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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2020, Volume 30, Issue 1, Pages 92–111 (Mi vuu712)  

MATHEMATICS

On totally global solvability of controlled second kind operator equation

A. V. Chernovab

a Nizhny Novgorod State University, pr. Gagarina, 23, Nizhny Novgorod, 603950, Russia
b Nizhny Novgorod State Technical University, ul. Minina, 24, Nizhny Novgorod, 603950, Russia

Abstract: We consider the nonlinear evolutionary operator equation of the second kind as follows $\varphi=\mathcal{F}[f[u]\varphi]$, $\varphi\in W[0;T]\subset L_q([0;T];X)$, with Volterra type operators $\mathcal{F}\colon L_p([0;\tau];Y)\to W[0;T]$, $f[u]$: $W[0;T]\to L_p([0;T];Y)$ of the general form, a control $u\in\mathcal{D}$ and arbitrary Banach spaces $X$, $Y$. For this equation we prove theorems on solution uniqueness and sufficient conditions for totally (with respect to set $\mathcal{D}$) global solvability. Under natural hypotheses associated with pointwise in $t\in[0;T]$ estimates the conclusion on univalent totally global solvability is made provided global solvability for a comparison system which is some system of functional integral equations (it could be replaced by a system of equations of analogous type, and in some cases, of ordinary differential equations) with respect to unknown functions $[0;T]\to\mathbb{R}$. As an example we establish sufficient conditions of univalent totally global solvability for a controlled nonlinear nonstationary Navier–Stokes system.

Keywords: nonlinear evolutionary operator equation of the second kind, totally global solvability, Navier–Stokes system.

DOI: https://doi.org/10.35634/vm200107

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Full text: http://vm.udsu.ru/.../2020-1-7
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Bibliographic databases:

UDC: 517.957, 517.988, 517.977.56
MSC: 47J05, 47J35, 47N10
Received: 23.08.2019

Citation: A. V. Chernov, “On totally global solvability of controlled second kind operator equation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 30:1 (2020), 92–111

Citation in format AMSBIB
\Bibitem{Che20}
\by A.~V.~Chernov
\paper On totally global solvability of controlled second kind operator equation
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2020
\vol 30
\issue 1
\pages 92--111
\mathnet{http://mi.mathnet.ru/vuu712}
\crossref{https://doi.org/10.35634/vm200107}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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