It is well known that the Krasnosel'skii-Mann algorithm and the CQ algorithm for a split feasibility problem are not strongly convergent. In this paper, we present a KM-CQ-like algorithm with strong convergence, which combines the KM algorithm with the CQ algorithm by introducing two parameter sequences for solving the split feasibility problem. Under some parametric controlling conditions, the strong convergence of the algorithm is shown. Finally, we propose a modified KM-CQ-like algorithm and establish its strong convergence theorem.

Inspired by the inertial proximal algorithms for finding a zero of a maximal monotone operator, in this paper, we propose two inertial accelerated algorithms to solve the split feasibility problem. One is an inertial relaxed-CQ algorithm constructed by applying inertial technique to a relaxed-CQ algorithm, the other is a modified inertial relaxed-CQ algorithm which combines the KM method with the inertial relaxed-CQ algorithm. We prove their asymptotical convergence under some suitable conditions. Numerical results are reported to show the effectiveness of the proposed algorithms.

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We propose two new double projection algorithms for solving the split feasibility problem (SFP). Different from the extragradient projection algorithms, the proposed algorithms do not require fixed stepsize and do not employ the same projection region at different projection steps. We adopt flexible rules for selecting the stepsize and the projection region. The proposed algorithms are shown to be convergent under certain assumptions. Numerical experiments show that the proposed methods appear to be more efficient than the relaxed-CQ algorithm.

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