On Common Fixed Point Theorem of Four Self Maps in a Fuzzy Metric Space

In the present paper, we show that there exists a unique common fixed point for four self maps in a fuzzy metric space where two of the maps are reciprocally continuous and the other two maps are z-asymptotically commuting.


Introduction
L. Zadeh's [1] investigation of the concept of fuzzy set in the year 1965, has led to a rich growth of fuzzy mathematics.Today, it has become a well-accepted system to embrace upon uncertainties springing in numerous physical situations.The theory of fixed point equations is one of the extrusive basic tools to exploit various physical formulations.Theorems on fixed points in fuzzy mathematics are emerging with flourishing hope and vital certainty.
Many authors have introduced the concept of fuzzy metric space in various ways and have shown that every metric induces a fuzzy metric.There have been several endeavors to formulate fixed point theorems in fuzzy mathematics.In 1975, Kramosil and Michalek [2] generalized the statistical metric space and defined the fuzzy metric space which was later modified by George and Veeramani [3] [4] by introducing the concept of continuous t-norms.Recently, many researchers [5]- [9] have enormously developed the theory by studying various aspects of the theory and extending the concept of fuzzy metric through applying several contractive, expansive, continuity and compatibility conditions on the fuzzy metric and producing different results.
Pant [10] introduced the notion of reciprocally continuous mappings and established a fixed point theorem.S. N. Mishra, Nilima Sharma, S. L. Singh [11] defined z-asymptotically commuting maps in fuzzy metric spaces which may be seen as a comparable formulation given by Trivari-Singh [12] in metric spaces.These mappings are more general than commuting and weakly commuting maps.
The aim of this paper is to show that the self maps in a fuzzy metric space satisfying certain properties and inequalities possess a common fixed point which is unique.

Preliminaries
Here, we shall recall some prefaces: Definition 2.1 ([13]): A binary operation x y t denotes the degree of nearness between x, y with respect to "t".X, M, * be a fuzzy metric space.If there exists ( ) M , , M , , x y qt x y t ≥ for all , X x y ∈ and t > 0, then x y = .Succeeding the Grabiec's approach to fuzzy contraction principle, Mishra.S. N., Nilima Sharma, Singh.S. L. [11] had obtained common fixed point theorem for asymptotically commuting maps in fuzzy metric spaces.Theorem 2.9 ([11]): Let ( ) X, M, * be a complete fuzzy metric space with t t t * ≥ , [ ] 0,1 t ∈ and P, Q : X X → .If there exist continuous maps S, T : X X → and a constant ( ) 0,1 k ∈ such that 1) ST = TS 2) {P, S} and {Q, T} are asymptotically commuting pairs 3) for all , X x y ∈ , t > 0 and ( ) 0, 2 α ∈ then P, Q, S, T have a unique common fixed point.

Main Results
Theorem 3.1: Let ( ) X, M, * be a complete fuzzy metric space & * be any of the continuous t-norms given in 2.1(α) and let A, B, S, T be self maps of X satisfying • The pair {A, S} is reciprocally continuous • The pair {B, T} is z-asymptotically commuting • The pairs {B, S} and {T, S} commute with each other

Definition 2 . 3 ( 5 ( 7 (
[3]): A sequence { } n x in a fuzzy metric space ( ) X, M, * said to converge to X x → as n → ∞ .Definition 2.4 ([3]): A sequence { } n x in a fuzzy metric space ( )X, M, * is said to be a Cauchy sequence if for[3]): If every Cauchy sequence in a fuzzy metric space X is convergent, then X is said to be complete.Definition 2.6 ([10]): Two self maps A and B of a fuzzy metric space ( ) X, M, * are said to be reciprocally continuous on[11]): Two self maps A and B of a fuzzy metric space X are said to be z-asymptotically commuting if and only if Lemma 2.8([14]): Let ( )