We present a new algorithm for solving the two-set split common fixed point problem with total quasi-asymptotically pseudocontractive operators and consider the case of quasi-pseudocontractive operators. Under some appropriate conditions, we prove that the proposed algorithms have strong convergence. The results presented in this paper improve and extend the previous algorithms and results of Censor and Segal (2009), Moudafi (2011 and 2010), Mohammed (2013), Yang et al. (2011), Chang et al. (2012), and others.

Let

In particular, if

Censor and Segal [

Inspired by the work of Censor and Segal, for

However, we found that the strong convergence of (

In order to reach the main results, we first recall the following facts.

Let

(i) Recalled that

(ii) Let

(iii) A mapping

(iv) A mapping

(i) Let

(ii) A mapping

(iii) A mapping

(iv) A mapping

Note that the classes of directed operators and attracting operators belong to the class of paracontracting operators. The class of paracontracting operators belongs to the class of demicontractive operators, while the class of quasi-pseudocontractive operators includes the class of demicontractive operators. Further, the class of total quasi-asymptotically pseudocontractive operators, with quasi-pseudocontractive operators as a special case, includes the class of total quasi-asymptotically strictly pseudocontractive operators.

Let

Consider

(i)

(ii)

Let

A mapping

Let

Let a sequence

For all

In this section, we will prove the strong convergence of (

Let

Let

(1) First of all, we show that, for

From (

Since

Next, from (

From (

Substituting (

(2) Next we prove

For each

which implies that

From (

Therefore, when we take limit on both sides of (

Then,

(3) Next we prove that

From (

By the same way, from (

Therefore, from (

(4) Finally, we prove that

Substituting (

This completes the proof.

The following theorem can be concluded from Theorem

Let

For each

Algorithm (

In this work, we develop the split common fixed point problem with more general classes of total quasi-asymptotically pseudocontractive and quasi-pseudocontractive operators; corresponding algorithms are improved based on the viscosity iteration; thus we can obtain strong convergence without more constraints on operators.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the associate editor and the referees for their comments and suggestions. This research was supported by the National Natural Science Foundation of China (11071053).