  In this note we consider the continuous Galerkin time stepping method of
arbitrary order as a possible discretization scheme of nonlinear initial value
problems. In addition, we develop and generalize a well known existing result
for the discrete solution by applying a general linearizing procedure to the
nonlinear discrete scheme including also the simplified Newton solution
procedure. In particular, the presented existence results are implied by
choosing sufficient small time steps locally. Furthermore, the established
existence results are independent of the local approximation order. Moreover,
we will see that the proposed solution scheme is able to significantly reduce
the number of iterations. Finally, based on existing and well known a priori
error estimates for the discrete solution, we present some numerical
experiments that highlight the proposed results of this note.
