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The purpose of this paper is to present the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\phi ,\varphi )$$\end{document}(ϕ,φ)-contractive mapping and to discuss the relation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}ψ-contractive mappings and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\phi ,\varphi )$$\end{document}(ϕ,φ)-contractive mappings. Furthermore, the generalized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\phi ,\varphi )$$\end{document}(ϕ,φ)-contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}ψ-contraction mapping principle has been obtained by using an ingenious method.

type mappings involving altering distances (say, ψ-contractive mappings) in Menger PMspaces and proved their theorem for such kind of mappings in the setting of G-complete Menger PM-spaces. On contributing to this study, In 2015, Marwan Amin Kutbi et al. weakened the notion of ψ-contractive mapping and establish some fixed point theorems in G-complete and M-complete Menger PM-spaces, besides discussing some related results and illustrative examples.
Next we shall recall some well-known definitions and results in the theory of probabilistic metric spaces which are used later on in this paper. For more details, we refer the reader to Chauhan et al. (2014a, b), Kutbi et al. (2015), Xu et al. (2015a, b), Chauhan and Pant (2014), Su and Zhang (2014), Su et al. (2015).
Definition 1 A triangular norm (shorter T-norm) is a binary operation T on [0, 1] which satisfies the following conditions: The following are the four basic T-norms: It is easy to check, the above four T-norms have the following relations: for any a, b ∈ [0, 1].
Definition 2 A function F (t) : (−∞, +∞) → [0, 1] is called a distance distribution function if it is non-decreasing and left-continuous with lim t→−∞ F (t) = 0, lim t→+∞ F (t) = 1. and F (0) = 0. The set of all distance distribution functions is denoted by D + . A special distance distribution function is given by Definition 3 A Menger probabilistic metric space is a triple (E, F, T) where E is a nonempty set, T is a continuous t-norm and F is a mapping from E × E into D + such that, if F x,y denotes the value of F at the pair (x, y), the following conditions hold: for all x, y ∈ E and t ∈ (−∞, +∞); (MPM-3) F x,y (t + s) ≥ T (F x,z (t), F z,y (s)) for all x, y, z ∈ E and t > 0, s > 0.
(1) A sequence {x n } in E is said to converge to x ∈ E if for any given ε > 0 and > 0 , there exists a positive integer N = N (ε, ) such that F x n ,x (ε) > 1 − whenever n > N.
A sequence {x n } in E is called a Cauchy sequence if for any ε > 0 and > 0, there exists a positive integer N = N (ε, ) such that F x n ,x m (ε) > 1 − , whenever n, m > N .
x n+m (t) = 0 for any given positive integer m and t > 0.
Example Let x n = n i=1 1 i , n = 1, 2, 3, . . .. It is easy to show, for any given m, that as n → ∞. Hence {x n } is a G-Cauchy sequence. But it is not a Cauchy sequence, since x n does not converge.
Definition 5 (Kutbi et al. 2015) A function φ : R + → R + is said to be a φ-function if it satisfies the following conditions: In the sequel, the class of all φ-functions will be denoted by . We denote by the class of all continuous non-decreasing functions ψ : R + → R + such that ψ(0) = 0 and ψ n (a n ) → 0, whenever a n → 0, as n → ∞. Kutbi et al. (2015) proved the two generalized contraction mapping principles for the following so-called ψ-contractive mapping T from a Menger probabilistic metric space (E, F, T) into it-self: where c ∈ (0, 1) and ψ(t), φ(t) are two functions with the suitable conditions. In socalled M-complete Menger probabilistic spaces, they have proved a generalized ψ-contraction mapping principle provided that F is triangular: for every x, y, z ∈ E and each t > 0. The purpose of this paper is to present the definition of (φ, ϕ)-contractive mapping and to discuss the relation of ψ-contractive mappings and (φ, ϕ)-contractive mappings. Furthermore, the generalized (φ, ϕ)-contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized ψ-contraction mapping principle has been obtained by using an ingenious method.
Definition 6 Let (E, F, T) be a Menger probabilistic space and f : E → E be a mapping satisfying the following inequality Definition 7 Let (E, F, T) be a Menger probabilistic space and f : E → E be a mapping satisfying the following inequality where c ∈ (0, 1), φ ∈ �, ψ ∈ � 2 . The mapping f satisfying condition (2) is called (φ, ψ)-contractive mapping.
Theorem 8 Let T be a (φ, ψ)-contractive mapping, then T is also a (φ, ϕ)-contractive mapping, where Proof We rewrite the (2) to the following form which can be rewritten to

That is
This completes the proof.
Theorem 9 Let T be a (φ, ϕ)-contractive mapping, then T is also a (φ, ψ)-contractive mapping, where Proof From the (3), we have We rewrite the (1) to the following form which can be rewritten to That is, This completes the proof.
In this paper, we prove the following contraction mapping principle for the (φ, ϕ)-contractive mappings in a G-complete Menger probabilistic space. Meanwhile, we do not need to add the uniqueness condition of fixed point (see Kutbi et al. 2015).
Theorem 10 Let (E, F, T) be a G-complete Menger probabilistic space and f : E → E be a (φ, ϕ)-contractive mapping. Assume that lim a n →1 ϕ n (a n ) = 1. Then f has a unique fixed point.
Proof For any x 0 ∈ E, we define a sequence {x n } by x n+1 = Tx n for all n ≥ 0. From (1) and the properties of φ and ϕ we know, for all t > 0, that F fx,fy (φ(ct)) ≥ ϕ(F x,y (φ(t))) ∀x, y ∈ E, ∀t > 0. (3) as n → ∞. Let ε > 0 be given, then by using the properties (i) and (iv) of a function φ we can find t > 0 such that ε > φ(t). It follows from (4) that By using the triangle inequality (MPM-3), we obtain Thus, letting n → ∞ and making use of (5), for any integer p, we get Hence {x n } is a G-Cauchy sequence. Since (E, F, T) is G-complete, there exists a point u ∈ E such that x n → u as n → ∞. For any ε > 0, choose φ(t) < ε 2 , we have as n → ∞, which in turn yields that fu = u. Next we show the uniqueness of the fixed point. If there exists v such that fv = v, by using (3) we hvae as n → ∞. It is easy to see u = v. The proof is completed. Kutbi et al. (2015) proved the following fixed point theorem for the (φ, ψ)-contractive mappings in a G-complete Menger probabilistic space. Meanwhile, they need to add the uniqueness condition of fixed point (see Xu et al. 2015). In order to clearly show the content of theorem, we use a clear form to write this theorem.
Theorem 11 (Kutbi et al. 2015) Let (E, F, T) be a G-complete Menger probabilistic space and f : E → E be a (φ, ψ)-contractive mapping. Assume that lim a n →0 ψ n (a n ) = 0 . Then f has a fixed point.
In order to get the uniqueness of fixed point, authors added the following condition: where F(f) denotes the set of all fixed points of a mapping f.
Theorem 12 (Kutbi et al. 2015) Adding condition ( * ) to the hypotheses of Theorem 11, we obtain uniqueness of the fixed point.
By using Theorem 10, we can get the following contraction mapping principle for the (φ, ψ)-contractive mappings in a G-complete Menger probabilistic space.

Theorem 13 Let (E, F, T) be a G-complete
Menger probabilistic space and f : E → E be a (φ, ψ)-contractive mapping. Assume that lim a n →1 ϕ n (a n ) = 1. Then f has a unique fixed point, where Proof From Theorem 8, we know that, T is also a (φ, ϕ)-contractive mapping, where Since lim a n →1 ϕ n (a n ) = 1, by using Theorem 8, we obtain the conclusion. This completes the proof.
Open question 14 Is the following property right?
where If the property (6) is right, then we can obtain the following result.
Theorem 15 Let (E, F, T) be a G-complete Menger probabilistic space and f : E → E be a (φ, ψ)-contractive mapping. Assume that lim a n →0 ψ n (a n ) = 0. Then f has a unique fixed point.
Proof It is not hard to show that, the property (6) is equivalent to the following proposition where a n = 1 b n − 1 and Next, we prove (7). Let ψ n (a n ) = 0, ψ n (a n ) = 0,

then we have
Now we prove Observe Because lim n→∞ ϕ n (b n ) = 1 and lim t→1 A(t) = 0, we have lim a n →0 ψ n (a n ) = 0. Now we prove

Examples
Theorem 17 Let (X, d) be a metric space, f : X → X be a mapping satisfying the following condition: ψ n (a n ) = 0.
Proof (1) We prove (X, F , T 4 ) is a Menger probabilistic metric space. The conditions (MPM-1) and (MPM-2) obviously hold. We prove the condition (MPM-3). For any x, y, z ∈ X and t > 0, s > 0, we claim that If not, we have which is equivalent to Adding the above two inequalities, we get which implies This is a contradiction which implies the condition (MPM-3) holds.
Theorem 18 Let (X, d) be a metric space, f : X → X be a nonexpansive mapping. Let
Proof (1) It is a conclusion of Theorem 17.