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Approximating fixed points of nonexpansive mappings. (English) Zbl 0966.47035

Let \(C\) be a weakly compact and convex subset of a Banach space \(X\), which satisfies Opial’s condition, \(T_{i}, i = 1, 2, \ldots, k\) nonexpansive selfmaps of \(C\). Let \(S=\sum_{i = 0}^k \alpha_{i}T_{i}\), where \( T_{0} = I, \alpha_{i} \geq 0, \alpha_{0}, \alpha_{1} > 0\) and \(\sum_{i = 0}^k \alpha_{i} = 1\). Then the authors prove that the Picard iterates of \(S\) converge weakly to a common fixed point of the family. They have a similar result if the family has a nonempty fixed point set and satisfies condition (A) as defined in H. F. Senter and W. G. Dotson jun. [Proc. Am. Math. Soc. 44, 375-380 (1974; Zbl 0299.47032)].

MSC:

47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 0299.47032
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