Geodesic
Pointwise Estimation of the Difference of the Solutions of a Controlled Functional Operator Equation in Lebesgue Spaces
Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 288-302

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For a functional operator equation in Lebesgue space, we prove a statement on the pointwise estimate of the modulus of the increment of its global (on a fixed set $\Pi\subset\mathbb R^n$) solution under the variation of the control function appearing in this equation. As an auxiliary statement, we prove a generalization of Gronwall's lemma to the case of a nonlinear operator acting in Lebesgue space. The approach used here is based on methods from the theory of stability of existence of global solutions to Volterra operator equations.
Keywords: functional operator equation, control function, initial boundary-value problem, Gronwall's lemma, Volterra operator equation
Mots-clés : Lebesgue space, increment of a solution. 
@article{MZM_2010_88_2_a10,
     author = {A. V. Chernov},
     title = {Pointwise {Estimation} of the {Difference} of the {Solutions} of a {Controlled} {Functional} {Operator} {Equation} in {Lebesgue} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {288--302},
     publisher = {mathdoc},
     volume = {88},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_2_a10/}
}
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A. V. Chernov. Pointwise Estimation of the Difference of the Solutions of a Controlled Functional Operator Equation in Lebesgue Spaces. Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 288-302. http://geodesic.mathdoc.fr/item/MZM_2010_88_2_a10/