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

Oinas, Janne (2007-05-31)
 
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Oinas, Janne
University of Oulu
31.05.2007
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https://urn.fi/URN:ISBN:9789514284878

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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
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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.

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