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Contractibility and generalized convexity. (English) Zbl 0733.54011
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 $\emptyset \ne A\subset B\in <Y>$ 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 $\{$ $y\in Y\vert$ $d(y,E)<\epsilon \}$ 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.

MSC:
54C60Set-valued maps (general topology)
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54C65Continuous selections
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References:
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