Oscillation properties of an Emden-Fowler type equation on discrete time scales. (English) Zbl 1038.39009

The authors consider the second order dynamic equation \(u^{\Delta ^{2}}+p(t)u^{\gamma }(\sigma (t))=0\), where \( \gamma\) is a quotient of odd positive integers. They establish some necessary and sufficient conditions for oscillations. When \(\gamma >1\) they prove that every solution oscillates if and only if \(\int_{a}^{\infty }\sigma (l)p(l)\Delta l=\infty \) and when \(0<\gamma <1\) they prove that every solution oscillates if and only if \(\int_{a}^{\infty }\left[ \sigma (l)\right] ^{\gamma }p(l)\Delta l=\infty \). The results are applied only to discrete time scales. Some examples illustrating the main results are given.


39A12 Discrete version of topics in analysis
93C70 Time-scale analysis and singular perturbations in control/observation systems
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