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

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.

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.

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, 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 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 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 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∈ 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 (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

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 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

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 and a sequence (x_n) _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 lim_n→∞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) = kε (k ∈ (0, 1)), one obtains the contractive mapping considered in [3].

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:

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∈ N} from M such that p_n+1∈ fp_n, 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

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 (x_n)_n∈N from M such that for every n ≥ 2, x_n ∈ fx_n-1and

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

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 (x_n)_n∈N is a Cauchy sequence. This means that for every ε₁ > 0 and every λ₁ ∈ (0, 1) there exists n₁(ε₁, λ₁) ∈ N such that

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),

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

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 (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 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

Since sup_{x< 1}T(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 F_pq ∈ 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 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 .

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

On the other hand the sequence (p_n) 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 lim_n→∞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 < 1}T (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 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 p₁ = x₁ ∈ fx₀, p_n+1∈ fp_n. Given t > 0 and λ ∈ (0, 1), we will show that

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 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 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 b_j(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) = b_j(ε)∈ fx. The proof is complete. □

The authors declare that they have no competing interests.

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

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

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

3. Mihet, D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431–439 (2004). Publisher Full Text

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

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

6. Pap, E, Hadžić, O, Mesiar, RA: Fixed point theorem in probabilistic metric space and an application. J Math Anal Appl. 202, 433–449 (1996). Publisher Full Text

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. 127, 333–344 (2002). Publisher Full Text

8. Žikić-Došenović, T: A multivalued generalization of Hicks C-contraction. Fuzzy Sets Syst. 151, 549–562 (2005). Publisher Full Text

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. 11(1), 79–81 Cluj-Napoca (2002)

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

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. Trend in Logic, 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. 304, 464–472 (2005). Publisher Full Text

15. Mihet, D: A class of contractions in fuzzy metric spaces. Fuzzy Sets Syst. 161, 1131–1137 (2010). Publisher Full Text