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Zbl 1232.39014
Bajo, Ignacio; Liz, Eduardo
Global behaviour of a second-order nonlinear difference equation.
(English)
[J] J. Difference Equ. Appl. 17, No. 10, 1471-1486 (2011). ISSN 1023-6198

The article deals with the Cauchy problem $$x_{n+1} = \frac{x_{n-1}}{a + bx_nx_{n-1}}, \qquad (x_{-1},x_0) \in {\Bbb R}^2, \ n = 0,1,2,\dots,$$ where $a, b$ are reals. It is proved that the general solution to this equation (beside some exceptional cases) is presented in the form $$x_{2k+2} = x_0 \, \prod_{i=0}^k h(2i + 1), \quad x_{2k+1} = x_{-1} \, \prod_{i=0}^k h(2i),$$ where $$h(n) =\cases \frac{a^n(1 - a) + \alpha(-a^n)}{a^{n+1}(1 - a) + \alpha(1 - a^{n+1}}, & \text{if} \quad a \ne 1, \\ \frac{1 + \alpha n}{1 + \alpha(n + 1)}, & \text{if} \quad a = 1.\endcases$$ ($\alpha = bx_{-1}x_0$). Further, a complete analysis of the asymptotic behavior of these solutions in different cases ($a = -1$; $|a| \ge 1, \ a \ne -1$; $|a| < 1$) is given, the behavior of these solutions near bifurcation points ($a = -1$ and $a = 1$) s described and the stability properties of nonzero periodic solutions is investigated.
[Peter Zabreiko (Minsk)]
MSC 2000:
*39A20 Generalized difference equations
39A30
39A28
39A22

Keywords: rational difference equations; asymptotic behaviour; stability; Cauchy problem

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