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Journal of Applied Mathematics and Computing
Published in: Journal of Applied Mathematics and Computing 1-2/2018

20-05-2017 | Original Research

Dynamics of nonlinear difference equation \(x_{n+1}=\frac{\beta x_{n}+\gamma x_{n-k}}{A+Bx_{n}+C x_{n-k}}\)

Authors: Amer Jafar, M. Saleh

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2018

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Abstract

The main goal of this paper is to investigate the boundedness, invariant intervals, semi-cycles and global attractivity of all nonnegative solutions of the equation
$$\begin{aligned} x_{n+1}=\frac{\beta x_{n}+\gamma x_{n-k}}{A+Bx_{n}+C x_{n-k}},\quad n\in \mathbb {N}_0 , \end{aligned}$$
where the parameters \(\beta , \gamma , A, B\) and C and the initial conditions \(x_{-k},x_{-k+1},\ldots ,x_0\) are non-negative real numbers, \(k=\{1,2,\ldots \}\). We give a detailed description of the semi-cycles of solutions, and determine conditions that satisfy the global asymptotic stability of the equilibrium points.
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Metadata
Title
Dynamics of nonlinear difference equation
Authors
Amer Jafar
M. Saleh
Publication date
20-05-2017
Publisher
Springer Berlin Heidelberg
Published in
Journal of Applied Mathematics and Computing / Issue 1-2/2018
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-017-1116-1

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