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A characterization of the existence of zeros for operators with Lipschitzian derivative and closed range
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 08 January 2025, pp. 1-6
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- Article
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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 
$0\in \overline {{\mathrm {conv}}}(\Phi (X))$.
