Geodesic
A Generalization of Bihari's Lemma to the Case of Volterra Operators in Lebesgue Spaces
Matematičeskie zametki, Tome 94 (2013) no. 5, pp. 757-769

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For operators acting in the Lebesgue space $L_q(\Pi)$, $1$, 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, Volterra operator, controlled functional-operator equation, Goursat–Darboux problem, Gronwall's lemma
Mots-clés : Lebesgue space, Volterra $\delta$-chain. 
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     author = {A. V. Chernov},
     title = {A {Generalization} of {Bihari's} {Lemma} to the {Case} of {Volterra} {Operators} in {Lebesgue} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {757--769},
     publisher = {mathdoc},
     volume = {94},
     number = {5},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_94_5_a10/}
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A. V. Chernov. A Generalization of Bihari's Lemma to the Case of Volterra Operators in Lebesgue Spaces. Matematičeskie zametki, Tome 94 (2013) no. 5, pp. 757-769. http://geodesic.mathdoc.fr/item/MZM_2013_94_5_a10/