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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.

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
Full Text: DOI
[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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