  We consider continuous and discontinuous Galerkin time stepping methods of
arbitrary order as applied to nonlinear initial value problems in real Hilbert
spaces. Our only assumption is that the nonlinearities are continuous; in
particular, we include the case of unbounded nonlinear operators. Specifically,
we develop new techniques to prove general Peano-type existence results for
discrete solutions. In particular, our results show that the existence of
solutions is independent of the local approximation order, and only requires
the local time steps to be sufficiently small (independent of the polynomial
degree). The uniqueness of (local) solutions is addressed as well. In addition,
our theory is applied to finite time blow-up problems with nonlinearities of
algebraic growth. For such problems we develop a time step selection algorithm
for the purpose of numerically computing the blow-up time, and provide a
convergence result.
