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

Let $H$ be a real Hilbert space and $\mathrm{\Phi }:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that ${\mathrm{\Phi }}^{-1}(0)\mathrm{\ne }\mathrm{\varnothing }$ 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 ${\sup }_{x\in X}(\mathrm{\parallel }x{\mathrm{\parallel }}^2+\psi (x))-{\inf }_{x\in X}\mathrm{\parallel }x{\mathrm{\parallel }}^2+\psi (x))<\epsilon$ and $0\in \overline{conv}(\mathrm{\Phi }(X))$.