The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed point set. Recently Censor and Segal proposed an efficient algorithm for solving such a problem. However, to employ their algorithm, one needs to know prior information on the norm of the bounded linear operator. In this paper we propose a new algorithm that does not need any prior information of the operator norm, and we establish the weak convergence of the proposed algorithm under some mild assumptions. MSC: 47J25; 47J20; 49N45; 65J15 Keywords: split common fixed point problem; directed operators; demicontractive operator; quasi-nonexpansive operator

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In their recent paper (Math. Model. Anal., 17(4):457–466, 2012), Tang, Peng and Liu proposed a cyclic algorithm for solving the split common fixed point problem and established its weak convergence under some certain conditions. In this paper, we shall present a simple proof of such a result and moreover we shall remove one condition, continuity of the mapping involved, ensuring the convergence of the algorithm.

The paper deals with the split common fixed-point problem (SCFP) introduced by Censor and Segal. Motivated by Eicke's damped projection method, we propose a cyclic iterative scheme and prove its strong convergence to a solution of SCFP under some mild assumptions. An application of the proposed method to multiple-set split feasibility problems is also included.