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**On the trichotomy character of \(x_{n+1}=(\alpha+\beta x_n+\gamma x_{n-1})/(A+x_n)\).**
*(English)*
Zbl 1005.39017

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.

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.

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).

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.

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).

Reviewer: N.C.Apreutesei (Iasi)

### MSC:

39A11 | Stability of difference equations (MSC2000) |

39B05 | General theory of functional equations and inequalities |

### 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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\textit{C. H. Gibbons} et al., J. Difference Equ. Appl. 8, No. 1, 75--92 (2002; Zbl 1005.39017)

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### References:

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