Existence and relaxation theorems for nonlinear multivalued boundary valueproblems

In this paper we consider a general nonlinear boundary value problem for se cond-order differential inclusions. We prove two existence theorems, one fo r the "convex" problem and the other for the "nonconvex" problem. Then we s how that the solution set of the latter is dense in the C-1(T, R-N)-norm to the solution set of the former (relaxation theorem). Subsequently for a Di richlet boundary value problem we prove the existence of extremal solutions and we show that they are dense in the solutions of the convexified proble m for the C-1(T, R-N)-norm. Our tools come from multivalued analysis and th e theory of monotone operators and our proofs are based on the Leray-Schaud er principle.