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

Alotaibi, A. M.; El-Moneam, M. A.; Noorani, M. S. M.

doi:10.22436/jnsa.011.01.07

Geodesic

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

Alotaibi, A. M. ¹ ; El-Moneam, M. A. ² ; Noorani, M. S. M. ¹

¹ School of mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Malaysia
² Mathematics Department, Faculty of Science, Jazan University, Kingdom of Saudi Arabia

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 1, p. 80-97.

Voir la notice de l'article provenant de la source International Scientific Research Publications

Résumé

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 $p$. Some numerical examples will be given to illustrate our result.

DOI : 10.22436/jnsa.011.01.07

Classification : 39A10
Keywords: Difference equation, boundedness, prime period two solution, global stability

Affiliations des auteurs :

Alotaibi, A. M. ¹ ; El-Moneam, M. A. ² ; Noorani, M. S. M. ¹

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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 its applications, Tome 11 (2018) no. 1, p. 80-97. doi : 10.22436/jnsa.011.01.07. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.01.07/

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