  In this paper we study the behavior of finite dimensional fixed point
iterations, induced by discretization of a continuous fixed point iteration
defined within a Banach space setting. We show that the difference between the
discrete sequence and its continuous analogue can be bounded in terms depending
on the mesh size of the discretization and the contraction factor, defined by
the continuous iteration. Furthermore, we show that the comparison between the
finite dimensional and the continuous fixed point iteration naturally paves the
way towards a General a posteriori error analysis that can be used within the
framework of a fully adaptive solution procedure. In order to demonstrate our
approach, we use the Galerkin approximation of singularly perturbed semilinear
monotone problems. Our scheme combines the fixed point iteration with an
adaptive finite element discretization procedure (based on a robust a
posteriori error analysis), thereby leading to a fully adaptive
Fixed-Point-Galerkin scheme. Numerical experiments underline the robustness and
reliability of the proposed approach.
