  In this paper strong dissipativity of generalized time-fractional derivatives
on Gelfand triples of properly in time weighted $L^p$-path spaces is proved. In
particular, the classical Caputo derivative is included as a special case. As a
consequence one obtains the existence and uniqueness of solutions to evolution
equations on Gelfand triples with generalized time-fractional derivatives.
These equations are of type \begin{equation*}
  \frac{d}{dt} (k * u)(t) + A(t, u(t)) = f(t), \quad 0<t<T, \end{equation*}
with (in general nonlinear) operators $A(t,\cdot)$ satisfying general weak
monotonicity conditions. Here $k$ is a non-increasing locally
Lebesgue-integrable nonnegative function on $[0, \infty)$ with
$\underset{s\rightarrow\infty}{\lim}k(s)=0$. Analogous results for the case,
where $f$ is replaced by a time-fractional additive noise, are obtained as
well. Applications include generalized time-fractional quasi-linear
(stochastic) partial differential equations. In particular, time-fractional
(stochastic) porous medium and fast diffusion equations with ordinary or
fractional Laplace operators or the time-fractional (stochastic) $p$-Laplace
equation are covered.
