Two Weighted Norm Dynamic Inequalities with Applications on Second Order Half-Linear Dynamic Equations

In this paper, we prove some new characterizations of two weighted functions u and v in norm inequalities of Hardy’s type, in the context of dynamic inequalities on time scales T . These norm inequalities studied the boundedness of the operator of Hardy’s type between the weighted spaces L v p ( T ) and L u q ( T ) . The paper covers the different cases when 1 < p ≤ q < ∞ and when 1 < q < p < ∞ . As special cases, when T = R , we obtain the corresponding previously known results from the literature, while for T = N we obtain some discrete results which are essentially new. In seeking applications, we will establish some non-oscillation results for second-order half-linear dynamic equations on time scales.