The purpose of this paper is to investigate the strong convergence problem of a modified mixed Ishikawa iterative sequence with errors for approximating the fixed points of an asymptotically nonexpansive mapping in the intermediate sense and an asymptotically quasi-pseudo-contractive-type mapping in an arbitrary real Banach space. The results here improve and extend the corresponding results reported by some other authors recently.

It is well known that fixed point theory has emerged as an important tool in studying a wide class of nonlinear elliptic systems and nonlinear parabolic systems, obstacle, unilateral, and equilibrium problems, optimization problems, theoretical mechanics, and control theory, which arise in several branches of pure and applied nonlinear sciences in a unified and general framework. This alternative formulation has been used to study the existence of a fixed point as well as develop several numerical methods. Using this idea, one can suggest some iterative methods for fixed points and study the convergence of their iterative sequences.

Throughout this paper, we assume that

Let

Let

The mapping

The mapping

The mapping

It is easy to see that if

In 1972, Goebel and Kirk [

Let

The concept of asymptotically pseudocontractive-type mapping was first introduced by Zeng [

Let

The mapping

The mapping

From the Definitions

(1) Let

(2) If

The modified Ishikawa and Mann iterative sequences with errors were studied by Zeng. He [

In this paper, motivated by the above results, we introduce a strong convergence theorem of the modified Ishikawa iterative sequence with errors for approximating fixed points of asymptotically nonexpansive mapping in the intermediate sense and asymptotically quasi- pseudo-contractive-type mapping in an arbitrary real Banach space. The results here generalize and improve the recent results announced by many other authors to a certain extent, such as [

In order to prove our main results, we need the following lemmas.

Let

Let

Let

Let

Since

Assume there exists a strictly increasing function

By the definition of infimum, there exists

Since

Now we claim that for all

Since

Next we make an estimation for the third term on the right side of (

Substituting (

Using (

Again, by (

Assume that

Let

Our research and results in this paper have the following advantages. (a) The iterative scheme is the modified mixed Ishikawa and Mann iterative scheme with error. (b) The research object is the very generalized asymptotically quasi-pseudo-contractive-type mappings. (c) The proof methods are very different from previous ones and we do not need the condition of “boundedness of

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

The authors would like to thank the editors and referees for many useful comments and suggestions for the improvement of the paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11271330) and the Natural Science Foundation of Zhejiang Province (Grant no. Y6100696).