Existence of solutions of generalized variational inequalities in reflexive Banach spaces

In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set KK in a reflexive Banach space EE with the dual E∗E∗, a multifunction T:K→2E∗T:K→2E∗, and a vector b∈E∗b∈E∗, find x̄∈K such that there exists ū∈T(x̄) satisfying 〈ū−b,y−x̄〉≥0,for all y∈K. By using generalized projection and the well-known Fan–KKM Theorem, we prove existence results for solutions of GVIP. Our results extend some recent results from the literature.