Abstract
In this paper we consider the common fixed point problem for a finite family of quasi-nonexpansive mappings Ui:???$U_{i}\colon \mathcal H\rightarrow \mathcal H$, where i ? I := {1,?, M}, M?1, and ?$\mathcal H$ is a real Hilbert space. This problem is defined as follows: find x??i?IFixUi??$x\in \bigcap _{i\in I}\text {Fix} U_{i}\neq \emptyset $, where FixUi:={z???z=Uiz}$U_{i}:=\{z\in \mathcal H\mid z=U_{i}z\}$. We propose the following iterative method: x0??,xk+1:=Tkxk,$$ x^{0}\in\mathcal H,\quad x^{k+1}:=T_{k} x^{k}, $$ where for each k=0,1,2,?, the operator Tk is defined by a certain amalgamation procedure called modular string averaging. The main idea of this procedure is to combine repeatedly and recursively three primal operations: relaxation, convex combination and composition of the given operators Ui. The modular string averaging procedure, when combined with the above iterative method, provides a very flexible framework which covers and fills the gap between different algorithmic approaches such as string averaging and block iterative schemes. Moreover, our framework enables us to construct many algorithmic schemes, the convergence of which has not been investigated so far. The aim of this paper is to establish both weak and strong convergence results for the above iterative method. Moreover, in the case of firmly nonexpansive Ui's, we show that convergence is preserved in the presence of inexact computations. In particular, this implies that the iterative scheme is resilient to bounded perturbations, which is important from the superiorization methodology point of view.
Published Version
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