Existence of solutions for generalized vector quasi-equilibrium problems in abstract convex spaces with applications

Abstract

In this paper, several kinds of generalized vector quasi-equilibrium problems are introduced and studied in abstract convex spaces. Using the properties of Γ-convex and $\mathfrak{KC}$-maps, some sufficient conditions are given to guarantee the existence of solutions in connection with these generalized vector quasi-equilibrium problems. As applications, some existence theorems of solutions for the generalized semi-infinite programs with vector quasi-equilibrium constraints are also given.

1 Introduction

It is well known that the vector quasi-equilibrium problem is an important generalization of the vector equilibrium problem which provides a unified model for vector quasi-variational inequalities, vector quasi-complementarity problems, vector optimization problems and vector saddle point problems. In 2000, Fu [1] established the existence theorems for the generalized vector quasi-equilibrium problems and the set-valued vector equilibrium problems. In 2003, Ansari and Fabián [2] considered a generalized vector quasi-equilibrium problem with or without involving Φ-condensing mappings and proved the existence of its solution in real topological vector spaces. In 2005, Li et al. [3] studied the existence of solutions for two classes of generalized vector quasi-equilibrium problems. Recently, Lin et al. [4] introduced and studied a class of generalized vector quasi-equilibrium problems involving pseudomonotonicity hemicontinuity mappings under different conditions in topological vector spaces. Lin et al. [5] proved the existence of equilibria for generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on a noncompact/nonparacompact strategy set. They also considered a systems of generalized vector quasi-equilibrium problems in topological vector spaces. Very recently, Yang and Pu [6] studied the existence and essential components in connection with the set of solutions for the system of strong vector quasi-equilibrium problems. Fu and Wang [7] considered the generalized strong vector quasi-equilibrium problems with domination structure. On the other hand, Ding [8] studied the existence of solutions for generalized vector quasi-equilibrium problems in locally G-convex spaces. Balaj and Lin [9] investigated existence of solutions for the generalized equilibrium problems in G-convex spaces.

The abstract convex space, introduced by Park [10] in 2006, includes the convex subset of a topological vector space, the convex space, the H-space, and the G-convex space as special cases. Moreover, Park [11] investigated the property of the abstract convex spaces and showed some applications. Recently, several authors have focused on the studies concerned with the set-valued maps and optimization problems in abstract convex spaces with applications. For instance, Cho et al. [12] studied some coincidence theorems and minimax inequalities in abstract convex spaces. Yang et al. [13] proved some maximal element theorems for set-valued maps in abstract convex spaces with applications. Yang and Huang [14] gave some coincidence theorems for compact and noncompact $\mathfrak{KC}$-maps in abstract convex spaces with applications. Lu and Hu [15] established a new collectively fixed point theorem in noncompact abstract convex spaces with applications to equilibria for generalized abstract economies. Park [16] gave some comments on fixed points, maximal elements, and equilibria of economies in abstract convex spaces. Yang and Huang [17] studied the existence of solutions for the generalized vector equilibrium problems in abstract convex spaces. At the end of the paper [17], Yang and Huang pointed out that it is an interesting and important work to study some types of generalized vector quasi-equilibrium problems with moving cones in topological spaces. To the best of our knowledge, it seems that there is no work concerned with the study of the generalized vector quasi-equilibrium problems in abstract convex spaces. Therefore, it is natural and interesting to study some generalized vector quasi-equilibrium problems in abstract convex spaces under some suitable conditions.

On the other hand, we know that semi-infinite programs are constrained optimization problems in which the number of decision variables is finite, but the number of constraints is infinite. Since John [18] initiated semi-infinite programming precisely to deduce important results about two such geometric problems: the problems of covering a compact body in finite dimensional spaces by the minimum-volume disk and the minimum-volume ellipsoid, many researchers have been investigated the theory, applications and methods for the semi-infinite programming (see, for example, [19–22]). As a generalization of semi-infinite programming, the generalized semi-infinite programming has been become a vivid field of active research in mathematical programming in recent years due to its important applications to numerous real-life problems such as Chebyshev approximation, design centering, robust optimization, optimal layout of an assembly line, time minimal control, and disjunctive optimization (see [23] and the references therein). Therefore, it is important and interesting to study the existence of solutions concerned with some generalized semi-infinite programs with vector quasi-equilibrium constraints in abstract convex spaces.

The main purpose of this paper is to study several classes of generalized vector quasi-equilibrium problems in abstract convex spaces with applications to generalized semi-infinite programs. We give some sufficient conditions to guarantee the existence of solutions for these generalized vector quasi-equilibrium problems in abstract convex spaces. As applications, we give some existence theorems of solutions for the generalized semi-infinite programs under suitable conditions.

2 Preliminaries

Let X, Y be two nonempty sets. A set-valued mapping $T:X\rightrightarrows Y$ is a mapping from X into the power set $2^{Y}$. The inverse $T^{-1}$ of T is the set-valued mapping from Y to X defined by

$$T^{-1}(y)= \bigl\{ x\in X: y\in T(x) \bigr\} . $$

An abstract convex space $(X,D,\Gamma)$ consists of a nonempty set X, a nonempty set D, and a set-valued mapping $\Gamma:\langle D\rangle\rightrightarrows X$ with nonempty values, where $\langle D\rangle$ denotes the set of all nonempty finite subset of a set D. If for each $A\in\langle D\rangle$ with the cardinality $|A|=n+1$, there exists a continuous function $\phi_{A}:\triangle _{n}\rightarrow\Gamma(A)$ such that $J\in\langle A\rangle$ implies $\phi_{A}(\triangle_{J})\subseteq\Gamma(J)$, where $\triangle_{n}$ is the standard n-simplex and $\triangle_{J}$ the face of $\triangle _{n}$ corresponding to $J\in\langle A\rangle$, then the abstract convex space reduces to the G-convex space. Let $\Gamma_{A}:=\Gamma (A)$ for $A\in \langle D\rangle$. When $D\subset X$, the space is defined by $(X\supseteq D,\Gamma)$. In this case, a subset M of X is said to be Γ-convex if, for any $A\in\langle M\cap D \rangle$, we have $\Gamma_{A}\subseteq M$. In the case $X=D$, let $(X,\Gamma):=(X,X,\Gamma)$.

It is easy to see that any vector space Y is an abstract convex space with $\Gamma:=\operatorname{co}$, where co denotes the convex hull in the vector space Y. Next we give more examples as follows.

Example 2.1

([10])

Let E be a topological vector space with a neighborhood system $\mathcal{V}$ of its origin. A subset X of E is said to be almost convex (see [24] for more details) if for any $V\in\mathcal{V}$ and for any finite subset $A=\{x_{1}, x_{2}, \ldots, x_{n}\}$ of X, there exists a subset $B=\{ y_{1}, y_{2}, \ldots,y_{n}\}$ of X such that $y_{i} - x_{i} \in V$ for all $i = 1, 2, \ldots, n$ and $\operatorname{co} B \subset X$. Let $\Gamma_{A} = \operatorname{co} B$ for any $A \in\langle X \rangle$. Then $(X, \Gamma)$ is a G-convex space and hence an abstract convex space.

Example 2.2

([10])

Usually, a convex space $(E, \mathcal{C})$ in the classical sense consists of a nonempty set E and a family $\mathcal{C}$ of subsets of E such that E itself is an element of $\mathcal{C}$ and $\mathcal{C}$ is closed under arbitrary intersection. For any given subset $X\subset E$, the $\mathcal{C}$-convex hull of X is defined as by

$$\operatorname{Co}_{\mathcal{C}}X =\bigcap \{Y \in\mathcal{C}: X\subset Y \}. $$

We say that X is $\mathcal{C}$-convex if $X=\operatorname{Co}_{\mathcal{C}}X$. Consider the mapping $\Gamma: \langle E\rangle\rightrightarrows E$ defined by $\Gamma_{A}=\operatorname{Co}_{\mathcal{C}}A$. Then $(E, \Gamma)$ is an abstract convex space.

Example 2.3

Let $(M,d)$ be a pseudo-metric space, that is, $d:M\times M\to [0,+\infty)$ such that, for every $x,y,z \in M$,

• (i) $d(x,x) = 0$;

• (ii) $d(x,y) = d(y,x)$;

• (iii) $d(x,z) \leq d(x,y) + d(y,z)$.

For any $A\in\langle M\rangle$, define a set-valued mapping $\Gamma: \langle M\rangle\rightrightarrows M$ by

$$\Gamma_{A}=\Gamma(A)=\bigcap\{B: B \mbox{ is a closed ball containing } A\}. $$

Then it is easy to see that $(M,\Gamma)$ is an abstract convex space.

As pointed out by Park [25], the abstract convex space includes many generalized convex spaces as special cases such as L-spaces, spaces having property (H), pseudo H-spaces, M-spaces, G-H-spaces, another L-spaces, FC-spaces and others. Some more examples of the abstract convex space and comments on it can be found in the literature [10, 25, 26] and the references therein.

Let $(X,\Gamma)$ be an abstract convex space and V be a real topological vector space. Let E be a nonempty subset of X. Assume that $S:E\rightrightarrows E$ and $B:E\rightrightarrows E$ are two set-valued mappings. Suppose that $F:X\times X\times X\rightrightarrows V$ and $C:X \rightrightarrows V$ are two set-valued mappings such that for each $x\in X$, $C(x)$ is a closed convex cone with $\operatorname{int} C(x)\neq\emptyset$, here $\operatorname{int} C(x)$ denotes the interior of $C(x)$. In this paper, we will consider the following generalized vector quasi-equilibrium problems in abstract convex spaces.

• (GVQEP1) Find $\tilde{x}\in E$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde {x},y,z)\subseteq C(\tilde{x}),\quad \forall y\in S(\tilde{x}), \forall z\in B(\tilde{x}). $$

We would like to mention that (GVQEP1) was considered by Lin et al. [4] in topological vector spaces. When $S(x)=B(x)=E$ for all $x\in E$, (GVQEP1) was considered by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP2) Find $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde{x},y,\tilde {z})\subseteq C( \tilde{x}),\quad \forall y\in S(\tilde{x}). $$

When $C(x)$ was replaced by $-C(x)$, (GVQEP2) was considered by Li and Li [27] in topological vector spaces. If $S(x)=E$ for all $x\in E$, then (GVQEP2) was investigated by Fu and Wang [7] in topological vector spaces.

• (GVQEP3) Find $\tilde{x}\in E$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde{x},y,z)\cap {-}\operatorname{int} C(\tilde{x})=\emptyset, \quad\forall y\in S(\tilde{x}), \forall z\in B(\tilde{x}). $$

We note that (GVQEP3) was considered by Lin et al. [4] in topological vector spaces. When $S(x)=B(x)=E$ for all $x\in E$, (GVQEP3) was studied by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP4) Find $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde{x},y,\tilde {z})\cap{-} \operatorname{int} C(\tilde{x})=\emptyset,\quad \forall y\in S(\tilde{x}). $$

We note that (GVQEP4) was considered by Li and Li [27] in topological vector spaces.

• (GVQEP5) Find $\tilde{x}\in E$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde {x},y,z)\nsubseteq{-} \operatorname{int} C(\tilde{x}),\quad \forall y\in S(\tilde {x}), \forall z\in B( \tilde{x}). $$

When $S(x)=B(x)=E$ for all $x\in E$, (GVQEP5) was investigated by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP6) Find $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde{x},y,\tilde {z})\nsubseteq{-} \operatorname{int} C(\tilde{x}),\quad \forall y\in S(\tilde{x}). $$

It is worth mentioning that (GVQEP6) was considered by Lin et al. [4], and Li and Li [27] in topological vector spaces, respectively. Moreover, some special cases of (GVQEP6) were considered by Ansari and Fabián [2] in topological vector spaces.

• (GVQEP7) Find $\tilde{x}\in E$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde{x},y,z)\cap C(\tilde{x})\neq \emptyset,\quad \forall y\in S(\tilde{x}), \forall z\in B(\tilde{x}). $$

When $S(x)=B(x)=E$ for all $x\in E$, (GVQEP7) was studied by Yang and Huang [17] in abstract convex spaces and by Balaj and Lin [9] in G-convex spaces, respectively.

• (GVQEP8) Find $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad F(\tilde{x},y,\tilde {z})\cap C(\tilde{x}) \neq\emptyset,\quad \forall y\in S(\tilde{x}). $$

When $S(x)=B(x)=E$ for all $x\in E$, (GVQEP8) was considered by Lin [28] in topological vector spaces.

We would like to point out that, for a suitable choice of the spaces E, X, V and the mappings S, B, F, C, one can obtain a number of well-known insights into the generalized vector quasi-equilibrium problem [2, 4, 5, 7, 8, 27], the generalized vector equilibrium problem [9, 17, 28], the vector equilibrium problem, and the vector variational inequality problem [29, 30] as special cases of the problems (GVQEP1)-(GVQEP8).

Furthermore, assume that $h: X\rightrightarrows L$ is a set-valued mapping, where L is a real topological vector space ordered by a closed convex pointed cone $H\subseteq L$ with $\operatorname{int} H\neq\emptyset$. It is clear that the existence of solutions for problems (GVQEP1)-(GVQEP8) is closely analogous to the existence of solutions in connection with the following generalized semi-infinite programs with generalized vector quasi-equilibrium constraints:

• (GSIP1) Generalized semi-infinite program with constraint (GVQEP1):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\cap{-}\operatorname{int} C(x)=\emptyset, \forall y\in S(x), \forall z\in B(x) \bigr\} . $$

When $S(x)=B(x)=E$ for all $x\in E$, (GSIP1) was considered by Yang and Huang [17] in abstract convex spaces.

• (GSIP2) Generalized semi-infinite program with constraint (GVQEP2):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\subseteq C(x), \forall y \in S(x) \bigr\} . $$

Some special cases of (GSIP2) were considered by Lin [28] in topological vector spaces.

• (GSIP3) Generalized semi-infinite program with constraint (GVQEP3):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\cap{-}\operatorname{int} C(x)=\emptyset, \forall y\in S(x), \forall z\in B(x) \bigr\} . $$

When $S(x)=B(x)=E$ for all $x\in E$, (GSIP3) was studied by Yang and Huang [17] in abstract convex spaces.

• (GSIP4) Generalized semi-infinite program with constraint (GVQEP4):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\cap{-}\operatorname{int} C(x)=\emptyset, \forall y\in S(x) \bigr\} . $$

We would like to mention that some special cases of (GSIP4) were studied by Lin [28] in topological vector spaces.

• (GSIP5) Generalized semi-infinite program with constraint (GVQEP5):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\nsubseteq{-}\operatorname{int} C(x), \forall y\in S(x), \forall z\in B(x) \bigr\} . $$

When $S(x)=B(x)=E$ for all $x\in E$, (GSIP5) was investigated by Yang and Huang [17] in abstract convex spaces.

• (GSIP6) Generalized semi-infinite program with constraint (GVQEP6):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\nsubseteq{-} \operatorname{int} C(x), \forall y\in S(x) \bigr\} . $$

We note that some special cases of (GSIP6) were considered by Lin [28] in topological vector spaces.

• (GSIP7) Generalized semi-infinite program with constraint (GVQEP7):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\cap C(x)\neq\emptyset, \forall y\in S(x), \forall z\in B(x) \bigr\} . $$

When $S(x)=B(x)=E$ for all $x\in E$, (GSIP7) was studied by Yang and Huang [17] in abstract convex spaces.

• (GSIP8) Generalized semi-infinite program with constraint (GVQEP8):

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\cap C(x)\neq \emptyset, \forall y\in S(x) \bigr\} . $$

It is worth mentioning that (GSIP8) can be considered as a generalization of the generalized vector semi-infinite programming introduced and studied by Lin [28] in topological vector spaces.

In brief, for suitable choice of the spaces L, V, X, E and the mappings S, B, F, C, h, one can obtain a number of known the generalized semi-infinite program [17], the mathematical program with equilibrium constraint [19], the generalized semi-infinite program [23], the generalized vector semi-infinite programming [28], and the vector optimization problem [30–32] as special cases from the problems (GSIP1)-(GSIP8).

Now, we recall some useful definitions and lemmas as follows.

Definition 2.1

Let $K\subseteq V$ be a nonempty set and $C\subseteq V$ be the closed convex pointed cone with $\operatorname{int} C\neq\emptyset$. The set of all weak minimal points of K with respect to the ordering cone C is defined as

$$\operatorname{wMin}_{C}(K)= \bigl\{ x\in K: (x-K)\cap \operatorname{int} C=\emptyset \bigr\} . $$

Definition 2.2

Let $(X,D,\Gamma)$ be an abstract convex space and Z be a set. For a set-valued mapping $T:X\rightrightarrows Z$ with nonempty values, if a set-valued mapping $G:D\rightrightarrows Z$ satisfies

$$T(\Gamma_{N})\subseteq G(N):=\bigcup_{y\in N}G(y) \quad\mbox{for all } N\in\langle D\rangle, $$

then G is called a KKM mapping with respect to T. A KKM mapping $G:D\rightrightarrows X$ is a KKM mapping with respect to the identity mapping $I_{X}$.

A set-valued mapping $F:X\rightrightarrows Z$ is called to be a $\mathfrak{KC}$-map if, for any closed-valued KKM mapping $G:D\rightrightarrows Z$ with respect to F, the family $\{G(y)\} _{y\in D}$ has the finite intersection property. We denote

$$\mathfrak{KC}(X,Z):=\{F: F \mbox{ is }\mathfrak{KC}\mbox{-map}\}. $$

Definition 2.3

([33])

Let X and Y be two topological spaces. A set-valued mapping $F:X\rightrightarrows Y$ is said to be

• (i) upper semicontinuous (u.s.c.) at $x_{0}$ if for any open set $V \supseteq F(x_{0})$, there is an open neighborhood $O_{x_{0}}$ of $x_{0}$ such that $F(x')\subseteq V$ for each $x'\in O_{x_{0}}$,

• (ii) lower semicontinuous (l.s.c.) at $x_{0}$ if for any open set $V\cap F(x_{0})\neq\emptyset$, there is an open neighborhood $O_{x_{0}}$ of $x_{0}$ such that $F(x')\cap V\neq\emptyset$ for each $x'\in O_{x_{0}}$,

• (iii) continuous at $x_{0}$ if it is both upper and lower semicontinuous at $x_{0}$,

• (iv) upper semicontinuous (lower semicontinuous or continuous) on X if it is upper semicontinuous (lower semicontinuous or continuous) at every $x\in X$,

• (v) closed if and only if its graph $\operatorname{Graph}(F):=\{(x,y)\in X\times Y:y\in F(x)\}$ is closed.

Lemma 2.1

([34])

Let X and Y be two topological spaces and $F:X\rightrightarrows Y$ a set-valued mapping.

• (i) If Y is compact, then F is closed if and only if it is upper semicontinuous,

• (ii) if X is a compact space and F is a u.s.c. mapping with compact values, then $F(X)$ is a compact subset of Y.

Lemma 2.2

([35])

Let X and Y be two topological spaces and $F:X\rightrightarrows Y$ be upper semicontinuous and $F(x)$ is compact. Then for any net $\{ x_{\alpha}\}\subset X$ with $x_{\alpha}\rightarrow x$ and $y_{\alpha}\in F(x_{\alpha})$, there exists a subnet $\{y_{\beta}\}\subset y_{\alpha}$ such that $y_{\beta}\rightarrow y\in F(x)$.

Lemma 2.3

([36])

Let X and Y be two topological spaces and $F:X\rightrightarrows Y$ be lower semicontinuous at $x\in X$ if and only if for any $y\in F(x)$ and any net $\{x_{\alpha}\}$ with $x_{\alpha}\rightarrow x$, there is a net $\{y_{\alpha}\}$ such that $y_{\alpha}\in F(x_{\alpha})$ and $y_{\alpha}\rightarrow y$.

Lemma 2.4

([10])

Let $(X,D,\Gamma)$ be an abstract convex space, Z a set, and $T:X\rightrightarrows Z$ a set-valued mapping. Then $F\in\mathfrak{KC}(X,Z)$ if and only for any $G:D\rightrightarrows Z$ satisfying

• (i) G is closed-values;

• (ii) $F(\Gamma_{N})\subseteq G(N)$ for any $N\in\langle D\rangle$,

we have

$$F(E)\bigcap \bigl\{ G(y):y\in N \bigr\} \neq\emptyset $$

for each $N\in\langle D\rangle$.

Lemma 2.5

([32])

Assume that A is a nonempty compact subset of a real topological vector space V and D is a closed convex cone in V with $D\neq V$. Then one has $\operatorname{wMin}_{D} A\neq\emptyset$.

An abstract convex space with any topology is called an abstract convex topological space. In the rest of this paper, let $(X,\Gamma)$ be an abstract convex Hausdorff topological space and E be a nonempty compact subset of X. Let V be a topological vector spaces. Assume that $T:X\rightrightarrows X$, $B:E\rightrightarrows E$, $S:E\rightrightarrows E$, $F:E\times E\times E\rightrightarrows V$ and $Q:E\rightrightarrows V$ are five set-valued mappings. Let ρ be a binary relation on $2^{V}$ and $\rho^{c}$ be the complementary relation of ρ. Let α be any of the quantifiers ∀, ∃, and $\bar {\alpha}$ be the other of the quantifiers ∀, ∃.

3 Main results

In order to show the existence of solutions for the vector quasi-equilibrium problems (GVQEP1)-(GVQEP8), we first give the following general result.

Theorem 3.1

Suppose that the following conditions are satisfied:

• (i) $T\in\mathfrak{K}\mathfrak{C}(X,X)$;

• (ii) for each $y\in E$, the set $\{x\in E:(\bar{\alpha})z\in B(x), \rho^{c}(F(x,y,z), Q(x))\}$ is open in E;

• (iii) $G_{0}=\{x\in E:x\notin S(x)\}$ is open in E;

• (iv) for each $x\in E$, $S(x)$ is nonempty Γ-convex, $S^{-1}(y)$ is open for all $y\in E$;

• (v) for each $(x_{0}, y_{0})\in E\times E$ with $x_{0}\in T(y_{0})$ such that $y_{0}\notin S(x_{0})$.

Then there exists $\tilde{x}\in S(\tilde{x})$ such that $(\alpha) z\in B(\tilde{x})$, $\rho(F(\tilde{x},y,z), Q(\tilde{x}))$ for any $y\in S(\tilde{x})$.

Proof

For any $x \in E$, define $A:E\rightrightarrows E$ by

$$A(x)= \bigl\{ y\in E:(\bar{\alpha})z\in B(x), \rho^{c} \bigl(F(x,y,z), Q(x) \bigr) \bigr\} . $$

From the definition of $A(x)$, one has

$$A^{-1}(y)= \bigl\{ x\in E:(\bar{\alpha})z\in B(x), \rho^{c} \bigl(F(x,y,z), Q(x) \bigr) \bigr\} . $$

Define $P:E\rightrightarrows E$ by

$$\begin{aligned} P(x)= \left \{ \begin{array}{@{}l@{\quad}l} S(x)\cap A(x), & x\in E\setminus G_{0};\\ S(x),& x\in G_{0}. \end{array} \right . \end{aligned}$$

(1)

Let $M(y)=E\setminus P^{-1}(y)$. We show that $M(y)$ is closed for all $y\in E$. In fact, it follows from (1) that

$$\begin{aligned} P^{-1}(y) =& \bigl\{ x\in E\setminus G_{0}:y\in S(x)\cap A(x) \bigr\} \cup \bigl\{ x\in G_{0}: y\in S(x) \bigr\} \\ =& \bigl\{ x\in E\setminus G_{0}:x\in S^{-1}(y)\cap A^{-1}(y) \bigr\} \cup \bigl\{ x\in G_{0}: x\in S^{-1}(y) \bigr\} \\ =& \bigl\{ (E\setminus G_{0})\cap S^{-1}(y)\cap A^{-1}(y) \bigr\} \cup \bigl\{ G_{0}\cap S^{-1}(y) \bigr\} \\ =&S^{-1}(y)\cap \bigl(G_{0}\cup A^{-1}(y) \bigr). \end{aligned}$$

Since $S^{-1}(y)$, $A^{-1}(y)$, and $G_{0}$ are open, we know that $P^{-1}(y)$ is open and so $M(y)$ is closed.

We show that M is a KKM mapping with respect to T. Suppose that M is not a KKM mapping with respect to T. Then there exist a finite subset N and a point $x_{0}\in E$ such that $x_{0}\in T(\Gamma _{N})\setminus M(N)$. This shows that there exists a point $y_{0}\in \Gamma_{N}$ such that $x_{0}\in T(y_{0})$, $x_{0}\in P^{-1}(y)$ for any $y\in N$, and so $N\subset P(x_{0})\subset S(x_{0})$. Since $S(x_{0})$ is Γ-convex and $N\in\langle S(x_{0})\rangle$, we know that $y_{0}\in\Gamma_{N}\subset S(x_{0})$, which is a contradiction. It follows that M is a KKM mapping with respect to T.

It follows from Lemma 2.4 that M has finite intersection property. From the facts that $M(y)\subset E$ is closed and E is compact, we know that $M(y)$ is compact for any $y\in E$ and so

$$\bigcap_{y\in E} M(y)\neq\emptyset. $$

Thus, there exists a point $\tilde{x}\in E$ such that

$$\tilde{x}\in\bigcap_{y\in E} M(y)=E\setminus\bigcup _{y\in E}P^{-1}(y). $$

This implies that $\tilde{x}\notin P^{-1}(y)$ for all $y\in E$ and so $P(\tilde{x})=\emptyset$.

If $\tilde{x}\in G_{0}$, then it is easy to see that $S(\tilde {x})=P(\tilde{x})=\emptyset$, which is a contradiction. Therefore, we have

$$\tilde{x}\in E\setminus G_{0} \quad\mbox{with } S(\tilde{x})\cap A( \tilde{x})=P (\tilde{x})=\emptyset $$

and so

$$\tilde{x}\in S(\tilde{x}) \quad\mbox{and}\quad y\notin A(\tilde{x}), \forall y\in S( \tilde{x}), $$

that is, $\tilde{x}\in S(\tilde{x})$, $(\alpha)z\in B(\tilde{x})$, $\rho (F(\tilde{x},y,z), C(\tilde{x}))$ for all $y\in S(\tilde{x})$. This completes the proof. □

Remark 3.1

By Lemma 2.1, it is easy to see that the condition (iii) can be replaced by the following condition:

(iii)′:

$S:E\rightrightarrows E$ is a u.s.c. set-valued mapping.

Next we give some existence theorems in connection with the solution of the vector quasi-equilibrium problems (GVQEP1)-(GVQEP8).

Theorem 3.2

Assume that the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is l.s.c. and C is closed;

• (b) B is l.s.c.

Then there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde {x})$ and $F(\tilde{x},y,z)\subset C(\tilde{x})$ for all $y\in S(\tilde{x})$ and $z\in B(\tilde{x})$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\exists z\in B(x), F(x,y,z)\nsubseteq C(x) \bigr\} . $$

We show that

$$A^{-1}(y)= \bigl\{ x\in E:\exists z\in B(x), F(x,y,z)\nsubseteq C(x) \bigr\} $$

is open. Let $\{x_{\alpha}\}\subseteq E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then

$$F \bigl(x_{\alpha},y,z' \bigr)\subseteq C(x_{\alpha}),\quad \forall z'\in B(x_{\alpha}). $$

Since B and $F(\cdot,y,\cdot)$ are l.s.c., by Lemma 2.3, for any $z\in B(x_{0})$ and $v\in F(x_{0},y,z)$, there exist $z_{\alpha}\in B(x_{\alpha})$ and $v_{\alpha}\in F(x_{\alpha},y,z_{\alpha})$ such that $z_{\alpha}\rightarrow z$ and $v_{\alpha}\rightarrow v$. Now the closedness of C with $v_{\alpha}\in C(x_{\alpha})$ shows that $v\in C(x)$ and so $F(x_{0},y,z)\subseteq C(x)$ for any $z\in B(x_{0})$. This shows that $x_{0}\in E\setminus A^{-1}(y)$ and so $E\setminus A^{-1}(y)$ is closed. Thus, $A^{-1}(y)$ is open. It follows from Theorem 3.1 that there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde{x},y,z)\subseteq C(\tilde{x})$ for all $y\in S(\tilde{x})$ and $z\in B(\tilde{x})$. This completes the proof. □

Remark 3.2

Theorem 3.2 can be considered as a generalization of Theorem 3.3 in [4] under different conditions from the topological vector space to the abstract convex space.

Corollary 3.1

Assume that the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied with $B=S$. Suppose that, for each $y\in E$, $F(\cdot,y)$ is l.s.c. and C is closed. Then there exists $\tilde {x}\in E$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde {x},y)\subseteq C(\tilde{x})$ for all $y\in S(\tilde{x})$.

Proof

The proof is similar to that of Theorem 3.2 and so we omit it here. □

Remark 3.3

When $S(x)=E$ for all $x\in E$, Corollary 3.1 was given by Theorem 1 of Yang and Huang [17] under quite different conditions.

Theorem 3.3

Assume that the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is l.s.c. and C is closed;

• (b) B is u.s.c. and $B(x)$ is compact for each $x\in E$.

Then there exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde{x},y,\tilde{z})\subseteq C(\tilde{x})$ for all $y\in S(\tilde{x})$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\forall z\in B(x), F(x,y,z)\nsubseteq C(x) \bigr\} . $$

We first show that

$$A^{-1}(y)= \bigl\{ x\in E:\forall z\in B(x), F(x,y,z)\nsubseteq C(x) \bigr\} $$

is open. Let $\{x_{\alpha}\}\subseteq E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then there exists $z_{\alpha}\in B(x_{\alpha})$ such that $F(x_{\alpha},y,z_{\alpha })\subseteq C(x_{\alpha})$. Since B is u.s.c. with compact values, by Lemma 2.2, there exists a subset net of $\{z_{\alpha}\}$, denoted again by $\{z_{\alpha}\}$, such that $z_{\alpha}\rightarrow z_{0}\in B(x_{0})$. The fact that $F(\cdot,y,\cdot)$ is l.s.c. together with Lemma 2.3 shows that, for any $v\in F(x_{0},y,z_{0})$, there exists $v_{\alpha}\in F(x_{\alpha},y,z_{\alpha})$ such that $v_{\alpha}\rightarrow v$. Since $v_{\alpha}\in C(x_{\alpha})$ and C is closed, we know that $v\in C(x_{0})$ and so $F(x_{0},y,z_{0})\subseteq C(x_{0})$ for some $z_{0}\in B(x_{0})$. This implies that $x_{0}\in E\setminus A^{-1}(y)$ and so $E\setminus A^{-1}(y)$ is closed. Thus, $A^{-1}(y)$ is open. It follows from Theorem 3.1 that there exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde {x})$ and $F(\tilde{x},y,\tilde{z})\subseteq C(\tilde{x})$ for any $y\in S(\tilde{x})$. This completes the proof. □

Remark 3.4

When $S(x)=E$ for all $x\in E$, the existence of the solutions for generalized vector quasi-equilibrium was studied in Theorem 3.1 of [7] in real Hausdorff topological vector spaces.

Theorem 3.4

Assume that the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is l.s.c., $C(x)$ is a set-valued mapping with a nonempty interior for each $x\in E$, the mapping $W: E\rightrightarrows V$, defined by $W(x)=V\setminus ({-}\operatorname{int} C(x))$, is closed;

• (b) B is l.s.c.

Then there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde {x})$ and $F(\tilde{x},y,z)\cap({-}\operatorname{int} C(\tilde{x}))=\emptyset$ for all $y\in S(\tilde{x})$ and $z\in B(\tilde{x})$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\exists z\in B(x), F(x,y,z)\cap \bigl({-}\operatorname{int} C(x) \bigr)\neq \emptyset \bigr\} . $$

We prove that

$$A^{-1}(y)= \bigl\{ x\in E:\exists z\in B(x), F(x,y,z)\cap \bigl({-} \operatorname{int} C(x) \bigr)\neq \emptyset \bigr\} $$

is open. Let $\{x_{\alpha}\}\subseteq E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then

$$F \bigl(x_{\alpha},y,z' \bigr)\cap \bigl({-} \operatorname{int} C(x_{\alpha}) \bigr)=\emptyset,\quad \forall z' \in B(x_{\alpha}) $$

and so

$$F \bigl(x_{\alpha},y,z' \bigr)\subseteq W(x_{\alpha})= V \setminus \bigl({-}\operatorname{int} C(x_{\alpha}) \bigr). $$

Similar to the proof of Theorem 3.2, we get

$$F(x_{0},y,z)\subseteq W(x_{0})= V\setminus \bigl({-} \operatorname{int} C(x_{0}) \bigr). $$

This shows that $x_{0}\in E\setminus A^{-1}(y)$ and so $E\setminus A^{-1}(y)$ is closed. Thus, $A^{-1}(y)$ is open. It follows from Theorem 3.1 that there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde{x})$ and

$$F(\tilde{x},y,z)\cap \bigl({-}\operatorname{int} C(\tilde{x}) \bigr)=\emptyset,\quad \forall y\in S(\tilde{x}), \forall z\in B(\tilde{x}). $$

This completes the proof. □

Remark 3.5

Theorem 3.4 can be considered as a generalization of Theorem 3.2 in [4] under different conditions from the topological vector space to the abstract convex space.

Corollary 3.2

Assume that the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied with $S=B$. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y)$ is l.s.c., $C(x)$ has a nonempty interior for each $x\in E$, the map $W: E\rightrightarrows V$, defined by $W(x)=V\setminus({-}\operatorname{int} C(x))$, is closed.

Then there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde {x})$ and $F(\tilde{x},y)\cap({-}\operatorname{int} C(\tilde{x}))=\emptyset$ for all $y\in S(\tilde{x})$.

Proof

The proof is similar to that of Theorem 3.4 and so we omit it here. □

Remark 3.6

When $S(x)=E$ for all $x\in E$, Corollary 3.2 was given by Theorem 2 of Yang and Huang [17] under quite different conditions.

Theorem 3.5

Suppose the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is l.s.c., $C(x)$ has a nonempty interior for each $x\in E$, and the mapping $W: E\rightrightarrows V$, defined by $W(x)=V\setminus({-}\operatorname{int} C(x))$, is closed;

• (b) B is u.s.c. and $B(x)$ is compact for each $x\in E$.

Then there exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde{x},y,\tilde{z})\cap ({-}\operatorname{int} C(\tilde{x}))=\emptyset$ for all $y\in S(\tilde{x})$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\forall z\in B(x), F(x,y,z)\cap \bigl({-}\operatorname{int} C(x) \bigr)\neq \emptyset \bigr\} . $$

We show that

$$A^{-1}(y)= \bigl\{ x\in E:\forall z\in B(x), F(x,y,z)\cap \bigl({-} \operatorname{int} C(x) \bigr)\neq \emptyset \bigr\} $$

is open. Let $\{x_{\alpha}\}\subseteq E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then

$$F(x_{\alpha},y,z_{\alpha})\cap \bigl({-}\operatorname{int} C(x_{\alpha}) \bigr)=\emptyset $$

for some $z_{\alpha}\in B(x_{\alpha})$, that is,

$$F(x_{\alpha},y,z_{\alpha})\subseteq V\setminus \bigl({-} \operatorname{int} C(x_{\alpha}) \bigr). $$

Using similar arguments to the proof of Theorem 3.3, we have

$$F(x_{0},y,z_{0})\subseteq W(x_{0})= V\setminus \bigl({-}\operatorname{int} C(x_{0}) \bigr) $$

for some $z_{0}\in B(x_{0})$. This shows that $x_{0}\in E\setminus A^{-1}(y)$ and so $E\setminus A^{-1}(y)$ is closed. Thus, $A^{-1}(y)$ is open. It follows from Theorem 3.1 that there exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde {x})$ and

$$F(\tilde{x},y,\tilde{z})\cap \bigl({-}\operatorname{int} C(\tilde{x}) \bigr)= \emptyset,\quad \forall y\in S(\tilde{x}). $$

This completes the proof. □

Remark 3.7

When E is a nonempty convex compact of a topological vector space, Li and Li [27] studied the existence of solutions for (GVQEP4).

Theorem 3.6

Assume that the conditions (i), (iii), (iv), and (v) are satisfied in Theorem  3.1. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is u.s.c. with compact valued on $E\times E\times E$ and $C(x)$ has a nonempty interior for each $x\in E$, the mapping $W: E\rightrightarrows V$, defined by $W(x)=V\setminus({-}\operatorname{int} C(x))$, is closed;

• (b) B is l.s.c.

Then there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde {x})$ and $F(\tilde{x},y,z)\nsubseteq{-}\operatorname{int} C(\tilde{x})$ for all $y\in S(\tilde{x})$ and $z\in B(\tilde{x})$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\exists z\in B(x), F(x,y,z)\subseteq{-}\operatorname{int} C(x) \bigr\} . $$

We prove that

$$A^{-1}(y)= \bigl\{ x\in E:\exists z\in B(x), F(x,y,z)\subseteq{-} \operatorname{int} C(x) \bigr\} $$

is open. Let $x_{\alpha}\in E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then

$$F \bigl(x_{\alpha},y,z' \bigr)\nsubseteq{-}\operatorname{int} C(x_{\alpha}) $$

for any $z'\in B(x_{\alpha})$ and so there exists $v_{\alpha}\in V$ such that

$$v_{\alpha}\in F \bigl(x_{\alpha},y,z' \bigr)\setminus \bigl({-}\operatorname{int} C(x_{\alpha}) \bigr). $$

Since B is l.s.c., by Lemma 2.3, for any $z\in B(x_{0})$, there exists $z_{\alpha}\in B(x_{\alpha})$ such that $z_{\alpha }\rightarrow z$. Since $F(\cdot,y,\cdot)$ is u.s.c. with compact valued, by Lemma 2.2, there exists a subset net of $\{v_{\alpha }\}$, denoted again by $\{v_{\alpha}\}$, such that $v_{\alpha }\rightarrow v_{0}\in F(x_{0},y,z)$. On the other hand, the fact that $v_{\alpha}\notin{-}\operatorname{int} C(x_{\alpha})$ shows that $v_{\alpha}\in W(x_{\alpha})$. Now the closedness of W shows that $v_{0}\in W(x_{0})$ and so $v_{0}\notin{-}\operatorname{int} C(x_{0})$. Thus $F(x_{0},y,z)\nsubseteq{-}\operatorname{int} C(x_{0})$ for any $z\in B(x_{0})$. This implies that $x_{0}\in E\setminus A^{-1}(y)$ and so $E\setminus A^{-1}(y)$ is closed. Thus, $A^{-1}(y)$ is open. It follows from Theorem 3.1 that there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde {x},y,z)\nsubseteq{-}\operatorname{int} C(\tilde{x})$ for all $y\in S(\tilde{x})$ and $z\in B(\tilde{x})$. This completes the proof. □

Corollary 3.3

Assume that the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied with $S=B$. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y)$ is u.s.c. with compact valued on $E\times E$ and $C(x)$ has a nonempty interior for each $x\in E$, the mapping $W: E\rightrightarrows V$, defined by $W(x)=V\setminus ({-}\operatorname{int} C(x))$, is closed.

Then there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde {x})$ and $F(\tilde{x},y)\nsubseteq{-}\operatorname{int} C(\tilde{x})$ for all $y\in S(\tilde{x})$.

Proof

The proof is similar to that of Theorem 3.6 and so we omit it here. □

Remark 3.8

When $S(x)=E$ for all $x\in E$, Corollary 3.3 was given by Theorem 4 of Yang and Huang [17] under quite different conditions.

Theorem 3.7

Assume that the conditions (i), (iii), (iv), and (v) are satisfied in Theorem  3.1. Moreover, suppose that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is u.s.c. with compact valued on $E\times E\times E$ and $C(x)$ has a nonempty interior for each $x\in E$, the mapping $W: E\rightrightarrows V$, defined by $W(x)=V\setminus({-}\operatorname{int} C(x))$, is closed.

• (b) B is u.s.c. and $B(x)$ is compact for each $x\in E$.

Then there exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde{x},y,\tilde {z})\nsubseteq{-}\operatorname{int} C(\tilde{x})$ for all $y\in S(\tilde{x})$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\forall z\in B(x), F(x,y,z)\subseteq{-}\operatorname{int} C(x) \bigr\} . $$

We prove that

$$A^{-1}(y)= \bigl\{ x\in E:\forall z\in B(x), F(x,y,z)\subseteq{-} \operatorname{int} C(x) \bigr\} $$

is open. Let $x_{\alpha}\in E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $F(x_{\alpha },y,z_{\alpha})\nsubseteq{-}\operatorname{int} C(x_{\alpha})$ for some $z_{\alpha}\in B(x_{\alpha})$ and so there exists $v_{\alpha}\in V$ such that

$$v_{\alpha}\in F(x_{\alpha},y,z_{\alpha})\setminus \bigl({-} \operatorname{int} C(x_{\alpha}) \bigr). $$

Since B is u.s.c. with compact valued, by Lemma 2.2, there exists a subnet of $\{z_{\alpha}\}$, denoted again by $\{z_{\alpha}\}$, such that $z_{\alpha}\rightarrow z_{0}\in B(x_{0})$. The fact that $F(\cdot,y,\cdot)$ is u.s.c. with compact valued together with Lemma 2.2 shows that there exists a subset net of $\{ v_{\alpha}\}$, denoted again by $\{v_{\alpha}\}$, such that $v_{\alpha }\rightarrow v_{0}\in F(x_{0},y,z_{0})$. On the other hand, it is easy to see that $v_{\alpha}\in W(x_{\alpha})$. Since W is closed, we know that $v_{0}\in W(x_{0})$ and so $v_{0}\notin{-}\operatorname{int} C(x_{0})$. Thus $F(x_{0},y,z_{0})\nsubseteq{-}\operatorname{int} C(x_{0})$ for some $z_{0}\in B(x_{0})$ and so $x_{0}\in E\setminus A^{-1}(y)$. This implies that $E\setminus A^{-1}(y)$ is closed and so $A^{-1}(y)$ is open. It follows from Theorem 3.1 that there exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde {x})$ and $F(\tilde{x},y,\tilde{z})\nsubseteq{-}\operatorname{int} C(\tilde{x})$ for all $y\in S(\tilde{x})$. This completes the proof. □

Remark 3.9

Theorem 3.7 can be considered as a generalization of Theorem 3.1 in [2, 4] under different conditions from the topological vector space to the abstract convex space.

Remark 3.10

When E is a nonempty convex compact of topological vector space, Li and Li [27] studied the existence of solutions for (GVQEP6).

Theorem 3.8

Suppose the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied. Moreover, assume that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is u.s.c. with compact valued on $E\times E\times E$ and C is closed;

• (b) B is l.s.c.

There exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde{x},y,z)\cap C(\tilde{x})\neq\emptyset$ for all $y\in S(\tilde{x})$ and $z\in B(x)$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\exists z\in B(x), F(x,y,z)\cap C(x)=\emptyset \bigr\} . $$

We show that

$$A^{-1}(y)= \bigl\{ x\in E:\exists z\in B(x), F(x,y,z)\cap C(x)=\emptyset \bigr\} $$

is open. Let $\{x_{\alpha}\}\subseteq E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then

$$F \bigl(x_{\alpha},y,z' \bigr)\cap C(x_{\alpha})\neq \emptyset,\quad \forall z'\in B(x_{\alpha}). $$

It follows that there exists $v_{\alpha}\in F(x_{\alpha},y,z')\cap C(x_{\alpha})$. Since B is l.s.c., by Lemma 2.3, there exists $z_{\alpha}\in B(x_{\alpha})$ such that $z_{\alpha}\rightarrow z$ for any $z\in B(x_{0})$. By the fact that $F(\cdot,y,\cdot)$ is u.s.c. with compact valued, there exists a subset of $\{v_{\alpha}\}$, denoted again by $\{v_{\alpha}\}$, such that $v_{\alpha}\rightarrow v_{0} \in F(x_{0},y,z)$. Since $v_{\alpha}\in C(x_{\alpha})$ and C is closed, we know that $v_{0}\in C(x_{0})$ and so $v_{0}\in F(x_{0},y,z)\cap C(x_{0})$. Thus,

$$F(x_{0},y,z)\cap C(x_{0})\neq\emptyset,\quad \forall z\in B(x_{0}). $$

This shows that $x_{0}\in E\setminus A^{-1}(y)$ and so $E\setminus A^{-1}(y)$ is closed. Thus, $A^{-1}(y)$ is open. By Theorem 3.1, there exists $\tilde{x}\in E$ such that $\tilde{x}\in S(\tilde{x})$ and

$$F(\tilde{x},y,z)\cap C(x)\neq\emptyset, \quad\forall y\in S(\tilde {x}), \forall z\in B(\tilde{x}). $$

This completes the proof. □

Corollary 3.4

Assume that the conditions (i), (iii), (iv), and (v) in Theorem  3.1 are satisfied with $S=B$. Moreover, suppose that, for each $y\in E$, $F(\cdot,y)$ is u.s.c. with compact valued on $E\times E$ and C is closed. Then there exists $\tilde{x}\in E$ such that $\tilde {x}\in S(\tilde{x})$ and $F(\tilde{x},y)\cap C(\tilde{x})\neq \emptyset$ for all $y\in S(\tilde{x})$.

Proof

The proof is similar to that of Theorem 3.8 and so we omit it here. □

Remark 3.11

When $S(x)=E$ for all $x\in E$, Corollary 3.4 was given by Theorem 3 of Yang and Huang [17] under some different conditions.

Theorem 3.9

Suppose the conditions (i), (iii), (iv), and (v) are satisfied in Theorem  3.1. Moreover, assume that

• (a) for each $y\in E$, $F(\cdot,y,\cdot)$ is u.s.c. with compact valued on $E\times E\times E$ and C is closed;

• (b) B is u.s.c. and $B(x)$ is compact for each $x\in E$.

There exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde{x})$ and $F(\tilde{x},y,\tilde{z})\cap C(\tilde{x})\neq\emptyset$ for all $y\in S(\tilde{x})$.

Proof

Let

$$A(x)= \bigl\{ y\in E:\forall z\in B(x), F(x,y,z)\cap C(x)=\emptyset \bigr\} . $$

We prove that

$$A^{-1}(y)= \bigl\{ x\in E:\forall z\in B(x), F(x,y,z)\cap C(x)=\emptyset \bigr\} $$

is open. Let $\{x_{\alpha}\}\subseteq E\setminus A^{-1}(y)$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then

$$F(x_{\alpha},y,z_{\alpha})\cap C(x_{\alpha})\neq\emptyset $$

for some $z_{\alpha}\in B(x_{\alpha})$, that is, there exists $v_{\alpha }\in F(x_{\alpha},y,z_{\alpha})\cap C(x_{\alpha})$. Since B is u.s.c. and $B(x)$ is compact, it follows from Lemma 2.2 that there exists a subset of $\{z_{\alpha}\}$, denoted again by $\{z_{\alpha }\}$, such that $z_{\alpha}\rightarrow z_{0}\in B(x_{0})$. Similar to the proof of Theorem 3.8, we can prove that $F(x_{0},y,z_{0})\cap C(x_{0})\neq\emptyset$ for some $z_{0}\in B(x_{0})$. This shows that $x_{0}\in E\setminus A^{-1}(y)$ and so $E\setminus A^{-1}(y)$ is closed. Thus, $A^{-1}(y)$ is open. It follows from Theorem 3.1 that there exist $\tilde{x}\in E$ and $\tilde{z}\in B(\tilde{x})$ such that $\tilde{x}\in S(\tilde {x})$ and

$$F(\tilde{x},y,\tilde{z})\cap C(\tilde{x})\neq\emptyset,\quad \forall y\in S( \tilde{x}). $$

This completes the proof. □

4 Applications to the generalized semi-infinite programs

In this section, by the results presented in Section 3, we give some existence theorems of solutions to the generalized semi-infinite programs. Let L be a real topological vector space ordered by a closed convex pointed cone $H\subseteq L$ with $\operatorname{int} H\neq\emptyset$ and $h: X\rightrightarrows L$ be a u.s.c. mapping with compact values.

Theorem 4.1

Suppose that all the conditions of Theorem  3.2 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ and S are l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\subseteq C(x), \forall y\in S(x), \forall z \in B(x) \bigr\} . $$

Proof

Theorem 3.2 shows that $K\neq\emptyset$. From Lemma 2.5, it is sufficient to show that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, by Lemma 2.1, we only need to prove that K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$ and

$$F \bigl(x_{\alpha},y',z' \bigr)\subseteq C(x_{\alpha}),\quad \forall y'\in S(x_{\alpha}), \forall z'\in B(x_{\alpha}). $$

Since S and B are l.s.c., for any $y\in S(x_{0})$ and $z\in B(x_{0})$, it follows from Lemma 2.3 that there exist $y_{\alpha }\in S(x_{\alpha})$ and $z_{\alpha}\in B(x_{\alpha})$ such that $y_{\alpha} \rightarrow y$ and $z_{\alpha}\rightarrow z$. By the lower semi-continuity of F and Lemma 2.3, for any $v\in F(x_{0},y,z)$, there exists $v_{\alpha}\in F(x_{\alpha},y_{\alpha },z_{\alpha})$ such that $v_{\alpha}\rightarrow v$. Now the closedness of C with $v_{\alpha}\in C(x_{\alpha})$ shows that $v\in C(x_{0})$ and so $F(x_{0},y,z)\subseteq C(x_{0})$ for all $y\in S(x_{0})$ and $z\in B(x_{0})$. Moreover, the closedness of $E\setminus G_{0}$ shows that $x_{0}\in S(x_{0})$. Thus, K is closed. This completes the proof. □

Corollary 4.1

Suppose that all the conditions of Corollary  3.1 are satisfied. Moreover, assume that $F(\cdot,\cdot)$ and S are l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y)\subseteq C(x), \forall y\in S(x) \bigr\} . $$

Remark 4.1

When $S(x)=E$ for all $x\in E$, Corollary 4.1 was given by Theorem 5 of Yang and Huang [17] under some different conditions.

Theorem 4.2

Suppose that all the conditions of Theorem  3.3 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ and S are l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\subseteq C(x), \forall y \in S(x) \bigr\} . $$

Proof

Obviously, Theorem 3.3 shows that $K\neq\emptyset$. By Lemma 2.5, it is sufficient to prove that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, from Lemma 2.1, we only need to show that K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$ and there exists $z_{\alpha}\in B(x_{\alpha})$ such that

$$F \bigl(x_{\alpha},y',z_{\alpha} \bigr)\subseteq C(x_{\alpha}),\quad \forall y'\in S(x_{\alpha}). $$

Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of $\{z_{\alpha}\}$, denoted again by $\{z_{\alpha}\}$, such that $z_{\alpha}\rightarrow z_{0}\in B(x_{0})$. For any $y\in S(x_{0})$, the lower semi-continuity of S together with Lemma 2.3 implies that there exists $y_{\alpha}\in S(x_{\alpha})$ such that $y_{\alpha} \rightarrow y$. For $v\in F(x_{0},y,z_{0})$, by the fact that F is l.s.c., it follows from Lemma 2.3 that there exists $v_{\alpha}\in F(x_{\alpha},y_{\alpha},z_{\alpha})$ such that $v_{\alpha}\rightarrow v$. Now the closedness of C with $v_{\alpha }\in C(x_{\alpha})$ shows that $v\in C(x_{0})$ and so there exists $z_{0}\in B(x_{0})$ such that $F(x_{0},y,z)\subseteq C(x_{0})$ for all $y\in S(x_{0})$. Moreover, the closedness of $E\setminus G_{0}$ shows that $x_{0}\in S(x_{0})$. Thus, K is closed. This completes the proof. □

Theorem 4.3

Suppose that all the conditions of Theorem  3.4 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ and S are l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\cap{-}\operatorname{int} C(x)=\emptyset, \forall y\in S(x), \forall z\in B(x) \bigr\} . $$

Proof

Theorem 3.4 shows that $K\neq\emptyset$. From Lemma 2.5, it is sufficient to show that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, by Lemma 2.1, we only need to show that K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$,

$$F \bigl(x_{\alpha},y',z' \bigr)\cap{-} \operatorname{int} C(x)=\emptyset,\quad \forall y'\in S(x_{\alpha}), \forall z'\in B(x_{\alpha}) $$

and so

$$F \bigl(x_{\alpha},y',z' \bigr)\subseteq W(x_{\alpha}),\quad \forall y'\in S(x_{\alpha}), \forall z'\in B(x_{\alpha}). $$

Similar to the proof of Theorem 4.1, we have $x_{0}\in S(x_{0})$,

$$F(x_{0},y,z)\subseteq W(x_{0}),\quad \forall y\in S(x_{0}), \forall z\in B(x_{0}) $$

and so

$$F(x_{0},y,z)\cap{-}\operatorname{int} C(x_{0})=\emptyset,\quad \forall y\in S(x_{0}), \forall z\in B(x_{0}). $$

Thus, K is closed. This completes the proof. □

Corollary 4.2

Suppose that all the conditions of Corollary  3.2 are satisfied. Moreover, assume that $F(\cdot,\cdot)$ and S are l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y)\cap C(x)=\emptyset, \forall y\in S(x) \bigr\} . $$

Remark 4.2

When $S(x)=E$ for all $x\in E$, Corollary 4.2 was given by Theorem 6 of Yang and Huang [17] under some different conditions.

Theorem 4.4

Suppose that all the conditions of Theorem  3.5 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ and S are l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\cap {-} \operatorname{int} C(x)=\emptyset, \forall y\in S(x) \bigr\} . $$

Proof

It follows from Theorem 3.5 that $K\neq\emptyset$. From Lemma 2.5, it is sufficient to show that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, by Lemma 2.1, we only need to show K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$ and there exists $z_{\alpha}\in B(x_{\alpha})$ such that

$$F \bigl(x_{\alpha},y',z_{\alpha} \bigr)\cap{-} \operatorname{int} C(x_{\alpha})=\emptyset, \quad\forall y'\in S(x_{\alpha}). $$

Thus, there exists $z_{\alpha}\in B(x_{\alpha})$ such that

$$F \bigl(x_{\alpha},y',z_{\alpha} \bigr)\subseteq C(x_{\alpha}), \quad\forall y'\in S(x_{\alpha}). $$

Similar to the proof of Theorem 4.2, we know that $x_{0}\in S(x_{0})$ and there exists $z_{0}\in B(x_{0})$ such that

$$F(x_{0},y,z_{0})\subseteq W(x_{0}),\quad \forall y\in S(x_{0}). $$

Thus, $x_{0}\in S(x_{0})$ and there exists $z_{0}\in B(x_{0})$ such that

$$F(x_{0},y,z_{0})\cap{-}\operatorname{int} C(x_{0})=\emptyset,\quad \forall y\in S(x_{0}). $$

It follows that K is closed. This completes the proof. □

Theorem 4.5

Suppose that all the conditions of Theorem  3.6 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\nsubseteq{-}\operatorname{int} C(x), \forall y\in S(x), \forall z\in B(x) \bigr\} . $$

Proof

Theorem 3.6 shows that $K\neq\emptyset$. From Lemma 2.5, it is sufficient to show that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, by Lemma 2.1, we only need to prove that K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$,

$$F \bigl(x_{\alpha},y',z' \bigr)\nsubseteq{-} \operatorname{int} C(x_{\alpha}),\quad \forall y'\in S(x_{\alpha}), \forall z'\in B(x_{\alpha}) $$

and so there exists $v_{\alpha}\in V$ such that

$$v_{\alpha}\in F \bigl(x_{\alpha},y',z' \bigr)\setminus \bigl({-}\operatorname{int} C(x_{\alpha}) \bigr). $$

By the lower semi-continuity of S and B, for any $y\in S(x_{0})$ and $z\in B(x_{0})$, it follows from Lemma 2.3 that there exist $y_{\alpha}\in S(x_{\alpha})$ and $z_{\alpha}\in B(x_{\alpha})$ such that $y_{\alpha} \rightarrow y$ and $z_{\alpha}\rightarrow z$. Since $F(\cdot,\cdot,\cdot)$ is a u.s.c. mapping with compact values, Lemma 2.2 shows that there exists a subnet of $\{v_{\alpha}\}$, denoted again by $\{v_{\alpha}\}$, such that $v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z)$. On the other hand, the fact that $v_{\alpha }\notin{-}\operatorname{int} C(x_{\alpha})$ shows that $v_{\alpha}\in W(x_{\alpha})$. Now the closedness of W shows that $v_{0}\in W(x_{0})$ and so $v_{0}\notin{-}\operatorname{int} C(x_{0})$. Moreover, the closedness of $E\setminus G_{0}$ shows that $x_{0}\in S(x_{0})$. Thus,

$$F(x_{0},y,z)\nsubseteq{-}\operatorname{int} C(x_{0}) $$

for all $y\in S(x_{0})$ and $z\in B(x_{0})$ and so K is closed. This completes the proof. □

Corollary 4.3

Suppose that all the conditions of Corollary  3.3 are satisfied. Moreover, assume that $F(\cdot,\cdot)$ is u.s.c. and S is l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y)\nsubseteq C(x), \forall y\in S(x) \bigr\} . $$

Remark 4.3

When $S(x)=E$ for all $x\in E$, Corollary 4.3 was given by Theorem 8 of Yang and Huang [17] under some different conditions.

Theorem 4.6

Suppose that all the conditions of Theorem  3.7 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\nsubseteq {-} \operatorname{int} C(x), \forall y\in S(x) \bigr\} . $$

Proof

Theorem 3.7 shows that $K\neq\emptyset$. By Lemma 2.5, it is sufficient to prove that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, from Lemma 2.1, we only need to show that K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$ and there exists $z_{\alpha}\in B(x_{\alpha})$ such that

$$F \bigl(x_{\alpha},y',z_{\alpha} \bigr)\nsubseteq{-} \operatorname{int} C(x_{\alpha}),\quad \forall y'\in S(x_{\alpha}). $$

Thus, there exists $v_{\alpha}\in V$ such that

$$v_{\alpha}\in F \bigl(x_{\alpha},y',z_{\alpha} \bigr)\setminus \bigl({-}\operatorname{int} C(x_{\alpha}) \bigr). $$

Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of $\{z_{\alpha}\}$, denoted again by $\{z_{\alpha}\}$, such that $z_{\alpha}\rightarrow z_{0}\in B(x_{0})$. By the lower semi-continuity of S, for any $y\in S(x_{0})$, Lemma 2.3 shows that there exists $y_{\alpha}\in S(x_{\alpha})$ such that $y_{\alpha} \rightarrow y$. Since $F(\cdot ,\cdot,\cdot)$ is a u.s.c. mapping with compact values, Lemma 2.2 implies that there exists a subnet of $\{v_{\alpha}\}$, denoted again by $\{v_{\alpha}\}$, such that $v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z_{0})$. Similar to the proof of Theorem 4.5, we can prove that K is closed. This completes the proof. □

Theorem 4.7

Suppose that all the conditions of Theorem  3.8 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y,z)\cap C(x)\neq\emptyset, \forall y\in S(x), \forall z\in B(x) \bigr\} . $$

Proof

Theorem 3.8 shows that $K\neq\emptyset$. From Lemma 2.5, it is sufficient to show that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, by Lemma 2.1, we only need to show K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$,

$$F \bigl(x_{\alpha},y',z' \bigr)\cap C(x_{\alpha})\neq\emptyset, \quad\forall y'\in S(x_{\alpha}), z\in B(x_{\alpha}) $$

and so there exists $v_{\alpha}\in V$ such that

$$v_{\alpha}\in F \bigl(x_{\alpha},y',z' \bigr)\cap C(x_{\alpha}). $$

By the lower semi-continuity of S and B, for any $y\in S(x_{0})$ and $z\in B(x_{0})$, Lemma 2.3 shows that there exist $y_{\alpha }\in S(x_{\alpha})$ and $z_{\alpha}\in B(x_{\alpha})$ such that $y_{\alpha} \rightarrow y$ and $z_{\alpha}\rightarrow z$. Since $F(\cdot ,\cdot,\cdot)$ is u.s.c. with compact values, by Lemma 2.2, there exists a subnet of $\{v_{\alpha}\}$, denoted again by $\{v_{\alpha }\}$, such that $v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z)$. Now the closedness of C with $v_{\alpha}\in C(x_{\alpha})$ shows that $v_{0}\in C(x_{0})$ and so

$$F(x_{0},y,z)\cap C(x_{0})\neq\emptyset,\quad \forall y\in S(x_{0}), \forall z\in B(x_{0}). $$

Moreover, the closedness of $E\setminus G_{0}$ shows that $x_{0}\in S(x_{0})$. Thus, K is closed. This completes the proof. □

Corollary 4.4

Suppose that all the conditions of Corollary  3.4 are satisfied. Moreover, assume that $F(\cdot,\cdot)$ and S are l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), F(x,y)\cap C(x)\neq\emptyset, \forall y\in S(x) \bigr\} . $$

Remark 4.4

When $S(x)=E$ for all $x\in E$, Corollary 4.4 was given by Theorem 7 of Yang and Huang [17] under some different conditions.

Theorem 4.8

Suppose that all the conditions of Theorem  3.9 are satisfied. Moreover, assume that $F(\cdot,\cdot,\cdot)$ is a u.s.c. mapping with compact values and S is l.s.c. Then there is a solution to the problem

$$\operatorname{wMin}_{H}h(K), $$

where

$$K= \bigl\{ x\in E: x\in S(x), \exists z\in B(x), F(x,y,z)\cap C(x)\neq\emptyset, \forall y\in S(x) \bigr\} . $$

Proof

Theorem 3.8 shows that $K\neq\emptyset$. From Lemma 2.5, it is sufficient to prove that $h(K)$ is compact. Since h is u.s.c. and $K\subseteq E$, by Lemma 2.1, we only need to show K is closed. Let $\{x_{\alpha}\}\subseteq K$ be a net with $x_{\alpha}\rightarrow x_{0}$. Then $x_{\alpha}\in S(x_{\alpha})$ and there exists $z_{\alpha}\in B(x_{\alpha})$ such that

$$F \bigl(x_{\alpha},y',z_{\alpha} \bigr)\cap C(x_{\alpha})\neq\emptyset, \quad\forall y'\in S(x_{\alpha}). $$

Thus, there exists $v_{\alpha}\in V$ such that

$$v_{\alpha}\in F \bigl(x_{\alpha},y',z_{\alpha} \bigr)\cap C(x_{\alpha}). $$

Since B is a u.s.c. mapping with compact values, it follows from Lemma 2.2 that there exists a subnet of $\{z_{\alpha}\}$, denoted again by $\{z_{\alpha}\}$, such that $z_{\alpha}\rightarrow z_{0}\in B(x_{0})$. By the lower semi-continuity of S, for any $y\in S(x_{0})$, Lemma 2.3 implies that there exists $y_{\alpha}\in S(x_{\alpha})$ such that $y_{\alpha} \rightarrow y$. Since $F(\cdot,\cdot,\cdot)$ is a u.s.c. mapping with compact values, by Lemma 2.2, there exists a subnet of $\{v_{\alpha}\}$, denoted again by $\{v_{\alpha}\}$, such that $v_{\alpha}\rightarrow v_{0}\in F(x_{0},y,z_{0})$. Now the closedness of C with $v_{\alpha}\in C(x_{\alpha})$ shows that $v_{0}\in C(x_{0})$ and so there exists $z_{0}\in B(x_{0})$ such that

$$F(x_{0},y,z_{0})\cap C(x_{0})\neq\emptyset,\quad \forall y\in S(x_{0}). $$

Moreover, the closedness of $E\setminus G_{0}$ shows that $x_{0}\in S(x_{0})$. Therefore, K is closed. This completes the proof. □