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
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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
Keywords
- Difference equation
- boundedness
- prime period two solution
- global stability
MSC
References
-
[1]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation \(x_{n+1}=ax_{n}-bx_{n}/\left( cx_{n}-dx_{n-1}\right) \), Adv. Difference Equ., 2006 (2006), 10 pages.
-
[2]
E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation \(x_{n+1}=\frac{\alpha x_{n-l}+\beta x_{n-k}}{ Ax_{n-l}+Bx_{n-k}}\), Acta Math. Vietnam., 33 (2008), 85–94.
-
[3]
M. A. El-Moneam, S. O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21 (2014), 89–109.
-
[4]
E. M. Elsayed, On the global attractivity and the periodic character of a recursive sequence, Opuscula Math., 30 (2010), 431–446.
-
[5]
E. A. Grove, G. Ladas, Periodicities in nonlinear difference equations, Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, FL (2005)
-
[6]
W.-T. Li, H.-R. Sun, Dynamics of a rational difference equation, Appl. Math. Comput., 163 (2005), 577–591.
-
[7]
M. A. Obaid, E. M. Elsayed, M. M. El-Dessoky, Global attractivity and periodic character of difference equation of order four, Discrete Dyn. Nat. Soc., 2012 (2012 ), 20 pages.
-
[8]
M. Saleh, S. Abu-Baha, Dynamics of a higher order rational difference equation, Appl. Math. Comput., 181 (2006), 84–102.
-
[9]
E. M. E. Zayed, M. A. EL-Moneam, On the rational recursive sequence \(x_{n+1}=\frac{D+\alpha x_{n}+\beta x_{n-1}+\gamma x_{n-2}}{Ax_{n}+Bx_{n-1}+Cx_{n-2}}\), Comm. Appl. Nonlinear Anal., 12 (2005), 15–28.
-
[10]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=\frac{\alpha x_{n}+\beta x_{n-1}+\gamma x_{n-2}+\delta x_{n-3}}{Ax_{n}+Bx_{n-1}+Cx_{n-2}+Dx_{n-3}}\) , J. Appl. Math. Comput., 22 (2006), 247–262.
-
[11]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence\(x_{n+1}=\left( A+\sum_{i=0}^{k}\alpha _{i}x_{n-i}\right) /\left(B+\sum_{i=0}^{k}\beta _{i}x_{n-i}\right)\), Int. J. Math. Math. Sci., 2007 (2007), 12 pages.
-
[12]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=\left( A+\sum_{i=0}^{k}\alpha _{i}x_{n-i}\right)/\sum_{i=0}^{k}\beta _{i}x_{n-i}\), Math. Bohem., 133 (2008), 225–239.
-
[13]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=ax_{n}-bx_{n}/\left( cx_{n}-dx_{n-k}\right)\), Comm. Appl. Nonlinear Anal., 15 (2008), 47–57.
-
[14]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=Ax_{n}+\left( \beta x_{n}+\gamma x_{n-k}\right) /\left(Bx_{n}+Cx_{n-k}\right)\), Comm. Appl. Nonlinear Anal., 16 (2009), 91–106.
-
[15]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(\ x_{n+1}=\left( \alpha +\beta x_{n-k}\right) /\left( \gamma-x_{n}\right)\), J. Appl. Math. Comput., 31 (2009), 229–237.
-
[16]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=Ax_{n}+Bx_{n-k}+\frac{\beta x_{n}+\gamma x_{n-k}}{ Cx_{n}+Dx_{n-k}}\), Acta. Appl. Math., 111 (2010), 287–301.
-
[17]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=\frac{\alpha _{0}x_{n}+\alpha _{1}x_{n-l}+\alpha _{2}x_{n-k}}{ \beta _{0}x_{n}+\beta _{1}x_{n-l}+\beta _{2}x_{n-k}}\), Math. Bohem., 135 (2010), 319–336.
-
[18]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=\gamma x_{n-k}+\left( ax_{n}+bx_{n-k}\right) /\left( cx_{n}-dx_{n-k}\right)\), Bull. Iranian Math. Soc., 36 (2010), 103–115.
-
[19]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive two sequences \(x_{n+1}=ax_{n-k}+bx_{n-k}/\left(cx_{n}+\delta dx_{n-k}\right)\), Acta. Math. Vietnam., 35 (2010), 355–369.
-
[20]
E. M. E. Zayed, M. A. El-Moneam, On the global asymptotic stability for a rational recursive sequence, Iran. J. Sci. Technol. Trans. A Sci., 35 (2011), 333–339.
-
[21]
E. M. E. Zayed, M. A. El-Moneam, On the global attractivity of two nonlinear difference equations, Translated from Sovrem. Mat. Prilozh., 70 (2011), J. Math. Sci. (N.Y.), 177 (2011), 487–499.
-
[22]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=\frac{ A+\alpha _{0}x_{n}+\alpha _{1}x_{n-\sigma }}{ B+\beta _{0}x_{n}+\beta _{1}x_{n-\tau }}\), Acta Math. Vietnam., 36 (2011), 73–87.
-
[23]
E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence \(x_{n+1}=\frac{\alpha _{0}x_{n}+\alpha _{1}x_{n-l}+\alpha _{2}x_{n-m}+\alpha _{3}x_{n-k}}{\beta _{0}x_{n}+\beta _{1}x_{n-l}+\beta _{2}x_{n-m}+\beta _{3}x_{n-k}}\), WSEAS. Trans. Math., 11 (2012), 373–382.
-
[24]
E. M. E. Zayed, M. A. El-Moneam, On the qualitative study of the nonlinear difference equation \(x_{n+1}=\frac{\alpha x_{n-\sigma }}{\beta+\gamma x_{n-\tau }^{p}}\), Fasc. Math., 50 (2013), 137–147.
