Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences