International Journal of Nonlinear Analysis and Applications

Dynamics of a system of higher order difference equations with a period-two coefficient

Document Type : Research Paper

Authors

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

Abstract

The aim of this paper is to study the dynamics of the system of two rational difference equations:
$$
 x_{n+1}=\alpha_{n}+\frac{y_{n-k}}{y_{n}},\quad y_{n+1}=\alpha_{n}+\frac{x_{n-k}}{x_{n}},\quad n=0, 1,\dots
 $$
where \(\left\{\alpha_n\right\}_{n\geq0}\) is a two periodic sequence of nonnegative real numbers and the initial conditions \(x_{i}, y_{i}\) are arbitrary positive numbers for \(i=-k, -k+1, -k+2,\dots, 0\) and $k\in\mathbb{N}$. We investigate the boundedness character of positive solutions. In addition, we establish some sufficient conditions under which the local asymptotic stability and the global asymptotic stability are assured. Furthermore, we determine the rate of the convergence of the solutions. Some numerical are considered in order to confirm our theoretical results.

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 2043-2058
  • Receive Date: 29 March 2022
  • Revise Date: 11 May 2022
  • Accept Date: 08 June 2022