Criterion for the solvability of a class of nonlinear two-point boundary value problems on the plane

We study the solvability of a class of nonlinear two-point boundary value problems for systems of ordinary second-order differential equations on the plane. In these boundary value problems, we single out the leading nonlinear terms, which are positively homogeneous mappings. On the basis of properties of the leading nonlinear terms, we prove a criterion for the solvability of boundary value problems under arbitrary perturbations in a given set by using methods for the computation of the winding number of vector fields.