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

Ricceri, Biagio

Canadian mathematical bulletin, Tome 68 (2025) no. 2, pp. 395-400

Résumé

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 $\psi :X\to \mathbf {R}$ such that $\sup _{x\in X}(\|x\|^2+\psi (x))-\inf _{x\in X}(\|x\|^2+\psi (x))<\epsilon $ and $0\in \overline {{\mathrm {conv}}}(\Phi (X))$.

Détail

DOI : 10.4153/S0008439524000821

Mots-clés : Nonlinear operator, Lipschitzian derivative, minimax theorem