On solutions of three-dimensional system of difference equations with constant coefficients

Year 2023, Volume: 72 Issue: 2, 462 - 481, 23.06.2023

Merve Kara Ömer Aktaş

https://doi.org/10.31801/cfsuasmas.1163955

Abstract

In this study, we show that the system of difference equations
\begin{align}
x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a+bx_{n-2}y_{n-3} \right) }, \nonumber \\
y_{n}=\frac{y_{n-2}z_{n-3}}{z_{n-1}\left(c+dy_{n-2}z_{n-3} \right) },~n\in\mathbb{N}_{0}, ~ \nonumber \\
z_{n}=\frac{z_{n-2}x_{n-3}}{x_{n-1}\left(e+fz_{n-2}x_{n-3} \right) }, \nonumber \\
\end{align}
where the initial values $x_{-i}, y_{-i}, z_{-i}$, $i=\overline{1,3}$ and the parameters $a$, $b$, $c$, $d$, $e$, $f$ are non-zero real numbers, can be solved in closed form. Moreover, we obtain the solutions of above system in explicit form according to the parameters $a$, $c$ and $e$ are equal $1$ or not equal $1$. In addition, we get periodic solutions of aforementioned system. Finally, we define the forbidden set of the initial conditions by using the acquired formulas.

Keywords

Explicit form, forbidden set, periodicity

Supporting Institution

Karamanoglu Mehmetbey University

Project Number

13-YL-22

Thanks

This paper was presented in 4th International Conference on Pure and Applied Mathematics (ICPAM - VAN 2022), Van-Turkey, June 22-23, 2022. This work is supported by the Scientific Research Project Fund of Karamanoglu Mehmetbey University under the project number 13-YL-22.

References

• Abo-Zeid, R., Kamal, H., Global behavior of two rational third order difference equations, Univers. J. Math. Appl., 2(4) (2019), 212-217. https://doi.org/10.32323/ujma.626465.
• Abo-Zeid, R., Behavior of solutions of a second order rational difference equation, Math. Morav., 23(1) (2019), 11-25. https://doi.org/10.5937/MatMor1901011A.
• Abo-Zeid, R., Global behavior and oscillation of a third order difference equation, Quaest. Math., 44(9) (2021), 1261-1280. https://doi.org/10.2989/16073606.2020.1787537.
• Ahmed, A. M., Elsayed, E. M., The expressions of solutions and the periodicity of some rational difference equations systems, J. Appl. Math. Inform., 34(1-2) (2016), 35-48. https://doi.org/10.14317/jami.2016.035.
• Cinar, C., Toufik, M., Yalcinkaya, I., On the difference equation of higher order, Util. Math., 92 (2013), 161–166.
• Cinar, C., On the positive solutions of the difference equation $x_{n+1}=\frac{x_{n-1}}{1+x_{n}x_{n-1}}$, Appl. Math. Comput., 150(1) (2004), 21-24. https://doi.org/10.1016/S0096-3003(03)00194-2.
• El-Metwally, H., Elsayed, E. M., Solution and behavior of a third rational difference equation, Util. Math., 88 (2012), 27-42.
• Elsayed, E. M., Ahmed, A. M., Dynamics of a three dimensional system of rational difference equations, Math. Methods Appl. Sci., 39(5) (2016), 1026–1038. https://doi.org/10.1002/mma.3540.
• Elsayed, E. M., Alotaibi, A., Almaylabi, H. A., On a solutions of fourth order rational systems of difference equations, J. Comput. Anal. Appl., 22(7) (2017), 1298-1308.
• Elsayed, E. M., On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath., 7(6) (2014), 1-26. https://doi.org/10.1142/S1793524514500673.
• Folly-Gbetoula, M., Nyirenda, D., A generalized two-dimensional system of higher order recursive sequences, J. Difference Equ. Appl., 26(2) (2020), 244-260. https://doi.org/10.1080/10236198.2020.1718667.
• Gelisken, A., Kara, M., Some general systems of rational difference equations, J. Difference Equ., 396757 (2015), 1-7. http://dx.doi.org/10.1155/2015/396757.
• Halim, Y., Touafek, N., Yazlik, Y., Dynamic behavior of a second-order nonlinear rational difference equation, Turk. J. Math., 39(6) (2015), 1004-1018. https://doi.org/10.3906/mat-1503-80.
• Halim, Y., Rabago, J. F. T., On the solutions of a second-order difference equation in terms of generalized Padovan sequences, Math. Slovaca., 68(3) (2018), 625–638. https://doi.org/10.1515/ms-2017-0130.
• Ibrahim, T. F., Touafek, N., On a third order rational difference equation with variable coefficients, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms., 20 (2013), 251-264.
• Kara, M., Yazlik, Y., On the system of difference equations $x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1}\left(a_{n}+b_{n}x_{n-2}y_{n-3} \right) },\ y_{n}=\frac{y_{n-2}x_{n-3}}{x_{n-1}\left(\alpha_{n}+\beta_{n}y_{n-2}x_{n-3} \right) }$, J. Math. Extension., 14(1) (2020), 41-59.
• Kara, M., Yazlik, Y., Solvability of a system of nonlinear difference equations of higher order, Turk. J. Math., 43(3) (2019), 1533-1565. https://doi.org/10.3906/mat-1902-24.
• Kara, M., Yazlik, Y., On a solvable system of non-linear difference equations with variable coefficients, J. Sci. Arts., 1(54) (2021), 145-162. https://doi.org/10.46939/J.Sci.Arts-21.1-a13.
• Kara, M., Yazlik, Y., Touafek, N., Akrour, Y., On a three-dimensional system of difference equations with variable coefficients, J. Appl. Math. Inform., 39(3-4) (2021), 381-403. https://doi.org/10.14317/jami.2021.381.
• Kara, M., Yazlik, Y., On the solutions of three-dimensional system of difference equations via recursive relations of order two and applications, J. Appl. Anal. Comput., 12(2) (2022), 736-753. https://doi.org/10.11948/20210305.
• Kara, M., Yazlik, Y., On a solvable system of rational difference equations of higher order, Turk. J. Math., 46 (2022), 587-611. https://doi.org/10.3906/mat-2106-1.
• Kara, M., Solvability of a three-dimensional system of non-liner difference equations, Math. Sci. Appl. E-Notes., 10(1) (2022), 1-15. https://doi.org/10.36753/mathenot.992987.
• Kara, M., Yazlik, Y., Solutions formulas for three-dimensional difference equations system with constant coefficients, Turk. J. Math. Comput. Sci., 14(1) (2022), 107-116. https://doi.org/10.47000/tjmcs.1060075.
• Elaydi, S., An Introduction to Difference Equations, Springer, New York, 1996.
• Taskara, N., Uslu, K., Tollu, D. T., The periodicity and solutions of the rational difference equation with periodic coefficients, Comput. Math. Appl., 62(4) (2011), 1807-1813. https://doi.org/10.1016/j.camwa.2011.06.024.
• Taskara, N., Tollu, D. T., Yazlik, Y., Solutions of rational difference system of order three in terms of Padovan numbers, J. Adv. Res. Appl. Math., 7(3) (2015), 18–29. https://doi.org/10.5373/jaram.2223.120914.
• Taskara, N., Tollu, D. T., Touafek, N., Yazlik, Y., A solvable system of difference equations, Comm. Korean Math. Soc., 35(1) (2020), 301-319. https://doi.org/10.4134/CKMS.c180472.
• Tollu, D. T., Yazlik, Y., Taskara, N., On fourteen solvable systems of difference equations, Appl. Math. Comput., 233 (2014), 310-319. https://doi.org/10.1016/j.amc.2014.02.001.
• Tollu, D. T., Yazlik, Y., Taskara, N., Behavior of positive solutions of a difference equation, J. Appl. Math. Inform., 35(3-4) (2017), 217–230. https://doi.org/10.14317/jami.2017.217.
• Tollu, D. T., Yazlik, Y., Taskara, N., On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Difference Equ., 174 (2013), 1-7. https://doi.org/10.1186/1687-1847-2013-174.
• Tollu, D. T., Yazlik, Y., Taskara, N., On a solvable nonlinear difference equation of higher order, Turk. J. Math., 42(4) (2018), 1765-1778. https://doi.org/10.3906/mat-1705-33.
• Tollu, D. T., Yalcinkaya, I., Global behavior of a three-dimensional system of difference equations of order three, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(1) (2019), 1-16. https://doi.org/10.31801/cfsuasmas.443530.
• Touafek, N., On a second order rational difference equation, Hacet. J. Math. Stat., 41 (2012), 867-874.
• Touafek, N., Elsayed, E. M., On a second order rational systems of difference equations, Hokkaido Math. J., 44(1) (2015), 29-45.
• Yalcinkaya, I., Cinar, C., Global asymptotic stability of a system of two nonlinear difference equations, Fasc. Math., 43 (2010), 171-180.
• Yalcinkaya, I., Hamza, A. E., Cinar, C., Global behavior of a recursive sequence, Selçuk J. Appl. Math., 14(1) (2013), 3-10.
• Yalcinkaya, I., Tollu, D. T., Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math., 26(4) (2016), 653-667.
• Yazlik, Y., Tollu, D. T., Taskara, N., Behaviour of solutions for a system of two higher-order difference equations, J. Sci. Arts., 4(45) (2018), 813-826.
• Yazlik, Y., Kara, M., On a solvable system of difference equations of fifthorder, Eskisehir Tech. Univ. J. Sci. Tech. B-Theoret. Sci., 7(1) (2019), 29-45. https://doi.org/10.20290/aubtdb.422910.
• Yazlik, Y., Gungor, M., On the solvable of nonlinear difference equation of sixth-order, J. Sci. Arts., 2(47) (2019), 399-414.
• Yazlik, Y., Kara, M., On a solvable system of difference equations of higher-order with period two coefficients, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2) (2019), 1675-1693. https://doi.org/10.31801/cfsuasmas.548262.