#### Abstract

We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.

#### 1. Introduction

Let be a normed linear space. Let be a self-mapping on . Then is said to be *asymptotically nonexpansive* if there exists a sequence with such that for each ,
The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [1] as an important generalization of the class of nonexpansive maps (i.e., mappings such that , for all ). We use to denote the set of fixed points of , that is, . A self-mapping is a contraction on if there exists a constant such that
We use to denote the collection of all contractions on . That is, is a contraction on .

A family of mappings of into itself is called a * strongly continuous semigroup of Lipschitzian mappings* on if it satisfies the following conditions: (i) for all ; (ii) for all ; (iii)for each , there exists a bounded measurable function such that , for all ; (iv)for all , the mapping is continuous.

A strongly continuous semigroup of Lipchitszian mappings is called *strongly continuous semigroup of nonexpansive mappings* if for all and *strongly continuous semigroup of asymptotically nonexpansive* if . Note that for asymptotically nonexpansive semigroup , we can always assume that the Lipchitszian constant is such that for each , is nonincreasing in , and ; otherwise we replace , for each , with . We denote by the set of all common fixed points of , that is,
is called * uniformly asymptotically regular * on [2, 3] if for all and any bounded subset of ,
and *almost uniformly asymtotically regular* on [4] if

Let . Then, for each and for a nonexpansive map , there exists a unique point satisfying the following condition: since the mapping is a contraction. When is a Hilbert space and is a self-map, Browder [5] showed that converges strongly to an element of which is nearest to as . This result was extended to more various general Banach space by Morales and Jung [6], Takahashi and Ueda [7], Reich [8], and a host of other authors.

Many authors (see, e.g., [9, 10]) have also shown the convergence of the path , in Banach spaces for asymptotically nonexpansive mapping self-map under some conditions on .

It is an interesting problem to extend the above results to a strongly continuous semigroup of nonexpansive mappings and a strongly continuous semigroup of asymptotically nonexpansive mappings.

Let be a strongly continuous semigroup of nonexpansive self-mappings. In 1998 Shioji and Takahashi [11] introduced, in Hilbert space, the implicit iteration where is a sequence in , is a sequence of positive real numbers divergent to . Under certain restrictions to the sequence , Shioji and Takahashi proved strong convergence of (1.7) to a member of . Recently, Zegeye et al. [4] introduced the implicit (1.7) and the following explicit iteration process for a semigroup of asymptotically nonexpansive mappings: where and in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm. Suppose, in addition, that is almost uniformly asymptotically regular. Then the implicit sequence (1.7) and explicit sequence (1.8) converge strongly to a point of .

On the other hand, 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 [12]). Browder [12] initiated the study of certain classes of nonlinear operators by means of the duality mapping . Following Browder [12], 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.

In a Banach space having a weakly continuous duality mapping with a gauge function , an operator is said to be * strongly positive *[13] if there exists a constant with the property
where is the identity mapping. If is a real Hilbert space, then the inequality (1.12) reduces to
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 2009, motivated and inspired by Marino and Xu [14], Li et al. [15] introduced the following general iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup on a nonempty closed convex subset in a Hilbert space:
where and are sequences in and , respectively, is a strongly positive bounded linear operator on , and is a contraction on . And their convergence theorems can be proved under some appropriate control conditions on parameter and . Furthermore, by using these results, they obtained two mean ergodic theorems for nonexpansive mappings in a Hilbert space.

All of the above brings us to the following conjectures.

*Question 1. *Could we obtain strong convergence theorems for the general class of strongly continuous semigroup of asymptotically nonexpansive mappings in more general Banach spaces? such as a reflexive Banach space which admits a weakly continuous duality mapping , where is a gauge function.

In this paper, inspired and motivated by Shioji and Takahashi [11], Zegeye et al. [4], Marino and Xu [14], Li et al. [15], and Wangkeeree et al. [13], we prove the strong convergence theorems of the iterative approximation methods (1.16) for the general class of the strongly continuous semigroup of asymptotically nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping , where is a gauge function and is a strongly positive bounded linear operator on a Banach space . The results in this paper generalize and improve some well-known results in Shioji and Takahashi [11], Li et al. [15], and many others.

#### 2. Preliminaries

Throughout this paper, let be a real Banach space and be 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 [16]). 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 .

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 [10].

Lemma 2.1 (see [10]). *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:
*

The next valuable lemma is proved for applying our main results.

Lemma 2.2 (see [13, Lemma 3.1]). *Assume that a Banach space has a weakly continuous duality mapping with gauge . Let be a strong positive linear bounded operator on with coefficient and . Then . *

In the following, we also need the following lemma that can be found in the existing literature [17, 18].

Lemma 2.3 (see [18, Lemma 2.1]). *Let be a sequence of a nonnegative real number satisfying the property
**
where and such that and . Then converges to zero, as . *

#### 3. Main Theorem

Theorem 3.1. *Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a strongly continuous semigroup of asymptotically nonexpansive mappings on with a sequence and . Let with coefficient , let be a strongly positive bounded linear operator with coefficient and , and let and be sequences of real numbers such that , . Then the following holds.*(i)*If , for all , then there exists a sequence defined by
*(ii)*Suppose, in addition, that is almost uniformly asymptotically regular and the real sequences and satisfy the following conditions: (B1) ; (B2); (B3). *

*Then converges strongly as to a common fixed point in which solves the variational inequality:*

*Proof. *We first show the uniqueness of a solution of the variational inequality (3.2). Suppose both and are solutions to (3.2), then
Adding (3.3), we obtain
Noticing that for any ,
Therefore and the uniqueness is proved. Below, we use to denote the unique solution of (3.2).

Since , we may assume, without the loss of generality, that .

For each integer , define a mapping by
We show that is a contraction mapping. For any ,
Since , we have
It then follows that . We have is a contraction map with coefficient . Then, for each , there exists a unique such that , that is,
Hence (i) is proved.

(ii) We first show that is bounded. Letting and using Lemma 2.2, we can calculate the following:
Thus, we get that
Calculating the right-hand side of the above inequality, we have
Thus, we get that
where . Thus, there exists such that , for all . Therefore, is bounded and hence and are also bounded.

Let . Then, from (3.1), we get
Since is almost uniformly asymptotically regular and (3.14), we have
It follows from the 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
For , from (3.15) we obtain
On the other hand, however,
It follows from (3.19) and (3.20) that
This implies that for all , and so . Next, we show that as . In fact, since , for all , and is a gauge function, then for , and
Following Lemma 2.1, we have
This implies that
also
where . Now observing that implies , we conclude from the above inequality that
Hence as . Next, we prove that solves the variational inequality (3.2). For any , we observe that
Since
we can derive that
Since is strictly increasing and for some , we have . Thus
Notice that
Now using (B3) and replacing with in (3.30) and letting , we have
So, is a solution of the variational inequality (3.2), and hence by the uniqueness. In a summary, we have shown that each cluster point of (at ) equals . Therefore, as . This completes the proof.

If , the identity mapping on , and , then Theorem 3.1 reduces to the following corollary.

Corollary 3.2. *Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a strongly continuous semigroup of asymptotically nonexpansive mappings on with a sequence and . Let with coefficient and let and be sequences of real numbers such that and . Then the following holds.*(i)*If , for all , then there exists a sequence defined by
*(ii)*Suppose, in addition, that is almost uniformly asymptotically regular and the real sequences and satisfy the following: (B1) ; (B2); (B3). *

*Then converges strongly as to a common fixed point in which solves the variational inequality:*

If is a Hilbert space and is a strongly continuous semigroup of nonexpansive mappings on , then we have and Theorem 3.1 reduces to the following corollary.

Corollary 3.3 (see [15, Theorem 3.1]). *Let be a real Hilbert space. Suppose that is a contraction with coefficient and a strongly continuous semigroup of nonexpansive mappings on such that . Let be a strongly positive bounded linear operator with coefficient and let and be sequences of real numbers such that , such that and , then for any , there is a unique in such that
**
and the iterative sequence converges strongly as to a common fixed point in which solves the variational inequality:
*

Theorem 3.4. *Let be a reflexive strictly convex Banach space which admits a weakly continuous duality mapping with gauge such that is invariant on . Let be a strongly continuous semigroup of asymptotically nonexpansive mappings on with a sequence and . Let with coefficient ; let be a strongly positive bounded linear operator with coefficient and . For any , let the sequence be defined by
**
Suppose, in addition, that is almost uniformly asymptotically regular. Let and be sequences of real numbers such that , , *(C1)*; *(C2)*, ; *(C3)*. **
Then converges strongly as to a common fixed point in which solves the variational inequality (3.2). *

*Proof. *First we show that is bounded. By condition (C3) and given there exists such that for all . Thus
for all . Since , we may assume, without the loss of generality, that .

Claim that , , where .

Let . Then from (3.56) we get that
By induction,
and hence is bounded, so are and . Let . Then, since as , we obtain that
For any , we have
it follows from (3.41) and is almost uniformly asymptotically regular that
Next, we prove that
Let be a subsequence of such that
It follows from the reflexivity of and the boundedness of 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 (3.43), for each , we obtain
On the other hand, however,
It follows from (3.49) and (3.50) that
This implies that for all , and so . Since the duality map is single valued and weakly continuous, we get that
as required. Finally, we show that as . It follows from Lemma 2.1(i) that
where such that . Put