University of Oulu

The degree theory and the index of a critical point for mappings of the type (S+)

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Author: Oinas, Janne1
Organizations: 1University of Oulu, Faculty of Science, Department of Mathematical Sciences
Format: ebook
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 1.4 MB)
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Language: English
Published: 2007
Publish Date: 2007-05-31
Thesis type: Doctoral Dissertation
Defence Note: Academic dissertation to be presented, with the assent of the Faculty of Science of the University of Oulu, for public defence in Raahensali (Auditorium L10), Linnanmaa, on June 9th, 2007, at 12 noon
Reviewer: Professor Gustaf Gripenberg
Docent Jari Taskinen


The dissertation considers a degree theory and the index of a critical point of demi-continuous, everywhere defined mappings of the monotone type.

A topological degree is derived for mappings from a Banach space to its dual space. The mappings satisfy the condition (S+), and it is shown that the derived degree has the classical properties of a degree function.

A formula for the calculation of the index of a critical point of a mapping A : XX* satisfying the condition (S+) is derived without the separability of X and the boundedness of A. For the calculation of the index, we need an everywhere defined linear mapping A' : XX* that approximates A in a certain set. As in the earlier results, A' is quasi-monotone, but our situation differs from the earlier results because A' does not have to be the Frechet or Gateaux derivative of A at the critical point. The theorem for the calculation of the index requires a construction of a compact operator T = (A' + Γ)-1Γ with the aid of linear mappings Γ : XX and A'. In earlier results, Γ is compact, but here it need only be quasi-monotone. Two counter-examples show that certain assumptions are essential for the calculation of the index of a critical point.

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Series: Acta Universitatis Ouluensis. A, Scientiae rerum naturalium
ISSN-E: 1796-220X
ISBN: 978-951-42-8487-8
ISBN Print: 978-951-42-8486-1
Issue: 488
Copyright information: © University of Oulu, 2007. This publication is copyrighted. You may download, display and print it for your own personal use. Commercial use is prohibited.