International Journal of Nonlinear Analysis and Applications

On the dynamics of a nonautonomous rational difference equation

Document Type : Research Paper

Authors

Laboratory of Applied Mathematics‎, ‎Badji Mokhtar-Annaba University‎, ‎P.O‎. ‎Box 12‎, ‎Annaba‎, ‎23000‎, ‎Algeria‎

Abstract

‎In this paper‎, ‎we study the following nonautonomous rational difference equation‎
‎\[‎
‎y_{n+1}=\frac{\alpha_n+y_n}{\alpha_n+y_{n-k}},\quad n=0,1,...‎,
‎\]‎
‎where $\left\{\alpha_n\right\}_{n\geq0}$ is a bounded sequence of positive numbers‎, ‎$k$ is a positive integer and the initial values $y_{-k},...,y_0$ are positive real numbers‎. ‎We give sufficient conditions under which the unique equilibrium $\bar{y}=1$ is globally asymptotically stable‎. ‎Furthermore‎, ‎we establish an oscillation result for positive solutions about the equilibrium point‎. ‎Our work generalizes and improves earlier results in the literature‎.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 203-213
  • Receive Date: 07 August 2020
  • Revise Date: 21 December 2020
  • Accept Date: 25 December 2020