results of [@Trostorff2012_nonlin_bd; @Trostorff2013_nonautoincl] and [@Waurick2015_nonauto] and provide a unified solution theory for a broad class of problems.

Of course the problem of non-autonomous differential inclusions was studied in the literature before. We just mention some classical approaches, to tackle this problem. A standard approach for inclusions given in form of a Cauchy problem (i.e. $\mathcal{M}=1$ and time dependent $\mathcal{A}$ in ) uses the concept of evolution families introduced by Kato [@Kato1953] for evolution equations and generalised to inclusions by Crandall and Pazy in [@Crandall1972]. Other approaches use approximations by replacing the time derivative in by the difference quotient and then proving that the corresponding solutions converge in a suitable sense (e.g. [@Evans1977; @Kobayasi1984]). A third approach uses the notion of integral solutions introduced by Bénilan in [@Benilan1972] for autonomous problems and generalised in [@Kobayasi1984] to the non-autonomous case.

In contrast to all these classical approaches, we assume the relation $\mathcal{A}$ to be time-independent. The non-autonomous behaviour enters the equation via the bounded space-time operators $\mathcal{M}$ and $\mathcal{N}.$ This has the big advantage that we do not have any problems with time-dependent domains.

The article is structured as follows: In the next section we present the underlying Hilbert space framework 