Timeline for 2, 3, and 4 (a possible fixed point result ?)
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Mar 18, 2023 at 17:03 | history | suggested | Lorenzo Pompili | CC BY-SA 4.0 |
Added reference for a cited theorem.
|
| Mar 18, 2023 at 11:17 | review | Suggested edits | |||
| S Mar 18, 2023 at 17:03 | |||||
| Dec 18, 2016 at 14:03 | comment | added | Will Sawin | $FP(p,p,\infty)$ is false for any $p$. We can let $(Tx)_i = x_{i-1}$ for $i<1$ and $(Tx)_0 =(1/2)( 1 - \sum_i x_i^2)$. Then $||Tx||_{\ell_2}^2 = (1/4)(1-||x||_{\ell_2}^2)^2+ x_{\ell_2}^2 \leq 1$ since $x_{\ell_2}\leq 1$ and, because each coordinate is a $1$-Lipschitz function in the $\ell_2$ norm, the whole thing is a $1$-Lipschitz transformation from $\ell^\infty$ to $\ell^2$. | |
| Apr 14, 2015 at 7:45 | comment | added | Dirk | As far as I understood, the BGK (Browder-Göhde/Göbel-Kirk) fixed point theorem states that every non-expansive self-mapping on a non-empty, closed and convex subset of a uniformly convex Banach space has a fixed point. | |
| Apr 14, 2015 at 3:48 | comment | added | Włodzimierz Holsztyński | What are the related known results? | |
| Aug 6, 2013 at 16:19 | comment | added | Suvrit | Any progress on this? | |
| Jun 21, 2013 at 0:39 | comment | added | Włodzimierz Holsztyński | Would someone please state the classical Browder-Goehde-Kirk fixed point theorem? | |
| Apr 15, 2010 at 17:17 | comment | added | Ady | @Fabrizio Polo I'm not claiming that. Just that, e.g., $FP(2,4,3)$ holds, via BGK. Please read carefully my comment. | |
| Apr 14, 2010 at 10:34 | comment | added | Fabrizio Polo | @Ady: Since your question is whether or not $FP(2,3,4)$ holds, I'm confused. You seem to be claiming that BGK resolves your question. | |
| Apr 8, 2010 at 21:31 | comment | added | Ady | BGK $\Longrightarrow$ FP$(p,q,r)$ for 1 < p $\leq$ r $\leq q< \infty $ , e.g. | |
| Apr 6, 2010 at 10:58 | comment | added | Fabrizio Polo | For fun, I started considering variants of this question but made no real progress on them either. Let ${\bf FP}(p,q,r)$ be the same statement with $2,3,4$ replaced by $p,q,r$. The BGK fixed point theorem gives us ${\bf FP}(p,p,p)$ for $1 < p < \infty.$ For what other values of $p,q,r$ can you prove or disprove this statement? Do you know of a counterexample to $FP(1,\infty, \infty)$? | |
| Mar 16, 2010 at 7:57 | history | edited | Yemon Choi |
added banach-spaces tag
|
|
| Mar 15, 2010 at 15:11 | comment | added | Steven Gubkin | Why did this question get a downvote? Seems interesting to me. | |
| Mar 15, 2010 at 13:48 | comment | added | Mark Meckes | I don't think it does. I just meant to make sure other readers could easily tell what you were asking; I didn't downvote and I hope I didn't encourage anyone else to. | |
| Mar 15, 2010 at 13:40 | comment | added | Ady | If that deserves a "-1", it's cool to me. | |
| Mar 15, 2010 at 13:36 | comment | added | Mark Meckes |
It would be easier to read if you just wrote $\|Tx−Ty\|_4 \le \|x−y\|_3$. It took me a minute to get the point of your question since I could barely see the difference between the supersubscripts
|
|
| Mar 15, 2010 at 13:26 | history | asked | Ady | CC BY-SA 2.5 |