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A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces. (English) Zbl 1101.54025
Let X be a paracompact topological space, Y a hyperconvex metric space and F:XY a multifunction with sub-admissible values. A subset BY is called sub-admissible if coCB for each finite CB, where coC denotes the intersection of all closed balls in Y containing C. It is proved that if F is quasi-lsc (i.e., for each xX and ε>0 there is a point yF(x) and a neighbourhood U(x) of x such that for each tU(x), F(t)B(y,ε)), then F admits a continuous selection. Two fixed-point theorems are deduced. Remark: It would be interesting to locate the generalized convexity used in the paper under review in the framework of Bielawski’s simplicial convexity, cf. R. Bielawski [J. Math. Anal. Appl, 127, 155–171 (1987; Zbl 0638.52002)].
54C65Continuous selections
26E25Set-valued real functions
52A01Axiomatic and generalized geometric convexity