solution of inclusion (103). The uniqueness of mild solutions holds provided that, in addition to (i)–(iii), 𝒜 is single-valued. Direct computation of coefficient B1 , established with the help of estimate (108), shows that B1 ⩽ Mγθ L[

1 1 + ]. γ(β − θ) γ

Finally, we would like to propose the problem of transferring our results established in this section to the case where the pivot space Y is no longer [D((−𝒜)θ )] but θ the interpolation space X𝒜 .

2.11 Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations The main aim of this section is to investigate the (asymptotical) Weyl almost periodic properties of finite and infinite convolution products, as well as degenerate solution operator families. We first state the following modification of Proposition 2.6.11, stated here again with two different pivot spaces X and Y: Proposition 2.11.1. (i) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family satisfying ∞ that ∫0 ‖R(s)‖ ds < ∞. If g : ℝ → Y is bounded and (equi-)Weyl-almost periodic, then the function G(⋅), given by (68), is bounded continuous and (equi-)Weyl-almost periodic. (ii) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that M = ∑∞ k=0 ‖R(⋅)‖L∞ [k,k+1] < ∞. If g : ℝ → X is equi-Weyl-almost periodic, then the function G(⋅), given by (68), is bounded continuous and equi-Weyl-almost periodic.

2.11 Solutions of abstract Volterra integro-differential equations | 121

Proof. Part (i) will be proved only in the case when g : ℝ → Y is bounded and Weylalmost periodic. It is clear that, for every t ∈ ℝ, we have that G(t) is well-defined and ∞

󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩G(t)󵄩󵄩󵄩 ⩽ ‖g‖∞ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds. 0

The above estimate implies in particular that G(⋅) belongs to the space L1loc (ℝ : X) with all its translations, so that (34) implies the existence of the limit lim DSl [G(⋅ + τ), G(⋅)]

l→∞

for any τ ∈ ℝ. To prove the continuity of G(⋅), we can use the identity ∞

G(t) = ∫ R0 (s)g(t − s) ds,

t ∈ ℝ,

−∞

where R0 (s) := R(s)χ[0,∞) (s), s ∈ ℝ, the inequality 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩G(t + h) − G(t)󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩R0 (⋅ + h) − R0 (⋅)󵄩󵄩󵄩L1 (ℝ) ‖g‖L∞ (ℝ) ; cf. also [29, p. 22]. Let a number ε > 0 be given in advance. Then we can find two finite numbers lε > 0 and Lε > 0 such that any subinterval I of ℝ of length Lε contains a number τ ∈ I such that x+l

sup x∈ℝ

1 󵄩󵄩 󵄩 ∫ 󵄩g(t + τ) − g(t)󵄩󵄩󵄩 dt ⩽ ε, l 󵄩 x

l ⩾ lε .

(112)

It remains to prove that for any such τ we have lim DSl [G(⋅ + τ), G(⋅)] ⩽ ε,

l→∞

validity of which immediately follows once we prove that x+l

sup x∈ℝ

1 󵄩󵄩 󵄩 ∫ 󵄩G(t + τ) − G(t)󵄩󵄩󵄩 dt ⩽ Const. ε, l 󵄩 x

l ⩾ lε .

(113)

To see that (113) holds, we can argue as follows. Applying Fubini theorem and (112), we get that, for every x ∈ ℝ and l ⩾ lε , x+l

x+l ∞

x

x

1 󵄩󵄩 1 󵄩 󵄩 󵄩󵄩 󵄩 ∫ 󵄩G(t + τ) − G(t)󵄩󵄩󵄩 dt ⩽ ∫ [ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩g(t + τ − s) − g(t − s)󵄩󵄩󵄩 ds ]dt l 󵄩 l 0

122 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations ∞

x+l

0

x

󵄩1 󵄩 󵄩 󵄩 ⩽ ∫ [󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ∫ 󵄩󵄩󵄩g(t + τ − s) − g(t − s)󵄩󵄩󵄩 dt ]ds l ∞

󵄩 󵄩 ⩽ ε ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds. 0

This completes the proof of (i). For the proof of (ii), we first recall that any equi-Weylalmost periodic function must be Stepanov (Weyl, equivalently) bounded, so that our assumption M < ∞ taken together with the proof of Proposition 2.6.11 shows that function G(⋅) is well-defined and bounded on the real line. The continuity of G(⋅) can be deduced as in Proposition 3.5.3 below, while the remaining part of proof is essentially the same as of part (i). Remark 2.11.2. If the function g : ℝ → X in (ii) is only Stepanov p-bounded, then the same argumentation as above shows that function G(⋅) given by (68) is bounded and continuous. Furthermore, if the function g : ℝ → X in (ii) is Stepanov p-bounded and Weyl-almost periodic, then function G(⋅) will be bounded continuous and Weyl-almost periodic. Example 2.11.3. Let us recall that the function g(⋅) := χ[0,1/2] (⋅) is (equi)-Weyl-p-almost periodic for any p ∈ [1, ∞) but not Stepanov almost periodic. If we take X = Y = ℂ and R(t) = e−t for t > 0, then function G(⋅) will be given by G(t) = 0 for t ⩽ 0, G(t) = 1 − e−t 1/2 for 0 ⩽ t ⩽ 1/2 and G(t) = e−t ∫0 es ds for t > 1/2. It can be easily seen that this function cannot be Stepanov almost periodic. Similarly, the Heaviside function g(⋅) := χ[0,∞) (⋅) p is Wap -almost periodic for any p ∈ [1, ∞) but not equi-Weyl-1-almost periodic. Taking again X = Y = ℂ and R(t) = e−t for t > 0, the function G(⋅) will be given by G(t) = 0 for t ⩽ 0 and G(t) = 1 − e−t for t ⩾ 0. This function cannot be equi-Weyl-1-almost periodic, so that our result is, in a certain sense, optimal concerning the scale of generalized almost periodic functions. Let us remind ourselves that the Stepanov p-almost periodicity of function g(⋅) implies that function G(⋅) is almost periodic. The case p > 1 is a little bit delicate and then it seems almost inevitable to assume that the solution operator family has certain growth orders at zero and infinity. The following result has been recently proved in [189]: Theorem 2.11.4. Let 1/p + 1/q = 1 and let (R(t))t>0 ⊆ L(X, Y) satisfy Mt β−1 󵄩 󵄩󵄩 , t > 0 for some finite constants γ > 1, β ∈ (0, 1], M > 0. 󵄩󵄩R(t)󵄩󵄩󵄩L(X,Y) ⩽ 1 + tγ

(114)

Let a function g : ℝ → X be (equi-)Weyl-p-almost periodic and Weyl p-bounded, and let q(β − 1) > −1 provided that p > 1, resp. β = 1, provided that p = 1. Then the function G : ℝ → Y, defined through (68), is bounded continuous and (equi-)Weyl-p-almost periodic.

2.11 Solutions of abstract Volterra integro-differential equations | 123

Proof. We will consider the case that g(⋅) is Weyl-p-almost periodic with p > 1 and explain the main differences in the case that p = 1. Without loss of generality, we may assume that X = Y. Since we have assumed that g(⋅) is Weyl p-bounded (equivalently, Stepanov p-bounded) and q(β − 1) > −1, we can repeat literally the arguments given in the proof of Proposition 2.6.11 to deduce that G(⋅) is bounded and continuous on the real line (similar argument works also in the case p = 1). Therefore, it remains to prove that G(⋅) is Weyl-p-almost periodic. In order to do that, fix a number ε > 0. By definition, we can find a real number L > 0 such that any interval I 󸀠 ⊆ I of length L contains a point τ ∈ I 󸀠 such that there exists a number l(ε, τ) > 0 so that DpS [g(⋅ + τ), g(⋅)] ⩽ ε, l

l ⩾ l(ε, τ).

(115)

On the other hand, it is clear that 0

󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩u(t + τ) − u(t)󵄩󵄩󵄩 ⩽ ∫ 󵄩󵄩󵄩R(−s)󵄩󵄩󵄩󵄩󵄩󵄩g(s + t + τ) − g(s + t)󵄩󵄩󵄩 ds −∞

0

󵄩 󵄩 ⩽ M ∫ |s|β−1 󵄩󵄩󵄩g(s + t + τ) − g(s + t)󵄩󵄩󵄩/(1 + |s|γ ) ds,

t ∈ ℝ.

(116)

−∞

Since γ > 1 and β ∈ (0, 1], we have the existence of a positive real number ζ > 0 satisfying 1 1 0. This completes the proof of the theorem in a routine manner. Remark 2.11.5. Let 1 ⩽ p < ∞. Then any equi-Weyl-p-almost periodic function is automatically Weyl p-bounded, which seems to be still unknown for Weyl-p-almost periodic functions (see [20]). For any locally integrable function q ∈ L1loc (ℝ : X) and for any strongly continuous ∞ operator family (R(t))t>0 ⊆ L(X, Y) satisfying ∫0 ‖R(s)‖ ds < ∞, we formally set x+t

1 J(t, l) := sup{ ∫ [ l x⩾0 0

x+t−r+l

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr},

x+t−r

t > 0, l > 0.

Consider the following two conditions: lim lim J(t, l) = 0

(117)

lim lim J(t, l) = 0.

(118)

t→∞ l→∞

and l→∞ t→∞

The main purpose of subsequent proposition is to investigate the asymptotically Weyl-almost periodic properties of finite convolution product.

2.11 Solutions of abstract Volterra integro-differential equations | 125

Proposition 2.11.6. (i) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that ∞ ∫0 ‖R(s)‖ ds < ∞. If g : ℝ → X is bounded and Weyl-almost periodic, as well as q ∈ W01 ([0, ∞) : X), resp. q ∈ e − W01 ([0, ∞) : X), satisfies (117), resp. (118), then the function F(⋅), given by t

F(t) := ∫ R(t − s)[g(s) + q(s)] ds,

t ⩾ 0,

(119)

0 1 1 is in class Waap ([0, ∞) : Y), resp. Weaap ([0, ∞) : Y). (ii) Let the requirements of part (i) hold with g : ℝ → X being bounded and equiWeyl-almost periodic, as well as with function q(⋅) satisfying the same conditions 1 as in (i). Then function F(⋅), given by (119), is in class e − Waap ([0, ∞) : X), resp. 1 ee − Waap ([0, ∞) : X). (iii) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that M = ∑∞ k=0 ‖R(⋅)‖L∞ [k,k+1] < ∞. If g : ℝ → X is equi-Weyl-almost periodic and function q(⋅) satisfies the same conditions as in (i), then function F(⋅), given by (119), is 1 1 in class e − Waap ([0, ∞) : X), resp. ee − Waap ([0, ∞) : X).

Proof. We will prove only (i). By Proposition 2.11.1, G(⋅) is bounded and Weyl-almost periodic. Define t

∞

F(t) := ∫ R(t − s)q(s) ds − ∫ R(s)g(t − s) ds, t

0

t ⩾ 0.

Since H(t) = G(t) + F(t) for all t ⩾ 0, and ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ R(s)g(t − s) ds󵄩󵄩󵄩 ⩽ ‖g‖∞ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds → 0, 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 t t

t → +∞,

t

it suffices to show that the function t → L(t) := ∫0 R(t − s)q(s) ds, t ⩾ 0 is in class W01 ([0, ∞) : X), resp. e − W01 ([0, ∞) : X), provided q ∈ W01 ([0, ∞) : X), resp. q ∈ e − W01 ([0, ∞) : X), satisfies (117), resp. (118). Clearly, for every x ⩾ 0 and l > 0, we have by an elementary argument involving Fubini theorem: x+t+l

x+t+l

s

x+t

0

1 1 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds ⩽ ∫ [∫󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] ds l l x+t

x+t

⩽ ∫[ 0

x+t+l

1 󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩 ds]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l x+t

126 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations x+t+l

x+t+l

x+t

r

1 󵄩 󵄩 󵄩 󵄩 + ∫ [ ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩 ds]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr. l For the estimation of the second term, we can use the inequality x+t+l

x+t+l

1 󵄩 󵄩 󵄩 󵄩 ∫ [ ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩 ds]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l r

x+t

x+t+l ∞

1󵄩 󵄩 󵄩 󵄩 ⩽ ∫ [ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv] 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l x+t

0

∞

x+t+l

0

x+t

1 󵄩 󵄩 󵄩 󵄩 ⩽ [ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv] ⋅ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr, l

x ⩾ 0, l > 0;

therefore, since q ∈ W01 ([0, ∞) : X), resp. q ∈ e − W01 ([0, ∞) : X), we have that the second term is in the same class as well, by the uniform integrability of ‖R(⋅)‖. For the first term, it suffices to observe that condition (117), resp. (118), holds for q(⋅). Example 2.11.7. It is very simple to prove that condition (118) holds provided (R(t))t⩾0 ⊆ L(X, Y) is exponentially decaying, as well as there exists a finite constant M ⩾ 1 such that t

󵄩 󵄩 ∫ eω(t−s) 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds ⩽ M,

t ⩾ 0,

0

where ω < 0 denotes the exponential growth bound of (R(t))t⩾0 . Example 2.11.8. Assume that there exist two numbers a ∈ (0, 1) and b ∈ (1, ∞) satisfying ‖R(t)‖ ⩽ Mt −a , t ∈ (0, 1) and ‖R(t)‖ ⩽ Mt −b , t ⩾ 1, so that Proposition 2.11.6(ii)–(iii) can be applied, provided condition (117), resp. (118), holds; here, M ⩾ 1 is a finite constant. Since for any l ⩾ 1 we have x+t

1 ∫[ l 0

x+t−r+l

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr

x+t−r

x+t

M 󵄩 󵄩 ⩽ ∫ {(x + t − r + 1)1−a − (x + t − r)1−a }󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l(1 − a) 0

+

x+t

M 󵄩 󵄩 ∫ {(x + t − r + 1)1−b − (x + t − r + l)1−b }󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l(b − 1) 0

x+t

⩽

M 󵄩 󵄩 ∫ (x + t − r)−a 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l(1 − a) 0

2.11 Solutions of abstract Volterra integro-differential equations | 127 x+t

M(l − 1) 󵄩 󵄩 + ∫ (x + t − r + 1)−b 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr, l(b − 1) 0

(118) holds if the following conditions are satisfied: t (i) The mapping t 󳨃→ ∫0 (t − r)−a ‖q(r)‖ dr, t > 0 is bounded as t → +∞. t

(ii) We have limt→+∞ ∫0 (t + 1 − r)−b ‖q(r)‖ dr = 0.

These conditions hold for a substantially large class of functions q(⋅). Example 2.11.9. Assume now that (R(t))t>0 ⊆ L(X, Y) is strongly continuous and satisfies the estimate ‖R(t)‖ ⩽ Me−ct t β−1 , t > 0 for some finite constants c, β, M > 0. x+t−r+l

Dividing the integral ∫x+t−r

x+t−r+1

into two parts ∫x+t−r

x+t−r+l

and ∫x+t−r+1 , for l ⩾ 1, and estimat-

ing the integrand e−cv vβ−1 on [x + t − r, x + t − r + 1] by vβ−1 (on [x + t − r + 1, x + t − r + l] by e−cv ), as it has been done in the previous example, it can be simply verified that (118) holds if the following conditions are satisfied: t (i) The mapping t 󳨃→ ∫0 (t − r)β−1 ‖q(r)‖ dr, t > 0 is bounded as t → +∞. t

(ii) The mapping t 󳨃→ ∫0 e−c(t−r) ‖q(r)‖ dr, t > 0 is bounded as t → +∞.

As above, the analysis of asymptotically (equi-)Weyl-p-almost periodic properties of finite convolution product for p > 1 is not trivial in the general case, and we need some extra conditions on the ergodic part of the function under consideration (denoted henceforth by q(⋅)) in order to obtain any relevant result in this direction; the situation is, unfortunately, similar if the resolvent family (R(t))t>0 ⊆ L(X, Y) satisfies the estimate 󵄩 󵄩󵄩 −ct β−1 󵄩󵄩R(t)󵄩󵄩󵄩L(X,Y) ⩽ Me t , t > 0 for some finite constants c > 0, β ∈ (0, 1], M > 0, or (114). We will prove the following general proposition with regards to this question: Proposition 2.11.10. Let q ∈ Lploc ([0, ∞) : X), 1/p+1/q = 1 and (R(t))t>0 ⊆ L(X, Y) satisfy (114). Let a function g : ℝ → X be (equi-)Weyl-p-almost periodic and Weyl p-bounded, and let q(β − 1) > −1 provided that p > 1, resp. β = 1, provided that p = 1. Suppose that the function t

t 󳨃→ Q(t) ≡ ∫ R(t − s)q(s) ds,

t⩾0

0

belongs to the space ℱY , which equals to C0 ([0, ∞) : Y), S0p ([0, ∞) : Y), e − W0p ([0, ∞) : Y) or W0p ([0, ∞) : Y). Then the function t

H(t) ≡ ∫ R(t − s)[g(s) + q(s)] ds, 0

t⩾0

128 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations [0,∞),p [0,∞),p is continuous and belongs to the class (e−)Wap (Y) + ℱY , where (e−)Wap (Y) stands for the space of all restrictions of Y-valued (equi-)Weyl-p-almost periodic functions from the real line to the interval [0, ∞).

Proof. Without loss of generality, we may assume that X = Y. As many times before, denote t

∞

F(t) = ∫ R(t − s)q(s) ds − ∫ R(s)g(t − s) ds, t

0

t ⩾ 0.

By Theorem 2.11.4, the function G : ℝ → X, considered above, is bounded continuous and (equi-)Weyl-p-almost periodic. Due to the facts that H(t) = G(t) + F(t), t ⩾ 0, ℱX + C0 ([0, ∞) : X) = ℱX and our assumption that function Q(⋅) belongs to the space ℱX , it suffices to show that the mapping Q(t) is continuous for t ⩾ 0, as well as that ∞ the mapping t 󳨃→ ∫t R(s)g(t − s) ds, t ⩾ 0 is in class C0 ([0, ∞) : X). The continuity of mapping Q(t) for t ⩾ 0 can be proved as in the final part of proof of Theorem 2.11.4, by t using the equality Q(t) = ∫0 R(s)q(t − s) ds, t ⩾ 0, inclusion q ∈ Lploc ([0, ∞) : X) and

Hölder inequality. To prove that the mapping t 󳨃→ ∫t R(s)g(t − s) ds, t ⩾ 0 is in class C0 ([0, ∞) : X), observe first that ∞

∞

∞

t+1

t

t+1

t

∫ R(s)g(t − s) ds = ∫ R(s)g(t − s) ds + ∫ R(s)g(t − s) ds,

t ⩾ 0.

t+k+2

The continuity of mapping t 󳨃→ ∫t+1 R(s)g(t − s) ds = ∑∞ k=0 ∫t+k+1 R(s)g(t − s) ds := F (t), t ⩾ 0 can be shown following the lines of the proof of Proposition 3.5.3 ∑∞ k=0 k below since the mapping Fk (⋅) is continuous by the dominated convergence theorem and the series ∑∞ k=0 Fk (t) converges uniformly in t ⩾ 0 due to Weierstrass criterion. To ∞

prove the continuity of mapping ∫⋅ R(s)g(⋅ − s) ds, fix a number t ⩾ 0 and a sequence (tn )n∈ℕ in [t, t + 1] converging to t as n → +∞. Then an elementary argument involving Hölder inequality shows that ⋅+1

t+1 󵄩󵄩 󵄩󵄩 tn +1 󵄩 󵄩󵄩 󵄩󵄩 ∫ R(s)g(tn − s) ds − ∫ R(s)g(t − s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 tn t tn

⩽ M[∫ t

t+1

+∫ tn

tn +1

sβ−1 󵄩󵄩 sβ−1 󵄩󵄩 󵄩 󵄩 󵄩󵄩g(t − s)󵄩󵄩󵄩 ds + ∫ 󵄩g(t − s)󵄩󵄩󵄩 ds γ 1+s 1 + sγ 󵄩 n t+1

β−1

s 󵄩 󵄩󵄩 󵄩g(t − s) − g(t − s)󵄩󵄩󵄩 ds] 1 + sγ 󵄩 n

󵄩󵄩 ⋅β−1 󵄩󵄩 󵄩󵄩 ⋅β−1 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩 p [0,t −t] + 󵄩 ⩽ M[󵄩󵄩󵄩 ‖g‖ ‖g‖ p 󵄩 󵄩 󵄩󵄩 L n 󵄩󵄩 1 + ⋅γ 󵄩󵄩󵄩Lq [t,tn ] 󵄩󵄩 1 + ⋅γ 󵄩󵄩󵄩Lq [t+1,tn +1] L [0,tn −t]

2.11 Solutions of abstract Volterra integro-differential equations | 129

󵄩󵄩 ⋅β−1 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩 󵄩g(t − ⋅) − g(t − ⋅)󵄩󵄩󵄩Lp [0,t+2] ] 󵄩 󵄩󵄩 1 + ⋅γ 󵄩󵄩󵄩Lq [0,t+2] 󵄩 n 󵄩󵄩 ⋅β−1 󵄩󵄩 󵄩󵄩 ⋅β−1 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩󵄩 p [0,t −t] + 󵄩 ‖g‖ ‖g‖ p ⩽ M[󵄩󵄩󵄩 󵄩󵄩 󵄩 L n 󵄩󵄩 1 + ⋅γ 󵄩󵄩󵄩Lq [t+1,t+2] L [0,tn −t] 󵄩󵄩 1 + ⋅γ 󵄩󵄩󵄩Lq [t,t+1] 󵄩󵄩󵄩 ⋅β−1 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩 󵄩g(t − ⋅) − g(t − ⋅)󵄩󵄩󵄩Lp [0,t+2] ]. 󵄩󵄩 1 + ⋅γ 󵄩󵄩󵄩Lq [0,t+2] 󵄩 n The right continuity of mapping ∫⋅ R(s)g(⋅−s) ds at point t follows from the equalities limn→+∞ ‖g‖Lp [0,tn −t] = 0 and limn→+∞ ‖g(tn − ⋅) − g(t − ⋅)‖Lp [0,t+2] = 0, while the left ∞ continuity can be proved analogously. The vanishing of function t 󳨃→ ∫t R(s)g(t − s) ds, t ⩾ 0 at plus infinity follows from the estimates ⋅+1

󵄩󵄩 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩 󵄩󵄩 ∫ R(s)g(t − s) ds󵄩󵄩󵄩 ⩽ ‖g‖Sp ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩 q 󵄩󵄩 󵄩L [t+k,t+k+1] , 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k=0 t

t>0

and ∞

󵄩 󵄩 lim ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [t+k,t+k+1] = 0; t→+∞ k=0

see e. g. the proof of [273, Proposition 2.5]. [0,∞) p Remark 2.11.11. The space (e−)Wap (Y) is contained in (e−)Wap ([0, ∞) : Y). It is not clear whether an (equi-)Weyl-p-almost periodic function defined on [0, ∞) can be extended to an (equi-)Weyl-p-almost periodic function defined on ℝ. Therefore, it is not p [0,∞) clear whether (e−)Wap ([0, ∞) : Y) ⊆ (e−)Wap (Y).

Remark 2.11.12. Suppose that q ∈ C0 ([0, ∞) : X) and (R(t))t>0 ⊆ L(X, Y) satisfies (114). The argumentation contained in the proof of [11, Lemma 2.13] combined with the fact ∞ that ∫0 sβ−1 /(1 + sγ ) ds < ∞ shows that limt→+∞ Q(t) = 0, so that Q ∈ C0 ([0, ∞) : Y).

Remark 2.11.13. Suppose that q ∈ Lploc ([0, ∞) : X), 1/p + 1/q = 1, the mapping a : (0, ∞) → (0, ∞) satisfies 0 < a(t) < t, t > 0 as well as that (R(t))t>0 ⊆ L(X, Y) satisfies (114). Let p > 1, let Bp (0) := 0 and β−1−γ

Bp (t) := a(t)1/q (t − a(t)) + (t − a(t))

β−1−γ+ q1

1/p

a(t)

󵄩p 󵄩 ( ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds) t

0

1/p

󵄩p 󵄩 ( ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds)

,

t > 0.

a(t)

t

a(t)

t

Since Q(t) = ∫0 R(t − s)q(s) ds = ∫0 R(s)q(t − s) ds + ∫a(t) R(s)q(t − s) ds for all t ⩾ 0, applying the Hölder inequality and (114) we may conclude that ‖Q(t)‖Y ⩽ Bp (t) for all

130 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations t > 0. In the case that p = 1, set B1 (0) := 0 and B1 (t) := (t − a(t))

β−1−γ

a(t)

t

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds + ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds, 0

t > 0.

a(t)

Then we can similarly prove that ‖Q(t)‖Y ⩽ B1 (t) for all t > 0. This information may come in handy to describe the long time behavior of function Q(⋅). For example, the function q(t) := ∑∞ n=0 χ[n2 ,n2 +1] (t), t ⩾ 0 is not Stepanov p-vanishing but it is equi-Weyl-p-vanishing for any finite number p ⩾ 1 (using a mollification, we can simply adapt this example to construct an example of an equi-Weyl-p-vanishing function that belongs to the space C ∞ ([0, ∞)) and that is not Stepanov p-vanishing). t Furthermore, ∫0 ‖q(s)‖p ds ⩽ 2 + √t, t ⩾ 0 and we can use this estimate as well as the estimate obtained in the first part of this remark (with a(t) = t/2, t > 0) to see that Q ∈ C0 ([0, ∞) : Y), provided that the inequality p1 + β − γ < 0 holds true. The interested reader may try to construct some examples in which we have that the function Q(⋅) belongs to the class e − W0p ([0, ∞) : Y) or W0p ([0, ∞) : Y). 2.11.1 Weyl C (n) -almost periodic properties of convolution products We start this subsection by introducing the following notions (as before, the derivatives appearing below will be taken in distributional sense). Definition 2.11.14. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (I : X). (i) It is said that the function f (⋅) is equi-Weyl-p-C (n) -almost periodic, f ∈ e − C (n) − p p Wap (I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ e − Wap (I : X). p (ii) It is said that the function f (⋅) is Weyl-p-C (n) -almost periodic, f ∈ C (n) − Wap (I : X) p for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Wap (I : X).

The spaces C (n) − AAPW p ([0, ∞) : X), e − C (n) − AAPW p ([0, ∞) : X), C (n) − p AAPSW p ([0, ∞) : X), e − C (n) − AAPSW p ([0, ∞) : X), e − C (n) − Waap ([0, ∞) : X),

p p p ee − C (n) − Waap ([0, ∞) : X), C (n) − Waap ([0, ∞) : X) and C (n) − Weaap ([0, ∞) : X) are introduced similarly. In order to proceed, we provide some illustrative examples.

Example 2.11.15. Define the function f : ℝ → ℂ by f (x) := χ(−∞,0] (x) + xχ[0,1/2] (x) + χ(1/2,∞) (x),

x ∈ ℝ.

Then it is easily verified that its first distributional derivative is the locally integrable function g : ℝ → ℂ given by g(x) := χ(0,1/2) (x), x ∈ ℝ, which is not Stepanov almost periodic but equi-Weyl-1-almost periodic and therefore Weyl-1-almost periodic. Denote by H(x) := χ(0,∞) (x), x ∈ ℝ the well known Heaviside function, for which we know

2.11 Solutions of abstract Volterra integro-differential equations | 131

that it is Weyl-1-almost periodic but not equi-Weyl-1-almost periodic. Hence, in order 1 1 to see that f ∈ C (1) − Wap (ℝ : X) ∖ e − C (1) − Wap (ℝ : X), we only need to prove that the function h(x) := xχ[0,1/2] (x), x ∈ ℝ is equi-Weyl-1-almost periodic. This simply follows from definition of an equi-Weyl-1-almost periodic function and the following obvious estimates: x+l

1 󵄩󵄩 󵄩 ∫ 󵄩h(t + τ) − h(t)󵄩󵄩󵄩 dt l 󵄩 x

x+l

x+l

x

x

1 󵄩 1 󵄩 󵄩 󵄩 ⩽ ∫ 󵄩󵄩󵄩h(t + τ)󵄩󵄩󵄩 dt + ∫ 󵄩󵄩󵄩h(t)󵄩󵄩󵄩 dt l l 1/2

⩽

1 2 󵄩󵄩 󵄩 ∫ 󵄩h(t)󵄩󵄩󵄩 dt ⩽ , l 󵄩 2l

l > 0, τ ∈ ℝ.

0

1 It is clear that f ∉ C (2) − Wap (ℝ : X) since the weak derivative of g(⋅) is not a regular distribution. At the end of this example, we would like to observe that we can construct many other function spaces by using a “Sobolev type idea” from Definition 2.11.14, allowing possibly that the function and its derivatives belong to different function 1 1 spaces (here, concretely, f ∈ Wap (ℝ : X) and f 󸀠 ∈ e − Wap (ℝ : X)).

Example 2.11.16. Define R := {f ∈ L∞ (I : X) : supp(f ) is compact}. 1 Then the computation used in the former example shows that R ⊆ e − Wap (I : X). On the other hand, any nontrivial function from R cannot be Stepanov almost periodic; if we suppose on the contrary, then Theorem 2.1.1(xi) will imply t+1

t+1

󵄩 󵄩 󵄩 󵄩 sup ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds = sup ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds, t∈I

t

t⩾t0

t

t0 ∈ I; t+1

by choosing t0 arbitrarily large, the above will imply supt∈I ∫t ‖f (s)‖ ds = 0 for all t ∈ I and therefore f (s) = 0 a. e. s ∈ I. This can be easily employed for the construction 1 of a function f ∈ e − C (1) − Wap (I : X) ∖ APS1 (I : X); a typical example is given by f (x) := χ(−∞,−1) (x) + 2(x + 1)χ[−1,−1/2) (x) + χ[−1/2,1/2] (x) − 2(x − 1)χ(1/2,1) (x) + χ(1,∞) (x),

x ∈ ℝ.

Example 2.11.17. Let f (⋅) be defined by the last formula, and let f [1] (⋅) be its first anti1 derivative, defined as usual. Then it can be simply verified that f [1] ∈ C (2) − Wap (ℝ : X) 1 but f [1] ∉ e − Wap (ℝ : X).

132 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Now we would like to illustrate the importance of function spaces introduced above in the qualitative analysis of solutions of abstract inhomogeneous Volterra integro-differential equations. For our investigation of Weyl-C (n) -almost periodicity of infinite convolution products, it will be very important to introduce the following subclasses of (equi-)Weyl-C (n) -almost periodic functions. Definition 2.11.18. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (I : X). p (i) It is said that a function f (⋅) is equi-Weyl-p-Cb(n) -almost periodic, f ∈ e−Cb(n) −Wap (I : p X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ e − Wap (I : X) ∩ L∞ (I : X).

p (ii) It is said that a function f (⋅) is Weyl-p-Cb(n) -almost periodic, f ∈ Cb(n) − Wap (I : X) p for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Wap (I : X) ∩ L∞ (I : X).

Now we are ready to state the following proposition. Proposition 2.11.19. (i) Suppose that n ∈ ℕ and (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family ∞ 1 (ℝ : X), resp. such that ∫0 ‖R(s)‖ ds < ∞. If g : ℝ → X is in class e − Cb(n) − Wap

1 Cb(n) − Wap (ℝ : X), then the function G(⋅), given by (68), is in the same class as well. Furthermore, G ∈ C n (ℝ : X). (ii) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that (n) 1 − Wap (I : X), then the M = ∑∞ k=0 ‖R(⋅)‖L∞ [k,k+1] < ∞. If g : ℝ → X is in class e − C 1 function G(⋅), given by (68), is in class e − Cb(n) − Wap (I : X) ∩ C n (ℝ : X).

Proof. By Proposition 2.11.1(i), we have that G(⋅) is bounded and belongs to the class 1 1 e − Wap (ℝ : X), resp. Wap (ℝ : X). The final conclusion follows from the fact that, for 0 ⩽ l ⩽ n, we have ∞

G (t) = ∫ R(s)g (l) (t − s) ds, (l)

t ∈ ℝ,

(120)

0

which can be simply proved by applying the dominated convergence theorem. We can prove that G ∈ C n (ℝ : X) by using (120) and the corresponding part of the proof of Proposition 2.11.1(i). To prove the second part, we first observe that Proposition 2.11.1(ii) implies that G(⋅) is bounded, continuous and equi-Weyl-1-almost periodic. Therefore, it suffices to show that equation (120) holds for 0 ⩽ l ⩽ n. This follows by induction and its validity for l = 1, which is a consequence of the following computation employing Fubini theorem: ∞

G(t) = G(0) + ∫ R(v)[g(t − v) − g(−v)] dv 0

∞ t

= G(0) + ∫ ∫ R(v)g 󸀠 (s − v) ds dv 0 0

2.11 Solutions of abstract Volterra integro-differential equations | 133 t ∞

= G(0) + ∫ ∫ R(v)g 󸀠 (s − v) ds dv,

t ∈ ℝ.

0 0

We continue by stating some results about the asymptotical Weyl-C (n) -almost periodicity of the finite convolution product. For this purpose, we need to recall the following notion used above: For any function q ∈ L1loc (ℝ : X) with q(k) ∈ L1loc (ℝ : X) for 0 ⩽ k ⩽ n, and for any strongly continuous operator family (R(t))t>0 ⊆ L(X, Y) ∞ satisfying ∫0 ‖R(s)‖ ds < ∞, we formally set x+t

Jk (t, l) := sup{ ∫ [ x⩾0

0

1 l

x+t−r+l

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv]󵄩󵄩󵄩q(k) (r)󵄩󵄩󵄩 dr},

x+t−r

for t > 0, l > 0, 0 ⩽ k ⩽ n. The subsequent conditions will play an important role for us (0 ⩽ k ⩽ n): lim lim Jk (t, l) = 0

(121)

lim lim Jk (t, l) = 0.

(122)

t→∞ l→∞

and l→∞ t→∞

The main purpose of the next proposition is to enquire into the asymptotically Weyl-C (n) -almost periodic properties of the finite convolution product. Proposition 2.11.20. Let n ∈ ℕ. (i) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that ∞ 1 (ℝ : X), as well as q(k) ∈ ∫0 ‖R(s)‖ ds < ∞. If g : ℝ → X is in class e − Cb(n) − Wap

W01 ([0, ∞) : X), resp. q(k) ∈ e − W01 ([0, ∞) : X) for 0 ⩽ k ⩽ n, (121), resp. (122), holds 1 for 0 ⩽ k ⩽ n, then the function F(⋅), given by (119), is in class C (n) −Waap ([0, ∞) : Y), 1 resp. C (n) − Weaap ([0, ∞) : Y), provided that

(g + q)(k) (0) = 0,

0 ⩽ k ⩽ n − 1.

(123)

1 (ii) Let the requirements of part (i) hold with g ∈ e − Cb(n) − Wap (ℝ : X), as well as with the function q(⋅) satisfying the same conditions as in (i). Let condition (123) hold. 1 Then the function F(⋅), given by (119), is in class e − C (n) − Waap ([0, ∞) : X), resp. 1 ee − C (n) − Waap ([0, ∞) : X). (iii) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that (n) 1 − Wap (ℝ : X) and M = ∑∞ k=0 ‖R(⋅)‖L∞ [k,k+1] < ∞. If g : ℝ → X is in class e − C the function q(⋅) satisfies the same conditions as in (i), then the function F(⋅), given 1 1 by (119), is in class e − C (n) − Waap ([0, ∞) : X), resp. ee − C (n) − Waap ([0, ∞) : X), provided that condition (123) holds.

134 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Proof. The proof of the proposition follows almost immediately by applying Proposition 2.11.6 and the identity t

F (k) (t) := ∫ R(t − s)[g (k) (s) + q(k) (s)] ds,

t ⩾ 0, 0 ⩽ k ⩽ n,

0

which can be proved by using the arguments contained in the proof of [29, Proposition 1.3.6] and condition (123). Conditions (121) and (122) are satisfied for a large class of functions q(⋅), provided (R(t))t>0 is locally integrable at zero and has a certain polynomial or exponential decaying growth order at infinity; see [253] for further information. Now we will investigate the situation in which condition (123) is no longer satisfied, with the integer n ∈ ℕ given in advance. It will be always assumed that the resolvent (R(t))t>0 ⊆ L(X, Y) is locally integrable at zero and (n − 1)-times continuously differentiable for t > 0. Assuming f , f 󸀠 ∈ L1loc ([0, ∞) : X), the proofs of [29, Propositions 1.3.4 and 1.3.6] show that the function t

u(t) := ∫ R(t − s)f 󸀠 (s) ds + R(t)f (0),

t⩾0

0

is locally integrable on [0, ∞) and continuous on (0, ∞), as well as that t

(d/dt) ∫ R(t − s)f (s) ds = u(t),

t > 0.

0

Repeating this argument, we obtain inductively that, for every t > 0, t

t

0

0

k−1 dk ∫ R(t − s)f (s) ds = ∫ R(t − s)f (k) (s) ds + ∑ R(k−1−j) (t)g (j) (0), k dt j=0

provided f , f 󸀠 , . . . , f (n) ∈ L1loc ([0, ∞) : X) and 0 ⩽ k ⩽ n. Keeping in mind this equality and our previous investigation of the case in which condition (123) is satisfied, the following results can be deduced: Proposition 2.11.21. Let n ∈ ℕ. (i) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that ∞ 1 (ℝ : X), as well as q(k) ∈ ∫0 ‖R(s)‖ ds < ∞. If g : ℝ → X is in class e − Cb(n) − Wap

W01 ([0, ∞) : X), resp. q(k) ∈ e − W01 ([0, ∞) : X) for 0 ⩽ k ⩽ n, (121), resp. (122), holds 1 for 0 ⩽ k ⩽ n, then the function F(⋅), given by (119), is in class C (n) −Waap ([0, ∞) : Y), 1 resp. C (n) − Weaap ([0, ∞) : Y), provided that (R(t))t>0 is (n − 1)-times continuously

1 differentiable for t > 0 and R(k−1−j) (⋅) is pointwise in class Waap ([0, ∞) : Y), resp.

1 Weaap ([0, ∞) : Y), for any 0 ⩽ k ⩽ n and 0 ⩽ j ⩽ k − 1 such that (g + q)(j) (0) ≠ 0.

2.11 Solutions of abstract Volterra integro-differential equations | 135

1 (ii) Let the requirements of part (i) hold with g ∈ e − Cb(n) − Wap (ℝ : X), as well as with the function q(⋅) satisfying the same conditions as in (i). Let (R(t))t>0 be (n − 1)-times continuously differentiable for t > 0, and let R(k−1−j) (⋅) be pointwise in class 1 1 e − Waap ([0, ∞) : X), resp. ee − Waap ([0, ∞) : X), for any 0 ⩽ k ⩽ n and 0 ⩽ j ⩽

k − 1 such that (g + q)(j) (0) ≠ 0. Then the function F(⋅), given by (119), is in class 1 1 e − C (n) − Waap ([0, ∞) : X), resp. ee − C (n) − Waap ([0, ∞) : X). (iii) Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that (n) 1 − Wap (ℝ : X) and the M = ∑∞ k=0 ‖R(⋅)‖L∞ [k,k+1] < ∞. If g : ℝ → X is in class e − C function q(⋅) satisfies the same conditions as in (i), then the function F(⋅), given by 1 1 (119), is in class e−C (n) −Waap ([0, ∞) : X), resp. ee−C (n) −Waap ([0, ∞) : X), provided that (R(t))t>0 is (n − 1)-times continuously differentiable for t > 0 and R(k−1−j) (⋅) is 1 1 pointwise in class e − Waap ([0, ∞) : Y), resp. ee − Waap ([0, ∞) : Y), for any 0 ⩽ k ⩽ n and 0 ⩽ j ⩽ k − 1 such that (g + q)(j) (0) ≠ 0.

Remark 2.11.22. The condition that (R(t))t>0 ⊆ L(X, Y) is (n − 1)-times continuously differentiable for t > 0 holds, in particular, if (R(t))t>0 is analytic in a sector around the positive real axis (or that n = 1). Then we can apply the Cauchy integral formula to see that all derivatives of R(⋅) decay polynomially or exponentially at infinity, so that the conditions stated in the formulations of the above two propositions hold for a large class of solution operator families (R(t))t>0 examined so far. In the case p > 1, similar statements can be established by using Theorem 2.11.4 and Proposition 2.11.10 as auxiliary tools. Now we will state and prove the following result about the convolution invariance of introduced function spaces belonging to the Weyl class: 1 1 Theorem 2.11.23. Let n ∈ ℕ, and let f ∈ Cb(n) − Wap (ℝ : X) (f ∈ Wap,b (ℝ : X)), resp. 1 1 f ∈ e − Cb(n) − Wap (ℝ : X) (f ∈ e − Wap,b (ℝ : X)). Then, for every g ∈ L1 (ℝ), we have

1 1 1 that g ∗ f ∈ Cb(n) − Wap (ℝ : X) (g ∗ f ∈ Wap (ℝ : X)), resp. f ∈ e − Cb(n) − Wap (ℝ : X) 1 (f ∈ e − Wap,b (ℝ : X)).

Proof. The proof is a consequence of the following simple calculation using Fubini theorem: x+l󵄩 ∞ 󵄩󵄩󵄩 1 󵄩󵄩󵄩󵄩 󵄩 ∫ 󵄩󵄩 ∫ [f (t + τ − y) − f (t − y)]g(y) dy󵄩󵄩󵄩 dt 󵄩󵄩 l 󵄩󵄩󵄩 󵄩 x −∞ x+l ∞

⩽

1 󵄩󵄩 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩f (t + τ − y) − f (t − y)󵄩󵄩󵄩󵄩󵄩󵄩g(y)󵄩󵄩󵄩 dy dt l x −∞

∞

x+l

−∞

x

1 󵄩 󵄩 󵄩 󵄩 = ∫ [ ∫ 󵄩󵄩󵄩f (t + τ − y) − f (t − y)󵄩󵄩󵄩 dt]󵄩󵄩󵄩g(y)󵄩󵄩󵄩 dy l

136 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations

∞

x+l−y

−∞

x−y

1 󵄩 󵄩 󵄩 󵄩 = ∫ [ ∫ 󵄩󵄩󵄩f (r + τ) − f (r)󵄩󵄩󵄩 dr]󵄩󵄩󵄩g(y)󵄩󵄩󵄩 dy l x+l

∞

1 󵄩 󵄩 󵄩 󵄩 ⩽ [ ∫ 󵄩󵄩󵄩g(y)󵄩󵄩󵄩 dy] ⋅ sup[ ∫ 󵄩󵄩󵄩f (r + τ) − f (r)󵄩󵄩󵄩 dr], l x∈ℝ

−∞

x

the well-known distributional equality (g ∗ f )(n) = g ∗ f (n) and elementary definitions. It could be of some importance to examine whether we can carry over the above assertion to Besicovitch–Doss Cb(n) -almost periodic classes of functions. 2.11.2 Weyl-almost periodic properties and asymptotically Weyl-almost periodic properties of degenerate solution operator families We start this subsection with the observation that Proposition 2.5.1 can be formulated for (equi)-Weyl almost periodic functions. In order to avoid repeating, we will skip all related details. Let f : [0, ∞) → X be Weyl-p-almost periodic. As in the case of Stepanov almost periodic functions, the Bohr–Fourier coefficients t+α

1 ∫ e−irs f (s) ds t→∞ t

Pr (f ) = lim

α

exist for all r ∈ ℝ, independently of α ∈ ℝ, and the assumption Pr (f ) = 0 for all r ∈ ℝ implies that f (t) = 0 for a. e. t ∈ ℝ. In particular, f (⋅) satisfies (P1) and the argument in the proof of Theorem 2.5.5 shows that the following result holds true: Theorem 2.11.24. Let 𝒜 be the integral generator of a Weyl-p-almost periodic (a, k)-regularized C-resolvent family (R(t))t⩾0 for some p ∈ [1, ∞), let R(C) = D(𝒜) = X, and let k(0) ≠ 0. Define ℛ through (58). Suppose that k(t) and |a|(t) satisfy (P1), ̃ = 0, as well as that (59) holds. Then we have limRe z→∞ a(z) 󸀠󸀠 R R ̃ (Q) Pr x ∈ 𝒜[a(ir)P r x], r ∈ ℛ, x ∈ X and the mapping R(t)PrR x = ℒ−1 (

̃ a(ir) ̃ k(z) )(t)CPrR x, ̃ ̃ a(ir) − a(z)

t ⩾ 0, x ∈ X,

is Weyl-p-almost periodic for all r ∈ ℛ and x ∈ X. Suppose, in addition, that (60) holds. Then the set D consisting of all eigenvectors of ̃ −1 : r ∈ ℛ, a(ir) ̃ operator 𝒜 corresponding to eigenvalues λ ∈ {0} ∪ {a(ir) ≠ 0} of operator 𝒜 is total in X.

2.11 Solutions of abstract Volterra integro-differential equations | 137

Suppose now that α ∈ (0, 2) ∖ {1} and r ∈ ℝ ∖ {0}. Then the function t 󳨃→ Eα ((ir)α t α ), t ⩾ 0 is bounded and uniformly continuous so that its Weyl-p-almost periodicity for some p ∈ [1, ∞) implies its almost periodicity (see Theorem 2.3.3), which contradicts Lemma 2.6.9. Keeping this observation in mind, as well as the fact that any two Weyl-p-almost periodic functions having the same Bohr–Fourier coefficients need to be identical almost everywhere (see, e. g., [97]), appealing to Theorem 2.11.24 in place of Theorem 2.6.8 in the proof of Theorem 2.6.10 yields the following result: Theorem 2.11.25. Let C ∈ L(X) be injective, 1 ⩽ p < ∞, let A be a closed singlevalued linear operator, and let R(C) = X. Suppose that α ∈ (0, 2) ∖ {1} and A generates a Weyl-p-almost periodic (gα , C)-resolvent family (R(t))t⩾0 . Then A = 0 ∈ L(X) and R(t) = C, t ⩾ 0. Repeating literally the arguments in the proof of Proposition 2.6.16, and applying Theorem 2.3.3, we can deduce the following: Proposition 2.11.26. Suppose (S(t))t⩾0 is a bounded C-regularized semigroup with the integral generator 𝒜. If x ∈ X is such that the mapping t 󳨃→ S(t)x, t ⩾ 0 is Weyl-p-almost periodic for some p ∈ [1, ∞), then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is almost periodic. We are also in a position to clarify the following slight extension of Theorem 2.6.17: Theorem 2.11.27. Suppose that 1 ⩽ p < ∞ and (S(t))t⩾0 is a C-regularized semigroup with the integral generator 𝒜. Then the following holds: (i) Let x ∈ X be such that the mapping t 󳨃→ S(t)x, t ⩾ 0 is W p -bounded. Then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is bounded. Suppose that the mapping t 󳨃→ S(t)x, t ⩾ 0 is W p -bounded for all x ∈ X. Then we have the following: (ii) The mapping t 󳨃→ S(t)C 2 x, t ⩾ 0 is bounded and uniformly continuous for all x ∈ X, and there exists a finite constant M ⩾ 0 such that ‖S(t)C‖ ⩽ M, t ⩾ 0. Therefore, if x ∈ X is such that the mapping t 󳨃→ S(t)C 2 x, t ⩾ 0 is Weyl-p-almost periodic, then it is almost periodic. (iii) If R(C) is dense in X, then the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is bounded and uniformly continuous for all x ∈ X. Therefore, if x ∈ X is such that the mapping t 󳨃→ S(t)Cx, t ⩾ 0 is Weyl-p-almost periodic, then it is almost periodic. The class of two-parameter equi-Weyl-p-almost periodic functions has been recently introduced and analyzed by F. Bedouhene, Y. Ibaouene, O. Mellah and P. Raynaud de Fitte in [55]. A useful characterization of equi-Weyl-p-almost periodic functions established by L. I. Danilov [121] has been essentially employed in [55] for proving the composition principle for equi-Weyl-p-almost periodic functions defined on the whole real axis (as observed in [273], this result continues to hold for equi-Weyl-p-almost periodic functions defined on the nonnegative real axis). Two-

138 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations parameter (equi-)Weyl-p-almost periodic type functions and composition principles for (equi-)Weyl-p-almost periodic functions have been recently analyzed in [282] by using the approach of W. Long and H.-S. Ding [317], where the author has also considered some composition principles for the class of Stepanov p-almost periodic type functions that are not comparable with Theorem 2.7.1 and Proposition 2.7.3 (it is also worth mentioning an interesting result of S. Abbas [1], concerning composition principles for the class of Weyl pseudo almost automorphic functions). For more details about equi-Weyl-p-almost periodic solutions and asymptotically equi-Weyl-p-almost periodic solutions of abstract semilinear differential equations and inclusions, we refer the reader to [55, 273, 274] and [282] (cf. also the paper [233] by M. Kamenskii, O. Mellah and P. Raynaud de Fitte).

2.12 Pseudo-almost periodic solutions of abstract semilinear Cauchy inclusions of first order The notion of a vector-valued pseudo-almost periodic function was introduced in the doctoral dissertation of C. Zhang [412] (1992), as already mentioned in the introductory part. The main aim of this section is to enquire into the existence and uniqueness of pseudo-almost periodic solutions of abstract (semilinear) Cauchy inclusions of first order and to explain how we can similarly prove some related results for fractional relaxation semilinear inclusions with Weyl–Liouville derivatives. We start by introducing the basic notations. By PAP0 (ℝ : X) we denote the space consisting of all bounded continuous functions Φ : ℝ → X such that r

1 󵄩󵄩 󵄩 ∫ 󵄩Φ(s)󵄩󵄩󵄩 ds = 0; r→∞ 2r 󵄩 lim

−r

PAP0 (ℝ×Y : X) denotes the space consisting of all continuous functions Φ : ℝ×Y → X such that {Φ(t, y) : t ∈ ℝ} is bounded for all y ∈ Y, and r

1 󵄩󵄩 󵄩 lim ∫ 󵄩Φ(s, y)󵄩󵄩󵄩 ds = 0, r→∞ 2r 󵄩 −r

uniformly in y ∈ Y. A function f ∈ Cb (ℝ : X) is said to be pseudo-almost periodic iff it can be written as f (t) = g(t) + Φ(t), t ∈ ℝ, where g ∈ AP(ℝ : X) and Φ ∈ PAP0 (ℝ : X). The parts g(⋅) and Φ(⋅) are called the almost periodic part of f (⋅) and the ergodic perturbation of f (⋅). The vector space consisting of such functions is usually denoted by PAP(ℝ : X); the sup-norm turns PAP(ℝ : X) into a closed subspace of Banach space Cb (ℝ : X). For the purpose of research of pseudo-almost periodic properties of solutions to semilinear Cauchy inclusions, we need to recall the following well-known definitions and results (see, e. g., R. Agarwal, B. de Andrade, C. Cuevas [11]):

2.12 Pseudo-almost periodic solutions of abstract semilinear Cauchy inclusions | 139

Definition 2.12.1. A function f : ℝ × Y → X is said to be pseudo-almost periodic iff it is continuous and admits a decomposition f = g + q, where g ∈ AP(ℝ × Y : X) and q ∈ PAP0 (ℝ × Y : X). Denote by PAP(ℝ × Y : X) the vector space consisting of all pseudo-almost periodic functions. Arguing as in the well known result of H.-X. Li, F.-L. Huang and J.-Y. Li [299, Theorem 2.1], we can prove the following auxiliary lemma (recall that, in contrast to [299], we assume the boundedness of almost periodic component a priori): Lemma 2.12.2. Let f ∈ PAP(ℝ × Y : X) and h ∈ PAP(ℝ : Y). Then the mapping t 󳨃→ f (t, h(t)), t ∈ ℝ belongs to the space PAP(ℝ : X) provided that the following conditions hold: (i) The set {f (t, y) : t ∈ ℝ, y ∈ B} is bounded for every bounded subset B ⊆ Y. (ii) f (t, y) is uniformly continuous in each bounded subset of Y uniformly in t ∈ ℝ. That is, for any ε > 0 and B ⊆ Y bounded, there exists δ > 0 such that x, y ∈ B and ‖x − y‖ ⩽ δ imply ‖f (t, x) − f (t, y)‖ ⩽ ε for all t ∈ ℝ. Let condition (P) hold, and let (T(t))t>0 be the degenerate semigroup generated by the multivalued linear operator 𝒜. We need the following lemma: Lemma 2.12.3. Suppose that f : ℝ → X is pseudo-almost periodic. Then the function t F(t) := ∫−∞ T(t − s)f (s) ds, t ∈ ℝ is well-defined and pseudo-almost periodic. Proof. By definition, there exist two functions g(⋅) ∈ AP(ℝ : X) and Φ ∈ PAP0 (ℝ : X) such that f (t) = g(t)+Φ(t), t ∈ ℝ. Owing to Proposition 2.6.11, we have that the function t

t 󳨃→ ∫ T(t − s)g(s) ds,

t∈ℝ

−∞

is almost periodic so that it suffices to show that the function t

t 󳨃→ Ψ(t) := ∫ T(t − s)Φ(s) ds,

t∈ℝ

−∞

belongs to the space PAP0 (ℝ : X). This can be verified to be true on the basis of information given in the proof of [11, Lemma 2.14]; for the sake of completeness, we will include all relevant details. Due to the boundedness of function Φ(⋅), we have that there exists a finite constant M 󸀠 > 0 such that t 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ T(t − s)Φ(s) ds󵄩󵄩󵄩 ⩽ M 󸀠 ∫ e−c(t−s) (t − s)β−1 ds = M 󸀠 Γ(β)c−β , 󵄩󵄩 󵄩󵄩 󵄩󵄩−∞ 󵄩󵄩 −∞ t

t ∈ ℝ.

Since ∫−∞ T(t − s)Φ(s) ds = ∫0 T(s)Φ(t − s) ds, t ∈ ℝ, we can apply the dominated convergence theorem in order to see that function Ψ(⋅) is continuous on ℝ. Therefore, ∞

140 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations it remains to prove that r

1 󵄩󵄩 󵄩 ∫ 󵄩Ψ(s)󵄩󵄩󵄩 ds = 0. r→∞ 2r 󵄩 lim

(124)

−r

Towards this end, observe that Fubini theorem implies r

r ∞

1 󵄩󵄩 1 󵄩 󵄩󵄩 󵄩 󵄩 ∫ 󵄩Ψ(s)󵄩󵄩󵄩 ds ⩽ ∫ ∫ 󵄩󵄩󵄩T(v)󵄩󵄩󵄩󵄩󵄩󵄩Φ(s − v)󵄩󵄩󵄩 dv ds 2r 󵄩 2r −r 0 ∞

−r

⩽

M ∫ e−cv vβ−1 Φr (s) ds, 2r 0

r

where Φr (s) := 1/(2r) ∫−r ‖Φ(s − v)‖ dv, s ⩾ 0. Since Φr (⋅) is bounded on [0, ∞) and limr→∞ Φr (s) = 0, s ⩾ 0, we can apply the dominated convergence theorem again to see that (124) holds true. Let us recall that, by a mild solution of (78), we mean any continuous function u(⋅) such that u(t) = (Λu)(t), t ∈ ℝ, where t

t 󳨃→ (Λu)(t) := ∫ T(t − s)f (s, u(s)) ds,

t ∈ ℝ.

−∞

Suppose that inequality (73) holds with I = ℝ, X = Y and some bounded nonnegative function Lf (⋅). Set, for every n ∈ ℕ, xn

x2

−∞ −∞

−∞

t

Mn,∞ := M sup ∫ ∫ ⋅ ⋅ ⋅ ∫ e−c(t−xn ) (t − xn )β−1 n

t∈ℝ

n

n

× ∏ e−c(xi −xi−1 ) (xi − xi−1 )β−1 ∏ Lf (xi ) dx1 dx2 ⋅ ⋅ ⋅ dxn . i=2

i=1

Arguing as before, we have 󵄩󵄩 n n 󵄩 󵄩󵄩(Λ u) − (Λ v)󵄩󵄩󵄩∞ ⩽ Mn,∞ ‖u − v‖∞ ,

u, v ∈ Cb (ℝ : X), n ∈ ℕ.

(125)

Now we can state the following result: Theorem 2.12.4. Suppose that the following conditions hold: (i) f ∈ PAP(ℝ × X : X) is pseudo-almost periodic. (ii) Inequality (73) holds with I = ℝ, X = Y and some bounded nonnegative function Lf (⋅). (iii) ∑∞ n=1 Mn,∞ < ∞.

2.13 On Besicovitch–Doss almost periodic solutions | 141

Then there exists a unique pseudo-almost periodic solution of inclusion (78). Proof. Using Lemmas 2.12.2 and 2.12.3, we get that the mapping Λ : PAP(ℝ : X) → PAP(ℝ : X) is well-defined. Making use of (125) and Weissinger’s fixed point theorem, we obtain the existence of a unique pseudo-almost periodic mild solution of inclusion (78), as claimed. If Lf ≡ L is constant in Theorem 2.12.4, then it can be simply verified by a direct calculation that Mn,∞ ⩽ M n Ln (Γ(β)c−β )n , n ∈ ℕ. Hence, we have the following corollary: Corollary 2.12.5. Suppose that function f (⋅, u(⋅)) is pseudo-almost periodic and (73) holds with Lf ≡ L ∈ [0, cβ M −1 Γ(β)−1 ). Then there exists a unique pseudo-almost periodic solution of inclusion (78). Remark 2.12.6. In [11, Proposition 3.3], the authors have analyzed a version of Corollary 2.12.5 by assuming that the Stepanov L1 -norm of function Lf ∈ Cb (ℝ : X) is sufficiently small. A similar assertion can be proved in our framework. We close the section with the observation that we can similarly consider the pseudo-almost periodic solutions of fractional relaxation semilinear inclusion (9) as well.

2.13 On Besicovitch–Doss almost periodic solutions of abstract Volterra integro-differential equations Let X and Y denote two nontrivial complex Banach spaces. In [254], we have recently introduced the class of Besicovitch–Doss almost periodic functions in Banach spaces and analyzed the Besicovitch–Doss almost periodic properties of the infinite convot lution product t 󳨃→ ∫−∞ R(t − s)g(s) ds, t ∈ ℝ and the finite convolution product t

t 󳨃→ ∫0 R(t −s)f (s) ds, t ⩾ 0. The strongly continuous operator family (R(t))t>0 ⊆ L(X, Y) appearing above is assumed to satisfy the condition ∞

󵄩 󵄩 ∫ (1 + s)󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds < ∞, 0

which clearly implies the impossibility to apply our results from [254] to certain classes of fractional differential equations with Weyl–Liouville or Caputo derivatives. On the other hand, the results obtained easily apply to a wide class of abstract (degenerate) differential equations of first order, as well as to some abstract higher-order differential equations and abstract Volterra integro-differential inclusions. It is also worth noting that, in [254], we have introduced the class of Besicovitch-p-vanishing functions

142 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations and showed that this class reduces to the class consisting of all p-locally integrable X-valued functions whose Besicovitch seminorm is equal to zero. Let 1 ⩽ p < ∞. The class of Besicovitch-p-almost periodic functions extends the classes of Stepanov p-almost periodic functions and Weyl-p-almost periodic functions. There are several possible ways to introduce the notion of a Besicovitch-p-almost periodic function with values in Banach space; cf. also L. I. Danilov [122] for the corresponding notion in complete metric spaces. The standard procedure goes as follows. Following A. S. Besicovitch [57], for every function f ∈ Lploc (ℝ : X), we define t

‖f ‖ℳp

1 󵄩 󵄩p := lim sup[ ∫󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds] 2t t→+∞

1/p

;

−t

if f ∈

Lploc ([0, ∞)

: X), then t

‖f ‖ℳp

1 󵄩 󵄩p := lim sup[ ∫󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds] t t→+∞

1/p

.

0

Here we use the abbreviation ‖f ‖ℳp because of later J. Marcinkiewicz’s investigations of Besicovitch class [322]; see also M. A. Picardello [343]. In both cases, ‖ ⋅ ‖ℳp is a seminorm on the space ℳp (I : X) consisting of those p Lloc (I : X)-functions f (⋅) for which ‖f ‖ℳp < ∞. Denote Kp (I : X) := {f ∈ ℳp (I : X) : ‖f ‖ℳp = 0} and Mp (I : X) := ℳp (I : X)/Kp (I : X). The seminorm ‖ ⋅ ‖ℳp on ℳp (I : X) induces the norm ‖ ⋅ ‖M p on M p (I : X) under which M p (I : X) is complete; in other words, (M p (I : X), ‖ ⋅ ‖M p ) is a Banach space. Definition 2.13.1. Let 1 ⩽ p < ∞. We say that a function f ∈ Lploc (I : X) is Besicovitch-palmost periodic iff there exists a sequence of X-valued trigonometric polynomials converging to f (⋅) in (M p (I : X), ‖ ⋅ ‖M p ). The vector space consisting of all Besicovitch-p-almost periodic functions I → X will be denoted by Bp (I : X). It is well known that Bp (I : X) is a closed subspace of M p (I : X) and therefore Banach space itself, equipped with the norm ‖ ⋅ ‖M p . The Besicovitch class can be also introduced in a Bohr-like manner, by using the notion of satisfactorily uniform sets (see, e. g., [57] and [20, Definitions 5.10 and 5.11]). We will not use this approach henceforth. It is also worth noting that R. Doss [177] has analyzed the notion of scalar-valued almost periodicity in the sense of Riemann– Stepanov, Riemann–Weyl and Riemann–Besicovitch. We define the Besicovitch ‘distance’ of functions f , g ∈ Lploc (I : X) by DBp [f (⋅), g(⋅)] := ‖f − g‖ℳp ;

2.13 On Besicovitch–Doss almost periodic solutions | 143

the Besicovitch “norm” of f ∈ Lploc (I : X) is defined by ‖f ‖Bp := DBp [f (⋅), 0] := ‖f ‖ℳp . Let us recall that (see, e. g., [57, p. 73] for scalar-valued case): ‖f − g‖∞ ⩾ DSp [f (⋅), g(⋅)] ⩾ DW p [f (⋅), g(⋅)] ⩾ DBp [f (⋅), g(⋅)], l

(126)

for 1 ⩽ p < ∞, l > 0 and f , g ∈ Lploc (I : X), as well as that the assumption ‖f ‖ℳp = 0 does not imply f = 0 a. e. on I. We introduce the notion of Besicovitch–Doss-p-almost periodic function following the fundamental characterization of scalar-valued Besicovitch almost periodic functions established by R. Doss in [175, 176] (cf. also [20, pp. 160–161] for further information on the subject): Definition 2.13.2. Let 1 ⩽ p < ∞. It is said that f ∈ Lploc (I : X) is Besicovitch–Doss-palmost periodic iff the following conditions hold: (i) (Bp -boundedness) We have ‖f ‖ℳp < ∞. (ii) (Bp -continuity) We have t

1 󵄩 󵄩p lim lim sup[ ∫󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds] τ→0 t→+∞ 2t

1/p

= 0,

−t

in the case I = ℝ, resp. t

1 󵄩 󵄩p lim lim sup[ ∫󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds] τ→0+ t→+∞ t

1/p

= 0,

0

in the case I = [0, ∞). (iii) (Doss-p-almost periodicity) For every ε > 0, the set of numbers τ ∈ I for which t

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds] 2t t→+∞

1/p

< ε,

(127)

−t

in the case I = ℝ, resp. t

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds] t t→+∞

1/p

< ε,

0

in the case I = [0, ∞), is relatively dense in I. (iv) For every λ ∈ ℝ, we have that p 1/p t 󵄩 x+l l 󵄩󵄩󵄩 1 1 󵄩󵄩󵄩󵄩 󵄩󵄩 iλs lim lim sup [ ∫󵄩󵄩( ∫ − ∫)e f (s) ds󵄩󵄩 dx] = 0, 󵄩󵄩 l→+∞ t→+∞ l 2t 󵄩 󵄩 󵄩 −t 󵄩 x 0

(128)

144 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations in the case I = ℝ, resp. t

x+l

0

x

l

1 1 lim lim sup [ ∫ ‖( ∫ − ∫)eiλs f (s) ds‖p dx] l→+∞ t→+∞ l t

1/p

= 0,

0

in the case I = [0, ∞). The vector space consisting of all Besicovitch–Doss-p-almost periodic functions I → X in the sense of Definition 2.13.2 will be denoted by Bp (I : X). In the case X = ℂ, an intriguing result of R. Doss says that Bp (I : X) = Bp (I : X). The argument from [175, 176] cannot be so easily transferred to the vector-valued case. Because of this, we would like to raise the following issue: Problem. Let 1 ⩽ p < ∞, and let X be a Banach space. Is it true that Bp (I : X) = Bp (I : X) in the set-theoretical sense? If f ∈ Lploc (ℝ : X), then its restriction to the nonnegative real axis f+ ∈ Lploc ([0, ∞) : X) and it is very elementary to prove that the supposition f ∈ Bp (ℝ : X), resp. f ∈ Bp (ℝ : X) implies f+ ∈ Bp ([0, ∞) : X), resp. f+ ∈ Bp ([0, ∞) : X); see, e. g., [20, p. 153] for more details given in the scalar-valued case. It is simply said that a (Besicovitch–)Doss-1-almost periodic function is (Besicovitch–)Doss almost periodic. We continue by stating the following notion: Definition 2.13.3. It is said that q ∈ Lploc (I : X) is Besicovitch-p-vanishing iff 󵄩 󵄩 lim 󵄩󵄩q(t, ⋅)󵄩󵄩󵄩ℳp = 0.

(129)

t→∞󵄩

For any q ∈ Lploc ([0, ∞) : X), we define the function ‖q‖(⋅) ∈ Lploc ([0, ∞)) as usual. Then it is clear that q(⋅) is Weyl-p-vanishing iff 󵄩 󵄩 lim 󵄩󵄩‖q‖(t + ⋅)󵄩󵄩󵄩W p = 0,

t→+∞󵄩

while q(⋅) is Besicovitch-p-vanishing iff 󵄩 󵄩 lim 󵄩󵄩‖q‖(t + ⋅)󵄩󵄩󵄩Bp = 0.

t→+∞󵄩

Hence, (126) immediately implies that the class consisting of Besicovitch-p-vanishing functions defined on the nonnegative real axis extends the corresponding class consisting of Weyl-p-vanishing functions. As in the case of Weyl-p-almost periodicity, we can replace the limits in (129), i. e., for any q ∈ Lploc ([0, ∞) : X) we can look upon the following condition: s

1 󵄩 󵄩p lim sup lim [ ∫󵄩󵄩󵄩q(t + r)󵄩󵄩󵄩 dr] t→+∞ s s→+∞ 0

1/p

= 0.

(130)

2.13 On Besicovitch–Doss almost periodic solutions | 145

If (130) holds, then there is a positive number s0 > 0 such that s

󵄩p 󵄩 lim ∫󵄩󵄩q(t + r)󵄩󵄩󵄩 dr t→+∞ 󵄩 0

exists for all s > s0 . Unfortunately, an equi-Weyl-p-vanishing function need not satisfy the last condition; a simple counterexample is given by the function q(t) := ∑∞ n=0 χ[n2 ,n2 +1] (t), t ⩾ 0. This is the main reason why we will not analyze the class consisting of p-locally integrable X-valued functions satisfying condition (130). As mentioned before, the class of Besicovitch-p-vanishing functions is equal to the class of p-locally integrable X-valued functions whose Besicovitch seminorm is equal to zero. This basically follows from the analysis of R. Doss [175, p. 478], established for the case I = ℝ and showing that, for every q ∈ Lploc (I : X), we have ‖q(t, ⋅)‖ℳp = ‖q‖ℳp , t ∈ I. We will give another proof of this fact for the sake of completeness (with I = [0, ∞)): Proposition 2.13.4. Let 1 ⩽ p < ∞, and let q ∈ Lploc (I : X). Then ‖q(t, ⋅)‖ℳp = ‖q‖ℳp for all t ∈ I. Proof. We will examine the case I = [0, ∞) only. Since for any nonnegative function φ : ℝ → [0, ∞) one has lim sup φ(s) = lim sup φ(y), s→+∞ y⩾s

s→+∞

it suffices to show that t+y

1 󵄩 󵄩p lim sup[ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s→+∞ y⩾s y

1/p

t

y

1 󵄩 󵄩p = lim sup[ ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s→+∞ y⩾s y 0

for all t ⩾ 0. Fix such a number t. Then we have 1/p

t+y

sup[ y⩾s

1 󵄩󵄩 󵄩p ∫ 󵄩q(r)󵄩󵄩󵄩 dr] y 󵄩 t

t+y

1 󵄩 󵄩p ⩽ sup[ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] y⩾s y 0

1/p

1/p

t+y

t 1 󵄩p 󵄩 + ) ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] ⩽ sup[( t + y s(t + y) y⩾s y

0

t 1 󵄩 󵄩p ⩽ (1 + ) sup[ ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s y⩾s y 0

1/p

,

1/p

146 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations showing that t+y

lim sup[

s→+∞ y⩾s

1 󵄩󵄩 󵄩p ∫ 󵄩q(r)󵄩󵄩󵄩 dr] y 󵄩

1/p

t

y

1 󵄩 󵄩p ⩽ lim sup[ ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s→+∞ y⩾s y

1/p

.

0

For the opposite inequality, observe that t+y

sup[ y⩾s

1 󵄩󵄩 󵄩p ∫ 󵄩q(r)󵄩󵄩󵄩 dr] y 󵄩 t

1/p

t+y

t

1 󵄩 1 󵄩 󵄩p 󵄩p ⩾ sup[ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr − ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] y y⩾s y 0

0 1/p

t+y

1 󵄩 󵄩p ⩾ sup[( ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr) y y⩾s 0

1/p

1/p

t

1 󵄩 󵄩p − ( ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr) y

].

0

Since for any y ⩾ s one has t

t

0

0

1 󵄩 1 󵄩󵄩 󵄩p 󵄩p ∫󵄩q(r)󵄩󵄩󵄩 dr ⩽ ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr, y 󵄩 s the above computation yields t+y

1 󵄩 󵄩p lim sup[ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s→+∞ y⩾s y

1/p

t

1/p

t+y

1 󵄩 󵄩p ⩾ lim sup[ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s→+∞ y⩾s y 0

The final conclusion follows by noticing that t+y

1 󵄩 󵄩p lim sup[ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s→+∞ y⩾s y

1/p

0

⩾ lim sup[ s→+∞ y⩾s

t+y

1 󵄩p 󵄩 ∫ 󵄩󵄩q(r)󵄩󵄩󵄩 dr] t+y 󵄩 y

0

1 󵄩 󵄩p = lim sup [ ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] s→+∞ y⩾s+t y 0 y

1 󵄩 󵄩p = lim sup[ ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] v→+∞ y⩾v y

1/p

1/p

1/p

,

0

where the last equality follows by using the substitution v = s + t.

.

2.13 On Besicovitch–Doss almost periodic solutions | 147

Since for any nonnegative function φ : ℝ → [0, ∞) we have lim sup φ(s) = 0 ⇐⇒ lim φ(s) = 0, s→+∞

s→+∞

Proposition 2.13.4 immediately implies: Corollary 2.13.5. Let 1 ⩽ p < ∞, and let q ∈ Lploc (I : X). Then q(⋅) is Besicovitch-pvanishing iff ‖q‖ℳp = 0 iff q ∈ Kp (I : X). Let 1 ⩽ p < ∞, and let q ∈ Lploc (I : X) be Besicovitch-p-vanishing. Then

Lemma 2.2.13 yields that q ∈ Lploc (I : X) is Besicovitch-p󸀠 -vanishing for all p󸀠 ∈ [1, p]. Denote by q ∈ Lploc (ℝ : X) the even extension of function q ∈ Lploc ([0, ∞) : X) to the whole real axis. Then 󸀠

s

1 󵄩󵄩 󵄩p lim ∫ 󵄩q(r)󵄩󵄩󵄩 dr = 0, s→+∞ 2s 󵄩 −s

see [20, p. 153], so that the space PAP0 (ℝ : X), which has been investigated earlier, is a subspace of the space {q(⋅) : q(⋅) is Besicovitch-1-vanishing}. Denote by Bp0 (I : X) the set consisting of Besicovitch-p-vanishing functions. Then it can be trivially shown that Bp0 (I : X) has a linear vector structure. Many new “asymptotically almost periodic function spaces” can be defined as the sum of space Bp0 (I : X) and corresponding spaces of (Stepanov, Weyl, Doss, Hartman) almost periodic functions: such sums are not direct, in general [253]. The complete analysis is outside the scope of this book, and we only want to note that it is ridiculous to introduce the space of asymptotically Besicovitch almost periodic functions since the sum of space Bp (I : X) and Bp0 (I : X), with the meaning clear, is again the space Bp (I : X) on account of Corollary 2.13.5. What we will show is that the sum of space Bp (I : X) and Bp0 (I : X) is Bp (I : X) as well. Proposition 2.13.6. Let 1 ⩽ p < ∞. Then we have Bp (I : X) + Bp0 (I : X) = Bp (I : X). Proof. As above, we will only examine the case I = [0, ∞). It suffices to show that, for every function q ∈ Bp0 ([0, ∞) : X) and for every two real numbers τ ⩾ 0 and λ ∈ ℝ, we have t

1 󵄩 󵄩p lim lim sup[ ∫󵄩󵄩󵄩q(s + τ) − q(s)󵄩󵄩󵄩 ds] τ→0+ t→+∞ t 0

t

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩q(s + τ) − q(s)󵄩󵄩󵄩 ds] t t→+∞

1/p

= 0,

1/p

= 0,

0

and t

x+l

l

0

x

0

1 1 lim lim sup [ ∫ ‖( ∫ − ∫)eiλs q(s) ds‖p dx] l→+∞ t→+∞ l t

1/p

= 0.

148 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations The first two equalities follow almost immediately from Proposition 2.13.4 and Corollary 2.13.5, so that we only need to prove the third equality. This follows from the next computation t

x+l

l

0

x

0

1 1 [ ∫ ‖( ∫ − ∫)eiλs q(s) ds‖p dx] l t t

x+l

l

0

x

0

t

x+l

l

0

x

0

1/p

1 1 󵄩 󵄩 ⩽ [ ∫ |( ∫ + ∫)󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds|p dx] l t

1/p

l

1 1 󵄩 󵄩 󵄩 󵄩 ⩽ [ ∫ |( ∫ − ∫)󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds + 2 ∫󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds|p dx] l t p−1/p

⩽

⩽

2

l

t

0

x+l

0

x

t

x+l

l

x

0

0

0

1/p 󵄨󵄨p 2p−1/p 1 󵄨󵄨󵄨󵄨 󵄩󵄩 󵄩 󵄨󵄨 [ ∫󵄨󵄨2 ∫󵄩󵄩q(s)󵄩󵄩󵄩 ds󵄨󵄨󵄨 dx] 󵄨󵄨 l t 󵄨󵄨󵄨 󵄨 t󵄨

0

t

x+l

l

0

x

0

2p−1/p 1 󵄩 󵄩 [ ∫ |( ∫ − ∫)󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds|p dx] = l t p−1/p

2

1/p

l

0 1/p

l

0

+2

p

󵄨󵄨 󵄨󵄨 l 1 󵄩 󵄨󵄨 󵄩󵄩 p 󵄨󵄨󵄨 󵄩󵄩 󵄩󵄩 [ ∫{|( ∫ − ∫)󵄩󵄩q(s)󵄩󵄩 ds| + 󵄨󵄨2 ∫󵄩󵄩q(s)󵄩󵄩󵄩 ds󵄨󵄨󵄨 } dx] 󵄨󵄨 󵄨󵄨 t 󵄨 󵄨 l

2p−1/p 1 󵄩 󵄩 [ ∫ |( ∫ − ∫)󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds|p dx] l t +

1/p

1/p

l

󵄩 󵄩 ∫󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds 0

t

x+l

l

0

x

0

2p−1/p 1 󵄩 󵄩 [ ∫ |( ∫ − ∫)󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds|p dx] ⩽ l t l

1 󵄩 󵄩p + 2 ⋅ 2p−1/p [ ∫󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds] l

1/p

1/p

,

0

where the last estimate follows from an application of Lemma 2.2.13. The final conclusion follows from Corollary 2.13.5 and the fact that ‖q(⋅)‖ ∈ Bp0 ([0, ∞) : ℂ) = Bp (I : ℂ) satisfies the fourth equality of Definition 2.13.2. Now we will enquire into the Besicovitch–Doss almost periodic properties of finite and infinite convolution products. We start by stating the following result: Theorem 2.13.7. Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator ∞ family such that ∫0 (1 + s)‖R(s)‖ ds < ∞. If g : ℝ → X is bounded and Besicovitch–

2.13 On Besicovitch–Doss almost periodic solutions | 149

Doss almost periodic, then the function G(⋅), given by (68), is bounded, continuous and Besicovitch–Doss almost periodic as well. Proof. Since G(t) = ∫0 R(s)g(t − s) ds, t ⩾ 0, it is evident that, for every t ∈ ℝ, we have ∞

that G(t) is well-defined and ‖G(t)‖ ⩽ ‖g‖∞ ∫0 ‖R(s)‖ ds. In particular, G(⋅) and all its translations are both locally integrable and B1 -bounded. The continuity of G(⋅) can be proved as many times before. Now we will verify that G(⋅) is Doss almost periodic. Let a number ε > 0 be given in advance. Then we can find a finite number lε > 0 such that any subinterval I of ℝ of length lε contains a number τ ∈ ℝ such that (127) holds with function f (⋅) replaced therein by g(⋅). Then, for every t > 0, we have: ∞

t

t ∞

1 󵄩󵄩 1 󵄩 󵄩 󵄩 󵄩 󵄩 ∫󵄩G(s + τ) − G(s)󵄩󵄩󵄩 ds ⩽ ∫ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩g(s + τ − v) − g(s − v)󵄩󵄩󵄩 dv ds 2t 󵄩 2t −t 0

−t

∞

t

0

−t

󵄩 󵄩 1 󵄩󵄩 󵄩 = ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ ∫󵄩g(s + τ − v) − g(s − v)󵄩󵄩󵄩 ds dv 2t 󵄩 ∞

t−v

0

−t−v

󵄩 󵄩 󵄩 1 󵄩 = ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ ∫ 󵄩󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds dv 2t t

∞

󵄩 󵄩 1 󵄩󵄩 󵄩 ⩽ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ ∫󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds dv 2t 󵄩 0

−t

∞

−t

0

−t−v

󵄩 1 󵄩 󵄩 󵄩 + ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ ∫ 󵄩󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds dv 2t t

∞

󵄩 󵄩 󵄩 1 󵄩 + ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ ∫ 󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds dv 2t 󵄩 0

t

∞

t−v

󵄩 󵄩 1 󵄩󵄩 󵄩 ⩽ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ ∫󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds dv 2t 󵄩 0

−t

∞

+

2‖g‖∞ 󵄩 󵄩 ∫ v󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv. t 0

Doss almost periodicity of G(⋅) immediately follows from the last inequality of previous ∞ calculation, Doss almost periodicity of g(⋅) and the fact that ∫0 v‖R(v)‖ dv < ∞. We proceed by proving that the fourth condition in Definition 2.13.2 holds true for G(⋅). Let λ ∈ ℝ and ε > 0 be fixed. Then we know that ∃l0 (ε) > 0 ∀l ⩾ l0 (ε) ∃tl > 0 ∀t > tl :

t 󵄩 x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩󵄩󵄩 ∫󵄩󵄩( ∫ − ∫)eiλs g(s) ds󵄩󵄩󵄩 dx < ε/5. 󵄩󵄩 l 2t 󵄩󵄩󵄩 󵄩 x −t 0

(131)

150 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Furthermore, t 󵄩 x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩󵄩󵄩 ∫󵄩󵄩( ∫ − ∫)eiλs G(s) ds󵄩󵄩󵄩 dx 󵄩󵄩 l 2t 󵄩󵄩󵄩 󵄩 x −t 0

=

t 󵄩 x+l l ∞ 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩󵄩󵄩 ∫󵄩󵄩( ∫ − ∫)eiλs ∫ R(v)g(s − v) dv ds󵄩󵄩󵄩 dx 󵄩 󵄩󵄩 l 2t 󵄩󵄩 󵄩 x −t 0 0

t 󵄩 ∞ x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩󵄩󵄩 = ∫󵄩󵄩 ∫ ( ∫ − ∫)eiλs R(v)g(s − v) ds dv󵄩󵄩󵄩 dx 󵄩󵄩 l 2t 󵄩󵄩󵄩 󵄩 x −t 0 0

t 󵄩∞ x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩󵄩󵄩 iλs = ∫󵄩 ∫ R(v)( ∫ − ∫)e g(s − v) ds dv󵄩󵄩󵄩 dx 󵄩󵄩 l 2t 󵄩󵄩󵄩󵄩 󵄩 x −t 0 0

⩽

=

t ∞ 󵄩󵄩 x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩 󵄩󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩( ∫ − ∫)eiλs g(s − v) ds󵄩󵄩󵄩 dv dx 󵄩󵄩 󵄩󵄩 l 2t 󵄩 x 0 󵄩 −t 0

t ∞ 󵄩󵄩 x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩 󵄩󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩( ∫ − ∫)eiλ(s−v) g(s − v) ds󵄩󵄩󵄩 dv dx 󵄩 󵄩󵄩 l 2t 󵄩󵄩 x 󵄩 −t 0 0

t ∞ 󵄩󵄩 󵄩󵄩 x+l−v l−v 󵄩󵄩 1 1 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 = ∫ ∫ 󵄩󵄩R(v)󵄩󵄩 ⋅ 󵄩󵄩( ∫ − ∫ )eiλs g(s) ds󵄩󵄩󵄩 dv dx 󵄩󵄩 󵄩 l 2t 󵄩󵄩 x−v −v 󵄩 −t 0

t ∞ 󵄩󵄩 x+l l 1 1 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 = ∫ ∫ 󵄩󵄩R(v)󵄩󵄩 ⋅ 󵄩󵄩( ∫ − ∫) 󵄩󵄩 l 2t 󵄩 x 0 −t 0

x−v+l l x 0 󵄩󵄩 󵄩󵄩 + (− ∫ + ∫ + ∫ − ∫ )eiλs g(s) ds󵄩󵄩󵄩 dv dx 󵄩󵄩 󵄩 x−v −v+l −v x+l

⩽

t ∞ 󵄩󵄩 x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩 󵄩󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩( ∫ − ∫)eiλs g(s) ds󵄩󵄩󵄩 dv dx 󵄩󵄩 󵄩󵄩 l 2t 󵄩 x 0 󵄩 −t 0 t ∞

1 1 󵄩 󵄩 + ∫ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 ⋅ 4‖g‖∞ v dv dx l 2t −t 0

󵄩󵄩 x+l l 󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 iλs ⩽ ∫ ∫ 󵄩󵄩R(v)󵄩󵄩 ⋅ 󵄩󵄩( ∫ − ∫)e g(s) ds󵄩󵄩󵄩 dv dx 󵄩󵄩 󵄩󵄩 l 2t 󵄩 x 0 󵄩 −t 0 t ∞

∞

4‖g‖∞ 󵄩 󵄩 + ∫ v󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv, l 0

t > 0.

2.13 On Besicovitch–Doss almost periodic solutions | 151

Setting l0 (ε)󸀠 := max{l0 (ε), 5ε−1 ‖g‖∞ ∫0 v‖R(v)‖ dv}, we have that (131) holds for any l ⩾ l0 (ε)󸀠 and the same tl > 0. The B1 -continuity of G(⋅) follows from the calculation used in proving Doss almost periodicity of G(⋅) and an elementary argument. ∞

For any strongly continuous operator family (R(t))t>0 ⊆ L(X, Y) satisfying ∫0 (1 + s)‖R(s)‖ ds < ∞ and any function q ∈ L1loc ([0, ∞) : X), the following condition holds: ∞

t

t+l

0

t

1 󵄩 󵄩 󵄩 󵄩 lim lim sup ∫[ ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩 ds]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr = 0, t→+∞ l→+∞ l because for any t > 0 we have t

t+l

0

t

1 󵄩 󵄩 󵄩 󵄩 ∫[ ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩 ds]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr = 0; l→+∞ l lim

this follows from the estimate t

t+l

0

t

1 󵄩 󵄩 󵄩 󵄩 ∫[ ∫ 󵄩󵄩󵄩R(s − r)󵄩󵄩󵄩 ds]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l ∞

t

0

0

1 󵄩 󵄩 󵄩 󵄩 ⩽ [ ∫ (1 + s)󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds] ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr, l

t, l > 0.

Arguing similarly as in the proof of Proposition 2.11.6, we can deduce the following result: Theorem 2.13.8. Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator ∞ family such that ∫0 (1 + s)‖R(s)‖ ds < ∞. If g : ℝ → X is bounded and Besicovitch– Doss almost periodic, as well as q ∈ L1loc ([0, ∞) : X) is Besicovitch-1-vanishing, then the function F(⋅), given by (119), is Besicovitch–Doss almost periodic as well. If p > 1, then it seems that the condition ∫0 (1 + s)‖R(s)‖ ds < ∞ is not sufficient to ensure the invariance of Doss-p-almost periodicity and Besicovitch–Doss-p-almost periodicity under the action of infinite convolution product (see Theorem 2.13.7 for the case p = 1, where we assume that the coefficient g : ℝ → X is bounded and Besicovitch–Doss almost periodic). In what follows, we will present our recent research results obtained in this direction, in which we have assumed the special growth rate of solution operator family (R(t))t>0 ⊆ L(X, Y) under our consideration (see [283]). By Dp (I : X) we denote the class consisting of all Doss-p-almost periodic functions I → X; set (+∞)a := +∞ for any finite number a > 0. We will use the following elementary lemma: ∞

152 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Lemma 2.13.9. Suppose that 1 ⩽ p < ∞ and φ : ℝ → [0, ∞) is a nonnegative function. Then we have lim sup[φ(s)1/p ] = [lim sup φ(s)] s→+∞

s→+∞

1/p

.

Proof. Clearly, with the common consent introduced above, we have lim sup[φ(s)1/p ] = lim sup[φ(y)1/p ] s→+∞ y⩾s

s→+∞

1/p

= lim [sup φ(y)] s→+∞ y⩾s

1/p

= [ lim sup φ(y)] s→+∞ y⩾s

1/p

= [lim sup φ(s)] s→+∞

.

Now we are ready to state the following result: Theorem 2.13.10. Let 1/p+1/q = 1 and let (R(t))t>0 ⊆ L(X, Y) satisfy (114). Let a function g : ℝ → X be Doss-p-almost periodic and Stepanov p-bounded, and let q(β − 1) > −1 provided that p > 1, resp. β = 1, provided that p = 1. Then the function G : ℝ → Y, defined through (68), is bounded, continuous and Doss-p-almost periodic. Furthermore, if g(⋅) is Bp -continuous, then G(⋅) is Bp -continuous as well. Proof. We primarily analyze the case that g(⋅) is Doss-p-almost periodic with p > 1 and explain certain differences in the proof provided that p = 1; the assumption X = Y can be made. Since g(⋅) is Stepanov p-bounded and q(β−1) > −1, a similar line of reasoning as in the proof of Theorem 2.11.4 shows that G(⋅) is bounded and continuous on the real line. Now we will prove that G(⋅) is Doss-p-almost periodic. Let a number ε > 0 be fixed. By definition, we can find a real number l > 0 such that any interval I ⊆ ℝ of length l contains a point τ ∈ I such that (127) holds with f = g therein. Furthermore, there exists of a positive real number ζ > 0 satisfying p1 < ζ < p1 +γ −β (in the case p = 1, take any number ζ ∈ (1, γ) and repeat the procedure). As in the proof of Theorem 2.11.4, we may conclude that the integral 0

p 󵄩p 󵄩 ∫ 󵄩󵄩󵄩g(s + t + τ) − g(s + t)󵄩󵄩󵄩 /(1 + |s|ζ ) ds

−∞

converges for any t ∈ ℝ, as well as that there exists an absolute constant D > 0 such that 0

󵄩 󵄩󵄩 󵄩p 󵄩 ζ p 󵄩󵄩G(s + τ) − G(s)󵄩󵄩󵄩 ⩽ D[ ∫ 󵄩󵄩󵄩g(v + s + τ) − g(v + s)󵄩󵄩󵄩 /(1 + |v| ) dv] −∞

Using this estimate, Fubini theorem and Lemma 2.13.9, we get that t

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩G(s + τ) − G(s)󵄩󵄩󵄩 ds] 2t t→+∞ −t

1/p

1/p

,

s ∈ ℝ.

2.13 On Besicovitch–Doss almost periodic solutions | 153 t

1/p

0

1 p 󵄩p 󵄩 ⩽ D lim sup[ ∫ ∫ 󵄩󵄩󵄩g(v + s + τ) − g(v + s)󵄩󵄩󵄩 /(1 + |v|ζ ) dv ds] 2t t→+∞ 0

= D lim sup[ ∫ t→+∞

−∞ 0

−t −∞ t

1 󵄩󵄩 p 󵄩p ∫󵄩g(v + s + τ) − g(v + s)󵄩󵄩󵄩 /(1 + |v|ζ ) ds dv] 2t 󵄩 −t t

1 󵄩󵄩 p 󵄩p = D[lim sup ∫ ∫󵄩󵄩g(v + s + τ) − g(v + s)󵄩󵄩󵄩 /(1 + |v|ζ ) ds dv] 2t t→+∞ −∞

1/p

1/p

.

−t

By the reverse Fatou lemma, the above implies t

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩G(s + τ) − G(s)󵄩󵄩󵄩 ds] 2t t→+∞ −t

1/p

0

t

1 󵄩 1 󵄩p lim sup{ ∫󵄩󵄩g(v + s + τ) − g(v + s)󵄩󵄩󵄩 ds} dv] ⩽ D[ ∫ (1 + |v|ζ )p t→+∞ 2t 󵄩 −∞

1/p

−t

Keeping in mind Proposition 2.13.4, we get that, for every v ⩽ 0, t

t

−t

−t

1 󵄩 1 󵄩 󵄩p 󵄩p lim sup ∫󵄩󵄩󵄩g(v + s + τ) − g(v + s)󵄩󵄩󵄩 ds = lim sup ∫󵄩󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds, 2t 2t t→+∞ t→+∞ so that t

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩G(s + τ) − G(s)󵄩󵄩󵄩 ds] 2t t→+∞ −t

0

⩽ D[ ∫ −∞

1/p

t

1 󵄩 1 󵄩p lim sup{ ∫󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds} dv] (1 + |v|ζ )p t→+∞ 2t 󵄩 −t

t

1 󵄩 󵄩p = D[lim sup ∫󵄩󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds] t→+∞ 2t

1/p

−t

0

dv [∫ ] (1 + |v|ζ )p

1/p

1/p

.

−∞

Implementing again Lemma 2.13.9, we finally get t

1 󵄩 󵄩p lim sup[ ∫󵄩󵄩󵄩G(s + τ) − G(s)󵄩󵄩󵄩 ds] 2t t→+∞ −t

t

1/p

1 󵄩 󵄩p ⩽ D lim sup[ ∫󵄩󵄩󵄩g(s + τ) − g(s)󵄩󵄩󵄩 ds] 2t t→+∞ 0

⩽ Dε[ ∫ −∞

−t

dv ] (1 + |v|ζ )p

1/p

.

1/p

0

1/p

dv [∫ ] (1 + |v|ζ )p −∞

.

154 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations This shows that G(⋅) is Doss-p-almost periodic. Since the above computations hold for each τ ∈ ℝ, it is clear that the Bp -continuity of function g(⋅) implies that of G(⋅). The proof of the theorem is thereby complete. Concerning the finite convolution product, we state the following result which immediately follows from Theorem 2.13.10 and the proof of Proposition 2.11.10: Proposition 2.13.11. Let q ∈ Lploc ([0, ∞) : X), 1/p+1/q = 1 and let (R(t))t>0 ⊆ L(X, Y) satisfy (114). Let a function g : ℝ → X be Doss-p-almost periodic and Stepanov p-bounded, and let q(β − 1) > −1 provided that p > 1, resp. β = 1, provided that p = 1. Suppose that the function t

t 󳨃→ Q(t) ≡ ∫ R(t − s)q(s) ds, t ⩾ 0

(132)

0

belongs to some space ℱY of functions [0, ∞) → Y, satisfying that ℱY + C0 ([0, ∞) : Y) = ℱY .

(133)

Then the function F(⋅), defined through (119), is continuous and belongs to the class D[0,∞),p (Y) + ℱY , where D[0,∞),p (Y) stands for the space of all restrictions of Y-valued Doss-p-almost periodic functions from the real line to the interval [0, ∞). Concerning the Besicovitch–Doss-p-almost periodic functions, the situation is a bit worse. It seems that estimate (114) alone is not sufficient enough to ensure the validity of condition (128) for the function G(⋅) defined through (68). In the following theorem, we impose the additional condition (134) for the inhomogeneity g(⋅) under our consideration to overcome this difficulty: Theorem 2.13.12. Let 1/p+1/q = 1 and let (R(t))t>0 ⊆ L(X, Y) satisfy (114). Let a function g : ℝ → X be Besicovitch–Doss-p-almost periodic and Stepanov p-bounded, and let q(β − 1) > −1 provided that p > 1, resp. β = 1, provided that p = 1. Suppose, additionally, that for every λ ∈ ℝ and ε > 0 there exists a number l0 > 0 such that 󵄩 l l−v 󵄩󵄩p 󵄩󵄩 1 󵄩󵄩󵄩󵄩 iλs 󵄩󵄩(∫ − ∫ )e g(s) ds󵄩󵄩󵄩 < ε, 󵄩 󵄩󵄩 l 󵄩󵄩 󵄩 0

l ⩾ l0 , v ⩾ 0.

(134)

−v

Then the function G : ℝ → Y, defined through (68), is bounded continuous and Besicovitch–Doss-p-almost periodic. Proof. Let λ ∈ ℝ. By Theorem 2.13.10, we only need to show that the function G(⋅) satisfies (128). In order to do that, choose a positive real number ζ > 0 satisfying p1
such that (134) holds. Furthermore, equation (128) holds with f = g so that there exists l1 > 0 such that 1/p ∞ t t 󵄩 x+l l 󵄩󵄩p 󵄩󵄩 1 1 1 󵄩󵄩󵄩 dv ] 0 ⊆ L(X, Y) satisfy (114). Let a function g : ℝ → X be Besicovitch–Doss-p-almost periodic and Stepanov p-bounded, and let q(β − 1) > −1 provided that p > 1, resp. β = 1, provided that p = 1. Suppose, additionally, that for every λ ∈ ℝ and ε > 0 there exists a number l0 > 0 such that (134) holds as well as that the function Q(⋅) defined through (132) belongs to some space ℱY of functions [0, ∞) → Y, such that (133) holds. Then the function F(⋅), defined through (119), is continuous and belongs to the class B[0,∞),p (Y) + ℱY , where B[0,∞),p (Y) stands for the space of all restrictions of Y-valued Besicovitch–Doss-p-almost periodic functions from the real line to the interval [0, ∞). Remark 2.13.14. Suppose that ℱY = Bp0 ([0, ∞) : Y). Then the resulting function F(⋅) belongs to the space B[0,∞),p (Y), in the case of consideration Proposition 2.13.13, resp. to the space D[0,∞),p (Y), in the case of consideration Proposition 2.13.11. As in Remark 2.11.11, we can raise the issue whether a (Besicovitch–)Doss-p-almost periodic function defined on [0, ∞) can be extended to a (Besicovitch–)Doss-p-almost periodic function defined on ℝ, as well. 2.13.1 Besicovitch–Doss C (n) -almost periodic solutions of abstract inhomogeneous Cauchy inclusions We introduce the notions of a Besicovitch C (n) -almost periodic function and a Besicovitch–Doss C (n) -almost periodic function similarly as in Definition 2.11.14.

2.13 On Besicovitch–Doss almost periodic solutions | 157

Definition 2.13.15. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (I : X).

(i) It is said that a function f (⋅) is Besicovitch-p-C (n) -almost periodic, f ∈ C (n) − Bp (I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Bp (I : X).

(ii) It is said that a function f (⋅) is Besicovitch–Doss-p-C (n) -almost periodic, f ∈ C (n) − Bp (I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Bp (I : X).

We also need the following notion, which is much like the corresponding one al-

ready introduced in Definition 2.11.18.

Definition 2.13.16. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (I : X). (i) It is said that a function f (⋅) is Besicovitch-p-Cb(n) -almost periodic, f ∈ Cb(n) − Bp (I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Bp (I : X) ∩ L∞ (I : X).

(ii) It is said that a function f (⋅) is Besicovitch–Doss-p-Cb(n) -almost periodic, f ∈ Cb(n) − Bp (I : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Bp (I : X) ∩ L∞ (I : X).

Keeping in mind Theorem 2.13.7 and the proof of the first part of Proposition 2.11.19,

we can establish the following result:

Proposition 2.13.17. Suppose that n ∈ ℕ and (R(t))t>0 ⊆ L(X, Y) is a strongly con-

tinuous operator family such that ∫0 (1 + s)‖R(s)‖ ds < ∞. If g : ℝ → X is in class ∞

Cb(n) − B1 (I : X), then the function G(⋅), given by (68), is in the same class as well. Furthermore, G ∈ C n (ℝ : X).

Using Theorem 2.13.18 in place of [253, Proposition 5.3] leads us to the following

result concerning the finite convolution products:

Proposition 2.13.18. Suppose that n ∈ ℕ, and (R(t))t>0 ⊆ L(X, Y) is a strongly con-

tinuous operator family such that ∫0 (1 + s)‖R(s)‖ ds < ∞. If g : ℝ → X is in class ∞

Cb(n) − B1 (I : X), as well as q(k) ∈ L1loc ([0, ∞) : X) is Besicovitch-1-vanishing for 0 ⩽ k ⩽ n,

then the function F(⋅), given by (119), is in class f ∈ C (n) − B1 (I : X), as long as condition

(123) holds.

If (123) is not satisfied, then we have the following result: Proposition 2.13.19. Suppose that n ∈ ℕ, and (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that ∫0 (1 + s)‖R(s)‖ ds < ∞. If g : ℝ → X is in class ∞

Cb(n) − B1 (I : X), as well as q(k) ∈ L1loc ([0, ∞) : X) is Besicovitch-1-vanishing for 0 ⩽ k ⩽ n,

then the function F(⋅), given by (119), is in class f ∈ C (n) − B1 (I : X), as long as (R(t))t>0

is (n − 1)-times continuously differentiable for t > 0 and R(k−1−j) (⋅) is pointwise in class

B1 (I : Y) for any 0 ⩽ k ⩽ n and 0 ⩽ j ⩽ k − 1 such that (g + q)(j) (0) ≠ 0.

158 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations We leave to the interested readers the problem of transferring the statements established in this subsection for the solution operator families (R(t))t>0 ⊆ L(X, Y) satisfying condition (114) (the case p > 1).

2.14 Generalized almost periodic solutions and generalized asymptotically almost periodic solutions of inhomogeneous evolution equations First of all, we will recall some basic definitions from the theory of evolution equations, focusing our attention especially on hyperbolic evolution systems and Green’s functions. For more details on the subject, we refer the reader to [7, 8, 50, 51, 99, 107, 321, 362] and [381]. Definition 2.14.1. A family {U(t, s) : t ⩾ s, t, s ∈ ℝ} of bounded linear operators on X is said to be an evolution system iff the following holds: (a) U(s, s) = I, U(t, s) = U(t, r)U(r, s) for t ⩾ r ⩾ s and t, r, s ∈ ℝ; (b) {(τ, s) ∈ ℝ2 : τ > s} ∋ (t, s) 󳨃→ U(t, s)x is continuous for any fixed element x ∈ X. Unless stated otherwise, it will be assumed that the family A(⋅) satisfies the following condition introduced by P. Acquistapace and B. Terreni in [8] (with ω = 0): (H1) There is a real number ω ⩾ 0 such that the family of closed linear operators A(t), t ∈ ℝ acting on X satisfies Σϕ ⊆ ρ(A(t) − ω), −1 󵄩 󵄩󵄩 󵄩󵄩R(λ : A(t) − ω)󵄩󵄩󵄩 = O((1 + |λ|) ), t ∈ ℝ, λ ∈ Σϕ , and 󵄩 󵄩󵄩 μ −ν 󵄩󵄩(A(t) − ω)R(λ : A(t) − ω)[R(ω : A(t)) − R(ω : A(s))]󵄩󵄩󵄩 = O(|t − s| |λ| ),

for any t, s ∈ ℝ, λ ∈ Σϕ , where ϕ ∈ (π/2, π), 0 < μ, ν ⩽ 1 and μ + ν > 1. Then there exists an evolution system U(⋅, ⋅) generated by A(⋅), satisfying the following: 1. U(⋅, s) ∈ C 1 ((s, ∞) : L(X)) for all s ∈ ℝ; 2. 𝜕t U(t, s) = A(t)U(t, s), s ∈ ℝ, t > s; 3. ‖A(t)k U(t, s)‖ ⩽ Const. ⋅ (t − s)−k , t > s, k ∈ ℕ0 ; 4. ‖A(t)U(t, s)R(ω : A(s))‖ ⩽ Const., t > s; 5. ‖U(t, s)(ω − A(s))α x‖ ⩽ Const. ⋅ (μ − α)−1 (t − s)−α ‖x‖, for 0 < t − s ⩽ 1, k = 0, 1, 0 ⩽ α < ν, x ∈ D((ω − A(s))α ); 6. 𝜕s+ U(t, s)x = −U(t, s)A(s)x, for s ∈ ℝ, t > s, x ∈ D(A(s)) and A(s)x ∈ D(A(s)). It will be also assumed that the following nontrivial condition holds (as pointed out in [321], it is very difficult to find appropriate conditions on A(⋅) implying the validity of this condition; see [321] and [362] for further information in this direction):

2.14 Generalized (asymptotically) almost periodic solutions | 159

(H2) The evolution system U(⋅, ⋅) generated by A(⋅) is hyperbolic (or, equivalently, has exponential dichotomy), i. e., there exist a family of projections (P(t))t∈ℝ ⊆ L(X), being uniformly bounded and strongly continuous in t, and constants M 󸀠 , ω > 0 such that the following holds, with Q := I − P and Q(⋅) := I − P(⋅): (a) U(t, s)P(s) = P(t)U(t, s) for all t ⩾ s; (b) the restriction UQ (t, s) : Q(s)X → Q(t)X is invertible for all t ⩾ s (here we set UQ (s, t) = UQ (t, s)−1 ); (c) ‖U(t, s)P(s)‖ ⩽ M 󸀠 e−ω(t−s) and ‖UQ (s, t)Q(t)‖ ⩽ M 󸀠 e−ω(t−s) for all t ⩾ s. If the choice P(t) = I for all t ∈ ℝ is possible, then U(⋅, ⋅) is called exponentially stable, while U(⋅, ⋅) is said to be (bounded) exponentially bounded iff there exist real constants M > 0 and (ω = 0) ω ∈ ℝ such that ‖U(t, s)P(s)‖ ⩽ Me−ω(t−s) for all t ⩾ s. In our concrete situation, (H1) and (H2) hold so that the constructed evolution system U(⋅, ⋅) is bounded by point 3 above, with k = 0. We define the associated Green’s function Γ(⋅, ⋅) by U(t, s)P(s), Γ(t, s) := { −UQ (t, s)Q(s),

t ⩾ s, t, s ∈ ℝ, t < s, t, s ∈ ℝ.

Let M 󸀠 be as in the formulation of (H2). Then we have that 󵄩 󵄩󵄩 󸀠 −ω|t−s| , 󵄩󵄩Γ(t, s)󵄩󵄩󵄩 ⩽ M e

t, s ∈ ℝ

(135)

and the function +∞

u(t) := ∫ Γ(t, s)f (s) ds,

t∈ℝ

−∞

is a unique mild solution of the abstract Cauchy problem u󸀠 (t) = A(t)u(t) + f (t),

t ∈ ℝ,

(136)

i. e., u(⋅) is a unique bounded continuous function on ℝ satisfying t

u(t) = U(t, s)u(s) + ∫ U(t, τ)f (τ) dτ, s

t ⩾ s;

see, e. g., [321] and [132, Lemma 9.11, p. 234]. In anything that follows in this section, it will be assumed that the condition [321, (H3)] holds. It is worth noting that condition (H3) of this section and condition (H3e) introduced later are slightly stronger than condition (H3) used in [321]. Let f : [0, ∞) → X be continuous. By a mild solution of the abstract Cauchy problem u󸀠 (t) = A(t)u(t) + f (t),

t > 0; u(0) = x

(137)

160 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations we mean the function t

u(t) := U(t, 0)x + ∫ U(t, s)f (s) ds,

t ⩾ 0.

(138)

0

For certain conditions under which mild solutions of (137) are classical ones, and vice versa, we refer the reader to [321, Section 5]. Of concern are also the following semilinear Cauchy problems: u󸀠 (t) = A(t)u(t) + f (t, u(t)),

t∈ℝ

(139)

t > 0; u(0) = x.

(140)

and u󸀠 (t) = A(t)u(t) + f (t, u(t)), We will use the following definition. Definition 2.14.2. (i) Let f ∈ APSp (I × X : X) with p ⩾ 1. A function u ∈ Cb (ℝ : X) is said to be a mild solution of (139) iff +∞

u(t) = ∫ Γ(t, s)f (s, u(s)) ds,

t ∈ ℝ.

−∞

(ii) Let f ∈ AAPSp ([0, ∞) × X : X) with p ⩾ 1. A function u ∈ Cb ([0, ∞) : X) is said to be a mild solution of (140) iff t

u(t) = U(t, 0)x + ∫ U(t, s)f (s, u(s)) ds,

t ⩾ 0.

0

2.14.1 Generalized almost periodic solutions of inhomogeneous evolution equations Let f ∈ L1loc (ℝ : X). We will first state a few results concerning the (equi-)Weyl-almost +∞ periodicity of function t 󳨃→ u(t) = ∫−∞ Γ(t, s)f (s) ds, t ∈ ℝ, for which we know that represents the unique mild solution of the abstract Cauchy problem (136). For this purpose, we need to introduce the following two conditions: (H3e) To every ε > 0, there exist two real numbers l > 0 and L > 0 such that any interval I ⊆ ℝ of length L contains a number τ ∈ I such that x+l

1 󵄩 󵄩 sup ∫ 󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 dt < ε x∈ℝ l x

(141)

2.14 Generalized (asymptotically) almost periodic solutions |

161

and that for each h > 0 we have the following: For every t, s ∈ ℝ with |t − s| ⩾ h, we have 󵄩 󵄩 sup󵄩󵄩󵄩Γ(t + τ + r, s + τ + r) − Γ(t + r, s + r)󵄩󵄩󵄩 < ε, r∈ℝ

(142)

(H3) To every ε > 0, there exists a real number L > 0 such that any interval I ⊆ ℝ of length L contains a number τ ∈ I such that x+l

lim sup

l→+∞ x∈ℝ

1 󵄩󵄩 󵄩 ∫ 󵄩f (t + τ) − f (t)󵄩󵄩󵄩 dt < ε l 󵄩 x

and that for each h > 0 inequality (142) holds for every t, s ∈ ℝ with |t − s| ⩾ h. Then we have the following result, with p = 1. Theorem 2.14.3. Let f ∈ L∞ (ℝ : X), and let the mapping t 󳨃→ u(t), t ∈ ℝ be defined as above. 1 (i) Suppose that (H3e) holds. Then u ∈ Cb (ℝ : X) ∩ e − Wap (ℝ : X). 1 (ii) Suppose that (H3) holds. Then u ∈ Cb (ℝ : X) ∩ Wap (ℝ : X). Proof. Because proofs are similar, we will only show part (i). It is clear that (135) and the assumption f ∈ L∞ (ℝ : X) yield that u ∈ L∞ (ℝ : X). Furthermore, we have u(t) = ∞ t u1 (t) + u2 (t), t ∈ ℝ, where u1 (t) := ∫−∞ Γ(t, s)f (s) ds and u1 (t) := ∫t Γ(t, s)f (s) ds (t ∈ ℝ). Evidently, u1 , u2 ∈ L∞ (ℝ : X). We will first prove that u1 ∈ C(ℝ : X). Set Ψk (t) := t−k

∫t−k+1 Γ(t, s)f (s) ds, t ∈ ℝ. Since ∑∞ k=1 Ψk (t) = u1 (t) uniformly for t ∈ ℝ (see, e. g., the proof of [166, Theorem 2.3]), it is sufficient to show that for any fixed integer k ∈ ℕ one has Ψk ∈ C(ℝ : X). Since, due to (135), t 󸀠 −k 󵄩󵄩 t−k 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Γ(t, s)f (s) ds − ∫ Γ(t 󸀠 , s)f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 t−k+1 t 󸀠 −k+1

󵄩󵄩 t−k 󵄩󵄩 󵄩󵄩 t−k+1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ⩽ 󵄩󵄩󵄩 ∫ Γ(t, s)f (s) ds󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∫ Γ(t 󸀠 , s)f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩t 󸀠 −k 󵄩 󵄩t 󸀠 −k+1 󵄩 t−k+1

󵄩 󵄩 + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds t 󸀠 −k

t−k+1

󵄩 󵄩 ⩽ 2M 󸀠 ‖f ‖∞ |t − tk | + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds,

(143)

t 󸀠 −k

the right continuity of Ψk (⋅) follows by applying the dominated convergence theorem and the fact that Γ(t 󸀠 , s)f (s) = U(t 󸀠 , s)P(s)f (s) converges to Γ(t, s)f (s) = U(t, s)P(s)f (s)

162 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations as t 󸀠 → t. For the left continuity of Ψk (⋅), we can apply the same argument as above, by observing the following consequence of (143): t 󸀠 −k 󵄩󵄩 t−k 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Γ(t, s)f (s) ds − ∫ Γ(t 󸀠 , s)f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 t−k+1 t 󸀠 −k+1 t 󸀠 −k+1

󵄩 󵄩 ⩽ M 󸀠 ‖f ‖∞ |t − tk | + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds t 󸀠 −k

󵄩󵄩 t −k+1 󵄩󵄩 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩 ∫ [Γ(t, s) − Γ(t 󸀠 , s)]f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩t−k+1 󵄩 󸀠

t 󸀠 −k+1

󵄩 󵄩 ⩽ 3M ‖f ‖∞ |t − tk | + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds. 󸀠

t 󸀠 −k

Let ε > 0 be given, and let 4hM 󸀠 ‖f ‖∞ < ε. Assume that the real numbers l > 0 and L > 0 are chosen in such a way that any interval I ⊆ ℝ of length L contains a number τ ∈ I such that (141) holds, as well as that (142) holds for every t, s ∈ ℝ with |t − s| ⩾ h; see (H3e). Let τ ∈ I be such a number. Arguing as in the proof of [321, Theorem 4.5], the validity of conditions (141) and (142) implies that, for every x ∈ ℝ and l > 0, the following holds: x+l

1 󵄩󵄩 󵄩 ∫ 󵄩u(t + τ) − u(t)󵄩󵄩󵄩 dt l 󵄩 x

x+l ∞

⩽

1 󵄩 󵄩 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l x −∞ x+l

1 + ∫ l

󵄩 󵄩 ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ) − Γ(t, s)󵄩󵄩󵄩‖f ‖∞ ds dt

x |t−s|⩾h

x+l

+

1 ∫ l

󵄩 󵄩 ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ) − Γ(t, s)󵄩󵄩󵄩‖f ‖∞ ds dt

x |t−s|⩽h

x+l t

⩽

1 󵄩 󵄩 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l x −∞

x+l ∞

1 󵄩 󵄩 󵄩 󵄩 + ∫ ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l x

t

2.14 Generalized (asymptotically) almost periodic solutions |

x+l

1 + ∫ ‖f ‖∞ l x

∫ ε ds dt |t−s|⩽h

x+l

2M 󸀠 + ∫ ‖f ‖∞ l x

∫ e−ωh ds dt |t−s|⩽h

x+l t

⩽

1 󵄩 󵄩 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l x −∞

x+l ∞

1 󵄩 󵄩 󵄩 󵄩 + ∫ ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l x

t

+ 2h‖f ‖∞ ε + 4hM 󸀠 ‖f ‖∞

⩽ I + II + 2h‖f ‖∞ ε + ε, where x+l t

I=

1 󵄩 󵄩 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l

⩽M

x −∞

󸀠1

l

x+l t

󵄩 󵄩 ∫ ∫ e−ω(t−s) 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt x −∞

x+l +∞

1 󵄩 󵄩 ∫ ∫ e−ωv 󵄩󵄩󵄩f (t − v + τ) − f (t − v)󵄩󵄩󵄩 dv dt l

= M󸀠

x

0

+∞ x+l

1 󵄩 󵄩 ∫ ∫ e−ωv 󵄩󵄩󵄩f (t − v + τ) − f (t − v)󵄩󵄩󵄩 dt dv l

= M󸀠

x

0

x−v+l

+∞

=M ∫ e 󸀠

−ωv

1 󵄩 󵄩 [ ∫ 󵄩󵄩󵄩f (r + τ) − f (r)󵄩󵄩󵄩 dr] dv l x−v

0

⩽ M ε/ω 󸀠

and x+l t

1 󵄩 󵄩 󵄩 󵄩 II = ∫ ∫ 󵄩󵄩󵄩Γ(t + τ, s + τ)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l x −∞

x+l t

⩽ M󸀠 = M󸀠

1 󵄩 󵄩 ∫ ∫ e−ω(t−s) 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds dt l x −∞

x+l +∞

1 󵄩 󵄩 ∫ ∫ e−ωv 󵄩󵄩󵄩f (t + v + τ) − f (t + v)󵄩󵄩󵄩 dv dt l x

0

163

164 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations

=M

+∞ x+l

󸀠1

󵄩 󵄩 ∫ ∫ e−ωv 󵄩󵄩󵄩f (t + v + τ) − f (t + v)󵄩󵄩󵄩 dt dv

l

0

x

x+v+l

+∞

= M 󸀠 ∫ e−ωv [ 0

1 󵄩 󵄩 ∫ 󵄩󵄩󵄩f (r + τ) − f (r)󵄩󵄩󵄩 dr] dv l x+v

⩽ M 󸀠 ε/ω. This concludes the proof of the theorem. Remark 2.14.4. By [321, Theorem 4.5] and Theorem 2.1.1(xiv), conditions (H3e) and (H3) automatically hold in the case (H1)–(H2) are satisfied and f (⋅) is almost periodic; an elementary argument shows that this is also the case when f (⋅) is Stepanov p-almost periodic for some p ∈ [1, ∞). It is not clear whether conditions (H3e) and (H3) hold in the case when f (⋅) is only (equi-)Weyl-almost periodic. Remark 2.14.5. Let f ∈ L∞ (ℝ : X). Arguing similarly as in the proof of Theorem 2.14.3, we can deduce the following: ∞ (i) Suppose that (H3e) holds. Then the mapping t 󳨃→ ∫t Γ(t, s)f (s), t ∈ ℝ is in class 1 Cb (ℝ : X) ∩ e − Wap (ℝ : X).

(ii) Suppose that (H3) holds. Then the mapping t 󳨃→ ∫t Γ(t, s)f (s), t ∈ ℝ is in class 1 Cb (ℝ : X) ∩ Wap (ℝ : X). ∞

Before we move ourselves to the next subsection, we want to note that H.-S. Ding, W. Long and G. M. N’Guérékata have not considered the case of p = 1 in [169, Theo+∞ rem 2.3]. If so, then we have that the function t 󳨃→ u(t) = ∫−∞ Γ(t, s)f (s) ds, t ∈ ℝ is Stepanov almost periodic, i. e., Stepanov 1-almost periodic: Theorem 2.14.6. Let f ∈ APS1 (ℝ : X). Then u ∈ APS1 (ℝ : X). Proof. Keeping in mind the proof of [169, Theorem 2.3] (cf. especially lines −1 and −3, p. 239), Remark 2.14.4 and the fact that the limit of uniformly convergent Stepanov almost periodic functions is likewise Stepanov almost periodic, it suffices to show that r+1

t

󵄩 󵄩 lim sup ∫ ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds dt = 0. h →0+ r∈ℝ

0

r t−h0

But, this simply follows from the estimates r+1

r

t

s+h0

󵄩 󵄩 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 ds dt ⩽ ∫ ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 dt ds + r t−h0

r−h0

r

r+1−h0 s+h0

∫ r

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 dt ds s

r+1−h0 s+h0

+

∫

r+1

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (s)󵄩󵄩󵄩 dt ds ⩽ 3h0 ‖f ‖S1 , s

h0 ∈ (0, 1).

2.14 Generalized (asymptotically) almost periodic solutions |

165

2.14.2 Generalized asymptotically almost periodic solutions of inhomogeneous evolution equations The existence and uniqueness of asymptotically almost periodic solutions of the abstract Cauchy problem (137) has been investigated in [321, Section 5] by L. Maniar and R. Schnaubelt, provided that conditions (H1󸀠 )–(H3󸀠 ) introduced there hold. Since any asymptotically almost periodic function is bounded, they have needed the condition ∞

Q(0)x = − ∫ UQ (0, s)Q(s)f (s) ds, 0

as well; cf. [321, Theorem 5.4] for more details. In this section, we will follow a slightly different approach in which we do not use the above condition. Strictly speaking, we will use the same conditions as in the previous subsection and impose an extra condition: (H4) There exist finite numbers M 󸀠󸀠 > 0 and γ > 1 such that M 󸀠󸀠 󵄩 󵄩󵄩 󵄩󵄩U(t, s)Q(s)󵄩󵄩󵄩 ⩽ 1 + (t − s)γ

for t ⩾ s ⩾ 0.

Our first result reads as follows: Theorem 2.14.7. (i) Let p > 1, g ∈ APSp (ℝ : X), and q ∈ C0 ([0, ∞) : X). Assume that the function u(⋅) is given by (138), with the function f = g + q. If (H4) holds, x ∈ P(0)X ∩ D(A(0)) and t the mapping t 󳨃→ ∫0 U(t, s)Q(s)g(s) ds, t ⩾ 0 is asymptotically almost periodic, then u ∈ AAP([0, ∞) : X). (ii) Let g ∈ APS1 (ℝ : X) and q ∈ C0 ([0, ∞) : X). Assume that the function u(⋅) is given by (138), with the function f = g + q. If (H4) holds, x ∈ P(0)X and the mapping t t 󳨃→ ∫0 U(t, s)Q(s)g(s) ds, t ⩾ 0 is in class AAPS1 ([0, ∞) : X), then u(⋅) is in class AAPS1 ([0, ∞) : X) as well. Proof. We will prove only the second part. Keeping in mind definition of Green’s function and the identity I = P(t) + Q(t) for all t ∈ ℝ, it is clear that we have the following decomposition: t

u(t) = U(t, 0)x + ∫ U(t, s)f (s) ds 0

t

t

= U(t, 0)x + ∫ Γ(t, s)g(s) ds + ∫ Γ(t, s)q(s) ds t

0

t

0

+ ∫ U(t, s)Q(s)g(s) ds + ∫ U(t, s)Q(s)q(s) ds 0

0

166 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations t

0

t

= U(t, 0)x + ∫ Γ(t, s)g(s) ds − ∫ Γ(t, s)g(s) ds + ∫ Γ(t, s)q(s) ds −∞

t

t

0

−∞

+ ∫ U(t, s)Q(s)g(s) ds + ∫ U(t, s)Q(s)q(s) ds, 0

t ⩾ 0.

0

The function t 󳨃→ U(t, 0)x, t ⩾ 0 is exponentially decaying and continuous due to (135) and inclusion x ∈ P(0)X ∩ D(A(0)); see [321]. Using the proofs of [11, Lemma 2.13], [127, Lemma 4.1], as well as (135) and the validity of condition (H4), we get that t

t

lim ∫ Γ(t, s)q(s) ds = 0

t→+∞

and

0

lim ∫ U(t, s)Q(s)q(s) ds = 0.

t→+∞

(144)

0

0

The term ∫−∞ Γ(⋅, s)g(s) ds is also exponentially decaying since (135) yields: 0 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Γ(t, s)g(s) ds󵄩󵄩󵄩 ⩽ M 󸀠 ∫ e−ω(t−s) 󵄩󵄩󵄩g(s)󵄩󵄩󵄩 ds 󵄩 󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩−∞ 󵄩 −∞

=M e

󸀠 −ωt

∞

∫e 0

∞ k+1

󵄩 󵄩 󵄩 󸀠 −ωt ∑ ∫ e−ωs 󵄩󵄩󵄩g(−s)󵄩󵄩󵄩 ds 󵄩󵄩g(−s)󵄩󵄩󵄩 ds = M e

−ωs 󵄩 󵄩

k=0 k

∞

⩽ M 󸀠 e−ωt ‖g‖S1 ∑ e−ωk , k=0

t ⩾ 0.

Moreover, the proofs of [166, Theorem 2.3] and Theorem 2.14.6 show that the function t t 󳨃→ ∫−∞ Γ(t, s)g(s) ds, t ⩾ 0 is in class u ∈ APS1 ([0, ∞) : X). Since the mapping t 󳨃→ t

∫0 U(t, s)Q(s)g(s) ds, t ⩾ 0 is in class AAPS1 ([0, ∞) : X) by our assumption, the proof of the theorem is thereby complete. t

Remark 2.14.8. The mapping t 󳨃→ ∫0 U(t, s)Q(s)g(s) ds, t ⩾ 0 is clearly in class AAPS1 ([0, ∞) : X) if the evolution system U(⋅, ⋅) is exponentially stable, when also condition (H4) trivially holds. Concerning the existence and uniqueness of mild solutions of the abstract Cauchy problem (137) which belong to the class of asymptotically (equi-)Weyl-almost periodic functions, we have the following result: Theorem 2.14.9. Let g, q ∈ L1loc (ℝ : X), g(⋅) be essentially bounded, let (H4) hold, and let condition (H3e) hold with f (⋅) replaced by g(⋅) therein. Assume that x ∈ P(0)X. Set t

u(t) := U(t, 0)x + ∫ U(t, s)[g(s) + q(s)] ds, 0

t ⩾ 0.

2.14 Generalized (asymptotically) almost periodic solutions |

167

(i) Suppose that q ∈ e − W01 ([0, ∞) : X), as well as that ∫0 ‖q(s)‖ ds < ∞ and ∞

t

󵄩 󵄩 t 󳨃→ ∫(t − s)−γ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds,

t ⩾ 0 tends to zero as t → +∞.

(145)

0

t

1 ([0, ∞) : X) + e − If the mapping t 󳨃→ ∫0 U(t, s)Q(s)g(s) ds, t ⩾ 0 is in class e − Wap 1 W0 ([0, ∞) : X), then u(⋅) is in the same class as well. ∞ (ii) Suppose that q ∈ W01 ([0, ∞) : X), as well as that ∫0 ‖q(s)‖ ds < ∞ and (145) holds. If t

1 ([0, ∞) : X)+W01 ([0, ∞) : the mapping t 󳨃→ ∫0 U(t, s)Q(s)g(s) ds, t ⩾ 0 is in class Wap X), then u(⋅) is in the same class as well.

Proof. We will prove only the first part. As in the proof of Theorem 2.14.7, we have the following decomposition: t

0

t

u(t) = U(t, 0)x + ∫ Γ(t, s)g(s) ds − ∫ Γ(t, s)g(s) ds + ∫ Γ(t, s)q(s) ds −∞

t

t

−∞

+ ∫ U(t, s)Q(s)g(s) ds + ∫ U(t, s)Q(s)q(s) ds, 0

0

t ⩾ 0.

0

Keeping in mind the prescribed assumptions and the argument in the proof of aforet mentioned theorem, it suffices to show that the mappings t 󳨃→ ∫0 Γ(t, s)q(s) ds, t ⩾ 0 t

and t 󳨃→ ∫0 U(t, s)Q(s)q(s) ds, t ⩾ 0 belong to the class e − W01 ([0, ∞) : X). The proof t

for the mapping t 󳨃→ ∫0 U(t, s)Q(s)q(s) ds, t ⩾ 0 can be deduced by using the fact that

q ∈ e − W01 ([0, ∞) : X), the assumptions ∫0 ‖q(s)‖ ds < ∞, (145) and the following computation involving the mean value theorem, which is valid for l, t > 1: ∞

x+t+l s 󵄩󵄩 t 󸀠󸀠 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩∫ U(t, s)Q(s)q(s) ds󵄩󵄩󵄩 ⩽ M ∫ ∫ ‖q(r)‖ dr ds 󵄩󵄩󵄩 󵄩󵄩󵄩 l 1 + (s − r)γ 󵄩 󵄩0 x+t 0 x+t x+t+1

x+t+l

‖q(r)‖ M 󸀠󸀠 ds dr ⩽ ∫[ ∫ + ∫ ] l 1 + (s − r)γ x+t

0

x+t+1

x+t+l r+1

x+t+l

M 󸀠󸀠 ‖q(r)‖ + ds dr ∫ [∫ + ∫ ] l 1 + (s − r)γ x+t

x+t

⩽

r

r+1

M 󸀠󸀠 󵄩󵄩 󵄩 ∫ 󵄩󵄩q(r)󵄩󵄩󵄩 dr l 0

M + l

󸀠󸀠

x+t

󵄩 󵄩 ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩[(x + t + l − r)1−γ − (x + t + 1 − r)1−γ ] dr 0

168 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations x+t+l

x+t+l

x+t

x+t

M 󸀠󸀠 M 󸀠󸀠 󵄩 󵄩 󵄩 󵄩 + ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr + ∫ [(x + t + l − r)1−γ − 1]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr l l x+t

⩽

x+t

M 󸀠󸀠 (γ − 1)(l − 1) 󵄩󵄩 M 󸀠󸀠 󵄩󵄩 󵄩 󵄩 ∫ 󵄩󵄩q(r)󵄩󵄩󵄩 dr + ∫ 󵄩󵄩q(r)󵄩󵄩󵄩(x + t + 1 − r)−γ dr l l 0

+

M l

󸀠󸀠

0

x+t+l

󵄩 󵄩 ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr

x+t

x+t+l

(γ − 1)(l − 1) 󵄩 󵄩 + Const. ⋅ ∫ [1 + (x + t + l − r)−γ ]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr. l 0

t

The proof for the mapping t 󳨃→ ∫0 Γ(t, s)q(s) ds, t ⩾ 0 can be deduced similarly, but much easier. 2.14.3 Almost periodic and asymptotically almost periodic solutions of semilinear evolution equations with Stepanov coefficients In this section, we will prove two results on the existence and uniqueness of almost periodic and asymptotically almost periodic solutions of semilinear evolution equations. We start by explaining how one can prove a slight extension of [166, Theorem 2.4], where the authors have considered the case p > 2 only. Keeping in mind Proposition 2.6.11 and the argument in the proof of this theorem, the case 1 < p ⩽ 2 can be also handled: Theorem 2.14.10. Let f ∈ APSp (ℝ × X : X) with p > 1, and let there exist a number r ⩾ max(p, p/p − 1) and a function Lf ∈ LrS (ℝ) such that r > p/p − 1 and (73) holds. Then q = pr/p + r ∈ (1, p) and the validity of condition 1/r 󸀠

‖Lf ‖Lr (ℝ) S

ωr 󸀠 1 − e−ω ( < 󸀠 ) 󸀠 2M 1 − e−ωr

,

where 1/r + 1/r 󸀠 = 1, implies that there exists a unique almost periodic solution of the abstract semilinear Cauchy problem (139). Now we will state and prove the following result about the existence and uniqueness of asymptotically almost periodic solutions of the abstract semilinear Cauchy problem (139). Theorem 2.14.11. Let I = [0, ∞). Suppose that (H4) holds, x ∈ P(0)X ∩ D(A(0)) and the following conditions are satisfied: (i) g ∈ APSp (ℝ × X : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lg ∈ LrS (ℝ) such that r > p/p − 1 and (73) holds with f = g and Lf (⋅) = Lg (⋅).

2.14 Generalized (asymptotically) almost periodic solutions |

169

(ii) f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q ∈ C0 ([0, ∞) × X : X) and q := pr/p + r. (iii) There exists a measurable nonnegative function Lq (⋅) such that (73) holds with the function f (⋅, ⋅) replaced by the function q(⋅, ⋅) therein. (iv) There exists a number λ ∈ (0, 1) such that, for every t ⩾ 0, t

t

M 󸀠 ∫ e−ω(t−s) [Lg (s) + Lq (s)] ds + M 󸀠󸀠 ∫ 0

0

1 [L (s) + Lq (s)] ds ⩽ λ. 1 + (t − s)γ g

(146)

Then there exists a unique asymptotically almost periodic solution of the abstract semilinear Cauchy problem (139). t+1

Proof. Let M 󸀠󸀠󸀠 := supt∈ℝ [∫t Define the operator

Lg (s)r ds]1/r < ∞. As before, we have that 1 < q < p. t

(Λu)(t) := U(t, 0)x + ∫ U(t, s)f (s, u(s)) ds,

t ⩾ 0.

0

We contend that the mapping Λ : AAP([0, ∞) : X) → AAP([0, ∞) : X) is well-defined. Towards this end, we first observe that the equality t

0

(Λu)(t) = U(t, 0)x + ∫ Γ(t, s)g(s, u(s)) ds − ∫ Γ(t, s)g(s, u(s)) ds t

−∞

−∞

+ ∫ Γ(t, s)q(s, u(s)) ds 0

t

t

+ ∫ U(t, s)Q(s)g(s, u(s)) ds + ∫ U(t, s)Q(s)q(s, u(s)) ds 0

0

holds true for all t ⩾ 0. Again, the function t 󳨃→ U(t, 0)x, t ⩾ 0 is exponentially decaying and continuous due to (135) and inclusion x ∈ P(0)X ∩ D(A(0)). Since the range of any X-valued asymptotically almost periodic function is relatively compact in X, our assumptions yield that q(s, u(s)) → 0 as s → +∞. Therefore, as in the proof of Theorem 2.14.7, we may conclude that t

lim ∫ Γ(t, s)q(s, u(s)) ds = 0

t→+∞

0

t

and

lim ∫ U(t, s)Q(s)q(s, u(s)) ds = 0.

t→+∞

0

The boundedness of function g(⋅, ⋅) and the arguments employed in the proof of The0 orem 2.14.7 imply that limt→+∞ ∫−∞ Γ(t, s)g(s, u(s)) ds = 0, as well. Assume that x(t) =

170 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations y(t) + z(t) for all t ⩾ 0, where y ∈ AP([0, ∞) : X) and z ∈ C0 ([0, ∞) : X). Owing to Proposition 2.7.3, we have that the function s 󳨃→ g(s, x(s)), s ⩾ 0 belongs to the class t APSq (I : X) so that the mapping t 󳨃→ ∫−∞ Γ(t, s)g(s, x(s)) ds, t ∈ ℝ is almost periodic by our considerations from Remark 2.14.5. Hence, it suffices to show that t

lim ∫ Γ(t, s)[g(s, x(s)) − g(s, y(s))] ds = 0.

t→+∞

−∞

Let ε > 0 be given, and let ‖z(t)‖ < ε for t ⩾ t0 (ε). The required equality is a consequence of the following calculations involving Hölder inequality, with 1/q󸀠 + 1/q = 1 and any t > 0 such that t − ⌊t/2⌋ − 1 ⩾ t0 (ε) (cf. also the proof of estimate [317, (2.12)]): 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Γ(t, s)[g(s,x(s)) − g(s, y(s))] ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩−∞ 󵄩󵄩 t

󵄩 󵄩 ⩽ M 󸀠 ∫ e−ω(t−s) 󵄩󵄩󵄩g(s, x(s)) − g(s, y(s))󵄩󵄩󵄩 ds −∞ ∞

t−k

󵄩 󵄩 ⩽ M ∑ ∫ e−ω(t−s) 󵄩󵄩󵄩g(s, x(s)) − g(s, y(s))󵄩󵄩󵄩 ds 󸀠

k=0 t−k−1 ∞

t−k

󵄩 󵄩 ⩽ M ∑ ∫ e−ω(t−s) Lg (s)󵄩󵄩󵄩z(s)󵄩󵄩󵄩 ds 󸀠

k=0 t−k−1

t−k

∞

󵄩q 󵄩 ⩽ M 󸀠 ∑ e−ωk [ ∫ Lg (s)q 󵄩󵄩󵄩z(s)󵄩󵄩󵄩 ds] k=0

t−k−1

1/r

t−k

∞

1/q

t−k

󵄩p 󵄩 ⩽ M 󸀠 ∑ e−ωk [ ∫ Lg (s)r ds] [ ∫ 󵄩󵄩󵄩z(s)󵄩󵄩󵄩 ds] k=0

t−k−1 ∞

k=0

k=⌊t/2⌋+1

=M (∑ + 󸀠

t−k−1

t−k

⌊t/2⌋

∑

)e

−ωk

r

t−k−1

⩽ M 󸀠 M 󸀠󸀠󸀠 ∑ e−ωk ε + M 󸀠 M 󸀠󸀠󸀠 ⩽ M 󸀠 M 󸀠󸀠󸀠 ∑ e−ωk ε + M 󸀠 M 󸀠󸀠󸀠 k=0

t−k

t−k−1

∞

∞

∑

e−ω(⌊t/2⌋+1) ∑ e−ωk

∑

e−ω(⌊t/2⌋+1) ∑ e−ωk .

k=⌊t/2⌋+1 ∞

k=⌊t/2⌋+1

k=0 ∞

k=0

Keeping in mind that t

0

t

∫ Γ(t, s)g(s, u(s)) ds − ∫ Γ(t, s)g(s, u(s)) ds = ∫ Γ(t, s)g(s, u(s)) ds −∞

−∞

1/p

󵄩p 󵄩 [ ∫ Lg (s) ds] [ ∫ 󵄩󵄩󵄩z(s)󵄩󵄩󵄩 ds]

⌊t/2⌋ k=0 ∞

1/r

1/p

0

2.15 Vector-valued almost periodic ultradistributions | 171

for all t ⩾ 0, we can simply prove by (135), (146) and (H4) that the mapping Λ(⋅) is a λ-contraction, so that the proof of the theorem is completed by a routine application of the Banach contraction principle. We would like to note once more that condition (H2) is very nontrivial and difficult to verify in many concrete situations. For example, let 0 ≠ Ω ⊆ ℝn be a bounded domain with C 2 boundary 𝜕Ω being locally on one side of Ω, Dt = d/dt, Dk = d/dxk , and n(x) be the outer unit normal vector. Of importance is the following abstract parabolic problem: n

Dt u(t, x) = ∑ Dk akl (t, x)Dl u(t, x) + a0 (t, x)u(t, x) + f (t, x), n

k,l=1

∑ nk (x)akl (t, x)Dl u(t, x) = 0,

k,l=1

t ∈ ℝ,

t ∈ ℝ, x ∈ Ω;

x ∈ 𝜕Ω,

in the Lebesgue space X = Lp (Ω), where 1 < p < ∞ as well as the coefficients a0 (⋅) and akl (⋅) for k, l = 1, . . . , n satisfy the assumptions employed by L. Maniar and R. Schnaubelt in [321, Example 4.7] (see also A. Yagi [407, Section 4] for the first results in this direction). Set n

A(t; x; D) = ∑ Dk akl (t, x)Dl + a0 (t, x). k,l=1

The operator A(t) := A(t, ⋅, D) has its domain n

D(A(t)) := {φ ∈ W 2,p (Ω) : ∑ nk (⋅)akl (t, ⋅)Dl φ = 0 on 𝜕Ω}. k,l=1

Then (H1) and [321, (H3)] hold, but we do not yet know whether condition (H2) holds here, which clearly implies serious trials and tribulations for applications of the abstract theory. Finally, it is worth noting that T. Diagana has considered, in several research papers (see, e. g., [132]), generalized almost periodic and generalized almost automorphic solutions for various classes of abstract integro-differential equations by passing to nonautonomous evolution matrix equations on product spaces (see, e. g., [132, Section 10]).

2.15 Vector-valued almost periodic ultradistributions In this section, we assume that (Mp ) is a sequence of positive real numbers satisfying M0 = 1 and the following conditions: Mp2 ⩽ Mp+1 Mp−1 ,

p ∈ ℕ,

(M.1)

172 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Mp ⩽ AH p sup Mi Mp−i , 0⩽i⩽p

∞

∑

p ∈ ℕ, for some A, H > 1,

Mp−1

p=1

Mp

(M.2) (M.3󸀠 )

< ∞.

Any employment of the condition ∞

sup ∑

p∈ℕ q=p+1

Mq−1 Mp+1 pMp Mq

(M.3)

< ∞,

which is slightly stronger than (M.3󸀠 ), will be explicitly emphasized.

Let s > 1. The Gevrey sequence (p!s ) satisfies the above conditions. The associated ρp function of (Mp ) is defined by M(ρ) := supp∈ℕ ln M , ρ > 0; M(0) := 0. If λ ∈ ℂ, set p

M

M(λ) := M(|λ|). Define mp := M p , p ∈ ℕ. p−1 The space of Beurling, resp., Roumieu ultradifferentiable functions, is defined by

𝒟

(Mp )

:= indlim 𝒟K

,

𝒟

{Mp }

:= indlim 𝒟K

,

M ,h

resp. 𝒟K

(Mp )

K⋐⋐ℝ

resp. {Mp }

K⋐⋐ℝ

where (Mp )

𝒟K

:= projlim 𝒟K p ,

M ,h 𝒟K p

h→∞

{Mp }

M ,h

:= indlim 𝒟K p , h→0

:= {ϕ ∈ C (ℝ) : supp(ϕ) ⊆ K, ‖ϕ‖Mp ,h,K < ∞} ∞

and ‖ϕ‖Mp ,h,K := sup{

hp |ϕ(p) (t)| : t ∈ K, p ∈ ℕ0 }. Mp

As of now, the asterisk ∗ is used to designate both the Beurling case (Mp ) and Roumieu case {Mp }. The space consisting of all continuous linear functions from 𝒟∗ into X, denoted by 𝒟󸀠∗ (X) := L(𝒟∗ : X), is said to be the space of all X-valued ultradistributions of ∗-class. p Recall [242–244] that an entire function of the form P(λ) = ∑∞ p=0 ap λ , λ ∈ ℂ, is of class (Mp ), resp. of class {Mp }, if there exist l > 0 and c > 0, resp. for every l > 0 there exists a constant c > 0, such that |ap | ⩽ clp /Mp , p ∈ ℕ. The corresponding p ultradifferential operator P(D) = ∑∞ p=0 ap D is of class (Mp ), resp. of class {Mp }. We introduce the topology of the above spaces, as well as the convolution of scalar-valued ultradistributions (ultradifferentiable functions), in the same way as in the case of

2.15 Vector-valued almost periodic ultradistributions | 173

corresponding distribution spaces (see [242]). The convolution of Banach space valued ultradistributions and scalar-valued ultradifferentiable functions will be taken in the sense of considerations given on page 685 of [244]. Let us recall that for any f ∈ 𝒟󸀠∗ (X) and φ ∈ 𝒟∗ we have f ∗ φ ∈ ℰ ∗ (X), as well as that the linear mapping φ 󳨃→ ⋅ ∗ φ : 𝒟󸀠∗ (X) → ℰ ∗ (X) is continuous. Here, the space ℰ ∗ (X) is defined as (see [244, p. 678]) (M ) M ,h ℰ ∗ (X) := indlimK⋐⋐ℝ ℰK∗ (X), where in Beurling case ℰK p (X) := projlimh→∞ ℰK p (X), resp. in Roumieu case ℰK

{Mp }

M ,h

(X) := indlimh→0 ℰK p (X), and

M ,h

ℰK p (X) := {ϕ ∈ C (ℝ : X) : sup ∞

p⩾0

hp ‖ϕ(p) ‖C(K:X) < ∞}. Mp

The space consisting of all linear continuous mappings ℰ ∗ (ℂ) → X is denoted by ℰ 󸀠∗ (X); ℰ 󸀠∗ := ℰ 󸀠∗ (ℂ). The convolution of an X-valued ultradistribution f (⋅) and an element g ∈ ℰ 󸀠∗ , defined by the identity [244, (4.9)], is an X-valued ultradistribution and the mapping g ∗ ⋅ : 𝒟󸀠∗ (X) → 𝒟󸀠∗ (X) is continuous. Set ⟨Th , φ⟩ := ⟨T, φ(⋅ − h)⟩, T ∈ 𝒟󸀠∗ (X), h > 0. If (Mp ) satisfies (M.1), (M.2) and (M.3), then Pl (x) = (1 + x2 ) ∏ (1 + p∈ℕ

x2

l2 m2p

),

resp. Prp (x) = (1 + x2 ) ∏ (1 + p∈ℕ

x2 ), r 2 rp2

defines an ultradifferential operator of class (Mp ), resp. of class {Mp }. Here, l > 0 and (rp ) ∈ R, where R denotes the family of all sequences of positive real numbers tending to infinity. The following spaces of tempered ultradistributions of Beurling, resp. Roumieu type, are defined by S. Pilipović [344] as duals of the corresponding test spaces 𝒮

(Mp )

:= projlim 𝒮 Mp ,h , h→∞

resp. 𝒮 {Mp } := indlim 𝒮 Mp ,h , h→0

where 𝒮

Mp ,h

:= {ϕ ∈ C ∞ (ℝ) : ‖ϕ‖Mp ,h < ∞}

‖ϕ‖Mp ,h := sup{

(h > 0),

α+β

h β/2 󵄨 󵄨 (1 + t 2 ) 󵄨󵄨󵄨ϕ(α) (t)󵄨󵄨󵄨 : t ∈ ℝ, α, β ∈ ℕ0 }. Mα Mβ

A continuous linear mapping 𝒮 (Mp ) → X, resp. 𝒮 {Mp } → X, is said to be an X-valued tempered ultradistribution of Beurling, resp. Roumieu type. The space consisted of all vector-valued tempered ultradistributions of Beurling, resp. Roumieu type, will be

174 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations denoted by 𝒮 󸀠(Mp ) (X), resp. 𝒮 󸀠{Mp } (X); the common abbreviation will be 𝒮 󸀠∗ (X). It is well known that 𝒮 󸀠(Mp ) (X) ⊆ 𝒟󸀠(Mp ) (X), resp. 𝒮 󸀠{Mp } (X) ⊆ 𝒟󸀠{Mp } (X). For any h > 0, we define 𝒟L1 ((Mp ), h) := {f ∈ 𝒟L1 ; ‖f ‖1,h := sup

p∈ℕ0

hp ‖f (p) ‖1 < ∞}. Mp

Then (𝒟L1 ((Mp ), h), ‖ ⋅ ‖1,h ) is a Banach space and the space of all X-valued bounded Beurling ultradistributions of class (Mp ), resp. X-valued bounded Roumieu ultradistributions of class {Mp }, is defined as the space consisting of all linear continuous mappings from 𝒟L1 ((Mp )), resp. 𝒟L1 ({Mp }), into X, where 𝒟L1 ((Mp )) := projlim 𝒟L1 ((Mp ), h), h→+∞

resp. 𝒟L1 ({Mp }) := indlim 𝒟L1 ((Mp ), h). h→0+

These spaces, carrying strong topologies, will be shortly denoted by 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X). It is well known that 𝒟(Mp ) , resp. 𝒟{Mp } , is a dense subspace of 𝒟L1 ((Mp )), resp. 𝒟L1 ({Mp }), as well as that 𝒟L1 ((Mp )) ⊆ 𝒟L1 ({Mp }) (see [81]). Since ‖φ‖1,h ⩽ ‖φ‖Mp ,h for any φ ∈ 𝒮 (Mp ) and h > 0, we have that 𝒮 (Mp ) , resp. 𝒮 {Mp } , is

a dense subspace of 𝒟L1 ((Mp )), resp. 𝒟L1 ({Mp }), and that f|𝒮 (Mp ) : 𝒮 (Mp ) → X, resp.

f|𝒮 {Mp } : 𝒮 {Mp } → X, is a tempered X-valued ultradistribution of class (Mp ), resp. of class {Mp }. Following I. Cioranescu [104], we say that a bounded X-valued ultradistribution f ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. f ∈ 𝒟L󸀠 1 ({Mp } : X), is almost periodic of Beurling class (Mp ), resp. almost periodic of Roumieu class {Mp }, iff there exists a sequence of X-valued trigonometric polynomials converging to f (⋅) in 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X). In the case that (Mp ) satisfies conditions (M.1), (M.2) and (M.3), the space of all bounded scalar-valued ultradistributions of Beurling class has been characterized in [104, Theorem 1], and the space of all almost periodic scalar-valued ultradistributions of Beurling class has been characterized in [104, Theorem 2]; condition (M.3) is essentially employed in the proof of [104, Lemma 2], which is no longer true if we assume only condition (M.3󸀠 ) and which is a fundamental tool for proving the implications [104, Theorem 1, (iii) ⇒ (iv)] and [104, Theorem 2, (iv) ⇒ (ii)]. The assertion of [104, Lemma 1] continues to hold if condition (M.3) is disregarded, in both Beurling and Roumieu case, p that is, if we suppose that P(D) = ∑∞ p=0 ap D is an ultradifferential operator of class (Mp ), resp. of class {Mp }. Then the induced mapping PB (D) : 𝒟L1 ((Mp )) → 𝒟L1 ((Mp )), resp. PR (D) : 𝒟L1 ({Mp }) → 𝒟L1 ({Mp }) is linear and continuous. The following theorem gives some new insights into the assertion of [104, Theorem 1]:

2.15 Vector-valued almost periodic ultradistributions | 175

Theorem 2.15.1. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟󸀠∗ (X). Consider the following assertions: p (i) There exists an ultradifferential operator P(D) = ∑∞ p=0 ap D of class (Mp ), resp. of class {Mp }, and functions f , g ∈ Cb (ℝ : X) such that T = P(D)f + g, i. e., ∞

p

∞

∞

⟨T, φ⟩ = ∑ (−1) ap ∫ f (t)φ (t) dt + ∫ g(t)φ(t) dt, p=0

(p)

−∞

(147)

−∞

for all φ ∈ 𝒟L1 ((Mp )), resp. φ ∈ 𝒟L1 ({Mp }). (ii) We have T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X). (iii) For every φ ∈ 𝒟∗ , we have T ∗ φ ∈ Cb (ℝ : X); furthermore, if B ⊆ 𝒟∗ is bounded, then there exists a finite constant M ⩾ 1 such that ‖T ∗ φ‖∞ ⩽ M, φ ∈ B. (iv) For each compact set K ⊆ ℝ there exists h > 0 in the Beurling case, resp. for each compact set K ⊆ ℝ and for every h > 0 in the Roumieu case, we have T ∗ φ ∈ Cb (ℝ : M ,h

X), φ ∈ 𝒟K p .

Then we have (i) ⇒ (ii) ⇒ (iii) ⇒ (iv). Proof. The implications (ii) ⇒ (iii) ⇒ (iv) follow from the proof of the above-mentioned theorem and elementary facts about topological properties of vector-valued ultradistributions. We will give a direct proof of the assertion (i) ⇒ (ii) here, in which we do not use condition (M.2); for the sake of brevity, we examine only the Roumieu case. We need to prove that for each h > 0 the mapping T : 𝒟L1 ((Mp ), h) → X is continuous. By our assumption, for every l ∈ (0, h), there exists cl > 0 such that |ap | ⩽ cl lp /Mp , p ⩾ 0. Hence, for every φ ∈ 𝒟L1 ((Mp ), h), we have: ∞

󵄩 (p) 󵄩 󵄩 󵄩󵄩 󵄩󵄩⟨T, φ⟩󵄩󵄩󵄩 ⩽ ∑ ‖f ‖∞ |ap |󵄩󵄩󵄩φ 󵄩󵄩󵄩1 + ‖g‖∞ ‖φ‖1 p=0 ∞

⩽ ∑ ‖f ‖∞ p=0

|cl | p −p l h ‖φ‖1,h Mp + ‖g‖∞ ‖φ‖1 Mp

⩽ cl ‖f ‖∞ ‖φ‖1,h

h + ‖g‖∞ ‖φ‖1 . h−l

This, in turn, yields (ii). Remark 2.15.2. (i) Assume that (Mp ) additionally satisfies (M.3). Then assertions (i)–(iv) are mutually equivalent for the Beurling class [104] and there exists l > 0 such that the choice P(D) = Pl (D) is possible in (i). It is not clear whether this statement holds for the Roumieu class, with the operator P(D) = Prp (D) and some (rp ) ∈ R; see also [104, Lemma 2] and [81, Lemma 3.1.1(ii)]. (ii) The assertion of [201, Theorem 3.2] continues to hold in the vector-valued case.

176 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Concerning [104, Theorem 2], the following result should be stated in vectorvalued case (in the Beurling case, the implication (iii) ⇒ (iv) follows from the fact that equation [345, (13)] holds in the vector-valued case and the proof of corresponding implication [104, (iii) ⇒ (iv), Theorem 2]; in the Roumieu case, the statement follows directly). Theorem 2.15.3. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X). Consider the following assertions: p (i) There exists an ultradifferential operator P(D) = ∑∞ p=0 ap D of class (Mp ), resp. of class {Mp }, and functions f , g ∈ AP(ℝ : X) such that T = P(D)f + g, i. e., (147) holds for all φ ∈ 𝒟L1 ((Mp )), resp. φ ∈ 𝒟L1 ({Mp }). (ii) T is almost periodic. (iii) For every φ ∈ 𝒟∗ , we have T ∗ φ ∈ AP(ℝ : X). (iv) There exists h > 0 such that for each compact set K ⊆ ℝ, in the Beurling case, resp. for each compact set K ⊆ ℝ and for each h > 0, in the Roumieu case, the following M ,h holds T ∗ φ ∈ AP(ℝ : X), φ ∈ 𝒟K p . Then we have (i) ⇒ (ii) ⇒ (iii) ⇒ (iv). Remark 2.15.4. (i) Assume that (Mp ) additionally satisfies (M.3). Then the above assertions are equivalent for the Beurling class, when there exists l > 0 such that the choice P(D) = Pl (D) is possible in (i), and it is not clear whether these assertions are equivalent for the Roumieu class, with the operator P(D) = Prp (D) and some (rp ) ∈ R. (ii) It is worth noting that [201, Theorem 4.2] continues to hold in the vector-valued case. Consider the following assertion: (ii)󸀠 The set of all translations {Th : h ∈ ℝ} is relatively compact in 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X). Using the same arguments as in the proof of [201, Theorem 4.2], we can deduce that (i) ⇒ (ii)󸀠 ⇒ (iii). Let us introduce the following space: ℰAP (X) := {ϕ ∈ ℰ (X) : ϕ ∗

∗

(i)

∈ AP(ℝ : X) for all i ∈ ℕ0 }.

∗ Then ℰAP (X) ⊆ 𝒟L󸀠∗1 (X) and, due to Theorem 2.1.1(viii) and the proof of [74, Proposi∗ ∗ ∗ tion 5(i)], ℰAP (X) = ℰ ∗ (X) ∩ AP(ℝ : X). Furthermore, ℰAP (X) ∗ L1 (ℝ) ⊆ ℰAP (X) and ∗ ∗ ℰAP (X) is the space consisting exactly of those elements f (⋅) from ℰ (X) for which f ∗ φ ∈ AP(ℝ : X), φ ∈ 𝒟∗ ; see, e. g., the proof of [74, Corollary 1] given in the distribution case, for almost automorphy. Consider now the following statement: 󸀠󸀠 (ii) T ∈ 𝒟L󸀠∗1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠∗1 ({Mp } : X), and there exists a sequence (ϕn ) in ∗ ℰAP (X) such that limn→∞ ϕn = T for the topology of 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X).

2.15 Vector-valued almost periodic ultradistributions | 177

Now we will state and prove the following assertion: Lemma 2.15.5. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X). Then (ii)󸀠󸀠 ⇔ (iii), with (iii) being the same as in the formulation of Theorem 2.15.3. Proof. The proof of implication (ii)󸀠 ⇒ (iii) can be deduced as in distribution case (see the proof of [74, Proposition 7]), while the implication (iii) ⇒ (ii󸀠 ) can be proved in the following way. Let ρ ∈ 𝒟∗ , supp(ρ) ⊆ [0, 1] and Tn := T ∗ ρn (n ∈ ℕ). Since ∗ ∗ ℰAP (X) = ℰ ∗ (X) ∩ AP(ℝ : X), (ii) yields that Tn ∈ ℰAP (X) for all n ∈ ℕ. Then it suffices to 󸀠∗ show that limn→∞ Tn = T in 𝒟L1 (X). For the sake of brevity, we will consider only the Roumieu class. Let h > 0 be fixed. Then there exists c > 0 such that ‖⟨T, φ⟩‖ ⩽ c‖φ‖1,h , φ ∈ 𝒟L1 ((Mp ), h). Furthermore, the estimate ‖[ρň ∗ φ − φ](p) ‖L1 ⩽ (1/n)‖φ(p+1) ‖L1 (n ∈ ℕ, p ∈ ℕ0 ) holds on account of the proof of [74, Proposition 7], so that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩⟨T ∗ ρn − T, φ⟩󵄩󵄩󵄩 = 󵄩󵄩󵄩⟨T, ρň ∗ φ − φ⟩󵄩󵄩󵄩 hp ‖[ρň ∗ φ − φ](p) ‖L1 ⩽ c sup Mp p⩾0 ⩽

hp ‖φ(p+1) ‖L1 c sup n p⩾0 Mp

(hH)p+1 ‖φ(p+1) ‖L1 c AM1 sup nh Mp+1 p⩾0 c ⩽ AM1 ‖φ‖1,h , φ ∈ 𝒟L1 ((Mp ), h), nh

⩽

which completes the proof by simply employing some elementary topological properties of spaces 𝒟L∗1 and 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X). Now we can prove the following result: Theorem 2.15.6. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X). Consider assertions (ii), (iii) and (iv) stated in the formulation of Theorem 2.15.3. Then we have (ii) ⇔ (ii)” ⇔ (iii) ⇔ (iv). Proof. By Theorem 2.15.3 and Lemma 2.15.5, we only need to prove that (ii) implies (ii)” and that (iv) implies (iii). The equivalence of (iv) and (iii) follows directly in Roumieu case, by definition, while the implication (iv) ⇔ (iii) in Beurling case follows from the M ,h (M ) fact that 𝒟K p = ⋂h>0 𝒟K p . To prove that (ii)” implies (ii), it suffices to observe that ∗ any function f ∈ AP(ℝ : X), and therefore any function f ∈ ℰAP (X), can be uniformly approximated by trigonometric polynomials. Suppose (Mp ) does not satisfy (M.3). The following unsolved problems arise immediately from our previous analyses: A. Theorem 2.15.1: Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟󸀠∗ (X). Is it true that assertion (iv) implies (ii)?

178 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations B. Theorem 2.15.3 and Remark 2.15.4: Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X). Is it true that (ii)󸀠 is equivalent to (ii)? For any f ∈ AP(ℝ : X), we define t

1 ∫ f (s) ds. t→∞ t

M(f ) := lim

0

Let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X) be almost periodic. Following the analyses from [101] and [201], we define the Bohr–Fourier coefficients aλ (T) of T by M(f ) :=

M(T ∗ φ)

∫−∞ φ(s) ds ∞

,

where φ ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. φ ∈ 𝒟L󸀠 1 ({Mp } : X) is fixed and satisfies ∫−∞ φ(s) ds ≠ 0. Then T = 0 iff aλ (T) = 0, λ ∈ ℝ; as in the case of almost periodic functions, we have that the spectrum of T, consisting of all real numbers λ for which aλ (T) ≠ 0, is at most countable (see [201, Proposition 4.5]). The assertions of [201, Theorem 4.6, Proposition 4.7], regarding Bochner–Féjer summation methods for almost periodic ω-ultradistributions, continue to hold in the vector-valued case and it is not difficult to see that these statements hold for Komatsu’s vector-valued almost periodic ultradistributions, even in the case when condition (M.3) is not satisfied. ∞

2.15.1 Generalizations of vector-valued almost periodic ultradistributions In this subsection, we will follow only Komatsu’s approach for the sake of brevity. Let 𝔸 ⊆ L1loc (ℝ : X). It is said that 𝔸 satisfies condition (△) iff for every f ∈ L1loc (ℝ : X) with the property that △s f := f (⋅ + s) − f (⋅) ∈ 𝔸 for all s > 0, we have f − Mh f ∈ 𝔸, h > 0, where h

Mh f (⋅) :=

1 ∫ f (⋅ + s) ds, h

h > 0.

0

If the assumption △s f ∈ 𝔸, s > 0 implies f − M1 f ∈ 𝔸, for any f ∈ L1loc (ℝ : X), then it is said that 𝔸 satisfies condition (△1 ). For more details about the importance of Doss conditions (△) and (△1 ) in the theory of almost periodic vector-valued functions, the reader may consult [46, 47] and references cited therein. Define M(𝔸) := {f ∈ L1loc (ℝ : X) : Mh f ∈ 𝔸 for all h > 0}.

2.15 Vector-valued almost periodic ultradistributions | 179

Following B. Basit and H. Güenzler [46, 47], we introduce the following space of vectorvalued ultradistributions: 𝒟𝔸 (X) := {T ∈ 𝒟 (X) : T ∗ φ ∈ 𝔸 for all φ ∈ 𝒟 }. 󸀠∗

󸀠∗

∗

If A ⊆ 𝒟󸀠∗ (X), then we similarly define the space 𝒟A (X) := {T ∈ 𝒟 (X) : T ∗ φ ∈ A for all φ ∈ 𝒟 }. 󸀠∗

󸀠∗

∗

󸀠∗ 󸀠∗ Note that 𝒟A󸀠∗ (X) = 𝒟𝔸 (X) = 𝒟𝔸 (X), where 𝔸 = A ∩ L1loc (ℝ : X) and 𝔸 ∞ = A ∩ ∞ ∞ C (ℝ : X). By Theorem 2.15.3, we have that the space consisting of almost peri󸀠∗ odic vector-valued ultradistributions is contained in the space 𝒟𝔸 (X), where 𝔸 = AP(ℝ : X). Let sh := (1/h)χ(−h,0) (h > 0) and sh := ((−1)/h)χ(0,−h) (h < 0). Then, for any f ∈ L1loc (ℝ : X), we have Mh f = f ∗ sh , h > 0. For vector-valued ultradistributions, we set

⟨Ts , φ⟩ := ⟨T, φ(⋅ − s)⟩, ̃h T := T ∗ sh , M

φ ∈ 𝒟∗ (s ∈ ℝ, T ∈ 𝒟󸀠∗ (X)), h ≠ 0, T ∈ 𝒟󸀠∗ (X)

and, for any subset A of 𝒟󸀠∗ (X), ̃h T ∈ A for all h > 0}. ̃(A) := {T ∈ 𝒟 (X) : M ℳ 󸀠∗

󸀠∗ It is said that T ∈ 𝒟󸀠∗ (X) satisfies (△) iff △s T := T−s − T ∈ 𝒟𝔸 (X) for all s > 0 implies 󸀠∗ 󸀠∗ ̃ T − Mh T ∈ 𝒟𝔸 (X) for all h > 0; if the assumption △s T ∈ 𝒟𝔸 (X) for all s > 0 implies ̃1 T ∈ 𝒟󸀠∗ (X), then it is said that T satisfies (△1 ). T −M 𝔸 The space 𝒟A󸀠∗ (X) is closed under the action of any ultradifferential operator P(D) of ∗-class because, due to [242, Theorem 2.12], we have

P(D)T ∗ φ = T ∗ P(D)φ ∈ A,

φ ∈ 𝒟∗ .

By [47, Theorem 2.10], we have that the closedness of A under addition implies that, for any vector-valued distribution T ∈ 𝒟󸀠 (X), we have T ∗ φ ∗ ψ ∈ A,

φ, ψ ∈ 𝒟 ⇒ T ∗ φ ∈ A,

φ ∈ 𝒟.

(148)

To the best knowledge of the author, it is still unknown whether an ultradifferentiable function of ∗-class can be written as a finite sum of functions like φ ∗ ψ, where φ, ψ are ultradifferentiable functions of ∗-class. Because of this, in the present situation, we are not able to say whether (148) holds in the ultradistribution case, if A is only closed under addition. We would like to note that the closedness of set A ∩ C(ℝ : X) under uniform convergence over ℝ also implies that, for any vector-valued distribution T ∈ 𝒟󸀠 (X), we have (148). The proof can be given as in the ultradistribution case, where we have the following statement:

180 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Proposition 2.15.7. Let A∩C(ℝ : X) be closed under uniform convergence over ℝ. Then, for any vector-valued ultradistribution T ∈ 𝒟󸀠∗ (X), we have T ∗ φ ∗ ψ ∈ A,

φ, ψ ∈ 𝒟∗ ⇒ T ∗ φ ∈ A,

φ ∈ 𝒟∗ .

(149)

∗ Proof. Let ρ ∈ 𝒟[0,1] such that ∫−∞ ρ(t) dt = 1. Set ρn (t) := nρ(nt), t ∈ ℝ, n ∈ ℕ. It suffices to show that, for every φ ∈ 𝒟∗ , we have ∞

lim (T ∗ ρň ∗ φ)(x) = (T ∗ φ)(x),

n→+∞

uniformly on ℝ (although this basically follows from the proof of implication [104, (iii) ⇒ (iv), Theorem 2], we want to present a direct and much simpler proof here). Since (T ∗ ρň ∗ φ)(x) = ⟨T, φ(x − ⋅) ∗ ρn ⟩ and (T ∗ φ)(x) = ⟨T, φ(x − ⋅)⟩ for all x ∈ ℝ and n ∈ ℕ, we need to prove that lim ⟨T, [φ(x − ⋅) ∗ ρn − φ(x − ⋅)]⟩ = 0,

n→+∞

uniformly for x ∈ ℝ. In the Beurling case, we have the existence of a positive real number h > 0 such that, due to a simple calculation involving condition (M.2), the (Mp ) mean value theorem and the continuity of T on 𝒟[−1,1] , 󵄩󵄩 󵄩 󵄩󵄩⟨T, [φ(x − ⋅) ∗ ρn − φ(x − ⋅)]⟩󵄩󵄩󵄩 hp supy∈[−1,1] |[φ(p) (x − ⋅) ∗ ρn − φ(p) (x − ⋅)](y)| ⩽ Const. ⋅ sup Mp p⩾0 1

⩽ Const. ⋅ sup

hp ∫0 |φ(p) (x − y + t) − φ(p) (x − y)|ρn (t) dt Mp

p⩾0 1

⩽ Const. ⋅ ∫ tρn (t) dt ⋅ sup 0

p⩾0

hp+1 ‖φ(p+1) ‖∞ Mp

1

(h/H)p+1 ‖φ(p+1) ‖∞ 1 1 . sup ⩽ Const. ⋅ ∫ tρ(t) dt ⋅ n AM1 h p⩾0 Mp+1 0

In the Roumieu case, the above holds for all positive real numbers h > 0, which simply completes the proof of proposition. If f : ℝ → X is uniformly continuous, f ∈ 𝒟A󸀠∗ (X) and A ∩ C(ℝ : X) is closed under uniform convergence, then we can use the fact that limn→∞ ρn ∗ f = f , uniformly on ℝ, to get that f ∈ A. The statements of [47, Lemma 2.3, Proposition 2.4] hold in the ultradistribution 󸀠∗ ̃h T ∈ case, so that the assumption △h T := T−h − T ∈ 𝒟𝔸 (X) for all h > 0 implies T − M

2.15 Vector-valued almost periodic ultradistributions | 181

󸀠∗ 𝒟𝔸 (X), as well as:

̃ 󸀠∗ (X) = 𝒟󸀠∗ (i) 𝒟A󸀠∗ (X) ⊆ ℳ𝒟 ̃ (X); and A ℳA 󸀠∗ 󸀠∗ ̃ (ii) 𝒟A (X) ⊆ ℳ𝒟A (X) if A is a cone (i. e., [0, ∞) ⋅ A + [0, ∞) ⋅ A ⊆ A) satisfying (△1 ). It is worth noting that [47, Corollaries 2.5 and 2.6], results from [47, Section 4] and [46, Proposition 1.1] can be formulated in the ultradistribution case as well. The following statements are in a close connection with [104, Theorem 2] and [47, Theorem 2.11]: p 1. Let there exist an ultradifferential operator P(D) = ∑∞ p=0 ap D of class (Mp ), resp. of 󸀠∗ class {Mp }, and f , g ∈ 𝒟A (X) such that T = P(D)f + g. If A is closed under addition, then T ∈ 𝒟A󸀠∗ (X). 2. If A ∩ C(ℝ : X) is closed under uniform convergence, T ∈ 𝒟L󸀠 1 ((Mp ) : X) and T ∗ φ ∈ A, φ ∈ 𝒟(Mp ) , then there exists h > 0 such that for each compact set K ⊆ ℝ we M ,h

3.

have T ∗ φ ∈ A, φ ∈ 𝒟K p . In our personal opinion, we need to assume here that T ∈ 𝒟L󸀠 1 ((Mp ) : X) since the set-theoretical equality appearing on the second line of proof of implication [104, (iii) ⇒ (iv)] is mistakenly written. Assume T ∈ 𝒟󸀠(Mp ) (X) and there exists h > 0 such that for each compact set K ⊆ ℝ M ,h we have T ∗ φ ∈ A, φ ∈ 𝒟K p . If (Mp ) additionally satisfies (M.3), then there exist l > 0 and two elements f , g ∈ A such that T = Pl (D)f + g.

The notion of space 𝒟A󸀠∗ (X) can be modified for tempered vector-valued ultradistribution as follows: 𝒮A (X) := {T ∈ 𝒮 (X) : T ∗ φ ∈ A for all φ ∈ 𝒮 }. 󸀠∗

󸀠∗

∗

Concerning some known results on the structure of space 𝒮A󸀠∗ (X), we would like to note that P. Dimovski, B. Prangoski and D. Velinov [163] have examined the convolutors and the space of multipliers of Beurling and Roumieu tempered scalar-valued ultradistributions. We feel duty bound to observe that [163, Proposition 3.2] can be formulated in the vector-valued case as well, and that some implications from the formulation of this result holds even in the case when the sequence (Mp ) does not satisfy (M.3). To make this precise, we introduce the space of vector-valued convolutors O󸀠C (X) of 𝒮 󸀠∗ (X) as the space consisting of all tempered vector-valued ultradistributions T ∈ 𝒮 󸀠∗ (X) such that, for every φ ∈ 𝒮 ∗ , we have T ∗ φ ∈ 𝒮 ∗ (X) and that the mapping φ 󳨃→ T ∗ φ, 𝒮 ∗ → 𝒮 ∗ (X) is continuous. We have the following result: Proposition 2.15.8. Suppose that (Mp ) satisfies conditions (M.1), (M.2) and (M.3󸀠 ). Let T ∈ 𝒮 󸀠∗ (X). Then we have (i) ⇒ (ii) ⇒ (iii) ⇒ (iv), where: (i) T ∈ O󸀠C (X). (ii) For every φ ∈ 𝒮 ∗ , we have T ∗ φ ∈ 𝒮 ∗ (X), i. e., T ∈ 𝒮𝒮󸀠∗∗ (X) (X). (iii) For every φ ∈ 𝒟∗ , we have T ∗ φ ∈ 𝒮 ∗ (X).

182 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations (iv) For every r > 0, resp. there exists r > 0, such that the set {eM(r|h|) Th : h ∈ ℝ} is bounded in 𝒟󸀠∗ (X). Moreover, if (Mp ) satisfies (M.3), then we have (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) ⇔ (v), where: (v) For every r > 0, resp. there exists r > 0, there exists l > 0, resp. there exists a sequence (rp ) of positive real numbers tending to infinity, and two functions f , g ∈ L∞ (ℝ : X) such that T = Pl (D)f + g, resp. T = Trp (D)f + g, and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 sup󵄩󵄩󵄩eM(x|h|) [󵄩󵄩󵄩f (x)󵄩󵄩󵄩 + 󵄩󵄩󵄩g(x)󵄩󵄩󵄩]󵄩󵄩󵄩 < ∞. x∈ℝ

It is almost impossible to summarize here all obtained results about the structure of spaces 𝒟A󸀠∗ (X) and 𝒮A󸀠∗ (X) in the general case. We would like to mention here only one more result in this direction, obtained recently by P. Dimovski, S. Pilipović, B. Prangoski and J. Vindas [162]. They have introduced the notion of a translationinvariant Banach space of tempered ultradistributions of ∗-class and structurally characterized the space 𝒟E󸀠∗󸀠 (ℂ), where E is a translation-invariant Banach space (see [162, Definition 4.1, Theorem 6.1]). It would be very interesting to prove a vector-valued version of this result, as well as vector-valued versions of [81, Theorems 2.3.1 and 2.3.2] and some results from [164] and [284]. In [47, Theorem 2.15], B. Basit and H. Güenzler have proved several equivalent conditions for a vector-valued distribution T ∈ 𝒟󸀠 (X) to belong the space 𝒟𝒮󸀠 󸀠 (X) (X), i. e., that T ∗ φ ∈ 𝒮 󸀠 (X), φ ∈ 𝒟. As an interesting problem for our readers, we would like to address the problem of a structural characterization of vector-valued ultradistributions T ∈ 𝒟󸀠∗ (X) for which T ∗ φ ∈ 𝒮 󸀠∗ (X), φ ∈ 𝒟∗ . This problem seems to be unsolved even in the scalar-valued case.

2.16 Notes and appendices Almost anti-periodic functions. Assume that I = ℝ or I = [0, ∞), as well as that f : I → X is continuous. Given ε > 0, we call τ > 0 an ε-antiperiod for f (⋅) iff 󵄩 󵄩󵄩 󵄩󵄩f (t + τ) + f (t)󵄩󵄩󵄩 ⩽ ε,

t ∈ I.

(150)

In what follows, by ϑap (f , ε) we denote the set of all ε-antiperiods for f (⋅). Motivated by the idea of our colleague and friend D. Velinov, we introduce the notion of an almost anti-periodic function as follows (cf. [279] for more details). Definition 2.16.1. It is said that f (⋅) is almost anti-periodic iff for each ε > 0 the set ϑap (f , ε) is relatively dense in I.

2.16 Notes and appendices | 183

Suppose that τ > 0 is an ε-antiperiod for f (⋅). Applying (150) twice, we get that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩f (t + 2τ) − f (t)󵄩󵄩󵄩 = 󵄩󵄩󵄩[f (t + 2τ) + f (t + τ)] − [f (t + τ) + f (t)]󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩f (t + 2τ) + f (t + τ)󵄩󵄩󵄩 + 󵄩󵄩󵄩f (t + τ) + f (t)󵄩󵄩󵄩 ⩽ 2ε,

t ∈ I.

Taking this inequality in account, we obtain almost immediately from elementary definitions that f (⋅) needs to be almost periodic. Further on, assume that f : I → X is anti-periodic, i. e., there exists ω > 0 such that f (t + ω) = −f (t), t ∈ I; see, e.g., [93] and references cited therein. Then we obtain inductively that f (t + (2k + 1)ω) = −f (t), k ∈ ℤ, t ∈ I. Since the set {(2k + 1)ω : k ∈ ℤ} is relatively dense in I, the above implies that f (⋅) is almost anti-periodic. Therefore, we have demonstrated the following theorem: Theorem 2.16.2. (i) Assume f : I → X is almost anti-periodic. Then f : I → X is almost periodic. (ii) Assume f : I → X is anti-periodic. Then f : I → X is almost anti-periodic. It is well known that any anti-periodic function f : I → X is periodic since, with the notation used above, we have that f (t + 2kω) = f (t), k ∈ ℤ ∖ {0}, t ∈ I. The constant nonzero function is a simple example of a periodic function (therefore, almost periodic function) that is neither anti-periodic nor almost anti-periodic. Example 2.16.3. (i) Consider the function f (t) := sin(πt) + sin(πt √2), t ∈ ℝ. This is an example of an almost anti-periodic function that is not a periodic function. This can be verified as worked out by A. S. Besicovitch [57, Introduction, p. ix]. (ii) The function g(t) := f (t) + 5, t ∈ ℝ, where f (⋅) is defined as above, is almost periodic, not almost anti-periodic and not periodic. We continue by noting the following simple facts. Let f : I → X be continuous, and let ε󸀠 > ε > 0. Then the following holds true: (i) ϑap (f , ε) ⊆ ϑap (f , ε󸀠 ). (ii) If I = ℝ and (150) holds with some τ > 0, then (150) holds with −τ. (iii) If I = ℝ and τ1 , τ2 ∈ ϑap (f , ε), then τ1 ± τ2 ∈ ϑ(f , ε). Furthermore, the argument in the proofs of structural results on [57, pp. 3–4] shows that the following holds: Theorem 2.16.4. Let f : I → X be almost anti-periodic. Then we have: (i) cf (⋅) is almost anti-periodic for any c ∈ ℂ. (ii) If X = ℂ and infx∈ℝ |f (x)| = m > 0, then 1/f (⋅) is almost anti-periodic. (iii) If (gn : I → X)n∈ℕ is a sequence of almost anti-periodic functions and (gn )n∈ℕ converges uniformly to a function g : I → X, then g(⋅) is almost anti-periodic.

184 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations On the other hand, products and sums of almost anti-periodic functions need not be almost anti-periodic: Example 2.16.5. (i) The product of two scalar almost anti-periodic functions need not be almost antiperiodic. To see this, consider the functions f1 (t) = f2 (t) = cos t, t ∈ ℝ, which are clearly (almost) anti-periodic. Then f1 (t) ⋅ f2 (t) = cos2 t, t ∈ ℝ, cos2 (t + τ) + cos2 t ⩾ cos2 t, τ, t ∈ ℝ and therefore ϑap (f1 ⋅ f2 , ε) = 0 for any ε ∈ (0, 1). (ii) The sum of two scalar almost anti-periodic functions need not be almost antiperiodic, so that the almost anti-periodic functions do not form a vector space. To see this, consider the functions f1 (t) = 2−1 cos 4t and f2 (t) = 2 cos 2t, t ∈ ℝ, which are clearly (almost) anti-periodic. Then 3 f1 (t) + f2 (t) = 4 cos4 t − , 2

t ∈ ℝ.

Assume that f1 +f2 is almost anti-periodic. Then the above identity implies that the function t 󳨃→ 8 cos4 t − 3, t ∈ ℝ is almost anti-periodic as well. This, in particular, implies that for any ε ∈ (0, 1) we can find τ ∈ ℝ such that 󵄨 󵄨󵄨 4 4 󵄨󵄨8 cos (t + τ) + 8 cos t − 6󵄨󵄨󵄨 ⩽ ε,

t ∈ ℝ.

Plugging t = π, we get that 8 cos4 τ + 2 ⩽ ε, which is a contradiction. Finally, we would like to point out that there exist many much simpler examples which can be used for verification of the statement clarified in this part; for example, the interested reader can easily check that the function t 󳨃→ cos t + cos 2t, t ∈ ℝ is not almost anti-periodic. Assume that f : I → X is almost anti-periodic. Then it can be easily seen that f (⋅ + a) and f (b⋅) are likewise almost anti-periodic, where a ∈ I and b ∈ I ∖ {0}. Denote now by ANP0 (I : X) the linear span of almost anti-periodic functions I 󳨃→ X. By Theorem 2.16.2(i), ANP0 (I : X) is a linear subspace of AP(I : X). Let ANP(I : X) be the linear closure of ANP0 (I : X) in AP(I : X). Then, clearly, ANP(I : X) is a Banach space. Furthermore, we have the following result: Theorem 2.16.6. ANP(I : X) = APℝ∖{0} (I : X). Proof. Since the mapping E : AP([0, ∞) : X) → AP(ℝ : X) is a linear surjective isometry, it suffices to consider the case in which I = ℝ. Assume first that f ∈ APℝ∖{0} (I : X). By the foregoing, we have that f ∈ span{eiμ⋅ x : μ ∈ σ(f ), x ∈ R(f )}, where the closure is taken in the space AP(ℝ : X). Since σ(f ) ⊆ ℝ∖{0} and the function t 󳨃→ eiμt , t ∈ ℝ (μ ∈ ℝ ∖ {0}) is anti-periodic, we have that span{eiμ⋅ x : μ ∈ σ(f ), x ∈

2.16 Notes and appendices | 185

R(f )} ⊆ ANP0 (ℝ : X). Hence, f ∈ ANP(ℝ : X). The converse statement immediately follows if we prove that, for any fixed function f ∈ ANP(ℝ : X), we have that P0 (f ) = 0, i. e., t

1 ∫ f (s) ds = 0. t→∞ t lim

(151)

0

By almost periodicity of f (⋅), the limit in (151) exists. Hence, it is enough to show that for any given number ε > 0 we can find a sequence (ωn )n∈ℕ of positive reals such that limn→∞ ωn = ∞ and 2ωn 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 ⩽ ε/2, 󵄩󵄩 f (s) ds ∫ 󵄩󵄩 󵄩󵄩 2ω 󵄩󵄩 󵄩󵄩 n 0

n ∈ ℕ.

(152)

By definition of almost anti-periodicity, we have the existence of a number l > 0 such that any interval In = [nl, (n + 1)l] (n ∈ ℕ) contains a number ωn that is anti-period for f (⋅). The validity of (152) is a consequence of the following computation: 2ωn 󵄩󵄩 󵄩󵄩 󵄩󵄩 ωn 󵄩󵄩 2ωn 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∫ f (s) ds󵄩󵄩 = 󵄩󵄩 ∫ f (s) ds + ∫ f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩0 󵄩0 ωn

󵄩󵄩 ωn 󵄩󵄩󵄩 󵄩󵄩 󵄩 = 󵄩󵄩󵄩 ∫ [f (s) + f (s + ωn )] ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩0 ωn

󵄩 󵄩 ⩽ ∫ 󵄩󵄩󵄩f (s) + f (s + ωn )󵄩󵄩󵄩 ds ⩽ εωn ,

n ∈ ℕ,

0

finishing the proof of theorem. Let f ∈ AP(I : X) and 0 ≠ Λ ⊆ ℝ. Since σ(f ) ⊆ Λ iff Pr (f ) = 0, r ∈ ℝ ∖ Λ

iff P0 (e−ir⋅ f (⋅)) = 0, r ∈ ℝ ∖ Λ,

we have the following corollary of Theorem 2.16.6 (see also [29, Corollary 4.5.9]): Corollary 2.16.7. Let f ∈ AP(I : X) and 0 ≠ Λ ⊆ ℝ. Then f ∈ APΛ (I : X) iff e−ir⋅ f (⋅) ∈ ANP(I : X) for all r ∈ ℝ ∖ Λ. Furthermore, Theorem 2.16.6, combined with the obvious equality σ(Ef ) = σ(f ), immediately implies that the unique ANP extension of a function f ∈ ANP([0, ∞) : X) to the whole real axis is Ef (⋅). As the next proposition shows, this also holds for almost anti-periodic functions: Proposition 2.16.8. Suppose that f : [0, ∞) → X is almost anti-periodic. Then Ef : ℝ → X is a unique almost anti-periodic extension of f (⋅) to the whole real axis.

186 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations Proof. The uniqueness of an almost anti-periodic extension of f (⋅) follows from the uniqueness of an almost periodic extension of f (⋅). It remains to prove that Ef : ℝ → X is almost anti-periodic. To see this, let ε > 0 be given. Then there exists l > 0 such that any interval I ⊆ [0, ∞) of length l contains a number τ ∈ I such that ‖f (s+τ)+f (s)‖ ⩽ ε, s ⩾ 0. We only need to prove that any interval I ⊆ ℝ of length 2l contains a number τ ∈ I such that 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩[Ef ](t + τ) + [Ef ](t)󵄩󵄩󵄩 = 󵄩󵄩󵄩[W(t + τ)f + W(t)f ](0)󵄩󵄩󵄩 ⩽ ε,

t ∈ ℝ.

If I ⊆ [0, ∞), then the situation is completely clear. Suppose now that I ⊆ (−∞, 0]. Then −I ⊆ [0, ∞) and there exists a number −τ ∈ −I such that sups⩾0 ‖f (s−τ)+f (s)‖ ⩽ ε. Then the conclusion follows from the computation 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩[W(t + τ)f + W(t)f ](0)󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩W(t + τ)f + W(t)f 󵄩󵄩󵄩L∞ ([0,∞)) 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩W(t + τ)󵄩󵄩󵄩L∞ ([0,∞)) 󵄩󵄩󵄩W(−τ)f + f 󵄩󵄩󵄩L∞ ([0,∞)) 󵄩 󵄩 = sup󵄩󵄩󵄩f (s − τ) + f (s)󵄩󵄩󵄩 ⩽ ε, t ∈ ℝ. s⩾0

Finally, if I = I1 ∪ I2 , where I1 = [a, 0] (a < 0) and I2 = [0, b] (b > 0), then |a| ⩾ l or b ⩾ l. In the case |a| ⩾ l, the conclusion follows similarly as in the previously examined case. If b ⩾ l, then the conclusion follows from the computation 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩[W(t + τ)f + W(t)f ](0)󵄩󵄩󵄩 ⩽ 󵄩󵄩󵄩W(t + τ)f + W(t)f 󵄩󵄩󵄩L∞ ([0,∞)) 󵄩 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩W(t)󵄩󵄩󵄩L∞ ([0,∞)) 󵄩󵄩󵄩W(τ)f + f 󵄩󵄩󵄩L∞ ([0,∞)) 󵄩 󵄩 = sup󵄩󵄩󵄩f (s + τ) + f (s)󵄩󵄩󵄩 ⩽ ε, t ∈ ℝ, s⩾0

where τ ∈ I2 is an ε-antiperiod of f (⋅). Since almost anti-periodic functions do not form a vector space, we will focus our attention here to the almost anti-periodic properties of the finite and infinite convolution product, which is undoubtedly a safe and sound way for providing certain applications to abstract PDEs. Proposition 2.16.9. Suppose that 1 ⩽ p < ∞, 1/p + 1/q = 1 and (R(t))t>0 ⊆ L(X) is a strongly continuous operator family such that M = ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞. If g : ℝ → X is Sp -almost anti-periodic, then the function G(⋅), given by (68), is well-defined and almost anti-periodic. Proof. By the foregoing, we have that function G(⋅) is well-defined and almost periodic. It remains to prove that G(⋅) is almost anti-periodic. Let a number ε > 0 be given in advance. Then we can find a finite number l > 0 such that any subinterval I of ℝ of t+1 length l contains a number τ ∈ I such that ∫t ‖g(s + τ) + g(s)‖p ds ⩽ εp , t ∈ ℝ. Applying Hölder inequality and this estimate, similarly as in the proof of above-mentioned

2.16 Notes and appendices | 187

proposition, we get that ∞

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩G(t + τ) + G(t)󵄩󵄩󵄩 ⩽ ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩g(t + τ − r) + g(t − r)󵄩󵄩󵄩 dr 0

∞ k+1

󵄩 󵄩 󵄩 󵄩 = ∑ ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩 ⋅ 󵄩󵄩󵄩g(t + τ − r) + g(t − r)󵄩󵄩󵄩 dr k=0 k ∞

1/p

k+1

󵄩p 󵄩 󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩g(t + τ − r) + g(t − r)󵄩󵄩󵄩 dr) k=0 ∞

k

t−k

󵄩 󵄩 󵄩p 󵄩 = ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩g(s + τ) + g(s)󵄩󵄩󵄩 ds) k=0 ∞

t−k−1

󵄩 󵄩 ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ε = Mε, k=0

1/p

t ∈ ℝ,

which clearly implies that the set of all ε-antiperiods of G(⋅) is relatively dense in ℝ. In order to relax our exposition, we shall introduce the notion of an asymptotically (S -)almost anti-periodic function in the following way: p

Definition 2.16.10. (i) Let f ∈ Cb ([0, ∞) : X). Then we say that f (⋅) is asymptotically almost anti-periodic iff there are two functions g : ℝ → X and q : [0, ∞) → X satisfying the following conditions: (a) g is almost anti-periodic; (b) q belongs to the class C0 ([0, ∞) : X); (c) f (t) = g(t) + q(t) for all t ⩾ 0. (ii) Let 1 ⩽ p < ∞, and let f ∈ Lploc ([0, ∞) : X). Then we say that f (⋅) is asymptotically Stepanov p-almost anti-periodic, asymptotically Sp -almost anti-periodic shortly, iff there are two locally p-integrable functions g : ℝ → X and q : [0, ∞) → X satisfying the following conditions: (a) g is Sp -almost anti-periodic; (b) q̂ belongs to the class C0 ([0, ∞) : Lp ([0, 1] : X)); (c) f (t) = g(t) + q(t) for all t ⩾ 0. Keeping in mind Proposition 2.16.9, we can immediately clarify that the assertion of Proposition 2.6.13 and some conclusions from Remark 2.6.14 continue to hold for asymptotical (Sp -)almost anti-periodicity. Almost periodic solutions of abstract higher-order Cauchy problems with integer order derivatives. As already mentioned, the almost periodic solutions of complete second order Cauchy problem

188 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations u󸀠󸀠 (t) + Au󸀠 (t) + Bu(t) = 0,

t ⩾ 0,

have been considered by T.-J. Xiao and J. Liang [402, Section 7.1.2] provided that this equation is strongly well-posed, i. e., that the requirements of [186, Chapter VIII, Theorem 3.2] hold. The authors have paid special attention to the equation u󸀠󸀠 (t) + (aA0 + bI)u󸀠 (t) + (cA0 + dI)u(t) = 0,

t ⩾ 0,

with a, b, c, d ∈ ℂ and A0 being a closed linear operator with domain and range contained in a Banach space X. A noteworthy application is made to the case in which X = L2 [0, 1] and A0 is the Dirichlet Laplacian. In [227], N. Iraniparast and L. Nguyen have considered the following abstract Cauchy problem of higher-order: n−1

u(n) (t) = ∑ Ak u(k) (t) + f (t), k=0

t ∈ ℝ,

(153)

with Ak being closed linear operators acting on a Banach space X and f (⋅) being an X-valued continuous function. They have firstly introduced a new notion of mild solutions of this equation and after that examined some sufficient conditions for the solvability of operator equation n−1

B( ∑ Bj XDj ) − XDn = C, j=0

(154)

considering particularly the case of D = d/dx, the differential operator on a function space, and C = −δ0 , roughly speaking. Suppose that M is a closed, translationinvariant subspace of BUC(ℝ : X). Then M is called regularly admissible with respect to (153) iff for every f ∈ M equation (153) has a unique mild solution u ∈ M. The authors of [227] have shown that the regular admissibility of M is equivalent to saying that the operator equation (154) has a unique bounded solution, providing also some consequences of this result in the study of existence and uniqueness of almost periodic solutions of equation (153); see [227, Section 5] and references cited in [227] for further information about the regular admissibility of translation-invariant subspace of BUC(ℝ : X) with respect to some other types of equations. We would like to note that a wide class of abstract higher-order differential equations with asymptotically almost periodic (automorphic) solutions can be uncovered by using the ideas of F. Neubrander [335]. In order to explain this fact for second-order differential equations, let us assume that the operator A generates a strongly continuous semigroup on X as well as that B is a closed densely defined operator on X with D(A) ⊆ D(B). Due to [335, Theorem 3], there exists a number ω > 0 such that, for every λ ∈ ℂ with Re λ > ω, the matrix operator D := [

A−λ B

I ] −λ

2.16 Notes and appendices | 189

generates an exponentially decaying strongly continuous semigroup (T(t))t⩾0 on X×X. Let u1 ∈ D(A), u2 ∈ X, let λ be as above, and let t 󳨃→ f1,2 (t), t ⩾ 0 be continuously differentiable and asymptotically almost periodic (automorphic). Making use of a simple computation with variation of parameters formula, we get that the abstract Cauchy problem u󸀠󸀠 (t) − (A − 2λ)u󸀠 (t) − [λ(A − λ) + B]u(t) = f1󸀠 (t) + λf1 (t) + f2 (t), u(0) = u1 ,

u (0) = u2 + (A − λ)u1 + f1 (0),

t ⩾ 0;

󸀠

has a unique asymptotically almost periodic (automorphic) solution t 󳨃→ u1 (t), t ⩾ 0, satisfying additionally that the mapping t 󳨃→ u󸀠1 (t)−(A−λ)u1 (t), t ⩾ 0 is asymptotically almost periodic (automorphic), as well; cf. [402, Definition 1.1] for the notion. Almost periodic and almost automorphic functions on time scales. Applications to Volterra integro-differential equations. Many authors have recently analyzed almost periodic and almost automorphic functions on time scales and their applications to the abstract Volterra integro-differential equations. For basic definitions and results of time scale calculus, we refer the reader to the monograph [65] by M. Bochner and A. Peterson. A time scale 𝕋 is nothing else but a nonempty closed subset of the real numbers. The forward and backward jump operators σ, ρ : 𝕋 → 𝕋 and the forward graininess ν : 𝕋 → [0, ∞) are defined respectively by σ(t) := inf{s ∈ 𝕋 : s > t}, ρ(t) := sup{s ∈ 𝕋 : s < t} and ν(t) := σ(t) − t. It is said that a point t is left-dense if t > inf 𝕋 and ρ(t) = t, right-dense if t < sup 𝕋 and ρ(t) = t, left-scattered if σ(t) < t and right-scattered if σ(t) > t. If 𝕋 has a left-scattered maximum m, then 𝕋κ := v ∖ {m}, otherwise 𝕋κ = 𝕋. If 𝕋 has a right-scattered minimum m, then 𝕋κ = ∖{m}, otherwise 𝕋κ = 𝕋. A function f : 𝕋 → X is said to be right-dense continuous or rd-continuous iff it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋. Assuming f (⋅) is continuous at each right-dense point and each left-dense point, then f (⋅) is said to be a continuous function on 𝕋. Definitions of delta differentiability and delta integral are introduced as follows. For f : 𝕋 → ℂ and t ∈ 𝕋κ , we define the delta-derivative f Δ (t) to be the number (if it exists) with the following property: given ε > 0, there exists a neighborhood U of t such that 󵄨 󵄨 󵄨 󵄨󵄨 Δ 󵄨󵄨f (σ(t)) − f (s) − f (t)[σ(t) − s]󵄨󵄨󵄨 < ε󵄨󵄨󵄨σ(t) − s󵄨󵄨󵄨,

s ∈ U.

Assume that f (⋅) is right-dense continuous and ϒΔ (t) = f (t). Roughly speaking, we define the delta integral by t

∫ f (s)Δs := ϒ(t) − ϒ(a). a

190 | 2 Almost periodic type solutions of abstract Volterra integro-differential equations We refer the reader to [65] for the notion of a regressive function, elementary operations with regressive functions, as well as for the notion of a generalized exponential function. A subset S of 𝕋 is called relatively dense iff there exists a positive number L > 0 such that [a, a + L] ∩ S ≠ 0 for all a ∈ 𝕋. The number L is called the inclusion length. Let 𝒞 be a collection of sets which is constructed by subsets of ℝ. A time scale 𝕋 is called an almost periodic time scale with respect to 𝒞 , iff 𝒞 := {±τ ∈ ⋂ c : t ± τ ∈ 𝕋, ∀t ∈ 𝕋} ≠ 0 ∗

c∈𝒞

and 𝒞 ∗ is called the smallest almost periodic set of 𝕋. Let 𝕋 be an almost periodic time scale with respect to 𝒞 . A function f : 𝕋 → X is said to be almost periodic iff for any given ε > 0, in advance, the set 󵄩 󵄩 E(ε, f ) := {τ ∈ 𝒞 ∗ : 󵄩󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 < ε, ∀t ∈ 𝕋} is relatively dense in 𝕋. The notion of an almost automorphic function on invariant under translation time scales has been introduced by C. Lizama and J. G. Mesquita in [314]. For various applications of almost periodic and almost automorphic functions on time scales in the theory of abstract Volterra integro-differential equations, one may refer, e. g., to [10, 34, 289, 301, 314, 315, 326, 383] and the references cited therein. Quasi-asymptotically almost periodic functions and their Stepanov generalizations. Let 1 ⩽ p < ∞. In our recent research papers [274] and [281], we have investigated the classes of quasi-asymptotically almost periodic functions and Stepanov p-quasiasymptotically almost periodic functions in Banach spaces. These classes extend the well known classes of asymptotically almost periodic functions and asymptotically Stepanov p-almost periodic functions with values in Banach spaces. Furthermore, we have proved that the class of Stepanov p-quasi-asymptotically almost periodic functions contains all asymptotically Stepanov p-almost periodic functions and forms a subclass of the class consisting of all Weyl p-almost periodic functions. In such a way, in [281] we have initiated the study of generalized (asymptotical) almost periodicity that intermediate Stepanov and Weyl concept. The definition goes as follows (see also A. S. Kovanko [285]): Definition 2.16.11. (i) Suppose that I = [0, ∞) or I = ℝ. Then we say that a bounded continuous function f : I → X is quasi-asymptotically almost periodic iff for each ε > 0 there exists a finite number L(ε) > 0 such that any interval I 󸀠 ⊆ I of length L(ε) contains at least one number τ ∈ I 󸀠 such that there exists a finite number M(ε, τ) > 0 for which 󵄩 󵄩󵄩 󵄩󵄩f (t + τ) − f (t)󵄩󵄩󵄩 ⩽ ε,

provided t ∈ I and |t| ⩾ M(ε, τ).

2.16 Notes and appendices | 191

(ii) Let f ∈ LpS (I : X). Then f (⋅) is called Stepanov p-quasi-asymptotically almost periodic iff for each ε > 0 there exists a finite number L(ε) > 0 such that any interval I 󸀠 ⊆ I of length L(ε) contains at least one number τ ∈ I 󸀠 such that there exists a finite number M(ε, τ) > 0 for which t+1

󵄩p 󵄩 ∫ 󵄩󵄩󵄩f (s + τ) − f (s)󵄩󵄩󵄩 ds ⩽ εp , provided t ∈ I and |t| ⩾ M(ε, τ). t

A quasi-asymptotically almost periodic function is not necessarily uniformly continuous and its range is not necessarily relatively compact in X. The (Stepanov) class of S-asymptotically ω-periodic functions, introduced by H. R. Henríquez, M. Pierri and P. Táboas in [222], is a subclass of the class consisting of the (Stepanov) class of quasi-asymptotically almost periodic functions (for further information concerning S-asymptotically ω-periodic functions and their Stepanov generalizations, we refer the reader to [63, 116, 160, 298] and [406]). Although possessing many nice properties, primarily with regards to their invariance under the actions of finite and infinite convolution products, the (Stepanov) quasi-asymptotically almost periodic functions do not form vector spaces equipped with the usual operations of addition and multiplication with scalars, unfortunately. In [281], we have also analyzed (Stepanov) quasi-asymptotically almost periodic functions depending on two parameters and related composition principles. Several applications to abstract nonautonomous differential equations of first order have been given. Asymptotically periodic and asymptotically almost periodic properties of Riemann– Liouville integral of order α ∈ (0, 1). In [24, 25], I. Area, J. Losada and J. J. Nieto have analyzed quasi-periodic properties of fractional integrals and fractional derivatives of scalar-valued periodic functions. Let α ∈ (0, 1). They have paid special attention on the study of asymptotically quasi-periodic properties of Riemann–Liouville integral of order α. More precisely, in [24, Theorem 1] and [25, Theorem 8], they have proved that the t Riemann–Liouville integral I α f (t) := ∫0 gα (t − s)f (s) ds, t ∈ ℝ of a nonzero T-periodic scalar-valued function f ∈ L1 (ℝ) cannot be T-periodic, for any T > 0, as well as that T I α f (⋅) is bounded on ℝ iff ∫0 f (s) ds = 0, for any T > 0. By [25, Theorems 2, 5, and 9], the boundedness of function I α f (⋅) implies that I α f (⋅) is both S-asymptotically T-periodic and asymptotically T-periodic function, for any continuous T-periodic scalar-valued function f (⋅), T > 0, as well as that I α f (⋅) cannot be an almost periodic function (cf. [25] for the notion). We close the chapter with the observation that asymptotically periodic and asymptotically almost periodic properties of Riemann–Liouville integrals are very unexplored in the vector-valued case and that almost nothing has been said about asymptotical almost automorphic properties of Riemann–Liouville integrals so far.

3 Almost automorphic type solutions of abstract Volterra integro-differential equations In this chapter, we continue our previous investigations by enquiring into the existence and uniqueness of various almost automorphic and asymptotically almost automorphic type solutions of the abstract degenerate Volterra integro-differential equations and abstract degenerate (multiterm) fractional differential equations in Banach spaces. The pivot space (X, ‖ ⋅ ‖) is assumed to be a complex Banach space; if we need to have two different pivot spaces, we will use the symbols Y, Z, . . . to denote them. The injectiveness of regularizing operator C ∈ L(X), if needed, will be explicitly emphasized.

3.1 Almost automorphic functions, asymptotically almost automorphic functions and their generalizations For the basic information about almost automorphic functions, asymptotically almost automorphic functions and their generalizations, we refer the reader to [4, 64, 84, 132, 140, 143–145, 147–149, 155, 206, 355, 382] and [387, 388]. Let f : ℝ → X be continuous. As it is well known, f (⋅) is called almost automorphic, a. a. for short, iff for every real sequence (bn ) there exist a subsequence (an ) of (bn ) and a map g : ℝ → X such that lim f (t + an ) = g(t) and

n→∞

lim g(t − an ) = f (t),

n→∞

(155)

pointwise for t ∈ ℝ. If this is the case, then it is well known that f ∈ Cb (ℝ : X) and that the limit function g(⋅) must be bounded on ℝ but not necessarily continuous on ℝ. Furthermore, it is clear that the uniform convergence of one of the limits appearing in (155) implies the convergence of the second one in this equation and that, in this case, the function f (⋅) has to be almost periodic and the function g(⋅) has to be continuous. If the convergence of limits appearing in (155) is uniform on compact subsets of ℝ, then we say that f (⋅) is compactly almost automorphic, c. a. a. for short. The vector space consisting of all almost automorphic, resp. compactly almost automorphic functions, is denoted by AA(ℝ : X), resp. AAc (ℝ : X). By Bochner’s criterion [132], any almost periodic function has to be compactly almost automorphic. The converse statement is not true, however [192]. It is also worth noting that P. Bender proved in his doctoral dissertation that a. a. function f (⋅) is c. a. a. iff it is uniformly continuous (1966, Iowa State University). It is well-known that the reflection at zero keeps the spaces AA(ℝ : X) and AAc (ℝ : X) unchanged, as well as that the function g(⋅) from (155) satisfies ‖f ‖∞ = ‖g‖∞ and R(g) ⊆ R(f ), later needed to be a compact subset of X. The concept of almost automorphy coincides with that of Levitan N-almost periodicity (see [44] and [295] for more https://doi.org/10.1515/9783110641851-003

194 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations details in this direction) and the almost automorphy of a function f : ℝ → X can be defined also in the following equivalent way: A function f : ℝ → X is said to be almost automorphic iff for every real sequence (bn ) there exist a subsequence (an ) of (bn ) such that lim lim f (t + an − am ) = f (t),

m→∞ n→∞

t ∈ ℝ.

An interesting example of an almost automorphic function that is not almost periodic has been constructed by W. A. Veech f (t) :=

2 + eit + eit

√2

|2 + eit + eit √2 |

,

t ∈ ℝ.

A continuous function f : ℝ → X is called asymptotically (compact) almost automorphic, a. (c.) a. a. for short, iff there exist a function h ∈ C0 ([0, ∞) : X) and a (compact) almost automorphic function q : ℝ → X such that f (t) = h(t) + q(t), t ⩾ 0. Using Bochner’s criterion again, it readily follows that any asymptotically almost periodic function [0, ∞) 󳨃→ X is asymptotically (compact) almost automorphic. It is well known that the spaces of almost periodic, almost automorphic, compactly almost automorphic functions, and asymptotically (compact) almost automorphic functions are closed subspaces of Cb (ℝ : X) equipped with the sup-norm. A function f ∈ Lploc (ℝ : X) is called Stepanov p-almost automorphic, Sp -almost automorphic or Sp -a. a. shortly (see, e. g., [211] and [185]), iff for every real sequence (an ), there exists a subsequence (ank ) and a function g ∈ Lploc (ℝ : X) such that t+1

󵄩p 󵄩 lim ∫ 󵄩󵄩󵄩f (ank + s) − g(s)󵄩󵄩󵄩 ds = 0 k→∞

(156)

t

and t+1

󵄩p 󵄩 lim ∫ 󵄩󵄩󵄩g(s − ank ) − f (s)󵄩󵄩󵄩 ds = 0 k→∞

(157)

t

for each t ∈ ℝ; a function f ∈ Lploc ([0, ∞) : X) is called asymptotically Stepanov p-almost automorphic, asymptotically Sp -almost automorphic or asymptotically Sp -a. a. shortly, iff there exist an Sp -almost automorphic function g(⋅) and a function q ∈ LpS ([0, ∞) : X) such that f (t) = g(t) + q(t), t ⩾ 0 and q̂ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)). It can be easily verified that the Sp -almost automorphy of f (⋅) implies the compact almost automorphy of the mapping f ̂ : I → Lp ([0, 1] : X) defined above, with the limit function being g(⋅)(s) := g(s + ⋅) for a. e. s ∈ [0, 1], so that any Sp -almost automorphic function f (⋅) has to be Sp -bounded (1 ⩽ p < ∞). The vector space consisting of all Sp -almost automorphic functions is closed under translations and reflections at zero of the argument, and any limit of Sp -almost automorphic functions (fk ) converging to

3.1 Almost automorphic functions, asymptotically almost automorphic functions | 195

some p-locally integrable X-valued function f (⋅) in Sp -norm has the property that f (⋅) is Sp -almost automorphic. The vector space consisting of all Sp -almost automorphic functions, resp. asymptotically Sp -almost automorphic functions, will be denoted by AASp (ℝ : X), resp. AAASp ([0, ∞) : X). By (asymptotical) Stepanov almost automorphy we mean (asymptotical) Stepanov 1-almost automorphy. If 1 ⩽ p < q < ∞ and f (⋅) is Stepanov q-almost automorphic, resp. Stepanov q-almost periodic, then f (⋅) is Stepanov p-almost automorphic, resp. Stepanov palmost periodic (see, e. g., [145, Remark 2.15]). Furthermore, the (asymptotical) Stepanov p-almost periodicity of f (⋅) for some p ∈ [1, ∞) implies the (asymptotical) Stepanov p-almost automorphy of f (⋅). It is a well-known fact that if f (⋅) is an almost periodic (resp. a. a. p., a. a., a. a. a.) function then f (⋅) is also Sp -almost periodic (resp. asymptotically Sp -a. p., Sp -a. a., asymptotically Sp -a. a.) for 1 ⩽ p < ∞. The converse statement is false, however. Example 3.1.1 ([167]). Let ε ∈ (0, 1/2), and let f (t) := sin(1/(2 + cos n + cos √2n)), provided that n ∈ ℤ and t ∈ (n − ε, n + ε). Otherwise, we define f (t) := 0. Then for each p ∈ [1, ∞) we have that f (⋅) is Sp -almost automorphic. To the best of our knowledge, in the existing literature concerning Stepanov almost automorphic functions, the authors have examined only such functions that are Stepanov p-almost automorphic for any exponent p ∈ [1, ∞). Therefore, it is natural to ask whether there exists a Stepanov almost automorphic function that is not Stepanov p-almost automorphic for a certain exponent p ∈ (1, ∞). The answer is affirmative and, without going into full problematic concerning this and similar questions, we would like to recall that H. Bohr and E. Fölner have constructed, for any given number p > 1, a Stepanov almost periodic function defined on the whole real axis that is Stepanov p-bounded and not Stepanov p-almost periodic (see [69, Example, pp. 70–73]). This function, denoted here by f (⋅), is clearly Stepanov almost automorphic and now we will prove that f (⋅) cannot be Stepanov p-almost automorphic (see [152] for more details). Consider, for the sake of simplicity, the case h1 = 2 in the afore-mentioned example and suppose on the contrary. Then the mapping f ̂ : ℝ → Lp ([0, 1] : X) is compactly almost automorphic. Since the class of almost automorphic functions coincides with the class of Levitan N-almost periodic functions, it readily follows that, for every ε > 0 and N > 0, there exists a finite number L > 0 such that any interval I ⊆ ℝ contains a number τ ∈ I such that ‖f ̂(t ± τ) − f ̂(t)‖Lp ([0,1]:X) < ε. In particular, with ε > 0 arbitrarily small and N = 3/2, we get the existence of a finite number L > 0 such that any interval I ⊆ ℝ ∖ [−1, 1] contains a number τ ∈ I such that x+1

󵄨p 󵄨 ∫ 󵄨󵄨󵄨f (s + τ) − f (s)󵄨󵄨󵄨 ds < εp , x

With x = −3/2, this implies

x ∈ [−3/2, 3/2].

196 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations 3/2

󵄨p 󵄨 ∫ 󵄨󵄨󵄨f (s + τ) − f (s)󵄨󵄨󵄨 ds < 2εp ,

−3/2

3/2

which is in contradiction with the estimate ∫−3/2 |f (s + τ) − f (s)|p ds ⩾ 2−p (see [69, p. 73, l.-9–l.-4]). The concepts of Weyl almost automorphy and Weyl pseudo almost automorphy, more general than those of Stepanov almost automorphy and Stepanov pseudo almost automorphy, were introduced by S. Abbas [1] in 2012: Definition 3.1.2. Let p ⩾ 1. Then we say that a function f ∈ Lploc (ℝ : X) is Weyl p-almost automorphic iff for every real sequence (sn ), there exist a subsequence (snk ) and a function f ∗ ∈ Lploc (ℝ : X) such that l

1 󵄩󵄩 󵄩p ∫󵄩󵄩f (t + snk + x) − f ∗ (t + x)󵄩󵄩󵄩 dx = 0 k→+∞ l→+∞ 2l lim lim

(158)

−l

and

l

1 󵄩󵄩 ∗ 󵄩p ∫󵄩󵄩f (t − snk + x) − f (t + x)󵄩󵄩󵄩 dx = 0 k→+∞ l→+∞ 2l lim lim

(159)

−l

for each t ∈ ℝ. The set of all such functions are denoted by W p AA(ℝ : X). The set W p AA(ℝ : X), equipped with the usual operations of pointwise addition of functions and multiplication of functions with scalars, has a linear vector structure. We can simply prove this fact in the following way. Let (sn ) be an arbitrary real sequence. Then there exist a subsequence (snk ) and a function f ∗ ∈ Lploc (ℝ : X) such that (158)–(159) hold. By the Weyl p-almost automorphy of g(⋅), we get the existence of subsequence (snk m ) of (snk ) and a function g ∗ ∈ Lploc (ℝ : X) such that l

1 󵄩󵄩 󵄩p lim lim ∫󵄩g(t + snk m + x) − g ∗ (t + x)󵄩󵄩󵄩 dx = 0 m→∞ l→+∞ 2l 󵄩

(160)

−l

and

l

1 󵄩󵄩 ∗ 󵄩p lim lim ∫󵄩g (t − snk m + x) − g(t + x)󵄩󵄩󵄩 dx = 0. m→∞ l→+∞ 2l 󵄩

(161)

−l

Since (160)–(161) hold with g and g ∗ replaced therein with f and f ∗ , we get that, for a linear combination αf + βg, we can choose a subsequence (snk m ) of (sn ) and a limit function αf ∗ + βg ∗ satisfying all the requirements from Definition 3.1.2 (α, β ∈ ℂ). As stated by S. Abbas [1, p. 5, l. 2–3], without giving a corresponding proof, Weyl-p-almost periodic functions form a linear submanifold of W p AA(ℝ : X). We continue by providing the following illustrative example.

3.1 Almost automorphic functions, asymptotically almost automorphic functions | 197

Example 3.1.3. Let f (x) := χ(0,1/2) (x), x ∈ ℝ, where χ(0,1/2) (⋅) denotes the characteristic function of (0, 1/2). Then we already know that this function is equi-Weyl-1-almost periodic and not Stepanov p-almost periodic for 1 ⩽ p < ∞. A very simple analysis shows that f (⋅) is not Stepanov p-almost automorphic for 1 ⩽ p < ∞ as well as that f (⋅) is Weyl 1-almost automorphic. For the sake of completeness, we will prove only that f (⋅) cannot be Stepanov p-almost automorphic for 1 ⩽ p < ∞. If this is the case, then there exist a subsequence of (an := n2 ) and a function g ∈ Lploc (ℝ : X) such that (156)–(157) hold pointwise for each t ∈ ℝ. But, for any t ∈ ℝ and for any k0 ∈ ℕ suft+1 t+1 ficiently large, we have that ∫t ‖f (ank + s) − g(s)‖p ds = ∫t ‖g(s)‖p ds. Due to (156), we get that g(s) = 0 for a. e. s ∈ ℝ. Coming back to (157), we get that f (s) = 0 for a. e. s ∈ ℝ. Thus we arrive at a contradiction. The class of Besicovitch almost automorphic functions has been analyzed by F. Bedouhene, N. Challali, O. Mellah, P. Raynaud de Fitte and M. Smaali in [54]. This class extends the class of Weyl almost automorphic functions and its full importance lies in the fact that we do allow now the possible nonexistence of the limit l

1 󵄩󵄩 󵄩p lim ∫󵄩󵄩f (t + snk + x) − f ∗ (t + x)󵄩󵄩󵄩 dx, l→+∞ 2l −l

resp. l

1 󵄩󵄩 ∗ 󵄩p ∫󵄩󵄩f (t − snk + x) − f (t + x)󵄩󵄩󵄩 dx l→+∞ 2l lim

−l

in (158), resp. (159). As it is well-known, the limit superiors of these functions always exist and this will be the heart of the matter for the proofs of Propositions 3.1.6 and 3.5.5 below to work: Definition 3.1.4. Let p ⩾ 1. Then we say that a function f ∈ Lploc (ℝ : X) is Besicovitch p-almost automorphic iff for every real sequence (sn ), there exist a subsequence (snk ) and a function f ∗ ∈ Lploc (ℝ : X) such that l

1 󵄩 󵄩p lim lim sup ∫󵄩󵄩󵄩f (t + snk + x) − f ∗ (t + x)󵄩󵄩󵄩 dx = 0 k→∞ l→+∞ 2l

(162)

−l

and l

lim lim sup

k→∞ l→+∞

1 󵄩󵄩 ∗ 󵄩p ∫󵄩f (t − snk + x) − f (t + x)󵄩󵄩󵄩 dx = 0 2l 󵄩 −l

for each t ∈ ℝ. The set of all such functions is denoted by Bp AA(ℝ : X).

(163)

198 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations As in the case of Weyl almost automorphic functions, we can prove that the set Bp AA(ℝ : X), equipped with the usual operations, has a linear vector structure. To the best knowledge of the author, it is not clear how we can prove that a Besicovitch p-almost periodic function is Besicovitch p-almost automorphic. We have already proved that any Besicovitch p-vanishing function is Besicovitchp-almost periodic on [0, ∞) (Besicovitch–Doss-p-almost periodic on [0, ∞)). A similar statement holds for the Besicovitch p-almost automorphy: Proposition 3.1.5. Let q ∈ Bp0 ([0, ∞) : X). Define qe (t) := q(t) for t ⩾ 0, and qe (t) := 0 for t < 0. Then qe ∈ Bp AA(ℝ : X). Proof. Let a real sequence (sn ) be given. It suffices to show that, for every t ∈ ℝ and for every subsequence (snk ) of (sn ), we have: l

1 󵄩󵄩 󵄩p lim ∫󵄩󵄩qe (t + snk + x)󵄩󵄩󵄩 dx = 0 l→+∞ 2l

(164)

−l

and l

1 󵄩󵄩 󵄩p lim ∫󵄩󵄩qe (t + x)󵄩󵄩󵄩 dx = 0. l→+∞ 2l −l

We will prove only (164). Towards this end, it suffices to observe that l

1 1 󵄩󵄩 󵄩p ∫󵄩q (t + snk + x)󵄩󵄩󵄩 dx = 2l 󵄩 e 2l −l

1 ⩽ 2l

⩽

l+|t+snk |

∫ −l−|t+snk |

∫ −l+t+snk

1 󵄩p 󵄩󵄩 󵄩󵄩qe (x)󵄩󵄩󵄩 dx = 2l

l + |t + snk | 2l

l+t+snk

1 l + |t + snk |

󵄩p 󵄩󵄩 󵄩󵄩qe (x)󵄩󵄩󵄩 dx

l+|t+snk |

∫ 0

󵄩p 󵄩󵄩 󵄩󵄩q(x)󵄩󵄩󵄩 dx

l+|t+snk |

∫ 0

󵄩p 󵄩󵄩 󵄩󵄩q(x)󵄩󵄩󵄩 dx

and use the fact that q ∈ Bp0 ([0, ∞) : X). For the sequel, let us recall that, if f ∈ AA(ℝ : X) and g ∈ L1 (ℝ), then the infinite ∞ convolution product t 󳨃→ (g ∗f )(t) := ∫−∞ g(t −s)f (s) ds, t ∈ ℝ is almost automorphic as well (see [132]). Due to [185, Theorem 3.1], a similar statement holds for the spaces of compactly almost automorphic functions and Stepanov p-almost automorphic functions (1 ⩽ p < ∞). Now we will prove the following simple assertion concerning the invariance of infinite convolution product for the class of Besicovitch 1-almost automorphic functions:

3.1 Almost automorphic functions, asymptotically almost automorphic functions | 199

Proposition 3.1.6. Let f ∈ B1 AA(ℝ : X) and let g ∈ L1 (ℝ) be a scalar-valued function with compact support. Then the function F(⋅) := (g∗f )(⋅) belongs to the class B1 AA(ℝ : X) as well. Proof. Let −∞ < a < b < ∞, and let supp(g) ⊆ [a, b]. Let [−l − r, l − r] ⊆ [−2l, 2l] for all r ∈ [a, b] and l ⩾ l0 . Our assumptions on g(⋅) imply that (g ∗ h)(⋅) is a welldefined X-valued locally integrable function for any function h ∈ L1loc (ℝ : X). Let (sn ) be a given sequence. Then we can extract a subsequence (snk ) of (sn ) and a function f ∗ ∈ L1loc (ℝ : X) such that (162)–(163) hold with p = 1. Set F ∗ (⋅) := (g ∗ f ∗ )(⋅). Then (162) for F(⋅) and F ∗ (⋅) follows from its validity for f (⋅) and f ∗ (⋅), and the following simple integral calculation using Fubini theorem (l ⩾ l0 , k ∈ ℕ, t ∈ ℝ): l+t

1 󵄩 󵄩 ∫ 󵄩󵄩󵄩F(snk + x) − F ∗ (x)󵄩󵄩󵄩 dx 2l −l−t

l+t ∞

1 󵄩󵄩 󵄩 󵄩 ⩽ ∫ ∫ 󵄩󵄩󵄩f (snk + x − r) − f ∗ (x − r)󵄩󵄩󵄩󵄩󵄩󵄩g(r)󵄩󵄩󵄩 dr dx 2l −l−t −∞

∞

l+t

−∞

−l+t

󵄩 󵄩 󵄩 1 󵄩 = ∫ 󵄩󵄩󵄩g(r)󵄩󵄩󵄩[ ∫ 󵄩󵄩󵄩f (snk + x − r) − f ∗ (x − r)󵄩󵄩󵄩 dx] dr 2l l+t−r

∞

󵄩 1 󵄩 = ∫ 󵄩󵄩󵄩g(r)󵄩󵄩󵄩[ 2l −∞

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (snk + x) − f ∗ (x)󵄩󵄩󵄩 dx] dr

−l+t−r

b

l−r

a

−l−r

b

2l

a

−2l

󵄩 󵄩 󵄩 1 󵄩 = ∫󵄩󵄩󵄩g(r)󵄩󵄩󵄩[ ∫ 󵄩󵄩󵄩f (snk + x + t) − f ∗ (x + t)󵄩󵄩󵄩 dx] dr 2l 󵄩 1 󵄩 󵄩 󵄩 ⩽ ∫󵄩󵄩󵄩g(r)󵄩󵄩󵄩[ ∫ 󵄩󵄩󵄩f (snk + x) − f ∗ (x)󵄩󵄩󵄩 dx] dr. 2l The proof of (163) for F(⋅) and F ∗ (⋅) is similar and therefore omitted, finishing the proof of the proposition. In the present situation, we do not know whether the assumption that g(⋅) has a compact support can be relaxed and whether we can consider the case p > 1 here (see also Proposition 3.3.5 and Section 3.7 below). The notion of a pseudo almost-automorphic function was introduced by T.-J. Xiao, J. Liang and J. Zhang in [404] (2008). Let us recall that the space of pseudo-almost automorphic functions, denoted shortly by PAA(ℝ : X), is defined as the direct sum of spaces AA(ℝ : X) and PAP0 (ℝ : X), where PAP0 (ℝ : X) denotes the space consisting r of all bounded continuous functions Φ : ℝ → X such that limr→∞ 2r1 ∫−r ‖Φ(s)‖ ds = 0. Equipped with the sup-norm, the space PAA(ℝ : X) becomes a Banach space.

200 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Let p ⩾ 1. We refer the reader to Definition 2.3.12 for the notion of spaces Wp PAA0 (ℝ : X). Following S. Abbas [1], we say that a function f ∈ Lploc (ℝ : X) is Weyl p-pseudo almost automorphic iff there exist a Weyl-p-almost automorphic function g(⋅) and a function q ∈ Wp PAA0 (ℝ : X) such that f (⋅) = g(⋅)+q(⋅). By Wp PAA(ℝ : X) we denote the collection of all such functions. Finally, we will introduce the notion of a Besicovitch p-pseudo almost automorphic function: Definition 3.1.7. Let p ⩾ 1. Then we say that a function f ∈ Lploc (ℝ : X) is Besicovitch p-pseudo almost automorphic iff f (⋅) = g(⋅) + q(⋅), where g(⋅) is Besicovitch p-almost automorphic and q ∈ Lploc (ℝ : X) satisfies T

x+l

1 1 󵄩 󵄩p lim ∫ [lim sup ∫ 󵄩󵄩󵄩q(t)󵄩󵄩󵄩 dt] T→+∞ 2T l→+∞ 2l

1/p

dx = 0.

x−l

−T

The set of all such functions is denoted by Bp PAA(ℝ : X). A very simple analysis shows that Bp PAA(ℝ : X) is a vector space, as well as that Wp PAA(ℝ : X) ⊆ Bp PAA(ℝ : X).

3.2 Two-parameter generalized almost automorphic functions Let (Y, ‖⋅‖Y ) be another pivot space over the field of complex numbers. Let us recall that C0 ([0, ∞) × Y : X) designates the space of all continuous functions h : [0, ∞) × Y → X such that limt→∞ h(t, y) = 0 uniformly for y in any compact subset of Y. A jointly continuous function F : ℝ × Y → X is said to be almost automorphic iff for every sequence of real numbers (s󸀠n ) there exists a subsequence (sn ) such that G(t, y) := lim F(t + sn , y) n→∞

is well defined for each t ∈ ℝ and y ∈ Y, and lim G(t − sn , y) = F(t, y)

n→∞

for each t ∈ ℝ and y ∈ Y. The vector space consisting of such functions will be denoted by AA(ℝ × Y : X). A bounded continuous function f : ℝ × Y → X is said to be pseudo-almost automorphic iff F = G + Φ, where G ∈ AA(ℝ × Y : X) and Φ ∈ PAP0 (ℝ × Y : X); let us recall that, here, PAP0 (ℝ × Y : X) denotes the space consisting of all continuous functions Φ : ℝ × Y → X such that {Φ(t, y) : t ∈ ℝ} is bounded for all y ∈ Y, and r limr→∞ 2r1 ∫−r ‖Φ(s, y)‖ ds = 0, uniformly in y ∈ Y. The collection of such functions will be denoted henceforth by PAA(ℝ × Y : X).

3.2 Two-parameter generalized almost automorphic functions | 201

The notion of a Stepanov two-parameter p-almost automorphic function was introduced by H.-S. Ding, J. Liang and T.-J. Xiao in [166] (2009). The definition goes as follows: Definition 3.2.1. Let 1 ⩽ p < ∞, and let f : ℝ × Y → X be such that for each y ∈ Y we have f (⋅, y) ∈ Lploc (ℝ : X). Then it is said that f (⋅, ⋅) is Stepanov p-almost automorphic iff for every y ∈ Y the mapping f (⋅, y) is Sp -almost automorphic; that is, for any real sequence (an ) there exist a subsequence (ank ) of (an ) and a map g : ℝ × Y → X such that g(⋅, y) ∈ Lploc (ℝ : X) for all y ∈ Y, as well as that 1

󵄩p 󵄩 lim ∫󵄩󵄩󵄩f (t + ank + s, y) − g(t + s, y)󵄩󵄩󵄩 ds = 0

k→∞

and

0

1

󵄩p 󵄩 lim ∫󵄩󵄩󵄩g(t + s − ank , y) − f (t + s, y)󵄩󵄩󵄩 ds = 0 k→∞ 0

for each t ∈ ℝ and for each y ∈ Y. We denote by AASp (ℝ × Y : X) the vector space consisting of all such functions. We continue our work by observing that the well-known results of Z. Fan et al. [184] and H.-S. Ding et al. [167] (see, e. g., [132, pp. 134–138]) continue to hold in the case that the pivot spaces X and Y are mutually different (cf. Theorems 2.7.1 and 2.7.2 for the corresponding statements in Stepanov almost periodic case): Theorem 3.2.2. Assume that 1 ⩽ p < ∞, and f ∈ AASp (ℝ × Y : X). If there exists a constant L > 0 such that for all x, y ∈ Lploc (ℝ : Y) 1

1

󵄩p 󵄩 󵄩p 󵄩 ∫󵄩󵄩󵄩f (t + s, x(s)) − f (t + s, y(s))󵄩󵄩󵄩 ds ⩽ L ∫󵄩󵄩󵄩x(s) − y(s)󵄩󵄩󵄩Y ds, 0

0

p

then for each x ∈ AAS (ℝ : Y) with relatively compact range in Y one has that f (⋅, x(⋅)) ∈ AASp (ℝ : X). Theorem 3.2.3. Suppose that the following conditions hold: (i) f ∈ AASp (ℝ × Y : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lf ∈ LrS (ℝ) such that (73) holds with I = ℝ; (ii) x ∈ AASp (ℝ : Y), and there exists a set E ⊆ ℝ with m(E) = 0 such that K := {x(t) : t ∈ ℝ ∖ E} is relatively compact in Y. Then q := pr/p + r ∈ [1, p) and f (⋅, x(⋅)) ∈ AASq (ℝ : X). The following composition principle is basically due to J. Liang et al. [305]. Its validity for class PAA(ℝ × Y : X), where Y ≠ X, can be proved similarly.

202 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Theorem 3.2.4. Suppose that f = g + ϕ ∈ PAA(ℝ × Y : X) with g ∈ AA(ℝ × Y : X), ϕ ∈ PAP0 (ℝ × Y : X) and the following holds: (i) the mapping (t, y) 󳨃→ g(t, y) is uniformly continuous in any bounded subset K ⊆ Y uniformly for t ∈ ℝ; (ii) the mapping (t, y) 󳨃→ ϕ(t, y) is uniformly continuous in any bounded subset K ⊆ Y uniformly for t ∈ ℝ. Then for each y ∈ PAA(ℝ : Y) one has f (⋅, y(⋅)) ∈ PAA(ℝ : X). In order to continue, we state two composition principles for asymptotically Stepanov almost automorphic functions. Keeping in mind Theorem 3.2.3 and the foregoing arguments, used in the analysis of almost periodic case, we can clarify the following result: Proposition 3.2.5. Let I = [0, ∞). Suppose that the following conditions hold: (i) g ∈ AASp (ℝ × Y : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lg ∈ LrS (I) such that (73) holds with the functions f (⋅, ⋅) and Lf (⋅) replaced by the functions g(⋅, ⋅) and Lg (⋅) therein. (ii) y ∈ AASp (ℝ : Y), and there exists a set E ⊆ ℝ with m(E) = 0 such that K = {y(t) : t ∈ ℝ ∖ E} is relatively compact in Y. (iii) f (t, y) = g(t, y)+q(t, y) for all t ⩾ 0 and y ∈ Y, where q̂ ∈ C0 ([0, ∞)×Y : Lq ([0, 1] : X)) and q := pr/p + r. (iv) x(t) = y(t) + z(t) for all t ⩾ 0, where ẑ ∈ C0 ([0, ∞) : Lp ([0, 1] : Y)). (v) There exists a set E 󸀠 ⊆ I with m(E 󸀠 ) = 0 such that K 󸀠 = {x(t) : t ∈ I ∖ E 󸀠 } is relatively compact in Y. Then q ∈ [1, p) and f (⋅, x(⋅)) ∈ AAASq (I : X). Using Theorem 3.2.2 in place of Theorem 3.2.3, we can similarly prove the following result: Proposition 3.2.6. Let I = [0, ∞). Suppose that the following conditions hold: (i) g ∈ AASp (ℝ × Y : X) with p ⩾ 1, and there exists a constant L > 0 such that for all x, y ∈ Lploc (ℝ : Y) we have that (74) holds with f = g. (ii) y ∈ AASp (ℝ : Y), and there exists a set E ⊆ ℝ with m(E) = 0 such that K = {y(t) : t ∈ ℝ ∖ E} is relatively compact in Y. (iii) f (t, y) = g(t, y)+q(t, y) for all t ⩾ 0 and y ∈ Y, where q̂ ∈ C0 ([0, ∞)×Y : Lq ([0, 1] : X)) and q := pr/p + r. (iv) x(t) = y(t) + z(t) for all t ⩾ 0, where ẑ ∈ C0 ([0, ∞) : Lp ([0, 1] : Y)). (v) There exists a set E 󸀠 ⊆ I with m(E 󸀠 ) = 0 such that K 󸀠 = {x(t) : t ∈ I ∖ E 󸀠 } is relatively compact in Y. Then f (⋅, x(⋅)) ∈ AAASp (I : X).

3.3 Weighted pseudo-almost periodic (and automorphic) functions | 203

3.3 Weighted pseudo-almost periodic functions, weighted pseudo-almost automorphic functions and their generalizations As already mentioned, the class of weighted pseudo-almost periodic functions with values in Banach spaces was introduced by T. Diagana in [134] (2006), while the class of weighted pseudo-almost automorphic functions with values in Banach spaces was introduced by J. Blot, G. M. Mophou, G. M. N’Guérékata and D. Pennequin in [64] (2009). Set 𝕌 := {ρ ∈ L1loc (ℝ) : ρ(t) > 0 a. e. t ∈ ℝ}, 𝕌∞ := {ρ ∈ 𝕌 : infx∈ℝ ρ(x) < T

∞ and ν(T, ρ) := limT→+∞ ∫−T ρ(t) dt = ∞} and 𝕌b := L∞ (ℝ) ∩ 𝕌∞ . Then it is clear that 𝕌b ⊆ 𝕌∞ ⊆ 𝕌. We say that weights ρ1 (⋅) and ρ2 (⋅) are equivalent, ρ1 ∼ ρ2 for short, iff ρ1 /ρ2 ∈ 𝕌b . By 𝕌T we denote the space consisting of all weights ρ ∈ 𝕌∞ such that ρ is equivalent with all its translations. Next, we introduce the spaces consisting of so-called weighted ergodic components, depending on one or two variables. Assume that ρ1 , ρ2 ∈ 𝕌∞ . Set PAP0 (ℝ, X, ρ1 , ρ2 ) := {f ∈ Cb (ℝ : X) : lim

T→+∞

1

T

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (t)󵄩󵄩󵄩ρ2 (t) dt = 0} T 2 ∫−T ρ1 (t) dt −T

and PAP0 (ℝ × Y, X, ρ1 , ρ2 ) := {f ∈ Cb (ℝ × Y : X) : lim

T→+∞

1

T

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (t, y)󵄩󵄩󵄩ρ2 (t) dt = 0, uniformly on bounded subsets of Y}. T 2 ∫−T ρ1 (t) dt −T

The class of weighted pseudo-almost periodic functions was introduced by T. Diagana in [134] (2006), while the class of so-called double-weighted pseudo almostperiodic functions was introduced by the same author in [141, 142] (2011): Definition 3.3.1. (i) A function f ∈ Cb (ℝ : X) is said to be weighted pseudo-almost periodic iff it admits a decomposition f (t) = g(t) + q(t), t ∈ ℝ, where g(⋅) is almost periodic and q(⋅) ∈ PAP0 (ℝ, X, ρ1 , ρ2 ). (ii) A function f (⋅, ⋅) ∈ Cb (ℝ × Y : X) is said to be weighted pseudo-almost periodic iff it admits a decomposition f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is almost periodic and q(⋅, ⋅) ∈ PAP0 (ℝ × Y, X, ρ1 , ρ2 ). Denote by WPAP(ℝ, X, ρ1 , ρ2 ), resp. WPAP(ℝ × Y, X, ρ1 , ρ2 ), the vector space consisting of such functions. If ρ = ρ1 = ρ2 , then we set WPAP(ℝ, X, ρ1 , ρ2 ) ≡ WPAP(ℝ, X, ρ) and WPAP(ℝ × Y, X, ρ1 , ρ2 ) ≡ WPAP(ℝ × Y, X, ρ).

204 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations The following slight generalization of weighted pseudo-almost automorphic functions is introduced by T. Diagana [141, 142] (see also T. Diagana, K. Ezzinbi, M. Miraoui [147] and S. Abbas, V. Kavitha, R. Murugesu [4]): Definition 3.3.2. (i) A function f ∈ Cb (ℝ : X) is said to be weighted pseudo-almost automorphic iff it admits a decomposition f (t) = g(t) + q(t), t ∈ ℝ, where g(⋅) is almost automorphic and q(⋅) ∈ PAP0 (ℝ, X, ρ1 , ρ2 ). (ii) A function f (⋅, ⋅) ∈ Cb (ℝ×Y : X) is said to be weighted pseudo-almost automorphic iff it admits a decomposition f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is almost automorphic and q(⋅, ⋅) ∈ PAP0 (ℝ × Y, X, ρ1 , ρ2 ). Denote by WPAA(ℝ, X, ρ1 , ρ2 ), resp. WPAA(ℝ × Y, X, ρ1 , ρ2 ), the vector space consisting of such functions. If ρ = ρ1 = ρ2 , then we set WPAA(ℝ, X, ρ1 , ρ2 ) ≡ WPAA(ℝ, X, ρ) and WPAA(ℝ × Y, X, ρ1 , ρ2 ) ≡ WPAA(ℝ × Y, X, ρ). It is well-known that, for every ρ1 , ρ2 ∈ 𝕌T , we have that WPAP(ℝ, X, ρ1 , ρ2 ) and WPAA(ℝ, X, ρ1 , ρ2 ) are Banach spaces endowed with the sup-norm. For the Stepanov class, we will use the following definition from [4] (see also Z. Xia, M. Fan [400] for the case that ρ1 = ρ2 ): Definition 3.3.3. Let 1 ⩽ p < ∞. (i) A Stepanov p-bounded function f (⋅) is said to be weighted Stepanov p-pseudo almost periodic (automorphic) iff it admits a decomposition f (t) = g(t) + q(t), t ∈ ℝ, where g(⋅) is Sp -almost periodic (automorphic) and q(⋅) ∈ Lploc (ℝ : X) satisfies lim

T→+∞

1

T

T

t+1

󵄩p 󵄩 ∫ [ ∫ 󵄩󵄩󵄩q(s)󵄩󵄩󵄩 ds]

2 ∫−T ρ1 (t) dt −T

1/p

t

ρ2 (t) dt = 0.

(165)

Denote by Sp WPAP(ℝ, X, ρ1 , ρ2 ) (Sp WPAA(ℝ, X, ρ1 , ρ2 )) the vector space of such functions, and by Sp WPAA0 (ℝ, X, ρ1 , ρ2 ) the vector space of locally p-integrable X-valued functions q(⋅) such that (165) holds. (ii) A function f : ℝ × Y → X is said to be weighted Sp -pseudo almost periodic (automorphic) iff for each y ∈ Y we have that f (⋅, y) ∈ Lploc (ℝ : X) and f (⋅, ⋅) admits a decomposition f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is Sp -almost periodic (automorphic) and lim

T→+∞

T

1

T

t+1

󵄩p 󵄩 ∫ [ ∫ 󵄩󵄩󵄩q(s, y)󵄩󵄩󵄩 ds]

2 ∫−T ρ1 (t) dt −T

uniformly on bounded subsets of Y.

t

1/p

ρ2 (t) dt = 0,

(166)

3.3 Weighted pseudo-almost periodic (and automorphic) functions | 205

Denote by Sp WPAP(ℝ × Y, X, ρ1 , ρ2 ) (Sp WPAA(ℝ × Y, X, ρ1 , ρ2 )) the vector space of such functions, and by Sp WPAA0 (ℝ × Y, X, ρ1 , ρ2 ) the vector space of functions such that for each y ∈ Y one has that q(⋅) is a locally p-integrable X-valued function and (166) holds. If ρ = ρ1 = ρ2 , then we write Sp WPAP(ℝ×Y, X, ρ1 , ρ2 ) ≡ Sp WPAP(ℝ×Y, X, ρ) (Sp WPAA(ℝ × Y, X, ρ1 , ρ2 ) ≡ Sp WPAA(ℝ × Y, X, ρ)). For instance, the function f (t) := sign(cos 2πθt) + e−|t| , t ∈ ℝ, where θ is an irrational number, is weighted Sp -pseudo almost automorphic with the weight functions ρ1 (t) := 1 + t 2 , ρ2 (t) := 1 (t ∈ ℝ); this function is also Sp -pseudo almost automorphic (cf. [4, Example 1]). The function g(t) := eαt + sin(1/(2 + cos t + cos √2t)), t ∈ ℝ is weighted pseudo almost automorphic with the weight function ρ(t) = ρ1 (t) = ρ2 (t) defined by ρ(t) := 1, t ⩾ 0 and ρ(t) := e−βt for β ⩾ α (cf. [2, Example, pp. 24–25]). Denote by 𝕍∞ the collection of all weights ρ1 , ρ2 ∈ 𝕌∞ such that ρ (t + τ) lim sup 2 0 we have: T

1

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (−s + t)󵄩󵄩󵄩ρ2 (t) dt T 2 ∫−T ρ1 (t) dt −T =

T 2 ∫−T

1

T−s

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (t)󵄩󵄩󵄩ρ2 (t − s) dt

ρ1 (t) dt −T−s

208 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations

⩽

⩽

g(s)

T 2 ∫−T

T−s

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (t)󵄩󵄩󵄩ρ2 (t) dt

ρ1 (t) dt −T−s

T 󵄨󵄨 󵄨󵄨 󵄨󵄨 T−s 󵄨󵄨 −T 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄨󵄨]. 󵄨 󵄨 󵄨 ρ (t) dt + [ ρ (t) dt ρ (t) dt + f (t) ∫ ∫ ∫ 󵄨 󵄨 󵄨 󵄩 󵄩 2 2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄩 2 󵄩 T 󵄨󵄨 󵄨 󵄨 󵄨 2 ∫−T ρ1 (t) dt −T 󵄨 󵄨T 󵄨−T−s

g(s)

Then the final conclusion follows by applying (171). In [156], T. Diagana and M. Zitane have analyzed weighted Stepanov-like pseudoalmost periodic functions in Lebesgue space with variable exponents Lp(x) , where p : ℝ → ℝ is a measurable function satisfying certain conditions (cf. [157] for corresponding classes of pseudo-almost automorphic functions); in this book, we will not investigate a similar notion for Weyl and Besicovitch classes. For that matter, it is worth mentioning our forthcoming joint researches with T. Diagana [151, 152], where we have continued the study of almost periodicity (automorphy) which intermediate the classical and Stepanov concept (the notion of an almost periodic function in view of the Lebesgue measure, shortly η-a. p. function, has been defined already in the foundational papers of Stepanov and it is well known that any a. p. function needs to be η-a. p. function as well as that any bounded η-a. p. function needs to be Sp -a. p. for any exponents p ⩾ 1; for more details, see [78] and [376, 377]). In order to analyze generalized weighted pseudo-almost automorphic solutions of semilinear (fractional) Cauchy inclusions, we need to repeat some known facts about composition principles for the classes of weighted pseudo-almost automorphic functions and Stepanov weighted pseudo-almost automorphic functions (weighted pseudo-almost periodic solutions can be analyzed in a similar fashion; see, e. g., [142, Theorem 5.9] and [418, Theorems 3.1 and 3.5]). The subsequent result has been proved by T. Diagana (cf. [142, Theorem 5.8]): Theorem 3.3.7. Assume that ρ1 , ρ2 ∈ 𝕌∞ , f : ℝ × Y → X is weighted pseudo-almost automorphic, and h : ℝ → Y is weighted pseudo-almost automorphic. Assume that there exists a finite constant Lf > 0 such that (74) holds with I = ℝ and L = Lf . Then f (⋅, h(⋅)) ∈ WPAA(ℝ × Y, X, ρ1 , ρ2 ). Arguing as in the proof of [400, Theorems 3.6 and 3.7], we can prove the following composition principles, stated here with two generally different pivot spaces (see also [4, Theorems 2.2, 2.3 and 2.4]). Theorem 3.3.8. Assume that ρ1 , ρ2 ∈ 𝕌∞ , 1 ⩽ p < ∞, f : ℝ × Y → X is weighted Sp -pseudo almost automorphic, f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is Sp -almost automorphic and q(⋅, ⋅) satisfies (166), uniformly on bounded subsets of Y. Assume, further, that the following conditions hold: (i) A function h : ℝ → Y is weighted Sp -pseudo almost automorphic, h(t) = g(t) + q(t), t ∈ ℝ, where g(⋅) is Sp -almost automorphic with relatively compact range in Y, and q(⋅) ∈ Lploc (ℝ : X) satisfies (165).

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 209

(ii) There exist two finite constants Lf > 0 and Lg > 0 such that (74) holds with I = ℝ and L = Lf , as well as that (74) holds with I = ℝ, the function f (⋅, ⋅) and the number L replaced therein with g(⋅, ⋅) and Lg . Then f (⋅, h(⋅)) ∈ Sp WPAA(ℝ × Y, X, ρ1 , ρ2 ). Theorem 3.3.9. Assume that ρ1 , ρ2 ∈ 𝕌∞ , 1 < p < ∞, f : ℝ × Y → X is weighted Sp -pseudo almost automorphic, f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is Sp -almost automorphic and q(⋅, ⋅) satisfies (166), uniformly on bounded subsets of Y. Assume, further, that the following conditions hold: (i) A function h : ℝ → Y is weighted Sp -pseudo almost automorphic, h(t) = g(t) + q(t), t ∈ ℝ, where g(⋅) is Sp -almost automorphic with relatively compact range in Y, and q(⋅) ∈ Lploc (ℝ : X) satisfies (165). (ii) r ⩾ max(p, p/p − 1) and there exist two Stepanov r-almost automorphic scalarvalued functions Lf (⋅) and Lg (⋅) such that (73) holds with I = ℝ, as well as that (73) holds with I = ℝ, the functions f (⋅, ⋅) and Lf (⋅) replaced therein with the functions g(⋅, ⋅) and Lg (⋅). Set q := pr/p + r. Then q ∈ [1, p) and f (⋅, h(⋅)) ∈ Sq WPAA(ℝ × Y, X, ρ1 , ρ2 ).

3.4 Generalized asymptotically almost periodic and generalized asymptotically almost automorphic solutions of abstract multiterm fractional differential inclusions Of concern is the following abstract multiterm fractional differential inclusion: α

n−1

α

Dt n u(t) + ∑ Ai Dt i u(t) ∈ 𝒜Dαt u(t) + f (t), i=1

u(k) (0) = uk ,

k = 0, . . . , ⌈αn ⌉ − 1,

t ⩾ 0, (173)

where n ∈ ℕ ∖ {1}, A1 , . . . , An−1 are bounded linear operators on a Banach space X, 𝒜 is a closed multivalued linear operator on X, 0 ⩽ α1 < ⋅ ⋅ ⋅ < αn , 0 ⩽ α < αn , f (⋅) is an X-valued function, and Dαt denotes the Caputo fractional derivative of order α. The boundedness of linear operators A1 , . . . , An−1 is crucial for applications of vector-valued Laplace transform in this section, and therefore, will be assumed a priori. Unless stated otherwise, in the continuation of this section, we will always assume that the function k(⋅) is a scalar-valued kernel on [0, τ), as well as that the operator C ∈ L(X) is injective. Besides the function spaces that we have already used so far, we will deal with the vector space PT ([0, ∞) : X) consisting of all bounded continuous T-periodic functions, denoted by PT ([0, ∞) : X), PT ([0, ∞) : X) := {f ∈ Cb ([0, ∞)) : f (t + T) = f (t), t ⩾ 0},

210 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations this is a vector subspace of AP([0, ∞) : X). Set APT ([0, ∞) : X) := PT ([0, ∞) : X) ⊕ C0 ([0, ∞) : X). We will use the following extension of [247, Theorem 1.2.4(i)], the proof of which can be left to the reader as an easy exercise (see also the proof of [188, Theorem 1.7, p. 24]). Lemma 3.4.1. Let B, D ∈ L(X) and let 𝔸 be an MLO. If BD = DB, B𝔸 ⊆ 𝔸B, D𝔸 ⊆ 𝔸D and (B − 𝔸)−1 D ∈ L(X), then we have (B − 𝔸)−1 D𝔸 ⊆ B(B − 𝔸)−1 D − D ⊆ 𝔸(B − 𝔸)−1 D.

3.4.1 k-regularized C-propagation families for (173) Henceforth, we always assume that k, k1 , k2 , . . . are scalar-valued kernels and a ≠ 0 in L1loc ([0, τ)). Set mj := ⌈αj ⌉, 1 ⩽ j ⩽ n, m := m0 := ⌈α⌉, A0 := 𝒜 and α0 := α. We will use the following definition. Definition 3.4.2. A function u ∈ C mn −1 ([0, ∞) : X) is called a (strong) solution of (173) α mn −1 uk gk+1 ) ∈ C mn ([0, ∞) : X) iff Ai Dt i u ∈ C([0, ∞) : X) for 0 ⩽ i ⩽ n − 1, gmn −αn ∗ (u − ∑k=0 and (173) holds. Integrating the both sides of (173) αn -times and employing the closedness of 𝒜, Theorem 1.2.2 and the equality [53, (1.21)], it readily follows that any strong solution u(t), t ⩾ 0 of (173) satisfies the following: mn −1

n−1

k=0

j=1

mj −1

u(⋅) − ∑ uk gk+1 (⋅) + ∑ gαn −αj ∗ Aj [u(⋅) − ∑ uk gk+1 (⋅)] k=0

m−1

∈ gαn −α ∗ 𝒜[u(⋅) − ∑ uk gk+1 (⋅)]. k=0

(174)

If i ∈ ℕ0mn −1 , then we define Di := {j ∈ ℕn−1 : mj − 1 ⩾ i}. Plugging uj = 0, 0 ⩽ j ⩽ mn − 1, j ≠ i, in (174), we get: [u(⋅; 0, . . . , ui , . . . , 0) − ui gi+1 (⋅)]

+ ∑ gαn −αj ∗ Aj [u(⋅; 0, . . . , ui , . . . , 0) − ui gi+1 (⋅)] j∈Di

+ ∈{

∑

j∈ℕn−1 ∖Di

[gαn −αj ∗ Aj u(⋅; 0, . . . , ui , . . . , 0)]

gαn −α ∗ 𝒜u(⋅; 0, . . . , ui , . . . , 0), gαn −α ∗ 𝒜[u(⋅; 0, . . . , ui , . . . , 0) − ui gi+1 (⋅)],

m − 1 < i, m − 1 ⩾ i,

(175)

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 211

where ui appears in the ith place (0 ⩽ i ⩽ mn − 1) starting from 0. Proceeding as in the nondegenerate case [246], inclusion (175) motivates us to introduce the following extension of [246, Definition 2.10.2] (cf. also [258, Definition 2.1] and [256, Definitions 3.6 and 3.7] for similar notions). Definition 3.4.3. Suppose that 0 < τ ⩽ ∞, k ∈ C([0, τ)), C, C1 , C2 ∈ L(X), C and C2 are injective. A sequence ((R0 (t))t∈[0,τ) , . . . , (Rmn −1 (t))t∈[0,τ) ) of strongly continuous operator families in L(X) is called a (local, if τ < ∞): (i) k-regularized C1 -existence propagation family for (173) iff the following holds: [Ri (⋅)x − (k ∗ gi )(⋅)C1 x] + ∑ Aj [gαn −αj ∗ (Ri (⋅)x − (k ∗ gi )(⋅)C1 x)] j∈Di

+ ∈{

∑

j∈ℕn−1 ∖Di

Aj (gαn −αj ∗ Ri )(⋅)x

𝒜(gαn −α ∗ Ri )(⋅)x, 𝒜[gαn −α ∗ (Ri (⋅)x − (k ∗ gi )(⋅)C1 x)](⋅),

m − 1 < i, x ∈ X, m − 1 ⩾ i, x ∈ X,

(176)

for any i = 0, . . . , mn − 1; (ii) k-regularized C2 -uniqueness propagation family for (173) iff the following holds: [Ri (⋅)x − (k ∗ gi )(⋅)C2 x] + ∑ gαn −αj ∗ [Ri (⋅)Aj x − (k ∗ gi )(⋅)C2 Aj x] j∈Di

+

∑

j∈ℕn−1 ∖Di

(gαn −αj ∗ Ri (⋅)Aj x)(⋅)

(g ∗ Ri (⋅)y)(⋅), = { αn −α gαn −α ∗ [Ri (⋅)y − (k ∗ gi )(⋅)C2 y](⋅),

m − 1 < i, m − 1 ⩾ i,

(177)

provided (x, y) ∈ 𝒜 and i ∈ ℕ0mn −1 ; (iii) k-regularized C-resolvent propagation family for (173), k-regularized C-propagation family for (173) shortly, iff ((R0 (t))t∈[0,τ) , . . . , (Rmn −1 (t))t∈[0,τ) ) is a k-regularized C-uniqueness propagation family for (173), and if for every t ∈ [0, τ), i ∈ ℕ0mn −1 and j ∈ ℕ0n−1 , one has Ri (t)Aj ⊆ Aj Ri (t), Ri (t)C = CRi (t) and CAj ⊆ Aj C.

When k(t) = gζ +1 (t), where ζ ⩾ 0, we also say that ((R0 (t))t∈[0,τ) , . . . , (Rmn −1 (t))t∈[0,τ) ) is a ζ -times integrated C-resolvent propagation family; 0-times integrated C-resolvent propagation family for (173) is simply called a C-resolvent propagation family for (173). For a k-regularized (C1 , C2 )-existence and uniqueness family ((R0 (t))t∈[0,τ) , . . . , (Rmn −1 (t))t∈[0,τ) ), we call it exponentially bounded iff each single operator family (R0 (t))t∈[0,τ) , . . . (Rmn −1 (t))t∈[0,τ) is such. The above terminological agreement is accepted for all other classes of k-regularized C-propagation families introduced so far. If Aj = cj I, where cj ∈ ℂ for 1 ⩽ j ⩽ n − 1, then it is also said that 𝒜 is a subgenerator of ((R0 (t))t∈[0,τ) , . . . , (Rmn −1 (t))t∈[0,τ) ). The notion of integral generator of ((R0 (t))t∈[0,τ) , . . . , (Rmn −1 (t))t∈[0,τ) ) is introduced as in the nondegenerate case [246].

212 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Hereafter, the following equality will play an important role in our analysis: n−1

[Ri (⋅)x − (k ∗ gi )(⋅)Cx] + ∑ Aj gαn −αj ∗ [Ri (⋅)x − (k ∗ gi )(⋅)Cx] j=1

+ ∈{

∑

j∈ℕn−1 ∖Di

Aj [gαn −αj +i ∗ k](⋅)Cx

𝒜[gαn −α ∗ Ri ](⋅)x, 𝒜[gαn −α ∗ (Ri (⋅)x − (k ∗ gi )(⋅)Cx)],

m − 1 < i, x ∈ X, m − 1 ⩾ i, x ∈ X,

for any i = 0, . . . , mn − 1. The basic properties of subgenerators and integral generators continue to hold, with appropriate changes, in the degenerate case; cf. [246] and [247, Section 3.2] for more details. We leave to the interested reader the problem of transferring the assertions of [246, Propositions 2.10.3–2.10.5, Theorem 2.10.7] to the degenerate case. The following is a degenerate version of [246, Definition 2.10.6]: Definition 3.4.4. Let f ∈ C([0, ∞) : X). Consider the following inhomogeneous Cauchy inclusion: n−1

u(t) + ∑ (gαn −αj ∗ Aj u)(t) ∈ f (t) + (gαn −α ∗ 𝒜u)(t), j=1

t ⩾ 0.

(178)

A function u ∈ C([0, ∞) : X) is said to be: (i) a strong solution of (178) iff there exists a continuous function u𝒜 ∈ C([0, ∞) : X) such that u𝒜 (t) ∈ 𝒜u(t) for all t ⩾ 0 and n−1

u(t) + ∑ (gαn −αj ∗ Aj u)(t) = f (t) + (gαn −α ∗ u𝒜 )(t), j=1

t ⩾ 0;

(ii) a mild solution of (178) iff n−1

u(t) + ∑ Aj (gαn −αj ∗ u)(t) ∈ f (t) + 𝒜(gαn −α ∗ u)(t), j=1

t ∈ [0, T].

Clearly, every strong solution of (178) is also a mild solution of the same problem while the converse statement is not true, in general. We similarly define the notion of a strong (mild) solution of problem (174). We have the following: (a) If ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a C1 -existence propagation family for (173), then m −1

the function u(t) := ∑i=0n Ri (t)xi , t ⩾ 0 is a mild solution of (174) with ui = C1 xi for 0 ⩽ i ⩽ mn − 1. (b) If ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a C2 -uniqueness propagation family for (173), and 0 0 Ri (t)Aj x = Aj Ri (t)x, t ⩾ 0, x ∈ ⋂n−1 j=0 D(Aj ), i ∈ ℕmn −1 , j ∈ ℕn−1 , then the funcm −1

tion u(t) := ∑i=0n Ri (t)C2−1 ui , t ⩾ 0 is a strong solution of (174), provided ui ∈ C2 (⋂n−1 j=0 D(Aj )) for 0 ⩽ i ⩽ mn − 1.

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 213

For our later purposes, it will be sufficient to characterize the introduced classes of k-regularized propagation families by vector-valued Laplace transform; the proofs are almost the same as in the nondegenerate case and we will only point out some details of the proof of Theorem 3.4.6 below because the formulation of [246, Theorem 2.10.9] is slightly misleading since the injectivity of operator Pλ for λ ∈ ℂ with Re λ > ω has not been clarified in a proper way and property (ii) in the formulation of this theorem is required to hold for all i ∈ ℕ0mn −1 : Theorem 3.4.5. Suppose k(t) satisfies (P1), ω ⩾ max(0, abs(k)), (Ri (t))t⩾0 is strongly continuous, and the family {e−ωt Ri (t) : t ⩾ 0} ⊆ L(X) is bounded, provided 0 ⩽ i ⩽ mn −1. Let 𝒜 be a closed MLO on X, let C1 , C2 ∈ L(X), and let C2 be injective. Set n−1

Pλ := λαn −α + ∑ λαj −α Aj − 𝒜, λ ∈ ℂ ∖ {0}. j=1

(179)

(i) Suppose Aj ∈ L(X), j ∈ ℕn−1 . Then ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a global k-regularized C1 -existence propagation family for (173) iff the following conditions hold: (a) The inclusion ∞

αj −α−i ̃ ̃ Pλ ∫ e−λt Ri (t)x dt ∋ λαn −α−i k(λ)C k(λ)Aj C1 x, 1x + ∑ λ j∈Di

0

(180)

holds provided x ∈ X, i ∈ ℕ0mn −1 , m − 1 < i and Re λ > ω. (b) The inclusion ∞

Pλ ∫ e−λt [Ri (t)x − (k ∗ gi )(t)C1 x] dt 0

∋−

∑

j∈ℕn−1 ∖Di

̃ λαj −α−i k(λ)A j C1 x,

(181)

holds provided x ∈ X, i ∈ ℕ0mn −1 , m − 1 ⩾ i and Re λ > ω. (ii) Suppose Aj ∈ L(X), j ∈ ℕn−1 . Then ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a global k-regularized C2 -uniqueness propagation family for (173) iff, for every λ ∈ ℂ with Re λ > ω, x ∈ D(𝒜) and y ∈ 𝒜x, the following equality holds: ∞

∫ e−λt [Ri (t)x − (k ∗ gi )(t)C2 x] dt 0

∞

+ ∑ λαj −αn ∫ e−λt [Ri (t)Aj x − (k ∗ gi )(t)C2 Aj x] dt j∈Di

+

∑

j∈ℕn−1 ∖Di

0

λ

αj −αn

∞

∫ e−λt Ri (t)Aj x dt 0

214 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations ∞

{ λα−αn ∫ e−λt Ri (t)y dt, { { { 0 ={ ∞ { α−α { n {λ ∫ e−λt [Ri (t)y − (k ∗ gi )(t)C2 y] dt, { 0

m − 1 < i, m − 1 ⩾ i.

Theorem 3.4.6. Suppose k(t) satisfies (P1), ω ⩾ max(0, abs(k)), (Ri (t))t⩾0 is strongly continuous, and the family {e−ωt Ri (t) : t ⩾ 0} ⊆ L(X) is bounded, provided 0 ⩽ i ⩽ mn −1. (I) Let the following two conditions hold: (i) CAj ⊆ Aj C, j ∈ ℕ0n−1 , Aj ∈ L(X), j ∈ ℕn−1 , Ai Aj = Aj Ai , i, j ∈ ℕn−1 and Aj 𝒜 ⊆ 𝒜Aj , j ∈ ℕn−1 . (ii) There exists an index i ∈ ℕ0mn −1 satisfying exactly one of the following two conditions: (a) m − 1 < i and the operator λαn −i + ∑j∈Di λαj −i Aj is injective for every λ ∈ ℂ ̃ with Re λ > ω and k(λ) ≠ 0, (b) m − 1 ⩾ i, ℕn−1 ∖ Di ≠ 0 and the operator ∑j∈ℕn−1 ∖Di λαj −i Aj is injective for ̃ every λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0. If ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a global k-regularized C-resolvent propagation family for (173), and (176) holds, then Pλ is injective for every λ ∈ ℂ with Re λ > ω ̃ and k(λ) ≠ 0, as well as equalities (180)–(181) are fulfilled. ̃ (II) Suppose that Pλ is injective for every λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0, as well as equalities (180)–(181) are fulfilled. If condition (I)(i) holds, then ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a global k-regularized C-resolvent propagation family for (173). Proof. Concerning assertion (I), we will only sketch the main details of the proof of ̃ the injectivity of operator Pλ for every λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0 (we know that (180)–(181) hold on account of Theorem 3.4.5). Observe that we do not need condition (I)(ii) for the proof of (II), where we only use an elementary argument, as well as Theorem 1.2.2 and Lemma 3.4.1 (the composition property (177) follows by applying the Laplace transform and Lemma 3.4.1, while the commutation of operator families Ri (⋅) with the operators C and Aj for 0 ⩽ j ⩽ n−1 is much simpler to show). The consideration is quite similar in the case when condition (II)(a) or (II)(b) holds and, because of this, ̃ ) ≠ 0 be fixed, we will consider only the first case. Let λ0 ∈ ℂ with Re λ0 > ω and k(λ 0 and let 0 ∈ Pλ0 x for some x ∈ X. Using the fact that ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a global k-regularized C-uniqueness propagation family for (173), we can simply prove that α −i

α −i

λ0n x + ∑ λ0j Aj x = 0 j∈Di

by performing the Laplace transform of both sides of the composition property (177). By the injectivity of the operator λαn −i + ∑j∈Di λαj −i Aj for λ = λ0 , we obtain that x = 0, and the claimed assertion follows.

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 215

These results enable one to simply clarify the Hille–Yosida type theorems for exponentially bounded k-regularized C-resolvent propagation families. The analytical properties of k-regularized C-resolvent propagation families can be analyzed similarly as in the nondegenerate case [246]. We will use the following definition: Definition 3.4.7. (i) Suppose that ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is a k-regularized C-resolvent propagation family for (173), and β ∈ (0, π]. Then it is said that ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is an analytic k-regularized C-resolvent propagation family of angle β, iff for each i ∈ ℕ0mn −1 there exists a function Ri : Σα → L(X) which is such that, for every x ∈ X, the mapping z 󳨃→ Ri (z)x, z ∈ Σβ is analytic, as well as that: (a) Ri (t) = Ri (t), t > 0 and (b) limz→0,z∈Σγ Ri (z)x = Ri (0)x for all γ ∈ (0, β) and x ∈ X. (ii) Suppose that ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is an analytic k-regularized C-resolvent propagation family of angle β. Then it is said that ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) is an exponentially bounded, analytic k-regularized C-resolvent propagation family of angle β, resp. bounded k-regularized C-resolvent propagation family, iff for every γ ∈ (0, β), there exists ωγ ⩾ 0, resp. ωγ = 0, such that the family {e−ωγ Re z Ri (z) : z ∈ Σγ } ⊆ L(X) is bounded for all i ∈ ℕ0mn −1 . Since there is no risk for confusion, we will identify in the sequel Ri (⋅) and Ri (⋅). For our purposes, the following result will be sufficient (cf. Theorem 3.4.6 and [29, Theorem 2.6.1, Proposition 2.6.3 b)]; we are under an obligation to say that the small inconsistencies in the formulation of [246, Theorem 2.10.11] have been made, see also [258]): Theorem 3.4.8. Assume k(t) satisfies (P1), ω ⩾ max(0, abs(k)), β ∈ (0, π/2] and, for every i ∈ ℕ0mn −1 , the function (k ∗ gi )(t) can be analytically extended to a function ki : Σβ → ℂ such that, for every γ ∈ (0, β), the set {e−ωz ki (z) : z ∈ Σγ } is bounded. Let the following three conditions hold: (i) CAj ⊆ Aj C, j ∈ ℕ0n−1 , Aj ∈ L(X), j ∈ ℕn−1 , Ai Aj = Aj Ai , i, j ∈ ℕn−1 and Aj 𝒜 ⊆ 𝒜Aj , j ∈ ℕn−1 . (ii) The operator Pλ is injective for all ω + Σβ+π/2 . (iii) Let qi : ω + Σ π +β → L(X) (0 ⩽ i ⩽ mn − 1) be such that, for every x ∈ X, the mapping 2

λ 󳨃→ qi (λ)x, λ ∈ ω + Σ π +β is analytic, as well as that for each i ∈ ℕ0mn −1 there exists 2 an operator Di ∈ L(X) such that qi (λ)x = kĩ (λ)Pλ−1 (λαn −α C + ∑ λαj −α Aj C)x, j∈Di

x ∈ X, λ ∈ Vi ,

provided m − 1 < i, qi (λ)x = −kĩ (λ)Pλ−1

∑

j∈ℕn−1 ∖Di

λαj −α Aj Cx,

x ∈ X, λ ∈ Vi ,

216 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations provided m − 1 ⩾ i, the family {(λ − ω)qi (λ) : λ ∈ ω + Σ π +γ } is equicontinuous for all γ ∈ (0, β), 2

and, in the case D(𝒜) ≠ X, lim λqi (λ)x = Di x,

λ→+∞

x ∉ D(𝒜).

Then there exists an exponentially bounded, analytic k-regularized C-resolvent propagation family ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ) for (173), of angle β. Furthermore, (176) holds, the family {e−ωz Ri (z) : z ∈ Σγ } is bounded for all i ∈ ℕ0mn −1 and γ ∈ (0, β), and Ri (z)Aj ⊆ Aj Ri (z), z ∈ Σβ , j ∈ ℕ0n−1 .

Remark 3.4.9. For the sequel, it will be very important to note that the notion from Definition 3.4.3(iii) and Definition 3.4.7 can be introduced for any single operator family (Ri (t))t⩾0 of the tuple ((R0 (t))t⩾0 , . . . , (Rmn −1 (t))t⩾0 ). The assertions of Theorems 3.4.6 and 3.4.8 can be simply reformulated for (Ri (t))t⩾0 ; for example, if the index i ∈ ℕ0mn −1 is given in advance, then in the formulation of Theorem 3.4.8 it suffices to assume that the function (k ∗gi )(⋅) can be analytically extended to a function ki : Σβ → ℂ such that, for every γ ∈ (0, β), the set {e−ωz ki (z) : z ∈ Σγ } is bounded, as well as that (i)–(ii) hold and (iii) holds only for this specified index i. It will be said that (Ri (t))t⩾0 is an (exponentially bounded, analytic/analytic) ki -regularized C-resolvent propagation family. All terminological agreements explained before will be accepted for ki -regularized Cresolvent propagation families; the classes of ki -regularized C1 -existence propagation families and ki -regularized C2 -uniqueness propagation families are introduced similarly.

3.4.2 Asymptotical behavior of ki -regularized C-propagation families for (173) The main aim of this subsection is to investigate polynomial decay of ki -regularized C-propagation families for (173) as time goes to infinity. Applications of Theorem 3.4.8 (see also Remark 3.4.9) will be crucial in our work, and we start by observing that it is not clear how one can prove the injectivity of operator Pλ , given by (179), in general. Because of this, we will first focus our attention towards the case Aj = cj I, where cj ∈ ℂ for 1 ⩽ j ⩽ n − 1, by exploring the generation of fractionally integrated C-propagation families for (173) only. Moreover, we will assume that the numbers cj are nonnegative for 1 ⩽ j ⩽ n − 1, as well as that m − 1 < i (the case m − 1 ⩾ i can be analyzed similarly) and the multivalued linear operator 𝒜 under our consideration is possibly not densely defined. Theorem 3.4.10. Suppose that cj ⩾ 0 and Aj = cj I for 1 ⩽ j ⩽ n − 1, ζ 󸀠 ⩾ 0, 𝒜 : X → P(X) is a closed MLO, C ∈ L(X) is injective, C 𝒜 ⊆ 𝒜C and the following condition holds:

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 217

(H) There exist finite constants c < 0, M > 0, 0 < θ < π and β ∈ (0, 1] such that c + Σπ−θ ⊆ ρC (𝒜) and 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − 𝒜) C 󵄩󵄩󵄩 ⩽

M , |λ − c|β

λ ∈ c + Σπ−θ .

Assume that the mapping λ 󳨃→ (λ − 𝒜)−1 C, λ ∈ c + Σπ−θ is strongly continuous. Assume also that i ∈ ℕ0mn −1 satisfies m − 1 < i, as well as αn − α − i − ζ 󸀠 − (αn − α)β ⩽ 0 and ν󸀠 :=

(182)

π π−θ − > 0. αn − α 2

(183)

Set ζ := ζ 󸀠 if 𝒜 is densely defined, ζ > ζ 󸀠 otherwise, and ki (⋅) := gζ +1 (⋅). Then there exists an exponentially bounded, analytic ki -regularized C-propagation family (Ri (t))t⩾0 for (173), of angle ν := min(ν󸀠 , π/2). Moreover, (176) holds and there exists a finite constant M 󸀠 > 0 such that 󵄩 󵄩󵄩 󸀠 α+ζ +i−αn + 󵄩󵄩Ri (t)󵄩󵄩󵄩 ⩽ M [t

∑

j∈Di ,cj =0 ̸

t α+ζ +i−αj ],

t > 0.

(184)

Proof. Since we have assumed that the mapping λ 󳨃→ (λ−𝒜)−1 C, λ ∈ c + Σπ−θ is strongly continuous, its restriction to c+Σπ−θ is strongly analytic on this region; see [247, Proposition 1.2.6(iii)]. Taking into account (183) and the inequality cj ⩾ 0 for 1 ⩽ j ⩽ n − 1, we get λαn −α + ∑ cj λαj −α ∈ Σ(αn −α)(π/2+ν) ⊆ Σ(αn −α)(π/2+ν) ⊆ Σπ−θ , j∈Di

λ ∈ Σν+ π . 2

(185)

It is clear that (182) implies αj − α − i − ζ 󸀠 − (αn − α)β ⩽ 0

for all j ∈ Di with cj ≠ 0.

(186)

Using an elementary argument, (182), (185) and (186), we can simply verify that the conditions of Theorem 3.4.8 hold with ω > 0 sufficiently large, k(t) = gζ +1 (t), t ⩾ 0 and Di = 0, in the case that the operator 𝒜 is not densely defined. Hence, 𝒜 is a subgenerator of an exponentially bounded, analytic ζ -times integrated C-propagation family (Ri (t))t⩾0 for (173), of angle ν := min(ν󸀠 , π/2), as claimed. It remains to prove the estimate (184). Fix the numbers t > 0 and 0 < γ < ν. By the proof of [29, Theorem 2.6.1] and Cauchy theorem, we have that, for every x ∈ X, Ri (t)x =

1 ∫ eλt λ−ζ −1 Pλ−1 [λαn −α−i Cx + ∑ cj λαj −α−i Cx] dλ, 2πi j∈D ,c =0 ̸ Γ

i

j

218 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations where Γ is oriented counterclockwise and consists of Γ± := {rei(π/2+γ) : r ⩾ t −1 } and Γ0 := {t −1 eiθ : |θ| ⩽ π/2 + γ}. Keeping in mind (185) and the estimate from condition (H), it readily follows that for each λ ∈ Γ we have ‖Pλ−1 C‖ ⩽ M/|c sin θ|β , so that β 󵄩 M/|c sin θ| 󵄩󵄩 󵄩󵄩Ri (t)󵄩󵄩󵄩 ⩽ 2π

× ∫ eRe(λt) |λ|−ζ −1 [|λ|αn −α−i + Γ

∑

j∈Di ,cj =0 ̸

|cj ||λ|αj −α−i ] |dλ|.

But, then estimate (184) follows from a simple integral computation which is very similar to that appearing in the proof of [29, Theorem 2.6.1]. Remark 3.4.11. (i) It is worth noting that the value of exponent β in (H) does not depend on the final estimate (184). In our proof, we only use the estimate ‖Pλ−1 C‖ ⩽ M/|c sin θ|β , λ ∈ Γ. (ii) As mentioned in the introductory part, the proof of an important result of E. Cuesta [113, Theorem 2.1] for the classical fractional oscillation resolvent families generated by densely defined linear operators [53], satisfying condition (H) with β = 1, C = I and ω = c < 0, follows completely different lines. Furthermore, in our approach, the case in which π/2 < θ < π or αn − α < 1 can occur, so that Theorem 3.4.10 is applicable in the qualitative analysis of fractional relaxation multiterm differential inclusions (but not in the analysis of generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions; see the equation (187) below). Let ℱ ∈ {APT ([0, ∞) : ℂ), AAP([0, ∞) : ℂ), AAA([0, ∞) : ℂ)}, for example, where the symbol AAA([0, ∞) : ℂ) denotes the space of scalar-valued asymptotically almost automorphic functions defined as in Section 3.5, and let ki (⋅) be defined as above. If (Ri (t))t⩾0 is the ki -regularized C-propagation family for (173), constructed with the help of Theorem 3.4.10, and f ∈ ℱ , then it can be easily checked that ((Ri ∗ f )(t))t⩾0 is a ki -regularized C-propagation family (Ri (t))t⩾0 for (173), satisfying additionally (176), where ki (⋅) = (gζ +1 ∗ f )(⋅). By Theorem 3.4.10, some known assertions concerning inheritance of asymptotical periodicity, almost asymptotical almost periodicity and asymptotical almost automorphy under the action of finite convolution products (see [206, 236]), and assertions (a)–(b) clarified above, this yields the following result (the uniqueness of solutions follows from the fact that [246, Theorem 2.10.7] holds in the degenerate case and that condition (⬦) stated in the formulation of [246, Proposition 2.10.3] holds true, which can be verified by performing the Laplace transform): Corollary 3.4.12. Let the requirements of Theorem 3.4.10 hold, let f ∈ ℱ , and let ki (⋅) = (gζ +1 ∗ f )(⋅). Set ux (t) := (Ri ∗ f )(t)x, t ⩾ 0, x ∈ X. Assume that α + ζ + i − αn < −1

and

α + ζ + i − αj < −1

for all j ∈ Di with cj ≠ 0.

(187)

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 219

Then, for every x ∈ X, ux (⋅) ∈ ℱ is a unique mild solution of the abstract Cauchy inclusion n−1

[u(⋅) − (gζ +1+i ∗ f )(⋅)Cx] + ∑ cj gαn −αj ∗ [u(⋅) − (gζ +1+i ∗ f )(⋅)Cx] j=1

+

∑

j∈ℕn−1 ∖Di

cj [gαn −αj +i+ζ +1 ∗ f ](⋅)Cx ∈ 𝒜[gαn −α ∗ u](⋅).

(188)

Furthermore, ux (⋅) is a strong solution of (188) provided that x ∈ D(𝒜). As for the Stepanov asymptotical almost periodicity and (equi-)Weyl asymptotical almost periodicity, we have already seen that some extra conditions on the vanishing part of function f (⋅) must be imposed if we want the solution ux (⋅) defined above to belong to the same class of functions as f (⋅). Denote by ℱ 󸀠 the set consisting of all generalized (asymptotically) almost periodic function spaces considered so far and all generalized (asymptotically) almost automorphic function spaces defined in Section 3.5. Let ℱ 󸀠󸀠 denote the class of function spaces obtained by removing from ℱ 󸀠 the sums of spaces of (equi-)Weyl-p-almost periodic (automorphic) functions and (equi-)Weyl-p-almost vanishing functions. The second part of the following proposition is very similar to Proposition 2.5.1. Proposition 3.4.13. Suppose that k(t) satisfies (P1) and (Ri (t))t⩾0 is a strongly Laplace transformable ki -regularized C-propagation family for (173). (i) For every λ ∈ ℂ, there exists a function fλ (⋅) satisfying (P1)-L(X) and −1

fλ (t) := ℒ ([(1 −

λ

z αn −α

n−1

)I + ∑ j=1

z

−1

Aj

αn −αj

]

∑

j∈Di

̃ k(z) CAj )(t), zi

t ⩾ 0,

provided that m − 1 < i, resp. fλ (t) := ℒ−1 ([(1 − × (∑

j∈Di

λ

z αn

n−1

)I + ∑ −α j=1

z

Aj

−1

αn −αj ]

̃ ̃ k(z) k(z) CAj − λ α −α+i C))(t), i z z n

t ⩾ 0,

provided that m − 1 ⩾ i. (ii) Denote by D the set consisting of all eigenvectors x of operator 𝒜 corresponding to eigenvalues λ ∈ ℂ of operator 𝒜 for which the mapping fλ,x (t) := fλ (t)x,

t ⩾ 0,

belongs to the space ℱ 󸀠 . Then the mapping t 󳨃→ Ri (t)x, t ⩾ 0 belongs to the space ℱ 󸀠 for all x ∈ span(D); furthermore, the mapping t 󳨃→ Ri (t)x, t ⩾ 0 belongs to the space ℱ 󸀠󸀠 for all x ∈ span(D) provided additionally that (Ri (t))t⩾0 is bounded.

220 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Proof. We will examine the case m − 1 < i only. The proof of (i) can be given following the lines of the proof of [402, Theorem 1.1.11], with appropriate changes briefly Aj described as follows. Since the operator (1 − z αnλ−α )I + ∑n−1 j=1 z αn −αj is invertible in L(X) for all z ∈ ℂ with |z| sufficiently large, because the norm of the bounded linear operator Aj λ I − ∑n−1 j=1 z αn −αj for such values of z is strictly less than 1, we get that the term z αn −α [(1 −

n−1

λ

z αn

)I + ∑ −α j=1

z

Aj

−1

αn −αj ]

∑

j∈Di

̃ k(z) CAj zi

is well-defined for all z ∈ ℂ with Re z > ω0 , for some ω0 > max(0, abs(k)). Set n−1

H0 (t) := λgαn −αj (t) + ∑ gαn −αj (t)Aj , j=1

t > 0,

as well as a := min(αn − α0 , . . . , αn − αn−1 ) and b := max(αn − α0 , . . . , αn − αn−1 ). Then it is clear that ∞

∫ e−zt H0 (t) dt = 0

λ

z αn

n−1

I−∑ −α j=1

Aj

z αn −αj

,

Re z > ω0 ,

as well as that there exists a finite constant c > 0 such that 󵄩󵄩 ∗k 󵄩󵄩 n 󵄩󵄩H0 (t)󵄩󵄩 ⩽ c gk(αn −b) (t),

󵄩 󵄩 t ∈ (0, 1] and 󵄩󵄩󵄩H0∗k (t)󵄩󵄩󵄩 ⩽ cn gk(αn −a) (t),

t ⩾ 1,

where H0∗k (⋅) denotes the kth convolution power of H0 (⋅). The function H(t) := ∗k 󸀠 ∑∞ k=1 H0 (t), t > 0 is well-defined since there exists a finite constant c > 0 such that ∞ k(αn −b)−1 󵄩 󵄩󵄩 k t 󵄩󵄩H(t)󵄩󵄩󵄩 ⩽ ∑ c Γ(k(α − b)) k=1

n

(ct αn −b )k Γ(k(αn − b) + αn − b) k=0 ∞

⩽ ct αn −b−1 ∑

⩽ ct αn −b−1 Eαn −b,αn −b (ct αn −b ) ⩽ c󸀠 t αn −b−1 , and, due to Theorem 1.4.1, ∞ k(αn −a)−1 󵄩 󵄩󵄩 k t 󵄩󵄩H(t)󵄩󵄩󵄩 ⩽ ∑ c Γ(k(α − a)) k=1

n

(ct αn −a )k Γ(k(αn − a) + αn − a) k=0 ∞

⩽ ct αn −a−1 ∑

t ∈ (0, 1]

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 221

⩽ ct αn −a−1 Eαn −a,αn −a (ct αn −a ) ⩽ c󸀠 t αn −a−1 [1 + (ct αn −a )

1−(αn −a)/αn −a c1/αn −a t

e

],

t ⩾ 1.

For the remaining part of proof of (i), it suffices to repeat literally the arguments from the proof of [402, Theorem 1.1.11]. For the proof of (ii), observe first that, if λx ∈ 𝒜x for some λ ∈ ℂ, then performing the Laplace transform of both sides of the composition property (177), as it has been done in our previous examinations, immediately yields that [(1 −

λ

z αn −α

n−1

)I + ∑ j=1

z

Aj

αn −αj

∞

] ∫ e−zt Ri (t)x dt = ∑ 0

j∈Di

̃ k(z) CAj x, zi

for Re z > 0 sufficiently large, and therefore Ri (t)x = fλ,x (t), t ⩾ 0. As a consequence, we have that the mapping t 󳨃→ Ri (t)x, t ⩾ 0 belongs to the space ℱ 󸀠 for all x ∈ span(D). The boundedness of (Ri (t))t⩾0 implies the uniform convergence of Ri (t)xn to Ri (t)x (t ⩾ 0) for any sequence (xn ) ∈ span(D) converging to some element x ∈ span(D); then the final result follows by combining the previously proved statement and the fact that the limit of a uniform convergent sequence of bounded continuous functions belonging to any space from ℱ 󸀠 belongs to this space again (see Proposition 2.3.5 for the class of (equi-)Weyl-almost periodic functions). Remark 3.4.14. If m − 1 < i and Aj = cj I for some cj ∈ ℂ (1 ⩽ i ⩽ n − 1), then a simple calculation shows that fλ,x (t) = ∑

j∈Di

αn −i ̃ k(z)z

−αj z αn − λz −α + ∑n−1 j=1 cj z

CAj x,

for x ∈ X satisfying λx ∈ 𝒜x (λ ∈ ℂ). To the best of our knowledge, in the handbooks containing tables of Laplace transforms, the explicit forms of functions like fλ,x (⋅) are not known, with the exception of some very special cases of the coefficients αj , cj (see, e. g., [246, Remark 3.3.10(vi)]). The following theorem is motivated by some pioneering results of W. M. Ruess and W. H. Summers concerning integration of asymptotically almost periodic functions [356]. Theorem 3.4.15. Let (Ri (t))t⩾0 be an exponentially bounded ki -regularized C-propagation family (Ri (t))t⩾0 for (173). Let m − 1 < i, and suppose there exists a number ν > 1 such that ‖Ri (t)‖ = O(t −ν ), t ⩾ 1. Assume, further, that the following conditions hold: (i) Let f ∈ C([0, ∞) : X) satisfy that there exists a function g ∈ AAP([0, ∞) : X) such that (C −1 f )(t) = (gi ∗ k ∗ g)(t), t ⩾ 0. (ii) Assume that g(t) = gap (t) + g0 (t), t ⩾ 0, where gap ∈ AP([0, ∞) : X) and g0 ∈ C0 ([0, ∞) : X). (iii) Let αn − αj ∈ ℕ for all j ∈ Di .

222 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations (iv) Assume that gk ∗ g ∈ L∞ ([0, ∞) : X) for all k ∈ ℕ and c0 ⊈ X, or R(gk ∗ g) is weakly relatively compact in X for all k ∈ ℕ. (v) For every k ∈ ℕ, we have ∞

∞∞

󵄩 󵄩 ∫ 󵄩󵄩󵄩g0 (t)󵄩󵄩󵄩 dt < ∞,

󵄩 󵄩 ∫∫ 󵄩󵄩󵄩g0 (s)󵄩󵄩󵄩 ds dt < ∞, . . . ,

0

∞∞ ∞

0t ∞

󵄩 󵄩 . . . , ∫∫ ∫ ⋅ ⋅ ⋅∫ 󵄩󵄩󵄩g0 (s)󵄩󵄩󵄩 ds dx1 ⋅ ⋅ ⋅ dxk dt < ∞. 0 t xk

x1

Then there exists a unique exponentially bounded mild solution u(⋅) of the abstract Cauchy inclusion (178). Furthermore, u ∈ AAP([0, ∞) : X). Proof. From our previous analyses of the nondegenerate case, it is well known that any mild solution of the abstract Cauchy inclusion (178) has to satisfy the following equality: (Ri ∗ f )(t) = (k ∗ gi ∗ Cu)(t) + ∑ (gαn −αj +i ∗ k ∗ CAj u)(t), j∈Di

t ⩾ 0;

see, e. g., [246, Theorem 2.10.7]. Taking the Laplace transform, we get that −1

−1 −1 f (z)z i [k(z)] ̃ ̃i (z)C̃ ̃ , u(z) = [I + ∑ z αj −αn Aj ] R j∈Di

̃ Re z > ω, k(z) ≠ 0.

(189)

Since (C −1 f )(t) = (gi ∗ k ∗ g)(t), t ⩾ 0, we have −1

̃i (z)g(z), ̃ ̃ u(z) = [I + ∑ z αj −αn Aj ] R j∈Di

̃ Re z > ω, k(z) ≠ 0.

By the proofs of Proposition 3.4.13 and [402, Theorem 1.1.11], the right-hand side of above equality is really the Laplace transform of a continuous exponentially bounded function u(⋅) given by ∗k

∞

u(t) = ∑ {Ri ∗ [− ∑ gαn −αj (⋅)Aj ] k=1

j∈Di

∗ g}(t) + (Ri ∗ g)(t),

t ⩾ 0.

The Laplace transform may stand us to verify, after a simple calculation, that the function u(⋅), whose Laplace transform is given by (189), is a mild solution of the abstract fractional inclusion (178). The growth order of Ri (⋅) implies that the function t 󳨃→ (Ri ∗ g)(t), t ⩾ 0 is in AAP([0, ∞) : X). Since AAP([0, ∞) : X) is closed in Cb ([0, ∞) : X) and {Ri ∗ [− ∑j∈Di gαn −αj (⋅)Aj ]∗k ∗ g}(t) converges uniformly to ∑∞ k=1 {Ri ∗

[− ∑j∈Di gαn −αj (⋅)Aj ]∗k ∗ g}(t) for t ⩾ 0, the asymptotical almost periodicity of u(⋅) immediately follows if we prove that the function Gk (t) = (gk ∗ g)(t), t ⩾ 0 is asymptotically

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 223

almost periodic for all k ∈ ℕ; see (iii). This can be shown by making use of (iv)–(v) and applying successively [356, Theorem 2.2.2]; here, we only want to observe that the van∞ ∞ ishing term of function G1 (t) is given by ∫t g0 (s) ds (t ⩾ 0), since ∫0 ‖g0 (s)‖ ds < ∞. The uniqueness of exponentially bounded mild solutions of (178) can be proved as follows. Let u(⋅) be such a solution. Taking the Laplace transform and multiplying after that with z αn −α , we get that n−1

̃ − z αn −α f ̃(z)) ∈ 𝒜, ̃ (u(z), [z αn −α + ∑ z αj −α Aj ]u(z) j=1

̃ Re z > ω, k(z) ≠ 0.

̃ ̃ ̃ Hence, −z αn −α f ̃(z) ∈ Pz u(z) and u(z) = −Pz−1 z αn −α f ̃(z) for Re z > ω, k(z) ≠ 0. By the uniqueness theorem for the Laplace transform, u(⋅) must be uniquely determined. The proof of the theorem is thereby complete. Remark 3.4.16. Concerning Theorem 3.4.15, the case m − 1 ⩾ i is not easy for consideration: it seems that the assumption Aj = cj I for some complex numbers cj ∈ ℂ (1 ⩽ j ⩽ n − 1) has to be imposed for establishing any relevant result. Details can be left to the interested reader. We also want to point out that our method can be successfully applied in the analysis of abstract semilinear integro-differential inclusion [u(⋅) − (gζ +1+i ∗ f (⋅, u(⋅)))(⋅)Cx] n−1

+ ∑ cj gαn −αj ∗ [u(⋅) − (gζ +1+i ∗ f (⋅, u(⋅)))(⋅)Cx] j=1

+

∑

j∈ℕn−1 ∖Di

cj [gαn −αj +i+ζ +1 ∗ f (⋅, u(⋅))](⋅)Cx ∈ 𝒜[gαn −α ∗ u](⋅);

(190)

see Corollary 3.4.12 and problem (188). Using our previous examinations, it seems reasonable to define the mild solution of (190) by u(t) := (Ri ∗ f (⋅, u(⋅))(t)x, t ⩾ 0. 3.4.3 Generalized asymptotically almost periodic and generalized asymptotically almost automorphic solutions of abstract multiterm fractional differential inclusions with Riemann–Liouville derivatives The results already presented in this section also apply to abstract multiterm fractional differential inclusions with Riemann–Liouville derivatives. First, we need to inscribe here the basic operational theoretical approach for seeking solutions of the following multiterm fractional differential inclusion with Riemann–Liouville derivatives: α

n−1

α

Dt n u(t) + ∑ Aj Dt j u(t) ∈ 𝒜Dαt u(t) + f (t), j=1

t ∈ (0, τ),

(191)

224 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations where n ∈ ℕ ∖ {1}, 𝒜 is a closed multivalued linear operator on a complex Banach space X, Aj ∈ L(X) for 1 ⩽ j ⩽ n − 1, 0 ⩽ α1 < ⋅ ⋅ ⋅ < αn , 0 ⩽ α < αn , 0 < τ ⩽ ∞, f (t) is an X-valued function, and Dαt denotes the Riemann–Liouville fractional derivative of order α (see [247, Section 2.4] for a similar approach employed). Recall that α0 = α, m = ⌈α⌉, A0 = 𝒜 and mi = ⌈αi ⌉ for 1 ⩽ i ⩽ n. We introduce the notion of a mild (strong) solution of (191) as in [262]. In order to subject initial values to (191), we first define 𝒯(191) := {

1, 0,

if there exists j ∈ ℕ0n such that αj ∈ ℕ, otherwise,

and S := {j ∈ ℕ0n : αj ∈ ℕ}. As in the nondegenerate case, we distinguish the following three subcases of (191): (SC1) αn > 1. Here for each integer i ∈ ℕmn −1 we set 𝒟i := {j ∈ ℕ0n : mj − 1 ⩾ i}, Si := {mj − αj : j ∈ 𝒟i } and si := card(Si ). Then we have Si ⊆ [0, 1) and Si = {ai,1 , . . . , ai,si }, where 0 ⩽ ai,1 < ⋅ ⋅ ⋅ < ai,si ⩽ 1 (i ∈ ℕmn −1 ). Define 𝒟il := {j ∈ 𝒟i : mj − αj = ai,l } (i ∈ ℕmn −1 , 1 ⩽ l ⩽ si ). Then for each integer i ∈ ℕmn −1 we introduce si initial values xi,1 , . . . , xi,si for terms (gmj −αj ∗ u)(i) (0), where j ∈ 𝒟i . Additionally, if there

exists j ∈ ℕ0n such that αj ∈ ℕ, i. e., if S ≠ 0, then we supplement a new initial value x0 for term (g0 ∗ u)(0) ≡ u(0). (SC2) αn = 1. In this case, we introduce only one initial value for term (g0 ∗ u)(0) ≡ u(0). (SC3) αn < 1. In this case, we consider equation (191) without initial conditions. Put s + ⋅ ⋅ ⋅ + smn −1 + 𝒯(191) , { { 1

ℬ(191) := {1, {

{0,

if αn > 1, if αn = 1, if αn < 1.

By the foregoing, there will be exactly ℬ(191) initial conditions for (191). The subcase (SC3) will be not examined henceforth. For the subcases (SC1) and (SC2), we will use the following definition (cf. [247, Section 2.4] for more details). Definition 3.4.17. Let 0 < τ ⩽ ∞, k ∈ C([0, τ)), C, C1 , C2 ∈ L(X), and let C and C2 be injective. (i) (SC1) Suppose that, for every i ∈ ℕmn −1 and l ∈ ℕsi , (Ri,l (t))t∈[0,τ) ⊆ L(X) is strongly continuous, as well as that, for every t ∈ [0, τ), x ∈ X, i ∈ ℕmn −1 and l ∈ ℕsi , the following functional equation: Ri,l (t)x − χ𝒟l (n)(k ∗ gαn +i−mn )(t)C1 x i

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 225 n−1

+ ∑ Aj [gαn −αj ∗ (Ri,l (⋅)x − χ𝒟l (j)(k ∗ gαj +i−mj )(⋅)C1 x)](t) i

j=1

∈ 𝒜[gαn −α ∗ (Ri,l (⋅)x − χ𝒟l (0)(k ∗ gα+i−m )(⋅)C1 x)](t) i

(192)

holds. If S ≠ 0, then we also introduce a strongly continuous family (R0,1 (t))t∈[0,τ) ⊆ L(X) satisfying that, for every t ∈ [0, τ) and x ∈ X, R0,1 (t)x − χS (n)k(t)C1 x

n−1

+ ∑ Aj [gαn −αj ∗ (R0,1 (⋅)x − χS (j)k(⋅)C1 x)](t) j=1

∈ 𝒜[gαn −α ∗ (R0,1 (⋅)x − χS (0)k(⋅)C1 x)](t).

(193)

Then the sequence ((Ri,l (t))t∈[0,τ) )1⩽i⩽mn −1,1⩽l⩽si if S = 0, resp. ((Ri,l (t))t∈[0,τ) , (R0,1 (t))t∈[0,τ) )1⩽i⩽mn −1,1⩽l⩽si if S ≠ 0, is said to be a (local, if τ < ∞) k-regularized C1 -existence propagation family for (191). (SC2) A strongly continuous family (R(t))t∈[0,τ) ⊆ L(X) such that, for every t ∈ [0, τ) and x ∈ X, n−1

R(t)x − k(t)C1 x + ∑ Aj (gαn −αj ∗ R(⋅)x)(t) ∈ 𝒜(gαn −α ∗ R(⋅)x)(t), j=1

is said to be a (local, if τ < ∞) k-regularized C1 -existence propagation family for (191). (ii) (SC1) Suppose that, for every i ∈ ℕmn −1 and l ∈ ℕsi , (Wi,l (t))t∈[0,τ) ⊆ L(X) is strongly continuous, as well as that Wi,l (⋅)x − χ𝒟l (n)(k ∗ gαn +i−mn )(⋅)C2 x i

n−1

+ ∑ gαn −αj ∗ [Wi,l (⋅)Aj x − χ𝒟l (j)(k ∗ gαj +i−mj )(⋅)C2 Aj x] i

j=1

= gαn −α ∗ [Wi,l (⋅)y − χ𝒟l (0)(k ∗ gα+i−m )(⋅)C2 y], i

for every i ∈ ℕmn −1 , l ∈ ℕsi and (x, y) ∈ 𝒜. If S ≠ 0, then we also introduce a strongly continuous family (W0,1 (t))t∈[0,τ) ⊆ L(X) such that, for every (x, y) ∈ 𝒜, W0,1 (⋅)x − χS (n)k(⋅)C2 x

n−1

+ ∑ gαn −αj ∗ [W0,1 (⋅)Aj x − χS (j)k(⋅)C2 Aj x] j=1

= gαn −α ∗ [W0,1 (⋅)y − χS (0)k(⋅)C2 y]. Then the sequence ((Wi,l (t))t∈[0,τ) )1⩽i⩽mn −1,1⩽l⩽si if S = 0, resp. ((Wi,l (t))t∈[0,τ) , (W0,1 (t))t∈[0,τ) )1⩽i⩽mn −1,1⩽l⩽si if S ≠ 0, is said to be a (local, if τ < ∞) k-regularized

226 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations C2 -uniqueness propagation family for (191). If, in addition to the above, 𝒜j for 0 ⩽ j ⩽ n, any operator family Wi,l (⋅), the operator family W0,1 (⋅) if S ≠ 0, and the operator C2 , all commute with each other, then ((Wi,l (t))t∈[0,τ) )1⩽i⩽mn −1,1⩽l⩽si if S = 0, resp. ((Wi,l (t))t∈[0,τ) , (W0,1 (t))t∈[0,τ) )1⩽i⩽mn −1,1⩽l⩽si if S ≠ 0, is said to be a k-regularized C2 -resolvent propagation family for (191). (SC2) A strongly continuous family (W(t))t∈[0,τ) ⊆ L(X) such that, for every (x, y) ∈ 𝒜 and t ∈ [0, τ), n−1

W(t)x − k(t)C2 x + ∑ (gαn −αj ∗ W(⋅)C2 Aj x)(t) = (gαn −α ∗ W(⋅)C2 y)(t), j=1

is said to be a (local, if τ < ∞) k-regularized C2 -uniqueness propagation family for (191). If, additionally, 𝒜j for 0 ⩽ j ⩽ n, W(⋅) and the operator C2 , all commute with each other, then (W(t))t∈[0,τ) is said to be a k-regularized C2 -resolvent propagation family for (191). In the case that k(t) = gζ +1 (t), where ζ ⩾ 0, then a k-regularized C1 -existence propagation family for (191) is also said to be ζ -times integrated C1 -existence propagation family for (191); a 0-times integrated C1 -existence propagation family for (191) is simply called C1 -existence propagation family for (191); similar notions will be used for the classes of C2 -uniqueness propagation families for (191) and C-resolvent propagation families for (191). The notions of k-regularized C2 -uniqueness propagation families for (191) and k-regularized C2 -resolvent propagation families for (191) coincide with the corresponding notions introduced in the single-valued linear case (cf. [261, Definition 2.3(ii)–(iii)]). Furthermore, the notion of a k-regularized C1 -existence propagation family for (191) is slightly weaker from that introduced in [261, Definition 2.3(i)] because, in the case when the operator ℬ = B is single-valued, we do not assume here the strong continuity of families (BRi,l (t))t∈[0,τ) ⊆ L(X) and (BR0,1 (t))t∈[0,τ) ⊆ L(X) (subcase (SC1)). In the case k(t) = gζ +1 (t), where ζ ⩾ 0, we say that a k-regularized C1 -existence propagation family for (191) is also a ζ -times integrated C1 -existence propagation family for (191); a 0-times integrated C1 -existence propagation family for (191) is further abbreviated to C1 -existence propagation family for (191). A similar language is used for the classes of C2 -uniqueness propagation families for (191) and C-resolvent propagation families for (191). A k-regularized C1 -existence propagation family for (191) is said to be an exponentially bounded (resp. bounded), analytic k-regularized C1 -existence propagation family for (191), of angle α ∈ (0, π/2], iff for each single operator family (R(t))t⩾0 of it, the following holds: (a) For every x ∈ X, the mapping t 󳨃→ R(t)x, t > 0 can be analytically extended to the sector Σα ; we denote this extension by the same symbol. (b) For every x ∈ X and β ∈ (0, α), we have limz→0,z∈Σβ R(z)x = R(0)x.

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 227

(c) For every β ∈ (0, α), there exists ωβ ⩾ max(0, abs(k)) (resp. ωβ = 0) such that the family {e−ωβ z R(z) : z ∈ Σβ } ⊆ L(X) is bounded. We similarly introduce the classes of exponentially bounded (resp. bounded), analytic k-regularized C2 -uniqueness propagation families for (191) and exponentially bounded (resp. bounded), analytic k-regularized C-resolvent propagation families for (191). Concerning the well-posedness of abstract inhomogeneous Cauchy problems, we want only to observe that assertions (A)–(B) clarified in [261] continue to hold in the multivalued linear operator setting without any terminological changes. We leave to the interested readers the problem of transferring the assertions clarified in [261, Theorem 2.6, Remark 2.7] to the multivalued linear operator case. We can profile the class of k-regularized C-resolvent propagation families for (191) by means of vector-valued Laplace transform. In order to present the main ideas and possible applications, hereafter we will consider only subcase (SC1) in which 0 ∉ 𝒟il ∪S (1 ⩽ i ⩽ mn − 1, 1 ⩽ l ⩽ si ). The following two results can be deduced by using the argument in the proofs of our structural results for multiterm problems with Caputo fractional derivatives. Theorem 3.4.18. Suppose k(t) satisfies (P1), ω ⩾ max(0, abs(k)), as well as that for every i ∈ ℕmn −1 and l ∈ ℕsi , (e−ωt Ri,l (t))t⩾0 ⊆ L(X) is a strongly continuous bounded family; in the case S ≠ 0, (e−ωt R0,1 (t))t⩾0 ⊆ L(X) is also assumed to be a strongly continuous bounded family. Let 0 ∉ 𝒟il ∪ S (1 ⩽ i ⩽ mn − 1, 1 ⩽ l ⩽ si ). (I) Let the following three conditions hold: (i) CAj ⊆ Aj C, j ∈ ℕ0n−1 , Aj ∈ L(X), j ∈ ℕn−1 , Ai Aj = Aj Ai , i, j ∈ ℕn−1 and Aj 𝒜 ⊆ 𝒜Aj , j ∈ ℕn−1 . (ii) If S = 0, there exist an integer i ∈ ℕmn −1 and an integer l ∈ [1, si ] such that the operator n−1

Zλ := χ𝒟l (n)λmn + ∑ χ𝒟l (j)λmj Aj i

j=1

i

̃ is injective for all λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0. (iii) If S = 0 and there do not exist integers i ∈ ℕmn −1 and l ∈ [1, si ] such that the ̃ operator Z is injective for all λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0, then there exists λ

j ∈ ℕn with αj ∈ ℕ. If ((Ri,l (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S = 0, resp. ((Ri,l (t))t⩾0 , (R0,1 (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S ≠ 0, is a k-regularized C-resolvent propagation family for (191), then Pλ is injective for ̃ every λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0, as well as the equalities ∞

n−1

−α−i ̃ Pλ ∫ e−λt Ri,l (t)x dt = k(λ)λ [χ𝒟l (n)λmn Cx + ∑ χ𝒟l (j)λmj Aj Cx] 0

i

j=1

i

(194)

228 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations and ∞

n

̃ ∑ χ (j)A Cx Pλ ∫ e−λt R0,1 (t)x dt = k(λ) S j j=1

0

(195)

̃ are fulfilled for Re λ > ω and k(λ) ≠ 0. ̃ (II) Suppose that Pλ is injective for every λ ∈ ℂ with Re λ > ω and k(λ) ≠ 0 as well as equalities (194)–(195) are fulfilled and condition (I)(i) holds. Then ((Ri,l (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S = 0, resp. ((Ri,l (t))t⩾0 , (R0,1 (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S ≠ 0, is a k-regularized C-resolvent propagation family for (191). Theorem 3.4.19. Assume k(t) satisfies (P1), ω ⩾ max(0, abs(k)), β ∈ (0, π/2] and, for every i ∈ ℕ0mn −1 , the function (k ∗ gi )(t) can be analytically extended to a function ki : Σβ → ℂ such that, for every γ ∈ (0, β), the set {e−ωz ki (z) : z ∈ Σγ } is bounded. Let 0 ∉ 𝒟il ∪ S (1 ⩽ i ⩽ mn − 1, 1 ⩽ l ⩽ si ), and let the following four conditions hold: (i) CAj ⊆ Aj C, j ∈ ℕ0n−1 , Aj ∈ L(X), j ∈ ℕn−1 , Ai Aj = Aj Ai , i, j ∈ ℕn−1 and Aj 𝒜 ⊆ 𝒜Aj , j ∈ ℕn−1 . (ii) The operator Pλ is injective for all ω + Σβ+π/2 . (iii) For every integers i ∈ ℕmn −1 and l ∈ [1, si ], there exist an operator Di,l ∈ L(X) and a strongly analytic mapping qi,l : ω + Σ π +β → L(X) satisfying the following: 2

n−1

qi,l (λ)x = kĩ (λ)Pλ−1 [χ𝒟l (n)λmn −α Cx + ∑ χ𝒟l (j)λmj −α Aj Cx], i

j=1

i

for any x ∈ X, Re λ > ω, the family {(λ − ω)qi,l (λ) : λ ∈ ω + Σ π +γ } is bounded for all γ ∈ (0, β), 2

and, in the case D(𝒜) ≠ X, lim λqi,l (λ)x = Di,l x,

λ→+∞

x ∉ D(𝒜).

(iv) If S = 0, then there exist an operator D ∈ L(X) and a strongly analytic mapping q : ω + Σ π +β → L(X) satisfying the following: 2

n

−1 ̃ q(λ)x = k(λ)P λ ∑ χS (j)Aj Cx, j=1

x ∈ X, Re λ > ω,

the family {(λ − ω)q(λ) : λ ∈ ω + Σ π +γ } is bounded for all γ ∈ (0, β), 2

and, in the case D(𝒜) ≠ X, lim λq(λ)x = Dx,

λ→+∞

x ∉ D(𝒜).

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 229

Then there exists an exponentially bounded, analytic k-regularized C-resolvent propagation family ((Ri,l (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S = 0, resp. ((Ri,l (t))t⩾0 , (R0,1 (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S ≠ 0, for (191), of angle β. Furthermore, the family {e−ωz R⋅,⋅ (z) : z ∈ Σγ } is bounded for all γ ∈ (0, β), (192)–(193) and R⋅,⋅ (z)𝒜j ⊆ 𝒜j R⋅,⋅ (z), z ∈ Σβ , j ∈ ℕ0n−1 are valid for any single operator family R⋅,⋅ (⋅). Observe that the notion from Definition 3.4.17 can be modified and introduced for single operator families. The former two theorems can be simply reworded in this context, which will be important in the sequel. For our investigation of generalized asymptotically almost periodic and generalized asymptotically almost automorphic solutions of (191), the following analogue of Theorem 3.4.10 is crucial to be stated. The proof is very similar to that of Theorem 3.4.10: Theorem 3.4.20. Suppose that cj ⩾ 0, ℬ = I and Aj = cj I for 1 ⩽ j ⩽ n − 1, ζ 󸀠 ⩾ 0, 𝒜 : X → P(X) is a closed MLO, C ∈ L(X) is injective, C 𝒜 ⊆ 𝒜C and condition (H) holds with certain finite constants c < 0, M > 0, 0 < θ < π and β ∈ (0, 1]. Assume that the mapping λ 󳨃→ (λ − 𝒜)−1 C, λ ∈ c + Σπ−θ is strongly continuous. Let the integers i ∈ ℕmn −1 and l ∈ [1, si ] be fixed, and let 0 ∉ 𝒟il . Set r := max{s ∈ ℕn : s ∈ 𝒟il }. Assume also that r − α − i − ζ 󸀠 − (αn − α)β ⩽ 0, and ν󸀠 :=

(196)

π π−θ − > 0. αn − α 2

(197)

Set ζ := ζ 󸀠 if 𝒜 is densely defined, ζ > ζ 󸀠 otherwise, and ki (⋅) := gζ +1 (⋅). Then there exists an exponentially bounded, analytic ki -regularized C-propagation family (Ri,l (t))t⩾0 for (191), of angle ν := min(ν󸀠 , π/2). Moreover, (192)–(193) hold and there exists a finite constant M 󸀠 > 0 such that n−1

󵄩 󵄩󵄩 󸀠 α+ζ +i−mn l α+ζ +i−mj l 𝒟i (n) + ∑ t 𝒟i (j)], 󵄩󵄩Ri (t)󵄩󵄩󵄩 ⩽ M [t j=1

t > 0.

(198)

Proof. As in the proof of Theorem 3.4.10, the mapping λ 󳨃→ (λ − 𝒜)−1 C, λ ∈ c + Σπ−θ is strongly continuous and its restriction to c + Σπ−θ is strongly analytic. Since cj ⩾ 0 for 1 ⩽ j ⩽ n − 1 and estimates (196)–(197) are valid, we can easily show that the conditions of Theorem 3.4.19 hold (for single operator families, see the short discussion above), with ω > 0 sufficiently large and ki (⋅). Hence, 𝒜 is a subgenerator of an exponentially bounded, analytic ζ -times integrated C-propagation family (Ri,l (t))t⩾0 for

230 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations (191), of angle ν := min(ν󸀠 , π/2), as claimed. The estimate (198) can be deduced as in the proof of Theorem 3.4.10, employing some estimates contained in the proof of [29, Theorem 2.6.1]. Assume that ℱ ∈ {APT ([0, ∞) : ℂ), AAP([0, ∞) : ℂ), AAA([0, ∞) : ℂ)}. Let the function ki (⋅) be defined as above, let f ∈ ℱ , and let (Ri (t))t⩾0 be the ki -regularized C-propagation family for (191), constructed with the help of previous theorem. Then it can be easily verified that ((Ri ∗ f )(t))t⩾0 is a ki -regularized C-propagation family for (191), satisfying additionally (192), where ki (⋅) = (gζ +1 ∗ f )(⋅). Assume that (α + ζ + i − mn )𝒟il (n) + (α + ζ + i − mj )𝒟il (j) < −1.

(199)

Applying Theorem 3.4.20, already proven statements concerning inheritance of asymptotical periodicity, almost asymptotical almost periodicity and asymptotical almost automorphy under the action of finite convolution products, we can establish the following result (the uniqueness of solutions is a simple consequence of the fact that [261, Theorem 2.5] holds in our framework): Corollary 3.4.21. Let the requirements of Theorem 3.4.20 hold, let f ∈ ℱ , and let ki (⋅) = (gζ +1 ∗ f )(⋅). Assume that (199) holds. Define ux (t) := (Ri ∗ f )(t)x, t ⩾ 0, x ∈ X. Then, for every x ∈ X, ux (⋅) ∈ ℱ is a unique mild solution of the abstract Cauchy inclusion [u(t)x − χ𝒟l (n)(f ∗ gαn +i−mn +ζ +1 )(t)Cx] i

n−1

+ ∑ cj [gαn −αj ∗ (u(⋅) − χ𝒟l (j)(f ∗ gαj +i−mj +ζ +1 )(⋅)Cx)](t) i

j=1

∈ 𝒜[gαn −α ∗ (u(⋅) − χ𝒟l (0)(f ∗ gα+i−m+ζ +1 )(⋅)Cx)](t), i

t ⩾ 0.

Furthermore, ux (⋅) is a strong solution of the above inclusion provided that x ∈ D(𝒜). As before, Corollary 3.4.21 is applicable in the study of certain abstract degenerate integral Cauchy problems involving the Poisson heat operator (see [188]). Semilinear Cauchy integral inclusions can be also examined, with the help of already established results and theorems from the fixed point theory. Remark 3.4.22. The assertions of Theorem 3.4.20 and Corollary 3.4.21 can be also formulated, with minor modifications, for exponentially bounded, analytic C-regularized solution operator families whose Laplace transform can be computed as ∞

s

0

j=1

∫ e−λt R(t)x dt = λ−ζ −1 Pλ−1 C[λas + ∑ cj λaj ]x,

x ∈ X, Re λ > ω,

where ω > 0, s ∈ ℕ, 0 ⩽ a1 < a2 < ⋅ ⋅ ⋅ < as , ζ ⩾ 0 and cj ⩾ 0, j ∈ ℕs .

3.4 Generalized asymptotically almost periodic (and automorphic) solutions | 231

Let ℱ 󸀠 and ℱ 󸀠󸀠 be the spaces of generalized asymptotically almost periodic (automorphic) functions already defined in this section. The following analogue of Proposition 3.4.13 holds in our new framework: Proposition 3.4.23. Suppose that k(t) satisfies (P1), i ∈ ℕmn −1 , 1 ⩽ l ⩽ si , ((Ri,l (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S = 0, resp. ((Ri,l (t))t⩾0 , (R0,1 (t))t⩾0 )1⩽i⩽mn −1,1⩽l⩽si if S ≠ 0, is a strongly Laplace transformable k-regularized C-resolvent propagation family for (191). (i) For every λ ∈ ℂ, there exists a function fλi,l (⋅) satisfying (P1)-L(X) and fλi,l (t)

−1

:= ℒ ([(1 − × (χ𝒟l (n) i

λ

z αn −α ̃ k(z)

n−1

)I + ∑ j=1

Aj

−1

z αn −αj

]

n−1

z αn +i−mn

C + ∑ χ𝒟l (j)Aj j=1

i

z

̃ k(z)

αn +i−mj

C − χ𝒟l (0) i

̃ λk(z) C))(t), α +i−m n z

for any t ⩾ 0, and a function fλ0,1 (⋅) satisfying (P1)-L(X) and fλ0,1 (t)

−1

:= ℒ ([(1 −

λ

n−1

z αn −α

)I + ∑ j=1

Aj

−1

z αn −αj

]

n−1

αj −αn α−αn ̃ + ∑ χ (j)A k(z)z ̃ ̃ × (χS (n)k(z) − χS (0)k(z)z ))(t), S j j=1

for any t ⩾ 0. (ii) Denote by D the set consisting of all eigenvectors x of operator 𝒜 corresponding to eigenvalues λ ∈ ℂ of operator 𝒜 (λx ∈ 𝒜x) for which the mapping i,l fλ,x (t) := fλi,l (t)x,

t ⩾ 0,

resp.

0,1 fλ,x (t) := fλ0,1 (t)x,

t⩾0

belongs to the space ℱ 󸀠 . Then the mapping t 󳨃→ Ri,l (t)x, t ⩾ 0, resp. t 󳨃→ R0,1 (t)x, t ⩾ 0, belongs to the space ℱ 󸀠 for all x ∈ span(D); furthermore, the mapping t 󳨃→ Ri,l (t)x, t ⩾ 0, resp. t 󳨃→ R0,1 (t)x, t ⩾ 0, belongs to the space ℱ 󸀠󸀠 for all x ∈ span(D) provided additionally that (Ri (t))t⩾0 is bounded. The assertion of Theorem 3.4.15 can be also rephrased for abstract multiterm fractional differential inclusions with Riemann–Liouville derivatives. We leave details to the interested readers.

232 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations

3.5 Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions We divide this section into two separate subsections. In the first one, we examine the generalized (asymptotically) almost automorphic properties of convolution products, while in the second one we analyze the existence and uniqueness of various types of almost automorphic and asymptotically almost automorphic solutions to semilinear Cauchy inclusions with multivalued linear operators.

3.5.1 Generalized (asymptotically) almost automorphic properties of convolution products Proposition 2.6.11 can be straightforwardly formulated for various classes of (asymptotically) almost automorphic (a, k)-regularized C-resolvent families in Banach spaces. On the other hand, numerous very nontrivial and unpleasant problems occur if we try to reconsider some known assertions on the (asymptotical) almost periodicity of (a, k)-regularized C-resolvent families in Banach spaces, provided that the results from the Bohr–Fourier analysis of almost periodic functions are needed for their proofs. In our first structural result, we observe that the assertion of [117, Lemma 3.1] can be formulated for strongly continuous operator families which do have integrable singularities at zero, with possibly two different pivot spaces (see also [76, Theorem 2.1]): Proposition 3.5.1. Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator ∞ family satisfying that ∫0 ‖R(t)‖ dt < ∞. If g : ℝ → X is almost automorphic, then the Y-valued function F(⋅) = G(⋅), given by (68), with f = g, is well-defined and almost automorphic. Remark 3.5.2. In [168, Lemma 2.2], H.-S. Ding, J. Liang and T.-J. Xiao have proved, under the assumption on strong continuity of (R(t))t⩾0 and the existence of nonincreasing continuous function ϕ ∈ L1 ([0, ∞)) satisfying ‖R(t)‖ ⩽ ϕ(t), t ⩾ 0 that the function F(⋅) is almost automorphic provided only the Stepanov 1-almost automorphy of function f : ℝ → X. Here we would like to observe that their result holds provided that the strong continuity of (R(t))t⩾0 is replaced by the strong continuity of (R(t))t>0 and the boundedness of supt∈(0,1] ‖R(t)‖ (this also holds provided that X ≠ Y in Proposition 3.5.1). Possible applications can be made, e. g., in the qualitative analysis of the Poisson heat equation in the space H −1 (Ω), where 0 ≠ Ω ⊆ ℝn is an open bounded domain with smooth boundary (see [188, Theorem 3.1, Proposition 3.2, p. 48; Remark, p. 52; Example 3.3, pp. 74–75], with β = 1, for further information in this direction).

3.5 Generalized almost (and asymptotically almost) automorphic solutions | 233

Our first original contribution reads as follows (see Proposition 2.6.11 for almost periodic case). Proposition 3.5.3. Suppose that 1 ⩽ p < ∞, 1/p + 1/q = 1 and (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family satisfying that M = ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞. If p f : ℝ → X is S -almost automorphic, then the Y-valued function F(⋅) = G(⋅), given by (68), with f = g, is well-defined and almost automorphic. Proof. We may assume that X = Y. It is clear that, for every t ∈ ℝ, we have F(t) = ∞ ∫0 R(s)f (t − s) ds. The measurability of integrand is a consequence of the proof of [29, Proposition 1.3.4], while the absolute convergence of integral follows similarly as in t−(k−1) R(s)f (t − s) ds, t ∈ ℝ (k ∈ ℕ). We claim that Proposition 2.6.11. Define Fk (t) := ∫t−k Fk (⋅) is continuous. Let numbers ε > 0 and t ∈ ℝ be given in advance, and let (tn ) be a real sequence converging to t. Then Hölder inequality yields k+1

󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩Fk (tn ) − Fk (t)󵄩󵄩󵄩 ⩽ ∫ 󵄩󵄩󵄩R(σ)󵄩󵄩󵄩󵄩󵄩󵄩f (tn − σ) − f (t − σ)󵄩󵄩󵄩 dσ k

1/p

k+1

󵄩p 󵄩 󵄩 󵄩 ⩽ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩f (tn + σ) − f (t + σ)󵄩󵄩󵄩 dσ) k

1/p

k+t+1

󵄩 󵄩 󵄩p 󵄩 = 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] ( ∫ 󵄩󵄩󵄩f (t − tn + σ) − f (σ)󵄩󵄩󵄩 dσ)

,

k ∈ ℕ.

(200)

k+t

Since f ∈ Lploc (ℝ : X), the last term in brackets tends to zero as n → ∞; for the sake of completeness, we will give a direct proof of this fact below. It is clear that there exists a sequence of infinitely differentiable functions (fl )l∈ℕ converging to f in Lp [t +k, t +k +2] (k ∈ ℕ). Then k+t+1

󵄩p 󵄩 ∫ 󵄩󵄩󵄩f (t − tn + σ) − f (σ)󵄩󵄩󵄩 dσ

k+t

p−1

⩽3

k+t+1

󵄩p 󵄩 [ ∫ 󵄩󵄩󵄩f (t − tn + σ) − fl (t − tn + σ)󵄩󵄩󵄩 dσ k+t

k+t+1

k+t+1

k+t

k+t

󵄩p 󵄩 󵄩p 󵄩 + ∫ 󵄩󵄩󵄩fl (t − tn + σ) − fl (σ)󵄩󵄩󵄩 dσ + ∫ 󵄩󵄩󵄩fl (σ) − f (σ)󵄩󵄩󵄩 dσ]. Let l0 = l0 (ε) ∈ ℕ be such that k+t+1

k+t+1

k+t

k+t

󵄩p 󵄩 󵄩p 󵄩 ∫ 󵄩󵄩󵄩f (t − tn + σ) − fl (t − tn + σ)󵄩󵄩󵄩 dσ + ∫ 󵄩󵄩󵄩fl (σ) − f (σ)󵄩󵄩󵄩 dσ ⩽ 2ε/3

234 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations for all n ∈ ℕ and l ⩾ l0 (ε). Then it is clear that there exists n0 (ε) ∈ ℕ such that k+t+1

k+t+1

∫k+t ‖fl0 (t − tn + σ) − fl0 (σ)‖p dσ ⩽ ε for all n ⩾ n0 (ε). Hence, ∫k+t ‖f (t − tn + σ) − f (σ)‖p dσ ⩽ ε, n ⩾ n0 (ε) and continuity of Fk (⋅) follows by implementing (200). Since we have assumed ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞, Weierstrass criterion implies that F (t) = F(t) uniformly in t ∈ ℝ, so that F(⋅) is continuous on ℝ as well. Since ∑∞ k=1 k AA(ℝ : X) is closed in Cb (ℝ : X), it suffices to show that Gk ∈ AA(ℝ : X) for all k ∈ ℕ. Fix an integer k ∈ ℕ. Since f ∈ AA(ℝ : X), for every real sequence (bn ) there exist a subsequence (an ) of (bn ) and a map g : ℝ → X such that (156) and (157) hold pointk

wise for t ∈ ℝ. Define gk,c : ℝ → X by gk,c (t) := ∫k−1 R(σ)g(t − σ) dσ, t ∈ ℝ. Clearly, k

Fk (t) = ∫k−1 R(σ)f (t − σ) dσ, t ∈ ℝ and, due to Hölder inequality, k

󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩Fk (t + tn ) − gk,c (t)󵄩󵄩󵄩 ⩽ ∫ 󵄩󵄩󵄩R(σ)[f (t + tn − σ) − g(t − σ)]󵄩󵄩󵄩 dσ k−1

k

󵄩 󵄩 󵄩 󵄩p ⩽ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k−1,k] ( ∫ 󵄩󵄩󵄩f (t + tn − σ) − g(t − σ)󵄩󵄩󵄩 dσ) k−1

t−(k−1)

󵄩p 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k−1,k] ( ∫ 󵄩󵄩󵄩f (σ + tn ) − g(σ)󵄩󵄩󵄩 dσ)

1/p

1/p

,

t ∈ ℝ.

t−k

This, in combination with (156), implies limn→∞ ‖Fk (t + tn ) − gk,c (t)‖ = 0 pointwise for t ∈ ℝ. We can similarly prove that limn→∞ ‖gk,c (t − tn ) − Fk (t)‖ = 0 pointwise for t ∈ ℝ, finishing the proof of the theorem. Keeping in mind Proposition 3.5.3 and our previous considerations, we can immediately state the following assertion: Proposition 3.5.4. Suppose that 1 ⩽ p < ∞, 1/p + 1/q = 1 and (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that, for every s ⩾ 0, we have ms = p ∑∞ k=0 ‖R(⋅)‖Lq [s+k,s+k+1] < ∞. Suppose, further, that g : ℝ → X is S -almost automorphic, as well as that the locally p-integrable function q : [0, ∞) → X satisfies q̂ ∈ C([0, ∞) : Lp ([0, 1] : X)) and f (t) = g(t) + q(t), t ⩾ 0. Let there exist a finite number M > 0 such that the following holds: t+1 s (i) limt→+∞ ∫t [∫M ‖R(r)‖‖q(s − r)‖ dr]p ds = 0. t+1

(ii) limt→+∞ ∫t

mps ds = 0.

Then the Y-valued function H(⋅), given by t

H(t) := ∫ R(t − s)f (s) ds,

t ⩾ 0,

0

is well-defined, bounded continuous and asymptotically Sp -almost automorphic.

3.5 Generalized almost (and asymptotically almost) automorphic solutions | 235

Concerning the class of Besicovitch p-almost automorphic functions, the following result seems to be satisfactory only for the abstract differential equations with integer order derivatives and nonautonomous differential equations (for fractional resolvent families, condition (201) stated below does not hold in practical situations): Proposition 3.5.5. Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that ∞

󵄩 󵄩 ∫ (1 + t)󵄩󵄩󵄩R(t)󵄩󵄩󵄩 dt < ∞.

(201)

0

Let f ∈ B1 AA(ℝ : X), and let f (⋅) be essentially bounded. Then the function F(⋅) = G(⋅), given by (68), with f = g, is bounded and belongs to the class B1 AA(ℝ : Y). Proof. Without loss of generality, we may assume that X = Y. The fact that function F(⋅) is bounded continuous and well-defined follows from the proofs of Propositions 3.5.3 and 2.11.1. It remains to prove that F ∈ B1 AA(ℝ : X). Towards this end, let (sn ) be an arbitrary real sequence. By definition and elementary changes of variables, we know that there exist a subsequence (snk ) of (sn ) and a function f ∗ ∈ L1loc (ℝ : X) such that l+t

lim lim sup

k→∞ l→+∞

1 󵄩 󵄩 ∫ 󵄩󵄩󵄩f (snk + x) − f ∗ (x)󵄩󵄩󵄩 dx = 0 2l

(202)

−l+t

and l+t

1 󵄩 󵄩 lim lim sup ∫ 󵄩󵄩󵄩f ∗ (x − snk ) − f (x)󵄩󵄩󵄩 dx = 0 k→∞ l→+∞ 2l −l+t

x

for each t ∈ ℝ. Set F ∗ (x) := ∫−∞ R(x − s)f ∗ (s) ds, x ∈ ℝ. Then F ∗ ∈ L1loc (ℝ : X). To see this, it suffices to observe that, for −∞ < a < b < ∞, we have b󵄩 x

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ∫󵄩󵄩󵄩 ∫ R(x − s)f ∗ (s) ds󵄩󵄩󵄩 dx 󵄩󵄩 󵄩󵄩 󵄩 a 󵄩−∞ b x

󵄩 󵄩󵄩 󵄩 ⩽ ∫ ∫ 󵄩󵄩󵄩R(x − s)󵄩󵄩󵄩󵄩󵄩󵄩f ∗ (s)󵄩󵄩󵄩 ds dx a −∞ b ∞

󵄩󵄩 󵄩 󵄩 = ∫ ∫ 󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩f ∗ (x − s)󵄩󵄩󵄩 ds dx a 0

∞ b

󵄩 󵄩󵄩 󵄩 = ∫ ∫󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩f ∗ (x − s)󵄩󵄩󵄩 dx ds 0 a

236 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations ∞

b−s

0

a−s

󵄩 1 󵄩 󵄩 󵄩 = ∫ (1 + s)󵄩󵄩󵄩R(s)󵄩󵄩󵄩[ ∫ 󵄩󵄩f ∗ (r)󵄩󵄩󵄩 dr] ds s+1 󵄩 b−s

as well as that the continuous mapping s 󳨃→ (s + 1)−1 ∫a−s ‖f ∗ (r)‖ dr, s ⩾ 0 is bounded since condition (202) with t = 0 and the essential boundedness of function f (⋅) together imply that there exists a number s0 such that [a − s, b − s] ⊆ [−2s, 2s], s ⩾ s0 2s and s−1 ∫−2s ‖f ∗ (r)‖ dr ⩽ 4‖f ‖∞ + 4, s ⩾ s0 ; here we also use (201). Therefore, we need to prove that l+t

1 󵄩 󵄩 lim lim sup ∫ 󵄩󵄩󵄩F(snk + x) − F ∗ (x)󵄩󵄩󵄩 dx = 0 k→∞ l→+∞ 2l −l+t

and l+t

lim lim sup

k→∞ l→+∞

1 󵄩 󵄩 ∫ 󵄩󵄩󵄩F ∗ (x − snk ) − F(x)󵄩󵄩󵄩 dx = 0 2l

(203)

−l+t

pointwise for t ∈ ℝ. The first of these equalities follows from the next computation involving Fubini theorem: l+t

1 󵄩 󵄩 ∫ 󵄩󵄩󵄩F(snk + x) − F ∗ (x)󵄩󵄩󵄩 dx 2l −l+t

l+t 󵄩 x+snk x 󵄩󵄩 󵄩󵄩󵄩 1 󵄩󵄩 󵄩 = ∫ 󵄩󵄩 ∫ R(x − s + snk )f (s) ds − ∫ R(x − s)f ∗ (s) ds󵄩󵄩󵄩 dx 󵄩󵄩 󵄩󵄩 2l 󵄩 󵄩 −∞ −l+t −∞ x l+t 󵄩 x 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 ∗ = ∫ 󵄩󵄩 ∫ R(x − s)f (s + snk ) ds − ∫ R(x − s)f (s) ds󵄩󵄩󵄩 dx 󵄩󵄩 󵄩󵄩 2l 󵄩 −∞ −l+t 󵄩−∞ l+t 󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 ∗ = ∫ 󵄩󵄩 ∫ R(r)[f (x − r + snk ) − f (x − r)] dr 󵄩󵄩󵄩 dx 󵄩󵄩 󵄩󵄩 2l 󵄩 −l+t 󵄩 0 l+t ∞

⩽

1 󵄩 󵄩󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩󵄩󵄩󵄩f (x − r + snk ) − f ∗ (x − r)󵄩󵄩󵄩 dr dx 2l −l+t 0

∞

󵄩1 󵄩 = ∫ 󵄩󵄩󵄩R(r)󵄩󵄩󵄩 2l 0 ∞

l+t−r

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (x + snk ) − f ∗ (x)󵄩󵄩󵄩 dx dr

−l+t−r

󵄩 󵄩 = ∫ (1 + r)󵄩󵄩󵄩R(r)󵄩󵄩󵄩 0

3.5 Generalized almost (and asymptotically almost) automorphic solutions | 237

1 2(l + r) 1 ×[ 1 + r 2l 2(l + r)

l+t+r

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (x + snk ) − f ∗ (x)󵄩󵄩󵄩 dx] dr.

(−l+r)+t

For any ε > 0 given in advance, we can find k0 (ε) > 0 such that for every k ⩾ k0 (ε) we can find y0 (ε, k) > 0 such that, for every r ⩾ 0 and l ⩾ y0 (ε, k), we have 1 2(l + r)

l+t+r

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (x + snk ) − f ∗ (x)󵄩󵄩󵄩 dx < ε.

(−l+r)+t

Since (201) is assumed, this proves the claimed. The proof of (203) is similar and therefore omitted. Remark 3.5.6. Let the requirements of the previous proposition hold, and let the function q ∈ L1loc ([0, ∞) : X) be Weyl-1-vanishing, resp. equi-Weyl-1-vanishing. Set, as before, x+t

1 J(t, l) := sup{ ∫ [ l x⩾0 0

x+t−r+l

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr},

x+t−r

t > 0, l > 0.

Assume that the condition limt→∞ liml→∞ J(t, l) = 0 holds provided that q(⋅) is Weyl-1-vanishing, resp. that the condition liml→∞ limt→∞ J(t, l) = 0 holds when∞ ever q(⋅) is equi-Weyl-1-vanishing. By the foregoing, we have that limt→∞ ∫t R(s)g(t − t

s) ds = 0 as well as that the function t 󳨃→ ∫0 R(t − s)q(s) ds, t ⩾ 0 is Weyl-1-vanishing, t

resp. equi-Weyl-1-vanishing. Hence, the function t 󳨃→ ∫0 R(t − s)[f (s) + q(s)] ds, t ⩾ 0 belongs to the class B1 AA(ℝ : Y) + B10 ([0, ∞) : Y), B1 AA(ℝ : Y) + e − W01 ([0, ∞) : Y), with the meaning clear. Here we would like to note only that condition (201) enables one to estimate the term appearing in definition of J(t, l) for x ⩾ 0 in the following way: x+t

∫[ 0

1 l

x+t−r+l

󵄩 󵄩 󵄩 󵄩 ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr

x+t−r x+t

1 ⩽ ∫[ l(1 + x + t − r) 0

∞

x+t−r+l

󵄩 󵄩 󵄩 󵄩 ∫ (1 + x + t − r)󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv]󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr

x+t−r x+t

󵄩 󵄩 ⩽ [ ∫ (1 + t)󵄩󵄩󵄩R(t)󵄩󵄩󵄩 dt] ∫ 0

0

1 󵄩 󵄩󵄩 󵄩q(r)󵄩󵄩󵄩 dr, l(1 + x + t − r) 󵄩

t > 0, l > 0.

Finally, it seems reasonable to ask whether the conclusions from Proposition 3.5.5, this remark and Proposition 3.6.2 below can be reexamined for the value p > 1 of almost automorphicity exponent, provided that the solution operator family (R(t))t>0 ⊆ L(X, Y) satisfies condition (114).

238 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations 3.5.2 Semilinear Cauchy inclusions This subsection will be written in the expository manner, without giving the proofs of our abstract results. Our aim here is to explain how the already proven statements on almost periodic and pseudo almost-periodic solutions of semilinear (fractional) Cauchy inclusions can be formulated for almost automorphy and pseudo almostautomorphy. Let M > 0 denote the constant from (A), let the sequence (Mn ) be defined through (106) and let the sequence (Bn ) be defined through (110) (cf. Section 2.10 for more details). Keeping in mind Proposition 3.5.3 and Theorems 3.2.2–3.2.3, it is straightforward to prove the following automorphic versions of Theorems 2.10.3–2.10.4 and Theorems 2.10.9–2.10.10 (in these reformulations, we assume that Y = [D((−𝒜)θ )]): Theorem 3.5.7. Suppose that (P) holds, β > θ > 1 − β and the following conditions are satisfied: (i) f ∈ AASp (ℝ × Y : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) as well as a locally bounded nonnegative function Lf ∈ LrS (ℝ) such that r > p/p − 1 and (73) holds. pr Set q := pr/p + r and q󸀠 := pr−p−r . Assume also that: (ii) q󸀠 (β − θ − 1) > −1. (iii) Mn < 1 for some n ∈ ℕ. Then there exists an almost automorphic mild solution of inclusion (78). The uniqueness of mild solutions holds in the case 𝒜 is single-valued. Theorem 3.5.8. Suppose that (P) holds, β > θ > 1 − β and the following conditions are satisfied: (i) f ∈ AASp (ℝ × Y : X) with p > 1, and there exists a constant L > 0 such that (74) holds. p (β − θ − 1) > −1. (ii) p−1 (iii) Mn < 1 for some n ∈ ℕ. Then there exists an almost automorphic mild solution of inclusion (78). The uniqueness of mild solutions holds provided that, in addition to (i)–(iii), 𝒜 is single-valued. Theorem 3.5.9. Suppose that (P) holds, β > θ > 1 − β and the following conditions are satisfied: (i) f ∈ AASp (ℝ × Y : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) as well as a locally bounded nonnegative function Lf ∈ LrS (ℝ) such that r > p/p − 1 and (73) holds. pr . Set q := pr/p + r and q󸀠 := pr−p−r Assume also that:

3.5 Generalized almost (and asymptotically almost) automorphic solutions | 239

(ii) q󸀠 (γ(β − θ) − 1) > −1. (iii) Bn < 1 for some n ∈ ℕ. Then there exists an almost automorphic mild solution of inclusion (103). The uniqueness of mild solutions holds provided that, in addition to (i)–(iii), 𝒜 is single-valued. Theorem 3.5.10. Suppose that (P) holds, β > θ > 1 − β and the following conditions are satisfied: (i) f ∈ AASp (ℝ × Y : X) with p > 1, and there exists a constant L > 0 such that (74) holds. p (ii) p−1 (γ(β − θ) − 1) > −1. (iii) Bn < 1 for some n ∈ ℕ. Then there exists an almost automorphic mild solution of inclusion (103). The uniqueness of mild solutions holds provided that, in addition to (i)–(iii), 𝒜 is single-valued. Connecting Theorem 3.2.4 and Proposition 3.5.3, we can simply clarify the following modification of Theorem 2.12.4, as well: Theorem 3.5.11. Suppose that the following conditions hold: (i) f ∈ PAA(ℝ × X : X) is pseudo-almost automorphic. (ii) Inequality (73) holds with I = ℝ, X = Y and some bounded nonnegative function Lf (⋅). (iii) ∑∞ n=1 Mn < ∞. Then there exists a unique pseudo-almost automorphic solution of inclusion (78). The existence and uniqueness of pseudo-almost automorphic solutions of semilinear Cauchy inclusion (103) can be analyzed similarly. The proofs of the following automorphic versions of Lemma 2.9.3, formulated here as a proposition, and Theorem 2.9.5, can be deduced by using the foregoing arguments. Proposition 3.5.12. Let f ∈ AAASq ([0, ∞) : X) with some q ∈ (1, ∞), let 1/q + 1/q󸀠 = 1, and let q󸀠 (γβ − 1) > −1. Define t

H(t) := ∫ Rγ (t − s)f (s) ds,

t ⩾ 0.

0

Then H ∈ AAA([0, ∞) : X). Theorem 3.5.13. Suppose that 1 ⩾ θ > 1 − β and x0 ∈ D((−𝒜)θ ), resp. 1 > θ > 1 − β and θ x0 ∈ X𝒜 , as well as there exists a constant σ > γ(1 − β) such that, for every T > 0, there exists a finite constant MT > 0 such that f : [0, ∞) → X satisfies 󵄩 󵄩󵄩 σ 󵄩󵄩f (t) − f (s)󵄩󵄩󵄩 ⩽ MT |t − s| ,

0 ⩽ t, s ⩽ T.

240 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Let 1 ⩾ θ > 1 − β, resp. 1 > θ > 1 − β, and let θ f ∈ L∞ loc ((0, ∞) : [D((−𝒜) )]),

resp.

θ f ∈ L∞ loc ((0, ∞) : X𝒜 ).

Then there exists a unique classical solution u(⋅) of problem (DFP)f ,γ . If, additionally, f ∈ AAASq ([0, ∞) : X) with some q ∈ (1, ∞), 1/q + 1/q󸀠 = 1 and q󸀠 (γβ − 1) > −1, then u ∈ AAA([0, ∞) : X). Keeping in mind Propositions 3.2.5–3.2.6 and our results clarified in the previous section, we can repeat almost literally the proofs of our structural results given in Subsections 2.9.1 and 2.9.2. In such a way, we can simply state the automorphic versions of all established results from these subsections. For the sake of completeness, we will reformulate the above-mentioned Theorem 2.9.10 in our new context, only: Theorem 3.5.14. Suppose that I = [0, ∞) and the following conditions hold: (i) g ∈ AASp (ℝ × X : X) with p > 1, and there exist a number r ⩾ max(p, p/p − 1) and a function Lg ∈ LrS (I) such that (73) holds with f = g and Lf = Lg . (ii) f (t, x) = g(t, x) + q(t, x) for all t ⩾ 0 and x ∈ X, where q̂ ∈ C0 (I × X : Lq ([0, 1] : X)) and q = pr/p + r. Set q󸀠 := ∞, provided r = p/p − 1 and

q󸀠 :=

pr , provided r > p/p − 1. pr − p − r

Assume also that: (iii) q󸀠 (γβ − 1) > −1. (iv) (73) holds for a. e. t > 0, with X = Y and a locally bounded positive function Lf (⋅) satisfying An < 1 for some n ∈ ℕ; recall t xn

x2

󵄩 󵄩 An = sup ∫ ∫ ⋅ ⋅ ⋅ ∫󵄩󵄩󵄩Rγ (t − xn )󵄩󵄩󵄩 t⩾0

n

0 0

0

n

󵄩 󵄩 × ∏󵄩󵄩󵄩Rγ (xi − xi−1 )󵄩󵄩󵄩 ∏ Lf (xi ) dx1 dx2 ⋅ ⋅ ⋅ dxn , i=2

i=1

n ∈ ℕ.

Then there exists a unique asymptotically almost automorphic solution of inclusion (DFP)f ,γ,s .

3.6 Generalized weighted pseudo-almost periodic solutions and generalized weighted pseudo-almost automorphic solutions of abstract Volterra integro-differential inclusions This section, whose organization is very similar to that of previous one, will be broken down into two separate subsections.

3.6 Generalized weighted pseudo-almost periodic (and automorphic) solutions | 241

3.6.1 Generalized weighted almost periodic (automorphic) properties of convolution products We start this section by stating the following result, expanding thus our earlier researches. Proposition 3.6.1. Suppose that 1 ⩽ p < ∞, 1/p + 1/q = 1 and (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that M = ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞. If the space PAP0 (ℝ, X, ρ1 , ρ2 ) is translation invariant (see Proposition 3.3.6) and f : ℝ → X is weighted Sp -almost periodic, resp. weighted Sp -almost automorphic, then function F(⋅) = G(⋅), given by (68), with f = g, is continuous and belongs to the class AP(ℝ, Y, ρ1 , ρ2 ) + Sp WPAA0 (ℝ, Y, ρ1 , ρ2 ), resp. AA(ℝ, Y, ρ1 , ρ2 ) + Sp WPAA0 (ℝ, Y, ρ1 , ρ2 ). Proof. We will prove the theorem only for weighted Sp -almost automorphy, with X = Y. Let f (⋅) = g(⋅) + q(⋅), where g(⋅) and q(⋅) satisfy conditions from Definition 3.3.3(i). We have already seen that, due to the Sp -almost automorphy of g(⋅), we have that the function G(⋅) obtained by replacing f (⋅) in (68) by g(⋅), is almost automorphic. Define k+1 Qk (t) := ∫k R(s)q(t − s) ds, t ∈ ℝ (k ∈ ℕ). Arguing as in the proof of Proposition 3.5.3, we can prove that Qk (⋅) is bounded and continuous on ℝ for all k ∈ ℕ, as well as ⋅ that Qk (⋅) converges uniformly to Q(⋅) := ∫−∞ R(⋅ − s)q(s) ds. Therefore, all we need to prove is that, for any integer k ∈ ℕ given in advance, (165) holds with the function q(⋅) replaced therein by Qk (⋅). By Hölder inequality and an elementary change of variables in the double integral, we have the existence of a positive finite constant ck > 0 such that T

T

1

󵄩p 󵄩 ∫ [ ∫ 󵄩󵄩󵄩Qk (s)󵄩󵄩󵄩 ds]

2 ∫−T ρ1 (t) dt −T ⩽

=

⩽

t+1 t

‖R(⋅)‖Lq [k,k+1] T

2 ∫−T ρ1 (t) dt ‖R(⋅)‖Lq [k,k+1] T

2 ∫−T ρ1 (t) dt ‖R(⋅)‖Lq [k,k+1] T

2 ∫−T ρ1 (t) dt +

T

ρ2 (t) dt 1/p

t+1k+1

󵄩p 󵄩 ∫ [ ∫ ∫ 󵄩󵄩󵄩q(s − v)󵄩󵄩󵄩 dv ds] t k

−T

t+1s−(k+1)

T

󵄩p 󵄩 ∫ [ ∫ ∫ 󵄩󵄩󵄩q(v)󵄩󵄩󵄩 dv ds] t

−T

s−k

∫[ ∫ −T t−(k+1)

2 ∫−T ρ1 (t) dt

T

ρ2 (t) dt

1/p

t−k r+(k+1)

T

‖R(⋅)‖Lq [k,k+1] T

1/p

ρ2 (t) dt 1/p

󵄩p 󵄩 ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 ds dr] t

t−k r−k

󵄩p 󵄩 ∫ [ ∫ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 ds dr]

−T t−(k−1)t+1

ρ2 (t) dt

1/p

ρ2 (t) dt

242 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations

⩽

T

2 ∫−T ρ1 (t) dt +

⩽

t−k

󵄩p 󵄩 ∫ [ ∫ |r + k + 1 − t|󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr] T

T

2 ∫−T ρ1 (t) dt T

2 ∫−T ρ1 (t) dt

T

1/p

ρ2 (t) dt

1/p

ρ2 (t) dt

−T t−(k+1)

T

2 ∫−T ρ1 (t) dt

T

t−k

󵄩p 󵄩 ∫ [ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr]

1/p

ρ2 (t) dt

−T t−(k−1) T

t

󵄩p 󵄩 ∫ [ ∫ 󵄩󵄩󵄩q(r − (k + 1))󵄩󵄩󵄩 dr]

1/p

−T t−1

ck ‖R(⋅)‖Lq [k,k+1] T

t−k

󵄩p 󵄩 ∫ [ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr]

ck ‖R(⋅)‖Lq [k,k+1]

2 ∫−T ρ1 (t) dt

t−k

−T t−(k−1)

ck ‖R(⋅)‖Lq [k,k+1]

+

ρ2 (t) dt

󵄩p 󵄩 ∫ [ ∫ |t + 1 − r + k|󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr]

‖R(⋅)‖Lq [k,k+1]

T

1/p

−T t−(k+1)

ck ‖R(⋅)‖Lq [k,k+1]

+

=

T

‖R(⋅)‖Lq [k,k+1]

2 ∫−T ρ1 (t) dt

T

t+1

−T

t

󵄩p 󵄩 ∫ [ ∫ 󵄩󵄩󵄩q(r − (k − 1))󵄩󵄩󵄩 dr]

ρ2 (t) dt 1/p

ρ2 (t) dt,

T > 0.

Now the final conclusion follows from the fact that (165) holds with the function q(⋅) replaced therein by Qk (⋅) and the translation invariance of PAP0 (ℝ, X, ρ1 , ρ2 ). In the following theorem, besides conditions (201) and p = 1, we use the essential boundedness of functions f (⋅) and g(⋅), as well as condition (204) on weights ρ1 (⋅) and ρ2 (⋅). Proposition 3.6.2. Suppose that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that (201) holds. Let g ∈ B1 AA(ℝ : X), and let g(⋅) be essentially bounded. Assume, further, that q ∈ B1 WPAA0 (ℝ, X, ρ1 , ρ2 ), q(⋅) is essentially bounded, and f (⋅) = g(⋅) + q(⋅). Suppose that T

lim

T→+∞

∫−T ρ2 (t) dt T

∫−T ρ1 (t) dt

= 0.

(204)

Then function F(⋅) = G(⋅), given by (68), with f = g, is bounded continuous and belongs to the class B1 AA(ℝ : Y) + B1 WPAA0 (ℝ, Y, ρ1 , ρ2 ). Proof. We have already proved that function G(⋅), given by (68) with function f (⋅) replaced by g(⋅) therein, is continuous and belongs to class B1 AA(ℝ : Y). By the foregoing, we also have that function F(⋅) is essentially bounded, so that it suffices to show that function Q(⋅), obtained by replacing function g(⋅) with q(⋅) in (68), satisfies (168).

3.6 Generalized weighted pseudo-almost periodic (and automorphic) solutions | 243

Towards this end, we note that

T

∫ [lim sup

2 ∫−T ρ1 (t) dt −T =

=

=

⩽

=

t+l

T

1

T

1

l→+∞

T

∫ [ lim sup

T

‖q‖∞

t+y󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩 ∫ 󵄩󵄩󵄩 ∫ R(v)q(s − v) dv󵄩󵄩󵄩 ds]ρ2 (t) dt 󵄩󵄩 2y 󵄩󵄩󵄩 󵄩Y t−y 0 ∞ t+y

T l→+∞ y⩾l

T

1 󵄩 󵄩󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩󵄩󵄩󵄩q(s − v)󵄩󵄩󵄩 ds dv]ρ2 (t) dt 2y 0 t−y ∞

t+y−v

0

t−y−v

∫ [ lim sup

1 󵄩󵄩 󵄩 󵄩 󵄩 ∫ 󵄩R(v)󵄩󵄩󵄩 ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr dv]ρ2 (t) dt 2y 󵄩

T

∞

2 ∫−T ρ1 (t) dt −T T 2 ∫−T

l→+∞ y⩾l

∫ [ lim sup

2 ∫−T ρ1 (t) dt −T 1

t−l

T

2 ∫−T ρ1 (t) dt −T 1

1 󵄩󵄩 󵄩 ∫ 󵄩Q(s)󵄩󵄩󵄩Y ds]ρ2 (t) dt 2l 󵄩

l→+∞ y⩾l

󵄩 󵄩 ∫ [ lim sup ∫ 󵄩󵄩󵄩R(v)󵄩󵄩󵄩 dv]ρ2 (t) dt l→+∞

ρ1 (t) dt −T

∞ T ‖q‖∞ ∫0 ‖R(v)‖ dv ∫ T 2 ∫−T ρ1 (t) dt −T

y⩾l

0

ρ2 (t) dt,

T > 0.

The proof of proposition follows by applying (204). Remark 3.6.3. Observe that the validity of (204) implies that for any q ∈ B1 WPAA0 (ℝ, X, ρ1 , ρ2 ) such that q(⋅) is essentially bounded, we have that the condition ∞ ∫0 ‖R(v)‖ dv < ∞ is sufficient to ensure that function Q(⋅) defined above is in class ∞

B1 WPAA0 (ℝ, X, ρ1 , ρ2 ). As established before, the condition ∫0 ‖R(v)‖ dv < ∞ implies that function G(⋅) defined above is almost automorphic provided that g(⋅) is almost automorphic.

For our investigations of the finite convolution product, we need to slightly adapt the notation used so far. Set 𝕌p := {ρ ∈ L1loc ([0, ∞)) : ρ(t) > 0 a. e. t ⩾ 0}, 𝕌b,p := {ρ ∈ T

L∞ ([0, ∞)) : ρ(t) > 0 a. e. t ⩾ 0} and 𝕌∞,p := {ρ ∈ 𝕌p : ν(T, ρ) := limT→+∞ ∫0 ρ(t) dt = ∞}. Then 𝕌b,p ⊆ 𝕌∞,p ⊆ 𝕌p . If ρ1 , ρ2 ∈ 𝕌∞,p , then we set PAP0 ([0, ∞), X, ρ1 , ρ2 ) := {f ∈ Cb ([0, ∞) : X) : lim

T→+∞

T ∫0

1 ρ1 (t) dt

T

󵄩 󵄩 ∫󵄩󵄩󵄩f (t)󵄩󵄩󵄩ρ2 (t) dt = 0} 0

and PAP0 ([0, ∞) × Y, X, ρ1 , ρ2 ) := {f ∈ Cb ([0, ∞) × Y : X) :

244 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations

lim

T→+∞

T ∫0

T

1 ρ1 (t) dt

󵄩 󵄩 ∫󵄩󵄩󵄩f (t, y)󵄩󵄩󵄩ρ2 (t) dt = 0, uniformly on bounded subsets of Y}; 0

see also [61] and [147]. Concerning the invariance of space PAP0 ([0, ∞), X, ρ1 , ρ2 ) under the action of the finite convolution product, we have the following result: Proposition 3.6.4. Assume that there exists a nonnegative measurable function g : [0, ∞) → [0, ∞) such that ρ2 (t) ⩽ g(s)ρ2 (t − s) for 0 ⩽ s ⩽ t < ∞. Assume that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family satisfying ∞

󵄩 󵄩 ∫ (1 + g(s))󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds < ∞.

(205)

0

t

Let f ∈ PAP0 ([0, ∞), X, ρ1 , ρ2 ). Define F(t) := ∫0 R(t − s)f (s) ds, t ⩾ 0. Then we have F ∈ PAP0 ([0, ∞), Y, ρ1 , ρ2 ). Proof. It can be easily verified that F ∈ Cb ([0, ∞) : Y); cf. [29, Chapter 1] for more details. The claimed statement follows from the prescribed assumptions and the next computation:

T

1

∫0 ρ1 (t) dt

T

󵄩 󵄩 ∫󵄩󵄩󵄩F(t)󵄩󵄩󵄩Y ρ2 (t) dt 0

⩽

=

⩽

=

1

T

∫0 ρ1 (t) dt T ∫0

1 ρ1 (t) dt

T

1

∫0 ρ1 (t) dt T

1

∫0 ρ1 (t) dt

T

t

󵄩 󵄩󵄩 󵄩 ∫[∫󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s)󵄩󵄩󵄩 ds]ρ2 (t) dt 0

0

T T

󵄩 󵄩󵄩 󵄩 ∫ ∫󵄩󵄩󵄩R(s)󵄩󵄩󵄩󵄩󵄩󵄩f (t − s)󵄩󵄩󵄩ρ2 (t) dt ds 0 s T

T

󵄩 󵄩 󵄩 󵄩 ∫ g(s)󵄩󵄩󵄩R(s)󵄩󵄩󵄩[∫󵄩󵄩󵄩f (t − s)󵄩󵄩󵄩ρ2 (t − s) dt] ds s

0

T

T−s

󵄩 󵄩 󵄩 󵄩 ∫ g(s)󵄩󵄩󵄩R(s)󵄩󵄩󵄩[ ∫ 󵄩󵄩󵄩f (r)󵄩󵄩󵄩ρ2 (r) dr] ds 0

0

∞

󵄩 󵄩 ⩽ [ ∫ g(s)󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds] ⋅ [ 0

T ∫0

1 ρ1 (t) dt

T

󵄩 󵄩 ∫󵄩󵄩󵄩f (r)󵄩󵄩󵄩ρ2 (r) dr],

T > 0.

0

Remark 3.6.5. Assume, in place of condition (205), that 1 ⩽ p < ∞, 1/p + 1/q = 1, p M = ∑∞ k=0 ‖R(⋅)‖Lq [k,k+1] < ∞ and g(⋅) additionally is S -bounded. Then we can use

3.6 Generalized weighted pseudo-almost periodic (and automorphic) solutions | 245

similar arguments as above and the following estimate: T

⌈T⌉

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ∫ g(s)󵄩󵄩󵄩R(s)󵄩󵄩󵄩 ds ⩽ ∑ 󵄩󵄩󵄩R(⋅)󵄩󵄩󵄩Lq [k,k+1] 󵄩󵄩󵄩g(⋅)󵄩󵄩󵄩Lp [k,k+1] ⩽ M‖g‖Sp , 0

k=0

T>0

to see that F ∈ PAP0 ([0, ∞), Y, ρ1 , ρ2 ). Combining Proposition 3.5.3, Proposition 3.6.4–Remark 3.6.5 and the argument in the proof of Proposition 2.6.11, we can clarify the following proposition (weighted Sp -almost periodic case can be considered similarly): Proposition 3.6.6. Assume that 1 ⩽ p < ∞, 1/p+1/q = 1, and there exists a Sp -bounded function g : [0, ∞) → [0, ∞) such that ρ2 (t) ⩽ g(s)ρ2 (t − s) for 0 ⩽ s ⩽ t < ∞. Assume, further, that (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family such that for each s ⩾ 0 we have Ms = ∑∞ k=0 ‖R(⋅)‖Lq [s+k,s+k+1] < ∞, as well as that the space PAA0 (ℝ, X, ρ1 , ρ2 ) is translation invariant and g : ℝ → X is weighted Sp -almost auto∞ morphic. If q ∈ PAP0 ([0, ∞), X, ρ1 , ρ2 ), then the function F(⋅), given by F(t) := ∫0 R(t − s)[g(s) + q(s)] ds, t ⩾ 0, is well-defined and belongs to the class AA[0,∞) (ℝ, Y, ρ1 , ρ2 ) + Sp WPAA[0,∞) (ℝ, Y, ρ1 , ρ2 ) 0

+ S0p ([0, ∞) : Y) + PAP0 ([0, ∞), Y, ρ1 , ρ2 ).

(ℝ, Y, ρ1 , ρ2 ) denote the spaces consisting of Here, AA[0,∞) (ℝ, Y, ρ1 , ρ2 ) and Sp WPAA[0,∞) 0 restrictions of functions belonging to AA(ℝ, Y, ρ1 , ρ2 ) and Sp WPAA0 (ℝ, Y, ρ1 , ρ2 ) to the nonnegative real axis, respectively.

3.6.2 Weighted pseudo-almost automorphic solutions of semilinear (fractional) Cauchy inclusions In this subsection, we will clarify a few results concerning the existence and uniqueness of weighted pseudo-almost automorphic solutions of semilinear (fractional) Cauchy inclusions. Because of some obvious complications appearing in the study of existence and uniqueness of weighted pseudo-almost periodic (automorphic) solutions defined only for nonnegative values of time t (see Proposition 3.6.6), henceforth we will limit ourselves to the study of abstract Cauchy inclusions (78) and (103), only. The proofs of structural results are omitted since they can be deduced by using composition principles clarified above and the argumentation contained in our previous analyses of the almost periodic case. We deal with the class of multivalued linear operators 𝒜 satisfying condition (P). Define the fractional power (−𝒜)θ for θ > β − 1 as usual. Let γ ∈ (0, 1). Set Y := [D((−𝒜)θ )] and ‖ ⋅ ‖Y := ‖ ⋅ ‖[D((−𝒜)θ )] ; then Y is a Banach space which is continuously

246 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations embedded in X. Define, further, the operator families Tν (⋅) for ν > 0, Tγ,ν (⋅) for ν > −β, Sγ (⋅), Rγ (⋅), the sequences (Mn ) and (Bn ), as it has been done before.

The following results can be stated for weighted pseudo-almost automorphy.

Theorem 3.6.7. Let ρ1 , ρ2 ∈ 𝕌T and 1 < p < ∞. Suppose that (P) holds, β > θ > 1 − β and the following conditions are satisfied:

(i) f : ℝ × Y → X is weighted Sp -pseudo almost automorphic, f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is Sp -almost automorphic and q(⋅, ⋅) satisfies (166), uniformly on bounded subsets of Y.

(ii) Assume that r ⩾ max(p, p/p − 1), r > p/p − 1 and there exist two Stepanov r-almost

automorphic scalar-valued functions Lf (⋅) and Lg (⋅) such that (73) holds with I =

ℝ and (73) holds with I = ℝ and the functions f (⋅, ⋅), Lf (⋅) replaced therein by the functions g(⋅, ⋅), Lg (⋅).

Set q :=

pr p+r

and q󸀠 :=

pr . pr−p−r

If q󸀠 (β − θ − 1) > −1 and Mn < 1 for some n ∈ ℕ, then

there exists a weighted pseudo-almost automorphic mild solution of inclusion (78). The uniqueness of mild solutions holds in the case 𝒜 is single-valued.

Theorem 3.6.8. Let ρ1 , ρ2 ∈ 𝕌T and 1 < p < ∞. Suppose that (P) holds, β > θ > 1 − β

and the following conditions are satisfied:

(i) f : ℝ × Y → X is weighted Sp -pseudo almost automorphic, f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is Sp -almost automorphic and q(⋅, ⋅) satisfies (166), uniformly on bounded subsets of Y.

(ii) There exist two finite constants L > 0 and Lg > 0 such that (74) holds with I = ℝ

and (74) holds with I = ℝ, and the function f (⋅, ⋅) and number L replaced therein by

the function g(⋅, ⋅) and number Lg . If

p (β p−1

− θ − 1) > −1 and Mn < 1 for some n ∈ ℕ, then there exists a weighted pseudo-

almost automorphic mild solution of inclusion (78). The uniqueness of mild solutions

holds provided 𝒜 is single-valued, additionally.

Theorem 3.6.9. Let ρ1 , ρ2 ∈ 𝕌T and 1 < p < ∞. Suppose that (P) holds, β > θ > 1 − β

and the following conditions are satisfied:

(i) f : ℝ × Y → X is weighted Sp -pseudo almost automorphic, f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is Sp -almost automorphic and q(⋅, ⋅) satisfies (166), uniformly on bounded subsets of Y.

(ii) Assume that r ⩾ max(p, p/p − 1), r > p/p − 1 and there exist two Stepanov r-almost

automorphic scalar-valued functions Lf (⋅) and Lg (⋅) such that (73) holds with I =

ℝ and (73) holds with I = ℝ and the functions f (⋅, ⋅), Lf (⋅) replaced therein by the functions g(⋅, ⋅), Lg (⋅).

3.7 Besicovitch almost automorphic solutions | 247

pr pr Set q := p+r and q󸀠 := pr−p−r . If q󸀠 (γ(β − θ) − 1) > −1 and Bn < 1 for some n ∈ ℕ, then there exists a weighted pseudo-almost automorphic mild solution of inclusion (78). The uniqueness of mild solutions holds in the case 𝒜 is single-valued.

Theorem 3.6.10. Let ρ1 , ρ2 ∈ 𝕌T and 1 < p < ∞. Suppose that (P) holds, β > θ > 1 − β and the following conditions are satisfied: (i) f : ℝ × Y → X is weighted Sp -pseudo almost automorphic, f (t, y) = g(t, y) + q(t, y), t ∈ ℝ, where g(⋅, ⋅) is Sp -almost automorphic and q(⋅, ⋅) satisfies (166), uniformly on bounded subsets of Y. (ii) There exist two finite constants L > 0 and Lg > 0 such that (74) holds with I = ℝ and (74) holds with I = ℝ, the function f (⋅, ⋅) and number L replaced therein by the function g(⋅, ⋅) and number Lg . p (γ(β − θ) − 1) > −1 and Bn < 1 for some n ∈ ℕ, then there exists a weighted pseudoIf p−1 almost automorphic mild solution of inclusion (78). The uniqueness of mild solutions holds provided 𝒜 is single-valued, additionally.

Theorem 3.6.11. Assume that ρ1 , ρ2 ∈ 𝕌T , 1 ⩽ p < ∞, and the following conditions hold: (i) f ∈ PAA(ℝ × X, X, ρ1 , ρ2 ) is weighted pseudo-almost automorphic. (ii) Inequality (73) holds with some bounded nonnegative function Lf (⋅), X = Y and I = ℝ. (iii) ∑∞ n=1 Mn < ∞. Then there exists a unique weighted pseudo-almost automorphic solution of inclusion (78).

3.7 Besicovitch almost automorphic solutions of nonautonomous differential equations of first order The main aim of this section is to analyze the existence and uniqueness of Besicovitch almost automorphic solutions and weighted Besicovitch pseudo-almost automorphic solutions of nonautonomous differential equations of first order. For the basic information on hyperbolic evolution systems and Green’s functions, we refer the reader to Section 2.14; with the exception of Remark 3.7.7, we assume here that conditions (H1) and (H2) clarified there hold true. Our first result reads as follows: Theorem 3.7.1. Let f ∈ B1 AA(ℝ : X) ∩ L∞ (ℝ : X). Then the function u(⋅), defined by +∞ u(t) := ∫−∞ Γ(t, s)f (s) ds, t ∈ ℝ is a unique mild solution of the abstract Cauchy problem

248 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations (136). Furthermore, if l

x

1 󵄩 󵄩 lim ∫ ∫ 󵄩󵄩󵄩Γ(x + α, s + α) − Γ(x, s)󵄩󵄩󵄩 ds dx = 0 l→+∞ 2l

(α ∈ ℝ),

(206)

−l −∞

then u ∈ B1 AA(ℝ : X) ∩ Cb (ℝ : X). Proof. Due to (135), we have x

x

−∞

−∞

󵄩 󵄩 ∫ 󵄩󵄩󵄩Γ(x + α, s + α) − Γ(x, s)󵄩󵄩󵄩 ds ⩽ 2M 󸀠 ∫ e−ω(x−s) ds = 2M 󸀠 /ω, l+t

l

l+t

for any α ∈ ℝ. Writing ∫−l+t ⋅ = ∫−l ⋅ + ∫l that l+t

−l+t

⋅ − ∫−l

x ∈ ℝ,

⋅, and using (206) after that, we get

x

1 󵄩 󵄩 lim ∫ ∫ 󵄩󵄩󵄩Γ(x + α, s + α) − Γ(x, s)󵄩󵄩󵄩 ds dx = 0, l→+∞ 2l

t ∈ ℝ (α ∈ ℝ).

(207)

−l+t −∞

t

Furthermore, we have u(t) = u1 (t) + u2 (t), t ∈ ℝ, where u1 (t) := ∫−∞ Γ(t, s)f (s) ds and u2 (t) :=

∞ ∫t Γ(t, s)f (s) ds

(t ∈ ℝ). It can be easily shown that u1 , u2 ∈ L∞ (ℝ : X). We will t−k

first prove that u1 ∈ C(ℝ : X). Define, for every k ∈ ℕ, Ψk (t) := ∫t−k+1 Γ(t, s)f (s) ds, t ∈ ℝ. Since ∑∞ k=1 Ψk (t) = u1 (t) uniformly for t ∈ ℝ (see, e. g., the proof of [166, Theorem 2.3]), it is sufficient to show that for any fixed number k ∈ ℕ one has Ψk ∈ C(ℝ : X). Since, due to (135), t 󸀠 −k 󵄩󵄩 󵄩󵄩 t−k 󵄩 󵄩󵄩 󵄩󵄩 ∫ Γ(t, s)f (s) ds − ∫ Γ(t 󸀠 , s)f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 t−k+1 t 󸀠 −k+1

󵄩󵄩 t−k 󵄩󵄩 󵄩󵄩 t−k+1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ⩽ 󵄩󵄩 ∫ Γ(t, s)f (s) ds󵄩󵄩󵄩 + 󵄩󵄩󵄩 ∫ Γ(t 󸀠 , s)f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩t 󸀠 −k 󵄩 󵄩t 󸀠 −k+1 󵄩 t−k+1

󵄩 󵄩 + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds t 󸀠 −k

t−k+1

󵄩 󵄩 ⩽ 2M ‖f ‖∞ |t − tk | + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds, 󸀠

(208)

t 󸀠 −k

the right continuity of Ψk (⋅) follows by applying the dominated convergence theorem and the fact that Γ(t 󸀠 , s)f (s) = U(t 󸀠 , s)P(s)f (s) converges to Γ(t, s)f (s) = U(t, s)P(s)f (s) as t 󸀠 → t. For the left continuity of Ψk (⋅), we can apply the same argument as above,

3.7 Besicovitch almost automorphic solutions | 249

by observing the following consequence of (208): t 󸀠 −k 󵄩󵄩 t−k 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Γ(t, s)f (s) ds − ∫ Γ(t 󸀠 , s)f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 t−k+1 t 󸀠 −k+1 t 󸀠 −k+1

󵄩 󵄩 ⩽ M ‖f ‖∞ |t − tk | + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds 󸀠

t 󸀠 −k

󵄩󵄩 t −k+1 󵄩󵄩 󵄩󵄩 󵄩󵄩 + 󵄩󵄩󵄩 ∫ [Γ(t, s) − Γ(t 󸀠 , s)]f (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩t−k+1 󵄩 󸀠

t 󸀠 −k+1

󵄩 󵄩 ⩽ 3M ‖f ‖∞ |t − tk | + ∫ 󵄩󵄩󵄩[Γ(t, s) − Γ(t 󸀠 , s)]f (s)󵄩󵄩󵄩 ds. 󸀠

t 󸀠 −k

Let us prove that u1 ∈ B1 AA(ℝ : X). Fix a real sequence (sn ) and a number t ∈ ℝ. Then there exist a subsequence (snk ) of (sn ) and a function f ∗ ∈ L1loc (ℝ : X) such that (162)–(163) hold with p = 1. First of all, we will prove that there exists l0 > 0 such that l

1 󵄩󵄩 ∗ 󵄩󵄩 ∫󵄩f (s)󵄩󵄩 ds ⩽ ‖f ‖∞ + 1, 2l 󵄩

l > l0 .

(209)

−l

To see this, it suffices to observe that there exist k0 ∈ ℕ and l0󸀠 > 0 such that for each l ⩾ l0󸀠 we have l

1 󵄩󵄩 󵄩 ∫󵄩f (s + s) − f ∗ (s)󵄩󵄩󵄩 ds < 1/2; 2l 󵄩 nk0 −l

cf. (162) with ε = 1/3 and t = 0. This yields l

l

l

1 󵄩 1 󵄩 1 󵄩󵄩 ∗ 󵄩󵄩 󵄩 󵄩 ∫󵄩f (s)󵄩󵄩 ds ⩽ ∫󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 ds + ∫󵄩󵄩󵄩f (snk + s)󵄩󵄩󵄩 ds 0 0 2l 󵄩 2l 2l −l

−l

1 ⩽ + ‖f ‖∞ , 2

−l

l⩾

l0󸀠 ,

so that (209) holds with l0 = l0󸀠 . As a simple consequence of (209), we have s

󵄩 󵄩 ∫󵄩󵄩󵄩f ∗ (t + r)󵄩󵄩󵄩 dr ⩽ Const. ⋅ (|t| + |s|),

s ⩾ 0.

(210)

0

Next, we will prove that for each s ⩾ 0 we have 1 lim l→+∞ l

−l+t

󵄩 󵄩 ∫ 󵄩󵄩󵄩f ∗ (s)󵄩󵄩󵄩 ds = 0.

−l+t−s

(211)

250 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Let ε > 0 be fixed. Then, owing to (162), there exist k(ε) ∈ ℕ and l1 (ε) > 0 such that 1 l

−l+t

󵄩 󵄩 ∫ 󵄩󵄩󵄩f ∗ (s)󵄩󵄩󵄩 ds

−l+t−s

1 ⩽ l ⩽

1 l

−l+t

1 󵄩 󵄩 ∫ 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 ds + l

−l+t−s

−l+t

󵄩 󵄩 ∫ 󵄩󵄩󵄩f (snk + s)󵄩󵄩󵄩 ds

−l+t−s

l+s+t

s 󵄩 󵄩 ∫ 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 ds + ‖f ‖∞ l

−l+t−s

l+s s ⩽ε + ‖f ‖∞ , l l

l ⩾ l1 (ε).

t

This gives (211). Define u∗1 (t) := ∫−∞ Γ(t, s)f ∗ (s) ds, t ∈ ℝ. Then u∗1 (⋅) is well-defined and locally bounded since the partial integration in combination with (135) and (210) implies 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Γ(t, s)f ∗ (s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 −∞ 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∫ Γ(t, t + s)f ∗ (t + s) ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩0 󵄩 ∞

󵄩 󵄩 ⩽ ∫ Γ(t, t + s)󵄩󵄩󵄩f ∗ (t + s)󵄩󵄩󵄩 ds 0

∞ 󸀠

󵄩 󵄩 ⩽ M ∫ e−ωs 󵄩󵄩󵄩f ∗ (t + s)󵄩󵄩󵄩 ds 0 ∞

s

󵄩 󵄩 ⩽ M 󸀠 ∫ ωe−ωs ∫󵄩󵄩󵄩f ∗ (t + r)󵄩󵄩󵄩 dr ds ⩽ Const. ⋅ (1 + |t|). 0

0

Furthermore, by Fubini theorem and a straightforward calculation, we have x+sn

l+t 󵄩 x k 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 ∫ 󵄩󵄩 ∫ Γ(x + snk , s)f (s) ds − ∫ Γ(x, s)f ∗ (s) ds󵄩󵄩󵄩 dx 󵄩󵄩 󵄩󵄩 2l 󵄩 −∞ −l+t 󵄩 −∞ l+t

x

1 󵄩󵄩 󵄩 󵄩 ⩽ ∫ ∫ 󵄩󵄩󵄩Γ(x + snk , s + snk ) − Γ(x, s)󵄩󵄩󵄩󵄩󵄩󵄩f (s + snk )󵄩󵄩󵄩 ds dx 2l −l+t −∞ l+t

x

1 󵄩 󵄩󵄩 󵄩 + ∫ ∫ 󵄩󵄩󵄩Γ(x, s)󵄩󵄩󵄩󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 ds dx 2l −l+t −∞

3.7 Besicovitch almost automorphic solutions | 251 l+t

x

‖f ‖∞ 󵄩 󵄩 ⩽ ∫ ∫ 󵄩󵄩󵄩Γ(x + snk , s + snk ) − Γ(x, s)󵄩󵄩󵄩 ds dx 2l −l+t −∞

l+t

x

1 󵄩 󵄩󵄩 󵄩 + ∫ ∫ 󵄩󵄩󵄩Γ(x, s)󵄩󵄩󵄩󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 ds dx 2l −l+t −∞ l+t

=

x

‖f ‖∞ 󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩Γ(x + snk , s + snk ) − Γ(x, s)󵄩󵄩󵄩 ds dx 2l −l+t −∞

−l+t l+t

+

1 󵄩 󵄩󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩Γ(x, s)󵄩󵄩󵄩󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 dx ds 2l −∞ −l+t l+t l+t

+

1 󵄩 󵄩󵄩 󵄩 ∫ ∫ 󵄩󵄩󵄩Γ(x, s)󵄩󵄩󵄩󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 dx ds. 2l −l+t s

Keeping in mind (135) and (207), it suffices to show that −l+t l+t

1 󵄩 󵄩 lim lim sup ∫ ∫ e−ω|x−s| 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 dx ds = 0 k→∞ l→+∞ 2l

(212)

−∞ −l+t

and

l+t l+t

1 󵄩 󵄩 lim lim sup ∫ ∫ e−ω|x−s| 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 dx ds = 0. k→∞ l→+∞ 2l −l+t s

The proof of (213) is almost trivial since (162) holds with p = 1 and l+t l+t

1 󵄩 󵄩 ∫ ∫ e−ω|x−s| 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 dx ds 2l −l+t s

∞

⩽ [ ∫ e−ωr dr][ 0

l+t

1 󵄩 󵄩 ∫ 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 ds]. 2l −l+t

The calculation l+t l+t

1 󵄩 󵄩 ∫ ∫ e−ω|x−s| 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 dx ds 2l −l+t s

−l+t

⩽ Const. ⋅

1 󵄩 󵄩 ∫ e−ω(−l+t−s) 󵄩󵄩󵄩f (snk + s) − f ∗ (s)󵄩󵄩󵄩 ds 2l −∞

−l+t

1 ⩽ Const. ⋅ ‖f ‖∞ ∫ e−ω(−l+t−s) ds 2l −∞

(213)

252 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations −l+t

1 󵄩 󵄩 + Const. ⋅ ∫ e−ω(−l+t−s) 󵄩󵄩󵄩f ∗ (s)󵄩󵄩󵄩 ds 2l −∞

∞

∞

0

0

1 1 󵄩 󵄩 ⩽ Const. ⋅ ‖f ‖∞ ∫ e−ωs ds + Const. ⋅ ∫ e−ωs 󵄩󵄩󵄩f ∗ (−l + t − s)󵄩󵄩󵄩 ds 2l 2l indicates that (212) holds. The first term in the last estimate clearly tends to zero as l → +∞. The situation is similar with the second term since the reverse Fatou’s lemma and (211) together imply ∞

0 ⩽ lim sup l→+∞

1 󵄩 󵄩 ∫ e−ωs 󵄩󵄩󵄩f ∗ (−l + t − s)󵄩󵄩󵄩 ds 2l 0

∞

s

0

0

1 󵄩 󵄩 = Const. ⋅ lim sup ∫ e−ωs ∫󵄩󵄩󵄩f ∗ (−l + t − r)󵄩󵄩󵄩 dr ds 2l l→+∞ = Const. ⋅ lim sup l→+∞

⩽ Const. ⋅

∞

−l+t

0

−l+t−s

1 󵄩 󵄩 ∫ e−ωs ∫ 󵄩󵄩󵄩f ∗ (r)󵄩󵄩󵄩 dr ds 2l

∞

−l+t

0

−l+t−s

1 󵄩 󵄩 ∫ e−ωs lim sup ∫ 󵄩󵄩󵄩f ∗ (r)󵄩󵄩󵄩 dr ds = 0, 2l l→+∞

so that actually ∞

1 󵄩 󵄩 ∫ e−ωs 󵄩󵄩󵄩f ∗ (−l + t − s)󵄩󵄩󵄩 ds = 0. l→+∞ 2l lim

0

Hence, u1 ∈ B1 AA(ℝ : X), as claimed; we can similarly prove that u2 ∈ C(ℝ : X) ∩ B1 AA(ℝ : X) (for the proof of continuity of u2 (⋅), it is only worth noting that the mapping (t, s) 󳨃→ UQ (t, s)Q(s) is strongly continuous for t < s; cf. [321]). The proof of the theorem is thereby complete. Observe that condition (206) has not been well explored within the existing theory of nonautonomous differential equations of first order (cf. [42, Lemma 3.2, Theorem 3.3] for two broad-sense results in this direction). Concerning the existence and uniqueness of mild solutions of the abstract Cauchy problem (137) belonging to the class of Besicovitch p-almost automorphic functions, we can state the following result with p = 1. Theorem 3.7.2. Let f ∈ B1 AA(ℝ : X) ∩ L∞ (ℝ : X), and let (206) hold. Suppose t that x ∈ P(0)X. Define u(t) := U(t, 0)x + ∫0 U(t, s)f (s) ds, t ⩾ 0. If the mapping t

t 󳨃→ ∫0 U(t, s)Q(s)f (s) ds, t ⩾ 0 is in class B1 AA(ℝ : X), then u(⋅) is in the same class, as well.

3.7 Besicovitch almost automorphic solutions |

253

Proof. Clearly, we have the following decomposition: 0

t

t

u(t) = U(t, 0)x + ∫ Γ(t, s)f (s) ds − ∫ Γ(t, s)f (s) ds + ∫ U(t, s)Q(s)f (s) ds, −∞

−∞

0

for any t ⩾ 0. Since x ∈ P(0)X, the function U(⋅, 0)x is exponentially decaying due to (135). Keeping in mind the prescribed assumption that the mapping t 󳨃→ t ∫0 U(t, s)Q(s)f (s) ds, t ⩾ 0 is in class B1 AA(ℝ : X), as well as Theorem 3.7.1, it suf0

fices to show that ∫−∞ Γ(t, s)f (s) ds → 0 as t → +∞. This simply follows from the next estimates (t ⩾ 0): 0 0 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∫ Γ(t, s)f (s) ds󵄩󵄩󵄩 ⩽ M 󸀠 ‖f ‖∞ ∫ e−ω|t−s| ds ⩽ M 󸀠 ‖f ‖∞ e−ωt ∫ eωs ds. 󵄩󵄩 󵄩󵄩 󵄩󵄩−∞ 󵄩󵄩 −∞ −∞

Remark 3.7.3. If x ∈ P(0)X ∩ D(A(0)), then the mapping t 󳨃→ U(t, 0)x, t ⩾ 0 is continu0 t ous (see, e. g., [321]). Since the mapping t 󳨃→ ∫−∞ Γ(t, s)f (s) ds − ∫−∞ Γ(t, s)f (s) ds, t ⩾ 0 is bounded and continuous by the proof of Theorem 3.7.1, we have that the assumpt tion that the mapping t 󳨃→ ∫0 U(t, s)Q(s)f (s) ds, t ⩾ 0 is in class B1 AA(ℝ : X) ∩ C(ℝ : X) (B1 AA(ℝ : X) ∩ Cb (ℝ : X)) implies that u(⋅) is in the same class. It is clear that the above condition holds if the evolution system U(⋅, ⋅) is exponentially stable. In the following theorem, we examine the weighted Besicovitch 1-pseudo almost automorphic solutions of abstract Cauchy problem (136). Theorem 3.7.4. Suppose that g ∈ B1 AA(ℝ : X) ∩ L∞ (ℝ : X), as well as q ∈ B1 WPAA0 (ℝ, X, ρ1 , ρ2 ) ∩ L∞ (ℝ : X). Set f (t) := g(t) + q(t), t ∈ ℝ. Then the function u(⋅), defined by +∞ u(t) := ∫−∞ Γ(t, s)f (s) ds, t ∈ ℝ, is a unique mild solution of the abstract Cauchy problem (136). Suppose that (204) and (206) hold. Then we have u ∈ B1 WPAA(ℝ, X, ρ1 , ρ2 ) ∩ Cb (ℝ : X). Proof. By Theorem 3.7.1 and its proof, it suffices to show that the mapping t 󳨃→ ∞ uq (t) := ∫−∞ Γ(t, s)q(s) ds, t ∈ ℝ belongs to the class B1 WPAA0 (ℝ, X, ρ1 , ρ2 ). It is t

clear that uq (t) = uq1 (t) + uq2 (t), t ∈ ℝ, where uq1 (t) := ∫−∞ Γ(t, s)q(s) ds and ∞

uq2 (t) := ∫t Γ(t, s)q(s) ds (t ∈ ℝ). We have T

1

t+l

1 󵄩 󵄩 ∫ [lim sup ∫ 󵄩󵄩󵄩uq1 (s)󵄩󵄩󵄩 ds]ρ2 (t) dt T 2l 2 ∫−T ρ1 (t) dt −T l→+∞ t−l t+l󵄩 ∞ 󵄩󵄩 󵄩󵄩 1 󵄩󵄩󵄩󵄩 󵄩󵄩 ds]ρ2 (t) dt Γ(s, s − v)q(s − v) dv [lim sup = ∫ ∫ ∫ 󵄩 󵄩󵄩 󵄩 T 󵄩 2l 󵄩󵄩 󵄩󵄩 2 ∫−T ρ1 (t) dt −T l→+∞ t−l 0

1

1

T

T

t+l ∞

1 󵄩 󵄩󵄩 󵄩 ⩽ ∫ [lim sup ∫ ∫ 󵄩󵄩󵄩Γ(s, s − v)󵄩󵄩󵄩󵄩󵄩󵄩q(s − v)󵄩󵄩󵄩 dv ds]ρ2 (t) dt T 2l 2 ∫−T ρ1 (t) dt −T l→+∞ t−l 0

254 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations

1

T

∞ t+l

T

∞

1 󵄩 󵄩󵄩 󵄩 ⩽ ∫ [lim sup ∫ ∫ 󵄩󵄩󵄩Γ(s, s − v)󵄩󵄩󵄩󵄩󵄩󵄩q(s − v)󵄩󵄩󵄩 ds dv]ρ2 (t) dt T 2l 2 ∫−T ρ1 (t) dt −T l→+∞ 0 t−l t+l−v

1 󵄩 󵄩 ⩽ ∫ [lim sup ∫ e−ωv ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩 dr dv]ρ2 (t) dt T 2l 2 ∫−T ρ1 (t) dt −T l→+∞ 0 t−l−v M󸀠

⩽

T

M 󸀠 ‖q‖∞

∫ ρ2 (t) dt,

T

2ω ∫−T ρ1 (t) dt −T

T > 0.

Therefore, (204) yields that uq1 ∈ B1 WPAA0 (ℝ, X, ρ1 , ρ2 ). We can similarly prove that uq2 ∈ B1 WPAA0 (ℝ, X, ρ1 , ρ2 ), finishing the proof of theorem. Concerning the abstract Cauchy problem (137), we have the following result: Theorem 3.7.5. Suppose that g ∈ B1 AA(ℝ : X) ∩ L∞ (ℝ : X), as well as q ∈ B1 WPAA0 ([0, ∞), X, ρ1 , ρ2 ), x ∈ P(0)X, (206) and the following conditions hold: t (i) The mapping t 󳨃→ ∫0 U(t, s)Q(s)f (s) ds, t ⩾ 0 is in class B1 AA(ℝ : X) + B1 WPAA0 ([0, ∞), X, ρ1 , ρ2 ). (ii) There exist finite numbers M 󸀠󸀠 > 0 and γ > 1 such that M 󸀠󸀠 󵄩 󵄩󵄩 󵄩󵄩U(t, s)Q(s)󵄩󵄩󵄩 ⩽ 1 + (t − s)γ

for t ⩾ s ⩾ 0.

(iii) There exists a nonnegative measurable function g : [0, ∞) → [0, ∞) such that ρ2 (t) ⩽ g(s)ρ2 (t − s) for 0 ⩽ s ⩽ t < ∞, and ∞

∫ 0

g(s) ds < ∞. 1 + sγ

t

Define u(t) := U(t, 0)x + ∫0 U(t, s)[g(s) + q(s)] ds, t ⩾ 0. Then u(⋅) is essentially bounded and belongs to the class B1 AA(ℝ : X) + B1 WPAA0 ([0, ∞), X, ρ1 , ρ2 ). Proof. The mapping t 󳨃→ U(t, 0)x, t ⩾ 0 is bounded since U(⋅, ⋅) is bounded. This is t also clear for the mapping t 󳨃→ ∫0 U(t, s)[g(s) + q(s)] ds, t ⩾ 0 since, due to (135) and (iii), 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩∫U(t, s)[g(s) + q(s)] ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 t

t

󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 ⩽ ∫󵄩󵄩󵄩Γ(t, s)󵄩󵄩󵄩󵄩󵄩󵄩g(s) + q(s)󵄩󵄩󵄩 ds + ∫󵄩󵄩󵄩U(t, s)Q(s)󵄩󵄩󵄩󵄩󵄩󵄩g(s) + q(s)󵄩󵄩󵄩 ds 0

t

0

󵄩 󵄩 󵄩 󵄩 ⩽ (‖g‖∞ + ‖q‖∞ ) ∫[󵄩󵄩󵄩Γ(t, s)󵄩󵄩󵄩 + 󵄩󵄩󵄩U(t, s)Q(s)󵄩󵄩󵄩] ds 0

3.7 Besicovitch almost automorphic solutions | 255 t

⩽ (‖g‖∞ + ‖q‖∞ ) ∫[M 󸀠 e−ω(t−s) + 0

⩽ Const. ⋅ (‖g‖∞ + ‖q‖∞ ),

M 󸀠󸀠 ] ds 1 + (t − s)γ

t ⩾ 0.

Summarizing, u(⋅) is essentially bounded. We have the following decomposition: t

0

t

u(t) = U(t, 0)x + ∫ Γ(t, s)g(s) ds − ∫ Γ(t, s)g(s) ds + ∫ Γ(t, s)q(s) ds −∞

0

−∞

t

t

+ ∫ U(t, s)Q(s)g(s) ds + ∫ U(t, s)Q(s)q(s) ds, 0

t ⩾ 0.

0

Since x ∈ P(0)X, the function U(⋅, 0)x is exponentially decaying due to (135). Keeping in mind (i) and the proof of Theorem 3.7.1, it suffices to show that the mappings t t t 󳨃→ ∫0 Γ(t, s)q(s) ds, t ⩾ 0 and t 󳨃→ ∫0 U(t, s)Q(s)q(s) ds, t ⩾ 0 belong to the class B1 WPAA0 ([0, ∞), X, ρ1 , ρ2 ). We will prove this only for the second mapping. By the foret going, t 󳨃→ ∫0 U(t, s)Q(s)q(s) ds, t ⩾ 0 is a bounded continuous mapping. Furthermore, we have T󵄩 t

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩ρ2 (t) dt U(t, s)Q(s)q(s) ds ∫ ∫ 󵄩 󵄩󵄩 󵄩󵄩 T 󵄩󵄩 ∫0 ρ1 (t) dt 0 󵄩󵄩0 1

⩽

=

T ∫0 T ∫0

1 ρ1 (t) dt 1 ρ1 (t) dt 1

T

t

󵄩 󵄩󵄩 󵄩 ∫[∫󵄩󵄩󵄩U(t, t − s)Q(t − s)󵄩󵄩󵄩󵄩󵄩󵄩q(t − s)󵄩󵄩󵄩 ds]ρ2 (t) dt 0

0

T T

󵄩 󵄩󵄩 󵄩 ∫ ∫󵄩󵄩󵄩U(t, t − s)Q(t − s)󵄩󵄩󵄩󵄩󵄩󵄩q(t − s)󵄩󵄩󵄩ρ2 (t) dt ds 0 s T

T

T

T−s

g(s) 󵄩 󵄩 [∫󵄩󵄩󵄩q(t − s)󵄩󵄩󵄩ρ2 (t − s) dt] ds ⩽ T ∫ γ ∫0 ρ1 (t) dt 0 1 + s s 1

g(s) 󵄩 󵄩 = T [ ∫ 󵄩󵄩󵄩q(r)󵄩󵄩󵄩ρ2 (r) dr] ds ∫ γ 1 + s ∫0 ρ1 (t) dt 0 0 ∞

⩽ [∫ 0

T

1 g(s) 󵄩 󵄩 ds] ⋅ [ T ∫󵄩󵄩󵄩q(r)󵄩󵄩󵄩ρ2 (r) dr], 1 + sγ ∫ ρ (t) dt 0

1

T > 0.

0

Since (iii) holds and q ∈ B1 WPAA0 ([0, ∞), X, ρ1 , ρ2 ), we have that the mapping t 󳨃→ t ∫0 U(t, s)Q(s)q(s) ds, t ⩾ 0 is in the same class, as claimed. The proof of the theorem is thereby complete.

256 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Remark 3.7.6. As in Remark 3.7.3, the continuity of mapping u(⋅) is stipulated by the t continuity of mapping t 󳨃→ ∫0 U(t, s)Q(s)f (s) ds, t ⩾ 0 and the validity of condition x ∈ P(0)X ∩ D(A(0)). Remark 3.7.7. All established results continue to hold in the case that the operator family (A(t))t∈ℝ generates an exponentially stable evolution family (U(t, s))t⩾s in the sense of [137, Definition 3.1] (therefore, condition (H1) need not be necessarily satisfied, and (H2) holds with P(t) = I and Q(t) = 0, t ∈ ℝ; Γ(t, s) ≡ U(t, s)). The only thing worth noting here is that, with the notation already employed, condition (iii) from the formulation of Theorem 3.7.5 can be slightly weakened by assuming that ∞ ∫0 e−ωs g(s) ds < ∞, which follows from the fact that we can estimate the term ‖U(t, t − s)‖ in the proof of this theorem by M 󸀠 e−ωs (t ⩾ s ⩾ 0).

3.8 Vector-valued almost automorphic distributions and vector-valued almost automorphic ultradistributions The main aim of this section is to continue the analysis of vector-valued almost periodic ultradistributions given in Section 2.15. We will follow Komatsu’s approach to vector-valued ultradistributions, employing the same notion and notation as in Section 2.15. We divide this section into two separate subsections. In Subsection 3.8.1, which is written in an expository manner, we transfer the results of C. Bouzar and Z. Tchouar [74] to the vector-valued case. In Subsection 3.8.2, we introduce the notion of a vector-valued almost automorphic ultradistribution and further analyze this concept. 3.8.1 Almost automorphy of vector-valued distributions We will use the following elementary notion that is already employed in Section 2.8 (see [365] for more details). By 𝒟(X) = 𝒟(ℝ : X) we denote the Schwartz space of test functions with values in X, by 𝒮 (X) = 𝒮 (ℝ : X) we denote the space of rapidly decreasing functions with values in X, and by ℰ (X) = ℰ (ℝ : X) we denote the space of all infinitely differentiable functions with values in X; 𝒟 ≡ 𝒟(ℂ), 𝒮 ≡ 𝒮 (ℂ) and ℰ ≡ ℰ (ℂ). The spaces of all linear continuous mappings from 𝒟, 𝒮 and ℰ into X will be designated by 𝒟󸀠 (X), 𝒮 󸀠 (X) and ℰ 󸀠 (X), respectively. The main purpose of this subsection is to verify that all structural results proved by C. Bouzar and F. Z. Tchouar [74] continue to hold in the vector-valued case. Let 1 ⩽ p ⩽ ∞. By 𝒟Lp (X) we denote the vector space consisting of all infinitely differentiable functions f : ℝ → X such that for each number j ∈ ℕ0 we have f (j) ∈ Lp (ℝ : X). The Fréchet topology on 𝒟Lp (X) is induced by the following system of seminorms: k

󵄩 󵄩 ‖f ‖k := ∑ 󵄩󵄩󵄩f (j) 󵄩󵄩󵄩Lp (ℝ) , j=0

f ∈ 𝒟Lp (X) (k ∈ ℕ).

3.8 Vector-valued almost automorphic distributions and ultradistributions | 257

In the case X = ℂ, the above space is simply denoted by 𝒟Lp . Let us recall that a continuous linear mapping f : 𝒟L1 → X is said to be a bounded X-valued distribution; the space consisting of such vector-valued distributions will be denoted by 𝒟L󸀠 1 (X) or B󸀠 (X). Endowed with the strong topology, B󸀠 (X) becomes a complete locally convex space. For every f ∈ B󸀠 (X), we have that f|𝒮 : 𝒮 → X is a tempered X-valued distribution. Set (i)

ℰAA (X) := {ϕ ∈ ℰ (X) : ϕ

∈ AA(ℝ : X) for all i ∈ ℕ0 }.

Since, for every ϕ ∈ ℰAA (X), we have ϕ ∈ AA(ℝ : X) ⊆ Cb (ℝ : X) and 󵄩󵄩 󵄩󵄩 ∞ 󵄩 󵄩󵄩 󵄩󵄩 ∫ ϕ(t)φ(t) dt 󵄩󵄩󵄩 ⩽ ‖ϕ‖L∞ (ℝ) ‖φ‖L1 , 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩−∞

φ ∈ 𝒟L1 ,

∞

the mapping ϕ 󳨃→ ∫−∞ ϕ(t)φ(t) dt, φ ∈ 𝒟L1 is linear and continuous so that ℰAA (X) ⊆ 𝒟L󸀠 1 (X). Using the fact that the first derivative of a differentiable almost automorphic function is almost automorphic iff it is uniformly continuous [132], it can be easily verified that we have ℰAA (X) = ℰ (X) ∩ AA(ℝ : X); furthermore, ℰAA (X) ∗ L1 (ℝ) ⊆ ℰAA (X) and ℰAA (X) is a closed subspace of 𝒟L∞ (X) (see [74, Proposition 5]). We have that ℰAA (X) is the space consisting of exactly those elements f (⋅) from 𝒟L∞ (X) for which f ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟; see [74, Corollary 1]. For any vector-valued distribution T ∈ 𝒟󸀠 (X), we define τh T := Th by ⟨Th , φ⟩ := ⟨T, φ(⋅ − h)⟩, φ ∈ 𝒟 (h ∈ ℝ). The following result is crucial: Theorem 3.8.1 (see [74, Theorem 1]). Let T ∈ 𝒟L󸀠 1 (X). Then the following assertions are equivalent: (i) T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟. (ii) There exist an integer k ∈ ℕ and almost automorphic functions fj (⋅) : ℝ → X (1 ⩽ j ⩽ k) such that T = ∑kj=0 fj . (j)

It is said that a distribution T ∈ 𝒟L󸀠 1 (X) is almost automorphic iff T satisfies any of the above two equivalent conditions. By B󸀠AA (X) we denote the space consisting of all almost automorphic distributions. The space B󸀠AA (X) is closed under differentiation and [74, Proposition 6] continues to hold in the vector-valued case. This is also the case with the assertions of [74, Propositions 7–8, Theorem 2, Propositions 9–10], so that we have the following theorem: Theorem 3.8.2. (i) Let T ∈ 𝒟L󸀠 1 (X). Then T is almost automorphic iff for every real sequence (bn ), there exist a subsequence (an ) of (bn ) and a vector-valued distribution S ∈ 𝒟󸀠 (X) such that limn→∞ Tan = S in 𝒟󸀠 (X) and limn→∞ S−an = T in 𝒟󸀠 (X) iff there exists a sequence of almost automorphic functions converging to T in 𝒟L󸀠 1 (X) iff for every real sequence

258 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations (bn ), there exists a subsequence (an ) of (bn ) such that liml→∞ limn→∞ τ−al τan T = T in 𝒟󸀠 (X). (ii) Let f ∈ AASp (ℝ : X) for some p ∈ [1, ∞). Then the regular distribution associated to f (⋅) is almost automorphic. 3.8.2 Almost automorphy of vector-valued ultradistributions Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ). We refer the reader to Section 2.15 for the notions of spaces 𝒟L1 ((Mp )) and 𝒟L1 ({Mp }). The spaces consisting of all linear continuous mappings from 𝒟L1 ((Mp )), resp. 𝒟L1 ({Mp }), into X, equipped with the strong topologies, will be shortly denoted by 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X). The space 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X), is closed under the action of ultradifferential operators of (Mp )-class, resp. {Mp }-class. We will use the shorthand 𝒟L󸀠∗1 (Mp : X) to denote either 𝒟L󸀠 1 ((Mp ) : X) or 𝒟L󸀠 1 ({Mp } : X); 𝒟L󸀠∗1 (Mp ) ≡ 𝒟L󸀠∗1 (Mp : ℂ). Assume that A ⊆ 𝒟󸀠∗ (X). Let us remind ourselves of the following notion: 󸀠∗

󸀠∗

∗

𝒟A (X) := {T ∈ 𝒟 (X) : T ∗ φ ∈ A for all φ ∈ 𝒟 }. 󸀠∗ It is worth noting that 𝒟A󸀠∗ (X) = 𝒟𝔸∩B (X), for any set B ⊆ L1loc (ℝ : X) that contains ∞ 󸀠∗ C (ℝ : X), as well as that the set 𝒟A (X) is closed under the action of an ultradifferential operator of ∗-class. We have already proved the following assertions: p (i) Suppose that there exist an ultradifferential operator P(D) = ∑∞ p=0 ap D of class (Mp ), resp. of class {Mp }, and f , g ∈ 𝒟A󸀠∗ (X) such that T = P(D)f + g. If A is closed under addition, then T ∈ 𝒟A󸀠∗ (X). (ii) If A ∩ C(ℝ : X) is closed under uniform convergence, T ∈ 𝒟L󸀠 1 ((Mp ) : X) and T ∗ φ ∈ A, φ ∈ 𝒟(Mp ) , then there exists h > 0 such that for each compact set K ⊆ ℝ we have M ,h

T ∗ φ ∈ A, φ ∈ 𝒟K p . (iii) Suppose that T ∈ 𝒟󸀠(Mp ) (X) and there exists h > 0 such that for each compact set M ,h K ⊆ ℝ we have T ∗ φ ∈ A, φ ∈ 𝒟K p . If (Mp ) additionally satisfies (M.3), then there exist l > 0 and two elements f , g ∈ A such that T = P(D)f + g.

Now we will consider the case that A = AA(ℝ : X). Then A is closed under the uniform convergence and addition, and we have A ⊆ 𝒟A󸀠∗ (X). Hence, as a special case of the above assertions, we have the following result: Theorem 3.8.3. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟󸀠(Mp ) (X), resp. T ∈ 𝒟󸀠{Mp } (X). Then the following holds: p (i) Suppose that there exist an ultradifferential operator P(D) = ∑∞ p=0 ap D of class 󸀠∗ (Mp ), resp. of class {Mp }, and f , g ∈ 𝒟AA(ℝ:X) (X) such that T = P(D)f + g. Then 󸀠∗ T ∈ 𝒟AA(ℝ:X) (X). (ii) If T ∈ 𝒟L󸀠 1 ((Mp ) : X) and T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟(Mp ) , then there exists h > 0 such M ,h

that for each compact set K ⊆ ℝ we have T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟K p .

3.8 Vector-valued almost automorphic distributions and ultradistributions | 259

(iii) Suppose that T ∈ 𝒟󸀠(Mp ) (X) and there exists h > 0 such that for each compact set M ,h K ⊆ ℝ we have T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟K p . If (Mp ) additionally satisfies (M.3), then there exist l > 0 and two elements f , g ∈ AA(ℝ : X) such that T = P(D)f + g. As an immediate corollary of Theorem 3.8.3, we have the following result: Corollary 3.8.4. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X). Consider now the following assertions: (i) There exist a number l > 0, resp. a sequence (rp ) ∈ R, and two functions f , g ∈ AA(ℝ : X) such that T = Pl (D)f + g, resp. T = Prp (D)f + g. p (ii) There exist an ultradifferential operator P(D) = ∑∞ p=0 ap D of class (Mp ), resp. {Mp }, and two functions f , g ∈ AA(ℝ : X) such that T = P(D)f + g. (iii) We have T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟∗ . (iv) There exists h > 0 such that for each compact set K ⊆ ℝ, resp. for each h > 0 and M ,h for each compact set K ⊆ ℝ, we have T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟K p . Then we have (i) ⇒ (ii) ⇒ (iii) ⇔ (iv). Furthermore, if (Mp ) additionally satisfies condition (M.3), then assertions (i)–(iv) are equivalent for the Beurling class. Let us introduce the following space: ∗

∗

(i)

ℰAA (X) := {ϕ ∈ ℰ (X) : ϕ

∈ AA(ℝ : X) for all i ∈ ℕ0 }.

∗ ∗ ∗ (X) ⊆ 𝒟L󸀠∗1 (X), ℰAA (X) = ℰ ∗ (X) ∩ AA(ℝ : X) and ℰAA (X) ∗ As in distribution case, ℰAA ∗ ∗ ∗ 1 L (ℝ) ⊆ ℰAA (X); furthermore, ℰAA (X) is the space of those elements f (⋅) from ℰ (X) for which f ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟∗ . Consider now the following statement: (ii)󸀠 T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X), and there exists a sequence (ϕn ) ∗ in ℰAA (X) such that limn→∞ ϕn = T for the topology of 𝒟L󸀠 1 ((Mp ) : X), resp. 𝒟L󸀠 1 ({Mp } : X).

The proof of following lemma is almost the same as that of Lemma 2.15.5: Proposition 3.8.5. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X). Then we have (iii) ⇔ (ii)’, with (iii) being the same as in the formulation of Corollary 3.8.4. It is said that a bounded ultradistribution T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X), is almost automorphic iff T satisfies any of the above two equivalent conditions. It can be simply verified that a regular distribution (ultradistribution of ∗-class) determined by an almost automorphic vector-valued function that is not almost periodic is an almost automorphic vector-valued distribution (ultradistribution of ∗-class) that cannot be almost periodic (cf. [74, Example 2]). It is also worth stating the following result:

260 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Theorem 3.8.6. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X). Then we have (i) ⇒ (ii) ⇒ (iii) ⇒ (iv), where: p (i) There exist an ultradifferential operator P(D) = ∑∞ p=0 ap D of class (Mp ), resp. {Mp }, and two functions f , g ∈ AA(ℝ : X) such that T = P(D)f + g. (ii) For every real sequence (bn ), there exist a subsequence (an ) of (bn ) and a vectorvalued ultradistribution S ∈ 𝒟󸀠∗ (X) such that limn→∞ ⟨Tan , φ⟩ = ⟨S, φ⟩, φ ∈ 𝒟∗ and limn→∞ ⟨S−an , φ⟩ = ⟨T, φ⟩, φ ∈ 𝒟∗ . (iii) For every real sequence (bn ), there exists a subsequence (an ) of (bn ) such that liml→∞ limn→∞ ⟨τ−al τan T, φ⟩ = ⟨T, φ⟩, φ ∈ 𝒟∗ . (iv) We have T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟∗ . Furthermore, if (Mp ) additionally satisfies condition (M.3), then assertions (i)–(iv) are equivalent for the Beurling class. Proof. The proof of (ii) ⇒ (iii) ⇒ (iv) can be deduced as in distribution case (see, e. g., the proof of [74, Proposition 9]). For the proof of implication (i) ⇒ (ii), observe that the almost automorphy of functions f (⋅) and g(⋅) implies the existence of essentially bounded functions F ∈ L∞ (ℝ : X) and G ∈ L∞ (ℝ : X) such that lim f (t + an ) = F(t) and lim F(t − an ) = f (t)

n→∞

n→∞

and lim g(t + an ) = G(t) and lim G(t − an ) = g(t),

n→∞

n→∞

pointwise for t ∈ ℝ. Using these equations, the dominated convergence theorem and the fact that, for every bounded subset B of 𝒟∗ and for every compact set K ⊆ ℝ, there M ,h exists h > 0 such that B is bounded in 𝒟K p ([242]), it readily follows that limn→∞ f (⋅ + an ) = F in 𝒟󸀠∗ (X) and limn→∞ g(⋅ + an ) = G in 𝒟󸀠∗ (X), so that limn→∞ ⟨Tan , φ⟩ = ⟨S, φ⟩, where S ∈ 𝒟󸀠∗ (X) is given by S := P(D)F + G. Similarly we can deduce that limn→∞ ⟨S−an , φ⟩ = ⟨T, φ⟩, φ ∈ 𝒟∗ , finishing the proof of the theorem. In the present situation, we cannot tell whether the implication (iv) ⇒ (ii) holds true in general case. In [74, Section 6], C. Bouzar and F. Z. Tchouar have continued the analysis of S. Bochner [66] concerning linear difference-differential operators p

q

Lh = ∑ ∑ aij i=0 j=0

di τh , dxi j

where aij are complex numbers (0 ⩽ i ⩽ p, 0 ⩽ j ⩽ q) and h = (hj )0⩽j⩽q ⊆ ℝq . Taking into account the fact that [66, Theorem 4(i)] holds in the vector-valued case, Theorems 2.15.1 and 2.15.3 and the proof of [74, Theorem 3], we can simply clarify that the assertions of [74, Theorem 3, Corollary 3] hold for the vector-valued distributions, as well as for the vector-valued ultradistributions:

3.9 Asymptotically almost automorphic vector-valued generalized functions | 261

p (ℝ : X) denote the vector space of all p-times differentiable Theorem 3.8.7. Let Cbuc uniformly continuous functions f ∈ BUC(ℝ : X) for which f (j) ∈ BUC(ℝ : X), 0 ⩽ j ⩽ p. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let S ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. S ∈ 𝒟L󸀠 1 ({Mp } : X), be almost automorphic. p (ℝ : X) of the homogeneous equation Lh f = 0 is almost (i) If every solution f ∈ Cbuc automorphic, then every solution T ∈ 𝒟L󸀠 1 ((Mp ) : X), resp. T ∈ 𝒟L󸀠 1 ({Mp } : X), of the inhomogeneous equation Lh T = S is almost automorphic. (ii) If S󸀠 is almost automorphic, then S is almost automorphic. (iii) Any translation Sh of S is almost automorphic (h ∈ ℝ).

We close the section with the observation that the assertions of [74, Theorem 4, Corollary 4] can be also formulated for the vector-valued distributions and ultradistributions.

3.9 Asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized functions The main aim of this section is to introduce the notions of an asymptotically almost periodic ultradistribution and an asymptotically almost automorphic ultradistribution in a Banach space, as well as to provide some applications in the qualitative analysis of the vector-valued distributional and ultradistributional solutions to systems of ordinary differential equations. In such a way, we expand and contemplate the results obtained by C. Bouzar, Z. Tchouar [74], C. Bouzar, M. T. Khalladi [70], D. Cheban [90], I. Cioranescu [101–103], I. K. Dontvi [173], A. Halanay, D. Wexler [213] and B. Stanković [374, 375]. In this section, we will use the following notions (slightly different from those we have already employed). It is said that a continuous function f : ℝ → X is asymptotically almost periodic (automorphic) iff there is a function q ∈ C0 ([0, ∞) : X) and an almost periodic (automorphic) function g : ℝ → X such that f (t) = g(t)+q(t), t ⩾ 0. By AAP(ℝ : X), resp. AAA(ℝ : X), we denote the vector space consisting of all asymptotically almost periodic, resp. asymptotically almost automorphic functions. A function f ∈ LpS (ℝ : X) is said to be asymptotically Stepanov p-almost periodic, asymptotically Sp -almost periodic shortly, iff there are two locally p-integrable functions g : ℝ → X and q : [0, ∞) → X satisfying the following conditions: (i) g(⋅) is Sp -almost periodic, ̂ belongs to the class C0 ([0, ∞) : Lp ([0, 1] : X)), (ii) q(⋅) (iii) f (t) = g(t) + q(t) for all t ⩾ 0. By AAPSp (ℝ : X) we denote the space consisting of all asymptotically Stepanov p-almost periodic functions. A function f ∈ Lploc (ℝ : X) is said to be asymptotically Stepanov p-almost automorphic iff there exist an Sp -almost automorphic func-

262 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations tion g(⋅) and a function q ∈ LpS (ℝ : X) such that f (t) = g(t) + q(t), t ⩾ 0 and q̂ ∈ C0 ([0, ∞) : Lp ([0, 1] : X)). The vector space consisting of all asymptotically Sp -almost automorphic functions will be denoted by AAASp (ℝ : X), as before. We refer the reader to Sections 2.15 and 3.8 for the notion of various types of distribution and ultradistribution spaces used below. We need some preliminaries concerning the first antiderivative of a vector-valued (ultra-)distribution (see also Section 2.8 ∗ ) for a similar notion used in the distributional case). Let η ∈ 𝒟[−2,−1] (η ∈ 𝒟[−2,−1] ∞ be a fixed test function satisfying ∫−∞ η(t) dt = 1. Then, for every fixed φ ∈ 𝒟 (φ ∈ 𝒟∗ ), we define I(φ) by replacing the function ζ (⋅) in (82) with the function η(⋅); set G−1 (φ) := −G(I(φ)), φ ∈ 𝒟 (φ ∈ 𝒟∗ ). Concerning these notions, it should be noted that all things already clarified in the distributional setting continue to hold in the ultradistributional setting. For example, for every φ ∈ 𝒟 (φ ∈ 𝒟∗ ) and n ∈ ℕ, we k

∞

d (k−1) (x) − η(k−1) (x) ∫−∞ φ(u) du, x ∈ ℝ, have I(φ) ∈ 𝒟 (I(φ) ∈ 𝒟∗ ), dx k I(φ)(x) = φ k ∈ ℕ. Then we have G−1 ∈ 𝒟󸀠 (L(X)) (G−1 ∈ 𝒟󸀠∗ (L(X))) and (G−1 )󸀠 = G; more precisely, −G−1 (φ󸀠 ) = G(I(φ󸀠 )) = G(φ), φ ∈ 𝒟 (φ ∈ 𝒟∗ ). Furthermore, if ⟨T, −φ󸀠 ⟩ = ⟨G, φ⟩, φ ∈ 𝒟0 (φ ∈ 𝒟0∗ ) for some T, G ∈ 𝒟󸀠 (X) (T, G ∈ 𝒟󸀠∗ (X)), then T = G−1 + Const. on ∞ [0, ∞), i. e., ⟨T, φ⟩ = ⟨G−1 , φ⟩ + Const. ⋅ ∫0 φ(t) dt, φ ∈ 𝒟0 (φ ∈ 𝒟0∗ ).

3.9.1 Asymptotical almost periodicity and asymptotical almost automorphy of vector-valued distributions In this subsection, we will use the same notation as in Sections 2.15 and 3.8. The space of bounded vector-valued distributions tending to zero at plus infinity, B󸀠+,0 (X) for short, is defined by B󸀠+,0 (X) := {T ∈ 𝒟L󸀠 1 (X) : lim ⟨Th , φ⟩ = 0 for all φ ∈ 𝒟}. h→+∞

It can be simply verified that the structural characterization for the space B󸀠+,0 (ℂ), proved in [103, Proposition 1], is still valid in the vector-valued case, as well that the space B󸀠+,0 (ℂ) is closed under differentiation. Definition 3.9.1. A distribution T ∈ 𝒟L󸀠 1 (X) is said to be asymptotically almost periodic, resp. asymptotically almost automorphic, iff there exist an almost periodic, resp. almost automorphic, distribution Tap ∈ B󸀠AP (X), resp. Taa ∈ B󸀠AA (X), and a bounded distribution tending to zero at plus infinity Q ∈ B󸀠+,0 (X) such that ⟨T, φ⟩ = ⟨Tap , φ⟩+⟨Q, φ⟩, φ ∈ 𝒟0 , resp. ⟨T, φ⟩ = ⟨Taa , φ⟩ + ⟨Q, φ⟩, φ ∈ 𝒟0 . By B󸀠AAP (X), resp. B󸀠AAA (X), we denote the vector space consisting of all asymptotically almost periodic, resp. asymptotically almost automorphic distributions. It is well known that the representation T = Tap + Q is unique in the almost periodic case. This is also the case for asymptotical almost automorphy since [74, Proposition 6, 2.] continues to hold in the vector-valued case (more precisely, the suppositions

3.9 Asymptotically almost automorphic vector-valued generalized functions | 263

1 2 1,2 1 2 T = Taa + Q1 = Taa + Q2 , where Taa ∈ B󸀠AA (X) and Q1,2 ∈ B󸀠+,0 (X), imply Taa = Taa and Q1 = Q2 ). Further on, we would like to observe that the following structural result holds in the vector-valued case:

Theorem 3.9.2. Let T ∈ 𝒟L󸀠 1 (X). Then the following assertions are equivalent: (i) T ∈ B󸀠AAP (X). (ii) T ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟0 . (iii) There exist an integer k ∈ ℕ and asymptotically almost periodic functions fj (⋅) :

ℝ → X (0 ⩽ j ⩽ k) such that T = ∑kj=0 fj on [0, ∞). (iv) There exist an integer k ∈ ℕ and bounded asymptotically almost periodic func(j) tions fj (⋅) : ℝ → X (0 ⩽ j ⩽ k) such that T = ∑kj=0 fj on [0, ∞). (j)

(v)

There exist S ∈ 𝒟L󸀠 1 (X), k ∈ ℕ and bounded asymptotically almost periodic func(j) tions fj (⋅) : ℝ → X (0 ⩽ j ⩽ k) such that S = ∑kj=0 fj on ℝ, and ⟨S, φ⟩ = ⟨T, φ⟩ for all φ ∈ 𝒟0 . (vi) There exists S ∈ 𝒟L󸀠 1 (X) such that ⟨S, φ⟩ = ⟨T, φ⟩ for all φ ∈ 𝒟0 and S∗φ ∈ AAP(ℝ : X), φ ∈ 𝒟. (vii) There exists a sequence (Tn ) in ℰ (X) ∩ AAP(ℝ : X) such that limn→∞ Tn = T in 𝒟L󸀠 1 (X). (viii) T ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟.

Proof. The equivalence of (i)–(iii) can be proved as in the scalar-valued case (see [103, Theorem I, Proposition 1]). It is clear that (iv) implies (iii), while the converse statement follows from the fact that there exist functions gj ∈ AP(ℝ : X) and qj ∈ C0 ([0, ∞) : X) for 0 ⩽ j ⩽ k such that k

∞

j=0

0 ∞

⟨T, φ⟩ = ∑ (−1)j ∫ fj (t)φ(j) (t) dt k

= ∑ (−1)j ∫ [hj (t) + qj,e (t)]φ(j) (t) dt, j=0

φ ∈ 𝒟0 ,

0

where qj,e (⋅) denotes the even extension of function qj (⋅) to the whole real axis (0 ⩽ j ⩽ k). Therefore, T = ∑kj=0 [hj + qj,e ](j) on [0, ∞) and (iii) follows. The implication

(iv) ⇒ (v) trivially follows from the fact that the expression S = ∑kj=0 fj defines an ele(j)

ment from 𝒟L󸀠 1 (X). Since the space A ≡ AAP(ℝ : X) ∩ Cb (ℝ : X) is uniformly closed (and therefore, C ∞ -uniformly closed), closed under addition and A ∗ 𝒟 ⊆ A (see [47] for the notion), an application of [47, Theorem 2.11] yields that (v) implies (vi). The implication (vi) ⇒ (ii) is trivial, hence we have the equivalence of assertions (i)–(vi). In order to see that (ii) implies (vii), arbitrarily choose a test function ρ ∈ 𝒟 with supp(φ) ⊆ [0, 1]. Set ρn (⋅) := nρ(n⋅), for n ∈ ℕ. Then (ii) implies T ∗ ρn ∈ ℰ (X) ∩ AAP(ℝ : X) for all n ∈ ℕ. Due to the fact that limn→∞ Tn = T in 𝒟L󸀠 1 (X) (see, e. g., the second part of proof of [74,

264 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Proposition 7]), we get that (vii) holds true. The implication (vii) ⇒ (viii) follows from the first part of proof of [74, Proposition 7], while the implication (viii) ⇒ (ii) is trivial, finishing the proof of the theorem. Let 1 ⩽ p < ∞, and let a function f ∈ Lploc (ℝ : X) be asymptotically Stepanov p-almost periodic. Then the regular distribution associated to f (⋅), denoted by f(⋅) henceforth, is asymptotically almost periodic. In order to see this, let us assume that p-integrable functions g : ℝ → X and q : [0, ∞) → X satisfy the conditions from the definition of asymptotical Stepanov p-almost periodicity. Since ∞

∞

∞

∫ f (t)φ(t) dt = ∫ [f (t) − g(t)]φ(t) dt + ∫ g(t)φ(t) dt, −∞

−∞

φ ∈ 𝒟,

−∞

and the regular distribution associated to g(⋅) is almost periodic [47], it suffices to show that the regular distribution associated to (f − g)(⋅) is in class B󸀠AAP (X). It can be easily seen that this distribution is bounded, so that Theorem 3.9.2 yields that it is enough to show that, for every fixed φ ∈ 𝒟0 , we have (f − g) ∗ φ ∈ AAP(ℝ : X). Let supp(φ) ⊆ [a, b] ⊆ [0, ∞), where a, b ∈ ℕ0 . Then [(f − g) ∗ φ](x) x−a

x−a+1

x−b+1

= ∫ q(t)φ(x − t) dt + ∫ q(t)φ(x − t) dt + ⋅ ⋅ ⋅ + ∫ q(t)φ(x − t) dt, x−a−1

x−a

x−b

for any x ⩾ b, and therefore, for any number ε > 0 given in advance, we can find a sufficiently large positive number x0 (ε) ⩾ b such that, for every x ⩾ x0 (ε), we have 󵄩 󵄩󵄩 󵄩󵄩[(f − g) ∗ φ](x)󵄩󵄩󵄩 ⩽ ε, due to the Sp -vanishing of function q(⋅) and Hölder inequality. Let us write down the above result as Proposition 3.9.3. Let 1 ⩽ p < ∞, and let f ∈ AAPSp (ℝ : X). Then f ∈ B󸀠AAP (X). The following analogue of Theorem 3.9.2 holds for asymptotical almost automorphy: Theorem 3.9.4. Let T ∈ 𝒟L󸀠 1 (X). Then the following assertions are equivalent: (i) T ∈ B󸀠AAA (X). (ii) T ∗ φ ∈ AAA(ℝ : X), φ ∈ 𝒟0 . (iii) There exist an integer k ∈ ℕ and asymptotically almost automorphic functions (j) fj (⋅) : ℝ → X (0 ⩽ j ⩽ k) such that T = ∑kj=0 fj on [0, ∞). (iv) There exist an integer k ∈ ℕ and bounded asymptotically almost automorphic (j) functions fj (⋅) : ℝ → X (0 ⩽ j ⩽ k) such that T = ∑kj=0 fj on [0, ∞).

3.9 Asymptotically almost automorphic vector-valued generalized functions | 265

(v)

There exist S ∈ 𝒟L󸀠 1 (X), k ∈ ℕ and bounded asymptotically almost automorphic (j) functions fj (⋅) : ℝ → X (0 ⩽ j ⩽ k) such that S = ∑kj=0 fj on ℝ, and ⟨S, φ⟩ = ⟨T, φ⟩ for all φ ∈ 𝒟0 . (vi) There exists S ∈ 𝒟L󸀠 1 (X) such that ⟨S, φ⟩ = ⟨T, φ⟩ for all φ ∈ 𝒟0 and S∗φ ∈ AAA(ℝ : X), φ ∈ 𝒟. (vii) There exists a sequence (Tn ) in ℰ (X) ∩ AAA(ℝ : X) such that limn→∞ Tn = T in 𝒟L󸀠 1 (X). (viii) T ∗ φ ∈ AAA(ℝ : X), φ ∈ 𝒟. Proof. The equivalence of (i)–(iii) follows similarly as in the proofs of [103, Theorem I, Proposition 1], given in almost periodic case. The equivalence of (iii) and (iv), as well as the fact that (iv) implies (v), can be proved as in the previous theorem. Since the space A ≡ AAA(ℝ : X) ∩ Cb (ℝ : X) is uniformly closed (C ∞ -uniformly closed), closed under addition and A∗𝒟 ⊆ A, we can apply again [47, Theorem 2.11] in order to see that (v) implies (vi). The implication (vi) ⇒ (ii) is trivial, so that we have the equivalence of assertions (i)–(vi). The remaining part of the proof can be carried through as in the almost periodic case. Let 1 ⩽ p < ∞, and let a function f ∈ Lploc (ℝ : X) be asymptotically Stepanov p-almost automorphic. Then the regular distribution associated to f (⋅) is asymptotically almost automorphic, which can be seen as in the case of asymptotical almost periodicity, by appealing to [74] in place of [47], for almost automorphic part: Proposition 3.9.5. Let 1 ⩽ p < ∞, and let f ∈ AAASp (ℝ : X). Then f ∈ B󸀠AAA (X). Concerning the assertions of Theorems 3.9.2 and 3.9.4, it is worth noting the following: Remark 3.9.6. (i) The validity of (vi) for some S ∈ 𝒟L󸀠 1 (X) implies its validity for S replaced therein with SQ = S + Q, where Q ∈ 𝒟L󸀠 1 (X) and supp(Q) ⊆ (−∞, 0]. For this, it suffices to observe that (Q ∗ φ)(x) = ⟨Q, φ(x − ⋅)⟩ = 0 for all x ⩾ sup(supp(φ)), φ ∈ 𝒟. (ii) Theorems 3.9.2 and 3.9.4 imply that the spaces B󸀠AAP (X) and B󸀠AAA (X) are closed under differentiation, as well as that B󸀠AAP (X) and B󸀠AAA (X) are closed subspaces of 𝒟L󸀠 1 (X) (for this, we can apply the equivalences of (i) and (ii) in the abovementioned theorems, the fact that the spaces AAP(ℝ : X) and AAA(ℝ : X) are uniformly closed, and the fact that the translations {φ(h − ⋅) : h ∈ ℝ} of a given function φ ∈ 𝒟 form a bounded subset of 𝒟L1 ).

266 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations 3.9.2 Asymptotical almost periodicity and asymptotical almost automorphy of vector-valued ultradistributions The space of bounded vector-valued ultradistributions tending to zero at plus infinity, B󸀠∗ +,0 (X) for short, is defined by 󸀠∗ ∗ B󸀠∗ +,0 (X) := {T ∈ 𝒟L1 (Mp : X) : lim ⟨Th , φ⟩ = 0 for all φ ∈ 𝒟 }. h→+∞

Let T ∈ 𝒟L󸀠∗1 (X). Then we say that T is almost periodic, resp. almost automorphic, iff T satisfies T ∗ φ ∈ AP(ℝ : X), φ ∈ 𝒟∗ , resp. T ∗ φ ∈ AA(ℝ : X), φ ∈ 𝒟∗ . By B󸀠∗ AP (X), resp. B󸀠∗ (X), we denote the vector space consisting of all almost periodic, resp. almost AA automorphic, ultradistributions of ∗-class. Definition 3.9.7. An ultradistribution T ∈ 𝒟L󸀠∗1 (X) is said to be asymptotically almost periodic, resp. asymptotically almost automorphic, iff there exist an almost periodic, 󸀠∗ resp. almost automorphic, ultradistribution Tap ∈ B󸀠∗ AP (X), resp. Taa ∈ BAA (X), and a bounded ultradistribution tending to zero at plus infinity Q ∈ B󸀠∗ +,0 (X) such that ⟨T, φ⟩ = ⟨Tap , φ⟩ + ⟨Q, φ⟩, φ ∈ 𝒟0∗ , resp. ⟨T, φ⟩ = ⟨Taa , φ⟩ + ⟨Q, φ⟩, φ ∈ 𝒟0∗ . 󸀠∗ By B󸀠∗ AAP (X), resp. BAAA (X), we denote the vector space consisting of all asymptotically almost periodic, resp. automorphic, ultradistributions of ∗-class. As in distribution case, decomposition of an asymptotically almost periodic (automorphic) ultradistribution of ∗-class into its almost periodic (automorphic) part and bounded, tending to zero at plus infinity part, is unique. The space B󸀠∗ AAP (X), resp. B󸀠∗ (X), is closed under the action of ultradifferential operators of ∗-class. This folAAA 󸀠∗ lows from the fact that this is true for the space B󸀠∗ (X), resp. B (X), as well as that, AP AA 󸀠∗ for every Q ∈ B+,0 (X) and for every ultradifferential operator P(D) of ∗-class, we have ⟨P(D)Q, φ(⋅ − h)⟩ = ⟨Q, [P(D)φ](⋅ − h)⟩, h ∈ ℝ. We will use the following lemma, which is a simple consequence of our analyses from Subsection 2.15.1: Lemma 3.9.8. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟󸀠(Mp ) (X), resp. T ∈ 𝒟󸀠{Mp } (X). Then the following holds: p (i) Suppose that there exist an ultradifferential operator P(D) = ∑∞ p=0 ap D of class 󸀠∗ 󸀠∗ (Mp ), resp. of class {Mp }, and f , g ∈ 𝒟AAP(ℝ:X) (X), resp. f , g ∈ 𝒟AAA(ℝ:X) (X), such 󸀠∗ 󸀠∗ that T = P(D)f + g. Then T ∈ 𝒟AAP(ℝ:X) (X). (X), resp. T ∈ 𝒟AAA(ℝ:X) 󸀠 (Mp ) (ii) If T ∈ 𝒟L1 ((Mp ) : X) and T ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟 , resp. T ∗ φ ∈ AAA(ℝ : X), φ ∈ 𝒟(Mp ) , then there is a number h > 0 such that for each compact set K ⊆ ℝ we M ,h

M ,h

have T ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟K p , resp. T ∗ φ ∈ AAA(ℝ : X), φ ∈ 𝒟K p . (iii) Assume that T ∈ 𝒟󸀠(Mp ) (X) and there is a number h > 0 such that for each compact M ,h set K ⊆ ℝ we have T ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟K p , resp. T ∗ φ ∈ AAA(ℝ : X), M ,h

φ ∈ 𝒟K p . If (Mp ) additionally satisfies (M.3), then there exist l > 0 and bounded functions f , g ∈ AAP(ℝ : X), resp. f , g ∈ AAA(ℝ : X), such that T = Pl (D)f + g.

3.9 Asymptotically almost automorphic vector-valued generalized functions | 267

We need the following lemma, which is probably known in the existing literature. Lemma 3.9.9 (Supremum formula). Let f ∈ AA(ℝ : X). Then for each a ∈ ℝ we have ‖f ‖∞ = supx⩾a ‖f (x)‖. Proof. Essentially, we only need to prove that for each a, y ∈ ℝ and ε > 0 we have 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩f (y)󵄩󵄩󵄩 ⩽ sup󵄩󵄩󵄩f (x)󵄩󵄩󵄩 + ε. x⩾a

(214)

By definition of almost automorphy, for the sequence (bn := n) we can extract a subsequence (an ) of it such that f (y) = limn→∞ limk→∞ f (y − ak + an ). This, in particular, implies that we can find two integers n0 (ε), k0 (ε) ∈ ℕ such that, for every k ⩾ k0 (ε), we have ‖f (y − ak + an ) − f (y)‖ ⩽ ε. This gives (214) and finishes the proof of the lemma. Now we would like to state the following result: Theorem 3.9.10. Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let T ∈ 𝒟L󸀠∗1 (Mp : X). Consider the following assertions: (i) There exist an element S ∈ 𝒟L󸀠∗1 (Mp : X), a number l > 0 in the Beurling case/a sequence (rp ) ∈ R in the Roumieu case, and bounded functions f , g ∈ AAP(ℝ : X), resp. f , g ∈ AAA(ℝ : X), such that S = Pl (D)f + g, resp. S = Prp (D)f + g, and S = T on [0, ∞). (ii) There exist a number l > 0, resp. a sequence (rp ) ∈ R, and bounded functions f , g ∈ AAP(ℝ : X), resp. f , g ∈ AAA(ℝ : X), such that T = Pl (D)f + g, resp. T = Prp (D)f + g, on [0, ∞). p (iii) There exist an ultradifferential operator P(D) = ∑∞ p=0 ap D of ∗-class and bounded functions f1 , f2 ∈ AAP(ℝ : X), resp. f1 , f2 ∈ AAA(ℝ : X), such that T = P(D)f1 + f2 on [0, ∞). (iv) There exist an element S ∈ 𝒟L󸀠∗1 (Mp : X), an ultradifferential operator P(D) = p ∑∞ p=0 ap D of ∗-class and bounded functions f1 , f2 ∈ AAP(ℝ : X), resp. f1 , f2 ∈ AAA(ℝ : X), such that S = P(D)f1 + f2 and S = T on [0, ∞). 󸀠∗ 󸀠∗ (v) T ∈ B󸀠∗ AAP (X), resp. T ∈ BAAA (X), there exist an element S ∈ 𝒟L1 (Mp : X), an ∞ p ultradifferential operator P(D) = ∑p=0 ap D of ∗-class and bounded functions f1 , f2 ∈ AAP(ℝ : X), resp. f1 , f2 ∈ AAA(ℝ : X), such that S = P(D)f1 + f2 and S = T on [0, ∞). 󸀠∗ 󸀠∗ (vi) T ∈ B󸀠∗ AAP (X), resp. T ∈ BAAA (X), and there exists an element S ∈ 𝒟L1 (Mp : X) such ∗ that S = T on [0, ∞) and S ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟 , resp. S ∗ φ ∈ AAA(ℝ : X), φ ∈ 𝒟∗ . 󸀠∗ (vii) T ∈ B󸀠∗ AAP (X), resp. T ∈ BAAA (X). (viii) T ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟0∗ , resp. T ∗ φ ∈ AAA(ℝ : X), φ ∈ 𝒟0∗ . (ix) There exists a sequence (Tn ) in ℰ ∗ (X) ∩ AAP(ℝ : X), resp. ℰ ∗ (X) ∩ AAA(ℝ : X), such that limn→∞ Tn = T in 𝒟L󸀠∗1 (X). (x) T ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟∗ , resp. T ∗ φ ∈ AAA(ℝ : X), φ ∈ 𝒟∗ .

268 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Then we have (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi) ⇔ (vii) ⇔ (viii) ⇔ (ix) ⇔ (x). Furthermore, if (Mp ) additionally satisfies condition (M.3), then the assertions (i)–(x) are equivalent for the Beurling class. Proof. For the sake of brevity, we will consider only the asymptotically almost periodic case for the Beurling class (in the case of almost automorphy, it is only worth noting that, for the proof of implication (ix) ⇒ (vii), we need to use the supremum formula deduced in Lemma 3.9.9). The implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) and (vi) ⇒ (vii) are trivial, while the implication (vii) ⇒ (viii) can be deduced as it has been done for the scalar-valued distributions (cf. the proof of implication (i) ⇒ (ii) of [103, Theorem I]). In order to see that (iv) implies (v), we only need to prove that the assumptions of (iv) 󸀠(M ) imply T ∈ BAAPp (X). Suppose that fi (⋅) = gi (⋅) + qi (⋅) on [0, ∞), where gi ∈ AP(ℝ : X) and qi ∈ C0 ([0, ∞) : X) (i = 1, 2). Then T1 := P(D)g1 + g2 is an almost periodic ultradistribution of Beurling class, and all that we need to show is T2 := P(D)(f1 − g1 ) + 󸀠(M ) (f2 − g2 ) ∈ BAAPp (X), i. e., lim ⟨P(D)(f1 − g1 ) + (f2 − g2 ), φ(⋅ − h)⟩ = 0,

h→+∞

φ ∈ 𝒟(Mp ) .

Towards this end, assume that −∞ < a < b < +∞ and supp(φ) ⊆ [a, b]. Let ε > 0 be given. Then there exist a sufficiently large finite number h0 (ε) > 0 and a sufficiently large finite number cφ > 0, independent of ε, such that, for every h ⩾ h0 (ε), we have the following: 󵄩 󵄩󵄩 󵄩󵄩⟨P(D)(f1 − g1 ) + (f2 − g2 ), φ(⋅ − h)⟩󵄩󵄩󵄩 ∞ ∞ 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩 ∑ (−1)p ap ∫ (f1 (t) − g1 (t))φ(p) (t − h) dt + ∫ (f2 (t) − g2 (t))φ(t − h) dt 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩p=0 󵄩 󵄩 −∞ −∞ b 󵄩󵄩 ∞ 󵄩󵄩 p 󵄩 = 󵄩󵄩 ∑ (−1) ap ∫(f1 (t + h) − g1 (t + h))φ(p) (t) dt 󵄩󵄩p=0 󵄩 a b 󵄩󵄩 󵄩󵄩 + ∫(f2 (t + h) − g2 (t + h))φ(t) dt 󵄩󵄩󵄩 󵄩󵄩 󵄩 a

b b 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 p (p) 󵄩 = 󵄩󵄩 ∑ (−1) ap ∫ q1 (t + h)φ (t) dt + ∫ q2 (t + h)φ(t) dt 󵄩󵄩󵄩 󵄩󵄩p=0 󵄩󵄩 󵄩 󵄩 a a ∞

󵄩 󵄩 ⩽ ε[ ∑ |ap |󵄩󵄩󵄩φ(p) 󵄩󵄩󵄩L1 + ‖φ‖L1 ] ⩽ εcφ . p=0

This yields (v). The implication (v) ⇒ (vi) follows by applying Lemma 3.9.8. In order to see that (vii) implies (vi), assume that T = Tap + Q on [0, ∞), where Tap and Q satisfy the requirements of Definition 3.9.7. Put S := Tap + Q on ℝ. Then we need to prove that S ∗ φ ∈ AAP(ℝ : X), φ ∈ 𝒟∗ , i. e., that limh→+∞ (Q ∗ φ)(h) = 0, φ ∈ 𝒟∗ .

3.9 Asymptotically almost automorphic vector-valued generalized functions | 269

̌ − h)⟩ = ⟨Qh , φ⟩ ̌ → 0 as h → +∞ This follows from the fact that (Q ∗ φ)(h) = ⟨Q, φ(⋅ 󸀠(M ) by definition of space B+,0 p (X). We can show that (viii) implies (ix) as in the proof of implication (ii) ⇒ (vii) in Theorem 3.9.2. The implication (ix) ⇒ (x) follows similarly as in the first part of the proof of [74, Proposition 7], while the implication (x) ⇒ (vii) is trivial. Now we will prove the implication (ix) ⇒ (vii). Let (Tn ) be a sequence in 󸀠(M ) ℰ (Mp ) (X) ∩ AAP(ℝ : X) such that limn→∞ Tn = T in 𝒟L1 p (X), and let Tn = Gn + Qn on [0, ∞) for some Gn ∈ AP(ℝ : X) and Qn ∈ Cb (ℝ : X) tending to zero at plus infinity 󸀠(M ) (n ∈ ℕ). Let B be a bounded subset of 𝒟L1 p (X), and let ε > 0 be given. It can be simply 󸀠(M )

shown that the set B󸀠 := B ∪ {φ(± ⋅ ±h) : h ∈ ℝ, φ ∈ B} is a bounded subset of 𝒟L1 p (X), as well. Then (Gn − Gm ) + (Qn − Qm ) = Tn − Tm , m, n ∈ ℕ, so that there exists an integer n0 (ε) ∈ ℕ such that, for every m, n ∈ ℕ with min(m, n) ⩾ n0 (ε), we have 󵄩 󵄩 sup󵄩󵄩󵄩⟨Gn − Gm , φ⟩ + ⟨Qn − Qm , φ⟩󵄩󵄩󵄩 ⩽ ε.

φ∈B󸀠

Specifically, 󵄩 󵄩 sup 󵄩󵄩󵄩⟨Gn − Gm , φ(h − ⋅)⟩ + ⟨Qn − Qm , φ(h − ⋅)⟩󵄩󵄩󵄩 ⩽ ε.

(215)

φ∈B󸀠 , h∈ℝ

By the foregoing, we have limh→+∞ ‖⟨Qn − Qm , φ(h − ⋅)⟩‖ = 0, for every m, n ∈ ℕ with min(m, n) ⩾ n0 (ε), so that there exists a sufficiently large number Mε > 0 such that 󵄩 󵄩󵄩 󵄩󵄩⟨Gn − Gm , φ(h − ⋅)⟩󵄩󵄩󵄩 ⩽ ε,

m, n ⩾ n0 (ε), h ⩾ Mε , φ ∈ B󸀠 .

By Lemma 3.9.9 and the almost periodicity of function on the left-hand side of the above inequality, we get 󵄩 󵄩󵄩 󵄩󵄩⟨Gn − Gm , φ(h − ⋅)⟩󵄩󵄩󵄩 ⩽ ε,

m, n ⩾ n0 (ε), h ∈ ℝ, φ ∈ B󸀠 .

(216)

This clearly implies that the sequence (⟨Gn , φ⟩) is Cauchy in X and therefore convergent (φ ∈ 𝒟L1 ). Set ⟨G, φ⟩ := limm→∞ ⟨Gm , φ⟩ for all φ ∈ 𝒟L1 . Letting m → ∞ in (216), we obtain 󵄩 󵄩󵄩 󵄩󵄩⟨Gn − G, φ(h − ⋅)⟩󵄩󵄩󵄩 ⩽ ε,

n ⩾ n0 (ε), h ∈ ℝ, φ ∈ B󸀠 .

(217) 󸀠(Mp )

This, in turn, implies that G is the limit of convergent sequence (Gn ) in 𝒟L1 therefore G ∈

󸀠(M ) 𝒟L1 p (X). Furthermore, (217) shows that limn→∞ Gn ∗φ (Mp )

in Cb (ℝ : X), so that G ∗ φ ∈ AP(ℝ : X), φ ∈ 𝒟

(X) and

= G∗φ, φ ∈ 𝒟(Mp ) 󸀠(M )

; hence, G ∈ BAP p (X). Define 󸀠(M )

⟨Q, φ⟩ := limm→∞ ⟨T − Gm , φ⟩ for all φ ∈ 𝒟L1 . Then Q ∈ 𝒟L1 p (X) and all that we need to show is that Q tends to zero at plus infinity. For this, observe first that Q has to be 󸀠(M ) the limit of a Cauchy sequence (Qn ) in 𝒟L1 p (X) and that combining (215) and (216)

270 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations yields 󵄩 󵄩󵄩 󵄩󵄩⟨Qn − Qm , φ(h − ⋅)⟩󵄩󵄩󵄩 ⩽ 2ε,

m, n ⩾ n0 (ε), h ∈ ℝ, φ ∈ B󸀠 .

(218)

Letting m → ∞ in (218), we obtain 󵄩 󵄩󵄩 󵄩󵄩⟨Qn − Q, φ(h − ⋅)⟩󵄩󵄩󵄩 ⩽ 2ε,

n ⩾ n0 (ε), h ∈ ℝ, φ ∈ B󸀠 .

With n = n0 (ε), the above simply yields the existence of a number h0 (ε) > 0 such that 󵄩 󵄩󵄩 󵄩󵄩⟨Q, φ(h − ⋅)⟩󵄩󵄩󵄩 ⩽ 3ε,

h ⩾ h0 , φ ∈ B󸀠 .

Hence, (vii) is proved. Finally, if (Mp ) satisfies condition (M.3), then the implication (x) ⇒ (i) follows by applying Lemma 3.9.8(ii)–(iii) with S = T, and therefore, assertions (i)–(x) are mutually equivalent for the Beurling class. Exploiting the equivalence of parts (vii) and (x) in Theorem 3.9.10, as well as the 󸀠∗ continuity properties of convolution, it readily follows that B󸀠∗ AAP (X), resp. BAAA (X), is 󸀠∗ a closed subspace of 𝒟L1 (Mp : X). Concerning Theorem 3.9.10, we have the following observations: Remark 3.9.11. The implication (vii) ⇒ (x) of Theorem 3.9.10, as well as the implication (i) ⇒ (viii) of Theorem 3.9.2 (Theorem 3.9.4), can be proved directly, as explained below. Let T = Tap + Q on [0, ∞), with Tap and Q being the same as in Definition 3.9.7. Since (T ∗ φ)(h) = (Tap ∗ φ)(h) + ([T − Tap ] ∗ φ)(h),

h ∈ ℝ, φ ∈ 𝒟∗ ,

and Tap ∗ φ ∈ AP(ℝ : X), φ ∈ 𝒟∗ , we need to prove that lim ([T − Tap ] ∗ φ)(h) = 0,

φ ∈ 𝒟∗

lim ⟨T − Tap , φ(h − ⋅)⟩ = 0,

φ ∈ 𝒟∗ .

h→+∞

i. e., that h→+∞

(219)

Since supp(φ(h − ⋅)) ⊆ [0, ∞) for h ⩾ sup(supp(φ)), we have that (219) is equivalent to lim ⟨Q, φ(h − ⋅)⟩ = 0,

h→+∞

φ ∈ 𝒟∗ .

̌ for all φ ∈ 𝒟∗ , and the fact that This follows from the equality ⟨Q, φ(h − ⋅)⟩ = ⟨Qh , φ⟩ 󸀠∗ Q ∈ B+,0 (X). The assertion of [103, Proposition 2] can be formulated for vector-valued almost periodic distributions and vector-valued almost periodic ultradistributions of Beurling class, with the corresponding sequence (Mp ) satisfying (M.1), (M.2) and (M.3). Now we would like to state the following automorphic version of [103, Proposition 2] (vector-valued case):

3.9 Asymptotically almost automorphic vector-valued generalized functions | 271

Proposition 3.9.12. (i) Consider the equation G󸀠 = f ⋅G+U in 𝒟󸀠 (X), where f ∈ ℰ ∩AA(ℝ : ℂ) and U ∈ B󸀠AA (X). If G ∈ B󸀠AAA (X) is a solution of this equation on [0, ∞), then its almost automorphic part is a solution of this equation on ℝ. (ii) Let (Mp ) satisfy (M.1), (M.2) and (M.3). Consider the equation G󸀠 = f ⋅ G + U in 󸀠(M )

󸀠(M )

𝒟󸀠(Mp ) (X), where f ∈ ℰ (Mp ) ∩ AA(ℝ : ℂ) and U ∈ BAA p (X). If G ∈ BAAAp (X) is a

solution of this equation on [0, ∞), then its almost automorphic part is a solution of this equation on ℝ.

Proof. Since the pointwise product of an almost automorphic X-valued function with a scalar-valued almost automorphic function is again an almost automorphic X-valued 󸀠(M ) function, we can use the structural characterization of B󸀠AA (X), resp. BAA p (X), in order to see that these spaces are closed under pointwise multiplication with a scalarvalued function f ∈ ℰ ∩ AA(ℝ : ℂ), resp. f ∈ ℰ (Mp ) ∩ AA(ℝ : ℂ); furthermore, the almost automorphy of a bounded distribution G, resp. bounded ultradistribution G of Beurling class, is equivalent to saying that, for every real sequence (bn ), there exist a subsequence (an ) of (bn ) and a vector-valued distribution S ∈ 𝒟󸀠 (X), resp. ultradistribution S ∈ 𝒟󸀠(Mp ) (X), such that limn→∞ ⟨Gan , φ⟩ = ⟨S, φ⟩ for all φ ∈ 𝒟, resp. φ ∈ 𝒟(Mp ) , and limn→∞ ⟨S−an , φ⟩ = ⟨G, φ⟩ for all φ ∈ 𝒟, resp. φ ∈ 𝒟(Mp ) . Using these facts, we can repeat almost literally the proof of [103, Proposition 2] to deduce the validity of (i) and (ii); the only thing worth noting is that we do not need Bochner’s criterion for the proof because the limits appearing on lines 10 and −4 on p. 258 of [103] follows, actually, from the almost automorphy of corresponding distributions. 3.9.3 An application to systems of ordinary differential equations in distribution and ultradistribution spaces Assume that n ∈ ℕ and A = [aij ]1⩽i,j⩽n is a given complex matrix such that σ(A) ⊆ {z ∈ ℂ : Re z < 0}. In this subsection, we analyze the existence of asymptotically almost periodic (automorphic) solutions of equation G󸀠 = AG + F, and equation G󸀠 = AG + F,

G ∈ 𝒟󸀠 (X n )

on [0, ∞)

G ∈ 𝒟󸀠∗ (X n ) on [0, ∞),

(220) (221)

where F is an asymptotically almost periodic (automorphic) X n -valued distribution in (220) and F is an asymptotically almost periodic (automorphic) X n -valued ultradistribution of ∗-class in (221). By a solution of (220), resp. (221), we mean any distribution G ∈ 𝒟󸀠 (X n ), resp. ultradistribution G ∈ 𝒟󸀠∗ (X n ), such that (220), resp. (221), holds in distributional, resp. ultradistributional, sense on [0, ∞). Our main result in this direction reads as follows:

272 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Theorem 3.9.13. (i) Let F = [F1 F2 ⋅ ⋅ ⋅ Fn ]T ∈ B󸀠AAP (X n ), resp. F = [F1 F2 ⋅ ⋅ ⋅ Fn ]T ∈ B󸀠AAA (X n ). Then there exists a solution G = [G1 G2 ⋅ ⋅ ⋅ Gn ]T ∈ B󸀠AAP (X n ), resp. G = [G1 G2 ⋅ ⋅ ⋅ Gn ]T ∈ B󸀠AAA (X n ) of (220). Furthermore, any distributional solution G of (220) belongs to the space B󸀠AAP (X n ), resp. B󸀠AAA (X n ). (ii) Let (Mp ) satisfy conditions (M.1), (M.2) and (M.3󸀠 ), and let F = [F1 F2 ⋅ ⋅ ⋅ Fn ]T ∈ 𝒟L󸀠∗1 (Mp : X n ) be such that, for every i ∈ ℕn , there exist an ultradifferential operator p Pi (D) = ∑∞ p=0 ai,p D of ∗-class and bounded functions f1,i , f2,i ∈ AAP(ℝ : X), resp. f1,i , f2,i ∈ AAA(ℝ : X), for 1 ⩽ i ⩽ n, such that Fi = Pi (D)f1,i + f2,i on [0, ∞). Then there exist ultradifferential operators Pij (D) of ∗-class, bounded functions hij ∈ AAP(ℝ : X), resp. hij ∈ AAA(ℝ : X) (1 ⩽ i, j ⩽ n), and bounded functions h2,i ∈ AAP(ℝ : X), n resp. h2,i ∈ AAA(ℝ : X) (1 ⩽ i ⩽ n), such that G = [G1 G2 ⋅ ⋅ ⋅ Gn ]T ∈ B󸀠∗ AAP (X ), resp. n T 󸀠∗ n G = [G1 G2 ⋅ ⋅ ⋅ Gn ] ∈ BAAA (X ), is a solution of (221), where Gi = ∑j=1 Pij (D)hij + h2,i for i ∈ ℕn . Furthermore, any ultradistributional solution G of ∗-class to the equation (221) has such a form. Proof. We will prove only (ii). Since σ(A) ⊆ {z ∈ ℂ : Re z < 0}, [71, Lemma 1] and a careful inspection of the proof of [71, Theorem 3, pp. 117–118] yield that it is sufficient to prove the required assertion in the one-dimensional case. So, let Re λ < 0 and let us investigate the equation G󸀠 = λG + P(D)f + g,

G ∈ 𝒟󸀠∗ (X) on [0, ∞),

(222)

with given bounded functions f , g ∈ AAP(ℝ : X), resp. f , g ∈ AAA(ℝ : X), and p −λ⋅ 󸀠 −λ⋅ −λ⋅ P(D) = ∑∞ p=0 ap D . Then (e G) = e [P(D)f + g] on [0, ∞), which implies ⟨e G, φ⟩ =

−⟨e−λ⋅ [P(D)f + g], I(φ)⟩ + ⟨Const., φ⟩, φ ∈ 𝒟0∗ . Therefore, there exists a constant c ∈ ℂ such that ⋅

⟨G, φ⟩ = ⟨P(D)f + g, e

−λ⋅

∞

λt

∫ [e φ(t) − η(t) ∫ eλu φ(u) du] dt⟩ −∞

−∞

∞

+ c ∫ eλu φ(u) du,

φ ∈ 𝒟0∗ .

−∞

Applying the product rule for the term containing function f (⋅), as well as Fubini theorem and partial integration for the term containing function g(⋅), we get: ∞

∞

∞

⟨G, φ⟩ = ∑ (−1)p+1 ap ∫ f (t)[e−λ⋅ I(eλ⋅ φ)(⋅)] (t) dt + c ∫ eλu φ(u) du p=0

(p)

−∞

−∞

∞

t

∞

−∞

−∞

−∞

+ ∫ g(t)e−λt ∫ [eλu φ(u) − η(u) ∫ eλv φ(v) dv] du dt

3.9 Asymptotically almost automorphic vector-valued generalized functions | 273 ∞

p

k−1 ∞ k − 1 k−j λt (j) p )λ e φ (t) = ∑ ap ∫ f (t) ∑ (−1)k+1 ( )λp−k e−λt [ ∑ ( j k p=0 j=0 k=1 −∞

∞

− η(k−1) (t) ∫ eλu φ(u) du] −∞ ∞

∞

p=0

−∞

p −λt

− ∑ ap ∫ f (t)λ e

t

∫ [e φ(u) − η(t) ∫ eλv φ(v) dv] du dt −∞

∞

∞

λu

∞

t

0

0

−∞

+ c ∫ eλu φ(u) du + ∫ [∫ eλ(t−s) g(s) ds]φ(t) dt 0 ∞

t

∞

− ∫ eλv [ ∫ η(t) ∫ e−λs g(s) ds dt]φ(v) dv, 0

0

−∞

φ ∈ 𝒟0∗ .

Referring back to Fubini theorem for the term containing the part η(k−1) (⋅) as well as Fubini theorem and partial integration for the term containing the part f (t)λp e−λt , we get ∞

p󸀠

∞

p󸀠 k − 1 p󸀠 −p (p) ⟨G, φ⟩ = ∫ f (t)[ ∑ ap󸀠 ∑ (−1)k+1 ( )( )λ ]φ (t) dt k p k=p p󸀠 =p+1 0

2

∞

p

∞ p − ∫ eλv [ ∑ ap ∫ f (t) ∑ (−1)k+1 ( )λp−k e−λt η(k−1) (t) dt]φ(v) dv k p=0 k=1 0

∞

∞

t

0

p=0

0

1

− ∫ [ ∑ ap λp ∫ eλ(t−s) f (s) ds]φ(t) dt ∞

λv

∞

∞

p

t

+ ∫ e [ ∑ ap λ ∫ η(t) ∫ e 0

p=0

∞

t

0

0

−∞

∞

−λs

f (s) ds dt]φ(v) dv + c ∫ eλu φ(u) du

0

0

+ ∫ [∫ eλ(t−s) g(s) ds]φ(t) dt ∞

λv

∞

t

− ∫ e [ ∫ η(t) ∫ e−λs g(s) ds dt]φ(v) dv, 0

−∞

0

φ ∈ 𝒟0∗ .

The series in the first and fourth term converge, whereas the functions ceλ⋅ and ⋅ ∫0 eλ(⋅−s) g(s) ds are asymptotically almost periodic, resp. asymptotically almost automorphic, due to our assumption Re λ < 0. It is not difficult to show that the series t −λs p ∞ f (s) ds dt converges uniformly for t ⩾ 0 and, to complete the ∑∞ p=0 ap λ ∫−∞ η(t) ∫0 e

274 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations whole proof, it suffices to show that the series 2

∞

p

p ∑ ap ∫ f (t) ∑ (−1)k+1 ( )λp−k e−λt η(k−1) (t) dt k p=0 k=1 1

is absolutely convergent. Towards this end, observe that (M.3󸀠 ) yields the existence of a finite constant c󸀠 > 0 such that the absolute value of 2

p

p ap ∫ f (t) ∑ (−1)k+1 ( )λp−k e−λt η(k−1) (t) dt k k=1 1

M

M

does not exceed ‖f ‖∞ Const.|ap | ∑pk=0 hkk ⩽ ‖f ‖∞ Const.|ap |(p + 1)mp hpp , where m = ∑∞ p=1 1/mp < ∞. This completes the proof in a routine manner. Remark 3.9.14. Assume that (Mp ) additionally satisfies (M.3). Then, for the Beurling class, the assumptions in (ii) are equivalent to saying that F and G are asymptotically almost periodic, resp. asymptotically almost automorphic (see Theorem 3.9.10). We feel duty bound to say that, in this case, it seems very plausible that the assertions of Corollary 3.8.4 and Theorems 3.8.6 and 3.9.10 can be slightly generalized for ω-ultradistributions (see [201] for more details). The interested reader may also try to prove analogues of [90, Theorem 4.3.1, p. 123; Theorem 4.3.2, p. 124; Theorem 4.5.1, p. 128] for asymptotically almost periodic (automorphic) ultradistributions of ∗-class.

3.10 Examples and applications In this section, we will present several illustrative examples and applications of our results established so far. First, we will continue the analysis of R. Ponce and M. Warma [348] concerning diffusion Volterra integro-differential equations with memory, proving the existence of some specific classes of exponentially decaying (a, k)-regularized resolvent families (possibilities for work exist even in the case when C ≠ I and X is not a Banach space). In the final part of these examples, we will consider asymptotically almost periodic solutions and Stepanov asymptotically almost periodic solutions. Example 3.10.1. Suppose that α ∈ ℝ, α ≠ 0, β ⩾ 0, 0 < ζ ⩽ 1 and ω ∈ ℝ. Let any of the following two conditions hold: (i) α > 0, 𝒜 is an MLO satisfying 󵄩 󵄩 ω + Σζπ/2 ⊆ ρ(𝒜) and 󵄩󵄩󵄩R(λ : 𝒜)󵄩󵄩󵄩 = O(|λ − ω|−1 ),

λ ∈ ω + Σζπ/2 .

(223)

(ii) α < 0, α + βζ ⩾ |α| and 𝒜 is an MLO satisfying (223). Then it is well-known that the operator A ≡ 𝒜|D(𝒜) is single-valued, linear and densely defined in the Banach space D(𝒜), as well as that (223) holds with the operator 𝒜

3.10 Examples and applications | 275

t

replaced with the operator A; see, e. g., [339, Lemma 4.1]. Set a(t) := 1 + ∫0 k(s) ds,

where k(t) := αe−βt gζ (t). Owing to the proof of [348, Theorem 2.1], we get that A generates an exponentially bounded (a, I)-resolvent family (Sω (t))t⩾0 in D(𝒜), provided that ω = 0. In the general case ω ≠ 0, the perturbation result [247, Theorem 3.7.4] and decomposition A = (A − ωI|D(𝒜) ) + ωI|D(𝒜) show that A generates an exponen-

tially bounded (a, I)-resolvent family (S(t))t⩾0 in D(𝒜), too. This extends the assertion of [348, Corollary 2.2], and can be applied in the analysis of Poisson heat equation with memory, in the space H −1 (Ω); see, e. g., [188, Example 3.3] and the analysis below. The proof of [348, Theorem 2.3] works in the degenerate case as well, and we may conclude the following: Let α ≠ 0, β ⩾ 0, 0 < ζ < ζ ̃ ⩽ 1, ω < 0 and β + ω ⩽ 0. If (i) holds with the number ζ replaced with the number ζ ̃ therein, then ‖S(t)‖ = O(e−βt ), t ⩾ 0; if (ii) holds with the number ζ replaced with the number ζ ̃ 1/(ζ +1)

)t therein, then ‖S(t)‖ = O((1 + αωt ζ +1 )e−(β−(αω) ), t ⩾ 0. −1 Consider now the case in which X := H (Ω), where 0 ≠ Ω ⊆ ℝn is a bounded domain with smooth boundary. Let m(x) ⩾ 0 a. e. be a given function in L∞ (Ω), and let Δ denote the Dirichlet Laplacian defined as usual. Let θ ∈ (−π, π), 0 < ε < π − |θ|, and 𝒜 := eiθ Δm(x)−1 . Then the analysis contained in [188, Example 3.3] shows that (223) holds with some number ω = −c < 0 and the number ζ replaced by the number ζ ̃ := 2(π − ε − |θ|)/π therein. Let α ≠ 0, 0 < β ⩽ c, 0 < ζ < ζ ̃ ⩽ 1, and let β − (−αc)1/(ζ +1) > 0 in the case (ii). Hence, A generates an exponentially decaying (a, I)-resolvent family (S(t))t⩾0 in D(𝒜). Suppose that f ∈ C 1 ([0, ∞) : D(𝒜)) and f 󸀠 ∈ AAP([0, ∞) : D(𝒜)). Then the variation of parameters formula [247, Theorem 3.2.9(i)] (cf. also [349, Proposition 1.2(ii)]) and the proof of [127, Lemma 4.1] show that the mapt ping t 󳨃→ S(t)f (0)+∫0 S(t−s)f 󸀠 (s) ds, t ⩾ 0 is an asymptotically almost periodic solution t

of the abstract Volterra inclusion u(t) ∈ 𝒜 ∫0 a(t − s)u(s) ds + f (t), t ⩾ 0. The corresponding analysis in the space X = L2 (Ω) will not be conducted here (it is clear that [127, Lemma 4.1] can be applied in the analysis of existence and uniqueness of asymptotically almost periodic solutions of a substantially large class of inhomogeneous abstract Cauchy problems whose solution operator families are exponentially decaying; for the degenerate case, see also [403, Theorem 2.2, 2.4], [247, Theorem 2.2.20] with α = 1, and [188, Theorems 3.7–3.8]). Suppose, after all is said and done, that f ∈ C 1 ([0, ∞) : D(𝒜)) and f 󸀠 : [0, ∞) → D(𝒜) is asymptotically Stepanov almost periodic. Then the mapping t 󳨃→ S(t)f (0) + t ∫0 S(t − s)f 󸀠 (s) ds, t ⩾ 0 is an asymptotically Stepanov almost periodic solution of the t

abstract Volterra inclusion u(t) ∈ 𝒜 ∫0 a(t − s)u(s) ds + f (t), t ⩾ 0, provided that function f 󸀠 (⋅) can be written as f 󸀠 (⋅) = g(⋅) + q(⋅) (cf. Lemma 2.2.6 with function f (⋅) replaced therein by function f 󸀠 (⋅)), and q(⋅) satisfying the condition (i) of Proposition 2.6.13. Clearly, Proposition 2.11.6 can be applied in the analysis of asymptotically Weyl-almost periodic solutions of the above Volterra inclusion. Almost automorphic and asymptotically almost automorphic solutions can be analyzed similarly.

276 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Example 3.10.2. The Barenblatt–Zheltov–Kochina equation (λ − Δ)ut (t, x) = ζ Δu(t, x),

t ∈ ℝ, x ∈ Ω,

(224)

was introduced in 1960 by G. I. Barenblatt, Yu. P. Zheltov and I. N. Kochina [41] with a view to profile fluid filtration in fissured rocks. The importance of Barenblatt–Zheltov– Kochina equation in modeling processes of moisture transfer in soil and processes of two-temperature heat conductivity was perceived by M. Hallaire [214] (1964) and P. J. Chen, M. E. Gurtin [95] (1968), respectively. Suppose that 0 ≠ Ω ⊆ ℝn is a bounded domain with smooth boundary 𝜕Ω. By {λk } [= σ(Δ)] we denote the eigenvalues of the Dirichlet Laplacian Δ in X := L2 (Ω) (recall that 0 < −λ1 ⩽ −λ2 ≤ ⋅ ⋅ ⋅ ⩽ −λk ⩽ ⋅ ⋅ ⋅ → +∞ as k → ∞; see [247] for further information), numbered in nonascending order with regard to multiplicities. By {ϕk } ⊆ C ∞ (Ω) we denote the corresponding set of mutually orthogonal [in the sense of L2 (Ω)] eigenfunctions. In [247] (see [378, Theorem 5.1.3] for a related result, as well as [379] and [328] for the studies of stochastic Barenblatt–Zheltov–Kochina equations and numerical approximation of solutions to the Barenblatt–Zheltov–Kochina equation, respectively), we have analyzed the entire solutions of equation (224) equipped with the following initial conditions: u(0, x) = u0 (x),

x ∈ Ω;

u(t, x) = 0,

(t, x) ∈ ℝ × 𝜕Ω,

(225)

where ζ ∈ ℝ ∖ {0} and λ = λk0 ∈ σ(Δ). We have constructed an entire, exponentially bounded, (1, t)-regularized I-resolvent family (W 1 (t))t⩾0 generated by A := ζ Δ, B := λ − Δ, which is additionally such that there exists a finite constant ω > 0 for which the operator families {e−ωz W 1 (z) : z ∈ ℂ} ⊆ L(X) and {e−ωz BW 1 (z) : z ∈ ℂ} ⊆ L(X) are bounded (here, by entireness we mean that, for every f ∈ X, the mappings z 󳨃→ W 1 (z)f , z ∈ ℂ and z 󳨃→ BW 1 (z)f , z ∈ ℂ are entire). (i) Let ζ ∈ ℝ ∖ {0} and λ = λk0 ∈ σ(Δ). Consider the equation (λ − Δ)ut (t, x) = iζ Δu(t, x),

t ∈ ℝ, x ∈ Ω,

(226)

equipped with the initial conditions of type (225). It can be easily seen that (i−1 W 1 (ti))t⩾0 is an entire, exponentially bounded, (1, t)-regularized I-resolvent family generated by iA, B. Furthermore, if r = −ζ −1 λk−1 (λ − λk ) for some k ∈ ℕ with k ≠ k0 , then (ir)−1 Bϕk = (iA)ϕk . Hence, Proposition 2.5.10 implies that the mapping t 󳨃→ i−1 W 1 (ti)Bf , t ⩾ 0, appearing in (T2), is almost periodic for any f ∈ span({ϕk : k ∈ ℕ, k ≠ k0 }). By the representation formula from [6, Theorem 2.3], it readily follows that there exists a sufficiently large real number λ0 > 0 such that for each f ∈ H 2 (Ω) ∩ H01 (Ω), the expression u(t) = (λ0 B − iA)−1 Bf + (λ0 (λ0 B − iA)−1 B − I)i−1 W 1 (ti)Bf ,

t⩾0

3.10 Examples and applications | 277

defines a unique strong, almost periodic, solution of problem (226), with the initial value u0 = (λ0 B − iA)−1 Bf in (225). In conclusion, we obtain that for all initial values of u0 ∈ span({ϕk : k ≠ k0 }), the unique strong solution of (226) is almost periodic. It is clear that we cannot expect the almost periodicity of mappings like t 󳨃→ W 1 (t)f , t ⩾ 0 for all f ∈ X because W 1 (⋅) is entire. (ii) In the following general example, we would like to exhibit an idea concerning the use of regularizing functions k(t) of the form k(t) = ℒ−1 (

1 )(t), (z − ia1 )(z − ia2 ) ⋅ ⋅ ⋅ (z − ian )

t ⩾ 0,

(227)

where a1 , a2 , . . . , an are real numbers. Suppose P(z), P1 (z), Q(z), Q1 (z) are nonzero complex polynomials, dg(P1 ) < dg(Q1 ), A := P(Δ), B := Q(Δ), there exist a nonempty set Ω ⊆ ℂ and a finite constant M > 0 such that the operator λB − A is not injective, as well as all complex roots of the polynomial z 󳨃→ Q1 (z) − λP1 (z), z ∈ ℂ belong to the interval [−iM, iM] and are pairwise disjoint (λ ∈ Ω). Let a(t) := ℒ−1 (P1 (z)/Q1 (z))(t), t ⩾ 0 (for example, a(t) = sin t or a(t) = cos t). Sup−1 ̃ pose, further, that there exist N ∈ ℕ and ω > 0 such that ‖(B − a(z)A) ‖ + ‖B(B − −1 N ̃ a(z)A) ‖ = O(1 + |z| ), Re z > ω. Let n ⩾ N + 2 and let |ai | > M (1 ⩽ i ⩽ n) in (227). Then the complex characterization theorem for the Laplace transform shows that there exists an exponentially bounded (a, k)-regularized resolvent family generated by A, B; furthermore, the function ϑ(⋅) defined above is almost periodic for all λ ∈ Ω. Hence, Proposition 2.5.10 can be applied here. The existence and uniqueness of almost periodic (automorphic) solutions to abstract degenerate higher order Cauchy problems with integer order derivatives have not attracted the attention of authors so far. In the subsequent example, we will underline a few relevant facts concerning (asymptotically) almost periodic solutions of the abstract linearized Boussinesq–Love equation; cf. the paper [380] by G. A. Sviridyuk and A. A. Zamyshlyaeva for more details. Example 3.10.3. Suppose that 0 ≠ Ω ⊆ ℝn is a bounded domain with smooth boundary 𝜕Ω, and X = L2 (Ω). Of concern is the following Cauchy–Dirichlet problem for linearized Boussinesq–Love equation: (λ − Δ)utt (t, x) − α(Δ − λ󸀠 )ut (t, x) = β(Δ − λ󸀠󸀠 )u(t, x) + f (t, x), u(0, x) = u0 (x),

ut (0, x) = u1 (x),

(t, x) ∈ ℝ × Ω;

u(t, x) = 0,

t ∈ ℝ, x ∈ Ω, (228) (t, x) ∈ ℝ × 𝜕Ω, (229)

where λ, λ󸀠 , λ󸀠󸀠 ∈ ℝ, α, β ∈ ℝ and α, β ≠ 0 (we are under an obligation to say that nonlinear Boussinesq–Love equations, which will not be examined here, are much more important from an application point of view since these equations are used in describing longitudinal waves in a thin elastic rod with regard to transverse inertia effects;

278 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations for more details about this subject, we refer the reader to the paper [411] by A. A. Zamyshlyaeva, E. V. Bychkov and O. N. Tsyplenkova). We will use the same terminology as in Example 3.10.2. In [380, Theorem 5.1], the authors have proved the well-posedness results for the degenerate Cauchy problem (228)–(229) under the following conditions: (i) λ ∈ ρ(Δ); or (ii) λ ∈ σ(Δ) ∧ λ = λ󸀠 ∧ λ ≠ λ󸀠󸀠 . As observed in [247, Example 2.3.48], [380, Theorem 5.1] is inapplicable in the following case: (iii) λ ∈ σ(Δ) ∧ λ ≠ λ󸀠 ∧ (α = 0 ⇒ λ ≠ λ󸀠󸀠 ). If so, the existence and uniqueness of entire solutions of problem (228)–(229) (cf. [247, Definition 2.3.45] for the notion) for all initial values of u0 , u1 ∈ H 2 (Ω) ∩ H01 (Ω) follows from an application of [247, Theorem 2.3.46]. In what follows, we will consider the well-posedness of homogeneous problem (228)–(229) in case (i), with u0 , u1 ∈ H 2 (Ω) ∩ H01 (Ω). By [380, Theorem 5.1(i)], there exists a unique solution of this problem having the following form: ∞

u(t) = ∑ [

1 k=1 μk

1 2 μ1k μ2 eμk t − 1 k 2 eμk t ]⟨ϕk , u0 ⟩ϕk 2 μk − μk − μk 1

2

e μk t − e μk t ⟨ϕk , u1 ⟩ϕk , 1 2 k=1 (λ − λk )(μk − μk ) ∞

+∑

t ∈ ℝ,

where μ1,2 k

:=

−α(λ󸀠 − λk ) ± √α2 (λ󸀠 − λk )2 − 4β(λ − λk )(λ󸀠󸀠 − λk ) 2(λ − λk )

,

k ∈ ℕ.

Suppose that the following condition holds: 2

α2 (λ󸀠 − λk ) ⩾ 4β(λ − λk )(λ󸀠󸀠 − λk ),

k ∈ ℕ.

Then μ1,2 ∈ ℝ, k ∈ ℝ and the function t 󳨃→ u(it), t ∈ ℝ is almost periodic for all k u0 , u1 ∈ span({ϕk : k ∈ ℕ}), which is clearly dense in X. This fact cannot be established in case (ii) since then there exists a strong solution of (228)–(229) only for initial values of u0 , u1 ∈ H 2 (Ω) ∩ H01 (Ω) that are orthogonal to the functions ϕk for λ = λk (cf. [380, Theorem 5.1(ii)]). ] ⩽ 0, k ∈ ℕ. This condition clearly implies that the Suppose now that Re[μ1,2 k function t 󳨃→ u(t), t ∈ ℝ is an asymptotically almost periodic solution of the homogeneous counterpart of problem (228)–(229) for all u0 , u1 ∈ span({ϕk : k ∈ ℕ}), which is dense in X. Concerning the inhomogeneous term in the representation of u(⋅), with u0 = u1 = 0, the most simplest case when it will be asymptotically almost periodic for

3.10 Examples and applications | 279

t ⩾ 0 is that there exists a finite subset L ⊆ ℕ such that R(f ) ⊆ span({ϕk : k ∈ L}), Re[μ1,2 ] < 0, k ∈ L and the mappings t 󳨃→ ⟨ϕk , f (t)⟩, t ⩾ 0 are asymptotically almost k periodic for all k ∈ L (by Proposition 2.6.13, the asymptotical Sp -almost periodicity of this term can also occur provided certain growth order of these mappings). We would like to stress that a similar analysis can be done for the abstract Barenblatt–Zheltov–Kochina equation, as well as for the linearized Benney–Luke equation

(P)f

2 {(λ − Δ)ut (t, x) = (αΔ − βΔ )u(t, x) + f (t, x), { { { 𝜕 ( u(t, x)) = u0 (x), { { { t=0 { 𝜕t {u(t, x) = Δu(t, x) = 0,

t ⩾ 0, x ∈ Ω, x ∈ Ω, t ⩾ 0, x ∈ 𝜕Ω,

which is important in evolution modeling of some problems appearing in the theory of liquid filtration (cf. [378, Theorem 5.3.2] and [247]). Concerning possible applications to nondegenerate semilinear differential equations with almost sectorial operators, we will remind ourselves of the following instructive result of W. von Wahl [395], which is most commonly used for applications in the existing literature. Example 3.10.4. Assume that α ∈ (0, 1), m ∈ ℕ, Ω is a bounded domain in ℝn with boundary of class C 4m and X := C α (Ω). Define the operator A : D(A) ⊆ C α (Ω) → C α (Ω) by D(A) := {u ∈ C 2m+α (Ω) : Dβ u|𝜕Ω = 0 for all |β| ⩽ m − 1} and Au(x) := ∑ aβ (x)Dβ u(x) for all x ∈ Ω. |β|⩽2m

Here, β ∈ ℕn0 , |β| = ∑ni=1 βj , Dβ = ∏ni=1 ( 1i 𝜕x𝜕 )βi , and aβ : Ω → ℂ satisfy the following i conditions: (i) aβ (x) ∈ ℝ for all x ∈ Ω and |β| = 2m; (ii) aβ ∈ C α (Ω) for all |β| ⩽ 2m; and (iii) there is a constant M > 0 such that M −1 |ξ |2m ⩽ ∑ aβ (x)ξ β ⩽ M|ξ |2m |β|=2m

for all ξ ∈ ℝn and x ∈ Ω.

Then there exists a sufficiently large number σ > 0 such that the single-valued operα ator 𝒜 ≡ −(A + σ) satisfies condition (P) with β = 1 − 2m and some finite constants c, M > 0 (recall that 𝒜 is not densely defined and that the value of exponent β in (P) is sharp). We can simply incorporate Theorems 2.9.10–2.9.11 in the analysis of existence and uniqueness of asymptotically almost periodic solutions of the following fractional semilinear equation with higher order differential operators in the Hölder

280 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations space X = C α (Ω): γ

β {Dt u(t, x) = − ∑ aβ (t, x)D u(t, x) − σu(t, x) + f (t, u(t, x)), |β|⩽2m { {u(0, x) = u0 (x),

t ⩾ 0, x ∈ Ω; x ∈ Ω.

Asymptotically almost automorphic solutions of the above fractional differential equation can be analyzed similarly. Let us recall that almost sectorial operators have invaluable importance in the analysis of the asymptotic dynamics of a dissipative reactions diffusion equation in a dumbbell domain as the channel shrinks to a line segment; for more details about this subject, we refer the reader to the papers [31–33] by J. M. Arrieta, A. N. Carvahlo and G. Lozada-Cruz. The question of the attractivity of solutions for fractional evolution equations with almost sectorial operators has been recently analyzed by Y. Zhou in [422], while abstract fractional differential equations with almost sectorial operators and finite/infinite delay have been investigated by E. Hernández [219] and F. Li [297]. Linear perturbations of almost sectorial operators have been analyzed by A. Ducrot, P. Magal and K. Prevost in [178], where the theory of integrated semigroups has been applied in the study of abstract parabolic differential equations. For some other references treating abstract differential equations with almost sectorial operators, the reader may consult [172, 246, 247] and [417]. Example 3.10.5 (A. Favini, A. Yagi [188, Example 3.6]; cf. also [247]). It is well known that the theory of multivalued linear operators can be successfully applied in the analysis of famous degenerate Poisson heat equation and its fractional analogues. Let Ω be a bounded domain in ℝn , b > 0, m(x) ⩾ 0 a. e. x ∈ Ω, m ∈ L∞ (Ω), 1 < p < ∞ and X := Lp (Ω). Suppose that the operator A := Δ − b acts on X with the Dirichlet boundary conditions, and that B is the multiplication operator by the function m(x). Then we know that the multivalued linear operator 𝒜 := AB−1 satisfies condition (P) with β = 1/p and some finite constants c, M > 0; recall also that the validity of additional condition [188, (3.42)] on the function m(x) enables us to get the better exponent β in (P), provided that p > 2. From this, we know well that Theorems 2.7.6–2.7.7 can be incorporated in the study of existence and uniqueness of almost periodic solutions of semilinear Poisson heat equation { 𝜕 [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)), { 𝜕t {v(t, x) = 0,

t ∈ ℝ, x ∈ Ω;

(t, x) ∈ [0, ∞) × 𝜕Ω,

and how we can apply Theorems 2.7.8–2.7.9 in the study of existence and uniqueness of asymptotically almost periodic solutions of semilinear Poisson heat equation 𝜕 { { { 𝜕t [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)), { {v(t, x) = 0, { {m(x)v(0, x) = u0 (x),

t ⩾ 0, x ∈ Ω;

(t, x) ∈ [0, ∞) × 𝜕Ω, x∈Ω

3.10 Examples and applications | 281

in the space X, by using the substitution u(t, x) = m(x)v(t, x) and passing to the corresponding semilinear differential inclusions of first order (Proposition 2.11.1 can be applied in the analysis of existence and uniqueness of Weyl-almost periodic solutions of the classical version of the first problem). Concerning degenerate semilinear fractional differential equations, it is clear that the possible applications of Theorems 2.9.10–2.9.11 can be given in the analysis of existence and uniqueness of asymptotically almost periodic (automorphic) solutions of the following fractional Poisson semilinear heat equation: γ

D [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)), { { t {v(t, x) = 0, { {m(x)v(0, x) = u0 (x),

t ⩾ 0, x ∈ Ω; (t, x) ∈ [0, ∞) × 𝜕Ω, x ∈ Ω,

where 0 < γ < 1. We can apply Theorems 2.10.9 and 2.10.10 in the analysis of existence and uniqueness of almost periodic (automorphic) solutions of the following fractional Poisson semilinear heat equation with Weyl–Liouville derivatives, as well: γ

Dt,+ [m(x)v(t, x)] = (Δ − b)v(t, x) + f (t, m(x)v(t, x)),

t ∈ ℝ, x ∈ Ω.

At the end of this example, we want to emphasize that, in the present situation, we cannot say anything relevant about the physical meaning of the fractional Poisson heat equations and clearly mark the importance of such equations in the application setting. The convergence of solutions of fractional Poisson heat equations with Caputo or Weyl–Liouville fractional derivatives of order γ, as γ tends to 1−, is outside the scope of this book; see [247, Section 3.8] for more details about approximation and convergence of degenerate (a, k)-regularized C-resolvent operator families. Continuing we would like to observe that B. de Andrade and C. Lizama [127, Theorem 4.7] (cf. also R. Agarwal, B. de Andrade and C. Cuevas [11, Theorem 3.5]) have applied the Leray–Schauder alternative for proving the existence of asymptotically almost periodic solutions of abstract nondegenerate third order semilinear Cauchy problem (81). The argumentation contained in the proof of this theorem can be applied in the analysis of existence of asymptotically almost periodic solutions for a large class of related semilinear Cauchy problems whose solutions are governed by exponentially decaying degenerate operator families that are strongly continuous for t ⩾ 0 (some examples of such operator families can be found in [188, Chapter II]). Example 3.10.6. Assume that k ∈ ℕ, aα ∈ ℂ, 0 ⩽ |α| ⩽ k, aα ≠ 0 for some α with |α| = k, P(x) = ∑|α|⩽k aα i|α| xα , x ∈ ℝn , c󸀠 := supx∈ℝn Re(P(x)) < 0, X is one of the spaces Lp (ℝn ) (1 ⩽ p ⩽ ∞), C0 (ℝn ), Cb (ℝn ), BUC(ℝn ), P(D) := ∑ aα f (α) |α|⩽k

and D(P(D)) := {f ∈ E : P(D)f ∈ E distributionally}.

Set nX := n| 21 − p1 |, if X = Lp (ℝn ) for some p ∈ (1, ∞) and nX > n2 , otherwise. Then it is well known (see, e. g., [245, Example 2.8.6]) that operator P(D) generates a global

282 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations (1 − Δ)−nX k/2 -regularized semigroup (T(t))t⩾0 satisfying the estimate ‖T(t)‖ = O(e−ct ), t ⩾ 0 for all c ∈ (c󸀠 , 0). This immediately implies that we can consider the existence and uniqueness of asymptotically almost periodic (automorphic) solutions of the following semilinear Cauchy problem: γ

t ⩾ 0, x ∈ ℝn ,

(α) {Dt u(t, x) = ∑ aα f (t, x) + f (t, u(t, x)), |α|⩽k { u(0, x) = u 0 (x), {

x ∈ ℝn .

The method established in the proof of Theorem 3.4.10 will be further employed for reconsideration and improving some known results recently established by V. Keyantuo, C. Lizama, M. Warma [236] and V. T. Luong [318]. The main aim of the following example is to explain how we can do this. Example 3.10.7. In [236], the authors have considered an abstract two-term fractional differential equation β󸀠

Dαt +1 u(t) + c1 Dt u(t) = Au(t) + f (t), 󸀠

(k)

u (0) = uk ,

t ⩾ 0,

k = 0, 1,

where c1 > 0, A is a densely defined linear operator satisfying condition (H) with β = 1, C = I and c < 0, 0 < α󸀠 ⩽ β󸀠 ⩽ 1, and f (t) is a given X-valued function; here, we have been forced to slightly change the notation used in [236]. For this, the notion of an (α󸀠 , β󸀠 )c1 -regularized family generated by A, which is a special case of

the notion introduced in Definition 1.5.3 with a(t) = ℒ−1 (1/λα +1 + c1 λβ )(λ), t ⩾ 0 and 󸀠 󸀠 󸀠 k(t) = ℒ−1 (λα /λα +1 + c1 λβ )(λ), t ⩾ 0 has been introduced; cf. [236, Lemma 2.5]. Our main contributions will be given in the case that 0 < α󸀠 < β󸀠 ⩽ 1, which is used in the formulations and proofs of [236, Theorems 4.3 and 5.3], the main results of aforementioned paper (although possible applications can be given in the study of twoterm fractional Poisson heat equations in the space H −1 (Ω), where Ω is a bounded domain in ℝn with smooth boundary [188], we will pay our attention to the case that 𝒜 = A is a single-valued linear operator). Let us define an exponentially bounded, analytic (α󸀠 , β󸀠 , C)c1 -regularized family (Sα󸀠 ,β󸀠 (t))t⩾0 of angle ν ∈ (0, π/2], subgenerated by A, as the exponentially bounded, analytic (a, k)-regularized C-resolvent family of angle ν, subgenerated by the same operator, with a(t) and k(t) being defined as above. Then we have 󸀠

∞

∫ e−λt Sα󸀠 ,β󸀠 (t)x dt = λα (λα +1 + c1 λβ − A) Cx, 󸀠

󸀠

󸀠

−1

󸀠

x ∈ X, λ > 0 suff. large

0

and the assertion of [236, Theorem 3.1] holds with the initial values x, y ∈ X and the inhomogeneity f (t) replaced therein with the initial values C −1 x, C −1 y ∈ X and the inhomogeneity C −1 f (t), respectively, with the meaning clear.

3.10 Examples and applications | 283

Assume that condition (H) holds with β = 1, c < 0 and θ = π/2 − γ 󸀠 π/2, where α < γ 󸀠 < β󸀠 . Then we can argue as follows. Similarly as in the proof of Theorem 3.4.10, we have that operator A is a subgenerator of an exponentially bounded, analytic 󸀠 +1) (α󸀠 , β󸀠 , C)c1 -regularized family (Sα󸀠 ,β󸀠 (t))t⩾0 of angle ν := π/2(γ − π/2; cf. also [246, α󸀠 +1 󸀠

Theorem 2.2.5]. Estimating the term (λα +1 + c1 λβ − A)−1 C uniformly, as in the proof of Theorem 3.4.10, and using the integral computation given in the final part of this theorem and [29, Theorem 2.6.1], we get that 󸀠

󸀠

󸀠 󵄩󵄩 󸀠 󸀠 󵄩󵄩 α󸀠 +1 α󸀠 +1 + t β ), 󵄩󵄩Sα ,β (t)󵄩󵄩 = O(1/1 + t ) = O(1/1 + t

t ⩾ 0;

(230)

here we would like to note that the arguments used in the course of the proof of [236, Theorem 4.1] are vastly more complicated than ours and that our estimate is better even in the case C = I because then, with the notion introduced in [236] accepted, we only require that operator A be of sectorial angle γ 󸀠 π/2, not of β󸀠 π/2, as the stronger estimate in [236] requires. By means of estimate (230), we can repeat to the letter the proofs of [236, Theorems 4.3 and 5.3] in order to see that their claims hold with C-sectorial operators of smaller angle γ 󸀠 π/2, with the meaning clear. In the case C ≠ I, we can make important applications to generators of analytic C-regularized semigroups generated by nonelliptic differential operators in Lp -spaces [420]. In [318], V. T. Luong has investigated the following abstract two-term fractional differential equation with nonlocal conditions in a Banach space X: β󸀠

Dαt +1 u(t) + c1 Dt u(t) = Au(t) + f (t), 󸀠

u(0) + g(u) = u0 ,

ut (0) + h(u) = y0 ,

t ⩾ 0,

where 0 < α󸀠 ⩽ β󸀠 ⩽ 1, functions g(⋅) and h(⋅) satisfy certain conditions, and A is of sectorial angle β󸀠 π/2. Improvements of the main results of this paper, Theorems 3.4 and 4.4, to C-sectorial operators of angle γ 󸀠 π/2, with γ 󸀠 being clarified above, can be proved straightforwardly (0 < α󸀠 < β󸀠 ⩽ 1). As mentioned before, Corollary 3.4.12 can be applied to the almost sectorial operators and multivalued linear operators used in the analysis of Poisson heat equations. Example 3.10.8. (i) Letting the requirements of Example 3.10.4 hold, we will use the same notation here. For a sufficiently large number σ > 0, the operator −Aσ ≡ −(A + σ) defined η in this example satisfies condition (H) with β = 1 − 2q , C = I, c < 0 and some θ ∈ (0, π2 ). Let us remind ourselves that A is not densely defined and that the value η of exponent β = 1 − 2q is sharp. Applications of Corollary 3.4.12 are clear and here we would like to illustrate just one of them: cj = 0 for j < n − 1, αn−1 = αn −, α = 0+, η i = 0 and αn 2q < ζ < αn − 1. (ii) Let X := Lp (Ω), where Ω is a bounded domain in ℝn , b > 0, m(x) ⩾ 0 a. e. x ∈ Ω, m ∈ L∞ (Ω) and 1 < p < ∞. Suppose that the operator A := Δ − b acts on X with

284 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations the Dirichlet boundary conditions, and that B is the multiplication operator by function m(x). By the analysis contained in [188, Example 3.6] (see also Example 3.10.5) condition (H) is satisfied for the multivalued linear operator 𝒜 := AB−1 with β = 1/p, C = I and some numbers c < 0, θ ∈ (0, π2 ). Applications in the study of existence and uniqueness of asymptotically almost periodic and asymptotically almost automorphic solutions of the following multiterm fractional integrodifferential Poisson heat equation: [u(t, x) − (gζ +1+i ∗ f )(t, x)φ(x)] n−1

+ ∑ cj gαn −αj ∗ [u(t, x) − (gζ +1+i ∗ f )(t, x)φ(x)] j=1

+

∑

j∈ℕn−1 ∖Di

v(t, x) = 0,

cj [gαn −αj +i+ζ +1 ∗ f ](t, x) = (Δ − b)v(t, x);

(t, x) ∈ [0, ∞) × 𝜕Ω,

t

where m(x)v(t, x) = ∫0 gαn −α (t − s)u(s, x) ds for all t ⩾ 0 and x ∈ Ω, are immediate (φ ∈ X). Example 3.10.9. In [246, Subsection 3.3.2], we have analyzed hypercyclic and topologically mixing properties of solutions of abstract multiterm fractional Cauchy problem (173) with 𝒜 = A being single-valued and Aj = cj I for some cj ∈ ℂ (1 ⩽ i ⩽ n−1). Let it be the case. With the help of Proposition 3.4.13, we can reconsider many examples given in the above-mentioned part of [246] and provide several interesting applications in the investigation of the existence of a dense linear subspace X 󸀠 of X such that the mapping t 󳨃→ Ri (t)x, t ⩾ 0 is asymptotically almost periodic for all x ∈ X 󸀠 ; here, (Ri (t))t⩾0 is a given ki -regularized C-propagation family for (173) with a subgenerator A. For the sake of illustration, we will consider only the situation of [246, Example 3.3.12(ii)]; see also L. Ji, A. Weber [231]. Suppose that X is a symmetric space of noncompact type and rank one, p > 2, a well as the parabolic domain Pp and the positive real number cp pos-

sess the same meaning as in [231]. Suppose, further, that P(z) = ∑kj=0 aj z j , z ∈ ℂ is a nonconstant complex polynomial with ak > 0, n = 2, 0 < a < 2, α2 = 2a, α1 = 0, α = a, |p−2| π |p−2| , 2 − n arctan 2√p−1 − π2 a). Then we alc1 > 0, i = 0 and |θ| < min( π2 − n arctan 2√p−1

ready know that the operator −eiθ P(ΔX,p ) is the integral generator of an exponentially bounded, analytic resolvent propagation family ((Rθ,P,0 (t))t⩾0 , . . . , (Rθ,P,⌈2a⌉−1 (t))t⩾0 ) of ♮

π−n arctan

|p−2|

−|θ|

2√p−1 − π2 , π2 ), as well as that (Rθ,P,0 (t))t⩾0 is topologically mixing, angle min( a provided the following condition holds:

−eiθ P(int(Pp )) ∩ {(it)a + c1 (it)−a : t ∈ ℝ ∖ {0}} ≠ 0;

(231)

cf. [246] for the notion. Our new assumption will be, instead of (231), that there exist a number λ0 ∈ ℂ ∖ (−∞, 0] and a sufficiently small number ε > 0 such that λa +

3.10 Examples and applications |

285

c1 λ−a ∈ −eiθ P(int(Pp )), as well as that λa ∉ Σaπ/2 for |λ − λ0 | ⩽ ε. Let |λ − λ0 | ⩽ ε. Since ♮ there exists an x ≠ 0 such that −eiθ P(ΔX,p )x = (λa + c1 λ−a )x due to our assumption

λa + c1 λ−a ∈ −eiθ P(int(Pp )) ⊆ σp (−eiθ P(ΔX,p )), we can employ [246, Lemma 3.3.7] in order to see that ♮

Rθ,P,0 (t)x =

λa t −a (E (λa t a ) − Ea,2−a (c1 λ−a t a ))x λ2a − c1 a,2−a λa [λa Ea (λa t a ) + (a − 1)λa Ea,2 (λa t a ) + 2a λ − c1

− c1 λ−a Ea (c1 λ−a t a ) − (a − 1)c1 λ−a Ea,2 (c1 λ−a t a )]x + (λa + c1 λ−a )

λa [E (λa t a ) − Ea (c1 λ−a t a )]x, − c1 a

λ2a

t > 0;

see also [246, p. 418]. Due to our assumption λa ∉ Σaπ/2 , the asymptotic expansion formulae (16)–(18), and the fact that the first term in the above expression can be continuously extended to the nonnegative real axis, we get that the mapping t 󳨃→ Rθ,P,0 (t)x, t ⩾ 0 is asymptotically almost periodic. Since the set {x ∈ X : ∃λ ∈ ℂ ∖ (−∞, 0] s. t. |λ − ♮ λ0 | ⩽ ε and − eiθ P(ΔX,p )x = (λa + c1 λ−a )x} is total in X, Proposition 3.4.13 implies that

there exists a dense linear subspace X 󸀠 of X such that the mapping t 󳨃→ Rθ,P,0 (t)x, t ⩾ 0 is asymptotically almost periodic for all x ∈ X 󸀠 .

Example 3.10.10. Assume that A, B and C are closed linear operators in X, D(B) ⊆ D(A)∩D(C), B−1 ∈ L(X) and the conditions [188, (6.4)–(6.5)] hold with certain numbers c > 0 and 0 < β ⩽ α = 1. In [188, Chapter VI], the following second order differential equation: d (Cu󸀠 (t)) + Bu󸀠 (t) + Au(t) = f (t), dt

t > 0;

u(0) = u0 ,

Cu󸀠 (0) = Cu1

has been analyzed by the usual converting into the first order matrix system d Mz(t) = Lz(t) + F(t), dt

t > 0;

Mz(0) = Mz0 ,

where I M=[ O

O ], C

L=[

O −A

I ], −B

u z0 = [ 0 ] u1

0 and F(t) = [ ] (t > 0). f (t)

By the proof of [188, Theorem 6.1] (see also [188, Theorem 1.14]), we know that the multivalued linear operator (L[D(B)]×X − ωM[D(B)]×X )(M[D(B)]×X )−1 satisfies condition (P) for a sufficiently large number ω > 0, in the pivot space [D(B)] × X. Therefore, this MLO generates a degenerate semigroup (T(t))t>0 in [D(B)] × X, having an integrable singularity at zero and exponentially decaying growth rate at infinity. This enables

286 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations one to apply [188, Theorems 3.8–3.9] in the analysis of existence and uniqueness of solutions of the problem d Mz(t) = (L − ωM)z(t) + F(t), dt

t > 0;

Mz(0) = Mz0 ,

(232)

cf. [188, Section 3.1] for more details, as well as [260, Theorem 4.3] in the analysis of existence and uniqueness of solutions of the fractional problem γ

Dt [Mz(t)] = (L − ωM)z(t) + F(t),

t > 0;

Mz(0) = Mz0 ,

γ

where the Caputo fractional derivative Dt is taken in a weak sense [260]. Consider first the case of the abstract Cauchy problem (232). Denoting the components of z(t) by u(t) and v(t), from (232) we get that v(t) = u󸀠 (t) + ωu(t), t ⩾ 0 and (d/dt)(Cv(t)) = −Au(t) − (B + ωC)v(t) + f (t), t ⩾ 0, so that we are ready to solve the following second order differential equation: d (Cu󸀠 (t)) + (2ωC + B)u󸀠 (t) + (A + ωB + ω2 C)u(t) = f (t), dt u(0) = u0 , C[u󸀠 (0) + ωu0 ] = Cu1 .

t > 0; (233)

Roughly speaking, if M[u0 u1 ]T belongs to the domain of continuity of (T(t))t>0 and f (⋅) is Hölder continuous with an appropriate Hölder index, then there exists a unique solution z(t) of (232), continuous for t ⩾ 0, and moreover, t

u(t) u 0 Mz(t) = M [ ] = T(t)M [ 0 ] + ∫ T(t − s) [ ] ds, v(t) u1 f (s)

t ⩾ 0.

0

Therefore, if f (⋅) additionally satisfies the requirements of Proposition 3.5.4 (here we can apply many similar assertions known for asymptotical almost periodicity or asymptotical almost automorphy), then the unique solution u(⋅) of (233) will be such that Mz(⋅) = [u(⋅) C(u󸀠 (⋅) + ωu(⋅))]T is bounded and asymptotically Sp -almost automorphic. We can simply apply this result in the analysis of existence and uniqueness of asymptotically Sp -almost automorphic solutions of the following damped Poissonwave type equation in the spaces X := H −1 (Ω) or X := Lp (Ω) 𝜕 𝜕u 𝜕u 2 { { { { 𝜕t (m(x) 𝜕t ) + (2ωm(x) − Δ) 𝜕t + (A(x; D) − ωΔ + ω m(x))u(x, t) = f (x, t), { {t ⩾ 0, x ∈ Ω; u = 𝜕u/𝜕t = 0, (x, t) ∈ 𝜕Ω × [0, ∞), { { {u(0, x) = u0 (x), m(x)[(𝜕u/𝜕t)(x, 0) + ωu0 ] = m(x)u1 (x), x ∈ Ω. Here, Ω ⊆ ℝn is a bounded open domain with smooth boundary, 1 < p < ∞, m(x) ∈ L∞ (Ω), m(x) ⩾ 0 a. e. x ∈ Ω, Δ is the Dirichlet Laplacian in L2 (Ω), acting with domain H01 (Ω) ∩ H 2 (Ω), and A(x; D) is a second order linear differential operator on Ω with coefficients continuous on Ω; see [188, Example 6.1] for more details. In the fractional

3.10 Examples and applications | 287

relaxation case, we can similarly consider the existence and uniqueness of asymptotically Sp -almost automorphic solutions of the following fractional damped Poissonwave type equation in the spaces X := H −1 (Ω) or X := Lp (Ω): γ

γ

γ

D (m(x)Dt u) + (2ωm(x) − Δ)Dt u + (A(x; D) − ωΔ + ω2 m(x))u(x, t) = f (x, t), { { t γ t ⩾ 0, x ∈ Ω; u = Dt u = 0, (x, t) ∈ 𝜕Ω × [0, ∞), { { γ {u(0, x) = u0 (x), m(x)[Dt u(x, 0) + ωu0 ] = m(x)u1 (x), x ∈ Ω. In a series of research papers, T. Diagana has examined the existence and uniqueness of almost periodic and almost automorphic type solutions of nonautonomous differential equations of first order (see, e. g., [132] and references cited therein). In the following example, we continue the analysis initiated in [137, Section 4], which nicely fit into the consideration of Besicovitch almost automorphic solutions of nonautonomous evolution equations of first order. Example 3.10.11. Assume that X := L2 [0, π] and Δ denotes the Dirichlet Laplacian in X, with the domain H 2 [0, π] ∩ H01 [0, π]. Then Δ generates a strongly continuous semigroup (T(t))t⩾0 on X, satisfying the estimate ‖T(t)‖ ⩽ e−t , t ⩾ 0. Consider the following problem: ut (t, x) = uxx (t, x) + q(t, x)u(t, x) + f (t, x), u(0) = u(π) = 0,

t ⩾ 0, x ∈ [0, π];

(234)

u(0, x) = u0 (x) ∈ X,

(235)

where q : ℝ × [0, π] → ℝ is a jointly continuous function satisfying that q(t, x) ⩽ −γ0 , (t, x) ∈ ℝ × [0, π], for some number γ0 > 0. Define A(t)φ := Δφ + q(t, ⋅)φ,

φ ∈ D(A(t)) := D(Δ) = H 2 [0, π] ∩ H01 [0, π],

t ∈ ℝ.

Then (A(t))t∈ℝ generates an exponentially stable evolution family (U(t, s))t⩾s in the sense of [137, Definition 3.1], which is given by t

U(t, s)φ := T(t − s)e∫s q(r,⋅) dr φ,

t ⩾ s.

It is clear that we can rewrite the initial value problem (234)–(235) in the following form: u󸀠 (t) = A(t)u(t) + f (t),

t ⩾ 0;

u(0) = u0 .

Hence, Theorems 3.7.2 and 3.7.5 are applicable provided that the following condition holds: l

v

v 1 󵄨 󵄨 v+α ∫ ∫ sup 󵄨󵄨󵄨e∫s+α q(r,x) dr − e∫s q(r,x) dr 󵄨󵄨󵄨 ds dv = 0 l→+∞ 2l x∈[0,π]

lim

−l −∞

(α ∈ ℝ);

(236)

288 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations see (206). Condition (236) holds for a large class of functions q(⋅, ⋅), and what we will shown here is that this condition particularly holds for the function q(t, x) = −γ0 − 3t 2 − f (x), t ∈ ℝ, x ∈ [0, π], where f : ℝ → [0, ∞) is a continuous function. Let α ∈ ℝ be fixed. In our concrete situation, we have the following estimate of the integrand: v 3 3 3 3 2 󵄨 󵄨 v+α sup 󵄨󵄨󵄨e∫s+α q(r,x) dr − e∫s q(r,x) dr 󵄨󵄨󵄨 ⩽ Const. ⋅ [es −v + es −v +3|α|(v−s) ].

x∈[0,π]

Using substitution x = v − s, the only thing we need to prove is that l ∞

2 2 3 2 1 ∫ ∫ e3x v−3xv −x +3|α|x dx dv = 0. l→+∞ 2l

lim

−l 0

For v ∈ [−l, 0], we have e3x 0∞

∫ ∫ e3x

2

2

2

v−3xv −x 3 +3|α|x 2

v−3xv2 −x 3 +3|α|x 2

⩽ e−3xv

2

−x 3 +3|α|x 2

, so that Fubini theorem yields

dx dv

−l 0 0∞

⩽ ∫ ∫ e−3xv

2

−x 3 +3|α|x 2

dx dv

−l 0 ∞

⩽ ∫ e−x 0

3

+3|α|x 2

0

2

[ ∫ e−3xv dv] dx −∞

∞

∞

0

0

2 1 −x3 +3|α|x2 = [∫ dx][ ∫ e−y dy], e √3x

l > 0.

Since the integrand is always bounded by the proof of Theorem 3.7.1, the above computation shows that we only need to prove that: l ∞

2 2 3 2 1 lim ∫ ∫ e3x v−3xv −x +3|α|x dx dv = 0. l→+∞ 2l

1 0

For x ∈ [0, 1], we have e3x

2

v−3xv2 −x 3 +3|α|x 2

l 1

∫ ∫e3x

2

⩽ Const. ⋅ e3xv(1−v) , so that

v−3xv2 −x 3 +3|α|x 2

dx dv

1 0

l 1

⩽ Const. ⋅ ∫ ∫ e3xv(1−v) dv 1 0

1 l−1

= Const. ∫ ∫ e−3xy(1+y) dy dx 0 0

3.11 Notes and appendices | 289 1 ∞

2

⩽ Const. ∫ ∫ e−3xy dy dx 1

= [∫ 0

0 0 ∞

2 dx ][ ∫ e−y dy] √3x

(l ⩾ 1),

0

so we need to prove l ∞

2 2 3 2 1 ∫ ∫ e3x v−3xv −x +3|α|x dx dv = 0. l→+∞ 2l

(237)

lim

1 1

Let ε ∈ (0, 1) be given. For a given v ⩾ 1, the function F(x) := 3x2 v − 3xv2 − (1 − ε)x3 , x ⩾ 1 attains its local minimum (resp. maximum) at the point x = v 1−√ε (resp. x = v 1+√ε ). 1−ε 1−ε We have l ∞

∫ ∫ e3x

2

v−3xv2 −x 3 +3|α|x 2

dx dv

1 1

1−√ε l v 1−ε

=∫ ∫ e

3x2 v−3xv2 −x 3 +3|α|x 2

∞

⩽ [ ∫ e−3v 1

∞

∞

dx dv + ∫ ∫ e3x

2

v−3xv2 −x 3 +3|α|x 2

dx dv

1 v 1−√ε 1−ε

1

1

l

2 1−√ε ( 1−ε −1)

∞

dv][ ∫ e−x ∞

3

+3|α|x 2

dx]

0

+ [ ∫ e−cε v dv][ ∫ e−εx 1

3

3

+3|α|x 2

dx],

0

where limε→0+ cε = 1. Hence, the mapping l ∞

l 󳨃→ ∫ ∫ e3x

2

v−3xv2 −x 3 +3|α|x 2

1 1

dx dv,

l⩾1

is bounded, finishing the proof of (237).

3.11 Notes and appendices Almost periodic and almost automorphic functions in vector topological spaces. Let (X, τ) be a topological vector space (TVS). The notion of an almost periodic function f : I → X, where I = ℝ or I = [0, ∞), was introduced already in the landmark 1934–1935 papers of J. von Neumann [336] and S. Bochner, J. von Neumann [67]. Let

290 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations 𝒲 be a basis of balanced neighborhoods of zero in X. Then a function f : I → X is called almost periodic iff it is continuous and, for every W ∈ 𝒲 , there exists a number l > 0 such that each subinterval I 󸀠 ⊆ I of length l contains a point τ ∈ I 󸀠 such that f (t + τ) − f (t) ∈ W. If this is the case, then the range of f (⋅) is relatively compact and bounded in X, and f (⋅) is uniformly continuous; furthermore, the space consisting of all almost periodic functions, denoted by AP(I : X), is translation invariant, closed under uniform convergence and closed under reflections at zero, if I = ℝ (see, e. g., the paper [238] by L. A. Khan, S. M. Alsulami and, especially, the proofs of [238, Theorems 2–3, Proposition 4]). Concerning Bochner’s criterion, it is worthwhile to mention that the sequential compactness of set {fτ : τ ∈ ℝ} in the uniform topology of Cb (ℝ : X), where f : ℝ → X is a given function, implies that f (⋅) is almost periodic; the converse statement holds in the case when X is a Fréchet space, AP(ℝ : X) has a linear vector structure and the assertion of Theorem 2.1.1(xiv) continues to hold (see, e. g., the proofs of [238, Theorems 7–10]). Basic information about weakly almost periodic functions and integration of almost periodic functions can be obtained by consulting Section 4 of the paper [208] by G. M. N’Guérékata; regarding almost periodic strongly continuous semigroups in Fréchet spaces and some applications to abstract differential equations, see [208, Section 5] and [209]. For the notion of a two-parameter almost periodic function f : ℝ × X → X and some applications to abstract differential and integral equations, where X is a Fréchet space, we refer the reader to the paper [77] by D. Bugajewski and G. M. N’Guérékata; for an excellent survey of results about almost periodic functions with values in locally convex spaces and their representations, we can sincerely recommend the paper [368] by A. I. Shtern. Finally, we would like to stress that the notion of an asymptotically almost periodic function in a general (TVS) was introduced in 2013 by L. A. Khan, S. M. Alsulami [237] (cf. also [77] for some earlier results known in Fréchet spaces). Almost automorphic functions in general vector topological spaces have not attracted so much attention of the authors by now (see, e. g., the investigation of C. G. Gal, S. G. Gal and G. M. N’Guérékata [199]) concerning almost automorphic functions with values in p-Fréchet spaces, p ∈ (0, 1), which are nonlocally convex TVSs. Concerning almost automorphic functions with values in Fréchet spaces and almost automorphic solutions of abstract integro-differential equations in Fréchet spaces, we would like to mention the papers by C. G. Gal [196] and C. G. Gal, S. G. Gal, G. M. N’Guérékata [197, 198]. Let X be a Fréchet space. Due to [197, Theorem 2.4], we have that the set AA(ℝ : X) consisting of all almost automorphic functions ℝ → X is a vector space under the operations of pointwise sum of functions and multiplication with complex scalars; furthermore, the space AA(ℝ : X) is translation invariant, closed under uniform convergence, closed under reflections at zero and the inclusion f ∈ AA(ℝ : X) implies that the range of f (⋅) is relatively compact and bounded in X, as in the almost periodic case. In [197], we have collected some results on differentiation and integration of almost automorphic functions, as well as almost automorphic strongly continuous groups and semigroups of operators in Fréchet spaces (cf. also [198]). The no-

3.11 Notes and appendices | 291

tions of a two-parameter almost automorphic function and an asymptotically almost automorphic function in Fréchet space have been introduced and analyzed in [197], as well. We have already mentioned that the notions of almost periodicity and almost automorphy can be analyzed for functions on general topological groups (see, e. g., [67, 336, 382] and [387, 388]). Before we go any further, we would like to recommend for the reader the articles [389, 390] by M. Veselý for a basic source of information about almost periodic sequences and almost periodic functions in pseudometric spaces. Almost periodic and almost automorphic type solutions of abstract functional differential equations. Regrettably, abstract functional differential equations have not attracted the attention of authors so far. For the basic information on the existence and uniqueness of almost periodic and almost automorphic type solutions of functional differential equations with finite or infinite delay, besides the references already quoted, we can recommend for the reader the papers by S. Abbas [2], Y.-K. Chang, G. M. N’Guérékata, R. Zhang [87], C. Cuevas, E. M. Hernández [115], J. P. C. dos Santos, C. Cuevas [174], Y. Hino, S. Murakami [223] and I. Mishra, D. Bahugana, S. Abbas [325]. Differential equations with piecewise constant argument. It is well known that nonlinear differential equations with piecewise constant argument (DEPCAs, for short) combine the properties of nonlinear differential equations and nonlinear difference equations (cf. [79] for foundational results in this direction). The equations of this kind are prototypes of certain sequential continuous models appearing in disease dynamics, and they also have invaluable importance in mathematical economy. The existence and uniqueness of various types of almost periodic, periodic and quasi-periodic solutions of neutral differential equations with piecewise constant arguments have been investigated by many authors so far (see, e. g., [123, 292] and [408]; we will not examine the existence and uniqueness of almost periodic and almost automorphic type solutions of abstract impulsive differential equations, see [3, 5, 150, 308– 310, 373, 384] and the monograph [372] by G. Tr. Stamov for more details in this direction). Concerning the existence and uniqueness of various types of almost automorphic solutions for differential equations with piecewise constant arguments, mention should be made of paper [324] by N. V. Minh and T. T. Dat, where some results about the abstract Cauchy problem u󸀠 (t) = Au(⌊t⌋) + f (t),

t⩾0

(A ∈ L(X))

have been established, as well of the paper [89] by A. Chávez, S. Castillo and M. Pinto, where the notion of a discontinuous almost automorphic function, appearing naturally in the study of almost automorphic solutions of differential equations with piecewise constant argument, has been introduced. The authors of [89] have analyzed the almost automorphicity of the bounded solutions of the following DEPCA: x󸀠 (t) = Ax(t) + Bx(⌊t⌋) + f (t),

t ∈ ℝ,

292 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations where A, B ∈ ℝn,n are real matrices of format n × n and f (⋅) is an almost automorphic function. For some other results, we refer the reader to the papers [159] by W. Dimbour and [171] by H.-S. Ding, S.-M. Wan. In what follows, we will present the main details of our recent joint research with D. Velinov [280], where we have analyzed asymptotically Bloch-periodic solutions of abstract fractional nonlinear differential inclusions with piecewise constant argument; see also the research studies of W. Dimbour, V. Valmorin [386] and M. F. Hasler, G. M. N’Guérékata [218], as well as [79, 96, 171, 212, 215–217, 324, 366] and [398]. More precisely, we will consider the following fractional differential Cauchy inclusion with piecewise constant argument: γ

Dt u(t) ∈ 𝒜u(t) + A0 u(⌊t⌋) + g(t, u(⌊t⌋)),

t > 0;

u(0) = u0 ,

(238)

where A0 ∈ L(X), ⌊⋅⌋ is the largest integer function, g : [0, ∞) × X → X is a given γ function, and Dt u(t) denotes the Caputo fractional derivative or order γ, defined as folγ lows: The Caputo fractional derivative Dt u(t) is defined for those continuous functions u : [0, ∞) → X such that for each T > 0 one has g1−γ ∗ (u(⋅) − u(0)) ∈ W 1,1 ((0, T) : X), γ d by Dt u(t) := dt [g1−γ ∗ (u(⋅) − u(0))](t), t ∈ (0, T]. If γ = 1, then it is worth observing γ that the Caputo fractional derivative Dt u(t) is well-defined for any locally absolutely continuous function u : [0, ∞) → X. Concerning Bloch-periodic functions and asymptotically Bloch-periodic functions in Banach spaces, the following should be said (for the existence and uniqueness of anti-periodic and Bloch periodic solutions of certain classes of abstract Volterra integro-differential equations, we refer the reader to [96, 161, 212, 215–217, 309, 312] and [327]). Let I = [0, ∞) or I = ℝ, let p ⩾ 0, and let k ∈ ℝ. A bounded continuous function f : I → X is said to be (Bloch) (p, k)-periodic, or Bloch-periodic with period p and Bloch wave vector or Floquet exponent k iff f (x + p) = eikp f (x),

x ∈ I.

The space of all functions f : I → X that are (p, k)-periodic will be denoted by 𝒫p,k (I : X). The Bloch-periodicity is a more general notion than those of periodicity and antiperiodicity. Indeed, if f (⋅) is (p, k)-periodic, such that kp = 2π, then the function f (⋅) is periodic. If kp = π, then f (⋅) is anti-periodic. Let f , f1 , f2 ∈ 𝒫p,k (I : X). Then we have the following: (a) f1 + f2 (⋅) is (p, k)-periodic; (b) cf (⋅) is (p, k)-periodic for c ∈ ℂ; (c) ft (⋅) := f (⋅ + t) is (p, k)-periodic for t ∈ I; (d) If D : X → X is a linear continuous operator, then Df (⋅) is (p, k)-periodic; ⋅+a (e) The function Fa (⋅) := ∫a f (s) ds is (p, k)-periodic for a ∈ I.

3.11 Notes and appendices | 293

It is well known that the space 𝒫p,k (I : X), endowed with the sup-norm, is a closed subspace of Cb (I : X). Example 3.11.1 ([218]). Let us consider the heat equation ut (t, x) = uxx (t, x), u(0, x) = f (x) for t ⩾ 0 and x ∈ ℝ. If f ∈ 𝒫p,k (ℝ : X), then the unique solution of this equation is given by ∞

u(t, x) = ∫ e−

(x−s)2 4t

f (s) ds,

t > 0, x ∈ ℝ.

−∞

It can be easily shown that for each t ⩾ 0 the function x 󳨃→ u(t, x), x ∈ ℝ is in class 𝒫p,k (ℝ : X), as well. Similarly, if we consider the wave equation utt (t, x) = uxx (t, x), u(0, x) = f (x), ut (0, x) = g(x) for t ⩾ 0 and x ∈ ℝ, then it is well known that the unique solution of this equation is given by the d’Alambert’s formula x+t

1 1 u(t, x) = [f (x + t) + f (x − t)] + ∫ g(s) ds, 2 2 x−t

t > 0, x ∈ ℝ.

If f , g ∈ 𝒫p,k (ℝ : X), then for each t ⩾ 0 the solution x 󳨃→ u(t, x), x ∈ ℝ is in the same class, as well. Now we will repeat the following well-known definition of an asymptotically (p, k)-Bloch-periodic function (see, e. g., [218]). Definition 3.11.2. Let p ⩾ 0 and let k ∈ ℝ. A bounded continuous function f : [0, ∞) → X is called asymptotically (p, k)-Bloch-periodic iff there are a Bloch (p, k)-periodic function g : ℝ → X and a function h ∈ C0 ([0, ∞) : X) such that f (x) = g(x) + h(x),

x ⩾ 0.

(239)

Functions g(⋅) and h(⋅) are called principal and corrective parts of f (⋅), respectively. The space of all asymptotically (p, k)-Bloch-periodic X-valued functions will be denoted by 𝒜P p,k ([0, ∞) : X). Notice that the decomposition of an asymptotically (p, k)-Bloch-periodic X-valued function into its principal and corrective part is unique (cf. (239)). Now we are ready to analyze asymptotically Bloch-periodic solutions for nonlinear fractional differential inclusions with piecewise constant argument. We first suppose that γ ∈ (0, 1] and 𝒜 is an MLO in X satisfying condition (P). Definition 3.11.3. We say that a function u ∈ C([0, ∞) : X) is a classical solution of the fractional differential inclusion (238) iff the following is satisfied: γ (i) The derivative Dt u(t) is well-defined; (ii) We have that u(0) = u0 , g(⋅, u(⌊⋅⌋)) ∈ L1loc ([0, ∞) : X) and (238) is satisfied for a. e. t > 0.

294 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations In contrast to [386, Definition 3.1], where the case γ = 1 has been considered, we allow the possibility for the nonexistence of one-sided derivatives at nonnegative integer points (from a strictly mathematical viewpoint, this clearly has a certain sense). Furthermore, in our analyses, we do not require a priori that the function g : [0, ∞) × X → X is jointly continuous. Example 3.11.4. Consider the case in which X = ℂ, 0 < ε < γ ⩽ 1, 𝒜 = A0 = 0, u0 = 0, u(t) = g1+ε (t), t ⩾ 0 and g(t, z) := g1+ε−γ (t), t > 0, z ∈ ℂ. Then u(⋅) is a classical solution γ of (238) in the sense of Definition 3.11.3, Dt u(t) = g1+ε−γ (t) is not continuous for t = 0, u(⋅) is not a classical solution of (238), with γ = 1, in the sense of [386, Definition 3.1] because the right derivative of u(⋅) does not exist for t = 0, and g(⋅, ⋅) is not continuous on [0, ∞) × X. Define the operator families Tγ,ν (⋅), Sγ (⋅), Pγ (⋅) and Rγ (⋅) as it has been worked out in our previous examinations. Definition 3.11.5. Let γ ∈ (0, 1], and let u0 ∈ X be such that Sγ (t)u0 → u0 , t → 0+. A function u ∈ C([0, ∞) : X) is said to be a mild solution of the fractional differential inclusion (238) iff g(⋅, u(⌊⋅⌋)) ∈ L1loc ([0, ∞) : X) and t

u(t) = Sγ (t)u0 + ∫ Rγ (t − s)[A0 u(⌊s⌋) + g(s, u(⌊s⌋))] ds,

t > 0.

(240)

0

In the case that γ = 1 and the function g : [0, ∞) × X → X is jointly continuous, the notion of mild solution of (238) is equivalent to that introduced in [386, Definition 3.2]. In the case that γ ∈ (0, 1), the argument used in the proof of [396, Lemma 4.1] shows that any classical solution of (238) has to satisfy equation (240), provided that the mapping g(⋅, u(⌊⋅⌋)) is Laplace transformable (cf. [29, Section 1.4]); here, we would like to observe that it is not clear how we can prove the above fact in the case when the mapping g(⋅, u(⌊⋅⌋)) does not satisfy this condition. The following two lemmas and Theorem 3.11.8 below can be deduced by using the growth rates of operator families Sγ (⋅) and Rγ (⋅), as well as the arguments contained in the proofs of [386, Lemmas 3.1–3.2, Theorem 3.3]; the only thing worth noting is that the semigroup property of T(⋅) has not been used in the proofs of the afore-mentioned results. For the sake of completeness, we will present here only the most important details of the proof of Lemma 3.11.6: Lemma 3.11.6. Let A0 ∈ L(X), let p ∈ ℕ, and let k ∈ ℝ. Define the nonlinear operator Λ1 on 𝒜P p,k ([0, ∞) : X) through t

(Λ1 φ)(t) := ∫ Rγ (t − s)A0 φ(⌊s⌋) ds,

t ⩾ 0,

0

for any φ ∈ 𝒜P p,k ([0, ∞) : X). Then Λ1 is well-defined and maps 𝒜P p,k ([0, ∞) : X) into itself.

3.11 Notes and appendices | 295

Proof. The function φ ∈ 𝒜P p,k ([0, ∞) : X) can be written as φ = ψ + θ, where ψ ∈ 𝒫p,k (ℝ : X) and θ ∈ C0 ([0, ∞) : X). Furthermore, we can write (Λ1 φ)(t) = Ψ(t) + Θ(t), t ⩾ 0, where t

Ψ(t) := ∫ Rγ (t − s)A0 ψ(⌊s⌋) ds,

t∈ℝ

−∞

and

t

0

Θ(t) := ∫ Rγ (t − s)A0 θ(⌊s⌋) ds − ∫ Rγ (t − s)A0 ψ(⌊s⌋) ds, 0

t ⩾ 0.

−∞

Using the proof of [386, Lemma 3.1], it can be easily checked that Θ ∈ C0 ([0, ∞) : X). Hence, it is enough to prove that Ψ ∈ 𝒫p,k (ℝ : X). It can be simply proved that Ψ ∈ Cb (ℝ : X), so that the final conclusion follows by using the fact that p ∈ ℕ and the following simple computation: t+p

t

Ψ(t + p) = ∫ Rγ (t + p − s)A0 ψ(⌊s⌋) ds = ∫ Rγ (t − s)A0 ψ(⌊s + p⌋) ds −∞

−∞

t

t

−∞

−∞

= ∫ Rγ (t − s)A0 ψ(⌊s⌋ + p) ds = eikp ∫ Rγ (t − s)A0 ψ(⌊s⌋) ds = eikp Ψ(t), which is valid for any t ∈ ℝ. Lemma 3.11.7. Let p ∈ ℕ, let k ∈ ℝ, and let g : ℝ × X → X be a bounded mapping such that: (i) For all (t, x) ∈ ℝ × X, we have g(t + p, eikp x) = eikp g(t, x); (ii) There exists L > 0 such that for all (t, x) ∈ ℝ×X, we have ‖g(t, x)−g(t, y)‖ ⩽ L‖x −y‖; (iii) The mapping t 󳨃→ g(t, ψ(⌊t⌋)), t ∈ ℝ is measurable for all ψ ∈ 𝒫p,k ([0, ∞) : X), and the mapping t 󳨃→ g(t, θ(⌊t⌋)), t ⩾ 0 is measurable for all θ ∈ 𝒜P p,k ([0, ∞) : X). Define

t

(Λ2 φ)(t) := ∫ Rγ (t − s)g(s, φ(⌊s⌋)) ds,

t ⩾ 0,

0

for any φ ∈ 𝒜P p,k ([0, ∞) : X). Then the nonlinear operator Λ2 is well-defined and maps 𝒜P p,k ([0, ∞) : X) into itself. Theorem 3.11.8. Let p ∈ ℕ, let k ∈ ℝ, and let g : ℝ × X → X be a bounded mapping such that conditions (i)–(iii) of Lemma 3.11.7 hold. If A0 ∈ L(X) and ∞

(‖A0 ‖ + L) ∫ Rγ (t) dt < 1, 0

then equation (238) has a unique asymptotically Bloch-periodic solution.

296 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Assume now that γ ∈ (0, 1] and a closed linear operator 𝒜 = A generates an exponentially decaying C-regularized semigroup (T(t))t⩾0 , where C ∈ L(X) is injective (with the meaning clear and a tolerable abuse of notation). We introduce the operator families Tγ,ν (⋅), Sγ (⋅), Pγ (⋅) and Rγ (⋅) as before. In the newly arisen situation, the notion of a mild solution of (238) is introduced as follows. Definition 3.11.9. Let γ ∈ (0, 1], and let u0 ∈ R(C). A function u ∈ C([0, ∞) : X) is said to be a mild solution of the fractional differential inclusion (238) iff g(⋅, u(⌊⋅⌋)) ∈ L1loc ([0, ∞) : X), t

∫ Rγ (t − s)[A0 u(⌊s⌋) + g(s, u(⌊s⌋))] ds ∈ R(C),

t>0

(241)

0

and

t

u(t) = Sγ (t)C −1 u0 + C −1 ∫ Rγ (t − s)[A0 u(⌊s⌋) + g(s, u(⌊s⌋))] ds,

t > 0.

(242)

0

In the case γ ∈ (0, 1), we can use again the argumentation contained in the proof of [396, Lemma 4.1] to see that any classical solution of (238) has to satisfy equation (240), provided that the mapping g(⋅, u(⌊⋅⌋)) is Laplace transformable and u0 ∈ R(C). In the case γ = 1, we can prove the following proposition: Proposition 3.11.10. Suppose that u0 ∈ R(C) and g(⋅, u(⌊⋅⌋)) ∈ L1loc ([0, ∞) : X). If u(⋅) is a classical solution of (238), then u(⋅) is a mild solution of (238). Proof. Let t > 0 be fixed. Set v(s) := T(t −s)u(s), s ∈ [0, t]. Function v(⋅) is differentiable for a. e. s ∈ [0, t] due to the fact that u(⋅) is a classical solution of (238); furthermore, since A generates (T(t))t⩾0 , we have dv(s) = −AT(t − s)u(s) + T(t − s)u󸀠 (s) = −AT(t − s)u(s) + T(t − s)Au(s) ds + T(t − s)A0 u(⌊s⌋) + T(t − s)g(s, u(⌊s⌋)), for a. e. s ∈ [0, t]; see also [386, p. 1728]. Hence, dv(s) = T(t − s)A0 u(⌊s⌋) + T(t − s)g(s, u(⌊s⌋)), for a. e. s ∈ [0, t]. ds

(243)

Integrating (243) over [0, t], we obtain t

Cu(t) − T(t)u(0) = ∫ T(t − s)[A0 u(⌊s⌋) + g(s, u(⌊s⌋))] ds. 0

Since u(0) = u0 ∈ R(C), this immediately implies (241)–(242), finishing the proof.

3.11 Notes and appendices | 297

Moreover, the following analogues of Lemmas 3.11.6–3.11.7 and Theorem 3.11.8 hold: Lemma 3.11.11. Let C −1 A0 ∈ L(X), let p ∈ ℕ, and let k ∈ ℝ. Define the nonlinear operator Λ1 on 𝒜P p,k ([0, ∞) : X) through t

(Λ1 φ)(t) := ∫ Rγ (t − s)C −1 A0 φ(⌊s⌋) ds,

t ⩾ 0,

0

for any φ ∈ 𝒜P p,k ([0, ∞) : X). Then Λ1 is well-defined and maps 𝒜P p,k ([0, ∞) : X) into itself. Lemma 3.11.12. Let p ∈ ℕ, let k ∈ ℝ, and let C −1 g : ℝ × X → X be a bounded mapping such that: (i) For all (t, x) ∈ ℝ × X, we have C −1 g(t + p, eikp x) = eikp C −1 g(t, x); (ii) There exists L > 0 such that for all (t, x) ∈ ℝ × X, we have ‖C −1 g(t, x) − C −1 g(t, y)‖ ⩽ L‖x − y‖; (iii) The mapping t 󳨃→ C −1 g(t, ψ(⌊t⌋)), t ∈ ℝ is measurable for all ψ ∈ 𝒫p,k ([0, ∞) : X), and the mapping t 󳨃→ C −1 g(t, θ(⌊t⌋)), t ⩾ 0 is measurable for all θ ∈ 𝒜P p,k ([0, ∞) : X). Define t

(Λ2 φ)(t) := ∫ Rγ (t − s)C −1 g(s, φ(⌊s⌋)) ds,

t ⩾ 0,

0

for any φ ∈ 𝒜P p,k ([0, ∞) : X). Then the nonlinear operator Λ2 is well-defined and maps 𝒜P p,k ([0, ∞) : X) into itself. Theorem 3.11.13. Let p ∈ ℕ, let k ∈ ℝ, and let C −1 g : ℝ × X → X be a bounded mapping such that conditions (i)–(iii) of Lemma 3.11.12 hold. If C −1 A0 ∈ L(X) and ∞

󵄩 󵄩 (󵄩󵄩󵄩C −1 A0 󵄩󵄩󵄩 + L) ∫ Rγ (t) dt < 1, 0

then equation (238) has a unique asymptotically Bloch-periodic solution. Remark 3.11.14. The boundedness of mapping C −1 g(⋅, ⋅) is taken only for the sake of convenience. In Theorem 3.11.13, we can assume that this mapping is Stepanov-like bounded, in a certain sense. A similar statement holds in the case of consideration of Theorem 3.11.8. Remark 3.11.15. Condition (iii) of Lemma 3.11.12 automatically holds if C −1 g(⋅, ⋅) is continuous. A similar statement holds in the case of Lemma 3.11.7.

298 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Taking into account the computation from [329, Lemma 6], it seems very reasonable to define the notion of a mild solution of the following nonlinear fractional Cauchy inclusion with piecewise constant argument: γ

Dt,+ u(t) ∈ −𝒜u(t) + A0 u(⌊t⌋) + g(t, u(⌊t⌋)), t ∈ ℝ,

(244)

where 𝒜 is an MLO satisfying condition (P), A0 ∈ L(X) and g : [0, ∞) × X → X is a function given by t

u(t) := ∫ Rγ (t − s)[A0 u(⌊s⌋) + g(s, u(⌊s⌋))] ds,

t ∈ ℝ.

−∞

Therefore, the results of W. Dimbour, V. Valmorin [386] and M. F. Hasler, G. M. N’Guérékata [218] enable one to analyze the existence and uniqueness of Bloch-periodic solutions of nonlinear fractional Cauchy inclusion (244), as well. In [218], M. F. Hasler and G. M. N’Guérékata have investigated the existence and uniqueness of (asymptotically) Bloch-periodic solutions of some semilinear integrodifferential equations in Banach spaces. Applications have been given in the study of the following semilinear integro-differential equation: t

u󸀠 (t) = Au(t) + ∫ a(t − s)Au(s) ds + f (t, Du(t)),

t ∈ ℝ,

−∞

where A is a closed linear operator satisfying certain conditions, D ∈ L(X), a ∈ L1loc ([0, ∞)) and f (⋅, ⋅) obeys some properties. We would like to note that the assertions of [218, Lemma 4.1, Theorem 4.2] hold even if the considered operator families are not strongly continuous at the point t = 0. We can simply apply this observation in the analysis of existence and uniqueness of Bloch-periodic solutions of the abstract Cauchy–Volterra inclusion (4) and its semilinear analogues. Generalized C (n) -almost automorphic functions and applications to abstract inhomogeneous Cauchy problems. The main aim of this part is to briefly explain how one can introduce various types of generalized C (n) -almost automorphic functions and enquire into generalized C (n) -almost automorphic properties of finite and infinite convolution products; for the sake of simplicity, we will not consider here the spaces of weighted C (n) -pseudo-almost automorphic (periodic) functions. These results, appearing in this monograph for the first time, can be easily incorporated in the analysis of existence and uniqueness of generalized C (n) -almost automorphic solutions of abstract inhomogeneous Cauchy problems. We start by quoting the following well-known definitions from the existing literature.

3.11 Notes and appendices | 299

Definition 3.11.16. Let n ∈ ℕ. (i) It is said that the function f (⋅) ∈ X ℝ is C (n) -almost automorphic, f ∈ C (n) −AA(ℝ : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AA(ℝ : X). (ii) It is said that the function f (⋅) ∈ X [0,∞) is asymptotically C (n) -almost automorphic, f ∈ C (n) − AAA([0, ∞) : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AAA([0, ∞) : X). Definition 3.11.17. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (ℝ : X). (i) It is said that the function f (⋅) is Stepanov-p-C (n) -almost automorphic, f ∈ C (n) − AASp (ℝ : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AASp (ℝ : X). (ii) It is said that the function f (⋅) is asymptotically Stepanov-p-C (n) -almost automorphic, f ∈ C (n) − AAASp ([0, ∞) : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AAASp ([0, ∞) : X). The symbol AASbp (ℝ : X) (AAASbp ([0, ∞) : X)) denotes the space consisting of all essentially bounded functions from AASp (ℝ : X) (AAASp ([0, ∞) : X)). Definition 3.11.18. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (ℝ : X). (i) It is said that the function f (⋅) is Stepanov-p-Cb(n) -almost automorphic, f ∈ Cb(n) − AASp (ℝ : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AASbp (ℝ : X). (ii) It is said that the function f (⋅) is asymptotically Stepanov-p-Cb(n) -almost automorphic, f ∈ Cb(n) − AAASp ([0, ∞) : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ AAASbp ([0, ∞) : X). For example, the function ∞

sin nt , 4 n=1 n

f (t) = ∑

t ∈ ℝ,

which has been already considered, is C (2) -almost periodic but not C (3) -almost automorphic. Furthermore, for any real-valued function g ∈ C (3) − AA(ℝ : ℂ) satisfying inft∈ℝ g 󸀠󸀠󸀠 (t) > 0, we have that the function ∞

g(nt) , 4 n=1 n

f (t) = ∑

t ∈ ℝ,

belongs to the space C (2) -AAS1 (ℝ : ℂ)∖C (3) -AAS1 (ℝ : ℂ); see, e. g., [155, Example 2.23]. A similar line of reasoning as in the almost periodic case shows that the following result on the convolution invariance of space Cb(n) − AAS1 (ℝ : X) holds. Proposition 3.11.19. Let n ∈ ℕ, and let f ∈ Cb(n) − AAS1 (ℝ : X), resp. f ∈ AASb1 (ℝ : X). Then, for every g ∈ L1 (ℝ), we have that (g ∗f )(⋅) := ∫ℝ f (⋅−y)g(y) dy ∈ Cb(n) −AAS1 (ℝ : X), resp. (g ∗ f )(⋅) ∈ AASb1 (ℝ : X).

300 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations In the following definition, we introduce the notion of a Weyl-p-C (n) -almost automorphic function. Definition 3.11.20. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (ℝ : X). It is said that the function f (⋅) is Weyl-p-C (n) -almost automorphic, f ∈ C (n) − W p AA(ℝ : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ W p AA(ℝ : X). This class is very difficult to work with since the infinite convolution product does not map the space W p AA(ℝ : X) into itself. Because of this, we deeply believe that the class of Besicovitch-p-C (n) -almost automorphic functions is much more appropriate for applications. Definition 3.11.21. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (ℝ : X). It is said that the function f (⋅) is Besicovitch-p-C (n) -almost automorphic, f ∈ C (n) − Bp AA(ℝ : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Bp AA(ℝ : X). In order to analyze Besicovitch-p-C (n) -almost automorphy of infinite convolution products, it will be crucial to introduce the following subclass of Besicovitch-C (n) almost automorphic functions. Definition 3.11.22. Let 1 ⩽ p < ∞, let n ∈ ℕ, and let f ∈ Lploc (ℝ : X). It is said that the function f (⋅) is Besicovitch-p-Cb(n) -almost automorphic, f ∈ Cb(n) − Bp AA(ℝ : X) for short, iff for each k = 0, 1, . . . , n, we have that f (k) ∈ Bp AA(ℝ : X) ∩ L∞ (ℝ : X). We have the following (see also Proposition 3.5.1): Proposition 3.11.23. Suppose that n ∈ ℕ and (R(t))t>0 ⊆ L(X, Y) is a strongly continu∞ ous operator family such that ∫0 ‖R(s)‖ ds < ∞. If g : ℝ → X is in class C (n) − AA(ℝ : X), then the function G(⋅), given by (68), is in the same class, as well. The proof of following analogues of Propositions 3.5.1 and 2.13.18 can be deduced along the same lines. Proposition 3.11.24. (i) Suppose that n ∈ ℕ and (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family ∞ satisfying ∫0 (1+s)‖R(s)‖ ds < ∞. If g : ℝ → X is in class Cb(n) −B1 AA(ℝ : X), then the function G(⋅), given by (68), is in the same class, as well. Furthermore, G ∈ C n (ℝ : X). (ii) Suppose that n ∈ ℕ, and (R(t))t>0 ⊆ L(X, Y) is a strongly continuous operator family ∞ satisfying that ∫0 (1 + s)‖R(s)‖ ds < ∞. If g : ℝ → X is in class Cb(n) − B1 AA(ℝ : X), as well as q(k) ∈ L1loc ([0, ∞) : X) is Besicovitch-1-vanishing for 0 ⩽ k ⩽ n, then the function F(⋅), given by (119), is in class f ∈ C (n) − B1 AA([0, ∞) : X), as long as condition (123) holds.

Proposition 2.13.19 admits a reformulation in automorphic context, too. The proof is similar and therefore omitted.

3.11 Notes and appendices | 301

Proposition 3.11.25. Suppose that n ∈ ℕ, and (R(t))t>0 ⊆ L(X, Y) is a strongly contin∞ uous operator family satisfying that ∫0 (1 + s)‖R(s)‖ ds < ∞. If g : ℝ → X is in class

Cb(n) − B1 AA(ℝ : X), as well as q(k) ∈ L1loc ([0, ∞) : X) is Besicovitch-1-vanishing for 0 ⩽ k ⩽ n, then the function F(⋅), given by (119), is in class f ∈ C (n) − B1 AAA([0, ∞) : X), as long as (R(t))t>0 is (n − 1)-times continuously differentiable for t > 0 and R(k−1−j) (⋅) is pointwise in class B1 (I : Y) for any 0 ⩽ k ⩽ n and 0 ⩽ j ⩽ k − 1 such that (g + q)(j) (0) ≠ 0. Regarding the convolution invariance of space Cb(n) − B1 AA(ℝ : X), we can state the following result; cf. also Proposition 3.1.6. Theorem 3.11.26. Let n ∈ ℕ, and let f ∈ Cb(n) − B1 AA(ℝ : X). Then, for every g ∈ L1 (ℝ) with compact support, we have that g ∗ f ∈ Cb(n) − B1 AA(ℝ : X). Finally, it should be mentioned that we can pose the problem of extending the assertions of Propositions 3.11.19, 3.11.24–3.11.25 and Theorem 3.11.26 for the exponent p > 1 of almost automorphicity. Massera type theorems for almost periodic and almost automorphic type solutions of abstract inhomogeneous Cauchy problems. A fundamental and landmark result of J. L. Massera [323] says that, in the finite-dimensional case, the equation u󸀠 (t) = A(t)u(t) + f (t),

t ∈ ℝ,

(245)

where A(t) is a linear 1-periodic operator for every t ∈ ℝ, has a 1-periodic solution iff it has a bounded solution. This result has been carried over to the infinite-dimensional case in a series of research papers not cited here. Recently, there has been a growing interest in the harmonic analysis of bounded solutions of (245), provided that f (⋅) is an almost periodic or an almost automorphic type function. For example, T. Naito, N. V. Minh and J. S. Shin [332] have raised the following problem: Let (U(t, s))t⩾s be a 1-periodic evolution system on a Banach space X, which means that U(t + 1, s + 1) = U(t, s), t ⩾ s. Let (245) have a bounded (uniformly continuous) solution u(⋅), with forcing term f (⋅) being almost periodic. In which cases, (245) has an almost periodic solution u1 (⋅), possibly different from u(⋅), such that eisp(u1 ) ⊆ eisp(f ) ? Here, sp(f ) denotes the Carleman spectrum of function f (⋅) ∈ L∞ (ℝ : X), defined as the union of all real numbers ζ such that the mapping (Fourier–Carleman transform of f (⋅)) ∞

∫ e−λt f (t) dt, f ̂(λ) := { 0∞ ∫0 eλt f (−t) dt,

Re λ > 0, Re λ < 0,

has no holomorphic extension to a neighborhood of iζ . It is well known that the Carleman spectrum of function f (⋅) coincides with the Beurling spectrum of function f (⋅),

302 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations defined by sp(f ) := {ζ ∈ ℝ : ∀ε > 0 ∃g ∈ L1 (ℝ) supp(ℱ g) ⊆ (ζ − ε, ζ + ε), g ∗ f ≠ 0}, as well as that the following statements hold true (see, e. g., [333]): Let f , gn ∈ BUC(ℝ : X) such that gn → f , n → ∞ uniformly on ℝ. Then: (i) sp(f ) is a closed subset of ℝ; (ii) sp(f ) = sp(αf ), α ∈ ℂ ∖ {0}; (iii) sp(f ) = sp(f (⋅ + h)), h ∈ ℝ; (iv) sp(f ) ⊆ ⋃n∈ℕ sp(gn ); (v) If f ∈ AP(ℝ : X), then sp(f ) = σ(f ), where σ(f ) denotes the Bohr spectrum of f (⋅); (vi) If A is a closed linear operator with domain and range contained in X, as well as Af ∈ BUC(ℝ : X), then sp(Af ) ⊆ sp(f ); (vii) sp(ψ ∗ f ) ⊆ sp(f ) ∩ supp(ℱ ψ); (viii) If Y is another Banach space and K ∈ BV(L(Y, X)), then sp(dK ∗ f ) ⊆ sp(f ); (ix) If f (⋅) is uniformly continuous and sp(f ) is discrete, then f (⋅) is almost periodic; (x) If sp(f ) is finite, then f (⋅) is a trigonometric polynomial. The monodromy operator associated to an evolution system (U(t, s))t⩾s is defined by Pt := U(t, t − 1), t ∈ ℝ. In the case that (U(t, s))t⩾s is 1-periodic, we have that Pt = P1 , t ∈ ℝ; particularly, if A(t) ≡ A is the generator of a strongly continuous semigroup (T(t))t⩾0 , then P1 = T(1). Let S1 denote the unit circle in the complex plane, and let σS1 (P) denote the part of σ(P) in S1 . In [332, Corollaries 4.2–4.3], the following assertions regarding the autonomous case have been proved: Assume that f (⋅) is almost periodic, σ(f ) and σS1 (P) satisfy the spectral separation condition (see [332] for the notion). Assume, further, that σ(f ) is countable and X does not contain any subspace isomorphic to c0 . If there exists a bounded uniformly continuous solution u(⋅) of (245), then there exists an almost periodic solution u1 (⋅) of (245) such that σ(u1 ) = σ(f ). Furthermore, if σS1 (P) is at most countable and there exists a bounded uniformly continuous solution u(⋅) of (245), then it has to be almost periodic. Massera type theorems for almost automorphic solutions of abstract differential equations have been investigated by J. Liu, G. M. N’Guérékata and N. V. Minh in [307], where the authors have examined both finite- and infinite-dimensional case. In [307, Theorem 4.2], the authors have shown that the compactness of monodromy operator P and the assumption f ∈ AA(ℝ : X) together imply that (245) has a mild solution u1 (⋅) with eisp(u1 ) ⊆ eisp(f ) provided that it has a bounded mild solution u(⋅) on [0, ∞). For further information about Massera type theorems for almost automorphic solutions of differential equations, we refer the reader to the papers by S. Murakami, T. Naito, N. V. Minh [331], T. Naito, N. V. Minh, J. S. Shin [333], Z. Xia, M. Fan [401] and references cited therein. Concerning the classical Bohr–Neugebauer result on the existence of almost periodic solutions for inhomogeneous linear almost periodic differential equations and

3.11 Notes and appendices | 303

various strengthenings, we refer the reader to the papers by B. Basit, H. Güenzler [47], D. N. Cheban [91], K. Ezzinbi, G. M. N’Guérékata [182] and L. Radová [352]. Almost periodic and almost automorphic sequences. Denote by l∞ (ℕ : X) the Banach space of all bounded X-valued sequences equipped with the sup-norm. By c0 (ℕ : X) we denote the closed subspace of l∞ (ℕ : X) consisting of sequences (xn )n∈ℕ such that limn→+∞ xn = 0. It is said that an X-valued sequence (xn )n∈ℕ is (Bohr) almost periodic iff for each ε > 0, there exists a natural number N0 (ε) such that among any N0 (ε) consecutive natural numbers, there exists at least one natural number τ ∈ ℕ such that ‖xn+τ − xn ‖ ⩽ ε,

n ∈ ℕ.

As in the case of functions, this number is said to be an ε-period of sequence (xn )n∈ℕ . The Bochner almost periodicity of X-valued sequences can be also introduced and after that we can prove that this concept is equivalent to the concept of Bohr almost periodicity of X-valued sequences, as expected. Any almost periodic X-valued sequence has to be bounded. An X-valued sequence (xn )n∈ℕ is said to be asymptotically almost periodic iff there exist an almost periodic X-valued sequence (xn )n∈ℕ and a sequence (yn )n∈ℕ ∈ c0 (ℕ : X) such that xn = yn + zn , n ∈ ℕ. The sum representing (xn )n∈ℕ in such a way has to be direct. Composition theorems for almost periodic sequences can be clarified, as it has been for almost periodic functions before. An X-valued sequence (xn )n∈ℤ is said to be almost automorphic iff for every sequence (h󸀠k )k∈ℤ there exists a subsequence (h󸀠k )k∈ℤ of it and an X-valued sequence (yn )n∈ℤ such that lim x n→∞ n+hk

= yn ,

n ∈ ℤ and

lim y n→∞ n−hk

= xn ,

n ∈ ℤ.

An X-valued sequence (xn )n∈ℤ is said to be asymptotically almost automorphic iff there exist an almost automorphic X-valued sequence (xn )n∈ℤ and a sequence (yn )n∈ℕ ∈ c0 (ℕ : X) such that xn = yn + zn , n ∈ ℕ. As in the almost periodic case, the sum representing (xn )n∈ℕ has to be direct. Composition theorems for almost automorphic sequences can be set up, too. For more details about almost periodic and almost automorphic sequences, as well as almost periodic and almost automorphic type solutions of abstract differences equations, see [23, 114, 144, 146, 165, 192, 226, 229, 330] and [369]. Almost periodic and almost automorphic stochastic differential equations. In this book, we will not consider stochastic differential equations in Banach and Hilbert spaces. For more details on the almost periodic and almost automorphic type solutions to stochastic differential equations, we refer the reader to the papers by H. Bezandry, T. Diagana [58–60], D. Cheban, Z. Liu [92], G. Da Prato, C. Tudor [126], M. Fu [194] and references cited therein.

304 | 3 Almost automorphic type solutions of abstract Volterra integro-differential equations Almost periodic and almost automorphic nonlinear differential equations with monotone operators. The analysis of abstract differential equations with maximaly monotone or dissipative nonlinear operators is also outside the scope of this book. Mention should be made of [35], where M. Ayachi, J. Blot and P. Cieutat have examined the existence and uniqueness of Besicovitch almost periodic solutions for the second-order nonlinear differential equation u󸀠󸀠 (t) = f (u(t)) + e(t),

t ∈ ℝ,

on a Hilbert space H, where the vector field f : H → H is monotone, continuous and the forcing term e : ℝ → H is almost periodic. For some other references in this direction, one may refer, e. g., to P. Cieutat, K. Ezzinbi [100], E. Ait Dads, P. Cieutat, S. Fatajou [14] and E. Ait Dads, P. Cieutat, L. Lhachimi [15, 16]. It is also worthwhile to mention that A. Shirikyan and L. Volevich [367] have examined the existence and uniqueness of bounded or almost periodic solutions for nonlinear hyperbolic equations of higher order, as well.

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Index ϵ-antiperiod 182 ϵ-period 25 ζ -times integrated C-resolvent propagation family for (173) 211 ζ -times integrated C1 -resolvent propagation family 226 (a, k)-regularized C-resolvent family – bounded 19 – exponentially bounded 19 (a, k)-regularized (C1 , C2 )-existence and uniqueness family 18 (a, k)-regularized C1 -existence family 18, 60 (a, k)-regularized C2 -uniqueness family 18, 52 abstract Cauchy problem – (ACP)1 88 – (ACP)2 89 – first order degenerate 46 – second order degenerate 46 abstract Cauchy problems of third order 70, 86 abstract degenerate Volterra inclusion 17 abstract higher-order Cauchy problems with integer order derivatives 187 abstract multi-term fractional differential inclusions 209 abstract two-term fractional differential equations 282 abstract two-term fractional differential equations with nonlocal conditions 283 abstract two-term fractional differential inclusions XX almost periodic Colombeau generalized functions XXXI Arendt–Widder theorem 12 asymptotic expansions 15, 16 asymptotical almost automorphy 194 asymptotical compact almost automorphy 194 Banach space – ANP(I : X ) 184 – AP(I : X ) 26 – c0 28, 47, 50 – weakly sequentially complete 47, 76 Besicovitch almost automorphy 197 Beurling spectrum 301 Beurling vector-valued ultradistributions 172 Bloch-periodicity 292

Bochner’s criterion 27 Bohr spectrum 27 Bohr transform 26 Bohr–Fourier coefficient 26, 64 bounded distribution tending to zero at plus infinity 262 bounded ultradistribution tending to zero at plus infinity 266 bounded vector-valued ultradistributions 174 C-distribution cosine function 88 – X̃ -asymptotical almost periodicity 97 – X̃ -(weakly) almost periodicity 90 – exponential 89 – generator 89 C-distribution semigroup 87 – X̃ -asymptotical almost periodicity 97 – X̃ -(weakly) almost periodicity 90 – exponential 89 – generator 87 – weak almost periodicity 97 C-regularized semigroups 281, 296 C-resolvent propagation family for (173) 211 C-resolvent set XII, 1 C1 -existence propagation family 226 calculus on time scales 189 Carleman spectrum 28, 301 class – (P1)-X 10 complex inversion theorem 12 composition principles 79, 139, 201, 202, 208 condition – Doss 178 – (F) XXVI – (H1) 158, 247 – (H2) 159, 247 – (H3) 161 – (H3e) 160 – (H4) 165 – (M.1) 172, 258 – (M.2) 172, 258 – (M.3) 172 – (M.3󸀠 ) 172, 258 – (M.3) 258 – (P) XXIV, 9, 82, 100 – (P1) 10 – (PW) XXIV, 9

326 | Index

– (Q)󸀠 65 – (Q)󸀠󸀠 136 continuous linear mapping 1 convolution invariance 27, 206 core 1 corrective term 29 d’Alambert functional equation 49 degenerate C-regularized cosine functions 49, 72 degenerate C-regularized groups 47 degenerate C-regularized semigroups 46, 72 delta differentiability 189 delta distribution 2 delta integral 189 delta-derivative 189 differential operators in Lp -spaces 281 Dirac distribution 86 Dirichlet Laplacian 275 dual space 1 elastic vibrations of flexible structures 70 equation – backward Poisson heat XX – Barenblatt–Zheltov–Kochina 276 – Benney–Luke 279 – Boussinesq–Love XL, 277 – damped Poisson-wave 286 – fractional damped Poisson-wave 287 – fractional Poisson semilinear heat 281 – multi-term fractional integro-differential Poisson heat 284 – Poisson heat XX – semilinear Poisson heat XLI, 280 evolution system 158 exponential vectors 74 fluid filtration 276 Fourier–Carleman transform 301 fractional calculus 13 fractional derivatives – Caputo 13, 101 – Riemann–Liouville 14 – Weyl–Liouville 14 fractional differential equations 13 fractional nonlinear differential inclusions with piecewise constant argument 292 fractional oscillation resolvent families 218 fractional relaxation inclusions with Caputo derivatives 99

fractional semilinear equations with higher order differential operators in Hölder spaces XL fractional Sobolev inclusions 17 fractionally integrated C-cosine function 87 fractionally integrated C-resolvent propagation family for (173) 211 fractionally integrated C-semigroup 87 function – absolutely continuous 4 – almost anti-periodic 182 – almost automorphic 193 – almost periodic 25 – anti-periodic 183 – associated of (Mp ) 172 – asymptotically almost anti-periodic 187 – asymptotically C (n) -almost automorphic 299 – asymptotically Stepanov almost anti-periodic 187 – asymptotically Stepanov almost periodic 31 – asymptotically Stepanov C (n) -almost automorphic 299 – asymptotically Stepanov C (n) -almost periodic 35 – asymptotically Stepanov-p-Cb(n) -almost automorphic 299 – asymptotically Stepanov-p-Cb(n) -almost periodic 36 – Besicovitch almost periodic 142 – Besicovitch vanishing 144 – Besicovitch–Doss almost periodic 143 – Besicovitch–Doss-p-C (n) -almost periodic 157 – Besicovitch–Doss-p-Cb(n) -almost periodic 157 – Besicovitch-p-C (n) -almost automorphic 300 – Besicovitch-p-C (n) -almost periodic 157 – Besicovitch-p-Cb(n) -almost automorphic 300 – Beurling ultradifferentiable 172 – C (n) -almost automorphic 299 – C (n) -almost periodic 34 – compactly almost automorphic 193 – completely monotonic 16 – Doss-p-almost periodic 151 – equi-Besicovitch-p-C (n) -almost automorphic 300 – equi-Besicovitch-p-Cb(n) -almost automorphic 300 – equi-W p -normal 38 – equi-Weyl-p-almost periodic 36 – equi-Weyl-p-C (n) -almost automorphic 300

Index | 327

– equi-Weyl-p-C (n) -almost periodic 130 – equi-Weyl-p-Cb(n) -almost periodic 132 – Gamma 1 – Mittag-Leffler 15 – of bounded variation 4 – Roumieu ultradifferentiable 172 – scalarly Stepanov almost periodic 33 – Stepanov bounded 31 – Stepanov C (n) -almost automorphic 299 – Stepanov C (n) -almost periodic 35 – Stepanov-p-Cb(n) -almost automorphic 299

– Stepanov-p-Cb(n) -almost periodic 35 – two-parameter – almost periodic 29 – asymptotically almost periodic 29 – asymptotically Stepanov almost periodic 33 – pseudo-almost periodic 139 – Stepanov almost periodic 33 – uniformly almost periodic 26 – W p -normal 38 – weakly almost in the sense of Eberlein 28 – weakly almost periodic 26 – Weyl-p-almost periodic 36 – Weyl-p-C (n) -almost automorphic 300 – Weyl-p-C (n) -almost periodic 130 – Weyl-p-Cb(n) -almost periodic 132 – Weyl-p-vanishing 39 – Wright 15, 16

generalizations of vector-valued almost periodic ultradistributions 178 Gevrey sequence 172 Green’s function 158 Hankel path 16 Haraux’s criterion 28 harmonious set 58 heat conduction XIX Hölder inequality 66, 82 Hölder space XL hyperbolic evolution system 158 inhomogeneous Poisson heat equation 71 integrated C-semigroups and cosine functions XIX integrated C2 -uniqueness family 51 integrated cosine functions 51 integrated semigroups 51 integration of asymptotically almost periodic functions 221

k-regularized C-propagation family for (173) 210 k-regularized C-resolvent propagation family for (173) 211 k-regularized C1 -existence propagation family 225, 226 k-regularized C1 -existence propagation family for (173) 211 k-regularized C2 -uniqueness propagation family 226 k-regularized C2 -uniqueness propagation family for (173) 211 lacunary sequence 58 Landau inequality 96 Laplace integral – abscissa of convergence 10 Laplacian XIX left-scattered maximum 189 Levitan N-almost periodicity 194 linear relation 6 liquid filtration 279 Maak’s criterion 28 materials with memory XIX measure – locally finite Borel 7 moisture transfer 276 monodromy operator 302 multivalued linear operator – adjoint 7 – C-resolvent 7 – closed 7 – fractional powers 8 – integer powers 6 – inverse 6 – kernel 6 – MLO 6 – product 6 – resolvent equation 8 – restriction 6 – section 7 – sum 6 norm 1 operator – adjoint 1 – closed 1 – countable spectrum 61 – forward and backward jump 189

328 | Index

– linear 1 – point spectrum 49 ⊕ (asymptotical) almost periodicity 46 ordinary differential equations in distribution and ultradistribution spaces 271 p-solution 17 part of operator 1 Poisson heat equation with memory 275 pre-solution 17, 18 principal term 29 problem 77, 144 – (DFP)L 17 – (DFP)R 17 – (RP) XIX problem (191) – subcase (SC1) 224 – subcase (SC2) 224 – subcase (SC3) 224 quasi-periodic properties of fractional integrals 191 range 1 Rayleigh problem of viscoelasticity XIX rd-continuity 189 removable singularity at zero XXIV, 9, 216 resolvent set 1 reverse Fatou’s lemma 206, 252 Riemann–Liouville fractional integral 13, 191 right-scattered minimum 189 Roumieu vector-valued ultradistributions 172 satisfactorily uniform set 142 semilinear differential equations with almost sectorial operators in Hölder spaces 279 semilinear evolution equations 168 sequentially complete locally convex space – Hausdorff 1 singular set 13, 60 solution 17, 18 – almost automorphic 238 – almost periodic 83, 84, 168 – asymptotically almost automorphic 240 – asymptotically almost periodic 84, 85, 99, 169 – asymptotically Bloch-periodic 293 – asymptotically Weyl-almost periodic 120 – Besicovitch almost automorphic 247 – classical solution of (DFP)f ,γ 101 – generalized C (n) -almost automorphic 298

– mild 212 – mild solution of (139) 160 – mild solution of (140) 160 – pseudo-almost automorphic 239 – pseudo-almost periodic 138, 141 – strong 17, 212 – weighted pseudo-almost automorphic 246, 247 – Weyl-almost periodic 120 space – locally compact 7 – Schwartz XII, 86 – separable metric 7 – Z2 (A) 89 spectral synthesis 27 spectrum 1 Stepanov almost automorphy 194 Stepanov boundedness 37 Stepanov distance 31 Stepanov metric 30 Stepanov norm 31 subordinated fractional resolvent families 100 subordination principles 62 theorem – bipolar 56 – Bohr–Neugebauer 303 – Hahn–Banach 7 – Liouville 31 – Lusternik 28 – Mackey’s 57 – Massera 302 – Tauberian 101, 117 – the Banach contraction principle 83 – the uniform boundedness 57 – the Weissinger’s fixed point 86 trigonometric polynomial 52 two-parameter almost automorphy 200 two-parameter Besicovitch pseudo-almost automorphy 200 two-parameter generalized almost automorphy 199 two-parameter pseudo-almost automorphy 200 two-parameter Stepanov almost automorphy 201 two-temperature heat conductivity 276 ultradifferential operators 172 unconditional boundedness 28

Index | 329

vector-valued – almost automorphic distributions 256 – almost automorphic ultradistributions 256 – almost periodic ultradistributions 171 – asymptotically almost automorphic distributions 261 – asymptotically almost automorphic ultradistributions 261 – distributions XII, 86, 87 – convolution 87 – integral 88 – Laplace transform 19, 20 – Sobolev space 4, 13 – tempered ultradistributions 173 Volterra integro-differential equations with memory 274 weighted – Besicovitch pseudo-almost automorphic 247 – ergodic components 203

– pseudo-almost automorphy 204 – pseudo-almost periodicity 203 – spaces – 𝕌 203 – 𝕌b 203 – 𝕌b,p 244 – 𝕌∞ 203 – 𝕌∞,p 244 – 𝕌p 244 – 𝕍∞ 205 – Stepanov pseudo almost automorphy 204 – Stepanov pseudo almost periodicity 204 – Weyl pseudo almost automorphy 205 – Weyl pseudo almost periodicity 205 Weyl – almost automorphy 196 – boundedness 37 – distance 31 – norm 31 – p-pseudo ergodic component 45