On the rational difference equation \(y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{{n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{\beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}}y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}}\)

Volume 12, Issue 2, pp 102--119 http://dx.doi.org/10.22436/jnsa.012.02.04
Publication Date: October 18, 2018 Submission Date: July 21, 2018 Revision Date: September 12, 2018 Accteptance Date: September 19, 2018

Authors

M. A. El-Moneam - Mathematics Department, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia. E. S. Aly - Mathematics Department, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia. M. A. Aiyashi - Mathematics Department, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia.


Abstract

In this paper, we examine and explore the boundedness, periodicity, and global stability of the positive solutions of the rational difference equation \[ y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{% {n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{% \beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}% }y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}}, \] where the coefficients \({ \alpha _{i},\beta _{i}\in (0,\infty ),\ i=0,1,2,3,4,5},\) and \(p,q,r,s,\) and \(t\) are positive integers. The initial conditions \(y_{-t} ,\) \(\ldots, y_{-s} ,\ldots, y_{-r} ,\ldots, y_{-q} ,\ldots, y_{{% -p}} ,\ldots, y_{-1} , y_{0}\) are arbitrary positive real numbers such that \(% p<q<r<s<t\). Some numerical examples will be given to illustrate our result.


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