On The Solutions of Three-Dimensional Difference Equation Systems Via Pell Numbers

Year 2022, Issue: 34, 433 - 440, 31.03.2022

Necati Taşkara , Hüseyin Büyük

https://doi.org/10.31590/ejosat.1082643

Abstract

In this study, we investigate the form of the solutions of the following rational difference equation system
x_n=(z_(n-1) z_(n-3))/(x_(n-2)+2z_(n-3) ),y_n=(x_(n-1) x_(n-3))/(〖-y〗_(n-2)+6x_(n-3) ),z_n=(y_(n-1) y_(n-3))/(z_(n-2)+14y_(n-3) ) ,n∈N_0
where initial values〖 x〗_(-3) 〖,x〗_(-2), x_(-1),y_(-3),y_(-2),y_(-1),〖 z〗_(-3),〖 z〗_(-2),〖 z〗_(-1) are nonzero real numbers, such that their solutions are associated with Pell numbers. We also give a relationships between Pell numbers and solutions of systems

Keywords

System of difference equations, Pell numbers, Representation of solutions, Binet formula, Solutions

References

• Cinar C., On the positive solutions of the difference equation x(n+1 )= x(n−1)/ 1 + ax(n)x(n−1), Applied Mathematics and Computation, 2004, 158 (3): 809-812.
• Tollu D.T., Yazlik Y., Taskara N., 2018, On a solvable nonlinear difference equations of higher order, Turkish Journal of Mathematics, 2004, 42: 1765-1778.
• Elsayed, E. M., On the solution of recursive sequence of order two, Fasciculi Mathematici, 2008a, 40: 6-13.
• Şahinkaya A.F., Yalçınkaya İ., Tollu D.T., A solvable system of nonlinear difference equations, Ikonion Journal of Mathematics, 2020, 2: 10-20.
• Halim,Y., Berkal,M., Khelifa, A., On a three dimensional solvable system of difference equations,Turk J Math, 2020, 44:2001-40.doi:10.3906.
• Yazlik,Y., Tollu,D. T., Taskara,N., On the solutions of difference equation systems with Padovan numbers, Appl. Math., 2013, 4:15-20.
• Tollu D.T., Yazlık Y., Taskara N., On fourteen solvable systems of difference equations, Applied Mathematics and Computation, 2014, 233: 310-319.
• Okumus I., Soykan Y., On the solutions of four rational difference equations associated to Tribonacci numbers, Hacettepe Journal of Mathematics & Statistics, 2019, DOI: 10.15672/HJMS.xx.
• Stević S., More on a rational recurrence relation, Applied Mathematics E-Notes; 2004, 4 (1): 80-85.
• Tollu D.T., Yazlık Y., Taskara N., On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Differ. Equ., 2013, 4: 174.
• Touafek N., On a second order rational difference equation, Hacettepe Journal of Mathematics and Statistics, 2012, 41: 867-874.
• Zeng, X. Y., Shi, B., Zhang, D. C., Stability of solutions for the recursive sequence, Journal of Computational and Applied Mathematics, 2005, 176(2): 283-291.
• Tollu, D. T., Yazlik, Y. and Taskara, N., On global behavior of a system of nonlinear difference equations of order two, Advanced Studies in Contemporary Mathematics, 2017, 27(3): 373-383.
• Elsayed, E. M., Alotaibi, A. and Almaylabi, A. H., On a solutions of fourth order rational systems of difference equations, Journal of Computational Analysis and Applications, 2017, 7(22): 1298-1308.
• Kurbanli, A. S., Çinar, C. and Yalçinkaya, I., On the behavior of positive solutions of the system of rational difference equations, Mathematical and Computer Modelling, 2011, 53(5-6):1261-1267.
• Akrour Y, Touafek N, Halim Y., On a system of difference equations of second order solved in closed-form. Miskolc Mathematical Notes 20, 2019, (2): 701-717.
• Kara M, Yazlik Y., Solvability of a system of nonlinear difference equations of higher order. Turkish Journal of Mathematics 43, 2019, (3): 1533-1565. doi: 10.3906/mat-1902-24.
• Kara M, Yazlik Y, Tollu DT., Solvability of a system of higher order nonlinear difference equations. Hacettepe Journal of Mathematics & Statistics , 2020, doi: 10.15672/HJMS.xx.
• Yılmazyıldırım B., Tollu D.T., Explicit solutions of a three-dimensional system of nonlinear difference equations, Hittite Journal of Science and Engineering, 2018, 5(2): 119-123.
• Aloqeili M., Dynamics of a rational difference equation. Applied Mathematics and Computation, 2006,176 (1): 768-774.
• Halim Y., A system of difference equations with solutions associated to Fibonacci numbers. International Journal of Difference Equations , 2016, 11: 65-77.
• Khelifa A, Halim Y, Berkal M., Solutions of a system of two higher-order difference equations in terms of Lucas sequence. Universal Journal of Mathematics and Applications , 2019, 2(4): 202-211.
• I. Okumus¸, Y. Soykan , A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers , Communications in Advanced Mathematical Sciences Vol. II, 2019,2:291/292.
• O. Ocalan, O. Duman, On solutions of the recursive equations via Fibonacci-type sequences, Elect. J. Math. Anal. Appl., 2018, 7(1): 102-115.
• Y. Halim, J. F. T. Rabago,On the solutions of a second-order difference equation in terms of generalized Padovan sequences, Math. Slovaca, 2018, 68(3): 625-638.
• Tollu D.T., Yazlık Y., Taskara N., The Solutions of Four Riccati Difference Equations with Fibonacci Numbers, Balkan Journal Of Mathematics, 2014, 02:163-172
• Taskara, N., Uslu, K. and Tollu, DT, The periodicity and solutions of the rational difference equation with periodic coefficients, Computers & Mathematics with Applications, 2011, 62 (4) : 1807-1813