A fixed point theorem for asymptotically nonexpansive type mappings in uniformly convex Banach spaces

Let us consider two nonempty subsets A and B of a uniformly convex Banach space X. Let T: A∪ B→ A∪ B be a mapping such that T(A)⊆A,T(B)⊆B and there is a sequence { kn} in [ 1 , ∞) , with kn→ 1 , satisfying ‖ Tnx- Tny‖ ≤ kn‖ x- y‖ , for all x∈ A and y∈ B. We investigate sufficient conditions for the existence of fixed points x in A and y in B in such a way that the distance between x and y is optimum in some sense. Our main result provides a natural and simple proof for a particular case of Rajesh and Veeramani (Numer Funct Anal Optim 37:80–91, 2016) fixed point theorem for asymptotically relatively nonexpansive mappings. © 2017, Forum D'Analystes, Chennai.