Abstract
In this article, we present a new definition of a class of contraction for multivalued case. Also we prove some fixed point theorems for multivalued (ψ, φ, ε, λ)contraction mappings in probabilistic metric space.
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1 Introduction
The class of (ε, λ)contraction as a subclass of Bcontraction in probabilistic metric space was introduced by Mihet [1]. He and other researchers achieved to some interesting results about existence of fixed point in probabilistic and fuzzy metric spaces [2–4]. Mihet defined the class of (ψ, φ, ε, λ)contraction in fuzzy metric spaces [4]. On the other hand, Hadzic et al. extended the concept of contraction to the multi valued case [5]. They introduced multi valued (ψ  C)contraction [6] and obtained fixed point theorem for multi valued contraction [7]. Also Žikić generalized multi valued case of Hick's contraction [8]. We extended (φ  k)  B contraction which introduced by Mihet [9] to multi valued case [10]. Now, we will define the class of (ψ, φ, ε, λ)contraction in the sense of multi valued and obtain fixed point theorem.
The structure of article is as follows: Section 2 recalls some notions and known results in probabilistic metric spaces and probabilistic contractions. In Section 3, we will prove three theorems for multi valued (ψ, φ, ε, λ) contraction.
2 Preliminaries
We recall some concepts from the books [11–13].
Definition 2.1. A mapping T : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (a tnorm) if the following conditions are satisfied:

(1)
T (a, 1) = a for every a ∈ [0, 1];

(2)
T (a, b) = T (b, a) for every a, b ∈ [0, 1];

(3)
a ≥ b, c ≥ d ⇒ T(a, c) ≥ T(b, d) a, b, c, d ∈ [0, 1];

(4)
T(T(a, b), c) = T(a, T(b, c)), a, b, c ∈ [0, 1].
Basic examples are, T_{ L }(a, b) = max{a + b  1, 0}, T_{ P } (a, b) = ab and T_{ M } (a, b) = min{a, b}.
Definition 2.2. If T is a tnorm and \left({x}_{1},\phantom{\rule{0.3em}{0ex}}{x}_{2}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{x}_{n}\right)\in {\left[0,1\right]}^{n}\phantom{\rule{0.3em}{0ex}}\left(n\ge 1\right),\phantom{\rule{0.3em}{0ex}}{\top}_{i=1}^{\mathrm{\infty}}{x}_{i} is defined recurrently by {\top}_{i=1}^{1}{x}_{i}={x}_{1} and {\top}_{i=1}^{n}{x}_{i}=T\phantom{\rule{0.3em}{0ex}}\left({\top}_{i=1}^{n1}{x}_{i},\phantom{\rule{0.3em}{0ex}}{x}_{n}\right) for all n ≥ 2. T can be extended to a countable infinitary operation by defining {\top}_{i=1}^{\mathrm{\infty}}{x}_{i} for any sequence {\left({x}_{i}\right)}_{i\in N*}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{as}}\phantom{\rule{0.3em}{0ex}}{lim}_{n\to \mathrm{\infty}}{\top}_{i=1}^{n}{x}_{i}.
Definition 2.3. Let Δ_{+} be the class of all distribution of functions F : [0, ∞] → [0, 1] such that:

(1)
F (0) = 0,

(2)
F is a nondecreasing,

(3)
F is left continuous mapping on [0, ∞].
D_{+} is the subset of Δ_{+} which lim_{x→∞}F(x) = 1.
Definition 2.4. The ordered pair (S, F) is said to be a probabilistic metric space if S is a nonempty set and F : S × S → D_{+} (F(p, q) written by F_{ pq } for every (p, q) ∈ S × S) satisfies the following conditions:

(1)
F_{ uv }(x) = 1 for every x > 0 ⇒ u = v (u, v ∈ S),

(2)
F_{ uv } = F_{ vu } for every u, v ∈ S,

(3)
F_{ uv } (x) = 1 and F_{ vw }(y) = 1 ⇒ F_{ u,w }(x + y) = 1 for every u, v,w ∈ S, and every x, y ∈ R^{+}.
A Menger space is a triple (S, F, T) where (S, F) is a probabilistic metric space, T is a triangular norm (abbreviated tnorm) and the following inequality holds F_{ uv }(x + y) ≥ T (F_{ uw }(x), F_{ wv }(y)) for every u, v, w ∈ S, and every x, y ∈ R^{+}.
Definition 2.5. Let φ : (0, 1) → (0, 1) be a mapping, we say that the tnorm T is φconvergent if
Definition 2.6. A sequence (x_{ n })_{n∈ N}is called a convergent sequence to x ∈ S if for every ε > 0 and λ ∈ (0, 1) there exists N = N(ε, λ) ∈ N such that {F}_{{x}_{n}x}\left(\epsilon \right)>1\lambda ,\phantom{\rule{0.3em}{0ex}}\forall n\ge N.
Definition 2.7. A sequence (x_{ n })_{n∈ N}is called a Cauchy sequence if for every ε > 0 and λ ∈ (0, 1) there exists N = N(ε, λ)∈ N such that {F}_{{x}_{n}{x}_{n+m}}\left(\epsilon \right)>1\lambda ,\phantom{\rule{0.3em}{0ex}}\forall n\ge N\phantom{\rule{0.3em}{0ex}}\forall m\in \mathbb{N}.
We also have
A probabilistic metric space (S, F, T) is called sequentially complete if every Cauchy sequence is convergent.
In the following, 2^{S} denotes the class of all nonempty subsets of the set S and C(S) is the class of all nonempty closed (in the Ftopology) subsets of S.
Definition 2.8 [14]. Let F be a probabilistic distance on S and M ∈ 2^{S}. A mapping f: S → 2^{S} is called continuous if for every ε > 0 there exists δ > 0, such that
Theorem 2.1 [14]. Let (S, F, T) be a complete Menger space, sup _{0≤ t < 1}T (t, t) = 1 and f : S → C(S) be a continuous mapping. If there exist a sequence (t_{ n })_{n∈N}⊂ (0, ∞) with {\sum}_{1}^{\mathrm{\infty}}{t}_{n}<\mathrm{\infty} and a sequence (x_{ n }) _{n∈N}⊂ S with the properties:
Where {g}_{n}:={F}_{{x}_{n}{x}_{n+1}}\left({t}_{n}\right), then f has a fixed point.
The concept of (ψ, φ, ε, λ)  B contraction has been introduced by Mihet [15]. We will consider comparison functions from the class ϕ of all mapping φ : (0, 1) → (0, 1) with the properties:

(1)
φ is an increasing bijection;

(2)
φ (λ) < λ ∀λ ∈ (0, 1).
Since every such a comparison mapping is continuous, it is easy to see that if φ ∈ ϕ, then lim_{n→∞}φ^{n}(λ) = 0 ∀λ ∈ (0, 1).
Definition 2.9[15]. Let (X, M, *) be a fuzzy Metric space. ψ be a map from (0, ∞) to (0, ∞) and φ be a map from (0, 1) to (0, 1). A mapping f: X → X is called (ψ, φ, ε, λ)contraction if for any x, y ∈ X, ε > 0 and λ ∈ (0, 1).
If ψ is of the form of ψ(ε) = kε (k ∈ (0, 1)), one obtains the contractive mapping considered in [3].
3 Main results
In this section we will generalize the Definition 2.9 to multi valued case in probabilistic metric spaces.
Definition 3.1. Let S be a nonempty set, φ ∈ ϕ, ψ be a map from (0, ∞) to (0, ∞) and F be a probabilistic distance on S. A mapping f : S → 2^{S} is called a multivalued (ψ, φ, ε, λ)contraction if for every x, y ∈ S, ε > 0 and for all λ ∈ (0, 1) the following implication holds:
Now, we need to define some conditions on the tnorm T or on the contraction mapping in order to be able to prove fixed point theorem. These two conditions are parallel. If one of them holds, Theorem 3.1 will obtain.
Definition 3.2[11]. Let (S, F) be a probabilistic metric space, M a nonempty subset of S and f : M → 2^{S}  {∅}, a mapping f is weakly demicompact if for every sequence (p_{ n })_{n∈ N}from M such that p_{n+1}∈ fp_{ n }, for every n ∈ N and lim {F}_{{p}_{n+1},{p}_{n}}\left(\epsilon \right)=1, for every ε > 0, there exists a convergent subsequence {\left({p}_{{n}_{j}}\right)}_{j\in \mathsf{\text{N}}}.
The other condition is mentioned in the Theorem 3.1.
Theorem 3.1. Let (S, F, T) be a complete Menger space with sup _{0 ≤ a < 1}T (a, a) = 1, M ∈ C(S) and f : M → C(M) be a multivalued (ψ, φ, ε, λ)contraction, where the series Σψ^{n}(ε) is convergent for every ε > 0 and φ ∈ ϕ. Let there exists x_{0} ∈ M and x_{1} ∈ fx_{0} such that {F}_{{x}_{0}{x}_{1}}\in {D}_{+}. If f is weakly demicompact or
then there exists at least one element x ∈ M such that x ∈ fx.
Proof. Since there exists x_{0} ∈ M and x_{1} ∈ fx_{0} such that {F}_{{x}_{0}{x}_{1}}\in {D}_{+}, hence for every λ ∈ (0, 1) there exists ε > 0 such that {F}_{{x}_{0}{x}_{1}}>1\lambda. The mapping f is a (ψ, φ, ε, λ)contraction and therefore there exists x_{2} ∈ fx_{1} such that
Continuing in this way we obtain a sequence (x_{ n })_{n∈N}from M such that for every n ≥ 2, x_{ n } ∈ fx_{n1}and
Since the series Σψ^{n}(ε) is convergent we have lim_{n→∞}ψ^{n}(ε) = 0 and by assumption φ ∈ ϕ, so lim_{n→∞}φ^{n}(λ) = 0. We infer for every ε_{0} > 0 that
Indeed, if ε_{0} > 0 and λ_{0} ∈ (0, 1) are given, and n_{0} = n_{0}(ε_{0}, λ_{0}) is enough large such that for every n ≥ n_{0}, ψ^{n}(ε) ≤ ε_{0} and φ^{n}(λ) ≤ λ_{0} then
If f is weakly demicompact (3) implies that there exists a convergent subsequence {\left({x}_{{n}_{k}}\right)}_{k\in N}.
Suppose that (1) holds and prove that (x_{ n })_{n∈N}is a Cauchy sequence. This means that for every ε_{1} > 0 and every λ_{1} ∈ (0, 1) there exists n_{1}(ε_{1}, λ_{1}) ∈ N such that
for every n_{1} ≥ n_{1}(ε_{1}, λ_{1}) and every p ∈ N.
Let n_{2}(ε_{1}) ∈ N such that {\sum}_{n\ge {n}_{2}\left({\epsilon}_{1}\right)}{\psi}^{n}\left(\epsilon \right)<{\epsilon}_{1}. Since {\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}\left(\epsilon \right) is convergent series such a natural number n_{2}(ε_{1}) exists. Hence for every p ∈ N and every n ≥ n_{2}(ε_{1}) we have that
and (2) implies that
for every n ≥ n_{2}(ε_{1}) and every p ∈ N.
For every p ∈ N and n ≥ n_{2}(ε_{1})
and therefore for every p ∈ N and n ≥ n_{2}(ε_{1}),
From (1) it follows that there exists n_{3}(λ_{1}) ∈ N such that
for every n ≥ n_{3}(λ_{1}). The conditions (5) and (6) imply that (4) holds for n_{1}(ε_{1}, λ_{1}) = max(n_{2}(ε_{1}), n_{3}(λ_{1})) and every p ∈ N. This means that (x_{ n })_{n∈N}is a Cauchy sequence and since S is complete there exists lim_{n→∞}x_{ n }. Hence in both cases there exists {\left({x}_{{n}_{k}}\right)}_{k\in N} such that
It remains to prove that x ∈ fx. Since fx=\overline{fx} it is enough to prove that x\in \overline{fx} i.e., for every ε_{2} > 0 and λ_{2} ∈ (0, 1) there exists {b}_{{\epsilon}_{2},{\lambda}_{2}}\in fx such that
Since sup_{x< 1}T(x, x) = 1 for λ_{2} ∈ (0, 1) there exists δ(λ_{2}) ∈ (0, 1) such that T(1  δ(λ_{2}), 1  δ(λ_{2})) > 1  λ_{2}.
If δ'(λ_{2}) is such that
and δ''(λ_{2}) = min(δ(λ_{2}), δ'(λ_{2})) we have that
Since {lim}_{k\to \mathrm{\infty}}{x}_{{n}_{k}}=x there exists k_{1} ∈ N such that {F}_{x,{x}_{{n}_{k}}}\left(\frac{\epsilon}{3}\right)>1{\delta}^{\u2033}\left({\lambda}_{2}\right) for every k ≥ k_{1}. Let k_{2} ∈ N such that
The existence of such a k_{2} follows by (3). Let ε ∈ R_{+} be such that \psi \left(\epsilon \right)<\frac{{\epsilon}_{2}}{3} and k_{3} ∈ N such that {F}_{{x}_{{n}_{k}},x}\left(\epsilon \right)>1\delta \prime \prime \left({\lambda}_{2}\right) for every k ≥ k_{3}. Since f is a (ψ, φ, ε, λ)contraction there exists {b}_{{\epsilon}_{2},{\lambda}_{2},k}\in fx such that
Therefore for every k ≥ k_{3}
If k ≥ max(k_{1}, k_{2}, k_{3}) we have
and (7) is proved for {b}_{{\epsilon}_{2},{\lambda}_{2}}={b}_{{\epsilon}_{2},{\lambda}_{2},k},\phantom{\rule{0.3em}{0ex}}k\ge max\left({k}_{1},{k}_{2},{k}_{3}\right). Hence x\in \overline{fx}=fx, which means x is a fixed point of the mapping f.
Now, suppose that instead of Σψ^{n}(ε) be convergent series, ψ is increasing bijection.
Theorem 3.2. Let (S, F, T) be a complete Menger space with sup _{0 ≤ a < 1}T (a, a) = 1 and f : S → C(S) be a multivalued (ψ, φ, ε, λ) contraction.
If there exist p ∈ S and q ∈ fp such that F_{ pq } ∈ D_{+}, ψ is increasing bijection and {lim}_{n\to \mathrm{\infty}}{\top}_{i=1}^{\mathrm{\infty}}\left(1{\phi}^{n+i1}\left(\lambda \right)\right)=1, for every λ ∈ (0, 1), then, f has a fixed point.
Proof. Let ε > 0 be given and δ ∈ (0, 1) be such that δ < min{ε, ψ^{1}(ε)} or ψ(δ) < ε since ψ is increasing bijection. If F_{ uv }(δ) > 1δ then, due to (ψ, φ, ε, λ) contraction for each x ∈ fu we can find y ∈ fv such that F_{ xy }(ψ(δ)) > 1  φ(δ), from where we obtain that F_{ xy }(ε) > F_{ xy }(ψ(δ)) > 1  φ(δ) > 1  δ > 1  ε. So f is continuous. Next, let p_{0} = p and p_{1} = q be in fp_{0}. Since F_{ pq } ∈ D_{+}, hence for every λ ∈ (0, 1) there exist ε > 0 such that F_{ pq }(ε) > 1  λ, namely {F}_{{p}_{0}{p}_{1}}\left(\epsilon \right)>1\lambda.
Using the contraction relation we can find p_{2} ∈ fp_{1} such that {F}_{{p}_{1}{p}_{2}}\left(\psi \left(\epsilon \right)\right)>1\phi \left(\lambda \right), and by induction, p_{ n } such that p_{ n } ∈ fp_{n1}and {F}_{{p}_{n1}{p}_{n}}\left({\psi}^{n1}\left(\epsilon \right)\right)>1{\phi}^{n1}\left(\lambda \right) for all n ≥ 1. Defining t_{ n } = ψ^{n}(ε), we have {g}_{j}={F}_{{p}_{j}{p}_{j+1}}\left({t}_{j}\right)\ge 1{\phi}^{j}\left(\lambda \right), ∀j, so {lim}_{n\to \mathrm{\infty}}{\top}_{i=1}^{\mathrm{\infty}}{g}_{n+i1}\ge {lim}_{n\to \mathrm{\infty}}{\top}_{i=1}^{\mathrm{\infty}}\left(1{\phi}^{n+i1}\left(\lambda \right)\right)=1.
On the other hand the sequence (p_{ n }) is a Cauchy sequense, that is:
Suppose that ε > 0, then:
Since the series {\sum}_{n=1}^{\mathrm{\infty}}{t}_{n} is convergent, there exists n_{2}(= n_{2}(ε)) such that {\sum}_{n={n}_{2}}^{\mathrm{\infty}}{t}_{n}<\epsilon.
Let n_{0} = max{n_{1}, n_{2}}, then for all n ≥ n_{0} and m ∈ N we have:
as desired.
Now we can apply Theorem 2.1 to find a fixed point of f. The theorem is proved. □
When ψ is increasing bijection and lim_{n→∞}ψ^{n}(λ) be zero, by using demicompact contraction we have another theorem.
Theorem 3.3. Let (S, F, T) be a complete Menger space, T a tnorm such that sup _{0 ≤ a < 1}T (a, a) = 1, M a nonempty and closed subset of S, f : M → C(M) be a multivalued (ψ, φ, ε, λ) contraction and also weakly demicompact. If there exist x_{0} ∈ M and x_{1} ∈ fx_{0} such that {F}_{{x}_{0}{x}_{1}}\in {D}_{+},\psi is increasing bijection and lim_{n→∞}ψ (λ) = 0 then, f has a fixed point.
Proof. We can construct a sequence (p_{ n })_{n∈ N}from M, such that p_{1} = x_{1} ∈ fx_{0}, p_{n+1}∈ fp_{ n }. Given t > 0 and λ ∈ (0, 1), we will show that
Indeed, since {F}_{{x}_{0}{x}_{1}}\in {D}_{+}, hence for every ξ > 0 there exist η > 0 such that {F}_{{x}_{0}{x}_{1}}\left(\eta \right)>1\xi, and by induction {F}_{{p}_{n1}{p}_{n}}\left({\psi}^{n}\left(\eta \right)\right)>1{\phi}^{n}\left(\xi \right) for all n ∈ N. By choosing n such that ψ^{n}(η) < t and φ^{n}(ξ) < λ, we obtain
Since t and λ are arbitrary, the proof of (1) is complete.
By Definition 3.2, there exists a subsequence {\left({p}_{{n}_{j}}\right)}_{j\in \mathsf{\text{N}}} such that {lim}_{j\to \mathrm{\infty}}{p}_{{n}_{j}} exists. We shall prove that x=\underset{j\to \mathrm{\infty}}{lim}{p}_{{n}_{j}} is a fixed point of f. Since fx is closed, fx=\overline{fx}, and therefore, it remains to prove that x=\overline{fx}, i.e., for every ε > 0 and λ ∈ (0, 1), there exist b(ε, λ) ∈ fx, such that F_{x,b(ε,λ)}(ε) > 1  λ. From the condition sup _{0 ≤ a < 1}T (a, a) = 1 it follows that there exists η(λ) ∈ (0, 1) such that
Let j_{1}(ε, λ) ∈ N be such that
Since x={lim}_{j\to \mathrm{\infty}}{p}_{{n}_{j}}, such a number j_{1}(ε, λ) exists. Since f is (ψ, φ, ε, λ)contraction and ψ is increasing bijection, for {p}_{{n}_{j}+1}\in f{p}_{{n}_{j}} there exists b_{ j }(ε)∈ fx such that
From (1) it follows that {lim}_{j\to \mathrm{\infty}}{p}_{{n}_{j}+1}=x and therefore, there exists j_{2}(ε, λ) ∈ N such that {F}_{x,{p}_{{n}_{j}+1}}\left(\frac{\epsilon}{2}\right)>1\frac{\eta \left(\lambda \right)}{2} for every j ≥ j_{2}(ε, λ). Let j_{3}(ε, λ) = max{j_{1}(ε, λ), j_{2}(ε, λ)}. Then, for every j ≥ j_{3}(ε, λ) we have {F}_{x,{b}_{j}\left(\epsilon \right)}\left(\epsilon \right)\ge T\left({F}_{x,{p}_{{n}_{j}+1}}\left(\frac{\epsilon}{2}\right),{F}_{{p}_{{n}_{j}+1},{b}_{j\left(\epsilon \right)}}\left(\frac{\epsilon}{2}\right)\right)>1\lambda. Hence, if j > j_{3}(ε, λ), then, we can choose b(ε, λ) = b_{ j }(ε)∈ fx. The proof is complete. □
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PA defined the definitions and wrote the introduction, preliminaries and abstract. AB proved the theorems. AB has approved the final manuscript. Also PA has verified the final manuscript
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Beitollahi, A., Azhdari, P. Multivalued (ψ, φ, ε, λ)contraction in probabilistic metric space. Fixed Point Theory Appl 2012, 10 (2012). https://doi.org/10.1186/16871812201210
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DOI: https://doi.org/10.1186/16871812201210