Geodesic
On solvability of class of nonlinear equations with small parameter in Banach space
Ufa mathematical journal, Tome 12 (2020) no. 3, pp. 60-68 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the solvability of one class of nonlinear equations with a small parameter in a Banach space. The main difficulty is that the principal linear part of this equation is non-invertible. To study the solvability of the considered class of equations we apply a new method combining the Pontryagin method from the theory of autonomous systems on the plane and the methods of calculating the rotations of vector fields. We also employ a scheme for matrix representations of split operators known in the bifurcation theory of solutions to nonlinear equations. In contrast to the Pontryagin method, we do not assume a differentiability for a nonlinear mapping and apply methods for calculating the rotations of vector fields. On the base of the proposed method we formulate and prove a theorem on solvability conditions for the considered class of nonlinear equations. As an application, we study two periodic problems for nonlinear differential equations with a small parameter, namely, a periodic problem for the system of ordinary differential equations in a resonance case and a periodic problem for a nonlinear elliptic equations with a non-invertible linear part.
Keywords: nonlinear equation with small parameter, Pontryagin method, rotation of vector field, periodic problem. 
@article{UFA_2020_12_3_a6,
     author = {E. M. Mukhamadiev and A. B. Nazimov and A. N. Naimov},
     title = {On solvability of class of nonlinear equations with small parameter in {Banach} space},
     journal = {Ufa mathematical journal},
     pages = {60--68},
     year = {2020},
     volume = {12},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2020_12_3_a6/}
}
TY  - JOUR
AU  - E. M. Mukhamadiev
AU  - A. B. Nazimov
AU  - A. N. Naimov
TI  - On solvability of class of nonlinear equations with small parameter in Banach space
JO  - Ufa mathematical journal
PY  - 2020
SP  - 60
EP  - 68
VL  - 12
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UFA_2020_12_3_a6/
LA  - en
ID  - UFA_2020_12_3_a6
ER  - 
%0 Journal Article
%A E. M. Mukhamadiev
%A A. B. Nazimov
%A A. N. Naimov
%T On solvability of class of nonlinear equations with small parameter in Banach space
%J Ufa mathematical journal
%D 2020
%P 60-68
%V 12
%N 3
%U http://geodesic.mathdoc.fr/item/UFA_2020_12_3_a6/
%G en
%F UFA_2020_12_3_a6
E. M. Mukhamadiev; A. B. Nazimov; A. N. Naimov. On solvability of class of nonlinear equations with small parameter in Banach space. Ufa mathematical journal, Tome 12 (2020) no. 3, pp. 60-68. http://geodesic.mathdoc.fr/item/UFA_2020_12_3_a6/

[1] V. A. Trenogin, Functional analysis, Fizmatlit, M., 2002 (in Russian)

[2] L. S. Pontryagin, “On dynamical systems close to Hamiltonian ones”, Zhurn. Exper. Teor. Fiz., 4:8 (1934), 234–236 (in Russian)

[3] N. N. Bautin, E. A. Leontovich, “Methods and rules for the qualitative study of dynamical systems on the plane”, Nauka, M., 1900 (in Russian)

[4] M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, V. Ya. Stetsenko, Approximate solution of operator equations, Wolters-Noordhoff Publ., Groningen, 1972 | MR | Zbl

[5] M. A. Krasnosel'skii, P. P. Zabreiko, Geometrical methods of nonlinear analysis, Springer-Verlag, Berlin, 1984 | MR | Zbl

[6] E. Mukhamadiev, A. B. Nazimov, A. N. Naimov, “Study of solvability of one class of nonlinear equations with small parameter”, Vestnik Vologod. Gosud. Univer. Ser. Tekh. Nauki, 3:1 (2019), 50–53 (in Russian)

[7] E. Mukhamadiev, A. N. Naimov, A. Kh. Sattorov, “Solvability of a nonlinear boundary value problem with a small parameter”, Differ. Equat., 55:8 (2019), 1094–1104 | DOI | MR | Zbl

[8] A. B. Nazimov, E. Mukhamadiev, V. A. Morozov, M. Mullodzhanov, Method of regularization by a shift. Theory and applications, Vologda Gosud. Tekh. Univ., Vologda, 2012 (in Russian)

[9] M. A. Krasnoselskii, A. I. Perov, A. I. Povolotskii, P. P. Zabreiko, Vector fields on plane, Gosud. Izdat. Fiz. Matem. Liter., M., 1963 (in Russian) | MR