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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.

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 [

There have also been a few papers to study more general situations where the solution sets of variational inequalities may be set valued. Robinson [

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 [

Motivated by the work reported in [

The rest of the paper is organized as follows. In Section

Throughout this paper, if not other specified, let

Let

For each

We first recall the following definitions and results which are needed in the sequel. Let

For each

Let

Note that when

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

If

If

The following Lemma, which is an extension of Lemma 1.1 in [

Assume that

We shall use the similar arguments of [

Since

Let

We claim that these

On the other hand, by virtue of (

Lemma 2.1 of [

In this section, we always assume that

For each

To obtain our main result, we introduce the following assumptions for a neighborhood

There exist

There exists

There exists

Assumption (H_{1}) states _{2}) is the requirement that

_{1}) and (H_{2}) collapse to the locally Lipschitz at

_{2}) is weaker than the following condition which was introduced in [

_{2}), the local solution mapping LS, in general, is not single valued; that is, LS may be a set-valued mapping.

_{1}) and (H_{2}). It should be noted that the example also illustrates that (H_{2}) is weaker than

Let

However, the set-valued mapping does not satisfy

From Lemma

For each

Let

Since the converse can be similarly proved, we omit it.

Suppose that assumptions (_{1})–(_{3}) hold. Then,

(a) for any

(b) for any

(a) For each

Using assumption (H_{1}) 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 (H_{1}) and (H_{2}) that

Combing (_{3}),

Using the same arguments, we can show that

(b) By (a) and the Nadler fixed point theorem in [

As an immediate consequence of Lemma

For any _{1})–(_{3}) are satisfied.

Then, for any

Now, we state our main result.

Suppose that _{1})–(_{3}) hold. Then, there exist a neighborhood

(a) for all

(b) for all

By the Pseudo-Hölder continuity of

Since _{3}) and

Fix

For any

Then, using assumption (H_{1}), Proposition

Noting that the arbitrariness of

Then, substituting

According to (

We claim that there exist a neighborhood _{1}) and (

Therefore, (

On the other hand, it follows from the symmetrical roles of

Thus, we have established (

The following example is given to illustrate that the local solution set of (

Let

Clearly,

Moreover, by virtue of Lemma

Let

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

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).