In 1956, E. Michael proved his famous convex-valued selection theorems for l.s.c. mappings
de ned on spaces with higher separation axioms (paracompact, collectionwise
normal, normal and countably paracompact, normal, and perfectly normal), [39]. In
1959, he generalized the convex-valued selection theorem for mappings de ned on paracompact
spaces by replacing \convexity" with \ -paraconvexity", for some xed constant
0 < 1 (see, [42]). In 1993, P.V. Semenov generalized this result by replacing
with some continuous function f : (0;1) ! [0; 1) (functional paraconvexity) satisfying
a certain property called (PS), [63]. In this thesis, we demonstrate that the classical
Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can
be reduced only to compact-valued mappings modulo Dowker's extension theorem for
such spaces. The idea used to achieve this reduction is also applied to get a simple
direct proof of that selection theorem of Michael's. Some other possible applications
are demonstrated as well. We also demonstrate that the -paraconvex-valued and the
functionally-paraconvex valued selection theorems remain true for C 0
(Y )-valued mappings
de ned on -collectionwise normal spaces, where is an in nite cardinal number.
Finally, we prove that these theorems remain true for C (Y )-valued mappings de ned
on -PF-normal spaces; and we provide a general approach to such selection theorems. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ukzn/oai:http://researchspace.ukzn.ac.za:10413/10608 |
| Date | January 2012 |
| Creators | Makala, Narcisse Roland Loufouma. |
| Contributors | Gutev, V. |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
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