## 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.

### MSC:

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

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