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NORMALIZED EIGENVECTORS OF A PERTURBED LINEAR OPERATOR VIA GENERAL BIFURCATION

Published online by Cambridge University Press:  01 May 2008

RAFFAELE CHIAPPINELLI ,
MASSIMO FURI  and
MARIA PATRIZIA PERA
RAFFAELE CHIAPPINELLI
Affiliation:
Dipartimento di Scienze Matematiche ed Informatiche, Pian dei Mantellini 44, I-53100 Siena, Italy - E-mail address: chiappinelli@unisi.it
MASSIMO FURI
Affiliation:
Dipartimento di Matematica Applicata ‘G. Sansone’, Via S. Marta 3, I-50139 Florence, Italy - E-mail address: massimo.furi@unifi.it
MARIA PATRIZIA PERA
Affiliation:
Dipartimento di Matematica Applicata ‘G. Sansone‘, Via S. Marta 3, I-50139 Florence, Italy - E-mail address: mpatrizia.pera@unifi.it
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Abstract

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Let X be a real Banach space, A: X → X a bounded linear operator, and B: X → X a (possibly nonlinear) continuous operator. Assume that λ = 0 is an eigenvalue of A and consider the family of perturbed operators A + ϵB, where ϵ is a real parameter. Denote by S the unit sphere of X and let SA = S ∩ Ker A be the set of unit 0-eigenvectors of A. We say that a vector x0 ∈ SA is a bifurcation point for the unit eigenvectors of A + ϵ B if any neighborhood of (0,0, x0) ∈ × × X contains a triple (ϵ, λ, x) with ϵ ≠ 0 and x a unit λ-eigenvector of A + ϵB, i.e. x ∈ S and (A + ϵ B)x = λx.

We give necessary as well as sufficient conditions for a unit 0-eigenvector of A to be a bifurcation point for the unit eigenvectors of A + ϵB. These conditions turn out to be particularly meaningful when the perturbing operator B is linear. Moreover, since our sufficient condition is trivially satisfied when Ker A is one-dimensional, we extend a result of the first author, under the additional assumption that B is of class C2.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 2 , May 2008 , pp. 303 - 318
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

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