## Biagio Ricceri, A remark on variational inequalities in small balls

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DOI: 10.23952/jnva.4.2020.1.03
Volume 4, Issue 1, 1 April 2020, Pages 21-26

Abstract. In this paper, we prove the following result. Let $(H,\langle\cdot,\cdot\rangle)$ be a real Hilbert space, $B$ a ball in $H$ centered at $0$ and $\Phi:B\to H$ a $C^{1,1}$ function with $\Phi(0)\neq 0$ such that the function $x\to \langle \Phi(x),x-y\rangle$ is weakly lower semicontinuous in $B$ for all $y\in B$. Then, for each $r>0$ small enough, there exists a unique point $x^*\in H$ with $\|x^*\|=r$ such that $\max\{\langle \Phi(x^*),x^*-y\rangle, \langle \Phi(y),x^*-y\rangle\}<0$ for all $y\in H\setminus \{x^*\}$ with $\|y\|\leq r$.