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
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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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ISRP Style
M. A. El-Moneam, E. S. Aly, M. A. Aiyashi, On the rational difference equation \(y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{{n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{\beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}}y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}}\), Journal of Nonlinear Sciences and Applications, 12 (2019), no. 2, 102--119
AMA Style
El-Moneam M. A., Aly E. S., Aiyashi M. A., On the rational difference equation \(y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{{n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{\beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}}y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}}\). J. Nonlinear Sci. Appl. (2019); 12(2):102--119
Chicago/Turabian Style
El-Moneam, M. A., Aly, E. S., Aiyashi, M. A.. "On the rational difference equation \(y_{{n+1}}={\frac{\alpha _{{0}}y_{{n}}+\alpha _{{1}}y_{{n-p}}+\alpha _{{2}}y_{{n-q}}+\alpha _{{3}}y_{{n-r}}+\alpha _{{4}}y_{{n-s}}+\alpha _{{5}}y_{{n-t}}}{\beta _{{0}}y_{{n}}+\beta _{{1}}y_{{n-p}}+\beta _{{2}}y_{{n-q}}+\beta _{{3}}y_{{n-r}}+\beta _{{4}}y_{{n-s}}+\beta _{{5}}y_{{n-t}}}}\)." Journal of Nonlinear Sciences and Applications, 12, no. 2 (2019): 102--119
Keywords
- Difference equation
- boundedness
- prime
- period two solution
- stability
MSC
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