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On the Rational Recursive Sequence \(x_{n+1}=Ax_{n}+Bx_{n-k}+\frac{\beta x_{n}+\gamma x_{n-k}}{Cx_{n}+Dx_{n-k}}\)

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Abstract

In this article, we study the global and asymptotic properties of the solutions of the difference equation

$$x_{n+1}=Ax_{n}+Bx_{n-k}+(\beta x_{n}+\gamma x_{n-k})/(Cx_{n}+Dx_{n-k}),\quad n=0,1,2,\ldots,$$

where the initial conditions x −k ,…,x −1,x 0 are arbitrary positive real numbers and the coefficients A,B,C,D,β and γ are positive constants, while k is a positive integer number. Some numerical examples will be given to illustrate our results.

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References

  1. Aboutaleb, M.T., El-Sayed, M.A., Hamza, A.E.: Stability of the recursive sequence x n+1=(α−β x n )/(γ+x n−1). J. Math. Anal. Appl. 261, 126–133 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  3. Amleh, A.M., Grove, E.A., Ladas, G., Georgiou, D.A.: On the recursive sequence x n+1=α+(x n−1/x n ). J. Math. Anal. Appl. 233, 790–798 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Clark, C.W.: A delayed recruitment model of population dynamics with an application to baleen whale populations. J. Math. Biol. 3, 381–391 (1976)

    MATH  MathSciNet  Google Scholar 

  5. Devault, R., Schultz, S.W.: On the dynamics of x n+1=(β x n +γ x n−1)/(Bx n +Dx n−2). Commun. Appl. Nonlinear Anal. 12, 35–40 (2005)

    MATH  MathSciNet  Google Scholar 

  6. Devault, R., Ladas, G., Schultz, S.W.: On the recursive sequence x n+1=α+(x n /x n−1). Proc. Am. Math. Soc. 126(11), 3257–3261 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. 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 Anal. 47, 4743–4751 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Elabbasy, E.M., El-Metwally, H., Elsayed, E.M.: On the difference equation x n+1=ax n −bx n /(cx n −dx n−1). In: Advances in Difference Equations, vol. 2006, pp. 1–10 (2006). Article ID 82579. doi:10.1155/2006/82579

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

    Article  MATH  MathSciNet  Google Scholar 

  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, 837–850 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. El-Owaidy, H.M., Ahmed, A.M., Mousa, M.S.: On asymptotic behavior of the difference equation x n+1=α+(x pn−1 /x p n ). J. Appl. Math. Comput. 12, 31–37 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. El-Owaidy, H.M., Ahmed, A.M., Elsady, Z.: Global attractivity of the recursive sequence x n+1=(α−β x n−k)/(γ+x n ). J. Appl. Math. Comput. 16, 243–249 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gibbons, C.H., Kulenovic, M.R.S., Ladas, G.: On the recursive sequence x n+1=(α+β x n−1)/(γ+x n ). Math. Sci. Res. Hot-Line 4(2), 1–11 (2000)

    MATH  MathSciNet  Google Scholar 

  15. Grove, E.A., Ladas, G.: Periodicities in Nonlinear Difference Equations, vol. 4. Chapman & Hall/CRC, London/Boca Raton (2005)

    MATH  Google Scholar 

  16. Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon, Oxford (1991)

    Google Scholar 

  17. Karakostas, G.: Convergence of a difference equation via the full limiting sequences method. Differ. Equ. Dyn. Syst. 1, 289–294 (1993)

    MATH  MathSciNet  Google Scholar 

  18. Karakostas, G., Stević, S.: On the recursive sequences x n+1=A+f(x n ,…,x n−k+1)/x n−1. Commun. Appl. Nonlinear Anal. 11, 87–100 (2004)

    MATH  Google Scholar 

  19. Kocic, V.L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht (1993)

    MATH  Google Scholar 

  20. Kulenovic, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, London/Boca Raton (2001)

    Book  Google Scholar 

  21. Kulenovic, M.R.S., Ladas, G., Sizer, W.S.: On the recursive sequence x n+1=(α x n +β x n−1)/(γ x n +δ x n−1). Math. Sci. Res. Hot-Line 2(5), 1–16 (1998)

    MATH  MathSciNet  Google Scholar 

  22. Kuruklis, S.A.: The asymptotic stability of x n+1−ax n +bx n−k=0. J. Math. Anal. Appl. 188, 719–731 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ladas, G.: On the rational recursive sequence x n+1=(α+β x n +γ x n−1)/(A+Bx n +Cx n−1). J. Differ. Equ. Appl. 1, 317–321 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ladas, G., Gibbons, C.H., Kulenovic, M.R.S.: On the dynamics of x n+1=(α+β x n +γ x n−1)/(A+Bx n ). In: Proceedings of the Fifth International Conference on Difference Equations and Applications, Temuco, Chile, 3–7 January 2000, pp. 141–158. Taylor and Francis, London (2002)

    Google Scholar 

  25. Ladas, G., Gibbons, C.H., Kulenovic, M.R.S., Voulov, H.D.: On the trichotomy character of x n+1=(α+β x n +γ x n−1)/(A+x n ). J. Differ. Equ. Appl. 8, 75–92 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ladas, G., Camouzis, E., Voulov, H.D.: On the dynamic of x n+1=(α+γ x n−1+δ x n−2)/(A+x n−2). J. Differ. Equ. Appl. 9, 731–738 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Li, W.T., Sun, H.R.: Global attractivity in a rational recursive sequence. Dyn. Syst. Appl. 11, 339–346 (2002)

    MATH  MathSciNet  Google Scholar 

  28. Li, W.T., Sun, H.R.: Dynamics of a rational difference equations. Appl. Math. Comput. 157, 713–727 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Mickens, R.E.: Difference Equations, Theory and Applications. Van Nostrand, New York (1990)

    MATH  Google Scholar 

  30. Saleh, M., Abu-Baha, S.: Dynamics of a higher order rational difference equation. Appl. Math. Comput. 181, 84–102 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Stević, S.: On the recursive sequences x n+1=x n−1/g(x n ). Taiwan. J. Math. 6, 405–414 (2002)

    MATH  Google Scholar 

  32. Stević, S.: On the recursive sequences x n+1=g(x n ,x n−1)/(A+x n ). Appl. Math. Lett. 15, 305–308 (2002)

    Article  MathSciNet  Google Scholar 

  33. Stević, S.: On the recursive sequences x n+1=α+(x pn−1 /x p n ). J. Appl. Math. Comput. 18, 229–234 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=(D+α x n +β x n−1+γ x n−2)/(Ax n +Bx n−1+Cx n−2). Commun. Appl. Nonlinear Anal. 12, 15–28 (2005)

    MATH  MathSciNet  Google Scholar 

  35. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=(α x n +β x n−1+γ x n−2+δ x n−3)/(Ax n +Bx n−1+Cx n−2+Dx n−3). J. Appl. Math. Comput. 22, 247–262 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=(A+∑ ki=0 α i x n−i)/(B+∑ ki=0 β i x n−i). Int. J. Math. Math. Sci. 2007. Article ID 23618, 12 p. (2007). doi:10.1155/2007/23618

    Article  MathSciNet  Google Scholar 

  37. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=(A+∑ ki=0 α i x n−i)/∑ ki=0 β i x n−i. Math. Bohemica 133(3), 225–239 (2008)

    MATH  MathSciNet  Google Scholar 

  38. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=ax n −bx n /(cx n −dx n−k). Commun. Appl. Nonlinear Anal. 15, 47–57 (2008)

    MATH  MathSciNet  Google Scholar 

  39. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=(ax n +bx n−k)/(cx n −dx n−k). Commun. Appl. Nonlinear Anal. 15, 67–76 (2008)

    MathSciNet  Google Scholar 

  40. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=(α+β x n−k)/(γ−x n ). J. Appl. Math. Comput. doi:10.1007/s12190-008-0205-6 (in press)

  41. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence x n+1=γ x n−k+(ax n +bx n−k)/(cx n −dx n−k). Bull. Iranian Math. Soc. (to appear)

  42. Zayed, E.M.E., El-Moneam, M.A.: On the global attractivity of two nonlinear difference equations, J. Math. Sci. (to appear)

  43. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive two sequences x n+1=ax n−k+bx n−k/(cx n +δ dx n−k). Acta Math. Vietnam. (to appear)

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  1. E. M. E. Zayed

    Present address: Mathematics Department, Faculty of Science, Taif University, El-Taif, EL-Hawiyah, P.O. Box 888, Kingdom of Saudi Arabia

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  1. Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt

    E. M. E. Zayed & M. A. El-Moneam

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  1. E. M. E. Zayed
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Correspondence to M. A. El-Moneam.

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Zayed, E.M.E., El-Moneam, M.A. 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, 287–301 (2010). https://doi.org/10.1007/s10440-009-9545-y

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