Journal of Applied MathematicsVolume 2012 (2012), Article ID 506976, 14 pagesdoi:10.1155/2012/506976

Research Article

A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

Kamonrat Nammanee,¹ Suthep Suantai,2,3 and Prasit Cholamjiak1,3

¹School of Science, University of Phayao, Phayao 56000, Thailand²Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand³Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 23 November 2011; Accepted 27 January 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Kamonrat Nammanee et al. 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.

We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mapping , where is a gauge function on . Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

1. Introduction

Let be a real Banach space and the dual space of . Let be a nonempty, closed, and convex subset of . A (one-parameter) nonexpansive semigroup is a family of self-mappings of such that(i) for all ,(ii) for all and ,(iii)for each , the mapping is continuous,(iv)for each , is nonexpansive, that is, We denote by the common fixed points set of , that is, .

In 1967, Halpern [1] introduced the following classical iteration for a nonexpansive mapping in a real Hilbert space: where and .

In 1977, Lions [2] obtained a strong convergence provide the real sequence satisfies the following conditions:

C1: ; C2: ; C3: .

Reich [3] also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces. However, both Halpern’s and Lion’s conditions imposed on the real sequence excluded the canonical choice .

In 1992, Wittmann [4] proved that the sequence converges strongly to a fixed point of if satisfies the following conditions:

C1: ; C2: ; C3: .

Shioji and Takahashi [5] extended Wittmann’s result to real Banach spaces with uniformly Gâteaux differentiable norms and in which each nonempty closed convex and bounded subset has the fixed point property for nonexpansive mappings. The concept of the Halpern iterative scheme has been widely used to approximate the fixed points for nonexpansive mappings (see, e.g., [6–12] and the reference cited therein).

Let be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping as follows: where . Xu [14] also studied the iteration process (1.3) in uniformly smooth Banach spaces.

Let be a strongly positive bounded linear operator on a real Hilbert space , that is, there is a constant such that

A typical problem is to minimize a quadratic function over the fixed points set of a nonexpansive mapping on a Hilbert space : where is the fixed points set of a nonexpansive mapping on and is a given point in .

In 2006, Marino and Xu [15] introduced the following general iterative method for a nonexpansive mapping in a Hilbert space : where , is a contraction on , and is a strongly positive bounded linear operator on . They proved that the sequence generated by (1.6) converges strongly to a fixed point which also solves the variational inequality which is the optimality condition for the minimization problem: , where is a potential function for (i.e., for ).

Suzuki [16] first introduced the following implicit viscosity method for a nonexpansive semigroup in a Hilbert space: where and . He proved strong convergence of iteration (1.8) under suitable conditions. Subsequently, Xu [17] extended Suzuki’s [16] result from a Hilbert space to a uniformly convex Banach space which admits a weakly sequentially continuous normalized duality mapping.

Motivated by Chen and Song [18], in 2007, Chen and He [19] investigated the implicit and explicit viscosity methods for a nonexpansive semigroup without integral in a reflexive Banach space which admits a weakly sequentially continuous normalized duality mapping: where .

In 2008, Song and Xu [20] also studied the iterations (1.9) and (1.10) in a reflexive and strictly convex Banach space with a Gâteaux differentiable norm. Subsequently, Cholamjiak and Suantai [21] extended Song and Xu’s results to a Banach space which admits duality mapping with a gauge function. Wangkeeree and Kamraksa [22] and Wangkeeree et al. [23] obtained the convergence results concerning the duality mapping with a gauge function in Banach spaces. The convergence of iterations for a nonexpansive semigroup and nonlinear mappings has been studied by many authors (see, e.g., [24–38]).

Let be a real reflexive Banach space which admits the duality mapping with a gauge . Let be a nonexpansive semigroup on . Recall that an operator is said to be strongly positive if there exists a constant such that where and .

Motivated by Chen and Song [18], Chen and He [19], Marino and Xu [15], Colao et al. [39], and Wangkeeree et al. [23], we study strong convergence of the following general iterative methods: where , is a contraction on and is a positive bounded linear operator on .

2. Preliminaries

A Banach space is called strictly convex if for all with and . A Banach space is called uniformly convex if for each there is a such that for with and holds. The modulus of convexity of is defined by for all . is uniformly convex if , and for all . It is known that every uniformly convex Banach space is strictly convex and reflexive. Let . Then the norm of is said to be Gâteaux differentiable if exists for each . In this case is called smooth. The norm of is said to be Fréchet differentiable if for each , the limit is attained uniformly for . The norm of is called uniformly Fréchet differentiable, if the limit is attained uniformly for . It is well known that (uniformly) Fréchet differentiability of the norm of implies (uniformly) Gâteaux differentiability of the norm of .

Let be the modulus of smoothness of defined by

A Banach space is called uniformly smooth if as . See [40–42] for more details.

We need the following definitions and results which can be found in [40, 41, 43].

Definition 2.1. A continuous strictly increasing function is said to be gauge function if and .

Definition 2.2. Let be a normed space and a gauge function. Then the mapping defined by is called the duality mapping with gauge function .

In the particular case , the duality mapping is called the normalized duality mapping.

In the case , the duality mapping is called the generalized duality mapping. It follows from the definition that and .

Remark 2.3. For the gauge function , the function defined by is a continuous convex and strictly increasing function on . Therefore, has a continuous inverse function .

It is noted that if , then . Further

Remark 2.4. For each in a Banach space , , where denotes the sub-differential.

We also know the following facts:(i) is a nonempty, closed, and convex set in for each ,(ii) is a function when is strictly convex,(iii)If is single-valued, then Following Browder [43], we say that a Banach space has a weakly continuous duality mapping if there exists a gauge for which the duality mapping is single-valued and continuous from the weak topology to the weak* topology, that is, for any with , the sequence converges weakly* to . It is known that the space has a weakly continuous duality mapping with a gauge function for all . Moreover, is invariant on .

Lemma 2.5 (See [44]). Assume that a Banach space has a weakly continuous duality mapping with gauge . (i)For all , the following inequality holds: In particular, for all , (ii)Assume that a sequence in converges weakly to a point . Then the following holds: for all .

Lemma 2.6 (See [23]). Assume that a Banach space has a weakly continuous duality mapping with gauge . Let be a strongly positive bounded linear operator on with coefficient and . Then .

Lemma 2.7 (See [12]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (a) ; (b) or . Then .

3. Implicit Iteration Scheme

In this section, we prove a strong convergence theorem of an implicit iterative method (1.12).

Theorem 3.1. Let be a reflexive which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a nonexpansive semigroup on such that . Let be a contraction on with the coefficient and a strongly positive bounded linear operator with coefficient and . Let and be real sequences satisfying , and . Then defined by (1.12) converges strongly to which solves the following variational inequality:

Proof. First, we prove the uniqueness of the solution to the variational inequality (3.1) in . Suppose that satisfy (3.1), so we have Adding the above inequalities, we get This shows that which implies by the strong positivity of Since is invariant on , It follows that Therefore since .We next prove that is bounded. For each , by Lemma 2.6, we have which yields Hence is bounded. So are and .We next prove that is relatively sequentially compact. By the reflexivity of and the boundedness of , there exists a subsequence of and a point in such that as . Now we show that . Put , and for , fix . We see that So we have On the other hand, by Lemma 2.5 (ii), we have Combining (3.11) and (3.12), we have This implies that . Further, we see that So we have By the definition of , it is easily seen that Hence Therefore as since is weakly continuous; consequently, as by the continuity of . Hence is relatively sequentially compact.Finally, we prove that is a solution in to the variational inequality (3.1). For any , we see that On the other hand, we have which implies Observe as . Replacing by and letting in (3.20), we obtain So is a solution of variational inequality (3.1); and hence by the uniqueness. In a summary, we have proved that is relatively sequentially compact and each cluster point of (as ) equals . Therefore as . This completes the proof.

4. Explicit Iteration Scheme

In this section, utilizing the implicit version in Theorem 3.1, we consider the explicit one in a reflexive Banach space which admits the duality mapping .

Theorem 4.1. Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on [0,1]. Let be a nonexpansive semigroup on such that . Let be a contraction on with the coefficient and a strongly positive bounded linear operator with coefficient and . Let and be real sequences satisfying , , and . Then defined by (1.13) converges strongly to which also solves the variational inequality (3.1).

Proof. Since , we may assume that and for all . First we prove that is bounded. For each , by Lemma 2.6, we have It follows from induction that Thus is bounded, and hence so are and . From Theorem 3.1, there is a unique solution to the following variational inequality: Next we prove that Indeed, we can choose a subsequence of such that Further, we can assume that by the reflexivity of and the boundedness of . Now we show that . Put and for , fix . We obtain It follows that . From Lemma 2.5 (ii) we have So we have and hence . Since the duality mapping is weakly sequentially continuous, Finally, we show that . From Lemma 2.5 (i), we have Note that and . Using Lemma 2.7, we have as by the continuity of . This completes the proof.

Remark 4.2. Theorems 3.1 and 4.1 improve and extend the main results proved in [15] in the following senses:(i)from a nonexpansive mapping to a nonexpansive semigroup,(ii)from a real Hilbert space to a reflexive Banach space which admits a weakly continuous duality mapping with gauge functions.

Acknowledgments

The authors wish to thank the editor and the referee for valuable suggestions. K. Nammanee was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. S. Suantai and P. Cholamjiak wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.

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