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
where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.
This work was supported by Directorate general for Scientific Research and Technological Development, Algeria.
Communicated by Michal Fečkan
References
[1] Elsayed, E. M.: On a system of two nonlinear difference equations of order two, Proc. Jangeon Math. Soc. 18 (2015), 353–369.Search in Google Scholar
[2] Elsayed, E. M.—Ibrahim, T. F.: Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. Stat. 44 (2015), 1361–1390.10.15672/HJMS.2015449653Search in Google Scholar
[3] Elsayed, E. M.: Solution for systems of difference equations of rational form of order two, Comp. Appl. Math. 33 (2014), 751–765.10.1007/s40314-013-0092-9Search in Google Scholar
[4] Halim, Y.—Rabago, J. F. T.: On the solutions of a second-order difference equation in terms of generalized Padovan sequences, Math. Slovaca 68(3) (2018), 625–638.10.1515/ms-2017-0130Search in Google Scholar
[5] Halim, Y.—Rabago, J. F. T.: On some solvable systems of difference equations with solutions associated to Fibonacci numbers, Electron. J. Math. Anal. Appl. 5(1) (2017), 166–178.10.21608/ejmaa.2017.310883Search in Google Scholar
[6] Halim, Y.—Bayram, M.: On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Math. Methods Appl. Sci. 39 (2016), 2974–2982.10.1002/mma.3745Search in Google Scholar
[7] Halim, Y.—Touafek, N.—Elsayed, E. M.: Closed form solution of some systems of rational difference equations in terms of Fibonacci numbers, Dyn. Contin. Discrete Impulsive Syst. Ser. A. 21 (2014), 473–486.Search in Google Scholar
[8] Halim, Y.: Global character of systems of rational difference equations, Electron. J. Math. Anal. Appl. 3 (2015), 204–214.Search in Google Scholar
[9] Halim, Y.: A system of difference equations with solutions associated to Fibonacci numbers, International Journal of Difference Equations 11 (2016), 65–77.Search in Google Scholar
[10] Koshy, T.: Fibonacci and Lucas Numbers with Applications, A Wiley Interscience Publication, 2001.10.1002/9781118033067Search in Google Scholar
[11] Matsunaga, H.—Suzuki, R.: Classification of global behavior of a system of rational difference equations, Appl. Math. Lett. 85(1) (2018), 57–63.10.1016/j.aml.2018.05.020Search in Google Scholar
[12] Rabago, J. F. T.—Halim, Y.: Supplement to the paper of Halim, Touafek and Elsayed: Part I, Dyn. Contin. Discrete and Impulsive Systems Ser. A. 24(2) (2017), 121–131.Search in Google Scholar
[13] Rabago, J. F. T.—Halim, Y.: Supplement to the paper of Halim, Touafek and Elsayed: Part II, Dyn. Contin. Dyn. Contin. Discrete and Impulsive Systems Ser. A. 24(5) (2017), 333–345.Search in Google Scholar
[14] Stevic, S.: Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. 67 (2014), 15 pages.10.14232/ejqtde.2014.1.67Search in Google Scholar
[15] Stevic, S.: More on a rational recurrence relation, Appl. Math. E-Notes 4(1) (2004), 80–85.Search in Google Scholar
[16] Stevic, S.: Representation of solutions of a solvable nonlinear difference equation of second order, Electron. J. Qual. Theory Differ. Equ. 95 (2018), 18 pages.10.14232/ejqtde.2018.1.95Search in Google Scholar
[17] Tollu, D. T.—Yazlik, Y.—Taskara, N.: On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Difference Equ. 174 (2013), 7 pages.10.1186/1687-1847-2013-174Search in Google Scholar
[18] Tollu, D. T.— Yazlik, Y.—Taskara, N. : The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan J. Math. 2 (2014), 163–172.Search in Google Scholar
[19] Tollu, D. T.—Yazlik, Y.—Taskara, N.: On fourteen solvable systems of difference equations, Appl. Math. Comput. 233 (2014), 310–319.10.1016/j.amc.2014.02.001Search in Google Scholar
[20] Touafek, N.: On some fractional systems of difference equations, Iranian J. Math. Sci. Info. 9 (2014), 303–305.Search in Google Scholar
[21] Touafek, N.—Halim, Y.: On max type difference equations: expressions of solutions, Int. J. Nonlinear Sci. 11 (2011), 396–402.Search in Google Scholar
[22] Touafek, N.—Elsayed, E. M.: On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roum. 55 (2012), 217–224.Search in Google Scholar
[23] Touafek, N.—Elsayed, E. M.: On the solutions of systems of rational difference equations, Math. Comput. Modelling 55 (2012), 1987–1997.10.1016/j.mcm.2011.11.058Search in Google Scholar
[24] Vajda, S.: Fibonacci and Lucas numbers and the golden section: Theory and applications, Ellis Horwood Limited, 1989.Search in Google Scholar
[25] Yazlik, Y.—Tollu, D. T.—Taskara, N.: On the solutions of difference equation systems with Padovan numbers, Appl. Math. 12(1) (2013), 15–20.10.4236/am.2013.412A002Search in Google Scholar
[26] Yazlik, Y.—Tollu, D. T.—Taskara, N.: Behaviour of solutions for a system of two higher-order difference equations, J. Sci. Arts 45(4) (2018), 813–826.Search in Google Scholar
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