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Mat. Zametki, 2013, Volume 94, Issue 5, Pages 757–769 (Mi mz8951)  

This article is cited in 1 scientific paper (total in 1 paper)

A Generalization of Bihari's Lemma to the Case of Volterra Operators in Lebesgue Spaces

A. V. Chernov

Institute of Radio Engineering and Information Technologies, Nizhniy Novgorod State Technical University

Abstract: For operators acting in the Lebesgue space $L_q(\Pi)$, $1<q<\infty$, an abstract analog of Bihari's lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat–Darboux problem.

Keywords: Bihari's lemma, Lebesgue space, Volterra operator, controlled functional-operator equation, Goursat–Darboux problem, Gronwall's lemma, Volterra $\delta$-chain.

DOI: https://doi.org/10.4213/mzm8951

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English version:
Mathematical Notes, 2013, 94:5, 703–714

Bibliographic databases:

UDC: 517.988+517.977.56
Received: 09.10.2010

Citation: A. V. Chernov, “A Generalization of Bihari's Lemma to the Case of Volterra Operators in Lebesgue Spaces”, Mat. Zametki, 94:5 (2013), 757–769; Math. Notes, 94:5 (2013), 703–714

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Chernov, “O totalno globalnoi razreshimosti upravlyaemogo uravneniya tipa Gammershteina s variruemym lineinym operatorom”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:2 (2015), 230–243  mathnet  elib
  • Математические заметки Mathematical Notes
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