On Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods

N. A. Sidorov ; R. Yu. Leontiev ; A. I. Dreglea

Matematičeskie zametki, Tome 91 (2012) no. 1, pp. 120-135.

Résumé

We consider the nonlinear operator equation $B(\lambda)x+R(x,\lambda)=0$ with parameter $\lambda$, which is an element of a linear normed space $\Lambda$. The linear operator $B(\lambda)$ has no bounded inverse for $\lambda=0$. The range of the operator $B(0)$ can be nonclosed. The nonlinear operator $R(x,\lambda)$ is continuous in a neighborhood of zero and $R(0,0)=0$. We obtain sufficient conditions for the existence of a continuous solution $x(\lambda)\to 0$ as $\lambda\to 0$ with maximal order of smallness in an open set $S$ of the space $\Lambda$. The zero of the space $\Lambda$ belongs to the boundary of the set $S$. The solutions are constructed by the method of successive approximations.

Détail

Keywords: nonlinear operator equation, Banach space, sectorial neighborhood, Fredholm operator, Schmidt's lemma, regularizer for a nonlinear operator.
Mots-clés : bifurcation

Bibliographie

[1] M. M. Vainberg, V. A. Trenogin, Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR | Zbl

[2] Nelineinyi analiz i nelineinye differentsialnye uravneniya, eds. V. A. Trenogin, A. F. Filippov, Fizmatlit, M., 2003 | MR | Zbl

[3] N. Sidorov. B. Loginov, A. Sinitsyn, M. Falaleev, Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications, Math. Appl., 550, Kluwer Acad. Publ., Dordrecht, 2002 | MR | Zbl

[4] N. A. Sidorov, A. V. Sinitsin, “Issledovanie tochek bifurkatsii i netrivialnykh vetvei reshenii statsionarnoi sistemy Vlasova–Maksvella”, Matem. zametki, 62:2 (1997), 268–292 | MR | Zbl

[5] N. A. Sidorov, “Yavnaya i neyavnaya parametrizatsiya pri postroenii razvetvlyayuschikhsya reshenii iteratsionnymi metodami”, Matem. sb., 186:2 (1995), 129–141 | MR | Zbl

[6] B. V. Loginov, N. A. Sidorov, “Gruppovaya simmetriya uravneniya razvetvleniya Lyapunova–Shmidta i iteratsionnye metody v zadache o tochke bifurkatsii”, Matem. sb., 182:5 (1991), 681–691 | MR | Zbl

[7] N. A. Sidorov, V. R. Abdullin, “Spletaemye uravneniya razvetvleniya v teorii nelineinykh uravnenii”, Matem. sb., 192:7 (2001), 107–124 | MR | Zbl

[8] V. A. Trenogin, Funktsionalnyi analiz, Fizmatlit, M., 2002 | MR | Zbl

[9] N. A. Sidorov, “Minimalnye vetvi reshenii nelineinykh uravnenii i asimptoticheskie regulyarizatory”, Nelineinye granichnye zadachi, 14, Institut prikladnoi matematiki i mekhaniki, Donetsk, 2004, 161–164

[10] A. N. Tikhonov, V. K. Ivanov, M. M. Lavrentev, “Nekorrektno postavlennye zadachi”, Differentsialnye uravneniya s chastnymi proizvodnymi, Trudy simpoziuma, posvyaschennogo 60-letiyu akademika S. L. Soboleva, Nauka, M., 1970, 224–238 | MR | Zbl

[11] V. A. Trenogin, N. A. Sidorov, “Regulyarizatsiya prostykh reshenii nelineinykh uravnenii v okrestnosti tochek vetvleniya”, Sib. matem. zhurn., 19:1 (1978), 180–185 | MR | Zbl

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

[13] A. I. Dreglya, “Nekotorye analiticheskie i tochnye resheniya sistem uravnenii v teorii modelirovaniya polimerov”, Sib. zhurn. industr. matem., 11:3 (2008), 61–70 | MR | Zbl

[14] A. I. Dreglea, G. I. Shishkin, “Robust numerical method for a singularly perturbed equation with unboundedly growing convective term at infinity”, Proceedings of the International Conference on Computational Mathematics, ICCM-2004, Novosibirsk, 2004, 835–838