fa.functional analysis - 2, 3, and 4 (a possible fixed point result ?)

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 $T:Krightarrow K$
be a mapping such that $Vert Tx-TyVert _{ell^{4}}leqVert x-yVert _{ell^{3}}$
for all $x,yin K$.

Is it true that $T$ has fixed points ?