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 $(X,\|\cdot\|)$ is a Banach space, whose norm has suitable convexity and differentiability properties, $B_\rho:=\{x\in X: \|x\|\le\rho\}$, and $\Phi:B_\rho\to X^*$ is a $C^1$ function with Lipschitzian derivative, with $\Phi(0)\ne0$, then for each $r>0$ small enough, there exists a unique $x^*\in B_r$, with $\|x\|=r$, such that $\max\,\{\langle \Phi(x^*), x^*-x\rangle, \langle \Phi(x), x^*-x\rangle \}<0$ for all $x\in B_r\setminus\{x^*\}$. Our results extend to the setting of Banach spaces some results previously obtained by B. Ricceri in the setting of Hilbert spaces.