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.
Highlights
In 1925, Hardy [1] proved that ∞ ∫ ( 1 t t f (s) p ds) dt ≤ (p p − p )
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
Our aim is to give an affirmative answer to this question in our present paper, which is organized as follows: In Section 2, we present some basic definition concerning the delta calculus on time scales
Summary
Where f is a positive integrable function over any finite interval (0, t), fp is an integrable function over (0, ∞), and p > 1. The natural question which is expected now is Is it possible to prove new characterizations of the weighted functions in dynamic Hardy’s and Opial’s type inequalities by reducing the problem to the solvability of dynamic equations on time scales?. Throughout this paper, we will assume that (t), s(t), and f(t) are nonnegative rd-continuous functions and the integrals considered are assumed to exist
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