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 ${\Phi }^{-1}(0)\ne \varnothing$ if and only if, for each $\epsilon >0$, there exist a convex set $X\subset H$ and a convex function $\psi :X\to \mathbf{R}$ such that ${\sup }_{x\in X}(\Vert x{\Vert }^2+\psi (x))-{\inf }_{x\in X}\Vert x{\Vert }^2+\psi (x))<\epsilon$ and $0\in \overline{conv}(\Phi (X))$.

Accepted in Canad. Math. Bull