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

Yıl 2022, Cilt: 11 Sayı: 1, 48 - 61, 30.04.2022

Ibraheem Alsulami , Elsayed Elsayed

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.

Anahtar Kelimeler

Periodicity, Stability, Boundedness, Difference equations

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.