In 1979, Robinson [1] studied the following parametric variational inclusion problem: given , find such that

where is a single-valued function and is a multivalued map. It is known that (1.1) covers variational inequality problems and a vast of variational system important in applications. Since then, various types of variational inclusion problems have been extended and generalized by many authors (see, e.g., [2–7] and the references therein).

On the other hand, Tarafdar [8] generalized the classical Himmelberg fixed point theorem [9] to locally -convex uniform spaces (or -spaces). Park [10] generalized the result of Tarafdar [8] to locally -convex spaces (or -spaces). Recently, Park [11] introduced the concept of abstract convex spaces which include -spaces and -convex spaces as special cases. With this new concept, he can study the KKM theory and its applications in abstract convex spaces. More recently, Park [12] introduced the concept of -spaces which include -spaces and -spaces as special cases. He also established the Himmelberg type fixed point theorem in -spaces. To see some related works, we refer to [13–21] and the references therein. However, to the best of our knowledge, there is no paper dealing with systems of generalized quasivariational inclusion problems in -spaces.

Motivated and inspired by the works mentioned above, in this paper, we first prove that the product of a family of -spaces is also an -space. Then, by using the Himmelberg type fixed point theorem due to Park [12], we establish existence theorems of solutions for systems of generalized quasivariational inclusion problems, systems of variational equations, and systems of generalized quasiequilibrium problems in -spaces. Applications of the existence theorem of solutions for systems of generalized quasiequilibrium problems to optimization problems are given in -spaces.

For a set , will denote the family of all nonempty finite subsets of . If is a subset of a topological space, we denote by int and the interior and closure of , respectively.

A multimap (or simply a map) is a function from a set into the power set of ; that is, a function with the values for all . Given a map , the map defined by for all , is called the (lower) inverse of . For any , . For any , . As usual, the set is called the graph of .

For topological spaces and , a map is called

(i)closed if its graph is a closed subset of ,

(ii)upper semicontinuous (in short, u.s.c.) if for any and any open set in with , there exists a neighborhood of such that for all ,

(iii)lower semicontinuous (in short, l.s.c.) if for any and any open set in with , there exists a neighborhood of such that for all ,

(iv)continuous if is both u.s.c. and l.s.c.,

(v)compact if is contained in a compact subset of .

Lemma 2.1 . (see [22]).

Let and be topological spaces, be a map. Then, is l.s.c. at if and only if for any and for any net in converging to , there exists a net in such that for each and converges to .

Lemma 2.2 . (see [23]).

Let and be Hausdorff topological spaces and be a map.

(i)If is an u.s.c. map with closed values, then is closed.

(ii) If is a compact space and is closed, then is u.s.c.

(iii) If is compact and is an u.s.c. map with compact values, then is compact.

In what follows, we introduce the concept of abstract convex spaces and map classes , and having certain KKM properties. For more details and discussions, we refer the reader to [11, 12, 24].

Definition 2.3 (see [11]).

An abstract convex space consists of a topological space , a nonempty set , and a map with nonempty values. We denote for .

In the case , let . It is obvious that any vector space is an abstract convex space with , where co denotes the convex hull in vector spaces. In particular, is an abstract convex space.

Let be an abstract convex space. For any , the -convex hull of is denoted and defined by

(co is reserved for the convex hull in vector spaces). A subset of is called a -convex subset of relative to if for any , we have ; that is, co. This means that itself is an abstract convex space called a subspace of . When , the space is denoted by . In such case, a subset of is said to be -convex if co; in other words, is -convex relative to . When , -convex subsets reduce to ordinary convex subsets.

Let be an abstract convex space and a set. For a map with nonempty values, if a map satisfies

then is called a KKM map with respect to . A KKM map is a KKM map with respect to the identity map . A map is said to have the KKM property and called a -map if, for any KKM map with respect to , the family has the finite intersection property. We denote

Similarly, when is a topological space, a -map is defined for closed-valued maps , and a -map is defined for open-valued maps . In this case, we have

Note that if is discrete, then three classes , and are identical. Some authors use the notation instead of .

Definition 2.4 . (see [24]).

For an abstract convex space , the KKM principle is the statement .

A KKM space is an abstract convex space satisfying the KKM principle.

Definition 2.5.

Let be an abstract convex space, be a real t.v.s., and a map. Then,

is -quasiconvex-like if for any and any there exists such that ,

is -quasiconvex if for any and any there exists such that .

Remark 2.6.

If is a nonempty convex subset of a t.v.s. with , then Definition 2.5 (i) and (ii) reduce to Definition 2.4 (iii) and (vi) in Lin [5], respectively.

Definition 2.7 . (see [25]).

A uniformity for a set is a nonempty family of subsets of satisfying the following conditions:

each member of contains the diagonal ,

for each , ,

for each , there exists such that ,

if , , then ,

if and , then .

The pair is called a uniform space. Every member in is called an entourage. For any and any , we define . The uniformity is called separating if . The uniform space is Hausdorff if and only if is separating. For more details about uniform spaces, we refer the reader to Kelley [25].

Definition 2.8 . (see [12]).

An abstract convex uniform space is an abstract convex space with a basis of a uniformity of .

Definition 2.9 . (see [12]).

An abstract convex uniform space is called an -space if

is dense in , and

for each and each -convex subset , the set is -convex.

Lemma 2.10 . (see [12, Corollary ]).

Let be a Hausdorff KKM -space and a compact u.s.c. map with nonempty closed -convex values. Then, has a fixed point.

Lemma 2.11 . (see [24, Lemma ]).

Let be any family of abstract convex spaces. Let and . For each , let be the projection. For each , define . Then, is an abstract convex space.

Lemma 2.12.

Let be any index set. For each , let be an -space. If one defines , for each and , where . Then, is also an -space.

Proof.

By Lemma 2.11, is an abstract convex space. It is easy to check that is a subbase of the product uniformity of . Since is the basis generated by , we obtain that is a basis of the product uniformity, and the associated uniform topology on .

Now, we prove that for each and each -convex subset , the set is -convex. Firstly, we show that for each , is a -convex subset of . For any , we can find some with . Since is a -convex subset of , we have . It follows that . Thus, we have shown that is a -convex subset of . Secondly, we show that the set is -convex. Since each has the form for some and , we have that

By the definition of -spaces, we obtain that for each , the set is -convex. It follows from (2.6) that the set is a -convex subset of . Therefore is an -space. This completes the proof.

Remark 2.13.

Lemma 2.12 generalizes [26, Theorem ] from locally -uniform spaces to -spaces. The proof of Lemma 2.12 is different with the proof of [26, Theorem ].

Let be any index set. For each , let be a topological vector space, be an -space, and be an -space with . Let , and be the product -space as defined in Lemma 2.12. Furthermore, we assume that is a KKM space. Throughout this paper, we use these notations unless otherwise specified, and assume that all topological spaces are Hausdorff.

The following theorem is the main result of this paper.

Theorem 3.1.

For each , suppose that

(i) is a compact u.s.c. map with nonempty closed -convex values,

(ii) is a compact continuous map with nonempty closed -convex values,

(iii) is a closed map with nonempty values,

(iv) for each , is -quasiconvex; for each , is -quasiconvex-like and .

Then, there exists with and such that for each , , and for all .

Proof.

For each , define by

Then, is nonempty for each . Indeed, fix any and , define by

First, we show that is a KKM map w.r.t. . Suppose to the contrary that there exists a finite subset such that . Hence, there exists satisfying for all . Since is -convex, we have . By for all , we know that for all . Since is -quasiconvex-like, there exists such that

This leads to a contradiction. Therefore, is a KKM map w.r.t. . Next, we show that is closed for each . Indeed, if , then there exists a net in such that . For each , we have and . By condition (ii), is closed, and hence . By condition (iii), is closed, and hence . It follows that . Therefore, is closed. Since and is -convex, we have that . Having that is compact, we can deduce that . That is is nonempty.

is closed for each . Indeed, if , then there exists a net in such that . One has and for all . By condition (ii), is closed, and hence . Let , since is l.s.c., there exists a net satisfying and . We have . Since is closed, we obtain . Thus, we have shown that . Hence, is closed.

is -convex for each and . Indeed, if , then we have that and for all and all . For any given , we have because is -convex. For each , since is -quasiconvex, there exists such that

Hence, for all . It follows that and is -convex.

Since and is compact. It follows from Lemma 2.2(ii) that is a compact u.s.c. map for each . Define by

It follows from the above discussions that for each , is a compact u.s.c. map with nonempty closed -convex values. Thus, is a compact u.s.c. map with nonempty closed -convex values. By Lemma 2.10, there exists such that . That is there exists with and such that for each , , and for all . This completes the proof.

For the special case of Theorem 3.1, we have the following corollary which is actually an existence theorem of solutions for variational equations.

Corollary 3.2.

For each , suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,

(iii)₁ is a continuous mapping;

(iv)₁ for each , is -quasiconvex; for each , is also -quasiconvex and .

Then, there exists with and such that for each , , and for all .

Theorem 3.3.

For each , suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,

(iii)₂ is a closed map with nonempty values and is an u.s.c. map with nonempty compact values;

(iv)₂ for each , is -quasiconvex; for each , is -quasiconvex-like and .

Then, there exists with and such that for each , , and for all .

Proof.

For each , define by

Obviously, has nonempty values. Now, we show that is closed. Indeed, if , then there exists a net in such that . Since

there exist and such that . Let

Then is a compact subset of , and are compact subsets of . By condition (iii)₂ and Lemma 2.2(iii), is a compact subset of . Thus, we can assume that . By condition (iii)₂, is closed, and hence . Since and is closed, we have . Letting , it follows that

and so is closed.

By the above discussions, we know that condition (iii) of Theorem 3.1 is satisfied. It is easy to check that condition (iv) of Theorem 3.1 is also satisfied. By Theorem 3.1, there exists with and such that for each , , and

for all . This completes the proof.

For the special case of Theorem 3.3, we have the following corollary which is actually an existence theorem of solutions for variational equations.

Corollary 3.4.

For each , suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,

(iii)₃ is a continuous map and is a continuous map;

(iv)₃ for each , is -quasiconvex; for each , is also -quasiconvex and .

Then, there exists with and such that for each , , and for all .

From Theorem 3.3, we establish the following corollary which is actually an existence theorem of solutions for systems of generalized vector quasiequilibrium problems.

Corollary 3.5.

For each , suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,

(iii)₄ is a closed map with nonempty values and is an u.s.c. map with nonempty compact values;

(iv)₄for each , is -quasiconvex; for each , is -quasiconvex-like and .

Then, there exists with and such that for each , , , and for all .

Proof.

Define by for all . Since is a closed map with nonempty values, we have that is a closed map with nonempty values. All the conditions of Theorem 3.3 are satisfied. The conclusion of Corollary 3.5 follows from Theorem 3.3. This completes the proof.

Let be a real topological vector space, a proper convex cone in . A point is called a vector minimal point of if for any , . The set of vector minimal point of is denoted by .

Lemma 4.1 . (see [27]).

Let be a Hausdorff t.v.s., be a closed convex cone in . If is a nonempty compact subset of , then .

Theorem 4.2.

For each , suppose that conditions (i), (ii) in Theorem 3.1 and conditions (iii), (iv) in Corollary 3.5 hold. Furthermore, let be an u.s.c. map with nonempty compact values, where is a real t.v.s. ordered by a proper closed convex cone in . Then, there exists a solution to:

where and such that for each , , , and for all .

Proof.

By Corollary 3.5, there exists with and such that for each , , and for all . For each , let

and . Then and . We show that is closed for each . Indeed, if , then there exists a net in such that . For each , implies that

By the closedness of and , we have that and . Now, we prove that for all . For any , since is l.s.c., there exists a net satisfying and . Let . Since is u.s.c. with nonempty compact values, we can assume that . By the closedness of and , we have that . Thus, . It follows that is closed. Hence, is closed. Note that . We know that is a nonempty compact subset of . It follows from Lemma 2.2(iii) that is a nonempty compact subset of . By Lemma 4.1, . That is there exists a solution of the problem: where . This completes the proof.

Theorem 4.3.

For each , suppose that is compact and condition (ii) in Theorem 3.1 holds. Moreover,

(iii)₅ is a continuous function;

(iv)₅ for each , is -quasiconvex; for each , is also -quasiconvex and .

Furthermore, let is a l.s.c. function. Then there exists a solution to:

where and such that for each , and for all .

Proof.

For each , define and by

respectively. It is easy to check that all the conditions of Corollary 3.5 are satisfied. For each , define

and . Then, by Corollary 3.5, there exists and hence . Arguing as Theorem 4.2, we can prove that is a nonempty compact subset of . Hence there exists a solution to the problem where . This completes the proof.

Remark 4.4.

Theorem 4.3 generalizes [28, Corollary 3.5] from locally convex topological vector spaces to -spaces.

Theorem 4.5.

For each , suppose that is compact and condition (ii) in Theorem 3.1 holds. Moreover,

(iii)₆ is a continuous function;

(iv)₆ for each , is -quasiconvex.

Furthermore, let be a l.s.c. function. Then, there exists a solution to the problem:

where and such that for each , is the solution of the problem .

Proof.

For each , define by

It is easy to check that all the conditions of Theorem 4.3 are satisfied. Theorem 4.5 follows immediately from Theorem 4.3. This completes the proof.