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Extremal Solutions and Relaxation for Second Order Vector Differential
Inclusions

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*
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

npapg@math.ntua.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