Differential Inclusions, Control and Optimization 21 (2001)127-148
doi: 10.7151/dmdico.1020

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NONLINEAR MULTIVALUED BOUNDARY VALUE PROBLEMS

Ralf Bader

Center of Mathematics
Technical University Muenchen
Arcisstr. 21, D-80333 Muenchen, Germany
e-mail: bader@appl-math.tu-muenchen.de

Nikolaos S. Papageorgiou

National Technical University, Department of Mathematics
Zografou Campus, Athens 15780, Greece

e-mail: npapg@math.ntua.gr

Abstract

In this paper, we study nonlinear second order differential inclusions with a multivalued maximal monotone term and nonlinear boundary conditions. We prove existence theorems for both the convex and nonconvex problems, when domA ≠ IRN and domA = IRN, with A being the maximal monotone term. Our formulation incorporates as special cases the Dirichlet, Neumann and periodic problems. Our tools come from multivalued analysis and the theory of nonlinear monotone operators.

Keywords and phrases: usc and lsc multifunction, measurable selection, Leray-Schauder alternative theorem, Sobolev space, compact embedding, maximal monotone map, coercive map, surjective map, convex and nonconvex problem, nonlinear boundary conditions.

2000 Mathematics Subject Classification: 34B15.

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Received 10 July 2000