{
  "id": "1012.2762",
  "version": "v1",
  "published": "2010-12-13T15:32:47.000Z",
  "updated": "2010-12-13T15:32:47.000Z",
  "title": "Justification of the Dynamical Systems Method (DSM) for global homeomorphisms",
  "authors": [
    "A. G. Ramm"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "The Dynamical Systems Method (DSM) is justified for solving operator equations $F(u)=f$, where $F$ is a nonlinear operator in a Hilbert space $H$. It is assumed that $F$ is a global homeomorphism of $H$ onto $H$, that $F\\in C^1_{loc}$, that is, it has a continuous with respect to $u$ Fr\\'echet derivative $F'(u)$, that the operator $[F'(u)]^{-1}$ exists for all $u\\in H$ and is bounded, $||[F'(u)]^{-1}||\\leq m(u)$, where $m(u)>0$ is a constant, depending on $u$, and not necessarily uniformly bounded with respect to $u$. It is proved under these assumptions that the continuous analog of the Newton's method $\\dot{u}=-[F'(u)]^{-1}(F(u)-f), \\quad u(0)=u_0, \\quad (*)$ converges strongly to the solution of the equation $F(u)=f$ for any $f\\in H$ and any $u_0\\in H$. The global (and even local) existence of the solution to the Cauchy problem (*) was not established earlier without assuming that $F'(u)$ is Lipschitz-continuous. The case when $F$ is not a global homeomorphism but a monotone operator in $H$ is also considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-13T15:32:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47J35"
    ],
    "keywords": [
      "dynamical systems method",
      "global homeomorphism",
      "justification",
      "nonlinear operator",
      "hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.2762R"
    }
  }
}