Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Space Using R-Weakly Commuting Mappings

In this paper, we prove a common fixed point theorem in Intuitionistic fuzzy metric space by using pointwise R-weak commutativity and reciprocal continuity of mappings satisfying contractive conditions.


Introduction
Atanassove [1] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets.In 2004, Park [2] defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norms and continuous t-conorms.Recently, in 2006, Alaca et al. [3] defined the notion of intuitionistic fuzzy metric space by making use of Intuitionistic fuzzy sets, with the help of continuous t-norm and continuous t-conorms as a generalization of fuzzy metric space due to Kramosil and Michalek [4].In 2006, Turkoglu [5] et al. proved Jungck's [6] common fixed point theorem in the setting of intuitionistic fuzzy metric spaces for commuting mappings.For more details on intuitionistic fuzzy metric space, one can refer to the papers [7][8][9][10][11][12].
The aim of this paper is to prove a common fixed point theorem in intuitionistic fuzzy metric space by using pointwise R-weak commutativity [5] and reciprocal continuity [9] of mappings satisfying contractive conditions.

Preliminaries
Definition 2.1 [13].A binary operation       : 0,1 0,1 0,1    is continuous t-norm if * satisfies the following conditions: 1) * is commutative and associative; 2) * is continuous; 3) for all *1 a  a   , , a b c, 0,1 .d  Alaca et al. [3] defined the notion of intuitionistic fuzzy metric space as: Definition 2.3 [3].A 5-tuple   , , ,*, X M N  is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, * is a continuous t-norm, ◊ is a continuous tconorm and , M N are fuzzy sets on X 2 × [0, ∞) satisfying the conditions: 1) x y X  and if and only if x y X  .
The functions  , ,  M x y t and denote the degree of nearness and the degree of non-nearness between x and y w.r.t.t respectively.

N x y t
Remark 2.1 [12].Every fuzzy metric space   , ,* X M is an intuitionistic fuzzy metric space of the form such that t-norm * and t-conorm Clearly, every pair of weakly commuting mappings is pointwise R-weakly commuting with .
is a sequence such that n Au z  , for some z in X. n If A and S are both continuous, then they are obviously reciprocally continuous but converse is not true.

Lemmas
The proof of our result is based upon the following lemmas of which the first two are due to Alaca et al. [12]: be intuitionistic fuzzy metric space and for all x y X  , and if and .Then the continuity of one of the mappings in compatible pair

Taking
, we get  as .n   This proves that A and S are reciprocally continuous on X.Similarly, it can be proved that B and T are reciprocally continuous if the pair is assumed to be compatible and T is continuous.

Main Result
The main result of this paper is the following theorem: , , ,*, X M N   be a complete intuitionistic fuzzy metric space with continuous t-norm * and continuous t-conorm defined by t t t   and Further, let   N y y kt N y y t 2), we have Now, again by taking 1 Therefore, for any n and t, we have      Then by (3.2), take 1   for all x X  .
Clearly, 1) either of pair (A, S) or (B, T) be continuous self-mappings on X; 2)        ,  A X T X B X S X   ; 3) {A, S} and {B, T} are R-weakly commuting pairs as both pairs commute at coincidence points; 4) {A, S} and {B, T} satisfies inequality (3.2), for all , x y X  , where   0,1 k  .Hence, all conditions of Theorem 4.1 are satisfied and x = 0 is a unique common fixed point of A, B, S and T.

,AS
and   , B T be pointwise R-weakly commuting pairs of self mappings of X satisfying (3.1), (3.2).If one of the mappings in compatible pair   , A S or   , B T is continuous, then A, B, S and T have a unique common fixed point.Proof.Let 0 x X  .By (3.1), we define the sequences   n x and   n y in X such that for all 0

Definition 2.5 [3]. An intuitionistic fuzzy metric space is said to be complete if and only if every Cauchy sequence in X is convergent.
 

 Example 2.1 [3]. Let
Let  assume that A and S are compatible and S is continuous.We show that A and S are reciprocally n Since S is continuous, we have n and n as and since , , , , ,