Existence and relaxation theorems for nonlinear multivalued boundary value problems

In this paper we consider a general nonlinear boundary value problem for second-order differential inclusions. We prove two existence theorems, one for the "convex" problem and the other for the "nonconvex" problem. Then we show 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 Dirichlet boundary value problem we prove the existence of extremal solutions and we show that they are dense in the solutions of the convexified problem for the C-1(T, R-N)-norm. Our tools come from multivalued analysis and the theory of monotone operators and our proofs are based on the Leray-Schauder principle.