Let E, F be real Banach spaces and S the unit sphere of E. We study a nonlinear eigenvalue problem of the type Lx+εN(x) = λCx, where ε,λ are real parameters, L: E → F is a Fredholm linear operator of index zero, C : E → F is a compact linear operator, and N : S → F is a compact map. Given a solution (x , ε, λ) ∈ S × R × R of this problem, we say that the first element x of the triple is a unit eigenvector corresponding to the eigenpair (ε, λ). Assuming that λ_0 ∈ R is such that the kernel of L − λ_0C is odd dimensional and that a natural transversality condition between the operators L − λ_0C and C is satisfied, we prove that, in the set of all the eigenpairs, the connected component containing (0, λ_0 ) is either unbounded or meets an eigenpair (0,λ_1), with λ_1 different from λ_0. Our approach is topological and based on the classical Leray–Schauder degree.
Global continuation of the eigenvalues of a perturbed linear operator / Pierluigi, Benevieri; Alessandro, Calamai; Massimo, Furi; Maria Patrizia Pera,. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 197:(2018), pp. 1131-1149. [10.1007/s10231-017-0717-5]
Global continuation of the eigenvalues of a perturbed linear operator
PERA, MARIA PATRIZIA
2018
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+εN(x) = λCx, where ε,λ are real parameters, L: E → F is a Fredholm linear operator of index zero, C : E → F is a compact linear operator, and N : S → F is a compact map. Given a solution (x , ε, λ) ∈ S × R × R of this problem, we say that the first element x of the triple is a unit eigenvector corresponding to the eigenpair (ε, λ). Assuming that λ_0 ∈ R is such that the kernel of L − λ_0C is odd dimensional and that a natural transversality condition between the operators L − λ_0C and C is satisfied, we prove that, in the set of all the eigenpairs, the connected component containing (0, λ_0 ) is either unbounded or meets an eigenpair (0,λ_1), with λ_1 different from λ_0. Our approach is topological and based on the classical Leray–Schauder degree.File | Dimensione | Formato | |
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