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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 ?

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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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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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BGK $\Longrightarrow$ FP$(p,q,r)$ for 1 < p $\leq$ r $\leq q< \infty $ , e.g. –  Ady Apr 8 '10 at 21:31
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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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