대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제35권1호
  • /
  • Pages.301-319
  • /
  • 2020
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

A SOLVABLE SYSTEM OF DIFFERENCE EQUATIONS

  • Taskara, Necati (Department of Mathematics Selcuk University) ;
  • Tollu, Durhasan T. (Department of Mathematics-Computer Science Necmettin Erbakan University) ;
  • Touafek, Nouressadat (LMAM Laboratory, Department of Mathematics Mohamed Seddik Ben Yahia University) ;
  • Yazlik, Yasin (Department of Mathematics Nevsehir Haci Bektas Veli University)
  • 투고 : 2018.11.15
  • 심사 : 2019.07.17
  • 발행 : 2020.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we show that the system of difference equations $x_n={\frac{ay^p_{n-1}+b(x_{n-2}y_{n-1})^{p-1}}{cy_{n-1}+dx^{p-1}_{n-2}}}$, $y_n={\frac{{\alpha}x^p_{n-1}+{\beta}(y_{n-2}x_{n-1})^{p-1}}{{\gamma}x_{n-1}+{\delta}y^{p-1}_{n-2}}}$, n ∈ ℕ0 where the parameters a, b, c, d, α, β, γ, δ, p and the initial values x-2, x-1, y-2, y-1 are real numbers, can be solved. Also, by using obtained formulas, we study the asymptotic behaviour of well-defined solutions of aforementioned system and describe the forbidden set of the initial values. Our obtained results significantly extend and develop some recent results in the literature.

키워드

참고문헌

  1. L. Brand, Classroom notes: a sequence defined by a difference equation, Amer. Math. Monthly 62 (1955), no. 7, 489-492. https://doi.org/10.2307/2307362
  2. M. Dehghan, R. Mazrooei-Sebdani, and H. Sedaghat, Global behaviour of the Riccati difference equation of order two, J. Difference Equ. Appl. 17 (2011), no. 4, 467-477. https://doi.org/10.1080/10236190903049017
  3. I. Dekkar, N. Touafek, and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 111 (2017), no. 2, 325-347. https://doi.org/10.1007/s13398-016-0297-z
  4. S. N. Elaydi, An Introduction to Difference Equations, second edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4757-3110-1
  5. M. M. El-Dessoky and E. M. Elsayed, On the solutions and periodic nature of some systems of rational difference equations, J. Comput. Anal. Appl. 18 (2015), no. 2, 206-218. http://doi.org/10.1166/jctn.2015.4263
  6. E. M. Elsayed, On the solutions and periodic nature of some systems of difference equations, Int. J. Biomath. 7 (2014), no. 6, 1450067, 26 pp. https://doi.org/10.1142/S1793524514500673
  7. N. Haddad, N. Touafek, and J. F. T. Rabago, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Math. Methods Appl. Sci. 40 (2017), no. 10, 3599-3607. https://doi.org/10.1002/mma.4248
  8. N. Haddad, N. Touafek, and J. F. T. Rabago, Well-defined solutions of a system of difference equations, J. Appl. Math. Com-put. 56 (2018), no. 1-2, 439-458. https://doi.org/10.1007/s12190-017-1081-8
  9. Y. Halim and M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Math. Methods Appl. Sci. 39 (2016), no. 11, 2974-2982. https://doi.org/10.1002/mma.3745
  10. Y. Halim, N. Touafek, and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math. 39 (2015), no. 6, 1004-1018. https://doi.org/10.3906/mat-1503-80
  11. T. F. Ibrahim and N. Touafek, On a third order rational difference equation with variable coefficients, DCDIS Series B: Applications & Algorithms 20, (2013), no. 2, 251-264.
  12. E. A. Grove, Y. Kostrov, G. Ladas, and S. W. Schultz, Riccati difference equations with real period-2 coefficients, Comm. Appl. Nonlinear Anal. 14 (2007), no. 2, 33-56.
  13. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, FL, 2002.
  14. H. Matsunaga and R. Suzuki, Classification of global behavior of a system of rational difference equations, Appl. Math. Lett. 85 (2018), 57-63. https://doi.org/10.1016/j.aml.2018.05.020
  15. L. C. McGrath and C. Teixeira, Existence and behavior of solutions of the rational equation $x_n+1=(ax_{n-1}+bx_n)/(cx_{n-1}+dx_n)x_n,\;n$ = 0, 1, 2, ..., Rocky Mountain J. Math. 36 (2006), no. 2, 649-674. https://doi.org/10.1216/rmjm/1181069472
  16. S. Reich and A. J. Zaslavski, Asymptotic behavior of a dynamical system on a metric space, J. Nonlinear Variational Anal. 3 (2019), 79-85.
  17. H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Difference Equ. Appl. 15 (2009), no. 3, 215-224. https://doi.org/10.1080/10236190802054126
  18. S. Selvarangam, S. Geetha, and E. Thandapani, Existence of nonoscillatory solutions to second order neutral type difference equations with mixed arguments, Int. J. Difference Equ. 13 (2018), no. 1, 55-69.
  19. S. Stevic, On a system of difference equations with period two coefficients, Applied Mathematics and Computation 218 (2011), no. 8, 4317-4324. https://doi.org/10.1016/j.amc.2011.10.005
  20. S. Stevic, On some solvable systems of difference equations, Appl. Math. Comput. 218 (2012), no. 9, 5010-5018. https://doi.org/10.1016/j.amc.2011.10.068
  21. S. Stevic, Representation of solutions of bilinear difference equations in terms of gen-eralized Fibonacci sequences, Electronic Journal of Qualitative Theory of Differential Equations 2014 (2014), no. 67, 1-15.
  22. S. Stevic, M. A. Alghamdi, N. Shahzad, and D. A. Maturi, On a class of solvable difference equations, Abstr. Appl. Anal. 2013 (2013), Art. ID 157943, 7 pp. https://doi.org/10.1155/2013/157943
  23. D. T. Tollu, Y. Yazlik, and N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Difference Equ. 2013 (2013), 174, 7 pp. https://doi.org/10.1186/1687-1847-2013-174
  24. D. T. Tollu, Y. Yazlik, and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput. 233 (2014), 310-319. https://doi.org/10.1016/j.amc.2014.02.001
  25. D. T. Tollu, Y. Yazlik, and N. Taskara, On a solvable nonlinear difference equation of higher order, Turkish J. Math. 42 (2018), no. 4, 1765-1778. https://doi.org/10.3906/mat-1705-33
  26. Q. Wang and Q. Zhang, Dynamics of a higher-order rational difference equation, J. Appl. Anal. Comput. 7 (2017), no. 2, 770-787. http://doi.org/10.11948/2017048
  27. Y. Yazlik, On the solutions and behavior of rational difference equations, J. Comput. Anal. Appl. 17 (2014), no. 3, 584-594.
  28. Y. Yazlik, D. T. Tollu, and N. Taskara, On the solutions of a three-dimensional system of difference equations, Kuwait J. Sci. 43 (2016), no. 1, 95-111.