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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 207845, 8 pages
http://dx.doi.org/10.1155/2013/207845
Research Article

On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxed Monotone Mapping

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12121, Thailand

Received 10 September 2013; Accepted 20 October 2013

Academic Editor: S. Romaguera

Copyright © 2013 Marwan Amin Kutbi and Wutiphol Sintunavarat. 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 two new concepts of weakly relaxed monotone mappings and weakly relaxed semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.