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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 \(T: K \rightarrow K\) 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.

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54H25 Fixed-point and coincidence theorems (topological aspects)
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References:

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