Geodesic
On the rational recursive sequence $ x_{n+1}=\dfrac {\alpha_0x_n+\alpha_1x_{n-l}+\alpha _2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}}$
Mathematica Bohemica, Tome 135 (2010) no. 3, pp. 319-336
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$ x_{n+1}=\frac {\alpha _0x_n+\alpha _1x_{n-l}+\alpha _2x_{n-k}} {\beta _0x_n+\beta _1x_{n-l}+\beta _2x_{n-k}}, \quad n=0,1,2,\dots $$ where the coefficients $\alpha _i,\beta _i\in (0,\infty )$ for $ i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $ x_{-k}, \dots , x_{-l}, \dots , x_{-1}, x_0 $ are arbitrary positive real numbers such that $l
The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$ x_{n+1}=\frac {\alpha _0x_n+\alpha _1x_{n-l}+\alpha _2x_{n-k}} {\beta _0x_n+\beta _1x_{n-l}+\beta _2x_{n-k}}, \quad n=0,1,2,\dots $$ where the coefficients $\alpha _i,\beta _i\in (0,\infty )$ for $ i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $ x_{-k}, \dots , x_{-l}, \dots , x_{-1}, x_0 $ are arbitrary positive real numbers such that $l$. Some numerical experiments are presented.
DOI : 10.21136/MB.2010.140707
Classification : 34C99, 39A10, 39A20, 39A22, 39A23, 39A30, 39A99, 65Q10
Keywords: difference equation; boundedness; period two solution; convergence; global stability 
@article{10_21136_MB_2010_140707,
     author = {Zayed, E. M. E. and El-Moneam, M. A.},
     title = {On the rational recursive sequence $ x_{n+1}=\dfrac {\alpha_0x_n+\alpha_1x_{n-l}+\alpha _2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}}$},
     journal = {Mathematica Bohemica},
     pages = {319--336},
     year = {2010},
     volume = {135},
     number = {3},
     doi = {10.21136/MB.2010.140707},
     mrnumber = {2683642},
     zbl = {1224.39015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140707/}
}
TY  - JOUR
AU  - Zayed, E. M. E.
AU  - El-Moneam, M. A.
TI  - On the rational recursive sequence $ x_{n+1}=\dfrac {\alpha_0x_n+\alpha_1x_{n-l}+\alpha _2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}}$
JO  - Mathematica Bohemica
PY  - 2010
SP  - 319
EP  - 336
VL  - 135
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140707/
DO  - 10.21136/MB.2010.140707
LA  - en
ID  - 10_21136_MB_2010_140707
ER  - 
%0 Journal Article
%A Zayed, E. M. E.
%A El-Moneam, M. A.
%T On the rational recursive sequence $ x_{n+1}=\dfrac {\alpha_0x_n+\alpha_1x_{n-l}+\alpha _2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}}$
%J Mathematica Bohemica
%D 2010
%P 319-336
%V 135
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140707/
%R 10.21136/MB.2010.140707
%G en
%F 10_21136_MB_2010_140707
Zayed, E. M. E.; El-Moneam, M. A. On the rational recursive sequence $ x_{n+1}=\dfrac {\alpha_0x_n+\alpha_1x_{n-l}+\alpha _2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}}$. Mathematica Bohemica, Tome 135 (2010) no. 3, pp. 319-336. doi: 10.21136/MB.2010.140707

[1] Aboutaleb, M. T., El-Sayed, M. A., Hamza, A. E.: Stability of the recursive sequence $x_{n+1}=(\alpha -\beta x_n)/(\gamma +x_{n-1})$. J. Math. Anal. Appl. 261 (2001), 126-133. | DOI | MR | Zbl

[2] Agarwal, R.: Difference Equations and Inequalities. Theory, Methods and Applications. Marcel Dekker New York (1992). | MR | Zbl

[3] Amleh, A. M., Grove, E. A., Ladas, G., Georgiou, D. A.: On the recursive sequence $x_{n+1}=\alpha +(x_{n-1}/x_n)$. J. Math. Anal. Appl. 233 (1999), 790-798. | MR | Zbl

[4] Clark, C. W.: A delayed recruitment model of population dynamics with an application to baleen whale populations. J. Math. Biol. 3 (1976), 381-391. | DOI | MR | Zbl

[5] Devault, R., Kosmala, W., Ladas, G., Schultz, S. W.: Global behavior of $y_{n+1}=(p+y_{n-k})/(qy_n+y_{n-k})$. Nonlinear Analysis 47 (2001), 4743-4751. | MR

[6] Devault, R., Ladas, G., Schultz, S. W.: On the recursive sequence $x_{n+1}=\alpha +(x_n/x_{n-1})$. Proc. Amer. Math. Soc. 126(11) (1998), 3257-3261. | DOI | MR

[7] Devault, R., Schultz, S. W.: On the dynamics of $x_{n+1}=(\beta x_n+\gamma x_{n-1})/(Bx_n+Dx_{n-2})$. Comm. Appl. Nonlinear Anal. 12 (2005), 35-39. | MR

[8] Elabbasy, E. M., El-Metwally, H., Elsayed, E. M.: On the difference equation $x_{n+1}=( \alpha x_{n-l}+\beta x_{n-k}) /( Ax_{n-l}+Bx_{n-k})$. Acta Mathematica Vietnamica 33 (2008), 85-94. | MR

[9] El-Metwally, H., Grove, E. A., Ladas, G.: A global convergence result with applications to periodic solutions. J. Math. Anal. Appl. 245 (2000), 161-170. | DOI | MR | Zbl

[10] El-Metwally, H., Ladas, G., Grove, E. A., Voulov, H. D.: On the global attractivity and the periodic character of some difference equations. J. Differ. Equ. Appl. 7 (2001), 837-850. | DOI | MR | Zbl

[11] El-Morshedy, H. A.: New explicit global asymptotic stability criteria for higher order difference equations. J. Math. Anal. Appl. 336 (2007), 262-276. | DOI | MR | Zbl

[12] EL-Owaidy, H. M., Ahmed, A. M., Mousa, M. S.: On asymptotic behavior of the difference equation $x_{n+1}=\alpha +(x_{n-1}^p/x_n^p)$. J. Appl. Math. Comput. 12 (2003), 31-37. | DOI | MR

[13] EL-Owaidy, H. M., Ahmed, A. M., Elsady, Z.: Global attractivity of the recursive sequence $x_{n+1}=(\alpha -\beta x_{n-k})/(\gamma +x_n)$. J. Appl. Math. Comput. 16 (2004), 243-249. | DOI | MR

[14] Gibbons, C. H., Kulenovic, M. R. S., Ladas, G.: On the recursive sequence $x_{n+1}=(\alpha +\beta x_{n-1})/(\gamma +x_n)$. Math. Sci. Res. Hot-Line 4(2) (2000), 1-11. | MR

[15] Grove, E. A., Ladas, G.: Periodicities in Nonlinear Difference Equations. Vol. 4. Chapman & Hall / CRC (2005). | MR

[16] Karakostas, G.: Convergence of a difference equation via the full limiting sequences method. Differ. Equ. Dyn. Syst. 1 (1993), 289-294. | MR | Zbl

[17] Karakostas, G., Stević, S.: On the recursive sequences $x_{n+1}=A+f(x_n,\dots,x_{n-k+1})/ x_{n-1}$. Comm. Appl. Nonlinear Anal. 11 (2004), 87-100. | MR

[18] Kocic, V. L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers Dordrecht (1993). | MR | Zbl

[19] Kulenovic, M. R. S., Ladas, G.: Dynamics of Second Order Rational Difference Equations with Open Problems and conjectures. Chapman & Hall / CRC (2001). | MR

[20] Kulenovic, M. R. S., Ladas, G., Sizer, W. S.: On the recursive sequence $x_{n+1}=(\alpha x_n+\beta x_{n-1})/(\gamma x_n+\delta x_{n-1})$. Math. Sci. Res. Hot-Line 2 (1998), 1-16. | MR | Zbl

[21] Kuruklis, S. A.: The asymptotic stability of $ x_{n+1}-ax_n+bx_{n-k}=0$. J. Math. Anal. Appl. 188 (1994), 719-731. | MR

[22] Ladas, G., Gibbons, C. H., Kulenovic, M. R. S., Voulov, H. D.: On the trichotomy character of $x_{n+1}=(\alpha +\beta x_n+\gamma x_{n-1})/(A+x_n)$. J. Difference Equ. Appl. 8 (2002), 75-92. | MR | Zbl

[23] Ladas, G., Gibbons, C. H., Kulenovic, M. R. S.: On the dynamics of $x_{n+1}=(\alpha +\beta x_n+\gamma x_{n-1})/(A+Bx_n)$. Proceeding of the Fifth International Conference on Difference Equations and Applications, Temuco, Chile, Jan. 3-7, 2000 Taylor and Francis London (2002), 141-158. | MR

[24] Ladas, G., Camouzis, E., Voulov, H. D.: On the dynamic of $x_{n+1}=(\alpha +\gamma x_{n-1}+\delta x_{n-2})/(A+x_{n-2})$. J. Difference Equ. Appl. 9 (2003), 731-738. | MR

[25] Ladas, G.: On the rational recursive sequence $x_{n+1}=(\alpha +\beta x_n+\gamma x_{n-1})/(A+Bx_n+Cx_{n-1})$. J. Difference Equ. Appl. 1 (1995), 317-321. | MR

[26] Li, W. T., Sun, H. R.: Global attractivity in a rational recursive sequence. Dyn. Syst. Appl. 11 (2002), 339-346. | MR | Zbl

[27] Stevi'c, S.: On the recursive sequence $x_{n+1}=x_{n-1}/g(x_n)$. Taiwanese J. Math. 6 (2002), 405-414. | DOI | MR

[28] Stevi'c, S.: On the recursive sequence $x_{n+1}=g(x_n,x_{n-1})/(A+x_n)$. Appl. Math. Letter 15 (2002), 305-308. | MR

[29] Stevi'c, S.: On the recursive sequence $x_{n+1}=(\alpha +\beta x_n)/(\gamma -x_{n-k})$. Bull. Inst. Math. Acad. Sin. 32 (2004), 61-70. | MR

[30] Stevi'c, S.: On the recursive sequences $x_{n+1}=\alpha +(x_{n-1}^p/x_n^p)$. J. Appl. Math. Comput. 18 (2005), 229-234. | MR

[31] Yang, X., Su, W., Chen, B., Megson, G. M., Evans, D. J.: On the recursive sequence $x_{n+1}=(ax_{n-1}+bx_{n-2})/(c+dx_{n-1}x_{n-2})$. J. Appl. Math. Comput. 162 (2005), 1485-1497. | MR

[32] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=(D+\alpha x_n+\beta x_{n-1}+\gamma x_{n-2})/(Ax_n+Bx_{n-1}+Cx_{n-2})$. Comm. Appl. Nonlinear Anal. 12 (2005), 15-28. | MR

[33] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=(\alpha x_n+\beta x_{n-1}+\gamma x_{n-2}+\delta x_{n-3})/(Ax_n+Bx_{n-1}+Cx_{n-2}+Dx_{n-3})$. J. Appl. Math. Comput. 22 (2006), 247-262. | DOI | MR

[34] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=\Bigl( A+ \sum_{i=0}^k\alpha _ix_{n-i}\Bigr) \Big/ \Bigl( B+\sum_{i=0}^k\beta _ix_{n-i}\Bigr)$. Int. J. Math. Math. Sci. 2007 (2007), 12, Article ID23618. | MR

[35] Zayed, E. M. E., El-Moneam, M. A.: 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. | MR

[36] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=\Bigl( A+ \sum_{i=0}^k\alpha _ix_{n-i}\Bigr) \Big/ \sum_{i=0}^k\beta _ix_{n-i}$. Math. Bohem. 133 (2008), 225-239. | MR

[37] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=Ax_n+( \beta x_n+\gamma x_{n-k}) /( Cx_n+Dx_{n-k})$. Comm. Appl. Nonlinear Anal. 16 (2009), 91-106. | MR

[38] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=( \alpha +\beta x_{n-k}) /( \gamma -x_n)$. J. Appl. Math. Comput. 31 (2009), 229-237. | DOI | MR

Cité par Sources :