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On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces

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Kozlowski W. M.

Commentationes Mathematicae

2012

tom 52

nr 2

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.

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1 Commentationes Mathematicae

1 Kozlowski W. M.

1 2012

Wyniki wyszukiwania

On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces