SEMI-COMPATIBILITY AND FIXED POINT THEOREM IN MENGER SPACE USING IMPLICIT RELATION

Abstract

Abstract. In this paper the concept of semi-compatibility hasbeen introduced in Menger space and it has been applied to proveresultson existence ofunique commonfixed point offourselfmapssatisfying an implicit relation. It results in a generalization ofBanachcontractionprinciple establishedby Sehgal and Bharucha-Reid in [8] All the result presented in this paper are new. 1. IntroductionThere have been a number of generalizations of metric space. Onesuch generalization is Menger space initiated by Menger [3]. It is aprobabilistic generalization in which we assign to any two points xand y, a distribution function F x,y . Schweizer and Sklar [7] studiedthis concept and gave some fundamental results on this space. Seh-gal and Bharucha-Reid [8] obtained Banach contraction principle ina complete Menger space, which is a milestone in developing fixedpoint theory in Menger space. Cho, Sharma and Sahu [1] introducedthe concept of semi-compatible maps in a d-topological space. Theydefine a pair of self maps (S,T) to be semi-compatible if conditions(i)Sy = Ty implies STy = TSy (ii){Sx