We introduce two new concepts of weakly relaxed monotone mappings and weakly relaxed semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.
"On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxed Monotone Mapping." Abstr. Appl. Anal. 2013 (SI09) 1 - 8, 2013. https://doi.org/10.1155/2013/207845