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 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 𝕋=ℝ , 𝕋=ℕ , and when 𝕋=q^ℕ_0 as special cases. Some applications are given for illustrations.