# Some Convergence Theorems for Contractive Type Mappings inCAT(0) Spaces

Authors: Kyung Soo Kim
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We establish theorems of strong convergence, for the Ishikawa-type (or two step;cf. Ishikawa, 1974) iteration scheme, to a fixed point of a uniformly L-Lipschitzian asymptotically demicontractive mapping and a uniformly L-Lipschitzian hemicontractive mapping in CAT(0) space. Moreover, we will propose some open problems.


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## Body

## 1. Introduction

Let(X,d) be a metric space. One of the most interesting aspects of metric fixed point theory is to extend a linear version of known result to the nonlinear case in metric spaces. To achieve this, Takahashi [1] introduced a convex structure in a metric space (X,d). A mapping W:X×X×[0,1]→X is a convex structure in X if
(1)d(u,W(x,y,λ))≤λd(u,x)+(1-λ)d(u,y)
for all x,y∈X and λ∈[0,1]. A metric space together with a convex structure W is known as a convex metric space. A nonempty subset K of a convex metric space is said to be convex if
(2)W(x,y,λ)∈K
for all x,y∈K and λ∈[0,1]. In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see, [1]).Example 1 (see [2]).
LetX={(x1,x2)∈ℝ2:x1>0,x2>0}, for all x=(x1,x2), y=(y1,y2)∈X, and λ∈[0,1]. We define a mapping W:X×X×[0,1]→X by
(3)W(x,y,λ)=(λx1+(1-λ)y1,λx1x2+(1-λ)y1y2λx1+(1-λ)y1),
and define a metric d:X×X→[0,∞) by
(4)d(x,y)=|x1-y1|+|x1x2-y1y2|.
Then we can show that (X,d,W) is a convex metric space, but it is not a normed linear space.A metric spaceX is a CAT(0) space (the term is due to Gromov [3] and it is an acronym for E. Cartan, A. D. Aleksandrov, and V. A. Toponogov) if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane (see, e.g., [4], page 159). It is well known that any complete, simply connected Riemannian manifold nonpositive sectional curvature is a CAT(0) space. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [4] or Burago  et al. [5].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 mapping c from a closed interval [0,l]⊂ℝ to X such that c(0)=x, c(l)=y, and d(c(t),c(t′))=|t-t′|, for all t,t′∈[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 it is unique, this geodesic is denoted by [x,y]. The space (X,d) is said to be a geodesic 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⊆X is said to be convex if Y includes every geodesic segment joining any two of its points.Ageodesic triangle  Δ(x1,x2,x3) is a geodesic metric space (X,d) that consists of three points x1,x2,x3∈X (the vertices of Δ) and is a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle Δ(x1,x2,x3) in (X,d) is a triangle Δ-(x1,x2,x3)=Δ(x-1,x-2,x-3) in ℝ2 such that dℝ2(x-i,x-j)=d(xi,xj) for i,j∈{1,2,3}. Such a triangle always exists (see, [4]).A geodesic metric space is said to be aCAT(0)  space if all geodesic triangles of appropriate size satisfy the following CAT(0) comparison axiom.LetΔ be a geodesic triangle in X and let Δ-⊂ℝ2 be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(0)  inequality if for all x,y∈Δ and all comparison points x-,y-∈Δ-,
(5)d(x,y)≤d(x-,y-).
Complete CAT(0) spaces are often called Hadamard spaces (see, [6]). If x,y1,y2 are points of a CAT(0) space and if y0 is the midpoint of the segment [y1,y2], which we will denote by y1⊕y2/2, then the CAT(0) inequality implies
(6)d2(x,y1⊕y22)≤12d2(x,y1)+12d2(x,y2)-14d2(y1,y2).
This inequality is the (CN) inequality of Bruhat and Tits [7]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality (cf. [4], page 163). The previous inequality has been extended by Khamsi and Kirk [8] as
(CN*)d2(z,αx⊕(1-α)y)≤αd2(z,x)+(1-α)d2(z,y)-α(1-α)d2(x,y),
for any α∈[0,1] and x,y,z∈X. The inequality (CN*) also appeared in [9].Let us recall that a geodesic metric space is aCAT(0) space if and only if it satisfies the (CN) inequality (see, [4], page 163). Moreover, if X is a CAT(0) metric space and x,y∈X, then for any α∈[0,1], there exists a unique point αx⊕(1-α)y∈[x,y] such that
(7)d(z,αx⊕(1-α)y)≤αd(z,x)+(1-α)d(z,y)
for any z∈X and [x,y]={αx⊕(1-α)y:α∈[0,1]}. In view of the previous inequality, CAT(0) space has Takahashi's convex structure W(x,y,α)=αx⊕(1-α)y. It is easy to see that for any x,y∈X and λ∈[0,1],
(8)d(x,(1-λ)x⊕λy)=λd(x,y),d(y,(1-λ)x⊕λy)=(1-λ)d(x,y).
As a consequence,
(9)1·x⊕0·y=x,(1-λ)x⊕λx=λx⊕(1-λ)x=x.
Moreover, a subset K of CAT(0) space X is convex if for any x,y∈K, we have [x,y]⊂K.Definition 2.
LetC be a nonempty subset of a metric space (X,d). Let F(T) denote the fixed point set of T. Let F(T)≠∅.(1)
A mappingT:C→C is said to be  k-strict asymptotically pseudocontractive with sequence {an} if limn→∞an=1 for some constant k, 0≤k<1 and
(10)d2(Tnx,Tny)≤an2d2(x,y)+k(d(x,Tnx)-d(y,Tny))2,for allx,y∈C,  n∈ℕ.Ifk=0, then T is said to be asymptotically nonexpansive with sequence {an}, that is,
(11)d(Tnx,Tny)≤and(x,y),∀x,y∈C.(2)
A mappingT:C→C is said to be asymptotically demicontractive with sequence {an} if limn→∞an=1 for some constant k, 0≤k<1, and
(12)d2(Tnx,p)≤an2d2(x,p)+k·d2(x,Tnx),d2(Tnx,p)≤an2d2(x,p)∀p∈F(T),for allx∈C,  n∈ℕ.Ifk=0, then T is said to be asymptotically quasi-nonexpansive with sequence {an}, that is,
(13)d(Tnx,p)≤and(x,p),∀x∈C,∀p∈F(T).(3)
A mappingT:C→C is said to be asymptotically pseudocontractive with sequence {an} if limn→∞an=1 and
(14)d2(Tnx,Tny)≤and2(x,y)+[d(x,Tnx)-d(y,Tny)]2,for allx,y∈C,  n∈ℕ.(4)
A mappingT:C→C is said to be asymptotically hemicontractive with sequence {an} if limn→∞an=1 and
(15)d2(Tnx,p)≤and2(x,p)+d2(x,Tnx),∀p∈F(T),for allx∈C,  n∈ℕ.(5)
A mappingT:C→C is said to be uniformly  L-Lipschitzian if for some constant L>0,
(16)d(Tnx,Tny)≤L·d(x,y),∀x,y∈C,for alln∈ℕ.Liu [10] has proved the convergence of Mann and Ishikawa iterative sequence for uniformly L-Lipschitzian asymptotically demicontractive and hemicontractive mappings in Hilbert space (cf. [11]). The existence of (common) fixed points of one mapping (or two mappings or family of mappings) is not known in many situations. So the approximation of fixed points of one or more nonexpansive, asymptotically nonexpansive, or asymptotically quasi-nonexpansive mappings by various iterations have been extensively studied in Banach spaces, convex metric spaces, CAT(0) spaces, and so on (see, [2, 6, 8, 9, 12–27]).In this paper, we establish theorems of strong convergence for the Ishikawa-type (or two step,cf. [28]) iteration scheme to a fixed point of a uniformly L-Lipschitzian asymptotically demicontractive mapping and a uniformly L-Lipschitzian asymptotically hemicontractive mapping in CAT(0) space. Moreover, we will propose some open problems.


## 2. Preliminaries

We introduce the following iteration process.LetC be a nonempty convex subset of a CAT(0) space (X,d) and let T:C→C be a given mapping. Let x1∈C be a given point.Algorithm 3.
The sequences{xn} and {yn} defined by the iterative process
(17)xn+1=(1-αn)xn⊕αnTnyn,yn=(1-βn)xn⊕βnTnxn,n≥1,
is called an Ishikawa-type iterative sequence (cf. [28]).Ifβn≡0, then Algorithm 3 reduces to the followingAlgorithm 4.
The sequence{xn} defined by the iterative process
(18)xn+1=(1-αn)xn⊕αnTnxn,n≥1,
is called a Mann-type iterative sequence (cf. [29]).Lemma 5 (see [10]).
Let sequences{an}, {bn} satisfy that
(19)an+1≤an+bn,an≥0, for all n∈ℕ, ∑n=1∞‍bn is convergent, and {an} has a subsequence {ank} converging to 0. Then, we must have
(20)limn→∞an=0.


## 3. Convergence Theorems

Lemma 6.
Let(X,d) be a CAT(0) space and let C be a nonempty convex subset of X. Let T:C→C be an uniformly L-Lipschitzian mapping and let {αn}, {βn} be sequence in [0,1]. Define the iteration scheme {xn} as Algorithm 3. Then
(21)d(xn,Txn)≤d(xn,Tnxn)d(xn,Txn)≤+L(1+2L+L2)d(xn-1,Tn-1xn-1),
for all n≥1.Proof.
LetCn=d(xn,Tnxn). We have
(22)d(xn-1,yn-1)=d(xn-1,(1-βn-1)xn-1⊕βn-1Tn-1xn-1)d(xn-1,yn-1)≤βn-1·d(xn-1,Tn-1xn-1)d(xn-1,yn-1)=βn-1Cn-1.
From (22), we get
(23)d(xn-1,Tn-1yn-1)≤d(xn-1,Tn-1xn-1)d(xn-1,Tn-1yn-1)≤+d(Tn-1xn-1,Tn-1yn-1)d(xn-1,Tn-1yn-1)≤Cn-1+L·d(xn-1,yn-1)d(xn-1,Tn-1yn-1)≤Cn-1+βn-1·L·Cn-1.
From (22) and (23), we get
(24)d(xn,Txn)≤d(xn,Tnxn)+d(Tnxn,Txn)≤Cn+L·d(Tn-1xn,xn)≤Cn+L{d(Tn-1xn,Tn-1xn-1)+d(Tn-1xn-1,xn)}≤Cn+L2·d(xn,xn-1)+L·d(Tn-1xn-1,xn)≤Cn+L2·d((1-αn-1)xn-1⊕αn-1Tn-1yn-1,xn-1)≤+L·d(Tn-1xn-1,(1-αn-1)xn-1⊕αn-1Tn-1yn-1)≤Cn+L2·αn-1·d(Tn-1yn-1,xn-1)≤+L{(1-αn-1)d(Tn-1xn-1,xn-1)≤+L+αn-1·d(Tn-1xn-1,Tn-1yn-1)}≤Cn+L2·αn-1(Cn-1+βn-1·L·Cn-1)≤+L(1-αn-1)Cn-1+L2·αn-1·βn-1·Cn-1≤Cn+L(1+2L+L2)Cn-1,n≥1.
This completes the proof of Lemma 6.Theorem 7.
Let(X,d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T:C→C be a completely continuous and uniformly L-Lipschitzian and asymptotically demicontractive with sequence {an}, an∈[1,∞), ∑n=1∞‍(an2-1)<∞, ε≤αn≤1-k-ε, for all n∈ℕ and some ε>0. Given x0∈C, define the iteration scheme {xn} by
(25)xn+1=(1-αn)xn⊕αnTnxn,n≥1.
Then {xn} converges strongly to some fixed point of T.Proof.
SinceT is a completely continuous mapping in a bounded closed convex subset C of complete metric space, from Schauder's theorem, F(T) is nonempty. It follows from (CN*) inequality that
(26)d2(xn+1,p)=d2((1-αn)xn⊕αnTnxn,p)d2(xn+1,p)≤(1-αn)d2(xn,p)+αnd2(Tnxn,p)d2(xn+1,p)-(1-αn)αnd2(xn,Tnxn),
for all p∈F(T). Since T is a asymptotically demicontractive, we get
(27)d2(xn+1,p)≤(1-αn)d2(xn,p)d2(xn+1,p)≤+αn{an2d2(xn,p)+k·d2(xn,Tnxn)}d2(xn+1,p)≤-αn(1-αn)d2(xn,Tnxn)d2(xn+1,p)=d2(xn,p)+αn(an2-1)d2(xn,p)d2(xn+1,p)≤-αn(1-αn-k)d2(xn,Tnxn),d2(xn+1,p)≤-αn(1-αn-k)∀p∈F(T).
Since 0<ε≤αn≤1-k-ε, we have 1-k-αn≥ε. Thus,
(28)αn(1-k-αn)≥ε2.
From (27), we have
(29)d2(xn+1,p)≤d2(xn,p)+αn(an2-1)d2(xn,p)d2(xn+1,p)≤-ε2·d2(xn,Tnxn),
for all p∈F(T). Since C is bounded and T is self-mapping in C, there exist some M>0 so that d2(xn,p)≤M, for all n∈ℕ. Since 0≤αn≤1, it follows from (29) that
(30)d2(xn+1,p)≤d2(xn,p)+(an2-1)M-ε2·d2(xn,Tnxn),d2(xn+1,p)≤d2(xn,p)+(an2-1)M-1∀p∈F(T).
Therefore,
(31)ε2·d2(xn,Tnxn)≤d2(xn,p)-d2(xn+1,p)+(an2-1)M.
So
(32)∑n=1m‍ε2·d2(xn,Tnxn)≤d2(x1,p)-d2(xm+1,p)+∑n=1m‍(an2-1)M≤2M+∑n=1∞‍(an2-1)M,
for all m∈ℕ. Since ∑n=1∞‍(an2-1)<∞, we get
(33)∑n=1∞‍ε2·d2(xn,Tnxn)<∞.
Therefore,
(34)limn→∞d2(xn,Tnxn)=0,limn→∞d(xn,Tnxn)=0.
Since T is a uniformly L-Lipschitzian, it follows from Lemma 6 that
(35)limn→∞d(xn,Txn)=0.
Since {xn} is a bounded sequence and T is completely continuous, there exist a convergent subsequence {Txnk} of {Txn}. Therefore, from (35), {xn} has a convergent subsequence {xnk}. Let limk→∞xnk=q. It follows from the continuity of T and (35), we have q=Tq. Therefore, {xn} has a subsequence which converges to the fixed point q of T. Let p=q in the inequality (30). Since ∑n=1∞‍(an2-1)<∞ and ∑n=1∞‍ε2·d2(xn,Tnxn)<∞, from (30) and Lemma 5, we have
(36)limn→∞d2(xn,q)=0.
Therefore,
(37)limn→∞xn=q.
This completes the proof of Theorem 7.Corollary 8.
Let(X,d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T:C→C be a completely continuous and uniformly L-Lipschitzian and k-strict asymptotically pseudocontractive with sequence {an}, an∈[1,∞), ∑n=1∞‍(an2-1)<∞, and ε≤αn≤1-k-ε, for all n∈ℕ and some ε>0. Given x0∈C, define the iteration scheme {xn} by
(38)xn+1=(1-αn)xn⊕αnTnxn,n≥1.
Then {xn} converges strongly to some fixed point of T.Proof.
By Definition2, T is k-strict asymptotically pseudocontractive; then T must be asymptotically demicontractive. Therefore, Corollary 8 can be proved by using Theorem 7.Lemma 9.
Let(X,d) be a CAT(0) space and let C be a nonempty convex subset of X. Let T:C→C be an uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {an}⊂[1,∞), for all n∈ℕ, and F(T) is nonempty. Define the iteration scheme {xn} as follows:
(39)xn+1=(1-αn)xn⊕αnTnyn,yn=(1-βn)xn⊕βnTnxn,n≥1.
Then the following inequality holds:
(40)d2(xn+1,p)≤[1+αn(an-1)(anβn+1)]d2(xn,p)d2(xn+1,p)≤-αnβn(1-βn-anβn-L2βn2)d2(xn,Tnxn)d2(xn+1,p)≤-αn(βn-αn)d2(xn,Tnyn),
for all p∈F(T).Proof.
It follows from(CN*) inequality that
(41)d2(xn+1,p)=d2((1-αn)xn⊕αnTnyn,p)d2(xn+1,p)≤(1-αn)d2(xn,p)+αnd2(Tnyn,p)d2(xn+1,p)=-(1-αn)αnd2(xn,Tnyn),(42)d2(yn,p)=d2((1-βn)xn⊕βnTnxn,p)d2(yn,p)≤(1-βn)d2(xn,p)+βnd2(Tnxn,p)d2(yn,p)=-(1-βn)βnd2(xn,Tnxn),
for all p∈F(T). Since T is asymptotically hemicontractive, we get
(43)d2(Tnyn,p)≤and2(yn,p)+d2(yn,Tnyn),(44)d2(Tnxn,p)≤and2(xn,p)+d2(xn,Tnxn).
From (42) and (44), we have
(45)d2(yn,p)≤(1-βn)d2(xn,p)d2(yn,p)≤+βn[and2(xn,p)+d2(xn,Tnxn)]d2(yn,p)≤-(1-βn)βnd2(xn,Tnxn)d2(yn,p)=[1+(an-1)βn]d2(xn,p)+βn2d2(xn,Tnxn).
From (CN*) inequality, we have
(46)d2(yn,Tnyn)=d2((1-βn)xn⊕βnTnxn,Tnyn)d2(yn,Tnyn)≤(1-βn)d2(xn,Tnyn)+βnd2(Tnxn,Tnyn)d2(yn,Tnyn)=-(1-βn)βnd2(xn,Tnxn).
Substituting (45) and (46) into (43), we get
(47)d2(Tnyn,p)≤an[1+(an-1)βn]d2(xn,p)+anβn2d2(xn,Tnxn)≤+(1-βn)d2(xn,Tnyn)+βnd2(Tnxn,Tnyn)≤-(1-βn)βnd2(xn,Tnxn).
From (41) and (47), we obtain
(48)d2(xn+1,p)≤(1-αn)d2(xn,p)+αnan(1+(an-1)βn)d2(xn,p)≤+αnanβn2d2(xn,Tnxn)+αn(1-βn)d2(xn,Tnyn)≤+αnβnd2(Tnxn,Tnyn)-αn(1-βn)βnd2(xn,Tnxn)≤-(1-αn)αnd2(xn,Tnyn)=[1+αn{(an-1)+an(an-1)βn}]d2(xn,p)≤-αnβn(1-βn-anβn)d2(xn,Tnxn)≤-αn(βn-αn)d2(xn,Tnyn)+αnβnd2(Tnxn,Tnyn).
Since T is uniformly L-Lipschitzian, we have
(49)d(Tnxn,Tnyn)≤L·d(xn,yn)d(Tnxn,Tnyn)=L·d(xn,(1-βn)xn⊕βnTnxn)d(Tnxn,Tnyn)≤Lβn·d(xn,Tnxn).
Substituting (49) into (48), we obtain
(50)d2(xn+1,p)≤[1+αn(an-1)(1+anβn)]d2(xn,p)d2(xn+1,p)≤-αnβn(1-βn-anβn-L2βn2)d2(xn,Tnxn)d2(xn+1,p)≤-αn(βn-αn)d2(xn,Tnyn),∀p∈F(T).
This completes the proof of Lemma 9.Lemma 10.
Let(X,d) be a CAT(0) space and let C be a nonempty bounded convex subset of X. Let T:C→C be a uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {an}⊂[1,∞), for all n∈ℕ and ∑n=1∞‍(an-1)<∞. Let F(T) be nonempty. Given x1∈C, define the iteration scheme {xn} by
(51)xn+1=(1-αn)xn⊕αnTnyn,yn=(1-βn)xn⊕βnTnxn,n≥1.
If ε≤αn≤βn≤b for some ε>0 and b∈(0,(1+L2-1)/L2), then
(52)limn→∞d(xn,Txn)=0.Proof.
First, we will provelimn→∞d(xn,Tnxn)=0. From Lemma 9 and 0≤αn≤βn, we have
(53)d2(xn+1,p)≤[1+αn(an-1)(1+anβn)]d2(xn,p)d2(xn+1,p)≤-αnβn(1-βn-anβn-L2βn2)d2(xn,Tnxn).
Thus
(54)d2(xn+1,p)-d2(xn,p)≤αn(an-1)(1+anβn)d2(xn,p)≤-αnβn(1-βn-anβn-L2βn2)d2(xn,Tnxn).
Since ∑n=1∞‍(an-1)<∞, we have limn→∞(an-1)=0. Hence, {an} is bounded. By boundedness of C and 0≤αn≤βn≤1, we obtain that {αn(1+anβn)d2(xn,p)} is bounded. Therefore, there exists a constant M>0 such that
(55)0≤αn(1+anβn)d2(xn,p)≤M.
From (54) and (55), we get
(56)d2(xn+1,p)-d2(xn,p)≤≤(an-1)M-αnβn(L2βn21-βn-anβn≤≤(an-1)M-αnβn(-L2βn2)d2(xn,Tnxn).
LetD=1-2b-L2b2>0. Since limn→∞an=1, there exists N∈ℕ such that
(57)1-βn-anβn-L2βn2≥1-b-anb-L2b2≥D2>0,
for all n≥N. Suppose that limn→∞d(xn,Tnxn)≠0, then there exist a ε0>0 and a subsequence {xni} of {xn} such that
(58)d2(xni,Tnixni)≥ε0.
Without loss of generality, we let n1≥N. From (56), we have
(59)αnβn(1-βn-anβn-L2βn2)d2(xn,Tnxn)≤(an-1)M+d2(xn,p)-d2(xn+1,p),
so
(60)∑l=1i‍αnlβnl(1-βnl-anlβnl-L2βnl2)d2(xnl,Tnlxnl)=∑m=n1ni‍αmβm(1-βm-amβm-L2βm2)d2(xm,Tmxm)≤∑m=n1ni‍(am-1)M+d2(xn1,p)-d2(xni+1,p).
From (57)–(60) and ε≤αn≤βn, we obtain
(61)i·ε2·D2·ε0≤∑m=n1ni‍(am-1)M+d2(xn1,p)-d2(xni+1,p).
Since ∑n=1∞‍(an-1)<∞ and the boundedness of C, the right side of (61) is bounded. However, if we have i→∞, then the left side of (61) is unbounded. This is a contradiction. Therefore,
(62)limn→∞d(xn,Tnxn)=0.
Since T is a uniformly L-Lipschitzian, from Lemma 6, we get
(63)limn→∞d(xn,Txn)=0.
This completes the proof of Lemma 10.Theorem 11.
Let(X,d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T:C→C be a completely continuous and uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {an}⊂[1,∞) satisfying ∑n=1∞‍(an-1)<∞, for all n∈ℕ. Given x1∈C, define the iterative scheme {xn} by
(64)xn+1=(1-αn)xn⊕αnTnyn,yn=(1-βn)xn⊕βnTnxn,n≥1.
If {αn},{βn}⊂[0,1] with ε≤αn≤βn≤b for some ε>0 and b∈(0,(1+L2-1)/L2), then {xn} converges strongly to some fixed point of T.Proof.
SinceT is a completely continuous mapping in a bounded closed convex subset C of complete metric space, from Schauder's theorem, F(T) is nonempty. Since T is completely continuous, there exist a convergent subset {Txni} of {Txn}. Let
(65)limi→∞Txni=p.
Since limn→∞d(xn,Txn)=0, from Lemma 10, we have
(66)limi→∞xni=p.
On the other hand, from the continuity of T, (66), and Lemma 10, we have
(67)d(p,Tp)=limi→∞d(xni,Txni)=0.
This means that p is a fixed point of T. From (55), (57), and αn≤βn, we obtain Lemma 9 that
(68)d2(xn+1,p)≤d2(xn,p)+(an-1)M.
From (66), there exists a subsequence {d2(xni,p)} of {d2(xn,p)} which converges to 0. Therefore, from Lemma 5 and (68),
(69)limn→∞d2(xn,p)=0.
Hence,
(70)limn→∞xn=p.
This completes the proof of Theorem 11.Corollary 12.
Let(X,d) be a complete CAT(0) space, let C be a nonempty bounded closed convex subset of X, and let T:C→C be a completely continuous and uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence {an}⊂[1,∞) satisfying ∑n=1∞‍(an2-1)<∞, for all n∈ℕ. Given x1∈C, define the iterative scheme {xn} by
(71)xn+1=(1-αn)xn⊕αnTnyn,yn=(1-βn)xn⊕βnTnxn,n≥1.
If {αn},{βn}⊂[0,1] with ε≤αn≤βn≤b for some ε>0 and b∈(0,(1+L2-1)/L2), then {xn} converges strongly to some fixed point of T.Proof.
By Definition2, T is an asymptotically pseudocontractive mapping, then T is an asymptotically hemicontractive mapping. Since an∈[1,∞), we have an2-1≥an-1≥0. Obviously, ∑n=1∞‍(an-1)≤∑n=1∞‍(an2-1)<∞. Therefore, Corollary 12 can be proved by using Theorem 11.


## 4. Some Remarks and Open Problems

LetS be a semigroup. We denote by B(S) the space of all bounded real-valued functions defined on S with supremum norm. For each s∈S, we define the left and right translation operators ls and rs on B(S) by
(72)(lsf)(t)=f(st),(rsf)(t)=f(ts),
for each t∈S and f∈B(S), respectively. Let X be a subspace of B(S) containing 1. An element μ in the dual space X* of X is said to be a mean on X if ∥μ∥=μ(1)=1. For s∈S, we can define a point evaluation δs by δs(f)=f(s) for each f∈X. It is well known that μ is mean on X if and only if
(73)infs∈Sf(s)≤μ(f)≤sups∈Sf(s)
for each f∈X. Each mean on X is the weak*-limit of convex combination of point evaluations.LetX be a translation invariant subspace of B(S) (i.e., lsX⊂X and rsX⊂X for each s∈S) containing 1. Then a mean μ on X is said to be left invariant (resp., right invariant) if
(74)μ(lsf)=μ(f)(resp.,μ(rsf)=μ(f)),
for each s∈S and f∈X. A mean μ on X is said to be invariant if μ is both left and right invariant ([30–34]). X is said to be left (resp., right) amenable if X has a left (resp., right) invariant mean. X is amenable if X is left and right amenable. In this case, we say that the semigroup S is an amenable semigroup (see [35, 36]). Moreover, B(S) is amenable when S is a commutative semigroup or a solvable group. However, the free group or semigroup of two generators is not left or right amenable.A net{μα} of means on X is said to be asymptotically left (resp., right) invariant if
(75)limα(μα(lsf)-μα(f))=0(resp.,limα(μα(rsf)-μα(f))=0),
for each f∈X and s∈S, and it is said to be left (resp., right) strongly asymptotically invariant (or strong regular) if
(76)limα∥ls*μα-μα∥=0(resp.,limα∥rs*μα-μα∥=0),
for each s∈S, where ls* and rs* are the adjoint operators of ls and rs, respectively. Such nets were first studied by Day in [35] where they were called weak*invariant and norm invariant, respectively.It is easy to see that if a semigroupS is left (resp., right) amenable, then the semigroup S′=S∪{e}, where es′=s′e=s′ for all s′∈S is also left (resp., right) amenable and conversely.A semigroupS is called left reversible if any two right ideals of S have nonvoid intersection, that is, aS∩bS≠∅ for a,b∈S. In this case, (S,⪯) is a directed system when the binary relation “⪯” on S is defined by a⪯b if and only if {a}∪aS⊇{b}∪bS for a,b∈S. It is easy to see that t⪯ts for all t,s∈S. Further, if t⪯s, then pt⪯ps for all p∈S. The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup S is left amenable, then S is left reversible. But the converse is not true ([31, 37–41]).LetS be a semigroup and F(T) denote the fixed point set of T. Then ℑ={Ts:s∈S} is called a representation ofS if Te=I and Tst=TsTt for each s,t∈S. We denote by F(ℑ) the set of common fixed points of {Ts:s∈S}, that is,
(77)F(ℑ)=⋂s∈S‍F(Ts)=⋂s∈S‍{x∈C:Tsx=x}.Open Problem 1.
It will be interesting to obtain a generalization of both Theorems7 and 11 to commutative, amenable, and reversible semigroups as in the case of Hilbert spaces or some Banach spaces (cf. [8, 30, 32, 42–45]).
For a real numberκ, a CAT(κ) space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding triangle in a model space with curvature κ.
Forκ=0, the 2-dimensional model space Mκ2=M02 is the Euclidean space ℝ2 with the metric induced from the Euclidean norm. For κ>0,  Mκ2 is the 2-dimensional sphere (1/κ)𝕊2 whose metric is length of a minimal great arc joining each two points. For κ<0, Mκ2 is the 2-dimensional hyperbolic space (1/-κ)ℍ2 with the metric defined by a usual hyperbolic distance. For more details about the properties of CAT(κ) spaces, see [4, 46–48].Open Problem 2.
It will be interesting to obtain a generalization of both Theorems7 and 11 to CAT(κ) space.



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