Abstract
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.
Highlights
In 1920 Hardy [13] proved the discrete inequality ∞ 1 n n ai p ≤ n=1 i=1 p p–1 p∞ apn, n=1 p > 1, (1.1)where an ≥ 0 for n ≥ 1
This inequality has been discovered in his attempt to give an elementary proof of Hilbert’s inequality for double series that was known at that time
In 1925 Hardy [14] proved the continuous inequality using the calculus of variations, which states that for f ≥ 0 integrable over any finite interval (0, x) and f p integrable and convergent over (0, ∞) and p > 1
Summary
We refer the reader to [3] for recent results of Hardy-type inequalities on time scales. Motivated by the above results, we naturally raise the question : Is the solvability of some nonlinear dynamic equations on time scales sufficient but in a certain sense necessary for the validity of some Hardy-type inequality?
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