Fixed Point Theorem in Fuzzy Metric Space

After Zadeh [5] introduced fuzzy sets in 1965, many researchers from various areas have developed the theory of fuzzy sets and its applications. Deng [8],Erceg [9], George and Veeramani [7] etc gave initial foundations of different forms of fuzzy metric spaces. Grebiec [3] extended Banach’s [11] and Edelstein’s [13] fixed point theorem in fuzzy metric space. Kramosil and Michalek [12] investigated common fixed point theorems for compatible maps. The investigation of fixed point theorems are going on fuzzy metric spaces.


Introduction
After Zadeh [5] introduced fuzzy sets in 1965, many researchers from various areas have developed the theory of fuzzy sets and its applications.Deng [8],Erceg [9], George and Veeramani [7] etc gave initial foundations of different forms of fuzzy metric spaces.Grebiec [3] extended Banach's [11] and Edelstein's [13] fixed point theorem in fuzzy metric space.Kramosil and Michalek [12] investigated common fixed point theorems for compatible maps.The investigation of fixed point theorems are going on fuzzy metric spaces.
In this paper we will study a fixed point theorem from view point of a new class of fuzzy metric defined on a fuzzy set.This concept came to exist when the author was investigating properties in a generalized closed set of bitopological space using topological ideal.Often topological ideal is simply stated as ideal.
A non-empty collection I of subsets of a set X is said to be an ideal if it follows following two conditions Fixed point theorems in any areas are most useful.Mathematical economists, physicists,computer scientists etc are using fixed point theorems in their respective research areas.Now a days fuzzy fixed point theorems are also playing crucial role in mathematical economics, social choices,auction theory.One remarkable 2000 Mathematics Subject Classification: 54H25, 47H10 Santanu Acharjee application of convex fixed point theory can be found in the 1994's Nobel laureate John Fr.Nash's classic seminal paper of equilibrium point of "Noncooperative games" [14].His proof is based on Kakutani's fixed point theorem [15], which is the generalization of Brouwer's fixed point theorem .

Preliminary definitions
In this section we discuss some existing definitions.Definition 2.2.( [12]) Let X be a non-empty set, * be a continuous t-norm and M : X 2 × [0, ∞) → [0, 1] be a fuzzy set.Consider the following conditions for all x, y, z ∈ X and t, s ∈ [0, ∞); Then (X, M, * ) is said to be a fuzzy metric space.

Main result
This section contains some new definitions, terminologies and they are used to prove one theorem in fuzzy fixed point theory.Definition 3.3.A sequence < x n > in a SF M S is said to be structure convergent if there exists x ∈ X such that lim n→∞ M (x n , x, t) = 1 ∀t > 0. Then x is said to be structure limit of < x n > and denoted by lim n→∞ x n = x.
Definition 3.4.A sequence < x n > in a SF M S (X, M, * ) is said to be structure Cauchy sequence if for each t > 0 and r ∈ N such that lim n→∞ M (x n+r , x n , t) = 1.
(X, M, * ) is said to be structure complete if every structure Cauchy sequence in it is structure convergent.Definition 3.5.Let (X, M, * ) be a SF M S, f and h are self maps on X.Then f and h are said to be normalized at x if and only if The functions f and h are said to be normalized on X if f and h are normalized at all points x of X. Definition 3.6.The functions f and h are said to be common domain normalized (CDN ) if they are normalized at the coincidence point of f and h.Remark 3.7.A SF M S has a unique limit point.
Proof: Proof is easy, so omitted.✷ Now we discuss the main theorem of this section.
Theorem 3.8.Let (X, M, * ) be a SFMS and let f, h : X → X be two mappings with the following conditions, (a) f (X) ⊂ h(X) (b) Either f (X) or h(X) is structure complete (c) M (f x, f y, kt) ≥ M (hx, hy, t) for all x, y ∈ X and 0 < k < 1,t ∈ [0, ∞) (d) lim t→∞ M (x, y, t) = 1 Then f and h have a coincidence point; moreover if f and h are CDN then f and h have a unique fixed point.