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

Publication Date: 2017-12-24

http://dx.doi.org/10.22436/jnsa.011.01.07

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.

Keywords

Difference equation, boundedness, prime period two solution, global stability

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