of non-autonomous differential inclusions which are non-local in space and time. A classical example would be integral operators of the form $$\left(\mathcal{M}u\right)(t)\coloneqq\int_{\mathbb{R}}k(t,s)u(s){\,\mathrm{d}}s\quad(t\in\mathbb{R})$$ with a suitable (possibly operator-valued) kernel $k$. Indeed, if $k$ is differentiable with respect to the first variable and satisfies suitable integrability conditions, one easily can show that (a) holds. To find the right conditions on $k$ to ensure (b) is more delicate and will be postponed to future research. In case of a kernel $k(t,s)=k(t-s)$, this was done in [@Trostorff2012_integro; @Trostorff_habil] even for operator-valued kernels.

We begin by proving in the case $\mathcal{A}=0.$ The proof follows the rationale of [@Waurick_habil Section 3.3]. For the readers convenience we recall the definition of a core of a closed operator.

Let $S:{\operatorname{dom}}(S)\subseteq X\to Y$ be a closed linear operator between two normed spaces $X$ and $Y$. A linear subspace $D\subseteq X$ is called a *core for $S$*, if $S=\overline{S|_{D}}.$

\[prop:max\_mon\_materiallaw\] $C_{c}^{\infty}({\mathbb{R}};H)$ is a core for $\partial_{t,\rho}\mathcal{M}$. Moreover, the operator $\partial_{t,\rho}\mathcal{M}+\mathcal{N}$ is $c$-maximal monotone.

First, we observe that $(1+\varepsilon\partial_{t,\rho})^{-1}\to1$ strongly as $\varepsilon\to0+.$ Indeed, since $\partial_{t,\rho}$ is maximal monotone by we have $\|(1+\varepsilon\partial_{t,\rho})^{-1}\|\leq1$ for each $\varepsilon>0.$ Thus, it suffices to prove the strong convergence on a dense subset of $L_{2,\rho}({\mathbb{R}};H).$ Since for 