A note on unbounded solutions of a class of second order rational difference equations. (English) Zbl 1107.39007

Kulenović, M. R. S.; Merino, O.

J. Difference Equ. Appl. 12, No. 7, 777-781 (2006).

The authors study the difference equation \[ x_{n+1}=\frac{\alpha+\beta x_n +\gamma x_{n-1}}{A+B x_n +C x_{n-1}}, \] where all coefficients and initial conditions are nonnegative, \(A+B x_n +C x_{n-1}>0\) for all \(n\). A characterization of unbounded solutions for this equation is presented.
The paper aswers two open problems posed by M. R. S. Kulenović and G. Ladas [Dynamics of second order rational difference equations. With open problems and conjectures, London: Chapman and Hall/CRC (2001; Zbl 0981.39011)]

Reviewer: Oleg Anashkin (Simferopol)

MSC:

39A11	Stability of difference equations (MSC2000)
39A20	Multiplicative and other generalized difference equations, e.g., of Lyness type

Keywords:

stable manifold

References:

[1]	DOI: 10.1016/S0362-546X(02)00294-8 · Zbl 1019.39006 · doi:10.1016/S0362-546X(02)00294-8
[2]	DOI: 10.1080/10236190211940 · Zbl 1005.39017 · doi:10.1080/10236190211940
[3]	Hirsch, M. and Smith, H. 2005.Monotone Dynamical Systems, Handbook of Differential Equations, Ordinary Differential Equations, 239–357. Amsterdam: Elsevier B.V. · Zbl 1094.34003
[4]	DOI: 10.1080/10236190412331285360 · Zbl 1064.39006 · doi:10.1080/10236190412331285360
[5]	DOI: 10.1201/9781420035384 · doi:10.1201/9781420035384
[6]	DOI: 10.1201/9781420035353 · doi:10.1201/9781420035353
[7]	M.R.S. Kulenović and Merino, O., Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete Cont. Dynamical Syst. Ser. B (to appear). · Zbl 1116.37030
[8]	DOI: 10.1155/JIA.2005.127 · Zbl 1086.39008 · doi:10.1155/JIA.2005.127
[9]	Robinson C., Stability, Symbolic Dynamics, and Chaos (1995)
[10]	DOI: 10.1080/10236199708808108 · Zbl 0907.39004 · doi:10.1080/10236199708808108