On probabilistic φ-contractions in Menger spaces

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Fuzzy Sets and Systems ••• (••••) •••–•••

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On probabilistic ϕ-contractions in Menger spaces

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Valentín Gregori 1 , Juan-José Miñana, Samuel Morillas

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Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n 46022 Valencia, Spain Received 8 January 2016; received in revised form 14 July 2016; accepted 16 July 2016

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Abstract In this note we show that the fixed point theorems given for Menger spaces by J. Jachymski (2010) [4, Theorem 1] and J.X. Fang (2015) [2, Theorem 3.1] are equivalent. © 2016 Published by Elsevier B.V.

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Keywords: Fixed point; Menger space; Probabilistic ϕ-contraction

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1. Introduction

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The problem of obtaining fixed point theorems for probabilistic ϕ-contractions (see Definition 2.1) in Menger spaces has been studied by different authors (see [1] and the references therein). In the former approaches to this topic the authors used conditions too much restrictive on the gauge function ϕ. Later, Jachymski in [4, Theorem 1] obtained a fixed point theorem for Menger spaces, defined for a continuous t -norm of Hadži´c type, weakening the conditions on ϕ which improved the applicability of these types of theorems. Recently, Fang [2, Theorem 3.1] has obtained a fixed point theorem that generalizes the Jachymski’s theorem (in the context of Menger spaces) in two senses. In fact, Fang does not demand the condition of continuity for the t -norm in his theorem, and, on the other hand, he demands a weaker condition on the gauge function ϕ. In this note, we prove that in the framework of Menger spaces a self-mapping T of X is a probabilistic ϕw -contraction in the sense of Fang if and only if it is a probabilistic ψ-contraction in the sense of Jachymski (Corollary 2.7). Consequently, the mentioned theorems are equivalent.

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Definition 2.1. Let (X, F, ∗) be a Menger space. A mapping T : X → X is called a probabilistic ϕ-contraction, for ϕ, if it satisfies

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2. Results

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E-mail addresses: [email protected] (V. Gregori), [email protected] (J.-J. Miñana), [email protected] (S. Morillas). 1 Valentín Gregori acknowledges the support of Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P. http://dx.doi.org/10.1016/j.fss.2016.07.005 0165-0114/© 2016 Published by Elsevier B.V.

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FT x,T y (ϕ(t)) ≥ Fx,y (t) for all x, y ∈ X and t > 0,

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where ϕ : [0, ∞[ → [0, ∞[ is a gauge function satisfying certain conditions.

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We will consider here three classes of gauge functions:

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(i)  will denote the class of gauge functions ψ satisfying: 0 < ψ(t) < t and lim ψ (t) = 0 for all t > 0.

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n→∞

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(ii)  will denote the class of gauge functions ϕ satisfying:

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lim ϕ n (t) = 0 for all t > 0.

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n→∞

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(iii) w will denote the class of gauge functions ϕw satisfying:

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n for each t > 0 there exists r ≥ t such that lim ϕw (r) = 0.

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n→∞

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Obviously,  ⊂  ⊂ w and it is easy to verify that these inclusions are strict. Recall that a t -norm ∗ is called of H -type (Hadžíc [3]) if the family of functions { m  at t = 1, where ∗t = t ∗ · · · ∗ t m-times.

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∗t : n ∈ N} is equicontinuous

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Theorem 2.3 (Jachymski [4]). Let (X, F, ∗) be a complete Menger space, where ∗ is a t-norm of H -type. If T : X → X is a probabilistic ψ -contraction, for ψ ∈ , then T has a unique fixed point x0 ∈ X, and the sequence {T n (x)} converges to x0 for all x ∈ X.

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Theorem 2.4 (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 ∈ w , then T has a unique fixed point x0 ∈ X, and the sequence {T n (x)} converges to x0 for all x ∈ X.

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Theorem 2.5. Let (X, F, ∗) be a Menger space and let T : X → X be a mapping. Then, T is a probabilistic ϕw -contraction, for some ϕw ∈ w , if and only if T is a probabilistic ϕ-contraction, for some ϕ ∈ .

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Fang asserts in [2] that Theorem 2.4 is more general than Theorem 2.3. In the next, we will see that this assertion is not true showing that both theorems are equivalent. For it we will show that in the framework of Menger spaces the class of probabilistic ϕw -contractions, for some ϕw ∈ w , coincides with the class of probabilistic ψ-contractions, for some ψ ∈  (Corollary 2.7). We start with the next theorem.

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On the other hand the Fang’s theorem is the next one.

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So, the Jachymski’s theorem can be written, really, as follows.

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Remark 2.2. In [4, Theorem 1], J. Jachymski demands that ∗ be a continuous t -norm of H -type. Nevertheless, the author does not use the continuity of ∗ in the proof of this theorem, but the continuity of ∗ at (1, 1). Now, the continuity m  of ∗ at (1, 1) is fulfilled due to the equicontinuity of the family of functions { ∗t : n ∈ N} at t = 1, since ∗ is a t -norm of H -type.

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Proof. Suppose that T : X → X is a probabilistic ϕw -contraction, for some ϕw ∈ w . Then, FT x,T y (ϕw (t)) ≥ Fx,y (t) for each x, y ∈ X and t > 0.   n (t) = 0 . Denote A := t > 0 : limn→∞ ϕw

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n (r) = 0}, which is a non-empty set since ϕ ∈  . For each t ∈ A we consider the set Bt = {r > t : limn→∞ ϕw w w Then, by the Axiom of Choice, we can take an element rt ∈ Bt , for each t ∈ A. Obviously, rt ∈ / A, for each t ∈ A. Now, we will construct a gauge function ϕ ∈ , such that T is a probabilistic ϕ-contraction, for ϕ, as follows. Define  ϕw (t), if t ∈ / A, ϕ(t) = ϕw (rt ), if t ∈ A.

Obviously, ϕ is well-defined. We will prove that limn ϕ n (t) = 0 for all t > 0 in three steps.

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n n n+1 ϕ n+1 (t) = ϕ(ϕ n (t)) = ϕ(ϕw (t)) = ϕw (ϕw (t)) = ϕw (t). n (t) = 0. limn ϕ n (t) = limn ϕw

Therefore, (III) Suppose that t ∈ A. n (r ) for all n ∈ N. The assertion is true for n = 1 by definition We will prove, by induction on n, that ϕ n (t) = ϕw t of ϕ. Suppose the assertion is true for n. By induction hypothesis and by (I ) we have that n n n+1 ϕ n+1 (t) = ϕ(ϕ n (t)) = ϕ(ϕw (rt )) = ϕw (ϕw (t)) = ϕw (t). n (r ) = 0. Therefore, limn ϕ n (t) = limn ϕw t

Thus, limn→∞ ϕ n (t) = 0 for each t > 0, and then ϕ ∈ . Finally, we will see that T is a probabilistic ϕ-contraction for this 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), since rt > t and Fx,y is non-decreasing. Therefore, T is a probabilistic ϕ-contraction, for ϕ ∈ . The converse is obvious since  ⊂ w . 2 To complete our purpose, we will show that the class of probabilistic ϕ-contractions, for ϕ ∈ , coincides with the class of probabilistic ψ -contractions, for ψ ∈ , in the next theorem.

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Theorem 2.6. Let (X, F, ∗) be a Menger space and let T : X → X be a mapping. Then, T is a probabilistic ϕ-contraction, for some ϕ ∈ , if and only if T is a probabilistic ψ -contraction, for some ψ ∈ .

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(II) Suppose that t ∈ / A. n (t) for all n ∈ N. Indeed, ϕ(t) = ϕ (t) since t ∈ We will prove, by induction on n, that ϕ n (t) = ϕw / A. Suppose w the assertion is true for n. By induction hypothesis and by (I ) we have that

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Now, we distinguish two cases:

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i (t) ∈ n (ϕ i (t)) = / A then ϕw / A for all i ∈ N. Indeed, let t ∈ / A and i ∈ N. Then, limn ϕw (I) First we prove that if t ∈ w n+i i limn ϕw (t) = 0, and so ϕw (t) ∈ / A.

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Proof. Let (X, F, ∗) be a Menger space and suppose that T : X → X is a probabilistic ϕ-contraction, for some ϕ ∈ . Then, FT x,T y (ϕ(t)) ≥ Fx,y (t) for each x, y ∈ X and t > 0. We claim that for all t > 0 we can find r > t such that 0 < ϕ(r) < t . Indeed, suppose there exists t0 > 0 such that ϕ(r) ≥ t0 for each r ≥ t0 . Then, for a fixed r0 ≥ t0 we have that ϕ(r0 ) ≥ t0 and so ϕ 2 (r0 ) ≥ t0 . Proceeding inductively on n we obtain ϕ n (r0 ) ≥ t0 for each n ∈ N and so, limn→∞ ϕ n (r0 ) ≥ t0 , a contradiction. Besides, ϕ(r) > 0. Indeed,

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if ϕ(r) = 0 we have that 0 = FT x,T x (ϕ(r)) ≥ Fx,x (r) = 1, since T is a probabilistic ϕ-contraction, a contradiction. Therefore, 0 < ϕ(r) < t. Now, for each t > 0 we consider the set Rt = {r ≥ t : 0 < ϕ(r) < t}, which is a non-empty set, as we have just seen. For each t > 0, we define t + inf{ϕ(r) : r ∈ Rt } ψ(t) = . 2 The function ψ is well-defined, since for each r ∈ Rt we have that ϕ(r) > 0 and clearly ψ(t) > 0, for each t > 0. Next, we will see that ψ ∈  in six steps.

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(II)

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(III)

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(IV) (V)

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t + ϕ(ra ) a + δ + a − δ < = a. 2 2 Choose t > a such that ψ(t) < a. Then, limn ψ n (t) ≥ a, since t > a. On the other hand, {ψ n (t)}n is a strictly decreasing sequence with ψ(t) < a, a contradiction. Therefore, limn ψ n (t) = 0 for each t > 0, and so ψ ∈ .

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Finally we will see that T is a probabilistic ψ -contraction for ψ ∈ . First, note that for all t > 0 we can find rt ∈ Rt such that 0 < ϕ(rt ) < ψ(t). Contrary, suppose that there exists t0 > 0 such that for all r ∈ Rt0 we have that ϕ(r) > ψ(t0 ). Then, t0 + inf{ϕ(r) : r ∈ Rt0 } 2 and so inf{ϕ(r) : r ∈ Rt0 } ≥ t0 . Now, by definition t0 > inf{ϕ(r) : r ∈ Rt0 }, a contradiction. Let x, y ∈ X and t > 0. By the last paragraph we can take rt ∈ Rt such that 0 < ϕ(rt ) < ψ(t). Then, inf{ϕ(r) : r ∈ Rt0 } ≥ ψ(t0 ) =

FT x,T y (ψ(t)) ≥ FT x,T y (ϕ(rt )) ≥ Fx,y (rt ) ≥ Fx,y (t). Therefore, T is a probabilistic ψ -contraction, for ψ ∈ . The converse is obvious since  ⊂ . 2

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t t +t ≤ ψ(t) < = t. 2 2 The function ψ is non-decreasing. Indeed, if we suppose that for some 0 < s < t we have that ψ(s) > ψ(t), then inf{ϕ(r) : r ∈ Rs } > inf{ϕ(r) : r ∈ Rt }. Therefore, we can find rt ∈ Rt such that 0 < ϕ(rt ) < inf{ϕ(r) : r ∈ Rs }. Taking into account that inf{ϕ(r) : r ∈ Rs } < s, then rt ∈ Rs , since rt ≥ t > s and 0 < ϕ(rt ) < s, and hence ϕ(rt ) > inf{ϕ(r) : r ∈ Rs }, a contradiction. The sequence {ψ n (t)}n is strictly decreasing for each t > 0. It is obvious, since for each t > 0 we have that 0 < ψ(t) < t. Consequently: {ψ n (t)}n converges to the infimum of the sequence in [0, ∞[ with respect to the usual topology of R. It is fulfilled that limn ψ n (s) ≤ limn ψ n (t) whenever 0 < s < t . This assertion is obvious since both limits exist and ψ is non-decreasing. We will see that limn ψ n (t) = 0 for each t > 0. Suppose that limn ψ n (t0 ) = a > 0 for some t0 > 0. We assert that limn ψ n (s) ≥ a for each s > a. Indeed, if s > a then we can find ns ∈ N such that ψ i (t0 ) ∈ ]a, s[ for all i ≥ ns . In particular a < ψ ns (t0 ) < s. Then, by (V ) we have that a = limn ψ n (ψ ns (t0 )) ≤ limn ψ n (s). We claim that there exists t > a such that ψ(t) < a. Indeed, given a > 0 we can find, as we have seen above, ra > a such that 0 < ϕ(ra ) < a. Now, note that for each t ∈ ]a, ra [ we have that ra ∈ Rt , since ra ≥ t and 0 < ϕ(ra ) < a < t. Take δ > 0 such that δ < min{a − ϕ(ra ), ra − a}. Then, for each t ∈ ]a, a + δ[ we have 0<

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(I) First, note that for each t > 0 we have that 0 < ψ(t) < t. Indeed, by definition, for each t > 0 we have that

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As a consequence of these two last theorems we obtain the next corollary.

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Corollary 2.7. Let (X, F, ∗) be a Menger space and let T : X → X be a mapping. They are equivalent:

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(i) T is a probabilistic ψ -contraction, for some ψ ∈ . (ii) T is a probabilistic ϕw -contraction, for some ϕw ∈ w .

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References

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[5] [7] [8] [9]

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Uncited references

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The authors are very grateful to the referees for their valuable suggestions.

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Acknowledgements

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Remark 2.8. The reader could find in this note a way of addressing the recent open question posed by Mihet in [6, Open question 1].

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Consequently, Theorem 2.3 and Theorem 2.4 are equivalent.

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´ c, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Anal. 72 (2010) [1] L. Ciri´ 2009–2018. [2] J.X. Fang, On ϕ-contractions in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst. 267 (2015) 86–99. [3] O. Hadžíc, Fixed point theorems for multivalued mappings in probabilistic metric spaces, Fuzzy Sets Syst. 88 (1997) 219–226. [4] J. Jachymski, On probabilistic ϕ-contractions on Menger spaces, Nonlinear Anal. 73 (2010) 2199–2203. [5] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28 (1942) 535–537. [6] D. Mihet, C. Zaharia, On some classes of nonlinear contractions in probabilistic metric spaces, Fuzzy Sets Syst. 300 (2016) 84–92, http://dx.doi.org/10.1016/j.fss.2016.04.005. [7] B. Schweizer, A. Sklar, Statistical metric spaces, Pac. J. Math. 10 (1960) 314–334. [8] B. Schweizer, A. Sklar, E.O. Thorp, The metrization of statistical metric spaces, Pac. J. Math. 10 (2) (1960) 673–675. [9] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North Holland Series in Probability and Applied Mathematics, New York, Amsterdam, Oxford, 1983.

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