Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems

Journal Article · Mon Mar 15 04:00:00 EST 1999 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S002459900106· OSTI ID:21067552

Avgerinos, E P ^[1]; Papageorgiou, N S ^[2]

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1. Department of Education, Mathematics Division, University of the Aegean, 1 Demokratias Avenue, Rhodes 85100 (Greece)
2. Department of Mathematics, National Technical University, Zografou Campus, Athens 157 80 (Greece)

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{sup 1}(T,R{sup 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{sup 1}(T,R{sup N}) -norm . Our tools come from multivalued analysis and the theory of monotone operators and our proofs are based on the Leray-Schauder principle.

• OSTI ID: 21067552
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2 Vol. 39; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOUNDARY-VALUE PROBLEMS
MATHEMATICAL SOLUTIONS
NONLINEAR PROBLEMS
RELAXATION