# Fixed Point Theory and Graph Theory

## 1st Edition

### Foundations and Integrative Approaches

**Editors:**Monther Alfuraidan Qamrul Ansari

**eBook ISBN:**9780128043653

**Hardcover ISBN:**9780128042953

**Imprint:**Academic Press

**Published Date:**10th June 2016

**Page Count:**442

## Description

*Fixed Point Theory and Graph Theory* provides an intersection between the theories of fixed point theorems that give the conditions under which maps (single or multivalued) have solutions and graph theory which uses mathematical structures to illustrate the relationship between ordered pairs of objects in terms of their vertices and directed edges.

This edited reference work is perhaps the first to provide a link between the two theories, describing not only their foundational aspects, but also the most recent advances and the fascinating intersection of the domains.

The authors provide solution methods for fixed points in different settings, with two chapters devoted to the solutions method for critically important non-linear problems in engineering, namely, variational inequalities, fixed point, split feasibility, and hierarchical variational inequality problems. The last two chapters are devoted to integrating fixed point theory in spaces with the graph and the use of retractions in the fixed point theory for ordered sets.

## Key Features

- Introduces both metric fixed point and graph theory in terms of their disparate foundations and common application environments
- Provides a unique integration of otherwise disparate domains that aids both students seeking to understand either area and researchers interested in establishing an integrated research approach
- Emphasizes solution methods for fixed points in non-linear problems such as variational inequalities, split feasibility, and hierarchical variational inequality problems that is particularly appropriate for engineering and core science applications

## Readership

All academicians and PhD students in the field of fixed point or graph theory interested in applications.

## Table of Contents

- Foreword
- Acknowledgments
- Preface
- About the Authors
- Chapter 1: Caristi-Browder Operator Theory in Distance Spaces
- Abstract
- 1.1 From the Caristi Fixed Point Theorems to Caristi, Caristi-Kirk and Caristi-Browder Operators
- 1.2 List of Notations
- 1.3 Weakly Picard Operators on L-Spaces
- 1.4 Caristi-Browder Operators on Metric Spaces
- 1.5 Caristi-Browder Operators on + m -Metric Spaces
- 1.6 Caristi-Browder Operators on s( +)-Metric Spaces
- 1.7 Caristi-Browder Operators on Kasahara Spaces
- 1.8 Research Directions in the Caristi-Browder Operator Theory

- Chapter 2: Iterative Approximation of Fixed Points of Single-valued Almost Contractions
- Abstract
- 2.1 Introduction
- 2.2 Fixed Point Theorems for Single-valued Self Almost Contractions
- 2.3 Implicit Almost Contractions
- 2.4 Common Fixed Point Theorems for Almost Contractions
- 2.5 Almost Contractive type Mappings on Product Spaces
- 2.6 Fixed Point Theorems for Single-valued Nonself Almost Contractions

- Chapter 3: Approximate Fixed Points
- Abstract
- 3.1 Introduction
- 3.2 Approximate Fixed Points of Mappings in Banach Spaces
- 3.3 Approximate Fixed Points of Mappings in Hyperbolic Spaces
- 3.4 Approximate Fixed Points of Monotone Mappings
- 3.5 Approximate Fixed Points of Nonlinear Semigroups
- 3.6 Approximate Fixed Points of Monotone Nonlinear Semigroups

- Chapter 4: Viscosity Methods for Some Applied Nonlinear Analysis Problems
- Abstract
- 4.1 Introduction
- 4.2 Viscosity Method for Fixed Point Problems
- 4.3 Viscosity Method for Split Common Fixed Point Problems
- 4.4 Viscosity Method for Split Equilibrium Problems

- Chapter 5: Extragradient Methods for Some Nonlinear Problems
- Abstract
- 5.1 Introduction
- 5.2 Preliminaries
- 5.3 Projection Gradient Method
- 5.4 Extragradient Method for Nonexpansive Mappings and Variational Inequalities
- 5.5 Modified Extragradient Method for Nonexpansive Mappings and Variational Inequalities
- 5.6 Extragradient Method for Hierarchical Variational Inequalities
- 5.7 Extragradient Methods for Split Feasibility Problems
- Acknowledgments

- Chapter 6: Iterative Methods for Nonexpansive Type Mappings
- Abstract
- 6.1 Introduction and Preliminaries
- 6.2 Nonexpansive Mappings in Uniformly Convex Banach Spaces
- 6.3 Nonexpansive Mappings in CAT(0) Spaces
- 6.4 An Algorithm of Asymptotically Nonexpansive Mappings
- 6.5 Existence and Approximation of Fixed Points
- 6.6 Viscosity Method for Generalized Asymptotically Nonexpansive Mappings
- Acknowledgments

- Chapter 7: Metric Fixed Point Theory in Spaces with a Graph
- Abstract
- 7.1 Introduction
- 7.2 Banach Contraction Principle
- 7.3 Basic Definitions and Properties
- 7.4 Banach Contraction Principle in Metric Spaces with a Graph
- 7.5 Caristi’s Fixed Point Theorem
- 7.6 The Contraction Principle in Modular Metric Spaces with a Graph
- 7.7 Monotone Pointwise Contractions in Banach Spaces with a Graph
- 7.8 Monotone Ćirić Quasi-Contraction Mappings
- 7.9 Monotone Nonexpansive Mappings in Banach Spaces with a Graph
- 7.10 Monotone Nonexpansive Mappings in Hyperbolic Metric Spaces with a Graph
- 7.11 The Contraction Principle for Monotone Multivalued Mappings
- 7.12 Monotone Nonexpansive Multivalued Mappings
- Acknowledgments

- Chapter 8: The Use of Retractions in the Fixed Point Theory for Ordered Sets
- Abstract
- 8.1 Introduction
- 8.2 Chain-Complete Ordered Sets
- 8.3 The Abian-Brown Theorem
- 8.4 Comparative Retractions
- 8.5 Irreducible Points
- 8.6 Constraint Propagation
- 8.7 Retractable Points
- 8.8 Verifying Connected Collapsibility Directly
- 8.9 Graphs
- 8.10 Simplicial Complexes
- 8.11 Topological Realizations
- 8.12 Iterated Clique Graphs
- 8.13 K-Null Comparability Graphs
- Acknowledgments

- Index

## Details

- No. of pages:
- 442

- Language:
- English

- Copyright:
- © Academic Press 2016

- Published:
- 10th June 2016

- Imprint:
- Academic Press

- eBook ISBN:
- 9780128043653

- Hardcover ISBN:
- 9780128042953

## About the Editor

### Monther Alfuraidan

Monther Rashed Alfuraidan is Associate Professor of Mathematics in the Department of Mathematics & Statistics at King Fahd University of Petroleum & Minerals at Dhahran, Saudi Arabia. He obtained his Ph.D. (Mathematics) from Michigan State University. He has written more than twenty articles on graph theory, algebraic graph theory and metric fixed point theory. He peer-reviewed many articles (among others) for: algebraic journal of combinatorics, Arabian Journal of Mathematics, Fixed Point Theory and Applications and Journal of Inequality and Applications.

### Affiliations and Expertise

King Fahd University, Saudi Arabia

### Qamrul Ansari

Qamrul Hasan Ansari is Professor of Mathematics at Aligarh Muslim University, Aligarh, India, and joint professor at King Fahd University of Petroleum \& Minerals, Dhahran, Saudi Arabia. He obtained his Ph.D. (Mathematics) from Aligarh Muslim University, India. He is an associate editor of Journal of Optimization Theory and Applications, Fixed Point Theory and Applications and Carpathian Journal of Mathematics. He also edited several special issues of several journals, namely, Journal of Global Optimization, Fixed Point Theory and Applications, Abstract and Applied Analysis, Journal of Inequalities and Applications, Applicable Analysis, Positivity, Filomat, etc. He has written more than 180 articles on variational inequalities, fixed point theory and applications, vector optimization, etc. in various international peer-reviewed journals. He has edited 6 books for Springer, Taylor & Francis and Narosa, India. He is an author of a book on Metric Spaces published by Narosa, India and has coauthored one book on Variational Inequalities and Nonsmooth Optimization for Taylor & Francis.

### Affiliations and Expertise

Aligarh Muslim University, India