Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness

Transcription

1 Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness

2 Józef Banaś Mohamed Jleli Mohammad Mursaleen Bessem Samet Calogero Vetro Editors Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness 123

3 Editors Józef Banaś Department of Nonlinear Analysis Rzeszów University of Technology Rzeszów Poland Mohamed Jleli Department of Mathematics King Saud University Riyadh Saudi Arabia Bessem Samet Department of Mathematics King Saud University Riyadh Saudi Arabia Calogero Vetro Dipartimento di Matematica e Informatica University of Palermo Palermo Italy Mohammad Mursaleen Department of Mathematics Aligarh Muslim University Aligarh, Uttar Pradesh India ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer Nature Singapore Pte Ltd This work is subject to copyright. All rights are reserved 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, express 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore , Singapore

4 Preface The theory of measures of noncompactness constitutes a very important branch of nonlinear functional analysis. It finds a lot of applications in operator theory. First of all, it allows us to select very significant class of operators being generalizations of compact operators. Those operators are known as operators satisfying the Darbo condition or contractions with respect to a measure of noncompactness as well as condensing operators. Measures of noncompactness are widely applied in fixed point theory and are especially useful in investigations connected with the theories of differential equations, integral equations, functional integral equations and in optimization theory. The present book contains a comprehensive treatment of the theory of measures of noncompactness and its applications. The book is addressed to the large audience of mathematical community applying the methods and tools of nonlinear analysis and investigating a lot of topics connected with that large branch of mathematics. This book can also serve as a source of examples, references and new approaches associated with the theory of measures of noncompactness and its numerous valuable suggestions. Apart from Preface, the present book consists of 13 chapters treating various topics of analysis in which the theory of measures of noncompactness is applicable. Each chapter is self-contained and contributed by specialists using in their research the methods and techniques associated with the theory of measures of noncompactness. All chapters contain several examples and complete references for the topics considered in them. In Chap. 1 Measures of Noncompactness in the Space of Continuous and Bounded Functions Defined on the Real Half-Axis the authors present a review of results concerning measures of noncompactness in the space of real functions defined, continuous and bounded on the real half-axis and equipped with the classical supremum norm. Moreover, measures of noncompactness in a more general space of functions defined, continuous on real half-axis and tempered by a given function, are investigated. A number of applications to the solvability of v

5 vi Preface nonlinear integral equations of measures of noncompactness in question are included. Chapter 2 entitled Measures of Noncompactness and their Applications provides a survey of some results concerning classical measures of noncompactness in miscellaneous spaces. In particular, some fixed point theorems are established for different classes of operators. Moreover, some applications of measures of noncompactness to functional equations, including nonlinear integral equations of fractional orders, implicit fractional integral equations and the so-called q-integral equations of fractional orders, are presented. On Some Results Using Measures of Noncompactness creates Chap. 3 of our book. It contains a review of some results on measures of noncompactness including the classical Kuratowski, Hausdorff, inner Hausdorff and separation measures of noncompactness as well as measures defined in an axiomatic way. Apart from this, the role of the mentioned measures of noncompactness in fixed point theory and operator theory is discussed. Further, the authors presented methods of characterization of compact matrix operators between some BK spaces, matrix domains of triangles in certain sequence spaces and spaces of strongly summable and bounded sequences. Next, Chap. 4 Space of Functions with Growths Tempered by a Modulus of Continuity discusses the space of functions defined on a compact metric space and having growths (increments) tempered by a given modulus of continuity. The authors present a measure of noncompactness defined in the space in question and show its applicability in establishing some existence results for nonlinear quadratic integral equations. A few examples illustrating the discussed results are included. Measure of Noncompactness in Functional Fractional Calculus constitutes the next Chap. 5 devoted to the application of measures of noncompactness in the study of the Cauchy problem for fractional differential equations. This chapter presents the extension of a few classical results from the theory of ordinary differential equations in Banach spaces to the theory of fractional differential equations. A review of such results is given. The subsequent Chap. 6 entitled Measures of Weak Noncompactness and Fixed Points is dedicated to exhibiting the importance of the use of measures of weak noncompactness in topological fixed point theory. This chapter shows simultaneously the applicability of the theory of measures of weak noncompactness in the theory of integral and partial differential equations. A few results concerning the existence of fixed points of operators acting in Banach algebras are also included. Chapter 7 The Class of F-Contraction Mappings with a Measure of Noncompactness deals with the discussion of the existence of fixed points for some classes of operators. The results contained in the chapter are obtained with the help of measures of noncompactness. The usefulness of the mentioned existence results in the theory of functional and integral equations is demonstrated. Next, Chap. 8 On the Measure of Noncompactness in Banach Spaces and Application to the Theory of Differential and Integral Equations presents a review of some important generalizations of Darbo s fixed point theorem. The results provided in the chapter are applied to the study of the existence and qualitative

6 Preface vii behaviour of solutions of some integral and differential equations with feedback control. In Chap. 9 Partial Hadamard-Stieltjes Fractional Integral Equations in Banach Spaces, a few existence results concerning some classes of functional partial integral equations with the use of Hadamard fractional operators are given. The main tool used in considerations is the combination of the technique of measures of noncompactness with fixed point theorems of Darbo and Mönch type. The next Chap. 10 entitled On the Aronszajn Property for Differential Equations of Fractional Order in Banach Spaces presents a survey of results concerning some topological properties of the set of solutions of differential and integro-differential equations of fractional order in Banach spaces. The so-called Aronszajn property creates the considerations of this chapter. The Kuratowski measure of noncompactness is the main tool exploited in the chapter in question. On the Qualitative Behaviours of Nonlinear Functional Differential Systems of Third Order is Chap. 11 that discusses new sufficient conditions for the boundedness and global asymptotic stability of solutions of some nonlinear delay differential systems of third order. The technique of Lyapunov Krasovskii functionals in combination with LaSalle s invariant principle is basic tools used for reasonings in this chapter. In Chap. 12 On the Approximation of Solutions to a Fixed Point Problem with Inequality Constraints in a Banach Space Partially Ordered by a Cone a few results are given on the approximation of solutions to a fixed point problem concerning mappings defined on a metric space furnished with two partial orders, under constraint inequalities. The last Chap. 13 A Short Survey on Dislocated Metric Spaces via Fixed Point Theory presents a short review of results on dislocated metric and b-metric spaces via the fixed point theory for operators acting on the mentioned dislocated metric spaces. The editors would like to express their gratitude to the contributors who have submitted chapters to this volume. Special thanks are also addressed to Mr. Shamim Ahmad, Editor of Mathematics in Springer India, for taking interest in publishing this book. Rzeszów, Poland Riyadh, Saudi Arabia Aligarh, India Riyadh, Saudi Arabia Palermo, Italy Józef Banaś Mohamed Jleli Mohammad Mursaleen Bessem Samet Calogero Vetro

7 Contents 1 Measures of Noncompactness in the Space of Continuous and Bounded Functions Defined on the Real Half-Axis... 1 Józef Banaś, Nelson Merentes and Beata Rzepka 2 Measures of Noncompactness and Their Applications Mohammad Mursaleen, Syed M.H. Rizvi and Bessem Samet 3 On Some Results Using Measures of Noncompactness Eberhard Malkowsky and Vladimir Rakočević 4 Space of Functions with Growths Tempered by a Modulus of Continuity I.J. Cabrera, J. Harjani, B. López and K.B. Sadarangani 5 Measure of Noncompactness in Functional Fractional Calculus Ehsan Pourhadi 6 Measures of Weak Noncompactness and Fixed Points Agnieszka Chlebowicz and Mohamed-Aziz Taoudi 7 The Class of F-Contraction Mappings with a Measure of Noncompactness Calogero Vetro and Francesca Vetro 8 On the Measure of Noncompactness in Banach Spaces and Application to the Theory of Differential and Integral Equations Neda Khodabakhshi and S. Mansour Vaezpour 9 Partial Hadamard-Stieltjes Fractional Integral Equations in Banach Spaces Saïd Abbas, Mouffak Benchohra and Johnny Henderson ix

8 x Contents 10 On the Aronszajn Property for Differential Equations of Fractional Order in Banach Spaces Aldona Dutkiewicz 11 On the Qualitative Behaviors of Nonlinear Functional Differential Systems of Third Order Cemil Tunç 12 On the Approximation of Solutions to a Fixed Point Problem with Inequality Constraints in a Banach Space Partially Ordered by a Cone Mohamed Jleli, Erdal Karapinar and Bessem Samet 13 A Short Survey on Dislocated Metric Spaces via Fixed-Point Theory Erdal Karapınar Index

9 Editors and Contributors About the Editors Józef Banaś is Full Professor of Mathematics and Chair of the Department of Nonlinear Analysis in Rzeszów University of Technology, Poland. Professor J. Banaś visited several scientific centres all over the world. He was a participant and speaker of numerous international conferences and congresses. He is member of the editorial committees of many journals of international repute. He was a lead guest editor and guest editor of many special issues for Abstract and Applied Analysis, Journal of Function Spaces, Scientific World Journal and Advances in Pure Mathematics. He has over 150 published research papers in international journals. He is also coauthor of three books: Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics 60 (Marcel Dekker, New York, 1980) (with K. Goebel); Bounded Variation and Around (De Gruyter Series in Nonlinear Analysis and Applications 17, Walter De Gruyter, Berlin, 2014) (with J. Appell and N. Merentes); Sequence Spaces and Measure of Noncompactness with Applications to Differential and Integral Equations (Springer, New Delhi, 2014) (with M. Mursaleen). Also, he is a supervisor of 10 Ph.D. theses and over 150 B.Sc. and M.Sc. theses in mathematics. Mohamed Jleli is Full Professor of Mathematics at King Saud University, Saudi Arabia. He obtained his Ph.D. degree in Pure Mathematics on the thesis entitled constant mean curvature hypersurfaces from Faculty of Sciences of Paris 12, France, in He has written several papers on differential geometry, partial differential equations, evolution equations, fractional differential equations and fixed point theory. He is on the editorial board of some international journals and acts as a referee for several international journals in mathematics. Mohammad Mursaleen is Full Professor & Chairman of Department of Mathematics, Aligarh Muslim University, India. As an active researcher, Prof. Mursaleen has authored seven books and five book chapters, in addition to his contributions to more than 200 research papers in ISI journals. He is referee of more than 100 scientific journals and member of the editorial board of various scientific journals, international scientific bodies and organizing committees. He has guided 15 Ph.D. students so far. He has visited several countries including USA and UK and delivered about 38 talks. His main research interests include summability, approximation theory and operator theory. He has been recipient of the Outstanding Researcher of the Year-2014 Award of Aligarh Muslim University. Bessem Samet is Full Professor of Applied Mathematics at King Saud University, Saudi Arabia. He obtained his Ph.D. degree in Applied Mathematics on the thesis entitled topological derivative method for Maxwell equations and its applications from Paul Sabatier University, France, in His topics of interest include different branches of nonlinear analysis, including fixed point xi

10 xii Editors and Contributors theory, partial differential equations, differential equations, fractional calculus, etc. He is the author or coauthor of more than 100 published papers in ISI journals. He is in the list of Thomson Reuters Highly Cited Researchers for the years 2015 and Calogero Vetro is Assistant Professor of Mathematical Analysis at University of Palermo, Italy, where he has been teaching since He received the Laurea Degree in Mechanical Engineering in 2000 and the Ph.D. degree in Engineering of Automation and Control Systems in His research interests include best approximation, fixed point theory, functional analysis, mathematical programming, operator theory and partial differential equations. He is the author or coauthor of more than 120 published papers. He is in the Thomson Reuters Highly Cited Researchers List for the years 2015 and Contributors Saïd Abbas Laboratory of Mathematics, University of Saïda, Saïda, Algeria Józef Banaś Department of Nonlinear Analysis, Rzeszów University of Technology, Rzeszów, Poland Mouffak Benchohra Laboratory of Mathematics, University of Sidi Bel-Abbès, Sidi Bel-Abbès, Algeria; Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia I.J. Cabrera Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain Agnieszka Chlebowicz Department of Nonlinear Analysis, Rzeszów University of Technology, Rzeszów, Poland Aldona Dutkiewicz Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, Poznań, Poland J. Harjani Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain Johnny Henderson Department of Mathematics, Baylor University, Texas, Waco, USA Mohamed Jleli Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia Erdal Karapinar Department of Mathematics, Atilim University, Ankara, Turkey; Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia Neda Khodabakhshi Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran B. López Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

11 Editors and Contributors xiii Eberhard Malkowsky Državni Univerzitet u Novom Pazaru, Vuka Karadžića bb, Novi Pazar, Serbia S. Mansour Vaezpour Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran Nelson Merentes Department of Mathematics, Central University of Venezuela, Caracas, Capital District, Venezuela Mohammad Mursaleen Department of Mathematics, Aligarh Muslim University, Aligarh, India Ehsan Pourhadi School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran Vladimir Rakočević University of Niš, Faculty of Mathematics and Sciences, Niš, Serbia Syed M.H. Rizvi Department of Mathematics, Aligarh Muslim University, Aligarh, India Beata Rzepka Department of Nonlinear Analysis, Rzeszów University of Technology, Rzeszów, Poland K.B. Sadarangani Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain Bessem Samet Department of Mathematics, College of Science, King Saud University, Riyadh, Saudi Arabia Mohamed-Aziz Taoudi Cadi Ayyad University, National School of Applied Sciences, Marrakech, Morocco Cemil Tunç Department of Mathematics, Yüzüncü Yıl University, Van, Turkey Calogero Vetro Department of Mathematics and Computer Sciences, University of Palermo, Palermo, Italy Francesca Vetro Department of Energy, Information Engineering and Mathematical Models (DEIM), University of Palermo, Palermo, Italy