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Characterizing conditions for the existence of nonoscillatory solutions of a discrete Emden-Fowler equation with summable coefficients. (English) Zbl 0971.39003

The author presents necessary and sufficient conditions for the existence of nonoscillatory solutions for the discrete Emden-Fowler differential equation \[ \Delta ^2x_{n-1}+a_n|x_n|^\gamma \text{ sign} x_n=0,\quad n=0,1,2,... \] in the case when \(\{ a_n\}\) is a sequence of real numbers satisfying the conditions \[ -\infty <\lim_{n\rightarrow \infty}\sum_{s=0}^{n-1}a_s<\infty, \] and \(A_n\equiv \sum_{s=n}^\infty a_s\geq 0\) for all \(n=0,1,2,...\). It is analysed the sublinear case \(0<\gamma <1\) as well as the superlinear case \(\gamma >1\).

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
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