Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications

by Afif Ben Amar, Donal O'Regan

Overview

This is a monograph covering topological fixed point theory for several classes of single and multivalued maps. The authors begin by presenting basic notions in locally convex topological vector spaces. Special attention is then devoted to weak compactness, in particular to the theorems of Eberlein–Šmulian, Grothendick and Dunford–Pettis. Leray–Schauder alternatives and eigenvalue problems for decomposable single-valued nonlinear weakly compact operators in Dunford–Pettis spaces are considered, in addition to some variants of Schauder, Krasnoselskii, Sadovskii, and Leray–Schauder type fixed point theorems for different classes of weakly sequentially continuous operators on general Banach spaces. The authors then proceed with an examination of Sadovskii, Furi–Pera, and Krasnoselskii fixed point theorems and nonlinear Leray–Schauder alternatives in the framework of weak topologies and involving multivalued mappings with weakly sequentially closed graph. These results are formulated in terms of axiomatic measures of weak noncompactness.

The authors continue to present some fixed point theorems in a nonempty closed convex of any Banach algebras or Banach algebras satisfying a sequential condition (P) for the sum and the product of nonlinear weakly sequentially continuous operators, and illustrate the theory by considering functional integral and partial differential equations. The existence of fixed points, nonlinear Leray–Schauder alternatives for different classes of nonlinear (ws)-compact operators (weakly condensing, 1-set weakly contractive, strictly quasi-bounded) defined on an unbounded closed convex subset of a Banach space are also discussed. The authors also examine the existence of nonlinear eigenvalues and eigenvectors, as well as the surjectivity of quasibounded operators. Finally, some approximate fixed point theorems for multivalued mappings defined on Banach spaces. Weak and strong topologies play a role here and both bounded and unbounded regions are considered. The authors explicate a method developed to indicate how to use approximate fixed point theorems to prove the existence of approximate Nash equilibria for non-cooperative games.

Fixed point theory is a powerful and fruitful tool in modern mathematics and may be considered as a core subject in nonlinear analysis. In the last 50 years, fixed point theory has been a flourishing area of research. As such, the monograph begins with an overview of these developments before gravitating towards topics selected to reflect the particular interests of the authors.

Product Details

ISBN-13:	9783319811628
Publisher:	Springer International Publishing
Publication date:	05/31/2018
Edition description:	Softcover reprint of the original 1st ed. 2016
Pages:	194
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Donal O'Regan is Professor in the School of Mathematics, Statistics, and Applied Mathematics at the National University of Ireland, Galway. His research interests include differential equations, nonlinear analysis, and fixed point theory. He is the author of several books, including An Introduction to Ordinary Differential Equations (with Ravi Agarwal), and Oscillation and Stability of Delay Models in Biology (with Ravi Agarwal and Samir Saker). His articles have appeared in journals such as Positivity, Boundary Value Problems, and Fixed Point Theory and Applications.

Afif Ben Amar is Professor in the Department of Mathematics, Faculty of Sciences, at the University Sfax, Tunisia. His research interests include operator theory, partial differential equations, integral equations, and applications of mathematics to the natural sciences. His work has been featured in the Arabian Journal of Mathematics, Afrika Mathematika, and Acta Applicandae Mathematicae.

Table of Contents

Basic Concepts.- Nonlinear Eigenvalue Problems in Dunford-Pettis Spaces.- Fixed Point Theory in Locally Convex Spaces.- Fixed Points for Maps with Weakly Sequentially-Closed.- Fixed Point Theory in Banach Algebras.- Fixed Point Theory for (ws)-Compact Operators.- Approximate Fixed Point Theorems in Banach Spaces.

Editorial Reviews

This monograph, written by experts in the field and reflecting their particular interests, deals with some developments of fixed point theory for several classes of singlevalued and multivalued maps. … Both graduate students and mathematicians working in fixed point theory, integral equations, ordinary and partial differential equations, game theory, and other related areas will find this book useful and highly instructive.” (Paola Rubbioni, zbMATH 1357.47001, 2017)

“The book, including original contributions of the authors, is addressed to researchers interested in applications of fixed point results (in functional analytic context) to integral equations, ordinary and partial differential equations, game theory, etc. The detailed exposition of the subject and the prerequisites make it appropriate for graduate courses in linear and in nonlinear functional analysis.” (S. Cobzaş, Studia Universitatis Babes-Bolyai, Mathematica, Vol. 61 (3), 2016)

From the Publisher