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The aim of this paper is to construct a novel implicit iterative algorithm for the split common fixed point problem for the demicontractive operators

The split feasibility problem (SFP) is to find a point

This problem was proposed by Censor and Elfving [

Since the SFP can extensively be applied in fields such as intensity-modulated radiation therapy, signal processing, and image reconstruction, then the SFP has received so much attention by so many scholars; see [

In 1994, Censor and Elfving [

As we know, the computation of the inverse

In 2002 and 2004, Byrne [

For the stepsize of algorithm (

The split common fixed point problem (SCFP) is to find a point

This problem was proposed by Censor and Segal [

In 2009, Censor and Segal [

The next two years, some extension results on the operators are obtained, such as Moudafi (2010) [

In order to overcome this disadvantage, Cui and Wang [

Recently, Byrne et al. [

Motivated by the viscosity idea of [

The research highlight of this paper is that the strong convergence of the SCFP (

Throughout this paper, we denote the set of all solutions of the SCFP (

Let

Some concepts and lemmas are given in the following and they are useful in proving our main results.

A operator

(i) nonexpansive if

(ii) quasi-nonexpansive if

(iii) directed if

(iv)

Note that (

Let

As we know, the nonexpansive mappings are demiclosed at zero [

Let

Let

Let

Let

In addition, for

where

Let

Based on the definitions in preliminaries, the classes of

The relations of

From Definition

The nonexpansive operator is quasi-nonexpansive operator.

The quasi-nonexpansive operator is

The directed operator is

Next, we give the novel implicit algorithm to solve the SCFP (

Both

Choose an initial guess

Assume the solution set of the SCFP (

The proof is divided into three steps.

Denote

By the reflexivity of Hilbert space

First, we show that

Next, we denote

For

Then, (

Second, we show that

For the case

Since

From (

Hence,

Moreover,

By

For

From (

From (

Third, we show that

Indeed, from (

take the limit through

To show that

Assuming the above conclusion does not hold, that is to say, there exists another subsequence

Replacing

Adding up the above variational inequality yields

Thus

The proof is completed.

In this section, we consider some special cases as the applications of Theorem

Based on the relations of

Let

Let

Let

Let

Let

In this paper, the research highlights that the strong convergence of the SCFP (

The authors declare that they have no competing interests.

This work was carried out by the three authors, in collaboration. Moreover, the three authors have read and approved the final manuscript.

This paper was funded by Fundamental Research Funds for the Central Universities (no. JB150703), National Science Foundation for Young Scientists of China (no. 11501431), and National Science Foundation for Tian Yuan of China (no. 11426167).