A fixed point theorem for mappings with a nonexpansive iterate
W. A. Kirk · Proceedings of the American Mathematical Society · 1971
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 → K T:K \to K has the property that, for some positive integer N , T N N,{T^N} is nonexpansive ( ‖ T N x − T N y ‖ ≦ ‖ x − y ‖ \left \| {{T^N}x - {T^N}y} \right \| \leqq \left \| {x - y} \right \| for all x , y ∈ K x,y \in K ). A function γ ( N ) \gamma (N) is determined, γ ( N ) > 1 \gamma (N) > 1 , such that if ‖ T j x − T j y ‖ ≦ k ‖ x − y ‖