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On measure of noncompactness in Lebesgue and Sobolev spaces with an application to the functional integro-differential equation

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Abstract

In this paper we attempt to define axiomatic measures of non-compactness for Sobolev spaces of integer order \( W^{n,p}(\Omega )\), where \(\Omega \subset \mathbb {R}^{d}\) (which is equivalent to \(\Omega \) being any set of infinite measure). We consider two cases, one with \(\Omega \) being an open subset with finite measure, and another when \(\Omega =\mathbb {R}^{d}\), and discuss basic features of the measure of non-compactness in each case. Next we define a partial measure of non-compactness on space \(L^{p}(\Omega ;B)\), where B is a Banach space. Furthermore, we give some application to solve the functional integro-differential equation in \(W^{1,p}(\Omega ).\)

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  • 25 March 2023

    10 March 2023

Notes

  1. a measurable function by definition is a.e. the limit of a sequence of countably-valued functions

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Authors and Affiliations

  1. Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

    M. Mursaleen

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    M. Mursaleen & S. M. H. Rizvi

  3. Department of Mathematics, Islamic Azad University, Sari Branch, Sari, Iran

    R. Arab & A. S. Haghighi

  4. Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran

    R. Allahyari

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  1. M. Mursaleen
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  2. S. M. H. Rizvi
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  3. R. Arab
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  4. A. S. Haghighi
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  5. R. Allahyari
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All the authors contributed equally and significantly in writing this paper.

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Correspondence to M. Mursaleen.

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The Original article is revised to update the author names ‘lebesgue’ and ‘sobolev”.

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Mursaleen, M., Rizvi, S.M.H., Arab, R. et al. On measure of noncompactness in Lebesgue and Sobolev spaces with an application to the functional integro-differential equation. Aequat. Math. 97, 199–217 (2023). https://doi.org/10.1007/s00010-022-00906-1

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