Journal of Inequalities and Applications 
Volume 2008 (2008), Article ID 678014, 7 pages
doi:10.1155/2008/678014
Research Article

Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems

Zhi-Bin Liu,1 Jong Kyu Kim,2 and Nan-Jing Huang3

 1Department of Applied Mathematics, Southwest Petroleum University, Chengdu, Sichuan 610500, China
2Department of Mathematics, Kyungnam University, Masan, Kyungnam 631701, South Korea
3Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Received 9 January 2008; Accepted 4 April 2008

Recommended by R. P. Gilbert

Abstract

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 nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.