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Variational Methods for One and Several Parameter Non-Linear Eigenvalue Problems

Published online by Cambridge University Press:  20 November 2018

Paul Binding
Paul Binding*
Affiliation:
University of Calgary, Calgary, Alberta
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We shall consider a multiparameter eigenvalue problem of the form

(1.1)

where λ ∈ Rk while Tn and Vn(λ) are self-adjoint linear operators on a Hilbert space Hn. If λ = (λ1, …, λk) ∈ Rk and satisfy (1.1) then we call λ an eigenvalue, x an eigenvector and (λ, x) an eigenpair. While our main thrust is towTards the general case of several parameters λn, the method ultimately involves reduction to a sequence of one parameter problems. Our chief contributions are (i) to generalise the conditions under which this reduction is possible, and (ii) to develop methods for the one parameter problem particularly suited to the multiparameter application. For example, we give rather general results on the magnitude and direction of the movement of non-linear eigenvalues under perturbation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 1 , 01 February 1981 , pp. 210 - 228
Copyright
Copyright © Canadian Mathematical Society 1981

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