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Let \(C\) be a bounded, closed, convex subset of a uniformly convex and uniformly smooth Banach space \(X\). We investigate the weak convergence of the generalized Krasnosel'skii-Mann and Ishikawa iteration processes to common fixed points of semigroups of nonlinear mappings \(T_t\colon C \to C\). Each of \(T_t\) is assumed to be pointwise Lipschitzian, that is, there exists a family of functions \(\alpha_t\colon C \to [0, \infty)\) such that \(\|T_t(x) - T_t (y)\| \leq\alpha_t (x)\|x -y\|\) for \(x, y \in C\). The paper demonstrates how the weak compactness of \(C\) plays an essential role in proving the weak convergence of these processes to common fixed points.
Słowa kluczowe
Fixed point
common fixed point
Lipschitzian mapping
pointwise Lipschitzian mapping
semigroup of mappings
asymptotic pointwise nonexpansive mapping
uniformly convex Banach space
uniformly smooth Banach space
Fréchet differentiable norm
weak compactness
fixed point iteration process
Krasnosel'skii-Mann process
Mann process
Ishikawa process
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Tom
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wydano
2012
online
2017-12-19
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bwmeta1.element.ojs-doi-10_14708_cm_v52i2_5331