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

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.


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
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