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

Authors: Withun Phuengrattana; Suthep Suantai
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We introduce a new modified Halpern iteration for a countable infinite family of
nonexpansive mappings{Tn} in convex metric spaces. We prove that the sequence {xn} generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping
when {Tn} 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.


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## Body

## 1. Introduction

LetC 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:(1.1)xn+1=αnu+(1-αn)Txn∀n∈N,
where x1,u∈C are arbitrarily chosen, and {αn} 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 {xn} generated as follows:(1.2)yn=αnxn+(1-αn)Txn,xn+1=βnu+(1-βn)yn∀n∈N,
where x1,u∈C are arbitrarily chosen and {αn}, {βn} 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:(1.3)xn+1=αnu+(1-αn)Tnxn∀n∈N,
where x1,u∈C are arbitrarily chosen, {αn} is a sequence in [0,1], and {Tn} is a sequence of nonexpansive mappings with some conditions. They proved that the sequence {xn} generated by (1.3) converges strongly to a common fixed point of {Tn}. 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], (1.4)d(z,W(x,y,λ))≤λd(z,x)+(1-λ)d(z,y),
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{Tn} of C into itself in a convex metric space as follows:(1.5)yn=W(u,Tnxn,αn),xn+1=W(yn,Tnyn,βn)∀n∈N,
where x1,u∈C are arbitrarily chosen, and {αn}, {βn} are two sequences in [0,1]. The main propose of this paper is to prove the convergence theorem of the sequence {xn} 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)
ifW(x,y,λ)=W(y,x,1-λ) for all x,y∈X and λ∈[0,1],(I)
ifd(W(x,y,λ1),W(x,y,λ2))≤|λ1-λ2|d(x,y) for all x,y∈X and λ1,λ2∈[0,1],(H)
ifd(W(x,y,λ),W(x,z,λ))≤(1-λ)d(y,z) for all x,y,z∈X and λ∈[0,1],(S)
ifd(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 structureW: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
(2.1)d(x,W(x,y,λ))=(1-λ)d(x,y),d(y,W(x,y,λ))=λd(x,y),d(x,W(y,x,1-λ))=(1-λ)d(x,y),d(y,W(y,x,1-λ))=λd(x,y).
We will show thatW(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
(2.2)d(x,W(z1,z2,12))≤r1(1-δ(ε1)),d(y,W(z1,z2,12))≤r2(1-δ(ε2)).
By λ∈(0,1), we get x≠y. Since δ(ε1)>0 and δ(ε2)>0, then
(2.3)d(x,y)≤d(x,W(z1,z2,12))+d(y,W(z1,z2,12))≤r1(1-δ(ε1))+r2(1-δ(ε2))<r1+r2=d(x,y).
This is a contradiction. Hence, W(x,y,λ)=W(y,x,1-λ).By Lemma2.2, 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.Ageodesic 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):=∆(x¯1,x¯2,x¯3) in the Euclidean plane 𝔼2 such that d𝔼2(x¯i,x¯j)=d(xi,xj) 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)≤d𝔼2(x¯,y¯).Ifz,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 (CN)d(z,m)2≤12d(z,x)2+12d(z,y)2-14d(x,y)2.
This is the (CN) inequality of Bruhat and Tits [13], which is equivalent to (CN*)d(z,λx⊕(1-λ)y)2≤λd(z,x)2+(1-λ)d(z,y)2-λ(1-λ)d(x,y)2,
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 that(2.4)d(z,λx⊕(1-λ)y)≤λd(z,x)+(1-λ)d(z,y),
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.Letμ be a continuous linear functional on l∞, the Banach space of bounded real sequences, and let (a1,a2,…)∈l∞. We write μn(an) instead of μ((a1,a2,…)). We call μ a Banach limit if μ satisfies ∥μ∥=μ(1,1,…)=1 and μn(an)=μn(an+1) for each (a1,a2,…)∈l∞. For a Banach limit μ, we know that liminfn→∞an≤μn(an)≤limsupn→∞an for all (a1,a2,…)∈l∞. So if (a1,a2,…)∈l∞ with limn→∞an=c, then μn(an)=c, see also [17].Lemma 2.3 ([4], Proposition 2).
Let(a1,a2,…)∈l∞ be such that μn(an)≤0 for all Banach limit μ. If limsupn→∞(an+1-an)≤0, then limsupn→∞an≤0.Lemma 2.4 ([6], Lemma 2.3).
Let{sn} be a sequence of nonnegative real numbers, let {αn} be a sequence of real numbers in [0,1] with ∑n=1∞αn=∞, let {δn} be a sequence of nonnegative real numbers with ∑n=1∞δn<∞, and let {γn}be a sequence of real numbers with limsupn→∞γn≤0. Suppose that
(2.5)sn+1≤(1-αn)sn+αnγn+δn∀n∈N.
Then limn→∞sn=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 ɛ and r, there exists η=η(ε)>0 such that
(2.6)d(z,W(x,y,λ))≤r(1-2min⁡{λ,1-λ}η),
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 {Tn} be a countable infinite family of mappings from C into itself. We say that {Tn} satisfies AKTT-condition if(3.1)∑n=1∞sup⁡{d(Tn+1z,Tnz):z∈B}<∞,
for each bounded subset B of C. If C is a closed subset and {Tn} satisfies AKTT-condition, then we can define a mapping T:C→C such that Tx=limn→∞Tnx for all x∈C. In this case, we also say that ({Tn},T) satisfies AKTT-condition. By using the same argument as in [6, Lemma 3.2], we have the following lemma.Lemma 3.1.
If({Tn},T) satisfies AKTT-condition, then limn→∞sup{d(Tz,Tnz):z∈B}=0 for all bounded subsets B of C.Theorem 3.2.
LetC be a nonempty closed convex subset of a complete convex metric space (X,d,W) with the properties (I) and (S). Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅. Suppose that {xn} is a sequence of C generated by (1.5), and let {αn} and {βn} be sequences in [0,1] which satisfy the conditions:  (C1)
0<αn<1, limn→∞αn=0, ∑n=1∞αn=∞ and ∑n=1∞|αn+1-αn|<∞,(C2)
βn∈(b,1] for some b∈(0,1) and ∑n=1∞|βn+1-βn|<∞. Suppose that ({Tn},T) satisfies AKTT-condition. Then limn→∞d(xn+1,xn)=0 and limn→∞d(Txn,xn)=0.Proof.
Letp∈⋂n=1∞F(Tn). By the definition of {xn} and {yn}, we have
(3.2)d(xn+1,p)=d(W(yn,Tnyn,βn),p)≤βnd(yn,p)+(1-βn)d(Tnyn,p)≤d(yn,p)=d(W(u,Tnxn,αn),p)≤αnd(u,p)+(1-αn)d(Tnxn,p)≤αnd(u,p)+(1-αn)d(xn,p)≤max⁡{d(u,p),d(xn,p)}.
By induction on n, we obtain that d(xn,p)≤max{d(u,p),d(x1,p)} for all n∈ℕ and all p∈⋂n=1∞F(Tn). Hence, the sequence {xn} is bounded and so {yn}, {Tnxn}, {Tnyn} are bounded.
It follows by condition(C1) that
(3.3)d(yn,Tnxn)=d(W(u,Tnxn,αn),Tnxn)=αnd(u,Tnxn)⟶0.
By the definition of {xn} and {yn}, we have
(3.4)d(yn,yn-1)=d(W(u,Tnxn,αn),W(u,Tn-1xn-1,αn-1))≤d(W(u,Tnxn,αn),W(u,Tnxn-1,αn))+d(W(u,Tnxn-1,αn),W(u,Tn-1xn-1,αn))+d(W(u,Tn-1xn-1,αn),W(u,Tn-1xn-1,αn-1))≤(1-αn)d(Tnxn,Tnxn-1)+(1-αn)d(Tnxn-1,Tn-1xn-1)+|αn-αn-1|d(u,Tn-1xn-1)≤(1-αn)d(xn,xn-1)+(1-αn)d(Tnxn-1,Tn-1xn-1)+|αn-αn-1|d(u,Tn-1xn-1)≤(1-αn)d(xn,xn-1)+d(Tnxn-1,Tn-1xn-1)+|αn-αn-1|d(u,Tn-1xn-1),d(xn+1,xn)=d(W(yn,Tnyn,βn),W(yn-1,Tn-1yn-1,βn-1))≤d(W(yn,Tnyn,βn),W(yn-1,Tn-1yn-1,βn))+d(W(yn-1,Tn-1yn-1,βn),W(yn-1,Tn-1yn-1,βn-1))≤βnd(yn,yn-1)+(1-βn)d(Tnyn,Tn-1yn-1)+|βn-βn-1|d(yn-1,Tn-1yn-1)≤βnd(yn,yn-1)+(1-βn)(d(Tnyn,Tnyn-1)+d(Tnyn-1,Tn-1yn-1))+|βn-βn-1|d(yn-1,Tn-1yn-1)≤βnd(yn,yn-1)+(1-βn)(d(yn,yn-1)+d(Tnyn-1,Tn-1yn-1))+|βn-βn-1|d(yn-1,Tn-1yn-1)≤d(yn,yn-1)+d(Tnyn-1,Tn-1yn-1)+|βn-βn-1|d(yn-1,Tn-1yn-1)≤(1-αn)d(xn,xn-1)+d(Tnxn-1,Tn-1xn-1)+|αn-αn-1|d(u,Tn-1xn-1)+d(Tnyn-1,Tn-1yn-1)+|βn-βn-1|d(yn-1,Tn-1yn-1)≤(1-αn)d(xn,xn-1)+(|αn-αn-1|+|βn-βn-1|)M+d(Tnxn-1,Tn-1xn-1)+d(Tnyn-1,Tn-1yn-1),
where M=max{supnd(u,Tn-1xn-1),supnd(yn-1,Tn-1yn-1)}.
Puttingδn=(|αn-αn-1|+|βn-βn-1|)M+d(Tnxn-1,Tn-1xn-1)+d(Tnyn-1,Tn-1yn-1), we have
(3.5)∑n=2∞δn≤M∑n=2∞(|αn-αn-1|+|βn-βn-1|)+∑n=2∞sup⁡{d(Tnz,Tn-1z):z∈{xk}}+∑n=2∞sup⁡{d(Tnz,Tn-1z):z∈{yk}}.
Hence, it follows from conditions (C1), (C2), AKTT-condition, and Lemma 2.4 that
(3.6)lim⁡n→∞d(xn+1,xn)=0.
Now, observe that
(3.7)d(xn+1,yn)=d(W(yn,Tnyn,βn),yn)=(1-βn)d(yn,Tnyn)≤(1-b)(d(yn,Tnxn)+d(Tnxn,Tnxn+1)+d(Tnxn+1,Tnyn))≤(1-b)(d(yn,Tnxn)+d(xn,xn+1)+d(xn+1,yn)).
We obtain
(3.8)d(xn+1,yn)≤1-bb(d(yn,Tnxn)+d(xn,xn+1)).
This implies by (3.3) and (3.6) that limn→∞d(xn+1,yn)=0. Therefore, we have
(3.9)d(xn,yn)≤d(xn,xn+1)+d(xn+1,yn)⟶0.
Since
(3.10)d(Tnxn,xn)≤d(Tnxn,yn)+d(yn,xn),
it follows by (3.3) and (3.9) that
(3.11)lim⁡n→∞d(Tnxn,xn)=0.
By (3.11) and Lemma 3.1, we get
(3.12)d(Txn,xn)≤d(Txn,Tnxn)+d(Tnxn,xn)≤sup⁡{d(Tz,Tnz):z∈{xk}}+d(Tnxn,xn)⟶0.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.
LetC 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 St:C→C defined by Stx=tu⊕(1-t)Tx for x∈C has a unique fixed point xt∈C, that is, xt=Stxt=tu⊕(1-t)Txt.Lemma 3.4.
LetC, T be as the preceding lemma. Then F(T)≠∅ if and only if {xt} remains bounded as t→0. In this case, the following statements hold:  (i)
{xt} converges to the unique fixed point z of T which is nearest to u;(ii)
d(u,z)2≤μnd(u,xn)2 for all Banach limit μ and all bounded sequences {xn} with limn→∞d(xn,Txn)=0.Previously, we know that CAT(0) spaces have convex structureW(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.
LetC be a nonempty closed convex subset of a complete CAT(0) space X. Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅. Suppose that u,x1∈C are arbitrarily chosen and {xn} is a sequence of C generated by
(3.13)yn=αnu⊕(1-αn)Tnxn,xn+1=βnyn⊕(1-βn)Tnyn∀n∈N,
where {αn} and {βn} are sequences in [0,1] which satisfy the conditions (C1) and (C2) as in Theorem 3.2. Suppose that ({Tn},T) satisfies AKTT-condition. Then limn→∞d(xn+1,xn)=0 and limn→∞d(Txn,xn)=0.Theorem 3.6.
LetC be a nonempty closed convex subset of a complete CAT(0) space X. Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅. Suppose that {xn} is a sequence of C generated by (3.13), and let {αn} and {βn} be sequences in [0,1] which satisfy the conditions (C1) and (C2) as in Theorem 3.2. Suppose that ({Tn},T) satisfies AKTT-condition and F(T)=⋂n=1∞F(Tn). Then {xn} converges strongly to a common fixed point of {Tn} which is nearest to u.Proof.
By Theorem3.5, we have limn→∞d(Txn,xn)=0. For each t∈(0,1), let zt be a unique point of C such that zt=tu⊕(1-t)Tzt. It follows from Lemma 3.4 that {zt} converges to a point z∈F(T) which is nearest to u, and
(3.14)d(u,z)2≤μnd(u,xn)2forallBanachlimitsμ,
that is, μn(d(u,z)2-d(u,xn)2)≤0. Moreover, by Theorem 3.5, we get limn→∞d(xn+1,xn)=0. It follows that
(3.15)limsup⁡n→∞((d(u,z)2-d(u,xn+1)2)-(d(u,z)2-d(u,xn)2))=0.
By limn→∞d(Tnxn,xn)=0 and Lemma 2.3, we obtain
(3.16)limsup⁡n→∞(d(u,z)2-(1-αn)d(u,Tnxn)2)=limsup⁡n→∞(d(u,z)2-d(u,xn)2)≤0.
Finally, we show that limn→∞d(xn,z)=0. By the definition of {xn} and {yn}, we have
(3.17)d(xn+1,z)2=d(βnyn⊕(1-βn)Tnyn,z)2≤(βnd(yn,z)+(1-βn)d(Tnyn,z))2≤d(yn,z)2=d(αnu⊕(1-αn)Tnxn,z)2≤αnd(u,z)2+(1-αn)d(Tnxn,z)2-αn(1-αn)d(u,Tnxn)2≤αnd(u,z)2+(1-αn)d(xn,z)2-αn(1-αn)d(u,Tnxn)2=(1-αn)d(xn,z)2+αn(d(u,z)2-(1-αn)d(u,Tnxn)2).
This implies by ∑n=1∞αn=∞, inequality (3.16), and Lemma 2.4 that limn→∞d(xn,z)2=0. Hence, {xn} converges to z∈F(T)=⋂n=1∞F(Tn) which is nearest to u.Corollary 3.7 (see [7], Theorem 8).
LetC be a nonempty closed convex subset of a complete CAT(0) space X. Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅. Suppose that u,x1∈C are arbitrarily chosen and {xn} is a sequence of C generated by
(3.18)xn+1=αnu⊕(1-αn)Tnxn∀n∈N,
where {αn} is a sequence in [0,1] which satisfies the condition (C1) as in Theorem 3.2. Suppose that ({Tn},T) satisfies AKTT-condition and F(T)=⋂n=1∞F(Tn). Then {xn} converges strongly to a common fixed point of {Tn} which is nearest to u.Proof.
By puttingβn=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 {Tn} is a family of nonexpansive mappings from C into itself with ⋂n=1∞F(Tn)≠∅. We say that {Tn} satisfies SZ-condition if, for any bounded subset K of C, there exists a nonexpansive mapping T of C into itself such that(3.19)lim⁡n→∞sup⁡{d(T(Tnx),Tnx):x∈K}=0,F(T)=⋂n=1∞F(Tn).Theorem 3.8.
LetC be a nonempty closed convex subset of a complete CAT(0) space X. Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅ and satisfies SZ-condition. Suppose that {xn} is a sequence of C defined by (3.13) with limn→∞d(xn+1,xn)=0. Let {αn} and {βn} be sequences in [0,1] which satisfy the following conditions: (C3)
0<αn<1, limn→∞αn=0, and ∑n=1∞αn=∞,(C4)
limn→∞βn=1. Then {xn} converges strongly to a common fixed point of {Tn} which is nearest to u.Proof.
As in the proof of Theorem3.2, we have that {xn} and {Tnxn} are bounded. Since {Tn} satisfies SZ-condition, there exists a nonexpansive mapping T of C into itself such that limn→∞sup{d(T(Tnx),Tnx):x∈{xk}}=0 and F(T)=⋂n=1∞F(Tn). By the definition of {xn} and {yn}, we have
(3.20)d(xn+1,Tnxn)=d(βnyn⊕(1-βn)Tnyn,Tnxn)≤βnd(yn,Tnxn)+(1-βn)d(Tnyn,Tnxn)≤βnd(yn,Tnxn)+(1-βn)d(yn,xn)=βnd(αnu⊕(1-αn)Tnxn,Tnxn)+(1-βn)d(αnu⊕(1-αn)Tnxn,xn)≤βnαnd(u,Tnxn)+(1-βn)(αnd(u,xn)+(1-αn)d(Tnxn,xn)).
It follows from condition (C3) and (C4) that
(3.21)lim⁡n→∞d(xn+1,Tnxn)=0.
Since
(3.22)d(xn+1,Txn+1)≤d(xn+1,Tnxn)+d(Tnxn,T(Tnxn))+d(T(Tnxn),Txn+1)≤2d(xn+1,Tnxn)+sup⁡{d(T(Tnx),Tnx):x∈{xk}},
this implies by (3.21) and SZ-condition, we have
(3.23)lim⁡n→∞d(xn,Txn)=0.
From limn→∞d(xn+1,xn)=0 and
(3.24)d(xn,Tnxn)≤d(xn,xn+1)+d(xn+1,Tnxn),
it follows that
(3.25)lim⁡n→∞d(xn,Tnxn)=0.
By using the same arguments and techniques as those of Theorem 3.6, we can show that {xn} converges to a common fixed point of {Tn} which is nearest to u.Corollary 3.9.
LetC be a nonempty closed convex subset of a complete CAT(0) space X. Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅ and satisfies SZ-condition. Suppose that {xn} is a sequence of C defined by (3.18) with limn→∞d(xn+1,xn)=0. Let {αn} be a sequence in [0,1] which satisfies the condition (C3) as in Theorem 3.8. Then {xn} converges strongly to a common fixed point of {Tn} which is nearest to u.Proof.
By puttingβn=1 for all n∈ℕ in Theorem 3.8, we obtain the desired result.


## 4. W-Mapping in Convex Metric Spaces

In Theorems3.2, 3.5, and 3.6 and Corollary 3.7, to obtain a convergence result, we have to assume that ({Tn},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 {Tn} of nonexpansive mappings satisfying AKTT-condition.Let{Tn} 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 Un;1,Un;2,…,Un;n and Sn as follows. For {λn} a sequence in [0,1] and x∈X, (4.1)Un;nx=W(Tnx,x,λn),Un;n-1x=W(Tn-1Un;nx,x,λn-1),Un;n-2x=W(Tn-2Un;n-1x,x,λn-2),⋮Un;kx=W(TkUn;k+1x,x,λk),Un;k-1x=W(Tk-1Un;kx,x,λk-1),⋮Un;2x=W(T2Un;3x,x,λ2),Snx=Un;1x=W(T1Un;2x,x,λ1).
Such a mapping Sn is called the W-mapping generated by T1,T2,…,Tn and λ1,λ2,…,λn.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.
LetC 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,…,TN be nonexpansive mappings of C into itself such that ⋂n=1NF(Tn)≠∅ and let λ1,λ2,…,λN be real numbers such that 0<λn<1 for every n=1,2,…,N. Let SN be the W-mapping of C into itself generated by T1,T2,…,TN and λ1,λ2,…,λN. Then F(SN)=⋂n=1NF(Tn).Next, we consider theW-mapping given by a countable infinite family of nonexpansive mappings in a uniformly convex metric space.Lemma 4.2.
LetC be a nonempty closed convex subset of a complete uniformly convex metric space (X,d,W) with the property (H). Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅, and let λ1,λ2,… be real numbers such that 0<λn≤b<1 for every n∈ℕ. Then for every x∈C, and k∈ℕ, limn→∞Un;kx exists.Proof.
Letx∈C and p∈⋂n=1∞F(Tn). Fix k∈ℕ. Then for any n∈ℕ with n>k, we have
(4.2)d(Un+1;kx,Un;kx)=d(W(TkUn+1;k+1x,x,λk),W(TkUn;k+1x,x,λk))≤λkd(TkUn+1;k+1x,TkUn;k+1x)≤λkd(Un+1;k+1x,Un;k+1x)=λkd(W(Tk+1Un+1;k+2x,x,λk+1),W(Tk+1Un;k+2x,x,λk+1))≤λkλk+1d(Un+1;k+2x,Un;k+2x)⋮≤λkλk+1⋯λn-1d(Un+1;nx,Un;nx)=λkλk+1⋯λn-1d(W(TnUn+1;n+1x,x,λn),W(Tnx,x,λn))≤λkλk+1⋯λnd(TnUn+1;n+1x,Tnx)≤λkλk+1⋯λnd(Un+1;n+1x,x)=λkλk+1⋯λnd(W(Tn+1x,x,λn+1),x)=λkλk+1⋯λn+1d(Tn+1x,x)≤λkλk+1⋯λn+1(d(Tn+1x,p)+d(p,x))≤2d(p,x)bn-k+2.
Thus for m>n,
(4.3)d(Um;kx,Un;kx)≤d(Um;kx,Um-1;kx)+d(Um-1;kx,Um-2;kx)+⋯+d(Un+1;kx,Un;kx)≤2d(p,x)b(m-1)-k+2+2d(p,x)b(m-2)-k+2+⋯+2d(p,x)bn-k+2=2d(p,x)∑j=nm-1bj-k+2.
It follows that {Un;kx} is a Cauchy sequence. Hence, limn→∞Un;kx exists.Using the above lemma, one can define mappingsU∞;k and S of C into itself as(4.4)U∞;kx=lim⁡n→∞Un;kx,Sx=lim⁡n→∞Snx=lim⁡n→∞Un;1x,
for every x∈C. Such a mapping S is called the W-mapping generated by T1,T2,… and λ1,λ2,….Lemma 4.3.
LetC be a nonempty closed convex subset of a complete uniformly convex metric space (X,d,W) with the property (H). Let {Tn} be a family of nonexpansive mappings of C into itself such that ⋂n=1∞F(Tn)≠∅, and let λ1,λ2,… be real numbers such that 0<λn≤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=1∞F(Tn).Proof.
First, we show thatS is a nonexpansive mapping. For x,y∈C, we have
(4.5)d(Snx,Sny)=d(W(T1Un;2x,x,λ1),W(T1Un;2y,y,λ1))≤λ1d(T1Un;2x,T1Un;2y)+(1-λ1)d(x,y)≤λ1d(Un;2x,Un;2y)+(1-λ1)d(x,y)⋮≤λ1λ2⋯λn-1d(Un;nx,Un;ny)+(1-λ1λ2⋯λn-1)d(x,y)=λ1λ2⋯λn-1d(W(Tnx,x,λn),W(Tny,y,λn))+(1-λ1λ2⋯λn-1)d(x,y)≤λ1λ2⋯λn-1λnd(Tnx,Tny)+λ1λ2⋯λn-1(1-λn)d(x,y)+(1-λ1λ2⋯λn-1)d(x,y)≤d(x,y).
This implies that Sn is a nonexpansive mapping, and we have d(Sx,Sy)=limn→∞d(Snx,Sny)≤d(x,y). Thus, S is also a nonexpansive mapping.
Finally, we show thatF(S)=⋂n=1∞F(Tn). Let p∈⋂n=1∞F(Tn). Then, it is obvious that Un;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=1∞F(Tn)⊆F(S). We now show that F(S)⊆⋂n=1∞F(Tn). Let x∈F(S) and let p∈⋂n=1∞F(Tn). Then we have
(4.6)d(Snp,Snx)=d(Un;1p,Un;1x)=d(p,W(T1Un;2x,x,λ1))≤λ1d(p,T1Un;2x)+(1-λ1)d(p,x)≤λ1d(p,Un;2x)+(1-λ1)d(p,x)⋮≤λ1λ2⋯λk-1d(p,Un;kx)+(1-λ1λ2⋯λk-1)d(p,x)=λ1λ2⋯λk-1d(p,W(TkUn;k+1x,x,λk))+(1-λ1λ2⋯λk-1)d(p,x)≤λ1λ2⋯λk-1λkd(p,TkUn;k+1x)+λ1λ2⋯λk-1(1-λk)d(p,x)+(1-λ1λ2⋯λk-1)d(p,x)=λ1λ2⋯λkd(p,TkUn;k+1x)+(1-λ1λ2⋯λk)d(p,x)≤λ1λ2⋯λkd(p,Un;k+1x)+(1-λ1λ2⋯λk)d(p,x)⋮≤λ1λ2⋯λn-1d(p,Un;nx)+(1-λ1λ2⋯λn-1)d(p,x)=λ1λ2⋯λn-1d(p,W(Tnx,x,λn))+(1-λ1λ2⋯λn-1)d(p,x)≤λ1λ2⋯λn-1λnd(p,Tnx)+λ1λ2⋯λn-1(1-λn)d(p,x)+(1-λ1λ2⋯λn-1)d(p,x)=λ1λ2⋯λnd(p,Tnx)+(1-λ1λ2⋯λn)d(p,x)≤d(p,x).
Taking n→∞, we obtain
(4.7)d(Sp,Sx)≤λ1λ2⋯λk-1d(p,W(TkU∞;k+1x,x,λk))+(1-λ1λ2⋯λk-1)d(p,x)≤λ1λ2⋯λk-1λkd(p,TkU∞;k+1x)+λ1λ2⋯λk-1(1-λk)d(p,x)+(1-λ1λ2⋯λk-1)d(p,x)=λ1λ2⋯λkd(p,TkU∞;k+1x)+(1-λ1λ2⋯λk)d(p,x)≤d(p,x).
Since p∈⋂n=1∞F(Tn)⊆F(S), we have d(Sp,Sx)=d(p,x). Then, for λn∈(0,1), n∈ℕ, we have
(4.8)d(p,TkU∞;k+1x)=d(p,x),d(p,W(TkU∞;k+1x,x,λk))=d(p,x),
for every k∈ℕ. Suppose that TkU∞;k+1x≠x. Then d(TkU∞;k+1x,x)>0. It follows by Lemma 2.5, we have
(4.9)d(p,W(TkU∞;k+1x,x,λk))<d(p,x).
This is a contradiction. Hence, TkU∞;k+1x=x. Since Un;k+1x=W(Tk+1Un;k+2x,x,λk+1), we have
(4.10)U∞;k+1x=lim⁡n→∞Un;k+1x=W(Tk+1U∞;k+2x,x,λk+1)=x.
So, we have x=TkU∞;k+1x=Tkx for every k∈ℕ. This implies that x∈⋂n=1∞F(Tn). Therefore, we have F(S)⊆⋂n=1∞F(Tn).Lemma 4.4.
Suppose thatX,C,{Tn},{λn} are as in Lemma 4.3. Let Sn and S be the W-mappings generated by T1,T2,…,Tn and λ1,λ2,…,λn, and T1,T2,… and λ1,λ2,…, respectively. Then ({Sn},S) satisfies AKTT-condition, and F(S)=⋂n=1∞F(Sn).Proof.
LetB be a bounded subset of C and x∈B. For p∈⋂n=1∞F(Tn), we have
(4.11)d(Sn+1x,Snx)=d(Un+1;1x,Un;1x)=d(W(T1Un+1;2x,x,λ1),W(T1Un;2x,x,λ1))≤λ1d(T1Un+1;2x,T1Un;2x)≤λ1d(Un+1;2x,Un;2x)⋮≤λ1λ2⋯λn-1d(Un+1;nx,Un;nx)=λ1λ2⋯λn-1d(W(TnUn+1;n+1x,x,λn),W(Tnx,x,λn))≤λ1λ2⋯λnd(Un+1;n+1x,x)=λ1λ2⋯λnd(W(Tn+1x,x,λn+1),x)≤λ1λ2⋯λn+1d(Tn+1x,x)≤λ1λ2⋯λn+1(d(Tn+1x,p)+d(p,x))≤2λ1λ2⋯λn+1d(p,x)≤2bn+1d(p,x).
This implies
(4.12)∑n=1∞sup⁡{d(Sn+1x,Snx):x∈B}<∞.
Thus, ({Sn},S) satisfies AKTT-condition. Moreover, from Lemmas 4.1–4.3, we obtain that F(S)=⋂n=1∞F(Sn).Remark 4.5.
Lemmas4.2 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 thatX,C,{Tn},{λn} are as in Lemma 4.3. Let Sn and S be the W-mappings generated by T1,T2,…,Tn and λ1,λ2,…,λn, and T1,T2,… and λ1,λ2,…, respectively. By Lemma 4.4, we know that ({Sn},S) satisfies the AKTT-condition and F(S)=⋂n=1∞F(Sn). Therefore, in Theorems 3.2, 3.5, and 3.6 and Corollary 3.7, the mapping Tn can be also replaced by Sn without assuming the AKTT-condition and F(S)=⋂n=1∞F(Sn).



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