{
  "id": "0812.3385",
  "version": "v1",
  "published": "2008-12-17T20:09:08.000Z",
  "updated": "2008-12-17T20:09:08.000Z",
  "title": "Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations",
  "authors": [
    "Sukanya Basu",
    "Orlando Merino"
  ],
  "comment": "23 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "For nonnegative real numbers $\\alpha$, $\\beta$, $\\gamma$, $A$, $B$ and $C$ such that $B+C>0$ and $\\alpha+\\beta+\\gamma >0$, the difference equation \\begin{equation*} x_{n+1}=\\displaystyle\\frac{\\alpha +\\beta x_{n}+\\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \\quad n=0,1,2,... %, \\quad x_{-1},x_{0}\\in [0,\\infty) \\end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \\medskip \\noindent Theorem 1. {\\it For every choice of positive parameters $\\alpha$, $\\beta$, $\\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \\begin{equation*} x_{n+1}=\\displaystyle\\frac{\\alpha +\\beta x_{n}+\\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \\quad n=0,1,2,..., \\quad x_{-1},x_{0}\\in [0,\\infty) \\end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \\medskip \\noindent Theorem 2. {\\it For every choice of positive parameters $\\alpha$, $\\beta$, $\\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \\begin{equation*} x_{n+1}= \\displaystyle\\frac{\\alpha +\\beta x_{n}+\\gamma x_{n-1}}{B x_{n}+C x_{n-1}}, \\quad n=0,1,2,..., \\quad x_{-1},x_{0}\\in (0,\\infty) \\end{equation*} converge to the positive equilibrium or to a prime period-two solution.}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-17T20:09:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39A05",
      "39A11"
    ],
    "keywords": [
      "second order rational difference equations",
      "global behavior",
      "prime period-two solution",
      "positive equilibrium",
      "positive parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.3385B"
    }
  }
}