Mixed Variational-Like Inequality for Fuzzy Mappings in Reflexive Banach Spaces

Abstract

Some existence theorems for the mixed variational-like inequality for fuzzy mappings (FMVLIP) in a reflexive Banach space are established. Further, the auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution. Consequently, not only the existence of solutions of the FMVLIP is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results.

1. Introduction

The concept of fuzzy set theory was introduced by Zadeh [1]. The applications of the fuzzy set theory can be found in many branches of mathematical and engineering sciences including artificial intelligence, control engineering, management sciences, computer science, and operations research [2]. On the other hand, the concept of variational inequality was introduced by Hartman and Stampacchia [3] in early 1960s. These have been extended and generalized to study a wide class of problems arising in mechanics, physics, optimization and control, economics and transportation equilibrium, and so forth. The generalized mixed variational-like inequalities, which are generalized forms of variational inequalities, have potential and significant applications in optimization theory [4, 5], structural analysis [6], and economics [4, 7]. Motivated and inspired by the recent research work going on these two different fields, Chang [8], Chang and Huang [9], Chang and Zhu [10] and Noor [11] introduced and studied the concept of variational inequalities and complementarity problems for fuzzy mappings in different contexts.

It is noted that there are many effective numerical methods for finding approximate solutions of various variational inequalities (e.g., the projection method and its variant forms, linear approximation, descent and Newton's methods), and there are very few methods for general variational-like inequalities. For example, among the most effective numerical technique is the projection method and its variant forms; however, the projection type techniques cannot be extended for constructing iterative algorithms for mixed variational-like inequalities, since it is not possible to find the projection of the solution. Thus, the development of an efficient and implementable technique for solving variational-like inequalities is one of the most interesting and important problems in variational inequality theory. These facts motivated Glowinski et al. [12] to suggest another technique, which does not depend on the projection. The technique is called the auxiliary principle technique.

Recently, the auxiliary principle technique was extended by Huang and Deng [13] to study the existence and iterative approximation of solutions of the set-valued strongly nonlinear mixed variational-like inequality, under the assumptions that the operators are bounded closed values. On the other hand, by using the concept of -strongly mixed monotone of a fuzzy mapping on a bounded closed convex set, the auxiliary principle technique was extended by Chang et al. [14] to study the existence and iterative approximation of solutions of the mixed variational-like inequality problem for fuzzy mappings in a Hilbert space.

In this paper, the mixed variational-like inequality problem for fuzzy mapping (FMVLIP) in a reflexive Banach space is studied, and some existence theorems for the problem are proved. We also prove the existence theorem for auxiliary problem of FMVLIP. Further, by exploiting the theorem, we construct and analyze an iterative algorithm for finding the solution of the FMVLIP. Finally, we discuss the convergence analysis of iterative sequence generated by the iterative algorithm.

2. Preliminaries

Throughout this paper, we assume that is a real Banach space with its topological dual , a nonempty convex subset of , is the generalized duality pairing between and , is the family of all nonempty bounded and closed subsets of , and is the Hausdorff metric on defined by

(2.1)

In the sequel we denote the collection of all fuzzy sets on by . A mapping from to is called a fuzzy mapping. If is a fuzzy mapping, then the set , for , is a fuzzy set in (in the sequel we denote by ) and , for each is the degree of membership of in .

A fuzzy mapping is said to be closed, if, for each , the function is upper semicontinuous; that is, for any given net satisfying , we have .

For and , the set

(2.2)

is called a -cut set of .

A closed fuzzy mapping is said to satisfy condition , if there exists a function such that for each the set

(2.3)

is a nonempty bounded subset of .

Remark 2.1.

It is worth mentioning that if is a closed fuzzy mapping satisfying condition , then for each , the set . Indeed, let be a net and , then for each . Since the fuzzy mapping is closed, we have

(2.4)

This implies that , and so .

Let be a real reflexive Banach space with the dual space . In this paper, we devote our study to a class of mixed variational-like inequality problem for fuzzy mappings, which is stated as follows.

Let are two closed fuzzy mappings satisfying the condition with functions , respectively. and are two single-valued mappings. Let be a real bifunction. We shall study the following problem :

(2.5)

The problem (2.5) is called a fuzzy mixed variational-like inequality problem, and we will denote by the solution set of the problem (2.5).

Now, let us consider some special cases of problem (2.5).

() Let be two ordinary set-valued mappings, and let be the mappings as in problem (2.5). Define two fuzzy mappings as follows:

(2.6)

where and are the characteristic functions of the sets and , respectively. It is easy to see that and both are closed fuzzy mappings satisfying condition with constant functions and , for all , respectively. Furthermore,

(2.7)

Thus, problem (2.5) is equivalent to the following problem:

(2.8)

This kind of problem is called the set-valued strongly nonlinear mixed variational-like inequality, which was studied by Huang and Deng [13], when .

() If is a Hilbert space, then problem (2.5) collapses to the following problem: Let are two closed fuzzy mappings satisfying the condition with functions , respectively. are two single-valued mappings. Let be a real bifunction. We consider the following problem:

(2.9)

The inequality of type (2.9) was studied by Chang et al. [14] under the additional condition that is a nonempty bounded closed subset of .

• (3) If is a Hilbert space and f(u,v)=0, then problem (2.5) is equivalent to the following problem:

(2.10)

This is also a class of special fuzzy variational-like inequalities, which has been studying by many authors.

Evidently, for appropriate and suitable choice of the fuzzy mappings , mappings , the bifunction , and the space , one can obtain a number of the known classes of variational inequalities and variational-like inequalities as special cases from problem (2.5) (see [1, 4, 5, 7–19]).

The following basic concepts will be needed in the sequel.

Definition 2.2.

Let be a nonempty subset of a Banach space . Let be two closed fuzzy mappings satisfying the condition with functions , respectively. Let be mappings. Then

(i) is said to be -cocoercive with respect to the first argument of , if there exists a constant , such that

(2.11)

for each , and for all ;

(ii) is Lipschitz continuous in the second argument with respect to the fuzzy mapping , if there exists a constant such that

(2.12)

for any and ;

(iii) is -strongly monotone in the first argument with respect to the fuzzy mapping if there exists a constant such that

(2.13)

for any and . Similarly, -strongly monotone of in the second argument with respect to the fuzzy mapping can be defined;

(iv) is said to be -Lipschitz continuous if there exists a constant such that

(2.14)

for any ;

(v) is Lipschitz continuous, if there exists a constant such that

(2.15)

for any .

Definition 2.3.

The bifunction is said to be skew-symmetric, if

(2.16)

for all .

Remark 2.4.

The skew-symmetric bifunctions have properties which can be considered as an analogs of monotonicity of gradient and nonnegativity of a second derivative for a convex function. As for the investigations of the skew-symmetric bifunction, we refer the reader to [20].

Definition 2.5 (see [15, 21]).

Let be a nonempty convex subset of a Banach space . Let be a Fréchet differentiable function and . Then is said to be

(i)-convex, if

(2.17)

for all

(ii)-strongly convex, if there exists a constant such that

(2.18)

for all

Note that if for all , then is said to be strongly convex.

Throughout this paper, we shall use the notations "" and "" for weak convergence and strong convergence, respectively.

Remark 2.6.

() Assume that for each fixed the mapping is continuous from the weak topology to the weak topology. Let and be fixed, and let be a functional defined by

(2.19)

Then, it is easy to see that is a weakly continuous functional on .

() Let be a Fréchet differentiable function, and let be a mapping such that . If is an -strongly convex functional with constant on a convex subset of then is -strongly monotone with constant (see [19], Proposition??2.1).

The following lemma due to Zeng et al. [19] will be needed in proving our results.

Lemma 2.7 (see [19, Lemma??2]).

Let be a nonempty convex subset of a topological vector space and let be such that

(i)for each is lower semicontinuous on each nonempty compact subset of ;

(ii)for each finite set and for each ;

(iii)there exists a nonempty compact convex subset of such that for some , there holds

(2.20)

Then there exists , such that ??for all .

We also need the following lemma.

Lemma 2.8 (see [22]).

Let be a complete metric space and let and be any real number. Then, for every there exists such that .

In the sequel, we assume that and satisfy the following assumption.

Assumption 2.9.

Let be two mappings satisfying the following conditions:

(a) for each ;

(b)for each fixed is a concave function;

(c)for each fixed , the functional is weakly lower semicontinuous function from to , that is,

(2.21)

Remark 2.10.

It follows from Assumption 2.9(a) that and .

3. The Existence Theorems

Theorem 3.1.

Let be a real reflexive Banach space with the dual space , and be a nonempty convex subset of . Let be two closed fuzzy mappings satisfying the condition with functions , respectively. Let , and . Let be skew-symmetric and weakly continuous such that and is a proper convex, for each . Suppose that

(i) is -cocoercive with respect to the first argument of with constant ;

(ii) is Lipschitz continuous with constant ;

(iii) is Lipschitz continuous and -strongly monotone in the second argument with respect to with constant and , respectively.

If Assumption 2.9 is satisfied, then .

Proof.

For any , we define a function by

(3.1)

where .

Observe that, by is weakly continuous functional and since each fixed the functional is weakly lower semicontinuous, we have the functional is weakly lower semicontinuous for each . This shows that condition (i) in Lemma 2.7 holds. Next, we claim that satisfies condition (ii) in Lemma 2.7. If it is not true, then there exist a finite set and , such that for all , that is,

(3.2)

This gives

(3.3)

Note that for each is a convex functional, that is Hence,

(3.4)

From Assumption 2.9, we obtain

(3.5)

which is a contradiction. Thus condition (ii) in Lemma 2.7 holds. Since for each is a proper convex weakly lower semicontinuous functional and int, the element intcan be found. Moreover, by Proposition I.2.6 of Pascali and Sburlan [23, page 27], is subdifferentiable at . This means

(3.6)

Since is skew-symmetric, it follows that

(3.7)

Letting and be fixed, by using conditions (ii) and (iv) and equality , we have

(3.8)

Define and . Then is a weakly compact convex subset of . Furthermore, it is easy to see that for all . Thus, condition (iii) of Lemma 2.7 is satisfied. By Lemma 2.7, there exists such that for all , this means that

(3.9)

where . Hence, is a solution of the fuzzy variational like inequality (2.5), that is, . This completes the proof.

Remark 3.2.

If all assumptions to Theorem 3.1 hold and is -strongly monotone in the first argument with respect to with constant , then the solution of problem (2.5) is unique up to the element . Indeed, supposing that and are elements in , we have

(3.10)

(3.11)

Taking in (3.10) and in (3.11) and adding two inequalities, since is skew-symmetric, we obtain

(3.12)

Using this one, in view of Remark 2.10, we have

(3.13)

Since is -strongly monotone in the first argument with respect to with the constant and -strongly monotone in the second argument with respect to with constant , we obtain

(3.14)

Since , we must have .

4. Convergence Analysis

4.1. Auxiliary Problem and Algorithm

In this section, we extend the auxiliary principle technique to study the fuzzy mixed variational-like inequality problem (2.5) in a reflexive Banach space . First, we give the existence theorem for the auxiliary problem for the problem (2.5). Consequently, we construct the iterative algorithm for solving the problem of type (2.5).

Let be a mapping, let be a given Fréchet differentiable -convex functional, and let be a given positive real number. Given , we consider the following problem : find such that

(4.1)

The problem is called the auxiliary problem for fuzzy mixed variational-like inequality problem (2.5).

Theorem 4.1.

If the conditions of Theorem 3.1 hold and for each fixed is continuous from the weak topology to the weak topology. If the function is -strongly convex with constant and the functional is weakly upper semicontinuous on for each , then the auxiliary problem has a unique solution.

Proof.

Let and be fixed. Define a functional by

(4.2)

Note that, for each fixed , the functional is weakly upper semicontinuous on , is continuous from the weak topology to the weak topology, and is weakly continuous. Thus, it is easy to see that for each fixed the function is weakly lower semicontinuous continuous on each weakly compact subset of and so condition (i) in Lemma 2.7 is satisfied. We claim that condition (ii) in Lemma 2.7 holds. If this is false, then there exist a finite set and a with and , such that

(4.3)

By Assumption 2.9, in light of Remark 2.10, together with the convexity of , we have

(4.4)

which is a contradiction. Thus, condition (ii) in Lemma 2.7 is satisfied. Note that the -strong convexity of implies that is -strongly monotone with constant ; see Remark 2.6(ii). By using the similar argument as in the proof of Theorem 3.1, we can readily prove that condition (iii) of Lemma 2.7 is also satisfied. By Lemma 2.7 there exists a point , such that for all . This implies that is a solution to the problem .

Now we prove that the solution of problem is unique. Let and be two solutions of problem (4.1). Then,

(4.5)

(4.6)

Taking in (4.5) and in (4.6), and adding these two inequalities, since and is skew-symmetric, we obtain

(4.7)

Thus, by is -strongly monotone, we have

(4.8)

This implies that and the proof is completed.

By virtue of Theorem 4.1, we now construct an iterative algorithm for solving the fuzzy mixed variational-like inequalities problem (2.5) in a reflexive Banach space .

Let be fixed. For given , from Theorem 4.1, there is such that

(4.9)

Since , by Lemma 2.8, there exist and such that

(4.10)

Again by Theorem 4.1, there is such that

(4.11)

Since , by Lemma 2.8, there exist and such that

(4.12)

Continuing in this way, we can obtain the iterative algorithm for solving problem (2.5) as follows.

Algorithm 4.2.

Let be fixed. For given there exist the sequences and such that

(4.13)

4.2. Convergence Theorems

Now, we shall prove that the sequences and generated by Algorithm 4.2 converge strongly to a solution of problem (2.5).

Theorem 4.3.

Suppose that conditions of Theorem 4.1 hold, and the mapping are Lipschitzian continuous fuzzy mappings with Lipschitzian constant and , respectively. If then the iterative sequences obtained from Algorithm 4.2 converge strongly to a solution of problem (2.5).

Proof.

Let . Define a function by

(4.14)

By the -strong convexity of , we have

(4.15)

Note that for all and is skew-symmetric. Since and , from the -strong convexity of , and Algorithm 4.2 with it follows that

(4.16)

where .

Consider

(4.17)

Therefore, we have

(4.18)

Since , the inequality (4.18) implies that the sequence is strictly decreasing (unless ) and is nonnegative by (4.15). Hence it converges to some number. Thus, the difference of two consecutive terms of the sequence goes to zero, and so the sequence converges strongly to . Further, from Algorithm 4.2, we have

(4.19)

These imply that and are Cauchy sequence in , since is a convergence sequence. Thus, we can assume that and (as ). Noting and , we have

(4.20)

Hence we must have . Similarly, we can obtain . Finally, we will show that . In regarded of Assumption 2.9(c), for each fixed we have that the functional is an upper semicontinuous functional; this together with the weak continuity of the function , we obtain

(4.21)

This implies that , and the proof is completed.

Remark 4.4.

() Theorems 3.1 and 4.3 are the extension of the results by Chang et al. [14], from Hilbert setting to a general reflexive Banach space, but it is worth noting that the bounded condition of the convex set is not imposed here.

() Since every set-valued mapping is the fuzzy mapping, hence, all results obtained in this paper are still hold for any set-valued mappings .

Thus, our results can be view as a refinement and improvement of the previously known results for variational inequalities.