# On a system of difference equations of third order solved in closed form

## Abstract

In this work, we show that the system of difference equations
xn+1=(ayn-2xn-1yn+bxn-1yn-2+cyn-2+d)/(yn-2xn-1yn),
yn+1=(axn-2yn-1xn+byn-1xn-2+cxn-2+d)/(xn-2yn-1xn),
where n belongs to the set of positive integer numbers, x-2, x-1, x0, y-2, y-1 and y0 are arbitrary nonzero real numbers, and the parameters a, b, c and d are arbitrary real numbers with d nonzero can be solved in a closed form.
We will see that when a = b = c = d = 1, the solutions are expressed using the famous Tetranacci numbers. In particular, the results obtained here extend those in our recent work.

## Article Details

How to Cite
[1]
AKROUR, Y., Touafek, N. and Halim, Y. 2021. On a system of difference equations of third order solved in closed form. Journal of Innovative Applied Mathematics and Computational Sciences. 1, 1 (Dec. 2021), 1–15. DOI:https://doi.org/10.58205/jiamcs.v1i1.8.
Section
Research Articles

## References

Y. Akrour, N. Touafek and Y. Halim, On a system of difference equations of second order solved in closed form, Miskolc Math. Notes 20(2) (2019), 701–717.

Y. Akrour, M. Kara, N. Touafek and Y. Yazlik, Solutions formulas for some general systems of difference equations, Miskolc Math. Notes in press (2021).

A. Asiri, M. M. El-Dessoky and E. M. Elsayed, Solution of a third order fractional system of difference equations, J. Comput. Anal. Appl. 24(3) (2018), 444–453.

R. Azizi, Global behaviour of the rational Riccati difference equation of order two: the general case, J. Difference Equ. Appl. 18(6) (2012), 947–961.

F. Belhannache, N. Touafek, and R. Abo-Zeid, Dynamics of a third-order rational difference equation, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 59(107) (2016), 13–22.

M. M. El-Dessoky, On a solvable for some systems of rational difference equations, J. Nonlinear Sci. Appl. 9(6) (2016), 3744–3759.

M. M. El-Dessoky, E. M. Elsayed, E. M. Elabbasy and Asim Asiri, Expressions of the solutions of some systems of difference equations, J. Comput. Anal. Appl. 27(7) (2019), 1161– 1172.

E. M. Elsayed, On a system of two nonlinear difference equations of order two, Proc. Jangjeon Math. Soc. 18(1) (2015), 353–368.

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.

E. M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math. 33(1) (2015), 751–765.

M. Gumus, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl. 24(1) (2018), 976–991.

M. Gumus and O. Ocalan, The qualitative analysis of a rational system of difference equations, J. Fract. Calc. Appl. 9(2) (2018), 113–126.

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(1) (2018), 129–143.

M. Gumus, Analysis of periodicity for a new class of nonlinear difference equations by using a new method, Electron. J. Math. Anal. Appl. 8(1) (2020), 109–116.

M. Gumus, The periodic character in a higher order difference equation with delays, Math. Methods Appl. Sci. 43(3) (2020), 1112–1123.

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(1) (2017), 3599–3607.

N. Haddad, N. Touafek and E. M. Elsayed, A note on a system of difference equations, An. ¸Stiin¸t. Univ. Al. I. Cuza Ia¸si, Ser. Nou˘a, Mat.(N.S.) 63(3) (2017), 599–606.

N. Haddad, N. Touafek and J. F. T. Rabago, Well-defined solutions of a system of difference equations, J. Appl. Math. Comput. 56(1) (2018), 439–458.

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.

Y. Halim and J. F. T. Rabago, On some solvable systems of difference equations with solutions associated to Fibonacci numbers, Electron. J. Math. Anal. Appl. 5(1) (2017), 166–178.

Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J. Differ. Equ. 11(1) (2016), 65–77.

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.

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.

Y. Halim, Global character of systems of rational difference equations, Electron. J. Math. Anal. Appl. 3(1) (2016), 204–214.

G. S. Hathiwala and D. V. Shah, Binet-type formula for the sequence of Tetranacci numbers by alternate methods, J. Interdiscip. Math. 6(1) (2017), 37–48.

M. Kara and Y. Yazlik, Solvability of a system of nonlinear difference equations of higher order, Turkish J. Math. 43(3) (2019), 1533–1565.

M. Kara and Y. Yazlik, Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients, Math. Slovaca. 71(5)( 2021), 1133–1148.

H.Matsunaga and R. Suzuki, Classification of global behavior of a system of rational difference equations, Appl. Math. Lett. 85(1) (2018), 57–63.

S. Stevic, Some representations of the general solution to a difference equation of additive type, Adv. Differ. Equ. 2019(431) (2019), 1–19.

S. Stevic, B. Iricanin and W. Kosmala, Representations of general solutions to some classes of nonlinear difference equations, Adv. Differ. Equ. 2019 (73) (2019), 1–21.

S. Stevic, B. Iricanin, W. Kosmala and Z. Smarda, Note on the bilinear difference equation with a delay, Math. Methods Appl. Sci. 41(18) (2018), 9349–9360.

S. Stevic, B. Iricanin,W. Kosmala and Z. Smarda, Representation of solutions of a solvable nonlinear difference equation of second order, Electron. J. Qual. theory Differ. Equ. 2015 (95) (2018), 1–18.

S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. theory Differ. Equ. 2014(67) (2014), 1–15.

D. T. Tollu, Y. Yazlik and N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Differ. Equ. 2013(174) (2013), 1–7.

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.

D. T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput. 233(1) (2014), 310–319.

N. Touafek, On some fractional systems of difference equations, Iran. J. Math. Sci. Inform. 9(2) (2014), 303–305.

N. Touafek, Y. Halim, On max type difference equations: expressions of solutions, Int. j. appl. nonlinear sci. 11(4) (2011), 396–402.

N. Touafek, E.M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. math. Soc. Sci. Math. Roum. 55(1) (2012), 217–224.

N. Touafek, E.M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Model. 55(1) (2012), 1987–1997.

C. Wang, X. Fang and R. Li, On the solution for a system of two rational difference equations, J. Comput. Anal. Appl. 20(1) (2016), 175–186.

C. Wang, X. Fang and R. Li, On the dynamics of a certain four-order fractional difference equations, J. Comput. Anal. Appl. 22(5) (2017), 968–976.

C. Wang, X. Fang and R. Li, On the periodicity of a max-type rational difference equation, J. Nonlinear Sci. Appl. 10(9) (2017), 4648–4661.

C. Wang, Y. Zhou, S. Pan, R. Li, On a system of three max-type nonlinear difference equations, J. Comput. Anal. Appl. 25(8) (2018), 1463–1479.

Y. Yazlik, D. T. Tollu, N. Taskara, On the solutions of difference equation systems with Padovan numbers, Appl. Math. 4 (2013), 15–20.

Y. Yazlik, M. Kara, On a solvable system of difference equations of higher-order with period two coefficients, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(1) (2019), 1675–1693.

M. E. Waddill, The Tetranacci sequence and generalizations, Fibonacci Quart. 30(1) (1992), 9–20.