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New method to obtain periodic solutions of period two and three of a rational difference equation

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Abstract

This paper aims to develop a new method to obtain periodic solutions with period two and three of the following recursive sequence

$$\begin{aligned} x_{n+1} =\alpha +\frac{\beta x_n }{x_{n-1} }+\frac{\gamma x_{n-1}}{x_n}, \end{aligned}$$

where the parameters \(\alpha ,\beta \) and \(\gamma \) are positive real numbers and the initial conditions \(x_{-1} ,\,x_{0}\) are positive real numbers. Also, it investigates the global convergence and boundedness of the aforementioned equation.

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References

  1. Agarwal, R.P.: Difference Equations and Inequalities, 1st edn. Marcel Dekker, New York (1992)

    MATH  Google Scholar 

  2. Agarwal, R.P.: Difference Equations and Inequalities, 2nd edn. Marcel Dekker, New York (2000)

    MATH  Google Scholar 

  3. Agarwal, R.P., Elsayed, E.M.: Periodicity and stability of solutions of higher order rational difference equation. Adv. Stud. Contemp. Math. 17(2), 181–201 (2008)

    MATH  MathSciNet  Google Scholar 

  4. Ahmed, A.M.: On the dynamics of a higher-order rational difference equation. Discrete Dyn. Nat. Soc., vol. 2011, Article ID 419789, 8 pp. (2011)

  5. Ahmed, E., Hegazi, A.S., Elgazzar, A.S.: On difference equations motivated by modelling the heart. Nonlinear Dyn. 46(1–2), 49–60 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Amleh, A.M., Grove, E.A., Ladas, G., Georgiou, D.A.: On the recursive sequence \(x_{n+1} =\alpha +\frac{x_{n-1} }{x_n }\). J. Math. Anal. Appl. 233, 790–798 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Battaloglu, N., Cinar, C., Yalcinkaya, I.: The dynamics of the difference equation. Ars Comb. 97, 281–288 (2010)

  8. DeVault, R., Dial, G., Kocic, V.L., Ladas, G.: Global behavior of solutions of \(x_{n+1} =ax_n +f\left( x_n, x_{n-1} \right)\). J. Differ. Equ. Appl. 3(3–4), 311–330 (1998)

    MATH  MathSciNet  Google Scholar 

  9. Elabbasy, E.M., El-Metwally, H., Elsayed, E.M.: On the difference equation \(x_{n+1} =ax_n -\frac{bx_n }{cx_n -dx_{n-1}}\). Adv. Differ. Equ., vol. 2006, Article ID 82579, pp. 1–10 (2006)

  10. Elabbasy, E.M., El-Metwally, H., Elsayed, E.M.: On the difference equations \(x_{n+1} =\frac{\alpha x_{n-k} }{\beta +\gamma \prod \nolimits _{i=0}^k x_{n-i}}\). J. Conc. Appl. Math. 5(2), 101–113 (2007)

    MATH  MathSciNet  Google Scholar 

  11. Elaiw, A.M.: Global properties of a class of virus infection models with multitarget cells. Nonlinear Dyn. 69(1–2), 423–435 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Elsayed, E.M.: Solution and attractivity for a rational recursive sequence. Discrete Dyn. Nat. Soc., vol. 2011, Article ID 982309, 17 pp. (2011)

  13. Emre Erdogana, M., Cinar, C., Yalcinkaya, I.: On the dynamics of the recursive sequence. Math. Comput. Model. 54(5–6), 1481–1485 (2011)

    Article  Google Scholar 

  14. Gumus, M., Ocalan, O.: Some notes on the difference equation \(x_{n+1} =\alpha +\frac{x_{n-1} }{x_n^k}\). Discrete Dyn. Nat. Soc. 2012, Article ID 258502, 12 pp. (2012)

  15. Hamza, A.E., Barbary, S.G.: Attractivity of the recursive sequence \(x_{n+1} =(\alpha -\beta x_n )F(x_{n-1},\ldots, x_{n-k} )\). Math. Comput. Model. 48(11–12), 1744–1749 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hamza, A.E., Morsy, A.: On the recursive sequence \(x_{n+1} =\alpha +\frac{x_{n-1} }{x_n^k }\). Appl. Math. Lett. 22(1), 91–95 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

  19. Li, T.Y., Yorke, J.A.: Period three implies chaos. Amer. Math. Mon. 82, 985–992 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ocalan, O.: Dynamics of the difference equation \(x_{n+1} =p_n +\frac{x_{n-k} }{x_n }\) with a period-two coefficient. Appl. Math. Comput. 228, 31–37 (2014)

    Article  MathSciNet  Google Scholar 

  21. Rafei, M., Van Horssen, W.T.: Solving systems of nonlinear difference equations by the multiple scales perturbation method. Nonlinear Dyn. 69(4), 1509–1516 (2012)

    Article  MATH  Google Scholar 

  22. 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 

  23. Saleh, M., Aloqeili, M.: On the difference equation \(x_{n+1} =A+\frac{x_n }{x_{n-k}}\). Appl. Math. Comput. 171, 862–869 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Saleh, M., Aloqeili, M.: On the difference equation \(y_{n+1} =A+\frac{y_n }{y_{n-k}}\) with \({\>}0\). Appl. Math. Comput. 176(1), 359–363 (2006)

  25. Sun, T., Xi, H.: On convergence of the solutions of the difference equation \(x_{n+1} =1+\frac{x_{n-1} }{x_n}\). J. Math. Anal. Appl. 325(2), 1491–1494 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang, C., Gong, F., Wang, S., Li, L., Shi, Q.: Asymptotic behavior of equilibrium point for a class of nonlinear difference equation. Adv. Differ. Equ., vol. 2009, Article ID 214309, 8 pp. (2009)

  27. Yalçınkaya, I., Cinar, C., Atalay, M.: On the solutions of systems of difference equations. Adv. Differ. Equa., vol. 2008, Article ID 143943, 9 pp. (2008)

  28. Yalçınkaya, I.: On the difference equation \(x_{n+1} =\alpha +\frac{x_{n-m} }{x_n^k }\). Discrete Dyn. Nat. Soc., vol. 2008, Article ID 805460, 8 pp. (2008). doi:10.1155/2008/805460

  29. Zayed, E.M.E., El-Moneam, M.A.: On the rational recursive sequence \(x_{n+1} =Ax_n +\frac{\beta x_n +\gamma x_{n-k}}{Cx_n +Dx_{n-k}}\). Commun. Appl. Nonlinear Anal. 16, 91–106 (2009)

    MATH  MathSciNet  Google Scholar 

  30. Zhang, F., Mu, C., Wang, X., Ahmed, I., Shu, Y. : Solution bounds of a new complex PMSM system. Nonlinear Dyn. (2013). doi:10.1007/s11071-013-1022-5

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Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (130-033-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia

    E. M. Elsayed

  2. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    E. M. Elsayed

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  1. E. M. Elsayed
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Correspondence to E. M. Elsayed.

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Elsayed, E.M. New method to obtain periodic solutions of period two and three of a rational difference equation. Nonlinear Dyn 79, 241–250 (2015). https://doi.org/10.1007/s11071-014-1660-2

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  • DOI: https://doi.org/10.1007/s11071-014-1660-2

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