Return to search
## Normality-like properties, paraconvexity and selections.

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 |

Page generated in 0.0069 seconds