#### Abstract

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappings on a nonempty closed convex subset of a Banach space with respect to a sequence of asymptotically left invariant means defined on an appropriate invariant subspace of , where is a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed points , where .

#### 1. Introduction

Let be a real Banach space with the topological dual and let be a closed and convex subset of . A mapping of into itself is called* nonexpansive* if for each .

Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Halpern [1] and is defined as follows: where is a sequence in . He pointed out that the conditions and are necessary in the sense that if the iteration (1) converges to a fixed point of , then these conditions must be satisfied. The second iteration process is known as Mann’s iteration process [2] which is defined as follows: where the initial is taken in arbitrary and the sequence is in .

The third iteration process is referred to as Ishikawa’s iteration process [3] which is defined as follows: where the initial is taken in arbitrary and and are sequences in .

In 2007, Lau et al. [4] proposed the following modification of Halpern’s iteration (1) for amenable semigroups of nonexpansive mappings in a Banach space.

Theorem 1. *Let be a left reversible semigroup and let be a representation of as nonexpansive mappings from a compact convex subset of a strictly convex and smooth Banach space into , let be an amenable and -stable subspace of , and let be a strongly left regular sequence of means on . Let be a sequence in such that and . Let and let be the sequence defined by**
Then converges strongly to , where denotes the unique sunny nonexpansive retraction of onto .*

Let be a closed and convex subset of and let be a mapping from into itself. We denote by the set of fixed points of . Point in is said to be an* asymptotic fixed point* of [5] if contains a sequence which converges weakly to such that the strong . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called* relatively nonexpansive* [6–8], if and for all and . The asymptotic behavior of relatively nonexpansive mappings was studied in [6, 7, 9].

Recently, Kim [10] proved a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the shrinking method.

Theorem 2. *Let be a left reversible semigroup and let be a representation of as relatively nonexpansive mappings from a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space into with . Let be a subspace of and let be a asymptotically left invariant sequence of means on . Let be a sequence in such that and . Let be a sequence generated by the following algorithm:
**
Then converges strongly to , where is the generalized projection from onto .*

Let be a semigroup. The purpose of this paper is to study modified Halpern type and Ishikawa type iterations for a semigroup of relatively nonexpansive mappings on a nonempty closed convex subset of a Banach space with respect to a sequence of asymptotically left invariant means defined on an appropriate invariant subspace of . We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed points , where .

#### 2. Preliminaries

A real Banach space is said to be* strictly convex* if for all with and . It is said to be* uniformly convex* if for any two sequences and in such that and . Let be the unit sphere of . Then the Banach space is said to be* smooth* if
exists for each . It is said to be* uniformly smooth* if the limit is attained uniformly for .

Let be a real Banach space with norm and let be the dual space of . Denote by the duality product. We denote by the normalized duality mapping from to defined by for . A Banach space is said to have the Kadec-Klee property if a sequence of satisfies that and and then , where and denote the weak convergence and the strong convergence, respectively.

We know the following:(1)the duality mapping is monotone, that is, whenever and ;(2)if is strictly convex, then is one-to-one; that is, if is nonempty, then ;(3)if is strictly convex, then is strictly monotone; that is, whenever , and ;(4)if is uniformly convex, then has the Kadec-Klee property;(5)if is uniformly convex, then is reflexive and strictly convex;(6)if is smooth, then is single-valued and norm-to- continuous;(7)if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of ;(8)if is reflexive, then is onto;(9)if is smooth and reflexive, then is norm-to-weak continuous; that is, whenever ;(10)if is smooth, strictly convex, and reflexive, then is single-valued, one-to-one and onto; in this case, the inverse mapping coincides with the duality mapping on ;(11)if is strictly convex, then is single-valued;(12)the norm of is Fréchet differentiable if and only if is strictly convex and reflexive Banach space which has the Kadec-Klee property.

For more details, see [11].

As well known, if is a nonempty, closed, and convex subset of a Hilbert space and is the metric projection of onto , then is nonexpansive (see, the reference therein). This fact actually characterizes Hilbert spaces. Consequently, it is not true to more general Banach spaces. In this connection, Alber [12] introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces. Consider the function defined by
for . Observe that, in a Hilbert space , (8) reduces to
for . The generalized projection is a mapping that assigns an arbitrary point to the minimum point of the functional ; that is, , where is the solution to the minimization problem:
The existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [12, 13]). In a Hilbert space, . It is obvious from the definition of the function that (*ϕ*_{1}) for all ,(*ϕ*_{2}) for all ,(*ϕ*_{3}) for all ,(*ϕ*_{4})if is a reflexive, strictly convex, and smooth Banach space, then, for all ,
For more details see [14].

Let be a semigroup. We denote by the Banach space of all bounded real-valued functionals on with supremum norm. For each , we define the left and right translation operators and on by
for each and , respectively. Let be a subspace of containing . An element in the dual space of is said to be a* mean* on if . For , we can define a point evaluation by for each . It is well known that is mean on if and only if
for each .

Let be a translation invariant subspace of (i.e., and for each ) containing . Then a mean on is said to be* left invariant* (resp.,* right invariant*) if
for each and . A mean on is said to be* invariant* if is both left and right invariant [15–19]. is said to be* left* (resp.,* right*)* amenable* if has a left (resp., right) invariant mean. is amenable if is left and right amenable. We call a semigroup * amenable* if is amenable. Further, amenable semigroups include all commutative semigroups and solvable groups. However, the free group or semigroup of two generators is not left or right amenable (see [20–22]).

A net of means on is said to be* asymptotically left* (resp.,* right*)* invariant* if
for each and , and it is said to be* left* (resp.,* right*)* strongly asymptotically invariant* (or* strong regular*) if
for each , where and are the adjoint operators of and , respectively. Such nets were first studied by Day in [20] where they were called * invariant* and* norm invariant*, respectively.

It is easy to see that if a semigroup is left (resp., right) amenable, then the semigroup , where for all , is also left (resp., right) amenable and converse.

From now on denotes a semigroup with an identity . is called* left reversible* if any two right ideals of have nonvoid intersection; that is, for . In this case, is a directed system when the binary relation “” on is defined by if and only if for . It is easy to see that for all . Further, if then for all . The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup is left amenable, then is left reversible. But the converse is not true [23–28].

Let be a semigroup and let be a closed and convex subset of . Let denote the fixed point set of . Then is called a* representation of ** as relatively nonexpansive mappings on * if is relatively nonexpansive with and for each . We denote by the set of common fixed points of ; that is,

We know that if is a mean on and if for each the function is contained in and is weakly compact, then there exists a unique point of such that for each . We denote such a point by . Note that is contained in the closure of the convex hull of for each . Note that for each ; see [29–31].

#### 3. Lemmas

We need the following lemmas for the proof of our main results.

Lemma 3 (see [9]). *Let be a strictly convex and smooth Banach space, let be a closed convex subset of , and let be a relatively nonexpansive mapping from into itself. Then is closed and convex.*

Lemma 4 (see [12, 32]). *Let be a reflexive, strictly convex, and smooth Banach space and let be a nonempty, closed, and convex subset of and . Then
**
for all .*

Lemma 5 (see [32]). *Let be a uniformly convex and smooth Banach space and let , be two sequences of . If and either or is bounded, then .*

Lemma 6 (see [4, 33]). *Let be a left invariant mean on . Then , where denotes the set of almost periodic elements in ; that is, all such that is relatively compact in the norm topology of .*

Lemma 7 (cf. [4, 10]). *Let be an asymptotically left invariant sequence of means on . If and , then is a common fixed point of .*

#### 4. Strong Convergence Theorems

In this section, we will establish two strong convergence theorems of various iterative sequences for finding common fixed point of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach spaces (cf. [34–36]).

We begin with a strong convergence theorem of modified Halpern’s type.

Theorem 8. *Let be a left reversible semigroup and let be a representation of as relatively nonexpansive mappings from a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space into itself. Let be a subspace of and let be an asymptotically left invariant sequence of means on . Let be a sequence in such that . Let be a sequence generated by the following algorithm:
**If the interior of is nonempty, then converges strongly to some common fixed point .*

*Proof. *We show first that the sequence converges strongly in .

From Lemma 3, we know is closed and convex. So, we can define the generalized projection onto . Most of all, we have

Then, from the definition of relatively nonexpansive, we have
for all . From the convexity of and (21), we get
So, we have
Since , we obtain
Therefore is bounded and exists. Then is also bounded. This implies that is bounded. Since the interior of is nonempty, there exist and such that
whenever . By , we have
for any . This implies
Also, we have
On the other hand, by (24) and (25), we have that
From (27), we get
Then, by (27), we have
for . Hence

Since with is arbitrary, by (24), we have

So, we have
We know that converges. Hence, is a Cauchy sequence. Since is complete, converges strongly to some point in . Since is uniformly convex, has a Fréchet differentiable norm. Then is continuous on . Hence converges strongly to some point in .

Now, we show that , where .

By (33) and the convergence of , we have
Since is uniformly norm-to-norm continuous on bounded sets, it follows that
Let . Then, we have
Since , we have
Since is uniformly norm-to-norm continuous on bounded sets, we get
From and Lemma 4, we have
Since
and , we have
From (40), we get
By Lemma 5, we obtain
Since , from (36), (39), and (44), we have
From Lemma 7, we have . Since is closed and , we have , where .

We now establish a convergence theorem of modified Ishikawa type.

Theorem 9. *Let be a left reversible semigroup and let be a representation of as relatively nonexpansive mappings from a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space into itself. Let be a subspace of and let be an asymptotically left invariant sequence of means on . Let and be sequences of real numbers such that and , . Let be a sequence generated by the following algorithm:
**
If the interior of is nonempty, then converges strongly to some common fixed point .*

*Proof. *Firstly, we show that converges strongly in .

From Lemma 3, we know is closed and convex. So, we can define the generalized projection onto . Let . From the definition of relatively nonexpansive and the convexity of , from (21), we have

for all . From (47), we obtain
Hence, is bounded and exists. This implies that , , and are bounded. Since the interior of is nonempty, similar to the proof of Theorem 8, we obtain that converges strongly to in .

Next, we show that , where .

Let
From Lemma 4, we have
Also,
From and (52), we have
Since is uniformly norm-to-norm continuous, we obtain
Hence,
By (53) and (54), we have
From (50) and (51), we obtain
From Lemma 5, we get
Since
and , we have
Since is uniformly norm-to-norm continuous, we obtain
Since and is uniformly norm-to-norm continuous,
By (46) and (49), we have
From (63), we obtain
Combining (53), (62), and (64), we get
Since is uniformly norm-to-norm continuous, we have
Since
therefore, by (58), (61), (66), and (67), we obtain
From Lemma 7, we have . Since is closed and , we have , where .

If we set , then the iteration (46) reduces modified Mann type. Hence we obtain the following corollary.

Corollary 10. *Let be a left reversible semigroup and let be a representation of as relatively nonexpansive mappings from a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space into itself. Let be a subspace of and let be an asymptotically left invariant sequence of means on . Let be a sequence of real number such that and . Let be a sequence generated by the following algorithm:
**
If the interior of is nonempty, then converges strongly to some common fixed point .*

In a Hilbert space, is the identity operator. Theorems 8 and 9 reduce to the following.

Corollary 11. *Let be a left reversible semigroup and let be a representation of as relatively nonexpansive mappings from a nonempty, closed, and convex subset of a Hilbert space into itself. Let be a subspace of and let be an asymptotically left invariant sequence of means on . Let be a sequence in such that . Let be a sequence generated by the following algorithm:
**
If the interior of is nonempty, then converges strongly to some common fixed point , where is a metric projection.*

Corollary 12. *Let be a left reversible semigroup and let be a representation of as relatively nonexpansive mappings from a nonempty, closed, and convex subset of a Hilbert space into itself. Let be a subspace of and let be an asymptotically left invariant sequence of means on . Let and be sequences of real numbers such that and , . Let be a sequence generated by the following algorithm:
**
If the interior of is nonempty, then converges strongly to some common fixed point , where is a metric projection.*

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author would like to thank Professor Anthony To-Ming Lau and Professor Jong Kyu Kim for their helpful suggestions. Also, special thanks are due to the referee for his/her deep insight which improved the paper. This work was supported by Kyungnam University Foundation Grant, 2013.