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The coincidence problem for compositions of set-valued maps. (English) Zbl 0727.54019

The author proves a host of results on fixed points and coincidences under various conditions imposed on the space and maps, using as main special tool approximate selections of the multivalued maps involved. The proofs can be called elementary if the absence of homology-theoretical arguments is the criterion. It is impossible to summarize the results due to the number of technical variations and their interrelations.
There is a section which deals with maps and the existence of selections for them and one on extension spaces. The main section deals with fixed points for compositions of maps. A typical example: Theorem 4.10. If X is a neighbourhood extension space of some compact space and the compact map A a composition of u.s.c. maps \(X\to X\) with convex compact values, then A ha a fixed point. The concluding coincidence results follow naturally via the link between the existence of coincidences of two maps and the existence of fixed points of the composite of the one map with the inverse of the other.
The subject matter of this paper is closely related to recent results of Gorniewicz, Granas and Lassonde.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
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