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).

MSC:

 39A11 Stability of difference equations (MSC2000) 39B05 General theory of functional equations and inequalities
Full Text:

References:

 [1] DOI: 10.1006/jmaa.1999.6346 · Zbl 0962.39004 [2] Grove, E. A., Kent, C. M., Levins, R., Ladas, G. and Valicenti, S. 1999. Global Stability in Some Population Models. Proceedings of the Fourth International Conference on Difference Equations and Applications. August27-311999, Poznan, Poland. pp.149–176. 2000: Gordon and Breach Science Publishers. · Zbl 0988.39018 [3] Gibbons C. H., Math. Sci. Res. Hot-Line 4 pp 1– (2000) [4] Gibbons, C. H., Kulenovic, M. R. S. and Ladas, G. 2000. On the Dynamics of xn+1=((a+ßxn+?xn-1)/(A+Bxn)). Proceedings of the Fifth International Conference on Difference Equations and Applications. Jan.3-72000, Temuco, Chile. Gordon and Breach Science Publishers. (to appear) [5] Kocic V. L., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001 [6] Kulenovié M. R. S., Dynamics of Second Order Rational Difference Equations [7] DOI: 10.1080/10236190008808246 · Zbl 0966.39003 [8] DOI: 10.1080/10236190008808249 · Zbl 0970.39007 [9] DOI: 10.1080/10236199508808030 · Zbl 0855.39006 [10] DOI: 10.1080/10236199908808181 · Zbl 0927.39001
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.