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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 643740, 21 pages
http://dx.doi.org/10.1155/2011/643740
Research Article

Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 4 February 2011; Accepted 22 March 2011

Academic Editor: Yonghong Yao

Copyright © 2011 Rattanaporn Wangkeeree. 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.

Abstract

Let be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonexpansive semigroup on such that , and is a contraction on with coefficient . Let be -strongly accretive and -strictly pseudocontractive with and a positive real number such that . When the sequences of real numbers and satisfy some appropriate conditions, the three iterative processes given as follows: , , , , and , converge strongly to , where is the unique solution in of the variational inequality , . Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others.

1. Introduction

Let be a real Banach space. A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . A mapping is called -contraction if there exists a constant such that for all . A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ; (ii) for all ; (iii) for all and ; (iv) for all , the mapping is continuous.

We denote by the set of all common fixed points of , that is,

In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space where is a sequence in and is a sequence of positive real numbers which diverges to . Under certain restrictions on the sequence , Shioji and Takahashi [1] proved strong convergence of the sequence to a member of . In [2], Shimizu and Takahashi studied the strong convergence of the sequence defined by in a real Hilbert space where is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset of a Banach space and . Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup with in a Banach space: where is a contraction. Note however that their iterate at step is constructed through the average of the semigroup over the interval . Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process: for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if satisfies an asymptotic regularity condition and fulfills the control conditions , , and , then both the implicit iteration process (1.5) and the explicit iteration process (1.6), converge to a same point of . In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping. Recently, Chen and He [7] introduced the viscosity approximation process: where is a contraction and is a sequence in and a nonexpansive semigroup . The strong convergence theorem of is proved in a reflexive Banach space which admits a weakly sequentially continuous duality mapping. In [8], Chen et al. introduced and studied modified Mann iteration for nonexpansive mapping in a uniformly convex Banach space.

On the other hand, iterative approximation methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [911] and the references therein. 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 [10] 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 (1.9) provided the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [12] introduced the following iterative process for nonexpansive mappings (see [13] for further developments in both Hilbert and Banach spaces). Let be a contraction on . Starting with an arbitrary initial , define a sequence recursively by where is a sequence in . It is proved [12, 13] that, under certain appropriate conditions imposed on , the sequence generated by (1.11) strongly converges to the unique solution in of the variational inequality Recently, Marino and Xu [14] mixed the iterative method (1.10) and the viscosity approximation method (1.11) and considered the following general iterative method: where is a strongly positive bounded linear operator on . They proved that if the sequence of parameters satisfies the certain conditions, then the sequence generated by (1.13) converges strongly to the unique solution in of the variational inequality which is the optimality condition for the minimization problem, , where is a potential function for for ).

Very recently, Li et al. [15] introduced the following iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup on a Hilbert space : where is a strongly positive bounded linear operator on .

Let and be two positive real numbers such that . Recall that a mapping with domain and range in is called -strongly accretive if, for each , there exists such that where is the normalized duality mapping from into the dual space . Recall also that a mapping is called -strictly pseudocontractive if, for each , there exists such that It is easy to see that (1.17) can be rewritten as see [16].

In this paper, motivated by the above results, we introduce and study the strong convergence theorems of the general iterative scheme defined by (1.19) in the framework of a reflexive Banach space which admits a weakly sequentially continuous duality mapping: where is -strongly accretive and -strictly pseudocontractive with , is a contraction on with coefficient , is a positive real number such that , and is a nonexpansive semigroup on . The strong convergence theorems are proved under some appropriate control conditions on parameters and . Furthermore, by using these results, we obtain strong convergence theorems of the following new general iterative schemes and defined by The results presented in this paper extend and improve the main results in Li et al. [15], Chen and He [7], and many others.

2. Preliminaries

Throughout this paper, it is assumed that is a real Banach space with norm and let denote the normalized duality mapping from into given by for each , where denotes the dual space of denotes the generalized duality pairing, and denotes the set of all positive integers. In the sequel, we will denote the single-valued duality mapping by , and consider . When is a sequence in , then (resp., , ) will denote strong (resp., weak, weak*) convergence of the sequence to . In a Banach space , the following result (the subdifferential inequality) is well known [17, Theorem  4.2.1]: for all , for all , for all , A real Banach space is said to be strictly convex if for all with and . It is said to be uniformly convex if, for all , there exits such that The following results are well known and can be founded in [17]:(i)a uniformly convex Banach space is reflexive and strictly convex [17, Theorems  4.2.1 and 4.1.6],(ii)if is a strictly convex Banach space and is a nonexpansive mapping, then fixed point set of is a closed convex subset of [17, Theorem  4.5.3].

If a Banach space admits a sequentially continuous duality mapping from weak topology to weak star topology, then from Lemma  1 of [18], it follows that the duality mapping is single-valued and also is smooth. In this case, duality mapping is also said to be weakly sequentially continuous, that is, for each with , then (see [18, 19]).

In the sequel, we will denote the single-valued duality mapping by . A Banach space is said to satisfy Opial's condition if, for any sequence in , as implies By Theorem  1 of [18], we know that if admits a weakly sequentially continuous duality mapping, then satisfies Opial’s condition and is smooth; for the details, see [18].

Now, we present the concept of uniformly asymptotically regular semigroup (also see [20, 21]). Let be a nonempty closed convex subset of a Banach space , a continuous operator semigroup on . Then, is said to be uniformly asymptotically regular (in short, u.a.r.) on if, for all and any bounded subset of , The nonexpansive semigroup defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in [20, Examples  17 and 18].

Lemma 2.1 (see [3, Lemma  2.7]). Let be a nonempty closed convex subset of a uniformly convex Banach space , a bounded closed convex subset of , and a nonexpansive semigroup on such that . For each , set , then

Example 2.2. The set defined by Lemma 2.1 is u.a.r. nonexpansive semigroup. In fact, it is obvious that is a nonexpansive semigroup. For each , we have Applying Lemma 2.1, we have
Let be a nonempty closed and convex subset of a Banach space and a nonempty subset of . A mapping is said to be sunny if whenever for and . A mapping is called a retraction if for all . Furthermore, is a sunny nonexpansive retraction from onto if is a retraction from onto which is also sunny and nonexpansive. A subset of is called a sunny nonexpansive retraction of if there exists a sunny nonexpansive retraction from onto . The following lemma concerns the sunny nonexpansive retraction.

Lemma 2.3 (see [22, 23]). Let be a closed convex subset of a smooth Banach space . Let be a nonempty subset of and be a retraction. Then, is sunny and nonexpansive if and only if for all and .

Lemma 2.4 (see [24, Lemma  2.3]). Let be a sequence of nonnegative real numbers satisfying the property where satisfy the restrictions (i); (ii); (iii).
Then, .

The following lemma will be frequently used throughout the paper and can be found in [25].

Lemma 2.5 (see [25, Lemma  2.7]). Let be a real smooth Banach space and a mapping.
(i)If is -strongly accretive and -strictly pseudocontractive with , then is contractive with constant .
(i)If is -strongly accretive and -strictly pseudocontractive with , then, for any fixed number , is contractive with constant .

3. Main Results

Now, we are in a position to state and prove our main results.

Theorem 3.1. Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping . Let be a u.a.r. nonexpansive semigroup on such that . Suppose that the real sequences , satisfy the conditions Let be -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and a positive real number such that . Then, the sequence defined by (1.19) converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Proof. Note that is a nonempty closed convex set. We first show that is bounded. Let . Thus, by Lemma 2.5, we have By induction, we get This implies that is bounded and, hence, so are and . This implies that Since is a u.a.r. nonexpansive semigroup and , we have, for all , Hence, for all , That is, for all , Let . Then, is a contraction on . In fact, from Lemma 2.5(i), we have Therefore, is a contraction on due to . Thus, by Banach contraction principle, has a unique fixed point . Then, using Lemma 2.3, is the unique solution in of the variational inequality (3.2). Next, we show that Indeed, we can take a subsequence of such that We may assume that as , since a Banach space has a weakly sequentially continuous duality mapping satisfying Opial's condition [13]. We will prove that . Suppose the contrary, , that is, for some . It follows from (3.8) and Opial’s condition that This is a contradiction, which shows that for all , that is, . In view of the variational inequality (3.2) and the assumption that duality mapping is weakly sequentially continuous, we conclude Finally, we will show that . For each , we have On the other hand, where is a constant satisfying . Substituting (3.15) in (3.14), we obtain where It is easily seen that . Since is bounded and , by (3.46), we obtain , applying Lemma 2.4 to (3.16) to conclude as . This completes the proof.

Using Theorem 3.1, we obtain the following two strong convergence theorems of new iterative approximation methods for a nonexpansive semigroup .

Corollary 3.2. Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping . Let be a u.a.r. nonexpansive semigroup on such that . Suppose that the real sequences , satisfy the conditions Let be -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and a positive real number such that . Then, the sequence defined by (1.20) converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Proof. Let be the sequence given by and Form Theorem 3.1, . We claim that . Indeed, we estimate It follows from , , and Lemma 2.4 that . Consequently, as required.

Corollary 3.3. Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping . Let be a u.a.r. nonexpansive semigroup on such that . Suppose that the real sequences , satisfy the conditions Let be -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and a positive real number such that . Then, the sequence defined by (1.21) converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Proof. Define the sequences and by Taking , we have It follows from induction that Thus, both and are bounded. We observe that Thus, Corollary 3.2 implies that converges strongly to some point . In this case, we also have Hence, the sequence converges strongly to some point . This complete the proof.

Using Theorem 3.1, Lemma 2.1, and Example 2.2, we have the following result.

Corollary 3.4. Let be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping . Let be a nonexpansive semigroup on such that . Suppose that the real sequences , satisfy the conditions Let be -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and a positive real number such that . Then, the sequence defined by converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Corollary 3.5. Let be a real Hilbert space. Let be a nonexpansive semigroup on such that . Suppose that the real sequences , satisfy the conditions Let be a contraction mapping with coefficient and a strongly positive bounded linear operator with coefficient and . Then, the sequence defined by converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Proof. Since is a strongly positive bounded linear operator with coefficient , we have Therefore, is -strongly accretive. On the other hand, Since is strongly positive if and only if is strongly positive, we may assume, without loss of generality, that , so that Hence, is 12-strongly pseudocontractive. Applying Corollary 3.4, we conclude the result.

Theorem 3.6. Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping . Let be a u.a.r. nonexpansive semigroup on such that . Let and be sequences of real number satisfying Let be -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and a positive real number such that . Then, the sequence defined by converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Proof. By the same argument as in the proof of Theorem 3.1, we can obtain that , , and are bounded and is a contraction on . Thus, by Banach contraction principle, has a unique fixed point . Then, using Lemma 2.3, is the unique solution in of the variational inequality (3.40). Next, we show that Indeed, we can take a subsequence of such that We may assume that as . Now, we show that . Put Fix , then we have Thus, for all , we obtain Since Banach space has a weakly sequentially continuous duality mapping satisfying Opial's condition [13], we can conclude that for all , that is, . In view of the variational inequality (3.2) and the assumption that duality mapping is weakly sequentially continuous, we conclude By the same argument as in the proof of Theorem 3.1, we conclude that as . This completes the proof.

Using Theorem 3.6 and the method as in the proof of Corollary 3.7, we have the following result.

Corollary 3.7. Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping . Let be a u.a.r. nonexpansive semigroup on such that . Let and be sequences of real number satisfying Let be a -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and is a positive real number such that . Then, the sequence defined by converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Using Theorem 3.6 and the method as in the proof of Corollary 3.8, we have the following result.

Corollary 3.8. Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping . Let be a u.a.r. nonexpansive semigroup on such that . Let and be sequences of real number satisfying Let be a -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and is a positive real number such that . Then, the sequence defined by converges strongly to , where is the unique solution in of the variational inequality or equivalently , where is the sunny nonexpansive retraction of onto .

Using Theorem 3.6, Lemma 2.1, and Example 2.2, we have the following result.

Corollary 3.9. Let be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping . Let be a nonexpansive semigroup on such that . Let and be sequences of real numbers satisfying Let be -strongly accretive and -strictly pseudocontractive with , a contraction mapping with coefficient , and a positive real number such that . Then, the sequence defined by converges strongly to , where is the unique solution in of the variational inequality