Systems of Generalized Quasivariational Inclusion Problems with Applications in -Spaces
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Research Article Systems of Generalized Quasivariational Inclusion Problems with Applications in LΓ-Spaces Ming-ge Yang,1, 2 Jiu-ping Xu,3 and Nan-jing Huang1, 3 1
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, China 3 College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China 2
Correspondence should be addressed to Nan-jing Huang, [email protected] Received 27 September 2010; Accepted 22 October 2010 Academic Editor: Yeol J. E. Cho Copyright q 2011 Ming-ge Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We first prove that the product of a family of LΓ-spaces is also an LΓ-space. Then, by using a Himmelberg type fixed point theorem in LΓ-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 LΓ-spaces. Applications of the existence theorem of solutions for systems of generalized quasiequilibrium problems to optimization problems are given in LΓ-spaces.
1. Introduction In 1979, Robinson 1 studied the following parametric variational inclusion problem: given x ∈ Rn , find y ∈ Rm such that 0 ∈ g x, y Q x, y ,
1.1
where g : Rn × Rm → Rp is a single-valued function and Q : Rn × Rm Rp 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 H-convex uniform spaces or LC-spaces. Park 10 generalized the result of Tarafdar 8 to locally G-convex spaces or LG-spaces. Recently, Park 11
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Fixed Point Theory and Applications
introduced the concept of abstract convex spaces which include H-spaces and G-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 LΓ-spaces which include LC-spaces and LG-spaces as special cases. He also established the Himmelberg type fixed point theorem in LΓ-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 LΓ-spaces. Motivated and inspired by the works mentioned above, in this paper, we first prove that the product of a family of LΓ-spaces is also an LΓ-space. Then, by using the Himmelberg type fixed point theorem due to Park 12, we establish existence theorems of soluti
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