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.
| Identifer | oai:union.ndltd.org:oulo.fi/oai:oulu.fi:isbn978-951-42-8487-8 |
| Date | 31 May 2007 |
| Creators | Oinas, J. (Janne) |
| Publisher | University of Oulu |
| Source Sets | University of Oulu |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/doctoralThesis, info:eu-repo/semantics/publishedVersion |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess, © University of Oulu, 2007 |
| Relation | info:eu-repo/semantics/altIdentifier/pissn/0355-3191, info:eu-repo/semantics/altIdentifier/eissn/1796-220X |
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