  We establish a fixed point theorem for mappings of square matrices of all
sizes which respect the matrix sizes and direct sums of matrices. The
conclusions are stronger if such a mapping also respects matrix similarities,
i.e., is a noncommutative function. As a special case, we prove the
corresponding contractive mapping theorem which can be viewed as a new version
of the Banach Fixed Point Theorem. This result is then applied to prove the
existence and uniqueness of a solution of the initial value problem for ODEs in
noncommutative spaces. As a by-product of the ideas developed in this paper, we
establish a noncommutative version of the principle of nested closed sets.
