Oscillation for Higher Order Dynamic Equations on Time Scales

Abstract

We investigate the oscillation of the following higher order dynamic equation: $\{a_n(t)[(a_{n-\mathrm{1}}(t)(\cdots (a_{\mathrm{1}}(t)x^{\mathrm{\Delta }}(t){)}^{\mathrm{\Delta }}\cdots {)}^{\mathrm{\Delta }}{)}^{\mathrm{\Delta }}{]}^{\alpha }{\}}^{\mathrm{\Delta }}+p(t)x^{\beta }(t)=\mathrm{0}$, on some time scale $\mathit{T}$, where $n\ge \mathrm{2}$, $a_k(t)$ $\mathrm{}\mathrm{}(\mathrm{1}\le k\le n)$ and $p(t)$ are positive rd-continuous functions on $\mathit{T}$ and $\alpha ,\beta$ are the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.