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 ρθ -pseudomonotonicity for bifunctions, and we consider some examples to show that ρ-θ pseudomonotonicity generalizes bothmonotonicity 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.