On the Recursive Sequence $x_{n+1}= \frac{x_{n-29}}{1+x_{n-4}x_{n-9}x_{n-14}x_{n-19}x_{n-24}}$

Year 2021, Volume: 4 Issue: 1, 46 - 54, 29.03.2021

Burak OĞUL Dağistan ŞİMŞEK

https://doi.org/10.33434/cams.814296

Cited By: 1

Abstract

In this paper, we are going to analyze the following difference equation $$x_{n+1}=\frac{x_{n-29}}{1+x_{n-4}x_{n-9}x_{n-14}x_{n-19}x_{n-24}} \quad n=0,1,2,...$$ where $x_{-29}, x_{-28}, x_{-27}, ..., x_{-2}, x_{-1}, x_{0} \in \left(0,\infty\right)$.

Keywords

difference equation, recursive sequence, period 30 solutions

References

• [1] A.M. Amleh, G.A. Grove, G. Ladas, D.A. Georgiou, On the recursive sequence $y_{n+1}=\alpha + \dfrac{y_{n-1}}{y_{n}}$ J. of Math. Anal. App., 233, (1999), 790-798.
• [2] C. Cinar, On the positive solutions of the difference equation $x_{n+1}=\dfrac{x_{n-1}}{-1+\alpha x_{n} x_{n-1}}$, J. of App. Math. Comp., 158(3), (2004), 793-797.
• [3] C. Cinar, T. Mansour, I. Yalcinkaya, On the difference equation of higher order, Utilitas Mathematica, 92, (2013), 161-166.
• [4] C.H. Gibbons, M.R.S. Kulenovic, G. Ladas, On the recursive sequence $\dfrac{\alpha+\beta x{n-1}}{\chi+\beta x{n-1}}$, Math. Sci. Res. Hot-Line, 4(2), (2000), 1-11.
• [5] D. Simsek, C. Cinar, I. Yalcinkaya, On the recursive sequence $x_{n+1}=\dfrac{x_{n-3}}{1+x_{n-1}}$, Int J. Contemp., 9(12), (2006), 475-480.
• [6] D. Simsek, C. Cinar, I. Yalcinkaya, On the recursive sequence $x_{n+1}=\dfrac{x_{n-5}}{1+x_{n-2}}$, Int J. Pure Appl. Math., 27, (2006), 501-507.
• [7] D. Simsek, C. Cinar, I. Yalcinkaya, On the recursive sequence $x_{n+1}=\dfrac{x_{n-5}}{1+x_{n-1}x_{n-3}}$, Int J. Pure Appl. Math., 28, (2006), 117-124.
• [8] D. Simsek, B. Ogul, C. Cinar, Solution of the rational difference equation $x_{n+1}=\dfrac{x_{n-17}}{1+x_{n-5}x_{n-11}}$, Filomat, 33(5), (2019), 1353-1359.
• [9] D. Simsek, B. Ogul, F. Abdullayev, Solution of the Rational Difference Equation $x_{n+1}=\dfrac{x_{n-13}}{1+x_{n-1}x_{n-3}x_{n-5}x_{n-7}x_{n-9}x_{n-11}}$, Applied Mathematics and Nonlinear Sciences, 5(1), (2020), 485-494.
• [10] E.M. Elsayed, On the solution of some difference equation, Europan Journal of Pure and Applied Mathematics, 4(3), (2011), 287-303.
• [11] E.M. Elsayed, On the Dynamics of a higher order rational recursive sequence, Communications in Mathematical Analysis, 12(1), (2012), 117-133.
• [12] E.M. Elsayed, Solution of rational difference system of order two, Mathematical and Computer Modelling, 5, (2012), 378-384.
• [13] E.M. Elsayed, Behavior and expression of the solutions of some rational difference equations, Journal of Computational Analysis Applications, 15(1), (2013), 73-81.
• [14] E.M. Elsayed, Solution of rational difference system of order two, Journal of Computational Analysis Applications, 33(3), (2014), 751-765.
• [15] E.M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society, 2011, (2011), 1-17.
• [16] E.M. Elabbasy, H. El-Metwally, E.M. Elsayed, On the difference equation $x_{n+1}=\dfrac{\alpha_{x} n_{k}}{\beta}+\gamma\pi=0$, Journal of Concrete Applicable Mathematics, 5(2), (2007),101-103.
• [17] E.M. Erdogan, C. Cinar, I. Yalcinkaya, On the dynamics of the recursive sequence, Mathematical and Computer Modelling, 54(5), (2011), 1481-1485.
• [18] F. Belhannache, N. Touafek, R. Abo-Zeid, On a higher-order rational difference equation J. Appl. Math. Informatics, 34(5-6), (2016), 369-382.
• [19] M. Lui, Z. Guo, Solvability of a higher-order nonlinear neutral delay difference equation, Advances in Difference Equations, 2010(1), (2010), 620-627.
• [20] M. Ari, A. Gelisken, Periodic and asymptotic behavior of a difference equation, Asian-European Journal of Mathematics, 12(6), (2019), 2040004.
• [21] M.R.S. Kulenovic, G. Ladas, W.S. Sizer, On the recursive sequence $\dfrac{\alpha x_{n}+\beta x_{n-1}}{\chi x_{n}+\beta x_{n-1}}$, Math. Sci. Res.Hot-Line, 2(5), (1998), 1-16.
• [22] O. Moaaz, D. Chalishajar, O. Bazighifan, Some qualitative behavior of solutions of general class of difference equations. Mathematics, 7(7), (2019), 585.
• [23] R. DeVault, G. Ladas, S.W. Schultz, On the recursive sequence$x_{n+1}=\dfrac{A}{x_{n}}+\dfrac{1}{x_{n-2}}$, Proc.Amer. Math. Soc., 126(11), (1998), 3257-3261.
• [24] R.P. Agarwal, E.M. Elsayed, On the solution of fourth-order rational recursive sequence, Advanced Studies in Contemporary Mathematics, 20(4),(2010), 525-545.
• [25] R. Karatas, C. Cinar, D. Simsek, On Positive Solutions of the Difference Equation $x_{n+1}=\dfrac{x_{n-5}}{1+x_{n-2}x_{n-5}}$, Int. J. Contemp. Math. Sci, 10(1), (2006), 495-500.
• [26] R. Karatas, Global behavior of a higher order difference equation, Int. J. Contemp. Math. Sci., 12(3), (2017), 133-138.
• [27] S.E. Das, M. Bayram, On a system of rational difference equations, World Applied Sciences Journal, 10(11), (2010), 1306-1312.
• [28] S. Stevic, B. Iricanin, Z. Smarda, On a product-type system of difference equations of second order solvable in closed form, Journal of Inequalities and Applications, 2015(1), (2012), 327-334.
• [29] T.F. Ibrahim, Behavior of some higher order nonlinear rational partial difference equations, Journal of the Egyptian Mathematical Society, 24(4), (2016), 532-537.
• [30] T.F. Ibrahim, Bifurcation and periodically semicycles for fractional difference equation of fifth order Journal of Nonlinear Sciences and Applications, 11(3), (2018), 375-382.