#### Abstract

We introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for finding common fixed point of nonexpansive semigroups in the framework of Banach spaces. We proved that under some mild conditions these iterative processes converge strongly to the common fixed point of , which is the unique solution of some variational inequality. The results obtained in this paper extend and improve on the recent results of Li et al. (2009), Chen and He (2007), and many others as special cases.

#### 1. Introduction

Let be a real Banach space, and let be a nonempty closed convex subset of . A mapping of into itself is said to be nonexpansive if for each . We denote as the set of fixed points of . We know that is nonempty if is bounded; for more detail see [1]. Throughout this paper we denote by and the set of all positive integers and all positive real numbers, respectively. A one-parameter family from of into itself is said to be a nonexpansive semigroup on if it satisfies the following conditions: (i),(ii) for all ,(iii)for each the mapping is continuous,(iv) for all and .

We denote by the set of all common fixed points of , that is . We know that is nonempty if is bounded; see [2]. Recall that a self-mapping is a contraction if there exists a constant such that for each .

Definition 1.1. A mapping is said to be weakly contractive if there exists a continuous and strictly increasing function such that

Remark 1.2. As a special case, if we consider , for each and , then we get the contraction mapping with the coefficient .

In the last ten years or so, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3โ5]. Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a strongly positive bounded linear operator on : that is, there is a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is the fixed point set of a nonexpansive mapping on and is a given point in .

In 2003, Xu [5] proved that the sequence defined by the iterative method below, with the initial guess chosen arbitrarily: converges strongly to the unique solution of the minimization problem (1.3) provided the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the following iterative process for nonexpansive mappings (see [7, 8] for further developments in both Hilbert and Banach spaces). Let be a contraction on . Starting with an arbitrary initial , we defined the sequence recursively by where is a sequence in . It is proved in [6, 8] that under certain appropriate conditions imposed on , the sequence generated by (1.5) strongly converges to a unique solution of the variational inequality Recently, Marino and Xu [9] combined the iterative method (1.4) with the viscosity approximation method (1.5) considering the following general iterative process: where . They proved that the sequence generated by (1.7) converges strongly to a unique solution of the variational inequality which is the optimality condition for the minimization problem: , where is a potential function for (i.e., for ).

On the other hand, Shioji and Takahashi [10] introduced in a Hilbert space the implicit iteration as follows: where is a sequence in is a positive real divergent sequence and for . They proved, under certain restrictions on the sequence , that the sequence defined by (1.9) converges strongly to a member of (see also [11]).

Chen and Song [12] studied the strong convergence of the following sequence (1.10) for a nonexpansive semigroup with in a uniformly convex Banach space: Suzuki [13] was the first to introduced again in a Hilbert space the following implicit iteration process: for a nonexpansive semigroup case. Xu [14] established a Banach space version of the sequence (1.11) of Suzuki [13].

In 2007, Chen and He [15] extended the result of Suzuki [13] and Xu [14] and studied the strong convergence theorem of viscosity implicit iteration process for a nonexpansive semigroup with , in Banach spaces as follows: Very recently, S. Li et al. [16] considered a general iterative process for a nonexpansive semigroup in a Hilbert space as follows: where and are two sequences satisfying certain conditions. They proved that the sequence defined by (1.13) converges strongly to , which solves the following variational inequality (1.8).

Question 1. Can Theorem of S. Li et al. [16] be extended from Hilbert space to a general Banach space?

Question 2. Can we extend the iterative method of algorithms (1.10) and (1.12) to general algorithms?

Question 3. We know that the weakly contractive mapping is more general than the contractive mapping. What happens if the contractive mapping is replaced by the weakly contractive mapping?

The purpose of this paper is to give affirmative answer to these questions mentioned above. Motivated by the iterative process (1.10), (1.12), and (1.13), we introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of these iterative processes in a Banach space which admits a weakly sequentially continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Chen and He [15] and S. Li et al. [16] and many others as special cases.

#### 2. Preliminaries

Throughout this paper, let be a real Banach space and a closed convex subset of . Let be a normalized duality mapping by , where denotes the dual space of and denotes the generalized duality paring. In the following, the notations and denote the weak and strong convergence, respectively. Also, a mapping denotes the identity mapping.

The norm of a Banach space is said to be Gรขteaux differentiable if the limit exists for each on the unit sphere of . In this case is smooth. Recall that the Banach space is said to be smooth if duality mapping is single valued. In a smooth Banach space, we always assume that is strongly positive (see [17]), that is, a constant with the property

Moreover, if for each in the limit (2.1) is uniformly attained for , we say that the norm is uniformly Gรขteaux differentiable. The norm of is said to be Frรชchet differentiable if, for each , the limit (2.1) is attained uniformly for . The norm of is said to be uniformly Frรชchet differentiable (or is said to be uniformly smooth) if the limit (2.1) is attained uniformly for . A Banach space is said to be strictly convex if , implies โand uniformly convex if for all , where is modulus of convexity of defined by , for all . A uniformly convex Banach space is reflexive and strictly convex (see Theorems , and of [18]) and every uniformly smooth Banach space is a reflexive Banach with uniformly Gรขteaux differentiable norm (see Theorems , and of [18] and also [19]).

In the sequel we will use the following lemmas, which will be used in the proofs for the main results in the next section.

Lemma 2.1 (see [17]). Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and . Then .

Lemma 2.2 (see [20]). Let be a complete metric space and a weakly contractive mapping. Then, has a unique fixed point in .

If a Banach space admits a sequentially continuous duality mapping from weak topology to weak star topology, then, by Lemma 1 of [21], we have that duality mapping is a single value. In this case, the duality mapping is said to be a weakly sequentially continuous duality mapping, that is, for each with , we have (see [21โ23] for more details).

A Banach space is said to be satisfying Opialโs condition if for any sequence for all implies

By Theorem 1 in [21], it is well known that, if admits a weakly sequentially continuous duality mapping, then satisfies Opialโs condition and is smooth.

Lemma 2.3 ([22] (Demiclosed Principle)). Let be a nonempty closed convex subset of a reflexive Banach space which satisfies Opial's condition, and that suppose is nonexpansive. Then the mapping is demiclosed at zero, that is, and imply that .

Lemma 2.4 (see [15]). Let be a closed convex subset of a uniformly convex Banach space and a nonexpansive semigroup on such that . Then, for each and ,

#### 3. Main Results

In this section, we prove our main results.

Theorem 3.1. Let be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping from into and a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself such that . Let be a weakly contractive mapping and a strongly positive linear bounded operator with a coefficient such that . Let be a sequence defined by where are two sequences in and is a positive real divergent sequence satisfying the following conditions: , . Then the sequence defined by (3.1) converges strongly to the common fixed point , where is the unique solution of the variational inequality

Proof. Firstly, we show that defined by (3.1) is well define. Since , we may assume, with no loss of generality, that for each . Define the mapping by Indeed, from Lemma 2.1, we have, for all , This shows that is weakly contractive. It follows from Lemma 2.2 that has a unique fixed point , that is, defined by (3.1) is well defined.
Next, we show that is bounded. Letting , we get and so Thus, It follows that . Hence, which implies that is bounded. Since is weakly contractive, we have Then, is bounded. From (3.1), we have Thus, is also bounded. We denoted . And since is also bounded by the boundedness of and .
Next, we show that as , for all . We note that It follows that By conditions and , we obtain Moreover, we note that, for all , Define the set ; then is a nonempty bounded closed convex subset of , which is-invariant for each (i.e., ). Since and is bounded, there exists such that , and it follows from Lemma 2.4 that Noting (3.14) and (3.16), then, from (3.15), we obtain
Next, we show that contains a subsequence converging strongly to . Since is bounded and Banach space is a uniformly convex, it is reflexive and there exists a subsequence , which converges weakly to some as . Again since Banach space has a weakly sequentially continuous duality mapping satisfying Opial's condition, noting (3.17) and by Lemma 2.3, we have . From (3.7), replace by to obtain Since is single valued and weakly sequentially continuous from to , we get that Thus, as .
Next, we show that is a solution of the variational inequality (3.2). Firstly, since by condition , we obtain . Since , then .
From (3.1), we derive that . Then, for each , Therefore, Since the duality mapping is single-valued and weakly sequentially continuous duality mapping from to , for each and , then, from (3.22), we obtain That is, is a solution of the variational inequality (3.2).
Next, we show the uniqueness of the solution of the variational inequality (3.2). Suppose that satisfy (3.2). Then, Adding up (3.24), we get Thus . By the property of , we must have and the uniqueness is proved.
Finally, we show that converges strongly to . Suppose that there exists another subsequence as . We note that is the solution of the variational inequality (3.2). Hence, by uniqueness. In summary, we have shown that is sequentially compact and each cluster point of the sequence is equal to . Therefore, we conclude that as . This proof is complete.

Remark 3.2. (1) Theorem 3.1 improves and generalizes Theorem 3.1 of S. Li et al. [16] from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.
(2) Theorem 3.1 also improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.

A strong mean convergence theorem for nonexpansive mapping was first established by Baillon [24], and it was generalized to that for nonlinear semigroups by Reich [25โ27]. It is clear that Theorem 3.1 is valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.3. Let be a real Hilbert space and a nonempty closed convex subset of such that . Let be a nonexpansive mapping from into itself such that . Let be a weakly contractive mapping and a strongly positive linear bounded operator with a coefficient such that . Let be a sequence defined by where are two sequences in satisfying the following conditions: , . Then the sequence defined by (3.26) converges strongly to the common fixed point , where is the unique solution of the variational inequality

Taking and in Theorem 3.1, we get the following corollary.

Corollary 3.4. Let be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping from into ,โand a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself such that . Let be a weakly contractive mapping. Let be a sequence defined by where , are two sequences in and is a positive real divergent sequence satisfying the following conditions: , . Then the sequence defined by (3.28) converges strongly to the common fixed point , where is the unique solution of the variational inequality

Next, we prove a strong convergence theorem under different conditions.

Theorem 3.5. Let be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping from into and a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself such that . Let be a weakly contractive mapping and a strongly positive linear bounded operator with a coefficient such that . Let be a sequence defined by where , are two sequences in and is a positive real sequence satisfying the following conditions: ,. Then, the sequence defined by (3.30) converges strongly to the common fixed point , where is the unique solution of the variational inequality

Proof. Firstly, we show that defined by (3.30) is well defined. Since , we may assume, with no loss of generality, that for each . Define the mapping by From Lemma 2.1, we have for all , This show that is weakly contractive. It follows from Lemma 2.2 that has a unique fixed point , that is, defined by (3.30) is well defined.
Next, we show that is bounded. Letting , we get and so Thus, It follows that . Hence, This implies that is bounded, so are , and .
Next, we show that contains a subsequence converging strongly to . By reflexivity of and boundedness of the sequence , there exists subsequence such that as . Now, we show that . Put ,โ, , , and for , and fix . We note that For all , we have By the assumption that Banach space has a weakly sequentially continuous duality mapping satisfying Opialโs condition, (3.39) implies that , and we get that . From (3.36), replace by to obtain Since is singlevalued and weakly sequentially continuous from to , we get that Thus, as , namely, there is a subsequence such that as .
Next, we show that is a solution of the variational inequality (3.31). Firstly, since by condition , we obtain . Since , then .
From (3.30), we derive that . Then, for each , Therefore, By using the same argument and techniques as those of Theorem 3.1, we note that the variational inequality (3.31) has a unique solution. We denoted by the unique solution of (3.31). Therefore, as . The proof is completed.

Remark 3.6. (1) Theorem 3.5 improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping, and from Hilbert spaces to Banach spaces.
(2) Theorem 3.5 also improves and generalizes Theorem 3.1 of Chen and He [15] from a contractive mapping to a weakly contractive mapping.

Taking and in Theorem 3.5, we get the following corollary.

Corollary 3.7. Let be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping from into and a nonempty closed convex subset of such that . Let be a nonexpansive semigroup from into itself such that . Let be a weakly contractive mapping. Let be a sequence defined by where ,โ are two sequences in and is a positive real sequence satisfying the following conditions: , . Then the sequence defined by (3.45) converges strongly to the common fixed point , where is the unique solution of the variational inequality

#### Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under CSEC project no. 54000267). The first author would like to give thanks to the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand for their financial support.