Solvability of a system of higher order nonlinear difference equations

Year 2020, Volume: 49 Issue: 5, 1566 - 1593, 06.10.2020

Merve Kara , Yasin Yazlik , Durhasan Turgut Tollu

https://doi.org/10.15672/hujms.474649

Cited By: 18

Abstract

In this paper we show that the system of difference equations

\[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-( k+l ) }}{b x_{n-(k+l)}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-(k+l)}}{\beta y_{n-(k+l)}}+\gamma x_{n-l}, \]  

where $n\in \mathbb{N}_{0},$ $k$ and $l$ are positive integers, the parameters $a$, $b$, $c$, $d$, $\alpha $, $\beta $, $\gamma $, $\delta $ are real numbers and the initial values $x_{-j}$, $y_{-j}$, $j=\overline{1,k+l}$, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case $l=1$ and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

Keywords

System of difference equations, Asymptotic behavior, Fibonacci sequence, Forbidden set

References

• [1] R.P. Agarwal, Difference Equations and Inequalities, New York USA, Marcel Dekker, 1992.
• [2] R.P. Agarwal and E.M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv. Stud. Contemp. Math. 20 (4), 525–545, 2010.
• [3] I. Bajo and E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Difference Equ. Appl. 17 (10), 1471–1486, 2011.
• [4] L. Brand, A sequence defined by a difference equation, Am. Math. Mon. 62 (7), 489– 492, 1955.
• [5] E. Camouzis and R. DeVault, The forbidden set of $x_{n+1}=p+\frac{x_{n-1}}{x_{n}}$, J. Difference Equ. Appl. 9 (8), 739–750, 2003.
• [6] C. Cinar, On the positive solutions of difference equation, Appl. Math. Comput. 150 (1), 21–24, 2004.
• [7] C. Cinar, S. Stevic, and I. Yalcinkaya, On positive solutions of a reciprocal difference equation with minimum, J. Appl. Math. Comput. 17 (1-2), 307–314, 2005.
• [8] M. Dehghan, R. Mazrooei-Sebdani, and H. Sedaghat, Global behaviour of the Riccati difference equation of order two, J. Difference Equ. Appl. 17 (4), 467–477, 2011.
• [9] I. Dekkar, N. Touafek, and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 111 (2), 325–347, 2017.
• [10] Q. Din, M.N. Qureshi and A.Q. Khan, Dynamics of a fourth-order system of rational difference equations, Adv. Difference Equ. 2012 (215), 1–15, 2012.
• [11] E.M. Elabbasy, H.A. El-Metwally, and E.M. Elsayed, Global behavior of the solutions of some difference equations, Adv. Difference Equ. 2011 (1), 1–16, 2011.
• [12] M.E. Elmetwally and E.M. Elsayed, Dynamics of a rational difference equation, Chin. Ann. Math. Ser. B 30 (2), 187–198, (2009).
• [13] E.M. Elsayed, Qualitative behavior of a rational recursive sequence, Indag. Math. 19 (2), 189–201, 2008.
• [14] E.M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta Appl. Math. 110 (2), 589–604, 2010.
• [15] E.A. Grove and G. Ladas, Periodicities In Nonlinear Difference Equations, Chapman & Hall, CRC Press, Boca Raton, 2005.
• [16] 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 (10), 3599–3607, 2017.
• [17] N. Haddad, N. Touafek, and J.F.T. Rabago, Well-defined solutions of a system of difference equations, J. Appl. Math. Comput. 56 (1-2), 439–458, 2018.
• [18] Y. Halim, N. Touafek, and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math. 39 (6), 1004–1018, 2015.
• [19] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, New York (2001).
• [20] M.R.S. Kulenovic and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, New York, NY, USA: CRC Press, 2002.
• [21] L.C. McGrath and C. Teixeira, Existence and behavior of solutions of the rational equation $x_{n+1}=\frac{ax_{n-1}+bx_{n}}{cx_{n-1}+dx_{n}}x_{n}$, Rocky Mountain J. Math. 36 (2), 649–674, 2006.
• [22] I. Okumus and Y. Soykan, Dynamical behavior of a system of three-dimensional nonlinear difference equations, Adv. Difference Equ. 2018 (223), 1–15, 2018.
• [23] Ö. Öcalan, Oscillation of nonlinear difference equations with several coefficients, Commun. Math. Anal. 4 (1), 35–44, 2008.
• [24] Ö. Öcalan and O. Akin, Oscillation properties for advanced difference equations, Novi Sad J. Math. 37 (1), 39–47, 2007.
• [25] G. Papaschinopoulos and C.J. Schinas, On a system of two difference equations, J. Math. Anal. Appl. 219 (2), 415–426, 1998.
• [26] G. Papaschinopoulos and G. Stefanidou, Asymptotic behavior of the solutions of a class of rational difference equations, Int. J. Difference Equ. 5 (2), 233–249, 2010.
• [27] A. Raouf, Global behaviour of the rational riccati difference equation of order two: the general case, J. Difference Equ. Appl. 18 (6), 947–961, 2012.
• [28] A. Raouf, Global behavior of the higher order rational riccati difference equation, Appl. Math. Comput. 230, 1–8, 2014.
• [29] J. Rubió-Massegú, On the existence of solutions for difference equations, J. Difference Equ. Appl. 13 (7), 655–664, 2007.
• [30] H. Sedaghat, Existence of solutions for certain singular difference equations, J. Difference Equ. Appl. 6 (5), 535–561, 2000.
• [31] H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Difference Equ. Appl. 15 (3), 215–224, 2009.
• [32] S. Stević, On some solvable systems of difference equations, Appl. Math. Comput. 218 (9), 5010–5018, 2012.
• [33] S. Stević, M.A. Alghamdi, N. Shahzad, and D.A. Maturi, On a class of solvable difference equations, Abstr. Appl. Anal. 2013, 1–7, 2013.
• [34] S. Stević, J. Diblík, B. Iričanin, and Z. Šmarda, On a solvable system of rational difference equations, J. Difference Equ. Appl. 20 (5-6), 811–825, 2014.
• [35] J. Sugie, Nonoscillation theorems for second-order linear difference equations- via the riccati-type transformation II, Appl. Math. Comput. 304, 142–152, 2017.
• [36] N. Taskara, K. Uslu, and D.T. Tollu, The periodicity and solutions of the rational difference equation with periodic coefficients, Comput. Math. Appl. 62 (4), 1807–1813, 2011.
• [37] D.T. Tollu, Y. Yazlik, and N. Taskara, On the solutions of two special types of riccati difference equation via Fibonacci numbers, Adv. Difference Equ. 1 (2013), 1–7, 2013.
• [38] D.T. Tollu, Y. Yazlik, and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput. 233, 310–319, 2014.
• [39] D.T. Tollu, Y. Yazlik, and N. Taskara, On a solvable nonlinear difference equation of higher order, Turkish J. Math. 42 (4), 1765–1778, 2018.
• [40] N. Touafek, On a second order rational difference equation, Hacet. J. Math. Stat. 41 (6), 867–874, 2012.
• [41] N. Touafek and E.M. Elsayed, On a second order rational systems of difference equations, Hokkaido Math. J. 44 (1), 29–45, 2015.
• [42] I. Yalcinkaya, On the difference equation $x_{n+1}=\alpha+\frac{x_{n-m}}{x_{n}^k}$, Discrete Dyn. Nat. Soc. 2008, 1–8, 2008.
• [43] X. Yang, On the system of rational difference equations $x_{n}=A+\frac{y_{n-1}}{x_{n-p}y_{n-q}}, y_{n}=A+\frac{x_{n-1}}{x_{n-r}y_{n-s}}$, J. Math. Anal. Appl. 307, 305–311, 2006.
• [44] Y. Yazlik, On the solutions and behavior of rational difference equations, J. Comput. Anal. Appl. 17 (3), 584–594, 2014.
• [45] Y. Yazlik, E.M. Elsayed, and N. Taskara, On the behaviour of the solutions of difference equation systems, J. Comput. Anal. Appl. 16 (5), 932–941, 2014.
• [46] Y. Yazlik, D.T. Tollu, and N. Taskara, On the behaviour of solutions for some systems of difference equations, J. Comput. Anal. Appl. 18 (1), 166–178, 2015.
• [47] Y. Yazlik, D.T. Tollu, and N. Taskara, On the solutions of a max-type difference equation system, Math. Methods Appl. Sci. 38 (17), 4388–4410, 2015.
• [48] Y. Yazlik, D.T. Tollu, and N. Taskara, On the solutions of a three-dimensional system of difference equations, Kuwait J. Sci. 43 (1), 95–111, 2016.