Abstract and Applied Analysis
Volume 2012 (2012), Article ID 310818, 26 pages
http://dx.doi.org/10.1155/2012/310818
Research Article

Minimax Theorems for Set-Valued Mappings under Cone-Convexities

Yen-Cherng Lin,1 Qamrul Hasan Ansari,2,3 and Hang-Chin Lai4

1Department of Occupational Safety and Health, College of Public Health, China Medical University, Taichung 404, Taiwan
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Received 7 September 2012; Accepted 27 October 2012

Academic Editor: Ondrej Dosly

Copyright © 2012 Yen-Cherng Lin 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.

Abstract

The aim of this paper is to study the minimax theorems for set-valued mappings with or without linear structure. We define several kinds of cone-convexities for set-valued mappings, give some examples of such set-valued mappings, and study the relationships among these cone-convexities. By using our minimax theorems, we derive some existence results for saddle points of set-valued mappings. Some examples to illustrate our results are also given.