Set-Valued and Variational Analysis

, Volume 26, Issue 3, pp 693–728 | Cite as

Perturbed Evolution Problems with Continuous Bounded Variation in Time and Applications

  • Dalila Azzam-Laouir
  • Charles Castaing
  • M. D. P. Monteiro Marques


This paper is devoted to the study of evolution problems of the form \(-\frac {du}{dr}(t) \in A(t)u(t) + f(t, u(t))\) in a new setting, where, for each t, A(t) : D(A(t)) → 2 H is a maximal monotone operator in a Hilbert space H and the mapping tA(t) has continuous bounded or Lipschitz variation on [0, T], in the sense of Vladimirov’s pseudo-distance. The measure dr gives an upper bound of that variation. The perturbation f is separately integrable on [0, T] and separately Lipschitz on H. Several versions and new applications are presented.


Bolza control problem Bounded variation Lipschitz mapping Maximal monotone operators Pseudo-distance Perturbations Skorokhod problem 

Mathematics Subject Classification (2010)

34H05 34K35 60H10 28A25 28C20 


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M. D. P. Monteiro Marques was partially supported by Fundação para a Ciência e a Tecnologia, grant UID/MAT/04561/2013.


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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Pures et Appliquées, FSEIUniversité Mohammed Seddik Benyahia-JijelJijelAlgeria
  2. 2.Département de MathématiquesUniversité Montpellier IIMontpellierFrance
  3. 3.CMAF-CIO, Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisboaPortugal

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