Explicit Solutions of a Three-dimensional System of Nonlinear Difference Equations
In this paper, we show that the system of difference equations xn+1=(xn+yn)/(1+xnyn), yn+1=(yn+zn)/(1+ynzn), zn+1=(zn+xn)/(1+znxn); n =0,1,..., where the initial values x0, y0, z0 are positive real numbers, are solvable in explicit form via some changes of variables and tricks. Also, we determine the forbidden set of the initial values x0, y0, z0 for the above mentioned system and investigate asymptotic behavior of the well-defined solutions by using these explicit formulas.
Asymptotic behavior; Explicit solution; Forbidden set; Rational difference equation; System.
2. I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Revista de la real academia de ciencias exactas, físicas y naturales. Serie A. Matemáticas, Doi:10.1007/s13398-016-0297-z.
3. E.M. Elsayed, Qualitative behavior of a rational recursive sequence, Indagationes Mathematicae 19(2)(2008), 189- 201.
4. E.M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta Applicandae Mathematicae 110(2)(2010), 589-604.
5. 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, Mathematical Methods in the Applied Sciences, (2016), doi: 10.1002/mma.4248.
6. N. Haddad, N. Touafek and J. F. T. Rabago, Well-defined solutions of a system of difference equations, Journal of Applied Mathematics and Computing, (2017), Doi:10.1007/ s12190 - 017-1081- 8 .
7. Y. Halim, N. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish Journal of Mathematics 39(6)(2015), 1004-1018.
8. A. S. Kurbanli, C. Çinar and D. Şimşek, On the periodicity of solutions of the system of rational difference equations, Applied Mathematics, 2 (2011), 410-413.
9. S. Stevic, On some solvable systems of difference equations, Applied Mathematics and Computation, 218 (2012) 5010- 5018.
10. S. Stevic, M. A. Alghamdi, N. Shahzad and D. A. Maturi, On a class of solvable difference equations, Abstract and Applied Analysis, Vol. 2013, Article ID 157943, 7 pages.
11. S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electronic Journal of Qualitative Theory of Differential
Equations, 2014, No. 67, (2014), 1-15.
12. S. Stevic, J. Diblík, B. Iricanin and Z. Šmarda, On a solvable system of rational difference equations, Journal of Difference Equations and Applications, 20(5-6)(2014): 811-82 5.
13. S. Stevic, M. A. Alghamdi, A. Alotaibi and E. M. Elsayed, Solvable product-type system of difference equations of second order, Electronic Journal of Differential Equations, 2015, No:169, (2015), 1-20.
14 . S. Stevic, New class of solvable systems of difference equations, Applied Mathematics Letters, 63(2017), 137- 144.
15. N. Taskara, K. Uslu and D. T. Tollu, The periodicity and solutions of the rational difference equation with periodic coefficients, Computers & Mathematics with Applications, 62 (2011), 1807-1813.
16. D. T. Tollu, Y. Yazlik and N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Advances in Difference Equations, 2013, 2013:174.
17. D. T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Applied Mathematics and Computation, 233 (2014), 310-319.
18. N. Touafek, On a second order rational difference equation, Hacettepe Journal of Mathematics and Statistics, 41(6) (2012), 867-874.
19. I. Yalcinkaya, C. Cinar and D. Simsek, Global asymptotic stability of a system of difference equations, Applicable Analysis, 87(6)(2008), 677-687, DOI: 10.1080/00036810802140657.
20. I. Yalcinkaya, On the global asymptotic stability of a second- order system of difference equations, Discrete Dynamics in Nature and Society, vol. 2008, Article ID 860152, 12 pages, 2008. doi:10.1155/2008/860152.
21. I. Yalcinkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, Ars Combinatoria, 95 (2010), 151-159.
22. I. Yalcinkaya and C. Cinar, Global asymptotic stability of two nonlinear difference equations, Fasciculi Mathematici, 43 (2010), 171-180.
23. I. Yalcinkaya and D. T. Tollu, Global behavior of a second- order system of difference equations, Advanced Studies in Contemporary Mathematics, 26(4)(2016), 653-667.
24. Y. Yazlik, On the solutions and behavior of rational difference equations, Journal of Computational Analysis & Applications, 17(3)(2014), 584-594.
25. Y. Yazlik, E. M. Elsayed and N. Taskara, On the behaviour of the solutions of difference equation systems, Journal of Computational Analysis & Applications, 16(5)(2014), 932- 941.
26. Y. Yazlik, D. T. Tollu and N. Taskara, On the behaviour of solutions for some systems of difference equations, Journal of Computational Analysis & Applications, 18(1)(2015), 166 -178 .
27. Y. Yazlik, D. T. Tollu and N. Taskara, On the solutions of a max-type difference equation system, Mathematical Methods in the Applied Sciences, 38(17)(2015), 4388-4410.
28. Y. Yazlik, D. T. Tollu and N. Taskara, On the solutions of a three-dimensional system of difference equations, Kuwait Journal of Science 43(1)(2016), 95-111.
29. L. Xianyi and Z. Deming, Global asymptotic stability in a rational equation, Journal of Difference Equations and Applications, 9(9)(2003), 833-839.
30. L. Xianyi and Z. Deming, Two rational recursive sequences, Computers & Mathematics with Applications, 47 (2004) 1487-1494.
31. X. Wang and Z. Li, Global asymptotic stability for two kinds of higher order recursive sequences, Journal of Difference Equations and Applications, 22(10)(2016), 1542-1553, DOI: 10.1080/10236198.2016.1216111.
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