{
  "id": "0901.4377",
  "version": "v1",
  "published": "2009-01-28T00:07:08.000Z",
  "updated": "2009-01-28T00:07:08.000Z",
  "title": "The Dynamical Systems Method for solving nonlinear equations with monotone operators",
  "authors": [
    "N. S. Hoang",
    "A. G. Ramm"
  ],
  "comment": "50pp",
  "categories": [
    "math.NA",
    "math.DS"
  ],
  "abstract": "A review of the authors's results is given. Several methods are discussed for solving nonlinear equations $F(u)=f$, where $F$ is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM) for solving the equation are formulated. These methods consist of a regularized Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation $F(u)=f$ is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to equation $F(u)=f$ is justified. New nonlinear differential inequalities are derived and applied to a study of large-time behavior of solutions to evolution equations. Discrete versions of these inequalities are established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-01-28T00:07:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47H05",
      "47J05",
      "47N20",
      "65J20",
      "65M30"
    ],
    "keywords": [
      "dynamical systems method",
      "solving nonlinear equations",
      "monotone operator",
      "posteriori choices",
      "nonlinear differential inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}