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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