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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Li-na Wang 1 and Xiao-bing Li 2

Academic Editor:Sheng-Jie Li

1, Chongqing Water Resources and Electric Engineering College, Chongqing 402160, China
2, College of Sciences, Chongqing Jiaotong University, Chongqing 400074, China

Received 17 January 2014; Accepted 10 March 2014; 22 April 2014

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.

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 ( _{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 [1-4] 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 E be a Hilbert space which is equipped with an inner product Y9; · , · YA; and with the norm || · || , respectively. Let Λ , M be two parameter sets of the normed spaces, and let B ( x , τ ) denote the closed ball with the center x and the radius τ . The Hausdorff metric between two nonempty subsets A , B of E is defined by [figure omitted; refer to PDF] where d ( a , B ) : = _{inf ... b ∈ B} || a - b || .

Let K : Λ ... E be a set-valued mapping with nonempty closed convex values, and let T : E × M ... E be a set-valued mapping with nonempty compact values. Consider the following parametric generalized variational inequality consisting of finding x ∈ K ( λ ) such that there exists ^{z *} ∈ T ( x , μ ) with [figure omitted; refer to PDF]

For each ( λ , μ ) ∈ Λ × M , the solution set of (2) is defined by [figure omitted; refer to PDF]

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

Lemma 1.

For each z ∈ E , x = _{P K} ( z ) if and only if [figure omitted; refer to PDF]

Definition 2.

Let K : Λ ... E be a set-valued mapping with nonempty closed convex values, λ ¯ ∈ Λ and x ¯ ∈ K ( λ ¯ ) . K ( · ) is said to be k . α -Pseudo-Hölder continuous at ( λ ¯ , x ¯ ) if and only if there exist a neighborhood V of λ ¯ , a neighborhood W of x ¯ such that [figure omitted; refer to PDF] where k > 0 , α > 0 .

Note that when α = 1 , Definition 2 reduces to the Aubin property in [12].

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

Lemma 3.

If α > 0 , then the norm mapping || · || : E [arrow right] R satisfies [figure omitted; refer to PDF]

Proof.

If x = 0 or y = 0 , the conclusion is trivial. Otherwise, || x + y || / ( || x || + || y || ) ...4; 1 and ^{( || x + y || / ( || x || + || y || ) ) α} ...4; 1 . Hence, ^{|| x + y || α} ...4; ^{( || x || + || y || ) α} .

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

Lemma 4.

Assume that K ( · ) is k . α -Pseudo-Hölder continuous at ( λ ¯ , x ¯ ) and X is a closed bounded convex neighborhood of x ¯ . Then, there exist a neighborhood _{X 1} of x ¯ , a neighborhood _{V 1} of λ ¯ such that ∀ y ∈ _{X 1} , ∀ λ , ^{λ [variant prime]} ∈ _{V 1} ∩ Λ , [figure omitted; refer to PDF]

Proof.

We shall use the similar arguments of [2, 13] to prove the result.

Since K ( · ) is k . α -Pseudo-Hölder continuous at ( λ ¯ , x ¯ ) , there exist a neighborhood W of x ¯ , a neighborhood V of λ ¯ which satisfy (5). Choose τ ∈ ( 0 , k / 6 ) such that B ( x ¯ , τ ) ⊂ X ∩ W , and fix a number δ > 0 satisfying [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF]

We claim that these _{X 1} , _{V 1} satisfy (7). Indeed, since B ( x ¯ , τ ) ⊂ X ∩ W and _{V 1} ⊂ V , then (5) implies that [figure omitted; refer to PDF] Furthermore, (5) implies [figure omitted; refer to PDF] Then, it follows from (8) and (10) that there exists x ∈ K ( λ ) such that [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] Let x ∈ K ( λ ) ∩ B ( x ¯ , τ / 8 ) . Then, it follows from (9) and the definition of projection operator that, for any y ∈ _{X 1} , λ ∈ _{V 1} ∩ Λ , [figure omitted; refer to PDF] which means [figure omitted; refer to PDF] Assume to the contrary that (7) is false. Then, there exist λ , ^{λ [variant prime]} ∈ _{V 1} ∩ Λ , y ∈ _{X 1} such that [figure omitted; refer to PDF] where a = _{P K ( λ ) ∩ X} ( y ) and b = _{P K ( ^{λ [variant prime]} ) ∩ X} ( y ) . From (16), we can easily see that [figure omitted; refer to PDF] Then, by virtue of (11), there exist _{a 1} ∈ K ( λ ) and _{b 1} ∈ K ( ^{λ [variant prime]} ) such that [figure omitted; refer to PDF] Therefore, (8), (10), (11), (18), and (19) and Lemma 3 together yield that [figure omitted; refer to PDF] This means that [figure omitted; refer to PDF] Similarly, we have [figure omitted; refer to PDF] Then, it follows from (19), (21), (22), and the definition of projection operator that [figure omitted; refer to PDF] Let u : = || y - _{a 1} || , v : = || y - a || , and w : = || _{a 1} - a || . From (19) and (23), we have [figure omitted; refer to PDF] Applying (17) and (19), we obtain [figure omitted; refer to PDF] It follows from (8) and Lemma 3 that [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] which together with (25) implies w > k ^{|| λ - λ [variant prime] || α / 2} . Since K ( λ ) is a convex subset of E and X is a convex neighborhood of x ¯ , we have [figure omitted; refer to PDF] Since a = _{P K ( λ ) ∩ X} ( y ) , by the property of the projection, we have [figure omitted; refer to PDF] Then, it follows that [figure omitted; refer to PDF] Hence, by (24), [figure omitted; refer to PDF] Since (17) implies that λ ...0; ^{λ [variant prime]} , we have [figure omitted; refer to PDF]

On the other hand, by virtue of (9), (18), and (21), we have [figure omitted; refer to PDF] As τ was chosen so that τ ∈ ( 0 , k / 6 ) , then [figure omitted; refer to PDF] which is a contradiction to (32). The proof is complete.

Remark 5.

Lemma 2.1 of [2] has (7) with α = 1 . When 0 < α ...4; 1 or α > 1 ; 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 x ¯ ∈ S ( λ ¯ , μ ¯ ) is a unique solution to (2) at given parameter ( λ ¯ , μ ¯ ) ∈ Λ × M . Let X be a closed bounded convex neighborhood of x ¯ , let V be a neighborhood of λ ¯ , and let U be a neighborhood of μ ¯ . In order to analyze the behavior of the set-valued mapping S ( λ , μ ) around S ( λ ¯ , μ ¯ ) when ( λ , μ ) is close to ( λ ¯ , μ ¯ ) , we need to consider the following restrict problem.

For each ( λ , μ ) ∈ ( V ∩ Λ ) × ( U ∩ M ) , find x ∈ K ( λ ) ∩ X such that there exists ^{z *} ∈ T ( x , μ ) with [figure omitted; refer to PDF] Similarly, for ( λ , μ ) ∈ ( V ∩ Λ ) × ( U ∩ M ) , the local solution set of (2) is defined by [figure omitted; refer to PDF]

To obtain our main result, we introduce the following assumptions for a neighborhood V of λ ¯ and a neighborhood U of μ ¯ .

(H₁ ): There exist h > 0 , l > 0 and β > 0 such that for any t ( · , · ) ∈ T ( · , · ) satisfying [figure omitted; refer to PDF] where t : E × M [arrow right] E is a vector-valued mapping.

(H₂ ): There exists m > 0 such that for any t ( · , · ) ∈ T ( · , · ) satisfying [figure omitted; refer to PDF] where t : E × M [arrow right] E is a vector-valued mapping.

(H₃ ) : There exists ρ > 0 such that [figure omitted; refer to PDF]

Assumption (H₁ ) states T ( · , · ) is locally strong h . 1 - l . β -Hölder continuous at ( x ¯ , μ ¯ ) , while assumption (H₂ ) is the requirement that T ( · , μ ) is locally weak m -monotone at x ¯ with a coefficient independent to μ ∈ U ∩ M .

Remark 6.

( 1 ) If T : E × M [arrow right] E is a single-valued mapping and β = 1 , then assumptions (H₁ ) and (H₂ ) collapse to the locally Lipschitz at ( x ¯ , μ ¯ ) and locally strongly monotone at x ¯ with a coefficient independent to μ ∈ U ∩ M of [2], respectively.

( 2 ) If T ( · , · ) is a set-valued mapping with nonempty compact values, then for any x , ^{x [variant prime]} ∈ X ∩ E , μ , ^{μ [variant prime]} ∈ U ∩ M , [figure omitted; refer to PDF]

( 3 ) Obviously, assumption (H₂ ) is weaker than the following condition which was introduced in [14].

( ^{H 2 [variant prime]} ) For all x , ^{x [variant prime]} ∈ X ∩ E , μ ∈ U ∩ M , there exists m > 0 such that ∀ ^{z *} ∈ T ( x , μ ) , ^{z [variant prime] *} ∈ T ( ^{x [variant prime]} , μ ) , [figure omitted; refer to PDF] It is well know that ( ^{H 2 [variant prime]} ) implies that the local solution mapping LS to (2) is single valued. However, when ( ^{H 2 [variant prime]} ) 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.

( 4 ) 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 ( ^{H 2 [variant prime]} ) in [14].

Example 7.

Let E = ... and M = [ 0,1 ] . Let T : E × M ... E be a set-valued mapping which is defined by [figure omitted; refer to PDF] where _{t 1} ( x , μ ) = x - μ and _{t 2} ( x , μ ) = x + μ . Obviously, T ( x , μ ) is not single-valued mapping; _{t 1} ( · , · ) or _{t 2} ( · , · ) is 1.1 -Lipschitz on E × M ; for any μ ∈ M , _{t 1} ( · , μ ) or _{t 2} ( · , μ ) is 1 -strongly monotone on E .

However, the set-valued mapping does not satisfy ( ^{H 2 [variant prime]} ) . Indeed, take x , y ∈ E : x > y and μ ...0; 0 . Then, we can easily see that Y9; _{t 1} ( x , μ ) - _{t 2} ( y , μ ) , x - y YA; = ^{( x - y ) 2} - 2 μ ( x - y ) < ^{( x - y ) 2} .

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

Lemma 8.

For each ( λ , μ ) ∈ ( V ∩ Λ ) × ( U ∩ M ) , the problem (2) has a local solution x ( λ , μ ) if and only if x : = x ( λ , μ ) is a fixed point of the set-valued mapping G : ( X ∩ E ) × ( _{V 1} ∩ Λ ) × ( U ∩ M ) ... X ∩ E defined by [figure omitted; refer to PDF] where ρ > 0 is a fixed number.

Proof.

Let x ( λ , μ ) be a local solution of (2) at parameters λ ∈ V ∩ Λ , μ ∈ U ∩ M . Then, x : = x ( λ , μ ) ∈ K ( λ ) ∩ X and t ¯ ( x , μ ) ∈ T ( x , μ ) exist such that [figure omitted; refer to PDF] Then, for any given ρ > 0 , [figure omitted; refer to PDF] which along with Lemma 1 implies that [figure omitted; refer to PDF] Therefore, x ( λ , μ ) is a fixed point of G ( · , λ , μ ) .

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

Proposition 9.

Suppose that assumptions (H ₁ )-(H ₃ ) hold. Then,

(a) for any λ ∈ V ∩ Λ , μ ∈ U ∩ M , G ( · , λ , μ ) defined by (43) is a set-valued θ -contradiction mapping on X ∩ E ; that is, for any x , ^{x [variant prime]} ∈ E ∩ X , [figure omitted; refer to PDF] where θ : = 1 - 2 ρ m + ^{ρ 2 h 2} < 1 ;

(b) for any λ ∈ V ∩ Λ , μ ∈ U ∩ M , the solution set to (2) is nonempty compact.

Proof.

(a) For each x ∈ X ∩ E , λ ∈ V ∩ Λ , and μ ∈ U ∩ M , since T ( x , μ ) is a compact valued mapping and the projection mapping _{P K ( λ ) ∩ X} ( · ) is continuous, then G ( x , λ , μ ) defined by (43) is a compact valued mapping. Therefore, the left part of (47) is well defined. Let a ∈ G ( x , λ , μ ) be arbitrarily given. Then, there exists t ¯ ( x , μ ) ∈ T ( x , μ ) such that [figure omitted; refer to PDF] Let b : = _{P K ( λ ) ∩ X} ( ^{x [variant prime]} - ρ t ¯ ( ^{x [variant prime]} , μ ) ) , which along with (43) yields that b ∈ G ( ^{x [variant prime]} , λ , μ ) .

Using assumption (H₁ ) and the property that the projection onto a fixed closed convex subset is a nonexpansive mapping, we obtain [figure omitted; refer to PDF]

On the other hand, it follows from assumptions (H₁ ) and (H₂ ) that [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF]

Combing (49) and (51), we obtain [figure omitted; refer to PDF] By assumption (H₃ ), θ < 1 and hence we have [figure omitted; refer to PDF]

Using the same arguments, we can show that [figure omitted; refer to PDF] which along with (53) implies that G ( · , λ , μ ) is a set-valued θ -contradiction mapping on X ∩ E .

(b) By (a) and the Nadler fixed point theorem in [15], G ( · , λ , μ ) has a fixed point x ( λ , μ ) for each ( λ , μ ) ∈ ( V ∩ Λ ) × ( U ∩ M ) . Hence, for any ( λ , μ ) ∈ ( V ∩ Λ ) × ( U ∩ M ) , LS ( λ , μ ) ...0; ∅ by Lemma 8. Moreover, we claim that LS ( λ , μ ) is a closed subset. Indeed, let _{x n} ∈ LS ( λ , μ ) with _{x n} [arrow right] _{x 0} as n [arrow right] ∞ . Then, it follows from Lemma 8 that [figure omitted; refer to PDF] Therefore, by (a), we obtain [figure omitted; refer to PDF] which implies that _{x 0} ∈ G ( _{x 0} , λ , μ ) and _{x 0} ∈ LS ( λ , μ ) by Lemma 8. Hence, LS ( λ , μ ) is a compact set since K ( λ ) ∩ X 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 ( λ , μ ) ∈ ( V ∩ Λ ) × ( U ∩ M ) , t ( · , μ ) ∈ T ( · , μ ) , the vector-valued mapping g : ( X ∩ E ) × ( _{V 1} ∩ Λ ) × ( U ∩ M ) [arrow right] X ∩ E defined by [figure omitted; refer to PDF] where ρ > 0 , is a fixed number. Furthermore, assume that assumptions (H ₁ )-(H ₃ ) are satisfied.

Then, for any λ ∈ V ∩ Λ , μ ∈ U ∩ M , g ( · , λ , μ ) is a vector-valued θ -contradiction mapping and has a unique fixed point x ( λ , μ ) .

Now, we state our main result.

Theorem 11.

Suppose that K ( · ) is k . α -Pseudo-Hölder continuous at ( λ ¯ , x ¯ ) . Furthermore, suppose that assumptions (H ₁ )-(H ₃ ) hold. Then, there exist a neighborhood U ¯ of μ ¯ , a neighborhood V ¯ of λ ¯ such that

(a) for all ( ^{λ [variant prime]} , ^{μ [variant prime]} ) , ( λ , μ ) ∈ ( V ¯ ∩ Λ ) × ( U ¯ ∩ M ) , [figure omitted; refer to PDF]

(b) for all ( λ , μ ) ∈ ( V ¯ ∩ Λ ) × ( U ¯ ∩ M ) , L S ( λ , μ ) ⊂ int ... X and for any x ( λ , μ ) ∈ L S ( λ , μ ) is a solution to (2) at parameters λ ∈ V ¯ ∩ Λ , μ ∈ U ¯ ∩ M .

Proof.

By the Pseudo-Hölder continuity of K ( · ) and Lemma 4, there exist a neighborhood _{V 1} of λ ¯ , a neighborhood _{X 1} of x ¯ such that [figure omitted; refer to PDF]

Since T ( x ¯ , μ ¯ ) is compact, then σ : = _{sup ... t ( x ¯ , μ ¯ ) ∈ T ( x ¯ , μ ¯ )} || t ( x ¯ , μ ¯ ) || < ∞ . Let _{ρ 1} satisfy assumption (H₃ ) and [figure omitted; refer to PDF] where int ... _{X 1} denotes the interior of _{X 1} .

Fix ρ ∈ ( 0 , _{ρ 1} ] . For any ( λ , μ ) ∈ ( _{V 1} ∩ Λ ) × ( U ∩ M ) , let x ∈ LS ( λ , μ ) be arbitrarily given. Then, there exists t ^ ( x , μ ) ∈ T ( x , μ ) such that [figure omitted; refer to PDF]

For any ( ^{λ [variant prime]} , ^{μ [variant prime]} ) ∈ ( _{V 1} ∩ Λ ) × ( U ∩ M ) , by Corollary 10, g ( · , ^{λ [variant prime]} , ^{μ [variant prime]} ) = _{P K ( ^{λ [variant prime]} ) ∩ X} ( · - ρ t ^ ( · , μ ) ) has a unique fixed point ^{x [variant prime]} = x ( ^{λ [variant prime]} , ^{μ [variant prime]} ) . Then, it follows from Lemma 8 that ^{x [variant prime]} = x ( ^{λ [variant prime]} , ^{μ [variant prime]} ) is a solution to (2) at parameters ( ^{λ [variant prime]} , ^{μ [variant prime]} ) ; that is, [figure omitted; refer to PDF]

Then, using assumption (H₁ ), Proposition 9, and the property that the projection onto a fixed convex subset is a nonexpansive mapping, we obtain [figure omitted; refer to PDF] where z ( x , μ ) : = x - ρ t ^ ( x , μ ) , z ( ^{x [variant prime]} , ^{μ [variant prime]} ) : = ^{x [variant prime]} - ρ t ^ ( ^{x [variant prime]} , ^{μ [variant prime]} ) , z ( x , ^{μ [variant prime]} ) : = x - ρ t ^ ( x , ^{μ [variant prime]} ) , and θ < 1 defined as in Proposition 9.

Noting that the arbitrariness of x ∈ LS ( λ , μ ) , we obtain that [figure omitted; refer to PDF]

Then, substituting ( ^{λ [variant prime]} , ^{μ [variant prime]} ) = ( λ ¯ , μ ¯ ) in (64) and by the uniqueness of LS ( λ ¯ , μ ¯ ) we obtain that [figure omitted; refer to PDF] where z ( x ¯ , μ ¯ ) : = x ¯ - ρ t ^ ( x ¯ , μ ¯ ) .

According to (60), we have z ( x ¯ , μ ¯ ) : = x ¯ - ρ t ^ ( x ¯ , μ ¯ ) ∈ int ... _{X 1} , which along with (59), (65) yields that [figure omitted; refer to PDF]

We claim that there exist a neighborhood V ¯ of λ ¯ , a neighborhood U ¯ of λ ¯ such that (58) is satisfied. Indeed, let ... > 0 be so small that z ( x ¯ , μ ¯ ) + ... B ( 0,1 ) ⊂ int ... _{X 1} . Using (66), assumption (H₁ ) and (5) of Remark 6, we can find a neighborhood V ¯ ⊂ _{V 1} of λ ¯ , a neighborhood U ¯ ⊂ U of μ ¯ such that for any ( λ , μ ) ∈ ( V ¯ ∩ Λ ) × ( U ¯ ∩ M ) , [figure omitted; refer to PDF]

Therefore, (60) and (67) together yield that, for all x ∈ LS ( λ , μ ) , t ( x , μ ) ∈ T ( x , μ ) , [figure omitted; refer to PDF] Hence, for any ( λ , μ ) , ( ^{λ [variant prime]} , ^{μ [variant prime]} ) ∈ ( V ¯ ∩ Λ ) × ( U ¯ ∩ M ) , it follows from (59), (64), and (68) that [figure omitted; refer to PDF]

On the other hand, it follows from the symmetrical roles of ( λ , μ ) , ( ^{λ [variant prime]} , ^{μ [variant prime]} ) that [figure omitted; refer to PDF] Therefore, (69) and (70) together yield that for any ( λ , μ ) , ( ^{λ [variant prime]} , ^{μ [variant prime]} ) ∈ ( V ¯ ∩ Λ ) × ( U ¯ ∩ M ) , [figure omitted; refer to PDF]

Thus, we have established (58) and obtained LS ( λ , μ ) ⊂ int ... X in (67), which implies that for all x ( λ , μ ) ∈ LS ( λ , μ ) is the solution in X to (2) at parameters λ ∈ V ¯ ∩ Λ , μ ∈ U ¯ ∩ M . 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 E = ^{... 2} , Λ = M = ... , ( λ ¯ , μ ¯ ) = ( 1,0 ) ∈ Λ × M be a given point and let V = Λ , U = M be a neighborhood of ( λ ¯ , μ ¯ ) . Let T : E × M ... E be a set-valued mapping which is defined by [figure omitted; refer to PDF] and K : Λ ... E defined by [figure omitted; refer to PDF]

Clearly, x ¯ = ( 1,1 ) is a unique solution to (2) at parameter ( λ ¯ , μ ¯ ) . Let X = ^{... 2} be a neighborhood of x ¯ and let t ( · , · ) ∈ T ( · , · ) be taken arbitrarily. Obviously, t ( · , · ) is 2.1 - 1.1 -Hölder continuous and strongly 1 -monotone on ( X ∩ E ) × ( U ∩ M ) ; there exists ρ satisfying 0 < ρ ...4; 1 / 4 ; the map K ( · ) is 1.1 -Hölder continuous with closed convex values on V ∩ Λ . Therefore, all assumptions of Theorem 11 are satisfied.

Moreover, by virtue of Lemma 8, we note that x ∈ K ( λ ) ∩ X is a local solution of (2) at parameter ( λ , μ ) if and only if x satisfies the inclusion [figure omitted; refer to PDF]

Let x = ( _{x 1} , _{x 2} ) ∈ E be any point. By a simple computation, we obtain [figure omitted; refer to PDF] Then, it follows that, for each ρ : 0 < ρ ...4; 1 / 4 and any t ( x , μ ) ∈ T ( x , μ ) , [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] By virtue of Lemma 8, direct computation shows that, for any ( λ , μ ) ∈ ( V ∩ Λ ) × ( U ∩ M ) , [figure omitted; refer to PDF] Then, the local solution set LS ( · , · ) of (2) satisfies (58) for any neighborhoods V of λ ¯ , U of μ ¯ . Hence, Theorem 11 is applicable.

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).

Conflict of Interests

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

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