Advances in Operations Research
Volume 2012 (2012), Article ID 648070, 15 pages
Research Article

Variational-Like Inequalities and Equilibrium Problems with Generalized Monotonicity in Banach Spaces

Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

Received 22 September 2011; Accepted 16 November 2011

Academic Editor: Abdellah Bnouhachem

Copyright © 2012 N. K. Mahato and C. Nahak. 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 the notion of relaxed (ρ-θ)-η-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variational-like inequality problems with relaxed (ρ-θ)-η-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (ρ-θ)-pseudomonotonicity for bifunctions, and we consider some examples to show that (ρ-θ)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (ρ-θ)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique.