Geodesic
Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case
Matematičeskie zametki, Tome 103 (2018) no. 3, pp. 392-403.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is devoted to the study of the solution of the Cauchy problem for a first-order differential equation in a Banach space with a small parameter on the right-hand side perturbing the equation. The coefficient of the derivative of the unknown function is a Fredholm operator with index zero and one-dimensional kernel. The case of a regular pair of operator coefficients is considered. An asymptotic expansion of the solution of the problem is constructed by using a method due to Vasil'eva, Vishik, and Lyusternik. In calculating the components of the regular and boundary-layer parts of the expansion, the cascade decomposition of the equations is used. It is proved that this expansion is asymptotic. Conditions for regular degeneration are found. The behavior of the solution as the parameter tends to zero is studied.
Keywords: differential equation, Fredholm operator, asymptotic expansion
Mots-clés : small perturbation, cascade decomposition. 
@article{MZM_2018_103_3_a5,
     author = {S. P. Zubova and V. I. Uskov},
     title = {Asymptotic {Solution} of the {Cauchy} {Problem} for a {First-Order} {Equation} with a {Small} {Parameter} in a {Banach} {Space.} {The} {Regular} {Case}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {392--403},
     publisher = {mathdoc},
     volume = {103},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a5/}
}
TY  - JOUR
AU  - S. P. Zubova
AU  - V. I. Uskov
TI  - Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case
JO  - Matematičeskie zametki
PY  - 2018
SP  - 392
EP  - 403
VL  - 103
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a5/
LA  - ru
ID  - MZM_2018_103_3_a5
ER  - 
%0 Journal Article
%A S. P. Zubova
%A V. I. Uskov
%T Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case
%J Matematičeskie zametki
%D 2018
%P 392-403
%V 103
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a5/
%G ru
%F MZM_2018_103_3_a5
S. P. Zubova; V. I. Uskov. Asymptotic Solution of the Cauchy Problem for a First-Order Equation with a Small Parameter in a Banach Space. The Regular Case. Matematičeskie zametki, Tome 103 (2018) no. 3, pp. 392-403. http://geodesic.mathdoc.fr/item/MZM_2018_103_3_a5/

[1] S. P. Zubova, E. V. Raetskaya, “Issledovanie zhestkosti deskriptornoi dinamicheskoi sistemy v banakhovom prostranstve”, Problemy matematicheskogo analiza, 79 (2015), 127–132

[2] S. P. Zubova, Svoistva vozmuschennogo fredgolmovskogo operatora. Reshenie differentsialnogo uravneniya s fredgolmovskim operatorom pri proizvodnoi, Dep. v VINITI No 2516-V91, Voronezh, 1991

[3] P. L. Christiansen, P. S Lomdahl, V. Muto, “On a Toda lattice model with a transversal degree of freedom”, Nonlinearity, 4:2 (1991), 477–501 | DOI | MR | Zbl

[4] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, 2003 | MR | Zbl

[5] Nguen Khak Diep, V. F. Chistyakov, “O modelirovanii s ispolzovaniem differentsialno-algebraicheskikh uravnenii v chastnykh proizvodnykh”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 6:1 (2013), 98–111

[6] S. M. Nikolskii, “Lineinye uravneniya v lineinykh normirovannykh prostranstvakh”, Izv. AN SSSR. Ser. matem., 7:3 (1943), 147–166 | MR | Zbl

[7] M. M. Vainberg, V. A. Trenogin, Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR | Zbl

[8] E. V. Raetskaya, Ob issledovanii povedeniya resheniya odnogo singulyarno vozmuschennogo uravneniya, Dep. v VINITI No 1039-V2002, Voronezh, 2002

[9] N. G. Chebotarev, Teoriya algebraicheskikh funktsii, Librokom, M., 2009 | MR

[10] A. B. Vasileva, V. F. Butuzov, Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973 | MR | Zbl

[11] M. I. Vishik, L. A. Lyusternik, “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, UMN, 12:5 (77) (1957), 3–122 | MR | Zbl

[12] S. G. Krein, Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967 | MR | Zbl

[13] V. I. Uskov, S. P. Zubova, “Asimptoticheskoe reshenie zadachi Koshi dlya deskriptornogo uravneniya s malym parametrom v banakhovom prostranstve”, Voronezhskaya zimnyaya matematicheskaya shkola S. G. Kreina, Nauchnaya kniga, Voronezh, 2016, 175–177