Global attractivity in a nonlinear difference equation
type: Item
instance of: scholarly article
title: Global attractivity in a nonlinear difference equation
zbMATH Open document ID: 0817.39005
DOI: 10.1016/0096-3003(94)90086-8
author: Ch. G. Philos
author: Y. G. Sficas
author: I. K. Purnaras
published in: Applied Mathematics and Computation
publication date: 1995-08-01T00:00:00Z
review text: Consider the nonlinear difference equation \[ \text{(E)} \qquad x_ n = a + \sum^ m_{k=1} {b_ k \over x_ n - k}\qquad(n = 0,1,2, \dots) \] where \(a, b_ 1,\dots,b_ m\) are nonnegative numbers with \(b = \sum^ m_{k=1} b_ k > 0\). The equation has a unique positive equilibrium point \(L = {a \over 2} + \sqrt {({a \over 2})^ 2 + B}\). Theorem: (i) Assume that \(a > 0\). Then \(L\) is a global attractor of all positive solutions of (E). (ii) Assume that \(a = 0\). Let \(\nu\), \(1 \leq \nu \leq m\), be an integer such that \(b_ \nu > 0\), and suppose that there exists a positive integer \(\mu\) with \(2 \mu \nu \leq m\) such that \(b_{2 \mu \nu} > 0\). Then \(L = \sqrt B\) is a global attractor of all positive solutions of (E).
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 637513
zbMATH Keywords: stability
zbMATH Keywords: nonlinear difference equation
zbMATH Keywords: positive equilibrium point
zbMATH Keywords: global attractor
zbMATH Keywords: positive solutions
reviewed by: Louis J. Grimm
MaRDI profile type: Publication
full work available at URL: 0096-3003(94)90086-8
OpenAlex ID: W1994842043
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description: scientific article; zbMATH DE number 637513