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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 651304, 11 pages

http://dx.doi.org/10.1155/2012/651304

## Strong Convergence Theorems for Asymptotically Weak -Pseudo--Contractive Non-Self-Mappings with the Generalized Projection in Banach Spaces

Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China

Received 28 August 2012; Revised 7 October 2012; Accepted 9 October 2012

Academic Editor: Xiaolong Qin

Copyright © 2012 Yuanheng Wang. 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.

#### Abstract

A new concept of the asymptotically weak -pseudo--contractive non-self-mapping is introduced and some strong convergence theorems for the mapping are proved by using the generalized projection method combined with the modified successive approximation method or with the modified Mann iterative sequence method in a uniformly and smooth Banach space. The proof methods are also different from some past common methods.

#### 1. Introduction

Let be a real Banach space with the norm , its dual space with the norm . As usually, we introduce a dual product in by , where and . Let be the normalized duality mapping in defined as It is clear that the operator is well defined in a Banach space by the famous Hahn-Banach theorem.

The concept of asymptotically nonexpansive mappings was first introduced by Goebel and Kirk [1] in 1972 and then Schu [2] introduced the asymptotically pseudocontractive mappings in 1991.

*Definition 1.1. *Let be a nonempty subset of a real Banach space and be a mapping. (1)The mapping is said to be asymptotically nonexpansive, if there exists a number sequence in with such that
for all and . (2)The mapping is said to be asymptotically pseudocontractive, if for all , there exists a number sequence in with and such that
(3)The mapping is said to be asymptotically demi-pseudocontractive, if for all , , there exists a number sequence in with and such that
where is the set of all fixed points of the mapping .

The iterative approximation problems for asymptotically nonexpansive and pseudocontractive mapping were studied by many authors and we always assume that the fixed point set of the operator is nonempty, such as see [1–6]. In 2011, Qin et al. [7] introduced a new concept of the asymptotically strict quasi--pseudocontractive mapping . They combined the generalized projection to give a new iterative sequence for the and proved that the sequence converges strongly to a point .

But, all these arguments are not enough if the operator acts from to , which we called non-self-mappings, and the iterative methods we used to be, such as Mann iterative method and its some modifications, can not be used. Under this condition, it is natural for us to try to consider the metric projection operator and the generalized projection operator , and some authors have given relevant results and applications of the operator and (see [8–11]).

Very recently, in 2012, Yao et al.[12] and Liou et al. [13] considered the non-self-mapping in a Hilbert space . They also proved their new iterative sequence for the combined with the metric projection converges strongly to a point and the unique solution of a variational inequality, respectively.

Motivated and inspired by the said above, we first introduce a new concept of the asymptotically weak -pseudo--contractive non-self-mapping . Then, in a uniformly convexand smooth Banach space, we prove some strong convergence theorems for the mapping by using the generalized projection method and the modified successive approximation method or the modified Mann iterative sequence method where is a sunny nonexpansive retraction. So, in some ways, our results extend and improve some results of other authors (such as, see [1–5, 7, 9–13]), from self mappings to non-self-mappings, from Hilbert spaces to Banach spaces.

#### 2. Preliminaries

In the sequel, we will assume that is a real uniformly convex and uniformly smooth (hence reflexive) Banach space, then will be the same. If we denote by the modulus of convexity of the Banach space and by its modulus of smoothness, then are all continuous and increasing on their domains, respectively, and (see [9]). Also, under the conditions the normalized duality operator is single-valued, strictly monotone, continuous, coercive, bounded, and homogeneous, but not addible. In a Hilbert space, is the Identity operator .

*Definition 2.1 (see [10, 11]). *The operator is called metric projection operator if it assigns to each its nearest point , that is, the solution for the minimization problem

The operator is called the generalized projection operator if it assigns to each a minimum point of the Lapunov function : that is, a solution of the following minimization problem:

Lemma 2.2 (see [10, 11]). *The point is the metric projection of on if and only if the following inequality is satisfied:
**
and the operator is nonexpansive in Hilbert spaces. ** The point is the generalized projection of on if and only if the following inequality is satisfied:
**
Furthermore, the inequality below also holds:
**
And thus, we have
*

Lemma 2.3 (see [8]). *For all , if and , then the following inequality is satisfied:
**
where is a constant. *

In general, the operator and are not nonexpansive in Banach spaces. It is easy to see in Hilbert spaces because of . In a uniformly convex and uniformly smooth Banach space, is well defined on a closed convex set and is also well defined on a closed convex set from the properties of the functional and strict monotonicity of the mapping . More properties of the mappings , , , and and some of their applications can be found in [8–11].

*Definition 2.4 (see [14]). *Let be a real Banach space, be a subset. The operator is called sunny nonexpansive retract if is nonexpansive, , and for any , , holds .

If is a uniformly smooth Banach space and , is a closed convex set, then the unique sunny nonexpansive retract exists.

*Definition 2.5. *Let be a real Banach space, be a nonempty subset of , and be a non-self-mapping. If there exists a sequence in with and a continuous increasing function for all with , , it is shown as follows, respectively:(1)The mapping is said to be asymptotically weak --contractive mapping, if
(2)The mapping is said to be asymptotically weak -quasi--contractive mapping, if
for all , and , where .(3)The mapping is said to be asymptotically weak --pseudocontractive mapping, if
for all .(4)The mapping is said to be asymptotically weak -quasi--pseudocontractive mapping, if
for all , , and , where .

*Remark 2.6. *It is clear that one can omit each operator in (2.11)–(2.14) if the mapping acts from to , that is, is a self-mapping. So, the class of asymptotically weak --contractive mappings contains that of asymptotically nonexpansive mappings and the class of asymptotically weak --pseudocontractive mappings contains that of asymptotically pseudocontractive mappings. Therefore, all the results and applications of asymptotically nonexpansive mappings can be as a part of the asymptotically weak --contractive mappings.

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

Lemma 2.7 (see [15]). *Let , , , and be sequences of nonnegative numbers satisfying the following conditions:
**
Suppose the following recurse inequality holds:
**
where is a continuous strictly increasing function for all with , . Then as .*

Lemma 2.8 (see [16]). * Let be a real Banach space and be the normalized duality mapping. Then
**
for all and .*

#### 3. Main Results

Theorem 3.1. * Let be a uniformly convex and uniformly smooth Banach space, be a closed convex subset of , be an asymptotically weak G-quasi--contractive mapping with a sequence , , and is its fixed point. Then the iterative sequence generated by the modified successive approximation method (1.5) is bounded for all and converges strongly to .*

* Proof. *If is the fixed point of in , that is, , then we get by (1.5) and (2.8) in Lemma 2.2 for ,
We use the condition (2.12) of asymptotically weak -quasi--contractive of the operator and get

Because , we know constant and is bounded, say by for all .

It is obviously that satisfies the inequality
Therefore by (3.2) and (3.3), we have
for all , that is, the sequence is bounded.

The sequence of positive number defined by are bounded and from (3.2) it satisfies the following inequality:
where , , , , . So, using Lemma 2.7 we get

Because is a constant and , we obtain from the left part of the estimate of (2.10) in Lemma 2.3 the following:
By the properties of , this implies
that is, the sequence converges strongly to fixed point .

Corollary 3.2. *Let be a closed convex set in , be an asymptotically weak G--contractive mapping with a sequence , , and its fixed point. Then the iterative sequence defined by modified successive approximation method (1.5) converges strongly to .*

* Proof. * If we take as the fixed point of , then we have

So the asymptotically weak --contractive mapping is also an asymptotically weak -quasi--contractive mapping and the results of Theorem 3.1 still hold.

Theorem 3.3. * Let be a real uniformly convex and uniformly smooth Banach space, be a nonempty closed convex subset of , be an asymptotically weak G-quasi--pseudocontractive mapping with a sequence , , and its fixed point. Consider the iterative sequence defined by the modified Mann iterative sequence method (1.6). Suppose the sequence and are bounded, is a number sequence in satisfing the conditions below:
**
where is a sunny nonexpansive retraction. Then the iterative sequence converges strongly to .*

* Proof. *By the virtue of (2.17) in Lemma 2.8, it follows that
Since is bounded, say by , we have

From (3.10) we know and then one gets By using the uniform continuity of in the uniformly convex and uniform smooth Banach space and the bound of the sequence , we have Substituting (3.14) into (3.11) and using (2.14), we get

Thus, the sequence of positive number defined by satisfies the recursive inequality where , , , as . Therefore by the virtue of Lemma 2.7, it is clear that the assertion as holds, that is,

Corollary 3.4. *Let be a real uniformly convex and uniformly smooth Banach space, be a nonempty closed convex subset of , be an asymptotically weak G--pseudocontractive mapping with a sequence , , and its fixed point. Consider the iterative sequence defined by the modified Mann iterative sequence (1.6). Suppose the sequence and are bounded, is a real number sequence in satisfying the conditions (3.10). Then one has
**
and the iterative sequence converges in norm to .*

* Proof. *Following Theorem 3.3, we can have the assertions of the corollary.

*Remark 3.5. *Because a Hilbert space must be a uniformly convex and uniformly smooth Banach space, the above results still hold in a Hilbert space. In fact, if we notice in Hilbert spaces, we can abate some conditions in Corollary 3.4 and have the following theorem.

Theorem 3.6. * Let be a closed convex set of a Hilbert space . is said to be an asymptotically weak G-quasi--pseudocontractive mapping with a sequence , , and its fixed point, if
**
where is a continuous increasing function for all with , . Consider the new modified Mann iterative sequence defined by the modified Mann iterative sequence (1.6). If the number sequence satisfies the conditions
**
then the iterative sequence converges strongly to .*

* Proof. *Because is nonexpansive in Hilbert spaces, satisfies (3.20) and the operator satisfies (3.19), we get

Denote and we have the following inequality:
where , , , , . Therefore we know as by using Lemma 2.7, that is,

*Remark 3.7. * It is clear that the above results, in some ways, extend and improve some results of other authors (such as, see [1–5, 7, 9–13]), from self mappings to non-self-mappings, from Hilbert spaces to Banach spaces. And in the proof process, our methods are different from some past common methods.

#### Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and the National Natural Science Foundation of China (11071169, 11271330).

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