The question below is related to the classical [Browder-Goehde-Kirk fixed point theorem][1].

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?


  [1]: https://www.researchgate.net/publication/38323269_An_elementary_proof_of_the_fixed_point_theorem_of_Browder_and_Kirk