A Solution Form of a Rational Difference Equation

Year 2023, Volume: 11 Issue: 1, 20 - 23, 30.04.2023

Ramazan Karataş

Abstract

This paper shows the solution form of the rational difference equation \begin{equation*} x_{n+1}=\frac{ax_{n-(2k+3)}}{-a\mp x_{n-(k+1)}x_{n-(2k+3)}}\text{ }% ,~n=0,1,... \end{equation*} where $k$ is a positive integer $a$ and initial conditions are non-zero real numbers with $x_{n-(k+1)}x_{n-(2k+3)}\neq \mp a$ for all $n\in N_{0}$.

Keywords

Rational difference equation, Solution, boundedness, periodicity

References

• [1] Gelis¸ken, A., On a system of rational difference equation, Journal of Computational Analysis and Applications, 23(4), 593-606, 2017.
• [2] Gelis¸ken, A., Karatas¸, R., On a solvable difference equation with sequence coefficients, Advances and Applications in Discrete Mathematics, 30, 27-33, 2022.
• [3] C¸ inar, G., Gelis¸ken, A., ”Ozkan, O., Well-defined solutions of the difference equation $x_{n}=\frac{x_{n-3k}x_{n-4k}x_{n-5k}}{x_{n-k}x_{n-2k}\left( \pm 1\pm x_{n-3k}x_{n-4k}x_{n-5k}\right) }$}, Asian-European Journal of Mathematics, 12(6), 2019.
• [4] Simsek, D., Abdullayev, F., On the recursive sequence $x_{n+1}=\frac{x_{n-(4k+3)}}{1+\prod\nolimits_{t=0}^{2}x_{n-(k+1)t-k}}$ , Journal of Mathematical Sciences, 6(222), 762-771, 2017.
• [5] Simsek, D., Abdullayev, F., On the recursive sequence $x_{n+1}=\frac{x_{n-(k+1)}}{1+x_{n}x_{n-1}...x_{n-k}}$ , Journal of Mathematical Sciences, 234(1), 73-81, 2018.
• [6] Elsayed, E. M., Alzahrani, F., Alayachi, H. S., Formulas and properties of some class of nonlinear difference equation, Journal of Computational Analysis and Applications, 24(8), 1517-1531, 2018.
• [7] Almatrafi, M. B., Elsayed, E. M., Alzahrani, F., Investigating some properties of a fourth order difference equation, Journal of Computational Analysis and Applications, 28(2), 243-253, 2020.
• [8] Ari, M., Gelis¸ken, A., Periodic and asymptotic behavior of a difference equation, Asian-European Journal of Mathematics, 12(6), 2040004, 10pp, 2019.
• [9] ”Ozkan, O., Kurbanlı, A. S., On a system of difference equations, Discrete Dynamics in Nature and Society, 2013.
• [10] Abo-Zeid, R., Behavior of solutions of higher order difference equation, Alabama Journal of Mathematics, 42, 1-10, 2018.
• [11] Karatas¸, R., Gelis¸ken, A., A solution form of a higher order difference equation, Korunalp Journal of Mathematics, 9(2), 316-323, 2021.
• [12] Karatas, R., Global behavior of a higher order difference equation, Computers and Mathematics with Applications, 60, 830-839, 2010.
• [13] Karatas, R., On the solutions of the recursive sequence $x_{n+1}=\frac{\alpha x_{n-(2k+1)}}{-a+x_{n-k}x_{n-(2k+1)}},$ ; Fasciculi Mathematici, 45, 37-45, 2010.
• [14] Ergin, S., Karatas, R., On the solutions of the recursive sequence $x_{n+1}=\frac{\alpha x_{n-k}}{a-\prod\nolimits_{i=0}^{k}x_{n-i}},$ ; Thai Journal of Mathematics, 14(2), 391-397, 2016.
• [15] Saary, T.I., Modern Nonlinear Equations, McGraw Hill, Newyork, 1967.
• [16] Kocic, V.L., Ladas, G., Global Behavior of Nonlinear Difference Equations of High Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.