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Journal of the London Mathematical Society Advance Access originally published online on April 3, 2008
Journal of the London Mathematical Society 2008 78(1):65-84; doi:10.1112/jlms/jdm083
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© 2008 London Mathematical Society

Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

D. E. Edmunds, W. D. Evans and D. J. Harris

School of Mathematics
Cardiff University
Senghennydd Road
Cardiff CF24 4AG
Wales
United Kingdom
DavidEEdmunds@aol.com
pryske@boyns.net

Let X and Y be reflexive Banach spaces with strictly convex duals, and let T be a compact linear map from X to Y. It is shown that a certain nonlinear equation, involving T and its adjoint, has a normalised solution (an ‘eigenvector’) corresponding to an ‘eigenvalue’, and that the same is true for each member of a countable family of similar equations involving the restrictions of T to certain subspaces of X. The action of T can be described in terms of these ‘eigenvectors’. There are applications to the p-Laplacian, the p-biharmonic operator and integral operators of Hardy type.

2000 Mathematics Subject Classification 47A75, 47B06, 47B40, 35P30.

Received January 11, 2007; revised January 24, 2008; published online April 3, 2008.


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