inverse relation $$\mathcal{S}_{\rho}\coloneqq\left(\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}\right)^{-1}:L_{2,\rho}({\mathbb{R}};H)\to L_{2,\rho}({\mathbb{R}};H)$$ is a Lipschitz-continuous mapping with $\left|\mathcal{S}_{\rho}\right|_{\mathrm{Lip}}\leq\frac{1}{c}$. Hence, for each $f\in L_{2,\rho}({\mathbb{R}};H)$ there exists a unique $u\in L_{2,\rho}({\mathbb{R}};H)$ with $$(u,f)\in\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}$$ and $u$ depends Lipschitz-continuously on $f$.

Before we come to the proof of this theorem, we illustrate the hypotheses by some concrete examples for the operators $\mathcal{M}$ and $\mathcal{N}$.

(a) If $\mathcal{N}=0$ and $\mathcal{M}$ is given in terms of an operator-valued analytic function of $\partial_{t,\rho}$, then condition (a) is trivially satisfied with $\mathcal{M}'=0$ and the resulting problem becomes autonomous. This class of operators was introduced in [@Picard] and allows the treatment of a broad class of differential equations arising in mathematical physics (see e.g. [@Picard; @Picard2010_poroelastic; @Picard2012_Impedance; @Picard2015_micropoloar]) and includes different types of equations, such as fractional differential equations (see [@Picard2013_fractional]), delay equations with finite and infinite delay (see [@Kalauch2011]), and integro-differential equations (see [@Trostorff2012_integro]).

(b) A second class of problems covered by the hypotheses is given by non-autonomous equations, which are local with respect to time. More precisely, $\mathcal{M},\mathcal{N}$ are given as operator-valued multiplication operators, such that $\mathcal{M}$ is Lipschitz-continuous. Then, Rademacher’s theorem implies that (a) holds. Such equations were studied in [@Picard2013_nonauto] and generalised to inclusions in [@Trostorff2013_nonautoincl].

(c) The hypotheses also allow the treatment 