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Zbl 1219.34115
Grace, Said R.; Agarwal, Ravi P.; Pinelas, Sandra
Comparison and oscillatory behavior for certain second order nonlinear dynamic equations.
(English)
[J] J. Appl. Math. Comput. 35, No. 1-2, 525-536 (2011). ISSN 1598-5865; ISSN 1865-2085/e

The authors consider the second order nonlinear dynamic equation $$\left(a(x^{\Delta})^{\alpha}\right)^{\Delta}(t)+q(t)x^{\beta}(t)=0$$ on an arbitrary time scale $\Bbb T$, where $\alpha$ and $\beta$ are ratios of positive odd integers, $a$ and $q$ are positive rd-continuous functions on $\Bbb T$. They establish comparison results with the inequality $$\left(a(x^{\Delta})^{\alpha}\right)^{\Delta}(t)+q(t)x^{\beta}(t)\leq 0$$ which are applied to neutral equations. A necessary and sufficient condition is obtained for the oscillation property of second order equations on time scales.
MSC 2000:
*34N05
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: dynamic equation; time scale; oscillation theory; non oscillations; comparison theorems

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