# Solvability of Some Perturbed Generalized Variational Inequalities in Reflexive Banach Spaces

Authors: Xue-ping Luo
Journal: Journal of Function Spaces
 (2018)

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## 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.


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## Body

## 1. Introduction

Throughout this paper, let us denoteX to be a reflexive Banach space with the dual space X∗, let x be the norm of x∈X, and let ϕ,x be the dual pair between ϕ∈X∗ and x∈X. Let K⊂X be a nonempty, closed, and convex subset and let F:K→2X∗ be a set-valued mapping with nonempty values. We consider the followinggeneralized variational inequality problem: to find x∈K and x∗∈F(x) such that(1)x∗,y-x≥0,∀y∈K,which has been extensively studied in the literature [1–9]. Here in this paper, we use GVIP(F,K) and S(F,K) to denote problem (1) and its solution set, respectively.IfF is single-valued, then (1) reduces to the following classical variational inequality VIP(F,K) [10, 11]: to find x∈K such that (2)Fx,y-x≥0,∀y∈K.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 F 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 X=Rn and F is single-valued, then a coercivity condition implies that a perturbed variational inequality VIP(F+q,K) has a solution.Assuming that the barrier cone ofK has nonempty interior, [28] showed a comprehensive study of the stability of the solution set of GVIP(F,K), where F is maximal monotone. Later, He [4] discussed the stability analysis of GVIP(F,K) with either the mapping or the constraint set perturbed when F 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 f-pseudomonotone mappings. All the research mentioned above on the stability results considered variational inequalities when the mapping F 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 F to be a set-valued mapping with a weaker coercivity condition and presented the solvability of perturbed generalized variational inequalities in Rn. Very recently, Li and Sun [30] generalized the main result of [6] from Rn 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 F and the constraint set K 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 F is single-valued.The rest of this paper is organized as follows. In Section2, 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 F and the constraint set K perturbed simultaneously in reflexive Banach spaces. Finally, we conclude this paper in Section 4.


## 2. Notations and Preliminaries

LetX, X∗, and K be as those in Section 1. The symbols “→” and “⇀” are used to denote the strong and weak convergence, respectively. Let(3)barrK≔x∗∈X∗:supx∈K⁡x∗,x<∞denoting the barrier cone of K. The recession cone of K is the closed convex cone defined by(4)K∞≔d∈X:∃tn↓0,∃xn∈K,tnxn⇀d.It is known that, given x0∈K,(5)K∞=d∈X:x0+λd∈K,∀λ>0.For a nonempty set D in X, D-≔x∗∈X∗:x∗,x≤0,∀x∈D and int⁡(D) denotes the interior of D. By Proposition 3.10 in [31], we have(6)barrK-=K∞.For a positive integerm∈N, we denote Km≔{x∈K:x≤m}, B(0,m)≔{x∈X:x<m} and B¯(0,m)≔{x∈X:x≤m}.Definition 1.
LetF:K→2X∗ be a set-valued mapping with nonempty values. F is said to be (i)
monotone onK if for each pair of points x,y∈K and for all x∗∈F(x) and y∗∈F(y), (7)y∗-x∗,y-x≥0;(ii)
quasimonotone onK if for each pair of points x,y∈K and for all x∗∈F(x) and y∗∈F(y), (8)x∗,y-x>0⟹y∗,y-x≥0;(iii)
stably quasimonotone onK with respect to a set U⊂X∗ if F and F(·)-u are quasimonotone on K for every u∈U;(iv)
upper semicontinuous atx0∈K if for any neighborhood N(F(x0)) of F(x0), there exists a neighborhood N(x0) of x0 such that (9)Fx⊂NFx0,∀x∈Nx0∩K;(v)
upper hemicontinuous onK if the restriction of F to every line segment of K is upper semicontinuous with respect to the weak topology in X∗.Some preliminary results are quoted below.Lemma 2 (see [4]).
LetK be a nonempty, closed, and convex subset in X with int⁡(barrK)≠∅; then there does not exist {xn}⊂K with each xn→∞ such that xn/xn⇀0. If additionally K is a cone, then there does not exist {dn}⊂K with each dn=1 such that dn⇀0.Lemma 3 (see [32]).
IfK is a nonempty and convex subset of a Hausdorff topological vector space E and G:K→2E is a set-valued mapping from K into E satisfying the following properties: (i)
G is a KKM mapping for every finite subset A of K, co⁡(A)⊂⋃x∈AG(x),(ii)
G(x) is closed in E for every x∈K,(iii)
G(x0) is compact in E for some x0∈K,
then⋂x∈KG(x)≠∅.Next we discuss the variational inequality property.Definition 4.
F is said to havevariational inequality property onK if for every nonempty, bounded, closed, and convex D of K, GVIP(F,D) 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 nonemptyw∗-compact convex values.(iii)
IfF:K→2X∗ is a weak-· upper semicontinuous set-valued mapping with nonempty compact convex values and q:K→X∗ is a strongly continuous mapping, then F+q has the variational inequality property.(iv)
IfF:K→2X∗ is an upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty w∗-compact convex values, then for any ζ∈X∗, F-ζ has the variational inequality property.Proof.
(i) is well known in the literature and (ii) is verified in [7]. Let us now prove (iii).
LetD be a bounded, closed, and convex subset of K. Let G:D→2D be a set-valued mapping defined by (10)Gy≔x∈D:supξ∈Fxξ+qx,y-x≥0,∀y∈D.We first prove that G(y) is a weakly closed subset of D. Indeed, let xn∈G(y) with xn⇀x0. This means that, for some ξn∈F(xn),(11)ξn+qxn,y-xn≥0,∀y∈D.Since F is weak-· upper semicontinuous with nonempty compact convex values, {ξn} is compact. Without loss of generality, we assume limn→∞⁡ξn=ξ for some ξ∈F(x0). Thus, for any y∈D, from q is strongly continuous and (11), we have (12)supξ∈Fx0⁡ξ+qx0,y-x0≥0,which shows that x0∈G(y) and so G(y) is sequentially weakly closed; being in addition a bounded subset of a Banach space, by virtue of the Eberlein-Šmulian theorem, G(y) is automatically weakly closed. Since X is reflexive Banach space, G(y) is weakly compact in D for any y∈D.
Next we prove thatG is a KKM map from D to D. If it not true, then there exist λ1,…,λn∈[0,1] with ∑i=1nλi=1, {y1,…,yn}⊂D and y¯=∑i=1nλiyi such that y¯∉⋃i=1nG(yi). It follows that (13)0=supξ∈Fy¯⁡ξ,∑i=1nλiyi-y¯+qx0,∑i=1nλiyi-y¯=∑i=1nλisupξ∈Fy¯⁡ξ,yi-y¯+qx0,yi-y¯<0.This contradiction shows that G is a KKM map. By Lemma 3, there exists x¯∈⋂y∈DG(y) and then x¯ solves GVIP(F+q,D). Thus, F+q has the variational inequality property.
Finally, (iv) can be proved similarly as Proposition5 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 mappingF and the set K as follows:(i)
The mappingF is perturbed by a nonlinear mapping and the set K is perturbed by a closed unit ball.(ii)
The mappingF is perturbed along a direction and the set K is perturbed along another direction.They are discussed in Sections3.1 and 3.2, respectively.


### 3.1. Nonlinear Perturbed Mapping and Ball Perturbed Set

In this subsection, we denoteKμ to be the set K+μB¯ for any μ>0 with B¯ being the closed unit ball in X and B(0;ε,(Kμ)m) to be the set of strongly continuous and monotone mappings q:Kμ→X∗ satisfying q(x)<ε for all x∈(Kμ)m.Theorem 6.
LetK be a nonempty, closed, and convex subset in X. Suppose that, for any μ>0, F:Kμ→2X∗ is a weak-· upper semicontinuous mapping with nonempty compact convex values. If the following coercivity condition holds:(14)∃r>0,∀x∈K∖Kr,∃y∈Krsuchthatsupx∗∈Fxx∗,y-x<0,then for every m>r, there exist ε>0 and δ>0 such that (15)∅≠SF+q,K+θB¯⊂B¯0,m,∀q∈B0;ε,Kδm,∀θ∈0,δ.Proof.
( 1 ) Firstly, we need to prove that, for every m>r, there exist ε>0 and δ>0, for any q∈B(0;ε,(Kδ)m) and θ∈(0,δ) such that S(F+q,K+θB¯)≠∅. If not, then there exists m>r, for every ε>0 and every δ>0, there exist qε∈B(0;ε,(Kδ)m) and θδ∈(0,δ) such that F is weak-· upper semicontinuous with nonempty compact convex values on Kθδ and S(F+qε,K+θδB¯)=∅.
DenoteLm≔x∈Kθδ:x≤m. By the fact that F is weak-· upper semicontinuous with nonempty compact convex values and Proposition 5(iii), F+qε has the variational inequality property on Kθδ. Since Lm is bounded, closed, and convex, there exists xεδ∈Lm such that(16)supξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ≥0,∀y∈Lm.
(i) If for someε>0 and some δ>0, xεδ<m, then for any y∈K+θδB¯, there exists t∈(0,1) such that zt≔xεδ+t(y-xεδ)∈Lm, because of the convexity of K+θδB¯. It follows from (16) that (17)0≤supξ∈Fxεδ⁡ξ+qεxεδ,zt-xεδ=tsupξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ.Since y∈K+θδB¯ is arbitrary, xεδ solves GVIP(F+qε,K+θδB¯). Thus, xεδ∈S(F+qε,K+θδB¯)≠∅.
(ii) If for eachε>0 and each δ>0, xεδ=m. Without loss of generality, we may assume that xεδ⇀d as ε→0+ and δ→0+. Since xεδ∈K+θδB¯, there exists xεδ~∈K such that (18)limδ→0+⁡xεδ-xεδ~=0.This together with the weak convergence of xεδ to d yield that xεδ~⇀d. Thus, we can obtain d∈K because K is closed and convex and hence weakly closed. Moreover, as xεδ-xεδ~≤xεδ-xεδ~ and m>r, we have xεδ~∉Kr for small enough δ>0. Thus, the coercivity condition (14) implies that, for small enough δ>0, there exists yεδ∈Kr such that supη∈Fxεδ~⁡η,yεδ-xεδ~<0. By yεδ≤r<m, without loss of generality, yεδ⇀y0 as ε→0+ and δ→0+. As F is weak-· upper semicontinuous and xεδ~⇀d, we have supζ∈Fd⁡ζ,y0-d≤0.
Next we claim that, forε,δ>0 small enough, supξ∈Fxεδ⁡ξ,yεδ-xεδ<0. Indeed, xεδ⇀d implies every weak neighborhood U of d such that xεδ∈U with sufficiently small ε>0 and δ>0. Moreover, as the set {ζ∈X∗:ζ,y0-d≤0} is a neighborhood of F(d) for the weakly∗ topology and F is weak-· upper semicontinuous, for any ξεδ∈F(xεδ), we have ξεδ,y0-d≤0. Since F has nonempty compact convex values, {ξεδ} is compact. Without of loss of generality, ξεδ→ξ0 for some ξ0∈F(d). It follows that (19)limε→0+,δ→0+⁡ξεδ,yεδ-xεδ=ξ0,y0-d≤0,and so for ε,δ>0 small enough, we have the claim.
Now, asxεδ∈Kθδ⊂Kδ and xεδ=m, we have xεδ∈(Kδ)m. Therefore, it follows from supx∈(Kδ)mqε(x)<ε that (20)limε→0+,δ→0+⁡qεxεδ,yεδ-xεδ=0.Thus, there exist ε1>0 and δ1>0 such that(21)supξ∈Fxεδ⁡ξ+qεxεδ,yεδ-xεδ<0,∀ε∈0,ε1,∀δ∈0,δ1.Since yεδ<m, for any y∈K+θδB¯, there exists λ∈(0,1) such that zλδ≔yεδ+λ(y-yεδ) and zλδ≤m. Now we claim that, for δ>0 small enough, zλδ∈K+θδB¯. Indeed, in view of yεδ∈K, there exist y1∈K+θδB¯ and b∈θδB¯ such that yεδ=y1-b. Thus, zλδ=y1+λ(y-y1)-(1-λ)b and zλδ converges to y1+λ(y-y1) in the norm topology as δ→0+. Since K+θδB¯ is convex, y1+λ(y-y1)∈K+θδB¯. It follows that there exists γ>0 such that (22)zλδ∈K+θδB¯,∀δ∈0,γ.Therefore, zλδ∈Lm for sufficiently small δ>0.
By (16) and (21), for any ε∈(0,ε1) and any δ∈(0,min⁡δ1,γ), we have (23)0≤supξ∈Fxεδ⁡ξ+qεxεδ,zλδ-xεδ=λsupξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ+1-λsupξ∈Fxεδ⁡ξ+qεxεδ,yεδ-xεδ<λsupξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ.Since y∈K+θδB¯ is arbitrary, xεδ solves GVIP(F+qε,K+θδB¯). It shows that xεδ∈S(F+qε,K+θδB¯)≠∅ for sufficiently small ε>0 and δ>0.
In either case, we obtain a contradiction toS(F+qε,K+θδB¯)=∅.
( 2 ) Now we need to prove that, for every m>r, there exist ε>0 and δ>0, for any q∈B(0;ε,(Kδ)m) and θ∈(0,δ) such that S(F+q,K+θB¯)⊂B¯(0,m). By the first part (1), S(F+q,K+θB¯)≠∅. Suppose that there exists m0>r, and for any ε^>0 and δ^>0, there exist qε^~∈B(0;ε^,(Kδ^)m0) and θδ^~∈(0,δ^) such that xε^δ^∈S(F+qε^~,K+θδ^~B¯); however, xε^δ^∉B¯(0,m0).
Sincexε^δ^∈K+θδ^~B¯ and xε^δ^>m0>r, applying the same method of proof as (1), there exist xε^δ^~∈K and xε^δ^~>r for small enough δ^>0. It follows from coercivity condition (14) that there exists yε^δ^∈Kr such that supζ∈F(xε^δ^~)ζ,yε^δ^-xε^δ^~<0. Similarly, we can get (24)supξ∈Fxε^δ^⁡ξ,yε^δ^-xε^δ^<0.
Byyε^δ^∈K⊂K+θδ^~B¯ and xε^δ^∈S(F+qε^~,K+θδ^~B¯), we have(25)0≤supξ∗∈Fxε^δ^⁡ξ∗+qε^~xε^δ^,yε^δ^-xε^δ^=supξ∗∈Fxε^δ^⁡ξ∗,yε^δ^-xε^δ^+qε^~xε^δ^,yε^δ^-xε^δ^<qε^~xε^δ^,yε^δ^-xε^δ^.
Moreover,yε^δ^≤r<m0 implies that supyε^δ^∈(Kδ^)m0⁡qε^~(yε^δ^)<ε. It follows that (26)limε^→0+,δ^→0+⁡qε^~yε^δ^,yε^δ^-xε^δ^=0.Thus, there exist small enough ε^>0 and δ^>0 such that(27)qε^~yε^δ^-qε^~xε^δ^,yε^δ^-xε^δ^<0.We obtain a contradiction from the monotonicity of qε^~. This completes the proof.Remark 7.
Li and He [6] gave a similar result where only the mapping was perturbed in Rn (Theorem 3.1 of [6]). Theorem 6 generalizes the result of [6] from Rn 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 mappingF is perturbed along a direction and the set K is perturbed along another direction. Here in this section, we use I to denote the set K-tK∞∖{0} for any t>0.Lemma 8.
IfK is a nonempty, closed, and convex subset in X, then K∞=I∞ and barr(K)=barr(I).Proof.
Letd∈I∞. Given x0∈K, in view of K≔I+tK∞∖{0}, there exist y0∈I and d¯∈K∞∖{0} such that x0=y0+td¯. It follows that, for all λ>0, (28)x0+λd=y0+td¯+λd=y0+λd+td¯∈I+tK∞∖0=K,and so d∈K∞. This implies that I∞⊂K∞.
Next we prove the converse inclusion. Letd1∈K∞. Given x1∈I, in view of I≔K-tK∞∖{0}, there exist y1∈K and d1¯∈K∞∖{0} such that x1=y1-td1¯. It follows that, for all λ1>0, (29)x1+λ1d1=y1-td1¯+λ1d1=y1+λd1-td1¯∈K-tK∞∖0=I,and so d1∈I∞, implying that K∞⊂I∞. Therefore, K∞=I∞.
Finally, by (6), we have barr(K)=barr(I).Theorem 9.
LetK be a nonempty, closed, and convex subset in X. Suppose that int⁡(barrK)≠∅ and that F:I→2X∗ is an upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty w∗-compact convex values. If the following coercivity condition holds:(30)∃r>0,∀x∈K∖Kr,∃y∈Krsuchthat∃y∗∈Fy∩barrK,y∗,y-x<0,then for any q∈int⁡(barrK), there exists k>r such that (31)∅≠SF-εq,K-tK∞∖0⊂B¯0,1ε,∀ε∈0,1k,∀t>0.Proof.
( 1 ) Firstly, we need to prove that, for any q∈int⁡(barrK), there exists m>r such that, for every ε∈(0,1/m), S(F-εq,K-tK∞∖{0})≠∅. Suppose that there exists q∈int⁡(barrK), and for any m>r, there exists εm∈(0,1/m) such that S(F-εmq,K-tK∞∖{0})=∅.
DenoteEm≔{x∈I:x≤1/εm}. By that Em is bounded, closed, and convex, F is upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty w∗-compact convex values, and Proposition 5(iv), GVIP(F-εmq,Em), has a solution. For each m∈N, there exists xm∈Em such that(32)supξ∈Fxm⁡ξ-εmq,y-xm≥0,∀y∈Em.
(i) If for somem>r, xm<1/εm, then for any y∈K-tK∞∖{0}, there exists λ∈(0,1) such that zλ≔xm+λ(y-xm)∈Em. This is because the convexity of K and K∞ implies that K-tK∞∖{0} is convex. It follows that (33)0≤supξ∈Fxm⁡ξ-εmq,zλ-xm=tsupξ∈Fxm⁡ξ-εmq,y-xm.Since y∈K-tK∞∖{0} is arbitrary, xm solves GVIPF-εmq,K-tK∞∖0. Thus, xm∈SF-εmq,K-tK∞∖0≠∅.
(ii) If for eachm>r, xm=1/εm>m. Since xm∈K-tK∞∖{0}, there exist xm~∈K and pm∈K∞∖{0} such that xm=xm~-tpm. We claim that xm~∉Kr. Indeed, if not, then xm~≤r. Thus, (34)tpm≤xm+xm~≤1εm+r<2εm.It follows that tpm<+∞ for each m. However, by pm∈K∞∖{0}, we have pm>0. Letting t→+∞, this implies that tpm has no upper bound for each m>r. Thus, we obtain a contradiction and so the claim is proved. The coercivity condition (30) implies that there exists ym∈Kr such that there exists ym∗∈F(ym)∩barr(K), ym∗,ym-xm~<0. Thus, by (6), we have (35)ym∗,ym-xm=ym∗,ym-xm~+ym∗,xm~-xm<tym∗,pm≤0,that is, for large enough m, infym∗∈F(ym)ym∗,ym-xm<0 holds. It follows from the quasimonotonicity of F that(36)supξ∈Fxm⁡ξ,ym-xm≤0.
Without loss of generality, we may assume thatxm/xm⇀d∈I∞. By Lemma 8, we have d∈K∞. As int⁡(barrI)=int⁡(barrK)≠∅, Lemma 2 implies d≠0. From q∈int⁡(barrK) and (6), we obtain q,d≤0. We claim that q,d<0. In fact, if not, then q,d=0. Since q∈int⁡(barrK), for any x∗∈X∗, there exists δ∈(0,1) such that δq+(1-δ)x∗∈barrK. It implies that δq+1-δx∗,d≤0 and so x∗,d≤0. Similarly, we can obtain -x∗,d≤0. Thus, x∗,d=0 which contradicts with d≠0. Therefore, q,d<0.
Sinceym<r<1/εm, then for any y∈K-tK∞∖{0}, there exists θ∈(0,1) such that zθ≔ym+θ(y-ym) and zθ≤1/εm. Let us now claim that zθ∈I. Indeed, in view of I≔K-tK∞∖{0} and ym∈K, there exist ym~∈I and pm∈K∞∖{0} such that ym=ym~+pm, and hence zθ=ym~+θ(y-ym~)+(1-θ)pm. Since I is convex, ym~+θ(y-ym~)∈I. From Lemma 8, we know pm∈K∞=I∞. By the definition of the recession cone and 1-θ>0, we get zθ∈I. Therefore, zθ∈Em.
It follows from (32) and (36) that (37)0≤supξ∈Fxm⁡ξ-εmq,zθ-xm=θsupξ∈Fxm⁡ξ-εmq,y-xm+1-θsupξ∈Fxm⁡ξ,ym-xm+εm1-θq,xm-q,ym≤θsupξ∈Fxm⁡ξ-εmq,y-xm+εm1-θq,xm-q,ym.Since εm=1/xm, εmq,xm=q,xm/xm→q,d<0. By sign-preserving theorem of limit, there exists M>0, when m>M, εmq,xm<(1/2)q,d<0.
Also noticing that{ym}⊂Kr, {ym} is bounded, we have εmq,ym→0. It follows that, for large enough m, (38)supξ∈Fxm⁡ξ-εmq,y-xm≥0.As y∈K-tK∞∖{0} is arbitrary, xm solves GVIP(F-εmq,K-tK∞∖{0}).
In either case, we obtain a contradiction toS(F-εmq,K-tK∞∖{0})=∅.
( 2 ) Now we need to prove that, for any q∈int⁡(barrK), there exists k≥m>r such that, for every ε∈(0,1/k), S(F-εq,K-tK∞∖{0})⊂B¯(0,1/ε). By the first part (1), there exists m>r, and for any k≥m and ε∈(0,1/k), we have S(F-εq,K-tK∞∖{0})≠∅. Suppose that there exists q∈int⁡(barrK), and for any k≥m, there exists εk∈(0,1/k) such that xk∈S(F-εkq,K-tK∞∖{0}); however, xk∉B¯(0,1/εk).
Sincexk∈I and xk>1/εk>k≥m>r, applying the same method of proof as (1), there exists xk~∈K and xk~>r. It follows from coercivity condition (30) that there exists yk∈Kr such that there exists yk∗∈F(yk)∩barr(K), yk∗,yk-xk~<0. Similarly, we can get (39)supξk∈Fxk⁡ξk,yk-xk≤0.
Moreover,yk∈K implies that yk∈I, because K=I+tK∞=I+tI∞⊂I. By xk∈S(F-εkq,K-tK∞∖{0}), we have(40)0≤supξk∗∈Fxk⁡ξk∗-εkq,yk-xk=supξk∗∈Fxk⁡ξk∗,yk-xk+εkq,xk-q,yk≤εkq,xk-q,yk.Similarly, we can assume that xk/xk⇀e. Thus, q,xk/xk→q,e<0. By sign-preserving theorem of limit, there exists K>0, when k>K, q,xk/xk<0. Since xk>1/εk, xkεk>1. When k>K, (41)εkq,xk=xkεkq,xkxk<0.Similarly, {yk}⊂Kr implies that {yk} is bounded, εkq,yk→0. For large enough k, εk(q,xk-q,yk)<0. It is a contradiction with (40). This completes the proof.Remark 10.
Li and He [6] gave a similar result of GVIP(F,K) where only the mapping was directional perturbed in Rn (Theorem 3.2 of [6]). Theorem 9 generalizes the result of [6] from Rn 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.


## 3.1. Nonlinear Perturbed Mapping and Ball Perturbed Set

In this subsection, we denoteKμ to be the set K+μB¯ for any μ>0 with B¯ being the closed unit ball in X and B(0;ε,(Kμ)m) to be the set of strongly continuous and monotone mappings q:Kμ→X∗ satisfying q(x)<ε for all x∈(Kμ)m.Theorem 6.
LetK be a nonempty, closed, and convex subset in X. Suppose that, for any μ>0, F:Kμ→2X∗ is a weak-· upper semicontinuous mapping with nonempty compact convex values. If the following coercivity condition holds:(14)∃r>0,∀x∈K∖Kr,∃y∈Krsuchthatsupx∗∈Fxx∗,y-x<0,then for every m>r, there exist ε>0 and δ>0 such that (15)∅≠SF+q,K+θB¯⊂B¯0,m,∀q∈B0;ε,Kδm,∀θ∈0,δ.Proof.
( 1 ) Firstly, we need to prove that, for every m>r, there exist ε>0 and δ>0, for any q∈B(0;ε,(Kδ)m) and θ∈(0,δ) such that S(F+q,K+θB¯)≠∅. If not, then there exists m>r, for every ε>0 and every δ>0, there exist qε∈B(0;ε,(Kδ)m) and θδ∈(0,δ) such that F is weak-· upper semicontinuous with nonempty compact convex values on Kθδ and S(F+qε,K+θδB¯)=∅.
DenoteLm≔x∈Kθδ:x≤m. By the fact that F is weak-· upper semicontinuous with nonempty compact convex values and Proposition 5(iii), F+qε has the variational inequality property on Kθδ. Since Lm is bounded, closed, and convex, there exists xεδ∈Lm such that(16)supξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ≥0,∀y∈Lm.
(i) If for someε>0 and some δ>0, xεδ<m, then for any y∈K+θδB¯, there exists t∈(0,1) such that zt≔xεδ+t(y-xεδ)∈Lm, because of the convexity of K+θδB¯. It follows from (16) that (17)0≤supξ∈Fxεδ⁡ξ+qεxεδ,zt-xεδ=tsupξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ.Since y∈K+θδB¯ is arbitrary, xεδ solves GVIP(F+qε,K+θδB¯). Thus, xεδ∈S(F+qε,K+θδB¯)≠∅.
(ii) If for eachε>0 and each δ>0, xεδ=m. Without loss of generality, we may assume that xεδ⇀d as ε→0+ and δ→0+. Since xεδ∈K+θδB¯, there exists xεδ~∈K such that (18)limδ→0+⁡xεδ-xεδ~=0.This together with the weak convergence of xεδ to d yield that xεδ~⇀d. Thus, we can obtain d∈K because K is closed and convex and hence weakly closed. Moreover, as xεδ-xεδ~≤xεδ-xεδ~ and m>r, we have xεδ~∉Kr for small enough δ>0. Thus, the coercivity condition (14) implies that, for small enough δ>0, there exists yεδ∈Kr such that supη∈Fxεδ~⁡η,yεδ-xεδ~<0. By yεδ≤r<m, without loss of generality, yεδ⇀y0 as ε→0+ and δ→0+. As F is weak-· upper semicontinuous and xεδ~⇀d, we have supζ∈Fd⁡ζ,y0-d≤0.
Next we claim that, forε,δ>0 small enough, supξ∈Fxεδ⁡ξ,yεδ-xεδ<0. Indeed, xεδ⇀d implies every weak neighborhood U of d such that xεδ∈U with sufficiently small ε>0 and δ>0. Moreover, as the set {ζ∈X∗:ζ,y0-d≤0} is a neighborhood of F(d) for the weakly∗ topology and F is weak-· upper semicontinuous, for any ξεδ∈F(xεδ), we have ξεδ,y0-d≤0. Since F has nonempty compact convex values, {ξεδ} is compact. Without of loss of generality, ξεδ→ξ0 for some ξ0∈F(d). It follows that (19)limε→0+,δ→0+⁡ξεδ,yεδ-xεδ=ξ0,y0-d≤0,and so for ε,δ>0 small enough, we have the claim.
Now, asxεδ∈Kθδ⊂Kδ and xεδ=m, we have xεδ∈(Kδ)m. Therefore, it follows from supx∈(Kδ)mqε(x)<ε that (20)limε→0+,δ→0+⁡qεxεδ,yεδ-xεδ=0.Thus, there exist ε1>0 and δ1>0 such that(21)supξ∈Fxεδ⁡ξ+qεxεδ,yεδ-xεδ<0,∀ε∈0,ε1,∀δ∈0,δ1.Since yεδ<m, for any y∈K+θδB¯, there exists λ∈(0,1) such that zλδ≔yεδ+λ(y-yεδ) and zλδ≤m. Now we claim that, for δ>0 small enough, zλδ∈K+θδB¯. Indeed, in view of yεδ∈K, there exist y1∈K+θδB¯ and b∈θδB¯ such that yεδ=y1-b. Thus, zλδ=y1+λ(y-y1)-(1-λ)b and zλδ converges to y1+λ(y-y1) in the norm topology as δ→0+. Since K+θδB¯ is convex, y1+λ(y-y1)∈K+θδB¯. It follows that there exists γ>0 such that (22)zλδ∈K+θδB¯,∀δ∈0,γ.Therefore, zλδ∈Lm for sufficiently small δ>0.
By (16) and (21), for any ε∈(0,ε1) and any δ∈(0,min⁡δ1,γ), we have (23)0≤supξ∈Fxεδ⁡ξ+qεxεδ,zλδ-xεδ=λsupξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ+1-λsupξ∈Fxεδ⁡ξ+qεxεδ,yεδ-xεδ<λsupξ∈Fxεδ⁡ξ+qεxεδ,y-xεδ.Since y∈K+θδB¯ is arbitrary, xεδ solves GVIP(F+qε,K+θδB¯). It shows that xεδ∈S(F+qε,K+θδB¯)≠∅ for sufficiently small ε>0 and δ>0.
In either case, we obtain a contradiction toS(F+qε,K+θδB¯)=∅.
( 2 ) Now we need to prove that, for every m>r, there exist ε>0 and δ>0, for any q∈B(0;ε,(Kδ)m) and θ∈(0,δ) such that S(F+q,K+θB¯)⊂B¯(0,m). By the first part (1), S(F+q,K+θB¯)≠∅. Suppose that there exists m0>r, and for any ε^>0 and δ^>0, there exist qε^~∈B(0;ε^,(Kδ^)m0) and θδ^~∈(0,δ^) such that xε^δ^∈S(F+qε^~,K+θδ^~B¯); however, xε^δ^∉B¯(0,m0).
Sincexε^δ^∈K+θδ^~B¯ and xε^δ^>m0>r, applying the same method of proof as (1), there exist xε^δ^~∈K and xε^δ^~>r for small enough δ^>0. It follows from coercivity condition (14) that there exists yε^δ^∈Kr such that supζ∈F(xε^δ^~)ζ,yε^δ^-xε^δ^~<0. Similarly, we can get (24)supξ∈Fxε^δ^⁡ξ,yε^δ^-xε^δ^<0.
Byyε^δ^∈K⊂K+θδ^~B¯ and xε^δ^∈S(F+qε^~,K+θδ^~B¯), we have(25)0≤supξ∗∈Fxε^δ^⁡ξ∗+qε^~xε^δ^,yε^δ^-xε^δ^=supξ∗∈Fxε^δ^⁡ξ∗,yε^δ^-xε^δ^+qε^~xε^δ^,yε^δ^-xε^δ^<qε^~xε^δ^,yε^δ^-xε^δ^.
Moreover,yε^δ^≤r<m0 implies that supyε^δ^∈(Kδ^)m0⁡qε^~(yε^δ^)<ε. It follows that (26)limε^→0+,δ^→0+⁡qε^~yε^δ^,yε^δ^-xε^δ^=0.Thus, there exist small enough ε^>0 and δ^>0 such that(27)qε^~yε^δ^-qε^~xε^δ^,yε^δ^-xε^δ^<0.We obtain a contradiction from the monotonicity of qε^~. This completes the proof.Remark 7.
Li and He [6] gave a similar result where only the mapping was perturbed in Rn (Theorem 3.1 of [6]). Theorem 6 generalizes the result of [6] from Rn 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 mappingF is perturbed along a direction and the set K is perturbed along another direction. Here in this section, we use I to denote the set K-tK∞∖{0} for any t>0.Lemma 8.
IfK is a nonempty, closed, and convex subset in X, then K∞=I∞ and barr(K)=barr(I).Proof.
Letd∈I∞. Given x0∈K, in view of K≔I+tK∞∖{0}, there exist y0∈I and d¯∈K∞∖{0} such that x0=y0+td¯. It follows that, for all λ>0, (28)x0+λd=y0+td¯+λd=y0+λd+td¯∈I+tK∞∖0=K,and so d∈K∞. This implies that I∞⊂K∞.
Next we prove the converse inclusion. Letd1∈K∞. Given x1∈I, in view of I≔K-tK∞∖{0}, there exist y1∈K and d1¯∈K∞∖{0} such that x1=y1-td1¯. It follows that, for all λ1>0, (29)x1+λ1d1=y1-td1¯+λ1d1=y1+λd1-td1¯∈K-tK∞∖0=I,and so d1∈I∞, implying that K∞⊂I∞. Therefore, K∞=I∞.
Finally, by (6), we have barr(K)=barr(I).Theorem 9.
LetK be a nonempty, closed, and convex subset in X. Suppose that int⁡(barrK)≠∅ and that F:I→2X∗ is an upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty w∗-compact convex values. If the following coercivity condition holds:(30)∃r>0,∀x∈K∖Kr,∃y∈Krsuchthat∃y∗∈Fy∩barrK,y∗,y-x<0,then for any q∈int⁡(barrK), there exists k>r such that (31)∅≠SF-εq,K-tK∞∖0⊂B¯0,1ε,∀ε∈0,1k,∀t>0.Proof.
( 1 ) Firstly, we need to prove that, for any q∈int⁡(barrK), there exists m>r such that, for every ε∈(0,1/m), S(F-εq,K-tK∞∖{0})≠∅. Suppose that there exists q∈int⁡(barrK), and for any m>r, there exists εm∈(0,1/m) such that S(F-εmq,K-tK∞∖{0})=∅.
DenoteEm≔{x∈I:x≤1/εm}. By that Em is bounded, closed, and convex, F is upper hemicontinuous and stably quasimonotone set-valued mapping with nonempty w∗-compact convex values, and Proposition 5(iv), GVIP(F-εmq,Em), has a solution. For each m∈N, there exists xm∈Em such that(32)supξ∈Fxm⁡ξ-εmq,y-xm≥0,∀y∈Em.
(i) If for somem>r, xm<1/εm, then for any y∈K-tK∞∖{0}, there exists λ∈(0,1) such that zλ≔xm+λ(y-xm)∈Em. This is because the convexity of K and K∞ implies that K-tK∞∖{0} is convex. It follows that (33)0≤supξ∈Fxm⁡ξ-εmq,zλ-xm=tsupξ∈Fxm⁡ξ-εmq,y-xm.Since y∈K-tK∞∖{0} is arbitrary, xm solves GVIPF-εmq,K-tK∞∖0. Thus, xm∈SF-εmq,K-tK∞∖0≠∅.
(ii) If for eachm>r, xm=1/εm>m. Since xm∈K-tK∞∖{0}, there exist xm~∈K and pm∈K∞∖{0} such that xm=xm~-tpm. We claim that xm~∉Kr. Indeed, if not, then xm~≤r. Thus, (34)tpm≤xm+xm~≤1εm+r<2εm.It follows that tpm<+∞ for each m. However, by pm∈K∞∖{0}, we have pm>0. Letting t→+∞, this implies that tpm has no upper bound for each m>r. Thus, we obtain a contradiction and so the claim is proved. The coercivity condition (30) implies that there exists ym∈Kr such that there exists ym∗∈F(ym)∩barr(K), ym∗,ym-xm~<0. Thus, by (6), we have (35)ym∗,ym-xm=ym∗,ym-xm~+ym∗,xm~-xm<tym∗,pm≤0,that is, for large enough m, infym∗∈F(ym)ym∗,ym-xm<0 holds. It follows from the quasimonotonicity of F that(36)supξ∈Fxm⁡ξ,ym-xm≤0.
Without loss of generality, we may assume thatxm/xm⇀d∈I∞. By Lemma 8, we have d∈K∞. As int⁡(barrI)=int⁡(barrK)≠∅, Lemma 2 implies d≠0. From q∈int⁡(barrK) and (6), we obtain q,d≤0. We claim that q,d<0. In fact, if not, then q,d=0. Since q∈int⁡(barrK), for any x∗∈X∗, there exists δ∈(0,1) such that δq+(1-δ)x∗∈barrK. It implies that δq+1-δx∗,d≤0 and so x∗,d≤0. Similarly, we can obtain -x∗,d≤0. Thus, x∗,d=0 which contradicts with d≠0. Therefore, q,d<0.
Sinceym<r<1/εm, then for any y∈K-tK∞∖{0}, there exists θ∈(0,1) such that zθ≔ym+θ(y-ym) and zθ≤1/εm. Let us now claim that zθ∈I. Indeed, in view of I≔K-tK∞∖{0} and ym∈K, there exist ym~∈I and pm∈K∞∖{0} such that ym=ym~+pm, and hence zθ=ym~+θ(y-ym~)+(1-θ)pm. Since I is convex, ym~+θ(y-ym~)∈I. From Lemma 8, we know pm∈K∞=I∞. By the definition of the recession cone and 1-θ>0, we get zθ∈I. Therefore, zθ∈Em.
It follows from (32) and (36) that (37)0≤supξ∈Fxm⁡ξ-εmq,zθ-xm=θsupξ∈Fxm⁡ξ-εmq,y-xm+1-θsupξ∈Fxm⁡ξ,ym-xm+εm1-θq,xm-q,ym≤θsupξ∈Fxm⁡ξ-εmq,y-xm+εm1-θq,xm-q,ym.Since εm=1/xm, εmq,xm=q,xm/xm→q,d<0. By sign-preserving theorem of limit, there exists M>0, when m>M, εmq,xm<(1/2)q,d<0.
Also noticing that{ym}⊂Kr, {ym} is bounded, we have εmq,ym→0. It follows that, for large enough m, (38)supξ∈Fxm⁡ξ-εmq,y-xm≥0.As y∈K-tK∞∖{0} is arbitrary, xm solves GVIP(F-εmq,K-tK∞∖{0}).
In either case, we obtain a contradiction toS(F-εmq,K-tK∞∖{0})=∅.
( 2 ) Now we need to prove that, for any q∈int⁡(barrK), there exists k≥m>r such that, for every ε∈(0,1/k), S(F-εq,K-tK∞∖{0})⊂B¯(0,1/ε). By the first part (1), there exists m>r, and for any k≥m and ε∈(0,1/k), we have S(F-εq,K-tK∞∖{0})≠∅. Suppose that there exists q∈int⁡(barrK), and for any k≥m, there exists εk∈(0,1/k) such that xk∈S(F-εkq,K-tK∞∖{0}); however, xk∉B¯(0,1/εk).
Sincexk∈I and xk>1/εk>k≥m>r, applying the same method of proof as (1), there exists xk~∈K and xk~>r. It follows from coercivity condition (30) that there exists yk∈Kr such that there exists yk∗∈F(yk)∩barr(K), yk∗,yk-xk~<0. Similarly, we can get (39)supξk∈Fxk⁡ξk,yk-xk≤0.
Moreover,yk∈K implies that yk∈I, because K=I+tK∞=I+tI∞⊂I. By xk∈S(F-εkq,K-tK∞∖{0}), we have(40)0≤supξk∗∈Fxk⁡ξk∗-εkq,yk-xk=supξk∗∈Fxk⁡ξk∗,yk-xk+εkq,xk-q,yk≤εkq,xk-q,yk.Similarly, we can assume that xk/xk⇀e. Thus, q,xk/xk→q,e<0. By sign-preserving theorem of limit, there exists K>0, when k>K, q,xk/xk<0. Since xk>1/εk, xkεk>1. When k>K, (41)εkq,xk=xkεkq,xkxk<0.Similarly, {yk}⊂Kr implies that {yk} is bounded, εkq,yk→0. For large enough k, εk(q,xk-q,yk)<0. It is a contradiction with (40). This completes the proof.Remark 10.
Li and He [6] gave a similar result of GVIP(F,K) where only the mapping was directional perturbed in Rn (Theorem 3.2 of [6]). Theorem 9 generalizes the result of [6] from Rn 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 mappingF and on the constraint set K. It is important to observe that the second solvability result for perturbed generalized variational inequality when the perturbation terms q and K∞ 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.



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