  Time-fractional parabolic equations with a Caputo time derivative of order
$\alpha\in(0,1)$ are discretized in time using continuous collocation methods.
For such discretizations, we give sufficient conditions for existence and
uniqueness of their solutions. Two approaches are explored: the Lax-Milgram
Theorem and the eigenfunction expansion. The resulting sufficient conditions,
which involve certain $m\times m$ matrices (where $m$ is the order of the
collocation scheme), are verified both analytically, for all $m\ge 1$ and all
sets of collocation points, and computationally, for all $ m\le 20$. The
semilinear case is also addressed.
