Solvability of a Nonlinear Boundary Value Problem with a Small Parameter

Мухамадиев, Э. М.; Наимов, А. Н.; Sattorov, A. Kh.

doi:10.1134/s001226611908010x

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“…In work [6], equation (1.1) was studied in a finitedimensional case as = R . In an infinite-dimensional, when is a Hilbert space, equation of form (1.1) was studied in work [7] at an example of one nonlinear boundary value problem with a small parameter.…”

Section: Introductionmentioning

confidence: 99%

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

2020

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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.

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Section: Introductionmentioning

confidence: 99%