zbMATH — the first resource for mathematics

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

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
stable manifold
Full Text: DOI
[1] DOI: 10.1016/S0362-546X(02)00294-8 · Zbl 1019.39006
[2] DOI: 10.1080/10236190211940 · Zbl 1005.39017
[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
[5] DOI: 10.1201/9781420035384
[6] 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
[9] Robinson C., Stability, Symbolic Dynamics, and Chaos (1995)
[10] DOI: 10.1080/10236199708808108 · Zbl 0907.39004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.