Applied Analysis, Optimization and Soft Computing: ICNAAO-2021, Varanasi, India, December 21–23 9819905966, 9789819905966

Table of contents :
Contents
About the Editors
Fixed Point Theory
Large Contractions and Surjectivity in Banach Spaces
1 Introduction and Preliminaries
2 A Surjectivity Theorem for Single-valued Large Contractions
3 Multivalued Large Contractions
References
On Hick's Contraction Using a Control Function
1 Introduction and Mathematical Preliminaries
2 Main Results
3 Conclusion
References
Coupled Fixed Points for Multivalued Feng–Liu-Type Contractions with Application to Nonlinear Integral Equation
1 Introduction and Mathematical Background
2 Main Results
3 Some Consequences
4 Application to Nonlinear Integral Equations
References
Fractals
Clifford-Valued Fractal Interpolation
1 Introduction
2 A Brief Introduction to Clifford Algebra and Analysis
3 Some Results from Fractal Interpolation Theory
4 Clifford-Valued Fractal Interpolation
5 Paravector-Valued Functions
6 Brief Summary and Further Research Directions
References
Optimal Quantizers for a Nonuniform Distribution on a Sierpiński Carpet
1 Introduction
2 Preliminaries
3 Optimal Sets of n-means for All n2
4 Some Results and Observations
References
Fractal Dimension for a Class of Complex-Valued Fractal Interpolation Functions
1 Introduction
1.1 Preliminaries
1.2 Fractal Interpolation Functions
1.3 α-Fractal Functions
2 Main Theorems
References
A Note on Complex-Valued Fractal Functions on the Sierpiński Gasket
1 Introduction
2 Technical Introduction
3 Fractal Interpolation Function on SG
4 Some Results Associated with the Fractal Operator
5 Conclusion
References
Dimensional Analysis of Mixed Riemann–Liouville Fractional Integral of Vector-Valued Functions
1 Introduction
1.1 Motivation
1.2 Delineation
2 Notations and Prelude
2.1 Fractal Dimension
2.2 Variation of a Function
2.3 Fractal Dimension and Bounded Variation
2.4 Mixed Riemann–Liouville Fractional Integral
3 Few Fundamental Concepts on Dimension of the Graph of a Vector-Valued Function
4 Riemann–Liouville Fractional Integral and Fractal Dimension
5 Conclusion and Future Scope
References
Fractional Operator Associated with the Fractal Integral of A-Fractal Function
1 Introduction and Preliminaries
2 Fractal Interpolation Function
2.1 Hidden Variable A-Fractal Function
3 Fractal Integral on A-Fractal Function
4 Fractional Operator
References
Mathematical Modeling
A Multi-strain Model for COVID-19
1 Introduction
2 Single-Strain Model
2.1 Controlled Reproduction Number
3 Two-Strain Model
3.1 Effective Reproduction Number
3.2 Multi-strain Model
3.3 Numerical Example
4 Model Validation with COVID-19 Data
5 Discussion
References
Effect of Nonlinear Prey Refuge on Predator–Prey Dynamics
1 Introduction
2 Mathematical Model Formulation with Constant Refuge
3 Mathematical Model Formulation with Nonlinear Refuge
4 Model Comparison
5 Discussion and Conclusion
References
Effects of Magnetic Field and Thermal Conductivity Variance on Thermal Excitation Developed by Laser Pulses and Thermal Shock
1 Introduction
2 Laser Pulse Heat Source
3 Basic Equations Using Dual-Phase Lag Thermoelasticity
4 Mathematical Modelling of the Problem
5 Initial and Boundary Conditions
6 Solution in Laplace Transform Domain
7 Quantitative Results
7.1 Effects of Magnetic Field
7.2 Effects of Changing Thermal Conductivity
8 Conclusion
References
Differential and Integral Equations
On Unique Positive Solution of Hadamard Fractional Differential Equation Involving p-Laplacian
1 Introduction
2 Preliminaries
3 Existence and Uniqueness
4 Conclusion
References
Eigenvalue Criteria for Existence and Nonexistence of Positive Solutions for α-Order Fractional Differential Equations on the Half-Line (2

Citation preview

Springer Proceedings in Mathematics & Statistics

Tanmoy Som · Debdas Ghosh · Oscar Castillo · Adrian Petrusel · Dayaram Sahu   Editors

Applied Analysis, Optimization and Soft Computing ICNAAO-2021, Varanasi, India, December 21–23

Springer Proceedings in Mathematics & Statistics Volume 419

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Tanmoy Som · Debdas Ghosh · Oscar Castillo · Adrian Petrusel · Dayaram Sahu Editors

Applied Analysis, Optimization and Soft Computing ICNAAO-2021, Varanasi, India, December 21–23

Editors Tanmoy Som Department of Mathematical Sciences Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh, India Oscar Castillo Research Chair of Graduate Studies Tijuana Institute of Technology Tijuana, Mexico

Debdas Ghosh Department of Mathematical Sciences Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh, India Adrian Petrusel Faculty of Mathematics and Computer Science Babes-Bolyai University Cluj-Napoca, Romania

Dayaram Sahu Department of Mathematics Banaras Hindu University Varanasi, India

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-99-0596-6 ISBN 978-981-99-0597-3 (eBook) https://doi.org/10.1007/978-981-99-0597-3 Mathematics Subject Classification: 90-xx, 37-xx, 03E72, 00A71, 68Qxx © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

Fixed Point Theory Large Contractions and Surjectivity in Banach Spaces . . . . . . . . . . . . . . . . M˘ad˘alina Moga and Adrian Petru¸sel

3

On Hick’s Contraction Using a Control Function . . . . . . . . . . . . . . . . . . . . . Vandana Tiwari, Binayak S. Choudhury, and Tanmoy Som

13

Coupled Fixed Points for Multivalued Feng–Liu-Type Contractions with Application to Nonlinear Integral Equation . . . . . . . . . . . . . . . . . . . . . Binayak S. Choudhury, N. Metiya, S. Kundu, and P. Maity

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Fractals Clifford-Valued Fractal Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter R. Massopust

33

Optimal Quantizers for a Nonuniform Distribution on a Sierpinski ´ Carpet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mrinal Kanti Roychowdhury

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Fractal Dimension for a Class of Complex-Valued Fractal Interpolation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manuj Verma, Amit Priyadarshi, and Saurabh Verma

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A Note on Complex-Valued Fractal Functions on the Sierpinski ´ Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Agrawal and T. Som

79

Dimensional Analysis of Mixed Riemann–Liouville Fractional Integral of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Megha Pandey, Tanmoy Som, and Saurabh Verma

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Contents

Fractional Operator Associated with the Fractal Integral of A-Fractal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 T. M. C. Priyanka and A. Gowrisankar Mathematical Modeling A Multi-strain Model for COVID-19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Samiran Ghosh and Malay Banerjee Effect of Nonlinear Prey Refuge on Predator–Prey Dynamics . . . . . . . . . . 143 Shilpa Samaddar, Mausumi Dhar, and Paritosh Bhattacharya Effects of Magnetic Field and Thermal Conductivity Variance on Thermal Excitation Developed by Laser Pulses and Thermal Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Rakhi Tiwari Differential and Integral Equations On Unique Positive Solution of Hadamard Fractional Differential Equation Involving p-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Ramesh Kumar Vats, Ankit Kumar Nain, and Manoj Kumar Eigenvalue Criteria for Existence and Nonexistence of Positive Solutions for α-Order Fractional Differential Equations on the Half-Line (2 < α ≤ 3) with Integral Condition . . . . . . . . . . . . . . . . . 183 Abdelhamid Benmezai, Souad Chentout, and Wassila Esserhan A Collocation Method for Solving Proportional Delay Riccati Differential Equations of Fractional Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Basharat Hussain and Afroz Afroz On the Solution of Generalized Proportional Hadamard Fractional Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Rahul and N. K. Mahato Optimization Theory and Applications An Invitation to Optimality Conditions Through Non-smooth Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Joydeep Dutta Solving Multiobjective Environmentally Friendly and Economically Feasible Electric Power Distribution Problem by Primal-Dual Interior-Point Method . . . . . . . . . . . . . . . . . . . . . . 259 Jauny, Debdas Ghosh, and Ashutosh Upadhayay Optimization Methods Using Music-Inspired Algorithm and Its Comparison with Nature-Inspired Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 271 Debabrata Datta

Contents

vii

On Mathematical Programs with Equilibrium Constraints Under Data Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Vivek Laha and Lalita Pandey A New Approach to Solve Fuzzy Transportation Problem . . . . . . . . . . . . . 301 Ashutosh Choudhary and Shiv Prasad Yadav The Best State-Based Development of Fuzzy DEA Model . . . . . . . . . . . . . . 315 Anjali Sonkariya and Shiv Prasad Yadav Performance Evaluation of DMUs Using Hybrid Fuzzy Multi-objective Data Envelopment Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Awadh Pratap Singh and Shiv Prasad Yadav Development of Intuitionistic Fuzzy Data Envelopment Analysis Model Based on Interval Data Envelopment Analysis Model . . . . . . . . . . . 345 Meena Yadav and Shiv Prasad Yadav Pricing Policy with the Effect of Fairness Concern, Imprecise Greenness, and Prices in Imprecise Market for a Dual Channel . . . . . . . . 357 Sanchari Ganguly, Pritha Das, and Manoranjan Maiti Fuzzy Set Theory A Similarity Measure of Picture Fuzzy Soft Sets and Its Application . . . . 381 V. Salsabeela and Sunil Jacob John Soft Almost s-Regularity and Soft Almost s-Normality . . . . . . . . . . . . . . . . 391 Archana K. Prasad and S. S. Thakur Algebraic Properties of Spherical Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . 403 P. A. Fathima Perveen and Sunil Jacob John Divergence Measures of Pythagorean Fuzzy Soft Sets . . . . . . . . . . . . . . . . . 411 T. M. Athira and Sunil Jacob John Fuzzy-Rough Optimization Technique for Breast Cancer Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 K. Anitha and Debabrata Datta

About the Editors

Tanmoy Som is Professor and former Head of the Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, India. He completed his PhD from Institute of Technology, Banaras Hindu University, India, in 1986. He also has served as the Head of the Department of Mathematics at Assam Central University, Silchar. His research interests are in functional analysis, optimization and mathematical modelling, fuzzy set theory, soft computing, and image processing. He has successfully guided more than 16 PhD students and published more than 100 papers and several book chapters in reputed national and international journals, proceedings, and edited books. He has authored several papers on metric fixed point theory, optimization modelling and recently on applied analysis, soft computing, and fuzzy geometry. He has completed a BRNS-BARC-funded project titled “Fractional Calculus Approached Solutions for Two-Dimensional Ground Water Contaminations in Unsaturated Media” during 2014-18 jointly with Prof. S. Das of IIT (BHU). He is an editorial board member and reviewer of a few reputed journals including IEEE Transactions, American Mathematical Society, and one of the volume editors of the book Mathematics and Computing: ICMC 2018, Varanasi, India, January 9–11 (Springer 2018). He has been Guest/Handling Editor of the International Journal of Forensic and Community Medicine. He has delivered several invited talks at national and international conferences, seminars, refresher courses, etc., and has organized three international events as organizing chair. He is the Vice-President of Calcutta Mathematical Society—one of the oldest mathematical societies in India. He has made short academic visits to Texas A&M University, Kingsville (2012), and the University of California, Berkeley (2016). Debdas Ghosh is Assistant Professor at the Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, India. He earned his PhD degree in mathematics from the Indian Institute of Technology (IIT) Kharagpur. He completed his MSc from IIT Kharagpur in 2009 and BSc from Ramakrishna Mission Vidyamandira, Belur Math, University of Calcutta, Kolkata, India. He is a recipient of the Professor J. C. Bose Memorial Gold Medal from IIT Kharagpur (2009). Dr Ghosh has ix

x

About the Editors

been awarded the Outstanding Potential for Excellence in Research and Academics Award (2014) from BITS-Pilani (Hyderabad Campus), where he worked as Assistant Professor of mathematics, for the period June 2014–June 2016. His broad research interest includes optimization theory, fuzzy geometry, computational multiobjective optimization, and robust optimization. With more than 40 papers and 12 conference papers, he has published five papers on fuzzy geometrical ideas on plane and space. He has authored/edited several books: An Introduction to Analytical Fuzzy Plane Geometry, Mathematics and Computing (Springer), and A Primer on Interval Optimization (all with Springer). Dr Ghosh has completed a research project entitled “On Characterizing and Obtaining the Complete Efficient Solution Set of an Interval Optimization Problem under a DDominance and a Variable Dominance Structure”, funded by Science and Engineering Research Board, India. He is now handling a research project entitled “On Developing Polynomial-time Interior-Point Methods for Robust Multiobjective Convex Optimization Problems”, funded by Science and Engineering Research Board, India. Oscar Castillo is Professor of Computer Science at the Graduate Division, Tijuana Institute of Technology, Tijuana, Mexico. In addition, he is serving as Research Director of Computer Science and head of the research group on Hybrid Fuzzy Intelligent Systems. Currently, he is President of Hispanic American Fuzzy Systems Association (HAFSA) and Past President of International Fuzzy Systems Association (IFSA). He holds a Doctor of Science degree (Doctor Habilitatus) in Computer Science from the Polish Academy of Sciences (with the dissertation “Soft Computing and Fractal Theory for Intelligent Manufacturing”). Professor Castillo is also Chair of the Mexican Chapter of the Computational Intelligence Society (IEEE). He also belongs to the Technical Committee on Fuzzy Systems of IEEE and to the Task Force on “Extensions to Type-1 Fuzzy Systems”. He is Member of NAFIPS, IFSA, and IEEE. He belongs to the Mexican Research System (SNI Level 3). His research interests are in type-2 fuzzy logic, fuzzy control, neuro-fuzzy, and genetic-fuzzy hybrid approaches. He has published over 300 papers in several journals, authored 10 books, edited 50 books, more than 300 papers in conference proceedings, and more than 300 chapters in edited books; in total, more than 1000 publications according to Scopus (H index = 66), and more than 1200 publications according to Google Scholar (H index = 80). He has been Guest Editor of several successful special issues of the following journals: Applied Soft Computing, Intelligent Systems, Information Sciences, Non-Linear Studies, Fuzzy Sets and Systems, JAMRIS, and Engineering Letters. He is currently Associate Editor of the Information Sciences Journal, Engineering Applications of Artificial Intelligence Journal, Complex and Intelligent Systems Journal, Granular Computing Journal, and the International Journal on Fuzzy Systems. Finally, he has been elected IFSA Fellow in 2015 and MICAI Fellow member in 2017. He has been recognized as a Highly Cited Researcher in 2017 and 2018 by Clarivate Analytics because of having multiple highly cited papers in the Web of Science.

About the Editors

xi

Adrian Petrusel is Professor at the Department of Mathematics, Babe¸s-Bolyai University Cluj-Napoca, Romania (since 2003). He has also been Visiting Researcher at the University of Seville, Spain (2004); Visiting Professor/Researcher at National Sun Yat-sen University, Kaohsiung, Taiwan (2007, 2016–2019); and Visiting Professor at King Saud University, Riyadh, Saudi Arabia (2015–2016). He earned his PhD degree from Babe¸s-Bolyai University Cluj-Napoca, Romania (1994). His areas of research include fixed point theory, differential equations, and multivalued analysis. He is the author of six books and more than 200 research papers in reputed journals, which have over 2500 citations and H-index 23, according to the Web of Science. He received the “Babes-Bolyai” University Prize in 2002 for his books: Fixed Point Theory (1950–2000): Romanian Contributions and was recently awarded the distinction “Bologna Professor”—AOSR, 2021. He is an invited researcher at Sevilla University, Spain (2003); Valencia University, Spain (2004); Chiang Mai University, Chiang Mai, Thailand; and King Mongkut’s University of Technology Thonburi, Bangkok, Thailand (2012) and National Sun Yat-sen University, Kaohsiung, Taiwan (2015-19). He has been an expert member of The International Research Grants body in Taiwan (2015-19). He is the Editor-in-Chief of the following journals: Fixed Point Theory, Fixed Point Theory and Algorithms for Science and Engineering, and Studia Univ. Babe¸s-Bolyai Mathematica. Moreover, he is on the editorial board of the journals: Studia Universitatis Babe¸s-Bolyai, Mathematica, FILOMAT, Miskolc Math. Notes, Applicable Analysis, and Discrete Mathematics, Discrete Dynamics in Nature and Society; Linear and Nonlinear Analysis, The Journal of Nonlinear Sciences and its Applications; Applied Analysis and Optimization; Journal of Nonlinear and Variational Analysis; Mathematical Analysis and Convex Optimization; Journal of Nonlinear Analysis and Optimization: Theory and Applications; International Journal of Mathematical Sciences; and Mathematica (Cluj). Dayaram Sahu is Senior Professor at the Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India. Earlier, he was Associate Professor and the Head of the Department of Applied Mathematics, at Shri Shankaracharya College of Engineering and Technology, Junwani, Bhilai, India. He completed his PhD degree from Ravishankar Shukla University, Raipur, India, in 1996. He has 25 years of teaching experience at undergraduate and graduate levels and research. His research interests include fixed point theory, computational operator theory, variational inequality theory, computational convex optimization, Newtonlike methods, and image processing. He has supervised nine research scholars for their PhD degrees, and other six students are pursuing their PhD. He is on the editorial board of several international journals including Fixed Point Theory, Fixed Point Theory and Algorithms for Science and Engineering, and Journal of Applied and Numerical Optimization. He also is a reviewer of many reputed journals. He has authored one book and published more than 140 research papers (over 2833 citations, H-index 22; to date). He has completed two SERC Fast Track projects as Principal Investigator. He has delivered several invited talks at several international conferences: some of them are ICFPTA-2012 in Taiwan (2012),

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About the Editors

ICFPTA-2019 in China (July 2019), and ICTP-2002 in Italy. He has visited for academic works IISc Bangalore (2002), India; South Korea (2008), Taiwan (2018); Romania (2018); Guru Ghasidas University, Chhattisgarh, India (2018); and Jamia Millia Islamia, New Delhi (2019).

Fixed Point Theory

Large Contractions and Surjectivity in Banach Spaces M˘ad˘alina Moga

and Adrian Petru¸sel

Abstract In 1996, T.A. Burton proposed a new concept of contraction-type mapping by introducing the notion of large contraction. In his paper, a fixed-point result for a single-valued large contraction in a complete metric space is given and some applications to integral equations are deduced. In this paper, we will continue the study of the above-mentioned mappings in the context of a Banach space X . More precisely, we will show that any large contraction t : X → X is a norm-contraction in the sense of A. Granas. Then, as an application, a surjectivity theorem for the field 1 X − t generated by t is proved. In the second part of this work, we extend the concept of large contraction to the multivalued case and we prove fixed-point theorems for multivalued large contractions T : X → P(X ) in a Banach space X . Additionally, some surjectivity results for the field 1 X − T generated by the multivalued operator T are given. The results of this paper extend and complement several fixed-point theorems and surjectivity results in the recent literature. Keywords Banach space · Large contraction · Multivalued large contraction · Fixed point · Surjectivity theorem

1 Introduction and Preliminaries Let X be a nonempty set and t : X → X be an operator. We will denote by Fi x(t) := {x ∈ X |x = t (x)} M. Moga · A. Petru¸sel (B) Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University Cluj-Napoca, Kog˘alniceanu Street, No. 1, Cluj-Napoca, Romania e-mail: [email protected] M. Moga e-mail: [email protected] A. Petru¸sel Academy of Romanian Scientists, Splaiul Independentei no. 54, Bucharest, Romania © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_1

3

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M. Moga and A. Petru¸sel

the fixed-point set of t. We denote by t n := t ◦ t ◦ · · · ◦ t for n-times, the nth iterate of t. The sequence (xn )n∈N defined by xn+1 = t (xn ), n ∈ N (i.e., xn = t n (u), u ∈ X and n ∈ N∗ ) is the sequence of Picard iterates for t starting from u ∈ X . Let (X,  · ) be a Banach space. For given u ∈ X and r > 0, we denote by  B(u; r ) := {x ∈ X | x − u ≤ r } the closed ball of radius r centered at u. By P(X ), we will denote the set of all nonempty subsets of X. We will also use the following notations: Pb (X ) := {A ∈ P(X )|A is bounded } , Pcp (X ) := {A ∈ P(X )|A is compact } , Pcl (X ) := {A ∈ P(X )|A is closed } , Pcv (X ) := {A ∈ P(X )|A is convex } . Definition 1 (Granas–Dugundji [5]) Let X, Y be two normed spaces. Then: (i) the operator t : X → Y is called compact if its range t (X ) is contained in a compact subset of Y (or equivalently, the range t (X ) is relatively compact). (ii) the operator t : X → Y is called completely continuous if t continuous and t (A) is relatively compact, for each A ∈ Pb (X ). Definition 2 (see, e.g., Rus [10]) Let (X,  · ) be a normed space. An operator t : X → X is called quasi-bounded if there exist two numbers a, b ∈]0, ∞[ such that t (x) ≤ ax + b, for all x ∈ X. (1) In the above context, if t is a quasi-bounded operator, then the quasi-norm of t is defined by |t| = inf {a > 0| there exists b > 0 such that the relation (1) holds} . Definition 3 (see, e.g., Rus [10]) Let (X,  · ) be a normed space and t : X → X be a quasi-bounded operator. If the quasi-norm of t is strictly less than one, then t is called norm-contraction. In 1996, T.A. Burton proposed a new concept of contraction-type mapping by introducing the notion of large contraction. In his paper, a fixed-point result for a single-valued large contraction in a complete metric space is given and some applications to integral equations are deduced. In this paper, we will continue the study of the above-mentioned mappings in the context of a Banach space X . More precisely, we will show that any large contraction t : X → X is a norm-contraction in the sense of A. Granas. Then, as an application, a surjectivity theorem for the field 1 X − t generated by t (here, 1 X denotes the identity operator of the Banach space X ) is proved. In the second part of this work, we extend the concept of large contraction to the multivalued case and we prove fixed-point theorems for multivalued large contractions T : X → P(X ) in a Banach space X . Additionally, some surjectivity results

Large Contractions and Surjectivity in Banach Spaces

5

for the field 1 X − T generated by the multivalued operator T are given. The results of this paper extend and complement fixed-point and surjectivity theorems (see [8, 9, 11]), as well as other related results (see lemma in [1]), in the recent literature.

2 A Surjectivity Theorem for Single-valued Large Contractions In this section, we will prove the property that the field 1 X − t is surjective, where t is a single-valued large contraction. The proof is based on T.A. Burton’s large contraction principle [1] and a fixed-point theorem of Granas [4]. Theorem 1 (Granas [4]) Let (X,  · ) be a Banach space and g : X → X be a complete continuous operator. If, additionally, g is a norm-contraction, then Fi x(g) = ∅. The following notion was introduced by T.A. Burton in 1996, in the frame of a metric space. We recall the definition in the context of a Banach space. Definition 4 (Burton [1]) Let (X,  · ) be a normed space. An operator t : X → X is said to be a large contraction if 1. t is contractive, i.e., t (x) − t (y) < x − y, ∀x, y ∈ X, with x = y, 2. for each ε > 0, there exists δ(ε) ∈]0, 1[ such that x, y ∈ X, x − y ≥ ε ⇒ t (x) − t (y) ≤ δ(ε)x − y Jachymski in [7] noted that the contractive condition in the above definition can be avoided and, as a consequence, an operator t : X → X is a large contraction if, for each ε > 0, there exists δ(ε) ∈]0, 1[ such that x, y ∈ X, x − y ≥ ε ⇒ t (x) − t (y) ≤ δ(ε)x − y. For related equivalences involving classes of generalized contractions, see [7]. The following concept, introduced by M.A. Krasnoselskii, is related to the above definition. Definition 5 (M.A. Krasnoselskii, see e.g., Chen [3]) Let (X,  · ) be a Banach space. An operator t : X → X is called a generalized contraction if for any 0 < a < b < ∞, there exists δ(a, b) ∈]0, 1[ such that t (x) − t (y) ≤ δ(a, b)x − y, for all x, y ∈ X satisfying a < x − y < b. On the one hand, a large contraction is a generalized contraction, see [3]. On the other hand, there exist large contractions which are not (Banach) contractions. The main fixed-point theorem for large contractions was given by T.A. Burton.

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Theorem 2 (Burton [1]) Let (X, d) be a complete metric space and t : X → X be a large contraction. Suppose there exist x ∈ X and L > 0, such that d(x, t n (x)) ≤ L for all n ≥ 1. Then t has a unique fixed point in X. As a consequence of the above theorem, we obtain the following result. Corollary 1 Let (X, d) be a complete metric space and t : X → X an operator for which there exists n 0 ∈ N, n 0 ≥ 2 such that t n 0 is a large contraction. Suppose there is an x ∈ X and L > 0 such that d(x, t n (x)) ≤ L for all n ≥ 1. Then t has a unique fixed point. Proof From Burton’s Theorem 2, we have that Fi x(t n 0 ) = {x ∗ }. Then, by x ∗ = t n 0 (x ∗ ), we obtain that t (x ∗ ) = t n 0 (t (x ∗ )), which implies that t (x ∗ ) ∈ Fi x(t n 0 ). Thus, x ∗ = t (x ∗ ), i.e., x ∗ ∈ Fi x(t). In order to obtain the uniqueness of the fixed point, we suppose, by contradiction, that there exists y ∗ ∈ X such that y ∗ = x ∗ and y ∗ ∈ Fi x(t). This implies that y ∗ ∈ Fi x(t n 0 ). Now, the uniqueness of the fixed point of t follows by uniqueness of the fixed point of t n 0 . Hence, Fi x(t) = {x ∗ }. The following example is related to some notions and results from the paper of Burton and Purnaras [2]. Let K be a closed interval of the real axis and SK be the set of continuous functions defined on K . Then (SK ,  · ) is a normed space, where  ·  is the sup-norm. Let g : [a, b] → R be a given function. We consider the subspace M ⊂ SK defined by M := {x ∈ SK : a ≤ x(s) ≤ b, s ∈ K } .

(2)

Notice that (M,  · ) is a complete metric space. Consider now the operator h : M → SK defined by h(x)(s) = g(x(s)), s ∈ K .

(3)

The following characterization theorem was given in [2]. Theorem 3 (Burton–Purnaras [2]) Let t : [a, b] → R and g : [a, b] → [a, b], g(x) = x − t (x), x ∈ [a, b]. Then h defined by (3) is a large contraction on M if and only if t satisfies the relation 0
0, x ∈ [0, 1]. 2e

Moreover, t  is nonnegative and increasing on I . Thus, 0 ≤ t  (0) ≤ t  (x) ≤ t  (1) = 1 ≤ 2, x ∈ I It follows that t satisfies the relation (4). From Theorem 3, we get that the function h(x) := x −

e x (x 2 − x + 1) , x ∈ [0, 1] 2e

is a large contraction on I . We observe that h is not a contraction on I since h  (x) = 1 −

xe x (1 + x) and h  (0) = 1. 2e

We will present now another relevant example of large contraction, using the concept of Meir–Keeler operator. Recall that if (X,  · ) is a normed space, then t : X → X is called a Meir–Keeler operator if for every ε > 0 there is δ > 0, such that x, y ∈ X, ε ≤ x − y < ε + δ ⇒ t (x) − t (y) < ε. It is easy to see that any Meir–Keeler operator is contractive. Moreover, the following result of Suzuki is well known. Theorem 4 (Suzuki [12]) Let (X,  · ) be a Banach space and C be a nonempty and convex subset of X . Let t : C → C be a Meir-Keeler operator. Then, for each ε > 0, there exists rε ∈]0, 1[ such that x − y ≥ ε implies t (x) − t (y) ≤ rε x − y. By the above considerations, one can conclude that any self-Meir–Keeler operator on a nonempty and convex subset of a Banach space is a large contraction. The first main result of this section is the following surjectivity theorem. Theorem 5 Let (X,  · ) be a Banach space and the operator t : X → X satisfying the following assumptions:

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(i) t is a large contraction; (ii) t (A) is relatively compact, for each A ∈ Pb (X ). Then the field 1 X − t : X → X generated by t is a surjective operator. Proof From the definition, it is clear that 1 X − t is continuous. We will prove first that t is a norm-contraction. Let u ∈ X and r > 0. Let us consider the closed ball  B(u; r ) ⊂ X . Two cases will be taken into account: (a) We will consider first x ∈  B(u; r ). Then, we have t (x) ≤ t (  B(u; r )). (b) Let x ∈ X \  B(u; r ). Then, we have that t (x) ≤ t (x) − t (u) + t (u). By the large contraction definition, since x − u ≥ ε, there exists δ < 1 such that t (x) − t (u) ≤ δx − u. Thus, we have t (x) ≤ δx − u + t (u) ≤ δx + δu + t (u). If we denote b := δu + t (u) > 0, then we get t (x) ≤ δx + b, for each x ∈ X \  B(u; R). Hence, in both cases, we have that   t (x) ≤ δx + max t (  B(u; r )), δu + t (u) , for each x ∈ X . Thus, t is a norm-contraction. To prove that 1 X − t is surjective we have to show that for any y ∈ X exists x ∈ X such that (1 X − t)(x) = y. Thus, we should prove that, for every y ∈ X , the equation x = t (x) + y has at least one solution x ∈ X . For the above conclusion, it is sufficient to prove that for each y ∈ X , the set Fi x(g y ) is nonempty, where g y : X → X is given by g y (x) = t (x) + y. Let y ∈ X . Then, because t is complete continuous, it follows that g y is complete continuous. Moreover, because t is norm-contraction, we immediately obtain that g y is a norm-contraction, too. Thus, from Theorem 1, we have that Fi x(g y ) = ∅. Thus, the operator 1 X − t is surjective and the proof is complete.

Large Contractions and Surjectivity in Banach Spaces

9

3 Multivalued Large Contractions Let (X,  · ) be a normed space. We recall first some necessary notations and notions, which will help us prove the main result of the section (see, e.g., [9, 11]). (1) The distance from a point a ∈ X to a set B ∈ P(X ) D(a, B) = inf {a − b|b ∈ B} . (2) The excess of a set A over a set B from X ρ(A, B) = sup {D(a, B)|a ∈ A} . (3) The Pompeiu–Hausdorff distance between two sets A and B from X H (A, B) = max {ρ(A, B), ρ(B, A)} . (4) The diameter between two sets A and B from X Δ(A, B) = sup {a − b|a ∈ A, b ∈ B} , If T : X → P(X ) is a multivalued operator, then its fixed-point set is denoted by Fi x(T ) := {x ∈ X |x ∈ T (x)}, while the graph of T is the set Graph(T ) := {(x, y) ∈ X × X : y ∈ T (x)}. The following notion is important in our approach. Definition 6 (Iannacci [6]) Let (X,  · ) be a Banach space and T : X → Pb (X ). Then the multivalued operator T is called quasi-bounded if there exists m, M ∈]0, ∞[ such that y ≤ mx + M, for each (x, y) ∈ Graph(T ). (5) The quasi-norm of a multivalued quasi-bounded operator T is defined by |T | = inf {m > 0| there exists M > 0 such that the relation (5) holds} . If the quasi-norm of T is less than one (i.e., |T | < 1), then T is called a multivalued norm-contraction. In the above setting, we will denote T (x) := H (T (x), {0}), for any x ∈ X. Definition 7 A multivalued operator T : X → P(X ) is called completely continuous if T is upper semicontinuous on X and the set T (A) is relatively compact, for each A ∈ Pb (X ). The following result was proved in 1978 by R. Iannacci.

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Theorem 6 (Iannacci [6]) Let (X,  · ) be a Banach space and let T : X → Pcv (X ) be a multivalued a completely continuous operator. Suppose that T is multivalued norm-contraction. Then, the field 1 X − T generated by T is surjective. Recall also that, in the same setting, a multivalued operator T : X → P(X ) is called contractive (see, e.g., [13]) if H (T (x), T (y)) < x − y, for all x, y ∈ X, x = y. Definition 8 Let (X,  · ) be a Banach space. Then T : X → P(X ) is said to be a multivalued large contraction if for all ε > 0, exists δ(ε) ∈]0, 1[ such that x, y ∈ X, x − y ≥ ε ⇒ H (T (x), T (y)) ≤ δ(ε)x − y. It is easy to see that a multivalued large contraction is contractive. Theorem 7 Let (X,  · ) be a Banach space and let T : X → Pcp,cv (X ) be a multivalued operator satisfying the following assumptions: 1. T is a multivalued large contraction; 2. T (U ) is relatively compact, for each U ∈ Pb (X ). Then, the field 1 X − T generated by T is surjective. Proof We will prove first that T is a multivalued norm-contraction on X . Let u ∈ X and r > 0. Let us consider the closed ball  B(u; r ). Two cases will be taken into account: (a) We take x ∈  B(u; r ). Then, we have T (x) ≤ T (  B(u; r )). (b) Let us consider now x ∈ X \  B(u; r ). Then, we have T (x) = H (T (x), {0}) ≤ H (T (x), T (u)) + H (T (u), {0}). Now, we can take into account the fact that T is a multivalued large contraction. Then, for ε > 0, there exists δ ∈ (0, 1) such that H (T (x), T (u)) ≤ δx − u ≤ δ (x + u) . As a consequence, we get T (x) ≤ δ (x + u) + T (u). Hence, for all x ∈ X , we get that   T (x) ≤ δx + max T (  B(u; r )), δu + T (u) .

Large Contractions and Surjectivity in Banach Spaces

11

Therefore, T is a multivalued norm-contraction. Moreover, since any contractive operator with compact values is upper semicontinuous, we can apply the above Theorem 6 and we get that 1 X − T is surjective. Remark 1 It is an open question to prove a fixed-point theorem for multivalued large contractions in complete metric spaces. Concerning the above open problem, we can prove the following partial answer. Theorem 8 Let (X, d) be a complete metric space and T : X → Pcl (X ) be a multivalued operator. Suppose that the following assertions hold: (i) H (T (x), T (y)) < d(x, y), for every distinct elements x, y ∈ X ; (ii) for every ε > 0, there exists δ ∈]0, 1[ such that x, y ∈ X, x − y ≥ ε ⇒ Δ(T (x), T (y)) ≤ δx − y. (iii) there exist K > 0, x ∈ X and a sequence {xn }n∈N of Picard type iterates for T starting from x0 := x (i.e., xn+1 ∈ T (xn ), for every n ∈ N), such that d(x, xn ) ≤ K , for every n ∈ N∗ . Then: (a) T has at least one fixed point in X ; (b) if, instead of (i), we suppose that (i) Δ(T (x), T (y)) < d(x, y), for every x, y ∈ X, x = y, then the fixed point is unique. Proof (a) For x ∈ X , we consider the sequence {xn }n∈N of Picard-type iterates for T starting from x0 := x. Suppose that this sequence is not Cauchy. Then, there exist ε > 0, 0 < Nk  ∞ and there exists m k , n k > Nk . m k > n k such that d(xm k , xn k ) ≥ ε. Then, by (i), we get ε ≤ d(xm k , xn k ) ≤ d(xm k −1 , xn k −1 ) ≤ · · · ≤ d(x, xm k −n k ). Thus, for these distances, using (ii) and then (iii), we obtain ε ≤ d(xm k , xn k ) ≤ δd(xm k −1 , xn k −1 ) ≤ · · · ≤ δ n−k K . Since δ < 1, these relations yields a contradiction. As a consequence, we obtain that the given sequence {xn }n∈N of Picard-type iterates for T starting from x0 := x is Cauchy. Hence, the sequence {xn }n∈N converges to an element x ∗ ∈ X . We will show that x ∗ is a fixed point for T . Indeed, since xn+1 ∈ T (xn ) and T has a closed graph (being contractive), we obtain that x ∗ ∈ T (x ∗ ).

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(b) The uniqueness of the fixed point follows by (i)’. Indeed, if x ∗ and x˜ are two distinct fixed points of T , then we can write ˜ ≤ Δ(T (x ∗ ), T (x)) ˜ < d(x ∗ , x), ˜ d(x ∗ , x) a contradiction.

References 1. Burton, T.-A.: Integral equations, implicit functions, and fixed points. Proc. Am. Math. Soc. 124(8), 2383–2390 (1996) 2. Burton, T.-A., Purnaras, I.-K.: Necessary and sufficient conditions for large contractions in fixed point theory, Electron. J. Qual. Theory Differ. Equ. 94, 1–24 (2019) 3. Chen, Y.-Z.: Inhomogeneous iterates of contraction mappings and nonlinear ergodic theorems. Nonlinear Anal. 39(1), 1–10 (2000) 4. Granas, A.: On a certain class of nonlinear mappings in Banach space. Bull. Acad. Pol. Sci. 9, 867–871 (1957) 5. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003) 6. Iannacci, R.: The spectrum for nonlinear multi-valued maps via approximations, Boll. Un. Mat. Ital. 15-B, 527–545 (1978) 7. Jachymski, J, Jó´zwik, I.: Nonlinear contractive conditions: a comparison and related problems. In: Fixed Point Theory and its Applications, Banach Center Publ. vol. 77, pp. 123–146 (2007) 8. Moga, M.: Meir-Keeler operators and applications to surjectivity theorems. J. Nonlinear and Convex Anal. 23(3), 625–634 (2022) 9. Petru¸sel, G.: Generalized multivalued contractions which are quasi-bounded. Demonstratio Math. 40, 639–648 (2007) 10. Rus, I.-A.: Normcontraction mappings outside a bounded set, Itinerant Seminar on Functional Equations Approximation and Convexity, Cluj-Napoca, pp. 257–260 (1986) 11. Rus, I.-A., Petru¸sel, A., Petru¸sel, G.: Fixed point theorems for set-valued Y-contractions, In: Fixed Point Theory and its Applications, Banach Center Publ. vol. 77, pp. 227–237 (2007) 12. Suzuki, T.: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 325, 342–352 (2007) 13. Xu, H.K.: Metric fixed point theory for multivalued mappings. Diss. Math. vol. 389 39 pp. (2000)

On Hick’s Contraction Using a Control Function Vandana Tiwari, Binayak S. Choudhury, and Tanmoy Som

Abstract In this paper, we use L-convergence criteria to establish a Hick’s type contraction mapping theorem in different probabilistic metric spaces. A theorem is established by using the control function which is a recent introduction in literature and is a generalization of many other such functions. The fixed point obtained in our theorem is unique. Keywords Menger space · Fixed point · ϕ-contraction

1 Introduction and Mathematical Preliminaries Metric space was probabilistically generalized in the work of K. Menger [1] as early as 1942. Its theory has developed over the years in different directions. An accent of these development has been described in Schweizer and Sklar [3]. Several aspects of this study still continue to be developed. Metric fixed point theory was extended to the probabilistic metric spaces by Sehgal and Bharucha-reid [2], after which many researchers have worked on this research area which has resulted in a very large literature making probabilistic fixed point theory into a subject by itself. The structure of probabilistic metric spaces being inherently flexible, there has been an extension of several results obtained on metric spaces in many ways. In particular, there is an extension of Banach’s contraction which is given by Hicks [4] and is very V. Tiwari · T. Som (B) Department of Mathematical Sciences, Indian Institute of Technology (B.H.U.), Varanasi 221005, U.P., India e-mail: [email protected] V. Tiwari Department of Mathematics, Gandhi Smarak PG College, Samodhpur, Jaunpur 223102, Uttar Pradesh, India B. S. Choudhury Department of Mathematics, Indian Institute of Engineering Sciences and Technology, Shibpur, Howrah 711103, West Bengal, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_2

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different from the form considered by Sehgal et al. [2]. This contraction is called Hick’s contraction or C-contraction which along with its several modifications and generalizations have been introduces in various research papers. In probabilistic fixed point theory, a new domain of study started with the introduction of control functions by Choudhury et al. [12]. This parallels a corresponding development in metric spaces which was initiated by Khan et al. [12] and elaborated through several works [5–7]. Other types of control functions have been used by several authors like Ciric [8], Fang [12], etc. Particularly, the result of Fang [12] is a culmination of a trend of development in this life. In the present work, we use another control function for obtaining a generalized C-contraction result in probabilistic metric spaces. The control function is different from that of Fang [12] and its use has warranted a new method of proof of the corresponding fixed point result. In the following, we dwell upon some aspects of the probabilistic metric space and some other concepts which are required for further discussion. Definition 1 ([1]) A distribution function is defined as a left-continuous and nondecreasing mapping F : R → [0, 1] with inf F (x) = 0. If F (0) = 0, then F is x∈R

termed as a distance distribution function. Definition 2 ([1]) Menger distance distribution function is defined as a distance distribution function F with lim F (t) = 1. The collection of all Menger distance t→∞

distribution function is denoted by D + . Here the space D + is partially ordered via the usual pointwise ordering of functions, that is, F ≤ G if and only if G (t) ≥ F (t) for all t ∈ [0, ∞]. Distance distribution function H is the maximal element for D + , given by  0 if t = 0 H (t) = 1 if t > 0. Definition 3 ([3]) A t-norm  is defined as a binary operation on [0, 1] if 1.  (a, 1) = a for all a ∈ [0, 1] 2.  is associative and commutative, 3. For b ≤ d, a ≤ c and for each a, b, c, d ∈ [0, 1] , we have  (c, d) ≥  (a, d). Some examples are the following: (a) The product t-norm,  P , such that  P (a, b) = a.b, are two basic t−norms; (b) The minimum t-norm,  M , such that  M (a, b) = min {a, b} are two basic tnorms. Generalization of metric spaces is known as Menger probabilistic metric spaces (Menger space), which was given in 1942 by Menger [1]. Definition 4 ([1, 3]) Let  be a t-norm and X be a nonempty set, then the triplet (X, F, ) is defined as a Menger space. Here F : X × X → D + satisfies the following ( F (x, y) is denoted by Fx,y for x, y ∈ X ):

On Hick’s Contraction Using a Control Function

15

(PM-1) Fx,y (t) = H (t) for all t > 0, x, y ∈ X if and only if x = y, (PM-2) Fy,x  all t > 0 and y, x ∈ X ,  (t) = Fx,y (t) for (PM-3)  Fx,z (t), Fz,y (s) ≤ Fx,y (t + s) for all t, s > 0, x, y, z ∈ X . Definition 5 ([3, 17]) 1. If lim Fxn ,x (t) = 1, for all t > 0, then we say that the sequence (xn ) in (X, F, ) n→∞ converges to x ∈ X , written as xn → x. 2. Sequence (xn ) in (X, F, ) is said to be a Cauchy sequence if for any given λ ∈ (0, 1] and ε > 0, there exists k ∈ N, depending on ε, λ, such that Fxn ,xm (ε) > 1 − λ, whenever m, n ≥ k. 3. If each Cauchy sequence (xn ) in X is convergent to some point x ∈ X, then the Menger PM-space (X, F, ) is said to be complete. Definition 6 The class of function  consists of all ϕ : R+ → R+ such that there exists r ≥ t with lim ϕn (r ) = 0 for each t > 0 . For example ϕ : [0, ∞) → [0, ∞), n→∞ ⎧ 4 ⎪ ⎨t − 3 , if 2 < t < ∞, such that ϕ (t) = −t + 43 , if 1 ≤ t ≤ 2, 3 ⎪ ⎩ t , if 0 ≤ t < 1. 1+t We define the following convergence criteria for the Menger space. Definition 7 A Menger space (X, F, ) satisfies L−convergence criteria if for a sequence (xn ) and a sequence of positive real numbers {tn }, Fxn ,xn+1 (tn ) → 1 as n → ∞ implies that (xn ) is convergent. Lemma 1 If ϕ ∈ . then there exists r ≥ t such that ϕ(r ) < t for each t > 0.

2 Main Results Theorem 1 Suppose (X, F, ) is a Menger space. Let  be a continuous tnorm satisfying L−convergence. Further, let T : X → X be a probabilistic ϕC−contraction, that is Fx,y (t) > 1 − t ⇒ FT x,T y (ϕ(t)) > 1 − kt, for all t > 0 and x, y ∈ X,

(1)

where ϕ ∈  and 0 < k < 1. Then x∗ ∈ X is a unique fixed point of T and also {T n (x0 )} converges to x∗ for any arbitrary x0 ∈ X . Proof For any x0 ∈ X,. we write xn = T n x0 = T xn−1 for all n ≥ 1. Let 0 < η < 1. For Fx0 ,x1 (t) → 1 as t → ∞, there exists r > 0, such that Fx0 ,x1 (r ) > 1 − η.

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Then from (1), FT x0 ,T x1 (ϕ(r )) > 1 − kη, that is, Fx1 ,x2 (ϕ(r )) > 1 − kη. Continuing in this way, we obtain Fx2 ,x3 (ϕ2 (r )) > 1 − k 2 η. In general, ∀ n ∈ N, we have Fxn ,xn+1 (ϕn (r )) > 1 − k n η. Again, Fx0 ,x1 (t) → 1 as t → ∞, for any ε ∈ (0, 1]. Hence, there exists t1 > 0 such that Fx0 ,x1 (t1 ) > 1 − ε. Here ϕ ∈ . Therefore, there exists t0 ≥ t1 such that ϕn (t0 ) → 0 as n → ∞. (2) ϕn (t0 ) → 0 as n → ∞. Now, Fx0 ,x1 (t0 ) ≥ Fx0 ,x1 (t1 ) > 1 − ε. This implies that FT x0 ,T x1 (ϕ(t0 )) > 1 − kε > 1 − ε, that is, Fx1 ,x2 (ϕ(t0 )) > 1 − kε > 1 − ε, Continuing in similar manner, we obtain Fxn ,xn+1 (ϕn (t0 )) > 1 − ε.

(3)

Now, from (2) and (3), by L-convergence criteria, xn becomes a convergent sequence. Let (4) xn → x as n → ∞. Let ε > 0. For ϕ ∈ , ∃ r ≥ ε with ϕ(r ) < ε. Now Fx,T x (ε) ≥ (Fx,xn (ε − ϕ(r )), Fxn ,T x (ϕ(r )). As {xn } converges to x,, we have that as n → ∞ Fxn−1 ,x (r ) → 1 .

(5)

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Thus, for arbitrary 0 < λ < 1, we obtain N1 such that for all n > N1 , Fxn−1 ,x (r ) > 1 − λ, which implies that FT xn−1 ,T x (ϕ(r )) = Fxn ,T x (ϕ(r )) > 1 − kλ > 1 − λ.

(6)

Again, as {xn } converges to x, it is possible to find N2 such that for all n > N2 , we obtain (7) Fxn ,x (ε − ϕ(r )) > 1 − λ. We choose N = max(N1 , N2 ). Then, ∀ n > N , from (5)–(7) we get Fx,T x (ε) ≥ (1 − λ, 1 − λ). Here,  is continuous t-norm and λ is arbitrary, hence for any ε > 0 we find, Fx,T x (ε) = 1, thus, x = T x. Next, to establish the uniqueness of the fixed point, let x and y be any two fixed points of T, that is, y = T y and x = T x. There exists t0 > 0 such that Fx,y (t0 ) > 1 − ε because Fx,y (t) → 1 as t → ∞, for any ε ∈ (0, 1]. Since ϕ ∈ , therefore, as n → ∞, there exists t1 ≥ t0 , such that ϕn (t1 ) → 0. Let t > 0 be arbitrary. Then we can obtain n 0 ∈ N, such that ϕn (t1 ) < t for all n ≥ n 0 . Thus, by monotonicity of the distribution function, we get Fx,y (ϕn (t1 )) ≤ Fx,y (t).

(8)

Now, Fx,y (t1 ) ≥ Fx,y (t0 ) > 1 − ε. This implies that Fx,y (t1 ) = FT x,T y (ϕ(t1 )) > 1 − kε, that is, Fx,y (ϕ(t1 )) > 1 − kε, Continuing in a similar manner, we obtain Fx,y (ϕn (t1 )) > 1 − k n ε.

(9)

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Now, (8) and (9) give Fx,y (t) > 1 − k n ε, for all n ≥ n 0 .

(10)

Letting n → ∞, Fx,y (t) = 1 for all t > 0, that is, x = y. For ϕ(x) = kx such that 1 > k > 0, we have the following result. Corollary 1 Suppose (X, F, ) is a Menger space. Here  denotes a continuous t-norm with 0 < k < 1 and L-convergence criteria. Let T : X → X be such that for all 0 < λ < 1, r > 0 and y, x ∈ X, 1 − λ < Fx,y (r ) =⇒ 1 − kλ < FT x,T y (kr ).

(11)

Then the mapping T has a unique fixed point.

3 Conclusion Incidentally, we may also mention that there is no unique way of defining probabilistic metrics. This flexibility makes it possible to suitably orient the definition to fulfil specific purposes as, for instance, in [20], the definition has been tailored to describe a nuclear fusion-related problem. In particular, fixed point-related studies have required considerations of t-norm, the different choices of which radically alter the characteristics of the space. It may be seen in future work how our result presented here is varied with different choices of t-norms. Acknowledgements The third author’s research is supported by the University Grant Commission (No. 19-06/2011(i)EU-IV), India. The research work of the second author is supported by DST-WB, India (624(sane)/ST/P/S & T/MISC-5/2012/Dated 27.08.2013).

References 1. Meneger, K.: Statistical metrics. Proc. Nat. Acad. Sci. USA 28, 535–537 (1942) 2. Sehgal, V.M., Bharucha-Reid, A.T.: Fixed points of contraction mappings on PM-spaces. Math. Syst. Theory 6, 97–102 (1972) 3. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier, New York (1983) 4. Hicks, T.L.: Fixed point theory in probabilistic metric spaces, Zb. Rad. Prir. Mat. Fak. Univ. Novom Sadu 13, 63–72 (1983) 5. Hicks, T.L.: Fixed point theory in probabilistic metric spaces II. Math. Japon- ica 44(3), 487– 493 (1996) 6. Mihet, D.: Generalized Hicks contractions: an extension of a result of Žiki´c. Fuzzy Sets Syst. 157, 2384–2393 (2006) 7. Mihe¸t, D.: Weak-Hicks contractions, 6, 71–78 (2005) 8. Ciric, L.: Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces,. Nonlinear Anal. 72, 2009–2018 (2010)

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9. Jachymski, J.: On probabilistic ϕ-contractions on Menger spaces. Nonlinear Anal. 73, 2199– 2203 (2010) 10. Schweizer, B., Sklar, A., Thorp, E.: The metrization of statistical metric spaces. Pac. J. Math. 10, 673–675 (1960) 11. O. Hadzic, Some theorems on the fixed points in probabilistic metric and random normed spaces. Boll. Unione Mat. Ital. Sez. B 1(6), 381–391 (1982), 12. Fang, J.-X.: On ϕ-contractions in probabilistic and fuzzy metric spaces. Fuzzy Sets Syst. 267, 86–99 (2015) 13. Xiao, J.Z., Zhu, X.H., Cao, Y.F.: Common coupled fixed point results for probabilistic ϕcontractions in Menger spaces. Nonlinear Anal. Theory Methods Appl. 74, 4589–4600 (2011) 14. Chauhan, S., Pant, B.D.: Fixed point theorems for compatible and subsequentially continuous mappings in Menger spaces. J. Nonlinear Sci. Appl. 7, 78–89 (2014) 15. Choudhury, B.S., Das, K.: A new contraction principle in Menger spaces. Acta Math. Sin. Engl. Ser 24(8), 1379–1386 (2008) 16. Dutta, P.N., Choudhury, B.S., Das, K.: Some fixed point results in Menger spaces using a control function. Surv. Math. Appl. 4, 41–52 (2009) 17. Kutbi, M.A., Gopal, D., Vetro, C., Sintunavarat, W.: Further generalization of fixed point theorems in Menger PM-spaces. Fixed Point Theory Appl. 2015(1), 32 (2015) 18. Hadzic, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Springer (2001) 19. Samet, B.: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 72, 4508–4517 (2010) 20. Verdoolaege, G., Karagounis, G., Murari, A., Vega, J., Van Oost, G.: JET-EFDA contributors, modeling fusion data in probabilistic metric spaces: applications to the identification of confinement regimes and plasma disruptions. Fusion Sci. Technol. 62, 356–365 (2012)

Coupled Fixed Points for Multivalued Feng–Liu-Type Contractions with Application to Nonlinear Integral Equation Binayak S. Choudhury, N. Metiya, S. Kundu, and P. Maity

Abstract In this paper, we establish existence of the coupled fixed point for setvalued Feng–Liu-type contractions in complete metric spaces under two different sets of conditions. Some consequences are obtained and an application to a nonlinear integral equation is included. Keywords Coupled fixed point · Cyclic admissible mapping · Hausdorff metric · Integral equations AMS Subject Classification 54H10 · 54H25 · 47H10

1 Introduction and Mathematical Background Multivalued nonlinear contractions appeared first in fixed point theory in the work of Nadler [14]. This work was followed by a development of the branch of fixed point theory in the domain of set-valued analysis through works like [8, 11, 12, 15, 17, 21]. Our aim in this paper is to establish the existence of fixed points of a coupled multivalued mapping satisfying a generalized Feng–Liu-type contraction and new admissibility condition which we defined in [3]. Further we apply our coupled fixed point result for solving a system of nonlinear integral equations. B. S. Choudhury · P. Maity Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India e-mail: [email protected] N. Metiya (B) Department of Mathematics, Sovarani Memorial College, Jagatballavpur, Howrah 711408, West Bengal, India e-mail: [email protected]; [email protected] S. Kundu Department of Mathematics, Government General Degree College, Salboni, Paschim Medinipur 721516, West Bengal, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_3

21

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Suppose CL(X ) denotes the collection of all nonempty closed subsets of a metric space (X, ρ). We use following notations and definitions when P, Q ∈ CL(X ): D(u, Q) = inf {ρ(u, v) : v ∈ Q}, where u ∈ X, D(P , Q) = inf {ρ(s, t) : s ∈ P , t ∈ Q}

and  H(P, Q) =

max {sup D( p, Q), sup D(q, P)}, if the infimum exists, p∈P

∞,

q∈Q

otherwise.

The pair (CL(X ), H) is a generalized metric space and H is called the generalized Hausdorff distance [6]. The idea of a coupled fixed point was introduced by Guo and Lakshmikantham [10] in 1987. But only after the publication of the work of Bhaskar and Lakshmikantham [9], a large number of papers have been written on this topic and on topics related to it [2, 3, 5, 17–19]. A coupled fixed point of a mapping S : X × X → X is a point (u, v) ∈ X × X such that u = S(u, v) and v = S(v, u). For a multivalued mapping S : X × X → CL(X ), a coupled fixed point is an element (u, v) ∈ X × X satisfying u ∈ S(u, v) and v ∈ S(v, u) Various admissibility criteria have been introduced in the study of fixed points of mappings. In particular, we refer the reader to [1, 3, 4, 7, 13, 20]. Definition 1.1 ([3])A coupled mapping T : X × X → X is called cyclic (α, β)admissible, where α, β : X → [0, ∞), if for (x, y) ∈ X × X , (i) α(x) ≥ 1 and β(y) ≥ 1 ⇒ β(T (x, y)) ≥ 1, (ii) β(x) ≥ 1 and α(y) ≥ 1 ⇒ α(T (x, y)) ≥ 1. Definition 1.2 ([3])A multivalued coupled mapping T : X × X → CL(X ) is called cyclic (α, β)- admissible, where α, β : X → [0, ∞), if for (x, y) ∈ X × X , (i) α(x) ≥ 1 and β(y) ≥ 1 ⇒ β(u) ≥ 1 for all u ∈ T (x, y), (ii) β(x) ≥ 1 and α(y) ≥ 1 ⇒ α(v) ≥ 1 for all v ∈ T (x, y). Example 1.1 Take the usual metric space X = [0, 1]. Define T : X × X → CL(X ) as T (x, y) = [0, x+y ], for x, y ∈ X and α, β : X → [0, ∞) as 16  α(x) =

e x , if x ∈ [0, 21 ], and β(x) = 0, otherwise



cosh x, if x ∈ [0, 21 ], 0, otherwise.

Suppose that (x, y) ∈ X × X such that α(x) ≥ 1 and β(y) ≥ 1. Then x, y ∈ [0, 21 ] and T (x, y) ⊆ [0, 21 ]. It follows that β(u) ≥ 1 for all u ∈ T (x, y). Similarly, one can show α(v) ≥ 1 for all v ∈ T (x, y) whenever (x, y) ∈ X × X with β(x) ≥ 1 and α(y) ≥ 1. The mapping T is here cyclic (α, β)-admissible.

Coupled Fixed Points for Multivalued Feng–Liu-Type Contractions …

23

Definition 1.3 A function f : X × X → R, where X is a metric space, is said to be lower semi-continuous if for any sequence {(xn , yn )} ⊆ X × X and (x, y) ∈ X × X, (xn , yn ) → (x, y) as n → ∞ implies f (x, y) ≤ lim inf n→∞ f (xn , yn ). By  we denote the class of all functions φb : [0, ∞) → [0, b), 0 < b < 1, satisfying lim supr −→t + φb (r ) < b, for each t ∈ [0, ∞). We use this class of functions in our theorems.

2 Main Results Theorem 2.1 Let (X, d) be a complete metric space and α, β : X → [0, ∞) be two mappings. Let T : X × X → CL(X ) be a continuous and cyclic (α, β)admissible mapping. Suppose that there exist b ∈ (0, 1) and φb ∈  such that for (x, y) ∈ X × X with (x, y) ≥ 1 there exist u ∈ T (x, y) and v ∈ T (y, x) for which b max{d(x, u), d(y, v)} ≤ f (x, y), f (u, v) ≤ φb (max {d(x, u), d(y, v)}) max {d(x, u), d(y, v)},

(2.1)

where f (s, t) = max{D(s, T (s, t)), D(t, T (t, s))} for s, t ∈ X and (x, y) = α(x) β(y) or (x, y) = α(y) β(x). If there exist x0 , y0 ∈ X such that α(x0 ) ≥ 1 and β(y0 ) ≥ 1, then T has a coupled fixed point. Proof If possible, suppose that f (x, y) = max {D(x, T (x, y)), D(y, T (y, x))} = 0 for some (x, y) ∈ X × X . Then D(x, T (x, y)) = 0 and D(y, T (y, x)) = 0, which imply that x ∈ T (x, y) and y ∈ T (y, x), that is, (x, y) is a coupled fixed point of T . Hence we shall assume that f (x, y) = 0 for every (x, y) ∈ X × X . Since b ∈ (0, 1), there exist u ∈ T (x, y) and v ∈ T (y, x) such that b max {d(x, u), d(y, v)} ≤ max {D(x, T (x, y)), D(y, T (y, x))} = f (x, y). (2.2) Starting with x0 , y0 ∈ X for which α(x0 ) ≥ 1 and β(y0 ) ≥ 1, we have (x0 , y0 ) = α(x0 )β(y0 ) ≥ 1. By (2.1) and (2.2), we can choose x1 ∈ T (x0 , y0 ) and y1 ∈ T (y0 , x0 ) such that b max {d(x0 , x1 ), d(y0 , y1 )} ≤ f (x0 , y0 ) and f (x1 , y1 ) ≤ φb (max {d(x0 , x1 ), d(y0 , y1 )}) max {d(x0 , x1 ), d(y0 , y1 )}. By the admissibility assumption of T , we have β(x1 ) ≥ 1 and α(y1 ) ≥ 1. Then (x1 , y1 ) = α(y1 )β(x1 ) ≥ 1. By (2.1) and (2.2), we can choose x2 ∈ T (x1 , y1 ) and y2 ∈ T (y1 , x1 ) such that

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b max {d(x1 , x2 ), d(y1 , y2 )} ≤ f (x1 , y1 ) and f (x2 , y2 ) ≤ φb (max {d(x1 , x2 ), d(y1 , y2 )}) max {d(x1 , x2 ), d(y1 , y2 )}. By the admissibility assumption of T , we have α(x2 ) ≥ 1 and β(y2 ) ≥ 1. Then (x2 , y2 ) = α(x2 )β(y2 ) ≥ 1. By (2.1) and (2.2), we can choose x3 ∈ T (x2 , y2 ) and y3 ∈ T (y2 , x2 ) such that b max {d(x2 , x3 ), d(y2 , y3 )} ≤ f (x2 , y2 ) and f (x3 , y3 ) ≤ φb (max {d(x2 , x3 ), d(y2 , y3 )})max {d(x2 , x3 ), d(y2 , y3 )}. Continuing this process, we construct two sequences {xn } and {yn } in X such that xn+1 ∈ T (xn , yn ) and yn+1 ∈ T (yn , xn ) with (xn , yn ) ≥ 1, for all n ≥ 0. (2.3) Also, ⎫ ⎬

b max {d(xn , xn+1 ), d(yn , yn+1 )} ≤ f (xn , yn ) and f (xn+1 , yn+1 ) ≤ φb (max{d(xn , xn+1 ), d(yn , yn+1 )}) max{d(xn , xn+1 ), d(yn , yn+1 )}.

⎭

(2.4) We shall show that f (xn , yn ) → 0 as n → ∞. Let rn = max {d(xn , xn+1 ), d(yn , yn+1 )}, for all n ≥ 0.

(2.5)

From (2.4), (2.5), we have b rn ≤ f (xn , yn ) and f (xn+1 , yn+1 ) ≤ φb (rn ) rn . Therefore, rn+1 ≤ and

⎫ ⎪ ⎬

f (xn+1 , yn+1 ) φb (rn ) ≤ rn b b

f (xn , yn ) − f (xn+1 , yn+1 ) ≥ b rn − φb (rn ) rn = [b − φb (rn )] rn .

⎪ ⎭

(2.6)

As, φb (rn ) < b, from (2.6), we have rn+1 ≤ and

φb (rn ) rn < rn b

f (xn , yn ) − f (xn+1 , yn+1 ) ≥ [b − φb (rn )] rn > 0.

⎫ ⎪ ⎬ ⎪ ⎭

(2.7)

Coupled Fixed Points for Multivalued Feng–Liu-Type Contractions …

25

It follows that {rn } and { f (xn , yn )} are strictly decreasing sequences of non-negative real numbers. From the property of φb there exists a q ∈ [0, b) such that lim sup φb (rn ) = q. n→∞

Hence, for any b0 ∈ (q, b), there exists n 0 ∈ N such that φb (rn ) < b0 , for all n > n 0 .

(2.8)

Consequently, we have from (2.7) that for all n > n 0 , f (xn , yn ) − f (xn+1 , yn+1 ) ≥ γ rn , where γ = b − b0 .

(2.9)

By (2.6) and (2.8), we have φb (rn ) f (xn , yn ) b φb (rn ) φb (rn−1 ) f (xn−1 , yn−1 ) b2 φb (rn ) φb (rn−1 ) φb (rn−2 ) f (xn−2 , yn−2 ) b3 ... φb (rn ) φb (rn−1 ) φb (rn−2 )...φb (r2 ) φb (r1 ) f (x1 , y1 ) bn   φb (rn 0 ) φb (rn 0 −1 ) ...φb (r1 ) φb (rn ) φb (rn−1 ) ...φb (rn 0 +1 ) f (x , y ) 1 1 bn−n 0 bn 0

n−n 0  b0 φb (rn 0 ) φb (rn 0 −1 ) ...φb (r1 ) f (x , y ) , for all n > n 0 . (2.10) 1 1 b bn 0

f (xn+1 , yn+1 ) ≤ φb (rn ) rn ≤ ≤ ≤ ≤ ≤ =
n > n 0 . By (2.5) and (2.9), we have m−1

m−1 1 d(xm , xn ) ≤ d(x j , x j+1 ) ≤ rj ≤ [ f (x j , y j ) − f (x j+1 , y j+1 )] γ j=n j=n j=n

=

m−1

f (xn , yn ) 1 [ f (xn , yn ) − f (xm , ym )] ≤ γ γ

and d(ym , yn ) ≤

m−1 j=n

d(y j , y j+1 ) ≤

m−1 j=n

rj ≤

m−1 1 [ f (x j , y j ) − f (x j+1 , y j+1 )] γ j=n

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=

1 f (xn , yn ) [ f (xn , yn ) − f (xm , ym )] ≤ . γ γ

Using (2.11), we have limm,n→∞ d(xm , xn ) = 0 and limm,n→∞ d(ym , yn ) = 0. Therefore, both {xn } and {yn } are Cauchy sequences in X . As the metric space X is complete, there exist points x, y ∈ X such that xn → x

and

yn → y, as n → ∞.

(2.12)

Then (xn , yn ) → (x, y) and (yn , xn ) → (y, x), as n → ∞. As T is continuous, we have H(T (xn , yn ), T (x, y)) → 0 and H(T (yn , xn ), T (y, x)) → 0, as n → ∞. Therefore, D(xn+1 , T (x, y)) → 0 and D(yn+1 , T (y, x)) → 0, as n → ∞, that is, D(x, T (x, y)) = 0 and D(y, T (y, x)) = 0. Since T (x, y), T (y, x) ∈ CL(X ), T (x, y) = T (x, y) and T (y, x) = T (y, x), where T (x, y) and T (y, x) denote the closures of T (x, y) and T (y, x), respectively. Now, D(x, T (x, y)) = 0 and D(y, T (y, x)) = 0 imply that x ∈ T (x, y) = T (x, y) and y ∈ T (y, x) = T (y, x), that is, (x, y) is a coupled fixed point of T . In our next result, we take the semi-continuity assumption of the function f (x, y) = max {D(x, T (x, y)), D(y, T (y, x))} instead of taking the continuity assumption on T . Theorem 2.2 Let (X, d) be a complete metric space and α, β : X → [0, ∞) be two mappings. Let T : X × X → CL(X ) be a cyclic (α, β)-admissible mapping. Suppose that there exist b ∈ (0, 1) and φb ∈  such that for (x, y) ∈ X × X with (x, y) ≥ 1 there exist u ∈ T (x, y) and v ∈ T (y, x) for which b max{d(x, u), d(y, v)} ≤ f (x, y) and also (2.1) of Theorem 2.1 is satisfied, where f (s, t) and (x, y) are as given in Theorem 2.1. If f is lower semi-continuous and there exist x0 , y0 ∈ X such that α(x0 ) ≥ 1 and β(y0 ) ≥ 1, then T has a coupled fixed point. Proof Assuming f (x, y) = 0 for every (x, y) ∈ X × X we construct the same sequences {xn } and {yn } as in the proof of Theorem 2.1. Then {xn } and {yn } satisfy (2.3) and (2.4). Like in the proof of Theorem 2.1, we prove that both {xn } and {yn } are Cauchy sequences in X and satisfy (2.12), that is, xn → x and yn → y, as n → ∞. As f is lower semi-continuous, we have 0 ≤ f (x, y) ≤ lim inf n→∞ f (xn , yn ) = 0, that is, f (x, y) = max {D(x, T (x, y)), D(y, T (y, x))} = 0. Then D(x, T (x, y)) = 0 and D(y, T (y, x)) = 0. Arguing similarly as in the proof of Theorem 2.1, we prove that (x, y) is a coupled fixed point of T .

3 Some Consequences Theorem 3.1 Let (X, d) be a complete metric space and T : X × X → CL(X ). Suppose there exist b ∈ (0, 1) and φb ∈  such that for (x, y) ∈ X × X there exist u ∈ T (x, y) and v ∈ T (y, x) for which b max{d(x, u), d(y, v)} ≤ f (x, y),

Coupled Fixed Points for Multivalued Feng–Liu-Type Contractions …

27

f (u, v) ≤ φb (max {d(x, u), d(y, v)}) max {d(x, u), d(y, v)}, where f (s, t) is as defined in Theorem 2.1. Then T has a coupled fixed point if either T is continuous or f is lower semi-continuous. Proof Define α, β : X → [0, ∞) as α(x) = β(x) = 1, for all x ∈ X . Then the proof follows from that of Theorem 2.1 if T is continuous and from that of Theorem 2.2 if f is lower semi-continuous. Theorem 3.2 Let (X, d) be a complete metric space and T : X × X → CL(X ). Suppose there exist b ∈ (0, 1) and φb ∈  such that for all (x, y), (u, v) ∈ X × X , H(T (x, y), T (u, v)) ≤ φb (max {d(x, u), d(y, v)}) max {d(x, u), d(y, v)}. T admits a coupled fixed point if either f (x, y) = max{D(x, T (x, y)), D(y, T (y, x))} is lower semi-continuous or T is continuous. Proof By the condition of the theorem for any u ∈ T (x, y) and v ∈ T (y, x), we have D(u, T (u, v)) ≤ H(T (x, y), T (u, v)) ≤ φb (max {d(x, u), d(y, v)}) max {d(x, u), d(y, v)}

and D(v, T (v, u)) ≤ H(T (y, x), T (v, u)) ≤ φb (max {d(x, u), d(y, v)}) max {d(x, u), d(y, v)}.

Combining these two inequalities, we have f (u, v) = max{D(u, T (u, v)), D(v, T (v, u))} ≤ φb (max{d(x, u), d(y, v)})max {d(x, u), d(y, v)}.

Then the proof follows from that of Theorem 3.1. Theorem 3.3 Let (X, d) be a complete metric space and T : X × X → CL(X ). Suppose there exists b ∈ (0, 1) such that for all (x, y), (u, v) ∈ X × X , H(T (x, y), T (u, v)) ≤

b max {d(x, u), d(y, v)}. 2

If either T is continuous or f (x, y) = max{D(x, T (x, y)), D(y, T (y, x))} is lower semi-continuous then T has a coupled fixed point. b Proof Take b ∈ (0, 1) and φb ∈ , where φb (t) = , for all t ∈ [0, ∞). Then the 2 proof follows from that of Theorem 3.2.

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4 Application to Nonlinear Integral Equations In this section, we present an application of our coupled fixed point results derived in Sect. 2 to solve a nonlinear integral equation. Every single-valued mapping T : X × X → X can be treated as a multivalued mapping T : X × X → CL(X ) in which case T (x, y) is a singleton set for (x, y) ∈ X × X . Taking α(x) = β(x) = 1, for all x ∈ X in Theorem 2.1, we have the following result which is a special case of Theorem 2.1. Theorem 4.1 Let (X, d) be a complete metric space and T : X × X → X be a continuous mapping. Suppose there exist b ∈ (0, 1) and φb ∈  such that for all (x, y) ∈ X × X max {d(T (x, y), T (T (x, y), T (y, x))), d(T (y, x), T (T (y, x), T (x, y)))} ≤ φb (max {d(x, T (x, y)), d(y, T (y, x))}) max {d(x, T (x, y)), d(y, T (y, x))}.

Then T has a coupled fixed point. Fixed point theorems for operators in metric spaces have found applications in differential and integral equations (see [16] and references therein). We consider here a system of nonlinear integral equation as follows: 

b x(t) = f (t) + a h(t, s, x(s), y(s))ds and

b y(t) = f (t) + a h(t, s, y(s), x(s))ds, where t, s ∈ [a, b] (4.1) and the unknown functions x(t) and y(t) are real valued. Let X = C([a, b]), where b > a be the space of all real-valued continuous functions defined on [a, b]. It is well known that C([a, b]) endowed with the metric d(x, y) = max | x(t) − y(t) | t∈[a, b]

(4.2)

is a complete metric space. Define a mapping T : X × X → X by  T (x, y)(t) = f (t) +

b

h(t, s, x(s), y(s))ds, for all t, s ∈ [a, b].

a

We designate the following assumptions by A1 and A2 :

(4.3)

Coupled Fixed Points for Multivalued Feng–Liu-Type Contractions …

29

A1 : f ∈ C([a, b]) and h : [a, b] × [a, b] × R × R → R is a continuous mapping; A2 : | h(t, s, x, y) − h(t, s, u, v) |≤ M(t, s, x, y, u, v), for all (x, y), (u, v) ∈ X × X 1 and for all t, s ∈ [a, b], where M(t, s, x, y, u, v) = [ | x − u | + | y − v | ]. 2 A3 : 2b − a < 1.

Theorem 4.2 Let (X, d) = (C([a, b]), d), T, h(t, s, x, y) satisfy the assumptions A1 , A2 and A3 . Then the system of integral equations (4.1) has a unique solution in C([a, b]) × C([a, b]). Proof Consider the mapping T : X × X → X defined by (4.3). Take c = 2 b − a and φc (t) = b − a, for all t ∈ [0, ∞). Then lim supr −→t + φc (r ) < 1, for all t ∈ [0, ∞). By A3 , φc ∈ . By assumptions A1 and A2 , for all (x, y) ∈ C([a, b]) × C([a, b]) and for u = T (x, y), v = T (y, x) with t, s ∈ [a, b], we have | u(t) − F(u, v)(t) |=| F(x, y)(t) − F(u, v)(t) |=| =

 b a

 b a

[h(t, s, x(s), y(s)) − h(t, s, u(s), v(s))]ds |

| [h(t, s, x(s), y(s)) − h(t, s, u(s), v(s))] | ds

 b 1 ≤ [ [ | x(s) − u(s) | + | y(s) − v(s) | ]ds a 2  d(x, u) + d(y, v) b (b − a) [d(x, u) + d(y, v)] = . 1 ds = 2 2 a

Similarly, we have | v(t) − F(v, u)(t) | = | F(y, x)(t) − F(v, u)(t) |≤ =

(b − a) [d(x, u) + d(y, v)] . 2

(b − a) [d(y, v) + d(x, u)] 2

Combining above two inequalities, we have max {| u(t) − F(u, v)(t) |, | v(t) − F(v, u)(t) |} ≤

(b − a) [d(x, u) + d(y, v)] 2

(b − a) [d(x, u) + d(y, v)] ≤ (b − a) [ max {d(x, u), d(y, v)}] 2 = φc ( max {d(x, u), d(y, v)} ) [max {d(x, u), d(y, v)}],

=

which implies that max {d(T (x, y), T (T (x, y), T (y, x))), d(T (y, x), T (T (y, x), T (x, y)))} = max {d(u, T (u, v)), d(v, T (v, u))} = max {| u(t) − F(u, v)(t) |, | v(t) − F(v, u)(t) |} ≤ φc (max {d(x, T (x, y)), d(y, T (y, x))}) max {d(x, T (x, y)), d(y, T (y, x))}.

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B. S. Choudhury et al.

Therefore, all the conditions of Theorem 4.1 are satisfied. Then there exists a point (x, y) in X × X such that x = T (x, y) and y = T (y, x), that is, (x, y) is a solution of the system of nonlinear integral equations (4.1).

References 1. Alizadeh, S., Moradlou, F., Salimi, P.: Some fixed point results for (α − β) - (ψ − ϕ) - contractive mappings. Filomat 28(3), 635–647 (2014) 2. Choudhury, B.S., Metiya, N., Kundu, S.: Existence, data-dependence and stability of coupled fixed point sets of some multivalued operators. Chaos, Soliton Fractals 133, 109678 (2020) 3. Choudhury, B.S., Gnana Bhaskar, T., Metiya, N., Kundu, S.: Existence and stability of coupled fixed point sets of multi-valued mappings. Fixed Point Theory 22(2), 571–586 (2021) 4. Choudhury, B.S., Metiya N., Kundu, S.: A multivalued fixed point result with associated data dependence and stability study. Prob. Analiza-Issue Anal. 11(1)(29), 45–57 (2022) ´ c, 5. Ciri ´ L.B., Lakshmikantham, V.: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stoch. Anal. Appl. 27(6), 1246–1259 (2009) 6. Covitz, H., Nadler, S.B.: Multi-valued contraction mappings in generalized metric spaces 8(1), 5–11 (1970) 7. Felhi, A., Aydi, H., Zhang, D.: Fixed points for α - admissible contractive mappings via simulation functions. J. Nonlinear Sci. Appl. 9, 5544–5560 (2016) 8. Feng, Y., Liu, S.: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 317, 103–112 (2006) 9. Gnana Bhaskar, T., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006) 10. Guo, D., Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11, 623–632 (1987) 11. Klim, D., Wardowski, D.: Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 334, 132–139 (2007) 12. Laha, A.K., Saha, M.: Fixed point for a class of set valued mappings on a metric space endowed with a graph. ROMAI J. 11(1), 115–129 (2015) 13. Laha, A.K., Saha, M.: Fixed point on α − ψ multivalued contractive mappings in cone metric space. Acta Comment. Univ. Tartu. Math. 20(1), 35–43 (2016) 14. Nadler, S.B., Jr.: Multivalued contraction mappings. Pac. J. Math. 30, 475–488 (1969) 15. Nicolae, A.: Fixed point theorems for multi-valued mappings of Feng-Liu type. Fixed Point Theory 12(1), 145–154 (2011) 16. Nieto, J.J., Rodrguez-Lopez, ´ R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005) 17. Petrusel, A., Petrusel, G., Samet, B., Yao, J.C.: Scalar and vectorial approaches for multi-valued fixed point and multi-valued coupled fixed point problems in b-metric spaces. J. Nonlinear Convex Anal. 10(17), 2049–2061 (2016) 18. Li, J., Petrusel, A.: Extended coupled fixed point problems for set-valued mappings on partially ordered Banach spaces and their applications to systems of Hammerstein integral equations. J. Nonlinear Convex Anal. 11(20), 2321–2333 (2019) 19. Samet, B., Vetro, C.: Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. Nonlinear Anal. 74(12), 4260–4268 (2011) 20. Samet, B., Vetro, C., Vetro, P.: Fixed point theorems for α − ψ-contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012) 21. Zhang, X.: Fixed point theorems of multivalued monotone mappings in ordered metric spaces. Appl. Math. Lett. 23(3), 235–240 (2010)

Fractals

Clifford-Valued Fractal Interpolation Peter R. Massopust

Abstract In this short note, we merge the areas of hypercomplex algebras with that of fractal interpolation and approximation. The outcome is a new holistic methodology that allows the modeling of phenomena exhibiting a complex self-referential geometry and which require for their description an underlying algebraic structure. Keywords Iterated function system (IFS) · Banach space · Fractal interpolation · Clifford algebra · Clifford analysis

1 Introduction In this short note, we merge two areas of mathematics: the theory of hypercomplex algebras as exemplified by Clifford algebras and the theory of fractal approximation or interpolation. In recent years, hypercomplex methodologies have found their way into many applications one of which is digital signal processing. See, for instance, [1, 36] and the references given therein. The main idea is to use the multidimensionality of hypercomplex algebras to model signals with multiple channels or images with multiple color values and to use the underlying algebraic structure of such algebras to operate on these signals or images. The results of these algebraic or analytic operations produce again elements of the hypercomplex algebra. This holistic approach cannot be performed in finite dimensional vector spaces as these do not possess an intrinsic algebraic structure. On the other hand, the concept of fractal interpolation has been employed successfully in numerous applied situations over the last decades. The main purpose of fractal interpolation or approximation is to take into account complex geometric https://www-m15.ma.tum.de/Allgemeines/PeterMassopust. P. R. Massopust (B) Department of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching b. Munich, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_4

33

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self-referential structures and to employ approximants that are well suited to model these types of structures. These approximants or interpolants are elements of vector spaces and cannot be operated on in an algebraic way to produce the same type of object. Hence, the need for an extension of fractal interpolation to the hypercomplex setting. An initial investigation into the novel concept of hypercomplex iterated function system was already undertaken in [32] albeit along a different direction. The structure of this paper is as follows. In Sect. 2, we give a brief introduction of Clifford algebras and mention a few items from Clifford analysis. In the third section, we review some techniques and state relevant results from the theory of fractal interpolation in Banach spaces. These techniques are then employed in Sect. 4 to a Clifford algebraic setting. The next section briefly mentions a special case of Cliffordvalued fractal interpolation, namely that based on paravector-valued functions. In the last section, we provide a brief summary and mention future research directions.

2 A Brief Introduction to Clifford Algebra and Analysis In this section, we provide a terse introduction to the concept of Clifford algebra and analysis and introduce only those items that are relevant to the purposes of this paper. For more details about Clifford algebra and analysis, the interested reader is referred to, for instance, [9, 10, 13, 18, 23, 24] and to, i.e., [11, 12, 19, 22] for its ramifications. To this end, denote by {e1 , . . . , en } the canonical basis of the Euclidean vector space Rn . The real Clifford algebra, Rn , generated by Rn is defined by the multiplication rules (1) ei e j + e j ei = −2δi j , i, j ∈ {1, . . . , n} =: Nn , where δi j is the Kronecker symbol.  x A e A with x A ∈ R and An element x ∈ Rn can be represented in the form x = A

{e A : A ⊆ Nn }, where e A := ei1 ei2 · · · eim , 1 ≤ i 1 < · · · < i m ≤ n, and e∅ =: e0 := 1. Thus, the dimension of Rn regarded as a real vector space is 2n . The rules defined in (1) make Rn into, in general, a noncommutative algebra, i.e., a real vector space together with a bilinear operation Rn × Rn → Rn .  A conjugation on Clifford numbers is defined by x := x A e A where e A := A

eim · · · ei1 with ei := −ei for i ∈ Nn , and e0 := e0 = 1. In this context, one also has (2) e0 e0 = e0 = 1 and e0 ei = ei e0 = ei . The Clifford norm of the Clifford number x =

 A

x A e A is defined by

Clifford-Valued Fractal Interpolation

35

⎛



|x| := ⎝

⎞1/2 |x A |2 ⎠

.

A⊆Nn

In the following, we consider Clifford-valued functions f : G ⊆ Rm → Rn , where G is a nonempty open domain. For this purpose, let X be G or any suitable subset of G. Denote by F(X ) any of the following functions spaces: s (X ), where C k (X ), C k,α (X ), L p (X ), W s, p (X ), B sp,q (X ), F p,q 1. C k (X ), k ∈ N0 := {0} ∪ N, is the Banach space of k-times continuously differentiable R-valued functions; 2. C k,α (X ), k ∈ N0 , 0 < α ≤ 1, is the Banach space of k-times continuously differentiable R-valued functions whose kth derivative is Hölder continuous with Hölder exponent α; 3. L p (X ), 1 ≤ p < ∞, are the Lebesgue spaces on X ; 4. W s, p (X ), s ∈ N or s > 0, 1 ≤ p < ∞, are the Sobolev–Slobodeckij spaces. 5. B sp,q (X ), 1 ≤ p, q < ∞, s > 0, are the Besov spaces; s (X ), 1 ≤ p, q < ∞, s > 0, are the Triebel–Lizorkin spaces. 6. F p,q The real vector space F(X, Rn ) of Rn -valued functions over X is defined by F(X, Rn ) := F(X ) ⊗R Rn . This linear space becomes a Banach space when endowed with the norm ⎛

f := ⎝



⎞1/2

f A 2F (X ) ⎠

.

A⊆Nn

It is known [20, Remark 2.2. and Proposition 2.3.] that f ∈ F(X, Rn ) iff f =



f AeA

(3)

A⊆Nn

with f A ∈ F(X ). Furthermore, functions in F(X, Rn ) inherit all the topological properties such as continuity and differentiability from the functions f A ∈ F(X ).

3 Some Results from Fractal Interpolation Theory In this section, we briefly summarize fractal interpolation and the Read–Bajrakterevi´c operator. For a more detailed introduction to fractal geometry and its subarea of fractal interpolation, the interested reader is referred to the following, albeit incomplete, list of references: [2–6, 8, 14–17, 21, 25, 27, 33, 34].

36

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To this end, let X be a nonempty bounded subset of the Banach space Rm . Suppose N of injective contractions X → X generating a we are given a finite family {L i }i=1 partition of X in the sense that ∀ i, j ∈ N N , i = j : L i (X ) ∩ L j (X ) = ∅; X=

N 

L i (X ).

(4) (5)

i=1

For simplicity, we write X i := L i (X ). Here and in the following, we always assume that 1 < N ∈ N. The purpose of fractal interpolation is to obtain a unique global function ψ:X=

N 

Xi → R

i=1

belonging to some prescribed Banach space of functions F(X ) and satisfying N functional equations of the form ψ(L i (x)) = qi (x) + si (x)ψ(x), x ∈ X, i ∈ N N ,

(6)

where for each i ∈ N N , qi ∈ F(X ) and si : X → R are given functions. In addition, we require that si is bounded and satisfies si · f ∈ F(X ) for any f ∈ F(X ), i.e., si is a multiplier for F(X ). It is worthwhile mentioning that Eq. (6) reflects the self-referential or fractal nature of the global function ψ. The idea behind obtaining ψ is to consider (6) as a fixed point equation for an associated affine operator acting on F(X ) and to show that the fixed point—should it exist—is unique. (Cf. also [33].) For this purpose, define an affine operator T : F(X ) → F(X ), called a Read– Bajractarevi´c (RB) operator, by T f := qi ◦ L i−1 + si ◦ L i−1 · f ◦ L i−1 ,

(7)

on X i , i ∈ N N , or, equivalently, by Tf =

N 

qi ◦

L i−1

i=1

= T (0) +

χXi +

N 

si ◦ L i−1 · f ◦ L i−1 χ X i

i=1 N 

si ◦ L i−1 · f ◦ L i−1 χ X i , x ∈ X,

i=1

where χ S denotes the characteristic or indicator function of a set S: χ S (x) = 1, if x ∈ S, and χ S (x) = 0, otherwise.

Clifford-Valued Fractal Interpolation

37

Then, (6) is equivalent to showing the existence of a unique fixed point ψ of T : T ψ = ψ. The existence of a unique fixed point follows from the Banach Fixed-Point Theorem once it has been shown that T is a contraction on F(X ). The RB operator T is a contraction on F(X ) if there exists a constant γF (X ) ∈ [0, 1) such that for all f, g ∈ F(X )

T f − T g F (X )

N  −1 −1 = si ◦ L i · ( f − g) ◦ L i χ X i i=1

F (X )

≤ γF (X ) f − g F (X ) holds. Here, · F (X ) denotes the norm on F(X ). Should such a unique fixed point ψ exist then is termed a fractal function of type F(X ) as its graph is in general a fractal set. Now, let F(X ), for an appropriate X , denote one of the following Banach space of functions: the Lebesgue spaces L p (X ), the smoothness spaces C k (X ), Hölder spaces C k,α (X ), Sobolev–Slobodeckij spaces W s, p (X ), Besov spaces B sp,q (X ), and s (X ). Triebel–Lizorkin spaces F p,q The following results were established in a series of papers [26–30]. Theorem 1 Let L i , i ∈ N N , be defined as in (4) and (5). Further, let qi ∈ F(X ) and let si : X → R be bounded and a pointwise multiplier for F(X ). Define T : F(X ) → F(X ) as in (7). Then there exists a constant γF ∈ [0, 1) depending on m, the indices defining F(X ), Lip(L i ), and si L ∞ such that T f ≤ γF f , for all f ∈ F(X ). Hence, T has a unique fixed point ψ ∈ F(X ) which is referred to as a fractal function of class F(X ).

4 Clifford-Valued Fractal Interpolation In this section, we introduce the novel concept of Clifford-valued fractal interpolation. To this end, we refer back to Sect. 2 and the definition of X and F(X ). We consider here only the case m = 1 and leave the extension to higher dimensions to the diligent reader. According to which function space F(X ) represents, X is either an open, half-open, or closed interval of finite length. Assume that there exist N , 1 < N ∈ N, nontrivial contractive injections L i : X → X such that {L 1 (X ), . . . , L N (X )} forms a partition of X , i.e., that (P1) L i (X )

∩ L j (X ) = ∅, for i = j; L i (X ). (P2) X = i∈N N

As above, we write X i := L i (X ), i ∈ N N .  On the spaces F(X,  Rn ), we define an RB operator T as follows. Let f ∈ f A e A , where f A ∈ F(X ), for all A ⊆ Nn . Let T : F(X ) → F(X, Rn ) with f = A⊆Nn

38

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F(X ) be an RB operator of the form (7). Then,  f := T



T ( f A )e A ∈ F(X, Rn ),

(8)

A⊆Nn

provided that T ( f A ) ∈ F(X ) for all A ⊆ Nn . Under the latter assumption and the supposition that T is contractive on F(X ) with Lipschitz constant γF (X ) , we obtain for f, g ∈ F(X, Rn )   g 2 =

T f A − T g A 2F (X ) T f −T A⊆Nn 2 ≤ γF (X )



f A − g A 2F (X )

A⊆Nn 2 2 = γF (X ) f − g .

 is also contractive on F(X, Rn ) and with the same Lipschitz constant Hence, T γF (X ) . The following diagram illustrates the above approach. T

F(X ) −−−−→ ⏐ ⏐⊗ R

R n

F(X ) ⏐ ⏐⊗ R

R n

(9)

 T

F(X, Rn ) −−−−→ F(X, Rn ) The next theorem summarizes the main result. Theorem 2 Let X ⊂ R be as mentioned above. Further, let nontrivial injective contractions L i : X → X , i ∈ N N , be given such that (P1) and (P2) are satisfied. Let F(X ) be any one of the function spaces defined in Sect. 2.  : F(X, Rn ) On the space F(X, Rn ) = F(X ) ⊗R Rn define an RB operator T → F(X, Rn ) by   f := T ( f A )e A , T A⊆Nn

where T : F(X ) → F(X ) be an RB operator of the form (7). If T : F(X ) → F(X ) is a contractive RB operator on F(X ) with Lipschitz con is also contractive on F(X, Rn ) with the same Lipschitz constant. stant γF (X ) , then T Furthermore, the unique fixed point ψ ∈ F(X, Rn ) satisfies the Clifford-valued self-referential equation ψ(L i (x)) = qi (x) + si (x)ψ(x), x ∈ X, i ∈ N N .

Clifford-Valued Fractal Interpolation

39

Proof The validity of these statements follows directly from the above elaborations. For the sake of completeness, we now list the Lipschitz constants γF (X ) for the functions spaces listed in Sect. 2 in the case m = 1. The conditions are γF (X ) < 1. Note that the expressions are different for the case m > 1. 1. C k (X ): γC k (X ) = max{Lip(L i )−(k+1) si L ∞ : i ∈ N N }. −(k+α)

si L ∞ : i ∈ N N }. 2. C k,α (X ): γC k,α (X ) = max{Lip(L i ) p p Lip(L i ) si L ∞ . 3. L (X ): γ L p (X ) = 

i∈N N

4. W s, p (X ): γW s, p (X ) = 5.

B sp,q (X ):

γ B sp,q (X )

p

Lip(L i )1−sp si L ∞ .

i∈N N  q = Lip(L i )(1/ p−s)q si L ∞ .



i∈N N s s (X ) = (X ): γ Fp,q 6. F p,q

p

Lip(L i )1−sp si L ∞ .

i∈N N

 lies at hand: Each of the functions f A is conThe geometric interpretation of T tracted by T along the direction in Rn determined by e A . There is no mixing taking place between different directions. This provides a holistic representation of features necessitating such a structure as, for instance, multichannel data or multicolored images.

5 Paravector-Valued Functions An important subspace of Rn is the space of paravectors. These are Clifford numbers n  xi ei . The subspace of paravectors is denoted by An+1 := of the form x = x0 + i=1

span R {e0 , e1 , . . . , en } = R ⊕ Rn . Given a Clifford number x ∈ Rn , we assign to x n  its paravector part by means of the mapping π : Rn → An+1 , x → x0 + xi ei . i=1

Note that each paravector x can be identified with an element (x0 , x1 , . . . , xn ) =: (x0 , x) ∈ R × Rn . For many applications in Clifford theory, one, therefore, identifies An+1 with Rn+1 . Although as point sets, these two sets are identical but differ considerably in their algebraic structures. For instance, every x ∈ An+1 has an inverse, whereas there is no such object for a vector v ∈ Rn+1 . We also notice that An+1 is not necessarily closed under multiplication unless a multiplication table [23] is defined or n = 3, in which case A4 = H, the noncommutative division algebra of quaternions. An+1 endowed with a multiplication table produces in general a nonassociative noncommutative algebra. See [1] for a suitability investigation of such algebras in the area of digital signal processing. n  xi ei The scalar part, Sc, and vector part, Vec, of a paravector An+1  x = x0 + is given by x0 and x =

n  i=1

i=1

xi ei , respectively.

40

P. R. Massopust

Given a Clifford number x ∈ Rn , we assign to x its paravector part, PV(x), by n  xi ei =: PV(x). means of the mapping π : Rn → An+1 , x → x0 + i=1

A function f : An+1 → An+1 is called a paravector-valued function. Any such function is of the form n  f i (x)ei , (10) f (x) = f 0 (x) + i=1

align where f a : R × Rn → R, a ∈ {0, 1, . . . , n}. The expression (10) for a paravector-valued function can also be written in the more succinct form f (x + x) = f 0 (x0 , |x|) + ω(x) f 1 (x0 , |x|), x ∈ Sn with Sn denoting the unit where now f 0 , f 1 : R × Rn → R and ω(x) := |x| n sphere in R . For some properties of paravector-valued functions, see, for instance, [20, 35]. Prominent examples of paravector-valued functions are, for instance, the exponential and sine functions [35] for x ∈ An+1 :

exp(x) = exp(x0 ) (cos |x| + ω(x) sin |x|) , sin(x) = sin x0 cosh |x| + ω(|x|) sinh |x| . A large class of paravector-valued functions is given by right-linear linear transformations. To this end, let Mk (An+1 ) be the right module of k × k-matrices over An+1 . Every element H = (Hi j ) of Mk (An+1 ) induces a right linear transformation L : k  Akn+1 → Rkn via L(x) = H x defined by L(x)i = Hi j x j , Hi j ∈ An+1 . To obtain j=1

an endomorphism L : Akn+1 → Akn+1 , we set L(x)i := π(L(x)i ), i = 1, . . . , k. In this case, we write L = π ◦ L. For example, if n := 3 (the case of real quaternions) L : Ak4 → Ak4 and thus L = L. Theorem 2 applies also to paravector-valued functions and thus provides a framework for paravector-valued fractal interpolation as well and relevant associated function spaces for appropriate X are defined in an analogous fashion as above. To this end, let F(X ) be, for instance, one of the function spaces listed in Sect. 2. Then, F(X, An+1 ) := F(X ) ⊗R An+1 and an element f of F(X, An+1 ) has therefore the form f =

n  k=0

f k ek .

Clifford-Valued Fractal Interpolation

41

Theorem 2 then asserts the existence of a paravector-valued function ψ ∈ F(X, An+1 of self-referential nature: ψ(L i (x)) = qi (x) + si (x)ψ(x), x ∈ X, i ∈ N N , where the functions qi and si have the same meaning as in Sect. 4.

6 Brief Summary and Further Research Directions In this short note, we have initiated the investigation of fractal interpolation into a hypercomplex setting. The main idea was to define fractal interpolants along the different directions defined by a Clifford algebra Rn and use the underlying algebraic structure to manipulate the hypercomplex fractal object to yield another hypercomplex fractal object. There are several extensions of this first initial approach: 1. Define—under suitable conditions—RB operators acting directly on appropriately defined function spaces F(X, Rn ) instead of resorting to the “component” RB operators. 2. Provide a local version of the defined hypercomplex fractal interpolation in the sense first defined in [7] and further investigated in, i.e., [5, 29]. 3. Construct nonstationary approaches to Clifford-valued fractal interpolation in the spirit of [31]. 4. Extend the notion of hypercomplex fractal interpolation to systems of function systems as described in [14, 25].

References 1. Alsmann, D.: On families of 2 N -dimensional hypercomplex algebras suitable for digital image processing. In: 14th European Signal Processing Conference (EUSIPCO 2006), pp. 1–4 (2006) 2. Barnsley, M.F.: Fractals Everywhere. Dover Publications Inc. (2012) 3. Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986) 4. Barnsley, M.F., Harding, B., Vince, C., Viswanathan, P.: Approximation of rough functions. J. Approx. Th. 209, 23–43 (2016) 5. Barnsley, M.F., Hegland, M., Massopust, P.R.: Numerics and fractals. Bull. Inst. Math. Acad. Sin. (N.S.) 9(3), 389–430 (2014) 6. Barnsley, M.F., Hegland, M., Massopust, P.R.: Self-referential functions. arXiv: 1610.01369 7. Barnsley, M.F., Hurd, L.P.: Fractal Image Compression. AK Peters Ltd., Wellesly (1993) 8. Bedford, T., Dekking, M., Keane, M.: Fractal image coding techniques and contraction operators. Delft University of Technology Report, pp. 92–93 (1992) 9. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman Books (1982) 10. Abłamowicz, R., Sobczyk, G.: Lectures on Clifford (Geometric) Algebras and Applications. Birkhäuser, New York (2004)

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11. Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions. Birkhäuser Verlag (2010) 12. Colombo, F., Sabadini, I., Struppa, D.C.: Entire Slice Regular Functions. Springer (2016) 13. Delanghe, R., Sommen, F., Souˇcek, V.: Clifford Algebra and Spinor-Valued Functions. A Function Theory for the Dirac Operator. Springer, Dordrecht (1992) 14. Dira, N., Levin, D., Massopust, P.: Attractors of trees of maps and of sequences of maps between spaces and applications to subdivision. J. Fixed Point Theory Appl. 22(14), 1–24 (2020) 15. Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114(1), 185–204 (1986) 16. Dubuc, S.: Interpolation fractale. In: Bélais , J., Dubuc, S. (eds.) Fractal Geomety and Analysis. Kluwer Academic Publishers, Dordrecht, The Netherlands (1989) 17. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley (2014) 18. Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic Functions in the Plane and nDimensional Space. Birkhäuser Verlag (2000) 19. Gürlebeck, K., Habetha, K., Sprößig, W.: Application of Holomorphic Functions in Two and Higher Dimensions. Birkhäuser Verlag (2016) 20. Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley (1997) 21. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981) 22. Huang, S., Qiao, Y.Y., Wen, G.C.: Real and Complex Clifford Analysis. Springer (2006) 23. Kantor, I.L., Solodovnik, A.S.: Hypercomplex Numbers. An Elementary Introduction to Algebras. Springer, New York (1989) 24. Kravchenko, V.: Applied Quaternionic Analysis. Heldermann Verlag, Lemgo, Germany (2003) 25. Levin, D., Dyn, N., Viswanathan, P.: Non-stationary versions of fixed-point theory, with applications to fractals and subdivision. J. Fixed Point Theory Appl. 21, 1–25 (2019) 26. Massopust, P.R.: Fractal functions and their applications. Chaos, Solitons Fractals 8(2), 171– 190 (1997) 27. Massopust, P.R.: Interpolation and Approximation with Splines and Fractals. Oxford University Press, Oxford, USA (2010) 28. Massopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets, 2nd edn. Academic, San Diego, USA (2016) 29. Massopust, P.R.: Local fractal functions and function spaces. Springer Proc. Math. Stat.: Fractals, Wavelets and their Appl. 92, 245–270 (2014) 30. Massopust, P.R.: Local fractal functions in Besov and Triebel-Lizorkin spaces. J. Math. Anal. Appl. 436, 393–407 (2016) 31. Massopust, P.R.: Non-stationary fractal interpolation. Mathematics 7(8), 1–14 (2019) 32. Massopust, P.R.: Hypercomplex iterated function systems. In: Cereijeras, P., Reissig, M., Sabadini, I., Toft, J. (eds.) Current Trends in Analysis, Its Applications and Computation. Proceedings of the 12th ISAAC Congress, Aveiro, Portugal. Birkhäuser (2019) 33. Serpa, C., Buescu, J.: Constructive solutions for systems of iterative functional equations. Constr. Approx. 45(2), 273–299 (2017) 34. Serpa, C., Buescu, J.: Compatibility conditions for systems of iterative functional equations with non-trivial contact sets. Results Math. 2, 1–19 (2021) 35. Sprössig, W.: On operators and elementary functions in clifford analysis. Zeitschrift für Analysis und ihre Anwendung. 19(2), 349–366 (1999) 36. Schutte, H.-D., Wenzel, J.: Hypercomplex numbers in digital signal processing. In: IEEE International Symposium on Circuits and Systems (1990). 10.1109ISCAS.1990.112431

Optimal Quantizers for a Nonuniform Distribution on a Sierpinski ´ Carpet Mrinal Kanti Roychowdhury

Abstract The purpose of quantization for a probability distribution is to estimate the probability by a discrete probability with finite support. In this paper, a nonuniform probability measure P on R2 which has support on the Sierpi´nski carpet generated by a set of four contractive similarity mappings with equal similarity ratios has been considered. For this probability measure, the optimal sets of n-means and the nth quantization errors are investigated for all n ≥ 2. Keywords Sierpi´nski carpet · Self-affine measure · Optimal quantizers · Quantization error

1 Introduction Quantization is a destructive process. Its purpose is to reduce the cardinality of the representation space, in particular when the input data is real-valued. It is a fundamental problem in signal processing, data compression, and information theory. We refer to [4, 9, 13] for surveys on the subject and comprehensive lists of references to the literature, see also [1, 5–7]. Let Rd denote the d-dimensional Euclidean space,  ·  denote the Euclidean norm on Rd for any d ≥ 1, and n ∈ N. Then, the nth quantization error for a Borel probability measure P on Rd is defined by Vn := Vn (P) = inf



 min x − a2 d P(x) : α ⊂ Rd , 1 ≤ card(α) ≤ n . a∈α

 If x2 d P(x) < ∞, then there is some set α for which the infimum is achieved (see [1, 5–7]). Such a set α for which the infimum occurs and contains no more than n points is called an optimal set of n-means, or optimal set of n-quantizers. The M. K. Roychowdhury (B) School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, 1201 West University Drive, Edinburg 78539-2999, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_5

43

44

M. K. Roychowdhury

collection of all optimal sets of n-means for a probability measure P is denoted by Cn := Cn (P). It is known that for a continuous probability measure an optimal set of n-means always has exactly n-elements (see [7]). Given a finite subset α ⊂ Rd , the Voronoi region generated by a ∈ α is defined by M(a|α) = {x ∈ Rd : x − a = min x − b} b∈α

i.e., the Voronoi region generated by a ∈ α is the set of all points in Rd which are closest to a ∈ α, and the set {M(a|α) : a ∈ α} is called the Voronoi diagram or Voronoi tessellation of Rd with respect to α. A Borel measurable partition {Aa : a ∈ α} of Rd is called a Voronoi partition of Rd with respect to α (and P) if P-almost k generated surely Aa ⊂ M(a|α) for every a ∈ α. Given a Voronoi tessellation {Mi }i=1 k by a set of points {z i }i=1 (called sites or generators), the mass centroid ci of Mi with respect to the probability measure P is given by 1 ci = P(Mi )





M xd P =  i Mi

Mi

xd P dP

.

The Voronoi tessellation is called the centroidal Voronoi tessellation (CVT) if z i = ci for i = 1, 2, · · · , k, that is, if the generators are also the centroids of the corresponding Voronoi regions. Let us now state the following proposition (see [4, 7]): Proposition 1.1 Let α be an optimal set of n-means and a ∈ α. Then, (i) P(M(a|α)) > 0, (ii) P(∂ M(a|α)) = 0, (iii) a = E(X : X ∈ M(a|α)), and (iv) P-almost surely the set {M(a|α) : a ∈ α} forms a Voronoi partition of Rd . Let α be an optimal set of n-means and a ∈ α, then by Proposition 1.1, we have 1 a= P(M(a|α))





M(a|α) xd P =  M(a|α)

xd P

M(a|α)

dP

,

which implies that a is the centroid of the Voronoi region M(a|α) associated with the probability measure P (see also [3, 11]). A transformation f : X → X on a metric space (X, d) is called contractive or a contraction mapping if there is a constant 0 < c < 1 such that d( f (x), f (y)) ≤ cd(x, y) for all x, y ∈ X . On the other hand, f is called a similarity mapping or a similitude if there exists a constant s > 0 such that d( f (x), f (y)) = sd(x, y) for all x, y ∈ X . Here s is called the similarity ratio of the similarity mapping f . Let C be the Cantor set generated by the two contractive similarity mappings S1 and S2 on R given by S1 (x) = r1 x and S2 (x) = r2 x + (1 − r2 ) where 0 < r1 , r2 < 1 and r1 + r2 < 21 . Let P = p1 P ◦ S1−1 + p2 P ◦ S2−1 , where P ◦ Si−1 denotes the image measure of P with respect to Si for i = 1, 2 and ( p1 , p2 ) is a probability vector with 0 < p1 , p2 < 1. Then, P is a singular continuous probability measure on R with support the Cantor set C (see [10]). For r1 = r2 = 13 and p1 = p2 = 21 , Graf

Optimal Quantizers for a Nonuniform Distribution …

45

and Luschgy gave a closed formula to determine the optimal sets of n-means for the probability distribution P for any n ≥ 2 (see [8]). For r1 = 41 , r2 = 21 , p1 = 14 , and p2 = 43 , L. Roychowdhury gave an induction formula to determine the optimal sets of n-means and the nth quantization error for the probability distribution P for any n ≥ 2 (see [12]). Let P be a Borel probability measure on R2 supported by the Cantor dusts generated by a set of 4u , u ≥ 1, contractive similarity mappings satisfying the strong separation condition. For this probability measure, Cömez and Roychowdhury determined the optimal sets of n-means and the nth quantization errors for all n ≥ 2 (see [2]). In addition, they showed that though the quantization dimension of the measure P is known, the quantization coefficient for P does not exist. In this paper, we have considered the probability distribution P given by P = 1 P ◦ S1−1 + 18 P ◦ S2−1 + 38 P ◦ S3−1 + 38 P ◦ S4−1 which has support on the Sierpi´nski 8 carpet generated by the four contractive similarity mappings given by S1 (x1 , x2 ) = 1 (x , x ), S2 (x1 , x2 ) = 13 (x1 , x2 ) + ( 23 , 0), S3 (x1 , x2 ) = 13 (x1 , x2 ) + (0, 23 ), and 3 1 2 S4 (x1 , x2 ) = 13 (x1 , x2 ) + ( 23 , 23 ) for all (x1 , x2 ) ∈ R2 . The probability distribution P considered in this paper is called ‘nonuniform’ to mean that all the basic squares at a given level that generate the Sierpi´nski carpet do not have the same probability. For this probability distribution in Propositions 3.1–3.3, first we have determined the optimal sets of n-means and the nth quantization errors for n = 2, 3, and 4. Then, in Theorem 1 we state and prove an induction formula to determine the optimal sets of n-means for all n ≥ 2. We also give some figures to illustrate the locations of the optimal points (see Fig. 1). In addition, using the induction formula, we obtain some results and observations about the optimal sets of n-means which are given in Sect. 4; a tree diagram of the optimal sets of n-means for a certain range of n is also given (see Fig. 2).

2 Preliminaries In this section, we give the basic definitions and lemmas that will be instrumental in our analysis. For k ≥ 1, by a word ω of length k over the alphabet I := {1, 2, 3, 4} it is meant that ω := ω1 ω2 · · · ωk , i.e., ω is a finite sequence of symbols over the alphabet I . Here k is called the length of the word ω. If k = 0, i.e., if ω is a word of length zero, we call it the empty word and is denoted by ∅. Length of a word ω is denoted by |ω|. I ∗ denotes the set of all words over the alphabet I including the empty word ∅. By ωτ := ω1 · · · ωk τ1 · · · τ it is meant that the word obtained from the concatenations of the words ω := ω1 ω2 · · · ωk and τ := τ1 τ2 · · · τ for k,  ≥ 0. The maps Si : R2 → R2 , 1 ≤ i ≤ 4, will be the generating maps of the Sierpi´nski carpet defined as before. For ω = ω1 ω2 · · · ωk ∈ I k , set Sω = Sω1 ◦ · · · ◦ Sωk and Jω = Sω ([0, 1] × [0, 1]). For the empty word ∅, by S∅ we mean the identity mapping on R2 , and write J = J∅ = S∅ ([0, 1] × [0, 1]) = [0, 1] × [0, 1]. The sets {Jω : ω ∈ {1, 2, 3, 4}k } are just the 4k squares in the kth level in the construction of the Sierpi´nski

46

M. K. Roychowdhury

Fig. 1 Configuration of the points in an optimal set of n-means for 1 ≤ n ≤ 16

carpet. The squares Jω1 , Jω2 , Jω3 and Jω4 into which Jω is split up at the (k + 1)th level are called the basic squares of Jω . The set S = ∩k∈N ∪ω∈{1,2,3,4}k Jω is the Sierpi´nski carpet and equals the support of the probability measure P given by P = 18 P ◦ S1−1 + 18 P ◦ S2−1 + 38 P ◦ S3−1 + 38 P ◦ S4−1 . Set s1 = s2 = s3 = s4 = 13 , p1 = p2 = 1 and p3 = p4 = 38 , and for ω = ω1 ω2 · · · ωk ∈ I k , write c(ω) := card({i : ωi = 8 3 or 4, 1 ≤ i ≤ k}), where card(A) of a set A represents the number of elements in the set A. Then, for ω = ω1 ω2 · · · ωk ∈ I k , k ≥ 1, we have sω =

1 3c(ω) and p = p p · · · p = . ω ω ω ω 1 2 k 3k 8k

Let us now give the following lemma.

Optimal Quantizers for a Nonuniform Distribution …

47

Fig. 2 Tree diagram of the optimal sets from α8 to α21

Lemma 1 Let f : R → R+ be Borel measurable and k ∈ N. Then,  f dP =



 pω

f ◦ Sω d P.

ω∈I k

Proof We know P= p1 P ◦ S1−1 + p2 P ◦ S2−1 + p3 P ◦ S3−1 + p4 P ◦ S4−1 , and so   by induction P = ω∈I k pω P ◦ Sω−1 , and thus the lemma is yielded. Let S(i1) , S(i2) be the horizontal and vertical components of the transformation Si for i = 1, 2, 3, 4. Then, for any (x1 , x2 ) ∈ R2 we have S(11) (x1 ) = 13 x1 , S(12) (x2 ) = 1 x , S(21) (x1 ) = 13 x1 + 23 , S(22) (x2 ) = 13 x2 , S(31) (x1 ) = 13 x1 , S(32) (x2 ) = 13 x2 + 23 , 3 2 and S(41) (x1 ) = 13 x1 + 23 , S(42) (x2 ) = 13 x2 + 23 . Let X := (X 1 , X 2 ) be a bivariate continuous random variable with distribution P. Let P1 , P2 be the marginal distributions of P, i.e., P1 (A) = P(A × R) for all A ∈ B, and P2 (B) = P(R × B) for all B ∈ B. Here B is the Borel σ -algebra on R. Then, X 1 has distribution P1 and X 2 has distribution P2 .

48

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Let us now state the following lemma. The proof is similar to Lemma 2.2 in [2]. Lemma 2 Let P1 and P2 be the marginal distributions of the probability measure P. Then, −1 −1 −1 −1 • P1 = 18 P1 ◦ S(11) + 18 P1 ◦ S(21) + 38 P1 ◦ S(31) + 38 P1 ◦ S(41) and −1 −1 −1 −1 1 1 3 3 • P2 = 8 P2 ◦ S(12) + 8 P2 ◦ S(22) + 8 P2 ◦ S(32) + 8 P2 ◦ S(42) .

Let us now give the following lemma. Lemma 3 Let E(X ) and V (X ) denote the expected vector and the expected squared distance of the random variable X . Then,  E(X ) = (E(X 1 ), E(X 2 )) =

1 3 , 2 4



 and V := V (X ) = EX −

1 3 2 7 ,  = . 2 4 32

Proof We have 

1 x d P1 = 8



1 + 8



−1 x d P1 ◦ x d P1 ◦ S(21)   3 3 −1 −1 + + x d P1 ◦ S(31) x d P1 ◦ S(41) 8 8       2 2 1 1 1 1 3 3 1 1 x d P1 + x+ d P1 + x d P1 + x+ d P1 , = 8 3 8 3 3 8 3 8 3 3

E(X 1 ) =

−1 S(11)

which after simplification yields E(X 1 ) = 21 , and similarly E(X 2 ) = 34 . Now,  E(X 12 ) = x 2 d P1     1 1 3 3 −1 −1 −1 −1 = + + + x 2 d P1 ◦ S(11) x 2 d P1 ◦ S(21) x 2 d P1 ◦ S(31) x 2 d P1 ◦ S(41) 8 8 8 8      2   1 2 2 1 1 2 2 1 1 3 3 1 x+ x x+ = d P1 + d P1 + d P1 ( x)2 d P1 + 8 3 8 3 3 8 3 8 3 3    1 2 4 4 1 1 1 2 x d P1 + x + x+ d P1 = 2 9 2 9 9 9 1 1 4 4 E(X 12 ) + E(X 12 ) + E(X 1 ) + = 18 18 18 18 1 1 = E(X 12 ) + . 9 3 21 This implies E(X 12 ) = 38 . Similarly, we can show that E(X 22 ) = 32 . Thus, V (X 1 ) = 3 1 1 3 2 2 E(X 1 ) − (E(X 1 )) = 8 − 4 = 8 , and similarly V (X 2 ) = 32 . Hence,

 EX −

  1 3 2 1 2 3 2 7  = E X1 − + E X2 − = V (X 1 ) + V (X 2 ) = , . 2 4 2 4 32

Optimal Quantizers for a Nonuniform Distribution …

49

Thus, the proof of the lemma follows.

 

Let us now give the following note. Note 1 From Lemma 3 it follows that the optimal set of one-mean is the expected vector and the corresponding quantization error is the expected squared distance of the random variable X . For words β, γ , . . . , δ in I ∗ , by a(β, γ , . . . , δ) we mean the conditional expected vector of the random variable X given Jβ ∪ Jγ ∪ · · · ∪ Jδ , i.e., a(β, γ , . . . , δ) = E(X |X ∈ Jβ ∪ Jγ ∪ · · · ∪ Jδ ) =

 1 xd P. (1) P(Jβ ∪ · · · ∪ Jδ ) Jβ ∪···∪Jδ

For ω ∈ I k , k ≥ 1, since a(ω) = E(X : X ∈ Jω ), using Lemma 1, we have    1 x d P(x) = x d P ◦ Sω−1 (x) = Sω (x) d P(x) = E(Sω (X )) P(Jω ) Jω Jω  1 3 . = Sω , 2 4

a(ω) =

2 For any (a, b) ∈ R2 , EX V + ( 21 , 43 ) − (a, b)2 . In fact, for any  − (a, b) = k 2 ω ∈ I , k ≥ 1, we have Jω x − (a, b) d P = pω (x1 , x2 ) − (a, b)2 d P ◦ Sω−1 , which implies

 Jω

 x − (a, b)2 d P = pω sω2 V + a(ω) − (a, b)2 .

(2)

The expressions (1) and (2) are useful to obtain the optimal sets and the corresponding quantization errors with respect to the probability distribution P. The Sierpi´nski carpet has the maximum symmetry with respect to the vertical line x1 = 21 , i.e., with respect to the line x1 = 21 the Sierpi´nski carpet is geometrically symmetric as well as symmetric with respect to the probability distribution: if the two basic rectangles of similar geometrical shape lie in the opposite sides of the line x1 = 21 , and are equidistant from the line x1 = 21 , then they have the same probability.

3 Optimal Sets of n-means for All n ≥ 2 In this section we determine the optimal sets of n-means for all n ≥ 2. First, prove the following proposition. Proposition 3.1 The set α = {a(1, 3), a(2, 4)}, where a(1, 3) = ( 61 , 43 ) and 31 a(2, 4) = ( 56 , 34 ), is an optimal set of two-means with quantization error V2 = 288 = 0.107639.

50

M. K. Roychowdhury

Proof Since the Sierpi´nski carpet has the maximum symmetry with respect to the vertical line x1 = 21 , among all the pairs of two points which have the boundaries of the Voronoi regions oblique lines passing through the point ( 21 , 34 ), the two points which have the boundary of the Voronoi regions the line x1 = 21 will give the smallest distortion error. Again, we know that the two points which give the smallest distortion error are the centroids of their own Voronoi regions. Let (a1 , b1 ) and (a2 , b2 ) be the centroids of the left half and the right half of the Sierpi´nski carpet with respect to the line x1 = 21 , respectively. Then using (1), we have (a1 , b1 ) = E(X : X ∈ J1 ∪ J3 ) =

1 P(J1 ∪ J3 )



 J1 ∪J3

xd P =

1 3 , 6 4



and 1 (a2 , b2 ) = E(X : X ∈ J2 ∪ J4 ) = P(J2 ∪ J4 )



 J2 ∪J4

xd P =

5 3 , . 6 4





Write α := { 16 , 34 , 56 , 43 }. Then, the distortion error is obtained as 

 min x − c2 d P = c∈α

J1 ∪J3

1 3 x − ( , )2 d P + 6 4

 J2 ∪J4

5 3 31 x − ( , )2 d P = = 0.107639. 6 4 288

Since V2 is the quantization error for two-means, we have 0.107639 ≥ V2 . We now show that the points in an optimal set of two means cannot lie on a vertical line. Suppose that the points in an optimal set of two-means lie on a vertical line. Then, we can assume that β = {( p, a), ( p, b)} is an optimal set of two-means with a ≤ b. Then, by the properties of centroids we have  ( p, a)P(M(( p, a)|β)) + ( p, b)P(M(( p, b)|β)) =

1 3 , , 2 4

which implies p P(M(( p, a)|β)) + p P(M(( p, b)|β)) = 21 and a P(M(( p, a)|β)) + b P(M(( p, b)|β)) = 43 . Thus, we see that p = 21 , and the two points ( p, a) and ( p, b) lie on the opposite sides of the point ( 21 , 43 ). Since the optimal points are the centroids of their own Voronoi regions, we have 0 ≤ a ≤ 43 ≤ b ≤ 1 implying 21 (0 + 43 ) = 38 ≤ 1 (a + b) ≤ 21 ( 43 + 1) = 78 < 89 , and so J33 ∪ J34 ∪ J43 ∪ J44 ⊂ M(( 21 , b)|β) and 2 5 . Then as a(33, 34, 43, 44) = E(X : J1 ∪ J2 ⊂ M(( 21 , a)|β). Suppose that a ≥ 12 1 35 X ∈ J33 ∪ J34 ∪ J43 ∪ J44 ) = 2 , 36 , we have 

 min x − c2 d P ≥ c∈α

J1 ∪J2

1 5 x − ( , )2 d P + 2 12

 J33 ∪J34 ∪J43 ∪J44

1 35 515 x − ( , )2 d P = = 0.111762, 2 36 4608

Optimal Quantizers for a Nonuniform Distribution …

51

which is a contradiction, as 0.111762 > 0.107639 ≥ V2 and α is an optimal set 5 5 . Since a < 12 and b ≤ 1, we of two-means. Thus, we can assume that a < 12 1 1 5 17 1 have 2 (a + b) ≤ 2 ( 12 + 1) = 24 , which yields that B ⊂ M(( 2 , b)|α) where B = J33 ∪ J34 ∪ J43 ∪ J44 ∪ J313 ∪ J314 ∪ J323 ∪ J324 ∪ J413 ∪ J414 ∪ J423 ∪ J424 . Using 503 503 (1), we have E(X : X ∈ B) = ( 21 , 540 ) which implies that b ≤ 540 . Now if a ≥ 13 , we have 



 min x − c2 d P ≥

x −

c∈α

J1 ∪J2

1 1 , 2 3





 2 d P +

x −

1 503 , 2 540

2 d P =

106847 = 0.128818 > V2 , 829440

B

which is a contradiction. So, we can assume that a < 13 . Then, J1 ∪ J1 ⊂ M(( 21 , a)|α) and J3 ∪ J4 ⊂ M(( 21 , b)|α), and so ( 21 , a) = E(X : X ∈ J1 ∪ J2 ) = ( 21 , 41 ) and 11 ( 21 , b) = E(X : X ∈ J3 ∪ J4 ) = ( 21 , 12 ), and 



 min x − c2 d P =

x −

c∈α

J1 ∪J2

1 1 , 2 4





 2 d P +

x − J3 ∪J4

1 11 , 2 12

2 d P =

13 = 0.135417 > V2 , 96

which leads to another contradiction. Therefore, we can assume that the points in an optimal set of two-means cannot lie on a vertical line. Hence, α = {( 16 , 34 ), ( 56 , 43 )} 31 forms an optimal set of two-means with quantization error V2 = 288 = 0.107639.   Remark 1 The set α in Proposition 3.1 forms a unique optimal set of two-means. Proposition 3.2 The set α = {a(1, 2), a(3), a(4)}, where a(1, 2) = E(X : X ∈ 11 J1 ∪ J2 ) = ( 21 , 41 ), a(3) = E(X : X ∈ J3 ) = ( 16 , 12 ) and a(4) = E(X : X ∈ J4 ) = 5 11 5 ( 6 , 12 ), forms an optimal set of three-means with quantization error V3 = 96 = 0.0520833. Proof Let us first consider the three-point set β given by β = {a(1, 2), a(3), a(4)}. Then, the distortion error is obtained as  min x − c2 d P c∈α   x − a(1, 2)2 d P + = J1 ∪J2

 J3

x − a(3)2 d P +

J4

x − a(4)2 d P = 0.0520833.

Since V3 is the quantization error for an optimal set of three-means, we have 0.0520833 ≥ V3 . Let α := {(ai , bi ) : 1 ≤ i ≤ 3} be an optimal set of three-means. Since the optimal points are the centroids of their own Voronoi regions, we have α ⊂ [0, 1] × [0, 1]. Then, by the definition of centroid, we have  (ai ,bi )∈α

 (ai , bi )P(M((ai , bi )|α)) =

1 3 , , 2 4

52

M. K. Roychowdhury

  which implies (ai ,bi )∈α ai P(M((ai , bi )|α)) = 21 and (ai ,bi )∈α bi P(M((ai , bi )|α)) = 34 . Thus, we conclude that all the points in an optimal set cannot lie in one side of the vertical line x1 = 21 or in one side of the horizontal line x2 = 43 . Without any loss of generality, due to symmetry we can assume that one of the optimal points, say (a1 , b1 ), lies on the vertical line x1 = 21 , i.e., a1 = 21 , and the optimal points (a2 , b2 ) and (a3 , b3 ) lie on a horizontal line and are equidistant from the vertical line x1 = 21 . Further, due to symmetry we can assume that (a2 , b2 ) and (a3 , b3 ) lie on the vertical lines x1 = 16 and x1 = 56 respectively, i.e., a2 = 16 and a3 = 56 . Suppose that ( 21 , b1 ) lies on or above the horizontal line x2 = 34 , and so ( 16 , b2 ) and ( 56 , b3 ) lie on or below the line x2 = 43 . Then, if 23 ≤ b2 , b3 ≤ 43 , we have 



 min x − c d P ≥ 2

min x −

2

c∈α

J1 ∪J31 ∪J33

which is a contradiction. If

1 2

2 3 3 ≤b≤ 4

1 , b 2 d P = 0.0820313 > V3 , 6

≤ b2 , b3 ≤ 23 ,

 min x − c2 d P c∈α    

1 1 2 , b 2 d P + x − , 2 d P + ≥2 min x − 1 2 6 6 3 2 ≤b≤ 3 J1 ∪J31 ∪J321

J33

6521 281 277  + + = 0.0649821 > V3 , =2 442368 18432 110592

which leads to a contradiction. If

1 3



 min x −

J342 ∪J344

3 4 ≤b≤1

 1 , b 2 d P 2

≤ b2 , b3 ≤ 21 , then

 min x − c2 d P c∈α    

1 1 1 1 , 2 d P + x − , 2 d P + ≥2 x − 6 2 6 3 J31 ∪J321 ∪J331

J1

811 1 78373  + + = 0.0546912 > V3 , =2 110592 256 4866048



 min x −

J34 ∪J334

3 4 ≤b≤1

 1 , b 2 d P 2

which gives a contradiction. If 0 ≤ b2 , b3 ≤ 13 , then  min x − c d P ≥ 2 2



c∈α

J1

=2



 x − a(1) d P +

min x −

2

7 109  + = 0.0770399 > V3 2304 3072

J33 ∪J34

3 4 ≤b≤1

 1 , b 2 d P 2

which leads to another contradiction. Therefore, we can assume that ( 21 , b1 ) lies on or below the horizontal line x2 = 43 , and ( 16 , b2 ) and ( 56 , b3 ) lie on or above the line x2 = 43 . Notice that for any position of ( 21 , b1 ) on or below the line x2 = 3 79 , always J31 ∪ J33 ∪ J34 ⊂ M(( 61 , b2 )|α) which implies that b2 ≤ 84 . Similarly, 4

Optimal Quantizers for a Nonuniform Distribution …

53

79 b3 ≤ 84 . Suppose that 21 ≤ b1 ≤ 43 . Then, writing A = J133 ∪ J321 ∪ J324 and B = J11 ∪ J12 ∪ J14 ∪ J132 , we have

 min x − c2 d P c∈α 

≥2 J31 ∪J33 ∪J34 ∪J323

 min x −

3 79 4 ≤b≤ 84

     1 1 3 2 1 1 , b 2 d P + x − ,  d P + x − , 2 d P 6 6 4 2 2 A

B

588517 5347 6601  =2 + + = 0.0529346 > V3 , 78299136 1327104 442368

which is a contradiction. So, we can assume that b1 < 21 . Suppose that 13 ≤ b1 < 21 . 79 Then, as 43 ≤ b2 ≤ 84 , we see that J31 ∪ J33 ∪ J34 ∪ J321 ∪ J323 ∪ J324 ⊂ M(( 61 , b2 ) |α). Then, writing A1 := J31 ∪ J33 ∪ J34 ∪ J321 ∪ J323 ∪ J324 and A2 := J322 ∪ J1331 ∪ J1333 ∪ J1334 ∪ J13323 ∪ J13324 and A3 := J11 ∪ J12 ∪ J14 ∪ J131 ∪ J132 ∪ J134 ∪ J13322 , we have  min x − c2 d P c∈α     

 1 1 3 2 1 1 2  ≥2 min x − , b 2 d P + x − ,  d P + x − ,  dP 3 79 6 6 4 2 3 4 ≤b≤ 84 A1

A2

A3

242191 4135547 31584803  =2 = 0.0521401 > V3 , + + 27869184 1146617856 2293235712

which gives a contradiction. So, we can assume that b1 ≤ 13 . Then, notice that J11 ∪ J12 ∪ J132 ∪ J141 ∪ J142 ∪ J144 ∪ J21 ∪ J22 ∪ J241 ∪ J231 ∪ J232 ∪ J233 ⊂ M(( 21 , b1 ) |α) which implies that b1 ≥ 13 . Thus, we have 13 ≤ b1 ≤ 13 . Suppose that 43 ≤ 68 68 5 b2 , b3 ≤ 6 . Then,  min x − c2 d P c∈α 

 1 ≥2 , b 2 d P + min x − 3 ≤b≤ 5 6 J3 4

6



 x −

+ J1331 ∪J1333 ∪J1334



 13

min

1

≤b≤ 3 J11 ∪J12 ∪J14 ∪J131 ∪J132 68

x −

1 , b 2 d P 2

  1 3 2 1 1 2  ,  dP + ,  dP x − 6 4 2 3 J134

3 147359 32969 3881  + + + = 0.054808 > V3 , =2 256 15261696 10616832 1327104

which leads to a contradiction. So, we can assume that 56 < b2 , b3 ≤ 1. Then, we have J1 ∪ J2 ⊂ M(( 21 , b1 )|α), J3 ⊂ M(( 61 , b2 )|α) and J4 ⊂ M(( 65 , b3 )|α) which yield that ( 21 , b1 ) = a(1, 2), ( 16 , b2 ) = a(3) and ( 56 , b3 ) = a(4), and the quantization error 5 is V3 = 96 = 0.0520833. Thus, the proof of the proposition is complete.   Proposition 3.3 The set α = {a(1), a(2), a(3), a(4)} forms an optimal set of four7 means with quantization error V4 = 288 = 0.0243056.

54

M. K. Roychowdhury

Proof Let us consider the four-point set β given by β := {a(1), a(2), a(3), a(4)}. Then, the distortion error is given by 

4  

min x − c d P = 2

c∈β

i=1

x − a(i)2 d P = Ji

7 = 0.0243056. 288

Since, V4 is the quantization error for four-means, we have 0.0243056 ≥ V4 . As the optimal points are the centroids of their own Voronoi regions, α ⊂ J . Let α be an optimal set of n-means for n = 4. By the definition of centroid, we know 

1 3 (a, b)P(M((a, b)|α)) = ( , ). 2 4 (a,b)∈α

(3)

If all the points of α are below the line x2 = 43 , i.e., if b < 43 for all (a, b) ∈ α, then by   (3), we see that 43 = (a,b)∈α b P(M((a, b)|α)) < (a,b)∈α 43 P(M((a, b)|α)) = 34 , which is a contradiction. Similarly, it follows that if all the points of α are above the line x2 = 43 , or left of the line x1 = 21 , or right of the line x1 = 21 , a contradiction will arise. Suppose that all the points of α are on the line x2 = 43 . Then, for (x1 , x2 ) ∈ 5 , and for (x1 , x2 ) ∈ ∪i,2 j=1 Ji j , we have ∪i,4 j=3 Ji j , we have minc∈α (x1 , x2 ) − c ≥ 36 23 minc∈α (x1 , x2 ) − c ≥ 36 , which implies that 

 min x − c d P ≥ 4

 min (x1 , x2 ) − c d P + 4

2

c∈α

c∈α

J33

=4

min (x1 , x2 ) − c2 d P

2

c∈α

J11

5 2

23 2 377 = 0.0363619 > V4 , P(J33 ) + 4 P(J11 ) = 36 36 10368

which is a contradiction. Thus, we see that all the points of α cannot lie on x2 = 34 . Similarly, all the points of α cannot lie on x1 = 21 . Recall that the Sierpi´nski carpet has maximum symmetry with respect to the line x1 = 21 . As all the points of α cannot lie on the line x1 = 21 , due to symmetry we can assume that the points of α lie either on the three lines x1 = 16 , x1 = 56 , and x1 = 21 , or on the two lines x1 = 16 and x1 = 56 . Suppose α contains points from the line x1 = 21 . As α cannot contain all the points from x1 = 21 , we can assume that α contains two points, say ( 21 , b1 ) and ( 21 , b2 ) with b1 < b2 , from the line x1 = 21 which are in the opposite sides of the centroid ( 21 , 34 ), and the other two points, say ( 61 , a1 ) and ( 56 , a2 ), from the lines x1 = 16 and x1 = 56 . Then, if α does not contain any point from J3 ∪ J4 , we have 



 min x − c2 d P ≥ 2

x −

c∈α

J31 ∪J33

1 2 2 25 ,  dP = = 0.0325521 > V4 , 6 3 768

Optimal Quantizers for a Nonuniform Distribution …

55

which leads to a contradiction. So, we can assume that ( 16 , a1 ) ∈ J3 and ( 65 , a2 ) ∈ J4 . Suppose 23 ≤ a1 , a2 ≤ 56 . Then, notice that J31 ∪ J33 ∪ J321 ∪ J323 ⊂ M(( 61 , a1 )|α) and similar is the expression for the point ( 56 , a2 ). Further, notice that J11 ∪ J12 ∪ J14 ∪ J21 ∪ J22 ∪ J23 ⊂ M(( 21 , 13 )|α). Therefore, under the assumption 23 ≤ a1 , a2 ≤ 5 , writing A1 := J31 ∪ J33 ∪ J321 ∪ J323 and A2 := J11 ∪ J12 ∪ J14 , we have the dis6 tortion error as  min x − c d P ≥ 2 2



c∈α

A1

 min x −

2 5 3 ≤b≤ 6

 1 , b 2 d P + 6

2051 2021  + = 0.0269833 > V4 , =2 331776 276480

A2

 1 min x − ( , b)2 d P 2 0≤b≤ 34

which leads to a contradiction. So, we can assume that 56 < a1 , a2 ≤ 1. Then, we see that J1 ∪ J2 ⊂ M(( 21 , b1 )|α) for b1 = 21 , and so the distortion error is 



 min x − c2 d P ≥ 2 c∈α

min x −

3 J1 0≤b≤ 4

1 13 , b 2 d P = = 0.0338542 > V4 2 384

which is a contradiction. All these contradictions arise due to our assumption that α contains points from the line x1 = 21 . So, we can assume that α cannot contain any point from the line x1 = 21 , i.e., we can assume that α contains two points from the line x1 = 16 and two points from the line x1 = 56 . Thus, we can take α := {( 61 , a1 ), ( 16 , b1 ), ( 56 , a2 ), ( 56 , b2 )} where a1 ≤ 43 ≤ b1 and a2 ≤ 43 ≤ b2 . Notice that the Voronoi region of ( 16 , a1 ) contains J1 and the Voronoi region of ( 56 , a2 ) contains J2 . If the Voronoi region of ( 16 , a1 ) contains points from J3 , we must have 21 (a1 + b1 ) ≥ 2 7 which yields a1 ≥ 43 − b1 ≥ 43 − 43 = 12 , and similarly if the Voronoi region of 3 5 7 ( 6 , a2 ) contains points from J4 , we must have a2 ≥ 12 . But, then 



 min x − c2 d P ≥ 2

x −

c∈α

J1

 1 7 , 2 d P + 2 6 12

x − a(33, 34)2 d P

J33 ∪J34

65 = 0.0423177 > V4 , = 1536 which is a contradiction. So, we can assume that the Voronoi regions of ( 16 , a1 ) and ( 65 , a2 ) do not contain any point from J3 ∪ J4 . Thus, we have ( 16 , a1 ) = a(1) = ( 16 , 14 ), 11 11 ), and ( 65 , b2 ) = a(4) = ( 56 , 12 ), ( 65 , a2 ) = a(2) = ( 65 , 41 ), ( 16 , b1 ) = a(3) = ( 16 , 12 7 and the quantization error is V4 = 288 = 0.0243056. Thus, the proof of the proposition is complete.   Proposition  3.4 Let n ≥ 4 and αn be an optimal set of n-means, and let 1 ≤ i ≤ 4. Then αn Ji = ∅, and αn ∩ (J \ J1 ∪ J2 ∪ J3 ∪ J4 ) is an empty set.

56

M. K. Roychowdhury

Proof Let αn be an optimal set of n-means for n ≥ 4. If n = 4, the proposition is true by Proposition 3.3. We now show that the proposition is true for n ≥ 5. Consider the set of five points β := {(a(1), a(2), a(3, 3), a(3, 4), a(4)}. The distortion error due to the set β is given by  min x − (a, b)2 d P =

(a,b)∈β

17 = 0.0196759. 864

Since Vn is the quantization error for n-means for n ≥ 5, we have Vn ≤ 0.0196759. As described in the proof of Proposition 3.2, we can assume that all the optimal points cannot lie in one side of the vertical line x1 = 21 or in one side of the horizontal line x2 = 43 .   Note 2 Let α be an optimal set of n-means for some n ≥ 2. Then, for a ∈ α, we have a = a(ω), a = a(ω1, ω3), or a = a(ω2, ω4) for some ω ∈ I ∗ . Moreover, if a ∈ α, then P-almost surely M(a|α) = Jω if a = a(ω), M(a|α) = Jω1 ∪ Jω3 if a = a(ω1, ω3), and M(a|α) = Jω2 ∪ Jω4 if a = a(ω2, ω4). For ω ∈ I ∗ , (i = 1 and j = 3), (i = 2 and j = 4), or (i = 1, j = 2) write 

 x − a(ω)2 d P, and E(ωi, ωj) :=

E(ω) :=

x − a(ωi, ωj)2 d P. (4) Jωi ∪Jωj

Jω

Let us now give the following lemma. Lemma 4 For any ω ∈ I ∗ , let E(ω), E(ω1, ω3), E(ω2, ω4), and E(ω1, ω2) be 31 13 E(ω), E(ω1, ω2) = 84 E(ω), defined by (4). Then, E(ω1, ω3) = E(ω2, ω4) = 126 1 1 E(ω1) = E(ω2) = 72 E(ω), and E(ω3) = E(ω4) = 24 E(ω). Proof By (2), we have 

 x − a(ω1, ω3)2 d P =

E(ω1, ω3) =  +

Jω1 ∪Jω3

x − a(ω1, ω3)2 d P Jω1

x − a(ω1, ω3)2 d P Jω3

2 2 = pω1 (sω1 V + a(ω1) − a(ω1, ω3)2 ) + pω3 (sω3 V + a(ω3) − a(ω1, ω3)2 ).

Notice that  

1 3 1 3  1 + pω3 Sω3 , , pω1 Sω1 a(ω1, ω3) = pω1 + pω3 2 4 2 4   3 1 3 1 3  1 1 + Sω3 , = 1 3 Sω1 , , 2 4 8 2 4 +8 8 8

Optimal Quantizers for a Nonuniform Distribution …

57

which implies a(ω1, ω3) = 41 Sω1 ( 21 , 43 ) + 34 Sω3 ( 21 , 43 ). Thus, we have 

  1 3 1 3 1 3 2 1 3 a(ω1) − a(ω1, ω3) =Sω1 , − Sω1 , − Sω3 ,  2 4 4 2 4 4 2 4 9 2 1 = sω2 (0, )2 = sω2 , 16 3 4 2

and similarly, a(ω3) − a(ω1, ω3)2 =

1 2 s (0, 23 )2 16 ω

=

1 2 s . 36 ω

Thus, we obtain,

 1 2 1 2 2 E(ω1, ω3) = + sω ) + pω3 sω3 V + sω 4 36  1 1 2 2 2 2 p1 + p3 = p ω sω V ( p 1 s1 + p 3 s3 ) + p ω sω 4 36  1 31 1 1 = pω sω2 V V = + E(ω), 18 24 126 2 pω1 (sω1 V

and similarly, we can prove the rest of the lemma. Thus, the proof of the lemma is complete.   Remark 2 From the above lemma it follows that E(ω1, ω3) = E(ω2, ω4) > E(ω1, ω2) > E(ω3) = E(ω4) > E(ω1) = E(ω2). The following lemma gives some important properties about the distortion error. Lemma 5 Let ω, τ ∈ I ∗ . Then (i) E(ω) > E(τ ) if and only if E(ω1, ω3) + E(ω2, ω4) + E(τ ) < E(ω) + E(τ 1, τ 3) + E(τ 2, τ 4); (ii) E(ω) > E(τ 1, τ 3)(= E(τ 2, τ 4)) if and only if E(ω1, ω3) + E(ω2, ω4) + E(τ 1, τ 3) + E(τ 2, τ 4) < E(ω) + E(τ 1, τ 2) + E(τ 3) + E(τ 4); (iii) E(ω1, ω3)(= E(ω2, ω4)) > E(τ 1, τ 3)(= E(τ 2, τ 4)) if and only if E(ω1, ω2) + E(ω3) + E(ω4) + E(τ 1, τ 3) + E(τ 2, τ 4) < E(ω1, ω3) + E(ω2, ω4) + E(τ 1, τ 2) + E(τ 3) + E(τ 4); (iv) E(ω1, ω3)(= E(ω2, ω4)) > E(τ ) if and only if E(ω1, ω2) + E(ω3) + E(ω4) + E(τ ) < E(ω1, ω3) + E(ω2, ω4) + E(τ 1, τ 3) + E(τ 2, τ 4); (v) E(ω1, ω2) > E(τ ) if and only if E(ω1) + E(ω2) + E(τ ) < E(ω1, ω2) + E(τ 1, τ 3) + E(τ 2, τ 4); (vi) E(ω1, ω2) > E(τ 1, τ 3)(= E(τ 2, τ 4)) if and only if E(ω1) + E(ω2) + E(τ 1, τ 3) + E(τ 2, τ 4) < E(ω1, ω2) + E(τ 1, τ 2) + E(τ 3) + E(τ 4); (vii) E(ω1, ω2) > E(τ 1, τ 2) if and only if E(ω1) + E(ω2) + E(τ 1, τ 2) < E(ω1, ω2) + E(τ 1) + E(τ 2); (viii) E(ω) > E(τ 1, τ 2) if and only if E(ω1, ω3) + E(ω2, ω4) + E(τ 1, τ 2) < E(ω) + E(τ 1) + E(τ 2).

58

M. K. Roychowdhury

Proof Let us first prove (iii). Using Lemma 4, we see that 5 E(ω) + 21 31 R H S = E(ω1, ω3) + E(ω2, ω4) + E(τ 1, τ 2) + E(τ 3) + E(τ 4) = E(ω) + 63 L H S = E(ω1, ω2) + E(ω3) + E(ω4) + E(τ 1, τ 3) + E(τ 2, τ 4) =

31 E(τ ), 63 5 E(τ ). 21

5 5 Thus, L H S < R H S if and only if 21 E(ω) + 31 E(τ ) < 31 E(ω) + 21 E(τ ), which 63 63 yields E(ω) > E(τ ), i.e., E(ω1, ω3) > E(τ 1, τ 3). Thus (iii) is proved. The other parts of the lemma can similarly be proved. Thus, the lemma follows.  

In the following theorem, we give the induction formula to determine the optimal sets of n-means for any n ≥ 2. Theorem 1 For any n ≥ 2, let αn := {a(i) : 1 ≤ i ≤ n} be an optimal set of nmeans, i.e., αn ∈ Cn := Cn (P). For ω ∈ I ∗ , let E(ω), E(ω1, ω3) and E(ω2, ω4) be defined by (4). Set ˜ E(a(i)) :=



E(ω) if a(i) = a(ω) for some ω ∈ I ∗ , E(ωk, ω) if a(i) = a(ωk, ω) for some ω ∈ I ∗ ,

where (k = 1,  = 3), or (k = 2,  = 4), or (k = 1,  = 2), and W (αn ) := {a( j) : ˜ ˜ j)) ≥ E(a(i)) for all 1 ≤ i ≤ n}. Take any a( j) ∈ W (αn ), and a( j) ∈ αn and E(a( write ⎧ (αn \ {a( j)}) ∪ {a(ω1, ω3), a(ω2, ω4)} if a( j) = a(ω), ⎪ ⎪ ⎨ (αn \ {a(ω1, ω3), a(ω2, ω4)}) ∪ {a(ω1, ω2), a(ω3), a(ω4)} αn+1 (a( j)) := if a( j) = a(ω1, ω3) or a(ω2, ω4), ⎪ ⎪ ⎩ (αn \ {a( j)}) ∪ {a(ω1), a(ω2)} if a( j) = a(ω1, ω2), Then αn+1 (a( j)) is an optimal set of (n + 1)-means, and the number of such sets is given by card



 {αn+1 (a( j)) : a( j) ∈ W (αn )} .

αn ∈Cn

Proof By Propositions 3.1–3.3, we know that the optimal sets of two-, three- , and four-means are respectively {a(1, 3), a(2, 4)}, {a(1, 2), a(3), a(4)}, and {a(1), a(2), a(3), a(4)}. Notice that by Lemma 4, we know E(1, 3) ≥ E(2, 4), and E(1, 2) ≥ E(3) = E(4). Thus, the lemma is true for n = 2 and n = 3. For any n ≥ 3, let us now assume that αn is an optimal set of n-means. Let αn := {a(i) : 1 ≤ i ≤ n}. ˜ / W (αn ), i.e., if Let E(a(i)) and W (αn ) be defined as in the hypothesis. If a( j) ∈ a( j) ∈ αn \ W (αn ), then by Lemma 5, the error

Optimal Quantizers for a Nonuniform Distribution …



59

˜ E(a(i)) + E(ω1, ω3) + E(ω2, ω4) if a( j) = a(ω),

a(i)∈(αn \{a( j)})



˜ E(a(i)) + E(ω1, ω2) + E(ω3) + E(ω4) if a( j) = a(ω1, ω3) or a(ω2, ω4),

a(i)∈(αn \{a(ω1,ω3), a(ω2,ω4)})



˜ E(a(i)) + E(ω1) + E(ω2) if a( j) = a(ω1, ω2),

a(i)∈(αn \{a( j)})

obtained in this case is strictly greater than the corresponding error obtained in the case when a( j) ∈ W (αn ). Hence for any a( j) ∈ W (αn ), the set αn+1 (a( j)), where ⎧ (αn \ {a( j)}) ∪ {a(ω1, ω3), a(ω2, ω4)} if a( j) = a(ω), ⎪ ⎪ ⎨ (αn \ {a(ω1, ω3), a(ω2, ω4)}) ∪ {a(ω1, ω2), a(ω3), a(ω4)} αn+1 (a( j)) := if a( j) = a(ω1, ω3) or a(ω2, ω4), ⎪ ⎪ ⎩ (αn \ {a( j)}) ∪ {a(ω1), a(ω2)} if a( j) = a(ω1, ω2), is an optimal set of (n + 1)-means, and the number of such sets is card



 {αn+1 (a( j)) : a( j) ∈ W (αn )} .

αn ∈Cn

Thus the proof of the theorem is complete (also see Note 3).

 

Remark 3 Once an optimal set of n-means is known, by using (2), the corresponding quantization error can easily be calculated. Remark 4 By Theorem 1, we note that to obtain an optimal set of (n + 1)-means one needs to know an optimal set of n-means. We conjecture that unlike the uniform probability distribution, i.e., when the probability measures on the basic rectangles at each level of the Sierpi´nski carpet construction are equal, for the nonuniform probability distribution considered in this paper, to obtain the optimal sets of nmeans a closed formula cannot be obtained. Running the induction formula given by Theorem 1 in computer algorithm, we obtain some results and observations about the optimal sets of n-means, which are given in the following section.

60

M. K. Roychowdhury

4 Some Results and Observations First, we explain about some notations that we are going to use in this section. Recall that the optimal set of one-mean consists of the expected value of the random variable X , and the corresponding quantization error is its variance. Let αn be an optimal set of n-means, i.e., αn ∈ Cn , and then for any a ∈ αn , we have a = a(ω), or a = a(ωi, ωj) for some ω ∈ I ∗ , where (i = 1, j = 3), (i = 2, j = 4), or (i = 1, j = 2). For ω = ω1 ω2 · · · ωk ∈ I k , k ≥ 1, in the sequel, we will identify the elements a(ω) and a(ωi, ωj) by the sets {{ω1 , ω2 , . . . , ωk }} and {{ω1 , ω2 , . . . , ωk , i}, {ω1 , ω2 , . . . , ωk , j}}, respectively. Thus, we can write α2 = {{{1}, {3}}, {{2}, {4}}}, α3 = {{{1}, {2}}, {{3}}, {{4}}}, α4 = {{{1}}, {{2}}, {{3}}, {{4}}}, and so on. For any n ≥ 2, if card(Cn ) = k, we write  Cn =

{αn,1 , αn,2 , · · · , αn,k } if k ≥ 2, if k = 1. {αn }

If card(Cn ) = k and card(Cn+1 ) = m, then either 1 ≤ k ≤ m, or 1 ≤ m ≤ k (see Table 1). Moreover, by Theorem 1, an optimal set at stage n can contribute multiple distinct optimal sets at stage n + 1, and multiple distinct optimal sets at stage n can contribute one common optimal set at stage n + 1; for example from Table 1, one can see that the number of α21 = 8, the number of α22 = 28, the number of α23 = 56, the number of α24 = 70, and the number of α25 = 56. By αn,i → αn+1, j , it is meant that the optimal set αn+1, j at stage n + 1 is obtained from the optimal set αn,i at stage n, similar is the meaning for the notations αn → αn+1, j , or αn,i → αn+1 , for example from Fig. 2: 

 α16 → α17,1 , α16 → α17,2 , α16 → α17,3 , α16 → α17,4 ;     { α17,1 → α18,1 , α17,1 → α18,2 , α17,1 → α18,4 , α17,2 → α18,1 , α17,2 → α18,3 , α17,2 → α18,5 ,     α17,3 → α18,2 , α17,3 → α18,3 , α17,3 → α18,6 , α17,4 → α18,4 , α17,4 → α18,5 , α17,4 → α18,6 };       { α18,1 → α19,1 , α18,1 → α19,2 , α18,2 → α19,1 , α18,2 → α19,3 , α18,3 → α19,1 , α18,3 → α19,4 ,       α18,4 → α19,2 , α18,4 → α19,3 , α18,5 → α19,2 , α18,5 → α19,4 , α18,6 → α19,3 , α18,6 → α19,4 ;   α19,1 → α20 , α19,2 → α20 , α19,3 → α20 , α19,4 → α20 .

Moreover, one can see that  α8 = {{1, 1}, {1, 3}}, {{1, 2}, {1, 4}}, {{2, 1}, {2, 3}}, {{2, 2}, {2, 4}}, {{3, 1}, {3, 3}},  31 = 0.0119599; {{3, 2}, {3, 4}}, {{4, 1}, {4, 3}}, {{4, 2}, {4, 4} with V8 = 2592  α9,1 = {{3, 3}}, {{3, 4}}, {{1, 1}, {1, 3}}, {{1, 2}, {1, 4}}, {{2, 1}, {2, 3}}, {{2, 2}, {2, 4}},

Optimal Quantizers for a Nonuniform Distribution …

61

Table 1 Number of αn in the range 5 ≤ n ≤ 82 n

card(Cn ) n

5

2

18

card(Cn ) n 6

31

card(Cn ) n 4

44

70

card(Cn ) n 57

card(Cn ) n 8

70

card(Cn ) 6

6

1

19

4

32

1

45

56

58

28

71

4

7

2

20

1

33

4

46

28

59

56

72

1

8

1

21

8

34

6

47

8

60

70

73

24

9

2

22

28

35

4

48

1

61

56

74

276

10

1

23

56

36

1

49

8

62

28

75

2024

11

2

24

70

37

4

50

28

63

8

76

10626

12

1

25

56

38

6

51

56

64

1

77

42504

13

2

26

28

39

4

52

70

65

4

78

134596

14

1

27

8

40

1

53

56

66

6

79

346104

15

2

28

1

41

8

54

28

67

4

80

735471

16

1

29

4

42

28

55

8

68

1

81

1307504

17

4

30

6

43

56

56

1

69

4

82

1961256

 {{3, 1}, {3, 2}}, {{4, 1}, {4, 3}}, {{4, 2}, {4, 4}} ,  α9,2 = {{4, 3}}, {{4, 4}}, {{1, 1}, {1, 3}}, {{1, 2}, {1, 4}}, {{2, 1}, {2, 3}}, {{2, 2}, {2, 4}},  25 = 0.00964506; {{3, 1}, {3, 3}}, {{3, 2}, {3, 4}}, {{4, 1}, {4, 2}} with V9 = 2592  α10 = {{3, 3}}, {{3, 4}}, {{4, 3}}, {{4, 4}}, {{1, 1}, {1, 3}}, {{1, 2}, {1, 4}}, {{2, 1}, {2, 3}},  19 = 0.00733025, {{2, 2}, {2, 4}}, {{3, 1}, {3, 2}}, {{4, 1}, {4, 2}} with V10 = 2592

and so on. Note 3 Notice that there is only one optimal set of n-means for n = 72. By the notations used in Theorem 1, we can write α72 = {a(i) : 1 ≤ i ≤ 72}. Then, W (α72 ) = {{{1, 3, 3}}, {{1, 3, 4}}, {{1, 4, 3}}, {{1, 4, 4}}, {{2, 3, 3}}, {{2, 3, 4}}, {{2, 4, 3}}, {{2, 4, 4}}, {{3, 1, 3}}, {{3, 1, 4}}, {{3, 2, 3}}, {{3, 2, 4}}, {{3, 3, 1}}, {{3, 3, 2}}, {{3, 4, 1}}, {{3, 4, 2}}, {{4, 1, 3}}, {{4, 1, 4}}, {{4, 2, 3}}, {{4, 2, 4}}, {{4, 3, 1}}, {{4, 3, 2}}, {{4, 4, 1}}, {{4, 4, 2}}}.

= Since card(W (α72 )) = 24, by the theorem, we have card(C73 ) = 24 1 24 24 24 24, card(C74 ) = 2 = 276, card(C75 ) = 3 = 2024, card(C76 ) = 4 = 10626, etc., for details see Table 1. Let us now conclude the paper with the following remark: Remark 5 Consider a set of four contractive affine transformations S(i, j) on R2 , such that S(1,1) (x1 , x2 ) = ( 41 x1 , 41 x2 ), S(2,1) (x1 , x2 ) = ( 21 x1 + 21 , 41 x2 ), S(1,2) (x1 , x2 ) = ( 41 x1 , 21 x2 + 21 ), and S(2,2) (x1 , x2 ) = ( 21 x1 + 21 , 21 x2 + 21 ) for all (x1 , x2 ) ∈ R2 . Let S be the limit set of these contractive mappings. Then, S is called the Sierpi´nski

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carpet generated by S(i, j) for all 1 ≤ i, j ≤ 2. Let P be the Borel probability mea−1 −1 −1 −1 1 3 3 9 P ◦ S(1,1) + 16 P ◦ S(2,1) + 16 P ◦ S(1,2) + 16 P ◦ S(2,2) . sure on R2 such that P = 16 Then, P has support the Siepi´nski carpet S. For this probability measure, the optimal sets of n-means and the nth quantization errors are not known yet for all n ≥ 2. Acknowledgements The research of the author was supported by U.S. National Security Agency (NSA) Grant H98230-14-1-0320

References 1. Abaya, E.F., Wise, G.L.: Some remarks on the existence of optimal quantizers. Stat. Prob. Lett. 2(6), 349–351 (1984) 2. Çömez, D., Roychowdhury, M.K.: Quantization for uniform distributions of Cantor dusts on R2 . Topol. Proc. 56, 195–218 (2020) 3. Du, Q., Faber, V., Gunzburger, M.: Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41(4), 637–676 (1999) 4. Gersho, A., Gray, R.M.: Vector Quantization and Signal Compression. Kluwer Academy publishers, Boston (1992) 5. Gray, R.M., Kieffer, J.C., Linde, Y.: Locally optimal block quantizer design. Inf. Control 45, 178–198 (1980) 6. György, A., Linder, T.: On the structure of optimal entropy-constrained scalar quantizers. IEEE Trans. Inf. Theory 48(2) (2002) 7. Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, vol. 1730. Springer, Berlin (2000) 8. Graf, S., Luschgy, H.: The Quantization of the Cantor Distribution. Math. Nachr. 183, 113–133 (1997) 9. Gray, R.M., Neuhoff, D.L.: Quantization. IEEE Trans. Inf. Theory 44(6), 2325–2383 (1998) 10. Hutchinson, J.: Fractals and self-similarity. Indiana Univ. J. 30, 713–747 (1981) 11. Roychowdhury, M.K.: Quantization and centroidal Voronoi tessellations for probability measures on dyadic Cantor sets. J. Fractal Geom. 4, 127–146 (2017) 12. Roychowdhury, L.: Optimal quantization for nonuniform Cantor distributions. J. Interdiscip. Math. 22(8), 1325–1348 (2019) 13. Zam, R.: Lattice Coding for Signals and Networks: A Structured Coding Approach to Quantization, Modulation, and Multiuser Information Theory. Cambridge University Press (2014)

Fractal Dimension for a Class of Complex-Valued Fractal Interpolation Functions Manuj Verma, Amit Priyadarshi, and Saurabh Verma

Abstract There are many research papers dealing with fractal dimension of realvalued fractal functions in the recent literature. The main focus of our paper is to study the fractal dimension of complex-valued functions. This paper also highlights the difference between dimensional results of the complex-valued and real-valued fractal functions. We study the fractal dimension of the graph of complex-valued function g(x) + i h(x), compare its fractal dimension with the graphs of functions g(x) + h(x) and (g(x), h(x)) and also obtain some bounds. Moreover, we study the fractal dimension of the graph of complex-valued fractal interpolation function associated with a germ function f , base function b, and scaling functions αk . Keywords Box dimension · Iterated function systems · Hausdorff dimension · Fractal interpolation functions · Packing dimension Mathematics Subject Classification Primary (28A80) · Secondary (41A30)

1 Introduction An important concept in fractal geometry is the fractal dimension. Computation of the fractal dimension of graphs and sets has received a lot of attention in the literature [9, 17]. In [2], Barnsley defined the notion of fractal interpolation functions (FIFs) and determined the box dimension (BD) of the affine FIF. Estimation of BD for a class of affine FIFs presented in [4, 5, 11]. Several authors [6, 12–14, 22] also M. Verma (B) · A. Priyadarshi Department of Mathematics, IIT Delhi, New Delhi 110016, India e-mail: [email protected] A. Priyadarshi e-mail: [email protected] S. Verma Department of Applied Sciences, IIIT Allahabad, Prayagraj 211015, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_6

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calculated the fractal dimension of the graphs of FIFs. In 1991, Massopust [16] determined BD of the graphs of vector-valued FIFs. Later, Hardin and Massopust [12] described the FIFs from Rn to Rm and determined the BD of its graph. The reader can see some recent works on fractal dimension of fractal functions defined on different domains such as Sierpinski gasket [1, 20], rectangular domain [8], and interval [13, 22]. To the best of our knowledge, we may say that there is no work available for the dimension of complex-valued fractal functions. Here, we give some basic results for complex-valued FIF and provide some results to convince the reader that there is some difference between the dimensional result of the complex-valued and real-valued fractal functions. In 1986, Mauldin and Williams [17] were the pioneers who studied the problem of decomposition of the continuous functions in terms of the Hausdorff dimension (HD). They proved the existence of decomposition of any continuous function on [0, 1] into a sum of two continuous functions, where each has HD one. Later in 2000, Wingren [23] gave a technique to construct the above decomposition of Mauldin and Williams. Moreover, he proved the same type of result as Mauldin and Williams for the lower BD. Bayart and Heurteaux [7] also proved the similar result for HD β = 2, and raised the question for β ∈ [1, 2]. Recently, in 2013, Jia Liu and Jun Wu [15] solved the question which was raised by Bayart and Haurteaux. More precisely, they showed that, for any given β ∈ [1, 2], we can decompose any continuous function on [0, 1] into a sum of two continuous functions, where each has HD β. Falconer and Fraser [10] proved that the upper BD of the graph of the sum of two continuous functions depends on BD of both graphs. In [10, 15], the authors determined that HD of the graph of g + h does not depend on HD of the graph of g and h, whereas the upper BD depends on both. Motivated by this, we think about the behavior of HD of the graph of g + i h, whether it depends on HD of graphs of g and h or not. We obtained an affirmative answer to this question. Also, the upper BD of g + i h depends on the upper BD of g and h which is quite different from the upper BD of g + h. Finally, we studied some relations between fractal dimensions of the graphs of g(x) + i h(x), g(x) + h(x), and (g(x), h(x)). The article is arranged as follows. In the upcoming Sect. 1.1, we provide some preliminary results and the required definition for the next section. Section 2 contains some results related to the dimension of the complex-valued continuous functions and the FIFs. In this section, first, we establish some results to form a connection between the fractal dimension of complex-valued and real-valued continuous functions. After that, we determine the fractal dimension of FIFs under some assumptions. We also obtain some conditions under which α-fractal function becomes Hölder continuous function and bounded variation function, and calculate its fractal dimension.

1.1 Preliminaries Definition 1 Let (Y, d) be a metric space and F ⊆ Y . The Hausdorff dimension (HD) dim H F of F is given by

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dim H F = inf{η > 0 : for each δ > 0, ∃ cover {Vi } of F with



|Vi |η < δ},

where |Vi | is the diameter of Vi . Definition 2 Let (Y, d) be a metric space and F ⊆ Y, F = ∅. The box dimension (BD) of the set F is defined as log Nδ (F) , δ→0 − log δ

dim B F = lim

where Nδ (F) is the minimum number of sets of diameter δ > 0 that can cover F. If this limit does not exist, then liminf and limsup are known as the lower and upper BDs, respectively.   l Definition 3 For any δ > 0 and l ≥ 0, let Pδl (F) := sup n |On | , where {On } is a set of the pairwise disjoint balls of the diameter less than or equal to 2δ with centers in F ⊆ Y . One can observe that Pδs decreases as δ decreases. Thus, limδ→0 Pδl (F) = P0l (F) exists. We define l-dimensional packing measure as P (F) = inf l



P0l (Fn )

n

:F⊂

∞ 

 Fn .

n=1

With the help of packing measure, we define the packing dimension (PD) as follows: dim P (F) = sup{l ≥ 0 : P l (F) = ∞} = inf{l ≥ 0 : P l (F) = 0}. Note The graph of function f will be denoted by G( f ) throughout this paper. For σ ∈ (0, 1], the Hölder space Hσ ([a, b], R), Hσ ([a, b], R) := { f : [a, b] → R : | f (t1 ) − f (t2 )| ≤ C f |t1 − t2 |σ , ∀ t1 , t2 ∈ [a, b], for some C f > 0}.

Remark 1 f = f 1 + i f 2 : [a, b] → C is a Hölder function with exponent σ if and only if f i ∈ Hσ ([a, b], R) for each i = 1, 2. Theorem 1 ([9]) If f ∈ Hσ ([a, b], R), then dim B (G( f )) ≤ 2 − σ. Let (Y, d) be a complete metric space. The class of all non-empty compact subsets of Y is denoted by H (Y ). Let A1 , A2 ∈ H (Y ). The Hausdorff metric D on H (Y ) is given by D(A1 , A2 ) = max{max min d(a1 , a2 ), max min d(a1 , a2 )}. a1 ∈A1 a2 ∈A2

Then the metric space (H (Y ), D) is complete.

a2 ∈A2 a1 ∈A1

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Definition 4 A mapping θ : (Y, d) → (Y, d) is said to be a contraction if d(θ(a), θ(b)) ≤ c d(a, b), ∀ a, b ∈ Y, for some c < 1.

  Definition 5 The system I = (Y, d); θ1 , θ2 , . . . , θ N is said to be an iterated function system (IFS), if for every i ∈ {1, 2, . . . , N }, θi : (Y, d) → (Y, d) is a contraction.   Note- Let I = (Y, d); θ1 , θ2 , . . . , θ N be an IFS. We define a mapping S : H (Y ) → H (Y ) as N θi (A). S(A) = ∪i=1 Thus, S is a contraction on (H (Y ), D). If (Y, d) is complete, then using Banach fixed N θi (E). This compact set E point theory; there is a unique E ∈ H (Y ) with E = ∪i=1 is said to be an attractor of the IFS, see for instance, [3, 9]. Definition 6 Let I = {(Y, d); θ1 , θ2 , . . . , θ N } be an IFS and E be an attractor of I. The IFS I satisfies the strong separation condition (SSC) if θi (E) ∩ θ j (E) = ∅ ∀ i = j. And if there is an open set O such that O = ∅, θi (O) ∩ θ j (O) = ∅ ∀ i = j and θi (O) ⊂ O ∀ i, then I satisfies the open set condition (OSC). Furthermore, if O ∩ E = ∅ then I satisfies the strong open set condition (SOSC) (see [21]).

1.2 Fractal Interpolation Functions Let us assume a finite data set {(xi , yi ) ∈ R × C : i = 1, 2, . . . , N } such that x1 < x2 < · · · < x N . Let L = [x1 , x N ] and T = {1, 2, ..., N − 1}. Set L k = [xk , xk+1 ] for every k ∈ T. For each k ∈ T , We define a contractive map Pk : L → L k such that Pk (x1 ) = xk , Pk (x N ) = xk+1 .

(1)

For every k ∈ T , we define a continuous map k : L × C → C such that |k (t, ξ1 ) − k (t, ξ2 )| ≤ sk |ξ1 − ξ2 |, k (x1 , y1 ) = yk , k (x N , y N ) = yk+1 , where 0 ≤ sk < 1 and (t, ξ1 ), (t, ξ2 ) ∈ L × C. For every k ∈ T , we can take particular choices of Pk and k as Pk (t) = ak t + dk , k (t, ξ) = αk ξ + qk (t). Since Pk satisfy Eq. (1), we can obtain the unique constants ak and dk . The constant multiplier αk is said to be a scaling factor and |αk | < 1. The map qk : L → C is a

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continuous with the property that qk (x1 ) = yk − αk y1 and qk (x N ) = yk+1 − αk y N . With the help of Pk and k , for every k ∈ T , we define a continuous function Wk : L × C → L × C as

Wk (t, ξ) = Pk (t), k (t, ξ) . Thus, J := {L × C; W1 , W2 , . . . , W N −1 } is an IFS. By [2, Theorem 1], J has a unique attractor and this attractor is the graph of a function h which satisfies:



h(t) = αk h Pk−1 (t) + qk Pk−1 (t) , where t ∈ L k and k ∈ T. The function h is called FIF.

1.3 α-Fractal Functions We can adapt the idea of the construction of FIF. The set of complex-valued continuous functions defined on L = [x1 , x N ] ⊂ R is denoted by C(L , C), with the sup norm. Let f be a given function in C(L , C), known as the germ function. For constructing the IFS, we consider the following assumptions: 1. Let  := {(x1 , x2 , . . . , x N ) : x1 < x2 < · · · < x N } be a partition of L = [x1 , x N ]. 2. Let αk ∈ C(L , C) with αk ∞ = max{|αk (t)| : t ∈ L} < 1, for all k ∈ T . These αk are called the scaling functions. 3. Let b ∈ C(L , C) with b = f and b(xi ) = f (xi ) for i ∈ {1, N } and, named as the base function. Motivated by [2, 3], Navascués [18] considered the following set of functions: Pk (t) = ak t + dk ,



k (t, ξ) = αk (t)ξ + f Pk (t) − αk (t)b(t).

(2)

Then the corresponding IFS J := {L × C; W1 , W2 , . . . , W N −1 }, where  Wk (t, ξ) = Pk (t), k (t, ξ) , α : has a unique attractor and this attractor is the graph of a continuous function f ,b α α α L → C with f ,b (xk ) = f (xk ), k ∈ T . For simplicity, we write f ,b by f . The real valued f α is widely known as α-fractal function, see, for instance, [1, 8, 13, 22]. Moreover, f α satisfies



f α (t) = f (t) + αk (Pk−1 (t)).( f α − b) Pk−1 (t) for all t ∈ L and k ∈ T . The function f α is a “fractal perturbation” of f .

(3)

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2 Main Theorems In the upcoming lemma, we determine a relation between HD of the graph of a complex-valued continuous function and its real and imaginary parts. Lemma 1 Suppose f ∈ C([a, b], C) and g, h : [a, b] → R is real and imaginary part of f , respectively, that is, f = g + i h. Then (1) dim H (G(g + i h)) ≥ max{dim H (G(g)), dim H (G(h))}. (2) dim H (G(g + i h)) = dim H (G(g)), whenever h is Lipschitz. Proof (1) We consider Φ : G( f ) → G(g) as Φ(t, g(t) + i h(t)) = (t, g(t)). We aim to show that Φ is a Lipschitz mapping. Using simple properties of norm, it follows that Φ(t1 , g(t1 ) + i h(t1 )) − Φ(t2 , g(t2 ) + i h(t2 ))2 = (t1 , g(t1 )) − (t2 , g(t2 ))2 = |t1 − t2 |2 + |g(t1 ) − g(t2 )|2 ≤ |t1 − t2 |2 + |g(t1 ) − g(t2 )|2 + |h(t1 ) − h(t2 )|2 = (t1 , g(t1 ) + i h(t1 )) − (t2 , g(t2 ) + i h(t2 )2 . That is, Φ is Lipschitz. Thus, from [9, Corollary 2.4], we obtain dim H (G( f )) ≥ dim H (G(g)). On similar lines, we obtain dim H (G( f )) ≥ dim H (G(h)). Combining both of the above inequalities, we get dim H (G( f )) ≥ max{dim H (G(g)), dim H (G(h))}, completing the proof of item (i). (2) From Part (1) of Lemma 1, it is obvious that Φ is a Lipschitz. Now, (t1 , g(t1 ) + i h(t1 )) − (t2 , g(t2 ) + i h(t2 )2 = |t1 − t2 |2 + |g(t1 ) − g(t2 )|2 + |h(t1 ) − h(t2 )|2 ≤ |t1 − t2 |2 + C12 |t1 − t2 |2 + |g(t1 ) − g(t2 )|2 ≤ (1 + C12 ){ |t1 − t2 |2 + |g(t1 ) − g(t2 )|2 }

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= (1 + C12 )(t1 , g(t1 )) − (t2 , g(t2 ))2 = (1 + C12 )Φ(t1 , g(t1 ) + i h(t1 )) − Φ(t2 , g(t2 ) + i h(t2 ))2 . Therefore, Φ is bi-Lipschitz. Thus, from [9, Corollary 2.4], we get dim H (G( f )) = dim H (G(g)). Thus, the proof is done. Next, we present similar results for some other dimensions. Proposition 1 Suppose f ∈ C([a, b], C) and g, h : [a, b] → R is real and imaginary part of f , respectively, that is, f = g + i h. Then dim P (G(g + i h)) ≥ max{dim P (G(g)), dim P (G(h))}, dim B (G(g + i h)) ≥ max{dim B (G(g)), dim B (G(h))}, dim B (G(g + i h)) ≥ max{dim B (G(g)), dim B (G(h))}. Proof Using the same idea as in part (1) of Lemma 1, one can easily prove this. Proposition 2 Suppose f ∈ C([a, b], C) and g, h : [a, b] → R is real and imaginary part of f , respectively, that is f = g + i h. If h is Lipschitz, then dim P (G( f )) = dim P (G(g)), dim B (G( f )) = dim B (G(g)), and dim B (G( f )) = dim B (G(g)). Proof Using the idea of part (2) in Lemma 1, one can obtain the required result. Lemma 2 Suppose f ∈ C([a, b], C) and g, h : [a, b] → R is real and imaginary part of f , respectively, that is, f = g + i h. If h is Lipschitz, then dim H (G(g + i h)) = dim H (G(g + h)) = dim H (G(g, h)) = dim H (G(g)), dim B (G(g + i h)) = dim B (G(g + h)) = dim B (G(g, h)) = dim B (G(g)), dim P (G(g + i h)) = dim P (G(g + h)) = dim P (G(g, h)) = dim P (G(g)). Proof We consider Φ : G(g + h) → G(g) as Φ(t, (g(t) + h(t))) = (t, g(t)) is a bi-Lipschitz map, see part(1) of Lemma 1. Now, from [9, Corollary 2.4], we get

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dim H (G(g + h)) = dim H (G(g)).

(4)

And we can show that Φ : G(g, h) → G(g) defined by Φ(t, (g(t), h(t))) = (t, g(t)), is a bi-Lipschitz map, see part (2) of Lemma 1. Thus, from [9, Corollary 2.4], we obtain (5) dim H (G(g, h)) = dim H (G(g)). Further, by Lemma 1, Eqs. 4 and 5, we get dim H (G(g + i h)) = dim H (G(g + h)) = dim H (G(g, h)) = dim H (G(g)). Since upper BD, lower BD, and PD satisfy bi-Lipschitz invariance property, the rest follows. Lemma 3 Suppose g, h : [a, b] → R are continuous functions. Then g + i h : [a, b] → C, (g, h) : [a, b] → R2 are continuous functions and dim H G(g + i h) = dim H G(g, h). Proof We consider Φ : G(g + i h) → G(g, h) as Φ(t, g(t) + i h(t)) = (t, (g(t), h(t))). We target to prove that Φ is bi-Lipschitz. Performing simple calculations, we have Φ(t1 , g(t1 ) + i h(t1 )), Φ(t2 , g(t2 ) + i h(t2 ))2 =(t1 , (g(t1 ), h(t1 ))), (t2 , (g(t2 ), h(t2 )))2 =|t1 − t2 |2 + |g(t1 ) − g(t2 )|2 + |h(t1 ) − h(t2 )|2 =(t1 , g(t1 ) + i h(t1 )), (t2 , g(t2 ) + i h(t2 ))2 . Therefore, Φ is bi-Lipschitz. By using [9, Corollary 2.4], we get dim H G(g + i h) = dim H G(g, h). Since upper BD, lower BD, and PD also fulfill the bi-Lipschitz invariance property, we complete the proof. Remark 2 The Peano space filling curve g : [0, 1] → [0, 1] × [0, 1] is a function, which is 21 -Hölder, see details [14]. The component functions satisfy dim H G(g1 ) =

dim H G(g2 ) = 1.5. Since g is a space filling curve, we have dim H G(g ≥ 2. Now, consider a complex-valued mapping f (x) = g1 (x) + ig2 (x), and using Lemma 3, dim H G( f ) ≥ 2. Now, we conclude this remark. If f ∈ Hσ ([0, 1], R) for σ ∈ (0, 1),

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then dim H G( f ) ≤ 2 − σ. But if f ∈ Hσ ([0, 1], C) for σ ∈ (0, 1). Is 2 − σ again the upper bound of dim H G( f )? From the above, we may not get a positive answer. From Remark 2, it is clear that, in general, the dimensional results for the complexvalued and real-valued functions are not same. Now, we are ready to give some dimensional results for the complex-valued FIFs. We define a metric D0 on L × C by D0 ((t1 , ξ1 ), (t2 , ξ2 )) = |t1 − t2 | + |ξ1 − ξ2 |

∀ (t1 , ξ1 ), (t2 , ξ2 ) ∈ L × C.

Then L × C, D0 is a complete metric space. Theorem 2 Let I := {L × C; W1 , W2 , . . . , W N −1 } be the IFS defined in the construction of f α such that ck D0 ((t1 , ξ1 ), (t2 , ξ2 )) ≤ D0 (Wk (t1 , ξ1 ), Wk (t2 , ξ2 )) ≤ Ck D0 ((t1 , ξ1 ), (t2 , ξ2 )), where 0 < ck , Ck < 1 ∀ k ∈ T

and (t1 , ξ1 ), (t2 , ξ2 ) ∈ L × C. Then r ≤ N −1 ckr = 1 and dim H (G( f α )) ≤ R, where r and R are uniquely determined by N −1 k=1

k=1

CkR

= 1, respectively.

Proof For upper bound of dim H (G( f α )), one may follow Proposition 9.6 in [9]. For the lower bound of dim H (G( f α )), we shall proceed in the following way. Let V = (x1 , x N ) × C. Thus, we have Wi (V ) ∩ W j (V ) = ∅, for all i = j ∈ T. Because



Pi (x1 , x N ) ∩ P j (x1 , x N ) = ∅,

∀ i = j ∈ T.

We can observe that V ∩ G( f α ) = ∅, thus the IFS I satisfies the SOSC. Then, there is an index i ∈ T ∗ with Wi (G( f α )) ⊂ V, where T ∗ := ∪n∈N {1, 2, . . . , N − 1}n . We denote Wi (G( f α )) by (G( f α ))i for any i ∈ T ∗ . Now, one can observe that for each n ∈ N, the sets {(G( f α )) ji : j ∈ T n } is disjoint. Then, for each n ∈ N, IFS Ln = {W ji : j ∈ T n } satisfies all the assumptions of Proposition 9.7 in [9]. Hence, by Proposition 9.7 in  [9], if A∗n is an attractor of Ln , then rn ≤ rn ∗ ∗ α dim H (An ), where rn is given by j∈T n c ji = 1. Since An ⊂ G( f ), we have ∗ α α rn ≤ dim H (An ) ≤ dim H (G( f )). Suppose that dim H (G( f )) < r. Thus, rn < r . Let cmax = max{c1 , c2 , . . . , c N −1 }. We have ci−rn =

 j∈T n

crjn ≥

 j∈T n

dim H (G( f α ))−r

crj c j

≥

 j∈T n

n(dim H (G( f crj cmax

α

))−r )

.

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This implies that

n(dim H (G( f ci−r ≥ cmax

α

))−r )

.

We have a contradiction for a large value of n ∈ N. Therefore, we get dim H (G( f α )) ≥ r, proving the assertion. Remark 3 In [19], Roychowdhury estimated HD and BD of the attractor of hyperbolic recurrent IFS consisting of bi-Lipschitz mappings under OSC using Bowen’s pressure function and volume argument. Note that recurrent iterated function is a generalization of the iterated function, hence so is Roychowdhury’s result. We should emphasize on the fact that, in the above, we provide a proof without using pressure function and volume argument. Our proof can be generalized to general complete metric spaces. Remark 4 Theorem 2 can be compared with Theorem 2.4 in [13]. The Hölder space is defined as follows: Hσ (L , C) := {h : L → C : h is a Hölder function with exponent σ}. We know that (Hσ (L , C), .H ) is a complete norm linear space, where hH := h∞ + [h]σ and [h]σ =

sup

t1 ,t2 ∈L ,t1 =t2

|h(t1 ) − h(t2 )| . |t1 − t2 |σ

Theorem 3 Let f, b, α ∈ Hσ (L , C). Set c := min{ak : k ∈ T }. If f α ∈ Hσ (L , C).

αH cσ


0, σ ∈ (0, 1]. Let f 1 , f 2 be component of f , b1 , b2 be component of b,α1j , α2j be component of α j and f 1α , f 2α be component of f α . Let M > 0 such that  f α ∞ = max{| f α (t)| : t ∈ L} ≤ M. Also, consider constants l fi , δ0 > 0 such that for all t1 ∈ L and δ < δ0 , there is a t2 ∈ L with |t1 − t2 | ≤ δ and | f i (t1 ) − f i (t2 )| ≥ l fi |t1 − t2 |σ for i ∈ {1, 2}. If αH < cσ min



−σ l −2(b +M)l c−σ 1 l f1 −2(b∞ +M)lα c , , f2 2(Cα,∞f,b +lb ) α N 2(Cα, f,b +lb )

 , then we have





1 ≤ dim H G( f iα ) ≤ dim B G( f iα ) = 2 − σ for i = 1, 2.

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Moreover, 1 ≤ dim H G( f α ) ≤ dim B G( f α ) ≥ 2 − σ. Proof Since αH
0 and for i = 1, 2. Firstly, we try to obtain the upper bound of the upper BD of G( f iα ) ∀ i ∈ {1, 2} as follows: For δ ∈ (0, 1), assume m =  1δ , where . denotes the ceiling function and Nδ (G( f iα )) is the least number of δ-squares that covers G( f iα ), we have Nδ (G( f iα )) ≤ 2m +

m−1  r =0

R fiα [(r δ, (r + 1)δ] δ



 m−1  R f α [r δ, (r + 1)δ] 1 i +1 + ≤2 δ δ r =0  m−1   1 +1 + Cα, f,b δ σ−1 . ≤2 δ r =0 

(7)

From this, we deduce that

log Nδ (G( f iα )) dim B G( f iα ) = lim ≤2−σ δ→0 − log δ

Next, we will prove that dim B G( f iα ) ≥ 2 − σ self-referential equation of f α , we obtain

∀ i = 1, 2.

∀ i = 1, 2. For this, using the





 f 1α (t) = f 1 (t) + αk1 Pk−1 (t) f 1α Pk−1 (t) − b1 Pk−1 (t)



 − αk2 Pk−1 (t) f 2α Pk−1 (t) − b2 Pk−1 (t)

(8)

for every t ∈ L k and k ∈ T. Let t1 , t2 ∈ L k such that |t1 − t2 | ≤ δ. From Eq. 8, we have 



 | f 1α (t1 ) − f 1α (t2 )| = f 1 (t1 ) − f 1 (t2 ) + αk1 Pk−1 (t1 ) f 1α Pk−1 (t1 )







− αk1 Pk−1 (t2 ) f 1α Pk−1 (t2 ) − αk1 Pk−1 (t1 b1 Pk−1 (t1 )





+ αk1 Pk−1 (t2 ) b1 Pk−1 (t2 ) − αk2 Pk−1 (t1 ) f 2α Pk−1 (t1 )





+ αk2 Pk−1 (t2 ) f 2α Pk−1 (t2 ) + αk2 Pk−1 (t1 ) b2 Pk−1 (t1 )



 − αk2 Pk−1 (t2 ) b2 Pk−1 (t2 )  

  ≥| f 1 (t1 ) − f 1 (t2 )| − α∞  f 1α Pk−1 (t1 ) − f 1α Pk−1 (t2 )  



  − α∞ b1 Pk−1 (t1 ) − b1 Pk−1 (t2 )  − b∞ +  f α ∞

Fractal Dimension for a Class of Complex-Valued Fractal …

75

 



 1 −1

 αk Pk (t1 ) − αk1 Pk−1 (t2 )  − α∞  f 2α Pk−1 (t1 ) 



  − f 2α Pk−1 (t2 )  − α∞ b2 Pk−1 (t1 ) − b2 Pk−1 (t2 ) 



 − b∞ +  f α ∞ αk2 Pk−1 (t1 ) − αk2 Pk−1 (t2 ) . With the help of Eq. (6), we get σ    | f 1α (t1 ) − f 1α (t2 )| ≥ l f 1 |t1 − t2 |σ − 2α∞ Cα, f,b Pk−1 (t1 ) − Pk−1 (t2 )  σ   − 2α∞ lb Pk−1 (t1 ) − Pk−1 (t2 ) σ

  − 2 b∞ + M lα Pk−1 (t1 ) − Pk−1 (t2 ) ≥ l f 1 |t1 − t2 |σ − 2α∞ Cα, f,b a −σ |t1 − t2 |σ − 2α∞ lb c−σ |t1 − t2 |σ

− 2 b∞ + M c−σ lα |t1 − t2 |σ 

= l f 1 − 2(Cα, f,b + lb )α∞ c−σ − 2 b∞ + M c−σ lα |t1 − t2 |σ .



Set M0 := l f1 − 2 b∞ + M c−σ lα − 2(Cα, f,b + lb )α∞ c−σ . Thus by the assumption M0 > 0. Let δ = cn for n ∈ N and w =  c1n . We estimate Nδ (G( f 1α )) ≥

w 



 max 1, c−n R f1α [r δ, (r + 1)δ]

r =0 w  −n

c R f1α [r δ, (r + 1)δ] ≥ r =0

≥

w 

M0 c−n cnσ

r =0

= M0 cn(σ−2) . Thus, we have



dim B G( f 1α ) = lim

δ→0

 log Nδ (G( f 1α )) − log(δ)

≥ lim

 log M0 cn(σ−2)

n→∞

= 2 − σ, Similarly, we get

establishing the result.



dim B G( f 2α ) ≥ 2 − σ,

−n log c

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Definition 7 Suppose h : L → C is a function. The total variation V (h, L , C) of h on L, m  V (h, L , C) = sup |h(γi ) − h(γi−1 )|. Q=(γ0 ,γ1 ,...,γm ) partition of L

i=1

We call h a function of bounded variation if V (h, L , C) < ∞. BV(L , C):={h : L → C : V (h, L , C) < ∞}. The space BV(L , C) is Banach under the norm hBV := |h(x0 )| + V (h, L , C). Theorem 5 If f, b ∈ BV(L , C) ∩ C(L , C) and αk ∈ BV(L , C) ∩ C(L , C) ∀ k ∈ T such that αBV < 2(N1−1) , then f α ∈ BV(L , C) ∩ C(L , C). Moreover, dim H (G( f α )) = dim B (G( f α )) = 1. Proof Following Theorem 3.11 and [13, Theorem 3.24 ], one may complete the proof. Remark 5 The above theorem will reduce to [13, Theorem 3.24] when all functions f, b and αk are real-valued.

References 1. Agrawal, V., Som, T.: Fractal dimension of α-fractal function on the Sierpi´nski Gasket. Eur. Phys. J. Spec. Top. 230(21), 3781–3787 (2021) 2. Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2(1), 303–329 (1986) 3. Barnsley, M.F.: Fractal Everywhere. Academic, Orlando, Florida (1988) 4. Barnsley, M.F., Elton, J., Hardin, D.P., Massopust, P.R.: Hidden variable fractal interpolation functions. SIAM J. Math. Anal. 20(5), 1218–1242 (1989) 5. Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. J. Approx. Theory 57(1), 14–34 (1989) 6. Barnsley, M.F., Massopust, P.R.: Bilinear fractal interpolation and box dimension. J. Approx. Theory 192, 362–378 (2015) 7. Bayart, F., Heurteaux, Y.: On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets. Further Developments in Fractals and Related Fields, pp. 25–34 (2013) 8. Chandra, S., Abbas, S.: The calculus of bivariate fractal interpolation surfaces. Fractals 29(3), 2150066 (2021) 9. Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (1999) 10. Falconer, K.J., Fraser, J.M.: The horizon problem for prevalent surfaces. Math. Proc. Camb. Philos. Soc. 151(2), 355–372 (2011) 11. Hardin, D.P., Massopust, P.R.: The capacity for a class of fractal functions. Commun. Math. Phys. 105(3), 455–460 (1986) 12. Hardin, D.P., Massopust, P.R.: Fractal interpolation functions from Rn to Rm and their projections. Zeitschrift für Analysis und ihre Anwendungen 12(3), 535–548 (1993) 13. Jha, S., Verma, S.: Dimensional analysis of α-fractal functions. RM 76(4), 1–24 (2021)

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14. Kono, N.: On self-affine functions. Jpn. J. Appl. Math. 3(2), 259–269 (1986) 15. Liu, J., Wu, J.: A remark on decomposition of continuous functions. J. Math. Anal. Appl. 401(1), 404–406 (2013) 16. Massopust, P.R.: Vector-valued fractal interpolation functions and their box dimension. Aequationes Math. 42(1), 1–22 (1991) 17. Mauldin, R.D., Williams, S.C.: On the Hausddorff dimension of some graphs. Trans. Am. Math. Soc. 298(2), 793–803 (1986) 18. Navascués, M.A.: Fractal polynomial interpolation. Zeitschrift für Analysis und ihre Anwendungen 24(2), 401–418 (2005) 19. Roychowdhury, M.K.: Hausdorff and upper box dimension estimate of hyperbolic recurrent sets. Israel J. Math. 201(2), 507–523 (2014) 20. Sahu, A., Priyadarshi, A.: On the box-counting dimension of graphs of harmonic functions on the Sierpi´nski gasket. J. Math. Anal. Appl. 487(2), 124036 (2020) 21. Schief, A.: Separation properties for self-similar sets. Proc. Am. Math. Soc. 122(1), 111–115 (1994) 22. Verma, S., Viswanathan, P.: A revisit to α-fractal function and box dimension of its graph. Fractals 27(6), 1950090 (2019) 23. Wingren, P.: Dimensions of graphs of functions and lacunary decompositions of spline approximations. Real Analysis Exchange, pp. 17–26 (2000)

A Note on Complex-Valued Fractal Functions on the Sierpinski ´ Gasket V. Agrawal and T. Som

Abstract Traditional non-recursive approximation methods are less versatile than fractal interpolation and approximation approaches. The concept of fractal interpolation functions (FIFs) have been found to be an effective technique for generating interpolants and approximants which can approximate functions generated by nature that exhibit self-similarity when magnified. Using an iterated function system (IFS), Barnsley discovered the FIFs, which is the most prominent approach for constructing fractals. In this article, we investigate some properties of the real-valued fractal operator and the complex-valued fractal operator defined on the Sierpi´nski gasket (SG in short). We also calculate the bound for the perturbation error on SG. Furthermore, we prove that the complex-valued fractal operator is bounded. In the last part, we establish the connection between the norm of the real-valued fractal operator and the complex-valued fractal operator. Keywords Fractal interpolation functions · Sierpi´nski gasket · Self-similar measure

1 Introduction Smooth functions are used for representing a collection of data or approximating a difficult function in classical approximation theory, which has a significant history. The idea of FIFs represents a significant step forward in the fields of numerical analysis and approximation theory. This is due to the fact that the functions that are used in FIFs are not always differentiable, and as a result, they represent the rough nature of signals that come from the real world. Approximating naturally existing V. Agrawal (B) · T. Som Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi 221005, India e-mail: [email protected] T. Som e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_7

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functions that exhibit self-similarity under magnification, Barnsley [5] proposed the idea of FIFs by using IFS on a compact interval of R, a number of his followers further explored the field of fractal interpolation and approximation. Over the past two decades, this field of study has experienced a tremendous expansion, and intriguing new possibilities have emerged. Celik et al. [7] have extended Barnsley’s work by introducing FIFs on the SG. Furthermore, Ruan [17] generalized the notion of FIFs on a fractal domain. We suggest the reader to [11] for a comprehensive development of the theory of IFS and fractal functions, and fractal surfaces. Massopust [11] has discussed local IFS as well as novel fractal interpolation. One of the most prominent example of post-critically finite fractal in fractal theory is SG, which was invented by Polish mathematician W. Sierpi´nski in 1915. Kigami [10] has explored fractal analysis on a self-similar set. FIFs and linear FIFs have been presented by Ruan [17] on a self-similar set. He furthermore showed that linear FIFs exhibit finite energy. On SG, Ri and Ruan [16] have examined some fundamental features of uniform FIFs, a specific family of FIFs. Verma et al. [20] have studied the fractal operator associated with the bivariate FIFs. In [23], through fractal dimensions, the authors have developed a novel concept of constrained approximation. On the SG, the dimensions of graphs of FIFs have been investigated by Sahu and Priyadarshi in [18]. Fractal dimension is a crucial aspect of fractal geometry, since it provides details about the geometric structure of the objects it examines. There are several notions of fractal dimension, with the Hausdorff dimension and the box dimension being the most prevalent and these dimensions of the graphs of FIFs have been analyzed in detail. In the oscillation spaces, Verma and Sahu [24] have investigated the fractal dimensions of various functions. They have also developed a certain bounded variation concepts for the SG, from which they deduced several dimensional conclusions. T. Bagby [4] has explored mean approximation in the plane C using complex analytic functions. Recently, FIFs have been produced by Prasad and Verma [14] on the products of two SG. Furthermore, they have gathered certain observations about the smoothness of the produced FIFs. The author of [15] has demonstrated that the graphs of FIFs formed on the SG are attractors of some IFS and provide the new nonlinear FIFs. Furthermore, Navascués et al. [13] have studied the vector-valued interpolation functions on the SG through a certain family of fractal functions. In [8, 9], dimensions of FIFs are investigated more rigorously by Jha and her collaborators. In [20]–[25], Verma and his collaborators have discussed the dimension of α-fractal functions in more detail. They have discussed the class of univariate and bivariate FIFs and constrained approximation in their research. In [1], Agrawal and Som have studied the fractal dimension of α-fractal function on the SG and the same authors in [2] have investigated the L p approximation using fractal functions on the SG. Agrawal et al. [3] have further introduced the concept of dimension-preserving approximation for bivariate continuous functions. In [12], Navascués has defined an α-fractal function associated with the square-integrable complex-valued function on the real compact interval. Furthermore, she has explored significant properties of the associated fractal operator. Motivated by her work, we define the α-fractal function associated with the square-integrable complex-valued function on SG. Furthermore, we study the properties of the associated fractal operator.

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This paper becomes more intriguing in many ways because it will use the definition of self-similar measures on SG to get most of the important results. The paper is organized as follows: In Sect. 2, we give a few preliminaries required for the paper. We denote the space of all the real-valued continuous functions defined on SG by C(SG) and space of all the complex-valued continuous functions defined on SG by C(SG, C). Let L2 (SG) = {h | h : SG → R and h2 < ∞} and L2 (SG, C) = {h|h : SG → C and h2 < ∞}. In Sect. 3, we determine the bounds of the real-valued fractal operator by imposing certain conditions. In Sect. 4, we establish some bounds of the complex-valued fractal operator FCα : C(SG, C) → C(SG, C) that maps to a complex-valued continuous function f to its fractal version f α .

2 Technical Introduction To obtain an attractor, we consider the following IFS: {H ; W j , j = 1, 2, . . . , k}, where (H, d) is a complete metric space (CMS) and W j : H → H are contractive mappings. The aforementioned IFS aids to build the mapping W : D(H ) → D(H ), which is defined as follows: W (F) = ∪kj=1 W j (F). The symbol D(H ) stands for the class of all non-empty compact subsets of H . The map W acting on D(H ) endowed with Hausdorff metric h d is a contraction mapping. The contraction ratio α of W is equal to max{α j : 1 ≤ j ≤ k}, where α j is the contraction ratio of W j .Then, by the Banach contraction principle, we get a unique non-empty compact subset F∗ , which satisfies F∗ = ∪kj=1 W j (F∗ ), the set F∗ is referred to be an attractor of the IFS. One can refer to [6] for more information. We use a self-similar measure to prove all the results in this paper and this measure arises from the IFS with probability vectors, which is the fixed point of the Markov operator, i.e., the invariant measure of the IFS with probability vectors. To understand the self-similar measure, we refer the reader to [6].

3 Fractal Interpolation Function on SG We begin with a brief review of the relevant definitions and preliminary information on the SG. The reader can find further information at [7, 10, 19]. SG is constructed with the help of a very important technique, which is known as the IFS. Here, we generate this system via three contraction mappings defined on the R2 plane. Let V0 = { p j : 1 ≤ j ≤ 3} is the collection of the equilateral triangle’s

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three vertices. Corresponding to these three points, contraction mappings are defined as follows: ψ j (t) =

1 (t + p j ). 2

The following IFS provides the SG as an attractor, SG = ψ1 (SG) ∪ ψ2 (SG) ∪ ψ3 (SG). For n, N ∈ N, let us represent the set of all words having a length of n by {1, 2, 3}n , that is, if i ∈ {1, 2, 3}n , then i = i 1 , i 2 , . . . , i n , where i j ∈ {1, 2, 3}. We define it for i ∈ {1, 2, 3} N and we further write S instead of {1, 2, 3} N . ψi = ψi1 ◦ ψi2 ◦ · · · ◦ ψi N . Let μs be a self-similar measure on SG. This can be written as 1 μs ◦ ψi−1 , 3 i=1 3

μs =

1 d(μs ◦ ψi−1 ). dμs = 3 i=1 3

(1)

Consider the set VN is vertices on N th level and defined by VN = { p, ψi ( p) : i ∈ S and p ∈ V0 }. Let us assume f : SG → R be a square-integrable function on SG. The following IFS arises an attractor for the graph of f α , which satisfies f α |VN = f |VN . Let us assume Y = SG × R and define the map Wi : Y → Y by   Wi (t, x) = ψi (t), E i (t, x) , i ∈ S, where E i (t, x) : SG × R → R is a contraction map in the 2nd coordinate, where i ∈ S with E i ( p j , f ( p j )) = f (ψi ( p j )). More precisely, we define E i (t, x) = αi (t)x + f (ψi (t)) − αi (t)b(t), where a square-integrable function b : SG → R is a base function, which satisfies b( p j ) = f ( p j ), 1 ≤ j ≤ 3, and for any i ∈ S, αi ∈ C(SG) holds αi ∞ < 1. We now have an IFS {Y ; Wi , i ∈ S}.     f α (t) = f (t) + αi (ψi−1 (t)) f α ψi−1 (t) − αi (ψi−1 (t)) b ψi−1 (t) , for each t ∈ ψi (SG), i ∈ S.

(2)

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In the research, it is proven that f α is non-differentiable and dependent on certain parameters. It has a Hausdorff dimension which is not an integer. One can consider f α to be a “fractal perturbation” of f and f α is referred to as the α-fractal function corresponding to f . Furthermore, using the Eq. 1 and the idea of a change of variables, we deduce  ψi (SG)

1 3 i=1 3

g ◦ ψi−1 (t) dμs (t) =

 ψi (SG)

g ◦ ψi−1 (t) d(μs ◦ ψi−1 )(t)

3  1 g(t˜)dμs (t˜) = 3 i=1 SG  = g(t˜) dμs (t˜),

(3)

SG

for any g ∈ C(SG). Theorem 1 Let f : SG → R be a square-integrable function on SG and  f 2 =   21 2 SG | f | dμs . The IFS {Y ; Wi , i ∈ S} defined above has a unique attractor graph( f α ). The set graph( f α ) = {(x, f α (x)) : x ∈ SG} is the graph of a squareintegrable function f α : SG → R which satisfies f α |VN = f |VN . If α∞ < 1N , 32 then f α satisfies the following functional equation:   f α (t) = f (t) + αi (ψi−1 (t))( f α − b) ψi−1 (t) ∀ t ∈ ψi (SG), i ∈ S.

(4)

Proof Let L2f (SG) = {g ∈ L2 (SG) : g|V0 = f |V0 }. One can derive directly that set L2f (SG) is a closed subset of L2 (SG) by ordinary calculations. Since (L2 (SG), .2 ) is a Banach space, we get L2f (SG) is a CMS endowed with the norm .2 . Consider the map T defined by T : L2f (SG) → L2f (SG) by (T g)(t) = f (t) + αi (ψi−1 (t)) (g − b)(ψi−1 (t)), for every t ∈ ψi (SG), where i ∈ S. It is easy to derive that T is well defined. Now, consider g, h ∈ L2f (SG) to obtain the following: (T g)(t) − (T h)(t) = αi (ψ −1 (t)) (g − b)(ψ −1 (t)) − αi (ψ −1 (t)) (h − b)(ψ −1 (t)) i i i i = |αi (ψi−1 (t)) (g − h)(ψi−1 (t))| = |αi (ψi−1 (t))| |(g − h)(ψi−1 (t))| ≤ α∞ |(g − h)(ψi−1 (t))|,

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∀ t ∈ ψi (SG) and ∀ i ∈ S. Further, one has   (T g)(t) − (T h)(t) 2 dμs (t) ≤ α2 ∞ SG

i∈S

ψi (SG)

|(g − h)(ψi−1 (t))|2 dμs (t),

from Eq. (3), one can determine that  (T g)(t) − (T h)(t) 2 dμs (t) ≤ α2



∞

SG

i∈S



≤ 3 α2∞

ψi (SG)

|(g − h)(t˜)|2 dμs (t˜)

|(g − h)(t˜)|2 dμs (t˜).

N

SG

N

Therefore, we obtain that T g − T h2 ≤ 3 2 α∞ g − h2 . Using α∞ = maxS N αi ∞ and 3 2 α∞ < 1, this implies that T is a contraction map on L2f (SG). Banach contraction principle is used to get a unique fixed point of T , namely f α ∈ L2f (SG).   Hence, T ( f α ) = f α , it immediately follows that f α (t) = E i ψi−1 (t), f α (ψi−1 (t)) ∀ t ∈ ψi (SG), i ∈ S. This is further represented as f α (ψi (t)) = E i (t, f α (t)) for i ∈ S. It can be verified that the graph( f α ) is an attractor of the IFS and hence ∪i∈S Wi (graph( f α )) = graph( f α ). Remark 1 Let us recall the following equation:   f α (t) = f (t) + αi (ψi−1 (t))( f α − b) ψi−1 (t) ∀ t ∈ ψi (SG), ∀ i ∈ S. Further, one can deduce   | f α (t) − f (t)| = |αi (ψi−1 (t))( f α − b) ψi−1 (t) |   = |αi (ψi−1 (t))| |( f α − b) ψi−1 (t) |   ≤ αi ∞ |( f α − b) ψi−1 (t) |. The aforementioned inequality leads to  f α − f 2 ≤ 3 2 α∞  f α − b2 . Now, N apply triangle inequality to get  f α − f 2 ≤ 3 2 α∞  f α − f 2 + α∞  f − N  2 b2 . Hence, one gets  f α − f 2 ≤ 3 Nα∞  f − b2 . N

Finally, we get the following:

1−3 2 α∞

N

 f α 2 −  f 2 ≤  f α − f 2 ≤

3 2 α∞ N

1 − 3 2 α∞

 f − b2 .

Let g ∈ C(SG), now define a real-valued fractal operator F α : C(SG) → C(SG) by F α (g) = g α , where g α is the fractal version of g ∈ C(SG). The following theorem

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85

includes some significant properties of the fractal operator F α . Let us consider a α sequence of a real-valued fractal operators {Fnα }∞ n=1 , where Fn : C(SG) → C(SG) α α defined by Fn (g) = gn . Let us assume a sequence of base function {bn }n=∞ n=1 such that bn = L n (g), where L n : C(SG) → C(SG) is an operator, which is bounded and linear satisfying (L n g)(x) = g(x) for all x ∈ V0 . Theorem 2 Let us assume f ∈ C(SG) and f be a square-integrable function over SG. Let us consider L n : C(SG) → C(SG) the sequence of bounded and linear operators, which satisfies for every f ∈ C(SG), (L n f )( p j ) = f ( p j ), where 1 ≤ j ≤ 3 and L n ( f ) − f 2 → 0 uniformly as n → ∞, then we have the following: N



f nα

− f 2 ≤

3 2 α∞ N

1 − 3 2 α∞

 f − L n ( f )2 ,

N

Fnα

− I d2 ≤

3 2 α∞ N

1 − 3 2 α∞

Fnα 2 ≤ 3 2 +

 ,

N 2

3 α∞

N

N

1 − 3 2 α∞

(5)  .

, Here, I d represents the identity operator, and .2 is the square-integrable operator norm and  = max{I d − L 1 2 }, I d − L 2 2 , . . . , I d − L N0 2 },  > 0. Proof Let us write the self-referential equation:   f nα (t) − f (t) = αi (ψi−1 (t))( f nα − L n ( f )) ψi−1 (t) ∀ t ∈ ψi (SG), i ∈ S. Note that

α f − f 2 ≤ n 2



α f (t) − f (t) 2 dμs (t) n SG  α f (t) − f (t) 2 dμs (t). = n i∈S

ψi (SG)

We can deduce the following from the self-referential equation:  α f − f 2 = n 2 ≤



i∈S

ψi (SG)

i∈S

ψi (SG)



≤ α2∞

  αi (ψ −1 (t))( f α − L n ( f )) ψ −1 (t) 2 dμs (t) n i i   2 αi 2∞ ( f nα − L n ( f )) ψi−1 (t) dμs (t)

 i∈S

Using Eqs. 6 and 3, one gets

ψi (SG)

α   ( f − L n ( f )) ψ −1 (t) 2 dμs (t). n i

(6)

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 f nα − f 22 ≤

 SG

i∈S

≤

α2∞

≤3

N

  2 αi 2∞ ( f nα − L n ( f )) t˜ dμs (t˜)



i∈S SG α2∞  f nα

α   ( f − L n ( f )) t˜ 2 dμs (t˜) n − L n ( f )22 .

Consequently, we get α f − f ≤ 3 N2 α∞  f α − L n ( f )2 . n n 2

(7)

Using this and the triangle inequality, α

 f − f ≤ 3 N2 α∞  f α − f 2 +  f − L n ( f )2 . n n 2 Hence, from the part above being bounded for the perturbation error, we have α f − f ≤ n 2 α f − f ≤ n 2

N

3 2 α∞ N

 f − L n ( f )2 .

(8)

 f 2 I d − L n 2 .

(9)

1 − 3 2 α∞ N

3 2 α∞ N

1 − 3 2 α∞

Since  f − L n ( f )2 → 0 whenever n → ∞, for any  > 0, we can get N0 ∈ N such that  f − L n ( f )2 <  ∀ n > N0 =⇒ I d − L n 2 <  ∀ n > N0 .

(10)

Therefore, we obtain I d − L n 2 <  ∀ n ∈ N.

(11)

Hence, Eq. (9) yields N

 f nα − f 2 ≤

3 2 α∞ N

1 − 3 2 α∞

  f 2 .

(12)

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Finally, applying the definition of a fractal operator, we have the following inequality: N

Fnα 2 − I d2 ≤ Fnα − I d2 ≤

3 2 α∞ N

1 − 3 2 α∞

 .

(13)

Therefore, N

Fnα 2

≤ I d2 +

3 2 α∞ N

1 − 3 2 α∞

 .

Since N

I d2 ≤ 3 2 , hence, we have N

Fnα 2

N 2

≤3 +

3 2 α∞ N

1 − 3 2 α∞

 .

This completes the proof. Moreover, figures, (Figs. 1 and 2) below give us a better understanding of how parameter changes affect graph( f α ).

4 Some Results Associated with the Fractal Operator In this section, we obtain the bound for the operator norm of FCα . Furthermore, we find the bounds of the perturbation error. Theorem 3 Consider a real-valued square-integrable fractal operator F α on C(SG). The complex-valued fractal operator FCα : C(SG, C) → C(SG, C) defined as FCα (A + i B) = F α (A) + iF α (B), for all A, B ∈ C(SG). Then FCα is bounded linear operator. Proof We know that F α is linear. Hence, FCα is linear. Let h = A + i B. To obtain the boundedness of fractal operator FCα , we apply the following process:

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graph(f α ) on level 1

graph(f α ) on level 2

graph(f α ) on level 3

graph(f α ) on level 4

Fig. 1 f (x, y) = (x 2 y 2 − 1), b(x, y) = (x 2 y 2 − 1) − x(x − 1)(2x y − 1.5) and α = 0.5 α (h)2 = FC 2

 

SG

SG

=

α (h)|2 dμ |FC s

 |F α (A)|2 + |F α (B)|2 dμs

= F α (A)22 + F α (B)22



 ≤ F α 2 A22 + B22 = F α 2 = F α 2







 SG

SG α 2 = F  h22 .

|A|2 dμs +  |h|2 dμs

SG

|B|2 dμs

A Note on Complex-Valued Fractal Functions on the Sierpi´nski Gasket

graph(f α ) on level 1

graph(f α ) on level 2

graph(f α ) on level 3

graph(f α ) on level 4

89

Fig. 2 f (x, y) = (x 2 y 2 − 1), b(x, y) = (x 2 y 2 − 1) − x(x − 1)(2x y − 1.5) and α = 0.9

Hence, FCα (h)22 ≤ F α 2 h22 . Finally, we get FCα  ≤ F α . This inequality proves that FCα is a bounded operator.



Theorem 4 Consider a real-valued square-integrable fractal operator F α on C(SG). For any h = h r e + i h im ∈ C(SG, C), let us define the complex-valued fractal operator FCα : C(SG, C) → C(SG, C) by

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FCα (h) = F α (h r e ) + i F α (h im ) for all h r e , h im ∈ C(SG). Then for all h ∈ C(SG, C), N

FCα (h) − h2 ≤

3 2 α∞ N

1 − 3 2 α∞

h − L C (h)2 ,

where bC : SG → C is a complex-valued square-integrable function, which satisfies bC = L C h for every h ∈ C(SG, C), where L C : C(SG, C) → C(SG, C) is bounded and linear operator satisfying (L C h)(x) = h(x) for all x ∈ V0 and defined by L C (h) = L(h r e ) + i L(h im ). Proof Since FCα (h) = F α (h r e ) + i F α (h im ), where Re(FCα (h)) = F α (h r e ) , I m(FCα (h)) = F α (h im ). We now use Theorem 2 and a few inequalities to obtain the desired result FCα (h) − h22 =

 

SG

SG

=

|FCα (h) − h|2 dμs

 |FCα (h r e ) − h r e |2 + |FCα (h im ) − h im |2 dμs

= F α (h r e ) − h r e 22 + F α (h im ) − h im 22



 ≤ 3 N α2∞ F α (h r e ) − L(h r e )22 + F α (h im ) − L(h im )22

 = 3 N α2∞ 

|F α (h r e ) − L(h r e )|2 dμs + SG



|F α (h im ) − L(h im )|2 dμs



SG

|FCα (h) − L C (h)|2 dμs SG 3 N α2∞ FCα (h) − L C (h)22 .

= 3 N α2∞ =

(14)

Finally, we get FCα (h) − h22 ≤ 3 N α2∞ FCα (h) − L C (h)22 . That is, FCα (h) − h2 ≤ 3 2 α∞ FCα (h) − L C (h)2 . Thanks to the triangle inequality, N

FCα (h) − h2 ≤ 3 2 α∞ FCα (h) − h2 + 3 2 α∞ h − L C (h)2 . N

This can be written as

N

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91

N

FCα (h) − h2 ≤

3 2 α∞ N

1 − 3 2 α∞

h − L C (h)2 .

This completes the assertion.

5 Conclusion We have explored some properties of the real-valued fractal operator and complexvalued fractal operator defined on SG. The bound for the perturbation error on SG has also been calculated. Under certain conditions, the fractal operator’s bounds are established. Furthermore, we have established relations between the complex-valued fractal operator and the real-valued fractal operator.

References 1. Agrawal, V., Som, T.: Fractal dimension of α-fractal function on the Sierpi´nski Gasket. Eur. Phys. J. Spec. Top. (2021). https://doi.org/10.1140/epjs/s11734-021-00304-9 2. Agrawal, V., Som, T.: L p approximation using fractal functions on the Sierpi´nski gasket. Results Math. 77(2), 1–17 (2021) 3. Agrawal, V., Som, T., Verma, S.: On bivariate fractal approximation. J. Anal. (2022). https:// doi.org/10.1007/s41478-022-00430-0 4. Bagby, T.: L p approximation by analytic functions. J. Approx. Theory 5, 401–404 (1972) 5. Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 301–329 (1986) 6. Barnsley, M.F.: Fractals Everywhere. Academic, Orlando, Florida (1988) 7. Celik, D., Kocak, S., Özdemir, Y.: Fractal interpolation on the Sierpi´nski Gasket. J. Math. Anal. Appl. 337, 343–347 (2008) 8. Jha, S., Verma, S., Chand, A. K. B.: Non-stationary zipper α-fractal functions and associated fractal operator. Fractional Calculus and Applied Analysis (2022). https://doi.org/10.1007/ s13540-022-00067-7 9. Jha, S., Verma, S.: Dimensional analysis of α-fractal functions. Results Math 76(4), 1–24 (2021) 10. Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge, UK (2001) 11. Massopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets, 2nd edn. Academic (2016) 12. Navascués, M.A.: Fractal approximation. Complex Anal. Oper. Theory 4, 953–974 (2010). https://doi.org/10.1007/s11785-009-0033-1 13. Navascués, M.A., Verma, S., Viswanathan, P.: Concerning the Vector-Valued Fractal Interpolation Functions on the Sierpi´nski Gasket. Mediterr. J. Math. 18(5), 1–26 (2021) 14. Prasad, S. A., Verma S.: Fractal Interpolation Function On Products of the Sierpi´nski Gaskets (2022). arXiv:2206.01920v1 15. Ri, S.: Fractal Functions on the Sierpi´nski Gasket. Chaos, Solitons Fractals 138, 110142 (2020) 16. Ri, S.G., Ruan, H.J.: Some properties of fractal interpolation functions on Sierpi´nski gasket. J. Math. Anal. Appl. 380, 313–322 (2011) 17. Ruan, H.J.: Fractal interpolation functions on post critically finite self-similar sets. Fractals 18, 119–125 (2010) 18. Sahu, A., Priyadarshi, A.: On the box-counting dimension of graphs of harmonic functions on the Sierpi´nski gasket. J. Math. Anal. Appl. 487, 124036 (2020)

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19. Strichartz, R.S.: Differential Equations on Fractals. Princeton University Press, Princeton, NJ (2006) 20. Verma, S., Viswanathan, P.: A fractal operator associated with bivariate fractal interpolation functions on rectangular grids. Result Math. 75, 25 (2020) 21. Verma, S., Viswanathan, P.: Katugampola fractional integral and fractal dimension of bivariate functions. Results Math. 76, 165 (2021). https://doi.org/10.1007/s00025-021-01475-6 22. Verma, S., Viswanathan, P.: A revisit to α-fractal function and box dimension of its graph. Fractals 27(06), 1950090 (2020) 23. Verma, S., Massopust, P. R.: Dimension preserving approximation, To appear in Aequationes Mathematicae, https://doi.org/10.48550/arXiv.2002.05061 24. Verma, S., Sahu, A.: Bounded variation on the Sierpi´nski Gasket. Fractals (2022). https://doi. org/10.1142/S0218348X2250147X 25. Verma, S., Viswanathan, P.: Parameter identification for a class of bivariate fractal interpolation functions and constrained approximation. Numer. Funct. Anal. Optim. 41(9), 1109–1148 (2020)

Dimensional Analysis of Mixed Riemann–Liouville Fractional Integral of Vector-Valued Functions Megha Pandey, Tanmoy Som, and Saurabh Verma

Abstract In this paper, we attempt to develop the concept of fractal dimension of the continuous bivariate vector-valued maps. We give few fundamental concepts of the dimension of the graph of bivariate vector-valued functions and prove some basic results. We prove that upper bound of the Hausdorff dimension of Hölder continuous function is 3 − σ. Because of its wide applications in many important areas, fractal dimension has become one of the most interesting parts of fractal geometry. Estimating the fractal dimension is one of the most fascinating works in fractal theory. It is not always easy to estimate the fractal dimension even for elementary real-valued functions. However, in this paper, an effort is made to find the fractal dimension of continuous bivariate vector-valued maps and, in particular, the fractal dimension of the Riemann–Liouville fractional integral of a continuous vector-valued bivariate map of bounded variation defined on a rectangular domain is also found. Keywords Bounded variation · Fractal dimension · Riemann–Liouville fractional integral · Hölder continuous

M. Pandey (B) · T. Som Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India e-mail: [email protected] T. Som e-mail: [email protected] S. Verma Department of Applied Sciences, Indian Institute of Information Technology Allahabad, Prayagraj 211015, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_8

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1 Introduction Jordan [10] was the one who first introduced bounded variation for the function, g : J → R, where J is a compact interval of R. The present paper is focused on those functions which are bivariate vector-valued bounded variation. Unlike univariate functions, there are several definitions which have been introduced in the literature on a bivariate function known to be of bounded variation, for example, the definitions by Vitali, Hardy, Arzelá, Pierpoint, Fréchet, Tonelli, and Hahn [1, 2, 6]. In this article, we have given the results for a function, which satisfies the bounded variation definition in Arzelá sense. Integration and differentiation have always played a significant role in mathematics. Fractional Calculus (FC), dealing with the basic as well as advanced theories on integration and differentiation of non-integer (fractional) order and their properties, is one of the wide subjects existed in the literature from the centuries. Several papers and books have been published so far, interested reader can follow: [4, 14–16, 19–22] to know more about FC and its applications in different areas. To define Fractional Integral (FI), many formulae have been given in the theory, for example, Hadamard FI [26], Riemann–Liouville Fractional Integral (RLFI) [4, 23], Katugampola FI [5, 25], etc. Fractal Dimension (FD) is one of the most entertained topics in fractal geometry. It has always gained the attention of researchers with its extensive applications in different areas. Although estimating the FD of the graph of elementary functions is not simple. However, many theories have been developed for finding FD of the graph of special classes of functions (see, for instance, [3, 7, 9, 13]).

1.1 Motivation Motivation for connecting FC with fractals is to explore the physical interpretations of the integration and differentiation of non-integer order. In this direction, Liang [12] calculated the box dimension of the graph of mixed RLFI of a bounded variation continuous map. In [23], some results related to RLFI and unbounded variation points of a function have been established. Motivated by Liang [12], Verma and Vishwanathan [24] introduced the concept of FD of the graph of bivariate function of bounded variation. With persuasion of the work of Verma and Vishwanathan [24], in this article, we introduce the theory of FD of the bivariate vector-valued functions and RLFI of such functions. Most of the results of this paper are simple extensions of the results of Verma and Vishwanathan [24], and in addition we are able to give an upper bound for FD of the graph of only coordinate functions of bivariate vector-valued Hölder continuous functions but not for vector-valued function (see Theorem 8). Some initial theories have been introduced in the direction of univariate vector-valued function by Pandey et al. [17].

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1.2 Delineation The proposed work is assembled as follows. The next section is reserved for background and preliminaries required for the study. In Sect. 3, we have proved some basic results for the bivariate vector-valued functions. We have concluded our paper in Sect. 4, where we have determined that FD of the graph of bivariate vector-valued bounded variation function defined on rectangular domain is 1 and so is the FD of the graph of RLFI of such functions. Also, we have given an upper bound for the graphs of coordinate functions of bivariate vector-valued Hölder continuous function and ended the section with a question that “ whether we can get an upper bound for bivariate vector-valued Hölder continuous function?”

2 Notations and Prelude The following are the notations which we shall be using in the paper: • • • •

R: collection of all real numbers. N: collection of natural numbers. [c, d] × [ p, q]: Rectangular domain in R2 . σ-HC: Hölder continuous function having exponent σ.

In what follows, we have collected the preliminary materials required for our study. To know more in detail, one can follow [7, 8].

2.1 Fractal Dimension Definition 1 (Diameter) Let U ⊆ Rn be a non-empty subset of Rn , then diameter of U is defined as |U| = sup {u − y2 : u, y ∈ U}. Definition 2 (δ-cover) Let B ⊂ Rn be non-empty set and {Ui } be a countable collec tion (or a finite collection) of subsets with diameter at most δ > 0 such that B ⊆ Ui , i∈N

then {Ui } is said to be a δ-cover of B.

Definition 3 (Hausdorff dimension) Let s ≥ 0 and δ > 0 be real numbers and B ⊆ Rn . Let ∞   s s Hδ (B) = inf |Ui | : {Ui } is a δ-cover of B , i=1

and s-dimensional Hausdorff measure of B is given by H s (B) = lim Hδs (B). Then, Hausdorff dimension of B is defined as

δ→0

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  dim H (B) = sup{s : H s (B) = ∞ = inf s : H s (B) = 0}. Definition 4 (Box Dimension) Let B ⊆ Rn be bounded and non-empty and Nδ (B) be lowest number of sets with at most δ diameter with covering B. Then, dim B (B) = lim inf δ→0

log Nδ (B) − log δ

is said to be lower box dimension of B. And dim B (B) = lim sup δ→0

log Nδ (B) − log δ

is known as upper box dimension of B. If both lower and upper box dimensions are same, then that quantity is known as box dimension of B and defined as log Nδ (B) . δ→0 − log δ

dim B (B) = lim

Definition 5 (Graph of Vector-valued functions) Let g : [c, d] × [ p, q] → Rn be a bivariate vector-valued map, then the graph of g is defined as the set {(u, w, g(u, w)) : (u, w) ∈ [c, d] × [ p, q]} and is denoted by Gr (g). Note For a vector-valued map g : [c, d] × [ p, q] → Rn , denote gi : [c, d] × [ p, q] → R as the coordinate functions of g. That is,  g(u, w) = g1 (u, w), g2 (u, w), . . . , gn (u, w) . Note The set of all continuous maps defined on [c, d] × [ p, q] is denoted by  C [c, d] × [ p, q], Rn . Definition 6 (Hölder Continuity) Consider g : [c, d] × [ p, q] → Rn be a vectorvalued function. For a positive constant K and 0 < σ ≤ 1 let g satisfy the condition g(u, w) − g(v, y)2 ≤ K(u, w) − (v, y)σ2 for all (u, w), (v, y) ∈ [c, d] × [ p, q], (1)

then g is known as Hölder continuous function. For σ = 1, g is known as Lipschitz continuous. Lemma 1 ([24]) Assume that g is a bivariate real-valued function, which satisfies (1), then dim H (Gr (g)) ≤ dim B (Gr (g)) ≤ dim B (Gr (g)) ≤ 3 − σ.

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2.2 Variation of a Function Definition 7 (Bounded Variation Bivariate function [6]) A bivariate function, g : [c, d] × [ p, q] → R, will be of bounded variation in Arzelá sense if for each set of points {(u 0 , w0 ), (u 1 , w1 ), . . . , (u m , wm )} satisfying c = u 0 ≤ u 1 ≤ u 2 ≤ · · · ≤ u m = d; p = w0 ≤ w1 ≤ w2 ≤ · · · ≤ wm = q, the sum m−1 

|Δf (u i , wi )|, where Δf (u i , wi ) = f (u i+1 , wi+1 ) − f (u i , wi )

i=0

is bounded. Theorem 1 ([1]) The below statements are equivalent: (i) g : [c, d] × [ p, q] → R is a real-valued bivariate function of bounded variation satisfying the Arzelá condition. (ii) There exist two bounded variation functions, φ1 : [c, d] × [ p, q] → R and φ2 : [c, d] × [ p, q] → R such that g can be written as difference of φ1 and φ2 , where φ1 and φ2 satisfy the relation Δ10 φi (u, w) ≥ 0, Δ01 φi (u, w) ≥ 0,

i = 1, 2, for each (u, w) ∈ [c, d] × [ p, q],

(2) where Δ10 φi (u, w) = φi (u + h, w) − φi (u, w), Δ01 φi (u, w) = φi (u, w + k) − φi (u, w), for arbitrary h, k > 0. Remark 1 A bounded variation function which satisfies (2) is known as a monotone function. Note 1 From now, instead of writing “function of bounded variation in Arzelá sense” we shall write “function of bounded variation”. Remark 2 If each of the coordinate functions of bivariate vector-valued function are of bounded variation, then the vector-valued function will also be of bounded variation and vice versa. BV([c, d] × [ p, q], Rn ) represents the collection of all bivariate vector-valued functions of bounded variation defined on [c, d] × [ p, q].

2.3 Fractal Dimension and Bounded Variation In this subsection, we shall give a few fundamental results on FD of a bounded variation function.

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Theorem 2 ([6, P. 827]) Surface area of a continuous bounded variation function is finite. Theorem 3 Consider g ∈ BV([c, d] × [ p, q], Rn ) ∩ C([c, d] × [ p, q], Rn ), then dim H (Gr (g)) = 2. Proof In context of Theorem 2, notice that two-dimensional Hausdorff measure of the graph of g is finite, hence the conclusion of the theorem.

2.4 Mixed Riemann–Liouville Fractional Integral Definition 8 Consider the integrable map g : [c, d] × [ p, q] → R, then Mixed RLFI of g is expressed as 

(ρ,μ) g (u, w) = (c, p) J

1 Γ (ρ)Γ (μ)



u



c

w

(u − t)ρ−1 (w − s)μ−1 g(t, s) dt ds,

p

where ρ, μ > 0 for all (u, w) ∈ [c, d] × [ p, q], and 0 ≤ c < d < ∞, 0 ≤ p < q < ∞. Definition 9 Consider the vector-valued integrable function g defined on [c, d] × [ p, q], then RLFI of g is determined as (c, p) J



(ρ,μ) g(u, w) = (ρ,μ) g (u, w), (ρ,μ) g (u, w), . . . , 1 2 (c, p) J (c, p) J

(c, p) J



(ρ,μ) g (u, w) , n

where (ρ,μ) gi (u, w) = (c, p) J

1 Γ (ρ)Γ (μ)

c

u



w

(u − t)ρ−1 (u − s)μ−1 gi (t, s) dt ds,

p

for i ∈ 1, n, and ρ, μ > 0 for all (u, w) ∈ [c, d] × [ p, q].

3 Few Fundamental Concepts on Dimension of the Graph of a Vector-Valued Function Some fundamental concepts concerning the Hausdorff dimension of the graph of vector-valued continuous functions have been proved in this section. Similar results are well recorded for real-valued functions. These vector-valued analogs appear to be folklore but we have been unable to track down complete proofs. So for the sake of reader convenience we decide to include the proofs. Lemma 2 Let g : [c, d] × [ p, q] → Rn be a continuous vector-valued function, and gi : [c, d] × [ p, q] → R be coordinate functions of g. Then, we have

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 (i) dim H Gr (g) ≥ max{dim H (Gr (gi )) : 1 ≤ i ≤ n}. (ii) dim H (Gr (g)) = dim H (Gr (gi )) for some i ∈ 1, n given that the coordinate maps g1 , g2 , . . . , gi−1 , gi+1 , . . . , gn are all Lipschitz. Proof

(i) Define a function Ψ : Gr (g) → Gr (gi ) such that   Ψ u 1 , u 2 , g(u 1 , u 2 ) = u 1 , u 2 , gi (u 1 , u 2 ) .

Observe that the map Ψ is surjective. Now for (u 1 , u 2 ), (y1 , y2 ) ∈ [c, d] × [ p, q], we have

 

Ψ u 1 , u 2 , g(u 1 , u 2 ) − Ψ y1 , y2 , g(y1 , y2 )

2 = (u 1 , u 2 , gi (u 1 , u 2 )) − (y1 , y2 , gi (y1 , y2 )) 2  = (u 1 − y1 )2 + (u 2 − y2 )2 + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2   n   2 2  ≤ (u 1 − y1 ) + (u 2 − y2 ) + (g j (u 1 , u 2 ) − g j (y1 , y2 ))2 j=1

 

= u 1 , u 2 , g(u 1 , u 2 ) − y1 , y2 , g(y1 , y2 ) . 2

 Hence, Ψ is a Lipschitz map. Therefore, dim H (Gr (gi )) = dim H Ψ (Gr (g) ≤ dim H (Gr (g)) for each i ∈ 1, n, proving the statement. (ii) Define a map ϕ : Gr (gi ) → Gr (g) such that ϕ(u 1 , u 2 , gi (u 1 , u 2 )) =  u 1 , u 2 , g(u 1 , u 2 ) . Notice that ϕ is an onto map. Let L g j be the Lipschitz constant of g j for j = i. For (u 1 , u 2 ), (y1 , y2 ) ∈ [c, d] × [ p, q], we have

 

ϕ u 1 , u 2 , gi (u 1 , u 2 ) − ϕ y1 , y2 , gi (y1 , y2 ) 2

  = u 1 , u 2 , g(u 1 , u 2 ) − y1 , y2 , g(y1 , y2 ) 2   n   =(u 1 − y1 )2 + (u 2 − y2 )2 + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2   2   = (u i − yi )2 +

i=1 n 

i=1

j=1, j =i

i=1

j=1

(g j (u 1 , u 2 ) − g j (y1 , y2 ))2 + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2

 ⎛ ⎞  2 n 2     ≤ (u i − yi )2 + L g j ⎝ (u i − yi )2 ⎠ + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2 i=1

  2  ≤K (u i − yi )2 + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2 i=1

 

=K u 1 , u 2 , gi (u 1 , u 2 ) − y1 , y2 , gi (y1 , y2 ) , 2

(3)

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where L = max{L g j : j = 1, 2, . . . , i − 1, i + 1, . . . , n} is Lipschitz constant  and K = 1 + (n − 1)L2 . Moreover,

 

ϕ u 1 , u 2 , gi (u 1 , u 2 ) − ϕ y1 , y2 , gi (y1 , y2 ) 2



= (u 1 , u 2 , g(u 1 , u 2 )) − (y1 , y2 , g(y1 , y2 )) 2

  = u 1 , u 2 , g1 (u 1 , u 2 ), . . . , gn (u 1 , u 2 ) − y1 , y2 , g1 (y1 , y2 ), . . . , gn (y1 , y2 ) 2   2 n   = (u − y )2 + (g (u , u ) − g (y , y ))2 i

i=1

 

i

i

1

2

i

1

2

i=1

K  =  (u i − yi )2 + K 2

n 

i=1

j=1, j=i

(g j (u 1 , u 2 ) − g j (y1 , y2 ))2 + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2

  2   1 ≥  1 + (n − 1)L2 (u i − yi )2 + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2 K

i=1

  2  ≥ (u i − yi )2 + (gi (u 1 , u 2 ) − gi (y1 , y2 ))2 i=1

 

= u 1 , u 2 , gi (u 1 , u 2 ) − y1 , y2 , gi (y1 , y2 ) . 2

(4)

Equations 3 and 4 together give (u 1 , u 2 , gi (u 1 , u 2 )) − (y1 , y2 , gi (y1 , y2 ))2 ≤ϕ(u 1 , u 2 , gi (u 1 , u 2 )) − ϕ(y1 , y2 , gi (y1 , y2 ))2 ≤K (u 1 , u 2 , gi (u 1 , u 2 )) − (y1 , y2 , gi (y1 , y2 ))2 .

Thus, ϕ is a bi-Lipschitz map, hence we have dim H (Gr (g)) = dim H (Gr (gi )). Remark 3 Note that in [11] the Peano space filling curve g : [c, d] → [0, 1] × [0, 1] is 21 -HC and coordinate functions g1 and g2 satisfy dim H (Gr (g1 )) = dim H (Gr (g2 )) = 1.5. Now define h : [c, d] × [ p, q] → Rn such that h(u, w) = (h 1 (u, w), h 2 (u, w), 0, . . . , 0), where h 1 , h 2 : [c, d] × [ p, q] → R are defined as h 1 (u,  w) = g1 (u) andh 2 (u, w) = Gr (h 1 ) = dim H Gr (h 2 ) = have dim g2 (u). Then, using [24, Lemma 3.7], we H  2.5. Therefore, using Lemma 2, dim H Gr (h) ≥ 2.5. This shows that the inequality present in (i) of Lemma 2 can be strict.

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Proposition 1 Let g : [c, d] × [ p, q] → Rn be σ-HC, then each of its coordinate maps is also σ-HC and vice versa. Proof First consider g is σ-HC, that is, g(u 1 , u 2 ) − g(y1 , y2 )2 ≤ L g (u 1 , u 2 ) − (y1 , y2 )σ2 for all (u 1 , u 2 ), (y1 , y2 ) ∈ [c, d] × [ p, q] and for some L g > 0. Then, we have

gi (u 1 , u 2 ) − gi (y1 , y2 ) ≤ g(u 1 , u 2 ) − g(y1 , y2 )2 ≤ L g (u 1 , u 2 ) − (y1 , y2 )σ . 2 2

Hence, gi are σ-HC. Conversely, assume that all the coordinate functions, gi , are σ-HC with Lipschitz constant L gi , then we have g(u 1 , u 2 ) − g(y1 , y2 )2 ≤

√ √ n max |gi (u 1 , u 2 ) − gi (y1 , y2 )| ≤ n L g (u 1 , u 2 ) − (y1 , y2 )σ2 , 1≤i≤n

where L g = max{L g1 , L g2 , . . . , L gn }. Therefore, g is σ-HC. Lemma 3 ([24, Corollory 3.11]) Let g : [c, d] × [ p, q] → R be a σ-HC function, then dim H (Gr (g)) ≤ 3 − σ.  Corollary 1 Let g : [c, d] × [ p, q] → Rn be σ-HC, then dim H Gr (gi ) ≤ 3 − σ for each i ∈ 1, n. Proof Since g is σ-HC, using Proposition 1 we have gi is also σ-HC for each i ∈ 1, n. Then, using Lemma 3 we get  dim H Gr (gi ) ≤ 3 − σ.

4 Riemann–Liouville Fractional Integral and Fractal Dimension We present our findings in this section. We back up our claims with straightforward arguments. Throughout this section, we consider 0 ≤ c < d < ∞ and 0 ≤ p < q < ∞. Theorem 4 Let g : [c, d] × [ p, q] → Rn be a bounded function, then is also bounded.

(ρ,μ) g (c, p) J

Proof As g is bounded, all coordinate functions g1 , . . . gn are bounded. That means there exists K > 0 such that max |gi (x1 , x2 )| ≤ K for all (x1 , x2 ) ∈ [c, d] × [ p, q]. 1≤i≤n

Now

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 √





(c, p) J(ρ,μ) g(x1 , x2 ) ≤ n max  (c, p) J(ρ,μ) gi (x1 , x2 ) 2 1≤i≤n  

x x   1 2 √ 1 = n max  (x1 − s)ρ−1 (x2 − t)μ−1 gi (s, t)ds dt  Γ (ρ)Γ (μ) 1≤i≤n c p

x x     1 2 √ 1  ≤ n max (x1 − s)ρ−1 (x2 − t)μ−1  gi (s, t) ds dt 1≤i≤n Γ (ρ)Γ (μ) c p

x x   1 2 √ K  ≤ n (x1 − s)ρ−1 (x2 − t)μ−1  ds dt Γ (ρ)Γ (μ) c p √ ≤ n

(x1 − a)ρ (x2 − c)μ K . Γ (ρ)Γ (μ) ρμ

That is,

√

(c, p) J(ρ,μ) g(x1 , x2 ) ≤ n 2

K (d − c)ρ (q − p)μ for all (x1 , x2 ) ∈ [c, d] × [ p, q]. Γ (ρ + 1)Γ (μ + 1)

This completes the proof. Theorem 5 If g ∈ C([c, d] × [ p, q], Rn ), then Rn ).

(ρ,μ) g (c, p) J

∈ C([c, d] × [ p, q],

Proof Since g is continuous, then each coordinate functions g1 , . . . , gn of g is continuous on [c, d] × [ p, q]. For 0 < c < x1 < x1 + h ≤ d and 0 < p < x2 < x2 + k ≤ q, we have

(c, p) J(ρ,μ) g(x1 + h, x2 + k) −(c, p) J(ρ,μ) g(x1 , x2 ) 2   √ ≤ n max1≤i≤n (c, p) J(ρ,μ) gi (x1 + h, x2 + k) −(c, p) J(ρ,μ) gi (x1 , x2 )   x1 +h  x2 +k √  1 = n max1≤i≤n  Γ (ρ)Γ (x1 + h − s)ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt p (μ) c   x1  x2  1 ρ−1 μ−1 − Γ (ρ)Γ (x − s) (x − t) g (s, t) ds dt  1 2 i p (μ) c   c+h  p+k √  1 = n max1≤i≤n  Γ (ρ)Γ (x1 + h − s)ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt p (μ) c  x1 +h  p+k 1 (x1 + h − s)ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt + Γ (ρ)Γ p (μ) c+h  c+h  x2 +k 1 ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt + Γ (ρ)Γ p+k (x 1 + h − s) (μ) c   x1 +h x2 +k 1 ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt + Γ (ρ)Γ p+k (x 1 + h − s) (μ) c+h   x1  x2  1 ρ−1 μ−1 − Γ (ρ)Γ (x − s) (x − t) g (s, t) ds dt  1 2 i p (μ) c √ = n max1≤i≤n |Ii1 + Ii2 + Ii3 + Ii4 − Ii5 |, where

Dimensional Analysis of Mixed Riemann–Liouville Fractional …

Ii1 = Ii2 = Ii3 = Ii4 = Ii5 =

103

 c+h  p+k

(x1 + h − s)ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt, c p  x1 +h  p+k 1 (x1 + h − s)ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt, p Γ (ρ)Γ (μ) c+h  c+h  x2 +k 1 ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt, p+k (x 1 + h − s) Γ (ρ)Γ (μ) c   x1 +h x2 +k 1 ρ−1 (x2 + k − t)μ−1 gi (s, t) ds dt, p+k (x 1 + h − s) Γ (ρ)Γ (μ) c+h 1 Γ (ρ)Γ (μ)

1 Γ (ρ)Γ (μ)

 x1  x2 c

p

(x1 − s)ρ−1 (x2 − t)μ−1 gi (s, t) ds dt.

Applying the change of variables z = s − h and w = t − k in the integral Ii4 , we get Ii1 + Ii2 + Ii3 + Ii4 − Ii5 = Ii1 + Ii2 + Ii3 + Ii6 , where Ii 6 =

x1 x2   1 (x1 − s)ρ−1 (x2 − t)μ−1 gi (s + h, t + k) − gi (s, t) ds dt. Γ (ρ)Γ (μ) c p

Because gi ’s are continuous on a compact subset of R2 , it is going to be bounded on [c, d] × [ p, q]. So, there exists M > 0 such that max |gi (s, t)| ≤ M for every 1≤i≤n

(s, t) ∈ [c, d] × [ p, q]. Using this, we get |Ii1 | ≤ M1 hk, for a suitable constant M1 . M(d−c)ρ , we get |Ii2 | ≤ M2 k. Taking, M2 = max (y + k − t)μ−1 Γ (ρ+1)Γ (μ+1) p≤t≤ p+k

Similarly, for a suitable constant M3 , we get |Ii3 | ≤ M3 h. Since gi is uniformly continuous for each i ∈ 1, n, for a given  > 0, there exists δ > 0 such that Γ (ρ + 1)Γ (μ + 1) . |gi (s, t) − gi (z, w)| < 4(d − c)ρ (q − p)μ Thus,   (c, p) J(ρ,μ) gi (x1 + h, x2 + k) −(c, p) J(ρ,μ) gi (x1 , x2 ) ≤ M2 k + M2 k + M3 h +  . 4 Hence, proof follows. The next result can be observed in [24] for bivariate real-valued bounded variation maps. But for completeness, we demonstrate the result here. Lemma 4 For a bounded variation function, g : [c, d] × [ p, q] → Rn the following will hold: (i) If gi (c, p) ≥ 0, then there will exist monotone functions h i and f i such that gi = f i − h i with f i (c, p) ≥ 0 and h i (c, p) = 0. (ii) If gi (c, p) < 0, then there will exist monotone functions h i and f i such that gi = f i − h i with f i (c, p) = 0 and h i (c, p) > 0.

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Proof Since g = (g1 , . . . , gn ) is of bounded variation on [c, d] × [ p, q], this yields gi for each i ∈ 1, n to be of bounded variation on [c, d] × [ p, q]. By using Theorem 1, we have gi = φi − ξi , where ξi and φi are monotone functions for each i ∈ 1, n. Now (i) Define f i (u, y) = φi (u, y) + gi (c, p) − φi (c, p) and h i (u, y) = ξi (u, y) + gi (c, p) − φi (c, p). Then, for all (u, y) ∈ [c, d] × [ p, q], we have   f i (u, y) − h i (u, y) = φi (u, y) + gi (c, p) − φi (c, p) − ξi (u, y) + gi (c, p) − φi (c, p) (u, y) − ξ( u, y) = gi (u, y),

this implies that gi = f i − h i , for each 1 ≤ i ≤ n. Also notice that f i (c, p) ≥ 0 and h i (c, p) = 0. Hence, the result. (ii) Consider f i (u, y) = φi (u, y) − gi (c, p) − ξi (c, p) and h i (u, y) = φi (u, y) − gi (c, p) − ξi (c, p), proof follows same as above. Next theorem is a well-known result for univariate real-valued function. One can refer [7] for detail study. In the interest of completeness, we include the demonstration of the theorem. Theorem 6 If g is of bounded variation on [c, d] × [ p, q], then dim H (Gr (g)) = 2. Proof Since g is a function of bounded variation, at most countable number of discontinuities can exist for g. Let {(x1 , y1 ), (x2 , y2 ), . . . } be that set of points of discontinuity of g, such that c ≤ x1 < x2 < · · · ≤ d and p ≤ y1 < y2 · · · ≤ q. Let us consider for a moment c = x0 and p = y0 and Gi be the graph of function g restricted to rectangle [xi , xi+1 ] × [yi , yi+1 ] for i ∈ N ∪ {0}. Obviously Gr (g) = ∞  Gi (g). Since Hausdorff dimension satisfies the countable stability, we obtain i=0 ∞   Gi (g) = sup dim H (Gi (g)). Note that on each rectdim H (Gr (g)) = dim H i=0

0≤i 0 such that max 1≤i≤n t∈[c,d]×[ p,q]

approximated as follows:

Dimensional Analysis of Mixed Riemann–Liouville Fractional …



u

|Ii7 | ≤ γ



c

y

107

  (u − s)ρ−1 (y − t)μ−1 − ((u + h) − s)ρ−1 (y + k − t)μ−1 .

p

With simple calculations, we get  γ  (u − c)ρ (y − p)μ − (u + h − c)ρ (y + k − p)μ + h ρ (y + k − p)μ + k μ (u + h − c)ρ − h ρ k μ , ρμ γ |Ii8 | ≤ h ρ k μ . ρμ |Ii7 | ≤

Then, using Bernoulli’s inequality, we get  |Ii7 | + |Ii8 | ≤ B( h 2 + k 2 )σ , for some suitable constant B. Hence, by (5), we conclude that  (c, p) J(ρ,μ) g(u + h, y + k) −

(ρ,μ) g(u, (c, p) J

 y)2 ≤ B( h 2 + k 2 )σ .

That is, (c, p) J(ρ,μ) g is σ-HC. Therefore, using Lemma 1 and Proposition 1, we get dim B (Gr ( (c, p) J(ρ,μ) gi )) ≤ 3 − σ, completing the proof. Remark 4 Comparing Theorem 8 with Theorem 5, where it is demonstrated that for 0 < μ, ρ, the RL integral (c, p) J(ρ,μ) g is continuous whenever g is continuous, we notice that (c, p) J(ρ,μ) g is also Hölder continuous on [c, d] × [ p, q] whenever g is a continuous function on [c, d] × [ p, q] with 0 < c < d < ∞, 0 < p < q < ∞, and ρ, μ > 0. Further the semigroup property of the RL integral for a “sufficiently good” function is there as a preamble to the next theorem. (ρ1 ,μ1 ) (ρ2 ,μ2 ) g (c, p) J (c, p) J

=(c, p) J(ρ1 ,μ1 )+(ρ2 ,μ2 ) g.

Now, the next theorem is the direct deduction of the previous theorem: Theorem 9 Let g be continuous on [c, d] × [ p, q], 0 < c < d < ∞, 0 < p < q < ∞ and 0 < ρ, μ ≤ 1. 1. If 0 < σ < 1, then     2 ≤ dim H Gr (c, p) J(ρ,μ) gi ≤ dim B Gr (c, p) J(ρ,μ) gi ≤ 3 − σ. 2. If σ ≥ 1, then     dim H Gr (c, p) J(ρ,μ) gi = dim B Gr (c, p) J(ρ,μ) gi = 2. Remark 5 In Remark 3, we notice that h is also a 21 -HC but dim H (Gr (h)) ≥ 3 − 1 = 2.5. It shows that even if a vector-valued function is σ-HC, we cannot provide 2 its upper bound like we do for a real-valued function.

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5 Conclusion and Future Scope We have given the concept of mixed RLFI of vector-valued functions (Definition 9), thereafter we have given dimensional bounds for a bivariate vector-valued functions (Lemma 2). In Sect. 4, we have explored the properties of the mixed RLFI of the vector-valued functions (Theorems 4, 5, Lemma 4) and proved some dimensional results for bivariate vector-valued functions (Theorem 6) and for mixed RLFI (Theorems 7, 8, and 9) In view of Remark 5, a question arises: can we find an upper bound for the dimension of the graph of (c, p) J(ρ,μ) g? One may ask a more general question on the fractal dimension of RLFI of a set-valued function. Recently, Pandey et al. [18] tried to initiate a study on fractal dimension of set-valued functions. From [18], one can notice that to establish results in terms of set-valued mappings, a different set of tools is required.

References 1. Adams, C.R., Clarkson, J.A.: Properties of functions f (x, y) of bounded variation. Trans. Amer. Math. Soc. 36(4), 711–730 (1934) 2. Adams, C.R., Clarkson, J.A.: A correction to “properties of functions f (x, y) of bounded variation.” Trans. Amer. Math. Soc. 46, 468 (1939) 3. Barnsley, M.: Fractals Everywhere, 2nd edn (1993) 4. Chandra, S., Abbas, S.: The calculus of bivariate fractal interpolation surfaces. Fractals 29(03), 2150066 (2021) 5. Chandra, S., Abbas, S.: Box dimension of mixed katugampola fractional integral of twodimensional continuous functions. Fract. Calcul. Appl. Anal. 1–15 (2022) 6. Clarkson, J.A., Adams, C.R.: On definitions of bounded variation for functions of two variables. Trans. Amer. Math. Soc. 35(4), 824–854 (1933) 7. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (2004) 8. Gordon, R.A.: Real Analysis: A First Course. Pearson College Division (2002) 9. Jha, S., Verma, S.: Dimensional analysis of α-fractal functions. Results Math. 76(4), 1–24 (2021) 10. Jordan, C.: Sur la series de fourier. CR Acad. Sci. Paris 92, 228–230 (1881) 11. Kôno, N.: On self-affine functions. Jpn. J. Appl. Math. 3(2), 259–269 (1986) 12. Liang, Y.: Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation. Nonlinear Anal.: Theory, Methods & Appl. 72(11), 4304–4306 (2010) 13. Massopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets. Academic, Cambridge (2016) 14. Moshrefi-Torbati, M., Hammond, J.: Physical and geometrical interpretation of fractional operators. J. Frank. Inst. 335(6), 1077–1086 (1998) 15. Nigmatullin, R.: Fractional integral and its physical interpretation. Theor. Math. Phys. 90(3), 242–251 (1992) 16. Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier, Amsterdam (1974) 17. Pandey, M., Som, T., Verma, S.: Fractal dimension of Katugampola fractional integral of vectorvalued functions. Eur. Phys. J. Spec. Topics 1–8 (2021) 18. Pandey, M., Som, T., Verma, S.: Set-valued α-fractal functions (2022). arXiv:2207.02635

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19. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, Amsterdam (1998) 20. Podlubny, I.: Geometric and Physical interpretation of fractional integration and fractional differentiation (2001). arXiv:math/0110241 21. Ross, B.: Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974, vol. 457. Springer, Berlin (2006) 22. Samko, S.G.: Fractional integrals and derivatives, theory and applications. Minsk, Nauka I Tekhnika (1987) 23. Verma, S., Liang, Y.: Effect of the Riemann-Liouville fractional integral on unbounded variation points. arXiv:2008.11113 24. Verma, S., Viswanathan, P.: Bivariate functions of bounded variation: fractal dimension and fractional integral. Indag. Math. 31(2), 294–309 (2020) 25. Verma, S., Viswanathan, P.: A note on Katugampola fractional calculus and fractal dimensions. Appl. Math. Comput. 339, 220–230 (2018) 26. Wu, X.E., Du, J.H.: Box dimension of Hadamard fractional integral of continuous functions of bounded and unbounded variation. Fractals 25(03), 1750035 (2017)

Fractional Operator Associated with the Fractal Integral of A-Fractal Function T. M. C. Priyanka and A. Gowrisankar

Abstract An advanced calculus called the fractal calculus is formulated as a generalization of ordinary calculus and it is being applied to functions with fractal support, where the standard calculus cannot be applied. In this paper, the concepts of fractal functions and fractal calculus have been interconnected by exploring the fractal integral of A-fractal function with predefined initial conditions. In addition, a fractional operator is presented, which takes each vector-valued continuous function to its fractal integral. Keywords Iterated function system · Fractal interpolation function · A-fractal function · Fractal integral · Fractional operator

1 Introduction and Preliminaries Fractal calculus has been introduced as a new framework to study the fractal curves and fractal like sets, which is different from fractional and classical calculus methods (refer [1, 2]). The derivatives and integrals involved in fractal calculus are, respectively, called as fractal derivatives and fractal integrals. As fractal functions are nondifferentiable, its fractional calculus have been investigated by many researchers in [3–12]. Besides, recently fractal calculus of fractal functions has been discussed in [13–15]. For more interesting results on types of fractal interpolation functions (FIFs) and its developments, refer [16–21]. The fractal integral of hidden variable FIF and α-fractal function has been investigated in [22]. So far, the fractal calculus of A-fractal function, which is the blend of α-fractal function and hidden variable FIF [23, 24] and [25, 26], has not been discussed. This literature gap instigated us to work on the fractal integral of A-fractal function. In addition, a fractional operator is

T. M. C. Priyanka · A. Gowrisankar (B) Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_9

111

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proposed, which gives the fractal integral of A-fractal function for each vector-valued continuous function. In this section, the general material required for defining the fractal integral is summarized. Section 2 discusses the construction of fractal interpolation function and A-fractal function. The fractal integral of A-fractal function is explored with the predefined initial conditions in Sect. 3. Fractional operator associated with the fractal integral of A-fractal function is presented in Sect. 4. This paper begins with the definition of mass function. A-fractal curve can be described as the image of R2 -valued continuous functions f defined on R, which are fractals. Let F ⊂ Rn be a fractal curve with sub-division D[a,b] . The fractal curve F is parameterizable if there exists a bijective continuous function w : R → F. Definition 1 The mass function γ α (F, a, b) is defined by γ α (F, a, b) = lim

inf

n−1  |w(yk ) − w(yk+1 )|α

→0 {P[a,b] :|P|≤}

k=1

Γ (α + 1)

,

where Γ (x) is the gamma function, |.| is the Euclidean metric on Rn , 1 < α ≤ 2 and |P| denotes the maximum of (yk+1 − yk ) for k = 0, 1, . . . , n − 1. Definition 2 For the fractal curve F and for some arbitrary fixed point r0 ∈ [a, b], the staircase function is defined by S Fα (t) =



γ α (F, r0 , y), y ≥ r0 , −γ α (F, y, r0 ), otherwise.

Let C(y1 , y2 ) denote the segment of the fractal curve, C(y1 , y2 ) = {w(y ∗ ) : y ∗ ∈ [y1 , y2 ]}.

Definition 3 For the function f , the upper and the lower f α -sum over the subdivision D are provided by U α [ f, F, D] =

n−1 

M[ f, C(yk , yk+1 )][S Fα (yk+1 ) − S Fα (yk )]

(1)

m[ f, C(yk , yk+1 )][S Fα (yk+1 ) − S Fα (yk )].

(2)

k=0

L α [ f, F, D] =

n−1  k=0

Define

 C(a,b)

f (x)d Fα x = sup L α [ f, F, D] P[a,b]

Fractional Operator Associated with the Fractal Integral …

and

 C(a,b)

113

f (x)d Fα x = inf U α [ f, F, D]. P[a,b]

Definition 4 For the continuous bounded function f and for x ∈ F, the F α -integral is given by  C(a,b)

f (x)d Fα x

 = C(a,b)

f (x)d Fα x

 = C(a,b)

f (x)d Fα x.

The following section describes the generation of fractal interpolation function and A-fractal function via the iterated function system defined on I × R and I × R2 , respectively.

2 Fractal Interpolation Function The following is the construction of the fractal interpolation function to interpolate the dataset {(xk , yk ) ∈ R2 : k = 0, 1, . . . , N , N ∈ N} with xk−1 < xk for all k = 1, 2, . . . , N . Suppose the closed intervals [x0 , x N ] and [xk−1 , xk ] are denoted as I and Ik , respectively, k = 0, 1, . . . , N . Define the contraction homeomorphisms L k : I → Ik by L k (x) = ak x + bk , where ak =

xk − xk−1 x N xk−1 − x0 xk , bk = , x N − x0 x N − x0

satisfying the contraction condition |L k (t) − L k (t  )| ≤ m k |t − t  |, for all t, t  ∈ I , m k ∈ (0, 1) and the maps L k obey L k (x0 ) = xk−1 , L k (x N ) = xk .

(3)

Let Fk be the real-valued continuous functions defined on I × R by FI (x, y) = αk y + qk (x), where αk is the vertical scaling factor such that |αk | < 1; qk is a continuous function on I ; and for all v, v  ∈ R and u ∈ I , Fk are contraction in the second variable, |Fk (u, v) − Fk (u, v  )| ≤ rk |v − v  |, where rk is the contraction factor such that rk ∈ (0, 1), i ∈ {1, 2, . . . , N }. In addition, Fk satisfy the join-up conditions

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Fk (x0 , y0 ) = yk−1 , Fk (x N , y N ) = yk .

(4)

Now, using the two contraction maps L k and Fk , a new contraction map wk : X → Ik × R, k = 1, 2, . . . , N is defined by wk (x, y) = (L k (x), Fk (x, y)).

(5)

The complete metric space X with the finite collection of contraction maps wk constitutes an Iterated Function System (IFS) denoted by {X ; wk : k = 1, 2, . . . , N }.

(6)

A Hutchinson self-map W is defined on the collection of all non-empty compact subsets of X , denoted by K(X ), as W (B ∗ ) =

N 

wk (B ∗ )

k=1

for any B ∗ ∈ K(X ). As W is contraction on the complete metric space K(X ) with respect to Hausdorff metric, by the Banach contraction theorem, there is a unique invariant set G ∗ for the map W satisfying G ∗ = W (G ∗ ). Therefore, the set G ∗ is known as the attractor or deterministic fractal for the IFS (6). Let C(I ) be the set of continuous functions g : I → R satisfying g(x0 ) = y0 , g(x N ) = y N and ρ be the uniform metric defined on C(I ) as follows: ρ(g, h) = max{|g(x) − h(x)| : x ∈ I }. Then, (C(I ), ρ) becomes a complete metric space. To obtain the fractal interpolation function, the Read–Bajraktarevi´c (RB) operator T is defined on C(I ) by   −1 T(h(t)) = Fk L −1 k (t), h ◦ L k (t) , t ∈ Ik , k = 1, 2, . . . , N .

(7)

By the Banach contraction principle, the contraction map T has a unique fixed point f ∈ C(I ) satisfying the functional equation,   −1 f (t) = T( f (t)) = Fk L −1 k (t), f ◦ L k (t) , k = 1, 2, . . . , N ,

(8)

for all t ∈ I . This function is referred to as the Fractal Interpolation Function (FIF) corresponding to the mappings wk for all k = 1, 2, . . . , N . The reader is encouraged to refer [27–33].

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115

2.1 Hidden Variable A-Fractal Function The definition of hidden variable A-fractal function is precisely described as follows. In order to approximate R2 -valued continuous functions, a special type of fractal function called the A-fractal function is constructed as follows. To provoke a family of fractal functions f[A] parametrized by a block matrix  αk βk , a continuous R2 -valued function A = [Ak ], k = 1, 2, . . . , N with Ak = 0 γk f : I → R2 is enunciated. When A = 0, f[A] equals the original function f. Consider the generalized dataset D = {(xk , f 1 (xk ), f 2 (xk )) : k = 1, 2, . . . , N }. Then the IFS corresponding to D is given by {I × R2 ; wk : k = 1, 2, . . . , N },

(9)

where wk (x, y) = (L k (x), Fk (x, y, z)), L k (x) = ak x + bk ,

Fk (x, y, z) = (αk y + βk z + pk (x), γk z + qk (x)).

The continuous functions pk and qk are given by pk (x) = f 1 ◦ L k (x) − αk b1 (x) − βk b2 (x), qk (x) = f 2 ◦ L k (x) − γk b2 (x), (10) where b = (b1 , b2 ) ∈ C(I, R2 ) obeys b(x0 ) = f(x0 ) and b(x N ) = f(x N ). The fixed point of the IFS (9) is the graph of the continuous vector-valued function f[A] = ( f 1 [A], f 2 [A]) which obeys the following fixed point equation: f[A](t) = f(t) + Ak (f[A] − b)(L −1 k (t)), t ∈ I, k = 1, 2, . . . , N .

(11)

This function f[A] is known as the hidden variable A-fractal function (or simply A-fractal function) approximating the given continuous function f with respect to x0 < x1 < · · · < x N and the base function b. Then, the two components of f[A], namely, f 1 [A] and f 2 [A] satisfy −1 −1 f 1 [A](x) = αk f 1 [A]L −1 k (x) + βk f 2 [A]L k (x) + pk (L k (x)), −1 f 2 [A](x) = γk f 2 [A]L −1 k (x) + qk (L k (x)).

(12)

For any choice of A and b satisfying the above-defined conditions, it is noticed that f[A](xk ) = f(xk ), ∀k = 0, 1, . . . , N . Hence, the function f[A] can be referred to as the fractal generalization of the continuous R2 -valued function f. For more information on A-fractal function, see [34, 35].

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3 Fractal Integral on A-Fractal Function The fractal integral of A-fractal function defined at the initial point is investigated in this section. The following is the definition of fractal integral of A-fractal function. Definition 5 Let f[A] be the A-fractal function corresponding to the IFS (6). For given yˆ0 and zˆ 0 , the fractal integral of order μ for f[A] is defined componentwise as follows:  x μ μ ˆ f 1 [A](x) = yˆ0 + S F ( f 1 [A](t))d F t, x  0x (13) μ μ ˆ f 2 [A](x) = zˆ 0 + S F ( f 2 [A](t))d F t. x0

The following theorem examines the resultant function of the fractal integral of A-fractal function. Theorem 1 Suppose f[A] is the (hidden variable) A-fractal function generated by the IFS {L k (x), Fk (x, y, z) : k = 1, 2, . . . , N }. The fractal integral of f[A], defined in (13), is denoted as ˆf[A]. Then, for given yˆ0 = 0 and zˆ 0 = 0, ˆf[A] = ( fˆ1 [A], fˆ2 [A]) is again an A-fractal function determined by the IFS {(L k (x), Fˆk (x, yˆ , zˆ )) : k ∈ 1, 2, . . . , N }, where Fˆ1k (x, yˆ , zˆ ) = ak αk yˆ + ak βk zˆ + pˆ k (x), Fˆ2k (x, zˆ ) = ak γk zˆ +

N

N ak αk = 1, k=1 ak γk = 1, qˆk (x) , with k=1 pˆ k (x) = yˆk−1 + ak yˆk =

k 

x0

μ

 x

μ

S F ( f 1 ◦ L k (t))d F t − ak αk

x0

μ

μ

S F (b1 (t))d F t − ak βk

 x x0

μ

μ

S F (b2 (t))d F t,

  x  x N μ N μ μ μ an αn yˆ N + βn zˆ N + S F ( f 1 ◦ L n (t))d F t − αn S F (b1 (t))d F t

n=1

− βn

 x

 x N x0

 μ μ S F (b2 (t))d F t ,

x0

x0

  x μ x μ x μ μ μ μ βn zˆ N + x0N S F ( f 1 ◦ L n (t))d F t − αn x0N S F (b1 (t))d F t − βn x0N S F (b2 (t))d F t yˆ N = ,

N 1 − n=1 an αn  x  x μ μ μ μ qˆk (x) = zˆ k−1 + ak S F ( f 2 ◦ L k (t))d F t − ak γk S F (b2 (t))d F t,

N

n=1 an

zˆ k =

k 

x0



xN

an γn zˆ N +

n=1

N zˆ N =

x0



n=1 an



x0

xN x0

μ

μ

μ

μ

S F ( f 2 ◦ L n (t))d F t − γn

N 1 − n=1 an γn

for k = 1, 2, . . . , N .



S F ( f 2 ◦ L n (t))d F t − γn xN x0

xN x0

μ

 μ μ S F (b2 (t))d F t , μ

S F (b2 (t))d F t

 ,

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Proof Using the definition of fractal integral for the function fˆ1 [A] provides fˆ1 [A](L k (x)) = yˆ0 + = yˆ0 +



L k (x)

x0  xk−1 x0

μ

μ

S F ( f 1 [A](t))d F t μ



μ

S F ( f 1 [A](t))d F t +



x

= yˆk−1 + ak

L k (x) xk−1

μ

μ

S F ( f 1 [A](t))d F t

μ μ S F ( f 1 [A]L k (t))d F t.

x0

The first functional equation in (12) gives = yˆk−1 + ak αk

 x

μ

μ

S F (αk f 1 [A](t) + βk f 2 [A](t) + pk (t))d F t  x  x μ μ μ μ = yˆk−1 + ak αk S F ( f 1 [A](t))d F t + ak βk S F ( f 2 [A](t))d F t + ak x0 x0 x0  x  x  x μ μ μ μ = yˆk−1 + ak αk S F ( f 1 [A](t))d F t + ak βk S F ( f 2 [A](t))d F t + ak x0 x0 x0  x  x μ μ μ μ S F (b1 (t))d F t − ak βk S F (b2 (t))d F t − a k αk x0  x

x0

x0

x0

x0

μ

μ

S F ( pk (t))d F t μ

μ

S F ( f 1 ◦ L k (t))d F t

 x μ μ = yˆk−1 + ak αk fˆ1 [A](x) + ak βk fˆ2 [A](x) + ak S F ( f 1 ◦ L k (t))d F t x0  x  x μ μ μ μ − a k αk S F (b1 (t))d F t − ak βk S F (b2 (t))d F t = ak αk fˆ1 [A](x) + ak βk fˆ2 [A](x) + pˆ k (x) = Fˆ1k (x, fˆ1 [A](x), fˆ2 [A](x)).

x μ x μ μ μ Denote pˆ k (x) = yˆk−1 + ak x0 S F ( f 1 ◦ L k (t))d F t − ak αk x0 S F (b1 (t))d F t − x μ μ ak βk x0 S F (b2 (t))d F t. Now, applying the definition of fractal integral for the function fˆ2 [A] provides fˆ2 [A](L k (x)) = zˆ 0 + = zˆ 0 +



L k (x)

x0  xk−1 x0

= zˆ k−1 + ak



μ

μ

S F ( f 2 [A](t))d F t μ

μ

S F ( f 2 [A](t))d F t + x x0



L k (x) xk−1

μ μ S F ( f 2 [A]L k (t))d F t.

The second functional equation in (12) gives

μ

μ

S F ( f 2 [A](t))d F t

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T. M. C. Priyanka and A. Gowrisankar 

x

= zˆ k−1 + ak αk = zˆ k−1 + ak γk = zˆ k−1 + ak γk

x0  x x0  x x0

μ

μ

S F (γk f 2 [A](t) + qk (t))d F t μ



μ

S F ( f 2 [A](t))d F t + ak μ

μ

S F ( f 2 [A](t))d F t + ak

= zˆ k−1 + ak γk fˆ2 [A](x) + ak



x x0

x

x0  x x0

μ

μ

S F (qk (t))d F t μ



μ

S F ( f 2 ◦ L k (t))d F t − ak γk

μ



μ

S F ( f 2 ◦ L k (t))d F t − ak γk

x x0

x x0

μ

μ

μ

S F (b2 (t))d F t

μ

S F (b2 (t))d F t

= ak γk fˆ2 [A](x) + qˆk (x) = Fˆ2k (x, fˆ2 [A](x)).

x μ x μ μ μ Denote qˆk (x) = zˆ k−1 + ak x0 S F ( f 2 ◦ L k (t))d F t − ak γk x0 S F (b2 (t))d F t. In order to find the new data points yˆk and zˆ k , take x = x N and L k (x N ) = xk , 

yˆk = yˆk−1 + ak αk fˆ1 [A](x N ) + ak βk fˆ2 [A](x N ) + ak 

xN

− ak αk

x0

μ



xN

yˆk − yˆk−1 = ak αk yˆ N + ak βk zˆ N + ak

x0



xN

− ak αk

x0

μ



x0

k 

  an αn yˆ N + βn zˆ N +

x0

n=1

zˆ k =

k 

xN





an γn zˆ N +

xN x0

n=1

μ

μ

 

xN

μ

xN

μ

S F (b2 (t))d F t

x0 μ

μ



μ

μ

k



μ

μ

μ



xN x0

xN x0

xN

μ

xN x0

μ

μ

S F (b2 (t))d F t

μ

S F (b2 (t))d F t.

x0

n=1 (yn+1

S F ( f 1 ◦ L n (t))d F t − αn

S F ( f 2 ◦ L n (t))d F t − γn



μ

S F ( f 2 ◦ L k (t))d F t − ak γk

S F ( f 2 ◦ L k (t))d F t − ak γk

Using the system of equations, yˆk = yˆ0 + be obtained as follows: yˆk =

μ

S F (b2 (t))d F t

μ

x0 xN

μ

μ

S F ( f 1 ◦ L k (t))d F t

S F ( f 1 ◦ L k (t))d F t

μ

zˆ k = zˆ k−1 + ak γk fˆ2 [A](x N ) + ak

xN x0

S F (b1 (t))d F t − ak βk

zˆ k − zˆ k−1 = ak γk zˆ N + ak

x0



μ

S F (b1 (t))d F t − ak βk

xN

− yn ), the new data points can

μ

μ

S F (b1 (t))d F t − βn



xN x0

μ

μ



S F (b2 (t))d F t

 μ μ S F (b2 (t))d F t .

For k = N , one can get

yˆ N =

N zˆ N =



x x x μ μ μ N μ N μ N μ n=1 an βn zˆ N + x0 S F ( f 1 ◦ L n (t))d F t − αn x0 S F (b1 (t))d F t − βn x0 S F (b2 (t))d F t

N

n=1 an



x

N

x0

N 1 − n=1 a n αn  μ μ μ μ S F ( f 2 ◦ L n (t))d F t − γn xx N S F (b2 (t))d F t 1−

N

n=1 an γn

0

.



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119

Since the continuous functions pˆ k (x) and qˆk (x) obey the below endpoint conditions: pˆ k (x0 ) = yˆk−1 , pˆ k (x N ) = yˆk − ak αk yˆ N − ak βk zˆ N and qˆk (x0 ) = zˆ k−1 , qˆk (x N ) = zˆ k − ak γk zˆ N which implies Fˆk (x0 , yˆ0 , zˆ 0 ) = ( yˆk−1 , zˆ k−1 ) and Fˆk (x N , yˆ N , zˆ N ) = ( yˆk , zˆ k ). Therefore, the function ˆf[A] = ( fˆ1 [A], fˆ2 [A]) is again an A-fractal function corresponding to the IFS {(L k (x), Fˆk (x, yˆ , zˆ )) : k ∈ 1, 2, . . . , N }.

4 Fractional Operator This section proposes a fractional operator associated with the fractal integral of A-fractal function and verifies its linearity. Let f ∈ C(I, R2 ) consider the base function b = f ◦ c, where c = (c1 , c2 ) ∈ C(I, R2 ) is not the identity function such that c(x0 ) = x0 and c(x N ) = x N . The fractional operator of order 0 < m < 1 for the vector-valued function, F m [A] :C(I, R2 ) → C(I, R2 ) f −→ ˆf[A] is defined by F m [A](f(x)) = yˆk−1 + ak αk fˆ1 [A](x) + ak βk fˆ2 [A](x) + ak − a k αk

 x x0

μ

μ

S F (b1 (t))d F t − ak βk

+ zˆ k−1 + ak γk fˆ2 [A](x) + ak

 x x0

 x x0

μ

 x x0

μ

μ

S F ( f 1 ◦ L k (t))d F t

μ

S F (b2 (t))d F t

μ

μ

S F ( f 2 ◦ L k (t))d F t − ak γk

 x x0

μ

μ

S F (b2 (t))d F t,

(14)

for all x ∈ Ik , k = 1, 2, . . . , N . For some u, v ∈ R and f, g ∈ C(I, R2 ), F m [A](uf(x)) = u yˆk−1 + ak αk u fˆ1 [A](x) + ak βk u fˆ2 [A](x) + ak u

x0

μ

μ

S F ( f 1 ◦ L k (t))d F t

 x μ μ μ μ S F (b1 (t))d F t − ak βk u S F (b2 (t))d F t x0 x0  x  x μ μ μ μ S F ( f 2 ◦ L k (t))d F t − ak γk u S F (b2 (t))d F t + u zˆ k−1 + ak γk u fˆ2 [A](x) + ak u x0 x0  x μ μ F m [A](vg(x)) = v yˆk−1 + ak αk v gˆ 1 [A](x) + ak βk v gˆ 2 [A](x) + ak v S F (g1 ◦ L k (t))d F t − a k αk u

 x

 x

x0

120

T. M. C. Priyanka and A. Gowrisankar − a k αk v

 x x0

μ

μ

S F (b1 (t))d F t − ak βk v

+ v zˆ k−1 + ak γk v gˆ 2 [A](x) + ak v

 x

 x x0

x0

μ

μ

S F (b2 (t))d F t

μ

μ

S F (g2 ◦ L k (t))d F t − ak γk v

 x x0

μ

μ

S F (b2 (t))d F t.

From the above equations, it can be easily verified that F m [A](uf + vg) = uF m [A](f) + vF m [A](g). Therefore, the operator F m [A] is a linear operator. Finally, it is concluded that the resultant function of the fractal integral of A-fractal function is again an A-fractal function. Furthermore, the fractional operator defined in Sect. 4 is demonstrated as a linear operator.

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Mathematical Modeling

A Multi-strain Model for COVID-19 Samiran Ghosh and Malay Banerjee

Abstract The main objective of this work is to propose and analyze a multicompartment ordinary differential equation model for multi-strain epidemic disease. The proposed model mainly focuses on the epidemic disease spread due to SARSCoV-2, and the recurrent outbreaks are due to the emergence of a new strain. The possibility of reinfection of the recovered individuals is considered in the model. The multi-strain model is validated with the help of strain-specific daily infection data from France and Italy. Keywords Epidemic model · Multi-strain · Reproduction number · Epidemic outbreak

1 Introduction In description of the dynamics of disease progression over a short or long period, two different types of compartmental epidemic models are used, namely, the model with demography and model without demography [10, 23]. Researchers have considered models with demography for established epidemic diseases which helps to perform some preliminary stability analysis of the model under consideration around the disease-free equilibrium (DFE) and the endemic equilibrium point. However, the possibility of an endemic equilibrium in the context of COVID-19 remains in vein. In some reported researches, the demographic terms are incorporated into the models by considering certain rate of recruitment in the susceptible compartment from the healthy compartment and the natural mortality within each of the compartments involved with the model. In reality, one can determine the recruitment rate in the susceptible compartment from healthy individuals only when we have a detailed history of epidemic progression and relevant data for 10 years or more [6, 12, 16]. Further, the natural mortality rate of each compartment is based on the average life S. Ghosh · M. Banerjee (B) Department of Mathematics and Statistics, IIT Kanpur, Kanpur 208016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_10

125

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span of a human, which can be considered as 70−75 years on average. However, when measured from the time since infection, the COVID-19-related mortality rate varies from 15 to 60 days at most. As a matter of fact, both the death rates cannot be included in a model which is described in terms of a single time scale. Compartmental models, defined in terms of ordinary differential equations, have played a crucial role in studying the transmission dynamics of infectious diseases. In the history of mathematical epidemiology, the conventional SIR model has been widely used to describe the epidemic disease progression [10, 23, 24]. The Spanish flu epidemic of 1918−1919, also known as the Great Influenza epidemic, was an exceptionally deadly epidemic across the globe caused by the N1H1 influenza-A virus. This Spanish flu resulted in approximately 40,000,000 death tolls [26]. It was challenging for the epidemiologist to find an adequate methodology to describe the epidemic’s progression. In this context, W. O. Kermack and A. G. McKendrick gave a general theory of epidemics in the year 1927, which is one of the early contributions to the field of mathematical modeling of epidemic diseases [17–19]. Kermack and McKendrick formulated a deterministic epidemic model consisting of susceptible, infected, and recovered compartments. Their model formulation included the age of infection, i.e., the time since infection. Kermack and McKendrick did not consider any demography pattern in their model, and they only focused on the single disease outbreak. After this novel mathematical formulation to describe epidemic spread, many developments have been observed in mathematical modeling and relevant study for epidemic diseases. Mathematical modeling of infectious diseases has gained more importance in the advent of the HIV epidemic outbreak in the 1980s [11, 14]. After that, the world has experienced several epidemic diseases, and currently the COVID19 since the end of 2019. For better modeling of various infectious diseases, a wide variety of mathematical models have been developed and analyzed thoroughly. In the context of COVID-19, there exists a large number of research works that studied and predicted the disease progression of COVID-19, based on the multi-compartment models (see, for example, [8, 21, 25, 27]). Continuous mutation of viruses, which are the main factor behind the spread and recurrent outbreaks, lead to many Variants of Concern (VOC). It is a matter of the fact that the individuals who are recovered from some particular variant can get infected further by another variant/strain. Thus, when a new variant arises, the size of susceptible to the new variant is larger compared to the size of susceptible to the existing strains. The two-strain model for TB, SIQR-type model with multiple strains, etc. is thoroughly discussed in [10, 23]. The SIQR stands for susceptibleinfected-quarantined-recovered compartments. In the context of COVID-19, we have witnessed that many VOCs appeared one after another over the last 2 years, e.g., Alpha, Beta, Gamma, Delta, and recently Omicron. As per the information available at [1], the earliest documented sample of Alpha variant was found in the United Kingdom in September 2020. The variant Beta was first reported in South Africa in May 2020. Gamma strain was first reported in Brazil in November 2020. Delta strain was first reported in India in October 2020, and very recently, the Omicron was registered simultaneously in many countries in the month of November 2021. We have noticed that in most countries, each VOC resulted in a significant sizeable new

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epidemic outbreak. Thus, the appearance of new variants needs to be incorporated into the modeling approach appropriately to capture successive outbreaks of the epidemic. In this work, we aimed to study the effect of the appearance of a new strain of SARS-CoV-2 through a multi-compartment multi-strain epidemic model. First, we consider a single-strain epidemic model and calculate the controlled reproduction in Sect. 2. Then, in Sect. 3, we will formulate a two-strain model based on the assumption that a new strain appears after a certain period counted from the peak of the first wave. Finally, we will extend our two-strain model to an n-strain model in Sect. 3.2. To validate the proposed model with real data, some parameters involved with the model are estimated with the help of strain-specific data for France and Italy. The number of daily infected obtained from the simulation of the proposed model is verified with the real data in Sect. 4. The significant outcomes of this study are summarized in the concluding section.

2 Single-Strain Model Existing mathematical models of epidemiology with multi-strain assume that healthy and susceptible individuals can get an infection from an infected individual who is infected through any one of the existing strains. In the contrary, multiple strains can emerge one by one due to some evolutionary aspects taking place over a shorter or longer time scale with respect to the duration of several outbreaks [9]. Here we consider a multi-compartment model to describe the epidemic disease progression due to the arrival of different strains one after another at some time intervals. This type of modeling approach is inspired by the successive outbreak of COVID-19 epidemic due to the mutation of SARS-CoV-2 viruses since the beginning of 2020. In order to start with, we assume that the first outbreak occurs due to initial strain and we consider SEIHR-type multi-compartment model. For simplicity of mathematical calculations as well as epidemiological justification, we consider the model without demography in order to take care of the fact that the duration of epidemic outbreak is significantly less compared to the life span of human. Our starting model consists of six compartments including susceptible individuals (S), exposed individuals (E), infected individuals with symptoms (I), asymptomatic infected individuals (A), hospitalized individuals (H), and recovered individuals (R). The governing equation for each compartment is described by the following set of ordinary differential equations:

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d S(t) dt d E(t) dt d I (t) dt d A(t) dt d H (t) dt d R(t) dt

β S(t) (I (t) + α A(t)), N β S(t) = (I (t) + α A(t)) − σ E(t), N =−

(1a) (1b)

= r σ E(t) − ηI (t) − (δ I + μ I )I (t),

(1c)

= (1 − r )σ E(t) − δ A A(t),

(1d)

= ηI (t) − (δ H + μ H )H (t),

(1e)

= δ I I (t) + δ A A(t) + δ H H (t),

(1f)

subject to non-negative initial condition with empty hospitalized and recovered compartments. Here β is the disease transmission rate by symptomatic infected individuals I (t); α is the ratio of the disease transmission rate by asymptomatic infected individuals A(t) over symptomatic infected individuals I (t); σ is the reciprocal of latent period; r is the rate at which exposed individuals become symptomatic; η is the rate of hospitalization of symptomatic individuals; δ I , δ A , and δ H are recovery rate from symptomatic, asymptomatic, and hospitalized compartments, respectively; μ I , μ H denote death rates in symptomatic and hospitalized compartments, respectively; and N is the total population size.

2.1 Controlled Reproduction Number First, we compute the controlled reproduction number of the system (1). The diseasefree equilibrium point is given by (S, E, I, A, H, R) ≡ P1 (N , 0, 0, 0, 0, 0). We use the next-generation matrix approach [13] to calculate the controlled reproduction number. We consider the compartments S, R as the non-infected compartments and the compartments E, I , A, and H as the infected compartments. Then the matrix F corresponding to the new infection and the matrix V corresponding to the outflow from the infected compartments are given by ⎛ βS ⎜ F =⎜ ⎝

N

⎞ (I + α A) ⎟ 0 ⎟, ⎠ 0 0

⎛

⎞ σE ⎜ ηI + (δ I + μ I )I − r σ E ⎟ ⎟. V=⎜ ⎝ δ A A − (1 − r )σ E ⎠ (δ H + μ H )H − ηI

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The Jacobian of F and V evaluated at the disease-free equilibrium point P1 is given by ⎛

0 ⎜0 F =⎜ ⎝0 0

β 0 0 0

αβ 0 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

⎛

σ 0 ⎜ −r σ (η + δ I + μI ) V =⎜ ⎝ −(1 − r )σ 0 0 −η

0 0 δA 0 (δ H

⎞ 0 ⎟ 0 ⎟. ⎠ 0 + μH )

The controlled reproduction number is the spectral radius of the matrix F V −1 and is given by βr αβ(1 − r ) + . Rc = η + δI + μI δA It is important to mention here that the reproduction number calculated above is known as controlled reproduction number as the expression involves parameters related to the hospitalized compartment. Substituting η = 0 in above expression one can find the basic reproduction number. In the context of epidemiology, the provision of hospitalization is considered as a control measure. In reality it reduces not only the death toll rather it can reduce the epidemic spread due to isolation. Using the approach described in [27, 28], one can calculate the important epidemic indicators like final size of the epidemic, maximum number of infected, and the day on which maximum number of infection will appear for the model (1). Here we omit those calculations to avoid the repetition of same mathematical calculations. However, some of these quantities are calculated numerically in the next section.

3 Two-Strain Model To formulate the two-strain model, we assume that only one strain is responsible for the initial outbreak of the epidemic until a new strain appears, and the second strain appears at time t1 say. Then the model (1) is valid for t ≤ t1 and for t > t1 we have to define the model with two strains. Since there are many literature which suggest that the re-infection may occur after 5–6 months after the earlier infection [20], the re-infection of the individuals recovered from new strain (R2 compartment) is ignored for the time being, and the same will be incorporated when another strain will emerge. So, for the time t > t1 and till another strain emerges, we will use the following model:

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β1 S(t) d S(t) β2 S(t) =− (I1 (t) + α1 A1 (t)) − (I2 (t) + α2 A2 (t)), dt N N β1 S(t) d E 1 (t) = (I1 (t) + α1 A1 (t)) − σ1 E 1 (t) dt N β1 +ξ11 (I1 (t) + α1 A1 (t))R1 (t), N d I1 (t) = r1 σ1 E 1 (t) − η1 I1 (t) − (δ1I + μ1I )I1 (t), dt d A1 (t) = (1 − r1 )σ1 E 1 (t) − δ1A A1 (t), dt d H1 (t) = η1 I1 (t) − (δ1H + μ1H )H1 (t), dt d R1 (t) = δ1I I1 (t) + δ1A A1 (t) + δ1H H1 (t) dt R1 (t) (ξ11 β1 (I1 (t) + α1 A1 (t)) + ξ12 β2 (I2 (t) + α2 A2 (t))), − N β2 S(t) d E 2 (t) = (I2 (t) + α2 A2 (t)) − σ2 E 2 (t) dt N β2 +ξ12 (I2 (t) + α2 A2 (t))R1 (t), N d I2 (t) = r2 σ2 E 2 (t) − η2 I2 (t) − (δ2I + μ2I )I2 (t), dt d A2 (t) = (1 − r2 )σ2 E 2 (t) − δ2 A A2 (t), dt d H2 (t) = η2 I2 (t) − (δ2H + μ2H )H2 (t), dt d R2 (t) = δ2I I2 (t) + δ2 A A2 (t) + δ2H H2 (t), dt

(2a)

(2b) (2c) (2d) (2e)

(2f)

(2g) (2h) (2i) (2j) (2k)

where ξ11 is a multiplicative factor such that ξ11 β1 represents the rate of infection of recovered individuals from strain-1 again by strain-1. Similarly, ξ12 is another multiplicative factor such that ξ12 β2 is the rate of infection of individuals of R1 compartment by the strain-2. All other parameters with subscript j ( j = 1, 2) bear the same meaning as they were for model (1) corresponding to two strains. The disease transmission rate of an individual is proportional to the viral load inside the body [15]. The viral load at the time of re-infection (by the same or different strain) may be lower than the viral load during earlier infection, due to the acquired immunity resulting from prior infection [5]. In case of re-infection by the same strain, the individuals will move from R1 to E 1 once they are exposed. Otherwise, the individuals will move from R1 to E 2 in case of re-infection due to the second (new) strain. To track the role of re-infection on new outbreak, recovered individuals are not transferred to the susceptible compartment S.

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3.1 Effective Reproduction Number In this section, our focus is to compute the effective reproduction number at the time t1 when a new strain emerges. We will find the effective reproduction number of the system (2) around the point P2 (S(t1 ), I1 (t1 ), A1 (t1 ), H1 (t1 ), R1 (t1 ), 0, 0, 0, 0, 0). We now order the infected compartments in the following order: E 1 , I1 , A1 , H1 , E 2 , I2 , A2 , H2 , and we can write two matrices as follows ⎛

⎛ β1

⎞ σ1 E 1 ⎜ η1 I1 + (δ1I + μ1I )I1 − r1 σ1 E 1 ⎟ ⎜ ⎟ ⎜ ⎟ δ1A A1 − (1 − r1 )σ1 E 1 ⎜ ⎟ ⎜ ⎟ (δ1H + μ1H )H1 − η1 I1 ⎟. V=⎜ ⎜ ⎟ σ2 E 2 ⎜ ⎟ ⎜ η2 I2 + (δ2I + μ2I )I2 − r2 σ2 E 2 ⎟ ⎜ ⎟ ⎝ ⎠ δ2 A A2 − (1 − r2 )σ2 E 2 (δ2H + μ2H )H2 − η2 I2

⎞ (I1 + α1 A1 )(S + ξ11 R1 ) ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ 0 ⎟ F =⎜ ⎜ β2 (I2 + α2 A2 )(S + ξ12 R1 ) ⎟ , ⎜N ⎟ ⎜ ⎟ 0 ⎜ ⎟ ⎝ ⎠ 0 0 N

The corresponding Jacobian matrices evaluated at the point P2 are given by  F=

F1 θ4×4 θ4×4 F2



 ,

V =

V1 θ4×4 θ4×4 V2

,

where ⎛

0 ⎜0 Fj = ⎜ ⎝0 0 ⎛

βj N

(S(t1 ) + ξ1 j R1 (t1 )) 0 0 0

0 σj ⎜ −r j σ j (η + δ j j I + μjI) Vj = ⎜ ⎝ −(1 − r j )σ j 0 0 −η j

αjβj N

⎞ (S(t1 ) + ξ1 j R1 (t1 )) 0 0 0⎟ ⎟, 0 0⎠ 0 0

0 0 δjA 0 (δ j H

⎞ 0 ⎟ 0 ⎟, ⎠ 0 + μjH)

j = 1, 2,

and θ4×4 is the 4 × 4 null matrix. The spectral radius of F j V j−1 is given by R j (t1 ) =

β j r j (S(t1 ) + ξ1 j R1 (t1 )) α j β j (1 − r j )(S(t1 ) + ξ1 j R1 (t1 )) + , j = 1, 2. N (η j + δ j I + μ j I ) NδjA

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The corresponding effective reproduction number for the system (2) calculated at time t1 is given by R(t1 ) = max{R1 (t1 ), R2 (t1 )}. Based on the magnitudes of the parameters and size of different compartments, one can have either one or both of the quantities (R1 (t1 ) and R2 (t1 )) greater than one. Here two parameters ξ and β play crucial role to determine R j (t1 ), j = 1, 2 greater than one or not. Further outbreak of the epidemic is indicated by at least one of them is greater than one.

3.2 Multi-strain Model In the context of COVID-19, we have observed a recurrent outbreaks of the epidemic due to the emergence of various strains since its first outbreak at the beginning of 2020 in most of the countries across the globe. It is quite difficult to track the appearance of a new strain in different countries as the determination of appropriate strain among the infected individuals pose a huge economic burden. At the initial time of the COVID-19 outbreak, the strain-specific infection data was not available. However, recent scientific efforts help us to track the spread of infection based on different strains of SARS-CoV-2 virus available for certain countries. There exist multiple variants of a single strain but in the context of large epidemic outbreaks, it is evident that the third wave was due to the delta strain and Omicron is responsible for the fourth wave. Following the approach described above, we can define a multicompartment model with the assumption that the nth strain has appeared at t = tn−1 . Susceptible individuals can get infection from the symptomatic or asymptomatic individuals infected through any one of the strains and hence their growth equation can be written as S(t)

d S(t) =− β j (I j (t) + α j A j (t)). dt N j=1 n

(3)

The inflow in the exposed compartment corresponding to jth strain comes from susceptible compartment and recovered from strain-1 to nth strain and hence their growth rate can be described as β j S(t) d E j (t) = (I j (t) + α j A j (t)) − σ j E j (t) dt N n−1

βj ξr j (I j (t) + α j A j (t))Rr (t), + N r =1

(4)

where η j is the rate of transfer from exposed to infected compartment once the incubation period is over, j = 1, 2, . . . , n. We can write the growth equations symptomatic

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infected, asymptomatic infected, and hospitalized compartments corresponding to strain- j as follows: d I j (t) = r j σ j E j (t) − η j I j (t) − (δ j I + μ j I )I j (t), dt

(5)

d A j (t) = (1 − r j )σ j E j (t) − δ j A A j (t), dt

(6)

d H j (t) = η j I j (t) − (δ j H + μ j H )H j (t), dt

(7)

where 1 ≤ j ≤ n, and the parameters have similar meaning as described earlier. Once the nth strain arrives, the individuals of all the recovered compartments R j , 1 ≤ j ≤ n − 1 can get re-infection from any one of the existing strains and hence we can write the growth equation for R j as follows: d R j (t) = δ j I I j (t) + δ j A A j (t) + δ j H H j (t) dt n

βr − ξ jr (Ir (t) + αr Ar (t))R j (t). N r =1

(8)

Finally, the growth equation of Rn compartment can be written as d Rn (t) = δn I In (t) + δn A An (t) + δn H Hn (t). dt

(9)

One can easily verify that the model (2) can be obtained from (3)–(9) with n = 2. The proposed modeling approach takes care of the possibility of re-infection without transferring the individuals from the recovered compartment to the susceptible compartment. One can calculate the effective reproduction number following the same approach as described above for the model with two strains. Explicit expression will be quite lengthy and hence we omit such calculations here.

3.3 Numerical Example Now we consider the numerical simulation results with hypothetical set of parameter values before proceeding to the model validation with realistic dataset. Throughout the numerical simulation, all the parameter values of β, σ , η, δ I , μ I , δ A , δ H , μ H in model (1) and the corresponding parameters in the model (3)–(9) are chosen in the unit of day −1 [29]. α and r are proportionality constants and hence they are dimensionless parameters. We consider the hypothetical set of parame-

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Fig. 1 Simulation corresponding to model (2). Panel-a: The strain-1 appears at time t = 0 and no new strain appears thereafter. The blue curve in this panel corresponds to only strain-1 with β1 = 0.6. Panel-b: The strain-1 appears at time t = 0 and the strain-2 appears at time t = t1 = 1000. The magenta curve extension and red curve extension in this panel correspond to β2 = 0.8 and β2 = 0.9, respectively. Other parameter values are: N = 107 , r1 = 0.7, r2 = 0.8, ξ11 = 0.01, ξ12 = 0.05, α j = 1, σ j = 1/5.2, η j = 0.1, δ j I = δ j A = 1/2.3, δ j H = 0.535, μ j I = μ j H = 0.03, for j = 1, 2. Initial conditions: S(1) = N − 3, E(1) = I (1) = A(1) = 1, H (1) = R(1) = 0

ter values: β = 0.6, α = 1, σ = 0.1923, r = 0.7, η = 0.1, δ I = δ A = 0.4348, μ I = μ H = 0.03 and simulate the model (1) with initial condition S(0) ≈ N , I (0) = 1, E(0) = A(0) = H (0) = R(0) = 0 where N = 107 . First we consider the case where only one strain is present in the system and the strain emerged at time t = 0 (shown in Fig. 1a). In that case, we find the maximum number of daily infected is around 16, 840, maximum number of daily infected appeared around 550th day from the onset of outbreak, and the speed of epidemic spread slows down gradually but one can observe that daily number of infected will be around 1700 even after 1,800 days (see Fig. 1a). Numerically one can calculate that total number of infected on 1,092th day is approximately 2.596 × 106 and the number of symptomatic active cases I (t) and asymptomatic active cases A(t) on 1,092th day are less than one, which implies that the epidemic is over theoretically, as max{I (t), A(t)} < 1, around t = 1, 092 days. Also we can calculate the final size of the epidemic as S f = 7.404 × 106 . Next we assume that a second strain appeared on 1,000th day and now we need to simulate the model (2). The choice of hypothetical parameter set is β1 = 0.6, β2 = 0.9, r1 = 0.7, r2 = 0.8, ξ11 = 0.01, ξ12 = 0.05, α j = 1, σ j = 0.1923, η j = 0.1, δ j I = δ j A = 0.4348, δ j H = 0.535, μ j I = μ j H = 0.03 for j = 1, 2 and t1 = 1, 000. Note that choices of initial conditions are S(0) ≈ N , I1 (0) = 1, I2 (0) = 0, E j (0) = A j (0) = H j (0) = R j (0) = 0 for j = 1, 2, N = 107 . Importantly, we have to introduce I2 (t = 1, 000) = 1 while no change in other compartments involved with the model (2). We assume that the second strain is more transmissible than the first strain and hence we choose the transmission rate of the second strain as β2 = 0.9. The re-growth of daily infected is shown by the red curve in Fig. 1b, and we find the maximum number of infected during the second peak as 37,457

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and the peak is observed around 1,310th day. To understand the sensitivity of the model parameters on the second outbreak, without any detailed mathematical insight, simply we decrease the transmission rate of the second strain slightly by choosing β2 = 0.8. Then we observe that the peak of the second outbreak (shown by the magenta curve in Fig. 1b) is significantly low. Also the maximum number of daily infected appears on 1,565th day approximately. For two different choices of β2 , the total number of infected on t = 2, 000 days are given by 4.2371 × 106 and 3.1721 × 106 corresponding to β2 = 0.9 (red curve in Fig. 1b) and β2 = 0.8 (magenta curve in Fig. 1b), respectively. These simulation results indicate that the transmission rate of the emerging strain affects the epidemic progression significantly. It is important to mention here that the proposed model is capable of capturing the multiple outbreaks of an epidemic disease due to the appearance of new strain. This scenario is quite relevant in the context of COVID-19. For better illustration and validation of our proposed model, we consider the model validation by estimating the parameters from real dataset in the next section.

4 Model Validation with COVID-19 Data In this section, we consider the strain-specific data from France during the time period March 1, 2020 to September 30, 2021, collected from [2]. During this span of 608 days for COVID-19 epidemic spread and recurrent outbreaks, continuous mutation of spike protein has led to many new variants of SARS-CoV-2 virus [3]. We have noticed that a new variant is responsible for behind each outbreak. Within the said time period, we have observed four major outbreaks due to four different strains which were identified as VOC. The first outbreak started roughly on March 1, 2020 followed by the successive outbreaks due to Alpha strain, Beta strain, and Delta strain, respectively. It is important to mention here that during the beginning of the COVID-19 epidemic spread, the strain-specific data source was not available, that is why we consider all the strains before Alpha strain by a single strain which is not identified with any specific name. As per information and data available at [4], the VOC Alpha (20I (Alpha, V1)) appeared in France through the mutation S:H69 around August 24, 2020. After that the VOC Beta (20H (Beta, V2)) appeared approximately around December 31, 2020. The proportion of Gamma variant (20J (Gamma, V3)) in France was negligible compared to some other countries. Further, the VOC Delta (21J (Delta)) was reported around May 3, 2021. For simplicity of numerical simulation and rapidity of convergence of the scheme, we kept most of the parameters are same for all the variants except the disease transmission rates (β j ) and the latency periods (1/r j ) for j = 1, 2, 3, 4. Here j = 1 corresponds to unidentified variant and then for France, Alpha, Beta, and Delta strains correspond to j = 2, 3, 4, respectively. We fit the 7-day moving average data of COVID-19 daily infection for France with the model (3)–(9) and estimate

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Fig. 2 Plots of 7 days moving average of number of daily infected individuals in France starting from March 1, 2020 to September 30, 2021 are plotted with the simulation results using the model (3)–(9) with n = 4. Data points are marked in black and the simulation results are presented in magenta color, a daily data and b cumulative data. The values of the parameters are given in the text, and the strain-specific transmission rates β j are mentioned in Table 1

the parameters β j and r j (see Fig. 2). Other parameter values used for simulation and fitting of the model with the data are chosen from the information available in literature and mentioned below. The estimated values of r j are r1 = 0.7, r2 = 0.8, r3 = 0.8, and r4 = 0.85. The values of β j are estimated over different time intervals in order to take care of change in the rate of disease progression due to several restrictions imposed and relaxed successively. Estimated values of β j over different time intervals are listed in Table 1. The other parameter values are α j = 1, σ j = 0.1923, η j = 0.1, δ j I = δ j A = 0.4348, δ j H = 0.5347, μ j I = μ j H = 0.03, for j = 1, . . . , 4, (see [29] for details). The value of N ≈ 6.74 × 107 and initial condition is S(0) ≈ N , E 1 (0) = 346, I1 (0) = 6, A1 (0) = 6, H1 (0) = 12 and R1 (0) = 0. Initial values of other compartments are considered as zero and here t = 0 corresponds to March 1, 2020. In case of France, t1 = (August 24, 2020), t2 = (January 1, 2021), and t3 = (March 3, 2020) as per the information available at [4]. The rates of re-infection with the same strain are comparatively less and hence ξii = 0.007 and ξik = 0.009, for i = 1, 2, 3 and k = 1, 2, 3, 4 and k = i. As ξik ’s are kept fixed throughout the simulation, the values of βk ’s are estimated from time to time to validate with strain-specific daily infection data. The proposed model can be validated with strain-specific data from any other country and the parameters can be estimated accordingly. To substantiate our claim, the model is fitted with the strain-specific data from Italy. Estimated values of β j ’s with associated time intervals are presented in Table 2. The other parameter values are α j = 1, σ j = 0.1923, η j = 0.1, δ j I = δ j A = 0.4348, δ j H = 0.5347, μ j I = μ j H = 0.03, for j = 1, . . . , 4, (see [29] for details). The value of N ≈ 5.96 × 107 and initial condition is S(0) ≈ N , E 1 (0) = 10, I1 (0) = 2, A1 (0) = 1, H1 (0) = 5, and

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Table 1 Estimated values of β j for France Date

β1

β2

β3

β4

01/03/20–10/04/20 11/04/20–15/05/20 16/05/20–29/06/20 30/06/20–23/08/20 24/08/20–26/11/20 27/11/20–31/12/20 01/01/21–24/02/21 25/02/21–05/04/21 06/04/21–02/05/21 03/05/21–24/07/21 25/07/21–02/09/21 03/09/21–30/10/21

0.95 0.76 0.28 0.575 0.575 0.37 0.61 0.42 0.25 0.2 0.2 0.06

– – – – 0.93 0.48 0.66 0.65 0.5 0.21 0.3 0.35

– – – – – – 0.98 0.94 0.85 0.42 0.45 0.3

– – – – – – – – – 0.96 0.95 0.52

Table 2 Estimated values of β j for Italy Date

β1

β2

β3

β4

01/02/20–22/03/20 23/03/20–30/06/20 1/07/20–19/08/20 20/08/20–8/10/20 9/10/20–12/11/20 13/11/20–27/12/20 28/12/21–8/01/21 9/01/21–05/02/21 06/02/21–13/03/21 14/03/21–26/04/21 27/04/21–31/05/21

0.98 0.41 0.52 0.8 0.75 0.45 0.64 0.48 0.64 0.5 0.36

– – – – 0.91 0.55 0.62 0.58 0.75 0.45 0.46

– – – – – – 0.97 0.85 0.9 0.61 0.42

– – – – – – – – – 0.85 0.66

R1 (0) = 0. The rates of re-infection with the same strain are comparatively less and hence ξii = 0.007 and ξik = 0.009, for i = 1, 2, 3 and k = 1, 2, 3, 4 and k = i. Simulation results along with the data are presented in Fig. 3. In [29], one can find the estimates of the parameters for four countries—France, Italy, Germany, and Spain. It is worthy to mention here that the proposed modeling approach is capable of capturing the recent outbreak due to Omicron strain but we refrain ourselves from such attempt as the outbreak is not over yet.

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Fig. 3 Number of daily infected and 7-day cumulative number of infected individuals in Italy starting from February 1, 2020 to May 31, 2021 are plotted with the simulation results using the model (3)–(9) with n = 4. The values of the parameters are given in the text, and the strain-specific transmission rates β j are presented in Table 2

5 Discussion In terms of ordinary differential equations, mathematical models of epidemic diseases are of two types: without and demographic terms. The fundamental challenge in mathematical modeling, especially in the context of COVID-19, is to capture the recurrence outbreaks of epidemic diseases. Researchers tried to explain the recurrent outbreaks through Hopf-bifurcation around endemic equilibrium; however, such an approach is not justified for the newly emerging epidemic diseases. There are few modeling approaches proposed so far in the context of COVID-19, which can successfully capture the recurrent outbreaks with the help of models without demographic terms. For example, in [7, 27, 28], we have explained that the division of entire population into two groups can lead to a further increase in the number of COVID-19 cases based on human behavior. Specific example of human behavior is one group of people who follow the COVID-related protocols and another group who is not obeying the restrictions. It was shown that even 20–30% defaulters could be responsible for the re-growth of the epidemic once the lockdown and other related restrictions are relaxed. Here we have proposed a multi-compartment model which can accommodate the successive outbreaks due to the appearance of new strains and validated with the COVID-19 daily infection data from France and Italy. The proposed model considers the possibility of re-infection of recovered individuals by the same strain or some other strain. Here we considered the data up to a time point when the Omicron strain did not arrive. A significant contribution of this work is that the proposed model can capture recurrent outbreaks and successful validation of the model depends upon the availability of accurate strain-specific data. It is essential to mention here that the parameters related to the re-infection are taken from [20, 22]; however, these can be

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estimated effectively once we find accurate re-infection data. We did not consider any isolated compartment and quarantined compartment but the estimation of the rate of infection over different time period implicitly takes care of social restrictions imposed and withdrawn from time to time [27]. In France, the first outbreak roughly started on March 1, 2020, and then several VOCs appeared afterward. From the model and estimated parameter values, we find the controlled reproduction number at the epidemic’s beginning was 1.833. The effective reproduction numbers at the time of emergence of Alpha, Beta, and Delta strains were 1.68, 1.71, and 1.63, respectively. During the calculation of effective reproduction numbers at different time points, it is observed that the corresponding new emerging strain dominates the effective reproduction number. Hence, the emerging strain at a particular time point plays a crucial role in determining the growth or decay of an epidemic. The disease transmission rate (β) and the latency period (1/r ) corresponding to a particular strain depend upon the within-body viral load dynamics. It is essential to incorporate within-body viral dynamics of a particular strain into the modeling approach to have a better result, like the immuno-epidemic model recently considered in [15]. In the context of COVID-19, it is also essential to incorporate vaccine-induced immunological responses into the model to understand the epidemic progression more accurately. The vaccinated compartment is overlooked here as the strain-specific effectiveness of the vaccines is not clear yet. Including a vaccinated compartment will only alter the estimates of infectivity, which can be carried out similarly as outlined here. In this work, we have fitted our model with the data from France and Italy for three prominent VOCs and one initial outbreak for which the strain is not identified. Before the appearance of Omicron strain, the infection is reported through some other strains apart from Alpha, Beta, and Delta in these two countries. Other strains are also responsible for the spread of the epidemic to a certain extent, but their numbers are negligible compared to the number of infections due to the VOCs. Of course, one can consider all these strains to develop a better model that can effectively capture the number of hospitalizations and deaths more accurately. Our primary motivation for this work is to propose and validate a relatively new model of epidemic spread which can capture the successive outbreaks and validate the model with the accurate strain-specific data for the COVID-19 epidemic. It is needless to say that the proposed modeling approach can be applied to other epidemic diseases also where more than one strain is responsible for successive outbreaks.

References 1. 2. 3. 4.

https://www.who.int/en/activities/tracking-SARS-CoV-2-variants/ https://covid.ourworldindata.org https://nextstrain.org/ncov/gisaid/global/6m https://ourworldindata.org/grapher/covid-cases-omicron?time=2022-01-24& country=GBR~FRA~BEL~DEU~ITA~ESP~USA~ZAF~BWA~AUS

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5. Abu-Raddad, L.J., Chemaitelly, H., Ayoub, H.H., Tang, P., Coyle, P., Hasan, M.R., Yassine, H.M., Benslimane, F.M., Al Khatib, H.A., Al Kanaani, Z., et al.: Effect of vaccination and of prior infection on infectiousness of vaccine breakthrough infections and reinfections. medRxiv (2021) 6. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford (1992) 7. Ani¸ta, S., Banerjee, M., Ghosh, S., Volpert, V.: Vaccination in a two-group epidemic model. Appl. Math. Lett. 119, 107197 (2021) 8. Banerjee, M., Ghosh, S., Volpert, V.: Epidemic progression in a heterogeneously distributed population (2021) 9. Banerjee, M., Lipniacki, T., d’Onofrio, A., Volpert, V.: Epidemic model with a strain-dependent transmission rate. Commun. Nonlinear Sci. Num. Simul. 114, 106641 (2022) 10. Brauer, F., Castillo-Chavez, C., Feng, Z.: Mathematical Models in Epidemiology, vol. 32. Springer, Berlin (2019) 11. Chintu, C., Athale, U.H., Patil, P.: Childhood cancers in zambia before and after the hiv epidemic. Arch. Dis. Childhood 73(2), 100–105 (1995) 12. Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, vol. 5. Wiley, New York (2000) 13. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002) 14. Fisher-Hoch, S., Hutwagner, L.: Opportunistic candidiasis: an epidemic of the 1980s. Clin. Infecti. Dis. 21(4), 897–904 (1995) 15. Ghosh, S., Banerjee, M., Volpert, V.: Immuno-epidemiological model-based prediction of further covid-19 epidemic outbreaks due to immunity waning. Math. Modell. Nat. Phenom. 17, 9 (2022) 16. Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton (2011) 17. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A Contain. Papers Math. Phys. Character 115(772), 700–721 (1927) 18. Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. ii.-the problem of endemicity. Proc. R. Soc. Lond. Ser. A Contain. Papers Math. Phys. Character 138(834), 55–83 (1932) 19. Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. iii.-further studies of the problem of endemicity. Proc. R. Soc. Lond. Ser. A Contain. Papers Math. Phys. Character 141(843), 94–122 (1933) 20. Kim, P., Gordon, S.M., Sheehan, M.M., Rothberg, M.B.: Duration of sars-cov-2 natural immunity and protection against the delta variant: a retrospective cohort study. Clin. Inf. Dis. (2021) 21. de León, U.A.P., Pérez, Á.G., Avila-Vales, E.: An seiard epidemic model for covid-19 in mexico: mathematical analysis and state-level forecast. Chaos Solitons Fractals 140, 110165 (2020) 22. Mahase, E.: Covid-19: past infection provides 83% protection for five months but may not stop transmission, study finds (2021) 23. Martcheva, M.: An Introduction to Mathematical Epidemiology, vol. 61. Springer, Berlin (2015) 24. Murray, J.: Mathematical Biology, vol. 1. Springer, Heidelberg (2002) 25. Parolini, N., Dede’, L., Antonietti, P.F., Ardenghi, G., Manzoni, A., Miglio, E., Pugliese, A., Verani, M., Quarteroni, A.: Suihter: a new mathematical model for covid-19. application to the analysis of the second epidemic outbreak in Italy. Proc. R. Soc. A 477(2253), 20210027 (2021) 26. Patterson, K.D., Pyle, G.F.: The geography and mortality of the 1918 influenza pandemic. Bull. Hist. Med. 65(1), 4–21 (1991)

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27. Sharma, S., Volpert, V., Banerjee, M.: Extended seiqr type model for covid-19 epidemic and data analysis. Math. Biosci. Eng. (2020) 28. Volpert, V., Banerjee, M., Sharma, S.: Epidemic progression and vaccination in a heterogeneous population. application to the covid-19 epidemic. Ecolog. Compl. 47, 100940 (2021) 29. Wang, K., Ding, L., Yan, Y., Dai, C., Qu, M., Jiayi, D., Hao, X.: Modelling the initial epidemic trends of covid-19 in Italy, Spain, Germany, and France. PLoS One 15(11), e0241743 (2020)

Effect of Nonlinear Prey Refuge on Predator–Prey Dynamics Shilpa Samaddar , Mausumi Dhar , and Paritosh Bhattacharya

Abstract A mathematical model on predator–prey dynamics is analyzed in this study. In traditional models, prey refuge is usually taken constant which is nearly impossible in real-life scenario. We have considered nonlinear prey refuge which depends on both prey and predator. We have performed various dynamical studies incorporating Holling type-II functional response. The system can perceive at most three equilibria. The boundedness of all the solutions, stability–instability conditions, and bifurcation analysis are demonstrated in this work. All the analytical findings are verified with numerical simulations. Additionally, a model comparison is performed which helps to understand the dynamical changes due to nonlinear refuge. Keywords Prey–Predator · Hopf bifurcation · Nonlinear prey refuge

1 Introduction Prey population is always threatened by predators. Prey obtains some protection from predators by hiding in a no predator accessible area. These preys are called prey refuge. Since some of the preys are inaccessible to predator attack, they help the prey from extinction [9, 10]. It is understandable that prey refuge is a common scenario in any ecological interaction. Till now most of the models on prey–predator interaction have considered constant prey refuge m [8, 10, 11]. Here (1 − m)x is the number of prey (x) available for hunting and m ∈ [0, 1). In this study, we have incorporated nonlinear refuge to the Holling type-II functional response. Functional response denotes the consumption rate of an individual predator. Holling has introduced three functional responses compatible to the ecology as Holling Types I, II, and III [1–3]. Among these type-II is widely used. Type-II Supported by organization National Institute of Technology Agartala. S. Samaddar (B) · M. Dhar · P. Bhattacharya National Institute of Technology Agartala, Tripura, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_11

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response first described the downtrend consumption rate. It elaborates that predator growth due to prey consumption is limited by its food processing capability. Mathbx where b and h, ematical expression for Holling type-II functional response is 1+hbx respectively, represent the prey attack rate and handling time of predators. By conb(1−m)x . Using this response sidering constant prey refuge, the function becomes 1+hb(1−m)x eminent works have been performed. Ji and Wu [4] have shown in their study how prey refuge is capable of sustaining the prey density from extinction as well as the dynamical stability behavior depending on refuge factor on harvesting efforts. Although there are marvelous results on constant prey refuge, for natural ecological scenario it is little unrealistic. In general, preys are seeking refuge due to the presence of predator. It is quite obvious that the refuge count cannot be exactly constant as refuge preys will have less foraging that can lead to starvation. It will force them to be exposed again. As the refuge prey depends on the existence of predators, it should depend on predator density also. Since prey refuge is a very important factor to any dynamics, consideration of nonlinear refuge which depends on both the species can solve this problem. In that aspect we have incorporated a nonlinear refuge in the functional response as described in the study [5–7]. The nonlinear prey mx y where m is the prey refuge count can be expressed as the function g(x, y) = a+y refuge coefficient, a is the half saturation constant of refuge prey, and y is the preda my x tor population. Then the density of prey available for hunting becomes 1 − a+y and the functional response turns to

  my x b 1− a+y  .  my x 1+hb 1− a+y

The rest of the paper is organized as follows: in Sect. 2, we have first described the mathematical model with constant prey refuge and find all the conditions for local stability, instability, and Hopf bifurcation. Here we also numerically represent all the analytical findings. Later, in Sect. 3, we formulated the model with nonlinear prey refuge and elaborated all the stability and bifurcation conditions analytically and numerically. To show the dynamical changes a system can achieve due to nonlinear prey, we perform a comparison of these two models in Sect. 4. Finally, in Sect. 5, we concluded all the findings of this study.

2 Mathematical Model Formulation with Constant Refuge First we will discuss about the mathematical model which consists of constant prey refuge in a Holling type-II functional response. The model can be represented as X B(1 − m)X Y dX = R X (1 − ) − , dT K 1 + H B(1 − m)X E B(1 − m)X Y dY = − DY, dT 1 + H B(1 − m)X X (0) > 0; Y (0) > 0.

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Here X, Y represent prey and predator density at any time T . The parameter used R, K , B, H, E, D, respectively, represent prey intrinsic growth rate, prey carrying capacity, predator attack rate, prey handing time of predators, conversion factor from prey biomass to predator biomass, and death rate of predator. Here m denotes the constant prey proportion and m ∈ [0, 1). Nondimensionalization of the system using the transformations X = K x, Y = E K y, and T = Rt provides b(1 − m)x y dx = x(1 − x) − , dt 1 + hx(1 − m) dy b(1 − m)x y = − dy, dt 1 + hx(1 − m) x(0) > 0, y(0) > 0, where b = B ER K , h = B H K , and d = rium points which are

D . R

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The system has three ecological equilib-

1. E 0 (0, 0) which is always unstable saddle. b(1−m) 2. E 1 (1, 0) which is stable if d > 1+h(1−m) else unstable. d ∗) 3. E 2 (x∗ , y∗ ) where x∗ = (1−m)(b−dh) and y∗ = (1−x∗ )(1+h(1−m)x . The interior equib(1−m) librium exists if b > dh. The system is locally asymptotically stable at E 2 if b+dh else unstable. (1 − m) < h(b−dh) We fix the parameter values as b = 2.5, h = 3, d = 0.4, m = 0.2. For these parameter values, the system (1) gets stability at (0.3846, 0.5917) which is presented in Fig. 1. Theorem 1 The system encounters Hopf bifurcation at m [h] = 1 −

b+dh . h(b−dh)

Proof The trace of the Jacobian matrix J of the system around E 2 is zero at m [h] and the determinant is positive. Additionally, d(b+dh) d T race(J ) |m=m [h] = − b(1−m [h] )2 (b−dh)  = 0. dm Hence the system counters Hopf bifurcation.

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At m [h] = 0.0513, the system undergoes Hopf bifurcation and generates limit cycle around (0.32433, 0.54785). Figure 2 denotes the bifurcation diagram of the system with respect to m. Figure 3a and b, respectively, represents the periodic behavior of prey and predator biomass and Fig. 3c denotes the limit cycle around the interior equilibrium point.

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3 Mathematical Model Formulation with Nonlinear Refuge In natural scenario, it is very rare to get a species which has constant refuge. Prey takes refuge seeking some protection from predator. It is pretty clear that prey refuge count must depend on predator density. For this purpose, we have incorporated a mx y , where m is the refuge constant and a nonlinear prey refuge function g(x, y) = a+y is the half saturation constant of prey refuge. The modified model becomes my b(1 − a+y )x y dx = x(1 − x) − my , dt 1 + hx(1 − a+y )

(2a)

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(2c)

The system has three ecological equilibria: 1. E 0 (0, 0) which is always unstable saddle. b else unstable. 2. E 1 (1, 0) which is stable if d > 1+h 3. Positive equilibrium point E 2 (x∗ , y∗ ). The positive equilibrium can be obtained from the two nullclines:

b{a + (1 − m)y}y =0 a + y + h{a + (1 − m)y}x b{a + (1 − m)y}y − d = 0. f2 ≡ a + y + h{a + (1 − m)y}x f1 ≡ 1−x −

(3a) (3b)

a[(b−dh)x∗ −d] Solving the nullcline f 2 = 0 we get y∗ = d−(1−m)(b−dh)x and putting the value in ∗ the nullcline f 1 = 0 we get x∗ is a solution of

(1 − m)(b − dh)x 3 − {d + (1 − m)(b − dh)}x 2 + d{1 − a(b − dh)}x + ad 2 = 0. (4) Remark 1 1. b > dh else y∗ < 0. 2. (b − dh)x∗ > d if not, d > (b − dh)x∗ and d > (1 − m)(b − dh)x∗ as m < 1 which implies y∗ < 0. d < 3. d > (1 − m)(b − dh)x∗ else y∗ < 0. Concluding above two cases we get b−dh d x∗ < (1−m)(b−dh) . Descartes’ rule of signs ensures that Eq. 4 has exactly one negative solution and at d . Since most two positive solutions. Now the sum of the solutions is 1 + (1−m)(b−dh) d x∗ < min{1, (1−m)(b−dh) }, if positive solution exists then there will be exactly one

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d solution in the range ( b−dh , equilibrium point.

d ). Hence, system exhibits exactly one positive (1−m)(b−dh)

Derivation of the nullcline f 2 gives a[(b − dh){d − (b − dh)(1 − m)x} + (1 − m)(b − dh){(b − dh)x − d}] dy = > 0. dx [a + y + h{a + (1 − m)y}x]2

Hence the nullcline f 2 is monotonically increasing. Theorem 2 The positive equilibrium is locally stable if x f x1 + y f y2 < 0 at E 2 (x∗ , y∗ ). Proof The Jacobian matrix of the system (2) at E 2 can be expressed as 

x f x1 x f y1 J= y f x2 y f y2

 , E2

where bh{a + (1 − m)y}2 y [a + y + h{a + (1 − m)y}x]2 b(1 − m)y 2 {1 + h(1 − m)x} + ab(1 + hx){a + 2(1 − m)y} f y1 = − 0 f x2 = [a + y + h{a + (1 − m)y}x]2 abmx f y2 = − < 0. [a + y + h{a + (1 − m)y}x]2 f x1 = −1 +

Using the implicit function theorem, the determinant of the Jacobian can be written as   1 ( f 2) dy ( f ) 1 2 dy Det (J ) = x y f y f y − . dx dx (x∗ ,y∗ )

dy ( f dx

2)

1

dy ( f ) can be positive or negative at E 2 . If dx 1 1 dy ( f ) dy ( f ) < 0, then Det (J ) is positive. If d x > 0, then from Fig. 4 it is clear that the dx ( f 2) ( f 1) slope of f 2 is greater than slope of f 1 . So dyd x is always greater than dyd x . Hence Det (J ) is always positive at E 2 (x∗ , y∗ ). By Routh–Hurwitz criterion, the system is stable at positive equilibrium if T race(J ) < 0, i.e., x f x1 + y f 2 y < 0 at E 2 .

Now

is always positive and

For numerical simulation purpose, we have taken the value of a = 1.

Theorem 3 The interior equilibrium point E 2 of the system (2) loses its stability when x f x1 + y f 2 y > 0. The equilibrium point may change its stability trough Hopf bifurcation at m = m [h] .

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4 Model Comparison The axial equilibrium point of the system (1) with constant prey refuge is stable b(1−m) while in case of the system (2) with nonlinear prey refuge it is when d > 1+h(1−m) b stable if d > 1+h . This means the stability of system (2) does not depend on prey refuge. Further, for the refuge count m = 0.7, we have chosen three initial values and compare the dynamics for both the systems. It is noted that for the first system the trajectories approach to (1, 0), which means predators are extinct but in the case of the system (2), the trajectories approach to an interior equilibrium point, which means predators survive in system (2).

5 Discussion and Conclusion In this study, we have analyzed a predator–prey dynamical system through a mathmy . ematical model incorporating with nonlinear prey refuge function g(x, y) = a+y The interaction between prey and predator is described by the functional response Holling type-II. The main purpose of the study is to identify the influence of nonlinear prey refuge on a system dynamics. For fulfilling the purpose, we have analyzed two

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models with constant prey refuge and nonlinear prey refuge. For both the models, we have described ecologically possible equilibrium points and derive all the stability and bifurcation conditions. Local stability nature of the positive equilibrium of the system (1) can be understood from Figs. 1, 2 and 3. On the other hand, the stability nature of the positive equilibrium of system (2) can be understood from Figs. 5, 6 and 7. Also a model comparison is performed (see Fig. 8) which confirms that at high refuge, predators can survive in the nonlinear prey refuge environment. At m = 0.7, predator extincts in the system (1) but survives in the system (2). Hence, the nonlinear prey refuge has an significant impact on any system and our study drives the system toward more natural scenario.

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References 1. Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the european pine sawfly1. Can. Entomol. 91(5), 293–320 (1959) 2. Holling, C.S.: Some characteristics of simple types of predation and parasitism1. Can. Entomol. 91(7), 385–398 (1959) 3. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 97(S45), 5–60 (1965)

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4. Ji, L., Wu, C.: Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge. Nonlinear Anal.: Real World Appl. 11(4), 2285–2295 (2010) 5. Molla, H., Sarwardi, S., Sajid, M.: Predator-prey dynamics with allee effect on predator species subject to intra-specific competition and nonlinear prey refuge. J. Math. Comput. Sci 25, 150– 165 (2021) 6. Mondal, S., Samanta, G.: Dynamics of a delayed predator-prey interaction incorporating nonlinear prey refuge under the influence of fear effect and additional food. J. Phys. A: Math. Theor. 53(29), 295601 (2020) 7. Mondal, S., Samanta, G., Nieto, J.J.: Dynamics of a predator-prey population in the presence of resource subsidy under the influence of nonlinear prey refuge and fear effect. Complexity 2021 (2021) 8. Samaddar, S., Dhar, M., Bhattacharya, P.: Effect of fear on prey–predator dynamics: Exploring the role of prey refuge and additional food. Chaos: Interdiscip. J. Nonlinear Sci. 30(6), 063129 (2020) 9. Samaddar, S., Dhar, M., Bhattacharya, P.: Supplement of additional food: dynamics of selfcompetitive prey-predator system incorporating prey refuge. Iran. J. Sci. Technol. Trans. A: Sci. 44(1), 143–153 (2020) 10. Samaddar, S., Dhar, M., Bhattacharya, P.: Impact of refuge to the heterogeneous interaction of species in food chain model: a holistic approach. Iran. J. Sci. Technol. Trans. A: Sci. 45(1), 221–233 (2021) 11. Wang, J., Cai, Y., Fu, S., Wang, W.: The effect of the fear factor on the dynamics of a predatorprey model incorporating the prey refuge. Chaos: Interdiscip. J. Nonlinear Sci. 29(8), 083109 (2019)

Effects of Magnetic Field and Thermal Conductivity Variance on Thermal Excitation Developed by Laser Pulses and Thermal Shock Rakhi Tiwari

Abstract The current investigation is aimed to execute the influence of the magnetic field on the transient outcomes inside a semi-infinite medium with dual-phase lag thermoelasticity. Properties of the considered material are taken to be variable, i.e. not constant. The boundary of the medium is exposed to a sudden heat input (thermic shock). Moreover, the bounded surface is assumed to be affected by a non-Gaussian laser beam-type heat source. Closed-form solutions are evaluated by adopting the concept of Kirchhoff transformation and Laplace transform. Impacts of the magnetic field as well as thermal conductivity variance are deduced on the important field quantities such as dimensionless displacement, dimensionless conductive temperature, as well as dimensionless stress through quantitative results. Auspicious outcomes are achieved and the prominent role of the magnetic field and variations of the thermal conductivity are observed on the field components. The author believes that the current theoretical study will be helpful in designing the various structures affected by the laser beam of a non-Gaussian pattern. Keywords Half-space · Magnetic field · Dual-phase lag thermal conductivity equation · Non-Gaussian laser pulse

1 Introduction In contrast to the conventional thermoelastic theory, the concept of generalized thermoelasticity is seeking attractions of academicians regarding its numerous significances in varied areas such as design and structures, engineering, geophysics, aeronautics, plasma physics, acoustics, stream turbine and missiles. Earlier, the traditional approach of thermoelasticity based on Fourier law was in a trend that predicts infinite velocity of thermal shivering which is observed to be unrealistic from the R. Tiwari (B) Department of Mathematics, Nitishwar College, a Constituent Unit of Babasaheb Bhimrao Ambedkar Bihar University, Bihar, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_12

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physical aspect. In the series of inventors of the generalized theory of thermoelasticity, Lord and Shulman [1] was the first person who included a relaxation time parameter/phase lag in the traditional heat expressions proposed by Fourier law. The second time, Green and Lindsay [2] presented the modification of the Fourier law of heat conduction. Afterwards, three distinct theories (GN-I, GN-II and GN-III) have been propounded by Green and Naghdi [3–5]. A few ongoing improvements vital to continuum mechanics are viewed as an application in the area of generalized thermoelasticity that has been very much accepted in different studies. On the other hand, significant studies on smart materials are noticed in structural engineering electronic gadgets, which are very important in our current daily life. Such gadgets have the property that obeys the mechanical surface wave rules for an elastic structural body. As far as increasing popularity of nano particles research problems, the above theories were observed to be inappropriate for predicting realistic results; then Tzou [6] and Chandrasekhariah [7] invented two phase-lag (DPL) theories of heat conduction where they added two relaxation time parameters in Fourier law. Researchers are applying dual-phase lag heat equation for studying the extensive variety of physical systems [8, 9]. Researchers observed that the material characteristic parameters like thermal conductivity, specific heat and elastic modulus are no longer constants, and they alter when extensive heat with ultra-high temperature propagates on the material. In these circumstances, the above-mentioned material parameters are taken to be the exponential or linear function of temperature. Several results predicting the transient effects inside the structures are achieved under the purview of the diverse theories of generalized thermoelasticity such as single-phase lag generalized theory, theory of fractional derivatives, thermic topics with two temperatures and generalized bio-thermoelasticity [10–17]. Moreover, the multi-fold significance in diverse areas such as pulsed laser technologies in material processing, dermatology, eye operations, test ultrasound and acoustics, heat, electricity, optics, physics and laser technology has become a powerful tool for practical applications. Moreover, apart from this laser beam has fascinating properties like it being coherent, fast, monochromatic, non-destructive, well-controlled and is a precisely directed beam of light. A huge number of articles describing the role of laser pulses have been published by researchers [18, 19]. The motive of the current article is to derive the influences of fluctuating thermal conductivity and magnetic field on the propagation of the magneto-thermoelastic waves inside a semi-infinite medium in the context of dual-phase lag thermoelasticity. Half-space medium is affected by the thermal shock heat input as well as the non-Gaussian laser pulse heat source. Closed-form solutions are obtained by adopting Laplace and Kirchhoff transformation techniques. The impact of magnetic field and changing thermal conductivity parameters is determined on the crucial field measures like displacement, conductive temperature and stress with the help of computational outcomes. Results are achieved and the prominent role of the magnetic field and variations of the thermal conductivity are observed on the field components. The author believes that the current theoretical study will be helpful in designing the

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various structures affected by the laser beam of a non-Gaussian pattern. Analytically, the governing equations are solved in Laplace transform domain and solved in the space–time domain numerically. The graphical illustration of distributions of physical quantities is studied due to a different function, time parameter and heat source.

2 Laser Pulse Heat Source The profile of laser pulse is mentioned in the following way [15]:   I0 t t . exp − t p2 tp

L(t) =

(1)

I0 denotes the laser intensity, and t p is the characteristic time of the laser pulse. Heat source Q(x, t) is stated as    x 1− R Ra I 0 t t x . Q(x, t) = exp − L(t) = exp − − b b bt p2 b tp

(2)

Ra = 1 − R. b, Ra denote absorption depth of heating energy and surface reflectivity, respectively.

3 Basic Equations Using Dual-Phase Lag Thermoelasticity The basic equations representing the generalized dual-phase lag theory of thermoelasticity are presented below. Equation of motion (body force is observed to be absent) [15]: σi j, j + μ0 (( J × H)i = ρ u¨ i .

(3)

μ0 (( J × H)i represents the Lorentz force as the half-space medium is suffering from the magnetic field. The strain–displacement relation is [15] ei j =

 1 u i, j + u j,i . 2

(4)

The constitutive equations are [15] σi j = λekk δi j + 2μei j − γ θ δi j ,

(5)

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ρη = γ ekk +

ρCv θ. T0

(6)

The thermal conductivity equation in DPL theory (in the presence of a heat source) [15]:      ∂ ∂  2 ∇ θ = 1 + τq ρCv θ˙ + γ T0 e˙ − Q . K 1 + τT ∂t ∂t

(7)

σi j —stress tensor, u i —displacement component vector, ei j —stress tensor, ekk — cubic dilation, ρ—density; Cv —specific heat at constant volume, η—entropy density, δi j —Kronecker delta, T0 —reference temperature; θ —temperature, λ, μ—Lame constants; γ = (3λ + 2μ)αT , J—current density vector, H—magnetic intensity vector; αT —thermal expansion coefficient, τq —phase lag of heat flux, τT —phase lag of temperature; gradient, qi —heat flux component, K —thermal conductivity and Q—strength of heat source.

4 Mathematical Modelling of the Problem For the present problem, we assume an isotropic, homogeneous, and thermoelastic problem of a half-space medium (x ≥ 0) permeated with the uniform magnetic field. Boundary of the half space is subjected to a sudden heat input. Additionally, a laser pulse type heat source has been imposed to the medium. Due to the magnetothermo-mechanical interactions among three diverse fields, magneto-thermoelastic waves are generated inside the medium. Considering that the direction of wave propagation is along the +ve direction of the x− axis, i.e. (x ≥ 0), hence, each field quantity is dependent on two coordinates— time t and the space coordinate x. Following the assumptions mentioned above, the displacement components are u x = u(x, t), u y = 0 and u z = 0.

(8)

The strain components are defined as ex x =

∂u . ∂x

(9)

Maxwell equations: The following Maxwell equations will take place in the mathematical modelling of the problem

Effects of Magnetic Field and Thermal Conductivity Variance …

curl h = J + 0

157

  ∂u ∂E ∂h , curl E = −μ0 , E = −μ0 × H , div h = 0. ∂t ∂t ∂t (10)

0 represents the electric permittivity of the medium. Initial magnetic field H is assumed to be homogeneous with its components (0, 0, H0 ). Induced electric field E and perturbed magnetic field h are mentioned as E = (0, E, 0), h = (0, 0, h), respectively. Components of the displacement considered the form (u(x, t), 0, 0). Maxwell equations take the following form on applying the assumptions:      ∂h ∂u ∂u ∂ 2u , J = 0, − − 0 μ0 h 0 2 , 0 . E = 0, μ0 h 0 , 0 , h = 0, 0, −H0 ∂t ∂x ∂x ∂t (11) 

Now the governing Eqs. (3), (5) and (7) obtain the following form: α

2 γ ∂θ ∂ 2u 2∂ u = c − , 0 2 2 ∂t ∂x ρ ∂x

∂u − γ θ, ∂x      ∂ ∂ 2θ ∂  · ˙ + γ T0 ekk K 1 + τT θ ρC = 1 + τ −Q . q v ∂t ∂ x 2 ∂t σx x = (λ + 2μ)

(12) (13)

(14)

  2  α0 μ0 (λ+2μ) 2 2 2 , α = H represents Alfven velocity, c = C +α , C = 0 0 1 2 0 1 0 c ρ ρ  1 represents the propagation of longitudinal wave and c = 0 μ0 , where c denotes the speed of light. α = 1+

5 Initial and Boundary Conditions Initial conditions (time t = 0) are considered as

u(x, 0) = u(x, ˙ 0) = 0, θ (x, 0) = θ˙ (x, 0) = 0.

(15)

Boundary conditions are

σ (0, t) = 0, θ (0, t) = θ0 H (t),

(16)

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H (t) denotes the Heaviside unit step function. Due to a realistic approach, stress and temperature fields disappear at an infinite distance from the boundary of the medium, i.e.

σ (x → ∞, t) = 0, θ (x → ∞, t) = 0.

(17)

Thermal conductivity (K ) is considered to be a linear function of thermodynamical temperature (θ ), hence, it is expressed in the following manner [15]: K = K (θ ) = K 0 (1 + K 1 · θ ), N=

K0 . ρCv

(18) (19)

K 0 , N are constants where N represents the thermal diffusivity of the material. K 1 denotes the parameter of variable thermal conductivity. Considering Kirchhoff’s transformation given below 1 ψ= K0

θ K (θ )dθ.

(20)

0

where ψ is the conductive temperature. With the help of Leibnitz’s rule of differentiation, we differentiate Eq. (20) with respect to x and we find K0

∂θ ∂ψ = K (θ ) . ∂x ∂x

(21)

Again differentiating Eq. (21) with respect to x, we obtain  ∂θ ∂ 2ψ ∂ K0 2 = K (θ ) . ∂x ∂x ∂x

(22)

Differentiating Eq. (20) with respect to t, we achieve K0

∂θ ∂ψ = K (θ ) . ∂t ∂t

(23)

With the help of Eqs. (20), (22), we obtain ψ =θ+

1 K1 · θ 2. 2

(24)

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Applying boundary condition (16b) in the above equation, we achieve ψ(0, t) = θ (0, t) +

1 K 1 · (θ (0, t))2 . 2

(25)

With the help of the equation mentioned above, the value of θ can be determined in terms of ψ in the following manner: θ=

−1 +

√ 2K 1 · ψ + 1 . K1

(26)

Using Eqs. (20)–(26) in Eqs. (12)–(14), we find the following equations (we have neglected second and higher order): ∂ 2u ∂ 2u γ ∂ψ , = c02 2 − 2 ∂t ∂x ρ ∂x       Q ∂ ψ γ T0 ∂ ∂ 2 1 + τT , ∇ θ = 1 + τq + ekk − ∂t ∂t ∂t N K0 K0 α

σx x = σ = (λ + 2μ)

∂u − γ ψ. ∂x

(27) (28) (29)

Non-dimensionalization: For the purpose of simplifying Eqs. ((27)–(29)), we convert these equations to non-dimensional form with the help of following nondimensional quantities: 



x  = c0 η0 x, u  = c0 η0 u, b = c0 η0 b, t  = c02 η0 t, t p = c02 η0 t p , τT = c02 η0 τT ,  τq = c02 η0 τq , γθ γψ η0 σ L 0 , θ  = λ+2μ , ψ  = λ+2μ , σ  = λ+2μ , c02 = λ+2μ , h  = Hh0 , L 0 = Ccv0ρT ρ 0 E  = μ0EH0 v , J  = H0Jvη . ∂ 2u ∂ψ ∂ 2u , (30) = β − ∂t 2 ∂x2 ∂x        x ∂ t ∂ ∂ 2ψ ∂ 1 + τT , .ex p − = 1 + τ e t exp − − N + N ) (ψ q 1 kk 2 ∂t ∂ x 2 ∂t ∂t b tp (31) α

∂u − ψ. ∂x

(32)

T0 γ 2 N γρCv T0 N Ra L 0 , N2 = . K 0 (λ + 2μ) K 0 (λ + 2μ)bt p2

(33)

σ = N1 =

For convenience, primes are suppressed from Eqs. (30)–(32). β =

c02 . C12

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6 Solution in Laplace Transform Domain ∞

Laplace transform is defined as L[ f (t)] = f (s) = ∫ e−st f (t)dt, Re(s) > 0. 0

Applying Laplace transform on both sides of Eqs. (30)–(32), we achieve

Here, D =

∂ ∂x

  2 β D − αs 2 u = Dψ,

(34)

     (1 + sτT )D 2 ψ = 1 + sτq s ψ + N1 Du − G(x, s) ,

(35)

σ = Du − ψ.

(36)

and G(x, s) =

N2  2 exp s+ t1p

 x −b .

Initial and boundary conditions are obtained in Laplace transform domain in the following way:

σ (0, s) = 0, θ(0, s) = 1s θ0 .

(38)

σ (x → ∞, s) = 0, θ(x → ∞, s) = 0.

(39)

Applying Laplace transform in Eq. (25), we achieve the following equation: ψ(0, s) =

1 1 1 θ0 + K 1 · (θ0 )2 . s 2 s

(40)

Eliminating ψ from Eqs. (34), (35), we obtain the following decoupled differential equation in terms of u:  x  D 4 − a1 D 2 + a2 u = N4 exp − . b     1 + sτq 1 s 3 1 + sτq a1 = (αs 2 + s (N1 + 1)), a2 = , β 1 + sτT β 1 + sτT   N2 1 + sτq N4 = 2 .  bβ(1 + sτT ) s + t1p 

(41)

The general solution of the differential equation with degree four (Eq. (41)) is obtained as x

u(x, s) = Ae−ξ1 x + Be−ξ2 x + Ce− b .

(42)

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ξi (i = 1, 2) denote the roots with positive real parts of Eq. (44) which are evaluated as        a + a 2 − 4a  a − a 2 − 4a  1  1 2 2 1 1 N4 b4 . ξ1 = , ξ2 = and C = 2 2 1 − a1 b 2 + a2 b 4 A, B are undetermined parameters. Substituting Eq. (42) in Eqs. (35) and (38), respectively, we obtain the following closed-form solutions of conductive temperature ψ and stress σ in the Laplace transform domain: ψ(x, s) = ψ1 e−ξ1 x + ψ2 e−ξ2 x + ψ3 e− b ,

(43)

σ (x, s) = σ1 e−ξ1 x + σ2 e−ξ2 x + σ3 e− b ,

(44)

x

x

where ψ1 =

σ1 =

1 ξ1

 2  αs − βξ12 A, ψ2 =

  −αs 2 + (β − 1)ξ12 A ξ1

1 ξ2

   2  αs − βξ22 B, ψ3 = αbs 2 − βb C,

 σ2 =

 −αs 2 + (β − 1)ξ22 B ξ2

, σ3 =

(β − 1)C − αbs 2 C. b

Using boundary conditions, we obtain M1 A + M2 B = αbs 2 C −

(β − 1)C , b

(45)

 θ0 K1 2 β 2 − αbs C + + θ . N1 A + N2 B = (46) b s 2s 0      2  2 −αs 2 + (β − 1)ξ12 αs − βξ12 αs − βξ22 , N2 = , M1 = N1 = , ξ1 ξ2 ξ1   −αs 2 + (β − 1)ξ22 . M2 = ξ2 

Solving Eqs. (45), (46), we obtain the following expressions of A and B: K 1 M2 θ0 M2 + θ2 s(N1 M2 − N2 M1 ) 2s(N1 M2 − N2 M1 ) 0      C β (β + 1) + N2 M2 −αbs 2 + + − αbs 2 , b b (N1 M2 − N2 M1 ) K 1 M1 θ0 M1 2 + θ B= s(N2 M1 − N1 M2 ) 2s(N2 M1 − N1 M2 ) 0      C β (β + 1) + N1 − αbs 2 . M1 −αbs 2 + + b b (N2 M1 − N1 M2 ) A=

(47)

(48)

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7 Quantitative Results The current section derives the variations of the dimensionless conductance temperature ψ, dimensionless displacement u and non-dimensional stress σ˜ for time t = 0.35 against the non-dimensional distance x by sketching graphical results. The ‘Bellmen Technique’ has been adopted in order to find the numerical inversion of the Laplace transform. We have divided numerical results into two subsections. The first subsection exhibits the impact of magnetic field on the variations of all physical fields, while the other subsection characterizes the role of changing thermal conductivity on the nature of field quantities. Copper material is selected for the purpose of the computational study. Physical data [15] for the material is λ = 7.76 × 1010 NM−2 , μ = 3.86 × 1010 NM−2 , ρ = 8954 kgm−3 , t p = 0.2ps 383.1J , T0 = 293 K, kg K L 0 = 1 × 102 JM−2 , Ra = 0.5, τT = 0.15, τq = 0.2,

K 0 = 386 Wm−1 K−1 , cv =

b = 0.01 m, θ0 = 1α = 0.01442, β = 1.

7.1 Effects of Magnetic Field The present subsection exhibits the impact of magnetic field intensity H on the variations in various dimensionless field components—conductive temperature. Here, the value of K 1 = 0.2. Figure 1 exhibits the variations of conductive temperature ψ with respect to the distance x at time t = 0.35. The trend of variation of plots is observed to be almost similar to the change in the magnetic field intensity. However, the values of the temperature field are altered significantly as we change the values of the magnetic field. Plots start from the same constant value and increase as the distance increases but after providing maximum value, they go down and vanish. It is noticed that the highest values of the temperature field are observed for the highest value of the magnetic field, and values of the temperature profile enhance as the magnetic field intensity increases. Further, it is notified that the peak point shifts far from the first boundary when the magnetic field intensity becomes smaller. Figures 2 and 3 demonstrate the disparity of the distributions of displacement and stress field, respectively, at time t = 0.35. Similar to the temperature field, the impact of the magnetic field is observed to be very prominent on the variations of displacement and stress fields. The stress field is most influenced by the variations of

Effects of Magnetic Field and Thermal Conductivity Variance …

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Fig. 1 Disparity of conductive temperature ψ versus distance x for time t = 0.35

Fig. 2 Disparity of displacement u versus distance x for time t = 0.35

magnetic field intensity. Additionally, we observe that changing the magnetic field affects the maximum value of the displacement field. Apart from this, it is noticed that as the value of the magnetic field goes beyond 2000, changes in the measures of the displacement fields become maximum compared to lower values of the magnetic field. For H = 3000, the nature of the stress curve is observed to be changed compared to the lower intensity of the magnetic field. It can be concluded from the above discussion that stress profiles suffer from the discontinuities unlike the temperature and displacement fields.

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Fig. 3 Disparity of stress σ versus distance x for time t = 0.35

7.2 Effects of Changing Thermal Conductivity In the current subsection, the act of the thermal conductivity variance is displayed on the variations in different dimensionless physical fields—displacement, conductive temperature and stress. The value of magnetic field intensity H is taken to be constant such as H = 3500. Figures 4, 5 and 6 demonstrate the changes in the dimensionless profiles of conductive temperature, displacement as well as stress with respect to the dimensionless distance x, (0 ≤ x ≤ 5) at time t = 0.35, respectively. Figure 4 states the behaviour of conductive temperature for the varied measures of changing thermal conductivity parameter K 1 = 0, 0.2, 0.4, 0.6. The variance pattern of the temperature line is found to be unaffected by the changes in the conductivity Fig. 4 Disparity of conductive temperature ψ versus distance x for time t = 0.35

Effects of Magnetic Field and Thermal Conductivity Variance …

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parameter K 1 . Influences of variable thermal conductivity parameters are noticed to be less effective on the variations of temperature field. Figure 5 exhibits the behaviour of the non-dimensional displacement field versus distance x. The trend of propagation of the displacement profile is found to be similar to Fig. 2. The influence of variable thermal conductivity parameter is more prominent on displacement profiles compared to the temperature profiles. Apart from this, the role of K 1 is found to be highly influential on the maximum value of the displacement field. Figure 6 derives the impact of changing the thermal conductivity parameter on the behaviour of dimensionless stress. Influences of thermal conductivity parameters are observed prominently on the variations of stress field also. This impact is highest on the jumps of the stress profile. Fig. 5 Disparity of displacement u versus distance x for time t = 0.35

Fig. 6 Disparity of stress σ versus distance x for time t = 0.35

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8 Conclusion On the basis of the present study, the following observations can be made: (1)

(2)

(3)

(4)

All field quantities attain non-zero values in the bounded region and fade beyond this region. This nature of magneto-thermoelastic waves shows that waves propagate with a finite speed. Alteration in the magnetic field intensity changes prominently the values of the profiles of all field quantities, and these changes occur prominently near the boundary of the half-space for stress and displacement fields; however, the temperature field is highly affected at the peak values. Stress curves suffer major changes at the boundary of the half-space. The high value of the magnetic field intensity increases the values of physical fields. Prominent changes among the values of physical fields are noticed for the range 2000 < H < 3000. The changing thermal conductivity affects the distributions of stress curves as well as displacement curves, while the temperature field is observed to be less effective with the alterations in the values of the thermal conductivity parameter stress and conductive temperature significantly. Influences of changing thermal conductivity parameters are noticed to be maximum at the jumps of the field quantities. Stress curves are affected by the discontinuities, while conductive temperature and displacement fields are observed to be smooth in nature.

The author strongly believes that the present work may help the scientific community in designing the structures and for the study of waves inside the various materials influenced by the heat source.

References 1. Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967) 2. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972) 3. Green, A.E., Naghdi, P.M.: A Re-Examination of the basic postulates of thermomechanics. Proc. R. Soc. A: Math. Phys. Eng. Sci. 432, 171–194 (1991) 4. Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253–264 (1992) 5. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993) 6. Tzou, D.Y.: A unified field approach for heat conduction from macro- to micro-scales. J. Heat Trans. 117, 9–16 (1995) 7. Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–529 (1998) 8. Ho, J.R., Kuo, C.P., Jiaung, W.S.: Study of heat transfer in multilayered structure within the framework of dual-phase-lag heat conduction model using lattice Boltzmann method. Int. J. Heat Mass Transf. 48, 55–69 (2003)

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9. Kumar, R., Tiwari, R., Singhal, A., Mondal, S.: Characterization of thermal damage of skin tissue subjected to moving heat source in the purview of dual phase lag theory with memory-dependent derivative. Waves Random Compl. Media (2021). https://doi.org/10.1080/ 17455030.2021.1979273 10. Youssef, H.M.: State-Space approach on generalized thermoelasticity for an infinite material with a spherical cavity and variable thermal conductivity subjected to ramp-type heating. Can. Appl. Math. Q. 13, 369–390 (2005) 11. Youssef, H.M.: Three-dimensional generalized thermoelasticity with variable thermal conductivity Int. J. Comput. Mater. Sci. Surf. Eng. 9(1) (2010) 12. Li, C., Guo, H., Tian, X., et al.: Transient response for a half-space with variable thermal conductivity and diffusivity under thermal and chemical shock. J. Therm. Stresses 40, 389–401 (2017) 13. Li, X., Xue, Z., Tian, X.: A modified fractional order generalized bio-thermoelastic theory with temperature-dependent thermal material properties. Int. J. Therm. Sci. 132, 249–256 (2018) 14. Li, X., Li, C., Xue, Z., Tian, X.: Analytical study of transient thermo-mechanical responses of dual-layer skin tissue with variable thermal material properties. Int. J. Therm. Sci. 124, 459–466 (2018) 15. Tiwari, R., Kumar, R.: Investigation of thermal excitation induced by laser pulses and thermal shock in the half space medium with variable thermal conductivity. Waves Random Compl. Media (2020). https://doi.org/10.1080/17455030.2020.1851067 16. Sherief, H., Abd El-Latief, A.M.: Effect of variable thermal conductivity on a half-space under the fractional order theory of thermoelasticity. Int. J. Mech. Sci. 74, 185–189 (2013) 17. Wang, Y., Liu, D., Wang, Q., et al.: Thermoelastic response of thin plate with variable material properties under transient thermal shock. Int. J. Mech. Sci. 104, 200–206 (2015) 18. Tiwari, R., Abouelregal, A.E.: Memory response on magneto-thermoelastic vibrations on a viscoelastic micro-beam exposed to a laser pulse heat source. Appl. Math. Model. 99, 328–245 (2021) 19. Abd-Elaziz, E.M., Othman, M.I.A:. Effect of Thomson and thermal loading due to laser pulse in a magneto-thermo-elastic porous medium with energy dissipation. J. Appl. Math. Mech. (2019)

Differential and Integral Equations

On Unique Positive Solution of Hadamard Fractional Differential Equation Involving p-Laplacian Ramesh Kumar Vats, Ankit Kumar Nain, and Manoj Kumar

Abstract In this paper, the authors have studied p-Laplacian Hadamard fractional differential equation with integral boundary condition. The sufficient condition for the existence and uniqueness of solution is developed using a new fixed point theorem (Zhai and Wang [21]) of ϕ − (h, e)-concave operator. Further, an iterative method is also given for approximating the solution corresponding to any arbitrary initial value taken from an appropriate set. Keywords Existence and uniqueness · Fractional differential equation · Fixed point theorems · p-Laplacian · Positive solution · ϕ − (h · e)-concave operator

1 Introduction Fractional derivatives have numerous applications, including simulating the mechanical and electrical properties of the materials, describing rheological features of rocks and many others. Also, it has been demonstrated in recent decades that fractionalorder models are more suited than previously utilized integer models because fractional derivatives are non-local operators by nature as compared to integer-order derivatives which are local, and because of this fact, they are a good tool for describing the hereditary and memory properties of many processes and materials where integer-order derivatives have failed (Zhou et al. [22]). The Hadamard fractional differential operator is taken into consideration in this paper which differs from other fractional derivatives as its definition consists of the logarithmic function in its kernel which is very helpful in studying the ultra-slow diffusion processes. Also, in the study of rheology and ultra-slow kinetics, the logarithmic creep law is used to describe the creep phenomenon of ingenious rock, which can be described using Hadamard fracR. K. Vats · A. K. Nain (B) National Institute of Technology, Hamirpur 177005, India e-mail: [email protected] M. Kumar R.K.S.D. (P.G.) College, Kaithal 136027, Haryana, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_13

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tional calculus (Garra et al. [6]). For further insight of fractional derivatives, one can go through (Diethelm [5], Kilbas et al. [9], Nain et al. [12, 13], Podlubny [16]) and references therein. Differential equations of fractional order with boundary conditions involving integral term have received much consideration of the researchers due to the applications in problems of thermal conduction, semiconductors, and hydrodynamic issues which all include integral boundary conditions (Henderson and Luca [8]). Also, equations consisting of p-Laplacian operator have been derived from the studies of non-Newtonian fluid theory and nonlinear elastic mechanics. In 1945, Leibenson [10] described the turbulent flow model in porous media using the following p-Laplacian equation: (φ p (ψ  (x))) = Φ(x, ψ(x), ψ  (x)), 1 1 where φ p (t) = |t| p−2 t is the p-Laplacian operator with φ−1 p (t) = φq (t), p + q = 1. Diaz and Tehlin [4] established a series of models involving the p-Laplacian operator, which emerges in the study of incompressible turbulent fluids flowing through porous surfaces and gases moving through pipes with uniform cross-sectional areas. Using the properties of green function and utilizing nonlinear alternative of Leray– Schauder, Li and Lin [11] established the existence of positive solution for the Hadamard fractional boundary value problem (FBVP): H

β

D1 (φ p ( H D1α ψ(x))) = Φ(x, ψ(x)), 1 < x < e,

ψ(1) = ψ  (1) = ψ  (e) = 0,

H

D1α ψ(1) = H D1α ψ(e) = 0,

where 2 < α ≤ 3, 1 < β ≤ 2 and H D1− is the Hadamard fractional derivative. Following the work of Li and Lin [11], Wang and Wang [18] derived the sufficient condition for the existence of solution using Schaefer’s fixed point theorem for FBVP with p-Laplacian operator with strip conditions. Recently, Wang and Zhai [19] studied the existence and uniqueness of solution for FBVP in partially ordered cone and established an iterative method for obtaining the solution. Taking into account their importance in analysis and applications, much more attention have been drawn to the analysis of p-Laplacian FBVP. For more information on the applications of pLaplacian operator in differential equations, one can go through (Benedikt et al. [1], Cheng and Wang [3], Wang et al. [17], Xue et al. [20]), and references therein. Also, further development of some new fixed point theorems like α-type F-contractive mappings, α-type F-contractions, etc. can be found in Gopal et al. [2, 7, 15] and references therein. Supposedly, there are not many articles that study the existence and uniqueness of positive solution for p-Laplacian FBVP. Therefore, followed by the work discussed above, the authors have considered the Hadamard p-Laplacian FBVP involving integral boundary conditions as follows:

On Unique Positive Solution of Hadamard Fractional Differential …

⎧ H d2 D1 (φ p ( H D1d1 ψ(x) − χ(x))) + Φ(x, ψ(x)) = 0, 1 < x < e, ⎪ ⎨ e ψ(1) = δψ(1) = 0, ψ(e) = 1 ρ(x)ψ(x) dx , x ⎪ ⎩ H d1 D1 ψ(1) = 0,

173

(1)

where 2 < d1 ≤ 3, 0 < d2 ≤ 1, ρ : [1, e] → [0, ∞) with ρ ∈ L 1 [1, e], χ ∈ C([1, e], R), H D1d2 is the Hadamard fractional derivative of order d2 given by H

D1d2 ψ(x) =

    x n−d2 −1 ψ(s) d n x 1 ln x ds, Γ (n − d2 ) dx s s 1

provided the integral exists, where n − 1 < d2 ≤ n , n = [d2 ] + 1, and ln(·) = loge (·) (see Kilbas et al. [9]) and H I1d2 is the Hadamard fractional integral of order d2 > 0 given by H d2 I1 ψ(t)

=

1 Γ (d2 )

t ln 1

t s

d2 −1

ψ(s) ds s

provided the integral exists, where ln(·) = loge (·) (see Kilbas et al. [9]).

2 Preliminaries This section presents the definitions, notations, and lemmas which supports the results presented in Sect. 3. A subset C(= ∅) of a real Banach space (U, · ), which is closed and convex is said to be a cone if it gratifies the following conditions: (i) ψ ∈ C, ξ ≥ 0 implies ξψ ∈ C, (ii) ψ ∈ C, −ψ ∈ C implies ψ = θ, where θ is zero element of U. Every cone C in U defines a partial ordering in U given by ψ ≤ ω ⇔ ω − ψ ∈ C. A cone C ⊂ U is said to be normal if there exists a constant M > 0 such that for all ψ, ω ∈ U, θ ≤ ψ ≤ ω implies ψ ≤ M ω . For ψ, ω ∈ U, the notation ψ ∼ ω means that there exist μ, ν > 0 such that μψ ≤ ω ≤ νψ. For θ < η ∈ U, denote Cη = {ψ ∈ U | ψ ∼ η}, with Cη ⊂ C. Let e ∈ C with θ ≤ e ≤ η, denote a set Cη,e = {ψ ∈ U | ψ + e ∈ Cη }. Then one can see that η ∈ Cη,e and for any ψ ∈ U there exists μ = μ(η, e, ψ) > 0 and ν = ν(η, e, ψ) > 0 such that μη ≤ ψ + e ≤ νη. An operator Υ : U → U is increasing if ψ ≤ ω implies Υ (ψ) ≤ Υ (ω).

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Definition 1 (Zhai and Wang [21]) Let Υ : Cη,e → U be a given operator. For any ψ ∈ Pη,e and ξ ∈ (0, 1), there exists ϕ(ξ) > ξ such that Υ (ξψ + (ξ − 1)e) ≥ ϕ(ξ)Υ (ψ) + (ϕ(ξ) − 1)e. Then Υ is called a ϕ − (η, e)-concave operator. Lemma 1 (Kilbas et al. [9]) Let d1 ≥ 0. Then H D1d1 ψ(x) = 0 has a solution of the form: n

c j (ln x)d1 − j , (2) ψ(x) = j=1

where c j ∈ R and n = [d1 ] + 1, in addition H d1 H d1 I1 D1 ψ(x)

= ψ(x) +

n

c j (ln x)d1 − j .

(3)

j=1

Lemma 2 (Zhai and Wang [21]) Let C be normal and Υ be an increasing ϕ − (η, e)concave operator, Υ (η) ∈ Cη,e , then Υ has a unique fixed point ψ ∗ in Cη,e . Moreover, for any ψ0 ∈ Cη,e , making the sequence ψn = Υ (ψn−1 ), n = 1, 2, . . . , then ψn − ψ ∗ → 0 as n → ∞. Lemma 3 (Nain et al. [14]) Let ϑ ∈ C[1, e]. Then the Hadamard FBVP 

D1d1 ψ(x)) + ϑ(x) = 0, e ψ(1) = δψ(1) = 0, ψ(e) = 1 H

ρ(x)ψ(x) dx x

(4)

has unique solution given by

e

ψ(x) =

H (x, w)ϑ(w)

1

dw , w

(5)

where H (x, w) = H1 (x, w) + H2 (x, w), H1 (x, w) =

1 Γ (d1 )



(6)

(ln x)d1 −1 (1 − ln w)d1 −1 − (ln wx )d1 −1 , 1 ≤ w ≤ x ≤ e, (7) 1 ≤ x ≤ w ≤ e, (ln x)d1 −1 (1 − ln w)d1 −1 ,

H2 (x, w) =

(ln x)d1 −1 1−a

a= 1

e



e

H1 (x, w) p(x)

1

(ln x)d1 −1 p(x)

dx > 1. x

dx , x

(8)

(9)

On Unique Positive Solution of Hadamard Fractional Differential …

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Lemma 4 (Nain et al. [14]) The function H1 (x, w) is a continuous function along with H1 (x, w) > 0 and (ln x)d1 −1 (1 − ln x)(1 − ln w)d1 −1 ln w ≤ Γ (d1 )H1 (x, w) ≤ (d1 − 1)(1 − ln w)d1 −1 ln w.

Furthermore, the function H (x, w) satisfies (ln x)d1 −1 ln w(1 − ln w)d1 −1 a (ln x)d1 −1 (1 − ln w)d1 −1 ≤ H (x, w) ≤ , Γ (d1 )(1 − a) Γ (d1 )(1 − a) where a =



e

(ln x)d1 −1 (1 − ln x)

1

(10)

dx ≥ 0. x

In order to study the FVBP (1), first consider the associated linear form of FVBP (1) as follows: ⎧ H d2 d ⎨ D1 (φ p ( H D1 1 ψ(x) − χ(x))) + ϑ(x)  e = 0, 1 < x < e, (11) dx, ψ(1) = δψ(1) = 0, ψ(e) = 1 ρ(x)ψ(x) x ⎩ H d1 D1 ψ(1) = 0, for ϑ ∈ C([1, e], R) and ϑ ≥ 0. Lemma 5 If χ ∈ C([1, e], R) with χ(1) = 0. Then (11) has the unique solution given by

e

ψ(x) = 1

Proof Let Θ =

H

dw H (x, w)φq ( H I1d2 ϑ(w)) w

−

e

H (x, w)χ(w)

1

dw . w

(12)

D1d1 ψ, ν = φ p (Θ − χ). Then the initial value problem



D1d2 (ν(x)) + ϑ(x) = 0, ν(1) = φ p ( H D1d1 ψ(1) − χ(1)) = φ p (0) = 0 H

has the solution

(13)

ν(x) = c1 (ln x)d2 −1 − H I1d2 ϑ(x).

Note that ν(1) = 0, 0 < d2 ≤ 1, we get c1 = 0. Therefore, ν(x) = − H I1d2 ϑ(x), 1 ≤ x ≤ e φ p (Θ − χ)(x) = − H I1d2 ϑ(x), Θ(x) = φq (− H I1d2 ϑ(x)) + χ(x), H

D1d1 ψ(x) = φq (− H I1d2 ϑ(x))  e+ χ(x) dx. ψ(1) = δψ(1) = 0, ψ(e) = 1 ρ(x)ψ(x) x

(14)

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Therefore, by following Lemma 3, the solution of equation (14) can be expressed as follows: e e − 1 H (x, w)χ(w) dw ψ(x) = 1 H (x, w)φq ( H I1d2 ϑ(w)) dw w w (q−1)   d2 −1     e w 1 w dt dw (15) ln t = Γ (d2 ) ϑ(t) t w 1 H (x, w)φq 1 e − 1 H (x, w)χ(w) w . w

3 Existence and Uniqueness Let U = C(J, R) be a Banach space of continuous functions, where J := [1, e] with norm ψ = sup{|ψ(x)| : x ∈ J} and this space is endowed with the following partial order: ψ, ζ ∈ C[1, e], ψ ≤ ζ ⇔ ψ(x) ≤ ζ(x), x ∈ J. Also, C = {ψ, ∈ U, ψ(x) ≥ 0, x ∈ J} forms a standard normal cone and set

e

e(x) =

H (x, w)χ(w)

1

dw . w

Theorem 1 Assume χ(x) ≥ 0, χ(x) ≡ 0, x ∈ [1, e], and (H1) Φ : [1, e] × [−e∗ , +∞) → (−∞, +∞) is increasing w.r.t. second variable, where e∗ = max{e(x), x ∈ [1, e]}. υ(ξ) 1 > (q−1) such that (H2) For any ξ ∈ (0, 1) there exists υ(ξ) ∈ (0, 1) with lnln(ξ) Φ(x, ξμ + (ξ − 1)ν) ≥ υ(ξ)Φ(x, μ); x ∈ [1, e], μ ∈ (−∞, +∞), ν ∈ [0, e∗ ] (H3) Φ(x, 0) ≥ 0, Φ(x, 0) ≡ 0, x ∈ [1, e]. Then the FBVP (1) has a unique nontrivial solution ψ ∗ in Cη,e , where η(x) = K (ln x)d1 −1 with e dw (1 − ln w)d1 −1 χ(w) . K ≥ w 1 Γ (d1 )(1 − a) Moreover, for arbitrary ψ0 ∈ Cη,e , constructing an iterative sequence  w  (q−1) e   w d2 −1 1 dt dw ln H (x, w)φq Φ(t, ψn−1 (t)) Γ (d2 ) t t w 1 1 e dw , n = 1, 2, . . . H (x, w)χ(w) − w 1

 ψn (x) =

and ψn (x) → ψ ∗ (x) as n → ∞.

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Proof In order to apply Lemma 2, construct the set Cη,e with respect to function η and e. First, we show that 0 ≤ e(x) ≤ η(x), x ∈ J. From, Lemma 4, H (x, w) ≥ 0 and χ(x) ≥ 0, thus for x ∈ J, e(x) =

e

H (x, w)χ(w)

1

dw ≥ 0, w

(16)

which means that e ∈ C. Again, with the help of Lemma 4, for x ∈ J, e(x) = ≤ =

e 1

e

H (x, w)χ(w) dw w

(ln x)d1 −1 (1−ln w)d1 −1 χ(w) dw 1 Γ (d1 )(1−a) w  e (1−ln w)d1 −1 dw d1 −1 1 Γ (d1 )(1−a) χ(w) w × (ln x) d1 −1

≤ K (ln x)

(17)

= η(x).

Therefore, concluding from (16) and (17), 0 ≤ e(x) ≤ η(x) and the set Cη,e = {ψ ∈ U, ψ + e ∈ Cη } is well defined. The FBVP (1) has the following integral representation, which is due to Lemma 5 is as follows:  w  (q−1) e   w d2 −1 dt dw 1 ln H (x, w)φq Φ(t, ψ(t)) Γ (d2 ) t t w 1 1 e dw H (x, w)χ(w) − w 1  w   q−1 e   w d2 −1 1 dt dw ln = − e(x), H (x, w)φq Φ(t, ψ(t)) Γ (d2 ) t t w 1 1

 ψ(x) =

where H (x, w) is given in (6). Using the above integral representation of solution, set an operator Υ : Cη,e → U which is defined as follows:  w  (q−1) e   w d2 −1 1 dt dw ln − e(x). (Υ ψ)(x) = H (x, w)φq Φ(t, ψ(t)) Γ (d2 ) t t w 1 1 

(18) Obviously, ψ(x) is the solution of the problem (1) if and only if ψ(x) = (Υ ψ)(x), i.e., Υ has a fixed point. The operator φq is monotone increasing in [0, +∞) and suppose ψ ∈ Cη,e , ξ ∈ (0, 1) Υ (ξψ + (ξ − 1)e)(x) =  (q−1) e    w  1 w d2 −1 dt dw − e(x) ln H (x, w)φq Φ(t, ξψ(t) + (1 − ξ)e(t)) Γ (d2 ) t t w 1 1   w  (q−1) e d2 −1  1 w dt dw ≥ (υ(ξ))(q−1) ln − e(x) H (x, w)φq Φ(t, ψ(t)) Γ (d2 ) t t w 1 1

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= (υ(ξ))(q−1)



 w  (q−1) e    1 w d2 −1 dt dw ln − e(x) H (x, w)φq Φ(t, ψ(t)) Γ (d2 ) t t w 1 1

+ (υ(ξ))(q−1) e(x) − e(x) = (υ(ξ))(q−1) Υ ψ(x) + [(υ(ξ))(q−1) − 1]e(x). υ(ξ) 1 Set ϕ(ξ) = (υ(ξ))(q−1) for ξ ∈ (0, 1). From (H2), lnln(ξ) > (q−1) for ξ ∈ (0, 1), which implies ln ϕ(ξ) = (q − 1) · ln(υ(ξ)) υ(ξ) · ln(ξ) = (q − 1) · lnln(ξ) 1 > (q − 1) · (q−1) · ln(ξ) = ln(ξ),

which gives ϕ(ξ) > ξ, for ξ ∈ (0, 1).Hence, for ψ ∈ Cη,e , ξ ∈ (0, 1), we obtain Υ (ξψ + (ξ − 1)e) ≥ ϕ(ξ) · Υ (ψ) + [ϕ(ξ) − 1] · e. Therefore,Υ : Cη,e → U is ϕ − (η, e)-concave operator. For ψ ∈ Cη,e , we have ψ + e ∈ Cη and thus there exists ρ > 0 such that ψ(x) + e(x) ≥ ρη(x), x ∈ J. Hence, using this argument, we obtain ψ(x) ≥ ρη(x) − e(x) ≥ −e(x) ≥ e∗ and using (H1), we deduce that Υ : Cη,e → U is increasing. Finally, we show that Υ (η) ∈ Cη,e , i.e., Υ (η) + e ∈ Cη . Let (Υ η)(x) + e(x)  w  (q−1) e    w d2 −1 dt dw 1 ln H (x, w)φq Φ(t, η(t)) = Γ (d2 ) t t w 1 1 (q−1) e    w   w d2 −1 1 d1 −1 dt dw ln H (x, w)φq Φ(t, K (ln t) ) = Γ (d2 ) t t w 1 1 (q−1) e d2 −1   w   d −1 d −1 1 1 w (ln x) (1 − ln w) dt dw 1 φq ln Φ(t, K ) ≤ Γ (d2 ) Γ (d1 )(1 − a) t t w 1 1 (q−1) d2 −1   w   e w 1 dt dw 1 ln (1 − ln w)d1 −1 φq Φ(t, K ) · (ln x)d1 −1 = Γ (d2 ) Γ (d1 )(1 − a) 1 t t w 1 (q−1)    w   e w d2 −1 1 1 dt dw ln = (1 − ln w)d1 −1 φq Φ(t, K ) · η(x), Γ (d2 ) K Γ (d1 )(1 − a) 1 t t w 1

and (Υ η)(x) + e(x) (q−1) e    w   w d2 −1 dt dw 1 ln H (x, w)φq Φ(t, K (ln t)d1 −1 ) = Γ (d2 ) t t w 1 1 (q−1) e    w   w d2 −1 (ln x)d1 −1 (ln w)(1 − ln w)d1 −1 a dt dw 1 ln Φ(t, 0) ≥ φq Γ (d2 ) Γ (d1 )(1 − a) t t w 1 1 (q−1) d2 −1   w   e dw w a dt 1 ln (ln w)(1 − ln w)d1 −1 φq Φ(t, 0) = · (ln x)d1 −1 Γ (d2 ) Γ (d1 )(1 − a) 1 t t w 1 (q−1)    w   e w d2 −1 a 1 dt dw · η(x). ln = (ln w)(1 − ln w)d1 −1 φq Φ(t, 0) Γ (d2 ) K Γ (d1 )(1 − a) 1 t t w 1

On Unique Positive Solution of Hadamard Fractional Differential …

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Fix (q−1) a 1 Γ (d2 ) K Γ (d1 )(1 − a)  (q−1) 1 1 ν= Γ (d2 ) K Γ (d1 )(1 − a)

 μ=

 w 

  w d2 −1 dt dw , Φ(t, 0) t t w 1 1  w  d2 −1  e dt dw w (1 − ln w)d1 −1 φq Φ(t, K ) ln , t t w 1 1 e

(ln w)(1 − ln w)d1 −1 φq

ln

and (q−1) 1 1 Γ (d2 ) K Γ (d1 )(1 − a) (q−1)  a 1 ≥ Γ (d2 ) K Γ (d1 )(1 − a) (q−1)  a 1 ≥ Γ (d2 ) K Γ (d1 )(1 − a)  (q−1) 1 a ≥ Γ (d2 ) K Γ (d1 )(1 − a) 

ν=

  w d2 −1 dt dw Φ(t, K ) t t w 1 1 d2 −1   w  e dw w dt ln (1 − ln w)d1 −1 φq Φ(t, 0) t t w 1 1 d2 −1   w  e w dt dw ln (1 − ln w)d1 −1 φq Φ(t, 0) t t w 1 1  w  d2 −1  e dt dw w (ln w)(1 − ln w)d1 −1 φq Φ(t, 0) ln t t w 1 1 e

(1 − ln w)d1 −1 φq

 w 

ln

=μ > 0.

Thus, there are two functions μ and ν such that μη ≤ Υ (η) + e ≤ νη which implies Υ (η) + e ∈ Cη . Therefore, Υ satisfies all the conditions of Lemma 2 and thus Υ has a unique fixed point ψ ∗ ∈ Cη,e and thus  ψ ∗ (x) =

 w  (q−1) e   w d2 −1 1 dt dw ln − e(x), x ∈ [1, e], H (x, w)φq Φ(t, ψ ∗ (t)) Γ (d2 ) t t w 1 1

i.e., ψ ∗ (x) is the solution of the problem (1). Moreover, for any ψ0 ∈ Cη,e , the sequence ψn = Υ ψn−1 , n = 1, 2, . . . , satisfies ψn → ψ ∗ as n → ∞.Namely, (q−1) e    w  w d2 −1 1 dt dw ln H (x, w)φq Φ(t, ψn−1 (t)) Γ (d2 ) t t w 1 1 e dw , n = 1, 2, . . . H (x, w)χ(w) − w 1

 ψn (x) =

and ψn (x) → ψ ∗ (x) as n → ∞. Remark 1 For some FBVP, we can derive functions e(x), η(x) and construct functions which satisfies the postulates of Theorem 1. For example, suppose Φ(x, ψ) =   13 e(x) ψ + e(x) , where θ ≤ e(x) ≤ η(x). Then Φ is increasing w.r.t. second variable e∗ and Φ ≡ 0. Set ϕ(ξ) = ξ 3 , then for ξ ∈ (0, 1), μ ∈ (−∞, ∞), ν ∈ [0, e∗ ], 1

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 13 e(x) [ξμ + (ξ − 1)ν] + e(x) e∗ 

  1 e(x) 1 e(x) 3 1 μ+ 1− ν + =ξ 3 e∗ ξ ξ   1 e(x) 1 e(x) e(x) 3 1 3 =ξ μ+ 1− ν+ e∗ ξ e∗ ξ   13 e(x) 1 ≥ξ 3 μ + e(x) = ϕ(ξ)Φ(x, μ). e∗

Φ(x, ξμ + (ξ − 1)ν) =

4 Conclusion The existence and uniqueness of solution together with an iterative method has been established for a Hadamard FBVP. The result has been established using a fixed point theorem for ϕ − (η, e)-concave operator in a partially ordered cone. The method for constructing the solution presented in Theorem 1 starts with a simple function which is useful for computational purpose.

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Eigenvalue Criteria for Existence and Nonexistence of Positive Solutions for α-Order Fractional Differential Equations on the Half-Line (2 < α ≤ 3) with Integral Condition Abdelhamid Benmezai, Souad Chentout, and Wassila Esserhan

Abstract This article concerns the nonexistence and existence of positive solutions to the fractional differential equations ⎧ α ⎨ D u(t) + f (t, u(t)) = 0 0 ≤ t < +∞ ⎩ u(0) = D α−2 u(0) = 0

lim D α−1 u(t) = A

t→+∞

η

u(s)dμ(s)

0

where η, A ∈ (0, +∞), and 2 < α ≤ 3, μ(t) is the continuous nondecreasing funcη

tion on (0, +∞), with μ(0) = 0 and

u(s)dμ(s) denotes the Riemann–Stieljes 0

integrals of u with respect to μ; D α is standard Riemann–Liouville derivative, f : R+ × R+ → R+ is a continuous function. Keywords Fractional differential equation · Fixed point index theory · Boundary value problems · Positive solutions · Green’s function · Integral condition

1 Introduction and Main Results Since fractional differential equations are considered as alternative models for the nonlinear differential equations, the study of the existence of positive solutions to boundary value problems associated with fractional differential equations has become a very important area of applied mathematics over the last few decades. A. Benmezai National Higher School of Mathematics, Algiers, Algeria S. Chentout (B) Faculty of Mathematics, USTHB, Algiers, Algeria e-mail: [email protected] W. Esserhan ENSSEA, Pole Universitaire KOLEA, 42003 Tipaza, Algeria © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_14

183

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Such a subject has been discussed in many recent papers; see for examples, [5, 6, 13], and references therein. We are concerned in this paper with the nonexistence and existence of positive solutions of the fractional boundary value problem (fbvp): 

D α u(t) + f (t, u(t)) = 0, 0 ≤ t < +∞ η u(o) = D α−2 u(o) = 0, limt→+∞ D α−1 u(t) = A 0 u(s)dμ(s)

(1)

where η, A ∈ (0, +∞),  η 2≺ α ≤ 3, μ(t) is continuous on (0, +∞), and non-decreasing with μ(0) = 0 and 0 u(s)dμ(s) denotes the Riemann–Stieljes integrals of u with respect to μ; D α is standard Riemann–Liouville derivative, f : R+ × R+ → R+ is a continuous function. Motivated by the works in [2–4, 12, 13], we want to establish nonexistence and existence results to the fbvp (1) under eigenvalue criteria. Set    Dα = q ∈ C R+ , R+ : q(s) > 0 a.e. s > 0 and

+∞

q(s)(1 + s)α−1 ds < ∞ .

0

A continuous function h : R+ × R+ → R+ is said to be Dα -Caratheodory if for all r > 0 there is Ψr ∈ Dα such that   (1 + t)α−1 h(t, u) ≤ a(t) + b(t) |u|ρ , for all t, u ∈ R+ .  where ρ ∈ (0, +∞) and a, b ∈ Dα ∈ C R+ with b(s) = (1 + s)(α−1)(ρ−1) b(s), is a typical Dα -Caratheodory function. Consider for q in Dα the linear fractional boundary value problem: D α u(t) + ρq(t)u(t) = 0 0 ≤ t < +∞ u(0) = D α−2 u(0) = 0, η . lim D α−1 u(t) = A 0 u(s)dμ(s)

(2)

t→+∞

where ρ is a real parameter. Proposition 1 See [2]. For all functions q in Dα , the fbvp (2) admits a unique positive eigenvalue ρα (q). Proposition 2 See [2]. Assume that the nonlinearity f is a Dα -Caratheodory function and there exists q ∈ Dα such that one of the following Hypotheses (3) or (4) ρα (q) < 1 ρα (q) > 1

and

and

f (t, u) ≥ q(t)u for all t, u ≥ 0 f (t, u) ≤ q(t)u

for all t, u ≥ 0

holds true. Then the fbvp (1) has no positive solutions.

(3)

(4)

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185

The existence result for positive solutions of the fbvp (1) needs the introduction of the following additional notations. Set for a Dα -Caratheodory function h : R+ × R+ → R+ , q ∈ Dα , and ν = 0, +∞

 h(t, (1 + t)α−1 u) , = lim sup max t≥0 (1 + t)α−1 q(t)u u→v

 h(t, (1 + t)α−1 u) min , h− (q) = lim inf ν u→v t≥0 (1 + t)α−1 q(t)u

h+ ν (q)

Theorem 1 Assume that the nonlinearity f is Dα -Caratheodory function and there exist two functions q0, q∞ in Dα such that one of the following hypotheses (5) or (6) holds true: + f − (q0 ) (q∞ ) f +∞ , and  denote, respectively, reverse situations. Definition 2 Let L be a compact operator. L is said to be 1. positive, if L(K ) ⊂ K ; 2. Strongly positive, if int (K ) = ∅ and L(K \ {0 X }) ⊂ int (K ); 3. lower bounded on the cone K , if inf { Lu : u ∈ K ∩ ∂ B(0 E , 1)} > 0.

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In all what follows, L K (X ) denotes the subset of all positive compact operators in L(X ) and we set for all L ∈ L K (X ) Λ L = {θ ≥ 0 : ∃u > 0 X such that Lu ≥ θu} Γ L = {θ ≥ 0 : ∃u > 0 X such that Lu ≤ θu} Definition 3 Let L be an operator in L K (X ) and ρ positive. The operatorL is said to have the strong index jump property (SIJP for short) at ρ if r (L) = sup Λ L = inf Γ L . Proposition 3 See [1]. Let L be an operator in L K (X ). If L is strongly positive, then L has the SIJP in r (L). Proposition 4 Let T : K → K be a continuous mapping and L ∈ L K (X ) having the SIJP at r (L). If either r (L) > 1 and T u  Lu for all u ∈ K

(7)

r (L) < 1 and T u  Lu for all u ∈ K ,

(8)

or

then T has no positive fixed point. Theorem 2 Let T : K → K be a completely continuous mapping and assume that there exist two operators L 1 , L 2 ∈ L c (B) and two functions F1 , F2 : K → K such that L 1 is lower bounded on K and has SIJP at r (L 1 ) r (L 2 ) < 1 < r (L 1 ) and for all u ∈ K L 1 u − F1 u  T u  L 2 u + F2 u. If either F1 u = ◦ ( u ) as u → ∞ and F2 u = ◦ ( u ) as u →

(9)

F1 u = ◦ ( u ) as u → 0 and F2 u = ◦ ( u ) as u → ∞,

(10)

or

then T has a positive fixed point. For a delaited presentation on fixed point theory see [8] and [14].

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3 Riemann–Liouville Derivative Now, let us recall some basic facts related to the theory of fractional differential equations. Let β be a positive real number, the Riemann–Liouville fractional integral of order β of a function f : (0, +∞) → R is defined by β I 0+

1 f (t) = Γ (β)



t

(t − s)β−1 f (s)ds,

(11)

0

where Γ (β) is the gamma function, provided that the right side is pointwise defined on (0, +∞). For example, we have where n = [β] + 1, [β] denotes the integer part of the number β, provided that the right side is pointwise defined on β (σ+1) σ−β (0, ∞). As a basic example, we quote for σ > −1, D0+ t σ = Γ Γ(σ−β+1) t . Thus, β

if u ∈ C (0, 1) ∩ L 1 (0, 1), then the fractional differential equation D0+ u(t) = 0 has i=[β]+1 β−i u(t) = i=1 ci t , ci ∈ R, as unique solution and if u has a fractional derivative of order β in C (0, 1) ∩ L 1 (0, 1), then β



i=[β]+1

β

I0+ D0+ u(t) = u(t) +

ci t β−i , ci ∈ R.

(12)

i=1

For a detailed presentation on fractional differential calculus, see [10] and [11].

4 Fixed Point Formulation Now, we introduce some spaces and operators needed for the proof of the main results. we let E be the linear space defined by

  u(t) E = u ∈ C R+ , R , lim α−1 exists t→+∞ t equipped with the norm u E =

|u(t)| α−1 t∈[0,+∞[ (1 + t) sup

(E, . E ) is a Banach space. In all what follows E+ denotes the cone of nonnegative functions in E and the subset P of E defined by P = {u ∈ E, u(t) ≥ γ1 (t) u E , for all t ≥ 0}

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where γ1 (t) = min(λ, λt α−1 ) and λ = 

is a cone in E. Lemma 1 If σ =

A Γ (α)

 η 0

1 γ(η)

+

1 (1−σ)Γ (α)

1 η 1



γ(t)

0

dμ(t)

(1 + η)α−1

t α−1 dμ(t) ≺ 1 and q(t) ∈ L 1 ([0, +∞)), then the fbvp

⎧ α ⎪ ⎨ D u(t) + q(t) = 0 0 ≤ t < +∞ u(0) = D α−2 u(0) = 0,  ⎪ ⎩ lim D α−1 u(t) = A 0η u(s)dμ(s). t→+∞

has a unique solution given by  u(t) =

+∞

G(t, s)q(s)ds 0

where G(t, s) is the Green’s function defined by G(t, s) = G 1 (t, s) + and G 1 (t, s) =

1 Γ (α)



At α−1 (1 − σ) Γ (α)

η

(G 1 (t, s)dμ(t))

0

  α−1 − (t − s)α−1 t t α−1

t ≥s t ≤s

Proof Applying the operator I α , we obtain from (13), that u is solution to D α u(t) + q(t) = 0 if and only if u(t) = C1 t α−1 + C2 t α−2 + C3 t α−3 −

1 Γ (α)



t

(t − s)α−1 q(s)ds

0

consequently, the boundary conditions u(o) = 0 lead C3 = 0 We have u(t) = C1 t α−1 + C2 t α−2 − and D α−2 u(t) = −

1 Γ (2)



Dα−2 u(0) = 0 =⇒ C2 = 0

t 0

1 Γ (α)



t

(t − s)α−1 q(s)ds

0

(t − s)q(s)ds + C1 Γ (α)t + C2 Γ (α − 1)

(13)

Eigenvalue Criteria for Existence and Nonexistence of Positive …

u(t) = C1 t α−1 −

1 Γ (α)

and D α−1 u(t) = −



t



t

189

(t − s)α−1 q(s)ds

0

q(s)ds + C1 Γ (α)

0

Lim t→+∞ D α−1 u(t) = −



+∞



0

hence C1 =

1 Γ (α)



η

q(s)ds + C1 Γ (α) = A

u(s)dμ(s) 0

+∞

q(s)ds +

0

A Γ (α)



η

u(s)dμ(s) 0

We obtain t α−1 u(t) = Γ (α)



+∞

0

At α−1 q(s)ds + Γ (α)



η

0

1 u(s)dμ(s) − Γ (α)



t

(t − s)α−1 q(s)ds

0

 t  +∞  α−1 1 1 α−1 u(t) = t q(s)ds + − (t − s) t α−1 q(s)ds Γ (α) 0 Γ (α) t  At α−1 η u(s)dμ(s) + Γ (α) 0  +∞  At α−1 η = G 1 (t, s)q(s)ds + u(s)dμ(s) Γ (α) 0 0 Integrating once again u(t) from 0 to η gives 

η

0



 G 1 (t, s)q(s)ds dμ(t) 0 0

 η  η A t α−1 dμ(s) u(s)dμ(s) + Γ (α) 0 0

 u(t)dμ(t) =

η

+∞

Which means

1−

A Γ (α)

 0

η

  t α−1 dμ(t)

η 0



η

u(t)dμ(t) = 0



+∞ 0

 G 1 (t, s)q(s)ds dμ(t)

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η

 η  +∞ u(t)dμ(t) =

0

0

 G 1 (t, s)q(s)ds dμ(t)   η  A α−1 dμ(s) 1 − Γ (α) t 0 0

so  η  +∞

 G (t, s)q(s)ds dμ(t) 1 0 0 G 1 (t, s)q(s)ds +  u(t) =   η α−1 A 0 1 − Γ (α) t dμ(s) 0   +∞   η α−1 At G 1 (t, s) + = (G 1 (t, s)dμ(t)) q(s)ds (1 − σ) Γ (α) 0 0  +∞ u(t) = G(t, s)q(s)ds 

+∞

0

where At α−1 G(t, s) = G 1 (t, s) + (1 − σ) Γ (α) σ=

A Γ (α)

 η 0



η

(G 1 (t, s)dμ(t))

0

t α−1 dμ(s) and G 1 (t, s) =

1 Γ (α)

  α−1 − (t − s)α−1 t t α−1

t ≥s t ≤s

Lemma 2 If 0 < σ < 1 1. The functions G 1 and

∂G 1 ∂t

are continuous and have the following properties: G 1 (0, s) = 0 for all s  0

2. For all t, s ∈ [0, +∞[ 0 < G 1 (t, s) ≤ 

3.

t α−1 Γ (α)

(t,s) 1 limt→0 Gt1α−1 = Γ (α) ≺∞ (t,s) limt→+∞ Gt1α−1 0

(14)

(15)

(16)

4. G 1 (t, s) ≥ γ(t)

G 1 (τ , s) , (1 + τ )α−1

for all t, τ , s ≥ 0 with γ(t) = min(1, t α−1 ) (17)

Eigenvalue Criteria for Existence and Nonexistence of Positive …

5.

191

∂G 1 (t, s) > 0 for all t, s ∈ [0, +∞[. ∂t

(18)

Proof See [2]. Lemma 3 G(t, s) has the following proprieties:

 α−1 t Aη α−1 G(t, s) ≤ 1 + μ(η) 1−σ Γ (α)

1.

2. G(t, s) ≥ γ1 (t) where γ1 (t) = λγ(t), and λ =

G(τ , s) (1 + τ )α−1

1 γ(η)

+

1 1−σ

for

1 η 0

t, s, τ ≥ 0

1 dμ(t) γ(t)

(19)

(20)

,

3.  η G(t, s) G(t, s) 1 A ≺ ∞, lim lim = = (G 1 (t, s)dμ(t)) > 0 t→+∞ t α−1 Γ (α) (1 − σ) Γ (α) 0 t→0 t α−1

(21)

∂G (t, s)  0 for all t, s ∈ [0, +∞[ ∂t

(22)

Proof Properties (19), (21), and (22) are easy to check. Let us prove property (20) η t α−1 G 1 (t, s) + (1−σ)Γ (t, s)dμ(t)) G(t, s) (α) 0 (G 1 =  α−1 η τ G(τ , s) G 1 (τ , s) + (1−σ)Γ (τ , s)dμ(τ )) (α) 0 (G 1

(23)

for (17), we have G 1 (t, s) ≥ γ(t)

G 1 (η, s) , (1 + η)α−1

G 1 (τ , s) ≤

(1 + τ )α−1 G 1 (η, s) γ(η)

(24)

and increasing of G 1 , we have G(t, s) ≥ G(r, s)

G 1 (η,s) γ(t) (1+η) α−1 + (1+τ )α−1 G 1 (η,s) γ(η)

+

t α−1 1−σ

τ α−1 1−σ

η

G 1 (η,s) γ(k) (1+η) α−1 dμ(k) γ(t)  ≥λ  η  (1+η)α−1 G 1 (η,s) (1 + τ )α−1 dμ(k) 0 γ(k) 0

(25)

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with λ= 

1 γ(η)

+

A (1−σ)Γ (α)

1 η 1 0

γ(t)

 dμ(t)

(1 + η)α−1

proving property (20) of the function G. In order to prove the compactness of operators, we use the following Lemma. Lemma 4 See [7]. A nonempty subset N of E is relatively compact if the following conditions hold: 1. N is bounded in E,   x(t) 2. the function belonging to u : u(t)= (1+t) are locally equicontinuous α−1 , x ∈ N

on [0, +∞[, that is, equicontinuous on every compact interval of R+ and   x(t) are equiconvergent at 3. the functions belonging to u : u(t) = (1+t) α−1 , x ∈ N +∞, that is, given ε  0, there corresponds T (ε)  0 such that |x(t) − x(+∞)| ≺ ε, for any t  T (ε) and x ∈ N .

Lemma 5 Let h : R+ × R+ → R+ be a Dα -Caratheodory function. The operη ator Th u(t) = 0 G(t, s)h(s, u(s))ds, is well defined and is completely continuous.Moreover, if h(t, u) ≥ 0  α D u(t) + h(t, u(t)) = 0 0 ≤ t < +∞ η (26) u(o) = D α−2 u(o) = 0, limt→+∞ D α−1 u(t) = A 0 u(s)dμ(s) for all t, u ≥ 0, then Th (E + ) ⊂ P and u ∈ E is a fixed point of Th if and if u is solution to (??). Proof We show that the operator T : E → Eis relatively compact. Let Ω ⊂ B(0 E , R) be subset of E, we have then for all u∈ Ω |T u| ≤ (1 + t)α−1



+∞

G(t, s) u(s) f (s, (1 + s)α−1 ds α−1 (1 + t) (1 + s)α−1 0   Aη α−1 

1 + (1−σ)Γ μ(η)  +∞ (α) u(s) ds (1 + s)α−1 Ψ ≤ Γ (α) (1 + s)α−1 0   Aη α−1  +∞ 1 + (1−σ)Γ μ(η) (α) ≤ (1 + s)α−1 Ψ R (s)ds ≺ ∞ Γ (α) 0

Hence, T Ω is uniformly bounded. To show that T Ω is equicontinuous on any compact interval of [0, +∞[, let θ  0, t1 , t2 ∈ [0, θ] , t2  t1 and u ∈ Ω. Then

Eigenvalue Criteria for Existence and Nonexistence of Positive …

193

   T u(t2 ) T u(t1 )   −  (1 + t )α−1 (1 + t1 )α−1  2   +∞    G(t1 , s) G(t2 , s)  − f (s, u(s))ds =   α−1 α−1 (1 + t2 ) (1 + t1 ) 0   +∞ 

   G(t2 , s) − G(t1 , s) 1 1  G(t1 , s) f (s, u(s))ds  + − = α−1 α−1 α−1 + t + t + t (1 ) (1 ) (1 ) 0 2 2 1

We have mean value theorem     1 1  ≤ (α − 1) |t1 − t2 | (1 + c)α−2  −  (1 + t )α−1 α−1  + t (1 ) 2 1    T u(t2 ) T u(t1 )    (1 + t )α−1 − (1 + t )α−1  ≤ 2 1 

+∞



|G(t2 , s) − G(t1 , s)| (1 + t2 )α−1

0



+∞

≤



+

|G(t2 , s) − G(t1 , s)| (1 + t2 )α−1

0



+∞

≤ 0



  (1 + t1 )α−1 _ (1 + t2 )α−1  (1 + t1 )α−1 (1 + t2 )α−1

+

G(t1 , s)

f (s, (1 + s)α−1 )

u(s) (1 + s)α−1

)ds

 u(s) (α − 1) |t1 − t2 | (1 + c)α−2 G(t , s) (1 + s)α−1 Ψ ( )ds 1 (1 + t1 )α−1 (1 + t2 )α−1 (1 + s)α−1

|G(t2 , s) − G(t1 , s)| (1 + t2 )α−1



+

 (α − 1) |t1 − t2 | (1 + c)α−2 G(t , s) (1 + s)α−1 Ψ R (s)ds 1 (1 + t1 )α−1 (1 + t2 )α−1

since for s ∈ [0, +∞[, the function t → G(t, s) is continuous on the compact interval [0, θ] then it is uniformly continuous on [0, θ] , Hence |G(t2 , s) − G(t1 , s)| → 0 uniformly as |t1 − t2 | → 0 so    T u(t2 ) T u(t1 )    (1 + t )α−1 − (1 + t )α−1  → 0 2 1 uniformly as |t1 − t2 | → 0 Thus, TΩ is locally equicontinuous on [0, +∞[. We show that T Ω is equiconvergent at infinity. For any u ∈ Ω, we get f ∈ L 1 ([0, +∞[). Moreover

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T u(t) (1+t)α−1   +∞ G(t,s) η t α−1 lim 0 α−1 + (1−σ)(1+t)α−1 0 (G(t, s)dμ(t)) f (s, u(s)ds (1+t) t→+∞  t t α−1 −(t−s)α−1  +∞ t α−1 1 f (s, u(s)ds + f (s, u(s)ds 0 Γ (α) Γ (α)(1+t)α−1 t (1+t)α−1

   lim   t t α−1 −(t−s)α−1  +∞ 1 t α−1 t→+∞ t α−1 η f (s, u(s)ds + f (s, u(s)ds dμ(t) α−1 α−1 t 1−σ 0 Γ (α) 0 Γ (α)(1+t) (1+t)

lim

t→+∞

= =

and we have |T u| ≤ (1 + t)α−1



+∞

0



+∞

≤ 0



+∞

≤ 0

G(t, s) u(s) f (s, (1 + s)α−1 ds α−1 (1 + t) (1 + s)α−1 G(t, s) u(s) (1 + s)α−1 Ψ ( )ds α−1 (1 + t) (1 + s)α−1 G(t, s) (1 + s)α−1 Ψ R (s)ds ≺ ∞ (1 + t)α−1

the property (21) of the function G and the dominated convergence theorem lead to 

   T u(t) T u(t)    (1 + t)α−1 − lim t→+∞ (1 + t)α−1  → 0 and then T Ω is equiconvergent. Now, we show that operatorT : X → X is continuous. Let u be a function in Ω and (un ) ⊂ Ω in such that limn→+∞ u n = u. Because of 

+∞

T u n − T u ≤ 0

≤ ≤ ≤ ≤

 1+  1+  1+  1+



G(t, s)

| f (s, u n (s)) − f (s, u(s))| ds

α−1

(1 + t) Aη α−1 1−σ

Γ (α) Aη α−1 1−σ

 μ(η) 

+∞

(1 + s)α−1 Ψ

0

 Aη α−1  1−σ μ(η)

+∞

u(s) (1 + s)α−1



+ (1 + s)α−1 Ψ

(1 + s)α−1 Ψ R (s)ds + (1 + s)α−1 Ψ R (s)ds

0



Aη α−1 1−σ μ(η)

Γ (α)

(| f (s, u n (s)| + | f (s, u(s))|) ds

0

 μ(η) 

Γ (α) Γ (α)

+∞



+∞

2

(1 + s)α−1 Ψ R (s)ds

0

and we have lim n→+∞ | f (s, u n (s)) − f (s, u(s))| = 0 for all s  0

u(s) (1 + s)α−1

 ds

Eigenvalue Criteria for Existence and Nonexistence of Positive …

195

by Lebesgue dominated convergence theorem limn→+∞ T u n − T u = 0. Proving the continuité of T, Thus, T (Ω) satisfies all conditions of Lemma 5 and the mapping T is completely continuous. At the end, assume that u ∈ E + ,and t ≥ 0, et τ ≥ 0, 

+∞

T u(t) =

 G(t, s) f (s, u(s)ds ≥

0

+∞

γ1 (t)

0

G(τ , s) f (s, u(s)ds (1 + τ )α−1

leading to  T u(t) ≥ γ1 (t) sup τ ≥0

+∞

0

G(τ , s) f (s, u(s)ds ≥ γ1 (t) T u (1 + τ )α−1

This prove that T (E + ) ⊂ P and in particular T (P) ⊂ P.

5 Proofs of Mean Results The main result of this section is the proof need to introduce additional notations, with a function q in Dα and T  0 are associated the linear operators L q in £(E), F F in £(E) and K q,T in£(FT ) defined by L qF , L q,T 

+∞

L q u(t) =

G(t, s)q(s)u(s)ds

for all u ∈ E

L qF u = L q u(t) for all u ∈ F  T F u(t) = G T (t, s)q(s)u(s)ds L q,T 0  T F u(t) = G T (t, s)q(s)u(s)ds K q,T

for all u ∈ F

0

for all u ∈ FT

0

where for T  0, G T : R+ × [0, T ] → R+ be such that  G T (t, s) =

G(t, s), if t, s ∈ [0, T ] G(T, s), if t ≥ T

and FT , FT1 are the Banach spaces defined by

 u(t) FT = u ∈ [0, T ] : lim α−1 = l ∈ R+ t→0 t

 u(t) FT1 = u ∈ [0, T ] : α−1 ∈ C 1 [0, T ] t

(27) (28)

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equipped with the norms |u(t)| , for all u ∈ FT α−1 t∈[0,T ] t    u(t)    = u FT + sup  α−1  , for all u ∈ FT1 t

u FT = sup u FT1

t∈[0,T ]

Set for T 0 

u(t) ST = u ∈ FT : u(t) > 0, for all t ∈ [0, T ] and lim α−1 > 0 t→0 t Lemma 6 See [2]. The set ST is open in the Banach space FT and S⊂ FT+ . F Lemma 7 For all functions q in Dα and all T > 0, the operator L q,T has the SIJP F at r (L q,T ). F F F has the SIJP at r (L q,T ) if and only if K q,T has the SIJP at Proof Observe that Lq,T F F r (K q,T ) = r (L q,T ). F To this aim, we will prove that the operator K q,T is strongly positive and we F ). conclude then by Proposition 4 that is has the SIJP at r( Kq,T F Let us prove first that Kq,T is compact. We have for all u∈ FT

    K F    +∞ G(t, s)  q,T    q(s)u(s)ds   α−1  =  α−1  t  t 0   +∞  G 1 (t, s) =  q(s)u(s)ds t α−1 0    +∞  η  A + (G 1 (t, s)dμ(t)) q(s)u(s)ds  (1 − σ) Γ (α) 0 0    α − 1  t s  s α−2 = 1− q(s)u(s)ds   2 Γ (α) 0 t t   α − 1  t s  s α−2  ≤ 1− ds  q ∞ u FT Γ (α)  0 t 2 t Leading to limt→0 Thus, the operator

where

 K F  q,T

t α−1

F = 0 and L q,T u ∈ FT1 . F q,T : FT → FT1 , K

F F q,T u(t) = K q,T u(t), for all u ∈ FT1 and for t ∈ [0, T ] , K

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F is well defined and K q,T ∈ L(FT , FT1 ). F F q,T Since the embedding j of FT1 in FT is compact and K q,T = jo K . We have F that K q,T is compact. Now, since for all u ∈ FT+ with u = 0

 F u(t) = K q,T

T

G T (t, s)q(s)u(s)ds > 0 for all t ∈ [0, T ]

0

and by dominated convergence theorem lim

F u(t) K q,T

t→0

t α−1

 +∞ G T (t, s) q(s)u(s)ds t→0 0 t α−1   T  η  T A q(s)u(s)ds + = (G 1 (t, s)dμ(t)) q(s)u(s)ds > 0, (1 − σ) Γ (α) 0 0 0 = lim

we have

F (FT+ \ {o}) ⊂ ST ⊂ FT+ K q,T

F and the operator K q,T is strongly positive. This ends the proof.

Theorem 3 For all functions q in Dα , the operator L q has the SIJP at r (L q ) and is lower bounded on the cone P. Proof First let us prove that L qF has the SIJP at r (L qF ). This will be obtained from F Theorem 2, whence we prove that L qF = lim T →+∞ L q,T in operator norm and T → F L q,T is increasing We have for all u ∈ F with u F = 1  F   +∞   T  L u(t) − L F u(t)    G(t, s) G T (t, s) q,T  q  q(s)u(s)ds − q(s)u(s)ds    =  α−1 α−1   t α−1 t t 0 0   T  ∞   G(t, s) − G T (t, s) G(t, s) q(s)u(s)ds + q(s)u(s)ds  ≤  α−1 α−1 t t 0 T   ∞  +∞  η G 1 (t, s) A α−1 ≤ q(s)s ds + (t, s)dμ(t)) q(s)ds (G 1 t α−1 (1 − σ) Γ (α) T T 0   +∞   +∞  η   A q(s)s α−1 ds  + ≤  (G 1 (t, s)dμ(t)) q(s)ds (1 − σ) Γ (α) T

T

0

since G T (t, s) = G(t, s), for t, s ≤ T, we have     +∞  L F u(t) − L F u(t)  Aμ(η) q,T  q  q(s)s α−1 ds, for all t ≤ T  ≤ 1+   t α−1 (1 − σ) Γ (α) T and since

∂ ∂t



G(t,s) t α−1



< o, for s ∈ (0, t). We have in the case of t ≥ T,

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 F   +∞  T  T  L u(t) − L F u(t)  G(t, s) G T (t, s) q,T  q  α−1 α−1 ≤ q(s)s ds + q(s)s ds + q(s)s α−1 ds     t α−1 t α−1 t α−1 T 0 0  T  T  +∞ G(t, s) G(T, s) q(s)s α−1 ds + q(s)s α−1 ds + q(s)s α−1 ds ≤ α−1 t T α−1 T 0 0   +∞

 T Aμ(η) G(T, s) ≤ 1+ q(s)s α−1 ds + 2 q(s)s α−1 ds T α−1 (1 − σ) Γ (α) T 0

The above estimates lead to  F   L u(t) − L F u(t)  q,T  q  sup sup    t α−1 u F =1 t>1 

  +∞  T Aμ(η) G(T, s) ≤ 1+ q(s)s α−1 ds + 2 q(s)s α−1 ds → 0 as T → ∞ T α−1 (1 − σ) Γ (α) T 0

   F F  L q − L q,T =



Then by  means of the dominated convergence theorem, we conclude that  F  F converge to L qF in operator norm. lim T →+∞ L qF − L q,T  = 0 and L q,T For T1 < T2 and u ∈ F + , we have  F F L q,T u(t) − L q,T u(t) = 2 1

T2

0



T1

= 0

 G T2 (t, s)q(s)u(s)ds − 

0

T1

G T2 (t, s)q(s)u(s)ds

G T2 (t, s) − G T1 (t, s) q(s)u(s)ds +



T2 T1

G T2 (t, s)q(s)u(s)ds

Because of

G T2 (t, s) − G T1 (t, s) =

⎧ ⎪ ⎪  ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

A (1−σ)Γ (α)



η 0

0,if t ≤ T1  α−1 , if T1 ≤ t ≤ T2 , ≥ 0 (G 1 (t, s)dμ(t)) t α−1 − T1

A (1−σ)Γ (α)

η 0

ϕT2 (s) − ϕ   T1 (s)+ α−1 , if T2 ≤ t (G 1 (t, s)dμ(t)) T2α−1 − T1

F F we have L q,T ≥ L q,T . 2 1 At this stage, we are able to prove that L q has the SIJP at r (L q ). We have Λ L qF ⊂ Λ L q and Γ L qF ⊂ Γ L q . So, let us prove that Λ L qF = Λ L q and Γ L qF = Γ L q . To this aim, let λ ≥ 0 and u∈ E + \ {0} be such that L q u ≥ λu. We have U = L q u ∈ F + \ {0} , L qF U = L q L q u ≥ λL q u = λU and λ ∈ Λ L qF . This proves that Λ L qF = Λ L q . In a similar way, we also obtain that Γ L qF = Γ L q . Thus, we have

 r (L qF ) = sup(Λ L qF ) = sup(Λ L q ) = inf(Γ L qF ) = inf Γ L q

Eigenvalue Criteria for Existence and Nonexistence of Positive …

199

and the operator Lq has the SIJP at r (L qF ). Furthermore, the cone E + is total in E. Claims that r (L qF ) is the unique positive eigenvalue of L q , we have r (L q ) = r (L qF ) and L q has the SIJP at r (L q ). It remains to show that the operator L q is lower bounded on the cone P.

L q u(t) =

 +∞ 0

G(t, s)q(s)u(s)ds ≥

 +∞ 0

 G(t, s)q(s)γ1 (s)(1 + s)α−1 ds u E

leading to    L q u  = sup

L q u(t) ≥ sup α−1 t≥0 (1 + t) t≥0

 +∞ 0

G(t, s) q(s)γ1 (s)(1 + s)α−1 ds (1 + t)α−1

 u E

This completes the proof.

5.1 Proof of Proposition 1 We have from Lemma 4 that ρ is a positive eigenvalue of the linear eigenvalue problem (2) if and only if ρ−1 is a positive eigenvalue of the compact operator L q . Since Proposition 3 claims that L q has the SIJP at r (L q ), we have from Propositions 3.14 and 3.15 in [1] that r (L q ) is the unique positive eigenvalue of L q . Therefore, we have that ρ = r (L1 q ) is the unique positive eigenvalue of the linear eigenvalue problem (2).

5.2 Proof of Proposition 2 Assume that hypothesis (3) holds true, we have then from Proposition 3 the operator L q has the SIJP at r (L q ) where r (L q ) =

1 ρα (q)

and for all u ∈ P  +∞  T f u(t) = G(t, s) f (s, u(s))ds ≥ 0

+∞

G(t, s)q(s)u(s)ds = L q u(t).

0

thus hypothesis (7) holds and Propositions 4 claims that the operator T f has no fixed point. At the end, we conclude by Lemma 5 that fbvp (1) has no positive solution.

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5.3 Proof of Theorem 1 Assume that hypothesis (5) holds true. Then we obtain from  + + (q∞ ) < ρ1 (q∞ ) that for ε ∈ 0, ρ1 (q∞ ) − f +∞ (q∞ ) f +∞ there is R large such that f (t, u(t)) ≤ (ρα (q∞ ) − ε) q∞ (1 + t)α−1 u, for all t ≥ 0 and u ≥ R. Since the nonlinearity f is Dα -Cartheodory, there is Ψ R ∈ Dα such that f (t, u) ≤ (ρα (q∞ ) − ε)q∞ (t)(1 + t)α−1 u + Ψ R (t)(1 + t)α−1 for all t,u ≥ 0 (29)  Also, we have from f 0− (q0 ) > ρα (q0 ) that for ε ∈ 0, f 0− (q0 ) − ρα (q∞ ) there is r > 0 such that f (t, u) ≥ (ρα (q0 ) + ε) q0 (t)(1 + t)α−1 u for all t ≥ 0 and u ∈ [0, r ] Thus, we have f (t, u) ≤ (ρα (q0 ) + ε) q0 (t)(1 + t)α−1 u f˜(t, u) where

for all t, u ≥ 0

 f (t, u) = sup 0, (ρα (q∞ + ε)q0 (t)(1 + t)α−1 u − f (t, u) .

Therefore, we obtain from (29) and (30) that: L q0 u − F0 u ≤ T f u ≤ L q∞ u + F∞ u, ∀u ∈ P, where  F0 u(t) =

+∞

˜ u(s))ds G(t, s) f (s,

0

 F∞ u(t) =

+∞

G(t, s)Ψ R (s)(1 + s)α−1 ds

0

r (L q0 ) =

ρα (q∞ ) +  ρα (q0 ) +  >1> = r (L q∞ ) ρα (q0 ) ρα (q∞)

Theorem 2 implies the fbvp (1) admits a positive solution.

(30)

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201

6 Examples Two examples are provided to illustrate our existence or no existence result.

6.1 Example 1 Consider the following boundary value problem on the half-line:  5 D 2 u + u 3 e−t = 0 (1+t) 2 1 1 3 u(o) = D 2 u(0) = 0 limt→+∞ D 2 u(t) = 0 u(s)ds μ(t) = t η = 1 f (t, u) =

u 3

(1+t) 2

(31)

e−t ≤ ue−t .

We find σ = 158√π < 1, q(t) = e−t . Hence, all conditions of Proposition 1 hold, then the fbvp (30) admit unique eigenvalue μα (q) < 1 then fbvp (31) has not positive solution.

6.2 Example 2 Consider the fbvp  ⎧ ⎨ D 5u + ⎩ where  f (t, u) =

2

|u| 3

(1+t) 2 1

e−t = 0

u(o) = D 2 u(0) = 0 |u| 3

(1+t) 2

3

limt→+∞ D 2 u(t) =

1 0

(32) u(s)ds

e−t

√ 3 f (t, (1 + t) 2 u(t)) = |u|e−t Hence, all conditions of Theorem 1 hold, then fpvp (32) admit a positive solution.

References 1. Benmezai, A., Boucheneb, B., Henderson, J., Mecherouk, S.: The index jump property for I-homogeneous positive maps and fixed point theorems in cones. J. Nonlinear Funct. Anal. 2017, Article ID 6 (2017) 2. Benmezai, A., Chentout, S.: Eigenvalue-criteria for existence and nonexistence of positive solutions for α−order fractional differential equations(2 0 of a function g(t), t ∈ (c, d) is defined as α R L Dc,t g(t) =

dζ 1 (ζ − α) dt ζ



t

(t − u)ζ−α−1 g(u)du,

(3)

0

ζ − 1 < α < ζ, ζ ∈ N. In particular, for 0 < α < 1, we have n = 1, and hence,

206

B. Hussain and A. Afroz α R L Dc,t g(t) =

d 1 (1 − α) dt



t

(t − u)−α g(u)du.

(4)

c

Definition 3 The Caputo fractional derivative of order α > 0 of a function g(t), t ∈ (c, d) is defined as α C Dc,t g(t)

=

1 (ζ − α)



t

(t − u)ζ−α−1 g (ζ) (u)du,

(5)

c

ζ − 1 < α < ζ, ζ ∈ N.In particular, for 0 < α < 1, we have n = 1, and hence, α C Dc,t g(t)

1 = (1 − α)



t

(t − u)−α g  (u)du.

(6)

c

2.2 Wavelet In 1982, Jean Morlet, a French geophysical engineer, first introduced the concept of wavelets as a family of functions generated by shifting and stretching of a single function known as the “mother wavelet”:ψ(t). When the stretch a and shift b varies continuously, we get the family of continuous wavelet as   1 t −b , a = 0, b ∈ R. ψa,b (t) = √ ψ a a −j

If we restrict a and b to discrete values as a = a0 , b = kb0 a − j , where a0 ≥ 1, b0 ≥ 0 and j, k ∈ N. We have the following family of discrete wavelets as 1 j ψ(a0 t − kb0 ), ψ j,k (t) =  −j a0 where {ψ j,k (t)} j,k∈N forms a wavelet basis for L 2 (R) − space. In particular, the choices a0 = 2 and b0 = 1 produced an orthonormal basis [17, 18].

2.3 Haar Wavelet ∞ To construct the Haar wavelet system {hi (t)}i=1 on [μi , μ f ] two basic functions are required, namely: (a). The Haar scaling function (father wavelet):

h1 (t) = I[μi ,μ f ) (t).

(7)

A Collocation Method for Solving Proportional …

207

(b). The mother wavelet: h2 (t) = I[μi ,(μi +μ f )/2) (t) − I[(μi +μ f )/2,μ f ) (t)

(8)

where I[μi ,μ f ] (t) is characteristic/indicator function. Now for generating the Haar wavelet series, let j be dilation and k be translation parameter. Then i-th Haar Wavelet is defined as ⎧ ⎪ for t ∈ [ϑ1 (i), ϑ2 (i)) ⎨1 (9) hi (t) = −1 for t ∈ [ϑ2 (i), ϑ3 (i)) ⎪ ⎩ 0 other wise, where j j ϑ1 (i) = μi + (μ f − μi )k/2 , ϑ2 (i) = μi + (μ f − μi )(k + 0.5)/2 , ϑ3 (i) = μi + j j (μ f − μi )(k + 1)/2 . The index i = 2 + k + 1, j = 0, 1, . . . , J where J is maxij mum level of wavelet and k = 0, 1, . . . , 2 − 1. (c). Define scaling function space and wavelet space as follows: 2 −1 , V j = span{2 j/2 h1 (2 j t − k), t ∈ [μi , μ f ]}k=0

(10)

2 j −1 [μi , μ f ]}k=0 .

(11)

j

W j = span{2

j/2

h2 (2 t − k), t ∈ j

Suppose 0 ≤ J0 < J , then following relation holds: V J = V J0 ⊗ W J0 ⊗ W J0 +1 · · · ⊗ W J −1 .

(12)

2 The spaces V j are such that V0 ⊂ V1 ⊂ V2 · · · ⊂ L 2 ([μi , μ f ]) and ∪∞ j=0 V j = L ([μi ,



∞ μ f ]). Hence, L 2 ([μi , μ f ]) = V0 ( j0 W j ) holds. It allows us to approximate any f ∈ L 2 ([μi , μ f ]) with following truncated Haar series: J +1

f appr ox (t) =

2 

ai hi (t).

(13)

i=1

To apply the Haar wavelet, the following integral is required: Piα (t) =

1 (α + 1)



t μi

(t − u)α hi (u)du.

The R-L integration of (9) yields ⎧ ⎪ ⎨φ1 (t), 1 α Pi (t) = φ2 (t), (α + 1) ⎪ ⎩ φ3 (t),

t ∈ [ϑ1 (i), ϑ2 (i)) t ∈ [ϑ2 (i), ϑ3 (i)) t ∈ [ϑ3 (i), 1),

(14)

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B. Hussain and A. Afroz

φ1 (t) = (t − ϑ1 (i))α , φ2 (t) = [(t − ϑ1 (i))α − 2(t − ϑ2 (i))α ], φ3 (t) = [(t − ϑ1 (i))α − 2(t − ϑ2 (i))α + (t − ϑ3 (i))α ], where ϑ1 (i), ϑ2 (i), ϑ3 (i) are same as defined in Eq. (9). Haar wavelet is considered an efficient tool in numerical analysis, image processing, signal processing, and has numerous other applications in mathematics and engineering. For details, readers may refer to [3, 15, 20, 21] and their further references.

3 Description of Method In this section, we present the Haar wavelet series method to find the approximate solution of the proportional delay Riccati differential equation of fractional order represented in Eq. (1). For that, we approximate χα (t) present in Eq. (1) by truncated Haar wavelet series as follows: J +1

α

χ (t) =

2 

ai hi (t).

(15)

ai Piα (t) + χ(0).

(16)

i=1

R-L Integration of (15) from 0 to t yields J +1

χ(t) =

2  i=1

Now, replace t by qt in Eq. (16), we get J +1

χ(qt) =

2 

ai Piα (qt) + χ(0).

(17)

i=1

Using Eqs. (15)–(17) in Eq. (1), we get J +1

2 

J +1

ai hi (t) = (t) + c1 (t)(

i=1

2 

ai Piα (t) + χ(0))

i=1

+c2 (t)(

2 J +1 

ai Piα (qt) + χ(0))(c3 (t)

i=1 J +1

−(

2  i=1

ai Piα (qt) + χ(0))).

(18)

A Collocation Method for Solving Proportional …

209

Discretize the system (18) with the chosen collocation points tl = J +1

2 

(l−0.5) , 2 J +1

we get

J +1

ai hi (tl ) = (tl ) + c1 (tl )(

i=1

2 

ai Piα (tl ) + χ(0))

i=1

+c2 (tl )(

2 J +1 

ai Piα (qtl ) + χ(0))(c3 (tl )

(19)

i=1 J +1

−(

2 

ai Piα (qtl ) + χ(0))).

i=1

Solve the above system for Haar wavelet coefficients ai, s. Plugging these coefficients into the Eq. (16) produces the approximate solution χ(tl ).

4 Applications and Numerical Results The combination of fractional calculus with the theory of delay differential equations has enhanced the mathematical description of a number of real-world phenomena during the past few years. On the other hand, several numerical treatments have been developed for solving fractional differential models. However, very few researchers have thoroughly investigated fractional differential equations with delay. In this section, we shall be concerned with numerical treatment of some fractional order delay differential equations using the Haar wavelet series method (HWSM). Before solving numerical examples, we shall state some real-world applications of HWSM from existing literature. In December 2019, a threatful outbreak called the novel corona virus-2019 disease brought the world to its knees and took daily life to a grinding halt in much of the world. The researchers claim that the virus was initiated in the Chinese city of Wuhan. Planet-wide research to identify the symptoms, to control its spread, and to cure and eradicate the disease is still in full swing. In an attempt, Shah et al. [24] studied the transmission dynamics of the novel coronavirus2019 and construct a fractional order differential mathematical model by considering three compartments including the susceptible population, infected population, and recovered population. Further, the solution of the model is computed using the Haar wavelet collocation method. Hence, the method is proven an efficient tool in infectious disease spread modeling. Several recent studies which have promoted Haar wavelet as a favorable mathematical tool are [4, 14, 16, 25]. Example 1 Consider the following fractional order PDRDE: y α (t) =

t t 1 y(t) + y( )(1 − y( )), 0 ≤ α ≤ 1, t ∈ (0, 1), 4 2 2

(20)

210

B. Hussain and A. Afroz 1.25 Approximate solution Exact solution

1.25

1.2

approxinate solution

1.2 1.15

1.15

1.1 1.1 1.05

1 1

1.05 0.8

1 0.6

0.9 0.8

0.4 0.7

0.2 t

1

0.6 0

0

0.5

0.1

0.2

0.3

0.4

(a)

0.5

0.6

0.7

0.8

0.9

1

(b)

Fig. 1 a Haar Solution at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1. b Solution Comparison at α = 1 Table 1 Approximate solution at α = 0.5, 0.7, 0.9, 1 (Example 1) tl α = 0.5 α = 0.7 α = 0.9 α=1 0.0313 0.1563 0.2813 0.4063 0.5313 0.6563 0.7813 0.9063

1.0455 1.0933 1.1173 1.1335 1.1456 1.1551 1.1627 1.1689

1.0235 1.0692 1.1000 1.1243 1.1442 1.1609 1.1751 1.1872

1.0113 1.0473 1.0784 1.1065 1.1322 1.1558 1.1775 1.1973

1.0078 1.0383 1.0677 1.0960 1.1232 1.1492 1.1739 1.1972

yexact 1.0078 1.0383 1.0677 1.0961 1.1232 1.1492 1.1739 1.1973

with initial condition y(0) = 1, and possesses the exact solution √ √ √ 2t 2 2t 1 1 )+ sin( ), y(t) = + cos( 2 2 4 2 4 when α = 1. The approximate solution of (20) is computed using HWSM. The Solution behavior at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1 is presented graphically in Fig. 1. Also, we have presented a solution for α = 0.5, 0.7, 0.9, 1 at selected collocation points in Table 1. Maximum absolute errors (MAEs) at different wavelet levels J are demonstrated in Table 4.

A Collocation Method for Solving Proportional …

211 2.8 Approximate solution Exact solution

2.6 5 2.4

Approximate solution

4.5 4

2.2

3.5 2

3 2.5

1.8 2 1.5

1.6

1 1

1.4 0.8

1 0.6

0.9

1.2

0.8

0.4 0.7

0.2 t

0.6 0

1 0

0.5

0.1

0.2

0.3

(a )

0.4

0.5

0.6

0.7

0.8

0.9

1

(b )

Fig. 2 a Haar Solution at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1. b Solution Comparison at α = 1

Example 2 Solve the following fractional order PDRDE: y α (t) =

t t 1 1 ex p( )y( ) + y(t), 0 ≤ α ≤ 1, t ∈ (0, 1), 2 2 2 2

(21)

with initial condition y(0) = 1, and exact solution y(t) = ex p(t)), when α = 1. Approximate solution of (21) is computed using HWSM. The solution behavior at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1 is presented graphically in Fig. 2. Also, we have presented a solution for α = 0.5, 0.7, 0.9, 1 at selected collocation points in Table 2. Maximum absolute errors (MAEs) at different wavelet levels J are demonstrated in Table 4. Example 3 Solve the following fractional order PDRDE: t t 1 y α (t) = − y(t) + y( )(1 − y( )), 0 ≤ α ≤ 1, t ∈ (0, 1), 8 2 2 with initial condition y(0) =

1 , 4

(22)

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Table 2 Approximate solution at α = 0.5, 0.7, 0.9, 1 (Example 2) tl α = 0.5 α = 0.7 α = 0.9 α=1 0.0313 0.1563 0.2813 0.4063 0.5313 0.6563 0.7813 0.9063

1.2426 1.6442 1.9987 2.3653 2.7606 3.1951 3.6775 4.2165

1.1065 1.3661 1.6148 1.8796 2.1698 2.4919 2.8520 3.2561

1.0480 1.2183 1.3998 1.6007 1.8255 2.0782 2.3632 2.6849

yexact

1.0323 1.1697 1.3255 1.5021 1.7021 1.9288 2.1857 2.4768

1.0317 1.1691 1.3248 1.5012 1.7011 1.9276 2.1842 2.4750

0.42 Approximate solution Exact solution

0.4 0.45

Approximate solution

0.38 0.4

0.36

0.35

0.34

0.32 0.3 0.3 0.25 1

0.28 0.8

1 0.6

0.9 0.8

0.4 t

0.26

0.7

0.2

0.6 0

0.5

0.24 0

0.1

0.2

0.3

0.4

(a)

0.5

0.6

0.7

0.8

0.9

1

(b)

Fig. 3 a Haar Solution at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1. b Solution Comparison at α = 1

and exact solution √ √ √ 5 5t 5t 1 1 )+ sin( ), y(t) = − cos( 2 4 8 4 8 when α = 1. Approximate solution of (22) is computed using HWSM. The solution behavior at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1 is presented graphically in Fig. 3. Also, we have presented a solution for α = 0.5, 0.7, 0.9, 1 at selected collocation points in Table 3. Maximum absolute errors (MAEs) at different wavelet levels J are demonstrated in Table 4. Example 4 Now for comparison, we choose the following fractional Riccati differential equation from literature [27]: y α (t) = t 3 y 2 (t) − 2t 4 y(t) + t 5 , 0 ≤ α ≤ 1, t ∈ (0, 1),

(23)

A Collocation Method for Solving Proportional …

213

Table 3 Approximate solution at α = 0.5, 0.7, 0.9, 1 (Example 3) tl α = 0.5 α = 0.7 α = 0.9 α=1 0.0313 0.1563 0.2813 0.4063 0.5313 0.6563 0.7813 0.9063

0.2825 0.3248 0.3522 0.3743 0.3935 0.4106 0.4263 0.4407

0.2655 0.2985 0.3243 0.3472 0.3684 0.3883 0.4073 0.4255

0.2572 0.2810 0.3031 0.3246 0.3457 0.3665 0.3871 0.4075

0.2549 0.2747 0.2947 0.3150 0.3355 0.3562 0.3770 0.3981

yexact 0.2549 0.2746 0.2947 0.3150 0.3355 0.3562 0.3770 0.3981

Table 4 Maximum Absolute Error(MAE) max|yappr ox − yexact | J

Example 1

Example 2

Example 3

3 4 5 6 7 8 9 10

3.0351e-05 7.6090e-06 1.9048e-06 4.7652e-07 1.1917e-07 2.9797e-08 7.4499e-09 1.8620e-09

1.8900e-03 4.8189e-04 1.2167e-04 3.0572e-05 7.6623e-06 1.9178e-06 4.7979e-07 1.1996e-07

9.4852e-06 2.3778e-06 5.9526e-07 1.4891e-07 3.7240e-08 9.3117e-09 2.3281e-09 5.8205e-10

with initial condition y(0) = 0, and exact solution y(t) = t, when α = 1. Approximate solution of (23) is computed using HWSM. The solution behavior at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1 is presented graphically in Fig. 4. Maximum absolute errors (MAEs) at different wavelet levels J are demonstrated in Table 4 and Fig. 5. Also, the obtained results and their comparison with the existing schemes [10, 27] are given in Tables 5, 6 and 7.

214

B. Hussain and A. Afroz Approximate solution Exact solution

0.9 1.2

0.8

1 Approximate solution

0.7 0.8 0.6 0.6 0.5 0.4 0.4 0.2 0.3 0 1

0.2

0.8

1 0.6

0.9

0.1

0.8

0.4 0.7

0.2 t

0.6 0

0 0

0.5

0.1

0.2

0.3

0.4

0.5

(a)

0.6

0.7

0.8

0.9

(b)

Fig. 4 a Haar solution at α = 0.5, 0.6, 0.7, 0.8, 0.9, 1. b Solution comparison at α = 1 10-16

1.2

1

Absolute error

0.8

0.6

0.4

0.2

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

Fig. 5 Absolute error at Wavelet level J = 4 (Example 4) Table 5 Comparison of MAE (Example 4) HWSM J = 3 HWSMJ = 4 2.8315e − 08

1.1102e − 16

IRKHSM N = 4[27] 2e − 07

1

1

A Collocation Method for Solving Proportional …

215

Table 6 Comparison of HWSM versus FDE solver [10] (Example 4) tl α = 0.5 α = 0.5 α = 0.7 α = 0.7 α = 0.9 HWSM FDE HWSM FDE HWSM 0.03125 0.09375 0.15625 0.21875 0.28125 0.34375 0.40625 0.46875 0.53125 0.59375 0.65625 0.71875 0.78125 0.84375 0.90625 0.96875

0.19947 0.34550 0.44610 0.52800 0.59904 0.66285 0.72146 0.77615 0.82774 0.87677 0.92362 0.96849 1.01150 1.05269 1.09202 1.12946

0.03125 0.31335 0.43027 0.52015 0.59620 0.66357 0.72493 0.78186 0.83537 0.88610 0.93447 0.98073 1.02502 1.06737 1.10774 1.14607

0.09727 0.20988 0.30012 0.37985 0.45297 0.52139 0.58626 0.64830 0.70803 0.76583 0.82196 0.87664 0.93000 0.98216 1.03317 1.08308

0.03125 0.18927 0.28797 0.37226 0.44840 0.51906 0.58568 0.64917 0.71014 0.76902 0.82612 0.88167 0.93583 0.98872 1.04041 1.09094

0.04595 0.12351 0.19560 0.26478 0.33198 0.39770 0.46224 0.52579 0.58851 0.65051 0.71187 0.77266 0.83294 0.89275 0.95213 1.01110

Table 7 MAEs: HWSM versus FDE Solver [10] at α = 1 (Example 4) tl HWSM FDE MAEs: HWSM 0.03125 0.09375 0.15625 0.21875 0.28125 0.34375 0.40625 0.46875 0.53125 0.59375 0.65625 0.71875 0.78125 0.84375 0.90625 0.96875

0.03125 0.09375 0.15625 0.21875 0.28125 0.34375 0.40625 0.46875 0.53125 0.59374 0.65624 0.71874 0.78124 0.84374 0.90624 0.96874

0.03125 0.09374 0.15624 0.21874 0.28124 0.34374 0.40624 0.46874 0.53124 0.59374 0.65624 0.71874 0.78124 0.84374 0.90624 0.96874

0 0 0 0 0 0 0 0 0 1.99840e-15 4.59632e-14 1.53699e-12 2.23649e-11 2.87682e-10 3.12901e-09 2.83157e-08

α = 0.9 FDE 0.03125 0.11699 0.19126 0.26173 0.32984 0.39626 0.46137 0.52540 0.58853 0.65089 0.71257 0.77366 0.83421 0.89426 0.95387 1.01306

MAEs: FDE 0 2.01164e-07 1.13245e-06 1.13245e-06 1.13245e-06 1.13245e-06 1.13246e-06 1.13246e-06 1.13247e-06 1.13251e-06 1.13259e-06 1.13278e-06 1.13318e-06 1.13398e-06 1.13550e-06 1.13826e-06

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5 Conclusion In this paper, the HWSM is employed to explore the solutions of fractional order proportional delay Riccati differential equations. We illustrate applicability and utility of the method by solving a few benchmark problems. The comparison against existing methods is presented in Tables 5, 6, and 7. Numerical simulations presented in the form of tables and graphs show that the obtained results are comparatively more promising and the method is well accurate for computing the solutions.

References 1. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model (2016). arXiv:1602.03408 2. Ali, K.K., El Salam, M.A.A., Mohamed, E.M.: Chebyshev operational matrix for solving fractional order delay-differential equations using spectral collocation method. Arab J. Basic Appl. Sci. 26(1), 342–353 (2019) 3. Abdullah, A., Rafiq, M.: A new numerical scheme based on Haar wavelets for the numerical solution of the Chen-Lee-Liu equation. Optik 226, 165847 (2021) 4. Hussain, B., Afroz., Jahan, S.: Approximate solution for proportional-delay Riccati differential equations by Haar wavelet method. Poincare J. Anal. Appl. 8(2):157–170(2021) 5. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press (2013) 6. Bah¸si, M.M., Çevik, M.: Numerical solution of pantograph-type delay differential equations using perturbation-iteration algorithms. J. Appl. Math. Art. ID 139821, 10 pp (2015) 7. Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 1–13 (2015) 8. Davaeifar, S., Rashidinia, J.: Solution of a system of delay differential equations of multi pantograph type. J. Taibah Univ. Sci. 11(6), 1141–1157 (2017) 9. Diethelm, K.: An application-oriented exposition using differential operators of Caputo type. In: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) 10. Garrappa, R.: Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics 6(2):16 (2018) 11. Ghomanjani, F., Shateyi, S.: Solving a Quadratic Riccati differential equation, multi-pantograph delay differential equations, and optimal control systems with pantograph delays. Axioms 9(3):82 (2020) 12. Izadi, M., Srivastava, H.M.: An efficient approximation technique applied to a non-linear LaneEmden pantograph delay differential model. Appl. Math. Comput. 401:126123,10pp (2021) 13. Jafari, H., Mahmoudi, M., Noori Skandari, M.H.: A new numerical method to solve pantograph delay differential equations with convergence analysis. Adv. Differ. Equ. 2021:129 (2021). https://doi.org/10.1186/s13662-021-03293-0 14. Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods. Math. Methods Appl. Sci. 43(8), 5564–5578 (2020) 15. Lepik Ü., Hein H.: Haar Wavelets. In: Haar Wavelets. Mathematical Engineering. Springer, Cham (2014) 16. Meng, L., Kexin, M., Ruyi, X., Mei, S., Cattani, C.: Haar wavelet transform and variational iteration method for fractional option pricing models. Math. Methods Appl. Sci. (2022) 17. Mehra, M., Mehra, W., Ahmad, M.: Wavelets Theory and Its Applications. Springer, Singapore (2018)

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18. Nievergelt, Y.: Wavelets Made Easy. Birkhauser ¨ Boston, Inc., Boston (1999) 19. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press Inc, San Diego, CA (1999) 20. Raza, A., Khan, A.: Haar wavelet series solution for solving neutral delay differential equations. J. King Saud Univ.-Sci. 31(4), 1070–1076 (2019) 21. Ray, S.S., Gupta, A.K.: Wavelet methods for solving partial differential equations and fractional differential equations. CRC Press, Boca Raton, FL (2018) 22. Rihan, F.A.: Delay Differential Equations and Applications to Biology. Springer, Singapore (2021) 23. Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integral transforms and special functions 1(4), 277–300 (1993) 24. Shah, K., Khan, Z.A., Ali, A., Amin, R., Khan, H., Khan, A.: Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative. Alexandria Engineering Journal 59(5), 3221–3231 (2020) 25. Srivastava, H.M., Shah, F.A., Irfan, M.: Generalized wavelet quasilinearization method for solving population growth model of fractional order. Mathematical Methods in the Applied Sciences 43(15), 8753–8762 (2020) 26. Smith, H.: An introduction to delay differential equations with applications to the life sciences. Texts in Applied Mathematics 57,Springer, New York (2011) 27. Sakar, M.G., Akgül, A., Baleanu, D.: On solutions of fractional Riccati differential equations. Advances in Difference Equations 2017(1), 1–10 (2017) 28. Shah, F.A., Abass, R., Debnath, L.: Numerical solution of fractional differential equations using Haar wavelet operational matrix method. International Journal of Applied and Computational Mathematics 3(3), 2423–2445 (2017) 29. Yüzba¸sı, S, ¸ Sezer, M.: An exponential approximation for solutions of generalized pantographdelay differential equations. Applied Mathematical Modelling 37(22), 9160–9173 (2013)

On the Solution of Generalized Proportional Hadamard Fractional Integral Equations Rahul and N. K. Mahato

Abstract In this article, we consider a new fractional integral equation, namely, generalized proportional Hadamard fractional (GPHF) integral equations. Then as an application of Darbo’s fixed point theory (DFPT), we establish the existence of the solution of above-mentioned GPHF integral equations, using a measure of noncompactness (MNC). At the end, we have provided a suitable example to verify our obtained results. Keywords GPHF integral equation · Modulus of contiunity · MNC · DFPT

1 Introduction The MNC for the very first time was initiated by Kuratowski [1] in 1930. This notion of MNC is generalized by Banas [2] for the convenience to solve functional equations, which is applicable to numerous mathematical problems. Using the notion of MNC, Darbo [3] ensures that the existence of fixed points, which is obtained by the generalization of Schauder and Banach’s fixed point theory. Fractional integral equations (FIE) play a very important role in different fields of mathematical analysis and still continue to earned the attention of researchers in various applications of functional calculus in science and technology. Fractional calculus is a very powerful tool to achieve differentiation and integration with real or complex number powers, which is adopted in the sixteenth century. For recent research on fractional calculus, we refer reader to (see [4–6]). In the present time, the fixed point theory (FPT) has applications in several fields of mathematics. Also, FPT can be applied for the existence of solutions of FIE. Rahul · N. K. Mahato (B) IIITDM, Jabalpur, India e-mail: [email protected] Rahul e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_16

219

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Rahul and N. K. Mahato

In 1892, Hadamard [7] have introduced the following Hadamard FIE: τ a

dσ1 σ1

σ1 a

dσ2 ... σ2

σn−1 a

f (σn ) 1 dσn = σn Γ (α)

τ  τ α−1 dσ log f (σ) , α > 0, τ > a. σ σ a

The above-mentioned Hadamard FIE is a generalization of the classical integral. In this article, we have generalized the above-mentioned Hadamard FIE as a GPHF integral equation of order  > 0 and defined as 

H ,ρ Ia z



(s) =

1 ρ Γ ()





s

exp a

 (ρ − 1) (log(s) − log(t)) z(t) dt, (log(s) − log(t))−1 ρ t

where ρ ∈ (0, 1] and s, t ∈ [a, b]. If we take ρ = 1, then we get the classical Hadamard FIE [7]. The present study focuses on the following FIE:   ,ρ   z(s) =  s, L(s, z(s)), H I1 z (s) ,

(1)

where  > 1, ρ ∈ (0, 1], s ∈ I = [a, b], a > 0, b = T and  : I × R2 → R, L : I × R → R are continuous functions.

2 Definition and Preliminaries We will be using the following notations, definitions, and theorems throughout this paper. – – – – – – – –

 . E : norm on the Banach space E ; N¯ : closure of N ; ConvN : convex closure of N ; ME : collection of all nonempty and bounded subsets of E; NE : collection of all relatively compact sets; R: (−∞, ∞); R+ = [0, ∞); N: the set of natural numbers.

Banas and Lecko [8] have given the following definition of MNC. Definition 1 A MNC is a mapping η : ME → R+ , if it fulfills the following constraints, for all N , N1 , N2 ∈ ME . (N1 ) (N2 ) (N3 ) (N4 )

The family ker η = {N ∈ ME : η (N ) = 0} = ∅ and ker η ⊂ NE N1 ⊂ N2 =⇒ η (N1 ) ≤ η (N2 ) η N¯ = η (N ) η (Conv N ) = η (N )

On the Solution of Generalized Proportional …

(N5 ) (N6 )

221

η (k N1 + (1 − k) N2 ) ≤ kη (N1 ) + (1 − k) η (N2 ) for k ∈ [0, 1] If Nn ∈ ME , Nn = N¯ n , Nn+1 ⊂ Nn for n = 1, 2, 3, ... and lim η (Nn ) =

0, then N∞ =

∞

n→∞

Nn = ∅.

n=1

Theorem 1 (Schauder [9]) A mapping Φ : N → N which is compact and continuous has at least one fixed point (FP), where N is a nonempty, bounded, closed, and convex (NBCC) subset of a Banach space E. Theorem 2 (Darbo [10]) Let Φ : N → N be a continuous mapping and η is a MNC. If for any nonempty subset M of N , there exists a k ∈ [0, 1) having the inequality η (Φ M) < k η(M), then the mapping Φ has a FP in N . Definition 2 [11] Let N be a bounded subset of metric space E. Then for bounded set N , the Hausdorff MNC η is defined as η (N ) = inf { > 0 : N has a finite  − net in E} .

2.1 MNC on C(I) Let E = C(I ) is the space of continuous functions on I = [a, b] with the norm  ϕ = sup {|ϕ(t)| : t ∈ I } , ϕ ∈ E. Let Υ (= Φ) ⊂ E, so for any ϕ ∈ Υ and  > 0, the modulus of the continuity of ϕ is denoted as ω(ϕ, ) and defined by ω(ϕ, ) = sup {|ϕ(t1 ) − ϕ(t2 )| : t1 , t2 ∈ I, |t1 − t2 | ≤ } . Next, one can write ω(Υ, ) = sup {ω(ϕ, ) : ϕ ∈ Υ } , ω0 (Υ ) = lim ω(Υ, ). →0

The function ω0 is an MNC in E, so that the Hausdorff MNC η is given by η(Υ ) = 1 ω (Υ ) (see [11]). 2 0

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3 Proposed Method We consider the following assumptions to solve the Eq. (1): (i)

 : I × R2 → R, L : I × R → R be continuous and there exists constants μ1 , μ2 , μ3 ≥ 0 with μ1 μ3 < 1 such that









(s, L, I1 ) − (s, L, ¯ I¯1 ) ≤ μ1 L − L¯ + μ2 I1 − I¯1 , s ∈ I, L, I1 , L, ¯ I¯1 ∈ R

and |L(s, P1 ) − L(s, P2 )| ≤ μ3 |P1 − P2 | , P1 , P2 ∈ R. (ii) Let B fo = {z ∈ E :  z ≤ f 0 } and  = sup{|(s, L, I1 | : s ∈ I, L ∈ [−L , L], I1 ∈ [−I, I]} ≤ f 0 , where L = sup{|L(s,  ,ρ z(s))| : s ∈ I, z(s) ∈ [− f 0 , f 0 ]} and I = sup{| H I1 z (s)| : s ∈ I, z(s) ∈ [− f 0 , f 0 ]. (iii) |(s, 0, 0)| = 0, L(s, 0) = 0. (iv) There is a positive solution f 0 satisfying:  μ1 μ3 f 0 + μ2

f 0 exp

(ρ−1) log T ρ

ρ Γ ( + 1)

 (log T ) ≤ f 0 .

Theorem 3 If the conditions (i) − (iv) are satisfied, then the Eq. (1) has a solution in E = C(I ). Proof Consider the operator Φ : B f0 → E is as   ,ρ   (Φz)(s) =  s, L(s, z(s)), H I1 z (s) . Step 1: First, we have to prove that Φ maps B f0 into B f0 . Let Φ ∈ B f0 , then, by using the assumptions, we have   ,ρ   |(Φz)(s)| = | s, L(s, z(s)), H I1 z (s) − (s, 0, 0)| + |(s, 0, 0)|  ,ρ  ≤ μ1 |L(s, z(s)) − 0| + μ2 | H I1 z − 0| + |(s, 0, 0)|. Also, ,ρ  I1 z (s) − 0|



   s

1 z(t)

(ρ − 1) (log(s) − log(t)) dt

=

 exp (log(s) − log(t))−1 ρ Γ () 1 ρ t   f 0 exp (ρ−1)ρlog T  s dt ≤ (log(s) − log(t))−1 ρ Γ () t 1

|

H

On the Solution of Generalized Proportional …

 ≤

f 0 exp

(ρ−1) log T ρ

ρ Γ ( + 1) ≤ f0 .

223

 (log T )

 f 0 exp

(ρ−1) log T ρ

Hence z < f 0 , gives Φz < μ1 μ3 f 0 + μ2 ρ Γ (+1) So by the assumption (iv), Φ maps B f0 into B f0 .



(log T ) .

Step 2: Now, we prove that Φ is continuous on B f0 . Let  > 0 and z, z¯ ∈ B f0 such that  z − z¯ < , we have |(Φz) (s) − (Φ z¯ ) (s)|

  ,ρ     ,ρ  

≤  s, L(s, z(s)), H I1 z (s) −  s, L(s, z¯ (s)), H I1 z¯ (s)

 ,ρ   ,ρ 

≤ μ1 |L(s, z(s)) − L(s, z¯ (s))| + μ2 H I1 z (s) − H I1 z¯ (s) . Also,

   

H ,ρ

,ρ

I1 z (s) − H I1 z¯ (s)



   s

(ρ − 1) (log(s) − log(t)) dt

1 exp =

 (log(s) − log(t))−1 (z(s) − z¯ (s))

ρ Γ () 1 ρ t    s dt (ρ − 1) (log(s) − log(t)) 1 exp ≤  (log(s) − log(t))−1 |z(t) − z¯ (t)| ρ Γ () 1 ρ t   (ρ−1) log T  exp ρ < (log T ) . ρ Γ ( + 1)

Hence, z − z¯  < , gives that  exp



(ρ−1) log T ρ



|(Φz) (s) − (Φ z¯ ) (s)| < μ1 μ3  + μ2 ρ Γ (+1) (log T ) . As  → 0, we get |(Φz) (s) − (Φ z¯ ) (s)| → 0. This shows that Φ is continuous on B fo . Step 3: Finally, an estimate of Φ w. r. t. ω0 . Suppose Υ be nonempty subset of B fo , then, for an arbitrary  > 0, choose z ∈ Υ and s1 , s2 ∈ I such that |s2 − s1 | ≤  and s2 ≥ s1 . Then we have |(Φz) (s2 ) − (Φz) (s1 )|

  ,ρ     ,ρ  

=  s2 , L(s2 , z(s2 )), H I1 z (s2 ) −  s1 , L(s1 , z(s1 )), H I1 z (s1 )

  ,ρ     ,ρ  

≤  s2 , L(s2 , z(s2 )), H I1 z (s2 ) −  s2 , L(s2 , z(s2 )), H I1 z (s1 )

  ,ρ     ,ρ  

+  s2 , L(s2 , z(s2 )), H I1 z (s1 ) −  s2 , L(s1 , z(s1 )), H I1 z (s1 )

  ,ρ     ,ρ  

+  s2 , L(s1 , z(s1 )), H I1 z (s1 ) −  s1 , L(s1 , z(s1 )), H I1 z (s1 )



 ,ρ   ,ρ  ≤ μ2 H I1 z (s2 ) − H I1 z (s1 ) + μ1 |L(s2 , z(s2 )) − L(s1 , z(s1 ))| + ω (I, )

 ,ρ   ,ρ  ≤ μ2 H I z (s2 ) − H I z (s1 ) + μ1 μ3 |z(s2 ) − z(s1 )| + ω (I, ), 1

1

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Rahul and N. K. Mahato

where ω (I, ) = sup {|(s2 , L, I1 ) − (s1 , L, I1 )| : |s2 − s1 | ≤ ; s1 , s2 ∈ I } . Also,

 H ,ρ   

I z (s2 ) − H I ,ρ z (s1 )

1 1

   s2

1 z(t) (ρ − 1) (log(s2 ) − log(t))

=  dt exp (log(s2 ) − log(t))−1 ρ Γ () 1 ρ t

   s1 1 z(t)

(ρ − 1) (log(s1 ) − log(t)) −  exp dt

(log(s1 ) − log(t))−1 ρ Γ () 1 ρ t

 s2  

1 z(t) (ρ − 1) (log(s2 ) − log(t))

≤  dt exp (log(s2 ) − log(t))−1 ρ Γ () 1 ρ t

   s1 z(t)

(ρ − 1) (log(s2 ) − log(t)) − dt

exp (log(s2 ) − log(t))−1 ρ t 1

 s1  

1 z(t) (ρ − 1) (log(s2 ) − log(t))

+  dt exp (log(s2 ) − log(t))−1 ρ Γ () 1 ρ t

   s1 z(t)

(ρ − 1) (log(s1 ) − log(t)) − dt

exp (log(s1 ) − log(t))−1 ρ t 1    s2 1 |z(t)| (ρ − 1) (log(s2 ) − log(t)) ≤  dt exp (log(s2 ) − log(t))−1 ρ Γ () s1 ρ t    s1 

(ρ − 1) (log(s2 ) − log(t)) 1

exp +  (log(s2 ) − log(t))−1 ρ Γ () 1

ρ

  z(t)

(ρ − 1) (log(s2 ) − log(t)) dt − exp (log(s1 ) − log(t))−1 ρ t

 exp (ρ−1)ρlog T ≤  z(log T ) ρ Γ ( + 1)    s1 

1

exp (ρ − 1) (log(s2 ) − log(t)) (log(s2 ) − log(t))−1 + z  ρ Γ () 1

ρ

 

(ρ − 1) (log(s2 ) − log(t)) −1 1

dt. − exp (log(s1 ) − log(t)) ρ t

 ,ρ 

 ,ρ  As  → 0, then s2 → s1 , and also H I1 z (s2 ) − H I1 z (s1 ) → 0.

On the Solution of Generalized Proportional …

225

Therefore, |(Φz) (s2 ) − (Φz) (s1 )|

 ,ρ   ,ρ  ≤ μ2 H I z (s2 ) − H I z (s1 ) + μ1 μ3 ω(z, ) + ω (I, ), 1

1

gives

 ,ρ   ,ρ  ω(Φz, ) ≤ μ2 H I1 z (s2 ) − H I1 z (s1 ) + μ1 μ3 ω(z, ) + ω (I, ). Since  is uniform continuity on I × [−L, L] × [−I, I], we get as  → 0 gives ω (I, ) → 0. Taking sup and  → 0, we get Φ∈Υ

ω0 (ΦΥ ) ≤ μ1 μ3 ω0 (Υ ). Thus, by DFPT Φ has a FP in Υ ⊆ B f0 . Hence the Eq. (1) has a solution in C(I ). Example 4 Consider the following FIE √ 3

z(s) = s 3 +



z(s) + 7 + s5 + s2

1 H 5, 5 I1 z



(s) (2)

3500

for s ∈ [1, 2] = I. Here, 

1 H 5, 5 I1 z



(s) =

3125 Γ (5)

 1

s

  z(t) dt. exp −4 (log(s) − log(t)) (log(s) − log(t))4 t

I1 Also, (s, L, I1 ) = s 3 + L + 3500 and L(s, z) = , L are the continuous functions satisfying

|L(s, P1 ) − L(s, P2 )| ≤ and

√ 3 z(s) . 7+s 5 +s 2

It can be seen that both

|P1 − P2 | 9





(s, L, I1 ) − (s, L, ¯ I¯ 1 ) ≤ L − L¯ +

1

I1 − I¯ 1 . 3500

1 Therefore, μ1 = 1, μ2 = 3500 , μ3 = 19 and μ1 μ3 = 19 < 1. If  z ≤ f 0 , then we have f0 55 f 0 exp[−4(log 2)](log 2)4 L= , I= . 9 Γ (6)

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Further, |(s, L, I1 | ≤

55 f 0 exp[−4(log 2)](log 2)4 ≤ f0 . 3500Γ (6)

If we choose f 0 = 5, then we have L=

56 exp[−4(log 2)](log 2)4 5 , I= 9 Γ (6)  ≤ 5, μ1 μ3 < 1.

We see that all the assumptions from (i)−(iv) of Theorem 3 are fulfilled. Hence, by the Theorem 3, we concluded that the Eq. (1) has a solution in C(I ).

4 Conclusion In the current paper, we have defined a new class of fractional integral operators, which can be reduced to other related operators by choosing suitable values. Then, we established the endurance of solution of a GPHF integral equation, using DFPT. Finally, the obtained result is illustrated by a suitable example.

References 1. Kuratowski, K.: Sur les espaces complets. Fundamenta Mathematicae 15, 301–309 (1930) 2. Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, Dekker, New York (1980) 3. Darbo, G.: Punti uniti in trasformazioni a codominio non compatto (Italian). Rendiconti Del Seminario Matematico Della Universit di Padova 24, 84–92 (1955) 4. Agarwal, R., Almeida, R., Hristova, S., O‘Regan, D.: Caputo fractional differential equation with state dependent delay and practical stability. Dyn. Syst. Appl. 28(3), 715–742 (2019) 5. Hristova, S., Ivanova, K.: Caputo fractional differential equations with non-instantaneous random erlang distributed impulses. Fractal Fract. 3(2), 28–37 (2019) 6. Rahul, Mahato, N.K.: Existence solution of a system of differential equations, using generalized Darbo’s fixed point theorem. AIMS Math. 6(12), 13358–13369 (2021) 7. Hadamard, J.: Essai sur l‘étude des fonctions données par leur développement de Taylor. J. Pure Appl. Math. 4(8), 101–186 (1892) 8. Banas, J., Lecko, M.: Solvability of infinite systems of differential equations in Banach sequence spaces. J. Comput. Appl. Math. 6(2), 363–375 (2001) 9. Aghajani, A., Banas, J., Sabzali, N.: Some generalizations of Darbo fixed point theorem and applications. Bull. Belgian Math. Soc-Simon Stevin 2(2), 345–358 (2013) 10. Altun, I., Turkoglu, D.: A Fixed point theorem for mappings satisfying a general condition of operator type. J. Comput. Anal. Appl. 22(1), 1–9 (2007) 11. Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes Pure Appl. Math. 60 (1980)

Optimization Theory and Applications

An Invitation to Optimality Conditions Through Non-smooth Analysis Joydeep Dutta

Abstract In this short article, we show the fundamental role that non-smooth analysis plays in devising optimality conditions. Written with the graduate students and young researchers in mind, this article aims to bring to the fore how non-smooth analysis lies at the core of modern optimization. Keywords Non-smooth analysis · Optimality conditions · Limiting subdifferentials · Subdifferential calculus Mathematics Subject Classification (2020 49J52 · MSC code2 · 90C29

1 Intoduction Optimality conditions are well-known aspects of modern theory of optimization. It may appear to one that one need not discuss this issue anymore. But the huge number of papers that appear on optimality conditions shows that it is still a study in progress though its basic framework has been long established. As modern applications produce complex optimization problems, it becomes important to develop tractable optimality conditions. However as has been the tradition is optimization while developing necessary optimality conditions aim has been to structure the optimality conditions along the lines of the two fundamental optimality rules in optimization, namely, the Lagrange multiplier rule and the Karush-Kuhn-Tucker conditions (KKT conditions for short). The Lagrange multiplier rule which is a key tool in optimization is built in the following way. Let us consider the following problem (P1): min f (x) subject to H (x) = 0,

J. Dutta (B) Department of Economic Sciences, Indian Institute of Technology, Kanpur 208016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_17

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where f : Rn → R and H : Rn → Rm are continuously differentiable functions. If x ∗ is the local minimizer of (P) and J H (x ∗ ), the Jacobian of H at x ∗ has full rank m, (i.e., m ≤ n), then there exists y ∗ ∈ Rn such that ∇ f (x ∗ ) + J H (x ∗ )T y ∗ = 0. This is the celebrated Lagrange multiplier rule and y ∗ is the Lagrange multiplier vector. What happens if we ignore the rank condition on the Jacobian, can we still have something to say about the local minimizer of (P). The answer in fact is yes. We can without any condition prove that there exists (y0∗ , y ∗ ) ∈ R × Rm such that, (y0∗ , y ∗ ) = 0, y0∗ ∇ f (x ∗ ) + J H (x ∗ )T y ∗ = 0. This can be viewed as a primitive form of the Lagrange multiplier rule. However, in the erstwhile in the Soviet union, this approach to the multiplier rule was fundamental. In this situation one might have y ∗ = 0, and thus the objective function gets removed from the process of computing a candidate minimizer of (P1). In fact in many problems from the hypothesis of the problem, one can deduce that y0∗ = 0. This issue has been beautifully dealt with in the text by Brinkhuis and Tikhomirov [2]. Further it goes without saying that if y ∗ = 0 then we can normalize to consider y0∗ = 1. Once we set y0∗ = 1 in the above equation it reduces to the usual Lagrange multiplier rule. Another approach to guarantee that y0∗ = 0 is to make certain assumptions on the Jacobian of H at x ∗ . The most natural assumption is to assume that J H (x ∗ ) has full row-rank m. Once this assumption is in place, once we have y0∗ = 0, we shall immediately conclude that the vector y ∗ = 0 and this contradicts the fact (y0∗ , y ∗ ) = 0. During the Second World War, new optimization problems arose. The hallmark of these problems was that they were having inequalities as constraints. Consider the problem (P2) min f (x) subject to G(x) ≤ 0, where G : Rn → Rm is a differentiable function, and “≤” implies component-wise ordering. In 1948, Fritz John, a specialist in partial differential equation first published a formal optimality condition for (P2), in a conference proceeding [3], after his attempt to publish in Duke Journal of Mathematics failed. It said that if x ∗ is a local minimizer of (P2), there exists vector (λ∗0 , λ∗ ) ∈ R+ × Rm + such that (i) λ∗0 ∇ f (x ∗ ) + J G(x ∗ )T λ∗ = 0 (ii) (λ∗0 , λ∗ ) = 0 (iii) λ∗ , G(x ∗ ) = 0. Here the key feature is (λ∗0 , λ∗ ) = 0. Thus the result can hold with λ∗0 = 0, which is an interesting issue since it may lead to situations where the Fritz John condition can get satisfied for arbitrary feasible points which are not local minimizers. In fact this can happen even in the case of linear programming. For more details on

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the issue see, for example, Bazaara, Sherali, and Shetty [1]. Let us now turn our attention to the condition (iii) in the Fritz John necessary optimality conditions. If we write the vector function G(x) = (g1 (x), . . . , gm (x)) and λ∗ = (λi∗ , . . . , λ∗m ), then the condition (iii) implies that λi∗ gi (x ∗ ) = 0 for all i = 1, . . . , m. Observe that if gi (x ∗ ) < 0 then λi∗ = 0 and if λi∗ > 0 then gi (x ∗ ) = 0. This condition is thus called the complementary slackness condition which means that both λi∗ and gi (x ∗ ) cannot hold with strict inequalities at the same time. Further we can also guarantee that λ∗0 > 0 by assuming certain additional conditions on the constraints. This was independently achieved by Kuhn and Tucker [4] in 1951. I had the great privilege to know how the now celebrated Karush-Kuhn-Tucker conditions or KKT conditions were developed by Harold Kuhn himself at the sidelines of the EURO-OR conference held in Lisbon in 2010. They approached the problem form a very different angle and not through the lens of the Fritz John conditions. In fact Kuhn, Tucker, and Gale had already established the necessary and sufficient condition for a linear programming problem. These conditions for the linear programming problems were their guide to prove a necessary and if possible sufficient condition for the problem (P2). Tucker then wanted to device an optimality conditions for quadratic programming problems David Gale however left the team to focus on game theory and it was Kuhn who suggested that they focus on the case of differentiable functions and that allowed them to provide the following results for (P2). If we assume a suitable condition satisfied by the constraints at a local minimizer x ∗ , then Kuhn and Tucker established the existence of λ∗ ∈ Rm + such that (i) ∇ f (x ∗ ) + J G(x ∗ )T λ∗ = 0 (ii) λ∗ , G(x ∗ ) = 0. Observe that the above conditions are the Fritz John conditions with λ∗0 = 1, and hence the condition (λ∗0 , λ∗ ) = 0 is automatically met. The question that is of crucial importance is that what is that condition which guarantees that λ∗0 = 0, i.e., λ∗0 > 0 in the case. In their famous paper of 1951, Kuhn and Tucker introduced a geometric condition now called the Kuhn-Tucker constraint qualification. However, a more natural condition can be provided as follows. For any λ ≥ 0 with J G(x ∗ )T λ = 0 =⇒ λ = 0. This is referred to in the current literature as the Basic constraints qualification holding at x ∗ . We leave it to reader to prove that if the basic constraints qualification (BCQ for short) holds at x ∗ , then in the Fritz John conditions it is easy to establish that λ∗0 > 0 and thus can consider λ∗0 = 1 without loss of generality. It is now a well-known fact that since 1980, the Kuhn-Tucker condition is known as the Karush-Kuhn-Tucker condition, since Kuhn discovered that W. Karush of Chicago proved a similar result in 1939 and wrote to Karush that this historic mistake should be corrected and these conditions would henceforth be known as Karush-KuhnTucker (KKT for short) conditions. A major development began in 1960s where it was found that it has become essential to deal with issue of non-differentiability. This happens precisely when convex problems are considered; since in such cases

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the non-differentiability of the function generically lay precisely at the point where the function has a minimum value. Traditionally, f (x) = |x|, x ∈ R is usually mentioned as a prototype model of such a function. Observe that x = 0 is the (global) minimizer over R and the function has no derivative there. A key notion was that of a subdifferential, which is a set-valued map playing the role of a derivative. An extensive calculus rule was developed for the subdifferential and it deviated from the usual calculus since the subdifferential could be calculated for convex functions which intrinsically has no derivative. The calculus of convex functions was extended for locally Lipschitz function by Clarke [5]. However the calculus rules for the now famous Clarke generalized gradient or Clarke subdifferential are weaker than those of convex functions. Later on in the setting of a lower-semicontinuous function, the limiting subdifferential played a key role. Boris Mordukhovich [7, 8] Alexander Kruger [9], Jon. Borwein and Zhu [10], Rockafellar and Wets [11] played a crucial role in the development of the calculus of limiting subdifferential. From the erstwhile Soviet union came two other different approaches to subdifferential for non-smooth functions. These are, namely, the tangential subdifferential of Pschenichny and the quasi-differential by Demyanov and Rubinov [12]. Our focus in this article will be a brief survey of role played by the limiting subdifferential as the uniting force in developing the necessary optimality conditions for various classes of optimization problems. We list below the headings of the various sections of the paper. Section 2: A non-smooth Analysis Tool Box Section 3: Basic Optimality Conditions Section 4: The geometry of BCQ In Sect. 2, we shall briefly present the main tools of non-smooth analysis, reflecting both the geometric and the analytic aspects. While discussing non-smooth geometry, our key focus would be to discuss the notion of the limiting normal cone and how such a robust object can be built up from more elementary notions of normal cones specially in a non-convex setting. In the analytical aspect, our key goal would be to elucidate the notion of the limiting subdifferential and tie it up with robust calculus rules associated with it. We shall also emphasize the role of the proximal normal cone and the proximal subdifferential in providing a more clear view of the limiting normal cone and the limiting subdifferential. In Sect. 3, our focus is on developing optimality conditions for the basic problem of minimizing a function f over the set C. We will show how a geometric condition called the transversality condition plays a crucial role in deriving the optimality conditions. In Sect. 4 we show that how non-smooth tools can be effectively used in developing the KKT optimality conditions for smooth optimization problems with both inequality and equality constraints. In this, we also show that the transversality conditions are actually the geometric version of the Basic Constraint Qualification (BCQ). In this section, we show an example of an optimistic bilevel programming problem where the Basic Constraint Qualification fails at a local minimizer. Our notations as we have already used some in this section are fairly standard. For example, for the inner product of two vectors in Rn , we use the notation x, y ,

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where x, y ∈ Rn . For any set A, intA and A¯ denote the interior and closure of A, respectively. Other notations denoting sequences and convergence are also fairly standard. A sequence of vectors in Rn , (n > 1) will be denoted by {v k }, where as a sequence in Rn is given as {tk }, k ∈ N. The convex hull of a set A is denoted as co A. We begin with the hope that the reader is fairly conversant with convex analysis and basic optimization. Our aim here is only to open the entrance of huge edifices called non-smooth analysis and optimization using the limiting subdifferential as the guide.

2 A Non-smooth Analysis Tool Box Let us note in the beginning that this article is more in the form of a research exposition or survey rather than an original work. Even in this exposition we want to keep ourselves at the simplest level. Our key tool will be the limiting normal cone and the limiting subdifferential. We intend to study variational geometry or non-smooth geometry first and then move on to subdifferentials and useful calculus rules. Our key source would be Rockafellar and Wets [11], Mordukhovich [7] and Loewen [13].

2.1 Non-smooth Geometry One of the key tools of non-smooth geometry is the notion of a Bouligand tangent cone introduced in 1938. Let S ⊆ Rn , and let x¯ ∈ Rn , then v is a tangent vector to S at x¯ ∈ Rn if there exists a sequence {v k } in Rn , such that v k → v and tk ↓ 0, (i.e., tk > 0 & tk → 0) such that x¯ + tk v k ∈ S. The collection of all tangents vectors of S ¯ at x¯ is called the Bouligand tangent cone and is denoted by TS (x). Given a convex set C ⊆ Rn , the normal cone to C at x¯ ∈ C, is given as the set ¯ = {v ∈ Rn : v, x − x ¯ ≤ 0, ∀x ∈ C}. NC (x) ¯ is a closed convex cone and when C is a convex set, TC (x) ¯ is also a closed NC (x) convex cone and both of these objects are connected by the polarity relation ¯ (TC (x¯0 ))◦ = NC (x) ¯ ◦ = TC (x). ¯ (NC (x)) This polarity relation is the key to the famous Rockafellar-Pschenichny conditions in convex optimization. The move away from convexity of a set is very nicely captured in the definition of the regular normal cone. Let C ⊆ Rn be non-empty, then v ∈ Rn is called a regular normal to C at x, ¯ if v, x − x ¯ ≤ o(x − x), ¯

(1)

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for all x ∈ C, where lim

x→x¯

o(x − x) ¯ =0 x − x ¯

Intuitively the right-hand side in (1) measures the possible derivation from convexity. The set of all regular normals to C at x¯ forms a cone called the regular normal cone to C at x¯ and is denoted as Nˆ C (x) ¯ = NC (x). ¯ The drawback of Nˆ C (x) ¯ is that it can just reduce to the trivial set {0}. To avoid such unpleasant situations the notion of the limiting normal cone was introduced. A vector v ∈ Rn is called a limiting normal to C at x, ¯ if there exists a sequence of vectors {x k } in C, such that x k → x¯ and sequence {v k } in Rn such that v k → v, as k → ∞ with v k ∈ Nˆ C (x k ) for each k ∈ N. The collection of all the limiting subdifferentials is called the limiting normal ¯ Further it is clear that Nˆ C (x) ¯ ⊂ NCL (x), ¯ but unlike cone and is denoted as NCL (x). ˆ ¯ the limiting normal cone to C at x¯ need not be convex even though it is NC (x), closed. Further if C is convex we have ¯ = Nˆ C (x) ¯ = NCL (x). ¯ NC (x) This has given rise to notion of regularity of sets. A set C ⊆ Rn is said to be regular ¯ = NCL (x). ¯ A convex set is thus regular at all its points. However if one at x¯ if Nˆ C (x) needs to visualize the limiting normal cone in a more effective way, we need to develop the idea of the limiting normal cone form the lens of a proximal normal cone. Assuming that the idea of the projection on to a closed convex set we define the notion of projection map. Given S ⊆ Rn , the projection map Pr ojS is a set-valued map form Rn to S, i.e., Pr ojS : Rn ⇒ S, given as Pr ojS (x) = argmin 21 x − s2 . S

If S is a closed set then Pr ojS (x) = ∅ for each x. Else there can be x ∈ Rn for which Pr ojS (x) = ∅. ¯ provided x¯ ∈ S, and there We say a vector v ∈ Rn is a proximal normal to S at x, ¯ for some λ > 0. The set exists y ∈ Rn such that x¯ ∈ Proj S (y) and v = λ(y − x) of all proximal normals forms a cone called the proximal normal cone, denoted as ¯ Thus NCP (x). ¯ = {v = λ(y − x) ¯ : λ > 0, x¯ ∈ Pr ojS (y)}. NCP (x) ¯ = NC (x). ¯ Further it can be shown that if v ∈ Of course if C is convex NCP (x) then there exists σ > 0, such that

¯ NCP (x),

v, x − x ¯ ≤ σx − x ¯ 2 for all x ∈ C.

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This shows that v ∈ Nˆ C (x). ¯ This shows that NCP (x) ¯ ⊂ Nˆ C (x), ¯ though the reverse inclusion may not hold. ¯ then there exists a sequence {x k } in C and {v k } in In fact if v ∈ NCL (x), n k R , with x → x¯ and v k → v such that v k ∈ NCP (x k ). This view of limiting normals allows us actually to visualize the limiting normal cone. Example 1 Let C ⊆ R2 , given as (Fig. 1) C = epi(−|x|), x ∈ R where epi f denotes the epigraph of the function. In Figs. 2 and 3 below the green coloured part denotes the epigraph of f (x) = −|x|.

y 4 3 2 1

x -4

-3

-2

-1

1 -1 -2 -3 -4

Fig. 1 Graph of f (x) = −|x|

Fig. 2 Epigraph of f (x) = −|x|

y=-|x|

2

3

4

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Fig. 3 Proximal normal and construction of limiting normal

Observe that outside C, there is no point whose projection is the point (0, 0), i.e., P ¯ = Nepi NCP (x) f (0, 0) = {(0, 0)}

From Fig. 3, we can now write L 2 2 Nepi f (0, 0) = {(x, y) ∈ R : y = −x, x ≥ 0} ∪ {(x, y) ∈ R : y = x, x ≤ 0} L It is clear that in this case Nepi f (0, 0) is a closed set but not a convex set. With this example, we shall end our discussion on variational geometry and move to subdifferentials. 

2.2 Subdifferentials ¯ where R ¯ = In this section, we consider extended valued function, f : Rn → R, R ∪ {+∞, −∞} and the arithmetic rules involving infinity is chosen in this article is along the lines of Rockafellar and Wets [11]. We say that f is proper if f never takes the value −∞ and the set dom f = {x ∈ Rn : f (x) < +∞} is non-empty. Our first point of focus would be the subdifferential of a convex function. ¯ be a proper convex function, and x¯ ∈ dom f . Then the subdifferLet f : Rn → R ential of f at x¯ is the set ¯ ≥ v, y − x , ¯ ∀y ∈ Rn }. ∂ f (x) ¯ = {v ∈ Rn : f (y) − f (x) If x¯ ∈ / dom f , then define ∂ f (x) ¯ = ∅. It is simple to observe that ∂ f (x) ¯ is a convex ¯ An important set. Further x¯ is a global minima of f over Rn , if and only if 0 ∈ ∂ f (x).

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example which we will need in what follows is the subdifferential of the indicator function δC of a set C ⊆ Rn . We have  δC (x) =

0, if x ∈ C ∞ if x ∈ / C.

If C is convex, then δC : Rn → R is a lower-semicontinuous proper convex function. ¯ = NC (x), ¯ if x¯ ∈ In fact it is simple to observe that if C is convex, then ∂δC (x) ¯ = ∅, if x ∈ / C. C and ∂δC (x) Further if x¯ ∈ intdom f , then ∂ f (x) ¯ = ∅, convex and compact. When f is no longer convex we focus on what is known as the regular subdifferential with an error ¯ which is proper we say that term being added to the right. Thus for f ∈ Rn → R, v ∈ Rn , is a regular subgradient of f at x¯ if f (y) − f (x) ¯ ≥ v, y − x ¯ + o(y − x), ¯ for all y ∈ Rn , where lim

y→x¯

o(y − x) ¯ = 0. y − x ¯

The set of all regular subgradients forms a set called the regular subdifferential ∂ˆ f (x). ¯ If f is convex, then ∂ˆ f (x) ¯ = ∂ f (x). ¯ • It is important to note that ∂ˆ f (x) ¯ need not be non-empty at each of dom f point even if f is locally Lipschitz. • Further there is an elegant geometrical representation of ∂ˆ f (x) ¯ given as ¯ f (x))}. ¯ ∂ˆ f (x) ¯ = {v ∈ Rn : (v, −1) ∈ Nˆ epi f (x, Now if f is differentiable, then for any v ∈ Rn and ξ ∈ ∂ˆ f (x) ¯ f (x¯ + λv) − f (x) ¯ ≥ ξ, x¯ + λv − x ¯ + o(λv) where, λ > 0. Thus o(λ) f (x¯ + λv) − f (x) ¯ ≥ ξ, v + . λ λ As λ ↓ 0, then ∇ f (x), ¯ v ≥ ξ, v ∇ f (x) ¯ − ξ, v ≥ 0, ∀v ∈ Rn .

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Hence ξ = ∇ f (x), ¯ showing that if f is differentiable at x, ¯ then ∂ˆ f (x) ¯ = {∇ f (x)}, ¯ if ∂ˆ f (x) ¯ = ∅. Next we shall come to the notion of a limiting subdifferential, which is obtained as ¯ be a proper function and a limit of a sequence of regular normals. Let f : Rn → R let x¯ ∈ dom f. Then v is said to be a limiting subgradient or basic subgradient at x, ¯ ¯ f (x k ) → f (x), ¯ and a sequence if there exists a sequence {x k }, such that x k → x, {v k }, with v k → v, with v k ∈ ∂ˆ f (x k ). ¯ and is called The collection of all limiting subgradients at x¯ is denoted by ∂ L f (x), ¯ is often referred the limiting subdifferential or basic subdifferential of f at x. ¯ ∂ L f (x) to as the Mordukhovich subdifferential. The limiting subdifferential is closed though need not be a convex set. In order to see this, we need to actually compute it for a specific case. This can be done using the elegant geometrical formulation, of the limiting subdifferential, i.e., L ¯ = {v ∈ Rn : (v, −1) ∈ Nepi ¯ f (x))}. ¯ ∂ L f (x) f ( x,

For more details on how to arrive at above form of the limiting subdifferential see, for example, [7, 11, 14]. Example 2 Let f (x) = −|x|, then we shall compute ∂ L f (0). Thus L ∂ L f (0) = {ξ ∈ R : (ξ, −1) ∈ Nepi f (0, 0)}.

We already know that L 2 2 Nepi f (0, 0) = {(x, y) ∈ R , y = −x, x ≥ 0} ∪ {(x, y) ∈ R , y = x, x ≤ 0}.

Observe that (1, −1) ∈ {(x, y), y = −x, x ≥ 0} (−1, −1) ∈ {(x, y) : y = x, x ≤ 0}. L Thus (1, −1) and (−1, −1) belongs to Nepi f (0, 0). Further observe that there are L no other v ∈ R such that (v, −1) ∈ Nepi f (0, 0). This can be seen from Fig. 3 in the previous subsection. Hence from (2.2) we conclude that ∂ L f (0) = {−1, +1} . It is  clear that ∂ L f (0) is closed though not convex.

However a more simpler way or rather a geometrical way of viewing the limiting normal is through the notion of a proximal subdifferential rather than the regular subdifferential. Looking at the geometrical representation of other subdifferentials it ¯ at x¯ ∈ dom f as is intuitive to define the proximal of a proper function f : Rn → R P ¯ = {v ∈ Rn : (v, −1) ∈ Nepi ¯ f (x))}. ¯ ∂ P f (x) f ( x,

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The set ∂ P f (x) ¯ is convex but may not be non-empty at each x¯ ∈ dom f . Observe that if f (x) = −|x|, then, P Nepi f (0, 0) = {(0, 0)}.

Hence ∂ P f (0, 0) = ∅. But the limiting subdifferential can be obtained as a limit ¯ there exists a of proximal subdifferentials in the following sense: If v ∈ ∂ L f (x), sequence x k → x with f (x k ) → f (x) and a sequence v k → v, with v k ∈ ∂ P f (x k ). More formally we can define the limiting subdifferential in terms of the proximal subdifferential as follows. Definition 3 Let f : Rn → R be a proper function and let x¯ ∈ dom f . Then we say that a vector v ∈ Rn is a limiting subdifferential of f at x¯ if there exists a sequence {v k } ∈ Rn converging to v, a sequence {x k } in Rn converging to x¯ and { f (x k )} converging to f (x) ¯ such that v k ∈ ∂ P f (x k ). Observe that for f (x) = −|x|, we have ∂ P f (0) = ∅ and ∂ L f (0) = {−1, +1}. Observe that if {x k } ⊆ R, x k ≥ 0, x k → 0, then ∂ P f (x k ) = {−1} while if we take {x k } ⊆ R, x k ≤ 0, x k → 0, then ∂ P f (x k ) = {+1} . Thus from the above 3 it is clear that ∂ L f (0) = {−1, +1}. Let us now provide an example of computing the limiting subdifferential of a function f : R2 → R and demonstrate that even for a simple function such a computation need not be very simple. Example 4 Consider the function φ : R2 → R such that φ(x, y) = |x| − |y|, (x, y) ∈ R2 . Our aim is to compute ∂ L φ(x, y), for any (x, y) ∈ R2 . Observe that along the x-axis and y-axis, φ is not differentiable. So let us have a look at the structure of φ in detail. Also note that if φ is continuously differentiable at (x, ¯ y¯ ), then ¯ y¯ ) = {∇φ(x, ¯ y¯ )} ∂ L φ(x, Now look at the following cases (a) (b) (c) (d)

If x If x If x If x

> 0, y < 0, y < 0, y > 0, y

> 0, φ(x, y) = x − y, ∂ L φ(x, y) = {(1, −1)} > 0, φ(x, y) = −x − y, ∂ L φ(x, y) = {(−1, −1)} < 0, φ(x, y) = −x + y, ∂ L φ(x, y) = {(−1, 1)} < 0, φ(x, y) = x + y, ∂ L φ(x, y) = {(1, 1)}

Now on the x-axis and y-axis the function φ is not differentiable. On the y-axis at each point we need to compute the ∂ L | · | at x = 0 and at each point of the x-axis ∂ L (−| · |) at y = 0. To compute this we shall use the following fact. Let f and g be two locally Lipschitz functions on Rn , and if h(x, y) = f (x) + g(y),

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then, ∂ L h(x, y) = ∂ L f (x) × ∂ L g(y). One can obtain this using the definition of the limiting subgradients in terms of proximal normal. We will provide the proof after we complete the example. Observe that ∂ L φ(0, 0) = ∂ L (|0|) × ∂ L (−|0|) = [−1, +1] × {−1, +1} ∂ L (|0|) = [−1, +1], since f (x) = |x| is convex and ∂ L (−|0|) = {−1, +1} has been shown to be {-1,+1} in Example 1. Hence ∂ L φ(0, 0) = {(v, −1), −1 ≤ v ≤ 1} ∪ {(v, 1) : −1 ≤ v ≤ +1} Now let us compute ∂ L φ(x, y) along the y-axis, at all points except the origin. If y>0 ∂ L φ(0, y) = ∂ L (|0|) × {−1} = [−1, +1] × {−1} = {(v, −1) : −1 ≤ v ≤ 1} If y < 0 ∂ L φ(0, y) = ∂ L (|0|) × {+1} = [−1, +1] × {+1} = {(v, 1) : −1 ≤ v ≤ 1} We will now compute along x-axis, except the origin. If x > 0, then ∂ L φ(x, 0) = {1} × ∂ L (−|0|) = {+1} × {−1, +1} = {(1, −1), (1, 1)} Further if x < 0, we have ∂ L φ(x, 0) = {−1} × ∂ L (−|0|) = {−1} × {−1, +1} = {(−1, −1), (−1, 1)}

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This completes the example. Our aim would be now to establish the fact that ∂ L φ(x, y) = ∂ L f (x) × ∂ L g(y)

¯ y¯ ) then using Proposition 4A.3 from It is important to note that if (ξ, η) ∈ ∂ P φ(x, Loewen [13], we conclude that ∃δ > 0 and μ > 0 such that φ(x, y) − φ(x, ¯ y¯ ) ≥ (ξ, η), (x, y) − (x, ¯ y¯ ) − μ(x, y) − (x, ¯ y¯ )2 , for any (x, y) such that (x, y) − (x, ¯ y¯ ) < δ, thus ( f (x) + g(y)) − ( f (x) ¯ + g( y¯ )) ≥ ξ, x − x ¯ + η, y − y¯ − μ[x − x ¯ 2 + y − y¯ 2 ]

for all (x, y) with (x, y) − (x, ¯ y¯ ) < δ. Note that (x, y¯ ) − (x, ¯ y¯ ) < δ., whenever x − x ¯ < δ Hence, ¯ < δ. f (x) − f (x) ¯ ≥ ξ, x − x ¯ + μx − x ¯ 2 , ∀x with x − x ¯ using Proposition 4.3A of Loewen [13]. We can in a similar way Hence ξ ∈ ∂ P f (x) ¯ × ∂ P g( y¯ ). Thus prove that η ∈ ∂ P g( y¯ ). Hence (ξ, η) ∈ ∂ P f (x) ¯ y¯ ) ⊆ ∂ P f (x) ¯ × ∂ P g( y¯ ). ∂ P φ(x, ¯ and η ∈ ∂ P g( y¯ ). Now again using Proposition 4A.3 in Consider now ξ ∈ ∂ P f (x) Loewen [13], we have that ∃δ > 0 and μ > 0 such that ¯ < f (x) − f (x) ¯ ≥ ξ, x − x ¯ − μx − x ¯ 2 , ∀x with x − x

δ . 2

Further g(y) − g( y¯ ) ≥ η, x − x ¯ − μy − y¯ 2 ∀y with y − y¯ 
0 and δ > 0, such that f (y) ≥ f (x) + ξ, y − x − σy − x2 , for all y such that y − x < δ. If x is a local minimizer, there exists δ  > 0, such that δ  < δ f (y) ≥ f (x), for all y such that y − x < δ  . Thus we have f (y) − f (x) ≥ 0 ≥ −σy − x2 , for any σ > 0 f (y) − f (x) ≥ 0 ≥ 0, y − x − σy − x2 , for all y in such y − x < δ  . This show that 0 ∈ ∂ p f (x) and hence 0 ∈ ∂ L f (x) as ∂ P f (x) ⊂ ∂ L f (x). Similar arguments will work for the regular normal cone. Note ¯ = {0} and hence the that in the problem (P), if f is locally Lipschitz then ∂ L∞ f (x) condition ¯ ∩ (−NC (x)) ¯ = {0} (3) ∂ L∞ f (x) automatically holds. The condition (3) is often referred to as the transversality condition and as we will see later is deeply linked with the basic constraint qualification. Clarke subdifferential of a locally Lipschitz finite valued function is the convex hull of its limiting subdifferential. This convexification kills several key properties. Observe that for f (x) = −|x|, x¯ = 0 is the global maximizer and not the minimizer, local or global. / ∂ L f (0) and Here the limiting subdifferential ∂ L f (0) = {−1, +1}, showing that 0 ∈

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hence zero cannot be a local minimizer. Thus the limiting subdifferential is a powerful tool, to analyze local minimizers. While ∂ ◦ f (0) = co∂ L f (0) = [−1, +1], and we have 0 ∈ ∂ ◦ f (0), though the construction of the Clarke subdifferential is such that it is geared towards analyzing local minimizers. The Clarke subdifferential does not provide more information about the nature of an optimizing point though the limiting subdifferential does. Let us have a look again at the transversality condition and we ¯ and v2 ∈ NCL (x). ¯ will observe that it is equivalent to, the following. If v1 ∈ ∂ L∞ f (x) Then the transversality condition holds at x¯ iff v1 + v2 = 0 =⇒ v1 = 0, v2 = 0. One of the key focuses of this exposition is to explore the transversality condition and its role in devising optimality conditions. Now consider the problem (P) with, C = C1 ∩ C2 , where C1 , C2 ⊂ Rn min f (x) subject to x ∈ C1 ∩ C2 If x ∗ be a local minimizer of (P), then the necessary condition for optimality is given as ¯ (4) 0 ∈ ∂ L ( f + δC1 ∩C2 )(x). Observe that for any x ∈ Rn δC1 ∩C2 (x) = δC1 + δC2 (x). Hence the necessary condition (4) can be re-written as ¯ 0 ∈ ∂ L ( f + δC1 + δC2 )(x). This motivates us to look into the extended form of the sum rule. Let f (x) = f 1 (x) + ¯ is proper and lower-semicontinuous. Let f 2 (x) . . . + f m (x), where each f i : Rn → R m ¯ i = 1, . . . , m and x¯ ∈ ∩i=1 dom f i . Further assume that whenever vi ∈ ∂ L∞ f i (x), v1 + . . . + vm = 0 we have vi = 0 for all i. Then ¯ ⊂ ∂ L f 1 (x) ¯ + . . . + ∂ L f m (x). ¯ ∂ L ( f 1 + . . . + f m )(x) In the problem (P) if we have C = C1 ∩ C2 , then the required transversality condition ¯ v2 ∈ NCL1 (x) ¯ and v3 ∈ NCL2 (x), ¯ then v1 = is, v1 + v2 + v3 = 0 with v1 ∈ ∂ L∞ f (x), v2 = v3 = 0. If f is locally Lipschitz in (P) and C = C1 ∩ C2 , then the transversality ¯ ∩ (−NC2 (x)) ¯ = {0}. condition reduces to NC1 (x) The reader can easily check this fact, We shall now turn our focus to the case when f is continuously differentiable around x, ¯ then f is locally Lipschitz, and the necessary optimality condition is given through the following generalized equation ¯ 0 ∈ ∇ f (x) ¯ + NCL (x).

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In fact continuous differentiability is really not a strict requirement. If we just assume differentiability then a local minimizer x¯ satisfies the necessary condition. ¯ ∇ f (x), ¯ d ≥ 0, ∀ d ∈ TC (x) By definition of a polar cone from convex analysis we have ¯ 0 −∇ f (x) ¯ ∈ (TC (x)) C (x) Further as (TC (x)) ¯ 0=N ¯ (see Chap. 6 in Rockafellan and Wets [11]), we have C (x). ¯ f (x) ¯ ∈N C (x) ¯ ⊂ NCL (x). ¯ We have Further as N ¯ 0 ∈ ∇ f (x) ¯ + NCL (x). Let us turn over attention to convex programming problems and more specifically for a convex programming with linear constraints. Consider the problem (CLP) min f (x) subject to Ax = b x ≥ 0, where f : Rn → R is a convex function, A is a m × n matrix, b ∈ Rm and x ≥ 0 means xi ≥ 0, ∀ i = 1, . . . , n. It is a well-known fact that the necessary and sufficient optimality conditions hold for this class of problems by application of the celebrated Farkas Lemma, (see, for example, Guler [16]). The problem (CLP) can be written as min f (x) x ∈ C1 ∩ C2 , where C1 = {x : Ax = b} C2 = {x : x ≥ 0} = Rn+ Thus the transversality condition at any x ∈ C1 ∩ C2 in this case can be written as   NC1 (x) ∩ −NC2 (x) = {0}. It is by now a well-known fact that

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NC1 (x) = Im A T ; ∀ x ∈ C1 . Hence, the transversality condition can be equivalently written as (Im A T ) ∩ (−NRn+ (x)) = {0}.

(5)

So the natural question is as follows: Does the condition (5) holds automatically for the problem (CLP)? The answer is surprisingly no. Example 5 Consider C1 = {(x1 , x2 ) ∈ R2 : x1 + x2 = 0} C2 = R2+ C1 ∩ C2 = {(0, 0)}. Thus NC1 (0, 0) = {(λ, λ) : λ ∈ R} NR2+ (0, 0) = −R2+ . Hence, NC1 (0, 0) ∩ (−NR2+ (0, 0)) = {(λ, λ) : λ ∈ R} ∩ {R2+ } = {0}. Thus, the transversality condition fails. This is indeed an intriguing fact.



4 The Geometry of BCQ By the term BCQ we mean Basic Constraint Qualification as we have mentioned earlier. The Basic Constraint Qualification (BCQ) appears to be the most natural condition under which one can derive the KKT conditions for an optimization problem with both equality and inequality constraints. In fact in this section we shall keep our focus first only on the smooth case and try to study intrinsic geometry of BCQ. Further in the second part of this section we shall consider a problem which is intrinsically non-smooth for which the BCQ fails for a local minimizer. In that context, we shall also show how the notion of BCQ can get extended to the non-smooth setting. We shall now formally define the notion of BCQ for the problem (MP), min f (x) gi (x) ≤ 0, i = 1, . . . , m h j (x) = 0, j = 1, . . . , k,

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where f, gi , h j are real-valued and continuously differentiable. Let x0 be feasible for (MP). Then, we say that BCQ holds at x0 if for λi ≥ 0, i = / I (x0 ) = {i : gi (x0 ) = 0}, and μ j ∈ R, j = 1, . . . , k 1, . . . , m, λi = 0, for i ∈ 0=

m 

λi ∇gi (x0 ) +

i=1

k 

μ j ∇h j (x0 )

j=1

implies that λi = 0 for all i = 1, . . . , m and μ j = 0 for j = 1, . . . , k. We will first derive a necessary optimality condition for the problem (MP) using tools of non-smooth analysis. After that we will show that BCQ actually is a geometric condition, and in fact it is the transversality condition. Theorem 3 Let x0 be the local minimizer of the problem (MP). Let the BCQ holds at x0 . Then, there exists λi ≥ 0, i = 1, . . . , m and μ j ∈ R, j = 1, . . . , k such that (i) 0 = ∇ f (x0 ) +

m 

λi ∇gi (x0 ) +

i=1

k  μ j ∇h j (x0 ). j=1

(ii) λi gi (x0 ) = 0, i = 1, . . . , m.

Proof We shall essentially provide a scheme of the proof. The reader is requested to fill up the gaps. Since x0 is a local minimizer of (MP) we observe that x0 is a local minimizer of F(x) over x ∈ X , where F(x) = max{ f (x) − f (x0 ), g1 (x), . . . , gm (x)} and X = {x ∈ Rn : h j (x) = 0, j = 1, . . . , k}. Since F is locally Lipschitz and hence 0 ∈ ∂ F(x0 ) + N X (x0 ). Hence, using the estimate of the limiting subdifferential of a max function in Sect. 2.2, we can conclude that there exists λ0 ≥ 0, λi ≥ 0, i ∈ I (x0 ) such that λ0 + λi = 1 and i∈I (x0 )



0 = λ0 ∇ f (x0 ) +

λi ∇gi (x0 ) + N XL (x0 ).

i∈I (x0 )

Since BCQ holds at x0 , {∇h 1 (x0 ), . . . , ∇h j (x0 )} are linearly independent. Hence, using Proposition 1.9 in Clarke et.al.[6], we have N XP (x0 ) ⊂

⎧ k ⎨ ⎩

j=1

⎫ ⎬ μ j ∇h j (x0 ) : μ j ∈ R . ⎭

If ξ ∈ N XL (x0 ), then there exists a sequence ξ n → ξ and x n → x0 ,, as n → ∞ such that ξ n ∈ N XP (x n ). Hence, we can write

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ξn =

k 

λnj ∇h j (x k )

251

(6)

j=1

Suppose λn = (λn1 , . . . , λnk ) is a bounded and without relabelling we consider λn → λ as n → ∞ Therefore, k  λ j ∇h j (x0 ) ξ= j=1

If {λn } is not bounded, then λn  → ∞ as n → ∞ without loss of generality. Hence setting λn ωn = n . λ  As n → ∞ we see from (6) that 0=

k 

ω j ∇h j (x0 ),

j=1

where ω n → ω  without loss of generality. Since ω   = 1, we see that BCQ is violated. Thus, we have μ j ∈ R, j = 1, . . . , k 0 = λ0 ∇ f (x0 ) +



λi ∇gi (x0 ) +

i∈I (x0 )

If λ0 = 0, then 0=

 i∈I (x0 )

as

i∈I (x0 )

λi ∇gi (x0 ) +

k 

μ j ∇h j (x0 ).

j=1

k 

μ j ∇h j (x0 ),

j=1

λi + λ0 = 1 it implies that there exists i ∈ I (x0 ) such that λi > 0 and thus

violating BCQ. Hence the result. The part (ii) is obtained by setting λi = 0, for i ∈ I (x0 ). Our next aim is to show that the satisfaction BCQ at a feasible point x0 , then the transversality condition NCL1 (x0 ) ∩ (−NCL2 (x0 )) = {0} holds with C1 = {x : gi (x) ≤ 0, i = 1, . . . , m} C2 = {x : h j (x) = 0, j = 1, . . . , k}

(7) (8)

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Theorem 4 Let C1 and C2 be defined above in (7) and (8). Let x0 ∈ C1 ∩ C2 . Let us assume that BCQ holds at x0 for the problem (MP). Then NCL1 (x0 ) ∩ (−NCL2 (x0 )) = {0} Proof We shall first prove that if BCQ holds, then  NCL1

⊂

m 

 λi ∇gi (x0 ) : λi ≥ 0, λi gi (x0 ) = 0

i=1

and NCL2 ⊂

⎧ k ⎨ ⎩

j=1

⎫ ⎬ μ j ∇h j (x0 ) : μ j ∈ R . ⎭

Note the BCQ implies that {∇h 1 (x0 ), . . . , ∇h j (x0 )} forms a linearly independent set of vectors. From the discussion in Theorem 3 we know that NCL2 can be estimated as above. Note the BCQ at x0 also implies that λi ≥ 0, λi = 0, if i ∈ I (x0 ) and

m 

λi ∇gi (x0 ) = 0 =⇒ λi = 0, ∀ i = 1, . . . , m.

i=1

Now this condition allows us to show that  m   P λi ∇gi (x0 ) : λi ≥ 0, λi gi (x0 ) = 0 . NC 1 ⊂ i=1

Now arguing in the way we did in the proof of the last theorem (Theorem 3) we conclude that  m   L λi ∇gi (x0 ) : λi ≥ 0, λi gi (x0 ) = 0 . NC 1 ⊂ i=1

Now let v1 ∈ NCL1 (x0 ) & v2 ∈ NCL2 (x0 ) and v1 + v2 = 0. Now ∃ λˆ i ≥ 0, λˆ i gi (x0 ) = 0 such that v1 =

m  λˆ i ∇gi (x0 ) and also there exists i=1

μˆ j ∈ R, such that v2 =

m  i=1

μˆ j ∇h j (x0 )

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since v1 + v2 = 0, we have m 

λˆ i ∇gi (x0 ) +

i=1

k 

μˆ j ∇h j (x0 ) = 0.

j=1

Since BCQ holds at x0 , we conclude that λˆ i = 0 i = 1, . . . , m and μˆ j = 0, j = 1, . . . , k. This proves that v1 = 0, v2 = 0, and hence the transversality condition holds. The converse is simple to construct, and we shall not prove it here but will just mention it and leaving the proof to the reader. Theorem 5 Let C1 and C2 be as given in (7) and (8). Let x0 ∈ C1 ∩ C2 . Assume that NC1 (x0 ) ∩ (−NC2 (x0 )) = {0}. Further, assume that {∇h 1 (x0 ), . . . , ∇h m (x0 )} are linearly independent and there exists d ∈ R such that ∇gi (x0 ), d < 0 for all i ∈ I (x0 ). Then, BCQ holds at x0 for (MP). Our next and final discussion is of particular class of problems, called bilevel programming problems for which the Basic Constraint qualification never holds at any feasible point. This was discussed in Dutta [17], where the proof appears to be incomplete. We supply the full proof here. Consider the following optimistic bilevel programming problem (OBP) min F(x, y) x,y

subject to y ∈ S(x) where S(x) = argmin{ f (x, y) : y ∈ K } y

We assume that F(x, y) is jointly convex in (x, y), f (x, y) is also jointly convex and y ∈ K is convex. Here F(x, y) is called the upper-level objective or the leader’s objective and f (x, y) is called the lower-level objective or follower’s objective. Even though the problem has fully convex data; the problem is intrinsically nonsmooth and non-convex. For more details on bilevel programming see, for example, Dempe [18] and the references therein. For optimistic bilevel programming see, for example, Dempe et al. [19] and the references there in. Here we assume the upperlevel objective function is smooth. In Dutta [17] it was shown that if (x, ¯ y¯ ) is a local minimizer, then there exists λ0 ≥ 0 and λ1 ≥ 0 such that (i) 0 ∈ λ0 ∇ F(x, ¯ y¯ ) + λ1 (∂ f (x, ¯ y¯ ) − (∂v(x) ¯ × {0}) + {0} × N K ( y¯ )) (ii) 0 = (λ0 , λ1 ) ∈ R+ × R+ Here v(x) = inf { f (x, y) : y ∈ K }, the value function associated with the lowery

level problem of the bilevel problem (OBP).

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The above Fritz John type necessary condition was possible since the bilevel problem is equivalent to the single-level problem min F(x, y) x,y

subject to f (x, y) − v(x) ≤ 0 y∈K Note that v(x) is also a convex function on Rn . Since we have not made any assumption on the differentiability of the convex functions and the single-level problem above is a non-convex and non-smooth optimization problem. So we shall now describe what should be the form of BCQ for the following problem (NSP), min f (x), subject to x ∈ X, gi (x) ≤ 0, i = 1, . . . , m. where f , gi , i = 1, . . . , m are real-valued functions on Rn which are locally Lipschitz. Let x0 be a feasible problem for (NSP). We will say the BCQ holds at x0 for (NSP) / I (x0 ) and if we have scalars λi ≥ 0, for all i = 1, . . . , m and λi = 0 if i ∈ 0∈

m 

λi ∂ L gi (x0 ) + N X (x0 )

1=1

implies that λi = 0 for all i = 1, . . . , m. Hence the Basic Constraint Qualification (BCQ) will be satisfied at (x, ¯ y¯ ) of (OBP) if: ¯ y¯ ) − (∂v(x) ¯ × {0}) + {0} × N K ( y¯ )) λ1 ≥ 0 0 ∈ λ1 (∂ f (x, implies λ1 = 0. Using partial subgradients of the convex function, the Basic Constraint Qualification can be written as ¯ y¯ ) − ∂v(x)) ¯ 0 ∈ λ1 (∂x f (x,   0 ∈ λ1 ∂ y f (x, ¯ y¯ ) + N K ( y¯ )

(9) (10)

implies λ1 = 0. ¯ y¯ ) means We shall show that (9) and (10) can hold with λ1 > 0. Here ∂x f (x, the subdifferential with respect to x for the convex function f (., y¯ ). The partial subdifferential with y can be defined similarly. Let w ∈ ∂v(x). ¯ Hence, for any x ∈ Rn v(x) − v(x) ¯ ≥ w, x − x . ¯

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But v(x) ¯ = f (x, ¯ y¯ ), as y¯ ∈ argmin f (x, ¯ y). Further, v(x) ≤ f (x, y¯ ); for any y∈K

x ∈ Rn . Hence, for any x ∈ Rn

f (x, y¯ ) − f (x, ¯ y¯ ) ≥ w, x − x . ¯ ¯ y¯ ). Hence, This shows that w ∈ ∂x f (x, ¯ y¯ ) − ∂v(x). ¯ 0 ∈ ∂x f (x, Hence, we can take λ1 > 0 in (9). ¯ y), the standard optimality conditions, namely, the Since y¯ ∈ argmin f (x, y∈K

Rockafellar-Pschenichny condition gives us 0 ∈ ∂ y f (x, ¯ y¯ ) + N K ( y¯ ). Thus for any λ1 > 0 we have ¯ y¯ ) + λ1 N K ( y¯ ). 0 ∈ λ1 ∂ y f (x, Since N K ( y¯ ) is a cone we have ¯ y¯ ) + N K ( y¯ ) 0 ∈ λ1 ∂ y f (x, Hence, we can take λ1 > 0 in (10). This shows us that BCQ never holds for the problem (OBP). The qualification condition BCQ has another deep link with the geometry of the set of Lagrange multipliers or if the reader prefers KKT multipliers of the problem at a local minimizer of (MP). The KKT multiplier set at a feasible point x0 of (MP) is given as   k K K T (x0 ) = (λ, μ) ∈ Rm + × R : ∇x L(x 0 , λ, μ) = 0, λ, g(x) = 0 . where L(x, λ, μ) = f (x) +

m  i=1

λi gi (x) +

k 

μ j ∇h j (x),

j=1

denotes the Lagrangian function associated with (MP). We shall present below the following result whose proof can be completed by the reader without much difficulty. Theorem 6 Let x0 be a local minimizer of the problem (MP). Then BCQ holds at x0 if and only if the set K K T (x0 ) is bounded.

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5 Conclusions This expository article has been written keeping in mind the young researchers and graduate students. The key aim was to show how non-smooth analysis lies at the heart of modern optimization. Of course a lot of studies have been carried out seeking connections between non-smooth analysis and optimality conditions (see, for example, Borwein and Zhu [10]) our aim here is to bring out the central theme of the issue like the constraint qualification BCQ and also it’s geometrical significance. We also show that though non-smooth analysis provides a unifying theme in the study of optimality conditions linear programming problems seem to be outside that unifying framework. There had been debates over which is the most important class of optimality conditions in modern optimization. One that has been given by Fritz John or that by Karush, Kuhn, and Tucker. Pourciau [20], for example, considers that the Fritz John conditions are more fundamental and thus of more value as the KKT conditions can be derived from them by assuming simple conditions on the constraints. However one needs to know that even for linear programming problems the Fritz John conditions can hold at points which are just feasible and not optimal. For linear programming problems as we know that KKT conditions always hold at optimal points and not at non-optimal feasible points. Thus the KKT conditions are more important from the practical point of view. The importance of the KKT conditions can be gauged from the following special class of nonlinear programming problems called mathematical programming problem with complementarity constraints called MPCC problems in short. An MPCC problem is given as follows. We shall now formally define the notion of BCQ for the problem (MP), min f (x) gi (x) ≤ 0, i = 1, . . . , m h j (x) = 0, j = 1, . . . , k, H (x) ≥ 0, G(x) ≥ 0, H (x), G(x) = 0 where f, gi , h j are real-valued and continuously differentiable while H, G : Rn → Rn are also continuously differentiable vector-valued function. The name MPCC comes from the last constraint of the problem. An important fact is that the Fritz John condition is satisfied as each feasible point of the above MPCC through the KKT conditions does not hold at all feasible points and may hold only at the local minimizers since BCQ never holds at any feasible point of the problem. Thus from the view of detecting optimal points the KKT conditions clearly score a point over the Fritz John conditions. The real use of Fritz John condition in constrained programming lies in the fact that it provides a negative certificate of optimality. If at a feasible point of a constrained optimization problem the Fritz John condition fails, then the point is definitely not a local minimizer. However the failure of the KKT

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conditions can lead us to one of the two conclusions or both. The feasible point is not a local minimizer or some underlying qualification condition like BCQ has failed or both. We hope that this exposition will provide the reader with some interesting insights into both the fundamentals of non-smooth analysis and its role in optimality conditions. Acknowledgements The author is grateful to the financial assistance from the project titled: “First Order Methods in Scalar and Vector Optimization” funded by the Science and Engineering research Board of the Government of India. The author is also grateful to the anonymous referees whose comments have led to the improvement in the presentation.

References 1. Bazaraa, M.S., Sherali, H.D., Shetty C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (1993) 2. Brinkhuis, J., Tikhomirov, V.: Optimization: Insights and Applications. Princeton University Press (2005) 3. John, F.: Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, 8 Jan. 1948, pp. 187-204 (1948) 4. Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: 2nd Berkeley Symposium, pp. 481– 492 (1951) 5. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983) 6. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski P.R.: Nonsmooth Analysis and Control Theory, vol. 178. Springer Science and Business Media (2008) 7. Mordukhovich, B.S.: Variational analysis and generalized differentation, vol. I: Basic Theory, vol. II: Applications (2006) 8. Chikrii, A.A.: A review of the book variational analysis and generalized differentiation (vol. 1, basic theory and vol. 2, applications) by BS Mordukhovich. Cybernetics and Systems Analysis, vol. 42, no. 5, p. 769 (2006) 9. Kruger, A.Y.: Properties of generalized differentials. Sib. Math. J. 26(6), 822–832 (1985) 10. Borwein, J.M., Qiji J.Z.: Variational techniques in convex analysis. Tech. Var. Anal. 111–163 (2005) 11. Rockafellar, R.T., Wets R.J.B. .: Variational Analysis, vol. 317. Springer Science and Business Media (2009) 12. Demyanov, V.F., Rubinov A.M.: Constructive Nonsmooth Analysis, vol. 7. Peter Lang Pub Incorporated (1995) 13. Loewen, P.D.: Optimal control via nonsmooth analysis, No. 2. American Mathematical Soc (1993) 14. Vinter, R.B.: Optimal Control, Birkhauser (2001) 15. Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer Science and Business Media (2004) 16. Güler, O.: Foundations of Optimization, vol. 258. Springer Science and Business Media (2010) 17. Dutta, J.: Optimality conditions for bilevel programming: an approach through variational analysis. In: Generalized Nash Equilibrium Problems. Bilevel Programming and MPEC, pp. 43–64. Springer, Singapore (2017) 18. Dempe, S.: Foundations of Bilevel Programming. Springer Science and Business Media (2002) 19. Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56(5–6), 577–604 (2007) 20. Pourciau, B.H.: Modern multiplier rules. Am. Math. Month. 87(6), 433–452 (1980)

Solving Multiobjective Environmentally Friendly and Economically Feasible Electric Power Distribution Problem by Primal-Dual Interior-Point Method Jauny, Debdas Ghosh, and Ashutosh Upadhayay

Abstract This paper introduces a primal-dual interior-point algorithm to obtain the Pareto optimal solutions for a multiobjective environmentally friendly and economically feasible distribution problem. This problem has two conflicting objectives—the fuel cost and the emission. A Pascoletti–Serafini scalarization technique is utilized to convert the multiobjective environmentally friendly and economically feasible distribution problem into a parametric scalar optimization problem. We derive the KKT conditions corresponding to the barrier problem of the parametric scalar optimization problem and solve it with the help of primal-dual interior-point method. In order to solve the KKT conditions, the primal-dual interior-point method uses the Newton method to calculate the direction. A merit function is also utilized to take the suitable steplength toward the direction. Successful numerical results of the multiobjective environmentally friendly and economically feasible distribution problem demonstrate the efficiency of the proposed method. Keywords Environmentally friendly and economically feasible distribution problem · Multiobjective opimization · Interior-point method · Merit function

1 Introduction Among some of the real-world multiobjective optimization problems, environmentally friendly and economically feasible electric power distribution problem [12] is crucial. In order to ensure safe, environmentally friendly, and economically feasible Jauny (B) · D. Ghosh · A. Upadhayay Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi 221 005, India e-mail: [email protected] D. Ghosh e-mail: [email protected] A. Upadhayay e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_18

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distribution (EFEFD) of electric power, the generating units must provide outputs that fulfill the demand, cause the least amount of pollution, and emit the least amount of atmospheric emissions. EFEFD problem of electric power systems aims to obtain generating unit outputs in which the operating system cost remains minimal. Furthermore, the system constraints must be met while reducing pollution and emissions. This type of application involves optimizing multiple conflicting nonlinear objectives simultaneously. Multiobjective optimization problems (MOPs) consider to optimize several conflicting objectives simultaneously. Therefore, most often, a single solution that performs well for each objective function does not exist. In solving MOP problems, sometimes decision makers will come up with a compromise solution by analyzing a set of points that are representative of the entire Pareto set. A feasible point is called Pareto optimal (nondominated point [2]) if it is impossible to improve one objective without sacrificing another. When solving a MOP, the goal is to identify all possible Pareto optimal solutions. MOPs have been solved through several scalarization techniques [7] over the last few years. The reputed classical methods such as weighted sum [5, 6, 8], ε-constraint [4], normal boundary intersection [1, 10], physical programming [9], cone method [3, 14], etc., are known to find the Pareto optimal solutions. However, these methods are not able to yield a complete Pareto front. Recently, Pascoletti–Serafini [13] technique has been established that can generate all Pareto solutions. In this article, we propose a primal-dual interior-point method (PDIPM) combined with the Pascoletti–Serafini [13] technique to find nondominated points of EFEFD problem. The proposed method exploits the efficiencies of the Pascoletti–Serafini [13] and PDIPM [11] in solving EFEFD problem. As a consequence, the proposed method captures the discrete set of nondominated points. This paper follows the following structure. In Sect. 2, we describe the EFEFD problem in detail. In Sect. 3, a brief review of Pascoletti–Serafini technique is discussed. Then we give a formulation of interior-point method (IPM) for a nonlinear problem, which is formulated into Sect. 2, and find the search direction formulas. In Sect. 4, a merit function is presented. In Sect. 5, numerical results of EFEFD problem are presented. Finally, Sect. 6 ends with a few concluding remarks.

2 EFEFD Problem In EFEFD problem, the objective is to minimize two conflicting objectives fuel cost and emissions subject to an equality and bound constraints. Let p1 , p2 , . . . , pn be the power outputs of the generators G 1 , G 2 , . . . , G n , and f c (P) denotes the cost function of the generators. Thus, the total fuel cost f c (P) is given by the following expression: n  Pi + Qi pi + Ri pi2 , (1) f c (P) = i=1

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where P = ( p1 , p2 , . . . , pn ) represents the power output vector of the generators and Pi , Qi , and Ri are ith-generator cost coefficients (see Table 1). Let f e (P) denote the total emission of atmospheric pollutants (Mixture of sulfur oxides (S Ox ) and nitrogen oxides (N Ox )) and can be expressed as f e (P) =

n 

  10−2 αi + βi pi + γi pi2 + ξi eλi pi ,

(2)

i=1

where αi , βi , γi , ξi and λi are ith-generator emission characteristic coefficients (see Table 2).

To ensure reliable operation, each generator power output is bounded, i.e., sup

piinf ≤ pi ≤ pi , i = 1, 2, . . . , n.

(3)

Also, the entire power generation will be equal to the sum of two quantities, PDemand and Ploss , where PDemand is the total demand and Ploss is the power loss in the transmission lines. Hence, n  pi = PDemand + Ploss . (4) i=1

In this paper, we have taken six generator data from [12]. The data of cost coefficients and emission characteristic coefficients for six generators are provided in sup Tables 1 and 2. Also, piinf = 10, pi = 120, and PDemand + Ploss = 283. Therefore, the mathematical EFEFD problem is minimize subject to

[ f c (P), f e (P)] n  pi − 283 = 0

(5)

i=1

10 ≤ pi ≤ 120,

i = 1, 2, . . . , n.

The above problem can be decoded in the following manner: minimize

F (y) = [F1 (y), F2 (y)]

subject to G (y) = 0 l ≤ y ≤ u,

(6)

where x = P, F1 (y) = f c (P), F2 (y) = f e (P), g(y) = ( p1 − 120, p2 − 120, . . . , pn − 120) , l = 10 and u = 120.

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In the next section, we describe a single objective formulation of the bi-objective optimization problem (6) by using a scalarization technique.

3 The Pascoletti and Serafini Scalarization Technique To obtain the weakly efficient and efficient solutions [13] of EFEFD problem (5), we formulate it into a parametric scalar optimization problem with the help of Pascoletti and Serafini scalarization technique [13]. The formulation of Pascoletti and Serafini scalar optimization problem with parameters a, r ∈ R2 , with respect to the ordering cone K = R2 : minimize y,t subject to

t G (y) = 0 a + tr − F (y) ≥ 0

(7)

l ≤ y ≤ u, t ∈ R. By solving the scalar optimization problem (6) for various values of a and r , one can be obtained the Pareto front of EFEFD problem. To simplicity, we take a = F 0 , where F 0 = (F10 , F20 ) is the ideal point of EFEFD problem. For EFEFD problem, the ideal point is (599.22, 0.19) . Now, problem (6) can be rewritten as minimize y¯ subject to

F( y¯ ) G ( y¯ ) = 0 H ( y¯ ) ≥ 0,

(8)

where y¯ = (y, t) , F( y¯ ) = t and h( y¯ ) = (a + tr − F (y), y − l, u − y, t) . In the next section, we apply primal-dual interior-point method to solve the problem (8).

4 Primal-Dual Interior-Point Method In the following section, a PDIPM is discussed to solve (8). We formulate problem (8) into barrier problem and then Karush–Kuhn–Tucker (KKT) conditions are derived. Thereafter, PDIPM takes advantage of the Newton method to obtain the solution of the KKT system. An overview of the method is described below. We formulate the problem (8) by introducing slack variables vector s = (s1 , s2 , . . . , s5 ) as follows :

Solving Multiobjective Environmentally Friendly and Economically …

F( y¯ ) G ( y¯ ) = 0

minimizex,s ¯ subject to

H ( y¯ ) − s = 0 s ≥ 0.

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(9)

The barrier problem corresponding to (9) is as follows: minimize y¯ ,s subject to

B( y¯ , s; μ) G ( y¯ ) = 0

(10)

H ( y¯ ) − s = 0, where B( y¯ , s; μ) = F( y¯ ) − μ rier parameter.

4 i=1

log si is a barrier function and μ > 0 is a bar-

The followings are the first-order KKT conditions for barrier problem (10): ⎫   ∇ y¯ F( y¯ ) − ∇ y¯ G ( y¯ ) y − ∇ y¯ (H ( y¯ ) − s) z = 0 ⎪ ⎪ ⎪ ⎪ ⎪ −μS −1 e + z = 0 ⎪ ⎬ G ( y¯ ) = 0 ⎪ ⎪ ⎪ H ( y¯ ) − s = 0 ⎪ ⎪ ⎪ ⎭ z ≥ 0,

(11)

where S is the diagonal matrix with entries from the vector s. For a fixed μ > 0, the step Δd = (Δ y¯ , Δs, Δx, Δz) at the point ( y¯ , s, x, z) is obtained by applying Newton method to the system (11) and solving the following primal-dual system: ⎡ 2 ∇ y¯ y¯ L ⎢ 0 ⎢ ⎣ A( y¯ ) B( y¯ )

0 Z 0 −I

−(A( y¯ )) 0 0 0

⎤⎡ ⎤ ⎤ ⎡ −(B( y¯ )) Δ y¯ ∇ y¯ F( y¯ ) − (A( y¯ )) x − (B( y¯ )) z ⎥ ⎢ Δs ⎥ ⎥ ⎢ Sz − μe S ⎥⎢ ⎥ = − ⎢ ⎥, ⎦ ⎣Δx ⎦ ⎦ ⎣ G ( y¯ ) 0 Δz H ( y¯ ) − s 0

(12) where L denotes the Lagrangian for (9): L ( y¯ , s, x, z) = F ( y¯ ) − x  G ( y¯ ) − z  (H ( y¯ ) − s) ,

(13)

A( y¯ ) and B( y¯ ) are the Jacobian matrix of the function g( y¯ ) and h( y¯ ) − s, respectively.

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The matrix on the left of (12) is not symmetric. However, it can be easily symmetrized by multiplying the first equation by −1 and the second equation by −S −1 . Accordingly, we get the following to modify primal-dual system (12) ⎡ −∇ y2¯ y¯ L ⎢ 0 ⎢ ⎣ A( y¯ ) B( y¯ )

0 −S −1 Z 0 −I

(A( y¯ )) 0 0 0

⎤⎡ ⎤ ⎡ ⎤ (B( y¯ )) Δ y¯ ∇ y¯ F( y¯ ) − (A( y¯ )) x − (B( y¯ )) z ⎥ ⎢ ⎥ ⎢ z − μS −1 e −I ⎥ ⎥ ⎢ Δs ⎥ = ⎢ ⎥. ⎦ ⎦ ⎣ ⎦ ⎣ Δx G ( y ¯ ) − 0 Δz −H ( y¯ ) + s 0

(14) We note that second equation of (14) can be used to eliminate Δs without producing any off-diagonal fill-in in the remaining system with the help of the following equation:   Δs = S Z −1 z − μS −1 e + Δz . (15) Accordingly, from (15), the resulting reduced KKT system is given by ⎤⎡ ⎤ ⎡ ⎤ −∇ y2¯ y¯ L (A( y¯ )) (B( y¯ )) Δ y¯ ∇ y¯ F( y¯ ) − (A( y¯ )) y − (B( y¯ )) z ⎦. ⎣ A( y¯ ) −G ( y¯ ) 0 0 ⎦ ⎣Δx ⎦ = ⎣ −1 −2 Δz −H ( y¯ ) + μZ e B( y¯ ) 0 − μS (16) ⎡

In order to solve reduced KKT system (16) one can apply a Cholesky factorization. However, due to the indefiniteness of the matrix ∇x2¯ x¯ L , we cannot apply the Cholesky factorization. In this case, symmetric indefinite factorization (see [15]) will be the best strategy to use. Therefore, we applied symmetric indefinite factorization to solve the system (16). After solving (12), we obtain the step Δd and then calculate the new iterate (x¯ + , s + , y + , z + ) as follows: ⎫ y¯ + = y¯ + ζs Δ y¯ ⎪ ⎪ ⎪ s + = s + ζs Δs ⎬ , x + = x + ζz Δx ⎪ ⎪ ⎪ ⎭ z + = z + ζz Δz where

 ζs = max{ζ ∈ (0, 1) : s + ζ Δs ≥ (1 − η)s} , ζz = max{ζ ∈ (0, 1) : z + ζ Δz ≥ (1 − η)z}

(17)

(18)

with η ∈ (0, 1). The steplength calculated by (18) ensures that the variables s and y remain positive at every iterate. But, there is no guarantee of the reduction in the objective function and convergence of the generated sequence to a minimum point. To evaluate the progress toward optimality of the algorithm, we take the advantage of the merit function.

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4.1 Merit Function With merit functions, steps are shortened so that an appropriate reduction toward optimality can be made along the search direction. IPM can be viewed as methods for solving the barrier problem (10). Therefore, we define the following merit function in terms of barrier functions:   φν ( y¯ , s) = B( y¯ , s; μ) + ν H ( y¯ − s)22 + G ( y¯ )22 ,

(19)

where ν > 0 is penalty parameter. The penalty parameter ν is updated such that the following inequality hold: ν≥

∇ F  Δp + (σ/2)Δp  ∇ y2¯ y¯ L Δp (1 − ρ) (H ( y¯ ) − s + G ( y¯ ))

,

(20)

where Δp = (Δ y¯ , Δs) , ρ ∈ (0, 1) and σ is defined as:  σ =

1 0

if Δp  ∇ y2¯ y¯ L Δp > 0, otherwise.

After computing the step Δd from (12), we compute the steplength αs and αz with the help of (18) and reach the next iteration by using (17). Now, if the new point ( y¯ + , s + , x + , z + ) is able to reduce the merit function φν ( y¯ + , s + ), then ( y¯ + , s + , x + , z + ) is accepted and the algorithm continues. If the merit function φν ( y¯ + , s + ) does not reduce, then the new point ( y¯ + , s + , x + , z + ) is not accepted. In this case, we choose α ∈ [0, α max ], where α max = min{αs , αz } so that the following Armijo condition satisfied:   φν ( y¯ + , s + ) − φν ( y¯ , s) ≤ δα ∇φν ( y¯ + , s + ) Δp,

(21)

where 0 < δ < 1 and Δp = (Δ y¯ , Δs).

4.2 Update of Barrier Parameter In IPMs, the choice of the barrier parameter μ is critical since optimality is attained when the barrier parameter approaches zero. In addition, if the value of μ is set to decline slowly, many iterations will be needed to reach convergence. However, if it is reduced rapidly, some slack variables or Lagrange multipliers will approach zero very soon. The following technique of updating μ has demonstrated the effectiveness in practice: s y , (22) μ=ς 6

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where σ is chosen as follows:  3 1−℘ min{s1 x1 , s2 x2 , . . . , s4 x4 } ς = 0.2 min 0.01 , 2 , where ℘ = . ℘ (s  x)/6

(23)

Algorithm 1 provides a detailed step-wise procedure to find the nondominated points of EFEFD problem with the help of the process described above. The following function is derived from the perturbed KKT system (11):   Ξ ( y¯ , s, x, z) = max ∇ y¯ F( y¯ ) − (A( y¯ )) y − (B( y¯ )) z, Sx − μe, H ( y¯ ) − s, G ( y¯ ) .

(24) Algorithm 1 PDIPM for EFEFD problem Inputs: (a) Given EFEFD (5) (b) Provide the number of subproblems to be solved, N 1: Initialization: Set Pareto set D ← ∅ Provide an initial point w (0) = ( y¯ (0) , s (0) , x (0) , z (0) ) with y¯ (0) > 0, s (0) > 0, x (0) > 0, z (0) > 0 Give the accuracy precision ε > 0 and choose δ ∈ (0, 1) Choose initial values for μ > 0 Set k ← 0 2: Main Part: 3: for i = 1 : N do 4: Choose randomly a ∈ R p , r ∈ R p \ {0} 5: while Ξ ( y¯ (k) , s (k) , x (k) , z (k) ) ≥ ε do 6: Compute Δd (k) by solving the system (12) 7: Compute steplength αs and αz with the help of (18) 8: Set y¯ (k+1) = y¯ (k) + ζs Δ y¯ (k) , s (k+1) = s (k) + ζs Δs (k) , (k) (k+1) (k) (k) ζz Δx , z = z + ζz Δz 9: Compute the penalty parameter by (20) 10: if φ( y¯ (k+1) , s (k+1) ) > φ( y¯ (k) , s (k) ) then 11: Compute ζ max = min{ζs , ζz } and backtrack α ∈ [0, ζ max ] until

x (k+1) = x (k) +

  φν ( y¯ (k+1) , s (k+1) ) − φν ( y¯ (k) , s (k) ) ≤ δζ ∇φν ( y¯ (k) , s (k) ) Δp (k) Set y¯ (k+1) = y¯ (k) + ζ Δ y¯ (k) , s (k+1) = s (k) + ζ Δs (k) , x (k+1) = x (k) + ζ Δx (k) , z (k+1) = z (k) + ζ Δz (k) 13: end if 14: Update barrier parameter μ by using (22) 15: k ←k+1 16: end while 17: Calculate F ( y¯ )  18: Update D ← D {F ( y¯ )} 19: end for 20: return The set D (a discrete approximation of the whole Pareto set) 12:

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4.3 Well-definedness of Algorithm 1 The well-definedness of Algorithm 1 depends on line numbers 6 and 7. Algorithm 1 solves the primal-dual system (16) with the help of symmetric indefinite factorization. Thereafter, Algorithm 1 recover Δs with the help of Eq. (15). Then, Algorithm 1 calculates the steplength αs and αz by (18).

5 Numerical Results In this section, a MATLAB implementation of Algorithm 1 is applied to EFEFD problem. The test was performed on a PC with Intel Core i5-11300H 3.10 GHz CPU and 8GB RAM in MATLAB 2020a. Table 1 contains the fuel cost coefficients and Table 2 contains the emission coefficients of EFEFD problem. The results of EFEFD problem are shown in Fig. 1.

Table 1 Fuel cost coefficients i Pi 1 2 3 4 5 6

10 10 20 10 20 10

Table 2 Emission coefficients i αi 1 2 3 4 5 6

4.091 2.543 4.258 5.326 4.258 6.131

sup

Qi

Ri

piinf

pi

200 150 180 100 180 150

100 120 40 60 40 100

0.05 0.05 0.05 0.05 0.05 0.05

0.5 0.6 1.00 1.00 1.00 0.6

βi

γi

ξi

λi

−5.554 −6.047 −5.094 −3.550 −5.0944 −5.555

6.490 5.638 4.586 3.380 4.586 5.151

2.0e − 4 5.0e − 4 1.0e − 6 2.0e − 3 1.0e − 6 1.0e − 5

2.857 3.333 8.000 2.000 8.000 6.667

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Fig. 1 Pareto front of EFEFD problem obtained by Algorithm 1

6 Conclusion In this paper, EFEFD problem has been solved using IPM. To solve EFEFD problem, we have used a Pascoletti and Serafini scalarization technique to transform EFEFD problem into a parametric scalar optimization problem. Thereafter, the parametric scalar optimization problem has been solved with the help of IPM by changing the parameters a and r . The results in Sect. 5 have shown that the proposed algorithm efficiently solves the EFEFD problem. Acknowledgements Authors are truly thankful to the reviewers for their comments on the paper. Jauny gratefully acknowledges a Senior Research Fellowship from the Council of Scientific and Industrial Research, India (File No. 09/1217(0025)2017-EMR-I), to perform this research work. Debdas Ghosh acknowledges the research grant MATRICS (MTR/2021/000696) from SERB, India, to carry out this research work.

References 1. Das, I., Dennis, J.E.: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998) 2. Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin, Heidelberg (2005) 3. Ghosh, D., Chakraborty, D.: A new Pareto set generating method for multi-criteria optimization problems. Oper. Res. Lett. 42(8), 514–521 (2014)

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4. Khalili-Damghani, K., Amiri, M.: Solving binary-state multi-objective reliability redundancy allocation series-parallel problem using efficient epsilon-constraint, multi-start partial bound enumeration algorithm, and DEA. Reliab. Eng. Syst. Saf. 103, 35–44 (2012) 5. Kim, I.Y., De Weck, O.L.: Adaptive weighted-sum method for bi-objective optimization: pareto front generation. Struct. Multidiscip. Optim. 29(2), 149–158 (2005) 6. Kim, I.Y., De Weck, O.L.: Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation. Struct. Multidiscip. Optim. 31(2), 105–116 (2006) 7. Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26(6), 369–395 (2004) 8. Marler, R.T., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multidiscip. Optim. 41(6), 853–862 (2010) 9. Messac, A., Ismail-Yahaya, A.: Multiobjective robust design using physical programming. Struct. Multidiscip. Optim. 23(5), 357–371 (2002) 10. Motta, R.D.S., Afonso, S.M., Lyra, P.R.: A modified NBI and NC method for the solution of N-multiobjective optimization problems. Struct. Multidiscip. Optim. 46(2), 239–259 (2012) 11. Nocedal, J., Stephen, W.: Numerical Optimization. Springer Science & Business Media (2006) 12. Oliveira, L.S.D., Saramago, S.F.: Multiobjective optimization techniques applied to engineering problems. J. Brazilian Soc. Mech. Sci. Eng. 32, 94–105 (2010) 13. Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42, 499–524 (1984) 14. Upadhayay, A., Ghosh, D., Ansari, Q.H., Jauny: Augmented Lagrangian cone method for multiobjective optimization problems with an application to an optimal control problem. Optim. Eng. (2022). https://doi.org/10.1007/s11081-022-09747-y 15. Vanderbei, R.J.: Symmetric quasidefinite matrices. SIAM J. Optim. 5, 100–113 (1995)

Optimization Methods Using Music-Inspired Algorithm and Its Comparison with Nature-Inspired Algorithm Debabrata Datta

Abstract Optimization problems of various categories either static or dynamic in one direction and single objective and multiobjective in another route always occurred in the industry. Risk-informed decision-making-based management of any industry depends on the optimal solution to problems faced by the industry. Classical optimization methods for handling industrial optimization problems fail and hence society looks for an efficient and simple technique. In this context, an emerging metaheuristic optimization algorithm named Harmony Search (HS) plays a major role in the field of engineering and medical science. The HS optimization method works on the basis of metaheuristics and is based on the harmony of music and is categorized as musicinspired optimization algorithm. We can apply HS for function optimization, pipe network optimization, and data optimization for its classifications. The paper will explore the fundamentals of HS algorithm and its applications for two case studies: (a) to optimize piping for water network system and (b) to optimize effective radiation dose delivered to affected cancerous target organ. Outcome of Harmony Search algorithm is further compared with BAT algorithm (nature inspired algorithm) and Bee Colony Optimization algorithm. The present paper will also explore the variation of HS algorithm for designing a harmony filter system in the field of signal processing in an optimized manner. Keywords Harmony search · Optimization · BAT algorithm · Bee colony · Network

D. Datta Former Scientist, Bhabha Atomic Research Centre, & Head, RP&AD, Mumbai 400085, India (B) Department of Information Technology, Heritage Institute of Technology, Kolkata, WB 700017, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_19

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1 Introduction The objective of an optimization problem is to minimize the cost, consumption of energy and to maximize the performance, efficiency and sustainability of the system under consideration. In general, optimization problems are highly nonlinear and are categorized as either single objective or multiobjective with specified constraints. Most of the time gradient based algorithm is followed for an optimization problem which yields local minimum if the target is set for minimization else it yields a local maximum if the target is set for maximization. Optimization is a procedure to make a system or design effective, especially the mathematical techniques involved known as metaheuristics. In the field of optimization, our aim is to find the Best Solution which can be either any of these types such as (a) minimal cost, (b) minimal error, (c) maximal profit, and (d) maximal utility. An optimization method can be classified as (i) deterministic and (ii) stochastic. The function to be minimized or maximized can be either single objective or multiobjective. In either of these classifications, the algorithms behind are grouped into two deterministic and stochastic. The deterministic optimization algorithm can be further divided into linear and nonlinear programming as well as either gradient-based or gradient free. Stochastic algorithm for optimization can be either heuristic or metaheuristic, and metaheuristic is further divided into population-based and trajectorybased. For example, Ant Colony Optimization, Bee Colony Optimization and Particle Swarm Intelligence are categorised as metaheuristic optimization algorithms and all of them are population based. These algorithms work on the basis of the collective behaviour of the decentralized self-organized agent in a population [1]. Details of all these optimization algorithms can be found elsewhere in [2, 3]. However, in the last two decades, optimization algorithms have shaped into a new dimension. Innovative algorithms such as the bat algorithm (BA) [3], and harmony search [4, 5], have become very popular in the field of engineering optimization. Most of the metaheuristic algorithms belong to evolutionary computation in general and have developed on the basis of inspiration from natural phenomena. However, all algorithms are not nature inspired. For example, the harmony search algorithm (HSA) has been developed by Geem et al. [5] on the basis of the improvisation characteristics of a musician, and therefore, the said algorithm is known as a music-inspired algorithm [5]. On the contrary, the bat algorithm is nature-inspired by the echolocation behaviour of bats while sensing distances. The diversity of these algorithms and their applications has opened a new path for the optimization process in the industry. In this article, we have explored music-inspired harmony search algorithm as an engineering optimization technique for use in industry and we have compared the outcome of HSA with nature-inspired BA. The remaining part of the article is presented in four sections. Section 2 presents a nature-inspired optimization algorithm (NIOA) in which we have presented BA with its pseudocode. Section 3 describes the musicinspired metaheuristic harmony search algorithm. Section 4 presents a few case studies exploring the practical applications of HSA and its comparison with BA. Section 5 draws conclusions from the article by summarizing the learned lessons.

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2 Nature-Inspired Optimization Algorithm Nature-inspired optimization algorithm (NIOA) is a cluster of algorithms. The physics of those algorithms is based on natural phenomena [5]. Particle swarm optimization (PSO), Bio-inspired Genetic Algorithm (BGA), Ant Colony Optimization (ACO), and Bee Colony Optimization (BCO) are a few examples of NIOA. The structure of most of the NIOAs is similar even though they are defined in various forms. The common process of most of the NIOAs consists of four steps: (1) Initialize population, (2) fitness computation with the conditions of termination, (3) Components operation and Update population, and (4) outcome; a flow chart of NIOA is as shown in Fig. 1. Theoretically common characteristics of all NIOAs are: (a) randomicity or probabilistic uncertainty and have the capacity to enhance the global search capability of individuals, (b) information interactivity which provides the direct or indirect exchange of information between individuals in a NIOA and (c) optimality meaning that the each candidate in NIOA proceed towards the best global solution through different mechanisms of information exchange.

2.1 Nature-Inspired BAT Algorithm The BAT algorithm is a metaheuristics algorithm and has been developed by Yang [6, 7]. The algorithm is based on the behaviour of bats like echolocation character and this characteristics is used to sense the distance. Bats typically emit short, loud sound impulses and listen to the bounced-back echo from an obstacle or prey while

Fig. 1 Flow chart of nature inspired optimization algorithm

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hunting at night. A bat can use its special auditory mechanism to identify the size and position of an object. The steps of the bat algorithm are as follows: Step 1: Initialize the algorithm parameters. Step 2: Update the best global position x ∗ , pulse frequency, velocity, and position of the ith bat using the expression f i = f min + ( f max − f min )β, wher e, β[0, 1],

(1)

  vit+1 = vit + xit + x ∗ f i ,

(2)

xit+1 = xit + vit

(3)

Step 3: If β(randomnumber ) is > ri , then write down the new solution as. xnew = xold +  At , where At represents the average loudness of all bats at time t and liesintherange[−1, 1]. Step 4: Accept the new solution, provided the random number is lower than Ai and f (xi ) > f (x ∗ ). Failure to the condition as prescribed, update Ai and ri using the expression Ait+1 = α Ait ,

(4)

  rit = ri0 1 − e−γ t

(5)

where, rit andri0 are the pulse rate at time t and at the initial phase. The constant γ [0, 1]. Step 5: On the basis of their fitness sort the bats and find the optimal solution, x ∗ of the present iteration. Step 6: Return to step 2 till the maximum iteration is reached. Finally, output the global optimal solution. A case study of the bat algorithm for function optimization is presented in Sect. 4.

3 Music-Inspired Harmony Search Algorithm Harmony search is a metaheuristic music-inspired optimization algorithm. The objective of this algorithm is to search for a perfect state of harmony and its basis is towards the improvisation characteristics of a musician [8–10]. This is the reason why harmony search is known as music-inspired algorithm. The harmony search algorithm being music-inspired is a function of harmony memory, pitch adjustment rate, and randomization, and these are labelled as parameters of harmony search. Following subsections describe these parameters.

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3.1 Harmony Memory Rate The quality of music is improved continuously by a musician by playing an iconic piece of music (e.g., a series of pitches is in harmony) from his/her memory implying that the best harmonies will replace the better harmony. Therefore, the effectiveness of more memory is attributed to harmony memory assigned by a parameter called harmony memory rate (HMR) denoted by raccept [0, 1]. Lower value of r selects the few best harmonies with a slow convergence and the extremely high value of r indicates the wrong solution because almost all harmonies in harmony memory (HM) are selected at that time which is not feasible at all.

3.2 Pitch Adjustment Rate (PAR) The PAR, r pa is determined by a pitch bandwidth (PB), brange . In practice, for simplicity of computation, the pitch is adjusted linearly by xcurr ent = x pr evious + brange ∗ ε

(6)

where x pr evious is the existing pitch or solution from HM and xnew is the new pitch after adjustment. Random number generator, ε lies in the range of [−1, 1]. We can say that pitch adjustment is equivalent to mutation operator in a genetic algorithm. The PAR can be assigned for controlling the degree of adjustment. A lower value of PAR with a narrow bandwidth can slow down the convergence of HS due to a limitation in the exploration of a small subspace of the complete search space.

3.3 Randomization In order to increase the diversity of the solution, we need randomization in HS. The usage of randomization drives the system further to explore various diverse solutions so as to find global optimality.

3.4 Algorithm of Harmony Search (Pseudocode) Harmony search in summary is constituted by several operators known as HS operators and these are (a) random playing, (b) memory considering, (c) pitch adjusting, (d) ensemble considering, and (e) dissonance considering. All the three components in harmony search can be summarized into an algorithm written in simple command with a corresponding pseudo code as

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Algorithm: 1. 2. 3. 4.

Construct a Harmony Memory Improvise a new Harmony with Experience (HM) or Randomness (rather than Gradient) If the new Harmony is better, include it in Harmony Memory Repeat Step 1 and Step 2 till goal is achieved

Pseudocode: Begin Objective function, f(x),

)

Generate initial harmonics [an array of real numbers] Assign value of PAR, pitch limits and bandwidth Define harmony memory accepting rate (raccept) While (k < Maximum iteration number) Generate new harmonies by accepting best harmonies Adjust pitch to get new harmonies (solutions) If (rand > raccept) choose an existing harmonic randomly Else if (rand > rpa) adjust the pitch randomly within limits Else Generate new harmonics via randomization End if Accept the new harmonics (solutions) if better

End while Find the current best solutions End

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4 Case Study 4.1 Optimization of Rosenbrock Function (Benchmark Function) Mathematically, the Rosenbrock function [11, 12] is written as f (x) =

N −1    2 100 xn+1 − xn2 + (1 − xn )2

(7)

n=1

The search domain of this function is −2.048 ≤ xn ≤ 2.048, n = 1, 2, . . . . . . . , 30. At every value of n, we have a local minimum and we apply musicinspired Harmony Search optimization algorithm to obtain the minimum value of the function, f at x = 1, which is f(1) = 0. Suppose we have N = 3; so we have local minima of the function, f(x) at n = 1 and at n = 2. The value of the decision variable results in x(1) = 9.55E-01 and x(2) = 9.13E-01. Finally, we obtain the best estimate of the function using the values of the decision variable as 2.03E-03. The graphical representation of the computational result is shown in Fig. 2 and the profile of the best cost of the function with iteration is shown in Fig. 3. Fig. 2 Optimum value of Rosenbrock function

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Best Cost

10 -1

10

-2

10 -3

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Iteration

Fig. 3 Profile of Best cost w.r.t iteration

4.2 Optimization of Beam Orientation for Identification of Target Volume One of the possible ways to treat cancer is radiation therapy. Treatment of cancer using radiation by Cobalt-60, proton, and X-ray is known as radiation oncology. In the field of radiation oncology, we use CT/MRI machines to allow radiation to pass through cancerous cells or affected organs of the patient. Typically, multiple beams of different radiation doses are used from different sides and different angles. In this kind of problem, the primary aim is to decide the strength of the radiation dose coming from the particular beam to use to achieve sufficient damage to the target tumour and limit the damage to healthy tissues. Figure 4 presents the orientation of beam therapy to treat the cancerous tumour. Fig. 4 Incidence of beam on tumour

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Table 1 Input data of radiation therapy Fraction of areal absorbed radiation dose (cGy) Region of applying beam

Radiation Bm1

Radiation Bm2

Prohibition on total average dose

Normal tissue

0.339

0.446

Minimization

Critical tissue

0.25

0.09

< = 2.7

Tumour region

0.5

0.5

6

Centre of tumour

0.6

0.4

> =6

Our goal is to optimize the orientation of the beam to identify the target volume. In practice, this concept is implemented in Bhabhatron, an indigenously designed Cobalt-60 therapeutic machine, and the resulting hardware is known as a multileaf collimator. The data used in radiotherapy is presented in Table 1. Decision variables D1 and D2 represent the dose strength for beam 1 and beam 2, respectively. Mathematical statement of the problem is as follows: Minimize 0.339D1 + 0.339D2

(Radiation Dose to normal tissue)

(8)

subject to 0.25D1 + 0.09D2 ≤ 2.7 (Radiation Dose to critical tissue)

(9)

0.5D1 + 0.5D2 = 6 (dose to tumor)

(10)

0.6D1 + 0.4D2 ≥ 6 (dose to tumor center) D1 ≥ 0, D2 ≥ 0

(11)

The harmony search algorithm is applied to solve the defined optimization problem and our optimal solution results at D1 = 7.5 and D2 = 4.5. Substituting the value of D1 and D2 in Eq. (8) that represents the dose to healthy anatomy, we obtain the optimal dose as 5.25 cGy. Figure 5 presents the outcome of the harmony search algorithm for the optimization of the target volume.

4.3 Case Study Using BAT Algorithm Bat Algorithm is proposed as a bio-inspired metaheuristics method. It is used to solve stochastic nonlinear optimization problems. It tries to mimic the behaviour of bats hunting for their prey. The pseudocode of the BAT algorithm is as follows:

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Fig. 5 Harmony research in oncology

where vi (t)—real-valued velocity vector of i-th bat, xi (t)—real-valued position vector of i-th bat, Q i —pulsation frequency of i-th bat, α, γ , Q min , Q max —constant. In this case study, we have used the BAT algorithm to optimize sphere and Rosenbrock functions with comparative results using HS. Results are tabulated in Tables 2 and 3. d We obtain the best estimate (optimal value) of sphere function f (x) = i=1 xi2 using HS as 1.2E + 1 with decision variables x(1), x(2), and x(3) are equal to 2.0 Results of optimization of Rosenbrock function using HS are good in agreement with that using BAT algorithm. However, the BAT algorithm is computationally intensive compared to the harmony search.

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Table 2 Optimization of sphere function Man value

n

D = 10

D = 20

D = 30

10

1800 · 10−6

25,200 · 10−6

1054 · 10−4

900 · 10−6

9370 · 10−6

346.8 · 10−4

230 ·

10−6

5540 ·

10−6

270 · 10−4

150 ·

10−6

1710 ·

10−6

78.5 · 10−4

Stand. dev Mean value

20

Stand. dev Mean value

0.137 · 10−6

50

Stand. dev

6.76 ·

10−6

353 · 10−6

33.3 · 10−4

10−6

11.1 · 10−4

127 ·

Table 3 Optimization of Rosenbrock function Mean value

n

D = 10

D = 20

D = 30

10

15.130

92.707

202.422

20.321

111.359

216.167

8.112

62.930

87.279

1.639

62.619

66.909

8.583

29.889

81.823

0.938

22.738

57.414

7.782

17.791

37.268

1.163

1.743

15.339

Stand. dev Mean value

20

Stand. dev Mean value

50

Stand. dev Mean value Stand. dev

100

5 Conclusions In this article, we have presented the harmony search algorithm and nature-inspired BAT algorithm. Nature-inspired algorithms are based on metaheuristics. In musicinspired optimization (harmony search), we have learned the importance of music (harmony, pitch adjustment) for solving optimization problems. In nature-inspired optimization, we have learned various social behaviour which can be converted into an optimization algorithm (metaheuristics). Music-inspired-based Harmony search innovates a new optimization algorithm. Several case studies are presented to illustrate the working methodology of the harmony search algorithm and bat algorithm. In future, our work will be towards the usage of the harmony search algorithm in data science rather than data optimization.

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References 1. Spall, J.C.: Introduction to Stochastic Search and Optimization: Estimation. Wiley, Simulation and Control (2003) 2. Cui, Z., Sun, B., Wang, G., Xue, Y., Chen, J.: A novel oriented cuckoo search algorithm to improve DV-hop per-formance for cyber-physical systems. J. Parallel Distrib. Comput. 103, 42–52 (2016) 3. Yang, X.S., Deb, S.: Engineering optimization by cuckoo search. Int. J. Math. Model. Numer. Optim. 1, 4, 330–343 (2010) 4. Yang, X.-S.: A new metaheuristic bat-inspired algorithm. Nat. Insp. Cooper. Strat. Optim. 284, 65–74 (2010) 5. Geem, Z.W., Kim, J.H., Loganathan, G.V.: A New Heuristic Optimization Algorithm: Harmony Search. SIMULATION 76, 60–68 (2001) 6. Yang, X.S.: Nature-inspired Metaheuristic Algorithms. Luniver Press, Backington, UK (2008) 7. Yang, X.S., Gandomi, A.H.: Bat algorithm: a novel approach for global engineering optimization. Eng. Comput. 29, 464–483 (2012) 8. Omran, M., Mahdavi: Global-best harmony search. Appl. Math. Comput. 198, 643–656 (2008) 9. Geem, Z.W.: Music-Inspired Harmony Search Algorithm. Springer, Heidelberg, Germany (2009) 10. Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1:67–82 (1997) 11. Back, T., Fogel, D., Michalewicz Z.: Handbook of Evolutionary Computation. Oxford University Press (1997) 12. Dervis, K., Bahriye, B.: A powerful and efficient algorithm for numerical function optimization: Artificial Bee Colony (ABC) algorithm J. Glob. Optim. 39, 459–471 (2007)

On Mathematical Programs with Equilibrium Constraints Under Data Uncertainty Vivek Laha

and Lalita Pandey

Abstract A mathematical program with equilibrium constraints (MPEC), a particular type of mathematical program, is one in which the decision variables must fulfil a limited set of constraints in addition to an equilibrium condition. An optimization problem with equilibrium constraints that appear in many practical applications is difficult to solve as they do not satisfy the standard regularity conditions. Moreover, due to prediction or measurement mistakes, the input data for the objective function and the restrictions in real-world problems are imprecise or lacking. In this article, we take into MPECs in the phase of data uncertainty of the feasible region within the framework of robust optimization. For a weak stationary point to be a global or local minimizer of the ambiguous MPECs, we construct optimality criteria. Keywords Mathematical programs with equilibrium constraints · Data uncertainty · Robust optimization · Stationary point · Generalized convexity

1 Introduction Extensions of bilevel optimisation is known as mathematical program with equilibrium constraints (MPEC) which is widely used in many different fields (see, e.g. [7, 8, 31]). Flegel and Kanzow [12] used standard Fritz-John (FJ) conditions to obtain new stationary concepts for MPECs. For the MPECs, a constraint qualification (CQ) of the Abadie type was created in [13] and it’s relation with other CQs was analysed to derive M-stationarity conditions in [14, 44]. In the context of MPECs, strong stationarity has been shown to be an essential optimality criterion under Guignard CQ (GCQ) in [14]. Flegel et al. [15] applied optimality conditions for disjunctive programs in MPECs. Liu et al. [27] worked on a partial exact penalty for MPECs. Movahedian and Nobakhtian [36] studied nonsmooth MPECs. Guo and Lin [17] V. Laha (B) · L. Pandey Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_20

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investigated several CQs for MPECs. Guo et al. [19] derived second-order optimality conditions for MPECs. Duality results for MPECs were produced by Pandey and Mishra [39] along with Guo et al. [18]. A mathematical modeling technique called robust optimization (RO) is used to address issues with the uncertain objectives or the uncertain data in the feasible region (see, e.g. [1–4, 16]). Jeyakumar et al. [20] studied robust duality under data uncertainty. To find efficient robust solutions, Kuroiwa and Lee [22] constructed the optimality theorems. Strong duality minimax results were obtained by Jeyakumar et al. [21] under robustness. Lee and Kim [24] developed Wolfe duality results for a RO problem. Soleimanian and Jajaei [40] worked on robust nonlinear optimization with conic representable uncertainty sets. Chuong [6] dealt with robust multiobjective optimization problems with nonsmooth nonlinear data. Lee and Kim [25] provided optimality and duality results for robust non-smooth multiobjective optimization. Fakhar et al. [9] analysed robust portfolio optimization. Theorems for robust semiinfinite optimization were found by Lee and Lee [26]. Chen et al. [5] achieved results for situations involving robust non-smooth multiobjective optimization with restrictions. For nonsmooth robust multiobjective optimization problems, Fakhar et al. [10] researched approximative solutions. Wang et al. [42] used image space analysis to study general robust dual problems. The objective is to solve MPECs with uncertainty using RO. The outline is: we review definitions and outcomes from RO in Sect. 2. In Sect. 3, we deal with an optimization problem with mixed assumptions of deterministic inequality and equality constraints along with inequality constraints with uncertainties denoted by MUP. We create robust FJ criteria for the given problem and utilize them to derive robust Karush-Kuhn-Tucker (KKT) conditions under the premise that there is no nonzero abnormal multiplier CQ (NNAMCQ). Additionally, we find that the MUP has sufficient optimality criteria when convexity assumptions are made. The results of Sect. 3 are used in Sect. 4 to derive robust optimality criterion for the MPECs with mixed assumptions of certain constraints and uncertain constraints in the feasible region denoted by UMPEC. We define a suitable constraint qualification using a nonlinear program connected to the UMPEC, from which we derive necessary and sufficient weak stationary optimality requirements. The definitions and relationships of a number of other stationary conditions connected to the UMPEC, such as the C-, A-, M-, and S-stationary conditions, are discussed. Section 5 wraps up the findings of this paper and explores several potential directions for further research.

2 Preliminaries Take into account the following problem: min f(ζ ) s.t. gi (ζ ) ≤ 0, ∀i ∈ P := {1, . . . , p},

(P)

where f and gi (i ∈ P) are continuously differentiable real-valued functions on Rn . An altered version of (P) under uncertainty of the constraints is:

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min f(ζ ) s.t. gi (ζ, ui ) ≤ 0, ∀i ∈ P,

(UP)

where for any i ∈ P, a convex compact collection Ui in Rl contains the uncertain variable ui and the real-valued functions gi (i ∈ P) on Rn × Rl are continuously differentiable wrt the first component. The robust analogue of (UP) is stated as min f(ζ ) s.t. ζ ∈ F := {ζ ∈ Rn : gi (ζ, ui ) ≤ 0, ∀ui ∈ Ui , ∀i ∈ P}.

(RP)

Definition 1 A point ζ ∗ ∈ F is a global minimizer of RP iff f(ζ ) ≥ f(ζ ∗ ) for all ζ ∈ F. A point ζ ∗ ∈ F is a local minimizer of RP iff there exists  > 0 such that f(ζ ) ≥ f(ζ ∗ ) for all ζ ∈ F ∩ B (ζ ∗ ) with B (ζ ∗ ) := {ζ ∈ Rn : ζ − ζ ∗  < }. For any ζ ∗ ∈ F, the index set P may be decomposed into P1 (ζ ∗ ) and P2 (ζ ∗ ) with P = P1 (ζ ∗ ) ∪ P2 (ζ ∗ ) and P1 (ζ ∗ ) ∩ P2 (ζ ∗ ) = ∅, where P1 (ζ ∗ ) := {i ∈ P : ∃ui ∈ Ui s.t. gi (ζ ∗ , ui ) = 0} and P2 (ζ ∗ ) := P \ P1 (ζ ∗ ). For any i ∈ P1 (ζ ∗ ), define Ui0 := {ui ∈ Ui : gi (ζ ∗ , ui ) = 0}. The extended Mangasarian-Fromovitz CQ (EMFCQ) for the RP is defined as follows by Jeyakumar et al. [20]: Definition 2 We say ζ ∗ ∈ F satisfy EMFCQ iff ∃v ∈ Rn :

1 gi (ζ ∗ , ui )T v < 0, ∀ui ∈ Ui0 , ∀i ∈ P1 (ζ ∗ ),

(EMFCQ)

where the 1 gi represents the derivative of gi wrt the first variable. Jeyakumar et al. [20] established robust KKT conditions for the RP using the robust Gordan’s theorem and linearization. Theorem 1 (KKT condition for RP) Let ζ ∗ ∈ F be a local minimizer of RP and n let gi (ζ, .) be concave on Ui for each  pζ ∈ gR and for each i ∈ P. Then, there exist g 0 ≥ 0, i ≥ 0(i ∈ P) with 0 + i=1 i = 1 and ui ∈ Ui (i ∈ P) such that ∗

0 f(ζ ) +

p 

g

i 1 gi (ζ ∗ , ui ) = 0,

i=1 g

i gi (ζ ∗ , ui ) = 0, ∀i ∈ P. Additionally, if we believe that EMFCQ is valid at ζ ∗ , then 0 > 0.

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3 Optimality Condition Involving Equality Constraints Consider a nonlinear programming problem with the mixed assumption of deterministic inequality and equality constraints along with some uncertain inequality constraints as follows: ⎧ ⎪ min f(ζ ) ⎪ ⎪ ⎪ ⎪ ⎪ s. ⎨ t. (MUP) gi (ζ, ui ) ≤ 0, ∀i ∈ P, ⎪ ⎪ ⎪ ∀i ∈ Q := {1, . . . , q} φi (ζ ) ≤ 0, ⎪ ⎪ ⎪ ⎩h (ζ ) = 0, ∀i ∈ R := {1, . . . , r }, i where φi and hi are continuously differentiable real-valued functions on Rn in addition to the components discussed for RP. The robust alternative to MUP is provided by (MRP) min f(ζ ) s. t. ζ ∈ F2 , where

F2 := {ζ ∈ Rn : gi (ζ, ui ) ≤ 0, ∀ui ∈ Ui , ∀i ∈ P, φi (ζ ) ≤ 0, ∀i ∈ Q, hi (ζ ) = 0, ∀i ∈ R}.

For any ζ ∗ ∈ F2 , define P1 (ζ ∗ ) := {i ∈ P : ∃ui ∈ Ui s. t. gi (ζ ∗ , ui ) = 0}, P2 := P \ P1 (ζ ∗ ), Q1 (ζ ∗ ) := {i ∈ Q : φi (ζ ∗ ) = 0}, Q2 := Q \ Q1 (ζ ∗ ), and Ui0 = {ui ∈ Ui : gi (ζ ∗ , ui ) = 0}, ∀i ∈ P1 (ζ ∗ ). The FJ condition by Mangasarian and Fromovitz [32] and by Lee and Son [29, Theorem 2.4] lead to the following FJ condition for MRP. Theorem 2 (FJ condition for MRP) Let ζ ∗ ∈ F2 be a local minimizer of MRP and let gi (ζ, ·) be concave on Ui for every ζ ∈ Rn and for every i ∈ P. Then, there φ g h exists 0 ≥ 0, i ≥ 0 (i ∈ P), i ≥ 0 (i ∈ Q), i ∈ R (i ∈ R), not all zero, and ui ∈ Ui (i ∈ P) such that 0 f(ζ ∗ ) +

p 

g

i 1 gi (ζ ∗ , ui ) +

i=1 g ∗ i gi (ζ , ui ) = 0, φ i φi (ζ ∗ ) = 0, ∀i

q  i=1

∀i ∈ P, ∈ Q.

φ

i φi (ζ ∗ ) +

r  i=1

h

i hi (ζ ∗ ) = 0,

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Proof Define Gi : F2 → R by Gi (ζ ) := supui ∈Ui gi (ζ, ui ) for every i ∈ P. Then, (MRP) may be re-written as ⎧ ⎪ ⎪ ⎪min f(ζ ) ⎪ ⎪ ⎪ ⎨s.t. Gi (ζ ) ≤ 0, ∀i ∈ P, ⎪ ⎪ ⎪φi (ζ ) ≤ 0, ∀i ∈ Q, ⎪ ⎪ ⎪ ⎩h (ζ ) = 0, ∀i ∈ R. i

(MRP2)

Since ζ ∗ ∈ F2 is a local minimizer of MRP, therefore ζ ∗ also minimizes MRP2 g locally. Hence, by the generalized FJ condition [32], there exist 0 ≥ 0, i ≥ 0 (i ∈ φ h P), i ≥ 0 (i ∈ Q), i ∈ R (i ∈ R), not all zero, such that 0 f(ζ ∗ ) +



g

i Gi (ζ ∗ ) +

i∈P



φ

i φi (ζ ∗ ) +

i∈Q



h

i hi (ζ ∗ ) = 0,

i∈R

g

i Gi (ζ ∗ ) = 0, ∀i ∈ P, φ

i φi (ζ ∗ ) = 0, ∀i ∈ Q. By [29, Theorem 2.4], one has   Gi (ζ ∗ ) = ∪ui ∈Ui (ζ ∗ ) 1 gi (ζ ∗ , ui ) , where Ui (ζ ∗ ) := {ui ∈ Ui : gi (ζ ∗ , ui ) = Gi (ζ ∗ )} for every i ∈ P, which implies that, there exist ui ∈ Ui (ζ ∗ ) (i ∈ P) such that 0 f(ζ ∗ ) +



g

i 1 gi (ζ ∗ , ui ) +

i∈P

 i∈Q

φ

i φi (ζ ∗ ) +



h

i hi (ζ ∗ ) = 0,

i∈R

g

i gi (ζ ∗ , ui ) = 0, ∀i ∈ P, φ

i φi (ζ ∗ ) = 0, ∀i ∈ Q. The evidence is now complete.



By the FJ condition for MRP, if 0 is never zero, then it can be taken as 1 which will lead to KKT-type robust optimality condition. We introduce the following extended NNAMCQ (ENNAMCQ) for MUP which will solve the purpose based on the NNAMCQ introduced by Ye [43].

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Definition 3 The MRP satisfies the ENNAMCQ at ζ ∗ ∈ F2 iff for any ui ∈ Ui (i ∈ P), one has ⎧ p q r φ g h ∗ ∗ ∗ ⎪   g (ζ , u ) +  φ (ζ ) + 1 i i i ⎪ i i=1 i=1 i=1 i hi (ζ ) = 0, i ⎪ ⎪ g ∗ ⎪ ⎪ ⎨i ≥ 0, i ∈ P1 (ζ ), φ i ≥ 0, i ∈ Q1 (ζ ∗ ) ⎪ ⎪ ⎪implies that ⎪ ⎪ ⎪ ⎩g = 0, ∀i ∈ P (ζ ∗ ), φ = 0, ∀i ∈ Q (ζ ∗ ), h = 0, ∀i ∈ R. 1 1 i i i (ENNAMCQ) In the view of the above ENNAMCQ and FJ conditions for MRP, the KKT criterion for MRP is as follows: Theorem 3 (KKT condition for MRP) Let ζ ∗ ∈ F2 be a local minimizer of MRP and let gi (ζ, ·) be concave on Ui for every ζ ∈ Rn and for every i ∈ P. If ENNAMCQ φ g is satisfied at ζ ∗ , then there exist ui ∈ Ui (i ∈ P), i ≥ 0 (i ∈ P), i ≥ 0 (i ∈ Q), h and i ∈ R (i ∈ R) such that f(ζ ) +

p 

g

i 1 gi (ζ ∗ , ui ) +

i=1 g ∗ i gi (ζ , ui ) = φ i φi (ζ ∗ ) = 0,

q 

φ

i φi (ζ ∗ ) +

i=1

r 

h

i hi (ζ ∗ ) = 0,

i=1

0, ∀i ∈ P, ∀i ∈ Q.

Proof According to the theorem’s premise, there will be scalars that satisfy the FJ condition of Theorem 2 at ζ ∗ ∈ F2 . Since ENNAMCQ is satisfied at ζ ∗ , for φ g g h 0 = 0, we will have i = 0 (i ∈ P), i = 0 (i ∈ Q), i = 0 (i ∈ R) which is inconsistent with the observation that all scalar multipliers are not zero. Hence, g  0 > 0 and this gives the required KKT condition at ζ ∗ . We specify the upcoming index sets that will be utilized to derive sufficient optimality condition for MU P : g

φ

P + := {i ∈ P : i > 0}, Q+ := {i ∈ Q : i > 0}, h

h

R+ := {i ∈ R : i > 0}, R− := {i ∈ R : i < 0}. Theorem 4 (Condition for robust sufficient optimality in MUP) Assume that for φ g h some ui ∈ Ui (i ∈ P), i ≥ 0 (i ∈ P), i ≥ 0 (i ∈ Q) and i ∈ R (i ∈ R) the ∗ KKT condition of Theorem 3 for MRP is fulfilled at ζ . If the functions f, gi (., ui )(i ∈ P + ), φi (i ∈ Q+ ), hi (i ∈ R+ ), −hi (i ∈ R− ) are convex at ζ ∗ ∈ F2 , then ζ ∗ is a global minimizer of MRP. Proof Let ζ ∈ F2 . Since f is convex, therefore f(ζ ) − f(ζ ∗ ) ≥ f(ζ ∗ )T (ζ − ζ ∗ ), ∀ζ ∈ F2 ,

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which implies that f(ζ ) − f(ζ ∗ )

p T q r  g   φ h ∗ ∗ ∗ ≥− i 1 gi (ζ , ui ) + i φi (ζ ) + i hi (ζ ) (ζ − ζ ∗ ), i=1

i=1

i=1

∀ζ ∈ F2 . By convexity of gi (., ui ) at (ζ ∗ , ui ), it follows that 1 gi (ζ ∗ , ui )T (ζ − ζ ∗ ) ≤ gi (ζ, ui ) − gi (ζ ∗ , ui ), ∀ζ ∈ F2 , ∀i ∈ P + . Since gi (ζ ∗ , ui ) = 0 (i ∈ P + ), therefore 1 gi (ζ ∗ , ui )T (ζ − ζ ∗ ) ≤ 0, ∀ζ ∈ F2 , ∀i ∈ P + .

(1)

φi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0, ∀ζ ∈ F2 , ∀i ∈ Q+ , hi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0, ∀ζ ∈ F2 , ∀i ∈ R+ , hi (ζ ∗ )T (ζ − ζ ∗ ) ≥ 0, ∀ζ ∈ F2 , ∀i ∈ R− .

(2) (3) (4)

Similarly

φ

g

h

Multiplying (1)–(4) by i > 0 (i ∈ P + ), i > 0 (i ∈ Q+ ), i > 0(i ∈ R+ ), h and i < 0 (i ∈ R− ), respectively, and adding, we get ⎞T ⎛ p q r    φ g h ∗ ∗ ∗ ⎝ i 1 gi (ζ , ui ) + i φi (ζ ) + i hi (ζ )⎠ (ζ − ζ ∗ ) ≤ 0, , ∀ζ ∈ F2 , i=1

i=1

i=1

therefore, we get f(ζ ) ≥ f(ζ ∗ ) for all ζ ∈ F2 .



The example below demonstrates the aforementioned outcome. Example 1 Consider an optimization problem with mixed assumptions of equality and inequality constraints with uncertainty in the feasible region as follows: min

ζ :=(ζ1 ,ζ2 ,ζ3 )∈R3

f(ζ ) := ζ12 − cos ζ2

s. t. g(ζ, v) := vζ12 + ζ22 + ζ32 − 1 ≤ 0, φ(ζ ) := ζ12 + ζ2 ζ3 ≤ 0, and h(ζ ) := ζ1 + ζ2 + ζ3 − 1 = 0, where v ∈ [0.5, 1]. The associated robust counterpart is

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min

ζ :=(ζ1 ,ζ2 ,ζ3 )∈R3

f(ζ ) := ζ12 − cos ζ2

s. t. g(ζ, v) := vζ12 + ζ22 + ζ32 − 1 ≤ 0, ∀v ∈ [0.5, 1], φ(ζ ) := ζ12 + ζ2 ζ3 ≤ 0, and h(ζ ) := ζ1 + ζ2 + ζ3 − 1 = 0. Now, suppose that for some ζ ∈ R3 , g ≥ 0, φ ≥ 0 and h ∈ R, one has g 1 g(ζ, v) + φ φ(ζ ) + h h(ζ ) = 0, which implies that

2vζ1 g + 2ζ1 φ + h = 0; 2ζ2 g + ζ3 φ + h = 0; 2ζ3 g + ζ2 φ + h = 0;

giving homogeneous linear equations in g , φ and h . The system will have an unique trivial solution g = φ = h = 0 iff the determinant   2vζ1 2ζ1 1    2ζ2 ζ3 1 = 0,    2ζ3 ζ2 1 which is possible iff ζ2 = ζ3 and (2 + v)ζ1 − ζ2 − ζ3 = 0. It is evident that ζ ∗ = (0, 1, 0) and ζ¯ = (0, 0, 1) are two feasible points with all constraints as active and which satisfies the above non-zero determinant condition. Hence, ENNAMCQ holds at both the points. Now f(ζ ∗ ) + g 1 g(ζ ∗ , v) + φ φ(ζ ∗ ) + h h(ζ ∗ ) = 0, gives g = −0.5 sin 1, φ = 0, h = 0, which implies that the situations of Theorem 3 are not satisfied at ζ ∗ . Now f(ζ¯ ) + g 1 g(ζ¯ , v) + g φ(ζ¯ ) + h h(ζ¯ ) = 0, gives g = φ = h = 0, and hence the robust KKT situations of Theorem 3 are fulfilled at ζ¯ . Since g = φ = h = 0, therefore the sets P + , Q+ , R+ and R− are empty, and hence to verify the sufficient optimality conditions of Theorem 4, we have to verify only the convexity of f at ζ¯ over the feasible region. Indeed, f is convex at ζ¯ , because its Hessian at ζ¯ is positive semi-definite. Thus, Theorem 4 approves that ζ¯ is a global minimizer of the robust problem.

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4 Application to Robust MPECs Consider a MPEC involving uncertainty in the feasible region: ⎧ min f(ζ ) ⎪ ⎪ ⎪ ⎪ ⎪ s. t. ⎪ ⎪ ⎪ ⎨g (ζ, u ) ≤ 0, i i ⎪ (ζ ) ≤ 0, φ i ⎪ ⎪ ⎪ ⎪ ⎪hi (ζ ) = 0, ⎪ ⎪ ⎩ Gi (ζ ) ≥ 0, Hi (ζ ) ≥ 0, Gi (ζ )Hi (ζ ) = 0,

∀i ∀i ∀i ∀i

∈ P, ∈ Q, ∈ R, ∈ S := {1, . . . , s},

(UMPEC)

where in addition to the symbols defined for MUP, we have Gi , Hi : Rn → R are continuously differentiable functions. The UMPEC’s robust equivalent is: min f(ζ ) s. t. ζ ∈ Ω,

(RMPEC)

where Ω := {ζ ∈ Rn : gi (ζ, ui ) ≤ 0, ∀ui ∈ Ui , ∀ i ∈ P, φi (ζ ) ≤ 0, ∀i ∈ Q, hi (ζ ) = 0, Gi (ζ ) ≥ 0,

∀i ∈ R, ∀i ∈ S,

Hi (ζ ) ≥ 0, ∀i ∈ S, Gi (ζ )Hi (ζ ) = 0, ∀i ∈ S}. . Definition 4 A point ζ ∗ ∈ Ω is a robust global minimizer of U M P EC iff ζ ∗ is a global minimizer of RMPEC, that is, f(ζ ) ≥ f(ζ ∗ ) for all ζ ∈ Ω. A point ζ ∗ is a robust local minimizer of U M P EC iff there exist  > 0 such that f(ζ ) ≥ f(ζ ∗ ) with ζ ∈ Ω ∩ B(ζ ∗ , ). In addition to the index sets defined earlier, we need the following index sets related to the equilibrium constraints P1 (ζ ∗ ) := {i ∈ P : ∃ui ∈ Ui , s. t. gi (ζ ∗ , ui ) = 0}, Q1 (ζ ∗ ) := {i ∈ Q : φi (ζ ∗ ) = 0}, I0+ (ζ ∗ ) := {i ∈ S : Gi (ζ ∗ ) = 0, Hi (ζ ∗ ) > 0}, I00 (ζ ∗ ) := {i ∈ S : Gi (ζ ∗ ) = 0, Hi (ζ ∗ ) = 0}, I+0 (ζ ∗ ) := {i ∈ S : Gi (ζ ∗ ) > 0, Hi (ζ ∗ ) = 0}, I++ (ζ ∗ ) := {i ∈ S : Gi (ζ ∗ ) > 0, Hi (ζ ∗ ) > 0}, Ui0 := {ui ∈ Ui : gi (ζ ∗ , ui ) = 0}, ∀i ∈ P1 (ζ ∗ ).

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The following tightened robust program depending upon ζ ∗ ∈ Ω, denoted by RTNLP(ζ ∗ ), will be useful to define a suitable CQ for RMPEC. ⎧ min f(ζ ) ⎪ ⎪ ⎪ ⎪ ⎪ s. t. ⎪ ⎪ ⎪ ⎪ ⎪ gi (ζ, ui ) ≤ 0, ⎪ ⎪ ⎪ ⎪ ⎪ φ ⎨ i (ζ ) ≤ 0, hi (ζ ) = 0, ⎪ ⎪ ⎪ G ⎪ i (ζ ) = 0, ⎪ ⎪ ⎪ ⎪ Gi (ζ ) ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ Hi (ζ ) = 0, ⎪ ⎪ ⎩ Hi (ζ ) ≥ 0,

∀ui ∈ Ui , ∀i ∈ P, ∀i ∈ Q, ∀i ∈ R, ∀i ∈ I0+ ∪ I00 , ∀i ∈ I+0 , ∀i ∈ I+0 ∪ I00 , ∀i ∈ I0+ .

(RTNLP(ζ ∗ ))

Since the feasible set of RTNLP(ζ ∗ ) is a subset of the feasible set of RMPEC. Therefore, if ζ ∗ is a local minimizer of RMPEC, then ζ ∗ is also a local minimizer of corresponding RTNLP(ζ ∗ ) and RTNLP(ζ ∗ ) may be used to define a suitable variant of ENNAMCQ for RMPEC. Definition 5 The R M P EC satisfies RMPEC-ENNAMCQ at ζ ∗ iff RTNLP(ζ ∗ ) satisfies the ENNAMCQ at ζ ∗ , that is, for any ui ∈ Ui (i ∈ P), one has ⎧  q r p φ g h ∗ ∗ ∗ ⎪ ⎪ i=1 i 1 gi (ζ , ui ) +  i=1 i φi (ζ ) + i=1 i hi (ζ ) ⎪ ⎪  ⎪ G G ∗ ∗ ⎪ − i∈I+0 ςi Gi (ζ ) − i∈I0+ ∪I00 ςi Gi (ζ ) ⎪ ⎪ ⎪   ⎪ ⎪ − i∈I0+ ςiH Hi (ζ ∗ ) − i∈I+0 ∪I00 ςiH Hi (ζ ∗ ) = 0, ⎪ ⎪ ⎪ φ ⎨ g i ≥ 0 (i ∈ P1 (ζ ∗ )), i ≥ 0 (i ∈ Q1 (ζ ∗ )), ςiG ≥ 0 (i ∈ I+0 ), ςiH ≥ 0, (i ∈ I0+ ) ⎪ =⇒ ⎪ ⎪ ⎪ ⎪ φ g h ⎪ ⎪ i = 0 (i ∈ P1 (ζ ∗ )), i = 0 (i ∈ Q1 (ζ ∗ )), i = 0 (i ∈ R), ⎪ ⎪ ⎪ ⎪ ςiG = 0 (i ∈ I+0 ), ςiG = 0 (i ∈ I0+ ∪ I00 ), ⎪ ⎪ ⎪ ⎩ ς H = 0 (i ∈ I ), ς H = 0 (i ∈ I ∪ I ). i

0+

i

+0

00

(RMPEC-ENNAMCQ)

We can now establish the KKT criteria for RMPEC. Theorem 5 (KKT condition for RMPEC) Let ζ ∗ ∈ Ω be a local minimizer of RMPEC and let gi (ζ, ·) be concave on Ui for every ζ ∈ Rn and for every i ∈ P. If RMPEC-ENNAMCQ holds at ζ ∗ ∈ Ω, then we can find (g , β, γ , ς G , ς H ) ∈ R p+q+r +2s and ui ∈ Ui (i ∈ P) such that

On Mathematical Programs with Equilibrium Constraints Under Data Uncertainty

⎧ p q  φ g h f(ζ ∗ ) + i=1 i 1 gi (ζ ∗ , ui ) + i=1 i φi (ζ ∗ ) + ri=1 i hi (ζ ∗ ) ⎪ ⎪ ⎪ s ⎪ ⎪ − i=1 [ςiG Gi (ζ ∗ ) + ςiH Hi (ζ ∗ )] = 0, ⎪ ⎪ ⎪ ⎨ g ≥ 0, g g (ζ ∗ , u ) = 0, ∀i ∈ P, i i i i ⎪ iφ ≥ 0, iφ φi (ζ ∗ ) = 0, ∀i ∈ Q, ⎪ ⎪ ⎪ ⎪ ⎪ ςiG ∈ R, ∀i ∈ I0+ ∪ I00 , ςiH ∈ R, ∀i ∈ I+0 ∪ I00 , ⎪ ⎪ ⎩ G ςi = 0, ∀i ∈ I+0 , ςiH = 0, ∀i ∈ I0+ .

293

(5)

Proof Since ζ ∗ ∈ Ω minimizes the RMPEC locally, therefore ζ ∗ also minimizes the RTNLP(ζ ∗ ) locally. Also, since RMPEC-ENNAMCQ holds at ζ ∗ , therefore ENNAMCQ also holds at ζ ∗ for the RTNLP(ζ ∗ ). Then, by Theorem 3, we have the required result.  Based on the above theorem and following the notion of stationary points for MPEC (see, e.g. [44]), we may now define weak stationary points for the RMPEC. Definition 6 (W-stationary point of the RMPEC) Any ζ ∗ ∈ Ω is W-stationary for the RMPEC iff we can find (, β, γ , ς G , ς H ) ∈ R p+q+r +2s and ui ∈ Ui (i ∈ P) satisfying (5). The sequel will use following indices depending on ζ ∗ ∈ Ω: g

P + := {i ∈ P(ζ ∗ ) : i > 0}, h

φ

Q+ := {i ∈ Q(ζ ∗ ) : i > 0}, h

R+ := {i ∈ R : i > 0},

R− := {i ∈ R : i < 0}.

+ := {i ∈ I00 : ςiG > 0, ςiH > 0}, I00

− := {i ∈ I00 : ςiG < 0, ςiH < 0}, I00

+ := {i ∈ I0+ : ςiG > 0}, I0+

− := {i ∈ I0+ : ςiG < 0}, I0+

+ I+0 := {i ∈ I+0 : ςiH > 0},

− I+0 := {i ∈ I+0 : ςiH < 0}.

Theorem 6 (Sufficiency of W-stationarity for RMPEC) Let ζ ∗ ∈ Ω be a W-stationary point. If the functions f, gi (·, ui )(i ∈ P + ), φi (i ∈ Q+ ), hi (i ∈ + + − − + ∪ I00 ), Gi (i ∈ I0+ ∪ I00 ), −Hi (i ∈ I+0 ∪ R+ ), −hi (i ∈ R− ), −Gi (i ∈ I0+ + − − ∗ I00 ), Hi (i ∈ I+0 ∪ I00 ) are convex at ζ over Ω, then − − − (a) ζ ∗ minimizes the RMPEC globally when I0+ ∪ I00 ∪ I+0 = ∅; − ∗ (b) ζ minimizes the RMPEC locally when I00 = ∅; (c) ζ ∗ minimizes the RMPEC locally when ζ ∗ happens to be an interior point wrt Ω ∩ {ζ : Gi (ζ ) = 0, Hi (ζ ) = 0} .

Proof (a) By the convexity of f at ζ ∗ over Ω, we have f(ζ ) − f(ζ ∗ ) ≥ f(ζ ∗ )T (ζ − ζ ∗ ), ∀ζ ∈ Ω. Since ζ ∗ is a W-stationary point of the RMPEC, therefore

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f(ζ ) − f(ζ ∗ )

p T q r  g   φ h ≥− i 1 gi (ζ ∗ , ui ) + i φi (ζ ∗ ) + i hi (ζ ∗ ) (ζ − ζ ∗ ) i=1

i=1

i=1

s T  + [ςiG Gi (ζ ∗ ) + ςiH Hi (ζ ∗ )] (ζ − ζ ∗ ), i=1

∀ζ ∈ Ω. (6)

By the convexity of gi (., ui ) at (ζ ∗ , ui ) for every i ∈ P + , one has 1 gi (ζ ∗ , ui )T (ζ − ζ ∗ ) ≤ gi (ζ, ui ) − gi (ζ ∗ , ui ), ∀ζ ∈ Ω ∀i ∈ P + . Since gi (ζ ∗ , ui ) = 0 (i ∈ P + ), therefore 1 gi (ζ ∗ , ui )T (ζ − ζ ∗ ) ≤ 0, ∀ζ ∈ Ω ∀i ∈ P + .

(7)

Similarly φi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0, hi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0,

∀ζ ∈ Ω ∀i ∈ Q+ , ∀ζ ∈ Ω, ∀i ∈ R+ ,

(8) (9)

hi (ζ ∗ )T (ζ − ζ ∗ ) ≥ 0, ∀ζ ∈ Ω, ∀i ∈ R− , + + ∪ I00 , Gi (ζ ∗ )T (ζ − ζ ∗ ) ≥ 0, ∀ζ ∈ Ω, ∀i ∈ I0+

(10) (11)

+ + ∪ I00 . Hi (ζ ∗ )T (ζ − ζ ∗ ) ≥ 0, ∀ζ ∈ Ω ∀i ∈ I+0

(12)

φ

g

h

Multiplying (7)–(12) by i > 0 (i ∈ P + ), i > 0 (i ∈ Q+ ), i > 0(i ∈ h + + ∪ I00 ), and ςiH > 0 (i ∈ R+ ), −i > 0 (i ∈ R− ), ςiG > 0 (i ∈ I0+ + + I+0 ∪ I00 ), respectively, and summing, we obtain

p 

g i 1 gi (ζ ∗ , ui )

i=1

+

q  i=1

φ i φi (ζ ∗ )

+

r 

T h i hi (ζ ∗ )

(ζ − ζ ∗ )

i=1

s T  G ∗ H ∗ − [ςi Gi (ζ ) + ςi Hi (ζ )] (ζ − ζ ∗ ) ≤ 0, ∀ζ ∈ Ω, i=1

which implies from (5) that f(ζ ) ≥ f(ζ ∗ ) for all ζ ∈ Ω. − , then Hi (ζ ∗ ) > 0 and Hi (ζ ) > 0 for ζ close enough to ζ ∗ , which (b) If i ∈ I0+ implies by the complementarity constraints that Gi (ζ ) = 0 for every ζ close − . Hence, for every ζ close enough to ζ ∗ , by enough to ζ ∗ and for every i ∈ I0+ − the convexity of Gi (i ∈ I0+ ), one has − Gi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0, ∀i ∈ I0+ .

(13)

On Mathematical Programs with Equilibrium Constraints Under Data Uncertainty

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− Similarly, for every ζ close enough to ζ ∗ , by the convexity of Hi (i ∈ I+0 ), one has − . (14) Hi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0, ∀i ∈ I+0 φ

g

h

Multiplying (7)–(14) by i > 0(i ∈ P + ), i > 0(i ∈ Q+ ), i > 0(i ∈ h + + + + ∪ I00 ), ςiH > 0(i ∈ I+0 ∪ I00 ), R+ ), −i > 0(i ∈ R− ), ςiG > 0(i ∈ I0+ − − H −ςiG > 0(i ∈ I0+ ) and −ςi > 0(i ∈ I+0 ), respectively, we get

p 

g i 1 gi (ζ ∗ , ui )

+

i=1

q 

φ i φi (ζ ∗ )

+

i=1

r 

T h i hi (ζ ∗ )

(ζ − ζ ∗ )

i=1

s T  G ∗ H ∗ − [ςi Gi (ζ ) + ςi Hi (ζ )] (ζ − ζ ∗ ) ≤ 0, ∀ζ ∈ Ω, i=1

which implies from (5) that f(ζ ) ≥ f(ζ ∗ ) for every ζ close enough to ζ ∗ . Hence, ζ ∗ is a local minimizer of the RMPEC. (c) Since ζ ∗ is an interior point wrt Ω ∩ {ζ : Gi (ζ ) = 0, Hi (ζ ) = 0} , for every ζ − − ) and Hi (i ∈ I00 ), one has close enough to ζ ∗ , by the convexity of Gi (i ∈ I00

and

− , Gi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0, ∀i ∈ I00

(15)

− . Hi (ζ ∗ )T (ζ − ζ ∗ ) ≤ 0, ∀i ∈ I00

(16)

φ

g

h

Multiplying (7)–(16) by i > 0 (i ∈ P + ), i > 0 (i ∈ Q+ ), i > 0(i ∈ h + + + + ∪ I00 ), ςiH > 0 (i ∈ I+0 ∪ I00 ), R+ ), −i > 0 (i ∈ R− ), ςiG > 0 (i ∈ I0+ − − − ), −ςiH > 0(i ∈ I+0 ), −ςiG > 0(i ∈ I00 ) and −ςiH > 0(i ∈ −ςiG > 0(i ∈ I0+ − ), respectively, and adding, we get I00

p 

g i 1 gi (ζ ∗ , ui )

i=1

+

q  i=1

φ i φi (ζ ∗ )

+

r 

T h i hi (ζ ∗ )

(ζ − ζ ∗ )

i=1

s T  − [ςiG Gi (ζ ∗ ) + ςiH Hi (ζ ∗ )] (ζ − ζ ∗ ) ≤ 0, ∀ζ ∈ Ω. i=1

which implies from (5) that f(ζ ) ≥ f(ζ ∗ ) for every ζ close enough to ζ ∗ . Hence, ζ ∗ is a local minimizer of the RMPEC. This concludes the proof. The results above are illustrated by the example below.

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Example 2 Consider a MPEC with uncertainty in the feasible region as follows: min

ζ :=(ζ1 ,ζ2 ,ζ3 )∈R3

f(ζ ) := eζ1 ζ3 + ζ22

s. t. g1 (ζ, v) := vζ12 + ζ22 − 1 ≤ 0, φ1 (ζ ) := ζ12 + ζ2 − 1 ≤ 0, and G1 (ζ ) = ζ1 ≥ 0, H1 (ζ ) = ζ3 ≥ 0, G1 (ζ )H1 (ζ ) = 0, where v ∈ [0.5, 1]. The associated robust counterpart is min

ζ :=(ζ1 ,ζ2 ,ζ3 )∈R3

f(ζ ) := eζ1 ζ3 + ζ22

s. t. g1 (ζ, v) := vζ12 + ζ22 − 1 ≤ 0, ∀v ∈ [0.5, 1], φ1 (ζ ) := ζ12 + ζ2 − 1 ≤ 0, and G1 (ζ ) = ζ1 ≥ 0, H1 (ζ ) = ζ3 ≥ 0, G1 (ζ )H1 (ζ ) = 0. Now, consider a point ζ ∗ := (1, 0, 0) in the feasible region where the constraints φ g g1 (., 1), φ1 and H1 are active with I+0 (ζ ∗ ) = {1} . Now, for any 1 ≥ 0, 1 ≥ 0 and H ς1 ≥ 0 with φ g 1 1 g1 (ζ ∗ , 1) + 1 φ1 (ζ ∗ ) − ς1H H1 (ζ ∗ ) = 0, g

φ

one has 1 = 1 = ς1H = 0, and hence RMPEC-ENNAMCQ is satisfied at ζ ∗ . Now φ

g

f(ζ ∗ ) + 1 1 g1 (ζ ∗ , v) + 1 φ1 (ζ ∗ ) − ς1H H1 (ζ ∗ ) = 0, g

φ

gives the existence of 1 = 0, ∀v ∈ [0.5, 1], 1 = 0, ς1H = 1, which implies that + (ζ ∗ ) is the robust KKT conditions are fulfilled from Theorem 5 at ζ ∗ . Since I+0 nonempty, therefore to verify the sufficient optimality conditions of Theorem 6, we have to verify the convexity of f and H1 at ζ ∗ . Indeed, f and −H1 are both convex at ζ ∗ over the feasible region, which implies by Theorem 6 that ζ ∗ is a robust global minimizer. Similarly, we can show that ζ¯ := (0, 0, 1) is also a robust global minimizer of the uncertain problem. In fact, any point of the type (t, 0, 0) or (0, 0, t) with t ∈ [0, 1] minimizes the RMPEC globally. Further, if we add another equality constraint like ζ1 + ζ2 + ζ3 = 0, then the origin will be an unique local minimizer. We can now define some other stationary points for the RMPEC as follows: Definition 7 (Different stationary concepts for the RMPEC) If there exist (g , φ , h , ς G , ς H ) ∈ R p+q+r +2s and ui ∈ Ui (i ∈ P) which satisfy (5) at ζ ∗ ∈ Ω and (a) ςiG ςiH ≥ 0 for every i ∈ I00 , then C-stationarity holds at ζ ∗ ; (b) either ςiG ≥ 0 or ςiH ≥ 0 for any i ∈ I00 , then A-stationarity holds at ζ ∗ ;

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Fig. 1 Relation among different stationary points for RMPEC

(c) either ςiG > 0, ςiH > 0 or ςiG ςiH = 0 for any i ∈ I00 , then M-stationarity holds at ζ ∗ ; (d) ςiG ≥ 0, ςiH ≥ 0 for any i ∈ I00 , then S-stationarity holds at ζ ∗ . Remark 1 If the uncertainty set Ui is a singleton set for every i ∈ P, then the stationary concepts given in Definitions 6 and 7 reduce to the corresponding stationary concepts for MPEC (see, e.g. [44]). We can derive sufficient optimality conditions for different stationary concepts for RMPEC similar to Theorem 6. Moreover, FJ and KKT type C-, A-, M- and S-stationary conditions can be derived using the approaches of Scheel and Scholtes [41], Flegel and Kanzow [11], Ye [44] along with Theorems 3 and 5. The relation among different stationary points are given in Fig. 1.

5 Conclusion We studied a nonlinear problem with mixed assumptions of deterministic inequality and equality constraints along with uncertain inequality constraints which is termed as MUP. We have derived FJ condition for the MUP under data uncertainty assumptions in the feasible region. We have introduced a variant of the NNAMCQ for the MUP and used it to find the KKT condition for the MUP. The sufficiency of the results are confirmed under convexity hypothesis. The results obtained for MUP has been used to study a MPEC with data uncertainty in the feasible region called RMPEC. An FJ condition has been developed for the RMPEC, and a robust tailored nonlinear programming problem RTNLP has been used to develop suitable NNAMCQ for the UMPEC, denoted by RMPECENNAMCQ. Finally, the KKT condition for the UMPEC were established by uti-

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lizing the assumptions of RMPEC-ENNAMCQ from the FJ type robust necessary conditions for the UMPEC. We also provide the sufficiency of the results for the UMPEC under convexity hypothesis. The findings may be used to develop robust optimality conditions by involving uncertainty in the equality constraints as well which will be a matter of future research work. Additionally, some other generalised convexity [23, 33–35] might be investigated to determine its sufficiency. Moreover, the results can be extended involving vanishing constraints with data uncertainty both for the smooth [28, 37, 38] and the nonsmooth [30] cases.

References 1. Ben-Tal, A., Nemirovski, A.: Robust optimization methodology and applications. Math. Program. 92, 453–480 (2002) 2. Beyer, H.G., Sendhoff, B.: Robust optimization-a comprehensive survey. Comput. Methods Appl. Mech. Eng. 196, 3190–3218 (2007) 3. Ben-Tal, A., El-Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press (2009) 4. Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53, 464–501 (2011) 5. Chen, J., Köbis, E., Yao, J.C.: Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 181, 411–436 (2019) 6. Chuong, T.D.: Optimality and duality for robust multiobjective optimization problems. Nonlinear Anal. 134, 127–143 (2016) 7. Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153, 235–256 (2007) 8. Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3), 333–359 (2003) 9. Fakhar, M., Mahyarinia, M.R., Zafarani, J.: On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization. Eur. J. Oper. Res. 265, 39–48 (2017) 10. Fakhar, M., Mahyarinia, M.R., Zafarani, J.: On approximate solutions for nonsmooth robust multiobjective optimization problems. Optimization 68(9), 1653–1683 (2019) 11. Flegel, M.L., Kanzow, C.: Optimality conditions for mathematical programs with equilibrium constraints: Fritz John and Abadie-type approaches. Institute of Applied Mathematics and Statistics (2002) 12. Flegel, M.L., Kanzow, C.: A Fritz John approach to first order optimality conditions for mathematical programs with equilibrium constraints. Optimization 52, 277–286 (2003) 13. Flegel, M.L., Kanzow, C.: Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124, 595–614 (2005) 14. Flegel, M.L., Kanzow, C.: On M-stationary points for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 310(1), 286–302 (2005) 15. Flegel, M.L., Kanzow, C., Outrata, J.V.: Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 15, 139– 162 (2007) 16. Gabrel, V., Murat, C., Thiele, A.: Recent advances in robust optimization: an overview. Eur. J. Oper. Res. 235, 471–483 (2014) 17. Guo, L., Lin, G.H.: Notes on some constraint qualifications for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 156, 600–616 (2013)

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18. Guo, L., Lin, G.H., Zhao, J.: Wolfe-type duality for mathematical programs with equilibrium constraints. Acta Math. Appl. Sin. Engl. Ser. 35, 532–540 (2019) 19. Guu, S.M., Mishra, S.K., Pandey, Y.: Duality for nonsmooth mathematical programming problems with equilibrium constraints. J. Inequalities Appl. 28, 1–5 (2016) 20. Jeyakumar, V., Li, G., Lee, G.M.: Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Anal. 75, 1362–1373 (2012) 21. Jeyakumar, V., Li, G.Y., Srisatkunarajah, S.: Strong duality for robust minimax fractional programming problems. Eur. J. Oper. Res. 228(2), 331–336 (2013) 22. Kuroiwa, D., Lee, G.M.: On robust multiobjective optimization. Vietnam J. Math. 40, 305–317 (2012) 23. Laha, V., Al-Shamary, B., Mishra, S.K.: On nonsmooth V-invexity and vector variational-like inequalities in terms of the Michel-Penot subdifferentials. Optim. Lett. 8(5), 1675–1690 (2014) 24. Lee, G.M., Kim, M.H.: On duality theorems for robust optimization problems. J. Chungcheong Math. Soc. 26, 723–734 (2013) 25. Lee, G.M., Kim, M.H.: Robust duality for nonsmooth multiobjective optimization problems. J. Chungcheong Math. Soc. 30, 31–40 (2017) 26. Lee, J.H., Lee, G.M.: On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems. Ann. Oper. Res. 269, 419–438 (2018) 27. Liu, G., Ye, J.J., Zhu, J.: Partial exact penalty for mathematical programs with equilibrium constraints. Set-Valued Anal. 16, 785–804 (2008) 28. Laha, V., Kumar, R., Singh, H.N., Mishra, S.K.: On minimax programming with vanishing constraints. In: Laha, V., Maréchal, P., Mishra, S.K. (eds.) Optimization, Variational Analysis and Applications. IFSOVAA 2020. Springer Proceedings in Mathematics & Statistics, vol 355. Springer, Singapore (2021) 29. Lee, G.M., Son, P.T.: On nonsmooth optimality theorems for robust optimization problems. Bull. Korean Math. Soc. 51(1), 287–301 (2014) 30. Laha, V., Singh, V., Pandey, Y., Mishra, S.K.: Nonsmooth mathematical programs with vanishing constraints in Banach spaces. In: Nikeghbali, A., Pardalos, P.M., Raigorodskii, A.M., Rassias, M.T. (eds.) High-Dimensional Optimization and Probability. Springer Optimization and Its Applications, vol 191. Springer, Cham (2022) 31. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) 32. Mangasarian, O.L., Fromovitz, S.: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 17(1), 37–47 (1967) 33. Mishra, S.K., Laha, V.: On V − r −invexity and vector variational-like inequalities. Filomat 26(5), 1065–1073 (2012) 34. Mishra, S.K., Laha, V.: On approximately star-shaped functions and approximate vector variational inequalities. J. Optim. Theory Appl. 156(2), 278–293 (2013) 35. Mishra, S.K., Laha, V.: On minty variational principle for nonsmooth vector optimization problems with approximate convexity. Optim. Lett. 10(3), 577–589 (2016) 36. Movahedian, N., Nobakhtian, S.: Constraint qualifications for nonsmooth mathematical programs with equilibrium constraints. Set-Valued Var. Anal. 17, 63–95 (2009) 37. Mishra, S.K., Singh, V., Laha, V.: On duality for mathematical programs with vanishing constraints. Ann. Oper. Res 243(1), 249–272 (2016) 38. Mishra, S.K., Singh, V., Laha, V., Mohapatra, R.N.: On constraint qualifications for multiobjective optimization problems with vanishing constraints. In: Xu, H., Wang, S., Wu, S.Y. (eds.) Optimization Methods, Theory and Applications. Springer, Berlin, Heidelberg (2015) 39. Pandey, Y., Mishra, S.K.: Duality of mathematical programming problems with equilibrium constraints. Pac. J. Optim. 13, 105–122 (2017) 40. Soleimanian, A., Jajaei, G.S.: Robust nonlinear optimization with conic representable uncertainty set. Eur. J. Oper. Res. 228(2), 337–344 (2013) 41. Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity. Math. Oper. Res. 25, 1–22 (2000)

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42. Wang, J., Li, S.J., Chen, C.R.: Generalized robust duality in constrained nonconvex optimization. Optimization 70(3), 591–612 (2021) 43. Ye, J.J.: Multiplier rules under mixed assumptions of differentiability and Lipschitz continuity. SIAM J. Control Optim. 39(5), 1441–1460 (2001) 44. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)

A New Approach to Solve Fuzzy Transportation Problem Ashutosh Choudhary and Shiv Prasad Yadav

Abstract In the real-life problem, the decision-maker (DM) faces uncertainty because of a number of uncontrolled circumstances, including the weather, k mthe state of the roads, and the price of diesel. Due to these uncontrollable factors, the DM hesitates to predict the actual situation. To handle these uncertainties, many authors showed their interest in intuitionistic fuzzy set to represent. This article develops a transportation problem (TP) where costs are triangular intuitionistic fuzzy number (TIFN) while supplies and demands are real numbers. To deal with uncertainty in the TP, an algorithm is developed to find out the optimal solution of TP in terms of TIFNs. The proposed method is demonstrated by an example. Keywords Triangular intuitionistic fuzzy number · Transportation problem · Ranking function · Optimal solution

1 Introduction The theory of fuzzy set (FS) was first established by Zadeh [25], by defining single membership degree. FS has been applied in several disciplines of mathematics, engineering, and management. The application of FS is more popular in the domain of optimization and in real-world decision-making problems after the groundbreaking work carried out by Bellman and Zadeh [5]. Atanassov [3] developed the theory of intuitionistic fuzzy set (IFS) in terms of generalization of FS, which extends the single membership degree of FS to two more logical terms, the membership degree and the non-membership degree, so that

A. Choudhary (B) · S. P. Yadav Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India e-mail: [email protected]; [email protected] S. P. Yadav e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_21

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their summation lies in [0, 1]. The IFS describes the vagueness of an element in the set more comprehensively than the FS and has been very helpful in dealing with uncertainty. In the real-life TPs, there are several instances where transportation cost (TC) may be unsure due to number of reasons (Dempe and Starostina [6]). Many authors (Nagoorgani and Razak [17], Dinager and Palanivel [7], Pandian and Natrajan [18]) have studied fuzzy transportation problems (FTPs) to address the ambiguity that arises in TP. Mohideen and Kumar [15] investigate a TP with all the parameters being trapezoidal fuzzy numbers. By utilizing the ranking of fuzzy numbers, Basirzadeh [4] suggested a simplistic but efficient parametric approach to solving the FTP. Kaur and Kumar [12] introduced a novel approach for determining the fuzzy optimum solution for an FTP where TC is expressed by generalized fuzzy numbers. Szmidt and Kacprzyk [22] have introduced the notion of distance between two IFSs with the assistance of membership and non-membership functions. Mukherjee and Basu [16] investigated the solution of the assignment problem using similarity measures under intuitionistic environment. In various real-world TP, DM does not assure about TCs due to uncertain quantities arising from variation in fuel prices, heavy traffic, temperature fluctuations, etc. In this circumstance, DM hesitates in prediction of TC. So, to cover hesitation factor, IFS is more suitable than the FS. Hussain and Kumar [10], Singh and Yadav [21] have used TFS in real-life TP. Antony et al. [1] have studied TP using TIFNs. Singh and Yadav [20] solved TP using intuitionistic fuzzy cost but crisp availabilities and demand. Kumar [13] suggested a novel approach to obtain a solution of intuitionistic FTP of type-2 where TC is expressed by TIFN. Hunwisai et al. [9] proposed a novel method to solving FTP using trapezoidal intuitionistic fuzzy numbers in generalized form. Traneva and Tranev [23] extended the fuzzy zero point method (FZPM) to intuitionistic FZPM and obtained the solution of intuitionistic FTP using the concept of IFSs and index matrices. Josephine et al. [11] proposed an efficient algorithm to find the optimal solution of a TP in which costs, supplies, and demands all are trapezoidal fuzzy numbers (TrFNs). Mishra and kumar [14] proposed a new method (named JMD method) to transform an unbalanced triangular intuitionistic fully IFTP into a balanced fully IFTP and then obtained the intuitionistic fuzzy optimal solution of unbalanced fully IFTP. Singh and Garg [19] proposed a family of Hamming, Euclidean, and utmost distance measures for type-2 IFSs. Further proposed a ranking method based on these measures for solving group decision-making problems. Anusha and Sireesha proposed [2], a new distance measure for type-2 IFSs and its application to multi-criteria group decision-making. Xue and Deng [24] presented the decision-making under measure-based granular uncertainty in intuitionistic fuzzy environment. Garg and Singh [8] presented a new group decision-making approach based on similarity measure between type-2 IFSs. In the present study, we have modeled a TP in which TCs are TIFNs, availabilities and requirements are real numbers. For ordering of TIFNs, we propose the accuracy function for TIFNs. To determine the optimal solution of FTP, we develop a new algorithm. The rest of the article is organized as follows: in Sect. 2, definitions and mathematical operations on TIFNs from the existing literature (Singh and Yadav [21],

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Atanassov [3]) are defined. In Sect. 3, ranking and ordering of TIFNs are defined. Section 4 deals with solution procedure of TIFTP. A numerical example is used in Sect. 5 to show how to use the approach to obtain the optimal solution.

2 Some Definitions 2.1 Intuitionistic Fuzzy Set (IFS) Let X be a non-empty set. Then an IFS A˜ I in X is defined by A˜ I = {(x, μ A˜I (x), ν A˜I (x)) : x ∈ X }, where μ A˜I , ν A˜I : X → [0, 1] are functions satisfying the relation 0 ≤ μ A˜I (x) + ν A˜I (x) ≤ 1 ∀x ∈ X . The values μ A˜I (x) and ν A˜I (x) represent the degrees of membership and non-membership of the element x ∈ X being in A˜ I . π A˜I (x) = 1 − μ A˜I (x) − ν A˜I (x) is called the degree of hesitation for an element x ∈ X being in A˜ I .

2.2 Intuitionistic Fuzzy Number (IFN) Let the collection of real numbers be R. Then an IFS A˜ I = {(x, μ A˜I (x), ν A˜I (x)) : x ∈ R} is called an IFN if the following two necessary conditions hold: 1. ∃ a unique point k1 ∈ R and ∃ at least one point k2 ∈ R so that μ A˜I (k1 ) = 1 and ν A˜I (k2 ) = 1. 2. μ A˜I , ν A˜I : R → [0, 1] are piece-wise continuous functions and 0 ≤ μ A˜I (x) + ν A˜I (x) ≤ 1 ∀x ∈ R, where for l, r, l  , r  ∈ R with l ≤ l  , r ≤ r  , we have ⎧ g1 (x), k1 − l ≤ x < k1 , ⎪ ⎪ ⎪ ⎨ 1, x = k1 , μ A˜I (x) = ⎪ (x), k g 2 1 < x ≤ k1 + r, ⎪ ⎪ ⎩ 0, other wise, and ⎧ h 1 (x), ⎪ ⎪ ⎪ ⎨0, ν A˜I (x) = ⎪ h 2 (x), ⎪ ⎪ ⎩ 1,



k1 − l ≤ x < k1 ; 0 ≤ g1 (x) + h 1 (x) ≤ 1, x = k1 ,  k1 < x ≤ k1 + r ; 0 ≤ g2 (x) + h 2 (x) ≤ 1, other wise.

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Here k1 is called the mean value of A˜ I ; l and r are called the left and right spreads of   μ A˜I , respectively; l and r are called the left and right spreads of ν A˜I , respectively; g1 and h 2 are piece-wise continuous and increasing functions in [k1 − l, k1 ) and  (k1 , k1 + r ], respectively; and g2 and h 1 are piece-wise continuous and decreasing  functions in (k1 , k1 + r ] and [k1 − l , k1 ), respectively. The IFN A˜ I is denoted by   A˜ I = (k1 ; l, r ; l , r ).

2.3 Triangular Intuitionistic Fuzzy Number (TIFN) An IFN A˜ I is said to be a TIFN if its membership function μ A˜I and non-membership function ν A˜I are defined by ⎧ x − ξ1 ⎪ ⎪ , ξ1 < x ≤ ξ2 , ⎪ ⎪ ⎨ ξ2 − ξ1 μ A˜I (x) = ξ3 − x , ξ ≤ x < ξ , 2 3 ⎪ ⎪ ξ3 − a2 ⎪ ⎪ ⎩0, other wise, and

⎧ ξ2 − x  ⎪ ⎪ ξ1 < x ≤ ξ2 ,  , ⎪ ⎪ ⎨ ξ2 − a1 x − ξ2  ν A˜I (x) = , ξ2 ≤ x < ξ3 , ⎪  ⎪ aξ3 − ξ2 ⎪ ⎪ ⎩ 1, other wise,

  where ξ1 ≤ ξ1 < ξ2 < ξ3 ≤ ξ3 . The TIFN A˜ I is denoted by   A˜ I = (ξ1 , ξ2 , ξ3 ; ξ1 , ξ2 , ξ3 ) and is shown in Fig. 1.

2.4 Arithmetic Operation on TIFNs     Let A˜ I = (ξ1 , ξ2 , ξ3 ; ξ1 , ξ2 , ξ3 ) and B˜ I = (ζ1 , ζ2 , ζ3 ; ζ1 , ζ2 , ζ3 ) be TIFNs. Then     Addition: A˜ I ⊕ B˜ I = (ξ1 + ζ1 , ξ2 + ζ2 , ξ3 + ζ3 ; aξ1 + ζ1 , ξ2 + ζ2 , aξ3 + ζ3 ),     Subtraction: A˜ I B˜ I = (ξ1 − ζ3 , ξ2 − ζ2 , ξ3 − ζ1 ; ξ1 − ζ3 , ξ2 − ζ2 , ξ3 − ζ1 ),   Multiplication: A˜ I ⊗ B˜ I = (m 1 , m 2 , m 3 ; m 1 , m 2 , m 3 ),

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  Fig. 1 TIFN A˜I = (ξ1 , ξ2 , ξ3 ; ξ1 , ξ2 , ξ3 )

where m 1 = min{ξ1 ζ1 , ξ1 ζ3 , ξ3 ζ1 , ξ3 ζ3 }, m 2 = ξ2 ζ2 , m 3 = max{ξ1 ζ1 , ξ1 ζ3 , ξ3 ζ1 , ξ3 ζ3 } 



































m 1 = min{ξ1 ζ1 , ξ1 ζ3 , ξ3 ζ1 , ξ3 ζ3 }, m 3 = max{ξ1 ζ1 , ξ1 ζ3 , ξ3 ζ1 , ξ3 ζ3 }. Scalar multiplication   1. λ A˜ I = (λξ1 , λξ2 , λξ3 ; λξ1 , λξ2 , λξ3 ) : λ ≥ 0.   2. λ A˜ I = (λξ3 , λξ2 , λξ1 ; λξ3 , λξ2 , λξ1 ) : λ < 0.

3 Rank and Ordering of TIFNs   The rank or ranking value of a TIFN A˜ I = (ξ1 , ξ2 , ξ3 ; ξ1 , ξ2 , ξ3 ) is denoted by r( A˜ I )   ξ1 + 4ξ2 + ξ3 + ξ1 + ξ3 . and is defined by r( A˜ I ) = 8 Let A˜ I and B˜ I be two TIFNs. Then ordering is defined as follows:

(a) A˜ I  B˜ I ⇐⇒ r( A˜ I ) ≤ r ( B˜ I ). (b) A˜ I ≈ B˜ I ⇐⇒ r( A˜ I ) = r ( B˜ I ). (c) A˜ I ≺ B˜ I ⇐⇒ r( A˜ I ) < r ( B˜ I ). (d) min{ A˜ I , B˜ I } = A˜ I ⇐⇒ A˜ I  B˜ I .

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r( A˜ I )=

Theorem 3.1 The ranking function r : F (R) → R defined by   ξ1 + ξ2 + ξ3 + ξ1 + ξ3 is linear, where F (R) is the set of all TIFNs. 8

    Proof Let A˜ I = (ξ1 , ξ2 , ξ3 ; ξ1 , ξ2 , ξ3 ) and B˜ I = (ζ1 , ζ2 , ζ3 ; ζ1 , ζ2 , ζ3 ) ∈ F (R) and α, β ∈ R. Case 1. Let α > 0, β > 0.     r (α A˜I + β B˜ I ) = r [(αξ1 , αξ2 , αξ3 ; αξ1 , αξ2 , αξ3 ) ⊕ (βζ1 , βζ2 , βζ3 ; βζ1 , βζ2 , βζ3 )] 





βζ2 , , αξ3 βζ3 )]



= r [(αξ1 + βζ1 , αξ2 + βζ2 , αξ3 + βζ3 ; αξ1 + βζ1 , αξ2 + 







αξ1 + βζ1 + 4(αξ2 + βζ2 ) + αξ3 + βζ3 + αξ1 + βζ1 + αξ3 + βζ3 8     αξ1 + 4αξ2 + αξ3 + αξ1 + αξ3 βζ1 + 4βζ2 + βζ3 + βζ1 + βζ3 = + + 8 8     ξ1 + 4ξ2 + ξ3 + ξ1 + ξ3 ζ1 + 4ζ2 + ζ3 + ζ1 + ζ3 =α +β 8 8 = α r ( A˜ I ) + β r ( B˜ I ). Case 2. Let α > 0, β < 0.     r (α A˜I + β B˜ I ) = r [(αξ1 , αξ2 , αξ3 ; αξ1 , αξ2 , αξ3 ) ⊕ (βζ3 , βζ2 , βζ1 ; βζ3 , βζ2 , βζ1 )] =







βζ2 , , αξ3 βζ1 )]



= r [(αξ1 + βζ3 , αξ2 + βζ2 , αξ3 + βζ1 ; αξ1 + βζ3 , αξ2 + 







αξ1 + βζ3 + 4(αξ2 + βζ2 ) + αξ3 + βζ1 + αξ1 + βζ3 + αξ3 + βζ1 8     βζ3 + 4βζ2 + βζ1 + βζ3 + βζ1 αξ1 + 4αξ2 + αξ3 + αξ1 + αξ3 + + = 8 8     ξ1 + 4ξ2 + ξ3 + ξ1 + ξ3 ζ1 + 4bζ2 + ζ3 + ζ1 + ζ3 =α +β 8 8 = α r ( A˜ I ) + β r ( B˜ I ). Case 3. Let α < 0, β < 0.     r (α A˜I + β B˜ I ) = r [(αξ3 , αξ2 , αξ1 ; αξ3 , αξ2 , αξ1 ) ⊕ (βζ3 , βζ2 , βζ1 ; βζ3 , βζ2 , βζ1 )] =







βζ2 , , αξ3 βζ1 )]



=



= r [(αξ3 + βζ3 , αξ2 + βζ2 , αξ1 + βζ1 ; αξ3 + βζ3 , αξ2 + 





αξ3 + βζ3 + 4(αξ2 + βζ2 ) + αξ1 + βζ1 + αξ3 + βζ3 + αξ1 + βζ1 8

A New Approach to Solve Fuzzy Transportation Problem 



307 



βζ3 + 4βζ2 + βζ1 + βζ3 + βζ1 αξ3 + 4αξ2 + αξ1 + αξ3 + αξ1 + + = 8 8     ξ1 + 4ξ2 + ξ3 + ξ1 + ξ3 ζ1 + 4ζ2 + ζ3 + ζ1 + ζ3 =α +β 8 8 ˜ ˜ I I = α r ( A ) + β r ( B ). Cases 1, 2, and 3 show that r is a linear function. Definition 3.2 A balanced IFTP (BIFTP) in which costs are TIFN but supplies and demands are crisp is defined as Min Z˜ I ≈ subject to

n m  

c˜iIj ⊗ xi j

i=1 j=1 n 

xi j ≈ ai i = 1, 2, 3, . . . , m,

j=1 m 

xi j ≈ b j j = 1, 2, 3, . . . , n,

i=1

xi j ≥ 0 i = 1, 2, 3, . . . , m; j = 1, 2, 3, . . . , n, where c˜iIj is the unit fuzzy transportation cost (IFTC) from the ith supply point to the jth destination, ai is the availability of the commodity at the ith supply point, b j is the requirement of the commodity at the jth destination, xi j is the number of units of the commodity transported from the ith supply point to the jth destination.

4 Proposed Method We present the following algorithmic program to find the optimal solution: Step1: Firstly, write the FIFTP in the matrix form and call it as the IF transportation cost (IFTC) matrix. Examine whether it is a BIFTP or not. If yes, go to Step 2. If not, introduce an additional row/column with IF zero cost and supply/demand equals to the IF positive difference of supply and demand. Step2: Find the smallest element using 3 (d) in each row and subtract it from all elements of that row. Step3: Repeat Step 2 for each column of the matrix obtained in Step 2. Call the matrix thus obtained as the reduced IFTC matrix. (Observe that each row and each column have IF zero cost).

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Step4: For each row (column) of the reduced IFTC matrix, find the difference between the smallest and next higher cost. Write them by the side of the matrix against the respective rows (column) and call them as row (column) IF penalties. Step5: Choose the row or column with the largest IF penalty. Make the maximum possible allocation to the IF zero cost cell and cross off the satisfied row or column. If there is a tie for the largest IF penalty or the IF zero cost cell, go to Step 7. If not, go to Step 6. Step6: Now consider modified matrix obtained in Step 5. Repeat Steps 2–5 until all the rows and columns are satisfied. Step7 (Tie) 1. If the largest IF penalty is not unique, then break the tie among them by choosing the row or column having the smallest IF cost, i.e., IF zero cost. 2. If in the chosen row or column, the smallest IF cost is not unique, then choose the cell from tied ones to which more allocation can be made.

5 Numerical Example Consider the following BIFTP. In this problem, there are four supply points S1 , S2 , S3 , S4 and four destinations D1 , D2 , D3 , D4 (Tables 1 and 2). Applying Step 2, the resultant IFTC is given in Table 3. Applying Step 3, the resultant IFTC is given in Table 4. In Table 4, calculate the row and column IF penalties and write them against the respective rows and columns. The resultant IFTC matrix is given in Table 5.

Table 1 IFTC matrix D1

D2

D3

D4

ai

S1

(2, 4, 5; 1, 4, 6)

(2, 5, 7; 1, 5, 8)

(4, 6, 8; 3, 6, 9)

(4, 7, 8; 3, 7, 9)

11

S2

(4, 6, 8; 3, 6, 9)

(3, 7, 12; 2, 7, 13)

(10, 15, 20; 8, 15, 22)

(11, 12, 13; 10, 12, 14)

11

S3

(3, 4, 6; 1, 4, 8)

(8, 10, 13; 5, 10, 16)

(2, 3, 5; 1, 3, 6)

(6, 10, 14; 5, 10, 15)

11

S4

(4, 6, 9; 3, 6, 9)

(3, 7, 9; 2, 7, 10)

(2, 4, 5; 1, 4, 6)

(9, 17, 23; 6, 17, 25)

12

bj

16

10

8

11

Table 2 IFTC matrix with ranking values D3

D4

ai

S1

(2, 4, 5; 1, 4, 6)(3.75) (2, 5, 7; 1, 5, 8)(4.75)

D1

D2

(4, 6, 8; 3, 6, 9)(6)

(4, 7, 8; 3, 7, 9)(6.5)

11

S2

(4, 6, 8; 3, 6, 9)(6)

(10, 15, 20; 8, 15, 22)(15) (11, 12, 13; 10, 12, 14)(12) 11

S3

(3, 4, 6; 1, 4, 8)(4.25) (8, 10, 13; 5, 10, 16)(10.25) (2, 3, 5; 1, 3, 6)(3.25)

(6, 10, 14; 5, 10, 15)(10)

S4

(4, 6, 9; 3, 6, 9)(4)

(3, 7, 9; 2, 7, 10)(7.87)

(2, 4, 5; 1, 4, 6)(6.37)

(9, 17, 23; 6, 17, 25)(4.25) 12

bj

16

10

8

11

(3, 7, 12; 2, 7, 13)(7.25)

11

A New Approach to Solve Fuzzy Transportation Problem

309

Table 3 Row reduced IFTC matrix D1

D2

D3

D4

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 1, 2; 0, 1, 2)(1)

(2, 2, 3; 2, 2, 3)(2.25)

(2, 3, 3; 2, 3, 3)(2.75)

ai

11

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 1, 4; −1, 1, 4)(1.25)

(6, 9, 12; 5, 9, 13)(9)

(7, 6, 5; 7, 6, 5)(6)

11

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 7, 8; 4, 7, 10)(7)

(0, 0, 0; 0, 0, 0)(0)

(4, 7, 9; 4, 7, 9)(6.75)

11

S4

(0, 0, 0; 0, 0, 0)(0)

(1, 5, 4; 1, 5, 5)(3.87)

(1, 2, 4; 1, 2, 5)(2.37)

(1, 0, −1; 1, 0, 1)(0.25)

12

bj

16

10

8

11

Table 4 Column reduced IFTC matrix D1

D2

D3

D4

ai

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

(2, 2, 3; 2, 2, 3)(2.25)

(1, 3, 4; 1, 3, 2)(2.50)

11

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25) (6, 9, 12; 5, 9, 13)(9)

(6, 6, 6; 6, 6, 4)(5.75)

11

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 6, 6; 4, 6, 8)(5.75)

(3, 7, 10; 3, 7, 8)(6.50) 11

S4

(0, 0, 0; 0, 0, 0)(0)

(1, 4, 2; 1, 4, 3)(2.62)

(1, 2, 4; 1, 2, 5)(2.37)

(0, 0, 0; 0, 0, 0)(0)

bj

16

10

8

11

(0, 0, 0; 0, 0, 0)(0)

12

Table 5 First reduced IFTC matrix with penalties S1

D1 (0, 0, 0; 0, 0, 0)(0)

D2 (0, 0, 0; 0, 0, 0)(0)

D3 (2, 2, 3; 2, 2, 3) (2.25)

D4 (1, 3, 4; 1, 3, 2) (2.50)

ai 11

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2) (0.25)

(6, 9, 12; 5, 9, 13) (9)

(6, 6, 6; 6, 6, 4) (5.75)

11

(−1, 0, 2; −1, 0, 2) (0.25)

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 6, 6; 4, 6, 8) (5.75)

(0, 0, 0; 0, 0, 0) (0)

(3, 7, 10; 3, 7, 8) (6.50)

11

(1, 1, 1; 0, 1, 2) (1)

S4

(0, 0, 0; 0, 0, 0)(0)

(1, 4, 2; 1, 4, 3) (2.62)

(1, 2, 4; 1, 2, 5) (2.37)

(0, 0, 0; 0, 0, 0) (0)

12

(0, 0, 0; 0, 0, 0) (0)

bj

16

10

8

11

I F penalties

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2) (0.25)

(2, 2, 3; 2, 2, 3) (2.25)

(1, 3, 4; 1, 3, 2) (2.50)

I F penalties (0,0,0;0,0,0) (0)

Table 6 First reduced IFTC matrix with allocation D1

D2

D3

D4

ai

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

(2, 2, 3; 2, 2, 3)(2.25)

−

11

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25)

(6, 9, 12; 5, 9, 13)(9)

−

11

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 6, 6; 4, 6, 8)(5.75)

(0, 0, 0; 0, 0, 0)(0)

−

11

S4

(0, 0, 0; 0, 0, 0)(0)

(1, 4, 2; 1, 4, 3)(2.62)

(1, 2, 4; 1, 2, 5)(2.27)

(11)

1

bj

16

10

8

(1, 3, 4; 1, 3, 2) is the largest penalty corresponding to column D4 . Allocate 11 to the cell (4, 4) and cross off the fourth column. The resultant IFTC matrix is shown in Table 6.

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Applying steps 2 and 3 to Table 6, we get the reduced IFTC matrix as given in Table 7. Calculate the new penalties. (1, 2, 4; 1, 2, 5) is the largest penalty corresponding to row S4 . Allocate 1 to the cell (4, 1) and cross off the fourth row. The resultant IFTC matrix is shown in Table 8. Applying steps 2 and 3 to Table 8, we get the reduced IFTC matrix as given in Table 9. Calculate the new penalties.

Table 7 Second reduced IFTC matrix with penalties D1

D2

D3

D4

ai

I F penalties

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

(2, 2, 3; 2, 2, 3) (2.25)

−

11

(0, 0, 0; 0, 0, 0)(0)

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2) (0.25)

(6, 9, 12; 5, 9, 13)(9)

−

11

(−1, 0, 2; −1, 0, 2) (0.25)

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 6, 6; 4, 6, 8) (5.75)

(0, 0, 0; 0, 0, 0)(0)

−

11

(1, 1, 1; 0, 1, 2)(1)

S4

(0, 0, 0; 0, 0, 0)(0)

(1, 4, 2; 1, 4, 3) (2.62)

(1, 2, 4; 1, 2, 5) (2.37)

(11)

1

(1, 2, 4; 1, 2, 5) (2.27)

bj

16

10

8

I F penalties

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2) (0.25)

(2, 2, 3; 2, 2, 3) (2.25)

(−)

Table 8 Second reduced IFTC matrix with allocation D1

D2

D3

D4

ai

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

(2, 2, 3; 2, 2, 3)(2.25)

−

11

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25)

(6, 9, 12; 5, 9, 13)(9)

−

11

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 6, 6; 4, 6, 8)(5.75)

(0, 0, 0; 0, 0, 0)(0)

−

11

S4

(1)

−

−

(11)

bj

15

10

8

Table 9 Third reduced IFTC matrix with penalties D1

D2

D3

D4

ai

I F penalties

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

(2, 2, 3; 2, 2, 3) (2.25)

−

11

0, 0, 0; 0, 0, 0)(0)

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2) (0.25)

(6, 9, 12; 5, 9, 13)(9) −

11

(−1, 0, 2; −1, 0, 2) (0.25)

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 6, 6; 4, 6, 8) (5.75)

(0, 0, 0; 0, 0, 0)(0)

−

11

(1, 1, 1; 0, 1, 2)(1)

S4

(0, 0, 0; 0, 0, 0)(0)

(1, 4, 2; 1, 4, 3) (2.62)

(1, 2, 4; 1, 2, 5) (2.37)

(11)

1

(1, 2, 4; 1, 2, 5) (2.27)

bj

16

10

8

I F penalties

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2) (0.25)

(2, 2, 3; 2, 2, 3) (2.25)

(−)

A New Approach to Solve Fuzzy Transportation Problem

311

Table 10 Third reduced IFTC matrix with allocation D1

D2

D3

D4

ai

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

−

−

11

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25)

−

−

11

S3

(1, 1, 1; 0, 1, 2)(1)

(6, 6, 6; 4, 6, 8)(5.75)

(8)

−

3

S4

(1)

−

−

(11)

bj

15

10

Table 11 Fourth reduced IFTC matrix with penalties D1

D2

D3

D4

ai

I F penalties

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

−

−

11

(0, 0, 0; 0, 0, 0)(0)

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25)

−

−

11

(−1, 0, 2; −1, 0, 2)(0.25)

S3

(0, 0, 0; 0, 0, 0)(0)

(5, 5, 5; 4, 5, 6)(4.75)

−

−

3

(5, 5, 5; 4, 5, 6)(4.75)

S4

(1)

−

−

(11)

8

bj

16

10

I F penalties

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25)

Table 12 Fourth reduced IFTC matrix with allocation D1 D2 S1 (0, 0, 0; 0, 0, 0)(0) (0, 0, 0; 0, 0, 0)(0) S2 (0, 0, 0; 0, 0, 0)(0) (−1, 0, 2; −1, 0, 2)(0.25) S3 (3) − S4 (1) − bj 12 10

D3 − − (8) −

D4 − − − (11)

ai 11 11

(2, 2, 3; 2, 2, 3) is the largest penalty corresponding to column D3 . Allocate 8 to the cell (3, 3) and cross off the third column. The resultant IFTC matrix is shown in Table 10. Applying steps 2 and 3 to Table 10, we get the reduced IFTC matrix as given in Table 11. Calculate the new penalties. (5, 5, 5; 4, 5, 6) is the largest penalty corresponding to row S3 . Allocate 3 to the cell (3, 1) and cross off the third row. The resultant IFTC matrix is shown in Table 12. Applying steps 2 and 3 to Table 12, we get the reduced IFTC matrix as given in Table 13. Calculate the new penalties. (−1, 0, 2; −1, 0, 2) is the largest penalty corresponding to row S2 and column D2 . Using step 7 allocate 11 to the cell (2, 1) and cross off the second row. The resultant IFTC matrix is shown in Table 14. Finally, we allocate 1 in (1, 1) cell and 10 in (1, 2) cell and we obtained the optimal solution matrix in Table 15.

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Table 13 Fifth reduced IFTC matrix with penalties D1

D2

D3

D4

ai

I F penalties

S1

(0, 0, 0; 0, 0, 0)(0)

(0, 0, 0; 0, 0, 0)(0)

−

−

11

(0, 0, 0; 0, 0, 0)(0)

S2

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25)

−

−

11

(−1, 0, 2; −1, 0, 2)(0.25)

S3

(3)

−

(8)

−

−

(11)

S4

(1)

−

bj

12

10

I F penalties

(0, 0, 0; 0, 0, 0)(0)

(−1, 0, 2; −1, 0, 2)(0.25)

Table 14 Fifth reduced IFTC matrix with allocation D1 D2 S1 (0, 0, 0; 0, 0, 0)(0) (0, 0, 0; 0, 0, 0)(0) S2 (11) − S3 (3) − S4 (1) − bj 1 10

Table 15 Optimal solution D1 S1 (1) S2 (11) S3 (3) S4 (1)

D2 (10) − − −

D3 − − (8) −

D3 − − (8) −

D4 − − − (11)

ai 11

D4 − − − (11)

Now we verify that the solution obtained in Table 15 is optimal with the help of MODI method [20]. I I I In Table 16, we observed that r (d˜i j ) ≤ 0 where d˜i j = u˜i I ⊕ v˜j I C˜i j . Hence Table 15 gives an optimal solution. The optimal solution is x11 = 1, x12 = 10, x21 = 11, x31 = 3, x33 = 8, x41 = 1, x44 = 11. The total IF cost = 1(2, 4, 5; 1, 4, 6) ⊕ 10(2, 5, 7; 1, 5, 8) ⊕ 11(4, 6, 8; 3, 6, 9) ⊕ 3(3, 4, 6; 1, 4, 8) ⊕ 1(2, 4, 6; 1, 4, 7) ⊕ 8(2, 3, 5; 1, 3, 6) ⊕ 11(3, 4, 5; 2, 4, 8) = (126, 204, 282; 78, 204, 352).

A New Approach to Solve Fuzzy Transportation Problem

313

Table 16 Optimality test u˜1 I = (−6, −2, 1; −8, −2, 3)

(2, 4, 5; 1, 4, 6)

1

10

(−18/8)

(−20/8)

u˜2 I = (0, 0, 0; 0, 0, 0)

(4, 6, 8; 3, 6, 9)

(3, 7, 12; 2, 7, 13)

(10, 15, 20; 8, 15, 22)

(11, 12, 13; 10, 12, 14)

u˜3 I = (−5, −2, 2; −8, −2, 5) u˜4 I = (−6, −2, 2; −8, −2, 4)

(2, 5, 7 : 1, 5, 8)

(4, 6, 8; 3, 6, 9)

(4, 7, 8; 3, 7, 9)

11

(−2/8)

(−80/8)

(−46/8)

(3, 4, 6; 1, 4, 8)

(8, 10, 13; 5, 10, 16)

(2, 3, 5; 1, 3, 6)

(6, 10, 14; 5, 10, 15)

3

(−40/8)

8

(−38/8)

(2, 4, 6; 1, 4, 7)

(3, 9, 10; 2, 9, 12)

(3, 6, 10; 2, 6, 12)

(3, 4, 5; 2, 4, 8)

1

(−23/8)

(−11/8)

11

v˜1 I = (4, 6, 8; 3, 6, 9)

v˜2 I = (1, 7, 13; −2, 7, 16)

v˜3 I = (0, 5, 10; −4, 5, 14)

v˜4 I = (1, 6, 11; −2, 6, 16)

6 Conclusion In this article, a novel approach to obtaining the optimal solution to the intuitionistic FTP is proposed, which is used for anticipating the transportation cost, which varies due to weather condition, diesel cost, traffic conditions, etc. The proposed method provides a simpler method compared to the solution available in the existing literature [1, 21]. The advantage of the suggested approach is that it is very easy to figure out the best way to solve the problem. The suggested approach provides solution which is very near to optimal solution. There is almost no scope of improvement of the solution and this can be seen when MODI method is applied in our problem. The TC based on suggested approach is less than the previous work by [1] and equal to [21]. The suggested approach would facilitate the DMs in solving logistical problem that arises in the actual world by helping them in their decision-making and providing an optimal solution in a simple and efficient manner. The proposed method can be extended for solving FTP by taking all the parameters of TIFNs in future.

References 1. Antony, R.J.P., Savarimuthu, S.J., Pathinathan, T.: Method for solving the transportation problem using triangular intuitionistic fuzzy number. Int. J. Comput. Algorithm 3(1), 590–605 (2014) 2. Anusha, V., Sireesha, V.: A new distance measure to rank type-2 intuitionistic fuzzy sets and its application to multi-criteria group decision making. Int. J. Fuzzy Syst. Appl. (IJFSA) 11(1), 1–17 (2022) 3. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 4. Basirzadeh, H.: An approach for solving fuzzy transportation problem. Appl. Math. Sci. 5(32), 1549–1566 (2011) 5. Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Manag. Sci. 17(4), B-141 (1970)

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6. Dempe, S., Starostina, T.: Optimal toll charges in a fuzzy flow problem. In: Computational Intelligence, Theory and Applications, pp. 405–413. Springer, Berlin, Heidelberg (2006) 7. Dinagar, D.S., Palanivel, K.: The transportation problem in fuzzy environment. Int. J. Algorithms, Comput. Math. 2(3), 65–71 (2009) 8. Garg, H., Singh, S.: Algorithm for solving group decision-making problems based on the similarity measures under type 2 intuitionistic fuzzy sets environment. Soft Comput. 24(10), 7361–7381 (2020) 9. Hunwisai, D., Kumam, P., Kumam, W.: A new method for optimal solution of intuitionistic fuzzy transportation problems via generalized trapezoidal intuitionistic fuzzy numbers. Fuzzy Inf. Eng. 11(1), 105–120 (2019) 10. Hussain, R.J., Kumar, P.S.: Algorithmic approach for solving intuitionistic fuzzy transportation problem. Appl. Math. Sci. 6(80), 3981–3989 (2012) 11. Josephine, F.S., Saranya, A., Nishandhi, I.F.: A dynamic method for solving intuitionistic fuzzy transportation problem. Eur. J. Mol. Clin. Med. 7(11), 2020 (2020) 12. Kaur, A., Kumar, A.: A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Appl. Soft Comput. 12(3), 1201–1213 (2012) 13. Kumar, P.S.: A note on’a new approach for solving intuitionistic fuzzy transportation problem of type-2’. Int. J. Logist. Syst. Manag. 29(1), 102–129 (2018) 14. Mishra, A., Kumar, A.: JMD method for transforming an unbalanced fully intuitionistic fuzzy transportation problem into a balanced fully intuitionistic fuzzy transportation problem. Soft Comput. 24(20), 15639–15654 (2020) 15. Mohideen, S.I., Kumar, P.S.: A comparative study on transportation problem in fuzzy environment. Int. J. Math. Res. 2(1), 151–158 (2010) 16. Mukherjee, S., Basu, K.: Solution of a class of intuitionistic fuzzy assignment problem by using similarity measures. Knowl.-Based Syst. 27, 170–179 (2012) 17. Nagoorgani, A., Razak, K.A.: Two stage fuzzy transportation problem. J. Phys. Sci. 10, 63–69 (2006) 18. Pandian, P., Natarajan, G.: A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems. Appl. Math. Sci. 4(2), 79–90 (2010) 19. Singh, S., Garg, H.: Distance measures between type-2 intuitionistic fuzzy sets and their application to multi-criteria decision-making process. Appl. Intell. 46(4), 788–799 (2017) 20. Singh, S.K., Yadav, S.P.: Efficient approach for solving type-1 intuitionistic fuzzy transportation problem. Int. J. Syst. Assur. Eng. Manag. 6(3), 259–267 (2015) 21. Singh, S.K., Yadav, S.P.: A new approach for solving intuitionistic fuzzy transportation problem of type-2. Ann. Oper. Res. 243(1), 349–363 (2016) 22. Szmidt, E., Kacprzyk, J.: Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst. 114(3), 505–518 (2000) 23. Traneva, V., Tranev, S.: Intuitionistic fuzzy transportation problem by zero point method. In: 2020 15th Conference on Computer Science and Information Systems (FedCSIS), pp. 349–358. IEEE (2020) 24. Xue, Y., Deng, Y.: Decision making under measure-based granular uncertainty with intuitionistic fuzzy sets. Appl. Intell. 51(8), 6224–6233 (2021) 25. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

The Best State-Based Development of Fuzzy DEA Model Anjali Sonkariya and Shiv Prasad Yadav

Abstract This paper studies fuzzy data envelopment analysis (FDEA) to measure the relative efficiencies of homogeneous decision-making units (DMUs) using the α-cut approach. The proposed model uses the same set of constraints for all DMUs. It gives a uniform environment for all DMUs. Furthermore, a ranking method based on lower and upper bound FDEA performance efficiencies is proposed to rank the DMUs. The proposed FDEA models and ranking approach are demonstrated using an example with triangular fuzzy numbers (TFNs), and the results are presented in Tables 2 and 3. Keywords α-cut · Data envelopment analysis · Fuzzy data envelopment analysis · Efficiency measurement · Fuzzy ranking

1 Introduction Data envelopment analysis (DEA) is a technique to measure the relative performance efficiencies of homogeneous DMUs using crisp data. Although in real-life scenarios, uncertainty is always there. The establishment of fuzzy theory is accomplished to handle uncertainty in real-life scenarios. Zadeh [16] introduced the fuzzy concept, in which a membership function is defined to express the fuzziness of data. When the inputs, outputs, and prior knowledge are inexact and imprecise, Sengupta [11] proposed FDEA models employing fuzzy entropy, fuzzy regression, and the fuzzy mathematical programme by using the α-cut technique. Cooper et al. [3] presented the imprecise DEA (IDEA) method, which allows mixtures of imprecisely and precisely known data to be transformed into standard linear programming forms, as well as the assurance region-IDEA (AR-IDEA) model, which combines imprecise data abilA. Sonkariya (B) · S. P. Yadav Indian Institute of Technology Roorkee, Roorkee 247667, India e-mail: [email protected]; [email protected] S. P. Yadav e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_22

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ities with cone-ratio envelopment theory and assurance region DEA models. Cooper et al. [4] applied these models to evaluate the efficiency of a telecommunication company’s branch offices in Korea. Kao and Liu [10] proposed a method for calculating DMU efficiency using fuzzy observations. The core concept uses the α-cut strategy to convert FDEA models into conventional DEA models with crisp data. The membership functions of the efficiency scores are constructed using a pair of parametric algorithms that characterize the class of crisp DEA models. Guo and Tanaka [8] developed the FDEA model to cope with the efficiency assessment problem using fuzzy data. Furthermore, considering the interaction between DEA and Regression analysis (RA), an augmentation of the FDEA model to a more generic form is provided. Despotis and Smirlis [5] used a basic formulation that differs from that used in IDEA to convert the non-linear DEA model to an LP. Unlike IDEA, they used variable transformations based on the original data set, and no scale modifications were applied to the data. Jahanshahloo et al. [9] developed the slack-based measure (SBM)—DEA model in fuzzy environment for assessing the relative efficiency scores and ranking technique using fuzzy input. Wen and Li [15] created a hybrid technique that combines fuzzy simulation and genetic algorithm to encounter fuzzy problems. The hybrid model can be changed to linear programming when all inputs and outputs are trapezoidal or triangular fuzzy variables. Dotoli et al. [6] introduced a new cross-efficiency FDEA approach for assessing distinct elements (DMUs) in an environment of uncertainty. A cross-assessment derived from a compromise between adequately specified objectives is used to assign a fuzzy triangular efficiency to each DMU. The DMUs are then ranked once the results have been defuzzified. The suggested technique is used to assess the performance of healthcare systems in a Southern Italian area. Arya and Yadav [1] used an α-cut strategy to create FDEA models in order to quantify the left-hand relative efficiency and right-hand relative efficiency for DMUs and suggested mechanism for ranking DMUs accordingly their obtained left-hand efficiency and right-hand efficiencies. Tavassoli et al. [12] examined four different types of supply chain supplier selection models and offered a decisionmaking framework for supplier selection with fuzzy, deterministic and stochastic settings. Using the α-cut approach, the suggested stochastic FDEA (SFDEA) model can be solved as a crisp programme. Wang et al. [13] measured the efficiency of DMUs under fixed production frontier using interval input–output data. In this study, FDEA models are developed to avert the usage of separate production boundaries to estimate the efficiency of distinct DMUs. For measuring both the lower bound efficiency and upper bound efficiency of each DMU, models employ interval arithmetic using the same constraints, resulting in a unified and consistent production frontier. The proposed FDEA model is applied to a numerical example with TFNs. Further, a ranking approach is proposed for ranking DMUs. Results are presented in Table 2. The framework of this paper is as follows. Section 2 presents the preliminaries of basic fuzzy set theory. The CCR efficiency model is stated in Sect. 3. In Sect. 4, the proposed FDEA model is demonstrated. Section 5 discusses numerical examples, and results (lower and upper bound efficiencies) are demonstrated in Table 2. The

The Best State-Based Development of Fuzzy DEA Model

317

proposed ranking approach and ranks of DMUs are presented in Sect. 6. The paper’s last Sect. 7 ends with conclusions.

2 Preliminaries Some key concepts and fuzzy arithmetic operations that are necessary for developing the FDEA model are provided in this section. Definition 1 (Fuzzy Set) [17] The definition of a fuzzy set (FS) in a universal set X is given as R˜ = {(x, μ R˜ (x)) : x ∈ X}, where μ R˜ : x → [0, 1]. Definition 2 (Convex Fuzzy Set) [17] The definition of a convex FS (CFS) R˜ in X is given as min {μ R˜ (x1 ), μ R˜ (x2 )} ≤ μ R˜ (t x1 + (1 − t)x2 ), ∀x1 , x2 ∈ X, and t ∈ [0, 1]. Definition 3 (Fuzzy number) [17] A fuzzy number (FN) R˜ is a CFS R˜ in R, the set of real numbers, if: (i) μ R˜ (xo ) = 1, for a unique xo ∈ R, (ii) μ R˜ is a piecewise continuous function in R. ˜ xo is said to be the mean value of R. L ˜ denoted by R=(r ˜ Definition 4 (Triangular Fuzzy Number) [17] A FN R, , r M , r U ), is defined as a triangular FN (TFN) if the membership function μ R˜ is defined as

⎧ x − rL ⎪ ⎪ , rL < x ≤ r M; ⎪ ⎪ ⎨r M − r L U μ R˜ (x) = r − x , r M ≤ x < rU ; ⎪ ⎪ ⎪ rU − r M ⎪ ⎩ 0 otherwise. ∀x ∈ R. Definition 5 (α − cut) [17] The α − cut of a FS R˜ in X is Rα defined as follows: Rα = {x ∈ X : μ R˜ (x) ≥ α}, 0 ≤ α ≤ 1.

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Remark 1 R0 = X. Definition 6 Arithmetic operations on TFNs. Let R˜ = (r L , r M , r U ) and S˜ = (s L , s M , s U ) be two positive TFNs, i.e., r L > 0, s L > 0. Then the arithmetic operations are defined below [14]: (i) (ii) (iii) (iv)

Addition: R˜ + S˜ = (r L + s L , r M + s M , r U + s U ), Subtraction: R˜ − S˜ = (r L − s U , r M − s M , r U − s L ), Multiplication: R˜ × S˜ ≈ (r L s L , r M s M , r U s U ), ˜ S˜ ≈ (r L /s U , r M /s M , r U /s L ). Division: R/

3 Development of the FDEA Model Suppose we have n DMUs and each DMU produces q outputs using p inputs. Let the quantity of ith input utilized and rth outputs produced by jth DMU are xi j and yr j respectively, where i = 1, 2, . . . , p and r = 1, 2, . . . q. Then DEA models developed by Arya and Yadav [1] to calculate efficiency for DMU j is given as below Input minimization lower bound DEA model 

min E kL =

p 

u xik u ik

i=1 q



subject to p 

r =1 q u xik u ik −



yrl k vr k ≥ 0,

r =1 q

i=1 p



yrl k vr k = 1,

xil j u ik −



yruj vr k ≥ 0 ∀ j, where j = k,

r =1

i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀ r. where u ik and vr k are the weights corresponding to the xik and yr k , respectively, and ε is non-archimedean infinitesimal. Input minimization upper bound DEA model 

min E kU =

p 

l xik u ik

i=1 q

subject to

 r =1

yruk vr k = 1,

The Best State-Based Development of Fuzzy DEA Model p 

l xik u ik −

q 

i=1

r =1

p 

q 

xiuj u ik −

319

yruk vr k ≥ 0, yrl j vr k ≥ 0 ∀ j, where j = k,

r =1

i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀ r. If we look closely at the lower and upper efficiency models, we can see that constraint sets in the above models to assess DMUs’ efficiencies differ from one DMU to the other. The most significant disadvantage of using multiple constraint sets to estimate DMUs’ efficiencies is that the efficiencies are not comparable because different production boundaries were used in the efficiency measurement process. So, we develop the models in which we use the same set of constraints. The conventional CCR DEA [2] model is presented as follows: Model 1—The CCR Model q 

max Ek =

r =1 p 

yr k vr k xik u ik

i=1

subject to q  r =1 p 

yr j vr k ≤ 1, xi j u ik

i=1

u ik , vr k ≥ 0 ∀ i, r respectively. Model 2—Linear Model max Ek =

q 

yr k vr k

r =1

subject to p 

xik u ik = 1,

i=1 q  r =1

yr j vr k −

p  i=1

xi j u ik ≤ 0,

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u ik , vr k ≥ 0 ∀ i, r respectively.

4 Proposed FDEA Model To avert utilizing separate production frontiers to assess the relative efficiencies of DMUs, the FDEA models are proposed (Models 6, 7 and 8). The models use interval arithmetic operations and always work on the alike constraint group, resulting in a consistent and consolidated production frontier ∀ DMUs. The FDEA model is presented as follows: Model 3—FDEA Model [14] max E˜k =

q 

y˜r j vr k

r =1

subject to p 

˜ x˜ik u ik = 1,

i=1 q 

y˜r j vr k −

r =1

p 

˜ x˜i j u ik ≤ 0,

i=1

u ik , vr k ≥ 0 ∀ i, r respectively. In Model 3, input and output quantities are FNs, so resulted efficiency score will also be a fuzzy quantity. Here, we take them as TFNs,  x˜i j = xiLj , xiMj , xiUj  y˜r j = yrLj , yrMj , yrUj and the efficiency score for DMUk  E˜ k = E kL , E kM , E kU and constants 0˜ and 1˜ can be represented as 0˜ = (0, 0, 0) and 1˜ = (1, 1, 1).

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Hence, Model 3 will be converted to Model 4 as follows: Model 4 max Ek =

q  

yrLk , yrMk , yrUk vr k

r =1

subject to p  

U xikL , xikM , xik = (1, 1, 1),

i=1 q  

 yrLj , yrMj , yrUj vr k − xiLj , xiMj , xiUj u ik ≤ (0, 0, 0), p

r =1

i=1

u ik , vr k ≥ 0 ∀ i, r respectively. The α-cuts of x˜i j and y˜r j for α ∈ (0, 1] are given by 

(x˜i j )α = [(xi j )αL , (xi j )Uα ] = αxiMj + (1 − α)xiLj , αxiMj + (1 − α)xiUj ,

 ( y˜rI j )α = [(yr j )αL , (yr j )Uα ] = αyrMj + (1 − α)yrLj , αyrMj + (1 − α)yrUj . Now, using α-cuts in Model 4, we get Model 5 as follows: Model 5

max (E˜k )α = [(Ek )αL , (Ek )Uα ] =

q 

 αyrMk + (1 − α)yrLk , αyrMk + (1 − α)yrUk vr k

r =1

subject to p 

M  U αxik + (1 − α)xikL , αxikM + (1 − α)xik u ik = [1, 1], i=1

q p

  αyrMj + (1 − α)yrLj , αyrMj + (1 − α)yrUj vrk − αxiMj + (1 − α)xiLj , αxiMj + (1 − α)xiUj u ik ≤ [0, 0], r=1

i=1

The 2nd constraint implies that

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q p    M  M αyr j + (1 − α)yrLj vr k − αxi j + (1 − α)xiUj u ik , r =1

i=1 q   r =1

αyrMj + (1 − α)yrUj



vr k −

p  

αxiMj + (1 − α)xiLj



⎤ u ik ⎦ ≤ [0, 0],

i=1

u ik , vr k ≥ 0 ∀ i, r respectively. Henceforth, Model 5 can be reformed as follows: Model 5’

max (E˜k )α = [(Ek )αL , (Ek )Uα ] =

q 

 αyrMk + (1 − α)yrLk , αyrMk + (1 − α)yrUk vr k

r =1

subject to p 

M  U αxik + (1 − α)xikL , αxikM + (1 − α)xik u ik = [1, 1], i=1



q p    M  M αyr j + (1 − α)yrLj vr k − αxi j + (1 − α)xiUj u ik , r =1

i=1 q   r =1

αyrMj + (1 − α)yrUj



vr k −

p  

αxiMj + (1 − α)xiLj



⎤ u ik ⎦ ≤ [0, 0],

i=1

u ik , vr k ≥ 0 ∀ i, r respectively. Now using interval arithmetic, Model 5’ can be written as following Model 6: Model 6 max (E˜k )α = [(Ek )αL , (Ek )Uα ] =

q 

 αyrMk + (1 − α)yrLk , αyrMk + (1 − α)yrUk vr k

r =1

subject to p 

M  U αxik + (1 − α)xikL , αxikM + (1 − α)xik u ik = [1, 1], i=1

The Best State-Based Development of Fuzzy DEA Model

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q p    M  M αyr j + (1 − α)yrUj vr k − αxi j + (1 − α)xiLj u ik ≤ 0, r =1

i=1

u ik , vr k ≥ 0 ∀ i, r respectively. Model 6 is the FDEA model in the best state. By extracting lower and upper bounds, we get two models (Models 7 and 8), the lower and upper bound efficiency, respectively. Model 7—Lower bound efficiency max (Ek )αL =

q   M αyr k + (1 − α)yrLk vr k r =1

subject to

p   M U αxik + (1 − α)xik u ik = 1, i=1

q p    M  M αyr j + (1 − α)yrUj vr k − αxi j + (1 − α)xiLj u ik ≤ 0, r =1

i=1

u ik , vr k ≥ 0 ∀ i, r respectively.

Model 8—Upper bound efficiency max

(Ek )Uα

q   M αyr k + (1 − α)yrUk vr k = r =1

subject to

p  

αxikM + (1 − α)xikL u ik = 1,

i=1 q p    M  M U αyr j + (1 − α)yr j vr k − αxi j + (1 − α)xiLj u ik ≤ 0, r =1

i=1

u ik , vr k ≥ 0 ∀ i, r respectively.

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Table 1 Fuzzy input-output data DMUs Fuzzy inputs Input 1 (x1L , x1M , x1U ) 1 2 3 4 5

(3.5, 4.0, 4.5) (2.9, 2.9, 2.9) (4.4, 4.9, 5.4) (3.4, 4.1, 4.8) (5.9, 6.5, 7.1)

Input 2 (x2L , x2M , x2U )

Fuzzy outputs Output 1 (y1L , y1M , y1U )

Output 2 (y2L , y2M , y2U )

(1.9, 2.1, 2.3) (1.4, 1.5, 1.6) (2.2, 2.6, 3.0) (2.1, 2.3, 2.5) (3.6, 4.1, 4.6)

(2.4, 2.6, 2.8) (2.2, 2.2, 2.2) (2.7, 3.2, 3.7) (2.5, 2.9, 3.3) (4.4, 5.1, 5.8)

(3.8, 4.1, 4.4) (3.3, 3.5, 3.7) (4.3, 5.1, 5.9) (5.5, 5.7, 5.9) (6.5, 7.4, 8.3)

(Source Guo and Tanaka [8]) Table 2 Efficiencies of DMUs α↓

DMU1

(E1 )αL , (E1 )U α

DMU2



DMU3

DMU4

DMU5







(E2 )αL , (E2 )U (E3 )αL , (E3 )U (E4 )αL , (E4 )U (E5 )αL , (E5 )U α α α α

0.1

[0.654,0.906]

[0.863,0.995]

[0.587,1]

[0.868,1]

[0.662,1]

0.2

[0.685,0.914]

[0.892,1]

[0.621,1]

[0.881,1]

[0.694,1]

0.3

[0.718,0.92]

[0.923,1]

[0.656,0.999]

[0.895,1]

[0.727,1]

0.4

[0.738,0.915]

[0.941,1]

[0.686,0.978]

[0.91,1]

[0.761,1]

0.5

[0.757,0.909]

[0.956,1]

[0.716,0.957]

[0.924,1]

[0.797,1]

0.6

[0.777,0.901]

[0.97,1]

[0.746,0.937]

[0.939,1]

[0.834,1]

0.7

[0.796,0.892]

[0.981,1]

[0.775,0.916]

[0.953,1]

[0.873,1]

0.8

[0.816,0.881]

[0.99,1]

[0.805,0.897]

[0.969,1]

[0.913,1]

0.9

[0.835,0.868]

[0.996,1]

[0.833,0.878]

[0.984,1]

[0.955,1]

1

[0.854,0.854]

[1,1]

[0.86,0.86]

[1,1]

[1,1]

We use the following numerical example to illustrate the evaluation of efficiencies and the ranking of DMUs based on the proposed lower and upper bound efficiency models (Model 7 and Model 8).

5 Numerical Example In this numerical illustration, we have taken five DMUs into consideration. Here, each DMU consumes 2 inputs and generates 2 outputs. Data is presented in fuzzy environment, specifically taken as TFNs presented in Table 1. After applying Models 7 and 8, each DMU’s lower and upper bound efficiencies for α ∈ (0, 1] are presented in Table 2.

The Best State-Based Development of Fuzzy DEA Model Table 3 Ranking of DMUs DMU Rk 1 2 3 4 5

8.2626 9.7503 8.2598 9.6550 9.0469

325

Rank

GR

4 1 5 2 3

5 1 4 2 3

6 Proposed Aggregated Accuracy Function In DEA models, we rank the DMUs according to their efficiency scores in decreasing order. But in FDEA models, it’s not possible because α-cut gives two bounds (lower and upper). Consequently, the resulted efficiency score is in interval form. So, we propose the aggregated accuracy function (Rk , k = 1, 2, . . . , n) as follows: Rk =

1  

(Ek )αL .(Ek )Uα ,

k = 1, 2, . . . , n.

α=0.1

To rank the DMUs, the steps are as follows: Step 1: Find the lower and upper bound efficiencies of DMUs using models 7 and 8, respectively. 

Step 2: From step 1, obtain the efficiency scores (E j )αL , (E j )Uα of DMUs j = 1, 2, . . . , 5 for α = 0.1, 0.2, . . . , 1.0. Step 3: With the help of efficiency scores, calculate the aggregated accuracy function Rk for all DMUs. Step 4: Rank the DMUs, in decreasing order of their Rk values. Applying the above ranking method to a numerical example (Table 1) of five DMUs, we calculated Rk scores for all DMUs by using efficiency scores listed in (Table 1). In Table 3, the Rk scores are presented . The highest Rk score is 9.7503, which is associated with DMU2 . Hence, DMU2 is the best performer in this set of five DMUs. In order to validate the proposed ranking, the resulted rank of DMUs is compared to Ghasemi et al. [7]’ ranking (GR). In both the approaches, the first rank is given to DMU2 . From Table 3, we observe that the ranks of DMUs at ranks 1, 2 and 3 are the same in both the methods. This validates the proposed ranking approach.

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7 Conclusion In this study, the FDEA model is developed to assess the relative efficiencies of DMUs in a uncertain environment. We employ the same set of constraints for all DMUs in the proposed model, resulting in a consistent production frontier for all DMUs. Using α-cut, lower and upper bound efficiency models are developed. Further, to rank the DMUs, a ranking technique based on lower and upper bound FDEA performance efficiency is given. An example shown in Table 1, in which five DMUs each uses two fuzzy inputs and produces two fuzzy outputs in the form of TFNs, is used to show the proposed FDEA model. Lower and upper bound efficiency results are presented in Table 2. Ranked DMUs, on the basis of the proposed ranking method, resulted in DMU2 > DMU4 > DMU5 > DMU1 > DMU3 . DMU2 is the best performer in this DMUs group and DMU3 is the worst performer. The presented model is limited to TFNs. In the future, we can extend it to L-R type fuzzy numbers, trapezoidal fuzzy numbers and intuitionistic fuzzy environments.

References 1. Arya, A., Yadav, S.P.: Development of FDEA models to measure the performance efficiencies of DMUs. Int. J. Fuzzy Syst. 20(1), 163–173 (2018) 2. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2(6), 429–444 (1978) 3. Cooper, W.W., Park, K.S., Yu, G.: IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Manag. Sci. 45(4), 597–607 (1999) 4. Cooper, W.W., Park, K.S., Yu, G.: An illustrative application of idea (imprecise data envelopment analysis) to a Korean mobile telecommunication company. Oper. Res. 49(6), 807–820 (2001) 5. Despotis, D.K., Smirlis, Y.G.: Data envelopment analysis with imprecise data. Eur. J. Oper. Res. 140(1), 24–36 (2002) 6. Dotoli, M., Epicoco, N., Falagario, M., Sciancalepore, F.: A cross-efficiency fuzzy data envelopment analysis technique for performance evaluation of decision making units under uncertainty. Comput. Ind. Eng. 79, 103–114 (2015) 7. Ghasemi, M.-R., Ignatius, J., Lozano, S., Emrouznejad, A., Hatami-Marbini, A.: A fuzzy expected value approach under generalized data envelopment analysis. Knowl.-Based Syst. 89, 148–159 (2015) 8. Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets Syst. 119(1), 149–160 (2001) 9. Jahanshahloo, G.R., Soleimani-Damaneh, M., Nasrabadi, E.: Measure of efficiency in DEA with fuzzy input-output levels: a methodology for assessing, ranking and imposing of weights restrictions. Appl. Math. Comput. 156(1), 175–187 (2004) 10. Kao, C., Liu, S.-T.: Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst. 113(3), 427–437 (2000) 11. Sengupta, J.K.: A fuzzy systems approach in data envelopment analysis. Comput. Math. Appl. 24(8–9), 259–266 (1992) 12. Tavassoli, M., Saen, R.F., Zanjirani, D.M.: Assessing sustainability of suppliers: a novel stochastic fuzzy DEA model. Sustain. Prod. Consum. 21, 78–91 (2020) 13. Wang, Y.-M., Greatbanks, R., Yang, J.-B.: Interval efficiency assessment using data envelopment analysis. Fuzzy Sets Syst. 153(3), 347–370 (2005)

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14. Wang, Y.-M., Luo, Y., Liang, L.: Fuzzy data envelopment analysis based upon fuzzy arithmetic with an application to performance assessment of manufacturing enterprises. Expert Syst. Appl. 36(3), 5205–5211 (2009) 15. Wen, M., Li, H.: Fuzzy data envelopment analysis (DEA): model and ranking method. J. Comput. Appl. Math. 223(2), 872–878 (2009) 16. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965) 17. Zimmermann, H.-J.: Fuzzy Set Theory-and Its Applications. Springer Science & Business Media (2011)

Performance Evaluation of DMUs Using Hybrid Fuzzy Multi-objective Data Envelopment Analysis Awadh Pratap Singh and Shiv Prasad Yadav

Abstract The current study aims to evaluate the performance of decision-making units (DMUs) with the help of the fuzzy multi-objective (FMO) data envelopment analysis (DEA) model. The performance of DMUs is measured based on the optimistic and pessimistic efficiencies of DMUs. FMO optimistic (FMOO) and FMO pessimistic (FMOP), DEA models are developed to serve the purpose. An algorithm is developed to solve FMOO and FMOP DEA models. The geometric average efficiency approach ranks the DMUs based on the efficiencies obtained by FMOO and FMOP DEA models. Finally, an education sector application is presented to validate the acceptability of the proposed methodology. Keywords Performance analysis · Efficiency · Fuzzy multi-objective data envelopment analysis (FMODEA) · Fuzzy multi-objective optimistic (FMOO) · Fuzzy multi-objective pessimistic (FMOP) · Ranking method · Education sector efficiencies

1 Introduction Data envelopment analysis (DEA) is a non-parametric, data-based approach for assessing the performance of decision-making units (DMUs) that handle multiple inputs and outputs. Educational institutions, hospitals, banks, airlines, and other governmental agencies are examples of DMUs. Charnes et al. [1] are known for inventing the DEA technique. The output-to-input ratio of a DMU is defined as efficiency (efficiency = output/input). DEA models assess efficiencies from both optimistic and pessimistic perspectives. The ratio of DMU’s efficiency to the largest efficiency under consideration is called the optimistic efficiency or the best relative efficiency, or simply relative efficiency. It is computed from an optimistic viewpoint and lies in [0, 1]. The DMUs are called the optimistic efficient if they have an efficiency score A. P. Singh (B) · S. P. Yadav Indian Institute of Technology Roorkee, Roorkee 247667, India e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_23

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of 1, and called the optimistic non-efficient if they have an efficiency score of less than 1. The ratio of DMU’s efficiency to the smallest efficiency under consideration is called the pessimistic efficiency or the worst relative efficiency. It is computed from a pessimistic viewpoint. Its value is greater than or equal to l. The DMUs are called the pessimistic inefficient if they have an efficiency score of 1 and the pessimistic noninefficient if they have a pessimistic efficiency score greater than 1. We must assess both optimistic and pessimistic efficiencies simultaneously for the total performance of DMUs because we have two types of efficiencies: optimistic and pessimistic. In literature, Entani et al. [2], Azizi [3, 4] evaluated the efficiencies from both optimistic and pessimistic viewpoints in a crisp environment while Arya and Yadav [5] in the fuzzy environment. Gupta et al. [6] developed intuitionistic fuzzy optimistic and pessimistic multi-period portfolio optimization models. In their investigation, Puri and Yadav [7] created intuitionistic fuzzy optimistic and pessimistic DEA models. The goal of these studies was to create an interval using optimistic and pessimistic efficiencies. Optimistic efficiency is the lower end of the interval in all of these studies, whereas pessimistic efficiency is the upper end. In all the researches mentioned above, the lower bound of optimistic efficiency and the upper bound of pessimistic efficiency are evaluated. The ranking is based on an interval created by combining the optimistic and pessimistic efficiency DEA models’ lower bound and upper bound efficiency. We are not in favour of considering only one bound for each optimistic and pessimistic efficiencies. To overcome this shortcoming, we suggest considering both bounds of optimistic and pessimistic efficiency intervals for performance assessment. Based on this idea, we propose fuzzy multi-objective optimistic (FMOO) and fuzzy multi-objective pessimistic DEA models to rank the DMUs. Awadh et al. [8] proposed a fuzzy multi-objective DEA model to evaluate the performance of DMUs in a fuzzy environment. The advantage of the methodology is that it provides the complete ranking of DMUs. There are several multi-objective optimization techniques that exist in the literature for solving DEA models [9–12]. But to the best of our knowledge, this is the first study using the fuzzy multi-objective technique for performance evaluation of DMUs in the optimistic and pessimistic environment. In this study, we develop FMOO and FMOP DEA models and use Wang et al’s [13] geometric average efficiency approach to rank the DMUs. The rest of the paper is organized as follows. Section 2 presents the preliminaries and some basic definitions. The proposed FMOO and FMOP DEA models along with the solving techniques are described in Sect. 3. The complete hybrid fuzzy multi-objective (FMO) DEA technique is explained in Sect. 4. Section 5 presents the numerical illustration of the proposed methodology. Finally, Sect. 6 gives the concluding remarks and future scope.

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2 Preliminary For developing fuzzy optimistic and pessimistic DEA models, some basic definitions and operation are given in this section. Definition 1 (Fuzzy Set (FS) [14]) Let X be a universal set. A FS A˜ can be defined by A˜ = {(x, μ A˜ (x)) : x ∈ X }, wher e μ A˜ : X → [0, 1]. Definition 2 (Convex Fuzzy Set (CFS) [14]) A FS A˜ is defined as a CFS if for all x1 , x2 in X , min {μ A˜ (x1 ), μ A˜ (x2 )} ≤ μ A˜ (λx1 + (1 − λ)x2 ), wher e λ ∈ [0, 1]. Definition 3 (Fuzzy number (FN) [14]) A CFS A˜ is defined as a FN on the real line R if (i) (ii)

∃ an unique xo ∈ R with μ A˜ (xo ) = 1; μ A˜ is a continuous function in piece-wise sense,

˜ xo is called the mean value of A. Definition 4 (Triangular Fuzzy Number (TFN) [14]) The TFN A˜ = (a L , a M , a U ) is defined by the membership function μ A˜ given by ⎧ x−a L ⎪ aL < x ≤ aM; ⎨ a M −a L , U −x μ A˜ (x) = aaU −a a M ≤ x < aU ; M , ⎪ ⎩ 0 otherwise. ∀x ∈ R. Definition 6 if a L > 0.

(Positive TFN [14]) A TFN A˜ = (a L , a M , a U ) is positive if and only

2.1 Arithmetic Operations on TFNs Let A˜ = (a L , a M , a U ) and B˜ = (b L , b M , bU ) be two positive TFNs. Then the arithmetic operations on TFNs are defined as follows [15]: (i) (ii) (iii) (iv)

Addition: A˜ + B˜ = (a L + b L , a M + b M , a U + bU ), Subtraction: A˜ − B˜ = (a L − bU , a M − b M , a U − b L ), Multiplication: A˜ × B˜ ≈ (a L b L , a M b M , a U bU ), ˜ B˜ ≈ (a L /bU , a M /b M , a U /b L ) Division: A/

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3 Proposed Fuzzy Multi-objective Optimistic and Pessimistic DEA Models Suppose that we wish to evaluate the efficiencies of n homogenous DMUs (D MU j ; j = 1, 2, 3, . . . , n). Assume that D MU j uses m inputs xi j , i = 1, 2, 3, . . . , m and produces s outputs yr j , r = 1, 2, 3, . . . , s. Let u ik and vr k be the weights associated with the ith input xik and the r th output yr k of D MUk (k = 1, 2, 3, . . . , n). Let E kO and E kP stand for the optimistic and pessimistic efficiencies, respectively. Entani et al. [2] proposed the optimistic and pessimistic DEA models given in Table 1. Definition 8 ([5]) Let the optimal values of the optimistic and pessimistic DEA models for D MUk be E kO∗ and E kP∗ , respectively. Then D MUk is said to be optimistic efficient if E kO∗ = 1; otherwise optimistic non-efficient. On the other hand, D MUk is said to be pessimistic inefficient if E kP∗ = 1; otherwise pessimistic non-inefficient. Due to the ambiguity and fluctuation of real-world data, it is challenging to get accurate and reliable input and output data. Assume that the fuzzy inputs and outputs for the D MU j are x˜i j and y˜r j , respectively. Then the fuzzy optimistic (FO) and fuzzy pessimistic (FP) DEA models are described (see Table 2) as follows. Assume that the fuzzy input x˜i j = (xiLj , xiMj , xiUj ), fuzzy output y˜r j = L (yr j , yrMj , yrUj ) for the D MU j , and 1˜ = (1, 1, 1) are taken as TFNs. Then FO and FP models can be transformed into triangular fuzzy optimistic (TFO) and triangular fuzzy pessimistic (TFP) DEA models as follows (see Table 3). Now, we will propose a methodology to solve TFO and TFP DEA models given in Table 3. In this methodology, we will use Awadh et. al’s FMODEA [8]. First, let us try to develop FMOO DEA model for the efficiency evaluation of DMUs in an optimistic sense. The TFO DEA model, described in Table 3 is re-written in Model 7.

Table 1 Optimistic and pessimistic DEA models Optimistic DEA model (Model 1) Pessimistic DEA model (Model 2) For k = 1, 2, 3, . . . , n, s  yr k vr k r =1 max E kO = m  xik u ik

For k = 1, 2, 3, . . . , n, s  yr k vr k r =1 min E kP = m  xik u ik

i=1

s 

subject to

r =1 m 

i=1

s 

yr j vr k ≤ 1 ∀ j, xi j u ik

i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0.

subject to

r =1 m 

yr j vr k ≥ 1 ∀ j, xi j u ik

i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ∀k, ε > 0.

Performance Evaluation of DMUs Using Hybrid Fuzzy Multi-objective Data … Table 2 FO and FP DEA models FO DEA model (Model 3) For k = 1, 2, 3, . . . , n, s  y˜r k vr k r =1 O ˜ max E k = m  x˜ik u ik

FP DEA model (Model 4) For k = 1, 2, 3, . . . , n, s  y˜r k vr k r =1 P ˜ min E k = m  x˜ik u ik

i=1

s 

subject to

r =1 m 

y˜r j vr k

i=1

s 

≤ 1˜ ∀ j,

subject to

x˜i j u ik

i=1

Table 3 TFO and TFP DEA models TFO DEA model (Model 5) For k = 1, 2, 3, . . . , n, max(E kO,L , E kO,M , E kO,U ) = s  (yrLk , yrMk , yrUk )vr k

y˜r j vr k

≥ 1˜ ∀ j,

x˜i j u ik

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ∀k, ε > 0.

TFP DEA model (Model 6) For k = 1, 2, 3, . . . , n, min(E kP,L , E kP,M , E kP,U ) = s  (yrLk , yrMk , yrUk )vr k

r =1 m 

M , x U )u (xiL , xik ik ik i=1 s  L M U (yr j , yr j , yr j )vr k r =1 subject to m  L M U (xi j , xi j , xi j )u ik i=1

r =1 m  i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0.

r =1 m 

L , x M , x U )u (xik ik ik ik

i=1

≤ (1, 1, 1) ∀ j, subject to

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0.

Model 7

333

s 

(yrLj , yrMj , yrUj )vr k

r =1 m 

≥ (1, 1, 1) ∀ j,

(xiLj , xiMj , xiUj )u ik

i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ∀k, ε > 0.

For k = 1, 2, 3, . . . , n,

   ( s vr k yrLk , rs =1 vr k yrMk , rs =1 vr k yrUk ) Max (E kO,L , E kO,M , E kO,U ) = rm=1 (1)   m m U L, M ( i=1 u ik xik i=1 u ik x ik , i=1 u ik x ik )    ( rs =1 vr k yrLj , rs =1 vr k yrMj , rs =1 vr k yrUj ) ≤ (1, 1, 1) ∀ j = 1, 2, 3, . . . , n; subject to m   m m ( i=1 u ik xiLj , i=1 u ik xiMj , i=1 u ik xiUj )

(2) u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0.

On applying division rule for two TFNs, Model 7 reduces to Model 8 as follows:

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A. P. Singh and S. P. Yadav

Model 8

For k = 1, 2, 3, . . . , n,

 s vr k y L s vr k y M s vr k y U  r =1 r =1 r =1 rk rk , rk , m U m u x M m u x L u x ik ik ik i=1 i=1 i=1 ik ik ik  s U M s  rs =1 vr k y L r =1 vr k yr j r =1 vr k yr j rj subject to m , , ≤ (1, 1, 1) ∀ j = 1, 2, 3, . . . , n; U m u x M m u x L i=1 ik i j i=1 ik i j i=1 u ik xi j Max (E kO,L , E kO,M , E kO,U ) =

(3) (4)

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0.

TFNs have the following property: s

L r =1 vr k yr j m U i=1 u ik x i j

s ≤

M r =1 vr k yr j m M i=1 u ik x i j

s

U r =1 vr k yr j L i=1 u ik x i j

≤ m

(5)

Now, using the property stated in Eq. 5, Model 8 can be transformed into FMOO DEA model as follows. Model 9

(Proposed FMOO DEA model): For k = 1, 2, 3, . . . , n, s v y L s v y M s v y U

rk rk rk rk rk rk Max rm=1 , rm=1 , rm=1 U M L u x u u x i=1 ik ik i=1 ik x ik i=1 ik ik s U r =1 vr k yr j subject to m ≤ 1 ∀ j = 1, 2, 3, . . . , n; L i=1 u ik x i j

(6) (7)

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0. Now, using Charnes–Cooper transformation [16], Model 9 can be converted into Model 10 as follows. Model 10

(Proposed FMOO DEA model): For k = 1, 2, 3, . . . , n, Max

s 

vr k yrLk ,

r =1

subject to

s 

vr k yrMk ,

r =1 m 

U u ik xik =1

s 

vr k yrUk

(8)

r =1

(9)

i=1 m  i=1 m  i=1

u ik xikM = 1

(10)

u ik xikL = 1

(11)

Performance Evaluation of DMUs Using Hybrid Fuzzy Multi-objective Data … s 

vr k yrUj −

r =1

m 

u ik xiLj ≤ 0 ∀ j = 1, 2, 3, . . . , n;

335

(12)

i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0. Model 10 is a deterministic FMOO DEA model which provides the efficiency of DMUs in optimistic sense. Similarly, we will convert the TFP DEA model into FMOP DEA model by using Awadh et al.’s [8] FMODEA model as follows. Model 11

(Proposed FMOP DEA model): For k = 1, 2, 3, . . . , n, s v y L s v y M s v y U

rk rk rk rk rk rk , rm=1 , rm=1 Min rm=1 U M L i=1 u ik x ik i=1 u ik x ik i=1 u ik x ik s L r =1 vr k yr j subject to m ≥ 1 ∀ j = 1, 2, 3, . . . , n; U i=1 u ik x i j

(13) (14)

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0. Now, using Charnes–Cooper transformation [16], Model 11 can be convereted into Model 12 as follows. Model 12

(Proposed FMOP DEA model): For k = 1, 2, 3, . . . , n, Min

s 

vr k yrLk ,

r =1

subject to

s 

vr k yrMk ,

r =1 m 

s 

vr k yrUk

(15)

r =1

U u ik xik =1

(16)

i=1 m  i=1 m  i=1 s  r =1

u ik xikM = 1

(17)

u ik xikL = 1

(18)

vr k yrLj

−

m 

u ik xiUj ≥ 0 ∀ j = 1, 2, 3, . . . , n;

(19)

i=1

u ik ≥ ε ∀i, vr k ≥ ε ∀r, ε > 0. Model 12 is a deterministic FMOP DEA model which provides the efficiency of DMUs in pessimistic sense.

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3.1 Algorithm for Solving the Proposed FMOO and FMOP DEA Models This section presents an algorithm for solving the proposed FMOO and FMOP DEA models. Algorithm is given as follows: Step 1: First of all, for each DMU, convert MOOP to SOOP (single objective optimization problem) with the help of the weighted sum method [17]. Step 2: In this step, random weights are generated. Suppose that there are d objectives to be optimized with p variables. Then generate (100 × p) set of d weights. There is no thumb rule for population selection. It depends upon the decisionmaker. For the computational purpose, we take a hundred times the number of variables present in the problem. Step 3: Solve the SOOP formulated in Step 1 with the help of the weights generated in Step 2 for each DMU. Step 4: 100 × p Pareto solutions are obtained from Step 3. Choose the most favourable solution among 100 × p. Step 5: The solution obtained in Step 4 is the efficiency of DMU. Step 6: Use above steps to determine E kO∗ and E kP∗ . After getting both optimistic and pessimistic efficiency scores we rank the DMUs by considering both the efficiencies simultaneously. To rank the DMUs we will the geometric average efficiency approach, proposed by Wang et al. [13]. According to Wang et al. [13], if E kO∗ and E kP∗ are the optimistic and pessimistic efficiencies, geometric respectively, for D MUk , then the geometric average efficiency E k is defined as follows: geometric = E k∗O × E k∗P (20) Ek Wang et al. [13] proposed geometric average efficiency approach for the DMUs with crisp input and output data. We extend this idea to the DMUs with fuzzy input and output data, particularly triangular fuzzy data.

4 Complete Hybrid FMODEA Performance Decision Process In complex real-world problems, the hybridization of DEA/FDEA/FMODEA using other techniques is very effective [2, 3, 5]. These researchers handled both the optimistic and pessimistic DEA models simultaneously. The hybridization process can be divided into four steps as follows: (i) Input-output data selection and collection phase: In this phase, the decisionmaker follows the following steps: (a) selection of the relevant input and output

Performance Evaluation of DMUs Using Hybrid Fuzzy Multi-objective Data …

337

data variables, (ii) Collection of input–output data quantitatively and qualitatively, (iii) classification of the data according to its nature, crisp or fuzzy, (iv) fuzzification of the data based on criteria and expert’s opinion. (ii) Efficiency measurement phase: During this phase, the decision-maker chooses the best method for evaluating the performances. The best and worst performance are obtained using the suggested fuzzy optimistic and pessimistic DEA models. Consequently, the suggested FMODEA strategy leads to the overall performance of DMUs. As a result, the hybrid FMODEA technique is ideal for a fair decision-making process. (iii) Ranking phase: Using the geometric average ranking approach, the complete ranking is obtained by selecting optimistic and pessimistic FMODEA models. (iv) Recommendation phase: This is the final stage of the decision-making process, in which policy-makers and experts give recommendations based on the ranking results obtained in the previous phase. The recommendations include suggestions for critical modifications that the management must undertake in order to increase the DMUs’ efficiencies. Figure 1, depicts the suggested hybrid FMODEA performance efficiency evaluation process.

5 Numerical Illustration In this section, we use an example to demonstrate the efficacy of the proposed models. This example is based on Guo and Tanaka’s study [18], which used five DMUs with two fuzzy inputs and two fuzzy outputs. We try to validate the proposed models and ranking approach using this scenario. A case study in the field of education is also provided.

5.1 Numerical Illustration: An Example Table 4 presents the fuzzy input-output data for the problem considered by Guo and Tanaka [18], which has five DMUs with 2 fuzzy inputs and 2 fuzzy outputs. The optimistic and pessimistic efficiency scores are calculated by using the proposed FMOO and FMOP DEA models, respectively. Table 5 presents the outcomes (E k∗O and E k∗P ) of both the models. Based on the efficiency scores (E k∗O , E k∗P ) obtained from FMOO and FMOP DEA models; the geometric efficiency score is obtained by using Eq. (20). The ranking is geometric . The ranking obtained from the proposed FMODEA done based on the E k model (see Table 7) is compared with Wang et al.’s [19] method. Since both the methods provide different levels of efficiency scores; so it will be appropriate to compare the ranks of DMUs with the help of Spearman’s rank correlation coefficient

338

A. P. Singh and S. P. Yadav Selection of the relevant input and output data variables for performance evaluation of the DMUs

Sources:

Literature review Experts’ opinion

Selection Approaches

Data Collection

Identify crisp/ fuzzy Data Variables Fuzzification of data using Expert’s opinion

Representation of crisp data in fuzzy form

Final input and output data set in the form of TFNs

Performance evaluation using FDEA approach

Selection of FMO DEA model Fuzzy pessimistic DEA model (worst performance)

Fuzzy optimistic DEA model (best performance)

Pessimistic efficiency

Optimistic efficiency

Use the geometric average ranking approaches

Rank the DMUs on the basis of the proposed ranking approach

Recommendations

Fig. 1 Schematic view of the proposed hybrid FMODEA performance decision model

Performance Evaluation of DMUs Using Hybrid Fuzzy Multi-objective Data … Table 4 Fuzzy input-output data for 5 DMUs DMUs Fuzzy inputs Input 1 Input 2 D MU A D MU B D MUC D MU D D MU E

(3.5, 4.0, 4.5) (2.9, 2.9, 2.9) (4.4, 4.9, 5.4) (3.4, 4.1, 4.8) (5.9, 6.5, 7.1)

(1.9, 2.1, 2.3) (1.4, 1.5, 1.6) (2.2, 2.6, 3.0) (2.1, 2.3, 2.5) (3.6, 4.1, 4.6)

339

Fuzzy outputs Output 1

Output 2

(2.4, 2.6, 2.8) (2.2, 2.2, 2.2) (2.7, 3.2, 3.7) (2.5, 2.9, 3.3) (4.4, 5.1, 5.8)

(3.8, 4.1, 4.4) (3.3, 3.5, 3.7) (4.3, 5.1, 5.9) (5.5, 5.7, 5.9) (6.5, 7.4, 8.3)

(Source Guo and Tanaka [18]) Table 5 Optimistic efficiency scores (E k∗O ) of DMUs (Example) DMUs

Most favourable weights

Most favourable solutions

Efficiency (E k∗O )

w1

w2

w3

v1

v2

u1

D MU A

0.0283

0.0414

0.9303

0.1017

0.0858

2.05E-05 0.4733

0.6579

D MU B

0.0065

0.0003

0.9932

3.68E-05 0.1987

3.45E-01 1.00E-05

0.7347

D MUC

0.0183

0.0315

0.9502

1.13E-01 0.0459

2.03E-01 1.00E-05

0.6822

D MU D

0.0094

0.1102

0.8804

0.0020

1.36E-01 1.00E-05 4.33E-01

0.8061

D MU E

0.0238

0.0189

0.9573

1.33E-01 0.0040

u2

1.53E-01 1.00E-05

0.8011

(Source Guo and Tanaka [18]) Table 6 Pessimistic efficiency scores (E k∗P ) of DMUs (Example) DMUs

Most favourable weights

Most favourable solutions v2

w1

w2

w3

v1

D MU A

0.9702

0.0136

0.0162

0.1056

D MU B

0.9711

0.0066

0.0223

5.87E-05 0.4330

D MUC

0.9257

0.0444

0.0298

0.1905

D MU D

0.9462

0.0401

0.0138

0.4805

1.43E-04 1.21E-05 4.33E-01

1.2151

D MU E

0.8932

0.1068

7.15E-05 0.0004

1.92E-01 1.53E-01 1.00E-05

1.2675

0.2639

u1

Efficiency (E k∗P ) u2

1.00E-05 0.4733

1.2611

0.3448

1.00E-05

1.4335

1.35E-01 2.03E-01 1.00E-05

1.1158

(Source Guo and Tanaka [18])

[20]. The correlation coefficient (ρ = 0.883) indicates that the efficiency obtained from both models is highly correlated. This indicates that the proposed methodology is quite efficient to rank the DMUs in fuzzy environment.

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Table 7 The proposed and geometric average efficiency scores and ranks of DMUs geometric DMUs E k∗O E k∗P Ek Proposed Wang et al. Rank rank D MU A D MU B D MUC D MU D D MU E

0.6579 0.7347 0.6822 0.8061 0.8011

1.2611 1.4335 1.1158 1.2151 1.2675

0.9108 1.0262 0.8725 0.9867 1.0076

4 1 5 3 2

1.090 1.154 1.092 1.154 1.154

3 1 2 1 1

5.2 Education Sector Application In this section, the acceptability of the proposed models is validated by considering the Indian Institutes of Management (IIMs) in India as an education sector application. In this real-life application, 13 IIMs are considered with 2 fuzzy inputs (Number of students (x1 ), Number of faculty members (x2 )) and 2 fuzzy outputs (Placements and higher studies (y1 ), Publications (y2 )). The input–output data is taken from Awadh et al.’s [8] study. The input–output data is in the form of TFNs and is given in Table 8. The E k∗O and E k∗P are calculated by using the proposed DEA models. The efficiencies are calculated by using the proposed algorithm (Sect. 3.1). The results for the optimistic and pessimistic efficiencies are shown in Tables 9 and 10, respectively.

Table 8 Input and output data for IIMs DMU

IIM Name

State

Inputs

Outputs

x˜1

x˜2

y˜1

y˜2

D1

IIM Bangalore

Karnataka

(424,682,955)

(91,104,113)

(393,410,449)

(72,136,212)

D2

IIM Ahmedabad

Gujarat

(461,715,992)

(91, 112,128)

(411,421,427)

(30,109,217)

D3

IIM Calcutta

West Bengal

(487,803,1042)

(86,94,105)

(483,505,535)

(29,109,207)

D4

IIM Lucknow

Uttar Pradesh

(455,725,990)

(81,88,95)

(440,456,506)

(8,65,126)

D5

IIM Indore

Madhya Pradesh

(549,1020,1657) (73,94,104)

(508,593,634)

(16,66,141)

D6

IIM Kozhikode

Kerala

(370,593,806)

(58,69,77)

(347,360,382)

(28,74,97)

D7

IIM Udaipur

Rajasthan

(120,260,419)

(21,46,101)

(120,136,171)

(14,37,67)

D8

IIM Tiruchirappalli

Tamilnadu

(108,228,387)

(25,37,52)

(102,121,172)

(5,16,24)

D9

IIM Raipur

Chhattisgarh

(160,270,438)

(21,33,46)

(111,140,193)

(12,35,52)

D10

IIM Rohtak

Haryana

(158,276,428)

(20,28,34)

(137,146,155)

(31,52,69)

D11

IIM Shillong

Meghalaya

(155,263,365)

(27,27,28)

(118,146,172)

(3,15,30)

D12

IIM Kashipur

Uttarakhand

(125,262,472)

(15,28,38)

(101,123,164)

(7,21,36)

D13

IIM Ranchi

Jharkhand

(189,300,452)

(16,29,40)

(156,169,178)

(11,22,51)

Performance Evaluation of DMUs Using Hybrid Fuzzy Multi-objective Data …

341

Table 9 Optimistic efficiency scores (E k∗O ) of IIMs DMUs

Most favourable weights

Most favourable solutions

Efficiency ( E k∗O )

w1

w2

w3

v1

v2

u1

u2

D1

0.01787

0.0778

0.9043

444.153

3324.761

2285.578

2.33E+03

D2

0.0478

0.1005

0.8518

1396.474

16176.274

9418.478

10935.333

0.6898

D3

0.0499

0.196

0.754

806.660

2433.738

1982.503

5583.592

0.5587

D4

0.0221

0.3803

0.5976

1546.952

2440.449

2880.0246

9311.586038 0.4327

D5

0.0477

0.1005

0.8518

365.137

820.407

480.326

9054.454

D6

0.2034

0.0314

0.7652

1602.965

3615.045

3725.355

20785.561

0.3215

D7

0.3551

0.1704

0.4744

642.319

1230.289

1288.628

2208.538

0.5495

D8

0.3664

0.2204

0.4132

425565.259 203161.842 647291.776 757192.279

0.4879

D9

0.0383

0.1884

0.7732

338.6579

0.3668

D10

0.1080

0.1712

0.7207

115.1186

423.1365

290.529

3449.675

0.3456

D11

0.2775

0.0378

0.6846

375.977

475.993

644.032

2112.405

0.3981

D12

0.0650

0.4640

0.4709

1410.191

3379.327

2175.496

32052.563

0.3053

D13

0.0845

0.3342

0.5813

283.888

1052.874

229.074

10865.027

0.3498

1003.258

701.589

7803.547

0.7226

0.4108

Table 10 Pessimistic efficiency scores (E k∗P ) of IIMs DMUs

Most favourable weights

Most favourable solutions

Efficiency (E k∗O )

w1

w2

w3

v1

v2

u1

u2

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13

0.9356 0.9683 0.9848 0.9077 0.942 0.9702 0.9903 0.971 0.9367 0.0247 0.9437 0.9515 0.9538

0.052 0.0217 0.0006 0.0701 0.0136 0.0161 0.0013 0.0065 0.0353 0.0294 0.0369 0.0105 0.0406

0.0123 0.0099 0.0144 0.0221 0.0443 0.0136 0.0083 0.0223 0.0278 2.25E-05 0.0192 0.0378 0.0054

0.0081 0.0074 0.0087 0.0004 0.00047 0.0121 0.0144 0.0065 0.023 1.00E-05 0.0098 0.0277 0.0131

1.01E-05 1.01E-05 1.01E-05 1.08E-01 1.03E-01 1.01E-05 3.55E-05 1.19E-01 1.36E-04 0.0349 1.79E-01 1.47E-04 1.99E-05

1.00E-05 1.00E-05 1.00E-05 1.00E-05 1.00E-05 1.00E-05 3.10E-03 1.00E-05 1.00E-05 4.033 1.00E-05 1.00E-05 2.52E-03

0.0095 0.0088 0.0103 0.0112 0.0107 0.0144 1.00E-05 0.0241 0.0273 4.2989 0.0364 0.0329 0.0036

3.2013 3.0711 4.2401 1.7888 2.5246 4.2338 1.7457 1.3322 2.6345 4.5642 1.8937 2.8733 2.0613

The ranking of DMUs depends upon the choice of the selection of input–output data. The aim of this piece of work is to provide a novel approach for evaluating DMU’s performance. The ranking is done based on the efficiencies obtained from Tables 9 and 10. The proposed ranking method is used and DMUs are ranked as presented in Table 11.

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Table 11 The proposed geometric average efficiency scores and ranks of DMUs geometric DMUs E k∗O E k∗P Ek Ranks D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D12 D13

0.7226 0.6898 0.5587 0.4328 0.4108 0.3215 0.5495 0.4879 0.3668 0.3456 0.3981 0.3050 0.3498

3.2013 3.0711 4.2401 1.7888 2.5246 4.2338 1.7457 1.3323 2.6345 4.0468 1.8937 2.8733 2.0613

1.5209 1.4554 1.5392 0.8798 1.1084 1.1667 0.9795 0.8063 0.9829 1.1826 0.8683 0.9366 0.8491

2 3 1 10 6 5 8 13 7 4 11 9 12

6 Conclusions In this paper, an FMODEA model is proposed to evaluate the performance of DMUs by simultaneously considering optimistic and pessimistic efficiencies. To calculate optimistic (E k∗O ) and pessimistic (E k∗P ) efficiencies of DMUs in a fuzzy environment, the FMOO and FMOP DEA models are developed. An algorithm is also proposed to solve the proposed FMOO and FMOP DEA models. The geometric average efficiency approach is adopted to combine both (E k∗O ) and (E k∗P ). The ranking is based on the geometric efficiency scores obtained by the proposed methodology. To validate the acceptability of the proposed methodology, an education sector application is presented in which 13 IIMs are considered. It is found that among 13 IIMs, IIM Calcutta is the best performing institute, while IIM Tiruchirappalli is the worst performing. The advantage of the proposed method is that it provides a complete ranking for DMUs under considering optimistic and pessimistic efficiencies simultaneously.

References 1. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2(6), 429–444 (1978) 2. Entani, T., Maeda, Y., Tanaka, H.: Dual models of interval DEA and its extension to interval data. Eur. J. Oper. Res. 136(1), 32–45 (2002) 3. Azizi, H.: The interval efficiency based on the optimistic and pessimistic points of view. Appl. Math. Model. 35(5), 2384–2393 (2011) 4. Azizi, H.: DEA efficiency analysis: a DEA approach with double frontiers. Int. J. Syst. Sci. 45(11), 2289–2300 (2014)

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5. Arya, A., Yadav, S.P.: Development of optimistic and pessimistic models using FDEA to measure performance efficiencies of DMUs. Int. J. Process. Manag. Benchmarking 9(3), 300– 322 (2019) 6. Gupta, P., Mehlawat, M.K., Yadav, S., Kumar, A.: Intuitionistic fuzzy optimistic and pessimistic multi-period portfolio optimization models. Soft Comput. 1–26 (2020) 7. Puri, J., Yadav, S.P.: Intuitionistic fuzzy data envelopment analysis: an application to the banking sector in India. Expert Syst. Appl. 42(11), 4982–4998 (2015) 8. Singh, A., Yadav, S., Singh, S.: A multi-objective optimization approach for DEA models in a fuzzy environment. Soft Comput. (2022). https://doi.org/10.1007/s00500-021-06627-y 9. Wang, W.-K., Lu, W.-M., Liu, P.-Y.: A fuzzy multi-objective two-stage DEA model for evaluating the performance of us bank holding companies. Expert Syst. Appl. 41(9), 4290–4297 (2014) 10. Chen, W., Li, S.-S., Zhang, J., Mehlawat, M.K.: A comprehensive model for fuzzy multiobjective portfolio selection based on DEA cross-efficiency model. Soft Comput. 24(4), 2515– 2526 (2020) 11. Boubaker, S., Do, D.T., Hammami, H., Ly, K.C.: The role of bank affiliation in bank efficiency: a fuzzy multi-objective data envelopment analysis approach. Ann. Oper. Res. 1–29 (2020) 12. Zamani, P.: The position of multiobjective programming methods in fuzzy data envelopment analysis. Int. J. Math. Model. Comput. 10(2) (SPRING), 95–101 (2020) 13. Wang, Y.-M., Chin, K.-S., Yang, J.-B.: Measuring the performances of decision-making units using geometric average efficiency. J. Oper. Res. Soc. 58(7), 929–937 (2007) 14. Zimmermann, H.-J.: Fuzzy Set Theory-and Its Applications. Springer Science & Business Media (2011) 15. Wang, J.-W., Cheng, C.-H., Huang, K.-C.: Fuzzy hierarchical TOPSIS for supplier selection. Appl. Soft Comput. 9(1), 377–386 (2009) 16. Charnes, A., Cooper, W., Lewin, A.Y., Seiford, L.M.: Data envelopment analysis theory, methodology and applications. J. Oper. Res. Soc. 48(3), 332–333 (1997) 17. Marler, R.T., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multidiscip. Optim. 41(6), 853–862 (2010) 18. Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets Syst. 119(1), 149–160 (2001) 19. Wang, Y.-M., Yang, J.-B., Xu, D.-L., Chin, K.-S.: On the centroids of fuzzy numbers. Fuzzy Sets Syst. 157(7), 919–926 (2006) 20. Zar, J.H.: Spearman rank correlation. Encycl. Biostat. 7 (2005)

Development of Intuitionistic Fuzzy Data Envelopment Analysis Model Based on Interval Data Envelopment Analysis Model Meena Yadav and Shiv Prasad Yadav

Abstract In the present study, we highlight the importance of taking hesitation into account when calculating the efficiencies of firms or decision-making units (DMUs) with intuitionistic fuzzy (IF) data. Most of the previous studies related to the calculation of efficiencies of DMUs with IF data have ignored the hesitation present in IF variables. The present study uses the technique of data envelopment analysis (DEA) to calculate the relative efficiencies of DMUs. It develops a new model for the calculation of IF efficiencies in the form of intervals. We conclude that the hesitation is indirectly related to the efficiencies of DMUs and deduce a formula to calculate the reduced efficiencies. We also present a method for ranking firms. Lastly, an example is illustrated to verify the working of the developed model. Keywords Intuitionistic fuzzy number · Intuitionistic fuzzy sets · Hesitation · Efficiency · Interval efficiency

1 Introduction The traditional set theory originated from the crisp set theory, where either an element belongs or does not belong to the set. There is no ambiguity regarding an elements’ belongingness/not belongingness in the set. For example, whether a student has done the assignment or not. It is a yes or no answer to whether a particular element (student) is contained in the set (of students that have done their assignments) or not. The function that defines the containment of elements in the crisp set is called the characteristic function. This concept of crisp set theory was extended by Zadeh [27] when he gave the concept of fuzzy set theory. It was based on the fact that there is always some ambiguity regarding the membership of elements in the set. For example, consider the set of intelligent students in a class. The answer to whether a student is intelligent could be a yes or no but it does not reveal all the information M. Yadav (B) · S. P. Yadav Department of Mathematics, IIT Roorkee, Roorkee, Uttarakhand, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_24

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regarding the question. However, a simple yes or no does not clarify the extent to which a student is intelligent. So, Zadeh [27] introduced the concept of membership function. For each element, the membership function defines a value between 0 to 1 to define its level of presence in the set. For example, based on the previous year’s grades, the teacher concludes that the fuzzy set of intelligence of students could be ˜ as given below. From the set, we get the information that Arti is intelligent, Rakhi A, might be intelligent, Sunil is better than average, Arun is not really intelligent, and Anil is not intelligent at all. A˜ = {(Aar ti, 1), (Rakhi, 0.5), (Sunil, 0.7), (Ar un, 0.2), (Anil, 0)}. Now, based on the class performance of students, the teacher also knows the weak points or what is lacking in the students, i.e., she can also estimate the level of non-intelligence present in students. This led to the extension of fuzzy sets (FS) to intuitionistic fuzzy sets (IFS) by introducing the concept of non-membership and hesitation present in a variable by Atanassov [4]. Like membership function, he defined the non-membership function to define the extent to which an element does not belong to the set. For example, the above set can be rewritten as A˜ I given below. The set informs that Aarti is brilliant and does not lack anything, Rakhi might be intelligent but she lacks a few qualities, Sunil is intelligent but lacks a few qualities; Arun is not really intelligent and needs a lot of improvement and Anil is not intelligent at all and needs improvement in everything. Here, the hesitation can be defined as the lack of knowledge of teacher about each student in defining the membership and non-membership grades. For Aarti, the hesitation is zero (1-1-0), for Rakhi, it is 0.2 (1-0.5-0.3), for Sunil, it is 0.1 (1-0.7-0.2), for Arun, it is 0.2 (1-0.2-0.6), and for Anil it is zero (1-0-1). A˜ I = {(Aar ti, 1, 0), (Rakhi, 0.5, 0.3), (Sunil, 0.7, 0.2), (Ar un, 0.2, 0.6), (Anil, 0, 1)}

The IFS A˜ I gives all the desired information about the set. There are well-defined operations for addition, subtraction, multiplication, and division of IFSs [26]. Now, the efficiency of a decision-making unit (DMU) under consideration is defined as the ratio of the sum of weighted outputs to the sum of weighted inputs. One such non-parametric technique to find the relative efficiency of homogeneous DMUs (with crisp data) and to identify the best performer in the DMUs is the technique of data envelopment analysis (DEA), proposed by Charnes et al. [8]. Wang et al. [24] proposed an interval efficiency model for crisp data using DEA and extended it to fuzzy sets. IF variables are preferred over fuzzy variables as they define linguistic variables mathematically in a more realistic way. IF variables convey the information of membership, non-membership values, and the lack of knowledge of decision maker in deciding these values. Fuzzy sets are unable to express this lack of knowledge. Being a hot topic of research in Operations research, many studies have been done to calculate the efficiency of DMUs with IF input and output variables [1, 11, 15, 16].

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Table 1 Example DMU A DMU B

Input

Output

(2,0.8,0.1) (2,0.6,0.1)

(2,0.8,0.1) (2,0.6,0.1)

Most of the recent studies have ignored the effect of hesitation on efficiency. Hesitation indirectly affects the efficiency of a DMU that utilizes IF inputs to produce IF outputs. This effect can differ from small to large depending on the extent of hesitation present in the DMU. For example, consider 2 DMUs A and B with single input and output as given in Table 1. Both the input and output of DMU A have a hesitation of 0.1. Hesitation of both the input and output of DMU B is 0.3. Now though both DMUs are using the same amount of input (2 units) to produce the same amount of output (2 units). The information about DMU A is more precise than DMU B. In other words, the ambiguity regarding the knowledge of input and output of DMU A is less than that of B. Lower hesitation in variables means higher precision in membership and non-membership values. Hence, DMU A should be more efficient than DMU B. In the present study, we propose a new IF efficiency incorporated with hesitation and identify the best performer among the set of homogeneous DMUs. We will also rank the DMUs based on the proposed efficiency. The rest of the paper is organized as follows. Section 2 deals with the recent developments in IFS theory and IF efficiency. Section 3 defines the preliminaries of IFSs and basic definitions of IFSs. Section 4 presents the methodology of the present study, followed by the merits of the proposed model in Sect. 5, a numerical example in Sect. 6, and conclusion in Sect. 7.

2 Literature Review IFSs have a wide range of applications (in medical diagnosis [9], career determination [12], pattern recognition [10], Clustering algorithm [6], multi-attribute information classification [21], banking [19], risk assessment methodology [7], health sector [3], transportation problem [22], and many more). The wide field of applications of IFSs has also led to the development of various IF models ([13, 25], and many others). Arya and Yadav [2] used the α − and β − cuts to calculate the efficiency of DMUs with IF variables and proposed input and output targets. Santos Arteaga et al. [20] developed an alphabetical approach to solve the IF efficiency model. Hajiagha et al. [14] developed an IF efficiency model based on an aggregation operator. Mohammadi Ardakani et al. [18] developed a model based on Stackelberg game theory and efficiency decomposition using the best and worst return approach of Azizi and Wang [5]. Szmidt et al. [23] gave a measure of the amount of knowledge conveyed by IFSs

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by finding a relationship between the positive information (membership value), negative information (non-membership value), and the lack of information (hesitation). Puri and Yadav [19] developed the max-min approach to calculate the efficiencies and ranks of DMUs with IF variables. To the best of our knowledge, no one has studied the effect of hesitation on the efficiency of DMUs.

3 Preliminaries 1. Efficiency: The efficiency of a DMU is its ability to judiciously use available inputs to produce the maximum outputs. It is defined as the ratio of the total output produced to the total input used. 2. CCR Model: Consider a homogeneous set of ‘n’ DMUs using ‘m’ number of inputs to produce ‘s’ number of outputs. The CCR model [8] to find the relative efficiency of DMUk , k = 1, 2, . . . , n, is given by Model 1 s vrk .yrk Max E k = rm=1 i=1 u ik .x ik s vrk .yrj subject to rm=1 ≤ 1, i=1 u ik .x ij

1 ≤ j ≤ n,

u ik ≥ 0,

1 ≤ i ≤ m,

vrk ≥ 0,

1 ≤ r ≤ s,

where u ik and vrk are unknown weights associated with the ith input and r th output of DMUk . The CCR model measures the relative efficiency of a homogeneous set of ’n’ DMUs in the interval (0,1]. DMUk is said to be efficient if the optimal value of Model 1 is E k ∗ = 1 and the optimal weights u ik ∗ and vrk ∗ are not all zero. 3. Fuzzy Set [27]: A fuzzy subset A˜ of a universal set X is defined by its membership function μ A˜ : X → [0, 1], where the value of μ A˜ (x) shows the degree of ˜ membership of x in A. 4. Intuitionistic fuzzy set [4]: An IFS A˜ I in the universal set X is defined by A˜ I = {< x, μ A˜I (x) , ν A˜I (x) > : x ∈ X }, where μ A˜I : X → [0, 1] is called the membership function and ν A˜I : X → [0, 1] is called the non-membership function of A˜ I . The function π A˜I : X → [0, 1] defined by π  A I (x) = 1 − μ  A I (x) − ν  A I (x) denotes the degree of hesitation asso-

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ciated with the element x in A˜ I . If the hesitation π A˜I (x) = 0 ∀ x ∈ X , then the IFS A˜ I is reduced to a fuzzy set A˜ in X . 5. Triangular Intuitionistic Fuzzy Number (TIFN) [2]: A TIFN A˜ I is written as (a, b, c; a’, b, c’), is an IFS with the membership function μ A (x) and nonmembership function ν A (x) given by μ A˜ I (x) =

x−a , b−a x−c , b−c

0,



b−x a < x ≤ b; , a < x ≤ b; b−a  x−b b ≤ x < c; andν A˜I (x) = c −b , b ≤ x < c ; other wise; 1, other wise.

6. Alpha cut [17] For α ∈ (0, 1], the α − cut of an IFS set A˜ I , denoted by A I α , is defined by A I α = {x ∈ X : μ A˜I (x) ≥ α}.

4 Methodology 1. Efficiency of TIFNs: Various studies are done to develop efficiency models for DMUs with TIFN data. We propose to develop a new efficiency model to calculate the efficiencies of DMUs using Wang’s efficiency model [24] with inputs and outputs in the form of interval data. Let the interval input data be [xik l , xik u ] and interval output data be [yrk l , yrk u ] for DMUk , k = 1, 2, . . . ,n. Then the interval DEA model as presented by Wang [24] is given by Model 2 Max [θ k l , θk u ]=

s v .[y l , y u ] rm=1 rk rkl rk u u ik .[xik , xik ] i=1 s vrk .[yrj l , yrj u ] rm=1 l u ≤ 1, u i=1 ik .[x ij , x ij ]

subject to 1 ≤ j ≤ n, 1 ≤ i ≤ m, u ik ≥ 0, 1 ≤ r ≤ s. vrk ≥ 0, Using interval arithmetic, we can split Model 2 into two linear models, namely, lower bound and upper bound models, given by [Model 3U] s vrk .yrk u Max θku = i=1 m subject to i=1  u ik .xik l = 1 s m u l i=1 vrk .yrj − i=1 u ik .x ij ≤ 0, 1 ≤ i ≤ m, u ik ≥ 0, 1 ≤ r ≤ s. vrk ≥ 0, [Model 3L] s vrk .yrk l Max θkl = i=1 m subject to i=1  u ik .xik u = 1 s m u l i=1 vrk .yrj − i=1 u ik .x ij ≤ 0,

1 ≤ j ≤ n,

1 ≤ j ≤ n,

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u ik ≥ 0, 1 ≤ i ≤ m, 1 ≤ r ≤ s. vrk ≥ 0, [θk l , θk u ] forms the efficiency interval of DMUk , k=1, 2, …, n. Now, suppose that the inputs and outputs are TIFNs. Let the r th output of DMUk be given by   y˜r k = (yrk l , yrk m , yrk u ; yrk l , yrk m , yrk u ) and the ith input of DMUk is given   l l m u m by x˜ik = (xik , xik , xik ; xik , xik , xik u ). For α ∈ (0, 1], the input α-cut is given by the closed interval (x˜ik )α = [αxik m + (1 − α)xik l , αxik m + (1 − α)xik u ]. Similarly, the output α − cut is given by (y˜rk )α = [α yrk m + (1 − α)yrk l , α yrk m + (1 − α)yrk u ]. So, the above interval efficiency model (Model 2) can be written as Model 4 given below. [Model 4] l u , E k,α ]= Max [E k,α

s m l m u v .[αyrk +(1−α)yrk , αyrk +(1−α)yrk ] mr =1 rk m l m u i=1 u ik . [αx ik +(1−α)x ik , αx ik +(1−α)x ik ] s m l m u v .[αyrk +(1−α)yrk , αyrk +(1−α)yrk ] mr =1 rk ≤ [1, 1] , m l m u u . [αx +(1−α)x , αx +(1−α)x ik i=1 ik ik ik ik ]

subject to 1 ≤ k ≤ n, u rk ≥ 0, 1 ≤ r ≤ s, 1 ≤ i ≤ m. vik ≥ 0, Note that Model 4 is an interval DEA model which can be rewritten as two linear programming problem given by Model 5U and Model 5L respectively. For comparability of different α − cuts and to keep the production possibility set same for all α, we use α = 0 in the constraint [24]. [Model 5U] u m u = rs =1 vrk .(αyrk + (1 − α)yrk ) Max E k,α subject to   m m l = 1, u . αx + − α) x (1 ik ik ik i=1 m s u l v .y − u .x ≤ 0, 1 ≤ j ≤ n, r =1 rk rj i=1 ik ij u ik ≥ 0, 1 ≤ i ≤ m, 1 ≤ r ≤ s. vrk ≥ 0, [Model 5L] l m u = rs =1 vrk .(αyrk + (1 − α)yrk ) Max E k,α subject to   m u ik . αxikm + (1 − α) xiku = 1, i=1 s m u l 1 ≤ j ≤ n, r =1 vrk .yrj − i=1 u ik .x ij ≤ 0, u ik ≥ 0, 1 ≤ i ≤ m, vr k ≥ 0, 1 ≤ r ≤ s. Model 5U and Model 5L give the upper and lower bounds of the membership l u , E k,α ] forms the memfunction of each DMU for α ∈ (0, 1]. The interval [E k,α bership interval of efficiency for the kth DMU for respective α.    2. Hesitation: Consider the TIFN A˜ I = a, b, c; a , b, c . The hesitation is defined by π A˜I (x) = 1 − μ A˜I (x) − ν A˜I (x), where

Development of Intuitionistic Fuzzy Data Envelopment Analysis Model …

μ A˜I (x) =

x−a , b−a c−x , c−b

0,

351

b−x a < x ≤ b; , a  < x ≤ b; b−a  x−b b ≤ x < c; and ν A˜I (x) = c −b , b ≤ x < c ; other wise; 1, other wise.

By simple calculations, π A˜I (x) can be written as

π A˜I (x) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x−a  , b−a   (a−a )(b−x) , (b−a  )(b−a)  (c −c)(x−b) , (c −b)(c−b)  (c −x) , (c −b)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0,

a  < x ≤ a; a ≤ x ≤ b; b ≤ x ≤ c; c ≤ x ≤ c ; other wise.

    One can easily observe that π A˜I (x) ∈ 0, max a−a , c −c ⊆ [0, 1). If b = a  b−a  c −b and c = b then π A˜I (x) = 0. Now, for the ith input variable of the kth DMU, the hesitation lies in closed interval [0, h ik ], and for the r th output, it lies in interval [0, h  rk ]. We can now define the hesitation of D MUk , k = 1, 2, . . . , n by appropriate method from the following methods:

(b) (c)

m

 h ik + rs =1 h  rk m+s m s  h ik r =1 h rk Maximum hesitation: Hk = max( i=1 , ) m s  m s  i=1 h ik r =1 h rk Minimum hesitation: Hk = min ( m , ) s

(a) Average hesitation: Hk =

i=1

3. Incorporating hesitation in efficiency: The efficiency intervals calculated above do not include the hesitation. Any ranking or result, that does not include the hesitation in IF variables, is doing partial justice to the essence of IF variable. IF efficiency should indirectly depend on hesitation. If a DMU under consideration possess high hesitation, then it should have lower efficiency as compared to the DMU with lower hesitation. Hesitation can be seen as a penalty on efficiency of a DMU. Higher hesitation should imply higher penalty. A DMU with no hesitation should face no penalty. Lower hesitation should have lower impact on efficiency of the DMU. Keeping these things in mind, we will incorporate hesitation in efficiency. For α ∈ (0, 1], let the interval l u , E k,α ] be the interval of efficiency of DMUk , k = 1, 2, . . . , n calculated [E k,α using Model 5 given in the above section. Let E k,α be efficiency incorporated with hesitation of DMUk . Then we define E k,α by E k,α =

l u E k,α + E k,α

2(1 + Hk )

The above defined efficiency incorporated with hesitation radially reduces the average of efficiency interval of each DMU by the extent of hesitation present in respective DMU. One can easily note that E k,α = 1.

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4. Ranking of DMUs: Now, we can compare the DMUs on the basis of their efficiency incorporated with hesitation, i.e., by taking E k,α as the ranking index. A DMU with higher ranking index will perform better than the DMU with lower ranking index. Now, rank the DMUs in decreasing order of their ranking index.

5 Advantages of the Proposed Method over the Existing Methods The proposed model calculates efficiency intervals of DMUs with IF variables. IF variables were introduced to mathematically model the lack of information or vagueness present in the linguistic variables. Many studies have been done to calculate efficiency of IF variables, but they have ignored this lack of information present in variables in calculating the efficiency. The most significant advantage of the proposed efficiency model is direct use of hesitation present in IF variables to compare the efficiency of DMUs. Another major advantage of the proposed method is the use of the same production possibility set for all DMUs. Comparison of DMUs with different production possibility sets makes the comparison unjustified. The use of the same production possibility set for different DMUs makes this comparison justified.

6 Numerical Example Consider 5 firms that use 2 IF inputs to produce 2 IF outputs. The input/output data for each firm is given in Table 2. To check which firm is performing best, let us first find the efficiency interval for each firm (or DMU) using Model 5. The calculated efficiency intervals for different values of α = 0, 0.25, 0.5, 0.75 and 1 are given in Table 3. We calculate the hesitation present in each firm using average hesitation. The hesitation in firm A, B, C, D, and E come out to be 0.443, 0.291, 0.294, 0.473, and 0.383, respectively. Firm D has the highest hesitation and firm B has the lowest hesitation. Now we calculate the efficiency incorporated with hesitation in Table 4. At α =0.25, the averages of

Table 2 Example 2 (Source Arya and Yadav [2]) Input 1

Input 2

Output 1

Output 2

A

(3.5,4,4.5;3.2,4,4.7)

(1.9,2.1,2.3;1.7;2.1;2.5)

(2.4,2.6,2.8;2.2,2.6,3)

(3.8,4.1,4.4;3.6,4.1,4.6)

B

(2.9,2.9,2.9;2.9,2.9,2.9)

(1.4,1.5,1.6;1.3,1.5,1.8)

(2.2,2.2,2.2;2.2,2.2,2.2)

(3.3,3.5,3.7;3.1,3.5,3.9)

C

(4.4,4.9,5.4;4.2,4.9,5.6)

(2.2,2.6,3;2.1,2.6,3.2)

(2.7,3.2,3.7;2.5,3.2,3.9)

(4.3,5.1,5.9;4.1,5.1,6.2)

D

(3.4,4.1,4.8;3.1;4.1;4.9)

(2.2,2.3,2.4;2.1,2.3,2.6)

(2.5,2.9,3.3;2.4,2.9,3.6)

(5.5,5.7,5.9;4.1,5.1,6.2)

E

(5.9,6.5,7.1;5.6,6.5,7.2)

(3.6,4.1,4.6;3.5,4.1,4.7)

(4.4,5.1,5.8;4.2,5.1,6.6)

(6.5,7.4,8.3;5.6,7.4,9.2)

Development of Intuitionistic Fuzzy Data Envelopment Analysis Model … Table 3 Efficiency interval values for different values of α α 0 0 0.25 0.25 0.5 0.5 U L U L U L A B C D E

0.910 0.985 1 1 0.867

0.627 0.858 0.562 0.854 0.638

0.868 0.955 0.931 0.98 0.854

0.656 0.863 0.605 0.871 0.68

0.827 0.926 0.868 0.961 0.842

Table 4 Efficiency incorporated with hesitation α 0 0.25 0.5 A B C D E

0.53271 0.713788 0.603555 0.628901 0.544107

0.528067 0.704105 0.593509 0.627883 0.554591

0.524602 0.695972 0.58694 0.627205 0.566161

0.687 0.871 0.651 0.888 0.724

353

0.75 U

0.75 L

1 U

1 L

0.789 0.897 0.808 0.942 0.831

0.719 0.878 0.7 0.906 0.771

0.753 0.886 0.752 0.924 0.82

0.753 0.886 0.752 0.924 0.82

0.75

1

Rank

0.522523 0.687452 0.582689 0.626866 0.579176

0.52183 0.68629 0.581144 0.626866 0.592914

5 1 4 2 3

efficiency interval for the firms A, B, C, D, and E are 0.762, 0.909, 0.768, 0.925, and 0.767, respectively, and the efficiencies incorporated with hesitation are 0.528, 0.704, 0.593, 0.627, and 0.554, respectively. The reductions in efficiencies due to incorporation of hesitation are 0.233, 0.204, 0.174, 0.297, and 0.212, respectively. Firm D has faced the largest reduction as it faces bigger penalty. Firm B has faced the smallest reduction as it faces comparably smaller penalty than other firms. Firm B is the best performer and firm A is worst performer. The rankings obtained in Table 4 are the same for all values of α = 0, 0.25, 0.5, 0.75 and 1.

7 Conclusion The hesitation in IF variables represent the level of doubt/ambiguity/imprecision present in the variable. Due to the presence of high ambiguity in the IF variables, the respective DMU should face high penalty as it makes it challenging for the decision maker to find the precise efficiency interval. DMUs with lower hesitation in their data should face lesser penalty. The present study highlights the importance of considering hesitation directly in efficiency calculation of DMUs with IF inputs and outputs. This study develops a model for calculation of efficiency intervals of DMUs with IF data. For this purpose, we use the efficiency model proposed by Wang [24] for DMUs with input and output present in the form of intervals. The proposed model incorporates the hesitation present in IF variables to calculate the efficiency intervals of DMUs. Finally, we calculate the crisp efficiency for DMUs and rank

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them on the basis of their efficiency scores. At last, we present an example to find the efficiency incorporated with hesitation and rank a set of 5 hypothetical DMUs. In the proposed model, we have used only membership function and hesitation to calculate efficiency intervals of DMUs with inputs and outputs in form of TIFNs. The model can be extended to DMUs with inputs and outputs as trapezoidal IFNs with appropriate changes. We can also use non-membership function or β− cuts to improve the efficiency calculated. Acknowledgements The first author is deeply thankful to the funding agency UGC, Govt. of India, for the financial support to carry out this research work.

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18. Mohammadi Ardakani, S., Babaei Meybodi, H., Sayyadi Tooranloo, H.: Development of a bounded two-stage data envelopment analysis model in the intuitionistic fuzzy environment. Adv. Oper. Res. 2022 (2022) 19. Puri, J., Yadav, S.P.: Intuitionistic fuzzy data envelopment analysis: an application to the banking sector in India. Expert Syst. Appl. 42(11), 4982–4998 (2015) 20. Santos Arteaga, F.J., Ebrahimnejad, A., Zabihi, A.: A new approach for solving intuitionistic fuzzy data envelopment analysis problems. Fuzzy Optim. Model. J. 2(2), 46–56 (2021) 21. Singh, P., Huang, Y.P., Wu, S.I.: An intuitionistic fuzzy set approach for multi-attribute information classification and decision-making. Int. J. Fuzzy Syst. 22, 1506–1520 (2020) 22. Singh, S.K., Yadav, S.P.: Efficient approach for solving type-1 intuitionistic fuzzy transportation problem. Int. J. Syst. Assur. Eng. Manag. 6(3), 259–267 (2015) 23. Szmidt, E., Kacprzyk, J., Bujnowski, P.: How to measure the amount of knowledge conveyed by Atanassov’s intuitionistic fuzzy sets. Inf. Sci. 257, 276–285 (2014) 24. Wang, Y.M., Greatbanks, R., Yang, J.B.: Interval efficiency assessment using data envelopment analysis. Fuzzy Sets Syst. 153(3), 347–370 (2005) 25. Xu, Z.: Models for multiple attribute decision making with intuitionistic fuzzy information. Int. J. Uncertain., Fuzziness Knowl.-Based Syst. 15(03), 285–297 (2007) 26. Yadav, M., Yadav, S.P.: Performance efficiency measurement of MGNREGA 2018-19 in Indian states and union territories based on DEA. Int. J. Syst. Assur. Eng. Manag. 1–8 (2022) 27. Zadeh, L.A.: Fuzzy sets. In: Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A Zadeh, pp. 394–432. World Scientific (1996)

Pricing Policy with the Effect of Fairness Concern, Imprecise Greenness, and Prices in Imprecise Market for a Dual Channel Sanchari Ganguly, Pritha Das, and Manoranjan Maiti

Abstract Fairness concern behavior, a well-known cognitive bias, refers to a person’s attitude of dissatisfaction for unequal pay-offs in someone’s favor. Against environmental pollution, many firms are focused on green manufacturing to maintain sustainable development. The adoption of online shopping by customers around the globe has altered the dynamics of competitiveness in the retail supply chain (SC). Considering the above facts, the effect of fairness concern on optimal prices and channel members’ profits in a two-level dual channel SC is investigated. Here, we consider an SC comprising a green manufacturer and a fair-minded retailer where the manufacturer produces and sells the green product through both offline and online channels to consumers. Due to uncertainty in the real world, the fuzziness is associated with market demand, price elasticity, and the coefficient of greenness. The fuzzy objectives and constraints are reduced to crisp ones using expectation and possibility measures, respectively. Both centralized and decentralized models (with and without cognitive bias) are formulated and solved by the Stackelberg game for the optimal prices and product greenness level. The models’ optimal solutions are analyzed and compared with the deterministic models numerically. The effects of the fairness concern coefficient on the optimal prices, product greenness level, and channel members’ profits are investigated. The sensitivity analyses are presented to study the effects of the fuzzy degree of customers’ sensitivity toward greenness on channel members’ profits. Retailer’s fairness is harmful to the manufacturer but beneficial for her own profit. Finally, some conclusions and managerial insights are presented. Keywords Game theory · Fuzzy theory · Fairness concern · Green supply chain

S. Ganguly (B) · P. Das Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, WB, India e-mail: [email protected] M. Maiti Department of Applied Mathematics, Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, WB, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_25

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1 Introduction The environmental consequences of global warming are becoming increasingly serious. It has put an enormous strain on human survival and has sparked significant worry around the world. As the quality of the environment deteriorates, consumer environmental consciousness grows, and more people are eager to buy environmentally friendly items. In the pursuit of product greening, industries increase their investments in product greening. SAIC General Motors established its Green Future Plan in 2008, concentrating on improving suppliers’ environmental performance and lean manufacturing. They had financed 420 million yuan in green initiative projects by 2015, with 380 million yuan in direct financial benefits to firms. In recent years, consumers are becoming accustomed to online purchasing as information technology advances and enterprises increase their public awareness. Many companies, including IBM, Dell, Nike, Eastman Kodak, and Apple, have begun to sell their products online. Also, E-platforms provide a broad customer flow and standardized low-cost sales services, which give green producers new sales options. A dual-channel distribution system is when a manufacturer sells through a retail store while also having a direct channel to consumers. According to a 2008 survey report by the ChinaASEAN Mobile Internet Industry Alliance, customers’ preferences for online and offline purchasing are nearly identical. However, the development of the e-channel makes the channel conflict prominent. In the actual world, due to a lack of historical data, the probability distribution of the parameters may not always be available to the decision maker. In this case, managers’ judgments, intuitions, and experience can be used to approximate the uncertainty parameters, which are characterized as fuzzy variables. Zadeh et al. [25] provided the fuzzy theory, which can be used as an alternative technique to deal with this type of uncertainty. The possibility and linguistic expressions can be depicted reasonably by fuzzy theory; for example, price elasticity can be expressed as ‘very sensitive’ or ‘not sensitive’ to make rough estimates. Fairness concerns play a crucial role in decision-making regarding unfair treatment by partners in profit allocation. In comparison to a traditional SC, the dualchannel SC is more likely to have unfair collaboration as a result of channel competition and other circumstances, which can create a conflict and even can break the partnership. For example, high-end liquor companies, Yibin Wuliangye Group Company Ltd., have launched online direct channels and provided discounted products on them in order to enhance sales. Because of the positive market outlook for Chinese liquor, the wholesale price has been raised. Meanwhile, liquor companies were attempting to safeguard their brand image by stabilizing offline retail prices and limiting the minimum selling price. Then some of their retail shops felt they were unfairly treated and used price-off promotions to destabilize the offline price system and harm Wuliangye’s interests. In general, each player in an SC aims to maximize her revenue. Individuals compare profits with their partners and believe they have been treated unfairly if their shares are less or more than expected. Here, we consider

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the retailer to be fairness concerned. If a retailer’s monetary profit is less or more than the expected profit, she is in disadvantageous/advantageous inequality (Haitao Cui et al. [9]). Considering the above facts, the following questions arise: (a) How does the retailer’s fairness affect the optimal prices, product greenness level, and expected profits of the channel members for decentralized SC with the green product? (b) What is the impact of the fuzzy degree of coefficient of product greenness on the profits of the SC members as well as the total SC’s expected profit for decentralized and centralized models, respectively? These inquiries encouraged us to consider the current investigation, which answers the above-mentioned questions. Here, we investigate an SC with one green manufacturer and one retailer, in which the manufacturer offers her product to customers through retail (offline) and online channels. The parameters of the SC, i.e., market demand, price elasticity, and product greenness, are uncertain (imprecise) in nature. At first, a centralized benchmark model without cognitive bias is evaluated. Then, with a fair-minded retailer and a rational manufacturer, a decentralized SC is created. We analyze the optimal pricing strategies for which SC members’ expected profits are the maximum using game-theoretic approach. The fuzzy objectives and corresponding constraints are reduced to crisp ones using expectation and possibility measures, respectively. Despite the fact that the two environments (deterministic and fuzzy) are distinct, these two fuzzy models are numerically compared with their respective deterministic cases. The effects of the fairness concern coefficient on the decentralized model’s optimal decisions and profits are demonstrated. In addition, the impact of the coefficient of product greenness on channel members’ expected profits for the decentralized model as well as the SC’s total expected profit for the centralized model is discussed. Finally, we conclude with some valuable findings. Novelties of this investigation are • Identification of cognitive bias in a dual-channel SC where demand parameters are uncertain. • Effects of the crisp cognitive bias (fairness concern) on optimal prices and profits for a dual-channel green SC. This investigation is outlined as follows: Sect. 1 begins with an introduction. Section 2 briefs the literature survey. In Sect. 3, the model formulation is discussed. In Sect. 4, we develop two game-theoretical models—centralized and decentralized with a retailer and a manufacturer selling green items through both offline and online channels, having (i) rational channel members and (ii) fair-minded retailer. The numerical outputs are examined in Sect. 5. In Sect. 6, sensitivity analyses of some parameters for the two proposed models are presented. Finally, some findings and managerial insights are summarized in Sect. 7.

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2 Literature Survey Our findings are relevant to three areas of research: green supply chain, game in a fuzzy environment, and fairness in supply chain.

2.1 Green Supply Chain Product greening in the supply chain has been studied in several research. Ghosh and Shah [8] studied an apparel supply chain that addressed product greening. They investigated the influence of greening expenses and customer sensitivity to green clothes using game-theoretical models. Parsaeifar et al. [17] examined a supply chain comprised of one manufacturer, many suppliers, and retailers, where the supplier and retailer compete horizontally in Nash equilibrium, while everyone competes vertically in Stackelberg equilibrium. Regarding green product decisions, they found that their profit increases as retailer market competition expands but declines while supplier market competition grows. Wang et al. [22] focuses on the selection and coordination of green e-commerce supply chains under the fairness concerns of green producers, taking into account the green degree of product and service provided by the e-commerce platform. A decision-making problem with three-level green SC is addressed by Das et al. [2] under various game structures. Das et al. [3] considered interconnected three-stage forward and reverse SC consisting of green products. They formulated and solved one centralized and decentralized models using a game.

2.2 Game in Fuzzy Supply Chain The majority of past game theory research has focused on deterministic demands. In recent works, several academics have employed fuzzy theory to express uncertainties in supply chain models. In a fuzzy supply chain with two competitive producers and a common retailer, Zhao et al. [27] examined the pricing policy of substitutable products. Li [14] delves into a new set of ideas and methods for making decisions by using intuitionistic fuzzy sets in real-world decisions and games. He researched the methodologies and models of interval-valued cooperative games. Yiang and Xiao [23] explored a green supply chain under several scenarios with government action. Under demand uncertainty, the price strategies for a dual-channel supply chain considering the sales effort and green investment were studied by Wang and Song [21]. Liu et al. [16] proposed one centralized and three decentralized models for a closedloop SC with fuzzy demand and a variety of quality levels for second-hand products. The optimal solutions are determined using the Stackelberg game and fuzzy cut-set method. De et al. [4] developed a cost-minimization problem in a pollution-sensitive production-transportation supply chain using binomial and Gaussian strategic fuzzy

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game approach. In our problem, the market demand, price elasticity, and product greenness coefficient are all considered uncertain.

2.3 Fairness Concern in Supply Chain Social preferences and disparity result in fairness concerns among the supply chain partners. Haitao Cui et al. [9] included fairness in a dyadic channel and investigated the effect of fairness on channel coordination using a linear demand function, claiming that a manufacturer can set the wholesale price higher than his marginal cost for channel coordination to achieve maximum channel utility. Taking private fairness concern as fuzzy, Liang and Qin [15] create an estimation model using fuzzy theory. Sharma [19] examined the pricing decisions of a dyadic supply chain with one fair-minded manufacturer and retailer under several gaming structures and found that for the Stackelberg game, the manufacturer’s (retailer’s) profit decreases (increases) with respect to her fairness and is uncertain for the Vertical Nash game. Yoshihara and Matsubayashi [24] suggested a setup with a single manufacturer and two competing fair-minded retailers. In a dual channel where manufacturers sell items through online retailer, Du and Zhao [6] studied the combined effects of fairness preference and channel preference on the firms’ operational strategies. Wang et al. [20] investigated the pricing decisions of two competitive manufacturers under horizontal and vertical fairness concerns. We find from the above literature that none of them examined cognitive bias in a green supply chain with online and offline channels in an uncertain environment. Table 1 illustrates our work in context with the above literature survey. Through the

Table 1 Literature survey Authors Fuzzy parameters He et al. [10] Zhang et al. [26] Ke et al. [13] Sharma [19] Jamali et al. [11] Zhao et al. [28] Chen et al. [1] Ranjan and Jha [18] Liu et al. [16] Wang et al. [20] Present study

No No Yes No Yes No No No Yes No Yes

Fairness concern

Green supply chain

Channel

Yes Yes No Yes No Yes No No No Yes Yes

No Yes No No Yes No Yes Yes No Yes Yes

Traditional channel Traditional channel Traditional channel Dual channel Dual channel Dual channel Online and offline channel Online and offline channel Traditional channel Traditional channel Offline and online channel

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current investigation, we have attempted to fill this gap. In this study, we examine the impact of retailer’s fairness concern on the optimal decisions of channel members when the parameters are defined by fuzzy variables.

3 Model Formulation In a non-cooperative market, a two-level SC system with a dual distribution channel including one manufacturer and one retailer is proposed. The manufacturer produces green product and sells it through the retailer as well as online (cf. Fig. 1). Both members make their strategies to maximize their expected profits. The market demand, price elasticities, and the coefficient of greenness are fuzzy in nature. The following notations are used to represent the proposed models. i : Channel index, i = r, e ˜ d˜ : Self-price/cross-price elasticity b/ γ˜i : Coefficient of product greenness in channel i λ : Fairness concern coefficient θ : Product greenness level I : Cost coefficient of product greenness

Fig. 1 Structure of the model

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c : Unit production cost of product w/w f : Wholesale price (decision variable) of product in centralized channel without fairness/in decentralized channel with fairness. θ/θ f : Product greenness level (decision variable) of product in centralized channel without fairness/in decentralized channel with fairness. pr / pe : Unit selling price (decision variable) of product in offline channel/online channel. Dr /De : Customer’s demand of product in offline/online channel. f

πr /πr : Profit function of retailer in centralized channel without fairness/in decentralized channel with fairness. f

πm /πm : Profit function of manufacturer in centralized channel/in decentralized channel. πc : Total supply chain’s profit in centralized channel. Abbreviation (Superscripts—‘ f ’ stands for fairness concern, ‘w.r.t.’ means with respect to, ‘r’ and ‘e’ denote offline and online channels respectively, and ‘SC’ means supply chain.) Similar to the majority of the studies, the demand function is taken as linear. With respect to the own and cross prices, the demand for green product decreases and increases, respectively. Furthermore, product greenness has a positive impact on demand. Demand for the green product for offline and online channels are D˜ r = a˜ r − b˜ pr + d˜ pe + γ˜r θ, D˜ e = a˜ e − b˜ pe + d˜ pr + γ˜e θ Now, for avoiding the channel conflict, the selling prices of the green product are considered the same in both offline and online channels, i.e., pr = pe = p. ˜ demand functions take the form: Taking β˜ = b˜ − d, D˜ r = a˜ r − β˜ p + γ˜r θ, D˜ e = a˜ e − β˜ p + γ˜e θ All parameters which are characterized by fuzzy are independent and non-negative ˜ > E[d] ˜ > 0. In the real world, customer by our assumption in this study. Also, E[b] demand for both channels is non-negative. Therefore, Pos({a˜ r − β˜ p + γ˜r θ } < 0) = 0 and Pos({a˜ e − β˜ p + γ˜e θ } < 0) = 0, where Pos(A) is the possibility that event A will occur. The profits of the manufacturer and retailer can be expressed as follows:

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πm = (w − c) D˜ r + ( p − c) D˜ e −

1 2 Iθ 2

(1)

πr = ( p − w) D˜ r

(2)

The greening cost is considered as 21 θ 2 in the profit function [12]. The total profit of the supply chain becomes πc = ( p − c)[(a˜ r + a˜ e ) − 2β˜ p + (γ˜r + γ˜e )θ ] −

1 2 Iθ 2

(3)

3.1 Preliminaries Let (ϑ, P(ϑ), Pos) be a possibility space, where ϑ is a non-empty set, P(ϑ) is its power set, and Pos is a possibility measure. Each element in P(ϑ) is referred to as a fuzzy event, and for each event A, Pos(A) denotes the possibility that event A occurs. • The credit index of A is defined as Cr(A)= 21 (1 + Pos(A) − Pos(Ac )), where Ac is the complement of A. • A fuzzy variable ξ has its expected value as 

+∞

 Cr ({ξ ≥ z} dz +

0

0

−∞

Cr ({ξ ≤ z} dz

provided at least one integral is finite and α ∈ (0, 1). • The α-pessimistic value and the α-optimistic value of a triangular fuzzy variable ξ = (x1 , x2 , x3 ): ξαL = x2 α + x1 (1 − α), ξαU = x2 α + x3 (1 − α) . • If ξ = (x1 , x2 , x3 ) is a triangular fuzzy variable, then the expected value of ξ is E[ξ ] = x1 +2x42 +x3 . 1 • If its expected value is finite, then E[ξ ] = 21 0 (ξαL + ξαU ) dα.

4 Formulation of Models Both centralized and decentralized models are formulated for the two-echelon SC (see Fig. 1).

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4.1 Centralized Model In this scenario, the manufacturer and the retailer act in an integrated way to maximize the total expected SC profit, or E[πc ( p, θ )]. The centralized model is described as follows: ˜ + θ(E[γ˜r ] + E[γ˜e ])] − max E[πc ( p, θ)] = max ( p − c)[E[a˜ r ] + E[a˜ e ] − 2 pE[β]

( p,θ)

( p,θ)

1 2 Iθ 2

subject to 

Pos({a˜ r − β˜ p + γ˜r θ } < 0) = 0, Pos({a˜ e − β˜ p + γ˜e θ } < 0) = 0

(4)

p − c > 0, p > 0 U Theorem 1 If the conditions Δ(a˜ i0 ˜i0U+ ) > β0L+ [Δc + I (Ar + Ae )] (i = r, e) + + θγ ˜ ˜ then the optimal retail hold, where Ar = E[a˜ r ] − E[β]c and Ae = E[a˜ e ] − E[β]c, price and greenness level are given respectively as p = c + ΔI (Ar + Ae ), θ = (E[γ˜r ]+E[γ˜e ]) (Ar + Ae ) Δ U Proof From the possibility constraints (4), we get Δ(a˜ i0 ˜i0U+ ) > β0L+ [Δc + + + θγ (E[γ˜r ]+E[γ˜e ])2 , the optimal retail price I (Ar + Ae )] (i = r, e) (Zhao et al. [27]). If I > ˜ 4E[β] and product greenness are obtained by solving the first-order conditions of the total supply chain, i.e.,

∂E[πc ] ˜ + θ (E[γ˜r ] + E[γ˜e ]) + 2E[β]c ˜ + E[a˜ r ] + E[a˜ e ] = 0 = 0 =⇒ −4 pE[β] ∂p (5) ∂E[πc ] = 0 =⇒ (E[γ˜r ] + E[γ˜e ]) p − I θ − (E[γ˜r ] + E[γ˜e ])c = 0 (6) ∂θ    −4E[β] ˜ E[γ˜r ] + E[γ˜e ]  ˜ The Hessian matrix obtained is H =   = [4I E[β] − −I E[γ˜r ] + E[γ˜e ] (E[γ˜r ] + E[γ˜e ])2 ] = Δ. For maintaining the concavity of the retailer’s expected profit, the first principal ˜ < 0 and the second minor of the Hessian matrix should be negative, i.e., −4E[β] ˜ − (E[γ˜r ] + E[γ˜e ])2 > 0. For feasiprincipal minor should be positive, i.e., 4I E[β] ]+E[γ˜e ])2 must be satisfied. By solving Eqs. (5) bility of the optimal decisions, I > (E[γ˜r4E[ ˜ β] and (6), p and θ are derived, provided the possibility conditions are satisfied and the theorem is proven.

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4.2 Decentralized Model This model is a leader-follower structure (Stackelberg game) with a green manufacturer and a fair-minded retailer, where the manufacturer and the retailer are the leader and follower, respectively. Here, the manufacturer announces the product’s wholesale price first, followed by the retailer’s retail price. Because the manufacturer has the power to set the wholesale price and the retailer is a follower of the manufacturer’s decisions, being fair-minded is justifiable for the retailer. When one of the channel participants exhibits fairness, her goal is to maximize her own utility function rather than her own monetary profit (Fehr and Schmidt [7]). Disadvantageous inequality occurs when her profit is less than her belief of equitable share; otherwise, advantageous inequality occurs. When the manufacturer is the Stackelberg leader, disadvantageous inequality is more relevant for the retailer than advantageous inequality (Du et al. [5]). Therefore, in this model, we consider the disadvantageous inequality only in the retailer’s utility function. Now, retailers pay more attention to the profit made by the manufacturer from the offline channel. Therefore, the utility function of the fair-minded retailer is given by πrf = πr − λ[πm,r − πr ]+ where [πm,r − πr ]+ = max(πm,r − πr , 0), λ > 0 is the retailer’s fairness concern intensity. The higher value of λ indicates that retailers care more about fairness. πm,r and πr are the manufacturer’s and retailer’s monetary profits from offline channels, respectively. Here, the profit function of the manufacturer becomes 1 πmf (w, θ ) = (w − c) D˜ r + ( p − c) D˜ e − I θ 2 2 Formulating as a Stackelberg game model, we optimize f

max(w,θ) E[πm (w, θ )] subject to 

Pos({a˜ r − β˜ p + γ˜r θ } < 0) = 0, Pos({a˜ e − β˜ p + γ˜e θ } < 0) = 0

(7)

and the optimal price is obtained from solving retailer’s problem 1 f ˜ p + E[γ˜r ]θ] + λI θ 2 max E[πr ( p)] = max[(1 + λ) p − (1 + 2λ)w + λc][E[a˜ r ] − E[β] p p 2

subject to Pos({a˜ r − β˜ p + γ˜r θ } < 0) = 0.

(8)

Lemma 1 Using the backward induction method, the retailer’s optimal decision (retail price) is obtained as

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pf =

1 ˜ + λ) 2E[β](1

367

˜ ˜ + 2λ)(w − c) + (1 + λ)(E[a˜ r ] + E[γ˜r ]θ + E[β]c)] [E[β](1

(9) ˜ + 2λ)(w − c) + (1 + λ)(E[a˜ r ] ˜ + λ)(a˜ U + + γ˜ U+ θ ) > β˜ L+ [E[β](1 provided 2E[β](1 0 r0 r0 ˜ holds. + E[γ˜r ]θ + E[β]c)] ˜ + Proof From the possibility constraint of the retailer’s problem, we get 2E[β](1 ˜ + 2λ)(w − c) + (1 + λ)(E[a˜ r ]+E[γ˜r ]θ +E[β]c)]. ˜ λ)(a˜ rU0+ + γ˜rU0+ θ ) > β˜0L+ [E[β](1 The retail price of the green product is obtained by solving the first-order condition of the retailer’s expected profit f

∂E[πr ] ˜ p + (1 + 2λ)E[β](w ˜ ˜ + θ E[γ˜r ] + E[a˜ r )] = 0 = 0 =⇒ −2(1 + λ)E[β] − c) + (1 + λ)(E[β]c ∂p

(10)

f

r ] = The second derivative of the expected profit function of the retailer is ∂ E[π ∂ p2 ˜ −2E[β](1 + λ) < 0. Hence, the retailer’s expected profit function is concave in p. Replacing this optimal price in the manufacturer’s problem, we evaluate the optimal decisions (wholesale price and greenness). 2

˜ 1 (a˜ U + − β˜ L+ c) + (G 1 Ar + (1 + 2λ)G 2 Ae )(2E[β] ˜ γ˜ U+ − Theorem 2 If 2E[β]Δ 0 i0 i0 β˜0L+ E[γ˜r ]) > β˜0L+ [(1 + 2λ)(G 3 Ar + G 4 Ae ) + Δ1 Ar ] (i = r, e) hold, then the optimal wholesale price and greenness level are given as wf = c +

1+λ 1 [G 3 Ar + G 4 Ae ] and θ f = [G 1 Ar + (1 + 2λ)G 2 Ae ] ˜ Δ E[β]Δ1 1

˜ 1 (a˜ U + − β˜ L+ c) + (G 1 Ar + Proof From possibility constraints (7), we get 2E[β]Δ i0 0 U L L ˜ γ˜ + − β˜ + E[γ˜r ]) > β˜ + [(1 + 2λ)(G 3 Ar + G 4 Ae ) + Δ1 Ar ] (1 + 2λ)G 2 Ae )(2E[β] 0 0 i0 for i = r, e. If Δ1 > 0, solving first-order conditions of the profit function of an overconfident manufacturer, the optimal wholesale price and product greenness level are obtained, i.e., f

∂E[πm ] ˜ + 2λ)(3 + 4λ)(w − c) + (1 + λ)B2 θ − λ(1 + λ)Ar + (1 + λ)(1 + 2λ)Ae = 0 = 0 =⇒ −E[β](1 ∂w

(11)

f

∂E[πm ] ˜ 2 (w − c) − (1 + λ)B1 θ + (1 + λ)(E[γ˜2 ] − E[γ˜1 ])Ar + (1 + λ)E[γ˜1 ]Ae = 0 = 0 =⇒ E[β]B ∂θ

(12)

The Hessian matrix of the profit function of the manufacturer is   ˜  −E[β](1+2λ)(3+4λ)  B   1 2 1 1+λ H1 = 4(1+λ) 2  −B1 (1+λ)  = 4 Δ1   B2 ˜ E[β]

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To maintain concavity of the profit function of the manufacturer, the first principal ˜ minor should be less than 0, i.e., −E[β](1+2λ)(3+4λ) < 0 and second principal minor 1+λ should be positive, i.e., Δ1 > 0 should be maintained. By solving Eqs. (11) and (12), w f and θ f are obtained, provided the possibility condition is satisfied and the theorem is proven. Particular case: When the retailer is not fairness sensitive about the offline channel, then the above-decentralized model reduces to the general model with all rational channel members where the previously mentioned parameters remain fuzzy in nature. ˜ 1 a˜ U + + Ar [6E[β]δ ˜ γ γ˜ U+ − β˜ L+ B3 ] + A2 [2E[β](3E[ ˜ Theorem 3 If 2E[β]Δ γ˜r ] + 0 i0 i0 U ˜ 1 cβ˜ L+ (i = r, e) hold, then the optimal wholesale E[γ˜e ])γ˜i0+ − β˜0L+ B4 ] > 2E[β]Δ 0 price and greenness level are given as w=c+

1 [E[γ˜e ]δγ Ar + (B1 + E[γ˜r ]E[γ˜e ])Ae ] ˜ 1 E[β]Δ

and θ=

1 [3δγ Ar + (3E[γ˜r ] + E[γ˜e ])Ae ] Δ1

provided Δ1 > 0 and p is obtained similarly as in Eq. 9 in Lemma 1. Proof This can be proved following the previous theorem.

5 Numerical Experiments In this section, numerical results of the centralized and decentralized models are provided against the same data. Experts’ expertise is frequently used to determine the relationship between linguistic variables and fuzzy triangular numbers for base market demand, self-price elasticity, cross-price elasticity, and coefficient of product greenness. For the numerical example, Table 2 shows linguistic variables as well as fuzzy triangular numbers. Based on Table 2, we have a˜ r = (1200, 1500, 1700), a˜ e = (1100, 1250, 1400), b˜ = (40, 50, 65), d˜ = (30, 40, 55). For these data, the possibility constraints of the models are satisfied. According to preliminaries, triangular fuzzy variable ξ = (x1 , x2 , x3 ) has expected value E[ξ ] = x1 +2x2 +x3 . Thus, the expected values of the fuzzy parameters in Table 2 are as follows: 4 ˜ = 10. E[a˜ r ] = 1475, E[a˜ e ] = 1250, E[β] We present the output values for optimal decisions and channel members’ expected profits for the decentralized model in Table 3. From Table 4, it is noticed that when E[γr ] increases, the expected profit of the manufacturer increases up to a certain limit, then decreases but for the fair-minded retailer and overall SC, the profits escalates. The level of product greenness and retail price increases when customers’ sensitivity to product greenness level grows in the

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Table 2 The values of the parameters Fuzzy parameter

Linguistic variable

Triangular fuzzy number

a˜ r

Big

(1700, 2000, 2200)

Medium

(1200, 1500, 1700)

Sensitive

(30, 40, 45)

Small

(700, 1000, 1200)

Not sensitive

(25, 30, 35)

Big

(1600, 1760, 1900)

Medium

(1100, 1250, 1400)

Sensitive

(20, 30, 35)

Small

(500, 650, 800)

Not sensitive

(15, 20, 25)

a˜ e

Fuzzy parameter b˜

Linguistic variable

Triangular fuzzy number

Very sensitive (40, 50, 65)

d˜

Very sensitive (30, 40, 55)

Table 3 Optimal values of decisions and profits E[γr ]

E[γe ]

λ

f

f

wf

θf

pf

E[πm ]

E[πr ]

E[πc ]

9.0

53.58

26.71

122.05

49571

102701

97398

10.2

53.45

27.03

123.72

49632

108157

100492

11.4

53.31

27.32

125.38

49655

113743

103904

12.6

53.12

27.56

127.03

49638

119425

107680

13.8

52.89

27.75

128.63

49577

125166

111873

15.0

52.63

27.88

130.2

49472

130922

116550

8.0

36.81

3.78

99.74

28189

66586

77868

10.0

37.25

5.51

101.38

28868

70010

80083

12.0

37.90

7.38

103.29

29825

73824

82629

14.0

38.80

9.41

105.53

31100

78146

85559

16.0

39.99

11.65

108.14

32740

83140

88937

18.0

41.52

14.13

111.19

34810

89038

92848

1.3

52.63

27.88

130.2

49472

130922

−

1.4

52.08

27.8

129.99

49280

138234

−

1.5

51.58

27.73

129.81

49109

145535

−

1.6

51.13

27.66

129.64

48955

152827

−

1.7

50.72

27.6

129.48

48816

160110

−

1.8

50.35

27.55

129.34

48689

167386

−

Parameters

offline channel but the wholesale price decreases. When customers’ sensitivity about product greenness increases more in the online channel, all the optimal decisions and corresponding expected profits of the channel members increase. Also, from Table 4, it is observed that the retailer’s fairness is harmful to the decisions of the channel members but beneficial for her own profit.

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Table 4 Comparison of solutions of fuzzy approach and deterministic approach Fuzzy approach (i) Deterministic Difference between (i) approach (ii) and (ii) (%) p = 120.26 θ = 41.11 Decentralized scenario p f = 130.20 with retailer’s fairness concern θ f = 27.88 w f = 52.63 Decentralized scenario p = 138.97 without retailer’s fairness concern θ = 31.87 w = 82.62 Centralized scenario

p = 127.36 θ = 45.93 p f = 135.54

5.9 11.72 4.10

θ f = 30.74 w f = 55.46 p = 144.93

10.25 5.37 4.28

θ = 35.25 w = 87.28

10.6 5.64

5.1 Comparison Between Optimal Decisions Between Fuzzy and Deterministic Approaches of the Decentralized Model In this section, we compare fuzzy solutions and deterministic solutions of the centralized and decentralized models (with and without retailer’s fairness) numerically, where we assumed the value of fuzzy parameters (expected value) near the deterministic values. These values are as per the expert’s opinions. The difference between the optimal values in percentage is shown in Table 3. It is noticed that the highest difference between the centralized scenario and decentralized scenario with and without the cognitive bias is in product greenness level. For the comparison, the deterministic decentralized models with and without the cognitive bias (fairness concern) are solved. The optimal decisions of the manufacturer for the deterministic model with retailer’s fairness are as follows: wf = c + 

  1+λ ˜ 2 [G 3 Ar E[β]Δ





+ G 4 Ae ] and θ f = 

  1 [G 1 Ar Δ2





+ (1 + 2λ)G 2 Ae ]

where Ar = ar − βc, Ae = ae − βc and Δ2 > 0 should be satisfied and the optimal retail price can be obtained similarly following previous section (Sect. 4).

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6 Sensitivity Analyses 6.1 Effect of Fairness Concern Coefficient on Optimal Decisions and Profits of the Channel Members for Decentralized Model with Fair-Minded Retailer 6.1.1

Effect of λ on Optimal Prices and Greenness Level

To illustrate the effect of λ for the offline and online channels, on optimal retail and wholesale prices of the green product, product greenness level as well as expected profits of the channel members, the following numerical values of the model parameters are considered: a˜ r = (1200, 1500, 1700), a˜ e = (1100, 1250, 1400), b˜ = (40, 50, 65), d˜ = (30, 40, 55), γ˜r = (10, 15, 20), γ˜e = (21, 26, 31), and c = 20, I = 100. We evaluate the optimal decisions presented in Fig. 2 w.r.t. λ. When the retailer exhibits more and more fairness about the offline channel w.r.t. the manufacturer from Fig. 2, it is observed that the wholesale price of the manufacturer decreases. Consequently, it is also realized that an increase in retailer’s

Fig. 2 Optimal prices and greenness level versus λ

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Fig. 3 Profits versus λ

fairness concern has an adverse impact on both channels’ retail prices as well as on the greening level of the product. The power of bargaining increases when the retailer’s fairness concern intensifies, so for the sake of a fair deal, she will force the manufacturer to reduce her wholesale price to such an extent that their profits are not jeopardized. Hence, the retailer is focused on her fairness more, and both channel members’ optimal prices descend. Also, the fairness of retailer is harmful to the environmental performance of the supply chain. Effect of λ on channel members’ profits: The effect of the fairness concern coefficient λ on the expected profits of the channel members is depicted in Fig. 3. From Fig. 3, it is clear that when a retailer’s fairness w.r.t. the manufacturer increases, the expected profit of the manufacturer diminishes but her own revenue gets benefited. Now, when the retailer is fair-minded, the manufacturer sacrifices her wholesale price to care about her concern for fairness. So, the expected profit of the manufacturer is declined. It is inferred that the manufacturer cannot be aggressive while choosing her strategy when the retailer is fair-minded. Hence, the retailer’s fairness concern about the offline channel w.r.t. the manufacturer is favorable for her expected profit but detrimental for the manufacturer.

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6.2 Effect of E[γ˜r ] and E[γ˜e ] on Expected Profits of the Channel Members in Retail and Online Channels The effect of the fuzzy degree of product greenness coefficient for offline and online channels on expected profits of the channel members (decentralized model) as well as on expected total SC’s profit (centralized model) is depicted in Figs. 4 and 5, respectively. From Fig. 4, it is noticed that with the increment of customer’s sensitivity to product greenness level in offline channel, the expected profit of the manufacturer increases first, until E[γ˜r ] exceeds a certain amount and when E[γ˜r ] > 11.4, her profit decreases. Also, the total expected SC’s profit in the centralized model as well as the retailer’s expected profit increases when E[γ˜r ] increases for our chosen range of E[γ˜r ]. Hence, when customers prefer product greenness more, the retailer (for decentralized channel) and overall supply chain profit (for centralized channel) get an advantage, but it is beneficial for the manufacturer up to a certain limit, then it is detrimental for her expected profit. It can also be seen from Fig. 5 that when the product greenness coefficient increases in the online channel, the channel members’ expected profits for the decentralized model as well as the total SC’s profit for the centralized model increase. Therefore, if E[γ˜e ] increases in the online channel, the channel members’ expected profits are enhanced. As the manufacturer’s and retailer’s expected profit increase with the coefficient of product greenness for a

Fig. 4 Profits versus E[γ˜r ]

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Fig. 5 Profit versus E[γ˜e ]

certain range, the total expected profit for SC (centralized model) also grows. In the market, where consumers are highly sensitive to greenness, they frequently choose products with high levels of greening. The development of new products lead to higher cost, resulting in increase in sales, so the overall expected profit of the SC increases.

6.3 Parametric Study of Cost Coefficient for Greening When the cost for product greening increases more, the expected profits of the channel members (decentralized model) when the retailer is fair-minded as well as the total SC’s profit (centralized model) decrease. This follows naturally and is justified by Fig. 6.

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Fig. 6 Profit versus I

7 Conclusion In this investigation, we considered an SC that includes a green manufacturer and a retailer, where the manufacturer sells green product to customers through both offline and online channels. Due to the uncertainty of the parameters in the real world, the problem is modeled using fuzzy theory. Fuzzy parameters are used to characterize market demand, price elasticity, and the coefficient of product greenness. We studied and solved for both centralized and decentralized models (with and without the retailer’s fairness concern). We compared the solutions of the decentralized model for fuzzy and deterministic approaches. Also, we analyzed the effect of fairness of the retailer on optimal wholesale and retail prices, product greenness, and expected profits of the channel members for the model. We also examined the effect of the product greenness coefficient on the total expected profit of SC and channel members’ profits for the centralized and decentralized scenarios, respectively. Analyzing our theoretical and numerical results, we have the following conclusions: (i) Retailer’s fairness concern has an adverse impact on optimal wholesale and retail prices for both channels in the decentralized model. (ii) The effect of the fairness concern of the retailer on the greening level of the product is negative. Hence, this cognitive bias is unfavorable for the environmental performance of the SC.

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(iii) Customer’s sensitivity toward product greening for the online channel has a beneficial impact on all channel members’ profits in the decentralized model as well as on total supply chain profit in the centralized model. (iv) Increment in customer’s sensitivity toward product greening for the retail channel is helpful for the total expected supply chain profit in the centralized model and the retailer’s expected profit in the decentralized model but has a positive effect on the manufacturer’s expected profit up to a certain limit. (v) The manufacturer’s expected profit decreases with the retailer’s fairness, but the retailer’s profit enhances. (vi) All channel members’ profits, as well as total SC’s profit, reduce when the cost of greening increases.

7.1 Managerial Insights This study considers game-theoretic models for a green product in a dual-channel scenario under an uncertain environment to examine the impact of fairness concerns. This is a real-life occurrence in the e-market. Thus, a manufacturing company with dual-channel SC for a green product might use this analysis along with the relevant information and expert opinions to derive the pricing strategies. If the management knows in advance that her retailer is an upright person and believes in fairness, the management should be prepared to share the profits in a reasonable way. To satisfy the retailer’s bias, management can reduce either the wholesale price sacrificing her own profit, or reducing the green level (cost), which has a detrimental effect on mankind.

7.2 Limitations and Future Extensions The present study has the following limitations: We have considered a two-level SC with two members—one manufacturer and one retailer only. However, an SC with several channel members can be considered. In this study, fairness concern is represented by crisp values, though this can be uncertain in nature.

Nomenclature ˜ − E[γ˜r ](2E[γ˜e ] − E[γ˜r ]), B2 = (1 + 2λ)E[γ˜e ] − λE[γ˜r ], B1 = 2E[β]I Δ1 = B1 (1 + 2λ)(3 + 4λ) − B22 , G 1 = (1 + 2λ)(3 + 4λ)(E[γ˜e ] − E[γ˜r ]) − λB2 ,

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G 2 = B2 + (3 + 4λ)E[γ˜r ], G 3 = B2 (E[γ˜e ] − E[γ˜r ]) − λB1 , G 4 = (1 + 2λ)B1 + E[γ˜r ]B2 δγ = E[γ˜e ], B3 = δγ (3E[γ˜r ] + E[γ˜e ]) + 1, B4 = B1 + E[γ˜r ](3E[γ˜r ] + 2E[γ˜e ]), 



B1 = 2β I − γr (2γe − γr ), B2 = (1 + 2λ)γe − λγr , 







Δ2 = B1 (1 + 2λ)(3 + 4λ) − (B2 )2 , G 1 = (1 + 2λ)(3 + 4λ)(γe − γr ) − λB2 , 















G 2 = B2 + (3 + 4λ)γr , G 3 = B2 γe − γr ) − λB1 , G 4 = (1 + 2λ)B1 + γr B2 . Declarations The authors have no conflict of interest. Acknowledgements Thanks to IIEST authority for giving scope for research.

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Fuzzy Set Theory

A Similarity Measure of Picture Fuzzy Soft Sets and Its Application V. Salsabeela and Sunil Jacob John

Abstract Picture Fuzzy Soft Sets (PFS f S) is an extension of Intuitionistic Fuzzy Soft Sets and has a wide range of applications. Within this work, we propose a similarity measure and a corresponding weighted similarity measure between two PFS f S. A numerical example is presented to demonstrate that the proposed method which can be effectively applied to problems in the field of medical diagnosis. Keywords Picture fuzzy sets · Soft sets · Picture fuzzy soft sets

1 Introduction In 1965, Zadeh [4] invented the notion of fuzzy set (FS), an approach for dealing with uncertainties. Researchers have given a lot of attention to the FS theory, and numerous experts have applied it to a variety of situations. In fuzzy sets, there is a function that returns the degree of membership of an object to a non-empty set from a non-empty set to the closed unit interval [0, 1]. It is a framework for dealing with ambiguity, exaggeration, and confusion, as well as assigning a degree of membership to each member of the nature of discussion to a subset of it. A soft set, which is a parameterized family of subsets of a crisp universal set, could handle more data which contains uncertainty. Using the parametrization of tools methodology, Molodtsov [1] offered the idea of soft sets in 1999. Molodtsov has effectively employed soft set theory in a variety of regions including game theory, operations research, and Reimann integration. Soft set theory offers a wide range of applications, only a few of which Molodtsov demonstrated in his pioneering work [1]. Yang et al. [5] defined picture fuzzy soft set (PFS f S) as an extended version of soft sets. V. Salsabeela (B) · S. J. John Department of Mathematics, National Institute Technology, Calicut 673601, India e-mail: [email protected] S. J. John e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_26

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Similarity measures are a crucial metric for analyzing the degree to which two things are alike. The idea of similarity measure is significant in practically every sector of research and engineering. Many researchers have now developed many similarity measures and applied them to MADM, pattern identification, mineral field recognition, building material recognition, strategy decision-making, and other domains. We frequently want to know if two patterns or images are alike or nearly alike, or at the very least to what extent they are alike. Various academics have explored the topic of measuring similarity between fuzzy sets, fuzzy numbers, and ambiguous sets, including Chen, [6, 7] and [8]. Li and Xu, [9] Hong and Kim, [10] C. P. Pappis, [11, 12], and others. However, in [13], Kharal provides counterexamples to demonstrate various inaccuracies, as well as introduce some set operations on soft set distances and similarity measurements based on distance. According to the Hamming distance, Euclidean distance, and their normalized correspondents, Atanassov [14], Szmidt, and Kacprzyk [15] suggested a similarity measure for intuitionistic fuzzy sets. Guiwu Wei and Yu Wei [16] suggested ten similarity measures in accordance with the cosine function for Pythagorean fuzzy sets. Guiwu Wei [17] has presented eight cosine-based similarity measurements between picture fuzzy sets. Kifayat et al. [18] recently discussed certain gray similarity measures, cosine similarity measures, and set-theoretic similarity measures that can be used to compare spherical fuzzy sets and T-spherical fuzzy sets. Cuong [19] recently suggested the picture fuzzy set (PFS) and examined a few of its fundamental operations and features. Here, put forward a similarity measure and a weighted similarity measure for PFS f S and explain with an illustrative example related with our real-life situation. The following is the format of the paper’s presentation. The concepts of soft sets, fuzzy soft sets, picture fuzzy sets, and similarity measures between two fuzzy soft sets are reviewed in segment 2. Part 3 introduces the concept of PFS f S similarity measure and proposes an equation for finding the similarity measure and a weighted similarity measure between two PFS f S. The mentioned similarity measure is applied to a problem related with the medical field in Sect. 4. Section 5 wraps up the project with some recommendations for the future.

2 Preliminaries In the present section, we will review certain basic concepts for soft sets, fuzzy soft sets, picture fuzzy sets, picture fuzzy soft sets, and similarity measures between fuzzy soft sets, that would be important in the following talks. Consider  = {1 , 2 , 3 , . . . n } as the universe of discourse and K = {k1 , k2 , k3 , . . . kn } is the set of parameters. Definition 1 ([1]) A pair (, ) is said to be a soft set over the universe , in which  is a mapping defined by  :  → P().

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383

Otherwise stated, a parameterized family of subsets of the universe  is said to be a soft set. Definition 2 ([2]) Consider  as the universe,  be the set of parameters and P() denote the collection of all fuzzy soft sets over . A pair (, ) is said to be a fuzzy soft set over the universe , in which  :  → P() is a function defined from  → P(). Definition 3 ([3]) A Picture Fuzzy Set (P F S), P on a universe of discourse  is an object in the form of, P = {(, P (), ϒP (), P ())| ∈ } in which, P () ∈ [0, 1] is called the “positive membership degree of  in ”, ϒP () ∈ [0, 1] is called the “neutral membership degree of  in ” and P () ∈ [0, 1] is called the “negative membership degree of  in ” and where P , ϒP and P satisfy the given criteria: ∀ ∈ , P () + ϒP () + P () ≤ 1. Then for  ∈ , πP () = 1 − (P () + ϒP () + P ()) is called the refusalmembership degree of  in . For simpleness, we call P(P (), ϒP (), P ()) a picture fuzzy number (PFN) denoted by k = P(k , ϒk , k ), where k , ϒe , k ∈ [0, 1], πk = 1 − (k + ϒk + k ), and k + ϒk + k ≤ 1. Definition 4 ([17]) Let P1 and P2 be two picture fuzzy sets defined on the universe of discourse . Then P1 ⊆ P2 , i f P1 () ≤ P2 (), ϒP1 () ≤ ϒP2 (), P1 ()) ≥ P2 ()) Definition 5 ([3]) Consider  as the universal of discourse, K is the set of all parameters and ⊆ K . A pair ,  is said to be a picture fuzzy soft set (PFS f S) over the universe , in which  is a mapping defined by  : → P F S(). For any parameter k ∈ K , (k) can be represented as a picture fuzzy soft set in such a way that (k) = {(, (k) (), ϒ(k) (), (k) ())| ∈ } where (k) () is the positive membership degree, ϒ(k) () is the neutral membership degree and (k) () is the negative membership degree function respectively with the condition, (k) () + ϒ(k) () + (e) () ≤ 1. If for any parameter k ∈ and for any  ∈ , ϒ(k) () = 0, then (k) will become a Pythagorean fuzzy set and ,  will turn into an intuitionistic fuzzy soft set if it is true for all k ∈ . Here, P F S f S() represents the set of all picture fuzzy soft sets over .

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Definition 6 Let ,  and ,  be two fuzzy soft sets, then ,  ⊆ , , if 1.  ⊆  2. ∀ t ∈ , (t) ⊆ (t) Definition 7 ([17]) Consider  as the universe of discourse and K is a set of all parameters. Let F S f S() represent the set of all fuzzy soft sets over the universe . A mapping Sm : F S f S() × F S f S() → [0, 1] is said to be a similarity measure between fuzzy soft sets ,  and , , represented as Sm ( , , , ), if it fulfills the following requirements: 1. 2. 3. 4.

0 ≤ Sm ( , , , ) ≤ 1; Sm ( , , , ) = Sm ( , , , ); Sm ( , , , ) = 1 iff  = ; Consider another fuzzy soft set , ∂, if ,  ⊆ ,  and

,  ⊆ , ∂, then Sm ( , , , ∂) ≤ Sm ( , , , ) Sm ( , , , ∂) ≤ Sm ( , , , ∂).

and

3 Similarity Measure Between Picture Fuzzy Soft Sets The idea of similarity measures between two PFS f Ss is discussed in this section, as well as its equivalent weighted similarity measure, is also mentioned here. Definition 8 Consider  as an initial universal set and K as the set of all parameters. Suppose P F S f S() represents the set of all picture fuzzy soft sets defined on the universe . A mapping Sm : P F S f S() × P F S f S() → [0, 1] is said to be a similarity measure in between fuzzy soft sets ,  and , , represented as Sm ( , , , ), if it satisfies the given conditions: 1. 2. 3. 4.

0 ≤ Sm ( , , , ) ≤ 1; Sm ( , , , ) = Sm ( , , , ); Sm ( , , , ) = 1 iff  = ; Let , ∂ be a picture fuzzy soft set, if ,  ⊆ ,  and

,  ⊆ , ∂, then Sm ( , , , ∂) ≤ Sm ( , , , ) Sm ( , , , ∂) ≤ Sm ( , , , ∂).

and

Now, for any PFS f S ,  ∈ PFS f S(),  ⊆ K , we will extend the PFS f S to ˆ  in which (k) = φ,∀k ∈ / , that is, (k) () = ϒ(k) () = the PFS f S , ˆ ˆ () = 0, ∀k ∈ / . As a result, we will now treat the parameter subset of every (k) ˆ PFS f S over  as if it were the set of parameter K , without sacrificing generality. Definition 9 Consider  = {u 1 , u 2 , . . . u n } as a universe with the parameter set K = {k1 , k2 , . . . km }. Then, a similarity measure between two PFS f S , K  and , K  can be found using the given formula:

A Similarity Measure of Picture Fuzzy Soft Sets and Its Application Sm (, ) =

385

  m n 1   1 − 21 min{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ(ki ) (u j )|   1 mn 1 + 2 max{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ(ki ) (u j )| i=1 j=1

(1) Example 1 Suppose that U = {u 1 , u 2 , u 3 } as a universal set K = {k1 , k2 , k3 } is the set of all parameters. Here we can examine the two PFS f S ⎛ ⎞ ⎛ ⎞ (0.5, 0.2, 0.1) (0.8, 0.1, 0) (0.3, 0.1, 0.2) (0.7, 0.1, 0) (0.4, 0.4, 0.1) (0.6, 0.2, 0.1)

, K  = ⎝(0.5, 0.3, 0.1) (0.4, 0.3, 0.1) (0.6, 0.1, 0.1)⎠ and , K  = ⎝(0.6, 0.1, 0.1) (0.3, 0.2, 0.1) (0.5, 0.2, 0.1)⎠ (0.6, 0.2, 0.1) (0.5, 0.3, 0) (0.7, 0.1, 0.1) (0.5, 0.3, 0.1) (0.4, 0.2, 0.2) (0.4, 0.3, 0.2)

Then Sm ,  = 0.7735.

Theorem 1 If , K , , K  and R, K  are three PFS f Ss defined over the universal set U. Then Sm fulfills the four features of similarity measures listed below: 1. 2. 3. 4.

0 ≤ Sm ( , ) ≤ 1; Sm ( , ) = Sm ( , ); Sm ( , ) = 1 iff  = ; If , K  ⊆ , K  and , K  ⊆ R, K , then Sm ( , R) ≤ Sm ( , ) and Sm ( , R) ≤ Sm ( , R).

Proof 1. Proof is straightforward. 2. Proof is straightforward. 3. Assume that  =  ⇒ (ki ) (u j ) = (ki ) (u j ), ϒ(ki ) (u j ) = ϒ(ki ) (u j ) and (ki ) (u j ) = (ki ) (u j ). ⇒ Sm (, ) = 1. Conversely  suppose that Sm (, ) = 1.  ⇒

1− 21 min{|(ki ) (u j )−(ki ) (u j )|,|(ki ) (u j )−(ki ) (u j )|}+|ϒ(ki ) (u j )−ϒ(ki ) (u j )|



 =1

1+ 21 max{|(ki ) (u j )−(ki ) (u j )|,|(ki ) (u j )−(ki ) (u j )|}+|ϒ(ki ) (u j )−ϒ(ki ) (u j )|  1 − 21 min{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} 

⇒ + |ϒ(ki ) (u j ) − ϒ(ki ) (u j )| =  1 + 21 max{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} +  |ϒ(ki ) (u j ) − ϒ(ki ) (u j )|   ⇒ 21 min{|(ki ) (u j )−(ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|}]+ 21 max{|(ki )  (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|}] + |ϒ(ki ) (u j ) − ϒ(ki ) (u j )| = 0 |ϒ(ki ) (u j ) − ϒ(ki ) (u j )| = 0 and ⇒ |(ki ) (u j ) − (ki ) (u j )| = 0, |(ki ) (u j ) − (ki ) (u j )| = 0 ⇒= 4. We have , K  ⊆ , K  ⊆ R, K . | ϒ(ki ) (u j ) − ⇒ | (ki ) (u j ) − (ki ) (u j ) |≤| (ki ) (u j ) −  R(ki ) (u j ) |, ϒ(ki ) (u j ) |≤| ϒ(ki ) (u j ) − ϒ R(ki ) (u j ) | and | (ki ) (u j ) − (ki ) (u j ) |≤| (ki ) (u j ) −  R(ki ) (u j ) |. ⇒ min{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ(ki ) (u j )| ≤ min{|(ki ) (u j ) −  R(ki ) (u j )|, |(ki ) (u j ) −  R(ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ R(ki ) (u j )| and max{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} + |ϒ(ki ) (u j ) −

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ϒ(ki ) (u j )| ≤ max{|(ki ) (u j ) −  R(ki ) (u j )|, |(ki ) (u j ) −  R(ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ R(ki ) (u j )|. max{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} +

|ϒ(ki ) (u j ) − ϒ(ki ) (u j )| ≥ 1 − 21 max{|(ki ) (u j ) −

 R(ki ) (u j )|, |(ki ) (u j ) −  R(ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ R(ki ) (u j )| and 1 − 21 max{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} +

|ϒ(ki ) (u j ) − ϒ(ki ) (u j )| ≤ 1 − 21 max{|(ki ) (u j ) −

 R(ki ) (u j )|, |(ki ) (u j ) −  R(ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ R(ki ) (u j )| . ⇒ Sm ( , R) ≤ Sm ( , ). 1−

1 2

Likewise, we can prove that Sm ( , R) ≤ Sm ( , R). The weight of the parameter ki ∈ K should be considered in a variety of situations. When the weights of ki are taken into account, a weighted similarity measure W P F S f S between , K  and , K  is defined like this: W P F S f S (, )

(2)

  m n  1 − 21 min{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ(ki ) (u j )| 1 = ∅i   1 n 1 + 2 max{|(ki ) (u j ) − (ki ) (u j )|, |(ki ) (u j ) − (ki ) (u j )|} + |ϒ(ki ) (u j ) − ϒ(ki ) (u j )| i=1 j=1

T where ∅ = (∅1 , ∅2 , . . . ∅ m ) is regarded as the weight vector of ki with ∅i ∈ m [0 1], (i = 1, 2, . . . m), i ∅i = 1. The weighted similarity measure of two P F S f S(, K ) and (, K ) also satisfies the following conditions:

1. 0 ≤ W P F S f S ( , ) ≤ 1; 2. W P F S f S ( , ) = W P F S f S ( , ) 3. W P F S f S ( , ) = 1 iff  = . We may prove these conditions 1 to 3 using a similar proof as in Theorem 2. Definition 10 Let P, K  and Q, K  are two P F S f S defined within the universe . These P F S f S are regarded to be significantly similar if S P F S f S (P, Q) ≥ 0.75.

4 The Application of Picture Fuzzy Soft Set Similarity Measures in Medical Diagnosis We offer a strategy for solving a medical diagnosis problem in accordance with the suggested similarity measure of P F S f Ss in this section. In the following application, a similarity measure between two P F S f Ss could be used to determine if a person has an illness or not.

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Diarrhea is a common issue characterized by loose, watery, and possibly more frequent bowel motions. It can occur on its own or in conjunction with other symptoms such as nausea, vomiting, stomach discomfort, or weight loss. Diarrhea is characterized by loose, watery feces occurring three or more times per day. Diarrhea can be acute, chronic, or recurrent. Acute diarrhea is far more common than chronic or persistent diarrhea. Dehydration and malabsorption are two side effects of diarrhea. If your diarrhea lasts fewer than four days, your doctor may not need to investigate the cause. If your diarrhea persists or you have other symptoms, your doctor may consult your medical and family history, perform a physical examination, or order tests to determine the cause. Here, we are attempting to determine how likely it is that a patient with particular apparent symptoms is suffering from diarrhea. With the help of a medical officer, here, we first create a P F S f S for the illness and a P F S f S for the sick person. After that, we calculate the similarity measures between these two P F S f Ss. If they are considerably comparable, we can deduce that the person is suffering from diarrhea. If they are not, we can rule out diarrhea. Example 2 Suppose that there are two patients P1 , P2 admitted to a hospital with those kinds of signs and symptoms of diarrhea. Suppose that universal set consists of the three elements “harsh” (ϕ1 ), “soft” (ϕ2 ), and “no” (ϕ3 ). That is ϕ = {ϕ1 , ϕ2 , ϕ3 }. Consider  = {1 , 2 , 3 , 4 , 5 , 6 } as the set of parameters which indicates definite symptoms of diarrhea, where 1 stands for watery, 2 stands for frequent bowel movements, 3 stands for cramping or pain in the abdomen, 4 stands for nausea, 5 stands for bloating, and 6 stands for bloody stools. Suppose F,  be a P F S f S over the universe ϕ for diarrhea, developed by utilizing the assistance of a civil medical officer provided in Table 1. The P F S f Ss is then constructed for two patients, as shown in Tables 2 and 3. Therefore, we get S P F S f S F, P1  = 0.8311 ≥ 0.75 and S P F S f S F, P2  = 0.5719 ≤ 0.75. In light of these findings, we can state that the patient P1 is possibly suffering from diarrhea. Table 1 P F S f S F,  for Diarrhea

F, 

1

2

3

4

5

6

ϕ1 ϕ2 ϕ3

(0.7,0,0.1) (0.6,0.2,0.1) (0.2,0.3,0.1) (0.4,0.2,0.2) (0.2,0.3,0.1) (0.4,0.3,0.1) (0.5,0.1,0.2) (0.4,0,0.5) (0.4,0.2,0.1) (0.8,0.1,0) (0.6,0.1,0.1) (0.5,0.2,0.1) (0.3,0.2,0.4) (0.4,0.2,0.3) (0.6,0.1,0.2) (0.4,0.2,0.2) (0.3,0.1,0.1) (0.6,0.1,0.1)

Table 2 P F S f S P1 ,  for the patient P1

P1 , 

1

ϕ1 ϕ2 ϕ3

(0.8,0.1,0) (0.5,0.3,0.1) (0.3,0.4,0.1) (0.4,0.1,0.1) (0.3,0.4,0.2) (0.3,0.2,0.2) (0.6,0.2,0.1) (0.3,0,0.6) (0.3,0.3,0.2) (0.7,0.2,0.1) (0.5,0.2,0.2) (0.6,0.1,0.1) (0.4,0.3,0.2) (0.3,0.3,0.2) (0.5,0.2,0.1) (0.3,0.3,0.1) (0.4,0.2,0.2) (0.5,0.2,0.2)

2

3

4

5

6

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Table 3 P F S f S P2 ,  for the patient P2

P2 , 

1

2

3

4

5

6

ϕ1 ϕ2 ϕ3

(0.3,0.3,0.2) (0.2,0.4,0.3) (0.5,0.1,0.3) (0.1,0.4,0.4) (0.5,0.1,0.3) (0.1,0.6,0.2) (0.2,0.4,0.3) (0.1,0.7,0.1) (0.5,0.5,0.3) (0.1,0.5,0.1) (0.1,0.4,0.3) (0.1,0.5,0.3) (0.1,0.5,0.1) (0.8,0.1,0) (0.3,0,0.5) (0.2,0.3,0.4) (0.5,0.3,0.3) (0.1,0.4,0.4)

5 Conclusion Similarity measure is a useful implement for calculating the extent of similarities between two things. It is frequently manufactured for the goal of pretending the object or document’s legitimacy. Practically, every sector of research and engineering, the idea of similarity measurement is significant. This work provides a similarity measure and the accompanying weighted similarity measure related with two picture fuzzy soft sets. Picture fuzzy soft set is a novel soft set paradigm that is more feasible and precise in certain circumstances than previous soft set structures. It can be considered as the expanded version of fuzzy soft sets. Finally, an application for a medical diagnosis problem is presented for determining wether a patient is suffering from an illness or not. In more applicable real-life situations, the suggested similarity measure as well as the weighted similarity measure could be used. These are also particularly useful for dealing with a variety of uncertainty problems in pattern recognition, decision-making, clustering, and other fields.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Molodtsov, D.: Soft set theory-first results. Comput. & Math. Appl. 37, 19–31 (1999) Maji, P.K., Biswas, R., Roy, A.R.: Fuzzy soft sets. J. Fuzzy Math. 9, 589–602 (2001) Cuong, B.C., Kreinovich, V.: Picture fuzzy sets. J. Comput. Sci. Cybern. 30, 409–420 (2014) Zadeh, L.A.: Fuzzy, sets. Inf. Control. 8, 338–353 (1965) Yang, Y., Liang, C., Ji, S., et al.: Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making. J. Int. Fuzzy Syst. 29(4), 1711–1722 (2015) Chen, S.M., et al.: A comparison of similarity measures of fuzzy values. Fuzzy Sets Syst. 72, 79–89 (1995) Chen, S.M.: Measures of similarity between vague sets. Fuzzy Sets Syst. 74, 217–223 (1995) Chen, S.M.: Similarity measures between vague sets and between elements. IEEE Trans. Syst. Man Cybern. (Part B). 27(1), 153–168 (1997) Li, F., Xu, Z.Y.: Similarity measure between vague sets. Chin. J. Softw. 12(6), 922–927 (2001) Hong, D.H., Kim, C.: A note on similarity measure between vague sets and elements. Inf. Sci. 115, 83–96 (1999) Pappis, C.P.: Value approximation of fuzzy systems variables. Fuzzy Sets Syst. 39, 111–115 (1991) Pappis, C.P., Karacapilidis, N.I.: A comparative assessment of measures of similarity of fuzzy values. Fuzzy Sets Syst. 56, 171–174 (1993)

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13. Majumdar, P., Samanta, S.K.: Similarity measure of soft sets. New Math. Nat. Comput. 4, 1–12 (2008) 14. Atanassov, K.: Intuitionistic Fuzzy Sets. Theory and Applications, studies in Fuzziness and Soft Computing, vol. 35, pp. 324–330. Physica-Verlag, Heidelberg (1999) 15. Szmidt, E., Kacprzyk, J.: Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst. 114, 505–518 (2000) 16. Wei, G., Wei, Y.: Similarity measures of pythagorean fuzzy sets based on the cosine function and their applications. Int. J. Intell. Syst. 33, 634–652 (2018) 17. Wei, G.: Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making. Inform. 28, 547–564 (2017) 18. Ullah, K., Mahmood, T., Jan, N.: Similarity measures for t-spherical fuzzy sets with applications in pattern recognition. Symmetry. 10, 193 (2018) 19. Cuong, B.: Picture fuzzy sets-first results. part 1, seminar neuro-fuzzy systems with applications, Institute of Mathematics, Hanoi (2013)

Soft Almost s-Regularity and Soft Almost s-Normality Archana K. Prasad and S. S. Thakur

Abstract The present paper introduces the axioms of soft almost s-regularity and soft almost s-normality and presents their studies in soft topological spaces. Keywords Soft sets · Soft topology · Soft almost s-regular · Soft almost s-normal spaces

1 Introduction Researchers introduce many concepts to deal with uncertainty and to solve complicated problems in economy, engineering, medicine, sociology and environment because of the unsuccessful use of classical methods. The well-known theories can be considered as a mathematical tool for dealing with uncertainty and imperfect knowledge of the theory of fuzzy sets, theory of intuitionistic fuzzy sets, theory of vague sets, theory of rough sets and theory of probability. In 1999, Molodtsov [13] initiated the theory of soft sets as a new mathematical tool to deal with uncertainty while modeling problems with incomplete information. He [13] applied successfully soft sets in many directions such as Smoothness of function, Game theory, Operation research, Riemann integration, Perron integration, Probability and Theory of measurement to model the emphasis information. Maji et al. [11] describe an application of soft set theory to decision-making problems and gave the operation of soft sets and their properties. Later on, Ali et al. [1] improved the work of Maji et al. [11] and Das et al. [5]. Pie and Miao [15] investigated the relationship between soft sets to information systems. They showed that soft sets are A. K. Prasad (B) Department of Mathematics, Swami Vivekanand Government College, Lakhnadon, Dist-Seoni, MP 480886, India e-mail: [email protected] S. S. Thakur Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, MP 482011, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_27

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classes of special information systems. In 2011, Shabir et al. [23] initiated the study of soft topological spaces as a generalization of topological spaces. They further studied the concepts of closure, interior and neighborhood of soft sets. Since the inception of soft topology, many authors such as Aygüno˘glu and Aygün [2], Hussain [9], Hussain and Ahmad [10], Min [12], Nazmul and Samanta [14], Hazra et al. [8], Senel and Cagman [22], Zorlutuna et al. [25] and others extended and studied many topological concepts to soft sets. The study of soft separation axioms was initiated by Shabir and Naz [23]. In the recent past, many authors such as Georgiou et al. [6], Guler and Kale [9], Prasad et al. [16–21], Ça˘gman et al. [3], Varol and Aygun [24] and others studied various soft separation axioms in soft topological spaces. The present paper introduces two new soft separation axioms called soft almost s-regularity and soft almost s-normality and obtains their characterizations and properties.

2 Preliminaries Let X be a nonempty set,  be the set of parameters and S(X, ) refer to the collection of all soft sets of X relative to . Definition 2.1 ([23]) A soft topology  on X is a sub-collection of S(X, ) satisfying the following conditions: ∼

∼

[ST1] , X∈ .  [ST2] (λi ) ∈  ⇒ i∈ (λi , ) ∈ , for every i ∈ . [ST3] (λ1 ,), (λ2 , ) ∈  ⇒ (λ1 , ) ∩ (λ2 , ) ∈ . 

The triplet (X, , ) is called a soft topological space (briefly written as

STS(X, , )). The members of  are known as soft open sets in X and their complements are called soft closed sets in X. The collection of all soft closed sets in 

a STS(X, , ) is denoted by SC(X, ). 

Definition 2.2 ([23]) Let (X, , ) be a STS and (ξ, ) ∈ S(X, ). Then the closure of (ξ, ) (denoted by Cl(ξ, )) and the interior of (ξ, ) (denoted by Int(ξ, )) are defined as follows: (a) Cl(ξ, ) = ∩{(σ, ) : (σ, ) ∈ SC(X, ) and (ξ, ) ⊆ (σ, )}. (b) Int(ξ, ) = ∪{(σ, ) : (σ, ) ∈  and (σ, ) ⊆ (ξ, )}. 

Lemma 2.1 ([23]) Let (X, , ) be a STS and (ξ, ),(σ, ) ∈ S(X, ). Then (a) (b) (c) (d) (e)

(ξ, ) ∈ SC(X, ) ⇐⇒ (ξ, ) = Cl(ξ, ). If (ξ, ) ⊆ (σ, ) ⇒ Cl(ξ, ) ⊆ Cl(σ, ). (ξ, ) ∈  ⇐⇒ (ξ, ) = Int(ξ, ). If (ξ, ) ⊆ (σ, ) ⇒ Int(ξ, ) ⊆ Int(σ, ). (Cl(ξ, ))c = Int((ξ, )c ).

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(f) (Int(ξ, ))c = Cl((ξ, )c ). In 2013, Chen [4] studied soft semi-open sets as follows. 

Definition 2.3 ([4, 9]) Let (X, , ) be a STS and (ξ, ) ∈ S(X, ) then (ξ, ) is called (a) (b) (c) (d)

Soft regular open if (ξ, ) =Int(Cl(ξ, )). Soft regular closed if (ξ, ) = Cl(Int(ξ, )). Soft semi-open if (ξ, ) ⊆ Cl(Int(ξ, )). Soft semi-closed if (ξ, )⊇ Int(Cl(ξ, )). 

In a STS(X, , ), the collection of SRO(X, )(resp SSO(X, )) refers to the class of soft regular open (resp. soft semi-open) and SRC(X, )(resp SSC(X, )) refers to the class of soft regular closed (resp. soft semi-closed) sets. Theorem 2.1 ([4, 9]) Let (ξ, ),(σ, ) ∈ S(X, ) then, (a) (ξ, ) ∈ SRO(X, ) ⇔ (ξ, )c ∈ SRC(X, ). (b) (ξ, ) ∈ SSO(X, ) ⇔ (ξ, )c ∈ SSC(X, ). 

Lemma 2.2 ([4, 9]) In a STS(X, , ), the next containments are true: (a) SRO(X, ) ⊆  ⊆ SSO(X, ). (b) SRC(X, ) ⊆ SC(X, ) ⊆ SSC(X, ). The reverse containments may be false. 

Definition 2.4 ([4]) Let (X, , ) be a STS and (ξ, ) ∈ S(X, ). Then the semiclosure of (ξ, ) (written as sCl(ξ, )) and semi-interior of (ξ, ) (written as sInt(ξ, )) are defined as follows: (a) sCl(ξ, ) = ∩{(σ, ) : (σ, ) ∈ SSC(X, ) and (ξ, ) ⊆ (σ, )}. (b) sInt(ξ, ) = ∪{(σ, ) : (σ, ) ∈ SSO(X, ) and (σ, ) ⊆ (ξ, )}.   ∼ Example 2.1 Let X= x1, x2 , x3 , = {β1 , β2 } and    (λ, ) = (β1 , {x1 }), (β2 , {x2 , x3 }) and (X, , ) be a STS where  =  ∼  φ, X, (λ, ) . Then SC(X, ) =  ∼ φ, X and

  ∼  ∼ φ, X, (λ, )c , SRO(X, ) = φ, X , SRC(X, ) = ∼

SSO(X, ) = {φ, X, (λ, ),(μ, ), (ν, ), (δ, )), (ψ, ), (ω, ), (χ , )}, where, (λ, )c = {(β1 , {x2 , x 3 }), (β2 , {x1 })}, (μ, ) = {(β1 , {x1 , x2 }), (β2 , {x2 , x3 })},

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(ν, ) = {({(β1 , {x1 , x3 }), (β2 , {x2 , x3 })}, ∼

(δ, ) = {(β1 , X), (β2 , {x2 , x3 })},  ∼

(ψ, ) = {(β1 , {x1 , x2 }), β2 , X },  ∼ (ω, ) = {(β1 , {x1 , x3 }), β2 , X },  ∼ (χ , ) = {(β1 , {x1 }), β2 , X }. Consider the soft set (μ, ). Then Int(μ, ) = (λ, ) and 

X . And so, (μ, ) ⊆ Cl(Int(μ, )) = Cl(λ, ) = Cl(Int(μ,)) and hence(μ, ) ∈ SSO(X, ). Similarly, we can verify for other members of SSO(X, ). Also, by Theorem 2.1b, SSC(X, )= ∼

{φ, X, (λ, )c , (μ, )c , (ν, )c , (δ, )c , (ψ, )c , (ω, )c , (χ , )c } where (λ, )c = {(β1 , {x2 , x 3 }), (β2 , {x1 })}, (μ, )c = {(β1 , {x3 }), (β2 , {x1 })}, (ν, )c = {(β1 , {x2 }), (β2 , {x1 })}, (δ, )c = {(β1 , φ), (β2 , {x1 })}, (ψ, )c = {(β1 , {x3 }), (β2 , φ)}, (ω, )c = {(β1 , {x2 }), (β2 , φ)}, x 3 }), (β (χ , )c = {(β1 , {x2 ,  

 2 , φ)}.  Take (ξ, ) = β1 , x2, x3 , (β2 , {x1 }) , then sInt(ξ, ) ∼

=

φ and

sCl(ξ, ) =X. Remark 2.1 ([4]) In a STS (X, , ), the following containments are true: (a) sCl(ξ, ) ⊆ Cl(ξ, ). (b) Int(ξ, ) ⊆ sInt(ξ, ), for every (ξ, ) ∈ S(X, ). Definition 2.5 ([25]) A soft set (ξ, ) ∈ S(X,) is said to be a soft point if , ∃ x ∈ X and β ∈  with ξ(β) = {x} and ξ (β c ) = φ for each β c ∈  − {β} and denoted by xβ . The family of all soft points over X is written as SP(X, ). Lemma 2.3 ([8]) Let (ξ, ), (σ, ) ∈ S(X, ) and xβ ∈ SP(X, ). Then we have (a) (b) (c) (d)

/ (ξ, )c . xβ ∈ (ξ, ) ⇔ xβ ∈ xβ ∈ (ξ, ) ∪ (σ, ) ⇔ xβ ∈ (ξ, ) or xβ ∈ (σ, ). xβ ∈ (ξ, ) ∩ (σ, ) ⇔ xβ ∈ (ξ, ) and xβ ∈ (σ, ). (ξ,  ⊆ (σ, ) ⇔ xβ ∈ (ξ, ) ⇒ xβ ∈ (σ, ). 

Definition 2.6 ([8]) A STS(X, , ) is soft regular if ∀(ξ, ) ∈ SC(X, ) and / (ξ, ), ∃(μ, ),(λ, ) ∈  such that xβ ∈ ∀xβ ∈ SP(X, ) such that xβ ∈ (μ, ),(ξ, ) ⊆ (λ, ) and (μ, )∩(λ, )=φ. 

Definition 2.7 ([16]) A STS(X, , ) is soft almost regular if ∀(λ, ) ∈ / (λ, ), ∃ soft sets (μ, ),(ν, ) ∈ SRC(X, ) and ∀xβ ∈ SP(X, ) such that xβ ∈  such that xβ ∈ (μ, ), (λ, ) ⊆ (ν, ) and (μ, ) ∩ (ν, )=φ.

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Definition 2.8 ([20]) A STS(X, , ) is soft s-regular if ∀(λ, ) ∈ SC(X, ) and / (λ, ), ∃(μ, ),(ν, ) ∈ SSO(X, ) such that xβ ∈ SP(X, ) such that xβ ∈ xβ ∈ (μ, ), (λ, ) ⊆ (ν, ) and (μ, ) ∩ (ν, )=φ. Remark 2.2 ([16, 20]) The axiom soft regularity implies soft s-regularity and soft almost regularity but any soft almost regular (resp. soft s-regular space) may fail to be soft regular. The axiom soft almost regularity and soft s-regularity are independent. 

Definition 2.9 ([8]) A STS(X, , ) is soft normal if ∀(ξ, ), (σ, ) ∈ SC(X, ) such that (ξ, ) ∩ (σ, )=φ, ∃(μ, ), (λ, ) ∈  such that (ξ, ) ⊆ (μ, ),(σ, ) ⊆ (λ, ) and (μ, )∩(λ, )=φ. 

Definition 2.10 ([21]) A STS(X, , ) is soft almost normal if ∀(ξ, ) ∈ SC(X, )and(σ, ) ∈ SRC(X, ) such that (ξ, )∩(σ, )=φ, ∃(μ, ), (λ, ) ∈  such that (ξ, ) ⊆ (μ, ),(σ, ) ⊆ (λ, ) and (μ, )∩(λ, )=φ. 

Definition 2.11 ([19]) A STS(X, , ) is soft s-normal if ∀(ξ, ), (σ, ) ∈ SC(X, ) such that (ξ, ) ∩ (σ, )=φ, ∃(μ, ), (λ, ) ∈ SSO(X, ) such that (ξ, ) ⊆ (μ, ),(σ, ) ⊆ (λ, ) and (μ, )∩(λ, )=φ. Remark 2.3 ([19, 21]) The axiom soft normality implies soft s-normality and soft almost normality but any soft almost normal (resp. soft s-normal space) may fail to be soft normal. The axiom soft almost normality and soft s-normality are independent. Definition 2.12 ([7]) Let (ξ, )∈ S(X, ) and xβ ∈ SP(X,). Then (ξ, ) is called the soft neighborhood of xβ , if ∃σ, )∈  such that xβ ∈ (σ, ) ⊆ (ξ, ).

3 Soft Almost s-Regular Spaces 

Definition 3.1 A STS(X, , ) is called soft almost s-regular if ∀(λ, ) ∈ / (λ, ), ∃(μ, ), (ν, ) ∈ SRC(X, ) and xβ ∈ SP(X, ) such that xβ ∈ SSO(X, ) such that xβ ∈ (μ, ), (λ, ) ⊆ (ν, ) and (μ, ) ∩ (ν, )=φ. 

Definition 3.2 A soft set (ξ, ) of a STS(X, , ) is said to be soft regular semiopen if (ξ, )=sInt(sCl(ξ, )). 

The family of all soft regular semi-open sets in a STS(X, , ) will be denoted by SRSO(X, ).

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Definition 3.3 In a STS(X, , ), (ξ, ) ∈ S(X, ) is called soft regular semiclosed if (ξ, )c ∈ SRSO(X, ). 

The family of all soft regular semi-closed sets in a STS(X, , ) is written as SRSC(X, ). 

Remark 3.1 In a STS(X, , ), the next statements are true: (a) SRO(X, ) ⊆ SRSO(X, ) ⊆ SSO(X, ). (b) SRC(X, ) ⊆ SRSC(X, ) ⊆ SSC(X, ). It can be easily verified from examples that the reverse containments may be false. ∼

Example 3.1 Let X= {x1, x2 , x3 , x4 }, ={β1 , β2 } and (λ, ), (μ, )and (ν, ) be defined as follows: (λ, ) = {(β  2 , {x2 , x3 })},  1 }), (β  1 , {x (β2 , {x1 }) , (μ, ) = β1 , x2, x3 ,

(ν, ) = { β1 , x1 , x2, x3 , (β2 , {x1 , x 2 , x3 })}.  ∼   Then STS(X, , )) where  = φ, X, (λ, ), (μ, ), (ν, ) , (ν, ) ∈ SSO(X, ) but (ν, ) ∈ / SRSO(X, ). Remark 3.2 The family  and SRSO(X, ) are independent each other. In  Example  3.1, (ν, ) ∈  but (ν, ) ), β SRSO(X, ) while (β , {x , x , ({x , x , x })} ∈ SRSO(X, ) 1 2 4 3 4 1 2   / . (β1 , {x 1 , x 4 ), β2 , ({x 2 , x 3 , x 4 })} ∈

of ∈ / but



Theorem 3.1 For every soft set (ξ, ) of a STS(X, , ), sInt(sCl(ξ, )) ∈ SRSO(X, ). Proof Since sCl(sInt(sCl(ξ, ))) ⊇ sInt(sCl(ξ, )), therefore, sInt(sCl(sInt(sCl(ξ, )))) ⊇ sInt(sInt(sCl(ξ, ))) = sInt(sCl(ξ, )). Also, sCl(ξ, ) ⊇ sInt(sCl(ξ, )). Therefore, sCl(ξ, ) = ⊇ sCl(sCl(ξ, )) ⊇ sCl(sInt(sCl(ξ, ))). And so, sInt(sCl(ξ, )) sInt(sCl(sInt(sCl(ξ, )))). Hence, sInt(sCl(ξ, )) ∈ SRSO(X, ). 

Theorem 3.2 For every soft set (ξ, ) of a STS(X, , ), sCl(sInt(ξ, )) ∈ SRSC(X, ). Remark 3.3 Theorem 2.1 and Remark 3.1. reveal that the axiom soft s-regularity as well as soft almost regularity imply soft almost s-regularity. However, a soft almost s-regular space may fail to be soft almost regular. For,

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Example 3.2 Let X= {x1, x2 , x3 }, ={β1 , β2 } and (λ, ), (μ, )and(ν, ) be defined as follows: (λ, ) = {(β1 , {x1 }), (β2 , {x2 })}, (μ, ) = {(β1 , {x2 }), (β2 , {x1 })}, (ν, ) = {(β1 , {x1 , x2 }), (β2 , {x1 , x2 })}.  ∼

Then STS (X, , ), where  = φ, X, (λ, ), (μ, ), (ν, ) is soft almost s-regular but not soft almost regular. ∼

Example 3.3 Let X={x1 , x 2 , x3 }, ={β1 , β2 } and (λ, ), (μ, ), (ν, )and(δ, ) be defined as follows: (λ, ) = {(β1 , {x1 }), (β2 , {x2 })}, (μ, ) = {(β1 , {x3 }), (β2 , {x3 })}, (ν, ) = {(β1 , {x1 , x3 }), (β2 , {x2, x 3 })}, (δ, ) = {(β1 , {x1 , x2 }), (β2 , {x1 , x 2 })}.  ∼

Then STS (X, , ), where  = φ, X, (λ, ), (μ, ), (ν, ), (δ, ) is soft almost regular but not soft s-regular. The following theorem gives several characterizations for soft almost s-regular spaces. 

Theorem 3.3 The following statements are equivalent for a STS(X, , ): (a) (X, , ) is soft almost s-regular. (b) For every xβ ∈ SP(X, ) and (λ, ) ∈ SRO(X, ) with xβ ⊆ (λ, ), ∃(μ,)  ∈ SRSO(X, ) such that   xβ ∈ (μ, ) ⊆ sCl(μ, ) ⊆ (λ, ). (c) (ξ, ) = SRSC (X, ) and (ξ, ) ⊆ (ν, ) , (ν, ) : (ν, ) ∈ SSO ∀(ξ, ) ∈ SRC  (X, ).  (d) (ξ, ) = SSC (X, ) and (ξ, ) ⊆ (ν, ) , (ν, ) : (ν, ) ∈ SSO ∀(ξ, ) ∈ SRC (X, ). (e) For every soft set (ν, )and(ψ, ) ∈ SRO(X, ) such that (ν, )∩(ψ, ) = φ, ∃(ρ, ) ∈ SSO(X, ) such that (ν, ) ∩ (ρ, ) = φ and sCl(ρ, ) ⊆ (ψ, ). (f) For every soft set (ν, ) = φ and (ψ, ) ∈ SRC(X, ) such that (ν, ) ∩ (ψ, ) = φ, ∃(ρ, ), (δ, ) ∈ SSO(X, ) such that (ρ, ) ∩ (δ, ) = φ, (ρ, ) ∩ (ν, ) = φ and (ψ, ) ⊆ (δ, ). Proof (a)⇒(b) Let (λ, ) ∈ SRO(X, ) such that xβ ⊆ (λ, ). Then / (λ, )c . Since (X, , ) is soft almost s-regular, (λ, )c ∈ SRC (X, ) and xβ ∈ ∃(χ , ), (ϑ, ) ∈ SSO(X, ) such that xβ ∈ (ϑ, ), (λ, )c ⊆ (χ , ) and (χ , )∩ (ϑ, ) = φ. Since (ϑ, )⊆ (χ , )c and (χ , )c ∈ SSC(X, ), sCl(ϑ, ) ⊆ (χ , )c . This gives xβ ∈ (ϑ, ) ⊆ sCl(ϑ, ) ⊆ (λ, ). Again, (ϑ, ) ⊆ sInt(sCl(ϑ, )) ⊆ sCl(ϑ, ) ⊆ (λ, ). Let (μ, ) = sInt(sCl(ϑ, )). Then (ϑ, ) ⊆ (μ, ) ⊆ sCl(μ, ) ⊆ sCl(ϑ, ) ⊆ (λ, ). Now (μ, ) ∈ SRSO(X, ) such that xβ ∈ (μ, ) ⊆ sCl(μ, ) ⊆ (λ, ). / (ξ, ). Then, (λ, ) = (b)⇒(c) Suppose (ξ, ) ∈ SRC (X, ) and xβ ∈ (ξ, )c ∈ SRO(X, ) and xβ ∈ (λ, ). By (b), ∃(ϑ, ) ∈ SRSO(X, ) such that xβ ∈ (ϑ, ) ⊆ sCl(ϑ, ) ⊆ (λ, ). The soft set (ϑ, )c ∈ SRSC(X, ) and

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(ϑ, )c ⊇ (sCl(ϑ, ))c ⊇ (ξ, ). Consequently, (ϑ, )c is a soft regular semi-closed / (ϑ, )c . Hence (c) holds. semi nbd of (ξ, ) with xβ ∈ (c)⇒(d) It follows from the fact that SRSC(X, ) ⊆ SSC(X, ). (d)⇒(e) Suppose (ν, ) ∩ (ψ, ) = φ where (ψ, ) ∈ SRO(X, ). Let xβ ∈ / (ψ, )c , ∃(λ, ) ∈ (ν, ) ∩ (ψ, ). Since (ψ, )c ∈ SRC (X, ) such that xβ ∈ c / (λ, ). Let SSC(X, ) ∩ SSO(X, ) such that (ψ, ) ⊆ (λ, ) and xβ ∈ (ϑ, ) ∈ SSO(X, ) for which (ψ, )c ⊆ (ϑ, ) ⊆ (λ, ). Then (ρ, ) = (λ, )c ∈ SSO(X, ) which contains xβ and so (ρ, ) ∩ (ν, ) = φ. Also, (ϑ, )c ∈ SSC(X, ) , sCl(ρ, ) = sCl(λ, )c ⊆ (ϑ, )c ⊆ (ψ, ). (e)⇒(f) Let (ν, ) ∩ (ψ, ) = φ where (ν, ) = φ and (ψ, ) ∈ SRC (X, ) then (ν, ) ∩ (ψ, )c = φ and (ψ, )c ∈ SRO(X, ).Therefore by(e), ∃(ρ, ) ∈ SSO(X, ) such that (ν, ) ∩ (ρ, ) = φ and(ρ, ) ⊆ sCl(ρ, ) ⊆ (ψ, )c . Put (δ, ) = (sCl(ρ, ))c . Then (ψ, ) ⊆ (δ, ).Consequently, (ρ, ), (δ, ) ∈ SSO(X, ) such that (ρ, )∩(δ, ) = φ, (ρ, )∩(ν, ) = φ and (ψ, ) ⊆ (δ, ). / (λ, ). (f) ⇒ (a)Let(λ, ) ∈ SRC(X, ) and xβ ∈ SP(X, ) such that xβ ∈ Clearly , x β ∩ (λ, )=φ. Therefore by (f), ∃(ρ, ), (δ, ) ∈ SSO(X, ) such that (ρ, )∩(δ, ) = φ, (ρ, )∩xβ = φ and(λ, ) ⊆ (δ, ). Clearly , (ρ, )∩xβ = φ 

implies xβ ∈ (ρ, ). Hence by Definition 3.1, STS(X, , ) is soft almost s-regular.

4 Soft Almost s-Normal Spaces 

Definition 4.1 A STS(X, , ) is called soft almost s-normal if ∀(λ, ) ∈ SC(X, ) and (μ, ) ∈ SRC (X, ) such that (λ, ) ∩ (μ, ) = φ, ∃(δ, ),(ν, ) ∈ SSO(X, ) such that (λ, ) ⊆ (δ, ),(μ, ) ⊆ (ν, ) and (δ, ) ∩ (ν, ) = φ. Remark 4.1 The axiom soft s-normality implies soft almost s-normality, but any soft almost s-normal space may fail to be soft s-normal. For, ∼

{β} = and {x1 , x 2 , x3 , x4 , x 5 },  Example 4.1 Let X= (λ, ), (μ, ), (ν, )(δ, ), (χ , ), (ψ, ), (ω, ) and (κ, ) be defined as follows: (λ, ) = {(β, {x1 , x 2 , x3 , x4 })}, (μ, ) = {(β, {x1 , x 2 , x3 })}, (ν, ) = {(β, {x1 , x 2 , x3 , x4 , x5 })}, (δ, ) = {(β, {x2 , x3 , x4 })}, (χ , ) = {(β, {x2 , x3 })}, (ψ, ) = {(β, {x2 , x4 , x5 })}, (ω, ) = {(β, {x2 , x4 })}, (κ, ) = {(β, {x2 })}.

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 ∼ Then the STS(X, , ), where  = φ, X, (λ, ), (μ, ), (ν, ), (δ, ),

(χ , ), (ψ, ), (ω, ), (κ, ) , is soft almost s-normal. For (X, , ), being the only soft regular closed set, each non-empty soft closed set in (X, , ) intersects 

with (X, , ). But the STS(X, , ) is not soft s-normal. For {β, {x 1 }} and {β, {x 5 }} are disjoint soft closed sets but each set containing {β, {x 1 }} meets   soft semi-open each soft semi-open set containing β, {x 5 } . Remark 4.2 The axiom soft almost normality implies soft almost s-normality but any soft almost s-normal space may fail to be soft almost normal. For, Example 4.2 Let X={x1 , x 2 , x3 , x4 },  = {β} and (λ, ), (μ, ), (ν, ), (δ, ), (ψ, ) and (ω, ) be defined as follows: (λ, ) = {(β, {x1 , x 2 , x3 })}, (μ, ) = {(β, {x1 , x 2 })}, (ν, ) = {(β, {x2 , x 3 })}, (δ, ) = {(β, {x2 })}, (ψ, ) = {(β, {x2 , x3 , x4 })}, (ω, ) = {(β, {x3 })}.  ∼ Then, STS(X, , ), where  = φ, X, (λ, ), (μ, ), (ν, ), (δ, ), (ψ, ),

(ω, ) is soft almost s-normal but not soft almost normal. Remark 4.3 The axioms of almost normality and soft s-normality are independent. For, Example 4.3 Let X={x1 , x 2 , x3 },  = {β1 , β2 } and (λ, ), (μ, )and(ν, ) be defined as follows: (λ, ) = {(β1 , {x1 }), (β2 , {x1 })}, (μ, ) = {(β1 , {x1 , x2 }), (β2 , {x1 , x2 })}, (ν, ) = {(β1 , {x1 , x3 }), (β2 , {x1 , x3 })}.  ∼   Then STS(X, , ) , where  = φ, X, (λ, ), (μ, ), (ν, ) , is soft almost normal but not soft s-normal. {β1 , β2 } Example 4.4 Let X = {x1 , x 2 , x3 , x4 },  = (λ, ), (μ, ), (ν, ), (δ, ),(ψ, )and(ω, ) be defined as follows: (λ, ) = {(β1 , {x1 }), (β2 , {x1 })},

and

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(μ, ) = {(β1 , {x1 , x2 , x3 }), (β2 , {x1 , x2 , x3 })}, (ν, ) = {(β1 , {x1 , x2 }), (β2 , {x1 , x2 })}, (δ, ) = {(β1 , {x2 }), (β2 , {x2 })}, (ψ, ) = {(β1 , {x2 , x3 , x4 }), (β2 , {x2 , x3 , x4 })}, (ω, ) = {(β1 , {x3 }), (β2 , {x3 })}.  ∼ Then STS(X, , ), where  = φ, X, (λ, ), (μ, ), (ν, ), (δ, ), (ψ, ),

(ω, ) , is soft s-normal but not soft almost normal. The following theorem gives several characterizations for soft almost s-normal spaces. 

Theorem 4.1 The following statements are equivalent for a STS(X, , ): (a) (X, , ) is soft almost s-normal. (b) For every (ν, ) ∈ SC(X, ) and (ψ, ) ∈ SRO(X, ) such that (ν, ) (ψ, ), ∃(ϑ, ) ∈ SSO (X, ) such that(ν, ) ⊆ (ϑ, ) ⊆ sCl(ϑ, ) (ψ, ). (c) For every (ν, ) ∈ SRC (X, ) and (ψ, ) ∈  such that (ν, ) (ψ, ), ∃(ϑ, ) ∈ SSO (X, ) such that (ν, ) ⊆ (ϑ, ) ⊆ sCl (ϑ, ) (ψ, ).

⊆ ⊆ ⊆ ⊆

Proof (a)⇒(b) Let (ν, ) ∈ SC(X, )and(ψ, ) ∈ SRO(X, ) such that (ν, ) ⊆ (ψ, ). Then (ν, ) ∩ (ψ, )c = φ, where (ν, ) ∈ SC(X, ) and (ψ, )c ∈ SRC (X, ). Therefore, ∃(ϑ, ), (λ, ) ∈ SSO(X, ) such that (ν, ) ⊆ (ϑ, ),(ψ, )c ⊆ (λ, ) and (ϑ, ) ∩ (λ, ) = φ. Then (ν, ) ⊆ (ϑ, ) ⊆ (λ, )c ⊆ (ψ, ). Now (λ, )c ∈ SSC(X, ) so it follows that (ν, ) ⊆ (ϑ, ) ⊆ sCl(ϑ, ) ⊆ (ψ, ). (b)⇒(c) Let (ν, ) ∈ SRC(X, ) and (ψ, ) ∈  such that (ν, ) ⊆ (ψ, ). Then (ψ, )c ⊆ (ν, )c . Now (ν, )c ∈ SRO(X, ) and (ψ, )c ∈ SC(X, ) such that (ψ, )c ⊆ (ν, )c , ∃(μ, ) ∈ SSO(X, ) such that (ψ, )c ⊆ (μ, ) ⊆ sCl(μ, ) ⊆ (ν, )c . Thus, (ν, ) ⊆ (sCl(μ, ))c ⊆ (μ, )c ⊆ (ψ, ). Let (sCl(μ, ))c = (ϑ, ). Then (ϑ, ) ∈ SSO(X, ). And so, (ν, ) ⊆ (ϑ, ) ⊆ sCl(ϑ, ) ⊆ (ψ, ), since (μ, )c ∈ SSC(X, ) such that (ϑ, ) ⊆ (μ, )c . (c)⇒(a) Let (ν, ) ∈ SC(X, ) and (ψ, ) ∈ SRC (X, ) such that (ν, ) ∩ (ψ, ) = φ. Then (ψ, ) ⊆ (ν, )c . And so ∃(ϑ, ) ∈ SSO(X, ) such that (ψ, ) ⊆ (ϑ, ) ⊆ sCl(ϑ, ) ⊆ (ν, )c . Let (sCl(ϑ, ))c = (λ, ). Then , (λ, ) ∈ SSO(X, ) such that (ν, ) ⊆ (λ, ) with (ψ, ) ⊆ (ϑ, ) and (ϑ, )∩(λ, ) = φ.

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Algebraic Properties of Spherical Fuzzy Sets P. A. Fathima Perveen and Sunil Jacob John

Abstract The spherical fuzzy set (SFS) is an advanced version of fuzzy set, intutionistic fuzzy set, Pythagorean fuzzy set, and picture fuzzy set. This generalized three dimensional fuzzy set model is more realistic and accurate. In this paper, we discuss the algebraic operations on SFSs such as union, intersection, complement, algebraic sum, algebraic product, exponentiation operation, and scalar multiplication operation. Also, we prove some fundamental algebraic properties of these operations. Keywords Fuzzy sets · Spherical fuzzy sets · Algebraic structures of spherical fuzzy sets

1 Introduction Fuzzy set theory, introduce by Zadeh [11], is one of the most powerful techniques for tackling multi-attribute decision-making problems due to the challenges of obtaining sufficient and accurate data for practical decision-making due to the vagueness and ambiguity of socioeconomics. However, sometimes there are flaws and limitations in fuzzy set theory when it comes to dealing with the task at hand in a more objective way. Fuzzy sets are a more advanced version of classical sets that have a membership grade for each element. To resolve some of the difficulties of fuzzy sets, Atanassov [2] developed intuitionistic fuzzy sets. Also, many other fuzzy set generalizations have been proposed, such as interval-valued intuitionistic fuzzy sets [3], Pythagorean fuzzy sets [10], picture fuzzy sets [4], and so on. Recently, Ashraf et al. [1] proposed spherical fuzzy set (SFS) as a generalization of picture fuzzy set, with each element having three membership degrees: positive, neutral, and negative. P. A. Fathima Perveen (B) · S. J. John Department of Mathematics, National Institute of Technology Calicut, Kerala Calicut-673 601, India e-mail: [email protected] S. J. John e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_28

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Nowadays, research on these sets has become extremely fruitful, yielding numerous significant theoretical and practical results. Luca and Termini [5] proposed algebraic properties of the class of fuzzy sets and stated that the class of generalized characteristic function equipped with the lattice operation suggested by Zadeh is a Brouwerian lattice. Tanaka et al. [7] studied the algebraic properties of fuzzy sets under the operations bounded sum and bounded difference. A new idea of complex intuitionistic fuzzy subgroups was proposed by Gulzar et al. [6]. Silambarasan et al. [9] researched the algebraic operations on Pythagorean fuzzy matrices. Silambarasan [8] also studied some algebraic operations of the Picture fuzzy sets. The objective of this paper is to present certain algebraic properties for SFSs. The spherical fuzzy model, which stands out for having a massive region of participation of admissible triplets, is more adaptable than the previous fuzzy set models in that it widens the field of uncertain and ambiguous information. Therefore, discussing algebraic properties on SFSs is critical for their theoretical and practical advancement. The remainder of the paper is organized in the following manner. Section 2 revisits some fundamental concepts of fuzzy sets and spherical fuzzy sets, as well as redefines the definitions of spherical fuzzy union and spherical fuzzy intersection. In Sect. 3, some algebraic properties of SFSs are proved based on the algebraic operations such as union, intersection, complement, algebraic sum, algebraic product, scalar multiplication, and exponentiation. Finally, Sect. 4 concludes the work with recommendations for future work.

2 Preliminaries We review the fundamental notions of spherical fuzzy sets in this section and provide new definitions for spherical fuzzy intersection and union. Definition 1 [11] Let Σ be the universal set of discourse. A fuzzy set ℵ on Σ is an object of the form ℵ = {(, μℵ ())| ∈ Σ}, where μℵ : Σ → [0, 1] is the membership function of ℵ, the value μℵ () is the grade of membership of  in ℵ. Definition 2 [1] A spherical fuzzy set (SFS) ℵ over Σ can be expressed as ℵ = {(, μℵ (), ηℵ (), ϑℵ ())|  ∈ Σ}, where μℵ (), ηℵ () and ϑℵ () are the functions defined from Σ to [0,1], are called the membership functions (positive, neutral, and negative respectively) of  ∈ Σ, with the condition 0 ≤ μ2ℵ () + ηℵ2 () + ϑℵ2 () ≤ 1, ∀ ∈ Σ. Definition 3 [1] Let ℵ and Ω be two SFSs over the universe Σ, where ℵ = {(, μℵ (), ηℵ (), ϑℵ ())|  ∈ Σ} and Ω = {(, μΩ (), ηΩ (), ϑΩ ())|  ∈ Σ}. Then ℵ is said to be a spherical fuzzy subset of Ω, denoted by ℵ ⊆ Ω, if μℵ () ≤ μΩ (), ηℵ () ≤ ηΩ (), ϑℵ () ≥ ϑΩ (), ∀ ∈ Σ. Definition 4 [1] Let ℵ = {(, μℵ (), ηℵ (), ϑℵ ())|  ∈ Σ} be a SFS over Σ. Then the complement of ℵ is a SFS, denoted by ℵc , is defined as ℵc = {(, ϑℵ (), ηℵ (), μℵ ())|  ∈ Σ}.

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Definition 5 Let ℵ = {(, μℵ (), ηℵ (), ϑℵ ())|  ∈ Σ} and Ω= {(, μΩ (), ηΩ (), ϑΩ ())|  ∈ Σ} be two spherical fuzzy sets over Σ. Then (1) The spherical fuzzy union (modified), denoted as ℵ ˜ Ω, is defined by  = ℵ ˜ Ω = {(, μ (), η (), ϑ ())|  ∈ Σ}, where μ () =μℵ () ∨ μΩ () ηℵ () ∨ ηΩ () ; if (μℵ () ∨ μΩ ())2 + (ηℵ () ∨ ηΩ ())2 + (ϑℵ () ∧ ϑΩ ())2 ≤ 1 η () = ηℵ () ∧ ηΩ () ; otherwise

ϑ () = ϑℵ () ∧ ϑΩ () (2) The spherical fuzzy intersection (modified), denoted as, ℵ ˜ Ω, is defined by  = ℵ ˜ Ω = {(, μ (), η (), ϑ ())|  ∈ Σ}, where μ () = μℵ () ∧ μΩ () ηℵ () ∨ ηΩ () ; if (μℵ () or μΩ ()) = 1 η () = ηℵ () ∧ ηΩ () ; otherwise ϑ () = ϑℵ () ∨ ϑΩ ()

Where the symbols “∨” and “∧” denote the maximum and minimum operations, respectively. Definition 6 [1] Let ℵ and Ω be two SFSs over Σ. Then the algebraic sum and algebraic product of ℵ and Ω, denoted by, ℵ  Ω and ℵ  Ω respectively, defined as follows 

1. ℵ  Ω = {(, μ2ℵ () + μ2Ω () − μ2ℵ ()μ2Ω (), ηℵ ()ηΩ (), ϑℵ ()ϑΩ ())| ∈ Σ}.  2 () − ϑ 2 ()ϑ 2 ()| ∈ Σ}. 2. ℵ  Ω = {(, μℵ ()μΩ (), ηℵ ()ηΩ (), ϑℵ2 () + ϑΩ Ω ℵ

Definition 7 [1] Let ℵ be any SFS over Σ. Then the scalar multiplication operation and the exponentiation operation of Σ, denoted by, nℵ and ℵn respectively, where n denotes the natural number, is defined as follows  1. nℵ = {(, 1 − (1 − μ2ℵ ())n , ηℵn (), ϑℵn ())| ∈ Σ}  2. ℵn = {(, μnℵ (), ϑℵn (), 1 − (1 − ϑℵ2 ())n )| ∈ Σ} Example 1 Let ℵ = {(1 , 0.8, 0.3, 0.2), (2 , 0.4, 0.5, 0.3), (3 , 0.9, 0.1, 0.2)} and Ω = {(1 , 0.7, 0.4, 0.5), (2 , 0.3, 0.6, 0.4), (3 , 0.7, 0.3, 0.4)} be two SFSs over the universe Σ = {1 , 2 , 3 }. Then ℵc = {(1 , 0.2, 0.3, 0.8), (2 , 0.3, 0.5, 0.4), (3 , 0.2, 0.1, 0.9)} ℵ ˜ Ω = {(1 , 0.8, 0.4, 0.2), (2 , 0.4, 0.6, 0.3), (3 , 0.9, 0.3, 0.2)} ℵ ˜ Ω = {(1 , 0.7, 0.3, 0.5), (2 , 0.3, 0.5, 0.4), (3 , 0.7, 0.1, 0.4)} ℵ  Ω = {(1 , 0.90, 0.12, 0.10), (2 , 0.49, 0.30, 0.12), (3 , 0.95, 0.03, 0.08)} ℵ  Ω = {(1 , 0.56, 0.12, 0.53), (2 , 0.12, 0.30, 0.49), (3 , 0.63, 0.03, 0.44)} Suppose n = 3, then 3ℵ = {(1 , 0.976, 0.027, 0.008), (2 , 0.638, 0.125, 0.027), (3 , 0.996, 0.001, 0.008)} ℵ3 = {(1 , 0.512, 0.027, 0.339), (2 , 0.064, 0.125, 0.496), (3 , 0.729, 0.001, 0.339)}

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3 The Algebraic Structures of Spherical Fuzzy Sets In this section, we establish some fundamental algebraic properties of spherical fuzzy sets (SFSs). Let S F S(Σ) be the collection of all SFSs over the universe Σ. Also, for simplicity we use ℵ = (μℵ , ηℵ , ϑℵ ) to denote ℵ = {(, μℵ (), ηℵ (), ϑℵ ())| ∈ Σ} Theorem 1 Suppose that ℵ, Ω ∈ S F S(Σ). Then ℵ  Ω ⊆ ℵ  Ω  Proof We have, ℵ  Ω = ( μ2ℵ + μ2Ω − μ2ℵ μ2Ω , ηℵ ηΩ , ϑℵ ϑΩ ) and ℵ  Ω =  (μℵ μΩ , ηℵ ηΩ , ϑℵ2 + ϑΩ2 − ϑℵ2 ϑΩ2 )  It can be easily verified that μ2ℵ + μ2Ω − μ2ℵ μ2Ω ≥ μℵ μΩ  That is, if μ2ℵ + μ2Ω − μ2ℵ μ2Ω ≤ μℵ μΩ ⇒ μ2ℵ + μ2Ω − μ2ℵ μ2Ω ≤ μ2ℵ μ2Ω ⇒ μ2ℵ + μ2Ω − μ2ℵ μ2Ω − μ2ℵ μ2Ω ≤ 0 (1 − μ2Ω ) + μ2Ω (1 − μ2ℵ ) ≤ 0, it is a contradiction. ⇒ μ2ℵ

μ2ℵ + μ2Ω − μ2ℵ μ2Ω ≥ μℵ μΩ   2 2 Similarly, ηℵ2 + ηΩ − ηℵ2 ηΩ ≥ ηℵ ηΩ and ϑℵ2 + ϑΩ2 − ϑℵ2 ϑΩ2 ≥ ϑℵ ϑΩ Therefore, μℵΩ μℵΩ , ηℵΩ ηℵΩ , and ϑℵΩ ≥ ϑℵΩ ⇒ℵΩ ⊆ℵΩ Thus

Theorem 2 Let ℵ, Ω,  ∈ S F S(Σ). Then (1) (2) (3) (4)

ℵΩ =Ω ℵ ℵΩ =Ω ℵ (ℵ  Ω)   = ℵ  (Ω  ) (ℵ  Ω)   = ℵ  (Ω  )

Proof Proof of (1) and (2) can be done directly. We prove only (3), (4) can be prove in similar way.  We know, ℵ  Ω = ( μ2ℵ + μ2Ω − μ2ℵ μ2Ω , ηℵ ηΩ , ϑℵ ϑΩ ) Now,

 (ℵ  Ω)   = ( μ2ℵ + μ2Ω − μ2ℵ μ2Ω + μ2 − (μ2ℵ + μ2Ω − μ2ℵ μ2Ω )μ2 , (ηℵ ηΩ )η , (ϑℵ ϑΩ )ϑ )  = ( μ2ℵ + μ2Ω + μ2 − μ2ℵ μ2Ω − μ2ℵ μ2 − μ2Ω μ2 + μ2ℵ μ2Ω μ2 , ηℵ ηΩ η , ϑℵ ϑΩ ϑ )  = ( μ2ℵ + μ2Ω + μ2 − μ2Ω μ2 − μ2ℵ (μ2Ω + μ2 − μ2Ω μ2 ), ηℵ (ηΩ η ), ϑℵ (ϑΩ ϑ )) = ℵ  (Ω  )

Theorem 3 Let ℵ, Ω ∈ S F S(Σ). Then (1) (ℵ  Ω)c = ℵc  Ω c (2) (ℵ  Ω)c = ℵc  Ω c

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(3) (ℵ  Ω)c ⊆ ℵc  Ω c (4) (ℵ  Ω)c ⊇ ℵc  Ω c Proof We prove only (1) and (3). Other proofs can be done in similar way.  (1) ℵ  Ω = ( μ2ℵ + μ2Ω − μ2ℵ μ2Ω , ηℵ ηΩ , ϑℵ ϑΩ )  (ℵ  Ω)c = (ϑℵ ϑΩ ), ηℵ ηΩ , μ2ℵ + μ2Ω − μ2ℵ μ2Ω ) = ℵc  Ω c  (3) (ℵ  Ω)c = (ϑℵ ϑΩ , ηℵ ηΩ , μ2ℵ + μ2Ω − μ2ℵ μ2Ω ), and  ℵc  Ω c = ( ϑℵ2 + ϑΩ2 − ϑℵ2 ϑΩ2 , ηℵ ηΩ , μℵ μΩ )   2 2 By Theorem 1, we have μ2ℵ + μ2Ω − μ2ℵ μ2Ω ≥ μℵ μΩ , ηℵ2 + ηΩ − ηℵ2 ηΩ ≥  ηℵ ηΩ , and ϑℵ2 + ϑΩ2 − ϑℵ2 ϑΩ2 ≥ ϑℵ ϑΩ ⇒ μ(ℵΩ)c ≤ μℵc Ω c , η(ℵΩ)c ≤ ηℵc Ω c , and ϑ(ℵΩ)c ≥ ϑℵc Ω c ⇒ (ℵ  Ω)c ⊆ ℵc  Ω c Theorem 4 Let ℵ, Ω ∈ S F S(Σ) and let n be any natural number. If ℵ ⊆ Ω, then (1) nℵ ⊆ nΩ (2) ℵn ⊆ Ω n Proof We prove only (1). (2) can be prove using similar method. Suppose that ℵ  μℵ ≤ μΩ , ηℵ ≤ ηΩ , and ϑℵ ≥ ϑΩ . ⊆ Ω, which means

n , and ϑℵ ≥ μℵ ≤ μΩ ⇒ 1 − (1 − μ2ℵ )n ≤ 1 − (1 − μ2Ω )n , ηℵ ≤ ηΩ ⇒ ηℵn ≤ ηΩ n n ϑΩ ⇒ ϑℵ ≥ ϑΩ . Therefore, nℵ ⊆ nΩ.

Theorem 5 Let ℵ, Ω ∈ S F S(Σ) and let n be any natural number. Then (1) n(ℵ  Ω) = nℵ  nΩ (2) (ℵ  Ω)n = ℵn  Ω n Proof The proof of (1) is given as follows. (2) can be proved in similar way. 2 n     μ2ℵ + μ2Ω − μ2ℵ μ2Ω n(ℵ  Ω) = 1− 1− , (ηℵ ηΩ )n , (ϑℵ ϑΩ )n   n = 1 − (1 − μ2ℵ − μ2Ω + μ2ℵ μ2Ω )n , ηℵn ηΩ , ϑℵn ϑΩn   n 1 − (1 − μ2ℵ )n (1 − μ2ℵ )n , ηℵn ηΩ , ϑℵn ϑΩn = = nℵ  nΩ Theorem 6 Let ℵ be a SFS over the universe Σ and let n 1 and n 2 be two natural numbers. Then (1) n 1 ℵ  n 2 ℵ = (n 1 + n 2 )ℵ (2) ℵn1  ℵn2 = ℵ(n 1 +n 2 )

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Proof We prove only (1). (2) can be proved in same method.  n1 ℵ  n2 ℵ =

1 − (1 − μ2ℵ )n 1 + 1 − (1 − μ2ℵ )n 2 − (1 − (1 − μ2ℵ )n 1 )(1 − (1 − μ2ℵ )n 2 ),

  = 1 − (1 − μ2ℵ )n 1 +n 2 , ηℵn 1 +n 2 , ϑℵn 1 +n 2 = (n 1 + n 2 )ℵ

ηℵn 1 ηℵn 2 , ϑℵn 1 ϑℵn 2



Theorem 7 Let ℵ, Ω ∈ S F S(Σ). Then ˜  Ω) = ℵ  Ω (1) (ℵ  Ω) (ℵ ˜  Ω) = ℵ  Ω (2) (ℵ  Ω) (ℵ Proof The proof is direct from Theorem 1. Theorem 8 Let ℵ, Ω,  ∈ S F S(Σ). Then ˜ (1) (ℵ ˜ Ω)   = (ℵ  ) (Ω  ) ˜ (2) (ℵ ˜ Ω)   = (ℵ  ) (Ω  ) Proof The proof of (1) is given as follows. (2) can be proved in similar way. Consider the following cases. Case 1 : If (μℵ () or μΩ ()) = 1, for every  ∈ Σ. Then ℵ ˜ Ω = (μℵ ∧ μΩ , ηℵ ∧ ηΩ , ϑℵ ∨ ϑΩ )   (μℵ ∧ μΩ )2 + μ2 − (μℵ ∧ μΩ )2 μ2 , (ηℵ ∧ ηΩ )η , (ϑℵ ∨ ϑΩ )ϑ (ℵ ˜ Ω)   =    = μ2ℵ + μ2 − μ2ℵ μ2 ∧ μ2Ω + μ2 − μ2Ω μ2 , ηℵ η ∧ ηΩ η , ϑℵ ϑ ∨ ϑΩ ϑ ˜ = (ℵ  ) (Ω  )

Case 2 : If (μℵ () or μΩ ()) = 1, for some  ∈ Σ, say 0 . Without loss of generality, assume that μℵ (0 ) = 1, then ηℵ (0 ) = ϑℵ (0 ) = 0 ˜ 0 = Ω0 ⇒ ((ℵ ˜ Ω)  )0 = (Ω  )0 Thus, ℵ0 Ω ˜ Also, (ℵ  )0 = ℵ0 ⇒ ((ℵ  ) (Ω  ))0 = (Ω  )0 ˜ Implies that ((ℵ ˜ Ω)  )0 = ((ℵ  ) (Ω  ))0 Note that, In an SFS ℵ, if μℵ () = 1, ηℵ () = 0, and ϑℵ () = 0 ∀ ∈ Σ, then also the result is true if we choose 0 as arbitrary in Case 2. ˜ Therefore, we can conclude that in any case (ℵ ˜ Ω)   = (ℵ  ) (Ω  ) Let S F S ∗ (Σ) be the collection of all SFSs over Σ, where for any two SFSs ℵ, Ω ∈ S F S ∗ (Σ), (μℵ () ∨ μΩ ())2 + (ηℵ () ∨ ηΩ ())2 + (ϑℵ () ∧ ϑΩ ())2 1,for every  ∈ Σ. Theorem 9 Let ℵ, Ω ∈ S F S ∗ (Σ). Then (1) (ℵ ˜ Ω)  (ℵ ˜ Ω) = ℵ  Ω (2) (ℵ ˜ Ω)  (ℵ ˜ Ω) = ℵ  Ω

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Proof The proof of (1) is given below. (2) can be proved using similar steps. Consider the following two cases. Case 1: If (μℵ () or μΩ ()) = 1, for every  ∈ Σ. Then  (ℵ ˜ Ω)  (ℵ ˜ Ω) = (μℵ ∨ μΩ )2 + (μℵ ∧ μΩ )2 − (μℵ ∨ μΩ )2 (μℵ ∧ μΩ )2 ,  (ηℵ ∨ ηΩ )(ηℵ ∧ ηΩ ), (ϑℵ ∧ ϑΩ )(ϑℵ ∨ ϑΩ )   = μ2ℵ + μ2Ω − μ2ℵ μ2Ω , ηℵ ηΩ , ϑℵ ϑΩ =ℵΩ Case 2 : If (μℵ () or μΩ ()) = 1, for some  ∈ Σ, say 0 . Without loss of generality, assume that μℵ (0 ) = 1, then ηℵ (0 ) = ϑℵ (0 ) = 0 ˜ 0 = ℵ0 and ℵ0 Ω ˜ 0 = Ω0 Thus, ℵ0 Ω ⇒ ((ℵ ˜ Ω)  (ℵ ˜ Ω))0 = (ℵ  Ω)0 Therefore, in any case (ℵ ˜ Ω)  (ℵ ˜ Ω) = ℵ  Ω Theorem 10 Let ℵ, Ω,  ∈ S F S ∗ (Σ). Then ˜ (1) (ℵ ˜ Ω)   = (ℵ  ) (Ω  ) ˜ (2) (ℵ ˜ Ω)   = (ℵ  ) (Ω  ) Proof We prove  only (1). (2) can be proved in same method. (ℵ ˜ Ω)   = =



(μℵ ∨ μΩ )2 + μ2 − (μℵ ∨ μΩ )2 μ , (ηℵ ∨ ηΩ )η , (ϑℵ ∧ ϑΩ )ϑ



(μ2ℵ + μ2 ) ∨ (μ2Ω + μ2 ) − (μ2ℵ μ2 ) ∨ (μ2Ω μ2 ), ηℵ η ∨ ηΩ η , ϑℵ ϑ ∧ ϑΩ ϑ    (μ2ℵ + μ2 ) − μ2ℵ μ2 ∨ (μ2Ω + μ2 ) − μ2Ω μ2 , ηℵ η ∨ ηΩ η , ϑℵ ϑ ∧ ϑΩ ϑ = ˜ = (ℵ  ) (Ω  )



Definition 8 Let ℵ be any SFS over the universe Σ. Then the concentration of ℵ is a SFS, denoted as ℵC O N , is defined by ℵC O N = ℵ2 . That is,    ℵC O N = μ2ℵ , ηℵ2 , 1 − (1 − ϑℵ2 )2 Definition 9 Let ℵ be any SFS over the universe Σ. Then the dilation of ℵ is a SFS, denoted as ℵ D I L , is defined by ℵ D I L = ℵ1/2 . That is,    1/2 1/2 ℵ D I L = μℵ , ηℵ , 1 − (1 − ϑℵ2 )1/2 Theorem 11 Let ℵ ∈ S F S(Σ), then ℵC O N ⊆ ℵ ⊆ ℵ D I L Proof We know that μℵ , ηℵ , ϑℵ ∈ [0, 1]. 1/2 Thus, μ2ℵ ≤ μℵ ≤ μℵ 1/2 Similarly, ηℵ2 ≤ ηℵ ≤ ηℵ 2 2 Also, (1 − ϑℵ ) ≤ (1 − ϑℵ2 ) ≤ (1 − ϑℵ2 )1/2 ⇒ 1− (1 − ϑℵ2 )2 ≥ 1 − (1 −ϑℵ2 ) ≥ 1 − (1 − ϑℵ2 )1/2 ⇒ 1 − (1 − ϑℵ2 )2 ≥ ϑℵ ≥ 1 − (1 − ϑℵ2 )1/2 Therefore, ℵC O N ⊆ ℵ ⊆ ℵ D I L

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4 Conclusion In this paper, we have proved some algebraic properties of spherical fuzzy sets, including commutativity, associativity, distributivity, absorption, etc. We also established several theorems and defined new concentration and dilation of SFSs. These results can be used in more applications of spherical fuzzy set theory. The results of this research will be useful in the creation of the Picture fuzzy semilattice and its algebraic property. The applicability of the suggested operators of SFSs in decision making, risk analysis, and many other uncertain and fuzzy environments can be recommended as future work.

References 1. Ashraf, S., Abdullah, S., Mahmood, T., Ghani, F., Mahmood, T.: Spherical fuzzy sets and their applications in multi-attribute decision making problems. J. Intell. & Fuzzy Syst. 36, 2829–2844 (2019) 2. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986) 3. Atanassov, K.T., Gargov, G.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343–349 (1989) 4. Cuong, B.C., Kreinovich, V.: Picture fuzzy sets-a new concept for computational intelligence problems. In: Proceedings of 2013 Third World Congress on Information and Communication Technologies (WICT 2013), pp. 1–6. IEEE (2013) 5. De Luca, A., Termini, S.: Algebraic properties of fuzzy sets. J. Math. Anal. Appl. 40(2), pp. 373–386 (1972) 6. Gulzar, M., Mateen, M.H., Alghazzawi, D., Kausar, N.: A novel applications of complex intuitionistic fuzzy sets in group theory. IEEE Access 8, 196075–196085 (2020) 7. Mizumoto, M., Tanaka, K.: Fuzzy sets and their operations. Inf. Control. 48(1), 30–48 (1981) 8. Silambarasan, I.: Some algebraic properties of picture fuzzy sets. Bull. Int. Math. Virtual Inst. 11(3), 429–442 (2021) 9. Silambarasan, I., Sriram, S.: Algebraic operations on Pythagorean fuzzy matrices. Math. Sci. Int. Res. J. 7(2), 406–414 (2018) 10. Yager, R.R.: Pythagorean fuzzy subsets. In: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), pp. 57-61. IEEE (2013) 11. Zadeh, L.A.: Fuzzy sets. Inf. Control. 8(3), 338–353 (1965)

Divergence Measures of Pythagorean Fuzzy Soft Sets T. M. Athira and Sunil Jacob John

Abstract This work initiates the study of divergence measures of Pythagorean fuzzy soft sets (PFSSs). A couple of expressions to find PFSS divergence measures are obtained, and thus we can quantify the deviation between any two PFSSs. Also, certain theorems based on the properties of proposed expressions are proved. Keywords Pythagorean fuzzy sets · Soft sets · Divergence measure

1 Introduction Comparing descriptions of two objects and identifying the similarities or differences between them are crucial in many real-life situations. The interrogation of modeling the process of such comparison was done by several researchers. For example, inspired by the concept of divergence of probability distributions, Bhandari et al. [1] introduced a notion of divergence for fuzzy sets. Divergence measures are the differences between two generalized sets. Divergence measures are widely applied in a variety of contexts, including decision-making, medical diagnosis, pattern recognition, and other applications. In pattern recognition problems, it is simpler to classify and group patterns using divergence measures. The literature presents numerous divergence measures for Fuzzy sets and Intuitionistic Fuzzy sets. The articles [2, 5, 10] study divergence measures in different contexts with relevant applications. The divergence measure of Intuitionistic fuzzy sets was initially presented by Monte et al. and characterizing the divergence for IFSs was crucial since it has applications in many fields [2, 12]. For intuitionistic fuzzy information clustering, Liu et al. [13] combined similarity and divergence measures. Thao [14] established divergence measures for Intuitionistic fuzzy sets using the Archimedean t-conorm operators.

T. M. Athira (B) · S. J. John Department of Mathematics, National Institute of Technology Calicut, Calicut 673 601, Kerala, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_29

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Zhou et al. [15] proposed a novel divergence measure for Pythagorean fuzzy set using the belief function from Dempster–Shafer evidence theory. Research on the theoretical and application point of view of soft sets and their various generalizations is growing rapidly. In [9] Peng et al. introduced Pythagorean fuzzy soft sets (PFSSs) as a blending of soft sets (SS) and Pythagorean fuzzy sets(PFS). Interestingly, PFSS generalizes both SS and PFS. Recently, Athira et al. [3, 4] initiated the study of the entropy and distance measures for PFSS. To the best of the authors’ knowledge, very little work has been done on the theory of the PFSS. Keeping the usefulness of PFSSs to look over the given data better than IFSS or FSS, this article focused on divergence measures of PFSS. Divergence measure quantifies the difference between any two PFSS. PFSS with lower divergence is “more similar”. The axiomatic definition and a couple of expressions to calculate PFSS divergence measures are proposed. Also, certain interesting theorems based on properties of PFSS divergence measures are proved. This paper contains two sections besides the introduction. Section 2 contains preliminary definitions and results required for our entire discussion. In Sect. 3, we introduce divergence measures for PFSSs. Some interesting properties of divergence measures are obtained, and we propose certain nice examples for divergence measures for PFSSs. The TOPSIS method based on PFSS divergence measure for selecting the best-fitting alternative is proposed.

2 Preliminaries This section explains the fundamental definitions needed for entire discussions. Here, U represents the universal sets and E is the set of parameters. Definition 1 [8] The pair (S, E) is said to be soft set over U if S is a mapping from E to P(U), where P(U) is power set of U. Definition 2 [11] A Pythagorean fuzzy set P on U is defined as the set {(u, μ P (u), v P (u)) : u ∈ U} where μ P : U → [0, 1] and v P : U → [0, 1] with 0 ≤ μ2P + v 2P ≤ 1. Definition 3 [9] A Pythagorean fuzzy soft set (PFSS) is defined as the pair (P, E) such that P : E → PFS(U) and PFS(U) is the collection of all Pythagorean fuzzy subsets of U. Definition 4 [9] Let (P1 , E1 ) and (P2 , E2 ) be two PFSSs on U. Then 1. (P1 , E1 ) is a subset of (P2 , E2 ) if, (a) E1 ⊆ E2 (b) for each ζ ∈ E1 , μP1 (ζ ) (u) ≤ μP2 (ζ ) (u) and vP1 (ζ ) (u) ≥ vP2 (ζ ) (u) ∀u ∈ U. 2. The complement of (P, E), i.e., (P, E)c = (Pc , E),

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3. The union of (P1 , E1 ) and (P2 , E2 ), i.e., (P1 , E1 ) ∪ (P2 , E2 ) = (P,  E1 ∪ E2) where P(ζ ) = u, max μP1 (u), μP2 (u) , min vP1 (u), vP2 (u) : u ∈ U for each ζ ∈ E1 ∪ E2 . 4. The intersection of (P1 , E1 ) and (P2 , E2), i.e., (P1 , E1 ) (P ,E ) = 2 2  (P, E1 ∩ E2 ), P(ζ ) = u, min μP1 (u), μP2 (u) , max vP1 (u), vP2 (u) : u ∈ U for each ζ ∈ E1 ∩ E2 .

3 Divergence Measure of Pythagorean Fuzzy Soft Sets In this section, an axiomatic definition of PFSS divergence measure is proposed, and a couple of expressions to find out divergence measures are identified. Definition 5 Let (P, E), (Q, E) be two PFSS on  with parameter set E. The PFSS divergence measure, denoted as Div of two PFSSs, is a function from P F SS() → R which satisfies the following conditions: 1. 2. 3. 4.

Div((P, E), (Q, E)) = Div((Q, E), (P, E)) Div((P, E), (Q, E)) = 0 iff (P, E) = (Q, E)

Div((P, E)  (R, E), (Q, E)  (R, E)) ≤ Div((P, E), (Q, E))∀(R, E) ∈ PFSS() Div((P, E) (R, E), (Q, E) (R, E)) ≤ Div((P, E), (Q, E))∀(R, E) ∈ PFSS()

Theorem 1 The PFSS divergence measures are non-negative. Proof From conditions 2 and 4, it is possible to obtain Div((P, E), (Q, E)) ≥ 0 for any (P, E), (Q, E) ∈ PFSS(). Because, when (R, E) = ∅, Div((P, E) (R, E), (Q, E) (R, E)) = Div(∅, ∅) = 0 from the condition 2. And so 0 ≤ Div((P, E), (Q, E)) from 4. Theorem 2 Let (P, E), (Q, E) are two PFSSs over . For a PFSS divergence measure Div, Div(P, E) (Q, E), (Q, E)) = Div(P, E), (P, E)  (Q, E)) Div(P, E)  (Q, E), (Q, E)) = Div(P, E), (P, E) (Q, E)) Proof From conditions 3 and 4 of definition, we have Div(P, E) (Q, E), (Q, E)) = Div(P, E) (Q, E), ((P, E)  (Q, E)) (Q, E)) ≤ Div(P, E), (P, E)  (Q, E)) = Div((P, E) (Q, E))  (P, E), (Q, E)  (P, E)) ≤ Div(P, E) (Q, E), (Q, E)) Thus, Div(P, E) (Q, E), (Q, E)) = Div(P, E), (P, E)  (Q, E)). Similarly,

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Div(P, E), (P, E) (Q, E)) = Div((P, E)  (Q, E)) (P, E), (Q, E) (P, E)) ≤ Div(P, E)  (Q, E), (Q, E)) = Div((P, E)  (Q, E)), (P, E) (Q, E)  (Q, E)) ≤ Div(P, E), (P, E) (Q, E)) Thus, Div(P, E)  (Q, E), (Q, E)) = Div(P, E), (P, E) (Q, E)) The following theorem gives some expressions for divergence measure. Theorem 3 Let  = {u 1 , u 2 , . . . u n } and E = {ζ1 , ζ2 , . . . , ζm }. Consider (P, E), (Q, E) are two PFSSs. Then the following are divergence measure of (P, E) and (Q, E).   ⎧ 2μ2 u ⎪ ⎪ P(ζ j ) i   ⎪ 2 ⎪ ⎪ μ u log    + ⎪ i P(ζ ) ⎪ 2 2 j ⎪ μ u +μ u ⎪ ⎪ P(ζ j ) i Q(ζ j ) i ⎪ ⎪ ⎪ ⎪   ⎪ 2 ⎪ 2ν u ⎪ ⎪ P(ζ j ) i ⎪   ⎪ 2 ⎪ νP(ζ u log ⎪    +   ⎪ 2 2 ⎪ j) i ν u + ν u ⎪ if μP(ζ ) u i = 0, P(ζ j ) i Q(ζ j ) i ⎪ j n m ⎪ 1  ⎨     Div1 ((P, E), (Q, E)) = &νP(ζ ) u i = 0 2μ2 u ⎪ 2mn j Q(ζ j ) i   ⎪ i=1 j=1 ⎪ 2 ⎪ ⎪ μ + u log     ⎪ Q(ζ j ) i ⎪ ⎪ μ2 u + μ2 u ⎪ Q(ζ j ) i P(ζ j ) i ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ 2 ⎪ 2ν u ⎪ ⎪ Q(ζ j ) i   ⎪ ⎪ 2 ⎪     ⎪ νQ(ζ ) u i log 2 ⎪ j ⎪ ν u + ν2 u ⎪ ⎪ Q(ζ j ) i P(ζ j ) i ⎪ ⎩ 0 otherwise

(1)



⎧ 2 μ2P(ζ ) (u i ) + μ2Q(ζ ) (u i ) ⎪ j j ⎪ μ2P(ζ j ) (u i ) − μ2Q(ζ j ) (u i ) + ⎪ ⎪ 2 2 ⎪ μ μ (u ) (u ) ⎪ i i if μP(ζ j ) (u i ) = 0, P(ζ j ) Q(ζ j ) n m ⎪ 1 ⎨ 

2 2 2 νP(ζ ) (u i ) + νQ(ζ  Div2 ((P, E), (Q, E)) = (u i ) νP(ζ j ) (u i ) = 0 j j) ⎪ 2 2 2mn ⎪ i=1 j=1 ⎪ ⎪ νP(ζ j ) (u i ) − νQ(ζ j ) (u i ) 2 2 ⎪ νP(ζ (u i ) νQ(ζ (u i ) ⎪ ) ) j j ⎪ ⎩ 0 otherwise

(2) Proof We will prove the necessary part of condition 2 and condition 3 for Divi , i = 1, 2. Condition 4 can be proved just like proof of condition 3, and the remaining parts are trivial. To prove necessary part of condition 2, Div1 ((P, E), (Q, E)) = 0 implies ⎛

⎞ 2μ2P(ζ j ) (u i ) 2 μ log (u ) ⎜ P (ζ j ) i μ2P(ζ j ) (u i ) + μ2Q(ζ j ) (u i ) ⎟ ⎜ ⎟ ⎜ ⎟=0 ⎜ ⎟ 2μ2Q(ζ j ) (u i ) ⎝ ⎠ 2 +μQ(ζ j ) (u i ) log 2 2 μP(ζ j ) (u i ) + μQ(ζ j ) (u i ) Suppose there exist an ζ j0 and u i0 such that μP(ζ j0 ) (u i0 )) = μQ(ζ j0 ) (u i0 )). Then

(u i 0 ) j0 ) > 0 and = 1. (u i0 )+μ2Q(ζ j0 ) (u i0 )

2μ2P(ζ μ2P(ζ ) j0

(3)

Divergence Measures of Pythagorean Fuzzy Soft Sets

We have the expression 1 − then, 1 −

μ2P(ζ

j0

2 ) (u i 0 )+μQ(ζ

2μ2P(ζ

That is,

j0 )

j0

(u i 0 )

1 a

415

< log a, ∀a > 0, a = 1. Take a =

) (u i 0 )

< log μ2

2μ2P(ζ

P(ζ j ) 0

j0

) (u i 0 )

(u i0 )+μ2Q(ζ j0 ) (u i0 )

μ2P(ζ

  2μ2P(ζ ) u i0 j   0 2   ) u i 0 +μQ(ζ ) u i 0

j0

j0

.

  2μ2P(ζ j ) u i0       1 2   0   μP(ζ j ) u i0 − μ2Q(ζ j ) u i0 < μ2P(ζ j ) u i0 log 2 0 0 0 2 μP(ζ j ) u i0 + μ2Q(ζ j ) u i0 0 0 (4) Similarly,   2μ2Q(ζ j ) u i0       1 2 2 2 0     μQ(ζ j ) u i0 − μP(ζ j ) u i0 < μQ(ζ j ) u i0 log 2 0 0 0 2 μP(ζ j ) u i0 + μ2Q(ζ j ) u i0 0 0 (5) By adding Eqs. 4 and 5, we get   ⎞ 2μ2P(ζ j ) u i0   0 2   +⎟ ⎜ μ2P(ζ j ) (u i ) log 2 ⎜ u i0 + μQ(ζ j ) u i0 ⎟ μ P(ζ ) j0 ⎜ ⎟ 0   ⎜ ⎟ > 0. 2 ⎜ ⎟ u 2μ i   0 Q(ζ j0 ) ⎝ μ2    ⎠ Q(ζ j0 ) u i 0 log 2 2 μP(ζ j ) u i0 + μQ(ζ j ) u i0 ⎛

0

0

Thus, from Eq. 3, we get a contradiction. Hence, μP(ζ j ) (u i )) = μQ(ζ j ) (u i )), ∀i, j. and similar way, we get νP(ζ j ) (u i )) = νQ(ζ j ) (u i )), ∀i, j. Thus Div1 ((P, E), (Q, E)) = 0 implies (P, E) = (Q, E). Now, Div2 ((P, E), (Q, E)) = 0. Since each term is positive   we can conclude that 2 2 2 2 μ2P(ζ j ) (u i ) − μ2Q(ζ j ) (u i ) = 0 and νP(ζ (u i ) − νQ(ζ (u i ) ∀i, j. j) j) Hence μP(ζ j ) (u i ) = μQ(ζ j ) (u i ) and νP(ζ j ) (u i ) = νQ(ζ j ) (u i ). Thus, Div2 ((P, E), (Q, E)) = 0 implies (P, E) = (Q, andE). Now, we will prove the condition 3, i.e., Divi ((P, E)  (R, E), (Q, E)  (R, E)) ≤ Divi ((P, E), (Q, E)) ∀(R, E) ∈ P F SS(U ), i =

1, 2 μ Let us denote Divi for the expression involving membership value only and ν Divi for the expression involving non-membership value only. Thus, Divi = μ Divi + Diviν , i = 1, 2. For the case of Div1 , the expression is symmetric. So, it is enough to consider three μ cases for Div1 and for Divν1 .

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1. Let μP(ζ j ) (u i ) ≤ μQ(ζ j ) (u i ) ≤ μ R (ζ j ) (u i ) for some u i and ζ j . Then, μ2PR ζ (u i ) = μ2R ζ (u i ) and μ2QR ζ (u i ) = μ2R(ζi ) (u i ). ( j) ( j) ( j) μ

μ

Div1 ((P, E)  (R, E), (Q, E)  (R, E)) = Div1 ((R, E), (R, E)) =0 μ

≤ Div1 ((P, E), (Q, E)) ∀(R, E) ∈ PFSS().

2. μP(ζ j ) (u i ) ≤ μ R (ζ j ) (u i ) ≤ μQ(ζ j ) (u i ) for some u i and ζ j . Then μ2PR ζ (u i ) = μ2R ζ (u i ) and μ2QR ζ (u i ) = μ2Q ζ (u i ). ( j) ( j) ( j) ( j) μ μ Div1 ((P, E)  (R, E), (Q, E)  (R, E)) = Div1 ((R, E), (Q, E)). 2 2 For a function f (x) = x 2 log a 22x+x 2 + a 2 log a 2a+x 2 , a, x ∈ (0, 1], x ≤ a, we have 2

f x = 2x log a 22x+x 2 ≤ 0. Thus for any y ∈ (0, 1] such that y ≤ x, f (y) ≥ f (x). Here, take a = μ2Q ζ (u i ), x = μ2R ζ (u i ) , and y = μ2P ζ (u i ) then we get ( j) ( j) ( j) the required inequality. 3. μ R (ζ j ) (u i ) ≤ μP(ζ j ) (u i ) ≤ μQ(ζ j ) (u i ) for some u i and ζ j μ2PR ζ (u i ) = μ2P ζ (u i ) and μ2QR ζ (u i ) = μ2Q ζ (u i ) ( ) ( ) ( ) ( ) j

j

j

j

Thus, Divμ1 ((P, E)  (R, E), (Q, E)  (R, E)) = Divμ1 ((P, E), (Q, E)).

For the case of Div2 , the expression is symmetric. So, it is enough to consider three cases for Divμ2 and for Divν2 . 1. μP(ζ j ) (u i ) ≤ μQ(ζ j ) (u i ) ≤ μ R (ζ j ) (u i ) for some u i and ζ j . Then, μ2PR ζ (u i ) = μ2R ζ (u i ) and μ2QR ζ (u i ) = μ2R(ζi ) (u i ) ( j) ( j) ( j) Thus, μ

μ

Div1 ((P, E)  (R, E), (Q, E)  (R, E)) = Div1 ((R, E), (R, E)) =0 μ

≤ Div1 ((P, E), (Q, E)) ∀(R, E) ∈ PFSS().

2. μP(ζ j ) (u i ) ≤ μ R (ζ j ) (u i ) ≤ μQ(ζ j ) (u i ) for some u i and ζ j . Then, μ2PR ζ (u i ) = μ2R ζ (u i ) and μ2QR ζ (u i ) = μ2Q ζ (u i ). ( j) ( j) ( j) ( j) Consider the function f (x) = a12 + x12 for a given a ∈ (0, 1] and for all x ∈ [a, 1]. Since f  (x) ≤ 0, f (x) is non-increasing function. Here, take a = μQ(ζ j ) (u i ) and we have μP(ζ j ) (u i ) ≤ μ R (ζ j ) (u i ) that implies  

2  μ (u i )+μ2 (u i ) μ2P(ζ ) (u i )+μ2Q(ζ ) (u i ) R (ζ j ) Q(ζ j ) j j ≤ μ2 μ2 (u )μ2 (u ) (u )μ2 R(ζ j )

i

Q(ζ j )

i

P(ζ j )

i

Q(ζ j )

 2  2 Also, μ2R ζ (u i ) − μ2Q ζ (u i ) ≤ μ2P ζ (u i ) − μ2Q(ζ j ) (u i ) . ( j) ( j) ( j) Thus, the required inequality holds in this case also. 3. μ R (ζ j ) (u i ) ≤ μP(ζ j ) (u i ) ≤ μQ(ζ j ) (u i ) for some u i and ζ j . Thus μ2PR ζ (u i ) = μ2P ζ (u i ) and μ2QR ζ (u i ) = μ2Q ζ (u i ) ( j) ( j) ( j) ( j) μ μ Div1 ((P, E)  (R, E), (Q, E)  (R, E)) = Div1 ((P, E), (Q, E)). Similarly, we μ can prove the case of Diviν ,i = 1, 2. Since Divi = Divi + Diviν , we can conclude the third condition of definition 5.

Divergence Measures of Pythagorean Fuzzy Soft Sets

417

Hence, Divi , i = 1, 2 satisfy all the four conditions in definition 5, and thus Divi , i = 1, 2 are divergence measure. Example 1 The PFSSs (P1 , E), (P2 , E) and (P3 , E) over  = {u 1 , u 2 , u 3 , u 4 , u 5 } and E = {ζ1 , ζ2 , ζ3 , ζ4 } are given below. ⎛

u1

⎛

u1

⎛

u1

(0.7, 0.3) ζ2⎜ (0.0, 0.9) (P1 , E) = ⎜ ζ3⎝ (0.4, 0.7) ζ4 (0.8, 0.2) ζ1

(0.4, 0.3) ζ2⎜ (0.0, 0.9) ⎜ (P2 , E) = ⎝ ζ3 (0.2, 0.2) ζ4 (0.8, 0.3) ζ1

(0.7, 0.1) ζ2⎜ (0.2, 0.3) (P3 , E) = ⎜ ζ3⎝ (0.1, 0.3) ζ4 (0.8, 0.1) ζ1

u2

u3

u4

(0.3, 0.8) (0.6, 0.6) (0.7, 0.3) (0.5, 0.7)

(0.1, 0.8) (0.2, 0.6) (0.9, 0.1) (0.2, 0.8)

(0.8, 0.6) (0.9, 0.3) (0.1, 0.3) (0.7, 0.3)

u2

u3

u4

(0.4, 0.8) (0.6, 0.4) (0.7, 0.7) (0.3, 0.7)

(0.1, 0.0) (0.2, 0.6) (0.6, 0.1) (0.2, 0.3)

(0.9, 0.2) (0.9, 0.3) (0.1, 0.9) (0.8, 0.3)

u2

u3

u4

(0.1, 0.9) (0.3, 0.8) (0.9, 0.3) (0.1, 0.7)

(0.5, 0.8) (0.8, 0.5) (0.9, 0.1) (0.4, 0.8)

(0.4, 0.6) (0.2, 0.4) (0.3, 0.3) (0.3, 0.7)

Div1 ((P1 , E), (P2 , E)) = 0.133 Div1 ((P1 , E), (P3 , E)) = 0.142 Div1 ((P2 , E), (P3 , E)) = 0.194

u5 ⎞ (0.7, 0.7) (0.2, 0.3) ⎟ ⎟ (0.5, 0.9) ⎠ (0.7, 0.5)

u5

⎞ (0.2, 0.7) (0.1, 0.3) ⎟ ⎟ (0.2, 0.9) ⎠ (0.8, 0.5) u5 ⎞ (0.7, 0.3) (0.4, 0.3) ⎟ ⎟ (0.5, 0.9) ⎠ (0.7, 0.5)

Div2 ((P1 , E), (P2 , E)) = 0.806 Div2 ((P1 , E), (P3 , E)) = 1.529 Div2 ((P2 , E), (P3 , E)) = 2.006

From this we can conclude that the difference between (P2 , E) and (P3 , E) is much larger than with other PFSSs.

4 TOPSIS Based on Divergence Measure of PFSSs This section explains an algorithm for the TOPSIS method based on the proposed divergence measure under the PFSS environment and a numerical example of knowledge management.

4.1 Algorithm for TOPSIS Method Consider the MCDM problem consisting of r variants V1 , V2 , . . . Vr that is characterized by s qualities Q 1 , Q 2 , . . . Q s . Also, there are t experts S1 , S2 , . . . St to evaluate

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T. M. Athira and S. J. John

each quality of variants individually. The experts are given their opinion of agreement and disagreement with each alternative in terms of PFSS. By considering the opinion of each specialist significantly, we have to identify which one is the best among the variants under study. The TOPSIS method based on the PFSS divergence measure for selecting the best-fitting alternative is summarized below. Step 1: Construct the decision matrices in terms of PFSSs for each {Vu : u = 1, 2, . . . r } with experts set {S1 , S2 , . . . St } is universal set and set of qualities {Q 1 , Q 2 , . . . Q s } as the parameter set. Step 2: Normalise the decision matrices according to which the attributes are benefittype or cost-type.  vi j =

vi j if j is benefit-type attribute vicj if j is cost-type attribute

where, vi j is i j th entry of decision matrix Vu and vicj is the compliment of vi j . Step 3: Calculate the Positive Ideal Solution(PID) and Negative Ideal Solution(NID) as given below. The positive ideal matrix V + is defined as, V + = [vi+j ]s×t where, vi+j =



 max {μVu (Q j ) (Si )}, min {νVu (Q j ) (Si )}

u=1,2,...,r

(6)

u=1,2,...,r

The negative ideal matrix V − is defined as, V − = [vi−j ]s×t where, vi−j

 =

 min {μVu (Q j ) (Si )}, max {νVu (Q j ) (Si )}

u=1,2,...,r

(7)

u=1,2,...,r

Step 4: Using Eqs. 1 or 2 frame Du+ and Du− , u = 1, 2, , . . . r. Du+ = Diva (V + , Vu ), u = 1, 2, . . . , r, a = 1, 2.

(8)

Du− = Diva (V − , Vu ), u = 1, 2, . . . , r, a = 1, 2.

(9)

Step 5: Calculate the relative closeness Rc according to the following equation. Rcu =

Du+

Du+ . + Du−

(10)

Step 6: Rank the variants Vu according to the relative closeness, the bigger is the Rcu , the better is the variant Vu .

Divergence Measures of Pythagorean Fuzzy Soft Sets

419

4.2 Numerical Example: The Knowledge Management The crucial target of knowledge management(KM) is guaranteeing an organization’s data. The information is gathered and saved to make it valid for employees or customers to find and use the needed data. Here, we are going to elucidate a numerical example of decision-making on business KM system. An entrepreneur would like to model a business trade and there are six trades T1 , T2 , T3 , T4 , T5 , T6 in his final list. He has preferred the five main qualities of the trades that are document management(DM), collaboration(CB), communication(CM), scalability(SC), and workflow management(WM). There are four well KM experts S1 , S2 , S3 , S4 to explain each quality of each trade. Here are step-by-step explanations of how the person makes the best decision by considering all the properties and opinions of the expert as equally important. The judgment of experts subject to five criteria of each trade is given in Tables 1, 2, 3, 4, 5, 6. Step 1: From the Tables 1, 2, 3, 4, 5, 6, it is easy to obtain the decision matrices in terms of PFSSs, where the universal set is {S1 , S2 , S3 , S4 } and parameter set is {D M, C B, C M, SC, W M}. Step 2: As it is benefit-type criteria, it is already normalized.

Table 1 Trade T1 DM S1 S2 S3 S4

(0.12,0.23) (0.77,0.05) (0.77,0.10) (0.56,0.14)

Table 2 Trade T2 DM S1 S2 S3 S4

(0.90,0.02) (0.89,0.13) (0.58,0.32) (0.68,0.11)

Table 3 Trade T3 DM S1 S2 S3 S4

(0.68,0.22) (0.79,0.12) (0.88,0.03) (0.86,0.10)

DM

CM

SC

WM

(0.76,0.36) (0.71,0.37) (0.40,0.33) (0.78,0.22)

(0.36,0.56) (0.82,0.55) (0.92,0.05) (0.71,0.62)

(0.36,0.56) (0.84,0.18) (0.89,0.34) (0.75,0.22)

(0.81,0.21) (0.57,0.45) (0.44,0.76) (0.27,0.65)

DM

CM

SC

WM

(0.23,0.46) (0.77,0.21) (0.37,0.32) (0.62,0.23)

(0.56,0.67) (0.77,0.34) (0.77,0.45) (0.78,0.46)

(0.28,0.56) (0.87,0.04) (0.89,0.13) (0.45,0.26)

(0.82,0.18) (0.11,0.69) (0.43,0.53) (0.90,0.10)

DM

CM

SC

WM

(0.30,0.64) (0.47,0.30) (0.73,0.22) (0.90,0.21)

(0.57,0.56) (0.70,0.55) (0.72,0.45) (0.82,0.46)

(0.29,0.69) (0.76,0.24) (0.89,0.38) (0.45,0.26)

(0.61,0.45) (0.56,0.70) (0.67,0.50) (0.87,0.21)

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T. M. Athira and S. J. John

Table 4 Trade T4 DM S1 S2 S3 S4

(0.28,0.56) (0.90,0.20) (0.78,0.12) (0.76,0.11)

Table 5 Trade T5 DM S1 S2 S3 S4

(0.58,0.34) (0.85,0.34) (0.47,0.57) (0.24,0.52)

Table 6 Trade T6 DM S1 S2 S3 S4

(0.81,0.22) (0.79,0.12) (0.67,0.24) (0.62,0.21)

DM

CM

SC

WM

(0.33,0.36) (0.71,0.21) (0.74,0.22) (0.67,0.27)

(0.52,0.76) (0.74,0.50) (0.81,0.44) (0.88,0.56)

(0.21,0.65) (0.64,0.34) (0.90,0.23) (0.58,0.21)

(0.71,0.47) (0.91,0.26) (0.87,0.25) (0.65,0.23)

DM

CM

SC

WM

(0.35,0.60) (0.54,0.32) (0.54,0.69) (0.61,0.25)

(0.51,0.46) (0.67,0.25) (0.23,0.68) (0.45,0.65)

(0.25,0.68) (0.26,0.64) (0.39,0.73) (0.35,0.27)

(0.37,0.81) (0.22,0.66) (0.34,0.54) (0.71,0.15)

DM

CM

SC

WM

(0.32,0.66) (0.76,0.33) (0.74,0.42) (0.56,0.23)

(0.56,0.67) (0.73,0.57) (0.74,0.24) (0.83,0.61)

(0.27,0.63) (0.67,0.46) (0.89,0.23) (0.57,0.26)

(0.35,0.67) (0.32,0.67) (0.74,0.23) (0.90,0.23)

Step 3: The positive ideal matrix V + is given by ⎛

V+

(0.90, 0.02) ⎜(0.90, 0.05) =⎜ ⎝(0.88, 0.03) (0.86, 0.11)

(0.76, 0.36) (0.77, 0.21) (0.74, 0.22) (0.90, 0.21)

(0.57, 0.46) (0.82, 0.25) (0.92, 0.05) (0.88, 0.46)

(0.36, 0.56) (0.87, 0.04) (0.90, 0.13) (0.75, 0.21)

⎞ (0.81, 0.18) (0.91, 0.26)⎟ ⎟ (0.87, 0.23)⎠ (0.90, 0.10)

(0.21, 0.69) (0.26, 0.64) (0.39, 0.73) (0.35, 0.27)

⎞ (0.35, 0.81) (0.11, 0.70)⎟ ⎟ (0.34, 0.76)⎠ (0.27, 0.65)

The negative ideal matrix V − is given by ⎛

V−

Step 4:

(0.12, 0.56) ⎜(0.77, 0.34) =⎜ ⎝(0.47, 0.57) (0.24, 0.52)

(0.23, 66) (0.47, 0.37) (0.37, 0.69) (0.56, 0.27)

(0.36, 0.76) (0.67, 0.57) (0.23, 0.68) (0.45, 0.65)

  D + = 0.0568 0.0489 0.0374 0.0378 0.1386 0.0511   D − = 0.0989 0.1110 0.1019 0.1081 0.0241 0.0882 .

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Step 5: The relative closeness are given by Rc1 = 0.3648 Rc2 = 0.3058 Rc3 = 0.2685 Rc4 = 0.2591 Rc5 = 0.8519 Rc6 = 0.3668

Step 6: The highest relative closeness is for Trade 5. Hence, we can conclude that Trade 5 is the best decision.

5 Conclusion We introduced and studied the measure of divergence for Pythagorean fuzzy soft sets. Some nice expressions to calculate PFSS divergence measures are obtained. Certain theorems that state the properties of PFSS divergence measures are also proven. As future work, we can use this PFSS divergence measurement in decision problems such as TOPSIS algorithm, pattern recognition etc.

References 1. Bhandari, D., Pal, N.R., Majumder, D.D.: Fuzzy divergence, probability measure of fuzzy events and image thresholding. Pattern Recognit. Lett. 13(12), 857–867 (1992) 2. Montes, I., Pal, N.R., Janiš, V., Montes, S.: Divergence measures for intuitionistic fuzzy sets. IEEE Trans. Fuzzy Syst. 23(2), 444–456 (2014) 3. Athira, T.M., John, S.J., Garg, H.: Entropy and distance measures of pythagorean fuzzy soft sets and their applications. J. Intell. & Fuzzy Syst. 37(3), 4071–4084 (2019) 4. Athira, T.M., John, S.J., Garg, H.: A novel entropy measure of pythagorean fuzzy soft sets. AIMS Math. 5(2), 1050–1061 (2020) 5. Garg, H., Rani, D.: Novel exponential divergence measure of complex intuitionistic fuzzy sets with an application to the decision-making process. Sci. Iran. 28(4), 2439–2456 (2021) 6. Hwang, C.L., Yoon, K.: Methods for multiple attribute decision making. Multiple Attribute Decision Making, pp. 58–191. Springer, Berlin, Heidelberg 7. Maji, P.K., Biswas, R., Roy, A.R.: Intuitionistic fuzzy soft sets. J. Fuzzy Math. 9(3), 677–692 (2001) 8. Molodtsov, D.: Soft set theory-first results. Comput. & Math. Appl. 37(4–5), 19–31 (1999) 9. Peng, X.D., Yang, Y., Song, J., Jiang, Y.: Pythagorean fuzzy soft set and its application. Comput. Eng. 41(7), 224–229 (2015) 10. Xiao, F., Ding, W.: Divergence measure of Pythagorean fuzzy sets and its application in medical diagnosis. Appl. Soft Comput. 79, 254–267 (2019) 11. Yager, R.R.: Pythagorean membership grades in multicriteria decision making. IEEE Trans. Fuzzy Syst. 22(4), 958–965 (2013) 12. Montes, I., Pal, N.R., Montes, S.: Entropy measures for Atanassov intuitionistic fuzzy sets based on divergence. Soft Comput. 22(15), 5051–5071 (2018) 13. Liu, B., Zhou, Q., Ding, R.X., Ni, W., Herrera, F.: Defective alternatives detection-based multi-attribute intuitionistic fuzzy large-scale decision making model. Knowl. Based Syst. 186, 104962 (2019) 14. Thao, N.X.: Some new entropies and divergence measures of intuitionistic fuzzy sets based on Archimedean t-conorm and application in supplier selection. Soft Comput. 25(7), 5791–5805 (2021) 15. Zhou, Q., Mo, H., Deng, Y.: A new divergence measure of pythagorean fuzzy sets based on belief function and its application in medical diagnosis. Math. 8(1), 142 (2020)

Fuzzy-Rough Optimization Technique for Breast Cancer Classification K. Anitha and Debabrata Datta

Abstract Breast cancer is one of the deadly diseases amid women. The survival rate can be increased through early detection. The classification model with high level of predictive performance will help the medical experts to early identification of this disease. To develop such types of robust and optimal classification model, computational approach will be useful in early identification. In this paper, we introduce hybrid intelligent fuzzy-rough classification method based on rule induction. At initial stage, irrelevant features are removed through weak gamma evaluator. Performance of this classification model is examined for Wisconsin Breast Cancer Database (WBCD) and classification accuracy evaluated through F-measure. Performance measure of fuzzy-rough set optimization technique is taken into account by measuring sensitivity, specificity, and accuracy of the applied technique. Verification and validation exercise of the applied technique is carried out on the basis of results obtained in the similar direction by various realistic breast cancer images captured by thermography. Keywords Set approximation · Indiscernibility relation · Decision rules MSC classification: 03E72 · 03E75 · 62A07 · 62C05

1 Introduction One of the most common life-threatening diseases among women is breast cancer. Age factor, consumption of alcohol, obesity, breast implants, hormonal imbalance, and uses of hormone-related medicines and non-breast feeding are root causes for this K. Anitha (B) Department of Mathematics, SRM Institute of Science and Technology, Ramapuram, Chennai, India e-mail: [email protected] D. Datta Former Nuclear Scientist, Bhabha Atomic Research Centre, Mumbai 400085, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 T. Som et al. (eds.), Applied Analysis, Optimization and Soft Computing, Springer Proceedings in Mathematics & Statistics 419, https://doi.org/10.1007/978-981-99-0597-3_30

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disease. It can be diagnosed through self-breast examination, mammography, ultrasonography, and thermography testing. As per the information from World Health Organization, 27 million new cases will be expected in the year 2030. Nowadays, medical scientists are showing interests on thermographic image study to detect breast cancer [1]. Indian cities especially metropolitan cities like Bangalore, Chennai, and Delhi encountered more cases than other places. 10% of breast cancers due to hereditary whereas 90% are based on life style. In the year 2018, India catalogued 1,62,468 new breast cancer cases and 87,090 listed deaths due to detection in higher stages. Early detection of breast cancer can reduce the mortality rate. Thermographic analysis plays lead role in early detection of this disease which is non-invasive and painless [2]. Thermographic images are high-resolution infrared 3D images. These images are having certain limitations in view of exact geometrical shape of breast, exact tumor size with respect to spatial location of breast, and the field of internal blood flow. Early identification process will reduce the death rate. In image analysis process, RGB images are converted into grayscale images. Then region of interest (ROI) has to be selected. From ROI relative frequency and classification process will be processed. Throughout this process denoising is more important. Denoising is the process of removing irrelevant variables or features. Several soft computing techniques based on fuzzy systems, rough set model, neural network, and supporting vector machines have been proposed for this process. In this paper, we have considered fuzzy and rough set-based image analysis technique based on rule generation. Zadeh fuzzy set model has graded membership values ranging from zero to one [3]. In image processing, these membership values represent the degree of pixel which belongs to either an edge or a uniform region. Let OI (x, y) be the original image. This image is to be mapped to fuzzy membership domain [0, 1]. Original image is being processed in this domain which is denoted as FI . Fuzzy image can be addressed through membership values only. But some variables in the image are in imprecise position which means that they are in boundary region. The set which is having non-empty boundary region is known as rough set which was introduced by Pawlak in 1982. Elements of fuzzy set are characterized by membership values whereas in rough set elements are addressed by indiscernibility relation. Indiscernibility relation is an equivalence relation in which elements are having same attribute values. Topological properties of rough set are discussed in [4]. Anitha and Thangeswari [5] analyzed rough set-based optimization techniques for attribute reduction without information loss. Algebraic structures of fuzzy set are demonstrated in [6]. There are several techniques available in breast cancer detection like fuzzy classification, neural network classification, etc. Statistical feature extraction can be done through rough sets, fuzzy-rough set, and fuzzy-intuitionistic rough set techniques. Image segmentation based on hybrid fuzzy-intuitionistic technique is done in [7]. Water Swirl Algorithm (WSA) was introduced in [8]. They developed this method to identify the potential gene which is associated in cancer diagnosis. Jaya Kumari [9] proposed fuzzy precognition c-means concept to confront breast cancer in multifarious cases. Various advancement techniques are being implemented to study thermographic images. Fuzzy c-means, k-means, and level set methods are developed for image segmentation in [10]. Thermographic image classification tools

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and techniques are discussed in [11–13]. CAD system was introduced through neutrosophic and optimized fuzzy c-means method in [14]. Fuzzy-rough set approach machine learning techniques are clearly explained in [15]. Correlation-based filter technique is used to reduce the high-dimensional cancer data. Maji [16] proposed IT2 fuzzy-rough set attribute reduction method by increasing the relevance and significance of features. In this paper, we have proposed fuzzy-rough feature extraction technique by rule generation based on similarity measure and it is being implemented on thermographic image. Preliminary concepts are discussed in Sect. 2, main work is demonstrated in Sect. 3, and experimental results are discussed in Sect. 4 followed by concluded remarks.

2 Preliminaries In knowledge acquisition process, rule induction and decision tree-based classification techniques are commonly used to handle high-dimensional data. Rule induction is concerned about the data with uncertain and inconsistent information. For handling such types of data, rough set plays a key role. This theory was introduced by Pawlak by means of equivalence relation between the objects. He gave the formal approximations for classical set named upper and lower approximations. In some cases, fuzzy sets may turn as approximations.

2.1 Information System Let Is = (Us , As ) be the Information System which is the function. Between Us and As non empty attribute value set such that Is : Us → Ca , where a ∈ As and Ca is the value of. Let us take E ⊆ As . And define Indiscernibility Relation ((I N D E ) as follows: INDE = {(x, y) ∈ Us 2 | attr (x) = attr (y)}. The family of all equivalence classes of INDE forms partition of Us and it is denoted by [x]E . Let X ⊆ Us be the target set which will represent E ⊆ As , then set approximations are defined by 1. Lower Approximation: E(X) = {x : [x]E ⊆ X}. 2. Upper Approximation: E(X) = {x : [x]E ∩ X = ∅}. 3. Boundary Region: E(X) − E(X). If E(X) − E(X) = ∅ then the set X is crisp or conventional set, otherwise it is rough set  4. Positive Region: E(X). 5. Negative Region: Us − E(X).

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Table 1 Information system Cases Symptoms (conditional attributes)

O1 O2 O3 O4 O5 O6

Decision attribute (Flu)

S1

S2

S3

0 1 1 0 1 0

1 0 1 1 0 1

High High Heavy Standard High Heavy

6. Rough Accuracy of Approximation:

Confirm Confirm Confirm Not confirm Not confirm Confirm

|E(X)| . |E(X)|

Example 1 The following information system consists of six objects, three conditional attributes, and a decision attribute. Here symptoms are conditional attributes, flu is decision attribute (Table 1). Let X = Target set = {flu confirm} = {O1, O2, O3, O6} Conditional Attributes E = {S1, S2, S3} E(X) = {O1, O3, O6} E(X) = {O1, O2, O3, O5, O6} Boundary Region(X) = {O2, O5} Hence X is rough set.

Reduct and Core Reduct(R) is the minimal representation of original information system or subset of attribute set R ⊆ As and it should satisfy following conditions: 1. [x]E = [x]R . 2. [x]R is the minimal representation such that [x]R−{a} = [x]E . We can construct many numbers of reducts for given data. Intersection of all reducts forms core.  Core = Ri . Finite number of deterministic rules can be generated through reducts for given dataset. The concepts of rule induction for knowledge discovery are developed in [17, 18].

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Let us take two attribute sets A1 & A2 (disjoint) from an information system and their equivalence classes are [x]A1 , [x]A2 , respectively. The dependency of attribute set A2 on A1 is given by d = γ(A1 , A2 ) =

n  |A1 (X)A2i | . U i=1

If d = 1 then A1 totally depends on A2 . If d < 1 then A1 depends on degree d with A2 . The reduct set R starts with an empty set and adds features one at a time. If γ(A1 U x) > γ(A1 ) then x ∈ R. Information system 2.1 has two reducts R1 and R2 . R1 = {S1, S2}, R2 = {S1, S3}  Cor e = Ri = R1 ∩ R2 = {S1}. S1 is the most important attribute to give the decision about whether the patient may get affected with flu or not. If we remove the variables from reduct and core, it will minimize the quality of data. The main disadvantage of rough set is that it will bring optimal result when the information system is discrete or the attribute values are precise. It cannot be possible for high-dimensional data, especially image data.

2.2 Fuzzy-Rough Set Classical rough set deals about attributes having precise (certain) values but, in some data, attributes are having fuzzy values. They are called hybrid attributes. For handling these types of attributes, hybridization technique is needed to get optimal result. Hybridization is the process of merging two or more than two optimization techniques into single system. When the attribute takes fuzzy values, the equivalence relation between attribute can be constructed through fuzzy-similarity relation. In this paper, we proposed fuzzy-rough hybridization technique to identify consistent attributes. For this technique, fuzzy-similarity relation can be considered as a base for rough approximation. In this process, rough set indiscernibility relation is extended to similarity relation. Let I be the indiscernibility relation which satisfies reflexive, transitive, and symmetric. If I satisfies the following properties, then it is said to be fuzzy-similarity relation (SR) 1. ∀x ∈ Us , I(x) ⊆ SR(x) 2. ∀x ∈ Us , ∀y ∈ SR(x), I(y) ⊆ SR(x). From this relation, fuzzy-rough approximations are defined as follows:

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SR(x) = {x ∈ X : SR(x) ⊆ X}  SR(x) = SR(x). x∈X

Distance-based fuzzy-similarity relation (SR) is defined by [19] the function ϕ: SR(X) × SR(X) → [0, 1] ⎧

d d ⎨ 1− ,