Nonlinear Boundary Value Problems for Ordinary Differential Equations with a Small Parameter

Abstract

The author is concerned with the solution of nonlinear boundary value problems. In particular, he deals with the problem of resonance for a nonselfadjoint system of real ordinary differential equations with homogeneous linear boundary conditions. An alternative method similar to the one used by J. K. Hale when studying periodic solutions is used to reduce the problem to solving a system of q equations in p unknowns where p is the number of independent solutions to the associated linear boundary value problem and q is the number of independent solutions to the associated adjoint linear boundary value problem. Conditions are given for solving the system of q equations in p unknowns by the implicit function theorem. Several examples are included to illustrate the method.