RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 3 scientific papers (total in 3 papers)

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
\begin{equation} x(t)=\theta(t)+ A[ f(.,x(.),u(.)) ](t), \quad t\in \Pi , \quad x\in\mathcal{X}^{\ell}, \tag{1} \label{eq1} \end{equation}
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.

Full text: PDF file (303 kB)
References: PDF file   HTML file
UDC: 517.957, 517.988, 517.977.56
MSC: 47J05, 47J35, 47N10
Received: 29.03.2015

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}


Linking options:
  • http://mi.mathnet.ru/eng/vuu479
  • http://mi.mathnet.ru/eng/vuu/v25/i2/p230

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  mathnet  crossref  elib
    2. A. V. Chernov, “Preservation of the solvability of a semilinear global electric circuit equation”, Comput. Math. Math. Phys., 58:12 (2018), 2018–2030  mathnet  crossref  crossref  isi  elib
    3. A. V. Chernov, “On preservation of global solvability of controlled second kind operator equation”, Ufa Math. J., 12:1 (2020), 56–81  mathnet  crossref  isi
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
    Number of views:
    This page:315
    Full text:72
    References:48

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020