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Asymptotic behaviour of a difference equation with complex-valued coefficients. (English) Zbl 1122.39006
Motivated by his previous results concerning the differential equation $$z'= f(t, z)$$, the author investigates the first order difference equation $$\Delta z_n = f(n, z_n)$$, where $$z_n \in \Omega$$, $$\Omega$$ being a simply connected region in $$\mathbb C$$ containing $$z=0$$. It is supposed that the function $$f(n, z)$$ is for every $$n\in \mathbb N$$ closed, in a certain sense, to a function $$h(z)$$ satisfying $$h'(0)\neq 0$$ and $$h(z)=0$$ iff $$z=0$$. Using the function $$h$$, a certain Lyapunov like function $$V$$ and certain subsets in $$\Omega$$ are defined and then used to study asymptotic properties of solutions of $$\Delta z_n = f(n, z_n)$$.
##### MSC:
 39A22 Growth, boundedness, comparison of solutions to difference equations
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