Abstract
For operators acting in the Lebesgue space L q (Π), 1 < q < ∞, 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.
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Original Russian Text © A. V. Chernov, 2013, published in Matematicheskie Zametki, 2013, Vol. 94, No. 5, pp. 757–769.
Sometimes it is also called the Gronwall-Bellman lemma or Bellman’s lemma.
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Chernov, A.V. A generalization of Bihari’s lemma to the case of Volterra operators in Lebesgue spaces. Math Notes 94, 703–714 (2013). https://doi.org/10.1134/S0001434613110114
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DOI: https://doi.org/10.1134/S0001434613110114