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Zbl 0802.39001
Kocic, V.Lj.; Ladas, G.
Global attractivity in a second-order nonlinear difference equation. (English)
[J] J. Math. Anal. Appl. 180, No.1, 144-150 (1993). ISSN 0022-247X

The following difference equation $x\sb{n+1}= x\sb n f(x\sb{n - 1})$, $n = 0, 1, 2,\dots$, is considered. The authors suppose that the function $f$ satisfies the following conditions:\par (i) $f \in C\bigl[[0,\infty],(0,\infty)\bigr]$ and $f(u)$ is nonincreasing in $u$;\par (ii) the equation $f(x)=1$ has a unique positive solution;\par (iii) if $\widetilde{x}$ denotes the unique positive solution of $f(x) = 1$ then $[x f(x) -\widetilde{x}] (x-\widetilde{x}) > 0$, $x \ne \widetilde{x}$.\par They prove that in these conditions $\widetilde{x}$ is a global attractor of all positive solutions of the equation.
[S.Balint (Timişoara)]

MSC 2000:: *39A10 Difference equations

Keywords: second-order nonlinear differential equation; global attractor; positive solutions

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