Some fixed point results in probabilistic Menger space

In this paper, we define the concept of α − β-contractive mapping in probabilistic Menger space and prove some fixed point theorems for such mapping. Some examples are given to support the obtained results.


Introduction and preliminaries
probabilistic metric space were introduced by Menger in 1942, by using the notion of distribution functions in place of non-negative real numbers [5]. Sehgal and Bharucha-Reid proved the probabilistic version of the classical Banach contraction principle for B-contraction mappings in 1972 [8]. After this initial work, the fixed point theory in probabilistic metric spaces has been developed in many works such as [7,12,13,14,15,16,17,18]. The concepts of α − ψ−type contractive and α− admissible mappings were introduced by Gopal et. al. [3], who also established some fixed point theorems for these mappings in complete Menger spaces. After that, Shams and Jafari generalized this concept to (α, β, ψ)-contractive and α − β−admissible mappings and proved some fixed point theorems for such maps [11]. In this paper, we give a generalization of concept of contractive mapping in [11] and introduce the notion of (F, h) − α − β-contractive mapping. Also we compare it with previous results in Menger space and prove some fixed point theorems for these contractive mappings. Our results generalize and improve the previous results in [3] and [11]. We first bring notion, definitions and known results, which are related to our work. For more details, we refer the reader to [4]. Moreover, inf t∈R F (t) = 0 and sup t∈R F (t) = 1. The set of all the distribution functions is denoted by D, and the set of those distribution functions such that F (0) = 0 is denoted by D + . We will denote the specific Heaviside distribution function by: The following are three basic continuous t-norms. (i) The minimum t-norm, say T M , defined by T M (a, b) = min{a, b}.
These t-norms are related in the following way: , where X is a nonempty set, T is a continuous t-norm, and F is a mapping from X × X into D + such that the following conditions hold: for all x, y, z ∈ X and s, t ≥ 0 According to [6], the (ǫ, λ)-topology in Menger space (X, F, T ) is introduced by the family of neighborhoods N x of a point x ∈ X given by The (ǫ, λ) -topology is a Hausdorff topology. In this topology, a function f is continuous in x 0 ∈ X if and only if f (x n ) → f (x 0 ), for every sequence x n → x 0 .
In the sequel, the class of all Φ-functions will be denoted by Φ.
Definition 1.7. [11] Let (X, F, T ) be a Menger space and let f : X → X be a given mapping. We say that f is a generalized α − βcontractive mapping if there exist two functions α, β : , for all x, y ∈ X and for all t > 0, where ϕ ∈ Φ and c ∈ (0, 1). Definition 1.8. [9,10]We say that the function h : Example 1.9. [9,10] Define h : for all x, y, s, t ∈ R + . Then the pair (F, h) is an upper class of type I. Definition 1.12. [9,10] We say that the function h : Example 1.13. [9,10] Define h : for all x, y, z ∈ R + . Then h is a function of subclass of type II.
) for all t > 0, then x = y. Definition 2.2. [3] Let (X, F, T ) be a Menger space and f : X → X be a given mapping. We say that f is a generalized β-type contractive mapping if there exists a function β : for all x, y ∈ X and for all t > 0, where ϕ ∈ Φ and c ∈ (0, 1).

Theorem 2.3. [1] Let (X, F, T ) be a complete Menger space with continuous t-norm T which satisfies
T (a, a) ≥ a for each a ∈ [0, 1]. Let c ∈ (0, 1) be fixed. If for a Φ-function ϕ and a self-mapping f on X, we have ,

2)
for all x, y ∈ X and for all t > 0, then f has a unique fixed point in X.
Now, we introduce the following definition: for all x, y ∈ X and for all t > 0, where pair (F, h) is an upper class of type I, ϕ ∈ Φ and c ∈ (0, 1).

Remark 2.5. If α(x, y, t) = β(x, y, t) = 1 for all x, y ∈ X and for all
x n+1 , t) ≥ 1 for all n ∈ N and for all t > 0, and x n → x as n → ∞, then β(x n , x, t) ≤ 1 and α(x n , x, t) ≥ 1 for all n ∈ N and for all t > 0.
Then f has a fixed point.
Proof. Since T is continuous and T (a, a) ≥ a, for all a ∈ [0, 1], then we have for all b ∈ [0, 1], and we can write F x,y (2t) ≥ min{F x,z (t), F z,y (t)}, for all x, y, z ∈ X. Now, Let x 0 ∈ X be such that (ii) holds and define a sequence {x n } in X such that x n+1 = f x n , for all n ∈ N. First, we suppose x n x n+1 for all n ∈ N, otherwise f has trivially a fixed point.
We shall prove that If we assume that F xn,xn+1 (ϕ( r c )) is the minimum, that from lemma 2.1, we get that x n = x n+1 , which leads to contradiction with the assumption that x n+1 x n and so F xn−1,xn (ϕ( r c )) is the minimum and therefore (2.4) holds true. Since ϕ is strictly increasing, we have that is, F xn,xn+1 (t) ≥ F x0,x1 (ϕ( r c n )) for arbitrary n ∈ N. Next, Let m, n ∈ N with m > n, then by (PM3) we have Since ϕ is strictly increasing and ϕ(t) → ∞ as t → ∞, then for any fixed ǫ ∈ (0, 1), so there exists n 0 ∈ N such that F x0,x1 (ϕ( r c n )) > 1 − ǫ, whenever n ≥ n 0 . This implies that, for every m > n ≥ n 0 , we get F xn,xm ((m − n)t) ≥ 1 − ǫ. Since t > 0 and ǫ ∈ (0, 1) is arbitrary, we deduce that {x n } is a Cauchy sequence in the complete Menger space (X, F, T ). Then, x n → u as n → ∞ for some u ∈ X. We will show that u is a fixed point of f . By (PM3), we have Notice that, if x n = f u for infinitely many values of n, then u = f u and hence the proof finishes. Assume that x n f u for all n ∈ N. Thus, since lim n→∞ x n = u, for any arbitrary ǫ ∈ (0, 1) and n large enough, we get F xn,u (t − ϕ(r)) > 1 − ǫ and hence, we have F u,f u (t) ≥ min{F f u,xn (ϕ(r)), 1 − ǫ}. Since ǫ > 0 is arbitrary, we can write F f u,u (t) ≥ F f u,xn (ϕ(r)). Next, we get Hence we have It follows that Finally, since ǫ ∈ (0, 1) is arbitrary, we have F f u,u (ϕ(r)) ≥ F f u,u (ϕ( r c )) and so, by Lemma 2.1, we deduce that u = f u. This completes the proof. Corollary 2.9. [11] Let (X, F, T ) be a complete Menger space with continuous t-norm T which satisfies T (a, a) ≥ a with a ∈ [0, 1], let f : X → X satisfy the following conditions: α(x n , x n+1 , t) ≥ 1 for all n ∈ N and for all t > 0, and x n → x as n → ∞, then β(x n , x, t) ≤ 1 and α(x n , x, t) ≥ 1 for all n ∈ N and for all t > 0.
Then f has a fixed point. (i) f is a β-type contractive maping, (ii) For any x, y ∈ X and for all t > 0, β(x, y, t)