Second order multivalued boundary value problems

Nikolaos Halidias and Nikolaos S. Papageorgiou

Address. National Technical University, Department of Mathematics, Zografou Campus, Athens 157 80, GREECE


Abstract. 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 field. We prove two existence theorems. One when the orientor field $F(t,x,y)$ 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.

AMSclassification. 34B15, 34B24

Keywords. Upper solution, lower solution, order interval, usc multifunction, lsc multifunction, decomposable set, truncation map, penalty function, extremal solutions, Sturm--Liouville boundary conditions, periodic solutions