Fixed point theorems for mappings in fuzzy type metric spaces

Abstract

According to Stefan Banach(1922), on a complete metric space, every contraction has a unique fixed point. In order to broaden the Banach fixed point theorem, several writers developed a number of contractive type constraints. By extending the concept of contraction from single-valued to multi-valued mappings, Nadler applied the Banach contraction principle. Several authors have now broadened the range of contractive type constraints to include multi-valued mappings. In order to be the first work to present fixed point theory in fuzzy metric spaces [in the sense of Kramosil and Michalek], Grabiec (1988) produced a fuzzy metric version of the Banach and Edelstein fixed point theorems(1975). One of the more appealing generalisations of the Banach contraction theorem, which establishes metric completeness, is Caristi’s(1976) fixed point theorem, which is well recognised. With regard to fuzzy metric spaces, an intriguing generalisation of the fixed point theorem by Caristi (1976) and the variational principle by Ekeland (1972) was re cently reported by Abbasi and Golshan (2016). However, their findings do not address the Kirk’s dilemma or the accompanying fuzzy metric’s completeness characterisation (1976). J. Martinez-Moreno et. al develops a class of Caristi type mappings with a fixed point and describes the completeness of the appropriate fuzzy metric to solve these problems. The results of fuzzy metric space on fuzzy b-metric space are expanded in this disser tation. Basic results and definitions that are required for later chapters are provided in chapter 1. In the second chapter, we define a class of Caristi type mappings with fixed points and describe the completeness of the related fuzzy metric. Some results from chapter 2 are extended in chapter 3 on fuzzy b-metric space