A Fixed Point Theorem for Contraction Type Mappings in Menger Spaces

We proved a common fixed point theorem for a sequence of self maps satisfying a new contraction type condition in Menger spaces, results extended and generalize some known results in metric spaces and fuzzy metric spaces.


INTRODUCTION
There have been a number of generalizations of metric space. One such generalization is Menger space introduced in 1942 by Menger [1] who was use distribution functions instead of nonnegative real numbers as values of the metric. Schweizer and Sklar [2] studied this concept and gave some fundamental results on this space. The important development of fixed-point theory in Menger spaces was due to Sehgal and Bharucha-Reid [3] . The study of common fixed points of maps satisfying some contractive type condition has been at the centre of vigorous research activity. It is observed by many authors [3,[4][5][6][7][8][9][10] that contraction condition in metric space may be translated into probabilistic metric space endoved with min norms. The purpose of this was to define and investigate a new class of self-maps satisfying a new contraction type condition in Menger spaces.

Definition 1:
A triangular norm * (shorty t-norm) is a binary operation on the unit interval [0,1] such that for all a, b, c, d ∈[0,1] the following conditions are satisfied: Some examples of t-norms are a * b = max{a+b-1,0} and a * b = min{a,b}. 1] which is left continuous on ℜ, non-decreasing and F(-∞) = 0, F(∞) = 1. If X is a nonempty set, F: X×X → ∆ is called a probabilistic distance on X and F(x,y) is usually detoned by F xy .
The ordered triple (X,F, * ) is called Menger probabilistic metric space (shortly Menger space) if (X,F) is a PM-space, * is a t-norm and the following condition is also satisfies: for all x, y, z ∈ X and t, s > 0, (PM-4) F xy (t+s) ≥ F xz (t) * F zy (s). For every PSM-space (X,F), we can consider the sets of the form U ε,λ = {(x,y) ∈ X×X : F xy (ε) > 1−λ }.
In [13] , it is proved if sup t<0 (t * t) = 1, then U F is a uniformity, called F-uniformity, which is metrizable. The F-topology is generated by the F-uniformity and is determined by the F-convergence:

Definition 4 ( [2] ):
A sequence {x n } in a Menger space (X,F, * ) is called converge to a point x in X (written as If there exists a constant α ∈ (0,1) such that for all t > 0 and n = 1,2..., then {x n } is a Cauchy sequence in X.
for all x, y ∈ X and t > 0, then x = y.
and it is well known that such t-norm is continuous.

RESULTS
Theorem 1: Let {T n }, n = 1, 2, ... be a sequence of mappings of a complete Menger space (X,F, * ) into itself with t * t ≥ t for all t∈ [0,1] and S : X → X be a continuous mapping such that T n (X) ⊆ S(X) and S is commuting with each T n . If there exists a constant α ∈ (0,1) such that for any two mappings T i and T j min{ holds for all x, y ∈ X and 0 < p,q < 1 and 0 ≤ a < 1 such that p+q−a = 1, then there exists a unique common fixed point for all T n and S.
Proof: Let x 0 be an arbitrary point of X and {x n } be a sequence defined by Sx n = T n x n-1 , n = 1,2,… Then for each t > 0 and 0 < α < 1, we have min{ Thus, it follows that min{ ) ( ) ( ), ( Thus, by Lemma 1, {Sx n } is a Cauchy sequence in X. Since X is complete, there exists some u∈X such that Sx n → u. Since Sx n = T n x n-1 , {T n x n-1 } also converges to u. Since S commutes with each T n , using (3.1), we have min{ Using the continuity of S and taking limits on both sides, we have min{ SuT k α ≥ 1 for all α ∈ (0,1) and t > 0.
Therefore Su = T k u for any fixed integer k. Moreover, min { Am. J. Appl. Sci., 4 (6): 371-373, 2007 Taking the limits on both sides, we have min{ Thus, it follows that and t > 0. Therefore u = Su = T k u for any fixed integer k. Thus u is a common fixed point of S and T n for n = 1,2,… For uniquenesses, let v be another common fixed point of S and T n for n = 1,2,… Using (3.1), we have min{ F uv α ≥ 1 for all α ∈ (0,1) and t > 0. Hence, by Lemma 2, u = v. This completes the proof. If we take a = 0 in the main Theorem, we have the following: Corollary 1: Let {T n }, n = 1,2, ... be a sequence of mappings of a complete Menger space (X,F, * ) into itself with t * t ≥ t for all t∈ [0,1] and S : X → X be a continuous mapping such that T n (X) ⊆ S(X) and S is commuting with each T n . If there exists a constant α ∈ (0,1) such that for any two mappings T i and T j min{ SxT j α holds for all x, y ∈ X and 0 < p,q < 1 such that p+q = 1, then there exists a unique common fixed point for all T n and S.
Proof: It is easy to verify from Theorem 1. If we take a = 0 and S = I X (the identity map on X) in the main Theorem, we have the following: α holds for all x, y ∈ X and 0 < p,q < 1 such that p+q = 1, then for any x 0 ∈ X the sequence {x n } = {T n x n-1 }, n = 1, 2,… converges and its limit is the unique common fixed for all T n .