On the Persistence of the Eigenvalues of a Perturbed Fredholm Operator of Index Zero under Nonsmooth Perturbations

Abstract

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Lx + εN (x) = λx, where ε, λ ∈ R, L : H → H is a bounded self-adjoint (linear) operator with nontrivial kernel and closed image, and N : H → H is a (possibly) nonlinear perturbation term. A unit eigenvectorx ∈ S ∩ Ker L of L (corresponding to the eigenvalue λ = 0) is said to be persistent if it is close to solutions x ∈ S of the above equation for small values of the parameters ε = 0 and λ. We give an affirmative answer to a conjecture formulated by R. Chiappinelli and the last two authors in an article published in 2008. Namely, we prove that if N is Lipschitz continuous and the eigenvalue λ = 0 has odd multiplicity, then the sphere S ∩Ker L contains at least one persistent eigenvector. We provide examples in which our results apply, as well as examples showing that if the dimension of Ker L is even, then the persistence phenomenon may not occur.where ε and λ are real parameters. Assume, for the moment, that the eigenvalue λ 0 is simple. Thus, when ε = 0 and λ = λ 0 , one gets exactly two vectors in H satisfying the above problem; namely, the two unit eigenvectors of L corresponding to the eigenvalue λ 0 . Denote by x 1 and x 2 these vectors. Considering small values of ε, R. Chiappinelli in [11] obtained a sort of persistence result of these eigenvectors as well as of the eigenvalue λ 0 . More

Nonlinear Rayleigh Quotients and Nonlinear Spectral Theory

“…There are useful relations between the various constants introduced so far, that can be easily obtained by the definitions and are shown for instance in [4,6,17]. We report here, for further use in the present paper, only the following:…”

“…We remark that in general, the study of the measure of noncompactness in Banach spaces forms a vast and active research field in Functional Analysis, that has received further interest and expansion from the axiomatic approach presented in [16]; for an updated overview of this, see for instance [18] and the references therein. While in particular, the importance of ω for the study of nonlinear operators, originally shown in [6], has been further demonstrated especially in works by M. Furi and his school, see for instance their recent paper [17]. One basic property of ω(F) that we shall use repeatedly in Section 3 is expressed by the following statement (see [6], Proposition 3.1.3), the proof of which will be given there for the reader's convenience: if ω(F) > 0, then F is proper on closed bounded sets; that is, given any compact K ⊂ X and any closed bounded…”

“…For the elementary properties of α we refer, for instance, to the books [4,16] and to the papers [6,17]. In particular we recall that α(A) = 0 if and only if A is relatively compact, meaning that the closure A of A is compact.…”

Global continuation of the eigenvalues of a perturbed linear operator

“…After the result of Chiappinelli, in a series of papers [6,[8][9][10][11] the above property of "local" persistence of the eigenvalues and eigenvectors was extended to the case in which the multiplicity of the eigenvalue λ 0 is bigger than one. In this case the set of unit eigenvectors of L corresponding to λ 0 is the (n − 1)-dimensional unit sphere S n−1 = S ∩ Ker(L − λ 0 I ), where S is the unit sphere in H , I is the identity of H and n is the multiplicity of λ 0 .…”

“…Actually, the results of the recent papers [6,11] are obtained in the more general context of real Banach spaces. To better explain these results, consider the system…”