#### Yaxuan Zhang, Yingying Li, Two modified relaxed CQ algorithms for the multiple-sets split feasibility problem, Vol. 2021 (2021), Article ID 39, pp. 1-14

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DOI: 10.23952/jnfa.2021.39

Received April 7, 2021; Accepted November 25, 2021; Published December 17, 2021

Abstract. The multiple-sets split feasibility problem is to find a point $x^{*}\in \bigcap_{i=1}^{t}C_{i}$ such that $Ax^{*}\in \bigcap_{j=1}^{r}Q_{j}$, where $C_{i}\subset \mathcal{H}_{1}$ and $Q_{j}\subset \mathcal{H}_{2}$ are nonempty, closed, and convex subsets, $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are Hilbert spaces, and $A: \mathcal{H}_1\to \mathcal{H}_2$ is a bounded and linear operator. In this paper, we present two modified relaxed CQ algorithms with the step size determined by the Armijo-line search. Under mild conditions, we establish the weak convergence, and provide numerical experiments to illustrate the effectiveness of the proposed algorithms.