Let be a weakly compact convex subset of a Banach space. Then, largely follwing L. P. Belluce and W. A. Kirk [Proc. Am. Math. Soc. 20, 141–146 (1969; Zbl 0165.16801)] and W. A. Kirk [J. Math. Anal. Appl. 277, No. 2, 645–650 (2003; Zbl 1022.47036)], a mapping is called an asymptotic pointwise contraction (APC) if there exists a function such that, for each integer ,
for each , where pointwise on .
The principal result of this paper states that an APC has a unique fixed point in , and the Picard sequence of iterates of converges to the fixed point. Further, the authors extend this result to pointwise asymptotically nonexpansive mappings when is a bounded closed convex subset of a uniformly convex Banach space. Two new questions are also posed.