- Research Article
- Open Access
© 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.
- Convex Subset
- Existence Theorem
- Uniform Space
- Nonempty Convex Subset
- Quasivariational Inclusion
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 .
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.
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).
- Robinson SM: Generalized equations and their solutions. I. Basic theory. Mathematical Programming Study 1979, 10: 128–141. 10.1007/BFb0120850MATHView ArticleMathSciNetGoogle Scholar
- Chang SS: Set-valued variational inclusions in Banach spaces. Journal of Mathematical Analysis and Applications 2000,248(2):438–454. 10.1006/jmaa.2000.6919MATHMathSciNetView ArticleGoogle Scholar
- Ding XP: Perturbed proximal point algorithms for generalized quasivariational inclusions. Journal of Mathematical Analysis and Applications 1997,210(1):88–101. 10.1006/jmaa.1997.5370MATHMathSciNetView ArticleGoogle Scholar
- Huang N-J: A new class of generalized set-valued implicit variational inclusions in Banach spaces with an application. Computers & Mathematics with Applications 2001,41(7–8):937–943. 10.1016/S0898-1221(00)00331-XMATHMathSciNetView ArticleGoogle Scholar
- Lin L-J: Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems. Journal of Global Optimization 2007,38(1):21–39. 10.1007/s10898-006-9081-5MATHMathSciNetView ArticleGoogle Scholar
- Lin L-J: Variational inclusions problems with applications to Ekeland's variational principle, fixed point and optimization problems. Journal of Global Optimization 2007,39(4):509–527. 10.1007/s10898-007-9153-1MATHMathSciNetView ArticleGoogle Scholar
- Lin L-J: Systems of variational inclusion problems and differential inclusion problems with applications. Journal of Global Optimization 2009,44(4):579–591. 10.1007/s10898-008-9359-xMATHMathSciNetView ArticleGoogle Scholar
- Tarafdar E: Fixed point theorems in locally -convex uniform spaces. Nonlinear Analysis 1997,29(9):971–978. 10.1016/S0362-546X(96)00174-5MATHMathSciNetView ArticleGoogle Scholar
- Himmelberg CJ: Fixed points of compact multifunctions. Journal of Mathematical Analysis and Applications 1972, 38: 205–207. 10.1016/0022-247X(72)90128-XMathSciNetView ArticleMATHGoogle Scholar
- Park S: Fixed point theorems in locally -convex spaces. Nonlinear Analysis 2002,48(6):869–879. 10.1016/S0362-546X(00)00220-0MATHMathSciNetView ArticleGoogle Scholar
- Park S: On generalizations of the KKM principle on abstract convex spaces. Nonlinear Analysis Forum 2006,11(1):67–77.MATHMathSciNetGoogle Scholar
- Park S: Fixed point theory of multimaps in abstract convex uniform spaces. Nonlinear Analysis 2009,71(7–8):2468–2480. 10.1016/j.na.2009.01.081MATHMathSciNetView ArticleGoogle Scholar
- Deng L, Yang MG: Coincidence theorems with applications to minimax inequalities, section theorem, best approximation and multiobjective games in topological spaces. Acta Mathematica Sinica (English Series) 2006,22(6):1809–1818. 10.1007/s10114-005-0743-xMATHMathSciNetView ArticleGoogle Scholar
- Deng L, Yang MG: Weakly R-KKM mappings-intersection theorems and minimax inequalities in topological spaces. Applied Mathematics and Mechanics 2007,28(1):103–109. 10.1007/s10483-007-0112-1MATHMathSciNetView ArticleGoogle Scholar
- Park S: Several episodes in recent studies on the KKM theory. Nonlinear Analysis Forum 2010, 15: 13–26.MATHMathSciNetGoogle Scholar
- Park S: Generalized convex spaces, -spaces, and -spaces. Journal of Global Optimization 2009,45(2):203–210. 10.1007/s10898-008-9363-1MATHMathSciNetView ArticleGoogle Scholar
- Park S: Remarks on fixed points, maximal elements, and equilibria of economies in abstract convex spaces. Taiwanese Journal of Mathematics 2008,12(6):1365–1383.MATHMathSciNetGoogle Scholar
- Park S: Remarks on the partial KKM principle. Nonlinear Analysis Forum 20009, 14: 51–62.MathSciNetMATHGoogle Scholar
- Park S: Comments on recent studies on abstract convex spaces. Nonlinear Analysis Forum 2008, 13: 1–17.MATHMathSciNetGoogle Scholar
- Park S: The rise and decline of generalized convex spaces. Nonlinear Analysis Forum 2010, 15: 1–12.MATHMathSciNetGoogle Scholar
- Yang M-G, Xu J-P, Huang N-J, Yu S-J: Minimax theorems for vector-valued mappings in abstract convex spaces. Taiwanese Journal of Mathematics 2010,14(2):719–732.MATHMathSciNetGoogle Scholar
- Tan NX: Quasi-variational inequalities in topological linear locally convex Hausdorff spaces. Mathematische Nachrichten 1985, 122: 231–224. 10.1002/mana.19851220123MathSciNetView ArticleGoogle Scholar
- Aubin JP, Cellina A: Differential Inclusion. Springer, Berlin, Germany; 1994.Google Scholar
- Park S: Equilibrium existence theorems in KKM spaces. Nonlinear Analysis 2008,69(12):4352–5364. 10.1016/j.na.2007.10.058MATHMathSciNetView ArticleGoogle Scholar
- Kelley JL: General Topology. Volume 2. Springer, New York, NY, USA; 1975:xiv+298.MATHGoogle Scholar
- Ding XP: The generalized game and the system of generalized vector quasi-equilibrium problems in locally -uniform spaces. Nonlinear Analysis 2008,68(4):1028–1036. 10.1016/j.na.2006.12.003MATHMathSciNetView ArticleGoogle Scholar
- Luc DC: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems. Volume 319. Springer, Berlin, Germany; 1989:viii+173.Google Scholar
- Lin L-J: Mathematical programming with system of equilibrium constraints. Journal of Global Optimization 2007,37(2):275–286. 10.1007/s10898-006-9049-5MATHMathSciNetView ArticleGoogle Scholar
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