Fixed points of asymptotically regular mappings in spaces with uniformly normal structure

summary:It is proved that: for every Banach space $X$ which has uniformly normal structure there exists a $k>1$ with the property: if $A$ is a nonempty bounded closed convex subset of $X$ and $T:A\rightarrow A$ is an asymptotically regular mapping such that $$ \liminf _{n\rightarrow \infty } |\kern -0.8pt|\kern -0.8pt|T^n|\kern -0.8pt|\kern -0.8pt|< k, $$ where $|\kern -0.8pt|\kern -0.8pt|T|\kern -0.8pt|\kern -0.8pt|$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$