Minimax Theory and its Applications 08 (2023), No. 2, 393--408
Copyright Heldermann Verlag 2023
Existence and Uniqueness of Common Solutions of Strict Stampacchia and Minty Variational Inequalities with Non-Monotone 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
[Abstract-pdf]
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.
Keywords: Saddle point, minimax theorem, Banach space, modulus of convexity, $C^1$ function, Stampacchia and Minty variational inequalities, ball, non-monotone operators.
MSC: 47J20, 49J35, 49J40.
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