A fixed point theorem for asymptotically nonexpansive mappings

Let K be a subset of a Banach space X. A mapping F : K → K F:K \to K is said to be asymptotically nonexpansive if there exists a sequence { k i } \{ {k_i}\} of real numbers with k i → 1 {k_i} \to 1 as i → ∞ i \to \infty such that ‖ F i x − F i y ‖ ≦ k i ‖ x − y ‖ , x , y ∈ K \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 → K 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.