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The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.

Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$ be a mapping such that $$\Vert Tx-Ty\Vert _{\ell^{4}}\leq\Vert x-y\Vert _{\ell^{3}}$$ for all $x,y\in K$.

Is it true that $T$ has fixed points?

Lorenzo Pompili
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Ady
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    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 – Mark Meckes Mar 15 '10 at 13:36
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    If that deserves a "-1", it's cool to me. – Ady Mar 15 '10 at 13:40
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    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. – Mark Meckes Mar 15 '10 at 13:48
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    Why did this question get a downvote? Seems interesting to me. – Steven Gubkin Mar 15 '10 at 15:11
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    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 $FP(1,\infty, \infty)$? – Fabrizio Polo Apr 06 '10 at 10:58
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    BGK $\Longrightarrow$ FP$(p,q,r)$ for 1 < p $\leq$ r $\leq q< \infty $ , e.g. – Ady Apr 08 '10 at 21:31
  • @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. – Fabrizio Polo Apr 14 '10 at 10:34
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    @Fabrizio Polo I'm not claiming that. Just that, e.g., $FP(2,4,3)$ holds, via BGK. Please read carefully my comment. – Ady Apr 15 '10 at 17:17
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    Would someone please state the classical Browder-Goehde-Kirk fixed point theorem? – Włodzimierz Holsztyński Jun 21 '13 at 00:39
  • Any progress on this? – Suvrit Aug 06 '13 at 16:19
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    What are the related known results? – Włodzimierz Holsztyński Apr 14 '15 at 03:48
  • 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. – Dirk Apr 14 '15 at 07:45
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    $FP(p,p,\infty)$ 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 $\ell^2$. – Will Sawin Dec 18 '16 at 14:03

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