Research Article | Open Access

# The Equivalence of Convergence Results between Mann and Multistep Iterations with Errors for Uniformly Continuous Generalized Weak -Pseudocontractive Mappings in Banach Spaces

**Academic Editor:**Jen-Chih Yao

#### Abstract

We prove the equivalence of the convergence of the Mann and multistep iterations with errors for uniformly continuous generalized weak -pseduocontractive mappings in Banach spaces. We also obtain the convergence results of Mann and multistep iterations with errors. Our results extend and improve the corresponding results.

#### 1. Introduction

Let be a real Banach space, be its dual space, and be the normalized duality mapping defined by where denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by .

*Definition 1.1. * A mapping is said to be(1) strongly accretive if for all , there exist a constant and such that
(2)strongly accretive if there exist and a strictly increasing function with such that
(3) generalized accretive if, for any , there exist and a strictly increasing function with such that

*Remark 1.2. *Let . If in the formulas of Definition 1.1 is replaced by , , then is called strongly quasi-accretive, strongly quasi-accretive, generalized quasi-accretive mapping, respectively.

Closely related to the class of accretive-type mappings are those of pseudocontractive type mappings.

*Definition 1.3. * A mapping with domain and range is said to be(1) strongly pseudocontractive if there exist a constant and such that for each ,
(2)strongly pseudocontractive if there exist and a strictly increasing function with such that
(3) generalized pseudocontractive if, for any , there exist and a strictly increasing function with such that

*Definition 1.4. *Let . The mapping is called strongly pseudocontractive, generalized -pseudocontractive, if, for all , , the formula (2), (3) in the above Definition 1.3 hold.

*Definition 1.5. *A mapping is said to be(1) generalized weak accretive if, for all , there exist and a strictly increasing function with such that
(2) generalized weak quasi-accretive if, for all , , there exist and a strictly increasing function with such that
(3) generalized weak pseudocontractive if, for any , there exist and a strictly increasing function with such that
(4) generalized weak hemicontractive if, for any , , there exist and a strictly increasing function with such that

It is very well known that a mapping is strongly pseudocontractive (hemicontractive), strongly pseudocontractive (strongly hemicontractive), generalized -pseudocontractive (generalized -hemicontractive), generalized weak -pseudocontractive (generalized weak -hemicontractive) if and only if are strongly accretive (quasi-accretive), strongly accretive (strongly quasi-accretive), is generalized accretive (generalized quasi-accretive), generalized weak accretive (weak quasi-accretive), respectively.

It is shown in [1] that the class of strongly pseudocontractive mappings is a proper subclass of -strongly pseudocontractive mappings. Furthermore, an example in [2] shows that the class of -strongly hemicontractive mappings with the nonempty fixed point set is a proper subclass of generalized -hemicontractive mappings. Obviously, generalized -hemicontractive mapping must be generalized weak -hemicontractive, but, on the contrary, it is not true. We have the following example.

*Example 1.6. *Let be real number space with usual norm and . defined by
Then has a fixed point . defined by is a strictly increasing function with . For all and , we have
Then is a generalized weak -hemicontractive map, but it is not a generalized -hemicontractive map; that is, the class of generalized weak -hemicontractive maps properly contains the class of generalized -hemicontractive maps. Hence the class of generalized weak -hemicontractive mappings is the most general among those defined above.

*Definition 1.7. *The mapping is called Lipschitz, if there exists a constant such that
It is clear that if is Lipschitz, then it must be uniformly continuous. Otherwise, it is not true. For example, the function is uniformly continuous but it is not Lipschitz.

Now let us consider the multi-step iteration with errors. Let be a nonempty convex subset of , and let be a finite family of self-maps of . For , the sequence is generated as follows: where are any bounded sequences in and are sequences in satisfying certain conditions.

If , (1.15) becomes the Ishikawa iteration sequence with errors defined iteratively by

If , for all , then from (1.16), we get the Mann iteration sequence with errors defined by where is bounded.

Recently, many authors have researched the iteration approximation of fixed points by Lipschitz pseudocontractive (accretive) type nonlinear mappings and have obtained some excellent results [3â€“12]. In this paper we prove the equivalence between the Mann and multi-step iterations with errors for uniformly continuous generalized weak -pseduocontractive mappings in Banach spaces. Our results extend and improve the corresponding results [3â€“12].

Lemma 1.8 (see [13]). *Let be a real normed space. Then, for all , the following inequality holds:
*

Lemma 1.9 (see [14]). * Let be nonnegative sequence which satisfies the following inequality:
**
where and . Then as .*

Lemma 1.10. *Let and be four nonnegative real sequences satisfying the following conditions: (i) ; (ii) ; (iii) . Let be a strictly increasing and continuous function with such that
**
If is bounded, then as .*

*Proof. *Since is bounded, we set , , then . Otherwise, we assume that , then there exists a constant with and a natural number such that
for .

Then, from (1.20), we get
Since , there exists a nature number , such that . Hence and (1.22) becomes
By Lemma 1.9, we obtain that as . Since is strictly increasing and continuous with . Hence , which is contradicting with the assumption . Then , there exists a subsequence of such that as . Let be any given. Since , , then there exists a natural number , such that
for all . Next, we will show that for all . First, we want to prove that . Suppose that it is not the case, then . Since is strictly increasing,
From (1.24) and (1.25), we obtain that
That is , which is a contradiction. Hence . Now we assume that holds. Using the similar way, it follows that . Therefore, this shows that as .

#### 2. Main Results

Theorem 2.1. *Let be a nonempty closed convex subset of a Banach space . Suppose that , and are uniformly continuous generalized weak hemicontractive mappings with . Let be a sequence in defined iteratively from some by (1.17), where is an arbitrary bounded sequence in and are two sequences in satisfying the following conditions: (i) , (ii) , (iii) , (iv) . Then the iteration sequence converges strongly to the unique fixed point of .*

*Proof. *Since , set . Since the mapping are generalized weak hemicontractive mappings, there exist strictly increasing functions with and such that

Firstly, we claim that there exists with such that . In fact, if , then we have done. Otherwise, there exists the smallest positive integer such that . We denote , then we will obtain that . Indeed, if , then . If with , then for , there exists a sequence such that as with , and we also obtain that the sequence is bounded. So there exists such that for , then we redefine , let .

Next we shall prove for . Clearly, holds. Suppose that , for some , then we want to prove . If it is not the case, then . Since is a uniformly continuous mapping, setting , there exists such that , whenever ; and are bounded operators, set . Since , , without loss of generality, let

From (1.17), we have

Since are uniformly continuous mappings, so .

Applying Lemma 1.8, the recursion (1.17), and the above inequalities, we obtain
Inequality (2.5) implies
which is a contradiction with the assumption . Then ; that is, the sequence is bounded. Let . From (2.4), we have
that is, . Since is on uniformly continuous, so
Again using (2.5), we have
where
By (2.8), the conditions (iii) and (iv), we get . So applying Lemma 1.10 on (2.9), we obtain .

Theorem 2.2. * Let be a Banach space and be a nonempty closed convex subset of , are as in Theorem 2.1. For , the sequence iterations are defined by (1.15) and (1.17), respectively. are sequences in satisfying the following conditions:*(i)*;
*(ii)*;
*(iii)*;
*(iv)*;
*(v)*. **Then the following two assertions are equivalent:*(I)*the iteration sequence strongly converges to the common point of ;*(II)*the sequence iteration strongly converges to the common point of .*

*Proof. *Since , set . If the iteration sequence strongly converges to , then setting , we obtain the convergence of the iteration sequence . Conversely, we only prove that (II)(I). The proof is divided into two parts.*Step 1*. We show that is bounded.

By the proof method of Theorem 2.1, there exists with such that . Setting , we have . Set , . Since are bounded mappings and are some bounded sequences in , we can set . Since are uniformly continuous mappings, given , such that whenever ,â€‰for all . Now, we define . Since the control conditions (iii)-(iv), without loss of generality, we let .

Now we claim that if , then .

From (1.15), we obtain that
we also obtain that

Now we suppose that holds. We will prove that . If it is not the case, we assume that . From (1.15), we obtain that
Consequently, by (2.11) and (2.12), we obtain
Since are uniformly continuous mappings, we get
Using (2.1), Lemma 1.8, and the recursion formula (1.15), we have
Which implies
which is a contradiction with the assumption , then ; that is, the sequence is bounded. Since , as , so the sequence is bounded.*Step **2*. We prove .

Since is bounded, again applying (2.11) and (2.12), we get the boundedness of . Since are bounded mappings, set , . From (1.15) and (1.17), we obtain

By the conditions (iii)â€“(v), we have

Since , so
That is:

By the uniform continuity of , we obtain

From (2.23) and the conditions (iii) and (v), (2.18) becomes

By Lemma 1.10, we get . Since , and the inequality , so .

From Theorems 2.1 and 2.2, we can obtain the following corollary.

Corollary 2.3. *Let be a Banach space and be a nonempty closed convex subset of , are as in Theorem 2.1. For , the sequence iterations is defined by (1.15). are sequences in satisfying the following conditions:*(i)*;
*(ii)*;
*(iii)*;
*(iv)*;
*(v)*. **Then the iteration sequence strongly converges to the common point of , .*

Corollary 2.4. *Let , are uniformly continuous generalized weak quasi-accretive mappings. Suppose , that is, there exists . Let be sequences in satisfying the following conditions:*(i)*;
*(ii)*;
*(iii)*;
*(iv)*;
*(v)*. **Let the sequence in be generated iteratively from some by
**
where for all and are any bounded sequences in .**Then defined by (2.25) converges strongly to .*

*Proof. * We simply observe that are uniformly continuous generalized weak hemicontractive mappings. The result follows from Corollary 2.3.

#### Acknowledgments

This project was supported by Hebei Province Natural Science Foundation (A2011210033); Shijiazhuang Tiedao University Foundation (Q64).

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#### Copyright

Copyright © 2012 Guiwen Lv and Haiyun Zhou. 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.