Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 609353, 9 pages
doi:10.1155/2009/609353
Research Article

The Solvability of a New System of Nonlinear Variational-Like Inclusions

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

1Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian Liaoning 116029, China
2Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea
3Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, South Korea

Received 23 November 2008; Accepted 1 April 2009

Academic Editor: Marlene Frigon

Copyright © 2009 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.

Abstract

We introduce and study a new system of nonlinear variational-like inclusions involving s-(G,η)-maximal monotone operators, strongly monotone operators, η-strongly monotone operators, relaxed monotone operators, cocoercive operators, (λ,ξ)-relaxed cocoercive operators, (ζ,φ,ϱ)-g-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with s-(G,η)-maximal monotone operators and Banach contraction principle, we demonstrate the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper improve and extend some known results in the literature.