## 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

Zbl 0148.13601
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### References:

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