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Zbl 1123.47044
Ceng, L.C.; Cubiotti, P.; Yao, J.C.
Strong convergence theorems for finitely many nonexpansive mappings and applications.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 67, No. 5, A, 1464-1473 (2007). ISSN 0362-546X

This article deals with new, more cumbersome and freakish, approximations $x_n$ to a fixed point for nonexpansive mappings of a nonempty closed convex subset of a Banach space $E$; the authors consider the case when $E$ is reflexive and has a weakly continuous duality mapping and the norm of $E$ is uniformly Gâteaux differentiable. Moreover, they consider a finite family of nonexpansive mappings ${\cal T} = \{T_1, \dots, T_r\}$ of $C$ into itself with nonempty set of common fixed points and a class of $W$-mappings generated by ${\cal T}$. These $W$-mappings are defined by chains $U_1 = \alpha_1T_1 + (1 - \alpha_1)I$, $U_2 = \alpha_2T_2U_1 + (1 - \alpha_2)I, \dots , U_{r-1} = \alpha_{r-1}T_{r-1}U_{r-2} + (1 - \alpha_r)I$, $W = U_r = \alpha_rT_rU_{r-1} + (1 - \alpha_r)I$, where $\alpha, \dots , \alpha_r$ are reals from $[0,1]$. The authors' approximations are $$x_{n+1} = \lambda_ny + (1 - \lambda_n)W_nx_n, \quad n = 1,2,\dots, \ y, x_1 \in C,$$ where $W_n$ is a sequence of $W$-mappings generated by ${\cal T}$ and $\lambda_n$ is a sequence from $(0,1)$ such that $$\lim_{n \to \infty} \ \lambda_n = 0, \quad \sum_{n=1}^\infty \lambda_n = \infty, \quad \lim_{n \to \infty} \ \frac{\lambda_{n-1}}{\lambda_n} = 1.$$ Theorems on the strong convergence of these approximations are proved. Based on these results, the problem of finding a common fixed point of finitely many mappings is also considered.
[Peter Zabreiko (Minsk)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: nonexpansive mapping; iterative scheme; common fixed point; strong convergence; Banach space; sunny nonexpansive retraction

Cited in: Zbl 1223.47098

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