Abstract: Let E be a real Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Let D be a nonempty bounded closed convex subset of $E$ and $T:D\rightarrow D be an asymptotically nonexpansive mapping. It is shown that under some suitable conditions,the modified Reich-Takahashi type iteration method converges strongly to a fixed point of T.

Key words: Fixed point, Asymptotically nonexpansive mapping, Modified Reich-Takahashi type iteration method, Uniform normal structure, Uniformaly Gateaux differentiable norm