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

We introduce the notion of relaxed (rho-theta)-n-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 (rho-theta)-n-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (rho-theta)-eudomonotonicity for bifunctions, and we consider some examples to show that (rho-theta)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (rho-theta)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique.