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March  2020, 9(1): 219-254. doi: 10.3934/eect.2020004

## Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators

 1 Laboratoire LAOTI, FSEI, Université Mohammed Seddik Benyahia de Jijel, Algérie 2 IMAG, Université Montpellier, CNRS, Montpellier Cedex 5, France 3 CMAFcIO, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Portugal

* Corresponding author: Dalila Azzam-Laouir

Received  December 2018 Revised  May 2019 Published  March 2020 Early access  August 2019

In this paper, we study the existence of solutions for evolution problems of the form $-\frac{du}{dr}(t) \in A(t)u(t) + F(t, u(t))+f(t, u(t))$, where, for each $t$, $A(t) : D(A(t)) \to 2 ^H$ is a maximal monotone operator in a Hilbert space $H$ with continuous, Lipschitz or absolutely continuous variation in time. The perturbation $f$ is separately integrable on $[0, T]$ and separately Lipschitz on $H$, while $F$ is scalarly measurable and separately scalarly upper semicontinuous on $H$, with convex and weakly compact values. Several new applications are provided.

Citation: Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing, M. D. P. Monteiro Marques. Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations & Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004
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