Stability of the third order rational difference equation

Year 2020, Volume: 8 Issue: 1, 68 - 76, 30.06.2020

Mehmet Emre Erdoğan

Abstract

In this paper, we examine the global stability and boundedness of the difference equation
\[ x_{n+1}=\frac{\alpha x_{n}x_{n-1}+\beta x_{n}x_{n-2}}{\gamma {x}_{n-1}+\theta {x}_{n-2}}\]
where the initial conditions are non zero real numbers and are positive constants such that
\[\alpha+\beta\leq \gamma+\theta.\]
Also, we discuss and illustrate the stability of the solutions of the considered equation via MATLAB at the end of study to support our results.

Keywords

Asymptotic Stability , Difference Equation , Global Behavior

References

• X. Yang, On the global asymptotic stability of the difference equation , Applied Mathematics and Computation, 171(2), pp.857–861, 2005.
• M.R.S. Kulenović, G. Ladas, W.S. Sizer, On the recursive sequence . Mathematical Sciences Research Hot-Line, 2(5), pp.1–16, 1998.
• E.M. Elabbasy, H.A. El-Metwally, E.M. Elsayed, Global behavior of the solutions of some difference equations. Advances in Difference Equations, 2011(1), 28, 2011.
• A. Khaliq, E.M. Elsayed, Qualitative study of a higher order rational difference equation, Hacettepe Journal of Mathematics and Statistics, 47(5), 1128–1143, 2018.
• M.E. Erdogan, C. Cinar, I. Yalcinkaya, On the dynamics of the recursive sequence , Computers & Mathematics with Applications, 61(3), pp.533-537, 2011.
• M.E. Erdogan, C. Cinar, I. Yalcinkaya, On the dynamics of the recursive sequence , Mathematical and Computer Modelling, 54(5), pp.1481-1485, 2011.
• M.E. Erdogan, C. Cinar, On the dynamics of the recursive sequence , Fasciculi Mathematici, 50, pp.59-66, 2013.
• R. Abo-Zeid M.A. Al-Shabi, Global Behavior of a third order difference equation, Tamkang Journal of Mathematics, 43(3), pp.375-383, 2012.
• R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Chapman \& Hall/CRC Pure and Applied Mathematics, 2000.
• G.L. Karakostas, Convergence of a Difference Equation Via The Full Limiting Sequences Method, Differential Equations and Dynamical Systems, 1(4), pp.289–294, 1993.
• F. Belhannache, N. Touafek, R. Abo-Zeid, Dynamics of a third-order rational difference equation. Bulletin Mathematique de La Societe Des Sciences Mathematiques de Roumanie, 59(1), pp.13–22, 2016.
• A. E. Hamza, A. M. Ahmed, A. M. Youssef, On the recursive sequence . Arab Journal of Mathematical Sciences, 17(1), pp.31–44, 2011.
• A. E. Hamza, E. M. Elsayed, Stability problem of some nonlinear difference eqauations, International Journal of Mathematics and Mathematical Sciences, 21(2), pp.331–340, 1998.
• A. E. Grove, G. Ladas, M. Predescu, M. Radin, On the Gloval Character of the Difference Equation . Journal of Difference Equations and Applications, 9(2), pp.171–199, 2003.
• E. Camouzis, G. Ladas, Dynamics of Third - Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, 2007.