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.
Existence and uniqueness of common solutions of strict Stampacchia and Minty variational inequalities with non-monotone operators in Banach spaces
FILIPPO CAMMAROTO;PAOLO CUBIOTTI
2023-01-01
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 $(X,\|\cdot\|)$ 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.Pubblicazioni consigliate
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