We study linear systems of ordinary differential equations on a finite interval with the most general (generic) inhomogeneous boundary conditions in Sobolev spaces. These boundary problems include all known types of classical and numerous nonclassical conditions. The latter may contain derivatives of integer and fractional order, which may exceed the order of the differential equation.
We investigate the character of solvability of inhomogeneous boundary-value
problems, prove their Fredholm properties, and find the indices, Fredholm numbers of these problems. These results are illustrated by examples.
We obtained necessary and sufficient conditions for continuity in a parameter of solutions to the introduced boundary-value problems in the Sobolev spaces. Some applications of these results to the solutions of multipoint boundary-value problems are also presented.
The theorem on the approximation of solutions to inhomogeneous generic boundary-value problem by solutions of the multipoint boundary-value problems is proved.
Adrien Petrov