A connection between weighted Hardy’s inequality and half-linear dynamic equations

In this paper, we give an affirmative answer to the following question: Is the solvability of some nonlinear dynamic equations on a time scale $\mathbb{T}$ not only sufficient but in a certain sense also necessary for the validity of some dynamic Hardy-type inequalities with two different weights? In fact, this answer will give a new characterization of the weights in a weighted Hardy-type inequality on time scales. The results contain the results when $\mathbb{T}=\mathbb{R}$ , $\mathbb{T}=\mathbb{N}$ , and when $\mathbb{T}=q^{\mathbb{N}_{0}}$ as special cases. Some applications are given for illustrations.