Strong and Weak Solutions to Second Order Differential Inclusions Governed by Monotone Operators

In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u ″( t ) ∈ Au ( t ) + f ( t ), where A is a maximal monotone operator in a Hilbert space H . We prove existence and uniqueness of weak solutions to two point boundary value problems associa...

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Bibliographic Details
Published in:Set-valued and variational analysis 2014-06, Vol.22 (2), p.521-531
Main Authors: Khatibzadeh, Hadi, Moroşanu, Gheorghe
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
Optimization
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Summary:In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u ″( t ) ∈ Au ( t ) + f ( t ), where A is a maximal monotone operator in a Hilbert space H . We prove existence and uniqueness of weak solutions to two point boundary value problems associated with such kind of equations. Furthermore, existence of (strong and weak) solutions to the equation above which are bounded on the positive half axis is proved under the optimal condition tf ( t ) ∈ L 1 (0, ∞; H ), thus solving a long-standing open problem (for details, see our comments in Section 3 of the paper). Our treatment regarding weak solutions is similar to the corresponding theory related to the first order differential inclusions of the form f ( t ) ∈ u ′( t ) + Au ( t ) which has already been well developed.
ISSN:1877-0533
1877-0541
DOI:10.1007/s11228-013-0270-3