Some Maximal Elements' Theorems in -Spaces
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Research Article Some Maximal Elements’ Theorems in FC-Spaces Rong-Hua He1, 2 and Yong Zhang1 1
Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610103, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Rong-Hua He, [email protected] Received 30 March 2009; Accepted 1 September 2009 Recommended by Nikolaos Papageorgiou Let I be a finite or infinite index set, let X be a topological space, and let Yi , ϕNi i∈I be a family of FC-spaces. For each i ∈ I, let Ai : X → 2Yi be a set-valued mapping. Some new existence theorems of maximal elements for a set-valued mapping and a family of set-valued mappings involving a better admissible set-valued mapping are established under noncompact setting of FC-spaces. Our results improve and generalize some recent results. Copyright q 2009 R.-H. He and Y. Zhang. 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.
1. Introduction It is well known that many existence theorems of maximal elements for various classes of set-valued mappings have been established in different spaces. For their applications to mathematical economies, generalized games, and other branches of mathematics, the reader may consult 1–12 and the references therein. In most of the known existence results of maximal elements, the convexity assumptions play a crucial role which strictly restrict the applicable area of these results. In this paper, we will continue to study existence theorems of maximal elements in general topological spaces without convexity structure. We introduce a new class of generalized GB -majorized mappings Ai : X → 2Yi for each i ∈ I which involve a set-valued mapping F ∈ BY, X. The notion of generalized GB -majorized mappings unifies and generalizes the corresponding notions of GB -majorized mappings in 4; LS -majorized mappings in 2, 13; H-majorized mappings in 14. Some new existence theorems of maximal elements for generalized GB -majorized mappings are proved under noncompact setting of FC-spaces. Our results improve and generalize the corresponding results in 2, 4, 14–16.
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Journal of Inequalities and Applications
2. Preliminaries Let X and Y be two nonempty sets. We denote by 2Y and X the family of all subsets of Y and the family of all nonempty finite subsets of X, respectively. For each A ∈ X, we denote by |A| the cardinality of A. Let Δn denote the standard n-dimensional simplex with the vertices {e0 , . . . , en }. If J is a nonempty subset of {0, 1, . . . , n}, we will denote by ΔJ the convex hull of the vertices {ej : j ∈ J}. Let X and Y be two sets, and let T : X → 2Y be a set-valued mapping. We will use the following notations in the sequel: i T x {y ∈ Y : y ∈ T x}, ii T A x∈A T x, iii T −1 y {x ∈ X : y ∈ T x}. For topological spaces X and Y , a subset A of
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