PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE
Glasgow mathematical journal, Tome 55 (2013) no. 3, pp. 629-638
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Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, 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 x0 ∈ 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 ∈ R and ε ≠ 0. In this paper, we prove that if B is a C1 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.
CHIAPPINELLI, RAFFAELE; FURI, MASSIMO; PERA, MARIA PATRIZIA. PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE. Glasgow mathematical journal, Tome 55 (2013) no. 3, pp. 629-638. doi: 10.1017/S0017089512000791
@article{10_1017_S0017089512000791,
author = {CHIAPPINELLI, RAFFAELE and FURI, MASSIMO and PERA, MARIA PATRIZIA},
title = {PERSISTENCE {OF} {THE} {NORMALIZED} {EIGENVECTORS} {OF} {A} {PERTURBED} {OPERATOR} {IN} {THE} {VARIATIONAL} {CASE}},
journal = {Glasgow mathematical journal},
pages = {629--638},
year = {2013},
volume = {55},
number = {3},
doi = {10.1017/S0017089512000791},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089512000791/}
}
TY - JOUR AU - CHIAPPINELLI, RAFFAELE AU - FURI, MASSIMO AU - PERA, MARIA PATRIZIA TI - PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE JO - Glasgow mathematical journal PY - 2013 SP - 629 EP - 638 VL - 55 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089512000791/ DO - 10.1017/S0017089512000791 ID - 10_1017_S0017089512000791 ER -
%0 Journal Article %A CHIAPPINELLI, RAFFAELE %A FURI, MASSIMO %A PERA, MARIA PATRIZIA %T PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE %J Glasgow mathematical journal %D 2013 %P 629-638 %V 55 %N 3 %U http://geodesic.mathdoc.fr/articles/10.1017/S0017089512000791/ %R 10.1017/S0017089512000791 %F 10_1017_S0017089512000791
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