
Minimax Theory and its Applications 08 (2023), No. 2, 393408 Copyright Heldermann Verlag 2023 Existence and Uniqueness of Common Solutions of Strict Stampacchia and Minty Variational Inequalities with NonMonotone Operators in Banach Spaces Filippo Cammaroto Dept. of Mathematical and Computer Sciences, University of Messina, Italy fdcammaroto@unime.it Paolo Cubiotti Dept. of Mathematical and Computer Sciences, University of Messina, Italy pcubiotti@unime.it [Abstractpdf] We study the existence of common solutions of the Stampacchia and Minty variational inequalities associated to nonmonotone operators in Banach spaces, as a consequence of a general saddlepoint 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. Keywords: Saddle point, minimax theorem, Banach space, modulus of convexity, $C^1$ function, Stampacchia and Minty variational inequalities, ball, nonmonotone operators. MSC: 47J20, 49J35, 49J40. [ Fulltextpdf (141 KB)] for subscribers only. 