Journal of Inequalities and Applications
Volume 2005 (2005), Issue 3, Pages 251-259

Applications of W. A. Kirk's fixed-point theorem to generalized nonlinear variational-like inequalities in reflexive Banach spaces

Zeqing Liu,1 Jeong Sheok Ume,2 and Shin Min Kang3

1Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China
2Department of Applied Mathematics, College of Natural Sciences, Changwon National University, Kyongnam, Changwon 641-773, Korea
3Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Korea

Received 10 September 2003

Copyright © 2005 Zeqing 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.


We introduce and study a new class of generalized nonlinear variational-like inequalities, which includes these variational inequalities and variational-like inequalities due to Bose, Cubiotti, Dien, Ding, Ding and Tarafdar, Noor, Parida, Sahoo, and Kumar, and Yao, and others as special cases. By applying Kirk's fixed-point theorem and Ding-Tan minimax inequality, we establish the existence theorems of solutions for the generalized nonlinear variational-like inequalities in reflexive Banach spaces.