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

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. 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 variational-like 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 variational-like 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