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A note on probabilistic φ-contractions in Menger spaces
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A note on probabilistic ϕ-contractions in Menger spaces ✩ Dingwei Zheng a , Xinhe Liu a , Pei Wang b,∗ a College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, PR China b School of Mathematics and Information Science, Yulin Normal University, Yulin, Guangxi 537000, PR China
Received 1 November 2018; received in revised form 10 May 2019; accepted 12 May 2019
Abstract In this note, we prove that each probabilistic (ϕ, ε − λ)-contraction in a complete Menger space (X, F, ∗) with ∗ of H -type is a Picard mapping, where ϕ : [0, ∞) → [0, ∞) be a function with the properties: i) ϕ((0, 1)) ⊂ (0, 1); ii) for each t > 0, there exists r ≥ t such that limn→∞ ϕ n (r) = 0. As a consequence, we answer an open question raised by Mihet and Zaharia affirmatively. © 2019 Elsevier B.V. All rights reserved. Keywords: Fixed point; Probabilistic ϕ-contraction; Probabilistic (ϕ, ε − λ)-contraction; Menger spaces
1. Introduction A large number of probabilistic contractions such as probabilistic ϕ-contractions [2,5], probabilistic (ϕ, ε − λ)-contractions [4] in Menger metric spaces have been defined and studied by several authors. And some fixed point theorems for these probabilistic contractions have been obtained (see [6] and the references therein). For convenience, we only give the notions of probabilistic ϕ-contraction and probabilistic (ϕ, ε − λ)-contraction. Other terminology can be found in [4,6]. Definition 1.1. [2,5] Let (X, F, ∗) be a Menger space. A mapping T : X → X is called a probabilistic ϕ-contraction if it satisfies the following condition: FT x,T y (ϕ(t)) ≥ Fx,y (t), ∀x, y ∈ X, ∀t > 0, where ϕ : [0, ∞) → [0, ∞) is a gauge function. According to [3,6], denote by the class of gauge functions ψ : [0, ∞) → [0, ∞) satisfying: 0 < ψ(t) < t and limn→∞ ψ n (t) = 0 for all t > 0. ✩ This research is supported by NSFC (Nos. 11861018, 11761011) and Guangxi Natural Science Foundation (2018GXNSFAA138171, 2017GXNSFAA198100). * Corresponding author. E-mail addresses: [email protected] (D. Zheng), [email protected] (X. Liu), [email protected] (P. Wang).
https://doi.org/10.1016/j.fss.2019.05.005 0165-0114/© 2019 Elsevier B.V. All rights reserved.
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Denote by the class of gauge functions ϕ : [0, ∞) → [0, ∞) satisfying: limn→∞ ϕ n (t) = 0 for all t > 0. And denote by ω the class of gauge functions ϕ : [0, ∞) → [0, ∞) satisfying: for each t > 0 there exists r ≥ t such that limn→∞ ϕ n (r) = 0. Obviously, ⊂ ⊂ ω and the inclusions are strict. For probabilistic ϕ-contractions, Jachymski in [5] obtained the following result. Theorem 1.2. [5] Let (X, F, ∗) be a complete Menger space with ∗ of H -type. If T is a probabilistic ϕ-contraction with ϕ ∈ , then T is a Picard mapping. In 2015, Fang [2] obtained the following theorem as a generalization of Theorem 1.2. Theorem 1.3. [2] Let (X, F, ∗) be a complete Menger space with ∗ of H -type. If T is a probabilistic ϕ-contraction with ϕ ∈ ω , then T is a Picard mapping. Definition 1.4. [6] Let (X, F, ∗) be a Menger space. A mapping T : X → X is called a probabilistic (ϕ, ε − λ)-contraction if the following implication holds for every ε > 0, λ ∈ (0, 1) and every x, y ∈ X: Fx,y (ε) > 1 − λ ⇒ FT x,T y (ϕ(ε)) > 1 − ϕ(λ), where ϕ : [0, ∞) → [0, ∞) is a gauge function. In 2016, Mihet and Zaharia [6] improved a fixed point result concerning probabilistic (ϕ, ε − λ)-contractions. Theorem 1.5. [6] Let (X, F, ∗) be a complete Menger space with ∗ of H -type and ϕ : [0, ∞) → [0, ∞) be a function with the properties: i) ϕ((0, 1)) ⊂ (0, 1); ii) ϕ ∈ , that is, limn→∞ ϕ n (t) = 0 for all t > 0. Then every probabilistic (ϕ, ε − λ)-contraction is a Picard mapping. Open question 1 in [6]: Does Theorem 1.5 remain true when ϕ ∈ ω ? Just recently, Gregori et al. [3] proved the following theorem which showed that Theorem 1.2 and Theorem 1.3 are equivalent. Theorem 1.6. [3] Let (X, F, ∗) be a Menger space and T : X → X be a mapping. Then T is a ψ -contraction for some ψ ∈ if and only if T is a ϕω -contraction for some ϕω ∈ ω . To obtain Theorem 1.6, the authors proved the following two theorems. Theorem 1.7. [3] Let (X, F, ∗) be a Menger space and T : X → X be a mapping. Then T is a ϕ-contraction for some ϕ ∈ if and only if T is a ϕω -contraction for some ϕω ∈ ω . Theorem 1.8. [3] Let (X, F, ∗) be a Menger space and T : X → X be a mapping. Then T is a ϕ-contraction for some ϕ ∈ if and only if T is a ψ -contraction for some ψ ∈ . Obviously, Theorem 1.6 depends strongly on Theorem 1.7 and Theorem 1.8. The original proof of Theorem 1.8 is constructive, complicated and quite long. Unfortunately, such a proof is not correct, as we will show by means of a counterexample (see Example 2.1). However, Theorem 1.6 remains valid as it was proved, from another point of view, by Alegre and Romaguera in [1]. Although, the authors in [3, Remark 2.8] wrote that the reader could find in that note a way of addressing Open question 1 in [6], we, finally, give an affirmative response to such a question based on the results provided in [1]. 2. Main results In this section, we first give an example.
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Example 2.1. Let X = [0, +∞) withthe metric d(x, y) = |x − y|, then (X, d) is a complete metric space. 0, if t ≤ d(x, y) , then (X, F, ) is a complete Menger space, where is a Let Fx,y (t) = H (t − d(x, y)) = 1, if t > d(x, y) t-norm of H-type. ⎧t if 0 ≤ t ≤ 1 ⎪ 2, ⎪ ⎪ ⎨1, if 1 < t < 2 Now, define T : X → X by T x = x2 and ϕ : [0, +∞) → [0, +∞) by ϕ(t) = . ⎪ 3, if t = 2 ⎪ ⎪ ⎩t if t > 2 2, We notice that ϕ ∈ / since ϕ(2) = 3 > 2, while ϕ ∈ . It is easy to see that T is a probabilistic ϕ-contraction for ϕ ∈ . In fact, 2t ≤ d( x2 , y2 ) ⇔ t ≤ d(x, y) and 2t > x y d( 2 , 2 ) ⇔ t > d(x, y). So FT x,T y ( 2t ) = F x , y ( 2t ) = Fx,y (t). Since ϕ(t) ≥ 2t for all t > 0, we have 2 2 FT x,T y (ϕ(t)) ≥ FT x,T y ( 2t ) = F x , y ( 2t ) = Fx,y (t). 2 2 Then T is a ϕ-contraction for ϕ ∈ . From the example, we can obtain the following two facts. (F1) It is worth noting that for t = 1, we have R1 = {r ≥ 1 : 0 < ϕ(r) < 1} = {1}. That is to say, for t = 1 we can only find one real number r = 1 such that r ≥ 1 and 0 < ϕ(r) < 1. t+3 t+7 t+2n −1 2 3 n (F2) If 1 < t < 2, then we have ψ(t) = t+1 by induction. It is 2 , ψ (t) = 4 , ψ (t) = 8 and ψ (t) = 2n obvious that limn→∞ ψ n (t) = 1 for all 1 < t < 2. From (F1) and (F2), we can see that the proof of Theorem 1.7 in [3] is not correct. However, Theorem 1.6 remains valid as it was proved, from another point of view, by Alegre and Romaguera in [1]. Similarly, one can see that the proof of Theorem 2.2 in [7] is not correct, but Theorem 2.2 in [7] is still valid. One can prove it by a dual method based on [1]. We, finally, give an affirmative response to Open question 1 based on the results provided in [1]. Theorem 2.2. Let (X, F, ∗) be a complete Menger space with ∗ of H -type and ϕ : [0, ∞) → [0, ∞) be a function with the properties: i) ϕ((0, 1)) ⊂ (0, 1); ii) ϕ ∈ ω . Then every probabilistic (ϕ, ε − λ)-contraction is a Picard mapping. Proof. Let (X, F, ∗) be a Menger probabilistic metric space and suppose that T : X → X is a probabilistic (ϕ, ε − λ)-contraction for some ϕ ∈ ω . Then, for every ε > 0, λ ∈ (0, 1) and every x, y ∈ X: Fx,y (ε) > 1 − λ ⇒ FT x,T y (ϕ(ε)) > 1 − ϕ(λ).
(2.1)
Let A = {t > 0| limn→∞ ϕ n (t) = 0}. If t ∈ A, denote by kt the first positive integer number such that ϕ kt (t) < t ≤ ϕ kt −1 (t) (recall that ϕ 0 (t) = t ). If t > 0 and t ∈ / A, take an rt > t such that rt ∈ A, and denote by kt the first positive integer number such that ϕ kt (rt ) < t ≤ ϕ kt −1 (rt ). Define a function ψ : [0, +∞) → [0, +∞) as follows: ψ(0) = 0, ψ(t) = ϕ kt (t) for t ∈ A, and ψ(t) = ϕ kt (rt ) for t > 0 and t ∈ / A. Obviously, 0 < ψ(t) < t for t > 0 and ψ ∈ . Next, we show that T is a probabilistic (ψ, ε − λ)-contraction with ψ ∈ . We distinguish four cases: (i) If ε, λ ∈ A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (ε)) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (λ). By (2.1), we have FT x,T y (ϕ kε (ε)) > 1 − ϕ kλ (λ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). (ii) If ε, λ ∈ / A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (rε )) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (rλ ). By (2.1), we have FT x,T y (ϕ kε (rε )) > 1 − ϕ kλ (rλ ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). (iii) If ε ∈ A, λ ∈ / A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (ε)) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (rλ ). By (2.1), we have FT x,T y (ϕ kε (ε)) > 1 − ϕ kλ (rλ ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). (iv) If ε ∈ / A, λ ∈ A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (rε )) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (λ).
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By (2.1), we have FT x,T y (ϕ kε (rε )) > 1 − ϕ kλ (λ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). From (i)–(iv), we know that T is a probabilistic (ψ, ε − λ)-contraction with ψ ∈ . By Theorem 1.5, it follows that T is a Picard mapping. 2 3. Conclusion In this paper, we firstly construct a counterexample in a complete Menger space through which we point out some mistakes in some literatures. However, the corresponding conclusions remains valid from another point of view, by Alegre and Romaguera in [1]. Finally, we give an affirmative answer to an open question raised by Mihet and Zaharia in [6] based on the techniques provided in [1]. What we need to study further is whether we can use similar techniques to solve another open problem raised by Mihet and Zaharia in [6]. References [1] [2] [3] [4] [5] [6] [7]
C. Alegre, S. Romaguera, A note on ϕ-contractions in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst. 313 (2017) 119–121. J.X. Fang, On ϕ-contractions in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst. 267 (2015) 86–99. V. Gregori, J.J. Miñana, S. Morillas, On probabilistic ϕ-contractions in Menger spaces, Fuzzy Sets Syst. 313 (2017) 114–118. O. Had˘zi´c, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, 2001. J. Jachymski, On probabilistic ϕ-contractions on Menger spaces, Nonlinear Anal. 73 (2010) 2199–2203. D. Mihet, C. Zaharia, On some classes of nonlinear contractions in probabilistic metric spaces, Fuzzy Sets Syst. 300 (2016) 84–92. D.W. Zheng, P. Wang, On probabilistic ψ -contractions in Menger probabilistic metric spaces, Fuzzy Sets Syst. 350 (2018) 107–110.
[m3SC+; v1.300; Prn:15/05/2019; 16:02] P.1 (1-4)
Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
Short communication
A note on probabilistic ϕ-contractions in Menger spaces ✩ Dingwei Zheng a , Xinhe Liu a , Pei Wang b,∗ a College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, PR China b School of Mathematics and Information Science, Yulin Normal University, Yulin, Guangxi 537000, PR China
Received 1 November 2018; received in revised form 10 May 2019; accepted 12 May 2019
Abstract In this note, we prove that each probabilistic (ϕ, ε − λ)-contraction in a complete Menger space (X, F, ∗) with ∗ of H -type is a Picard mapping, where ϕ : [0, ∞) → [0, ∞) be a function with the properties: i) ϕ((0, 1)) ⊂ (0, 1); ii) for each t > 0, there exists r ≥ t such that limn→∞ ϕ n (r) = 0. As a consequence, we answer an open question raised by Mihet and Zaharia affirmatively. © 2019 Elsevier B.V. All rights reserved. Keywords: Fixed point; Probabilistic ϕ-contraction; Probabilistic (ϕ, ε − λ)-contraction; Menger spaces
1. Introduction A large number of probabilistic contractions such as probabilistic ϕ-contractions [2,5], probabilistic (ϕ, ε − λ)-contractions [4] in Menger metric spaces have been defined and studied by several authors. And some fixed point theorems for these probabilistic contractions have been obtained (see [6] and the references therein). For convenience, we only give the notions of probabilistic ϕ-contraction and probabilistic (ϕ, ε − λ)-contraction. Other terminology can be found in [4,6]. Definition 1.1. [2,5] Let (X, F, ∗) be a Menger space. A mapping T : X → X is called a probabilistic ϕ-contraction if it satisfies the following condition: FT x,T y (ϕ(t)) ≥ Fx,y (t), ∀x, y ∈ X, ∀t > 0, where ϕ : [0, ∞) → [0, ∞) is a gauge function. According to [3,6], denote by the class of gauge functions ψ : [0, ∞) → [0, ∞) satisfying: 0 < ψ(t) < t and limn→∞ ψ n (t) = 0 for all t > 0. ✩ This research is supported by NSFC (Nos. 11861018, 11761011) and Guangxi Natural Science Foundation (2018GXNSFAA138171, 2017GXNSFAA198100). * Corresponding author. E-mail addresses: [email protected] (D. Zheng), [email protected] (X. Liu), [email protected] (P. Wang).
https://doi.org/10.1016/j.fss.2019.05.005 0165-0114/© 2019 Elsevier B.V. All rights reserved.
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D. Zheng et al. / Fuzzy Sets and Systems ••• (••••) •••–•••
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Denote by the class of gauge functions ϕ : [0, ∞) → [0, ∞) satisfying: limn→∞ ϕ n (t) = 0 for all t > 0. And denote by ω the class of gauge functions ϕ : [0, ∞) → [0, ∞) satisfying: for each t > 0 there exists r ≥ t such that limn→∞ ϕ n (r) = 0. Obviously, ⊂ ⊂ ω and the inclusions are strict. For probabilistic ϕ-contractions, Jachymski in [5] obtained the following result. Theorem 1.2. [5] Let (X, F, ∗) be a complete Menger space with ∗ of H -type. If T is a probabilistic ϕ-contraction with ϕ ∈ , then T is a Picard mapping. In 2015, Fang [2] obtained the following theorem as a generalization of Theorem 1.2. Theorem 1.3. [2] Let (X, F, ∗) be a complete Menger space with ∗ of H -type. If T is a probabilistic ϕ-contraction with ϕ ∈ ω , then T is a Picard mapping. Definition 1.4. [6] Let (X, F, ∗) be a Menger space. A mapping T : X → X is called a probabilistic (ϕ, ε − λ)-contraction if the following implication holds for every ε > 0, λ ∈ (0, 1) and every x, y ∈ X: Fx,y (ε) > 1 − λ ⇒ FT x,T y (ϕ(ε)) > 1 − ϕ(λ), where ϕ : [0, ∞) → [0, ∞) is a gauge function. In 2016, Mihet and Zaharia [6] improved a fixed point result concerning probabilistic (ϕ, ε − λ)-contractions. Theorem 1.5. [6] Let (X, F, ∗) be a complete Menger space with ∗ of H -type and ϕ : [0, ∞) → [0, ∞) be a function with the properties: i) ϕ((0, 1)) ⊂ (0, 1); ii) ϕ ∈ , that is, limn→∞ ϕ n (t) = 0 for all t > 0. Then every probabilistic (ϕ, ε − λ)-contraction is a Picard mapping. Open question 1 in [6]: Does Theorem 1.5 remain true when ϕ ∈ ω ? Just recently, Gregori et al. [3] proved the following theorem which showed that Theorem 1.2 and Theorem 1.3 are equivalent. Theorem 1.6. [3] Let (X, F, ∗) be a Menger space and T : X → X be a mapping. Then T is a ψ -contraction for some ψ ∈ if and only if T is a ϕω -contraction for some ϕω ∈ ω . To obtain Theorem 1.6, the authors proved the following two theorems. Theorem 1.7. [3] Let (X, F, ∗) be a Menger space and T : X → X be a mapping. Then T is a ϕ-contraction for some ϕ ∈ if and only if T is a ϕω -contraction for some ϕω ∈ ω . Theorem 1.8. [3] Let (X, F, ∗) be a Menger space and T : X → X be a mapping. Then T is a ϕ-contraction for some ϕ ∈ if and only if T is a ψ -contraction for some ψ ∈ . Obviously, Theorem 1.6 depends strongly on Theorem 1.7 and Theorem 1.8. The original proof of Theorem 1.8 is constructive, complicated and quite long. Unfortunately, such a proof is not correct, as we will show by means of a counterexample (see Example 2.1). However, Theorem 1.6 remains valid as it was proved, from another point of view, by Alegre and Romaguera in [1]. Although, the authors in [3, Remark 2.8] wrote that the reader could find in that note a way of addressing Open question 1 in [6], we, finally, give an affirmative response to such a question based on the results provided in [1]. 2. Main results In this section, we first give an example.
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Example 2.1. Let X = [0, +∞) withthe metric d(x, y) = |x − y|, then (X, d) is a complete metric space. 0, if t ≤ d(x, y) , then (X, F, ) is a complete Menger space, where is a Let Fx,y (t) = H (t − d(x, y)) = 1, if t > d(x, y) t-norm of H-type. ⎧t if 0 ≤ t ≤ 1 ⎪ 2, ⎪ ⎪ ⎨1, if 1 < t < 2 Now, define T : X → X by T x = x2 and ϕ : [0, +∞) → [0, +∞) by ϕ(t) = . ⎪ 3, if t = 2 ⎪ ⎪ ⎩t if t > 2 2, We notice that ϕ ∈ / since ϕ(2) = 3 > 2, while ϕ ∈ . It is easy to see that T is a probabilistic ϕ-contraction for ϕ ∈ . In fact, 2t ≤ d( x2 , y2 ) ⇔ t ≤ d(x, y) and 2t > x y d( 2 , 2 ) ⇔ t > d(x, y). So FT x,T y ( 2t ) = F x , y ( 2t ) = Fx,y (t). Since ϕ(t) ≥ 2t for all t > 0, we have 2 2 FT x,T y (ϕ(t)) ≥ FT x,T y ( 2t ) = F x , y ( 2t ) = Fx,y (t). 2 2 Then T is a ϕ-contraction for ϕ ∈ . From the example, we can obtain the following two facts. (F1) It is worth noting that for t = 1, we have R1 = {r ≥ 1 : 0 < ϕ(r) < 1} = {1}. That is to say, for t = 1 we can only find one real number r = 1 such that r ≥ 1 and 0 < ϕ(r) < 1. t+3 t+7 t+2n −1 2 3 n (F2) If 1 < t < 2, then we have ψ(t) = t+1 by induction. It is 2 , ψ (t) = 4 , ψ (t) = 8 and ψ (t) = 2n obvious that limn→∞ ψ n (t) = 1 for all 1 < t < 2. From (F1) and (F2), we can see that the proof of Theorem 1.7 in [3] is not correct. However, Theorem 1.6 remains valid as it was proved, from another point of view, by Alegre and Romaguera in [1]. Similarly, one can see that the proof of Theorem 2.2 in [7] is not correct, but Theorem 2.2 in [7] is still valid. One can prove it by a dual method based on [1]. We, finally, give an affirmative response to Open question 1 based on the results provided in [1]. Theorem 2.2. Let (X, F, ∗) be a complete Menger space with ∗ of H -type and ϕ : [0, ∞) → [0, ∞) be a function with the properties: i) ϕ((0, 1)) ⊂ (0, 1); ii) ϕ ∈ ω . Then every probabilistic (ϕ, ε − λ)-contraction is a Picard mapping. Proof. Let (X, F, ∗) be a Menger probabilistic metric space and suppose that T : X → X is a probabilistic (ϕ, ε − λ)-contraction for some ϕ ∈ ω . Then, for every ε > 0, λ ∈ (0, 1) and every x, y ∈ X: Fx,y (ε) > 1 − λ ⇒ FT x,T y (ϕ(ε)) > 1 − ϕ(λ).
(2.1)
Let A = {t > 0| limn→∞ ϕ n (t) = 0}. If t ∈ A, denote by kt the first positive integer number such that ϕ kt (t) < t ≤ ϕ kt −1 (t) (recall that ϕ 0 (t) = t ). If t > 0 and t ∈ / A, take an rt > t such that rt ∈ A, and denote by kt the first positive integer number such that ϕ kt (rt ) < t ≤ ϕ kt −1 (rt ). Define a function ψ : [0, +∞) → [0, +∞) as follows: ψ(0) = 0, ψ(t) = ϕ kt (t) for t ∈ A, and ψ(t) = ϕ kt (rt ) for t > 0 and t ∈ / A. Obviously, 0 < ψ(t) < t for t > 0 and ψ ∈ . Next, we show that T is a probabilistic (ψ, ε − λ)-contraction with ψ ∈ . We distinguish four cases: (i) If ε, λ ∈ A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (ε)) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (λ). By (2.1), we have FT x,T y (ϕ kε (ε)) > 1 − ϕ kλ (λ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). (ii) If ε, λ ∈ / A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (rε )) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (rλ ). By (2.1), we have FT x,T y (ϕ kε (rε )) > 1 − ϕ kλ (rλ ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). (iii) If ε ∈ A, λ ∈ / A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (ε)) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (rλ ). By (2.1), we have FT x,T y (ϕ kε (ε)) > 1 − ϕ kλ (rλ ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). (iv) If ε ∈ / A, λ ∈ A, suppose that Fx,y (ε) > 1 − λ, then Fx,y (ϕ kε −1 (rε )) ≥ Fx,y (ε) > 1 − λ ≥ 1 − ϕ kλ −1 (λ).
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By (2.1), we have FT x,T y (ϕ kε (rε )) > 1 − ϕ kλ (λ), that is, FT x,T y (ψ(ε)) > 1 − ψ(λ). From (i)–(iv), we know that T is a probabilistic (ψ, ε − λ)-contraction with ψ ∈ . By Theorem 1.5, it follows that T is a Picard mapping. 2 3. Conclusion In this paper, we firstly construct a counterexample in a complete Menger space through which we point out some mistakes in some literatures. However, the corresponding conclusions remains valid from another point of view, by Alegre and Romaguera in [1]. Finally, we give an affirmative answer to an open question raised by Mihet and Zaharia in [6] based on the techniques provided in [1]. What we need to study further is whether we can use similar techniques to solve another open problem raised by Mihet and Zaharia in [6]. References [1] [2] [3] [4] [5] [6] [7]
C. Alegre, S. Romaguera, A note on ϕ-contractions in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst. 313 (2017) 119–121. J.X. Fang, On ϕ-contractions in probabilistic and fuzzy metric spaces, Fuzzy Sets Syst. 267 (2015) 86–99. V. Gregori, J.J. Miñana, S. Morillas, On probabilistic ϕ-contractions in Menger spaces, Fuzzy Sets Syst. 313 (2017) 114–118. O. Had˘zi´c, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, 2001. J. Jachymski, On probabilistic ϕ-contractions on Menger spaces, Nonlinear Anal. 73 (2010) 2199–2203. D. Mihet, C. Zaharia, On some classes of nonlinear contractions in probabilistic metric spaces, Fuzzy Sets Syst. 300 (2016) 84–92. D.W. Zheng, P. Wang, On probabilistic ψ -contractions in Menger probabilistic metric spaces, Fuzzy Sets Syst. 350 (2018) 107–110.