On Certain Contraction Mappings in a Partially Ordered Vector Space1

G. Birkhoff [l] and H. Samelson [4] have shown that a means of solving problems concerning the existence and uniqueness of eigenvectors of positive operators is given by introducing a suitable metric on a subset of the cone with respect to which the operators are contractions. More specifically, they have proved the Perron theorem for matrices with positive elements by intersecting the positive quadrant with a hyperplane and by using the Hubert metric (see, for example, [3]) on this section. Birkhoff was able to extend this method (using the same metric) to certain positive linear operators in a more general setting. Since the contraction mapping principle is essentially nonlinear it seemed likely that this method could be used for a class of nonlinear operators. In this paper we use a slightly different distance function the domain of definition of which is not restricted to such a section of the cone and we obtain a theorem for a class of nonlinear mappings which contract this metric. After giving necessary preliminaries the metric is defined in §2 and the completeness of certain subsets is proved. This is followed by a theorem on nonlinear operators and two examples.