Strong and weak solutions to second order differential inclusions governed by monotone operators

TitleStrong and weak solutions to second order differential inclusions governed by monotone operators
Publication TypeJournal Article
AuthorsKhatibzadeh, Hadi, and Gheorghe Morosanu
Journal titleSet-Valued and Variational Analysis
Year2014
Pages521-531
Volume22
Issue2
Abstract

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.

ISSN1877-0533
LanguageEnglish
DOI10.1007/s11228-013-0270-3
Publisher linkhttp://link.springer.com/article/10.1007/s11228-013-0270-3
Unit: 
Department of Mathematics and its Applications