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Remarks on recent results in analytical fixed point theory. (English) Zbl 1119.47054
Takahashi, Wataru (ed.) et al., Nonlinear analysis and convex analysis. Proceedings of the 4th international conference (NACA 2005), Okinawa, Japan, June 30–July 4, 2005. Yokohama: Yokohama Publishers (ISBN 978-4-946552-27-4/hbk). 517-525 (2007).
R. Cauty [Fund. Math. 170, 231–246 (2001; Zbl 0983.54045)] proved the Schauder fixed point theorem in topological vector spaces without assuming local convexity (see also [T. Dobrowolski, Abstr. Appl. Anal. 7, 407–433 (2003; Zbl 1022.54029)] for an expanded version which is more easily accessible). The present author uses this result to obtain some more or less obvious generalizations to the case of set-valued mappings. Here, he reports on rumors that both Cauty’s and Dobrowolski’s proofs might contain gaps (without giving any evidence for this claim) and he reconsiders his results under the assumption that the rumors should prove correct.
For the entire collection see [Zbl 1104.47002].

47H10 Fixed-point theorems
47H04 Set-valued operators
54H25 Fixed-point and coincidence theorems (topological aspects)