Irreducible continua of solutions for Birkhoff-Kellogg type equations

Abstract

BIRKHoFF-KELLOGG THEOREM. Let E be an infinite dimensional Banach space and let aB(O, 1) be the boundary of its unit ball B(0, 1). Let h: dB(0, l)+ E be a compact map such that llh(x)ll 2 a> 0 for all x E aB(O, 1). Then, there exist z E dZ3(0, 1) and A > 0 such that x = Ah(x), i.e. h has an invariant point, in the terminology adopted in [l]. Sometimes in the literature the Birkhoff-Kellogg Theorem is reported in slightly different terms. Namely, the compact map h is assumed to be defined on the closure B(O, 1) of B(0, 1) and it satisfies the Birkhoff-Kellogg condition on dB(0, 1). The rotation of the unit circle (of the unit disk) in lw2 provides an example of a compact map (over a finite dimensional Banach space) satisfying the Birkhoff-Kellogg condition and not having invariant points. On the other hand, it is not hard to prove that if the compact map h: B(O, 1) --, E satisfies /h(x)ll 2 LY > 0 for all x E B(O, l), then there exists an invariant point for the map h, regardless if E is finite or infinite dimensional. The point is that one can use this strengthened version to prove the Birkhoff-Kellogg Theorem in infinite dimensional Banach spaces by suitably extending the compact map h: dB(0, l)+ E to the whole closed ball B(O, 1). In the first part of this paper, we will see that the situation described above carries over to a much more general context. The first part will also contain two results regarding the existence of unbounded components of eigenvectors (and of solutions) for nonlinear equations with compact “right hand sides” satisfying Birkhoff-Kellogg type assumptions in a neighbourhood of infinity (i.e. on the complement of an open and bounded subset). These two are the main theorems of the paper, which improve and refine some recent results obtained in [7] and in [W