Geodesic
Bounded solutions of families of systems of differential equations and their approximation
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 29-47.

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

In the paper, we consider the problem of finding a bounded solution of a one-parametric family of systems of ordinary differential equations. Using the parametrization method, we prove necessary and sufficient conditions for the existence of a unique solution of the problem considered that is bounded on the whole axis in terms of a two-sided infinite block-band matrix composed with respect to integrals over intervals of length $h>0$ the matrix of the system of differential equations. Also, we construct a family of two-point boundary-value problems on a finite interval that approximate the problem of finding the bounded solution and finds an interconnection between the correct solvability of the initial and approximating problems. 
@article{FPM_2006_12_5_a3,
     author = {D. S. Dzhumabaev},
     title = {Bounded solutions of families of systems of differential equations and their approximation},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {29--47},
     publisher = {mathdoc},
     volume = {12},
     number = {5},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a3/}
}
TY  - JOUR
AU  - D. S. Dzhumabaev
TI  - Bounded solutions of families of systems of differential equations and their approximation
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2006
SP  - 29
EP  - 47
VL  - 12
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a3/
LA  - ru
ID  - FPM_2006_12_5_a3
ER  - 
%0 Journal Article
%A D. S. Dzhumabaev
%T Bounded solutions of families of systems of differential equations and their approximation
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2006
%P 29-47
%V 12
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a3/
%G ru
%F FPM_2006_12_5_a3
D. S. Dzhumabaev. Bounded solutions of families of systems of differential equations and their approximation. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 5, pp. 29-47. http://geodesic.mathdoc.fr/item/FPM_2006_12_5_a3/

[1] Abramov A. A., Konyukhova N. B., Balla K., “Ustoichivye nachalnye mnogoobraziya i singulyarnye kraevye zadachi dlya sistem obyknovennykh differentsialnykh uravnenii”, Comput. Math. Banach Center Publ., 13, PWN Polish Scient. Publs., Warsaw, 1984, 319–351 | MR

[2] Anosov D. V., “Geodezicheske potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny uravnenii”, Trudy Mat. in-ta im. V. A. Steklova AN SSSR, 90, 1967, 1–210 | MR

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

[4] Dzhumabaev D. S., “Priznaki odnoznachnoi razreshimosti lineinoi kraevoi zadachi dlya obyknovennogo differentsialnogo uravneniya”, ZhVM i MF, 29:1 (1989), 50–66 | MR | Zbl

[5] Dzhumabaev D. S., “Approksimatsiya ogranichennogo resheniya lineinogo obyknovennogo differentsialnogo uravneniya resheniyami dvukhtochechnykh kraevykh zadach”, ZhVM i MF, 30:3 (1990), 388–404 | MR | Zbl

[6] Dzhumabaev D. S., “Approksimatsiya ogranichennogo resheniya i eksponentsialnaya dikhotomiya na osi”, ZhVM i MF, 30:11 (1990), 1646–1660 | MR

[7] Dzhumabaev D. S., “Ob ogranichennosti na polose resheniya sistem giperbolicheskikh uravnenii”, Dokl. RAN, 395:2 (2004), 157–159 | MR | Zbl

[8] Kiguradze I. T., “Kraevye zadachi dlya sistem obyknovennykh differentsialnykh uravnenii”, Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Noveishie dostizheniya, 30, VINITI, M., 1987, 3–103 | MR

[9] Konyukhova N. B., Gladkie mnogoobraziya Lyapunova i singulyarnye kraevye zadachi, Soobscheniya po prikladnoi matematike, VTs RAN, M., 1996

[10] Massera Kh., Sheffer Kh., Lineinye differentsialnye uravneniya i funktsionalnye prostranstva, Mir, M., 1970 | MR | Zbl

[11] Mitropolskii Yu. A., Samoilenko A. M., Kulik V. L., Issledovanie dikhotomii lineinykh sistem differentsialnykh uravnenii s pomoschyu funktsii Lyapunova, Naukova dumka, Kiev, 1990 | MR

[12] Mukhamadiev E., “Issledovaniya po teorii periodicheskikh i ogranichennykh reshenii differentsialnykh uravnenii”, Mat. zametki, 30:3 (1981), 443–460 | MR | Zbl

[13] Pliss V. A., “Ogranichennye resheniya neodnorodnykh lineinykh sistem differentsialnykh uravnenii”, Problemy asimptoticheskoi teorii nelineinykh kolebanii, Naukova dumka, Kiev, 1977, 168–173 | MR

[14] Pliss V. A., Integralnye mnozhestva periodicheskikh sistem differentsialnykh uravnenii, Naukua, M., 1977 | MR | Zbl

[15] Trenogin V. A., Funktsionalnyi analiz, Naukua, M., 1984 | Zbl

[16] Markowich P. A., “A theory for the approximation of solutions of boundary values problems on infinite intervals”, SIAM J. Math. Anal., 13:13 (1982), 484–513 | DOI | MR | Zbl