Regularized continuous analog of the Newton method for monotone equations in the Hilbert space

I. P. Ryazantseva

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2016), pp. 53-67.

Résumé

We construct regularized continuous analog of the Newton method in the Hilbert space for nonlinear equation with Frechét differentiable and monotone operator. We obtain sufficient conditions of its strong convergence to normal solution of the given equation under approximate assignment of an operator and right-hand of an equation.

Détail

Keywords: Hilbert space, monotone operator, Newton method, continuous method
Mots-clés : convergence.

Bibliographie

[1] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972 | MR

[2] Kantorovich L. V., Akilov G., Funktsionalnyi analiz, Nauka, M., 1984 | MR

[3] Gavurin M. K., “Nelineinye funktsionalnye uravneniya i nepreryvnye analogi iterativnykh metodov”, Izv. vuzov. Matem., 1958, no. 5, 18–31 | MR | Zbl

[4] Zhidkov E. P., Makarenko G. I., Puzynin I. V., “Nepreryvnyi analog metoda Nyutona v nelineinykh zadachakh fiziki”, Fizika elementarnykh chastits i atomnogo yadra, 4:1 (1973), 127–166 | MR

[5] Zhidkov E. P., Kozlova O. V., “Nepreryvnyi analog metoda Nyutona v obratnoi zadache teorii rasseyaniya pri nalichii sobstvennykh funktsii i znachenii”, Matem. modelirovanie, 18:2 (2006), 120–128 | MR | Zbl

[6] Alber Ya., Ryazantseva I., Nonlinear ill-posed problems of monotone type, Springer, Dordrecht, 2006 | MR | Zbl

[7] Ryazantseva I. P., Izbrannye glavy teorii operatorov monotonnogo tipa, NGTU, Nizhnii Novgorod, 2008

[8] Kachurovskii R. I., “Nelineinye monotonnye operatory v banakhovykh prostranstvakh”, UMN, 23:2 (1968), 121–168 | MR | Zbl

[9] Bakushinskii A. B., Goncharskii A. V., Nekorrektnye zadachi. Chislennye metody i prilozheniya, Izd-vo Mosk. un-ta, M., 1989

[10] Antipin A. S., “Nepreryvnye i iterativnye protsessy s operatorami proektirovaniya i tipa proektirovaniya”, Vychisl. voprosy analiza bolshikh sistem, Vopr. kibernetiki, 154, Nauchn. sovet po kompleksnoi probleme “Kibernetika” AN SSSR, M., 1989, 5–43 | MR

[11] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981 | MR

[12] Ryazantseva I. P., “O nepreryvnykh metodakh pervogo poryadka i ikh regulyarizovannykh variantakh dlya smeshannykh variatsionnykh neravenstv”, Differents. uravneniya, 45:7 (2012), 1020–1032 | MR

[13] Ryazantseva I. P., “Nepreryvnye metody pervogo poryadka dlya monotonnykh vklyuchenii v gilbertovom prostranstve”, Zhurn. vychisl. matem. i matem. fiziki, 53:8 (2013), 1241–1248 | DOI | MR | Zbl

[14] Tikhonov A. N., Goncharskii A. V., Stepanov V. V., Yagola A. G., Regulyarizovannye algoritmy i apriornaya informatsiya, Nauka, M., 1983

[15] Tikhonov A. N., Goncharskii A. V., Stepanov V. V., Yagola A. G., Chislennye metody resheniya nekorrektnykh zadach, Nauka, M., 1990 | MR

[16] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1988 | MR

[17] Ryazantseva I. P., “O nekotorykh metodakh nepreryvnoi regulyarizatsii dlya monotonnykh uravnenii”, Zhurn. vychisl. matem. i matem. fiziki, 34:1 (1994), 3–11 | MR | Zbl

[18] Ramm A. G., Dynamical systems method for solving operator equations, Elsevier, Amsterdam, 2007 | MR | Zbl

[19] Kokurin M. Yu., “Nepreryvnye metody ustoichivoi approksimatsii reshenii nelineinykh uravnenii v gilbertovom prostranstve na osnove regulyarizovannoi skhemy Gaussa–Nyutona”, Zhurn. vychisl. matem. i matem. fiziki, 44:1 (2004), 8–17 | MR | Zbl