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Global attractivity in a second-order nonlinear difference equation. (English) Zbl 0802.39001
The following difference equation \(x_{n+1}= x_ n f(x_{n - 1})\), \(n = 0, 1, 2,\dots\), is considered. The authors suppose that the function \(f\) satisfies the following conditions:
(i) \(f \in C\bigl[[0,\infty],(0,\infty)\bigr]\) and \(f(u)\) is nonincreasing in \(u\);
(ii) the equation \(f(x)=1\) has a unique positive solution;
(iii) if \(\widetilde{x}\) denotes the unique positive solution of \(f(x) = 1\) then \([x f(x) -\widetilde{x}] (x-\widetilde{x}) > 0\), \(x \neq \widetilde{x}\).
They prove that in these conditions \(\widetilde{x}\) is a global attractor of all positive solutions of the equation.

MSC:
39A10 Additive difference equations
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