A System of Nonlinear Operator Equations for a Mixed Family of Fuzzy and Crisp Operators in Probabilistic Normed Spaces

Abstract

By using a random version of the theory of contractor introduced by Altman, we introduce and study a system of nonlinear operator equations for a mixed family of fuzzy and crisp operators in probabilistic normed spaces. We construct some new iterative algorithms for solving this kind of nonlinear operator equations. We also prove some new existence theorems of solutions of a new system of nonlinear operator equations for a mixed family of fuzzy and crisp operators and some new convergence results of sequences generated by iterative algorithms under joint orbitally complete conditions.

1. Introduction

Altman [1, 2] introduced the theory of contractor and contractor direction, which has a very strong significant for the study of existence and uniqueness for solving nonlinear operator equations in Banach spaces. The theory of contractor offers a unified approach to a very large class of iterative methods including the most important ones. Chang [3] introduced the concept of probabilistic contractor and studied the existence and uniqueness of solution for nonlinear operator equations with probabilistic contractor in Menger PN-spaces. By using the theory of countable extension of -norms [4–6] and the results from [7, 8], many results for the more general classes of -norms have been proved (see [9] and the references therein).

On the other hand, since then, several kinds of variational inequalities, variational inclusions, complementarity problems, and nonlinear equations with fuzzy mappings were introduced and studied by many authors (see, e.g., [8–15]). Sharma et al. [15] considered two nonfuzzy mappings and a sequence of fuzzy mappings to define a hybrid -compatible condition. They also showed the existence of common fixed points under such condition, where the range of the one of the two nonfuzzy mappings is joint orbitally complete. Furthermore, Cho et al. [10] introduced the concept of probabilistic contractor couple in probabilistic normed spaces and discuss the solution for nonlinear equations of fuzzy mappings and the convergence of sequences generated by the algorithms in Menger probabilistic normed spaces. Very recently, Hadžić and Pap [16] introduced some new classes of probabilistic contractions in probabilistic metric spaces. They also obtained a new fixed point theorem for the -probabilistic contraction and gave some applications to random operators.

Motivated and inspired by the above works, in this paper, by using a random version of the theory of contractor introduced by Altman, we introduce and study a system of nonlinear operator equations for a mixed family of fuzzy and crisp operators in probabilistic normed spaces. We construct some new iterative algorithms for solving this kind of nonlinear operator equations. We also prove some new existence theorems of solution for the system of nonlinear operator equations for a mixed family of fuzzy and crisp operators and new convergence results of sequences generated by the iterative algorithms under joint orbitally complete condition. The results presented in this paper improve and generalize corresponding results of [9, 15–17].

2. Preliminaries

Let be the set of all distribution functions such that ( is a nondecreasing, left continuous mapping from into such that The special distribution function is defined by

(2.1)

The ordered pair is said to be a probabilistic metric space if is a nonempty set and is written by for all satisfying the following conditions:

(i) for all for all

(ii) for all

(iii) and for all and

A Menger space (see [18]) is a triple where is a probabilistic metric space, is a triangular norm (abbreviated -norm), and the following inequality holds:

(2.2)

Recall that a mapping is a triangular norm (shortly, a -norm) if the following conditions are satisfied:

(i) for all ;

(ii) for all ;

(iii)if and , then for all ;

(iv) for all .

Example 2.1.

The following are the four basic examples.

(1)The minimum -norm is defined by

(2.3)

(2)The product -norm is defined by

(2.4)

(3)The Lukasiewicz -norm is defined by

(2.5)

(4)The weakest -norm, the drastic product , is defined by

(2.6)

The -topology in is introduced by the family of neighbourhoods , where

(2.7)

If a -norm is such that then is a metrizable topological space under the -topology.

Let be a vector space over the real or complex number field , , and a -norm. The ordered triple is a Menger probabilistic normed space (briefly, a Menger PN-space) if and only if the following conditions are satisfied, where for all :

(i) for all and ( is a neutral element for in );

(ii) for all and ();

(iii) for all and .

If is a Menger PN-space and, for all , is defined by

(2.8)

then is a Menger space.

The following definition can be found in [7].

Definition 2.2 (see [7]).

is called a non-Archimedean Menger PN-space (briefly, an N.A. Menger PN-space) if is a Menger PN-space satisfying the following condition:

(2.9)

If is a Menger PN-space and is a -norm which satisfies the condition:

(2.10)

then is a Hausdorff topological vector space in the topology induced by the base of neighborhoods of

(2.11)

where

Example 2.3 (see [9]).

It is easy to see that an ultra-metric space belongs to the class of N.A. Menger PN-spaces, where and satisfying the following conditions:

(i) for all ;

(ii) for all ;

(iii) for all .

A fuzzy set in is a function from into . If , then the function value is called the grade of membership of in . The -level set of , denoted by , is defined by

(2.12)

Let denote the collection of all fuzzy sets in such that is compact and convex for all and . For any , means for all .

Let be an arbitrary set and any linear metric space. A function is called fuzzy operator. Now, we define an orbit for mixed operators and a joint orbitally complete space as follows.

Definition 2.4 (see [15]).

Let be two operators from into itself and a sequence of fuzzy operators from into . If, for some , there exist sequences and in such that

(2.13)

then is called an orbit for the mixed operators .

Definition 2.5 (see [15]).

is called -joint orbitally complete if every Cauchy sequence of each orbit at is convergent in .

Remark 2.6 (see [15]).

Clearly, if is an any complete space and , then is -joint orbitally complete, while the converse is not necessarily true.

3. Some Countable -Norms

Let be a -norm and, for each , and a mapping let defined in the following way:

(3.1)

A -norm is of -type if the family is equicontinuous at (see [19]).

Each -norm can be extended (by the associativity) in a unique way to the -ary operation taking the values for any , which is defined by

(3.2)

A -norm can be extended to countable infinitely operations taking the value

(3.3)

for any sequence in . Also, the sequence is nonincreasing and bounded from below and hence the limit exists.

By (3.3) and fixed point theory in the book by Hadžić and Pap [4], it is interested to investigate the classes of -norms and sequences in the interval such that and

(3.4)

It is obvious that

(3.5)

for and .

The important classes of -norms are given in the following example.

Example 3.1 (see [16]).

() The Dombi family of -norms is defined by

(3.6)

 () The Aczél-Alsina family of -norms is defined by

(3.7)

  () The family of Sugeno-Weber -norms is given by

(3.8)

  () The Schweizer-Sklar family of -norms is defined by

(3.9)

  The condition is fulfilled by the families , .

There exists a member of the family which is incomparable with and there exists a member of the family which is incomparable with .

In [4], the following results and proposition are obtained.

(1)If is the Dombi family of -norms and is a sequence of elements from such that , then we have the following equivalence:

(3.10)

(2)If is the Sugeno-Weber family of -norms and is a sequence of elements from such that , then we have the following equivalence:

(3.11)

(3)The equivalence (3.10) holds also for the family , that is,

(3.12)

Proposition 3.2.

Let be a sequence of numbers from such that and a -norm of -type. Then

4. The Main Results

Let be a Menger PN-space with the -norm satisfying condition and a nonempty subset of . If , where

(4.1)

then is called a probabilistically bounded set. Let be the collection of all nonempty closed probabilistically bounded subsets of . For any , define the distribution functions and by

(4.2)

(4.3)

respectively.

Let be two fuzzy operators satisfying the following condition (I).

(I)There exist two mappings such that, for all , the set and .

We note that

(4.4)

where is a real number and is a fuzzy set in decided by the fuzzy operator at . By using each pair of fuzzy operators and , we can define two set-valued mappings and as follows:

(4.5)

In the sequel, for some , and are called the set-valued mappings induced by the fuzzy mappings and , respectively.

We need the following lemma and definitions.

Lemma 4.1 (see [7]).

Let be a Menger PN-space with a -norm satisfying and let . Then we have the following.

(1).

(2) for all if and only if .

(3) for all with .

(4)If , then we have for all

Definition 4.2.

Let and be two Menger PN-spaces. A set-valued mapping is said to be -closed if, for any and , whenever and , we have and .

Definition 4.3.

A function is said to satisfy the condition if it is nondecreasing and for all .

It is easy to prove that if satisfies the condition , then for all .

Definition 4.4 (see [7]).

Let and be two Menger PN-spaces and , be two set-valued mappings. Let be two mappings, where denotes the space of all linear operators from to . is called a probabilistic -contractor couple of and if there exists a function satisfying the condition such that

(4.6)

for all , , and

(4.7)

for all , , and

Now, we introduce two algorithms for our main results as follows.

Algorithm 1.

Let be an N.A. Menger PN-space with a -norm and be a Menger PN-space with a -norm . Let , be two operators from into itself, a sequence of fuzzy operators from into satisfying the condition (I), and the -closed set-valued operators induced by the fuzzy operators for all . Let and satisfy the condition . Suppose that

(i) and for all ;

(ii) for all and , for all and ;

(iii) is a probabilistic -contractor couple of and ;

(iv)for all and , there exists such that

(4.8)

and, for all and , there exists such that

(4.9)

For any and , put , where is a real number. It follows from the assumption (ii) that . Replacing and by and in (4.6), respectively, from (3.11) of Lemma 4.1, the assumption (iii), and , it follows that

(4.10)

Since and , it follows that and so

(4.11)

Since is left continuous, now we have

(4.12)

and so for all In fact, if there exists such that , then it follows from (4.12) that

(4.13)

which is a contradiction. Therefore, for all . Thus, from (4.12), we have

(4.14)

By the assumption (iv) and (4.14), for any , there exists such that

(4.15)

Let , where is a real number which satisfies inequality . By the assumption (ii), we know that . Similarly, since , it follows from (4.6) that

(4.16)

It is easy to check that for all and so it follows from (4.15) that

(4.17)

Now, for any , the assumption (iv) implies that there exists such that

(4.18)

Inductively, we can get two sequences in and in , respectively, as follows:

(4.19)

where is a real monotone decreasing sequence in and as the sequence in is defined by (2.13) and satisfies the following:

(4.20)

Algorithm 2.

Let be a N.A. Menger PN-space with a -norm , a Menger PN-space with a -norm , and be two operators. Let a sequence of fuzzy operators from into satisfying the condition (I) and the set-valued operators induced by the fuzzy operators for all . Let and satisfy the condition . Suppose that the conditions (ii)–(iv) in Algorithm 1 are satisfied. If

and for all , then, for any and , we have two sequences in and in , respectively, defined as follows:

(4.21)

where the sequence in is defined by (2.13).

Now, we state our main results by using the similar ideas as in [9].

Theorem 4.5.

Let be an N.A. Menger PN-space with a t-norm and be a Menger PN-space with a t-norm . Let , , , , , , and be the same as in Algorithm 1. Suppose that the conditions (i)–(iv) in Algorithm 1 hold and the following conditions are satisfied:

is -joint orbitally complete for some ;

there exists a constant such that, for any constant ,

(4.22)

there exist and such that the -norm satisfies the following condition:

(4.23)

where is a real nonnegative number.

Then the following system of nonlinear operator equations:

(4.24)

has a solution such that . Further, -converges to a solution of (4.24) and -converges to , where in and in are two sequences generated by Algorithm 1.

Proof.

By (4.19), (4.20), and the assumption (vi), since is a monotone decreasing sequence with , we have

(4.25)

which imply that

(4.26)

Since is N.A. Menger PN-space, it follows from (4.26) that, for any positive integers ,

(4.27)

Since , it follows that, for all and , there exists a positive integer such that, for all and ,

(4.28)

and so

(4.29)

Hence is a -Cauchy sequence in . Since is -joint orbitally complete, we can assume that Moreover, by (4.20), it is easy to see that for all and so . Since and are -closed, it follows from (4.19) and the assumption (i) that

(4.30)

that is, is a solution of (4.24) and . This completes the proof.

From Theorem 4.5, we have the following.

Corollary 4.6.

Let be an N.A. Menger PN-space with a t-norm and a Menger PN-space with a -norm . Let , be two operators from into itself, a sequence of fuzzy operators from into satisfying the condition (I), and the -closed set-valued operators induced by the fuzzy operators for all . Let and satisfy the condition . Suppose that the conditions (i)-(iv) in Algorithm 1 and (v)-(vi) in Theorem 4.5 are satisfied. If -norm is of -type, then the conclusions of Theorem 4.5 still hold.

Proof.

By Proposition 3.2, we know that all the conditions of Theorem 4.5 are satisfied. Thus the conclusions of Theorem 4.5 still hold.

Corollary 4.7.

Let for some be a N.A. Menger PN-space and be a Menger PN-space. Let , be two operators from into itself, be a sequence of fuzzy operators from into satisfying the condition (I) and be the -closed set-valued operators induced by the fuzzy operators for all . Let and satisfy the condition . Suppose that the conditions (i)–(iv) in Algorithm 1 and (v)-(vi) of Theorem 4.5 are satisfied. If there exist and for some such that for all , where is a constant, then the conclusions of Theorem 4.5 still hold.

Proof.

From the equivalence (3.10), we have

(4.31)

Corollary 4.8.

Let for some be an N.A. Menger PN-space. Let , , , , , , , and be the same as in Theorem 4.5. Suppose that the conditions (i)–(iv) in Algorithm 1 and (v)-(vi) in Theorem 4.5 are satisfied. If there exist and for some such that for all , where is a constant, then the conclusions of Theorem 4.5 still hold.

Proof.

From the equivalence (3.11), we have

(4.32)

Remark 4.9.

Since

(4.33)

it is easy to see that Corollary 4.8 is a generalization of the corresponding result in Fang [9].

Corollary 4.10.

Let for some be an N.A. Menger PN-space with a -norm . Let , , , ,, , , and be the same as in Theorem 4.5. Suppose that the conditions (i)–(iv) in Algorithm 1 and (v)-(vi) of Theorem 4.5 are satisfied. If there exist and for some such that for all , where is a constant, then the conclusions of Theorem 4.5 still hold.

Proof.

From the equivalence (3.12), we have

(4.34)

Corollary 4.11.

Let be an N.A. Menger PN-space and satisfy the following condition:

(4.35)

where a mapping satisfies the condition . Suppose that the conditions (i) in Algorithm 1 and (v) in Theorem 4.5 are satisfied and there exists and such that -norm satisfies the following condition:

(4.36)

and, for all and , there exists such that

(4.37)

Then there exists a point such that , that is, is a fixed point of .

Proof.

Putting for any fixed and , the mappings and satisfy all the hypotheses of Theorem 4.5. Therefore, there exists a point such that , which means that is a fixed point of . This completes the proof.

Theorem 4.12.

Let be an N.A. Menger PN-space with a -norm and be a Menger PN-space with a -norm . Let , , , , , , and be the same as in Algorithm 2. Suppose that the conditions (ii)–(iv) in Algorithm 1 and in Algorithm 2 are satisfied. If

is -joint orbitally complete for some ,

there exists a constant such that

(4.38)

there exist and such that the -norm satisfies the following condition:

(4.39)

then the following system of nonlinear operator equations:

(4.40)

has a solution such that . Further, -converges to a solution of (4.40) and -converges to , where the sequences in and in are defined by Algorithm 2.

Proof.

Let and for all . It is obvious that all the conditions of Theorem 4.5 are satisfied. Therefore, the conclusion of Theorem 4.12 follows from Theorem 4.5 immediately.

Remark 4.13.

Similarly, we can obtain the conclusions of Theorem 4.12 if we replace the condition (iii) in Theorem 4.12 by the corresponding condition in Proposition 3.2 and the equivalences (3.10)–(3.12), respectively.