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An implicit iteration process for nonexpansive mappings. (English) Zbl 0999.47043

Let \(C\) be a closed convex subset of a Hilbert space \(H\) and \(T: C\to C\) be a nonexpansive mapping (i.e., \(\|Tx-Ty\|\leq \|x-y\|\), \(x,y\in C\)). 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 \(x_t\). 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 \(T\).
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:
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 \(\bigcap^N_{i=1} \text{Fix}(T_i)\neq\emptyset\). 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: \[ x_n= t_n x_{n-1}+ (1- t_n) T_n x_n,\quad n\geq 1, \] 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 \(T_1,T_2,\dots, T_N\).

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65J15 Numerical solutions to equations with nonlinear operators
47H10 Fixed-point theorems

Citations:

Zbl 0148.13601
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References:

[1] DOI: 10.1006/jmaa.1996.0308 · Zbl 0956.47024
[2] DOI: 10.1007/BF00251595 · Zbl 0148.13601
[3] DOI: 10.1090/S0002-9904-1967-11864-0 · Zbl 0177.19101
[4] Lions P. L., C.R. Acad. Sci. Paris, Sér. A 284 pp 1357– (1977)
[5] DOI: 10.1090/S0002-9904-1967-11761-0 · Zbl 0179.19902
[6] DOI: 10.1016/0022-247X(80)90323-6 · Zbl 0437.47047
[7] DOI: 10.1007/BF01190119 · Zbl 0797.47036
[8] DOI: 10.1016/0362-546X(94)E0059-P · Zbl 0826.47038
[9] Xu H. K., C. R. Acad. Sci. Paris, Sér. I 325 pp 151– (1997) · Zbl 0888.47036
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