we can prove our main theorem.

We have that $\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}$ is $c$-monotone by and hence, $$\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}-c=\left(\overline{\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}}-c\right)+(\mathcal{M}'+\mathcal{N}-\delta)$$ is maximal monotone by and .

Causality and independence of $\rho$
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The aim of this section is twofold. First, we prove that the solution operator $\mathcal{S}_{\rho}$ associated with the differential inclusion $$(u,f)\in\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}$$ is causal under suitable additional conditions on $\mathcal{M},\mathcal{N}$ and $\mathcal{A}.$ And second, we prove that the solution operator is independent of the choice of the parameter $\rho$, if $\mathcal{M},\mathcal{N}$ and $\mathcal{A}$ are independent of $\rho$ in some sense.

We start with the causality and define, what we mean by a causal operator.

Let $F:L_{2,\rho}({\mathbb{R}};H)\to L_{2,\rho}({\mathbb{R}};H).$ Then, $F$ is called *causal*, if for each $f,g\in L_{2,\rho}({\mathbb{R}};H)$ when $f=g$ on $]-\infty,a]$ for some $a\in\mathbb{R}$, also $F(f)=F(g)$ on $]-\infty,a]$ holds.

(a) Causality is a crucial property for operators describing a temporal evolution, as it says that the solution $u$ up to some time $a\in{\mathbb{R}}$ does not depend on the behaviour of the right hand side $f$ on $[a,\infty[.$

(b) For mappings not defined on the whole space $L_{2,\rho}({\mathbb{R}};H)$ one has to adopt the definition of causality in a way, that it is preserved under closure (i.e. $\overline{F}$ is causal if $F$ is causal) and 