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# On parametric implicit vector variational inequality problems

- A Farajzadeh
^{1}and - S Plubteing
^{2}Email author

**2014**:151

https://doi.org/10.1186/1029-242X-2014-151

© Farajzadeh and Plubteing; licensee Springer. 2014

**Received:**13 August 2013**Accepted:**28 March 2014**Published:**2 May 2014

## Abstract

Recently Huang *et al.* (Math. Comput. Model. 43:1267-1274, 2006)introduced a class of parametric implicit vector equilibrium problems (for shortPIVEP) and they presented some existence results for a solution of PIVEP. Also,they provided two theorems about upper and lower semi-continuity of the solutionset of PIVEP in a locally convex Hausdorff topological vector space. The paperextends the corresponding results obtained in the setting of topological vectorspaces with mild assumptions and removing the notion of locally non-positivenessat a point and lower semi-continuity of the parametric mapping.

## Keywords

- topological vector space
- equilibrium problem
- KKM mapping

## 1 Introduction and preliminaries

*et al.*[5] considered the implicit vector equilibrium problem (for short IVEP) whichconsists of finding $x\in E$ such that

*X*and

*Y*are twoHausdorff topological vector spaces,

*E*is a nonempty closed convex subsetof

*X*and $C:E\to {2}^{Y}$ be a set-valued mapping such that for any$x\in E$, $C(x)$ is a closed and convex cone with$C(x)\cap -C(x)=\{0\}$, that is pointed, with nonempty interior. Theycontinued their research and introduced the parametric implicit vector equilibriumproblem, which consists of finding ${x}^{\ast}\in \mathbf{K}(\lambda )$, for each given $(\lambda ,\u03f5)\in {\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}$ such that

where ${\mathrm{\Lambda}}_{i}$ ($i=1,2$) are Hausdorff topological vector spaces (theparametric spaces), $\mathbf{K}:{\mathrm{\Lambda}}_{1}\to {2}^{X}$ a set-valued mapping such that for any$\lambda \in {\mathrm{\Lambda}}_{1}$, $\mathbf{K}(\lambda )$ is a nonempty, closed and convex subset of *X*with $\mathbf{K}({\mathrm{\Lambda}}_{1})={\bigcup}_{\lambda \in {\mathrm{\Lambda}}_{1}}\mathbf{K}(\lambda )\subseteq E$ and $f:{\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}\times E\to Y$. They obtained some existence results for a solutionof PIVEP and further they studied upper and lower semi-continuity of the solution ofPIVEP in locally convex Hausdorff topological vector spaces. This paper is motivatedand inspired by the recent paper [5] and its aim is to extend the results to the setting of Hausdorfftopological vector spaces with mild assumptions and removing the condition of beinglocally non-positive at a point has been applied in Proposition 3.3 of [6] and lower semi-continuity of the parametric mapping used inTheorem 3.2 of [6]. More precisely, we first establish an existence result for a solution ofIVEP and then by using it we will deal with the behavior of the solution set ofPIVEP when the parameters $(\lambda ,\u03f5)$ start to change. In fact we will show that thesolution set as a mapping $S:{\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}\to {2}^{X}$ is upper semi-continuous and lower semi-continuousunder special conditions. In the rest of this section we recall some definitions andresults that we need in the next section.

*P*of

*Y*is called a pointed and convex cone if and only if$P+P\subseteq P$, $tP\subseteq P$, for all $t\ge 0$, and $P\cap -P=\{0\}$. The domain of a set-valued mapping$W:X\to {2}^{Y}$ is defined as $D(W)=\{x\in X:W(x)\ne \mathrm{\varnothing}\}$ and its graph is defined as

Also *W* is said to be closed if its graph, that is, $Graph(W)$, is a closed subset of $X\times Y$. A set-valued mapping $T:X\to {2}^{Y}$ is called upper semi-continuous (u.s.c.) at$x\in X$ if for every open set *V* containing$T(x)$ there exists an open set *U* containing*x* such that $T(u)\subseteq V$, for all $u\in U$. The mapping *T* is said to be lowersemi-continuous (l.s.c.) if for every open set *V* with$T(x)\cap V\ne \mathrm{\varnothing}$ there exists an open set *U* containing*x* such that $T(u)\cap V\ne \mathrm{\varnothing}$. The mapping *T* is continuous at *x*if it is both u.s.c. and l.s.c. at *x*. Moreover, *T* is u.s.c.(l.s.c.) on *X* if *T* is u.s.c. (l.s.c.) at each point of*X*.

We need the following lemma in the sequel.

**Lemma 1.1** ([7])

*Let*

*X*

*and*

*Y*

*be topological spaces and*$T:X\to {2}^{Y}$

*be a mapping*.

*The following statements are true*:

- (i)
*If for any*$x\in X$, $T(x)$*is compact*,*then**T**is u*.*s*.*c*.*at*$x\in X$*if and only if for any net*$\{{x}_{i}\}\subseteq X$*such that*${x}_{i}\to x$*and for every*${y}_{i}\in T({x}_{i})$,*there exist*$y\in T(x)$*and a subnet*$\{{y}_{j}\}$*of*$\{{y}_{i}\}$*such that*${y}_{j}\to y$. - (ii)
*T**is l*.*s*.*c*.*at*$x\in X$*if and only if for any net*$\{{x}_{i}\}\subseteq X$*with*${x}_{i}\to x$*and for any*$y\in T(x)$,*there exists a net*$\{{y}_{i}\}$*such that*${y}_{i}\in T({x}_{i})$*and*${y}_{i}\to y$.

*X*be a topological vector space. A mapping $F:K\subseteq X\to {2}^{X}$ is said to be a KKM mapping, if, for any finite set$A\subseteq K$,

where co*A* denotes the convex hull of *A*.

The following lemma plays a crucial rule in this paper.

**Lemma 1.3** ([8])

*Let* *K* *be a nonempty subset of a topological vector space* *X* *and*$F:K\to {2}^{X}$*be a*$KKM$*mapping with closed values in* *K*. *Assume that there exists a nonempty compact convex subset* *B* *of* *K* *such that*${\bigcap}_{x\in B}F(x)$*is compact*. *Then*${\bigcap}_{x\in K}F(x)\ne \mathrm{\varnothing}$.

## 2 Main results

The next result provides an existence result for a solution of IVEP.

**Theorem 2.1**

*Let*

*K*

*be closed convex subset of a t*.

*v*.

*s*.

*X*

*and*$f:K\times K\to Y$

*and*$g:K\to K$

*be two mappings*.

*If the following assumptions are satisfied*:

- (a)
$f(g(x),x)\notin -intC(x)$, $\mathrm{\forall}x\in K$,

- (b)
*the mapping*$x\to f(g(x),y)$*is continuous*,*for all*$y\in K$, - (c)
*for each*$x\in K$,*the set*$\{y\in K:f(g(x),y)\in -intC(x)\}$*is convex*, - (d)
*the mapping*$W:K\to {2}^{Y}$*defined by*$W(x)=Y\mathrm{\setminus}(-intC(x))$*is closed*, - (e)
*there exist subsets**M**and**N**of**K*,*compact convex and compact*,*respectively*,*such that for all*$x\in K\mathrm{\setminus}N$*there is*$y\in M$*such that*$f(g(x),y)\in -intC(x)$,

*then the solution set of IVEP is nonempty and compact*.

*Proof*Define the set-valued mapping $F:K\to {2}^{K}$ by

*F*satisfies all the assumptions of Lemma 1.3. By (b) and(d), $F(y)$ is a closed subset of

*K*for all$y\in K$. It follows from (c) and (a) that

*F*is a KKMmapping. Indeed, on the contrary of the assertion if there exist${y}_{1},{y}_{2},\dots ,{y}_{n}$ in

*K*and $z={\sum}_{i=1}^{n}{\lambda}_{i}{y}_{i}\in co\{{y}_{1},{y}_{2},\dots ,{y}_{n}\}\mathrm{\setminus}{\bigcup}_{i=1}^{n}F({y}_{i})$, then $f(g(z),{y}_{i})\in -intC(z)$ and so by (c) we deduce that

*F*is a KKM mapping. Also, it isobvious from (e) that ${\bigcap}_{y\in M}F(y)\subseteq N$ and so ${\bigcap}_{y\in M}F(y)$ is compact (note that $F(y)$ is closed for each $y\in Y$ and

*M*is compact). Hence by Lemma 1.3there exists $\overline{x}\in K$ such that

and it is easy to see that the solution set of IVEP is equal to the set${\bigcap}_{x\in K}F(x)$ and hence $\overline{x}$ is a solution of IVEP and further it is compact (note${\bigcap}_{x\in K}F(x)\subseteq {\bigcap}_{x\in M}F(x)\subseteq N$) and hence the proof is complete. □

*g*is continuous and

*f*is continuous with respect tothe first variable then the mapping $x\to f(g(x),y)$ is continuous and so condition (b) holds while thesimple example $g(x)=1$ if

*x*is rational, and$g(x)=0$ if

*x*is irrational, and$f(x,y)=1$, for

*x*rational, and $f(x,y)=0$ if

*x*is irrational, shows that it is easy tocheck that the mapping $x\to f(g(x),y)$ is continuous; nevertheless, neither

*g*nor

*f*is continuous, which shows that the converse does not hold ingeneral. Moreover, in the example, if we take $K=[0,1]$ then

*f*and

*g*satisfy all theassumptions of Theorem 2.1 and so the solution set of IVEP is nonempty andcompact but the example cannot fulfill all the conditions of Proposition 3.1 in [6]. Hence Theorem 2.1 extends Proposition 3.1 in [6]. Also one can easily see the

*C*-convexity of

*f*at thesecond variable, that is, for each $x\in K$,

which implies condition (c) of Theorem 2.1, while if we take$X=\mathrm{\Re}$ and let *K* be any nonempty convex and compactsubset of *X* and define $f(x,y)=-{y}^{2}$, for all $x,y\in K$ and we let *g* be an arbitrary mapping, wetake the example $f(x,y)=-{y}^{2}$, for $x,y\in K$ and *g* an arbitrary mapping, then thisexample fulfills condition (c) (note that it satisfies all the assumptions ofTheorem 2.1) but *f* is not convex at the second variable and hencecondition (c) improves condition (3) in Proposition 3.1 of [6].

**Definition 2.2** ([6])

where $\partial V({x}_{0})$ is the boundary of $V({x}_{0})$. In the case that *g* is the identity mapping,the mapping *f* is called locally non-positive at ${x}_{0}\in E$.

The following corollary is an extension of Proposition 3.3 in [6] for topological vector spaces. Furthermore, the condition that *f*is locally non-positive at ${x}_{0}\in K$ has been omitted.

**Corollary 2.3**

*Let*

*K*

*be a nonempty closed and convex subset of a Hausdorff topological vectorspace*

*X*

*and let*$f:K\times K\to Y$

*and*$C:K\to Y$

*be two mappings such that*:

- (a)
$f(g(x),x)=0$, $\mathrm{\forall}x\in K$,

- (b)
*the mapping*$x\to f(g(x),y)$*is continuous*,*for all*$y\in K$, - (c)
*for each*$x\in K$*the mapping*$y\to f(g(x),y)$*is*$C(x)$-*convex*, - (d)
*the mapping*$W:K\to {2}^{Y}$*defined by*$W(x)=Y\mathrm{\setminus}(-intC(x))$*is closed*, - (e)
*there exist a nonempty compact and convex subset**D**of*$K\cap V({x}_{0})$*and*${y}_{0}\in D$*such that for all*$x\in (K\cap V({x}_{0}))\mathrm{\setminus}D$$f(g(x),y)\in -intC(x).$

*Then IVEP has a solution in the neighborhood*$V({x}_{0})$

*of*${x}_{0}$,

*that is*,

*there exists*${x}^{\ast}\in (K\cap V({x}_{0}))$

*such that*

*Moreover*, *the solution set is a compact subset of*$K\cap V({x}_{0})$.

*Proof*There is neighborhood

*U*of ${x}_{0}$ such that $coU\subseteq V({x}_{0})$ (see, for example, [10]). Hence by Theorem 2.1, IVEP has a solution on$B=K\cap \overline{co(coU\cup \{{y}_{0}\})}$. Then there exists ${x}^{\ast}\in B$ such that

*t*that is small enough. Then bycondition (c) we have

which is a contradiction. Hence ${x}^{\ast}$ is a solution of IVEP. The second part follows fromcondition (e). This completes the proof. □

The next theorem is an extension of Theorems 3.1, 3.3 and Corollary 3.3 in [6] with mild assumptions for mappings which do not need to satisfy thelocally non-positive condition. In fact this condition has been removed.

**Theorem 2.4**

*Let*$F:{\mathrm{\Lambda}}_{2}\times E\times E\to Y$

*and*$g:E\to E$

*be two mappings*.

*If the following assumptions hold*:

- (i)
$\mathbf{K}:{\mathrm{\Lambda}}_{1}\to {2}^{E}$

*is a continuous mapping with nonempty convex compact values*; - (ii)
$(\epsilon ,x,y)\to F(\epsilon ,g(x),y)$

*is continuous*; - (iii)
*the set*$\{y:F(\epsilon ,g(x),y)\in -intC(x)\}$*is convex and*$F(\epsilon ,g(x),x)=0$,*for each*$(\u03f5,x)\in {\mathrm{\Lambda}}_{1}\times E$; - (iv)
*the mapping*$W:E\to {2}^{Y}$*defined by*$W(x)=Y\mathrm{\setminus}-intC(x)$*is closed*;

*then*

- (i)
*for each*$(\lambda ,\epsilon )\in {\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}$,*the solution set*$S(\lambda ,\epsilon )=\{x\in \mathbf{K}(\lambda ):F(\epsilon ,g(x),y)\notin -intC(x),\mathrm{\forall}y\in \mathbf{K}(\lambda )\},$

*is nonempty and compact*,

- (ii)
*the solution set mapping*$S:{\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}\to {2}^{X}$*defined by*$(\lambda ,\epsilon )\to S(\lambda ,\epsilon )$

*is continuous*.

*Proof*The first part, that is, (i), follows from Theorem 2.1 by taking,for each $(\lambda ,\epsilon )\in {\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}$, $M=N=\mathbf{K}(\lambda )$ and defining $f(x,y)=F(\epsilon ,x,y)$, for all $(x,y)\in \mathbf{K}(\lambda )\times \mathbf{K}(\lambda )$. To prove (ii), let ${\{({\lambda}_{i},{\u03f5}_{i})\}}_{i\in I}\subseteq {\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}$ be a net with $({\lambda}_{i},{\u03f5}_{i})\to (\lambda ,\u03f5)$ and ${z}_{i}\in S({\lambda}_{i},{\u03f5}_{i})\subset K({\lambda}_{i})$. Since

**K**is u.s.c., ${\lambda}_{i}\to \lambda $ and ${z}_{i}\in \mathbf{K}({\lambda}_{i})$, using Lemma 1.1(i), there exist$z\in \mathbf{K}(\lambda )$ and a subnet $\{{z}_{{i}_{j}}\}$ of $\{{z}_{i}\}$ which converges to

*z*. So

*S*will be an u.s.c.). If the claim is not truethen there is $y\in \mathbf{K}(\lambda )$ such that

**K**is l.s.c., there exists net ${w}_{j}\in \mathbf{K}({\lambda}_{j})$ such that ${w}_{j}\to y$. Then it follows from (1) that

*W*we get$f(\u03f5,g(x),y)\in W(x)$, which is a contradiction, and then the solution setmapping

*S*is u.s.c. Now we show that

*S*is l.s.c. Let${\{({\lambda}_{i},{\u03f5}_{i})\}}_{i\in I}\subseteq {\mathrm{\Lambda}}_{1}\times {\mathrm{\Lambda}}_{2}$ be a net with $({\lambda}_{i},{\u03f5}_{i})\to (\lambda ,\u03f5)$ and

*z*an arbitrary element of$S(\lambda ,\u03f5)\subseteq \mathbf{K}(\lambda )$. Put $\mathrm{\Xi}=\{V:V\text{is a neighborhood of}z\}$ (note by the relation $V\u2aafW$ if and only if $V\supseteq W$, the set Ξ is a directed set). Then for each$(V,i)\in \mathrm{\Xi}\times I$, there is a closed and convex neighborhood${H}_{V,i}$ of

*z*such that ${H}_{V,i}\subset V$ (see [10]) and so it follows from Theorem 2.1 that there is${z}_{i}\in {H}_{V,i}\cap \mathbf{K}({\lambda}_{i})$ such that

and hence ${z}_{i}\in S({\u03f5}_{i},{\lambda}_{i})$. Consequently, for each point $(V,i)\in \mathrm{\Xi}\times I$ there is ${z}_{i}\in S({\u03f5}_{i},{\lambda}_{i})$, and so ${z}_{i}\to z$. Hence it follows from Lemma 1.1(ii) that*S* is l.s.c. and the proof is completed. □

Inspired by the proof of the second part of the previous theorem we can deduce thelower semi-continuity of the solution set mapping. Indeed the next theorem is animprovement of Theorem 3.2 in [6] without using the lower semi-continuity of the mapping$\mathbf{K}:{\mathrm{\Lambda}}_{1}\to {2}^{E}$; its proof is similar to the proof presented for thesecond part of Theorem 2.4 and so we omit the proof.

**Theorem 2.5**

*Let*$F:{\mathrm{\Lambda}}_{2}\times E\times E\to Y$

*and*$g:E\to E$

*be two mappings*.

*For a given*$({\lambda}_{0},{\u03f5}_{0})\in ({\mathrm{\Lambda}}_{1},{\mathrm{\Lambda}}_{2})$

*there exist neighborhoods*$U({\lambda}_{0})$

*of*${\lambda}_{0}$

*and*$M({\u03f5}_{0})$

*of*${\u03f5}_{0}$

*such that the following assumptions are satisfied*:

- (i)
$\mathbf{K}:U({\lambda}_{0})\to {2}^{X}$

*is a mapping with nonempty convex and compact values*; - (ii)
*mapping*$(\epsilon ,x,y)\to F(\epsilon ,g(x),y)$*is continuous*,*for each*$(\epsilon ,x,y)\in M({\u03f5}_{0})\times E\times E$; - (iii)
*for each*$x\in E$*the mapping*$y\to F(\epsilon ,g(x),y)$*is*$C(x)$-*convex and*$F(\epsilon ,g(x),x)=0$,*for each*$(\u03f5,x)\in M({\u03f5}_{0})\times E$; - (iv)
*the mapping*$W:E\to {2}^{Y}$*defined by*$W(x)=Y\mathrm{\setminus}-intC(x)$*is closed*.

*Then*

- (i)
$S(\lambda ,\epsilon )$,

*for each*$(\lambda ,\epsilon )\in U({\lambda}_{0})\times M({\epsilon}_{0})$*is nonempty and compact*; - (ii)
*the mapping**S**is l*.*s*.*c*.*at*$({\lambda}_{0},{\epsilon}_{0})$.

## Declarations

## Authors’ Affiliations

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