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

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