Abstract

In this paper, first we prove a common fixed point theorem using weakly compatible mapping in 2- Menger space which generalize the well known results. Secondly, we prove a common fixed point theorem using (S-B) property along with weakly compatible maps. (S-B) property defined by Sharma and Bamoria [16] via implicit relation. Keywords : Common fixed points, Metric space, S-B property, 2-Menger space, weakly compatible mapping and implicit relation. AMS subject classification – 47H10, 54H25. DOI : 10.7176/MTM/9-5-01 Publication date :May 31 st 2019

Highlights

  • AND PRELIMINARIESIn 1922, Banach proved the principal contraction result [4]

  • The study of 2-metric spaces was initiated by Gahler[7] and some fixed point theorems in 2-metric spaces were proved in [8],[9], [10] and [15]

  • X and a mapping F from X × X to L, where L is the collection of all distribution functions (a distribution function F is non decreasing and left continuous mapping of reals in to [0,1] with properties, inf F(x) = 0 and sup F(x) = 1). 1

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Summary

AND PRELIMINARIES

In 1922, Banach proved the principal contraction result [4]. As we know, there have been published many works about fixed point theory for different kinds of contractions on some spaces such as quasi-metric spaces, cone metric spaces, convex metric spaces, partially ordered metric spaces, G-metric spaces, partial metric spaces, quasi-partial metric spaces, fuzzy metric spaces and Menger spaces. Metric as follows; A probabilistic metric space shortly PM-Space, is an ordered pair (X, F) consisting of a non empty set. A Menger space is a triplet (X, F, T), where (X, F) is a PM-Space, X is a non-empty set and a t − norm satisfying instead of (FM-8) a stronger requirement. Definition 1.2 :- Let (X, F,∗) be a Menger space and be a continuous t-norm. Remark 1.3:- If is a continuous t-norm, it follows from (FM − 4) that the limit of sequence in Menger space is uniquely determined. Definition 1.4:- Self maps A and B of a Menger space (X, F,∗) are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if Ax = Bx for some x ∈ X

Weakly Compatible Maps
Bxn and we deduce that lim n
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