  We give a simple proof of the existence of response solutions in some
quasi-periodically forced systems with a degenerate fixed points. The same
questions were answered in \cite{ss18} using two versions of KAM theory.
  Our method is based on reformulating the existence of response solutions as a
fixed point problem in appropriate spaces of smooth functions. By algebraic
manipulations, the fixed point problem is transformed into a contraction.
  Compared to the KAM method, the present method does not incur a loss of
regularity. That is, the solutions we obtain have the same regularity as the
forcing. Moreover, the method here applies when problems are only finitely
differentiable. It also weakens slightly the non-degeneracy conditions. Since
the method is based on the contraction mapping principle, we also obtain
automatically smooth dependence on parameters and, when studying complex
versions of the problem we discover the new phenomenon of monodromy. We also
present results for higher dimensional systems, but for higher dimensional
systems, the concept of degenerate fixed points is much more subtle than in one
dimensional systems.
  To illustrate the power of the method, we also consider two problems not
studied in \cite{ss18}: the forcing with zero average and second order
oscillators. We show that in the zero average forcing case, the solutions are
qualitatively different, but the second order oscillators is remarkably
similar.
