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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 2, Pages 230–243 (Mi vuu479)

MATHEMATICS

On the totally global solvability of a controlled Hammerstein type equation with a varied linear operator

A. V. Chernovab

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

Abstract: Let $n,m,\ell,s\in\mathbb{N}$ be given numbers, $\Pi\subset\mathbb{R}^n$ be a measurable bounded set, $\mathcal{X}, \mathcal{Z}, \mathcal{U}$ be Banach ideal spaces of functions measurable on the set $\Pi$, $\mathcal{D}\subset\mathcal{U}^{s}$ be a convex set, $\mathcal{A}$ be some class of linear bounded operators $A:\mathcal{Z}^{m} \to\mathcal{X}^{\ell}$. We study the controlled Hammerstein type functional operator equation as follows
$$x(t)=\theta(t)+ A[ f(.,x(.),u(.)) ](t), \quad t\in \Pi , \quad x\in\mathcal{X}^{\ell}, \tag{1} \label{eq1}$$
where $\{ u,\theta,A\}\in \mathcal{D}\times \mathcal{X}^{\ell}\times \mathcal{A}$ is the set of controlled parameters; $f(t,x,v): \Pi\times\mathbb{R}^{\ell}\times\mathbb{R}^{s}\to\mathbb{R}^{m}$ is a given function measurable with respect to $t\in\Pi$, continuous with respect to $\{x,v\}\in\mathbb{R}^\ell\times\mathbb{R}^s$ and satisfying to certain natural hypotheses. Eq. \eqref{eq1} is a convenient form of representation of the broad class of controlled distributed systems. For the equation under study we prove a theorem concerning sufficient conditions of global solvability for all $u\in\mathcal{D}$, $A\in\mathcal{A}$ and $\theta$ from a pointwise bounded set. For the original equation we define some majorant and minorant inequalities obtaining them from Eq. \eqref{eq1} with the help of upper and lower estimates of the right-hand side. The theorem is proved providing global solvability of the majorant and minorant inequalities. As an application of obtained general results we prove a theorem concerning the total (with respect to the whole set of admissible controls) global solvability of the mixed boundary value problem for a system of hyperbolic equations of the first order with controlled higher coefficients.

Keywords: totally global solvability, functional operator equation of the Hammerstein type, pointwise estimate of solutions, system of hyperbolic equations of the first order with controlled higher coefficients.

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

Citation: A. V. Chernov, “On the totally global solvability of a controlled Hammerstein type equation with a varied linear operator”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:2 (2015), 230–243

Citation in format AMSBIB
\Bibitem{Che15} \by A.~V.~Chernov \paper On the totally global solvability of a controlled Hammerstein type equation with a varied linear operator \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2015 \vol 25 \issue 2 \pages 230--243 \mathnet{http://mi.mathnet.ru/vuu479} \elib{http://elibrary.ru/item.asp?id=23681104} 

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This publication is cited in the following articles:
1. A. V. Chernov, “Mazhorantnyi priznak pervogo poryadka totalno globalnoi razreshimosti upravlyaemogo funktsionalno-operatornogo uravneniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:4 (2018), 531–548
2. A. V. Chernov, “Preservation of the solvability of a semilinear global electric circuit equation”, Comput. Math. Math. Phys., 58:12 (2018), 2018–2030
3. A. V. Chernov, “On preservation of global solvability of controlled second kind operator equation”, Ufa Math. J., 12:1 (2020), 56–81
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