This article is cited in 2 scientific papers (total in 2 papers)
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
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.
method of similar operator, spectrum, mixed problem, group of operators, differential operator
PDF file (585 kB)
Ufa Mathematical Journal, 2018, 10:3, 11–34 (PDF, 495 kB); https://doi.org/10.13108/2018-10-3-11
MSC: 34L15, 34B09, 47E05
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
\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.
\jour Ufa Math. J.
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N. B. Uskova, “Matrichnyi analiz spektralnykh proektorov vozmuschennykh samosopryazhennykh operatorov”, Sib. elektron. matem. izv., 16 (2019), 369–405
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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