Contractibility and generalized convexity
The author defines various generalizations of convexity which are strong enough to give rise to selection theorems for set-valued mappings. He then applies these selection theorems to obtain fixed point results for set-valued mappings. A c-structure on the topological space Y is given by a mapping F from the non-empty finite subsets \(<Y>\) into the non-empty contractible subsets of Y such that \(\emptyset \neq A\subset B\in <Y>\) implies F(A)\(\subset F(B)\). A set \(Z\subset Y\) is called an F-set if F(A)\(\subset Z\) whenever \(A\in <Z>\). A c-structure (Y,F) is called an l.c. metric space if (Y,d) is a metric space and \(\{\) \(y\in Y|\) \(d(y,E)<\epsilon \}\) is an F-set whenever \(\epsilon >0\) and E is an F-set and if open balls are F-sets. Michael's theorem, in this context, reads as follows: Let X be a paracompact space, (Y,F) an l.c. complete metric space, and let T be a lower semicontinuous mapping from X into the non- empty closed F-sets of Y. Then there is a continuous selection for T. The author then derives fixed point theorems and provides a wealth of examples.
scientific article; zbMATH DE number 4212978
scientific article