#### Abstract

We introduce a new modified Halpern iteration for a countable infinite family of
nonexpansive mappings in convex metric spaces. We prove that the sequence generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping
when satisfies the AKTT-condition, and strong convergence theorems of the proposed iteration
to a common fixed point of a countable infinite family of nonexpansive mappings in CAT(0)
spaces are established under AKTT-condition and the SZ-condition. We also generalize the concept
of *W*-mapping for a countable infinite family of nonexpansive mappings from a Banach space
setting to a convex metric space and give some properties concerning the common fixed point
set of this family in convex metric spaces. Moreover, by using the concept of *W*-mappings, we
give an example of a sequence of nonexpansive mappings defined on a convex metric space which
satisfies the AKTT-condition. Our results generalize and refine many known results in the current
literature.

#### 1. Introduction

Let be a nonempty closed convex subset of a metric space , and let be a mapping of into itself. A mapping is called *nonexpansive* if for all . The set of all fixed points of is denoted by , that is, .

In 1967, Halpern [1] introduced the following iterative scheme in Hilbert spaces which was referred to as *Halpern iteration* for approximating a fixed point of :
where are arbitrarily chosen, and is a sequence in . Wittmann [2] studied the iterative scheme (1.1) in a Hilbert space and obtained the strong convergence of the iteration. Reich [3] and Shioji and Takahashi [4] extended Wittmann's result to a real Banach space.

The modified version of Halpern iteration was investigated widely by many mathematicians. For instance, Kim and Xu [5] studied the sequence generated as follows: where are arbitrarily chosen and , are two sequences in . They proved the strong convergence of iterative scheme (1.2) in the framework of a uniformly smooth Banach space. In 2007, Aoyama et al. [6] introduced a Halpern iteration for finding a common fixed point of a countable infinite family of nonexpansive mappings in a Banach space as follows: where are arbitrarily chosen, is a sequence in , and is a sequence of nonexpansive mappings with some conditions. They proved that the sequence generated by (1.3) converges strongly to a common fixed point of . In 2010, Saejung [7] extended the results of Halpern [1], Wittmann [2], Reich [3], Shioji and Takahashi [4], and Aoyama et al. [6] to the case of a CAT(0) space which is an example of a convex metric space. Recently, Cuntavepanit and Panyanak [8] extended the result of Kim and Xu [5] to a CAT(0) space.

Takahashi [9] introduced the concept of convex metric spaces by using the convex structure as follows. Let be a metric space. A mapping is said to be a *convex structure* on if for each and ,
for all . A metric space together with a convex structure is called a *convex metric space* which will be denoted by . A nonempty subset of is said to be *convex* if for all and . Clearly, a normed space and each of its convex subsets are convex metric spaces, but the converse does not hold.

Motivated by the above results, we introduce a new iterative scheme for finding a common fixed point of a countable infinite family of nonexpansive mappings of into itself in a convex metric space as follows: where are arbitrarily chosen, and , are two sequences in . The main propose of this paper is to prove the convergence theorem of the sequence generated by (1.5) to a common fixed point of a countable infinite family of nonexpansive mappings in convex metric spaces and CAT(0) spaces under certain suitable conditions.

#### 2. Preliminaries

We recall some definitions and useful lemmas used in the main results.

Lemma 2.1 (see [9, 10]). *Let be a convex metric space. For each and , we have the following. *(i)* and .*(ii)* and . *(iii)*. *(iv)*.*

We say that a convex metric space has the property: (C)if for all and , (I)if for all and , (H)if for all and , (S)if for all and .

From the above properties, it is obvious that the property (C) and (H) imply continuity of a convex structure . Clearly, the property (S) implies the property (H). In [10], Aoyama et al. showed that a convex metric space with the property (C) and (H) has the property (S).

In 1996, Shimizu and Takahashi [11] introduced the concept of uniform convexity in convex metric spaces and studied some properties of these spaces. A convex metric space is said to be *uniformly convex* if for any , there exists such that for all and with , and imply that . Obviously, uniformly convex Banach spaces are uniformly convex metric spaces. In fact, the property (I) holds in uniformly convex metric spaces, see [12].

Lemma 2.2. *Property (C) holds in uniformly convex metric spaces.*

*Proof. *Suppose that is a uniformly convex metric space. Let and . It is obvious that the conclusion holds if or . So, suppose . By Lemma 2.1(ii), we have

We will show that . To show this, suppose not. Put and . Let , , , and . It is easy to see that . Since is uniformly convex, we have
By , we get . Since and , then
This is a contradiction. Hence, .

By Lemma 2.2, it is clear that a uniformly convex metric space with the property (H) has the property (S), and the convex structure is also continuous.

Next, we recall the special space of convex metric spaces, namely, CAT(0) spaces. Let be a metric space. A *geodesic path* joining to (or, more briefly, a *geodesic* from to ) is a map from a closed interval to such that and for all . In particular, is an isometry and . The image of is called a *geodesic* (or *metric*) *segment* joining and . When unique, this geodesic is denoted . The space is said to be a *geodesic metric space* if every two points of are joined by a geodesic, and is said to be *uniquely geodesic* if there is exactly one geodesic joining and for each . A subset of is said to be *convex* if includes every geodesic segment joining any two of its points.

A *geodesic triangle * in a geodesic metric space consists of three points in (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A *comparison triangle* for geodesic triangle in is a triangle in the Euclidean plane such that for .

A geodesic metric space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom. Let be a geodesic triangle in , and let be a comparison triangle for . Then is said to satisfy the CAT(0) inequality if for all and all comparison points , .

If are points in a CAT(0) space and if is the midpoint of the segment , then the CAT(0) inequality implies This is the (CN) inequality of Bruhat and Tits [13], which is equivalent to for any , where denotes the unique point in . The (CN*) inequality has appeared in [14]. By using the (CN) inequality, it is easy to see that the CAT(0) spaces are uniformly convex. In fact [15], a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality. Moreover, if is CAT(0) space and , then for any , there exists a unique point such that for any . It follows that CAT(0) spaces have convex structure . It is clear that the properties (C), (I), and (S) are satisfied for CAT(0) spaces, see [15, 16]. This is also true for Banach spaces.

Let be a continuous linear functional on , the Banach space of bounded real sequences, and let . We write instead of . We call a *Banach limit* if satisfies and for each . For a Banach limit , we know that for all . So if with , then , see also [17].

Lemma 2.3 ([4], Proposition 2). *Let be such that for all Banach limit . If , then .*

Lemma 2.4 ([6], Lemma 2.3). *Let be a sequence of nonnegative real numbers, let be a sequence of real numbers in with , let be a sequence of nonnegative real numbers with , and let be a sequence of real numbers with . Suppose that
**
Then .*

Lemma 2.5 ([18], Lemma 1). *Let be a uniformly convex metric space with a continuous convex structure . Then for arbitrary positive number and , there exists such that
**
for all , , , , and .*

*Remark 2.6. *The above lemma also holds for a uniformly convex metric space with the property (H).

#### 3. Main Results

The following condition was introduced by Aoyama et al. [6]. Let be a subset of a complete convex metric space , and let be a countable infinite family of mappings from into itself. We say that satisfies *AKTT-condition* if
for each bounded subset of . If is a closed subset and satisfies AKTT-condition, then we can define a mapping such that for all . In this case, we also say that satisfies AKTT-condition. By using the same argument as in [6, Lemma 3.2], we have the following lemma.

Lemma 3.1. *If satisfies AKTT-condition, then for all bounded subsets of .*

Theorem 3.2. *Let be a nonempty closed convex subset of a complete convex metric space with the properties (I) and (S). Let be a family of nonexpansive mappings of into itself such that . Suppose that is a sequence of generated by (1.5), and let and be sequences in which satisfy the conditions: *(C1)*, , and ,*(C2)* for some and . ** Suppose that satisfies AKTT-condition. Then and .*

*Proof. *Let . By the definition of and , we have
By induction on , we obtain that for all and all . Hence, the sequence is bounded and so , , are bounded.

It follows by condition that
By the definition of and , we have
where .

Putting , we have
Hence, it follows from conditions , , AKTT-condition, and Lemma 2.4 that
Now, observe that
We obtain
This implies by (3.3) and (3.6) that . Therefore, we have
Since
it follows by (3.3) and (3.9) that
By (3.11) and Lemma 3.1, we get

Next, we consider a convergence theorem in CAT(0) spaces. The following two lemmas obtained by Saejung [7] are useful for our main results.

Lemma 3.3. *Let be a closed convex subset of a complete CAT(0) space , and let be a nonexpansive mapping. Let be fixed. For each , the mapping defined by for has a unique fixed point , that is, .*

Lemma 3.4. *Let , be as the preceding lemma. Then if and only if remains bounded as . In this case, the following statements hold: *(i)* converges to the unique fixed point of which is nearest to ; *(ii)* for all Banach limit and all bounded sequences with . *

Previously, we know that CAT(0) spaces have convex structure and also have the properties (C), (I), and (S). Thus, we have the following result.

Theorem 3.5. *Let be a nonempty closed convex subset of a complete CAT(0) space . Let be a family of nonexpansive mappings of into itself such that . Suppose that are arbitrarily chosen and is a sequence of generated by
**
where and are sequences in which satisfy the conditions and as in Theorem 3.2. Suppose that satisfies AKTT-condition. Then and .*

Theorem 3.6. *Let be a nonempty closed convex subset of a complete CAT(0) space . Let be a family of nonexpansive mappings of into itself such that . Suppose that is a sequence of generated by (3.13), and let and be sequences in which satisfy the conditions and as in Theorem 3.2. Suppose that satisfies AKTT-condition and . Then converges strongly to a common fixed point of which is nearest to .*

*Proof. *By Theorem 3.5, we have . For each , let be a unique point of such that . It follows from Lemma 3.4 that converges to a point which is nearest to , and
that is, . Moreover, by Theorem 3.5, we get . It follows that
By and Lemma 2.3, we obtain
Finally, we show that . By the definition of and , we have
This implies by , inequality (3.16), and Lemma 2.4 that . Hence, converges to which is nearest to .

Corollary 3.7 (see [7], Theorem 8). *Let be a nonempty closed convex subset of a complete CAT(0) space . Let be a family of nonexpansive mappings of into itself such that . Suppose that are arbitrarily chosen and is a sequence of generated by
**
where is a sequence in which satisfies the condition as in Theorem 3.2. Suppose that satisfies AKTT-condition and . Then converges strongly to a common fixed point of which is nearest to .*

*Proof. *By putting for all in Theorem 3.6, we obtain the desired result.

In 2009, Song and Zheng [19] introduced a condition in Banach spaces for a countable infinite family of nonexpansive mappings which is different from AKTT-condition and also give some examples of a family of mappings that satisfies this condition. Now, we state this condition in CAT(0) spaces, and it is referred as SZ-condition as follows. Let be a nonempty closed convex subset of a complete CAT(0) space . Suppose that is a family of nonexpansive mappings from into itself with . We say that satisfies *SZ-condition* if, for any bounded subset of , there exists a nonexpansive mapping of into itself such that

Theorem 3.8. *Let be a nonempty closed convex subset of a complete CAT(0) space . Let be a family of nonexpansive mappings of into itself such that and satisfies SZ-condition. Suppose that is a sequence of defined by (3.13) with . Let and be sequences in which satisfy the following conditions: *(C3)*, , and , *(C4)*. ** Then converges strongly to a common fixed point of which is nearest to .*

*Proof. *As in the proof of Theorem 3.2, we have that and are bounded. Since satisfies SZ-condition, there exists a nonexpansive mapping of into itself such that and . By the definition of and , we have
It follows from condition and that
Since
this implies by (3.21) and SZ-condition, we have
From and
it follows that
By using the same arguments and techniques as those of Theorem 3.6, we can show that converges to a common fixed point of which is nearest to .

Corollary 3.9. *Let be a nonempty closed convex subset of a complete CAT(0) space . Let be a family of nonexpansive mappings of into itself such that and satisfies SZ-condition. Suppose that is a sequence of defined by (3.18) with . Let be a sequence in which satisfies the condition as in Theorem 3.8. Then converges strongly to a common fixed point of which is nearest to .*

*Proof. *By putting for all in Theorem 3.8, we obtain the desired result.

#### 4. W-Mapping in Convex Metric Spaces

In Theorems 3.2, 3.5, and 3.6 and Corollary 3.7, to obtain a convergence result, we have to assume that satisfies AKTT-condition. In general, one cannot apply these results for a sequence of nonexpansive mappings. However, we give an example of a sequence of nonexpansive mappings satisfying AKTT-condition.

Let be a family of nonexpansive mappings of into itself, where is a convex subset of a convex metric space . We now define mappings and as follows. For a sequence in and ,
Such a mapping is called the *-mapping* generated by and .

In 2007, Shimizu [18] generalized -mapping which was introduced by Takahashi [20] from Banach spaces to convex metric spaces. Then, the following result is obtained by using the same proof as in of [18, Lemma 2].

Lemma 4.1. *Let be a nonempty closed convex subset of a uniformly convex metric space with a continuous convex structure . Let be nonexpansive mappings of into itself such that and let be real numbers such that for every . Let be the -mapping of into itself generated by and . Then .*

Next, we consider the -mapping given by a countable infinite family of nonexpansive mappings in a uniformly convex metric space.

Lemma 4.2. *Let be a nonempty closed convex subset of a complete uniformly convex metric space with the property (H). Let be a family of nonexpansive mappings of into itself such that , and let be real numbers such that for every . Then for every , and , exists.*

*Proof. *Let and . Fix . Then for any with , we have
Thus for ,
It follows that is a Cauchy sequence. Hence, exists.

Using the above lemma, one can define mappings and of into itself as for every . Such a mapping is called the -mapping generated by and .

Lemma 4.3. *Let be a nonempty closed convex subset of a complete uniformly convex metric space with the property (H). Let be a family of nonexpansive mappings of into itself such that , and let be real numbers such that for every . Let be the -mapping generated by and . Then, is a nonexpansive mapping and .*

*Proof. *First, we show that is a nonexpansive mapping. For , we have
This implies that is a nonexpansive mapping, and we have . Thus, is also a nonexpansive mapping.

Finally, we show that . Let . Then, it is obvious that for all with . So we have for all . Therefore, we have , and hence, . We now show that . Let and let . Then we have
Taking , we obtain
Since , we have . Then, for , , we have
for every . Suppose that . Then . It follows by Lemma 2.5, we have
This is a contradiction. Hence, . Since , we have
So, we have for every . This implies that . Therefore, we have .

Lemma 4.4. *Suppose that are as in Lemma 4.3. Let and be the -mappings generated by and , and and , respectively. Then satisfies AKTT-condition, and .*

*Proof. *Let be a bounded subset of and . For , we have
This implies
Thus, satisfies AKTT-condition. Moreover, from Lemmas 4.1โ4.3, we obtain that .

*Remark 4.5. *Lemmas 4.2 and 4.3 were proved in Banach spaces by Shimoji and Takahashi [21], and Lemma 4.4 was proved in Banach spaces by Peng and Yao [22].

*Remark 4.6. *Suppose that are as in Lemma 4.3. Let and be the -mappings generated by and , and and , respectively. By Lemma 4.4, we know that satisfies the AKTT-condition and . Therefore, in Theorems 3.2, 3.5, and 3.6 and Corollary 3.7, the mapping can be also replaced by without assuming the AKTT-condition and .

#### Acknowledgments

The authors would like to thank the referees for valuable suggestions on the paper and the National Research University Project under Thailand's Office of the Higher Education Commission, the Commission on Higher Education for financial support. The first author is supported by the Office of the Higher Education Commission and the Graduate School of Chiang Mai University, Thailand.