Fixed Points of Nonlinear Operators: Iterative Methods 9783110667097, 9783110663976

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Haiyun Zhou and Xiaolong Qin Fixed Points of Nonlinear Operators

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Haiyun Zhou and Xiaolong Qin

Fixed Points of Nonlinear Operators |

Mathematics Subject Classification 2010 47H05, 47H06, 47H09, 47H10, 65K15 Authors Prof. Haiyun Zhou Shijiazhuang Mechanical Engineering University Shijia Zhuang China

Prof. Xiaolong Qin Hangzhou Normal University Hangzhou China [email protected]

ISBN 978-3-11-066397-6 e-ISBN (PDF) 978-3-11-066709-7 e-ISBN (EPUB) 978-3-11-066401-0 Library of Congress Control Number: 2019957139 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. © 2020 National Defense Industry Press and Walter de Gruyter GmbH, Berlin/Boston Cover image: stevanovicigor / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Fixed point theory has a long history. In 1895–1900, a French mathematician H. Poincaré attributed the existence problem of periodic solutions of the restricted three-body problem to the existence problem of fixed points of a continuous transformation under certain conditions. This can be viewed as the origin of the notion of fixed points. In 1910, L. E. Brouwer proved that a continuous mapping on a polyhedron in a finitedimensional space has at least one fixed point, which initiated the study of fixed point theory. In 1922, a Polish mathematician S. Banach, based on the Picard iteration method, introduced the well-known contraction mapping principle. In 1930, a Polish mathematician J. Schauder extended the Brouwer fixed point theorem from continuous mappings in finite-dimensional spaces to completely continuous mappings in infinite-dimensional spaces. Then, in 1935, A. Tychonoff extended Schauder fixed point theorem to locally convex topological linear spaces. In 1965, F. E. Browder, W. A. Kirk, and D. Göhde independently established a fixed point theorem for nonexpansive mappings in uniformly convex Banach spaces or reflexive Banach spaces with the normal structure. In 1973, R. H. Martin established a fixed point theorem for continuous pseudocontractive mappings. One year later, K. Deimling improved R.H. Martin’s results. Then, in 1999, C. H. Morales and C. E. Chidume further improved the results of Deimling. In 1976, J. Caristi generalized the well-known contraction mapping principle in complete metric spaces, and he proved an amazing fixed point theorem. It is well known that Caristi fixed point theorem is equivalent to the famous Ekeland variational principle. F. E. Browder, a former president of the American Mathematics Association, always highly appraised the Caristi fixed point theorem as one of the most important results in nonlinear analysis. Under the premise of the existence of a fixed point, the method of constructing the fixed point is an important theme in the research of fixed point theory. Historically, there have been many iteration methods for different mappings, among which the most classical methods are the Picard iterative method, steepest descent methods, the normal Mann iterative method, Ishikawa iterative method, Halpern-type iterative method, Bruck regularization iterative method, the gradient projection method, the extragradient method, hybrid projection method, hybrid steepest descent methods, asymptotic iterative method, Moudafi viscosity iterative method, and Rockafellar approximate proximal method, to name only a few. Mathematicians all over the world apply these iteration methods to successfully construct fixed or zero points of various mappings, and a large number of excellent results with scientific significance and application value have emerged. These achievements are still scattered in mathematics journals at home and abroad, and have not been sorted out yet. This is our original intention of compiling this book. Fixed point iteration methods can be applied to solve practical problems with different physical engineering background, such as reshttps://doi.org/10.1515/9783110667097-201

VI | Preface olution recognition and signal synthesis in signal processing; image restoration and reconstruction in image processing; power control, broadband allocation, and beam imaging in CDMA data networks; image denoising based on wavelet transform; video coding technology; radar antenna mode synthesis; control design of a rocket launch tower; pattern recognition of nuclear submarine, and machine learning and artificial intelligence, etc. These problems can be modeled by convex optimization frameworks, feasible frameworks, or split feasibility frameworks, so they can be solved by fixed point iteration methods or variational inequality iteration methods. We are fully convinced that the fixed point iteration method will be more widely used in the fields of science and technology. Most of this book material is collected, screened, processed, and sorted out from the vast literature, the proofs of many theorems in the book have been refined and simplified by the authors of the book, and some of the results have not yet been published. Systematic collation of the increasingly expanding literature is indeed one of the purposes of this book, but we absolutely made no attempt to exhaustively survey all the research results. The research of iteration methods is gradually developing in depth, and new ideas, methods, and achievements emerge endlessly, changing with each passing day; it is impossible to include all of them in one book. Therefore, we only select the most fundamental and important results so that readers could appreciate excellent works. The book is divided into six chapters. In Chapter 1, we give basic knowledge required for this book. In Chapter 2, we discuss iterative methods for fixed points of nonexpansive mappings in Hilbert spaces. In Chapter 3, we consider the iterative methods of fixed points of pseudocontractive mappings and zeros of monotone mappings in Hilbert spaces. In Chapter 4, we study the iteration methods for fixed points of nonexpansive mappings in Banach spaces. In Chapter 5, we investigate the iterative methods for zeros of accretive operators and fixed points of pseudocontractive mappings in Banach spaces. Finally, in Chapter 6, we introduce three kinds of iteration methods for zeros of maximal monotone operators and their corresponding applications in Banach spaces. To read the book, readers need some prerequisite knowledge of nonlinear functional analysis and convex analysis, especially some knowledge on nonlinear mapping theory and Banach space geometry theory would be helpful. These preliminaries are given directly (without proof) in Chapter 1, and the details where to find proofs are provided in the references. Due to the limitations of our knowledge, and since the proofs of some theorems are modified by the authors of the book, there may be some inevitably mistakes and shortcomings in the book. We sincerely hope that the readers can notify us. Shijiazhuang and Hangzhou, October 2019

Haiyun Zhou Xiaolong Qin

Contents Preface | V 1 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.5 1.6 1.7 1.8 1.8.1 1.8.2 1.8.3 1.8.4 1.8.5 1.8.6 1.9 1.9.1 1.9.2 1.9.3 1.9.4 1.10 1.11

Introduction and preliminaries | 1 Partial order relations, partial order sets, and Zorn lemma | 1 Metric spaces | 2 Topological spaces | 4 Point sets in topological spaces | 4 Topological bases, subbases and neighborhood bases | 8 Directed sets, nets, and convergence of nets | 10 Continuous mappings and homeomorphism mappings | 11 Topological linear spaces | 14 Linear spaces | 14 Normed linear spaces and Banach spaces | 17 Hilbert spaces | 23 Topological linear spaces | 27 Lower semicontinuity, convexity, and conjugate functions | 29 Differential calculus on Banach spaces | 32 Subdifferentials of convex functions | 34 Geometry of Banach spaces | 37 Convexity of Banach spaces | 37 Duality mappings | 40 Differentiability of norms and smoothness of spaces | 43 Geometry constants in Banach spaces | 47 Banach limits | 48 Projection mappings | 49 Some classes of nonlinear mappings | 52 Nonexpansive mappings | 54 Accretive operators | 58 Monotone mappings | 62 Pseudocontractive mappings | 75 Some useful lemmas | 76 Exercises | 77

2

Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces | 81 Basic properties for nonexpansive mappings and their subclasses | 81 Opial conditions and asymptotic centers | 86 The demiclosedness principle and fixed point theorems | 91 Iterative methods of fixed points | 92

2.1 2.2 2.3 2.4

VIII | Contents 2.4.1 2.4.2 2.5 2.6 2.7 2.7.1 2.7.2 2.7.3 2.8 2.9 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.2 3.3 3.4 3.5 3.6 3.6.1 3.6.2 3.6.3 3.7 3.8 4 4.1 4.2 4.3 4.4 4.5 4.6

Weak convergence theorems | 93 Strong convergence theorems | 98 Fixed points of nonexpansive nonself-mappings | 118 Iterative methods for fixed points of quasi-nonexpansive mappings | 123 Applications | 130 Convex minimization problems | 131 Monotone variational inequality problems | 132 Split feasibility problems | 134 Remark | 138 Exercises | 138 Iterative methods for zeros of monotone mappings and fixed points of pseudocontractive mappings in Hilbert spaces | 143 Basic properties of monotone mappings | 143 Local boundedness and hemicontinuity | 143 Characteristic description for monotone mappings | 147 Demiclosedness principle for monotone mappings | 148 Resolvents and Yosida approximations | 151 Criteria of maximal monotone mappings | 157 Acute angle principle for monotone mappings | 162 Monotone variational inequalities | 170 Fixed point theory of pseudocontractive mappings | 184 Iterative methods of fixed points for pseudocontractive mappings and zeros for monotone mappings | 190 Normal Mann iterative method | 190 Ishikawa iterative method | 193 Bruck regularization iterative method | 197 Remarks | 205 Exercises | 205 Fixed point theory and iterative methods for fixed points of nonexpansive mappings in Banach spaces | 209 Several celebrated fixed point theorems | 209 Normal Mann iterative method and Reich’s weak convergence theorem | 211 Halpern iterative method and its strong convergence theorems | 215 Moudafi’s viscosity iterative method | 228 Iterative methods for common fixed points of a family of nonexpansive mappings in Banach spaces | 236 Iterative methods of common fixed points for a nonexpansive semigroup in Banach spaces | 248

Contents | IX

4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.6.1 5.6.2 5.6.3 5.7 5.7.1 5.7.2 5.7.3 5.8 5.9 5.10 5.11 5.12 5.13 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Iterative methods of fixed points for nonexpansive nonself-mappings | 253 Remarks | 259 Exercises | 259 Iterative methods for zeros for accretive operators and fixed points of pseudocontractive mappings in Banach spaces | 263 Characterizations of accretive operators | 263 Nonlinear semigroups of ω-type | 264 Zero point theorems of accretive operators | 265 Demiclosedness of accretive operators | 269 The existence and convergence of paths for accretive operators | 272 Iterative methods of zero points for accretive operators | 277 The steepest decent method | 277 The Bruck regularization iterative method | 280 The iterative methods based on APPA | 288 Iterative methods for variational inequalities with accretive operators | 296 Two kinds of variational inequality problems | 296 Tools to solve the variational inequality | 299 Strong convergence theorems | 301 Fixed points of strongly pseudocontractive mappings | 312 Demiclosedness principles for pseudocontractive mappings | 315 Fixed point theorems for pseudocontractive mappings | 316 Iterative methods for fixed points of pseudocontractive mappings | 318 Remarks | 327 Exercises | 327 Iterative methods for zeros of maximal monotone operators in Banach spaces | 331 Lyapunov functional and generalized projection | 331 Rockafellar–Mann iterative method and its weak convergence theorem | 335 Rockafellar–Halpern iterative method and its strong convergence theorem | 338 Rockafellar–Haugazeau iterative method and its strong convergence theorem | 342 Minimizers of convex functionals and monotone variational inequalities | 345 Remark | 348 Exercises | 348

X | Contents Bibliography | 353 Index | 359 Nomenclature | 365

1 Introduction and preliminaries The purpose of this chapter is to give some essential definitions, symbols, and terminology. We focus on the geometric theory of Hilbert and Banach spaces. Some fundamental concepts in convex analysis are stated. Some important properties of several classes of nonlinear operators are discussed.

1.1 Partial order relations, partial order sets, and Zorn lemma The purpose of this section is to state the famous Zorn lemma, which is an important tool in many branches of modern mathematics for existence results, such as the Hahn–Banach extension theorem, the maximal extension of monotone or accretive operators, the Kirk fixed point theorem, and some other important results in algebra or topology. In order to state the Zorn lemma, we first review some basic concepts. Definition 1.1.1. Let X be a nonempty set and let R be a relation on X. If R is (i) reflexive, i. e., ⟨x, x⟩ ∈ R, ∀x ∈ X; (ii) antisymmetric, i. e., if ⟨x, y⟩ ∈ R and ⟨y, x⟩ ∈ R, then x = y, ∀x, y ∈ X; (iii) transitive, i,e., if ⟨x, y⟩ ∈ R and ⟨y, z⟩ ∈ R, then ⟨x, z⟩ ∈ R, ∀x, y, z ∈ X, then R is said to be a partial order relation on X, denoted by ⪯. If ⟨x, y⟩ ∈⪯, we denote it as x ⪯ y. A set X with a partial order relation ⪯ is called a partially ordered set, denoted by ⟨X, ⪯⟩. Also x ≺ y if and only if x ⪯ y, and x ≠ y. We say that x is comparable to y if either x ⪯ y or y ⪯ x holds. For a partially ordered set ⟨X, ⪯⟩ and A ⊂ X, if ∀x, y ∈ A, x is comparable to y, then A is a totally ordered set, which is also called a chain. Definition 1.1.2. Let ⟨X, ⪯⟩ be a partially ordered set, A ⊂ X, and y ∈ X. (i) If x ∈ A ⇒ x ⪯ y, ∀x ∈ X, then y is an upper bound of A. (ii) If x ∈ A ⇒ y ⪯ x, ∀x ∈ X, then y is a lower bound of A. Definition 1.1.3. Let ⟨X, ⪯⟩ be a partially ordered set, A ⊂ X, and y ∈ A. (i) If x ∈ A ⇒ y ⪯ x, ∀x ∈ X, then y is the least element of A. (ii) If x ∈ A ⇒ x ⪯ y, ∀x ∈ X, then y is the greatest element of A. (iii) If x ∈ A ∧ x ⪯ y ⇒ x = y, ∀x ∈ X, then y is the minimal element of A. (iv) If x ∈ A ∧ y ⪯ x ⇒ x = y, ∀x ∈ X, then y is the maximal element of A. Remark 1.1.1. (i) A maximal element (or a minimal element) is not necessary the greatest element (or the least element), and a maximal element (or a minimal element) is not necessarily comparable to any element in X. (ii) For a finite partial order ⟨X, ⪯⟩, a maximal element (or a minimal element) must exist, but the greatest element (or the least element) need not exist. https://doi.org/10.1515/9783110667097-001

2 | 1 Introduction and preliminaries (iii) If the greatest element (or the least element) exists, it must be unique. Generally speaking, a maximal element (or a minimal element) is not unique, however, if there is a maximal element (or a minimal element) in X, then it must be the greatest element (or the least element) in X. Lemma 1.1.1 (Zorn Lemma). Let ⟨X, ⪯⟩ be a partially ordered set. If X has upper bounds for every chain, then it has one maximal element.

1.2 Metric spaces In the 1870s, Cantor studied some concepts, which have a close relation with metrics, such as accumulation, interior, exterior, and boundary points, open and closed sets, and so on, in Euclidean spaces. With the development of the classical analysis, Ascoli, Volterra, Alzera, Hadamard, Borel, and many famous mathematicians studied various convergence notions in various function classes. Their convergence analysis is based on the distance between functions. In 1906, Fréchet introduced metric spaces and investigated some basic concepts related to the distance based on Cantor’s ideas. From now on, we use ℝ, ℝ+ , and ℝ to denote (−∞, +∞), [0, +∞), and [−∞, +∞], respectively; ℕ stands for the set of natural numbers. Definition 1.2.1. Let X be a nonempty set. Let d : X × X → ℝ+ be a bivariate function satisfying the following three conditions, ∀x, y, z ∈ X, (d1) d(x, y) ≥ 0 and d(x, y) = 0 ⇔ x = y; (d2) d(x, y) = d(y, x); (d3) d(x, y) ≤ d(x, z) + d(z, y). Then d is called a distance on X, and (X, d) is called a metric space. The open ball Br (x0 ) with center x0 and radius r is defined as follows: Br (x0 ) = {x ∈ X : d(x0 , x) < r}. The closed ball Br [x0 ] with center x0 and radius r is defined as follows Br [x0 ] = {x ∈ X : d(x0 , x) ≤ r}. The sphere Sr (x0 ) with center x0 and radius r is defined as follows: Sr (x0 ) = {x ∈ X : d(x0 , x) = r}. Let A be a subset of X. The diameter of set A is defined by Diam A = sup{d(x, y) : x, y ∈ A}. Recall that A is open if there exists an open ball Brx ⊂ A, where rx > 0, ∀x ∈ A. If the complementary set ℘(A) = X\A of A is an open set, then A is said to be a closed set; A is said to be bounded if there exists a closed ball Br [0] such that A ⊂ Br [0].

1.2 Metric spaces | 3

Let (X, d1 ) and (Y, d2 ) be metric spaces. Let f : X → Y be a mapping. Let Dom(f ) denote the domain of f and let x0 ∈ Dom(f ) be a fixed element. Recall that f is said to be continuous at x0 if ∀ε > 0 there exists δ = δ(ε, x0 ) > 0 such that d2 (f (x), f (x0 )) < ε whenever d1 (x, x0 ) < δ. Recall that f : Dom(f ) ⊂ X → Y is said to be continuous if f is continuous at every point in its domain. Recall that f : Dom(f ) ⊂ X → Y is said to be L-Lipschitz if there exists L > 0 such that d2 (f (x), f (y)) ≤ Ld1 (x, y),

∀x, y ∈ Dom(f ).

Recall that f : Dom(f ) ⊂ X → Y is said to be nonexpansive if the above inequality holds with L = 1; also f : Dom(f ) ⊂ X → Y is said to be contractive if the above inequality holds with L ∈ (0, 1). Furthermore, f : Dom(f ) ⊂ X → Y is said to be Meir–Keeler contractive if, for any ε > 0, there exists δ > 0 such that d1 (x, x0 ) < δ + ε implies d2 (f (x), f (x0 )) < ε,

∀x, y ∈ Dom(f ).

Let {xn } be a sequence in a metric space (X, d). Recall that {xn } is said to be convergent in X if, for any ε > 0, there exists a natural number n0 ≥ 1 such that d(xn , x) < ε, where x a point in X, for all n ≥ n0 . We denote it by xn → x or limn→∞ xn = x. Recall that {xn } is said to be a Cauchy sequence if, for any ε > 0, there exists a natural number n0 such that d(xm , xn ) < ε whenever m, n ≥ n0 . A convergent sequence is a Cauchy sequence and a Cauchy sequence is always bounded. If every Cauchy sequence in X is convergent, then a metric space (X, d) is said to be complete. Theorem 1.2.1 (Cantor Intersection Theorem). Let (X, d) be a complete metric space and let {Fn } be a countable family of decreasing nonempty closed subsets of X such that δ(Fn ) = sup{d(x, y) : x, y ∈ Fn } → 0 as n → ∞. Then ⋂∞ n=1 Fn contains exactly one point. Using Theorem 1.2.1, one can obtain the following celebrated Banach contraction mapping principle. Theorem 1.2.2 (Banach Contraction Mapping Principle, 1922). Let (X, d) be a complete metric space and let f : X → X be a contraction mapping. Then f has a unique fixed point in X, that is, there exists a unique element x∗ ∈ X such that x∗ = f (x ∗ ). Remark 1.2.1. The Banach contraction mapping principle is one of the most significant and fundamental fixed point theorems and it has been widely applied to many scientific fields. There are a number of generalization and extensions of the Banach contraction mapping principle, one of which is the following Meir–Keeler fixed point theorem.

4 | 1 Introduction and preliminaries Theorem 1.2.3 (Meir–Keeler Fixed Point Theorem, 1969). Let (X, d) be a complete metric space, and let f : X → X be a Meir–Keeler contraction mapping. Then f processes a unique fixed point in X. Recall that a function φ : X → ℝ is said to be lower semicontinuous if ∀x ∈ X, {xn } ⊂ X and xn → x imply that φ(x) ≤ lim inf φ(xn ). n→∞

Using Theorem 1.2.1, one can obtain the following celebrated Caristi fixed point theorem. Theorem 1.2.4 (Caristi Fixed Point Theorem, 1976). Let (X, d) be a complete metric space. Let f : X → X be a mapping (not necessarily continuous) and let φ : X → ℝ+ be a lower semicontinuous and bounded below function such that d(x, f (x)) ≤ φ(x) − φ(f (x)),

∀ x ∈ X.

Then f has at least one fixed point.

1.3 Topological spaces The basic concepts in modern analysis are the “limit” and “continuity” of mappings in abstract spaces. It is well known that the concept of “limit” is induced by a “distance” in metric spaces. Based on the “distance”, we have open sets, neighborhoods, interior points, accumulation points, subspaces, etc. Convergence with respect to a distance is essential in modern analysis, however, there exist convergence notions, which cannot be characterized by a distance. Can one directly define open sets, neighborhoods, etc., to obtain the “limit” and “continuity”? The answer is positive. There are a number of methods to induce a topology on sets. In 1914, Hausdorff introduced a topology based on neighborhood systems. Since then, many methods, including open set systems, closed set systems, closure operators, etc., were developed to induce a topology. These topologies are uniform. Based on some basic properties of open sets in metric spaces, the theory of topological spaces can be established.

1.3.1 Point sets in topological spaces Definition 1.3.1. Let X be a nonempty set. Let ζ be a family of subsets of X such that (O1 ) 0, X ∈ ζ ; (O2 ) If G1 , G2 ∈ ζ , then G1 ∩ G2 ∈ ζ ; (O3 ) If Gi ∈ ζ (i ∈ Λ), where Λ is an arbitrary index set, then ⋃i∈Λ Gi ∈ ζ .

1.3 Topological spaces |

5

Then ζ is said to be a topology on X, (X, ζ ) is said to be a topological space, and the elements of ζ are said to be open sets of X. Example 1.3.1. Let (X, d) be a metric space. Then the family of all open sets of X satisfies Axioms (O1 )–(O3 ) in Definition 1.3.1. Hence, every metric space is a topological space. Example 1.3.2. Let X be an arbitrary set and let ζ∞ = {A : A ⊂ X}. Then (X, ζ∞ ) is a topological space and ζ∞ is a “discrete topology”. Let ζ0 = {X, 0}. Then (X, ζ0 ) is also a topological space and ζ0 is called a “dense topology”. This shows that one can induce different topologies on the same set to obtain different topological spaces. Definition 1.3.2. Let ζ1 and ζ2 be two topologies on X. If ζ2 ⊂ ζ1 , then the topology ζ2 is said to be a coarser (weaker or smaller) topology than ζ1 , and ζ1 is said to be a finer (stronger or larger) topology than ζ2 . Definition 1.3.3. Let (X, ζ ) be a topological space and A ⊂ X. Let ζA = {A ∩ G : G ∈ ζ }. Then ζA is a topology on A. And we call ζA a relative topology, and (A, ζA ) a subspace of (X, ζ ). Definition 1.3.4. Let (X, ζ ) be a topological space. Let A ⊂ X and U ⊂ X. If there exists G ∈ ζ such that A ⊂ G ⊂ U, then U is said to be a “neighborhood” of A. We use £(A) to denote all the neighborhoods of A. If A = {x}, U is the neighborhood system of x. In space (X, ζ ), the set of all the neighborhood systems of points is said to be the system of neighborhoods of X, which is denoted by £, that is, £ = {£(x); x ∈ X}. Remark 1.3.1. The neighborhoods of x need not be open subsets containing x. Let X = {a, b, c} and ζ = {0, {a}, X}. Then (X, ζ ) is a topological space. It is obvious that U = {a, b} is the neighborhood of a, but it is not an open set. Indeed, U is also not a neighborhood of b. However, an open set is a neighborhood of its any point. The reverse is also true. Theorem 1.3.1. Let (X, ζ ) be a topological space and let £ be a family of neighborhoods of X, reduced by ζ . Then £ satisfies the following conditions: (i) £(x) ≠ 0, ∀x ∈ X; (ii) x ∈ U, ∀U ∈ £(x); (iii) U ∩ V ∈ £(x), ∀U, V ∈ £(x);

6 | 1 Introduction and preliminaries (iv) V ∈ £(x), ∀V ⊃ U ∈ £(x); (v) For any U ∈ £(x), there exists V ∈ £(x) such that U ⊃ V and V ∈ £(x) for any y ∈ V. Definition 1.3.5. Let (X, ζ ) be a topological space and let F ⊂ X. If ℘F = X\F ∈ ζ , where ℘F is the complementary set of F, then F is closed. Theorem 1.3.2. Let (X, ζ ) be a topological space and let ϑ be the family of all closed sets of X. Then ϑ satisfies the following conditions: (i) 0, X ∈ ϑ; (ii) if F1 , F2 ∈ ϑ, then F1 ∪ F2 ∈ ϑ; (iii) if Fi ∈ ϑ (i ∈ Λ), then ⋂i∈Λ Fi ∈ ϑ. Definition 1.3.6. Let (X, ζ ) be a topological space and let A ⊂ X. If x ∈ A has a neighborhood V of x such that V ⊂ A, then x is an interior point of A. The set of all interior points is called the interior of A, denoted by A0 or Int A; x ∈ X is said to be an exterior point of A if x is an interior point of ℘A. The set of all exterior points is called the external of A, denoted by Ae . Definition 1.3.7. Let (X, ζ ) be a topological space and let A ⊂ X. (i) If x ∈ X, for all U ∈ £(x), one has U ∩ A ≠ 0. Then x is called a closure point of A and the set of all closure points is called the closure of A, denoted by A or cl A. (ii) x ∈ X is said to be an accumulation point of A if ∀U ∈ £(x), U ∩ (A \ {x}) ≠ 0. The set of all accumulation points is called a derived set, denoted by Ad . If A = Ad , then A is a completed set. (iii) x ∈ X is said to be a boundary point of A if ∀U ∈ £(x), U ∩ A ≠ 0 and U ∩ ℘A ≠ 0. The set of all boundary points is called the boundary of A, denoted by 𝜕A. (iv) x is said to be an isolated point of A if there exists U ∈ £(x) such that U ∩ A = {x}. (v) If B ⊂ A, then A is dense in B. If A = X, then A is dense everywhere. (vi) If there exists a countable dense subset in X, then X is separable. Indeed, ℝn , C[a, b], lp , and Lp (p ≥ 1) are all separable. (vii) If Int(A) = 0, then A is a nowhere dense set. (viii) A subset of a topological space X is said to be of the first category in X if it is a union of countably many nowhere dense subsets. A subset of a topological space X is said to be of the second category in X if it is not of the first category in X. It is known that every complete metric space is of the second category. Definition 1.3.8. Let (X, ζ ) be a topological space. Every mapping S : ℕ → X is called a sequence in X, denoted by {xi }, where xi = S(i) ∈ X, i ∈ ℕ. In particular, S(i) = a ∈ X, ∀i ∈ ℕ, where a ∈ X is a fixed point. Then S is a constant sequence, denoted by {a}.

1.3 Topological spaces |

7

Definition 1.3.9. Let {xi } be a sequence in the topological space (X, ζ ). If for every neighborhood U ∈ L(x) of a point x ∈ X, there exists i0 ∈ ℕ such that xi ∈ U, when i ≥ i0 , then xi is said to be the limit of {xi }, denoted by limi xi = x. If {xi } has at least one limit, then {xi } is a convergent sequence. Definition 1.3.10. Let (X, ζ ) be a topological space and let S, S1 : ℕ → X be sequences in X. If there exists a strictly increasing mapping φ : ℕ → ℕ, such that S1 = S ∘ φ, then S1 is said to be a subsequence. Remark 1.3.2. There is a huge difference regarding the convergence of sequences between topological and metric spaces. For example, any sequence is convergent in a dense topological space. This is not true in metric spaces. However, they also have some similarities. Theorem 1.3.3. Let {xi } be a sequence in a topological space (X, ζ ). Then (i) If {xi } is a constant sequence that takes value x ∈ X, then limi xi = x; (ii) If {xi } converges to x ∈ X, then every subsequence of {xi } converges to x. Theorem 1.3.4. Let (X, ζ ) be a topological space and let A ⊂ X. If there exists a sequence {xi } in A\{x} such that {xi } converges to x ∈ X, then x is an accumulation point of A. The reverse may not be true. Definition 1.3.11. (i) If the collection {Gλ : λ ∈ Λ} of X satisfies X = ⋃λ∈Λ Gλ , then {Gλ : λ ∈ Λ} is a cover of X. In particular, if every Gλ is an open set, then {Gλ : λ ∈ Λ} is said to be an open cover of X. (ii) Let A be a subset of a topological space (X, ζ ). We say A is compact if every open cover of A has a finite subcover. (iii) A is said to be countably compact if every countable open cover has a finite subcover. (iv) A is said to be sequentially compact if every infinite sequence has a convergent subsequence. In particular, if A = X, it is said to be a compact space, a countably compact space, and a sequentially compact space, respectively. Sequentially compact and compact spaces are mutually independent notions. Both sequentially compact and compact spaces are countably compact, however, the reverses are not true. Definition 1.3.12. A collection ß of subsets of X is said to have the finite intersection property if the intersection of any finite subcollection of A is nonempty. Theorem 1.3.5. The topological space (X, ζ ) is compact if and only if any collection of closed subsets of X with the finite intersection property has a nonempty intersection.

8 | 1 Introduction and preliminaries A compact subset in a topological space is not necessarily closed, for example, any nonempty subset in the dense topological space is compact, but not necessarily closed. Theorem 1.3.6. Let (X, ζ ) be a topological space and let A, B ⊂ X. Then (i) 0 = 0; (ii) A ⊂ A; (iii) A = A; (iv) A ∪ B = A ∪ B; (v) A = ℘(℘A)∘ ; (vi) A = A ∪ Ad ; (vii) A is closed if and only if A = A; (viii) A is the minimum closed set containing A. Theorem 1.3.7. Let (X, ζ ) be a topological space and let A, B ⊂ X. Then (i) X ∘ = X; (ii) A∘ ⊂ A; (iii) A∘∘ = A; (iv) (A ∩ B)∘ = A∘ ∪ B∘ ; (v) A∘ = ℘(℘A); (vi) A is open if and only if A∘ = A; (vii) A∘ is the maximum open set included in A. Theorem 1.3.8. Let (X, ζ ) be a topological space and let A ⊂ X. Then (i) A∘ ∩ 𝜕A = Ae ∩ 𝜕A = 0; (ii) X = A∘ ∪ Ae ∪ 𝜕A; (iii) A = A∘ ∪ 𝜕A = A ∪ 𝜕A = A ∪ Ad ; (iv) A is closed if and only if 𝜕A ⊂ A; (v) A is open if and only if A ∩ 𝜕A = 0; (vi) A∘ = A\(℘A)d = A\𝜕A; (vii) ℘A = ℘A∘ ; (viii) 𝜕A = A ∪ (℘A) = A\A∘ ; (ix) ℘(𝜕A) = A∘ ∪ (℘A)∘ . 1.3.2 Topological bases, subbases and neighborhood bases Based on open sets in usual topological spaces, the set of real numbers has the celebrated “open set construction theorem”, that is, any open set in ℝ1 is of the form G = ⋃m i=1 (αi , βi ), where m is finite or ∞ and (αi , βi ) are open intervals with empty intersections. This shows that the open interval as a special open set in ℝ1 plays a basic role in the construction of collections of open sets. Usually, it is called a collection

1.3 Topological spaces |

9

of “basic open sets” or “open set bases”. We can also define “topological bases” in a general topological space. Definition 1.3.13. Let (X, ζ ) be a topological space and let B be a family of subsets of ζ . If G = ⋃B∈B B, ∀G ∈ ζ , then B is said to be a “topological base” of (X, ζ ). Let Φ be a nonempty subset of ζ . If the collection of the intersections of any finite elements in Φ is a topological base of ζ , then Φ is said to be a subbase of ζ . Theorem 1.3.9. Let X be a nonempty set and let B be a family of subsets of X. Then B is some topological base on X if and only if B has the following properties: (i) ∀x ∈ X, there exists G ∈ B such that x ∈ G; (ii) If x ∈ G1 ∩ G2 , where Gi ∈ B(i = 1, 2), then there exists G3 ∈ B such that x ∈ G3 ⊂ G1 ∩ G2 . Theorem 1.3.10. A family of sets B ⊂ ζ is a topological base for the given topology ζ if and only if ∀G ∈ ζ , x ∈ G, there exists Gx ∈ B such that x ∈ Gx ⊂ G. Definition 1.3.14. Let (X, ζ ) be a topological space and let B(x) be a family of subsets of £(x). Family B(x) is said to be a “neighborhood base” (or “local base” or “fundamental system of neighborhoods”) if, for all U ∈ £(x), there exists V ∈ B(x) such that V ⊂ U. Definition 1.3.15. Let (X, ζ ) be a topological space. Let £ be a neighborhood system of X and let B be a subfamily of £. This B is said to be a neighborhood base of (X, ζ ) if, ∀U ∈ ζ , x ∈ U, there exists B ∈ B such that x ∈ B ⊂ U. Theorem 1.3.11. Let B be a topological base of a topological space (X, ζ ). Then V ⊂ X is a neighborhood of x if and only if there exists B ∈ B such that x ∈ B ⊂ V. Theorem 1.3.12. Let B be a family of open sets of a topological space (X, ζ ). Then B is a topological base of X if and only if, for all x ∈ X, B(x) = {B ∈ B : x ∈ B} is a neighborhood base of x. Theorem 1.3.13. A family of sets B is some topological base on the set X = ∪{B : B ∈ B} if and only if ∀U, V ∈ B, ∀x ∈ U ∩ V, there exists W ∈ B such that x ∈ W and W ⊂ U ∩ V. Theorem 1.3.14. Let φ be a family of nonempty sets. Then the family of all finite intersections of subsets of φ is a topological base of X = ∪{S : S ∈ φ}. Theorem 1.3.15. Let ζ1 , ζ2 be two topologies on the set X. Also ∀x ∈ X, let B1 (x) and B2 (x) be neighborhood bases of x in (X, ζ1 ) and (X, ζ2 ), respectively. Then ζ2 ⊂ ζ1 if and only if, ∀x ∈ X, ∀B2 ∈ B2 (x), there exists B1 ∈ B1 (x) such that B1 ⊂ B2 . Theorem 1.3.16. Let x ∈ X and B(x) be a family of subsets of X. Then there exists a topology ζ on X such that B(x) is a neighborhood base of x if and only if the following axioms of neighborhood bases hold:

10 | 1 Introduction and preliminaries (i) For B1 , B2 , . . . , Bm ∈ B(x), there exists B ∈ B(x) such that B ⊂ ∩m i=1 Bi ; (ii) If B ∈ B(x), then x ∈ B; (iii) If V ∈ B(x), then there exists W ∈ B(x) such that W ⊂ V, and ∀y ∈ W there exists B ∈ B(y) such that B ⊂ V. 1.3.3 Directed sets, nets, and convergence of nets Definition 1.3.16. Let (S, ⪯) be a partially ordered set. If for all x, y ∈ S, there exists z ∈ S such that z ⪰ x, z ⪰ y, then (S, ⪯) is said to be a directed set. Definition 1.3.17. Let (X, ζ ) be a topological space and let (S, ⪯) be a directed set. Then a mapping from S to X is called a net (or directed point sequence) on X, denoted by {xz }z∈S , where xz ∈ X. Definition 1.3.18. Let (X, ζ ) be a topological space and let (S, ⪯) be a directed set. Let {xz }z∈S be a net on X. If ∃x0 ∈ X, ∀U ∈ £(x0 ), there exists z0 ∈ S such that xz ∈ U whenever z ≥ z0 , then {xz }z∈S is said to be convergent to x0 (net convergence) and x0 is said to be a limit of net {xz }z∈S , denoted by xz → x0 (z ∈ S) or limz∈S xz = x0 . Definition 1.3.19. Let (S, ⪯) be a directed set and let C ⊂ S. If C is also a directed set on “⪯”, then C is said to be a directed subset of S. If ∀δ ∈ S, there exists α ∈ C, such that α ⪰ δ, then C is said to be a cofinal directed subset of S. Definition 1.3.20. Let {xz }z∈S and {yα }α∈Δ be two nets. If there exists J : △ → S such that ∀α ∈ △, yα = xJ(α) and it satisfies the following conditions: (i) ∀α1 , α2 ∈ △, α1 ⪯ α2 ⇒ J(α1 ) ⪯ J(α2 ); (ii) ∀α ∈ S, there exists δ ∈ △, such that J(δ) ⪰ α, then {yα }β∈Δ is called a subnet of {xz }z∈S . Remark 1.3.3. A net on a directed set is not necessarily a subnet. A sequence is always a net, but a net is not necessarily a sequence. A subsequence is always a subnet, but a subnet is not necessarily a subsequence. A sequence may have a subnet which is not its subsequence. Utilizing the concept of net convergence, one can give characterizations of accumulation and closure points, open, closed, and compact sets, as well as many other important concepts in a topological space (X, ζ ). Theorem 1.3.17. Let A be a subset of topological space (X, ζ ) and x0 ∈ X. Then the following statements are true: (i) x0 ∈ Ad if and only if there exists a net {xδ } in A\{x0 } such that xδ → x0 ; (ii) x0 ∈ A if and only if there exists a net {xδ } in A such that xδ → x0 ; (iii) A is open if and only if there does not exist any net converging to a point in A;

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11

(iv) A is closed if and only if for any net {xδ } ⊂ A, if xδ → x, then x ∈ A; (v) A is compact if and only if each net in A has a subnet converging to a point in A.

1.3.4 Continuous mappings and homeomorphism mappings Definition 1.3.21. Let (X, ζ1 ) and (Y, ζ2 ) be two topological spaces. Let f : X → Y be a mapping and let x0 ∈ X. If, for any neighborhood Uy0 ∈ L(y0 ) of y0 = f (x0 ), there exists a neighborhood Vx0 ∈ L(x0 ) such that f (Vx0 ) ⊂ L(y0 ), then f is continuous at x0 . If f is continuous at each point x ∈ X, then f is a continuous function. If f is bijection, and both f and f −1 are all continuous, then f is a homeomorphism mapping. In such a case, we say that X and Y are homeomorphic. For some property P in a topological space, if it remains unchanged under homeomorphisms, then P is called a topologically invariant property. Theorem 1.3.18. Let (X, ζ ) and (Y, τ) be two topological spaces. Let f : X → Y be a mapping and let B(x) be a neighborhood base of x ∈ X. Let B1 (f (x)) be a neighborhood base of f (x). Then f is continuous at x if and only if ∀C ∈ B1 (f (x)), there exists B ∈ B(x) such that f (B) ⊂ C. Theorem 1.3.19. Let (X, ζ1 ) and (Y, ζ2 ) be topological spaces and let f : X → Y be a mapping. Then the following statements are equivalent: (i) f is continuous; (ii) For any open set G ⊂ Y, the preimage set f −1 (G) in X is open; (iii) For any closed set F ⊂ Y, the preimage set f −1 (F) in X is closed; (iv) For any set B ⊂ Y, f −1 (B) ⊂ f −1 (B); (v) For any A ⊂ X, f (A) ⊂ f (A); (vi) For any net {xδ } ⊂ X, if xδ → x, then f (xδ ) → f (x) in Y. Theorem 1.3.20. Let (X, ζ1 ) and (Y, ζ2 ) be topological spaces and let f : X → Y be a continuous mapping. If A is a compact subset in X, then f (A) is a compact subset in Y. In particular, if Y = ℝ, there exist x0 , x1 ∈ A such that f (x0 ) = inf{f (x) : x ∈ A} = min{f (x) : x ∈ A} and f (x1 ) = sup{f (x) : x ∈ A} = max{f (x) : x ∈ A}. Theorem 1.3.21. Let (X, ζ1 ), (Y, ζ2 ), (Z, ζ3 ) be topological spaces. If f : X → Y and g : Y → Z are continuous mappings, then the composite function h = gf : X → Z is also a continuous mapping.

12 | 1 Introduction and preliminaries In order to make a topological space more concrete, as close as possible to metric spaces, we need to give some restrictions on the topology. The separation axiom is one of the restrictions. Definition 1.3.22. Let (X, ζ ) be a topological space and let x, y be two distinct points in X. If there exists a neighborhood U of x which does not contain y and there exists a neighborhood V of y which does not contain x, then (X, ζ ) is said to be a T1 topological space. Theorem 1.3.22. A topological space (X, ζ ) is T1 if and only if any single point set in X is a closed set. Definition 1.3.23. Let (X, ζ ) be a topological space. If ∀x, y ∈ X, x ≠ y, there exist neighborhoods U ∈ £(x) of x and V ∈ £(y) of y such that U ∩ V = 0, then (X, ζ ) is said to be a T2 space or a Hausdorff space. Theorem 1.3.23. A topological space (X, ζ ) is a T2 space if and only if every limit of a convergent net in X is unique. Definition 1.3.24. Let (X, ζ ) be a topological space. If for any given closed set F and any given point x ∉ F, there exists a neighborhood U of x and a neighborhood V of F such that U ∩ V = 0, then (X, ζ ) is said to be a regular space. Theorem 1.3.24. A topological space (X, ζ ) is regular if and only if for all x ∈ X, U ∈ £(x), there exists V ∈ £(x) such that x ∈ V ⊂ V ⊂ U. Definition 1.3.25. A topological space (X, ζ ) is said to be normal if for any disjoint closed sets E and F, there are neighborhoods U of E and V of F such that U ∩ V = 0. Theorem 1.3.25. A topological space (X, ζ ) is normal if and only if for any closed set F ⊂ X and U ∈ £(x), there exists V ∈ £(F) such that F ⊂ V ⊂ V ⊂ U. If a regular space satisfies the T1 axiom, then it is said to be a T3 space. If a normal space satisfies the T1 axiom, then it is said to be a T4 space. It is easy to see that T4 space ⇒ T3 space ⇒ T2 space ⇒ T1 space, however, the reverse is not true. Theorem 1.3.26. Every metric space is a normal space. Definition 1.3.26. Let (X, ζ ) be a topological space. If ζ is defined by some distance on X, then (X, ζ ) is “metrizable”. Another restriction imposed on topological spaces is the “countability axiom”. Axiom A1 (The First Axiom of Countability). Let (X, ζ ) be a topological space. ∀x ∈ X, every neighborhood system of x has an at most countable neighborhood base. Axiom A2 (The Second Axiom of Countability). Let (X, ζ ) be a topological space. X has an at most countable neighborhood base.

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13

Spaces that satisfy the second axiom of countability also satisfy the first. However, the converse may not be true. Indeed, uncountable spaces with the discrete topology do not satisfy the second axiom of countability. A metric space (X, d) is an A1 space, and Euclidean space ℝn is an A2 space. Theorem 1.3.27. Let (X, ζ ) be an A2 space. The space X is metrizable if and only if X is an A3 space. Theorem 1.3.28. Let (X, ζ ) be an A1 space. Then ∀x ∈ X, there exists a countable neighborhood base {Vn } (n = 1, 2, . . . ) of x such that Vn+1 ⊂ Vn (n = 1, 2, . . . ). Theorem 1.3.29. Let (X, ζ ) be an A1 space which satisfies the T1 axiom of separation and let A ⊂ X. Then x0 ∈ Ad if and only if there exists a sequence {xn }, which is generated by different points in A\{x0 }, converging to x0 . Theorem 1.3.30. Let (X, ζ ) be an A1 space. Then (X, ζ ) is Hausdorff if and only if each limit of a convergent sequence is unique in X. Theorem 1.3.31. Let (X, ζ ) be an A1 space and let (Y, τ) be any topological space. Let f : X → Y be a mapping from X to Y. Then f is a continuous mapping if and only if for any sequence xn → x0 in X, we have f (xn ) → f (x0 ) in Y. Definition 1.3.27. Let Γ be an index set, {Xα }α∈Γ a family of sets, and x : Γ → ⋃α∈Γ Xα a mapping such that x(α) = xα ∈ Xα . Then the set {x | x : Γ → ⋃α∈Γ Xα , xα ∈ Xα , α ∈ Γ} is called a product set of Xα (α ∈ Γ), denoted by ∏α∈Γ Xα . Each Xα (α ∈ Γ) is called a factor set of ∏α∈Γ Xα , the mapping Pα : ∏α∈Γ Xα → Xα defined by Pα (x) = xα is said to be α-projective, and xα is called an α-component of x. Let {(xα , Tα )}α∈Γ be a family of topological spaces and X = ∏α∈Γ Xα a product set. We can introduce a topology on X by means of topology Tα on Xα as follows: For any index α ∈ Γ and any open set Gα in (X, Tα ), one can construct a family of sets in such manner: φ = {Pα−1 (Gα ) | Gα ∈ Tα , α ∈ Γ}, which induces a topology T whose topological base is given by ℬ = {Pα1 (Gα1 ) ∩ ⋅ ⋅ ⋅ ∩ Pαn (Gαn ) | Pαi (Gαi ) ∈ T, i = 1, 2, . . . , n} −1

−1

−1

The topology T is called the product topology on X and (X, T) is called the product topological space, in short, product space. It is clear that the product topology is the weakest topology such that each projective mapping Pα : X → Xα is continuous. Now we state some basic facts on product spaces. Let {Xα }α∈Γ be a family of topological spaces and let X = ∏α∈Γ Xα be the product space. Then the following facts are known: (f1 ) Each projective mapping Pα : X → Xα (α ∈ Γ) is a continuous open mapping, that is, Pα is continuous and Pα maps any open set of X into some open set of Xα (α ∈ Γ).

14 | 1 Introduction and preliminaries (f2 ) Let {xδ } ⊂ X be a net. Then xδ → x ∈ X if and only if for each α ∈ Γ, the net {(xα )δ } converges to xα in Xα . (f3 ) Let Y be a topological space and f : Y → X a mapping. Then f is continuous if and only if for every α ∈ Γ, the composite mapping fα = Pα f : Y → Xα is continuous. (f4 ) X = ∏α∈Γ Xα is a T2 -space if and only if for every α ∈ Γ, Xα is a T2 space. (f5 ) (Tychonoff theorem) X = ∏α∈Γ Xα is compact if and only if for every α ∈ Γ, Xα is compact.

1.4 Topological linear spaces When we deal with problems in function spaces, an algebraic operation is also our concern. However, there are only topological structures in topological spaces. In addition to the topological structures, we usually also need other structures, for example, algebraic structures, to deal with many problems in modern analysis.

1.4.1 Linear spaces Definition 1.4.1. Let X be a nonempty set and let 𝕂 be complex field (or real number field). The set X is said to be a linear space if the following conditions hold: (i) X is an additive commutative group, that is, for all x, y ∈ X, there exists u ∈ X, denoted by u = x + y and called the sum of x, y such that: (1) x + y = y + x; (2) (x + y) + z = x + (y + z); (3) There exists a unique element θ ∈ X such that for all x ∈ X, x + θ = θ + x; (4) For all x ∈ X, there exists a unique element x 󸀠 ∈ X such that x +x 󸀠 = θ, denoted by x󸀠 = −x. (ii) A scalar multiplication with the number field 𝕂 is defined, that is, for all (α, x) ∈ 𝕂 × X, there is a product denoted by αx and called α scalar-multiplication with x such that (1) α(βx) = (αβ)x (for all α, β ∈ 𝕂 and all x ∈ X); (2) 1 ⋅ x = x; (3) (α + β)x = αx + βx (for all α, β ∈ 𝕂 and all x ∈ X); (4) α(x + y) = αx + αy (for all α ∈ 𝕂 and all x, y ∈ X). An element in a linear space is called a vector. Thus a linear space is also called a vector space. Linear isomorphism Let X, Y be linear spaces and let T : X → Y be a mapping. T : X → Y is said to be a linear isomorphism if

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(i) T is both injective and surjective; (ii) For all x, y ∈ X and α, β ∈ 𝕂, T(αx + βy) = αTx + βTy. Linear subspace Let V ⊂ X. Then V is said to be a linear subspace of X if it is a linear space with respect to addition and scalar multiplication on X. It is known that V is a linear subspace of X if and only if V is closed under addition and scalar multiplication, that is, for all x, y ∈ V and α, β ∈ 𝕂, we have αx + βy ∈ V. Both X and {θ} are subspaces of X. They are trivial subspaces. Other subspaces are called proper subspaces. Linear manifold Let E ⊂ X. If there exist x0 ∈ X and a linear subspace E0 ⊂ X such that E = E0 + x0 ≜ {x + x0 : x ∈ E0 }, then E is said to be a linear manifold. Indeed, a linear manifold is just a shift of a subspace. Linear dependence A set of vectors x1 , x2 , . . . , xn ∈ X is said to be linearly dependent if there exist a finite number of distinct vectors λ1 , λ2 , . . . , λn ∈ K, not all zero, such that ∑ni=1 λi xi = θ. The set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the others. These concepts are central to the definition of dimension. Dimension The dimension of a vector space is the cardinality (i. e., the number of vectors of a basis) of the space over its base field, denoted by dim X. Linear base If A is a maximal independent set in X, then A is said to be a set of linear bases of X. Linear closure Let A be an index set and let {xλ | λ ∈ Λ} be a collection of vectors of X. The set {y = ∑ni=1 αi xλi | λi ∈ Λ, αi ∈ K, i = 1, 2, . . . , n}, which is generated by finite linear combinations of {xλ | λ ∈ Λ}, is said to be the linear closure of {xλ | λ ∈ Λ}. A linear closure is a subspace. Indeed, it is an intersection of all linear subspaces containing {xλ | λ ∈ Λ}, and we call the linear closure the linear subspace spanned by {xλ | λ ∈ Λ}, denoted by Span{xλ | λ ∈ Λ}. Linear sum and direct sum Let X1 , X2 be subspaces of X. The set {x1 + x2 | xi ∈ X, i = 1, 2} is said to be a linear sum of X1 and X2 , denoted by X1 + X2 . If, in addition, X1 ∩ X2 = {θ}, then the linear sum is said to be a direct sum, denoted by X1 ⊕ X2 .

16 | 1 Introduction and preliminaries Definition 1.4.2. Let X be a linear space and let A ⊂ X. (i) If for all x ∈ X, there exists a positive number λ0 such that for every λ satisfying |λ| ≤ λ0 , λx ∈ A, then A is called an absorbing set. In other words, for any direction in X, A must contain a segment with that direction and center θ. (ii) If λ satisfies |λ| ≤ 1 and is such that λA ⊂ A, then A is a balanced set. For a real linear space X, A is called a balanced set if for all x ∈ A, [−x, x] ≜ {λx| |λ| ≤ 1} ⊂ A. (iii) If A = −A, then A is called a symmetric set. Remark 1.4.1. A balanced set is a symmetric set that contains θ. A set A is said to be convex if, for all x and y in A and all t in the interval (0, 1), the point (1 − t)x + ty also belongs to A. In other words, every point on the line segment connecting x and y lies in A. The intersection of all convex sets containing A is called the convex hull of A, denoted by co(A). Remark 1.4.2. (i) The convex hull co(A) of A is the minimum convex set containing A. (ii) The set {y|y = αx, |α| ≤ 1} is said to be a balance closure of A, denoted by Ab . Remark 1.4.3. The balance hull Ab of A is the minimum balance set containing A. We next list some basic properties about convex sets, convex hulls, and balance hulls. Property P1. Let A be a convex set. Let xi ∈ A, ∀i = 1, 2, . . . , n. Then ∑ni=1 αi xi , where αi ≥ 0 and ∑ni=1 αi = 1, belongs to A. Property P2. Let A1 , A2 be convex sets. Then λ1 A1 + λ2 A2 , λ1 , λ2 ∈ 𝕂, is also a convex set. This is also true for any finite collection of convex sets. Property P3. A is a convex set if and only if the shift x0 + A is a convex set. Property P4. Let A be a convex set. Let α and β be nonnegative real numbers. Then αA + βA = (α + β)A. Property P5. Let Λ be an index set and let {Aλ | λ ∈ Λ} be a family of convex sets. Then ⋂λ∈Λ Aλ is also a convex set. Property P6. Let X, Y are two linear spaces. Let T : X → Y be a linear mapping and let A be a convex subset of X. Then T(A) is a convex set of Y. The converse is also true. Property P7. A is a balance convex set (absolutely convex) if and only if for all x, y ∈ A, |α| + |β| ≤ 1, αx + βy ∈ A. Property P8. n

n

i=1

i=1

co(A) = {x|x = ∑ λi xi , xi ∈ A, λi ≥ 0, ∑ λi = 1, n is arbitrary}. Property P9. The convex hull co(A) of a balance set A is a balance set.

1.4 Topological linear spaces | 17

Property P10. For all α, β ∈ 𝕂, A1 , A2 ⊂ X, we have (i) co(αA1 + βA2 ) = αco(A1 ) + βco(A2 ); (ii) co(A1 ∪ A2 ) = ∪{[x, y] | x ∈ co(A1 ), y ∈ co(A2 )}. Property P11. The convex hull co(Ab ) of a balance hull Ab of A is called a balance convex set, and n

n

i=1

i=1

co(Ab ) = {y|y = ∑ αi xi , ∑ |αi | ≤ 1, xi ∈ A, n is arbitrary}.

1.4.2 Normed linear spaces and Banach spaces Definition 1.4.3. Let X be a linear space and let ‖ ⋅ ‖ : X → ℝ+ = [0, ∞) be a mapping. Assume that the following conditions hold: (i) ‖x‖ ≥ 0, and ‖x‖ = 0 if and only if x = θ (for all x ∈ X); (ii) ‖αx‖ = |α|‖x‖ (for all α ∈ K, x ∈ X); (iii) ‖x + y‖ ≤ ‖x‖ + ‖y‖ (for all x, y ∈ X). Then ‖ ⋅ ‖ is called a norm on X. A linear space X equipped with a norm is called a normed linear space, denoted by (X, ‖ ⋅ ‖). Define d(x, y) = ‖x − y‖, ∀x, y ∈ X. Then d : X × X → ℝ+ is a distance on X. If X is also complete, then (X, ‖ ⋅ ‖) is called a Banach space, that is, a complete normed linear space is called a Banach space. Let (X, ‖ ⋅ ‖1 ) and (Y, ‖ ⋅ ‖2 ) be two normed spaces and let T : X → Y be a linear mapping. Then T is said to be bounded if there exists a constant M ≥ 0 such that, for all x ∈ X, ‖Tx‖ ≤ M‖x‖. Theorem 1.4.1. Let (X, ‖ ⋅ ‖1 ) and (Y, ‖ ⋅ ‖2 ) be two normed spaces. Let T : X → Y be a linear mapping. Then the following statements are equivalent: (i) T is continuous; (ii) T is continuous at original point θ; (iii) T is bounded. If T : X → Y is a bounded linear mapping, then we can define a norm ‖T‖ = sup ‖Tx‖. ‖x‖≤1

If X ≠ {θ}, then it is equivalent to ‖T‖ = sup ‖Tx‖. ‖x‖=1

It is not hard to see that ‖T‖ = inf{M : M ≥ 0, ‖Tx‖ ≤ M‖x‖, ∀x ∈ X}.

18 | 1 Introduction and preliminaries Then ‖Tx‖ ≤ ‖T‖‖x‖, ∀x ∈ X. B(X, Y) ≜ {T | T : X → Y is a continuous (or bounded) linear mapping}. For ∀T, S ∈ B(X, Y), we define linear operations: (T + S)x = Tx + Sx,

(αT)x = αTx.

Then B(X, Y) is a linear space with the above “sum” and “scalar-multiplication”. Theorem 1.4.2. If X, Y are normed spaces, then B(X, Y) is also a normed space. If Y is a Banach space, then B(X, Y) is also a Banach space. If X is a normed space and Y = K, then B(X, K) is a topological dual (or adjoint) space of X, denoted by X ∗ . From the above theorem, we see that B(X, K) is a Banach space. For T ∈ B(X, Y), define the adjoint operator T ∗ of T by (T ∗ f )(x) = f (Tx),

∀ f ∈ Y ∗ , x ∈ X.

It is known that (i) T ∗ : Y ∗ → X ∗ is a bounded linear operator and ‖T ∗ ‖ = ‖T‖; (ii) For all α ∈ K, (αT)∗ = αT ∗ ; (iii) For all T1 , T2 ∈ B(X, Y), (T1 + T2 )∗ = T1∗ + T2∗ ; (iv) For all T1 ∈ B(X, Y), T2 ∈ B(Y, X), (T1 T2 )∗ = T2∗ T1∗ ; (v) If T has a bounded inverse operator, then T ∗ also has a bounded inverse operator, and (T ∗ )−1 = (T −1 )∗ . Theorem 1.4.3 (Hahn–Banach Extension Theorem). Let V be a subspace of a normed space X and let f be a continuous linear function on V. Then f can be extended to a continuous linear function f0 on X such that ‖f ‖ = ‖f0 ‖. From the Hahn–Banach Extension Theorem, we can obtain the following results. Corollary 1.4.1. Let X be a normed space and x0 ∈ X\{θ}. Then there exists a function f ∈ X ∗ , which is continuous with f (x0 ) = ‖x0 ‖ and ‖f ‖ = 1. If f0 = ‖x0 ‖f , then ‖f0 ‖ = ‖x0 ‖ and ⟨f0 , x0 ⟩ = ‖x0 ‖2 . Corollary 1.4.2. Let V be a subspace of a normed space X and x0 ∉ V. Then there exists a continuous function f ∈ X ∗ such that f (V) = {θ},

f (x0 ) = 1,

and ‖f ‖ =

1 , d

where d = inf{‖x0 − y‖ : y ∈ V} > 0. Corollary 1.4.3. For all x ∈ X, 󵄨 󵄨 󵄨 󵄨 ‖x‖ = sup{󵄨󵄨󵄨⟨f , x⟩󵄨󵄨󵄨 : f ∈ X ∗ , ‖f ‖ ≤ 1} = max{󵄨󵄨󵄨⟨f , x⟩󵄨󵄨󵄨 : f ∈ X ∗ , ‖f ‖ ≤ 1}.

1.4 Topological linear spaces | 19

The normed space X is said to be strictly convex if, for all x, y ∈ X, ‖x‖ = ‖y‖ = 1 and x ≠ y, 󵄩 󵄩󵄩 󵄩󵄩tx + (1 − t)y󵄩󵄩󵄩 < 1,

∀t ∈ (0, 1).

Remark 1.4.4. If X ∗ is strictly convex, then the functional f0 given by the Hahn– Banach extension theorem is unique. Definition 1.4.4. Let X be a real normed space. Let f ; X → ℝ1 be a linear functional and f ≢ θ, β ∈ ℝ1 . Then H = {x ∈ X : f (x) = α} is called a hyperplane in X. Also f ≤ (α) = {x ∈ X : f (x) ≤ α},

f ≥ (α) = {x ∈ X : f (x) ≥ α}, f < (α) = {x ∈ X : f (x) < α}, and f > (α) = {x ∈ X : f (x) > α} are called half-spaces. Indeed, they are convex sets. If f ∈ X ∗ , then H, f ≤ , and f ≥ are all closed convex sets. In this case, H is said to be a closed hyperplane, and f ≤ and f ≥ are said to be closed half-spaces. Since f < and f > are open sets, we call f < and f > open half-spaces. It is easy to check that if H is a hyperplane of X, then there exists x0 ∉ H and a nontrivial subspace L of X such that H = x0 + L. Hence, closed hyperplanes do not contain interior points. Let X be a linear space. Let A and B be nonempty subsets of X, and let α ∈ ℝ1 . If both A ⊂ f ≤ (α) and B ⊂ f ≥ (α) or both A ⊂ f ≥ (α) and B ⊂ f ≤ (α) (both A ⊂ f < (α) and B ⊂ f > (α) or both A ⊂ f > (α) and B ⊂ f < (α)), then hyperplane H separates (or strictly separates) A and B. A hyperplane H separates (or strictly separates) sets A and B if and only if for x ∈ A, we have f (x) ≤ α(f (x) < α) and for x ∈ B, we have f (x) ≥ α(f (x) > α) ⇔ supx∈A f (x) ≤ infx∈B f (x) (supx∈A f (x) < infx∈B f (x)). Theorem 1.4.4 (Separating Hyperplane Theorem I). Let X be a real normed space. Let A and B be two nonempty convex subsets of X. Let A0 ≠ 0 and A0 ∩ B = 0. Then there exists a closed hyperplane H, which separates A and B. In addition, if both A and B are open, then H strictly separates A and B. Theorem 1.4.5 (Separating Hyperplane Theorem II). Let X be a real normed space. Let A and B be two nonempty convex subsets of X such that A is compact, B is closed and A ∩ B = 0. Then there exists a closed hyperplane H, which strictly separates A and B. Corollary 1.4.4. Let V be a subspace of a real normed space X. If V ≠ X, then there exists f ∈ X ∗ , f ≠ θ such that ⟨f , x⟩ = 0, ∀x ∈ V.

20 | 1 Introduction and preliminaries Corollary 1.4.5. Let X be a real normed space and let C be a nonempty convex subset of X. If x ∉ C, then there exists f ∈ X ∗ such that f (x) < inf{f (y) : y ∈ C}. Theorem 1.4.6 (The Resonance Theorem). Let X be a Banach space and let Y be a normed linear space. Let {Tλ }λ∈Λ be a collection of continuous linear operators from X to Y. If there exists a dense subset E of X such that supλ∈Λ ‖Tλ (x)‖ < +∞, ∀x ∈ E, then there exists a positive real number M such that ‖Tλ ‖ ≤ M, ∀λ ∈ Λ. Theorem 1.4.7 (The Closed Graph Theorem). Let X and Y be Banach spaces. Let T : X → Y be a continuous linear bijection. Then T −1 ∈ B(Y, X) and T : X → Y is a homeomorphism mapping. Theorem 1.4.8 (The Open Mapping Theorem). Let X and Y be Banach spaces. Let T be a surjective continuous linear operator. Then T is an open mapping, i. e., if U is an open set in X, then T(U) is open in Y. Theorem 1.4.9. Let {Tp,q } (p = 1, 2, . . . ) be a collection of continuous mappings from a Banach space X to a normed linear space Yq , q = 1, 2, . . . If, for every p, there exists xp ∈ X such that limq→∞ ‖Tp,q xp ‖ = ∞, then B = {x ∈ X : lim ‖Tp,q xp ‖ = ∞, p = 1, 2, . . . } q→∞

is of the second category set in X. Theorem 1.4.10 (Bishop–Phelps Theorem). Let C be a bounded closed convex subset of a Banach space X and let A = {f ∈ X ∗ : f (x) = sup f (C), x ∈ C}. Then A = X ∗ . Definition 1.4.5. Let X be a Banach space and let X ∗ be a dual space of X. Define a mapping C : X → X ∗∗ by f → ⟨f , x⟩, ∀x ∈ X. Then it is a continuous function from X ∗ to K, denoted by Cx, i. e., Cx ∈ X ∗∗ such that ⟨Cx, f ⟩ = ⟨f , x⟩,

∀x ∈ X, f ∈ X ∗ .

The mapping C : X → X ∗∗ is called standard (or “canonical”) embedding; C(X) is a closed subspace of X ∗∗ . The mapping C : X → X ∗∗ may not be surjective. If C(X) = X ∗∗ , then X is said to be reflexive. Theorem 1.4.11. Let X be a Banach space. Then X is reflexive if and only if X ∗ is reflexive. Next, we denote the unit spheres of X and X ∗ by S(X) = {x ∈ X : ‖x‖ = 1} and S(X ∗ ) = {f ∈ X ∗ : ‖f ‖ = 1}. Theorem 1.4.12 (James Theorem). Let X be a Banach space. Then X is reflexive if and only if for all f ∈ S(X ∗ ), there exists x ∈ S(X) such that f (x) = 1.

1.4 Topological linear spaces | 21

Definition 1.4.6. Let X be a normed space. A topology on X is called a strong topology (or norm topology) if it is induced by a distance d(x, y) = ‖x − y‖, denoted by τs (X). Definition 1.4.7. Let X be a normed space and let X ∗ be the dual space of X. For all f ∈ X ∗ , let φf : X → K be a mapping determined by φf (x) = f (x) and let {φf }f ∈X ∗ be a collection of mappings from X to K for each f ∈ X ∗ . The weak topology on X is induced by {φf }f ∈X ∗ . We denote the weak topology by σ(X, X ∗ ), which is the weakest topology such that all mappings {φf }f ∈X ∗ are continuous. Definition 1.4.8. For all x ∈ X, let φf : X ∗ → K be a mapping determined by φf (x) = ⟨f , x⟩ and let {φx }x∈X be a collection of mappings from X ∗ to K for each x ∈ X. The weak topology on X ∗ is induced by {φx }x∈X . We denote this weak topology by σ(X ∗ , X), which is the weakest topology such that all mappings {φx }x∈X are continuous. Remark 1.4.5. (i) σ(X ∗ , X) ⊆ σ(X ∗ , X ∗∗ ) ⊂ τs (X ∗ ); (ii) σ(X, X ∗ ) ⊆ τs (X); (iii) Both σ(X, X ∗ ) and σ(X ∗ , X) are Hausdorff; (iv) Let x0 ∈ X. Then the neighborhood base of x0 in the topology σ(X, X ∗ ) can be characterized by the collection of sets V = {x ∈ X : ⟨fi , x − x0 ⟩ < ε,

∀ i ∈ I},

where I is a finite set, fi ∈ X ∗ and ε > 0; (v) Let f0 ∈ X ∗ . Then the neighborhood base of f0 in the topology σ(X ∗ , X) can be characterized by the collection of sets V = {f ∈ X ∗ : ⟨f − f0 , xi ⟩ < ε,

∀i ∈ I},

where I is a finite set, xi ∈ X and ε > 0. Theorem 1.4.13. Let X be a normed space and let {xn } be a sequence in X. Then w

xn 󳨀→ x in topology σ(X, X ∗ ) if and only if f (xn ) → f (x), ∀f ∈ X ∗ ; s

w

If xn 󳨀→ x in norm, then xn 󳨀→ x in topology σ(X, X ∗ ); w

If xn 󳨀→ x in topology σ(X, X ∗ ), then {‖xn ‖} is bounded and ‖x‖ ≤ lim infn→∞ ‖xn ‖; w

s

If xn 󳨀→ x in topology σ(X, X ∗ ), and fn 󳨀→ f in norm, then fn (xn ) → f (x) (n → ∞). Theorem 1.4.14. Let X be a Banach space and {fn } a sequence in X ∗ . Then w∗

fn 󳨀→ f in topology σ(X ∗ , X) if and only if fn (x) → f (x) (n → ∞), ∀x ∈ X; s

w

If fn 󳨀→ f in norm, then fn 󳨀→ f ; w∗

If fn 󳨀→ f in topology σ(X ∗ , X), then {‖fn ‖} is bounded and ‖f ‖ ≤ lim infn→∞ ‖fn ‖; w∗

s

If fn 󳨀→ f in topology σ(X ∗ , X), and xn 󳨀→ x in norm, then fn (xn ) → f (x) (n → ∞).

22 | 1 Introduction and preliminaries Remark 1.4.6. w w (i) With {fn } ⊂ X ∗ , f ∈ X ∗ , fn 󳨀→ f (n → ∞), and {xn } ⊂ X, x ∈ X, xn 󳨀→ x, we cannot conclude fn (xn ) → f (x); (ii) If dim X < +∞, then the three topologies on X ∗ coincide, that is, τs (X ∗ ) = σ(X ∗ , X) = σ(X ∗ , X ∗∗ ). Theorem 1.4.15. Let X be a normed space and let X ∗ be the dual space of X. The weak topology σ(X, X ∗ ) is equivalent to the strong topology τs (X) if and only if dim X < +∞. Theorem 1.4.16. Let X be a Banach space and let X ∗ be the dual space of X. Then the weak topology σ(X ∗ , X ∗∗ ) on X ∗ is equivalent to the weak∗ topology σ(X ∗ , X) if and only if X is reflexive. Theorem 1.4.17. Let X be a normed space and let X ∗ be the dual space of X. Then (X, σ(X, X ∗ ))∗ = X ∗ and (X ∗ , σ(X ∗ , X))∗ = X. Next, we use BX {x ∈ X : ‖x‖ ≤ 1} and BX ∗ = {f ∈ X ∗ : ‖f ‖ ≤ 1} to denote the closed unit balls of X and X ∗ , respectively. Definition 1.4.9. Let X be a normed space and let X ∗ be the dual space of X. Let A be a subset of X and let B be a subset of X ∗ . Recall that A is bounded in the norm topology if there exists a closed ball Br [θ] such that A ⊂ Br [θ], r > 0; A is w-bounded if for any neighborhood V of the origin in σ(X, X ∗ ), there exists a constant λ > 0 such that λV ⊃ A; B is w∗ -bounded if for any neighborhood V ∗ of the origin in σ(X ∗ , X), there exists a constant s > 0 such that sV ∗ ⊃ B. Theorem 1.4.18. Let X be a normed space and let X ∗ be the dual space of X. Let A be a subset of X. Then, A is bounded in the norm topology if and only if A is w-bounded. Let X be a Banach space and B a subset of X ∗ . Then, B is bounded in norm topology if and only if B is w∗ -bounded. Theorem 1.4.19. Let C be a convex subset of a Banach space X. Then, C is strongly closed ⇔ C is weakly closed ⇔ C is sequentially weakly closed. Theorem 1.4.20. Let X be a normed space and let BX be a closed unit ball in X. Then BX is compact in the norm topology if and only if dim X < +∞. Theorem 1.4.21 (Alaoglu Theorem). Let X be a Banach space and let X ∗ be the topological conjugate space of X. Then the closed unit ball BX ∗ of X ∗ is weak∗ compact. Theorem 1.4.22 (Eberlein–Smulian Theorem). Let X be a Banach space and let A be a subset of X. Then A is weakly compact if and only if A is weakly sequentially compact. Theorem 1.4.23. Let X be a Banach space. Then (i) the closed unit ball BX ∗ in X ∗ is metrizable in topology σ(X ∗ , X) if and only if X is separable;

1.4 Topological linear spaces | 23

(ii) the closed unit ball BX in X is metrizable in topology σ(X, X ∗ ) if and only if X ∗ is separable. Theorem 1.4.24. Let X be a Banach space. Then the following statements are equivalent: (i) X is reflexive; (ii) any bounded closed convex subset of X is weakly compact; (iii) any bounded sequence in X has a weakly convergent sequence; (iv) the closed unit ball BX in X is weakly compact; (v) the closed unit ball BX in X is weakly sequentially compact; (vi) every closed subspace in X is reflexive; (vii) every closed convex subset in X has a minimum-norm element; (viii) every closed convex subset of X is approximable, that is, for every x ∈ X, there exists y ∈ A such that ‖x − y‖ = d(x, A); (ix) every decreasing sequence of bounded closed convex sets of X has a nonempty intersection; (x) every w-closed set is w∗ -closed; (xi) every pair of disjoint closed convex sets, one of which is bounded, can be strictly separated by a hyperplane; (xii) every pair of disjoint bounded closed convex sets can be strictly separated by a hyperplane; (xiii) X ∗∗ = (X ∗ , σ(X ∗ , X))∗ ; (xiv) for all f ∈ X ∗ , f attains its norm on BX ; (xv) every nontrivial quotient space of X is reflexive; (xvi) M and X/M are reflexive, where M is any closed subspace of X; (xvii) M and X/M are reflexive, where M is some closed subspace X.

1.4.3 Hilbert spaces Definition 1.4.10. Let H be a linear space on a number field 𝕂 and let ⟨⋅, ⋅⟩ : H ×H → 𝕂 be a mapping. Assume that (i) ⟨x, x⟩ ≥ 0 and ⟨x, x⟩ = 0 ⇔ x = θ, ∀x ∈ H; (ii) ⟨x, y⟩ = ⟨y, x⟩, ∀x, y ∈ H; (iii) ⟨αx + βy, z⟩ = α⟨x, z⟩ + β⟨y, z⟩, ∀x, y, z ∈ H, and ∀α, β ∈ 𝕂. Then ⟨x, y⟩ is said to be an inner product of x and y. A linear space equipped with the inner product is said to be an inner product space. Let H be an inner product space. Define ‖x‖ = √⟨x, x⟩,

∀x ∈ H.

24 | 1 Introduction and preliminaries Then ‖ ⋅ ‖ : H → ℝ+ is said to be a norm on H and the norm is induced by the inner product. If the inner product space H is complete in the norm topology, then H is called a Hilbert space. If ⟨x, y⟩ = 0, then x and y are orthogonal, denoted by x ⊥ y. Let M ⊂ H, x ∈ H. If x ⊥ y, ∀y ∈ M then x and M are orthogonal, denoted by x ⊥ M. Put M ⊥ = {x ∈ H : x ⊥ M}. The set M ⊥ is said to be the orthogonal complement of M. Let M, L ⊂ H. If ∀x ∈ M, x ⊥ L, then M and L are orthogonal, denoted by M ⊥ L. Theorem 1.4.25. Let H be an inner product space. Then (i) |⟨x, y⟩|2 ≤ ⟨x, x⟩⟨y, y⟩, for any x, y ∈ H. The “=” holds if and only if x and y are linearly dependent. The above inequality is called the Cauchy–Schwarz inequality; (ii) ⟨⋅, ⋅⟩ : H × H → 𝕂 is continuous, that is, ∀{xn }, {yn } ⊂ H, xn → x, yn → y ⇔ ⟨xn , yn ⟩ → ⟨x, y⟩. If xn ⇀ x and yn → y (or xn → x and yn ⇀ y), then ⟨xn , yn ⟩ → ⟨x, y⟩; (iii) ‖x ± y‖2 = ‖x‖2 ± 2 Re⟨x, y⟩ + ‖y‖2 , ∀x, y ∈ H; (iv) (Parallelogram Law) ‖x + y‖2 + ‖x − y‖2 = 2(‖x‖2 + ‖y‖2 ), ∀x, y ∈ H; (v) (Pythagorean Theorem) If x ⊥ y, then ‖x + y‖2 = ‖x‖2 + ‖y‖2 , ∀x, y ∈ H; (vi) (Polarization Identity) ⟨x, y⟩ =

1 (‖x + y‖2 − ‖x − y‖2 + i‖x + iy‖2 − i‖x − iy‖2 ), ∀x, y ∈ H, 4 1 Re⟨x, y⟩ = (‖x + y‖2 − ‖x − y‖2 ), ∀x, y ∈ H; 4

(vii) ‖(1 − λ)y + λx‖2 = (1 − λ)‖y‖2 − λ(1 − λ)‖x − y‖2 + λ‖x‖2 , ∀x, y ∈ H and ∀λ ∈ ℝ; w (viii) (Opial Condition [60]) Let {xn } be a sequence in a Hilbert space with xn 󳨀→ x. Then, for all y ≠ x, lim infn→∞ ‖xn − x‖ < lim infn→∞ ‖xn − y‖, which is also equivalent to lim supn→∞ ‖xn − x‖ < lim supn→∞ ‖xn − y‖; (ix) Let H be a Hilbert space. For any y ∈ H, define a functional f by fy (x) = ⟨x, y⟩,

x ∈ H.

Then fy : H → ℝ is a bounded linear functional and ‖fy ‖ = ‖y‖. On the contrary, for any f ∈ H ∗ , there exists a unique y ∈ H such that f (x) = ⟨x, y⟩, ∀x ∈ H; (x)

Let H1 and H2 be two Hilbert spaces. Let ⟨⋅, ⋅⟩1 and ⟨⋅, ⋅⟩2 be the inner products of H1 and H2 , respectively. For any x = (x1 , x2 ), y = (y1 , y2 ) ∈ H1 × H2 , define the sum and the scalar product in H1 × H2 as follows: x + y = (x1 , x2 ) + (y1 , y2 ) = (x1 + y1 , x2 + y2 ), αx = α(x1 , x2 ) = (αx1 , αx2 ),

and ⟨x, y⟩ = ⟨x1 , y1 ⟩1 + ⟨x2 , y2 ⟩2 Then H1 × H2 is a Hilbert space.

1.4 Topological linear spaces | 25

Let g : H1 × H2 → ℝ be a bounded linear functional. From the above, we find that there exists a unique (z1 , z2 ) ∈ H1 × H2 such that, for all (x1 , x2 ) ∈ H1 × H2 , g⟨x1 , x2 ⟩ = ⟨(x1 , x2 ), (z1 , z2 )⟩ = ⟨x1 , z1 ⟩1 + ⟨x2 , z2 ⟩2 . Theorem 1.4.26. Let X be a normed space. If the norm satisfies the parallelogram law, then we can define an inner product such that the new norm induced by the inner product coincides the original norm of X. Theorem 1.4.27. Let C be a nonempty closed convex subset of a Hilbert space H. Then, for any x ∈ H, there exists a unique x0 ∈ C such that ‖x − x0 ‖ = inf{‖x − y‖} = d(x, C). y∈C

Theorem 1.4.28. Let C be a nonempty closed convex subset of a Hilbert space H. Let x ∈ H and x0 ∈ C. Then ‖x − x0 ‖ = d(x, C) ⇔ ⟨x − x0 , y − x0 ⟩ ≤ 0,

∀y ∈ C.

Let x0 = PC x. Then PC : H → C is said to be the metric projection from H onto C. If C is simple enough, then PC has an analytic expression. (i) If C = {x ∈ H : ‖x − x‖ ≤ r} is a closed ball centered at x ∈ H with a radius r > 0, then PC = {

x−x x + r ‖x−x‖ ,

x,

x ∉ C,

x ∈ C.

(ii) If C = [a, b], where a = (a1 , a2 , . . . , an )T and b = (b1 , b2 , . . . , bn )T are closed rectangles in an n-dimensional Euclidean space, then ai , xi < ai , { { { (PC x)i = {xi , xi ∈ [ai , bi ], { { {bi , xi > bi . (iii) If C = {y ∈ H : ⟨u, y⟩ = α}, where u ≠ θ and α ∈ ℝ is a hyperplane, then PC x = x +

α − ⟨u, x⟩ u. ‖u‖2

(iv) If C = {y ∈ H : ⟨u, y⟩ ≤ α}, where u ≠ θ and α ∈ ℝ is a closed half-space, then x+ PC x = { x,

α−⟨u,x⟩ u, ‖u‖2

⟨u, x⟩ > α,

⟨u, x⟩ ≤ α.

26 | 1 Introduction and preliminaries (v) If C = Ran(A), where A ∈ ℝm×n is an m × n matrix and r(A) = n, then PC x = AA+ x, where A+ = (AT A)−1 AT is a generalized P–M inverse of A. Theorem 1.4.29. Let C be a nonempty closed convex subset of a Hilbert space H. The metric projection PC has the following properties: (i) ‖PC x − PC y‖2 ≤ ⟨x − y, PC x − PC y⟩, ∀x, y ∈ H, 󵄩󵄩 󵄩2 󵄩󵄩(I − PC )x − (I − PC )y󵄩󵄩󵄩 ≤ ⟨x − y, (I − PC )x − (I − PC )y⟩, ∀x, y ∈ H. In particular, 󵄩󵄩 󵄩 󵄩󵄩(I − PC )x − (I − PC )y󵄩󵄩󵄩 ≤ ‖x − y‖,

∀x, y ∈ H,

and ‖PC x − PC y‖ ≤ ‖x − y‖,

∀x, y ∈ H.

(ii) 󵄩 󵄩2 ‖PC x − PC y‖2 ≤ ‖x − y‖2 − 󵄩󵄩󵄩(I − PC )x − (I − PC )y󵄩󵄩󵄩 ,

∀x, y ∈ H.

In particular, ‖y − PC x‖2 + ‖x − PC x‖2 ≤ ‖x − y‖2 ,

∀x ∈ H, y ∈ C.

w

(iii) Let {xn } ⊂ H and x ∈ H. If xn 󳨀→ x and PC xn → y, then PC x = y. (iv) PC = 21 I + 21 S, where S : H → H is a nonexpansive mapping. Theorem 1.4.30. Let M be a closed subspace of a Hilbert space H. Then the projection PM : H → M is a bounded linear operator and ‖PM ‖ = 1. Definition 1.4.11. Let G be a closed subspace of a Banach space H and let L be a subspace of X. Subspace L is said to be a complementary subspace of G if (i) L is a closed set; (ii) X = G ⊕ L. Remark 1.4.7. (i) Any finite dimensional subspace G has a topologically complementary subspace. (ii) In a Hilbert space H, any closed subspace has a topologically complementary subspace.

1.4 Topological linear spaces | 27

(iii) Any closed subspace G with a finite codimension codim(G) = dim(X⟋G) has a topologically complementary subspace. Theorem 1.4.31 (Lindenstrauss and Tzafriri [41]). Let X be a Banach space. If every nonempty closed subspace of X is topologically complementary, then Banach space X is isomorphic to some Hilbert space. Theorem 1.4.32 (Kakutani [34]). Let X be a Banach space with dim X ≥ 3. If every closed subspace with the complementary dimension 1 is equal to the range of the projection operator with the norm 1, then X is an inner product space. Theorem 1.4.33 (Bruck [14]). Let X be a Banach space. Every nonempty closed convex subset of X is a nonexpansive retract of X if and only if either dim X = 2 or X is a Hilbert space. Corollary 1.4.6. Let X be a Banach space with dim X > 3. Then every nonempty closed convex subset of X is a nonexpansive retract of X if and only if X is a Hilbert space.

1.4.4 Topological linear spaces Definition 1.4.12. Let 𝕂 be a number field and let X be a linear space over 𝕂. Let ζ be a topology on X. If the following conditions are satisfied: (i) X is a T1 space, that is, every single point set is closed in X; (ii) The linear operation in X is continuous with respect to ζ , that is, the addition operation (x, y) → x + y as a mapping of (X, ζ ) × (X, ζ ) → (X, ζ ) is continuous; the multiplication operation (α, x) → αx as a mapping of (𝕂, τK ) × (X, ζ ) → (X, ζ ), where τK is a topology according to the usual distance, is continuous. Then (X, ζ ) is said to be a topological linear space. If every neighborhood of the zero point contains a convex neighborhood, then the topological space (X, ζ ) is said to be locally convex. Normed linear spaces (X, ‖ ⋅ ‖), (X, σ(X, X ∗ )), (X ∗ , σ(X ∗ , X)) all are locally convex, regular, and Hausdorff linear spaces. Definition 1.4.13. Let (X, T) be a topological linear space and let A be a subset of X. A is said to be T-bounded if and only if for any neighborhood V of A, there exists a constant λ > 0 such that A ⊂ λV. Theorem 1.4.34. Let (X, T) be a topological linear space and let A be a subset of X. A is T

T-bounded if and only if ∀λn → 0 and ∀{xn } ⊂ A, λn xn 󳨀→ θ (n → ∞).

Definition 1.4.14. Let (X, T1 ) and (Y, T2 ) be two topological linear spaces. Let T : Dom(T) ⊂ X → Y be a mapping. This T is said to be bounded if and only if T maps any bounded subset of Dom(T) into some bounded subset of Y. In particular, T is said to be a bounded functional on X if and only if Y = ℝ or 𝕂.

28 | 1 Introduction and preliminaries Theorem 1.4.35. Let (X, T1 ) and (Y, T2 ) be two topological linear spaces. Let T : Dom(T) ⊂ X → Y be a linear mapping. If T is continuous, then it must be bounded. In particular, if T : Dom(T) ⊂ X → Y, where Y = ℝ or 𝕂, is continuous, then it must be a bounded linear functional. We also remark here that the inverse of the above theorem may be not true. Theorem 1.4.36. If (X, T) is an A1 topological linear space, then every bounded linear functional is continuous. Hence, the boundedness and continuity of a bounded linear functional are equivalent in the framework of topological spaces satisfying the first axiom of countability. Theorem 1.4.37 (Separation Theorem). Let X be a topological linear space. Let A and B be two disjoint nonempty convex sets of X. (i) If A is an open set, then there exist f ∈ X ∗ and r ∈ ℝ such that Re f (x) < r ≤ Re f (y),

∀x ∈ A, y ∈ B.

(ii) If A is compact, B is closed, and X is locally convex, then there exist f ∈ X ∗ and r1 , r2 ∈ ℝ such that Re f (x) < r1 < r2 < Re f (y),

∀x ∈ A, y ∈ B.

Theorem 1.4.38 (Krein–Milman Theorem). Let X be a topological linear space and let X ∗ be the conjugate space of X. Assume that there exists a point f ∈ X ∗ such that f (x1 ) ≠ f (x2 ), ∀x1 , x2 ∈ X, x1 ≠ x2 . Let K be a nonempty compact convex subset of X. Then K = co(ext K), that is, K is a closed convex hull of its extreme point set. Theorem 1.4.39 (Tychonoff Fixed Point Theorem). Let X be a locally convex topological vector space. Let C be a nonempty compact convex subset of X and let T : C → C be a continuous mapping. Then T has a fixed point in C. From the Tychonoff fixed point theorem, we can prove the Schauder and Brouwer fixed point theorems. Theorem 1.4.40 (Schauder Fixed Point Theorem). Let X be a normed linear space. Let C be a nonempty compact convex subset of X and let T : C → C be a continuous mapping. Then T has a fixed point in C. Theorem 1.4.41 (Brouwer Fixed Point Theorem). Let ℝn be an n-dimensional Euclidean space. Let C be a nonempty bounded closed convex subset of ℝn and let T : C → C be a continuous mapping. Then T has a fixed point in C. Theorem 1.4.42 (Banach–Alaoglu Theorem). Let X be a topological linear space and let V be a neighborhood of θ in X. Let X ∗ be the topological conjugate space of X. Then the set K = {f ∈ X ∗ : |f (x)| ≤ 1, x ∈ V} is w∗ -compact.

1.5 Lower semicontinuity, convexity, and conjugate functions | 29

1.5 Lower semicontinuity, convexity, and conjugate functions Definition 1.5.1. Let X be a topological space and let f : X → ℝ = (−∞, ∞] be a function. Then f is said to be lower semicontinuous if ∀α ∈ ℝ, the level set {x ∈ X : f (x) ≤ a} is a closed set in X. If f ≢ ∞, then f is said to be proper. It is clear that f is proper if and only if Dom(f ) ≠ 0. Theorem 1.5.1. Let X be a compact topological space and let f : X → ℝ be a lower semicontinuous function. Then there exists x0 ∈ X such that f (x0 ) = min{f (x) : x ∈ X}. Let X be a topological space and let f : X → ℝ be a function. Let {xα } be a net in X. Define lim infα f (xα ) = supα infα≤β f (xβ ) and lim supα f (xα ) = infα supα≤β f (xβ ). Theorem 1.5.2. Let X be a topological space and let f : X → ℝ be a function. Then f : X → ℝ is lower semicontinuous if and only if xα → x0 ⇒ f (x0 ) ≤ lim inf f (xα ), α

∀x0 ∈ X.

Hence, if f is lower semicontinuous and xn → x, then f (x) ≤ lim infn→∞ f (xn ). Definition 1.5.2. The set epi(f ) = {[x, λ] ∈ X × ℝ : f (x) ≤ λ} is said to be the epigraph of f . Theorem 1.5.3. Let X be a topological space and let f : X → ℝ be a function. Then f is lower semicontinuous if and only if epi(f ) is a closed subset of X × ℝ. Theorem 1.5.4. Let X be a topological space and let {fλ }λ∈Λ be a family of lower semicontinuous functions from X to ℝ. Define a function g : X × ℝ by g(x) = sup fλ (x), λ∈Λ

∀x ∈ X.

Then g is lower semicontinuous and g is called a coenvelope of {fλ }λ∈Λ . Theorem 1.5.5. Let X be a topological space. Let f , g : X → ℝ be lower semicontinuous function. Then, ∀α, β ∈ ℝ+ , αf + βg is lower semicontinuous. Theorem 1.5.6. Let X, Y be topological spaces. Let β : X → ℝ+ = [0, ∞) be a continuous function and let f : Y → ℝ be a lower semicontinuous function. Define a function βf : X × Y → ℝ by (βf )(x, y) = β(x)f (y), Then βf is lower semicontinuous.

(x, y) ∈ X × Y.

30 | 1 Introduction and preliminaries Definition 1.5.3. Let X be a linear space. A function f : Dom(f ) ⊂ X → ℝ is said to be convex if f [tx + (1 − t)y] ≤ tf (x) + (1 − t)f (y),

∀x, y ∈ Dom(f ), ∀t ∈ [0, 1].

If the above inequality strictly holds for x ≠ y, then f is said to be strictly convex. Recall that f is uniformly convex on Dom(f ) if there exists a function μ : ℝ+ → ℝ+ with μ(t) = 0 ⇔ t = 0 and f [tx + (1 − t)y] ≤ tf (x) + (1 − t)f (y) − t(1 − t)μ(‖x − y‖), ∀t ∈ [0, 1], ∀x, y ∈ Dom(f ). If the above inequality holds for t = 21 , then f is said to be uniformly convex at the center. It is well known that f is uniformly convex at the center if and only if f is uniformly convex on Dom(f ). Recall also that f is said to be quasi-convex if the following inequality holds for ∀t ∈ [0, 1] and ∀x, y ∈ Dom(f ): f [tx + (1 − t)y] ≤ max{f (x), f (y)}. Furthermore, if the above inequality strictly holds for x ≠ y, then f is said to be strictly quasi-convex. Clearly, if f is uniformly convex on Dom(f ), then it is strictly convex on Dom(f ); if f is (strictly) convex on Dom(f ), then it is (strictly) quasi-convex on Dom(f ). Theorem 1.5.7. Let X be a linear space and f : X → ℝ a function. Then the following assertions are true: (i) f is a convex function on X if and only if epi(f ) is a convex subset of X × ℝ+ ; (ii) If f is a convex function on X, then ∀λ ∈ ℝ the level set {x ∈ X : f (x) ≤ λ} is convex. But the converse is not true; (iii) If f1 and f2 are convex functions on X, then for any c1 ≥ 0, c2 ≥ 0, c1 f1 + c2 f2 is also convex; (iv) If {fλ }λ∈Λ is a family of convex functions on X, then the coenvelope of {fλ }λ∈Λ is also convex; (v) Let f : X → ℝ be a convex function and g : ℝ → ℝ an increasing convex function. Then the composite function of f and g, h(x) = g(f (x)), is also a convex function, here we require that g(+∞) = +∞; (vi) If X is a real normed space and f : X → ℝ a convex function, then every local minimum is also global one; (vii) f : X → ℝ is convex if and only if for any fixed x, y ∈ Dom(f ), the univariate function ϕ(t) = f (tx + (1 − t)y) is convex in t ∈ [0, 1]. Next, we assume that X is a normed linear space. Definition 1.5.4. Let f : X → ℝ be a proper function. The function defined by f ∗ (x ∗ ) = supx∈X {⟨x∗ , x⟩ − f (x)}, x∗ ∈ X ∗ is said to be the conjugate function of f .

1.5 Lower semicontinuity, convexity, and conjugate functions | 31

By means of Definition 1.5.4, it is easy to check that the following are true. (1) f ∗ : X ∗ → ℝ is lower semicontinuous and convex. (2) If f , g : X → ℝ satisfy the relation f ≤ g, then f ∗ ≥ g ∗ . (3) (Young inequality) For all x ∈ X and x∗ ∈ X ∗ , f (x) + f ∗ (x∗ ) ≥ ⟨x ∗ , x⟩. Theorem 1.5.8. Let f : X → ℝ be a proper and lower semicontinuous convex function. Then f ∗ ≢ ∞. The function f ∗∗ : X → ℝ defined by f ∗∗ (x) = sup {⟨x∗ , x⟩ − f ∗ (x ∗ )} x∗ ∈X ∗

is said to be the second conjugate function of f . We denote by co(f ) the function whose epigraph is co(epi f ). Example 1.5.1. Let X be a normed space and let p > 1 be a fixed number. Then the function f : X → ℝ defined by f (x) =

1 ‖x‖p , p

x ∈ X,

f ∗ (x∗ ) =

1 󵄩󵄩 ∗ 󵄩󵄩q 󵄩x 󵄩 , q󵄩 󵄩

x∗ ∈ X ∗ ,

is a convex function,

and f ∗∗ = f , where

1 p

+

1 q

= 1.

Theorem 1.5.9 (Fenchel–Moreau–Rockafellar). Let f : X → ℝ be a function. Then f ∗∗ = cof , in particular, if f is a proper and lower semicontinuous convex function, then f ∗∗ = f . Theorem 1.5.10 (Fenchel–Rockafellar). Let f , g : X → ℝ be two convex functions. Assume that there exists x0 ∈ X such that f (x0 ) < +∞ and g(x0 ) < +∞. If f is continuous at x0 , then inf {f (x) + g(x)} = sup {−f ∗ (−x∗ ) − g ∗ (x ∗ )}

x∈X

x∗ ∈X ∗

= max {−f ∗ (−x ∗ ) − g ∗ (x ∗ )}. ∗ ∗ x ∈X

Definition 1.5.5. Let C be a nonempty closed convex subset of X and define +∞, x ∉ C,

IC (x) = {

0,

Then IC is an indicator function of C.

x ∈ C.

32 | 1 Introduction and preliminaries It is easy to see that IC is a convex, proper, lower semicontinuous function, and its conjugate function IC∗ is called the supporting function of C, denoted by SC , that is, SC (x∗ ) = sup{⟨x ∗ , x⟩ : x ∈ C, x ∗ ∈ X ∗ }. It is clear that SC∗ (x) = IC (x). Theorem 1.5.11. d(x0 , C) = maxx∗ ∈X ∗ ,‖x∗ ‖≤1 {⟨x ∗ , x0 ⟩ − IC∗ (x ∗ )}, ∀x0 ∈ X. Theorem 1.5.12. Let C be a nonempty closed convex subset of a Banach space X. Let f : X → ℝ be a convex function. Then f is lower semicontinuous in the norm topology if and only if f is lower semicontinuous in the weak topology. Theorem 1.5.13. Let C be a nonempty weakly compact convex subset of a Banach space X. Let f : X → ℝ be a proper, lower semicontinuous, and convex function. Then there exists x0 ∈ X such that f (x0 ) < +∞ and f (x0 ) = min{f (x) : x ∈ C}. Theorem 1.5.14. Let C be a nonempty closed convex subset of a reflexive Banach space X. Let f : X → ℝ be a proper, lower semicontinuous, and convex function. Assume that f (x) → +∞ as ‖x‖ → +∞. Then there exists x0 ∈ X such that f (x0 ) < +∞ and f (x0 ) = min{f (x) : x ∈ C}.

1.6 Differential calculus on Banach spaces Let X and Y be two normed linear spaces. Let T : Dom(T) ⊂ X → Y be a mapping. Definition 1.6.1. A mapping T : Dom(T) ⊂ X → Y is said to be (i) bounded if T maps any bounded subset of Dom(T) to a bounded subset of Y; (ii) locally bounded if ∀x0 ∈ Dom(T) there exists a neighborhood U of x0 such that T is bounded on U; (iii) continuous at x0 ∈ Dom(T) if xn → x0 ⇒ Txn → Tx0 (n → ∞),

∀{xn } ⊂ Dom(T);

(iv) uniformly continuous on Dom(T) if xn − yn → θ ⇒ Txn − Tyn → θ (n → ∞), (v)

∀{xn }, {yn } ⊂ Dom(T);

demicontinuous at x0 ∈ Dom(T) if w

xn → x0 ⇒ Txn 󳨀→ Tx0 (n → ∞),

∀{xn } ⊂ Dom(T);

(vi) weakly continuous at x0 ∈ Dom(T) if w

w

xn 󳨀→ x0 ⇒ Txn 󳨀→ Tx0 (n → ∞),

∀{xn } ⊂ Dom(T);

1.6 Differential calculus on Banach spaces | 33

(vii) strongly continuous at x0 ∈ Dom(T) if w

xn 󳨀→ x0 ⇒ Txn → Tx0 (n → ∞),

∀{xn } ⊂ Dom(T).

Let Y = X ∗ and T : Dom(T) ⊂ X → X ∗ . Then T is said to be hemicontinuous at w∗

x0 ∈ Dom(T) if, for all h ∈ X, tn → 0 as tn → 0, and x0 + tn h ∈ Dom(T), T(x0 + tn h) 󳨀→ Tx0 (n → ∞). The implication relations for various kinds of continuity are listed as follows: ↗ strongly continuous ↘

continuous

↘

weakly continuous ↗

demicontinuous → hemicontinuous

The implication relations for uniform continuity, continuity, boundedness, and local boundedness are listed as follows: Continuous 󳨀→ Locally bounded

↑↓ dim X < +∞

↑

Uniformly continuous

󳨀→

Bounded

Dom(T) is convex Definition 1.6.2. Let X be a real linear space and let Y be a real normed linear space. Let T : Dom(T) ⊂ X → Y be a mapping and x0 ∈ Dom(T). For all h ∈ X, there exists T(x +th)−Tx0 αh > 0 such that x0 + h ∈ Dom(T) when |t| < αh . If limt→0 0 t exists, then T is said to be Gâteaux differentiable at x0 and the limit is called the Gâteaux differential of T at x0 along direction h, denoted by D[Tx0 , h]. If there exists a bounded linear operator B : X → Y such that Gâteaux differential can be represented as D[Tx0 , h] = Bh, then T has a bounded linear Gâteaux differential (G differential) at x0 and B is called a Gâteaux derivative (G derivative) at x0 , denoted by T 󸀠 (x0 ). It is clear that if T is Gâteaux differentiable at x0 , then T is hemicontinuous at x0 , but not necessarily continuous at x0 . Definition 1.6.3. Let X be a real normed linear space and let Ω be an open set of X. Let f : Ω → ℝ1 be a function. If f has a bounded linear Gâteaux differential on Ω, denoted by F(x) = f 󸀠 (x), then the mapping F : Ω → X ∗ is said to be the gradient mapping of f , denoted by F(x) = grad f (x) or F = grad f = ∇f . We say that f is the potential of F, and the gradient mapping satisfies lim

t→0

f (x + th) − f (x) = F(x)h, t

∀h ∈ X, x ∈ Ω.

Definition 1.6.4. Let X, Y be real normed linear spaces and let T : Dom(T) ⊂ X → Y be a mapping. Let x0 be an interior point of Dom(T). Then T is said to be Fréchet differentiable at x0 if there exists a bounded linear operator B : X → Y such that lim

‖h‖→0

‖T(x0 + h) − Tx0 − Bh‖ = 0. ‖h‖

34 | 1 Introduction and preliminaries Operator B is said to be the Fréchet derivative at x0 , denoted by T 󸀠 (x0 ), and Bh is the F differential of T at x0 . If T is F differentiable at each point of Dom(T), then T is said to be F differentiable on Dom(T). It is clear that if T is Fréchet differentiable at x0 , then T is continuous at x0 . Theorem 1.6.1. Let Ω be an open set in X and T : Dom(T) → Y. Let x0 ∈ Dom(T). Then the following assertions hold: (i) If T is F differentiable at x0 , then T must have a bounded linear G differential at x0 , and the G derivative of T coincides with the F derivative of T at x0 ; (ii) If T has a bounded linear G derivative everywhere in some neighborhood of x0 , and the G derivative T 󸀠 (x) is continuous at x = x0 , then T is F differentiable at x0 . Theorem 1.6.2 (Chain rule). Let X, Y, Z be real normed linear spaces and let U ⊂ X, V ⊂ Y be open subsets. Let T : U → V be F differentiable at x0 and let S : V → Z be F differentiable at T(x0 ). Then S ∘ T : U → Z is F differentiable at x0 and (S ∘ T)󸀠 (x0 ) = S󸀠 ∘ T(x0 )T 󸀠 (x0 ). Example 1.6.1. Let X be a real reflexive Banach space. Let A ∈ L(X, X ∗ ) be a positive operator, that is, ⟨Ax, x⟩ ≥ 0, ∀x ∈ X, denoted by φ(x) = ⟨Ax, x⟩. Then grad φ(x) = (A + A∗ )x, ∀x ∈ X, where A∗ is the conjugate operator of A. In particular, If X = H, a Hilbert space, and A is an identity mapping, then grad ‖x‖2 = 2x, x ∈ H,

grad ‖x‖ =

x (x ≠ θ). ‖x‖

Example 1.6.2. Let Ω be a Lebesgue measurable set in ℝn and X = Lp (Ω), p > 1. Then 󵄨 󵄨p−2 grad ‖x‖p = p󵄨󵄨󵄨x(s)󵄨󵄨󵄨 x(s),

x ≠ θ

and 󵄨 󵄨p−2 grad ‖x‖ = ‖x‖1−p 󵄨󵄨󵄨x(s)󵄨󵄨󵄨 x(s),

x ≠ θ.

Example 1.6.3. Let X = lp , where p > 1. Then grad ‖x‖ = ‖x‖1−p z, x ≠ θ, where x = (x1 , x2 , . . . ) ∈ lp , z = (|x1 |p−2 x1 , |x2 |p−2 x2 , . . . ) ∈ lq , p1 + q1 = 1.

1.7 Subdifferentials of convex functions Definition 1.7.1. Let X be a Banach space and let φ : X → ℝ be a proper convex function. Let x0 ∈ X be fixed vector. Then x∗ ∈ X ∗ is said to be a subgradient of φ at x0 if φ(x0 ) + ⟨x∗ , x − x0 ⟩ ≤ φ(x), for any x ∈ X.

1.7 Subdifferentials of convex functions | 35

The family of all the subgradients of φ at x0 is said to be the subdifferential of φ at x0 , denoted by 𝜕φ(x0 ). If 𝜕φ(x0 ) ≠ 0, then φ is said to be subdifferentiable at x0 . If 𝜕φ(x0 ) = 0, then φ is said to be not subdifferentiable at x0 . From the definition of the subdifferential, one has 𝜕φ(x0 ) = ⋂ {x ∗ ∈ X ∗ : φ(x0 ) + ⟨x∗ , x − x0 ⟩ ≤ φ(x)}, x∈X

∀x0 ∈ X.

Generally speaking, 𝜕φ is multi-valued. For convex functions on ℝ, the subgradient at x0 contains the tangent slopes through x0 . Example 1.7.1. Let a, b ∈ ℝ with a < b. Let f : (a, b) → ℝ be convex. Then 𝜕f (x0 ) = [f−󸀠 (x0 ), f+󸀠 (x0 )],

∀x0 ∈ (a, b).

Example 1.7.2. Let X = ℝ and consider +∞,

φ(x) = {

0,

|x| ≥ 1, |x| < 1.

If |x0 | ≥ 1, then 𝜕φ(x0 ) = 0. Example 1.7.3. Let φ(x) = |x|, x ∈ ℝ. Then x , 𝜕φ(x) = { |x| [−1, 1],

x ≠ θ,

x = θ.

The following theorems show the relation between the subdifferential and the G differential. Theorem 1.7.1. Let φ : X → ℝ be a proper, convex function and continuous at x ∈ X. If φ is G differentiable at x ∈ X, then φ is subdifferentiable at x ∈ X, and 𝜕φ(x) = {φ󸀠 (x)}. Theorem 1.7.2. Let φ : X → ℝ be a convex function. Assume that φ is continuous at x ∈ Dom(φ) and 𝜕φ(x) contains only one subgradient. Then φ is G differentiable at x ∈ Dom(φ). Example 1.7.4. Let X = H be a Hilbert space. Let φ(x) = 21 ‖x‖2 , ∀x ∈ H. Then 𝜕φ(x) = {∇φ(x)} = {x}. Example 1.7.5. Let X = H be a Hilbert space. Let φ(x) = ‖x‖, ∀x ∈ H. Then x , x ≠ θ, 𝜕‖x‖ = { ‖x‖ B1 [θ], x = θ,

where B1 [θ] = {x ∈ H : ‖x‖ ≤ 1} is a closed unit ball in H.

36 | 1 Introduction and preliminaries Definition 1.7.2. Let C be a closed convex subset of a real Banach space X and x ∈ C. The set NC (x) = {x∗ ∈ X : ⟨x∗ , y − x⟩ ≤ 0, ∀y ∈ C} is said to be a normal cone of C at x. Example 1.7.6. 𝜕IC (x) = NC (x), ∀x ∈ C. Next, we list some basic properties of the subdifferential of φ in the following theorem. Theorem 1.7.3. Let φ : X → ℝ be a convex function. Then (i) 𝜕φ(x0 ) is a w∗ -closed convex subset of X ∗ , ∀x0 ∈ X. (ii) Dom(𝜕φ) ⊂ Dom(φ) and Ran(𝜕φ) ⊂ Dom(φ∗ ). (iii) φ attains its minimum at x ∈ Dom(φ) if and only if θ ∈ 𝜕φ(x). (iv) 𝜕(λφ) = λ𝜕φ, ∀λ > 0. (v) If φ : X → ℝ is proper, lower semicontinuous and convex, and (Dom(φ))0 ≠ 0, then φ is continuous and subdifferentiable on (Dom(φ))0 , and 𝜕φ(x0 ) is w∗ -compact, ∀x0 ∈ (Dom(φ))0 . (vi) If φ1 , φ2 : X → ℝ are convex functions satisfying Dom(φ1 ) ∩ Dom(φ2 ) ≠ 0 and there exists x0 ∈ Dom(φ1 ) ∩ Dom(φ2 ) such that φ1 or φ2 is continuous at x0 , then 𝜕(φ1 + φ2 ) = 𝜕φ1 + 𝜕φ2 . (vii) x∗ ∈ 𝜕φ(x0 ) ⇔ φ(x0 ) + ⟨x ∗ , x − x0 ⟩ = 0 is a supporting hyperplane of epi(φ) at (x0 , φ(x0 )). (viii) Let φ : X → ℝ be a proper, lower semicontinuous convex function. Then φ(x) + φ∗ (x ∗ ) = ⟨x∗ , x⟩ ⇔ x ∗ ∈ 𝜕φ(x). (ix) Let φ : X → ℝ be a proper and lower semicontinuous convex function. Then x ∗ ∈ 𝜕φ(x) ⇔ x ∈ 𝜕φ∗ (x∗ ), which implies 𝜕φ∗ = (𝜕φ)−1 . (x) Let X be a real reflexive Banach space, f : X → ℝ and g : ℝ+ → ℝ+ be two proper, lower semicontinuous convex functions. Then the following are equivalent: (a) ∀x ∗ ∈ 𝜕f (x), y ∈ Dom(f ), f (y) ≥ f (x) + ⟨y − x, x ∗ ⟩ + g(‖y − x‖); (b) ∀x ∗ ∈ 𝜕f (x), y∗ ∈ Dom(f ∗ ), 󵄩 󵄩 f ∗ (y∗ ) ≤ f ∗ (x ∗ ) + ⟨x, y∗ − x ∗ ⟩ + g ∗ (󵄩󵄩󵄩y∗ − x ∗ 󵄩󵄩󵄩). (xi) (Mean-Value Formula) Let f : X → ℝ be proper, lower semicontinuous and convex. Let x, y ∈ X be two distinct points. Write xt := ty + (1 − t)x for all t ∈ [0, 1] and define the univariate function φ(t) := f (xt ). Then the following expressions are valid: (1) 𝜕φ(t) = {⟨s, y − x⟩ : s ∈ 𝜕f (xt )}.

1.8 Geometry of Banach spaces | 37

(2) There exist t ∈ (0, 1) and s ∈ 𝜕f (xt ) such that f (y) − f (x) = ⟨s, y − x⟩. (3) For any st ∈ 𝜕f (xt ), 1

f (y) − f (x) = ∫⟨st , y − x⟩dt. 0

(xii) (Moreau–Rockafellar Theorem) Let X, Y be Banach spaces. Let f : X → ℝ, g : Y → ℝ be proper, lower semicontinuous, and convex functions. Let A : X → Y be a bounded linear operator and F(x) = f (x) + g(Ax). Assume that the regularity condition θ ∈ Int{A(Dom(f )) − Dom(g)} holds. Then, for all x0 ∈ Dom(F), 𝜕F(x0 ) = 𝜕f (x0 ) + A∗ [𝜕g(Ax0 )], where A∗ is the adjoint operator of A. Example 1.7.7. Let X = H be a Hilbert space and let f (x) = 21 ‖(I − PC )x‖2 , ∀x ∈ H. Then 𝜕f (x) = {󳶚f (x)}, 󳶚f (x) = (I − PC )x, ∀x ∈ H. Example 1.7.8. Let X = H be a Hilbert space. Let A : H → H be a bounded linear operator and g(x) = 21 ‖(I −PC )Ax‖2 , ∀x ∈ H. Then 𝜕g(x) = {∇g(x)}, ∇g(x) = A∗ (I −PC )Ax, x ∈ H.

1.8 Geometry of Banach spaces 1.8.1 Convexity of Banach spaces Definition 1.8.1. Let X be a Banach space. Then X is said to be strictly convex if for any x, y ∈ X, which are linearly independent, ‖x + y‖ < ‖x‖ + ‖y‖; X is said to be uniformly convex if for any {xn }, {yn } ⊂ X with ‖xn ‖ = ‖yn ‖ = 1 and ‖xn + yn ‖ → 2, ‖xn − yn ‖ → 0 as n → ∞; X is said to be locally uniformly convex if for any ε > 0 and a given point x ∈ X, ‖ x ∈ S(X) = {x ∈ X : ‖x‖ = 1}, there exists δ = δ(x, ε) > 0, such that ‖x − y‖ ≥ ε ⇒ ‖ x+y 2 ≤ 1 − δ, ∀y ∈ S(X). Theorem 1.8.1. Let X be a Banach space. Then the following assertions are equivalent: (i) X is uniformly convex; (ii) ∀{xn }, {yn } ⊂ X, ‖xn ‖ → 1 and ‖yn ‖ → 1, ‖xn + yn ‖ → 2 ⇒ xn − yn → θ as n → ∞; (iii) ∀ε ∈ (0, 2], there exists δ = δ(ε) > 0 such that 21 ‖x + y‖ ≤ 1 − δ. The following theorem is not hard to derive.

38 | 1 Introduction and preliminaries Theorem 1.8.2. Let X be a Banach space. Then X is locally uniformly convex if and only if, for any x ∈ S(X) and {xn } ⊂ S(X), ‖xn − x‖ → 0 as ‖xn + x‖ → 2; X is strictly convex if and only if, for any f ∈ X ∗ , f attains its maximum at most one point on S(X) ⇔ ‖ x+y ‖ < 1, 2 ∀x, y ∈ S(X), x ≠ y. We have the following implications: uniformly convex 󳨐⇒ locally uniformly convex 󳨐⇒ strictly convex. Note that any inverse of above implications is not true in general. However, for a finite-dimensional normed linear space X, they are true, that is, X is uniformly convex if and only if X is locally uniformly convex if and only if X is strictly convex. 1

1

Example 1.8.1. Fix μ > 0 and consider the norm ‖x‖μ = ‖x‖0 + μ(∫0 x 2 (t)dt) 2 on C[0, 1], where ‖x‖0 is the usual maximum norm. Then, for any x ∈ C[0, 1], ‖x‖0 ≤ ‖x‖μ ≤ (1 + μ)‖x‖0 , and (C[0, 1], ‖ ⋅ ‖μ ) is strictly convex, but not uniformly convex. Example 1.8.2. l1 , l∞ , c0 , L1 , L∞ , C[a, b] are not strictly convex. Hence, they are also not uniformly convex. Example 1.8.3. Both lp and Lp , where p > 1, are uniformly convex. Hence, they are also locally uniformly convex and strictly convex. Definition 1.8.2. Let X be a normed linear space with dim X ≥ 2. Define the modulus of convexity δX : (0, 2] → [0, 1] of X by δX (ε) = inf{1 − ‖ x+y ‖ : ‖x‖ = ‖y‖ = 1; ε = ‖x − y‖}, 2 where δX (0) = 0. It is obvious that δX : [0, 2] → [0, 1] is continuous and monotonically increasing, while δXε(ε) is nondecreasing on (0, 2]. In addition, we have the following equivalent expressions: 󵄩󵄩 x + y 󵄩󵄩 󵄩󵄩 󵄩 δX (ε) = inf{1 − 󵄩󵄩󵄩 󵄩 : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ε ≤ ‖x − y‖} 󵄩󵄩 2 󵄩󵄩󵄩 󵄩󵄩 x + y 󵄩󵄩 󵄩 󵄩󵄩󵄩 : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ε = ‖x − y‖}. = inf{1 − 󵄩󵄩󵄩 󵄩󵄩 2 󵄩󵄩󵄩 Definition 1.8.3. Let X be a normed space and let p > 1 be a fixed real number. Then X is said to be p-uniformly convex if there exists a constant c > 0 such that δ(ϵ) ≥ cϵp ,

∀ϵ ∈ [0, 2].

Theorem 1.8.3. Let X be a Banach space and let p > 1 be a fixed real number. Then X is p-uniformly convex if and only if there exists a constant c ∈ (0, 1] such that 1 (‖x + y‖p + ‖x − y‖p ) ≥ ‖x‖p + cp ‖y‖p , 2

∀x, y ∈ X.

1.8 Geometry of Banach spaces | 39

Theorem 1.8.4 (Xu [98], 1991). Let X be a Banach space and let p > 1 be a fixed real number. Then X is p-uniformly convex if and only if there exists a constant cp > 0 such that, ∀x, y ∈ X, λ ∈ [0, 1], 󵄩p 󵄩󵄩 p p p 󵄩󵄩(1 − λ)y + λx󵄩󵄩󵄩 ≤ (1 − λ)‖y‖ − cp wp (λ)‖x − y‖ + λ‖x‖ , where wp (λ) = λp (1 − λ) + λ(1 − λ)p . Example 1.8.4. Let X = H be a Hilbert space. Then δH (ε) = 1 − √1 −

ε2 1 ≥ ϵ2 . 4 8

For any Banach space, one has δX (ε) ≤ δH (ε). In addition, if δX (ε) ≥ cεp , c > 0, then p ≥ 2. Example 1.8.5. Let X = Lp where p > 1. Then 1

{1 − [1 − ( ε2 )p ] p ≥ p1 ( ϵ2 )p , δ (ε) = { 1 [1 − ( ε2 )q ] q ≥ p−1 ϵ2 , 8 { Lp

where p, q are conjugate numbers with

1 p

+

1 q

p ≥ 2,

p ∈ (1, 2],

= 1.

From the above example, we see that Lp is max{p, 2}-uniformly convex and Hilbert space is 2-uniformly convex, in particular, L2 is 2-uniformly convex. Theorem 1.8.5. Let X be a Banach space. Then X is uniformly convex if and only if δX (ε) > 0, ∀ε > 0. Theorem 1.8.6. Let X be a uniformly convex Banach space. Then, ∀r ≥ ε > 0, ‖x‖ ≤ r, ‖y‖ ≤ r and ‖x − y‖ ≥ ε > 0 imply δX ( εr ) > 0 and ε 1 ‖x + y‖ ≤ r{1 − δX ( )}. 2 r Theorem 1.8.7 (Bruck [13], 1978). Let X be a uniformly convex Banach space. Then, ∀x, y ∈ B1 [θ], t ∈ [0, 1], 󵄩󵄩 󵄩 󵄩󵄩tx + (1 − t)y󵄩󵄩󵄩 ≤ 1 − 2 min{t, 1 − t}δX (‖x − y‖). Theorem 1.8.8 (Zeidle [112], 1985). Let X be a uniformly convex Banach space. Let {tn } be a real number sequence such that 0 < a ≤ tn ≤ b < 1 and let {xn } and {yn } be sequences in X such that lim supn→∞ ‖xn ‖ ≤ c, lim supn→∞ ‖yn ‖ ≤ c and 󵄩 󵄩 lim󵄩󵄩󵄩tn xn + (1 − tn )yn 󵄩󵄩󵄩 = c. n Then xn − yn → θ as n → ∞.

40 | 1 Introduction and preliminaries Theorem 1.8.9 (Xu [98], 1991). Let X be a uniformly convex Banach space and let r and p be two real numbers such that r > 0 and p > 1. Then there exists a continuous strictly increasing convex function g : ℝ+ → ℝ+ with g(0) = 0, such that, ∀x, y ∈ Br [θ], ∀λ, μ ∈ [0, 1], λ + μ = 1, ‖λx + μy‖p ≤ λ‖x‖p + μ‖y‖p − ωp (λ)g(‖x − y‖), where ωp (λ) = λp μ + λμp . In particular, if p = 2, then ‖λx + μy‖2 ≤ λ‖x‖2 + μ‖y‖2 − λ(1 − λ)(λ)g(‖x − y‖). Let X be a Banach space and let {xn } be a sequence in X. Then X is said to have the Kadec–Klee property if xn ⇀ x and ‖xn ‖ → ‖x‖ imply xn → x as n → ∞. Theorem 1.8.10. Let X be a locally uniformly convex Banach space. Then X has the Kadec–Klee property. In particular, uniformly convex Banach spaces have the Kadec–Klee property. Theorem 1.8.11. Uniformly convex Banach spaces are reflexive. So, the class of uniformly convex Banach spaces is a class of significant spaces between the class of Hilbert spaces and the class of reflexive spaces. Theorem 1.8.12. Let X be a uniformly convex Banach space and let C be a nonempty closed subset of X. Then, ∀x ∈ X, there exists unique point x0 ∈ C such that ‖x − x0 ‖ = d(x, C) ≜ inf{‖x − y‖ : y ∈ C}. 1.8.2 Duality mappings Most results in this subsection are taken from [1, 87, 98]. Definition 1.8.4. Let φ be a continuous and strictly increasing function with φ(0) = 0 and φ(t) → +∞ (t → ∞). Then φ : ℝ+ → ℝ+ is said to be a gauge function. t

Theorem 1.8.13. Let φ : ℝ+ → ℝ+ be a gauge function and let ψ(t) = ∫0 φ(s)ds. Then ψ is a convex function. Definition 1.8.5. Let φ be a gauge function. Define a mapping Jφ : X → 2X by ∗

󵄩 󵄩 󵄩 󵄩 Jφ x := {x∗ ∈ X ∗ : ⟨x ∗ , x⟩ = 󵄩󵄩󵄩x ∗ 󵄩󵄩󵄩‖x‖; 󵄩󵄩󵄩x ∗ 󵄩󵄩󵄩 = φ(‖x‖)}. Then Jφ is said to be a generalized duality mapping with gauge function φ. In particular, duality mapping J = Jφ is called the normal duality mapping if φ(t) = t.

1.8 Geometry of Banach spaces | 41

From the Hahn–Banach theorem, one sees that Jφ x ≠ 0, ∀x ∈ X. So Dom(Jφ ) = X. Next, we use jφ (x) to denote any element of Jφ (x) and list some basic properties of Jφ as follows: (i) For any x ∈ X, Jφ x is a ω∗ -closed convex subset of X ∗ . (ii) For any x, y ∈ X, x∗ ∈ Jφ x, y∗ ∈ Jφ y, one has ⟨x − y, x ∗ − y∗ ⟩ ≥ 0. (iii) For any x ≠ θ ∈ X, λ ∈ ℝ, one has Jφ (λx) = sign(λ)

φ(|λ|‖x‖) J (x), φ(‖x‖) φ

and Jφ (θ) = {θ}. In particular, one has J(λx) = λJx and J(θ) = {θ}, ∀x ∈ X, λ ∈ ℝ. (iv) J is a bounded operator, that is, J maps any bounded subset of X into a bounded subset of X ∗ . t (v) For any x ∈ X, Jφ (x) = 𝜕ψ(‖x‖), where ψ(t) = ∫0 φ(s)ds. In particular, for the generalized duality mapping Jφ (x) = ‖x‖p−2 Jx, x ≠ 0, and the normal duality mapping with gauge functions φ(t) = t p−1 and φ(t) = t, respectively, one has Jφ (x) = 𝜕( p1 ‖x‖p ), { J(x) = 𝜕( 21 ‖x‖2 ).

∀p > 1,

(vi) For any x, y ∈ X, y∗ ∈ Jy, ‖x‖p − ‖y‖p ≥ p⟨x − y, y∗ ⟩, ‖x‖2 − ‖y‖2 ≥ 2⟨x − y, y∗ ⟩, we find that ‖x + y‖p ≤ ‖x‖p + p⟨y, j(x + y)⟩,

{

2

2

‖x + y‖ ≤ ‖x‖ + 2⟨y, j(x + y)⟩,

∀j(x + y) ∈ J(x + y),

∀j(x + y) ∈ J(x + y),

which are called the subdifferential inequalities. (vii) Let X = LP (or lp ), where p > 1. Then the normal duality mappings on X are Jx = ‖x‖2−p |x(s)|p−2 x(s), { Jx = ‖x‖2−p z, x ∈ lp ,

x ∈ Lp ,

where x = (x1 , x2 , x3 , . . . ), z = (|x1 |p−2 x1 , |x2 |p−2 x2 , |x3 |p−3 x3 , . . . ) ∈ lq , p1 + q1 = 1. (viii) Let X be a Banach space with a weakly continuous duality Jφ , that is, there exists some gauge function φ : ℝ+ → ℝ+ such that Jφ is single-valued and sequentially continuous in the ω-topology of X and the ω∗ -topology of X ∗ . Then, ∀{xn } ⊂ X, w

y ∈ X, xn 󳨀→ x,

lim sup ψ(‖xn − y‖) = lim sup ψ(‖xn − x‖) + ψ(‖x − y‖). n→∞

n→∞

So X satisfies the Opial condition, lim sup ‖xn − y‖ > lim sup ‖xn − x‖, n→∞

n→∞

∀y ≠ x.

This also shows that Banach spaces with the weakly continuous duality Jφ must satisfy the Opial condition, however, the inverse may not be true.

42 | 1 Introduction and preliminaries (ix) Let X be a real smooth Banach space. Then, ∀x, y ∈ X, 1

ψ(‖x + y‖) = ψ(‖x‖) + ∫⟨y, jφ (xt )⟩dt ,

∀jφ (xt ) ∈ Jφ (xt ),

0 t

where xt = ty + (1 − t)x, t ∈ [0, 1] and ψ(t) = ∫0 φ(s)dx We remark here that lp has the weakly continuous duality Jφ with φ(t) = t p−1 , where p > 1, however, Lp has no weakly continuous duality mapping Jφ unless p = 2. So, lp satisfies the Opial condition, however, Lp does not satisfy the Opial condition if p ≠ 2. Theorem 1.8.14 (Takahashi [87], 2000). Let X be a Banach space. Then the following assertions hold: (i) If X is strictly convex, then Jφ is injective, that is, x ≠ y ⇒ Jφ (x) ∩ Jφ (y) = 0. (ii) If X is reflexive, then Jφ is surjective, that is, Jφ (X) = X ∗ . (iii) If X ∗ is strictly convex, then Jφ is single-valued. Theorem 1.8.15 (Takahashi [87], 2000). Let X be a Banach space. Then X is strictly convex ⇔ x∗ ∈ Jx, y∗ ∈ Jy, and x ≠ y ⇒ ⟨x − y, x ∗ − y∗ ⟩ > 0. Definition 1.8.6. Let ω : ℝ+ → ℝ+ be a function. It is said to belong to the Γ class if it satisfies the following conditions: (i) ω(0) = 0; (ii) ω(r) > 0, ∀r > 0; (iii) t ≤ s ⇒ ω(t) ≤ ω(s). Theorem 1.8.16 (Takahashi [87], 2000). Let X be a uniformly convex Banach space. Then ∀R > 0 there exists a function ωR ∈ Γ such that, ∀x, y ∈ BR [θ] with x ∗ ∈ Jx and y∗ ∈ Jy, ⟨x − y, x∗ − y∗ ⟩ ≥ ωR (‖x − y‖)‖x − y‖. By means of the generalized duality mapping, we can give the following characteristic inequalities for p-uniformly convex Banach spaces. Theorem 1.8.17 (Xu [98], 1991). Let p > 1 be a given real number and let X be a Banach space. Then the following statements are equivalent: (i) X is p-uniformly convex. (ii) There exists a constant c > 0 such that, for any x, y ∈ X, jp (x) ∈ Jp (x) and jp (y) ∈ Jp (y), the following inequality holds: ‖x + y‖p ≥ ‖x‖p + p⟨y, jp (x)⟩ + c‖y‖p .

1.8 Geometry of Banach spaces | 43

(iii) There exists a constant c1 > 0 such that, for any x, y ∈ X, jp (x) ∈ Jp (x) and jp (y) ∈ Jp (y), the following inequality holds: ⟨x − y, jp (x) − jp (y)⟩ ≥ c1 ‖x − y‖p .

1.8.3 Differentiability of norms and smoothness of spaces Let X be a Banach space and let S(X) = {x ∈ X : ‖x‖ = 1}. Definition 1.8.7. (i) The norm of X is said to be G-differentiable (or X is said to be smooth) if the limit lim

t→0

‖x + ty‖ − ‖x‖ t

(L)

exists for each x, y ∈ S(X). (ii) The norm of X is said to be uniformly G-differentiable if, for each y ∈ S(X), limit (L) is attained uniformly over x ∈ S(X). (iii) The norm of X is said to be F-differentiable, if for each x ∈ S(E), limit (L) is attained uniformly over y ∈ S(E). (iv) The norm of X is said to be uniformly F-differentiable (or X is said to be uniformly smooth), limit (L) is attained uniformly over x, y ∈ S(E). Theorem 1.8.18. A Banach space is smooth if and only if Jφ is single-valued. Theorem 1.8.19. Let X ∗ be a uniformly convex Banach space. Then Jφ is single-valued and is uniformly continuous on bounded subsets of X. Theorem 1.8.20. (i) Let X be a Banach space. If the norm of X is F-differentiable, then Jφ is continuous. (ii) Let X be a reflexive Banach space and let X ∗ be strictly convex. Then Jφ : X → X ∗ is demicontinuous. (iii) If X is reflexive and X ∗ is locally uniformly convex, then Jφ : X → X ∗ is continuous. Theorem 1.8.21. Let X be a Banach space. If the norm of X is uniformly G-differentiable, then Jφ is ‖ ⋅ ‖-ω∗ uniformly continuous on bounded subsets of X. Theorem 1.8.22. Space X ∗ is uniformly convex if and only if the norm of X is uniformly F-differentiable. Theorem 1.8.23. Let X be a Banach space. Then (i) If X ∗ is strictly convex, then X is smooth; (ii) If X ∗ is smooth, then X is strictly convex.

44 | 1 Introduction and preliminaries Theorem 1.8.24. Let X be a reflexive Banach space. Then (i) X is strictly convex if and only if X ∗ is smooth; (ii) X is smooth if and only if X ∗ is strictly convex. Theorem 1.8.25. l1 , l∞ , L1 and L∞ are not smooth. Definition 1.8.8. Let X be a Banach space. Then X is said to be uniformly smooth if ∀ε > 0 there exists δ = δ(ε) > 0 such that, ∀x, y ∈ X, x ∈ S(X) and ‖y‖ ≤ δ imply ‖x − y‖ + ‖x + y‖ − 1 < ε‖y‖. 2 Definition 1.8.9. Let X be a Banach space. Define a function ρX (⋅) : ℝ+ → ℝ+ by ρX (τ) = sup{

‖x − y‖ + ‖x + y‖ − 1 : x ∈ S(X), ‖y‖ ≤ τ}. 2

Function ρX is said to be the smoothness module of X and has the following properties: (i) ρX (0) = 0, ρX (τ) ≤ τ; (ii) ρX (⋅) is continuous, convex, and monotonically increasing; ρ (τ) (iii) Xτ is monotonically increasing; (iv) ∀0 < τ ≤ η ⇒ 2

ρη η2

≤c

ρX (τ) , τ2

(v) ρX (cτ) ≤ c ρX (τ), ∀τ > 0.

where c > 0 is fixed constant;

Remark 1.8.1. (i) ‖x + y‖ + ‖x − y‖ − 1 : ‖x‖ ≤ 1, ‖y‖ ≤ τ} 2 ‖x + y‖ + ‖x − y‖ = sup{ − 1 : ‖x‖ = 1, ‖y‖ = τ}. 2

ρX (τ) = sup{

(ii) Finite-dimensional Banach spaces are smooth if and only if they are uniformly smooth. Theorem 1.8.26. Let X be a Banach space. Then X is uniformly smooth if and only if ρX (τ) → 0 as τ → 0. τ Theorem 1.8.27. Let X be a Banach space. Then the following assertions are mutually equivalent: (i) X is uniformly smooth; (ii) X ∗ is uniformly convex; (iii) Jφ : S(X) → S(X ∗ ) is uniformly continuous in the norm topology; (iv) The norm of X is uniformly F-differentiable. Hence, if X is uniformly smooth, then X is smooth and reflexive.

1.8 Geometry of Banach spaces | 45

Theorem 1.8.28 (Reich [72], 1978). Let X be a real uniformly smooth Banach space. Then there exists a continuous monotonically increasing function b : ℝ+ → ℝ+ with b(0) = 0 and b(ct) ≤ cb(t), ∀c ≥ 1, such that ‖x + y‖2 ≤ ‖x‖2 + max{‖x‖, 1}‖y‖b(‖y‖) + 2⟨y, Jx⟩,

∀x, y ∈ X.

(RI)

Remark 1.8.2. The function b appearing in (RI) indeed is defined by b(t) = sup{

‖x + ty‖2 − 1 − 2⟨y, J(x)⟩ : x, y ∈ S(X)}. t

Further, we can require that function b in Reich inequality (RI) be strictly increasing. Theorem 1.8.29 (Xu and Roach [106], 1991). Let X be a real uniformly smooth Banach space. Then there exist two constants k, c > 0 such that c ‖x + y‖2 ≤ ‖x‖2 + 2⟨y, Jx⟩ + k max{‖x‖ + ‖y‖, }ρX (‖y‖), 2

∀x, y ∈ X,

(XRI)

where ρX (⋅) is the smoothness module of X. Theorem 1.8.30 (Alber [2], 1996). Let X be a real uniformly smooth Banach space. Then the following inequality holds: ‖x + y‖2 ≤ ‖x‖2 + cρX (‖y‖) + 2⟨y, Jx⟩,

∀x, y ∈ X,

(AI)

where c = 48 max{L, ‖x‖, ‖y‖} and L ∈ (1, 1.7) is the Figiel constant. Theorem 1.8.31 (Xu [98], 1991). Let X be a real Banach space. Let q > 1 and r > 0. Then X is uniformly smooth Banach space if and only if there exists a continuous strictly increasing convex function β : ℝ+ → ℝ+ with β(0) = 0 such that ‖x + y‖q ≤ ‖x‖q + q⟨y, Jq (x)⟩ + β(‖y‖), where 󵄩 󵄩 Jq (x) = {x ∗ ∈ X ∗ : ⟨x, x∗ ⟩ = ‖x‖q , 󵄩󵄩󵄩x ∗ 󵄩󵄩󵄩 = ‖x‖q−1 } is the generalized duality mapping. Theorem 1.8.32. Let X be a Banach space. Then ρX ∗ (τ) = sup{

τε − δX (ε) : 0 ≤ ε ≤ 2} 2

and ρX (τ) = sup{

τε − δX ∗ (ε) : 0 ≤ ε ≤ 2} 2

(XI)

46 | 1 Introduction and preliminaries Example 1.8.6. 1

ρH (τ) = (1 + τ2 ) 2 − 1, 1

(1 + τp ) p − 1 < p1 τp , 1 < p < 2, ρlp (τ) = ρLp (τ) = ρWmp (τ) = { p−1 2 p−1 2 2 τ + o(τ ) < 2 τ , p ≥ 2. 2 It is clear that Hilbert spaces are 2-uniformly smooth and lp , Lp and Wmp are min{p, 2}-uniformly smooth. Theorem 1.8.33. For any Banach space X, we have ρX (τ) ≥ ρH (τ) = √1 + τ2 − 1. If ρX (τ) ≤ cτq , then q ≤ 2. Definition 1.8.10. Let X be a Banach space and q > 1 be a given real number. Then X is said to be q-uniformly smooth if there exists some constant c > 0 such that ρX (τ) ≤ cτq . Theorem 1.8.34 (Xu [98], 1991). Let X be a real Banach space. Then the following assertions are mutually equivalent: (i) X is q-uniformly smooth; (ii) There exists a constant c > 0 such that ‖x + y‖q ≤ ‖x‖q + q⟨y, Jq (x)⟩ + c‖y‖q ,

∀x, y ∈ X.

(iii) There exists a constant c1 > 0 such that ⟨x − y, Jq (x) − Jq (y)⟩ ≤ c1 ‖x − y‖q ,

∀x, y ∈ X.

Theorem 1.8.35. Let X = Lp (or lp ), p > 1. Then we have the following conclusions: (i) If 2 < p < +∞, for all x, y ∈ X, λ ∈ [0, 1], we have (1) ‖(1 − λ)y + λx‖p ≤ (1 − λ)‖y‖p − wp (λ)cp ‖x − y‖p + λ‖x‖p ; (2) ‖(1 − λ)y + λx‖2 ≥ (1 − λ)‖y‖2 − λ(1 − λ)(p − 1)‖x − y‖2 + λ‖x‖2 ; (3) ‖x + y‖p ≥ ‖x‖p + p⟨y, Jp (x)⟩ + cp ‖y‖p ; (4) ‖x + y‖2 ≤ ‖x‖2 + 2⟨y, Jp (x)⟩ + (p − 1)‖y‖2 ; (5) ⟨Jp (x) − Jp (y), x − y⟩ ≥ 2p−1 cp ‖x − y‖p ; (6) ⟨Jx − Jy, x − y⟩ ≤ (p − 1)‖x − y‖2 . (ii) If 1 < q ≤ 2, for all x, y ∈ Lq , λ ∈ [0, 1], then (1) ‖(1 − λ)y + λx‖2 ≤ (1 − λ)‖y‖2 − λ(1 − λ)(q − 1)‖x − y‖2 + λ‖x‖2 ; (2) ‖(1 − λ)y + λx‖q ≥ (1 − λ)‖y‖q − wq (λ)cq ‖x − y‖q + λ‖x‖q ; (3) ‖x + y‖2 ≥ ‖x‖2 + 2⟨y, Jq (x)⟩ + cq ‖y‖2 ; (4) ‖x + y‖q ≤ ‖x‖q + q⟨y, Jq (x)⟩ + cq ‖y‖q ; (5) ⟨Jx − Jy, x − y⟩ ≥ (q − 1)‖x − y‖2 ; (6) ⟨Jq (x) − Jq (y), x − y⟩ ≤ 2q−1 cq ‖x − y‖q , where cp , cq > 0 are fixed constants.

1.8 Geometry of Banach spaces | 47

1.8.4 Geometry constants in Banach spaces Geometric constants of Banach spaces play an important role in fixed point problems of nonlinear Lipschitz mappings. In this subsection, we consider asymptotic centers, normal structures, and uniform normal structure constants. Let X be a Banach space. Let C be a nonempty subset of X and let {xn } be a bounded sequence in X. Define a functional r : X → ℝ+ by r(x) = lim sup ‖xn − x‖, n→∞

∀x ∈ X,

where r(C, {xn }) := inf{r(y) : y ∈ C}, A(C, {xn }) := {z ∈ C : r(z) = min{r(x) : x ∈ C}}. Definition 1.8.11. r(C, {xn }) is called the asymptotic radius of {xn } on C. If z ∈ A(C, {xn }), then z ∈ C is called the asymptotic center of {xn } on C. It is worth mentioning that A(C, {xn }) may be an empty set or a singleton, or a set with infinitely many points. In fact, if xn → x as n → ∞, then A(C, {xn }) = {x} for x ∈ C and A(C, {xn }) = {y ∈ C : ‖x − y‖ = d(x, C)} for x ∉ C. What we are mainly concerned is the case when A(C, {xn }) is a singleton. Theorem 1.8.36. Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let {xn } be a bounded sequence in X. Then A(C, {xn }) is a singleton. Let X be a uniformly convex Banach space. From the above theorem, we see that any bounded sequence has a unique asymptotic center with respect to a nonempty closed convex subset C of X. Next theorem further characterizes the property of asymptotic centers. Theorem 1.8.37. Let C be a nonempty closed convex subset of a uniformly convex Banach space X. Then each bounded sequence {xn } with respect to C has a unique asymptotic center z ∈ C, {z} = A(C, {xn }), and lim sup ‖xn − z‖ < lim sup ‖xn − x‖, n→∞

n→∞

∀x ≠ z.

From the above theorem, one can immediately conclude the following result. Theorem 1.8.38. Let X be a uniformly convex Banach space satisfying the Opial condition. Let C be a nonempty closed convex subset of X. Let {xn } be a sequence in C such w

that xn 󳨀→ z. Then {z} = A(C, {xn }).

Definition 1.8.12. Let C be a nonempty bounded subset of a Banach space X. diam(C) = sup{‖x − y‖ : x, y ∈ C} is said to be the diameter of C. We say that x0 ∈ C is the (i) diameter point of C if sup{‖x0 − x‖ : x ∈ C} = diam(C); (ii) non-diameter point of C if sup{‖x0 − x‖ : x ∈ C} < diam(C).

48 | 1 Introduction and preliminaries Definition 1.8.13. Let C be a nonempty convex subset of a Banach space X. Then C is said to have the normal structure if each bounded subset D of C that contains at least two points has a non-diameter point in C, that is, there exists x0 ∈ D such that sup{‖x0 − x‖ : x ∈ D} < diam(D). We say that X has the normal structure if each any nonempty bounded convex subset C with a positive diameter (diam(C) > 0) has the normal structure in X. Theorem 1.8.39. (i) Each compact convex subset C of a Banach space X has the normal structure. (ii) Each bounded closed convex subset of a uniformly convex Banach space X has the normal structure. (iii) Any uniformly convex Banach space has the normal structure. (iv) If a Banach space X satisfies the Opial condition, then X has the normal structure, however, the inverse is not true. Definition 1.8.14. Let X be a Banach space. Let C be a bounded closed convex subset of X and diam(C) > 0. Then N(X) := inf{

diam(C) : diam(C) > 0} r(C)

is said to be the normal structure coefficient of X, where r(C) := {rx (C) : x ∈ C}, and rx (C) := sup{‖x − y‖ : y ∈ C}. It is obvious that N(X) ≥ 1. Also X is said to have the uniformly normal structure if N(X) > 1. For a Hilbert space H, one has N(H) = √2. Theorem 1.8.40. Let X be a uniformly smooth Banach space. Then X has the uniform normal structure. Hence, it also has the normal structure.

1.8.5 Banach limits It is known that the limit operation of real number sequences is linear, but, the limit of bounded sequences does not always exist. This motivates us to consider superior and inferior limits. It is not true that both superior limit and inferior limit are not linear. Can one find a linear operator to replace the superior/inferior limit operation? The answer is positive. The Banach limit is such an operation that keeps the usual linear structure and is more efficient than the superior/inferior limit operation. Definition 1.8.15. Let μ be a bounded linear function on l∞ . If it has the following properties: (L1) ‖μ‖ = μ(e) = 1, where e = (1, 1, 1, . . . ) ∈ l∞ ; (L2) μ(xn+1 ) = μ(xn ), for all xn = (x1 , x2 , x3 , . . . ) ∈ l∞ ,

1.8 Geometry of Banach spaces | 49

then μ is called a Banach limit. Sometimes, we also use μn , or LIM, to denote the Banach limit. Using the Tychonoff fixed point theorem or the Hahn–Banach theorem, we can show the existence of the Banach limit. Theorem 1.8.41. Let μ be a Banach limit. Then lim inf xn ≤ μ(x) ≤ lim sup xn , n→∞

n→∞

∀x = (x1 , x2 , . . . ) ∈ l∞ .

In particular, if xn → a, then μ(x) = a. But, the converse may not be true. Example 1.8.7. Let x = (1, 0, 1, 0, . . . ) ∈ l∞ . Then lim infn→∞ xn = 0 and lim supn→∞ xn = 1. Hence, limn→∞ xn does not exist, however, for any Banach limit μ, one has μ(xn ) = 21 . Theorem 1.8.42. Let x = (xn ) ∈ l∞ and a ∈ ℝ. If for all Banach limits μ, μ(xn ) ≤ a, and lim supn→∞ (xn+1 − xn ) ≤ 0, then lim supn→∞ xn ≤ a. Theorem 1.8.43. Let X be a reflexive Banach space and let {xn } be a bounded sequence in X. Let μ be a Banach limit. Then there exists x0 ∈ X such that μn ⟨xn , x∗ ⟩ = ⟨x0 , x∗ ⟩,

∀x ∗ ∈ X ∗ .

Theorem 1.8.44. Let X be a reflexive Banach space and let C be a nonempty closed convex subset of X. Let {xn } be a bounded sequence in C and let μn be a Banach limit. Define φ(x) = μn ‖xn − x‖2 , ∀x ∈ C. Then the set defined by K = {z ∈ C : μn ‖xn − z‖ = inf φ(x)}, x∈C

is a nonempty, bounded, closed, and convex subset of C. If X is a uniformly convex Banach space, then K is a singleton. Theorem 1.8.45. Let X be a Banach space with a uniformly G-differentiable norm. Let C be a nonempty, closed, and convex subset of X, and let {xn } be a bounded sequence in C. Let μn be a Banach limit, and let z ∈ C be fixed element. Then z ∈ K ⇔ μn ⟨y − z, J(xn − z)⟩ ≤ 0,

∀y ∈ C.

We remark here that most of the results presented in this subsection are taken from Takahashi [87]. 1.8.6 Projection mappings Projection mappings play an important role in various iterative methods. It is necessary to understand their basic properties. This subsection is devoted to four kinds

50 | 1 Introduction and preliminaries of projection mappings: the metric projection, the generalized projection, the sunny nonexpansive projection, and the sunny generalized nonexpansive projection. Let X be a real reflexive, smooth, and strictly convex Banach space. Let C be a nonempty, closed, and convex subset of X. Then ∀x ∈ X there exists unique x0 ∈ C such that ‖x − x0 ‖ = min{‖x − y‖ : y ∈ C}. Definition 1.8.16. Let x0 = PC (x). Then PC : X → C is uniquely defined and it is said to be the metric projection from X onto C. Using Ju =

1 2

grad ‖u‖2 , we have the following characterization.

Theorem 1.8.46. Let x ∈ X and x0 ∈ C. Then x0 = PC (x) if and only if ⟨x0 − y, J(x − x0 )⟩ ≥ 0, ∀y ∈ C. Consider a bivariate function φ(x, y) = ‖x‖2 − 2⟨x, Jy⟩ + ‖y‖2 ,

∀x, y ∈ X.

Fixing y ∈ X, we see that φ(⋅, y) is a continuous convex function with φ(x, y) → +∞ (‖x‖ → +∞). Hence, for any x ∈ X, there exists x0 ∈ C such that φ(x0 , y) = min{φ(x, y) : x ∈ C}. The uniqueness of x0 is guaranteed by the strict convexity of X. Definition 1.8.17. Let x0 = ΠC x. Then ΠC : X → C is uniquely defined and it is said to be generalized projection from X onto C. Theorem 1.8.47. Let x ∈ X and x0 ∈ C. Then x0 = ΠC x if and only if ⟨x0 −y, J(x−x0 )⟩ ≥ 0, ∀y ∈ C. It is known that the metric projection PC , mapping Hilbert space H onto a nonempty closed convex subset C, has the following property: PC [λx + (1 − λ)PC (x)] = PC x,

∀x ∈ H, λ ∈ [0, 1].

This shows that each point on the segment seg{x, PC x} can be the best approximation of PC x. This motives us to introduce the following general concepts. Definition 1.8.18. Let X be a Banach space. Let C and D be subsets of X. Let P : C → D be a mapping. Then P is said to be sunny if P[tx + (1 − t)Px] = Px, for all x ∈ C, t ≥ 0 and tx + (1 − t)Px ∈ C. Let D ⊂ C. We say that a continuous mapping P : C → D is a retraction if Px = x, ∀x ∈ D, that is, P 2 = P. The set D is called a retract of C. In addition, if P is nonexpansive, then P is said to be a nonexpansive retraction, and the set D is called a nonexpansive retract.

1.8 Geometry of Banach spaces | 51

Example 1.8.8. Every closed convex subset of a Euclidean space ℝn (or a Hilbert space H) is a retract of ℝn (or H). Example 1.8.9. Every closed convex subset of a uniformly convex Banach space X is a retract of X, however, it may not be a nonexpansive retract. Theorem 1.8.48. Let C be a nonempty convex subset of a smooth Banach space X. Let D be a nonempty subset of C and let P : C → D be a retraction mapping such that ⟨x − Px, J(y − Px)⟩ ≤ 0,

∀x ∈ C, y ∈ D.

Then P : C → D is a sunny nonexpansive retraction. Theorem 1.8.49. Let C be a nonempty convex subset of a smooth Banach space X. Let D be a nonempty subset of C and let P : C → D be a retraction mapping. Then the following assertions are mutually equivalent: (i) P is a sunny nonexpansive retraction mapping; (ii) ⟨x − Px, J(y − Px)⟩ ≤ 0, ∀x ∈ C, y ∈ D; (iii) ⟨x − y, J(Px − Py)⟩ ≥ ‖Px − Py‖2 , ∀x, y ∈ C. Remark 1.8.3. (i) There is at most one sunny nonexpansive retraction mapping in a smooth Banach space X. (ii) The concept of the nonexpansive retract generalizes two results: one is a linear ergodic theorem in the framework of reflexive Banach spaces, and the other is the metric projection in the framework of Hilbert spaces. Example 1.8.10. Let X be a reflexive Banach space and let L : X → X be a linear operator with ‖L‖ ≤ 1. Then the mean ergodic theorem guarantees that the mean-value operator Ln = n−1 ∑nj=1 Lj pointwise converges to projection P from X to Ker(I − L) with ‖P‖ ≤ 1 and Ker(I − L) = Fix(L). Example 1.8.11. Let H be a Hilbert space and let C be a nonempty closed and convex subset of H. Let T : C → C be a nonexpansive mapping, that is, ‖Tx−Ty‖ ≤ ‖x−y‖, for all x, y ∈ C. Denote the fixed point set of T by Fix(T). Then Fix(T) is a closed convex subset of X. If Fix(T) ≠ 0, then PFix(T) is nonexpansive. Indeed, PFix(T) is firmly nonexpansive, that is, ‖PFix(T) x − PFix(T) y‖2 ≤ ⟨x − y, PFix(T) x − PFix(T) y⟩,

∀ x, y ∈ C.

Definition 1.8.19. Let D be a nonempty closed convex subset of a Banach space X. Let R : X → D be a mapping. Then R is said to be generalized nonexpansive if Fix(R) ≠ 0, and φ(Rx, y) ≤ φ(x, y), ∀x ∈ X, y ∈ Fix(R), where φ(x, y) = ‖x‖2 − 2⟨y, Jx⟩ + ‖y‖2 ,

∀x, y ∈ X.

52 | 1 Introduction and preliminaries Theorem 1.8.50. Let X be a reflexive, smooth, and strictly convex Banach space. Let C be a nonempty closed subset of X and let RC : X → C be a retraction mapping. Then RC : X → C is a sunny generalized nonexpansive mapping if and only if ⟨x − RC x, J(RC x) − J(y)⟩ ≥ 0,

∀x ∈ X, y ∈ X.

In a reflexive, smooth, and strictly convex Banach space X, there are four kinds of projection mappings from X to a nonempty closed convex subset C of X. They are metric projection PC , generalized projection ΠC , sunny nonexpansive projection QC , and sunny generalized nonexpansive projection Rc . Also ∀x ∈ X, x0 ∈ C, we have (i) x0 = PC x ⇔ ⟨J(x − x0 ), x0 − y⟩ ≥ 0, ∀y ∈ C; (ii) x0 = ΠC x ⇔ ⟨Jx − Jx0 , x0 − y⟩ ≥ 0, ∀y ∈ C; (iii) x0 = QC x ⇔ ⟨x − x0 , J(x0 − y)⟩ ≥ 0, ∀y ∈ C; (iv) x0 = RC x ⇔ ⟨x − x0 , J(x0 − Jy)⟩ ≥ 0, ∀y ∈ C. If X = H is a Hilbert space, then PC = ΠC = QC = RC . In addition, PC and ΠC uniquely exist, however, QC and RC may not exist.

1.9 Some classes of nonlinear mappings In this section, we introduce several classes of important nonlinear mappings: the class of nonexpansive mappings, the class of accretive operators, the class of monotone mappings, and the class of pseudocontractive mappings. Several demiclosed principles and fixed point theorems are also investigated in this section. From Bruck’s viewpoint, the class of nonexpansive mappings, which is a natural extension of the class of contractive mappings, is one of most important nonlinear mappings. (i) The concept of a nonexpansive mapping has a close relation with the monotone methods that were investigated in 1960s. Browder–Kirk–Göhde fixed point theorem is the first fixed point theorem based on geometric properties of Banach spaces instead of compactness. (ii) Nonexpansive mappings have an intimate relation with the following initial value problem of the differential inclusion: θ∈

du + T(t)u, dt

where {T(t)}t≥0 is a family of dissipative mappings or multi-valued continuous accretive operators. On the other hand, nonexpansive mappings have an intimate relation with accretive mappings, monotone mappings, nonlinear semigroups, and nonlinear evolution equations.

1.9 Some classes of nonlinear mappings | 53

(i) Let T : Dom(T) ⊂ X → X be a nonexpansive mapping. Let U = I − T and Dom(U) = Dom(T). Then U : Dom(U) → X is accretive. (ii) Let {U(t)}t≥0 be a nonlinear semigroup and T its generator. Then U(t) is nonexpansive ⇔ −T is accretive ⇔ T is dissipative. The class of pseudocontractive mappings is a generalization of the class of nonexpansive mappings. It also has an intimate relation with the class of accretive operators, that is, T is pseudocontractive if and only if I − T is accretive. Since Fix(T) = U −1 (θ), we can convert zero point problems of accretive operators to fixed point problems of pseudocontractive mappings. Now, we give some necessary notations, which will be used in the subsequent discussion and statements. In Sections 1.7 and 1.8, we discussed two multi-valued mappings 𝜕φ and Jφ . The reasons for studying multi-valued mappings are aspective and sufficient. Let X and Y be two real normed linear spaces and let X ∗ be the topological conjugate of X. Let A : X → 2Y be a multi-valued mapping. Then we use the following notations: Dom(A) = {x ∈ X : Ax ≠ 0}: the effective domain of A, Ran(A) = ⋃{Ax : x ∈ X}: the range of A, Graph(A) = {[x, y] ∈ X × Y : x ∈ Dom(A), y ∈ Ran(A)}: the graph of A. We may view a multi-valued mapping A : X → 2Y as the Graph(A) of A, and use the notation A ⊂ X × Y to denote the multi-valued mapping A : X → 2Y . Let A, B : X → 2Y be multi-valued mappings and let λ ∈ ℝ. Then A + B := {[x, y + z] ∈ X × Y : [x, y] ∈ A, [x, z] ∈ B}, λA := {[x, λy] ∈ X × Y : [x, y] ∈ A}, A−1 := {[x, y] ∈ X × Y : [y, x] ∈ A}. Let X, Y and Z be real normed linear spaces. Let A ⊂ X × Y and B ⊂ Y × Z be multi-valued mappings. Then define a multi-valued mapping BA ⊂ X × Z by BA := {[x, z] ∈ X × Z : there exists y ∈ Y such that [x, y] ∈ A and [y, z] ∈ B}. From above notations, we have the following conclusions: (i) Let A, B ⊂ X × Y. Then (a) Dom(A + B) = Dom(A) ∩ Dom(B), (b) Dom(λA) = Dom(A), (c) Dom(A−1 ) = Ran(A). (ii) Let A ⊂ X × Y and B ⊂ Y × Z. Then Dom(BA) = {x ∈ X : x ∈ Dom(A), Ax ∈ Dom(B)}. (iii) Let A, B ⊂ X × Y. Then (A + B)x = Ax + Bx = {y + z ∈ Y : y ∈ Ax, z ∈ Bx},

∀x ∈ Dom(A + B),

54 | 1 Introduction and preliminaries (iv) (λA)x = {λy ∈ Y : y ∈ Ax}, ∀x ∈ Dom(λA), (v) A−1 (x) = {y ∈ X : x ∈ Ay}, ∀x ∈ Dom(A−1 ), (vi) Let A ⊂ X × Y and B ∈ Y × Z. Then (BA)x = B(Ax) = {z ∈ Z : ∃y ∈ Dom(B), y ∈ Ax, z ∈ By}

∀x ∈ Dom(BA).

Let A ⊂ X × Y be multi-valued mapping and let T : X → X be a single-valued mapping. We also use the following notations: (i) A−1 (θ) = N(A) := {x ∈ Dom(A) : θ ∈ Ax}: the set of zeros of A; (ii) Fix(T) := {x ∈ Dom(A) : x = Tx}: the set of fixed points of T;

1.9.1 Nonexpansive mappings Definition 1.9.1. Let X be a normed space and let C be a nonempty subset of X. Let T : C ⊂ X → X be a mapping. Then T is said to be (i) L-Lipschitz if there exists L > 0 such that ‖Tx − Ty‖ ≤ L‖x − y‖,

∀x, y ∈ C;

(ii) strictly contractive if L ∈ [0, 1); (iii) nonexpansive if L = 1; (iv) averaged nonexpansive (or averaged) if T = (1 − λ)I + λS, (v)

where S is a nonexpansive mapping on C and λ is a real number in (0, 1); firmly nonexpansive if 󵄩 󵄩 ‖Tx − Ty‖ ≤ 󵄩󵄩󵄩r(x − y) + (1 − r)(Tx − Ty)󵄩󵄩󵄩,

∀x, y ∈ C, r > 0;

(vi) quasi-nonexpansive if Fix(T) ≠ 0 and ‖Tx − y‖ ≤ ‖x − y‖,

∀x ∈ C, y ∈ Fix(T);

(vii) strongly nonexpansive if lim ((xn − yn ) − (Txn − Tyn )) = θ,

n→∞

whenever {xn } and {yn } are sequences in C such that {xn − yn } is bounded and limn→∞ (‖xn − yn ‖ − ‖Txn − Tyn ‖) = 0.

1.9 Some classes of nonlinear mappings | 55

Proposition 1.9.1. (i) T is firmly nonexpansive if and only if there exists j(Tx − Ty) ∈ J(Tx − Ty) such that ⟨x − y, j(Tx − Ty)⟩ ≥ ‖Tx − Ty‖2 , (ii)

∀x, y ∈ C.

T is strongly nonexpansive if and only if ∀M > 0 there exists a strictly increasing function γ : [0, 2M] → [0, M] such that, ∀r > 0, γ(r) > 0, and ∀x, y ∈ C, ‖x −y‖ ≤ M, 󵄩 󵄩 γ(󵄩󵄩󵄩x − y − (Tx − Ty)󵄩󵄩󵄩) + ‖Tx − Ty‖ ≤ ‖x − y‖.

(iii) A linear projection mapping P with ‖P‖ = 1 is firmly nonexpansive. The metric projection PC : H → C from a Hilbert space H onto its nonempty closed and convex subset C is firmly nonexpansive and it is also a 21 -averaged mapping. (iv) If X is uniformly convex, then firmly nonexpansive mappings and averaged mappings are strongly nonexpansive. (v) The composition of a finite family of nonexpansive mappings (if well defined) is still nonexpansive. (vi) The composition of a finite family of averaged mappings (if well defined) is still averaged. In particular, let T1 be a λ1 -averaged mapping, where λ1 ∈ (0, 1) and let T2 be a λ2 -averaged mapping, where λ2 ∈ (0, 1). If both T1 T2 and T2 T1 are well defined, then both T1 T2 and T2 T1 are λ-averaged, where λ = λ1 + λ2 − λ1 λ2 . (vii) Let C be nonempty subset of a strictly convex Banach space X. Let T1 , T2 , T3 , . . . , Tr , where r is some positive integer, be averaged mappings from C to itself with a nonempty common fixed point set F = ⋂ri=1 Fix(Ti ) ≠ 0. Then F = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) = Fix(T1 Tr ⋅ ⋅ ⋅ T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 Tr−2 ⋅ ⋅ ⋅ Tr ). (viii) Let X be a Banach space and let C be a nonempty subset of X. Let T1 , T2 , T3 , . . . , Tr , where r is some positive integer, be strongly nonexpansive mappings from C to itself with a nonempty common fixed point set F = ⋂ri=1 Fix(Ti ) ≠ 0. Then F = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) = Fix(T1 Tr ⋅ ⋅ ⋅ T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 Tr−2 ⋅ ⋅ ⋅ Tr ). (ix) A convex combination of a nonexpansive mapping and an averaged mapping is an averaged mapping. (x) Let X be a uniformly convex Banach space. Let T0 be a strongly nonexpansive mapping and let T1 be a nonexpansive mapping. Let S = (1 − c)T0 + cT1 , where c is a real number in (0, 1). Then S is strongly nonexpansive. (xi) Let X be a strictly convex Banach space and let {Ti }∞ i=1 be an infinite family of nonexpansive mappings. Let Tx = ∑∞ λ T x, ∀x ∈ X, where λi ∈ (0, 1) for each i=1 i i i = 1, 2, . . . with ∑∞ λ = 1. Then T is nonexpansive. If ⋂∞ i=1 i i=1 Fix(Ti ) ≠ 0, then ∞ ⋂i=1 Fix(Ti ) = Fix(T). For nonexpansive mappings, we have the following known result.

56 | 1 Introduction and preliminaries Theorem 1.9.1. Let X be a Banach space and let X ∗ be the dual space of X. Let C be a nonempty, closed, and convex subset of X, and let T : C → C be a nonexpansive mapping. Then there exists some f ∈ S(X ∗ ) such that lim f (

n→∞

󵄩󵄩 T n x 󵄩󵄩 T nx 󵄩 󵄩󵄩 ) = lim 󵄩󵄩󵄩 󵄩󵄩 = inf{‖y − Ty‖} ≜ d, n→∞󵄩 n 󵄩 n 󵄩󵄩 y∈C

∀x ∈ C.

For averaged mappings, we have the following known result. Theorem 1.9.2. Let X be a Banach space and let C be a nonempty, closed, and convex subset of X. Let T : C → C be an averaged mapping. Then n+k n 󵄩󵄩 T n x 󵄩󵄩 󵄩 󵄩 limn ‖T x − T x‖ 󵄩 󵄩󵄩 lim󵄩󵄩󵄩T n+1 x − T n x󵄩󵄩󵄩 = = lim󵄩󵄩󵄩 󵄩󵄩 = d, n n 󵄩 󵄩󵄩 k n 󵄩

∀k ≥ 1, x ∈ C,

where d is defined as above. If {T n x} is bounded, then T n+1 x − T n x → θ

as n → ∞.

In this case, we say that T is asymptotically regular at x ∈ C. If T is asymptotically regular at every point of C, then T is asymptotically regular on C. The asymptotical regularity is a necessary, but not sufficient condition for convergence of {T n x}. Example 1.9.1. Let X = l1 and let C = {xn ∈ l1 : xn = (1, 21 , . . . , n1 , 0, . . . )}. Define Txn = xn+1 . Then T : C → C is a nonexpansive mapping, which is also asymptotically regular. Since n

‖xn ‖ = ∑ i=1

1 → ∞, i

we see that {T n x} is not convergent for all x ∈ C. Theorem 1.9.3 (Browder [7], 1965). Let C be a nonempty convex and closed subset of a uniformly convex Banach space X. Let T : C → X be a nonexpansive mapping. Then, for any {xn } ⊂ C satisfying xn ⇀ x and (I − T)xn → y ∈ X, we have x ∈ C and (I − T)x = y. In addition, (I − T)(C) is a closed subset of X. We say that I − T is demiclosed at y ∈ X. In particular, if y = θ, then I − T is demiclosed at the origin θ. Using the above demiclosedness principle, the following celebrated Brower fixedpoint theorem is not hard to derive. Theorem 1.9.4 (Browder [7], 1965). Let X be a uniformly convex Banach space. Let C be a nonempty, bounded, closed, and convex subset of X, and let T : C → C be a nonexpansive mapping. Then Fix(T) is a nonempty and closed convex subset of C. In 1965, Kirk proved the following existence result.

1.9 Some classes of nonlinear mappings | 57

Theorem 1.9.5 (Kirk [38], 1965). Let X be a reflexive Banach space which has the normal structure. Let C be a nonempty, bounded, closed, and convex subset of X, and let T : C → C be a nonexpansive mapping. Then T has a fixed point. Since every nonempty, closed, convex, and bounded subset of a uniformly convex Banach space has the normal structure, Kirk’s existence theorem generalizes the Browder’s existence theorem. Let D be a nonempty set of X. Indeed, compact convex sets always have the normal structure, however, Alspach’s example [3] shows that weakly compact convex sets need not have the normal structure. Recall that D ⊂ C is said to have the fixed point property (in short, f. p. p.) if every nonexpansive mapping defined on weakly compact convex subsets of D has fixed points. Uniformly convex Banach spaces have the fixed point property and reflexive Banach spaces with the normal structure also have the fixed point property. A Banach space X is said to have the weak fixed point property if for each weakly compact convex subset D ⊂ E, and a nonexpansive mapping T : D → D, D contains a fixed point for T. Theorem 1.9.6. Let X be a Banach space and let C be a nonempty weakly compact convex subset of X, which also has the normal structure. Let T = {T1 , T2 , . . . , Tr } be a finite commutative family of nonexpansive mappings from C to itself. Then there exists x0 ∈ C such that x0 = Ti x0 , i = 1, 2, . . . , r, that is, ⋂ri=1 Fix(Ti ) ≠ 0. Theorem 1.9.7. Let X be a strictly convex Banach space and let C be a weakly compact convex subset of X, which also has the normal structure. Let ℜ = {Ti : i ∈ Λ} be a commutative family of nonexpansive mappings from C to itself. Then there exists x0 ∈ C such that x0 = Ti x0 , i ∈ Λ, that is, ⋂i∈Λ Fix(Ti ) ≠ 0. is,

Let {xn } be a sequence in X. We use ωw (xn ) to denote the weak limit set of {xn }, that ωw (xn ) = {x ∈ X : ∃ {xni } ⊂ {xn } such that xni ⇀ x}.

Let X be a Banach space and let C be a nonempty, closed, and convex subset of X. Let T : C → C is an averaged nonexpansive mapping with Fix(T) ≠ 0. From the above theorems, we see that, for any x ∈ C, T n+1 x → T n x → θ. Putting xn = T n x, we have Txn −xn → θ. Sequence {xn } is called a sequence of an approximate fixed points of T. Let X be a uniformly convex Banach space. Form the Browder demiclosedness principle, we see that ωw (xn ) ⊂ Fix(T). If ωw (xn ) is a singleton, then {xn } converges weakly to some fixed point of T. Hence, it is necessary to give conditions that guarantee ωw (xn ) is a singleton. Theorem 1.9.8. Let C be a nonempty closed convex subset of a Banach space X. Let {Tn }n≥1 : C → C be a family of Lipschitz mappings with the Lipschitz coefficient Ln ≥ 1, for each n ≥ 1. Assume that ⋂n≥1 Fix(Tn ) is nonempty and ∑n≥1 (Ln − 1) < ∞. Let {xn } be a sequence generated by the iterative process x1 ∈ C, xn+1 = Tn xn . Then

58 | 1 Introduction and preliminaries (i) if X is a uniformly convex Banach space, then, ∀u, v ∈ ⋂n≥1 Fix(Tn ), ∀t ∈ [0, 1], limn ‖txn + (1 − t)u − v‖ exists; (ii) if X is a uniformly convex Banach space with the F-differentiable norm, then ∀u, v ∈ ⋂n≥1 Fix(Tn ), limn ⟨xn , j(u−v)⟩ exists. In particular, ⟨p−q, j(u−v)⟩ = 0, ∀p, q ∈ ωw (xn ). If ωw (xn ) ⊂ ⋂n≥1 Fix(Tn ), then p = q. So ωw (xn ) is a singleton. Theorem 1.9.9. Let X be a reflexive Banach space and let X ∗ be the dual space. Assume that X ∗ has the Kadec–Klee property and {xn } is a bounded sequence in X. If, for some u, v ∈ ωw (xn ), limn→∞ ‖txn + (1 − t)u − v‖, ∀t ∈ [0, 1] exists, then u = v. Let X be a uniformly convex Banach space and let X ∗ be its dual space. Assume that X ∗ has the Kadec–Klee property and ωw (xn ) ⊂ ⋂n≥1 Fix(Tn ). Then ωw (xn ) must be a singleton. Definition 1.9.2. Let C be a nonempty closed convex subset of a Banach space X. If ‖xn+1 − u‖ ≤ ‖xn − u‖, ∀u ∈ C, then the sequence {xn } ⊂ X is said to be Fejér monotone on C. Theorem 1.9.10. Let X be a Banach space satisfying the Opial condition and let C be a nonempty, closed, and convex subset of X. Assume that {xn } ⊂ K is Fejér monotone on C, and ωw (xn ) ⊂ C. Then ωw (xn ) is a singleton. 1.9.2 Accretive operators The class of accretive operators was introduced independently by Browder [8] and Kato [37] in 1967. Interest in accretive operators stems mainly from their connections with the existence theory of solutions for nonlinear evolution equations in Banach spaces. It is well known that many real problems with physical background can be modeled in terms of an initial value problem of the form: du + Au = θ, { dt u(0) = u0 ,

(IVP)

where A is an accretive operator in an appropriate Banach space. Typical examples of such evolution equations may be found in modes involving the heat, wave or Schrödinger equations (see, e. g., [112, 113] and the references therein). An early significant and fundamental result in the theory of accretive operators, due to Browder [8], states that equation (IVP) is solvable when A is locally Lipschitz continuous and accretive on a Banach space X. Utilizing the existence result for equation (IVP), Browder [8] further proved that if A : X → X is locally Lipschitz continuous and accretive, then A is m-accretive. Afterwards, Martin [48] extended the result of Browder to the case where A is continuous and accretive. Deimling [27], Kartstos [36],

1.9 Some classes of nonlinear mappings | 59

Morales [51, 52], and others extensively developed the results of Browder and Martin. Within the past 50 years or so, the accretive operator theory has been developed rapidly and extensively. The achievements of the accretive operator theory are rich and colorful. It is very difficult for one to cover every aspect of the theory. Now we pick up some important and basic results which will be needed in the subsequent chapters. Let X be a Banach space. Let A : Dom(A) ⊂ X → 2X be a set-valued mapping. Sometimes, we write set-valued mapping by A ⊂ X × X. Dom(A) = {x ∈ X : Ax ≠ 0},

Ran(A) = ∪{Ax : x ∈ Dom(A)},

Graph(A) = {[x, y] ∈ X × X : x ∈ Dom(A), y ∈ A(x)}

Definition 1.9.3. Let A ⊂ X × X be a set-valued mapping. Then A is said to be (i) accretive if, for all x, y ∈ Dom(A), 󵄩 󵄩 ‖x − y‖ ≤ 󵄩󵄩󵄩x − y + s(u − v)󵄩󵄩󵄩,

∀s > 0, ∀u ∈ Ax, ∀v ∈ Ay.

Equivalently, A is accretive if, for all x, y ∈ Dom(A) there exists some j(x − y) ∈ J(x − y) such that ⟨u − v, j(x − y)⟩ ≥ 0,

∀u ∈ Ax, ∀ v ∈ Ay.

(ii) η-strongly accretive if there exist a positive real number η and j(x − y) ∈ J(x − y) such that, for all x, y ∈ Dom(A), ⟨u − v, j(x − y)⟩ ≥ η‖x − y‖2 ,

∀u ∈ Ax, ∀v ∈ Ay.

(iii) v-inverse-strongly accretive if there exist a positive real number v and j(x − y) ∈ J(x − y) such that, for all x, y ∈ Dom(A), ⟨u − v, j(x − y)⟩ ≥ v‖u − v‖2 ,

∀u ∈ Ax, ∀v ∈ Ay.

(iv) g-strongly accretive if there exist a function g : ℝ+ → ℝ+ with g(0) = 0, ∀r0 > 0, lim infr→r0 g(r) > 0, lim supr→∞ g(r) > 0 and j(x − y) ∈ J(x − y) such that, for all x, y ∈ Dom(A), ⟨u − v, j(x − y)⟩ ≥ g(‖x − y‖)‖x − y‖,

∀u ∈ Ax, ∀ v ∈ Ay.

(v) m-accretive if A is accretive and there exists some λ > 0 such that Ran(I + λA) = X. (vi) maximal accretive if A is accretive and for any accretive operator B ⊃ A, we have A = B. Theorem 1.9.11. A set-valued mapping A is m-accretive if and only if A is accretive and Ran(I + λA) = X, ∀λ > 0.

60 | 1 Introduction and preliminaries Example 1.9.2. Let C be a nonempty, closed, and convex subset of a Banach space X, and let T : C → C be a nonexpansive operator. Let A := I − T. Then A : C → X is an accretive mapping and C = Dom(A) ⊂ ⋂ Ran(I + rA). r>0

In such a case, we say that A satisfies the “range condition”. Example 1.9.3. Let H be a Hilbert space and let f : H → ℝ be a proper, lower semicontinuous convex function. Then 𝜕f : H → 2H is an m-accretive operator. Let A ⊂ X × X be an accretive operator and let Jλ := (I + λA)−1 , where λ is a positive real number. Then Jλ : Ran(I + λA) → Dom(A) is said to be the resolvent of A, denoted by Aλ := λ−1 (I − Jλ ), ∀λ > 0; Aλ : Ran(I + λA) → Dom(A) is said to the Yosida approximation. Both Jλ and Aλ are single-valued. Also Dom(Jλ ) = Ran(I + λA) and Ran(Jλ ) = Dom(A). We list some basic properties of Jλ and Aλ as follows. Theorem 1.9.12. (i) ‖Jλ x − Jλ y‖ ≤ ‖x − y‖, ∀x, y ∈ Ran(I + λA). (ii) Aλ is accretive and ‖Aλ x − Aλ y‖ ≤ (iii) (iv) (v) (vi) (vii)

2 ‖x − y‖, λ

∀x, y ∈ Ran(I + λA).

Aλ x ∈ AJλ x, ∀x ∈ Ran(I + λA). ‖Aλ x‖ ≤ |Ax|, ∀x ∈ Dom(A) ∩ Ran(I + λA), where |Ax| = inf{‖z‖ : z ∈ Ax}. A−1 (θ) = {x ∈ Dom(A) : θ ∈ A(x)} = Fix(Jλ ), ∀λ > 0. For all 0 < μ ≤ λ, and x ∈ Ran(I − λA), we have ‖Aλ x‖ ≤ ‖Aμ x‖. For all λ, μ > 0 and x ∈ Dom(Jλ ), we have λ−μ μ x+ J x ∈ Dom(Jμ ), λ λ λ

μ λ−μ Jλ x = Jμ ( x + J x). λ λ λ

(viii) Let c2 ≥ c1 > 0. Then ‖Jc1 x − x‖ ≤ ‖Jc2 x − x‖, ∀x ∈ Dom(Jλ ). (ix) Let Dom(Jλ ) ⊂ ⋂λ>0 Ran(I − λA). Then ∀x ∈ Dom(A), λ, μ > 0, and m, n > 0, 1 󵄩󵄩 n m 󵄩 2 2 2 󵄩󵄩Jλ x − Jμ x󵄩󵄩󵄩 ≤ [(nλ − mμ) + nλ + mμ ] 2 |Ax|.

Theorem 1.9.13 (Reich [74], 1980; Takahashi [88], 1984). Let X be a reflexive Banach space whose norm is uniformly G-differentiable. Let A ⊂ X × X be an accretive operator satisfying the range condition Dom(A) ⊂ ⋂r>0 Ran(I + rA). Let C be a nonempty, closed, and convex subset of X satisfying Dom(A) ⊂ C ⊂ ⋂r>0 Ran(I + rA). Suppose that, for nonexpansive mappings, each weakly compact convex subset of C has the f. p. p., θ ∈ Ran(A). Then, for any x ∈ C, limt→∞ Jt x exists and belongs to A−1 (θ). In addition, if we define Qx = limt→∞ Jt x, then Q is a unique sunny nonexpansive retraction from Q onto A−1 (θ).

1.9 Some classes of nonlinear mappings | 61

Theorem 1.9.14 (Morales and Chidume [53], 1999). Let X be a Banach space and let C be a nonempty, closed, and convex subset of X. Let A : C → X be a continuous g-strongly accretive operator satisfying the “flow-invariance” condition: lim

λ→0+

d(x − λAx, C) = 0. λ

If there exists some point x0 ∈ C such that lim infr→r0 g(r) > ‖Ax0 ‖, then there exists a unique z ∈ C such that Az = θ. Theorem 1.9.15 (Martin [47], 1973). Let X be a Banach space and let D be a nonempty closed set of X. Let B : D → X be a continuous η-strongly accretive mapping. If B satisfies the “flow-invariance” condition lim+

h→0

d(x − hBx, D) = 0, h

∀x ∈ D,

then there exists a unique x∗ ∈ D such that Bx∗ = θ. Theorem 1.9.16 (Deimling [27], 1974). Let X be a Banach space and let D be a nonempty closed set of X. Let A : D → X be a continuous mapping. If ⟨Ax − Ay, j(x − y)⟩ ≥ α(‖x − y‖)‖x − y‖,

∀x, y ∈ D, j(x − y) ∈ J(x − y),

where α : ℝ+ → ℝ+ is a continuous function, and (i) α(0) = 0; (ii) α(r) > 0, ∀r > 0; (iii) α(r) → ∞ as r → ∞; (iv) lim infr→0 α(r)/r > 0. If lim

h→0+

d(x − hBx, D) = 0, h

∀x ∈ D,

then there exists a unique x∗ ∈ D such that Ax ∗ = θ. Theorem 1.9.17 (Ray [71], 1980). Let X be a Banach space and let D be a nonempty, closed, and convex subset of X. Assume that D has the f. p. p. for nonexpansive mappings, and A : D → X is a continuous accretive operator. If A satisfies the “flow-invariance” condition, then θ ∈ A(D). Generally speaking, the sum of two accretive operators may not be accretive in a Banach space since we cannot find the same j(x − y) ∈ J(x − y) for the two accretive operators at the same time. Morales [54] proved the following result. Theorem 1.9.18 (Morales [54], 2007). Let X be a Banach space and let C be a nonempty convex subset of X. Let A : C → X be a continuous accretive operator. If A satisfies the

62 | 1 Introduction and preliminaries “flow-invariance” condition: lim+

h→0

d(x − hAx, C) = 0, h

∀x ∈ C,

then ⟨Ax − Ay, j(x − y)⟩ ≥ 0, ∀x, y ∈ C, ∀j(x − y) ∈ J(x − y). Theorem 1.9.19 (Garcéa-Falset and Morales [29], 2005). Let X be a Banach space. Let A ⊂ X × X be an m-accretive operator and let B : Dom(B) = X → X be a continuous g-strongly accretive operator. Then, ∀μ, λ > 0, both A + μB and B + λA are surjective, that is, Ran(A + μB) = Ran(B + λA) = X. Hence, for any λ > 0, there exists a unique continuous path {xλ } ⊂ X such that, ∀z ∈ X, z ∈ Bxλ + λAxλ . Theorem 1.9.20. Let X be a uniformly smooth Banach space. Let A ⊂ X × X be an m-accretive operator and let B : Dom(B) = X → X be a demicontinuous accretive operator. Then A + B is m-accretive. 1.9.3 Monotone mappings The terminology of mappings of monotone type was introduced by Browder, Minty, and Brézis et al. in the early 1960s. The main motivation for the study of mappings of monotone type was inspired by a lot of problems from pure mathematics and applied sciences. The interest and importance of this class of mappings lies in the fact that both theoretical and practical problems can be modeled in terms of mappings of monotone type, and solving these problems usually reduces to finding a zero of a maximal monotone mapping. Examples of mappings of monotone type are rich and countless, for instance, under appropriate assumptions on a Banach space X, the normalized ∗ duality mapping J : X → 2X is monotone, strictly monotone, η-strongly monotone, φ-strongly monotone, and maximal monotone. Another important example is the subdifferential 𝜕f of a lower semicontinuous proper convex function f : X → ℝ. It is well known that if a Banach space X is reflexive, then 𝜕f is maximal monotone. More examples of mappings of monotone type can be found in Rockafellar [76, 77], Browder [8], Pascali and Sburlan [61], and Zeidle [112, 113]. The achievements of the theory of mappings of monotone type are flourishing and perfected, especially in reflexive Banach spaces. Indeed, the theory of monotone mappings, which has a close relation with optimization theory, variational inequalities, and equilibrium problems, finds lots of applications in the boundary value problems of elliptic partial differential equations and parabolic partial differential equations, in the solubility of the Hammerstein nonlinear integral equations. In this subsection, we pick up some basic concepts and important results for mappings of monotone type, which will be needed in the subsequent chapters. All of the following concepts and results can be found in the references mentioned above.

1.9 Some classes of nonlinear mappings | 63

In the sequel, we always use X to denote a real Banach space and X ∗ the topological conjugate of X, unless otherwise indicated. Also, we use A ⊂ X × X ∗ to denote a ∗ multi-valued mapping from X into 2X , and A : X → X ∗ a single-valued mapping from X into X ∗ . Definition 1.9.4. Let X be a real Banach space and let X ∗ be the conjugate space of X. Let T ⊂ X × X ∗ be a mapping. Then T is said to be (i) monotone if, for all x, y ∈ Dom(T) and u ∈ Tx, v ∈ Ty, ⟨u − v, x − y⟩ ≥ 0. (ii)

If ⟨u − v, x − y⟩ = 0 implies x = y, then T is said to be strictly monotone. η-strongly monotone if there exists a positive real number η > 0 such that, for all x, y ∈ Dom(T) and u ∈ Tx, v ∈ Ty, ⟨u − v, x − y⟩ ≥ η‖x − y‖2 .

(iii) φ-strongly monotone if there exists a strictly increasing function φ : ℝ+ → ℝ+ with φ(0) = 0 such that ⟨u − v, x − y⟩ ≥ φ(‖x − y‖)‖x − y‖ for all x, y ∈ Dom(T), u ∈ Tx, v ∈ Ty; (iv) ν-inverse-strongly monotone if there exists a positive real number ν > 0 such that, for all x, y ∈ Dom(T) and u ∈ Tx, v ∈ Ty, ⟨u − v, x − y⟩ ≥ ν‖u − v‖2 . (v)

maximal monotone if [x, y] ∈ X × X ∗ , ⟨x − u, y − v⟩ ≥ 0, for all [u, v] ∈ Graph(T) ⇒ [x, y] ∈ Graph(T). (vi) generalized pseudomonotone if [xn , fn ] ∈ Graph(T), xn ⇀ x, fn ⇀ f and lim sup⟨fn , xn − x⟩ ≤ 0 implies that [x, f ] ∈ Graph(T) n→∞

and

lim ⟨f , x ⟩ n→∞ n n

= ⟨f , x⟩;

(vii) of type (M) if [xn , fn ] ∈ Graph(T), xn ⇀ x, fn ⇀ f , and lim supn→∞ ⟨fn , xn − x⟩ ≤ 0 imply [x, f ] ∈ Graph(T); Clearly, the zero mapping defined on the whole space X is maximal monotone, however, the zero mapping defined on a proper subset of X is monotone but not necessarily maximal monotone. In order to introduce the concept of pseudomonotone mappings, we give the socalled upper semicontinuity concept for a multi-valued mapping in a general topological space. Definition 1.9.5. Let X and Y be topological spaces and let T ⊂ X ×Y be a multi-valued mapping. Then T is said to be upper semicontinuous at x ∈ Dom(T) if, for any neighborhood V of Tx ⊂ Y, there exists some neighborhood U of X such that T(U) = {f ∈ Y : f ∈ Ty, y ∈ Dom(T) ∩ U} ⊂ V.

64 | 1 Introduction and preliminaries Furthermore, T is said to be upper semicontinuous if T is upper semicontinuous at every x ∈ Dom(T). We remark here that a single-valued mapping is upper semicontinuous if and only if it is continuous. Definition 1.9.6. Let X be a real reflexive Banach space and let X ∗ be the topological conjugate of X. A multi-valued mapping T ⊂ X × X ∗ is said to be pseudomonotone if the following conditions are satisfied: (C1) for every x ∈ Dom(T), Tx is a closed convex subset of X ∗ ; (C2) {xn } ⊂ Dom(T), xn ⇀ x, fn ∈ Txn and lim supn→∞ ⟨fn , xn − x⟩ ≤ 0 imply that, ∀y ∈ Dom(T), there exists f (y) ∈ Tx such that ⟨f (y), x − y⟩ ≤ lim inf⟨fn , xn − y⟩; n→∞

(C3) for every subspace W of X with dim W < ∞, the restriction T|W : Dom(T) ∩ W → ∗ 2X is upper semicontinuous in the w-topology σ(X ∗ , X). Mappings defined above are said to be of monotone type according to their usage. By means of above definitions, it is not difficult to show the following implications: (here we assume that X is a real reflexive Banach space, and T ⊂ X × X ∗ is a mapping such that Dom(T) = X): (R1 ) T being maximal monotone implies that T is pseudomonotone; (R2 ) T being pseudomonotone implies that T is generalized pseudomonotone; (R3 ) T being generalized pseudomonotone implies that T is of type (M); (R4 ) T being maximal monotone implies that T is monotone; (R5 ) T being η-strongly monotone implies that T is φ-strongly monotone; furthermore, being both η-strongly monotone and φ-strongly monotone implies being strictly monotone; (R6 ) T being ν-inverse-strongly monotone implies that T is 1/ν-Lipschitz continuous and monotone; (R7 ) T being strictly monotone implies that T is monotone. We remark in passing that the converse implications above do not hold in general. Now we present some examples of monotone type mappings. Example 1.9.4. Let φ : ℝ → ℝ be a monotonically increasing function. Then, the mapping T : ℝ → 2ℝ defined by Tx = [φ(x − 0), φ(x + 0)],

∀x ∈ ℝ,

is maximal monotone, where φ(x − 0) and φ(x + 0) stand for the left- and right-limit of φ at x, respectively.

1.9 Some classes of nonlinear mappings | 65

Example 1.9.5. Let X = H = l2 and define a mapping T : l2 → l2 by Txn = αn xn ,

∀{xn } ∈ l2 ,

where αn > 0 and limn→∞ αn = 0. Then T is strictly monotone, but not strongly monotone. Example 1.9.6. Let X = ℝ and define a mapping T : ℝ → ℝ by t,

Tx = {

t + 1,

t ≤ 0, t > 0.

Then T : ℝ → ℝ is monotone, but not maximal monotone. Example 1.9.7. Let A be a semipositive definite matrix of order n, n ∈ ℕ. Then A is monotone, and A is η-strongly monotone if A is a positive definite matrix of order n, where η = min1≤i≤n {λi }, {λi } are all eigenvalues of A. Example 1.9.8. Let X be a real reflexive Banach space. Let φ : X → ℝ be a lower semicontinuous, proper, and convex function. Then 𝜕φ ⊂ X×X ∗ is maximal monotone. Example 1.9.9. Let X = H be a real Hilbert space and let T : H → H be a nonexpansive mapping. Set A = I − T. Then A ⊂ H × H is a 2-Lipschitz continuous and 21 -inversestrongly monotone mapping. Example 1.9.10. Let X be a real Banach space and let J ⊂ X × X ∗ be the normalized duality mapping. Then the following assertions are known: (a1 ) J is monotone; (a2 ) if X is reflexive, then J is surjective, i. e., JX = X ∗ ; (a3 ) if X is smooth, then J : X → X ∗ is single-valued; (a4 ) if X is strictly convex, then J is strictly monotone; (a5 ) if X is reflexive and smooth, then J is maximal monotone; (a6 ) if X is real uniformly convex, then J is ωR -strongly monotone on any closed ball BR [θ] = {x ∈ X : ‖x‖ ≤ R},

R > 0,

where ωR : ℝ → ℝ is a function given in Theorem 1.8.16. (a7 ) let X = Lp , 1 < p ≤ 2. Then J is η-strongly monotone with η = p − 1 ∈ (0, 1]. For the generalized duality mapping Jp , we have similar conclusions. Example 1.9.11. Let X be a real Banach space. Let C be a nonempty, closed, and convex subset of X, and let A : C → X ∗ be a hemicontinuous and monotone mapping. Then the mapping T ⊂ X × X ∗ defined by NC (x) + Ax,

Tx = {

0,

x ∈ C, x∉C

66 | 1 Introduction and preliminaries is maximal monotone. Furthermore, N(T) = VI(C, A), where NC (x) = {x ∈ X ∗ : ⟨y − x, x ∗ ⟩ ≤ 0, y ∈ C} denotes the normal cone of C at x, VI(C, A) = {u ∈ C : ⟨Au, y − u⟩ ≥ 0 ∀y ∈ C} denotes the solution set of the variational inequality problem ⟨Au, y − u⟩ ≥ 0 ∀y ∈ C. Now we have the following observations: (O1 ) under translation and positive multiplicative transformations, the monotonicity of a mapping of monotone type stays unchanged; more precisely, if A enjoys some kind monotonicity, then the mapping  ⊂ X × X ∗ defined by ̂ = A(x + x0 ), Ax

x ∈ Dom(A)̂ = Dom(A) − x0

and x0 ∈ Dom(A) is a fixed vector, enjoys the same kind monotonicity; for a fixed vector z ∈ X ∗ , the mapping à ⊂ X × X ∗ defined by ̃ = Ax − z, Ax

x ∈ Dom(A)̃ = Dom(A),

enjoys the same kind monotonicity; for λ > 0, λA enjoys same kind monotonicity. (O2 ) if Ai ⊂ X × X ∗ (i = 1, 2, . . . , n) are monotone (pseudomonotone), so is ∑ni=1 Ai ; (O3 ) if X is a real reflexive Banach space, then A is monotone (maximal monotone) if and only if A−1 is monotone (maximal monotone). In order to study and understand various properties of monotone type mappings, it is necessary to give some related concepts. Definition 1.9.7. A mapping A ⊂ X × X ∗ is said to be locally bounded at x0 ∈ Dom(A) if there exists some neighborhood U of x0 such that A(U) = {f ∈ X ∗ : [x, f ] ∈ Graph(A), x ∈ U} is a bounded subset of X ∗ ; A is said to be locally bounded on Dom(A) if A is locally bounded at every x ∈ Dom(A); A−1 ⊂ X ∗ ×X is said to be locally bounded on X ∗ if for any f ∈ X ∗ there exist some r > 0 and open ball Br (f ) such that set {x ∈ X : Ax ∩ Br (f ) ≠ 0} is a bounded subset of X. It is evident that A is locally bounded at x0 ∈ Dom(A) if and only if for any {fn } ⊂ X ∗ , fn ∈ Ax0 , if xn → x0 ∈ Dom(A), then {fn } remains bounded. It is also clear that A−1 is locally bounded on X ∗ if and only if for any {xn } ⊂ X, yn ∈ Axn , if yn → f ∈ X ∗ , then {xn } remains bounded. We remark here that a mapping which is locally bounded on Dom(A) is certainly bounded in a normed linear space X with dimX < ∞.

1.9 Some classes of nonlinear mappings | 67

Definition 1.9.8. Let A, B ⊂ X × X ∗ be multi-valued mappings. Then B is said to be an extension of A, denoted by A ⊂ B, if Dom(A) ⊂ Dom(B) and Ax ⊂ Bx for all x ∈ Dom(A). Definition 1.9.9. A set M ⊂ X × X ∗ is called monotone if ⟨f − g, x − y⟩ ≥ 0,

∀[x, f ], [y, g] ∈ M.

Furthermore, M is called a maximal monotone set if M is a monotone set and it is not a proper subset of any monotone set in X × X ∗ . It is evident that A ⊂ X × X ∗ is monotone (maximal monotone) if and only if Graph(A) is a monotone (maximal monotone) set of X × X ∗ . Definition 1.9.10. Let C be a nonempty subset of X. Then a set G ⊂ C × X ∗ is said to be maximal monotone if it is monotone and it is not a proper subset of any monotone set ∗ in C × X ∗ ; T : Dom(T) ⊂ C → 2X is said to be maximal monotone with respect to C if Graph(T) is a maximal monotone set of C × X ∗ . Definition 1.9.11. A mapping T ⊂ X × X ∗ is said to be coercive if there exists a function c : ℝ+ → ℝ with c(r) → ∞ as r → ∞ such that ⟨f , x⟩ ≥ c(‖x‖)‖x‖,

∀[x, f ] ∈ Graph(T).

Furthermore, T is said to be coercive with respect to h ∈ X ∗ if there exists r > 0 such that ⟨f − h, x⟩ > 0,

for all [x, f ] ∈ Graph(T) and ‖x‖ > r.

Using these concepts, we can state some fundamental and important results from the theory of mappings of monotone type. Theorem 1.9.21. Let A ⊂ X × X ∗ be a monotone mapping. Then the following assertions hold true: (a1 ) A is locally bounded on int(Dom(A)); (a2 ) if A : Dom(A) ⊂ X → X ∗ is single-valued and hemicontinuous, then A is demicontinuous on int(Dom(A)); (a3 ) A has a maximal monotone extension A;̃ if X is a real reflexive Banach space and C a nonempty, closed, and convex subset of X, then there exists a maximal monotone extension à of A|C such that Dom(A)̃ ⊂ C, where A|C is the restriction of A on C, i. e., Graph(A|C ) = {[x, f ] ∈ Graph(A) : x ∈ C ∩ Dom(A)}; (a4 ) if A is maximal monotone, then (i) A is s-w∗ -upper semicontinuous on int(Dom(A)); (ii) Ax = ⋂[y,g]∈Graph(A) {f ∈ X ∗ : ⟨f − g, x − y⟩ ≥ 0} for all x ∈ Dom(A);

68 | 1 Introduction and preliminaries (iii) Ax is w∗ -closed convex for all x ∈ Dom(A); (iv) Ax is bounded for all x ∈ int(Dom(A)); (v) Ax is w∗ -compact for all x ∈ int(Dom(A)); (vi) Graph(A) is demiclosed, i. e., either (dc1 ) for any {xn } ⊂ Dom(A), xn ⇀ x, yn ∈ Axn and yn → y imply [x, y] ∈ Graph(A) or (dc2 ) for any {xn } ⊂ Dom(A), xn → x, yn ∈ Axn and yn ⇀ y imply [x, y] ∈ Graph(A). In the following theorems, X is supposed to be a real reflexive Banach space. Theorem 1.9.22. Let A ⊂ X × X ∗ be a pseudomonotone mapping. Then Ax is bounded for all x ∈ int(Dom(A)). The next fundamental result is closely related to the variational inequality problems with multi-valued mappings of monotone type. Theorem 1.9.23. Let C be a closed convex subset of X. Let A ⊂ X × X ∗ be a monotone mapping with [θ, θ∗ ] ∈ Graph(A|C ) and let P : C → X ∗ be pseudomonotone, bounded, and coercive with respect to h ∈ X ∗ . Then there exists x0 ∈ C such that ⟨f + Px0 − h, x − x0 ⟩ ≥ 0, for all [x, f ] ∈ Graph(A|C ). Thinking of (O1 ), we see that condition “[θ, θ∗ ] ∈ Graph(A|C )” can be replaced by Dom(A) ∩ C ≠ 0. Theorem 1.9.24. Let A ⊂ X × X ∗ be a maximal monotone mapping and let C a closed convex subset of X with θ ∈ Dom(A) and Dom(A) ⊂ C. Let P : C → X ∗ be pseudomonotone, bounded, and coercive. Then, for any h ∈ X ∗ , there exists x0 ∈ C such that h ∈ (A + P)x0 , which implies Graph(A + P) = X ∗ . We remark that condition “θ ∈ Dom(A)” can be dropped. In the following theorems, X is supposed to be real reflexive and smooth. Theorem 1.9.25. Let A ⊂ X × X ∗ be a maximal monotone mapping. Then Ran(A + λJ) = X ∗ ,

∀λ > 0,

where J : X → X ∗ is the normalized duality mapping. Theorem 1.9.26. Let A ⊂ X×X ∗ be maximal monotone and coercive. Then Ran(A) = X ∗ .

1.9 Some classes of nonlinear mappings | 69

Theorem 1.9.27. Let A : Dom(A) = X → X ∗ be monotone hemicontinuous. Then A is maximal monotone. Theorem 1.9.28. Let A : Dom(A) = X → X ∗ be hemicontinuous and φ-strongly monotone such that φ(r) → ∞ as r → ∞. Then Ran(A) = X ∗ . Theorem 1.9.29. Let A, B ⊂ X×X ∗ be monotone. Then A is maximal monotone whenever A + B is maximal monotone. In the following theorems, X is supposed to be real reflexive, smooth and strict convex. Theorem 1.9.30. Let A ⊂ X × X ∗ be monotone. If Ran(A + λJ) = X ∗ for λ > 0, then (A + λJ)−1 : X ∗ → X is single-valued, demicontinuous, and maximal monotone. The next theorem presents a criterion for a monotone mapping to be maximal monotone. Theorem 1.9.31. Let A ⊂ X × X ∗ be a monotone mapping. Then A is maximal monotone if and only if there exists λ > 0 such that Ran(A + λJ) = X ∗ . By means of Theorem 1.9.31, we can prove the following important result. Theorem 1.9.32. Let A ⊂ X × X ∗ be a maximal monotone mapping. Then the following conclusions hold true: (c)1 for arbitrarily given x ∈ X and λ > 0, there exists a unique solution Rλ x ∈ Dom(A) of the inclusion θ ∈ J(y − x) + λAy, which yields Rλ x = (I + λJ −1 A) x. −1

Write Aλ x := λ−1 J(x − Rλ )x. Then Aλ x = (λJ −1 + A−1 ) x, −1

Aλ x ∈ ARλ x

and J(Rλ x − x) + λAλ x = θ. The single-valued mappings Rλ : X → Dom(A) and Aλ : X → X ∗ are called resolvent and Yosida approximation of A, respectively. (c)2 Rλ x → x on co(Dom(A)) as λ → 0+ , Dom(A) and Ran(A) are convex, ‖Aλ x‖ → ∞ as λ → 0+ if x ∈ ̸ Dom(A) and Aλ x ⇀ A∘ x on Dom(A) as λ → 0+ , where A∘ x is the unique element of Ax having minimal norm, i. e., ‖A∘ x‖ = d(θ, Ax). Furthermore, Aλ is a bounded demicontinuous maximal monotone mapping with Aλ (θ) = θ if θ ∈ Aθ. If X ∗ is locally uniformly convex, then Aλ x → A∘ x on Dom(A) as λ → 0+ .

70 | 1 Introduction and preliminaries Theorem 1.9.33. Let X be a real reflexive, smooth, and strictly convex Banach space whose topological conjugate space X ∗ is locally uniformly convex. Let A ⊂ X × X ∗ be maximal monotone. Then the following statement holds: for any xλ , x ∈ Dom(A), if xλ → x as λ → 0+ , A∘ xλ → A∘ x as λ → 0+ , then Aλ xλ → A∘ x as λ → 0+ . Next theorem presents a criterion for a maximal monotone to be surjective. Theorem 1.9.34. Let X be a real reflexive, smooth, and strictly convex Banach space and let A ⊂ X × X ∗ be a maximal monotone mapping. Then Ran(A) = X ∗ if and only if A−1 is locally bounded on X ∗ . Theorem 1.9.35. Let X be a real reflexive, smooth, and strictly convex Banach space. Let A ⊂ X × X ∗ be a maximal monotone mapping. Suppose that there exists r > 0 and x̄ ∈ Dom(A) such that ⟨f , x − x⟩̄ ≥ 0,

∀[x, f ] ∈ Graph(A) with ‖x − x‖̄ ≥ r.

Then there exists x0 ∈ Dom(A) ∩ Br [x]̄ satisfying θ∗ ∈ Ax0 . Usually, Theorem 1.9.35 is called the acute angle principle for monotone mappings. As a corollary of Theorem 1.9.35, we have the following acute angle principle for single-valued monotone mappings. Theorem 1.9.36. Let X be real reflexive, smooth, and strictly convex Banach space. Let A : Dom(A) = X → X ∗ be monotone and hemicontinuous. Suppose that there exists r > 0 and x̄ ∈ X such that ⟨Ax, x − x⟩̄ ≥ 0,

̄ ∀x ∈ 𝜕Br (x).

Then there exists x0 ∈ Br [x]̄ satisfying Ax0 = θ∗ . Generally speaking, the sum of two maximal monotone mappings is not necessarily maximal monotone, however, we have the following known result. Theorem 1.9.37 (Rockafellar [77]). Let X be a real reflexive, smooth, and strictly convex Banach space with the topological conjugate space X ∗ . Let A, B ⊂ X × X ∗ be maximal monotone mappings. If one of the following conditions holds: (1) Dom(A) ∩ int(Dom(B)) ≠ 0; (2) there exists x ∈ Dom(A) ∩ Dom(B) such that B is locally bounded at x, then A + B is maximal monotone. The following three results, which are from Takahashi [87], are closely related to the monotone type single-valued variational inequality problems.

1.9 Some classes of nonlinear mappings | 71

Theorem 1.9.38. Let C be a nonempty convex subset of a topological linear space X, and let T be a hemicontinuous and monotone mapping of C into X ∗ . Let x0 ∈ C be an element of C. Then the following statements are equivalent each other: (1) ⟨Tx, x − x0 ⟩ ≥ 0, ∀x ∈ C; (2) ⟨Tx0 , x − x0 ⟩ ≥ 0, ∀x ∈ C. Theorem 1.9.39. Let C be a nonempty compact convex subset of a topological linear space X, and let T be a monotone mapping of C into X ∗ . Then there exists x0 ∈ C such that ⟨Tx, x − x0 ⟩ ≥ 0, ∀x ∈ C. Theorem 1.9.40. Let C be a nonempty compact convex subset of a topological linear space X, and let T be a monotone and hemicontinuous mapping of C into X ∗ . Then there exists x0 ∈ C such that ⟨Tx0 , x − x0 ⟩ ≥ 0, ∀x ∈ C. As a consequence of Theorem 1.9.40, we have the following known result. Theorem 1.9.41. Let C be a nonempty bounded and closed convex subset of a real reflexive Banach space X, and let T be a monotone and hemicontinuous mapping of C into X ∗ . Then there exists x0 ∈ C such that ⟨Tx0 , x − x0 ⟩ ≥ 0 ∀x ∈ C. Applying Theorem 1.9.41 to a closed ball Br [θ], r > 0, we can obtain the following interesting result. Theorem 1.9.42. Let X be a real reflexive and smooth Banach space. Let T : Br [θ] → X ∗ be monotone and hemicontinuous. Then the following conclusions hold: (1) there exists x0 ∈ Br [θ] such that ⟨Tx0 , x − x0 ⟩ ≥ 0

∀x ∈ Br [θ];

(2) if x0 ∈ Br (θ), then Tx0 = θ∗ ; (3) if x0 ∈ 𝜕Br [θ] and Tx0 ≠ θ∗ , then there exists λ > 0 such that Tx0 = −λJx0 , equivalently, if for any t ≥ 0 and x ∈ 𝜕Br [θ], one has Tx + tJx ≠ θ∗ , then Tx0 = θ∗ . When C is not bounded, we have at least two ways to prove the existence of solutions for the above monotone type variational inequality problem. One way is to apply the following theorems and the other is to apply the previous Example 1.9.11. Theorem 1.9.43 (Baiocchi and Capelo [5], 1984). Let C be a nonempty, closed, and convex subset of a real reflexive Banach space X, and let T be a monotone and hemicontinuous mapping of C into X ∗ . Suppose that T is coercive in the following sense: there exists w ∈ C such that ⟨Tx, x − w⟩ →∞ ‖x‖

as ‖x‖ → ∞,

where w ∈ C is a fixed element. Then the following conclusions hold:

72 | 1 Introduction and preliminaries (1) for any h ∈ X ∗ , there exists x0 ∈ C such that ⟨Tx0 − h, x − x0 ⟩ ≥ 0,

∀x ∈ C;

(2) if T is strictly monotone, then above x0 ∈ C is a unique. Theorem 1.9.44 (Baiocchi and Capelo [5], 1984). Let C be a nonempty, closed, and convex subset of a real reflexive Banach space X, and let T be a η-strongly monotone and hemicontinuous mapping of C into X ∗ . Then for any h ∈ X ∗ , there exists a unique x0 ∈ C such that ⟨Tx0 − h, x − x0 ⟩ ≥ 0,

∀x ∈ C.

In 1966, Browder [9] established the following known result. In 1967, Mosco [55] gave a new proof to this fact by combining the previous a result due to Browder with the product-space technique. Theorem 1.9.45. Let T be a monotone hemicontinuous mapping from a real reflexive Banach space X to its dual space X ∗ . Let f be a proper, convex, and lower semicontinuous function from X to ℝ with f (θ) = 0 such that ⟨Tu, u⟩ + f (u) →∞ ‖u‖

as ‖u‖ → ∞.

Then for any given h ∈ X ∗ , there exists a solution u0 ∈ X of the variational inequalities ⟨Tu0 − h, x − u0 ⟩ ≥ f (u0 ) − f (x)

for all x ∈ X.

In particular, If f ≡ θ, then h = Tu0 . Hence Ran(T) = X ∗ . If T ≡ θ and h = θ∗ , then f (u0 ) = min f (x). x∈X

Finally, we present more examples of monotone type mappings to support our conclusions. Example 1.9.12. Let X = ℝ2 . Then the mapping defined by φ(x, y) = xy2 (x 2 + y4 ) , −1

(x, y) ∈ ℝ2 ,

is hemicontinuous, but not demicontinuous. Example 1.9.13. Let X = ℝ2 , n ≥ 2, Dom(A) = B1 (θ), {xi : i ∈ ℕ} ⊂ 𝜕B1 (θ). Then the mapping A : B1 (θ) → ℝn by x,

Ax = { is unbounded on 𝜕B1 (θ).

(i + 1)xi ,

x ≠ x i ,

x = xi ,

1.9 Some classes of nonlinear mappings | 73

Example 1.9.14. Let X = ℝn and let f : ℝn → ℝ be a function defined by 1

{ − (1 − ‖x‖2 ) 2 , f (x) = { {∞,

‖x‖ ≤ 1,

‖x‖ > 1.

Then f is proper, lower semicontinuous, and convex function. It is also clear that f (x) is differentiable as long as ‖x‖ < 1 and x . 󳶚f (x) = √1 − ‖x‖2 Furthermore, 𝜕f (x) = 0 as long as ‖x‖ ≥ 1. Thus

x { {{ √1−‖x‖2 }, 𝜕f (x) = { { {0,

‖x‖ < 1, ‖x‖ ≥ 1.

Notice that Dom(f ) = B1 [θ] and Dom(𝜕f ) = B1 (θ). We see that 𝜕f : B1 (θ) → ℝn is maximal monotone, unbounded, and Ran(𝜕f ) = ℝn . Example 1.9.15. Let X = L2 (ℝ), Dom(A) = {x ∈ C(ℝ) : limt→∞ x(t) = 0 and x 󸀠 ∈ X}, Ax = x󸀠 , Dom(B) = Dom(A), and Bx = −x 󸀠 . Then both A and B are maximal monotone, however, the sum A + B is not maximal monotone. Example 1.9.16. Let X be a real Banach space and let f : X → ℝ+ be a function defined by f (x) = ‖x‖,

∀x ∈ X.

Then 𝜕f (x) = argmax{⟨x∗ , x⟩ : x∗ ∈ BX ∗ } = {x∗ ∈ BX ∗ : ⟨x∗ , x⟩ = ‖x‖}. Clearly, 𝜕‖θ‖ = BX ∗ and 󵄩 󵄩 𝜕f (x) = {x ∗ ∈ X ∗ : 󵄩󵄩󵄩x∗ 󵄩󵄩󵄩 = 1, ⟨x ∗ , x⟩ = ‖x‖},

x ≠ θ.

BX ∗ , 𝜕‖x‖ = { ∗ 󵄩 󵄩 {x ∈ X ∗ : 󵄩󵄩󵄩x∗ 󵄩󵄩󵄩 = 1, ⟨x∗ , x⟩ = ‖x‖},

x = θ,

Hence, x ≠ θ.

Thus, 𝜕f : X → 2X is a bounded maximal monotone mapping with Dom(𝜕f ) = X. If X ∗ is strictly convex, then ∗

󵄩 󵄩 {x∗ ∈ X ∗ : 󵄩󵄩󵄩x∗ 󵄩󵄩󵄩 = 1, ⟨x∗ , x⟩ = ‖x‖} is a singleton. In particular, if X is a real Hilbert space, then BH ,

𝜕‖x‖ = {

x , ‖x‖

x = θ,

x ≠ θ.

74 | 1 Introduction and preliminaries Example 1.9.17. Let X be a real reflexive, smooth, and strictly convex Banach space. If A : Dom(A) = X → X ∗ is a hemicontinuous monotone mapping such that ‖Ax‖ → ∞ as ‖x‖ → ∞, then Ran(A) = X ∗ . We remark that the condition “‖Ax‖ → ∞ as ‖x‖ → ∞” is weaker than the coercitivity condition ⟨Ax, x⟩ → ∞ as ‖x‖ → ∞. ‖x‖ Conversely, we have the following interesting fact. Example 1.9.18. Let A : ℝn → ℝn be monotone and Ran(A) = ℝn . Then ‖Ax‖ → ∞ as ‖x‖ → ∞. Example 1.9.19. Let H be a real Hilbert space and let A : Dom ⊂ H → H be a bounded maximal monotone mapping. Then Dom(A) = H. Example 1.9.20. Let H be a real Hilbert space and let L : Dom(L) ⊂ H → H be linear and monotone. Then L is maximal monotone if and only if Dom(L) = H and L is maximal in the family of all linear monotone K : Dom(K) → H, where Dom(K) is a subspace of H. Example 1.9.21. Let X be a real reflexive Banach space and let A ⊂ X × X ∗ be a linear and monotone mapping, i. e., Graph(A) is a subspace of X × X ∗ and ⟨f , x⟩ ≥ 0, ∀[x, f ] ∈ Ax. Then the following assertions hold: (1) if Dom(A) = X, then A is single-valued; (2) A is maximal monotone if and only if Graph(A) is closed and A∗ is monotone, ∗ where A∗ : Dom(A∗ ) ⊂ X → 2X is the adjoint of A, defined by x ∗ ∈ A∗ x if and only if ⟨x ∗ , y⟩ = ⟨f , x⟩ for all [x, f ] ∈ Graph(A). Example 1.9.22. Let H = L2 (0, 2π). Define a mapping A : Dom(A) ⊂ H → H by (Ax)(t) = x󸀠 (t),

Dom(A) = {x ∈ H : x 󸀠 (t) exists and x 󸀠 (t) ∈ H}

and mapping B : Dom(B) ⊂ H → H by (Bx)(t) = x󸀠 (t),

Dom(B) = {x ∈ H : x(0) = x(2π), x 󸀠 (t) exists and x 󸀠 ∈ H}.

Then Dom(B) ⊂ Dom(A), Ax = Bx if x ∈ Dom(B), Graph(B) ⊂ Graph(A), A and B are all unbounded maximal monotone mappings.

1.9 Some classes of nonlinear mappings | 75

1.9.4 Pseudocontractive mappings The terminology and concept of pseudocontractive mappings were introduced and used by Browder [8] in 1967. A characterization of this class of mappings is given by Browder [8]. He observed that T is pseudocontractive if and only if A := I − T is accretive. Apart from being a generalization of nonexpansive mappings, interest in the pseudocontractive mapping theory stems mainly from their firm connection with the class of accretive operators. The achievements of the fixed point theory for pseudocontractive mappings are fruitful, and so it is very difficult to cover every aspect of the theory. Therefore, we only collect and reorganize some important and useful concepts and facts of the fixed point theory for pseudocontractive mappings. Definition 1.9.12. Let X be a Banach space and let T : Dom(T) ⊂ X → X be a mapping. Then T is said to be (i) pseudocontractive if there exists j(x − y) ∈ J(x − y) such that ⟨Tx − Ty, j(x − y)⟩ ≤ ‖x − y‖2 ,

∀x, y ∈ Dom(T);

(ii) β-strongly pseudocontractive if there exist β ∈ [0, 1) and j(x − y) ∈ J(x − y) such that ⟨Tx − Ty, j(x − y)⟩ ≤ β‖x − y‖2 ,

∀x, y ∈ Dom(T);

(iii) k-strictly pseudocontractive if there exist k ∈ [0, 1) and j(x − y) ∈ J(x − y) such that 󵄩 󵄩2 ⟨Tx − Ty, j(x − y)⟩ ≤ ‖x − y‖2 − k 󵄩󵄩󵄩(I − T)x − (I − T)y󵄩󵄩󵄩 ,

∀x, y ∈ Dom(T);

(iv) g-strongly pseudocontractive if there exist g : ℝ+ → ℝ+ with g(0) = 0, ∀r0 > 0, lim infr→r0 g(r) > 0, lim supr→∞ g(r) > 0, and j(x − y) ∈ J(x − y) such that ⟨Tx − Ty, j(x − y)⟩ ≤ ‖x − y‖2 − g(‖x − y‖)‖x − y‖,

∀x, y ∈ Dom(T).

Remark 1.9.1. (i) Let A := I − T. Then T is g-strongly pseudocontractive if and only if A is g-strongly accretive. (ii) If T is k-strictly pseudocontractive, then T is L-Lipschitz, where L = 1+k ≥ 1. k Definition 1.9.13. Let C be a nonempty convex subset of a Banach space X. Then IC (x) := {y ∈ X : y = x + λ(u − x), u ∈ C, λ ≥ 0}, ∀x ∈ C, is said to be an inward set. Let T : C → X be a mapping. If Tx ∈ IC (x), ∀x ∈ C, mapping T is said to satisfy the inward condition. If Tx ∈ IC (x), ∀x ∈ C, mapping T : C → X is said to satisfy the weak inward condition. In 1976, Caristi [17] established the equivalence between the “weak inward” condition (WIC) and the “flow-invariance” condition (FIC).

76 | 1 Introduction and preliminaries Theorem 1.9.46 (Caristi [17], 1976). Let C be a nonempty, closed, and convex subset of a Banach space X, and let A : C → X be a mapping. Then A satisfies the “flowinvariance” condition if and only if T = I − A satisfies the “weak inward” condition. One can give a equivalent characterization of the “weak inward” condition. It is useful to see whether a mapping T : C → X satisfies the “weak inward” condition. Theorem 1.9.47 (Caristi [17], 1976). Let C be a real nonempty closed convex subset of a Banach space X. Let T : C → X be a mapping. Then T satisfies the “weak inward” condition if and only if x ∈ 𝜕C, x∗ ∈ X ∗ ,

x∗ (x) = sup{x∗ (y) : y ∈ C} ⇒ x ∗ (x − Tx) ≥ 0.

From Theorems 1.9.46 and 1.9.47, the following fixed point theorems are not hard to derive. Theorem 1.9.48 (Morales–Chidume [53], 1999). Let C be a nonempty closed convex subset of a Banach space X. Let T : C → X be a continuous g-strongly pseudocontractive mapping satisfying the “weak inward” condition. If there exists some point x0 ∈ C such that lim infr→∞ g(r) > ‖x0 − Tx0 ‖, then T has unique fixed point in X. Theorem 1.9.49 (Caristi [17], 1976). Let X be a Banach space. Let C be a nonempty closed convex subset such that it has the f. p. p. for nonexpansive mappings. If T : C → X is a continuous and pseudocontractive mapping satisfying the “weak inward” condition, then T at least has one fixed point in X. Theorem 1.9.50 (Zhou [114], 2008). Let X be a uniformly convex Banach space and let C be a nonempty closed convex subset of X. Let T : C → X be a continuous and pseudocontractive mapping satisfying the “weak inward” condition. Then I − T is demiclosed at the origin. Theorem 1.9.51 (Zhou [114], 2008). Let X be a reflexive Banach space satisfying the Optial condition, and let C be a nonempty closed convex subset of X. Let T : C → X be a continuous and pseudocontractive mapping satisfying the “weak inward” condition. Then I − T is demiclosed at the origin.

1.10 Some useful lemmas Lemma 1.10.1 (Tan and Xu [92], 1993). Let {an }, {bn }, and {σn } be nonnegative sequences of real numbers. Assume that an+1 ≤ (1 + σn )an + bn ,

∀n ≥ 1.

∞ If ∑∞ n=1 bn < ∞ and ∑n=1 σn < ∞, then limn→∞ an exists. If sequence {an } has a subsequence {ani } such that ani → 0 as i → ∞, then an → 0 as n → ∞.

1.11 Exercises | 77

Lemma 1.10.2 (Liu [43], 1995; Xu [99], 2002). Let {an } be a nonnegative sequence of real numbers. Assume that an+1 ≤ (1 − tn )an + tn δn + bn ,

∀n ≥ 1,

where tn ∈ [0, 1], δn ∈ ℝ and bn ∈ ℝ+ satisfy (i) tn → 0 and ∑∞ n=1 tn = ∞; (ii) lim supn→∞ δn ≤ 0; (iii) ∑∞ n=1 bn < ∞. Then an → 0 as n → ∞. Lemma 1.10.3 (Suzuki [82], 2005). Let {xn } and {yn } be bounded sequences of a Banach space X. Let {λn } be a real number sequence in (0, 1) such that 0 < lim inf λn ≤ lim sup λn < 1. n→∞

n→∞

Let xn+1 = (1 − λn )xn + λn yn , ∀n ≥ 1. If lim infn→∞ (‖yn+1 − yn ‖ − ‖xn+1 − xn ‖) ≤ 0, then limn→∞ ‖yn − xn ‖ = 0. Lemma 1.10.4 (Maingé [45], 2010). Let {Γn } be a real number sequence. Assume that there exists some subsequence {Γnj }j≥0 ⊂ {Γn } satisfying Γnj < Γnj +1 , ∀j ≥ 0. Define a sequence {τn }n≥n0 by τ(n) := max{k ≤ n | Γk < Γk+1 }. Then the following assertions hold: (i) {τn }n≥n0 is a nondecreasing sequence with τ(n) → ∞ as n → ∞; (ii) Γτ(n) ≤ Γτ(n)+1 , ∀n ≥ n0 ; (iii) Γn ≤ Γτ(n)+1 , ∀n ≥ n0 . Lemma 1.10.5 (Maingé [46], 2014). Let {an } be a nonnegative sequence of real numbers such that an+1 ≤ (1 − γn )an + γn rn ,

∀n ≥ 1,

where {rn } ⊂ (−∞, +∞) is bounded above and {γn } ⊂ [0, 1] satisfies ∑∞ n=1 γn = ∞. Then lim infn→∞ an ≤ lim supn→∞ rn .

1.11 Exercises 1.

Let X be a real normed linear space with the topological conjugate space X ∗ . Let A be a bounded subset of X in the weak topology σ(X, X ∗ ). Prove that A is bounded in the norm topology.

78 | 1 Introduction and preliminaries 2.

Let X be a real Banach space with the topological conjugate space X ∗ . Let B be a bounded subset of X ∗ in the weak star topology σ(X ∗ , X). Prove that B is bounded in the norm topology. Try to give an example to show that the above conclusion is not necessarily true in a normed linear space X. 3. Let X be a real Banach space with the topological conjugate space X ∗ . Let A be a subset of X ∗ . Prove that A is bounded in the norm topology ⇔ A is bounded in the weak topology σ(X ∗ , X) ⇔ A is bounded in the weak star topology σ(X ∗ , X). 4. Let X be a real Banach space with the topological conjugate space X ∗ . Prove that every w∗ -bounded and w∗ -closed subset of X ∗ is w∗ -compact. Present an example to show that above conclusion is not necessarily true for the weak topology. 5. Let X be a real Banach space and X ∗ the topological conjugate space of X. Let A be a subset of X ∗ . Prove that if A is w∗ -sequentially compact, then it is w∗ -compact. Try to give an example to show that a w∗ -compact subset in X ∗ is not necessarily w∗ -sequentially compact. w 6. Let X be a real Banach space and S(X) denote the unit sphere. Prove that S(X) = w BX = {x ∈ X : ‖x‖ ≤ 1}, where S(X) denotes the weak closure in the weak topology σ(X, X ∗ ). 7. Use Theorem 1.8.28 to establish the relation between b(t) and ρX (t). ∗ 8. Let X be a real Banach space and T : X → 2X a maximal monotone mapping. Prove that the function defined by φ(x) = inf{‖f ‖ : f ∈ Tx} 9.

is lower semicontinuous. Let X be a real reflexive and smooth Banach space, T : Br [θ] → X ∗ a hemicontinuous and monotone mapping. Prove that the following conclusions hold: (i) There exists some x0 ∈ Br [θ] such that ⟨Tx0 , x − x0 ⟩ ≥ 0, ∀x ∈ Br [θ].

(ii) If x0 ∈ Br (θ), then Tx0 = θ∗ . (iii) If ‖x0 ‖ = r, and Tx0 ≠ θ∗ , then there exists some s > 0 such that Tx0 = −sJx0 . 10. Let X be a real Banach space and C a nonempty closed convex subset of X. Define a function d : X → ℝ+ by d(x) = dist (x, C) = inf{‖x − y‖ : y ∈ C}. Prove that the following assertions hold: (i) d : X → ℝ+ is nonexpansive and convex. ̄ If X is (ii) If X is reflexive, then ∀x ∈ X, there exists x̄ ∈ C such that d(x) = ‖x − x‖. also strictly convex, then above x̄ ∈ C is unique. ̄ where NC (x) = {x∗ ∈ X ∗ : (iii) 𝜕d(x0 ) = NC (x)̄ ∩ {x ∗ ∈ X ∗ : x∗ ∈ 𝜕‖x0 − x‖}, ∗ ⟨x , y − x⟩ ≤ 0, ∀y ∈ C} is the normal cone of C at x.

1.11 Exercises | 79

(iv) If X = H, a real Hilbert space, then NC (x0 ) ∩ BH ,

𝜕d(x0 ) = {

x0 −PC x0 , ‖x0 −PC x0 ‖

if x0 ∈ C, if x0 ∈ ̸ C.

11. Let f : ℝn → ℝ be a convex function. Prove that ∀x ∈ ℝn , 𝜕f (x) exists and 𝜕f : ℝ → n 2ℝ is a bounded operator, that is, 𝜕f (A) is bounded subset of ℝn whenever A is a bounded subset of ℝn . 12. Consider a function f : ℝn → ℝ given by 1

{ − (1 − ‖x‖2 ) 2 , f (x) = { { + ∞,

‖x‖ ≤ 1, ‖x‖ > 1.

Compute 𝜕f (x). 13. Let X be a real Banach space and let φ : X → ℝ be a continuous convex function. Prove that ∀x ∈ X, 𝜕φ(x) ≠ 0 and 𝜕φ(x) is w∗ -compact. 14. Let X be a real Hilbert space and C a nonempty closed convex subset of H. Define a function f : H → ℝ by 1󵄩 󵄩2 f (x) = 󵄩󵄩󵄩(I − PC )x󵄩󵄩󵄩 , 2

x ∈ H.

Compute ∇f (x). Let A : H → H be a bounded linear operator. Define a function g : H → ℝ by 1󵄩 󵄩2 g(x) = 󵄩󵄩󵄩(I − PC )Ax󵄩󵄩󵄩 , 2

x ∈ H.

Compute ∇g(x). 15. Let X be a real Banach space. For p > 1, define a function φ : X → ℝ by φ(x) =

1 ‖x‖p , p

x ∈ X.

Compute φ∗ (x∗ ). 16. Let H be a real Hilbert space and let f : H → ℝ be a convex function such that ∇f (x) exists for all x ∈ H and ∇f (x) is L-Lipschitz continuous. Prove that ∇f : H → H is 1 -inverse-strongly monotone. L 17. Let X = lp (p > 1) and Jφ the generalized duality mapping with the gauge function φ(t) = t p−1 . Prove that Jφ : lp → lq is weakly sequentially continuous. Present an example to show that above conclusion does not hold for Lp (p ≠ 2). 18. Let X be a real Banach space and let f : X → ℝ be a proper lower semicontinuous convex function. Show that there exist x∗ ∈ X ∗ and μ ∈ ℝ such that f (x) ≥ ⟨x, x ∗ ⟩+ μ for all x ∈ X.

80 | 1 Introduction and preliminaries 19. Let H be a real Hilbert space, let f : H → ℝ be a proper lower semicontinuous convex function, and let x0∗ ∈ H. If g : H → ℝ is defined by 1 g(x) = ‖x‖2 + f (x) − ⟨x, x0∗ ⟩, 2

∀x ∈ H,

then show that g is proper, lower semicontinuous and convex. Further, show that ‖zn ‖ → ∞ ⇒ g(zn ) → ∞. 20. Let φ : ℝ+ → ℝ+ satisfy the following conditions: (i) t1 > t2 ⇒ φ(t1 ) ≥ φ(t2 ); (ii) φ(t) = 0 ⇔ t = 0. Let {bn } ⊂ ℝ+ satisfy limn→∞ bn = 0. Then show that any sequence {an } ⊂ ℝ+ defined by an+1 ≤ an − φ(an ) + bn+1 ,

n = 0, 1, 2, . . .

converges to 0. 21. Let (X, d) be a metric space and let K be a nonempty compact subset of X. Let T : K → K be a strictly nonexpansive mapping, that is, d(Tx, Ty) ≤ d(x, y), for all x, y ∈ K, and “=” holds if and only if x = y. Prove that T has a unique fixed point in K. 22. Let H be a real Hilbert space and let n ≥ 2, xi ∈ H and λi ∈ [0, 1] with ∑ni=1 λi = 1. Prove that 󵄩󵄩 n 󵄩󵄩2 n 󵄩󵄩 󵄩 󵄩󵄩∑ λi xi 󵄩󵄩󵄩 = ∑ λi ‖xi ‖2 − ∑ λi λj ‖xi − xj ‖2 . 󵄩󵄩󵄩 󵄩󵄩󵄩 i=1 1≤i 0, rT is υr -inverse strongly monotone. (3) For any λ ∈ (0, 1), T is λ-averaged nonexpansive if and only if A := I − T is 2λ1 -inverse strongly monotone. (4) A convex combination of finitely many averaged nonexpansive mappings is averaged nonexpansive. Furthermore, if S : H → H is β-averaged nonexpansive, V is nonexpansive and α ∈ (0, 1), then T = (1 − α)S + αV is λ-averaged nonexpansive, where λ = α + β − αβ. If F = Fix(S) ∩ Fix(V) ≠ 0, then Fix(T) = F. (5) A composition of finitely many averaged nonexpansive mappings is averaged nonexpansive. In particular, if Ti is λi -averaged nonexpansive for each i = 1, 2, then T1 T2 and T2 T1 are λ-averaged nonexpansive, where λ = λ1 + λ2 − λ1 λ2 . (6) Let Ti : H → H be averaged nonexpansive mappings for all i = 1, . . . , r. If F = ⋂ri=1 Fix(Ti ) ≠ 0, then F = Fix(T1 T2 ⋅ ⋅ ⋅ Tr ) = Fix(Tr T1 ⋅ ⋅ ⋅ Tr−1 ) = ⋅ ⋅ ⋅ = Fix(T2 T3 ⋅ ⋅ ⋅ T1 ).

2.1 Basic properties for nonexpansive mappings and their subclasses | 85

Proof. (1) The conclusion can be obtained directly from (2) of Proposition 2.1.1. (2) The conclusion can be verified directly by the definition of an inverse-strongly monotone mapping. (3) From Proposition 2.1.2, it follows that 1−λ ‖Ax − Ay‖2 ≤ ‖x − y‖2 − ‖Tx − Ty‖2 λ = ‖x − y‖2 − ‖Ax − Ay‖2 + 2⟨Ax − Ay, x − y⟩ − ‖x − y‖2 = 2⟨Ax − Ay, x − y⟩ − ‖Ax − Ay‖2 . Hence ⟨Ax − Ay, x − y⟩ ≥

1 ‖Ax − Ay‖2 . 2λ

If A := I − T is 2λ1 -inverse strongly monotone, then it follows from conclusion (2) of this proposition that λ1 A is 21 -inverse strongly monotone. Define a mapping S by S = (1 − λ1 )I + λ1 S. Then T = (1 − λ)I + λS and A = I − T = λ(I − S). It follows that 1 A = I − S is 21 -inverse strongly monotone. It follows from conclusion (1) of this λ proposition that S is nonexpansive. Hence T is λ-averaged nonexpansive. (4) We only prove the case of two mappings. Since S is β-averaged nonexpansive, one finds that there exist a constant β ∈ (0, 1) and a nonexpansive mapping U such that S = (1 − β)I + βU. Denote λ = α + β − αβ = α + β(1 − α) ∈ (0, 1) and W=

(1 − α)β α U + V. λ λ

Then W is nonexpansive and T = (1 − α)S + αV = (1 − α)[(1 − β)I + βU] + αV = (1 − α − β − αβ)I + (1 − α)βU + αV = (1 − λ)I + λW. Thus T is λ-averaged nonexpansive.

86 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces (5) We only prove the case of two mappings. Denote T1 = (1 − λ1 )I + λ1 S1 ,

T2 = (1 − λ2 )I + λ2 S2 ,

where λ1 and λ2 are two real numbers in (0, 1), and S1 and S2 are two nonexpansive mappings. Then T1 T2 = [(1 − λ1 )I + λ1 S1 ][(1 − λ2 )I + λ2 S2 ] = (1 − λ1 )[(1 − λ2 )I + λ2 S2 ] + λ1 S1 T2

= (1 − λ1 )(1 − λ2 )I + λ2 (1 − λ1 )S2 + λ1 S1 T2 = (1 − λ)I + λ[

λ2 (1 − λ1 ) λ S2 + 1 S1 T2 ]. λ λ

Setting B=

λ λ2 (1 − λ1 ) S2 + 2 S1 T2 , λ λ

one sees that B is nonexpansive. Hence T1 T2 is λ-averaged nonexpansive. Similarly, we can prove that T2 T1 is λ-averaged nonexpansive. (6) It is obvious that F ⊂ Fix(T1 T2 ⋅ ⋅ ⋅ Tr ). So, we only need to prove the converse. Suppose that x = T1 T2 ⋅⋅⋅Tr x and p ∈ F. In view of conclusion (5) of this proposition, one sees that Tr is averaged nonexpansive. Hence, there exist a constant α ∈ (0, 1) and a nonexpansive mapping S such that Tr = (1 − α)I + αS such that ‖x − p‖2 = ‖T1 T2 ⋅ ⋅ ⋅ Tr x − p‖2

≤ ‖Tr x − p‖2 󵄩 󵄩2 = 󵄩󵄩󵄩(1 − α)(x − p) + α(Sx − p)󵄩󵄩󵄩

= (1 − α)‖x − p‖2 + α‖Sx − p‖2 − α(1 − α)‖x − Sx‖2 ≤ ‖x − p‖2 − α(1 − α)‖x − Sx‖2 . Therefore, x = Sx 󳨐⇒ Tr x = x. Similarly, we can prove that Tr−1 x = x, . . . , T1 x = x, that is, x ∈ F. We can prove other equalities in a similar way. This completes the proof.

2.2 Opial conditions and asymptotic centers In this section, we prove that Hilbert spaces satisfy the Opial condition, which is indeed an important property of Hilbert spaces. The condition plays an important role in proving the existence of fixed points for nonexpansive mappings and convergence of iterative sequences in Hilbert spaces. With the aid of asymptotic centers, we can establish the sufficient and necessary conditions of the weak convergence of Banach– Picard iterative sequences.

2.2 Opial conditions and asymptotic centers | 87

Proposition 2.2.1. Let {xn } be a bounded sequence in H with xn ⇀ x. Then (i) lim sup ‖xn − y‖2 = lim sup ‖xn − x‖2 + ‖x − y‖2 , n→∞

∀y ∈ H;

n→∞

(2.6)

(ii) lim inf ‖xn − y‖2 = lim inf ‖xn − x‖2 + ‖x − y‖2 , n→∞

∀y ∈ H.

n→∞

(2.7)

Proof. It follows from the norm and scalar product properties on H that ‖xn − y‖2 = ‖xn − x + x − y‖2

= ‖xn − x‖2 + 2⟨xn − x, x − y⟩ + ‖x − y‖2 ,

∀y ∈ H.

(2.8)

From xn ⇀ x, one has ⟨xn − x, x − y⟩ → θ as n → ∞. Taking the superior and inferior limits, respectively, we obtain (2.6) and (2.7) immediately. This completes the proof. Theorem 2.2.1 (Opial [60], 1967). Let {xn } be a sequence in H with xn ⇀ x as n → ∞. Then (1) For any y ∈ H with y ≠ x, lim sup ‖xn − y‖ > lim sup ‖xn − x‖. n→∞

(2.9)

n→∞

(2) For any y ∈ H with y ≠ x, lim inf ‖xn − y‖ > lim inf ‖xn − x‖. n→∞

(2.10)

n→∞

Proof. We prove (2.10) only. In a similar way, we can prove (2.9). Letting a := lim infn→∞ ‖xn − x‖, one has a ≥ 0. If a = 0, then we obtain from the weak lower semicontinuity of the norm that lim inf ‖xn − x‖ = 0 < ‖x − y‖ ≤ lim inf ‖xn − y‖, n→∞

n→∞

∀y ≠ x.

Hence we obtain (2.10). If a > 0, then, for all ε ∈ (0, a), there exists n0 ≥ 1 such that, for all n ≥ n0 , ‖xn − x‖ > (a − ε) 󳨐⇒ lim inf ‖xn − x‖2 ≥ (a − ε)2 . n→∞

(2.11)

Using (2.7) and (2.11), one has lim inf ‖xn − y‖2 = lim inf ‖xn − x‖2 + ‖x − y‖2 n→∞

n→∞

≥ (a − ε)2 + ‖x − y‖2 .

(2.12)

88 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces From the arbitrariness of ε ∈ (0, a), we find from (2.12) that 1 lim inf ‖xn − y‖2 ≥ a2 + ‖x − y‖2 > a2 + ‖x − y‖2 , n→∞ 2

∀y ≠ x.

(2.13)

For a sufficiently large n ∈ ℕ, we have 1 ‖xn − y‖2 > a2 + ‖x − y‖2 , ∀y ≠ x 2 1 󳨐⇒ ‖xn − y‖ ≥ √a2 + ‖x − y‖2 , ∀y ≠ x 2 > a = lim inf ‖xn − x‖, ∀y ≠ x. n→∞

This completes the proof. Remark 2.2.1. In 1967, Opial proved inequality (2.9) and inequality (2.10) in the framework of Hilbert spaces. There are equivalent to each other. Opial conditions (2.9) and (2.10) show that proper, continuous and convex functions f (y) = lim sup ‖xn − y‖ n→∞

and g(y) = lim inf ‖xn − y‖ n→∞

attain their minima at the weak limit point of the sequence {xn }. Definition 2.2.1. Let {xn } be a bounded sequence in H and let C be a nonempty closed convex subset of H. Define a function r : H → ℝ+ by r(x) := lim sup ‖xn − x‖, n→∞

∀x ∈ H,

(2.14)

r(C, {xn }) := inf{r(y) : y ∈ C},

(2.15)

A(C, {xn }) := {z ∈ C : r(z) = min{r(x) : x ∈ C}}.

(2.16)

and

Then r(C, {xn }) is called the asymptotic center of {xn } with respect to C and A(C, {xn }) is called the set of asymptotic centers of {xn } with respect to C. Proposition 2.2.2. Let {xn } be a bounded sequence in H and let C be a nonempty closed convex subset of H. Then {xn } has a unique asymptotic center with respect to C. Proof. Since r : C → ℝ+ is proper, continuous, and convex, and r(x) → ∞ (‖x‖ → ∞), one sees that there exists z ∈ C such that r(z) = min{r(x) : x ∈ C}. Hence A(C, {xn }) ≠ 0.

2.2 Opial conditions and asymptotic centers | 89

Next, we prove that A(C, r(x)) is a singleton. To achieve this, for all z1 , z2 ∈ A(C, {xn }) and t ∈ (0, 1), we let zt = tz1 + (1 − t)z2 . Then r(z1 ) ≤ r(zt ),

r(z2 ) ≤ r(zt ).

(2.17)

It follows that 󵄩2 󵄩 ‖xn − zt ‖2 = 󵄩󵄩󵄩t(xn − z1 ) + (1 − t)(xn − z2 )󵄩󵄩󵄩 = t‖xn − z1 ‖2 + (1 − t)‖xn − z2 ‖2 − t(1 − t)‖z1 − z2 ‖2 .

(2.18)

Letting n → ∞ in (2.18), one concludes that 2

lim sup ‖xn − zt ‖2 ≤ t(lim sup ‖xn − z1 ‖) + (1 − t)(lim sup ‖xn − z2 ‖) n→∞

n→∞

2

n→∞

− t(1 − t)‖z1 − z2 ‖2

= tr 2 (z1 ) + (1 − t)r 2 (z2 ) − t(1 − t)‖z1 − z2 ‖2 ≤ tr 2 (zt ) + (1 − t)r 2 (zt ) − t(1 − t)‖z1 − z2 ‖2

= r 2 (zt ) − t(1 − t)‖z1 − z2 ‖2 .

(2.19)

If z1 ≠ z2 , then t(1 − t)‖z1 − z2 ‖2 > 2t (1 − t)‖z1 − z2 ‖2 . It follows from (2.19) that t lim sup ‖xn − zt ‖2 < r 2 (zt ) − (1 − t)‖z1 − z2 ‖2 . 2 n→∞

(2.20)

In view of the property of the lim supn→∞ , one finds that there exists n0 ≥ 1 such that t ‖xn − zt ‖2 < r 2 (zt ) − (1 − t)‖z1 − z2 ‖2 , ∀n ≥ n0 , 2 t 󳨐⇒ ‖xn − zt ‖ < √r 2 (zt ) − (1 − t)‖z1 − z2 ‖2 2 t 󳨐⇒ r(zt ) = lim sup ‖xn − zt ‖ ≤ √r 2 (zt ) − (1 − t)‖z1 − z2 ‖2 < r(zt ), 2 n→∞ which is a contradiction. Hence z1 = z2 . Therefore, A(C, {xn }) is a singleton. This completes the proof. Proposition 2.2.3. Let C be a closed convex subset of H and let {xn } be a bounded sequence in H. Let A(C, xn ) = {z} and {yn } ⊂ C with r(yn ) → r(z) as n → ∞. Then yn → z as n → ∞. Proof. From the definition of r(⋅), r(yn ) ≥ r(z) for each n ≥ 1, since r(yn ) → r(z) as n → ∞, we see that, for arbitrary ε > 0, there exists M ≥ 1 such that r(yn ) < r(z) + ε,

∀m ≥ M.

(2.21)

90 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Letting um = tz + (1 − t)ym ,

∀t ∈ (0, 1),

(2.22)

we arrive at ‖xn − um ‖2 = t‖xn − z‖2 + (1 − t)‖xn − ym ‖2 − t(1 − t)‖ym − z‖2 .

(2.23)

Combining this with (2.21), we obtain that 2

lim sup ‖xn − um ‖2 ≤ t(lim sup ‖xn − z‖) + (1 − t)(lim sup ‖xn − ym ‖2 ) n→∞

n→∞

n→∞

− t(1 − t)‖ym − z‖2

= tr 2 (z) + (1 − t)r 2 (yn ) − t(1 − t)‖ym − z‖2 ≤ r 2 (yn ) − t(1 − t)‖ym − z‖2 2

< (r(z) + ε) − t(1 − t)‖ym − z‖2 ,

∀m ≥ M.

(2.24)

Therefore, for all m ≥ M, there exists N(m) ≥ 1 such that, for all N ≥ N(m), 2

‖xn − um ‖ < √(r(z) + ε) − t(1 − t)‖ym − z‖2 2

󳨐⇒ r(z) ≤ r(um ) = lim sup ‖xn − um ‖ ≤ √(r(z) + ε) − t(1 − t)‖ym − z‖2 n→∞ 2

󳨐⇒ r 2 (z) ≤ (r(z) + ε) − t(1 − t)‖ym − z‖2 ,

∀m ≥ M,

2

󳨐⇒ t(1 − t)‖ym − z‖2 ≤ (r(z) + ε) − r 2 (z) = ε[2r(z) + ε],

∀m ≥ M.

(2.25)

Hence yn → z as n → ∞. This completes the proof. Proposition 2.2.4. Let C be a closed convex subset of H. Let {xn } be a sequence in C satisfying xn ⇀ x as n → ∞. Then A(C, {xn }) = {x}. Proof. Since C is weakly closed, we conclude that x ∈ C. In view of Theorem 2.2.1, we find, for all y ∈ C, that r(x) = lim sup ‖xn − x‖ ≤ lim sup ‖xn − y‖ = r(y) n→∞

n→∞

󳨐⇒ r(x) ≤ inf{r(y) : y ∈ C} ≤ r(x) 󳨐⇒ r(x) = inf{r(y) : y ∈ C}. Hence x ∈ A(C, {xn }). It follows from Proposition 2.2.2 that A(C, {xn }) = {x}. This completes the proof. Proposition 2.2.5. Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping. Let {xn } be a bounded sequence in C defined by xn = T n x for all x ∈ C and A(C, {xn }) = {z}. If there exists a subsequence {xnj } ⊂ {xn } such that xnj ⇀ x0 as j → ∞ and x0 ∈ Fix(T), then x0 = z.

2.3 The demiclosedness principle and fixed point theorems | 91

Proof. Note that 󵄩 󵄩 r(z) ≤ r(x0 ) = lim sup ‖xn − x0 ‖ = lim sup󵄩󵄩󵄩T n x − x0 󵄩󵄩󵄩 n→∞

n→∞

and 󵄩 󵄩 r(z) ≤ r(x0 ) = lim sup ‖xn − x0 ‖ = lim sup󵄩󵄩󵄩T n x − x0 󵄩󵄩󵄩 n→∞

n→∞

󵄩 󵄩 = lim sup󵄩󵄩󵄩T n−nj T nj x − T n−nj x0 󵄩󵄩󵄩, n→∞

∀n ≥ nj ,

󵄩 󵄩 󵄩 󵄩 ≤ lim sup󵄩󵄩󵄩T nj x − x0 󵄩󵄩󵄩 ≤ lim sup󵄩󵄩󵄩T nj x − z 󵄩󵄩󵄩 j→∞

j→∞

󵄩 󵄩 ≤ lim sup󵄩󵄩󵄩T n x − z 󵄩󵄩󵄩 = r(z). n→∞

Therefore, r(z) = r(x0 ) = r(z). In view of A(C, {xn }) = {z}, one finds that x0 = z. This completes the proof.

2.3 The demiclosedness principle and fixed point theorems In this subsection, we establish the demiclosedness principle and some fixed point theorems for nonexpansive mappings. The demiclosedness principle is not only a technique for the existence of fixed points, but also an essential tool for the convergence of iterative methods. Theorem 2.3.1 (Demiclosedness principle of nonexpansive mappings). Let C be a nonempty, closed, and convex subset of H, and let T : C → C be a nonexpansive mapping. Then I − T is demiclosed at y ∈ H, that is, for any {xn } ⊂ C with xn ⇀ x and xn − Txn → y ∈ H, it follows that x ∈ C and x − Tx = y. In particular, if y = θ, then I − T is said to be demiclosed at the origin. Proof. Since C is a closed convex subset of H, we see that C is weakly closed, and so x ∈ C. In view of the (2) of Proposition 2.1.1, we have 1󵄩 󵄩2 ⟨(I − T)xn − (I − T)x, xn − x⟩ ≥ 󵄩󵄩󵄩(I − T)xn − (I − T)x 󵄩󵄩󵄩 , 2

∀n ≥ 1.

Letting n → ∞, one has 1󵄩 󵄩2 0 ≥ 󵄩󵄩󵄩y − (I − T)x󵄩󵄩󵄩 󳨐⇒ (I − T)x = y. 2 This completes the proof. Theorem 2.3.2 (Fixed Point Theorem of Nonexpansive Mappings). Assume that C is a nonempty, bounded, closed, and convex subset of H, and let T : C → C be a nonexpansive mapping. Then Fix(T) is a nonempty, closed, and convex subset of H.

92 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Proof. Fix u ∈ C. For every n ≥ 1, we define a mapping Tn : C → C by Tn x =

1 1 u + (1 − )Tx. n+1 n+1

(2.26)

Then Tn : C → C is a contraction mapping. Using the Banach’s Contraction Principle, there exists a unique xn ∈ C such that xn = Tn xn =

1 1 u + (1 − )Txn , n+1 n+1

∀n ≥ 1.

(2.27)

Since C is bounded, {xn } ⊂ C, and {Txn } ⊂ C, one concludes that both {xn } and {Txn } are bounded sequences. It follows from (2.27) that xn − Txn =

1 (u − Txn ) → θ n+1

(n → ∞).

Suppose that xn ⇀ x as n → ∞. Using the demiclosedness principle of nonexpansive mappings, we have x = Tx. Next, we show that Fix(T) is closed and convex. First, we prove that Fix(T) is closed. Let {pn } be a sequence in Fix(T) with pn → p as n → ∞. Then pn = Tpn implies p = Tp. Hence p ∈ Fix(T). Let pi ∈ Fix(T) for each i = 1, 2 and t ∈ (0, 1). Let pt = (1 − t)p1 + tp2 . Since C is convex, we have pt ∈ C. It follows that 󵄩 󵄩2 ‖pt − Tpt ‖2 = 󵄩󵄩󵄩(1 − t)(p1 − Tpt ) + t(p2 − Tpt )󵄩󵄩󵄩

= (1 − t)‖p1 − Tpt ‖2 + t‖p2 − Tpt ‖2 − t(1 − t)‖p1 − p2 ‖2

≤ (1 − t)‖p1 − pt ‖2 + t‖p2 − pt ‖2 − t(1 − t)‖p1 − p2 ‖2

= [t 2 (1 − t) + t(1 − t)2]‖p1 − p2 ‖2 − t(1 − t)‖p1 − p2 ‖2 . = 0.

Thus pt = Tpt , that is, pt ∈ Fix(T). This completes the proof.

2.4 Iterative methods of fixed points Let C be a nonempty, closed, and convex subset of H, and let T : C → C be a nonexpansive mapping. Now, we introduce some iterative methods for fixed points of T as follows: (1) The Banach–Picard iterative method: x1 ∈ C, xn+1 = Txn ,

∀n ≥ 1.

(BPIM)

(2) The Krasnosel’skiǐ–Man iterative method: x1 ∈ C, xn+1 = (1 − αn )xn + αn Txn ,

∀n ≥ 1,

where {αn } is a sequence in [0, 1] satisfying certain conditions.

(KMIM)

2.4 Iterative methods of fixed points | 93

(3) The Halpern iterative method: x1 , u ∈ C, xn+1 = αn u + (1 − αn )Txn ,

∀n ≥ 1,

(HIM)

where {αn } is a sequence in [0, 1] satisfying certain conditions; (4) The Moudafi–Halpern iterative method: x1 ∈ C, xn+1 = αn f (xn ) + (1 − αn )Txn ,

∀n ≥ 1,

(MHIM)

where f : C → C is a contraction with the contraction coefficient ρ ∈ (0, 1). 2.4.1 Weak convergence theorems The general methodology to obtain the weak convergence of the iterative sequence {xn } generated by (BPIM) or (KMIM) is to prove that the iterative sequence is a bounded approximate fixed point sequence, that is, xn − Txn → θ as n → ∞ and then to prove that ωω (xn ) ⊂ Fix(T) with the aid of the demiclosedness principle of nonexpansive mappings. Since xn ⇀ x ⇐⇒ all subsequences of {xn } converge weakly to the same point ⇐⇒ ωω (xn ) is a singleton set, it is, therefore, essential to find the conditions to guarantee that ωω (xn ) is a singleton. Thus, we need the following definition. Definition 2.4.1. Let {xn } be a sequence in H and let S be a subset of H. Then the sequence {xn } is said to be Fejér monotone with respect to S if ‖xn+1 − p‖ ≤ ‖xn − p‖,

∀p ∈ S.

(2.28)

Example 2.4.1. Let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Let {xn } be an iterative sequence generated by (BPIM) or (KMIM). Then {xn } is Fejér monotone with respect to Fix(T). From (2.28), we see that limn→∞ ‖xn − p‖ exists, but is not necessarily zero. Example 2.4.2. Let {en } be a standard orthogonal basis in H. Then {en } is Fejér monotone with respect to the singleton set S = {θ} and en ⇀ θ, but it does not converge strongly to θ. However, we have the following proposition. Proposition 2.4.1. Let {xn } be a sequence in H. Let C be a nonempty, closed, and convex subset of H, and let {xn } be a Fejér monotone sequence with respect to C. Then limn→∞ PC xn exists. Furthermore, if ωω (xn ) ⊂ C, then ωω (xn ) is a singleton set and {xn } converges weakly to a point in C. Note that Proposition 2.4.1 is called the Browder’s convergence theorem. Proof. It follows from the definition of the metric projection and Fejér monotone sequence that ‖xn+1 − PC xn+1 ‖ ≤ ‖xn+1 − PC xn ‖ ≤ ‖xn − PC xn ‖,

∀n ≥ 1

94 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces implies that limn→∞ ‖xn − PC xn ‖ does exist. Using the property of PC , for all n, m ≥ 1, we derive ‖PC xn+m − PC xn ‖2 ≤ ‖xn+m − PC xn ‖2 − ‖xn+m − PC xn+m ‖2 ≤ ‖xn − PC xn ‖2 − ‖xn+m − PC xn+m ‖2

→0

(n, m → ∞),

(2.29)

which shows that {PC xn } is a Cauchy sequence in H. Hence limn→∞ PC xn exists. We denote the limit by limn→∞ PC xn = y ∈ H. Since C is closed, we see that y ∈ C. If ωω (xn ) ⊂ C, then, for all x̂ ∈ ωω (xn ), there is a subsequence {xnj } ⊂ {xn } such that xnj ⇀ x̂ ∈ C. The properties of PC imply that ⟨xn − PC xn , z − PC xn ⟩ ≤ 0,

∀z ∈ C.

(2.30)

⟨xnj − PC xnj , z − PC xnj ⟩ ≤ 0,

∀z ∈ C.

(2.31)

In particular, we have

Letting j → ∞ in (2.31), we find that ⟨x̂ − y, z − y⟩ ≤ 0,

∀z ∈ C.

(2.32)

Letting z = x̂ ∈ C in (2.32), we find that 0 ≤ ‖x̂ − y‖2 = ⟨x̂ − y, x̂ − y⟩ ≤ 0 󳨐⇒ x̂ = y = lim PC xn , n→∞

which implies that ωω (xn ) is a singleton set. Hence {xn } converges weakly to a point in C. This completes the proof. Theorem 2.4.1. Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. If, for any x ∈ C, T n+1 x − T n x → θ as n → ∞, then T n x ⇀ p ∈ Fix(T) as n → ∞. Proof. Let xn = T n−1 x for each n ≥ 1. Then x1 ∈ C and xn+1 = Txn , that is, {xn } is a Banach–Picard iterative sequence. Fixing p ∈ Fix(T), we have ‖xn+1 − p‖ = ‖Txn − p‖ ≤ ‖xn − p‖. Hence {xn } is a Fejér monotone sequence with respect to Fix(T). Note that T n+1 x − T n x → θ ⇐⇒ Txn − xn → θ. It follows from the demiclosedness principle of nonexpansive mappings that ωω (xn ) ⊂ Fix(T). Therefore, it follows from Proposition 2.4.1 that xn ⇀ p ∈ Fix(T), where p = limn→∞ PFix(T) (xn ). This completes the proof.

2.4 Iterative methods of fixed points | 95

Using the technique of the asymptotic center, we can improve Theorem 2.4.1. Theorem 2.4.2. Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Then, for any x ∈ C, T n x ⇀ z ∈ Fix(T) ⇐⇒ T n+1 x − T n x ⇀ θ

(n → ∞).

Proof. “󳨐⇒” is obvious. So we only show “⇐󳨐”. Suppose that {T nj x} is a subsequence of {T n x}. Since Fix(T) ≠ 0, we see that {T n x} is a bounded sequence. Assume that T nj x ⇀ y as j → ∞. Then, for all m ≥ 1, T nj +m x ⇀ y as j → ∞. It follows from Proposition 2.2.5 that A(C, {T nj +m x}) = {y}. For all m > s ≥ 1, letting ys = T s y yields 󵄩󵄩 󵄩 s n +m 󵄩 n +m 󵄩 󵄩󵄩ys − T j x󵄩󵄩󵄩 = 󵄩󵄩󵄩T y − T j x 󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩T s y − T s T nj +m−s x 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩y − T nj +m−s x 󵄩󵄩󵄩.

(2.33)

Fixing s, nj and letting m → ∞, we find from (2.33) that r(y) ≤ r(ys ) ≤ r(y).

(2.34)

Letting s → ∞ in (2.34), we obtain r(ys ) → r(y)

(s → ∞).

From Proposition 2.2.3, we see that ys → y (s → ∞). Since T is continuous, we have Tys → Ty (s → ∞). Note that ys+1 = T s+1 y = TT s y = Tys → Ty

(s → ∞).

By the uniqueness of the limit, one has y = Ty. Finally, we see from Proposition 2.2.5 that y = z and {z} = A(C, {T n x}). Hence n T x ⇀ z as n → ∞. This completes the proof. Corollary 2.4.1. Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. If, for any λ ∈ (0, 1), we define a mapping Tλ : C → C by Tλ x = (1 − λ)x + λTx,

∀x ∈ C,

then, for any x ∈ C, Tλn x ⇀ p ∈ Fix(T) as n → ∞. Proof. It follows from Proposition 2.1.3 that Tλn+1 x − Tλn x → θ as n → ∞. From Theorem 2.4.2, it follows that Tλn x ⇀ p ∈ Fix(Tλ ) = Fix(T). This completes the proof.

96 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Theorem 2.4.3. Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Suppose that {αn } is a sequence in [0, 1] satisfying the following conditions: ∞

∑ αn (1 − αn ) = ∞.

n=1

If {xn } is the sequence generated by Krasnosel’skiǐ–Man iterative method (KMIM), then the following results hold: (1) {xn } is Fejér monotone with respect to Fix(T); (2) xn − Txn → θ as n → ∞; (3) xn ⇀ p ∈ Fix(T) as n → ∞. Proof. (1) Fixing p ∈ Fix(T), it follows from the iterative method (KMIM) that ‖xn+1 − p‖ ≤ (1 − αn )‖xn − p‖ + αn ‖Txn − p‖ ≤ ‖xn − p‖. This shows that {xn } is Fejér monotone with respect to Fix(T). Hence limn→∞ ‖xn − p‖ exists. In particular, we have that {xn } is bounded. (2) By the properties of the norm and scalar product on H, we have 󵄩 󵄩2 ‖xn+1 − p‖2 = 󵄩󵄩󵄩(1 − αn )(xn − p) + αn (Txn − p)󵄩󵄩󵄩 󵄩 󵄩2 = (1 − αn )‖xn − p‖2 + αn 󵄩󵄩󵄩(Txn − p)󵄩󵄩󵄩 − αn (1 − αn )‖Txn − xn ‖2 ≤ ‖xn − p‖2 − αn (1 − αn )‖Txn − xn ‖2 .

(2.35)

This implies that ∞

∑ αn (1 − αn )‖Txn − xn ‖2 ≤ ‖x1 − p‖2 < ∞.

n=1

(2.36)

Using the assumption that ∑∞ n=1 αn (1 − αn ) = ∞, we see that lim inf ‖xn − Txn ‖2 = 0. n→∞

It follows that lim inf ‖xn − Txn ‖ = 0. n→∞

Observe that

(2.37)

󵄩 󵄩 ‖xn+1 − Txn+1 ‖ = 󵄩󵄩󵄩(1 − αn )(xn − Txn+1 ) + αn (Txn − Txn+1 )󵄩󵄩󵄩 ≤ (1 − αn )‖xn − Txn ‖ + ‖Txn − Txn+1 ‖

≤ (1 − αn )‖xn − Txn ‖ + ‖xn − xn+1 ‖

= (1 − αn )‖xn − Txn ‖ + αn ‖xn − Txn ‖

= ‖xn − Txn ‖,

∀n ≥ 1.

(2.38)

2.4 Iterative methods of fixed points | 97

So that limn→∞ ‖xn − Txn ‖ exists. Using (2.37), we obtain lim ‖xn − Txn ‖ = 0.

n→∞

(3) It follows from the demiclosedness principle of nonexpansive mappings that ωω (xn ) ⊂ Fix(T). Therefore, we conclude from 2.4.1 that xn ⇀ p ∈ Fix(T), where p = lim PFix(T) (xn ). n→∞

This completes the proof. Corollary 2.4.2. Let λ be a real number in (0, 1) and let T : C → C be a λ-averaged nonexpansive mapping with Fix(T) ≠ 0. Suppose that {αn } is a sequence in [0, λ1 ] satisfying the following condition: ∞

∑ αn (1 − λαn ) = ∞.

n=1

If {xn } is the sequence generated by Krasnosel’skiǐ–Man iterative method (KMIM), then the following results hold: (1) {xn } is Fejér monotone sequence with respect to Fix(T); (2) xn − Txn → θ as n → ∞; (3) xn ⇀ p ∈ Fix(T) as n → ∞. Proof. Let S = (1 − λ1 )I + λ1 T and μn = λαn . Then S is nonexpansive, Fix(S) = Fix(T), μn ∈ [0, 1], ∑∞ n=1 μn (1 − μn ) = ∞, and xn+1 = (1 − αn )xn + αn Txn = xn − αn (I − T)xn

= xn − λαn (I − S)xn

= (1 − μn )xn + μn Sxn ,

∀n ≥ 1.

(2.39)

By Theorem 2.4.3, we obtain the three conclusions immediately. This completes the proof. Corollary 2.4.3. Let T : C → C be a firmly nonexpansive mapping with Fix(T) ≠ 0. Suppose that {αn } is a sequence in [0, 2] satisfying the following condition: ∞

∑ αn (2 − αn ) = ∞.

n=1

If {xn } is the sequence generated by Krasnosel’skiǐ–Man iterative method (KMIM), then the following results hold: (1) {xn } is Fejér monotone sequence with respect to Fix(T);

98 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces (2) xn − Txn → θ as n → ∞; (3) xn ⇀ p ∈ Fix(T) as n → ∞. Proof. Corollary 2.1.1 implies that T is a 21 -averaged nonexpansive mapping. Taking λ = 21 in Corollary 2.4.2, we obtain the conclusions immediately. This completes the proof.

2.4.2 Strong convergence theorems Generally speaking, Banach–Picard iterative method (BPIM) and Krasnoselskiǐ–Man iterative method (KMIM) are only weakly convergent in infinite-dimensional Hilbert spaces. Of course, if T : C → C is a linear nonexpansive mapping or T(C) is contained in a compact set, then they can be strongly convergent. In order to obtain the strong convergence of iterative sequences, Halpern proposed his iterative method (HIM) in 1967 without the assumption that T : C → C is a linear nonexpansive mapping or T(C) is contained in a compact set and studied the conditions imposed on control sequences for fixed points of nonexpansive mappings in Hilbert spaces. One decade later, Lions [42] gave another control condition for the iterative method (HIM), however, his conditions exclude the canonical choice of the control sequences, for example, the sequence {αn } defined by αn = n1 for each n ≥ 1. In 1992, Wittmann [97] overcame this flaw with the sequence αn = n1 . In 2002, Xu [99] gave another weak condition. In 2007, Suzuki [84] used new techniques to prove that Halpern iterative method (HIM) for averaged nonexpansive mappings is strongly convergent in case {αn } satisfies the following conditions: (C1) αn → 0; (C2) ∑∞ n=1 αn = ∞ In recent two decades, Halpern iterative method (HIM) was extensively investigated by many authors, and many important results were established both in Hilbert and Banach spaces. Next, we list some recent important and celebrated results. Theorem 2.4.4 (Browder [7], 1965). Let C be a nonempty, bounded, closed, and convex subset of H, and let T : C → C be a nonexpansive mapping. For any x0 ∈ C and n ≥ 1, define a mapping Tn : C → C by Tn x = (1 −

1 1 )Tx + x , n+1 n+1 0

∀x ∈ C, n ≥ 1.

Then the following results hold: (1) For each n ≥ 1, Tn has a unique fixed point un ∈ C; (2) {un } converges strongly to z = PFix(T) x0 .

(2.40)

2.4 Iterative methods of fixed points | 99

Proof. (1) It is a direct corollary of the Banach’s contraction principle. (2) Note that un = (1 −

1 1 )Tun + x , n+1 n+1 0

n ≥ 1.

(2.41)

Since both {un } and {Tun } are bounded sequences, we have un − Tun =

1 (x − Tun ) → θ n+1 0

(n → ∞).

(2.42)

It follows from Theorem 2.3.2 that Fix(T) ≠ 0. Fix p ∈ Fix(T). Using (2.41), we have ‖un − p‖2 = ⟨un − p, un − p⟩

1 1 )⟨Tun − p, un − p⟩ + ⟨x − p, un − p⟩ n+1 n+1 0 1 1 )‖un − p‖2 + ⟨x − p, un − p⟩. ≤ (1 − n+1 n+1 0 = (1 −

(2.43)

This implies that ‖un − p‖2 ≤ ⟨x0 − p, un − p⟩,

∀p ∈ Fix(T),

(2.44)

⟨un − x0 , un − p⟩ ≤ 0, n ≥ 1,

∀p ∈ Fix(T).

(2.45)

and

Since {un } ⊂ C is bounded, there exists a subsequence {unj } ⊂ {un } such that unj ⇀ z as j → ∞. It follows from the demiclosedness principle of nonexpansive mappings that z ∈ Fix(T). In (2.44), taking p = z ∈ C and n := nj , we obtain ‖unj − z‖2 ≤ ⟨x0 − z, unj − z⟩ → 0 󳨐⇒ unj → z

(j → ∞).

Letting j → ∞ in (2.45), we see that ⟨z − x0 , z − p⟩ ≤ 0,

󳨐⇒ z = PFix(T) x0 ,

∀p ∈ Fix(T) un → z (n → ∞).

This completes the proof. We remark that the conclusions of Browder’s strong convergence theorem still hold if the boundedness assumption of C is replaced by the assumption that Fix(T) ≠ 0.

100 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Theorem 2.4.5 (Halpern [30], 1967). Let C be a closed convex subset of H and let T : 1 C → C be a nonexpansive mapping with Fix(T) ≠ 0. Let αn = (n+1) a , where a ∈ (0, 1). Let {xn } be a sequence generated by (HIM). Then {xn } converges strongly to a fixed point of T which is closest to u. Proof. It follows from Theorem 2.4.4 that there exists a unique yn ∈ C such that yn = αn u + (1 − αn )Tyn .

(2.46)

Using (HIM) and (2.46), we have xn+1 − yn = (1 − αn )(Txn − Tyn ).

(2.47)

Since T : C → C is nonexpansive, it follows that ‖xn+1 − yn ‖ ≤ (1 − αn )‖xn − yn ‖.

(2.48)

‖xn+1 − yn ‖ ≤ (1 − αn )‖xn − yn−1 ‖ + ‖yn − yn−1 ‖.

(2.49)

Hence,

Now we estimate ‖yn − yn−1 ‖. It follows from (2.46) that 󵄩 󵄩 ‖yn − yn−1 ‖ = 󵄩󵄩󵄩αn u + (1 − αn )Tyn − αn u − (1 − αn−1 )Tyn−1 󵄩󵄩󵄩

≤ |αn − αn−1 |(‖u‖ + ‖Tyn−1 ‖) + (1 − αn )‖yn − yn−1 ‖.

Hence

󵄨󵄨 α 󵄨󵄨󵄨 󵄨 ‖yn − yn−1 ‖ ≤ 󵄨󵄨󵄨1 − n−1 󵄨󵄨󵄨(‖u‖ + ‖Tyn−1 ‖). 󵄨󵄨 αn 󵄨󵄨

(2.50)

It follows from Theorem 2.4.4 that yn → PFix(T) (u) as n → ∞. In particular, {yn } is a bounded sequence. So is {Tyn }. Denote M = ‖u‖ + supn≥0 {‖Tyn ‖}. Then (2.50) can be rewritten as follows: 󵄨󵄨 α 󵄨󵄨󵄨 󵄨 ‖yn − yn−1 ‖ ≤ M 󵄨󵄨󵄨1 − n−1 󵄨󵄨󵄨. (2.51) 󵄨󵄨 αn 󵄨󵄨 Since 1 −

αn−1 αn

= 1 − ( n+1 )a and n a

αn − αn−1 n+1 = (n + 1)a [1 − ( ) ]→0 αn n

(n → ∞),

we have ‖yn − yn−1 ‖ = o(αn ). In view of (2.49), we obtain ‖xn+1 − yn ‖ ≤ (1 − αn )‖xn − yn−1 ‖ + o(αn ), Using Lemma 1.10.2, we obtain xn+1 − yn → θ

(n → ∞).

Hence xn → PFix(T) (u) as n → ∞. This completes the proof.

∀n ≥ 1.

(2.52)

2.4 Iterative methods of fixed points | 101

In 1977, Lions [42] further generalized Theorem 2.4.5 as follows. Theorem 2.4.6 (Lions [42], 1977). Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Suppose that {αn } in [0, 1] satisfies the following conditions: (C1) αn → 0(n → ∞); (C2) ∑∞ n=0 αn = ∞; α −αn (C3) n+1 → 0. α2 n+1

Let {xn } be a sequence generated by (HIM). Then {xn } converges strongly to PFix(T) u. Since the proof is the same as that of Theorem 2.4.5, we omit the proof here. Remark 2.4.1. Halpern [30] pointed out that conditions (C1) and (C2) are necessary for the convergence of iterative sequences generated in iterative method (KMIM). Indeed, he also put forward the following open question: Are conditions (C1) and (C2) sufficient to ensure that the sequence generated by (HIM) converges strongly to a fixed point of T in C? Recently, Suzuki [85] gave an answer to the Halpern’s question, however, what are the sufficient and necessary conditions concerning the control sequence {αn } are still not clear. Remark 2.4.2. Theorems 2.4.5 and 2.4.6 exclude the canonical choice of the control 1 for each n ≥ 0. sequence {αn }, namely αn = n+1 In 1992, Wittmann [97] overcame this flaw and he introduced the following condition: (C4) ∑∞ n=0 |αn+1 − αn | < ∞. With the aid of condition (C4), he established the following strong convergence theorem. Theorem 2.4.7 (Wittmann [97], 1992). Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Suppose that {αn } ∈ [0, 1] satisfies the control conditions (C1), (C2), and (C4). Let {xn } be the sequence generated by (HIM). Then {xn } converges strongly to PFix(T) (u). Proof. The proof can be split into five steps as follows: Step 1. Show that {xn } is bounded. Fixing p ∈ Fix(T), one has ‖xn+1 − p‖ = αn ‖u − p‖ + (1 − αn )‖Txn − p‖ ≤ αn ‖u − p‖ + (1 − αn )‖xn − p‖

≤ max{‖u − p‖, ‖x0 − p‖} = M,

∀n ≥ 0.

102 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Step 2. Show that xn+1 − xn → θ as n → ∞. Note that 󵄩 󵄩 ‖xn+1 − xn ‖ = 󵄩󵄩󵄩(αn − αn−1 )u + (1 − αn )Txn − (1 − αn−1 )Txn−1 󵄩󵄩󵄩 󵄨 󵄨 ≤ 󵄨󵄨󵄨αn − αn−1 ‖u‖󵄨󵄨󵄨 + |1 − αn |‖xn − xn−1 ‖ + |αn − αn−1 |‖Txn−1 ‖ ≤ (1 − αn )‖xn − xn−1 ‖ + M1 |αn − αn−1 |, where M1 = ‖u‖ + supu≥0 {‖Txn ‖}. Using Lemma 1.10.2, we have xn+1 − xn → θ as n → ∞. Step 3. Show that xn − Txn → θ as n → ∞. It follows from (HIM) that xn+1 − Txn = αn (u − Txn ) → θ as n → ∞. From Step 2, we see that xn − Txn = (xn − xn+1 ) + (xn+1 − Txn ) → θ

(n → ∞).

Step 4. Show that lim supn→∞ ⟨u − PFix(T) u, xn − PFix(T) u⟩ ≤ 0. Let {xnk } be a subsequence of {xn } such that lim sup⟨u − PFix(T) u, xn − PFix(T) u⟩ n→∞

= lim ⟨u − PFix(T) u, xnk − PFix(T) u⟩. k→∞

Since {xnk } is bounded, we may, without loss of generality, assume that xnk ⇀ y as k → ∞. It follows from the Browder’s demiclosedness principle 1.9.3 that y ∈ Fix(T). By the property of PFix(T) , we obtain lim ⟨u − PFix(T) u, xnk − PFix(T) u⟩ = ⟨u − PFix(T) u, y − PFix(T) u⟩ ≤ 0.

k→∞

Step 5. Show that xn → PFix(T) u as n → ∞. Note that ‖xn+1 − PFix(T) u‖2 = αn2 ‖u − PFix(T) u‖2 + (1 − αn )2 ‖Txn − PFix(T) u‖2 + 2αn (1 − αn )⟨u − PFix(T) u, Txn − PFix(T) u⟩

≤ (1 − αn )‖xn − PFix(T) u‖2 + αn2 ‖u − PFix(T) u‖2 + 2αn (1 − αn )⟨u − PFix(T) u, Txn − xn ⟩

+ 2αn (1 − αn )⟨u − PFix(T) u, xn − PFix(T) u⟩

≤ (1 − αn )‖xn − PFix(T) u‖2 + 2αn ‖u − PFix(T) u‖‖xn − Txn ‖

+ 2αn (1 − αn )⟨u − PFix(T) u, xn − PFix(T) u⟩ + αn2 ‖u − PFix(T) u‖2

≤ (1 − αn )‖xn − PFix(T) u‖2 + o(αn ),

where o(αn ) = 2αn ‖u − PFix(T) u‖‖xn − Txn ‖ + 2αn (1 − αn )⟨u − PFix(T) u, xn − PFix(T) u⟩ + αn2 ‖u − PFix(T) u‖2 .

2.4 Iterative methods of fixed points | 103

By Lemma 1.10.2, we have xn → PFix(T) u as n → ∞. This completes the proof. In 2002, Xu [100] changed condition (C4) into the following condition in Theorem 2.4.7: α (C5) αn−1 → 1 as n → ∞. n

To be more precise, Xu proved the following result. Theorem 2.4.8 (Xu [100], 2002). Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Let {xn } be a sequence generated by (HIM), where {αn } is a sequence in (0, 1) satisfying conditions (C1), (C2), and (C5). Then {xn } converges strongly to PFix(T) (u). Proof. Note that condition (C4) is only used for the second step in Theorem 2.4.7. It guarantees that xn+1 − xn → θ as n → ∞. Note that ‖xn+1 − xn ‖ ≤ (1 − αn )‖xn − xn−1 ‖ + M1 |αn − αn−1 | = (1 − αn )‖xn − xn−1 ‖ + o(αn ).

By Lemma 1.10.2, it follows that xn+1 − xn → θ as n → ∞. The rest of the proof is similar to that of Theorem 2.4.7. This completes the proof. Remark 2.4.3. Xu pointed out that conditions (C4) and (C5) are mutually independent. Comparing conditions (C3) and (C5), we have (C5)

αn−1 →1 αn

(n → ∞) ⇐⇒

αn+1 − αn →0 αn+1

(n → ∞).

Observe that |αn+1 − αn | |αn+1 − αn | ≤ . 2 αn+1 αn+1 Therefore, if {αn } satisfies condition (C3), then it satisfies condition (C5). In addition, 1 condition (C5) is satisfied by the canonical choice αn = n+1 for each n ≥ 0, however, condition (C3) is not. Therefore, Theorem 2.4.8 is an improvement of Theorem 2.4.7. We can further improve Theorems 2.4.7 and 2.4.8 by using the Banach limit technique. By using Suzuki’s lemma 1.10.3, we can prove the following more general convergence theorem. The advantage is that it only needs to satisfy conditions (C1) and (C2) for the strong convergence of the iterative sequence {xn }. Theorem 2.4.9. Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Suppose that {αn }, {βn }, and {γn } are three sequences in [0, 1] such that (1) αn + βn + γn = 1 for all n ≥ 0; (2) 0 < lim infn→∞ αn ≤ lim supn→∞ αn < 1;

104 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces (3) βn → 0 as n → ∞; (4) ∑∞ n=0 βn = ∞. For an arbitrary initial value x0 ∈ C and a fixed anchor point u ∈ C, define a sequence {xn } iteratively in C as follows: xn+1 = βn u + αn xn + γn Txn ,

∀n ≥ 1.

Then {xn } converges strongly to PFix(T) (u). Proof. The proof is split into six steps as follows. Step 1. Show that {xn } is bounded. Fixing p ∈ Fix(T), one finds from the iterative method (A) that ‖xn+1 − p‖ = βn ‖u − p‖ + αn ‖xn − p‖ + γn ‖Txn − p‖ ≤ βn ‖u − p‖ + (αn + γn )‖xn − p‖

≤ max{‖u − p‖, ‖x0 − p‖}

= M,

∀n ≥ 0.

Hence {xn } is bounded, so is {Txn }. Step 2. Show that xn+1 − xn → θ as n → ∞. Define a sequence {yn } by yn =

xn+1 − αn xn , 1 − αn

∀n ≥ 0.

Then xn+1 = αn xn + (1 − αn )yn for all n ≥ 0. Note that 󵄩󵄩 x − α x x − αn xn 󵄩󵄩󵄩󵄩 󵄩 n+1 n+1 ‖yn+1 − yn ‖ = 󵄩󵄩󵄩 n+2 − n+1 󵄩 󵄩󵄩 1 − αn+1 1 − αn 󵄩󵄩󵄩 󵄩󵄩 β u − γ + Tx β u − λn Txn 󵄩󵄩󵄩󵄩 󵄩 n n+1 − n = 󵄩󵄩󵄩 n+1 󵄩 󵄩󵄩 1 − αn+1 1 − αn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 β βn 󵄩 󵄩 )u󵄩󵄩󵄩 + ‖Txn+1 − xn ‖ ≤ 󵄩󵄩󵄩( n+1 − 󵄩󵄩 1 − αn+1 1 − αn 󵄩󵄩 󵄩󵄩 β Tx β Tx 󵄩󵄩󵄩 󵄩 + 󵄩󵄩󵄩 n+1 n+1 − n n 󵄩󵄩󵄩 󵄩󵄩 1 − αn+1 1 − αn 󵄩󵄩 󵄨󵄨 β βn 󵄨󵄨󵄨󵄨 󵄨 ≤ 󵄨󵄨󵄨 n+1 − 󵄨(‖u‖ + ‖Txn ‖) 󵄨󵄨 1 − αn+1 1 − αn 󵄨󵄨󵄨

βn+1 )‖xn+1 − xn ‖ 1 − αn+1 󵄨󵄨 β βn+1 βn 󵄨󵄨󵄨󵄨 󵄨 ≤ 󵄨󵄨󵄨 n+1 − )‖xn+1 − xn ‖, 󵄨M + (1 + 󵄨󵄨 1 − αn+1 1 − αn 󵄨󵄨󵄨 1 1 − αn+1 + (1 +

where M1 = ‖u‖ + supn≥0 {‖Txn ‖}. Since βn → 0 as n → ∞, one has lim sup(‖yn+1 − yn ‖ − ‖xn+1 − xn ‖) ≤ 0. n→∞

(A)

2.4 Iterative methods of fixed points | 105

From Suzuki’s lemma 1.10.3, we see that yn − xn → θ (n → ∞). Therefore, we conclude xn+1 − xn = (1 − αn )(yn − xn ) → θ

(n → ∞).

Step 3. Show that xn − Txn → θ (n → ∞). It follows from conditions (1), (2), and (3) that lim inf γn > 0. n→∞

Since xn+1 − xn = βn (u − xn ) + γn (Txn − xn ) → θ, βn → 0 and xn+1 − xn → θ as n → ∞, we find that xn − Txn → θ (n → ∞). Step 4. Show that ωω (xn ) ⊂ Fix(T), where ωω (xn ) = {x ∈ C : ∃ {xnj } ⊂ {xn } such that xnj ⇀ x

(j → ∞)}

is a weak-ω limit set of {xn }. For all x ∈ ωω (xn ), since xnj ⇀ x as j → ∞ and xnj − Txnj → θ as j → ∞, one concludes from Theorem 1.9.3 that x = Tx. Step 5. Show that lim supn→∞ ⟨u − z, xn − z⟩ ≤ 0, where z = PFix(T) u. Let {xnj } be a subsequence of {xn } such that lim sup⟨u − z, xn − z⟩ = lim sup⟨u − z, xnj − z⟩. n→∞

j→∞

Since {xnj } is bounded, we may, without loss of generality, assume that xnj ⇀ x as j → ∞. From (4), we see that x = Tx. Hence lim sup⟨u − z, xn − z⟩ = ⟨u − z, x − z⟩ ≤ 0. n→∞

Letting σn = max{⟨u − z, xn − z⟩, 0}, we have σn > 0 and σn → 0 as n → ∞. Step 6. Show that xn → z as n → ∞. Note that ‖xn+1 − z‖2 = ⟨xn+1 − z, xn+1 − z⟩

= βn ⟨u − z, xn+1 − z⟩ + αn ⟨xn − z, xn+1 − z⟩ + γn ⟨Txn − z, xn+1 − z⟩ α α ≤ βn σn+1 + n ‖xn − z‖2 + n ‖xn+1 − z‖2 2 2 γn γn 2 + ‖xn − z‖ + ‖xn+1 − z‖2 2 2 1 1 = βn σn+1 + (1 − βn )‖xn − z‖2 + (1 − βn )‖xn+1 − z‖2 . 2 2

This implies that ‖xn+1 − z‖2 = (1 − βn )‖xn − z‖2 + 2βn σn+1 = (1 − βn )‖xn − z‖2 + o(βn ).

106 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces It follows from Lemma 1.10.2 that xn → PFix(T) u as n → ∞. This completes the proof. Corollary 2.4.4 ([84]). Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Suppose that {αn } and {λn } are two sequences in [0, 1] satisfying the following control conditions: (C1) αn → 0(n → ∞); (C2) ∑∞ n=0 αn = ∞; (C3) 0 < lim infn→∞ λn ≤ lim supn→∞ λn < 1. For an arbitrary initial value x0 ∈ C and a fixed anchor u ∈ C, define a sequence {xn } iteratively in C as follows: xn+1 = αn u + (1 − αn )[(1 − λn )xn + λn Txn ],

∀n ≥ 1.

(A1)

Then {xn } converges strongly to PFix(T) (u). Proof. It follows from iterative method (A1) that xn+1 = αn u + (1 − αn )(1 − λn )xn + (1 − αn )λn Txn ,

∀n ≥ 1.

Using Theorem 2.4.9, we obtain the desired conclusion immediately. Corollary 2.4.5. Let C be a closed convex subset of H and let {Ti }ri=1 : C → C be a finite family of r nonexpansive mappings with F = ⋂ri=1 Fix(Ti ) ≠ 0. Suppose that {αn } and {λn } are two sequences in [0, 1] satisfying the following control conditions: (C1) αn → 0 as n → ∞; (C2) ∑∞ n=0 αn = ∞; (C3) 0 < lim infn→∞ λn ≤ lim supn→∞ λn < 1. For an arbitrary initial value x0 ∈ C and a fixed anchor u ∈ C, define a sequence {xn } iteratively in C as follows: r

xn+1 = αn u + (1 − αn )[(1 − λn )xn + λn ∑ ωi Ti xn ], i=1

∀n ≥ 1,

(A2)

where ωi ∈ (0, 1), i = 1, . . . , r, satisfy ∑ri=1 ωi = 1. Then {xn } converges strongly to PF (u). Proof. Let T = ∑ri=1 ωi Ti . Then T : C → C is a nonexpansive mapping with F = Fix(T). By Corollary 2.4.4, we obtain the conclusion immediately. This completes the proof. Theorem 2.4.10 (Bauschke [6], 1996). Let C be a closed convex subset of H and let {Ti }ri=1 : C → C be a finite family of r nonexpansive mappings satisfying the following

2.4 Iterative methods of fixed points | 107

condition: r

F = ⋂ Fix(Ti ) = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) i=1

= Fix(T1 Tr Tr−1 ⋅ ⋅ ⋅ T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 ⋅ ⋅ ⋅ T1 Tr ) ≠ 0.

(CC)

Suppose that {αn } is a real number sequence in [0, 1] satisfying the control conditions (C1), (C2), and αn (C3) ∑∞ → 1 as n → ∞. n=1 |αn+r − αn | < +∞ or (C4) α n+r

For an arbitrary initial value x0 ∈ C and a fixed element u ∈ C, define a sequence {xn } iteratively in C as follows: xn+1 = αn u + (1 − αn )Tn+1 xn ,

∀n ≥ 0,

(B)

where Tn = Tn mod r . Then xn → z as n → ∞, where z = PF u and PF is the metric projection from C onto F. Proof. In order to prove xn → PF u as n → ∞, we estimate ‖xn+1 − z‖2 as follows: 󵄩 󵄩2 ‖xn+1 − z‖2 = 󵄩󵄩󵄩(1 − αn )(Tn+1 xn − Tn+1 z) + αn (u − z)󵄩󵄩󵄩 = (1 − αn )2 ‖Tn+1 xn − Tn+1 z‖2

+ 2αn (1 − αn )⟨u − z, Tn+1 xn − z⟩ + αn2 ‖u − z‖2

≤ (1 − αn )‖xn − z‖2 + 2αn (1 − αn )⟨u − z, Tn+1 xn − z⟩ + o(αn ).

(2.53)

Thus we need to prove lim sup⟨u − z, Tn+1 xn − z⟩ ≤ 0. n→∞

(2.54)

From the assumption F ≠ 0, it follows that, for any p ∈ F, ‖xn+1 − p‖2 = (1 − αn )‖Tn+1 xn − Tn+1 p‖ + αn ‖u − z‖ ≤ (1 − αn )‖xn − p‖ + αn ‖u − z‖ ≤ max{‖x0 − p‖, ‖u − z‖} = M.

Hence {xn } is a bounded sequence, so is {Tn+1 xn }. By (B) and (C1), we obtain xn+1 − Tn+1 xn = αn (u − Tn+1 xn ) → θ

(n → ∞).

(2.55)

Thus we only need to prove lim sup⟨xn+1 − z, u − z⟩ ≤ 0, n→∞

(2.56)

108 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces that is, lim sup⟨xn − z, u − z⟩ ≤ 0.

(2.57)

lim sup⟨u − z, xn − z⟩ = lim ⟨u − z, xnj − z⟩.

(2.58)

n→∞

Let j→∞

n→∞

Without loss of generality, we may assume that xnj ⇀ p as j → ∞. Then (2.58) can be rewritten as follows: lim sup⟨u − z, xn − z⟩ = ⟨u − z, p − z⟩. n→∞

(2.59)

If we can prove p ∈ F, then we may conclude from the property of the distance projection PF that ⟨u − z, p − z⟩ ≤ 0.

(2.60)

Thus we now need to prove p ∈ F. In fact, ‖xn+2 − Tn+2 Tn+1 xn ‖ = ‖xn+2 − Tn+2 xn+1 + Tn+2 xn+1 − Tn+2 Tn+1 xn ‖ ≤ ‖xn+2 − Tn+2 xn+1 ‖ + ‖xn+1 − Tn+1 xn ‖,

which implies from (2.55) that limn→∞ ‖xn+2 − Tn+2 xn ‖ = 0. Note that xn+r − Tn+r Tn+r−1 ⋅ ⋅ ⋅ Tn+1 xn → θ

(n → ∞).

(2.61)

Using (B), we have ‖xn+r+1 − xn+1 ‖ ≤ (1 − αn )‖xn − xn+r ‖ + M1 |αn − αn+r |,

(2.62)

where M1 = sup{|u‖ + ‖Tn+1 xn+r ‖ : n ≥ 1}. Using assumption (C3) and Lemma 1.10.2, we see that xn+r − xn → θ as n → ∞. In view of (2.61), we obtain xn − Tn+r Tn+r−1 ⋅ ⋅ ⋅ Tn+1 xn → θ

(n → ∞).

Let n = k mod r, where k ∈ {1, 2, . . . , r}. Hence (2.63) can be rewritten as follows: xn − Tk+r Tk+r−1 ⋅ ⋅ ⋅ Tk+1 xn → θ

(n → ∞).

It follows from the demiclosedness principle of nonexpansive mappings that p ∈ Fix(Tk+r Tk+r−1 ⋅ ⋅ ⋅ Tk+1 ) = F. This completes the proof.

(2.63)

2.4 Iterative methods of fixed points | 109

Remark 2.4.4. In fact, in Theorem 2.4.10, we only need to assume r

F = ⋂ Fix(Ti ) = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) ≠ 0. i=1

Then condition (CC) can be obtained. To weaken condition (C3), we first prove the following proposition. Proposition 2.4.2. Let a ∈ ℝ and r ∈ ℕ. Suppose that {an } ∈ l∞ satisfies condition μn (an ) ≤ a for any Banach limit μn . If lim supn→∞ (an+r −an ) ≤ 0, then lim supn→∞ an ≤ a. Proof. Set bn+i = an+ir (i = 0, 1, 2, . . . , p − 1). Then μn (bn ) = μn (an ) ≤ a for any Banach limit μn . For any ε > 0, there exist m ≥ 1 and p ≥ max{2, m} such that bn + bn+1 + ⋅ ⋅ ⋅ + bn+p+1 p

ε 0, there exists δ > 0 with 0 < t < δ such that ε 󵄩󵄩 (n) 󵄩 , 󵄩󵄩xt − PF u󵄩󵄩󵄩 ≤ 2M

∀n ≥ 0,

(2.80)

112 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces where M ≥ max{‖u − xn+r ‖, ‖xn+r − PF u‖} for all n ≥ 0 and t ∈ (0, 1). Note that ⟨u − PF (u), xn+r − PF (u)⟩ = ⟨u − xt(n) , xn+r − xt(n) ⟩ + ⟨u − xt(n) , xt(n) − PF (u)⟩ + ⟨xt(n) − PF (u), xn+r − PF (u)⟩ 󵄩 󵄩 ≤ ⟨u − xt(n) , xn+r − xt(n) ⟩ + 2M 󵄩󵄩󵄩xt(n) − PF (u)󵄩󵄩󵄩

≤ ⟨u − xt(n) , xn+r − xt(n) ⟩ + ε,

n ≥ 0, ∀t ∈ (0, δ).

(2.81)

Taking the Banach limit μn for n ≥ 1, we obtain μn ⟨u − PF (u), xn+r − PF (u)⟩ ≤ μn ⟨u − xt(n) , xn+r − xt(n) ⟩ + ε.

(2.82)

Letting t → 0 in (2.82), we see that μn ⟨u − PF (u), xn+r − PF (u)⟩ ≤ lim sup μn ⟨u − xt(n) , xn+r − xt(n) ⟩ + ε < ε. t→0

From the arbitrariness of ε > 0, we have μn ⟨u − PF (u), xn+r − PF (u)⟩ ≤ 0. By the property of the Banach limit μn , it follows that μn ⟨u − PF (u), xn − PF (u)⟩ = μn ⟨u − PF (u), xn+r − PF (u)⟩ ≤ 0. This proves (2.75). Denote an = ⟨u − PF (u), xn − PF (u)⟩. Then, for any Banach limit μn , one has μn (αn ) ≤ 0 and lim sup(an+r − an ) = lim sup⟨u − PF (u), xn+r − xn ⟩ n→∞

n→∞

= lim sup⟨u − PF (u), xnj +r − xnj ⟩. j→∞

From the assumption that xnj +r − xnj ⇀ θ as j → ∞, one obtains that lim sup⟨u − PF (u), xnj +r − xnj ⟩ = 0. j→∞

Hence lim supn→∞ (an+r − an ) ≤ 0. By Proposition 2.4.2, we see that lim supn→∞ an ≤ 0. Define rn = max{⟨u − PF (u), xn − PF (u)⟩, 0}. Then rn → 0 as n → ∞. Let z = PF u. Since 󵄩 󵄩2 ‖xn+1 − z‖2 = 󵄩󵄩󵄩(1 − αn )(Tn+1 xn − Tn+1 z) + αn (u − z)󵄩󵄩󵄩 = (1 − αn )2 ‖Tn+1 xn − Tn+1 z‖2 + 2αn (1 − αn )⟨u − z, xn − z⟩ + αn 2 ‖u − z‖2 ≤ (1 − αn )‖xn − z‖2 + 2αn rn + αn 2 ‖u − z‖2

= (1 − αn )‖xn − z‖2 + o(αn ),

and using Lemma 1.10.2, we obtain xn → z as n → ∞. This completes the proof.

2.4 Iterative methods of fixed points | 113

In order to study common fixed points of a countable family of nonexpansive mappings, we give the following definition of the W-mapping. Definition 2.4.2 ([80]). Let {Ti }∞ i=1 : C → C be a countable family of nonlinear mappings and let {γn } be a sequence in [0, 1]. For all n ≥ 1, define a mapping Wn as follows: Un,n+1 = I,

Un,n = γn Tn Un,n+1 + (1 − γn )I,

Un,n−1 = γn−1 Tn−1 Un,n + (1 − γn−1 )I, ..., Un,k = γk Tk Un,k+1 + (1 − γk )I,

Un,k−1 = γk−1 Tk−1 Un,k + (1 − γk−1 )I, ..., Un,2 = γ2 T2 Un,3 + (1 − γ2 )I,

Wn = Un,1 = γ1 T1 Un,2 + (1 − γ1 )I. Then Wn is called the W-mapping generated by Tn , Tn−1 , . . . , T1 and γn , γn−1 , . . . , γ1 . Next, we list two importance properties of the W-mappings as follows. Proposition 2.4.3. Let {Ti }ni=1 : C → C be a finite family of nonexpansive mappings with ⋂ni=1 Fix(Ti ) ≠ 0. Let γi a real number in (0, 1) for each i = 1, 2, . . . , n and let Wn be the W-mapping generated by Tn , Tn−1 , . . . , T1 and γn , γn−1 , . . . , γ1 . Then Wn is averaged nonexpansive and n

Fix(Wn ) = ⋂ Fix(Ti ). i=1

If γi ∈ (0, b), where b is a real number in 0 < b < 1, then, for all k ≥ 1 and x ∈ C, the limit limn→∞ Un,k x exists. Furthermore, if D is a bounded subset of C, then the limit limn→∞ Un,k x uniformly exists for all x ∈ D, that is, for all ε > 0, there exists n0 ≥ k (n0 is independent of x) such that ‖Un,k x − Um,k x‖ < ε,

∀m, n ≥ n0 , x ∈ D.

Hence, if we define W : C → C as follows: Wx = lim Wn x = lim Un,1 x, n→∞

n→∞

∀x ∈ C,

(2.83)

then W is called the W-mapping generated by T1 , T2 , . . . and γ1 , γ2 , . . . Proposition 2.4.4. Let {Ti }∞ i=1 : C → C be a countable family of nonexpansive mappings ∞ with F = ⋂i=1 Fix(Ti ) ≠ 0. Let γi ∈ (0, b] for each i = 1, 2, . . . , n, where 0 < b < 1, and let W be the W-mapping generated by T1 , T2 , . . . and γ1 , γ2 , . . . Then W : C → C is an averaged nonexpansive mapping with Fix(W) = F.

114 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Since the proof of the above two propositions can be derived from the definition of the W-mapping and some properties of averaged mappings, we here omit the proof. Theorem 2.4.12. Let {Ti }∞ i=1 : C → C be a countable family of nonexpansive mappings with F = ⋂∞ Fix(T ) = ̸ 0. Let γi ∈ (0, b] for each i = 1, 2, . . . , n, where 0 < b < 1, and let i i=1 Wn be the W-mapping generated by Tn , Tn−1 , . . . , T1 and γn , γn−1 , . . . , γ1 . Let {αn } be a real sequence in [0, 1] satisfying the following conditions: (C1) αn → 0 as n → ∞; (C2) ∑∞ n=0 αn = ∞. For an arbitrary initial value x0 ∈ C and a fixed anchor u ∈ C, define an iterative sequence {xn } as follows: xn+1 = αn u + (1 − αn )Wn xn ,

∀n ≥ 1.

(WNM)

Then {xn } converges strongly to PF u. Proof. For any initial value y0 ∈ C and a fixed anchor u ∈ C, define another iterative sequence {yn } as follows: yn+1 = αn u + (1 − αn )Wyn ,

∀n ≥ 1,

(WM)

where W is defined by (2.83). It follows from Proposition 2.4.4 that W : C → C is an averaged mapping. From (WM) and Corollary 2.4.4, it follows that yn → PFix(W) u as n → ∞. Using Proposition 2.4.4, we have Fix(W) = F. Hence {yn } converges strongly to PF u. Now, we only need to prove that yn − xn → θ as n → ∞. To achieve this, it follows from (WNM) and (WM) that ‖yn+1 − xn+1 ‖ ≤ (1 − αn )‖Wyn − Wn xn ‖

≤ (1 − αn )‖Wyn − Wxn ‖ + ‖Wxn − Wn xn ‖ ≤ (1 − αn )‖yn − xn ‖ + σn ,

∀n ≥ 0,

where σn = ‖Wxn − Wn xn ‖. Now, we estimate σn as follows: σn = lim ‖Wm xn − Wn xn ‖ m→∞

󵄩󵄩m−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 = lim 󵄩󵄩󵄩 ∑ (Wj+1 xn − Wj xn )󵄩󵄩󵄩 m→∞󵄩 󵄩󵄩 󵄩󵄩 j=n 󵄩 m−1

≤ lim ∑ ‖Wj+1 xn − Wj xn ‖ m→∞

j=n

m−1

j+1

j=n

i=1

≤ ∑ (∏ γi )‖Tj+1 xn − xn ‖

(2.84)

2.4 Iterative methods of fixed points | 115

∞

≤ 2M ∑ bn+1 j=n

2M n+1 = b , 1−b where M ≥ sup{‖xn − p‖ : n ≥ 0} and p ∈ Fix(T). Hence we have ∑∞ n=0 σn < ∞. Let an = ‖yn − xn ‖. Then (2.84) can be rewritten as follows: an ≤ (1 − αn )an + σn ,

∀n ≥ 0.

(2.85)

From Lemma 1.10.2, we obtain an → 0, that is, yn − xn → θ ⇐⇒ xn → PF u (n → ∞). This completes the proof. Since the convergence of the iterative method (MHIM) can be obtained via the convergence of the iterative method (HIM), we will not discuss the convergence of the iterative method (MHIM) here. Now, we prove a theorem to show how the convergence result of the iterative method (MHIM) is obtained. Theorem 2.4.13. Let C be a closed convex subset of H. Let T : C → C be a λ-averaged nonexpansive mapping with Fix(T) ≠ 0 and let f : C → C be a ρ-contractive mapping. Suppose that {αn } in [0, 1] satisfies the following conditions: (C1) αn → 0 as n → ∞; (C2) ∑∞ n=0 αn = ∞. Let {xn } be a sequence generated by (MHIM). Then {xn } converges strongly to a fixed point x ∗ ∈ Fix(T) of T. In addition, x ∗ is also a unique solution of the following variational inequality: ⟨(I − f )x∗ , x − x∗ ⟩ ≥ 0,

∀x ∈ Fix(T).

(VIP)

Proof. Since PFix(T) : C → Fix(T) is a nonexpansive mapping and f : C → C is a ρ-contractive mapping, we find that PFix(T) f : C → C is a ρ-contractive mapping. By the Banach’s contraction principle, there exists a unique x∗ ∈ Fix(T) such that x∗ = PFix(T) fx∗ , which is equivalent to the fact that x∗ is the unique solution of the variational inequality (VIP). For any y1 ∈ C, define an iterative sequence {yn } as follows: yn+1 = αn f (x ∗ ) + (1 − αn )Tyn ,

∀n ≥ 1.

(2.86)

116 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces By Corollary 2.4.4, we find that yn → PFix(T) fx ∗ = x ∗ ∈ Fix(T). From the iterative methods (MHIM) and (2.86), we see that 󵄩 󵄩 ‖yn+1 − xn+1 ‖ ≤ αn 󵄩󵄩󵄩fx ∗ − fxn 󵄩󵄩󵄩 + (1 − αn )‖Tyn − Txn ‖ 󵄩 󵄩 ≤ ραn 󵄩󵄩󵄩yn − x∗ 󵄩󵄩󵄩 + ραn ‖yn − xn ‖ + (1 − αn )‖yn − xn ‖ 󵄩 󵄩 = [1 − (1 − ρ)αn ]‖yn − xn ‖ + ραn 󵄩󵄩󵄩yn − x ∗ 󵄩󵄩󵄩 = [1 − (1 − ρ)αn ]‖yn − xn ‖ + o(αn ),

∀n ≥ 1.

It follows from Lemma 1.10.2 that yn − xn → θ as n → ∞. Hence xn → x ∗ ∈ Fix(T) as n → ∞. This completes the proof. Remark 2.4.5. If T : C → C is nonexpansive and the sequence {αn } satisfies condition (C4) or (C5) in Theorem 2.4.13, then the conclusion still holds. In recent years, hybrid projection methods have attracted much attention since they are useful when we investigate the strong convergence of iterative sequences; see [59, 65–69, 86, 81, 91, 95, 116–121]. Next, we give Nakajo–Takahashi’s result here. Theorem 2.4.14. Let C be a closed convex subset of H and let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Define a sequence {xn } as follows: x0 ∈ C, C1 = C, { { { { { {yn = αn xn + (1 − αn )Txn , { { { Cn = {z ∈ C : ‖yn − z‖ ≤ ‖xn − z‖}, { { { { { { {Qn = {z ∈ C : ⟨xn − z, x0 − xn ⟩ ≥ 0}, { { {xn+1 = PCn ∩Qn (x0 ), ∀n ≥ 0,

(HPM)

where {αn } ⊂ [0, 1] satisfies 0 ≤ αn < a < 1. Then {xn } converges strongly to PFix(T) (x0 ). Proof. The proof is split into five steps as follows: Step 1. Show that, for all n ≥ 0, Cn ∩ Qn are closed and convex. It is obvious that Cn ∩ Qn are closed. Note that ‖yn − z‖ ≤ ‖xn − z‖ ⇐⇒ ‖xn − yn ‖2 + 2⟨yn − xn , xn − z⟩ ≤ 0, which shows that Cn are convex. Hence Cn and Qn are closed and convex. Thus Cn ∩ Qn are also closed and convex. Step 2. Show that Fix(T) ⊂ Cn ∩ Qn for all n ≥ 0. This can be done by induction. It is obvious that Fix(T) ⊂ Cn . Since Q0 = C, we have Fix(T) ⊂ C0 ∩ Q0 . Assume that, for some k ≥ 0, Fix(T) ⊂ Ck ∩ Qk . It follows from the assumption Fix(T) ≠ 0 that Ck ∩ Qk ≠ 0 is closed and convex. Hence xk+1 = PCk ∩Qk (x0 ) is well-defined. From the property of the metric projection, we see that ⟨xk+1 − z, x0 − xn+1 ⟩ ≥ 0,

∀z ∈ Ck ∩ Qk .

(2.87)

2.4 Iterative methods of fixed points | 117

Since Fix(T) ⊂ Ck ∩ Qk , we obtain ⟨xk+1 − u, x0 − xn+1 ⟩ ≥ 0,

∀u ∈ Fix(T)

󳨐⇒ Fix(T) ⊂ Qk+1

󳨐⇒ Fix(T) ⊂ Ck+1 ∩ Qk+1 .

(2.88)

Hence Fix(T) ⊂ Cn ∩ Qn for all n ≥ 0. So Cn ∩ Qn ≠ 0 are closed and convex for all n ≥ 0. This implies that (HPM) is well-defined. Step 3. Show that {xn } is bounded. To achieved this, we denote z0 by z0 = PFix(T) (x0 ). Since xn+1 = PCn ∩Qn (x0 ), we have ‖xn+1 − x0 ‖ ≤ ‖z − x0 ‖,

∀z ∈ Cn ∩ Qn , n ≥ 0.

(2.89)

In particular, since z0 ∈ Fix(T) ⊂ Cn ∩ Qn , we obtain ‖xn+1 − x0 ‖ ≤ ‖z0 − x0 ‖,

∀n ≥ 0,

(2.90)

which shows that {xn } is a bounded sequence. Step 4. Show that xn − Txn → θ as n → ∞. Since xn+1 ⊂ Cn ∩ Qn ⊂ Qn and xn = PQn (x0 ), we see that ‖xn − x0 ‖ ≤ ‖xn+1 − x0 ‖,

n≥0

󳨐⇒ lim ‖xn − x0 ‖ exists n→∞

󳨐⇒ ‖xn+1 − x0 ‖2 − ‖xn − x0 ‖2 → 0

(n → ∞).

(2.91)

Since xn+1 ∈ Qn , we have ⟨xn − xn+1 , x0 − xn ⟩ ≥ 0,

∀n ≥ 0.

By the identical equality on H, (2.91) and (2.92), we derive 󵄩 󵄩2 ‖xn − xn+1 ‖2 = 󵄩󵄩󵄩(xn − x0 ) − (xn+1 − x0 )󵄩󵄩󵄩 = ‖xn − x0 ‖2 − 2⟨xn − x0 , xn+1 − x0 ⟩ + ‖xn+1 − x0 ‖2

= ‖xn+1 − x0 ‖2 − ‖xn − x0 ‖2 − 2⟨xn − xn+1 , x0 − xn ⟩ ≤ ‖xn+1 − x0 ‖2 − ‖xn − x0 ‖2 → 0

(n → ∞).

From (HPM) and xn+1 ∈ Cn , one arrives at 1 ‖y − xn ‖ 1 − αn n 1 ≤ (‖y − xn+1 ‖ + ‖xn+1 − xn ‖) 1 − αn n 1 ≤ (‖x − xn+1 ‖ + ‖xn+1 − xn ‖) 1 − αn n 2 = ‖x − xn ‖ → 0 (n → ∞). 1 − αn n+1

‖xn − Txn ‖ =

(2.92)

118 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Here we used the condition 0 ≤ αn < a < 1. This proves that xn − Txn → θ as n → ∞. Step 5. We prove that xn → z0 = PFix(T) (x0 ) as n → ∞. From Step 4 and the demiclosedness principle for nonexpansive mappings, we see that ωω (xn ) ⊂ Fix(T). It follows from (2.90) that lim sup ‖xn − x0 ‖ ≤ ‖x0 − z0 ‖. n→∞

(2.93)

Assume that lim sup⟨xn − z0 , x0 − z0 ⟩ = lim sup⟨xnj − z0 , x0 − z0 ⟩ n→∞

j→∞

and xnj → x.̂ Then x̂ ∈ Fix(T). It follows that lim sup⟨xn − z0 , x0 − z0 ⟩ = ⟨x̂ − z0 , x0 − z0 ⟩ ≤ 0. n→∞

(2.94)

By the properties of norm and scalar product on H, we obtain ‖xn − z0 ‖2 = ‖xn − x0 + x0 − z0 ‖2

= ‖xn − x0 ‖2 + 2⟨xn − x0 , x0 − z0 ⟩ + ‖x0 − z0 ‖2

= ‖xn − x0 ‖2 + 2⟨xn − z0 , x0 − z0 ⟩ − ‖x0 − z0 ‖2 .

(2.95)

Letting n → ∞ in (2.95), and combining (2.93) with (2.94), we have lim sup ‖xn − z0 ‖2 n→∞

≤ lim sup ‖xn − x0 ‖2 + 2 lim sup⟨xn − z0 , x0 − z0 ⟩ − ‖x0 − z0 ‖2 n→∞

2

n→∞

≤ (lim sup ‖xn − x0 ‖) − ‖x0 − z0 ‖2 ≤ 0. n→∞

Hence xn → z0 = PFix(T) (x0 ) as n → ∞. This completes the proof.

2.5 Fixed points of nonexpansive nonself-mappings Browder’s fixed point theorem (Theorem 1.9.4) claims that a nonexpansive selfmapping T defined on a nonempty, bounded, convex, and closed subset C of a Hilbert space H has fixed points. However, T may have no fixed points if it is a nonselfmapping. To ensure that nonexpansive nonself-mappings have fixed points, it is necessary to add some boundary conditions on the mappings. Recently, some boundary conditions have been proposed in different ways. Next, we introduce some popular boundary conditions. (1) Rothe’s condition: T(𝜕C) ⊂ C, where 𝜕C denotes the boundary of C.

2.5 Fixed points of nonexpansive nonself-mappings | 119

(2) Inward condition: Tx ∈ IC (x) for any x ∈ C, where IC (x) = {y ∈ H : y = x + a(z − x), z ∈ C, a ≥ 1}. (3) Weak inward condition: Tx ∈ IC (x) for any x ∈ C. (4) Generalized weak inward condition: d(Tx, C) < ‖Tx − x‖ for any x ∈ C and x ≠ Tx. (5) Nowhere normal–outward condition: Tx ∈ ℘Sx for any x ∈ C, where Sx = {y ∈ H : y ≠ x, PC y = x} and PC is the metric projection from H onto C. Proposition 2.5.1. For the above boundary conditions, the following implication relations hold: (1) 󳨐⇒ (2) 󳨐⇒ (3) 󳨐⇒ (4) 󳨐⇒ (5). Proof. (1) 󳨐⇒ (2). For any x ∈ C, if x ∈ 𝜕C, then Tx ∈ C and C ⊂ IC (x). Hence Tx ∈ IC (x). It is obvious that (2) 󳨐⇒ (3). (3) 󳨐⇒ (4). Assume that there exists x ∈ C with x ≠ Tx and d(Tx, C) = ‖Tx − x‖ > 0. Then ‖Tx − PC (Tx)‖ = ‖Tx − x‖. From the existence and the uniqueness of the metric projection PC (Tx), we see that x = PC (Tx) 󳨐⇒ ⟨Tx − x, y − x⟩ ≤ 0,

∀y ∈ C.

(2.96)

Since, for any x ∈ C, Tx ∈ IC (x), there exists yn ∈ TC (x) such that yn → Tx as n → ∞, where yn = x + an (zn − x),

an ≥ 1,

zn ∈ C.

(2.97)

Substituting (2.97) into (2.96), we obtain ⟨Tx − x, zn − x⟩ ≤ 0

󳨐⇒ ⟨Tx − x, an (zn − x)⟩ ≤ 0 󳨐⇒ ⟨Tx − x, yn − x⟩ ≤ 0,

∀x ≥ 1.

(2.98)

Letting n → ∞ in (2.98), we have ⟨Tx − x, Tx − x⟩ ≤ 0 󳨐⇒ Tx = x, which contradicts the assumption that x ≠ Tx. (4) 󳨐⇒ (5). Assume that there exists x ∈ C such that Tx ∉ ℘Sx . Then Tx ∈ Sx implies Tx ≠ x and PC (Tx) = x. From (4), we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Tx − PC (Tx)󵄩󵄩󵄩 = d(Tx, C) < ‖Tx − x‖ = 󵄩󵄩󵄩Tx − PC (Tx)󵄩󵄩󵄩, which is a contradiction. This completes the proof

120 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Remark 2.5.1. The above relations “󳨐⇒” are not reversible, the readers can illustrate this fact easily. Proposition 2.5.2. Let C be a closed convex subset of H and let T : C → H be a nonexpansive nonself-mapping which satisfies the nowhere normal–outward condition. Then Fix(T) = Fix(PC T) = Fix(TPC ). In addition, if C is bounded and T is nonexpansive, then Fix(T) is a nonempty closed convex subset of H. Proof. In order to establish the first equality, we only need to prove the inclusion relation Fix(PC T) ⊂ Fix(T). To achieve this, we assume that x = PC (Tx). Note that Tx ∈ ℘Sx 󳨐⇒ Tx ∉ Sx 󳨐⇒ Tx = x. To establish the second equality, we assume that x = TPC x. Note that PC x = PC T(PC x) 󳨐⇒ PC x ∈ Fix(PC T) = Fix(T), that is, TPC x = PC x. Since x = TPC x, we see that x = PC x 󳨐⇒ x = Tx. If C is a nonempty, bounded, closed, and convex subset of real Hilbert space H, then Fix(T) is a nonempty, closed, and convex subset of H. Indeed, since the self-mapping PC T : C → C is nonexpansive, it follows from Browder’s fixed point theorem (Theorem 1.9.4) that Fix(PC T) ≠ 0 is closed and convex. Noting that Fix(PC T) = Fix(T), we have the desired conclusion. This completes the proof. Proposition 2.5.3. Let C be a closed convex subset of H and let T : C → H be a nonexpansive nonself-mapping. If Fix(T) ≠ 0, then T satisfies the nowhere normal–outward condition. Proof. Suppose that there exists x0 ∈ C such that Tx0 ∈ Sx0 󳨐⇒ Tx0 ≠ x0 and PC Tx0 = x0 . Taking p ∈ Fix(T), we have ‖Tx0 − p‖2 = ‖Tx0 − x0 + x0 − p‖2

= ‖Tx0 − x0 ‖2 + 2⟨Tx0 − x0 , x0 − p⟩ + ‖x0 − p‖2

= ‖Tx0 − x0 ‖2 + 2⟨Tx0 − PC Tx0 , PC Tx0 − p⟩ + ‖x0 − p‖2 > ‖x0 − p‖2 ⇒ ‖Tx0 − p‖ > ‖x0 − p‖,

which contradicts ‖Tx0 − p‖ < ‖x0 − p‖. Hence, Tx ∈ ℘Sx , ∀x ∈ C. This completes the proof. Theorem 2.5.1. Let C be a closed convex subset of H and let T : C → H be a nonexpansive nonself-mapping satisfying Fix(T) ≠ 0. For each t ∈ (0, 1] and u ∈ C, define mappings Tt : C → C and St : C → C by Tt (x) = tu + (1 − t)PC Tx,

∀x ∈ C,

(2.99)

2.5 Fixed points of nonexpansive nonself-mappings | 121

and St (y) = PC [tu − (1 − t)Ty],

y ∈ C.

(2.100)

Then Tt and St have unique fixed points xt ∈ C and yt ∈ C, respectively. In addition, both {xt } and {yt } converge strongly to PFix(T) u. Proof. The existence and uniqueness of {xt } and {yt } follows from the Banach’s contraction principle. From Proposition 2.5.3, we see that T : C → H satisfies the nowhere normal–outward condition. It follows from Proposition 2.5.2 that Fix(T) = Fix(PC T). Denote U = PC T. Then xt = tu + (1 − t)PC Txt can be rewritten as follows: xt = tu + (1 − t)Uxt ,

∀t ∈ (0, 1].

(2.101)

From the Browder’s strong convergence theorem for U, it follows that {xt } converges strongly to PFix(U) u = PFix(T) u as t → 0. Since yt is the unique fixed point of St , we have yt = PC [tu − (1 − t)Tyt ],

∀t ∈ (0, 1].

(2.102)

Let zt = tu + (1 − t)Tyt . Then (2.102) can be rewritten as yt = PC zt . So zt = tu + (1 − t)TPC zt ,

∀t ∈ (0, 1].

(2.103)

Denote V = TPC . It follows from Proposition 2.5.2 that Fix(V) = Fix(TPC ) = Fix(T). Then (2.103) can be rewritten as follows: zt = tu + (1 − t)Vzt ,

∀t ∈ (0, 1].

(2.104)

Again, from the Browder’s strong convergence theorem for V, it follows that {zt } converges strongly to PFix(V) u = PFix(T) u as t → 0. Therefore, we have yt → PFix(T) u as t → 0. This completes the proof. Theorem 2.5.2. Let C be a closed convex subset of a Hilbert space H and let T : C → H be a nonexpansive nonself-mapping with Fix(T) ≠ 0. Let {αn } be a sequence in [0, 1] satisfying the following conditions: (C1) αn → 0; (C2) ∑∞ n=1 αn = ∞; (C3) ∑∞ n=1 |αn+1 − αn | < ∞ or α (C4) limn→∞ α n = 1. n+1

Define an iterative sequence {xn } as follows: x1 ∈ C, u ∈ C,

xn+1 = αn u + (1 − αn )PC Txn ,

Then {xn } converges strongly to PFix(T) u.

∀n ≥ 1.

(2.105)

122 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Proof. Denote U = PC T. It follows from Propositions 2.5.2 and 2.5.3 that Fix(U) = Fix(PC T) = Fix(T). Note that (2.105) can be rewritten as follows: x1 ∈ C, u ∈ C,

xn+1 = αn u + (1 − αn )Uxn ,

∀n ≥ 1.

(2.106)

Thus, by Wittmann’s convergence theorem (Theorem 2.4.7) and Xu’s convergence theorem (Theorem 2.4.8), we claim that {xn } converges strongly to a point PFix(U) u = PFix(T) u. This completes the proof. Theorem 2.5.3. Let C be a closed convex subset of a Hilbert space H and let T : C → H be a nonexpansive nonself-mapping with Fix(T) ≠ 0. Let {αn } be a sequence in [0, 1] satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=1 αn = ∞; (C3) ∑∞ n=1 |αn+1 − αn | < ∞ or α (C4) limn→∞ α n = 1. n+1

Define an iterative sequence {xn } as follows: x1 ∈ C, u ∈ C,

xn+1 = PC [αn u + (1 − αn )Txn ],

∀n ≥ 1.

(2.107)

Then {xn } converges strongly to PFix(T) u. Proof. Let yn = αn u + (1 − αn )Txn for each ≥ 1. Then (2.107) can be rewritten as xn+1 = PC yn . Hence yn+1 = αn+1 u + (1 − αn+1 )TPC yn ,

∀n ≥ 1.

(2.108)

Denote tn = αn+1 and V = TPC . Then (2.108) can be rewritten as follows: yn+1 = tn u + (1 − tn )Vyn ,

∀n ≥ 1.

(2.109)

From Propositions 2.5.2 and 2.5.3, we see that Fix(V) = Fix(PC T) = Fix(T). It is easy to see that {tn } satisfies conditions (C1), (C2) and (C3) or (C4). Thus, from Wittmann’s or Xu’s convergence theorem, it follows that {yn } converges strongly to a point PFix(V) u = PFix(T) u. Therefore, {xn } converges strongly to PFix(T) u. This completes the proof. Theorem 2.5.4. Let C be a closed convex subset of a Hilbert space H and let T : C → H be a nonexpansive nonself-mapping with Fix(T) ≠ 0. Let {αn } be a sequence in [0, 1] satisfying the following conditions:

2.6 Iterative methods for fixed points of quasi-nonexpansive mappings | 123

(C1) limn→∞ αn = 0; (C2) ∑∞ n=1 αn = ∞ For any λ ∈ (0, 1), denote Tλ (x) = λT +(1−λ)I and define an iterative sequence as follows: x1 ∈ C, u ∈ C,

xn+1 = PC [αn u + (1 − αn )Tλ xn ],

∀n ≥ 1.

(2.110)

Then {xn } converges strongly to PFix(T) u. Proof. Let un = αn u + (1 − αn )Tλ xn for each n ≥ 1. Then (2.110) can be rewritten as xn+1 = PC un . Thus (2.110) can be written as follows: un+1 = αn+1 u + (1 − αn+1 )Tλ PC un ,

∀n ≥ 1.

Denote W = Tλ PC . Then, from Proposition 2.1.4 (5), we see that W is nonexpansive and

(2.111) 1+λ -averaged 2

Fix(W) = Fix(Tλ ) ∩ Fix(PC ) = Fix(T) ∩ C = Fix(T). Using Suzuki’s convergence theorem [84] (see also Corollary 2.4.4) for W, we see that {un } converges strongly to a point PFix(W) u = PFix(T) u. Therefore, {xn } converges strongly to PFix(T) u. This completes the proof.

2.6 Iterative methods for fixed points of quasi-nonexpansive mappings The class of quasi-nonexpansive mappings is a class of mappings which is much more general than the class of nonexpansive mappings. Due to the word “quasi”, this class of mappings loses a number of good properties such as the Lipschitz continuity, the demiclosedness principle, and other important properties. For a long time, we did not know whether the Halpern iterative method (HIM) was valid for the class of quasinonexpansive mappings. In 2010, Maingé [45] solved this problem by using new techniques, and he proved that the Halpern-type viscosity iterative method is still valid for quasi-nonexpansive mappings. In this subsection, we discuss this method in a more general framework. Let us recall the definitions of quasi-nonexpansive and nonspreading mappings as follows. Definition 2.6.1. Let C be a nonempty subset of a real Hilbert space H and let T : C → H be a mapping. Then T is said to be (1) quasi-nonexpansive if Fix(T) ≠ 0 and ‖Tx − p‖ ≤ ‖x − p‖,

∀x ∈ C, p ∈ Fix(T);

124 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces (2) nonspreading if 2‖Tx − Ty‖2 ≤ ‖Tx − y‖2 + ‖x − Ty‖2 ,

∀x, y ∈ C.

Remark 2.6.1. (1) Every nonexpansive mapping with a nonempty fixed-point set is quasi-nonexpansive. (2) Every nonspreading mapping with a nonempty fixed-point set is quasi-nonexpansive. (3) Every firmly nonexpansive mapping is nonspreading. (4) Nonexpansive and nonspreading mappings are mutually independent. Example 2.6.1. Let E = {x ∈ H : ‖x‖ ≤ 1}. Let D = {x ∈ H : ‖x‖ ≤ 2}, and C = {x ∈ H : ‖x‖ ≤ 3}. Define a mapping T : C → H by θ, Tx = { PE x,

x ∈ D, x ∈ C\D.

Then T is a nonspreading mapping, but not a nonexpansive mapping. Example 2.6.2 ([88]). Define a mapping T : C → C, where C = [−1, 1], by Tx = −x, ∀x ∈ C. Then T is nonexpansive, but not nonspreading. Example 2.6.3. Take C = ℝ and define a mapping T : C → C by x sin x1 , Tx = { 2 0,

x ≠ 0, x = 0.

Then T is a quasi-nonexpansive mapping but not nonspreading. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T : C → H be a nonlinear mapping and let f : C → H be a ρ-contractive mapping. Let PC : H → C be the metric projection. Let {αn }, {βn }, and {γn } be three real number sequences in [0, 1] satisfying the following conditions: (1) αn + βn + γn = 1 for all n ∈ ℕ; (2) αn → 0 (n → ∞); (3) ∑∞ n=1 αn = ∞; (4) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1; Now, we introduce the following three Halpern-type viscosity iterative methods: x1 ∈ C,

x1 ∈ C,

xn+1 = αn f (xn ) + βn xn + γn PC Txn ,

n ≥ 1,

xn+1 = βn xn + (1 − βn )PC [αn f (xn ) + (1 − αn )Txn ],

x1 ∈ C,

xn+1 = PC [αn f (xn ) + βn xn + γn Txn ],

∀n ≥ 1.

(I) n ≥ 1,

(II) (III)

To show the convergence of the above three methods, we need the following tools.

2.6 Iterative methods for fixed points of quasi-nonexpansive mappings | 125

Proposition 2.6.1. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let T : C → H be a quasi-nonexpansive mapping. Then Fix(T) is a nonempty, closed, and convex subset of H. Proof. Let {pn } ⊂ Fix(T) such that pn → p. Then Tpn = pn and p ∈ C. It follows from the definition of a quasi-nonexpansive mapping T that pn → Tp, which implies that Tp = p, that is, Fix(T) is closed. Next we show that Fix(T) is also convex. To this end, let pi ∈ Fix(T) and pt = tp1 + (1 − t)p2 , for t ∈ (0, 1). Then it follows that 󵄩 󵄩2 ‖Tpt − pt ‖2 = 󵄩󵄩󵄩t(Tpt − p1 ) + (1 − t)(Tpt − p2 )󵄩󵄩󵄩

= t‖Tpt − p1 ‖2 + (1 − t)‖Tpt − p2 ‖2 − t(1 − t)‖p1 − p2 ‖2 ≤ t‖pt − p1 ‖2 + (1 − t)‖pt − p2 ‖2 − t(1 − t)‖p1 − p2 ‖2

≤ [t(1 − t)2 + t 2 (1 − t)]‖p1 − p2 ‖2 − t(1 − t)‖p1 − p2 ‖2

= 0.

This implies that pt = Tpt . So, Fix(T) is convex. Proposition 2.6.2 ([126]). If T : C → H is a quasi-nonexpansive mapping, then Fix(PC T) = Fix(T) = Fix(TPC ). Proof. It is obvious that Fix(T) ⊂ Fix(PC T). Next we show the converse inclusion. Assume that x = PC Tx. Then, for all x ∈ C, it follows from the property of PC that ‖x − p‖2 = ‖PC Tx − p‖2

≤ ‖Tx − p‖2 − ‖x − Tx‖2

≤ ‖x − p‖2 − ‖x − Tx‖2 . This implies that x = Tx. So, Fix(PC T) = Fix(T). It is obvious that Fix(T) ⊂ Fix(TPC ). Conversely, we assume x = TPC x, then PC x = PC T(PC x), which shows that PC x is a fixed point of PC T. It follows that PC x = T(PC x), which implies that PC x = x. Therefore, x = Tx. Hence the second equality holds. This completes the proof. Proposition 2.6.3 ([89]). Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let T : C → H be a nonspreading mapping. Then I − T is demiclosed at the origin. Theorem 2.6.1. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T : C → H be a quasi-nonexpansive mapping and let f : C → C be a ρ-contraction mapping, where ρ ∈ (0, 1). If I − T is demiclosed at the origin, then the iterative sequence {xn } generated by (I) strongly converges to a fixed point p ∈ Fix(T) of T and p is the unique solution of the following variational inequality: ⟨(I − f )x, y − x⟩ ≥ 0,

∀y ∈ Fix(T).

(VIP)

126 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Proof. Letting S = PC T, we see that S : C → C is a quasi-nonexpansive mapping. In fact, from Proposition 2.6.2, we have Fix(S) = Fix(T) ≠ 0. For all x ∈ C, p ∈ Fix(S), it follows from the nonexpansiveness of PC that ‖Sx − p‖ = ‖PC Tx − PC p‖ ≤ ‖Tx − p‖ ≤ ‖x − p‖. This shows that S : C → C is quasi-nonexpansive. Next, we prove that I − S is demiclosed at the origin. To achieve this, let {xn } be a sequence in C and assume that xn ⇀ x and xn − Sxn → θ as n → ∞. Since C is weakly closed, we have x ∈ C. For all p ∈ Fix(T), it follows from the firm nonexpansiveness of PC and the quasi-nonexpansiveness of T that ‖Sxn − p‖2 = ‖PC Txn − p‖2

≤ ‖Txn − p‖2 − ‖Txn − PC Txn ‖2 ≤ ‖xn − p‖2 − ‖Txn − Sxn ‖2 .

It follows that ‖Txn − Sxn ‖2 ≤ ‖xn − p‖2 − ‖Sxn − p‖2

≤ 2‖xn − Sxn ‖ ⋅ ‖xn − p‖2 ≤ M‖xn − Sxn ‖ → 0,

where M ≥ 2 sup{‖xn − p‖ : n ≥ 1}. Hence Txn → Sxn → θ as n → ∞. Note that ‖xn − Txn ‖ ≤ ‖xn − Sxn ‖ + ‖Sxn − Txn ‖, which implies that xn − Txn → θ as n → ∞. It follows from the assumption that I − T is demiclosed at the origin that x = Tx. Therefore, x = PC Tx = Sx. From [129, Theorem 2.1], we see that {xn } converges to a fixed point of S in norm and p ∈ Fix(S) = Fix(T). This completes the proof. Theorem 2.6.2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T : C → H be a quasi-nonexpansive mapping and let f : C → C be a ρ-contraction mapping, where ρ ∈ (0, 1). Suppose that I − T is demiclosed at the origin. Then the iterative sequence {xn } generated by (II) converges to a fixed point p ∈ Fix(T) of T in norm and p is the unique solution of variational inequality problem (VIP). Proof. Let yn = αn f (xn ) − (1 − αn )Txn for each n ≥ 1. Then it follows from (II) that, for all p ∈ Fix(T), 󵄩 󵄩 ‖yn − p‖ ≤ αn 󵄩󵄩󵄩f (xn ) − p󵄩󵄩󵄩 + (1 − αn )‖Txn − p‖ 󵄩 󵄩 󵄩 󵄩 ≤ αn 󵄩󵄩󵄩f (xn ) − f (p)󵄩󵄩󵄩 + αn 󵄩󵄩󵄩f (p) − p󵄩󵄩󵄩 + (1 − αn )‖xn − p‖ 󵄩 󵄩 ≤ [1 − (1 − ρ)αn ]‖xn − p‖ + αn 󵄩󵄩󵄩f (p) − p󵄩󵄩󵄩

2.6 Iterative methods for fixed points of quasi-nonexpansive mappings | 127

and ‖xn − p‖ ≤ βn ‖xn − p‖ + (1 − βn )‖yn − p‖ 󵄩 󵄩 󵄩 󵄩 ≤ αn 󵄩󵄩󵄩f (xn ) − f (p)󵄩󵄩󵄩 + αn 󵄩󵄩󵄩f (p) − p󵄩󵄩󵄩 + (1 − αn )‖xn − p‖ ≤ {βn + (1 − βn )[1 − (1 − ρ)αn ]}‖xn − p‖ + (1 − βn )(1 − ρ)αn ≤ max{‖x1 − p‖,

‖f (p) − p‖ 1−ρ

‖f (p) − p‖ } = Mp , 1−ρ

∀n ≥ 1.

This shows that {xn } is bounded. Hence both {Txn } and {xn } are bounded. It follows from Proposition 2.6.1 that Fix(T) ≠ 0 and it is closed convex. Thus, PFix(T) f is a ρ-contractive mapping. By Banach’s contraction principle, there exists a unique p ∈ Fix(T) such that p = PFix(T) f (p), which is equivalent to the fact that the variational inequality problem (VIP) has a unique solution p. From (II), we find that 󵄩 󵄩2 ‖xn+1 − p‖2 = 󵄩󵄩󵄩βn (xn − p) + (1 − βn )(PC yn − p)󵄩󵄩󵄩 󵄩 󵄩2 = βn ‖xn − p‖2 + (1 − βn )󵄩󵄩󵄩PC yn − p󵄩󵄩󵄩 − βn (1 − βn )‖xn − PC yn ‖2 ≤ βn ‖xn − p‖2 + (1 − βn )‖yn − p‖2 − (1 − βn )‖yn − PC yn ‖2 − βn (1 − βn )‖xn − PC yn ‖2

= βn ‖xn − p‖2 + (1 − βn )[αn2 ‖fxn − p‖2 + 2αn (1 − αn )⟨fxn − p, Txn − p⟩

+ (1 − αn )2 ‖Txn − p‖2 ] − (1 − βn )‖yn − PC yn ‖2 − βn (1 − βn )‖xn − PC yn ‖2

≤ βn ‖xn − p‖2 + (1 − βn )[1 − (1 − ρ)αn ‖xn − p‖2

+ 2αn (1 − αn )⟨fp − p, Txn − p⟩ + αn2 ‖fxn − p‖2 ]

− (1 − βn )‖yn − PC yn ‖2 − βn (1 − βn )‖xn − PC yn ‖2

≤ [1 − (1 − ρ)αn (1 − βn )]‖xn − p‖2

+ 2αn (1 − αn )(1 − βn )⟨fp − p, Txn − p⟩

+ (1 − βn )αn2 ‖fxn − p‖2 − (1 − βn )‖yn − PC yn ‖2 − βn (1 − βn )‖xn − PC yn ‖2 ,

which implies that ‖xn+1 − p‖2 − ‖xn − p‖2 + (1 − ρ)αn (1 − βn )‖xn − p‖2 + (1 − βn )‖yn − PC yn ‖2 + βn (1 − βn )‖xn − PC yn ‖2

≤ 2αn (1 − αn )(1 − βn )⟨fp − p, Txn − p⟩ + (1 − βn )αn2 ‖fxn − p‖2 ,

∀n ≥ 1.

128 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Denote Γn = ‖xn − p‖2 . The above inequality can be rewritten as Γn+1 − Γn + (1 − ρ)αn (1 − βn )Γn + (1 − βn )‖yn − PC yn ‖2 + βn (1 − βn )‖xn − PC yn ‖2

≤ 2αn (1 − αn )(1 − βn )⟨fp − p, Txn − p⟩ + (1 − βn )αn2 ‖fxn − p‖2 ,

∀n ≥ 1.

(2.112)

Now, we discuss (2.112) in two cases: Case 1. Γn+1 ≤ Γn for all n ≥ n0 . In this case, limn→∞ Γn exists. Taking the limit in (2.112), we have xn − PC yn → θ, yn − PC yn → θ as n → ∞. Therefore, we have xn − yn → θ as n → ∞. Since yn = αn fxn + (1 − αn )Txn for each n ≥ 1, we conclude from the boundedness of {fxn } and {Txn } that αn → 0 as n → ∞. So yn − Txn = αn f (xn − Txn ) → θ

(n → ∞)

and xn − Txn = xn − yn + yn − Txn → θ

(n → ∞).

From the assumption that I − T is demiclosed at the origin, we have ωW ⊂ Fix(T). It follows from the property of PC that lim sup⟨fp − p, Txn − p⟩ = lim sup⟨fp − p, xn − p⟩ n→∞

n→∞

= lim ⟨fp − p, xnj − p⟩ j→∞

= ⟨fp − p, x − p⟩ ≤ 0, where xnj ⇀ x ∈ Fix(T). Then (2.112) can be rewritten as follows: Γn+1 ≤ [1 − (1 − ρ)αn (1 − βn )]Γn + σn ,

∀n ≥ 1,

(2.113)

σ

where {σn } satisfies the condition lim supn→∞ αn ≤ 0. Using Lemma 1.10.2, we have n Γn → 0 as n → ∞, that is, xn → p as n → ∞. Case 2. Γnk ≤ Γnk +1 for all k ≥ 1. Define τ : ℕ → ℕ by τ(n) = max{k ≤ n : Γk ≤ Γk+1 }. Using Lemma 1.10.4, we find that τ(n) → ∞ as n → ∞ and Γτ(n) ≤ Γτ(n)+1 ,

Γn ≤ Γτ(n)+1 ,

∀n ≥ n0 .

It follows from (2.112) that Γτ(n)+1 −Γn → 0, xτ(n) −PC yτ(n) → θ and yτ(n) −PC yτ(n) → θ as n → ∞. These imply that xτ(n) − yτ(n) → θ 󳨐⇒ xτ(n) − Txτ(n) → θ

󳨐⇒ lim sup⟨fp − p, Txτ(n) − p⟩ ≤ 0. n→∞

(2.114)

2.6 Iterative methods for fixed points of quasi-nonexpansive mappings | 129

Using (2.112) again, we have (1 − ρ)ατ(n) (1 − βτ(n) )Γτ(n) ≤ 2ατ(n) (1 − ατ(n) )(1 − βτ(n) ) + ⟨fp − p, Txτ(n) − p⟩

2 + (1 − βτ(n) )ατ(n) ‖fxτ(n) − p‖2 .

Hence (1 − ρ)(1 − βτ(n) )Γτ(n) ≤ 2(1 − ατ(n) )(1 − βτ(n) )

+ ⟨fp − p, Txτ(n) − p⟩ + (1 − βτ(n) )ατ(n) ‖fxτ(n) − p‖2

≤ 2(1 − ατ(n) )(1 − βτ(n) )⟨fp − p, Txτ(n) − p⟩ + ατ(n) M1 ,

(2.115)

where M1 ≥ max{‖fxτ(n) ‖2 : n ≥ 1}. It follows from (2.114) and (2.115) that Γτ(n) → 0 (n → ∞) 󳨐⇒ Γτ(n)+1 → 0 (n → ∞) 󳨐⇒ Γn → 0 (n → ∞), that is, xn → p as n → ∞. This completes the proof. To obtain the convergence of the iterative method (III), we now establish a more general result. Theorem 2.6.3. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let S, T : C → C be two quasi-nonexpansive mappings with F = Fix(S) ∩ Fix(T) ≠ 0 and let f : C → C be a ρ-contractive mapping. Let {αn }, {βn }, {γn }, and {μn } be four real number sequences in [0, 1] satisfying the following conditions: (1) αn + βn + γn + μn = 1 for each n ∈ N; (2) αn → 0 as n → ∞ and ∑∞ n=1 αn = ∞; (3) 0 < lim infn→∞ γn ≤ lim supn→∞ γn < 1; (4) lim infn→∞ γn μn > 0. Define a sequence {xn } in C by x1 ∈ C,

xn+1 = αn fxn + βn xn + γn Sxn + μn Txn ,

∀n ≥ 1.

(IV)

If both I − S and I − T are demiclosed at the origin, then the sequence {xn } generated by (IV) converges to a common fixed point p ∈ F of S and T in norm, and p is the unique solution of the variational inequality problem ⟨(I − f )p, y − p⟩ ≥ 0,

∀y ∈ F.

(VIP’)

Proof. Using the proof method of Theorem 2.6.3, we can establish that {xn } is bounded, so are {fxn }, {Sxn }, and {Txn }. Since Fix(S) and Fix(T) are (nonempty) closed convex subsets, we find that F is closed and convex. It follows from the assumption that F ≠ 0 and F is a (nonempty) closed convex subset of H. Therefore, for all x ∈ H, PF x exists

130 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces uniquely. Since PF f : C → C is a ρ-contractive mapping, there exists a unique p ∈ F such that p = PF fp. Hence p is the unique solution of (VIP’). Since the rest of the proof is similar to the corresponding part of the proof of Theorem 2.6.2, we omit it here. This completes the proof. Theorem 2.6.4. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T : C → H be a quasi-nonexpansive mapping and let f : C → H be a ρ-contractive mapping. If I − T is demiclosed at the origin, then the sequence {xn } generated by (III) converges to a particular fixed point p ∈ Fix(T) of T in norm and p is the unique solution of the variational inequality problem (VIP). Proof. Denote zn = αn fxn + βn xn + γn Txn for each n ≥ 1. Then (III) can be rewritten as xn = PC zn for each n ≥ 1. So, zn+1 = αn+1 (fPC )zn + βn+1 PC zn + γn+1 TPC zn 1 1 = αn+1 (fPC )zn + βn+1 zn + βn+1 Szn + γn+1 TPC zn , 2 2

∀n ≥ 1,

(2.116)

where S is a nonexpansive mapping satisfying 2PC = I+S. Denote τn = αn+1 , ηn = 21 βn+1 , sn = γn+1 for each n ≥ 1, g = fPC , and V = TPC . Then (2.116) can be rewritten as zn+1 = τn gzn + ηn zn + ηn Szn + sn Vzn ,

∀n ≥ 1.

From Theorem 2.6.3, we see that {xn } converges to p = PFix(S)∩Fix(T) g(p) in norm. By Proposition 2.6.2, one has Fix(V) = Fix(T) and Fix(S) = C. It follows that Fix(S) ∩ Fix(V) = C ∩ Fix(T) = Fix(T), which implies that p = PFix(T) g(p) = PFix(T) fPC (p) = PFix(T) f (p). This shows that p is the unique solution of the variational inequality problem (VIP). Thus it follows from xn+1 = PC zn that {xn } converges to p ∈ Fix(T) in norm and p is the unique solution of the variational inequality problem (VIP). This completes the proof. Remark 2.6.2. Applying Theorems 2.6.1, 2.6.2, 2.6.3, and 2.6.4 to nonexpansive mappings with fixed points and using Proposition 2.6.3, we can obtain strong convergence results for fixed points of nonspreading mappings.

2.7 Applications In this subsection, we give some applications of the convergence theorems in the previous subsections to optimization problems, variational inequality problems, and split feasibility problems.

2.7 Applications | 131

2.7.1 Convex minimization problems Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let φ : C → ℝ be a continuous Fréchet differentiable convex functional. Consider the following convex minimization problem: Find a point x̃ ∈ C such that φ(x̃) = min{φ(x) : x ∈ C}.

(2.117)

We use Ω to denote the solution set of problem (2.117) and use ∇φ(x) to denote the gradient of φ at x ∈ C. Assume that Ω ≠ 0. Note that x̃ ∈ Ω ⇐⇒ ⟨∇φ(x), x − x̃⟩ ≥ 0,

∀x ∈ C ⇐⇒ x̃ = PC (I − γ∇φ)x̃,

∀γ > 0.

It follows from the Baillon–Haddad’s theorem that, if ∇φ(x) is L-Lipschitz continuous, then ∇φ is L1 -inverse strongly monotone. Thus, from Proposition 2.1.4 (3), if γ ∈ (0, L2 ), then I −γ∇φ is Lγ -averaged nonexpansive. Denote Tγ := PC (I −γ∇φ). Then, by Proposi2 tion 2.1.4 (5), we see that Tγ is 2+Lγ -averaged nonexpansive. Hence there exists a non4 expansive mapping S : H → H such that Tγ = (1 −

2 + Lγ 2 + Lγ )I + S 4 4

(2.118)

and Fix(Tγ ) = Fix(S) = Ω. Now, we consider the following iterative methods: x1 ∈ C is an initial point chosen arbitrarily and xn+1 = Tγ xn = Tγn x1 , u ∈ H, u ∈ C,

∀n ≥ 1;

xn+1 = PC [αn u + (1 − αn )Tγ xn ], xn+1 = αn u + (1 − αn )PC Tγ xn ,

(2.119) ∀n ≥ 1; ∀n ≥ 1.

(2.120) (2.121)

Theorem 2.7.1. Assume that Ω ≠ 0 and γ ∈ (0, L2 ). Let {αn } be a real number sequence in [0, 1] satisfying the following conditions: (C1) αn → 0 as n → ∞; (C2) ∑∞ n=1 αn = ∞. Then the sequence {xn } generated by (2.119) converges weakly to p = limn→∞ PΩ xn . In addition, the sequences generated by (2.120) and (2.121) converge strongly to x∗ = PΩ u. Proof. The first conclusion can be derived directly from Corollary 2.4.1. The second conclusion can be derived directly from Theorem 2.5.4. Note that Fix(V) = Fix(PC ) ∩ Fix(Tγ ) = C ∩ Fix(Tγ ) = Fix(Tγ ) = Ω. Applying Corollary 2.4.4 to V := PC Tγ , we can obtain the third conclusion immediately. This completes the proof.

132 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces 2.7.2 Monotone variational inequality problems Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let A : C → H be a k-Lipschitz continuous and η-strongly monotone mapping. We consider the following variational inequality problem: Find a x∗ ∈ C such that ⟨Ax∗ , x − x∗ ⟩ ≥ 0,

∀x ∈ C.

(VIP)

Letting VI(C, A) denote the solution set of the variational inequality problem, we see that x∗ ∈ VI(C, A) ⇐⇒ x∗ = PC (x∗ − μAx∗ ),

∀μ > 0.

2η

If μ ∈ (0, k ), then PC (I − μA) is a Banach contractive mapping from C to itself. Indeed, for all x, y ∈ C, 󵄩󵄩 󵄩 󵄩󵄩PC (I − μA)x − PC (I − μA)y󵄩󵄩󵄩 ≤ √1 − μ(2η − μk 2 )‖x − y‖. From Banach’s contraction principle, we see that PC (I − μA) has a unique fixed point x∗ ∈ C. Hence VI(C, A) = {x∗ }. The popular method for solving the variational inequality problem is the projected gradient method: x1 ∈ C,

xn+1 = PC (I − μA)xn ,

∀n ≥ 1.

Here the projected gradient method is actually the famous Banch–Picard iterative method: x1 ∈ C,

n

xn+1 = [PC (I − μA)] x1 ,

∀n ≥ 1.

It follows from Banach’s Contraction Principle that xn → x ∗ as n → ∞. These methods require us to compute the metric projection at every iteration, however, there are no analytic expressions for the metric projection operator in most cases. Recently, Yamada [108] introduced the following hybrid steepest descent method: x1 ∈ C,

xn+1 = (I − μλn A)Txn ,

∀n ≥ 1,

(HSDM)

where T : H → H is a nonexpansive mapping with Fix(T) ≠ 0, {λn } ⊂ (0, 1) and A : H → H is a k-Lipschitz continuous and η-strongly monotone mapping. Theorem 2.7.2. Suppose that μ ∈ (0, satisfying the following conditions: (C1) λn → 0 as n → ∞; (C2) ∑∞ n=1 λn = ∞;

2η ) k2

and {λn } is a real number sequence in [0, 1]

2.7 Applications | 133

(C3) ∑∞ n=1 |λn+1 − λn | < ∞ or λ (C4) limn→∞ λ n = 1. n+1

Then the sequence {xn } generated by (HSDM) converges strongly to x ∗ , which is the unique solution of the variational inequality problem (VIP). Proof. Note that (HSDM) can be rewritten as xn+1 = λn (I − μA)Txn + (1 − λn )Txn ,

∀n ≥ 1.

(2.122)

Denote f (x) = (I − μA)Tx for all x ∈ H. Then (2.122) can be rewritten as xn+1 = λn f (xn ) + (1 − λn )Txn ,

∀n ≥ 1,

(2.123)

where f : H → H is a ρ-contractive mapping. In fact, for all x, y ∈ H, one has 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩f (x) − f (y)󵄩󵄩󵄩 = 󵄩󵄩󵄩(I − μA)Tx − (I − μA)Ty󵄩󵄩󵄩 󵄩 󵄩2 = 󵄩󵄩󵄩Tx − Ty − μ(ATx − ATy)󵄩󵄩󵄩 = ‖Tx − Ty‖2 − 2μ⟨Tx − Ty, ATx − ATy⟩ + μ2 ‖ATx − ATy‖2 ≤ ‖Tx − Ty‖2 + 2ημ‖Tx − Ty‖2 + μ2 k 2 ‖Tx − Ty‖2 = [1 − μ(2η − μk 2 )]‖Tx − Ty‖2 ≤ [1 − μ(2η − μk 2 )]‖x − y‖2 . It follows that ‖fx − fy‖ ≤ ρ‖x − y‖,

∀x, y ∈ H,

where ρ = √1 − μ(2η − μk 2 ) ∈ (0, 1). By Remark 2.4.5, we see that {xn } converges strongly to a fixed point x∗ ∈ Fix(T), where x∗ = PFix(T) fx ∗ of T, which is the unique solution of the following variational inequality: ⟨fx∗ − x∗ , y − x∗ ⟩ ≤ 0,

∀y ∈ Fix(T)

⇐⇒ ⟨(I − μA)Tx − x , y − x∗ ⟩ ≤ 0, ∗

∗

⇐⇒ ⟨Ax , y − x ⟩ ≥ 0, ∗

∗

∀y ∈ Fix(T)

∀y ∈ Fix(T).

This completes the proof. Theorem 2.7.3. Let T : H → H be a nonexpansive mapping with Fix(T) ≠ 0 and let A : C → H be a k-Lipschitz continuous and η-strongly monotone mapping. For any 2η λ ∈ (0, 1), define a mapping Tλ = (1 − λ)I + λT. Suppose that μ ∈ (0, k2 ) and {λn } is a real number sequence in (0, 1) satisfying the following conditions: (C1) λn → 0 as n → ∞; (C2) ∑∞ n=1 λn = ∞.

134 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Define an iterative sequence {xn } as follows: x1 ∈ C,

xn+1 = (I − μλn A)Tλ xn ,

∀n ≥ 1.

(HSDM)

Then {xn } converges strongly to x∗ ∈ C, which is a unique solution of the variational inequality (VIP). Proof. By a similar proof to that of Theorem 2.6.2, (HSDM) can be rewritten as follows: xn+1 = λn g(xn ) + (1 − λn )Tλ xn ,

∀n ≥ 1,

(2.124)

where g(x) = (I − μA)Tλ x is a ρ-contractive mapping. By Theorem 2.4.13, it follows that {xn } converges strongly to a fixed point x∗ of Tλ , where x ∗ ∈ Fix(Tλ ) = Fix(T), and x∗ = PFix(Tλ ) gx∗ = PFix(T) gx∗ , which is equivalent to ⟨gx ∗ − x∗ , y − x∗ ⟩ ≤ 0,

∀y ∈ Fix(T)

⇐⇒ ⟨(I − μA)Tx − x , y − x ∗ ⟩ ≤ 0, ∗

∗

⇐⇒ ⟨Ax , y − x ⟩ ≥ 0, ∗

∗

∀y ∈ Fix(T)

∀y ∈ Fix(T).

This completes the proof. 2.7.3 Split feasibility problems Let C and Q be two nonempty, closed, and convex subsets of Hilbert spaces H1 and H2 , respectively. Let A : H1 → H2 be a bounded linear operator. Now, we consider the following split feasibility problem (SFP): Find a point x̂ ∈ C

such that Ax̂ ∈ Q.

(SFP)

The split feasibility problem is a special convex feasibility problem. It has very wide applications in signal processing, computer tomography, and radiation therapy treatment planning. For more details, we refer the readers to [15, 16, 18–22, 102]. Denote A−1 (Q) = {x ∈ H1 : Ax ∈ Q} and Γ = C ∩ A−1 (Q). Let L := ρ(A∗ A) denote the spectral radius of A∗ A and let A∗ be the adjoint operator of A. Let 1 f (x) = ‖I − PQ Ax‖2 , 2

∀x ∈ C.

Then we have the gradient of f : ∇f (x) = A∗ (I − PQ )Ax,

∀x ∈ C,

(2.125)

and 󵄩󵄩 󵄩 2 󵄩󵄩∇f (x) − ∇f (y)󵄩󵄩󵄩 ≤ ‖A‖ ‖x − y‖ = L‖x − y‖,

∀x, y ∈ C.

2.7 Applications | 135

Note that x∗ ∈ argminx∈C f (x) ⇐⇒ ⟨∇f (x∗ ), y − x ∗ ⟩ ≥ 0,

∀y ∈ C

⇐⇒ x ∈ PC [(I − γ∇f )x ]. ∗

(2.126)

∗

Under the assumption that Γ ≠ 0, we know that x∗ ∈ Γ ⇐⇒ x∗ ∈ argminx∈C f (x).

(2.127)

In fact, it follows from (2.126) that x∗ ∈ argminx∈C f (x) ⇐⇒ ⟨∇f (x∗ ), y − x ∗ ⟩ ≥ 0,

∀y ∈ C.

By (2.125), we obtain ⟨A∗ (I − PQ )Ax∗ , y − x∗ ⟩ ≥ 0,

∀y ∈ C

󳨐⇒ ⟨(I − PQ )Ax , Ay − Ax ⟩ ≥ 0, ∗

∗

∀y ∈ C.

(2.128)

From the assumption that Γ ≠ 0, we have that if z ∈ Γ, then z ∈ C and, if Az ∈ Q, then (I − PQ )Az = θ. Taking y = z ∈ C in (2.128), we conclude from (I − PQ )Az = θ that ⟨(I − PQ )Ax ∗ − (I − PQ )Az, Az − Ax∗ ⟩ ≥ 0

󳨐⇒ ⟨(I − PQ )Ax∗ − (I − PQ )Az, Ax∗ − Az⟩ ≤ 0.

(2.129)

Since I − PQ is firmly nonexpansive, it follows from (2.129) that 󵄩󵄩 ∗ 󵄩2 ∗ ∗ ∗ 󵄩󵄩(I − PQ )Ax 󵄩󵄩󵄩 ≤ 0 󳨐⇒ Ax = PQ Ax ∈ Q 󳨐⇒ x ∈ Γ. A popular method for solving problem (SFP) is the Byrne’s CQ-method: x1 ∈ H1 ,

xn+1 = PC [xn − γA∗ (I − PQ )Axn ],

∀n ≥ 1.

(CQ)

Indeed, it is the discretization of fixed point equation (2.126) and belongs to the class of projection gradient methods. In addition, the CQ-method has its specific structure and hence it is necessary to investigate the CQ-method extensively. Setting Tγ := I − γ∇f , for any γ ∈ (0, L2 ), one sees that Tγ is averaged nonexpansive. So PC Tγ is also averaged nonexpansive and Fix(PC Tγ ) = Fix(PC ) ∩ Fix(Tγ ) = C ∩ A−1 (Q) = Γ. Theorem 2.7.4. Let {xn } be the sequence generated by the CQ-method. Then {xn } converges weakly to a solution x∗ ∈ Γ of problem (SFP). Proof. Setting U = PC Tγ , we see that the CQ-method can be rewritten as follows: xn+1 = PC Tγ xn = Uxn ,

∀n ≥ 1.

From Corollary 2.4.1, we derive the conclusion immediately. This completes the proof.

136 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Now, we consider the following methods: for any x1 ∈ C and for any fixed u ∈ C, (1) xn+1 = αn u + (1 − αn )PC Tγ xn , ∀n ≥ 1; (2) xn+1 = PC [αn u + (1 − αn )Tγ xn ], ∀n ≥ 1. If {αn } is a real number sequence in [0, 1] satisfying the following conditions: (C1) αn → 0 as n → ∞; (C2) ∑∞ n=1 αn = ∞, then both methods (1) and (2) converge to a particular solution of the (SFP) in norm, that is, x∗ = PΓ u. In particular, for method (2), by taking u = θ ∈ H1 , we can show that {xn } converges strongly to the minimum norm solution of problem (SFP). This is not available in method (1). Further, for more results on problem (SFP), we refer the readers to [15, 58, 62–64, 94, 57, 58, 102, 104, 109–111, 124, 127]. Theorem 2.7.5. Let C and Q be two nonempty, closed, and convex subsets of Hilbert spaces H1 and H2 , respectively. Let A : H1 → H2 be a bounded linear operator. Suppose that the sequences {αn } and {βn } satisfy the following conditions: (1) αn → 0 and βn → 0 as n → ∞; (2) ∑∞ n=1 αn = ∞; α (3) βn → 0 as n → ∞. n

Let {xn } be a sequence generated in the following process: for any u, x1 ∈ H1 , xn+1 = PC [αn u + (1 − αn )(xn − βn A∗ (I − PQ )Axn )],

∀n ≥ 1.

(2.130)

If Γ ≠ 0, then the sequence {xn } generated by (2.130) converges to x ∗ = PΓ u in norm. In particular, if we take u = θ, then {xn } converges to x ∗ = PΓ θ in norm, that is, the minimum norm solution of problem (SFP). Proof. Since Γ ≠ 0 is closed convex, it follows that PΓ u is uniquely determined; denote it by x∗ = PΓ u. Denote vn = xn − βn A∗ (I − PQ )Axn for each n ≥ 1. Then it follows that, for all z ∈ Γ, 󵄩 󵄩2 ‖vn − z‖2 = 󵄩󵄩󵄩xn − z − βn A∗ (I − PQ )Axn 󵄩󵄩󵄩

󵄩 󵄩2 = ‖xn − z‖2 − 2βn ⟨xn − z, A∗ (I − PQ )Axn ⟩ + βn2 󵄩󵄩󵄩A∗ (I − PQ )Axn 󵄩󵄩󵄩 󵄩 󵄩2 = ‖xn − z‖2 − 2βn ⟨Axn − Az, (I − PQ )Axn ⟩ + βn2 󵄩󵄩󵄩A∗ (I − PQ )Axn 󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 ≤ ‖xn − z‖2 − 2βn 󵄩󵄩󵄩(I − PQ )Axn 󵄩󵄩󵄩 + βn2 ‖A‖2 󵄩󵄩󵄩(I − PQ )Axn 󵄩󵄩󵄩 󵄩 󵄩2 = ‖xn − z‖2 − βn (2 − βn ‖A‖2 )󵄩󵄩󵄩(I − PQ )Axn 󵄩󵄩󵄩 ≤ ‖xn − z‖2 , 󵄩󵄩 󵄩 ∗󵄩 ∗󵄩 󵄩󵄩xn+1 − x 󵄩󵄩󵄩 = 󵄩󵄩󵄩PC [αn u + (1 − αn )vn ] − PC x 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩αn (u − x∗ ) + (1 − αn )(vn − x ∗ )󵄩󵄩󵄩

2.7 Applications | 137

󵄩 󵄩 󵄩 󵄩 ≤ αn 󵄩󵄩󵄩u − x∗ 󵄩󵄩󵄩 + (1 − αn )󵄩󵄩󵄩vn − x∗ 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ αn 󵄩󵄩󵄩u − x∗ 󵄩󵄩󵄩 + (1 − αn )󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 ≤ max{󵄩󵄩󵄩u − x∗ 󵄩󵄩󵄩, 󵄩󵄩󵄩x1 − x∗ 󵄩󵄩󵄩} = M, and 󵄩󵄩 󵄩 ∗ 󵄩2 ∗ 󵄩2 󵄩󵄩xn+1 − x 󵄩󵄩󵄩 = 󵄩󵄩󵄩PC [αn u + (1 − αn )vn ] − PC x 󵄩󵄩󵄩 󵄩 󵄩2 ≤ 󵄩󵄩󵄩αn (u − x∗ ) + (1 − αn )(vn − x∗ )󵄩󵄩󵄩 󵄩 󵄩2 = (1 − αn )2 󵄩󵄩󵄩vn − x∗ 󵄩󵄩󵄩

󵄩 󵄩2 + 2αn (1 − αn )⟨u − x∗ , vn − x ∗ ⟩ + αn2 󵄩󵄩󵄩u − x ∗ 󵄩󵄩󵄩 󵄩 󵄩2 ≤ (1 − αn )󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩 󵄩2 󵄩 − (1 − αn )2 βn (2 − βn ‖A‖2 )󵄩󵄩󵄩(I − PQ )Axn 󵄩󵄩󵄩 󵄩 󵄩2 + 2αn (1 − αn )⟨u − x∗ , vn − x ∗ ⟩ + αn2 󵄩󵄩󵄩u − x ∗ 󵄩󵄩󵄩 .

(2.131)

Letting an = ‖xn − x∗ ‖2 for each n ≥ 1, we have rn =

βn 󵄩 󵄩2 (1 − αn )2 (2 − βn ‖A‖2 )󵄩󵄩󵄩(I − PQ )Axn 󵄩󵄩󵄩 αn 󵄩 󵄩2 − 2(1 − αn )⟨u − x∗ , vn − x∗ ⟩ − αn 󵄩󵄩󵄩u − x ∗ 󵄩󵄩󵄩 .

Then (2.131) can be rewritten by an+1 ≤ (1 − αn )an + αn (−rn ),

∀n ≥ 1.

Since {rn } is bounded below, we see that {−rn } is bounded above. Using Lemma 1.10.5, we have lim inf an ≤ lim sup(−rn ) = − lim inf rn < ∞. n→∞

n→∞

n→∞

Assume that lim inf(rn ) = lim (rnj ). n→∞

j→∞

Then {xnj } is a subsequence of {xn }. So there exists a constant c1 > 0 such that βnk

αnk

󵄩 󵄩2 (1 − αnk )2 (2 − βnk ‖A‖2 )󵄩󵄩󵄩(I − PQ )Axnk 󵄩󵄩󵄩 ≤ c1

αnk 󵄩 󵄩2 󳨐⇒ 󵄩󵄩󵄩(I − PQ )Axnk 󵄩󵄩󵄩 ≤ c1 . βnk (1 − αnk )2 (2 − βnk ‖A‖2 )

(2.132)

138 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces Since

αnk βnk

→ 0 and αnk → 0 and βnk → 0 as k → ∞, one has 󵄩2 󵄩󵄩 󵄩󵄩(I − PQ )Axnk 󵄩󵄩󵄩 → θ

(k → ∞).

Assume that xnk ⇀ x as k → ∞. Note that x ∈ C and Axnk ⇀ Ax as k → ∞. From the demiclosedness principle of I − PQ it follows that Ax = PQ Ax ∈ Q 󳨐⇒ x ∈ Γ 󳨐⇒ ⟨u − x ∗ , x − x ∗ ⟩ ≤ 0.

(2.133)

Since vn − xn = −βn A∗ (I − PQ )Axn → θ as n → ∞, we find that vnk ⇀ x as k → ∞. From the definition of {rn } and (2.133), we obtain lim inf rn = lim rnk ≥ −2⟨u − x ∗ , x − x ∗ ⟩ ≥ 0, n→∞

k→∞

which implies from (2.132) that lim supn→∞ an ≤ 0. However, we have lim infn→∞ an ≥ 0. Therefore, an → 0 as n → ∞, that is, {xn } converges to x ∗ = PΓ u in norm. This completes the proof. Remark 2.7.1. Take {αn } and {βn } as follows: αn =

1 , na

βn =

1 , nb

∀n ≥ 1,

where 0 < b < a ≤ 1. Then the sequences {αn } and {βn } satisfy conditions (C1)–(C3).

2.8 Remark All results presented in this chapter were obtained by various authors in different periods, however, most of them are reproved and simplified by the authors of the book.

2.9 Exercises We use H to denote a real Hilbert space. 1. Let {Ti }ri=1 : H → H be a finite family of mappings. Assume that K = ⋂ri=1 Fix(Ti ) ≠ 0 and K = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ). Show that K = Fix(T1 Tr ⋅ ⋅ ⋅ T3 T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 Tr−2 ⋅ ⋅ ⋅ T1 Tr ). 2.

Let C be a nonempty closed convex subset of H and x0 ∈ H. Prove that (i) If C = Br [x0 ], then {x, PC x = { r(x−x ) 0 , { ‖x−x0 ‖

if ‖x − x0 ‖ ≤ r, if ‖x − x0 ‖ ≥ r;

(ii) PC : H → Br [x0 ] is firmly nonexpansive.

2.9 Exercises | 139

3.

Let T : Br [θ] → H is a nonexpansive mapping. Prove that the following alternative law holds: (i) T has a fixed point; (ii) There exists some x ∈ 𝜕Br [θ] and λ ∈ (0, 1) such that x = λTx. 4. Let C be a closed subspace of a real Hilbert space H and let PC be the metric projection of H onto C. Then, show that the following (1), (2), (3), and (4) hold: (1) PC is a linear mapping of H onto C; (2) PC2 = P; (3) ‖PC x‖ ≤ ‖x‖ ∀x ∈ H; (4) ‖PC ‖ = 1. 5. Let C be a closed subspace of a real Hilbert space H and let x ∈ C. Define NC (x) = {z ∈ H : ⟨u − x, z⟩ ≤ 0, ∀u ∈ C}. 6.

Show that NC (x) is a closed convex cone of H. Let C be a nonempty convex and closed subset of a real Hilbert space H. Let A be a mapping of C into H and let PC be the metric projection of H onto C. Show Fix(PC A) = VI(C, I − A), where VI(C, I − A) denotes the solution set of the variational inequality ⟨x − Ax, y − x⟩ ≥ 0, ∀y ∈ C.

7.

Let C be a closed convex subset of a real Hilbert space H and let T : C → C be a quasi-nonexpansive mapping. Prove that Fix(T) = ⋂ {x ∈ C : 2⟨y − Ty, x⟩ ≤ ‖y‖2 − ‖Ty‖2 }. y∈C

By means of the above expression, prove that Fix(T) is a closed convex subset of H. 8. Let T : H → H be a quasi-nonexpansive mapping with K = Fix(T) ≠ 0. Let A : H → H be k-Lipschitz and η-strongly monotone on H. Let {λn } be a sequence of numbers 2η in (0,1) satisfying (C1) limn→∞ λn = 0, and (C2) ∑∞ n=1 λn = ∞. For μ ∈ (0, k 2 ) and σ ∈ (0, 1), define a sequence {xn } iteratively in H by (HIM)

x1 ∈ H, xn+1 = λn (I − μA)xn + (1 − λn )[σxn + (1 − σ)Txn ],

n ≥ 1.

Prove that the sequence {xn } generated by (HIM) converges strongly to the unique solution x∗ of the variational inequality problem: ⟨Ax, y − x⟩ ≥ 0, ∀y ∈ K.

(VIP)

140 | 2 Iterative methods for fixed points of nonexpansive mappings in Hilbert spaces 9.

Let T : H → H be a quasi-nonexpansive mapping with K = Fix(T) ≠ 0. Let A : H → H be k-Lipschitz and η-strongly monotone on H. Let {λn } be a sequence of numbers 2η in (0,1) satisfying (C1) limn→∞ λn = 0, and (C2) ∑∞ n=1 λn = ∞. For μ ∈ (0, k 2 ) and σ ∈ (0, 1), define a sequence {xn } iteratively in H by (HSDIM)

x1 ∈ H, xn+1 = σxn + (1 − σ)(Txn − μλATxn ),

n ≥ 1.

Prove that the sequence {xn } generated by (HSDIM) converges strongly to the unique solution x∗ of the variational inequality problem: ⟨Ax, y − x⟩ ≥ 0,

∀y ∈ K.

(VIP)

10. Let C be a nonempty closed convex subset of H. Let A : C → H be an α-inverse strongly-monotone mapping and let S : C → C be a nonexpansive mapping such that K = Fix(S) ∩ VI(C, A) ≠ 0. For arbitrary initial value x1 ∈ C, define a sequence {xn } iteratively in C by (MIM)

1 1 xn+1 = αn xn + (1 − αn )[ Sxn + PC (xn − λn Axn )], 2 2

n ≥ 1.

where λn ∈ [a, 2α], αn ∈ [c, d] for a > 0, c > 0, d < 1. Show that the sequence {xn } generated by (MIM) converges weakly to some x ∗ ∈ K, where x ∗ = limn→∞ PK xn . 11. Let C be a nonempty closed convex subset of a real Hilbert space H, let A : C → H be an α-inverse strongly monotone mapping and S : C → C be a nonexpansive mapping such that K = Fix(S) ∩ VI(C, A) ≠ 0. For arbitrary initial value x1 ∈ C and a fixed anchor u ∈ C, define a sequence {xn } iteratively in C by (HIM)

1 1 xn+1 = αn u + (1 − αn )[ Sxn + PC (xn − λn Axn )], 2 2

n ≥ 1,

where {αn } is a sequence of numbers in (0, 1) satisfying (C1) limn→∞ αn = 0, (C2) αn ∞ ∑∞ n=1 αn = ∞, and (C3) ∑n=1 |αn+1 − αn | < ∞ or (C4) αn+1 → 1; where {λn } is another sequence of numbers in (0, 1) satisfying λn ∈ [a, 2α] and ∑∞ n=1 |λn+1 − λn | < ∞ for some a > 0. Prove that the sequence {xn } generated by (HIM) converges strongly to the specific point z ∈ K, where z = PK u. 12. Let A : H → H be η-strongly monotone and k-Lipschitz mapping. Let {Ti }Ni=1 be nonexpansive self-mappings of H such that C = ⋂Ni=1 Fix(Ti ) ≠ 0. Let {λk } and {βki } be two sequences of number in (0,1) satisfying conditions: (C1) limk→∞ λk = 0, (C2) i i ∑∞ k=1 λk = ∞, (C3) βk ∈ (α, β), for some α, β ∈ (0, 1), and (C4) limk→∞ |βk+1 − βk | = 0, for i = 1, 2, . . . , N. For any initial value x0 ∈ H, define a sequence {xn } iteratively in H by k xn+1 = (I − λk μA)TNk TN−1 ⋅ ⋅ ⋅ T1k xk ,

k ≥ 0,

2.9 Exercises | 141

2η

where μ ∈ (0, k2 ) and Tik := (1 − βki )I + βki Ti , for i = 1, 2, . . . , N. Prove that the sequence {xk } converges strongly to the unique solution x ∗ of the variational inequality problem ⟨Ax, y − x⟩ ≥ 0,

∀y ∈ C.

3 Iterative methods for zeros of monotone mappings and fixed points of pseudocontractive mappings in Hilbert spaces In this chapter, we systematically discuss the theory of monotone mappings in Hilbert spaces, in particular, the theory of maximal monotone mappings. With the aid of the theory of monotone mappings, we establish the existence and unique theorems of solutions of variational inequalities with strongly monotone hemicontinuous mappings. We also obtain zero theorems of strongly monotone hemicontinuous mappings, from which we derive the existence, uniqueness, and convergence of paths, and some fixed point theorems for hemicontinuous pseudocontractive mappings in Hilbert spaces. Based on these results, we focus on the construction of iterative methods for zeros of monotone mappings, fixed points of pseudocontractive mappings, and solutions of monotone variational inequalities. All iterative methods and convergence theorems discussed in this chapter include some classical results and some of the authors’ recent results. Let H be a real Hilbert space. By ⟨⋅, ⋅⟩ and ‖ ⋅ ‖ we denote the inner product and norm induced by the inner product in H, respectively. For a multi-valued mapping A : H → 2H , we denote the effective domain, range, graph, and preimage of A as follows: (1) Dom(A) = {x ∈ H : Ax ≠ 0}; (2) Ran(A) = {y ∈ Ax : x ∈ Dom(A)}; (3) Graph(A) = {[x, y] : x ∈ Dom(A), y ∈ Ax} and A−1 (y) = {x ∈ H : y ∈ Ax}. For the sake of convenience, we use A ⊂ H × H to denote a multi-valued mapping A : H → 2H , while A : H → H denotes a single-valued mapping, and we use I to denote the identity mapping in H. Let → and ⇀ denote the strong and weak convergence, respectively.

3.1 Basic properties of monotone mappings In this subsection, we discuss some fundamental properties of (maximal) monotone mappings such as the local boundedness, demiclosedness, surjectivity, and so on.

3.1.1 Local boundedness and hemicontinuity Definition 3.1.1. Let A ⊂ H × H be a mapping. Then A is said to be a locally bounded at x0 ∈ Dom(A) if there exists a neighborhood U of x0 such that A(U) = {f : y ∈ U, f ∈ Ay} is bounded in H. https://doi.org/10.1515/9783110667097-003

144 | 3 Monotone and pseudocontractive mappings Definition 3.1.2. (1) Let A ⊂ H × H be a mapping. Then A is said to be hemicontinuous if ∀x ∈ Dom(A), ∀h ∈ H, tn → 0, A(x + tn h) ⇀ Ax (n → ∞) provided x + tn h ∈ Dom(A). (2) A ⊂ H × H is said to be demicontinuous if ∀x ∈ Dom(A), ∀{xn } ⊂ Dom(A), xn → x ⇒ Axn ⇀ Ax (n → ∞). Remark 3.1.1. If A is demicontinuous, then A must be hemicontinuous. The converse is not true. Example 3.1.1. Let H = ℝ2 , and consider the function φ : ℝ2 → ℝ defined by xy2 (x2 + y4 )−1 ,

φ(x, y) = {

0,

(x, y) ∈ ℝ2 , x = y = 0.

Then φ : ℝ2 → ℝ is hemicontinuous, but it is not demicontinuous. Remark 3.1.2. For mappings in finite-dimensional normed linear spaces, demicontinuity and continuity are equivalent, and local boundedness and boundedness are also equivalent. For linear mappings in infinite-dimensional normed linear spaces, demicontinuity and continuity are equivalent, and local boundedness and boundedness are also equivalent. In order to establish the local boundedness results of monotone mappings, we first prove a useful lemma. Lemma 3.1.1. Let {xn } be a sequence in H such that xn → θ and let {fn } be a sequence in H such that ‖fn ‖ → ∞ as n → ∞. Then, for any ρ > 0, there exist z ∈ Bρ [θ] and subsequences {xnj } and {fnj } of {xn } and {fn }, respectively, such that ⟨fnj , xnj − z⟩ → ∞

(j → ∞).

(3.1)

Proof. Assume that the conclusion is not true. Then ∀z ∈ Bρ [θ] there exists a constant Cz ∈ R such that ⟨fn , xn − z⟩ ≥ Cz ,

∀n ≥ 1.

(3.2)

For all k ≥ 1, let Ek = {u ∈ Bρ [θ] : ⟨fn , xn − u⟩ ≥ −k, n ≥ 1}. Then Ek is closed and Bρ [θ] = ⋃k≥1 Ek . Since Bρ (θ) is a complete metric space, it is of the second category. It follows from the Baire category theorem that there exists k0 ∈ N satisfying int(Ek0 ) ≠ 0, that is, there exists r > 0, y ∈ Bρ [θ] such that Br (y) ⊂ Ek0 . Note that −y ∈ Bρ [θ]. It follows from (3.2) that ⟨fn , xn + y⟩ ≥ C−y .

(3.3)

3.1 Basic properties of monotone mappings | 145

For any u ∈ Br (y), we have u ∈ Ek0 󳨐⇒ u ∈ Bρ [θ] and ⟨fn , xn − u⟩ ≥ −k0 , which implies from (3.3) that ⟨fn , 2xn + y − u⟩ ≥ C−y − k0 .

(3.4)

From the assumption xn → θ as n → ∞, we see that there exists n0 ∈ N such that ‖xn ‖ < 4r for all n ≥ n0 . For all z ∈ H with ‖z‖ < 2r , letting u = 2xn + y − z, we obtain ‖u − y‖ = ‖2xn − z‖ ≤

r r + = r, 2 2

∀n ≥ n0 .

This shows that u = 2xn + y − z ∈ Br (y). Using (3.4), we find that ⟨fn , z⟩ ≥ C−y − k0 , Since ‖ − z‖
0 such that, for all t ∈ (0, th ), ht := x + th ∈ int(Dom(A)) ⊂ Dom(A). Replacing u with ht in (3.6), we have ⟨y − Aht , x − ht ⟩ ≥ 0

󳨐⇒ ⟨y − Aht , −th⟩ ≥ 0, 󳨐⇒ ⟨y − Aht , h⟩ ≤ 0,

∀h ∈ H

∀h ∈ H.

(3.7)

Letting t → 0 in (3.7), one concludes from the hemicontinuity of A at x that ⟨y − Ax, h⟩ ≤ 0, 󳨐⇒ y = Ax,

∀h ∈ H

w

that is, Axnj 󳨀→ Ax

(j → ∞).

Since Ax is unique, we obtain that Axn ⇀ Ax as n → ∞. This verifies that A is demicontinuous at any point x ∈ Dom(A). Hence A is demicontinuous in int(Dom(A)). This completes the proof. Remark 3.1.5. Theorem 3.1.2 shows that, for single-valued monotone mappings defined on the whole space, demicontinuity is equivalent to hemicontinuity. Corollary 3.1.1. Let H be a finite-dimensional space and let A : H → H be a hemicontinuous monotone mapping. Then A is continuous on H. Proof. From Theorem 3.1.2, we find that A is demicontinuous in H. Hence ∀{xn } ⊂ H, xn → x ⇒ Axn ⇀ Ax (n → ∞). In view of dim H < ∞, we obtain that Axn → Ax. This completes the proof. Remark 3.1.6. Corollary 3.1.1 shows that, for single-valued monotone mappings defined on finite-dimensional spaces, demicontinuity, hemicontinuity, and continuity are equivalent. Corollary 3.1.2. If A : H → H be a linear monotone mapping, then A is continuous monotone. Indeed, it is ‖A‖-Lipschitz monotone. Proof. Since A is linear, we find that A is hemicontinuous. From Theorem 3.1.2, A is hemicontinuous, which implies that A is continuous. Next, we assume that A is discontinuous at the origin θ. Then ∃ε0 > 0, {xn } ⊂ H, xn → θ, but ‖Axn ‖ ≥ ε0 , n ∈ N. Tak1 ing tn = ‖xn ‖− 2 , yn = tn xn , we find that yn → θ. But ‖Ayn ‖ = tn ‖Axn ‖ ≥ tn ε0 → ∞, which

3.1 Basic properties of monotone mappings | 147

contradicts the hemicontinuity of A. Hence A is continuous, therefore, it is bounded. Also ‖Ax‖ ≤ ‖A‖‖x‖, which shows that A is ‖A‖-Lipschitz monotone. This completes the proof. Remark 3.1.7. Corollary 3.1.2 shows that, for linear monotone mappings defined on the whole space, hemicontinuity, demicontinuity, continuity, and Lipschitz continuity are equivalent.

3.1.2 Characteristic description for monotone mappings Proposition 3.1.1. Let A ⊂ H × H be a mapping. Then A is monotone if and only if, for all [x, f ], [y, g] ∈ Graph(A), 󵄩󵄩 󵄩 󵄩󵄩(x + tf ) − (y + tg)󵄩󵄩󵄩 ≥ ‖x − y‖,

∀t > 0.

(3.8)

Proof. Using the properties of the norms and the scalar products on H, we have 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩(x + tf ) − (y + tg)󵄩󵄩󵄩 = 󵄩󵄩󵄩x − y + t(f − g)󵄩󵄩󵄩

= ‖x − y‖2 + 2t⟨f − g, x − y⟩ + t 2 ‖f − g‖2 .

(3.9)

(󳨐⇒) If A is monotone, then ⟨f − g, x − y⟩ ≥ 0. Using (3.9), we find that 󵄩󵄩 󵄩 󵄩󵄩(x + tf ) − (y + tg)󵄩󵄩󵄩 ≥ ‖x − y‖. (⇐󳨐) Suppose that (3.8) holds. Then, we find from (3.8) and (3.9) that 2⟨f − g, x − y⟩ + t‖f − g‖2 ≥ 0,

∀t > 0.

Letting t → 0, we have ⟨f − g, x − y⟩ ≥ 0. This completes the proof. Remark 3.1.8. Proposition 3.1.1 describes a characteristic of a monotone mapping A, namely, for all t > 0, (I + tA) is an expansive mapping. So, (I + tA)−1 always exists on Ran(I + tA). Hence we can define a resolvent operator by Jt x = (I + tA)−1 x for any x ∈ Ran(I + tA) and Yosida approximation by At = 1t (I − Jt ) for any t > 0. Proposition 3.1.2. Let A ⊂ H ×H be a monotone mapping. Then A is maximal monotone if and only if ∀[x, f ] ∈ H × H, [y, g] ∈ Graph(A), ⟨f − g, x − y⟩ ≥ 0 󳨐⇒ [x, f ] ∈ Graph(A). Proof. (󳨐⇒) Suppose that A ⊂ H × H is maximal monotone. For any [x, f ] ∈ H × H, [y, g] ∈ Graph(A), ⟨f − g, x − y⟩ ≥ 0. We need to confirm [x, f ] ∈ Graph(A). If [x, f ] ∉ Graph(A), we construct a set M = Graph(A) ⋃{[x, f ]}, and then M is a monotone set and Graph(A) is a proper subset of M, which contradict the definition of the maximal monotonicity of A.

148 | 3 Monotone and pseudocontractive mappings (⇐󳨐) Suppose that M is an arbitrary monotone set containing Graph(A). Then, for any [x, f ] ∈ H × H and [y, g] ∈ Graph(A), ⟨f − g, x − y⟩ ≥ 0, which implies that [x, f ] ∈ Graph(A) 󳨐⇒ M ⊆ Graph(A) 󳨐⇒ M = Graph(A). Hence, that A is maximal monotone. This completes the proof. Remark 3.1.9. The above proposition is often used to prove that a monotone mapping is maximal monotone. 3.1.3 Demiclosedness principle for monotone mappings Definition 3.1.3. Let A ⊂ H×H be a mapping. Then A is said to be demiclosed if, for any {[xn , fn ]} ⊂ Graph(A), xn → x and fn ⇀ f or xn ⇀ x and fn → f imply [x, f ] ∈ Graph(A). To establish the demiclosedness principle of monotone hemicontinuous mappings, we first give the famous Minty’s lemma. Lemma 3.1.2 (Minty’s lemma). Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A : C → H be a monotone hemicontinuous mapping and x0 ∈ C. Then the following assertions are equivalent: (1) ⟨Ax, x − x0 ⟩ ≥ 0, ∀x ∈ C. (2) ⟨Ax0 , x − x0 ⟩ ≥ 0, ∀x ∈ C. Proof. (1) 󳨐⇒ (2). For any y ∈ C, t ∈ (0, 1), let yt = (1 − t)x0 + ty. Since C is convex, we have yt ∈ C. Replacing x with yt , we have ⟨Ayt , y − x0 ⟩ ≥ 0,

∀t ∈ (0, 1).

(3.10)

From the hemicontinuity of A, we obtain Ayt ⇀ Ax0 as t → 0+ . Letting t → 0+ in (3.10), we find that ⟨Ax0 , y − x0 ⟩ ≥ 0, ∀y ∈ C, that is, (2) holds. (2) 󳨐⇒ (1). For any x ∈ C, we conclude from the monotonicity of A that ⟨Ax − Ax0 , x − x0 ⟩ ≥ 0,

∀x ∈ C

󳨐⇒ ⟨Ax, x − x0 ⟩ ≥ ⟨Ax0 , x − x0 ⟩ ≥ 0,

∀x ∈ C,

that is, (1) holds. This completes the proof. Proposition 3.1.3. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let A : C → H be a monotone hemicontinuous mapping satisfying the “flow-invariance” condition (FIC): lim h−1 d((I − hA)x, C) = 0,

h→0+

Then A is demiclosed.

∀ ∈ C.

3.1 Basic properties of monotone mappings | 149

Proof. Suppose that {xn } is a sequence with xn ⇀ x ∈ H and Axn → θ as n → ∞. Now, we need to prove that x ∈ C and Ax = θ. Since C is closed and convex, we see that C is w-closed and hence x ∈ C. In order to prove Ax = θ, we write yh = x − hAx, ∀h > 0. It follows from (FIC) that, for any ε > 0, there exists δ > 0 with 0 < h < δ such that d(yh , C) = d((I − hA)x, C) < hε

󳨐⇒ ∃uh ∈ C s. t. ‖yh − uh ‖ < hε 󵄩󵄩 u − x 󵄩󵄩 󵄩 󵄩 󳨐⇒ 󵄩󵄩󵄩 h + Ax 󵄩󵄩󵄩 < ε 󵄩󵄩 h 󵄩󵄩 uh − x 󳨐⇒ → −Ax (h → 0+ ). h

(3.11)

On the other hand, we obtain from the monotonicity of A that ⟨Au − Axn , u − xn ⟩ ≥ 0,

∀u ∈ C, n ∈ ℕ.

(3.12)

⟨Au, u − x⟩ ≥ 0,

∀u ∈ C.

(3.13)

⟨Ax, u − x⟩ ≥ 0,

∀u ∈ C.

(3.14)

Letting n → ∞ in (3.12), one has

By Minty’s lemma, we derive

Taking u = uh in (3.14), we have ⟨Ax, uh − x⟩ ≥ 0, 󳨐⇒ ⟨Ax,

∀h > 0

uh − x ⟩ ≥ 0, h

∀h > 0.

(3.15)

Letting h → 0+ in (3.15) and combining with (3.11), we have −‖Ax‖2 ≥ 0 󳨐⇒ Ax = θ. If xn → x and fn ⇀ f as n → ∞, we can prove the conclusion in a similar way. This completes the proof. Proposition 3.1.4. Let A ⊂ H × H be a maximal monotone mapping. Then (i) Ax is a w-closed subset of H, ∀x ∈ Dom(A), (ii) A is demiclosed. Proof. (1) First, we prove that, for any x ∈ Dom(A), Ax is a convex subset of H. Take f1 , f2 ∈ Ax and t ∈ (0, 1) arbitrarily, and denote f = (1 − t)f1 + tf2 . Then, for any [y, g] ∈ Graph(A), ⟨f − g, x − y⟩ = (1 − t)⟨f1 − g, x − y⟩ + t⟨f2 − g, x − y⟩ ≥ 0.

150 | 3 Monotone and pseudocontractive mappings By Proposition 3.1.2, we see that [x, f ] ∈ Graph(A), that is, f ∈ Ax. Next, we prove that Ax is w-closed. In fact, since Ax =

{f ∈ H : ⟨f − g, x − y⟩ ≥ 0}

⋂ [y,g]∈Graph(A)

and the set {f ∈ H : ⟨f − g, x − y⟩ ≥ 0} is w-closed, one sees that their intersection Ax is also w-closed. (2) Let {[xn , fn ]} ⊂ Graph(A), xn → x and fn ⇀ f as n → ∞. Then ⟨fn − g, xn − y⟩ ≥ 0,

∀[y, g] ∈ Graph(A),

n ≥ 1.

Taking n → ∞, we have ⟨f − g, x − y⟩ ≥ 0,

∀[y, g] ∈ Graph(A).

By Proposition 3.1.2, we see that, for any [x, f ] ∈ Graph(A), if xn ⇀ x and fn → f as n → ∞, then the above conclusion still holds. Hence, Ax is demiclosed. This completes the proof. In fact, maximal monotone mappings have more general w-closed property as follows. Proposition 3.1.5. Let A ⊂ H × H be a maximal monotone mapping. Let {[xn , fn ]} ⊂ Graph(A), xn ⇀ x, fn ⇀ f and lim sup⟨fn , xn ⟩ ≤ ⟨f , x⟩. n→∞

Then [x, f ] ∈ Graph(A) and ⟨fn , xn ⟩ → ⟨f , x⟩ Proof. From the monotonicity of A, we derive that ⟨fn − g, xn − y⟩ ≥ 0,

∀[y, g] ∈ Graph(A),

n ≥ 1.

(3.16)

From lim supn→∞ ⟨fn , xn ⟩ ≤ ⟨f , x⟩ it follows that lim sup(⟨fn , xn ⟩ − ⟨f , x⟩) ≤ 0. n→∞

Hence lim inf⟨fn , x − xn ⟩ ≥ 0. n→∞

(3.17)

Using (3.16), we arrive at ⟨f − g, x − y⟩ = ⟨f − fn , x − y⟩ + ⟨fn − g, x − y⟩

= ⟨f − fn , x − y⟩ + ⟨fn − g, x − xn ⟩ + ⟨fn − g, xn − y⟩ ≥ ⟨f − fn , x − y⟩ + ⟨fn , x − xn ⟩ − ⟨g, x − xn ⟩.

(3.18)

3.1 Basic properties of monotone mappings | 151

Taking n → ∞ in (3.18), we obtain ⟨f − g, x − y⟩ ≥ lim inf⟨fn , x − xn ⟩ ≥ 0, n→∞

∀[y, g] ∈ Graph(A).

It follows from the monotonicity of A that [x, f ] ∈ Graph(A). Taking y = x and g = f in (3.16), respectively, we have ⟨fn − f , xn − x⟩ ≥ 0,

∀n ≥ 1.

(3.19)

Taking the limit inferior in (3.19), we derive lim inf⟨fn , xn ⟩ ≥ ⟨f , x⟩. n→∞

Combining with lim supn→∞ ⟨fn , xn ⟩ ≤ ⟨f , x⟩, we have lim ⟨f , x ⟩ n→∞ n n

= ⟨f , x⟩.

This completes the proof. Remark 3.1.10. Browder called the mapping with the property in Proposition 3.1.5 a “generalized pseudo-monotone mapping”. Compared with the class of (maximal) monotone mappings, this class of mappings has more applications. If we remove the restriction ⟨fn , xn ⟩ → ⟨f , x⟩, then this class of mappings is said to be of M-type. Remark 3.1.11. If xn → x and fn ⇀ f or xn ⇀ x and fn → f as n → ∞, then ⟨fn , xn ⟩ → ⟨f , x⟩. If xn ⇀ x and fn ⇀ f as n → ∞, ⟨fn , xn ⟩ → ⟨f , x⟩ is not necessarily true. Hence, we can not directly take the limit in (3.16) to obtain ⟨f − g, x − y⟩ ≥ 0. 3.1.4 Resolvents and Yosida approximations In this subsection, we discuss some fundamental properties of resolvent operators and the Yosida approximation of monotone mappings. They play an important role in constructing iterative methods for zeros of monotone mappings. In particular, the asymptotic behavior of the resolvent Jt is the basis of Bruck regularization method. Suppose that A ⊂ H × H is a monotone mapping. Then, for any t > 0, the resolvent Jt = (I + tA)−1 of A is well defined on Ran(I + tA) and the Yosida approximation At = t −1 (I −Jt ) of A is also well defined on Ran(I +tA). In particular, if A ⊂ H ×H is a maximal monotone mapping, then Dom(Jt ) = H, that is, Jt is well defined on Hilbert space H. Next, we list some fundamental properties. Proposition 3.1.6. For any t > 0, we have (1) Dom(Jt ) = Ran(I+tA), Ran(Jt ) = Dom(A), Jt and At are single-valued and monotone; (2) ⟨Jt x − Jt y, x − y⟩ ≥ ‖Jt x − Jt y‖2 , ∀x, y ∈ Dom(Jt ). In particular, ‖Jt x − Jt y‖ ≤ ‖x − y‖.

152 | 3 Monotone and pseudocontractive mappings (3) (4) (5) (6) (7)

‖Jt x − Jt y‖2 ≤ ‖x − y‖2 − ‖(I − Jt )x − (I − Jt )y‖2 , ∀x, y ∈ Dom(Jt ). [Jt x, At x] ∈ Graph(A), ∀x ∈ Dom(Jt ). Dom(At ) = Dom(Jt ) and ⟨At x − At y, x − y⟩ ≥ t‖At x − At y‖2 , ∀x, y ∈ Dom(At ). ‖At x‖ ≤ |Ax|, ∀x ∈ Dom(A) ∩ Dom(At ), where |Ax| = d(θ, Ax) = inf{‖y‖ : y ∈ Ax}. J x ∈ Dom(Js ), For any t, s > 0, x ∈ Dom(Jt ) and st x + t−s t t s t−s Jt x = Js ( x + J x). t t t

(8) If 0 < t < s, then ‖Jt x − x‖ ≤ 2‖Js x − x‖, ∀x ∈ Dom(Jt ) ∩ Dom(Js ). (9) If 0 < t < s, then ‖As x‖ ≤ ‖At x‖, ∀x ∈ Dom(At ) ∩ Dom(As ) and hence limt→0+ ‖At x‖ exists. Proof. (1) From the definition of Jt and At , it is easy to find the conclusion. (2) From the definition of Jt , ∀x, y ∈ Dom(Jt ), it follows that x ∈ Jt x + tAJt x 󳨐⇒ (I − Jt )x ∈ tAJt x, y ∈ Jt y + tAJt y 󳨐⇒ (I − Jt )y ∈ tAJt y. It follows that ⟨Jt x − Jt y, x − y⟩ = ⟨Jt x − Jt y, Jt x − Jt y⟩

+ ⟨Jt x − Jt y, (I − Jt )x − (I − Jt )y⟩

= ‖Jt x − Jt y‖2 + t⟨Jt x − Jt y, xt − yt ⟩ ≥ ‖Jt x − Jt y‖2 ,

where xt ∈ AJt x, yt ∈ AJt y. Thus, we conclude from the monotonicity of A that ⟨Jt x − Jt y, xt − yt ⟩ ≥ 0. (3) From (2), we have 󵄩󵄩 󵄩2 2 2 󵄩󵄩(I − Jt )x − (I − Jt )y󵄩󵄩󵄩 = ‖x − y‖ − 2⟨x − y, Jt x − Jt y⟩ + ‖Jt x − Jt y‖ ≤ ‖x − y‖2 − ‖Jt x − Jt y‖2 , that is, 󵄩 󵄩2 ‖Jt x − Jt y‖2 ≤ ‖x − y‖2 − 󵄩󵄩󵄩(I − Jt )x − (I − Jt )y󵄩󵄩󵄩 . (4) From the definition of Jt , we have At x = t −1 (I − Jt )(x) ∈ AJt (x),

∀x ∈ Dom(Jt ).

3.1 Basic properties of monotone mappings | 153

(5) It follows from the definition of Jt that Dom(At ) = Dom(Jt ), ∀x, y ∈ Dom(At ), t > 0. It follows from At x ∈ AJt x, At y ∈ AJt y and the monotonicity of A that ⟨At x − At y, x − y⟩ = ⟨At x − At y, tAt x + Jt x − tAt y − Jt y⟩

= t‖At x − At y‖2 + ⟨At x − At y, Jt x − Jt y⟩ ≥ t‖At x − At y‖2 .

(6) For all x ∈ Dom(A) ∩ Dom(At ) and y ∈ Ax, we have x + ty ∈ (I + tA)x 󳨐⇒ Jt (x + ty) = x. Hence

1 1󵄩 󵄩 ‖At x‖ = ‖x − Jt x‖ = 󵄩󵄩󵄩Jt (x + ty) − Jt x 󵄩󵄩󵄩 t t 1 ≤ ‖x + ty − x‖ = ‖y‖, t which implies that ‖At x‖ ≤ inf{‖y‖ : y ∈ Ax} = |Ax|. (7) For all t, s > 0 and x ∈ Dom(Jt ), there exists [x1 , y1 ] ∈ Graph(A) such that x = x1 + ty1 . Hence

and

Jt x = x1 , t−s s t−s s x+ J x = (x1 + ty1 ) + x = x1 + sy1 ∈ Ran(I + sA) = Dom(Js ) t t t t t 1 s t−s Js ( x + J x) = Js (x1 + sy1 ) = x1 = Jt x. t t t

(8) For all x ∈ D(Jt ) ∩ Dom(Js ) and 0 < t < s, it follows from (2) and (7) that 󵄩󵄩 t 󵄩󵄩 s−t 󵄩 󵄩 |Js x − Jt x| = 󵄩󵄩󵄩Jt ( x + Js x) − Jt x 󵄩󵄩󵄩 󵄩󵄩 s 󵄩󵄩 s t ≤ (1 − )‖x − Js x‖ s ≤ ‖x − Js x‖.

Hence ‖Jt x − x‖ ≤ ‖Jt x − Js x‖ + ‖Js x − x‖ ≤ 2‖x − Js x‖. (9) For all 0 < t < s and x ∈ Dom(At ) ∩ Dom(As ), it follows from (2) and (7) that 󵄩󵄩 󵄩󵄩󵄩 t t 󵄩 ‖Js x − x‖ = 󵄩󵄩󵄩Jt ( x + (1 − )Js x) − x 󵄩󵄩󵄩 s 󵄩󵄩 s 󵄩󵄩 󵄩󵄩 󵄩󵄩 t t 󵄩 󵄩 ≤ 󵄩󵄩󵄩Jt ( x + (1 − )Js x) − Jt x 󵄩󵄩󵄩 + ‖Jt x − x‖ 󵄩󵄩 󵄩󵄩 s s t ≤ (1 − )‖Js x − x‖ + ‖Jt x − x‖. s

154 | 3 Monotone and pseudocontractive mappings It follows that t‖Js x − x‖ ≤ s‖Jt x − x‖, that is, s−1 ‖Js x − x‖ ≤ t −1 |Jt x − x‖. Therefore, ‖As x‖ ≤ ‖At x‖. Hence limt→0+ ‖At x‖ exists. This completes the proof. For maximal monotone mappings, we have the following better conclusions. Proposition 3.1.7. Let A : H × H be a maximal monotone mapping. Then the following conclusions hold: (1) For any t > 0, At : H → H is a maximal monotone mapping and At = (A−1 + tI) , −1

(2) (3) (4) (5)

(At )s = At+s .

For any x ∈ H, limt→0+ Jt x = Pco Dom(A) x. Dom(A) and Ran(A) are closed convex subsets of H. A−1 (θ) = Fix(Jt ) and so the zero set A−1 (θ) of A is closed and convex. For any x ∈ Dom(A) and t > 0, limt→0+ At (x) = A0 (x), where A0 is the minimum section of A defined by A0 x = {y ∈ Ax : |Ax| = ‖y‖}.

(6) If θ ∈ Ran(A), then, for all x ∈ H, limt→∞ Jt x exists and belongs to the zero set A−1 (θ) of A, more precisely, limt→∞ Jt x = PA−1 (θ) x, where PA−1 (θ) x is the metric projection from H onto the zero set A−1 (θ) of A. Proof. (1) The conclusion can be directly obtained from the definition and properties of monotone mappings. So we omit it here. (2) For any x ∈ H, t > 0, since A ⊂ H × H is maximal monotone, we have Ran(I + tA) = H. Hence Dom(Jt ) = H and xt = Jt x, which imply that [xt , t −1 (x − xt )] ∈ Graph(A). From the monotonicity of A, we see that, ∀[x 󸀠 , y󸀠 ] ∈ Graph(A), ⟨t −1 (x − xt ) − y󸀠 , xt − x 󸀠 ⟩ ≥ 0

󳨐⇒ ⟨x − xt − ty󸀠 , xt − x󸀠 ⟩ ≥ 0

󳨐⇒ ⟨xt − x + ty󸀠 , xt − x󸀠 ⟩ ≤ 0,

∀t > 0.

This implies that {xt } is bounded. If tn → 0 and xn := xtn ⇀ x0 as n → ∞, then ‖xn ‖2 ≤ ⟨x − tn y󸀠 , xn − x 󸀠 ⟩ + ⟨xn , x 󸀠 ⟩

󳨐⇒ ‖x0 ‖2 = lim inf ‖xn ‖2 ≤ ⟨x, x0 − x 󸀠 ⟩ + ⟨x0 , x⟩ n→∞

󳨐⇒ ⟨x0 − x, x0 − x󸀠 ⟩ ≤ 0, 󳨐⇒ ⟨x0 − x, x0 − x󸀠 ⟩ ≤ 0,

∀x 󸀠 ∈ Dom(A)

∀x 󸀠 ∈ co Dom(A).

3.1 Basic properties of monotone mappings | 155

Hence x0 = Pco Dom(A) , which shows that {xt } converges to x0 weakly. On the other hand, for any x󸀠 ∈ co Dom(A), one has lim sup ‖xt ‖2 ≤ ⟨x, x0 − x󸀠 ⟩ + ⟨x0 , x 󸀠 ⟩. t→0+

Taking x󸀠 = x0 in the inequality above, we have lim sup ‖xt ‖2 ≤ ‖x0 ‖2 󳨐⇒ lim sup ‖xt ‖ ≤ ‖x0 ‖. t→0+

t→0+

w

Since xtn 󳨀→ x0 as n → ∞, we conclude from the weak lower semicontinuity of the norm that ‖x0 ‖ ≤ lim inf ‖xt ‖ ≤ lim sup ‖xt ‖ ≤ ‖x0 ‖ 󳨐⇒ ‖xt ‖ → ‖x0 ‖ + t→0

t→0+

(t → 0+ ).

Hence xt → x0 as t → 0+ . (3) For any x ∈ co Dom(A), it follows from (1) that xt → x0 as t → 0+ . Hence x ∈ Dom(A) and x ∈ Dom(A). Thus co Dom(A) ⊆ Dom(A). On the contrary, it is obvious that Dom(A) ⊆ co Dom(A). Hence Dom(A) = co Dom(A) is closed and convex. Since A is maximal monotone, one finds that A−1 is also maximal monotone. So, Ran(A) = Dom(A−1 ) is closed and convex. (4) Note that x ∈ A−1 (θ) ⇐⇒ θ ∈ Ax ⇐⇒ x = Jt x,

∀t > 0.

Since Jt is nonexpansive, we find that Fix(Jt ) is closed and convex. (5) For any x ∈ Dom(A) and t > 0, since A is monotone, we have ⟨A0 x − At x, x − Jt x⟩ ≥ 0 and 󵄩 󵄩 ‖At x‖2 ≤ ⟨A0 x, At x⟩ ≤ 󵄩󵄩󵄩A0 x 󵄩󵄩󵄩‖At x‖, which imply that ‖At x‖ ≤ ‖A0 x‖. This shows that {At x} is bounded. Take tn → 0 arbitrarily and denote xn := Atn x. Suppose that xn ⇀ y as n → ∞. It follows from (2) that Jtn x → x as n → ∞, xn ∈ Atn x and xn ⇀ y as n → ∞. Thus, we conclude

156 | 3 Monotone and pseudocontractive mappings from the demiclosedness of A that y ∈ Ax. Using the weak lower semicontinuity of the norm, we have 󵄩 󵄩 󵄩󵄩 0 󵄩󵄩 inf ‖xn ‖ ≤ lim sup ‖xn ‖ ≤ 󵄩󵄩󵄩A0 x 󵄩󵄩󵄩 󵄩󵄩A x󵄩󵄩 ≤ ‖y‖ ≤ lim n→∞ n→∞ 󵄩 󵄩 󳨐⇒ lim ‖xn ‖ = 󵄩󵄩󵄩A0 x 󵄩󵄩󵄩 = ‖y‖ n→∞

󳨐⇒ y = A0 x.

Thus limt→0+ ‖At x‖ = ‖A0 x‖. Note that 󵄩󵄩 󵄩 0 󵄩2 0 󵄩2 2 0 󵄩󵄩At x − A x󵄩󵄩󵄩 = ‖At x‖ − 2⟨At x, A x⟩ + 󵄩󵄩󵄩A x 󵄩󵄩󵄩 󵄩 󵄩2 ≤ ‖At x‖2 − 2‖At x‖2 + 󵄩󵄩󵄩A0 x 󵄩󵄩󵄩 󵄩 󵄩2 = 󵄩󵄩󵄩A0 x󵄩󵄩󵄩 − ‖At x‖2 → 0 (t → 0+ ). So, we obtain that limt→0+ At x = A0 x. (6) Since A ⊂ H × H is maximal monotone, we see that Ran(I + tA) = H, ∀t > 0. Hence, for any x ∈ H and t > 0, Jt x is well defined. For any x ∈ H, there exists a unique xt ∈ Dom(A) such that x ∈ xt + tAxt ,

(R1)

that is, xt = Jt x, ∀t > 0. For the sake of simplicity, we suppose that x = θ. If not, we consider a maximal ̂ = A(⋅ − x). monotone mapping A(⋅) Now, we are in a position to show that limt→∞ Jt θ = z, where z = PA−1 (θ) (θ). First, we prove that PA−1 (θ) (θ) is well defined. By the assumption θ ∈ Ran(A), we see that A−1 (θ) ≠ 0. It follows from (4) that A−1 (θ) is closed and convex. Hence PA−1 (θ) (θ) is uniquely determined. For any x∗ ∈ A−1 (θ) and yt ∈ Axt , we obtain from (R1) that θ = xt + tyt ,

∀t > 0.

This implies that 0 = ⟨xt , xt − x∗ ⟩ + t⟨yt , xt − x ∗ ⟩ 󵄩 󵄩2 = 󵄩󵄩󵄩xt − x∗ 󵄩󵄩󵄩 + ⟨x ∗ , xt − x ∗ ⟩ + t⟨yt − θ, xt − x ∗ ⟩ 󵄩 󵄩2 ≥ 󵄩󵄩󵄩xt − x∗ 󵄩󵄩󵄩 + ⟨x ∗ , xt − x ∗ ⟩. It follows that 󵄩󵄩 ∗ 󵄩2 ∗ ∗ 󵄩󵄩xt − x 󵄩󵄩󵄩 ≤ ⟨x , x − xt ⟩,

∀x∗ ∈ A−1 (θ), t > 0,

(R2)

and ⟨xt , xt − x∗ ⟩ ≤ 0,

∀x∗ ∈ A−1 (θ), t > 0.

(R3)

3.2 Criteria of maximal monotone mappings | 157

It follows from (R2) that ‖xt − x∗ ‖ ≤ ‖x ∗ ‖. Hence {xt } is bounded. For any tn > 0 with tn → ∞ as n → ∞, one denotes xn := xtn and yn := ytn . Since sequence {xn } is bounded and space H is reflexive, we may assume, without loss of generality, that xn ⇀ z as n → ∞. Then, it follows from θ = xn + tn yn that yn = − t1 xn → θ as n → ∞. Using the demiclosedness of Graph(A), we have θ ∈ Az. n Taking x∗ = z in (R2), we obtain xn → z as n → ∞. From (R3), we see that ⟨xn , xn − x∗ ⟩ ≤ 0, ∀x∗ ∈ A−1 (θ) and n ≥ 1. Taking n → ∞, we have ⟨z, z − x∗ ⟩ ≤ 0,

∀x ∗ ∈ A−1 (θ).

It follows from some properties of the metric projection that z = PA−1 (θ) θ. This proves that limt→∞ Jt θ = z. This completes the proof. Remark 3.1.12. One knows that some properties of maximal monotone mappings are better than those of general monotone mappings. Hence, we expect to study general monotone mappings via maximal monotone mappings. Indeed, this is possible since ̃ ⊂ H×H every monotone mapping A ⊂ H × H has a maximal monotone extension A ̃ ̃ satisfying Dom(A) ⊂ co Dom(A) and Graph(A) ⊂ G(A) when Dom(A) = C is closed and ̃ = Dom(A) = C and Ran(A) ⊂ Ran(A). ̃ For example, in order to study convex, Dom(A) monotone operator equations θ ∈ Ax, one first studies the case when A is maximal monotone. Based on this, one further ̃ With the aid of properties of A, we can considers its maximal monotone extension A. ̃ obtain some results, which are valid for A, also for A.

3.2 Criteria of maximal monotone mappings It is an important question whether a monotone mapping is maximal monotone. Theorem 3.2.1. If A : Dom(A) = H → H is a monotone hemicontinuous mapping, then A is maximal monotone. Proof. For any [x, f ] ∈ H × H, ⟨f − Ay, x − y⟩ ≥ 0,

∀y ∈ H.

(3.20)

Next, we prove Ax = f . If Ax ≠ f , then there exists z ∈ H such that ⟨f − Ax, z⟩ ≠ 0. Assume that ⟨f − Ax, z⟩ > 0. Letting yt := x + tz, t > 0 and taking y = yt in (3.20), we have ⟨f − Ayt , −tz⟩ ≥ 0,

∀t > 0.

158 | 3 Monotone and pseudocontractive mappings It follows that ⟨f − Ayt , z⟩ ≤ 0.

(3.21)

Note that A : H → H is hemicontinuous. Letting t → 0 in (3.21), we arrive at ⟨f − Ax, z⟩ ≤ 0, which contradicts ⟨f − Ax, z⟩ > 0. This completes the proof. Theorem 3.2.2. Let A : Dom(A) = H → H be a maximal monotone mapping. Then A is demicontinuous. Equivalently, if A is not maximal monotone, then A is not demicontinuous. Proof. For any {xn } ⊂ H with xn → x as n → ∞, we need to prove Axn ⇀ Ax as n → ∞. In fact, we see from Theorem 3.2.1 that {Axn } is a bounded sequence in H. Since H is reflexive, {Axn } has a subsequence {Axnj } such that Axnj ⇀ y as j → ∞. It follows that ⟨Axnj − g, xnj − y⟩ ≥ 0,

∀[y, g] ∈ Graph(A), n ≥ 1.

Letting j → ∞ in the above inequality, we have ⟨y − g, x − y⟩ ≥ 0,

∀[y, g] ∈ Graph(A).

It follows from the maximal monotonicity of A that y = Ax. Therefore, it follows that w w Axnj 󳨀→ Ax as j → ∞. Since Ax is unique, we must have Axn 󳨀→ Ax as n → ∞. This completes the proof. Remark 3.2.1. The above theorem gives another important property of a maximal monotone mapping. (1) If A : H → H is not demicontinuous, then A : H → H is not maximal monotone. (2) We know that the sum of finitely many monotone mappings is still monotone, however, the sum of two maximal monotone mappings is not necessarily maximal monotone. The following theorems give the necessary and sufficient conditions for the sum of two maximal monotone mappings still be maximal monotone. Theorem 3.2.3. Let A ⊂ H × H be a set-valued maximal monotone mapping and let B : H → H be single-valued monotone mapping. If T = A + B ⊂ H × H is maximal monotone, then A must be maximal monotone. Proof. Suppose that, for any [x, f ] ∈ H × H, ⟨f − g, x − y⟩ ≥ 0,

∀[y, g] ∈ Graph(A).

Denote u = Bx and v = By. Since g ∈ Ay is arbitrary, the elements in Ty take the form as w = g + v. So, ∀[y, g] ∈ Graph(T), we obtain ⟨w − u − f , y − x⟩ = ⟨g − f , y − x⟩ + ⟨v − u, y − x⟩ ≥ 0.

3.2 Criteria of maximal monotone mappings | 159

Since T is maximal monotone, we have [x, f + u] ∈ Graph(T), that is, x ∈ Dom(T) and f + u ∈ Tx = Ax + Bx 󳨐⇒ f ∈ Ax. So A is maximal monotone. This completes the proof. Theorem 3.2.4. Let A ⊂ H × H be a monotone mapping and λ > 0. If Ran(A + λI) = H, then (A+λI)−1 : H → H is single-valued, λ1 -Lipschitz continuous and maximal monotone. Proof. By Proposition 3.1.1, for any [x, f ], [y, g] ∈ Graph(A), one has 󵄩󵄩 󵄩 󵄩󵄩(λx + f ) − (λy + g)󵄩󵄩󵄩 ≥ λ‖x − y‖. This implies that (A + λI)−1 : H → H is single-valued, Dom((A + λI)−1 ) = H and 1 -Lipschitz continuous. By Theorem 3.2.1, we see that (A + λI)−1 : H → H is maxiλ mal monotone. This completes the proof. Thus we have the following criterion. Theorem 3.2.5. Let A ⊂ H × H be a monotone mapping. If there exists λ > 0 such that Ran(A + λI) = H, then A ⊂ H × H is a maximal monotone mapping. Proof. By Theorem 3.2.4, we see that (A+λI)−1 : H → H is maximal monotone, so is (A+ λI) ⊂ H ×H. In view of B = λI is monotone mapping, one concludes from Theorem 3.2.3 that A ⊂ H × H must be maximal monotone. This completes the proof. Indeed, the converse of the above theorem is also true. Theorem 3.2.6. Let A ⊂ H × H be a monotone mapping. If A is maximal monotone, then there exists λ0 > 0 such that R(A + λ0 I) = H.

(3.22)

Proof. Suppose that A ⊂ H × H is a monotone mapping. In order to prove (3.22), we prove, without loss of generality, that Ran(A + I) = H. To this end, for any y ∈ H, we need to prove that there exists x ∈ Dom(A) such that y ∈ (I + A)x. By the maximal monotonicity of A, we only need to prove that there exists x ∈ H such that ⟨v − (y − x), u − y⟩ ≥ 0,

∀[u, v] ∈ Graph(A).

(3.23)

Fixing [u, v] ∈ Graph(A), we let M(u, v) = {x ∈ H : ⟨v + x, u − x⟩ ≥ ⟨y, u − x⟩}. Now, we are in a position to show that M(u, v) is a nonempty, bounded, closed, and convex subset of H. Since u ∈ M(u, v), we have M(u, v) ≠ 0. By the continuity of the inner product ⟨⋅, ⋅⟩, we see that M(u, v) is closed. Then the construction of M(u, v) implies

160 | 3 Monotone and pseudocontractive mappings that ‖x‖2 − ⟨y, x⟩ + ⟨v, x⟩ − ⟨u, x⟩ ≤ ⟨u, v⟩,

∀x ∈ M(u, v)

󳨐⇒ ‖x‖2 ≤ ‖u‖‖v‖ + (‖y‖ + ‖v‖ + ‖u‖)‖x‖.

(3.24)

Thus M(u, v) is bounded. Note that ‖ ⋅ ‖2 is convex and ⟨y, ⋅⟩, ⟨v, ⋅⟩, ⟨u, ⋅⟩ are linear. From (3.24), we see that M(u, v) is convex and then M(u, v) is w-compact. Thus, in order to prove (3.23), we only need to prove that ⋂{M(u, v) : [u, v] ∈ Graph(A)} ≠ 0. From the Heine–Borel theorem on compact sets, we only need to prove, for any finite number of [ui , vi ] ∈ Graph(A) (i = 1, 2, . . . , m), that m

⋂ M(ui , vi ) ≠ 0. i=1

To achieve this, one considers the bounded convex subset K of ℝm given by m

K = {λ = (λ1 , λ2 , . . . , λm )T : λi ≥ 0, ∑ λi = 1} i=1

and the functional f : K × K → ℝ. Then m

f (λ, μ) = ∑ μi ⟨x(λ) + vi − y, x(λ) − ui ⟩, i=1

∀λ, μ ∈ K,

where λ = (λ1 , λ2 , . . . , λm )T , μ = (μ1 , μ2 , . . . , μm )T and x(λ) = ∑m i=1 λi μi . It is not hard to see that f is continuous and convex in the first variable λ and is linear in the second variable μ. It satisfies all the conditions of the Minimax Theorem [87]. Using the Minimax Theorem, one sees that there exists λ0 ∈ K such that f (λ0 , μ) ≤ max f (λ, λ), λ∈K

∀μ ∈ K.

m m m Since ∑m i=1 ∑j=1 aij = ∑j=1 ∑i=1 aij and due to the monotonicity of A, we have m m

f (λ, λ) = ∑ ∑ λi λj ⟨vi , uj − ui ⟩ = i=1 j=1

1 m m ∑ ∑ λ λ ⟨v − vj , uj − ui ⟩ ≤ 0. 2 i=1 j=1 i j i

It follows that m

∑ μi ⟨x(λ0 ) + vi − y, x(λ0 ) − ui ⟩ ≤ 0, i=1

∀μ ∈ K.

Hence, ⟨vi + x(λ0 ) − y, ui − x(λ0 )i ⟩ ≥ 0,

∀i = 1, 2, . . . , m.

Therefore, x(λ0 ) ∈ ⋂m i=1 M(ui , vi ) ≠ 0 and Ran(A + I) = H. This completes the proof.

3.2 Criteria of maximal monotone mappings | 161

Combining Theorems 3.2.5 and 3.2.6, we can obtain the following criterion. Theorem 3.2.7. Let A ⊂ H × H be a monotone mapping. Then the following statements are equivalent: (1) A is maximal monotone, (2) Ran(A + λI) = H, ∀λ > 0, (3) there exists λ0 > 0 such that Ran(A + λ0 I) = H. Proof. If A ⊂ H ×H is maximal monotone, then λ1 A, ∀λ > 0, is also maximal monotone. By Theorem 3.2.6, we see that Ran( λ1 A + I) = H. So, Ran(A + λI) = H, ∀λ > 0. In particular, there exists λ0 > 0 such that Ran(A + λ0 I) = H. If there exists λ0 > 0 such that Ran(A + λ0 I) = H, then we can conclude from Theorem 3.2.5 that A ⊂ H × H is maximal monotone. This completes the proof. As an application of Theorem 3.2.7, we give the following very useful conclusion. Theorem 3.2.8. Let f : H → ℝ be a proper, lower semicontinuous, and convex functional. Then the subdifferential 𝜕f ⊂ H × H of f is a maximal monotone mapping. Proof. From Example 1.9.8, we see that 𝜕f ⊂ H × H is monotone. From Theorem 3.2.7, we only need to prove that Ran(I + 𝜕f ) = H,

(3.25)

that is, for any y0 ∈ H, the equation y0 ∈ x + 𝜕f (x) has at least one solution x0 ∈ Dom(𝜕f ). To this end, we consider the functional φ : H → ℝ defined by 1 φ(x) = ‖x‖2 + f (x) − ⟨x, y0 ⟩, 2

∀x ∈ H.

(3.26)

It is easy to verify that φ : H → ℝ is proper, lower semicontinuous, and convex. Since Dom(𝜕f ) ≠ 0, there exist z ∈ Dom(f ) and x∗ ∈ H such that f (x) ≥ f (z) + ⟨x ∗ , x − z⟩,

∀x ∈ H.

Substituting (3.27) into (3.26), we obtain 1 φ(x) ≥ ‖x‖2 + f (z) + ⟨x∗ , x − z⟩ − ⟨x, y0 ⟩ 2 1 󵄩 󵄩 ≥ ‖x‖2 + f (z) + 󵄩󵄩󵄩x∗ 󵄩󵄩󵄩‖x − z‖ − ‖x‖‖y0 ‖. 2 It follows that lim φ(x) = ∞.

‖x‖→∞

(3.27)

162 | 3 Monotone and pseudocontractive mappings Using the existence of the functional minimum theorem, there exists x0 ∈ Dom(f ) such that φ(x0 ) ≤ φ(x),

∀x ∈ H.

(3.28)

By some properties of the subdifferential, we have θ ∈ 𝜕φ(x0 ). Note that 1 𝜕φ(x) = 𝜕 ‖x‖2 + 𝜕f (x) + 𝜕(⟨x, −y0 ⟩) 2 = {x} + 𝜕f (x) − {y0 } = {x − y0 } + 𝜕f (x).

Hence, y0 − x0 ∈ 𝜕f (x0 ) and so y0 ∈ x0 + 𝜕f (x0 ). This completes the proof. Remark 3.2.2. Using Theorem 3.2.8, one sees that the convex optimization problem min f (x) x∈H

can be transformed into a zero point problem of a maximal monotone mapping A = 𝜕f : θ ∈ Ax = 𝜕f (x). This shows that the theory of maximal monotone mappings is a powerful tool for solving convex optimization problems.

3.3 Acute angle principle for monotone mappings The acute angle principle of monotone mappings is an important constituent of monotone mappings. We can establish a series of important results, such as the surjectivity of maximal monotone mappings, from the acute angle principle. Theorem 3.3.1. Let A ⊂ H × H be a maximal monotone mapping. Suppose that there exists a fixed constant α > 0 such that ⟨x∗ , x⟩ ≥ 0,

∀x ∈ Dom(A) with ‖x‖ > α, x ∗ ∈ Ax.

(3.29)

Then there exists x ∈ Bα [θ] ⋂ Dom(A) such that θ ∈ Ax.

(3.30)

Proof. From Theorem 3.2.7, we see that, for any ε > 0, Ran(A+ αε I) = H. So, there exists a unique xε ∈ Dom(A) such that θ ∈ (A +

ε I)xε . α

3.3 Acute angle principle for monotone mappings | 163

Letting xε∗ = − αε xε , one has xε∗ ∈ Axε and 0 = ⟨xε∗ , x⟩ +

ε ε ⟨x , x ⟩ = ⟨xε∗ , x⟩ + ‖xε ‖2 . α ε ε α

Therefore, ε ⟨xε∗ , x⟩ = − ‖xε ‖2 . α

(3.31)

It follows from (3.29) that ‖xε ‖ ≤ α and 󵄩󵄩 ∗ 󵄩󵄩 ε 󵄩󵄩x 󵄩󵄩 = ‖xε ‖ ≤ ε, α

∀ε > 0.

(3.32)

Taking εn > 0 with εn → 0 as n → ∞, and setting xn := xεn and xn∗ := xε∗n , we obtain 󵄩 󵄩 ‖xn ‖ ≤ α, 󵄩󵄩󵄩xn∗ 󵄩󵄩󵄩 ≤ εn ,

∀n ≥ 1.

Since {xn } is a bounded sequence in H and H is reflexive, one sees that there exists a subsequence {xnj } ⊂ {xn } such that xnj ⇀ x as j → ∞. By the w-lower semicontinuity of the norm, we see that ‖x‖ ≤ lim inf ‖xnj ‖ ≤ α. j→∞

Hence x ∈ Bα [θ]. Since xnj ⇀ x as j → ∞, xn∗j → x as j → ∞ and A ⊂ H × H is maximal monotone, we conclude from Proposition 3.1.4 that x ∈ Dom(A) and θ ∈ Ax. Hence, (3.30) holds. This completes the proof. Proposition 3.3.1. Let A : Dom(A) = H → H be a hemicontinuous mapping. Suppose there exists a constant α > 0 such that ⟨Ax, x⟩ ≥ 0,

∀x ∈ H with ‖x‖ > α.

Then the equation Ax = θ has at least one solution in Bα [θ]. Proof. From Theorem 3.2.1, we see that A : Dom(A) = H → H is maximal monotone. From Theorem 3.3.1, we can obtain the desired conclusion immediately. This completes the proof. Next, we establish the surjectivity results for maximal monotone mappings. First, let us discuss the surjectivity under some coercitivity conditions. Definition 3.3.1. A mapping A ⊂ H × H is said to be coercive if there exists a function c : ℝ+ → ℝ such that c(r) → ∞ as r → ∞ and ⟨f , x⟩ ≥ c(‖x‖)‖x‖,

∀[x, f ] ∈ Graph(A).

(3.33)

164 | 3 Monotone and pseudocontractive mappings It is obvious that if A ⊂ H × H is an η-strongly monotone mapping, then A is coercive. Theorem 3.3.2 ([10]). Let A ⊂ H × H be a maximal monotone mapping satisfying the coercitivity condition (3.33). Then Ran(A) = H. Proof. Since translation does not change the maximal monotonicity and coercitivity of A, we only need to prove θ ∈ Ran(A). Using coercitivity condition (3.33), we see that there exists a constant α > 0 such that ⟨f , x⟩ ≥ c(‖x‖)‖x‖ ≥ 0 as soon as ‖x‖ > α. Using Theorem 3.3.1, we see that there exists x ∈ Bα (θ) ∩ Dom(A) such that θ ∈ A(x). This completes the proof. Corollary 3.3.1 ([5]). Let A : Dom(A) = H → H be hemicontinuous, monotone, and coercive. Then Ran(A) = H. Proof. Using Theorem 3.2.1, we see that A : Dom(A) = H → H is maximal monotone. Using Theorem 3.3.2, we can derive the desired conclusion immediately. This completes the proof. We know that local boundedness is a fundamental property of monotone mappings. Next, we prove that local boundedness is a characteristic property related to the surjectivity of a maximal monotone mapping. Definition 3.3.2. Let A ⊂ H × H be a mapping. Then A−1 is said to be locally bounded if for any y ∈ H, there exists r > 0 such that the set {x ∈ H : Br (y) ∩ Ax ≠ 0} is bounded in H. It is easy to see that A−1 is locally bounded if and only if, for any {xn } ⊂ H and {[xn , fn ]} ⊂ Graph(A), if fn → f as n → ∞, then {xn } is bounded. Remark 3.3.1. For any coercive mapping A, A−1 is locally bounded. Indeed, if ⟨fn , xn ⟩ ≥ c(‖xn ‖)‖xn ‖, then c(‖xn ‖) ≤ ‖fn ‖. If fn → f (n → ∞), then ‖fn ‖ → ‖f ‖ (n → ∞). This shows that {c(‖xn ‖)} is a bounded sequence, so is {xn }. Therefore, A−1 is locally bounded. Theorem 3.3.3. Let A ⊂ H × H be a maximal monotone mapping. Then the following are equivalent: (1) Ran(A) = H. (2) A−1 is locally bounded on H.

3.3 Acute angle principle for monotone mappings | 165

Proof. (1) 󳨐⇒ (2). Since A ⊂ H × H is maximal monotone, one sees that A−1 is also maximal monotone. Since Ran(A) = H, one has Dom(A−1 ) = Ran(A) = H. Using Theorem 3.1.1, one sees that A−1 is locally bounded on H. (2) 󳨐⇒ (1). Since H is connected, we only need to prove that Ran(A) is open and closed in H. First, we shows that Ran(A) is closed. For any y ∈ Ran(A), there exists {yn } ⊂ Ran(A) such that yn → y as n → ∞. Since A−1 is locally bounded on H, there exists r > 0 such that ‖xn ‖ ≤ r for all n ≥ 1 and xn ∈ A−1 (yn ), that is, yn ∈ Axn . Since H is reflexive, there exists a subsequence {xnj } ⊂ {xn } such that xnj ⇀ x as j → ∞. Using the monotonicity of A, we obtain ⟨xnj − v, ynj − u⟩ ≥ 0,

∀[u, v] ∈ Graph(A−1 ).

Taking j → ∞ in the above inequality, we have ⟨x − v, y − u⟩ ≥ 0,

∀[u, v] ∈ Graph(A−1 ).

Using the maximal monotonicity of A−1 , we see that [y, x] ∈ Graph(A−1 ), that is, x ∈ A−1 (y). It follows that y ∈ Ax ⊂ Ran(A). Next, we prove that Ran(A) is open. Suppose that y0 ∈ Ran(A). Then there exists x0 ∈ Dom(A), such that x0 ∈ A−1 (y0 ), because A−1 is locally bounded, we see that there exist r > 0 and an open ball B(y0 , r) ⊂ H such that A−1 (B(y0 , r) ∩ Ran(A)) is bounded. For all y ∈ B(y0 , r), ε > 0 and x0 + ε1 y ∈ H, it follows from Ran(I + ε1 A) = H that there exists xε ∈ Dom(A) such that 1 1 x0 + y ∈ xε + Axε . ε ε It follows then that εx0 + y ∈ εxε + Axε . Denote zε := y + ε(x0 − xε ). It follows from the monotonicity of A that ⟨y0 − zε , x0 − xε ⟩ ≥ 0 󳨐⇒ ⟨y0 − zε , zε − y⟩ ≥ 0

󳨐⇒ ‖y0 − zε ‖2 ≤ ⟨y0 − zε , y0 − y⟩ ≤ ‖y0 − zε ‖‖y0 − y‖

󳨐⇒ ‖y0 − zε ‖ ≤ ‖y0 − y‖ < r,

which implies zε ∈ B(y0 , r). Since xε ∈ A−1 (zε ) and A−1 (B(y0 , r) ∩ Ran(A)) is bounded, we see that {xε } is bounded. Take εn > 0 with εn → 0 as n → ∞. Setting xn := xεn and zn := zεn and letting xn ⇀ x as n → ∞, we obtain from zn = y +εn (x0 −xn ) that zn → y as n → ∞. Since A−1 is maximal monotone, we conclude from the demiclosedness that y ∈ Dom(A−1 ) and x ∈ A−1 (y), that is, y ∈ Ax ⊂ Ran(A). It follows that B(y0 , r) ⊂ Ran(A), which shows that y0 ∈ int(Ran(A)). Hence Ran(A) is an open set of H. This completes the proof.

166 | 3 Monotone and pseudocontractive mappings Corollary 3.3.2. Let A ⊂ H × H be a maximal monotone mapping. If A−1 is bounded, then Ran(A) = H. In particular, if Dom(A) is bounded, then Ran(A) = H. In the following, we discuss sufficient conditions under which the sum of two maximal monotone mappings is maximal monotone. Theorem 3.3.4 (Rockafellar [77], 1970). Let A, B ⊂ H × H be two maximal monotone mappings. If they satisfy one of the following conditions: (1) Dom(A) ∩ int(Dom(B)) ≠ 0. (2) There exists x ∈ Dom(A) ∩ Dom(B) such that B is locally bounded at x, then A + B ⊂ H × H is maximal monotone. Remark 3.3.2. By Rockafellar’s Theorem [77], we see that conditions (1) and (2) are equivalent. Proof. By Remark 3.3.2, we only consider condition (1). Since the translation does not change the (maximal) monotonicity, we may assume, without loss of generality, that θ ∈ Aθ and θ ∈ int(Dom(B)). Now, consider the following two cases. Case I. Dom(B) ⊂ H is bounded. To prove that A + B is maximal monotone, we find from Theorem 3.2.7 that we only need to prove Ran(A + B + I) = H. That is, for any y ∈ H, we only need to prove y ∈ (A + B + I). Since the translation does not change the maximal monotonicity, we may assume y = θ. Then we only need to prove that there exists x ∈ H such that θ ∈ (A + B + I)(x).

(3.34)

It is obvious that x satisfies (3.34) if and only if there exists x∗ ∈ H such that 1 − x∗ ∈ (A + I)(x), 2

1 x ∗ ∈ (B + I)(x). 2

(3.35)

Define two mappings S1 and S2 : H → H by 1 S1 (y) = −(A + I) (−y) 2

(3.36)

1 S2 (y) = (B + I) (y). 2

(3.37)

x∗ satisfies (3.43) ⇐⇒ θ = S1 (x ∗ ) + S2 (x∗ ).

(3.38)

−1

and −1

Then

3.3 Acute angle principle for monotone mappings | 167

To prove (3.34), we only need to prove θ ∈ Ran(S1 + S2 ).

(3.39)

By Theorem 3.2.4, we see that S1 and S2 are two single-valued, 2-Lipschitz continuous, maximal monotone, and Dom(S1 ) = H = Dom(S2 ). Hence S1 + S2 : H → H is single-valued, 2-Lipschitz continuous maximal monotone, and Dom(S1 + S2 ) = H. By Theorem 3.2.1, we see that S1 + S2 is maximal monotone. Using the assumption that θ ∈ Aθ, we have S1 (θ) = θ. By the monotonicity of S1 , we obtain that ⟨S1 (y), y⟩ ≥ 0,

∀y ∈ H.

(3.40)

1 I) 2

Since Ran(S2 ) = Dom(B + = Dom(B), we see that Ran(S2 ) is bounded and θ ∈ int(Ran(S2 )). Hence we claim that there exists α > 0 such that ⟨S2 (y), y⟩ ≥ 0,

∀y ∈ H with ‖y‖ > α.

(3.41)

If (3.41) holds, then we derive from (3.40) that ⟨(S1 + S2 )y, y⟩ ≥ 0,

∀y ∈ H with ‖y‖ > α.

By Theorem 3.3.1, we see that θ ∈ Ran(S1 + S2 ), that is, (3.39) holds. Next, we prove that (3.41) is true. For all y, z ∈ H, we conclude from the monotonicity of S2 that ⟨S2 (y) − S2 (z), y − z⟩ ≥ 0

󳨐⇒ ⟨S2 (y), y⟩ ≥ ⟨S2 (z), y⟩ + ⟨S2 (y) − S2 (z), z⟩.

(3.42)

Since Ran(S2 ) is bounded, there exists α1 > 0 such that 󵄨󵄨 󵄨 󵄨󵄨⟨S2 (y) − S2 (z), z⟩󵄨󵄨󵄨 ≤ 2α1 ‖z‖.

(3.43)

Note that Dom(S2−1 ) = Ran(S2 ). In view of (3.40), one sees that θ ∈ int(Dom(S2−1 )). By Theorem 3.1.1, S2−1 is locally bounded at the origin θ and thus there exist ε > 0 and α2 > 0 such that {S2 (z) : ‖z‖ ≤ α2 } ⊃ {u ∈ H : ‖u‖ ≤ ε}.

(3.44)

By (3.42) and (3.43), it follows that, for all z ∈ H, ‖z‖ ≤ α2 , ⟨S2 (y), y⟩ ≥ ⟨S2 (z), y⟩ − 2α1 α2 .

(3.45)

By (3.44) and (3.45), we see that ⟨S2 (y), y⟩ ≥ sup {⟨u, y⟩ − 2α1 α2 } = ε‖y‖ − 2α1 α2 . ‖u‖≤ε

(3.46)

Take α = 2αε1 α2 . Then, if ‖y‖ > α, then we see from (3.46) that ⟨S2 (y), y⟩ ≥ 0, which implies that (3.41) holds.

168 | 3 Monotone and pseudocontractive mappings Case II. Dom(B) ⊂ H is unbounded. In this case, using the perturbation techniques of monotone mappings and Case I, we can derive the desired conclusion. To complete the proof, let us collect some necessary and known conclusions. Let C be a nonempty, closed, and convex subset of a Hilbert space H. Recall that the indicator function of C is defined as follows: 0,

iC (x) = {

x ∈ C,

∞, x ∉ C.

It is well known that iC : H → ℝ is proper, lower semicontinuous, and convex. Further, the subdifferential 𝜕iC ⊂ H × H of iC is maximal monotone. Recall also that the normal cone of C at x is defined as follows: NC (x) = {z ∈ H : ⟨u − x, z⟩ ≤ 0,

∀y ∈ C}.

(3.47)

It is clear that NC (x) is a closed cone and 𝜕iC (x) = NC (x).

(3.48)

Let C = Bα [θ] = {x ∈ H : ‖x‖ ≤ α}, ∀α > 0. Then {θ}, ‖x‖ < α, NC (x) = { {λx : λ > 0}, ‖x‖ = α. Thus, we have that {θ}, ‖x‖ < α, { { { 𝜕iBα [θ] (x) = {{λx : λ > 0}, ‖x‖ = α, { { ‖x‖ > α. {0, Denote φα (x) = 𝜕iBα [θ] (x), ∀x ∈ H and α > 0. If T ⊂ H × H is monotone, then T + φα is monotone and (T + φα )(x) = Tx,

∀x ∈ Bα [θ],

Dom(T + φα ) = Dom(T) ∩ Bα [θ]. Now, we in a position to prove Theorem 3.3.4 under Case II. Assume that θ ∈ Dom(A) ∩ int(Dom(B)) Since Dom(φα ) = Bα [θ], we have int(Dom(φα )) = Bα [θ].

(3.49) (3.50)

3.3 Acute angle principle for monotone mappings | 169

It follows that θ ∈ Dom(B) ∩ Dom(φα ) ≠ 0. Furthermore, Dom(φα ) is bounded. Since Dom(B+φα ) = Dom(B)∩Bα [θ], we derive that θ ∈ Dom(A) ∩ int(Dom(B + φα )) ≠ 0, where Dom(B + φα ) is bounded. Letting T = B in (3.49), we have (B + φα )x = Bx, ∀x ∈ Bα [θ]. If ‖x‖ = α, then (B + φα )x = Bx + φα (x) = Bx + λx = (B + λI)x. Note that B and φα are maximal monotone. From Case I, we can prove that B + φα is maximal monotone. Since A + (B + φα ) = (A + B) + φα , we see that A + (B + φα ) is maximal monotone. Hence (A + B) + φα is maximal monotone. By Theorem 3.2.3, we see that A + B is maximal monotone. This completes the proof. Theorem 3.3.5 (Rockafellar [77], 1970). Let C be a nonempty convex and closed subset of a real Hilbert space H and let A : C → H be a monotone and hemicontinuous mapping. Define a mapping T : H → 2H by Ax + NC (x), x ∈ C, Tx = { 0, x ∉ C, where NC (x) is the normal cone of C at x. Then T ⊂ H × H is maximal monotone. Proof. Note that the effective domain of T is Dom(T) = C. If x ∈ C, then NC (x) = 𝜕iC (x). If x ∈ Dom(T) = C, then Tx = Ax + 𝜕iC (x) and A and 𝜕iC are monotone. Thus T is also monotone. Next, we prove that T is maximal monotone. From Proposition 3.1.2, we only need to prove, for any [x, f ] ∈ H × H, that ⟨y − x, g − f ⟩ ≥ 0,

∀[y, g] ∈ Graph(A) 󳨐⇒ [x, f ] ∈ Graph(A).

Since [y, g] ∈ Graph(A), g ∈ Ty = Ay + NC (y), we have g = Ay + z and z ∈ NC (y). It follows that ⟨y − x, z⟩ + ⟨y − x, Ay − f ⟩ ≥ 0,

∀y ∈ C, z ∈ NC (y).

(3.51)

Since NC (y) is a closed cone, we see that z ∈ NC (y). So, for any λ ≥ 0, one has λz ∈ NC (y). By (3.51), we obtain ⟨y − x, λz⟩ + ⟨y − x, Ax − f ⟩ ≥ 0,

∀y ∈ C,

(3.52)

170 | 3 Monotone and pseudocontractive mappings which implies from (3.52) that ⟨y − x, z⟩ ≥ 0,

∀z ∈ NC (y).

(3.53)

If (3.53) does not hold, then there exists z ∈ NC (y) such that ⟨y − x, z⟩ < 0. For any λ > 0, letting λ → ∞, one has λ⟨y − x, z⟩ = ⟨y − x, λz⟩ → −∞,

∀y ∈ C,

which contradicts (3.52). Since NC (y) = 𝜕iC (y), we see from (3.53) that ⟨y − x, z − θ⟩ ≥ 0,

∀z ∈ 𝜕iC (y).

Note that iC is maximal monotone. From Proposition 3.1.2, we have θ ∈ 𝜕iC (x). Hence x ∈ C. Letting xt = tu + (1 − t)x, u ∈ C, ∀t ∈ (0, 1), we have xt ∈ C. Taking y = xt and z = θ in (3.51), we obtain ⟨xt − x, θ⟩ + ⟨xt − x, Axt − f ⟩ ≥ 0, that is, ⟨xt − x, Axt − f ⟩ ≥ 0 󳨐⇒ t⟨u − x, Axt − f ⟩ ≥ 0, 󳨐⇒ ⟨u − x, Axt − f ⟩ ≥ 0,

∀t ∈ (0, 1)

∀t ∈ (0, 1).

Letting t → 0+ , we find from the hemicontinuity of A that ⟨u − x, Ax − f ⟩ ≥ 0,

∀u ∈ C 󳨐⇒ f − Ax ∈ NC (x) 󳨐⇒ Ax + NC (x) = Tx.

Therefore, T ⊂ H × H is maximal monotone. This completes the proof.

3.4 Monotone variational inequalities In this subsection, we study the existence and uniqueness of solutions for a class of monotone variational inequalities via the monotone mapping theory. First, we establish existence and uniqueness of solutions for variational inequality problems (in short, (VIP)) involving η-strongly monotone hemicontinuous mappings. Then we drive some zero point theorems of monotone hemicontinuous mappings. Using the existence theorems of solutions of variational inequalities, we introduce some iterative methods for finding the solutions of the monotone variational inequality problems.

3.4 Monotone variational inequalities | 171

Theorem 3.4.1. Let C be a nonempty closed convex subset of a real Hilbert space H and let A : C → H be an η-strongly monotone hemicontinuous mapping. Then ⟨Ax, y − x⟩ ≥ 0,

∀y ∈ C,

(3.54)

has a unique solution x∗ ∈ C. Proof. Denote the solution set of (3.54) by VI(C, A) and define T ⊂ H × H as in Theorem 3.3.5 by Ax + NC (x),

Tx = {

0,

x ∈ C, x ∉ C,

where NC (x) is the normal cone of C at x. By Theorem 3.3.5, T ⊂ H × H is maximal monotone and VI(C, A) = T −1 (θ) = {z ∈ C : θ ∈ Tz}.

(3.55)

In fact, we have z ∈ T −1 (θ) ⇐⇒ θ ∈ Az + NC (z)

⇐⇒ ⟨u − z, −Az⟩ ≤ 0, ⇐⇒ ⟨u − z, Az⟩ ≥ 0, ⇐⇒ z ∈ VI(C, A).

∀u ∈ C

∀u ∈ C

To prove that (3.54) has a unique solution, we only need to prove that T has a unique zero point. From Theorem 3.3.1, we just need to prove that there exists α > 0 such that ⟨x∗ , x⟩ ≥ 0, 1.

∀x ∈ C with ‖x‖ > α, x ∗ ∈ Tx.

(3.56)

Now, we consider the following two cases. θ ∈ C. Since A : C → H is η-strongly monotone, one has ⟨Ax − Aθ, x⟩ ≥ η‖x‖2 , 2

∀x ∈ C

󳨐⇒ ⟨Ax, x⟩ ≥ η‖x‖ − ‖Aθ‖‖x‖,

∀x ∈ C.

(3.57) (3.58)

On the other hand, we have x∗ ∈ Tx = Ax + NC (x) and x∗ = Ax + z,

∀z ∈ NC (x),

(3.59)

⟨−x, z⟩ ≤ 0 󳨐⇒ ⟨x, z⟩ ≥ 0.

(3.60)

z ∈ NC (x) 󳨐⇒ ⟨u − x, z⟩ ≤ 0,

∀u ∈ C.

Letting u = θ ∈ C yields

172 | 3 Monotone and pseudocontractive mappings From (3.58)–(3.60), we see that ⟨x ∗ , x⟩ = ⟨Ax + z, x⟩ = ⟨Ax, x⟩ + ⟨x, z⟩ ≥ η‖x‖2 − ‖Aθ‖‖x‖

= ‖x‖(η‖x‖ − ‖Aθ‖). Set α =

‖Aθ‖+1 η

> 0. If ‖x‖ > α, then ⟨x ∗ , x⟩ ≥ 0,

2.

∀x ∈ C, x∗ ∈ Tx.

θ ∉ C. ̃ = C − x . This shows θ ∈ C. ̃ Define a mapping Fixing x0 ∈ C, we obtain a shift C 0 ̃ (that is, x + x ∈ C). This implies that T̃ ⊂ H × H is ̃ T(x) = T(x + x0 ), ∀x ∈ C 0 ̃ From N ̃ (x) = ̃ = A(x + x0 ), ∀x ∈ C. still maximal monotone. In fact, we define A C ̃ NC (x + x0 ), ∀x ∈ C, one has ̃ ̃ + N ̃ (x), T(x) = T(x + x0 ) = A(x + x0 ) + NC (x + x0 ) = Ax C

̃ ∀x ∈ C.

̃ → H is η-strongly monotone and hemicontinuous. So, ̃ :C It is easy to verify that A it follows from Theorem 3.3.5 that T̃ ⊂ H × H is maximal monotone. From Case I, ̃ such that one sees that there exists x̃ ∈ Bα [θ] ⋂ C ̃ x̃), θ ∈ T( where α > 0 is a fixed constant, and Bα [θ] is a closed ball with the center origin and the radius α. Hence we obtain θ ∈ T(x̃ + x0 ). Setting z = x̃ + x0 , we have z ∈ C and θ ∈ Tz. Hence T −1 θ ≠ 0. The uniqueness is guaranteed by the η-strong monotonicity of A. This completes the proof. Corollary 3.4.1. Let C be a nonempty bounded closed convex subset of a real Hilbert space H and let A : C → H be a monotone hemicontinuous mapping. Then VI(C, A) is nonempty, closed, and convex. Proof. Define a mapping An : C → H by An (x) = Ax +

1 x, n

∀x ∈ C, n ≥ 1.

Then An : C → H is n1 -strongly monotone and hemicontinuous. From Theorem 3.4.1, there exists a unique xn ∈ C such that xn ∈ VI(C, A), that is, ⟨An xn , y − xn ⟩ ≥ 0,

∀y ∈ C, n ≥ 1.

From Minty’s lemma, it follows that ⟨An y, y − xn ⟩ ≥ 0, ∀y ∈ C and ∀n ≥ 1, that is, ⟨Ay +

1 y, y − xn ⟩ ≥ 0, n

∀y ∈ C, n ≥ 1.

(3.61)

3.4 Monotone variational inequalities | 173

Since C is bounded, one sees that {xn } is a bounded sequence. We may assume that xn ⇀ x as n → ∞. Letting n → ∞ in (3.61), we arrive at ⟨Ay, y − x⟩ ≥ 0,

∀y ∈ C.

(3.62)

In view of Minty’s lemma, we obtain ⟨Ax, y − x⟩ ≥ 0,

∀y ∈ C,

that is, VI(C, A) ≠ 0. It follows from (3.62) that VI(C, A) is closed and convex. This completes the proof. Corollary 3.4.2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let A : C → H be an η-strongly monotone hemicontinuous mapping satisfying the “flow-invariance” condition (FIC): lim t −1 d((I − tA)x, C) = 0,

∀x ∈ C.

t→0+

Then there exists a unique x∗ ∈ C such that Ax∗ = θ. Proof. From Theorem 3.4.1, we see that there exists a unique x ∗ ∈ C such that ⟨Ax∗ , x − x∗ ⟩ ≥ 0,

∀x ∈ C.

(3.63)

By using the “flow-invariance” condition, we see that, for any ε > 0, there exists δ > 0 with 0 < t < δ such that 󵄩 󵄩 d((I − tA)x∗ , C) < tε 󳨐⇒ ∃xt ∈ C such that 󵄩󵄩󵄩(I − tA)x ∗ − xt 󵄩󵄩󵄩 < tε 󵄩󵄩 󵄩󵄩󵄩 x − x∗ 󵄩 + Ax ∗ 󵄩󵄩󵄩 < ε 󳨐⇒ 󵄩󵄩󵄩 t 󵄩󵄩 󵄩󵄩 t xt − x∗ 󳨐⇒ → −Ax ∗ (t → 0+ ). t Letting x = xt ∈ C in (3.63), we obtain ⟨Ax∗ , xt − x∗ ⟩ ≥ 0,

∀t > 0 󳨐⇒ ⟨Ax∗ ,

xt − x ∗ ⟩ ≥ 0, t

∀t > 0

󳨐⇒ ⟨Ax∗ , −Ax ∗ , ⟩ ≥ 0 󳨐⇒ Ax∗ = θ. This completes the proof.

Corollary 3.4.3. Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let A : C → H be a monotone hemicontinuous mapping satisfying the “flow-invariance” condition (FIC). Then there exists some x ∗ ∈ C such that Ax ∗ = θ.

174 | 3 Monotone and pseudocontractive mappings Proof. From Corollary 3.4.1, we see that VI(C, A) ≠ 0, that is, there exists x ∗ ∈ C such that ⟨Ax∗ , x − x∗ ⟩ ≥ 0,

∀x ∈ C.

The rest of the proof is similar to the proof of Corollary 3.4.2, so we omit it here. This completes the proof. Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let A : C → H be a monotone hemicontinuous mapping and let R : H → H be an η-strongly monotone hemicontinuous mapping. Take u ∈ H and a sequence {rn }, where rn > 0 and rn → 0 as n → ∞. Then, for any n ∈ ℕ, rn (R − u) + A : C → H is ηrn -strongly monotone and hemicontinuous. From Theorem 3.4.1, we have that the following variational inequality ⟨rn (Ry − u) + Ay, x − y⟩ ≥ 0,

∀x ∈ C,

(3.64)

has a unique solution yn ∈ C, ∀n ∈ ℕ. α Letting rn = βn in (3.64), we see that (3.64) can be rewritten as n

⟨αn (Ryn − u) + βn Ayn , x − yn ⟩ ≥ 0,

∀x ∈ C,

(3.65)

which is equivalent to ⟨yn − αn (Ryn − u) − βn Ayn − yn , x − yn ⟩ ≤ 0,

∀x ∈ C.

(3.66)

From the property of PC and (3.66), we have yn = PC [(I − αn R)yn + αn u − βn Ayn ],

∀n ≥ 1.

(3.67)

α

Now, we are in a position to prove that if βn → 0 as n → ∞ and VI(C, A) ≠ 0, then n {yn } converges in norm to a special solution of variational inequality (3.54). Theorem 3.4.2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A : C → H be a monotone hemicontinuous mapping and let R : H → H be α an η-strongly monotone hemicontinuous mapping. Assume that βn → 0 as n → ∞ and n VI(C, A) ≠ 0. Let {yn } be a sequence generated by (3.67). Then {yn } converges in norm to a special solution x ∗ = PVI(C,A) [(I − R)x∗ + u] of the variational inequality (3.54). In particular, if R = I and u = θ, then {yn } converges in norm to the minimum norm solution x∗ = PVI(C,A) θ of variational inequality (3.54). Proof. From the assumption VI(C, A) ≠ 0, it follows from Minty’s lemma that VI(C, A) is closed and convex. Hence VI(C, A) is a nonempty, closed, and convex subset of H. Therefore, for any h ∈ H, PVI(C,A) h is well defined. Since (3.67) and (3.65) are equivalent, we find from Minty’s lemma that ⟨αn Rx − αn u + βn Ax, x − yn ⟩ ≥ 0,

∀x ∈ C.

(3.68)

3.4 Monotone variational inequalities | 175

For all x∗ ∈ VI(C, A), we derive ⟨Ax∗ , x − x∗ ⟩ ≥ 0,

∀x ∈ C.

(3.69)

∀x ∈ C.

(3.70)

In view of Minty’s lemma, we obtain ⟨Ax, x − x∗ ⟩ ≥ 0, Taking x = yn in (3.70), we see that ⟨Ayn , yn − x∗ ⟩ ≥ 0,

∀n ≥ 1.

(3.71)

Taking x = x∗ in (3.65), we have ⟨αn Ryn − αn u + βn Ayn , yn − x∗ ⟩ ≤ 0,

∀n ≥ 1.

(3.72)

It follows from (3.71), (3.72) and the strong monotonicity of R that 0 ≥ ⟨αn Ryn − αn u + βn Ayn , yn − x∗ ⟩

= αn ⟨Ryn − Rx∗ , yn − x∗ ⟩ + αn ⟨Rx∗ − u, yn − x ∗ ⟩ + βn ⟨Ayn , yn − x ∗ ⟩ 󵄩 󵄩2 ≥ ηαn 󵄩󵄩󵄩yn − x∗ 󵄩󵄩󵄩 + αn ⟨Rx∗ − u, yn − x∗ ⟩,

which implies that 1 󵄩󵄩 ∗ 󵄩2 ∗ ∗ 󵄩󵄩yn − x 󵄩󵄩󵄩 ≤ ⟨Rx − u, x − yn ⟩, η

∀x ∗ ∈ VI(C, A).

(3.73)

∀x ∗ ∈ VI(C, A), n ≥ 1,

(3.74)

In particular, we have 1󵄩 ∗ 󵄩2 󵄩󵄩 ∗ 󵄩2 󵄩󵄩yn − x 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Rx − u󵄩󵄩󵄩 , η

which shows that {yn } is bounded. We may assume, without loss of generality, that yn ⇀ y as n → ∞. Then y ∈ VI(C, A). In fact, it follows from (3.68) that ⟨

αn (Rx − u) + Ax, x − yn ⟩ ≥ 0, βn

∀x ∈ C.

(3.75)

Letting n → ∞ in (3.75), we obtain ⟨Ax, x − y⟩ ≥ 0,

∀x ∈ C.

It follows from Minty’s lemma that ⟨Ay, x − y⟩ ≥ 0,

∀x ∈ C,

(3.76)

176 | 3 Monotone and pseudocontractive mappings that is, y ∈ VI(C, A). Letting x∗ = y, we have ‖yn − y‖2 ≤

1 ⟨Ry − u, y − yn ⟩, η

∀n ≥ 1.

(3.77)

Since yn ⇀ y, it follows from (3.77) that yn ⇀ y as n → ∞. From (3.73), we arrive at ⟨Rx∗ − u, x∗ − yn ⟩ ≥ 0,

∀x∗ ∈ VI(C, A).

Since yn ⇀ y as n → ∞, we obtain ⟨Rx ∗ − u, x∗ − y⟩ ≥ 0,

∀x ∗ ∈ VI(C, A).

By Minty’s lemma, we have ⟨Ry − u, x∗ − y⟩ ≥ 0,

∀x∗ ∈ VI(C, A).

(3.78)

It follows from Theorem 3.4.1 that y is the unique solution of variational inequality (3.78). Hence {yn } converges in norm to y ∈ VI(C, A). From (3.78), we see that y = PVI(C,A) [(I − R)y + u]. This completes the proof. In view of the convergence of iterative method (3.67), it is natural to consider the following explicit form: x1 ∈ C,

xn+1 = PC [(I − αn R)xn + αn u − βn Axn ],

n ≥ 1.

(3.79)

Recall that a mapping T : C → H is said to be generalized Lipschitz continuous if there exists a constant L > 0 such that ‖Tx − Ty‖ ≤ L(1 + ‖x − y‖),

∀x, y ∈ C.

It is obvious that a Lipschitz continuous mapping must be a generalized Lipschitz continuous mapping, but the converse is not true. Generalized Lipschitz continuous mappings may not be continuous, for example, sign function is a simple example. Theorem 3.4.3. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A : C → H be a hemicontinuous and generalized Lipschitz continuous monotone mapping, and let R : H → H be a hemicontinuous and generalized Lipschitz continuous, η-strongly monotone mapping. Let {αn } and {βn } be two sequences in (0, 1) satisfying the following conditions: β2

α

(1) βn → 0 and αn → 0 as n → ∞; n n (2) αn → 0 as n → ∞ and ∑∞ n=1 αn = ∞; |α −α |+|β −β | (3) n+1 n α2 n+1 n → 0 as n → ∞. n

3.4 Monotone variational inequalities | 177

If VI(C, A) ≠ 0, then the sequence {xn } generated by (3.79) converges to x ∗ = PVI(C,A) [(I − R)x∗ + u] in norm. Proof. Assume that {yn } is the sequence defined by (3.67). It follows from Theorem 3.4.2 that yn → x∗ = PVI(C,A) [(I − R)x∗ + u]. Hence, we only need to prove that xn+1 − yn → θ as n → ∞. First, we prove that {xn } is bounded. In fact, for any p ∈ VI(C, A), one has p = PC [(1 − αn )p + αn p − βn Ap].

(3.80)

From (3.69), (3.80) and the property of PC , we have 󵄩 󵄩2 ‖xn+1 − p‖2 = 󵄩󵄩󵄩PC [(I − αn R)xn + αn u − βn Axn ] − PC [(1 − αn )p + αn p − βn Ap]󵄩󵄩󵄩 󵄩 󵄩2 ≤ 󵄩󵄩󵄩xn − p − αn (Rxn − p) + αn (u − p) − βn (Axn − Ap)󵄩󵄩󵄩 = ‖xn − p‖2 − 2αn ⟨Rxn − p, xn − p⟩ − 2αn ⟨Axn − Ap, xn − p⟩ + 2αn ⟨u − p, xn − p⟩ − 2βn ⟨Axn − Ap, xn − p⟩ 󵄩 󵄩2 + 󵄩󵄩󵄩αn (Rxn − u) + βn (Axn − Ap)󵄩󵄩󵄩

≤ (1 − 2ηαn )‖xn − p‖2 − 2αn ⟨Rp − u, xn − p⟩ + 2αn2 ‖Rxn − u‖2 + 2βn2 ‖Axn − Ap‖2 .

(3.81)

Since A and R are generalized Lipschitz continuous and ‖Rp − u‖ 󵄨 󵄨 2󵄨󵄨󵄨⟨Rp − u, xn − p⟩󵄨󵄨󵄨 ≤ 2η ‖xn − p‖ η ≤ η(

‖Rp − u‖2 + ‖xn − p‖2 ), η2

(3.82)

we have ‖Axn − Ap‖2 ≤ L2 (1 + ‖xn − p‖2 ) ≤ 2L2 (1 + ‖xn − p‖2 ),

(3.83)

‖Rxn − Rp‖ ≤ L (1 + ‖xn − p‖ ) ≤ 2L (1 + ‖xn − p‖ ),

(3.84)

‖Rxn − u‖2 ≤ 2(‖Rxn − Rp‖2 + ‖Rp − u‖2 ).

(3.85)

2

2

2

2

2

and

From condition (1) and (2), we may assume, without loss of generality, that 1 8L2 αn2 + 4L2 βn2 ≤ ηαn , 2

∀n ≥ 1.

(3.86)

178 | 3 Monotone and pseudocontractive mappings Substituting (3.82)–(3.86) into (3.81), we obtain ‖xn+1 − p‖2 ≤ (1 − ηαn + 8L2 αn2 + 4L2 βn2 )‖xn − p‖2 1 L2 2 + ηαn ( 2 + 4 + )‖Rp − u‖2 2 η η

1 2 L2 1 ≤ (1 − ηαn )‖xn − p‖2 + ηαn ( 2 + 4 + )‖Rp − u‖2 2 2 η η ≤ max{‖x1 − p‖2 , (

2 L2 + 4 + )‖Rp − u‖2 } 2 η η

= M, which shows that {xn } is bounded. Since R is η-strongly monotone and A is monotone, one has ⟨Rxn − Ryn , xn − yn ⟩ ≥ η‖xn − yn ‖2 ,

∀n ≥ 1,

(3.87)

⟨Axn − Ayn , xn − yn ⟩ ≥ η‖xn − yn ‖2 ,

∀n ≥ 1.

(3.88)

and

Since R and A are both generalized Lipschitz continuous, we have ‖Rxn − Ryn ‖2 ≤ 2L2 (1 + ‖xn − yn ‖2 ),

∀n ≥ 1,

(3.89)

‖Axn − Ayn ‖2 ≤ 2L2 (1 + ‖xn − yn ‖2 ),

∀n ≥ 1.

(3.90)

and

Further, we conclude from (3.67), the property of PC , monotonicity of A and strongly monotonicity of R that ‖yn+1 − yn ‖2 ≤ ⟨yn+1 − yn , (I − αn R)yn+1 − (I − αn R)yn + (αn+1 − αn )u⟩ − βn+1 Ayn+1 − βn Ayn

= ‖yn+1 − yn ‖2 − αn+1 ⟨Ryn+1 − Ryn , yn+1 − yn ⟩

− (αn+1 − αn )⟨Ryn − u, yn+1 − yn ⟩ − βn+1 ⟨Ayn+1 − Ayn , yn+1 − yn ⟩

− (βn+1 − βn )⟨Ayn , yn+1 − yn ⟩

≤ (1 − ηαn+1 )‖yn+1 − yn ‖2 + |αn+1 − αn |‖Ryn ‖‖yn+1 − yn ‖

+ |αn+1 − αn |‖u‖‖yn+1 − yn ‖ + |βn+1 − βn |‖Ayn ‖‖yn+1 − yn ‖.

(3.91)

M |αn+1 − αn | + |βn+1 − βn | , η αn+1

(3.92)

It follows that ‖yn+1 − yn ‖ ≤

∀n ≥ 1,

3.4 Monotone variational inequalities | 179

where M ≥ max{‖Ryn ‖ + ‖u‖, ‖Ayn ‖}. Using conditions (1) and (2), we may assume that αn2 + βn2 ≤

η α , 4L2 n

∀n ≥ 1.

(3.93)

It follows from the property of PC , and (3.87)–(3.93) that 󵄩2 󵄩 ‖xn+2 − yn+1 ‖2 ≤ 󵄩󵄩󵄩xn+1 − yn+1 − αn+1 (Rxn+1 − Ryn+1 ) − βn (Axn+1 − Ayn+1 )󵄩󵄩󵄩 = ‖xn+1 − yn+1 ‖2 − 2αn+1 ⟨Rxn+1 − Ryn+1 , xn+1 − yn+1 ⟩ − 2βn+1 ⟨Axn+1 − Ayn+1 , xn+1 − yn+1 ⟩ 󵄩 󵄩2 + 󵄩󵄩󵄩αn+1 (Rxn+1 − Ryn+1 ) + βn+1 (Axn+1 − Ayn+1 )󵄩󵄩󵄩 2 ≤ (1 − 2ηαn+1 )‖xn+1 − yn+1 ‖2 + 2αn+1 ‖Rxn+1 − Ryn+1 ‖2 + 2β2 ‖Axn+1 − Ayn+1 ‖2

2 ≤ (1 − 2ηαn+1 )‖xn+1 − yn+1 ‖2 + 4L2 αn+1 (1 + ‖xn+1 − yn+1 ‖2 ) 2 + 4L2 βn+1 (1 + ‖xn+1 − yn+1 ‖2 )

2 2 2 2 = [1 − 2ηαn+1 + 4L2 (αn+1 + βn+1 )]‖xn+1 − yn+1 ‖2 + 4L2 (αn+1 + βn+1 )

2 2 ≤ (1 − ηαn+1 )‖xn+1 − yn+1 ‖2 + 4L2 (αn+1 + βn+1 )

≤ (1 − ηαn+1 )[‖xn+1 − yn ‖2 + 2‖xn+1 − yn ‖‖yn+1 − yn ‖ 2 2 + ‖yn+1 − yn ‖2 ] + 4L2 (αn+1 + βn+1 )

≤ (1 − ηαn+1 )‖xn+1 − yn ‖2 + M1 2 2 + 4L2 (αn+1 + βn+1 )

|αn+1 − αn | + |βn+1 − βn | αn+1

≤ (1 − ηαn+1 )‖xn+1 − yn ‖2 + o(ηαn+1 ). In view of Lemma 1.10.2, we have xn+1 − yn → θ

(n → ∞).

This completes the proof. and 0 < b < a < Remark 3.4.1. Taking αn = n1a and βn = n1b for all n ≥ 1, where a < b+1 2 2b, we see that {αn } and {βn } satisfy conditions (1)–(3) in Theorem 3.4.3. Let A be a mapping on a Hilbert space H. Recall that A is said to be pseudomonotone if 0 ≤ ⟨Ax, y − x⟩ 󳨐⇒ 0 ≤ ⟨Ay, y − x⟩,

∀x, y ∈ H.

Also A is said to be γ-strongly pseudomonotone iff there exists γ > 0 such that 0 ≤ ⟨Ax, y − x⟩ ⇒ γ‖x − y‖2 ≤ ⟨Ay, y − x⟩,

∀x, y ∈ H.

180 | 3 Monotone and pseudocontractive mappings If A is L-Lipschitz continuous and monotone, Korpelevich [40] proposed the following well known extragradient method (with double projections) that reduces the monotonicity of operator A: x0 ∈ H, { { { y = PC (xn − τAxn ), { { n { {xn+1 = PC (xn − τAyn ),

∀n ≥ 0,

where τ is a real number in (0, L1 ). It was proved that sequence {xn } generated in the above extragradient algorithm converges weakly to a solution of the variational inequality. However, the price is that one needs to calculate two projections from H onto the feasibility set C. In most situations, there are no analytic expressions for the metric projection. Hence, the extragradient method is not very convenient and efficient in practical calculations. In 2011, Censor, Gibali, and Reich [22] studied a subgradient–extragradient method for solving the following variational inequality in a Hilbert space find x∗ ∈ C such that ⟨Ax ∗ , x − x ∗ ⟩ ≥ 0,

∀x ∈ C,

(3.94)

where A is a monotone and Lipschitz continuous mapping. Algorithm 3.4.7. Step 0. Select a starting point x0 ∈ H and τ > 0, and set n = 0. Step 1. Given the current iterate xn , compute yn = PC (xn − τAxn ), construct the half-space Tn , the bounding hyperplane of which supports C at yn , as Tn := {w ∈ H : ⟨(xn − τAxn ) − yn , w − yn ⟩ ≤ 0}, and calculate the next iterate xn+1 = PTn (xn − τAxn ). Step 2. If xn = yn , then stop. Otherwise, set k ← (k + 1) and return to Step 1. Theorem 3.4.4 (Censor, Gibali, and Reich [22], 2011). Let C be a nonempty, closed, and convex subset of a Hilbert space H. Assume A is L-Lipschitz continuous and monotone on C and let τ ≤ L1 . If the solution set of (3.94) is not empty, then the sequences {xn } and {yn } generated by Algorithm 3.4.7 weakly converge to the same point in the variational inequality (3.94). Recently, convergence rate problems received much attention. One of acceleration methods is the inertial method which is a two-step iterative method. Its feature is that the next iterate is defined by making use of the previous two iterates.

3.4 Monotone variational inequalities | 181

Next, we discuss an inertial subgradient–extragradient algorithm to solve variational inequality problem (3.94) with A being strongly pseudomonotone; see [28] for more details. Algorithm 3.4.8. Step 0. Choose x0 , x1 ∈ H arbitrarily. Step 1. Given the current iterate xn , compute w = xn + αn (xn − xn−1 ), { n yn = PC (wn − τn Awn ), construct the half-space Tn , the bounding hyperplane of which supports C at yn , as Tn = {x ∈ H | ⟨wn − τn Awn − yn , x − yn ⟩ ≤ 0}, and calculate the next iterate xn+1 = λn PTn (wn − τn Ayn ) + (1 − λn )wn , where {αn } is nondecreasing with α1 = 0 and 0 ≤ αn ≤ α < 1 (α is a fixed real number in (0, 1)), {τn } is a real sequence such that ∑∞ n=1 τn = ∞ and limn→∞ τn = 0, and λ, σ, and δ are three positive real numbers such that δ>

4α[σ + α(α + 1)] 1 − α2

and 0 < λ ≤ λn ≤

δ − 4α[σ + α(α + 1) + 41 αδ] 4δ[σ + α(α + 1) + 41 αδ]

.

Step 2. If xn = yn , then stop. Otherwise, set k ← (k + 1) and return to Step 1. Theorem 3.4.5. Let C be a nonempty, convex, and closed subset of a real Hilbert space H, and let A : H → H be L-Lipschitz continuous and strongly γ-pseudomonotone. Let {xn } be a sequence generated by Algorithm 3.4.8. If VI(C, A), the solution set, is not empty, then the sequence {xn } converges strongly to a solution of variational inequality (3.94). Proof. Let zn = wn − τn Ayn . Fixing p ∈ VI(C, A), we have 󵄩 󵄩2 ‖xn+1 − p‖2 = (1 − λn )‖wn − p‖2 + λn (‖zn − p‖2 + 󵄩󵄩󵄩PTn (zn ) − zn 󵄩󵄩󵄩 + 2⟨PTn (zn ) − zn , zn − p⟩).

In view of p ∈ C ⊂ Tn , we conclude 󵄩 󵄩2 2󵄩󵄩󵄩PTn (zn ) − zn 󵄩󵄩󵄩 + 2⟨zn − p, PTn (zn ) − zn ⟩ = 2⟨PTn (zn ) − zn , PTn (zn ) − p⟩ ≤ 0.

182 | 3 Monotone and pseudocontractive mappings Hence 󵄩2 󵄩2 󵄩 󵄩 −󵄩󵄩󵄩PTn (zn ) − zn 󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩PTn (zn ) − zn 󵄩󵄩󵄩 + 2⟨PTn (zn ) − zn , zn − p⟩. It follows that 󵄩2 󵄩 ‖xn+1 − p‖2 ≤ (1 − λn )‖wn − p‖2 + λn (‖wn − p‖2 − 󵄩󵄩󵄩PTn (zn ) − wn 󵄩󵄩󵄩 + 2τn ⟨Ayn , p − PTn (zn )⟩).

From the fact that p is in VI(C, A), one concludes 0 ≤ ⟨Ap, x − p⟩, ∀x ∈ C. Since A is strongly pseudomonotone, one obtains −γ‖x − p‖2 ≤ ⟨Ax, x − p⟩, ∀x ∈ C. By setting x = yn ∈ C, one has γ‖yn − p‖2 ≤ ⟨Ayn , yn − p⟩. It follows that ⟨Ayn , p − PTn (zn )⟩ ≤ ⟨Ayn , yn − PTn (zn )⟩ − γ‖yn − p‖2 . Using the above two inequalities, we obtain 󵄩2 󵄩 ‖xn+1 − p‖2 ≤ λn ‖wn − p‖2 + (1 − λn )‖wn − p‖2 − λn 󵄩󵄩󵄩PTn (zn ) − wn 󵄩󵄩󵄩 − 2λn τn γ‖yn − p‖2 + 2λn τn ⟨Ayn , yn − PTn (zn )⟩ 󵄩 󵄩2 = ‖wn − p‖2 − λn 󵄩󵄩󵄩PTn (zn ) − yn 󵄩󵄩󵄩 − λn ‖yn − wn ‖2

− 2λn τn γ‖yn − p‖2 + 2λn ⟨wn − yn − τn Ayn , PTn (zn ) − yn ⟩.

From the fact that yn = PC (wn − τn Awn ), one finds that 2λn ⟨PTn (zn ) − yn , wn − yn − τn Ayn ⟩ 󵄩 󵄩2 ≤ λn τn L󵄩󵄩󵄩PTn (zn ) − yn 󵄩󵄩󵄩 + λn τn L‖wn − yn ‖2 . This further yields 󵄩 󵄩2 ‖xn+1 − p‖2 ≤ ‖wn − p‖2 − λn ‖yn − wn ‖2 − λn 󵄩󵄩󵄩PTn (zn ) − yn 󵄩󵄩󵄩 󵄩 󵄩2 + λn τn L󵄩󵄩󵄩PTn (zn ) − yn 󵄩󵄩󵄩 − 2λn τn γ‖yn − p‖2 + λn τn L‖wn − yn ‖2 λ (1 − τn L) 󵄩󵄩 󵄩2 ≤ ‖wn − p‖2 − n 󵄩󵄩PTn (zn ) − wn 󵄩󵄩󵄩 . 2 From the restriction that τn → 0 as n → ∞, we find that there exists n0 ∈ ℕ with 1−τ L 1 for all n ≥ n0 . Hence, 41 ≤ 2n . For all n ≥ n0 , one has τn ≤ 2L ‖xn+1 − p‖2 ≤ ‖wn − p‖2 −

λn 󵄩󵄩 󵄩2 󵄩P (z ) − wn 󵄩󵄩󵄩 . 4 󵄩 Tn n

This further implies 1 󵄩󵄩 󵄩 󵄩󵄩PTn (zn ) − wn 󵄩󵄩󵄩 = ‖wn − xn+1 ‖. λn

3.4 Monotone variational inequalities | 183

Hence, ‖xn+1 − p‖2 ≤ ‖wn − p‖2 −

1 ‖w − xn+1 ‖2 . 4λn n

One also has ‖wn − p‖2 = αn (1 + αn )‖xn − xn−1 ‖2 + (1 + αn )‖xn − p‖2 − αn ‖xn−1 − p‖2 and ‖xn+1 − wn ‖2 ≥ (αn2 − where ρn =

1 . δλn +αn

αn )‖xn − xn−1 ‖2 + (1 − αn ρn )‖xn − xn+1 ‖2 , ρn

This shows that

‖xn+1 − p‖2 ≤ (1 + αn )‖xn − p‖2 − αn ‖xn−1 − p‖2 1 (1 − αn ρn )‖xn − xn+1 ‖2 + γn ‖xn − xn−1 ‖2 , − 4λn where γn = αn (αn + 1) + In view of δ =

1−ρn αn , λn ρn

γn =

1 1 α ( − αn ) > 0. 4λn n ρn

one has 1 1 1 α ( − αn ) + αn (αn + 1) ≤ αδ + α(α + 1). 4λn n ρn 4

Putting Γn = ‖xn − p‖2 + γn ‖xn − xn−1 ‖2 − αn ‖xn−1 − p‖2 , we can obtain Γn+1 − Γn ≤ ‖xn+1 − p‖2 − (αn + 1)‖xn − p‖2

+ αn ‖xn−1 − p‖2 + γn+1 ‖xn+1 − xn ‖2 − γn ‖xn − xn−1 ‖2

≤ (γn+1 + We can prove that γn+1 +

1 (α ρ − 1))‖xn+1 − xn ‖2 . 4λn n n

1 (α ρ 4λn n n

γn+1 +

− 1) ≤ −σ. Since

1 (α ρ − 1) ≤ −σ 4λn n n

⇐⇒ (αn ρn − 1) + 4λn (γn+1 + σ) ≤ 0 ⇐⇒ 4(αn + δλn )(γn+1 + σ) ≤ δ, one has 4(αn + δλn )(γn+1 + σ) ≤ 4[σ + α(α + 1) +

1 αδ](α + δλn ) ≤ δ. 4

184 | 3 Monotone and pseudocontractive mappings Hence, {Γn } is a nonincreasing sequence. In view of −α‖xn−1 − p‖2 ≤ ‖xn − p‖2 − α‖xn−1 − p‖2 ≤ Γn ≤ Γ1 , one has n−1

‖xn − p‖2 ≤ αn ‖x0 − p‖2 + Γ1 ∑ αk ≤ αn ‖x0 − p‖2 + k=1

Γ1 , 1−α

where Γ1 = ‖x1 − p‖2 ≥ 0. In view of the above three inequalities, we obtain that n

σ ∑ ‖xk+1 − xk ‖2 ≤ Γ1 − Γn+1 k=1

≤ Γ1 + α‖xn − p‖2 ≤

Γ1 + αn+1 ‖x0 − p‖2 . 1−α

This further yields that limn→∞ ‖xn − xn+1 ‖ = 0. In view of αn ≤ α, one arrives at ‖wn − xn+1 ‖2 = ‖xn − xn+1 ‖2 + 2αn ⟨xn − xn+1 , xn − xn−1 ⟩ + αn2 ‖xn − xn−1 ‖2 → 0. As in Alvarez and Attouch [4], one derives that limn→∞ ‖xn − p‖2 = l. Hence, limn→∞ ‖wn − p‖2 = l. This further implies 0 ≤ ‖xn − wn ‖ ≤ ‖xn − xn+1 ‖ + ‖xn+1 − wn ‖ → 0 as n → ∞. It follows that limn→∞ ‖wn − yn ‖ = 0 and limn→∞ ‖yn − xn ‖ = 0. Finally, one will show that {xn } converges strongly to p. Note that k

k

n=1

n−1

∑ 2λn τn γ‖yn − p‖2 ≤ α‖xk − p‖2 + ‖x1 − p‖2 + 2α ∑ ‖xn − xn−1 ‖2 ≤ M,

where M > 0 is some constant. It is not hard to find that lim infn→∞ ‖yn − p‖ = 0. Since {yn } is bounded, there exists a subsequence {ynk } of {yn } such that ‖ynk − p‖ → 0 as k → ∞. So, one also has limk→∞ ‖xnk − p‖ = 0. Using limn→∞ ‖xn − p‖2 = l, we obtain limn→∞ ‖xn − p‖ = 0, that is, xn → p as n → ∞. This completes the proof.

3.5 Fixed point theory of pseudocontractive mappings In this subsection, using the conclusions of the previous sections, we prove some fixed point theorems for hemicontinuous pseudocontractive mappings. In the following, we always assume that C is a nonempty, closed, and convex subset of a Hilbert space H. Setting A := I − T, we have the following result. Proposition 3.5.1. Let T : C → H be a mapping. Then the following conclusions hold: (1) T is pseudocontractive if and only if A is monotone;

3.5 Fixed point theory of pseudocontractive mappings | 185

(2) T is η-strongly pseudocontractive if and only if A is a (1 − η)-strongly monotone. (3) T is k-strictly pseudocontractive if and only if A is k-inverse strongly monotone. Proof. (1) T is pseudocontractive ⇐⇒ ⟨Tx − Ty, x − y⟩ ≤ ‖x − y‖2

⇐⇒ ‖x − y‖2 − ⟨Tx − Ty, x − y⟩ ≥ 0

⇐⇒ ⟨(I − T)x − (I − T)y, x − y⟩ ≥ 0

⇐⇒ ⟨Ax − Ay, x − y⟩ ≥ 0,

∀x, y ∈ C.

(2) T is η-strongly pseudocontractive ⇐⇒ ⟨Tx − Ty, x − y⟩ ≤ η‖x − y‖2

⇐⇒ ⟨Tx − Ty, x − y⟩ ≤ ‖x − y‖2 − (1 − η)‖x − y‖2 ⇐⇒ ‖x − y‖2 − ⟨Tx − Ty, x − y⟩ ≥ (1 − η)‖x − y‖2

⇐⇒ ⟨(I − T)x − (I − T)y, x − y⟩ ≥ (1 − η)‖x − y‖2 ⇐⇒ ⟨Ax − Ay, x − y⟩ ≥ (1 − η)‖x − y‖2 ,

∀x, y ∈ C.

(3) The claim follows from the definitions of strictly pseudocontractive mappings and inverse strongly monotone mappings. This completes the proof. Remark 3.5.1. The class of k-strictly pseudocontractive mappings, which is an important class of mappings between the class of nonexpansive mappings and the class of pseudocontractive mappings, is a subclass of Lipschitz continuous mappings. In general, pseudocontractive and strongly pseudocontractive mappings are not necessarily continuous. The above three types of pseudocontractive mappings are independent of each other. Example 3.5.1. Let C = [0, 1] and define a mapping T : C → ℝ by Tx = −2x, ∀x ∈ C. Then T : C → ℝ is 31 -strictly pseudocontractive, but not nonexpansive. Example 3.5.2 (Chidume and Mutangadura [24], 2001). Let H = ℝ2 , C = {x ∈ ℝ2 : ‖x‖ ≤ 1}, C1 = {x ∈ C : ‖x‖ ≤ 21 }, and C2 = {x ∈ C :

1 ≤ ‖x‖ ≤ 1}, 2

∀x = (a, b) ∈ ℝ2 .

Set x⊥ = (b, −a) and define a mapping T : C → C by x + x⊥ ,

Tx = {

x ‖x‖

−x+x , ⊥

x ∈ C1 ,

x ∈ C2 .

Then T is 5-Lipschitz pseudocontractive, but not strictly pseudocontractive.

186 | 3 Monotone and pseudocontractive mappings 2

x Example 3.5.3. Let C = (0, ∞) and define a mapping T : C → C by Tx = 1+x , ∀x ∈ C. Then T is strictly pseudocontractive, but not strongly pseudocontractive.

Example 3.5.4. Let H = ℝ and define a mapping T : H → H by 1, { { { { { {√1 − (x + 1)2 , Tx = { { −√1 − (x − 1)2 , { { { { {−1,

x ∈ (−∞, −1),

x ∈ [−1, 0), x ∈ [0, 1],

x ∈ (1, ∞).

Then T is strongly pseudocontractive, but not strictly pseudocontractive. Based on the results on hemicontinuous monotone mappings, we establish some fixed point theorems for hemicontinuous pseudocontractive mappings. They act as the theoretical basis for iterative methods when finding fixed points of such mappings. To this end, we recall the concepts of weak inward conditions. Definition 3.5.1. (1) A mapping T : C → H is said to satisfy the inward condition (IC) if Tx ∈ IC (x),

∀x ∈ C.

(2) A mapping T : C → H is said to satisfy the weak inward condition (WIC) if Tx ∈ IC (x),

∀x ∈ C,

where IC (x) = {x + λ(u − x) : λ ≥ 0, u ∈ C} and C is called an inward set at x. It is easy to verify the following basic properties: (1) C ⊂ IC (x), ∀x ∈ C; (2) if x ∈ int(C) ≠ 0, then IC (x) = H; (3) if C is a convex set, then IC (x) is also convex and IC (x) = {x + λ(u − x) : λ ≥ 1, u ∈ C}. Let A := I − T. The following proposition provides a close relation between weak inward condition and flow-invariance condition: Proposition 3.5.2 (Caristi [17], 1976). Let C be a nonempty, closed, and convex subset of a Hilbert space H, and let T : C → H be a mapping. Then T satisfies the weak inward condition if and only if A satisfies the flow-invariance condition (FIC): limh→0+ h−1 d((I − hA)x, C) = 0, ∀x ∈ C. Proof. (󳨐⇒) Suppose that x ∈ C and Tx ∈ IC (x). Then, for any ε > 0, there exists y ∈ IC (x) such that ‖y − Tx‖ < ε. Since y ∈ IC (x), we see that there exist u ∈ C and λ0 ≥ 1

3.5 Fixed point theory of pseudocontractive mappings | 187

such that y = x + h−1 (u − x),

∀h ∈ (0, λ0 ]

󳨐⇒ x + h(y − x) = u ∈ C,

∀h ∈ (0, λ0 ]

󵄩 󳨐⇒ h d((I − hA)x, C) ≤ h 󵄩󵄩(1 − h)x + hTx − [x + h(y − x)]󵄩󵄩󵄩 = ‖y − Tx‖ < ε, −1 󵄩 󵄩

−1

that is, (FIC) holds. (⇐󳨐) For any x ∈ C and ε > 0, we find h ∈ (0, 1) and y ∈ C such that 󵄩󵄩 󵄩 󵄩󵄩x − h(Tx − x) − y󵄩󵄩󵄩 ≤ d(x + h(Tx − x), C) + hε 󵄩 󵄩 󳨐⇒ 󵄩󵄩󵄩Tx − [(1 − h−1 )x + h−1 y]󵄩󵄩󵄩 ≤ h−1 d(x + h(Tx − x), C) + ε 󳨐⇒ Tx ∈ IC (x).

This completes the proof. Theorem 3.5.1. Let C be a nonempty, convex, and closed subset of a real Hilbert space H. Let T : C → H be a hemicontinuous pseudocontractive mapping satisfying the weak inward condition (WIC). Then I − T is demiclosed at the origin, that is, if, for any {xn } ⊂ C with xn ⇀ x, xn − Tx → θ as n → ∞, then x ∈ C and x = Tx. Proof. Letting A := I − T, we see that A : C → H is hemicontinuous and monotone. From Proposition 3.5.2, we see that A satisfies the flow-invariance condition (FIC). Thus, by Proposition 3.1.3, we see that A is demiclosed. Hence x ∈ C and Ax = θ 󳨐⇒ x = Tx. This completes the proof. Theorem 3.5.2. Let C be a nonempty, convex, and closed subset of a real Hilbert space H. Let T : C → H be a hemicontinuous η-strongly pseudocontractive mapping satisfying the weak inward condition (WIC). Then T has a unique fixed point in C. Proof. Letting A := I − T, we see that A : C → H is hemicontinuous and (1 − η)-strongly monotone. By Proposition 3.5.2, we see that A satisfies the flow-invariance condition (FIC). Thus it follows from Corollary 3.4.2 that there exists a unique x ∗ ∈ C such that Ax∗ = θ, that is, T has a unique fixed point in C. This completes the proof. Theorem 3.5.3. Let C be a nonempty, convex, and closed subset of a real Hilbert space H. Let T : C → H be a hemicontinuous pseudocontractive mapping satisfying the weak inward condition (WIC). Then Fix(T) ≠ 0. Proof. Letting A := I − T, we see that A : C → H is hemicontinuous and monotone. By Proposition 3.5.2, we see that A satisfies the flow-invariance condition (FIC). It follows from Corollary 3.4.3 that there exists a unique x∗ ∈ C such that Ax ∗ = θ, that is, x ∗ ∈ Fix(T) ≠ 0. This completes the proof. Theorem 3.5.4. Let C be a nonempty, convex, and closed subset of a real Hilbert space H. Let T : C → H be a mapping. Let PC : H → C be the metric projection from H onto C and let A := I − T. Then the following conclusions hold:

188 | 3 Monotone and pseudocontractive mappings (1) Fix(PC T) = VI(C, A); (2) if T satisfies the weak inward invariant condition (WIC), then Fix(PC T) = Fix(T) = Fix(TPC ). Proof. (1) u ∈ Fix(PC T) ⇐⇒ u = PC Tu

⇐⇒ ⟨Tu − u, y − u⟩ ≤ 0, ⇐⇒ ⟨Au, y − u⟩ ≥ 0, ⇐⇒

∀y ∈ C

∀y ∈ C

u ∈ VI(C, A).

(2) We first prove Fix(PC T) = Fix(T). Letting x = PC Tx, we find from (1) that x ∈ VI(C, A) 󳨐⇒ ⟨Ax, y − x⟩ ≥ 0,

∀y ∈ C.

(3.95)

Since Tx ∈ IC (x), there exists {yn } ⊂ IC (x) such that yn → Tx as n → ∞. From the definition of IC (x), we see that there exist un ∈ C and λn ≥ 1 such that yn = x + λn (un − x) for all n ≥ 1. Taking y := un ∈ C in (3.95), we have ⟨Ax, un − x⟩ ≥ 0,

∀n ≥ 1

󳨐⇒ ⟨Ax, λn (un − x)⟩ ≥ 0, 󳨐⇒ ⟨Ax, yn − x⟩ ≥ 0,

∀n ≥ 1

∀n ≥ 1.

(3.96)

Letting n → ∞ in (3.96), we obtain ⟨Ax, Tx − x⟩ ≥ 0 󳨐⇒ −‖Ax‖2 ≥ 0 󳨐⇒ Ax = θ, that is, x = Tx. Hence Fix(PC T) ⊆ Fix(TPC ). It is obvious to see that the inverse is also true. Therefore, we have Fix(PC T) = Fix(T). Next, we prove Fix(T) = Fix(TPC ). Letting x = TPC x, we have PC x = (PC T)PC x, which shows that PC x ∈ Fix(PC T). It follows that TPC x = PC x 󳨐⇒ PC x = x 󳨐⇒ x = Tx. This completes the proof. Corollary 3.5.1. Let C be a nonempty, convex, and closed subset of a real Hilbert space H. If T : C → H is a hemicontinuous pseudocontractive mapping, then Fix(PC T) is closed and convex in H. Furthermore, if T : C → H satisfies the weak inward condition (WIC), then Fix(T) is closed and convex in H. Theorem 3.5.5. Let C be a nonempty, convex, and closed subset of a real Hilbert space H. Let T : C → H be a hemicontinuous pseudocontractive mapping satisfying the weak

3.5 Fixed point theory of pseudocontractive mappings | 189

inward condition (WIC) and Fix(T) ≠ 0. Then, for any u ∈ C, there exists a continuous path {xt }t>0 such that xt → z as t → 0+ , where z = PFix(T) u and {xt } satisfies the following equation: tu + (1 − t)Txt = xt ,

∀t ∈ (0, 1).

(3.97)

Proof. For any t ∈ (0, 1) and u ∈ C, we define a mapping Tt : C → H by Tt (x) = tu + (1 − t)Tx,

∀x ∈ C.

(3.98)

Then Tt : C → H is hemicontinuous and (1 − t)-strongly pseudocontractive, and satisfies the weak inward condition (WIC). So, we conclude from Theorem 3.5.2 that there exists a unique xt ∈ C such that xt = Tt xt = tu + (1 − t)Txt ,

∀t ∈ (0, 1).

Hence (3.97) holds. Next, we prove that the net {xt }t>0 is continuous. To this end, we see from (3.97) that ‖xt − xt0 ‖2 = ⟨xt − xt0 , xt − xt0 ⟩

= ⟨tu + (1 − t)Txt − t0 u − (1 − t0 )Txt0 , xt − xt0 ⟩

= (t − t0 )⟨u − Txt0 , xt − xt0 ⟩ + (1 − t)⟨Txt − Txt0 , xt − xt0 ⟩

≤ |t − t0 |‖u − Txt0 ‖‖xt − xt0 ‖ + (1 − t)‖xt − xt0 ‖2 ,

∀t ∈ (0, 1).

This implies that ‖xt − xt0 ‖ ≤

|t − t0 | ‖u − Txt0 ‖. t

If t → t0 , then xt → xt0 , which verifies the continuity of the path {xt }t>0 . Next, we prove the convergence of {xt }. For any p ∈ Fix(T), it follows from (3.97) and the definition of pseudocontractive mappings that ‖xt − p‖2 = ⟨xt − p, xt − p⟩

= ⟨tu + (1 − t)Txt − p, xt − p⟩

= t⟨u − p, xt − p⟩ + (1 − t)⟨Txt − p, xt − p⟩ ≤ t⟨u − p, xt − p⟩ + (1 − t)‖xt − p‖2 ⇒ ‖xt − p‖2 ≤ ⟨u − p, xt − p⟩,

∀t ∈ (0, 1), p ∈ Fix(T).

(3.99)

In particular, one has ‖xt − p‖ ≤ ‖u − p‖, ∀t ∈ (0, 1), which shows that the net {xt }t>0 is bounded. From (3.97), we see that {Txt } is also bounded. Hence xt − Txt = t(u − Txt ) → θ

(t → 0).

(3.100)

190 | 3 Monotone and pseudocontractive mappings It follows from (3.99) that ⟨xt − p − u + p, xt − p⟩ ≤ 0 ⇒ ⟨xt − u, xt − p⟩ ≤ 0,

∀t ∈ (0, 1), p ∈ Fix(T).

(3.101)

Take a sequence {tn } with tn → ∞ arbitrarily and put xn := xtn . Since H is reflexive and sequence {xn } is bounded in H, we see that there exists a subsequence {xni } ⊂ {xn } such that xnk ⇀ z as k → ∞. It follows from (3.100) that xnk − Txnk → θ as k → ∞. From Theorem 3.5.1, we see that z ∈ C and z = Tz. Using (3.99), we derive ‖xnk − z‖2 ≤ ⟨u − z, xnk − z⟩.

(3.102)

Letting k → ∞ in (3.102), we have xnk → z as k → ∞. It follows from (3.101) that ⟨xnk − u, xnk − p⟩ ≤ 0,

∀p ∈ Fix(T).

(3.103)

Letting k → ∞ in (3.103), we have ⟨z − u, z − p⟩ ≤ 0,

∀p ∈ Fix(T),

which shows that z = PFix(T) u. Hence xn → z as n → ∞. Therefore, it follows that xt → z as t → 0. This completes the proof.

3.6 Iterative methods of fixed points for pseudocontractive mappings and zeros for monotone mappings This section is devoted a discussion of the iterative methods of fixed points of pseudocontractive mappings and zeros of monotone mappings, including the normal Mann’s iterative method, Ishikawa’s iterative method, Bruck regularization iterative method, hybrid projection iterative methods, and Ishikawa–Halpern-type iterative methods. By the transformation A := I − T or T := I − A, every convergence result of fixed points for pseudocontractive mappings implies a convergence result of zeros for monotone mappings. 3.6.1 Normal Mann iterative method For η-strongly pseudocontractive mappings and k-strictly pseudocontractive mappings defined on a Hilbert space H, we can find their fixed points via the normal Mann iterative method. Theorem 3.6.1. Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let T : C → C be a hemicontinuous η-strongly pseudocontractive mapping. Let {αn } be a sequence in [0, 1] satisfying the following conditions: αn → 0

3.6 Iterative methods of fixed points | 191

as n → ∞ and ∑∞ n=1 αn = ∞. Let {xn } be a sequence generated in the following normal Mann iterative method (NMIM): x1 ∈ C and xn+1 = (1 − αn )xn + αn Txn ,

∀n ≥ 1.

Then {xn } converges strongly to the unique fixed point of T. Proof. From Theorem 3.5.2, we see that T has a unique fixed point in C; denote it by q. Note that 󵄩 󵄩2 ‖xn+1 − q‖2 = 󵄩󵄩󵄩(1 − αn )(xn − q) + αn (Txn − q)󵄩󵄩󵄩 = (1 − αn )2 ‖xn − q‖2 + 2αn (1 − αn )⟨Txn − q, xn − q⟩ + αn2 ‖Txn − q‖2 ≤ (1 − αn )2 ‖xn − q‖2 + 2ηαn (1 − αn )‖xn − q‖2 + αn2 M 2 ≤ [1 − (1 − η)αn ]‖xn − q‖2 + αn2 M 2 ,

where M = diam(C). By Lemma 1.10.2, we have xn → q as n → ∞. This completes the proof. Corollary 3.6.1. Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let A : C → H be a hemicontinuous σ-strongly monotone mapping satisfying the condition that T := I − A : C → C is a self-mapping. Let {tn } be a sequence in (0, 1) satisfying the following conditions: tn → ∞ as n → ∞ and ∑∞ n=1 tn = ∞. Let {xn } be a sequence generated in the steepest descent method (SDM): x1 ∈ C and xn+1 = xn − tn Axn ,

∀n ≥ 1.

Then {xn } converges strongly to the unique zero point x∗ of A. Proof. Since T : C → C is hemicontinuous and (1 − σ)-strongly pseudocontractive, one sees that T has a unique fixed point x∗ . So, A has a unique zero point x ∗ . One can rewrite (SDM) as xn+1 = xn − tn (I − T)xn = (1 − tn )xn + tn Txn ,

∀n ≥ 1.

From Theorem 3.6.1, we can derive the conclusion immediately. This completes the proof. Theorem 3.6.2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let T : C → C be a k-strictly pseudocontractive mapping with Fix(T) ≠ 0. Let {αn } be a sequence satisfying the following conditions: αn ∈ (k, 1) and ∑∞ n=1 (αn −k)(1−αn ) = ∞. Let {xn } be a sequence generated in the normal Mann iterative method (NMIM): x1 ∈ C and xn+1 = (1 − αn )xn + αn Txn , ∀n ≥ 1. Then {xn } converges weakly to a fixed point p = limn→∞ PFix(T) (xn ) of T.

192 | 3 Monotone and pseudocontractive mappings Proof. Letting βn =

αn −k , 1−k

we have αn = βn + k(1 − βn ) and 1 − αn = (1 − k)(1 − βn ).

It follows that xn+1 = αn xn + (1 − αn )Txn

= [βn + k(1 − βn )]xn + [(1 − k)(1 − βn )]Txn = βn xn + (1 − βn )[kxn + (1 − k)Txn ],

∀n ≥ 1.

(3.104)

Define a mapping S : C → C by Sx = kx+(1−k)Tx, ∀x ∈ C. Then (3.104) can be rewritten as xn+1 = βn xn + (1 − βn )Sxn ,

∀n ≥ 1.

(3.105)

We know that S : C → C is a nonexpansive mapping. Indeed, 󵄩 󵄩2 ‖Sx − Sy‖2 = 󵄩󵄩󵄩k(x − y) + (1 − k)(Tx − Ty)󵄩󵄩󵄩

󵄩 󵄩2 = k‖x − y‖2 + (1 − k)‖Tx − Ty‖2 − k(1 − k)󵄩󵄩󵄩(I − T)x − (I − T)y󵄩󵄩󵄩 󵄩 󵄩2 ≤ k‖x − y‖2 + (1 − k)‖x − y‖2 + k(1 − k)󵄩󵄩󵄩(I − T)x − (I − T)y󵄩󵄩󵄩 󵄩 󵄩2 − k(1 − k)󵄩󵄩󵄩(I − T)x − (I − T)y󵄩󵄩󵄩 = ‖x − y‖2 .

It is easy to check that Fix(S) = Fix(T). Hence, ∞

∑ βn (1 − βn ) =

n=1

∞ 1 (α − k)(1 − αn ) = ∞. ∑ (1 − k)2 n=1 n

From Reich’s weak convergence theorem, we see that {xn } weakly converges to a fixed point p of S, where p = limn→∞ PFix(T) (xn ). Furthermore, Fix(S) = Fix(T). Hence {xn } weakly converges to a fixed point p of T. This completes the proof. Theorem 3.6.3. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A : C → H be a v-inverse strongly monotone mapping such that A−1 (θ) ≠ 0 and T := I − A : C → C is a self-mapping. Let {tn } be a sequence in (0, 1) satisfying the following conditions: tn ∈ (k, 1) and ∑∞ n=1 (tn −k)(1−tn ) = ∞, where k = 1−2v. Let {xn } be a sequence generated in the steepest descent method (SDM): x1 ∈ C and xn+1 = xn − tn Axn , ∀n ≥ 1. Then {xn } converges weakly to a zero point x∗ of A, where x∗ = lim PA−1 (θ) (xn ). n→∞

Proof. It follows from Proposition 3.5.2 (3) that T : C → C is hemicontinuous and k-strictly pseudocontractive, where k = 1 − 2v, and Fix(T) ≠ 0. Note that the (SDM) method can be rewritten as xn+1 = (1 − tn )xn + tn Txn , ∀n ≥ 1. We can obtain from Theorem 3.6.2 the desired conclusion immediately. This completes the proof.

3.6 Iterative methods of fixed points | 193

Remark 3.6.1. The above theorems show that, in the framework of Hilbert spaces, we can transform k-strictly pseudocontractive mappings into nonexpansive mappings via an appropriate convex combination of the identity mapping and a nonexpansive mapping. Since the normal Mann iterative method is valid for nonexpansive mappings, we see that it is also valid for k-strictly pseudocontractive mappings. However, for more general pseudocontractive mappings, even Lipschitz continuous, the normal Mann iterative method is not valid. In 2001, Chidume and Mutangadura gave a counterexample (see [24]). 3.6.2 Ishikawa iterative method In 1974, Ishkikawa introduced a two-step iterative method for fixed points of Lipschitz pseudocontractive mappings. Lemma 3.6.1 (Ishikawa [32], 1974). Let C be a convex subset of a real Hilbert space H and let T : C → C be an L-Lipschitz pseudocontractive mapping with Fix(T) ≠ 0. Let {αn } and {βn } be two sequences in [0, 1] satisfying the following condition: αn ≤ βn , ∀n ≥ 1. Let {xn } be a sequence generated in the following Ishikawa iterative method (IIM): x1 ∈ C and yn = (1 − βn )xn + βn Txn ,

{

xn+1 = (1 − αn )xn + αn Tyn ,

∀n ≥ 1.

Then, ‖xn+1 − p‖2 ≤ ‖xn − p‖2 − αn βn (1 − 2βn − L2 βn2 )‖xn − Txn ‖2 ,

∀p ∈ Fix(T).

(3.106)

Proof. Fixing p ∈ Fix(T), we have 󵄩 󵄩2 ‖xn+1 − p‖2 = 󵄩󵄩󵄩(1 − αn )(xn − p) + αn (Tyn − p)󵄩󵄩󵄩 = (1 − αn )‖xn − p‖2 + αn ‖Tyn − p‖2 − αn (1 − αn )‖xn − Tyn ‖2 , 2

2

(3.107)

2

‖Tyn − p‖ ≤ ‖yn − p‖ + ‖yn − Tyn ‖

= βn ‖Tyn − p‖2 + (1 − βn )‖xn − p‖2 − βn (1 − βn )‖xn − Txn ‖2 ,

2

+ ‖yn − Tyn ‖2 , 2

(3.108) 2

2

‖yn − Tyn ‖ = βn ‖Txn − Tyn ‖ + (1 − βn )‖xn − Tyn ‖ − βn (1 − βn )‖xn − Txn ‖ ≤ [L2 βn2 − βn (1 − βn )]‖xn − Txn ‖2 + (1 − βn )‖xn − Txn ‖2 ,

(3.109)

and ‖Txn − p‖2 ≤ ‖xn − p‖2 + ‖xn − Txn ‖2 .

(3.110)

194 | 3 Monotone and pseudocontractive mappings Substituting (3.108)–(3.110) into (3.107), we find from condition αn ≤ βn that ‖xn+1 − p‖2 ≤ ‖xn − p‖2 − αn βn (1 − 2βn − L2 βn2 )‖xn − Txn ‖2 . This completes the proof. Theorem 3.6.4 (Zhou [116], 2008). Let C be a closed convex subset of a real Hilbert space H and let T : C → C be an L-Lipschitz pseudocontractive mapping with Fix(T) ≠ 0. Let {αn } and {βn } be two real sequence satisfying the following conditions: (1) lim infn→∞ αn > 0; (2) αn ≤ βn , ∀n ≥ 1; (3) βn ≤ β
0 such that αn βn (1 − 2βn − L2 βn2 ) ≥ c. It follows from (3.111) that c‖xn − Txn ‖2 ≤ ‖xn − p‖2 − ‖xn+1 − p‖2 , ∀p ∈ Fix(T), 󳨐⇒ ‖xn+1 − p‖ ≤ ‖xn − p‖,

∀p ∈ Fix(T).

(3.112)

This shows that {xn } is Fejér monotone. Hence limn→∞ ‖xn −p‖ exists. This implies that {xn } is bounded. Letting n → ∞ in (3.111), we obtain xn − Txn → θ. By Theorem 3.5.1, we have ωω (xn ) ⊂ Fix(T). It follows from Browder’s convergence principle that xn ⇀ p as n → ∞, where p = limn→∞ PFix(T) (xn ). This completes the proof. Corollary 3.6.2. Let C be a closed convex subset of a real Hilbert space H. Let A : C → H be an L-Lipschitz monotone mapping such that A−1 (θ) ≠ 0 and T := I − A is a selfmapping on C. Let {xn } be a sequence generated in the following generalized steepest descent method (GSDM): x1 ∈ C and yn = xn − βn Axn ,

{

xn+1 = xn − αn (xn − Tyn ),

∀n ≥ 1,

where {αn } and {βn } are two sequences in (0, 1) satisfying the following conditions: (1) lim infn→∞ αn > 0;

3.6 Iterative methods of fixed points | 195

(2) αn ≤ βn for all n ≥ 1; (3) βn ≤ β
0; (2) αn ≤ βn , ∀n ≥ 1; 2

−1 (3) βn ≤ β < 1+L for all n ≥ 1; L2 (4) λn → 0 as n → ∞; (5) ∑∞ n=1 λn = ∞. √

Let {xn } be a sequence generated by (HIIM). Then xn → p as n → ∞, where p = PFix(T) u. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a mapping and f : C → C a ρ-strict contraction. Consider x1 ∈ C chosen arbitrarily, { { { { { {yn = (1 − βn )xn + βn Txn , { {zn = (1 − αn )xn + αn Tyn , { { { { {xn+1 = λn f (xn ) + (1 − λn )zn ,

(MHIIM) n ≥ 1,

where {αn }, {βn }, and {λn } are three real sequences in (0, 1) satisfying certain restrictions. We call above iterative method Moudafi–Halpern–Ishikawa iterative method. Theorem 3.6.7. Let C be a closed convex subset of a real Hilbert space H and let T : C → C be an L-Lipschitz pseudocontractive mapping with Fix(T) ≠ 0. Let f : C → C be a ρ-strict contraction. Let {αn }, {βn }, and {λn } be three real sequence in (0, 1) satisfying the following conditions: (1) lim infn→∞ αn > 0;

3.6 Iterative methods of fixed points | 197

(2) αn ≤ βn , ∀n ≥ 1; 2

−1 (3) βn ≤ β < 1+L for all n ≥ 1; L2 (4) λn → 0 as n → ∞; (5) ∑∞ n=1 λn = ∞. √

Let {xn } be a sequence generated by (MHIIM). Then xn → p as n → ∞, where p = PFix(T) f (p), and p is the unique solution of the variational inequality: ⟨(I − f )p, y − p⟩ ≥ 0,

∀y ∈ Fix(T).

As a consequence of the above theorem, we have the following interesting result. Theorem 3.6.8. Let H be a real Hilbert space and let T : H → H be an L-Lipschitz pseudocontractive mapping with Fix(T) ≠ 0. Let A : H → H be a k-Lipschitz continuous and η-strongly monotone operator. Let {αn }, {βn }, and {λn } be three real sequence in (0, 1) satisfying the following conditions: (1) lim infn→∞ αn > 0; (2) αn ≤ βn , ∀n ≥ 1; 2

−1 for all n ≥ 1; (3) βn ≤ β < 1+L L2 (4) λn → 0 as n → ∞; (5) ∑∞ n=1 λn = ∞. √

2η

For μ ∈ (0, k2 ) and initial value x1 ∈ H chosen arbitrarily, define a sequence {xn } iteratively in H by (MHIIM) with f (x) = (I − μA)x. Then xn → x ∗ as n → ∞, where x∗ = PFix(T) f (x∗ ), and x∗ is the unique solution of the monotone variational inequality: ⟨Ax∗ , x − x∗ ⟩ ≥ 0,

∀x ∈ Fix(T).

3.6.3 Bruck regularization iterative method Note that there is a strong restriction that the domain of the mapping is compact in Ishikawa convergence theorem (Theorem 3.6.5). Is there an iterative method that does not require the compactness so that the sequence generated by the Ishikawa iteration method is still convergent? The answer is positive. In 1974, Bruck [11] introduced the following regularization iteration method.

198 | 3 Monotone and pseudocontractive mappings Let A ⊂ H × H be a mapping. Let x1 be an arbitrary initial point in Dom(A) and let z be a fixed element in H. Let {xn } be a sequence generated by xn+1 = xn − λn (vn + θn (xn − z)),

n ≥ 1, vn ∈ Axn ,

(BRIM-1)

where {λn } and {θn } are two nonnegative real number sequences satisfying certain conditions. Then (BRIM-1) is called the Bruck regularization iterative method. Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let T : C → C be a mapping. Let x1 be an arbitrary initial point in C and let z be a fixed element in H. Let {xn } be a sequence generated by xn+1 = [1 − λn (1 + θn )]xn + λn Txn + λn θn z,

n ≥ 1,

(BRIM-2)

where {λn } and {θn } are two nonnegative real number sequences satisfying some certain conditions. Bruck regularization iterative methods are motivated at least by the following two aspects. One is the steepest descent method, which is for zeros of η-strongly monotone mappings, and the other is the Halpern iterative method, which is for fixed points of nonexpansive mappings. For η-strongly monotone mappings, steepest descent methods are effective, however, they are not available for general monotone mappings. Thanks to the fact that the perturbations A + θn I of A are ηn -strongly monotone, it is natural to replace A in the steepest descent method with A + θn I. This leads to (BRIM-1). On the other hand, for a maximal monotone mapping A ⊂ H × H if θ ∈ Ran(A), then the resolvent Jt of A has the following asymptotic property: limt→∞ Jt x = PA−1 (θ) x, ∀x ∈ H. Letting tn → ∞ and θn = tn−1 → 0, we find that there exists a unique yn ∈ Dom(A) such that θ ∈ θn yn + Ayn ,

n≥1

(3.113)

and yn → x ∗ = PA−1 (θ) θ (n → ∞). If we can find a subsequence {xnj } of {xn } such that xnj → x ∗ and prove that xn − ynj → θ (j → ∞), then we can prove xn → x ∗ (n → ∞). Therefore, it is essential to choose {λn } and {θn } to guarantee that sequence {xn } has a subsequence {xnj } which strongly converges to x∗ . To this end, Bruck [11] introduced the definition of admissible pairs. Definition 3.6.1. A pair of nonnegative sequences {λn } and {θn } is said to be an admissible pair if the following conditions hold: (1) θn > 0 and {θn } monotonically and decreasingly converges to 0; (2) There exists a strictly increasing subsequence {n(i)}∞ i=1 satisfying: n(i+1) lim infi→∞ θn(i) ∑n(i+1) λ > 0, lim [θ − θ = 0, and i→∞ n(i) n(i+1) ] ∑j=n(i) λj j=n(i) j 2 limi→0 ∑n(i+1) j=n(i) λj = 0.

3.6 Iterative methods of fixed points | 199

Examples of admissible pairs are as follows: 1 (1) λn = n1 , θn = log log , n ≥ 2, n(i) = ii ; n 1

1

(2) λn = n− 2 , θn = n− 4 , n ≥ 1, n(i) = i6 .

Bruck [11] first established the convergence of method (BRIM-1). As a direct consequence, the convergence of method (BRIM-2) can be derived. Next, we first prove the convergence of method (BRIM-2). As a corollary, the convergence of method (BRIM-1) can be easily obtained. Theorem 3.6.9. Let C be a nonempty, bounded, closed, and convex subset of a Hilbert space H. Let T : C → C be a single-valued hemicontinuous pseudocontractive mapping. Then T has at least one fixed point in C. Let {λn } and {θn } be a admissible pair and λn (1 + θn ) ≤ 1, ∀n ∈ N. Let {xn } be a sequence generated by (BRIM-2). Then {xn } converges strongly to p = PFix(T) z. Proof. From Theorem 3.5.3, we see that Fix(T) ≠ 0. It follows from Corollary 3.5.1 that θ Fix(t) is a closed and convex subset of H. Hence PFix(T) z is well defined. Let ti = 1+θi → i 0 as i → ∞. Using Theorem 3.5.5, we see that there exists a unique yi ∈ C such that yi = (1 − ti )Tyi + ti z,

i ≥ 1,

(3.114)

and yi → x∗ = PFix(T) z as i → ∞. Letting A := I − T, we find from (3.114) that θi (z − yi ) = Ayi ,

i ≥ 1.

(3.115)

Note that (BRIM-2) can be rewritten as (BRIM-1): xn+1 = xn − λn [Axn + θn (xn − z)],

i ≥ 1.

(3.116)

For n ≥ i ≥ 2, one has xn − yi = xn−1 − yi − λn−1 [Axn−1 + θn−1 (xn−1 − z)],

(3.117)

which implies that ‖xn − yi ‖2 = ‖xn−1 − yi ‖2 − 2λn−1 ⟨Axn−1 + θn−1 (xn−1 − z), xn−1 − yi ⟩ 󵄩 󵄩2 2 󵄩 + λn−1 󵄩󵄩Axn−1 + θn−1 (xn−1 − z)󵄩󵄩󵄩 = ‖xn−1 − yi ‖2 + 2λn−1 (θi − θn−1 )⟨xn−1 − z, xn−1 − yi ⟩ − 2λn−1 ⟨Axn−1 + θi (xn−1 − z), xn−1 − yi ⟩ 2 󵄩 󵄩󵄩Axn−1 + θn−1 (xn−1 − z)󵄩󵄩󵄩2 . + λn−1 󵄩 󵄩

(3.118)

From (3.115) and the monotonicity of A, we find that ⟨Axn−1 + θi (z − yi ), xn−1 − yi ⟩ ≥ 0.

(3.119)

200 | 3 Monotone and pseudocontractive mappings Hence, ⟨Axn−1 + θi (xn−1 − z), xn−1 − yi ⟩

= ⟨Axn−1 + θi (z − yi ), xn−1 − yi ⟩ + ⟨θi (z − yi ) + θi (xn−1 − z), xn−1 − yi ⟩ ≥ θi ‖xn−1 − yi ‖2 .

(3.120)

Substituting (3.120) into (3.118), we obtain ‖xn − yi ‖2 ≤ (1 − 2λn−1 θi )‖xn−1 − yi ‖2

+ 2λn−1 (θi − θn−1 )⟨xn−1 − z, xn−1 − yi ⟩ 󵄩2 2 󵄩 + λn−1 󵄩󵄩󵄩Axn−1 + θn−1 (xn−1 − z)󵄩󵄩󵄩 .

(3.121)

Since C is bounded, we find that {xn } is bounded, and so is {Axn }. There exists a constant M > 0 such that ∀n ≥ i ≥ 2, 󵄩 󵄩2 2⟨xn−1 − z, xn−1 − yi ⟩ ≤ M, 󵄩󵄩󵄩Axn−1 + θn−1 (xn−1 − z)󵄩󵄩󵄩 ≤ M. In view of 1 − 2λn−1 θi ≤ exp{−2λn−1 θi }, we find from (3.121) that 2 ‖xn − yi ‖2 ≤ exp(−2λn−1 θi )‖xn−1 − yi ‖2 + Mλn−1 (θi − θn−1 ) + Mλn−1 .

(3.122)

It follows that n−1

n−1

n−1

j=i

j=i

j=i

‖xn − yi ‖2 ≤ exp(−2θi ∑ λj )‖xi − yi ‖2 + M ∑ (θi − θj )λj + M ∑ λj2 . If j ≤ n, θi − θj ≤ θi − θn , then n−1

n

n

j=i

j=i

j=i

‖xn − yi ‖2 ≤ exp(−2θi ∑ λj )‖xi − yi ‖2 + M(θi − θn ) ∑ λj + M ∑ λj2 .

(3.123)

From (3.123), we first prove that xn(k) → x ∗ as k → ∞. In fact, letting i = n(k) and n = n(k + 1) in (3.123), one has n(k+1)

‖xn(k+1) − yn(k) ‖2 ≤ exp(−2θn(k) ∑ λj ) exp(2θn(k) λn(k+1) )‖xn(k) − yn(k) ‖2 j=n(k)

n(k+1)

n(k+1)

j=n(k)

j=n(k)

+ M(θn(k) − θn(k+1) ) ∑ λj + M ∑ λj2 . From the definition of admissible pairs, one concludes that λn → 0. Thus exp(θn(k) λn(k) ) → 1.

(3.124)

3.6 Iterative methods of fixed points | 201

Let r ∈ (0, 1) such that lim sup ‖xn(k+1) − yn(k) ‖2 ≤ r lim sup ‖xn(k) − yn(k) ‖2 . k→∞

k→∞

(3.125)

In view of limk→∞ yn(k) = x∗ , one has 󵄩 󵄩2 lim sup ‖xn(k+1) − yn(k) ‖2 = lim sup󵄩󵄩󵄩xn(k+1) − x ∗ 󵄩󵄩󵄩 k→∞

k→∞

= lim sup ‖xn(k) − yn(k) ‖2 . k→∞

(3.126)

Combining (3.125) and (3.126), and using the fact that r ∈ (0, 1), we claim that lim supk→∞ ‖xn(k) − x∗ ‖2 = 0, that is, xn(k) → x∗ (k → ∞). Now, we are in a position to show that xn − yn(k) → θ. To this end, ∀n > n(1), pick a k such that n(k) ≤ n ≤ n(k + 1) and let i = k in (3.123). It follows that n

n

‖xn − yn(k) ‖2 ≤ ‖xn(k) − yn(k) ‖2 + M(θn(k) − θn ) ∑ λj + M ∑ λj2 j=n(k)

j=n(k)

n(k+1)

n(k+1)

j=n(k)

j=n(k)

≤ ‖xn(k) − yn(k) ‖2 + M(θn(k) − θn(k+1) ) ∑ λj + M ∑ λj2 .

(3.127)

From the definition of admissible pairs, one obtains xn(k) → x ∗ as k → ∞. Note that k → ∞ if n → ∞. Letting n → ∞ in (3.127), we obtain xn −yn(k) → θ. Hence, yn(k) → x∗ . It follows that xn → x∗ as n → ∞. This completes the proof. Theorem 3.6.10. Let A ⊂ H × H be a maximal monotone mapping and θ ∈ Ran(A). Let {λn } and {θn } be an admissible pair and z ∈ H. Let {xn } ⊂ Dom(A) be a sequence generated by (BRIM-1). If {xn } and {vn } are bounded, then {xn } converges strongly to x∗ ∈ A−1 (θ) and x ∗ = PA−1 (θ) (z). Proof. Letting ti = θi−1 , x = z in Proposition 3.1.4 (2), we have limi→∞ Jti z = PA−1 (θ) (z). Without loss of generality, we may assume z = θ (otherwise, we consider A󸀠 (⋅) = A󸀠 (⋅ − z)). Putting yi = Jti θ, we have yi ⊂ Dom(A), yi → PA−1 (θ) as i → ∞. {yi } is the unique solution satisfying equation θ = θi yi +Ayi (i = 1, 2, . . . ). From the proof of Theorem 3.6.9, we can obtain the desired conclusion immediately. This completes the proof. From the examples of admissible pairs given by Bruck, we see that his choice is difficult. As mentioned above, n(i) = ii . Is it possible to give control sequences that are relatively easier to choose? The answer is positive! The following theorem gives another choice that guarantees the convergence of (BRIM-2). Theorem 3.6.11. Let C be a nonempty, bounded, closed, and convex subset of a Hilbert space H, and let T : C → C be a single-valued hemicontinuous pseudocontractive mapping. Then T has at least one fixed point. Suppose that the nonnegative real number sequences {λn } and {θn } satisfy the following conditions:

202 | 3 Monotone and pseudocontractive mappings (1) (2) (3) (4) (5)

λn (1 + θn ) ≤ 1 for all n ≥ 1; θn → 0 as n → ∞ (not necessarily monotonically decreasing); ∑∞ n=1 λn θn = ∞; θn−1 − 1 = o(λn θn ); θn λn = o(θn ).

Let {xn } be the sequence generated by (BRIM-2). Then {xn } converges strongly to p = PFix(T) (z). Proof. From Theorem 3.5.3, we see that Fix(T) ≠ 0. From Corollary 3.5.1, we see that θ Fix(T) is closed and convex. Hence PFix(T) (z) is well defined. Letting tn = 1+θn , one has n tn → 0 as n → ∞. From Theorem 3.5.5, there exists a unique {yn } ⊂ C such that yn = tn z + (1 − tn )Tyn ,

∀n ≥ 1,

(3.128)

and yn → p = PFix(T) (z) as n → ∞. From (3.128), we have (1 + θn )yn = θn z + Tyn ,

∀n ≥ 1.

(3.129)

It follows that λn (1 + θn )yn = λn θn z + λn Tyn ,

∀n ≥ 1.

(3.130)

If we can prove xn −yn → θ as n → ∞, then xn → p as n → ∞, which yields the desired conclusion. Therefore, we only need to prove xn − yn → θ as n → ∞. First, we estimate ‖yn − yn−1 ‖. From (3.129) and the definition of pseudocontractive mappings, we derive ‖yn − yn−1 ‖2 = ⟨yn − yn−1 , yn − yn−1 ⟩

= ⟨θn (z − yn ) − θn−1 (z − yn−1 ) + Tyn − Tyn−1 , yn − yn−1 ⟩

= ⟨θn (z − yn−1 ) + θn (yn − yn−1 ) − θn−1 (z − yn−1 ), yn − yn−1 ⟩ + ⟨Tyn − Tyn−1 , yn − yn−1 ⟩

= (θn − θn−1 )⟨z − yn−1 , yn − yn−1 ⟩

− θn ‖yn − yn−1 ‖2 + ⟨Tyn − Tyn−1 , yn − yn−1 ⟩

≤ (θn − θn−1 )⟨z − yn−1 , yn − yn−1 ⟩ + (1 − θn )‖yn − yn−1 ‖2

≤ |θn − θn−1 |‖z − yn−1 ‖‖yn − yn−1 ‖ + (1 − θn )‖yn − yn−1 ‖2 .

Then ‖yn − yn−1 ‖2 ≤

|θn − θn−1 | ‖z − yn−1 ‖‖yn − yn−1 ‖. θn

This implies that ‖yn − yn−1 ‖ ≤

|θn − θn−1 | |θ − θn−1 | ‖z − yn−1 ‖ ≤ M n , θn θn

(3.131)

3.6 Iterative methods of fixed points | 203

where M = sup{‖z − yn−1 ‖ : n ≥ 1}. For (BRIM-2) and (3.130), we have xn+1 − yn = [1 − λn (1 + θn )](xn − yn ) + λn (Txn − Tyn ).

(3.132)

Therefore, 2

‖xn+1 − yn ‖2 ≤ [1 − λn (1 + θn )] + 2λn [1 − λn (1 + θn )]

+ ⟨xn − yn , Txn − Tyn ⟩ + λn2 ‖Txn − Tyn ‖2 2

≤ {[1 − λn (1 + θn )] + 2λn [1 − λn (1 + θn )]}‖xn − yn ‖2 + Dλ2 ≤ (1 − 2λn θn )‖xn − yn ‖2 + Dλ2 ,

∀n ≥ 1,

(3.133)

where D = sup{‖Txn − Tyn ‖2 }. Since ‖xn − yn ‖ ≤ ‖xn − yn−1 ‖ + ‖yn − yn−1 ‖, we find from (3.131) that ‖xn − yn ‖2 ≤ ‖xn − yn−1 ‖2 + 2‖xn − yn−1 ‖‖yn − yn−1 ‖ + ‖yn − yn−1 ‖2 ≤ ‖xn − yn−1 ‖2 + M1

|θn − θn−1 | . θn

(3.134)

Substituting (3.134) into (3.133), we obtain ‖xn+1 − yn ‖2 ≤ (1 − 2λn θn )‖xn − yn−1 ‖2 + M1

|θn − θn−1 | + Dλn2 . θn

(3.135)

From conditions (2)–(5), we see that (3.135) can be rewritten as ‖xn+1 − yn ‖2 ≤ (1 − 2λn θn )‖xn − yn−1 ‖2 + o(λn θn ).

(3.136)

Setting an = ‖xn − yn−1 ‖2 and tn = 2λn θn , we see that (3.136) can be rewritten as an+1 ≤ (1 − tn )an + o(tn ), where {tn } is a sequence in [0, 1] satisfying ∑∞ n=1 tn = ∞. It follows from Lemma 1.10.2 that an → 0 as n → ∞, that is, xn → p = PFix(T) (z) as n → ∞. This completes the proof. The control sequences satisfying the above theorem are: λn =

1 , (n + 1)a

θn =

1 , (n + 1)b

where 0 < b < a and a + b < 1. It is obvious that the above two sequences {λn } and {θn } are relatively simple.

204 | 3 Monotone and pseudocontractive mappings Corollary 3.6.4. Let C be a nonempty, bounded, closed, and convex subset of a Hilbert space H. Let A : C → H be a monotone hemicontinuous mapping such that T = I − A : C → C is a self-mapping. Suppose that nonnegative real number sequences {λn } and {θn } satisfy the conditions of Theorem 3.6.11. Let {xn } be a sequence generated in (BRIM-1). Then {xn } converges strongly to a zero point x∗ = PA−1 (θ) (z) of A. Proof. It is clear that T : C → C is a hemicontinuous pseudocontractive mapping. From Theorem 3.5.3 we see that Fix(T) ≠ 0. It follows that A−1 (θ) ≠ 0. Using Corollary 3.5.1, we find that Fix(T) is closed and convex. Hence A−1 (θ) is nonempty, closed, and convex. This shows that PA−1 (θ) (z) is well defined. From T = I − A, one sees that (BRIM-1) can be rewritten as (BRIM-2). Thus the assumptions of Theorem 3.6.11 all are satisfied. From Theorem 3.6.11, we see that {xn } converges strongly to a fixed point x∗ = PFix(T) z = PA−1 (θ) z, that is, x∗ ∈ A−1 (θ). This completes the proof. The following theorem removes the boundedness requirement of set C, however, the price is the continuity of T. Theorem 3.6.12. Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let T : C → C be an L-Lipschitz pseudocontractive mapping such that Fix(T) ≠ 0. Let {xn } be the sequence generated by (BRIM-2), where {λn } and {θn } satisfy conditions (1)–(5) of Theorem 3.6.11. Then {xn } converges strongly to a fixed point x∗ = PFix(T) (z) of T. Proof. It follows from (BRIM-2) and (3.130) that 󵄩 󵄩2 ‖xn+1 − yn ‖2 = 󵄩󵄩󵄩[1 − λn (1 + θn )](xn − yn ) + λn (Txn − Tyn )󵄩󵄩󵄩 2

= [1 − λn (1 + θn )] ‖xn − yn ‖2 + λn2 ‖Txn − Tyn ‖2 + 2λn [1 − λn (1 + θn )]⟨xn − yn , Txn − Tyn ⟩ 2

≤ {[1 − λn (1 + θn )] + 2λn [1 − λn (1 + θn )] + L2 λn2 }‖xn − yn ‖2 ≤ [1 − 2λn θn + (4 + L2 )λn2 ]‖xn − yn ‖2 ≤ (1 − λn θn )‖xn − yn ‖2 .

For large enough n ≥ 1, we have ‖xn+1 − yn ‖ ≤ √1 − λn θn ‖xn − yn ‖ 1 ≤ (1 − λn θn )‖xn − yn ‖. 2 Using (3.131), we obtain 1 ‖xn+1 − yn ‖ ≤ (1 − λn θn )‖xn − yn−1 ‖ + ‖yn − yn−1 ‖ 2 󵄨󵄨 θ 󵄨󵄨 1 󵄨 󵄨 ≤ (1 − λn θn )‖xn − yn−1 ‖ + M 󵄨󵄨󵄨 n−1 − 1󵄨󵄨󵄨 2 󵄨󵄨 θn 󵄨󵄨 1 = (1 − λn θn )‖xn − yn−1 ‖ + o(λn θn ). 2

3.7 Remarks | 205

By Lemma 1.10.2, we derive that xx+1 − yn → θ as n → ∞. Hence xn → PFix(T) (z) as n → ∞. This completes the proof. Corollary 3.6.5. Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let A : C → H be an L-Lipschitz monotone mapping such that A−1 (θ) ≠ 0 and T = I − A : C → C is a self-mapping. Let {xn } be a sequence generated by (BRIM-1), where {λn } and {θn } satisfy conditions (1)–(5) of Theorem 3.6.11. Then {xn } converges strongly to a zero point x∗ = PA−1 (θ) (z) of A. Proof. Note that T : C → C is (L+1)-Lipschitz pseudocontractive with Fix(T) = A−1 (θ) ≠ 0 and {xn } can be rewritten as (BRIM-2). All the conditions of Theorem 3.6.12 are satisfied. Hence {xn } converges strongly to a fixed point x∗ = PFix(T) (z) = PA−1 (θ) (z), which is a zero point of A. This completes the proof.

3.7 Remarks (1) Theorem 3.3.5 is due to Rochafellar [77]. It establishes the connection between variational inequalities and maximal monotone operators, and it is a powerful tool for the solvability of monotone variational inequalities. (2) Theorem 3.4.1 is due to the authors [127]. It establishes the solvability of a class of variational inequalities with hemicontinuous strongly monotone operators and it is a powerful tool for iterative solutions of monotone variational inequalities. (3) Theorems 3.4.2 and 3.4.3 are both due to the authors [127]. Theorem 3.6.8 is due to the author of [116]. It improves the convergence theorem established by Ishikawa [32] in 1974. (4) Theorem 3.6.9 improves the convergence theorem established by Bruck [11] in 1974. It weakens the continuity assumption. Theorem 3.6.11 seems to be a new result compared with the results of Chidume and Zegeye [25]. And the requirement on the continuity of mappings is also reduced.

3.8 Exercises We use H to denote a real Hilbert space. 2 1. Show that T : [0, 1] → ℝ defined by Tx = 1 − x 3 is a continuous pseudocontractive mapping, which is not nonexpansive. 2. Let K be a closed, bounded, and convex subset of H, and T : K → K a k-strictly pseudocontractive mapping. Show that there exist values of λ ∈ (0, 1) such that the averaged mapping Tλ (x) = (1 − λ)x + λTx, is nonexpansive.

x ∈ K,

206 | 3 Monotone and pseudocontractive mappings 3.

Let K be a closed, bounded, and convex subset of H, T : K → K a k-strictly pseudocontractive mapping, and let {αn } be a sequence of real numbers satisfying the following conditions: (i) αn ∈ (0, 1) for all n ≥ 1; (ii) ∑∞ n=1 αn = ∞, and (iii) limn→∞ αn = α < 1 − k. Show that the Mann iteration method generated from an arbitrary x1 ∈ K by xn+1 = (1 − αn )xn + αn Txn ,

n ≥ 1,

converges weakly to a fixed point of T. 4. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and T : C → C be k-Lipschitz continuous and quasi-pseudocontractive. Show that Fix(T) is a nonempty, closed, and convex subset of C. 5. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and T : C → C be a demicontinuous pseudocontraction. Show that Fix(T) is a nonempty, closed, and convex subset of C, and I − T is demiclosed at zero. 6. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and T : C → C be a hemicontinuous pseudocontraction. Show that F(T) is a nonempty, closed, and convex subset of C, and I − T is demiclosed at zero. 7. Let T : H → H be mapping. Write A := I − T. Prove that T is pseudocontractive ⇐⇒ A is monotone ⇐⇒ ‖Tx − Ty‖2 ≤ ‖x − y‖2 + ‖Ax − Ay‖2 , ∀x, y ∈ H. 8. Let T : H → H be mapping. Prove that if T is k-strictly pseudocontractive, then T is Lipschitz continuous and pseudocontractive. 9. Let C be a nonempty, closed, and convex subset of H. Then T : C → C is said to be hemicontractive if ‖Tx − p‖2 ≤ ‖x − p‖2 + ‖x − Tx‖2 for all x ∈ H and all p ∈ Fix(T). Prove that if T : C → C is hemicontinuous and hemicontractive, then Fix(T) is a nonempty, closed, and convex subset of H. 10. Let C be a nonempty, closed, and convex subset of H, and T : C → C be an L-Lipschitz pseudocontraction such that Fix(T) ≠ 0. Let {αn }, {βn }, and {λn } be three sequences of numbers in (0,1) satisfying the following conditions: 2

−1 (1) 0 < a ≤ αn ≤ βn < b < 1+L , for all n ≥ 1; L ∞ (2) λn → 0 (n → ∞) and ∑n=1 λn = ∞. For an arbitrary initial value x1 ∈ C and a fixed anchor u ∈ C, define a sequence {xn } iteratively in C by √

x1 , u ∈ C, { { { { { {yn = (1 − βn )xn + βn Txn , { { un = (1 − αn )xn + αn Tyn , { { { { {xn+1 = λn u + (1 − λn )un , n ≥ 1.

(IHIM)

Prove that the sequence {xn } defined by (IHIM) converges strongly to a fixed point of T, x∗ = PFix(T) u.

3.8 Exercises | 207

11. In the above method (IHIM), if u is replaced by a contraction f : C → C, that is, u is replaced by f (xn ), then the corresponding sequence converges strongly to a fixed point of T, x∗ = PFix(T) f (x∗ ). Try to prove it. 12. Let A : H → H be an η-strongly monotone and k-Lipschitz mapping. Let T : H → H be an L-Lipschitz continuous pseudocontractive mapping such that Fix(T) ≠ 0. 2η Let {αn }, {βn }, and {λn } be the same as in Exercise 10. For μ ∈ (0, k2 ), define a sequence {xn } iteratively in H by x1 ∈ H, { { { { { {yn = (1 − βn )xn + βn Txn , { { {un = (1 − αn )xn + αn Tyn , { { { {xn+1 = λn (I − μA)xn + (1 − λn )un ,

(VIHIM) n ≥ 1.

Prove that the sequence {xn } defined by (VIHIM) converges strongly to the unique solution x∗ of the variational inequality: ⟨Ax, y − x⟩ ≥ 0,

∀y ∈ Fix(T).

13. Let A, T, {αn }, {βn }, {λn }, and μ be the same as in Exercise 12. Define a sequence {xn } iteratively by x1 ∈ H, { { { { { {yn = (1 − βn )xn + βn Txn , { {un = (1 − αn )xn + αn Tyn , { { { { {xn+1 = (I − μA)un , n ≥ 1.

(HSDM)

Prove that the sequence {xn } defined by (HSDM) converges strongly to the unique solution x∗ of the variational inequality: ⟨Ax, y − x⟩ ≥ 0,

∀y ∈ Fix(T).

14. Try to extend the result of Exercise 13 to a finite family of L-Lipschitz continuous pseudocontractive mappings such that K = ⋂ri=1 Fix(Ti ) ≠ 0. 15. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and A : C → H be an L-Lipschitz and monotone mapping such that the solution SOL(C, A) of the variational inequality VI(C, A) is nonempty. Define a sequence {xn } ⊂ C by x1 ∈ C, { { { yn = PC (xn − λAxn ), { { { {xn+1 = PC (xn − λAyn ),

(KIM) n ≥ 1,

where λ ∈ (0, 1/L) and PC denotes the metric projection from H onto C. Prove that the sequence {xn } defined by (KIM) converges weakly to some point x∗ = lim PSOL(C,A) xn . n→∞

208 | 3 Monotone and pseudocontractive mappings 16. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and A : C → H be an L-Lipschitz continuous and monotone mapping such that the solution SOL(C, A) ≠ 0. Also ∀x1 , u ∈ C, define a sequence {xn } ⊂ C by x1 , u ∈ C, { { { { { {yn = PC (xn − τAxn ), { { {zn = PC (xn − τAyn ), { { { {xn+1 = λn u + (1 − λn )zn ,

(MEGIM)1 n ≥ 1,

where τ ∈ (0, 1/L) and {λn } satisfies the following conditions: (C1) λn → 0 (n → ∞), and (C2) ∑∞ n=1 λn = ∞. Prove that the sequence {xn } defined by (MEGIM)1 converges strongly to some point x ∗ = PSOL(C,A) (u). 17. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and A : C → H be an L-Lipschitz continuous and monotone mapping such that the solution SOL(C, A) ≠ 0. Let S : C → C be a nonexpansive mapping such that K = Fix(S) ∩ SOL(C, A) ≠ 0. Also ∀x1 ∈ C, define a sequence {xn } iteratively in C by x1 ∈ C, { { { { { {yn = PC (xn − τAxn ), { {zn = PC (xn − τAyn ), { { { { {xn+1 = αn xn + (1 − αn )Szn ,

(MEGIM)2 n ≥ 1,

where τ ∈ (0, 1/L) and {αn } is a sequence of numbers satisfying the following condition: (C1) αn ∈ [a, b] for a, b ∈ (0, 1). Prove that the sequence {xn } defined by (MEGIM)2 converges weakly to some point x∗ = lim PK (xn ). n→∞

18. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and A : C → H be an L-Lipschitz continuous and monotone mapping such that the solution SOL(C, A) ≠ 0. Let S : C → C be a nonexpansive mapping such that K = Fix(S) ∩ SOL(C, A) ≠ 0. Also ∀x1 , u ∈ C, define a sequence {xn } iteratively in C by x1 , u ∈ C, { { { { { {yn = PC (xn − τAxn ), { { zn = PC (xn − τAyn ), { { { { {xn+1 = αn u + (1 − αn )Szn ,

(MEGIM)1 n ≥ 1,

where τ ∈ (0, 1/L) and {αn } is a sequence of numbers satisfying the following conditions: (C1) αn → 0 (n → ∞) and (C2) ∑∞ n=1 αn = ∞. Prove that the sequence {xn } defined by (MEGIM)1 converges strongly to a specific point x ∗ = PK u.

4 Fixed point theory and iterative methods for fixed points of nonexpansive mappings in Banach spaces In this chapter, we present some known fixed point theorems in the framework of general Banach spaces. We focus on the generalized Browder fixed point theorem proposed by Caristi in 1976. We also discuss the normal Mann iterative method and the Halpern iterative method for fixed points of nonexpansive mappings.

4.1 Several celebrated fixed point theorems Theorem 4.1.1 (Banach–Caristi [17], 1976). Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a contractive mapping satisfying the weak inward condition (WIC). Then T has a unique fixed point in C. If, in addition, T is a self-mapping on C, then, for any initial point x1 ∈ C, the Banach–Picard iterative sequence {T n x1 } converges to the fixed point. In 1969, Meir and Keeler improved the above result and established the following theorem. Theorem 4.1.2 (Meir–Keeler [50], 1969). Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be an MK contraction, that is, for any ϵ > 0, there exists δ > 0 such that ϵ ≤ ‖x − y‖ < ϵ + δ 󳨐⇒ ‖Tx − Ty‖ < ϵ,

∀x, y ∈ C.

(4.1)

Then T has a unique fixed point in C. Remark 4.1.1. (i) It is easy to check that (4.1) is equivalent to the following fact: for any ϵ > 0, there exists δ > 0 such that ‖x − y‖ < ϵ + δ 󳨐⇒ ‖Tx − Ty‖ < ϵ,

∀x, y ∈ C.

(4.2)

(ii) An MK mapping is a mapping in-between contractions and nonexpansive mappings. (iii) Theorem 4.1.2 may be still valid for the MK type nonself-mappings that satisfy the weak inward condition (WIC). In 1965, Browder and Göhde obtained the following known fixed point theorem independently. In the same year, Kirk also proved a fixed point theorem in reflexive Banach spaces with normal structures. In 1976, Caristi extended Browder–Göhde fixed point theorem from self-mappings to nonself-mappings. https://doi.org/10.1515/9783110667097-004

210 | 4 Nonexpansive mappings in Banach spaces Theorem 4.1.3 (Browder–Göhde–Caristi [17], 1976). Let E be a real uniformly convex Banach space and let C be a nonempty, bounded, closed, and convex subset of E. Let T : C → E be a nonexpansive mapping satisfying the weak inward condition (WIC). Then T has a fixed point in C. Moreover, Fix(T) = {x ∈ C : Tx = x} is nonempty, closed, and convex. Proof. Fixing u ∈ C, for any n ≥ 1, we define a mapping Tn : C → E by Tn x =

1 n u+ Tx, 1+n 1+n

∀x ∈ C.

(4.3)

Then, for any n ≥ 1, Tn : C → E is a contractive mapping satisfying the weak inward condition (WIC). Using Theorem 4.1.1, we see that, for any n ≥ 1, Tn has a unique fixed point xn in C, that is, xn = Tn xn =

1 n u+ Tx , 1+n 1+n n

n ≥ 1.

(4.4)

Due to {Txn } ⊂ C and the boundedness of C, we derive that {Txn } is bounded. From (4.4), we see that xn − Txn → θ

(n → ∞).

(4.5)

The uniform convexity of E asserts that E is reflexive. From the boundedness of {xn }, we assume that xn ⇀ x. By applying Theorem 1.9.3, we find that x ∈ Fix(T). Next, we show that Fix(T) is closed and convex. We first show the closedness of Fix(T). To this end, we fix {pn } ⊂ Fix(T) with pn → p as n → ∞. Since Tpn = pn and T is continuous, we find that Tp = p, that is, p ∈ Fix(T). Now, we are in a position to show that Fix(T) is convex. Let p1 , p2 ∈ Fix(T) and t ∈ (0, 1) and put pt = tp1 + (1 − t)p2 . It follows that pt ∈ C and ‖p1 − Tpt ‖ = ‖Tp1 − Tpt ‖ ≤ ‖p1 − pt ‖ = (1 − t)‖p1 − p2 ‖

(4.6)

‖p2 − Tpt ‖ = ‖Tp2 − Tpt ‖ ≤ ‖p2 − pt ‖ = t‖p1 − p2 ‖.

(4.7)

and

Combining (4.6) with (4.7), we find that ‖p1 − p2 ‖ ≤ ‖p1 − Tpt ‖ + ‖Tpt − p2 ‖ ≤ ‖p1 − pt ‖ + ‖pt − p2 ‖ = ‖p1 − p2 ‖. This implies that ‖p1 − p2 ‖ = ‖p1 − Tpt ‖ + ‖Tpt − p2 ‖. Set x = p1 − Tpt , y = Tpt − p2 . It follows that ‖x + y‖ = ‖x‖ + ‖y‖.

(4.8)

4.2 Normal Mann iterative method and Reich’s weak convergence theorem

| 211

Since E is uniformly convex, we see that it is also strictly convex. Using (4.8), there exists λ > 0 such that x = λy. Hence Tpt =

1 λ p1 + p. 1+λ 1+λ 2

(4.9)

It follows that λ ‖p − p2 ‖ = ‖Tpt − p1 ‖ ≤ ‖pt − p1 ‖ = (1 − t)‖p1 − p2 ‖ 1+λ 1 and

Hence t =

1 ‖p − p2 ‖ = ‖Tpt − p2 ‖ ≤ ‖pt − p2 ‖ = t‖p1 − p2 ‖. 1+λ 1 1 , 1+λ

(4.10)

that is, Tpt = pt . This completes the proof.

4.2 Normal Mann iterative method and Reich’s weak convergence theorem Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a mapping and let {αn } be a sequence in (0, 1). The normal Mann iterative method (NMIM) generates a sequence in the following manner: x1 ∈ C,

xn+1 = (1 − αn )xn + αn Txn .

If {αn } is a fixed constant in (0, 1), then the (NMIM) is also called the Krasnosel’skiĭ– Mann iterative method. Theorem 4.2.1 (Reich [73], 1979). Let E be a real uniformly convex Banach space such that the norm of E is Fréchet differentiable. Let C be a nonempty, closed, and convex subset of E, and let T : C → C be a nonexpansive mapping with a nonempty fixed point set Fix(T) ≠ 0. Let {xn } be a sequence generated by (NMIM), where {αn } is a real sequence such that ∑∞ n=1 αn (1 − αn ) = ∞. Then {xn } converges weakly to a fixed point of T. Proof. Define a mapping Tn : C → C by Tn x = (1 − αn )x + αn Tx,

∀x ∈ C, n ≥ 1.

(4.11)

Then {Tn }n≥1 : C → C is a family of nonexpansive mappings satisfying the condition Fix(T) ⊂ ⋂∞ i=1 Fix(Tn ). It follows from (4.11) that xn+1 = Tn xn ,

∀n ≥ 1.

(4.12)

Using Theorem 1.9.8, we need to prove ωw (xn ) ⊂ Fix(T). To this end, we first show that limn→∞ ‖xn − p‖ exists for all p ∈ Fix(T). Indeed, using either (NMIA) or (4.12), we find that ‖xn+1 − p‖ = ‖Tn xn − Tp‖ ≤ ‖xn − p‖,

∀p ∈ Fix(T).

212 | 4 Nonexpansive mappings in Banach spaces This shows that {‖xn −p‖} is decreasing. This further implies that limn→∞ ‖xn −p‖ exists for all p ∈ Fix(T). In particular, we see that sequence {xn } is bounded. Fix p ∈ Fix(T) and let r > 0 be a sufficiently large number such that ‖xn − p‖ ≤ r. Hence, ‖Txn − p‖ ≤ r. Using Theorem 1.8.9, we find that ‖xn+1 − p‖2 ≤ (1 − αn )‖xn − p‖2 + αn ‖Txn − p‖2 − (1 − αn )αn g(‖Txn − xn ‖) ≤ ‖xn − p‖2 − (1 − αn )αn g(‖Txn − xn ‖).

(4.13)

It follows that (1 − αn )αn g(‖Txn − xn ‖) ≤ ‖xn − p‖2 − ‖xn+1 − p‖2 .

(4.14)

Hence, ∞

∑ αn (1 − αn )g(‖Txn − xn ‖) < ∞.

n=1

(4.15)

In view of ∑∞ n=1 αn (1 − αn ) = ∞ and (4.15), we find that lim infn→∞ g(‖Txn − xn ‖) = 0. Hence, lim infn→∞ ‖Txn −xn ‖ = 0. From (NMIM), we also have ‖xn+1 −Txn+1 ‖ ≤ ‖xn −Txn ‖. This shows that limn→∞ ‖xn − Txn ‖ exists. It follows that limn→∞ ‖xn − Txn ‖ = 0. Using Theorem 1.9.3, we see that ωw (xn ) ⊂ Fix(T). With the aid of Theorem 1.9.8, we find the desired conclusion immediately. Remark 4.2.1. If E is a Hilbert space, the sequence {xn } defined by (NMIM) in Theorem 4.2.1 converges weakly to z ∈ Fix(T) and z is uniquely defined by limn→∞ PFix(T) xn = z. If E is not Hilbert, we are not sure whether sequence {xn } converges weakly to z ∈ Fix(T). Such a z can be defined by the limit of a generalized projection, which is an interesting problem. Recently, Takahashi and Yao investigated this problem; see [90] and the references therein. Remark 4.2.2. Generally speaking, the sequence {xn } generated by Theorem 4.2.1 is weakly convergent. Under an additional condition, it is also strongly convergent. To illustrate this, we introduce the following definitions. Definition 4.2.1. Let E be a Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a mapping. Then T is said to satisfy Condition (A) if Fix(T) ≠ 0 and there exits a nondecreasing function f : ℝ+ → ℝ+ with f (0) = 0, f (t) > 0, ∀t > 0 such that ‖x − Tx‖ ≥ f (d(x, Fix(T))),

∀x ∈ C,

where d(x, Fix(T)) = inf{‖x − z‖ : z ∈ Fix(T)}. Also T : C → C is said to satisfy Condition (B) if Fix(T) ≠ 0 and there exits a constant r > 0 such that ‖x − Tx‖ ≥ rd(x, Fix(T)),

∀x ∈ C.

4.2 Normal Mann iterative method and Reich’s weak convergence theorem

| 213

It is obvious mappings, which satisfy Condition (B), also satisfy Condition (A). Example 4.2.1. Let E = ℝ2 and ‖x‖ = √⟨x, x⟩, ∀x ∈ E. Let C = {(r, θ) : 0 ≤ r ≤ 1, −

π π ≤ θ ≤ − }. 2 4

Define a mapping T : C → C by T(r, θ) = (r, − π2 ), (r, θ) ∈ C. Then Fix(T) = {(r, − π2 ) : 0 ≤ r ≤ 1} ≠ 0 and ‖x − Tx‖ ≥ d(x, Fix(T)),

∀x ∈ C.

This shows that T satisfies Condition (B), which further implies that T also satisfies Condition (A). Definition 4.2.2. A mapping T is said to be semicompact if any bounded sequence {xn } ⊂ Dom(T), such that {xn − Txn } converges, has a convergent subsequence. Remark 4.2.3. (i) Let E be a Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a mapping with fixed points. If I − T maps bounded closed subsets of C to closed subsets of E, then T satisfies Condition (A). (ii) If T : C → C is continuous semicompact, then T satisfies Condition (A). Theorem 4.2.2 (Senter and Dotson [78], 1974). Let E be a real uniformly convex Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping with a nonempty fixed point set Fix(T) ≠ 0. Let {xn } be a sequence generated by (NMIM). If T satisfies Condition (A) and control sequence {αn } satisfies ∑∞ n=1 αn (1 − αn ) = ∞, then {xn } converges strongly to a fixed point of T in norm. Proof. From the proof of Theorem 4.2.1, we find that xn − Txn → θ as n → ∞. Using Condition (A), we conclude that f (d(xn , Fix(T))) → 0 as n → ∞, which implies that d(xn , Fix(T)) → 0 as n → ∞. Thus, a standard argument yields that xn → p ∈ Fix(T) as n → ∞. In 1976, Ishikawa [33] obtained the following remarkable result. Theorem 4.2.3 (Ishikawa [33], 1976). Let E be a Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping. Let {xn } be generated by the iterative method (NMIM), where {αn } is such that 0 ≤ αn ≤ c < 1 and ∑∞ n=1 αn = ∞. If, in addition, {xn } is bounded, then xn − Txn → θ (n → ∞). Using Theorem 4.2.3 and Condition (A), we establish the following strong convergence result. Theorem 4.2.4 (Ishikawa [33], 1976). Let E be a Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping with a

214 | 4 Nonexpansive mappings in Banach spaces nonempty fixed-point set Fix(T) ≠ 0. Let {xn } be the sequence generated by (NMIM). If T satisfies Condition (A), then {xn } converges strongly to a fixed point of T. Proof. It follows from Theorem 4.2.3 that xn − Txn → θ as n → ∞. Using Condition (A), we find that f (d(xn , Fix(T))) → 0 as n → ∞, which implies that d(xn , Fix(T)) → 0 as n → ∞. Thus, a standard argument yields that xn → p ∈ Fix(T) as n → ∞. Theorem 4.2.5. Let E be a reflexive Banach space. Let T : E → E be a linear nonexpansive mapping, that is, ‖Tx‖ ≤ ‖x‖, ∀x ∈ E. Fix λ ∈ (0, 1) and define a mapping Sλ : E → E by Sλ = λI + (1 − λ)T, ∀λ ∈ (0, 1). Then, for any x1 ∈ E, Sλn x1 → y ∈ Fix(T) as n → ∞. Proof. Letting xn = Sλn x1 , one has ‖xn ‖ = ‖Sλn x1 ‖ ≤ ‖x1 ‖ and xn+1 = Sλn+1 x1 = Sλ xn = λxn + (1 − λ)Txn ,

n ≥ 1.

(4.16)

From Theorem 4.2.3, we obtain that xn − Txn → θ as n → ∞, that is, (T − I)Sλn x1 → θ as n → ∞.

(4.17)

In view of Sλn x1 = [λI + (1 − λ)T]n x1 = ∑nj=0 Cnj λj (1 − λ)n−j T n−j x1 , we assert that (T − I)Sλn x1 = Sλn (T − I)x1 → θ

as n → ∞.

(4.18)

Next, we prove that R(T − I) = {x ∈ E : lim Sλn x = 0}. n→∞

(4.19)

Let z ∈ R(T − I). For any ε > 0, there exists ω ∈ R(I − T) such that ‖ z − ω ‖< ε 󳨐⇒‖ Sλn (z − ω) ‖≤‖ z − ω ‖< ε. It follows that Sλn (z − ω) = Sλn z − Sλn ω → θ

as n → ∞.

However, we find from (4.18) that Sλn ω → θ as n → ∞, which in turn implies that Sλn z → θ as n → ∞. This shows that z ∈ {x ∈ E : lim Sλn x1 = θ}. n→∞

Conversely, assume that limn→∞ Sλn x = θ. Then there exists N ≥ 1 such that, for any ε > 0, 󵄩󵄩 󵄩 n 󵄩 n 󵄩 󵄩󵄩x − (x − Sλ x)󵄩󵄩󵄩 = 󵄩󵄩󵄩Sλ x 󵄩󵄩󵄩 < ε,

∀n ≥ N.

(4.20)

4.3 Halpern iterative method and its strong convergence theorems | 215

In view of the equality n

x − Sλn x = [I − (λI + (1 − λ)T) ]x = (1 − λ)(I − T)(I + Sλ + Sλ2 + ⋅ ⋅ ⋅ + Sλn−1 )x ∈ R(I − T), we assert that x ∈ R(T − I). This establishes (4.19). Due to the facts that E is reflexive and {Sλn x1 } is bounded, we may assume, without loss of generality, that Sλn x1 ⇀ y (n → ∞). Since x1 − Sλn x1 ∈ R(I − T) and R(I − T) is weakly closed, we find that x1 − Sλn x1 ⇀ x1 − y ∈ R(I − T). It follows from (4.19) that limn→∞ Sλn (x1 − y) = θ, which further implies that Sλn x1 − Sλn y → θ

(n → ∞).

(4.21)

On the other hand, from Sλn x1 ⇀ y and (I − T)Sλn x1 → θ, we see that y = Ty. So, = y, ∀n ≥ 1. It follows from (4.21) that Sλn x1 → y (n → ∞). This completes the proof.

Sλn y

Remark 4.2.4. Theorem 4.2.5 shows that the Krasnosel’skiĭ–Mann iterative method is strongly convergent for linear nonexpansive mappings. It is an interesting problem whether normal Mann iterative method (NMIM) is also strongly convergent. Recently, Takahashi and Yao [90] investigated this problem in a real uniformly convex Banach space; see [90] for more details and the references therein.

4.3 Halpern iterative method and its strong convergence theorems Theorem 4.3.1. Let E be a real reflexive Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E such that every nonempty, bounded, closed, and convex subset of C has the f. p. p. for nonexpansive selfmappings. Let T : C → E be a nonexpansive mapping satisfying the weak inward condition (WIC) and Fix(T) ≠ 0. Then, for any fixed u ∈ C, for any t ∈ (0, 1], there exists a unique bounded continuous path {xt } ⊂ C such that xt = tu + (1 − t)Txt ,

t ∈ (0, 1]

(4.22)

and xt → p = QFix(T) u = limt→0 xt , where QFix(T) : C → Fix(T) is a unique sunny nonexpansive retraction mapping from C onto Fix(T). Proof. For any fixed u ∈ C and any t ∈ (0, 1], we define Tt : C → E by Tt x = tu + (1 − t)Tx,

∀x ∈ C.

(4.23)

Then, for any t ∈ (0, 1], one sees that Tt : C → E is contractive. Applying Banach– Caristi theorem (Theorem 4.1.1), one concludes that Tt has a unique fixed point xt ∈ C, that is, xt = Tt xt , t ∈ (0, 1]. From (4.23), one has xt = tu + (1 − t)Txt ,

∀t ∈ (0, 1].

(4.24)

216 | 4 Nonexpansive mappings in Banach spaces This shows the existence and uniqueness of the path {xt }, which satisfies (4.22). From the assumption that Fix(T) ≠ 0 and (4.22), we find that ‖xt − p‖2 = ⟨xt − p, j(xt − p)⟩

= t⟨u − p, j(xt − p)⟩ + (1 − t)⟨Txt − p, j(xt − p)⟩ ≤ t⟨u − p, j(xt − p)⟩ + (1 − t)‖xt − p‖2 ,

∀p ∈ Fix(T).

It follows that ‖xt − p‖2 ≤ ⟨u − p, j(xt − p)⟩,

∀p ∈ Fix(T).

(4.25)

Hence ‖xt − p‖ ≤ ‖u − p‖, ∀t ∈ (0, 1), which implies that {xt } is bounded. This yields that {Txt } is also bounded. Next, we prove that {xt } is continuous. To this end, ∀t, s ∈ (0, 1), we see that there exist unique paths {xt } and {xs }, respectively, satisfying (4.22). Thus, xt − xs = (t − s)u + (1 − t)Txt − (1 − s)Txs .

(4.26)

It follows that ‖xt − xs ‖ ≤ |t − s|‖u‖ + (1 − t)‖Txt − Txs ‖ + |t − s|‖Txs ‖ ≤ |t − s|(‖u‖ + ‖Txs ‖) + (1 − t)‖xt − xs ‖.

This implies that ‖xt − xs ‖ ≤

|t − s| (‖u‖ + ‖Txs ‖) t

(4.27)

Fix s ∈ (0, 1) and let t → s. Since {Txs } is bounded, we find that xt → xs as t → s. This shows that the path {xt } is continuous. Next, we show that xt → p ∈ Fix(T) as t → 0. Denote the Banach limit by μt and define a function φ : C → ℝ+ by φ(y) = μt ‖xt − y‖,

∀y ∈ C.

(4.28)

Then φ : C → ℝ+ is continuous convex and φ(y) → +∞ (‖y‖ → +∞). From the assumption that E is reflexive, while C is closed and convex in E, we see that there exists z ∈ C such that φ(z) = min φ(y). y∈C

Let K = {z ∈ C : φ(z) = min φ(y)}. y∈C

(4.29)

4.3 Halpern iterative method and its strong convergence theorems | 217

Then K ≠ 0. Further, K is bounded and closed convex. Indeed, K = ⋂ {z ∈ C : φ(z) ≤ φ(y)}. y∈C

From the above equality, one easily verifies that K is bounded, closed, and convex by means of the continuity, convexity, and the property φ(z) → ∞ as ‖z‖ → ∞. It follows from Banach contraction principle that C ⊂ (2I − T)(C). Write S = (2I − T)−1 . Then S : C → C is nonexpansive such that Fix(S) = Fix(T). We claim that S(K) ⊆ K. To see this, in view of (4.22), we have xt − Txt → 0 as t → θ, which yields that xt − Sxt → 0 as t → θ. By virtue of the definition of φ, we have for any z ∈ K that φ(Sz) = μt ‖xt − Sz‖ ≤ μt ‖xt − z‖ ≤ φ(y)

for all y ∈ C. This shows that S : K → K is a self-mapping. From the assumption that there exits z0 ∈ K such that z0 = Sz0 , that is, z0 ∈ Fix(T), and from Takahashi’s result, one has μt (⟨x − z0 , j(xt − z0 )⟩) ≤ 0, ∀x ∈ C. In particular, letting x = u ∈ C, we arrive at μt (⟨u − z0 , j(xt − z0 )⟩) ≤ 0,

∀t ∈ (0, 1].

(4.30)

On the other hand, one concludes from (4.25) that ⟨xt − u, j(xt − z0 )⟩ ≤ 0,

∀t ∈ (0, 1].

Taking the Banach limit in the above inequality, we arrive at μt (⟨xt − u, j(xt − z0 )⟩) ≤ 0.

(4.31)

Adding (4.30) and (4.31), we obtain μt (⟨xt − z0 , j(xt − z0 )⟩) ≤ 0. So, μt (‖xt − z0 ‖2 ) = 0. Thus, for any tn → 0, there exits a subsequence of {xtn } such that xtn → z0 as n → ∞. If sn → 0 and xsn → p ∈ Fix(T) as n → ∞, then p = z0 . It follows from (4.25) that ⟨z0 − u, j(z0 − p)⟩ ≤ 0

(4.32)

⟨p − u, j(p − z0 )⟩ ≤ 0.

(4.33)

and

Adding (4.32) and (4.33), we find that ‖z0 − p‖2 = ⟨z0 − p, j(z0 − p)⟩ ≤ 0, that is, p = z0 . Hence, {xt } strongly converges to p ∈ Fix(T). Defining QFix(T) : C → Fix(T) by QFix(T) u = limt→0 xt , one easily verifies that QFix(T) : C → Fix(T) is a unique sunny nonexpansive retraction from C onto Fix(T). This completes the proof.

218 | 4 Nonexpansive mappings in Banach spaces Corollary 4.3.1 (Reich [73], 1979). Let E be a uniformly smooth Banach space and let C be a closed convex subset of E. Let T : C → E be a nonexpansive mapping satisfying the weak inward condition (WIC) and Fix(T) ≠ 0. Then, for any fixed u ∈ C and for any t ∈ (0, 1], there exists a unique bounded continuous path {xt } ⊂ C satisfying (4.22) and the path {xt } strongly converges to p = QFix(T) u as t → 0. Corollary 4.3.2. Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. Let C be a closed and convex subset of E. Let T : C → E be a nonexpansive mapping satisfying the weak inward condition (WIC) and Fix(T) ≠ 0. Then, for any fixed u ∈ C and for any t ∈ (0, 1], there exists a unique bounded continuous path {xt } ⊂ C satisfying (4.22) and the path {xt } strongly converges to p = QFix(T) u, as t → 0. In order to establish strong convergence theorems of Halpern iterative methods, we first give the following result. Lemma 4.3.1. Let E be a real reflexive Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E such that every nonempty, bounded, closed, and convex subset of C has the f. p. p. for every nonexpansive self-mapping. Let T : C → E be a nonexpansive mapping satisfying the weak inward condition (WIC) and Fix(T) ≠ 0. Let {xn } ⊂ C be a bounded approximate fixed point sequence of T, that is, {xn } is bounded and xn → Txn → θ as n → ∞. Then lim sup⟨u − p, j(xn − p)⟩ ≤ 0 n→∞

where p = QFix(T) u, QFix(T) : C → F(T) is the unique sunny nonexpansive retraction from C onto Fix(T). Proof. From Theorem 4.3.1, one sees that there exists a unique sunny nonexpansive retraction QFix(T) : C → F(T), where QFix(T) u = limt→0 xt = p. On the other hand, {xt } satisfies (4.22), that is, xt = tu + (1 − t)Txt ,

∀t ∈ (0, 1].

(4.34)

Let M = sup{‖xn − xt ‖, t ∈ (0, 1), n ≥ 1}. It follows that ‖xt − xn ‖2 = ⟨xt − xn , j(xt − xn )⟩

= t⟨u − xn , j(xt − xn )⟩ + (1 − t)⟨Txt − xn , j(xt − xn )⟩ = t⟨u − xt , j(xt − xn )⟩ + t‖xt − xn ‖2

+ (1 − t)⟨Txt − Txn , j(xt − xn )⟩ + (1 − t)⟨Txn − xn , j(xt − xn )⟩

≤ t⟨u − xt , j(xt − xn )⟩ + ‖xt − xn ‖2 + (1 − t)M‖xn − Txn ‖. It follows that ⟨u − xt , j(xn − xt )⟩ ≤

M ‖x − Txn ‖. t n

(4.35)

4.3 Halpern iterative method and its strong convergence theorems | 219

Fixing t ∈ (0, 1) and letting n → ∞, it follows from (4.35) that lim sup⟨u − xt , j(xn − xt )⟩ ≤ 0. n→∞

(4.36)

Note that ⟨u−xt , j(xn −xt )⟩ = ⟨u−p, j(xn −p)⟩+⟨u−p, j(xn −xt )−j(xn −p)⟩+⟨p−xt , j(xn −xt )⟩. (4.37) Due to the fact that xt → p as t → 0, we find from the boundedness of {xn } and {xt } that ⟨p − xt , j(xn − xt )⟩ → θ

as t → θ.

(4.38)

Since the norm of E is uniformly Gâteaux differentiable, we have that j : E → E ∗ is s-w uniformly continuous on any bounded subset of E. Hence, ⟨u − p, j(xn − xt ) − j(xn − p)⟩ → θ

as t → θ.

(4.39)

From (4.38) and (4.39), for any ε > 0, we see that there exists δ = δ(ε) > 0 such that, for any n ≥ 1, 󵄨󵄨 󵄨 ε 󵄨󵄨⟨p − xt , j(xn − xt )⟩󵄨󵄨󵄨 < 2 and 󵄨󵄨 󵄨 ε 󵄨󵄨⟨u − p, j(xn − xt ) − j(xn − p)⟩󵄨󵄨󵄨 < 2 as 0 < t < δ. Taking the limits on both sides of (4.37), we find that lim sup⟨u − p, j(xn − p) ≤ n→∞

ε ε + = ε. 2 2

Since ε > 0 is chosen arbitrarily, we find that lim sup⟨u − p, j(xn − p) ≤ 0. n→∞

This completes the proof. Remark 4.3.1. (i) If path {xt } exists and strongly converges to p, then the assumptions that E is reflexive and C has the f. p. p. can be removed. (ii) If E is strictly convex, then the assumption that C has the f. p. p. can be removed. (iii) If E satisfies Opial condition and the duality mapping J : E → E ∗ is a weakly sequentially continuous at zero, then the assumptions that the norm of E is uniformly Gâteaux differentiable and C has the f. p. p. can be removed.

220 | 4 Nonexpansive mappings in Banach spaces Theorem 4.3.2. Let E be a real reflexive Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E such that every nonempty, bounded, closed, and convex subset of C has the f. p. p. for every nonexpansive self-mapping. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {αn } be a sequence in [0, 1] satisfying the control conditions: (i) limn→∞ αn = 0; (ii) ∑∞ n=1 αn = ∞; (iii) ∑∞ n=1 |αn+1 − αn | < ∞ or α (iv) limn→∞ αn+1 = 1. n

Let u ∈ C be a fixed element. For any initial value x1 ∈ C, define a sequence {xn } in following manner: xn+1 = αn u + (1 − αn )Txn ,

n ≥ 1.

(HIM)

Then the sequence {xn } converges strongly to some point p = QFix(T) u, where QFix(T) : C → Fix(T) is a unique sunny nonexpansive retraction mapping from C onto Fix(T). Proof. The proof is split into three steps. Step 1. Show that {xn } is a bounded sequence. Fixing p ∈ Fix(T), we have 󵄩 󵄩 ‖xn+1 − p‖ = 󵄩󵄩󵄩αn (u − p) + (1 − αn )(Txn − p)󵄩󵄩󵄩 ≤ αn ‖u − p‖ + (1 − αn )‖xn − p‖

≤ max{‖u − p‖, ‖x1 − p‖} ≜ M. Then {xn } is bounded, so is {Txn }. Since αn → 0 as n → ∞, one has xn+1 − Txn = αn (u − Txn ) → θ

as n → ∞.

(4.40)

Step 2. Show that xn − Txn → θ as n → ∞. Note that xn+1 − xn = (αn − αn−1 )u + (1 − αn )Txn − (1 − αn−1 )Txn−1 = (αn − αn−1 )u + (1 − αn )Txn − (1 − αn )Txn−1 + (1 − αn )Txn−1 − (1 − αn−1 )Txn−1

= (αn − αn−1 )(u − Txn−1 ) + (1 − αn )(Txn − Txn−1 ). It follows that ‖xn+1 − xn ‖ ≤ (1 − αn )‖xn − xn−1 ‖ + |αn − αn−1 |(‖u‖ + ‖Txn−1 ‖) ≤ (1 − αn )‖xn − xn−1 ‖ + M|αn − αn−1 |,

(4.43)

where M = sup{‖u‖ + ‖Txn ‖, n ≥ 1}. Using Lemma 1.10.2, we have xn+1 − xn → θ as n → ∞. In view of (4.40), we find that xn − Txn = xn − xn+1 + xn+1 − Txn → θ as n → ∞.

4.3 Halpern iterative method and its strong convergence theorems | 221

Step 3. Show that xn → p = QFix(T) u (n → ∞). From Lemma 4.3.1, we find that lim sup⟨u − p, j(xn − p)⟩ ≤ 0. n→∞

(4.41)

It follows that ‖xn+1 − p‖2 = αn ⟨u − p, j(xn+1 − p)⟩ + (1 − αn )⟨Txn − p, j(xn+1 − p)⟩

≤ (1 − αn )‖xn − p‖‖xn+1 − p‖ + αn ⟨u − p, j(xn+1 − p)⟩ 1 − αn 1 − αn ≤ ‖xn − p‖2 + ‖xn+1 − p‖2 + αn ⟨u − p, j(xn+1 − p)⟩. 2 2

This yields that ‖xn+1 − p‖2 ≤ (1 − αn )‖xn − p‖2 + 2αn ⟨u − p, j(xn+1 − p)⟩.

(4.42)

Setting an = ‖xn − p‖2 , tn = αn , and bn = 2αn ⟨u − p, j(xn+1 − p)⟩, we obtain from (4.42) that an+1 ≤ (1 − tn )an + bn , where {tn } and {bn } satisfy the following conditions: ∞ (1) Σ∞ n=1 tn = Σn=1 αn = ∞; b (2) lim supn→∞ t n = 2 lim supn→∞ ⟨u − p, j(xn+1 − p)⟩ ≤ 0. n By virtue of Lemma 1.10.2, we find that an → 0 as n → ∞, that is, xn → p = QFix(T) u as n → ∞. This completes the proof. Corollary 4.3.3. Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {xn } be a sequence defined in Theorem 4.3.2. Then xn → p = QFix(T) u as n → ∞. Corollary 4.3.4. Let E be a real uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {xn } be a sequence defined in Theorem 4.3.2. Then xn → p = QFix(T) u as n → ∞. In 2009, Suzuki [85] pointed out that conditions (i) and (ii) of Theorem 4.3.2 are not sufficient to guarantee the strong convergence of (HIA). Example 4.3.1. Let E = ℝ, C = [−1, 1], and u = 1. Define a nonexpansive mapping T : C → C by Tx = −x, ∀x ∈ C. Then Fix(T) = {0}. Let {αn } be a real sequence in [0, 1] defined by 0, n is an odd number, αn = { 1 , n is an even number. n

222 | 4 Nonexpansive mappings in Banach spaces Then {αn } satisfies conditions (i) and (ii) of Theorem 4.3.2. However, xn = (−1)(n+1) is not convergent for the initial value x1 = 1. This further implies that {xn } does not converge to the fixed point of T. In 2007, Suzuki [84] introduced the following iterative method: x1 , u ∈ C,

xn+1 = αn u + (1 − αn )[λTxn + (1 − λ)xn ],

(HSIM)

where λ is a real number in (0, 1). This method is now called the Halpern–Suzuki iterative method. Suzuki [84], based on new analysis techniques, proved that the sequence generated by (HSIM) converges to a fixed point of T under conditions (i) and (ii) of Theorem 4.3.2. Theorem 4.3.3 (Suzuki [84], 2007). Let E be a real Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {αn } be a real sequence satisfying conditions (i) and (ii) of Theorem 4.3.2. Let {xn } be a sequence defined by (HSIM) and let {xt } be a path defined by (4.22). If {xt } converges strongly to some p ∈ Fix(T) as t → 0, then xn → p = QFix(T) u (n → ∞). Proof. The key step is to prove that {xn } is a bounded approximate fixed point sequence of T. (i) {xn } is bounded. Let Tλ = λT + (1 − λ)T, λ ∈ (0, 1). For any P ∈ Fix(T), one has 󵄩 󵄩 ‖xn+1 − p‖ = 󵄩󵄩󵄩αn (u − p) + (1 − αn )(Tλ xn − p)󵄩󵄩󵄩 ≤ αn ‖u − p‖ + (1 − αn )‖xn − p‖

≤ max{‖u − p‖, ‖x1 − p‖} ≜ M, which shows that {xn } is bounded, so is {Tλ xn }. Since αn → 0 as n → ∞, we have that xn+1 − Tλ xn = αn (u − Tλ xn ) → θ

(n → ∞).

(4.43)

(ii) limn→∞ ‖xn − Txn ‖ = 0. Write λn = (1 − λ)(1 − αn ) and define a sequence {yn } by yn =

αn λ(1 − αn ) u+ Txn . 1 − λn 1 − λn

Then (HSIM) reduces to xn+1 = λn xn + (1 − λn )yn ,

∀n ≥ 1.

(4.44)

4.3 Halpern iterative method and its strong convergence theorems | 223

It follows that 󵄨󵄨 α λ(1 − αn+1 ) αn 󵄨󵄨󵄨󵄨 󵄨 ‖yn+1 − yn ‖ ≤ 󵄨󵄨󵄨 n+1 − ‖xn+1 − xn ‖ 󵄨‖u‖ + 󵄨󵄨 1 − λn+1 1 − λn 󵄨󵄨󵄨 1 − λn+1 󵄨󵄨 λ(1 − α ) λ(1 − α ) 󵄨󵄨 󵄨 n+1 n 󵄨󵄨 − + 󵄨󵄨󵄨 󵄨‖Tx ‖. 󵄨󵄨 1 − λn+1 1 − λn 󵄨󵄨󵄨 n Since αn → 0, we see that 1 − λn → λ. This, together with the boundedness of {Txn }, implies that lim sup(‖yn+1 − yn ‖ − ‖xn+1 − xn ‖) ≤ 0. n→∞

From Lemma 1.10.3, we find that yn − xn → θ as n → ∞. Hence xn+1 − xn = (1 − λn )(yn − xn ) → θ as n → ∞. It follows from (4.43) that xn − Tλ xn → θ as n → ∞, which yields xn −Txn → θ as n → ∞. From Lemma 4.3.1 and Remark 4.3.1, we have lim supn→∞ ⟨u − p, j(xn − p)⟩ ≤ 0. (iii) xn → p = QFix(T) u (n → ∞). From (HSIM), we have ‖xn+1 − p‖2 = αn ⟨u − p, j(xn − p)⟩ + (1 − αn )⟨Tλ xn − p, j(xn+1 − p)⟩ ≤ (1 − αn )‖xn − p‖‖xn+1 − p‖ + αn ⟨u − p, j(xn+1 − p)⟩ 1 − αn 1 − αn ≤ ‖xn − p‖2 + ‖xn+1 − p‖2 2 2 + αn ⟨u − p, j(xn+1 − p)⟩,

which yields ‖xn+1 − p‖2 ≤ (1 − αn )‖xn − p‖2 + 2αn ⟨u − p, j(xn+1 − p)⟩.

(4.45)

Putting an = ‖xn+1 − p‖2 , αn = tn , and bn = 2αn ⟨u − p, j(xn+1 − p)⟩, we conclude from (4.45) that an+1 ≤ (1 − tn )an + bn ,

∀n ≥ 1.

It follows from Lemma 1.10.2 that an → 0 as n → ∞, that is, xn → p = QFix(T) u (n → ∞). Corollary 4.3.5. Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {αn } be a real sequence satisfying conditions (i) and (ii) of Theorem 4.3.2. Let {xn } be generated by (HSIM). Then xn → p = QFix(T) u as n → ∞.

224 | 4 Nonexpansive mappings in Banach spaces Corollary 4.3.6. Let E be a real uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {αn } be a real sequence satisfying conditions (i) and (ii) of Theorem 4.3.2. Let {xn } be defined by (HSIM). Then xn → p = QFix(T) u(n → ∞) as n → ∞. Recently, several authors also investigated the following so-called general Halpern iterative method: xn+1 = αn u + βn xn + γn Txn ,

n ≥ 1,

where u is a fixed element in set C, and {αn }, {βn }, and {γn } are real sequences in (0, 1) such that αn +βn +γn = 1. With some conditions imposed on the control sequences {αn }, {βn }, and {γn }, we can show that the sequence generated by the above iterative method is also strongly convergent. Setting λ := λn in (HSIM), the above general Halpern iterative method is equivalent to the (HSIM). In 2010, Xu [103] investigated Reich-type implicit iterative methods and Halperntype explicit iterative method. Theorem 4.3.4 (Xu [103], 2010). Let E be a real uniformly smooth Banach space. Let C be a closed convex subset of E. Let T : C → E be a nonexpansive mapping satisfying the weak inward condition (WIC). Then, for any u ∈ C and t ∈ (0, 1], there exists a unique bounded continuous path {zt } ⊂ C satisfying zt = T(tu + (1 − t)zt ),

t ∈ (0, 1].

(4.46)

Further, {zt } is bounded as t → 0 if and only if Fix(T) ≠ 0. If, in addition, Fix(T) ≠ 0, then {zt } strongly converges to z = QFix(T) u as t → 0, where QFix(T) u : C → Fix(T) is a unique sunny nonexpansive retraction from C onto Fix(T). Proof. Fix u ∈ C. For any t ∈ (0, 1], we define the mapping Tt : C → C by Tt x = T(tu + (1 − t)x),

∀x ∈ C.

(4.47)

Then, for any t ∈ (0, 1], Tt : C → E is a contractive mapping satisfying the weak inward condition (WIC). From Banach–Caristi theorem (Theorem 4.1.1), we find that for any t ∈ (0, 1], there exists a unique continuous path {zt } ⊂ C satisfying (4.46). Let {xt } be defined by (4.22). Thus 󵄩 󵄩 ‖zt − Txt ‖ = 󵄩󵄩󵄩T(tu + (1 − t)zt ) − Txt 󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩tu + (1 − t)zt − xt 󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩tu + (1 − t)zt − tu − (1 − t)Txt 󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩(1 − t)(zt − Txt )󵄩󵄩󵄩 = (1 − t)‖zt − Txt ‖,

∀t ∈ (0, 1).

4.3 Halpern iterative method and its strong convergence theorems | 225

It follows that zt = Txt ,

∀t ∈ (0, 1].

(4.48)

From Theorem 4.3.1, {xt } is bounded as t → 0 if and only if Fix(T) ≠ 0. From the fact that {xt } is bounded if and only if {Txt } is bounded, we find from (4.48) that {zt } is bounded as t → 0 if and only if Fix(T) ≠ 0. If Fix(T) ≠ 0, it follows from Theorem 4.3.1 that xt → QFix(T) u as t → 0. Hence Txt → QFix(T) u as t → 0, that is, zt → QFix(T) u as t → 0. This completes the proof. Remark 4.3.2. In fact, the paths defined by (4.45) and (4.22) are equivalent to each other. If zt → QFix(T) u (t → 0), it follows from (4.48) that Txt → QFix(T) u (t → 0). Hence xt = tu + (1 − t)Txt → QFix(T) u (t → 0). Theorem 4.3.5 (Xu [103], 2010). Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {αn } be a sequence satisfying conditions (i)–(iv) of Theorem 4.3.2. For any initial value x1 ∈ C and any fixed element u ∈ C, we define a sequence {xn } iteratively by xn+1 = T[αn u + (1 − αn )xn ],

n ≥ 1.

(XIM-1)

Then xn → z = QFix(T) u (n → ∞), where QFix(T) : C → Fix(T) is a unique sunny nonexpansive retraction mapping from C onto Fix(T). Proof. For any initial value y1 ∈ C and given anchor u ∈ C, we define another sequence {yn } iteratively by yn+1 = αn u + (1 − αn )Tyn ,

n ≥ 1.

(4.49)

By applying Theorem 4.3.2, we have yn → QFix(T) u as n → ∞, so that Tyn+1 → QFix(T) u as n → ∞, since T is continuous. It follows from (4.49) and (XIM-1) that ‖xn+1 − Tyn+1 ‖ ≤ (1 − αn )‖xn − Tyn ‖. By using condition Σ∞ n=1 αn = ∞, we conclude that xn − Tyn → θ as n → ∞, that is, xn → QFix(T) u as n → ∞. This completes the proof. Remark 4.3.3. In fact, the convergence of sequence {xn } defined by (XIM-1) and sequence {yn } defined by (4.49) implies one another. If xn → QFix(T) u (n → ∞), then Tyn → QFix(T) u (n → ∞). From (4.49), one has yn+1 = αn u + (1 − αn )Tyn → QFix(T) u (n → ∞).

226 | 4 Nonexpansive mappings in Banach spaces Theorem 4.3.6 (Xu [103], 2010). Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {αn } be a real sequence satisfying the control conditions (i) and (ii) of Theorem 4.3.2. Fix u ∈ C. For any initial value x1 ∈ C, we define a sequence {xn } iteratively by xn+1 = λxn + (1 − λ)T[αn u + (1 − αn )xn ],

λ ∈ (0, 1), n ≥ 1.

(XIM-2)

Then xn → z = QFix(T) u as n → ∞, where QFix(T) : C → Fix(T) is a unique sunny nonexpansive retraction mapping from C onto Fix(T). Proof. The key is to prove xn − Txn → θ as n → ∞. Step 1. Show that {xn } is bounded. For any p ∈ Fix(T), it follows from (XIM-2) that ‖xn+1 − p‖ ≤ λ‖xn − p‖ + (1 − λ)(αn ‖u − p‖ + (1 − αn )‖xn+1 − p‖) = (1 − (1 − λ)αn )‖xn+1 − p‖ + (1 − λ)αn ‖u − p‖ ≤ {‖u − p‖, ‖x1 − p‖} ≜ M,

∀n ≥ 1,

which shows that {xn } is bounded, and so is {Txn }. Step 2. Show that xn − Txn → θ as n → ∞. Letting yn = T[αn u + (1 − αn ]xn ), one sees that (XIM-2) reduces to xn+1 = λxn + (1 − λ)yn ,

n ≥ 1.

(4.50)

Letting M = sup{‖u‖ + ‖xn ‖ : n ≥ 1}, one has 󵄩 󵄩 ‖yn+1 − yn ‖ = 󵄩󵄩󵄩T[αn+1 u + (1 − αn+1 )xn+1 ] − T[αn u + (1 − αn )xn ]󵄩󵄩󵄩 ≤ |αn+1 − αn |(‖u‖ + ‖xn ‖) + (1 − αn+1 )‖xn+1 − xn ‖

≤ M|αn+1 − αn | + ‖xn+1 − xn ‖.

This implies that lim supn→∞ (‖yn+1 − yn ‖ − ‖xn+1 − xn ‖) ≤ 0. Using (1.10.3), we find that yn − xn → θ as n → ∞. Observe that ‖xn − Txn ‖ ≤ ‖xn − yn ‖ + ‖yn − Txn ‖ ≤ ‖xn − yn ‖ + αn ‖u − xn ‖, we deduce that xn − Txn → θ as n → ∞. Step 3. Show that xn → z = QFix(T) u as n → ∞. Lemma 4.3.1 ensures that lim supn→∞ ⟨u − z, j(xn − z)⟩ ≤ 0. From (XIM-2), the convexity of ‖ ⋅ ‖2 , and Reich inequality (RI), for some fixed positive constant M, we

4.3 Halpern iterative method and its strong convergence theorems | 227

find that 󵄩2 󵄩 ‖xn+1 − z‖2 ≤ λ‖xn − z‖2 + (1 − λ)󵄩󵄩󵄩T(αn u + (1 − αn )xn ) − z 󵄩󵄩󵄩 󵄩2 󵄩 ≤ λ‖xn − z‖2 + (1 − λ)󵄩󵄩󵄩αn (u − z) + (1 − αn )(xn − z)󵄩󵄩󵄩 ≤ λ‖xn − z‖2 + (1 − λ)(1 − αn )‖xn − z‖2 + 2(1 − λ)αn (1 − αn )⟨u − z, j(xn − z)⟩

+ (1 − λ) max{‖xn − z‖, 1}αn ‖u − z‖β(αn ‖u − z‖) 󵄩 󵄩2 ≤ [1 − (1 − λ)αn ]󵄩󵄩󵄩(xn − z)󵄩󵄩󵄩 + 2(1 − λ)αn (1 − αn )⟨u − z, j(xn − z)⟩ + Mαn β(αn ).

(4.51)

Let tn = (1 − λ)αn , bn = 2(1 − λ)αn (1 − αn )⟨u − z, j(xn − z)⟩ + Mαn β(αn ), and an = ‖(xn − z)‖2 . Then (4.51) yields an ≤ (1 − tn )an + bn ,

n ≥ 1,

where {tn } and {bn } satisfy conditions (i) and (ii) of Lemma 1.10.2. From Lemma 1.10.2, one see that an → 0 (n → ∞), that is xn → z = QFix(T) u as n → ∞. This completes the proof. Remark 4.3.4. It is not clear whether the convergence of (XIM-2) can be deduced from the convergence of (HSIM). However, (XIM-2) can be modified so that its convergence can be deduced from the convergence of the (HSIM). Theorem 4.3.7. Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping with Fix(T) ≠ 0. Let {αn } be a sequence satisfying the control conditions (i) and (ii) of Theorem 4.3.2. Let λ be a real number in (0, 1) and define Tλ = λI + (1 − λ)T. Fix u ∈ C. For any initial value x1 ∈ C, we define a sequence {xn } iteratively by xn+1 = Tλ [αn u + (1 − αn )xn ],

n ≥ 1.

(MXIM)

Then xn → z = QFix(T) u as n → ∞, where QFix(T) : C → Fix(T) is a unique sunny nonexpansive retraction mapping from C onto Fix(T). Proof. Letting yn = αn u + (1 − αn )xn , we find from (MXIM) that xn+1 = Tλ yn . Hence yn+1 = αn+1 u + (1 − αn+1 )Tλ yn ,

n ≥ 1.

Using Theorem 4.3.4, we conclude that yn → z = QFix(T) u as n → ∞, that is, xn → z = QFix(T) u as n → ∞.

228 | 4 Nonexpansive mappings in Banach spaces Remark 4.3.5. Indeed, the convergence of (MXIM) is equivalent to the convergence of (HSIM). Let {yn } be a sequence generated by (HSIM), that is, yn+1 = αn u + (1 − αn )Tλ yn ,

n ≥ 1.

(4.52)

Then Tλ yn+1 = Tλ (αn u + (1 − αn )Tλ yn ),

n ≥ 1.

(4.53)

Combining (MXIM) with (4.52), we find that ‖xn+1 − Tλ yn+1 ‖ ≤ (1 − αn )‖xn − Tλ yn ‖,

n ≥ 1.

(4.54)

It follows from condition Σ∞ n=1 αn = ∞ that xn − Tλ yn → θ as n → ∞. If xn → z = QFix(T) u as n → ∞, we have Tλ yn → z as n → ∞. Using (4.52), we conclude that yn → z = QFix(T) u as n → ∞.

4.4 Moudafi’s viscosity iterative method In 2000, Moudafi [56] first introduced a viscosity iterative method with respect to nonexpansive mappings in a Hilbert space and established strong convergent theorems for this kind of iterative method. In 2004, Xu [100] revisited Moudafi’s viscosity iterative method in the framework of uniformly smooth Banach spaces. He established strong convergent theorems of fixed points, which are solutions of some monotone variational inequality. In 2007, Suzuki [83] extended the Moudafi’s viscosity iterative method to the case with Meir–Keeler contractive mappings, which is called the Moudafi-like viscosity iterative method, and established the convergence criteria. In view of Suzuki’s results, it is trivial to extend the Halpern-like iterative methods to the Moudafi-like viscosity iterative methods. In this section, we first introduce Suzuki’s results. Then, we derive Xu’s results based on Suzuki’s results. It is obvious that a contractive mapping f : C → C is a Meir–Keeler contractive mapping, however, a Meir–Keeler contractive mapping is only a conditional contraction. To be more clear, it is contractive if we put some restriction on the domain of f . Proposition 4.4.1. Let E be a real Banach space. Let C be a nonempty, closed, and convex subset of E. Let Φ : C → C be a Meir–Keeler contractive mapping. Then, for any ε > 0, there exists r ∈ (0, 1) such that ‖x − y‖ ≥ ε 󳨐⇒ ‖Φx − Φy‖ ≤ r‖x − y‖,

∀x, y ∈ C.

(4.55)

This implies that the restriction of Φ on the set of C ∩ {(x, y) | ‖x − y‖ ≥ ε} is contractive.

4.4 Moudafi’s viscosity iterative method | 229

Proof. Fixing ε > 0, one sees that there exists δ ∈ (0, ε) satisfying ‖u − v‖
0, there exists r ∈ (0, 1) satisfying (4.55). Set δ :=

tε(1 − r) >0 1 − t + tr

and fix x, y ∈ C such that ‖x − y‖ < ε + δ. Next, we divide the proof into two cases. (1) If ‖x − y‖ ≥ ε, then we conclude from Proposition 4.4.1 that ‖Tt x − Tt y‖ ≤ (1 − t)‖Tx − Ty‖ + t‖Φx − Φy‖ ≤ (1 − t)‖x − y‖ + tr‖x − y‖ = (1 − t + tr)‖x − y‖ < (1 − t + tr)(ε + δ) = ε.

230 | 4 Nonexpansive mappings in Banach spaces (2) If ‖x − y‖ < ε, then ‖Tt x − Tt y‖ ≤ (1 − t)‖Tx − Ty‖ + t‖Φx − Φy‖ ≤ (1 − t)‖x − y‖ + t‖x − y‖ = ‖x − y‖ < ε. So, for any ε > 0, there exists δ > 0 such that ‖Tt x − Tt y‖ < ε, provided that ‖x − y‖ < ε + δ. Therefore, Tt : C → C is Meir–Keeler contractive. This completes the proof. Proposition 4.4.3. Let E be a real smooth Banach space. Let C be a nonempty, closed, and convex subset of E, and let K be a subset of C. Let P : C → K be the unique sunny nonexpansive retraction and let Φ : C → C be a mapping. For any z ∈ K, we have the following assertions, which are equivalent to each other: (i) z = P ∘ Φz; (ii) ⟨Φz − z, j(z − y)⟩ ≥ 0, ∀y ∈ K. Proof. (i) 󳨐⇒ (ii). Due to the fact that P : C → K is the unique sunny nonexpansive retractive mapping, we have ⟨x − Px, j(Px − y)⟩ ≥ 0,

∀x ∈ C, y ∈ K.

(4.56)

Letting x = Φz ∈ C in the above inequality and using (i), we have ⟨Φz − z, j(z − y)⟩ ≥ 0,

∀y ∈ K.

(ii) 󳨐⇒ (i). Let y = P ∘ Φz in (ii). Then ⟨Φz − z, j(z − P ∘ Φz)⟩ ≥ 0.

(4.57)

Letting x = Φz and y = z in (4.56), we arrive at ⟨Φz − P ∘ Φz, j(P ∘ Φz − z)⟩ ≥ 0.

(4.58)

Combining (4.57) with (4.58), we have ⟨P ∘ Φz − z, j(z − P ∘ Φz)⟩ ≥ 0, that is, −‖P ∘ Φz − z‖2 ≥ 0 and z = P ∘ Φz. This completes the proof.

(4.59)

4.4 Moudafi’s viscosity iterative method | 231

Let C be a nonempty, closed, and convex subset of a real Banach space E. Let {Sn } be a family of nonexpansive mappings in C. Let {αn } be a real sequence in (0, 1] satisfying condition (i) of Theorem 4.3.2. We say that (E, C, {Sn }, αn ) has the Browder’s property if, for any u ∈ C, the sequence {yn } defined by yn = (1 − αn )Sn yn + αn u,

n ≥ 1,

(BIM)

is strongly convergent. Let {αn } ⊂ (0, 1] be a sequence satisfying conditions (i) and (ii) of Theorem 4.3.2. We say that (E, C, {Sn }, αn ) has the Halpern’s property if, for any u ∈ C, the sequence {yn } defined by y1 ∈ C,

yn+1 = (1 − αn )Sn yn + αn u,

n ≥ 1,

(HIM)

is strongly convergent. Example 4.4.1. Let C be a nonempty, closed, and convex subset of a real uniformly smooth Banach space E. Let T : C → E be a nonexpansive mapping with Fix(T) ≠ 0. Let {αn } be a real sequence in (0, 1] satisfying condition (1) of Theorem 4.3.2. Let Sn = T, ∀n ≥ 1. Then (E, C, {Sn }, αn ) has the Browder’s property. Furthermore, {αn } ⊂ (0, 1] also satisfies condition (2) of Theorem 4.3.2. For any λ ∈ (0, 1), put Tλ = λI + (1 − λ)T and Sn = Tλ , ∀n ≥ 1. Then (E, C, {Sn }, αn ) has the Halpern’s property. Suzuki also gave the following result. Theorem 4.4.1 (Suzuki [83], 2007). Let C be a nonempty, closed, and convex subset of a real Banach space E, and let {Sn } be a family of nonexpansive mappings on C. Let {αn } a real sequence in (0, 1] satisfying condition (i) of Theorem 4.3.2 and let Φ : C → C be a Meir–Keeler contractive mapping. Assume that the system (E, C, {Sn }, αn ) has the Browder’s property. Let {yn } be generated by (BIM). Let P : C → C be defined by Pu = limn→∞ yn and define a sequence {xn } ⊂ C by xn = (1 − αn )Sn xn + αn Φxn ,

n ≥ 1.

(VBIM)

Then {xn } strongly converges to the unique point z = P ∘ Φz. Proof. In fact, it follows from (ii) of Proposition 4.4.2, we have Tn x = (1 − αn )Sn x + αn Φx,

∀x ∈ C,

which is a Meir–Keeler contractive mapping. From the Meir–Keeler fixed point theorem, we have Tn has a unique fixed point in C, denoted by xn ∈ C, that is, xn = Tn xn . This shows that (VBIM) is well defined. Next, we prove that P : C → C is a nonexpansive mapping. Since (E, C, {Sn }, αn ) has the Browder’s property, one has that yn = (1 − αn )Sn xn + αn u, ∀u ∈ C is strongly convergent. Since {yn } ⊂ C, and C is closed, we

232 | 4 Nonexpansive mappings in Banach spaces have Pu ∈ C. Also ∀u, v ∈ C, let {un } and {un } be two sequences defined by (BIM), that is, un = (1 − αn )Sn un + αn u, vn = (1 − αn )Sn vn + αn v,

n ≥ 1, n ≥ 1.

It follows that ‖un − vn ‖ ≤ (1 − αn )‖Sn un − Sn vn ‖ + αn ‖u − v‖ ≤ (1 − αn )‖un − vn ‖ + αn ‖u − v‖.

Thus, ‖un − vn ‖ ≤ ‖u − v‖. Taking the limit, we find that ‖Pu − Pv‖ ≤ ‖u − v‖. This shows that P : C → C is nonexpansive. From (i) of Proposition 4.4.2, we have P ∘ Φ is Meir– Keeler contractive. Using the Meir–Keeler fixed point theorem, we conclude that there exists a unique z ∈ C such that z = P ∘ Φ(z). Define another sequence {yn } ⊂ C by yn = (1 − αn )Sn yn + αn Φz,

n ≥ 1.

(4.60)

Since (E, C, {Sn }, αn ) has the Browder’s property, one has that sequence {yn } ⊂ C strongly converges to z = P ∘ Φz. Combining (VBIM) with (4.60), we have ‖xn − yn ‖ ≤ (1 − αn )‖Sn xn − Sn yn ‖ + αn ‖Φxn − Φz‖ ≤ (1 − αn )‖xn − yn ‖ + αn ‖Φxn − Φz‖.

It follows that ‖xn − yn ‖ ≤ ‖Φxn − Φz‖,

∀n ≥ 1.

(4.61)

If {xn } does not converge to z, then there exists some ε > 0 and a subsequence {xnk } such that ‖xnk − z‖ ≥ ε for any k ≥ 1. By applying Proposition 4.4.1, we see that there exists r ∈ (0, 1) such that ‖Φxnk − Φz‖ ≤ r‖xnk − z‖,

∀k ≥ 1.

(4.62)

It follows from (4.61) and (4.62) that ‖xnk − z‖ ≤ ‖xnk − ynk ‖ + ‖ynk − z‖

≤ ‖Φxnk − Φz‖ + ‖ynk − z‖ ≤ r‖xnk − z‖ + ‖ynk − z‖.

This further implies ‖xnk − z‖ ≤

1 ‖y − z‖ → 0 1 − r nk

(k → ∞),

which yields a contradiction to ‖xnk − z‖ ≥ ε. Consequently, xn → z (n → ∞). This completes the proof.

4.4 Moudafi’s viscosity iterative method | 233

By virtue of Theorem 4.4.1, we obtain the following result. Theorem 4.4.2. Let E be a real uniformly smooth Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a nonexpansive mapping such that Fix(T) ≠ 0. Let {Sn } be a family of nonexpansive mappings on C and let Φ : C → C be a Meir–Keeler contractive mapping. Then there exists a unique path {xt } satisfying the equation: xt = (1 − t)Txt + tΦxt ,

t ∈ (0, 1].

(VRIM)

Further, as t → 0, {xt } strongly converges to z = P ∘ Φz, where P : C → Fix(T) is the unique sunny nonexpansive retraction mapping from C onto Fix(T), and z is a unique solution of the following variational inequality problem: ⟨Φz − z, j(z − y)⟩ ≥ 0,

∀y ∈ Fix(T).

Proof. Claim (ii) of Proposition 4.4.2 guarantees the existence and uniqueness of path {xt }. The convergence of {xt } can be derived from Theorem 4.4.1. Finally, using Theorem 4.4.3, one concludes that z is the unique solution of the above variational inequality. This completes the proof. Proposition 4.4.4. Suppose that a system (E, C, {Sn }, αn ) has the Halpern’s property. Define Pu = limn→∞ yn , where {yn } is a sequence generated by y1 ∈ C,

yn+1 = (1 − αn )Sn yn + αn u,

∀n ≥ 1, u ∈ C.

Then the following conclusions hold: (i) Pu is independent of the initial value y1 ∈ C; (ii) P : C → C is nonexpansive. Proof. Fix u ∈ C and define {un } and {yn } as u1 ∈ C,

y1 ∈ C,

un+1 = (1 − αn )Sn un + αn u, yn+1 = (1 − αn )Sn yn + αn u,

n ≥ 1, n ≥ 1.

Therefore, ‖un+1 − yn+1 ‖ ≤ (1 − αn )‖Sn un − Sn yn ‖ ≤ (1 − αn )‖un − yn ‖,

which implies limn→∞ ‖un − yn ‖ = 0 󳨐⇒ Pu = limn→∞ yn = limn→∞ un , that is, Pu is independent of y1 ∈ C. Fix v ∈ C and define {vn } as v1 = v, vn+1 = (1 − αn )Sn vn + αn v,

n ≥ 1.

234 | 4 Nonexpansive mappings in Banach spaces Thus ‖un+1 − vn+1 ‖ ≤ (1 − αn )‖Sn un − Sn vn ‖ + αn ‖u − v‖ ≤ (1 − αn )‖un − vn ‖ + αn ‖u − v‖,

n ≥ 1.

(4.63)

By induction, we find that ‖un − vn ‖ ≤ ‖u − v‖. It follows that ‖Pu − Pv‖ ≤ ‖u − v‖, ∀n ≥ 1, which implies that P : C → C is nonexpansive. This completes the proof. Theorem 4.4.3 (Suzuki [83], 2007). Let C be a nonempty, closed, and convex subset of a real Banach space E. Let {Sn } be a family of nonexpansive mappings on C. Let {αn } be a real sequence in (0, 1] satisfying conditions (i) and (ii) of Theorem 4.3.2. Let Φ : C → C be a Meir–Keeler contractive mapping. Assume that the system (E, C, {Sn }, αn ) has the Halpern’s property. Let P : C → C be defined in Proposition 4.4.4. Let {xn } ⊂ C be a sequence defined by x1 ∈ C,

xn+1 = (1 − αn )Sn xn + αn Φxn ,

n ≥ 1.

(VHIS)

Then {xn } strongly converges to the unique point z = P ∘ Φz. Proof. It follows from Proposition 4.4.4 that P : C → C is nonexpansive. Using (i) of Proposition 4.4.2, we find that P ∘ Φ : C → C is Meir–Keeler contractive. From the Meir–Keeler fixed point theorem, one concludes that P ∘ Φ : C → C has a unique fixed point z. Define {yn } ⊂ C by y1 ∈ C,

yn+1 = (1 − αn )Sn yn + αn Φz,

n ≥ 1.

(4.64)

So, {yn } strongly converges to the point z = P ∘ Φz. We next prove that xn − yn → θ as n → ∞. Assume that lim supn→∞ ‖xn − yn ‖ > 0. We choose an ε such that 0 < ε < lim supn→∞ ‖xn − yn ‖. Theorem 4.4.1 asserts that there exists r ∈ (0, 1) such that r‖yn −z‖ < ε. Using the ‖x − y‖ ≥ ε, ∀x, y ∈ C. Letting v1 ≥ 1, we find that, for any n ≥ v1 , 1−r definition of Φ, we have ‖Φx − Φy‖ ≤ max{r‖x − y‖, ε}, ∀x, y ∈ C. Next the proof will be split into two cases. (1) There exist v2 ≥ 1 such that v2 ≥ v1 and ‖xv2 − xv1 ‖ ≤ ε; (2) ‖xn − yn ‖ > ε, ∀n ≥ v1 . In Case (1), we have ‖xv2 +1 − yv2 +1 ‖ ≤ (1 − αv2 )‖Sv2 xv2 − Sv2 yv2 ‖ + αv2 ‖Φxv2 − Φz‖ ≤ (1 − αv2 )‖xv2 − yv2 ‖ + αv2 max{‖xv2 − z‖, ε}

≤ max{(1 − αv2 + rαv2 )‖xv2 − yv2 ‖ + rαv2 ‖yv2 − z‖, (1 − αv2 )‖xv2 − yv2 ‖ + αv2 ε}

≤ max{(1 − αv2 + rαv2 )‖xv2 − yv2 ‖ + (αv2 − rαv2 ) (1 − αv2 )‖xv2 − yv2 ‖ + αv2 ε}. ≤ ε.

r‖yv2 − z‖ 1−r

,

4.4 Moudafi’s viscosity iterative method | 235

Thus ‖xn − yn ‖ ≤ ε, ∀n ≥ v2 . This yields a contradiction to 0 < ε < lim supn→∞ ‖xn − yn ‖. In Case (2), we have ‖xn+1 − yn+1 ‖ ≤ (1 − αn )‖Sn xn − Sn yn ‖ + αn ‖Φxn − Φz‖

≤ (1 − αn )‖xn − yn ‖ + αn ‖Φxn − Φyn ‖ + αn ‖Φyn − Φz‖

≤ (1 − αn + rαn )‖xn − yn ‖ + (αn − rαn )

r‖yn − z‖ . 1−r

From Lemma 1.10.2, one has xn − yn → θ as n → ∞. This yields a contradiction to 0 < ε < lim supn→∞ ‖xn − yn ‖. Thus xn − yn → θ as n → ∞. It follows from ‖xn − z‖ ≤ ‖xn − yn ‖ + ‖yn − z‖ → 0 that xn → z (n → ∞). This completes the proof. Remark 4.4.1. If, in addition, the sequence {αn } ⊂ (0, 1] is assumed to satisfy conditions (i)–(iv) of Theorem 4.3.2 for (E, C, {Sn }, αn ), then (E, C, {Sn }, αn ) has the Halpern’s property: for any u ∈ C, the sequence {yn } ⊂ C defined by y1 ∈ C,

yn+1 = (1 − αn )Sn yn + αn u,

n ≥ 1,

is strongly convergent. Then the result of Theorem 4.4.3 is still true. Theorem 4.4.4. Let C be a nonempty, closed, and convex subset of a real Banach space E. Let {Sn } be a family of nonexpansive mappings on C. Let {αn } be a real sequence in (0, 1] satisfying conditions (i)–(iv) of Theorem 4.3.2. Let Φ : C → C be a Meir–Keeler contractive mapping. Assume that (E, C, {Sn }, αn ) has the Halpern’s property. Let P : C → C be defined in Proposition 4.4.4. Let {xn } ⊂ C be a sequence defined by x1 ∈ C,

xn+1 = (1 − αn )Sn xn + αn Φxn ,

n ≥ 1.

Then {xn } strongly converges to the unique point z = P ∘ Φz. Theorem 4.4.5. Let C be a nonempty, closed, and convex subset of a real uniformly smooth Banach space E. Let T : C → E be a nonexpansive mapping with Fix(T) ≠ 0. Let {αn } be a real sequence in (0, 1] satisfying conditions (i)–(iii) or (iv) of Theorem 4.3.2. Let Φ : C → C be a Meir–Keeler contractive mapping. Let {xn } ⊂ C be a sequence defined by x1 ∈ C,

xn+1 = (1 − αn )Txn + αn Φxn ,

n ≥ 1.

Then {xn } strongly converges to the unique point z = P ∘ Φz, where P : C → Fix(T) is the unique sunny nonexpansive retraction and z is the unique solution of the following variational inequality problem: ⟨Φz − z, j(z − y)⟩ ≥ 0,

∀y ∈ Fix(T).

Proof. From Theorem 4.3.2, we find that (E, C, {T}, αn ) has the Halpern’s property. Using Theorem 4.4.4, we obtain that {xn } strongly converges to the unique point z = P∘Φz. From Proposition 4.4.3, we conclude that z is the unique solution of the above variational inequality.

236 | 4 Nonexpansive mappings in Banach spaces Theorem 4.4.6. Let E, C, T, P, Φ, and {xn } be as in Theorem 4.4.5. Let {αn } be a sequence real sequence in (0, 1] satisfying (i) and (ii) of Theorem 4.3.6. For any λ ∈ (0, 1), let Tλ = λI + (1 − λ)T. Let {xn } ⊂ C be a sequence defined by x1 ∈ C,

xn+1 = (1 − αn )Tλ xn + αn Φxn ,

n ≥ 1.

Then {xn } strongly converges to the unique point z = P ∘ Φz, where P : C → Fix(T) is the unique sunny nonexpansive retraction and z is the unique solution of the above variational inequality. Proof. Note that (E, C, {T}, αn ) has the Halpern’s property. Using Theorem 4.4.4, we find that {xn } strongly converges to the unique point z = P ∘ Φz. From Proposition 4.4.3, we find that z is the unique solution of the above variational inequality. Corollary 4.4.1. Let C be a closed convex subset of a real uniformly smooth Banach space E. Let T : C → E be a nonexpansive mapping with Fix(T) ≠ 0. Let {αn } be a real sequence in (0, 1] satisfying conditions (i)–(iii) or (iv) of Theorem 4.3.2. Let f : C → C be a contractive mapping. Define a sequence {xn } ⊂ C by x1 ∈ C,

xn+1 = (1 − αn )Txn + αn fxn ,

n ≥ 1.

Then {xn } strongly converges to the unique point z = P ∘ fz, where z is the unique solution of the following variational inequality ⟨fz − z, j(z − y)⟩ ≥ 0,

∀y ∈ Fix(T).

Corollary 4.4.2. Let E, C, T, P, and f be as in Theorem 4.4.5. Let {αn } be a real sequence in (0, 1] satisfying (i) and (ii) of Theorem 4.3.2. Let λ ∈ (0, 1) and Tλ = λI + (1 − λ)T. Let {xn } ⊂ C be a sequence defined by x1 ∈ C,

xn+1 = (1 − αn )Tλ xn + αn fxn ,

n ≥ 1.

Then {xn } strongly converges to the unique point z = P ∘ fz, where z is the unique solution of the above variational inequality.

4.5 Iterative methods for common fixed points of a family of nonexpansive mappings in Banach spaces In this section, we introduce some iterative methods based on the cyclic and the Wmapping techniques. Let C be a nonempty convex subset of a Banach space E. Let {Ti }ri=1 be a finite family of nonexpansive mappings. Let αi be a real number in (0, 1). Define a family of

4.5 Iterative methods for common fixed points |

237

nonexpansive mappings {Ui }ri=1 by U1 = α1 T1 + (1 − α1 )I,

U2 = α2 T2 U1 + (1 − α2 )I, .. .

W = Ur = αr Tr Ur−1 + (1 − αr )I. The mapping Ur is a called a W-mapping generated by T1 , T2 , T3 , . . . , Tr and α1 , α2 , α3 , . . . , αr . Proposition 4.5.1. Let C be a nonempty, closed, and convex subset of a real strictly convex Banach space E. Let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings with F = ⋂ri=1 Fix(Ti ) ≠ 0. Let {αi }ri=1 ⊂ (0, 1) be a sequence of real numbers such that 0 < αi < 1 (i = 1, 2, . . . , r − 1), and 0 < αr ≤ 1. Let W be a W-mapping generated by T1 , T2 , T3 , . . . , Tr and α1 , α2 , α3 , . . . , αr . Then each Ui , where 1 ≤ i ≤ r, is nonexpansive and Fix(W) = ⋂ri=1 Fix(Ti ). Proof. From the construction, one sees that Ui : C → C is averaged for i = 1, 2, . . . , r − 1. However, Ur : C → C is nonexpansive. It is obvious that ⋂ri=1 Fix(Ti ) ⊂ Fix(W). For any x ∈ Fix(W), we fix p ∈ F = ⋂ri=1 Fix(Ti ) ≠ 0. It follows that x = Wx = Ur x = αr Tr Ur−1 x + (1 − αr )x. This implies x = αr Tr Ur−1 x and ‖x − p‖ = ‖Tr Ur−1 x − p‖

.. .

≤ ‖Ur−1 x − p‖ 󵄩 󵄩 = 󵄩󵄩󵄩αr−1 Tr−1 Ur−2 x + (1 − αr−1 )x − p󵄩󵄩󵄩 ≤ αr−1 ‖Ur−2 x − p‖ + (1 − αr−1 )‖x − p‖ 󵄩 󵄩 = αr−1 󵄩󵄩󵄩αr−2 Tr−2 Ur−3 x + (1 − αr−2 )x − p󵄩󵄩󵄩 + (1 − αr−1 )‖x − p‖ ≤ αr−1 αr−2 ‖Ur−3 x − p‖ + (1 − αr−1 αr−2 )‖x − p‖

≤ αr−1 αr−2 ⋅ ⋅ ⋅ α2 α1 ‖T1 x − p‖ + (1 − αr−1 αr−2 ⋅ ⋅ ⋅ α2 α1 )‖x − p‖

≤ ‖x − p‖. It follows that

󵄩 󵄩 ‖x − p‖ = ‖T1 x − p‖ = ‖U1 x − p‖ = 󵄩󵄩󵄩α1 T1 x + (1 − α1 )x − p󵄩󵄩󵄩. Since E is strictly convex, one has x −p = T1 x −p. Hence, x = T1 x and x = U1 x. Similarly, one has 󵄩 󵄩 ‖x − p‖ = ‖T2 U1 x − p‖ = ‖U2 x − p‖ = 󵄩󵄩󵄩α2 T2 U1 x + (1 − α2 )x − p󵄩󵄩󵄩.

238 | 4 Nonexpansive mappings in Banach spaces Since E is strictly convex, we find that x − p = T2 x − p. This shows that x = T2 x and x = U2 x. In a similar way, one also see that x = Tk x, x = Uk x (k = 3, 4, . . . , r − 1). In view of x = Tr Uk−1 x, we find that x = Tr x. Hence x ∈ ⋂ri=1 Fix(Ti ). Theorem 4.5.1. Let E be a real uniformly convex Banach space with a Fréchet differentiable norm. Let C be a closed convex subset of E. Let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings with F = ⋂ri=1 Fix(Ti ) ≠ 0. Let {αi }ri=1 ⊂ (0, 1) be a real sequence such that 0 < αi < 1 (i = 1, 2, . . . , r − 1), and 0 < αr ≤ 1. Let W be a W-mapping generated by T1 , T2 , T3 , . . . , Tr and α1 , α2 , α3 , . . . , αr . Let {βn } ⊂ [0, 1] be a real sequence such that 0 ≤ βn < 1 (n ≥ 1) and Σ∞ n=1 βn (1 − βn ) = ∞. Define a sequence {xn } by x1 = x ∈ C,

xn+1 = βn xn + (1 − βn )Wxn ,

n ≥ 1.

Then {xn } converges weakly to a common fixed point of {Ti }ri=1 . Proof. From Proposition 4.5.1 and Reich’s convergence theorem, we can obtain the desired conclusion immediately. Theorem 4.5.2. Let E be a real uniformly convex Banach space with a Gâteaux differentiable norm. Let C be a closed convex subset of E. Let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings with F = ⋂ri=1 Fix(Ti ) ≠ 0. Let {αi }ri=1 ⊂ (0, 1) be a real sequence such that 0 < αi < 1 (i = 1, 2, . . . , r − 1) and 0 < αr ≤ 1. Let W be a W-mapping generated by T1 , T2 , T3 , . . . , Tr and α1 , α2 , α3 , . . . , αr . Let {βn } ⊂ [0, 1] be a real sequence such that (1) βn → 0 (n → ∞); (2) ∑∞ n=1 βn = ∞; (3) ∑∞ n=1 |βn−1 − βn | < ∞; or β (4) β n → 1 (n → ∞). n+1

Define a sequence {xn } by u, x1 ∈ C,

xn+1 = βn u + (1 − βn )Wxn ,

n ≥ 1.

Then {xn } converges strongly to z = QF u, a common fixed point of {Ti }ri=1 . Proof. From Proposition 4.5.1 and Xu’s convergence theorem, we can obtain the desired result easily. Theorem 4.5.3. Let E, C, {Ti }ri=1 , αi , and W be defined as in Theorem 4.5.2. Let {βn } ⊂ [0, 1] be a real sequence satisfying limn→∞ βn = 0 and ∑∞ n=1 βn = ∞. Define a sequence {xn } by u, x1 ∈ C,

xn+1 = βn u + (1 − βn )Wλ xn ,

n ≥ 1,

where λ ∈ (0, 1) is a real constant and Wλ = λW + (1 − λ)I. Then {xn } converges strongly to z = QF u, a common fixed point of {Ti }ri=1 .

4.5 Iterative methods for common fixed points |

239

Proof. From Proposition 4.5.1 and Suzuki’s convergence theorem, we can obtain the desired result immediately. We next introduce cyclic iterative methods x1 ∈ C,

xn+1 = (1 − αn )xn + αn Tn xn ,

n ≥ 1,

(I)

and x0 , u ∈ C,

xn+1 = αn+1 u + (1 − αn+1 )Tn+1 xn ,

n ≥ 0,

(II)

where Tn := Tn mod r , where modular function “mod r” takes the values in {1, 2, . . . , r}. Theorem 4.5.4. Let E be a real uniformly convex Banach space with a Fréchet differentiable norm. Let C be a closed convex subset of E. Let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings with F = ⋂ri=1 Fix(Ti ) ≠ 0. Let {αn } ⊂ (0, 1) be a real sequence such that lim infn αn (1 − αn ) > 0. Let {xn } be a sequence generated by (I). Then {xn } converges weakly to a common fixed point of {Ti }ri=1 . Proof. First, we show that {xn } is bounded. Fixing p ∈ F, we find that ‖xn+1 − p‖ ≤ (1 − αn )‖xn − p‖ + αn ‖Tn xn − p‖ ≤ (1 − αn )‖xn − p‖ + αn ‖xn − p‖ = ‖xn − p‖.

Thus limn→∞ ‖xn − p‖ exists, which in turn shows that {xn } is bounded. Let r = supn≥1 {‖xn − p‖}. From Theorem 1.8.9, we find that 󵄩 󵄩2 ‖xn+1 − p‖2 = 󵄩󵄩󵄩(1 − αn )(xn − p) + αn (Tn xn − p)󵄩󵄩󵄩 ≤ (1 − αn )‖xn − p‖2 + αn ‖Tn xn − p‖2 − αn (1 − αn )g(‖xn − Tn xn ‖)

≤ ‖xn − p‖2 − αn (1 − αn )g(‖xn − Tn xn ‖). This implies that αn (1 − αn )g(‖xn − Tn xn ‖) ≤ ‖xn − p‖2 − ‖xn+1 − p‖2 . In view of lim infn αn (1 − αn ) > 0, we have g(‖xn − Tn xn ‖) → 0 as n → ∞. From the property of g, we find that xn − Tn xn → θ as n → ∞. From (I), one sees that xn+1 − xn = αn (Tn xn − xn ) → θ as n → ∞. This shows that xn+1 − xn → θ as n → ∞, and xn+i − Tn+i xn+i → θ as n → ∞. Hence xn − Tn+i xn → θ (n → ∞),

∀i = 1, 2, . . . , r.

240 | 4 Nonexpansive mappings in Banach spaces For any l = 1, 2, . . . , r, there exists i = 1, 2, . . . , r such that n + i = l mod r ∀n ≥ 1. This yields xn − Tl xn = xn − Tn+i xn → θ as n → ∞. Due to the reflexivity of E and the boundedness of {xn }, we may assume that xn ⇀ p as n → ∞. From Theorem 1.9.3, we have that p = Tl p, where l = 1, 2, . . . , r, that is, p ∈ F. In view of Lemma 4.3.1, one concludes that {xn } weakly converges to a common fixed point of {Ti }ri=1 . Theorem 4.5.5. Let E be a real uniformly convex Banach space with a Gâteaux differentiable norm. Let C be a closed convex subset of E. Let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings such that F = ⋂ri=1 Fix(Ti ) ≠ 0 and F = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) = Fix(T1 Tr ⋅ ⋅ ⋅ T3 T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 Tr−2 ⋅ ⋅ ⋅ T1 Tr ). Let {αi } ⊂ (0, 1) be a real sequence satisfying the following conditions: (1) αn → 0 (n → ∞); (2) ∑∞ n=1 αn = ∞; (3) ∑∞ n=1 |αn+r − αn | < ∞; or α (4) α n → 1 (n → ∞). n+r

Let {xn } be a sequence generated by (II). Then {xn } converges strongly to p = QF u, a common fixed point of {Ti }ri=1 . Proof. The proof is split into five steps. Step 1. Show that {xn } is bounded. For any p ∈ F, it follows that ‖xn+1 − p‖ ≤ αn+1 ‖u − p‖ + (1 − αn+1 )‖Tn+1 xn − p‖ ≤ αn+1 ‖u − p‖ + (1 − αn+1 )‖xn − p‖ ≤ max{‖u − p‖, ‖x1 − p‖} ≜ M,

∀n ≥ 1.

Thus {xn } is bounded, so is {Tn+1 xn }. This shows that xn+1 − Tn+1 xn → θ as n → ∞. Step 2. Show that xn+r − xn → θ as n → ∞. Note that 󵄩 󵄩 ‖xn+r − xn ‖ = 󵄩󵄩󵄩(αn+r − αn )u + (1 − αn+r )Tn+r xn+r−1 − (1 − αn−1 )Tn xn−1 󵄩󵄩󵄩 ≤ |αn+r − αn |(‖u‖ + ‖Tn xn−1 ‖) + (1 − αn+r )‖xn+r−1 − xn−1 ‖

≤ (1 − αn+r )‖xn+r−1 − xn−1 ‖ + M|αn+r − αn |.

From Lemma 1.10.2, we conclude that xn+r − xn → θ as n → ∞. Step 3. Show that xn − Tn+r ⋅ ⋅ ⋅ Tn+1 xn → θ as n → ∞. In view of assumption that αn → 0 as n → ∞, we have, as n → ∞, xn+r − Tn+r xn+r−1 → θ.

(4.65)

Taking into account the nonexpansivity of Tn+r yields Tn+r xn+r−1 − Tn+r Tn+r−1 xn+r−2 → θ.

(4.66)

4.5 Iterative methods for common fixed points | 241

Similarly, one has Tn+r Tn+r−1 xn+r−2 − Tn+r Tn+r−1 Tn+r−2 xn+r−3 → θ

(4.67)

Tn+r ⋅ ⋅ ⋅ Tn+2 Tn+1 xn+1 − Tn+r ⋅ ⋅ ⋅ Tn+1 xn → θ.

(4.68)

.. .

Using (4.65)–(4.68), we find that xn+r − Tn+r ⋅ ⋅ ⋅ Tn+1 xn → θ as n → ∞. From Step 2, we get that xn − Tn+r ⋅ ⋅ ⋅ Tn+1 xn → θ as n → ∞. Step 4. Show that lim supn ⟨u − p, j(xn − p)⟩ ≤ 0, where p = QF u. Pick subsequences of {xn } as follows: {nk } : 1, r + 1, 2r + 1, . . . ,

{mj } : 2, r + 2, 2r + 2, . . . , .. .

{si } : r, 2r, 3r, . . . . Thus, xnk − T1 Tr Tr−1 Tn+r ⋅ ⋅ ⋅ T2 xnk → θ xmj − T2 T1 Tr ⋅ ⋅ ⋅ T3 xmj → θ xsi − Tr Tr−1 ⋅ ⋅ ⋅ T1 xsi → θ

(k → ∞),

(j → ∞), (i → ∞).

Since Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) = Fix(T1 Tr ⋅ ⋅ ⋅ T3 T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 Tr−2 ⋅ ⋅ ⋅ T1 Tr ), one concludes from Lemma 4.3.1 that lim sup⟨u − p, j(xnk − p)⟩ ≤ 0, k

lim sup⟨u − p, j(xmj − p)⟩ ≤ 0, j

lim sup⟨u − p, j(xsi − p)⟩ ≤ 0. i

Thus we find that lim supn ⟨u − p, j(xn − p)⟩ ≤ 0. Step 5. Show that xn → p = QF u as n → ∞. From (II), we obtain that ‖xn+1 − p‖2 = αn+1 ⟨u − p, j(xn+1 − p)⟩ + (1 − αn+1 )⟨Tn+1 xn − p, j(xn+1 − p)⟩

≤ αn+1 ‖xn − p‖‖xn+1 − p‖ + αn+1 ⟨u − p, j(xn+1 − p)⟩ 1 1 ≤ (1 − αn+1 )‖xn − p‖2 + ‖xn+1 − p‖2 + αn+1 ⟨u − p, j(xn+1 − p)⟩. 2 2

242 | 4 Nonexpansive mappings in Banach spaces This yields ‖xn+1 − p‖2 ≤ (1 − αn+1 )‖xn − p‖2 + 2αn+1 ⟨u − p, j(xn+1 − p)⟩ ≤ (1 − αn+1 )‖xn − p‖2 + o(αn+1 ).

From Lemma 1.10.2, we conclude that xn → p = QF u as n → ∞. Next, we discuss the iterative construction problem of common fixed points of a countable infinite family of nonexpansive mappings. We introduce a more general W-mapping as follows. Let C be a convex subset of a real Banach space E. Let {Ti }∞ i=1 : C → C be a family ∞ of nonexpansive mappings. Let {αi }i=1 be a real sequence in [0, 1]. For any n ≥ 1, define a mapping Wn : C → C as Wn = Un,1 = α1 T1 Un,2 + (1 − α1 )I, where Un,2 = α2 T2 Un,3 + (1 − α2 )I, .. .

Un,k−1 = αk−1 Tk−1 Un,k + (1 − αk−1 )I, Un,k = αk Tk Un,k+1 + (1 − αk )I, .. .

Un,n−1 = αn−1 Tn−1 Un,n + (1 − αn−1 )I, Un,n = αn Tn Un,n+1 + (1 − αn )I, Un,n+1 = I.

Then Wn is called the W-mapping generated by Tn , Tn−1 , . . . , T1 and αn , αn−1 , . . . , α1 . Proposition 4.5.2. Let C be a nonempty, closed, and convex subset of a real Banach space E. Let {Ti }∞ i=1 : C → C be an infinite countable family of nonexpansive mappings ∞ with F = ⋂∞ Fix(T i ) ≠ 0. Let {αi }i=1 be real sequence in [0, 1] and Wn be the W-mapping i=1 generated by Tn , Tn−1 , . . . , T1 and αn , αn−1 , . . . , α1 . If E is strictly convex, then Fix(Wn ) = ∞ ⋂ni=1 Fix(Ti ), where ⋂∞ n=1 Fix(Wn ) = ⋂i=1 Fix(Tn ). Proposition 4.5.3. Let C be a real Banach space. Let {Ti }∞ i=1 : C → C be an infinite ∞ countable family of nonexpansive mappings with ⋂∞ Fix(T i ) ≠ 0. Let {αi }i=1 be a real i=1 sequence in [0, 1] satisfying 0 < αi ≤ b < 1. Then, for any x ∈ C, k ≥ 1, limn→∞ Un,k x exists. Furthermore, if D is a bounded subset of C, then ∀k ≥ 1, limn→∞ Un,k x exists for any x ∈ D.

4.5 Iterative methods for common fixed points | 243

Proof. For ∀x ∈ C, fixing ω ∈ ⋂ri=1 Fix(Ti ) with x ≠ ω and k ≥ 1, n ≥ 1, one has ‖Un+1,k x − Un,k x‖ 󵄩 󵄩 = 󵄩󵄩󵄩αk Tk Un+1,k+1 x + (1 − αk )x − αk Tk Un,k+1 x − (1 − αk )x󵄩󵄩󵄩 = αk ‖Tk Un+1,k+1 x − Tk Un,k+1 x‖ ≤ αk ‖Un+1,k+1 x − Un,k+1 x‖ 󵄩 = αk 󵄩󵄩󵄩αk+1 Tk+1 Un+1,k+2 x + (1 − αk+1 )x 󵄩 − αk+1 Tk+1 Un,k+2 x − (1 − αk+1 )x󵄩󵄩󵄩 = αk αk+1 ‖Tk+1 Un+1,k+2 x − Tk+1 Un,k+2 x‖

.. .

≤ αk αk+1 ‖Un+1,k+2 x − Un,k+2 x‖

≤ (Πni=k αi )‖Un+1,n+1 x − Un,n+1 x‖ 󵄩 󵄩 ≤ (Πni=k αi )󵄩󵄩󵄩αn+1 Tn+1 Un+1,n+2 x + (1 − αn+1 )x − x 󵄩󵄩󵄩 ≤ (Πn+1 i=k αi )‖Tn+1 x − x‖ ≤ (Πn+1 i=k αi )(‖Tn+1 x − ω‖ + ‖ω − x‖) ≤ (Πn+1 i=k αi )(‖x − ω‖ + ‖ω − x‖)

= 2(Πn+1 i=k αi )‖x − ω‖ ≤ 2bn−k+2 ‖x − ω‖.

For any ε > 0, there exists n0 ≥ k ≥ 1 such that ε(1 − b) , 2‖x − ω‖

bn0 −k+2 < then

m−1

‖Um,k x − Un,k x‖ ≤ ∑ ‖Uj+1,k x − Uj,k x‖ j=n

m−1

≤ 2‖x − ω‖ ∑ bj−k+2 j=n

n−k+2

≤

2b

< ε,

‖x − ω‖ 1−b ∀m > n > n0 ,

which implies that {Un,k x} is a Cauchy sequence. This yields that limn→∞ Un,k x exists. Let M = sup{‖x − ω‖ : x ∈ D}. Then, for any ε > 0, x ∈ D, one finds that there exists n0 ≥ k ≥ 1, where n0 is independent of x ∈ D such that if bn0 −k+2
n > n0 .

This implies that limn→∞ Un,k x is uniform with respect to x ∈ D. For any k ≥ 1, we define U∞,k , W : C → C by U∞,k x := lim Un,k x, Wx := lim Wn x = lim Un,1 x, n→∞

n→∞

n→∞

∀x ∈ C.

We say that W is a W-mapping generated by T1 , T2 , . . . and α1 , α2 , . . . Proposition 4.5.4. Let C be a nonempty, closed, and convex subset of a real strictly convex Banach space E. Let {Ti }∞ i=1 : C → C be an infinite countable family of nonexpan∞ sive mappings with F = ⋂∞ i=1 Fix(Ti ) ≠ 0. Let {αi }i=1 be a real sequence in [0, 1] with ∞ 0 < αi ≤ b < 1. Then Fix(W) = ⋂i=1 Fix(Ti ). Proof. Fixing ω ∈ ⋂∞ i=1 Fix(Ti ), one has Un,k ω = ω, ∀n ≥ k ≥ 1, which implies U∞,k ω = ω, ∀k ≥ 1, which further implies that W∞ ω = U∞,1 ω = ω. Hence, ⋂∞ i=1 Fix(Ti ) ⊂ Fix(W). For any x ∈ Fix(W) and any y ∈ ⋂∞ Fix(T ), we have i i=1 ‖Wn x − Wn y‖ 󵄩 󵄩 = 󵄩󵄩󵄩α1 T1 Un,2 x + (1 − α1 )x − α1 T1 Un,2 y − (1 − α1 )y󵄩󵄩󵄩 󵄩 󵄩 ≤ α1 ‖T1 Un,2 x − T1 Un,2 y‖ + (1 − α1 )󵄩󵄩󵄩(x − y)󵄩󵄩󵄩 󵄩 󵄩 ≤ α1 ‖Un,2 x − Un,2 y‖ + (1 − α1 )󵄩󵄩󵄩(x − y)󵄩󵄩󵄩 .. . 󵄩󵄩 󵄩󵄩 k−1 ≤ (Πk−1 i=1 αi )‖Un,k x − Un,k y‖ + (1 − Πi=1 αi )󵄩 󵄩(x − y)󵄩󵄩 󵄩󵄩 ≤ (Πk−1 󵄩αk Tk Un,k+1 x − (1 − αk )x − αk Tk Un,k+1 y i=1 αi )󵄩 󵄩 󵄩󵄩 󵄩󵄩 − (1 − αk )y󵄩󵄩󵄩 + (1 − Πk−1 i=1 αi )󵄩 󵄩(x − y)󵄩󵄩 󵄩󵄩 ≤ (Πk−1 󵄩αk (Tk Un,k+1 x − Tk Un,k+1 y) i=1 αi )󵄩 󵄩 󵄩󵄩 󵄩󵄩 + (1 − αk )(x − y)󵄩󵄩󵄩 + (1 − Πk−1 i=1 αi )󵄩 󵄩(x − y)󵄩󵄩 󵄩󵄩 󵄩󵄩 k−1 ≤ (Πk−1 i=1 αi )‖Tk Un,k+1 x − Tk Un,k+1 y‖ + (1 − Πi=1 αi )󵄩 󵄩(x − y)󵄩󵄩 󵄩󵄩 󵄩 k−1 ≤ (Πk−1 󵄩(x − y)󵄩󵄩󵄩. i=1 αi )‖Un,k+1 x − Un,k+1 y‖ + (1 − Πi=1 αi )󵄩 .. . 󵄩 󵄩 ≤ 󵄩󵄩󵄩(x − y)󵄩󵄩󵄩.

4.5 Iterative methods for common fixed points |

245

Taking the limit as n → ∞, we find that ‖Wn x − Wn y‖ 󵄩󵄩 ≤ (Πk−1 i=1 αi )󵄩 󵄩αk (Tk U∞,k+1 x − Tk U∞,k+1 y) 󵄩󵄩 󵄩󵄩 󵄩 + (1 − αk )(x − y)󵄩󵄩󵄩 + (1 − Πk−1 i=1 αi )󵄩 󵄩(x − y)󵄩󵄩

󵄩󵄩 󵄩󵄩 ≤ (Πki=1 αi )‖Tk U∞,k+1 x − Tk U∞,k+1 y‖ + (1 − Πk−1 i=1 αi )󵄩 󵄩(x − y)󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(x − y)󵄩󵄩󵄩.

Due to ‖Wn x − Wn y‖ = ‖x − y‖ and 0 < αi < 1, one has 󵄩󵄩 󵄩 󵄩󵄩αk (Tk U∞,k+1 x − Tk U∞,k+1 y) + (1 − αk )(x − y)󵄩󵄩󵄩 = ‖Tk U∞,k+1 x − Tk U∞,k+1 y‖. = ‖x − y‖,

∀k ≥ 1.

Since space E is strictly convex and y = Ti y, ∀i = 1, 2, . . . , one has x − y = Tk U∞,k+1 x − Tk U∞,k+1 y = Tk U∞,k+1 x − y

󳨐⇒ x = Tk U∞,k+1 x.

From Un,k+1 x = αk+1 Tk+1 Un,k+2 x + (1 − αk+1 )x, one has U∞,k+1 x = lim Un,k+1 x = αk+1 Tk+1 U∞,k+2 x + (1 − αk+1 )x n

= αk+1 x + (1 − αk+1 )x

= x.

Then x = Tk U∞,k+1 x = Tk x, ∀k ≥ 1, that is, x ∈ ⋂∞ i=1 Fix(Ti ). This implies that Fix(W) = Fix(T ). ⋂∞ i i=1 Remark 4.5.1. If αi ∈ (0, 1), then both Un,k and Wn are averaged for any k ≥ 1 and any n ≥ k. If αi ∈ (0, b] and b < 1, then W is also averaged. Theorem 4.5.6. Let E be a real uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. Let C be a closed convex subset of E. Let {Ti }∞ i=1 : C → C be an infinite countable family of nonexpansive mappings with F = ⋂∞ i=1 Fix(Ti ) ≠ 0. Let {αi }∞ i=1 ∈ [0, 1] be a real sequence satisfying 0 < αi ≤ b < 1 ∀i = 1, 2, . . . Let Wn be the W-mapping generated by Tn , Tn−1 , . . . , T1 and αn , αn−1 , . . . , α1 and let W be the W-mapping generated by T1 , T2 , . . . and α1 , α2 , . . . Let {βn } ⊂ [0, 1] be a real sequence such that limn→∞ βn = 0 and ∑∞ n=1 βn = ∞. Define a sequence {xn } by u, x1 ∈ C,

xn+1 = βn u + (1 − βn )Wn xn ,

Then {xn } converges to QF u in norm.

n ≥ 1,

246 | 4 Nonexpansive mappings in Banach spaces Proof. Define a new sequence {yn } ⊂ C by u, y1 ∈ C,

yn+1 = βn u + (1 − βn )Wyn ,

n ≥ 1.

Since W is averaged, one concludes from Suzuki’s convergence theorem that yn → QF u as n → ∞. Since Wn is nonexpansive, one has ‖xn+1 − yn+1 ‖ ≤ (1 − βn )‖Wn xn − Wn yn ‖

≤ (1 − βn )‖Wn xn − Wn yn ‖ + ‖Wn yn − Wn yn ‖ ≤ (1 − βn )‖xn − yn ‖ + ‖Wn yn − Wn yn ‖.

Since {yn } is convergent, one finds that {yn } is a bounded sequence. Letting D = supn≥1 {‖yn − ω‖}, ω ∈ F, ∀m > n ≥ 1, we have ‖Wn yn − Wn yn ‖ = lim ‖Wn yn − Wn yn ‖ m

m−1

≤ lim ∑ ‖Wj+1 yn − Wj yn ‖ m

j=n

∞

≤ ∑ 2Dbj+1 j=n

=

2Dbn+1 , 1−b

since the limit Wx = limn Wn x is uniform in x. This implies ∞

∑ ‖Wyn − Wn yn ‖ ≤

j=n

2D ∞ n+1 2Db2 < +∞. ∑b = 1 − b n=1 (1 − b)2

Using Lemma 1.10.2, one concludes that xn − yn → θ as n → ∞. Thus, xn → QF u as n → ∞. This completes the proof. Theorem 4.5.7. Let E be a real uniformly convex Banach space whose norm is Fréchet differentiable. Let C be a closed convex subset of E. Let {Ti }∞ i=1 : C → C be an infinite ∞ countable family of nonexpansive mappings with F = ⋂i=1 Fix(Ti ) ≠ 0. Let {αi }∞ i=1 ⊂ [0, 1] be a real sequence satisfying 0 < αi ≤ b < 1 (i = 1, 2, . . . ) and ∑∞ α (1 − α ) n = ∞. Let n=1 n Wn be the W-mapping generated by T1 , T2 , . . . and α1 , α2 , . . . Define a sequence {xn } by x1 ∈ C,

xn+1 = (1 − αn )xn + αn Wn xn ,

n ≥ 1.

Then {xn } weakly converges to a common fixed point of {Ti }∞ i=1 . Proof. (i) Show that {xn } is bounded.

4.5 Iterative methods for common fixed points | 247

Fixing p ∈ F, one gets ‖xn+1 − p‖ ≤ (1 − αn )‖xn − p‖ + αn ‖Wn xn − p‖ ≤ (1 − αn )‖xn − p‖ + αn ‖xn − p‖.

= ‖xn − p‖.

It follows that limn ‖xn − p‖ exists. This proves that {xn } is bounded. (ii) Show that xn − Wn xn → θ as n → ∞. Using Theorem 1.8.9, we have that ‖xn+1 − p‖2 ≤ (1 − αn )‖xn − p‖2 + αn ‖Wn xn − p‖2 − αn (1 − αn )g(‖xn − Wn xn ‖)

≤ ‖xn − p‖2 − αn (1 − αn )g(‖xn − Wn xn ‖). It follows that ∞

∑ αn (1 − αn )g(‖xn − Wn xn ‖) < +∞.

n=1

Since ∑∞ n=1 αn (1 − αn ) = ∞, we obtain that lim infn→∞ g(‖xn − Wn xn ‖) = 0. From the property of g, we find that lim inf ‖xn − Wn xn ‖ = 0. n→∞

On the other hand, one has ‖xn+1 − Wn+1 xn+1 ‖

≤ (1 − αn )‖xn − Wn xn ‖ + ‖Wn xn − Wn+1 xn+1 ‖

≤ (1 − αn )‖xn − Wn xn ‖ + ‖Wn xn − Wn+1 xn+1 ‖+

≤ (1 − αn )‖xn − Wn xn ‖ + ‖xn+1 − xn ‖ + ‖Wn+1 xn − Wn xn ‖ ≤ ‖xn − Wn xn ‖ + ‖Wn+1 xn − Wn xn ‖.

In view of the fact that ∑∞ n=1 ‖Wn+1 xn − Wn xn ‖ < +∞, one concludes from Lemma 1.10.1 that limn→∞ ‖xn − Wn xn ‖ exists. Hence xn − Wn xn → θ as n → ∞. Taking into account that ‖Wn xn − Wxn ‖ ≤ M1 bn−1 → 0 (n → ∞), one obtains ‖xn − Wxn ‖ ≤ ‖xn − Wn xn ‖ + ‖Wn xn − Wxn ‖ → 0

(n → ∞).

From Theorem 1.9.3, one sees that ω(xn ) ⊂ Fix(W). Using Theorem 1.9.8, one concludes xn ⇀ p ∈ Fix(W) as n → ∞. Using Proposition 4.5.4, one concludes that Fix(W) = ⋂∞ i=1 Fix(Ti ) = F. Therefore, xn ⇀ p ∈ F as n → ∞. This completes the proof.

248 | 4 Nonexpansive mappings in Banach spaces

4.6 Iterative methods of common fixed points for a nonexpansive semigroup in Banach spaces Definition 4.6.1. Let C be a nonempty, convex, and closed subset of a real Banach space. Let S = {T(s) : 0 ≤ s < +∞} be a family of mappings defined on C. Then S is said to be a one-parameter nonexpansive semigroup if it satisfies the following conditions: (i) T(0)x = x, ∀x ∈ C; (ii) T(s + t) = T(s)T(t), ∀s, t ∈ ℝ+ ; (iii) ‖T(s)x − T(s)y‖ ≤ ‖x − y‖, ∀x, y ∈ C, s ∈ ℝ+ ; (iv) For any x ∈ C, s 󳨃→ T(s)x is continuous. In this subsection, let Fix(S) = ⋂0≤t0 . Proof. Fixing x1 ∈ C and p ∈ F, let r = ‖x1 − p‖ and K = {x ∈ E, ‖x − p‖ ≤ r} ∩ C. Then K is a nonempty, bounded, closed, and convex subset of C and T(t)K ⊂ K, ∀t ∈ ℝ+ . Without loss of generality, we may assume that C is bounded. Using Theorem 1.8.9, we have tn 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩 1 󵄩 ‖xn+1 − ω‖ ≤ 󵄩󵄩αn (xn − ω) + (1 − αn )( ∫ T(s)xn ds − ω)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 tn 󵄩 󵄩 0 2

2 󵄩󵄩 tn 󵄩󵄩󵄩 󵄩󵄩 1 󵄩󵄩 󵄩 ≤ αn ‖xn − ω‖ + (1 − αn )󵄩󵄩 ∫ T(s)xn ds − ω󵄩󵄩 󵄩󵄩 󵄩󵄩 tn 󵄩 0 󵄩 2

4.6 Iterative methods of common fixed points | 249

󵄩󵄩 tn 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 − αn (1 − αn )g(󵄩󵄩󵄩 ∫ T(s)xn ds − xn 󵄩󵄩󵄩), 󵄩󵄩 tn 󵄩󵄩 󵄩 0 󵄩

ω ∈ Fix(S).

From αn ≤ a, we have that 󵄩󵄩 tn 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 αn (1 − a)g(󵄩󵄩󵄩 ∫ T(s)xn ds − xn 󵄩󵄩󵄩) 󵄩󵄩 tn 󵄩󵄩 󵄩 0 󵄩

󵄩󵄩 󵄩󵄩 tn 󵄩󵄩 󵄩󵄩 1 ≤ αn (1 − αn )g(󵄩󵄩󵄩 ∫ T(s)xn ds − xn 󵄩󵄩󵄩) 󵄩󵄩 󵄩󵄩 tn 󵄩 󵄩 0

󵄩󵄩 tn 󵄩󵄩2 󵄩󵄩 1 󵄩󵄩 󵄩 ≤ αn ‖xn − ω‖ + (1 − αn )󵄩󵄩 ∫ T(s)xn ds − ω󵄩󵄩󵄩 − ‖xn+1 − ω‖2 󵄩󵄩 tn 󵄩󵄩 󵄩 0 󵄩 2

≤ αn ‖xn − ω‖2 + (1 − αn )‖xn − ω‖2 − ‖xn+1 − ω‖2

= ‖xn − ω‖2 − ‖xn+1 − ω‖2 .

This implies that limn→∞ ‖xn − ω‖2 exists. It follows that 󵄩󵄩 󵄩󵄩 tn 󵄩󵄩 󵄩󵄩 1 lim αn g(󵄩󵄩󵄩 ∫ T(s)xn ds − xn 󵄩󵄩󵄩) = 0. n→∞ 󵄩󵄩 󵄩󵄩 tn 󵄩 󵄩 0 From the property of g, we have 󵄩󵄩 󵄩󵄩 tn 󵄩󵄩 󵄩󵄩 1 lim αn 󵄩󵄩󵄩 ∫ T(s)xn ds − xn 󵄩󵄩󵄩 = 0. n→∞ 󵄩󵄩 󵄩󵄩 tn 󵄩 󵄩 0 From the definition of {xn }, we find that xn+1 −

tn

tn

0

0

1 1 ∫ T(s)xn ds = αn (xn − ∫ T(s)xn d) → θ tn tn

as n → ∞. Note that tn 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩T(t)xn+1 − xn+1 󵄩󵄩 ≤ 󵄩󵄩T(t)xn+1 − T(t)( ∫ T(s)xn ds)󵄩󵄩󵄩 󵄩󵄩 tn 󵄩󵄩󵄩 󵄩 0

tn tn 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 1 󵄩 + 󵄩󵄩T(t)( ∫ T(s)xn ds) − ∫ T(s)xn ds󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 tn tn 󵄩 󵄩 0 0

󵄩󵄩 󵄩󵄩 tn 󵄩󵄩 󵄩󵄩 1 + 󵄩󵄩󵄩 ∫ T(s)xn ds − xn+1 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 tn 󵄩 󵄩 0

250 | 4 Nonexpansive mappings in Banach spaces 󵄩󵄩 tn 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 ≤ 2󵄩󵄩󵄩 ∫ T(s)xn ds − xn+1 󵄩󵄩󵄩 󵄩󵄩 tn 󵄩󵄩 󵄩 0 󵄩

tn tn 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 1 󵄩 + 󵄩󵄩T(t)( ∫ T(s)xn ds) − ∫ T(s)xn ds󵄩󵄩󵄩, 󵄩󵄩 󵄩󵄩 t t n n 󵄩 󵄩 0 0

∀t ∈ ℝ+ .

Using Lemma 4.6.1, we find that T(t)xn − xn → θ, ∀t ∈ ℝ+ , as n → ∞. From Theorem 1.9.3, we obtain that wω (xn ) ⊂ Fix(s). Define a mapping Tn : C → C by tn

Tn x = αn x + (1 − αn )

1 ∫ T(s)xds. tn 0

Then xn+1 = Tn xn and Tn : C → C is a nonexpansive mapping with Fix(s) ⊂ ⋂∞ n=1 Fix(Tn ) ≠ 0. It follows from Theorem 1.9.8 that {xn } converges weakly to a common fixed point of {Tt }t>0 . Definition 4.6.2. Let C be a nonempty, convex, and closed subset of a real Banach space. Let S = {T(s) : 0 ≤ s < +∞} be a family of mappings defined on C. Then S is said to be uniformly asymptotically regular on C if for any bounded subsequence K ⊂ C, we have 󵄩 󵄩 lim sup󵄩󵄩T(h)(T(t)x) − T(t)x 󵄩󵄩󵄩 = 0,

t→∞ x∈K 󵄩

∀h ∈ ℝ+ .

Theorem 4.6.2. Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Let {Tt }t≥0 be a uniformly asymptotically regular nonexpansive semigroup on C such that F = ⋂t≥0 Fix(T(t)) ≠ 0. Suppose that tn → ∞, {αn } ⊂ (0, 1), αn → 0 as n → ∞. Then, for any u ∈ C, there exists a unique sequence {xn } ⊂ C such that xn = αn u + (1 − αn )T(tn )xn , and xn → p = QF u as n → ∞. Proof. Define a mapping Tn : C → C by Tn x = αn u + (1 − αn )T(tn )x,

∀x ∈ C.

Then 󵄩 󵄩 ‖Tn x − Tn y‖ = (1 − αn )󵄩󵄩󵄩T(tn )x − T(tn )y󵄩󵄩󵄩 ≤ (1 − αn )‖x − y‖,

∀x, y ∈ C,

which shows that Tn is contractive. From the Banach fixed point theorem, there exists a unique xn ∈ C such that Tn xn = xn , that is, the iterative method is converges. We next prove that {xn } is bounded. Indeed, for any z ∈ F, one has 󵄩 󵄩 󵄩 󵄩 ‖xn − z‖ = 󵄩󵄩󵄩αn (u − z)󵄩󵄩󵄩 + (1 − αn )󵄩󵄩󵄩T(tn )xn − z 󵄩󵄩󵄩 ≤ αn ‖u − z‖ + (1 − αn )‖xn − z‖.

4.6 Iterative methods of common fixed points |

251

This implies that ‖xn − z‖ ≤ ‖u − z‖. It shows that {xn } is bounded. Hence, {T(tn )xn } is also bounded. It follows that xn − T(tn )xn = αn (u − T(tn )xn ) → θ as n → ∞. From the hypothesis that tn → ∞ and {T(t)} is uniformly asymptotically regular, we find that, ∀h > 0, 󵄩 󵄩 󵄩 󵄩 lim 󵄩󵄩T(h)(T(tn )xn ) − T(tn )xn 󵄩󵄩󵄩 ≤ lim sup󵄩󵄩󵄩T(h)(T(tn )x) − T(tn )x 󵄩󵄩󵄩 = 0, n→∞

n→∞󵄩

x∈K

where K is a bounded set containing {xn }. It follows that 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩xn − T(h)xn 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn − T(tn )xn 󵄩󵄩󵄩 + 󵄩󵄩󵄩T(tn )xn − T(h)(T(tn )xn )󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩T(h)(T(tn )xn ) − T(h)xn 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 2󵄩󵄩󵄩xn − T(tn )xn 󵄩󵄩󵄩 + 󵄩󵄩󵄩T(h)(T(tn )xn ) − T(tn )xn 󵄩󵄩󵄩 → 0 as n → ∞. This shows that xn − T(tn )xn = αn (u − T(tn )xn ) → θ, ∀h > 0, as n → ∞. Let g : C → ℝ+ be a functional defined by g(y) = μn ‖xn − y‖,

∀y ∈ C.

Then g is convex, continuous, and g(y) → ∞ as ‖y‖ → ∞. There also exists z ∈ C such that g(z) = min{g(y) : y ∈ C}. Putting K = {x ∈ C : g(x) = min{g(y) : y ∈ C}}, one finds that K ≠ 0 is closed and convex, and T(h)K ⊂ K, ∀h > 0. Indeed, 󵄩 󵄩 ∀x ∈ K, g(T(h)x) = μn 󵄩󵄩󵄩xn − T(h)x󵄩󵄩󵄩 󵄩 󵄩 = μn 󵄩󵄩󵄩xn − T(h)xn + T(h)xn − T(h)x󵄩󵄩󵄩 󵄩 󵄩 ≤ μn 󵄩󵄩󵄩T(h)xn − T(h)x󵄩󵄩󵄩 ≤ μn ‖xn − x‖ = g(x). It follows that T(h)x ∈ K, ∀h > 0. Fixing p ∈ F, one finds that there exists a unique v ∈ K such that ‖p − v‖ = min{‖p − x‖ : x ∈ K}. It follows from p = T(h)p and T(h)v ∈ K that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩p − T(h)v󵄩󵄩󵄩 = 󵄩󵄩󵄩T(h)p − T(h)v󵄩󵄩󵄩 ≤ ‖p − v‖.

252 | 4 Nonexpansive mappings in Banach spaces From the uniqueness of v ∈ K, we have that T(h)v = v, ∀h ∈ ℝ+ . From Theorem 1.8.45, we have μn ⟨y − v, j(xn − v)⟩ ≤ 0,

∀y ∈ C.

In particular, one has μn ⟨u − v, j(xn − v)⟩ ≤ 0. This yields μn ‖xn − v‖2 ≤ μn ⟨u − v, j(xn − v)⟩ ≤ 0. Thus μn ‖xn − v‖2 = 0. This shows that there exists a subsequence {xnk } of {xn } such that xnk → v as k → ∞. If {xn } has another subsequence {xmj } such that xmj → p ∈ F as j → ∞, then p = v. Indeed, we have ‖xn − y‖2 ≤ ⟨u − y, j(xn − y)⟩,

∀y ∈ F.

Thus ⟨xn − u, j(xn − y)⟩ ≤ 0, ∀y ∈ F. Consequently, ⟨xnk − u, j(xnk − y)⟩ ≤ 0 and ⟨xmj − u, j(xmj − y)⟩ ≤ 0. Taking the limits as k → ∞, j → ∞ in the above inequalities, respectively, we find that ⟨v − u, j(v − p)⟩ ≤ 0 and ⟨p − u, j(p − v)⟩ ≤ 0. It follows that ‖v − p‖2 ≤ ⟨v − p, j(v − p)⟩ ≤ 0. This shows p = v. Hence, xn → p = QF u as n → ∞, completing the proof. Theorem 4.6.3. Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let C be a closed convex subset of E. Let {Tt }t≥0 be a uniformly asymptotically regular nonexpansive semigroup on C with F = ⋂t≥0 Fix(T(t)) ≠ 0. Assume that tn → ∞, {αn } ⊂ (0, 1) such that limn→∞ αn = 0 and ∑∞ n=1 αn = ∞. Let {xn } be a sequence defined by u, x1 ∈ C,

xn+1 = αn u + (1 − αn )T(tn )xn ,

Then xn → p = QF u as n → ∞.

∀n ≥ 1.

4.7 Iterative methods of fixed points for nonexpansive nonself-mappings | 253

Proof. First, we show that {xn } is bounded. Fixing z ∈ F, we have ‖xn+1 − z‖ ≤ αn ‖u − z‖ + (1 − αn )‖xn − z‖

≤ max{‖u − z‖, ‖x1 − z‖} = M.

This shows {xn } is bounded, so is {T(tn )xn }. Since limn→∞ αn = 0, we find that xn − T(tn )xn = αn (u − T(tn )xn ) → θ as n → ∞. Taking into account the fact that {Tt }t>0 is uniformly asymptotically regular, we have 󵄩 󵄩 lim 󵄩󵄩T(h)(T(tn )xn ) − T(tn )xn 󵄩󵄩󵄩 = 0,

n→∞󵄩

∀h > 0.

Note that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩xn+1 − T(h)xn+1 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn+1 − T(tn )xn 󵄩󵄩󵄩 + 󵄩󵄩󵄩T(tn )xn − T(h)T(tn )xn 󵄩󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩T(h)T(tn )xn − T(h)xn+1 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 2󵄩󵄩󵄩xn+1 − T(tn )xn 󵄩󵄩󵄩 + 󵄩󵄩󵄩T(h)T(tn )xn − T(tn )xn 󵄩󵄩󵄩,

∀h > 0.

It follows that xn − T(h)xn → θ, ∀h > 0, as n → ∞. From Lemma 4.3.1, we find that lim sup ⟨u − p, j(xn − p)⟩ ≤ 0, n→∞

where p = QF u. Finally we show that xn → p (n → ∞). Note that ‖xn+1 − p‖2 = αn ⟨u − p, j(xn+1 − p)⟩ + (1 − αn )⟨T(tn )xn − p, j(xn+1 − p)⟩ ≤ (1 − αn )‖xn − p‖‖xn+1 − p‖ + αn ⟨u − p, j(xn+1 − p)⟩ 1 − αn 1 ≤ ‖xn − p‖2 + ‖xn+1 − p‖2 + αn ⟨u − p, j(xn+1 − p)⟩. 2 2

This implies ‖xn+1 − p‖2 ≤ (1 − αn )‖xn − p‖2 + 2αn ⟨u − p, j(xn+1 − p)⟩. Using Lemma 1.10.2, one concludes that xn → p as n → ∞.

4.7 Iterative methods of fixed points for nonexpansive nonself-mappings Let C be a nonempty, closed, and convex subset of a real Banach space E. Let QC : E → C be a sunny nonexpansive retraction and let T : C → E be a nonexpansive mapping. Let Φ : C → E be a Meir–Keeler (MK) contractive mapping. For any t ∈ (0, 1), define a mapping TtΦ : C → C by TtΦ x = QC [tΦx + (1 − t)Tx],

∀x ∈ C.

254 | 4 Nonexpansive mappings in Banach spaces From Proposition 4.4.2, we have that TtΦ : C → C is a Meir–Keeler contractive mapping. Using Theorem 4.1.2, we find that TtΦ has a unique fixed point xt in C, that is, xt = QC [tΦ(xt ) + (1 − t)T(xt )],

t ∈ (0, 1).

(4.69)

Discretizing (4.69) yields x1 ∈ C,

xn+1 = QC [αn Φ(xn ) + (1 − αn )T(xn )],

n ≥ 1.

(4.70)

We also consider another iterative method x1 ∈ C,

xn+1 = λxn + (1 − λ)QC [αn Φ(xn ) + (1 − αn )T(xn )],

n ≥ 1,

(4.71)

where λ is a real number in (0, 1) and {αn } is a real sequence in (0, 1) which satisfies certain conditions. In 2008, Matsushita and Takahashi [49] introduced a new boundary condition, which is called the MT condition in order to study fixed points of nonexpansive nonself-mappings in Banach spaces. Definition 4.7.1. Let C be a nonempty, closed, and convex subset of a real smooth Banach space E. Let T : C → E be a nonself-mapping. Then T is said to satisfy the MT condition if Tx ∈ ℘Sx ,

∀x ∈ C,

where Sx = {y ∈ E : y ≠ x, QC y = x}, and QC : E → C is a sunny nonexpansive retraction from E onto C. Matsushita and Takahashi [49] established the following useful results. Proposition 4.7.1. Let E be a real smooth Banach space. Let C be a nonempty, closed, and convex subset of E such that C is a sunny nonexpansive retract. Let T : C → E be a nonself-mapping. If T : C → E satisfies the weak inward condition (WIC), then T satisfies the MT condition. Proposition 4.7.2. Let E be a real smooth and strictly convex Banach space. Let C be a nonempty, closed, and convex subset of E such that C is a sunny nonexpansive retract. Let T : C → E be a nonself-mapping. If Fix(T) ≠ 0, then T satisfies the MT condition. The importance of the MT condition is explained by the following propositions. Proposition 4.7.3. Let E be a real smooth Banach space. Let C be a nonempty, closed, and convex subset of E such that C is a sunny nonexpansive retract. Let T : C → E be a nonself-mapping. If T satisfies the MT condition, then Fix(T) = Fix(QC T).

4.7 Iterative methods of fixed points for nonexpansive nonself-mappings |

255

In 2014, Zhou and Wang [123] improved Proposition 4.7.3 and obtained the following more useful result. Proposition 4.7.4. Let E be a real smooth Banach space. Let C be a nonempty, closed, and convex subset of E such that C is a sunny nonexpansive retract. Let T : C → E be a nonself-mapping. If T satisfies the MT condition, then Fix(T) = Fix(QC T) = Fix(TQC ). Theorem 4.7.1. Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E such that C is a sunny nonexpansive retract. Let T : C → E be a nonexpansive nonself-mapping such that Fix(T) ≠ 0. Let Φ : C → E be an MK contractive mapping. If T : C → E satisfies the MT condition, then the path {xt } generated by (4.69) converges in norm to p ∈ Fix(T) and p is also the unique solution of the variational inequality problem ⟨(I − Φ)p, j(y − p)⟩ ≥ 0,

∀y ∈ Fix(T).

(VIP)

Proof. Letting yt = tΦ(xt ) + (1 − t)T(xt ), one finds that (4.69) reduces to xt = QC yt , which gives yt = t(ΦQC )yt + (1 − t)(TQC )yt .

(4.72)

Since ΦQC : E → E is MK contractive, we find that TQC : E → E is nonexpansive. From Theorem 4.4.2, {yt } converges in norm to a fixed point p ∈ Fix(TQC ) as t → 0. Using Proposition 4.7.4, we have Fix(TQC ) = Fix(T). In view of the facts that p ∈ C and QC p = p, we have ⟨(I − Φ)p, j(y − p)⟩ ≥ 0,

∀y ∈ Fix(T).

From the continuity of QC , as t → 0, we conclude that {xt } converges in norm to p ∈ Fix(T) and p is the unique solution of the variational inequality (VIP). This completes the proof. Theorem 4.7.2. Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E such that C is a sunny nonexpansive retract. Let T : C → E be a nonexpansive nonself-mapping such that Fix(T) ≠ 0 and satisfies the MT condition. Let Φ : C → E be an MK contractive mapping. Let {xn } be a sequence defined by (4.70). If the control sequence {αn } satisfies the following conditions: (i) αn → 0 (n → ∞); (ii) ∑∞ n=1 αn = ∞; αn (iii) ∑∞ → 1 (n → ∞). n=1 |αn+1 − αn | < ∞ or α n+1

Then {xn } converges in norm to p ∈ Fix(T) and p is the unique solution of the variational inequality (VIP).

256 | 4 Nonexpansive mappings in Banach spaces Proof. Letting yn = αn Φ(xn ) + (1 − αn )T(xn ), we find that (4.70) reduces to xn+1 = QC yn , which yields yn+1 = αn+1 (ΦQC )yn + (1 − αn+1 )(TQC )yn .

(4.73)

Since ΦQC : E → E is MK contractive, we have that TQC : E → E is nonexpansive. From Theorem 4.4.5, {yn } converges in norm to p ∈ Fix(TQC ) as n → ∞. Using Proposition 4.7.4, we obtain that Fix(TQC ) = Fix(T), which together with p ∈ C, QC p = p yields that p ∈ Fix(T) is the unique solution of the variational inequality (VIP). From the continuity of QC , we find that {xn } converges in norm to p ∈ Fix(T). This completes the proof. Theorem 4.7.3. Let E be a real uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of E such that C is a sunny nonexpansive retract. Let T : C → E be a nonexpansive nonself-mapping such that Fix(T) ≠ 0 and satisfies the MT condition. Let Φ : C → E be an MK contractive mapping. Let {xn } be a sequence defined by (4.71). If the control sequence {αn } satisfies conditions limn→∞ αn = 0 and ∑∞ n=1 αn = ∞, then {xn } converges in norm to p ∈ Fix(T) and p is the unique solution of the variational inequality (VIP). Proof. First, we show that there exists a unique point z in Fix(T) such that z = QFix(T) Φz.

(4.74)

Indeed, for any u ∈ E, t ∈ (0, 1), we can define a mapping Tt : E → E by Tt x = tu + (1 − t)QC TQC x,

∀x ∈ E.

(4.75)

It is obvious that Tt is contractive. By using the Banach contractive mapping principle, there exists a unique path {xt } such that xt = tu + (1 − t)QC TQC xt ,

∀t ∈ (0, 1).

(4.76)

From Theorem 4.3.1 or Corollary 4.3.1, we obtain that {xt } converges in norm to p = QFix(QC TQC )u, that is, limt→0 xt = p. Since T satisfies the MT condition, we assert that QC T : C → C also satisfies the MT condition. If not, then there exists x ∈ C such that QC Tx ∉ ℘Sx 󳨐⇒ QC Tx ∈ Sx 󳨐⇒ QC Tx ≠ x and QC QC Tx = x. But QC QC = QC . Hence QC Tx = x, which produces a contradiction. From Proposition 4.7.4, we find that Fix(QC TQC ) = Fix(QC T) = Fix(T).

257

4.7 Iterative methods of fixed points for nonexpansive nonself-mappings |

Thus limt→0 xt = QFix(T) u defines a unique sunny nonexpansive retraction from E onto Fix(T). Since QFix(T) Φ : C → C is an MK contractive self-mapping, we find from Theorem 4.1.2 that there exists a unique element z in Fix(T) such that z = QFix(T) Φ(z). Fix λ ∈ (0, 1) and define a sequence {yn } by y1 ∈ C,

yn+1 = λyn + (1 − λ)QC [αn Φ(z) + (1 − αn )Tyn ],

n ≥ 1.

(4.77)

It is easy to prove that {yn } converges to z in norm. Next, we plan to prove that xn −yn → θ as n → ∞. Assume that a = lim supn→∞ ‖xn − yn ‖ > 0; then for any ε ∈ (0, a), we can choose η > 0 such that lim sup ‖xn − yn ‖ > ε + η.

(4.78)

n→∞

For the above chosen η > 0, we find from [83, Proposition 2] that there exists γ ∈ (0, 1) such that ‖Φx − Φy‖ ≤ γ‖x − y‖,

∀x, y ∈ C, ‖x − y‖ ≥ ε,

(4.79)

which implies that ‖Φx − Φy‖ ≤ max{γ‖x − y‖, ε},

∀x, y ∈ C.

(4.80)

Since yn → z as n → ∞, we find that there exists n0 ∈ N such that ‖yn − z‖ ≤ (1 − γ)η,

∀n ≥ n0 .

(4.81)

Next, we consider the following two possible cases: (i) There exists ν1 ≥ n0 such that ‖xν1 − yν1 ‖ ≤ ε + η.

(4.82)

From (4.70), (4.77), (4.80), (4.81), and (4.82), we find that ‖xν1 +1 − yν1 +1 ‖ ≤ λ‖xν1 − yν1 ‖ + (1 − λ)[αν1 ‖Φxν1 − Φz‖ + (1 − αν1 )‖xν1 − yν1 ‖] = [1 − (1 − λ)αν1 ]‖xν1 − yν1 ‖ + (1 − λ)αν1 ‖Φxν1 − Φz‖

≤ [1 − (1 − λ)αν1 ]‖xν1 − yν1 ‖ + (1 − λ)αν1 ‖Φxν1 − Φyν1 ‖ + (1 − λ)αν1 ‖Φyν1 − Φz‖

≤ [1 − (1 − λ)αν1 ]‖xν1 − yν1 ‖ + (1 − λ)αν1 max{γ‖xν1 − yν1 ‖, ε} + (1 − λ)αν1 ‖yν1 − z‖

≤ max{[1 − (1 − γ)(1 − λ)αν1 ]ε + (1 − γ)(1 − λ)αν1 η,

[1 − (1 − λ)αν1 ]ε + (1 − λ)αν1 (ε + η)} ≤ ε + η.

258 | 4 Nonexpansive mappings in Banach spaces In a similar way, we have ‖xν1 +2 − yν1 +2 ‖ ≤ ε + η. By induction, we find that ‖xν1 +m − yν1 +m ‖ ≤ ε + η,

∀m ∈ ℕ.

(4.83)

It follows that lim sup ‖xn − yn ‖ ≤ ε + η, n→∞

which yields a contradiction, implying that xn − yn → θ(n → ∞). Hence xn → z as n → ∞. (ii) ‖xn − yn ‖ > ε + η, ∀n ≥ ν1 . Suppose that Case (ii) holds. It follows from (4.79) that ‖Φxn − Φyn ‖ ≤ γ‖xn − yn ‖,

∀n ≥ ν1 .

(4.84)

From (4.71), (4.77), and (4.84), we find that ‖xn+1 − yn+1 ‖ ≤ λ‖xn − yn ‖ + (1 − λ)[αn ‖Φxn − Φz‖ + (1 − αn )‖xn − yn ‖]

= [1 − (1 − λ)αn ]‖xn − yn ‖ + (1 − λ)αn γ‖xn − yn ‖ + (1 − λ)αn ‖yn − z‖

≤ [1 − (1 − γ)(1 − λ)αn ]‖xn − yn ‖ + o(αn ).

Using Lemma 1.10.2, we obtain that ‖xn − yn ‖ → 0 as n → ∞, which yields that 0 ≥ ε+η, i. e., a contradiction. It further implies that Case (ii) is impossible, completing the proof. By using Propositions 4.7.1 and 4.7.2, and the methods used in Theorems 4.7.1, 4.7.2, and 4.7.3, we can obtain the following results (see Zhou and Wang [123] for more details). Theorem 4.7.4. Let E be a real reflexive and strictly convex Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E such that it is also a sunny nonexpansive retract. Let T : C → E be a nonexpansive nonself-mapping with Fix(T) ≠ 0 and let Φ : C → E be an MK contractive mapping. Let {xt } be a sequence generated by (4.69). Then {xt } converges in norm to p ∈ Fix(T) and p is the unique solution of the variational inequality (VIP). Theorem 4.7.5. Let E be a real reflexive and strictly convex Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E such that it is also a sunny nonexpansive retract. Let T : C → E be a nonexpansive nonself-mapping with Fix(T) ≠ 0 and let Φ : C → E be an MK contractive mapping. Let {xn } be a sequence generated by (4.70). If the control sequence {αn } satisfies the conditions of Theorem 4.7.2, then {xn } converges in norm to p ∈ Fix(T) and p is the unique solution of the variational inequality (VIP).

4.8 Remarks | 259

Theorem 4.7.6. Let E be a real reflexive, strictly convex Banach space whose norm is uniformly Gâteaux differentiable. Let C be a nonempty, closed, and convex subset of E such that it is also a sunny nonexpansive retract. Let T : C → E be a nonexpansive nonself-mapping with Fix(T) ≠ 0 and let Φ : C → E be an MK contractive mapping. Let {xn } be a sequence generated by (4.71). If the control sequence {αn } satisfies the conditions of Theorem 4.7.3, then {xn } converges in norm to p ∈ Fix(T) and p is the unique solution of the variational inequality (VIP).

4.8 Remarks Theorem 4.1.3, which extends Browder’s fixed point theorem to the setting of nonselfmappings, is due to Caristi [17]. Theorem 4.2.1, which is a remarkable result in the field of fixed point theory, is due to Reich [73]. Theorem 4.2.3, which provides us with a method to obtain approximate fixed points, is due to Ishikawa [33]. Theorem 4.2.5 shows that, under some additional restriction, Krasnosel’skiĭ–Mann iterative method can be strongly convergent. The origin of Theorem 4.3.1 can be traced back to the Browder’s convergence theorem. Lemma 4.3.1, which is applicable in various situations, was given by the authors of the book. Theorem 4.3.2 is indeed due to several authors. Example 4.2 is to Suzuki [84]. Theorem 4.3.3 is to Suzuki [84]. Theorems 4.3.4, 4.3.5, and 4.3.6 are due to Xu [103]. Most results in Section 4.4 are due to Suzuki [84], and the results in Section 4.5 are essentially due to Shioji and Takahashi [79]. The results in Section 4.6 are standard in convergence theorems of nonexpansive semigroups, and the results in Section 4.7 are due to Zhou and Wang [123].

4.9 Exercises 1.

Let X be the space ℝ2 equipped with the Euclidean norm and, with (r, θ) denoting the polar coordinates. Let C = {(r, θ) : 0 ≤ r ≤ 1,

π π ≤ θ ≤ }. 4 2

Define a mapping T : C → C by T(r, θ) = (r,

π ), 2

for each point (r, θ) ∈ C.

Prove that the following assertions hold true: (a1) T is a nonexpansive mapping; (a2) Fix(T) = {(r, π2 ) : 0 ≤ r ≤ 1};

260 | 4 Nonexpansive mappings in Banach spaces (a3) Take x0 = (r, θ0 ) = {(r, Mann sequence {xn } by

π )} 4

and αn ∈ [0, 1] for all n ≥ 0, and construct the

xn+1 = (1 − αn )xn + αn Txn ,

n ≥ 0.

Then rn = r0 = 1, n ≥ 0 and θn+1 = αn

π + (1 − α)θn , 2

n ≥ 0 and θ0 =

π . 4

(a4) If one takes αn ≡ 1, then xn → (1,

π ) ∈ Fix(T); 2

(a5) If one takes α ≡ 0, then xn → (1, 2.

π ) ∈ ̸ C. 4

Let X be a Banach space and {Ti }ri=1 : X → X be a finite family of nonexpansive mappings such that K = ⋂ri=1 Fix(Ti ) = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) ≠ 0. Show that K = Fix(T1 Tr ⋅ ⋅ ⋅ T3 T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 Tr−2 ⋅ ⋅ ⋅ T1 Tr ).

3.

Let C be a convex subset of a Banach space X and {Ti }∞ i=1 : C → C be a countable family of nonexpansive mappings. Let {αn } be a sequence of numbers in (0,1). Let Wn be the W-mappings generated by Tn , Tn−1 , . . . , T1 and αn , αn−1 , . . . , α1 , n ≥ 1. Show that Wn are averaged nonexpansive for all n ≥ 1. 4. Let C be a nonempty, closed, and convex subset of a strictly convex Banach space X. Let {Sk } be a sequence nonexpansive mappings of C into X, and βk be a sequence ∞ of positive real numbers such that ∑∞ k=1 βk = 1. Assume that ℱ = ⋂k=1 Fix(Sk ) ≠ ∞ 0. Then, show that the mapping T := ∑k=1 βk Sk is well defined and Fix(T) = ⋂∞ k=1 Fix(Sk ). 5. Let C be a nonempty, closed, and convex subset of a strictly convex and 2-uniformly smooth Banach space X with the best smoothness constant K. Let A be an α-inverse strongly monotone mapping of C into X, and let S : C → C be a nonexpansive mapping such that ℱ = Fix(S) ∩ SOL(C, A) ≠ 0. For arbitrary x1 = u ∈ C, define a sequence {xn } iteratively in C by 1 1 xn+1 = αn u + (1 − αn )[ Sxn + QC (xn − λAxn )], 2 2

n ≥ 1,

where QC : X → C is a sunny nonexpansive retraction from X onto C, {αn } and {λn } are sequences of real numbers in (0,1) satisfying the following conditions: (i) λn ∈ [a, b] for a, b ∈ (0, 1), and λn ≤ Kα2 ;

4.9 Exercises | 261

6.

(ii) ∑∞ n=1 |λn+1 − λn | < ∞; (iii) limn→∞ αn = 0; (iv) ∑∞ n=1 |αn+1 − αn | < ∞. Show that the sequence {xn } converges strongly to some point x ∗ = Pℱ (u). Let X be a uniformly convex and uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of X. Let T1 , T2 , . . . , Tr be a finite family of nonexpansive self-mappings of C, with ℱ = ⋂ri=1 Fix(Ti ) = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) ≠ 0. Let {λn } be a sequence in (0,1) satisfying the following conditions: (C1) limn→∞ λn = 0; (C2) ∑∞ n=1 λn = ∞. For a fixed δ ∈ (0, 1), define Sn : C → C by Sn x := (1−δ)x+δTn x, ∀x ∈ C, where Tn = Tn mod r . For arbitrary fixed u, x1 ∈ C, define a sequence {xn } iteratively in C by xn+1 = λn+1 u + (1 − λn+1 )Sn+1 xn ,

7.

n ≥ 1.

Assume that limn→∞ ‖Tn xn − Tn+1 xn ‖ = 0. Then, show that the sequence {xn } converges strongly to a common fixed point of the family {T1 , T2 , . . . , Tr }. Also, give an example showing that the assumption limn→∞ ‖Tn xn − Tn+1 xn ‖ = 0 holds. Let X be a uniformly convex and uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of X and let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings such that ℱ = ⋂ri=1 Fix(Ti ) ≠ 0. Let {βi }ri=1 be r positive numbers in (0,1) such that ∑ri=1 βi = 1. Define a mapping T : C → C by r

Tx = ∑ βi Ti x, i=1

x ∈ C.

Let {λn } be a sequence of positive numbers in (0,1) satisfying conditions: (C1) limn→∞ λn = 0; (C2) ∑∞ n=1 λn = ∞, and (C3) ∑i=1∞ |λn+1 − λn | < ∞, or (C4) λn → 1 (n → ∞). For arbitrary x1 , u ∈ C, define a sequence {xn } iteratively in λn+1 C by xn+1 = λn u + (1 − λn )Txn ,

n ≥ 1.

Prove that the sequence {xn } converges strongly to the some point x ∗ ∈ ℱ , furthermore, x∗ = Qℱ u. 8. Let X be a uniformly convex and uniformly smooth Banach space. Let C be a nonempty, closed, and convex subset of X and let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings such that ℱ = ⋂ri=1 Fix(Ti ) ≠ 0. Let {βi }ri=1 be r positive numbers in (0,1) such that ∑ri=1 βi = 1. Define a mapping T : C → C by r

Tx = ∑ βi Ti x, i=1

x ∈ C.

Let {λn } be a sequence of positive numbers in (0,1) satisfying conditions: (C1) limn→∞ λn = 0, and (C2) ∑∞ n=1 λn = ∞. For a fixed number σ ∈ (0, 1), and for

262 | 4 Nonexpansive mappings in Banach spaces arbitrary x1 , u ∈ C, define a sequence {xn } iteratively in C by xn+1 = λn u + (1 − λn )[(1 − σ)xn + σTxn ],

9.

n ≥ 1.

Prove that the sequence {xn } converges strongly to the some point x ∗ ∈ ℱ , furthermore, x∗ = Qℱ u. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space X. Let {Ti }ri=1 : C → C be a finite family of averaged nonexpansive mappings such that ℱ = ⋂ri=1 Fix(Ti ) ≠ 0. Define a mapping T : C → C by Tx = Tr Tr−1 ⋅ ⋅ ⋅ T1 x,

x ∈ C.

Let {λn } be a sequence of positive numbers in (0,1) satisfying conditions: (C1) limn→∞ λn = 0, and (C2) ∑∞ n=1 λn = ∞. For arbitrary x1 , u ∈ C, define a sequence {xn } iteratively in C by xn+1 = λn u + (1 − λn )Txn ,

n ≥ 1.

(HTIM)

Prove that the sequence {xn } defined by (HTIM) converges strongly to the some point x∗ ∈ ℱ , furthermore, x∗ = Qℱ u. 10. Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X. Let {Ti }∞ i=1 : C → C be a finite family of nonexpansive ∞ mappings such that ℱ = ⋂i=1 Fix(Ti ) ≠ 0. Let {λi } be a sequence of numbers in ∞ ∞ (0,1) which satisfies conditions: (i) ∑∞ i=1 λi = 1 and (ii) ∑n=1 ∑i=n+1 λi < ∞. Define another family of nonexpansive mappings by n

Sn x := ∑ i=1

λi Ti x, ∑ni=1 λi

x ∈ C, n ≥ 1.

Let {μn } be a sequence of positive numbers in (0,1) which satisfies conditions: (C1) limn→∞ μn = 0, and (C2) ∑∞ n=1 μn = ∞. For arbitrary x1 , u ∈ C and δ ∈ (0, 1), define a sequence {xn } iteratively in C by xn+1 = δxn + (1 − δ)[μn xn + (1 − μn )Sn xn ],

n ≥ 1.

Prove that the sequence {xn } converges strongly to the some point x ∗ ∈ ℱ , and x∗ = Qℱ u.

5 Iterative methods for zeros for accretive operators and fixed points of pseudocontractive mappings in Banach spaces The purpose of this chapter is to investigate zeros of accretive operators and fixed points of pseudocontractive mappings in the framework of Banach spaces via iterative methods including the steepest descent iterative method, the normal Mann iterative method, the Bruck regularization iterative method, and resolvent iterative methods.

5.1 Characterizations of accretive operators Lemma 5.1.1 (Kato [37], 1967). Let E be a real Banach space. For any x, y ∈ E, we have the following two conclusions, which are equivalent: (i) ‖x‖ ≤ ‖x + λy‖, ∀λ > 0, (ii) there exists f ∈ Jx such that ⟨y, f ⟩ ≥ 0. Using Lemma 5.1.1, we give the following characterization for accretive operators. Theorem 5.1.1. Let A ⊂ E × E. Then following two conclusions are equivalent (i) A is accretive, (ii) ∀λ > 0, (x1 , y1 ) ∈ A and (x2 , y2 ) ∈ A, we have 󵄩 󵄩 ‖x1 − x2 ‖ ≤ 󵄩󵄩󵄩x1 − x2 + λ(y1 − y2 )󵄩󵄩󵄩.

(5.1)

Remark 5.1.1. Kato lemma, which characterizes accretive operators, was first established by Kato [37] in 1967. From the viewpoint of geometry, an accretive operator A ⊂ E × E has the following properties: the range of the accretive operator I + λA increases, that is, I + λA is expansive. If it reaches the whole space E, that is, Ran(I + λA) = E, we say that A is m-accretive. This may be the meaning of “accretive”. For ∀λ > 0, I + λA is an expansive operator. We see that (I + λA)−1 is well defined on Ran(I + λA). So, we can consider the following for accretive operators: Jλ x = {z ∈ E : x ∈ z + λAz} and Aλ x = λ−1 (I − Jλ ),

∀λ > 0.

Then Jλ and Aλ are said to be the resolvent and Yosida approximation, respectively. They play an important role when treating operators which are accretive. https://doi.org/10.1515/9783110667097-005

264 | 5 Accretive operators and pseudocontractive mappings

5.2 Nonlinear semigroups of ω-type Definition 5.2.1. Let C be a closed subset of a Banach space E. Then {S(t) : t ≥ 0} is said to be a nonlinear semigroup of ω-type if and only if there exists ω ∈ ℝ such that (i) S(0)x = x, ∀x ∈ C; (ii) S(t + s)x = S(t)S(s)x, ∀x ∈ C, t, s ≥ 0; (iii) S(t)x is continuous in t, ∀x ∈ C; (iv) ‖S(t)x − S(t)y‖ ≤ eωt ‖x − y‖, ∀x ∈ C, t ≥ 0. In particular, a nonlinear semigroup of 0-type on C is said to be a nonlinear contraction semigroup. If {S(t) : t ≥ 0} only satisfies (i), (ii), and (iii), it is said to be a nonlinear semigroup. Definition 5.2.2. Let {S(t) : t ≥ 0} be a nonlinear semigroup on C. Let Ah x =

1 (S(h)x − x), h

∀h > 0

and Dom(A) = {x ∈ C : lim+ Ah x exists}. h→0

Then Ah is said be a generator of {S(t) : t ≥ 0}. Theorem 5.2.1. Let A ⊂ E × E be an accretive operator satisfying the following range condition (RC): Dom(A) ⊂ Ran(I + λA),

∀λ > 0.

Then T(t)x = lim (I + n→∞

t A) x, n −n

∀t ≥ 0, ∀x ∈ Dom(A)

(EF)

exists. Furthermore, limit (EF) is uniformly attained for t ∈ [0, τ], ∀τ > 0. Also {T(t) : t ≥ 0}, which is defined by limit (EF), is a nonlinear contraction semigroup on Dom(A). Remark 5.2.1. Limit (EF), which is now called the exponential formula, was established by Crandall and Ligget [26] in 1971. Also (EF) reveals the essential relation between nonlinear contraction semigroups and accretive operators and acts as an efficient tool for studying the solvability of differential equations in the framework of Banach spaces. Theorem 5.2.2. Let C be a closed subset of a Banach space E. Let T : C → E be a continuous operator. Then the following assertions are equivalent: (i) T is a generator of a nonlinear semigroup {S(t) : t ≥ 0};

5.3 Zero point theorems of accretive operators | 265

(ii) ∀x, y ∈ C, there exists j(x − y) ∈ J(x − y) such that ⟨Tx − Ty, j(x − y)⟩ ≤ ω‖x − y‖2 , lim h−1 d(x + hTx, C) = 0,

∀x ∈ C;

h→0+

(iii) ϵω < 1 󳨐⇒ C ⊂ (I − ϵT)(C), ∀ϵ > 0. Remark 5.2.2. Theorem 5.2.2 was established by Martine [47] in 1973. It is an efficient tool to study zero points of accretive operators and fixed points of pseudocontractive mappings in Banach spaces. From (ii) and (iii) in Theorem 5.2.2, we see that continuous accretive operators defined on closed and convex subsets satisfy the range condition if they satisfy the “flow-invariance” condition.

5.3 Zero point theorems of accretive operators Theorem 5.3.1. Let C be a nonempty, closed, and convex subset of a real Banach space E. Let A : C → E be a continuous g-strongly accretive operator satisfying the following “flow-invariance” condition (FIC): lim h−1 d(x − hAx, C) = 0,

h→0+

∀x ∈ C.

If, in addition, one of the following conditions holds: (i) there exists x0 ∈ C such that lim infr→∞ g(r) > ‖Ax0 ‖; (ii) there exists R > 0 such that ∀x ∈ E\BR (θ), jx ∈ Jx satisfying ⟨Ax, jx⟩ ≥ 0; (iii) ‖Ax‖ → ∞ as ‖x‖ → ∞, then A has a unique zero point in C. Proof. Letting T = −A, we see that T is dissipative. Indeed, there exists j(x−y) ∈ J(x−y) such that ⟨Tx − Ty, j(x − y)⟩ = −⟨Ax − Ay, j(x − y)⟩ ≤ −g(‖x − y‖)‖x − y‖ ≤ 0,

∀x, y ∈ C,

which shows that T is dissipative. From (FIC), we have lim h−1 d(x − hTx, C) = 0,

h→0+

∀x ∈ C.

Using Theorem 5.2.2, we have ∀ϵ > 0, C ⊂ (I + ϵA)(C).

(5.2)

266 | 5 Accretive operators and pseudocontractive mappings Letting ϵn → ∞ (n → ∞) in (5.2), we have C ⊂ (I − ϵn A)(C).

(5.3)

Fixing y ∈ C, we see that there exists {xn } ⊂ C such that y = xn + ϵn Axn .

(5.4)

Under one of conditions (i), (ii), and (iii), we are in a position to show that {xn } is bounded. We just take (i) as an example. Indeed, y − x0 = xn − x0 + ϵn (Axn − Ax0 ) + ϵn Ax0 . It follows that ⟨y − x0 , j(xn − x0 )⟩

= ‖xn − x0 ‖2 + ϵn ⟨Axn − Ax0 , j(xn − x0 )⟩ + ϵn ⟨Ax0 , j(xn − x0 )⟩ ≥ ‖xn − x0 ‖2 + ϵn g(‖xn − x0 ‖) − ϵn ‖Ax0 ‖‖xn − x0 ‖ ≥ ϵn g(‖xn − x0 ‖) − ϵn ‖Ax0 ‖‖xn − x0 ‖,

which is equivalent to g(‖xn − x0 ‖) ≤ ‖Ax0 ‖ +

1 ‖y − xn ‖. ϵn

(5.5)

If {xn } is unbounded, we may, without loss of generality, assume that ‖xn − x0 ‖ → ∞ as n → ∞. From condition (i), we have lim inf g(‖xn − x0 ‖) > ‖Ax0 ‖. n→∞

(5.6)

Using (5.5), we obtain lim inf g(‖xn − x0 ‖) ≤ ‖Ax0 ‖, n→∞

which contradicts (5.6). This proves that {xn } is a bounded sequence. From (5.6), we y−x see that Axn = ϵ n → θ as n → ∞. Since A : C → E is g-strongly accretive, we find n that g(‖xn − xm ‖)‖xn − xm ‖ ≤ ⟨Axn − Axm , j(xn − xm )⟩ ≤ ‖Axn − Axm ‖‖xn − xm ‖,

which implies that g(‖xn − xm ‖) ≤ ‖Axn − Axm ‖.

5.3 Zero point theorems of accretive operators | 267

Since xn − xm → θ as n, m → ∞, we find that g(‖xn − xm ‖) = 0 as n, m → ∞. Using the property of g, we see that {xn } is a Cauchy sequence. Let xn → x as n → ∞. From the continuity of A, we obtain that Axn → Ax as n → ∞. So, Ax = θ. This proves that A has a zero point in C. Next, we assume that there exists another zero point, say x ∗ ∈ C. It follows that 󵄩 󵄩 󵄩󵄩 g(󵄩󵄩󵄩x − x∗ 󵄩󵄩󵄩)󵄩󵄩󵄩x − x∗ 󵄩󵄩󵄩 ≤⟩Ax − Ax∗ , j(x − x ∗ )⟩ = 0. This implies that x = x∗ . This completes the proof. From Theorem 5.3.1, we have the following important result. Corollary 5.3.1. Let C be a nonempty, closed, and convex subset of a Banach space E. Let A : C → E be a continuous η-strongly accretive operator satisfying (FIC). Then A has a unique zero point in C. Proof. Note that g(r) = ηr → ∞ as r → ∞. Conditions (i), (ii) and (iii) are all satisfied. Using Theorem 5.3.1, we obtain the desired conclusion immediately. Theorem 5.3.2 (Ray [70], 1979). Let C be a nonempty, convex, and closed subset of a Banach space E, and assume that it has the fixed point property for nonexpansive selfmappings. Let A : C → E be a continuous accretive operator satisfying (FIC). Then θ ∈ A(C). Proof. Letting T = −A, we see that T is dissipative. Indeed, there exists j(x−y) ∈ J(x−y) such that ⟨Tx − Ty, j(x − y)⟩ = −⟨Ax − Ay, j(x − y)⟩ ≤ −g(‖x − y‖)‖x − y‖ ≤ 0,

∀x, y ∈ C,

which shows that T is dissipative. From (FIC), we have lim h−1 d(x − hTx, C) = 0,

h→0+

∀x ∈ C.

Using Theorem 5.2.1, we have ∀ϵ > 0, C ⊂ (I + ϵA)(C). Hence, (I + A)−1 exists. Putting f = (I + A)−1 , we see that f : C → C is nonexpansive. From the assumption that f has fixed points in C. Hence, we find that A has zero points in C, that is, θ ∈ A(C). This completes the proof. Remark 5.3.1. Theorem 5.3.2 was originally established by Ray in 1979. The proof given above is simpler than Ray’s. If C is only a closed subset of E, that is, there is no any convexity restriction on C, we still have the following result which needs more requirements on the “accretiveness”.

268 | 5 Accretive operators and pseudocontractive mappings Theorem 5.3.3 (Martin [47], 1973). Let C be a closed subset of a Banach space E. Let A : C → E be continuous accretive operator satisfying the following restrictions: (i) ∃η > 0, for ∀x, y ∈ C, ∀j(x − y) ∈ J(x − y), we have ⟨Ax − Ay, j(x − y)⟩ ≥ η‖x − y‖2 , (ii) limh→0+ h−1 d(x − hAx, C) = 0, ∀x ∈ C. Then A has a unique zero point in C. In 1974, Deimling [27] extended Theorem 5.3.3. To be more precise, he established the following theorem. Theorem 5.3.4 (Deimling [27], 1974). Let C be a closed subset of a Banach space E. Let A : C → E be a continuous operator satisfying the following conditions: (i) there exists a continuous function α : ℝ+ → ℝ+ , α(0) = 0, α(r) > 0, ∀r > 0, α(r) → ∞ as r → ∞ and lim infr→∞ α(r) > 0, and ∀x, y ∈ C, ∀j(x − y) ∈ J(x − y), we r have ⟨Ax − Ay, j(x − y)⟩ ≥ η‖x − y‖2 , (ii) limh→0+ h−1 d(x − hAx, C) = 0, ∀x ∈ C. Then A has a unique zero point in C. Remark 5.3.2. Theorem 5.3.3 is a direct consequence of Martin’s result [47, Proposition 3], and Theorem 5.3.4 was obtained by Deimling [27, Theorem 1], via the theory of ordinary differential equations in the framework of Banach spaces. Theorem 5.3.5 (Deimling [27], 1974). Let C be a closed subset of a Banach space E. Let A : C → E be a continuous strongly accretive operator with lim infr→∞ α(r) > 0. Then A(C) is an open set. Corollary 5.3.2. Let E be a Banach space and let A : E → E be a continuous α-strongly accretive operator with lim infr→∞ α(r) > 0 or ‖Ax‖ → ∞ as ‖x‖ → ∞. Then A : E → E is a homeomorphism. Theorem 5.3.6 (Kartsatos [36], 1985). Let E be a uniformly smooth Banach space and let U be a open set of E. Let A : U → E be a demicontinuous φ-strongly accretive operator. Then A(U) is a closed subset of E and A(U) is an open subset of E. Corollary 5.3.3. Let E be a uniformly smooth Banach space. Let A : E → E be a continuous φ-strongly accretive operator. Then A : E → E is a homeomorphism. Proof. From Theorem 5.3.6, one knows that A(E) is both closed and open, that is, A(E) = E. This shows that A : E → E is surjective. In view of ⟨Ax − Ay, j(x − y)⟩ ≥ φ(‖x − y‖)‖x − y‖,

∀x, y ∈ E,

5.4 Demiclosedness of accretive operators | 269

one concludes ‖Ax − Ay‖ ≥ φ(‖x − y‖),

∀x, y ∈ E.

If x ≠ y, one finds that φ(‖x − y‖) ≥ 0 󳨐⇒ Ax ≠ Ay. This shows that A : E → E is one-to-one. Hence, A : E → E is bijective. This implies that A−1 exists. Note that 󵄩 󵄩 ‖x − y‖ ≥ φ(󵄩󵄩󵄩A−1 x − A−1 y󵄩󵄩󵄩),

∀x, y ∈ E.

Fixing y ∈ E, one finds that φ(‖A−1 x − A−1 y‖) as x → y. From the property of φ, one obtains that A−1 x → A−1 y as x → y. This implies that A−1 : E → E is continuous. This proves that A : E → E is a homeomorphism. This completes the proof. From Corollaries 5.3.2 and 5.3.3, we find the following result. Theorem 5.3.7. Let E be a Banach space and let A : E → E be a continuous accretive operator. Then A is m-accretive. Theorem 5.3.8. Let E be a uniformly smooth Banach space and let A : E → E be a demicontinuous accretive operator. Then A is m-accretive. Theorem 5.3.9 (Kobayashi [39], 1975). Let E be a Banach space and let A ⊂ E × E be an m-accretive operator. Let B : E → E be a continuous accretive operator such that Dom(A) ⊂ D(B). Then A + B is m-accretive. Theorem 5.3.10 (Gracía-Falset, Morales [29], 2005). Let E be a Banach space and let A ⊂ E × E be an m-accretive operator. Let B : E → E be a continuous accretive operator such that B is φ-strongly accretive on Dom(A). Then (i) both A + μB and B + λA are surjective, that is, Ran(A + μB) = E and Ran(B + λA), (ii) ∀λ > 0, equation z ∈ Bx + λAx has a unique solution xλ and λ 󳨃→ xλ is continuous on ℝ1+ .

5.4 Demiclosedness of accretive operators Theorem 5.4.1. Let E be a Banach space such that its duality mapping J : E → E ∗ is weakly sequentially continuous. Let C be a nonempty, closed, and convex subset of a Banach space E. Let A : C → E be a hemicontinuous accretive operator satisfying the flow-invariance condition (FIC): lim h−1 d(x − hAx, C) = 0,

h→0+

∀x ∈ C.

Then A is demiclosed at the origin. That is, ∀{xn } ⊂ C with xn ⇀ x and Axn → θ as n → ∞, we have that x ∈ C and Ax = θ.

270 | 5 Accretive operators and pseudocontractive mappings Proof. Let {xn } be a sequence in C such that xn ⇀ x and Axn → θ as n → ∞. Since C is closed and convex, we find that x ∈ C. Since A : C → E is accretive, we have ⟨Au − Axn , J(u − xn )⟩ ≥ 0,

∀u ∈ C.

(5.7)

Letting n → ∞ in (5.7) and using the continuity of J, we arrive at ⟨Au, J(u − x)⟩ ≥ 0,

∀u ∈ C.

(5.8)

Let yt = (1 − t)x + ty, ∀y ∈ C, t ∈ (0, 1). It follows that yt ∈ C. Letting u = yt in (5.8), we see that ⟨Ayt , J(yt − x)⟩ ≥ 0.

(5.9)

Substituting yt − x = t(y − x) into (5.9) yields that ⟨Ayt , J(y − x)⟩ ≥ 0,

∀y ∈ C.

(5.10)

Since A : C → E is a hemicontinuous, we have yt → x 󳨐⇒ Ayt ⇀ Ax,

t → 0+ .

Letting t → 0+ in (5.10), we obtain ⟨Ax, J(y − x)⟩ ≥ 0,

∀y ∈ C.

(5.11)

∀ϵ > 0, ∃δ > 0, if 0 < h < δ, then we find from (FIC) that d(x − hAx, C) < hϵ. Hence, there exists uh ∈ C such that ‖x − hAx − uh ‖ < hϵ 󵄩󵄩 u − x 󵄩󵄩 󵄩 󵄩 󳨐⇒ 󵄩󵄩󵄩 h + Ax 󵄩󵄩󵄩 < ϵ 󵄩󵄩 h 󵄩󵄩 uh − x → −Ax, ∀h → 0+ . 󳨐⇒ h It follows that J(

uh − x ) → −J(Ax), h

∀h → 0+ .

Letting y = uh in (5.11), one sees that ⟨Ax, J(uh − x)⟩ ≥ 0, ⟨Ax, J(

h > 0,

uh − x )⟩ ≥ 0, h

h > 0.

(5.12)

5.4 Demiclosedness of accretive operators | 271

Letting h → 0+ in (5.12), one finds that ⟨Ax, −J(Ax)⟩ ≥ 0 󳨐⇒ −‖Ax‖2 ≥ 0 󳨐⇒ Ax = θ. This completes the proof. Theorem 5.4.2. Let E be a uniformly convex Banach space and let C be a nonempty, closed, and convex subset of E. Let A : C → E be a continuous accretive operator satisfying the flow-invariance condition (FIC): lim h−1 d(x − hAx, C) = 0,

h→0+

∀x ∈ C.

Then A is demiclosed at the origin. That is, ∀{xn } ⊂ C with xn ⇀ x and Axn → θ as n → ∞, we have that x ∈ C and Ax = θ. Proof. Let {xn } be a sequence in C with xn ⇀ x and Axn → θ as n → ∞. Since C is closed and convex, we find that x ∈ C. Letting T = −A, we find from Theorem 5.2.1 that C ⊂ (I + A)(C), which guarantees that g = (I + A)−1 is well defined on C, g : C → C is nonexpansive and Fix(g) = A−1 (θ). Since 󵄩󵄩 󵄩 󵄩 −1 󵄩 󵄩󵄩xn − g(xn )󵄩󵄩󵄩 = 󵄩󵄩󵄩g(g (xn )) − g(xn )󵄩󵄩󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩g −1 (xn ) − xn 󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩(I + A)xn − xn 󵄩󵄩󵄩 = ‖Axn ‖ → 0,

n → ∞,

and I − g is demiclosed at the origin, we find that x = g(x), that is, Ax = θ. This completes the proof. Theorem 5.4.3. Let E be a reflexive Banach space that satisfies Opial condition and let C be a nonempty, closed, and convex subset of E. Let A : C → E be a continuous accretive operator satisfying the flow-invariance condition (FIC): lim h−1 d(x − hAx, C) = 0,

h→0+

∀x ∈ C.

Then A is demiclosed at the origin. That is, ∀{xn } ⊂ C with xn ⇀ x and Axn → θ as n → ∞, we have that x ∈ C and Ax = θ. Proof. Let {xn } be a sequence in C with xn ⇀ x and Axn → θ as n → ∞. Since C is closed and convex, we find that x ∈ C. As the proof in Theorem 5.4.2, we find that xn − g(xn ) → θ as n → ∞.

272 | 5 Accretive operators and pseudocontractive mappings Next, we prove x = g(x). Assume that the inverse is true, that is, x ≠ g(x). Using Opial condition, we see that 󵄩 󵄩 lim inf ‖xn − x‖ < lim inf󵄩󵄩󵄩xn − g(x)󵄩󵄩󵄩 n→∞

n→∞

󵄩 󵄩 = lim inf󵄩󵄩󵄩xn − g(xn ) + g(xn ) − g(x)󵄩󵄩󵄩 n→∞

󵄩 󵄩 󵄩 󵄩 ≤ lim inf(󵄩󵄩󵄩xn − g(xn )󵄩󵄩󵄩 + 󵄩󵄩󵄩g(xn ) − g(x)󵄩󵄩󵄩) n→∞

≤ lim inf ‖xn − x‖, n→∞

which is a contradiction. This shows that x = g(x), that is, Ax = θ. This completes the proof.

5.5 The existence and convergence of paths for accretive operators Theorem 5.5.1. Let E be a Banach space and let C be a nonempty, convex, and closed subset of E. Let A : C → E be a continuous accretive operator, and let R : C → E be a continuous and η-strongly accretive operator. Assume that both A and R satisfy flowinvariance conditions: lim h−1 d(x − hAx, C) = 0,

∀x ∈ C,

lim h−1 d(x − hRx, C) = 0,

∀x ∈ C.

h→0+

and h→0+

Then (i) There exists a unique continuous path {xt } ⊂ C such that θ = tAxt + (1 − t)Rxt ,

∀t ∈ [0, 1).

(5.13)

(ii) Let {xn } ⊂ C be a bounded sequence with Axn → θ as n → ∞ and let {Rxn } be a bounded sequence in E. Then {xt } is bounded. In particular, if A−1 (θ) ≠ 0, then {xt } is bounded. (iii) If A−1 (θ) ≠ 0, then there exists j(xt − z) ∈ J(xt − z) such that ⟨Rxt , j(xt − z)⟩ ≤ 0,

∀z ∈ A−1 (θ).

Proof. (i) ∀t ∈ [0, 1), define a mapping At : C → E by At x = tAx + (1 − t)Rx,

∀x ∈ C.

(5.14)

5.5 The existence and convergence of paths for accretive operators | 273

Using the assumption, we see that At : C → E is a continuous mapping. Using Theorem 1.9.19, we obtain that At : C → E is (1−t)η-strongly accretive and satisfies the flow-invariance condition. Using Theorem 5.3.1 or Corollary 5.3.1, we find that At has a unique zero point xt in C, that is, θ = tAxt + (1 − t)Rxt , t ∈ [0, 1). Next, we show that path {xt } is continuous. Fix t0 ∈ [0, 1). Then there exists j(xt − xt0 ) ∈ J(xt − xt0 ) such that θ = t⟨Axt − Axt0 , j(xt − xt0 )⟩ + (t − t0 )⟨Axt0 , j(xt − xt0 )⟩

+ (1 − t)⟨Rxt − Rxt0 , j(xt − xt0 )⟩ + (t0 − t)⟨Rxt0 , j(xt − xt0 )⟩

≥ (1 − t)η‖xt − xt0 ‖2 − |t − t0 |‖Axt0 − Rxt0 ‖‖xt − xt0 ‖,

∀t ∈ [0, 1),

which implies that ‖xt − xt0 ‖ ≤

|t − t0 | ‖Axt0 − Rxt0 ‖. (1 − t)η

(5.15)

Letting t → t0 in (5.15), we find that xt → xt0 , that is, the path is continuous at t0 . Since t0 is chosen arbitrarily, we find that t 󳨃→ xt is continuous. (ii) Since A : C → E is continuous accretive and R : C → E is continuous and η-strongly accretive, we find from Theorem 1.9.19 that there exist j(xt − xn ) ∈ J(xt − xn ) such that θ = t⟨Axt − Axn , j(xt − xn )⟩ + t⟨Axn , j(xt − xn )⟩

+ (1 − t)⟨Rxt − Rxn , j(xt − xn )⟩ + (1 − t)⟨Rxn , j(xt − xn )⟩

≥ (1 − t)η‖xt − xn ‖2 − t‖Axn ‖‖xt − xn ‖ − (1 − t)‖Rxn ‖‖xt − xn ‖, which implies that lim sup ‖xt − xn ‖ ≤ n→∞

1 lim sup ‖Rxn ‖ < ∞. η n→∞

t (iii) Using (5.13), we find that Rxt = − 1−t Axt . It follows that

t ⟨Axt , j(xt − z)⟩ 1−t t ⟨Axt − Az, j(xt − z)⟩ =− 1−t ≤ 0, ∀z ∈ A−1 (θ).

⟨Rxt , j(xt − z)⟩ = −

This completes the proof. Theorem 5.5.2. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E and have the fixed point property for nonexpansive self-mappings. Let A : C → E be a continuous accretive

274 | 5 Accretive operators and pseudocontractive mappings operator and let R : C → E be a continuous and η-strongly accretive operator. Assume that both A and R satisfy the following flow-invariance condition: lim h−1 d(x − hAx, C) = 0,

∀x ∈ C,

lim h−1 d(x − hRx, C) = 0,

∀x ∈ C.

h→0+

and h→0+

If both sets, X = {x ∈ C : Ax = (1 − λ)Rx, λ > 1} and R(X), are bounded, then the path {xt : t ∈ [0, 1)} defined in (5.13) converges strongly to a zero point z ∗ of A and z ∗ uniquely solves the following variational inequality: ⟨Rz ∗ , j(z − z ∗ )⟩ ≥ 0,

∀z ∈ A−1 (θ).

Proof. Letting {tn } be a real positive real sequence with tn → 1 as n → ∞ and xn = xtn , we find from (5.13) that {xn } ⊂ X. Note that both {xn } and {Rxn } are bounded. It follows t −1 that Axn = nt Rxn → θ as n → ∞. Using Theorem 5.2.2, we find that C ⊂ (I + A)(C), n

which shows that (I + A)−1 exists on C. Let g(x) = (I + A)−1 x. It follows that g : C → C is a nonexpansive mapping with F(g) = A−1 (θ). Let μn be the Banach space. Since {xn } is bounded, we may define a real function f : E → ℝ+ such that f (x) = μn ‖xn − x‖2 ,

∀x ∈ E.

Since f is continuous, convex, and satisfies f (x) → ∞ as ‖x‖ → ∞, we see that there exists u ∈ C such that f (u) = min{f (x) : x ∈ C}. Letting C0 = {u ∈ C : f (u) = min{f (x) : x ∈ C}}, we see that C0 is nonempty, bounded, closed, and convex subset of C with g(C0 ) ⊂ C0 . Indeed, it is easy to find that C0 is nonempty, bounded, closed, and convex subset of C. We here only prove g(C0 ) ⊂ C0 . To this end, ∀y ∈ C0 , we find that y ∈ C and f (y) = min{f (x) : x ∈ C}. From the definition of g that 󵄩 󵄩2 f (g(y)) = μn 󵄩󵄩󵄩xn − g(y)󵄩󵄩󵄩 󵄩 󵄩2 = μn 󵄩󵄩󵄩g(g −1 (xn )) − g(y)󵄩󵄩󵄩 󵄩 󵄩2 ≤ μn 󵄩󵄩󵄩g −1 (xn ) − y󵄩󵄩󵄩 = μn ‖xn + Axn − y‖2 ≤ μn ‖xn − y‖2 = f (y).

5.5 The existence and convergence of paths for accretive operators | 275

On the other hand, f (y) ≤ f (g(y)). It follows that g(y) ∈ C0 . This proves that g(C0 ) ⊂ C0 . From the assumption, there exists z ∗ ∈ C0 such that g(z ∗ ) = z ∗ 󳨐⇒ Az ∗ = θ. Using the (iii) of Theorem 5.5.1, we find that ⟨Rxn , j(xn − z ∗ )⟩ ≤ 0. It follows that μn ⟨Rxn , j(xn − z ∗ )⟩ ≤ 0.

(5.16)

In view of z ∗ ∈ C0 , we obtain that 󵄩 󵄩2 f (z ∗ ) = μn 󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 = min μn ‖xn − x‖2 . x∈C

Using Theorem 1.8.36, we find that μn ⟨y − z ∗ , j(xn − z ∗ )⟩ ≤ 0,

∀y ∈ C.

(5.17)

Note that R : C → E satisfies the flow-invariance condition. For every ϵ > 0, there exists δ > 0 such that if 0 < h < δ, then d(z ∗ − hRz ∗ , C) ≤ ϵh. Hence, there exists uh ∈ C such that 󵄩󵄩 ∗ 󵄩 ∗ 󵄩󵄩z − hRz − uh 󵄩󵄩󵄩 ≤ ϵh 󵄩󵄩 u − z ∗ 󵄩󵄩 󵄩 󵄩 󳨐⇒ 󵄩󵄩󵄩 h + Rz ∗ 󵄩󵄩󵄩 ≤ ϵ. 󵄩󵄩 h 󵄩󵄩 This implies that uh − z ∗ → −Rz ∗ , h

as

h → 0+ .

(5.18)

Letting y = uh ∈ C in (5.17), we obtain that μn ⟨

uh − z ∗ , j(xn − z ∗ )⟩ ≤ 0. h

(5.19)

Combining (5.18) with (5.19), we arrive at μn ⟨−Rz ∗ , j(xn − z ∗ )⟩ ≤ 0.

(5.20)

It follows from (5.16) and (5.20) that μn ⟨Rxn − Rz ∗ , j(xn − z ∗ )⟩ ≤ 0.

(5.21)

Since R : C → E is η-strongly accretive, we have 󵄩 󵄩2 ⟨Rxn − Rz ∗ , j(xn − z ∗ )⟩ ≥ η󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 .

(5.22)

Taking the Banach limit in (5.22), we find from (5.21) that μn ‖xn − z ∗ ‖2 ≤ 0. This implies that there exists a subsequence {xnk } of {xn } such that xnk → z ∗ as k → ∞. Using the (iii) of Theorem 5.5.1, we find that ⟨Rz ∗ , j(z − z ∗ )⟩ ≥ 0, ∀z ∈ A−1 (θ). Since z ∗ is unique, we obtain that {xn } converges strongly to z ∗ . This completes the proof.

276 | 5 Accretive operators and pseudocontractive mappings Remark 5.5.1. The restrictions that both A and R are continuous and satisfy the flowinvariance condition ensure the existence of the path. If we assume that accretive operator A ⊂ E × E also satisfies the following range condition: Dom(A) ⊂ C ⊂ ⋂ Ran(I + rA), r>0

where C is a closed and convex subset of E, r > 0, we find from Theorem 5.5.2 the following result. Theorem 5.5.3. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Let A ⊂ E × E be a continuous accretive operator with the range condition Dom(A) ⊂ C ⊂ ⋂r>0 Ran(I + rA). Assume that every weakly compact convex subset of C has fixed points for every nonexpansive self-mapping. If θ ∈ Ran(A), then limr→∞ Jr x exists, ∀x ∈ C. Furthermore, if we define Qx = limr→∞ Jr x, ∀x ∈ C, then Q : C → A−1 (θ) is a sunny nonexpansive retraction. Proof. Since x ∈ C ⊂ ⋂r>0 R(I + rA), we see that there exists xr ∈ Dom(A) ⊂ C such that x ∈ (I + rA)xr ,

∀r > 0,

(5.23)

r , we see that t ∈ (0, 1) and r → ∞ ⇐⇒ t → 1− . Define a that is, xr = Jr x. Letting t = 1+r mapping R : C → E by Ru = u − x, ∀u ∈ C. Then R is 1-strongly accretive. From (5.23), we have

θ∈

1 r Ax + Rx , 1+r r 1+r r

∀r > 0.

(5.24)

Letting xr := yt , we see that (5.24) is reduced to θ ∈ tAyt + (1 − t)Ryt ,

∀t ∈ (0, 1).

Using Theorem 5.5.2, we find that {yt } converges strongly to zero point z ∗ of A as t → 1− and z ∗ uniquely solves the following variational inequality: ⟨Rz ∗ , j(z − z ∗ )⟩ ≥ 0,

∀z ∈ A−1 (θ),

that is, ⟨z ∗ − x, j(z − z ∗ )⟩ ≥ 0, ∀z ∈ A−1 (θ). Define Qx = z ∗ = limr→∞ xr = limr→∞ Jr x. It follows that ⟨x − Qx, j(Qx − z)⟩ ≥ 0,

∀z ∈ A−1 (θ).

Using Theorem 1.8.49, we find that Q : C → A−1 (θ) is a sunny nonexpansive retraction. This completes the proof. Corollary 5.5.1. Let E be a uniformly smooth Banach space and let A ⊂ E × E be an m-accretive operator. If θ ∈ Ran(A), then limit limr→∞ Jr x exists, for any x ∈ E. Furthermore, if we define Qx = limr→∞ Jr x, ∀x ∈ C, then Q : C → A−1 (θ) is a unique sunny nonexpansive retraction from C onto A−1 (θ).

5.6 Iterative methods of zero points for accretive operators | 277

Corollary 5.5.2. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Let A ⊂ E × E be a continuous accretive operator with the range condition Dom(A) ⊂ C ⊂ ⋂r>0 Ran(I + rA), Assume that every weakly compact convex subset of E has fixed points for every nonexpansive self-mapping. If Dom(A) is a convex subset of E and A−1 (θ) ≠ 0, then limr→∞ Jr x exists, ∀x ∈ Dom(A). Furthermore, if we define Qx = limr→∞ Jr x, ∀x ∈ Dom(A), then Q : Dom(A) → A−1 (θ) is a unique sunny nonexpansive retraction from Dom(A) onto A−1 (θ).

5.6 Iterative methods of zero points for accretive operators 5.6.1 The steepest decent method Define a sequence by x0 ∈ E,

xn+1 = xn − tn Axn ,

n ≥ 0,

(SDM)

where {tn } ⊂ (0, 1) is a real number sequence and A : E → E is a mapping. Theorem 5.6.1. Let E be a Banach space. Let A : E → E be an L-Lipschitz continuous g-strongly accretive mapping with the restriction that there exists x0 ∈ E such that lim infr→∞ g(r) > ‖Ax0 ‖. Assume that {tn } is a sequence in (0, 1) such that (i) limn→∞ tn = ∞ 2 0, (ii) ∑∞ n=1 tn = ∞, and (iii) ∑n=1 tn < ∞. Let {xn } be a sequence generated by (SDM). Then {xn } converges to the unique zero point of A in norm. Proof. Using Theorem 5.3.1, we see that A has a unique zero point z ∗ in E. Since xn+1 − z ∗ = xn − z ∗ − tn Axn , we find that 󵄩󵄩 󵄩 ∗ 󵄩2 ∗ 󵄩2 ∗ 󵄩󵄩xn+1 − z 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn − z 󵄩󵄩󵄩 − 2tn ⟨Axn , j(xn+1 − z )⟩ 󵄩 󵄩2 = 󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 − 2tn ⟨Axn − Axn+1 , j(xn+1 − z ∗ )⟩ − 2tn ⟨Axn+1 − Az ∗ , j(xn+1 − z ∗ )⟩ 󵄩 󵄩2 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 + 2Ltn ‖xn − xn+1 ‖󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 − 2tn g(󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 ≤ 󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 + L2 tn2 (󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 ) 󵄩 󵄩󵄩 󵄩 − 2tn g(󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩󵄩 󵄩 ≤ (1 + L2 tn2 )󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 + L2 tn2 󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 − 2tn g(󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩. Without loss of generality, we may assume that tn ≤ 1/2L for every n ≥ 1. It follows that 2tn 󵄩󵄩 󵄩 󵄩󵄩 󵄩 ∗ 󵄩2 2 2 󵄩 ∗ 󵄩2 g(󵄩󵄩x − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩. 󵄩󵄩xn+1 − z 󵄩󵄩󵄩 ≤ (1 + 3L tn )󵄩󵄩󵄩xn − z 󵄩󵄩󵄩 − 1 − L2 tn2 󵄩 n+1

(5.25)

2 ∗ ∗ Since ∑∞ n=1 tn < ∞, we find that limn→∞ ‖xn − z ‖ exists. This implies that {‖xn − z ‖} is bounded. In view of (5.25), we have

2tn 󵄩 󵄩󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩2 g(󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 + 3L2 tn2 M 2 , 2 2 1 − L tn

278 | 5 Accretive operators and pseudocontractive mappings where M = supn≥1 {‖xn − z ∗ ‖}. This implies that 󵄩2 󵄩2 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 tn g(󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 − 󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 + 3L2 tn2 M 2 , ∗ ∗ which implies that ∑∞ n=1 tn g(‖xn+1 − z ‖)‖xn+1 − z ‖ < ∞. In view of the assumption that ∞ ∑n=1 tn = ∞, we find that

󵄩 󵄩 󵄩󵄩 lim inf g(󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 = 0. n→∞

It follows that there exists a subsequence {xnj } of {xn } such that xnj → z ∗ as j → ∞. Using Lemma 1.10.1, we find the desired conclusion immediately. This completes the proof. If A : E → E is η-strongly accretive, then the requirements imposed on {tn } can be relaxed. Theorem 5.6.2. Let E be a Banach space. Let A : E → E be an L-Lipschitz continuous η-strongly accretive mapping. Assume that {tn } is a sequence in (0, 1) such that (i) limn→∞ tn = 0 and (ii) ∑∞ n=1 tn = ∞. Let {xn } be a sequence generated by (SDM). Then {xn } converges to the unique zero point of A in norm. Proof. Using Corollary 5.3.1, we see that A has a unique zero point z ∗ in E. Since xn+1 − z ∗ = xn − z ∗ − tn Axn , we find that 󵄩󵄩 󵄩 ∗ 󵄩2 ∗ 󵄩2 ∗ 󵄩󵄩xn+1 − z 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn − z 󵄩󵄩󵄩 − 2tn ⟨Axn , j(xn+1 − z )⟩ 󵄩 󵄩2 󵄩 󵄩 ≤ 󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 + 2Ltn ‖xn − xn+1 ‖󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 󵄩 󵄩󵄩 󵄩 − 2tn g(󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 ≤ (1 + L2 tn2 )󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 + tn (L2 tn − 2η)󵄩󵄩󵄩xn+1 − z ∗ 󵄩󵄩󵄩 . For sufficiently large n, we have 1 + L2 tn2 󵄩󵄩 󵄩󵄩 ∗ 󵄩2 ∗ 󵄩2 󵄩x − z 󵄩󵄩󵄩 󵄩󵄩xn+1 − z 󵄩󵄩󵄩 ≤ 1 − tn (L2 tn − 2η) 󵄩 n 󵄩 󵄩2 ≤ (1 − ηtn )󵄩󵄩󵄩xn − z ∗ 󵄩󵄩󵄩 . Using Lemma 1.10.2, we find the desired conclusion immediately. This completes the proof. Theorem 5.6.3. Let E be a uniformly smooth Banach space. Let A : E → E be a bounded demicontinuous g-strongly accretive mapping with the restriction that there exists x0 ∈ E such that lim infr→∞ g(r) > ‖Ax0 ‖. Assume that {tn } is a sequence in (0, 1) such that (i) limn→∞ tn = 0, (ii) ∑∞ n=1 tn = ∞. Then there exists a constant a > 0 such that tn < a, ∀n ≥ 1, and the sequence {xn } generated by (SDM) converges to the unique zero point of A in norm.

5.6 Iterative methods of zero points for accretive operators | 279

Proof. Using Theorem 5.3.1, we see that A has a unique zero point x ∗ in E. Define a0 = sup{r : g(r) ≤ ‖Ax0 ‖}. If a0 = +∞, then there exists rn → ∞ such that g(rn ) ≤ ‖Ax0 ‖. Hence ‖Ax0 ‖ < lim inf g(rn ) ≤ ‖Ax0 ‖. n→∞

This a contradiction. Hence, a0 is a finite positive number. Define m0 = sup{‖Ax0 ‖ : ‖xn − x0 ‖ ≤ 3a0 } + 1. Then 1 < m0 < +∞. Define a=

2a0 ‖Ax0 ‖ 1 min{a0 , h−1 ( )}, m0 (2a0 + 1)m0 em0 +1

where h : ℝ+ → ℝ+ is a strictly increasing continuous function, h(t) = et b(t), and b(t) is the function in Reich inequality (RI). We are in a position to show that {xn } is bounded under the restriction that tn < a. In view of 󵄩 󵄩 g(󵄩󵄩󵄩x0 − x∗ 󵄩󵄩󵄩) ≤ ‖Ax0 ‖, we see that ‖x0 − x∗ ‖ ≤ a0 ≤ 2a0 . If ‖x0 − x∗ ‖ < 2a0 , then 󵄩 󵄩 󵄩 󵄩 ‖xn − x0 ‖ ≤ 󵄩󵄩󵄩x0 − x∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩xn − x ∗ 󵄩󵄩󵄩 < 3a0 . It follows from the definition of m0 that ‖Axn ‖ ≤ m0 .

(5.26)

󵄩󵄩 ∗󵄩 󵄩󵄩xn+1 − x 󵄩󵄩󵄩 < 2a0 .

(5.27)

Next, we show

If not, then 󵄩󵄩 󵄩 ∗󵄩 ∗󵄩 󵄩󵄩xn − x 󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩xn+1 − x 󵄩󵄩󵄩 − tn ‖Axn ‖ ≥ 2a0 − tn m0 a ≥ 2a0 − 0 m0 m0 = a0 .

Using the definition of a0 , we see that 󵄩 󵄩 g(󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩) > ‖Ax0 ‖.

(5.28)

280 | 5 Accretive operators and pseudocontractive mappings Using the modified Reich inequality [72], we find from (5.25), (5.26), and (5.28) that 󵄩 󵄩󵄩 ∗ 󵄩2 ∗ ∗ 󵄩2 󵄩󵄩xn+1 − x 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn − x 󵄩󵄩󵄩 − 2tn ⟨Axn , J(xn − x )⟩ 󵄩 󵄩 + max{󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩, 1}tn ‖Axn ‖h(tn ‖Axn ‖) 󵄩 󵄩 󵄩󵄩 ≤ 4a20 − 2tn g(󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn − x ∗ 󵄩󵄩󵄩 + (2a0 + 1)tn m0 h(tn m0 ) ≤ 4a20 − 2a0 tn ‖Axn ‖ + 2a0 tn ‖Ax0 ‖ ≤ 4a20 ,

which yields ‖xn+1 − x∗ ‖ < 2a0 , and this is a contradiction. So ‖xn − x ∗ ‖ < 2a0 , ∀n ≥ 0. Since ‖xn − x0 ‖ ≤ ‖xn − x∗ ‖ + ‖x0 − x∗ ‖ < 3a0 , we have ‖Axn ‖ ≤ m0 , ∀n ≥ 0. Letting δ = lim infn→∞ ‖xn − x∗ ‖, we see that δ = 0. If not, then there exists n0 ∈ ℕ such that ‖xn − x∗ ‖ > δ2 , ∀n ≥ n0 . Also ∀r0 > 0, lim infr→r0 g(r) > 0 implies lim infn→∞ g(‖xn − x∗ ‖) = a∗ > 0. Then there exists n1 > n0 such that a∗ 󵄩 󵄩 g(󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩) ≥ , 2

∀n ≥ n1 .

Since h(tn ) → 0 as n → ∞, there exists n2 ≥ n1 such that h(tn ) ≤

a∗ δ , 4(2a0 + 1)m20 em0 −1

∀n ≥ n2 .

(5.29)

Using the modified Reich inequality [72] again, we have that 󵄩󵄩 󵄩 ∗ 󵄩2 ∗ 󵄩2 ∗ 󵄩󵄩xn+1 − x 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩xn − x 󵄩󵄩󵄩 − 2tn ⟨Axn , J(xn − x )⟩ 󵄩 󵄩 + max{󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩, 1}tn ‖Axn ‖h(tn ‖Axn ‖) 󵄩 󵄩2 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩 − 2tn g(󵄩󵄩󵄩xn − x ∗ 󵄩󵄩󵄩)󵄩󵄩󵄩xn − x ∗ 󵄩󵄩󵄩 + (2a0 + 1)tn m0 h(tn m0 ) 1 󵄩 󵄩2 1 ≤ 󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩 − a∗ δtn + a∗ δtn 2 2 1 ∗ 2 󵄩󵄩 ∗󵄩 󵄩 = 󵄩󵄩xn − x 󵄩󵄩 − a δtn , ∀n ≥ n2 . 4 This implies that 1 ∗ ∞ 󵄩 󵄩2 a δ ∑ tn ≤ 󵄩󵄩󵄩xn2 − x ∗ 󵄩󵄩󵄩 < ∞. 4 n≥n0 It, however, contradicts the assumption ∑∞ n=1 tn = ∞. Therefore, there exists a subsequence {xnk } of {xn } such that xnk → x∗ as k → ∞. Repeating above arguments, we find that xn → x∗ as n → ∞. This completes the proof. 5.6.2 The Bruck regularization iterative method Theorem 5.6.4. Let E be a uniformly smooth Banach space. Let A ⊂ E×E be an accretive operator satisfying the range condition Dom(A) ⊂ ⋂r>0 Ran(I + rA), and Dom(A) is a

5.6 Iterative methods of zero points for accretive operators | 281

convex subset of E. Let {xn } be a sequence in Dom(A) generated by the following Bruck regularization iterative method x1 ∈ Dom(A),

xn+1 ∈ xn − λn (Axn + θn (xn − z)),

n ≥ 1,

(BRIM)

where z is a fixed element in Dom(A), {λn } and {θn } are two real sequences in [0, 1] such that (i) θn → 0 as n → ∞, (ii) ∑∞ n=1 λn θn = ∞, θ (iii) θn−1 − 1 = o(λn θn ), n (iv) b(λn ) = o(θn ), where b : ℝ+ → ℝ+ is the function in the Reich inequality (RI). Assume that there exists constant c > 1 such that ‖un ‖ ≤ c(1 + ‖x‖),

∀un ∈ Axn , n ≥ 1.

(5.30)

If θ ∈ Ran(A), then {xn } converges to the unique zero point of A in norm. Proof. Without loss of generality, we assume z = θ. Otherwise, we consider Dom(A)̃ = ̃ From Corollary 5.5.2, we see that Dom(A) − z, Ã = A(x + z), x ∈ Dom(A). θ ∈ (I + θn−1 A)x

(5.31)

has a unique solution yn ∈ Dom(A), that is, θ ∈ θn yn + Ayn

(5.32)

and {yn } converges to x∗ = Q(θ) ∈ A−1 (θ) as n → ∞. In particular, {yn } is bounded. From (5.32), we find that there exists vn ∈ Ayn such that θn yn + vn = θ. Hence, {vn } is also bounded. Put M = max{sup{‖yn ‖ : n ≥ 1}, sup{‖yn ‖ : n ≥ 1}}. Using the linear growth condition (5.30), we find that that ‖un − vn ‖ M ≤c+ , 1 + ‖xn ‖ 1 + ‖xn ‖

‖un ‖ 1+‖xn ‖

≤ c, ∀un ∈ Axn . It follows

∀v ∈ Ayn .

In view of Theorem 5.5.1, we find from (5.32) that 󵄩󵄩 󵄩󵄩 1 󵄩 󵄩 (vn − vn−1 )󵄩󵄩󵄩 ‖yn − yn−1 ‖ ≤ 󵄩󵄩󵄩yn − yn−1 + 󵄩󵄩 󵄩󵄩 θn−1 󵄩󵄩 󵄩󵄩 1 1 󵄩 󵄩 ≤ 󵄩󵄩󵄩(yn + vn ) − (yn−1 + vn−1 ) + θn−1 vn − θn vn 󵄩󵄩󵄩 󵄩󵄩 θn θn−1 󵄩󵄩

(5.33)

282 | 5 Accretive operators and pseudocontractive mappings 󵄨󵄨 1 1 󵄨󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨 − 󵄨‖v ‖ 󵄨󵄨 θn θn−1 󵄨󵄨󵄨 n 󵄨󵄨 1 1 󵄨󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨 − 󵄨󵄨θn ‖yn ‖ 󵄨󵄨 θn θn−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 θ 󵄨 󵄨 = 󵄨󵄨󵄨 n − 1󵄨󵄨󵄨‖yn ‖ 󵄨󵄨 󵄨󵄨 θn−1 󵄨󵄨 󵄨󵄨 θ 󵄨 󵄨 ≤ M 󵄨󵄨󵄨 n − 1󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 θn−1

(5.34)

Using the Reich inequality (RI), we find from (5.33) that 󵄩󵄩 x − y 󵄩󵄩2 󵄩󵄩 n+1 n󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 1 + ‖xn ‖ 󵄩󵄩󵄩 󵄩󵄩2 󵄩󵄩 λn x − yn 󵄩 󵄩 − (un − vn )󵄩󵄩󵄩 = 󵄩󵄩󵄩(1 − λn θn ) n 󵄩󵄩 󵄩󵄩 1 + ‖xn ‖ 1 + ‖xn ‖ 2 󵄩󵄩󵄩 x − yn 󵄩󵄩󵄩 󵄩󵄩 − 2λn (1 − λn θn ) ⟨un − vn , J(xn − yn )⟩ ≤ (1 − λn θn )2 󵄩󵄩󵄩 n 󵄩 (1 + ‖xn ‖)2 󵄩󵄩 1 + ‖xn ‖ 󵄩󵄩 ‖x − yn ‖ ‖u − vn ‖ ‖un − vn ‖ + max{ n , 1}λn n b( ) 1 + ‖xn ‖ 1 + ‖xn ‖ 1 + ‖xn ‖ 2 󵄩󵄩 x − y 󵄩󵄩2 󵄩 n 󵄩 󵄩󵄩 + (M + 1)(c + M ) λn b(λn ). ≤ (1 − λn θn )2 󵄩󵄩󵄩 n 󵄩 󵄩󵄩 1 + ‖xn ‖ 󵄩󵄩 1 + ‖xn ‖ This implies that 2

‖xn+1 − yn ‖2 ≤ (1 − λn θn )‖xn − yn ‖2 + (M + 1)(c(1 + ‖xn ‖) + M) λn b(λn )

≤ (1 − λn θn )‖xn − yn ‖2 + 4(M + 1)(c2 (1 + ‖xn ‖2 ) + M 2 )λn b(λn )

= (1 − λn θn )‖xn − yn ‖2 + 4(M + 1)(c2 + M 2 )λn b(λn ) + 4(M + 1)c2 ‖xn − yn + yn ‖2 λn b(λn )

≤ (1 − λn θn )‖xn − yn ‖2 + 4(M + 1)(c2 + M 2 )λn b(λn ) + 8(M + 1)c2 ‖xn − yn ‖2 λn b(λn ) + 8(M + 1)c2 M 2 λn b(λn )

≤ (1 − λn θn + 8(M + 1)c2 λn b(λn ))‖xn − yn ‖2 + M1 λn b(λn ),

(5.35)

where M1 is an appropriate constant. From (iv), we see that there exists n0 ≥ 1 such that λn b(λn ) ≤

λn θn , 16(M + 1)c2

∀n ≥ n0 .

Hence, (5.35) is reduced to 1 ‖xn+1 − yn ‖2 ≤ (1 − λn θn )‖xn − yn ‖2 + M1 λn b(λn ). 2

5.6 Iterative methods of zero points for accretive operators | 283

Using (5.34), for sufficiently large n, we have 1 ‖xn+1 − yn ‖2 ≤ (1 − λn θn )‖xn − yn ‖2 + M1 λn b(λn ) 2 1 ≤ (1 − λn θn )(‖xn − yn−1 ‖2 + ‖yn−1 − yn ‖2 2 + 2‖yn−1 − yn ‖‖xn − yn−1 ‖) + M1 λn b(λn )

󵄨󵄨 θ 󵄨󵄨 1 󵄨 󵄨 ≤ (1 − λn θn )‖xn − yn−1 ‖2 + 2M‖xn − yn−1 ‖󵄨󵄨󵄨 n − 1󵄨󵄨󵄨 󵄨󵄨 θn−1 󵄨󵄨 2 2 󵄨󵄨 θ 󵄨󵄨 󵄨 󵄨 + M 2 󵄨󵄨󵄨 n − 1󵄨󵄨󵄨 + M1 λn b(λn ) 󵄨󵄨 θn−1 󵄨󵄨 󵄨󵄨 θ 󵄨󵄨 1 󵄨 󵄨 ≤ (1 − λn θn )‖xn − yn−1 ‖2 + M‖xn − yn−1 ‖2 󵄨󵄨󵄨 n − 1󵄨󵄨󵄨 󵄨󵄨 θn−1 󵄨󵄨 2 󵄨󵄨 θ 󵄨󵄨 󵄨󵄨 󵄨󵄨 θ 󵄨 󵄨 󵄨 󵄨 + M 󵄨󵄨󵄨 n − 1󵄨󵄨󵄨 + M 2 󵄨󵄨󵄨 n − 1󵄨󵄨󵄨 + M1 λn b(λn ) 󵄨󵄨 θn−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 θn−1 󵄨 󵄨 󵄨 󵄨 1 󵄨 θ 󵄨 = (1 − λn θn + M 󵄨󵄨󵄨 n − 1󵄨󵄨󵄨)‖xn − yn−1 ‖2 + o(λn θn ) 2 󵄨󵄨 θn−1 󵄨󵄨 1 ≤ (1 − λn θn )‖xn − yn−1 ‖2 + o(λn θn ). 4 Using Lemma 1.10.2, we obtain that xn+1 − yn → θ as n → ∞. It follows that xn → x ∗ = Q(θ) ∈ A−1 (θ) as n → ∞. This completes the proof. Remark 5.6.1. (1) If {un }, un ∈ Axn , is bounded, then there exists c ≥ 1 such that ‖un ‖ ≤ c ≤ c + ‖xn ‖ = c(1 + ‖xn ‖),

∀n ≥ 1.

(2) If A is Lipschitz continuous and A−1 (θ) ≠ 0, then 󵄩 󵄩 󵄩 󵄩 ‖un ‖ ≤ L󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩 ≤ L‖xn ‖ + L󵄩󵄩󵄩x ∗ 󵄩󵄩󵄩 = c(1 + ‖xn ‖), where L is the Lipschitz constant and c = max{L‖x∗ ‖, L} = L max{‖x∗ ‖, 1}. In particular, if A is single-valued, linear, and bounded, then ‖Axn ‖ ≤ ‖A‖‖xn ‖ ≤ ‖A‖(1 + ‖xn ‖). (3) If A is bounded, then M(r) = sup{‖u‖ : u ∈ Ax, x ∈ Dom(A), ‖x − x1 ‖ ≤ 2r} < ∞ for sufficiently large r ≥ 1. By mathematical induction, we find ‖xn − Q(θ)‖ ≤ r, ∀n ≥ 1 provided that b(λn ) 2r ≤ . θn (M(r) + 32 r)2

284 | 5 Accretive operators and pseudocontractive mappings It follows that ‖un ‖ ≤ c(1 + ‖xn ‖), ∀un ∈ Axn , n ≥ 1. Chidume and Zegeye [25] extended the celebrated Bruck’s theorems [11] to the framework of uniformly smooth Banach spaces via the Xu’s inequality. Definition 5.6.1. Let E be a uniformly smooth Banach space. Two nonnegative real sequences {λn } and {θn } are said to be a compatible pair if and only if {θn } converges to 0 monotonically and there exists a strictly increasing subsequence {n(i)}∞ i=1 such that n(i+1) (i) lim infn→∞ θn(i) ∑j=n(i) λj > 0, (ii) lim infn→∞ (θn(i) − θn(i+1) ) ∑n(i+1) j=n(i) λj = 0, (iii) limn→∞ ∑n(i+1) j=n(i) ρE (λj ) = 0,

where ρE is the smoothness modulus of E. Remark 5.6.2. Let E be a q-uniformly smooth Banach space. Two nonnegative real sequences {λn } and {θn } are said to be a compatible pair if and only if the above (iii) is replaced with (iii)󸀠

n(i+1)

lim ∑ λjq = 0.

n→∞

j=n(i)

We here give an example: λn = n−1 , θn = (log log n)−1 , n(i) = ii , which satisfies (i), (ii), and (iii). Theorem 5.6.5. Let E be a uniformly smooth Banach space. Let A ⊂ E × E be an m-accretive operator with 0 ∈ Ran(A). Let {λn } and {θn } be a compatible pair. Let {xn } be a sequence in (BRIM). If both {un } and {vn } are bounded, then {xn } converges to a zero point of A in norm. Proof. Without loss of generality, we assume z = θ. Since A is m-accretive, we see that ξ −1 A, ξ > 0 is also m-accretive, and Ran(A + ξ −1 A) = E, ∀ξ > 0. Hence, there exists a unique yi ∈ E such that θ ∈ θi yi + Ayi , ∀i ≥ 1. It follows that lim Jθ−1 (θ) = lim yi = x ∗ ∈ A−1 (θ).

θi−1 →∞

i

i→∞

Moreover, ∀n ≥ i ≥ 2, un ∈ Axn , we have xn − yi = xn−1 − yi − λn−1 (un−1 + θn−1 xn−1 ). Using (XRI), we obtain ‖xn − yi ‖2 ≤ ‖xn−1 − yi ‖2 − 2λn−1 ⟨un−1 + θi xn−1 , J(xn−1 − yi )⟩ + 2λn−1 (θi − θi−1 )⟨xn−1 , J(xn−1 − yi )⟩ + D max{‖xn−1 − yi ‖ c + λn−1 ‖un−1 + θn−1 xn−1 ‖, }ρE (λn−1 ‖un−1 + θn−1 xn−1 ‖). 2

(5.36)

5.6 Iterative methods of zero points for accretive operators | 285

In view of −θi yi ∈ Ayi , un−1 ∈ Axn−1 , we find from ⟨un−1 + θi xn−1 , J(xn−1 − yi )⟩ ≥ 0 that ⟨un−1 + θi xn−1 , J(xn−1 − yi )⟩

= ⟨un−1 + θi yi , J(xn−1 − yi )⟩ + θi ‖xn−1 − yi ‖2 ≥ θi ‖xn−1 − yi ‖2 .

(5.37)

It follows from (5.36) and (5.37) that ‖xn − yi ‖2 ≤ (1 − 2λn−1 θi )‖xn−1 − yi ‖2 + 2λn−1 (θi − θi−1 )‖xn−1 ‖‖xn−1 − yi ‖ + D max{‖xn−1 − yi ‖ c + λn−1 ‖un−1 + θn−1 xn−1 ‖, }ρE (λn−1 ‖un−1 + θn−1 xn−1 ‖). 2

(5.38)

Since both {xn } and {un } are bounded, we find that there exist constants c󸀠 > 0, D󸀠 > 0, and M 󸀠 > 0 such that 2‖xn−1 ‖‖xn−1 − yi ‖ ≤ c󸀠 , c D max{‖xn−1 − yi ‖ + λn−1 ‖un−1 + θn−1 xn−1 ‖, } ≤ D󸀠 2 and ‖un−1 + θn−1 xn−1 ‖ ≤ M 󸀠 . In view of 1 − 2λn−1 θi ≤ exp(−2λn−1 θi ), we find from (5.38) that ‖xn − yi ‖2 ≤ exp(−2λn−1 θi )‖xn−1 − yi ‖2 + c󸀠 λn−1 (θi − θi−1 ) + D󸀠 ρE (λn−1 M 󸀠 ) 2

≤ exp(−2λn−1 θi )‖xn−1 − yi ‖2 + c󸀠 λn−1 (θi − θi−1 ) + D󸀠 (M 󸀠 ) ρE (λn−1 ). By induction on n, we obtain that θi − θj ≤ θi − θn ,

∀j ≤ n.

It follows that n−1

n

j=1

j=i

‖xn − yi ‖2 ≤ exp(−2θi ∑ λj )‖xn−1 − yi ‖2 + c󸀠 (θi − θi−1 ) ∑ λj 2

n

+ D󸀠 (M 󸀠 ) ∑ ρE (λj ). j=i

(5.39)

286 | 5 Accretive operators and pseudocontractive mappings Since {λn } and {θn } is a compatible pair, we see that n

2

n

‖xn − yi ‖2 ≤ δ‖xi − yi ‖2 + c󸀠 (θi − θn ) ∑ λj + D󸀠 (M 󸀠 ) ∑ ρE (λj ). j=i

j=i

(5.40)

Letting n = n(k + 1), i = n(k) in (5.40), we find that n(k+1)

‖xn(k+1) − yn(k) ‖2 ≤ δ‖xn(k) − yn(k) ‖2 + c󸀠 (θn(k) − θn(k+1) ) ∑ λj j=n(i)

2

+ D󸀠 (M 󸀠 )

n(k+1)

∑ ρE (λj ).

j=n(k)

Since xn(k) → x∗ as k → ∞, we find that xn − yn(k) → θ as n → ∞. Since yn(k) → x ∗ as k → ∞, we obtain that xn → x∗ as n → ∞. This completes the proof. Remark 5.6.3. For the function b in the Reich inequality (RI), we have that b(t) ≥ t, ∀t ≥ 0. If b(λn ) = o(θn ), then λn = o(θn ). The converse may not be true. When λn = o(θn ), 1 1 letting λn = (1+n) , we find that {λn } and {θn } satisfy (i), (ii), and (iii) in a and θn = (1+n)b Theorem 5.6.4, and (iv)’ λn = o(θn ). We do not know if they satisfy (iv) b(λn ) = o(θn ). However, if A ⊂ E × E is assumed to be Lipschitz continuous, we can find {λn } and {θn }, which satisfy (i), (ii), and (iii) in Theorem 5.6.4 and (iv)’ λn = o(θn ). Theorem 5.6.4 holds for such {λn } and {θn }. Theorem 5.6.6. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E, and have the fixed point property for nonexpansive self-mappings. Let A : C → E be an L-Lipschitz continuous accretive operator such that I − A maps C to C, and let R : C → E be an η-strongly accretive operator such that I − R maps C to itself. Assume that {λn } and {θn } satisfy the following restrictions: (i) λn (1 + θn ) ≤ 1, ∀n ≥ 1, (ii) θn → 0 as n → ∞, (iii) ∑∞ n=1 λn θn = ∞, θ (iv) θn−1 − 1 = o(λn θn ), n (v) λn = o(θn ). Let {xn } be a sequence generated in the following iterative method: x1 ∈ C,

xn+1 = xn − λn (Axn + θn Rxn ),

∀un ∈ Axn , n ≥ 1.

(VBRIM)

If θ ∈ Ran(A), then {xn } converges to some zero point x ∗ of A in norm and x ∗ also is a unique solution to the following variational inequality: ⟨Rx∗ , J(z − x∗ )⟩ ≥ 0,

∀z ∈ A−1 (θ).

(5.41)

5.6 Iterative methods of zero points for accretive operators | 287

Proof. Since both I − A and I − R are self-mappings on C, we see that (VBRIM) is well defined. It also easy to check that both A and R satisfy the flow-invariance condition (FIC). Since A−1 (θ) is not empty, we find from Theorem 5.5.1 (ii) that set X is bounded. It follows that R(X) is also bounded. Using Theorem 5.5.2, we find that path {xt : t ∈ [0, 1)} defined in (5.13) converges to some zero point x∗ of A in norm as t → 1− and x ∗ is the 1 unique solution to (5.41). Letting tn = 1+θ and setting yn = xtn , we have n

θ=

θn 1 Ay + Ry . 1 + θn n 1 + θn n

(5.42)

It follows that θ = Ayn + θn Ryn .

(5.43)

Using Theorem 5.5.2, we see that {yn } converges to some zero point x ∗ of A in norm and x∗ is the unique solution to (5.41). Next, we show that xn+1 − yn → θ as n → ∞. Using (5.42), we find that ‖yn − yn−1 ‖2 = ⟨yn − yn−1 , J(yn − yn−1 )⟩ − ⟨Ayn − Ayn−1 , J(yn − yn−1 )⟩ − ⟨θn Ryn − θn−1 Ryn−1 , J(yn − yn−1 )⟩

≤ ‖yn − yn−1 ‖2 − θn ⟨Ryn − Ryn−1 , J(yn − yn−1 )⟩ + (θn−1 − θn )⟨θn Ryn−1 , J(yn − yn−1 )⟩

≤ ‖yn − yn−1 ‖2 − ηθn ‖yn − yn−1 ‖2

+ |θn−1 − θn |‖Ryn−1 ‖‖yn − yn−1 ‖.

This implies that 󵄨󵄨 θ 󵄨󵄨 󵄨 󵄨 ‖yn − yn−1 ‖ ≤ η−1 󵄨󵄨󵄨 n−1 − 1󵄨󵄨󵄨M, 󵄨󵄨 θn 󵄨󵄨

(5.44)

where M is some appropriate constant. Using (5.43), we find that xn+1 − yn = xn − yn − λn (Axn − Ayn + θn (Rxn − Ryn )).

(5.45)

Using (VBRIM), we obtain that 󵄩 󵄩 ‖yn − xn+1 ‖2 = 󵄩󵄩󵄩λn (Axn − Ayn ) + λn θn (Rxn − Ryn )󵄩󵄩󵄩 󵄩 󵄩 ≤ Lλn ‖xn − yn ‖ + Lλn θn 󵄩󵄩󵄩(xn − yn )󵄩󵄩󵄩 ≤ 2Lλn ‖xn − yn ‖.

(5.46)

288 | 5 Accretive operators and pseudocontractive mappings Combining (5.45) with (5.46), we find from the Petryshyn inequality that ‖yn − xn+1 ‖2 ≤ ‖xn − yn ‖2 − 2λn ⟨Axn − Ayn , J(xn+1 − yn )⟩ − 2λn θn ⟨Rxn − Ryn , J(xn+1 − yn )⟩

≤ ‖xn − yn ‖2 − 2λn ⟨Axn − Axn+1 , J(xn+1 − yn )⟩ − 2λn ⟨Axn+1 − Ayn , J(xn+1 − yn )⟩

− 2λn θn ⟨Rxn − Rxn+1 , J(xn+1 − yn )⟩ − 2λn θn ⟨Rxn+1 − Ryn , J(xn+1 − yn )⟩

≤ ‖xn − yn ‖2 + 8L2 λn2 ‖xn − yn ‖‖xn+1 − yn ‖ − 2ηλn θn ‖xn+1 − yn ‖2 = ‖xn − yn ‖2 + 4L2 λn2 ‖xn − yn ‖2 + 4L2 λn2 ‖xn+1 − yn ‖ − 2ηλn θn ‖xn+1 − yn ‖2

= (1 + 4L2 λn2 )‖xn − yn ‖2 + (4L2 λn2 − 2ηλn θn )‖xn+1 − yn ‖2 . For sufficiently large n, we have ‖yn − xn+1 ‖2 ≤

1 + 4L2 λn2 ‖x − yn ‖2 . 1 − 4L2 λn2 + 2ηλn θn n

This further implies from (5.44) that 1 ‖yx − xn+1 ‖ ≤ (1 − ηλn θn )‖xn − yn ‖ 2 1 ≤ (1 − ηλn θn )‖xn − yn−1 ‖ + ‖yn − yn−1 ‖ 2 󵄨󵄨 θ 󵄨󵄨 1 󵄨 󵄨 ≤ (1 − ηλn θn )‖xn − yn−1 ‖ + η−1 󵄨󵄨󵄨 n−1 − 1󵄨󵄨󵄨M. 󵄨󵄨 θn 󵄨󵄨 2 Using Lemma 1.10.2, we find that xn+1 − yn → θ as n → ∞. Hence, xn → x ∗ as n → ∞. This completes the proof.

5.6.3 The iterative methods based on APPA The approximation proximity point algorithm (APPA), which was introduced by Rockafellar, is a popular and efficient approximation iterative method. Combining Rockafellar’s approximation proximity point algorithm with the normal Mann iterative method, we obtain a Mann–Rockafellar iterative method. Combining Rockafellar’s approximation proximity point algorithm with the Halpern iterative method, we obtain a Halpern–Rockafellar iterative method. Next, we investigate the convergence of the two new iterative methods.

5.6 Iterative methods of zero points for accretive operators | 289

Let E be a Banach space and let A : E × E be an m-accretive operator. Let Jr = (I + rA)−1 be the resolvent operator of A and let Ar = r −1 (I − Jr ) be the Yosida approximation of A, where r > 0. Let x1 be an element chosen arbitrarily in E and let {xn } be a sequence generated by x1 ∈ E, { { { yn ≈ Jrn xn , { { { {xn+1 = (1 − αn )xn + αn yn ,

(MRIM) n ≥ 1,

where {αn } ⊂ (0, 1) and {rn } ⊂ ℝ+ are two real sequences, {yn } is a sequence in E such that ‖yn − Jrn xn ‖ ≤ δn , ∀n ≥ 1, and ∑∞ n=1 δn < ∞. We call (MRIM) a Mann–Rockafellar iterative method. Let x1 and u be elements chosen arbitrarily in E and let {xn } be a sequence generated by x1 ∈ E, { { { yn ≈ Jrn xn , { { { {xn+1 = tn u + (1 − tn )yn ,

(HRIM) n ≥ 1,

where {tn } ⊂ (0, 1) and {rn } ⊂ ℝ+ are two real sequences, {yn } is sequence in E such that ‖yn − Jrn xn ‖ ≤ δn , ∀n ≥ 1, and ∑∞ n=1 δn < ∞. We call (HRIM) a Halpern–Rockafellar iterative method. Theorem 5.6.7. Let E be a uniformly convex Banach space with a Fréchet differentiable norm. Let A ⊂ E ×E be an m-accretive operator such that A−1 (θ) ≠ 0. Let {αn } ⊂ (0, 1) and ∞ {rn } ⊂ ℝ+ be two real sequences such that ∑∞ n=1 αn (1 − αn ) = ∞, ∑n=1 |rn+1 − rn | < ∞, and lim infn→∞ rn > 0. Let {xn } be a sequence generated by (MRIM). Then {xn } converges weakly to some zero point x∗ of A. Proof. Fixing z ∈ A−1 (θ), we find that ‖xn+1 − z‖ ≤ (1 − αn )‖xn − z‖ + αn ‖yn − z‖

≤ (1 − αn )‖xn − z‖ + αn ‖yn − Jrn xn ‖ + αn ‖z − Jrn xn ‖

≤ ‖xn − z‖ + αn δn . Since ∑∞ n=1 δn

< ∞, we find from Lemma 1.10.1 that limn→∞ ‖xn −z‖ exist. Using a result due to Xu [98], we find that ‖xn+1 − z‖2 ≤ (1 − αn )‖xn − z‖2 + αn ‖yn − z‖2 − αn (1 − αn )g(‖xn − yn ‖) ≤ (1 − αn )‖xn − z‖2 + αn ‖yn − Jrn xn ‖2 + αn ‖z − Jrn xn ‖2

+ 2αn ‖z − Jrn xn ‖‖yn − Jrn xn ‖ − αn (1 − αn )g(‖xn − yn ‖)

≤ ‖xn − z‖2 + αn δn2 + 2αn δn ‖z − xn ‖ − αn (1 − αn )g(‖xn − yn ‖) ≤ ‖xn − z‖2 + δn M − αn (1 − αn )g(‖xn − yn ‖),

290 | 5 Accretive operators and pseudocontractive mappings where M is an appropriate constant. It follows that ∞

∞

n=1

n=1

∑ αn (1 − αn )g(‖xn − yn ‖) ≤ ‖x1 − z‖2 + M ∑ δn < +∞.

Since ∑∞ n=1 αn (1 − αn ) = ∞, we find that lim infn→∞ g(‖xn − yn ‖) = 0. This implies that lim infn→∞ ‖xn − yn ‖ = 0. Hence, lim infn→∞ ‖xn − Jrn xn ‖ = 0. Note that ‖xn+1 − Jrn+1 xn+1 ‖

≤ (1 − αn )‖xn − Jrn+1 xn+1 ‖ + αn ‖yn − Jrn+1 xn+1 ‖

≤ (1 − αn )‖xn − Jrn xn ‖ + (1 − αn )‖Jrn xn − Jrn+1 xn+1 ‖ + αn ‖yn − Jrn xn ‖ + αn ‖Jrn xn − Jrn+1 xn+1 ‖

≤ (1 − αn )‖xn − Jrn xn ‖ + ‖Jrn xn − Jrn+1 xn+1 ‖ + δn

≤ (1 − αn )‖xn − Jrn xn ‖ + ‖Jrn xn − Jrn+1 xn ‖ + ‖Jrn+1 xn − Jrn+1 xn+1 ‖ + δn 󵄨󵄨 r 󵄨󵄨󵄨 󵄨 ≤ (1 − αn )‖xn − Jrn xn ‖ + 󵄨󵄨󵄨1 − n 󵄨󵄨󵄨‖xn − Jrn+1 xn ‖ + ‖xn − xn+1 ‖ + δn 󵄨󵄨 rn+1 󵄨󵄨 󵄨󵄨 󵄨 r 󵄨󵄨 󵄨 ≤ ‖xn − Jrn xn ‖ + 󵄨󵄨󵄨1 − n 󵄨󵄨󵄨M1 + 2δn , 󵄨󵄨 rn+1 󵄨󵄨

where M1 = sup{‖xn − Jrn+1 xn ‖ : n ≥ 1}. r

∞ n Since ∑∞ n=1 |1− rn+1 | < ∞ and ∑n=1 δn < ∞, we find from Lemma 1.10.1 that limn→∞ ‖xn − Jrn xn ‖ exists. In view of lim infn→∞ ‖xn −Jrn xn ‖ = 0, we obtain that limn→∞ ‖xn −Jrn xn ‖ = 0. Since lim infn→∞ rn > 0, we may, without loss of generality, assume that there exists r > 0 such that rn ≥ r for every n ≥ 1. Taking a fixed positive r > ϵ > 0 and using the resolvent equality, we see that

󵄩󵄩 󵄩󵄩 ϵ ϵ 󵄩 󵄩 ‖Jrn xn − Jϵ xn ‖ = 󵄩󵄩󵄩Jϵ ( xn + (1 − )Jrn xn ) − Jϵ xn 󵄩󵄩󵄩 󵄩󵄩 rn 󵄩󵄩 rn 󵄩 󵄩󵄩 ϵ 󵄩󵄩 ϵ 󵄩 ≤ 󵄩󵄩󵄩( xn + (1 − )Jrn xn ) − xn 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 rn rn ≤ (1 −

ϵ )‖Jrn xn − xn ‖ rn

≤ ‖Jrn xn − xn ‖. It follows that

‖xn − Jϵ xn ‖ ≤ ‖xn − Jrn xn ‖ + ‖Jϵ xn − Jrn xn ‖ ≤ 2‖xn − Jrn xn ‖.

5.6 Iterative methods of zero points for accretive operators | 291

Hence, we obtain that limn→∞ ‖xn − Jϵ xn ‖ = 0. Since Jϵ is nonexpansive, we find that ω(xn ) ⊂ Fix(Jϵ ) = A−1 (θ). Using Tan–Xu’s Lemma [93], we find that p = q, ∀p, q ∈ ω(xn ). This shows that ω(xn ) is a singleton. Hence, {xn } converges weakly to some zero point x∗ of A. This completes the proof. Theorem 5.6.8. Let E be a uniformly smooth Banach space. Let A ⊂ E × E be an m-accretive operator such that A−1 (θ) ≠ 0. Let {tn } ⊂ (0, 1) and {rn } ⊂ ℝ+ be two real setn ∞ quences such that limn→∞ tn = 0, ∑∞ n=1 tn = ∞, ∑n=1 |tn+1 − tn | < ∞ (or limn→∞ tn+1 = 1), lim infn→∞ rn > 0, and ∑∞ n=1 |rn+1 − rn | < ∞. Let {xn } be a sequence generated by (HRIM). Then {xn } converges strongly to some zero point x∗ of A. Proof. Fixing z ∈ A−1 (θ), we find that ‖xn+1 − z‖ ≤ (1 − tn )‖u − z‖ + tn ‖yn − z‖

≤ (1 − tn )‖u − z‖ + tn ‖Jrn xn − z‖ + tn ‖yn − Jrn xn ‖

≤ (1 − tn )‖u − z‖ + tn ‖xn − z‖ + δn .

Since ∑∞ n=1 δn < ∞, by induction, we find that {xn } is bounded, so are {Jrn xn } and {yn }. Note that ‖xn+1 − xn ‖ ≤ |tn−1 − tn |(‖u‖ + ‖yn−1 ‖) + (1 − tn )‖yn − yn−1 ‖ ≤ |tn−1 − tn |M1 + (1 − tn )‖yn − yn−1 ‖,

(5.47)

where M1 = ‖u‖ + sup{‖yn ‖ : n ≥ 1}. Using the resolvent equality, we find that ‖yn−1 − yn ‖ ≤ ‖yn−1 − Jrn−1 xn−1 ‖ + ‖Jrn xn − Jrn−1 xn−1 ‖ + ‖Jrn xn − yn ‖ ≤ ‖Jrn xn − Jrn−1 xn−1 ‖ + δn + δn−1

≤ ‖xn − xn−1 ‖ + δn + δn−1 󵄩󵄩 󵄩󵄩 r r 󵄩 󵄩 + 󵄩󵄩󵄩Jrn−1 ( n−1 xn−1 + (1 − n−1 )Jrn xn−1 ) − Jrn−1 xn−1 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 rn rn 󵄩󵄩 r 󵄩󵄩 r 󵄩 󵄩 ≤ ‖xn − xn−1 ‖ + 󵄩󵄩󵄩( n−1 xn−1 + (1 − n−1 )Jrn xn−1 ) − xn−1 󵄩󵄩󵄩 + δn + δn−1 󵄩󵄩 󵄩󵄩 rn rn 󵄨 󵄨󵄨 r 󵄨󵄨 󵄨 ≤ ‖xn − xn−1 ‖ + 󵄨󵄨󵄨1 − n−1 󵄨󵄨󵄨M2 + δn + δn−1 , (5.48) 󵄨󵄨 rn 󵄨󵄨

where M2 = sup{‖Jrn xn−1 − xn−1 ‖ : n ≥ 1}.

292 | 5 Accretive operators and pseudocontractive mappings Substituting (5.48) into (5.47), we find that 󵄨󵄨 r 󵄨󵄨󵄨 󵄨 ‖xn+1 − xn ‖ ≤ |tn−1 − tn |M1 + (1 − tn )‖xn − xn−1 ‖ + (1 − tn )󵄨󵄨󵄨1 − n−1 󵄨󵄨󵄨M2 + δn + δn−1 . 󵄨󵄨 rn 󵄨󵄨 Using Lemma 1.10.2, we find that limn→∞ ‖xn+1 − xn ‖ = 0. This implies that limn→∞ ‖Jrn xn − xn ‖ = 0. Since lim infn→∞ rn > 0, we may, without loss of generality, assume that there exists r > 0 such that rn ≥ r for every n ≥ 1. Fix r > ϵ > 0. From the proof of Theorem 5.6.7, we find that limn→∞ ‖xn − Jϵ xn ‖ = 0. We are now in a position to show that lim sup⟨u − x∗ , J(yn − x ∗ )⟩ ≤ 0, n→∞

where x∗ = QA−1 (θ) u and QA−1 (θ) : E → A−1 (θ) is the unique sunny nonexpansive retraction from E onto A−1 (θ). Consider mapping Ct : E → E by Ct x = tu + (1 − t)Jϵ x,

∀x ∈ E, t ∈ (0, 1).

Then Ct : E → E is (1−t)-contractive. Using the Banach contractive mapping principle, we find that there exists a unique point yt in E such that yt = Ct (yt ) = tu + (1 − t)Jϵ yt ,

∀t ∈ (0, 1).

It is not hard to see that {yt } converges to x∗ = QA−1 (θ) u in norm and QA−1 (θ) : E → A−1 (θ) is the unique sunny nonexpansive retraction from E onto A−1 (θ). Note that ‖yt − xn ‖2 ≤ t⟨u − xn , J(yt − xn )⟩ + (1 − t)⟨Jϵ yt − xn , J(yt − xn )⟩

≤ t⟨u − xn , J(yt − xn )⟩ + (1 − t)⟨Jϵ yt − Jϵ xn , J(yt − xn )⟩ + (1 − t)⟨Jϵ xn − xn , J(yt − xn )⟩

≤ t⟨u − yt , J(yt − xn )⟩ + t⟨yt − xn , J(yt − xn )⟩

+ (1 − t)‖yt − xn ‖2 + (1 − t)‖Jϵ xn − xn ‖‖yt − xn ‖.

Hence, we have ⟨yt − u, J(yt − xn )⟩ ≤

1−t ‖Jϵ xn − xn ‖‖yt − xn ‖. t

(5.49)

Letting n → ∞, we find that lim sup⟨yt − u, J(yt − xn )⟩ ≤ 0, n→∞

∀t ∈ (0, 1).

Since {yt } converges to x∗ = QA−1 (θ) u as t → 0, we see that there exists κ1 > 0 such that ξ 󵄩󵄩 ∗󵄩 , 󵄩󵄩yt − x 󵄩󵄩󵄩 ≤ 3M1

∀0 < t < κ1 .

5.6 Iterative methods of zero points for accretive operators | 293

Since E is uniformly smooth, we see that J is uniformly continuous on any bounded subset of E. Hence, we have J(yt −xn ) → J(x∗ −xn ) as n → ∞. Hence, there exists κ2 > 0 such that ξ 󵄩 󵄩󵄩 ∗ , 󵄩󵄩J(yt − xn ) − J(x − xn )󵄩󵄩󵄩 ≤ 3(‖x∗ − u‖ + 1)

0 < t < κ2 .

Letting κ = min{κ1 , κ2 }, we see from (5.49) that there exists N ≥ 1 such that ⟨yt − u, J(yt − xn )⟩
0, there exists N ≥ 1 such that ⟨x∗ − u, J(x∗ − xn )⟩ 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩x∗ − u󵄩󵄩󵄩󵄩󵄩󵄩J(x∗ − xn ) − J(yt − xn )󵄩󵄩󵄩 + M1 ‖yt − xn ‖ + ⟨yt − u, J(yt − xn )⟩ ξ ξ ξ + + 3 3 3 = ξ , ∀n ≥ N.
0, limn→∞ |rn+1 − rn | = 0 and 0 < lim infn→∞ λn ≤ lim supn→∞ λn < 1. Let {xn } be generated by (MHRIM-1). Then {xn } converges strongly to some zero point x∗ of A. Proof. Set zn := λn xn + (1 − λn )yn . Fixing z ∈ A−1 (θ), we find that ‖yn − z‖ ≤ λn ‖xn − z‖ + (1 − λn )‖yn − z‖

≤ λn ‖xn − z‖ + (1 − λn )‖yn − Jrn xn ‖ + (1 − λn )‖Jrn xn − z‖ ≤ ‖xn − z‖ + δn .

It follows that ‖xn+1 − z‖ ≤ tn ‖u − z‖ + (1 − tn )‖yn − z‖

≤ tn ‖u − z‖ + (1 − tn )‖xn − z‖ + δn

Since ∑∞ n=1 δn < ∞, by induction, we find that {xn } is bounded, so are {Jrn xn }, {yn } and {zn }. Define wn =

xn+1 − λn xn . 1 − λn

Using the resolvent equality, we have tn+1 t + n )‖u‖ + ‖yn+1 − yn ‖ 1 − λn+1 1 − λn t t + n+1 ‖zn+1 ‖ + n ‖zn ‖ 1 − λn+1 1 − λn

‖wn+1 − wn ‖ ≤ (

≤(

tn+1 t + n )M + ‖yn+1 − Jrn+1 xn+1 ‖ 1 − λn+1 1 − λn

5.6 Iterative methods of zero points for accretive operators | 295

+ ‖Jrn+1 xn+1 − Jrn xn ‖ + ‖Jrn xn − yn ‖ ≤(

tn+1 t + n )M + δn+1 + δn + ‖xn+1 − xn ‖ + ‖Jrn+1 xn − Jrn xn ‖ 1 − λn+1 1 − λn

tn+1 t + n )M + δn+1 + δn + ‖xn+1 − xn ‖ 1 − λn+1 1 − λn 󵄨󵄨 r 󵄨󵄨󵄨 󵄨 + 󵄨󵄨󵄨1 − n 󵄨󵄨󵄨‖Jrn+1 xn − xn ‖. rn+1 󵄨󵄨 󵄨󵄨

≤(

Using the restrictions imposed on {tn }, {λn } and {rn }, we obtain that lim sup(‖wn+1 − wn ‖ − ‖xn+1 − xn ‖) ≤ 0. n→∞

In view of Lemma 1.10.3, we find that limn→∞ ‖wn − xn ‖ = 0. It follows that limn→∞ ‖xn+1 − xn ‖ = 0. This further implies limn→∞ ‖zn − xn ‖ = 0 and limn→∞ ‖Jrn xn − xn ‖ = 0. Since lim infn→∞ rn > 0, we may, without loss of generality, assume that there exists r > 0 such that rn ≥ r for every n ≥ 1. Fix r > ϵ > 0. From the proof of Theorem 5.6.7, we find that limn→∞ ‖xn − Jϵ xn ‖ = 0. From the proof in Theorem 5.6.8, we obtain the desired conclusion immediately. We now consider another modification of (HRIM). Let x1 and u be elements chosen arbitrarily in E, and let {xn } be a sequence generated by x1 ∈ E, { { { yn ≈ Jrn xn , { { { {xn+1 = λn xn + (1 − λn )(tn u + (1 − tn )yn ),

(MHRIM-2) n ≥ 1,

where {tn } ⊂ (0, 1), {λn } ⊂ (0, 1), and {rn } ⊂ ℝ+ are three real sequences, {yn } is sequence in E such that ‖yn − Jrn xn ‖ ≤ δn , ∀n ≥ 1, and ∑∞ n=1 δn < ∞. We also call (MHRIM-2) a modified Halpern–Rockafellar iterative method. Theorem 5.6.10. Let E be a uniformly smooth Banach space. Let A ⊂ E × E be an m-accretive operator such that A−1 (θ) ≠ 0. Let {tn } ⊂ (0, 1), {λn } ⊂ (0, 1), and {rn } ⊂ ℝ+ be three real sequences such that limn→∞ tn = 0, ∑∞ n=1 tn = ∞, lim infn→∞ rn > 0, limn→∞ |rn+1 − rn | = 0, and 0 < lim infn→∞ λn ≤ lim supn→∞ λn < 1. Let {xn } be a sequence generated by (MHRIM-2). Then {xn } converges strongly to some zero point x∗ of A. Let E be a Banach space and let C be closed convex subset of E. One knows that A is accretive if and only if T := I − A, where I is the identity, is pseudocontractive. From the Caristi fixed point theorem, we see that accretive operator A satisfies the flow-invariance condition (FIC) if and only if pseudocontractive mapping T satisfies the weak inward condition (WIC). So, every zero point theorem of accretive operators corresponds with a fixed point theorem of pseudocontractive mappings.

296 | 5 Accretive operators and pseudocontractive mappings

5.7 Iterative methods for variational inequalities with accretive operators In this section, we investigate some iterative methods for solving a class of variational inequalities with accretive operators. First, we recall and review shortly classical gradient projection method and Yamada’s hybrid steepest decent method for solving monotone-type variational inequality problems. We then point out some gaps appearing in Chidume’s monograph [23] and extend most of the results of Chidume [23] to more general cases. In particular, we extend the results of Yamada [108], Xu and Kim [105], Iemoto and Takahashi [31], and others from real Hilbert spaces to the more general setting of uniformly smooth Banach spaces. 5.7.1 Two kinds of variational inequality problems Let K be a nonempty, closed, and convex subset of a real Hilbert space H, and let A be a monotone mapping from K into H. The classical monotone type variational inequality problem is formulated as follows: Find a point x ∗ ∈ K such that ⟨Ax∗ , y − x∗ ⟩ ≥ 0, ∀y ∈ K.

(MTVIP)

We use VIP(A, K) to denote the set of solutions for (MTVIP). It is known that solving (MTVIP) is equivalent to finding a fixed point of the mapping PK (I − δA) : K → K, where δ ∈ (0, 1) is a fixed constant and PK is the nearest point projection mapping from H onto K, that is, PK x = y where ‖x − y‖ = inf{‖x − u‖ : u ∈ K},

for x ∈ H.

Due to the fact that PK : H → K is nonexpansive, under appropriate conditions on A and δ ∈ (0, 1), fixed point methods can be used to find a solution of (MTVIP). For instance, if A is η-strongly monotone and k-Lipschitz, then a mapping G : H → H 2η defined by Gx = PK (x − δAx), x ∈ H with δ ∈ (0, k2 ) is contractive. As a result of this fact, the known Picard iteration xn ∈ H, xn+1 = Gxn , n ≥ 0, of the classical Banach contractive mapping principle converges strongly to the unique solution of (MTVIP). On the other hand, the projection mapping PK may make the computation of the iterates difficult due to possible complexity of the convex set K. In order to reduce the possible difficulty with the use of PK , Yamada [108] introduced the so-called hybrid steepest descent methods for solving (MTVIP): x0 ∈ H,

xn+1 = Txn − λn+1 δA(Txn ),

n≥0

(HSDM1)

n ≥ 0,

(HSDM2)

and x0 ∈ H,

xn+1 = T[n+1] xn − λn+1 δA(T[n+1] xn ),

5.7 Iterative methods for variational inequalities with accretive operators | 297

where T[k] = Tk mod r , for k ≥ 1, with the mod r function taking values in the set {1, 2, . . . , r}. Yamada [108] proved the following results. Theorem 5.7.1. Let T : H → H be a nonexpansive mapping such that K := {x ∈ H : Tx = 2η x} ≠ 0. Let A be an η-strongly monotone and k-Lipschitz mapping on H. Let δ ∈ (0, k2 ) be an arbitrary but fixed real number and let {λn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ λn = 0; (C2) ∑∞ n=0 λn = ∞; λ (C3) limn→∞ λn+1 2 = 0. n

Then the sequence {xn } defined by (HSDM1) converges strongly to the unique solution of VIP(A, K). Theorem 5.7.2. Let {Ti }ri=1 : H → H be a finite family of nonexpansive mappings such that K = ⋂ri=1 F(Ti ) ≠ 0. Let A be an η-strongly monotone and k-Lipschitz mapping on H. Let {λn } be a sequence of real numbers in (0, 1) satisfying conditions (C1), (C2), and (C4) ∑∞ n=0 |λn − λn+r | < ∞. Then the sequence {xn } defined by (HSDM2) converges strongly to the unique solution of VIP(A, K). Afterwards, Xu and Kim [105] improved on the results of Yamada with condition |λ −λn | → 0 (n → ∞) and with condition (C4) replaced by (C6) (C3) replaced by (C5) n+1 λ |λn+r −λn | λn+r

n+1

→ 0 (n → ∞). Namely, they proved that following theorems.

Theorem 5.7.3. Let T : H → H be a nonexpansive mapping such that K = {x ∈ H : x = 2η Tx} ≠ 0. Let A be η-strongly monotone and k-Lipschitz on H. Let δ ∈ (0, k2 ) and let {λn } be a sequence in (0, 1) satisfying the following conditions: (C1) λn → 0 (n → ∞); (C2) ∑∞ n=0 λn = ∞; and |λn −λn+1 | (C5) → 0 (n → ∞). λ n+1

Then, the sequence {xn } defined by (HSDM1) converges strongly to the unique solution of VIP(A, K). Theorem 5.7.4. Let {Ti }ri=1 : H → H be a finite family of nonexpansive mappings such that K = ⋂ri=1 Fix(Ti ) = Fix(T1 T2 ⋅ ⋅ ⋅ Tr ) = Fix(Tr T1 ⋅ ⋅ ⋅ Tr−1 ) = ⋅ ⋅ ⋅ = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) ≠ 0. 2η Let A be η-strongly monotone and k-Lipschitz on H. Let δ ∈ (0, k2 ) and let {λn } be a sequence in (0, 1) satisfying the following conditions: (C1) λn → 0 (n → ∞); (C2) ∑∞ n=0 λn = ∞; and |λ −λ | (C6) nλ n+r → 0 (n → ∞). n+r

298 | 5 Accretive operators and pseudocontractive mappings Then the sequence {xn } defined by (HSDM2) converges strongly to the unique solution of VIP(A, K) In 2008, Iemoto and Takahashi [31] considered the case of K = ⋂∞ i=1 Fix(Ti ), where is a countable family of nonexpansive mappings on Hilbert spaces. Zhou and Wang [125] considered the case of K = ⋂ri=1 Fix(Ti ), where {Ti }ri=1 is a finite family of Lipschitz quasi-pseudocontractive mappings on Hilbert spaces. An interesting problem arises naturally: Can the gradient projection methods and the hybrid steepest decent methods be used to find a solution of some variational inequality problems with accretive operators in a real Banach space? Chidume considered this problem in [23]. There he attempted to extend some of the results mentioned above to the more general Banach spaces, however, there are some gaps in Chapter 7 of his monograph. We remark that if X is a q-uniformly smooth Banach space, then q ≤ 2. In Lemma 7.10 of Chidume [23], he requires that q ≥ 2. Thus q = 2. Consequently, the space appearing in Lemma 7.10 is 2-uniformly smooth. In other words, Lemma 7.10 of Chidume [23] holds true only for 2-uniformly smooth Banach spaces. Moreover, we point out that Lemma 7.30 of Chidume [23] holds true only for p = 2. Let K be a nonempty, closed, and convex subset of a real Banach space X, and A : K → X an accretive operator. Let J be the normalized duality mapping on X. The variational inequality problem with accretive operators is formulated as follows: Find a point x∗ ∈ K such that {Ti }∞ i=1

⟨Ax∗ , j(y − x∗ )⟩ ≥ 0, ∀y ∈ K,

(VIPAM1)

where j(y − x∗ ) ∈ J(y − x∗ ). Another kind of variational inequality problem with accretive operators is to find a point x∗ ∈ K such that ⟨y − x∗ , j(Ax∗ )⟩ ≥ 0,

∀y ∈ K.

(VIPAM2)

where j(Ax∗ ) ∈ J(Ax ∗ ). If X = H, a real Hilbert space, the variational inequality problems both (VIPAM1) and (VIPAM2) are reduced to (MTVIP). It is known that if X is a real reflexive, smooth, and strictly convex Banach space and K is a nonempty, closed, and convex subset of X, then ∀x ∈ X, there exists a unique x0 ∈ K such that ‖x − x0 ‖ = min{‖x − y‖ : y ∈ K}. Define a mapping PK : X → K by PK x = x0 . Then PK : X → K is called the metric projection mapping from X onto K.

5.7 Iterative methods for variational inequalities with accretive operators | 299

Using Theorem 1.8.49, we know that x0 = PK x is and only if ⟨y − x0 , J(x0 − x)⟩ ≥ 0,

∀y ∈ K.

Let A : K → X be a mapping and let δ > 0 be a fixed number. Observe that there is some x∗ ∈ K such that ⟨y − x∗ , J(Ax∗ )⟩ ≥ 0,

∀y ∈ K

⇐⇒ ⟨y − x , J(δAx )⟩ ≥ 0, ∗

∗

∀y ∈ K, δ > 0

⇐⇒ ⟨y − x , J(x − (x − δAx∗ ))⟩ ≥ 0, ∗

∗

∗

⇐⇒ x = PK (x − δAx ), ∗

∗

∀y ∈ K, δ > 0

δ > 0.

∗

Consequently, solving (VIPAM2) is equivalent to finding a fixed point of mapping PK (I− δA). Unfortunately, the metric projection mapping PK in a general Banach space is no longer nonexpansive, and thus the gradient projection method cannot be used to find a solution of variational inequality problem (VIPAM2). Let us consider (VIPAM1). Let K be a nonempty, closed, and convex subset of a real Banach space X. Assume that there exists a sunny nonexpansive retraction QK from X onto K. Observe that (VIPAM1) has a solution x∗ ∈ K such that ⟨j(y − x∗ ), Ax∗ ⟩ ≥ 0,

∀y ∈ K

⇐⇒ ⟨j(y − x ), δAx ⟩ ≥ 0, ∗

∗

∀y ∈ K, δ > 0

⇐⇒ ⟨j(y − x ), x − (x − δAx∗ )⟩ ≥ 0, ∗

∗

∗

⇐⇒ x = QK (x − δAx ), ∗

∗

∗

∀y ∈ K, δ > 0

δ > 0.

Therefore, (VIPAM1) has a unique solution provided that there exists a sunny nonexpansive retraction QK from X onto K and the mapping (I − δA) is a strict contraction on X. Furthermore, both fixed point methods and hybrid steepest decent methods can be used to find the unique solution of (VIPAM1). We present several convergence theorems which extend the results of Chidume [23] to the more general setting of uniformly smooth Banach spaces. In particular, the convergence theorems to be presented will be applicable in Lp , where 1 < p < ∞.

5.7.2 Tools to solve the variational inequality In this subsection, we list some important lemmas for variational inequality (VIPAM1). Lemma 5.7.1 ([107]). Let X be a real Banach space. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Then there exists some τ ∈ (0, 1) such that I − τA is contractive on X.

300 | 5 Accretive operators and pseudocontractive mappings Lemma 5.7.2. Let X be a real Banach space. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let T : X → X be a mapping such that K := Fix(T) = {x ∈ X : Tx = x} ≠ 0. Assume that there exists a sunny nonexpansive retraction Q : X → K from X onto K. Then the variational inequality (VIPAM1) has a solution x∗ ∈ K. Proof. Using Lemma 5.7.1, we know that there exists τ ∈ (0, 1) such that I − τA is contractive on X. Since QK : X → K is nonexpansive, we see that the mapping QK (I −τA) is contractive on X. It follows from the Banach contractive mapping principle that there exists a unique x∗ ∈ X such that x∗ = QK (I − τA)x∗ .

(5.50)

Since QK : X → K is a sunny nonexpansive retraction, we find from Theorem 1.8.49 and (5.50) that ⟨x∗ − (I − τA)x∗ , j(y − x ∗ )⟩ ≥ 0, ⇐⇒ ⟨Ax , j(y − x )⟩, ∗

∗

∀y ∈ K

∀y ∈ K.

Hence x∗ ∈ K is the unique solution of (VIPAM1). This completes the proof. By using the Reich’s inequality, we can deduce the following result. Lemma 5.7.3. Let X be a real uniformly smooth Banach space and let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Then, if δ ∈ (0, k −1 b−1 (2ηk −1 )), the mapping f := I − δA : X → X is contractive. Proof. Define f : X → X by f (x) = (I − δA)x,

x ∈ X.

(5.51)

For x ≠ y, x, y ∈ X, we have x−y Ax − Ay f (x) − f (y) = −δ . ‖x − y‖ ‖x − y‖ ‖x − y‖ By using Theorem 1.8.28, we have ‖f (x) − f (y)‖2 Ax − Ay x−y ≤ 1 − 2δ⟨ , j( )⟩ ‖x − y‖ ‖x − y‖ ‖x − y‖2 + max {1,1}δ

‖Ax − Ay‖ ‖Ax − Ay‖ b(δ ) ‖x − y‖ ‖x − y‖

≤ 1 − 2ηδ + δkb(δk), which implies

󵄩󵄩 󵄩2 2 󵄩󵄩f (x) − f (y)󵄩󵄩󵄩 ≤ (1 − 2ηδ + δkb(δk))‖x − y‖ .

(5.52)

5.7 Iterative methods for variational inequalities with accretive operators | 301

This yields 󵄩 󵄩󵄩 󵄩󵄩f (x) − f (y)󵄩󵄩󵄩 ≤ ρ‖x − y‖,

∀x, y ∈ X,

(5.53)

where ρ := √1 − 2ηδ + δkb(δk) ∈ (0, 1), completing the proof. It is well known that if X is a q-uniformly smooth Banach space, then there exists some constant c1 ≥ 1, such that b(t) ≤ c1 t q−1 ,

t ∈ [0, 1].

(5.54)

Thus, we have the following result. Lemma 5.7.4. Let X be a q-uniformly smooth Banach space and let A : X → X be an 2η

1

η-strongly accretive and k-Lipschitz mapping. Then, if δ ∈ (0, ( kq c ) q−1 ), the mapping 1 f := (I − δA) : X → X is contractive. It is well known that if X = Lp , p ≥ 2, then we have b(t) ≤ (p − 1)t,

t ≥ 0.

(5.55)

Using (5.55), we have the following result. Lemma 5.7.5. Let X = Lp , p ≥ 2. Let A : X → X be an η-strongly accretive and 2η k-Lipschitz mapping. Then, if δ ∈ (0, k2 (p−1) ), the mapping f := (I − δA) : X → X is contractive. 5.7.3 Strong convergence theorems Theorem 5.7.5. Let X be a real reflexive Banach space whose norm is uniformly Gâteaux differentiable. Suppose that every nonempty, closed, convex, and bounded subset of X has a fixed point property for nonexpansive self-mappings. Let T : X → X be a nonexpansive mapping such that K := Fix(T) ≠ 0. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let τ ∈ (0, 1) be the constant appearing in Lemma 5.7.1. Also ∀x0 ∈ X, define a sequence {xn } iteratively as follows: xn+1 = αn f (xn ) + (1 − αn )Txn ,

n ≥ 0,

(5.56)

where f := I − τA and {αn } is a sequence of real numbers that satisfies the following conditions: (C1) limn→∞ αn = 0;

302 | 5 Accretive operators and pseudocontractive mappings (C2) ∑∞ n=0 αn = ∞; (C3) ∑∞ n=1 |αn+1 − αn | < ∞; or α (C4) limn→∞ αn+1 = 1. n

Then, the sequence {xn } defined by (5.56) converges strongly to the unique solution x ∗ of (VIPAM1). Proof. From Theorem 4.3.1, we see that there exists a unique sunny nonexpansive retraction QFix(T) from X onto Fix(T). By Lemma 5.7.1, we know that f := (I − τA) : X → X is contractive. By Lemma 5.7.2, we conclude that there exists a unique solution x∗ ∈ Fix(T) that satisfies (VIPAM1). We now define an auxiliary sequence {yn } by y0 ∈ X,

yn+1 = αn f (x ∗ ) + (1 − αn )Tyn ,

n ≥ 0.

(5.57)

Then, by using Theorem 4.3.2, we know that {yn } converges strongly to p = QFix(T) f (x∗ ), where QFix(T) : X → Fix(T) is the sunny nonexpansive retraction from X onto Fix(T). By Theorem 1.8.49, we have that x∗ = QFix(T) (x ∗ − τAx ∗ ) = QFix(T) f (x ∗ ). Hence p = x ∗ . Since f is strictly contractive, we see that there is a constant ρ ∈ (0, 1) such that 󵄩󵄩 󵄩 󵄩󵄩f (x) − f (y)󵄩󵄩󵄩 ≤ ρ‖x − y‖,

∀x, y ∈ X.

(5.58)

Combining (5.56), (5.57), and (5.58), we have 󵄩 󵄩 ‖xn+1 − yn+1 ‖ ≤ αn 󵄩󵄩󵄩f (xn ) − f (x ∗ )󵄩󵄩󵄩 + (1 − αn )‖Txn − Tyn ‖ 󵄩 󵄩 ≤ αn ρ‖xn − yn ‖ + ραn 󵄩󵄩󵄩yn − x ∗ 󵄩󵄩󵄩 + (1 − αn )‖xn − yn ‖ = [1 − (1 − ραn )]‖xn − yn ‖ + o(αn ). It follows from Lemma 1.10.2 that xn − yn → θ as n → ∞. Consequently, {xn } converges strongly to the unique solution x∗ of (VIPAM1). Theorem 5.7.6. Let X, T, A, τ, and {αn } be the same as those in Theorem 5.7.5. Also ∀x0 ∈ X, define a sequence {xn } iteratively as follows: xn+1 = Txn − ταn A(Txn ),

n ≥ 0.

(5.59)

Then, the sequence {xn } defined by (5.59) converges strongly to the unique solution x ∗ of (VIPAM1). Proof. Notice that (5.59) can be rewritten as follows: xn+1 = (1 − αn )Txn + αn g(xn ),

n ≥ 0,

(5.60)

where g(x) = (I − τA)Tx, x ∈ X. It is clear that g : X → X is strictly contractive, since I − τA is strictly contractive and T is nonexpansive. By a similar argument as that used to prove Theorem 5.7.5, we can prove that {xn } converges strongly to the unique solution x∗ of (VIPAM1). This completes the proof.

5.7 Iterative methods for variational inequalities with accretive operators | 303

Theorem 5.7.7. Let X be a real uniformly smooth Banach space. Let T : X → X be a nonexpansive mapping such that K := Fix(T) ≠ 0. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let {αn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=0 αn = ∞; (C3) ∑∞ n=1 |αn+1 − αn | < ∞; or α (C4) limn→∞ αn+1 = 1. n

For δ ∈ (0, k −1 b−1 (2k −1 η)), define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = αn f (xn ) + (1 − αn )Txn ,

n ≥ 0,

(5.61)

where f (x) = x − δAx, ∀x ∈ X. Then, the sequence {xn } defined by (5.96) converges strongly to the unique solution x∗ of (VIPAM1). Proof. Using Lemma 5.7.3, we know that f : X → X is contractive. From Corollary 4.4.1, we see that {xn } converges strongly to the unique x ∗ ∈ Fix(T) that satisfies the variational inequality: ⟨x ∗ − f (x ∗ ), j(y − x∗ )⟩ ≥ 0, ⇐⇒ ⟨Ax , j(y − x )⟩ ≥ 0, ∗

∗

∀y ∈ Fix(T)

∀y ∈ Fix(T).

Hence x∗ is the unique solution of (VIPAM1). This completes the proof. Theorem 5.7.8. Let X, T, A, δ, and {αn } be the same as those in Theorem 5.7.7. Define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = Txn − δαn A(Txn ),

n ≥ 0.

(5.62)

Then, {xn } converges strongly to the unique solution x∗ of (VIPAM1). Proof. Notice that (5.62) can be rewritten as follows: xn+1 = (1 − αn )Txn + αn g(xn ),

n ≥ 0,

(5.63)

where g(x) = (I − δA)Tx, ∀x ∈ X. It is clear that g : X → X is strictly contractive by Lemma 5.7.3. Now it follows from Corollary 4.4.1 that {xn } converges strongly to the unique solution x∗ ∈ Fix(T) that satisfies variational inequality: ⟨x ∗ − g(x∗ ), j(y − x ∗ )⟩ ≥ 0, ⇐⇒ ⟨Ax , j(y − x )⟩ ≥ 0, ∗

This completes the proof.

∗

∀y ∈ Fix(T)

∀y ∈ Fix(T).

304 | 5 Accretive operators and pseudocontractive mappings Theorem 5.7.9. Let X be a real q-uniformly smooth Banach space. Let T : X → X be a nonexpansive mapping such that K := Fix(T) ≠ 0. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let {xn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=0 αn = ∞; (C3) ∑∞ n=0 |αn+1 − αn | < ∞; or α (C4) limn→∞ α n = 1. n+1

1

2η

For δ ∈ (0, ( kq c ) q−1 ), define a sequence {xn } iteratively in X by x0 ∈ X, 1

xn+1 = αn f (xn ) + (1 − αn )Txn ,

n ≥ 0.

(5.64)

where f (x) = x − δAx, ∀x ∈ X. Then, the sequence {xn } defined by (5.64) converges strongly to the unique solution x∗ of (VIPAM1). Proof. By Lemma 5.7.4, we know that f : X → X is contractive. Now the conclusion of Theorem 5.7.9 follows from Corollary 4.4.1. This completes the proof. Theorem 5.7.10. Let X, T, A, δ, and {xn } be the same those in Theorem 5.7.9. Define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = Txn − δαn A(Txn ),

n ≥ 0,

(5.65)

Then, the sequence {xn } defined by (5.65) converges strongly to the unique solution x ∗ of (VIPAM1). Proof. Its proof is the same as that of Theorem 5.7.9. Using a similar method, we can prove the following results. Theorem 5.7.11. Let X = Lp , p ≥ 2. Let T : X → X be a nonexpansive mapping such that K := Fix(T) ≠ 0. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let {αn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=0 αn = ∞; ∞ (C3) ∑n=0 |αn+1 − αn | < ∞; or α (C4) limn→∞ α n = 1. n+1

2η

Choose δ ∈ (0, k2 (p−1) ) and define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = αf (xn ) + (1 − αn )Txn ,

n ≥ 0,

(5.66)

where f (x) = (I − δA)x, ∀x ∈ X. Then, the sequence {xn } defined by (5.66) converges strongly to the unique solution x∗ of (VIPAM1).

5.7 Iterative methods for variational inequalities with accretive operators | 305

Theorem 5.7.12. Let X, T, A, δ, and {αn } be the same as those in Theorem 5.7.11. Define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = Txn − δαn A(Txn ),

n ≥ 0.

(5.67)

Then the sequence {xn } defined by (5.67) converges strongly to the unique solution x ∗ of (VIPAM1). Remark 5.7.1. For λ ∈ (0, 1), write Tλ := λI + (1 − λ)T. If we replace T by Tλ in all iterative methods mentioned above, then the corresponding convergence theorems are still true under conditions (C1) and (C2). Next, we deal with the case of K := ⋂ri=1 Fix(T) ≠ 0, and when {Ti }ri=1 : X → X is a finite family of nonexpansive mappings. Theorem 5.7.13. Let X be a uniformly convex and uniformly smooth Banach space. Let {Ti }ri=1 : X → X be a finite family of nonexpansive mappings such that K := ⋂ri=1 Fix(Ti ) ≠ 0. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let μi ∈ (0, 1) be fixed numbers (i = 1, 2, . . . , r). Let W be the W-mapping generated by T1 , T2 , . . . , Tr and μ1 , μ2 , . . . , μr . Let {αn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=0 αn = ∞. Choose δ ∈ (0, k −1 b−1 (2k −1 η)) and define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = αn f (xn ) + (1 − αn )Wxn ,

n ≥ 0,

(5.68)

where f (x) = (I − δA)x, ∀x ∈ X. Then, the sequence {xn } defined by (5.68) converges strongly to the unique solution x∗ of (VIPAM1). Proof. It is clear that W : X → X is averaged nonexpansive and Fix(W) = ⋂ri=1 Fix(T) = K ≠ 0. By Lemma 5.7.3, we know that f : X → X is contractive. By Corollary 4.4.1, we see that {xn } converges strongly to the unique solution x∗ ∈ Fix(W) that satisfies the variational inequality: ⟨x ∗ − f (x ∗ ), J(y − x∗ )⟩ ≥ 0, ⇐⇒ ⟨Ax , J(y − x )⟩ ≥ 0, ∗

∗

∀y ∈ K

∀y ∈ K.

Hence x∗ is the unique solution of the (VIPAM1) with K := Fix(W) = ⋂ri=1 Fix(Ti ). This completes the proof. Theorem 5.7.14. Let X, {Ti }ri=1 , A, W, δ, and {αn } be the same as those in Theorem 5.7.13. Define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = Wxn − δαn A(Wxn ),

n ≥ 0.

(5.69)

306 | 5 Accretive operators and pseudocontractive mappings Then, the sequence {xn } defined by (5.69) converges strongly to the unique solution x∗ of (VIPAM1). Proof. Notice that (5.69) can be rewritten as follows: x0 ∈ X,

xn+1 = (1 − αn )Wxn + αn g(xn ),

n ≥ 0,

where g(x) = (I − δA)Wx, ∀x ∈ X. It is clear that g : X → X is contractive. Now the conclusion follows from Corollary 4.4.1. This completes the proof. Theorem 5.7.15. Let X be a real uniformly convex and q-uniformly smooth Banach space. Let {Ti }ri=1 : X → X be a finite family of nonexpansive mappings such that K := ⋂ri=1 Fix(Ti ) ≠ 0. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let μi ∈ (0, 1) be fixed numbers (i = 1, 2, . . . , r). Let W be the W-mapping generated by T1 , T2 , . . . , Tr and μ1 , μ2 , . . . , μr . Let {αn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=0 αn = ∞. 1

2η

Choose δ ∈ (0, ( kq c ) q−1 ), and define a sequence {xn } in X by 1

x0 ∈ X,

xn+1 = αn f (xn ) + (1 − αn )Wxn ,

n ≥ 0,

(5.70)

where f (x) = (I − δA)x, ∀x ∈ X. Then, the sequence {xn } defined by (5.70) converges strongly to the unique solution x∗ of (VIPAM1). Proof. By Lemma 5.7.4, we know that f : X → X is contractive. It is clear that W : X → X is averaged nonexpansive and Fix(W) = ⋂ri=1 Fix(Ti ) = K ≠ 0. Now the conclusion follows from Corollary 4.4.1. This competes the proof. In the same way as when proving Theorem 5.7.14, we can establish the following results. Theorem 5.7.16. Let X, {Ti }ri=1 , A, W, δ, and {αn } be the same as those in Theorem 5.7.15. Define a sequence {Xn } in X by x0 ∈ X,

xn+1 = Wxn − δαn A(Wxn ),

n ≥ 0.

(5.71)

Then, the sequence {xn } defined by (5.71) converges strongly to the unique solution x ∗ of (VIPAM1). Consequently, we have the following results. Theorem 5.7.17. Let X = Lp , p ≥ 2. Let {Ti }ri=1 , A, W, and {αn } as in Theorem 5.7.15. 2η Choose δ ∈ (0, k2 (p−1) ) and define a sequence {xn } in X by x0 ∈ X,

xn+1 = αn f (xn ) + (1 − αn )Wxn ,

n ≥ 0,

(5.72)

5.7 Iterative methods for variational inequalities with accretive operators | 307

where f (x) = (I − δA)x, ∀x ∈ X. Then, the sequence {xn } defined by (5.72) converges strongly to the unique solution x∗ of the (VIPAM1). Theorem 5.7.18. Let X = Lp , p ≥ 2. Let {Ti }ri=1 , A, W, and {αn } as those in Theorem 5.7.17. 2η Choose δ ∈ (0, k2 (p−1) ) and define a sequence {xn } in X by x0 ∈ X,

xn+1 = αn f (xn ) + (1 − αn )Wxn ,

n ≥ 0.

(5.73)

Then, the sequence {xn } defined by (5.73) converges strongly to the unique solution x∗ of (VIPAM1). Before presenting further convergence theorems, we point out that Theorem 4.5.5 holds true if E is either uniformly smooth or E is a real reflexive Banach space whose norm is uniformly Gâteaux differentiable and every nonempty, closed, convex, and bounded subset of E has the fixed point property for nonexpansive self-mappings. Furthermore, if F = ⋂ni=1 Fix(Ti ) = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) ≠ 0, then F = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ) = Fix(T1 Tr ⋅ ⋅ ⋅ T1 T2 ) = ⋅ ⋅ ⋅ = Fix(Tr−1 Tr−2 ⋅ ⋅ ⋅ T1 Tr ). Now we state these revisions of Theorem 4.5.5 as follows. Theorem 5.7.19. Let X be a real uniformly smooth Banach space and let C be a nonempty, closed, and convex subset of X. Let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings such that F = ⋂ri=1 Fix(Ti ) ≠ 0 and F = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ). Let {αn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=0 αn = ∞; (C3) ∑∞ n=0 |αn+r − αn | < ∞; or α (C4) limn→∞ α n = 1. n+r

For an initial value x0 ∈ C and a fixed anchor u ∈ C, define a sequence {xn } iteratively in C by x0 ∈ C, u ∈ C,

xn+1 = αn+1 u + (1 − αn+1 )Tn+1 xn ,

n ≥ 0,

(5.74)

where Tk = Tk mod r , for k ≥ 1, with the mod r function taking values in the set {1, 2, . . . , r}. Then the sequence {xn } defined by (5.74) converges strongly to the specific common fixed point x ∗ = QF u, where QF : C → F is the unique sunny nonexpansive retraction from C onto F. Theorem 5.7.20. Let X be a real reflexive Banach space whose norm is uniformly Gâteaux differentiable and C a nonempty, closed, and convex subset of X. Suppose that every nonempty, closed, convex, and bounded subset of C has the fixed point property for nonexpansive self-mappings. Let {Ti }ri=1 : C → C be a finite family of nonexpansive mappings such that F = ⋂ri=1 Fix(Ti ) ≠ 0 and F = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ). Let {αn } and {xn } be

308 | 5 Accretive operators and pseudocontractive mappings given as in Theorem 5.7.19. Then, the sequence {xn } defined by (5.74) converges strongly to the unique common fixed point p = QF u, where QF : C → F is the sunny nonexpansive retraction from C onto F. Theorem 5.7.21. Let X be a real reflexive Banach space whose norm is uniformly Gâteaux differentiable. Suppose that every nonempty, closed, convex, and bounded subset of X has the fixed point property for nonexpansive self-mappings. Let {Ti }ri=1 : X → X be a finite family of nonexpansive mappings such that K = ⋂ri=1 Fix(Ti ) ≠ 0 and K = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ). Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let τ ∈ (0, 1) be the constant appearing in Lemma 5.7.1. Let {αn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ αn = 0; (C2) ∑∞ n=0 αn = ∞; (C3) ∑∞ n=0 |αn+r − αn | < ∞; or α (C4) limn→∞ α n = 1. n+r

Define a sequence {xn } in X by x0 ∈ X,

xn+1 = αn+1 f (xn ) + (1 − αn+1 )Tn+1 xn ,

n ≥ 0,

(5.75)

where f = (I − τA), Tk = Tk mod r , for k ≥ 1, and the mod r function takes values in the set {1, 2, . . . , r}. Then, the sequence {xn } defined by (5.75) converges strongly to the unique solution x∗ of (VIPAM1). Proof. It follows from Lemmas 5.7.1 and 5.7.2 that there exists a unique x ∗ ∈ K such that ⟨Ax∗ , j(y − x∗ )⟩ ≥ 0,

∀y ∈ K.

(5.76)

Notice that (5.76) is equivalent to ⟨x ∗ − f (x∗ ), j(y − x∗ )⟩ ≥ 0,

∀y ∈ K.

(5.77)

Hence x∗ = QF f (x∗ ) by using Theorem 1.8.49. We now consider an auxiliary sequence {yn } defined by y0 ∈ X,

yn+1 = αn+1 f (x ∗ ) + (1 − αn+1 )Tn+1 yn ,

n ≥ 0.

(5.78)

By Theorem 5.7.20, we assert that yn converges strongly to the unique common fixed point p = QF f (x∗ ) = x∗ . By Lemma 5.7.1, f (x) = (I − τA)x, ∀x ∈ X, is contractive. Then there is a ρ ∈ (0, 1) such that 󵄩󵄩 󵄩 󵄩󵄩f (x) − f (y)󵄩󵄩󵄩 ≤ ρ‖x − y‖,

∀x, y ∈ X.

(5.79)

5.7 Iterative methods for variational inequalities with accretive operators | 309

Combining (5.75) and (5.78) with (5.79), we have 󵄩 󵄩 ‖xn+1 − yn+1 ‖ ≤ αρ󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩 + (1 − αn+1 )‖xn − yn ‖

≤ (1 − αn+1 )‖xn − yn ‖ + ραn+1 ‖xn − yn ‖ 󵄩 󵄩 + ραn+1 󵄩󵄩󵄩yn − x∗ 󵄩󵄩󵄩 = [1 − (1 − ρ)αn+1 ]‖xn − yn ‖ + o(αn+1 ).

It follows from Lemma 1.10.2 that xn − yn → θ as n → ∞. Hence xn → x∗ as n → ∞, which completes the proof. Theorem 5.7.22. Let X be a real uniformly smooth Banach space. Let {Ti }ri=1 : X → X be a finite family of nonexpansive mappings such that K = ⋂ri=1 Fix(Ti ) ≠ 0 and K = Fix(Tr Tr−1 ⋅ ⋅ ⋅ T1 ). Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let δ ∈ (0, k −1 b−1 (2ηk −1 )). Let {αn } and {xn } be the same as in Theorem 5.7.21. Then, {xn } converges strongly to the unique solution x∗ ∈ K of (VIPAM1). Its proof lines are almost the same as those in the proof of Theorem 5.7.22. The only difference is that f (x) = x −δAx, x ∈ X, is contractive by using Lemma 5.7.3. So, we omit the detailed proof. In a similar way, we can prove the following results. Theorem 5.7.23. Let q > 1. Let X be a real q-uniformly smooth Banach space. Let {Ti }ri=1 2η

1

and A be the same as in Theorem 5.7.22. Let δ ∈ (0, ( kq c ) q−1 ). Let {αn } and {xn } be given 1 as in Theorem 5.7.22. Then, {xn } converges strongly to the unique solution x∗ ∈ K of (VIPAM1). Theorem 5.7.24. Let X = Lp , p ≥ 2. Let {Ti } and A be the same as in Theorem 5.7.22. 2η Let δ ∈ (0, k2 (p−1) ). Let {αn } and {xn } be given as in Theorem 5.7.22. Then, {xn } converges strongly to the unique solution x∗ ∈ K of (VIPAM1). Theorem 5.7.25. Let X be a real uniformly convex and uniformly smooth Banach space. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let {Ti }∞ i=1 : X → X be a countable family of nonexpansive mappings such that K = ⋂∞ Fix(T i ) ≠ 0. i=1 Assume that {αi }∞ is a sequence in (0, 1) satisfying the condition 0 < α ≤ b < 1 (i = i i=1 1, 2, . . . ). Let Wn be the W-mapping generated by Tn , Tn−1 , . . . , T1 and αn , αn−1 , . . . , α1 . Let {λn } be a sequence in (0, 1) satisfying the following conditions: (C1 ) λn → 0 (n → ∞) and −1 −1 −1 (C2 ) ∑∞ n=0 λn = ∞. For δ ∈ (0, k b (2ηk )), define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = λn f (xn ) + (1 − λn )Wn xn ,

n ≥ 0,

(5.80)

where f (x) = (I − δA)x, ∀x ∈ X. Then, the sequence {xn } defined by (5.80) converges strongly to the unique solution x∗ of (VIPAM1). Proof. By Lemmas 5.7.1 and 5.7.1, we know that there exists a unique x ∗ ∈ K such that ⟨Ax∗ , j(y − x∗ )⟩ ≥ 0,

∀y ∈ K,

310 | 5 Accretive operators and pseudocontractive mappings which implies that x∗ = QK f (x∗ ). Consider an auxiliary sequence {yn } in X defined by y0 ∈ X,

yn+1 = λn f (x ∗ ) + (1 − λn )Wn yn ,

n ≥ 0.

(5.81)

By using Theorem 4.5.6, we assert that {yn } converges strongly to the unique common fixed point p = Qk f (x∗ ) = x∗ . By Lemma 5.7.3, we know that f (x) = (I − δA)x, x ∈ X, is contractive, i. e., there exists some ρ ∈ (0, 1) such that 󵄩󵄩 󵄩 󵄩󵄩f (x) − f (y)󵄩󵄩󵄩 ≤ ρ‖x − y‖,

∀x, y ∈ X.

(5.82)

Using (5.80), (5.81), and (5.82), we have 󵄩 󵄩 ‖xn+1 − yn+1 ‖ ≤ ρλn 󵄩󵄩󵄩xn − x∗ 󵄩󵄩󵄩 + (1 − λn )‖Wn xn − Wn yn ‖ 󵄩 󵄩 ≤ [1 − (1 − ρ)λ]‖xn − yn ‖ + ρλ󵄩󵄩󵄩yn − x ∗ 󵄩󵄩󵄩 = [1 − (1 − ρ)λ]‖xn − yn ‖ + o(λn ).

(5.83) (5.84) (5.85)

Using Lemma 1.10.2, we have xn − yn → θ as n → ∞, implying xn → x ∗ (n → ∞). This completes the proof. In a similar way, we can prove the following results. Theorem 5.7.26. Let X be a real uniformly convex and q-uniformly smooth Banach 2η

1

q−1 ), space. Let A, {Ti }∞ i=1 , Wn , and {λn } be the same as in Theorem 5.7.21. For δ ∈ (0, ( k q c1 ) define a sequence {xn } iteratively in X by

x0 ∈ X,

xn+1 = λn f (xn ) + (1 − λn )Wn xn ,

n ≥ 0,

(5.86)

where f (x) = (I − δA)x, ∀x ∈ X. Then, the sequence {xn } defined by (5.86) converges strongly to the unique solution x∗ ∈ K of (VIPAM1). Theorem 5.7.27. Let X = Lp , p ≥ 2. Let A, {Ti }∞ i=1 , Wn , and {λn } be the same as in Theo2η rem 5.7.21. For δ ∈ (0, k2 (p−1) ), define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = λn f (xn ) + (1 − λn )Wn xn ,

n ≥ 0,

(5.87)

where f (x) = (I − δA)x, ∀x ∈ X. Then, the sequence {xn } defined by (5.87) converges strongly to the unique solution x∗ ∈ K of (VIPAM1). Finally, we present several convergence theorems involving the hybrid steepest decent methods and W-mapping techniques. Theorem 5.7.28. Let X be a real uniformly convex and uniformly smooth Banach space. Let A : X → X be an η-strongly accretive and k-Lipschitz mapping. Let {Ti }∞ i=1 : X → X be a countable family of nonexpansive mappings such that K = ⋂∞ Fix(T i ) ≠ 0. i=1 Assume that {αi }∞ is a sequence in (0, 1) satisfying the condition 0 < α ≤ b < 1 (i = i i=1

5.7 Iterative methods for variational inequalities with accretive operators |

311

1, 2, . . . ). Let Wn be the W-mapping generated by Tn , Tn−1 , . . . , T1 and αn , αn−1 , . . . , α1 , and W be W-mapping generated by T1 , T2 , . . . and α1 , α2 , . . . Let {λn } be a sequence in (0, 1) satisfying the following conditions: (C1) limn→∞ λn = 0 and (C2) ∑∞ n=1 λn = ∞. For δ ∈ (0, k −1 b−1 (2ηk −1 )), define a sequence {xn } iteratively in X by x0 ∈ X,

xn+1 = Wn xn − δλn A(Wn xn ),

n ≥ 0.

(5.88)

Then, the sequence {xn } defined by (5.88) converges strongly to the unique solution x ∗ of (VIPAM1). Proof. We consider another sequence {yn } defined by y0 ∈ X,

yn+1 = Wyn − δλn A(Wyn ),

n ≥ 0.

(5.89)

Since W : X → X is averaged nonexpansive, we obtain from Theorem 5.7.8 that {yn } converges strongly to the unique solution x∗ of (VIPAM1). For λ ∈ (0, 1), define fλ : X → X by fλ (x) = (I − λδA)x,

∀x ∈ X.

By using Lemma 5.7.3, we have 󵄩󵄩 󵄩 󵄩󵄩fλ (x) − fλ (y)󵄩󵄩󵄩 ≤ √1 − λδ(2η − kb(δk))‖x − y‖ 1 ≤ [1 − λδ(2η − kb(δk))]‖x − y‖ 2 1 = (1 − λδτ)‖x − y‖, 2

(5.90)

where τ := 2η − kb(δk) > 0. From (5.88), (5.89), and (5.90), we have 󵄩 󵄩 ‖xn+1 − yn−1 ‖ = 󵄩󵄩󵄩(I − δλn A)Wn x− (I − δλn A)Wyn 󵄩󵄩󵄩

δ ≤ [1 − λn (2η − kb(δk))]‖Wn xn − Wyn ‖ 2 δτλn = (1 − )‖Wn xn − Wyn ‖ 2 δτλn ≤ (1 − )‖Wn xn − Wn yn ‖ + ‖Wn yn − Wyn ‖ 2 δτλn ≤ (1 − )‖xn − yn ‖ + ‖Wn yn − Wyn ‖. 2

(5.91)

Since {yn } converges, one finds that {yn } is bounded. Write D = supn≥1 {‖yn − w‖}, where w ∈ K is some fixed element. Since the limit Wx = limm→∞ Wm x is uniform with re-

312 | 5 Accretive operators and pseudocontractive mappings spect to x ∈ D, ∀m > n ≥ 1, we have ‖Wyn − Wn yn ‖ = lim ‖Wm yn − Wn yn ‖ m→∞

m−1

≤ lim ∑ ‖Wj+1 yn − Wj yn ‖ m→∞

∞

j=n

≤ 2D ∑ bj+1 = j=n

2Db+1 , 1−b

which shows that ∞

∑ ‖Wyn − Wn yn ‖ ≤

n=1

2D ∞ n+1 2Db2 < ∞. ∑b = 1 − b n=1 1−b

It follows from Lemma 1.10.2 that xn − yn → θ as n → ∞, which yields xn → x∗ as n → ∞. This completes the proof. In a similar way, we can establish the following results. Theorem 5.7.29. Let X be a real uniformly convex and q-uniformly smooth Banach space. Let A, {Ti }∞ i=1 , Wn , W, and {λn } be the same as in Theorem 5.7.28. For δ ∈ 1

2η

(0, ( kq c ) q−1 ), define a sequence {xn } iteratively in X by 1

x0 ∈ X,

xn+1 = Wn xn − δλn A(Wn xn ),

n ≥ 0.

(5.92)

Then, the sequence {xn } defined by (5.92) converges strongly to the unique solution x ∗ ∈ K of (VIPAM1). Theorem 5.7.30. Let X = Lp , p ≥ 2. Let A, {Ti }∞ i=1 , Wn , W, and {λn } be the same as in 2η Theorem 5.7.28. For δ ∈ (0, k2 (p−1) ), define a sequence {xn } iteratively in X by x0 ∈ X

xn+1 = Wn xn − δλn A(Wn xn ),

n ≥ 0.

(5.93)

Then, the sequence {xn } defined by (5.93) converges strongly to the unique solution x ∗ ∈ K of (VIPAM1).

5.8 Fixed points of strongly pseudocontractive mappings Based on [47, Theorem 6], we establish the following fixed point theorem for continuous strongly pseudocontractive mappings. Theorem 5.8.1. Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous k-strongly pseudocontractive mapping satisfying the weak inward condition (WIC). Then T has a unique fixed point in C.

5.8 Fixed points of strongly pseudocontractive mappings | 313

Proof. Letting A = T −I, where I is the identity, we see that there exists j(x −y) ∈ J(x −y) such that ⟨Ax − Ay, j(x − y)⟩ = ⟨Tx − Ty, j(x − y)⟩ − ‖x − y‖2 ≤ (k − 1)‖x − y‖2 . Since T satisfies the weak inward condition, we find from the Caristi’s theorem that lim h−1 d(x + hAx, C) = lim+ h−1 d(x + h(Tx − x), C)

h→0+

h→0

= lim+ h−1 d((1 − h)x + hT), C) = 0. h→0

This shows that A satisfies condition (ii) of Theorem 6 in [47]. From condition (iii) of Theorem 6 in [47], we find that C ⊂ (I − ϵA)(C), ∀ϵ > 0, that is, C ⊂ ((1 + ϵI) − ϵT)(C),

∀ϵ > 0.

Letting ϵ = n → ∞, one sees that C ⊂ ((1 + nI) − nT)(C), ∀n ≥ 1. Fixing y ∈ C, we have y = (1 + n)xn − nTxn . It follows that 1 n y+ Tx = xn . 1+n 1+n n

(5.94)

Hence, we have 1 1 n 1 y − y, j(xn − x1 )⟩ + ⟨ Tx − Tx , j(xn − x1 )⟩ n+1 2 n+1 n 2 1 1 n 1 )‖y‖‖xn − x1 ‖ + ⟨Txn − Tx1 , j(xn − x1 )⟩ ≤( − 2 n+1 n+1 1 n − )Tx1 , j(xn − x1 )⟩. + ⟨( n+1 2

‖xn − x1 ‖2 = ⟨

This implies that 1 1 n 1 kn ‖xn − x1 ‖ ≤ (( − )‖y‖ + ( − )‖Tx1 ‖)(1 − ) . 2 n+1 n+1 2 n+1 −1

Letting n → ∞, we find that 1 lim sup ‖xn − x1 ‖ ≤ (‖y‖ + ‖Tx1 ‖)(1 − k)−1 < ∞. 2 n→∞ This shows that {xn } is bounded. From (5.94), we see that {Txn } is also bounded. Hence, xn − Txn =

y − Txn →θ n+1

314 | 5 Accretive operators and pseudocontractive mappings as n → ∞. It follows that xn − Txn − xm + Txm → θ as n, m → ∞. From the fact that ⟨xn − Txn − xm + Txm , j(xn − xm )⟩

= ‖xn − xm ‖2 − ⟨Txn − Txm , j(xn − xm )⟩

≥ (1 − k)‖xn − xm ‖2 ,

we have ‖xn − xm ‖ ≤ ‖xn − Txn − xm + Txm ‖. Letting n, m → ∞, we obtain that {xn } is a Cauchy sequence. Let {xn } converge to x in norm. Since C is closed, we see that x ∈ C. Since T is continuous, we find that limn→∞ Txn = Tx. This implies Tx = x. Next, we prove that x is unique. Assume that y ≠ x is also a fixed pint of T. Then ‖x − y‖2 = ⟨x − y, j(x − y)⟩ = ⟨Tx − Ty, j(x − y)⟩ ≤ k‖x − y‖2 , that is, (1 − k)‖x − y‖2 ≤ 0. Since 0 < k < 1, we obtain a contradiction. This proves x = y. This shows that T has a unique fixed point in C, completing the proof. Based on the results above, we establish the following fixed point theorem for continuous g-strongly pseudocontractive mappings. Theorem 5.8.2. Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous g-strongly pseudocontractive mapping satisfying the weak inward condition (WIC). If lim infr→∞ g(r) > 0, then T has a unique fixed point in C. Proof. Fixing xn ∈ C, ∀n ≥ 1, we define Tn : C → E by 1 1 Tn x = xn + + Tx, 2 2

∀x ∈ C.

Then Tn : C → E is a continuous 21 -strongly pseudocontractive mapping satisfying the weak inward condition. Using Theorem 5.8.1, for every n ≥ 1, we find that Tn has a unique fixed point in C, denoted this unique fixed point by xn+1 , that is, 1 1 xn+1 = xn + Txn+1 , 2 2

∀n ≥ 1.

This implies that xn+1 − xn = Txn+1 − xn+1 . It follows that g(‖xn+1 − xn ‖)‖xn+1 − xn ‖ ≤ ⟨xn+1 − Txn+1 − xn + Txn , j(xn+1 − xn )⟩

= ⟨xn − xn−1 , j(xn+1 − xn )⟩ − ‖xn+1 − xn ‖2 1 1 ≤ ‖xn − xn−1 ‖2 + ‖xn+1 − xn ‖2 − ‖xn+1 − xn ‖2 2 2 1 1 = ‖xn − xn−1 ‖2 − ‖xn+1 − xn ‖2 . 2 2

(5.95)

This implies that ‖xn+1 − xn ‖ ≤ ‖xn − xn−1 ‖. Hence limn→∞ ‖xn+1 − xn ‖ exists. Using (5.95), we arrive at lim g(‖xn+1 − xn ‖)‖xn+1 − xn ‖ = 0.

n→∞

5.9 Demiclosedness principles for pseudocontractive mappings | 315

This implies that xn+1 − xn → θ as n → ∞. If not, we assume that ‖xn+1 − xn ‖ → d > 0. From the property of g, we obtain that lim inf g(‖xn+1 − xn ‖) > 0, n→∞

which is a contradiction. This implies that Txn − xn → θ as n → ∞. It follows that xn − Txn − xm − Txm → θ as n, m → ∞. Note that g(‖xn − xm ‖)‖xn − xm ‖ ≤ ⟨xn − Txn − xm + Txm , j(xn − xm )⟩ ≤ ‖xn − Txn − xm + Txm ‖‖xn − xm ‖.

This implies g(‖xn − xm ‖) → 0

as n, m → ∞.

Since lim infr→r0 g(r) > 0, ∀r0 > 0 and lim infr→∞ g(r) > 0, we have ‖xn − xm ‖ → 0 as n, m → ∞. This shows that {xn } is a Cauchy sequence. Assume that {xn } converges to x in norm; then x ∈ C, since C is closed. Because T is continuous, we find that limn→∞ Txn = Tx, which implies Tx = x. Next, we prove that x is unique. Assume that y ≠ x is also a fixed pint of T. Then ‖x − y‖2 = ⟨x − y, j(x − y)⟩ = ⟨Tx − Ty, j(x − y)⟩ ≤ ‖x − y‖2 − g(‖x − y‖)‖x − y‖, that is, g(‖x − y‖)‖x − y‖ = 0. From the property of g, we find that x = y. This shows that T has a unique fixed point in C, completing the proof.

5.9 Demiclosedness principles for pseudocontractive mappings Theorem 5.9.1 (Demiclosedness principle I). Let E be a real uniformly convex Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping satisfying the weak inward condition (WIC). Then I − T is demiclosed at the origin and Fix(T) is closed and convex. Proof. Let {xn } be a sequence in C such that xn ⇀ x and xn − Txn → θ. Using Theorem 6 of Martin [47], we find that C ⊂ (2I − T)(C).

(5.96)

Since T : C → E is a continuous pseudocontractive mapping, we have ⟨(2I − T)x − (2I − T)y, j(x − y)⟩ = 2‖x − y‖2 − ⟨Tx − Ty, j(x − y)⟩ ≥ ‖x − y‖2 ,

∀x, y ∈ C,

316 | 5 Accretive operators and pseudocontractive mappings which yields 󵄩 󵄩󵄩 󵄩󵄩(2I − T)x − (2I − T)y󵄩󵄩󵄩 ≥ ‖x − y‖,

∀x, y ∈ C,

implying that the mapping (2I − T) : C → E is injective, and consequently (2I − T)−1 exists on Ran(2I − T), in particular, (2I − T)−1 exists on C, since C ⊂ (2I − T). Putting U := (2I − T)−1 , we see that U : C → C is nonexpansive such that Fix(U) = Fix(T). Observe that 󵄩 󵄩 󵄩 󵄩 ‖xn − Uxn ‖ = 󵄩󵄩󵄩UU −1 xn − Uxn 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩U −1 xn − xn 󵄩󵄩󵄩 = ‖Txn − xn ‖ → 0,

n → ∞.

From the demiclosedness principle for nonexpansive mappings, we find that x = Ux, and then, x = Tx. In view of the facts that Fix(U) = Fix(T) and Fix(U) is closed and convex, we find that Fix(T) is a closed and convex subset of E. This completes the proof. Theorem 5.9.2 (Demiclosedness principle II). Let E be a reflexive and strictly convex Banach space that satisfies Opial condition, and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping satisfying the weak inward condition (WIC). Then I − T is demiclosed at the origin and Fix(T) is closed and convex. Proof. Letting {xn } be a sequence in C such that xn ⇀ x and xn − Txn → θ, we have x ∈ C. Next, we prove x = Tx. Let U be defined as in Theorem 5.9.1. Then xn − Uxn → θ as n → ∞. Assume that x ≠ Ux. Using Opial condition, we arrive at lim sup ‖xn − x‖ < lim sup ‖xn − Ux‖ n→∞

n→∞

= lim sup ‖Uxn − Ux‖ n→∞

≤ lim sup ‖xn − x‖. n→∞

This shows, in fact, that x = Ux, yielding x = Tx. In view of the facts that Fix(U) = Fix(T) and Fix(U) is closed and convex, we find that Fix(T) is a closed and convex subset of E. This completes the proof.

5.10 Fixed point theorems for pseudocontractive mappings Theorem 5.10.1. Let E be a real uniformly convex Banach space, and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping satisfying the weak inward condition (WIC). If T − I is unbounded on any unbounded subset of C, then T has a fixed point in C.

5.10 Fixed point theorems for pseudocontractive mappings | 317

Proof. Fixing u ∈ C, we define Tn : C → E by Tn x =

1 n u+ Tx, n+1 n+1

∀x ∈ C.

It follows that ⟨Tn x − Tn y, j(x − y) ≤

n ‖x − y‖2 , n+1

∀x, y ∈ C.

n -strongly pseudocontractive. Note that Tn is also This shows that Tn : C → E is n+1 continuous and satisfies the weak inward condition. Using Theorem 5.8.1, we find that Tn has a unique fixed point in C. We denote the fixed point by xn , that is,

xn =

1 n u+ Tx , n+1 n+1 n

∀n ≥ 1.

(5.97)

Next, we show that {xn } is bounded. If {xn } is unbounded, we may assume that n θ ∈ C. Letting u = 0 in (5.97), we find that xn = n+1 Txn , ∀n ≥ 1. This implies that n ⟨Txn , j(xn )⟩ n+1 n n ⟨Txn − Tθ, j(xn )⟩ + ⟨Tθ, j(xn )⟩ = n+1 n+1 n n ≤ ‖x ‖2 + ‖Tθ‖‖xn ‖, n+1 n n+1

‖xn ‖2 =

that is,

‖xn ‖ n

≤ ‖Tθ‖. Hence, we arrive at ‖xn − Txn ‖ =

‖Txn ‖ ‖xn ‖ = , n+1 n

∀n ≥ 1.

This is a contradiction, showing that {xn } is bounded, so is {Txn }. Hence, xn − Txn = 1 − n+1 Txn → θ as n → ∞. Without loss of generality, we may assume xn ⇀ x as n → ∞. Using Demiclosedness principle I, we obtain that x = Tx, completing the proof. From the proof of the above theorem, we see that the following conclusion holds. Theorem 5.10.2. Let E be a real uniformly convex Banach space, and let C be a nonempty, bounded, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping satisfying the weak inward condition (WIC). Then T has at least one fixed point in C. By means of Demiclosedness principle II for pseudocontractive mappings, we easily deduce the following existence result. Theorem 5.10.3. Let E be a real reflexive Banach space that satisfies Opial condition, and let C be a nonempty, bounded, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping satisfying the weak inward condition (WIC). Then T has at least one fixed point in C.

318 | 5 Accretive operators and pseudocontractive mappings Theorem 5.10.4. Let E be a Banach space. Let C be a nonempty, closed, and convex subset of E that has the fixed point property for nonexpansive self-mapping. Let T : C → E be a continuous pseudocontractive mapping satisfying the weak inward condition (WIC). Then T has a fixed point in C. Proof. Letting U = (2I − T)−1 be defined in Theorem 5.9.1, we see that U : C → C is nonexpansive. By our assumption on C, we see that there exists x ∈ C such that x = Ux. Hence, x = Tx. This completes the proof.

5.11 Iterative methods for fixed points of pseudocontractive mappings Theorem 5.11.1. Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → C be a L-Lipschitz continuous k-strongly pseudocontractive mapping. Let {xn } be a sequence generated in the following normal Mann iterative method x1 ∈ C,

xn+1 = (1 − αn )xn + αn Txn ,

n ≥ 1,

where {αn } is a sequence in (0, 1) such that (C1) limn→∞ αn = 0 and (C2) ∑∞ n=1 αn = ∞. Then {xn } converges to the unique fixed point of T in norm. Proof. From Theorem 5.9.1, we see that T has a unique fixed point in C. Denote the unique fixed point by p. It follows that ‖xn+1 − p‖2 = (1 − αn )⟨xn − p, j(xn+1 − p)⟩ + αn ⟨Txn − p, j(xn+1 − p)⟩

= (1 − αn )‖xn − p‖‖xn+1 − p‖ + αn ‖Txn − Txn+1 ‖‖xn+1 − p‖

+ αn ⟨Txn+1 − p, j(xn+1 − p)⟩ 1 − αn 1 − αn ≤ ‖xn − p‖2 + ‖xn+1 − p‖2 + αn L‖xn − xn+1 ‖‖xn+1 − p‖ 2 2 + αn k‖xn+1 − p‖2 ≤

1 − αn 1 − αn ‖xn − p‖2 + ( + αn k)‖xn+1 − p‖2 2 2

+ αn2 L‖xn − Txn ‖‖xn+1 − p‖ ≤

1 − αn 1 − αn ‖xn − p‖2 + ( + αn k)‖xn+1 − p‖2 2 2

+ αn2 L(1 + L)‖xn − p‖‖xn+1 − p‖ ≤

1 − αn + αn2 L(1 + L) ‖xn − p‖2 2 1 − αn + αn2 L(1 + L) +( + αn k)‖xn+1 − p‖2 . 2

5.11 Iterative methods for fixed points of pseudocontractive mappings | 319

This in turn implies that ‖xn+1 − p‖2 ≤

1 − αn + αn2 L(1 + L) ‖x − p‖2 1 + αn (1 − 2k) − αn2 L(1 + L) n

≤ (1 − 2αn

1 − k − αn L(1 + L) )‖xn − p‖2 . 1 + αn (1 − 2k) − αn2 L(1 + L)

(5.98)

From (C1), we see that there exists n0 ≥ 1 such that 1 − k − αn L(1 + L) 1−k ≥ , 2 2 1 + αn (1 − 2k) − αn L(1 + L)

∀n ≥ n0 .

(5.99)

Combining (5.98) with (5.99), we obtain ‖xn+1 − p‖2 ≤ (1 − (1 − k)αn )‖xn − p‖2 ,

∀n ≥ n0 .

Using Lemma 1.10.2, we obtain xn → p as n → ∞. This completes the proof. Theorem 5.11.2. Let E be a real Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping and let h : C → E be a g-strongly pseudocontractive mapping satisfying the weak inward condition (WIC). Assume that lim infr→∞ g(r) > 0. Define a mapping Tth : C → E by Tth = th(x) + (1 − t)Tx,

∀x ∈ C, t ∈ (0, 1].

Then (i) Tth has a unique fixed point yt ∈ C and the path t 󳨃→ yt is continuous; (ii) if both {yt } and {Tyt } are bounded, then yt − Tyt → θ as t → 0+ ; (iii) if Fix(T) ≠ 0 and lim infr→∞ g(r) > ‖p − h(p)‖ for some p ∈ Fix(T), then {yt } is bounded and ⟨yt − h(yt ), j(yt − y)⟩ ≤ 0,

∀y ∈ Fix(T).

Proof. Using [52, Corollary 2], we find that Tth : C → E is continuous pseudocontractive which satisfies the weak inward condition. Indeed, ⟨Tth x − Tth y, j(x − y)⟩ = t⟨h(x) − h(y), j(x − y)⟩ + (1 − t)⟨Tx − Ty, j(x − y)⟩ ≤ t‖x − y‖2 − tg(‖x − y‖)‖x − y‖ + (1 − t)‖x − y‖2 = ‖x − y‖2 − tg(‖x − y‖)‖x − y‖. This shows that Tth : C → E is a continuous tg-strongly pseudocontractive mapping which also satisfies the weak inward condition. Using Theorem 5.8.2, we find that Tth : C → E has a unique fixed point in C. Denote the fixed point by yt . This means that there exists a unique path t 󳨃→ yt satisfying yt = th(yt ) + (1 − t)Tyt ,

∀t ∈ (0, 1].

(5.100)

320 | 5 Accretive operators and pseudocontractive mappings Fix t0 ∈ (0, 1]. For ∀t ∈ (0, 1], we find that there exists j(yt − yt0 ) ∈ J(yt − yt0 ) such that ‖yt − yt0 ‖2 = t⟨h(yt ) − h(yt0 ), j(yt − yt0 )⟩ + (1 − t)⟨Tyt − Tyt0 , j(yt − yt0 )⟩ + (t − t0 )⟨h(y0 ) − Tyt0 , j(yt − yt0 )⟩

󵄩 󵄩 ≤ ‖yt − yt0 ‖2 − tg(‖yt − yt0 ‖)‖yt − yt0 ‖ + |t − t0 |󵄩󵄩󵄩h(y0 ) − Tyt0 󵄩󵄩󵄩‖yt − yt0 ‖. Hence, g(‖yt − yt0 ‖)‖yt − yt0 ‖ ≤

|t − t0 | 󵄩󵄩 󵄩 󵄩h(y ) − Tyt0 󵄩󵄩󵄩‖yt − yt0 ‖. t 󵄩 0

This implies that yt → yt0 as t → t0 . (i) Using (5.100), we have yt − Tyt = t(h(yt ) − Tyt ). Since both {yt } and {h(yt )} are bounded, we find that {Tyt } is also bounded. It follows that yt − Tyt → θ as t → 0+ . (ii) For some p ∈ Fix(T), we find from (5.100) that ‖yt − p‖2 = ⟨yt − p, j(yt − p)⟩

= t⟨h(yt ) − h(p), j(yt − p)⟩ + t⟨h(p) − p, j(yt − p)⟩ + (1 − t)⟨Tyt − p, j(yt − p)⟩

≤ t‖yt − p‖2 − tg(‖yt − p‖)‖yt − p‖ + t⟨h(p) − p, j(yt − p)⟩ + (1 − t)‖yt − p‖2

= ‖yt − p‖2 − tg(‖yt − p‖)‖yt − p‖ + t⟨h(p) − p, j(yt − p)⟩, which yields g(‖yt − p‖)‖yt − p‖ ≤ ⟨h(p) − p, j(yt − p)⟩,

∀t ∈ (0, 1].

(5.101)

This shows that {yt } is bounded. Using (5.100), we find that ⟨yt − h(yt ), j(yt − y)⟩

= (1 − t)⟨Tyt − h(yt ), j(yt − y)⟩

= (1 − t)⟨Tyt − yt , j(yt − y)⟩ + (1 − t)⟨yt − h(yt ), j(yt − y)⟩

= (1 − t)⟨Tyt − y, j(yt − y)⟩ + (1 − t)⟨y − yt , j(yt − y)⟩ + (1 − t)⟨yt − h(yt ), j(yt − y)⟩

≤ (1 − t)⟨yt − h(yt ), j(yt − y)⟩,

∀y ∈ Fix(T).

This implies that ⟨yt − h(yt ), j(yt − y)⟩ ≤ 0, ∀y ∈ Fix(T), completing the proof. Theorem 5.11.3. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Suppose that every nonempty, bounded, closed, and convex subset of C has the fixed point property for nonexpansive self-mappings. Let T : C → E be a continuous pseudocontractive mapping

5.11 Iterative methods for fixed points of pseudocontractive mappings | 321

and let h : C → E be a continuous g-strongly pseudocontractive mapping satisfying the weak inward condition (WIC). Assume that lim infr→∞ g(r) > 0. Let {yt } be a sequence defined in Theorem 5.11.2. If both {yt } and {h(yt )} are bounded, then {yt } converges to some fixed point p of T as t → 0+ and p is the unique solution of the following variational inequality: ⟨p − h(p), j(y − p)⟩ ≥ 0,

∀y ∈ Fix(T).

Proof. From (i) and (ii) of Theorem 5.11.2, there exists a unique path {yt }, which satisfies (5.100) and yt −Tyt → θ as t → 0+ . From [47, Theorem 6], we see that C ⊂ (2I −T)(C). Letting U = (2I − T)−1 , we have that U : C → C is nonexpansive, Fix(U) = F(T), and yt − Uyt → θ as t → 0+ . Define a functional φ by φ(x) = μt ‖yt − x‖2 , ∀x ∈ E, where μt is a Banach limit. It is clear that φ is a proper, continuous, and convex function such that φ(x) → ∞ as ‖x‖ → ∞. Therefore, there exists at least a point x ∈ C such that φ(x) = miny∈C φ(y). Let C1 := {x ∈ C : φ(x) = inf φ(y)}. y∈C

Then C1 is a nonempty, bounded, closed, and convex subset of C. Observe that φ(Ux) = μt ‖yt − Ux‖2

= μt ‖Uyt − Ux‖2

≤ μt ‖yt − x‖2 = φ(x),

∀x ∈ C1 .

This shows that Ux ∈ C1 , that is, U(C1 ) ⊂ C1 . By our assumption on C, we see that U has a fixed point in C1 . Let p be a fixed point of U. Hence, p = Tp. From (iii) of Theorem 5.11.2, we see that ⟨yt − h(yt ), j(yt − p)⟩ ≤ 0. It follows that μt ⟨yt − h(yt ), j(yt − p)⟩ ≤ 0.

(5.102)

Using Theorem 5.11.2, we find that μt ⟨y − p, j(yt − p)⟩ ≤ 0,

∀y ∈ C.

(5.103)

Since h : C → E satisfies the weak inward condition, we obtain that h(p) ∈ IC (p). Hence, there exists zn ∈ IC (p) such that zn → h(p) as n → ∞. There exists un ∈ C such that zn = p + λn (un − p),

λn ≥ 1, n ≥ 1.

(5.104)

322 | 5 Accretive operators and pseudocontractive mappings Letting y = un in (5.103), we find that μt ⟨un − p, j(yt − p)⟩ ≤ 0. It follows that μt ⟨zn − p, j(yt − p)⟩ ≤ 0.

(5.105)

Since zn → h(p) as n → ∞, taking the limit in (5.105), we find that μt ⟨h(p) − p, j(yt − p)⟩ ≤ 0.

(5.106)

Adding (5.102) and (5.106), we have μt ‖yt − p‖2 ≤ μt ⟨h(yt ) − h(p), j(yt − p)⟩

≤ μt ‖yt − p‖2 − μt (g(‖yt − p‖)‖yt − p‖).

It follows that μt (g(‖yt − p‖)‖yt − p‖) = 0. This implies that there exists a subnet {ytα } of {yt } such that ytα → p. Using (iii) of Theorem 5.11.2, we find that ⟨ytα − h(ytα ), j(ytα − y)⟩ ≤ 0,

∀y ∈ Fix(T).

It follows that ⟨p − h(p), j(p − y)⟩ ≤ 0,

∀y ∈ Fix(T).

(5.107)

Next, we prove that variational inequality (5.107) has a unique solution. We assume that q is another solution of variational inequality (5.107), that is, ⟨q − h(q), j(q − y)⟩ ≤ 0,

∀y ∈ Fix(T).

Taking y = q in (5.107) and y = p in (5.108), we find that ⟨p − h(p), j(p − q)⟩ ≤ 0 and ⟨q − h(q), j(q − p)⟩ ≤ 0. Adding the two inequalities, we obtain ‖p − q‖2 ≤ ⟨h(p) − h(q), j(p − q)⟩ ≤ ‖p − q‖2 − g(‖p − q‖)‖p − q‖. This implies p = q, completing the proof.

(5.108)

5.11 Iterative methods for fixed points of pseudocontractive mappings | 323

If the space is strictly convex in Theorem 5.11.3, we may remove the restriction that subset C has the fixed point property. If Fix(T) ≠ 0, lim infr→∞ g(r) = ∞, and h : C → E is bounded continuous g-strongly pseudocontractive, we may remove the restrictions that both {yt } and {h(yt )} are bounded. Theorem 5.11.4. Let E be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping and let h : C → E be a bounded, continuous, g-strongly pseudocontractive mapping satisfying the weak inward condition (WIC). Assume that Fix(T) ≠ 0 and lim infr→∞ g(r) = ∞. Let {yt } be the path defined in Theorem 5.11.2. Then {yt } converges to some fixed point p of T as t → 0+ and p is the unique solution of the following variational inequality: ⟨p − h(p), j(y − p)⟩ ≤ 0,

∀y ∈ Fix(T).

Proof. From (i) of Theorem 5.11.2, there exists a unique continuous path {yt }, which satisfies (5.100). From (iii) of Theorem 5.11.2, we see that {yt } is bounded. Hence, yt − Tyt → θ as t → 0+ . This implies yt − Uyt → θ as t → 0+ , where U is defined in Theorem 5.11.3. Define a functional φ by φ(x) = μt ‖yt − x‖2 , ∀x ∈ E, where μt is a Banach limit. Let C1 := {x ∈ C : φ(x) = inf φ(y)}. y∈C

It is easy to see that C1 ≠ 0 is a bounded, closed, and convex subset of C, and U(C1 ) ⊂ C1 . Fix v ∈ Fix(T) and define ψ : C1 → ℝ+ by ψ(u) = ‖u − v‖. Define C2 := {u ∈ C1 : ψ(u) = inf ψ(y)}. y∈C1

Then C2 = {u0 }, u0 ∈ C1 . In view of v = Uv, we have ‖Uu0 − v‖ = ‖Uu0 − Uv‖ ≤ ‖u0 − v‖. This implies u0 = Uu0 and then u0 = Tu0 . Following the proof of Theorem 5.11.3, we obtain the desired conclusion immediately. This completes the proof. Furthermore, if the duality mapping is weakly sequentially continuous, the restrictions that E is strictly convex and has a uniformly Gâteaux differentiable norm are not needed. Theorem 5.11.5. Let E be a reflexive Banach space and let C be a nonempty, closed, and convex subset of E. Let T : C → E be a continuous pseudocontractive mapping and let h : C → E be a continuous g-strongly pseudocontractive mapping satisfying the weak inward condition (WIC). Assume that lim infr→∞ g(r) = ∞. Let {yt } be the path defined in Theorem 5.11.2. If the duality mapping J is weakly sequentially continuous, {yt } and

324 | 5 Accretive operators and pseudocontractive mappings {h(yt )} are both bounded, then {yt } converges to some fixed point p of T as t → 0+ and p is the unique solution of the following variational inequality: ⟨p − h(p), j(y − p)⟩ ≥ 0,

∀y ∈ Fix(T).

Proof. Since the duality mapping J : E → E ∗ is weakly sequentially continuous, we see that J is single valued. Next, we use j to denote the single-valued duality mapping. From (i) of Theorem 5.11.2, we see that there exists a unique continuous path {yt }, which satisfies (5.100). Note that both {yt } and {h(yt )} are bounded. From (ii) of Theorem 5.11.2, we see that yt − Tyt → θ as t → 0+ . Choose tn → 0+ and denote yn by yn = ytn . It follows that yn −Tyn → θ as n → ∞. Since E is reflexive and {yn } is bounded, we see that there exists a subsequence of {yn } converging to some point p in C. Without loss of generality, we may assume that yn ⇀ p as n → ∞. Using Demiclosedness principle II, we find that p = Tp, that is, Fix(T) ≠ 0. Using (5.102), we find that g(‖yn − p‖)‖yn − p‖ ≤ ⟨h(p) − p, j(yn − p)⟩,

∀n ≥ 1.

Letting n → ∞, we find that yn → p as n → ∞. From Theorem 5.11.2, we see that ⟨yn − h(yn ), j(yn − y)⟩ ≤ 0,

∀y ∈ Fix(T).

Letting n → ∞, we find that ⟨p − h(p), j(y − p)⟩ ≥ 0,

∀y ∈ Fix(T).

Since the above variational inequality has a unique solution, we find that {yt } converges to some fixed point p of T in norm and p is the unique solution of the variational inequality. This completes the proof. Based on the convergence of the path given in Theorem 5.11.2, we are now in a position to establish a convergence theorem of fixed points via a Bruck-like regularization iterative method. To this end, we consider the following control sequences. Let {θn } and {λn } be two sequences in [0, 1] satisfying the following conditions: (i) λn (1 + θn ) ≤ 1, ∀n ≥ 1, (ii) θn → 0 as n → ∞, (iii) ∑∞ n=1 λn θn = ∞, θ (iv) θn−1 − 1 = o(λn θn ), (v)

n

λn θn

≤

1−k , 8L(1+L)(1+2L)

∀n ≥ 1, where k ∈ (0, 1) and L ≥ 1 are two known constants.

It is easy to see that λn =

1 , (n + 1)a

satisfy the above conditions.

θn =

1 , (n + 1)b

0 < b < a, a + b < 1

5.11 Iterative methods for fixed points of pseudocontractive mappings | 325

Assume that all the conditions in Theorem 5.11.3 are satisfied. Let tn = denote yn by yn = ytn . Then yn = tn h(yn ) + (1 − tn )Tyn ,

(1 + θn )yn = θn h(yn ) + Tyn ,

θn 1+θn

and

n ≥ 1,

(5.109)

n ≥ 1,

(5.110)

λn (1 + θn )yn = λn θn h(yn ) + λn Tyn ,

n ≥ 1,

(5.111)

and yn = (1 − λn (1 + θn ))yn + λn θn h(yn ) + λn Tyn ,

n ≥ 1.

(5.112)

These motivate us to introduce the following iterative method: x1 ∈ C,

xn+1 = (1 − λn (1 + θn ))xn + λn θn h(xn ) + λn Txn ,

n ≥ 1.

(VBRIM)

We call (VBRIM) a viscosity Bruck regularization iterative method. Theorem 5.11.6. Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed, and convex subset of E with the fixed point property for nonexpansive mappings. Let T : C → E be an L-Lipschitz continuous pseudocontractive mapping with a nonempty fixed point set, and let h : C → E be an L-Lipschitz continuous k-strongly pseudocontractive mapping. Let {xn } be generated by (VBRIM). Then {xn } converges to some fixed point p of T in norm and p is the unique solution of the following variational inequality: ⟨p − h(p), j(y − p)⟩ ≥ 0,

∀y ∈ Fix(T).

(5.113)

Proof. Let {yn } be defined in (5.109). From Theorem 5.11.3, we see that {yn } converges to some fixed point p of T in norm and p is the unique solution to (5.113). Now it is sufficient to show that xn − yn → θ as n → ∞. Using (5.110), we arrive at ‖yn − yn−1 ‖2 = ⟨Tyn − Tyn−1 + θn (h(yn ) − yn ) − θn (h(yn−1 ) − yn−1 ), j(yn − yn−1 )⟩ + (θn − θn−1 )⟨h(yn−1 ) − yn−1 , j(yn − yn−1 )⟩

≤ ‖yn − yn−1 ‖2 + kθn ‖yn − yn−1 ‖2

+ (θn − θn−1 )⟨h(yn−1 ) − yn−1 , j(yn − yn−1 )⟩ − θn ‖yn − yn−1 ‖2 .

This implies that ‖yn − yn−1 ‖2 ≤

θn − θn−1 ⟨h(yn−1 ) − yn−1 , j(yn − yn−1 )⟩. (1 − k)θn

Hence, we have ‖yn − yn−1 ‖ ≤

θ 󵄨󵄨󵄨 M 󵄨󵄨󵄨󵄨 󵄨󵄨1 − n−1 󵄨󵄨󵄨, (1 − k) 󵄨󵄨 θn 󵄨󵄨

(5.114)

326 | 5 Accretive operators and pseudocontractive mappings where M = sup{‖h(yn−1 ) − yn−1 ‖ : n ≥ 1}. Using (VBRIM) and (5.112), we arrive at 󵄩 󵄩 ‖xn − xn+1 ‖2 ≤ λn 󵄩󵄩󵄩Txn − (1 + θn )xn + θn h(xn )󵄩󵄩󵄩 󵄩 󵄩 = λn 󵄩󵄩󵄩Txn − Tyn + (1 + θn )(yn − xn ) + θn (h(xn ) − h(yn ))󵄩󵄩󵄩 ≤ λn (L‖xn − yn ‖ + (1 + θn )‖yn − xn ‖ + Lθn ‖xn − yn ‖) ≤ 2(1 + L)λn ‖xn − yn ‖.

(5.115)

Again, using (VBRIM) and (5.112), we arrive at 󵄩 󵄩 ‖xn+1 − yn ‖ = 󵄩󵄩󵄩(1 − λn (1 + θn ))(xn − yn ) + λn (Txn − Tyn ) + λn θn (h(xn ) − h(yn ))󵄩󵄩󵄩 ≤ (1 − λn (1 + θn ))‖xn − yn ‖ + Lλn ‖xn − yn ‖ + Lλn θn ‖xn − yn ‖

≤ ((1 − λn (1 + θn )) + 2Lλn )‖xn − yn ‖.

(5.116)

Using (5.115) and (5.116), we obtain that 󵄩 󵄩2 ‖xn+1 − yn ‖2 = 󵄩󵄩󵄩(1 − λn (1 + θn ))(xn − yn ) + λn (Txn − Tyn ) + λn θn (h(xn ) − h(yn ))󵄩󵄩󵄩 2

≤ (1 − λn (1 + θn )) ‖xn − yn ‖2 + 2λn ⟨Txn − Tyn , j(xn+1 − yn ) + 2λn θn ⟨h(xn ) − h(yn ), j(xn+1 − yn ) 2

≤ (1 − λn (1 + θn )) ‖xn − yn ‖2 + 2Lλn ‖xn+1 − xn ‖‖xn+1 − yn ‖ + 2λn ‖xn+1 − yn ‖2 + 2Lλn θn ‖xn+1 − xn ‖‖xn+1 − yn ‖ + 2kλn θn ‖xn+1 − yn ‖2 2

≤ (1 − λn (1 + θn )) ‖xn − yn ‖2 + 4L(1 + L)(1 + 2L)λn2 ‖xn − yn ‖2 + 2λn (1 + kθn )‖xn+1 − yn ‖2 .

Using condition (v), we obtain that ‖xn+1 − yn ‖2 ≤

1 − 2λn (1 + θn ) + λn2 (1 + θn )2 ‖xn − yn ‖2 1 − 2λn (1 + kθn ) +

4L(1 + L)(1 + 2L)λn2 ‖xn − yn ‖2 1 − 2λn (1 + kθn )

≤ (1 − 2λn θn

1−k−

λn (1 2θn

+ θn )2

1 − 2λn (1 + kθn ) λ

+ 2λn θn

2L(1 + L)(1 + 2L) θn 1 − 2λn (1 + kθn )

≤ (1 − 2λn θn + 2λn θn

1−k−

1−k 4

1−k 4

1 − 2λn (1 + kθn )

‖xn − yn ‖2

n

1 − 2λn (1 + kθn )

)‖xn − yn ‖2

)‖xn − yn ‖2

‖xn − yn ‖2

≤ (1 − (1 − k)λn θn )‖xn − yn ‖2 .

5.12 Remarks | 327

This implies that 1−k λ θ )‖xn − yn ‖ 2 n n 1−k ≤ (1 − λ θ )‖xn − yn−1 ‖ + ‖yn − yn−1 ‖. 2 n n

‖xn+1 − yn ‖ ≤ (1 −

(5.117)

Substituting (5.114) into (5.117), we find that ‖xn+1 − yn ‖ ≤ (1 − ≤ (1 −

θ 󵄨󵄨󵄨 M 󵄨󵄨󵄨󵄨 1−k λn θn )‖xn − yn−1 ‖ + 󵄨󵄨1 − n−1 󵄨󵄨󵄨 2 (1 − k) 󵄨󵄨 θn 󵄨󵄨

1−k λ θ )‖xn − yn−1 ‖ + o(λn θn ). 2 n n

Using Lemma 6.1.4, we find that xn+1 − yn → θ as n → ∞. Hence, xn → p as n → ∞ and p is the unique solution to (5.113). This completes the proof.

5.12 Remarks The requirement that g is strongly accretive in Theorem 5.3.1 is weaker than the requirement that g is α-strongly accretive; see Deimling [27]. From the strong accretiveness of mapping g, one sees that Theorem 5.3.1 is an improvement of Deimling [27]. To the best of our knowledge, Theorem 5.4.1 is a new result. Theorems 5.5.1 and 5.5.2 are extensions of Morales results [54]. Theorem 5.6.3 is due to Zhou [128], Theorem 5.6.4 is due to Zhou and Shi [115], and Theorem 5.6.5 is due to Zhou and Shi [122]. Theorems 5.6.7, 5.6.8, 5.6.9, and 5.6.10 are due to Xu [101]. Most of the results in Section 5.7 are due to the authors of the book. Theorem 5.8.2 is an extension of Morales and Chidume [53]. Theorems 5.9.1 and 5.9.2 are due to Zhou [114]. We gave a new proof of Theorem 5.10.1. Theorem 5.10.4 was given by Ray [70], but our proof is more concise. Theorem 5.11.1 is due to Liu [44]. Theorems 5.11.2, 5.11.3, 5.11.4, 5.11.5, and 5.11.6 are new and due to the authors of the book.

5.13 Exercises 1.

2.

Let X be a real Banach space and T : Dom(T) → X be a mapping. Write A := I − T, where I is the identity mapping on X. Show that T is pseudocontractive ⇐⇒ A is accretive. In particular, if T is a nonexpansive mapping, then A is accretive. Let C be a nonempty, convex, and closed subset of a q-uniformly smooth Banach space X. Suppose that the generalized duality mapping Jq : X → X ∗ is weakly sequentially continuous at zero. Let T : C → X be k-strictly pseudocontractive with k ∈ (0, 1). Show that for any sequence {xn } ⊂ C, if xn ⇀ x, and xn − Txn → y ∈ X, then x − Tx = y, in particular, if y = θ, then x = Tx.

328 | 5 Accretive operators and pseudocontractive mappings 3.

Let C be a nonempty, convex, and closed subset of a reflexive Banach space X. Suppose that X satisfies Opial condition. Let T : C → X be a continuous pseudocontractive mapping that satisfies the weak inward condition. Show that I − T is demiclosed at zero, i. e., ∀{xn } ⊂ C with xn ⇀ x and xn − Txn → θ 󳨐⇒ x = Tx. 4. Let C be a nonempty, convex, and closed subset of a uniformly convex Banach space X, and let T : C → X be a continuous pseudocontractive mapping that satisfies the weak inward condition. Show that I − T is demiclosed at zero. 5. Let C be a nonempty, convex, and closed subset of a Banach space X. Let QC : X → C be a sunny nonexpansive retraction from X onto C. Let A : C → X be an η-strongly accretive and Lipschitz continuous mapping. Show that the accretivetype variational inequality, ⟨Ax∗ , j(x − x∗ )⟩ ≥ 0,

6.

∀x ∈ C,

(ATVIP)

has a unique solution x∗ ∈ C. We use the symbol SOL(C, A) to denote the set of solutions for A. Let X be a 2-uniformly smooth Banach space with the uniform smoothness constant K and let C be a nonempty, closed, and convex subset of X. Let QC be a sunny nonexpansive retraction from X onto C and let A be an η-inverse strongly accretive mapping of C into X such that SOL(C, A) ≠ 0. Let {αn }, {βn }, {γn }, and {λn } be four sequences in (0,1) satisfying the following conditions: (i) αn + βn + γn = 1, n ≥ 1; (ii) limn→∞ αn = 0 and ∑∞ n=1 αn = ∞; (iii) 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1; η (iv) a ≤ λn < b for some a, b ∈ (0, K 2 ]; (v) limn→∞ (λn+1 − λn ) = 0. For a fixed anchor u ∈ C and arbitrarily initial value x1 ∈ C, define a sequence {xn } iteratively by yn = QC (xn − λn Axn ), { xn+1 = αn u + βn xn + γn yn ,

n ≥ 1.

(HTIM)

Show that the sequence {xn } defined by (HTIM) converges strongly to the specific solution x∗ of the accretive type variational inequality (ATVIP): ⟨Ax∗ , j(x − x∗ )⟩ ≥ 0, 7.

∀x ∈ C.

Let X be a uniformly convex and 2-uniformly smooth Banach space with the uniform smoothness constant K, and let C be a nonempty, closed, and convex subset of X. Let QC be a sunny nonexpansive retraction from X onto C. Let A : C → X be an α-inverse strongly accretive mapping such that SOL(C, A) ≠ 0. Assume {λn } and {αn } are chosen so that λn ∈ [a, α/K 2 ] for some a > 0 and αn ∈ [b, c] for some b, c

5.13 Exercises | 329

with 0 < b < c < 1. Define a sequence {xn } iteratively in C by x1 ∈ C, { { { yn = QC (xn − λn Axn ), { { { {xn+1 = αn xn + (1 − αn )yn ,

(MTIM) n ≥ 1.

Prove that the sequence {xn } defined by (MTIM) converges weakly to some solution x∗ of (ATVIP). 8. Let X be a 2-uniformly smooth Banach space with the uniform smoothness constant K, and let C be a nonempty, closed, and convex subset of X. Let QC be a sunny nonexpansive retraction from X onto C. Let A : C → X be an α-inverse-strongly accretive mapping such that SOL(C, A) ≠ 0. Assume {λn } and {αn } are chosen so that λn ∈ [a, α/K 2 ] for some a > 0 and λn+1 − λn → 0. Let αn ∈ (0, 1) satisfy conditions: (C1) limn→∞ αn = 0 and (C2) ∑∞ n=1 αn = ∞. Let σ ∈ (0, 1) be a fixed number. For a fixed anchor u ∈ C and arbitrary initial value x1 ∈ C, define a sequence {xn } iteratively in C by u, x1 ∈ C, { { { yn = QC (xn − λn Axn ), { { { {xn+1 = σxn + (1 − σ)[αn u + (1 − αn )yn ],

9.

(HTIM) n ≥ 1.

Prove that the sequence {xn } defined by (HTIM) converges strongly to some solution x∗ of (ATVIP). Let X be a 2-uniformly smooth Banach space with the uniform smoothness constant K, and let C be a nonempty, closed, and convex subset of X. Let T be an η-strictly pseudocontractive mapping of C into itself such that Fix(T) ≠ 0. Assume {λn } and {αn } are chosen so that λn ∈ [a, η/K 2 ] for some a > 0 and λn+1 − λn → 0 (n → ∞). Let αn ∈ (0, 1) be a sequence satisfying conditions: (C1) limn→∞ αn = 0 and (C2) ∑∞ n=1 αn = ∞. Let σ ∈ (0, 1) be a fixed number. For a fixed anchor u ∈ C and arbitrary initial value x1 ∈ C, define a sequence {xn } iteratively in C by u, x1 ∈ C, { { { yn = (1 − λn )xn + λn Txn , { { { {xn+1 = σxn + (1 − σ)[αn u + (1 − αn )yn ],

(HTIM) n ≥ 1.

Prove that the sequence {xn } defined by (HTIM) converges strongly to a specific fixed point of T. 10. Let X be a 2-uniformly smooth Banach space and C a nonempty, closed, and convex subset of X. Let T : C → C be an η-strongly pseudocontractive and L-Lipschitz continuous mapping. Let {tn } be a sequence in (0,1) satisfying conditions: (C1) limn→∞ tn = 0 and (C2) ∑∞ n=1 tn = ∞. Prove that the normal Mann iteration {xn }

330 | 5 Accretive operators and pseudocontractive mappings defined by x1 ∈ C,

{

xn+1 = (1 − tn )xn + tn Txn ,

n ≥ 1,

converges strongly to the unique fixed point of T. Compare the convergence rate of algorithms appearing in Exercises 9 and 10, and write your comments on this matter. 11. Let X be a uniformly smooth Banach space, C a nonempty, closed, and convex subset of X, and T : C → C a continuous pseudocontractive mapping such that K := Fix(T) ≠ 0. Let A : C → X be an η-strongly accretive and L-Lipschitz continuous mapping. Show that the accretive type variational inequality problem (ATVIP) has a unique solution in K. 12. Let X be a uniformly smooth Banach space, C a nonempty, closed, and convex subset of X, and T : C → C an L-Lipschitz continuous pseudocontractive mapping such that K := Fix(T) ≠ 0. Let A : C → X be an η-strongly accretive and L-Lipschitz continuous mapping such that h := I − A : C → C is a self-mapping from C into itself. Let {λn } and {θn } be two sequences in (0,1) satisfying the following conditions: (i) λn (1 + θn ) ≤ 1, ∀n ≥ 1; (ii) θn → 0 (n → ∞); (iii) ∑∞ n=1 λn θn = ∞; θn−1 (iv) θ − 1 = o(λn θn ); and λ

n

(v) θn → 0 (n → ∞). n Define a sequence {xn } iteratively in C by x1 ∈ C,

xn+1 = (1 − λn )xn + λn Txn − λn θn Axn ,

n ≥ 1.

(HRIM)

Show that the sequence {xn } defined by (HRIM) converges strongly to the unique solution x∗ of the accretive type variational inequality problem (ATVIP).

6 Iterative methods for zeros of maximal monotone operators in Banach spaces The purpose of this chapter is to introduce three kinds of iterative methods for zero points of maximal monotone operators in the framework of Banach spaces. These methods are modifications to the celebrated Rockafellar approximate proximal point algorithm. With the aid of Lyapunov functionals, we obtain the convergence analysis of these iterative methods based on a generalized projection operator.

6.1 Lyapunov functional and generalized projection Let E be a real Banach space and let E ∗ be the dual space of E. Let ⟨⋅, ⋅⟩ denote the ∗ pairing between E and E ∗ . The normalized duality mapping J : E → 2E is defined by J(x) = {f ∈ E ∗ : ⟨x, y⟩ = ‖x‖2 = ‖f ‖2 } for all x ∈ E. Next, we use ⟨x, f ⟩ to denote the value of f at x. First, let us briefly recall some important concepts and conclusions on the generalized projection operator introduced in Chapter 1. Definition 6.1.1. Let E be a real smooth Banach space. The Lyapunov functional ϕ : E × E → ℝ+ is defined by ϕ(x, y) = ‖x‖2 + ‖y‖2 − 2⟨x, Jy⟩,

∀x, y ∈ E.

(6.1)

Lemma 6.1.1. Let E be a reflexive, smooth, and strictly convex Banach space. Let C be a nonempty, closed, and convex subset of E. Then there exists a unique x̄ ∈ C such that ϕ(x,̄ x) = min ϕ(y, x).

(6.2)

y∈C

The generalized projection ΠC : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(x, y), that is, ΠC x = x,̄ where x̄ is the solution to the minimization problem ϕ(x,̄ x) = miny∈C ϕ(y, x). It is clear that the generalized projection operator is an analogue of the metric projection in Hilbert spaces. The existence and uniqueness of the operator ΠC follow from the properties of functional ϕ(x, y) and the strict monotonicity of the mapping J. It is obvious from the definition of ϕ that 2

2

(‖y‖ − ‖x‖) ≤ ϕ(y, x) ≤ (‖y‖ + ‖x‖) ,

∀x, y ∈ E,

(6.3)

and ϕ(x, y) = ϕ(x, z) + ϕ(z, y) + 2⟨x − z, Jz − Jy⟩, https://doi.org/10.1515/9783110667097-006

∀x, y, z ∈ E.

(6.4)

332 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces We also remark here that, in the framework of reflexive, strictly convex, and smooth Banach spaces, for all x, y ∈ E, ϕ(x, y) = 0 if and only if x = y. It is sufficient to show that if ϕ(x, y) = 0, then x = y. From (6.3), we have ‖x‖ = ‖y‖. This implies that ⟨x, Jy⟩ = ‖x‖2 = ‖Jy‖2 . From the definition of J, we see that Jx = Jy. It follows that x = y. In [2], Alber gave the following results. Lemma 6.1.2. Let E be a smooth Banach space and let C be a nonempty, convex, and closed subset of E. Let x ∈ E and x̄ ∈ C. Then x̄ = ΠC x if and only if ⟨x̄ − y, Jx − J x⟩̄ ≥ 0,

∀y ∈ C.

(6.5)

Lemma 6.1.3. Let E be a reflexive, strictly convex, and smooth Banach space, and let C be a nonempty, convex, and closed subset of E. Let x ∈ E. Then ϕ(y, ΠC x) + ϕ(ΠC x, x) ≤ ϕ(y, x),

∀y ∈ C.

(6.6)

In [35], Kamimura and Takahashi gave the following results in the framework of uniformly convex and smooth Banach spaces. Lemma 6.1.4. Let E be a uniformly convex and smooth Banach space. Let {xn } and {yn } be two sequences in E. If ϕ(xn , yn ) → θ and either {xn } or {yn } is bounded, then xn −yn → θ as n → ∞. Next, we introduce another important operator QAr . Definition 6.1.2. Let E be a reflexive, strictly convex, and smooth Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator. For any r > 0, define an operator QAr : E → E by QAr x = (J + rA)−1 Jx, ∀x ∈ E. Since A ⊂ E × E ∗ is maximal monotone, one sees that Ran(J + rA) = E ∗ , ∀r > 0. Since the space is strictly convex and smooth, one sees that J is single-valued and oneto-one. So (J + rA)−1 is well-defined on E ∗ . It follows that QAr : E → E is a single-valued operator. We also remark here that operator QAr plays an important role in designing algorithms and in convergence analysis. The following two lemmas show the basic properties of QAr . Lemma 6.1.5. Let E be a reflexive, strictly convex, and smooth Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. For any x ∈ E, y ∈ A−1 (θ) and r > 0, one has ϕ(y, QAr x) + ϕ(QAr x, x) ≤ ϕ(y, x).

(6.7)

Proof. From the definition of QAr , one sees that there exists ur ∈ AQAr x such that JQAr x + rur = Jx.

(6.8)

6.1 Lyapunov functional and generalized projection

| 333

Using the monotonicity of A, one has ⟨QAr x − y, ur ⟩ ≥ 0,

∀y ∈ A−1 (θ).

(6.9)

Combining (6.8) with (6.9), one finds form the definition of ϕ that ϕ(y, x) − ϕ(QAr x, x) − ϕ(y, QAr x)

= ‖y‖2 − 2⟨y, Jx⟩ + ‖x‖2 − ‖QAr x‖2 + 2⟨QAr x, Jx⟩ − ‖x‖2 − ‖y‖2 󵄩 󵄩2 + 2⟨y, JQAr x⟩ − 󵄩󵄩󵄩QAr x󵄩󵄩󵄩 = 2⟨y, JQAr x − Jx⟩ − 2⟨QAr x, JQAr x − Jx⟩ = 2r⟨QAr x − y, ur ⟩ ≥ 0.

This completes the proof. Lemma 6.1.6. Let E be a reflexive, locally uniformly convex, and smooth Banach space. Let A ⊂ E × E ∗ be maximal monotone operator with A−1 (θ) ≠ 0. For any x ∈ E, one has limr→∞ QAr x = ΠA−1 (θ) x. Proof. Since A ⊂ E × E ∗ is maximal monotone, one sees that A−1 (θ) is nonempty, closed, and convex subset of E. Using Lemma 6.1.1, one obtains the existence and uniqueness of the generalized projection ΠA−1 (θ) x. First, we show that limr→∞ QAr x exists and belongs to A−1 (θ). To this end, we choose rn ∈ (0, ∞) with rn → ∞ as n → ∞. Letting xn = QArn x, one has Jx ∈ Jxn + rn Axn .

(6.10)

Since A−1 (θ) ≠ 0, we may choose a fixed element v in A−1 (θ). In view of (6.10), we have yn ∈ Axn such that Jx = Jxn + rn yn .

(6.11)

⟨Jxn , xn − v⟩ + rn ⟨yn , xn − v⟩ = ⟨Jx, xn − v⟩.

(6.12)

It follows that

From the fact that rn > 0 and ⟨yn , xn − v⟩ ≥ 0, one has ⟨Jxn , xn − v⟩ ≤ ⟨Jx, xn − v⟩,

(6.13)

⟨Jxn − Jx, xn − v⟩ ≤ 0.

(6.14)

which implies that

334 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces This further implies ⟨Jxn − Jv, xn − v⟩ = ⟨Jxn − Jx, xn − v⟩ + ⟨Jx − Jv, xn − v⟩ ≤ ⟨Jx − Jv, xn − v⟩

≤ ‖Jx − Jv‖(‖xn ‖ + ‖v‖).

(6.15)

On the other hand, one has 2

⟨Jxn − Jv, xn − v⟩ ≥ (‖xn ‖ − ‖v‖) .

(6.16)

Combining (6.15) with (6.16), one arrives at 2

(‖xn ‖ − ‖v‖) ≤ ‖Jx − Jv‖(‖xn ‖ + ‖v‖).

(6.17)

This shows that {xn } is bounded. Without loss of generality, we may assume that xn ⇀ p as n → ∞. Since rn → ∞ as n → ∞, we find from (6.11) that yn =

Jx − Jxn →θ rn

as n → ∞. Since the graph of A is demiclosed, we find that p ∈ Dom(A) and θ ∈ Ap, that is, p ∈ A−1 (θ). Using (6.13), we have ⟨Jxn − Jp, xn − p⟩ ≤ ⟨Jx − Jp, xn − p⟩.

(6.18)

lim sup⟨Jxn − Jp, xn − p⟩ ≤ 0.

(6.19)

It follows that n→∞

On the other hand, we have 2

⟨Jxn − Jp, xn − p⟩ ≥ (‖xn ‖ − ‖p‖) .

(6.20)

This implies that 2

lim sup(‖xn ‖ − ‖p‖) ≤ 0. n→∞

Hence, we have ‖xn ‖ → ‖p‖ as n → ∞. Since E is locally uniformly convex, we see that E has the Kadec–Klee property. It follows that ‖xn − p‖ → 0 as n → ∞. We are now in a position to show p = ΠA−1 (θ) x. Since J : E → E ∗ is demicontinuous, we have Jxn ⇀ Jp as n → ∞. Using the monotonicity of A, we find from (6.11) that ⟨Jxn − Jx, xn − y⟩ = −rn ⟨yn , xn − y⟩ ≤ 0,

∀y ∈ A−1 (θ).

This further implies that ⟨Jp − Jx, p − y⟩ ≤ 0,

∀y ∈ A−1 (θ).

Using Lemma 6.1.2, one sees that p = ΠA−1 (θ) x. This shows that {xn } converges to ΠA−1 (θ) x in norm. For any sn ∈ (0, ∞) with sn → ∞ as n → ∞, denoting zn = QAsn x = (J + sn A)−1 Jx and repeating the above proof, we see that {zn } converges to ΠA−1 (θ) x in norm. So, {QAr x} converges to ΠA−1 (θ) x as r → ∞ in norm. This completes the proof.

6.2 Rockafellar–Mann iterative method and its weak convergence theorem

| 335

6.2 Rockafellar–Mann iterative method and its weak convergence theorem Let E be a real reflexive, smooth, and locally uniformly convex Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Consider the following Rockafellar–Mann iterative method x1 ∈ E, xn+1 = J −1 (αn Jxn + βn JQArn xn + γn Jen ),

n ≥ 1,

(GRMIM)

where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , {en } is a bounded error sequence in E. We call (GRMIM) a Rockafellar–Mann iterative method. In order to obtain the weak convergence of (GRMIM), we prove the following lemma first. Lemma 6.2.1. Let E be a real, smooth, and uniformly convex Banach space. Let A ⊂ E×E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Let {xn } be a sequence generated −1 by (GRMIM). If ∑∞ n=1 γn < ∞, then {ΠA−1 (θ) xn } converges to v ∈ A (θ) in norm and v is also a unique solution to the following minimum problem: lim ϕ(v, xn ) = min lim ϕ(y, xn ).

n→∞

y∈A−1 (θ) n→∞

(6.21)

Proof. The proof is split into seven steps. Step 1. Show that {xn } is a bounded sequence. Fix p ∈ A−1 (θ). Since ‖ ⋅ ‖2 is convex, we find from Lemma 6.1.5 that ϕ(p, xn+1 ) = ‖p‖2 − 2⟨p, Jxn+1 ⟩ + ‖Jxn+1 ‖2

= αn ‖p‖2 + βn ‖p‖2 + γn ‖p‖2 − 2αn ⟨p, Jxn ⟩ − 2βn ⟨p, JQArn xn ⟩ − 2γn ⟨p, Jen ⟩ 󵄩 󵄩2 + 󵄩󵄩󵄩αn Jxn + βn JQArn xn + γn Jen 󵄩󵄩󵄩

≤ αn ‖p‖2 − 2αn ⟨p, Jxn ⟩ + αn ‖Jxn ‖2 + βn ‖p‖2 󵄩 󵄩2 − 2βn ⟨p, JQArn xn ⟩ + βn 󵄩󵄩󵄩JQArn xn 󵄩󵄩󵄩 + γn ‖p‖2 − 2γn ⟨p, Jen ⟩ + γn ‖Jen ‖2

= αn ϕ(p, xn ) + βn ϕ(p, QArn xn ) + γn ϕ(p, Jen )

≤ αn ϕ(p, xn ) + βn ϕ(p, xn ) − βn ϕ(QArn xn , xn ) + γn ϕ(p, Jen ) ≤ (1 − γn )ϕ(p, xn ) + γn ϕ(p, Jen ) ≤ max{ϕ(p, x1 ), Mp }, ∀n ≥ 1, where Mp = sup{ϕ(p, Jen ) : n ≥ 1}.

(6.22)

336 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces This shows ϕ(p, xn ) is a bounded sequence. From the definition of ϕ, we find that {xn } is also bounded. Step 2. Show that limn→∞ ϕ(p, xn ), ∀p ∈ A−1 (θ) exits. From Step 1, we have ϕ(p, xn+1 ) ≤ ϕ(p, xn ) + γn ϕ(p, en ) ≤ ϕ(p, xn ) + γn Mp . Since ∑∞ n=1 γn < ∞, by virtue of Lemma 1.10.1, we conclude that limn→∞ ϕ(p, xn ) exists ∀p ∈ A−1 (θ). Step 3. Show that the existence and uniqueness of the solution to (6.21). Define h(y) = limn→∞ ϕ(y, xn ), ∀y ∈ A−1 (θ). Then h : A−1 (θ) → ℝ+ is a proper, lower semicontinuous and convex function such that h(y) → ∞ as ‖y‖ → ∞. So, there exists v ∈ A−1 (θ) such that h(v) = miny∈A−1 (θ) h(y). Since E is uniformly convex, we see that h(y) is strictly convex. Hence, we get the uniqueness of v. Step 4. Show that limn→∞ ϕ(ΠA−1 (θ) xn , xn ) exists. Since A ⊂ E × E ∗ is maximal monotone and A−1 (θ) ≠ 0, we find that A−1 (θ) is nonempty, closed, and convex. So, the existence and uniqueness of the generalized projection operator ΠA−1 (θ) xn is ensured. From the definition of ΠA−1 (θ) , we have ϕ(ΠA−1 (θ) xn+1 , xn+1 ) ≤ ϕ(ΠA−1 (θ) xn , xn+1 ).

(6.23)

From Step 1, we find that ϕ(ΠA−1 (θ) xn , xn+1 ) ≤ ϕ(ΠA−1 (θ) xn , xn ) + γn ϕ(ΠA−1 (θ) xn , en ).

(6.24)

In view of Lemma 6.1.3, we arrive at ϕ(ΠA−1 (θ) xn , xn ) ≤ ϕ(v, xn ) − ϕ(v, ΠA−1 (θ) xn ),

(6.25)

where v ∈ A−1 (θ) satisfies (6.21). From Step 1, we see that ϕ(v, xn ) is bounded. Using (6.25), we find that ϕ(v, ΠA−1 (θ) xn ) is bounded, and then we find that {ΠA−1 (θ) xn } is also bounded in view of (6.3). It follows that {ϕ(ΠA−1 (θ) xn , en )} is also bounded. Setting M = sup{ϕ(ΠA−1 (θ) xn , en ) : n ≥ 1}, one finds that (6.24) is reduced to ϕ(ΠA−1 (θ) xn , xn+1 ) ≤ ϕ(ΠA−1 (θ) xn , xn ) + γn M.

(6.26)

Substituting (6.26) into (6.23), one arrives at ϕ(ΠA−1 (θ) xn+1 , xn+1 ) ≤ ϕ(ΠA−1 (θ) xn , xn ) + γn M. By virtue of Lemma 1.10.1, we conclude that limn→∞ ϕ(ΠA−1 (θ) xn , xn ) exists.

(6.27)

6.2 Rockafellar–Mann iterative method and its weak convergence theorem

| 337

Step 5. Show that {ΠA−1 (θ) xn } is Cauchy. From Step 1, by a simple induction, we have m−1

ϕ(p, xn+m ) ≤ ϕ(p, xn ) + M ∑ γn+j , j=0

∀p ∈ A−1 (θ),

(6.28)

for all m, n ≥ 1. By virtue of Lemma 6.1.3 and (6.28), we obtain ϕ(ΠA−1 (θ) xn , ΠA−1 (θ) xn+m )

≤ ϕ(ΠA−1 (θ) xn , xn+m ) − ϕ(ΠA−1 (θ) xn+m , xn+m )

m−1

≤ ϕ(ΠA−1 (θ) xn , xn ) − ϕ(ΠA−1 (θ) xn+m , xn+m ) + M ∑ γn+j j=0

(6.29)

for all n, m ≥ 1. From Step 5 and (6.29), it follows that {ΠA−1 (θ) xn } is Cauchy. Assume that limn→∞ ΠA−1 (θ) xn = p; then p ∈ A−1 (θ), since A−1 (θ) is closed. It is clear that limn→∞ ϕ(ΠA−1 (θ) xn , xn ) = limn→∞ ϕ(p, xn ) = h(p). Step 6. Show that {ΠA−1 (θ) xn } converges to v ∈ A−1 (θ), where v solves (6.21). By virtue of Lemma 6.1.3, we have ϕ(v, ΠA−1 (θ) xn ) ≤ ϕ(v, xn ) − ϕ(ΠA−1 (θ) xn , xn ).

(6.30)

Taking the lim sup on both sides of the above inequality, we have lim sup ϕ(v, ΠA−1 (θ) xn ) ≤ h(v) − h(p) ≤ 0. n→∞

On the other hand, we have lim infn→∞ ϕ(v, ΠA−1 (θ) xn ) ≥ 0. This shows that lim ϕ(v, ΠA−1 (θ) xn ) = 0.

n→∞

Form Lemma 6.1.4, we obtain that ΠA−1 (θ) xn → v as n → ∞. This completes the proof. Theorem 6.2.1. Let E be a real, uniformly smooth, and uniformly convex Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Let J : E → E ∗ be weakly sequentially continuous. Let {xn } be a sequence generated by (GRMIM). If lim infn→∞ rn > 0, lim infn→∞ βn > 0, ∑∞ n=1 γn < ∞, then {xn } converges weakly to v ∈ A−1 (θ), where v = limn→∞ ΠA−1 (θ) xn . Proof. Denote the weak-limit set of {xn } by ωω (xn ), that is, ωω (xn ) = {x ∈ E : ∃xni ⇀ x}. Since {xn } is bounded and the space is reflexive, we have ωω (xn ) ≠ 0. For any x ∈ ωω (xn ), we see that there exists a subsequence {xnj } of {xn } such that xnj ⇀ x as j → ∞. From Step 1 of Lemma 6.2.1, we have ϕ(p, xn+1 ) ≤ αn ϕ(p, xn ) − βn ϕ(QArn xn , xn ) + γn ϕ(p, en ),

∀p ∈ A−1 (θ).

338 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces It follows that βn ϕ(QArn xn , xn ) ≤ ϕ(p, xn ) − ϕ(p, xn+1 ) + γn Mp ,

∀p ∈ A−1 (θ).

Since limn→∞ ϕ(p, xn ) exits, lim infn→∞ βn > 0, ∑∞ n=1 γn < ∞, we have lim ϕ(QArn xn , xn ) = 0.

n→∞

Using Lemma 6.1.4, we find limn→∞ ‖QArn xn − xn ‖ = 0. Denoting un = QArn xn , there exists ωn ∈ Aun with Jun + rn ωn = Jxn .

(6.31)

Since E is a uniformly smooth Banach space, we see that J is uniformly continuous on bounded sets. In view of that limn→∞ ‖un −xn ‖ = 0, we find that limn→∞ ‖Jun −Jxn ‖ = 0. Since lim infn→∞ rn > 0, we find from 6.31 that ωn =

Jxn − Jun → θ, rn

n → ∞.

Since xnj ⇀ x, and xnj − unj → θ as j → ∞, unj ⇀ x. Since the graph of A, which

is maximal monotone, is demiclosed, we find that x ∈ Dom(A) and x ∈ A−1 (θ). This shows that ωω (xn ) ⊂ A−1 (θ). Next, we prove x = v = limn→∞ ΠA−1 (θ) xn . Let yn = ΠA−1 (θ) xn . Using Lemma 6.1.2 and x ∈ A−1 (θ), one has ⟨ynj − x, Jxnj − Jynj ⟩ ≥ 0.

(6.32)

From Lemma 6.2.1, one sees that ynj − v → θ as j → ∞. It follows that Jynj − Jv → θ as j → ∞. From the assumption, we see that Jxnj ⇀ Jx as j → ∞. It follows from (6.32) that ⟨v − x, Jx − Jv⟩ ≥ 0. Since J is monotone, we have ⟨v − x, Jx − Jv⟩ ≤ 0. Hence, we have ⟨v − x, Jx − Jv⟩ = 0. This implies x = v. This shows that xn ⇀ v ∈ A−1 (θ), where v = limn→∞ ΠA−1 (θ) xn , completing the proof. Remark 6.2.1. The restriction that J is weakly sequentially continuous is unsatisfactory. Indeed, even the space lp , where p > 1, does not satisfy this condition. It is of interest to further modify (GRMIM) to get the weak convergence in more general Banach spaces.

6.3 Rockafellar–Halpern iterative method and its strong convergence theorem Let E be a real reflexive, smooth, and strictly convex Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Consider the following Rockafellar– Halpern iterative method x1 ∈ E, xn+1 = J −1 (αn Jx1 + βn JQArn xn + γn Jen ),

n ≥ 1,

(GRHIM)

6.3 Rockafellar–Halpern iterative method and its strong convergence theorem

| 339

where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , {en } is a bounded error sequence in E. We call (GRHIM) a Rockafellar–Halpern iterative method. Theorem 6.3.1. Let E be a real, uniformly smooth, and uniformly convex Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Let {xn } be a sequence ∞ generated by (GRHIM). If rn → ∞ and αn → 0 as n → ∞, ∑∞ n=1 γn < ∞ and ∑n=1 αn = ∞, −1 then {xn } converges strongly to p ∈ A (θ) and p = ΠA−1 (θ) x1 . Proof. The proof is split into four steps. Step 1. Show that {xn } is bounded. Since ‖ ⋅ ‖2 is convex, we find from the definition of ϕ that ϕ(p, xn+1 ) = ‖p‖2 − 2⟨p, Jxn+1 ⟩ + ‖Jxn+1 ‖2

= αn ‖p‖2 + βn ‖p‖2 + γn ‖p‖2 − 2αn ⟨p, Jx1 ⟩ − 2βn ⟨p, JQArn xn ⟩ − 2γn ⟨p, Jen ⟩ 󵄩 󵄩2 + 󵄩󵄩󵄩αn Jx1 + βn JQArn xn + γn Jen 󵄩󵄩󵄩

≤ αn ‖p‖2 − 2αn ⟨p, Jx1 ⟩ + αn ‖Jx1 ‖2 + βn ‖p‖2 󵄩 󵄩2 − 2βn ⟨p, JQArn xn ⟩ + βn 󵄩󵄩󵄩JQArn xn 󵄩󵄩󵄩 + γn ‖p‖2 − 2γn ⟨p, Jen ⟩ + γn ‖Jen ‖2

= αn ϕ(p, x1 ) + βn ϕ(p, QArn xn ) + γn ϕ(p, Jen ).

(6.33)

Using Lemma 6.1.5, we find that ϕ(p, QArn xn ) ≤ ϕ(p, xn ).

(6.34)

Substituting (6.34) into (6.33), we obtain that ϕ(p, xn+1 ) ≤ αn ϕ(p, x1 ) + βn ϕ(p, xn ) + γn Mp , where Mp = sup{ϕ(p, Jen ) : n ≥ 1}. This implies that n−1

ϕ(p, xn ) ≤ ϕ(p, x1 ) + Mp ∑ γi . i=1

It follows that ∞

ϕ(p, xn ) ≤ ϕ(p, x1 ) + Mp ∑ γi , i=1

which shows that ϕ(p, xn ) is a bounded sequence. From the definition of ϕ, we find that {xn } is also bounded.

340 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces Step 2. Show that lim supn→∞ ⟨Jx1 − Jp, yn − p⟩ ≤ 0, where yn = QArn xn . For any r > 0, we see that ur ∈ AQAr x1 and

JQAr x1 + rur = Jx1 .

(6.35)

For any n ≥ 1, we see that ωn ∈ Ayn and Jyn + rn ωn = Jxn .

(6.36)

From (6.35) and (6.36), we find from the monotonicity of A that ⟨Jx1 − JQAr x1 , yn − QAr x1 ⟩ = r⟨ur , yn − QAr x1 ⟩

= r⟨ur − ωn , yn − QAr x1 ⟩ + r⟨ωn , yn − QAr x1 ⟩

≤ r⟨ωn , yn − QAr x1 ⟩ r = ⟨Jxn − Jyn , yn − QAr x1 ⟩ rn r ≤ ‖Jxn − Jyn ‖‖yn − QAr x1 ‖. rn

(6.37)

From (6.34), we see that {ϕ(p, yn )} is bounded. Hence, {yn } is a bounded sequence. Since rn → ∞ as n → ∞, we find from (6.37) that lim sup⟨Jx1 − JQAr x1 , yn − QAr x1 ⟩ ≤ 0. n→∞

(6.38)

Assume that there exists a subsequence {ynj } of {yn } such that ynj ⇀ y as j → ∞ and lim sup⟨JQAr x1 − Jp, yn − p⟩ = lim ⟨JQAr x1 − Jp, ynj − p⟩ j→∞

n→∞

= ⟨JQAr x1 − Jp, y − p⟩.

(6.39)

Note that ⟨Jx1 − Jp, yn − p⟩

= ⟨Jx1 − JQAr x1 , yn − p⟩ + ⟨JQAr x1 − Jp, yn − p⟩

= ⟨Jx1 − JQAr x1 , yn − QAr x1 ⟩ + ⟨Jx1 − JQAr x1 , QAr x1 − p⟩ + ⟨JQAr x1 − Jp, yn − p⟩.

(6.40)

It follows from (6.38), (6.39), and (6.40) that lim sup⟨Jx1 − Jp, yn − p⟩ ≤ ⟨Jx1 − JQAr x1 , QAr x1 − p⟩ + ⟨JQAr x1 − Jp, y − p⟩. n→∞

(6.41)

From Lemma 6.1.6, we see that QAr x1 → p as r → ∞. Using the continuity of J, we find that JQAr x1 → Jp as r → ∞. Therefore, we obtain from (6.41) that lim sup⟨Jx1 − Jp, yn − p⟩ ≤ 0. n→∞

6.3 Rockafellar–Halpern iterative method and its strong convergence theorem

| 341

Step 3. Show that max{⟨Jx1 − Jp, yn − p⟩, 0} → 0 as n → ∞. For any ϵ > 0, there exists a natural number n0 ≥ 1. When n ≥ n0 , one has ⟨Jx1 − Jp, yn − p⟩ ≤ ϵ. So, 0 ≤ max{⟨Jx1 − Jp, yn − p⟩, 0} ≤ ϵ. Hence, we have max{⟨Jx1 − Jp, yn − p⟩, 0} → 0 as n → ∞. Step 4. Show that xn → p as n → ∞. Note that ‖αn Jx1 + βn Jyn + γn Jen ‖2

≤ ‖αn Jx1 + βn Jyn ‖2 + 2γn ‖αn Jx1 + βn Jyn ‖‖Jen ‖ + γn2 ‖Jen ‖2

≤ ‖αn Jx1 + βn Jyn ‖2 + γn M 󸀠 ,

(6.42)

where M 󸀠 = sup{2‖αn Jx1 + βn Jyn ‖‖en ‖ + ‖en ‖2 : n ≥ 1}. Since the space E is uniformly smooth, we find from the Reich inequality that ‖αn Jx1 + βn Jyn ‖2

≤ βn2 ‖Jyn ‖2 + 2αn βn ⟨Jx1 , yn ⟩ + max{βn ‖yn ‖, 1}αn ‖x1 ‖b(αn ‖x1 ‖)

≤ βn2 ‖Jyn ‖2 + 2αn βn ⟨Jx1 , yn ⟩ + max{βn ‖yn ‖, 1}αn ‖x1 ‖b(αn max{‖x1 ‖, 1}) ≤ (1 − αn )2 ‖yn ‖2 + 2αn βn ⟨Jx1 , yn ⟩ + αn b(αn )M 󸀠󸀠 ,

(6.43)

where M 󸀠󸀠 = ‖x1 ‖ max{sup{‖yn ‖ : n ≥ 1}, 1} max{‖x1 ‖, 1}. It follows from (6.42) and (6.43) that ϕ(p, xn+1 ) = ‖p‖2 − 2⟨p, Jxn+1 ⟩ + ‖Jxn+1 ‖2

= αn ‖p‖2 + βn ‖p‖2 + γn ‖p‖2 − 2αn ⟨p, Jx1 ⟩

− 2βn ⟨p, JQArn xn ⟩ − 2γn ⟨p, Jen ⟩ + ‖αn Jx1 + βn JQArn xn + γn Jen ‖2

≤ αn ‖p‖2 + βn ‖p‖2 + γn ‖p‖2 − 2αn ⟨p, Jx1 ⟩ − 2βn ⟨p, JQArn xn ⟩ − 2γn ⟨p, Jen ⟩ + (1 − αn )2 ‖yn ‖2 + 2αn βn ⟨Jx1 , yn ⟩ + αn b(αn )M 󸀠󸀠 + γn M 󸀠

≤ 2αn ⟨p, Jp⟩ + 2αn ⟨Jp, yn ⟩ − 2αn ⟨Jp, yn ⟩ − 2αn ⟨p, Jx1 ⟩ + (1 − αn )‖yn ‖2 − 2(1 − αn − γn )⟨p, Jyn ⟩ + 2(1 − αn − γn )⟨Jx1 , yn ⟩ − αn (1 − αn )‖yn ‖2

− αn ‖p‖2 + (1 − αn )‖p‖2 + 2γn ‖p‖‖en ‖ + αn b(αn )M 󸀠󸀠 + γn M 󸀠

≤ (1 − αn )ϕ(p, yn ) + 2αn ⟨Jx1 − Jp, yn − p⟩ − αn ϕ(yn , p) + αn2 ϕ(yn , x1 )

+ 2αn γn ‖x1 ‖‖yn ‖ + 2γn ‖p‖‖yn ‖ + 2γn ‖p‖‖en ‖ + αn b(αn )M 󸀠󸀠 + γn M 󸀠

≤ (1 − αn )ϕ(p, xn ) + 2αn ⟨Jx1 − Jp, yn − p⟩ + αn2 ϕ(yn , x1 )

+ αn b(αn )M 󸀠󸀠 + γn (2αn ‖x1 ‖‖yn ‖ + 2‖p‖‖yn ‖ + 2‖p‖‖en ‖ + M 󸀠 )

≤ (1 − αn )ϕ(p, xn ) + o(αn ) + Cγn ,

342 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces where C is some fixed positive constant. It follows from Lemma 1.10.2 that ϕ(p, xn ) → 0 as n → ∞. Using Lemma 6.1.4, we find that xn → p as n → ∞. This completes the proof. Remark 6.3.1. The restriction that ∑∞ n=1 αn = ∞ is unsatisfactory because it may reduce the convergence rate of (GRHIM). It is of interest to develop more efficient iterative methods for approximating the zero points of maximal monotone operators in general Banach spaces.

6.4 Rockafellar–Haugazeau iterative method and its strong convergence theorem Let E be a real reflexive, smooth, and strictly convex Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Consider the following iterative method: x1 ∈ E, chosen arbitrarily, { { { { { { yn = J −1 (αn Jxn + βn JQArn xn + γn Jen ), { { { Cn = {z ∈ E : ϕ(z, yn ) ≤ ϕ(z, xn ) + γn B}, { { { { { Q { n = {z ∈ E : ⟨xn − z, Jx1 − Jxn ⟩ ≥ 0}, { { { {xn+1 = ΠCn ∩Qn x1 , n ≥ 1,

(RHIM)

where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , {en } is a bounded error sequence in E such that B = sup{ϕ(z, en ) : z ∈ A−1 (θ), n ≥ 1} < ∞. We call (RHIM) a Rockafellar–Haugazeau iterative method. Theorem 6.4.1. Let E be a real uniformly smooth and uniformly convex Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Let {xn } be a sequence generated by (RHIM). If lim infn→∞ βn > 0, lim infn→∞ rn > 0, γn → 0 as n → ∞, and {en } is a bounded error sequence in E such that B = sup{ϕ(z, en ) : z ∈ A−1 (θ), n ≥ 1} < ∞, then {xn } converges to ΠA−1 (θ) x1 in norm. Proof. First, we show that ΠA−1 (θ) x1 is well defined. Indeed, since A ⊂ E × E ∗ is a maximal monotone operator with A−1 (θ) ≠ 0, we find that A−1 (θ) is nonempty, closed, and convex. In view of Lemma 6.1.1, we find that ΠA−1 (θ) x1 is well defined. Next, we show that Cn ∩ Qn is closed and convex for every n ≥ 1. It is obvious that Qn is closed and convex. Note that ϕ(z, yn ) ≤ ϕ(z, xn ) + γn B is equivalent to ‖yn ‖2 − ‖xn ‖2 − 2⟨z, Jyn − Jxn ⟩ ≤ γn B. This implies set Cn is convex. It is also obvious that Cn is closed. This proves that Cn ∩Qn is closed and convex.

6.4 Rockafellar–Haugazeau iterative method and its strong convergence theorem | 343

that

Next, we show that A−1 (θ) ⊂ Cn ∩ Qn . For any z ∈ A−1 (θ), we find from Lemma 6.1.5 󵄩 󵄩2 ϕ(z, yn ) = ‖z‖2 − 2⟨z, αn Jxn + βn JQArn xn + γn Jen ⟩ + 󵄩󵄩󵄩αn Jxn + βn JQArn xn + γn Jen 󵄩󵄩󵄩 ≤ ‖z‖2 − 2αn ⟨z, Jxn ⟩ − 2βn ⟨z, JQArn xn ⟩ − 2γn ⟨z, Jen ⟩ 󵄩 󵄩2 + αn ‖xn ‖2 + βn 󵄩󵄩󵄩QArn xn 󵄩󵄩󵄩 + γn ‖en ‖2

≤ αn ϕ(z, xn ) + βn ϕ(z, xn ) − βn ϕ(QArn xn , xn ) + γn ϕ(z, en )

≤ (1 − γn )ϕ(z, xn ) + γn ϕ(z, en ) ≤ (1 − γn )ϕ(z, xn ) + γn B.

This implies that A−1 (θ) ⊂ Cn . Now we are in a position to show that A−1 (θ) ⊂ Cn ∩Qn by mathematical induction. For n = 1, since Q1 = E, we have A−1 (θ) ⊂ C1 ∩Q1 ⊂ C1 . Assume that A−1 (θ) ⊂ Cm ∩ Qm , m ∈ ℕ. It follows that xm+1 = ΠCm ∩Qn x1 . Using Lemma 6.1.2, we find that ⟨xm+1 − z, Jx1 − Jxm+1 ⟩ ≥ 0,

∀z ∈ Cm ∩ Qm .

It follows from A−1 (θ) ⊂ Cm ∩ Qm that ⟨xm+1 − z, Jx1 − Jxm+1 ⟩ ≥ 0,

∀z ∈ A−1 (θ),

which shows that z ∈ Qm+1 . It follows that z ∈ Cm+1 ∩ Qm+1 . Hence, we have A−1 (θ) ⊂ Cn ∩ Qn . This also shows (RHIA) is well defined. From the constructions of Qn and ΠQn , we have xn = ΠQn x1 . Using Lemma 6.1.3, we find ϕ(xn , x1 ) ≤ ϕ(z, x1 ) − ϕ(z, xn ) ≤ ϕ(z, x1 ),

∀z ∈ A−1 (θ),

which implies that {ϕ(xn , x1 )} is a bounded sequence, so are {xn } and QArn xn . In view of xn+1 = ΠCn ∩Qn x1 ∈ Cn ∩ Qn ⊂ Qn and xn = ΠQn x1 , one has ϕ(xn , x1 ) ≤ ϕ(xn+1 , x1 ). This proves that {ϕ(xn , x1 )} is nondecreasing. Hence, limn→∞ ϕ(xn , x1 ) exists. Using Lemma 6.1.3, we have ϕ(xn+1 , xn ) ≤ ϕ(xn+1 , ΠQn x1 ) ≤ ϕ(xn+1 , x1 ) − ϕ(xn , x1 ). Hence, we obtain limn→∞ ϕ(xn+1 , xn ) = 0. Since xn+1 = ΠCn ∩Qn x1 ∈ Cn , we obtain ϕ(xn+1 , yn ) ≤ ϕ(xn+1 , xn ) + γn B.

344 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces Hence, limn→∞ ϕ(xn+1 , yn ) = 0. It follows from Lemma 6.1.4 that lim ‖xn+1 − yn ‖ = lim ‖xn+1 − xn ‖ = 0.

n→∞

n→∞

This implies that limn→∞ ‖yn − xn ‖ = 0. Since E is a uniformly smooth Banach space, we see that J is uniformly continuous on bounded sets. Hence, limn→∞ ‖Jyn − Jxn ‖ = 0. Using the facts that lim infn→∞ βn > 0 and γn → 0 as n → ∞, we find that limn→∞ ‖JQArn xn − Jxn ‖ = 0. Since E ∗ is a uniformly smooth Banach space, we see that

J −1 is uniformly continuous on bounded sets. Hence, limn→∞ ‖QArn xn − xn ‖ = 0. Since E is reflexive and {xn } is bounded, we may, without loss of generality, assume that xn ⇀ x̄ as n → ∞. It follows that un ⇀ x̄ as n → ∞. Putting un = QArn xn , we see that there exists ωn ∈ Aun with Jun + rn ωn = Jxn , This implies that ωn =

Jxn − Jxn →θ rn

as n → ∞. Since the graph Graph(A) of A is demiclosed, we find that x̄ ∈ A−1 (θ). Setting z ∗ = ΠA−1 (θ) x1 , we have z ∗ ∈ A−1 (θ) ⊂ Cn ∩ Qn . It follows that ϕ(xn+1 , x1 ) ≤ ϕ(z ∗ , x1 ). From the weak lower semicontinuity of norms, we see that ϕ(z ∗ , x1 ) ≥ lim sup ϕ(xn , x1 ) n→∞

≥ lim inf ϕ(xn , x1 ) n→∞

= lim inf(‖xn ‖2 − 2⟨xn , Jx1 ⟩ + ‖x1 ‖2 ) n→∞ 2

≥ ‖x‖̄ − 2⟨x,̄ Jx1 ⟩ + ‖x1 ‖2

= ϕ(x,̄ x1 ),

which implies that limn→∞ ϕ(xn , x1 ) = ϕ(z ∗ , x1 ). Hence 0 = lim (ϕ(xn , x1 ) − ϕ(z ∗ , x1 )) n→∞

󵄩 󵄩2 = lim (‖xn ‖2 − 2⟨xn − z ∗ , Jx1 ⟩ − 󵄩󵄩󵄩z ∗ 󵄩󵄩󵄩 ) n→∞

󵄩 󵄩2 = lim (‖xn ‖2 − 󵄩󵄩󵄩z ∗ 󵄩󵄩󵄩 ), n→∞

that is, limn→∞ ‖xn ‖ = ‖z ∗ ‖. Since E has the Kadec–Klee property, we see that xn → z ∗ . From the uniqueness of z ∗ , we obtain that {xn } converges to ΠA−1 (θ) x1 in norm. In a similar way, we can prove the following result. Theorem 6.4.2. Let E be a real uniformly smooth and uniformly convex Banach space. Let A ⊂ E × E ∗ be a maximal monotone operator with A−1 (θ) ≠ 0. Let {xn } be a sequence

6.5 Minimizers of convex functionals and monotone variational inequalities | 345

generated by the following iterative method: x1 ∈ E, chosen arbitrarily, { { { { −1 A { { {yn = J (αn Jxn + βn JQrn xn + γn Jen ), { { C1 = E, { { { { { Cn+1 = {z ∈ Cn : ϕ(z, yn ) ≤ ϕ(z, xn ) + γn ϕ(z, en )}, { { { { {xn+1 = ΠCn+1 x1 , n ≥ 1,

(SRHIM)

where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , and {en } is a bounded error sequence in E such that B = sup{ϕ(z, en ) : z ∈ A−1 (θ), n ≥ 1} < ∞. Assume that lim infn→∞ βn > 0, lim infn→∞ rn > 0, and γn → 0 as n → ∞. Then {xn } converges to ΠA−1 (θ) x1 in norm. Remark 6.4.1. We call (SRHIM) a shrinking hybrid projection iterative method. Although its form is simpler than that of the classical method, it is not easy for one to determine Cn .

6.5 Minimizers of convex functionals and monotone variational inequalities Let E be a real Banach space and let f : E → ℝ be a proper, lower semicontinuous, and convex functional. The subdifferential mapping 𝜕f ⊂ E × E ∗ of f is defined by 𝜕f (x) = {x ∗ ∈ E ∗ : f (y) ≥ f (z) + ⟨y − z, x ∗ ⟩, ∀y ∈ E},

∀x ∈ E.

Rockafellar [76] proved that 𝜕f is a maximal monotone operator and 0 ∈ 𝜕f (v) if and only if f (v) = minx∈E f (x). Theorem 6.5.1. Let E be a real uniformly smooth and uniformly convex Banach space. Let f : E → ℝ be a proper, lower semicontinuous, and convex functional such that (𝜕f )−1 (θ) ≠ 0. Assume that normal duality map J : E → E ∗ is weakly sequentially continuous. Let {xn } be a sequence generated by the following iterative method: {x1 ∈ E, chosen arbitrarily, { { ⟨y,Jxn ⟩ ‖y‖2 {yn = arg miny∈E {f (y) + 2rn − rn }, { { −1 {xn+1 = J (αn Jxn + βn Jyn + γn Jen ), n ≥ 1,

(6.44)

where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , and {en } is a bounded error sequence in E. Assume that lim infn→∞ βn > 0, −1 lim infn→∞ rn > 0, and ∑∞ n=1 γn < ∞. Then {xn } converges weakly to v ∈ (𝜕f ) (θ), where v = limn→∞ Π(𝜕f )−1 (θ) xn .

346 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces Proof. Notice that ‖y‖2 ⟨y, Jxn ⟩ − } y∈E 2rn rn Jx Jy ⇐⇒ θ ∈ 𝜕f (yn ) + n − n rn rn

yn = arg min{f (y) +

⇐⇒ Jxn ∈ rn 𝜕f (yn ) + Jyn

⇐⇒ yn = (J + rn 𝜕f )−1 Jxn = Q𝜕f rn xn . Using Theorem 6.2.1, we find that {xn } converges weakly to v ∈ (𝜕f )−1 (θ) and v = limn→∞ Π(𝜕f )−1 (θ) xn . This completes the proof. Theorem 6.5.2. Let E be a real uniformly smooth and uniformly convex Banach space. Let f : E → ℝ be a proper, lower semicontinuous, and convex functional such that (𝜕f )−1 (θ) ≠ 0. Let {xn } be a sequence generated by the following iterative method: x1 ∈ E, chosen arbitrarily, { { { 2 ⟨y,Jx ⟩ yn = arg miny∈E {f (y) + ‖y‖ − r n }, { 2r n n { { −1 {xn+1 = J (αn Jx1 + βn Jyn + γn Jen ), n ≥ 1, where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , and {en } is a bounded error sequence in E. Assume that rn → ∞ and αn → 0 as ∞ n → ∞, ∑∞ n=1 γn < ∞ and ∑n=1 αn = ∞. Then {xn } converges to Π(𝜕f )−1 (θ) x1 in norm. Proof. From Theorem 6.3.1, we find the desired conclusion immediately. Theorem 6.5.3. Let E be a real uniformly smooth and uniformly convex Banach space. Let f : E → ℝ be a proper, lower semicontinuous, and convex functional such that (𝜕f )−1 (θ) ≠ 0. Let {xn } be a sequence generated by the following iterative method: x1 ∈ E, chosen arbitrarily, { { { 2 { { {yn = arg miny∈E {f (y) + ‖y‖ − ⟨y,Jxn ⟩ }, { 2r rn { n { { { {zn = J −1 (αn Jxn + βn Jyn + γn Jen ) { { { Cn = {z ∈ E : ϕ(z, zn ) ≤ ϕ(z, xn ) + γn B}, { { { { { Qn = {z ∈ E : ⟨xn − z, Jx1 − Jxn ⟩ ≥ 0}, { { { { {xn+1 = ΠCn ∩Qn x1 , n ≥ 1, where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , and {en } is a bounded error sequence in E such that B = sup{ϕ(z, en ) : z ∈ (𝜕f )−1 (θ), n ≥ 1} < ∞. Assume that lim infn→∞ βn > 0, lim infn→∞ rn > 0, γn → 0 as n → ∞. Then {xn } converges to Π(𝜕f )−1 (θ) x1 in norm. Proof. From Theorem 6.4.1, we find the desired conclusion immediately.

6.5 Minimizers of convex functionals and monotone variational inequalities | 347

Let C be a nonempty, closed, and convex subset of a Banach space E. Let A : C → E ∗ be a monotone operator which is hemicontinuous, that is, continuous along each line segment in C with respect to the weak∗ topology of E ∗ . Recall that the variational inequality problem with mappings of monotone type is to find a point x ∈ C such that ⟨y − x, Ax⟩ ≥ 0,

∀y ∈ C.

We use VI(C, A) to denote the solution set of the variational inequality. Recall that symbol NC (x) stands for the normal cone for C at a point x ∈ C, that is, NC (x) = {x ∗ ∈ E ∗ : ⟨y − x, x∗ ⟩ ≤ 0,

∀y ∈ C}.

From Rockafellar [75], we see that operator Ax + NC (x),

Tx = {

0,

x ∈ C, x ∉ C,

is maximal monotone and T −1 (0) = VI(C, A). So, we can study the monotone variational inequality via a maximal monotone operator equation. Theorem 6.5.4. Let E be a real uniformly smooth and uniformly convex Banach space. Let C be a nonempty, closed, and convex subset of E. Let A : C → E ∗ be a hemicontinuous monotone operator, and let normalized normal duality map J : E → E ∗ be weakly sequentially continuous. Assume that VI(C, A) ≠ 0. Let {xn } be a sequence generated by the following iterative method: {x1 ∈ E, chosen arbitrarily, { { J−Jxn {yn ∈ VI(C, A + rn ), { { −1 {xn+1 = J (αn Jxn + βn Jyn + γn Jen ),

(6.45) n ≥ 1,

where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , and {en } is a bounded error sequence in E. Assume that lim infn→∞ βn > 0, lim infn→∞ rn > 0, ∑∞ n=1 γn < ∞. Then {xn } converges weakly to v ∈ VI(C, A) and v = limn→∞ ΠVI(C,A) xn . Proof. Notice that yn ∈ VI(C, A + ⇐⇒ ⟨Ayn +

J − Jxn ) rn

Jyn − Jxn , y − yn ⟩ ≥ 0, rn

∀y ∈ C

⇐⇒ yn = (J + rn T)−1 Jxn = QTrn xn .

Using Theorem 6.5.1, we find that {xn } converges weakly to v ∈ T −1 (0) = VI(C, A) and v = limn→∞ ΠVI(C,A) xn . This completes the proof.

348 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces Theorem 6.5.5. Let E be a real uniformly smooth and uniformly convex Banach space. Let C be a nonempty, closed, and convex subset of E. Let A : C → E ∗ be a hemicontinuous monotone operator and assume that VI(C, A) ≠ 0. Let {xn } be a sequence generated by the following iterative method: x1 ∈ E, chosen arbitrarily, { { { J−Jx y ∈ VI(C, A + r n ), { n { n { −1 {xn+1 = J (αn Jx1 + βn Jyn + γn Jen ),

n ≥ 1,

where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , and {en } is a bounded error sequence in E. Assume that rn → ∞ and αn → 0 as ∞ n → ∞, ∑∞ n=1 γn < ∞ and ∑n=1 αn = ∞. Then {xn } converges to ΠVI(C,A) x1 in norm. Proof. From Theorem 6.3.1, we find the desired conclusion immediately. Theorem 6.5.6. Let E be uniformly smooth and uniformly convex Banach space. Let C be a nonempty, closed, and convex subset of E. Let A : C → E ∗ be a hemicontinuous monotone operator and assume that VI(C, A) ≠ 0. Let {xn } be a sequence generated by the following iterative method: x1 ∈ E, chosen arbitrarily, { { { { J−Jx { { yn ∈ VI(C, A + r n ), { { n { { { {zn = J −1 (αn Jxn + βn Jyn + γn Jen ) { { { Cn = {z ∈ E : ϕ(z, zn ) ≤ ϕ(z, xn ) + γn B}, { { { { { Qn = {z ∈ E : ⟨xn − z, Jx1 − Jxn ⟩ ≥ 0}, { { { { {xn+1 = ΠCn ∩Qn x1 , n ≥ 1, where {αn }, {βn }, and {γn } are three real sequences in (0, 1) such that αn + βn + γn = 1, rn ⊂ ℝ+ , and {en } is a bounded error sequence in E such that B = sup{ϕ(z, en ) : z ∈ VI(C, A), n ≥ 1} < ∞. Assume that lim infn→∞ βn > 0, lim infn→∞ rn > 0, γn → 0 as n → ∞. Then {xn } converges to ΠVI(C,A) x1 in norm. Proof. From Theorem 6.4.1, we find the desired conclusion immediately.

6.6 Remark The iterative methods and convergence theorems in this chapter are based on [96]. Here, we prove them with different methods.

6.7 Exercises 1.

Let C be a nonempty, compact, and convex subset of a topological linear space X, and let T be a monotone mapping of C into X ∗ . Show that there exists x0 ∈ C

6.7 Exercises | 349

such that ⟨Tx, x − x0 ⟩ ≥ 0, 2.

for all x ∈ C.

Let C be a nonempty convex subset of a topological linear space X and let T be a hemicontinuous mapping of C into X ∗ . Let x0 ∈ C be an element of C such that ⟨Tx, x − x0 ⟩ ≥ 0,

for all x ∈ C.

⟨Tx0 , x − x0 ⟩ ≥ 0,

for all x ∈ C.

Prove that

3.

Let C be a nonempty, compact, and convex subset of a topological linear space X, and let T be a monotone and hemicontinuous mapping of C into X ∗ . Prove that there exists x0 ∈ C such that ⟨Tx0 , x − x0 ⟩ ≥ 0,

for all x ∈ C.

4. Let C be a nonempty, bounded, closed, and convex subset of a real reflexive Banach space X, and let T be a monotone and hemicontinuous mapping of C into X ∗ . Then, prove that there exists x0 ∈ C such that ⟨Tx0 , x − x0 ⟩ ≥ 0, 5.

for all x ∈ C.

Let C be a nonempty, closed, and convex subset of a real reflexive Banach space X, and let T be a monotone and hemicontinuous mapping of C into X ∗ . Suppose that T is coercive, i. e., ⟨Tx, x − w⟩ →∞ ‖x‖

as ‖x‖ → ∞,

(*)

where w ∈ C is a fixed element. Prove that for all h ∈ X ∗ , there exists x0 ∈ C such that ⟨Tx0 − h, x − x0 ⟩ ≥ 0,

for all x ∈ C.

Furthermore, if T : C → X is also strictly monotone, then SOL(C, T) = {x0 }, in particular, if T : C → X is η-strongly monotone, then T satisfies the coercitivity condition (*), and hence SOL(C, T) = {x0 }. 6.

Let X be a real, reflexive, strictly convex, and smooth Banach space, and let T : ∗ X → 2X be a monotone mapping. Prove that T is maximal monotone ⇐⇒ there exists some λ > 0 such that Ran(T + λJ) = X ∗ ⇐⇒ ∀λ > 0, Ran(T + λJ) = X ∗ .

350 | 6 Iterative methods for zeros of maximal monotone operators in Banach spaces 7.

Let X be a real, reflexive, strictly convex, and smooth Banach space, let T : X → ∗ 2X be a maximal monotone mapping, and P : X → X ∗ be a bounded hemicontinuous and monotone mapping. Prove that T + P is maximal monotone. 8. Let C be a nonempty, closed, and convex subset of a real reflexive Banach space X, and let A be a monotone and hemicontinuous mapping of C into X ∗ . Let NC x = {x ∗ ∈ X ∗ : ⟨y−x, x∗ ⟩ ≤ 0, y ∈ C} be the normal cone of C at x ∈ C. Define a mapping T ⊂ X × X ∗ as follows: Ax + NC x, Tx = { 0,

x ∈ C, x ∉ C.

Prove that T is maximal monotone mapping and T −1 θ = SOL(C, A). 9. Let X be a real reflexive Banach space and φ : X → ℝ be a proper lower semicontinuous convex function. Prove that 𝜕φ is a maximal monotone mapping. 10. Let X be a real, reflexive, strictly convex, and smooth Banach space, and let T be a maximal monotone mapping of Dom(T) ⊆ X into X ∗ that satisfies the following angle condition: there exist some r > 0 and x0 ∈ Dom(T) such that ⟨Tx, x − x0 ⟩ ≥ 0,

for all x ∈ Dom(T) ∩ ℘(Br (x0 )).

Prove that there exists x∗ ∈ Dom(T) such that Tx∗ = θ∗ . 11. Let X be a real reflexive strictly convex and smooth Banach space. Let T : Dom(T) = X → X ∗ be a be a hemicontinuous and monotone mapping. Suppose that there exist some r > 0 and x0 ∈ X such that ⟨Tx, x − x0 ⟩ ≥ 0,

for all x ∈ X ∩ ℘(Br (x0 )).

Then, prove that there exists x∗ ∈ Br (x0 ) satisfying Tx∗ = θ∗ . 12. Let X be a real reflexive strictly convex and smooth Banach space. Let T : Dom(T) = X → X ∗ be a be a hemicontinuous φ-strongly monotone mapping. Show that Ran(T) = X ∗ . 13. Let X = Lp , 1 < p ≤ 2, and A : X → X ∗ be a L-Lipschitz continuous η-strongly monotone mapping. Define a sequence {xn } iteratively in X by x1 ∈ X,

{

xn+1 = J −1 (Jxn − tn Axn ),

n ≥ 1,

(GSDM)

where {tn } is a sequence in (0,1) satisfying conditions: (i) tn → 0 (n → ∞), (ii) ∑∞ n=1 tn = ∞. Prove that the sequence {xn } defined by (GSDM) converges strongly to the unique solution of the equation Ax = θ∗ . 14. Let X = Lp , 2 < p < ∞, and A : X → X ∗ be a L-Lipschitz continuous η-strongly monotone mapping. Assume that there exists k ∈ (0, 1) such that p

⟨Ax − Ay, x − y⟩ ≥ k‖x − y‖ p−1 ,

∀x, y ∈ X.

6.7 Exercises | 351

Define a sequence {xn } iteratively in X by x1 ∈ X,

{

xn+1 = J −1 (Jxn − tn Axn ),

n ≥ 1,

(GSDM)

where {tn } is a sequence in (0,1) satisfying conditions: (i) tn → 0 (n → ∞), (ii) ∑∞ n=1 tn = ∞. Prove that the sequence {xn } defined by (GSDM) converges strongly to the unique solution of the equation Ax = θ∗ . 15. Let X be a 2-uniformly convex, uniformly smooth Banach space whose duality mapping J is weakly sequentially continuous, and C is a nonempty, closed, and convex subset of X. Assume that A is an α-inverse strongly monotone mapping such that SOL(C, A) ≠ 0 and satisfies condition: ‖Ay‖ ≤ ‖Ay − Au‖ for all y ∈ C and u ∈ SOL(C, A). Let {λn } be a sequence in [a, b] for some a, b with 0 < a < b < c2 /2, where 1/c is the 2-uniform convexity constant of X. Define a sequence {xn } iteratively in C by x1 ∈ C,

{

xn+1 = ΠC J −1 (Jxn − tn Axn ),

n ≥ 1,

(GPSDM)

where ΠC is the generalized projection from X onto C and J is the duality mapping from X into X ∗ . Prove that the sequence {xn } defined by (GPSDM) converges weakly to some element z in SOL(C, A), furthermore, z = limn→∞ ΠSOL(C,A) xn . 16. Let X be a real, reflexive, locally uniformly convex, and smooth Banach space. Let A ⊂ X × X ∗ be a maximal monotone mapping. Let {αn }, {βn }, {βn󸀠 }, and {an } be four sequences of positive numbers satisfying the following conditions: (i) limn→∞ αn = 0; (ii) βn󸀠 = o(αn ); (iii) βn = αn−1 βn󸀠 , for all n ≥ 1; ‖xn ‖ + r ≤ an ≤ ‖xn ‖ + δ, n ≥ 1, where r and δ are arbitrary positive integers, while {xn } is a sequence defined in such a pattern: x1 ∈ X,

{

xn+1 = (αn J + A)−1 J(βn xn ),

n ≥ 1.

(GRIA)

Prove that the sequence {xn } defined by (GRIA) converges strongly to the minimum-norm solution of the equation θ ∈ Ax ⇐⇒ A−1 (θ) ≠ 0 ⇐⇒ {xn } is bounded.

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Index 2-uniformly convex 39 2-uniformly smooth 46 A1 space 13 A2 space 13 A3 space 13 CQ-method 135 F differentiable 34 F differential 34 g-strongly accretive 59, 75 g-strongly pseudocontractive 75 G derivative 34 G differential 34 k-strictly pseudocontractive 75, 185 L-Lipschitz 3, 54 m-accretive 59 p-uniformly convex 38 T-bounded 27 T1 space 12 T1 topological space 12 T2 space 12 T3 space 12 T4 space 12 v-inverse-strongly accretive 59 v-inverse-strongly monotone 63 w-bounded 22 W -mapping 113, 242 w ∗ -bounded 22 w ∗ -closed convex subset 37 α-component 13 α-projective 13 β-strongly pseudocontractive 75 η-strongly accretive 59 η-strongly monotone 63 η-strongly pseudocontractive 185 φ-strongly monotone 63 λ-averaged nonexpansive 81 ωR -strongly monotone 65 ρ-contractive mapping 115 absolutely convex 17 absorbing set 16 accretive 59 accumulation point 6, 7 add operation 27 additive commutative group 14 adjoint operator 18 adjoint space 18

Alaoglu theorem 22 all continuous 11 approximable 23 approximation proximity point algorithm (APPA) 288 asymptotic center 47, 88 asymptotic radius 47 asymptotically regular 56 averaged 54 averaged nonexpansive 54 axiom A1 12 axiom A2 12 balance closure 16 balance convex set 17 balance hull 16 balanced set 16 Banach contraction mapping principle 3 Banach limit 49 Banach space 17 Banach–Alaoglu theorem 28 Banach–Caristzi theorem 209 Banach–Picard iteration 82 Banach–Picard iterative method 93, 132 BIS 232 boundary point 6 bounded 3, 32 bounded functional 27 bounded inverse operator 18 bounded linear functional 28 bounded linear mapping 18 bounded linear operator 18 Brouwer fixed point theorem 28 Browder–Göhed’s–Caristi theorem 210 Browder–Kirk–Göhde fixed point theorem 52 Browder’s convergence theorem 93 Browder’s existence theorem 57 Brower fixed-point theorem 56 Bruck lemma 248 Bruck regularization iterative method 198 Bruck regularization iterative method (BRIM) 281 Byrne’s CQ-method 135 canonical embedding 20 Cantor’s intersection theorem 3 cardinality 15

360 | Index

Caristi fixed point theorem 4 Cauchy sequence 3 Cauchy–Schwarz inequality 24 chain 1 chain rule 34 closed ball 2 closed half-space 19, 26 closed hyperplane 19 closed rectangle 26 closed subspace 20, 23 closed unit ball 22 closure point 6 coarser (weaker or smaller) topology 5 coenvelope 29 coercive 67 cofinal directed subset 10 compact 7 compact space 7 comparable 1 complete 3 complete normed linear space 17 complete set 6 composite function 11 composite mapping 14 conjugate function 30 conjugate space 28 constant sequence 7 continuous 3, 32 continuous function 11 contractive 3 contractive mapping 110 convergent 3 convergent sequence 7 convex 16 convex function 30 convex hull 16 convex minimization problem 131 convex optimization problem 162 countability axiom 12 countably compact 7 cover 7 Deimling theorem 268 demiclosed principal II theorem 316 demiclosedness principle 56, 91 demicontinuous 33, 144 dense set 6 dense subset 6 dense topology 5

derived set 6 diameter 47 diameter point 47 differentiable 34 dimension 15 direct sum 15 directed point sequence 10 directed set 10 directed subset 10 discrete topology 5 distance 2 domain 3 dual space 20 duality mapping 41 Eberlein–Smulian theorem 22 eigenvalue 65 epigraph 29 exterior point 6 extreme point set 28 factor set 13 Fejér monotone 58 Fejér monotone sequence 93 Figiel constant 45 finer (stronger or larger) topology 5 finite intersection property 7 finite set 21 firmly nonexpansive 51, 54, 81 fixed point 3 fixed point theorem 91 flow invariance condition 61 flow-invariance condition (FIC) 265 Fréchet derivative 34 Fréchet differential 34 fundamental system of neighborhoods 9 Gâteaux derivative 33 Gâteaux differential 33 Gauge function 40 generalized duality mapping 45 generalized nonexpansive 51 generalized projection 50 generalized pseudomonotone 63 generalized weak inward condition 119 gradient mapping 33 greatest element 1 GRHIM 339 GRMIM 335

Index |

Hahn–Banach extension theorem 18 half-space 19 Halpern iterative method 93, 288 Halpern iterative method (HIM) 220 Halpern–Rockafellar iterative method (HRIM) 289 Halpern–Suzuki iterative algorithm (HSIM) 222 Halpern’s property 233 Hausdorff space 12 hemicontinuous 33, 144 Hilbert space 24 homeomorphism 11 homeomorphism mapping 11 hybrid projection algorithm 116 hybrid steepest descent method 132 hyperplane 19, 26 identity mapping 34 index set 5 indicator function 31 injective 42 inner product 23 inner product space 23 interior point 6 inward condition 119 Ishikawa iterative method 193 Ishikawa theorem 213 isolated point 6 isomorphic 27 James theorem 20 Kadec–Klee property 40, 334 Kartsatos theorem 268 Kato lemma 263 Kirk’s existence theorem 57 Kobayashi theorem 269 Krasnosel’skiǐ–Man iterative method 93 Krein–Milman theorem 28 least element 1 Lebesgue measurable set 34 left-limit 64 level set 29 limit of net 10 linear base 15 linear bijection 20 linear closure 15 linear combination 15

361

linear correlation 15 linear ergodic theorem 51 linear isomorphism 15 linear manifold 15 linear mapping 17 linear operation 18 linear space 14 linear subspace 15 linear sum 15 linearly correlative 15 linearly dependent 15, 24 Lipcshitz coefficient 58 local base 9 locally bounded 32, 66 locally convex 27 locally uniformly convex 37 lower bound 1 lower semicontinuous 29 Mann iterative method 191 Mann–Rockafellar iterative method 288 Mann–Rockafellar iterative method (MRIM) 289 mapping 3 Martin theorem 268 maximal accretive 59 maximal element 1 maximal independent set 15 maximal monotone 63 maximum open set 8 Meir–Keeler contractive 3 Meir–Keeler contractive mapping 228 Meir–Keeler fixed point theorem 4 Meir–Keeler theorem 209 metric projection 26, 50 metric space 2 metrizable 12, 23 MHRIM-2 295 minimal element 1 minimum balance set 16 minimum closed set 8 minimum convex set 16 minimum-norm element 23 Minty’s Lemma 148 MK 209 modulus of convexity 38 monotone 63 Moreau–Rockafellar theorem 37 Moudafi–Halpern iterative method 93 MRHIM 345

362 | Index

MT condition 254 multiplication operation 27 MXIM 227

pseudocontractive 75, 185 pseudocontractive mapping 53 pseudomonotone 64

neighborhood 5, 9 neighborhood base 9 net 10 net convergence 10 non-diameter point 47 nonexpansive 3, 50, 81 nonexpansive contract 50 nonexpansive retraction 50 nonlinear semigroup 53 nonspreading 124 nontrivial quotient space 23 nontrivial subspace 19 norm 17 norm topology 21 normal 12 normal cone 36, 66 normal duality mapping 41 normal space 12 normal structure 48 normed linear space 17 normed space 17 nowhere normal-outward condition 119

quasi-convex 30 quasi-nonexpansive 54, 124

open ball 2 open cover 7 open half-space 19 open mapping 20 Opial condition 24 orthogonal 24 orthogonal complement 24 parallelogram law 24 partial order relation 1 partially ordered set 1 Petryshyn inequality 42, 288 pointwise convergence 51 polarization Identity 24 positive operator 34 potential 33 preimage set 11 product set 13 product topology 13 projected gradient method 132 proper 29 proper subspace 15

radius 2 range condition 60 Ray theorem 267 reflexive 20 regular 12 regular space 12 regularity condition 37 Reich corollary 218 Reich inequality 45, 281 Reich theorem 211 relative topology 5 resolvent 60 retract 50 retraction 50 RI 226 right-limit 64 Rothe’s condition 119 scalar multiplication 14 Schauder fixed point theorem 28 Schrödinger equation 58 second category set 20 Senter and Dotson theorem 213 separable 6 separate 19 separating hyperplane theorem I 19 separating hyperplane theorem II 19 separation theorem 28 sequentially compact 7 singleton 47 smooth 43 smoothness module 44 spectral radius 134 sphere 2 split feasibility problem 134 standard embedding 20 steepest descent method 191 strictly contractive 54 strictly convex 19, 30, 37 strictly monotone 63 strictly separate 19 strong convergence theorem 98

Index |

strong topology 21 strongly closed 22 strongly continuous 33 strongly nonexpansive 54 subbase 9 subcover 7 subdifferentiable 35 subdifferential 35 subgradient 35 subnet 10 subsequence 7 subspace 5 sum 14 sunny generalized nonexpansive 52 sunny nonexpansive projection 52 sunny nonexpansive retraction 51 support function 32 supporting hyperplane 37 surjective 42 Suzuki lemma 77 Suzuki theorem 231, 234 symmetric set 16 tangent slopes 35 the Bishop–Phelps theorem 20 the closed graph theorem 20 the first fixed point theorem 52 the mean ergodic theorem 51 the monotone method 52 the open mapping theorem 20 the Pythagorean theorem 24 the resonance theorem 20 the set of asymptotic centers 88 topological base 9 topological conjugate 53 topological invariant property 11 topologically complementary subspace 27 topology 5 topology dual space 18 topology space 5 totally ordered set 1 trivial subspace 15

363

Tychonoff fixed point theorem 28 Tychonoff theorem 14 uniformly continuous 32 uniformly convex 30, 37, 43 uniformly F -differentiable 43 uniformly G-differentiable 43 uniformly normal structure 48 uniformly smooth 43 unit spheres 20 upper bound 1 upper semicontinuous 64 usual maximum norm 38 variational inequality problem 66, 132 VBIS 231 VBRIM 286, 287 vector 14 vector space 14 viscosity Bruck regularization iterative method (VBRIM) 325 weak convergence theorem 93 weak inward condition 75, 119 weak lower semicontinuity 156 weak topology 21 weak∗ compact 22 weakest topology 13 weakly closed 22 weakly compact 22 weakly continuous 33 weakly convergent sequence 23 weakly inward condition (WIC) 209 weakly sequentially compact 22 XIA-2 226 XRI 284 Xu theorem 224 Yosida approximation 60, 69 Zorn lemma 2

Nomenclature (S, ⪯) (X, T) (X, ζ ) (X, d) (X, ‖ ⋅ ‖) ‖⋅‖ ∩ ∪ dim X 0 ⟨⋅, ⋅⟩ ⟨X, ⪯⟩ ⇔ lim limt→0+ limt→0− lim inf ℕ ℝ+ ℝ T μn ∇f ωw (xn ) ℝ A 𝜕A 𝜕φ ΠC £ ⪯ ρX ⇒ → ⇀ σ(X, X ∗ ) s 󳨀→

directed set product topology space topological space X with topology ζ metric space X with distance d normed linear space with norm ‖ ⋅ ‖ norm intersection union the dimension of a vector space X empty set inner product partially ordered set X with partial order relation ⪯ if and only if limit the right limit of t → 0 the left limit of t → 0 inferior limit the natural numbers (0, +∞) the real numbers product topology the Banach limit the gradient mapping of f the weak limit set of {xn } [−∞, +∞] the closure of A the boundary of A the subdifferential of φ the generalized projection onto C the system of neighbourhoods partial ordered relation the smooth module of X imply the strong convergence the weak convergence the weak topology on X ∗ the strong convergence

󳨀→ w 󳨀→

the weak∗ convergence the weak convergence

w∗

https://doi.org/10.1515/9783110667097-008

366 | Nomenclature τs (X) grad f ℘(A) ζ0 ζA ζ∞ {a} {xz }z∈S A(C, {xn }) A∗ A0 Ad Ae Ab At Br (x) Br [x] cl A co(A) Dom(A) epi(f ) ext K f −1 Fix(T) Graph(A) I IC Int A J Jt Jφ jφ (x) Jq M⊥ N(X) NC (x) PC r(C, {xn }) Ran(A) Ran(T) S(X) Sr (x)

the strong topology on X the gradient mapping of f the complementary set of A dense topology relative topology discrete topology constant sequence of a net the asymptotic centre of {xn } on C the adjoint operator of A the set of all interior points of A the derived set of A the set of all exterior points of A the balance closure of A the Yosida approximation of A the open ball with center x and radius r the closed ball with center x and radius r the closure of A the convex hull of A the effective domain of A the epigraph of f extreme point set of K preimage of f the fixed points set of T the graph of A the identity mapping the indicator function of C the set of all interior points of A the normal duality mapping the resolvent the duality mapping with gauge function φ any element of Jφ (x) the generalized duality mapping the orthogonal complement of M the normal structure coefficient of X the normal cone of C at x the metric projection onto C the asymptotic radius of {xn } on C the range of A the range of T the unit spheres of X the sphere with center x and radius r

Nomenclature

x⊥y X∗ X ∗∗ X1 + X2 X1 ⊕ X2 B inf LIM max min sup

x and y are orthogonal, i. e., ⟨x, y⟩ = 0 the dual space of X the dual space of X ∗ linear sum of X1 and X2 direct sum of X1 and X2 with X1 ∩ X2 = {0} topological base infimum the Banach limit maximum minimum supremum

| 367