On probabilistic φ contractions in Menger spaces

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(1)A note on probabilistic ϕ-contractions in Menger spaces Valentı́n Gregori1 and Juan-José Miñana2 1. Instituto de Matemàtica Pura y Aplicada, Universitat Politècnica de València, Valencia, SPAIN vgregori@mat.upv.es 2 joesjumi@gmail.com. Abstract In this paper we show that probabilistic ϕ-contractions in the sense of J.X. Fang (see [2]) coincide with the class of probabilistic ϕ-contractions in the sense of J. Jachymski (see [4]). Keywords: Fixed point, Menger space, probabilistic ϕ-contraction 2010 MSC: 54A40, 54D35, 54E50 1. Introduction The problem of obtaining fixed point theorems for probabilistic ϕ-contractions (see Definition 2.4) on Menger spaces has been studied by different authors (see the references in [1]). In the former approaches to this topic the authors used conditions too much restrictive on the gauge function ϕ. Later, Jachymski in [4] obtained a fixed point theorem for Menger spaces, given by a continuous t-norm of Hadžić type, weakening the conditions on ϕ which improves the applicability of these type of theorems. Recently, Fang [2] has obtained a fixed point theorem for a wider class of Menger spaces than Jachymski and also, for weaker conditions on ϕ. In his paper, Fang asserts that he generalizes Jachymski’s theorem in two senses. Indeed, he enunciates his theorem for Menger spaces given by t-norms of H-type (without assuming continuity on the t-norm) and he demands weaker conditions on the gauge function ϕ. In this paper, we will show that Fang’s theorem only generalizes Jachymski’s theorem in one sense, since, as we will see, the class of gauge functions used by Fang is wider than the class used by Jachymski, but probabilistic ϕ-contractions are the same in both senses.. Preprint submitted to Nonlinear Analysis. October 1, 2015.

(2) 2. Preliminaries We begin this section recalling the concepts of distribution function and t-norm. A distribution function F [6] is a non-decreasing left-continuous mapping from the set of real numbers R into [0, 1] such that inf t∈R F (t) = 0 and supt∈R F (t) = 1. We denote by D the set of distribution functions, and D+ the subset of D consisting of those distribution functions F such that F (0) = 0. A continuous t-norm is a binary operator ∗ : [0, 1] × [0, 1] → [0, 1] satisfying the following conditions: (i) ∗ is associative and commutative; (ii) a ∗ 1 = a for each a ∈ [0, 1]; (iii) a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d, for f ora, b, c, d ∈ [0, 1]. Definition 2.1 (Hadžı́c [3]). A t-norm ∗ is called of H-type if the family of m m Q Q functions { ∗t : n ∈ N} is equicontinuous at t = 1, where ∗t = t ∗ · · · ∗ t m-times. A well-known example of t-norm of H-type is the minimum t-norm, which we denote by ∧, i.e. a ∧ b = min{a, b} for all a, b ∈ [0, 1]. The concept of probabilistic metric space was introduced by Menger in [5]. Nowadays, after the study of the axiomatic of probabilistic metric spaces in [6], particularly, on the triangle inequality, the next definition are commonly assumed. Definition 2.2 (Menger [5], Schweizer and Sklar [6]). A triplet (X, F, ∗) is called a Menger probabilistic metric space (for short, Menger space) if X is a non-empty set, ∗ is a t-norm and F is a mapping from X × X into D+ , whose value F (x, y) (denoted by Fx,y ) satisfies the following conditions for x, y ∈ X and t, s ≥ 0: (PM1) Fx,y (t) = 1 for all t > 0 if and only if x = y; (PM2) Fx,y = Fy,x ; (PM3) Fx,z (t + s) ≥ Fx,y (t) ∗ Fy,z (s). 2.

(3) Schweizer et al. proved in [7] that a Menger space (X, F, ∗), where the t-norm satisfies sup{t ∗ t : t ∈]0, 1[} = 1, generates a Hausdorff topology TF on X which has as a base the family B = {Np (, λ) : p ∈ X, λ ∈]0, 1], > 0}, where Np (, λ) = {q ∈ X : Fp,q () > 1 − λ}. Definition 2.3 (Schweizer and Sklar [8]). A sequence {xn } in a Menger space (X, F, ∗) is said to be F-Cauchy if for each > 0 and each λ ∈]0, 1] there is n0 ∈ N such that Fxn ,xm () > 1 − λ for all n, m ≥ n0 . X is called F -complete if every Cauchy sequence in X is convergent with respect to TF . Definition 2.4. Let (X, F, ∗) be a Menger space. A mapping T : X → X is called a probabilistic ϕ-contraction if it satisfies FT x,T y (ϕ(t)) ≥ Fx,y (t) for all x, y ∈ X and t > 0, where ϕ : [0, ∞[→ [0, ∞[ is a gauge function satisfying certain conditions. We will consider three classes of gauge functions: (i) Φ will denote the class of gauge functions satisfying lim ϕn (t) = 0 for all t > 0.. n→∞. (ii) Φw will denote the class of gauge functions satisfying for each t > 0 there exists r ≥ t such that lim ϕnw (r) = 0. n→∞. (iii) Ψ will denote the class of gauge functions satisfying: 0 < ψ(t) < t and lim ψ n (t) = 0 for all t > 0. n→∞. Obviously, Ψ ⊂ ΦΦw and it is easy to verify that these inclusions are strict. Theorem 2.5 (Jachymski [4]). Let (X, F, ∗) be a complete Menger space, where ∗ is a continuous t-norm of H-type. If T : X → X is a probabilistic ψ-contraction for ψ ∈ Ψ, then T has a unique fixed point x0 ∈ X, and the sequence {T n (x)} converges to x0 for all x ∈ X. Theorem 2.6 (Fang [2]). Let (X, F, ∗) be a complete Menger space, where ∗ is a t-norm of H-type. If T : X → X is a probabilistic ϕw -contraction for ϕw ∈ Φw , then T has a unique fixed point x0 ∈ X, and the sequence {T n (x)} converges to x0 for all x ∈ X. 3.

(4) 3. Results Fang asserts in [2] that Theorem 3.1 (in the same paper) is more general than Jachymski’s theorem [4, Theorem 1]. For it, he provided the next example. Example 3.1. ([2, Example 5.1].) Let X = [0, ∞[ and define F : X × X → D+ as follows: t , if |x − y| ≥ t, t+|x−y| F (x, y)(t) = Fx,y (t) = 1, if |x − y| < t. In [2] is showed that (X, F, ∧) is a complete Menger space. x Let T : X → X given by T x = 1+x , and let ϕw :]0, ∞[→]0, ∞[ given by:  t 0 < t < 1,  1+t , t 4 ϕw (t) = − + , 1 ≤ t ≤ 2,  3 4 3 t > 2. t − 3, Also, in [2] is showed that T is a probabilistic ϕw -contraction and ϕw ∈ Φw . Then by Theorem 2.6 we have that T has a unique fixed point x0 ∈ X, and the sequence {T n (x)} converges to x0 for all x ∈ X. Obviously, ϕw in the last example is not included in the class Ψ, as was observed by the author in [2]. By this reason the author asserted that Theorem 2.5 cannot be applied to the last example. But this argument is t for not correct. Indeed, if we consider ψ :]0, ∞[→]0, ∞[ given by ψ(t) = 1+t all t > 0. It is easy to verify that ψ ∈ Ψ and using a similar argument of [2, Example 5.1] one can verify that T is a probabilistic ψ-contraction, for ψ ∈ Ψ. Then, taking into account that ∧ is a continuous t-norm, we can apply Theorem 2.5 to the last example to show that T has a unique fixed point x0 ∈ X, and the sequence {T n (x)} converges to x0 for all x ∈ X. Moreover, in the next, we will show that the class of probabilistic ϕw contractions, for some ϕw ∈ Φw , coincides with the class of probabilistic ψ-contractions, for some ψ ∈ Ψ. We begin showing that the class of probabilistic ϕw -contractions, for some ϕw ∈ Φw , coincides with the class of probabilistic ϕ-contractions, for some ϕ ∈ Φ. Obviously, every probabilistic ϕ-contraction, for some ϕ ∈ Φ, is a probabilistic ϕ-contraction, for (the same) ϕ ∈ Φw , since Φ ⊂ Φw . The next proposition shows that the converse of the last affirmation is also true. 4.

(5) Proposition 3.2. Every probabilistic ϕw -contraction, for some ϕw ∈ Φw , is a probabilistic ϕ-contraction, for some ϕ ∈ Φ. Proof Let (X, F, ∗) be a Menger space and suppose that T : X → X is a probabilistic ϕw -contraction, for some ϕw ∈ Φw . Then, FT x,T y (ϕw (t)) ≥ Fx,y (t) for each x, y ∈ X and t > 0. Denote A := {t > 0 : limn→∞ ϕnw (t) 6= 0}. Now, we will construct ϕ ∈ Φ, such that T is a probabilistic ϕ-contraction, as follows: ϕw (t), if t ∈ / A, ϕ(t) = ϕw (rt ), if t ∈ A, for some rt ≥ t such that limn→∞ ϕnw (rt ) = 0. ϕ is well-defined. Indeed, for each t > 0 we have that t ∈ A or t ∈ / A. Obviously, if t ∈ / A, then ϕ(t) = ϕw (t) and so it is well-defined. Now, if t ∈ A, since ϕw ∈ Φw , we can find rt ≥ t such that limn→∞ ϕnw (rt ) = 0 and so ϕ(t) = ϕw (rt ) is also well-defined. By its definition ϕ ∈ Φ, since limn→∞ ϕn (t) = 0 for each t > 0. Finally, we will see that T is a probabilistic ϕ-contraction for the last ϕ ∈ Φ. Let x, y ∈ X and t > 0. We distinguish two cases: 1. If t ∈ / A, then FT x,T y (ϕ(t)) = FT x,T y (ϕw (t)) ≥ Fx,y (t). 2. If t ∈ A, then FT x,T y (ϕ(t)) = FT x,T y (ϕw (rt )) ≥ Fx,y (rt ) ≥ Fx,y (t). This last inequality is given because rt ≥ t and Fx,y is non-decreasing. Therefore, T is a probabilistic ϕ-contraction for ϕ ∈ Φ. To complete our purpose, we will show that the class of probabilistic ϕ-contractions, for some ϕ ∈ Φ, coincides with the class of probabilistic ψ-contractions, for some ψ ∈ Ψ. As before, it is obvious that every probabilistic ψ-contraction, for some ψ ∈ Ψ, is a probabilistic ψ-contraction, for the same ψ ∈ Φ, since Ψ ⊂ Φ. Now, we will see that the converse of this affirmation is also true. Proposition 3.3. Every probabilistic ϕ-contraction, for some ϕ ∈ Φ, is a probabilistic ψ-contraction, for some ψ ∈ Ψ.. 5.

(6) Proof Let (X, F, ∗) be a Menger space and suppose that T : X → X is a probabilistic ϕ-contraction, for some ϕ ∈ Φ. Then, FT x,T y (ϕ(t)) ≥ Fx,y (t) for each x, y ∈ X and t > 0. Denote B := {t > 0 : ϕ(t) ≥ t}. Let t ∈ B. Then, we can find rt ≥ t such that 0 < ϕ(rt ) < t. Indeed, if for each r ≥ t we have that ϕ(r) ≥ t, given r0 ≥ t we have that ϕ(r0 ) ≥ t and so ϕ2 (r0 ) ≥ t. Then, by induction on n we have that ϕn (r0 ) ≥ t for each n ∈ N and so, limn→∞ ϕn (r0 ) ≥ t, a contradiction. Besides ϕ(rt ) > 0, because if ϕ(rt ) = 0, since T is a probabilistic ϕ-contraction, we have that 0 = FT x,T x (ϕ(rt )) ≥ Fx,x (rt ) = 1, a contradiction. Therefore, 0 < ϕ(rt ) < t. Taking into account the last observation, we will construct a gauge function ψ ∈ Ψ, such that T is a probabilistic ψ-contraction, as follows: ϕ(t), if t ∈ / B, ψ(t) = ϕ(rt ), if t ∈ B, for some rt ≥ t such that 0 < ϕ(rt ) < t.. ψ is well-defined by the above observation. Besides, by its definition ψ ∈ Ψ, since 0 < ψ(t) < t and limn→∞ ψ(t) = 0 for each t > 0. Finally, we will see that T is a probabilistic ψ-contraction for the last ψ ∈ Ψ. Let x, y ∈ X and t > 0. We distinguish two cases: 1. If t ∈ / B, then FT x,T y (ψ(t)) = FT x,T y (ϕ(t)) ≥ Fx,y (t). 2. If t ∈ B, then FT x,T y (ψ(t)) = FT x,T y (ϕ(rt )) ≥ Fx,y (rt ) ≥ Fx,y (t). This last inequality is given because rt ≥ t and Fx,y is non-decreasing. Therefore, T is a probabilistic ψ-contraction for ψ ∈ Ψ. 4. Conclusions In this paper we have shown that the class of probabilistic ϕ-contractions, in the sense of Jachymski, coincide with the class of probabilistic ϕ-contractions in the sense of Fang. Therefore, when we talk about probabilistic ϕ-contractions we refer to anyone of them. Further, this result allows us to assert that Fang only generalized Jachymski’s theorem in one sense, since he didn’t demand continuity in the t-norm. Unfortunately, he didn’t justify, by means of an example, his generalization in this sense. Therefore, an interesting question related to these two theorems is the next one. 6.

(7) Problem 1. Finding a probabilistic ϕ-contraction in a Menger space (X, F, ∗), where ∗ is a t-norm of H-type, and it is not a probabilistic ϕ-contraction in any Menger space defined for a continuous t-norm of H-type. [1] L. Ćirić, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Analysis 72 (2010) 2009-2018. [2] J.X. Fang, On ϕ-contractions in probabilistic and fuzzy metric spaces, Fuzzy Sets and systems 267 (2015) 86-99. [3] O. Hadžı́c, Fixed point theorems for multivalued mappings in probabilistic metric spaces, Fuzzy Sets and Systems 88 (1997) 219-226. [4] J. Jachymski, On probabilistic ϕ-contractions on Menger spaces [5] K. Menger, Statistical metrics, Pro. Natl. Acad. Sci. USA 28 (1942) 535-537. [6] B. Schweizer, A. Sklar, Statistical metric spaces, Pacific Journal of Mathematics 10 (1960) 314-334. [7] B. Schweizer, A. Sklar, E.O. Thorp, The metrization of statistical metric spaces, Pacific Journal of Mathematics 10 (2) (1960) 673-675. [8] B. Schweizer, A. Sklar, Probabilistic metric spaces, North Holland Series in Probability and Applied Mathematics, New York, Amsterdam, Oxford, 1983.. 7.

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