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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 194509, 38 pages
http://dx.doi.org/10.1155/2012/194509
Research Article

Well-Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces

1Department of Mathematics, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China
2Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan

Received 24 September 2011; Accepted 3 November 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Lu-Chuan Ceng and Ching-Feng Wen. 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

We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well-posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well-posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

1. Introduction

Let be a real Banach space and a real-valued functional on . In 1966, Tikhonov [1] first introduced the classical notion of well-posedness for a minimization problem , which has been known as the Tikhonov well-posedness. A minimization problem is said to be Tikhonov well-posed if it has a unique solution toward which every minimizing sequence of the problem converges. It is obvious that the notion of Tikhonov well-posedness is inspired by the numerical methods producing optimizing sequences for optimization problems and plays a crucial role in the optimization theory. The notion of generalized Tikhonov well-posedness is also introduced for a minimization problem having more than one solution, which requires the existence of solutions and the convergence of some subsequence of every minimizing sequence toward some solution. Another important notion of well-posedness for a minimization problem is the well-posedness by perturbations or extended well-posedness due to Zolezzi [2, 3]. The notion of well-posedness by perturbations establishes a form of continuous dependence of the solutions upon a parameter. There are many other notions of well-posedness in optimization problems. For more details, we refer the readers to [17] and the references therein.

On the other hand, the concept of well-posedness has been generalized to other variational problems, such as variational inequalities [4, 814], saddle point problems [15], Nash equilibrium problems [14, 1618], equilibrium problems [19], inclusion problems [20, 21], and fixed point problems [2022]. An initial notion of well-posedness for a variational inequality is due to Lucchetti and Patrone [4]. 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 papers have been devoted to the extensions of well-posedness of minimization problems to various variational inequalities. Lignola and Morgan [12] generalized the notion of well-posedness by perturbations to a variational inequality and established the equivalence between the well-posedness by perturbations of a variational inequality and the well-posedness by perturbations of the corresponding minimization problem. Lignola and Morgan [14] introduced the concepts of -well-posedness for variational inequalities. Del Prete et al. [13] further proved that the -well-posedness of variational inequalities is closely related to the well-posedness of minimization problems. Recently, Fang et al. [9] generalized the notions of well-posedness and -well-posedness to a mixed variational inequality. In the setting of Hilbert spaces, Fang et al. [9] proved that under suitable conditions the well-posedness of a mixed variational inequality is equivalent to the existence and uniqueness of its solution. They also showed that the well-posedness of a mixed variational inequality has close links with the well-posedness of the corresponding inclusion problem and corresponding fixed point problem in the setting of Hilbert spaces. Subsequently, the notions of well-posedness and -well-posedness for a mixed variational inequality in [9] are extended by Ceng and Yao [11] to a generalized mixed variational inequality in the setting of Hilbert spaces. Very recently, Fang et al. [10] generalized the notion of well-posedness by perturbations to a mixed variational inequality in Banach spaces. In the setting of Banach spaces, they established some metric characterizations and showed that the well-posedness by perturbations of a mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. They also derived some conditions under which the well-posedness by perturbations of the mixed variational inequality is equivalent to the existence and uniqueness of its solution.

In this paper, we further extend the notion of well-posedness by perturbations to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities in [10]. Under very mild conditions, we establish some metric characterizations for the well-posed generalized mixed variational inequality and show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. We also derive some conditions under which the well-posedness by perturbations of the generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

2. Preliminaries

Throughout this paper, unless stated otherwise, we always suppose that is a real reflexive Banach space with its dual and the duality pairing between and . For convenience, we denote strong (resp., weak) convergence by (resp., ). Let be a nonempty-valued multifunction, a single-valued mapping, and a proper, convex, and lower semicontinuous functional. Denote by the domain of , that is,

The generalized mixed variational inequality associated with is formulated as follows: which has been studied intensively (see, e.g., [11, 2325]).

In the following, we give some special cases of .(i) Whenever , the identity mapping of , reduces to the following mixed variational inequality associated with : which has been considered in [811, 26].(ii) Whenever , reduces to the following classical variational inequality: where denotes the indicator functional of a convex subset of .(iii) Whenever , reduces to the global minimization problem:

Suppose that is a parametric normed space, is a closed ball with positive radius, and is a fixed point. The perturbed problem of is always given by where is such that and is such that .

Now we recall some concepts and results.

Definition 2.1 (see [26]). A mapping is said to be(i) monotone if (ii) maximal monotone if is monotone and where denotes the family of all subsets of and .

Definition 2.2 (see [11]). A nonempty-valued multifunction is said to be monotone with respect to a single-valued mapping if, for all ,

Proposition 2.3 (Nadler’s Theorem [27]). Let be a normed vector space and the Hausdorff metric on the collection of all nonempty, closed, and bounded subsets of , induced by a metric in terms of , which is defined by for and in , where with . If and lie in , then, for any and any , there exists such that . In particular, whenever and are compact subsets in , one has .

Definition 2.4. 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 selection . For more details on this topic, the reader is referred to [28].

Definition 2.5 (see [29]). A mapping is said to be(i) coercive if (ii) bounded if is bounded for every bounded subset of ;(iii) hemicontinuous if, for any , the function from into is continuous at ;(iv) uniformly continuous if, for any neighborhood of 0 in , there exists a neighborhood of 0 in such that for all .

Clearly, the uniform continuity implies the continuity, and the continuity implies the hemicontinuity, but the converse is not true in general.

Definition 2.6. (i) A nonempty weakly compact-valued multifunction is said to be -hemicontinuous if, for any , the function from into is continuous at , where is the Hausdorff metric defined on .
(ii) A nonempty weakly compact-valued multifunction is said to be -continuous at a point if, for any , there exists such that, for all with , one has , where is the Hausdorff metric defined on . If this multifunction is -continuous at each , then one says that is -continuous.
(iii) A nonempty weakly compact-valued multifunction is said to be -uniformly continuous if, for any , there exists such that, for all with , one has , where is the Hausdorff metric defined on .

Remark 2.7. If a real Hilbert space, then Definition 2.6(i)–(iii) reduce to Definition  2.3 (ii)–(iv) in [11], respectively.

Lemma 2.8. Let be weakly continuous (i.e., continuous from the weak topology of to the weak topology of ), let be a nonempty weakly compact-valued multifunction which is -hemicontinuous and monotone with respect to , and let be proper and convex. Then, for a given , the following statements are equivalent:(i) there exists such that , for all ;(ii), for all .

Proof. Suppose that, for some , Since is monotone with respect to , one has Consequently,
Conversely, suppose that the last inequality is valid. Given any , we define for all . Replacing by in the left-hand side of the last inequality, one derives, for each , which hence implies that Since is a nonempty weakly compact-valued multifunction, both and are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in implies the strong closedness and that the weak boundedness of sets in is equivalent to the strong boundedness. Thus, it is known that both and lie in . From Proposition 2.3, it follows that, for each and each fixed , there exists a such that Since is weakly compact, it follows from the net that there exists some subnet which converges weakly to a point of . Without loss of generality, we may assume that as . Since is -hemicontinuous, one deduces that as Observe that, for each , that is, as . Since is weakly continuous, and hence, for , Thus, letting in the left-hand side of (2.15), we obtain that Finally let us show that the vector in the last inequality is not dependent on , that is, Indeed, take a fixed arbitrarily, and define for all . Utilizing Proposition 2.3, for each and , there exists such that Since is -hemicontinuous, we deduce that as Thus, one has, for each , This shows that as . Since is weakly continuous, and hence, for , Replacing , , and in (2.15) by , , and , respectively, one concludes that This immediately implies that inequality (2.21) is valid. This completes the proof.

Corollary 2.9 (see [11, Lemma  2.2]). Let be a real Hilbert space. Let be weakly continuous (i.e., continuous from the weak topology of to the weak topology of ), let be a nonempty weakly compact-valued multifunction which is -hemicontinuous and monotone with respect to , and let be proper and convex. Then, for a given , the following statements are equivalent:(i) there exists such that , for all ;(ii), for all .

Definition 2.10 (see [30]). Let be a nonempty, closed, and convex subset of . One says that is well-positioned if there exist and such that

Remark 2.11 (see [10, Remark  2.1]). (i) If is well-positioned, then is well-positioned for all .
(ii) As pointed out in [30, Remark  2.2], every nonempty compact convex set of a finite-dimensional space is well-positioned. Some useful properties and interesting applications have been discussed in [30, 31]. The following result is exacted from Proposition  2.1 of [30]. Also see [31, Proposition  2.1].

Lemma 2.12. Let be a nonempty, closed, and convex subset of a reflexive Banach space . If is well-positioned, then there is no sequence with such that origin is a weak limit of .

Definition 2.13 (see [28]). Let be a nonempty subset of . The measure of noncompactness of the set is defined by where means the diameter of a set.

Lemma 2.14 (see [10, Lemma  2.3]). Let , be nonempty, closed, and convex subsets of a real reflexive Banach space , and let be well-positioned. Suppose that as and . Then, there is no sequence with such that origin is a weak limit of .

3. Well-Posedness by Perturbations and Metric Characterizations

In this section, we generalize the concepts of well-posedness by perturbations to the generalized mixed variational inequality and establish their metric characterizations. In the sequel we always denote by and the strong convergence and weak convergence, respectively. Let be a fixed number.

Definition 3.1. Let be with . A sequence is called an -approximating sequence corresponding to for if there exists a sequence with (for all ) and a sequence of nonnegative numbers with such that Whenever , we say that is an approximating sequence corresponding to for . Clearly, every -approximating sequence corresponding to is -approximating corresponding to provided .

Definition 3.2. One says that is strongly (resp., weakly) -well-posed by perturbations if has a unique solution and, for any with , every -approximating sequence corresponding to converges strongly (resp., weakly) to the unique solution. In the sequel, strong (resp., weak) 0-well-posedness by perturbations is always called strong (resp., weak) well-posedness by perturbations. If , then strong (resp., weak) -well-posedness by perturbations implies strong (resp., weak) -well-posedness by perturbations.

Remark 3.3. (i) When is a Hilbert space and (for all ), Definitions  3.1 and 3.2 coincide with Definitions  3.1 and 3.2 of [11], respectively. (ii) When and the identity mapping of , Definitions 3.1 and 3.2 reduce to the definitions of approximating sequences of the classical variational inequality (see [12, 13]).

Definition 3.4. One says that is strongly (resp., weakly) generalized -well-posed by perturbations if has a nonempty solution set and, for any with , every -approximating sequence corresponding to has some subsequence which converges strongly (resp., weakly) to some point of . Strong (resp., weak) generalized 0-well-posedness by perturbations is always called strong (resp., weak) generalized well-posedness by perturbations. Clearly, if , then strong (resp., weak) generalized -well-posedness by perturbations implies strong (resp., weak) generalized -well-posedness by perturbations.

Remark 3.5. (i) When is a Hilbert space and (for all ), Definition 3.4 coincides with Definition  3.3 of [11]. (ii) When and the identity mapping of , Definition 3.4 reduces to the definition of strong (resp., weak) parametric -well-posedness in the generalized sense for the classical variational inequality (see [1214]). (iii) When and , Definition 3.4 coincides with the definition of well-posedness by perturbations introduced for a minimization problem [2, 3].

To derive the metric characterizations of -well-posedness by perturbations, we consider the following approximating solution set of : where denotes the closed ball centered at with radius . In this section, we always suppose that is a fixed solution of . Define It is easy to see that is the radius of the smallest closed ball centered at containing . Now, we give a metric characterization of strong -well-posedness by perturbations by considering the behavior of when .

Theorem 3.6. is strongly -well-posed by perturbations if and only if as .

Proof. Repeating almost the same argument as in the proof of [10, Theorem  3.1], we can easily obtain the desired result.

Remark 3.7. Theorem 3.6 improves Proposition  2.2 of [13], Theorem  3.1 of [9], and Theorem  3.1 of [10].

Now, we give an example to illustrate Theorem 3.6.

Example 3.8. Let , , , , , , and for all , . Clearly, is a solution of . For any , it follows that where Observe that Thus, we obtain Therefore, for sufficiently small . By trivial computation, we have By Theorem 3.6, is 2-well-posed by perturbations.

To derive a characterization of strong generalized -well-posedness by perturbations, we need another function which is defined by where is the solution set of and is defined as in Proposition 2.3.

Theorem 3.9. is strongly generalized -well-posed by perturbations if and only if is nonempty compact and as .

Proof. Repeating almost the same argument as in the proof of [10, Theorem  3.2], we can readily derive the desired result.

Example 3.10. Let , , , , , , and for all , . Clearly, is a solution of . Repeating the same argument as in Example 3.8, we obtain that, for any , for sufficiently small . By trivial computation, we have By Theorem 3.9, is generalized -well-posed by perturbations.

The strong generalized -well-posedness by perturbations can be also characterized by the behavior of the noncompactness measure .

Theorem 3.11. Let be finite dimensional, weakly continuous (i.e., continuous from the product of the norm topology of and weak topology of to the weak topology of ), a nonempty weakly compact-valued multifunction which is -continuous, and a continuous functional such that is proper and convex. Then, is strongly generalized -well-posed by perturbations if and only if , for all and as .

Proof. First, we will prove that is closed for all . Let with . Then, there exist and with (for all ) such that Without loss of generality, we may assume since is finite dimensional. Since is a nonempty weakly compact-valued multifunction, and are nonempty weakly compact and hence are nonempty, weakly closed, and weakly bounded. Note that the weak closedness of sets in implies the strong closedness and that the weak boundedness of sets in is equivalent to the strong boundedness. Thus, it is known that and lie in . According to Proposition 2.3, for each and , there exists such that Since is -continuous, one deduces that Also, since is weakly compact, it follows from that there exists some subsequence of which converges weakly to a point of . Without loss of generality, we may assume that Consequently, one has, for each , This implies that as . Taking into account the weak continuity of , we immediately obtain that and hence, for each , that is, Therefore, it follows from (3.13) and the continuity of that This shows that and so is closed.
Second, we show that It is obvious that . Let . Let be a sequence of positive numbers such that . Then, and so there exist and such that It is clear that as . Since is weakly compact, it follows from that there exists some subsequence of which converges weakly to a point of . Without loss of generality, we may assume that Note that is weakly continuous. Thus, and hence, letting in the last inequality, we get For any and , putting in (3.26), we have This implies that Letting in the last inequality, we get Consequently, and so (3.22) is proved.
Now, we suppose that is strongly generalized -well-posed by perturbations. By Theorem 3.9, is nonempty compact and . Then, since for all . Observe that, for all , Taking into account the compactness of , we get
Conversely, we suppose that , for all , and as . Since is increasing with respect to , by the Kuratowski theorem [28, page 318], we have from (3.22) and is nonempty compact. By Theorem 3.9, is strongly generalized -well-posed by perturbations.

Remark 3.12. Theorem  3.3 of [10] generalizes Theorem  3.2 of [9]. Theorem 3.11 generalizes Theorem  3.2 of [11] from the case of strong generalized -well-posedness in the setting of Hilbert spaces to the case of strong generalized -well-posedness by perturbations in the setting of Banach spaces. Furthermore, Theorem 3.11 improves, extends, and develops [10, Theorem  3.3] in the following aspects.(i) The mixed variational inequality problem (MVI) in [10, Theorem  3.3] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem (GMVI) with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. Moreover, the concept of strong generalized -well-posedness by perturbations for MVI in [10, Theorem  3.3] is extended to develop the concept of strong generalized -well-posedness by perturbations for GMVI.(ii) Since the generalized mixed variational inequality problem (GMVI) is more general and more complicated than the mixed variational inequality problem (MVI), the assumptions in Theorem 3.11 are very different from the ones in [10, Theorem  3.3]; for instance, in Theorem 3.11, let be finite dimensional, weakly continuous, and a nonempty weakly compact-valued multifunction which is -continuous, but, in [10, Theorem  3.3], let be finite dimensional, a continuous mapping.(iii) The technique of proving strong generalized -well-posedness by perturbations for GMVI in Theorem 3.11 is very different from the one for MVI in [10, Theorem  3.3] because our technique depends on the well-known Nadler’s Theorem [27], the -continuity of nonempty weakly compact-valued multifunction and the property of the Hausdorff metric .

Remark 3.13. Clearly, any solution of is a solution of the problem: find such that, for some , but the converse is not true in general. To show this, let , , , and for all . It is easy to verify that the solution set of is empty and 0 is the unique solution of the corresponding problem with . If is proper and convex, then and problem have the same solution (this fact has been shown in the proof of Theorem 3.11).

4. Links with the Well-Posedness by Perturbations of Inclusion Problems

Lemaire et al. [20] introduced the concept of well-posedness by perturbations for an inclusion problem. In this section, we will show that the well-posedness by perturbations of a generalized mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem. Let us recall some concepts. Let . The inclusion problem associated with is defined by The perturbed problem of is given by where is such that .

Definition 4.1 (see [20]). Let be with . A sequence is called an approximating sequence corresponding to for if for all and , or, equivalently, there exists such that as .

Definition 4.2 (see [20]). One says that is strongly (resp., weakly) well-posed by perturbations if it has a unique solution and, for any with , every approximating sequence corresponding to converges strongly (resp., weakly) to the unique solution of . is said to be strongly (resp., weakly) generalized well posed by perturbations if the solution set of is nonempty and, for any with , every approximating sequence corresponding to has a subsequence which converges strongly (resp., weakly) to a point of .

Let be a proper, convex, and lower semicontinuous functional. Denote by and the subdifferential and -subdifferential of , respectively, that is, It is known that is maximal monotone and for all and for all . In terms of , is equivalent to the following inclusion problem:

In other words, we have the following lemma.

Lemma 4.3. Let be a fixed point, and let be a proper, convex, and lower semicontinuous functional. Then, the following statements are equivalent:(i) for all and some ;(ii).

Proof. Observe that for all and some . The desired result follows immediately from the above relations.

Naturally, we consider the perturbed problem of as follows: where is such that is proper, convex, and lower semicontinuous for all , and .

The following theorems establish the relations between the strong (resp., weak) well-posedness by perturbations of generalized mixed variational inequalities and the strong (resp., weak) well-posedness by perturbations of inclusion problems.

Theorem 4.4. Let be continuous for each , let be weakly continuous, let be a nonempty weakly compact-valued multifunction which is -hemicontinuous and monotone with respect to for each , and let be a continuous functional with respect to the product of the norm topology of and weak topology of such that the following conditions hold:(i)   is proper and convex for all ;(ii) is well-positioned and ;(iii) whenever , where is defined as in Proposition 2.3.
Then, is weakly well-posed by perturbations whenever has a unique solution.

Proof. Suppose that has a unique solution . Let be with , and let be an approximating sequence corresponding to for . Then, there exists such that . Further, there exists such that with . It follows that We claim that is bounded. Indeed, if is unbounded, without loss of generality, we may assume that . Let By conditions (i)-(ii), we get . Note that So, is bounded. Since is reflexive, it follows from the boundedness of that there exists some subsequence of which converges weakly to a point of . Hence, without loss of generality, we may assume that and . It follows from Lemma 2.14 and conditions (ii)-(iii) that . For any , observe that, for all , Since is the unique solution of , there exists some such that Also, since is monotone with respect to , we deduce that, for , , and , In addition, we have by virtue of the convexity of . It follows from (4.7)–(4.13) that Moreover, it is easy to see from that . Further, since is continuous for each and , it is known that and is bounded. Consequently, In the meantime, since is a continuous functional with respect to the product of the norm topology of and weak topology of , we conclude from and that and as . Now, letting in (4.14) we get Since is the unique solution of , from (4.16) we get which implies that Note that is weakly continuous, that is proper and convex, and that is a nonempty weakly compact-valued multifunction which is -hemicontinuous and monotone with respect to . Hence, all conditions of Lemma 2.8 are satisfied. Thus, it follows from Lemma 2.8 that there exists such that Therefore, is a solution of , a contradiction. This shows that is bounded.
Let be any subsequence of such that as . It follows from (4.7) that Since is continuous for each and , it is known that and is bounded. Consequently, Moreover, since is a continuous functional with respect to the product of the norm topology of and weak topology of , we conclude from and that and as . Note that is monotone with respect to . Hence, it follows that for, and , This together with Lemma 2.8 yields that there exists such that Consequently, solves . We must have since has a unique solution . Therefore, converges weakly to and so is weakly well-posed by perturbations.

Remark 4.5. Theorem 4.4 improves, extends, and develops [10, Theorem  4.1] in the following aspects. (i) The mixed variational inequality problem (MVI) in [10, Theorem  4.1] is extended to develop the more general problem, that is, the generalized mixed variational inequality problem with a nonempty weakly compact-valued multifunction in the setting of Banach spaces. Moreover, the inclusion problem corresponding to MVI in [10, Theorem  4