## 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.

### 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
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