On globally asymptotic stability of a fourth-order rational difference equation

Volume 27, Issue 2, pp 176--183 http://dx.doi.org/10.22436/jmcs.027.02.07

Publication Date: April 13, 2022 Submission Date: October 23, 2021 Revision Date: November 08, 2021 Accteptance Date: December 08, 2021

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Authors

L. Sh. Aljoufi - Deanship of Common First Year, Jouf University, P.O. Box 2014, Sakaka, Jouf, Saudi Arabia. - Basic Sciences Research Unit, Jouf University, P.O. Box 2014, Sakaka, Jouf, Saudi Arabia. A. M. Ahmed - Department of Mathematics, Faculty of Science, Al Azhar University, Nasr City 11884, Cairo, Egypt. S. A. Mohammady - Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka, Jouf, Saudi Arabia. - Department of Mathematics, Faculty of Science, Helwan University, Helwan 11795, Egypt.

Abstract

In this paper, we investigate the behavior of solutions of the difference equation \[ x_{n+1}=\frac{\alpha \left( x_{n-2}+x_{n-3}\right) +\left( \alpha -1\right) x_{n-2}x_{n-3}}{x_{n-2}x_{n-3}+\alpha },\;\ \ \ n=0,1,2,\ldots, \] where the initial conditions \(x_{-3},x_{-2},x_{-1},x_{0}\) are arbitrary non-negative real numbers and the parameter \(\alpha \in \lbrack 1,\infty ).\) More precisely, we study the boundedness, stability, and oscillation of the solutions of this equation.

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Keywords

• Difference equations
• stability
• oscillation

MSC

•  39A10
•  39A21
•  39A30

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