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.