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

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
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