# Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems

Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems

, Volume 39 (2) – Apr 1, 2024
23 pages

/lp/springer_journal/existence-and-relaxation-theorems-for-nonlinear-multivalued-boundary-pvrJYLtP3Z
Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900106
Publisher site
See Article on Publisher Site

### 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.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2024

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