Geodesic
On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces
Commentationes Mathematicae, Tome 52 (2012) no. 2, pp. 113-136.

Voir la notice de l'article provenant de la source Annales Societatis Mathematicae Polonae Series

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.
DOI : 10.14708/cm.v52i2.5331
Mots-clés : 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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W.M. Kozlowski. On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces. Commentationes Mathematicae, Tome 52 (2012) no. 2, pp.  113-136. doi : 10.14708/cm.v52i2.5331. http://geodesic.mathdoc.fr/articles/10.14708/cm.v52i2.5331/

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