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Zbl 1005.39017
Gibbons, C.H.; KulenoviÄ‡, M.R.S.; Ladas, G.; Voulov, H.D.
On the trichotomy character of $x_{n+1}=(\alpha+\beta x_n+\gamma x_{n-1})/(A+x_n)$.
(English)
[J] J. Difference Equ. Appl. 8, No.1, 75-92 (2002). ISSN 1023-6198

An analysis of the periodicity, convergence and boundedness of the solutions of the second order difference equation in the title is presented. The parameters $\alpha$, $\beta$, $\gamma$, $A$ and the initial conditions $x_{-1}$ and $x_{0}$ are nonnegative and the denominator is always positive. The main result is the following. \par Theorem 1. (a) Assume that $\gamma =\beta +A.$ Then every solution of the given equation converges to a period two solution. (b) If $\gamma >\beta +A$, then the equation possesses unbounded solutions. (c) If $\gamma <\beta +A,$ then every solution of the equation has a finite limit. \par In the case (b), the difference equation has a positive equilibrium which is a saddle point. In case (c), the equation may possess a unique equilibrium (zero or positive) which is globally asymptotically stable, or two equilibrium points of which one is zero (and this is unstable) and the other is positive (locally asymptotically stable).
[N.C.Apreutesei (Iasi)]
MSC 2000:
*39A11 Stability of difference equations
39B05 General theory of functional equations

Keywords: global attractivity; global asymptotic stability; period two solutions; trichotomy of solutions; bounded solution; convergence; unbounded solutions; positive equilibrium; saddle point

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