Second Order Multivalued Boundary

In this paper we use the method of upper and lower solutions to study multivalued Sturm{Liouville and periodic boundary value problems , with Caratheodory orientor eld. We prove two existence theorems. One when the orientor eld F (t; x; y) is convex{valued and the other when F (t; x; y) is nonconvex valued. Finally we show that the \convex" problem has extremal solutions in the order interval determined by an upper and a lower solution. 1. Introduction The method of upper and lower solutions has been successfully applied to study the existence of multiple solutions for initial and boundary value problems of the rst and second order. This method generates solutions of the problem, located in an order interval with the upper and lower solutions serving as bounds. Moreover, this method coupled with some monotonicity type hypotheses, leads to monotone iterative techniques which generate in a constructive way (amenable to numerical treatment) the extremal solutions within the order interval determined by the upper and lower solutions. This method has been used only in the context of single{valued diierential equations and the majority of the works assume that the vector eld is continuous in all variables and so they look for solutions in the Banach spaces C