Global continuation of the eigenvalues of a perturbed linear operator

Abstract

Let E, F be real Banach spaces and S the unit sphere of E. We study a nonlinear eigenvalue problem of the type $$Lx + \varepsilon N(x) = \lambda Cx$$ , where $$\varepsilon ,\lambda $$ are real parameters, $$L:E \rightarrow F$$ is a Fredholm linear operator of index zero, $$C:E \rightarrow F$$ is a compact linear operator, and $$N:S \rightarrow F$$ is a compact map. Given a solution $$(x,\varepsilon ,\lambda ) \in S \times \mathbb {R}\times \mathbb {R}$$ of this problem, we say that the first element x of the triple is a unit eigenvector corresponding to the eigenpair $$(\varepsilon ,\lambda )$$ . Assuming that $$\lambda _0 \in \mathbb {R}$$ is such that the kernel of $$L -\lambda _0C$$ is odd dimensional and a natural transversality condition between the operators $$L -\lambda _0C$$ and C is satisfied, we prove that, in the set of all the eigenpairs, the connected component containing $$(0,\lambda _0)$$ is either unbounded or meets an eigenpair $$(0,\lambda _1)$$ , with $$\lambda _1 \not = \lambda _0$$ . Our approach is topological and based on the classical Leray–Schauder degree.