Viscosity approximation methods for the split equality common fixed point problem of quasi-nonexpansive operators

Abstract

Let H1, H2, H3 be real Hilbert spaces, let A: H1 → H3, B: H2 → H3 be two bounded linear operators. The split equality common fixed point problem (SECFP) in the infinite-dimensional Hilbert spaces introduced by Moudafi (Alternating CQ-algorithm for convex feasibility and split fixed-point problems. Journal of Nonlinear and Convex Analysis) is (1)to find x∈F(U),y∈F(T) such that Ax=By,where U: H1 → H1 and T: H2 → H2 are two nonlinear operators with nonempty fixed point sets F(U) = {x ∈ H1: Ux = x} and F(T) = {x ∈ H2: Tx = x}. Note that, by taking B = I and H2 = H3 in (1), we recover the split fixed point problem originally introduced in Censor and Segal. Recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms with weak convergence for the SECFP (1) of firmly quasi-nonexpansive operators. In this paper, we introduce two viscosity iterative algorithms for the SECFP (1) governed by the general class of quasi-nonexpansive operators. We prove the strong convergence of algorithms. Our results improve and extend previously discussed related problems and algorithms.