Half-linear dynamic equations on time scales: IVP and oscillatory properties

In this paper we show how the basic results of oscillation theory of the Sturm--Liouville linear differential equation $$ (r(t)y')'+p(t)y=0 $$ can be extended to the half-linear dynamic equation $$ (r(t)\Phi(y^\Delta))^\Delta+p(t)\Phi(y^\sigma)=0 \tag{HL$^\Delta$E} $$ on an arbitrary time scale, where $\Phi(x)=|x|^{\alpha-1}\sgn x$ with $\alpha>1$. In particular, the generalization of the so called Roundabout theorem is proved for equation (HL$^\triangle$E), which provides powerful tools for the investigation of oscillatory properties of this equation, namely the Riccati technique and variational principle. As an application we present Sturmian theory, oscillation and nonoscillation criteria for (HL$^\Delta$E). The questions concerning the existence and uniqueness of a solution of initial value problem are also discussed.