On a solvable system of difference equations of higher-order with period two coefficients

Yıl 2019, Cilt: 68 Sayı: 2, 1675 - 1693, 01.08.2019

Yasin Yazlik , Merve Kara

https://doi.org/10.31801/cfsuasmas.548262

Cited By: 30

Öz

We show that the next difference equations system

x_{n+1}=((a_{n}x_{n-k+1}y_{n-k})/(y_{n}-α_{n}))+β_{n+1}, y_{n+1}=((b_{n}y_{n-k+1}x_{n-k})/(x_{n}-β_{n}))+α_{n+1}, n∈N₀,

where N₀=N∪{0}, the sequences (a_{n})_{n∈N₀}, (b_{n})_{n∈N₀}, (α_{n})_{n∈N₀}, (β_{n})_{n∈N₀} are two periodic and the initial conditions x_{-i}, y_{-i}  i∈{0,1,…,k}, are non-zero real numbers, can be solved. Also, we investigate the behavior of solutions of above mentioned system when a₀=b₁ and a₁=b₀.

Anahtar Kelimeler

Periodic solution, solution in closed form system of difference equations

Kaynakça

• Belhannache, F., Touafek, N. and Abo-Zeid, R., On a higher order rational difference equations, Journal of Applied Mathematics and Informatics, 34(5-6), (2016), 369--382.
• Dekkar, I., Touafek, N. and Yazlik, Y., 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, 111, (2017), 325--347.
• Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M., Qualitative behavior of higher order difference equation, Soochow J. Math, 33, (2007), 861--873.
• Cinar, C., Toufik, M. and Yalcinkaya, I., On the difference equation of higher order, Utilitas Mathematica, 92, (2013), 161--166.
• Cinar, C., On the positive solutions of the difference equation x_{n+1}=((x_{n-1})/(1+x_{n}x_{n-1})), Applied Mathematics and Computation, 150(1), (2004), 21--24.
• Elaydi, S., An Introduction to Difference Equations, Springer, New York, 1996.
• El-Dessoky, M. M., The form of solutions and periodicity for some systems of third-order rational difference equations, Mathematical Methods in the Applied Sciences, 39(5), (2016), 1076--1092.
• Elsayed, E. M. and Ahmed, A. M., Dynamics of a three dimensional system of rational difference equations, Mathematical Methods in the Applied Sciences, 39(5), (2016), 1026--1038.
• Elsayed, E. M., Alotaibi, A. and Almaylabi, H. A., On a solutions of fourth order rational systems of difference equations, Journal of Computational Analysis and Applications, 22(7), (2017), 1298-1308.
• Elsayed, E. M., On the solutions and periodic nature of some systems of difference equations, International Journal of Biomathematics, 7(6), (2014), 1--26.
• Gelisken, A. and Kara, M., Some general systems of rational difference equations, Journal of Difference Equations, 396757,(2015), 1--7.
• Grove, E. A. and Ladas, G., Periodicities in nonlinear difference equations, Advances in Discrete Mathematics and Applications, vol. 4. Chapman & Hall/CRC, London, 2005.
• Haddad, N., Touafek, N. and Rabago, J. F. T., Well-defined solutions of a system of difference equations, Journal of Appl. Math. and Comput., DOI: 10.1007/s12190-017-1081-8, (2017), 1--20.
• Haddad, N., Touafek, N. and Rabago, J.F.T., Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Mathematical Methods in the Applied Science, 40(10), (2017), 3599--3607.
• Halim, Y., Touafek, N. and Yazlik, Y., Dynamic behavior of a second-order nonlinear rational difference equation, Turk. J. Math, 39(6), (2015), 1004--1018.
• Ibrahim, T.F. and Touafek, N., On a third order rational difference equation with variable coefficients, Dynamics of Continuous Discrete and Impulsive Systems Series B, 20, (2013), 251--264.
• Kulenovic, M. R. S. and Ladas, G., Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall, CRC Press, London, 2001.
• Psarros, N., Papaschinopoulos, G. and Schinas, C.J., Semistability of two systems of difference equations using centre manifold theory, Mathematical Methods in the Applied Sciences, 39(18), (2016), 5216--5222.
• Rabago, J.F.T. and Bacani, J.B., On a nonlinear difference equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 24, (2017), 375--394.
• Stević, S., More on a rational recurrence relation, Appl. Math, E-Notes 4, (2004), 80--85.
• Stević, S., On a system of difference equations with period two coefficients, Applied Mathematics and Computation, 218, (2011), 4317--4324.
• Stević, S., On the difference equation x_{n}=((x_{n-2})/(b_{n}+c_{n}x_{n}x_{n-2})), Appl. Math. Comput., 218, (2011), 4507--4513.
• Stević, S., On a solvable rational system of difference equations, Appl. Math. Comput. 219, (2012), 2896--2908.
• Stević, S., Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electronic Journal of Qualitative Theory of Differantial Equations, (67), (2014), 1--15.
• Stević, S., Alghamdi, M. A., Shahzad, N. and Maturi, D. A., On a class of solvable difference equations, Abstract and Applied Analysis, (2013), Article ID: 157-943, 7 pages.
• Stević, S., On a solvable system of difference equations of kth order, Appl. Math. Comput., 219, (2013), 7765--7771.
• Stević, S., Diblik, J., Iricanin, B. and Smarda, Z., On a solvable system of rational difference equations, Journal of Difference Equations and Applications, 20(5-6), (2014), 811--825.
• Taskara, N., Tollu, D. T. and Yazlik, Y., Solutions of rational difference system of order three in terms of Padovan numbers, Journal of Advanced Research in Applied Mathematics, 7(3), (2015), 18--29.
• Tollu, D. T., Yazlik, Y. and Taskara, N., On fourteen solvable systems of difference equations, Applied Mathematics and Computation, 233, (2014), 310--319.
• Tollu, D. T., Yazlik, Y. and Taskara, N., Behavior of positive solutions of a difference equation, Journal of Applied Mathematics and Informatics, 35, (2017), 217--230.
• Tollu, D.T., Yazlik, Y. and Taskara, N., On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Advances in Difference Equations, 2013:174, (2013), 1-7.
• Tollu D. T., Yazlik Y. and Taskara N., On a solvable nonlinear difference equation of higher-order, Turkish Journal of Mathematics, 42(4), (2018), 1765-1778.
• Tollu, D. T. and Yalcinkaya, I., Global behavior of a three-dimensional system of difference equations of order three, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(1), (2019), 1--16.
• Touafek, N., On a second order rational difference equation, Hacet. J. Math. Stat., 41, (2012), 867--874.
• Yalcinkaya, I., Hamza, A. E. and Cinar, C., Global behavior of a recursive sequence, Selçuk Journal of Applied Mathematics, 14(1), (2013), 3--10.
• Yalcinkaya, I. and Tollu, D. T., Global behavior of a second order system of difference equations, Advanced Studies in Contemporary Mathematics, 26(4), (2016), 653--667.
• Yazlik, Y., On the solutions and behavior of rational difference equations, J. Comput. Anal. Appl., 17(3), (2014), 584--594.
• Yazlik, Y., Elsayed, E. M. and Taskara, N., On the behaviour of the solutions the solutions of difference equation system, J. Comput. Anal. Appl., 16(5), (2014), 932--941.
• Yazlik, Y., Tollu, D. T. and Taskara, N., On the behaviour of solutions for some systems of difference equations, J. Comput. Anal. Appl., 18(1), (2015), 166--178.
• Yazlik, Y., Tollu, D. T. and Taskara, N., On the solutions of a three-dimensional system of difference equations, Kuwait Journal of Science, 43(1), (2015), 95--111.