Existence and Uniqueness of Common Solutions of Strict Stampacchia and Minty Variational Inequalities with Non-Monotone Operators in Banach Spaces

Abstract

 We study the existence of common solutions of the Stampacchia and Minty variational inequalities associated to non-monotone operators in Banach spaces, as a consequence of a general saddle-point theorem. We prove, in particular, that if (X,∥ · ∥) is a Banach space, whose norm has suitable convexity and differentiability properties, Bρ := {x ∈ X : ∥x∥ ≤ ρ}, and Φ : Bρ → X∗ is a C1 function with Lipschitzian derivative, with Φ(0)= 0, then for each r > 0 small enough, there exists a unique x∗ ∈ Br, with ∥x∥ = r, such that max{⟨Φ(x∗),x∗ −x⟩,⟨Φ(x),x∗ −x⟩} < 0 for all x ∈Br\{x∗}. Our results extend to the setting of Banach spaces some results previously obtained by B. Ricceri in the setting of Hilbert spaces.