Geodesic
Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 11, pp. 1889-1901 
A. A. Belolipetskii; A. M. Ter-Krikorov. Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 11, pp. 1889-1901. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_11_a4/
@article{ZVMMF_2016_56_11_a4,
     author = {A. A. Belolipetskii and A. M. Ter-Krikorov},
     title = {Modified {Kantorovich} theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1889--1901},
     year = {2016},
     volume = {56},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_11_a4/}
}
TY  - JOUR
AU  - A. A. Belolipetskii
AU  - A. M. Ter-Krikorov
TI  - Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2016
SP  - 1889
EP  - 1901
VL  - 56
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_11_a4/
LA  - ru
ID  - ZVMMF_2016_56_11_a4
ER  - 
%0 Journal Article
%A A. A. Belolipetskii
%A A. M. Ter-Krikorov
%T Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2016
%P 1889-1901
%V 56
%N 11
%U http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_11_a4/
%G ru
%F ZVMMF_2016_56_11_a4

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

The functional equation $f(x,\varepsilon)=0$ containing a small parameter $\varepsilon$ and admitting regular and singular degeneracy as $\varepsilon\to0$ is considered. By the methods of small parameter, a function $x_n^0(\varepsilon)$ satisfying this equation within a residual error of $O(\varepsilon^{n+1})$ is found. A modified Newton's sequence starting from the element $x_n^0(\varepsilon)$ is constructed. The existence of the limit of Newton's sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton's iterative sequence). The deviation of the limit of Newton's sequence from the initial approximation $x_n^0(\varepsilon)$ has the order of $O(\varepsilon^{n+1})$, which proves the asymptotic character of the approximation $x_n^0(\varepsilon)$. The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.

[1] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz v normirovannykh prostranstvakh, Fizmatgiz, M., 1959 | MR

[2] Ter-Krikorov A. M., Nelineinyi analiz i asimptoticheskie metody malogo parametra, Uch. posobie, MFTI, M., 2007

[3] Belolipetskii A. A., Ter-Krikorov A. M., “O reshenii odnoi singulyarno vozmuschennoi nachalno-kraevoi zadachi dlya lineinogo parabolicheskogo uravneniya”, Tr. MFTI, 3:1 (2011), 14–17

[4] Belolipetskii A. A., Ter-Krikorov A. M., “Ob odnoi singulyarno vozmuschennoi zadache Koshi dlya lineinogo parabolicheskogo uravneniya s nelineinymi kraevymi usloviyami”, Zh. vychisl. matem. i matem. fiz., 54:1 (2014), 80–88 | DOI | Zbl

[5] Vazov V., Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1968

[6] Butuzov V. F., Vasileva A. B., Fedoryuk M. V., “Asimptoticheskie metody v teorii obyknovennykh differentsialnykh uravnenii”, Itogi nauki. Matem. analiz, VINITI, M., 1969, 5–73 | MR

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

[8] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vyssh. shkola, M., 1990 | MR

[9] Schitov I. N., Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh zadach dlya obyknovennykh differentsialnykh uravnenii, Fizmatlit, M., 2013