Article
Ufa Mathematical Journal
Volume 12, Number 3, pp. 60-68
On solvability class of nonlinear equations with small parameter in Banach space
Mukhamadiev E.M., Naimov A.N., Nazimov A.B.
DOI:10.13108/2020-12-3-60
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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. At that we employ a scheme for matrix representations of split operators known in the bifurcation theory for solutions of nonlinear equations. In contrast to the Pontryagin method, we do not assume the 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 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.