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Zbl 0895.47048
Tan, Kok-Keong; Xu, Hong-Kun
Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.
(English)
[J] J. Math. Anal. Appl. 178, No.2, 301-308 (1993). ISSN 0022-247X

It is the object of the present paper to show that if $X$ is a uniformly convex Banach space which satisfies Opial's condition or whose norm is Fréchet differentiable, $C$ is a bounded closed convex subset of $X$, and $T:C\to C$ is a nonexpansive mapping, then for any initial data $x_0$ in $C$ the Ishikawa iterates $\{x_n\}$ defined by $$x_{n+1}= t_nT (s_nTx_n+ (1-s_n)x_n)+ (1-t_n)x_n, \qquad n=0,1,2,\dots,$$ where $\{t_n\}$ and $\{s_n\}$ are chosen so that $\sum_n t_n(1-t_n)$ diverges, $\sum_n s_n(1-t_n)$ converges, and $\varlimsup_n s_n<1$, converge weakly to a fixed point of $T$. This generalizes a theorem of Reich.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: Opial's condition; norm is Fréchet differentiable; nonexpansive mapping; converge weakly to a fixed point

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