Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces using Implicit Relation

In this paper, we use the notion of property E.A. in an intuitionistic fuzzy metric space to prove a common fixed point theorem which generalizes Theorem-2 of Turkoglu et al. (2006).


InTRoduCTIon
I n 1986, Jungck introduced the notion of compatible maps for a pair of self mappings.Several papers involving compatible maps proved the existence of common fixed points in the classical and fuzzy metric spaces (Grorge andVeeramani, 1994, Kramosil andMichalek, 1975).Aamri and Moutawakil (2002) generalized the concept of non compatibility by defining the notion of property E.A. and proved common fixed point theorems under strict contractive conditions.Atanassove (1986) introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets and later there has been much progress in the study of intuitionistic fuzzy sets by many authors (Alaca, 2006;Atanassov, 1986;Coker, 1997;Manro et al., 2010Manro et al., , 2012;;Park, 2004;Park et al. 2005;Saadati and Park, 2006).In 2004, Park defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norms and continuous t-conorms as a generalization of fuzzy metric space due to George and Veeramani (1994).Fixed point theory has important applications in diverse disciplines of mathematics, statistics, engineering, and economics in dealing with problems arising in: Approximation theory, potential theory, game theory, mathematical economics, etc.Several authors (George and Veeramani 1994; Kramosil and Michalek, 1975) proved some fixed point theorems for various generalizations of contraction mappings in probabilistic and fuzzy metric space.Turkoglu et al. (2006) gave a generalization of Jungck's common fixed point theorem (Jungck, 1976) to intuitionistic fuzzy metric spaces.In this paper, we use the notion of property E.A. in an intuitionistic fuzzy metric space to prove a common fixed point theorem for a quadruplet of self mappings in intuitionistic fuzzy metric space.

PRelIMInARIeS.
The concepts of triangular norms (t -norm) and triangular conorms (tconorm) are were originally introduced by Schweizer and Sklar (1960) in the study of statistical metric spaces.Alaca et al. (2006) using the idea of Intuitionistic fuzzy sets, defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norm and continuous t-conorms as a generalization of fuzzy metric space due to Kramosil and Michalek (1975) as : definition 2.3.(Alaca et al., 2006) 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 t-conorm and M, N are fuzzy sets on X 2 × [0, ∞) satisfying the following conditions: (i) M(x, y, t) + N(x, y, t) ≤ 1 for all x, y ∈ X and t > 0; (ii) M(x, y, 0) = 0 for all x, y ∈ X; (iii) M(x, y, t) = 1 for all x, y ∈ X and t > 0 if and only if x = y; (iv) M(x, y, t) = M(y, x, t) for all x, y ∈ X and t > 0; (v) M(x, y, t) * M(y, z, s) ≤ M(x, z, t + s) for all x, y, z ∈ X and s, t > 0; (vi) for all x, y ∈ X, M(x, y, .)and t > 0 if and only if x = y;  (x) N(x, y, t) = N(y, x, t) for all x, y ∈ X and t > 0;  (xi) N(x, y, t) ◊ N(y, z, s) ≥ N(x, z, t + s) for all x, y, z ∈ X and s, t > 0 Then (M, N) is called an intuitionistic fuzzy metric space on X.The functions M(x, y, t) and N(x, y, t) denote the degree of nearness and the degree of non-nearness between x and y w.r.t.t respectively.
definition 2.6.A pair of self mappings ( f , g ) of an intuitionistic fuzzy metric space (X, M, N, *, ◊) is said to be commuting if   et al. (2006) proved the following Theorem:

Turkoglu
Theorem 3.1.Let (X, M, N, *, ◊) be a complete intuitionistic fuzzy metric space.Let f and g be self mappings of X satisfying the following conditions: (c) f is continuous.
Then f and g have a unique common fixed point provided f and g commute.
Now, we prove a common fixed point theorem using property E.A. in an intuitionistic fuzzy metric space, which is a generalization of Theorem 3.1 in the following way: to relax the continuity requirement of maps completely, (ii) property E.A buys containment of ranges.
Proof: In view of (3.1), there exists a sequence {x n } in X such that lim n→∞ gx n = lim n→∞ fx n = u for some u ∈ X. Suppose that f(X) is complete subspace of X, therefore, every convergent sequence of points of f(X) has a limit point in f(X) implies lim n→∞ fx n = fa = u = lim n→∞ gx n , for some a ∈ X , which implies that u = fa ∈ f(X).Now, we prove that ga = fa.From (3.2) take x = x n , y = a , we get M(gx n , ga, kt) ≥ M(fx n , fa, t) and N(gx n , ga, kt) ≤ N(fx n , fa, t).
This implies by Lemma 2.1, fa = ga.Therefore, u = fa = ga.This shows that 'a' is coincident point of g and f.
This proves that ga is the common fixed point of g and f.Now, we prove the uniqueness of common fixed point of g and f.If possible, let x 0 and y 0 be two common fixed points of f and g.Then by condition (3.Then by Lemma 2.1, we have x 0 = y 0 .Therefore, the mappings f and g have a unique common fixed point.This completes the proof.
. Consider (X, M, N, *, ◊) be an intuitionistic fuzzy metric space as in Example 2.1.Define fx = x 4 and gx = x 12 for all x ∈ X.Clearly, f and g are weakly compatible mappings on X, Also, (i) f and g satisfy the property E.A for the sequence Thus all the conditions of Theorem 3.2 are satisfied and so f and g have the common fixed point x = 0.
Then f and g have a unique common fixed point.