Solution of a Solvable System of Difference Equation

Year 2022, Volume: 4 Issue: 1, 1 - 8, 23.09.2022

Ali Gelişken , Murat Arı

https://doi.org/10.54286/ikjm.1050493

Abstract

In this study we give solutions for the following difference equation sytem
x_{n+1}= (a.x_{n}y_{n-3}/y_{n-2}-\alpha)+\beta y_{n+1}=(b.x_{n-3}y_{n}/x_{n-2}-\beta) +\alpha n ∈N0
where the parameters a,b,, and initial values x_{-i}, y_{-i}, i=0,1,2,3 are non-zero real numbers. We show the asymptotic behavior of the system of equation.

Keywords

Difference equations, Periodicity, System of difference equations, Asymptotic behavior

References

• Elabbasy, E.M., El-Metwally, H., Elsayed, E.M. (2007) Qualitative behavior of higher order difference equation. Soochow J. Math. 33, 861–873. Elaydi, S. (1999) An Introduction to Difference Equations. Springer, New York.
• Papaschinopoulos, G., Fotiades, N., Schinas, C.J. (2014) On a system of difference equations including negative exponential terms. J. Differ. Equ. Appl. 20, 717–732.
• Haddad, N., Touafek, N. and Rabago, JFT. (2018) “Well-defined solutions of a system of difference equations”, Journal of Appl. Math. and Comput., 56(1-2), 439-458.
• Kara, M. , Touafek, N. , Yazlik, Y. (2020) Well-Defined Solutions of a Three-Dimensional System of Difference Equations. Gazi University Journal of Science 33, 767-778.
• Stevi´c, S. (2012) On a solvable rational system of difference equations.Appl.Math. Comput. 219, 2896–2908.
• Stevi´c, S. (2013) On a solvable system of difference equations of kth order. Appl. Math. Comput. 219 , 7765–7771.
• Stevi´c, S. (2011) On a system of difference equations. Appl. Math. Comput. 218 , 3372–3378.
• Stevi´c, S. (2011) On a system of difference equations with period two coefficients. Appl.Math. Comput. 218, 4317–4324.
• Yazlik, Y., Elsayed, E.M., Taskara, N. (2014) On the behaviour of the solutions the solutions of difference equation system. J. Comput. Anal. Appl. 16(5), 932–941.
• Tollu DT, Yalçınkaya İ, Ahmad H, Yao SW. (2021) A detailed study on a solvable system related to the linear fractional difference equation. Math Biosci Eng. Jun 17;18(5):5392-5408
• Şahinkaya, A. , Yalçınkaya, İ. & Tollu, D. T. (2020) A solvable system of nonlinear difference equations . Ikonion Journal of Mathematics , 2 (1) , 10-20 .
• Simsek, D., Ogul,B. and Abdullayev,F. (2020) Solution of the Rational Difference Equation x_{n+1}= 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) 485-494.
• Simsek, D., Ogul,B and Abdullayev,F. (2017) Solutions of the rational difference equations x_{n+1}= x_{n-11} /(1+ x_{n-2} x_{n-5} x_{n-8}) , AIP Conference Proceedings 1880, 040003 Karatas, R., & Gelisken, A. (2011) Qualitative behavior of a rational difference equation. Ars Combinatoria, 100, 321-326.
• Karatas, R. (2017) Global behavior of a higher order difference equation International Journal of Contemporary Mathematical Sciences, Vol. 12, no. 3, 133-138
• Kurbanli, A. S. (2011). On the behavior of solutions of the system of rational difference equations. Advances in Difference Equations. 10.1186/1687-1847-2011-40.
• Kurbanli, A. S., Çinar, C. & Şimşek, D. (2011) On the Periodicity of Solutions of the System of Rational Difference Equations. Applied Mathematics. 02. 410-413. 10.4236/am.2011.24050.