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.
📌 3 Maddede Bu Makale
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Amaç & Hipotez: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.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı: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.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Asymptotic Stability · Difference Equation · Global Behavior
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.
ERDOGAN, M. E. (2020). Stability of the third order rational difference equation. Manas Journal of Engineering, Cilt: 8 - Sayı: 1, 68–76. https://ajindex.com/articles/281452
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.