On generalized variational-like inequalities with generalized monotone multivalued mappings

Let E be a reflexive Banach space with the dual space E ∗ and K be a nonempty closed convex subset of E . Let us have Ψ : K × K × E ∗ → R and A : E ∗ → E ∗ . We introduce the class of generalized α -monotone multifunctions T : K → 2 E ∗ with respect to Ψ and A where α : 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 E .