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Various Perturbations of Time Dependent Maximal Monotone/Accretive Operators in Evolution Inclusions with Applications

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Abstract

We are concerned in the present work with diverse perturbations of time dependent monotone/accretive evolution. New applications such as integral–differential Volterra perturbation, periodicity, relaxation, asymptotic properties are presented.

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Notes

  1. Actually it is possible to prove this inclusion by using the compactness of L and the maximal monotone extension of \( \mathcal A : D(\mathcal A) \subset L_H^2(I) \rightrightarrows L_H^2(I)\) given in Lemma 3.1

  2. If \(H= \mathbb {R}^d\), one may invoke a classical fact that on bounded subsets of \(L^\infty _H\) the topology of convergence in measure coincides with the topology of uniform convergence on uniformly integrable sets, i.e. on relatively weakly compact subsets, alias the Mackey topology. This is a lemma due to Grothendieck [31] [Chap. 5 Sect. 4 no 1 Prop. 1 and exercice]

  3. If \(H= \mathbb {R}^d\), one may invoke a classical fact that on bounded subsets of \(L^\infty _H\) the topology of convergence in measure coincides with the topology of uniform convergence on uniformly integrable sets, i.e. on relatively weakly compact subsets, alias the Mackey topology. This is a lemma due to Grothendieck [31] [Ch. 5 Sect. 4 no 1 Prop. 1 and exercice]

  4. If \(H= \mathbb {R}^d\), one may invoke a classical fact that on bounded subsets of \(L^\infty _H\) the topology of convergence in measure coincides with the topology of uniform convergence on uniformly integrable sets, i.e. on relatively weakly compact subsets, alias the Mackey topology. This is a lemma due to Grothendieck [31] [Ch. 5 Sect. 4 no 1 Prop. 1 and exercice]

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Funding

M.D.P. Monteiro Marques was partially supported by the Fundação para a Ciência e a Tecnologia, grant UID/MAT/04561/2019.

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Authors and Affiliations

  1. IMAG, Univ Montpellier, CNRS, Montpellier II, Case courrier 051, 34095, Montpellier Cedex 5, France

    Charles Castaing

  2. Université de Bretagne Occidentale, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS UMR 6205, 6, Avenue Victor Le Gorgeu, CS 9387, 29238, Brest Cedex3, France

    Christiane Godet-Thobie

  3. LMPA Laboratory, Department of Mathesmatics, Jijel University, Jijel, Algeria

    Soumia Saïdi

  4. CMAF, Departemento de Matematica, Faculdade de ciencas de Lisboa, Campo Grande, 1749-016, Lisbon, Portugal

    Manuel D. P. Monteiro Marques

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  1. Charles Castaing
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Correspondence to Christiane Godet-Thobie.

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Castaing, C., Godet-Thobie, C., Saïdi, S. et al. Various Perturbations of Time Dependent Maximal Monotone/Accretive Operators in Evolution Inclusions with Applications. Appl Math Optim 87, 24 (2023). https://doi.org/10.1007/s00245-022-09898-5

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