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

Arman Beitollahi1* and Parvin Azhdari2

Author Affiliations

1 Department of Statistics, Roudehen Branch, Islamic Azad University, Roudehen, Iran

2 Department of Statistics, North-Tehran Branch, Islamic Azad University, Tehran, Iran

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Fixed Point Theory and Applications 2012, 2012:10 doi:10.1186/1687-1812-2012-10

 Received: 28 October 2011 Accepted: 8 February 2012 Published: 8 February 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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.

##### Keywords:
probabilistic metric space; (ψ, φ, ε, λ)-contraction; fixed point

### 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 [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 t-norm) 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, TL(a, b) = max{a + b - 1, 0}, TP (a, b) = ab and TM (a, b) = min{a, b}.

Definition 2.2. If T is a t-norm and is defined recurrently by and for all n ≥ 2. T can be extended to a countable infinitary operation by defining for any sequence .

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 non-decreasing,

(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 × S D+ (F(p, q) written by Fpq for every (p, q) ∈ S × S) satisfies the following conditions:

(1) Fuv(x) = 1 for every x > 0 ⇒ u = v (u, v S),

(2) Fuv = Fvu for every u, v S,

(3) Fuv (x) = 1 and Fvw(y) = 1 ⇒ Fu,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 Fuv(x + y) ≥ T (Fuw(x), Fwv(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 (xn)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

Definition 2.7. A sequence (xn)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

We also have

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

Theorem 2.1 [14]. Let (S, F, T) be a complete Menger space, sup 0≤ t < 1T (t, t) = 1 and f : S C(S) be a continuous mapping. If there exist a sequence (tn)n∈N ⊂ (0, ∞) with and a sequence (xn) n∈N S with the properties:

Where 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 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: 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) = (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:

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 (pn)n∈ N from M such that pn+1fpn, for every n ∈ N and lim , for every ε > 0, there exists a convergent subsequence

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 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 . 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 , hence for every λ ∈ (0, 1) there exists ε > 0 such that . The mapping f is a (ψ, φ, ε, λ)-contraction and therefore there exists x2 fx1 such that

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

(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

(3)

Indeed, if ε0 > 0 and λ0 ∈ (0, 1) are given, and n0 = n0(ε0, λ0) is enough large such that for every n n0, ψn(ε) ≤ ε0 and φn(λ) ≤ λ0 then

If f is weakly demicompact (3) implies that there exists a convergent subsequence .

Suppose that (1) holds and prove that (xn)nN is a Cauchy sequence. This means that for every ε1 > 0 and every λ1 ∈ (0, 1) there exists n1(ε1, λ1) ∈ N such that

(4)

for every n1 n1(ε1, λ1) and every p N.

Let n2(ε1) ∈ N such that Since is convergent series such a natural number n2(ε1) exists. Hence for every p N and every n n2(ε1) we have that

and (2) implies that

for every n n2(ε1) and every p N.

For every p N and n n2(ε1)

and therefore for every p N and n n2(ε1),

(5)

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

(6)

for every n n3(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 (xn)nN is a Cauchy sequence and since S is complete there exists limn→∞xn. Hence in both cases there exists such that

It remains to prove that x fx. Since it is enough to prove that i.e., for every ε2 > 0 and λ2 ∈ (0, 1) there exists such that

(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

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

Since there exists k1 N such that for every k k1. Let k2 N such that

The existence of such a k2 follows by (3). Let ε R+ be such that and k3 N such that for every k k3. Since f is a (ψ, φ, ε, λ)-contraction there exists such that

Therefore for every k k3

If k ≥ max(k1, k2, k3) we have

and (7) is proved for Hence 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 < 1 T (a, a) = 1 and f : S C(S) be a multi-valued (ψ, φ, ε, λ)- contraction.

If there exist p S and q fp such that Fpq D+, ψ is increasing bijection and , 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 Fuv(δ) > 1-δ then, due to (ψ, φ, ε, λ)- contraction for each x fu we can find y fv such that Fxy(ψ(δ)) > 1 - φ(δ), from where we obtain that Fxy(ε) > Fxy(ψ(δ)) > 1 - φ(δ) > 1 - δ > 1 - ε. So f is continuous. Next, let p0 = p and p1 = q be in fp0. Since Fpq D+, hence for every λ ∈ (0, 1) there exist ε > 0 such that Fpq(ε) > 1 - λ, namely .

Using the contraction relation we can find p2 fp1 such that , and by induction, pn such that pn fpn-1and for all n ≥ 1. Defining tn = ψn(ε), we have , ∀j, so

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

Suppose that ε > 0, then:

Since the series is convergent, there exists n2(= n2(ε)) such that .

Let n0 = max{n1, n2}, then for all n n0 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 : M C(M) be a multi-valued (ψ, φ, ε, λ)- contraction and also weakly demicompact. If there exist x0 M and x1 fx0 such that is increasing bijection and limn→∞ψ (λ) = 0 then, f has a fixed point.

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

(11)

Indeed, since , hence for every ξ > 0 there exist η > 0 such that , and by induction for all n ∈ N. By choosing n such that ψn(η) < t and φn(ξ) < λ, we obtain

Since t and λ are arbitrary, the proof of (1) is complete.

By Definition 3.2, there exists a subsequence such that exists. We shall prove that is a fixed point of f. Since fx is closed, , and therefore, it remains to prove that , 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

Let j1(ε, λ) ∈ N be such that

Since , such a number j1(ε, λ) exists. Since f is (ψ, φ, ε, λ)-contraction and ψ is increasing bijection, for there exists bj(ε)∈ fx such that

From (1) it follows that and therefore, there exists j2(ε, λ) ∈ N such that for every j j2(ε, λ). Let j3(ε, λ) = max{j1(ε, λ), j2(ε, λ)}. Then, for every j j3(ε, λ) we have . Hence, if j > j3(ε, λ), then, we can choose b(ε, λ) = bj(ε)∈ fx. The proof is complete. □

### 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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