In this paper we study differential inclusions with boundary conditions in
which the vector field F(t,x,y) is a multifunction with Caratheodory type
conditions. We consider, first, the case which F has values in ℝ and we
establish the existence of extremal solutions in the order interval determined by the lower and the upper solution. Then we prove the existence of
solutions for a Dirichlet problem in the case in which F takes their values
in a Hilbert space.