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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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