# Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Mappings

- L. C. Ceng
^{1}, - David S. Shyu
^{2}and - J. C. Yao
^{3}Email author

**2009**:402602

**DOI: **10.1155/2009/402602

© L. C. Ceng et al. 2009

**Received: **26 November 2008

**Accepted: **28 May 2009

**Published: **16 June 2009

## Abstract

We propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established.

## 1. Introduction and Preliminaries

where is the generalized duality pairing between and . If is smooth, then is single valued and continuous from the norm topology of to the weak* topology of .

for all and all . In (1.2) and (1.3), the positive number is said to be a strictly pseudocontractive constant.

and hence where . It is clear that in Hilbert spaces the important class of nonexpansive mappings (mappings for which ) is a subclass of the class of strictly pseudocontractive maps.

where . Moreover, they proved the following convergence theorem in a Hilbert space.

Theorem 1.1 ([11]).

Let be a Hilbert space, and let be a nonempty closed convex subset of . Let be nonexpansive self-maps of such that where . Let , and let be a sequence in , such that . Then the sequence defined implicitly by (1.6) converges weakly to a common fixed point of the mappings .

Subsequently, Osilike [12] extended their results from nonexpansive mappings to strictly pseudocontractive mappings and derived the following convergence theorems in Hilbert and Banach spaces.

Theorem 1.2 ([12]).

Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where . Let , and let be a sequence in such that . Then the sequence defined by (1.6) converges weakly to a common fixed point of the mappings .

Theorem 1.3 ([12]).

Let be a real Banach space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where , and let be a real sequence satisfying the conditions:

(i)

(ii) ;

(iii) .

Let , and let be defined by (1.6). Then

(i) exists for all ;

(ii) .

where and . First, they established the following convergence theorem.

Theorem 1.4 ([13, Theorem ]).

Let be a real Banach space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where , and let be two real sequences satisfying the conditions:

(i) ;

(ii) ;

(iii) ;

(iv) , where is common Lipschitz constant of .

For , let be defined by (1.8). Then

(i) exists for all ;

(ii) .

Second, they derived the following result by using Theorem 1.4.

Theorem 1.5 ([13, Theorem ]).

Let be a nonempty closed convex subset of a real Banach space , let be a semicompact strictly pseudocontractive self-map of such that , where , and let be a real sequence satisfying the conditions:

(i) ;

(ii) .

converges strongly to a fixed point of .

On the other hand, Zeng and Yao [14] introduced a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive self-maps of a real Hilbert space and established some convergence theorems for this implicit iteration scheme. To be more specific, let be a finite family of nonexpansive self-maps of , and let be a mapping such that for some constants is a -Lipschitz and -strongly monotone mapping. Let and and take a fixed number . The authors proposed the following implicit iteration process with perturbed mapping .

It is clear that if , then the implicit iteration scheme (1.11) with perturbed mapping reduces to the implicit iteration process (1.6).

Theorem 1.6 ([14, Theorem ]).

Let be a real Hilbert space, and let be a mapping such that for some constants ; is -Lipschitz and -strongly monotone. Let be nonexpansive self-maps of such that . Let , let , , and let satisfying the conditions: and , for some . Then the sequence defined by (1.11) converges weakly to a common fixed point of the mappings .

The above Theorem 1.6 extends Theorem 1.1 from the implicit iteration process (1.6) to the implicit iteration scheme (1.11) with perturbed mapping.

for all and all .

Proposition 1.7.

Let be a real Banach space, and let be a mapping:

(i)if is -strictly pseudocontractive, then is a Lipschitz mapping with constant .

(ii)if is both -strictly pseudocontractive and -strongly accretive with , then is nonexpansive.

Proof.

and hence is nonexpansive.

where , and . In particular, whenever , it is easy to see that (1.15) reduces to (1.8).

maps into itself.

Therefore, if , then the composite implicit iteration process (1.15) with perturbed mapping can be employed for the approximation of common fixed points of strictly pseudocontractive self-maps of .

The purpose of this paper is to investigate the problem of approximating common fixed points of strictly pseudocontractive mappings of Browder-Petryshyn in an arbitrary real Banach space by this general implicit iteration process (1.15). To this end, we need the following lemma and definition.

Lemma 1.8 ([8]).

then exists.

The following definition can be found, for example, in [13].

Definition 1.9.

Let be a closed subset of a real Banach space , and let be a mapping. is said to be semicompact if, for any bounded sequence in such that , there must exist a subsequence such that .

## 2. Main Results

We are now in a position to prove our main results in this paper.

Theorem 2.1.

Let be a real Banach space, and let be a nonempty closed convex subset of such that . Let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . Let be strictly pseudocontractive self-maps of such that , where , and let , and be three real sequences in satisfying the conditions:

(i) ;

(ii) ;

(iii) ;

(iv) ;

(v) , where is the common Lipschitz constant of .

where , then

(i) exists for all ;

(ii) .

Proof.

Thus, in terms of Lemma 1.8 we deduce that exists, and hence is bounded.

This completes the proof of Theorem 2.1.

The conclusion of Theorem 2.1 remains valid for the iteration processes (2.33) and (2.34). Furthermore, we have the following result.

Theorem 2.2.

Let be a real Banach space, and let be a nonempty closed convex subset of such that . Let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . Let be a semicompact strictly pseudocontractive self-map of such that , where , and let and be two real sequences in satisfying the conditions:

(i) ;

(ii) ;

(iii) .

Then Mann iteration process (2.34) converges strongly to a fixed point of .

Proof.

This completes the proof of Theorem 2.2.

## Declarations

### Acknowledgments

The first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author was partially supported by the Grant NSF 97-2115-M-110-001

## Authors’ Affiliations

## References

- Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleMATHGoogle Scholar - Hicks TL, Kubicek JD:
**On the Mann iteration process in a Hilbert space.***Journal of Mathematical Analysis and Applications*1977,**59**(3):498–504. 10.1016/0022-247X(77)90076-2MathSciNetView ArticleMATHGoogle Scholar - Măruşter Ş:
**The solution by iteration of nonlinear equations in Hilbert spaces.***Proceedings of the American Mathematical Society*1977,**63**(1):69–73.MathSciNetView ArticleMATHGoogle Scholar - Osilike MO, Udomene A:
**Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type.***Journal of Mathematical Analysis and Applications*2001,**256**(2):431–445. 10.1006/jmaa.2000.7257MathSciNetView ArticleMATHGoogle Scholar - Osilike MO:
**Strong and weak convergence of the Ishikawa iteration method for a class of nonlinear equations.***Bulletin of the Korean Mathematical Society*2000,**37**(1):153–169.MathSciNetMATHGoogle Scholar - Rhoades BE:
**Comments on two fixed point iteration methods.***Journal of Mathematical Analysis and Applications*1976,**56**(3):741–750. 10.1016/0022-247X(76)90038-XMathSciNetView ArticleMATHGoogle Scholar - Rhoades BE:
**Fixed point iterations using infinite matrices.***Transactions of the American Mathematical Society*1974,**196:**161–176.MathSciNetView ArticleMATHGoogle Scholar - Osilike MO, Aniagbosor SC, Akuchu BG:
**Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces.***PanAmerican Mathematical Journal*2002,**12**(2):77–88.MathSciNetMATHGoogle Scholar - Zeng L-C, Lee GM, Wong NC:
**Ishikawa iteration with errors for approximating fixed points of strictly pseudocontractive mappings of Browder-Petryshyn type.***Taiwanese Journal of Mathematics*2006,**10**(1):87–99.MathSciNetMATHGoogle Scholar - Zeng L-C, Wong N-C, Yao J-C:
**Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type.***Taiwanese Journal of Mathematics*2006,**10**(4):837–849.MathSciNetMATHGoogle Scholar - Xu H-K, Ori RG:
**An implicit iteration process for nonexpansive mappings.***Numerical Functional Analysis and Optimization*2001,**22**(5–6):767–773. 10.1081/NFA-100105317MathSciNetView ArticleMATHGoogle Scholar - Osilike MO:
**Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps.***Journal of Mathematical Analysis and Applications*2004,**294**(1):73–81. 10.1016/j.jmaa.2004.01.038MathSciNetView ArticleMATHGoogle Scholar - Su Y, Li S:
**Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps.***Journal of Mathematical Analysis and Applications*2006,**320**(2):882–891. 10.1016/j.jmaa.2005.07.038MathSciNetView ArticleMATHGoogle Scholar - Zeng L-C, Yao J-C:
**Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2006,**64**(11):2507–2515. 10.1016/j.na.2005.08.028MathSciNetView ArticleMATHGoogle Scholar - Chang S-S:
**Some problems and results in the study of nonlinear analysis.***Nonlinear Analysis: Theory, Methods & Applications*1997,**30**(7):4197–4208. 10.1016/S0362-546X(97)00388-XMathSciNetView ArticleMATHGoogle Scholar - Sun Z-H:
**Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2003,**286**(1):351–358. 10.1016/S0022-247X(03)00537-7MathSciNetView ArticleMATHGoogle Scholar

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