Solutions of a System of Two Higher-Order Difference Equations in Terms of Lucas Sequence

Year 2019, Volume: 2 Issue: 4, 202 - 211, 26.12.2019

Yacine Halim , Amira Khelifa Massaoud Berkal

https://doi.org/10.32323/ujma.610399

Cited By: 11

Abstract

In this paper we give some theoretical explanations related to the representation for the general solution of the   system of the  higher-order rational difference equations $$ x_{n+1} = \frac{5 y_{n-k}-5}{y_{n-k}}, \qquad y_{n+1} = \frac{5 x_{n-k}-5}{x_{n-k}} ,\qquad n, k\in \mathbb{N}_0, $$ where  $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$,  and the initial conditions $x_{-k}$, $x_{-k+1},\ldots$, $x_{0}$, $y_{-k}$, $y_{-k+1},\ldots$, $y_{0}$ are non zero real numbers such that their solutions are associated to Lucas numbers. We also study the  stability character and asymptotic behavior of this system.

Keywords

General solution, Lucas numbers, stability, system of difference equations

References

• [1] E. M. Elsayed, On a system of two nonlinear difference equations of order two, Proc. Jangeon Math. Soc., 18(1)(2015), 353-368.
• [2] E. M. Elsayed and T. F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat., 44(1)(2015), 1361-1390.
• [3] E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Comp. Appl. Math., 33(1)(2014), 751-765.
• [4] E. Camouzis and G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, Vol5, CRC Press, (2007).
• [5] M. Gumus, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24(6)(2018), 976-991.
• [6] M. Gumus and R. Abo-Zeid, On the solutions of a (2k+2)th order difference equation, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, 25(2)(2018), 129-143.
• [7] Y. Halim and J. F. T. Rabago, On the solutions of a second-order difference equation in terms of generalized Padovan sequences, Math. Slovaca, 68(3)(2018), 625-638.
• [8] Y. Halim and J. F. T. Rabago,On some solvable systems of difference equations with solutions associated to Fibonacci numbers, Electron J. Math. Analysis Appl, 5(1)(2017), 166-178.
• [9] Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J. Difference Equ., 11(1)(2016), 65-77.
• [10] 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(1)(2016), 2974-2982.
• [11] Y. Halim, N. Touafek and E. M. Elsayed, Closed form solution of some systems of rational difference equations in terms of Fibonacci numbers, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal., 21(6)(2014), 473-486.
• [12] Y. Halim, Global character of systems of rational difference equations, Electron. J. Math. Analysis Appl., 3(1)(2015), 204-214.
• [13] Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J Difference Equ., 11(2016), 65–77.
• [14] V. L. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993).
• [15] T. Koshy, Fibonacci and Lucas numbers with applcations, Departement of mathematics, Framingham State University, (2017).
• [16] M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations: With open problems and conjectures, Chapman and Hall/CRC, (2001).
• [17] H. Matsunaga and R. Suzuki, Classification of global behavior of a system of rational difference equations, Appl. Math. Lett., 85(1)(2018), 57–63.
• [18] O. Ocalan and O. Duman, on solutions of the recursive equations $x_{n+1}=x_{n-1}^{p}/x_{n}^{p}(p>0)$ via Fibonacci-type sequences, Electron. J. Math. Analysis Appl., 7(1)(2019), 102-115.
• [19] S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ., 67(1)(2014), 15 pages.
• [20] S. Stevic, More on a rational recurrence relation, Appl. Math. E-Notes, 4(1)(2004), 80-85.
• [21] S. Stevic, Representation of solutions of a solvable nonlinear difference equation of second order, Electron. J. Qual. Theory Differ. Equ., 95(1)(2018), 18 pages.
• [22] D. T. Tollu, Y. Yazlik and N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci umbers, 174(1)(2013), 7 pages.
• [23] D. T. Tollu, Y. Yazlik and N. Taskara, The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan J. Math., 2(1)(2014), 163-172.
• [24] D. T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. & Comp., 233(1)(2014), 310-319.
• [25] N. Touafek, On some fractional systems of difference equations, Iran. J. Math. Sci. Inform., 9(2)(2014), 303-305.
• [26] N. Touafek and Y. Halim, On max type difference equations: expressions of solutions, Int. J. Appl. Nonlinear Sci., 11(4)(2011), 396-402.
• [27] N. Touafek and E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roum., Nouv. Ser., 55(1)(2012), 217-224.
• [28] N. Touafek and E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Modelling, 55(1)(2012), 1987-1997.
• [29] S. Vajda, Fibonacci and Lucas numbers and the golden section : Theory and applications, Department of Mathematics, University of Sussex, Ellis Horwood Limited, (1989).
• [30] Y. Yazlik, D. T. Tollu and N. Taskara, On the solutions of difference equation systems with Padovan numbers, Appl Math., 12(1)(2013), 15-20.
• [31] Y. Yazlik, D. T. Tollu and N. Taskara, behaviour of solutions for a system of two higher-order difference equations , J. Sci. Arts,45(4)(2018), 813-826.