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

Author:  Oinas, Janne 
Organizations:  University of Oulu, Faculty of Science, Department of Mathematical Sciences 
Format:  eBook 
Online Access:  PDF Full Text (PDF, 1.4 MB) 
Persistent link:  http://urn.fi/urn:isbn:9789514284878 
Language:  English 
Published: 
2007

Publish Date:  20070531 
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 
Description: 
AbstractThe dissertation considers a degree theory and the index of a critical point of demicontinuous, 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 quasimonotone, 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)^{1}Γ with the aid of linear mappings Γ : X→X and A'. In earlier results, Γ is compact, but here it need only be quasimonotone. Two counterexamples show that certain assumptions are essential for the calculation of the index of a critical point. 
Series: 
Acta Universitatis Ouluensis. A, Scientiae rerum naturalium 
ISSNE:  1796220X 
ISBN:  9789514284878 
ISBN Print:  9789514284861 
Issue:  488 
Subjects:  
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