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

• Autores: David E. Edmunds, W. Desmond Evans, D. J. Harris
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 78, Nº 1, 2008, págs. 65-84
• Idioma: inglés
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• Resumen

  • 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.