In this work, we demonstrate that the Leray-Schauder topological degree theory can be used for the development of a topological degree theory for maximal monotone perturbations of demicontinuous operators of type (S+) in separable reflexive Banach spaces. This is an extension of Berkovits’ degree development for operators as the perturbations above.
Berkovits has developed a topological degree for demicontinuous mappings of type (S+), and has shown that the degree mapping is unique under the assumption that it satisfies certain general properties. He proved that if f is a bounded demicontinous mapping of type (S+), G is an open bounded subset of X, and 0 ∈/ f(∂G), then there exists ε0 > 0 such that for every ε ∈ (0, ε0) we have 0 ∈/ (I+ (1/ε)QQ∗ (f))(∂G). Here, Q is a compact linear injection from a Hilbert space H into X, such that Q(H) is dense in X, and Q∗ its adjoint. The map I+ 1 εQQ∗ (f) is a compact displacement of the identity, for which the Leray-Schauder degree is well defined. The Berkovits degree is obtained as the limit of this Leray-Schauder degree as ε tends to zero. We utilize a demicontinuous (S+)-approximation of the form Tt + f, where Tt is the Yosida approximant of T. Namely, we show that if G is an open bounded set in X and 0 ∈/ (T + f)(∂G), then there exist ε0 > 0, t0 > 0, such that for every ε ∈ (0, ε0), t ∈ (0, t0), we have 0 ∈/ (I + (1/ε)QQ∗ (Tt + f))(∂G). Our degree is the limit of the Leray-Schauder degree of the compact displacement of the identity I + (1/ε)QQ∗ (Tt + f) as ε, t → 0. Various extension of the degree has been considered. Finally some properties and applications in invariance of domain, eigenvalue and surjectivity results have also been discussed.
| Identifer | oai:union.ndltd.org:USF/oai:scholarcommons.usf.edu:etd-1640 |
| Date | 08 June 2007 |
| Creators | Boubakari, Ibrahimou |
| Publisher | Scholar Commons |
| Source Sets | University of South Flordia |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Graduate Theses and Dissertations |
| Rights | default |
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