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 \(T_1,T_2,\dots, T_N\).


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


Zbl 0148.13601
Full Text: DOI


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.