MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
14  
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
5  
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
2  
Why did this question get a downvote? Seems interesting to me. – Steven Gubkin Mar 15 '10 at 15:11
1  
BGK $\Longrightarrow$ FP$(p,q,r)$ for 1 < p $\leq$ r $\leq q< \infty $ , e.g. – Ady Apr 8 '10 at 21:31
3  
@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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.