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 Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 678014 7 pages doi 2008 678014 Research Article Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems Zhi-Bin Liu 1 Jong Kyu Kim 2 and Nan-Jing Huang3 1 Department of Applied Mathematics Southwest Petroleum University Chengdu Sichuan 610500 China 2 Department of Mathematics Kyungnam University Masan Kyungnam 631701 South Korea 3 Department of Mathematics Sichuan University Chengdu Sichuan 610064 China Correspondence should be addressed to Jong Kyu Kim jongkyuk@ Received 9 January 2008 Accepted 4 April 2008 Recommended by R. P. Gilbert We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the noncon-vex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions. Copyright 2008 Zhi-Bin Liu 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. 1. Introduction The concept of vector variational inequality was first introduced by Giannessi 1 in 1980. Since then existence theorems for solution of general versions of the vector variational inequality have been studied by many authors see . 2-9 and the references therein . Recently vector variational inequalities and their generalizations have been used as a tool to solve vector optimization problems see 7 10-14 . Chen and Craven 11 obtained a sufficient condition for the existence of weakly efficient solutions for differentiable vector optimization
