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The degree theory and the index of a critical point for mappings of the type (<em>S</em><sub>+</sub>)

Abstract
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 : X→X* 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' : X→X* 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 : X→X 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.

Identiferoai:union.ndltd.org:oulo.fi/oai:oulu.fi:isbn978-951-42-8487-8
Date31 May 2007
CreatorsOinas, J. (Janne)
PublisherUniversity of Oulu
Source SetsUniversity of Oulu
LanguageEnglish
Detected LanguageEnglish
Typeinfo:eu-repo/semantics/doctoralThesis, info:eu-repo/semantics/publishedVersion
Formatapplication/pdf
Rightsinfo:eu-repo/semantics/openAccess, © University of Oulu, 2007
Relationinfo:eu-repo/semantics/altIdentifier/pissn/0355-3191, info:eu-repo/semantics/altIdentifier/eissn/1796-220X

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