Geodesic
A parametrization method for solving nonlinear two-point boundary value problems
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 1, pp. 39-63 
D. S. Dzhumabaev; S. M. Temesheva. A parametrization method for solving nonlinear two-point boundary value problems. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 1, pp. 39-63. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_1_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.

[1] Shamanskii V. E., Metody chislennogo resheniya kraevykh zadach na ETsVM, Ch. 1, Nauk. dumka, Kiev, 1963

[2] Bellman R., Kalaba R., Kvazilinearizatsiya i nelineinye kraevye zadachi, Mir, M., 1968 | Zbl

[3] Keller H. B., Numerical methods for two-point boundary-value problems, Walthan, Blaisdell, 1968 | MR | Zbl

[4] Roberts S. M., Shipman J. S., Two-point boundary-value problems: Shooting methods, Elsevier, N.Y., 1972 | MR

[5] Bakhvalov N. S., Chislennye metody, Fizmatgiz, M., 1973

[6] Keller H. B., White A. B., “Difference methods for boundary value problems in ordinary differential equations”, SIAM J. Numer. Analys., 12:5 (1975), 791–802 | DOI | MR | Zbl

[7] Dzh. Kholl, Dzh. Uatt (red.), Sovremennye chislennye metody resheniya obyknovennykh differentsialnykh uravnenii, Mir, M., 1979

[8] Monastyrnyi P. I., “O skhodimosti metoda intervalnoi pristrelki”, Zh. vychisl. matem. i matem. fiz., 18:6 (1978), 1139–1145 | MR | Zbl

[9] Monastyrnyi P. I., “O svyazi izolirovannosti reshenii so skhodimostyu metodov pristrelki”, Differents. ur-niya, 16:4 (1980), 732–740 | MR

[10] Nesterenko L. I., “O suschestvovanii i edinstvennosti resheniya dvukhtochechnoi granichnoi zadachi dlya sistemy obyknovennykh differentsialnykh uravnenii”, Nelineinye kraevye zadachi, In-t matem. AN USSR, Kiev, 1980

[11] Babenko K. I., Osnovy chislennogo analiza, Nauka, M., 1986 | MR

[12] Kiguradze I. T., “Kraevye zadachi dlya sistem obyknovennykh differentsialnykh uravnenii”, Sovrem. probl. matem. Noveishie dostizheniya, 30, VINITI AN SSSR, M., 1987, 3–103 | MR

[13] Samoilenko A. M., Ronto H. I., Chislenno-analiticheskie metody issledovaniya reshenii kraevykh zadach, Nauk. dumka, Kiev, 1986 | MR

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

[15] Ortega Dzh., Reinboldt V., Iteratsionnye metody resheniya nelineinykh sistem uravnenii so mnogimi neizvestnymi, Mir, M., 1975 | MR

[16] Dzhumabaev D. S., “Skhodimost iteratsionnykh metodov dlya neogranichennykh operatornykh uravnenii”, Matem. zametki, 41:5 (1987), 637–645 | MR

[17] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl