Research Article
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Year 2019, Volume: 68 Issue: 1, 1 - 16, 01.02.2019
https://doi.org/10.31801/cfsuasmas.443530

Abstract

References

  • Din, Q., Qureshi M. N. and Khan, A. Q., Dynamics of a fourth-order system of rational difference equations, Advances in Difference Equations, (2012), 2012:215.
  • Din, Q. and Elsayed, E. M., Stability analysis of a discrete ecological model, Computational Ecology and Software, 4(2)(2014), 89-103.
  • Din, Q., Ibrahim, T. F. and Khan, A. Q., Behavior of a competitive system of second-order difference equations, The Scientific World Journal, vol. 2014, Article ID 283982, 9 pages.
  • Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M., Some properties and expressions of solutions for a class of nonlinear difference equation, Utilitas Mathematica, 87(2012), 93-110.
  • El-Metwally, H. and Elsayed, E. M., Solution and behavior of a third rational difference equation, Utilitas Mathematica, 88(2012), 27--42.
  • El-Metwally, H., Yalcinkaya, I. and Cinar, C., Global stability of an economic model, Utilitas Mathematica, 95(2014), 235-244.
  • El-Owaidy, H. M., Ahmed, A. M. and Youssef, A. M., The dynamics of the recursive sequence x_{n+1}=((αx_{n-1})/(β+γx_{n-2}^{p})), Applied Mathematics Letters, 18(9)(2005), 1013-1018.
  • Elaydi, S., An introduction to difference equations, third edition, undergraduate texts in mathematics, Springer, New York, 1999.
  • Elsayed, E. M., El-Dessoky, M. M. and Alotaibi, A., On the solutions of a general system of difference equations, Discrete Dynamics in Nature and Society, Article ID 892571, (2012).
  • Gumus, M. and Soykan, Y., Global character of a six dimensional nonlinear system of difference equations, Discrete Dynamics in Nature and Society, vol. 2016, Article ID 6842521, 7 pages.
  • Kocic, V. L., and Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic, Dordrecht, 1993.
  • Kulenović, M. R. S. and Nurkanović, Z., Global behavior of a three-dimensional linear fractional system of difference equations, Journal of Mathematical Analysis and Applications, 310(2)(2005), 673-689.
  • Kurbanli, A. S., Cinar, C. and Yalcinkaya, I., On the behavior of positive solutions of the system of rational difference equations x_{n+1}=x_{n-1}/(y_{n}x_{n-1}+1), y_{n+1}=y_{n-1}/(x_{n}y_{n-1}+1), Mathematical and Computer Modelling, 53(5-6)(2011), 1261-1267.
  • Kurbanli, A. S., On the behavior of solutions of the system of rational difference equations x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1), y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), z_{n+1}=1/(y_{n}z_{n}), Advances in Difference Equations, 2011: 40.
  • Kurbanli, A. S., On the behavior of solutions of the system of rational difference equations x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1), y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), z_{n+1}=z_{n-1}/(y_{n}z_{n-1}-1), Discrete Dynamics in Nature and Society, vol. 2011, Article ID 932362, 12 pages.
  • Ozkan, O. and Kurbanli, A. S., On a system of difference equations, Discrete Dynamics in Nature and Society, vol. 2013, Article ID 970316, 7 pages.
  • Papaschinopoulos, G., Ellina, G. and Papadopoulos, K. B., Asymptotic behavior of the positive solutions of an exponential type system of difference equations, Applied Mathematics and Computation, 245(2014), 181-190
  • Papaschinopoluos, G., Psarros, N. and Papadopoulos, K. B., On a cyclic system of m difference equations having exponential terms, Electronic Journal of Qualitative Theory of Differential Equations, 5(2015), 1-13.
  • Taskara, N., Uslu, K. and Tollu, D.T., The periodicity and solutions of the rational difference equation with periodic coefficients, Computers & Mathematics with Applications, 62(2011), 1807-1813.
  • Taskara, N., Tollu, D. T. and Yazlik, Y., Solutions of rational difference system of order three in terms of Padovan numbers, Journal of Advanced Research in Applied Mathematics, 7(3)(2015), 18-29.
  • Tollu, D. T., Yazlik Y. and Taskara, N., On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Advances in Difference Equations, 2013:174, (2013).
  • Tollu, D. T., Yazlik Y. and Taskara, N., On fourteen solvable systems of difference equations, Applied Mathematics and Computation, 233(2014), 310-319.
  • Tollu, D. T., Yazlik Y. and Taskara, N., The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan Journal Of Mathematics, 2(2014), 163-172.
  • Touafek, N. and Elsayed, E. M., On the periodicity of some systems of nonlinear difference equations, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 55(103), No: 2, (2012), 217-224.
  • Van Khuong, V. and Nam Phong, M., On a system of two difference equations of exponential form, International Journal of Difference Equations, 8(2)(2013), 215-223.
  • Yalcinkaya, I., Cinar, C. and Simsek, D., Global asymptotic stability of a system of difference equations, Applicable Analysis, 87(6)(2008), 689-699.
  • Yalcinkaya, I., On the global asymptotic stability of a second-order system of difference equations, Discrete Dynamics in Nature and Society, vol. 2008, Article ID 860152, 12 pages.
  • Yalcinkaya, I., Cinar, C. and Atalay, M., On the solutions of systems of difference equations, Advances in Difference Equations, 9(2008), Article ID 143943.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the solutions of difference equation systems with Padovan numbers, Applied Mathematics, 4(2013), 15-20.
  • Yazlik, Y., On the solutions and behavior of rational difference equations, Journal of Computational Analysis and Applications, 17(3)(2014), 584-594.
  • Yazlik, Y., Elsayed, E. M. and Taskara, N., On the behaviour of the solutions of difference equation systems, Journal of Computational Analysis and Applications, 16(5)(2014), 932-941.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the behaviour of solutions for some systems of difference equations, Journal of Computational Analysis & Applications, 18(1)(2015), 166-178.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the solutions of a max-type difference equation system, Mathematical Methods in the Applied Sciences, 38(17)(2015), 4388--4410.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the solutions of a three-dimensional system of difference equations, Kuwait Journal of Science, 43(1)( 2016), 95-111.

Global behavior of a three-dimensional system of difference equations of order three

Year 2019, Volume: 68 Issue: 1, 1 - 16, 01.02.2019
https://doi.org/10.31801/cfsuasmas.443530

Abstract

In this paper, we investigate the global behavior of the positive solutions of the system of difference equations

u_{n+1}=((α₁u_{n-1})/(β₁+γ₁v_{n-2}^{p})), v_{n+1}=((α₂v_{n-1})/(β₂+γ₂w_{n-2}^{q})), w_{n+1}=((α₃w_{n-1})/(β₃+γ₃u_{n-2}^{r}))

for n∈ℕ₀ where the initial conditions u_{-i},v_{-i},w_{-i} (i=0,1,2) are non-negative real numbers and the parameters  α_{j},β_{j},γ_{j} (j=1,2,3) and p,q,r are positive real numbers, by extending some results in the literature.

References

  • Din, Q., Qureshi M. N. and Khan, A. Q., Dynamics of a fourth-order system of rational difference equations, Advances in Difference Equations, (2012), 2012:215.
  • Din, Q. and Elsayed, E. M., Stability analysis of a discrete ecological model, Computational Ecology and Software, 4(2)(2014), 89-103.
  • Din, Q., Ibrahim, T. F. and Khan, A. Q., Behavior of a competitive system of second-order difference equations, The Scientific World Journal, vol. 2014, Article ID 283982, 9 pages.
  • Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M., Some properties and expressions of solutions for a class of nonlinear difference equation, Utilitas Mathematica, 87(2012), 93-110.
  • El-Metwally, H. and Elsayed, E. M., Solution and behavior of a third rational difference equation, Utilitas Mathematica, 88(2012), 27--42.
  • El-Metwally, H., Yalcinkaya, I. and Cinar, C., Global stability of an economic model, Utilitas Mathematica, 95(2014), 235-244.
  • El-Owaidy, H. M., Ahmed, A. M. and Youssef, A. M., The dynamics of the recursive sequence x_{n+1}=((αx_{n-1})/(β+γx_{n-2}^{p})), Applied Mathematics Letters, 18(9)(2005), 1013-1018.
  • Elaydi, S., An introduction to difference equations, third edition, undergraduate texts in mathematics, Springer, New York, 1999.
  • Elsayed, E. M., El-Dessoky, M. M. and Alotaibi, A., On the solutions of a general system of difference equations, Discrete Dynamics in Nature and Society, Article ID 892571, (2012).
  • Gumus, M. and Soykan, Y., Global character of a six dimensional nonlinear system of difference equations, Discrete Dynamics in Nature and Society, vol. 2016, Article ID 6842521, 7 pages.
  • Kocic, V. L., and Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic, Dordrecht, 1993.
  • Kulenović, M. R. S. and Nurkanović, Z., Global behavior of a three-dimensional linear fractional system of difference equations, Journal of Mathematical Analysis and Applications, 310(2)(2005), 673-689.
  • Kurbanli, A. S., Cinar, C. and Yalcinkaya, I., On the behavior of positive solutions of the system of rational difference equations x_{n+1}=x_{n-1}/(y_{n}x_{n-1}+1), y_{n+1}=y_{n-1}/(x_{n}y_{n-1}+1), Mathematical and Computer Modelling, 53(5-6)(2011), 1261-1267.
  • Kurbanli, A. S., On the behavior of solutions of the system of rational difference equations x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1), y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), z_{n+1}=1/(y_{n}z_{n}), Advances in Difference Equations, 2011: 40.
  • Kurbanli, A. S., On the behavior of solutions of the system of rational difference equations x_{n+1}=x_{n-1}/(y_{n}x_{n-1}-1), y_{n+1}=y_{n-1}/(x_{n}y_{n-1}-1), z_{n+1}=z_{n-1}/(y_{n}z_{n-1}-1), Discrete Dynamics in Nature and Society, vol. 2011, Article ID 932362, 12 pages.
  • Ozkan, O. and Kurbanli, A. S., On a system of difference equations, Discrete Dynamics in Nature and Society, vol. 2013, Article ID 970316, 7 pages.
  • Papaschinopoulos, G., Ellina, G. and Papadopoulos, K. B., Asymptotic behavior of the positive solutions of an exponential type system of difference equations, Applied Mathematics and Computation, 245(2014), 181-190
  • Papaschinopoluos, G., Psarros, N. and Papadopoulos, K. B., On a cyclic system of m difference equations having exponential terms, Electronic Journal of Qualitative Theory of Differential Equations, 5(2015), 1-13.
  • Taskara, N., Uslu, K. and Tollu, D.T., The periodicity and solutions of the rational difference equation with periodic coefficients, Computers & Mathematics with Applications, 62(2011), 1807-1813.
  • Taskara, N., Tollu, D. T. and Yazlik, Y., Solutions of rational difference system of order three in terms of Padovan numbers, Journal of Advanced Research in Applied Mathematics, 7(3)(2015), 18-29.
  • Tollu, D. T., Yazlik Y. and Taskara, N., On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Advances in Difference Equations, 2013:174, (2013).
  • Tollu, D. T., Yazlik Y. and Taskara, N., On fourteen solvable systems of difference equations, Applied Mathematics and Computation, 233(2014), 310-319.
  • Tollu, D. T., Yazlik Y. and Taskara, N., The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan Journal Of Mathematics, 2(2014), 163-172.
  • Touafek, N. and Elsayed, E. M., On the periodicity of some systems of nonlinear difference equations, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 55(103), No: 2, (2012), 217-224.
  • Van Khuong, V. and Nam Phong, M., On a system of two difference equations of exponential form, International Journal of Difference Equations, 8(2)(2013), 215-223.
  • Yalcinkaya, I., Cinar, C. and Simsek, D., Global asymptotic stability of a system of difference equations, Applicable Analysis, 87(6)(2008), 689-699.
  • Yalcinkaya, I., On the global asymptotic stability of a second-order system of difference equations, Discrete Dynamics in Nature and Society, vol. 2008, Article ID 860152, 12 pages.
  • Yalcinkaya, I., Cinar, C. and Atalay, M., On the solutions of systems of difference equations, Advances in Difference Equations, 9(2008), Article ID 143943.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the solutions of difference equation systems with Padovan numbers, Applied Mathematics, 4(2013), 15-20.
  • Yazlik, Y., On the solutions and behavior of rational difference equations, Journal of Computational Analysis and Applications, 17(3)(2014), 584-594.
  • Yazlik, Y., Elsayed, E. M. and Taskara, N., On the behaviour of the solutions of difference equation systems, Journal of Computational Analysis and Applications, 16(5)(2014), 932-941.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the behaviour of solutions for some systems of difference equations, Journal of Computational Analysis & Applications, 18(1)(2015), 166-178.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the solutions of a max-type difference equation system, Mathematical Methods in the Applied Sciences, 38(17)(2015), 4388--4410.
  • Yazlik, Y., Tollu, D. T. and Taskara, N., On the solutions of a three-dimensional system of difference equations, Kuwait Journal of Science, 43(1)( 2016), 95-111.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Review Articles
Authors

Durhasan Turgut Tollu 0000-0002-3313-8829

İbrahim Yalçınkaya 0000-0003-4546-4493

Publication Date February 1, 2019
Submission Date January 1, 2017
Acceptance Date November 27, 2017
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Tollu, D. T., & Yalçınkaya, İ. (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. https://doi.org/10.31801/cfsuasmas.443530
AMA Tollu DT, Yalçınkaya İ. Global behavior of a three-dimensional system of difference equations of order three. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):1-16. doi:10.31801/cfsuasmas.443530
Chicago Tollu, Durhasan Turgut, and İbrahim Yalçınkaya. “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, no. 1 (February 2019): 1-16. https://doi.org/10.31801/cfsuasmas.443530.
EndNote Tollu DT, Yalçınkaya İ (February 1, 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.
IEEE D. T. Tollu and İ. Yalçınkaya, “Global behavior of a three-dimensional system of difference equations of order three”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 1–16, 2019, doi: 10.31801/cfsuasmas.443530.
ISNAD Tollu, Durhasan Turgut - Yalçınkaya, İbrahim. “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 (February 2019), 1-16. https://doi.org/10.31801/cfsuasmas.443530.
JAMA Tollu DT, Yalçınkaya İ. Global behavior of a three-dimensional system of difference equations of order three. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1–16.
MLA Tollu, Durhasan Turgut and İbrahim Yalçınkaya. “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, vol. 68, no. 1, 2019, pp. 1-16, doi:10.31801/cfsuasmas.443530.
Vancouver Tollu DT, Yalçınkaya İ. Global behavior of a three-dimensional system of difference equations of order three. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):1-16.

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