# Multi-valued (ψ, φ, ε, λ)-contraction in probabilistic metric space

## Abstract

In this article, we present a new definition of a class of contraction for multi-valued case. Also we prove some fixed point theorems for multivalued (ψ, φ, ε, λ)-contraction mappings in probabilistic metric space.

## 1 Introduction

The class of (ε, λ)-contraction as a subclass of B-contraction 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 [24]. 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 [1113].

Definition 2.1. A mapping T : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (a t-norm) if the following conditions are satisfied:

1. (1)

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

2. (2)

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

3. (3)

ab, cd T(a, c) ≥ T(b, d) a, b, c, d [0, 1];

4. (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 t-norm 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 ${\top }_{i=1}^{n}{x}_{i}=T\phantom{\rule{0.3em}{0ex}}\left({\top }_{i=1}^{n-1}{x}_{i},\phantom{\rule{0.3em}{0ex}}{x}_{n}\right)$ 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. (1)

F (0) = 0,

2. (2)

F is a non-decreasing,

3. (3)

F is left continuous mapping on [0, ∞].

D+ is the subset of Δ+ which limx→∞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 × SD+ (F(p, q) written by F pq for every (p, q) S × S) satisfies the following conditions:

1. (1)

F uv (x) = 1 for every x > 0 u = v (u, v S),

2. (2)

F uv = F vu for every u, v S,

3. (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 t-norm) 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 t-norm T is φ-convergent if

Definition 2.6. A sequence (x n )n Nis 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 Nis called a Cauchy sequence if for every ε > 0 and λ (0, 1) there exists N = N(ε, λ) N such that ${F}_{{x}_{n}{x}_{n+m}}\left(\epsilon \right)>1-\lambda ,\phantom{\rule{0.3em}{0ex}}\forall n\ge N\phantom{\rule{0.3em}{0ex}}\forall m\in ℕ.$

We also have

${x}_{n}{\to }^{F}x⇔{F}_{{x}_{n}x}\left(t\right)\to 1\phantom{\rule{2.77695pt}{0ex}}\forall t>0.$

A probabilistic metric space (S, F, T) is called sequentially complete if every Cauchy sequence is convergent.

In the following, 2S denotes the class of all nonempty subsets of the set S and C(S) is the class of all nonempty closed (in the F-topology) subsets of S.

Definition 2.8 [14]. Let F be a probabilistic distance on S and M 2S. A mapping f: S → 2S is called continuous if for every ε > 0 there exists δ > 0, such that

${F}_{uv}\left(\delta \right)>1-\phantom{\rule{2.77695pt}{0ex}}\delta ⇒\forall x\in fu\phantom{\rule{0.3em}{0ex}}\exists y\in fv\phantom{\rule{2.77695pt}{0ex}}:{F}_{xy}\left(\epsilon \right)>1\phantom{\rule{2.77695pt}{0ex}}-\epsilon .$

Theorem 2.1 [14]. Let (S, F, T) be a complete Menger space, sup 0≤ t < 1T (t, t) = 1 and f : SC(S) be a continuous mapping. If there exist a sequence (t n )nN (0, ∞) with ${\sum }_{1}^{\mathrm{\infty }}{t}_{n}<\mathrm{\infty }$ and a sequence (x n ) nN S with the properties:

${x}_{n+1}\in f{x}_{n}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{all}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}n\phantom{\rule{0.3em}{0ex}}\mathsf{\text{and}}\phantom{\rule{0.3em}{0ex}}\underset{n\to \mathrm{\infty }}{lim}{\top }_{i=1}^{\mathrm{\infty }}{g}_{n+i-1}=1,$

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. (1)

φ is an increasing bijection;

2. (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 limn→∞φn(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: XX is called (ψ, φ, ε, λ)-contraction if for any x, y X, ε > 0 and λ (0, 1).

$M\left(x,y,\epsilon \right)>1\phantom{\rule{2.77695pt}{0ex}}-\lambda ⇒M\left(f\left(x\right),f\left(y\right),\phantom{\rule{0.3em}{0ex}}\psi \left(\epsilon \right)\right)>1-\phi \left(\lambda \right).$

If ψ is of the form of ψ(ε) = (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 → 2S is called a multi-valued (ψ, φ, ε, λ)-contraction if for every x, y S, ε > 0 and for all λ (0, 1) the following implication holds:

${F}_{xy}\left(\epsilon \right)>1\phantom{\rule{2.77695pt}{0ex}}-\lambda ⇒\forall p\in fx\phantom{\rule{0.3em}{0ex}}\exists q\in fy:\phantom{\rule{2.77695pt}{0ex}}{F}_{pq}\left(\psi \left(\epsilon \right)\right)>1-\phi \left(\lambda \right).$

Now, we need to define some conditions on the t-norm 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 → 2S - {}, a mapping f is weakly demicompact if for every sequence (p n )n Nfrom M such that pn+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 < 1T (a, a) = 1, M C(S) and f : MC(M) be a multi-valued (ψ, φ, ε, λ)-contraction, where the series Σψn(ε) is convergent for every ε > 0 and φ ϕ. Let there exists x0 M and x1 fx0 such that ${F}_{{x}_{0}{x}_{1}}\in {D}_{+}$. If f is weakly demicompact or

(1)

then there exists at least one element x M such that x fx.

Proof. Since there exists x0 M and x1 fx0 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 x2 fx1 such that

${F}_{{x}_{2}{x}_{1}}\left(\psi \left(\epsilon \right)\right)>1-\phi \left(\lambda \right)$

Continuing in this way we obtain a sequence (x n )nNfrom M such that for every n ≥ 2, x n fxn-1and

${F}_{{x}_{n},{x}_{n-1}}\left({\psi }^{n-1}\left(\epsilon \right)\right)>1-{\phi }^{n-1}\left(\lambda \right).$
(2)

Since the series Σψn(ε) is convergent we have limn→∞ψn(ε) = 0 and by assumption φ ϕ, so limn→∞φn(λ) = 0. We infer for every ε0 > 0 that

$\underset{n\to \mathrm{\infty }}{lim}{F}_{{x}_{n}{x}_{n-1}}\left({\epsilon }_{0}\right)=1.$
(3)

Indeed, if ε0 > 0 and λ0 (0, 1) are given, and n0 = n0(ε0, λ0) is enough large such that for every nn0, ψn(ε) ≤ ε0 and φn(λ) ≤ λ0 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 )nNis a Cauchy sequence. This means that for every ε1 > 0 and every λ1 (0, 1) there exists n1(ε1, λ1) N such that

${F}_{{x}_{n+p}{x}_{n}}\left({\epsilon }_{1}\right)>1-{\lambda }_{1}$
(4)

for every n1n1(ε1, λ1) and every p N.

Let n2(ε1) 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 n2(ε1) exists. Hence for every p N and every nn2(ε1) we have that

${F}_{{x}_{n+p+1},{x}_{n}}\left({\epsilon }_{1}\right)\phantom{\rule{2.77695pt}{0ex}}\ge \phantom{\rule{2.77695pt}{0ex}}{\top }_{i=1}^{p+1}{F}_{{x}_{n+i},{x}_{n+i-1}}\left({\psi }^{n+i-1}\left(\epsilon \right)\right),$

and (2) implies that

${F}_{{x}_{n+p+1},{x}_{n}}\left({\epsilon }_{1}\right)\phantom{\rule{2.77695pt}{0ex}}\ge \phantom{\rule{2.77695pt}{0ex}}{\top }_{i=1}^{p+1}\left(1-\phantom{\rule{2.77695pt}{0ex}}{\phi }^{n+i-1}\left(\lambda \right)\right)$

for every nn2(ε1) and every p N.

For every p N and nn2(ε1)

${\top }_{i=1}^{p+1}\left(1-\phantom{\rule{2.77695pt}{0ex}}{\phi }^{n+i-1}\left(\lambda \right)\right)\ge \phantom{\rule{2.77695pt}{0ex}}{\top }_{i=1}^{\mathrm{\infty }}\left(1-\phantom{\rule{2.77695pt}{0ex}}{\phi }^{n+i-1}\left(\lambda \right)\right)$

and therefore for every p N and nn2(ε1),

${F}_{{x}_{n+p+1},{x}_{n}}\left({\epsilon }_{1}\right)\ge {\top }_{i=1}^{\mathrm{\infty }}\left(1-\phantom{\rule{2.77695pt}{0ex}}{\phi }^{n+i-1}\left(\lambda \right)\right).$
(5)

From (1) it follows that there exists n3(λ1) N such that

${\top }_{i=1}^{\mathrm{\infty }}\left(1-\phantom{\rule{2.77695pt}{0ex}}{\phi }^{n+i-1}\left(\lambda \right)\right)>1-{\lambda }_{1}$
(6)

for every nn3(λ1). The conditions (5) and (6) imply that (4) holds for n1(ε1, λ1) = max(n2(ε1), n3(λ1)) and every p N. This means that (x n )nNis a Cauchy sequence and since S is complete there exists limn→∞x n . Hence in both cases there exists ${\left({x}_{{n}_{k}}\right)}_{k\in N}$ such that

$\underset{k\to \mathrm{\infty }}{lim}{x}_{{n}_{k}}=x.$

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 ε2 > 0 and λ2 (0, 1) there exists ${b}_{{\epsilon }_{2},{\lambda }_{2}}\in fx$ such that

${F}_{x,{b}_{{\epsilon }_{2},{\lambda }_{2}}}\left({\epsilon }_{2}\right)>1-{\lambda }_{2}.$
(7)

Since supx< 1T(x, x) = 1 for λ2 (0, 1) there exists δ(λ2) (0, 1) such that T(1 - δ(λ2), 1 - δ(λ2)) > 1 - λ2.

If δ'(λ2) is such that

$T\left(1-{\delta }^{\prime }\left({\lambda }_{2}\right),1\phantom{\rule{2.77695pt}{0ex}}-{\delta }^{\prime }\left({\lambda }_{2}\right)\right)>1-\phantom{\rule{2.77695pt}{0ex}}\delta \left({\lambda }_{2}\right)$

and δ''(λ2) = min(δ(λ2), δ'(λ2)) we have that

$\begin{array}{cc}\hfill T\left(1-{\delta }^{″}\left({\lambda }_{2}\right),T\left(\left(1-\phantom{\rule{2.77695pt}{0ex}}{\delta }^{″}\left({\lambda }_{2}\right),1-\phantom{\rule{2.77695pt}{0ex}}{\delta }^{″}\left({\lambda }_{2}\right)\right)\right)& \ge T\left(1-\delta \left({\lambda }_{2}\right),T\left(\left(1-{\delta }^{\prime }\left({\lambda }_{2}\right),1\phantom{\rule{2.77695pt}{0ex}}-\delta \left({\lambda }_{2}\right)\right)\right)\hfill \\ \ge T\left(1-\delta \left({\lambda }_{2}\right),1-\delta \left({\lambda }_{2}\right)\right)\hfill \\ >1\phantom{\rule{2.77695pt}{0ex}}-{\lambda }_{2}.\hfill \end{array}$

Since ${lim}_{k\to \mathrm{\infty }}{x}_{{n}_{k}}=x$ there exists k1 N such that ${F}_{x,{x}_{{n}_{k}}}\left(\frac{\epsilon }{3}\right)>1-{\delta }^{″}\left({\lambda }_{2}\right)$ for every kk1. Let k2 N such that

The existence of such a k2 follows by (3). Let ε R+ be such that $\psi \left(\epsilon \right)<\frac{{\epsilon }_{2}}{3}$ and k3 N such that ${F}_{{x}_{{n}_{k}},x}\left(\epsilon \right)>1-\delta \prime \prime \left({\lambda }_{2}\right)$ for every kk3. Since f is a (ψ, φ, ε, λ)-contraction there exists ${b}_{{\epsilon }_{2},{\lambda }_{2},k}\in fx$ such that

Therefore for every kk3

$\begin{array}{cc}\hfill {F}_{{x}_{{n}_{k+1,}}{b}_{{\epsilon }_{2},{\lambda }_{2},k}}\left(\frac{{\epsilon }_{2}}{2}\right)& \ge {F}_{{x}_{{n}_{k}+1},{b}_{{\epsilon }_{2},{\lambda }_{2},k}}\left(\psi \left(\epsilon \right)\right)\hfill \\ >1-\phi \left({\delta }^{″}\left({\lambda }_{2}\right)\right)\hfill \\ >1-{\delta }^{″}\left({\lambda }_{2}\right)\hfill \end{array}$

If k ≥ max(k1, k2, k3) we have

$\begin{array}{c}{F}_{x,{b}_{{\epsilon }_{2},{\lambda }_{2},k}}\left({\epsilon }_{2}\right)\ge T\left({F}_{x,{x}_{{n}_{k}}}\left(\frac{{\epsilon }_{2}}{3}\right),\phantom{\rule{0.3em}{0ex}}T\left({F}_{{x}_{{n}_{k},}{x}_{{n}_{k}+1}}\left(\frac{{\epsilon }_{2}}{3}\right),\phantom{\rule{0.3em}{0ex}}{F}_{{x}_{{n}_{k}+1},{b}_{{\epsilon }_{2},{\lambda }_{2},k}}\left(\frac{{\epsilon }_{2}}{3}\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}T\left(1-{\delta }^{″}\left({\lambda }_{2}\right),\phantom{\rule{0.3em}{0ex}}T\left(1-{\delta }^{″}\left({\lambda }_{2}\right),\phantom{\rule{0.3em}{0ex}}1-{\delta }^{″}\left({\lambda }_{2}\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}>1-{\lambda }_{2}\end{array}$

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 Σψn(ε) be convergent series, ψ is increasing bijection.

Theorem 3.2. Let (S, F, T) be a complete Menger space with sup 0 ≤ a < 1T (a, a) = 1 and f : SC(S) be a multi-valued (ψ, φ, ε, λ)- contraction.

If there exist p S and q fp such that F pq D+, ψ is increasing bijection and ${lim}_{n\to \mathrm{\infty }}{\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)=1$, 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) > 1-δ 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 p0 = p and p1 = q be in fp0. 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 p2 fp1 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 fpn-1and ${F}_{{p}_{n-1}{p}_{n}}\left({\psi }^{n-1}\left(\epsilon \right)\right)>1-{\phi }^{n-1}\left(\lambda \right)$ for all n ≥ 1. Defining t n = ψn(ε), we have ${g}_{j}={F}_{{p}_{j}{p}_{j+1}}\left({t}_{j}\right)\ge 1-{\phi }^{j}\left(\lambda \right)$, j, so ${lim}_{n\to \mathrm{\infty }}{\top }_{i=1}^{\mathrm{\infty }}{g}_{n+i-1}\ge {lim}_{n\to \mathrm{\infty }}{\top }_{i=1}^{\mathrm{\infty }}\left(1-{\phi }^{n+i-1}\left(\lambda \right)\right)=1.$

On the other hand the sequence (p n ) is a Cauchy sequense, that is:

$\forall \epsilon >0\phantom{\rule{0.3em}{0ex}}\exists {n}_{0}=n0\left(\epsilon \right)\in N:{F}_{{p}_{n}{p}_{n+m}}\left(\epsilon \right)>1-\in ,\forall n\ge {n}_{0},\forall m\in N.$

Suppose that ε > 0, then:

$\underset{n\to \mathrm{\infty }}{lim}{\top }_{i=1}^{\mathrm{\infty }}{g}_{n+i+1}=1⇒\exists {n}_{1}={n}_{\mathsf{\text{1}}}\left(\epsilon \right)\in N\phantom{\rule{2.77695pt}{0ex}}:\phantom{\rule{2.77695pt}{0ex}}{\top }_{i=1}^{m}{g}_{n+i-1}>1-\epsilon ,\phantom{\rule{1em}{0ex}}\forall n\ge {n}_{1}\mathsf{\text{,}}\phantom{\rule{1em}{0ex}}\forall m\in N.$

Since the series ${\sum }_{n=1}^{\mathrm{\infty }}{t}_{n}$ is convergent, there exists n2(= n2(ε)) such that ${\sum }_{n={n}_{2}}^{\mathrm{\infty }}{t}_{n}<\epsilon$.

Let n0 = max{n1, n2}, then for all nn0 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 limn→∞ψn(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 t-norm such that sup 0 ≤ a < 1T (a, a) = 1, M a non-empty and closed subset of S, f : MC(M) be a multi-valued (ψ, φ, ε, λ)- contraction and also weakly demicompact. If there exist x0 M and x1 fx0 such that ${F}_{{x}_{0}{x}_{1}}\in {D}_{+},\psi$ is increasing bijection and limn→∞ψ (λ) = 0 then, f has a fixed point.

Proof. We can construct a sequence (p n )n Nfrom M, such that p1 = x1 fx0, pn+1 fp n . Given t > 0 and λ (0, 1), we will show that

$\underset{n\to \mathrm{\infty }}{lim}{F}_{{p}_{n+1}{p}_{n}}\left(t\right)=1.$
(11)

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 ${F}_{{p}_{n-1}{p}_{n}}\left({\psi }^{n}\left(\eta \right)\right)>1-{\phi }^{n}\left(\xi \right)$ for all n N. By choosing n such that ψn(η) < t and φn(ξ) < λ, we obtain

${F}_{{p}_{n+1}{p}_{n}}\left(t\right)>1-\lambda .$

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 Fx,b(ε,λ)(ε) > 1 - λ. From the condition sup 0 ≤ a < 1T (a, a) = 1 it follows that there exists η(λ) (0, 1) such that

$u>1-\eta \left(\lambda \right)⇒T\left(u,\phantom{\rule{0.3em}{0ex}}u\right)>1-\lambda .$

Let j1(ε, λ) N be such that

Since $x={lim}_{j\to \mathrm{\infty }}{p}_{{n}_{j}}$, such a number j1(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

${F}_{{p}_{{n}_{j}+1},{b}_{j\left(\epsilon \right)}}\left(\frac{\epsilon }{2}\right)>1-\phi \left(\frac{\eta \left(\lambda \right)}{2}\right)>1-\frac{\eta \left(\lambda \right)}{2}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{for}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{every}}\phantom{\rule{0.3em}{0ex}}j\ge {j}_{1}\left(\epsilon ,\lambda \right).$

From (1) it follows that ${lim}_{j\to \mathrm{\infty }}{p}_{{n}_{j}+1}=x$ and therefore, there exists j2(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 jj2(ε, λ). Let j3(ε, λ) = max{j1(ε, λ), j2(ε, λ)}. Then, for every jj3(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 > j3(ε, λ), then, we can choose b(ε, λ) = b j (ε) fx. The proof is complete. □

## References

1. Mihet D: A class of Sehgal's contractions in probabilistic metric spaces. An Univ Vest Timisoara Ser Mat Inf 1999, 37: 105–110.

2. Hadžić O, Pap E: New classes of probabilistic contractions and applications to random operators. In Fixed Point Theory and Application. Edited by: YJ, Cho, JK, Kim, SM, Kong. Nova Science Publishers, Hauppauge, New York; 2003:97–119.

3. Mihet D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst 2004, 144: 431–439. 10.1016/S0165-0114(03)00305-1

4. Mihet D: A note on a paper of Hadzic and Pap. In Fixed Point Theory and Applications. Volume 7. Edited by: YJ, Cho, JK, Kim, SM, Kang. Nova Science Publishers, New York; 2007:127–133.

5. Hadžić O, Pap E: Fixed point theorem for multi-valued probabilistic ψ -contractions. Indian J Pure Appl Math 1994, 25(8):825–835.

6. Pap E, Hadžić O, Mesiar RA: Fixed point theorem in probabilistic metric space and an application. J Math Anal Appl 1996, 202: 433–449. 10.1006/jmaa.1996.0325

7. Hadžić O, Pap E: A fixed point theorem for multivalued mapping in probabilistic Metric space and an application in fuzzy metric spaces. Fuzzy Sets Syst 2002, 127: 333–344. 10.1016/S0165-0114(01)00144-0

8. Žikić-Došenović T: A multivalued generalization of Hicks C-contraction. Fuzzy Sets Syst 2005, 151: 549–562. 10.1016/j.fss.2004.08.011

9. Mihet D: A fixed point theorem in probabilistic metric spaces. The Eighth International Conference on Applied Mathematics and Computer Science, Automat. Comput. Appl. Math 2002, 11(1):79–81. Cluj-Napoca

10. Beitollahi A, Azhdari P: Multi-valued contractions theorems in probabilistic metric space. Int J Math Anal 2009, 3(24):1169–1175.

11. Hadžić O, Pap E: Fixed point theory in PM spaces. Kluwer Academic Publishers, Dordrecht; 2001.

12. Klement EP, Mesiar R, Pap E: Triangular Norm. In Trend in Logic. Volume 8. Kluwer Academic Publishers, Dordrecht; 2000.

13. Schweizer B, Sklar A: Probabilistic Metric Spaces. North-Holland, Amesterdam; 1983.

14. Mihet D: Multi-valued generalization of probabilistic contractions. J Math Anal Appl 2005, 304: 464–472. 10.1016/j.jmaa.2004.09.034

15. Mihet D: A class of contractions in fuzzy metric spaces. Fuzzy Sets Syst 2010, 161: 1131–1137. 10.1016/j.fss.2009.09.018

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Correspondence to Arman Beitollahi.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

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. Multi-valued (ψ, φ, ε, λ)-contraction in probabilistic metric space. Fixed Point Theory Appl 2012, 10 (2012). https://doi.org/10.1186/1687-1812-2012-10