A variable Krasnosel’skiĭ–Mann algorithm and the multiple-set split feasibility problem. (English) Zbl 1126.47057

Xu, Hong-Kun

Inverse Probl. 22, No. 6, 2021-2034 (2006).

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.

Reviewer: Zhen Mei (Toronto)

MSC:

47J25	Iterative procedures involving nonlinear operators
47H10	Fixed-point theorems
65J10	Numerical solutions to equations with linear operators
49J53	Set-valued and variational analysis

Keywords:

fixed point iteration; Krasnosel’skij-Mann algorithm; Banach space; Hilbert space