A Fixed Point Theorem for Asymptotically Nonexpansive Mappings
K. Goebel, W. A. Kirk · Proceedings of the American Mathematical Society · 1972
Let K be a subset of a Banach space X. A mapping $F:K \to K$ is said to be asymptotically nonexpansive if there exists a sequence $\{ {k_i}\}$ of real numbers with ${k_i} \to 1$ as $i \to \infty$ such that $\left \| {{F^i}x - {F^i}y} \right \| \leqq {k_i}\left \| {x - y} \right \|,x,y \in K$. It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if $F:K \to K$ 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.