Oscillation Properties for Second-Order Half-Linear Dynamic Equations on Time Scales

Daxue Chen

Abstract


In this paper, we are concerned with the oscillation of the second-order half-linear dynamic equation
\[
\Big(a(t)|x^\Delta(t)|^{\gamma-1}x^\Delta (t)\Big)^\Delta+q(t)|x(t)|^{\gamma-1}x(t)=0
\]
on an arbitrary time scale $\mathbb{T}$, where $\gamma>0$ is a constant. By using a generalized Riccati substitution, the P\"{o}tzsche chain rule and a Hardy-Littlewood-P\'{o}lya inequality, we obtain some sufficient conditions for the oscillation of the equation and improve and extend some known results in which $\gamma>0$ is a quotient of odd positive integers. We also give some examples to illustrate our main results.


Full Text: PDF DOI: 10.5539/jmr.v4n2p90

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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