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 Ufimsk. Mat. Zh., 2020, Volume 12, Issue 3, Pages 62–70 (Mi ufa528)

On solvability of class of nonlinear equations with small parameter in Banach space

E. M. Mukhamadiev, A. B. Nazimov, A. N. Naimov

Vologda State University, Lenin str., 15, 160000, Vologda, Russia

Abstract: 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.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-01-00103_à18-47-350001_ð_à The work is partially supported by Russian Foundation for Basic Researches (projects nos. 18-47-350001r-a, 19-01-00103a).

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English version:
Ufa Mathematical Journal, 2020, 12:3, 60–68 (PDF, 330 kB); https://doi.org/10.13108/2020-12-3-60

Bibliographic databases:

UDC: 517.988.63+517.927.4+517.956.25
MSC: 46T20, 47A55,34B15, 35J66

Citation: E. M. Mukhamadiev, A. B. Nazimov, A. N. Naimov, “On solvability of class of nonlinear equations with small parameter in Banach space”, Ufimsk. Mat. Zh., 12:3 (2020), 62–70; Ufa Math. J., 12:3 (2020), 60–68

Citation in format AMSBIB
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