Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Dynamical Behavior of a Third-Order Difference Equation with Arbitrary Powers

  • Gumus, Mehmet (Department of Mathematics, Faculty of Science and Arts, Bulent Ecevit University) ;
  • Abo-Zeid, Raafat (Department of Basic Sciences, The Egyptian Academy for Engineering and Advanced Technology) ;
  • Ocalan, Ozkan (Department of Mathematics, Faculty of Science and Arts, Akdeniz University)
  • Received : 2016.12.19
  • Accepted : 2017.05.04
  • Published : 2017.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

The aim of this paper is to investigate the dynamical behavior of the difference equation $$x_{n+1}={\frac{{\alpha}x_n}{{\beta}+{\gamma}x^p_{n-1}x^q_{n-2}}},\;n=0,1,{\ldots}$$, where the parameters ${\alpha}$, ${\beta}$, ${\gamma}$, p, q are non-negative numbers and the initial values $x_{-2}$, $x_{-1}$, $x_0$ are positive numbers. Also, some numerical examples are given to verify our theoretical results.

Keywords

References

  1. R. P. Agarwal, Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics, 155, Marcel Dekker, Inc., New York, 1992.
  2. R. Abo Zeid, Global attractivity of a higher-order difference equation, Discrete Dyn. Nat. Soc., 2012, Art. ID 930410, 11 pages.
  3. A. M. Ahmed, On the dynamics of the recursive sequence $x_{n+1}=(bx_{n-1})/(A+Bx^p_nx^q_{n-2})$, J. Pure Appl. Math. Adv. Appl., 1(2)(2009), 215-223.
  4. A. M. Ahmed, On the dynamics of a higher-order rational difference equation, Discrete Dyn. Nat. Soc., 2011, Article ID 419789, 8 pages.
  5. E. Camouzis and G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, Advances in Discrete Mathematics and Applications, 5, Chapman & Hall/CRC, Boca Raton, FL, 2008.
  6. D. Chen, X. Li and Y. Wang, Dynamics for nonlinear difference equation $x_{n+1}=({\alpha}x_{n-k})/({\beta}+{\gamma}x^p_{n-l})$, Adv. Difference Equ., 2009, Art. ID 235691, 13 pages.
  7. H. M. El-Owaidy, A. M. Ahmed and A. M. Youssef, The dynamics of the recursive sequence $x_{n+1}=({\alpha}x_{n-1})/({\beta}+{\gamma}x^p_{n-2})$, Appl. Math. Lett., 18(9)(2005), 1013-1018. https://doi.org/10.1016/j.aml.2003.09.014
  8. E. M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dynam., 79(1)(2015), 241-250. https://doi.org/10.1007/s11071-014-1660-2
  9. E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011, Art. ID 982309, 17 pages.
  10. M. E. Erdogan, C. Cinar and I. Yalcinkaya, On the dynamics of the recursive sequence $x_{n+1}=({\alpha}x_{n-1})/({\beta}+{\gamma}{\Sigma}^t_{k=1}x_{n-2k}{\Pi}^t_{k=1}x_{x-2k})$, Math. Comput. Modelling, 54(5-6)(2011), 1481-1485. https://doi.org/10.1016/j.mcm.2011.04.020
  11. M. E. Erdogan and C. Cinar, On the dynamics of the recursive sequence $x_{n+1}=({\alpha}x_{n-1})/({\beta}+{\gamma}{\Sigma}^t_{k=1}x^p_{n-2k}{\Pi}^t_{k=1}x^q_{x-2k})$, Fasc. Math., 50(2013), 59-66.
  12. E. A. Grove and G. Ladas, Periodicities in nonlinear difference equations, Advances in Discrete Mathematics and Applications, 4, Chapman & Hall/CRC, Boca Raton, FL, 2005.
  13. M. Gumus, The periodicity of positive solutions of the nonlinear difference equation $x_{n+1}={\alpha}+(x^p_{n-k})/(x^q_n)$, Discrete Dyn. Nat. Soc., 2013, Art. ID 742912, 3 pages.
  14. A. E. Hamza and R. Khalaf-Allah, On the recursive sequence $x_{n+1}=(A{\Pi}^k_{i=l}x_{n-2i-1})/(B+C{\Pi}^{k-1}_{i=l}x_{n-2i})$, Comput. Math. Appl., 56 (2008), 1726-1731. https://doi.org/10.1016/j.camwa.2008.04.014
  15. G. Karakostas, Convergence of a difference equation via the full limiting sequences method, Differential Equations Dynam. Systems, 1(4)(1993), 289-294.
  16. R. Karatas, Global behavior of a higher order difference equation, Comput. Math. Appl., 60(3)(2010), 830-839. https://doi.org/10.1016/j.camwa.2010.05.030
  17. N. Kruse and T. Nesemann, Global asymptotic stability in some discrete dynamical systems, J. Math. Anal. Appl., 253(1)(1999), 151-158.
  18. V. Kocic. and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Mathematics and its Applications, 256, Kluwer Academic Publishers Group, Dordrecht, 1993.
  19. M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations, Chapman & Hall/CRC, Boca Raton, FL, 2002.
  20. H. Levy and F. Lessman, Finite Difference Equations, Dover Publications, Inc., New York, 1992.
  21. O. Ocalan, H. Ogunmez and M. Gumus, Global behavior test for a nonlinear difference equation with a period-two coefficient, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 21(2014), 307-316.
  22. H. Sedaghat, Nonlinear difference equations. Theory with applications to social science models, Mathematical Modelling: Theory and Applications, 15, Kluwer Academic Publishers, Dordrecht, 2003.
  23. I. Yalcinkaya and C. Cinar, On the dynamics of the difference equation $x_{n+1}=(ax_{n-k})/(b+cx^p_n)$, Fasc. Math., 42(2009), 141-148.