Existence of Fixed Points for Various Mappings in Abstract Spaces

Abstract

Fixed point theory is an important branch of non-linear analysis. Many problems, occurring in different branches of mathematics, such as differential equations, optimization theory and variational analysis, can be converted into the equation T x = x, where T is some non-linear operator defined on a certain space X. Solutions of this equation are called fixed point of T. Fixed point theory can be classified into three major areas: Metric fixed point theory, Topological fixed point theory and Discrete fixed point theory. The principal findings in these areas are Banach’s fixed point theorem, Brouwer’s fixed point theorem, and Tarski’s fixed point theorem respectively. Abstract space is a set of elements satisfying certain axioms. In 1906, the French mathematician Fr´echet introduced the first abstract space, called metric space. In 1922, Polish mathematician Stefan Banach gave the first fixed point theorem for contraction mappings in metric spaces, and this theorem is famous as the Banach contraction principle. This principle states that every contraction self-mapping defined on a complete metric space has a unique fixed point. This result has become one of the most popular and effective tools in solving existence problem in many branches of mathematics. Banach contraction principle has been generalized in several directions. There are two ways to extend or improve this principle. One way is to extend/improve the condition of contraction mappings, and the second approach is to replace complete metric space with a more general abstract space. In the first direction, there are numerous results in the literature proved by Kannan, Chatterjea, Reich, Hardy and Rogers, C´iri´c, Wang et al., Alber and Delabriae, Samet et al., Shahi et al., Wardowski and many more. In the second direction, we have several abstract spaces in the literature such as partial metric spaces, b-metric spaces, cone metric spaces, metric-like spaces, partial b-metric spaces, b-metric-like spaces, generalized metric spaces, F-metric spaces, 2-metric spaces, D-metric spaces, G-metric spaces, GPmetric spaces, generalized b-metric spaces etc. In the present thesis, we will work in both directions. The present thesis consists of six chapters. Chapter 1 is about the introduction related to our work. From the literature, a brief about various mappings related to fixed point theory and different abstract spaces are discussed in this chapter. At the end of the chapter, a brief plan of the results presented in the subsequent chapters is given. In Chapter 2, inspired by the concept of b-metric space, G-metric space, and generalized b-metric space, a new abstract space (named generalized Gb-metric space) has been introduced. Some basic concepts and properties of new space have been studied. Various fixed point theorems in the framework of generalized Gbmetric space has been proved. Multiple examples have been presented for the authenticity of the main results. After that, an application of one of our main results has also been given. In Chapter 3, another new class of abstract spaces (named G∗ -metric space) has been introduced as a generalization of generalized Gb-metric spaces and GP-metric spaces. Some basic concepts in G∗ -metric space have been studied. Some new types of Cauchy sequences have been noticed in this new abstract space. Various examples have also been presented for these new concepts. Chapter 4 deals with fixed point results for contraction and quasi-contraction type mappings in G∗ -metric space. As a consequence of these results, some fixed point theorems have been deduced in the framework of generalized Gb-metric space. Some examples have also been presented in support of the main results and consequences. In Chapter 5, a new class of functions has been introduced. With the help of this new class of functions, some new contractive mappings in b-metric spaces have been introduced. We establish some fixed point results for these new contractive mappings in b-metric spaces. As a consequence of the main results, various fixed point theorems have also been presented. An example has also been illustrated in support of our results. At the end of the chapter, an application has been provided to prove the uniqueness of the solution to a system of simultaneous linear equations. The aim of Chapter 6 is to extend the main results of Chapter 5 in the framework of b-metric-like spaces. Also, some common fixed point theorems have presented for weakly compatible mappings. Suitable examples have been provided for the main results. Related to the main results, some corollaries have also been presented at the end of this chapter.