Fixed point theorems with cyclical contractive conditions in b-Menger spaces

In this work, we prove a fixed point theorems in b-Menger spaces (Mbarki et al., Probabilistic b-metric spaces and nonlinear contractions, Fixed Point Theory Appl. 2017, Paper No. 29, 15 p. (2017)) using probabilistic contraction [13] with cyclical conditions. We support our results by an example.


Introduction
An interesting and important generalization of the notion of metric space was introduced, in 1942, by Menger [11] under the name of statistical metric space, which is now called probabilistic metric space, in this theory, the concept of the distance between two points has a probabilistic nature, i.e., it is exhibited by distribution functions.
Fixed point theory is one of the most famous mathematical theories with application in several branches of sciences, especially in chaos theory, game theory, theory of differential equations etc.The first result from the fixed point theory in probabilistic metric spaces was obtained by Sehgal and Bharucha-Reid [9] in 1972 and their fixed point theorem is further generalized by many authors, for example see [3,5,8,9].
The notion of b-metric spaces, as a generalization of metric spaces, was introduced by Bakhtin [1] in 1989 and an extension of Banach's contraction [2] in these spaces was showed by Czerwik [4].
Recently, Mbarki et al., [10] introduced the probabilistic b-metric spaces (b-Menger spaces), as a generalization of probabilistic metric spaces (Menger spaces) and b-metric spaces; and they studied topological structures and properties and showed the fixed point property for nonlinear contractions in these spaces.
On the other hand, cyclic contractions and cyclic contractive type mapping have appeared in several works.This line of research was initiated, in 2003, by Kirk, Srinivasan and Veeramani [7].
In this paper, we prove the existence and uniqueness of the fixed point for the cyclic probabilistic contraction in b-Menger spaces and we give an example which support the main results.

Preliminaries
We begin by briefly recalling some definitions and notions from probabilistic b-metric spaces theory that we will use in the sequal.For more details, we refer the reader to [10].
A nonnegative real function f defined on R + ∪ {∞} is called a distance distribution function (briefly, a d.d.f) if it is nondecreasing, left continuous on (0, ∞), with f (0) = 0 and f (∞) = 1.The set of all d.d.f's will be noted by ∆ + ; and the set of all F in ∆ + for which lim A commutative, associative and nondecreasing mapping T : As examples we mention the three typical examples of continuous t-norms as follows: number, and the following conditions are satisfied: for all x, y, z ∈ X and α, β > 0, The strong topology [12] in a probabilistic semimetric space (i.e., (1), ( 2) and (3) of Definition 2.1 are satisfied) is introduced by the family of neighborhoods ℘ x of a point x ∈ X given by where In b-Menger space, the convergence of sequence is defined as follows Definition 2.2.Let {p n } be a sequence in a b-Menger space (X, F, T, s).
1.A sequence {p n } is said to be convergent to p ∈ M, if for every > 0, there exists a positive integer N( ) ) is said to be complete if every Cauchy sequence has a limit.
Let f be a self map on X.Power of f at x ∈ X are defined by f 0 x = x and f n+1 x = f ( f n x), n ≥ 0. We will use the notation x n = f n x, in particular x 0 = x, x 1 = f x.
The letter Ψ denotes the set of all function ϕ : [0, ∞) −→ [0, ∞) such that 0 < ϕ(x) < x and lim n−→∞ ϕ n (x) = 0 f or each x > 0. Definition 2.3.[6] We say that a t-norm T is of H-type if the family {T n (t)} is equicontinuous at t = 1, that is, The t-norm T M is a trivial example of t-norm of H-type.Definition 2.4.[10] Let ϕ : [0, ∞) → [0, ∞) be a function such that ϕ(t) < t for t > 0, and f be a selfmap of a probabilistic b-metric space (X, F, τ, s).We say that f is ϕ-probabilistic contraction if for all p, q ∈ X and t > 0, Lemma 2.1.[10] Let (X, F, τ T , s) be a complete probabilistic b-metric space under a continuous t-norm T of H-type such that RanF ⊂ D + .Let f : X −→ X be a ϕ-probabilistic contraction where ϕ ∈ Ψ.Then f has a unique fixed point u, and, for any u ∈ X, lim Befor stating the main fixed point theorems, we need the following concepts.Definition 3.1.Let X be a nonempty set, m a positive integer, and f : X −→ X an operator.Y = m i=1 U i is a cyclic representation with respect to f if 1. U i , i = 1, 2, ..., m are nonempty subsets of X, Definition 3.2.Let (X, F, T, s) be a b-Menger space and f : X −→ X an operator.If 1. Y = m i=1 U i is a cyclic representation with respect to f , 2. F f x f y (γt) ≥ F xy (t), for any x ∈ U i , y ∈ U i+1 , and t > 0 where U m+1 = U 1 and γ ∈ (0, 1 s ).
Then f is called cyclic probabilistic contraction over Y in the b-Menger space (X, F, T, s).
In the proof of our theorems, we use the following lemma: Lemma 3.1.Let (X, F, T, s) be a complete b-Menger space under a continuous t-norm T of H-type such that RanF ⊂ D + .Let U and V be two nonempty closed subsets of X and let f : X −→ X be a cyclic probabilistic contraction over U ∪ V.
Then {x n } is a Cauchy sequence for each x ∈ U ∪ V.
Proof.Let x ∈ U ∪ V, n ∈ N and t > 0. We claim that For each n ∈ N, either x n ∈ U and x n+1 ∈ V or, x n ∈ V and x n+1 ∈ U. Then we have Next, we show that for each t > 0, n ≥ 0 Since T is a t-norm of H-type, then for arbitrary ∈ (0, 1), there exists θ = θ( ) ∈ (0, 1) such that if We have And since It follows that there exists N ∈ N such that So Hence {x n } is a Cauchy sequence in U ∪ V.
We now state our main results: Theorem 3.1.Let (X, F, T, s) be a complete b-Menger space under a continuous t-norm T of H-type such that RanF ⊂ D + .Let U and V be two nonempty closed subsets of X and let f : X −→ X be a cyclic probabilistic contraction over U ∪ V.

Conclusion
In this paper, we presented the notion of cyclic probabilistic contraction and proved the existence and uniqueness of fixed point for this type mapping in b-Menger space.An example is constructed to support our results.

Remark 2 . 1 .
In b-Menger spaces we have the following assertions[10] :(a) (M, F, T, s) is endowed with the strong topology is a Hausdorff topological space provided that T is continuous.(b)The function F is in general not continuous.

Proof.
For x ∈ U ∪ V.By Lemma 3.1, {x n } is a Cauchy sequence.Consequently {x n } converges to some point y ∈ X.The subsequences {x 2n } and {x 2n+1 } of {x n } also converge to y.Now, either {x 2n } ⊂ U and {x 2n+1 } ⊂ V, or {x 2n } ⊂ V and {x 2n+1 } ⊂ U, and since U and V are closed, then y ∈ U ∩ V, so U ∩ V = ∅.The condition (2) of Definition 3.1 implies that f : U ∩ V −→ U ∩ V and the condition (2) of Definition 3.2 implies that f restricted to U ∩ V is a probabilistic contraction mapping.By Lemma 2.1 with ϕ(t) = γst, because γs ∈ (0, 1), f has a unique fixed point in U ∩ V.
xy (t) f or all x ∈ U 1 and y ∈ U 2 .