# Upper Semicontinuity of Solution Mappings to Parametric Generalized Vector Quasiequilibrium Problems

Authors: Shu-qiang Shan; Yu Han; Nan-jing Huang
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We establish the upper semicontinuity of solution mappings for a class of parametric generalized vector quasiequilibrium problems. As applications, we obtain the upper semicontinuity of solution mappings to several problems, such as
parametric optimization problem, parametric saddle point problem, parametric Nash equilibria problem, parametric variational inequality, and parametric equilibrium problem.


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## Body

## 1. Introduction

It is well known that the vector equilibrium problem provides a unified model of several problems, such as the vector optimization problem, the vector saddle point problem, the vector complementarity problem, and the vector variational inequality problem [1, 2]. In recent years, the existence of solutions for various types of vector equilibrium problems has been investigated intensively by many authors under different conditions (see, e.g., [3–8] and the references therein).On the other hand, the stability analysis of the solution mapping to vector equilibrium problems is an important topic in vector optimization theory. In recent years, the lower semicontinuity and the upper semicontinuity of of the solution mappings to parametric optimization problems, parametric vector variational inequalities, and parametric vector equilibrium problems have been intensively studied in the literature; for instance, we refer the reader to [9–17]. Recently, Anh and Khanh [18] obtained the semicontinuity of the solution mapping to parametric vector quasiequilibrium problems. Khanh and Luu [19] discussed the upper semicontinuity of solution mapping to parametric vector quasivariational inequalities involving multifunctions without monotonicity assumptions. Fang and Huang [20] established upper semicontinuity of the solution maps to the vector homogeneous quasiequilibrium problems. By using a scalarization method, Cheng and Zhu [21] investigated the upper semicontinuity and lower semicontinuity of the solution mapping to a parametric weak vector variational inequality in finite-dimension Euclidean spaces. Li and Fang [22] studied the lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality by using a key assumption that includes the information about the solutions set. By virtue of a density result and scalarization technique, Gong and Yao [23] first discussed the lower semicontinuity of the set of efficient solutions to parametric vector equilibrium problems. Li et al. [24] investigated the upper semicontinuity and lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem. By using the Hölder relation, Zhang et al. [25] obtained the lower semicontinuity of the efficient solution mapping to a parametric vector equilibrium problem. Fan et al. [26] studied the continuity of the solution mapping concerned with a class of vector quasiequilibrium problems with an application to traffic network problems. Xu and Li [27] established the lower semicontinuity of solution mappings to a parametric generalized strong vector equilibrium problem by using a scalarization method. Very recently, by using a new proof method which is different from the ones used in the literature, Han and Gong [28] established the lower semicontinuity of the solution mappings to parametric generalized strong vector equilibrium problems without the assumptions of monotonicity and compactness.The aim of this paper is to establish the upper semicontinuity of solution mappings for a class of parametric generalized vector quasiequilibrium problems under some suitable conditions. We provide a uniform method to deal with the upper semicontinuity of solution mappings for several problems, such as parametric optimization problem, parametric saddle point problem, parametric Nash equilibria problem, parametric variational inequality, and parametric equilibrium problem. The rest of the paper is organized as follows. In Section2, we recall some basic concepts and known lemmas. In Section 3, we show a main result in connection with the upper semicontinuity of the solution mapping for the parametric generalized vector quasiequilibrium problem. Some applications of the main result are given in Section 4.


## 2. Preliminaries

Throughout this paper, unless otherwise specified, letΛ, W, X, and Y be four normed vector spaces. Let D be a nonempty closed subset of Y. Let F:X×X×W→2Y and K:Λ→2X be two set-valued mappings. For u,λ∈W×Λ, we consider that the following parametric generalized vector quasiequilibrium problems consist of finding x0∈Kλ such thatPGVQEPFx0,y,u⊆D,∀y∈Kλ.We define a solution mappingS to PGVQEP by(1)Su,λ=x∈Kλ:Fx,y,u⊆D,∀y∈Kλ,hhhhhhhhhhhhhhhhhhhhhhhu,λ∈W×Λ.Definition 1 (see [29]).
LetT and Q be two topological vector spaces. A set-valued mapping G:T→2Q is said to be (i)
upper semicontinuous (u.s.c.) atu0∈T if, for any neighborhood V of Gu0, there exists a neighborhood Uu0 of u0 such that, for every u∈Uu0, Gu⊆V;(ii)
lower semicontinuous (l.s.c.) atu0∈T if, for any x∈Gu0 and any neighborhood V of x, there exists a neighborhood Uu0 of u0 such that, for every u∈Uu0, Gu∩V≠∅.A set-valued mapping G is said to be u.s.c. and l.s.c. on T, if it is u.s.c. and l.s.c. at each u∈T, respectively. We say that G is continuous on T, if it is both u.s.c and l.s.c on T.Definition 2 (see [15]).
LetT and Q be two topological vector spaces and let C⊆Q be a cone. A set-valued mapping G:T→2Q is said to be C-lower semicontinuous (C-l.s.c.) at u0∈T if, for any x∈Gu0 and any neighborhood V of x, there exists a neighborhood Uu0 of u0 such that(2)Gu∩V-C≠∅,∀u∈Uu0.A set-valued mapping G is said to be C-l.s.c. on T, if it is C-l.s.c. at each u∈T.Remark 3.
It is easy to see that ifG is l.s.c. at u0∈T, then it is C-l.s.c. at u0∈T. In fact, since 0∈C, we have V⊆V-C. It follows from Gu∩V≠∅ that (3)Gu∩V-C≠∅.The following example shows that the reverse is not true in general.Example 4.
LetC=R+=x∈R:x≥0. We define a set-valued mapping G:R→2R as follows: (4)Gx=cos⁡x,sinx+2,x≠0,1,3,x=0.It is easy to see that G is C-l.s.c. at 0, but G is not l.s.c. at 0.Lemma 5 (see [29]).
A set-valued mappingG:T→2Q is l.s.c. at u0∈T if and only if, for any sequence un⊆T with un→u0 and for any x0∈Gu0, there exists xn∈Gun such that xn→x0.Lemma 6 (see [30]).
Assume thatG:T→2Q is a set-valued mapping. If Gu0 is compact for some u0∈T, then G is u.s.c. at u0∈T if and only if, for any sequence un⊆T with un→u0 and for any xn∈Gun, there exist x0∈Gu0 and a subsequence xnk of xn such that xnk→x0.


## 3. Upper Semicontinuity of Solution Mapping toPGVQEP

In this section, we establish the upper semicontinuity ofS at u0,λ0∈W×Λ.Theorem 7.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and F·,·,· is l.s.c. on Kλ0×Kλ0×u0. Then S is u.s.c. at u0,λ0.Proof.
Suppose to the contrary thatS·,· is not u.s.c. at u0,λ0. Then there exists a neighborhood W0 of Su0,λ0; for any neighborhood Uu0×Uλ0 of u0,λ0, there exists u′,λ′∈Uu0×Uλ0 such that Su′,λ′⊄W0. Hence, there exists a sequence un,λn with un,λn→u0,λ0 such that (5)Sun,λn⊄W0,∀n∈N.Then there exists(6)xn∈Sun,λnsuch that(7)xn∉W0,∀n∈N.Since K· is u.s.c. at λ0 and xn∈Kλn, by Lemma 6, there exist x0∈Kλ0 and a subsequence xnk of xn such that xnk→x0. Without loss of generality, we can assume that xn→x0.
We claim thatx0∈Su0,λ0. In fact, suppose that x0∉Su0,λ0. Then there exists y0∈Kλ0 such that(8)Fx0,y0,u0⊄D.Hence there exists z0∈Fx0,y0,u0 such that(9)z0∉D.Since K· is l.s.c. at λ0 and y0∈Kλ0, by Lemma 5, there exists yn∈Kλn such that yn→y0. Since F·,·,· is l.s.c. at x0,y0,u0 and z0∈Fx0,y0,u0, by Lemma 5, there exists zn∈Fxn,yn,un such that zn→z0. Noting (9) and that D is closed, we know that zn∉D for n large enough, which contradicts with (6). Therefore, x0∈Su0,λ0. It is easy to see that xn→x0∈W0, which contradicts with (7).Remark 8.
We would like to point out that the assumptions of Theorem7 are quite natural and easy to be verified.We give an example to illustrate Theorem7.Example 9.
LetY=R2, D=x1,x2∈R2:x1≤0,x2≥0, and Λ=W=X=R. Let B be the closed unit ball of R2. Assume that F:X×X×W→2Y is defined by(10)Fx,y,u=f1x,y,u,f2x,y,u+B,where(11)f1x,y,u=ln⁡x-xy-2y-y2cos⁡u+3y-x+2sinu-3,f2x,y,u=ln⁡x2+2xy+y2+2x+cos⁡u.
We define a set-valued mappingK:Λ→2X as follows: (12)Kλ:=x∈R:-3+sinλ≤x≤3+cos⁡λ,λ∈Λ.Then it is easy to see that 1∈S0,0 and so S0,0≠∅. Moreover, it is easy to check that all the assumptions of Theorem 7 are satisfied. Thus, it follows from Theorem 7 that S is u.s.c. at 0,0.


## 4. Some Applications

In this section, we give some applications of Theorem7 to the optimization problem, the saddle point problem, the Nash equilibria problem, the variational inequality, the variational inequality with set-valued mappings, the equilibrium problem, the generalized strong vector equilibrium problem, and the generalized weak vector equilibrium problem.Optimization Problem. Let f:X→R be a mapping. Let K be a nonempty subset of X. A point x0∈K is called a solution of optimization problem if and only if (13)fx0≤fy,∀y∈K.Let f:X×W→R be a mapping and let K:Λ→2X be a set-valued mapping. For u,λ∈W×Λ, we consider that the following parametric optimization problem consists of finding x0∈Kλ such thatPOPfx0,u≤fy,u,∀y∈Kλ.Define a solution mappingO:W×Λ→2X to POP by(14)Ou,λ=x∈Kλ:fx,u≤fy,u,∀y∈Kλ.From Theorem7, we can get the following corollary.Corollary 10.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and f·,· is continuous on Kλ0×u0. Then the mapping O is u.s.c. at u0,λ0.Saddle Point Problem. Let φ:X×X→R be a mapping. Let K be a nonempty subset of X. x¯1,x¯2 is called a saddle point on K if and only if(15)x¯1,x¯2∈K×K,φx¯1,y2≤φy1,x¯2,∀y1,y2∈K×K.Let φ:X×X×W→R be a mapping and let K:Λ→2X be a set-valued mapping. For u,λ∈W×Λ, we consider that the following parametric saddle point problem consists of finding x¯1,x¯2∈Kλ×Kλ such thatPSPPφx¯1,y2,u≤φy1,x¯2,u,∀y1,y2∈Kλ×Kλ.Define a solution mappingP:W×Λ→2X×X to PSPP by(16)Pu,λ=x1,x2∈Kλ×Kλ:φx1,y2,uh≤φy1,x2,u,∀y1,y2∈Kλ×Kλ.From Theorem7, we can get the following corollary.Corollary 11.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and φ·,·,· is continuous on Kλ0×Kλ0×u0. Then the mapping P is u.s.c. at u0,λ0.Nash Equilibria Problem. Let I be a finite index set. For every i∈I, let Ki be a nonempty subset of X and let fi:∏i∈IX→R be a mapping. Let K:=∏i∈IKi. For x=xii∈I∈K, we define xi=xjj∈I,j≠i. A point x¯=x¯ii∈I∈K is called a Nash equilibrium if and only if, for any i∈I, we have(17)fix¯≤fix¯i,yi,∀yi∈Ki.For everyi∈I, let fi:∏i∈IX×W→R be a mapping and let Ki:Λ→2X be a set-valued mapping. Let Kλ:=∏i∈IKiλ for λ∈Λ. For u,λ∈W×Λ, we consider that the following parametric Nash equilibria problem consists of finding x¯=x¯ii∈I∈Kλ such thatPNEP∀i∈I,fix¯,u≤fix¯i,yi,u,∀yi∈Kiλ.Define a solution mappingN:W×Λ→2∏i∈IX to PNEP by(18)Nu,λ=x=xii∈I∈Kλ:∀i∈I,hifix,u≤fixi,yi,u,∀yi∈Kiλ.From Theorem7, we can get the following corollary.Corollary 12.
Letu0,λ0∈W×Λ. For every i∈I, assume that Kiλ0 is nonempty compact, Ki· is continuous at λ0, and fi is continuous on ∏i∈IKiλ0×u0. Then the mapping N is u.s.c. at u0,λ0.Variational Inequality. Let X∗ be the topological dual space of X. Let T:X→X∗ be a mapping. Let K be a nonempty subset of X. A point x0∈K is called a solution of variational inequality if and only if(19)Tx0,y-x0≥0,∀y∈K.Let T:X×W→X∗ be a mapping and let K:Λ→2X be a set-valued mapping. For u,λ∈W×Λ, we consider that the following parametric variational inequality consists of finding x0∈Kλ such thatPVITx0,u,y-x0≥0,∀y∈Kλ.Define a solution mappingV:W×Λ→2X to PVI by(20)Vu,λ=x∈Kλ:Tx,u,y-x≥0,∀y∈Kλ.From Theorem7, we can get the following corollary.Corollary 13.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and T·,· is continuous on Kλ0×u0. Then the mapping V is u.s.c. at u0,λ0.Variational Inequality with Set-Valued Mappings. Let X∗ be the topological dual space of X. Let T:X→2X∗ be a set-valued mapping. Let K be a nonempty subset of X. A point x0∈K is called a solution of variational inequality with set-valued mappings if and only if(21)∃ξ∈Tx0,ξ,y-x0≥0,∀y∈K.Let T:X×W→2X∗ and K:Λ→2X be two set-valued mappings. For u,λ∈W×Λ, we consider that the following parametric variational inequality with set-valued mappings consists of finding x0∈Kλ such thatPVISM∃ξ∈Tx0,u,ξ,y-x0≥0,∀y∈Kλ.Define a solution mappingVs:W×Λ→2X to PVISM by(22)Vsu,λ=ξ,y-x≥0,∀y∈Kλx∈Kλ:∃ξ∈Tx,u,hhhiξ,y-x≥0,∀y∈Kλ.The proof of the following corollary is similar to the proof of Theorem7. For the convenience of the readers, we also give the proof.Corollary 14.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and T·,· is u.s.c. on Kλ0×u0 with nonempty compact values. Then the mapping Vs is u.s.c. at u0,λ0.Proof.
Suppose to the contrary thatVs·,· is not u.s.c. at u0,λ0. Then there exists a neighborhood W0 of Vsu0,λ0; for any neighborhood Uu0×Uλ0 of u0,λ0, there exists u′,λ′∈Uu0×Uλ0 such that Vsu′,λ′⊄W0. Hence, there exists a sequence un,λn with un,λn→u0,λ0 such that(23)Vsun,λn⊄W0,∀n∈N.Then there exists(24)xn∈Vsun,λnsuch that(25)xn∉W0,∀n∈N.Since K· is u.s.c. at λ0 and xn∈Kλn, by Lemma 6, there exist x0∈Kλ0 and a subsequence xnk of xn such that xnk→x0. Without loss of generality, we can assume that xn→x0.
By (24), there exists ξn∈Txn,un such that(26)ξn,y-xn≥0,∀y∈Kλn.Since ξn∈Txn,un, by Lemma 6, there exist ξ0∈Tx0,u0 and a subsequence {ξnk} of ξn such that ξnk→ξ0. Without loss of generality, we can assume that ξn→ξ0.
For anyy¯∈Kλ0, by Lemma 5, there exists yn∈Kλn such that yn→y¯. Noting (26), we have(27)ξn,yn-xn≥0.It follows from ξn,yn-xn→ξ0,y¯-x0 that(28)ξ0,y¯-x0≥0.Therefore, x0∈Vsu0,λ0. We can see that xn→x0∈W0, which contradicts with (25).Equilibrium Problem. Let f:X×X→R be a mapping. Let K be a nonempty subset of X. A point x0∈K is called a solution of equilibrium problem if and only if (29)fx0,y≥0,∀y∈K.Let f:X×X×W→R and K:Λ→2X be two set-valued mappings. For u,λ∈W×Λ, we consider that the following parametric equilibrium problem consists of finding x0∈Kλ such thatPEPfx0,y,u≥0,∀y∈Kλ.Define a solution mappingE:W×Λ→2X to PEP by(30)Eu,λ=x∈Kλ:fx,y,u≥0,∀y∈Kλ.From Theorem7, we can get the following corollary.Corollary 15.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and f·,·,· is continuous on Kλ0×Kλ0×u0. Then the mapping E is u.s.c. at u0,λ0.Generalized Strong Vector Equilibrium Problem. Assume that C⊆Y is a closed cone. Let F:X×X→2Y be a set-valued mapping. Let K be a nonempty subset of X. A point x0∈K is called a solution of generalized strong vector equilibrium problem if and only if(31)Fx0,y⊆C,∀y∈K.Let F:X×X×W→2Y and K:Λ→2X be two set-valued mappings. For u,λ∈W×Λ, we consider that the following parametric generalized strong vector equilibrium problem consists of finding x0∈Kλ such thatPGSVEPFx0,y,u⊆C,∀y∈Kλ.Define a solution mappingM:W×Λ→2X to PGSVEP by(32)Mu,λ=x∈Kλ:Fx,y,u⊆C,∀y∈Kλ.From Theorem7, we can get the following corollary.Corollary 16.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and F·,·,· is l.s.c. on Kλ0×Kλ0×u0. Then the mapping M is u.s.c. at u0,λ0.Generalized Weak Vector Equilibrium Problem. Assume that C⊆Y is a cone with nonempty interior. Let F:X×X→2Y be a set-valued mapping. Let K be a nonempty subset of X. A point x0∈K is called a solution of generalized weak vector equilibrium problem if and only if(33)Fx0,y∩-intC=∅,∀y∈K.Let F:X×X×W→2Y and K:Λ→2X be two set-valued mappings. For u,λ∈W×Λ, we consider that the following parametric generalized weak vector equilibrium problem consists of finding x0∈Kλ such thatPGWVEPFx0,y,u∩-intC=∅,∀y∈Kλ.Define a solution mappingG:W×Λ→2X to PGWVEP by(34)Gu,λ=x∈Kλ:Fx,y,u∩-intC=∅,∀y∈Kλ.From Theorem7, we can get the following corollary.Corollary 17.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and F·,·,· is l.s.c. on Kλ0×Kλ0×u0. Then the mapping G is u.s.c. at u0,λ0.Remark 18.
Corollary17 is similar to Theorem 3.4 of [18].From Corollary17, it is easy to get the following corollary.Corollary 19.
Letu0,λ0∈W×Λ. Assume that Kλ0 is nonempty compact, K· is continuous at λ0, and F·,·,· is C-l.s.c. on Kλ0×Kλ0×u0. Then G is u.s.c. at u0,λ0.Remark 20.
In the proof of upper semicontinuity of solution mapping, Corollary19 improves Theorem 3.1 of [15].



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