# Viscosity Approximation of Common Fixed Points for -Lipschitzian Semigroup of Pseudocontractive Mappings in Banach Spaces

- Xue-song Li
^{1}, - Jong Kyu Kim
^{2}and - Nan-jing Huang
^{1}Email author

**2009**:936121

https://doi.org/10.1155/2009/936121

© Xue-song Li et al. 2009

**Received: **14 January 2009

**Accepted: **5 March 2009

**Published: **8 March 2009

## Abstract

We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007) and the pseudocontractive mapping in (Zegeye et al., 2007) to the pseudocontractive semigroup in Banach spaces under different conditions.

## Keywords

## 1. Introduction

where denotes the generalized duality pairing. It is well known that (see, e.g., [1, pages 107–113])

(i) is single-valued if is strictly convex;

(ii) is uniformly smooth if and only if is single-valued and uniformly continuous on any bounded subset of .

Let be a nonempty closed convex subset of . A mapping is said to be

It is easy to see that the pseudocontractive mapping is more general than the nonexpansive mapping.

(3) is pseudocontractive for each ;

(4)for each , the mapping from into is continuous.

If the mapping in condition (3) is replaced by

then is said to be a nonexpansive semigroup on .

In the sequel, we always assume that .

where , , and obtained the convergence theorem as follows.

Theorem 1 (see [11]).

then generated by (1.9) converges strongly to a member of .

Xu [11] also proposed the following problem.

Problem 1 (see [11]).

We do not know if Theorem X holds in a uniformly convex and uniformly smooth Banach (e.g., for ).

This problem has been solved by Li and Huang [15] and Suzuki [8], respectively.

Moudafi's viscosity approximation method has been recently studied by many authors (see, e.g., [2, 3, 5, 10, 13, 15–17] and the references therein). Chen and He [3] studied the convergence of (1.8) constructed from a nonexpansive semigroup and a contractive mapping in a reflective Banach space with a weakly sequentially continuous duality mapping. Zegeye et al. [13] studied the convergence of (1.8) constructed from a pseudocontractive mapping and a contractive mapping.

where , and . Chen and He [3] studied the convergence of (1.12) constructed from a nonexpansive semigroup and obtained some convergence results.

An interesting work is to extend some results involving nonexpansive mapping, nonexpansive semigroup, and pseudocontractive mapping to the semigroup of pseudocontractive mappings. Li and Huang [15] generalized some corresponding results to pseudocontractive semigroup in Banach spaces. Some further study concerned with approximating common fixed points of the semigroup of pseudocontractive mappings in Banach spaces, we refer to Li and Huang [16].

Motivated by the works mentioned above, in this paper, we study the convergence of implicit viscosity iteration process (1.8) constructed from the pseudocontractive semigroup and -strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we obtain the convergence of the implicit iteration process for approximating the common fixed point of the nonexpansive semigroup in certain Banach spaces. We also study the convergence of the explicit viscosity iteration process (1.12) constructed from the pseudocontractive semigroup and -strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differential norms. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in [3] and the pseudocontractive mapping in [13] to the pseudocontractive semigroup in Banach spaces under different conditions.

## 2. Preliminaries

A real Banach space is said to have a weakly continuous duality mapping if is single-valued and weak-to- sequentially continuous (i.e., if each is a sequence in weakly convergent to , then converges to ). Obviously, if has a weakly continuous duality mapping, then is norm-to- sequentially continuous. It is well known that posses duality mapping which is weakly continuous (see, e.g., [11]).

A mapping with domain and range in is said to be demiclosed at a point if whenever is a sequence in which converges weakly to and converges strongly to , then .

For the sake of convenience, we restate the following lemmas that will be used.

Lemma 2.1 (see [18]).

Let be a Banach space, be a nonempty closed convex subset of , and be a strongly pseudocontractive and continuous mapping. Then has a unique fixed point in .

Lemma 2.2 (see [19]).

Lemma 2.3 (see [12]).

Lemma 2.4 (see [9]).

## 3. Main Results

We first discuss the convergence of implicit viscosity iteration process (1.8) constructed from a pseudocontractive semigroup .

Theorem 3.1.

Proof.

we know that is strongly pseudocontractive and strongly continuous. It follows from Lemma 2.1 that has a unique fixed point (say) , that is, generated by (1.8) is well defined.

This completes the proof.

Theorem 3.2.

Let be a uniformly convex Banach space with the uniformly Gâteaux differential norm and be a nonempty closed convex subset of . Let be an -Lipschitzian semigroup of pseudocontractive mappings satisfying (3.1) and let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that is a sequence generated by (1.8) and

Proof.

From Theorem 3.1, we know that is bounded and . It is easy to see that is a nonempty bounded closed convex subset of (see, e.g., [10]).

This implies and so is a singleton. Therefore, (3.11) implies that there exists such that .

This completes the proof.

- (1)
Theorem 3.2 extends and generalizes Theorem 3.1 of [3] from nonexpansive semigroup to Lipschitzian pseudocontractive semigroup in Banach spaces with different conditions; (2) If is a pseudocontractive mapping, then condition (3.1) is trivial.

If is a nonexpansive semigroup, then is an -Lipschitzian semigroup of pseudocontractive mappings, condition of Theorem 3.2 holds trivially. From Theorem 3.2, we have the following result.

Corollary 3.4.

then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Theorem 3.5.

then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Proof.

For the nonexpansive semigroup , condition of Theorem 3.2 is trivial and so formula (3.11) holds. Since uniformly smooth Banach space has the fixed point property for nonexpansive mapping (see, e.g., [10]), has a fixed point . The rest proof is similar to the proof of Theorem 3.2 and so we omit it. This completes the proof.

Theorem 3.6.

Proof.

It follows from [20, Theorem 3.18b] that is demiclosed at zero for each , where is an identity mapping. This implies that .

This implies that converges strongly to . Similar to the proof of Theorem 3.2, it is easy to show that converges strongly to that is also the unique solution to VI (3.27). This completes the proof.

Now we turn to discuss the convergence of explicit viscosity iteration process (1.12) for approximating the common fixed point of the pseudocontractive semigroup .

Theorem 3.7.

Let be a nonempty closed convex subset of a real Banach space . Let be an -Lipschitzian semigroup of pseudocontractive mappings with such that (3.1) holds. Let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and the following conditions hold:

Proof.

This completes the proof.

Remark 3.8.

for all , where is an any given positive real number. It is easy to see that the conditions with regard to and in Theorem 3.7 hold. If the mapping is Lipschitz continuous for any , then condition (iv) in Theorem 3.7 also holds.

Theorem 3.9.

Let be a uniformly convex Banach space with the uniformly Gâteaux differential norm and be a nonempty closed convex subset of . Let be an -Lipschitzian semigroup of pseudocontractive mappings with such that (3.1) holds. Let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Assume further that condition (2) of Theorem 3.2 holds, where is generated by (1.8) with . Then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Proof.

By Theorem3.2, we know that converges strongly to a fixed point of that is the unique solution in to VI (3.9), where is generated by (1.8) with . It follows from (3.46) that . This completes the proof.

- (1)
Theorem 3.9 extends Theorem 4.1 of [13] from Lipschitzian pseudocontractive mapping to Lipschitzian pseudocontractive semigroup in Banach spaces under different conditions; (2) Theorem 3.9 also extends Theorem 3.2 of [3] from nonexpansive semigroup to Lipschitzian pseudocontractive semigroup in Banach spaces under different conditions.

If is a nonexpansive semigroup, then is an -Lipschitzian semigroup of pseudocontractive mappings, condition of Theorem 3.2 holds trivially. Therefore, Theorem 3.9 gives the following result.

Corollary 3.11.

Let be a uniformly convex Banach space with the uniformly Gâteaux differential norm and be a nonempty closed convex subset of . Let be a nonexpansive semigroup satisfying (3.1) and be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Then converges strongly to a common fixed point of that is the unique solution in F(T) to VI (3.9).

Theorem 3.12.

Let be a uniformly smooth Banach space and be a nonempty closed convex subset of . Let be a nonexpansive semigroup satisfying (3.1) and let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Proof.

Let denote the sequence defined as in (1.8) with . By Theorem 3.5, we know that converges strongly to a fixed point of that is the unique solution in to VI (3.9). It follows from (3.46) that . This completes the proof.

Theorem 3.13.

Let be a real Hilbert space and be a nonempty closed convex subset of . Let be an -Lipschitzian semigroup of pseudocontractive mappings with such that (3.1) holds. Let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Then converges strongly to a common fixed point of that is the unique solution in to VI (3.27).

Proof.

Let denote the sequence defined as in (1.8) with . By Theorem 3.6, we know that converges strongly to a fixed point of that is the unique solution in to VI (3.27). It follows from (3.46) that . This completes the proof.

## Declarations

### Acknowledgments

This work was supported by the National Science Foundation of China (10671135, 70831005), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the Open Fund (PLN0703) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

## Authors’ Affiliations

## References

- Takahashi W:
*Nonlinear Functional Analysis. Fixed Point Theory and Its Applications*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Chang S-S:
**Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*2006,**323**(2):1402–1416. 10.1016/j.jmaa.2005.11.057MATHMathSciNetView ArticleGoogle Scholar - Chen R, He H:
**Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space.***Applied Mathematics Letters*2007,**20**(7):751–757. 10.1016/j.aml.2006.09.003MATHMathSciNetView ArticleGoogle Scholar - Huang N-j, Bai M-R:
**A perturbed iterative procedure for multivalued pseudo-contractive mappings and multivalued accretive mappings in Banach spaces.***Computers & Mathematics with Applications*1999,**37**(6):7–15. 10.1016/S0898-1221(99)00072-3MATHMathSciNetView ArticleGoogle Scholar - Jung JS:
**Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications. Series A*2006,**64**(11):2536–2552. 10.1016/j.na.2005.08.032MATHView ArticleMathSciNetGoogle 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)90323-6MATHMathSciNetView ArticleGoogle Scholar - Suzuki T:
**On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces.***Proceedings of the American Mathematical Society*2003,**131**(7):2133–2136. 10.1090/S0002-9939-02-06844-2MATHMathSciNetView ArticleGoogle Scholar - Suzuki T:
**Browder's type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces.***Israel Journal of Mathematics*2007,**157**(1):239–257. 10.1007/s11856-006-0010-6MATHMathSciNetView ArticleGoogle Scholar - Weng X:
**Fixed point iteration for local strictly pseudo-contractive mapping.***Proceedings of the American Mathematical Society*1991,**113**(3):727–731. 10.1090/S0002-9939-1991-1086345-8MATHMathSciNetView 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 - Xu H-K:
**A strong convergence theorem for contraction semigroups in Banach spaces.***Bulletin of the Australian Mathematical Society*2005,**72**(3):371–379. 10.1017/S000497270003519XMATHMathSciNetView ArticleGoogle 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-100105317MATHMathSciNetView ArticleGoogle Scholar - Zegeye H, Shahzad N, Mekonen T:
**Viscosity approximation methods for pseudocontractive mappings in Banach spaces.***Applied Mathematics and Computation*2007,**185**(1):538–546. 10.1016/j.amc.2006.07.063MATHMathSciNetView ArticleGoogle Scholar - Zeng W-Y, Huang N-j, Zhao C-W:
**Viscosity approximation methods for generalized mixed equilibrium problems and fixed points of a sequence of nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-15.Google Scholar - Li X-s, Huang N-j:
**Strong convergence theorems for fixed points of pseudo-contractive semigroup.***Bulletin of the Australian Mathematical Society*2007,**76**(3):441–452.MATHMathSciNetView ArticleGoogle Scholar - Li X-s, Huang N-j:
**Viscosity approximation methods for pseudo-contractive semigroup in Banach spaces.**In*Research Report*. Sichuan University, Chengdu, China; 2007.Google 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 - Deimling K:
**Zeros of accretive operators.***Manuscripta Mathematica*1974,**13**(4):365–374. 10.1007/BF01171148MATHMathSciNetView ArticleGoogle Scholar - Petryshyn WV:
**A characterization of strict convexity of Banach spaces and other uses of duality mappings.***Journal of Functional Analysis*1970,**6:**282–291. 10.1016/0022-1236(70)90061-3MATHMathSciNetView ArticleGoogle Scholar - Browder FE:
**Nonlinear mappings of nonexpansive and accretive type in Banach spaces.***Bulletin of the American Mathematical Society*1967,**73:**875–882. 10.1090/S0002-9904-1967-11823-8MATHMathSciNetView ArticleGoogle 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.