Hölder Continuity of a Parametric Generalized Variational Inequality

Abstract

By using the classic metric projection method, we obtain sufficient conditions for Hölder continuity of the nonunique solution mapping for a parametric generalized variational inequality with respect to data perturbation. The result is different from the recent ones in the literature and has a strong geometric flavor.

1. Introduction

Variational inequality is a very general mathematical format, which embraces the formats of several disciplines, as those for equilibrium problems of mathematical physics, those from game theory, and those for transportation equilibrium problems. Thus, it is important to derive results for parametric variational inequality concerning the properties of the solution mapping when the problem’s data vary.

It is well known that the Hölder continuity of the perturbed solution mapping of variational inequalities is one aspect of stability analysis. In general, the stability analysis of solution mappings for parametric variational inequalities includes semicontinuity, Lipschitz continuity, and Hölder continuity of solution mappings. Most of the research in the area of stability analysis for variational inequalities has been performed under assumptions which implied the local uniqueness of perturbed solutions so that the solution mapping was single valued. By using the metric projection method, Dafermos [1] first derived sufficient conditions for the local uniqueness, continuity (or Lipschitz continuity), and differentiability of a perturbed solution of variational inequalities. Using the same techniques, Yen [2] and Yen and Lee [3] later obtained uniqueness of the solution for a classical perturbed variational inequality and showed that the solution mapping is Hölder continuous with respect to parameters. Then, Domokos [4] extended these results of [1–3] to the case of reflexive Banach spaces.

There have also been a few papers to study more general situations where the solution sets of variational inequalities may be set valued. Robinson [5] investigated characterizations and existences of solutions for a generalized equation involving set-valued mappings under certain metric regularity hypotheses. As applications, he also studied some Lipschitz-type continuity property of the solution mapping for perturbed variational inequalities defined on closed convex sets. Ha [6] used the degree theory to derive some sufficient conditions, which guarantee the existence of nonunique perturbed solutions of nonlinear complementarity problems in a neighborhood of a reference point. Under the Hausdorff metric and the strong quasimonotonicity, Lee et al. [7] showed that nonunique solution mapping for a perturbed vector variational inequality is Hölder continuous with respect to parameters. Based on the scalarization technique and degree theoretic method, Wong [8] recently discussed the lower semicontinuity of the nonunique solution mapping for a perturbed vector variational inequality, where the operator may not be strongly monotone.

Although there have been many papers to study solution stability of perturbed variational inequalities, very few papers focus on such a study for perturbed generalized variational inequalities. Recently, by virtue of the strong quasimonotonicity, Ait Mansour and Aussel [9] have obtained a result on Hölder continuity of the nonunique solution mapping of a perturbed generalized variational inequality defined by strongly quasimonotone set-valued maps in the case of finite dimensional spaces. Without conditions related to the degree theory and the metric projection, Kien [10] derived sufficient conditions for the lower semicontinuity of nonunique perturbed solutions of a perturbed generalized variational inequality in reflexive Banach spaces.

Motivated by the work reported in [1–4, 9, 10], our main interest is to investigate the Hölder continuity of nonunique perturbed solution mapping for a perturbed generalized variational inequality defined on perturbed feasible sets. We first introduce a locally strong monotone set-valued operator, which is weaker than the corresponding ones in [1–4, 9, 10] and use the projection techniques of [1, 2, 4] to derive some sufficient conditions, which guarantee the Hölder continuity of the locally nonunique solution sets for a perturbed generalized variational inequality with respect to parameters.

The rest of the paper is organized as follows. In Section 2, we introduce the parametric generalized variational inequality and recall the definitions and corresponding results which are needed in this paper. Then, we derive a relation between the Pseudo-Hölder property of a set-valued mapping and the Hölder property of projection mapping. In Section 3, we first introduce the key assumption which is weaker than the corresponding ones in [1–4, 9, 10] and the relative assumptions. Under these assumptions, we follow the projection technique of [1–4] mainly to study the behavior and Hölder property of the nonunique perturbed solution mapping for a parametric generalized variational inequality without the differentiability assumption and the degree theory. An example is also given to illustrate that our main result is applicable.

2. Preliminaries

Throughout this paper, if not other specified, let be a Hilbert space which is equipped with an inner product and with the norm , respectively. Let , be two parameter sets of the normed spaces, and let denote the closed ball with the center and the radius . The Hausdorff metric between two nonempty subsets , of is defined by where .

Let be a set-valued mapping with nonempty closed convex values, and let be a set-valued mapping with nonempty compact values. Consider the following parametric generalized variational inequality consisting of finding such that there exists with

For each , the solution set of (2) is defined by

We first recall the following definitions and results which are needed in the sequel. Let be a nonempty closed convex subset of , and let denote the projection of onto . It is well known that the projection operator is a nonexpansive operator. From [11], we have the following result.

Lemma 1. For each , if and only if

Definition 2. Let be a set-valued mapping with nonempty closed convex values, and . is said to be -Pseudo-Hölder continuous at if and only if there exist a neighborhood of , a neighborhood of such that where , .
Note that when , Definition 2 reduces to the Aubin property in [12].

From the definition of norm, we can easily obtain the following result.

Lemma 3. If , then the norm mapping satisfies

Proof. If or , the conclusion is trivial. Otherwise, and . Hence, .

The following Lemma, which is an extension of Lemma 1.1 in [2], plays an important role in this paper.

Lemma 4. Assume that is -Pseudo-Hölder continuous at and is a closed bounded convex neighborhood of . Then, there exist a neighborhood of , a neighborhood of such that , , ,

Proof. We shall use the similar arguments of [2, 13] to prove the result.
Since is -Pseudo-Hölder continuous at , there exist a neighborhood of , a neighborhood of which satisfy (5). Choose such that , and fix a number satisfying
Let
We claim that these , satisfy (7). Indeed, since and , then (5) implies that Furthermore, (5) implies Then, it follows from (8) and (10) that there exists such that which implies Let . Then, it follows from (9) and the definition of projection operator that, for any , , which means Assume to the contrary that (7) is false. Then, there exist , such that where and . From (16), we can easily see that Then, by virtue of (11), there exist and such that Therefore, (8), (10), (11), (18), and (19) and Lemma 3 together yield that This means that Similarly, we have Then, it follows from (19), (21), (22), and the definition of projection operator that Let , , and . From (19) and (23), we have Applying (17) and (19), we obtain It follows from (8) and Lemma 3 that Then, which together with (25) implies . Since is a convex subset of and is a convex neighborhood of , we have Since , by the property of the projection, we have Then, it follows that Hence, by (24), Since (17) implies that , we have
On the other hand, by virtue of (9), (18), and (21), we have As was chosen so that , then which is a contradiction to (32). The proof is complete.

Remark 5. Lemma 2.1 of [2] has (7) with . When or ; Lemma 4 is always satisfied. Therefore, Lemma 4 generalizes and improves Lemma 2.1 of [2].

3. Main Results

In this section, we always assume that is a unique solution to (2) at given parameter . Let be a closed bounded convex neighborhood of , let be a neighborhood of , and let be a neighborhood of . In order to analyze the behavior of the set-valued mapping around when is close to , we need to consider the following restrict problem.

For each , find such that there exists with Similarly, for , the local solution set of (2) is defined by

To obtain our main result, we introduce the following assumptions for a neighborhood of and a neighborhood of .(H₁)There exist , and such that for any satisfying where is a vector-valued mapping.(H₂)There exists such that for any satisfying where is a vector-valued mapping.(H₃) There exists such that

Assumption (H₁) states is locally strong --Hölder continuous at , while assumption (H₂) is the requirement that is locally weak -monotone at with a coefficient independent to .

Remark 6. If is a single-valued mapping and , then assumptions (H₁) and (H₂) collapse to the locally Lipschitz at and locally strongly monotone at with a coefficient independent to of [2], respectively.
If is a set-valued mapping with nonempty compact values, then for any , , , ,
Obviously, assumption (H₂) is weaker than the following condition which was introduced in [14].
For all , , there exists such that , , It is well know that implies that the local solution mapping LS to (2) is single valued. However, when is replaced by assumption (H₂), the local solution mapping LS, in general, is not single valued; that is, LS may be a set-valued mapping.
The following example is given to illustrate the existence of a class of set-valued mapping satisfying (H₁) and (H₂). It should be noted that the example also illustrates that (H₂) is weaker than in [14].

Example 7. Let and . Let be a set-valued mapping which is defined by where and . Obviously, is not single-valued mapping; or is -Lipschitz on ; for any , or is -strongly monotone on .
However, the set-valued mapping does not satisfy . Indeed, take and . Then, we can easily see that .

From Lemma 1 and the definition of the local solution for (2), we can get the following result.

Lemma 8. For each , the problem (2) has a local solution if and only if is a fixed point of the set-valued mapping defined by where is a fixed number.

Proof. Let be a local solution of (2) at parameters , . Then, and exist such that Then, for any given , which along with Lemma 1 implies that Therefore, is a fixed point of .
Since the converse can be similarly proved, we omit it.

Proposition 9. Suppose that assumptions (H₁)–(H₃) hold. Then,
(a) for any , , defined by (43) is a set-valued -contradiction mapping on ; that is, for any , where ;
(b) for any , , the solution set to (2) is nonempty compact.

Proof. (a) For each , , and , since is a compact valued mapping and the projection mapping is continuous, then defined by (43) is a compact valued mapping. Therefore, the left part of (47) is well defined. Let be arbitrarily given. Then, there exists such that Let , which along with (43) yields that .
Using assumption (H₁) and the property that the projection onto a fixed closed convex subset is a nonexpansive mapping, we obtain
On the other hand, it follows from assumptions (H₁) and (H₂) that which implies that
Combing (49) and (51), we obtain By assumption (H₃), and hence we have
Using the same arguments, we can show that which along with (53) implies that is a set-valued -contradiction mapping on .
(b) By (a) and the Nadler fixed point theorem in [15], has a fixed point for each . Hence, for any , by Lemma 8. Moreover, we claim that is a closed subset. Indeed, let with as . Then, it follows from Lemma 8 that Therefore, by (a), we obtain which implies that and by Lemma 8. Hence, is a compact set since is compact. The proof is complete.

As an immediate consequence of Lemma 8, Proposition 9, and the Banach fixed point theorem, we have the following result.

Corollary 10. For any , , the vector-valued mapping defined by where , is a fixed number. Furthermore, assume that assumptions (H₁)–(H₃) are satisfied.
Then, for any , , is a vector-valued -contradiction mapping and has a unique fixed point .

Now, we state our main result.

Theorem 11. Suppose that is -Pseudo-Hölder continuous at . Furthermore, suppose that assumptions (H₁)–(H₃) hold. Then, there exist a neighborhood of , a neighborhood of such that
(a) for all ,
(b) for all , and for any is a solution to (2) at parameters , .

Proof. By the Pseudo-Hölder continuity of and Lemma 4, there exist a neighborhood of , a neighborhood of such that
Since is compact, then . Let satisfy assumption (H₃) and where denotes the interior of .
Fix . For any , let be arbitrarily given. Then, there exists such that
For any , by Corollary 10, has a unique fixed point . Then, it follows from Lemma 8 that is a solution to (2) at parameters ; that is,
Then, using assumption (H₁), Proposition 9, and the property that the projection onto a fixed convex subset is a nonexpansive mapping, we obtain where , , , and defined as in Proposition 9.
Noting that the arbitrariness of , we obtain that
Then, substituting in (64) and by the uniqueness of we obtain that where .
According to (60), we have , which along with (59), (65) yields that
We claim that there exist a neighborhood of , a neighborhood of such that (58) is satisfied. Indeed, let be so small that . Using (66), assumption (H₁) and (5) of Remark 6, we can find a neighborhood of , a neighborhood of such that for any ,
Therefore, (60) and (67) together yield that, for all , , Hence, for any , , it follows from (59), (64), and (68) that
On the other hand, it follows from the symmetrical roles of , that Therefore, (69) and (70) together yield that for any , ,
Thus, we have established (58) and obtained in (67), which implies that for all is the solution in to (2) at parameters , . The proof is complete.

The following example is given to illustrate that the local solution set of (2) is not single valued and Theorem 11 is applicable.

Example 12. Let , , be a given point and let , be a neighborhood of . Let be a set-valued mapping which is defined by and defined by
Clearly, is a unique solution to (2) at parameter . Let be a neighborhood of and let be taken arbitrarily. Obviously, is --Hölder continuous and strongly -monotone on ; there exists satisfying ; the map is -Hölder continuous with closed convex values on . Therefore, all assumptions of Theorem 11 are satisfied.
Moreover, by virtue of Lemma 8, we note that is a local solution of (2) at parameter if and only if satisfies the inclusion
Let be any point. By a simple computation, we obtain Then, it follows that, for each and any , Hence, By virtue of Lemma 8, direct computation shows that, for any , Then, the local solution set of (2) satisfies (58) for any neighborhoods of , of . Hence, Theorem 11 is applicable.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their deep gratitude to the anonymous referees and the associate editor for their valuable comments and suggestions which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 11201509).