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Zbl 1211.65065
Dang, Yazheng; Gao, Yan
The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. (English)
[J] Inverse Probl. 27, No. 1, Article ID 015007, 9 p. (2011). ISSN 0266-5611

Let $C$ and $Q$ be nonempty closed convex subsets of the real Hilbert spaces $H^1$ and $H^2$, respectively, and let $A: H^1 \to H^2$ be a bounded linear operator. The split feasibility problem is to find a point $x$ satisfying $x \in C$, $Ax \in Q$, if such point exists. For solving the problem, the authors suggest a strongly convergent algorithm which combines the CQ-method $x^{k+1}=P_C (I-\gamma A^T(I-P_Q)A)x^k$, $0<\gamma <2/\rho(A^TA)$ with the Krasnosel'skii-Mann (KM) iterative process $x^{k+1}=(1-\alpha_k)x^k+\alpha_k T x^k$, $\alpha_k \in (0,1)$, where $T$ is a nonexpansive operator.
[Mikhail Yu. Kokurin (Yoshkar-Ola)]

MSC 2000:: *65J22 Inverse problems
65J10 Equations with linear operators (numerical methods)
47A50 Equations and inequalities involving linear operators

Keywords: nonexpansive mapping; fixed point; convex set; iterative process; strong convergence; Hilbert spaces; bounded linear operator; split feasibility problem; CQ-method; Krasnosel'skii-Mann iterative process

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