Geodesic
Cauchy operator of a non-stationary linear differential equation with a small parameter at the derivative
Sbornik. Mathematics, Tome 196 (2005) no. 8, pp. 1165-1208 
K. I. Chernyshov. Cauchy operator of a non-stationary linear differential equation with a small parameter at the derivative. Sbornik. Mathematics, Tome 196 (2005) no. 8, pp. 1165-1208. http://geodesic.mathdoc.fr/item/SM_2005_196_8_a3/
@article{SM_2005_196_8_a3,
     author = {K. I. Chernyshov},
     title = {Cauchy operator of a~non-stationary linear differential equation with a~small parameter at the derivative},
     journal = {Sbornik. Mathematics},
     pages = {1165--1208},
     year = {2005},
     volume = {196},
     number = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_8_a3/}
}
TY  - JOUR
AU  - K. I. Chernyshov
TI  - Cauchy operator of a non-stationary linear differential equation with a small parameter at the derivative
JO  - Sbornik. Mathematics
PY  - 2005
SP  - 1165
EP  - 1208
VL  - 196
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/SM_2005_196_8_a3/
LA  - en
ID  - SM_2005_196_8_a3
ER  - 
%0 Journal Article
%A K. I. Chernyshov
%T Cauchy operator of a non-stationary linear differential equation with a small parameter at the derivative
%J Sbornik. Mathematics
%D 2005
%P 1165-1208
%V 196
%N 8
%U http://geodesic.mathdoc.fr/item/SM_2005_196_8_a3/
%G en
%F SM_2005_196_8_a3

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

A diagonalization algorithm for a matrix pencil depending on a variable and a parameter in the cases when the limiting matrix has a simple spectrum or a multiple eigenvalue for all values of the variable is put forward. The algorithm uses an exhaustive superposition of special similarity transformations. Formulae for the Cauchy operator of a linear non-stationary equation with a small parameter at the derivative and with a matrix pencil are obtained for various degeneracy orders of the structure matrix.

[1] Rapoport I. M., O nekotorykh asimptoticheskikh metodakh v teorii differentsialnykh uravnenii, Izd-vo AN USSR, Kiev, 1954

[2] Fedoryuk M. V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | Zbl

[3] Fedoryuk M. V., “Asimptoticheskie metody v analize”, Sovr. problemy matem. Fundament. napravleniya, 13, VINITI, M., 1986, 93–210 | MR

[4] Lomov S. A., Vvedenie v obschuyu teoriyu singulyarnykh vozmuschenii, Nauka, M., 1981 | MR

[5] Konyaev Yu. A., Teoriya vozmuschenii v prikladnykh zadachakh, Izd-vo MEI, M., 1990

[6] Feschenko S. F., Shkil N. I., Nikolenko L. D., Asimptoticheskie metody v teorii lineinykh differentsialnykh uravnenii, Naukova dumka, Kiev, 1966 | MR | Zbl

[7] Moiseev N. N., Asimptoticheskie metody nelineinoi mekhaniki, Nauka, M., 1969 | MR | Zbl

[8] Sotnichenko N. A., Feschenko S. F., Asimptoticheskoe integrirovanie differentsialnykh uravnenii, Preprint No 80.3, IM AN USSR, Kiev, 1980 | MR

[9] Eliseev A. G., “Teoriya singulyarnykh vozmuschenii dlya sistem differentsialnykh uravnenii v sluchae kratnogo spektra predelnogo operatora. I, II”, Izv. AN SSSR. Ser. matem., 48:5 (1984), 999–1020 | MR

[10] Eliseev A. G., Lomov S. A., “Asimptoticheskoe integrirovanie singulyarno vozmuschennnykh zadach”, UMN, 43:3 (1988), 3–53 | MR | Zbl

[11] Shkil N. I., Starun I. I., Yakovets V. P., Asimptoticheskoe integrirovanie lineinykh sistem obyknovennykh differentsialnykh uravnenii, Vischa shkola, Kiev, 1989

[12] Vazov V., Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1968

[13] Abgaryan K. A., Matrichnye i asimptoticheskie metody v teorii lineinykh sistem, Nauka, M., 1973 | MR

[14] Chernyshov K. I., “Metod standartnogo rasschepleniya singulyarno vozmuschennykh differentsialnykh uravnenii”, Dokl. AN SSSR, 311:6 (1990), 1311–1316 | MR | Zbl

[15] Chernyshov K. I., Asimptoticheskii analiz uravneniya s fredgolmovym operatorom pri proizvodnoi, Dis. ... dokt. fiz.-matem. nauk, IM NAN Ukrainy, Kiev, 1992

[16] Chernyshov K. I., “Asimptoticheskie razlozheniya resheniya zadachi Koshi dlya odnogo lineinogo differentsialnogo uravneniya”, Differents. uravneniya, 28:5 (1992), 765–779 | MR | Zbl

[17] Chernyshov K. I., “Ob eksponente matrichnogo puchka, zavisyaschego ot parametra, v sluchae kratnogo spektra predelnoi matritsy”, Proc. of the Thirteenth Crimean Autumn Math. School-Symposium (18–29 sentyabrya 2003 g., Simferopol, Ukraina), Spectral and evolution problems, 13, Tavrich. nats. un-t, Simferopol, 2003, 65–81 | MR

[18] Kolesov Yu. S., Kubyshkin E. P., “Minimalnyi algoritm issledovaniya ustoichivosti lineinykh sistem”, Issledovaniya po ustoichivosti v teorii kolebanii, Izd-vo YarGU, Yaroslavl, 1977, 142–155 | MR

[19] Arnold V. I., “O matritsakh, zavisyaschikh ot parametrov”, UMN, 26:2 (1971), 101–114 | MR

[20] Bogaevskii V. N., Povzner A. Ya., Algebraicheskie metody v nelineinoi teorii vozmuschenii, Nauka, M., 1987 | MR | Zbl

[21] Daletskii Yu. L., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970 | MR

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

[23] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovom prostranstve, Nauka, M., 1966 | MR | Zbl

[24] Territin Kh. L., “Asimptoticheskoe razlozhenie reshenii sistem obyknovennykh lineinykh differentsialnykh uravnenii, soderzhaschikh parametr”, Matematika. Sb. per., 1:2 (1957), 29–59