In this paper the author gives necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach's fixed point theorem a fixed point theorem can be proved for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = In +A* f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, oo). This will give conditions on A and f under which the equation has a unique solution in a certain set. The author considers two examples off in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A. (Author/publisher)
Samenvatting