## Nonlinear Analysis: Algorithm, Convergence, and Applications 2014

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# A Modified Mixed Ishikawa Iteration for Common Fixed Points of Two Asymptotically Quasi Pseudocontractive Type Non-Self-Mappings

**Academic Editor:**Rudong Chen

#### Abstract

A new modified mixed Ishikawa iterative sequence with error for common fixed points of two asymptotically quasi pseudocontractive type non-self-mappings is introduced. By the flexible use of the iterative scheme and a new lemma, some strong convergence theorems are proved under suitable conditions. The results in this paper improve and generalize some existing results.

#### 1. Introduction

Let be a real Banach space with its dual and let be a nonempty, closed, and convex subset of . The mapping is the normalized duality mapping defined by

Let be a mapping. We denote the fixed point set of by ; that is, . Recall that a mapping is said to be nonexpansive if, for each ,

is said to be asymptotically nonexpansive if there exists a sequence with as such that A sequence of self-mappings on is said to be uniform Lipschitzian with the coefficient if, for any , the following holds:

is said to be asymptotically pseudocontractive if there exist with as and such that

It is obvious to see that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically pseudocontractive. Goebel and Kirk [1] introduced the class of asymptotically nonexpansive mappings in 1972. The class of asymptotically pseudocontractive mappings was introduced by Schu [2] and has been studied by various authors for its generalized mappings in Hilbert spaces, Banach spaces, or generalized topological vector spaces by using the modified Mann or Ishikawa iteration methods (see, e.g.,[3–21]).

In 2003, Chidume et al. [22] studied fixed points of an asymptotically nonexpansive non-self-mapping and the strong convergence of an iterative sequence generated by where is a nonexpansive retraction.

In 2011, Zegeye et al. [23] proved a strong convergence of Ishikawa scheme to a uniformly L-Lipschitzian and asymptotically pseudocontractive mappings in the intermediate sense which satisfies the following inequality (see [24]): where with as .

Motivated and inspired by the above results, in this paper, we introduce a new modified mixed Ishikawa iterative sequence with error for common fixed points of two more generalized asymptotically quasi pseudocontractive type non-self-mappings. By the flexible use of the iterative scheme and a new lemma (i.e., Lemma 6 in this paper), under suitable conditions, we prove some strong convergence theorems. Our results extend and improve many results of other authors to a certain extent, such as [6, 8, 14–23].

#### 2. Preliminaries

*Definition 1. *Let be a nonempty closed convex subset of a real Banach space . is said to be a nonexpansive retract (with ) of if there exists a nonexpansive mapping such that, for all , . And is called a nonexpansive retraction.

Let be a non-self-mapping (maybe self-mapping). is called uniformly L-Lipschitzian (with ) if there exists a constant such that

is said to be asymptotically pseudocontractive (with ) if there exist with as and , such that

is said to be an asymptotically pseudocontractive type (with ) if there exist with as and , such that

is said to be an asymptotically quasi pseudocontractive type (with ) if , for , there exist with as , and, , such that

*Remark 2. *It is clear that every asymptotically pseudocontractive mapping (with ) is asymptotically pseudocontractive type (with ) and every asymptotically pseudocontractive type (with ) is asymptotically quasi pseudocontractive type (with ). If is a self-mapping, then we can choose as the identical mapping and we can get the usual definition of asymptotically pseudocontractive mapping, and so forth.

*Definition 3. *Let be a nonexpansive retract (with ) of , let be two uniformly L-Lipschitzian non-self-mappings and let be an asymptotically quasi pseudocontractive type (with ).

The sequence is called the new modified mixed Ishikawa iterative sequence with error (with ), if is generated by
where is arbitrary, and are bounded, and , .

If , (12) turns to
and it is called the new modified mixed Mann iterative sequence with error (with ).

If , (12) becomes
and it is called the new modified mixed Ishikawa iterative sequence (with ).

If , (14) turns to
and it is called the new mixed Ishikawa iterative sequence (with ).

If is a self-mapping and is the identical mapping, then (15) is just the modified Ishikawa iterative sequence
If , (15) becomes (6), obviously. So, iterative method (12) is greatly generalized.

The following lemmas will be needed in what follows to prove our main results.

Lemma 4 (see [19]). *Let be a real Banach space. Then, for all , , the following inequality holds:
*

Lemma 5 (see [6, 7]). *Let , , be three sequences of nonnegative numbers satisfying the recursive inequality:
**
where is some nonnegative integer. If , , then exists.*

Lemma 6. *Suppose that is a strictly increasing function with . Let be four sequences of nonnegative numbers satisfying the recursive inequality:
**
where is some nonnegative integer. If , , , then .*

*Proof. *From (19), we get
By Lemma 5, we know that exists. Let . Now we show . Otherwise, if , then , such that when . Because is a strictly increasing function, so . From (19) again, we have
This is a contradiction with the given condition . Therefore .

Lemma 7. *Suppose that is a strictly increasing function with . Let be five sequences of nonnegative numbers satisfying the recursive inequality:
**
where is some nonnegative integer. If , , , , then .*

*Proof. *Firstly, we show . If , then, for arbitrary , , such that when . Because is a strictly increasing function and , so and when . From (22), we have
By Lemma 6, we get . This is contradictory. So, .

Secondly, , from the given conditions in Lemma 7, , when , we have

On the other hand, since , such that . Now we claim
In fact, when , (25) holds. Suppose that (25) holds for dose not for . Then
Furthermore, , . But by (22), (24), and the inductive hypothesis, we have
This is a contradiction with (26). So, (25) holds. Whereupon,
Therefore, .

#### 3. Main Results

Now, we are in a position to state and prove the main results of this paper.

Theorem 8. *Let be nonexpansive retract (with ) of a real Banach space . Assume that are two uniformly L-Lipschitzian non-self-mappings (with ) and is an asymptotically quasi pseudocontractive type with coefficient numbers satisfying . Suppose that are two bounded sequences; are six number sequences satisfying the following:*(C1)*, ,;*(C2)*, , ;*(C3)*, , **If is arbitrary, then the iterative sequence generated by (12) converges strongly to the fixed point if and only if there exists a strictly increasing function with such that
*

*Proof. **(Adequacy)*. Let
Then there exists such that
From (29), we know that . So, .

Now, from the given conditions and (12), we can let
and . Then
where
So, by Lemma 4,
For the third in (36), we have
where
Substituting (35) into (36), we get
Let , , and
Then (39) becomes
By using , we have
From (40), (41), and the given conditions, we know
Then, . Therefore , when , . Let
So, when , we get
From (44) and the given conditions, we have , . On the other hand, from (43), we have
By Lemma 7, we at last get
for example, .*(Necessity).* Suppose that . Then we can choose an arbitrary continuous strictly increasing function with , such as . We can get .

Because is an asymptotically quasi pseudocontractive type (with ), by (11) in Definition 1, for any , we have
So,
that is, (29) holds. This completes the proof of Theorem 8.

Combining with Theorem 8 and Definition 3, we have some results as follows.

Theorem 9. *Let be nonexpansive retract (with ) of a real Banach space . Assume that are two uniformly L-Lipschitzian non-self-mappings (with ) and is an asymptotically quasi pseudocontractive type with coefficient numbers satisfying . Suppose that are four number sequences satisfying the following:*(C1)*, , ;*(C2)*, .**If is arbitrary, then the iterative sequence generated by (14) converges strongly to the fixed point if and only if there exists a strictly increasing function with such that (29) holds.*

Theorem 10. *Let be nonexpansive retract (with ) of a real Banach space . Assume that are two uniformly L-Lipschitzian non-self-mappings (with ) and is an asymptotically quasi pseudocontractive type with coefficient numbers satisfying . Suppose that are two number sequences satisfying the following:*(C1)*, , ;*(C2)*.**If is arbitrary, then the iterative sequence generated by (15) converges strongly to the fixed point if and only if there exists a strictly increasing function with such that (29) holds.*

Theorem 11. *Let be a nonempty closed convex subset of a real Banach space . Assume that is uniformly L-Lipschitzian self-mappings and asymptotically quasi pseudocontractive type with coefficient numbers satisfying . Suppose that are two number sequences satisfying the following:*(C1)*, , ;*(C2)*. **If is arbitrary, then the iterative sequence generated by (16) converges strongly to the fixed point if and only if there exists a strictly increasing function with such that (29) holds.*

*Remark 12. *Our research and results in this paper have the following several advantaged characteristics. (a) The iterative scheme is the new modified mixed Ishikawa iterative scheme with error on two mappings . (b) The common fixed point is studied. (c) The research object is the very generalized asymptotically quasi pseudocontractive type (with ) non-self-mapping. (d) The tool used by us is the very powerful tool: Lemma 7. So, our results here extend and improve many results of other authors to a certain extent, such as [6, 8, 14–23], and the proof methods are very different from the previous.

#### 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 their many useful comments and suggestions for the improvement of the paper. This work was supported by the National Natural Science Foundations of China (Grant no. 11271330) and the Natural Science Foundations of Zhejiang Province (Grant no. Y6100696).

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#### Copyright

Copyright © 2014 Yuanheng Wang and Huimin Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.