# 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 . He and other researchers achieved to some interesting results about existence of fixed point in probabilistic and fuzzy metric spaces . Mihet defined the class of (ψ, φ, ε, λ)-contraction in fuzzy metric spaces . On the other hand, Hadzic et al. extended the concept of contraction to the multi valued case . They introduced multi valued (ψ - C)-contraction  and obtained fixed point theorem for multi valued contraction . Also Žikić generalized multi valued case of Hick's contraction . We extended (φ - k) - B contraction which introduced by Mihet  to multi valued case . 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 .

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 . 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 . 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 . 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. 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 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. 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

## Author information

Authors

### Corresponding author

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

## Rights and permissions

Reprints and Permissions

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 