Let H1, H2, H3 be real Hilbert spaces, let A : H1 ? H3, B : H2 ? H3 be two
bounded linear operators. The general multiple-set split common fixed-point
problem under consideration in this paper is to find x ??p,i=1F(Ui), y ??r,j=1 F(Tj)
such that Ax = Bym, (1) where p, r ? 1 are integers, Ui : H1 ? H1
(1 ? i ? p) and Tj : H2 ? H2 (1 ? j ? r) are quasi-nonexpansive mappings
with nonempty common fixed-point sets ?p,i=1 F(Ui) = ?p,i=1 {x ? H1 : Uix = x}
and ?r,j=1F(Tj) = ?r,j=1 {x ? H2 : Tjx = x}. Note that, the above problem (1)
allows asymmetric and partial relations between the variables x and y. If H2
= H3 and B = I, then the general multiple-set split common fixed-point
problem (1) reduces to the multiple-set split common fixed-point problem
proposed by Censor and Segal [J. Convex Anal. 16(2009), 587-600]. In this
paper, we introduce simultaneous parallel and cyclic algorithms for the
general split common fixed-point problems (1). We introduce a way of
selecting the stepsizes such that the implementation of our algorithms does
not need any prior information about the operator norms. We prove the weak
convergence of the proposed algorithms and apply the proposed algorithms to
the multiple-set split feasibility problems. Our results improve and extend
the corresponding results announced by many others.
Weak Convergence Theorems on the Split Common Fixed Point Problem for Demicontractive Continuous Mappings