Abstract: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 $$ \liminf _{n\rightarrow \infty } |||T_n|||< k, $$ where $|||T|||$ is the Lipschitz constant (norm) of $T$, then $T$ has a fixed point in $A$.
Keywords: asymptotically regular mappings, uniformly normal structure, fixed points
AMS Subject Classification: 47H10