On the periodicity of the solution of a rational difference equation

Yıl 2019, Cilt: 68 Sayı: 2, 1427 - 1434, 01.08.2019

A. M. Alotaibi M. S. M. Noorani M. A. El-moneam

https://doi.org/10.31801/cfsuasmas.535865

Cited By: 1

Öz

In this paper, some cases on the periodicity of the rational difference equation

S_{n+1}=S_{n-p}(((aS_{n-q}+bS_{n-r}+cS_{n-s})/(dS_{n-q}+eS_{ n-r}+fS_{n-s}))),

are investigated, where a, b, c, d, e, f ∈(0,∞). The initial conditions S_{-p}, S_{-p+1},...,S_{-q},  S_{-q+1},...,S_{-r},  S_{-r+1},...,S_{-s},...,S_{-s+1},...,S₋₁ and S₀ are arbitrary positive real numbers such that p>q>r>s≥0. Some numerical examples are provided to illustrate the theoretical discussion.

Anahtar Kelimeler

Rational difference equations, periodicity, higher-order

Kaynakça

• Abdul, Khaliq, Faris, Alzahrani and Elsayed, E. M., Global attractivity of a rational difference equation of order ten, Journal of Nonlinear Sciences and Applications, 9 (2016), 4465--4477.
• Alotaibi, A. M., El-Moneam, M. A. and Noorani, M. S. M., On the rational difference equation y_{n+1}=((α₀y_{n}+α₁y_{n-p}+α₂y_{n-q}+α₃y_{ n-r}+α₄y_{n-s})/(β₀y_{n}+β₁y_{n-p}+β₂y_{n-q}+β₃y_{n-r}+β₄y_{n-s})), Journal of Nonlinear Sciences and Applications, 11 (2018), 80--97.
• Alotaibi, A. M., Noorani M. S. M. and El-Moneam, M. A., On the Asymptotic Behavior Of Some Nonlinear Difference Equations, Journal of Computational Analysis and Applications, 26 (2019), 604--627.
• El-Moneam, M. A. and Alamoudy, S. O., On study of the asymptotic behavior of some rational difference equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysi, 21 (2014), 89--109.
• El-Moneam, M. A., On the dynamics of the higher order nonlinear rational difference equation, Mathematical Sciences Letter, 3 (2014) 121--129.
• Elsayed, E. M., On the global attractivity and periodic character of a recursive sequence, Opuscula Mathematica, 30 (2010), 431--446.
• Grove, E. A. and Ladas, G., Periodicities in nonlinear difference equations, Vol.4, Chapman & Hall / CRC, 2005.
• Ibrahim, T. F., Periodicity and global attractivity of difference equation of higher order. Journal of Computational Analysis and Applications, 16 (2014).
• Ocalan, O., Global dynamics of a non-autonomous rational difference equation. Journal of Applied Mathematics &Informatics, 32, (2014) 843--848.
• Touafek, N. and Elsayed, E. M., On the periodicity of some systems of nonlinear difference equations, Bulletin mathematique de la Sociat des Sciences Mathematiques de Roumanie, 55, (2012), 217--224.