Open Access

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

Fixed Point Theory and Applications20122012:10

https://doi.org/10.1186/1687-1812-2012-10

Received: 28 October 2011

Accepted: 8 February 2012

Published: 8 February 2012

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)-contractionfixed 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 [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 ( x 1 , x 2 , . . . , x n ) [ 0 , 1 ] n ( n 1 ) , i = 1 x i is defined recurrently by i = 1 1 x i = x 1 and i = 1 n x i = T i = 1 n - 1 x i , x n for all n ≥ 2. T can be extended to a countable infinitary operation by defining i = 1 x i for any sequence ( x i ) i N * as lim n 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
δ ( 0 , 1 ) λ ( 0 , 1 ) s = s ( δ , λ ) i = 1 n ( 1 - φ s + i ( δ ) ) > 1 - λ , n 1 .

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 ( ε ) > 1 - λ , n 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 ( ε ) > 1 - λ , n N m .

We also have
x n F x F x n x ( t ) 1 t > 0 .

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 F-topology) 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
F u v ( δ ) > 1 - δ x f u y f v : F x y ( ε ) > 1 - ε .
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 1 t n < and a sequence (x n ) nN S with the properties:
x n + 1 f x n for all n and lim n i = 1 g n + i - 1 = 1 ,

Where g n : = F x n x n + 1 ( t n ) , 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→∞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: XX is called (ψ, φ, ε, λ)-contraction if for any x, y X, ε > 0 and λ (0, 1).
M ( x , y , ε ) > 1 - λ M ( f ( x ) , f ( y ) , ψ ( ε ) ) > 1 - φ ( λ ) .

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 → 2 S 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 x y ( ε ) > 1 - λ p f x q f y : F p q ( ψ ( ε ) ) > 1 - φ ( λ ) .

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 → 2 S - {}, 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 ( ε ) = 1 , for every ε > 0, there exists a convergent subsequence ( p n j ) j 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 Σψ n (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 D + . If f is weakly demicompact or
lim n i = 1 ( 1 - φ n + i - 1 ( ε ) ) = 1 for every ε > 0
(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 D + , hence for every λ (0, 1) there exists ε > 0 such that F x 0 x 1 > 1 - λ . The mapping f is a (ψ, φ, ε, λ)-contraction and therefore there exists x2 fx1 such that
F x 2 x 1 ( ψ ( ε ) ) > 1 - φ ( λ )
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 ( ψ n - 1 ( ε ) ) > 1 - φ n - 1 ( λ ) .
(2)
Since the series Σψ n (ε) is convergent we have limn→∞ψ n (ε) = 0 and by assumption φ ϕ, so limn→∞φ n (λ) = 0. We infer for every ε0 > 0 that
lim n F x n x n - 1 ( ε 0 ) = 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, ψ n (ε) ≤ ε0 and φ n (λ) ≤ λ0 then
F x n + 1 x n ( ε 0 ) F x n + 1 x n ( ψ n ( ε ) ) > 1 - φ n ( λ ) > 1 - λ 0  for every  n n 0 .

If f is weakly demicompact (3) implies that there exists a convergent subsequence ( x n k ) k 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 ( ε 1 ) > 1 - λ 1
(4)

for every n1n1(ε1, λ1) and every p N.

Let n2(ε1) N such that n n 2 ( ε 1 ) ψ n ( ε ) < ε 1 . Since n = 1 ψ n ( ε ) 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 ( ε 1 ) i = 1 p + 1 F x n + i , x n + i - 1 ( ψ n + i - 1 ( ε ) ) ,
and (2) implies that
F x n + p + 1 , x n ( ε 1 ) i = 1 p + 1 ( 1 - φ n + i - 1 ( λ ) )

for every nn2(ε1) and every p N.

For every p N and nn2(ε1)
i = 1 p + 1 ( 1 - φ n + i - 1 ( λ ) ) i = 1 ( 1 - φ n + i - 1 ( λ ) )
and therefore for every p N and nn2(ε1),
F x n + p + 1 , x n ( ε 1 ) i = 1 ( 1 - φ n + i - 1 ( λ ) ) .
(5)
From (1) it follows that there exists n3(λ1) N such that
i = 1 ( 1 - φ n + i - 1 ( λ ) ) > 1 - λ 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 ( x n k ) k N such that
lim k x n k = x .
It remains to prove that x fx. Since f x = f x ¯ it is enough to prove that x f x ¯ i.e., for every ε2 > 0 and λ2 (0, 1) there exists b ε 2 , λ 2 f x such that
F x , b ε 2 , λ 2 ( ε 2 ) > 1 - λ 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 ( 1 - δ ( λ 2 ) , 1 - δ ( λ 2 ) ) > 1 - δ ( λ 2 )
and δ''(λ2) = min(δ(λ2), δ'(λ2)) we have that
T ( 1 - δ ( λ 2 ) , T ( ( 1 - δ ( λ 2 ) , 1 - δ ( λ 2 ) ) ) T ( 1 - δ ( λ 2 ) , T ( ( 1 - δ ( λ 2 ) , 1 - δ ( λ 2 ) ) ) T ( 1 - δ ( λ 2 ) , 1 - δ ( λ 2 ) ) > 1 - λ 2 .
Since lim k x n k = x there exists k1 N such that F x , x n k ε 3 > 1 - δ ( λ 2 ) for every kk1. Let k2 N such that
F x n k , x n k + 1 ε 2 3 > 1 - δ ( λ 2 ) for every k k 2 .
The existence of such a k2 follows by (3). Let ε R+ be such that ψ ( ε ) < ε 2 3 and k3 N such that F x n k , x ( ε ) > 1 - δ ( λ 2 ) for every kk3. Since f is a (ψ, φ, ε, λ)-contraction there exists b ε 2 , λ 2 , k f x such that
F x n k + 1 , b ε 2 , λ 2 , k ( ψ ( ε ) ) > 1 - φ ( δ ( λ 2 ) ) for every k k 3 .
Therefore for every kk3
F x n k + 1 , b ε 2 , λ 2 , k ε 2 2 F x n k + 1 , b ε 2 , λ 2 , k ( ψ ( ε ) ) > 1 - φ ( δ ( λ 2 ) ) > 1 - δ ( λ 2 )
If k ≥ max(k1, k2, k3) we have
F x , b ε 2 , λ 2 , k ( ε 2 ) T F x , x n k ε 2 3 , T F x n k , x n k + 1 ε 2 3 , F x n k + 1 , b ε 2 , λ 2 , k ε 2 3 T ( 1 - δ ( λ 2 ) , T ( 1 - δ ( λ 2 ) , 1 - δ ( λ 2 ) ) ) > 1 - λ 2

and (7) is proved for b ε 2 , λ 2 = b ε 2 , λ 2 , k , k max ( k 1 , k 2 , k 3 ) . Hence x f x ¯ = f x , 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 < 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 i = 1 ( 1 - φ n + i - 1 ( λ ) ) = 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 ( ε ) > 1 - λ .

Using the contraction relation we can find p2 fp1 such that F p 1 p 2 ( ψ ( ε ) ) > 1 - φ ( λ ) , and by induction, p n such that p n fpn-1and F p n - 1 p n ( ψ n - 1 ( ε ) ) > 1 - φ n - 1 ( λ ) for all n ≥ 1. Defining t n = ψ n (ε), we have g j = F p j p j + 1 ( t j ) 1 - φ j ( λ ) , j, so lim n i = 1 g n + i - 1 lim n i = 1 ( 1 - φ n + i - 1 ( λ ) ) = 1 .

On the other hand the sequence (p n ) is a Cauchy sequense, that is:
ε > 0 n 0 = n 0 ( ε ) N : F p n p n + m ( ε ) > 1 - , n n 0 , m N .
Suppose that ε > 0, then:
lim n i = 1 g n + i + 1 = 1 n 1 = n 1 ( ε ) N : i = 1 m g n + i - 1 > 1 - ε , n n 1 , m N .

Since the series n = 1 t n is convergent, there exists n2(= n2(ε)) such that n = n 2 t n < ε .

Let n0 = max{n1, n2}, then for all nn0 and m N we have:
F p n p n + m ( ε ) F p n p n + m i = n n + m - 1 t i i = 1 m F p n + i - 1 p n + 1 ( t n + i - 1 )  =  i = 1 m g n + i - 1 > 1 - ε ,

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→∞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 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 D + , ψ 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
lim n F p n + 1 p n ( t ) = 1 .
(11)
Indeed, since F x 0 x 1 D + , hence for every ξ > 0 there exist η > 0 such that F x 0 x 1 ( η ) > 1 - ξ , and by induction F p n - 1 p n ( ψ n ( η ) ) > 1 - φ n ( ξ ) for all n N. By choosing n such that ψ n (η) < t and φ n (ξ) < λ, we obtain
F p n + 1 p n ( t ) > 1 - λ .

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

By Definition 3.2, there exists a subsequence ( p n j ) j N such that lim j p n j exists. We shall prove that x = lim j p n j is a fixed point of f. Since fx is closed, f x = f x ¯ , and therefore, it remains to prove that x = f x ¯ , 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 - η ( λ ) T ( u , u ) > 1 - λ .
Let j1(ε, λ) N be such that
F p n j , x ψ - 1 ε 2 > 1 - η ( λ ) 2 for every  j j 1 ( ε , λ ) .
Since x = lim j 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 f p n j there exists b j (ε) fx such that
F p n j + 1 , b j ( ε ) ε 2 > 1 - φ η ( λ ) 2 > 1 - η ( λ ) 2 for every j j 1 ( ε , λ ) .

From (1) it follows that lim j p n j + 1 = x and therefore, there exists j2(ε, λ) N such that F x , p n j + 1 ε 2 > 1 - η ( λ ) 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 ( ε ) ( ε ) T F x , p n j + 1 ε 2 , F p n j + 1 , b j ( ε ) ε 2 > 1 - λ . Hence, if j > j3(ε, λ), then, we can choose b(ε, λ) = b j (ε) fx. The proof is complete. □

Declarations

Authors’ Affiliations

(1)
Department of Statistics, Roudehen Branch, Islamic Azad University
(2)
Department of Statistics, North-Tehran Branch, Islamic Azad University

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© Beitollahi and Azhdari; licensee Springer. 2012

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