Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces

Abstract

We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings {𝑇_𝑛} in convex metric spaces. We prove that the sequence {𝑥_𝑛} generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when {𝑇_𝑛} 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 𝐶 be a nonempty closed convex subset of a metric space (𝑋,𝑑), and let 𝑇 be a mapping of 𝐶 into itself. A mapping 𝑇 is called nonexpansive if 𝑑(𝑇𝑥,𝑇𝑦)≤𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝐶. The set of all fixed points of 𝑇 is denoted by 𝐹(𝑇), that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}.

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 𝑇:𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛∀𝑛∈ℕ,(1.1) where 𝑥₁,𝑢∈𝐶 are arbitrarily chosen, and {𝛼_𝑛} is a sequence in [0,1]. 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 {𝑥_𝑛} generated as follows:𝑥𝑦_𝑛𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,=𝛽_𝑛𝑢+1−𝛽_𝑛𝑦_𝑛∀𝑛∈ℕ,(1.2) where 𝑥₁,𝑢∈𝐶 are arbitrarily chosen and {𝛼_𝑛}, {𝛽_𝑛} are two sequences in [0,1]. 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:𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛∀𝑛∈ℕ,(1.3) where 𝑥₁,𝑢∈𝐶 are arbitrarily chosen, {𝛼_𝑛} is a sequence in [0,1], and {𝑇_𝑛} is a sequence of nonexpansive mappings with some conditions. They proved that the sequence {𝑥_𝑛} generated by (1.3) converges strongly to a common fixed point of {𝑇_𝑛}. 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 (𝑋,𝑑) be a metric space. A mapping 𝑊∶𝑋×𝑋×[0,1]→𝑋 is said to be a convex structure on 𝑋 if for each 𝑥,𝑦∈𝑋 and 𝜆∈[0,1], 𝑑(𝑧,𝑊(𝑥,𝑦,𝜆))≤𝜆𝑑(𝑧,𝑥)+(1−𝜆)𝑑(𝑧,𝑦),(1.4) for all 𝑧∈𝑋. A metric space (𝑋,𝑑) together with a convex structure 𝑊 is called a convex metric space which will be denoted by (𝑋,𝑑,𝑊). A nonempty subset 𝐶 of 𝑋 is said to be convex if 𝑊(𝑥,𝑦,𝜆)∈𝐶 for all 𝑥,𝑦∈𝐶 and 𝜆∈[0,1]. 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 {𝑇_𝑛} of 𝐶 into itself in a convex metric space as follows:𝑥𝑦_𝑛𝑛+1=𝑊𝑢,𝑇_𝑛𝑥_𝑛,𝛼_𝑛,=𝑊𝑦_𝑛,𝑇_𝑛𝑦_𝑛,𝛽_𝑛∀𝑛∈ℕ,(1.5) where 𝑥₁,𝑢∈𝐶 are arbitrarily chosen, and {𝛼_𝑛}, {𝛽_𝑛} are two sequences in [0,1]. The main propose of this paper is to prove the convergence theorem of the sequence {𝑥_n} 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 (𝑋,𝑑,𝑊) be a convex metric space. For each 𝑥,𝑦∈𝑋 and 𝜆,𝜆₁,𝜆₂∈[0,1], we have the following. (i)𝑊(𝑥,𝑥,𝜆)=𝑥,𝑊(𝑥,𝑦,0)=𝑦 and 𝑊(𝑥,𝑦,1)=𝑥.(ii)𝑑(𝑥,𝑊(𝑥,𝑦,𝜆))=(1−𝜆)𝑑(𝑥,𝑦) and 𝑑(𝑦,𝑊(𝑥,𝑦,𝜆))=𝜆𝑑(𝑥,𝑦). (iii)𝑑(𝑥,𝑦)=𝑑(𝑥,𝑊(𝑥,𝑦,𝜆))+𝑑(𝑊(𝑥,𝑦,𝜆),𝑦). (iv)|𝜆₁−𝜆₂|𝑑(𝑥,𝑦)≤𝑑(𝑊(𝑥,𝑦,𝜆₁),𝑊(𝑥,𝑦,𝜆₂)).

We say that a convex metric space (𝑋,𝑑,𝑊) has the property: (C)if 𝑊(𝑥,𝑦,𝜆)=𝑊(𝑦,𝑥,1−𝜆) for all 𝑥,𝑦∈𝑋 and 𝜆∈[0,1], (I)if 𝑑(𝑊(𝑥,𝑦,𝜆₁),𝑊(𝑥,𝑦,𝜆₂))≤|𝜆₁−𝜆₂|𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝜆₁,𝜆₂∈[0,1], (H)if 𝑑(𝑊(𝑥,𝑦,𝜆),𝑊(𝑥,𝑧,𝜆))≤(1−𝜆)𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋 and 𝜆∈[0,1], (S)if 𝑑(𝑊(𝑥,𝑦,𝜆),𝑊(𝑧,𝑤,𝜆))≤𝜆𝑑(𝑥,𝑧)+(1−𝜆)𝑑(𝑦,𝑤) for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝜆∈[0,1].

From the above properties, it is obvious that the property (C) and (H) imply continuity of a convex structure 𝑊∶𝑋×𝑋×[0,1]→𝑋. 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 (𝑋,𝑑,𝑊) is said to be uniformly convex if for any 𝜀>0, there exists 𝛿=𝛿(𝜀)>0 such that for all 𝑟>0 and 𝑥,𝑦,𝑧∈𝑋 with 𝑑(𝑧,𝑥)≤𝑟, 𝑑(𝑧,𝑦)≤𝑟 and 𝑑(𝑥,𝑦)≥𝑟𝜀 imply that 𝑑(𝑧,𝑊(𝑥,𝑦,1/2))≤(1−𝛿)𝑟. 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 (𝑋,𝑑,𝑊) is a uniformly convex metric space. Let 𝑥,𝑦∈𝑋 and 𝜆∈[0,1]. It is obvious that the conclusion holds if 𝜆=0 or 𝜆=1. So, suppose 𝜆∈(0,1). By Lemma 2.1(ii), we have 𝑑𝑑((𝑥𝑥,,𝑊𝑊((𝑥𝑦,,𝑥𝑦,,1𝜆)−)=𝜆)()1=−(1𝜆)−𝑑𝜆(𝑥)𝑑,𝑦()𝑥,,𝑦),𝑑(𝑦,𝑑𝑊(𝑦,(𝑊𝑥,𝑦(,𝑦𝜆,)𝑥),=1−𝜆𝑑𝜆()𝑥),=𝑦)𝜆,𝑑(𝑥,𝑦).(2.1)
We will show that 𝑊(𝑥,𝑦,𝜆)=𝑊(𝑦,𝑥,1−𝜆). To show this, suppose not. Put 𝑧₁=𝑊(𝑥,𝑦,𝜆) and 𝑧₂=𝑊(𝑦,𝑥,1−𝜆). Let 𝑟₁=(1−𝜆)𝑑(𝑥,𝑦)>0, 𝑟₂=𝜆𝑑(𝑥,𝑦)>0, 𝜀₁=𝑑(𝑧₁,𝑧₂)/𝑟₁, and 𝜀₂=𝑑(𝑧₁,𝑧₂)/𝑟₂. It is easy to see that 𝜀₁,𝜀₂>0. Since (𝑋,𝑑,𝑊) is uniformly convex, we have 𝑑𝑥,𝑊𝑧₁,𝑧₂,12≤𝑟¹1−𝛿𝜀₁,𝑑𝑦,𝑊𝑧₁,𝑧₂,12≤𝑟²1−𝛿𝜀₂.(2.2) By 𝜆∈(0,1), we get 𝑥≠𝑦. Since 𝛿(𝜀₁)>0 and 𝛿(𝜀₂)>0, then 𝑑(𝑥,𝑦)≤𝑑𝑥,𝑊𝑧₁,𝑧₂,12+𝑑𝑦,𝑊𝑧¹,𝑧₂,1≤𝑟₁2<𝑟₁1−𝛿𝜀₁+𝑟₂1−𝛿𝜀₂+𝑟₂=𝑑(𝑥,𝑦).(2.3) This is a contradiction. Hence, 𝑊(𝑥,𝑦,𝜆)=𝑊(𝑦,𝑥,1−𝜆).

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

Next, we recall the special space of convex metric spaces, namely, CAT(0) spaces. Let (𝑋,𝑑) be a metric space. A geodesic path joining 𝑥∈𝑋 to 𝑦∈𝑋 (or, more briefly, a geodesic from 𝑥 to 𝑦) is a map 𝑐 from a closed interval [0,𝑙]⊂ℝ to 𝑋 such that 𝑐(0)=𝑥,𝑐(𝑙)=𝑦 and 𝑑(𝑐(𝑡₁),𝑐(𝑡₂))=|𝑡₁−𝑡₂| for all 𝑡₁,𝑡₂∈[0,𝑙]. In particular, 𝑐 is an isometry and 𝑑(𝑥,𝑦)=𝑙. The image 𝛼 of 𝑐 is called a geodesic (or metric) segment joining 𝑥 and 𝑦. When unique, this geodesic is denoted [𝑥,𝑦]. The space (𝑋,𝑑) is said to be a geodesic metric space if every two points of 𝑋 are joined by a geodesic, and 𝑋 is said to be uniquely geodesic if there is exactly one geodesic joining 𝑥 and 𝑦 for each 𝑥,𝑦∈𝑋. A subset 𝑌 of 𝑋 is said to be convex if 𝑌 includes every geodesic segment joining any two of its points.

A geodesic triangle ▵(𝑥₁,𝑥₂,𝑥₃) in a geodesic metric space (𝑋,𝑑) consists of three points 𝑥₁,𝑥₂,𝑥₃ in 𝑋 (the vertices of ▵) and a geodesic segment between each pair of vertices (the edges of ▵). A comparison triangle for geodesic triangle ▵(𝑥₁,𝑥₂,𝑥₃) in (𝑋,𝑑) is a triangle ▵(𝑥₁,𝑥₂,𝑥₃)∶=▵(𝑥₁,𝑥₂,𝑥₃) in the Euclidean plane 𝔼² such that 𝑑_𝔼²(𝑥_𝑖,𝑥_𝑗)=𝑑(𝑥_𝑖,𝑥_𝑗) for 𝑖,𝑗∈{1,2,3}.

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 𝑋, and let ▵ be a comparison triangle for ▵. Then ▵ is said to satisfy the CAT(0) inequality if for all 𝑥,𝑦∈▵ and all comparison points 𝑥,𝑦∈▵, 𝑑(𝑥,𝑦)≤𝑑_𝔼²(𝑥,𝑦).

If 𝑧,𝑥,𝑦 are points in a CAT(0) space and if 𝑚 is the midpoint of the segment [𝑥,𝑦], then the CAT(0) inequality implies 𝑑(𝑧,𝑚)²≤12𝑑(𝑧,𝑥)²+12𝑑(𝑧,𝑦)²−14𝑑(𝑥,𝑦)².(CN) This is the (CN) inequality of Bruhat and Tits [13], which is equivalent to 𝑑(𝑧,𝜆𝑥⊕(1−𝜆)𝑦)²≤𝜆𝑑(𝑧,𝑥)²+(1−𝜆)𝑑(𝑧,𝑦)²−𝜆(1−𝜆)𝑑(𝑥,𝑦)²,(CN∗) for any 𝜆∈[0,1], where 𝜆𝑥⊕(1−𝜆)𝑦 denotes the unique point in [𝑥,𝑦]. 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 𝑋 is CAT(0) space and 𝑥,𝑦∈𝑋, then for any 𝜆∈[0,1], there exists a unique point 𝜆𝑥⊕(1−𝜆)𝑦∈[𝑥,𝑦] such that𝑑(𝑧,𝜆𝑥⊕(1−𝜆)𝑦)≤𝜆𝑑(𝑧,𝑥)+(1−𝜆)𝑑(𝑧,𝑦),(2.4) for any 𝑧∈𝑋. It follows that CAT(0) spaces have convex structure 𝑊(𝑥,𝑦,𝜆)=𝜆𝑥⊕(1−𝜆)𝑦. 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 𝜇 be a continuous linear functional on 𝑙^∞, the Banach space of bounded real sequences, and let (𝑎₁,𝑎₂,…)∈𝑙^∞. We write 𝜇_𝑛(𝑎_𝑛) instead of 𝜇((𝑎₁,𝑎₂,…)). We call 𝜇 a Banach limit if 𝜇 satisfies ‖𝜇‖=𝜇(1,1,…)=1 and 𝜇_𝑛(𝑎_𝑛)=𝜇_𝑛(𝑎_𝑛+1) for each (𝑎₁,𝑎₂,…)∈𝑙^∞. For a Banach limit 𝜇, we know that liminf_𝑛→∞𝑎_𝑛≤𝜇_𝑛(𝑎_𝑛)≤limsup_𝑛→∞𝑎_𝑛 for all (𝑎₁,𝑎₂,…)∈𝑙^∞. So if (𝑎₁,𝑎₂,…)∈𝑙^∞ with lim_𝑛→∞𝑎_𝑛=𝑐, then 𝜇_𝑛(𝑎_𝑛)=𝑐, see also [17].

Lemma 2.3 ([4], Proposition 2). Let (𝑎₁,𝑎₂,…)∈𝑙^∞ be such that 𝜇_𝑛(𝑎_𝑛)≤0 for all Banach limit 𝜇. If limsup_𝑛→∞(𝑎_𝑛+1−𝑎_𝑛)≤0, then limsup_𝑛→∞𝑎_𝑛≤0.

Lemma 2.4 ([6], Lemma 2.3). Let {𝑠_𝑛} be a sequence of nonnegative real numbers, let {𝛼_𝑛} be a sequence of real numbers in [0,1] with ∑^∞𝑛=1𝛼_𝑛=∞, let {𝛿_𝑛} be a sequence of nonnegative real numbers with ∑^∞𝑛=1𝛿_𝑛<∞, and let {𝛾_𝑛}be a sequence of real numbers with limsup_𝑛→∞𝛾_𝑛≤0. Suppose that 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛾_𝑛+𝛿_𝑛∀𝑛∈ℕ.(2.5) Then lim_𝑛→∞𝑠_𝑛=0.

Lemma 2.5 ([18], Lemma 1). Let (𝑋,𝑑,𝑊) be a uniformly convex metric space with a continuous convex structure 𝑊∶𝑋×𝑋×[0,1]→𝑋. Then for arbitrary positive number 𝜀 and 𝑟, there exists 𝜂=𝜂(𝜀)>0 such that 𝑑(𝑧,𝑊(𝑥,𝑦,𝜆))≤𝑟(1−2min{𝜆,1−𝜆}𝜂),(2.6) for all 𝑥,𝑦,𝑧∈𝑋, 𝑑(𝑧,𝑥)≤𝑟, 𝑑(𝑧,𝑦)≤𝑟, 𝑑(𝑥,𝑦)≥𝑟𝜀, and 𝜆∈[0,1].

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 𝐶 be a subset of a complete convex metric space (𝑋,𝑑,𝑊), and let {𝑇_𝑛} be a countable infinite family of mappings from 𝐶 into itself. We say that {𝑇_𝑛} satisfies AKTT-condition if^∞_𝑛=1sup𝑑𝑇_𝑛+1𝑧,𝑇_𝑛𝑧∶𝑧∈𝐵<∞,(3.1) for each bounded subset 𝐵 of 𝐶. If 𝐶 is a closed subset and {𝑇_𝑛} satisfies AKTT-condition, then we can define a mapping 𝑇∶𝐶→𝐶 such that 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶. In this case, we also say that ({𝑇_𝑛},𝑇) satisfies AKTT-condition. By using the same argument as in [6, Lemma 3.2], we have the following lemma.

Lemma 3.1. If ({𝑇_𝑛},𝑇) satisfies AKTT-condition, then lim_𝑛→∞sup{𝑑(𝑇𝑧,𝑇_𝑛𝑧)∶𝑧∈𝐵}=0 for all bounded subsets 𝐵 of 𝐶.

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a complete convex metric space (𝑋,𝑑,𝑊) with the properties (I) and (S). Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Suppose that {𝑥_𝑛} is a sequence of 𝐶 generated by (1.5), and let {𝛼_𝑛} and {𝛽_𝑛} be sequences in [0,1] which satisfy the conditions: (C1)0<𝛼_𝑛<1, lim_𝑛→∞𝛼_𝑛=0, ∑^∞𝑛=1𝛼_𝑛=∞ and ∑^∞𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞,(C2)𝛽_𝑛∈(𝑏,1] for some 𝑏∈(0,1) and ∑^∞𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞. Suppose that ({𝑇_𝑛},𝑇) satisfies AKTT-condition. Then lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=0 and lim_𝑛→∞𝑑(𝑇𝑥_𝑛,𝑥_𝑛)=0.

Proof. Let 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛). By the definition of {𝑥_𝑛} and {𝑦_𝑛}, we have 𝑑𝑥_𝑛+1,𝑝=≤𝛽𝑑_𝑛𝑊𝑦_𝑛,𝑇_𝑛𝑦_𝑛,𝛽_𝑛,𝑝≤𝑑𝑑𝑦_𝑛𝑦_𝑛,𝑝+1−𝛽_𝑛𝑑𝑇_𝑛𝑦_𝑛,𝑝=𝑑𝑊,𝑝𝑢,𝑇_𝑛≤𝛼_𝑛𝑥_𝑛,𝛼_𝑛,𝑝≤𝛼_𝑛𝑑(𝑢,𝑝)+1−𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑝≤ma𝑑x(𝑢,𝑑𝑝)(𝑢+,𝑝)1,−𝑑𝛼𝑥_𝑛_𝑛𝑑𝑥_𝑛,𝑝,𝑝.(3.2) By induction on 𝑛, we obtain that 𝑑(𝑥_𝑛,𝑝)≤max{𝑑(𝑢,𝑝),𝑑(𝑥₁,𝑝)} for all 𝑛∈ℕ and all 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛). Hence, the sequence {𝑥_𝑛} is bounded and so {𝑦_𝑛}, {𝑇_𝑛𝑥_𝑛}, {𝑇_𝑛𝑦_𝑛} are bounded.
It follows by condition (𝐶1) that 𝑑𝑦_𝑛,𝑇_𝑛𝑥_𝑛=𝑑𝑊𝑢,𝑇_𝑛𝑥_𝑛,𝛼_𝑛,𝑇_𝑛𝑥_𝑛=𝛼_𝑛𝑑𝑢,𝑇_𝑛𝑥_𝑛⟶0.(3.3) By the definition of {𝑥_𝑛} and {𝑦_𝑛}, we have 𝑑𝑦_𝑛,𝑦_𝑛−1=≤𝑑𝑑𝑊𝑊𝑢𝑢,,𝑇𝑇_𝑛𝑛𝑥_𝑛,𝛼_𝑛,𝑊𝑢,𝑇_𝑛−1𝑥_𝑛−1,𝛼_𝑛−1+𝑑𝑊𝑢,𝑇𝑥_𝑛𝑛,𝛼_𝑛,𝑊𝑢,𝑇_𝑛𝑥_𝑛−1,𝛼_𝑛+𝑑𝑊𝑢,𝑇𝑥_{𝑛−1𝑛−1},𝛼_𝑛,𝑊𝑢,𝑇_𝑛−1𝑥_𝑛−1,𝛼_𝑛≤1−𝛼_𝑛𝑥_𝑛−1,𝛼_𝑛,𝑊𝑢,𝑇_𝑛−1𝑥_𝑛−1,𝛼_𝑛−1+||𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛−1,𝑇_𝑛−1𝑥_𝑛−1≤1−−𝛼_𝑛𝛼_𝑛−1||𝑑𝑢,𝑇_𝑛−1𝑥_𝑛−1+||𝛼_𝑛𝑑𝑥_𝑛,𝑥_𝑛−1+1−𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛−1,𝑇_𝑛−1𝑥_𝑛−1≤1−−𝛼_𝑛𝛼_𝑛−1||𝑑𝑢,𝑇_𝑛−1𝑥_𝑛−1+||𝛼_𝑛𝑑𝑥_𝑛,𝑥_𝑛−1+𝑑𝑇_𝑛𝑥_𝑛−1,𝑇_𝑛−1𝑥_𝑛−1𝑑𝑥_𝑛+1−𝛼_𝑛−1||𝑑𝑢,𝑇_𝑛−1𝑥_𝑛−1,,𝑥_𝑛=≤𝑑𝑑𝑊𝑊𝑦𝑦_𝑛𝑛,𝑇_𝑛𝑦_𝑛,𝛽_𝑛,𝑊𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1,𝛽_𝑛−1+𝑑𝑊𝑦,𝑇_𝑛𝑛−1𝑦_𝑛,𝛽_𝑛,𝑊𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1,𝛽_𝑛≤𝛽_𝑛,𝑇_𝑛−1𝑦_𝑛−1,𝛽_𝑛,𝑊𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1,𝛽_𝑛−1+||𝛽𝑑_𝑛𝑦_𝑛,𝑦_𝑛−1+1−𝛽_𝑛𝑑𝑇_𝑛𝑦_𝑛,𝑇_𝑛−1𝑦_𝑛−1≤𝛽_𝑛−𝛽_𝑛−1||𝑑𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1+||𝛽𝑑𝑦_𝑛,𝑦_𝑛−1+1−𝛽_𝑛𝑑𝑇_𝑛𝑦_𝑛,𝑇_𝑛𝑦_𝑛−1+𝑑𝑇_𝑛𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1_𝑛≤𝛽_𝑛−𝛽_𝑛−1||𝑑𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1+||𝛽𝑑𝑦_𝑛,𝑦_𝑛−1+1−𝛽_𝑛𝑑𝑦_𝑛,𝑦_𝑛−1+𝑑𝑇_𝑛𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1_𝑛≤𝑑𝑦_𝑛−𝛽_𝑛−1||𝑑𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1≤1−,𝛼𝑦_𝑛−1𝑛+𝑑𝑇_𝑛𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1+||𝛽_𝑛−𝛽_𝑛−1||𝑑𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1+||𝛼_𝑛𝑑𝑥_𝑛,𝑥_𝑛−1+𝑑𝑇_𝑛𝑥_𝑛−1,𝑇_𝑛−1𝑥_𝑛−1+||𝛽_𝑛−𝛼_𝑛−1||𝑑𝑢,𝑇_𝑛−1𝑥_𝑛−1+𝑑𝑇_𝑛𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1≤1−−𝛼_𝑛𝛽_𝑛−1||𝑑𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1+𝑑𝑇_𝑛𝑑𝑥_𝑛,𝑥_𝑛−1+||𝛼_𝑛−𝛼_𝑛−1||+||𝛽_𝑛−𝛽_𝑛−1||𝑀𝑥_𝑛−1,𝑇_𝑛−1𝑥_𝑛−1+𝑑𝑇_𝑛𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1,(3.4) where 𝑀=max{sup_𝑛𝑑(𝑢,𝑇_𝑛−1𝑥_𝑛−1),sup_𝑛𝑑(𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1)}.
Putting 𝛿_𝑛=(|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|)𝑀+𝑑(𝑇_𝑛𝑥_𝑛−1,𝑇_𝑛−1𝑥_𝑛−1)+𝑑(𝑇_𝑛𝑦_𝑛−1,𝑇_𝑛−1𝑦_𝑛−1), we have ^∞_𝑛=2𝛿_𝑛≤𝑀^∞_𝑛=2||𝛼_𝑛−𝛼_𝑛−1||+||𝛽_𝑛−𝛽_𝑛−1||+^∞_𝑛=2+sup𝑑𝑇_𝑛𝑧,𝑇_𝑛−1𝑧∶𝑧∈𝑥_𝑘_∞_𝑛=2sup𝑑𝑇_𝑛𝑧,𝑇_𝑛−1𝑧∶𝑧∈𝑦_𝑘.(3.5) Hence, it follows from conditions (𝐶1), (𝐶2), AKTT-condition, and Lemma 2.4 that lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛=0.(3.6) Now, observe that 𝑑𝑥_𝑛+1,𝑦_𝑛==𝑑1−𝑊𝛽_𝑛𝑦_𝑛,𝑇_𝑛𝑦_𝑛,𝛽_𝑛,𝑦_𝑛≤(1−𝑏)𝑑𝑑𝑦𝑦_𝑛𝑛,𝑇_𝑛𝑦_𝑛≤(1−𝑏)𝑑𝑦_𝑛,𝑇_𝑛𝑥_𝑛+𝑑𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑥_𝑛+1+𝑑𝑇_𝑛𝑥_𝑛+1,𝑇_𝑛𝑦_𝑛,𝑇_𝑛𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑦_𝑛.(3.7) We obtain 𝑑𝑥_𝑛+1,𝑦_𝑛≤1−𝑏𝑏𝑑𝑦^𝑛,𝑇_𝑛𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1.(3.8) This implies by (3.3) and (3.6) that lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑦_𝑛)=0. Therefore, we have 𝑑𝑥_𝑛,𝑦_𝑛≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑦_𝑛⟶0.(3.9) Since 𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛≤𝑑𝑇_𝑛𝑥_𝑛,𝑦_𝑛+𝑑𝑦_𝑛,𝑥_𝑛,(3.10) it follows by (3.3) and (3.9) that lim_𝑛→∞𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛=0.(3.11) By (3.11) and Lemma 3.1, we get 𝑑𝑇𝑥_𝑛,𝑥_𝑛≤≤𝑑sup𝑇𝑥_𝑛𝑑,𝑇_𝑛𝑇𝑥𝑧_𝑛,𝑇+_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛𝑧∶𝑧∈𝑥_𝑘+𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛⟶0.(3.12)

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 𝐶 be a closed convex subset of a complete CAT(0) space 𝑋, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Let 𝑢∈𝐶 be fixed. For each 𝑡∈(0,1), the mapping 𝑆_𝑡∶𝐶→𝐶 defined by 𝑆_𝑡𝑥=𝑡𝑢⊕(1−𝑡)𝑇𝑥 for 𝑥∈𝐶 has a unique fixed point 𝑥_𝑡∈𝐶, that is, 𝑥_𝑡=𝑆_𝑡𝑥_𝑡=𝑡𝑢⊕(1−𝑡)𝑇𝑥_𝑡.

Lemma 3.4. Let 𝐶, 𝑇 be as the preceding lemma. Then 𝐹(𝑇)≠∅ if and only if {𝑥_𝑡} remains bounded as 𝑡→0. In this case, the following statements hold: (i){𝑥_𝑡} converges to the unique fixed point 𝑧 of 𝑇 which is nearest to 𝑢; (ii)𝑑(𝑢,𝑧)²≤𝜇_𝑛𝑑(𝑢,𝑥_𝑛)² for all Banach limit 𝜇 and all bounded sequences {𝑥_𝑛} with lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0.

Previously, we know that CAT(0) spaces have convex structure 𝑊(𝑥,𝑦,𝜆)=𝜆𝑥⊕(1−𝜆)𝑦 and also have the properties (C), (I), and (S). Thus, we have the following result.

Theorem 3.5. Let 𝐶 be a nonempty closed convex subset of a complete CAT(0) space 𝑋. Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Suppose that 𝑢,𝑥₁∈𝐶 are arbitrarily chosen and {𝑥_𝑛} is a sequence of 𝐶 generated by 𝑥𝑦_𝑛𝑛+1=𝛼_𝑛𝑢⊕1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,=𝛽_𝑛𝑦_𝑛⊕1−𝛽_𝑛𝑇_𝑛𝑦_𝑛∀𝑛∈ℕ,(3.13) where {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] which satisfy the conditions (𝐶1) and (𝐶2) as in Theorem 3.2. Suppose that ({𝑇_𝑛},𝑇) satisfies AKTT-condition. Then lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=0 and lim_𝑛→∞𝑑(𝑇𝑥_𝑛,𝑥_𝑛)=0.

Theorem 3.6. Let 𝐶 be a nonempty closed convex subset of a complete CAT(0) space 𝑋. Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Suppose that {𝑥_𝑛} is a sequence of 𝐶 generated by (3.13), and let {𝛼_𝑛} and {𝛽_𝑛} be sequences in [0,1] which satisfy the conditions (𝐶1) and (𝐶2) as in Theorem 3.2. Suppose that ({𝑇_𝑛},𝑇) satisfies AKTT-condition and 𝐹(𝑇)=⋂^∞𝑛=1𝐹(𝑇_𝑛). Then {𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑛} which is nearest to 𝑢.

Proof. By Theorem 3.5, we have lim_𝑛→∞𝑑(𝑇𝑥_𝑛,𝑥_𝑛)=0. For each 𝑡∈(0,1), let 𝑧_𝑡 be a unique point of 𝐶 such that 𝑧_𝑡=𝑡𝑢⊕(1−𝑡)𝑇𝑧_𝑡. It follows from Lemma 3.4 that {𝑧_𝑡} converges to a point 𝑧∈𝐹(𝑇) which is nearest to 𝑢, and 𝑑(𝑢,𝑧)²≤𝜇_𝑛𝑑𝑢,𝑥_𝑛²forallBanachlimits𝜇,(3.14) that is, 𝜇_𝑛(𝑑(𝑢,𝑧)²−𝑑(𝑢,𝑥_𝑛)²)≤0. Moreover, by Theorem 3.5, we get lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=0. It follows that limsup_𝑛→∞𝑑(𝑢,𝑧)²−𝑑𝑢,𝑥_𝑛+1²−𝑑(𝑢,𝑧)²−𝑑𝑢,𝑥_𝑛²=0.(3.15) By lim_𝑛→∞𝑑(𝑇_𝑛𝑥_𝑛,𝑥_𝑛)=0 and Lemma 2.3, we obtain limsup_𝑛→∞𝑑(𝑢,𝑧)²−1−𝛼_𝑛𝑑𝑢,𝑇_𝑛𝑥_𝑛²=limsup_𝑛→∞𝑑(𝑢,𝑧)²−𝑑𝑢,𝑥_𝑛²≤0.(3.16) Finally, we show that lim_𝑛→∞𝑑(𝑥_𝑛,𝑧)=0. By the definition of {𝑥_𝑛} and {𝑦_𝑛}, we have 𝑑𝑥_𝑛+1,𝑧²=𝑑𝛽_𝑛𝑦_𝑛⊕1−𝛽_𝑛𝑇_𝑛𝑦_𝑛,𝑧²≤𝛽_𝑛𝑑𝑦_𝑛,𝑧+1−𝛽_𝑛𝑑𝑇_𝑛𝑦_𝑛,𝑧²≤𝑑𝑦_𝑛,𝑧²=𝑑𝛼_𝑛𝑢⊕1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑧²≤𝛼_𝑛𝑑(𝑢,𝑧)²+1−𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑧²−𝛼_𝑛1−𝛼_𝑛𝑑𝑢,𝑇_𝑛𝑥_𝑛²≤𝛼_𝑛𝑑(𝑢,𝑧)²+1−𝛼_𝑛𝑑𝑥_𝑛,𝑧²−𝛼_𝑛1−𝛼_𝑛𝑑𝑢,𝑇_𝑛𝑥_𝑛²=1−𝛼_𝑛𝑑𝑥_𝑛,𝑧²+𝛼_𝑛𝑑(𝑢,𝑧)²−1−𝛼_𝑛𝑑𝑢,𝑇_𝑛𝑥_𝑛².(3.17) This implies by ∑^∞𝑛=1𝛼_𝑛=∞, inequality (3.16), and Lemma 2.4 that lim_𝑛→∞𝑑(𝑥_𝑛,𝑧)²=0. Hence, {𝑥_𝑛} converges to 𝑧∈𝐹(𝑇)=⋂^∞𝑛=1𝐹(𝑇_𝑛) which is nearest to 𝑢.

Corollary 3.7 (see [7], Theorem 8). Let 𝐶 be a nonempty closed convex subset of a complete CAT(0) space 𝑋. Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. Suppose that 𝑢,𝑥₁∈𝐶 are arbitrarily chosen and {𝑥_𝑛} is a sequence of 𝐶 generated by 𝑥_𝑛+1=𝛼_𝑛𝑢⊕1−𝛼_𝑛𝑇_𝑛𝑥_𝑛∀𝑛∈ℕ,(3.18) where {𝛼_𝑛} is a sequence in [0,1] which satisfies the condition (𝐶1) as in Theorem 3.2. Suppose that ({𝑇_𝑛},𝑇) satisfies AKTT-condition and 𝐹(𝑇)=⋂^∞𝑛=1𝐹(𝑇_𝑛). Then {𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑛} which is nearest to 𝑢.

Proof. By putting 𝛽_𝑛=1 for all 𝑛∈ℕ 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 𝐶 be a nonempty closed convex subset of a complete CAT(0) space 𝑋. Suppose that {𝑇_𝑛} is a family of nonexpansive mappings from 𝐶 into itself with ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅. We say that {𝑇_𝑛} satisfies SZ-condition if, for any bounded subset 𝐾 of 𝐶, there exists a nonexpansive mapping 𝑇 of 𝐶 into itself such thatlim_𝑛→∞sup𝑑𝑇𝑇_𝑛𝑥,𝑇_𝑛𝑥∶𝑥∈𝐾=0,𝐹(𝑇)=^∞_𝑛=1𝐹𝑇_𝑛.(3.19)

Theorem 3.8. Let 𝐶 be a nonempty closed convex subset of a complete CAT(0) space 𝑋. Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅ and satisfies SZ-condition. Suppose that {𝑥_𝑛} is a sequence of 𝐶 defined by (3.13) with lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=0. Let {𝛼_𝑛} and {𝛽_𝑛} be sequences in [0,1] which satisfy the following conditions: (C3)0<𝛼_𝑛<1, lim_𝑛→∞𝛼_𝑛=0, and ∑^∞𝑛=1𝛼_𝑛=∞, (C4)lim_𝑛→∞𝛽_𝑛=1. Then {𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑛} which is nearest to 𝑢.

Proof. As in the proof of Theorem 3.2, we have that {𝑥_𝑛} and {𝑇_𝑛𝑥_𝑛} are bounded. Since {𝑇_𝑛} satisfies SZ-condition, there exists a nonexpansive mapping 𝑇 of 𝐶 into itself such that lim_𝑛→∞sup{𝑑(𝑇(𝑇_𝑛𝑥),𝑇_𝑛𝑥)∶𝑥∈{𝑥_𝑘}}=0 and 𝐹(𝑇)=⋂^∞𝑛=1𝐹(𝑇_𝑛). By the definition of {𝑥_𝑛} and {𝑦_𝑛}, we have 𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛=≤𝑑𝛽_𝑛𝛽_𝑛𝑦_𝑛⊕1−𝛽_𝑛𝑇_𝑛𝑦_𝑛,𝑇_𝑛𝑥_𝑛≤𝛽_𝑛𝑑𝑦_𝑛,𝑇_𝑛𝑥_𝑛+1−𝛽_𝑛𝑑𝑇_𝑛𝑦_𝑛,𝑇_𝑛𝑥_𝑛=𝛽_𝑛𝑑𝑦_𝑛,𝑇_𝑛𝑥_𝑛+1−𝛽_𝑛𝑑𝑦_𝑛,𝑥_𝑛≤𝛽_𝑛𝑑𝛼_𝑛𝑢⊕1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑇_𝑛𝑥_𝑛+1−𝛽_𝑛𝑑𝛼_𝑛𝑢⊕1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,𝑥_𝑛𝛼_𝑛𝑑𝑢,𝑇_𝑛𝑥_𝑛+1−𝛽_𝑛𝛼_𝑛𝑑𝑢,𝑥_𝑛+1−𝛼_𝑛𝑑𝑇_𝑛𝑥_𝑛,𝑥_𝑛.(3.20) It follows from condition (𝐶3) and (𝐶4) that lim_𝑛→∞𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛=0.(3.21) Since 𝑑𝑥_𝑛+1,𝑇𝑥_𝑛+1≤≤𝑑2𝑑𝑥𝑥_𝑛+1,_𝑛+1𝑇_𝑛𝑥_𝑛+𝑑𝑇_𝑛𝑥_𝑛,𝑇𝑇_𝑛𝑥_𝑛+𝑑𝑇𝑇_𝑛𝑥_𝑛,𝑇𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛+sup𝑑𝑇𝑇_𝑛𝑥,𝑇_𝑛𝑥∶𝑥∈𝑥_𝑘,(3.22) this implies by (3.21) and SZ-condition, we have lim_𝑛→∞𝑑𝑥_𝑛,𝑇𝑥_𝑛=0.(3.23) From lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=0 and 𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑇_𝑛𝑥_𝑛,(3.24) it follows that lim_𝑛→∞𝑑𝑥_𝑛,𝑇_𝑛𝑥_𝑛=0.(3.25) By using the same arguments and techniques as those of Theorem 3.6, we can show that {𝑥_𝑛} converges to a common fixed point of {𝑇_𝑛} which is nearest to 𝑢.

Corollary 3.9. Let 𝐶 be a nonempty closed convex subset of a complete CAT(0) space 𝑋. Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅ and satisfies SZ-condition. Suppose that {𝑥_𝑛} is a sequence of 𝐶 defined by (3.18) with lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑥_𝑛)=0. Let {𝛼_𝑛} be a sequence in [0,1] which satisfies the condition (𝐶3) as in Theorem 3.8. Then {𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑛} which is nearest to 𝑢.

Proof. By putting 𝛽_𝑛=1 for all 𝑛∈ℕ 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 ({𝑇_𝑛},𝑇) satisfies AKTT-condition. In general, one cannot apply these results for a sequence of nonexpansive mappings. However, we give an example of a sequence {𝑇_𝑛} of nonexpansive mappings satisfying AKTT-condition.

Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself, where 𝐶 is a convex subset of a convex metric space (𝑋,𝑑,𝑊). We now define mappings 𝑈_𝑛;1,𝑈_𝑛;2,…,𝑈_𝑛;𝑛 and 𝑆_𝑛 as follows. For {𝜆_𝑛} a sequence in [0,1] and 𝑥∈𝑋, 𝑈𝑈_{𝑛;𝑛𝑛;𝑛−1}𝑥=𝑊𝑇_𝑛𝑥,𝑥,𝜆_𝑛,𝑈_{𝑛;𝑛−2}𝑥=𝑊𝑇_𝑛−1𝑈_𝑛;𝑛𝑥,𝑥,𝜆_𝑛−1,𝑈_𝑛;𝑘𝑥=⋮𝑊𝑇_𝑛−2𝑈_{𝑛;𝑛−1}𝑥,𝑥,𝜆_𝑛−2,𝑈_{𝑛;𝑘−1}𝑥=𝑊𝑇_𝑘𝑈_𝑛;𝑘+1𝑥,𝑥,𝜆_𝑘,𝑈_𝑛;2𝑥=⋮𝑊𝑇_𝑘−1𝑈_𝑛;𝑘𝑥,𝑥,𝜆_𝑘−1,𝑆_𝑛𝑥=𝑊𝑇₂𝑈_𝑛;3𝑥,𝑥,𝜆₂,x=𝑈_𝑛;1𝑥=𝑊𝑇₁𝑈_𝑛;2𝑥,𝑥,𝜆₁.(4.1) Such a mapping 𝑆_𝑛 is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜆₁,𝜆₂,…,𝜆_𝑛.

In 2007, Shimizu [18] generalized 𝑊-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 𝐶 be a nonempty closed convex subset of a uniformly convex metric space (𝑋,𝑑,𝑊) with a continuous convex structure 𝑊∶𝑋×𝑋×[0,1]→𝑋. Let 𝑇₁,𝑇₂,…,𝑇_𝑁 be nonexpansive mappings of 𝐶 into itself such that ⋂^𝑁𝑛=1𝐹(𝑇_𝑛)≠∅ and let 𝜆₁,𝜆₂,…,𝜆_𝑁 be real numbers such that 0<𝜆_𝑛<1 for every 𝑛=1,2,…,𝑁. Let 𝑆_𝑁 be the 𝑊-mapping of 𝐶 into itself generated by 𝑇₁,𝑇₂,…,𝑇_𝑁 and 𝜆₁,𝜆₂,…,𝜆_𝑁. Then 𝐹(𝑆_𝑁)=⋂^𝑁𝑛=1𝐹(𝑇_𝑛).

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

Lemma 4.2. Let 𝐶 be a nonempty closed convex subset of a complete uniformly convex metric space (𝑋,𝑑,𝑊) with the property (H). Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅, and let 𝜆₁,𝜆₂,… be real numbers such that 0<𝜆_𝑛≤𝑏<1 for every 𝑛∈ℕ. Then for every 𝑥∈𝐶, and 𝑘∈ℕ, lim_𝑛→∞𝑈_𝑛;𝑘𝑥 exists.

Proof. Let 𝑥∈𝐶 and 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛). Fix 𝑘∈ℕ. Then for any 𝑛∈ℕ with 𝑛>𝑘, we have 𝑑𝑈_𝑛+1;𝑘𝑥,𝑈_𝑛;𝑘𝑥≤=𝜆𝑑_𝑘𝑊𝑇_𝑘𝑈_{𝑛+1;𝑘+1}𝑥,𝑥,𝜆_𝑘,𝑊𝑇_𝑘𝑈_𝑛;𝑘+1𝑥,𝑥,𝜆_𝑘≤𝜆_𝑘𝑑𝑇_𝑘𝑈_{𝑛+1;𝑘+1}𝑥,𝑇_𝑘𝑈_𝑛;𝑘+1𝑥=𝜆_𝑘𝑑𝑈_{𝑛+1;𝑘+1}𝑥,𝑈_𝑛;𝑘+1𝑥≤𝜆_𝑘𝑑𝑊𝑇_𝑘+1𝑈_{𝑛+1;𝑘+2}𝑥,𝑥,𝜆_𝑘+1,𝑊𝑇_𝑘+1𝑈_𝑛;𝑘+2𝑥,𝑥,𝜆_𝑘+1≤⋮𝜆_𝑘𝜆_𝑘+1𝑑𝑈_{𝑛+1;𝑘+2}𝑥,𝑈_𝑛;𝑘+2𝑥=𝜆_𝑘𝜆_𝑘+1⋯𝜆_𝑛−1𝑑𝑈_𝑛+1;𝑛𝑥,𝑈_𝑛;𝑛𝑥≤𝜆_𝑘𝜆_𝑘+1⋯𝜆_𝑛−1𝑑𝑊𝑇_𝑛𝑈_{𝑛+1;𝑛+1}𝑥,𝑥,𝜆_𝑛,𝑊𝑇_𝑛𝑥,𝑥,𝜆_𝑛≤𝜆_𝑘𝜆_𝑘+1⋯𝜆_𝑛𝑑𝑇_𝑛𝑈_{𝑛+1;𝑛+1}𝑥,𝑇_𝑛𝑥=𝜆_𝑘𝜆_𝑘+1⋯𝜆_𝑛𝑑𝑈_{𝑛+1;𝑛+1}𝑥,𝑥=𝜆_𝑘𝜆_𝑘+1⋯𝜆_𝑛𝑑𝑊𝑇_𝑛+1𝑥,𝑥,𝜆_𝑛+1,𝑥≤𝜆_𝑘𝜆_𝑘+1⋯𝜆_𝑛+1𝑑𝑇_𝑛+1𝑥,𝑥≤2𝑑𝜆(𝑝,_𝑘+1𝑥⋯)𝑏𝜆_{𝑛+1𝑛−𝑘+2}𝑑𝑇_𝑛+1𝑥,𝑝+𝑑(𝑝,𝑥).(4.2) Thus for 𝑚>𝑛, 𝑑𝑈_𝑚;𝑘𝑥,𝑈_𝑛;𝑘𝑥≤≤2𝑑𝑑𝑈(𝑝,𝑥_𝑚;𝑘𝑥)𝑏,𝑈_{𝑚−1;𝑘}𝑥_{(𝑚−1)−𝑘+2}+𝑑𝑈_{𝑚−1;𝑘}𝑥,𝑈_{𝑚−2;𝑘}𝑥+⋯+𝑑𝑈_𝑛+1;𝑘𝑥,𝑈_𝑛;𝑘𝑥+2𝑑(𝑝,𝑥)𝑏^{(𝑚−2)−𝑘+2}+⋯+2𝑑(𝑝,𝑥)𝑏^{𝑛−𝑘+2}=2𝑑(𝑝,𝑥)^𝑚−1_𝑗=𝑛𝑏^{𝑗−𝑘+2}.(4.3) It follows that {𝑈_𝑛;𝑘𝑥} is a Cauchy sequence. Hence, lim_𝑛→∞𝑈_𝑛;𝑘𝑥 exists.

Using the above lemma, one can define mappings 𝑈_∞;𝑘 and 𝑆 of 𝐶 into itself as𝑈_∞;𝑘𝑥=lim_𝑛→∞𝑈_𝑛;𝑘𝑥,𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥=lim_𝑛→∞𝑈_𝑛;1𝑥,(4.4) for every 𝑥∈𝐶. Such a mapping 𝑆 is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,… and 𝜆₁,𝜆₂,….

Lemma 4.3. Let 𝐶 be a nonempty closed convex subset of a complete uniformly convex metric space (𝑋,𝑑,𝑊) with the property (H). Let {𝑇_𝑛} be a family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞𝑛=1𝐹(𝑇_𝑛)≠∅, and let 𝜆₁,𝜆₂,… be real numbers such that 0<𝜆_𝑛≤𝑏<1 for every 𝑛∈ℕ. Let 𝑆 be the 𝑊-mapping generated by 𝑇₁,𝑇₂,… and 𝜆₁,𝜆₂,…. Then, 𝑆 is a nonexpansive mapping and 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑇_𝑛).

Proof. First, we show that 𝑆 is a nonexpansive mapping. For 𝑥,𝑦∈𝐶, we have 𝑑𝑆_𝑛𝑥,𝑆_𝑛𝑦=≤𝑑𝜆₁𝑊𝑇₁𝑈_𝑛;2𝑥,𝑥,𝜆₁,𝑊𝑇₁𝑈_𝑛;2𝑦,𝑦,𝜆₁≤𝜆₁𝑑𝑇₁𝑈_𝑛;2𝑥,𝑇₁𝑈_𝑛;2𝑦+1−𝜆₁𝑑(𝑥,𝑦)≤⋮𝜆₁𝑑𝑈_𝑛;2𝑥,𝑈_𝑛;2𝑦+1−𝜆₁𝑑(𝑥,𝑦)=𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑𝑈_𝑛;𝑛𝑥,𝑈_𝑛;𝑛𝑦+1−𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑(𝑥,𝑦)≤𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑𝑊𝑇_𝑛𝑥,𝑥,𝜆_𝑛,𝑊𝑇_𝑛𝑦,𝑦,𝜆_𝑛+1−𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑(𝑥,𝑦)+𝜆1₂−⋯𝜆𝜆_𝑛−11𝜆_𝑛𝑑𝑇_𝑛𝑥,𝑇_𝑛𝑦+𝜆₁𝜆₂⋯𝜆_𝑛−11−𝜆_𝑛𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑦).𝜆₂⋯𝜆_𝑛−1𝑑(𝑥,𝑦)(4.5) This implies that 𝑆_𝑛 is a nonexpansive mapping, and we have 𝑑(𝑆𝑥,𝑆𝑦)=lim_𝑛→∞𝑑(𝑆_𝑛𝑥,𝑆_𝑛𝑦)≤𝑑(𝑥,𝑦). Thus, 𝑆 is also a nonexpansive mapping.
Finally, we show that 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑇_𝑛). Let 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛). Then, it is obvious that 𝑈_𝑛;𝑘𝑝=𝑝 for all 𝑛,𝑘∈ℕ with 𝑛>𝑘. So we have 𝑈_∞;𝑘𝑝=𝑝 for all 𝑘∈ℕ. Therefore, we have 𝑆𝑝=𝑈_∞;1𝑝=𝑝, and hence, ⋂^∞𝑛=1𝐹(𝑇_𝑛)⊆𝐹(𝑆). We now show that 𝐹(𝑆)⊆⋂^∞𝑛=1𝐹(𝑇_𝑛). Let 𝑥∈𝐹(𝑆) and let 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛). Then we have 𝑑𝑆_𝑛𝑝,𝑆_𝑛𝑥==𝑑𝑑𝑝𝑈,𝑊_𝑛;1𝑝,𝑈𝑇_𝑛;11𝑥≤𝜆₁𝑈_𝑛;2𝑥,𝑥,𝜆₁≤𝜆₁𝑑𝑝,𝑇₁𝑈_𝑛;2𝑥+1−𝜆₁𝑑(𝑝,𝑥)≤⋮𝜆₁𝑑𝑝,𝑈_𝑛;2𝑥+1−𝜆₁𝑑(𝑝,𝑥)=𝜆₁𝜆₂⋯𝜆_𝑘−1𝑑𝑝,𝑈_𝑛;𝑘𝑥+1−𝜆₁𝜆₂⋯𝜆_𝑘−1𝑑(𝑝,𝑥)≤𝜆₁𝜆₂⋯𝜆_𝑘−1𝑑𝑝,𝑊𝑇_𝑘𝑈_𝑛;𝑘+1𝑥,𝑥,𝜆_𝑘+1−𝜆₁𝜆₂⋯𝜆_𝑘−1𝑑(𝑝,𝑥)+𝜆1₂−⋯𝜆𝜆_𝑘−11𝜆_𝑘𝑑𝑝,𝑇_𝑘𝑈_𝑛;𝑘+1𝑥+𝜆₁𝜆₂⋯𝜆_𝑘−11−𝜆_𝑘𝑑(𝑝,𝑥)=𝜆₁𝜆₂⋯𝜆_𝑘−1𝑑(𝑝,𝑥)≤𝜆₁𝜆₂⋯𝜆_𝑘𝑑𝑝,𝑇_𝑘𝑈_𝑛;𝑘+1𝑥+1−𝜆₁𝜆₂⋯𝜆_𝑘𝑑(𝑝,𝑥)≤⋮𝜆₁𝜆₂⋯𝜆_𝑘𝑑𝑝,𝑈_𝑛;𝑘+1𝑥+1−𝜆₁𝜆₂⋯𝜆_𝑘𝑑(𝑝,𝑥)=𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑𝑝,𝑈_𝑛;𝑛𝑥+1−𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑(𝑝,𝑥)≤𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑𝑝,𝑊𝑇_𝑛𝑥,𝑥,𝜆_𝑛+1−𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑(𝑝,𝑥)+𝜆1₂−⋯𝜆𝜆_𝑛−11𝜆_𝑛𝑑𝑝,𝑇_𝑛𝑥+𝜆₁𝜆₂⋯𝜆_𝑛−11−𝜆_𝑛𝑑(𝑝,𝑥)=𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑(𝑝,𝑥)≤𝑑(𝜆𝑝₂,𝑥⋯).𝜆_𝑛𝑑𝑝,𝑇_𝑛𝑥+1−𝜆₁𝜆₂⋯𝜆_𝑛𝑑(𝑝,𝑥)(4.6) Taking 𝑛→∞, we obtain 𝑑(𝑆𝑝,𝑆𝑥)≤≤𝜆𝜆₁₁𝜆₂⋯𝜆_𝑘−1𝑑𝑝,𝑊𝑇_𝑘𝑈_∞;𝑘+1𝑥,𝑥,𝜆_𝑘+1−𝜆₁𝜆₂⋯𝜆_𝑘−1𝑑(𝑝,𝑥)+𝜆1₂−⋯𝜆𝜆_𝑘−11𝜆_𝑘𝑑𝑝,𝑇_𝑘𝑈_∞;𝑘+1𝑥+𝜆₁𝜆₂⋯𝜆_𝑘−11−𝜆_𝑘𝑑(𝑝,𝑥)=𝜆₁𝜆₂⋯𝜆_𝑘−1𝑑(𝑝,𝑥)≤𝑑(𝜆𝑝₂,𝑥⋯).𝜆_𝑘𝑑𝑝,𝑇_𝑘𝑈_∞;𝑘+1𝑥+1−𝜆₁𝜆₂⋯𝜆_𝑘𝑑(𝑝,𝑥)(4.7) Since 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛)⊆𝐹(𝑆), we have 𝑑(𝑆𝑝,𝑆𝑥)=𝑑(𝑝,𝑥). Then, for 𝜆_𝑛∈(0,1), 𝑛∈ℕ, we have 𝑑𝑝,𝑇_𝑘𝑈_∞;𝑘+1𝑥=𝑑(𝑝,𝑥),𝑑𝑝,𝑊𝑇_𝑘𝑈_∞;𝑘+1𝑥,𝑥,𝜆_𝑘=𝑑(𝑝,𝑥),(4.8) for every 𝑘∈ℕ. Suppose that 𝑇_𝑘𝑈_∞;𝑘+1𝑥≠𝑥. Then 𝑑(𝑇_𝑘𝑈_∞;𝑘+1𝑥,𝑥)>0. It follows by Lemma 2.5, we have 𝑑𝑝,𝑊𝑇_𝑘𝑈_∞;𝑘+1𝑥,𝑥,𝜆_𝑘<𝑑(𝑝,𝑥).(4.9) This is a contradiction. Hence, 𝑇_𝑘𝑈_∞;𝑘+1𝑥=𝑥. Since 𝑈_𝑛;𝑘+1𝑥=𝑊(𝑇_𝑘+1𝑈_𝑛;𝑘+2𝑥,𝑥,𝜆_𝑘+1), we have 𝑈_∞;𝑘+1𝑥=lim_𝑛→∞𝑈_𝑛;𝑘+1𝑥=𝑊𝑇_𝑘+1𝑈_∞;𝑘+2𝑥,𝑥,𝜆_𝑘+1=𝑥.(4.10) So, we have 𝑥=𝑇_𝑘𝑈_∞;𝑘+1𝑥=𝑇_𝑘𝑥 for every 𝑘∈ℕ. This implies that 𝑥∈⋂^∞𝑛=1𝐹(𝑇_𝑛). Therefore, we have 𝐹(𝑆)⊆⋂^∞𝑛=1𝐹(𝑇_𝑛).

Lemma 4.4. Suppose that 𝑋,𝐶,{𝑇_𝑛},{𝜆_𝑛} are as in Lemma 4.3. Let 𝑆_𝑛 and 𝑆 be the 𝑊-mappings generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜆₁,𝜆₂,…,𝜆_𝑛, and 𝑇₁,𝑇₂,… and 𝜆₁,𝜆₂,…, respectively. Then ({𝑆_𝑛},𝑆) satisfies AKTT-condition, and 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑆_𝑛).

Proof. Let 𝐵 be a bounded subset of 𝐶 and 𝑥∈𝐵. For 𝑝∈⋂^∞𝑛=1𝐹(𝑇_𝑛), we have 𝑑𝑆_𝑛+1𝑥,𝑆_𝑛𝑥==𝑑𝑑𝑈𝑊𝑇_𝑛+1;11𝑥,𝑈_𝑛;1𝑥≤𝜆₁𝑈_𝑛+1;2𝑥,𝑥,𝜆₁,𝑊𝑇₁𝑈_𝑛;2𝑥,𝑥,𝜆₁≤𝜆₁𝑑𝑇₁𝑈_𝑛+1;2𝑥,𝑇₁𝑈_𝑛;2𝑥≤⋮𝜆₁𝑑𝑈_𝑛+1;2𝑥,𝑈_𝑛;2𝑥=𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑𝑈_𝑛+1;𝑛𝑥,𝑈_𝑛;𝑛𝑥≤𝜆₁𝜆₂⋯𝜆_𝑛−1𝑑𝑊𝑇_𝑛𝑈_{𝑛+1;𝑛+1}𝑥,𝑥,𝜆_𝑛,𝑊𝑇_𝑛𝑥,𝑥,𝜆_𝑛=𝜆₁𝜆₂⋯𝜆_𝑛𝑑𝑈_{𝑛+1;𝑛+1}𝑥,𝑥≤𝜆₁𝜆₂⋯𝜆_𝑛𝑑𝑊𝑇_𝑛+1𝑥,𝑥,𝜆_𝑛+1,𝑥≤𝜆₁𝜆₂⋯𝜆_𝑛+1𝑑𝑇_𝑛+1𝑥,𝑥≤2𝜆𝜆₂₁⋯𝜆_𝑛+1𝑑𝑇_𝑛+1𝑥,𝑝+𝑑(𝑝,𝑥)≤2𝑏𝜆_2𝑛+1⋯𝜆_𝑛+1𝑑(𝑝,𝑥)𝑑(𝑝,𝑥).(4.11) This implies ^∞_𝑛=1sup𝑑𝑆_𝑛+1𝑥,𝑆_𝑛𝑥∶𝑥∈𝐵<∞.(4.12) Thus, ({𝑆_𝑛},𝑆) satisfies AKTT-condition. Moreover, from Lemmas 4.1–4.3, we obtain that 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑆_𝑛).

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 𝑋,𝐶,{𝑇_𝑛},{𝜆_𝑛} are as in Lemma 4.3. Let 𝑆_𝑛 and 𝑆 be the 𝑊-mappings generated by 𝑇₁,𝑇₂,…,𝑇_𝑛 and 𝜆₁,𝜆₂,…,𝜆_𝑛, and 𝑇₁,𝑇₂,… and 𝜆₁,𝜆₂,…, respectively. By Lemma 4.4, we know that ({𝑆_𝑛},𝑆) satisfies the AKTT-condition and 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑆_𝑛). Therefore, in Theorems 3.2, 3.5, and 3.6 and Corollary 3.7, the mapping 𝑇_𝑛 can be also replaced by 𝑆_𝑛 without assuming the AKTT-condition and 𝐹(𝑆)=⋂^∞𝑛=1𝐹(𝑆_𝑛).

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.