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Yıl 2024, Cilt: 6 Sayı: 2, 45 - 62, 18.12.2024

Durhasan Turgut Tollu , Merve Kayhan

https://doi.org/10.54286/ikjm.1562737

Öz

Kaynakça

  • Abu-Saris, R., Cinar, C. and Yalcinkaya, I., 2008, On the asymptotic stability of xn+1 = (xnxn−k + a)/(xn + xn−k), Computers and Mathematics with Applications, 56(5), 1172-1175.
  • Boulouh, M., Touafek, N., and Tollu, D. T., 2021, On the behavior of the solutions of an abstract system of difference equations. Journal of Applied Mathematics and Computing, 1-33. https://doi.org/10.1007/s12190- 021-01641-7
  • Camouzis E., Chatterjee, E., Ladas, G., 2007, On the dynamics of xn+1 = δxn−2 + xn−3/A + xn−3, Journal Mathematical Analysis And Applications, 331, 230-239.
  • Camouzis, E. and Ladas, G., 2008, Dynamics of third-order rational difference equations with open problems and conjectures, Volume 5 of Advances in Discrete Mathematics and Applications, Chapman and Hall/CRC, Boca Raton, FL
  • Clark D., Kulenovi´c M.R.S., 2002, A coupled system of rational difference equations, An International Journal Computers and Mathematics with Applications, 43, 849-867.
  • Das, S. E., Bayram, M., 2010, On a system of rational difference equations, World Applied Sciences Journal, 10(11), 1306-1312.
  • Dekkar, I., Touafek, N. and Yazlik, Y., 2017, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales- Serie A: Matematicas, 111, 325-347.
  • Din, Q., Ibrahim, T. F., Khan, K. A., 2014, Behavior of a competitive system of second-order difference equations, The Scientific World Journal, 2014, Article ID 283982, 9 pages. https://doi.org/10.1155/2014/283982
  • Din, Q. and Elsayed E. M., 2014, Stability analysis of a discrete ecological model, Computational Ecology and Software, 4(2), 89–103.
  • Din, Q., 2016, Asymptotic behavior of an anti-competitive system of second-order difference equations, Journal of the Egyptian Mathematical Society, 24, 37-43.
  • Elaydi, S., 1995, An Introduction to Difference Equations, Springer-Verlag, New York.
  • El-Metwally, H., 2013, Solutions form for some rational systems of difference equations, Discrete Dynamics in Nature and Society,, 2013, Article ID 903593, 10 pages.
  • Elsayed, E. M. and Ahmed, A. M., 2016, Dynamics of a three-dimensional systems of rational difference equations, Mathematical Methods in the Applied Sciences, 39, 1026-1038.
  • Elsayed, E. M. and Alghamdi, A., 2016, The form of the solutions of nonlinear difference equations systems, Journal of Nonlinear Sciences and Applications, 9(5), 3179-3196.
  • Elsayed, E. M., Alotaibi, A. and Almaylabi, A. H., 2017, On a solutions of fourth order rational systems of difference equations, Journal of Computational Analysis and Applications, 7(22), 1298-1308.
  • Gibbons, C. H., Kulenovi´c, M. and Ladas, G., 2000, On the recursive sequence yn+1 = α+βyn−1 γ+yn , Mathematical Sciences Research Hot-Line, 4(2), 1-11.
  • Gümüş, M., Abo-Zeid, R., 2020, Global behavior of a rational second order difference equation, Journal of Applied Mathematics and Computing, 62, 119–133. https://doi.org/10.1007/s12190-019-01276-9.
  • Gümüş, M., Abo-Zeid, R., 2020, An explicit formula and forbidden set for a higher order difference equation, Journal of Applied Mathematics and Computing, 63, 133–142. https://doi.org/10.1007/s12190-019-01311-9.
  • Haddad, N., Touafek, N. and Rabago, J. F. T., 2017, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Mathematical Methods in the Applied Sciences, 40(10), 3599-3607.
  • Halilm, Y., Touafek, N. and Yazlik, Y., 2015, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish Journal of Mathematics, 39, 1004-1018.
  • Kara, M., Yazlık, Y., Touafek, N., and Akrour, Y., 2021, 1. On a three-dimensional system of difference equations with variable coefficients, Journal of Applied Mathematics & Informatics, 39(3-4), 381–403.
  • Khan, A. Q. and Din, Q., Qureshi, M. N. and Ibrahim, T.F., 2014, Global behavior of an anti-competitive system of fourth-order rational difference equations, Computational Ecology and Software, 4(1), 35-46.
  • Kocic, V. L. and Ladas, G., 1993, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, London.
  • Kulenovi´c, M. R. S. and Ladas, G., 2002, Dynamics of second order rational difference equations, Chapman & Hall/CRC , Boca Raton, Fla, USA, 232s.
  • Kurbanli, A. S., Çınar, C. and Şimşek, D., 2011, On the periodicity of solutions of the system of rational difference equations, Applied Mathematics, 2, 410-413.
  • Kurbanli, A. S., Cinar C. and Yalcinkaya, I., 2011, On the behavior of positive solutions of the system of rational difference equations xn+1 = xn−1/(ynxn+1 + 1), yn+1 = yn−1/(xnyn+1 + 1), Mathematical and Computer Modelling, 53, 1261-1267.
  • Moaaz, O., Chalishajar, D. and Bazighifan, O., 2019, Some qualitative behavior of solutions of general class of difference equations, Mathematics, 7, Article 585, 12 pages.
  • Ozkan, O. and Kurbanli, A. S., 2013, On a system of difference equations, Discrete Dynamics in Nature and Society, 2013, Article ID 970316, 7 pages.
  • Papaschinopoulos, G., Radin, M. A. and Schinas, C. J., 2011, On the system of two difference equations of exponential form: xn+1 = a + βxn−1e−xn , Mathematical and Computer Modelling, 54(11-12), 2969–2977.
  • Papaschinopoulos, G. and Schinas, C. J., 2012, On the dynamics of two exponential type systems of difference equations, Computers and Mathematics with Applications, 64(7), 2326–2334.
  • Pituk, M., 2002, More on Poincare’s and Perron’s theorems for difference equations, Journal of Difference Equations and Applications, 8(3), 201–216.
  • Şahinkaya, A. F., Yalçınkaya, İ. and Tollu, D. T., 2020, A solvable system of nonlinear difference equations, Ikonion Journal of Mathematics, 2(1), 10-20.
  • Thai, T. H., and Khuong, V. V., 2016. Stability analysis of a system of second-order difference equations, Mathematical Methods in the Applied Sciences, 39(13), 3691-3700.
  • Tollu, D. T., Yazlik, Y. and Taskara, N., 2017, On global behavior of a system of nonlinear difference equations of order two, Advanced Studies in Contemporary Mathematics, 27(3), 373-383.
  • Tollu D. T. and Yalcinkaya, I. 2019, Global behavior of a three-dimensional system of difference equations of order three, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 68(1), 1-16.
  • Tollu, D. T. YalÇınkaya, İ., Ahmad H. and Yao, S. W., 2021, A detailed study on a solvable system related to the linear fractional difference equation, Mathematical Bioscienses and Engineering, 18(5), 5392–5408.
  • N. Touafek, D. T. Tollu and Y. Akrour, 2021, On a general homogeneous three-dimensional system of difference equations, Electronic Research Archive, 29(5), 2841-2876.
  • Yalcinkaya, I., Cinar, C. and Simsek D., 2008, Global asymptotic stability of a system of difference equations, Applicable Analysis, 87(6), 677-687.
  • Yalcinkaya, I. and Tollu, D. T., 2016, Global behavior of a second-order system of difference equations, Advanced Studies in Contemporary Mathematics, 26(4), 653-667.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., 2013, On the solutions of difference equation systems with padovan numbers,Applied Mathematics,, 4, 15-20.
  • Yazlik, Y., Elsayed, E. M., and Taskara, N., 2014, On the behaviour of the solutions of difference equation systems, Journal of Computational Analysis & Applications, 16(5), 932-941.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., 2015, On the behaviour of solutions for some systems of difference equations, Journal of Computational Analysis and Applications, 18(1), 166-178.
  • Yazlik, Y. and Kara, M., 2019, On a solvable system of difference equations of higher-order with period two coefficients, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 68, 1675-1693.
  • Yıldırım, A. and Tollu, D. T., 2022, Global behavior of a second order difference equation with two-period coefficient, Journal of Mathematical Extension, 16(4), 1-21.

Qualitative behavior of solutions of a two-dimensional rational system of difference equations

Yıl 2024, Cilt: 6 Sayı: 2, 45 - 62, 18.12.2024

Durhasan Turgut Tollu , Merve Kayhan

https://doi.org/10.54286/ikjm.1562737

Öz

In this study, the rational system
\begin{equation*}
x_{n+1}=\frac{\alpha _{1}+\beta _{1}y_{n-1}}{a_{1}+b_{1}y_{n}}, \quad y_{n+1}=\frac{\alpha _{2}+\beta_{2}x_{n-1}}{a_{2}+b_{2}x_{n}}, \quad n\in\mathbb{N}_{0},
\end{equation*}
where $\alpha_{i}$, $\beta_{i}$, $a_{i}$, $b_{i}$, $(i=1,2)$, and $x_{-j}$, $y_{-j}$, $(j=0,1)$, are positive real numbers, is defined and its qualitative behavior is discussed. The system in question is a two-dimensional extension of an old difference equation in the literature. The results obtained generalize the results in the literature on the equation in question.

Anahtar Kelimeler

Boundedness and persistence equilibrium point globally asymptotically stability periodicity rate of convergence system of difference equations.

Kaynakça

  • Abu-Saris, R., Cinar, C. and Yalcinkaya, I., 2008, On the asymptotic stability of xn+1 = (xnxn−k + a)/(xn + xn−k), Computers and Mathematics with Applications, 56(5), 1172-1175.
  • Boulouh, M., Touafek, N., and Tollu, D. T., 2021, On the behavior of the solutions of an abstract system of difference equations. Journal of Applied Mathematics and Computing, 1-33. https://doi.org/10.1007/s12190- 021-01641-7
  • Camouzis E., Chatterjee, E., Ladas, G., 2007, On the dynamics of xn+1 = δxn−2 + xn−3/A + xn−3, Journal Mathematical Analysis And Applications, 331, 230-239.
  • Camouzis, E. and Ladas, G., 2008, Dynamics of third-order rational difference equations with open problems and conjectures, Volume 5 of Advances in Discrete Mathematics and Applications, Chapman and Hall/CRC, Boca Raton, FL
  • Clark D., Kulenovi´c M.R.S., 2002, A coupled system of rational difference equations, An International Journal Computers and Mathematics with Applications, 43, 849-867.
  • Das, S. E., Bayram, M., 2010, On a system of rational difference equations, World Applied Sciences Journal, 10(11), 1306-1312.
  • Dekkar, I., Touafek, N. and Yazlik, Y., 2017, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales- Serie A: Matematicas, 111, 325-347.
  • Din, Q., Ibrahim, T. F., Khan, K. A., 2014, Behavior of a competitive system of second-order difference equations, The Scientific World Journal, 2014, Article ID 283982, 9 pages. https://doi.org/10.1155/2014/283982
  • Din, Q. and Elsayed E. M., 2014, Stability analysis of a discrete ecological model, Computational Ecology and Software, 4(2), 89–103.
  • Din, Q., 2016, Asymptotic behavior of an anti-competitive system of second-order difference equations, Journal of the Egyptian Mathematical Society, 24, 37-43.
  • Elaydi, S., 1995, An Introduction to Difference Equations, Springer-Verlag, New York.
  • El-Metwally, H., 2013, Solutions form for some rational systems of difference equations, Discrete Dynamics in Nature and Society,, 2013, Article ID 903593, 10 pages.
  • Elsayed, E. M. and Ahmed, A. M., 2016, Dynamics of a three-dimensional systems of rational difference equations, Mathematical Methods in the Applied Sciences, 39, 1026-1038.
  • Elsayed, E. M. and Alghamdi, A., 2016, The form of the solutions of nonlinear difference equations systems, Journal of Nonlinear Sciences and Applications, 9(5), 3179-3196.
  • Elsayed, E. M., Alotaibi, A. and Almaylabi, A. H., 2017, On a solutions of fourth order rational systems of difference equations, Journal of Computational Analysis and Applications, 7(22), 1298-1308.
  • Gibbons, C. H., Kulenovi´c, M. and Ladas, G., 2000, On the recursive sequence yn+1 = α+βyn−1 γ+yn , Mathematical Sciences Research Hot-Line, 4(2), 1-11.
  • Gümüş, M., Abo-Zeid, R., 2020, Global behavior of a rational second order difference equation, Journal of Applied Mathematics and Computing, 62, 119–133. https://doi.org/10.1007/s12190-019-01276-9.
  • Gümüş, M., Abo-Zeid, R., 2020, An explicit formula and forbidden set for a higher order difference equation, Journal of Applied Mathematics and Computing, 63, 133–142. https://doi.org/10.1007/s12190-019-01311-9.
  • Haddad, N., Touafek, N. and Rabago, J. F. T., 2017, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Mathematical Methods in the Applied Sciences, 40(10), 3599-3607.
  • Halilm, Y., Touafek, N. and Yazlik, Y., 2015, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish Journal of Mathematics, 39, 1004-1018.
  • Kara, M., Yazlık, Y., Touafek, N., and Akrour, Y., 2021, 1. On a three-dimensional system of difference equations with variable coefficients, Journal of Applied Mathematics & Informatics, 39(3-4), 381–403.
  • Khan, A. Q. and Din, Q., Qureshi, M. N. and Ibrahim, T.F., 2014, Global behavior of an anti-competitive system of fourth-order rational difference equations, Computational Ecology and Software, 4(1), 35-46.
  • Kocic, V. L. and Ladas, G., 1993, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, London.
  • Kulenovi´c, M. R. S. and Ladas, G., 2002, Dynamics of second order rational difference equations, Chapman & Hall/CRC , Boca Raton, Fla, USA, 232s.
  • Kurbanli, A. S., Çınar, C. and Şimşek, D., 2011, On the periodicity of solutions of the system of rational difference equations, Applied Mathematics, 2, 410-413.
  • Kurbanli, A. S., Cinar C. and Yalcinkaya, I., 2011, On the behavior of positive solutions of the system of rational difference equations xn+1 = xn−1/(ynxn+1 + 1), yn+1 = yn−1/(xnyn+1 + 1), Mathematical and Computer Modelling, 53, 1261-1267.
  • Moaaz, O., Chalishajar, D. and Bazighifan, O., 2019, Some qualitative behavior of solutions of general class of difference equations, Mathematics, 7, Article 585, 12 pages.
  • Ozkan, O. and Kurbanli, A. S., 2013, On a system of difference equations, Discrete Dynamics in Nature and Society, 2013, Article ID 970316, 7 pages.
  • Papaschinopoulos, G., Radin, M. A. and Schinas, C. J., 2011, On the system of two difference equations of exponential form: xn+1 = a + βxn−1e−xn , Mathematical and Computer Modelling, 54(11-12), 2969–2977.
  • Papaschinopoulos, G. and Schinas, C. J., 2012, On the dynamics of two exponential type systems of difference equations, Computers and Mathematics with Applications, 64(7), 2326–2334.
  • Pituk, M., 2002, More on Poincare’s and Perron’s theorems for difference equations, Journal of Difference Equations and Applications, 8(3), 201–216.
  • Şahinkaya, A. F., Yalçınkaya, İ. and Tollu, D. T., 2020, A solvable system of nonlinear difference equations, Ikonion Journal of Mathematics, 2(1), 10-20.
  • Thai, T. H., and Khuong, V. V., 2016. Stability analysis of a system of second-order difference equations, Mathematical Methods in the Applied Sciences, 39(13), 3691-3700.
  • Tollu, D. T., Yazlik, Y. and Taskara, N., 2017, On global behavior of a system of nonlinear difference equations of order two, Advanced Studies in Contemporary Mathematics, 27(3), 373-383.
  • Tollu D. T. and Yalcinkaya, I. 2019, Global behavior of a three-dimensional system of difference equations of order three, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 68(1), 1-16.
  • Tollu, D. T. YalÇınkaya, İ., Ahmad H. and Yao, S. W., 2021, A detailed study on a solvable system related to the linear fractional difference equation, Mathematical Bioscienses and Engineering, 18(5), 5392–5408.
  • N. Touafek, D. T. Tollu and Y. Akrour, 2021, On a general homogeneous three-dimensional system of difference equations, Electronic Research Archive, 29(5), 2841-2876.
  • Yalcinkaya, I., Cinar, C. and Simsek D., 2008, Global asymptotic stability of a system of difference equations, Applicable Analysis, 87(6), 677-687.
  • Yalcinkaya, I. and Tollu, D. T., 2016, Global behavior of a second-order system of difference equations, Advanced Studies in Contemporary Mathematics, 26(4), 653-667.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., 2013, On the solutions of difference equation systems with padovan numbers,Applied Mathematics,, 4, 15-20.
  • Yazlik, Y., Elsayed, E. M., and Taskara, N., 2014, On the behaviour of the solutions of difference equation systems, Journal of Computational Analysis & Applications, 16(5), 932-941.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., 2015, On the behaviour of solutions for some systems of difference equations, Journal of Computational Analysis and Applications, 18(1), 166-178.
  • Yazlik, Y. and Kara, M., 2019, On a solvable system of difference equations of higher-order with period two coefficients, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 68, 1675-1693.
  • Yıldırım, A. and Tollu, D. T., 2022, Global behavior of a second order difference equation with two-period coefficient, Journal of Mathematical Extension, 16(4), 1-21.
Toplam 44 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Durhasan Turgut Tollu 0000-0002-3313-8829

Merve Kayhan 0000-0001-9601-8384

Yayımlanma Tarihi 18 Aralık 2024
Gönderilme Tarihi 7 Ekim 2024
Kabul Tarihi 11 Aralık 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 6 Sayı: 2

Kaynak Göster

APA Tollu, D. T., & Kayhan, M. (2024). Qualitative behavior of solutions of a two-dimensional rational system of difference equations. Ikonion Journal of Mathematics, 6(2), 45-62. https://doi.org/10.54286/ikjm.1562737
AMA Tollu DT, Kayhan M. Qualitative behavior of solutions of a two-dimensional rational system of difference equations. ikjm. Aralık 2024;6(2):45-62. doi:10.54286/ikjm.1562737
Chicago Tollu, Durhasan Turgut, ve Merve Kayhan. “Qualitative Behavior of Solutions of a Two-Dimensional Rational System of Difference Equations”. Ikonion Journal of Mathematics 6, sy. 2 (Aralık 2024): 45-62. https://doi.org/10.54286/ikjm.1562737.
EndNote Tollu DT, Kayhan M (01 Aralık 2024) Qualitative behavior of solutions of a two-dimensional rational system of difference equations. Ikonion Journal of Mathematics 6 2 45–62.
IEEE D. T. Tollu ve M. Kayhan, “Qualitative behavior of solutions of a two-dimensional rational system of difference equations”, ikjm, c. 6, sy. 2, ss. 45–62, 2024, doi: 10.54286/ikjm.1562737.
ISNAD Tollu, Durhasan Turgut - Kayhan, Merve. “Qualitative Behavior of Solutions of a Two-Dimensional Rational System of Difference Equations”. Ikonion Journal of Mathematics 6/2 (Aralık 2024), 45-62. https://doi.org/10.54286/ikjm.1562737.
JAMA Tollu DT, Kayhan M. Qualitative behavior of solutions of a two-dimensional rational system of difference equations. ikjm. 2024;6:45–62.
MLA Tollu, Durhasan Turgut ve Merve Kayhan. “Qualitative Behavior of Solutions of a Two-Dimensional Rational System of Difference Equations”. Ikonion Journal of Mathematics, c. 6, sy. 2, 2024, ss. 45-62, doi:10.54286/ikjm.1562737.
Vancouver Tollu DT, Kayhan M. Qualitative behavior of solutions of a two-dimensional rational system of difference equations. ikjm. 2024;6(2):45-62.
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