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Zbl 0999.47043
Xu, Hong-Kun; Ori, Ramesh G.
An implicit iteration process for nonexpansive mappings. (English)
[J] Numer. Funct. Anal. Optimization 22, No. 5-6, 767-773 (2001). ISSN 0163-0563; ISSN 1532-2467/e

Let $C$ be a closed convex subset of a Hilbert space $H$ and $T: C\to C$ be a nonexpansive mapping (i.e., $\|Tx-Ty\|\le \|x-y\|$, $x,y\in C$). It is well known, that for each $t\in (0,1)$, the contraction $T_t: C\to C$ defined by $T_t(x)= tu+(1-t)Tx$, $x\in C$ ($u\in C$ is a fix point) has a unique fixed point $x_t$. {\it F. E. Browder} [Arch. Ration. Mech. Anal. 24, 82-90 (1967; Zbl 0148.13601)] proved: $\{x_t\}$ converges in norm, as $t\to 0$, to a fixed point of $T$.\par In the present paper the authors study the convergence of an implicit iteration process to a fixed point of a finite family of nonexpansive mappings. The main result of the paper is the following:\par Theorem 2. Let $C$ be a closed convex subset of a Hilbert space $H$ and $T_1,T_2,\dots, T_N$ be $N$ nonexpansive self-mappings of $C$ such that $\bigcap^N_{i=1} \text{Fix}(T_i)\ne\emptyset$. Let $x_0\in C$ and $\{t_n\}$ be a sequence in $(0,1)$ such that $\lim_{n\to\infty} t_n= 0$. Then the sequence $\{x_n\}$ defined in the following way: $$x_n= t_n x_{n-1}+ (1- t_n) T_n x_n,\quad n\ge 1,$$ where $T_k= T_{k\text{ mod }N}$ (here the $\text{mod }N$ function takes values in $\{1,2,\dots, N\}$) converges weakly to a common fixed point of the mappings $T_1,T_2,\dots, T_N$.
[J.Górnicki (Rzeszów)]

MSC 2000:: *47H09 Mappings defined by "shrinking" properties
65J15 Equations with nonlinear operators (numerical methods)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: implicit iteration process; finite family of nonexpansive mappings; common fixed point

Citations: Zbl 0148.13601

Cited in: Zbl pre06090191 Zbl 1223.47097 Zbl 1199.47287 Zbl 1199.47305 Zbl 1184.47057 Zbl 1179.47059 Zbl 1163.47051 Zbl 1175.47062 Zbl 1174.47058 Zbl 1171.47052 Zbl 1170.47055 Zbl 1167.47305 Zbl 1145.47055 Zbl 1218.47125 Zbl 1164.47370 Zbl 1160.47322 Zbl 1140.47325 Zbl 1135.47053 Zbl 1126.65054 Zbl 1138.47045 Zbl 1086.47046 Zbl 1096.47059 Zbl 1090.47055 Zbl 1076.47057 Zbl 1045.47056 Zbl 1086.47045 Zbl 1025.47044

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