# Convergence of Iterative Sequences for Common Zero Points of a Family of -Accretive Mappings in Banach Spaces

- Yuan Qing
^{1}, - SunYoung Cho
^{2}Email author and - Xiaolong Qin
^{1}

**2011**:216173

https://doi.org/10.1155/2011/216173

© Yuan Qing et al. 2011

**Received: **21 November 2010

**Accepted: **8 February 2011

**Published: **1 March 2011

## Abstract

## 1. Introduction

where denotes the dual space of and denotes the generalized duality pairing. In the sequel, we denote a single-valued normalized duality mapping by .

*contraction*if there exists a constant such that

*fixed*

*point*of provided . Denote by the set of fixed points of , that is, . Given a real number and a contraction , we define a mapping

We use to denote the unique fixed point of , which yields that . In 1967, Browder [1] proved the following theorem.

Theorem B.

In a Hilbert space, as , converges strongly to a fixed point of , that is, closet to , that is, the nearest point projection of onto .

In [2], Moudafi proposed a viscosity approximation method which was considered by many authors [2–8]. If is a Hilbert space, is a nonexpansive mapping and is a contraction, he proved the following theorems.

Theorem M1.

where is a sequence of positive numbers tending to zero.

Theorem M2.

For each , we denote by the resolvent of , that is, . Note that if is -accretive, then is nonexpansive and , for all . We also denote by the Yosida approximation of , that is, . It is known that is a nonexpansive mapping from to .

where is a real sequence , is a contractive mapping, and is a nonexpansive mapping with a fixed point. Strong convergence theorems of fixed points are obtained in a uniformly smooth Banach space; see [10] for more details.

Very recently, Zegeye and Shahzad [11] studied the common zero problem of a family of -accretive mappings. To be more precise, they proved the following result.

Theorem ZS.

where is a real sequence which satisfies the following conditions: ; ; or and with for for and . If every nonempty, closed, bounded convex subset of has the fixed point property for a nonexpansive mapping, then converges strongly to a common solution of the equations for .

where with for , and is a real sequence in . It is proved that the sequence generated in the iterative algorithms (1.17) and (1.18) converges strongly to a common zero point of a finite family of -accretive mappings in reflexive Banach spaces, respectively.

## 2. Preliminaries

exists for each
in its unit sphere
. It is said to be *uniformly Fréchet differentiable* (and
is said to be *uniformly smooth*) if the limit in (2.1) is attained uniformly for
.

for , , , where , then (see [21]).

*weakly continuous duality map*if there exists a gauge for which the duality map is single valued and

*weak*-to-weak* sequentially continuous (i.e., if is a sequence in weakly convergent to a point , then the sequence converges weakly* to ). It is known that has a weakly continuous duality map for all with the gauge . In the case where for all , we write the associated duality map as and call it the (normalized) duality map. Set

where denotes the subdifferential in the sense of convex analysis. It also follows from (2.5) that is convex and .

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

The first part of the next lemma is an immediate consequence of the subdifferential inequality, and the proof of the second part can be found in [23].

Lemma 2.1.

Assume that has a weakly continuous duality map with the gauge .

Lemma 2.2 (see [24]).

Let be a Banach space satisfying a weakly continuous duality map, let be a nonempty, closed, convex subset of , and let be a nonexpansive mapping with a fixed point. Then, is demiclosed at zero, that is, if is a sequence in which converges weakly to and if the sequence converges strongly to zero, then .

Lemma 2.3 (see [11]).

Let be a nonempty, closed, convex subset of a strictly convex Banach space . Let , , be a family of -accretive mappings such that . Let be real numbers in such that and , where . Then, is nonexpansive and .

Lemma 2.4 (see [25]).

## 3. Main Results

Theorem 3.1.

Proof.

It follows from Lemma 2.1 that is a positive-scalar multiple of . We, therefore, obtain that is a solution to (3.1).

which guarantees . So, (3.1) can have at most one solution. This completes the proof.

We need the strong convergence of the implicit algorithm (1.17) to prove the strong convergence of the explicit algorithm (3.14).

Theorem 3.2.

Let be a strictly convex and reflexive Banach space which has a weakly continuous duality map with the gauge . Lek be a nonempty, closed, convex subset of and a contractive mapping. Let be a family of -accretive mappings with . Let for each . For any , let be generated by the algorithm (1.18), where with for , , and is a sequence in which satisfies the following conditions: and . Assume also that

(ii) converges strongly to , where is the sequence generated by the implicity algorithm (1.17).

Then, converges strongly to , which solves the variational inequality (3.1).

Proof.

where is a appropriate constant such that . In view of Lemma 2.4, we can obtain the desired conclusion easily. This completes the proof.

As an application of Theorems 3.1 and 3.2, we have the following results for a single mapping.

Corollary 3.3.

Then, converges strongly to a solution of the equations .

Corollary 3.4.

where is a sequence in which satisfies the following conditions: and . Also assume that

(ii) converges strongly to , where is the sequence generated by the implicity scheme (3.27) and .

## Authors’ Affiliations

## References

- Browder FE:
**Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces.***Archive for Rational Mechanics and Analysis*1967,**24:**82–90.MATHMathSciNetView ArticleGoogle Scholar - Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MATHMathSciNetView ArticleGoogle Scholar - Cho YJ, Qin X:
**Viscosity approximation methods for a family of -accretive mappings in reflexive Banach spaces.***Positivity*2008,**12**(3):483–494. 10.1007/s11117-007-2181-8MATHMathSciNetView ArticleGoogle Scholar - Cho YJ, Kang SM, Qin X:
**Some results on -strictly pseudo-contractive mappings in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):1956–1964. 10.1016/j.na.2008.02.094MATHMathSciNetView ArticleGoogle Scholar - Cho YJ, Kang SM, Qin X:
**Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces.***Computers & Mathematics with Applications*2008,**56**(8):2058–2064. 10.1016/j.camwa.2008.03.035MATHMathSciNetView ArticleGoogle Scholar - Ceng L-C, Xu H-K, Yao J-C:
**The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(4):1402–1412. 10.1016/j.na.2007.06.040MATHMathSciNetView ArticleGoogle Scholar - Qin X, Cho YJ, Kang SM:
**Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(1):99–112. 10.1016/j.na.2009.06.042MATHMathSciNetView ArticleGoogle 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.2004.04.059MATHMathSciNetView ArticleGoogle 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.2004.11.011MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**Strong convergence of an iterative method for nonexpansive and accretive operators.***Journal of Mathematical Analysis and Applications*2006,**314**(2):631–643. 10.1016/j.jmaa.2005.04.082MATHMathSciNetView ArticleGoogle 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.2006.01.012MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Zhou H, Cho YJ, Kang SM:
**Zeros and mapping theorems for perturbations of -accretive operators in Banach spaces.***Computers & Mathematics with Applications*2005,**49**(1):147–155. 10.1016/j.camwa.2005.01.012MATHMathSciNetView ArticleGoogle Scholar - Dominguez Benavides T, Lopez Acedo G, Xu H-K:
**Iterative solutions for zeros of accretive operators.***Mathematische Nachrichten*2003,**248/249:**62–71. 10.1002/mana.200310003MathSciNetView ArticleMATHGoogle Scholar - Kirk WA:
**On successive approximations for nonexpansive mappings in Banach spaces.***Glasgow Mathematical Journal*1971,**12:**6–9. 10.1017/S0017089500001063MATHMathSciNetView ArticleGoogle Scholar - Liu G, Lei D, Li S:
**Approximating fixed points of nonexpansive mappings.***International Journal of Mathematics and Mathematical Sciences*2000,**24**(3):173–177. 10.1155/S0161171200003252MATHMathSciNetView ArticleGoogle Scholar - Maiti M, Saha B:
**Approximating fixed points of nonexpansive and generalized nonexpansive mappings.***International Journal of Mathematics and Mathematical Sciences*1993,**16**(1):81–86. 10.1155/S0161171293000092MATHMathSciNetView ArticleGoogle Scholar - Park S:
**Fixed point theorems in locally -convex spaces.***Nonlinear Analysis: Theory, Methods & Applications*2002,**48**(6):869–879. 10.1016/S0362-546X(00)00220-0MATHMathSciNetView ArticleGoogle Scholar - Park S:
**Fixed point theory of multimaps in abstract convex uniform spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(7–8):2468–2480. 10.1016/j.na.2009.01.081MATHView ArticleMathSciNetGoogle Scholar - Qin X, Su Y:
**Approximation of a zero point of accretive operator in Banach spaces.***Journal of Mathematical Analysis and Applications*2007,**329**(1):415–424. 10.1016/j.jmaa.2006.06.067MATHMathSciNetView ArticleGoogle Scholar - Yao Y, Liou Y-C:
**Strong convergence to common fixed points of a finite family of asymptotically nonexpansive map.***Taiwanese Journal of Mathematics*2007,**11**(3):849–865.MATHMathSciNetGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point theory and Its Applications*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Browder FE:
**Convergence theorems for sequences of nonlinear operators in Banach spaces.***Mathematische Zeitschrift*1967,**100:**201–225. 10.1007/BF01109805MATHMathSciNetView ArticleGoogle Scholar - Xu ZB, Roach GF:
**Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces.***Journal of Mathematical Analysis and Applications*1991,**157**(1):189–210. 10.1016/0022-247X(91)90144-OMATHMathSciNetView ArticleGoogle Scholar - Lim T-C, Xu HK:
**Fixed point theorems for asymptotically nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*1994,**22**(11):1345–1355. 10.1016/0362-546X(94)90116-3MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar

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