Fixed Point Theorems for Mappings in Some Abstract Spaces

Abstract

The aim of this work is to extend, generalize and unify various results in different abstract spaces for various mappings such as compatible mappings, R-weakly commuting mappings and their different variants. The thesis consists of seven chapters. Chapter-I is introductory in nature, where we fix notations and terminology to be used in the subsequent chapters. In this chapter, some basic definitions, classical and recent results related to metric fixed point theory have been presented. Some of these results have been extended and generalized in the subsequent chapters. A brief summary of each chapter has been given towards the end of this chapter. In Chapter-II, the concepts of compatible mappings and weakly compatible mappings of type (A) in G-metric spaces have been introduced. Some properties related to these mappings have also been proved in this chapter. The notion of the property (E.A) has been used to prove a common fixed point theorem for weakly compatible mappings. The results presented in this chapter have been applied to obtain the solution of an integral equation and the bounded solution of a functional equation arising in dynamic programming. In Chapter-III, some new common fixed point theorems in G-metric spaces have been established by using the notions of compatibility, variants of R-weakly commutativity and weakly reciprocal continuity. Consequently, the results presented in this chapter have improved and sharpened many related results present in the existing literature. Also, some examples in support of these results have been given. The aim of Chapter-IV is to introduce a new property called “common limit in the range” for four self-mappings in fuzzy metric spaces and using this property, some common fixed point theorems have been established. The results presented in this chapter have generalized many known related results. Further, some common fixed theorems for R-weakly commuting mappings and some of their variants have been proved in fuzzy metric spaces. The objective of Chapter-V is to prove a common fixed point theorem in intuitionistic fuzzy metric spaces by using pointwise R-weak commutativity and reciprocal continuity of mappings satisfying some contractive conditions. Towards the end of this chapter, some common fixed point theorems for weakly compatible mappings along with common (E.A) like property have been proved by using some implicit relations. In Chapter-VI, some common fixed point theorems for pairs of compatible and subsequentially continuous mappings in intuitionistic fuzzy metric spaces via an implicit relation have been proved. The results proved in this chapter have improved many known related common fixed point theorems in these spaces. Further, some common fixed theorems for weakly compatible mappings in intuitionistic fuzzy metric spaces via an implicit relation and the common property (E.A) have been established. Chapter-VII is devoted to the extension of the work done by Saadati et al. [112] in the frame work of modified intuitionistic fuzzy metric spaces. In this chapter, some common fixed point theorems have been established using weakly reciprocal continuous, non-compatible self-mappings via some implicit relations. Finally, based on the present study, relevant topics for further research have been suggested.