{
  "id": "1405.5183",
  "version": "v1",
  "published": "2014-05-20T18:40:26.000Z",
  "updated": "2014-05-20T18:40:26.000Z",
  "title": "On the stability of the existence of fixed points for the projection-iterative methods with relaxation",
  "authors": [
    "Andrzej Komisarski",
    "Adam Paszkiewicz"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We consider an $\\alpha$-relaxed projection $P_A^\\alpha:H\\to H$ given by $P_A^\\alpha(x)=\\alpha P_A(x)+(1-\\alpha)x$ where $\\alpha\\in[0,1]$ and $P_A$ is the projection onto a non-empty, convex and closed subset $A$ of the real Hilbert space $H$. We characterise all the sets $F\\subset[0,1]$ such that for some non-empty, convex and closed subsets $A_1,A_2,\\dots,A_k\\subset H$ the composition $P_{A_k}^\\alpha P_{A_{k-1}}^\\alpha\\dots P_{A_1}^\\alpha$ has a fixed point iff $\\alpha\\in F$. It proves, that if $\\dim H\\geq 3$ and $k\\geq3$ then the class of the derscribed above sets $F$ of coefficients $\\alpha$ is exactly the class of $F_\\sigma$ subsets of $[0,1]$ containing $0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-20T18:40:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47H10",
      "47H09",
      "46C05",
      "52A15",
      "90C25"
    ],
    "keywords": [
      "fixed point",
      "projection-iterative methods",
      "relaxation",
      "real hilbert space",
      "closed subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.5183K"
    }
  }
}