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Zbl 1126.47057
Xu, Hong-Kun
A variable Krasnosel'skiĭ--Mann algorithm and the multiple-set split feasibility problem. (English)
[J] Inverse Probl. 22, No. 6, 2021-2034 (2006). ISSN 0266-5611

This paper is about a variable Krasnosel'skij--Mann algorithm $x_{n+1}=(1-\alpha_n) x_n + \alpha_n T_n x_n$ in Banach spaces and its weak convergence to a fixed point of the mapping $T$. Here, $\{\alpha_n\}$ is a sequence in $[0,1]$ and $\{T_n\}$ is a sequence of nonexpansive mappings such that $T_n x$ converges to $Tx$ for every $x$. Furthermore, the author applies his result to solve the split feasibility problem, i.e., finding a point $x$ such that $x\in C$ and $Ax\in Q$, where $C$ and $Q$ are closed convex convex subsets of Hilbert spaces. The algorithm is also generalized for solving multiple-set split feasibility problems. It would have been helpful if some examples had been used to illustrate the process.
[Zhen Mei (Toronto)]

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
65J10 Equations with linear operators (numerical methods)
49J53 Set-valued and variational analysis

Keywords: fixed point iteration; Krasnosel'skij-Mann algorithm; Banach space; Hilbert space

Cited in: Zbl 1232.49017 Zbl 1219.90185

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