A remark on variational inequalities in small balls

TL;DR

This paper establishes the existence and uniqueness of a special point on small balls in a Hilbert space where a variational inequality involving a $C^{1,1}$ function holds, under certain semicontinuity conditions.

Contribution

It proves a new result on variational inequalities in small balls in Hilbert spaces, extending previous understanding with specific semicontinuity and smoothness assumptions.

Findings

Existence of a unique point satisfying the inequality for small radii.

The point maximizes a certain variational inequality condition.

The result applies to functions with weakly lower semicontinuous properties.

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 $\|y\|\leq r$.