New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

Abstract

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases.

Hardy-type operators with general kernels and characterizations of dynamic weighted inequalities

“…In [37] the authors established a new weight characterization of (1.11) on an unbounded time scale interval [a, ∞) T , and proved that if…”

“…Also in [37] the authors considered the case when 1 < q < p < ∞ and 1/r = 1/q − 1/p and proved that the inequality (1.14) holds if and only if…”

“…For related dynamic inequalities, we refer the reader to the books [1,2] and the papers [8,10,33,32,34,35,36,39] and the references therein. Our aim in this paper is to prove some norm inequalities for a general operator of the form…”