Research Article | Open Access

# Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

**Academic Editor:**Jong Kyu Kim

#### 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.

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Copyright © 2014 Kyung Soo Kim. 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.