A characterization of the existence of zeros for operators with Lipschitzian derivative and closed range

Abstract

Let $H$ be a real Hilbert space and $\Phi:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $\Phi^{-1}(0)\neq \emptyset$ if and only if, for each $\epsilon>0$, there exist a convex set $X\subset H$ and a convex function $\psi:X\to {\bf R}$ such that $\sup_{x\in X}(\|x\|^2+\psi(x))-\inf_{x\in X}\|x\|^2+\psi(x))<\epsilon$ and $0\in \overline{conv}(\Phi(X))$.