Solvability of a system of higher order nonlinear difference equations
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.
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- [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.