Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems |
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| Authors: | E P Avgerinos N S Papageorgiou |
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| Institution: | (1) Department of Education, Mathematics Division, University of the Aegean, 1 Demokratias Avenue, Rhodes 85100, Greece , GR;(2) Department of Mathematics, National Technical University, Zografou Campus, Athens 157 80, Greece , GR |
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| 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
1
(T,R
N
) -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
1
(T,R
N
) -norm . Our tools come from multivalued analysis and the theory of monotone operators and our proofs are based on the Leray—Schauder
principle.
Accepted 18 September 1997 |
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| Keywords: | , 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, |
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