A relaxation theorem for a Sturm-Liouville-type of a boundary value problem for differential inclusions is proved. Namely, the author considers the following two problems: (1)
is a set-valued map and
denotes the set of extreme points of
. Using Green function it is proved that solutions of (2) do exist provided
is measurable in
and continuous in
and, moreover, if
is Lipschitz continuous they are dense in the solution set of (1). A special condition on the Green function which gives a priori bounds on the solutions is assumed throughout the paper. An application to control systems is given.