On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxed $\eta \text{-}\alpha $ Monotone Mapping

We introduce two new concepts of weakly relaxed $\eta \text{-}\alpha $ monotone mappings and weakly relaxed $\eta \text{-}\alpha $ semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed $\eta \text{-}\alpha $ monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed $\eta \text{-}\alpha $ semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.