{
  "id": "2405.08000",
  "version": "v1",
  "published": "2024-05-06T05:08:16.000Z",
  "updated": "2024-05-06T05:08:16.000Z",
  "title": "A characterization of the existence of zeros for operators with Lipschitzian derivative and closed range",
  "authors": [
    "Biagio Ricceri"
  ],
  "categories": [
    "math.FA"
  ],
  "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 $0\\in \\overline{conv}(\\Phi(X))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-06T05:08:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "closed range",
      "lipschitzian derivative",
      "characterization",
      "real hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}