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Inspired by the Moudafi (2010), we propose an algorithm for solving the split common fixed-point problem for a wide class of asymptotically quasi-nonexpansive operators and the weak and strong convergence of the algorithm are shown under some suitable conditions in Hilbert spaces. The algorithm and its convergence results improve and develop previous results for split feasibility problems.

Fixed-point problem is a classical problem in nonlinear analysis and it has application in a wide spectrum of fields such as economics, physics, and applied sciences. In this paper, We are concerned with the split common fixed point problem (SCFP). In fact, the SCFP is an extension of the split feasibility problem (SFP) and the convex feasibility problem (CFP), see [

Throughout this paper, we assume that both

The split common fixed point problem for quasi-nonexpansive mapping in the setting of Hilbert space was first introduced and studied by Moudafi [

The paper is organized as follows. In Section

Recall that a mapping

A mapping

A mapping

A mapping

A mapping

A mapping

Let

Let

Suppose

We now give a description of an algorithm.

Initialization: let

Iterative step: for

In what follows, we establish the weak convergence and strong convergence of Algorithm

Let

For every

Any weak cluster point of the sequence

Then, there exists

Let

any sequence

if

First, we prove that for each

By using (1) in Lemma

it follows that,

Noticing

Since there is no more than one weak cluster point, the weak convergence of the whole sequence

In this paper, we have proposed an algorithm for solving the SCFP in the wide class of asymptotically quasi-nonexpansive operators and obtained its weak and strong convergence in general Hilbert spaces in a new way. Next, we will improve the algorithm to solve the multiple split common fixed point problem in infinite Hilbert spaces.

This work was supported by the National Science Foundation of China (under Grant 11171221), Shanghai Leading Academic Discipline Project (under Grant XTKX2012), Basic and Frontier Research Program of Science and Technology Department of Henan Province (under Grant 112300410277), Innovation Program of Shanghai Municipal Education Commission (under Grant 14YZ094), Doctoral Program Foundation of Institutions of Higher Education of China (under Grant 20123120110004), Doctoral Starting Projection of the University of Shanghai for Science and Technology (under Grant ID-10-303-002), and Young Teacher Training Projection Program of Shanghai for Science and Technology.