The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and
accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a ll-valued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies :
(D1) D(I, Ω, y) = 1 if y Є Ω.
(D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ).
(D3) D(I - H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X.
An important property that follows from these three properties is (D4) F-1(y) ≠ Ø if D(F, Ω, y) ≠ 0.
This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0.
Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero. / Thesis (M.Sc.)-University of Durban-Westville, 1989.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ukzn/oai:http://researchspace.ukzn.ac.za:10413/9795 |
| Date | 21 October 2013 |
| Creators | Pillay, Paranjothi. |
| Contributors | Hill, C. K. |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
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