Component-wise Krasnosel'skii type fixed point theorem in product spaces and applications
Laura M Fernández-Pardo, Jorge Rodríguez-López
TL;DR
The paper develops a vector version of Krasnosel'skiĭ's cone-compression/expansion fixed point theory in product spaces to guarantee coexistence fixed points with nontrivial components. It provides a component-wise fixed point index analysis on Cartesian product cones and a main theorem covering all combinations of compressive/expansive behavior per component, including a mechanism to convert expansive cases to the compressive setting. The results yield existence (and, under strict inequalities, multiplicity) of positive, nontrivial solutions for systems of second-order differential equations and extend to singular and hybrid nonlinearities in the right-hand sides, via a Hammerstein formulation with Green's functions of fixed sign. The work thereby generalizes scalar Krasnosel'skiĭ results to coupled systems and supplies concrete criteria for positive periodic solutions with sublinear/superlinear and singular behaviors, relevant to nonlinear boundary value problems and dynamical systems.
Abstract
We present a version of Krasnosel'skii fixed point theorem for operators acting on Cartesian products of normed linear spaces, under cone-compression and cone-expansion conditions of norm type. Our approach, based on the fixed point index theory in cones, guarantees the existence of a coexistence fixed point - that is, one with nontrivial components. As an application, we prove the existence of periodic solutions with strictly positive components for a system of second-order differential equations. In particular, we address cases involving singular nonlinearities and hybrid terms, characterized by sublinear behavior in one component and superlinear behavior in the other.