Appl Math Optim 39:257–279 (1999)
1999 Springer-Verlag New York Inc.
Existence and Relaxation Theorems for Nonlinear Multivalued
Boundary Value Problems
E. P. Avgerinos
and N. S. Papageorgiou
Department of Education, Mathematics Division, University of the Aegean,
1 Demokratias Avenue, Rhodes 85100, Greece
Department of Mathematics, National Technical University,
Zografou Campus, Athens 157 80, Greece
Communicated by R. Conti
Abstract. 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
(T , R
)-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
(T , R
tools come from multivalued analysis and the theory of monotone operators and our
proofs are based on the Leray–Schauder principle.
Key Words. Maximal monotone operator, Coercive operator, Leray–Schauder
principle, Integration by parts, Compact embedding, Extremal solution, Continuous
selection, Weak norm, Strong relaxation.
AMS Classiﬁcation. 34A60, 34B15.
The ﬁrst author’s research was supported by Research Grant PENED 678(96) of the Greek Secretariat
of Science and Technology