Geodesic
A multiparametric family of solutions to the Volterra linear integral equation of the first kind
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2024), pp. 47-62 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the Volterra integral equation of the first kind with an integral operator of order $n$, a singularity and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms on the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term, we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows one to construct a particular and general solutions to the integro-differential equation in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multi-parameter family of solutions is being constructed for the original integral equation.
Keywords: integral equation, operator, operator pencil, spectrum. 
@article{IVM_2024_5_a4,
     author = {I. V. Sapronov},
     title = {A multiparametric family of solutions to the {Volterra} linear integral equation of the first kind},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {47--62},
     year = {2024},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2024_5_a4/}
}
TY  - JOUR
AU  - I. V. Sapronov
TI  - A multiparametric family of solutions to the Volterra linear integral equation of the first kind
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2024
SP  - 47
EP  - 62
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/IVM_2024_5_a4/
LA  - ru
ID  - IVM_2024_5_a4
ER  - 
%0 Journal Article
%A I. V. Sapronov
%T A multiparametric family of solutions to the Volterra linear integral equation of the first kind
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2024
%P 47-62
%N 5
%U http://geodesic.mathdoc.fr/item/IVM_2024_5_a4/
%G ru
%F IVM_2024_5_a4
I. V. Sapronov. A multiparametric family of solutions to the Volterra linear integral equation of the first kind. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2024), pp. 47-62. http://geodesic.mathdoc.fr/item/IVM_2024_5_a4/

[1] Magnitskii N.A., “O suschestvovanii mnogoparametricheskikh semeistv reshenii integralnogo uravneniya Volterra $1$-go roda”, Dokl. AN SSSR, 235:4 (1977), 772–774 | MR | Zbl

[2] Magnitskii N.A., “Mnogoparametricheskie semeistva reshenii integralnykh uravnenii Volterra”, Dokl. AN SSSR, 240:2 (1978), 268–271 | MR | Zbl

[3] Magnitskii N.A., “Lineinye integralnye uravneniya Volterra I i III rodov”, Zhurn. vychisl. matem. i matem. fiz., 19:4 (1979), 970–988 | MR | Zbl

[4] Krein S.G., Sapronov I.V., “O polnote sistemy reshenii integralnogo uravneniya Volterra s osobennostyu”, Dokl. RAN, 355:4 (1997), 450–452 | MR | Zbl

[5] Krein S.G., Sapronov I.V., “Ob integralnykh uravneniyakh Volterra s osobennostyami”, UMN, 50:(4) 340 (1995), 140

[6] Krein S.G., “Singular integral Volterra equations”, Abstracts of Internat. Congress of Math. (Zurich, 1994), 125

[7] Sapronov I.V., “Ob odnom klasse reshenii uravneniya Volterra II roda s regulyarnoi osobennostyu v banakhovom prostranstve”, Izv. vuzov. Matem., 2004, no. 6, 48–58 | Zbl

[8] Sapronov I.V., “Mnogoparametricheskoe semeistvo reshenii integralnogo uravneniya Volterra s osobennostyu v banakhovom prostranstve”, Izv. vuzov. Matem., 2005, no. 2, 81–83 | Zbl

[9] Sapronov I.V., “Uravnenie Volterra s osobennostyu v banakhovom prostranstve”, Izv. vuzov. Matem., 2007, no. 11, 45–55 | Zbl

[10] Sapronov I.V., “Mnogoparametricheskoe semeistvo reshenii integralnogo uravneniya Volterra s osobennostyu v banakhovom prostranstve”, Izv. vuzov. Matem., 2011, no. 1, 59–71 | Zbl

[11] Sapronov I.V., “Lineinoe integralnoe uravnenie Volterra I roda”, Vestn. VGU. Ser. Fiz. Matem., 2022, no. 1, 87–96 | MR

[12] Glushko V.P., Lineinye vyrozhdayuschiesya differentsialnye uravneniya, Izd-vo Voronezhsk. gos. un-ta, Voronezh, 1972