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 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 3, Pages 11–34 (Mi ufa433)

Fourier method for first order differential equations with involution and groups of operators

A. G. Baskakova, N. B. Uskovab

a Voronezh State University, Universitetskaya sq. 1, 394018, Voronezh, Russia
b Natalia Borisovna Uskova, Voronezh State Technical University, Moskovsky av. 14, 394016, Voronezh, Russia

Abstract: In the paper we study a mixed problem for a first-order differential equation with an involution. It is written by means of a differential operator with an involution acting in the space functions square integrable on a finite interval. We construct a similarity transform of this operator in an operator being an orthogonal direct sum of an operator of finite rank and operators of rank 1. The method of our study is the method of similar operators. Theorem on similarity serves as the basis for constructing groups of operators, whose generator is the original operator. We write out asymptotic formulae for groups of operators. The constructed group allows us to introduce the notion of a mild solution, and also to describe the mild solutions to the considered problem.
This serves to justify the Fourier method. Almost periodicity of bounded mild solutions is established. The proof of almost periodicity is based on the asymptotic representation of the spectrum of a differential operator with an involution.

Keywords: method of similar operator, spectrum, mixed problem, group of operators, differential operator with involution.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.3464.2017/4.6 Russian Foundation for Basic Research 16-01-00197_à The work of the first author is supported by the Ministry of Equcation and Science of Russia in the framework of the project part of state task (project no. 1.3464.2017/4.6). The work of the second author is supported by RFBR (project no. 16-01-00197).

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English version:
Ufa Mathematical Journal, 2018, 10:3, 11–34 (PDF, 495 kB); https://doi.org/10.13108/2018-10-3-11

Bibliographic databases:

UDC: 517.927
MSC: 34L15, 34B09, 47E05

Citation: A. G. Baskakov, N. B. Uskova, “Fourier method for first order differential equations with involution and groups of operators”, Ufimsk. Mat. Zh., 10:3 (2018), 11–34; Ufa Math. J., 10:3 (2018), 11–34

Citation in format AMSBIB
\Bibitem{BasUsk18} \by A.~G.~Baskakov, N.~B.~Uskova \paper Fourier method for first order differential equations with involution and groups of operators \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 3 \pages 11--34 \mathnet{http://mi.mathnet.ru/ufa433} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 3 \pages 11--34 \crossref{https://doi.org/10.13108/2018-10-3-11} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000457365400002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85057040985} 

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This publication is cited in the following articles:
1. N. B. Uskova, “Matrichnyi analiz spektralnykh proektorov vozmuschennykh samosopryazhennykh operatorov”, Sib. elektron. matem. izv., 16 (2019), 369–405
2. A. G. Baskakov, E. E. Dikarev, “Spectral theory of functions in studying partial differential operators”, Ufa Math. J., 11:1 (2019), 3–18
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