Previous |  Up |  Next

Article

Keywords:
difference equation; boundedness; period two solution; convergence; global stability
Summary:
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 $l<k$. Some numerical experiments are presented.
References:
[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 10.1006/jmaa.2001.7481 | MR 1850961 | Zbl 0990.39009
[2] Agarwal, R.: Difference Equations and Inequalities. Theory, Methods and Applications. Marcel Dekker New York (1992). MR 1155840 | Zbl 0925.39001
[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 1689579 | Zbl 0962.39004
[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 10.1007/BF00275067 | MR 0429174 | Zbl 0337.92011
[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 1975867
[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 10.1090/S0002-9939-98-04626-7 | MR 1473661
[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 2129054
[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 2418690
[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 10.1006/jmaa.2000.6747 | MR 1756582 | Zbl 0971.39004
[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 10.1080/10236190108808306 | MR 1870725 | Zbl 0993.39008
[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 10.1016/j.jmaa.2006.12.049 | MR 2348505 | Zbl 1186.39022
[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 10.1007/BF02936179 | MR 1976801
[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 10.1007/BF02936165 | MR 2080567
[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 1742735
[15] Grove, E. A., Ladas, G.: Periodicities in Nonlinear Difference Equations. Vol. 4. Chapman & Hall / CRC (2005). MR 2193366
[16] Karakostas, G.: Convergence of a difference equation via the full limiting sequences method. Differ. Equ. Dyn. Syst. 1 (1993), 289-294. MR 1259169 | Zbl 0868.39002
[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 2069821
[18] Kocic, V. L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic Publishers Dordrecht (1993). MR 1247956 | Zbl 0787.39001
[19] Kulenovic, M. R. S., Ladas, G.: Dynamics of Second Order Rational Difference Equations with Open Problems and conjectures. Chapman & Hall / CRC (2001). MR 1935074
[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 1623643 | Zbl 0960.39502
[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 1305480
[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 1884593 | Zbl 1005.39017
[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 2016061
[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 1992906
[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 1350447
[26] Li, W. T., Sun, H. R.: Global attractivity in a rational recursive sequence. Dyn. Syst. Appl. 11 (2002), 339-346. MR 1941754 | Zbl 1019.39007
[27] Stevi'c, S.: On the recursive sequence $x_{n+1}=x_{n-1}/g(x_n)$. Taiwanese J. Math. 6 (2002), 405-414. MR 1921603
[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 1891551
[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 2037745
[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.
[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 2113984
[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 2163175
[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. MR 2248455
[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 2295740
[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 2414364
[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 2494777
[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 2554552
[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 10.1007/s12190-008-0205-6 | MR 2545724
Partner of
EuDML logo