On certain comparison theorems for half-linear dynamic equations on time scales

Abstract

We obtain comparison theorems for the second-order half-linear dynamic equation $\big[r(t)\Phi \big(y^{\Delta}\big)\big]^{\Delta}+p(t)\Phi\big(y\sig\big)=0,$, where $\Phi(x)=|x|^{\alpha-1}\mathrm{sgn} x$ with $\alpha>1$. In particular, it is shown that the nonoscillation of the previous dynamic equation is preserved if we multiply the coefficient $p(t)$ by a suitable function $q(t)$ and lower the exponent $\alpha$ in the nonlinearity $\Phi$, under certain assumptions. Moreover, we give a generalization of Hille-Wintner comparison theorem. In addition to the aspect of unification and extension, our theorems provide some new results even in the continuous and the discrete case.