Contractive maps on normed linear spaces and their applications to nonlinear matrix equations (Q852653)

Necessary and sufficient conditions are given under which a map is a contraction on a positive cone of a normed linear space. (A positive cone is a subset closed under taking sums of elements and products of elements by nonnegative numbers and such that if the vectors \(x\) and \(-x\) belong to the set, then \(x=0\).) The author deduces from these conditions a fixed point theorem which can be applied to the matrix equation \(X=I_n+A^*f(X)A\) where \(A\) and \(I_n\) (the identity matrix) are \(n\times n\) and \(f\) is a map on the set of the \(n\times n\) positive definite matrices induced by a real valued function on \((0,\infty )\). This gives conditions on \(A\) and \(f\) under which the equation has a unique solution on a certain set. Two examples are studied in detail.