On parametric implicit vector variational inequality problems

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.

1 Introduction and preliminaries

Equilibrium problems have been extensively studied in recent years, the origin ofwhich can be traced back to Takahashi [[1], Lemma 1], Blum and Oettli [2], and Noor and Oettli [3]. It is well known that vector equilibrium problems provide a unifiedmodel for several classes of problems, for example, vector variational inequalityproblems, vector complementarity problems, vector optimization problems, and vectorsaddle point problems; see [2–4] and the references therein. In 2003, Huang et al.[5] considered the implicit vector equilibrium problem (for short IVEP) whichconsists of finding $x\in E$ such that

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

where $f:E\times E\to Y$ and $g:E\to E$, are mappings, 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 $(\lambda ,ϵ)\in {\mathrm{\Lambda }}_1\times {\mathrm{\Lambda }}_2$ such that

$$f(ϵ,g\left(x^{\ast }\right),y)\notin -intC\left(x^{\ast }\right),\mathrm{\forall }y\in \mathbf{K}(\lambda ),$$

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 Xwith $\mathbf{K}({\mathrm{\Lambda }}_1)={\bigcup }_{\lambda \in {\mathrm{\Lambda }}_1}\mathbf{K}(\lambda )\subseteq E$ 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 $(\lambda ,ϵ)$ 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.

A subset 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

$$Graph(W)=\{(x,z)\in X\times Y:z\in W(x)\}.$$

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 containingx 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 containingx such that $T(u)\cap V\ne \mathrm{\varnothing }$. The mapping T is continuous at xif 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 ofX.

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 $y_i\to y$.

Definition 1.2 ([8, 9])

Let 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$A\subseteq K$,

$$coA\subseteq F(A)=\bigcup _{x\in A}F(x),$$

where coA 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

$$F(y)=\{x\in K:f(g(x),y)\notin -intC(x)\}.$$

We show that 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

$$f(g(z),z)\in -intC(z),$$

which is a contradiction (by (a)). Then 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

$$\overline{x}\in \bigcap _{x\in K}F(x)$$

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. □

We note that if 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 norf 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 $x\in K$,

$$tf(x,y_1)+(1-t)f(x,y_2)-f(x,t{y}_1+(1-t)y_2)\in C(x),\mathrm{\forall }t\in [0,1],$$

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])

A mapping $f:E\times E\to Y$ is said to be locally non-positive at$x_0\in E$ with respect to a mapping $g:E\to E$ if there exist a neighborhood $V(x_0)$ of $x_0$ and a point $z_0\in E\cap intV(x_0)$ such that

$$f(g(x),z_0)\in -C(x),\mathrm{\forall }x\in E\cap \partial V(x_0),$$

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 fis 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

$$f(g\left(x^{\ast }\right),y)\notin intC\left(x^{\ast }\right),\mathrm{\forall }y\in K.$$

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

$$f(g\left(x^{\ast }\right),y)\notin -intC\left(x^{\ast }\right),\mathrm{\forall }y\in B.$$

We claim that

$$f(g\left(x^{\ast }\right),y)\notin -intC\left(x^{\ast }\right),\mathrm{\forall }y\in K.$$

Indeed, if the sentence is not true then there is $y\in K$ so that

$$f(g\left(x^{\ast }\right),y)\in -intC\left(x^{\ast }\right).$$

Put $y_t=x^{\ast }+t(y-x^{\ast })$, for $t>0$. It is clear that $y_t\in B$, for t that is small enough. Then bycondition (c) we have

$$f(g\left(x^{\ast }\right),y_t)\in (1-t)f(g\left(x^{\ast }\right),y)+tf(g\left(x^{\ast }\right),x^{\ast })\in -intC\left(x^{\ast }\right)+0=-intC\left(x^{\ast }\right),$$

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 $(ϵ,x)\in {\mathrm{\Lambda }}_1\times E$;

• (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 ${\{({\lambda }_i,{ϵ}_i)\}}_{i\in I}\subseteq {\mathrm{\Lambda }}_1\times {\mathrm{\Lambda }}_2$ be a net with $({\lambda }_i,{ϵ}_i)\to (\lambda ,ϵ)$ and $z_i\in S({\lambda }_i,{ϵ}_i)\subset K({\lambda }_i)$. 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

$$F({ϵ}_{i_j},g(z_{i_j}),y)\in Y\mathrm{\setminus }(-intC(z_{i_j}))=W(z_{i_j}),\mathrm{\forall }y\in \mathbf{K}({\lambda }_{i_j}).$$

(1)

We claim that $z\in S(\lambda ,ϵ)$ (note that if we show the claim then according toLemma 1.1 the mapping S will be an u.s.c.). If the claim is not truethen there is $y\in \mathbf{K}(\lambda )$ such that

$$F(ϵ,g(z),y)\in -intC(z).$$

(2)

So, since 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

$$F({ϵ}_{i_j},g(z_{i_j}),w_j)\in W(z_{i_j}),$$

and so by (ii) and (iv) we get

$$F(ϵ,g(z),y)\in Y\mathrm{\setminus }-intC(z)=W(z),$$

which is contradicted by (2) and so $z\in S(\lambda ,ϵ)$. It follows from $x_{i_j}\in S({\lambda }_{i_j},{ϵ}_{i_j})$ that $f({ϵ}_{i_j},g(x_{i_j}),{ϵ}_{i_j})\notin -intC(x_{i_j})$ and so $f({ϵ}_{i_j},g(x_{i_j}),y_{i_j})\in W(x_{i_j})$ and by the closedness of W we get$f(ϵ,g(x),y)\in W(x)$, which is a contradiction, and then the solution setmapping S is u.s.c. Now we show that S is l.s.c. Let${\{({\lambda }_i,{ϵ}_i)\}}_{i\in I}\subseteq {\mathrm{\Lambda }}_1\times {\mathrm{\Lambda }}_2$ be a net with $({\lambda }_i,{ϵ}_i)\to (\lambda ,ϵ)$ and z an arbitrary element of$S(\lambda ,ϵ)\subseteq \mathbf{K}(\lambda )$. Put $\mathrm{\Xi }=\{V:V\text{ is a neighborhood of }z\}$ (note by the relation $V⪯W$ 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

$$F({ϵ}_i,g(z_i),y)\notin -intC(z_i),\mathrm{\forall }y\in H_{V,i}\cap \mathbf{K}({\lambda }_i).$$

(3)

Now if there exists $y\in \mathbf{K}({\lambda }_i)$ such that

$$F({ϵ}_i,g(z_i),y)\in -intC(z_i),$$

it follows from $F({ϵ}_i,g(z_i),z_i)=0$ (see condition (i)) and (iii) that

$$F({ϵ}_i,g(z_i),z_i+t(y-z_i))\in -intC(z_i),\mathrm{\forall }t\in [0,1],$$

which is a contradiction, for $t\in [0,1]$ small enough, by (3) (note $H_{V,i}$ is an open set and $z_i\in H_{V,i}$). So

$$F({ϵ}_i,g(z_i),y)\notin -intC(z_i),\mathrm{\forall }y\in \mathbf{K}({\lambda }_i),$$

and hence $z_i\in S({ϵ}_i,{\lambda }_i)$. Consequently, for each point $(V,i)\in \mathrm{\Xi }\times I$ there is $z_i\in S({ϵ}_i,{\lambda }_i)$, and so $z_i\to z$. Hence it follows from Lemma 1.1(ii) thatS 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$({\lambda }_0,{ϵ}_0)\in ({\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2)$there exist neighborhoods$U({\lambda }_0)$of${\lambda }_0$and$M({ϵ}_0)$of${ϵ}_0$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 $(\epsilon ,x,y)\in M({ϵ}_0)\times E\times E$;

• (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 $(ϵ,x)\in M({ϵ}_0)\times E$;

• (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 $({\lambda }_0,{\epsilon }_0)$.