A fixed point theorem for mappings with a nonexpansive iterate

Abstract:

Let X be a reflexive Banach space which has strictly convex norm and suppose K is a nonempty, bounded, closed and convex subset of X. Suppose $T:K \to K$ has the property that, for some positive integer $N,{T^N}$ is nonexpansive ($\left \| {{T^N}x - {T^N}y} \right \| \leqq \left \| {x - y} \right \|$ for all $x,y \in K$). A function $\gamma (N)$ is determined, $\gamma (N) > 1$, such that if $\left \| {{T^j}x - {T^j}y} \right \| \leqq k\left \| {x - y} \right \|$ for all $x,y \in K,1 \leqq j \leqq N - 1$, where $k < \gamma (N)$, then T has a fixed point in K.