On Dynamics of A Higher-Order Rational Difference Equation

Year 2020, Volume: 8 Issue: 2, 223 - 228, 27.10.2020

Memiş Güler İbrahim Yalçınkaya Farida Belhannache

Abstract

In this paper, we study the dynamical behavior of the positive solutions of the difference equation $ y_{n+1}=\frac{A+By_{n}}{C+D\prod_{i=1}^{k}y_{n-i}^{q_{i}}},\ n\in \mathbb{N}_{0} $ where $\mathbb{N}_{0}=\mathbb{N} \cup \left\{ 0\right\} $, the initial conditions and the parameters $A,$ $B$ are non-negative real numbers, the parameters $C$, $D$\ are positive real numbers, $q_{i}$ for $i\in \{1,2,...k\}$ are fixed positive integers and $% 1\leq k$.                                                                                

Keywords

Difference equation, stability, global behavior

References

• [1] A. E. Hamza, A. M. Ahmed and A. M. Youssef, On the recursive sequence $x_{n+1}=(a+bx_{n})/(A+Bx_{n-1}^{k})$, Arab Journal of Mathematical Sciences., 17(2011), 31-44.
• [2] A. Khaliq and E. M. Elsayed, Qualitative study of a higher order rational difference equation, Hacettepe Journal of Mathematics and Statistics, 47(5)(2018), 1128-1143.
• [3] A. Q. Khan, Q. Din, M. N. Qureshi and T F. Ibrahim, Global behavior of an anti-competive system of fourth-order rational difference equations, Computational Ecology and Software, 4(1)(2014), 35-46.
• [4] D. T. Tollu, Y. Yazlik and N. Taskara, On a solvable nonlinear difference equation of higher order, Turkish Journal of Mathematics, 42(4)(2018), 1765-1778.
• [5] D. T. Tollu and I. Yalcinkaya, Global behavior of a three-dimensional system of difference equations of order three, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 681(2019), 1-16.
• [6] E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications, Volume 4, Chapman and Hall, CRS Press, 2005.
• [7] E. M. Elabbasy and S. M Elaissawy, Global behavior of a higher-order rational difference equation, Fasciculi Mathematici, 53(2014), 39-52.
• [8] E. M. Elsayed, On the dynamics of a higher-order rational recursive sequence, Communications in Mathematical Analysis, 12(2012), 117-133.
• [9] E. M. Elsayed and T. F. Ibrahim, Solutions and periodicity of a rational recursive sequences of order five, Bulletin of the Malaysian Mathematical Sciences Society, 38(2015), 95-112.
• [10] F. Belhannache, N. Touafek and R. Abo-Zeid, Dynamics of a third-order rational difference equation, Bulletin Mathematique de la Societe des Sciences Math´ematiques de Roumanie, 59(1)(2016), 13-22.
• [11] F. Belhannache, N. Touafek and R. Abo-Zeid, On a higher order rational difference equation, Journal of Applied Mathematics and Informatics, 5-6(34)(2016), 369-382.
• [12] I. Okumus and Y. Soykan, On the solutions of four rational difference equations associated to tribonacci numbers, Konuralp Journal of Mathematics, 8(1)(2020), 79-90.
• [13] M. E. Erdogan and C. Cinar, On the dynamics of the recursive sequence $x_{n+1}=\alpha x_{n-1}/(\beta +\gamma \Sigma _{k=1}^{t}x_{n-2k}^{p}\prod_{k=1}^{t}x_{n-2k}^{q})$, Fasciculi Mathematici, 50(2013), 59-66.
• [14] M. Gümüş¸ and Ö . Öcalan, Global asymptotic stability of a nonautonomous difference equation, Journal of Applied Mathematics, Article ID 395954, (2014), 5 pages.
• [15] M. Gümüş, R. Abo-Zeid and Ö . Öcalan, Dynamical behavior of a third-order difference equation with arbitrary powers, Kyungpook Mathematical Journal, 57(2017), 251-263.
• [16] M. Gümüş¸ and Y. Soykan, The dynamics of positive solutions of a higher order fractional difference equation with arbitrary powers, Journal of Applied Mathematics and Informatics, 35(2017), 267-276.
• [17] M. Gümüş and R. Abo-Zeid, On the solutions of a (2k+2)th order difference equation, Dynamics of Continuous Discrete and Impulsive Systems Series B: Applications & Algorithms, 25(2018), 129-143.
• [18] M. Shojaei, R. Saadati and H. Adbi, Stability and periodic character of a rational third order difference equation, Chaos Solitons and Fractals 39(2009), 1203-1209.
• [19] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall, CRC Press, 2001.
• [20] Q. Din, T. F. Ibrahim and K. A. Khan, Behavior of a competitive system of second-order difference equations, The Scientific World Journal, ID 283982, (2014), 9 pages.
• [21] R. Abo-Zeid, Global behavior of a higher order difference equation, Mathematica Slovaca, 64(4)(2014), 931-940.
• [22] S. Elaydi, An Introduction to Difference Equations, Springer, New York, 1999.
• [23] T. F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, Journal of Computational Analysis and Applications, 16(3)(2014), 552-564.
• [24] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
• [25] Y. Yazlik, D. T. Tollu and N. Taskara, On the behaviour of solutions for some systems of difference equations, Journal of Computational Analysis and Applications, 18(1)(2015), 166-178.
• [26] Y. Yazlik, D. T. Tollu and N. Taskara, Behaviour of solutions for a system of two higher-order difference equations, Journal of Science and Arts, 4(45)(2018), 813-826.