Alternating mann iterative algorithms for the split common fixed-point problem of quasi-nonexpansive mappings

Abstract

Very recently, Moudafi (Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal. ) introduced an alternating CQ-algorithm with weak convergence for the following split common fixed-point problem. Let H 1 , H 2 , H 3 be real Hilbert spaces, let A: H 1 → H 3 , B: H 2 → H 3 be two bounded linear operators. Find x∈F(U),y∈F(T) such that Ax=By, (1) where U: H 1 → H 1 and T: H 2 → H 2 are two firmly quasi-nonexpansive operators with nonempty fixed-point sets F(U)={x∈ H 1 :Ux=x} and F(T)={x∈ H 2 :Tx=x}. Note that by taking H 2 = H 3 and B=I, we recover the split common fixed-point problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009) and used to model many significant real-world inverse problems in sensor net-works and radiation therapy treatment planning. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce two alternating Mann iterative algorithms and prove the weak convergence of algorithms. At last, we provide some applications. Our results improve and extend the corresponding results announced by many others.MSC:47H09, 47H10, 47J05, 54H25.