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

Set-Valued and Variational Analysis volume 22, pages 521–531 (2014)Cite this article

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.

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Authors and Affiliations

  1. Department of Mathematics, Zanjan University, Zanjan, Iran

    Hadi Khatibzadeh

  2. Department of Mathematics and its Applications, Central European University, Nador u. 9, 1051, Budapest, Hungary

    Gheorghe Moroşanu

Authors
  1. Hadi Khatibzadeh
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  2. Gheorghe Moroşanu
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Correspondence to Gheorghe Moroşanu.

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Khatibzadeh, H., Moroşanu, G. Strong and Weak Solutions to Second Order Differential Inclusions Governed by Monotone Operators. Set-Valued Var. Anal 22, 521–531 (2014). https://doi.org/10.1007/s11228-013-0270-3

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  • DOI: https://doi.org/10.1007/s11228-013-0270-3

Keywords

  • Differential inclusion
  • Maximal monotone operator
  • Boundary value problem
  • Strong solution
  • Weak solution
  • Bounded solution

Mathematics Subject Classifications (2010)

  • 34G25
  • 47H05