Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + εB(x) = δx, where A : H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B : H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x_0 ∈ S ∩ Ker A of A (thus corresponding to the eigenvalue δ = 0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S ∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ and ε different from 0. In this paper, we prove that if B is a C^1 gradient mapping and the eigenvalue δ = 0 has finite multiplicity, then the sphere S ∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.
Persistence of the normalized eigenvectors of a perturbed operator in the variational case / R. Chiappinelli; M.Furi; M.P.Pera. - In: GLASGOW MATHEMATICAL JOURNAL. - ISSN 0017-0895. - STAMPA. - 55:(2013), pp. 629-638. [10.1017/S0017089512000791]
Persistence of the normalized eigenvectors of a perturbed operator in the variational case
M. Furi;M. P. Pera
2013
Abstract
Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + εB(x) = δx, where A : H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B : H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x_0 ∈ S ∩ Ker A of A (thus corresponding to the eigenvalue δ = 0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S ∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ and ε different from 0. In this paper, we prove that if B is a C^1 gradient mapping and the eigenvalue δ = 0 has finite multiplicity, then the sphere S ∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.File | Dimensione | Formato | |
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