Fixed point theorems in intuitionistic fuzzy contraction mappings in intuitionistic fuzzy generalized metric spaces

: In this paper, we introduce intuitionistic fuzzy contraction mappings in intuitionistic fuzzy generalized metric spaces. The presented theorems, extend, generalize and improve the corresponding result which given in the literature. Some fixed point theorems in intuitionistic fuzzy generalized metric space in the sense of George and Veeramani [2].


Introduction
In 1965, the concept of fuzzy set was introduced by Zadeh [14] in domain X and [0, 1]. In 1986, Atanassov [1] introduced the notion of an intuitionistic fuzzy sets. Afterward, Park [8] gave the notion of an intuitionistic fuzzy metric space and generalized the notion of a fuzzy metric space due to George and Veeramani. In 2008, Saadati et al. [10] modified the idea of an intuitionistic fuzzy metric space and presented the new notion of an intuitionistic fuzzy metric space.
On the other hand, in 1981, Heilpern [3] developed fixed point theory in fuzzy metric spaces, introduced the concept of fuzzy contraction mappings and proved some fixed point theorems for fuzzy contraction mappings. Afterward, in 2006, Rafi and Noorani [9] introduced the concept of intuitionistic fuzzy contraction mappings and proved the existence fixed point in intuitionistic fuzzy generalized metric spaces for an intuitionistic fuzzy contraction mappings. We introduce intuitionistic fuzzy contraction mappings in intuitionistic fuzzy generalized metric spaces. The presented theorems, extend, generalize and improve the corresponding result which given in the literature. , if X is an arbitrary set, * is a continuous t-norm, is a continuous t-conorm and ℳ, ࣨ are fuzzy sets on X 3 × (0, ∞) satisfying for all x, y, z, a ∈ X and s, t > 0, the following conditions The above definition, the triangular inequality (IFGM-5) and (IFGM-10) are replaced by ࣨ(x, y, z, a, t) ࣨ(a, z, z, s) ≥ ࣨ(x, y, z, min {t, s}).
Then the 5-tuple (X, ℳ, ࣨ, *, ◊) is called a non-Archimedean intuitionistic fuzzy generalized metric space. It is easy to check that the triangular inequality (NA) implies (IFGM-5) and (IFGM-10), that is every non-Archimedean intuitionistic fuzzy generalized metric space is itself an intuitionistic fuzzy generalized metric space. Definition 1.3. Let (X, ℳ, ࣨ, *, ◊) be an intuitionistic fuzzy generalized metric space. Then a) A sequence {xn} in X is said to be converges to a point x ∈ X, if for all t > 0, X is complete if every Cauchy sequence is converges in X.
Then {xn} is Cauchy sequence in X.

Main results
Definition 2.1. Let (X, ℳ, ࣨ, *, ◊) be an intuitionistic fuzzy generalized metric space. A mapping T : X → X is an intuitionistic fuzzy generalized contractive mapping, if there exists k ∈ (0, 1), such that and ࣨ(Tx, Ty, Tz, t) ≥ k, ࣨ(x, y, z, t), for each x, y, z ∈ X and t > 0 (k is called the contractive constant of T). A intuitionistic fuzzy generalized metric space in which every G-Cauchy sequence is convergent is called G-Complete.
Uniqueness: Suppose there exist v ∈ X such that Tv = v and v ≠ u. Now, we consider Since k < 1, we have M (u, v, v, t) = 1 and ࣨ(u, v, v, t) = 0. Then u = v. Therefore u is fixed point of T.