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 ${\left[r\left(t\right)\Phi \left(y^{\Delta }\right)\right]}^{\Delta }+p\left(t\right)\Phi \left(y^{\sigma }\right)=0$, where $\Phi \left(x\right)={\left|x\right|}^{\alpha -1}\text{sgn}\hspace{0.17em}x$ with $\alpha >1$. In particular, it is shown that the nonoscillation of the previous dynamic equation is preserved if we multiply the coefficient $p\left(t\right)$ by a suitable function $q\left(t\right)$ 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.