A remark on variational inequalities in small balls

Abstract

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