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Basin of attraction through invariant curves and dominant functions. (English) Zbl 1418.39013
Summary: We study a second-order difference equation of the form $$z_{n + 1} = z_n F(z_{n - 1}) + h$$, where both $$F(z)$$ and $$z F(z)$$ are decreasing. We consider a set of invariant curves at $$h = 1$$ and use it to characterize the behaviour of solutions when $$h > 1$$ and when $$0 < h < 1$$. The case $$h > 1$$ is related to the Y2K problem. For $$0 < h < 1$$, we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.
##### MSC:
 39A30 Stability theory for difference equations 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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##### References:
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