On the reduction of Krasnoselskii’s theorem to Schauder’s theorem

Krasnoselskii noted that many problems in analysis can be formulated as a mapping which is the sum of a contraction and compact map. He proved a theorem covering such cases which is the union of the contraction mapping principle and Schauder’s second fixed point theorem. In putting the two results together he found it necessary to add a condition which has been difficult to fulfill, although a great many problems have been solved using his result and there have been many generalizations and simplifications of his result. In this paper we point out that when the mapping is defined by an integral plus a contraction term, the integral can generate an equicontinuous map which is independent of the smoothness of the functions. Because of that, it is possible to set up that mapping, not as a sum of contraction and compact map, but as a continuous map on a compact convex subset of a normed space. An application of Schauder’s first fixed point theorem will then yield a fixed point without any reference to that difficult condition of Krasnoselskii. Finite and infinite intervals are handled separately. For the class of problems considered, application is parallel to the much simpler Brouwer fixed point theorem.