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Global attractivity of the equilibrium of \(x_{n+1}=\frac{px_n+x_{n-1}}{qx_n+x_{n-1}}\) for \(q < p\). (English) Zbl 1099.39007
The authors prove the global attractivity of the positive equilibrium \(\overline{x}=(p+1)/(q+1)\) of the second-order difference equation in the title with respect to positive initial values \(x_{-1}\), \(x_0\). The proof is based on the properties of a map on the plane associated to the equation. The approach is innovative and may be used to study similiar difference equations.

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
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References:
[1] Enciso G.A., Discrete Continuous Dynamical Systems · Zbl 1215.81040
[2] DOI: 10.1155/S168718390430806X · Zbl 1079.39007
[3] DOI: 10.1201/9781420035384
[4] Kulenović M.R.S., Mathematical Sciences Research Hot-Line 2 pp 1– (1998)
[5] Smith, H.L., The discrete dynamics of monotonically decomposable maps (to appear). · Zbl 1118.65057
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