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Asymptotic pointwise contractions. (English) Zbl 1172.47038

Let K 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 T:KK is called an asymptotic pointwise contraction (APC) if there exists a function α:K[0,1) such that, for each integer n1,

T n x-T n yα n (x)x-y,

for each x,yK, where α n α pointwise on K.

The principal result of this paper states that an APC T has a unique fixed point in K, and the Picard sequence of iterates of T converges to the fixed point. Further, the authors extend this result to pointwise asymptotically nonexpansive mappings T:KK when K is a bounded closed convex subset of a uniformly convex Banach space. Two new questions are also posed.

47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
54H25Fixed-point and coincidence theorems in topological spaces