“A Study Of Some Coincidence And Common Fixed Point Theorem In Fuzzy Metric Spaces”

Abstract

The research report in this thesis deals mainly with fixed point theorems and their applications. The F-type fuzzy topological spaces and quasi fuzzy metric spaces are introduced. Also, the variational principle and Christi’s fixed point theorem in F-type fuzzy topological spaces are established, the results of which are utilized to obtain a fixed point theorem for Manger probabilistic metric space. The compatible pair of reciprocally continuous mappings is defined and a fixed point theorem in a fuzzy metric space is obtained which generates a fixed point but does not force the map to be continuous. Further, V|/-compatible mapping is introduced in a fuzzy metric space and established the altering distances between the points using certain control functions which differ from the previous works. R-weakly commuting of type (A) and non compatible mappings in fuzzy metric space are introduced which leads to the proof of fixed point theorem without assuming completeness of the space or continuity. In addition, the common fixed point theorem for sequence of self mappings is proved by using the notion of families of functions. The concept of compatibility is introduced in generalized fuzzy metric space and obtained common fixed point theorems for compatible mappings. Further, we prove some relations between compatible mappings and compatible maps of type (a) and (3) in generalized fuzzy metric spaces. Besides, applications of fixed point theorems in various fuzzy differential equations are discussed.