Geodesic
On the properties of finite-dimensional systems of nonlinear equations with multiple solutions
Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 11 (2009), pp. 20-36 
M. V. Bulatov; V. F. Chistyakov. On the properties of finite-dimensional systems of nonlinear equations with multiple solutions. Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 11 (2009), pp. 20-36. http://geodesic.mathdoc.fr/item/VCHGU_2009_11_a2/
@article{VCHGU_2009_11_a2,
     author = {M. V. Bulatov and V. F. Chistyakov},
     title = {On the properties of finite-dimensional systems of nonlinear equations with multiple solutions},
     journal = {Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika},
     pages = {20--36},
     year = {2009},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VCHGU_2009_11_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Systems of nonlinear equations of the form $F(x)=0$, whose Jacobean is degenerate on the solution, are considered. Definitions of the value of solution's multiplicity are introduced for such problems. A relationship between the values of solution's multiplicity and the values of the index of matrix pencils has been determined; comparing with known definitions of multiplicity has been conducted. Possible applications of the concepts introduced in the theory of differential-algebraic equations, as well as in the process of analysis of non-Morse stationary point of the function and the values of the system's solution multiplicity, which satisfy the stationary point, are discussed.

[1] M. A. Krasnoselskii, G. M. Vainikko, P. P. Zabreiko , Ya. B. Rutitskii, V. Ya. Stetsenko, Priblizhennoe reshenie operatornykh uravnenii, Nauka, M., 1969 | MR

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

[3] A. V. Arutyunov, Usloviya ekstremuma. Anormalnye i vyrozhdennye zadachi, Faktorial, M., 1997 | MR

[4] A. F. Izmailov, A. A. Tretyakov, 2-regulyarnye resheniya nelineinykh zadach, Fizmatlit, M., 1999 | MR | Zbl

[5] O. A. Brezhneva, A. F. Izmailov, “O postroenii opredelyayuschikh sistem dlya otyskaniya osobykh reshenii nelineinykh uravnenii”, ZhVMMF, 42:1 (2002), 10—22 | MR | Zbl

[6] O. A. Brezhneva, Yu. G. Evtushenko, A. A. Tretyakov, “Teoriya $p$-regulyarnosti i malogo parametra Puankare”, Dokl. RAN, 411:6 (2006), 727—731 | MR | Zbl

[7] M. V. Bulatov, V. F. Chistykov, “On multiple solutions nonlinear equations”, Vychislitelnye i informatsionnye tekhnologii v nauke, tekhnike i obrazovanii (7—9 okt.), Vychislitelnye tekhnologii i Vestnik KazNU im. Al-Farabi. Ser. Matematika, mekhanika, informatika, 9, Sovmestn. vyp. po materialam mezhdunar. konf. “Vychislitelnye i informatsionnye tekhnologii v nauke, tekhnike i obrazovanii”. (7—9 okt.), 2004, 22–29

[8] Yu. E. Boyarintsev, Regulyarnye i singulyarnye sistemy lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, Novosibirsk, 1980 | MR

[9] G. I. Marchuk, V. V. Shaidurov, Povyshenie tochnosti raznostnykh skhem, Nauka, M., 1979 | MR

[10] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1967 | MR

[11] M. V. Bulatov, “Reduktsiya vyrozhdennykh sistem integralnykh uravnenii tipa Volterra k nevyrozhdennym”, Izv. vuzov. Matematika, 1998, no. 11 (438), 14—21 | Zbl

[12] M. V. Bulatov, “O preobrazovanii algebro-differentsialnykh sistem uravnenii”, ZhVMMF, 34:3 (1994), 360—372 | MR | Zbl

[13] V. I. Arnold, A. N. Varchenko, S. M. Gusein-Zade, Osobennosti differentsiruemykh otobrazhenii, MTsNMO, M., 2004 | MR

[14] I. R. Shafarevich, Osnovy algebraicheskoi geometrii: v 2 t., v. 1, Algebraicheskie mnogoobraziya v proektivnom prostranstve, Nauka, M., 1988 | MR

[15] L. Rall, “Convergence of the process to multiple solutions”, Numer. Math., 9 (1966), 33—37 | DOI | MR

[16] V. F. Chistyakov, Algebro-differentsialnye operatory s konechnomernym yadrom, Nauka, Novosibirsk, 1996 | MR