On solvability of class of nonlinear equations with small parameter in Banach space
Ufa mathematical journal, Tome 12 (2020) no. 3, pp. 60-68
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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},
publisher = {mathdoc},
volume = {12},
number = {3},
year = {2020},
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 PB - mathdoc 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 %I mathdoc %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/