© Ming-ge Yang et al. 2011
Received: 27 September 2010
Accepted: 22 October 2010
Published: 27 October 2010
We first prove that the product of a family of -spaces is also an -space. Then, by using a Himmelberg type fixed point theorem in -spaces, 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.
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  generalized the classical Himmelberg fixed point theorem  to locally -convex uniform spaces (or -spaces). Park  generalized the result of Tarafdar  to locally -convex spaces (or -spaces). Recently, Park  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  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 , 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.
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 .
Lemma 2.1 . (see ).
Lemma 2.2 . (see ).
Definition 2.3 (see ).
(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.
Definition 2.4 . (see ).
A KKM space is an abstract convex space satisfying the KKM principle.
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 , respectively.
Definition 2.7 . (see ).
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 .
Definition 2.8 . (see ).
Definition 2.9 . (see ).
Lemma 2.10 . (see [12, Corollary ]).
Lemma 2.11 . (see [24, Lemma ]).
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 .
3. Existence Theorems of Solutions for Systems of Generalized Quasivariational Inclusion Problems
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.
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.
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.
For the special case of Theorem 3.3, we have the following corollary which is actually an existence theorem of solutions for variational equations.
From Theorem 3.3, we establish the following corollary which is actually an existence theorem of solutions for systems of generalized vector quasiequilibrium problems.
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.
4. Applications to Optimization Problems
Lemma 4.1 . (see ).
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.
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.
Theorem 4.3 generalizes [28, Corollary 3.5] from locally convex topological vector spaces to -spaces.
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.
This work was supported by the Key Program of NSFC (Grant no. 70831005) and the Open Fund (PLN0904) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).
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