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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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