Fixed Point Theorems for Occasionally Weakly Compatible Maps in Fuzzy Metric Spaces

Abstract

The aim of this paper is to establish some common fixed point theorems under occasionally weakly compatible maps using implicit functions.

Common fixed point results for various mappings in fuzzy metric spaces with application

“…The following notions were given by Jain et al [18], which extend the definitions of variants of weakly commuting mappings from ordinary fixed point theory to coupled fixed point theory in the setup of fuzzy metric spaces: 18]). The mappings F : X × X → X and g : X → X are said to be weakly commuting if…”

“…Recently, Abbas et al [1], introduced the concept of w-compatible mappings, following which, some authors established coupled common fixed point results for the similar notion of weakly compatible mappings. Works noted in [15,17,18] are some examples in this direction.…”

“…Definition 1.13 ( [18]). An element (x, y) ∈ X × X, is called a coupled common fixed point of the mappings A : 18]). An element x ∈ X, is called a common fixed point of the mappings A : X × X → X, B : X × X → X, S : X → X and T : X → X if A(a, a) = B(a, a) = S(a) = T (a) = a.…”

“…Unfortunately, Zhu and Xiao [27] proved the falsity of the work presented by Sedghi et al [25] and thereby they presented a correct modification of the results proved by Sedghi et al [25]. On the other hand, Hu [14] presented a coupled common fixed point theorem for a pair of compatible mappings under a φ-contraction in fuzzy metric spaces, which was followed by the works of Choudhury et al [4,5], Jain et al [17,18], Hu et al [15], etc. Subsequently, Abbas et al [1] introduced the notion of w-compatible mappings as a generalization of compatible mappings.…”

“…Subsequently, Abbas et al [1] introduced the notion of w-compatible mappings as a generalization of compatible mappings. Recently, in order to obtain the existence and uniqueness of the coupled common fixed points for mappings in the setup of fuzzy metric spaces Jain et al [18] introduced the notions of weakly commuting mappings and their variants, that is, R-commuting mappings, R-weakly commuting mappings of type (A F ), R-weakly commuting mappings of type (A g ), R-weakly commuting mappings of type (P ). On the other hand, Sumitra and Masmali [7] studied the notions of variants of compatible mappings that includes compatible mappings of type (A), compatible mappings of type (B), compatible mappings of type (C), compatible mappings of type (P ), compatible mappings of type (A F ), compatible mappings of type (A g ) in the context of coupled fixed point problems in fuzzy metric spaces.…”