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

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, parallel to.parallel to) is a Banach space, whose norm has suitable convexity and differentiability properties, B-rho := {x is an element of X : parallel to x parallel to <= rho}, and Phi : B-rho -> X* is a C-1 function with Lipschitzian derivative, with Phi(0) not equal 0, then for each r > 0 small enough, there exists a unique x* is an element of B-r, with parallel to x parallel to = r, such that max {<Phi(x*), x* - x >, <Phi(x), x* - x >} < 0 for all x is an element of B-r \ {x*}. Our results extend to the setting of Banach spaces some results previously obtained by B. Ricceri in the setting of Hilbert spaces.