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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 $T:K\to K$ is called an asymptotic pointwise contraction (APC) if there exists a function $\alpha :K\to \left[0,1\right)$ such that, for each integer $n\ge 1$,

$\parallel {T}^{n}x-{T}^{n}y\parallel \le {\alpha }_{n}\left(x\right)\parallel x-y\parallel ,$

for each $x,y\in K$, where ${\alpha }_{n}\to \alpha$ 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 $T:K\to K$ when $K$ is a bounded closed convex subset of a uniformly convex Banach space. Two new questions are also posed.

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties 54H25 Fixed-point and coincidence theorems in topological spaces