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Zbl 1095.47034
Chidume, C.E.; Chidume, C.O.
Iterative approximation of fixed points of nonexpansive mappings. (English)
[J] J. Math. Anal. Appl. 318, No. 1, 288-295 (2006). ISSN 0022-247X

Let $K$ be a nonempty closed convex subset of a real Banach space $E$ which has a uniformly Gâteaux differentiable norm and let $T: K\rightarrow K$ be a nonexpansive mapping with nonempty set of all fixed points $F(T)$. For a sequence $\{\alpha_n\}$ of real numbers in $[0,1]$ and an arbitrary $u\in K$, the sequence $\{x_n\}$ in $K$ defined by $x_0\in K$ and $$ x_{n+1}=\alpha_n u+ (1-\alpha_n) T x_n,\,\,n\geq 0,$$ was introduced by {\it B.~Halpern} [Bull.\ Am.\ Math.\ Soc.\ 73, 957--961 (1967; Zbl 0177.19101)] and subsequently studied by several authors in order to approximate the fixed points of $T$, see, e.g., the reviewer's recent monograph [{\it V.~Berinde}, ``Iterative approximation of fixed points'' (Efemeride, Baia Mare) (2002; Zbl 1036.47037)]. In the present paper, the authors use the same kind of iterative scheme but with the averaged map $S$, given by $Sx:=(1-\delta)x+\delta Tx,\,\,\delta\in (0,1)$, instead of $T$, and prove a strong convergence theorem for approximating the fixed points of $T$.
[Vasile Berinde (Baia Mare)]

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
54H25 Fixed-point theorems in topological spaces

Keywords: uniformly smooth real Banach space; uniformly Gâteaux differentiable norm; nonexpansive mapping; fixed point; Halpern iteration; strong convergence theorem

Citations: Zbl 0177.19101; Zbl 1036.47037

Cited in: Zbl 1165.65356 Zbl 1220.47121

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