Two weighted norm dynamic inequalities with applications on second order half-linear dynamic equations. (English) Zbl 1482.26042
Summary: 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 \(\mathbb{T}\). These norm inequalities study the boundedness of the operator of Hardy’s type between the weighted spaces \(L_v^p(\mathbb{T})\) and \(L_u^q(\mathbb{T)}\). The paper covers the different cases when \(1<p\le q<\infty\) and when \(1<q<p<\infty \). As special cases, when \(\mathbb{T}=\mathbb{R}\), we obtain the corresponding previously known results from the literature, while for \(\mathbb{T}=\mathbb{N}\) 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.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |
| 26E70 | Real analysis on time scales or measure chains |
| 34N05 | Dynamic equations on time scales or measure chains |
Keywords:
Hardy inequality; time scales; half-linear dynamic equation; oscillation; Reid roundabout theorem; variational methodReferences:
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