#### Abstract

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.

#### 1. Introduction and Preliminaries

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 is a real Banach space, is its dual space, is the dual pair between and , and denote the domain of a mapping and the set of all fixed points of , respectively.

Let be the normalized duality mapping defined by

*Definition 1. *Let be a nonempty subset of a real Banach space and a mapping.(1)The mapping is said to be asymptotically nonexpansive, if there exists a number sequence with , such that
(2)The mapping is said to be asymptotically pseudocontractive, if there exists a number sequence with and , such that
(3)The mapping is said to be uniformly -Lipschitzian, if there exists a constant , such that

*Remark 2. *It is easy to see that if is an asymptotically nonexpansive mapping, then is uniformly -Lipschitzian (); and if is an asymptotically nonexpansive mapping, then is an asymptotically pseudocontractive mapping. But the converse is not true in general. This can be seen from the example in [1].

In 1972, Goebel and Kirk [2] introduced the concept of asymptotically nonexpansive mappings which was closely related to the theory of fixed points of mappings in Banach spaces. Nine years later, the concept of asymptotically pseudocontractive mapping was introduced by Schu [3] in 1991. The iterative approximation problems for asymptotically nonexpansive mappings and asymptotically pseudocontractive mappings were studied extensively by many authors; see [1, 4–11]. The concept of asymptotically nonexpansive mapping in the intermediate sense was introduced and studied by Bruck et al [12]. When is bounded, the class of asymptotically nonexpansive self-maps is the special case of the class of asymptotically nonexpansive self-maps in the intermediate sense.

*Definition 3 (see [12]). *Let be a mapping, if for each , , there holds the inequality
then is called an asymptotically nonexpansive mapping in the intermediate sense.

The concept of asymptotically pseudocontractive-type mapping was first introduced by Zeng [13] in 2004. On this basis, the asymptotically quasi-pseudo-contractive-type mapping can be given as follows.

*Definition 4. *Let be a mapping.(1)The mapping is said to be of asymptotically pseudocontractive type, if there exists a number sequence with , such that
(2)The mapping is said to be of asymptotically quasi-pseudo-contractive type, if there exists a number sequence with , such that

*Remark 5. *From the Definitions 1 and 4, it is easily known that the class of asymptotically quasi-pseudo-contractive-type mappings contains that of the asymptotically nonexpansive mappings with fixed points, the asymptotically pseudocontractive mappings with fixed points, and asymptotically pseudocontractive-type mappings with fixed points.

*Definition 6 (see [4, 5]). *(1) Let be a mapping, a nonempty convex subset of , a given point, and , , , and four number sequences in . Then the sequence defined by
is called the modified Ishikawa iterative sequence with errors of , where and are two bounded sequences in .

(2) If and in (8), then , the sequence defined by
is called the modified Mann iterative sequence with errors of .

The modified Ishikawa and Mann iterative sequences with errors were studied by Zeng. He [4] proved the strong convergence of the modified Ishikawa iterative sequence with errors for the uniformly -Lipschitzian asymptotically pseudocontractive mapping in an arbitrary real Banach space with the bounded range of . Zeng [4] investigated the strong convergence of the modified Ishikawa iterative sequence with errors for the non-Lipschitzian asymptotically pseudocontractive mapping in an arbitrary real Banach space and gave the necessary and sufficient condition that is bounded and .

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 [1, 4, 5, 12, 13].

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

Lemma 7 (see [14]). *Let be a real Banach space and the normalized duality mapping. Then
**
holds for all , and .*

Lemma 8 (see [15]). *Let be a strictly increasing function with , and let , and be nonnegative real sequences, such that , , and . Suppose that
**
then .*

#### 2. Main Results

Theorem 9. *Let G be a nonempty convex subset of a real Banach space E, be an asymptotically nonexpansive mapping in the intermediate sense and an asymptotically quasi- pseudo-contractive-type mapping with sequence , . Assume that , , , and are four number sequences in satisfying the following conditions:*(i)

*;*(ii)

*, ;*(iii)

*.*

*Let be a given point and the sequence the modified Ishikawa iterative sequence with errors defied by (8). Then converges strongly to a fixed point if only if there exists a strictly increasing function with such that*

*Proof (sufficiency). *Since is an asymptotically quasi-pseudo-contractive-type mapping, we know and we can choose a point .

Assume there exists a strictly increasing function with such that the inequality (12) is satisfied. We can let , , for all .

By the definition of infimum, there exists such that
By using (12), we have . Hence .

Since and are two bounded sequences, we can let
From (8) and (14), we have
Since is an asymptotically nonexpansive in the intermediate sense, and , , , , there exists , such that for all ,
From (15), (16), we have
By virtue of (13), we have
Thus there exists , such that for all ,
Hence
Since is a strictly increasing function, we obtain
where .

Now we claim that for all ,
Indeed, when , it is easy to know that the result has been established by (21). Assume if for , holds, we want to prove . Reduction to absurdity, assume that . Then, since is a strictly increasing function, we have
where . Using (17), we obtain for all ,
From Lemma 7 and (8), we have
Now we consider the second term on the right side of (25),
Using (8) and for all , we have

Since is an asymptotically nonexpansive mapping in the intermediate sense, and , , , , , we obtain
Substituting (29) in (27), we have . By virtue of (26), we know for all ,
where .

Next we make an estimation for the third term on the right side of (25). Using (13), we have
Finally we estimate the last term on the right side of (25). Using (24) and , we assume that , , , and we have
where , .

Substituting (30), (31), and (32) in (25), we obtain

Using (33) and , we have
Since , , there exists nonnegative integer , such that when , . Without loss of generality, for all . Let , . From (34), for all , we have
Substituting (23) in (35), we have
Since , , there exists , for all , . Without loss of generality, for all , . So, from (36) and the condition (iii), we have
This is a contradiction. Therefore for all , we have .

Again, by (35) and the conditions (ii), (iii), we have
Consider , and . So by Lemma 8, we obtain ; that is, .

*Proof (necessity). *Assume that . Since and is an asymptotically quasi- pseudo-contractive-type mapping, we have
For any given strictly increasing function with , such as , we have
From (39) and (40), we know that (12) holds.

Theorem 10. *Let be a nonempty convex subset of a real Banach space , and let be an asymptotically nonexpansive in the intermediate sense and an asymptotically quasi- pseudo-contractive-type mapping with sequence , . Assume that and are two sequences in satisfying the following conditions:*(i)*;*(ii)*;*(iii)*.**Let be a given point and let the sequence be the modified Mann iterative sequence with errors defined by (9). Then converges strongly to a fixed point if only if there exists a strictly increasing function with such that the condition (12) of Theorem (12) holds.*

*Remark 11. *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 ” (e.g., [4]). So, the results here generalize and improve the recent results announced by many other authors to a certain extent, such as [1, 4, 5, 12, 13].

#### Conflict of Interests

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

#### Acknowledgments

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).