On Krasnosel'skii-Precup fixed point theorem and eigenvalue criteria for existence of positive solutions

We provide an alternative approach to Krasnosel'skii-Precup fixed point theorem for operator systems based on the Leray-Schauder fixed point index in cones. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second--order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion.