
VariationalLike Inequalities and Equilibrium Problems with Generalized Monotonicity in Banach SpacesDOI: 10.1155/2012/648070 Abstract: We introduce the notion of relaxed (ρθ)ηinvariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variationallike inequality problems with relaxed (ρθ)ηinvariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (ρθ)pseudomonotonicity for bifunctions, and we consider some examples to show that (ρθ)pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (ρθ)pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique. 1. Introduction Let be a nonempty subset of a real reflexive Banach space , and let be the dual space of . Consider the operator and the bifunction . Then the variationallike inequality problem (in short, VLIP) is to find , such that where denote the pairing between and . If we take , then (1.1) becomes to find , such that which is classical variational inequality problems (VIPs). These problems have been studied in both finite and infinite dimensional spaces by many authors [1–3]. VIP has numerous applications in optimization, nonlinear analysis, and engineering sciences. In the study of VLIP and VIP, monotonicity is the most common assumption for the operator . Recently many authors established the existence of solutions for (VIP) and VLIP under generalized monotonicity assumptions, such as quasimonotonicity, relaxed monotonicity, densely pseudomonotonicity, relaxed  monotonicity, and relaxed  pseudomonotonicity (see [1, 3–6] and the references therein). In 2008 [7], Behera et al. defined various concepts of generalized (  ) invariant monotonicities which are proper generalization of generalized invariant monotonicity introduced by Yang et al. [8]. Chen [9] defined semimonotonicity and studied semimonotone scalar variational inequalities problems in Banach spaces. Fang and Huang [3] obtained the existence of solution for VLIP using relaxed  monotone mappings in the reflexive Banach spaces. In [1], Bai et al. extended the results of [3] with relaxed  pseudomonotone mappings and provided the existence of solution of the variationallike inequalities problems in reflexive Banach spaces. Bai et al. [10] studied variational inequalities problems with the setting of densely relaxed pseudomonotone operators and relaxed quasimonotone operators, respectively. Inspired and motivated by [1, 3, 10], we introduce the concept of relaxed ( ) invariant pseudomonotone mappings. Using
