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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2405.08000 |
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| _version_ | 1866908341426454528 |
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| author | Ricceri, Biagio |
| author_facet | Ricceri, Biagio |
| contents | Let $H$ be a real Hilbert space and $Φ:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $Φ^{-1}(0)\neq \emptyset$ if and only if, for each $ε>0$, there exist a convex set $X\subset H$ and a convex function $ψ:X\to {\bf R}$ such that $\sup_{x\in X}(\|x\|^2+ψ(x))-\inf_{x\in X}\|x\|^2+ψ(x))<ε$ and $0\in \overline{conv}(Φ(X))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08000 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of the existence of zeros for operators with Lipschitzian derivative and closed range Ricceri, Biagio Functional Analysis Let $H$ be a real Hilbert space and $Φ:H\to H$ be a $C^1$ operator with Lipschitzian derivative and closed range. We prove that $Φ^{-1}(0)\neq \emptyset$ if and only if, for each $ε>0$, there exist a convex set $X\subset H$ and a convex function $ψ:X\to {\bf R}$ such that $\sup_{x\in X}(\|x\|^2+ψ(x))-\inf_{x\in X}\|x\|^2+ψ(x))<ε$ and $0\in \overline{conv}(Φ(X))$. |
| title | A characterization of the existence of zeros for operators with Lipschitzian derivative and closed range |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2405.08000 |