Hölder Continuity of a Parametric Generalized Variational Inequality

By using the classic metric projection method, we obtain sufficient conditions for H¨older 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.


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][2][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 (H 2 ) 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][2][3][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.

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 Λ, 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 ( , ) := inf ∈ ‖ − ‖.
Let : Λ be a set-valued mapping with nonempty closed convex values, and let : × be a setvalued mapping with nonempty compact values. Consider the following parametric generalized variational inequality consisting of finding ∈ ( ) such that there exists * ∈ ( , ) with For each ( , ) ∈ Λ × , 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.
From the definition of norm, we can easily obtain the following result.
The following Lemma, which is an extension of Lemma 1.1 in [2], plays an important role in this paper.

Main Results
In this section, we always assume that ∈ ( , ) is a unique solution to (2) at given parameter ( , ) ∈ Λ × . 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.
To obtain our main result, we introduce the following assumptions for a neighborhood of and a neighborhood of .
→ is a single-valued mapping and = 1, then assumptions (H 1 ) and (H 2 ) collapse to the locally Lipschitz at ( , ) and locally strongly monotone at with a coefficient independent to ∈ ∩ of [2], respectively.
(H 2 ) For all , ∈ ∩ , ∈ ∩ , there exists > 0 such that ∀ * ∈ ( , ), * ∈ ( , ), It is well know that (H 2 ) implies that the local solution mapping LS to (2) is single valued. However, when (H 2 ) is replaced by assumption (H 2 ), the local solution mapping LS, in general, is not single valued; that is, LS may be a set-valued mapping.
(4) The following example is given to illustrate the existence of a class of set-valued mapping satisfying (H 1 ) and (H 2 ). It should be noted that the example also illustrates that (H 2 ) is weaker than (H 2 ) in [14].
However, the set-valued mapping does not satisfy (H 2 ). Indeed, take , ∈ : > and ̸ = 0. Then, we can easily see that From Lemma 1 and the definition of the local solution for (2), we can get the following result.
Since the converse can be similarly proved, we omit it.