COUPLED FIXED POINT THEOREMS FOR OCCASIONALLY WEAKLY COMPATIBLE MAPPINGS IN FUZZY METRIC SPACE

In this research paper we prove some coupled fixed point theorems for occasionally weakly compatible mappings in fuzzy metric space.


Introduction
Fuzzy set was defined by Zadeh [11].Fuzzy metric space was introduced by Kramosil and Michalek [7], George and Veermani [3] modified the notion and gave a new notion with the help of continuous t-norms of fuzzy metric spaces.Many researchers have obtained common fixed point theorems for mappings satisfying different types of commutativity conditions.The concepts of coupled fixed points and mixed monotone property was recently introduced by Bhaskar and Lakshmikantham [1] they illustrated these 2 SANDHYA SHUKLA AND PRIYANKA NIGAM results by proving the existence and uniqueness of the solution for a periodic boundary value problem.
Later these results were extended and generalized by Sedghi et al. [9], Fang [2] and Xin-Qi Hu [10] etc.Fixed point theorems, involving four self-maps, began with the assumption that they are commuted.Sessa [8] weakened the condition of commutativity to that of pairwise weakly commuting.Jungck generalized the notion of weak commutativity to that of pairwise compatible [4] and then pairwise weakly compatible maps [5].Jungck and Rhoades [6] introduced the concept of occasionally weakly compatible maps.
In this paper we introduce some coupled fixed point theorems for occasionally weakly compatible mappings in fuzzy metric space.(i) is an commutative and associative;

Preliminaries
(ii) is continuous; (iii) for all a [0,1]; (iv) whenever and Definitions 2.3 A 3-tuple is said to be a fuzzy metric space if X is an arbitrary set is a continuous and M is a fuzzy set on satisfying the following conditions, for all OCCASIONALLY WEAKLY COMPATIBLE MAPPINGS is continuous.
Then M is called a on X.Then Thus f and g are owc but not weakly compatible.

Main results
Theorem: 3.1 Let be a fuzzy metric space with for all .Let and be four self-mappings satisfying the following conditions:

Definition 2 . 1
A fuzzy set A in X is a function with domain X and values in [0, 1].Definition 2.2 A binary operation is a continuous t-norm if is satisfying conditions:

Lemma 2 . 5 Definition 2 . 6 Definition 2 . 7 Definition 2 . 8 Definition 2 . 9
Let (X, M,*) be a fuzzy metric space.If there exists (x, y, qt)  M(x ,y ,t) for all x, y  X and t>0, then x = y.An element is called a (i) Coupled fixed point of the mapping if .(ii) Coupled coincidence point of the mapping and if .(iii) Common Coupled coincidence point of the mapping and if .An element is called a common coupled fixed point of the mappings and if .Let and be four mappings.Then, the pair of maps and are said to have Common Coupled coincidence point if there exist in such that and .The mappings and of a set X are occasionally weakly compatible iff there is a point which is a coincidence point of f and g at which f and g commute i.e. are occasionally weakly compatible maps iff implies for .Example 2.10 Let be a fuzzy metric space, where with and Let be defined by Here, (0,0) and (1,1) are two coincidence points of f and g.That is .Thus f and g are owc but not weakly compatible.Example 2.11 Let be a fuzzy metric space, where with and OCCASIONALLY WEAKLY COMPATIBLE MAPPINGS Let be defined by Here, (1,0), (0,1), (2,4) and (4,2) are coincidence points of f and g.That is .
then there exists a unique point in such that Proof: Since the pairs (A,S) and (B,T) are owc so there are points in X pairs and are owc, then there exists a unique point in such that Proof: Since the pairs (A,S) and (B,T) are owc so there are points in pairs and are owc, then there exists a unique point in such that Proof: Since the pairs (A,S) and (B,T) are owc so there are points in pairs and are owc, then there exists a unique point in such that Proof: Since the pairs (A,S) and (B,T) are owc so there are points in Moreover if the pairs and are owc, then there exists a unique point in such that Proof: Since the pairs (A,S) and (B,T) are owc so there are points in