Fixed Point Theorems in Fuzzy Metric Spaces Using Implicit Relations

Abstract

ABSTRACT The aim of this paper is to present common fixed point theorem in fuzzy metric spaces, for four self maps, satisfying implicit relations. The results of B.Singh and M.S.Chauhan[16] are generalized in this paper. Also, the application of fixed points is studied for the Product spaces. Keywords Fuzzy metric space, H -chainable fuzzy metric space, compatible mappings, weakly compatible mappings, implicit relation and common fixed point [0,1] 1. INTRODUCTION L.Zadeh’s[18] investigation of the notion of fuzzy sets has led to the growth of fuzzy mathematics. The theory of fixed point equations is one of the preeminent basic tools to handle various physical formulations. Fixed point theorems in fuzzy mathematics has got a direction of vigorous hope and vital trust with the study of Kramosil and Michalek[10], who introduced the concept of fuzzy metric space. Later on, this concept of fuzzy metric space was modified by George and Veeramani[4 ]. Sessa[15] initiated the tradition of improving commutative condition in fixed point theorems by introducing the notion of weak commuting property . Further, Jungck[8] gave a more generalized condition defined as compatibility in metric spaces. Recently in 2006, Jungck and Rhoades [9] introduced the concept of weakly compatible maps which were found to be more generalized than compatible maps. Grabiec[5] followed Kramosil and Michalek[10] and he obtained the fuzzy version of Banach contraction principle. Recently in 2000, B.Singh and M.S.Chauhan[16] brought forward the concept of compatibility in fuzzy metric space. Popa[11] proved some fixed point theorems for weakly compatible noncontinous mappings using implicit relations. His work was extended by Imdad[6] who used implicit relations for coincidence commuting property. Singh and Jain[14] extended the result of Popa[11] in fuzzy metric spaces. This paper offers the fixed point theorems on fuzzy metric spaces, which generalize, extend and fuzzify several known fixed point theorems for compatible maps on metric space, by making use of implicit relations. The condition of