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# The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Zhiqun Xue1*, Guiwen Lv1 and BE Rhoades2

Author Affiliations

1 Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, P.R. China

2 Department of Mathematics, Indiana University, Bloomington, IN, 47405-7106, USA

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Fixed Point Theory and Applications 2012, 2012:206  doi:10.1186/1687-1812-2012-206

 Received: 11 May 2012 Accepted: 29 October 2012 Published: 22 November 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and be a generalized Lipschitz Φ-hemi-contractive mapping with . Let , , , be four real sequences in and satisfy the conditions (i) as and ; (ii) . For some , let , be any bounded sequences in D, and be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping.

MSC: 47H10.

##### Keywords:
generalized Lipschitz mapping; Φ-hemi-contractive mapping; Ishikawa iterative sequence with errors; uniformly smooth real Banach space

### 1 Introduction and preliminary

Let E be a real Banach space and be its dual space. The normalized duality mapping is defined by

where denotes the generalized duality pairing. It is well known that

(i) If E is a smooth Banach space, then the mapping J is single-valued;

(ii) for all and ;

(iii) If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the single-valued normalized duality mapping by j.

Definition 1.1 ([1])

Let D be a nonempty closed convex subset of E, be a mapping.

(1) T is called strongly pseudocontractive if there is a constant such that for all ,

(2) T is called ϕ-strongly pseudocontractive if for all , there exist and a strictly increasing continuous function with such that

(3) T is called Φ-pseudocontractive if for all , there exist and a strictly increasing continuous function with such that

It is obvious that Φ-pseudocontractive mappings not only include ϕ-strongly pseudocontractive mappings, but also strongly pseudocontractive mappings.

Definition 1.2 ([1])

Let be a mapping and .

(1) T is called ϕ-strongly-hemi-pseudocontractive if for all , , there exist and a strictly increasing continuous function with such that

(2) T is called Φ-hemi-pseudocontractive if for all , , there exist and the strictly increasing continuous function with such that

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

Definition 1.3 ([1])

Let . The mapping is called strongly quasi-accretive if for all , , there exist and a constant such that ; T is called ϕ-strongly quasi-accretive if for all , , there exist and a strictly increasing continuous function with such that ; T is called Φ-quasi-accretive if for all , , there exist and a strictly increasing continuous function with such that .

Definition 1.4 ([2])

For arbitrary given , the Ishikawa iterative process with errors is defined by

(1.1)

where , are any bounded sequences in D; , , , are four real sequences in and satisfy , , for all . If , then the sequence defined by

(1.2)

is called the Mann iterative process with errors.

Definition 1.5 ([3,4])

A mapping is called generalized Lipschitz if there exists a constant such that , .

The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φ-hemi-contractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.

Lemma 1.6 ([5])

LetEbe a uniformly smooth real Banach space, and letbe a normalized duality mapping. Then

(1.3)

for all.

Lemma 1.7 ([6])

Letbe a nonnegative sequence which satisfies the following inequality:

(1.4)

wherewith, . Thenas.

### 2 Main results

Theorem 2.1LetEbe an arbitrary uniformly smooth real Banach space, Dbe a nonempty closed convex subset ofE, andbe a generalized Lipschitz Φ-hemi-contractive mapping with. Let, , , be four real sequences inand satisfy the conditions (i) asand; (ii) . For some, let, be any bounded sequences inD, andbe an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed pointqofT.

Proof Since is a generalized Lipschitz Φ-hemi-contractive mapping, there exists a strictly increasing continuous function with such that

i.e.,

and

for any and .

Step 1. There exists and such that (range of Φ). Indeed, if as , then ; if with , then for , there exists a sequence in D such that as with . Furthermore, we obtain that is bounded. Hence, there exists a natural number such that for , then we redefine and .

Step 2. For any , is bounded. Set , then from Definition 1.2(2), we obtain that . Denote , . Since T is generalized Lipschitz, so T is bounded. We may define . Next, we want to prove that . If , then . Now, assume that it holds for some n, i.e., . We prove that . Suppose it is not the case, then . Since J is uniformly continuous on a bounded subset of E, then for , there exists such that when , . Now, denote

Owing to as , without loss of generality, assume that for any . Since , denote . So, we have

(2.1)

(2.2)

(2.3)

(2.4)

and

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

Therefore,

Using Lemma 1.6 and the above formulas, we obtain

(2.10)

and

(2.11)

Substitute (2.11) into (2.10)

(2.12)

this is a contradiction. Thus, , i.e., is a bounded sequence. So, , , are all bounded sequences.

Step 3. We want to prove as . Set .

By (2.10), (2.11), we have

(2.13)

and

(2.14)

where , and as .

Taking (2.14) into (2.13),

(2.15)

where as .

Set , then . If it is not the case, we assume that . Let , then , i.e., . Thus, from (2.15) it follows that

(2.16)

This implies that

(2.17)

Let , , . Then we get that

Applying Lemma 1.7, we get that as . This is a contradiction and so . Therefore, there exists an infinite subsequence such that as . Since , then as . In view of the strictly increasing continuity of Φ, we have as . Hence, as . Next, we want to prove as . Let , there exists such that , , , , for any . First, we want to prove . Suppose it is not the case, then . Using (1.1), we may get the following estimates:

(2.18)

(2.19)

Since Φ is strictly increasing, then (2.19) leads to . From (2.15), we have

(2.20)

which is a contradiction. Hence, . Suppose that holds. Repeating the above course, we can easily prove that holds. Therefore, for any m, we obtain that , which means as . This completes the proof. □

Theorem 2.2LetEbe an arbitrary uniformly smooth real Banach space, andbe a generalized Lipschitz Φ-quasi-accretive mapping with. Let, , , be four real sequences inand satisfy the conditions (i) asand; (ii) . For some, let, be any bounded sequences inE, andbe an Ishikawa iterative sequence with errors defined by

(2.21)

whereis defined byfor any. Thenconverges strongly to the unique solution of the equation (or the unique fixed point ofS).

Proof Since T is a generalized Lipschitz and Φ-quasi-accretive mapping, it follows that

i.e.,

i.e.,

for all , . The rest of the proof is the same as that of Theorem 2.1. □

Corollary 2.3LetEbe an arbitrary uniformly smooth real Banach space, Dbe a nonempty closed convex subset ofE, andbe a generalized Lipschitz Φ-hemi-contractive mapping with. Let, be two real sequences inand satisfy the conditions (i) asand; (ii) . For some, letbe any bounded sequence inD, andbe the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed pointqofT.

Corollary 2.4LetEbe an arbitrary uniformly smooth real Banach space, andbe a generalized Lipschitz Φ-quasi-accretive mapping with. Let, be two real sequences inand satisfy the conditions (i) asand; (ii) . For some, letbe any bounded sequence inE, andbe the Mann iterative sequence with errors defined by

(2.22)

whereis defined byfor any. Thenconverges strongly to the unique solution of the equation (or the unique fixed point ofS).

Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. This result provided significant improvements of some recent important results. Their result is as follows.

Theorem CC ([[1], Theorem 3.1])

LetEbe a uniformly smooth real Banach space andbe a mapping with. SupposeAis a generalized Lipschitz Φ-quasi-accretive mapping. Let, andbe real sequences insatisfying the following conditions: (i) ; (ii) ; (iii) ; (iv) . Letbe generated iteratively from arbitraryby

(2.23)

whereis defined by, andis an arbitrary bounded sequence inE. Then, there existssuch that if, , the sequenceconverges strongly to the unique solution of the equation.

However, there exists a gap in the proof process of above Theorem CC. Here, () does not hold in line 14 of Claim 2 on page 248, i.e., is a wrong case. For instance, set the iteration parameters: , where , , ;  . Then , but . Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally in writing this paper. All authors read and approved the final manuscript.

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