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 Coincidence Theorems, Generalized Variational Inequality Theorems, and Minimax Inequality Theorems for the Φ-Mapping on G-Convex Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 78696 13 pages doi 2007 78696 Research Article Coincidence Theorems Generalized Variational Inequality Theorems and Minimax Inequality Theorems for the O-Mapping on G-Convex Spaces Chi-Ming Chen Tong-Huei Chang and Ya-Pei Liao Received 14 December 2006 Revised 27 February 2007 Accepted 5 March 2007 Recommended by Simeon Reich We establish some coincidence theorems generalized variational inequality theorems and minimax inequality theorems for the family G-KKM X Y and the O-mapping on G-convex spaces. Copyright 2007 Chi-Ming Chen 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 and preliminaries In 1929 Knaster et al. 1 had proved the well-known KKM theorem on M-simplex. In 1961 Fan 2 had generalized the KKM theorem in the infinite-dimensional topological vector space. Later the KKM theorem and related topics for example matching theorem fixed point theorem coincidence theorem variational inequalities minimax inequalities and so on had been presented in grand occasions. Recently Chang and Yen 3 introduced the family KKM X Y and got some results about fixed point theorems coincidence theorems and some applications on this family. Later Ansari et al. 4 and Lin and Chen 5 studied the coincidence theorems for two families of set-valued mappings and they also gave some applications of the existence of minimax inequality and equilibrium problems. In this paper we establish some coincidence theorems generalized variational inequality theorems and minimax inequality theorems for the family G-KKM X Y and the O-mapping on G-convex spaces. 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 e Y y e T x ii T A ÚxcaT x iii T-1 y x e
