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 K in a reflexive Banach space E with the dual E ∗ , a multifunction T : K → 2 E ∗ , and a vector b ∈ E ∗ , find x ̄ ∈ K such that there exists u ̄ ∈ T ( x ̄ ) satisfying 〈 u ̄ − b , y − x ̄ 〉 ≥ 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.