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 ?

`$\|Tx−Ty\|_4 \le \|x−y\|_3$`

. 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