On the theory of the Cauchy problem for an ordinary differential equation with discontinuous right-hand side

V. V. Filippov

Sbornik. Mathematics, Tome 83 (1995) no. 2, pp. 383-403

Résumé

An apparatus is developed for proving theorems on the existence of a solution of the Cauchy problem and on continuous dependence of solutions on the initial conditions and parameters, applicable to situations not covered by known results on this subject.

Bibliographie

[1] Filippov V. V., “Ob obyknovennykh differentsialnykh uravneniyakh s osobennostyami v pravoi chasti”, Matem. zametki, 38:6 (1985), 832–851 | MR | Zbl

[2] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR | Zbl

[3] Fedorchuk V. V., Filippov V. V., Obschaya topologiya. Osnovnye konstruktsii, Izd-vo MGU, M., 1988 | Zbl

[4] Filippov V. V., “Aksiomaticheskaya teoriya prostranstv reshenii obyknovennykh differentsialnykh uravnenii”, DAN SSSR, 280:2 (1985), 304–308 | MR | Zbl

[5] Filippov V. V., “K teorii prostranstv reshenii obyknovennykh differentsialnykh uravnenii”, DAN SSSR, 285:5 (1985), 1073–1077 | MR | Zbl

[6] Filippov V. V., “O teoreme suschestvovaniya resheniya obyknovennogo differentsialnogo uravneniya”, Matem. zametki, 40:6 (1986), 743–748 | MR | Zbl

[7] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR

[8] Filippov A. F., “Differentsialnye uravneniya s razryvnoi pravoi chastyu”, Matem. sb., 51:1 (1960), 99–128 | MR | Zbl

[9] Vrkoc I., “Scorza-Dragoni property of Filippov mappings”, Tohoku math. J., Ser. 2, 32:2 (1980), 309–316 | DOI | MR | Zbl

[10] Shilov G. E., Matematicheskii analiz. Spetsialnyi kurs, FM, M., 1961

[11] Edvards R., Funktsionalnyi analiz. Teoriya i prilozheniya, Mir, M., 1969

[12] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[13] Davy J. L., “Properties of solution set of a generalized differentional equation”, Bull. Austral. Math. Soc., 6:3 (1972), 379–398 | DOI | MR | Zbl

[14] Filippov V. V., “Dva zamechaniya k teorii prostranstv reshenii obyknovennykh differentsialnykh uravnenii”, Vestn. MGU. Ser. 1, matem., mekh., 1988, no. 4, 53–55

[15] Scorza-Dragoni G., “Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un'altra variabile”, Rend. Semin. mat. Univ. Padova, 17 (1948), 102–106 | MR | Zbl

[16] Filippov V. V., “O teoreme Luzina i pravykh chastyakh differentsialnykh vklyuchenii”, Matem. zametki, 37:1 (1985), 93–98 | Zbl

[17] Filippov V. V., “O teoremakh Luzina i Skortsa-Dragoni”, Vestn. MGU. Ser. 1, matem., mekh., 1987, no. 3, 66–68 | Zbl

[18] Ricceri B., Villani A., “Separability and Scorza–Dragoni property”, Matematische, 37:1 (1982), 156–161 | MR

[19] Bokshtein M. F., “Teoremy suschestvovaniya i edinstvennosti reshenii dlya sistem obyknovennykh differentsialnykh uravnenii.”, Uchenye zapiski MGU. Matem., 1939, no. 15, 3–72

[20] Koddington E. A., Levinson N., Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1949

[21] Filippov V. V., “Teoriya zadachi Koshi dlya obyknovennogo differentsialnogo uravneniya s tochki zreniya obschei topologii”, Obschaya topologiya. Otobrazheniya topologicheskikh prostranstv, Izd-vo MGU, M., 1988 | Zbl

[22] Filippov V. V., “Ob asimptoticheskom integrirovanii obyknovennykh differentsialnykh uravnenii s razryvami v pravoi chasti”, DAN SSSR, 321:3 (1991), 482–485 | MR | Zbl

[23] Filippov V. V., “O statsionarnykh tochkakh i nekotorykh geometricheskikh svoistvakh reshenii obyknovennykh differentsialnykh uravnenii”, DAN SSSR, 323:6 (1992), 1043–1047 | Zbl

[24] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR