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On the dynamical behaviors and periodicity of difference equation of order three

Yıl 2022, Cilt: 11 Sayı: 1, 48 - 61, 30.04.2022
https://doi.org/10.54187/jnrs.1037024

Öz

The major target of our research paper is to demonstrate the boundedness, stability and periodicity of the solutions of the following third- order difference equation
$$
w_{n+1} = \alpha w_{n} +\frac {\beta+ \gamma w_{n_-2} }{\delta+\zeta w_{n-2}} , \;\;\;\; n = 0,1,2,\dots
$$
where $w_{-2}$, $w_{-1}$, and $w_{0} $ are arbitrary real numbers and the values $\alpha$, $\beta$, $\gamma$, $\delta$, and $\zeta$ are defined as positive constants.

Kaynakça

  • H. S. Alayachi, M. S. M. Noorani, A. Q. Khan, M. B. Almatrafi, Analytic solutions and stability of sixth order difference equations, Mathematical Problems in Engineering, 2020, (2020) Article ID: 1230969, 1–12.
  • E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Global attractivity and periodic character of a fractional difference equation of order three, YokohamaMathematical Journal, 53(2), (2007) 89–100.
  • X. Yang, W. Su, B. Chen, G. M. Megson, D. J. Evans, On the recursive sequence $x_n= ax_{n-1} +bx_{n-2} c +dx_{n-1} x_{n-2}$, Applied Mathematics and Computation, 162(3), (2005) 1485–1497.
  • H. S. Alayachi, M. S. M. Noorani, E. M. Elsayed, Qualitative analysis of a fourth order difference equation, Journal of Applied Analysis & Computation, 10(4), (2020) 1343–1354.
  • M. B. Almatrafi, E. M. Elsayed, F. Alzahrani, Investigating some properties of a fourth order difference equation, Journal of Computational Analysis and Applications, 28(2), (2020) 243–253.
  • E. M. Elsayed, M. M. El-Dessoky, A. Asiri, Dynamics and behavior of a second order rational difference equation, Journal of Computational Analysis and Applications, 16(4), (2014) 794–807.
  • A. Alshareef, F. Alzahrani, A. Q. Khan, Dynamics and solutions’ expressions of a higher-order nonlinear fractional recursive sequence,Mathematical Problems in Engineering, 2021, (2021) Article ID: 1902473, 1–12.
  • E. M. Elsayed, F. Alzahrani, H. S. Alayachi, Global attractivity and the periodic nature of third order rational difference equation, Journal of Computational Analysis and Applications, 237, (2017) 1230– 1241.
  • M. Aloqeili, Dynamics of a rational difference equation, Applied Mathematics and Computation, 176(2), (2006) 768–774.
  • C. Çınar, On the positive solutions of the difference equation $x_{n+1}=\frac{ax_{n-1}}{1+bx_{n} x_{n-1}} $, Applied Mathematics and Computation, 156(2), (2004) 587–590.
  • M. M. El-Dessoky,M. A. El-Moneam, On the higher order difference equation, Journal of Computational Analysis and Applications, 25(2), (2018) 342.
  • E. M. Elsayed, A. Alotaibi, H. A. Almaylabi, The behavior and closed form of the solutions of some difference equations, Journal of Computational and Theoretical Nanoscience, 13(11), (2016) 8642–8651.
  • T. F. Ibrahim, Solvability and attractivity of the solutions of a rational difference equation, Journal of Pure and AppliedMathematics: Advances and Applications, 2, (2009) 227–237.
  • T. F. Ibrahim, A. Q. Khan, A. Ibrahim, Qualitative behavior of a nonlinear generalized recursive sequence with delay, Mathematical Problems in Engineering, 2021, (2021) Article ID: 6162320, 1–8.
  • R. Karataş, Global behavior of a higher order difference equation, International Journal of Contemporary Mathematical Sciences, 12(3), (2017) 133–138.
  • A. Khaliq, F. Alzahrani, E. M. Elsayed, Global attractivity of a rational difference equation of order ten, Journal of Nonlinear Sciences and Applications, 9(6), (2016) 4465–4477.
  • A. Q. Khan, M. S. M. Noorani, H. S. Alayachi, Global dynamics of higher-order exponential systems of difference equations, Discrete Dynamics in Nature and Society, 2019, (2019) Article ID: 3825927, 1–19.
  • Y. Kostrov, Z. Kudlak, On a second-order rational difference equation with a quadratic term, International Journal of Difference Equations, 11(2), (2016) 179–202.
  • K. Liu, P. Li, F. Han, W. Zhong, Global dynamics of nonlinear difference equation $x_{n+1}= A +\frac{x_{n}}{x_{n-1} x_{n-2} }$, Journal of Computational Analysis & Applications, 24(6), (2018) 1125–1132.
  • M. Saleh, M. Aloqeili, On the difference equation $y_{n+1}=A +\frac{y_n}{y_{n-k}}$ with $A < 0$, Applied Mathematics and Computation, 176(1), (2006) 359–363.
  • D. Şimşek, C. Çınar, R. Karataş, İ. Yalçınkaya, On the recursive sequence, International Journal of Pure and AppliedMathematics, 28(1), (2006) 117–124.
  • Y. H. Su, W. T. Li, Global asymptotic stability of a second-order nonlinear difference equation, Applied Mathematics and Computation, 168(2), (2005) 981–989.
  • D. T. Tollu, Y. Yazlık, N. Ta¸skara, Behavior of positive solutions of a difference equation, Journal of Applied Mathematics & Informatics, 35(3–4), (2017) 217–230.
  • E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence $x_{n+1}=Ax_{n}+Bx_{n-k}+ \frac{\beta x_{n}+\gamma x_{n-k}}{Cx_{n}+Dx_{n-k}}$, Acta Applicandae Mathematicae, 111(3), (2010) 287–301.
  • N. Touafek, D. T. Tollu, Y. Akrour, On a general homogeneous three-dimensional system of difference equations, Electronic Research Archive, 29(5), (2021) 2841–2876.
  • A. S. Kurbanlı, C. Çınar, ˙I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations,Mathematical and ComputerModelling, 53(5-6), (2011) 1261–1267.
  • İ. Yalçınkaya, D. T. Tollu, Global behavior of a second order system of difference equations, Advanced Studies in Contemporary Mathematics, 26(4), (2016) 653–667.
Yıl 2022, Cilt: 11 Sayı: 1, 48 - 61, 30.04.2022
https://doi.org/10.54187/jnrs.1037024

Öz

Kaynakça

  • H. S. Alayachi, M. S. M. Noorani, A. Q. Khan, M. B. Almatrafi, Analytic solutions and stability of sixth order difference equations, Mathematical Problems in Engineering, 2020, (2020) Article ID: 1230969, 1–12.
  • E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, Global attractivity and periodic character of a fractional difference equation of order three, YokohamaMathematical Journal, 53(2), (2007) 89–100.
  • X. Yang, W. Su, B. Chen, G. M. Megson, D. J. Evans, On the recursive sequence $x_n= ax_{n-1} +bx_{n-2} c +dx_{n-1} x_{n-2}$, Applied Mathematics and Computation, 162(3), (2005) 1485–1497.
  • H. S. Alayachi, M. S. M. Noorani, E. M. Elsayed, Qualitative analysis of a fourth order difference equation, Journal of Applied Analysis & Computation, 10(4), (2020) 1343–1354.
  • M. B. Almatrafi, E. M. Elsayed, F. Alzahrani, Investigating some properties of a fourth order difference equation, Journal of Computational Analysis and Applications, 28(2), (2020) 243–253.
  • E. M. Elsayed, M. M. El-Dessoky, A. Asiri, Dynamics and behavior of a second order rational difference equation, Journal of Computational Analysis and Applications, 16(4), (2014) 794–807.
  • A. Alshareef, F. Alzahrani, A. Q. Khan, Dynamics and solutions’ expressions of a higher-order nonlinear fractional recursive sequence,Mathematical Problems in Engineering, 2021, (2021) Article ID: 1902473, 1–12.
  • E. M. Elsayed, F. Alzahrani, H. S. Alayachi, Global attractivity and the periodic nature of third order rational difference equation, Journal of Computational Analysis and Applications, 237, (2017) 1230– 1241.
  • M. Aloqeili, Dynamics of a rational difference equation, Applied Mathematics and Computation, 176(2), (2006) 768–774.
  • C. Çınar, On the positive solutions of the difference equation $x_{n+1}=\frac{ax_{n-1}}{1+bx_{n} x_{n-1}} $, Applied Mathematics and Computation, 156(2), (2004) 587–590.
  • M. M. El-Dessoky,M. A. El-Moneam, On the higher order difference equation, Journal of Computational Analysis and Applications, 25(2), (2018) 342.
  • E. M. Elsayed, A. Alotaibi, H. A. Almaylabi, The behavior and closed form of the solutions of some difference equations, Journal of Computational and Theoretical Nanoscience, 13(11), (2016) 8642–8651.
  • T. F. Ibrahim, Solvability and attractivity of the solutions of a rational difference equation, Journal of Pure and AppliedMathematics: Advances and Applications, 2, (2009) 227–237.
  • T. F. Ibrahim, A. Q. Khan, A. Ibrahim, Qualitative behavior of a nonlinear generalized recursive sequence with delay, Mathematical Problems in Engineering, 2021, (2021) Article ID: 6162320, 1–8.
  • R. Karataş, Global behavior of a higher order difference equation, International Journal of Contemporary Mathematical Sciences, 12(3), (2017) 133–138.
  • A. Khaliq, F. Alzahrani, E. M. Elsayed, Global attractivity of a rational difference equation of order ten, Journal of Nonlinear Sciences and Applications, 9(6), (2016) 4465–4477.
  • A. Q. Khan, M. S. M. Noorani, H. S. Alayachi, Global dynamics of higher-order exponential systems of difference equations, Discrete Dynamics in Nature and Society, 2019, (2019) Article ID: 3825927, 1–19.
  • Y. Kostrov, Z. Kudlak, On a second-order rational difference equation with a quadratic term, International Journal of Difference Equations, 11(2), (2016) 179–202.
  • K. Liu, P. Li, F. Han, W. Zhong, Global dynamics of nonlinear difference equation $x_{n+1}= A +\frac{x_{n}}{x_{n-1} x_{n-2} }$, Journal of Computational Analysis & Applications, 24(6), (2018) 1125–1132.
  • M. Saleh, M. Aloqeili, On the difference equation $y_{n+1}=A +\frac{y_n}{y_{n-k}}$ with $A < 0$, Applied Mathematics and Computation, 176(1), (2006) 359–363.
  • D. Şimşek, C. Çınar, R. Karataş, İ. Yalçınkaya, On the recursive sequence, International Journal of Pure and AppliedMathematics, 28(1), (2006) 117–124.
  • Y. H. Su, W. T. Li, Global asymptotic stability of a second-order nonlinear difference equation, Applied Mathematics and Computation, 168(2), (2005) 981–989.
  • D. T. Tollu, Y. Yazlık, N. Ta¸skara, Behavior of positive solutions of a difference equation, Journal of Applied Mathematics & Informatics, 35(3–4), (2017) 217–230.
  • E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence $x_{n+1}=Ax_{n}+Bx_{n-k}+ \frac{\beta x_{n}+\gamma x_{n-k}}{Cx_{n}+Dx_{n-k}}$, Acta Applicandae Mathematicae, 111(3), (2010) 287–301.
  • N. Touafek, D. T. Tollu, Y. Akrour, On a general homogeneous three-dimensional system of difference equations, Electronic Research Archive, 29(5), (2021) 2841–2876.
  • A. S. Kurbanlı, C. Çınar, ˙I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations,Mathematical and ComputerModelling, 53(5-6), (2011) 1261–1267.
  • İ. Yalçınkaya, D. T. Tollu, Global behavior of a second order system of difference equations, Advanced Studies in Contemporary Mathematics, 26(4), (2016) 653–667.
Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Ibraheem Alsulami 0000-0002-2174-6915

Elsayed Elsayed 0000-0003-0894-8472

Erken Görünüm Tarihi 30 Nisan 2022
Yayımlanma Tarihi 30 Nisan 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 11 Sayı: 1

Kaynak Göster

APA Alsulami, I., & Elsayed, E. (2022). On the dynamical behaviors and periodicity of difference equation of order three. Journal of New Results in Science, 11(1), 48-61. https://doi.org/10.54187/jnrs.1037024
AMA Alsulami I, Elsayed E. On the dynamical behaviors and periodicity of difference equation of order three. JNRS. Nisan 2022;11(1):48-61. doi:10.54187/jnrs.1037024
Chicago Alsulami, Ibraheem, ve Elsayed Elsayed. “On the Dynamical Behaviors and Periodicity of Difference Equation of Order Three”. Journal of New Results in Science 11, sy. 1 (Nisan 2022): 48-61. https://doi.org/10.54187/jnrs.1037024.
EndNote Alsulami I, Elsayed E (01 Nisan 2022) On the dynamical behaviors and periodicity of difference equation of order three. Journal of New Results in Science 11 1 48–61.
IEEE I. Alsulami ve E. Elsayed, “On the dynamical behaviors and periodicity of difference equation of order three”, JNRS, c. 11, sy. 1, ss. 48–61, 2022, doi: 10.54187/jnrs.1037024.
ISNAD Alsulami, Ibraheem - Elsayed, Elsayed. “On the Dynamical Behaviors and Periodicity of Difference Equation of Order Three”. Journal of New Results in Science 11/1 (Nisan 2022), 48-61. https://doi.org/10.54187/jnrs.1037024.
JAMA Alsulami I, Elsayed E. On the dynamical behaviors and periodicity of difference equation of order three. JNRS. 2022;11:48–61.
MLA Alsulami, Ibraheem ve Elsayed Elsayed. “On the Dynamical Behaviors and Periodicity of Difference Equation of Order Three”. Journal of New Results in Science, c. 11, sy. 1, 2022, ss. 48-61, doi:10.54187/jnrs.1037024.
Vancouver Alsulami I, Elsayed E. On the dynamical behaviors and periodicity of difference equation of order three. JNRS. 2022;11(1):48-61.


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