  In this note we develop a framework which allows to prove an abstract
existence result for non-linear evolution equations involving so-called
non-induced operators, i.e., operators which are not prescribed by a
time-dependent family of operators. Apart from this, we introduce the notion of
$C^0$-Bochner pseudo-monotonicity, and $C^0$-Bochner coercivity, which are
appropriate adaptions of the standard notion to the case of evolutionary
problems.
