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

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Abstract

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 \({\mathbb {T}}\). These norm inequalities studied 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 \(\mathbb {T=R}\), we obtain the corresponding previously known results from the literature, while for \(\mathbb {T=N}\) 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.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

    Samir H. Saker & Mahmoud M. Osman

  2. Department of Mathematics, Faculty of Science, New Mansoura University, New Mansoura City, Egypt

    Samir H. Saker

  3. Department of Mathematics, Concordia College, Moorhead, MN, 56562, USA

    Douglas R. Anderson

Authors
  1. Samir H. Saker
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  2. Mahmoud M. Osman
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  3. Douglas R. Anderson
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Correspondence to Douglas R. Anderson.

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Saker, S.H., Osman, M.M. & Anderson, D.R. Two Weighted Norm Dynamic Inequalities with Applications on Second Order Half-Linear Dynamic Equations. Qual. Theory Dyn. Syst. 21, 4 (2022). https://doi.org/10.1007/s12346-021-00534-1

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