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Monotone properties of certain classes of solutions of second-order difference equations. (English) Zbl 0933.39014

Summary: The authors consider the difference equations \[ \Delta(a_n\Delta x_n)=q_nx_{n+1} \tag{*} \] and \[ \Delta(a_n\Delta x_n)= q_nf(x_{n+1}), \tag{**} \] where \(a_n>0\), \(q_n>0\), and \(f:\mathbb{R} \to\mathbb{R}\) is continuous with \(uf(u)>0\) for \(u\neq 0\). They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.

MSC:

39A11 Stability of difference equations (MSC2000)
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References:

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