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.
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).
Citation:
A. G. Baskakov, N. B. Uskova, “Fourier method for first order differential equations with involution and groups of operators”, Ufa Math. J., 10:3 (2018), 11–34
\Bibitem{BasUsk18}
\by A.~G.~Baskakov, N.~B.~Uskova
\paper Fourier method for first order differential equations with involution and groups of operators
\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 3
\pages 11--34
\mathnet{http://mi.mathnet.ru/eng/ufa433}
\crossref{https://doi.org/10.13108/2018-10-3-11}
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Linking options:
https://www.mathnet.ru/eng/ufa433
https://doi.org/10.13108/2018-10-3-11
https://www.mathnet.ru/eng/ufa/v10/i3/p11
This publication is cited in the following 19 articles:
A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “O sglazhivanii operatornogo koeffitsienta differentsialnogo operatora pervogo poryadka v banakhovom prostranstve”, Materialy Voronezhskoi mezhdunarodnoi zimnei matematicheskoi shkoly «Sovremennye metody teorii funktsii i smezhnye problemy», Voronezh, 28 yanvarya – 2 fevralya 2021 g. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 206, VINITI RAN, M., 2022, 3–14
A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “Method of Similar Operators in the Study of Spectral Properties of Perturbed First-Order Differential Operators”, J Math Sci, 263:5 (2022), 599
M. Sh. Burlutskaya, “Some properties of functional-differential operators with involution ν(x)=1−x and their applications”, Russian Math. (Iz. VUZ), 65:5 (2021), 69–76
D. V. Belova, “Ob odnoi smeshannoi zadache s involyutsiei”, Materialy Voronezhskoi vesennei matematicheskoi shkoly
«Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniya–XXX». Voronezh, 3–9 maya 2019 g. Chast 5, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 194, VINITI RAN, M., 2021, 46–54
G. V. Garkavenko, N. B. Uskova, “Ob usloviyakh diagonalizuemosti vozmuschennogo raznostnogo operatora v nekotorykh prostranstvakh”, Mezhdunar. nauch.-issled. zhurn., 2021, no. 7(109), 6–14
G. V. Garkavenko, N. B. Uskova, “Ob otsenkakh sobstvennykh znachenii beskonechnykh blochnykh trekhdiagonalnykh matrits”, Materialy mezhdunarodnoi konferentsii po matematicheskomu modelirovaniyu v prikladnykh naukakh “International Conference on Mathematical Modelling in Applied Sciences — ICMMAS'19”. Belgorod, 20–24 avgusta 2019 g., Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 195, VINITI RAN, M., 2021, 118–126
G. V. Garkavenko, N. B. Uskova, “O spektralnykh svoistvakh odnoi trekhdiagonalnoi beskonechnoi matritsy”, Materialy 20 Mezhdunarodnoi Saratovskoi zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya», Saratov, 28 yanvarya — 1 fevralya 2020 g. Chast 1, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 199, VINITI RAN, M., 2021, 31–42
B. Kh. Turmetov, V. V. Karachik, “O razreshimosti kraevykh zadach Dirikhle i Neimana dlya uravneniya Puassona s mnozhestvennoi involyutsiei”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 31:4 (2021), 651–667
B. Turmetov, V. Karachik, M. Muratbekova, “On a boundary value problem for the biharmonic equation with multiple involutions”, Mathematics, 9:17 (2021), 2020
G Garkavenko, N Uskova, “Spectral analysis of one class perturbed first order differential operators”, J. Phys.: Conf. Ser., 1902:1 (2021), 012035
A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “On the spectral analysis of a differential operator with an involution and general boundary conditions”, Eurasian Math. J., 11:2 (2020), 30–39
A. G. Baskakov, D. M. Polyakov, “Fourier method for a mixed problem with the hill operator”, Differ. Equ., 56:6 (2020), 679–684
K. Zh. Nazarova, B. Kh. Turmetov, K. I. Usmanov, “O razreshimosti nekotorykh kraevykh zadach s involyutsiei”, Vestn. SamU. Estestvennonauchn. ser., 26:3 (2020), 7–16
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
I. A. Krishtal, N. B. Uskova, “Spektralnye svoistva differentsialnykh operatorov pervogo poryadka s involyutsiei i gruppy operatorov”, Sib. elektron. matem. izv., 16 (2019), 1091–1132
A. G. Baskakov, I. A. Krishtal, N. B. Uskova, “Metod podobnykh operatorov v issledovanii spektralnykh svoistv vozmuschennykh differentsialnykh operatorov pervogo poryadka”, Materialy Voronezhskoi zimnei matematicheskoi shkoly «Sovremennye metody teorii funktsii i smezhnye problemy». 28 yanvarya–2 fevralya 2019 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 171, VINITI RAN, M., 2019, 3–18
N. B. Uskova, “Spectral properties of the Dirac operator with a nonsmooth potential of the general form and operator groups”, Differ. Equ., 55:8 (2019), 1120–1124
G V Garkavenko, A R Zgolich, N B Uskova, “Spectral analysis of one class of the integro-differential operators”, J. Phys.: Conf. Ser., 1203 (2019), 012102
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