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Zbl 1243.39009
Berg, Lothar; SteviÄ‡, Stevo
On some systems of difference equations.
(English)
[J] Appl. Math. Comput. 218, No. 5, 1713-1718 (2011). ISSN 0096-3003

The authors consider the following three systems of nonlinear difference equations $$u_{n+1}=\frac{v_n}{1+v_n},\quad v_{n+1}=\frac {u_n}{1+u_n},$$ $$u_{n+1}=\frac{v_n}{1+u_n},\quad v_{n+1}=\frac {u_n}{1+v_n},$$ and $$u_{n+1}=\frac{u_n}{1+v_n},\quad v_{n+1}=\frac {v_n}{1+u_n},$$ where $n\in {\mathbb N}_0$ and the initial values $u_0$ and $v_0$ are given complex numbers. By help of the auxiliary Riccati equation $$x_{n+1}=\frac{x_n}{a+bx_n}, \quad a,\,b\in \mathbb C,\,\,n\in {\mathbb N}_0,$$ the general solution of each system is explicitly given. As a consequence, the asymptotic behaviour of the solutions is analysed in detail as $n\rightarrow \infty$.
[Vu Hoang Linh (Hanoi)]
MSC 2000:
*39A20 Generalized difference equations
39A22
39A23
39A30

Keywords: system of rational difference equations; asymptotics; Riccati equation

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