A Fixed Point Theorem for Asymptotically Nonexpansive Mappings

Let K be a subset of a Banach space X. A mapping is said to be asymptotically nonexpansive if there exists a sequence <!-- MATH $\{ {k_i}\}$ --> of real numbers with <!-- MATH ${k_i} \to 1$ --> as <!-- MATH $i \to \infty$ --> such that <!-- MATH $\left\| {{F^i}x - {F^i}y} \right\| \leqq {k_i}\left\| {x - y} \right\|,x,y \in K$ --> . It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.