Extremal Solutions and Relaxation for Second Order Vector Differential Inclusions

Evgenios P. Avgerinos and Nikolas S. Papageorgiou

Address. Evgenios P. Avgerinos, University of the Aegean, Mathematics Division, Department of Education, Rhodes 851 00, GREECE

Nikolas S. Papageorgiou, National Technical University, Department of Mathematics, Zografou Campus, Athens 157 80, GREECE

E-mail: eavger@rhodes.aegean.gr


Abstract. In this paper we consider periodic and Dirichlet problems for second order vector differential inclusions. First we show the existence of extremal solutions of the periodic problem (i.e. solutions moving through the extreme points of the multifunction). Then for the Dirichlet problem we show that the extremal solutions are dense in the $C^1(T,R^N)$-norm in the set of solutions of the ''convex'' problem (relaxation theorem).\newline

AMSclassification. 34A60, 34B15

Keywords. Lower semicontinuous multifunctions, continuous embedding, compact embedding, continuous selector, extremal solution, relaxation theorem