Birkhoff-Kellogg type results in product spaces and their application to differential systems

TL;DR

The paper develops a component-wise variant of the Birkhoff–Kellogg invariant-direction theorem in product spaces, producing eigenpairs $(\lambda_1,\lambda_2)$ with nontrivial eigenvectors $(x,y)$ satisfying $x=\lambda_1 T_1(x,y)$ and $y=\lambda_2 T_2(x,y)$, and achieving localization of each component by its norm. It extends the classical scalar theory to systems by considering the product of cones or cones with infinite-dimensional spaces, using a fixed-point framework and a suitable auxiliary map to guarantee a fixed point on the boundary. The theoretical results are then applied to differential systems, including a PDE system with Dirichlet boundary conditions and a coupled ODE system rewritten as Hammerstein integral equations, yielding positive component-wise eigenvalues with prescribed norms and, in the ODE case, two distinct eigenvalue families (one with a negative second eigenvalue) and nontrivial eigenfunctions. These results provide qualitative insights into the structure of nonlinear eigenvalue problems for differential systems and illustrate the added richness of component-wise spectral data over classical scalar eigenvalues. The work thus broadens nonlinear spectral theory in product spaces and informs the qualitative analysis of coupled PDE/ODE models.

Abstract

We provide a new version of the well-known Birkhoff-Kellogg invariant-direction Theorem in product spaces. Our results concern operator systems and give the existence of component-wise eigenvalues, instead of scalar eigenvalues as in the classical case, that have corresponding eigenvectors with all components nontrivial and localized by their norm. We also show that, when applied to nonlinear eigenvalue problems for differential equations, this localization property of the eigenvectors provides, in turn, qualitative properties of the solutions. This is illustrated in two contexts of systems of PDEs and ODEs. We show the applicability of our theoretical results with two explicit examples.

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