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Báo cáo hóa học: " Research Article Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 936428 15 pages doi 10.1155 2011 936428 Research Article Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application Ren-You Zhong 1 Nan-Jing Huang 1 and Yeol Je Cho2 1 Department of Mathematics Sichuan University Chengdu Sichuan 610064 China 2 Department of Mathematics Education and the RINS Gyeongsang National University Chinju 660-701 Republic of Korea Correspondence should be addressed to YeolJe Cho yjcho@gsnu.ac.kr Received 25 October 2010 Accepted 19 January 2011 Academic Editor K. Teo Copyright 2011 Ren-You Zhong 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. This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces when both the mapping and the constraint set are perturbed by different parameters. By using the properties of recession cones several equivalent characterizations are given for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. As an application the stability of solution set for the set-valued vector equilibrium problem in a reflexive Banach space is also given. The results presented in this paper generalize and extend some known results in Fan and Zhong 2008 He 2007 and Zhong and Huang 2010 . 1. Introduction Let X and Y be reflexive Banach spaces. Let K be a nonempty closed convex subset of X. Let F K X K 2Y be a set-valued mapping with nonempty values. Let P be a closed convex pointed cone in Y with int P 0. The cone P induces a partial ordering in Y which was defined by y1 Py2 if and only if y2 - y1 e P. We consider the following set-valued vector equilibrium problem denoted by SVEP F K .