- Research Article
- Open Access

# Strong Convergence of a New Iteration for a Finite Family of Accretive Operators

- Liang-Gen Hu
^{1}Email author and - Jin-Ping Wang
^{1}

**2009**:491583

https://doi.org/10.1155/2009/491583

© L.-G. Hu and J.-P.Wang. 2009

**Received:**9 March 2009**Accepted:**17 May 2009**Published:**16 June 2009

## Abstract

The viscosity approximation methods are employed to establish strong convergence of the modified Mann iteration scheme to a common zero of a finite family of accretive operators on a strictly convex Banach space with uniformly Gâteaux differentiable norm. Our work improves and extends various results existing in the current literature.

## Keywords

- Banach Space
- Nonexpansive Mapping
- Strong Convergence
- Nonempty Closed Convex Subset
- Convex Banach Space

## 1. Introduction

*nonexpansive*if , for all . A mapping is called

*-contraction*if there exists a constant such that

In the last ten years, many papers have been written on the approximation of fixed point for nonlinear mappings by using some iterative processes (see, e.g.,[1–20]).

An operator
is said to be *accretive* if
, for all
,
and
. If
is accretive and
is identity mapping, then we define, for each
, a nonexpansive single-valued mapping
by
, which is called the *resolvent* of
. we also know that for an accretive operator
,
, where
and
. An accretive operator
is said to be
*-accretive*, if
for all
. If
is a Hilbert space, then accretive operator is monotone operator. There are many papers throughout literature dealing with the solution of
(
) by utilizing certain iterative sequence (see [1–3, 8–10, 13, 16, 20]).

*Let*

*be a nonempty closed convex subset of*

*. For any*

*, the sequence*

*is generated by*

*where*
*and*
, *for some*
, *satisfy the following conditions*:

(C1) ,

(C2) ,

(C3) , and

(C4) .

They proved that the iterative sequence converges strongly to a zero of .

*Let*

*be a nonempty closed convex subset of*

*. For any*

*, the sequence*

*is generated by*

*where *
*with*
,
, *for*
,
, *and *
*satisfies the conditions*: (C1), (C2), (C3), or (
).
).

*Let*

*be a nonempty closed convex subset of*

*. For any*

*, the sequence*

*is generated by*

They proved that the sequence converges strongly to a common zero of .

*Let*

*be a nonempty closed convex subset of*

*and*

*a*

*-contraction. For any*

*, the sequence*

*is defined by*

*where*
*with*
, *for*
,
*and*
,
*and*
. The iterative sequence (1.7) is a natural generalization of all the above mentioned iterative sequences.

(i)In contrast to the iterations (1.3)–(1.5), the convex composition of the iteration (1.7) deals with only instead of and .

converges weakly to a zero of . However, the Mann iteration scheme has only weak convergence for nonexpansive mappings even in a Hilbert space (see [4]).

Our main purpose is to prove strong convergence theorems for a finite family of accretive operators on a strictly convex Banach space with uniformly G teaux differentiable norm by using viscosity approximation methods. Our theorems extend the comparable results in the following three aspects.

(1)In contrast to weak convergence results on a Hilbert Space in [9], strong convergence of the iterative sequence is obtained in the general setup of a Banach space.

(2)The restrictions (C3), ( ), and (C4) on the results in [10, 20] are dropped.

(3)A single mapping of the results in [3] is replaced by a finite family of mappings.

## 2. Preliminaries and Lemmas

exists for each
, where
. The norm of
is *uniformly G*
*teaux differentiable * if for each
, the limit is attained uniformly for
. The norm of
is *uniformly Fréchet differentiable* (
is also called *uniformly smooth*) if the limit is attained uniformly for each
. It is well known that if
is uniformly G
teaux differentiable norm, then the duality mapping
is single-valued and
uniformly continuous on each bounded subset of
.

A Banach space
is called *strictly convex* if for
,
, and
, we have
for
,
and
for
. In a strictly convex Banach space
, we have that if
, for
,
,
and
, then
.

Lemma 2.1 (The Resolvent Identity).

for each . In general, we use instead of . Let with , and let be a Banach limit on . Then . Further, we know the following result.

Let be a nonempty closed convex subset of a Banach space with uniformly G teaux differentiable norm. Assume that is a bounded sequence in . Let , and let a Banach limit. Then if and only if , .

Let
be a closed convex and, let
a mapping of
onto
. Then
is said to be *sunny* [12, 13] if
for all
and
. A mapping
of
onto
is said to be *retraction* if
; If a mapping
is a retraction then
for any
, the range of
. A subset
of
is said to be a *sunny nonexpansive retraction* of
if there exists a sunny nonexpansive retraction of
onto
, and it is said to be a *nonexpansive retraction* of
if there exists a nonexpansive retraction of
onto
. In a smooth Banach space
, it is known ([5, Page 48]) that
is a sunny nonexpansive retraction if and only if the following condition holds:
,
and
.

Lemma 2.3 (see [14]).

Then .

Lemma 2.4.

Lemma 2.5 (see [18]).

where is a sequence in and is a sequence in satisfying the following conditions:

(i) ;

(ii) or .

Then .

Lemma 2.6 ([8]).

Let be real numbers in with and , where and . Then is nonexpansive and .

## 3. Main Results

For the sake of convenience, we list the assumptions to be used in this paper as follows.

(i) is a strictly convex Banach space which has uniformly G teaux differentiable norm, and is a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings.

(ii)The real sequence satisfies the conditions: (C1). and (C2). .

We will employ the viscosity approximation methods [11, 19] to obtain a strong convergence theorem. The method of proof is closely related to [2, 3, 19].

Theorem 3.1.

Proof.

Thus the net is well defined.

and there exists a subsequence which is still denoted by such that .

for all .

Remark 3.2.

In addition, if
is a uniformly smooth Banach space in Theorem 3.1 and we define
*, * then we obtain from Theorem 3.1 and [19, Theorem 4.1] that the net
converges strongly to
*, * as
*, * where
and
is a sunny nonexpansive retraction of
onto
*.*

Theorem 3.3.

Then the sequence converges strongly to , where is the unique solution of a variational inequality .

Proof.

From (3.47), (3.48), (C1), (C2), and , it follows that and . Consequently applying Lemma 2.5 to (3.50), we conclude that .

If we take , for all , in the iteration (1.7), then, from Theorem 3.3, we have what follows

Corollary 3.4.

where with , for , and . Then the sequence converges strongly to .

Remark 3.5.

Theorem 3.3 and Corollary 3.4 prove strong convergence results of the new iterative sequences which are different from the iterative sequences (1.4) and (1.5). In contrast to [20], the restriction: (C3)*. *
or
is removed*.*

If we consider the case of an accretive operator , then as a direct consequence of Theorem 3.1 and Theorem 3.3, we have the following corollaries.

Corollary 3.6 ([3, Theorem 3.1]).

where . If , for some , then converges strongly to , as , where is the unique solution of a variational inequality:

Corollary 3.7.

where . Then the sequence converges strongly to , where is the unique solution of a variational inequality .

Remark 3.8.

(i)Corollary 3.7 describes strong convergence result in Banach spaces for a modification of Mann iteration scheme in contrast to the weak convergence result on Hilbert spaces given in [9, Theorem 3].

(ii)In contrast to the result [10, Theorem 4.2], the iterative sequence in Corollary 3.7 is different from the iteration (1.3), and the conditions and are not required.

## Declarations

### Acknowledgments

The work was supported partly by NNSF of China (no. 60872095), the NSF of Zhejiang Province (no. Y606093), K. C. Wong Magna Fund of NIngbo University, NIngbo Natural Science Foundation (no. 2008A610018), and Subject Foundation of Ningbo University (no. XK109050).

## Authors’ Affiliations

## References

- Bruck RE Jr.:
**A strongly convergent iterative solution of for a maximal monotone operator in Hilbert space.***Journal of Mathematical Analysis and Applications*1974,**48:**114–126. 10.1016/0022-247X(74)90219-4MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Khan AR, Ansari QH, Yao J-C:
**Strong convergence of composite iterative schemes for zeros of -accretive operators in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):1830–1840. 10.1016/j.na.2008.02.083MathSciNetView ArticleMATHGoogle Scholar - Chen R, Zhu Z:
**Viscosity approximation method for accretive operator in Banach space.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(4):1356–1363. 10.1016/j.na.2007.06.033MathSciNetView ArticleMATHGoogle Scholar - Genel A, Lindenstrauss J:
**An example concerning fixed points.***Israel Journal of Mathematics*1975,**22**(1):81–86. 10.1007/BF02757276MathSciNetView ArticleMATHGoogle Scholar - Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 83*. Marcel Dekker, New York, NY, USA; 1984:ix+170.Google Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Hu L-G:
**Strong convergence of a modified Halpern's iteration for nonexpansive mappings.***Fixed Point Theory and Applications*2008, Article ID 649162,**2008:**-9.Google Scholar - Hu L-G, Liu L:
**A new iterative algorithm for common solutions of a finite family of accretive operators.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(6):2344–2351. 10.1016/j.na.2008.03.016MathSciNetView ArticleMATHGoogle Scholar - Kamimura S, Takahashi W:
**Approximating solutions of maximal monotone operators in Hilbert spaces.***Journal of Approximation Theory*2000,**106**(2):226–240. 10.1006/jath.2000.3493MathSciNetView ArticleMATHGoogle Scholar - Kim T-H, Xu H-K:
**Strong convergence of modified Mann iterations.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(1–2):51–60. 10.1016/j.na.2004.11.011MathSciNetView ArticleMATHGoogle Scholar - Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Asymptotic behavior of contractions in Banach spaces.***Journal of Mathematical Analysis and Applications*1973,**44:**57–70. 10.1016/0022-247X(73)90024-3MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Strong convergence theorems for resolvents of accretive operators in Banach spaces.***Journal of Mathematical Analysis and Applications*1980,**75**(1):287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces.***Fixed Point Theory and Applications*2005,**2005**(1):103–123. 10.1155/FPTA.2005.103View ArticleMathSciNetMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Application*. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Takahashi W, Ueda Y:
**On Reich's strong convergence theorems for resolvents of accretive operators.***Journal of Mathematical Analysis and Applications*1984,**104**(2):546–553. 10.1016/0022-247X(84)90019-2MathSciNetView ArticleMATHGoogle Scholar - Wang C, Zhu J:
**Convergence theorems for common fixed points of nonself asymptotically quasi-non-expansive mappings.***Fixed Point Theory and Applications*2008, Article ID 428241,**2008:**-11.Google Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Zegeye H, Shahzad N:
**Strong convergence theorems for a common zero for a finite family of -accretive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(5):1161–1169. 10.1016/j.na.2006.01.012MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.