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On the recursive sequence $$x_{n+1}={\alpha+\beta x_ n+\gamma x_{n-1}\over A+Bx_ n+Cx_{n-1}}$$. (English) Zbl 0855.39006
The author presents five open problems and three conjectures about the recursive sequence in the title. We give two examples: Conjecture 3.3. Assume that $$B\in[0,\infty)$$ and $$\alpha,\beta, A, C\in (0,\infty)$$. Then the positive equilibrium of the equation in the title (with $$\gamma = 0)$$ is globally asymptotically stable. Open problem 3.3. Obtain necessary and sufficient conditions in terms of $$\alpha,\beta,\gamma, A, B, C$$, for every positive solution of the equation in the title to be bounded.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 39A10 Additive difference equations
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##### References:
 [1] Brand L., A Sequence Defined by a Difference Equation. Amer. Math. Monthly 62 pp 489– (1955) [2] Jaroma J. H., Global Asymptotic Stability of a Delay Difference Equation · Zbl 0860.39016 [3] Kocic V. L., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001 [4] Kocic V. L., Communications on Pure and Applied Mathematics [5] DOI: 10.1006/jmaa.1993.1057 · Zbl 0777.39002 · doi:10.1006/jmaa.1993.1057
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