The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. (English) Zbl 1211.65065

Dang, Yazheng; Gao, Yan

Inverse Probl. 27, No. 1, Article ID 015007, 9 p. (2011).

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.

Reviewer: Mikhail Yu. Kokurin (Yoshkar-Ola)

MSC:

65J22	Numerical solution to inverse problems in abstract spaces
65J10	Numerical solutions to equations with linear operators
47A50	Equations and inequalities involving linear operators, with vector unknowns

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