Forced Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

Abstract

We will establish a new interval oscillation criterion for second-order half-linear dynamic equation $\left(r\right(t\left)\right[x^{\Delta }\left(t\right){]}^{\alpha }{)}^{\Delta }+p\left(t\right)x^{\alpha }\left(\sigma \right(t\left)\right)=f\left(t\right)$ on a time scale $\mathbb{T}$ which is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second-order half-linear differential equation $\left(\right[x^{\prime }\left(t\right){]}^{\alpha }{)}^{\prime }+c\sin t{x}^{\alpha }\left(t\right)=\cos t$, where $\alpha =p/q$, $p$, $q$ are odd positive integers.