Stability of the third order rational difference equation
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
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.