On the rational difference equation \(y_{{n+1}}={\frac {\alpha_{{0}}y_{{n}}+\alpha_{{1}}y_{{n-p}}+\alpha_{{2}}y_{{n-q}} +\alpha_{{3}}y_{{n-r}}+\alpha_{{4}}y_{{n-s}}}{\beta_{{0}}y_{{n}}+\beta_{{1}}y_{{n-p} }+\beta_{{2}}y_{{n-q}}+\beta_{{3}}y_{{n-r}}+\beta_{{4}}y_{{n-s}}}}\)

Volume 11, Issue 1, pp 80--97 http://dx.doi.org/10.22436/jnsa.011.01.07
Publication Date: December 24, 2017 Submission Date: October 02, 2017 Revision Date: November 09, 2017 Accteptance Date: November 15, 2017

Authors

A. M. Alotaibi - School of mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Malaysia. M. A. El-Moneam - Mathematics Department, Faculty of Science, Jazan University, Kingdom of Saudi Arabia. M. S. M. Noorani - School of mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Malaysia.


Abstract

In this paper, we examine and explore the boundedness, periodicity, and global stability of the positive solutions of the rational difference equation \[ y_{{n+1} }={\frac {\alpha_{{0}}y_{{n}}+\alpha_{{1}}y_{{n-p}}+\alpha_{{2}}y_{{n-q}} +\alpha_{{3}}y_{{ n-r}}+\alpha_{{4}}y_{{n-s}}}{\beta_{{0}}y_{{n}}+\beta_{{1}}y_{{n-p} }+\beta_{{2}}y_{{n-q} }+\beta_{{3}}y_{{n-r}}+\beta_{{4}}y_{{n-s}}}}, \] where the coefficients \({\alpha_{i},\beta_{i}\in (0,\infty ),\ i=0,1,2,3,4},\) and \(p,q,r\), and \(s\) are positive integers. The initial conditions \(y_{-s},...,y_{-r},..., y_{-q},..., y_{{-p }},..., y_{-1},y_{0}\) are arbitrary positive real numbers such that \(p<q<r<s\). Some numerical examples will be given to illustrate our result.


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ISRP Style

A. M. Alotaibi, M. A. El-Moneam, M. S. M. Noorani, On the rational difference equation \(y_{{n+1}}={\frac {\alpha_{{0}}y_{{n}}+\alpha_{{1}}y_{{n-p}}+\alpha_{{2}}y_{{n-q}} +\alpha_{{3}}y_{{n-r}}+\alpha_{{4}}y_{{n-s}}}{\beta_{{0}}y_{{n}}+\beta_{{1}}y_{{n-p} }+\beta_{{2}}y_{{n-q}}+\beta_{{3}}y_{{n-r}}+\beta_{{4}}y_{{n-s}}}}\), Journal of Nonlinear Sciences and Applications, 11 (2018), no. 1, 80--97

AMA Style

Alotaibi A. M., El-Moneam M. A., Noorani M. S. M., On the rational difference equation \(y_{{n+1}}={\frac {\alpha_{{0}}y_{{n}}+\alpha_{{1}}y_{{n-p}}+\alpha_{{2}}y_{{n-q}} +\alpha_{{3}}y_{{n-r}}+\alpha_{{4}}y_{{n-s}}}{\beta_{{0}}y_{{n}}+\beta_{{1}}y_{{n-p} }+\beta_{{2}}y_{{n-q}}+\beta_{{3}}y_{{n-r}}+\beta_{{4}}y_{{n-s}}}}\). J. Nonlinear Sci. Appl. (2018); 11(1):80--97

Chicago/Turabian Style

Alotaibi, A. M., El-Moneam, M. A., Noorani, M. S. M.. "On the rational difference equation \(y_{{n+1}}={\frac {\alpha_{{0}}y_{{n}}+\alpha_{{1}}y_{{n-p}}+\alpha_{{2}}y_{{n-q}} +\alpha_{{3}}y_{{n-r}}+\alpha_{{4}}y_{{n-s}}}{\beta_{{0}}y_{{n}}+\beta_{{1}}y_{{n-p} }+\beta_{{2}}y_{{n-q}}+\beta_{{3}}y_{{n-r}}+\beta_{{4}}y_{{n-s}}}}\)." Journal of Nonlinear Sciences and Applications, 11, no. 1 (2018): 80--97


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