Journal of Function Spaces
Volume 2018 (2018), Article ID 3897495, 7 pages
https://doi.org/10.1155/2018/3897495

Research Article

Solvability of Some Perturbed Generalized Variational Inequalities in Reflexive Banach Spaces

Xue-ping Luo

College of Computer Science and Technology, Southwest University for Nationalities, Chengdu 610041, China

Correspondence should be addressed to Xue-ping Luo; xuepingluo2014@outlook.com

Received 15 March 2018; Accepted 23 April 2018; Published 3 June 2018

Academic Editor: Adrian Petrusel

Copyright © 2018 Xue-ping Luo. 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.

This paper aims to discuss the solvability of some perturbed generalized variational inequalities with both the mapping and the constraint set perturbed simultaneously in reflexive Banach spaces, under some coercivity conditions. In particular, a new result that the set is directional perturbed is presented. The main results generalize and extend some known results in this area.

1. Introduction

Throughout this paper, let us denote to be a reflexive Banach space with the dual space , let be the norm of , and let be the dual pair between and Let be a nonempty, closed, and convex subset and let be a set-valued mapping with nonempty values. We consider the following generalized variational inequality problem: to find and such that

which has been extensively studied in the literature [1–9]. Here in this paper, we use GVIP() and to denote problem (1) and its solution set, respectively.

If is single-valued, then (1) reduces to the following classical variational inequality [10, 11]: to find such that

The variational inequalities have been widely studied by many authors in recent years; see [12–18]. One of the most interesting subjects in the theoretical aspect is the research of properties for variational inequalities under data perturbation [19–27]. Most of the research in the literature considered variational inequality where space is finite dimensional and the mapping is single-valued and continuous, and the mail tool for analyzing is the theory of topological degree. In particular, Corollary 5.5.12 in [11] applied topological degree as a theoretical tool to prove that if and is single-valued, then a coercivity condition implies that a perturbed variational inequality has a solution.

Assuming that the barrier cone of has nonempty interior, [28] showed a comprehensive study of the stability of the solution set of , where is maximal monotone. Later, He [4] discussed the stability analysis of with either the mapping or the constraint set perturbed when is pseudomonotone, which is weaker than (maximal) monotone. In [3], Fan and Zhong further extended the main results of [4] to the case where the perturbation was imposed on the mapping and the constraint set simultaneously. Zhong and Huang [29] extended the results of [3, 4] to the mixed variational inequalities with -pseudomonotone mappings. All the research mentioned above on the stability results considered variational inequalities when the mapping is set-valued and the theory of topological degree is not used in infinite dimensional spaces. Recently, Li and He [6] extended the aforementioned result of [11] by allowing to be a set-valued mapping with a weaker coercivity condition and presented the solvability of perturbed generalized variational inequalities in Very recently, Li and Sun [30] generalized the main result of [6] from to an infinite dimensional space and gave a stronger conclusion than Theorem 1.1 of [6].

Inspired and motivated by the works of [6, 30], in this paper, we establish the solvability of some perturbed generalized variational inequalities when both the mapping and the constraint set are perturbed simultaneously in reflexive Banach spaces. Our work improves and extends the main results of [6, 30] which considered only the mapping is perturbed. It is worth mentioning that Theorem 9 of this paper in which both the mapping and the constraint set are perturbed along some certain directions is new even if the mapping is single-valued.

The rest of this paper is organized as follows. In Section 2, we recall some notations and present some basic results. In Section 3, we consider the solvability of two different kinds of perturbed generalized variational inequalities with both the mapping and the constraint set perturbed simultaneously in reflexive Banach spaces. Finally, we conclude this paper in Section 4.

2. Notations and Preliminaries

Let , , and be as those in Section 1. The symbols “” and “” are used to denote the strong and weak convergence, respectively. Let

denoting the barrier cone of . The recession cone of is the closed convex cone defined by

It is known that, given ,

For a nonempty set in , and denotes the interior of . By Proposition 3.10 in [31], we have

For a positive integer , we denote , and

Definition 1. Let be a set-valued mapping with nonempty values. is said to be

(i)monotone on if for each pair of points and for all and ,

(ii)quasimonotone on if for each pair of points and for all and ,

(iii)stably quasimonotone on with respect to a set if and are quasimonotone on for every ;(iv)upper semicontinuous at if for any neighborhood of , there exists a neighborhood of such that

(v)upper hemicontinuous on if the restriction of to every line segment of is upper semicontinuous with respect to the weak topology in

Some preliminary results are quoted below.

Lemma 2 (see [4]). Let be a nonempty, closed, and convex subset in with ; then there does not exist with each such that . If additionally is a cone, then there does not exist with each such that .

Lemma 3 (see [32]). If is a nonempty and convex subset of a Hausdorff topological vector space and is a set-valued mapping from into satisfying the following properties:

(i) is a KKM mapping for every finite subset of , ,(ii) is closed in for every ,(iii) is compact in for some ,

then .

Next we discuss the variational inequality property.

Definition 4. is said to have variational inequality property on if for every nonempty, bounded, closed, and convex of , has a solution.

Proposition 5. The following classes of mappings have the variational inequality property:

(i)Every weak- upper semicontinuous set-valued mapping with nonempty compact convex values.(ii)Every upper hemicontinuous and quasimonotone set-valued mapping with nonempty -compact convex values.(iii)If is a weak- upper semicontinuous set-valued mapping with nonempty compact convex values and is a strongly continuous mapping, then has the variational inequality property.(iv)If is an upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty -compact convex values, then for any , has the variational inequality property.

Proof. (i) is well known in the literature and (ii) is verified in [7]. Let us now prove (iii).
Let be a bounded, closed, and convex subset of . Let be a set-valued mapping defined by

We first prove that is a weakly closed subset of Indeed, let with This means that, for some ,

Since is weak- upper semicontinuous with nonempty compact convex values, is compact. Without loss of generality, we assume for some Thus, for any , from is strongly continuous and (11), we have

which shows that and so is sequentially weakly closed; being in addition a bounded subset of a Banach space, by virtue of the Eberlein-Šmulian theorem, is automatically weakly closed. Since is reflexive Banach space, is weakly compact in for any
Next we prove that is a KKM map from to If it not true, then there exist with , and such that It follows that

This contradiction shows that is a KKM map. By Lemma 3, there exists and then solves . Thus, has the variational inequality property.
Finally, (iv) can be proved similarly as Proposition 5 of (ii).

3. Solvability of Some Perturbed Generalized Variational Inequalities

This section is devoted to the main results in this paper, which include the solvability of perturbed generalized variational inequalities under two classes of perturbations allowed for the mapping and the set as follows:

(i)The mapping is perturbed by a nonlinear mapping and the set is perturbed by a closed unit ball.(ii)The mapping is perturbed along a direction and the set is perturbed along another direction.

They are discussed in Sections 3.1 and 3.2, respectively.

3.1. Nonlinear Perturbed Mapping and Ball Perturbed Set

In this subsection, we denote to be the set for any with being the closed unit ball in and to be the set of strongly continuous and monotone mappings satisfying for all

Theorem 6. Let be a nonempty, closed, and convex subset in Suppose that, for any , is a weak- upper semicontinuous mapping with nonempty compact convex values. If the following coercivity condition holds:

then for every , there exist and such that

Proof. Firstly, we need to prove that, for every , there exist and , for any and such that If not, then there exists , for every and every , there exist and such that is weak- upper semicontinuous with nonempty compact convex values on and
Denote By the fact that is weak- upper semicontinuous with nonempty compact convex values and Proposition 5(iii), has the variational inequality property on Since is bounded, closed, and convex, there exists such that

(i) If for some and some , , then for any , there exists such that , because of the convexity of . It follows from (16) that

Since is arbitrary, solves Thus,
(ii) If for each and each , Without loss of generality, we may assume that as and Since , there exists such that

This together with the weak convergence of to yield that Thus, we can obtain because is closed and convex and hence weakly closed. Moreover, as and , we have for small enough . Thus, the coercivity condition (14) implies that, for small enough , there exists such that . By , without loss of generality, as and . As is weak- upper semicontinuous and , we have
Next we claim that, for small enough, Indeed, implies every weak neighborhood of such that with sufficiently small and Moreover, as the set is a neighborhood of for the topology and is weak- upper semicontinuous, for any , we have Since has nonempty compact convex values, is compact. Without of loss of generality, for some It follows that

and so for small enough, we have the claim.
Now, as and , we have Therefore, it follows from that

Thus, there exist and such that

Since , for any , there exists such that and Now we claim that, for small enough, Indeed, in view of , there exist and such that Thus, and converges to in the norm topology as Since is convex, It follows that there exists such that

Therefore, for sufficiently small
By (16) and (21), for any and any , we have

Since is arbitrary, solves It shows that for sufficiently small and
In either case, we obtain a contradiction to .
Now we need to prove that, for every , there exist and , for any and such that By the first part , Suppose that there exists , and for any and , there exist and such that ; however,
Since and , applying the same method of proof as , there exist and for small enough It follows from coercivity condition (14) that there exists such that Similarly, we can get

By and , we have

Moreover, implies that It follows that

Thus, there exist small enough and such that

We obtain a contradiction from the monotonicity of This completes the proof.

Remark 7. Li and He [6] gave a similar result where only the mapping was perturbed in (Theorem 3.1 of [6]). Theorem 6 generalizes the result of [6] from to a reflexive Banach space, gives a stronger conclusion, and discusses the case that the mapping is perturbed by a nonlinear mapping and the constraint set is perturbed by a closed unit ball simultaneously.

3.2. Directional Perturbed Mapping and Directional Perturbed Set

Let us turn to the case that the mapping is perturbed along a direction and the set is perturbed along another direction. Here in this section, we use to denote the set for any

Lemma 8. If is a nonempty, closed, and convex subset in , then and

Proof. Let Given , in view of , there exist and such that It follows that, for all ,

and so This implies that
Next we prove the converse inclusion. Let Given , in view of , there exist and such that It follows that, for all ,

and so , implying that Therefore,
Finally, by (6), we have

Theorem 9. Let be a nonempty, closed, and convex subset in Suppose that and that is an upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty -compact convex values. If the following coercivity condition holds:

then for any , there exists such that

Proof. Firstly, we need to prove that, for any , there exists such that, for every , Suppose that there exists , and for any , there exists such that
Denote By that is bounded, closed, and convex, is upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty -compact convex values, and Proposition 5(iv), , has a solution. For each , there exists such that

(i) If for some , , then for any , there exists such that This is because the convexity of and implies that is convex. It follows that

Since is arbitrary, solves Thus,
(ii) If for each , Since , there exist and such that We claim that Indeed, if not, then Thus,

It follows that for each However, by , we have Letting , this implies that has no upper bound for each Thus, we obtain a contradiction and so the claim is proved. The coercivity condition (30) implies that there exists such that there exists , Thus, by (6), we have

that is, for large enough , holds. It follows from the quasimonotonicity of that

Without loss of generality, we may assume that By Lemma 8, we have As , Lemma 2 implies From and (6), we obtain We claim that In fact, if not, then Since , for any , there exists such that It implies that and so Similarly, we can obtain Thus, which contradicts with Therefore,
Since , then for any , there exists such that and Let us now claim that Indeed, in view of and , there exist and such that , and hence Since is convex, From Lemma 8, we know By the definition of the recession cone and , we get Therefore,
It follows from (32) and (36) that

Since , By sign-preserving theorem of limit, there exists , when ,
Also noticing that , is bounded, we have It follows that, for large enough ,

As is arbitrary, solves
In either case, we obtain a contradiction to .
Now we need to prove that, for any , there exists such that, for every , By the first part , there exists , and for any and , we have Suppose that there exists , and for any , there exists such that ; however,
Since and , applying the same method of proof as , there exists and It follows from coercivity condition (30) that there exists such that there exists , Similarly, we can get

Moreover, implies that , because By , we have

Similarly, we can assume that Thus, By sign-preserving theorem of limit, there exists , when , Since , When ,

Similarly, implies that is bounded, For large enough , It is a contradiction with (40). This completes the proof.

Remark 10. Li and He [6] gave a similar result of where only the mapping was directional perturbed in (Theorem 3.2 of [6]). Theorem 9 generalizes the result of [6] from to a reflexive Banach space, gives a stronger conclusion, and discusses the case that the mapping is perturbed along a direction and the constraint set is perturbed along another direction simultaneously. Theorem 9 also improves Theorem 3.1 of [30] to the case that the set is also directional perturbed simultaneously.

4. Conclusions

The aim of this paper is to improve the solvability results for perturbed generalized variational inequalities to the case that the perturbations act both on the mapping and on the constraint set It is important to observe that the second solvability result for perturbed generalized variational inequality when the perturbation terms and are along some certain directions simultaneously is new and interesting. Further research works should be carried out to develop the similar solvability results for some other optimization problems with the perturbation terms perturbed along some certain directions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant 11701480), Fundamental Research Funds for the Central Universities, Southwest University for Nationalities (Grant 2018HQZZ23), Key Projects of the Education Department of Sichuan Province (Grant 18ZA0511), Innovation Team Funds of Southwest University for Nationalities (Grant 14CXTD03), Innovative Research Team of the Education Department of Sichuan Province (Grant 15TD0050), and Sichuan Youth Science and Technology Innovation Research Team (Grant 2017TD0028).

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