Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T02:36:27.531Z Has data issue: false hasContentIssue false
  • English
  • Français

On the null spaces of linear Fredholm operators depending on several parameters

Published online by Cambridge University Press:  24 October 2008

M. Shearer
M. Shearer
Affiliation:
Fluid Mechanics Research Institute, University of Essex, Colchester
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X, Y be real Banach spaces, and let = {f(λ):λ ∈ m} be an m-parameter family of bounded linear operators from X to Y, with f(λ) depending continuously on λ. The cases m = 1 and m = 2 are considered, and conditions on are found which determine the null space of f(λ) for all λ near a given λ0 such that f(λ0): X → Y is a Fredholm operator. The results obtained are shown to be of particular interest in perturbed bifurcation theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 131 - 142
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bauer, L., Keller, H. B. and Riess, E. L.Multiple eigenvalues lead to secondary bifurcation. SIAM Review 17 (1975), 101–122.CrossRefGoogle Scholar
(2)Crandall, M. G. and Rabinowitz, P. H.Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971), 321–340.CrossRefGoogle Scholar
(3)Kato, T.Perturbation theory for linear operators (Springer, 1966).Google Scholar
(4)Keener, J. P.Perturbed bifurcation theory at multiple eigenvalues. Arch. Rat. Mech. Anal. 56 (1974), 348–366.CrossRefGoogle Scholar
(5)Magnus, R. J.On the local structure of the zero set of a Banach space valued mapping. J. Funct. Anal. 22 (1976), 58–72.CrossRefGoogle Scholar
(6)Mallet-Paret, J. Buckling of cylindrical shells with small curvature. (To appear.)Google Scholar
(7)Shearer, M. Bifurcation in the neighbourhood of a non-isolated singular point. (To appear.)Google Scholar