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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 {\it L. P.\thinspace Belluce} and {\it W. A.\thinspace Kirk} [Proc. Am. Math. Soc. 20, 141--146 (1969; Zbl 0165.16801)] and {\it W. A.\thinspace Kirk} [J. Math. Anal. Appl. 277, No. 2, 645--650 (2003; Zbl 1022.47036)], a mapping $T: K \rightarrow K$ is called an asymptotic pointwise contraction (APC) if there exists a function $\alpha:K\rightarrow [0,1)$ such that, for each integer $n\geq 1$, $$\|T^{n}x - T^{n}y\|\leq \alpha_{n}(x) \| x - y \|,$$ for each $x, y \in K$, where $\alpha_{n} \rightarrow \alpha$ 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 $T: K \rightarrow K$ 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
Full Text: DOI
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