Variational-like inequalities with generalized monotone mappings in Banach spaces

In this paper, we introduce two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed eta-alpha monotone mappings in reflexive Banach spaces. We present also the solvability of variational-like inequalities with relaxed eta-alpha semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve the corresponding results of Refs. 1-6.