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

Yıl 2023, Cilt: 72 Sayı: 2, 462 - 481, 23.06.2023
https://doi.org/10.31801/cfsuasmas.1163955

Öz

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.

Destekleyen Kurum

Karamanoglu Mehmetbey University

Proje Numarası

13-YL-22

Teşekkür

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.

Kaynakça

  • 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.
Yıl 2023, Cilt: 72 Sayı: 2, 462 - 481, 23.06.2023
https://doi.org/10.31801/cfsuasmas.1163955

Öz

Proje Numarası

13-YL-22

Kaynakça

  • 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.
Toplam 41 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Research Article
Yazarlar

Merve Kara 0000-0001-8081-0254

Ömer Aktaş Bu kişi benim 0000-0002-5763-0308

Proje Numarası 13-YL-22
Yayımlanma Tarihi 23 Haziran 2023
Gönderilme Tarihi 18 Ağustos 2022
Kabul Tarihi 26 Ekim 2022
Yayımlandığı Sayı Yıl 2023 Cilt: 72 Sayı: 2

Kaynak Göster

APA Kara, M., & Aktaş, Ö. (2023). On solutions of three-dimensional system of difference equations with constant coefficients. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(2), 462-481. https://doi.org/10.31801/cfsuasmas.1163955
AMA Kara M, Aktaş Ö. On solutions of three-dimensional system of difference equations with constant coefficients. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Haziran 2023;72(2):462-481. doi:10.31801/cfsuasmas.1163955
Chicago Kara, Merve, ve Ömer Aktaş. “On Solutions of Three-Dimensional System of Difference Equations With Constant Coefficients”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, sy. 2 (Haziran 2023): 462-81. https://doi.org/10.31801/cfsuasmas.1163955.
EndNote Kara M, Aktaş Ö (01 Haziran 2023) On solutions of three-dimensional system of difference equations with constant coefficients. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 2 462–481.
IEEE M. Kara ve Ö. Aktaş, “On solutions of three-dimensional system of difference equations with constant coefficients”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 72, sy. 2, ss. 462–481, 2023, doi: 10.31801/cfsuasmas.1163955.
ISNAD Kara, Merve - Aktaş, Ömer. “On Solutions of Three-Dimensional System of Difference Equations With Constant Coefficients”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/2 (Haziran 2023), 462-481. https://doi.org/10.31801/cfsuasmas.1163955.
JAMA Kara M, Aktaş Ö. On solutions of three-dimensional system of difference equations with constant coefficients. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:462–481.
MLA Kara, Merve ve Ömer Aktaş. “On Solutions of Three-Dimensional System of Difference Equations With Constant Coefficients”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 72, sy. 2, 2023, ss. 462-81, doi:10.31801/cfsuasmas.1163955.
Vancouver Kara M, Aktaş Ö. On solutions of three-dimensional system of difference equations with constant coefficients. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(2):462-81.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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