Solvability of a Nonlinear Boundary Value Problem with a Small Parameter

Abstract

We study the solvability of a nonlinear boundary value problem for a partial differential equation with a small parameter multiplying the nonlinearity. The solvability conditions are first derived for the corresponding linear problem by the Fourier method and then used to state and prove theorems about the solvability of the nonlinear boundary value problem. If the corresponding homogeneous linear boundary value problem has nonzero solutions, then the solvability of the nonlinear boundary value problem is established using ideas of the Pon-tryagin method and the methods and means of the theory of rotation of completely continuous vector fields.