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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 264721, 22 pages
http://dx.doi.org/10.1155/2012/264721
Research Article

Metric Characterizations of -Well-Posedness for a System of Mixed Quasivariational-Like Inequalities in Banach Spaces

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3Department of Occupational Safety and Health, College of Public Health, China Medical University, Taichung 404, Taiwan

Received 2 October 2011; Accepted 9 October 2011

Academic Editor: Yonghong Yao

Copyright © 2012 L. C. Ceng and Y. C. Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to investigate the problems of the well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept of -well-posedness to the system of mixed quasivariational-like inequalities, which includes symmetric quasi-equilibrium problems as a special case. Second, we establish some metric characterizations of -well-posedness for the system of mixed quasivariational-like inequalities. Under some suitable conditions, we prove that the -well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-like inequalities. The corresponding concept of -well-posedness in the generalized sense is also considered for the system of mixed quasivariational-like inequalities having more than one solution. The results presented in this paper generalize and improve some known results in the literature.

1. Introduction

The classical notion of well-posedness for a minimization problem (MP) is due to Tykhonov [1], which has already been known as the Tykhonov well-posedness. The so-called Tykhonov well-posedness means the existence and uniqueness of solution, and the convergence of every minimizing sequence toward the unique solution. Taking into account that in many practical situations the solution may not be unique for a minimization problem, ones naturally introduced the concept of well-posedness in the generalized sense, which means the existence of minimizers and the convergence of some subsequence of every minimizing sequence toward a minimizer. Obviously, the concept of well-posedness is inspired by numerical methods producing optimizing sequences for optimization problems. In the following years, the well-posedness has received much attention because it plays a crucial role in the stability theory for optimization problems. A large number of results about well-posedness have appeared in the literature; see, for example, [210], where the work in [2, 3, 5, 7, 10] is for the class of scalar optimization problems, and the work in [4, 6, 8, 11] is for the class of vector optimization problems.

On the other hand, the concept of well-posedness has been generalized to other related problems, such as variational inequalities [9, 1222], Nash equilibrium problems [16, 2325], inclusion problems [12, 14, 26, 27], and fixed-point problems [12, 14, 26, 28, 29]. An initial notion of well-posedness for variational inequalities is due to Lucchetti and Patrone [20]. They introduced the notion of well-posedness for variational inequalities and proved some related results by means of Ekeland’s variational principle. Since then, many authors have been devoted to generating the concept of well-posedness from the minimization problem to various variational inequalities. Lignola and Morgan [19] introduced the parametric well-posedness for a family of variational inequalities. Lignola [15] further introduced two concepts of well-posedness and -well-posedness for quasivariational-like inequalities and derived some metric characterizations of well-posedness. At the same time, Del Prete et al. [18] introduced the concept of -well-posedness for a class of variational inequalities. Recently, Fang et al. [14] generalized the concept of well-posedness to a class of mixed variational inequalities in Hilbert spaces. They obtained some metric characterizations of its well-posedness and established the links with the well-posedness of inclusion problems and fixed-point problems. Furthermore, Ceng and Yao [12] generalized the results of Fang et al. [14] to a class of generalized mixed variational inequalities in Hilbert spaces. Ceng et al. [13] investigated the well-posedness for a class of mixed quasivariational-like inequalities in Banach spaces. For the well-posedness of variational inequalities with functional constraints, we refer to Huang and Yang [9] and Huang et al. [17]. In 2006, Lignola and Morgan [23] presented the notion of -well-posedness for the Nash equilibrium problem and gave some metric characterizations of this type well-posedness. Petruşel et al. [29] and Llorens-Fuster et al. [28] discussed the well-posedness of fixed-point problems for multivalued mappings in metric spaces.

It is obvious that the equilibrium problem plays a very important role in the establishment of a general mathematical model for a wide range of practical problems, which include as special cases optimization problems, Nash equilibria problems, fixed-point problems, variational inequality problems, and complementarity problems (see, e.g, [30, 31]), and has been studied extensively and intensively. It is well known that each equilibrium problem can equivalently be transformed into a minimizing problem by using gap function, and some numerical methods have been extended to solve the equilibrium problem (see, e.g., [32]). This fact motivates the researchers to study the well-posedness for equilibrium problems. Recently, Fang et al. [33] introduced the concepts of parametric well-posedness for equilibrium problems and derived some metric characterizations for these types of well-posedness. For the well-posedness of equilibrium problems with functional constraints, we refer the readers to [34]. In 2009, Long and Huang [35] generalized the concept of -well-posedness to symmetric quasiequilibrium problems in Banach spaces, which includes equilibrium problems, Nash equilibrium problems, quasivariational inequalities, variational inequalities, and fixed-point problems as special cases. Under some suitable conditions, they established some metric characterizations of -well-posedness for symmetric quasiequilibrium problems. Moreover, they gave some examples to illustrate their results. Their results represent the generalization and improvement of some previously known results in the literature, for instance, [1215, 23, 33]. It is worth pointing out that up to the publication of [35] there are no results concerned with the problems of the well-posedness for symmetric quasiequilibrium problems in Banach spaces.

In this paper, we consider and study the problems of the well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept of -well-posedness to the system of mixed quasivariational-like inequalities, which include symmetric quasiequilibrium problems as a special case. Second, some metric characterizations of -well-posedness for the system of mixed quasivariational-like inequalities are given under very mild conditions. Furthermore, it is also proven that under quite appropriate conditions, the -well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-like inequalities. At the same time, the corresponding concept of -well-posedness in the generalized sense is also considered for the system of mixed quasivariational-like inequalities having more than one solution. In addition, we give some examples to illustrate our results. The results presented in this paper generalize and improve Long and Huang’s results in [35].

2. Preliminaries

Throughout this paper, unless specified otherwise, let and be two real Banach spaces, let their dual spaces be denoted by and , respectively, and let the duality pairing between and and the one between and be denoted by the same . We write to indicate that the sequence converges weakly to . However, implies that converges strongly to . Let and be two nonempty closed and convex subsets. Let and be two set-valued mappings, let , , and be four single-valued mappings, and let be two real-valued functions. Suppose that is a nonnegative real number and .

In this paper, we consider the system of mixed quasivariational-like inequalities (SMQVLIs), which is to find a point such that

Remark 2.1. Whenever , , , and , the Problem (2.1) reduces to the following symmetric quasiequilibrium problem (in short, SQEP) of finding a point such that This problem was first considered by Noor and Oettli [21], which includes equilibrium problems [30], Nash equilibrium problems [36], quasivariational inequalities [37], variational inequalities [38], and fixed-point problems [28, 29] as special cases. It is worth mentioning that Noor and Oettli [21] only established the existence of solutions for SQEP (2.2). Subsequently, Long and Huang [35] investigated the -well-posedness for SQEP (2.2) in Banach spaces.

Denote by the solution set of SMQVLI (2.1). In what follows, we introduce the notions of -approximating sequence and -well-posedness for SMQVLI (2.1).

Definition 2.2. A sequence is called an -approximating sequence for SMQVLI (2.1) if there exists a sequence with such that where denotes the ball of radius around , that is, the set . Whenever , one says that the sequence is an approximating sequence for SMQVLI (2.1).

We remark that if , , , and , the notions of -approximating sequence and approximating sequence for SMQVLI (2.1) reduce to the ones of -approximating sequence and approximating sequence for SQEP (2.2) in [35, Definition  2.1], respectively.

Definition 2.3. SMQVLI (2.1) is said to be -well-posed if it has a unique solution and every -approximating sequence converges strongly to . Whenever , we say that SMQVLI (2.1) is well-posed.

We remark that if , , , and , the notions of -well-posedness and well-posedness for SMQVLI (2.1) reduce to the ones of -well-posedness and well-posedness for SQEP (2.2) in [35, Definition  2.2], respectively.

Definition 2.4. SMQVLI (2.1) is said to be -well-posed in the generalized sense if the solution set of SMQVLI (2.1) is nonempty and every -approximating sequence has a subsequence which converges strongly to some element of . Whenever , one says that SMQVLI (2.1) is well-posed in the generalized sense.

We remark that if , , , and , the notions of -well-posedness in the generalized sense and well-posedness in the generalized sense for SMQVLI (2.1) reduce to the ones of -well-posedness in the generalized sense and well-posedness in the generalized sense for SQEP (2.2) in [35, Definition  2.3], respectively.

In order to investigate the -well-posedness for SMQVLI (2.1), we need the following definitions.

Definition 2.5 (see [39]). The Painleve-Kuratowski limits of a sequence are defined by

Definition 2.6 (see [39]). A set-valued mapping from a topological space to a topological space is called
(i)-closed if for every , for every sequence -converging to , and for every sequence -converging to a point , such that , one has , that is, (ii)-lower semicontinuous if for every , for every sequence -converging to , and for every , there exists a sequence -converging to , such that for sufficiently large, that is, (iii)-subcontinuous on , if for every sequence -converging in , every sequence , such that , has a -convergent subsequence.

Definition 2.7 (see [39]). Let be a nonempty subset of . The measure of noncompactness of the set is defined by where means the diameter of a set.

Definition 2.8 (see [39]). Let be a metric space and let , be nonempty subsets of . The Hausdorff metric between and is defined by where with . Let be a sequence of nonempty subsets of . One says that converges to in the sense of Hausdorff metric if . It is easy to see that if and only if for all section . For more details on this topic, the readers refered one to [39].

Now, we prove the following lemma.

Lemma 2.9. Suppose that set-valued mappings and are nonempty convex-valued, the function is convex on for any , and the function is convex on for any . Then if and only if the following two conditions hold: where both and are affine in the second variable such that and for all .

Proof. The necessity is obvious. For the sufficiency, suppose that (2.9) holds. Now let us show that . Indeed, let and for any , . Since is convex, and so Also, since is convex for any and is affine in the second variable with , , we have Thus, dividing by in the above inequality, we have By the similar argument, The combination of (2.12) and (2.13) implies, for tending to zero, that is a solution of SMQVLI (2.1). This completes the proof.

Corollary 2.10 (i.e., [35, Lemma  2.1]). Suppose that set-valued mappings and are nonempty convex-valued, the function is convex on for any , and the function is convex on for any . Then solves SQEP (2.2) if and only if the following two conditions hold:

Proof. Put , , , and in Lemma 2.9. Then, utilizing Lemma 2.9 we get the desired result.

3. Metric Characterizations of -Well-Posedness for SMQVLI

In this section, we will investigate some metric characterizations of -well-posedness for SMQVLI (2.1).

For any , the -approximating solution set of SMQVLI (2.1) is defined by

Theorem 3.1. SMQVLI (2.1) is -well-posed if and only if the solution set of SMQVLI (2.1) is nonempty and

Proof . Suppose that SMQVLI (2.1) is -well-posed. Then, is a singleton, and for any , since . Suppose by contraction that Then there exists with , and such that where the norm in the product space is defined as follows: Since , and SMQVLI (2.1) is -well-posed, the sequences and , which are both -approximating sequences for SMQVLI (2.1), converge strongly to the unique solution , and this leads to a contraction. Therefore, (3.2) holds.
Conversely, let (3.2) hold and let be any -approximating sequence for SMQVLI (2.1). Then, there exists a sequence with such that This implies that . Since the solution set of SMQVLI (2.1) is nonempty, we can take two elements in arbitrarily, denoted by and , respectively. Note that for all . Hence both and lie in for all . This fact together with (3.2) yields Utilizing (3.7) and the uniqueness of the limit, we conclude that . This means that is a singleton. Thus, it is known that SMQVLI (2.1) has the unique solution and converges strongly to . This shows that SMQVLI (2.1) is -well-posed. This completes the proof.

Corollary 3.2 (i.e., [35, Theorem  3.1]). SQEP (2.2) is -well-posed if and only if the solution set of SQEP (2.2) is nonempty and where

Proof. Put , , , and in Theorem 3.1. Then, utilizing Theorem 3.1 we get the desired result.

In the sequel, the following concept will be needed to apply to our main results.

Definition 3.3. Let be a nonempty, closed convex subset of . A single-valued mapping is said to be Lipschitz continuous if there exists a constant such that

We remark that whenever a Hilbert space and a nonempty closed convex subset of , the Lipschitz continuous mapping has been introduced and considered in Ansari and Yao [40]. In their main result for the existence of solutions and convergence of iterative algorithm (i.e., [40, Theorem 3.1]), the Lipschitz continuous mapping satisfies the following conditions:(a) for all ,(b) for all ,(c) is affine in the first variable,(d)for each fixed , is sequentially continuous from the weak topology to the weak topology (-continuous).

Inspired by the above restrictions imposed on the Lipschitz continuous mapping , we give the following theorem.

Theorem 3.4. Assume that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, -closed, -lower semicontinuous and -subcontinuous on ;(ii)single-valued mappings and are -continuous on ;(iii)single-valued mappings and are Lipschitz continuous with constants and respectively, such that(a) for all and for all ,(b) and both are affine in the second variable;(iv)functions and are continuous on ;(v)for any , the function is convex on ; for any , the function is convex on .
Then, SMQVLI (2.1) is -well-posed if and only if

Proof . First, utilizing condition (iii) (a), we can readily obtain that
The necessity has been proved in Theorem 3.1. For the sufficiency, let condition (3.11) hold. Let be any -approximating sequence for SMQVLI (2.1). Now let us show that is a singleton and converges strongly to the unique element of . As a matter of fact, since is -approximating sequence for SMQVLI (2.1), there exists a sequence with such that This means . It follows from (3.11) that is a Cauchy sequence in Banach space and hence converges strongly to a point . By the definition of the norm in Banach space , we deduce that On account of the closedness of and we conclude from and that and . In order to show , we start to prove that Indeed, suppose that the left inequality does not hold. Then there exists a positive number such that or equivalently, there exist an increasing sequence and a sequence , such that Since the set-valued mapping is -closed and -subcontinuous, the sequence has a subsequence, denoted still by , converging weakly to a point . From the weak lower semicontinuity of the norm, it follows that which leads to a contradiction. Thus we must have and hence . Similarly, we can prove .
To complete the proof, we take a point arbitrarily. Since is -lower semicontinuous, there exists a sequence converging strongly to , such that for sufficiently large. Furthermore, utilizing condition (iii) () and the Lipschitz continuity of we deduce that Since is -continuous, it is known that converges weakly to , that is, for each , the real sequence converges to the real number . This implies that is a bounded sequence of real numbers for each . Thus is bounded in the norm topology according to the uniform boundedness principle [41], that is, .
Now observe that Consequently, it follows from condition (iv) that Analogously, we have It follows from Lemma 2.9 that . Therefore, SMQVLI (2.1) is -well-posed. This completes the proof.

Corollary 3.5 (i.e., [35, Theorem  3.2]). Assume that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, -closed, -lower semicontinuous, and -subcontinuous on ;(ii)functions and are continuous on ;(iii)for any , the function is convex on ; for any , the function is convex on . Then, SQEP (2.2) is -well-posed if and only if

To illustrate Theorem 3.4, we give the following two examples.

Example 3.6. Let and . Let , , , , , , and for all and . Obviously, the conditions (i)–(v) of Theorem 3.4 are satisfied. Note that It follows that and so as . By Theorem 3.4, SMQVLI (2.1) is -well-posed.

Example 3.7. Let and . Let , , , , , and for all and . It is easy to see that the conditions (i)–(v) of Theorem 3.4 are satisfied, and . But, SMQVLI (2.1) is not -well-posed, since as .

Whenever , we have the following result.

Theorem 3.8. Assume that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, -closed, -lower semicontinuous, and -subcontinuous on ;(ii)single-valued mappings and are -continuous on ;(iii)single-valued mappings and are Lipschitz continuous with constants and , respectively, such that for all and (iv)functions and are continuous on . Then, SMQVLI (2.1) is well-posed if and only if

Corollary 3.9 (i.e., [35, Corollary  3.1]). Assume that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, -closed, -lower semicontinuous, and -subcontinuous on ;(ii)functions and are continuous on . Then, SQEP (2.2) is well-posed if and only if

The following theorem shows that under some suitable conditions, the -well-posedness of SMQVLI (2.1) is equivalent to the existence and uniqueness of its solutions.

Theorem 3.10. Let and be two finite-dimensional spaces. Suppose that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, closed, lower semicontinuous, and subcontinuous on ;(ii)single-valued mappings and are continuous on ;(iii) single-valued mappings and are Lipschitz continuous with constants and respectively, such that(a) for all and for all ,(b) and both are affine in the second variable;(iv) the functions and are continuous on ;(v)for any , the function is convex on ; for any , the function is convex on ;(vi) is nonempty bounded for some . Then, SMQVLI (2.1) is -well-posed if and only if SMQVLI (2.1) has a unique solution.

Proof. The necessary of the theorem is obvious. In order to show the sufficiency, let be the unique solution of SMQVLI (2.1) and let be any -approximating sequence for SMQVLI (2.1). Then there exists a sequence with such that which means . Let be such that is nonempty bounded. Then there exists such that for all . Thus, is bounded and so the sequence has a subsequence which converges to . Reasoning as in Theorem 3.4, one can prove that solves SMQVLI (2.1). The uniqueness of the solution implies that , and so the whole sequence converges to . Thus, SMQVLI (2.1) is -well-posed. This completes the proof.

Example 3.11. Let and . Let , , , , , , and for all and . Clearly, the conditions (i)–(vi) of Theorem 3.8 are satisfied, and SMQVLI (2.1) has a unique solution . By Theorem 3.8, SMQVLI (2.1) is -well-posed.

Corollary 3.12 (i.e., [35, Theorem  3.3]). Let and be two finite-dimensional spaces. Suppose that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, closed, lower semicontinuous and subcontinuous on ;(ii)the functions and are continuous on ;(iii) for any , the function is convex on ; for any , the function is convex on ;(iv) is nonempty bounded for some . Then, SQEP (2.2) is -well-posed if and only if SQEP (2.2) has a unique solution.

4. Metric Characterizations of -Well-Posedness in the Generalized Sense for SMQVLI

In this section, we derive some metric characterizations of -well-posedness in the generalized sense for SMQVLI (2.1) by considering the noncompactness of approximate solution set.

Theorem 4.1. SMQVLI (2.1) is -well-posed in the generalized sense if and only if the solution set of SMQVLI (2.1) is nonempty compact and

Proof . Suppose that SMQVLI (2.1) is -well-posed in the generalized sense. Then is nonempty. To show the compactness of , let . Clearly, if is an approximation sequence of SMQVLI (2.1), then it is also -approximation sequence. Since SMQVLI (2.1) is -well-posed in the generalized sense, it contains a subsequence converging strongly to an element of . Thus, is compact. Now, we prove that (4.1) holds. Suppose by contradiction that there exist , , and such that Being is an -approximating sequence for SMQVLI (2.1). Since SMQVLI (2.1) is -well-posed in the generalized sense, there exists a subsequence of converging strongly to some element of . This contradicts (4.2) and so (4.1) holds.
To prove the converse, suppose that is nonempty compact and (4.1) holds. Let be an -approximating sequence for SMQVLI (2.1). Then , and so . This implies that there exists a sequence such that where the norm in the product space is defined as follows:
Since is compact, there exists a subsequence of converging strongly to . Hence the corresponding subsequence of converges strongly to . Therefore, SMQVLI (2.1) is -well-posed in the generalized sense.

We give the following example to illustrate that the compactness condition of is necessary.

Example 4.2. Let and . Let , , , , , and for all and . Then . It is clear that as . It is easy to see that the diverging sequence is an -approximating sequence, but it has no convergent subsequence. Therefore, SMQVLI (2.1) is not -well-posed in the generalized sense.

Corollary 4.3 (i.e., [35, Theorem  4.1]). SQEP (2.2) is -well-posed in the generalized sense if and only if the solution set of SQEP (2.2) is nonempty compact and

Theorem 4.4. Assume that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, -closed, -lower semicontinuous, and -subcontinuous on ;(ii)single-valued mappings and are -continuous on ;(iii) single-valued mappings and are Lipschitz continuous with constants and respectively, such that(a) for all and for all ,(b) and both are affine in the second variable;(iv) functions and are continuous on ;(v)for any , the function is convex on ; for any , the function is convex on . Then, SMQVLI (2.1) is -well-posed in the generalized sense if and only if

Proof . Suppose that SMQVLI (2.1) is -well-posed in the generalized sense. By the same argument as in Theorem 4.1, is nonempty compact, and as . Clearly for any , because . Observe that for any , we have Since is compact, and the following relation holds (see, e.g, [2]): It follows that (4.6) holds.
Conversely, suppose that (4.6) holds. It is easy to prove that , for any , is closed. Note that whenever , their intersection is nonempty compact and satisfies [39, page 412], where By Lemma 2.9, we obtain that coincides with the solution set of SMQVLI (2.1). Thus, is compact.
Let be any -approximating sequence for SMQVLI (2.1). Then there exists a sequence with such that which means . It follows from (4.6) that there exists a sequence such that Since is compact, there exists a subsequence of converging strongly to . Hence, the corresponding subsequence of converges strongly to . Thus, SMQVLI (2.1) is -well-posed in the generalized sense.

Example 4.5. Let and . Let , , , , and for all and . Obviously, the conditions (i)–(v) of Theorem 4.4 are satisfied, and . By Theorem 4.4, SMQVLI (2.1) is -well-posed in the generalized sense.

Corollary 4.6 (i.e., [35, Theorem  4.2]). Assume that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, -closed, -lower semicontinuous, and -subcontinuous on ;(ii)functions and are continuous on ;(iii)for any , the function is convex on ; for any , the function is convex on . Then, SQEP (2.2) is -well-posed in the generalized sense if and only if

We now give a sufficient condition for the -well-posedness in the generalized sense of SMQVLI (2.1) in finite-dimensional spaces.

Theorem 4.7. Let and be two finite-dimensional spaces. Suppose that the following conditions hold:(i)set-valued mappings and are nonempty convex-valued, closed, lower semicontinuous, and subcontinuous on ;(ii)single-valued mappings and are continuous on ;(iii) single-valued mappings and are Lipschitz continuous with constants and respectively, such that(a)() for all and