Subject: Re: fixed points response Date: Mon, 19 Nov 2001 15:28:24 -0600 From: Philip Reny To: Don Davis Dear Don: I've now received several terrific examples in response to my questions, below. (Both questions have negative answers.) So, this might be a good time to remove my questions from your website. Thanks very much for providing this helpful and extremely efficient service. Phil Reny At 12:59 PM 11/19/01 -0500, you wrote: >Subject: Re: question about fixed points >Date: Mon, 19 Nov 2001 18:10:37 +0100 (CET) >From: Marek Golasinski > >You wrote: >=========== >Suppose that X is a nonempty, compact, contractible, metrizable subset >of a locally convex linear topological space. >(i) Is X an absolute neighborhood retract? (If so, is there a reference >for such a result?) > >(ii) If the answer to (i) is "not necessarily," then suppose in addition > >that F is a point to set mapping on X with a closed graph such that F(x) > >is nonempty and contractible for every x in X. Must F possess a fixed >point? >(Obviously an affirmative answer to (i) implies an affirmative answer to > >(ii). >=========== > >In the paper by Ronald J. Knill, "Cones, product and fixed points", >Fund.Math. LX (1967), 35-46 >on the page 43 there is constructed a compact and contractible >subset $B\in R^3$ and Theorem 3.4 says: >Neither the cone $C(v;B)($ nor the product $B\times I$ have >the fixed point. >Therefore the answer for the question (ii) is negative even for >one-valeued mappings. > >Marek Golasinski *************************** Philip J. Reny Department of Economics University of Chicago 1126 East 59th Street Chicago, IL 60637 http://www.src.uchicago.edu/users/preny/ (O) 773 - 702 - 8192 (F) 773 - 702 - 8490