Existence Results for Variational Inequalities with Surjectivity Consequences Related to Generalized Monotone Operators

We present existence results for variational inequalities given by generalized monotone operators. As a consequence, we deduce the existence of zeros, or even more, the surjectivity of some classes of set-valued operators. We show that by strengthening the continuity assumptions, similar surjectivity results can be obtained without any monotonicity assumption. In the framework of reflexive Banach spaces, we extend a related result due to Inoan and Kolumbán (Nonlinear Anal. 68:47–53, 2008).