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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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