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

TL;DR

This paper characterizes the existence of zeros for Lipschitzian derivative operators with closed range in Hilbert spaces, linking zero existence to convex sets and functions through approximation conditions.

Contribution

It provides a necessary and sufficient condition for the existence of zeros of such operators using convex analysis and approximation in Hilbert spaces.

Findings

Zero exists if and only if certain convex approximation conditions are met.

Characterizes zeros via convex sets and functions related to the operator.

Links the existence of solutions to geometric and convex properties of the operator.

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 $0\in \overline{conv}(\Phi(X))$.