Journal of the
Korean Mathematical Society JKMS

In this paper, we introduce two classes of generalized variational-like inequalities with compositely monotone multifunctions in Banach spaces. Using the KKM-Fan lemma and the Nadler's result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed $\eta-\alpha$ monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the solvability of generalized variational-like inequalities with compositely relaxed $\eta-\alpha$ semimonotone multifunctions in arbitrary Banach spaces by virtue of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve some earlier and recent results in the literature.

Keywords: generalized variational-like inequalities, compositely (semi) monotone multifunctions, KKM mappings, Hausdorff metric, $\widetilde H$-hemicontinuity, coercivity

MSC numbers: 49J40, 90C29, 47H10, 47H17