- Research Article
- Open Access

# 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

- Banach Space
- Nonexpansive Mapping
- Real Banach Space
- Common Fixed Point
- Nonempty Closed Convex Subset

## 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.

of self-mappings on such that

(1) for all ;

(2) for all and ;

(3) is pseudocontractive for each ;

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

If the mapping in condition (3) is replaced by

is nonexpansive for each ;

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]).

for each .

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]).

for all , , where .

Lemma 2.4 (see [9]).

where , , is fixed, , and . Then .

## 3. Main Results

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

Theorem 3.1.

then for any .

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

(1) ;

(2) , where with for all .

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:

(i) , , for all ;

Then for any .

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

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