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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the convergence of some iteration processes in uniformly convex Banach spaces

Author: J. Gwinner
Journal: Proc. Amer. Math. Soc. 71 (1978), 29-35
MSC: Primary 47H10
MathSciNet review: 0477899
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Abstract: For the approximation of fixed points of a nonexpansive operator T in a uniformly convex Banach space E the convergence of the Mann-Toeplitz iteration $ {x_{n + 1}} = {\alpha _n}T({x_n}) + (1 - {\alpha _n}){x_n}$ is studied. Strong convergence is established for a special class of operators T. Via regularization this result can be used for general nonexpansive operators, if E possesses a weakly sequentially continuous duality mapping. Furthermore strongly convergent combined regularization-iteration methods are presented.

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Keywords: Nonexpansive, $ \varphi $-accretive, duality mapping, fixed point, iteration, regularization
Article copyright: © Copyright 1978 American Mathematical Society