On generalized variational-like inequalities with generalized monotone multivalued mappings

Let EE be a reflexive Banach space with the dual space E∗E∗ and KK be a nonempty closed convex subset of EE. Let us have Ψ:K×K×E∗→RΨ:K×K×E∗→R and A:E∗→E∗A:E∗→E∗. We introduce the class of generalized αα-monotone multifunctions T:K→2E∗T:K→2E∗ with respect to ΨΨ and AA where α:E×E→Rα:E×E→R. By using the KKM technique and the concept of the Hausdorff metric, we establish some existence results for generalized variational-like inequalities with generalized monotone multivalued mappings in EE.