Review Article | Open Access

Rabian Wangkeeree, "The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces", *Abstract and Applied Analysis*, vol. 2011, Article ID 854360, 19 pages, 2011. https://doi.org/10.1155/2011/854360

# The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces

**Academic Editor:**Ljubisa Kocinac

#### Abstract

We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).

#### 1. Introduction

Let be a nonempty subset of a normed linear space . Recall that a mapping is called *nonexpansive* if
We use to denote the set of fixed points of ; that is, . A self-mapping is a contraction on if there exists a constant and such that

One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [1–3]. More precisely, take and define a contraction by where is a fixed point. Banach's contraction mapping principle guarantees that has a unique fixed point in . It is unclear, in general, what is the behavior of as , even if has a fixed point. However, in the case of having a fixed point, Browder [1] proved that if is a Hilbert space, then converges strongly to a fixed point of . Reich [2] extended Browder's result to the setting of Banach spaces and proved that if is a uniformly smooth Banach space, then converges strongly to a fixed point of and the limit defines the (unique) sunny nonexpansive retraction from onto . Xu [3] proved Reich's results hold in reflexive Banach spaces which have a weakly continuous duality mapping.

The iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [4–7] 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 a nonexpansive mapping on and is a given point in . In 2003, Xu [6] 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 (1.5) provided the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [8] introduced the following iterative iterative process for nonexpansive mappings (see [9, 10] 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 [8, 10] that under certain appropriate conditions imposed on , the sequence generated by (1.7) strongly converges to the unique solution in of the variational inequality Recently, Marino and Xu [11] mixed the iterative method (1.6) and the viscosity approximation method (1.7) 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 following appropriate conditions: , , and either or , then the sequence generated by (1.9) 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, Wangkeeree et al. [12] extended Marino and Xu's result to the setting of Banach spaces and obtained the strong convergence theorems in a reflexive Banach space which admits a weakly continuous duality mapping. Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a nonexpansive mapping with , a contraction with coefficient and a strongly positive bounded linear operator with coefficient and . Define the net by It is proved in [12] that converges strongly as to a fixed point of which solves the variational inequality

On the other hand, Ceng et al. [13] introduced the iterative approximation method for solving the variational inequality generated by two strongly positive bounded linear operators on a real Hilbert space . Let be a contraction with coefficient , and let be two strongly positive bounded linear operators with coefficient and , respectively. Assume that , is a sequence in , is a sequence in . Starting with an arbitrary initial , define a sequence recursively by It is proved in [13, Theorem 3.1] that if the sequences and satisfy the following conditions: (C1),(C2), (C3) or ,(C4),

then the sequence generated by (1.13) converges strongly to the unique solution in of the variational inequality Observe that if and for all , then algorithm (1.13) reduces to (1.9). Moreover, the variational inequality (1.14) reduces to (1.10). Furthermore, the applications of these results to constrained generalized pseudoinverse are studied.

In this paper, motivated by Marino and Xu [11], Wangkeeree et al. [12], and Ceng et al. [13], we introduce two general iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu [11], Wangkeeree et al. [12], and Ceng et al. [13], and many others.

#### 2. Preliminaries

Throughout this paper, let be a real Banach space and its dual space. We write (resp. ) to indicate that the sequence weakly (resp. weak) converges to ; as usual, will symbolize strong convergence. Let . A Banach space is said to *uniformly convex* if, for any , there exists such that, for any , implies . It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [14]). A Banach space is said to be *smooth* if the limit exists for all . It is also said to be *uniformly smooth* if the limit is attained uniformly for .

By a gauge function , we mean a continuous strictly increasing function such that and as . Let be the dual space of . The duality mapping associated to a gauge function is defined by

In particular, the duality mapping with the gauge function , denoted by , is referred to as the normalized duality mapping. Clearly, there holds the relation for all (see [15]). Browder [15] initiated the study of certain classes of nonlinear operators by means of the duality mapping . Following Browder [15], 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 has a weakly continuous duality mapping with a gauge function for all . Set

then

where denotes the subdifferential in the sense of convex analysis.

Now, we collect some useful lemmas for proving the convergence result of this paper.

The first part of the next lemma is an immediate consequence of the subdifferential inequality, and the proof of the second part can be found in [16].

Lemma 2.1 (see [16]). *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 identity holds: ** *

Lemma 2.2 (see [7]). *Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence such that *(a)*,
*(b)* or .** Then, .*

In a Banach space having a weakly continuous duality mapping with a gauge function , an operator is said to be * strongly positive * [12] if there exists a constant with the property
where is the identity mapping. If is a real Hilbert space, then (2.8) reduces to (1.4). The next valuable lemma can be found in [12].

Lemma 2.3 (see [12, Lemma 3.1]). *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, .*

#### 3. Main Results

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

Lemma 3.1. *Let be a Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on ; that is, . Let be a nonexpansive mapping and a contraction with coefficient . Let and be two strongly positive bounded linear operators with coefficients and , respectively. Let and be two constants satisfying the condition **
Then, for any , the mapping defined by
**
is a contraction with coefficient , where .*

*Proof. *Observe that
This shows that . Using Lemma 2.3, we obtain
Hence, is a contraction with coefficient .

Applying the Banach contraction principle to Lemma 3.1, there exists a unique fixed point of in ; that is,

*Remark 3.2. *For each , space has a weakly continuous duality mapping with a gauge function which is invariant on .

Theorem 3.3. *Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a nonexpansive mapping with , a contraction with coefficient , and , two strongly positive bounded linear operators with coefficients and , respectively. Let and be two constants satisfying the condition . Then, the net defined by (3.5) converges strongly as to a fixed point of which solves the variational inequality
*

*Proof. *We first show that the uniqueness of a solution of the variational inequality (3.6). Suppose that both and are solutions to (3.6), then
Adding (3.7), we obtain
On the other hand, we observe that
It then follows that for any ,
Applying (3.10) to (3.8), we obtain that and the uniqueness is proved. Below, we use to denote the unique solution of (3.6). Next, we will prove that is bounded. Take a , and denote the mapping by
From Lemma 3.1, we have
where . It follows that
Hence, is bounded, so are , and . The definition of implies that
If follows from reflexivity of and the boundedness of sequence that there exists which is a subsequence of converging weakly to as . Since is weakly sequentially continuous, we have by Lemma 2.1 that
Let
It follows that
Since
We obtain
On the other hand, however,
It follows from (3.19) and (3.20) that
which gives us, . Next, we show that as . In fact, since and is a gauge function, then for , and
Following Lemma 2.1, we have
Thus,
Now, observing that implies , we conclude from the last inequality that
Hence, as . Next, we prove that solves the variational inequality (3.6). For any , we observe that
Since
we can derive that
That is
Using (3.26), for each , we have
where is a constant satisfying . Noticing that
It follows from (3.30) that
So, is a solution of the variational inequality (3.6), and hence, by the uniqueness. In a summary, we have shown that each cluster point of (at ) equals . Therefore, as . This completes the proof.

According to the definition of strongly positive operator in a Banach space having a weakly continuous duality mapping with a gauge function , an operator is said to be * strongly positive * [12] if there exists a constant with the property

where is the identity mapping. We may assume, without loss of generality, that . Therefore, if , then we have the Corollary 3.4 immediately. Indeed, putting and , we have

Taking in Theorem 3.3, we obtain the following result.

Corollary 3.4 (see [12, Lemma 3.3]). *Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a nonexpansive mapping with , a contraction with coefficient , and a strongly positive bounded linear operator with coefficient and . Then, the net defined by
**
converges strongly as to a fixed point of which solves the variational inequality:
*

Corollary 3.5 (see [11, Theorem 3.6]). *Let be a real Hilbert space. Let be a nonexpansive mapping with , a contraction with coefficient , and a strongly positive bounded linear operator with coefficient and . Then, the net defined by
**
converges strongly as to a fixed point of which solves the variational inequality
*

Theorem 3.6. *Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a nonexpansive mapping with , a contraction with coefficient , and and two strongly positive bounded linear operators with coefficients and , respectively. Let be arbitrary and the sequence generated by the following iterative scheme:
**
where and are two constants satisfying the condition and is a real sequence in satisfying the following conditions: *(C1)* and , *(C2)* or . ** Then, the sequence defined by (3.39) converges strongly to a fixed point of that is obtained by Theorem 3.3.*

*Proof. *We first prove that is bounded. Take a , and denote
Using Lemma 3.1, we have
where . By induction, it is easy to see that
Thus, is bounded, and hence so are , , , and . Now, we show that
From the definition of , it is easily seen that
It follows that
where is a constant satisfying . From condition (C2), we deduce that either or . Therefore, it follows from Lemma 2.2 that . It then follows that
Next, we prove that
Let be a subsequence of such that
If follows from reflexivity of and the boundedness of a sequence that there exists which is a subsequence of converging weakly to as . Since is weakly continuous, we have by Lemma 2.1 that
Let
It follows that
From (3.46), we obtain
On the other hand, however,
It follows from (3.52) and (3.53) that
This implies that . Since the duality map is single valued and weakly continuous, we get that
as required. Finally, we show that as
where is a constant satisfying . It then follows that