Metric Structures and Fixed Point Theory

It is an indisputable argument that the formulation of metrics (by Fréchet in the early 1900s) opened a new subject in mathematics called non-linear analysis after the appearance of Banach?s fixed point theorem. Because the underlying space of this theorem is a metric space, the theory that developed following its publication is known as metric fixed point theory. It is well known that metric fixed point theory provides essential tools for solving problems arising in various branches of mathematics and other sciences such as split feasibility problems, variational inequality problems, non-linear optimization problems, equilibrium problems, selection and matching problems, and problems of proving the existence of solutions of integral and differential equations are closely related to fixed point theory. For this reason, many people over the past seventy years have tried to generalize the definition of metric space and corresponding fixed point theory. This trend still continues. A few questions lying at the heart of the theory remain open and there are many unanswered questions regarding the limits to which the theory may be extended.

Metric Structures and Fixed Point Theory provides an extensive understanding and the latest updates on the subject. The book not only shows diversified aspects of popular generalizations of metric spaces such as symmetric, b-metric, w-distance, G-metric, modular metric, probabilistic metric, fuzzy metric, graphical metric and corresponding fixed point theory but also motivates work on existing open problems on the subject. Each of the nine chapters?contributed by various authors?contains an Introduction section which summarizes the material needed to read the chapter independently of the others and contains the necessary background, several examples, and comprehensive literature to comprehend the concepts presented therein. This is helpful for those who want to pursue their research career in metric fixed point theory and its related areas.

Features

• Explores the latest research and developments in fixed point theory on the most popular generalizations of metric spaces
• Description of various generalizations of metric spaces
• Very new topics on fixed point theory in graphical and modular metric spaces
• Enriched with examples and open problems

This book serves as a reference for scientific investigators who need to analyze a simple and direct presentation of the fundamentals of the theory of metric fixed points. It may also be used as a text book for postgraduate and research students who are trying to derive future research scope in this area.

1 Symmetric Spaces and Fixed Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Pradip Ramesh Patle and Deepesh Kumar Patel

2 Fixed Point Theory in b-Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Nguyen Van Dung and Wutiphol Sintunavarat

3 Basics of w-Distances and Its Use in Various Types of Results . . . . . . . . . . . . . . . . . . . . . 67

Dhananjay Gopal and Mohammad Hasan

4 G-Metric Spaces: From the Perspective of F-Contractions and Best

Proximity Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Vishal Joshi and Shilpi Jain

5 Fixed Point Theory in Probabilistic Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Juan Martínez-Moreno

6 Fixed Point Theory For Fuzzy ContractiveMappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Dhananjay Gopal and Tatjana Došenovi´c

7 Set-Valued Maps and Inclusion Problems inModular Metric Spaces. . . . . . . . . . . . . 245

Poom Kumam

8 Graphical Metric Spaces and Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Satish Shukla

9 Fixed Point Theory in Partial Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Dhananjay Gopal and Shilpi Jain

Dhananjay Gopal has a doctorate in Mathematics from Guru Ghasidas University, Bilaspur, India and is currently Associate Professor of Mathematics in Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur (C.G.), Bilaspur, Gujarat, India. He is author and co-author of several papers in journals, proceedings and a monograph on Background and Recent Developments of Metric Fixed Point Theory and a book on Introduction to metric spaces. He is devoted to general research on the theory of Nonlinear Analysis and Fuzzy Metric Fixed Point Theory. D. Gopal has active research collaborations with KMUTT, Bangkok, Thammasat University Bangkok, Jaen University Spain and in his research pursuits he has visited South Africa, Thailand, Japan and Iran.

Dr. Praveen Agarwal was born in Jaipur (India) on August 18, 1979. After completing his schooling, he earned his Master’s degree from Rajasthan University in 2000. In 2006, he earned his Ph. D. (Mathematics) at the Malviya National Institute of Technology (MNIT) in Jaipur, India, which is one of the highest ranking universities in India.

Dr. Agarwal has been actively involved in research as well as pedagogical activities for the last 20 years. His major research interests include Special Functions, Fractional Calculus, Numerical Analysis, Differential and Difference Equations, Inequalities, and Fixed Point Theorems. He is an excellent scholar, dedicated teacher, and prolific researcher. He has published 7 research monographs and edited volumes and more than 150 publications (with almost 100 mathematicians all over the world) in prestigious national and international mathematics journals. Dr. Agarwal worked previously either as a regular faculty or as a visiting professor and scientist in universities in several countries, including India, Germany, Turkey, South Korea, UK, Russia, Malaysia and Thailand. He has held several positions including Visiting Professor, Visiting Scientist, and Professor at various univers