The Mann algorithm in a complete geodesic space with curvature bounded above

R E S E A R C H

Open Access

The Mann algorithm in a complete geodesic

space with curvature bounded above

Yasunori Kimura

1

_{, Satit Saejung}

2,3*

_{and Pongsakorn Yotkaew}

2

*_{Correspondence:} [email protected]

2_{Department of Mathematics,} Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

3_{Centre of Excellence in} Mathematics, CHE, Sriayudthaya Road, Bangkok, 10400, Thailand Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to prove two

-convergence theorems of the Mann algorithm to a common fixed point for a countable family of mappings in the case of a complete geodesic space with curvature bounded above by a positive number. The first one for nonexpansive mappings improves the recent result of Heet al.(Nonlinear Anal. 75:445-452, 2012). The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

MSC: 49J53

Keywords: CAT(

κ

) space; fixed point; Mann algorithm; nonexpansive mapping; quasi-nonexpansive mapping;

-convergence

1 Introduction

For a real numberκ, aCAT(κ) space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding comparison triangle in a model space with curvatureκ. The concept of these spaces has been studied by a large num-ber of researchers. We know that anyCAT(κ) space is aCAT(κ) space forκ>κ(see []), thus all results forCAT() spaces can immediately be applied to anyCAT(κ) withκ≤. Moreover,CAT(κ) spaces with positiveκcan be treated asCAT() spaces by changing the scale of the space. So we are interested inCAT() spaces.

One of the most important analytical problems is the existence of fixed points for non-linear mappings. In the case for nonexpansive mappings in aCAT(κ) space was proved by Kirk [, ] forκ≤, and by Espánola and Fernández-León [] forκ> . In the cases when at least one fixed point exists, it is natural to wonder whether such a fixed point can be approximated by iterations. There are many methods for approximating fixed points of a nonexpansive mappingT. One of the most successful methods is the Mann algorithm [] which is defined in a geodesic spaceXbyx∈Xand

xn+=tnxn⊕( –tn)Txn for alln∈N, (.)

where{tn}is a sequence in [, ]. By using this algorithm, Heet al.[] proved the following

result.

Theorem . Let X be a completeCAT()space and T:X→X be a mapping.Let{xn}be

the Mann algorithm(.)in X.Suppose that

(C) Tis nonexpansive withF(T)=∅; (C) d(x, F(T)) <π/;

(C) ∞_n=tn( –tn) =∞.

Then the sequence{xn}-converges to a fixed point of T.

Motivated by these results, we prove two-convergence theorems of the Mann algo-rithm to a common fixed point for a countable family of mappings in completeCAT() spaces. The first one for nonexpansive mappings improves Theorem .. The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

2 Preliminaries

LetXbe a metric space with a metricdand letx,y∈Xwithd(x,y) =l. Ageodesic path fromxto yis an isometryc: [,l]→Xsuch thatc() =xandc(l) =y. The image of a geodesic path fromx to yis called ageodesic segment joiningx andy. Let r∈(,∞]. If for everyx,y∈Xwithd(x,y) <r, a geodesic fromxtoyexists, then we say thatXis r-geodesic. Moreover, if such a geodesic is unique for each pair of points, thenXis said to ber-uniquely geodesic.

A geodesic segment joiningx andyis not necessarily unique in general. When it is unique, this geodesic segment is denoted by [x,y]. We writez∈[x,y] if and only if there ex-istst∈[, ] such thatd(z,x) = ( –t)d(x,y) andd(z,y) =td(x,y). In this case, we will write z=tx⊕(–t)yfor simplicity. Ageodesic triangle (x,y,z) consists of three pointsx,y,z∈X and geodesic segments [y,z], [z,x] and [x,z] joining two of them. We writew∈ (x,y,z) if w∈[y,z]∪[z,x]∪[x,y].

To define aCAT(κ) space, we use the following notation calledmodel space. Forκ= , the two-dimensional model spaceMκ =M is the Euclidean spaceR with the metric

induced from the Euclidean norm. Forκ> ,M

κis the two-dimensional sphere (/ √

κ)S

whose metric is a length of a minimal great arc joining each two points. Forκ < ,M

κ

is the two-dimensional hyperbolic space (/√–κ)H _{with the metric defined by a usual}

hyperbolic distance. The diameter ofM

κ is denoted byDκ, that is,Dκ=π/√κifκ>  andDκ=∞ifκ≤.

We know thatM

κis aDκ-uniquely geodesic space for eachκ∈R.

Letκ∈R. For (x,y,z) in a geodesic spaceXsatisfying thatd(x,y)+d(y,z)+d(x,z) < Dκ,

there exist pointsx,y,z∈M

κ such thatd(x,y) =dMκ(x,y),d(y,z) =dMκ(y,z) andd(x,z) =

d_M

κ(x,z). We call the triangle having verticesx,yandzinM



κ acomparison triangleof

(x,y,z). Notice that it is unique up to an isometry ofM

κ. For a specific choice of

com-parison triangles, we denote it by (x,y,z). A pointp∈[x,y] is called acomparison point forp∈[x,y] ifd(x,p) =d_M

κ(x,p).

Letκ∈RandXbe aDκ-geodesic space. If for anyx,y,z∈Xwithd(x,y)+d(y,z)+d(x,z) <

Dκ, for anyp,q∈ (x,y,z), and for their comparison pointsp,q∈ (x,y,z), the inequality

d(p,q)≤d_M

κ(p,q)

holds, then we callXaCAT(κ)space. It is easy to see that allCAT(κ) spaces areDκ-uniquely

Let {xn} be a bounded sequence in a metric space X. For x ∈ X, let r(x,{xn}) :=

lim sup_n→∞d(x,xn) and define theasymptotic radius r({xn}) of{xn}by

r{xn}

:=inf

x∈Xr

x,{xn}

.

An elementzofXis said to be anasymptotic centerof{xn}ifr(z,{xn}) =r({xn}). We say

that{xn}is-convergenttox∈Xifxis the unique asymptotic center of any subsequence

of{xn}. The concept of-convergence introduced by Lim in  was shown by Kirk and

Panyanak [] inCAT() spaces to be very similar to the weak convergence in Banach space setting.

Remark .

() If{xn}is a sequence in a completeCAT(κ)space such thatr({xn}) <Dκ/, then its asymptotic center consists of exactly one point.

() Every sequence{xn}whose asymptotic radius is less thanDκ/has a-convergent subsequence (see [, , ]), that is,ω({xn}) :={x∈X:there exists{xnk} ⊂ {xn}

such that{xnk}-converges tox} =∅.

We note that Remark .() was proved by Dhompongsaet al.[] for the caseκ≤, and by Espánola and Fernández-León [] for the caseκ> .

LetXbe a metric space with a metricd. A mappingT:X→Xis callednonexpansiveif

d(Tx,Ty)≤d(x,y) for allx,y∈X.

A pointx∈Xis called afixed pointofTifx=Tx. We denote by F(T) the set of fixed points ofT. The mappingTis calledquasi-nonexpansiveif F(T)=∅and

d(Tx,p)≤d(x,p) for allx∈Xandp∈F(T).

The following lemmas are essentially needed for our main results.

Lemma .([, Lemma ]) Let{an}and{bn}be two sequences of nonnegative real

num-bers such that

an+≤an+bn for all n∈N.

If∞_n=bn<∞,thenlimn→∞anexists.

Lemma .([, Lemma .]) Let (x,y,z)be a geodesic triangle in aCAT()space such that d(x,y) +d(x,z) +d(y,z) < π.Let u=tz⊕( –t)x and v=tz⊕( –t)y for some t∈[, ]. If d(x,z)≤M,d(y,z)≤M,andsin(( –t)M)≤sinM for some M∈(,π),then

d(u,v)≤sin( –t)M sinM d(x,y).

Lemma .([, Corollary .]) Let (x,y,z)be a geodesic triangle in aCAT()space such that d(x,y) +d(x,z) +d(y,z) < π.Let u=tx⊕( –t)y for some t∈[, ].Then

Lemma .([, Lemma .]) Let (x,y,z)be a geodesic triangle in aCAT()space such that d(x,y) +d(x,z) +d(y,z) < π,and let t∈[, ].Then

cosdtx⊕( –t)y,z≥tcosd(x,z) + ( –t)cosd(y,z).

3 Main results

We start with some propositions which are common tools for proving the main results in the next two subsections.

Proposition . Let{xn}be a sequence of a completeCAT()space X such that r({xn}) < π/.Suppose thatlimn→∞d(xn,z)exists for all z∈ω({xn}).Then{xn}-converges to an

element ofω({xn}).Moreover,ω({xn})consists of exactly one point.

Proof Letxbe the asymptotic center of{xn}and let{xnk}be any subsequence of{xn}with

the asymptotic centery. We show thaty=xand hence{xn}-converges toxas desired.

Sincer({xnk})≤r({xn}) <π/, there exists a subsequence{xn_kl}of{xnk}such that{xn_kl}

-converges tozfor somez∈X. Clearly,z∈ω({xn}) and it follows from the assumption

thatlimn→∞d(xn,z) exists. Letu∈X. Then

lim

n→∞d(xn,z) =klim→∞d(xnk,z)

=lim

l→∞d(xnkl,z)

≤lim sup

l→∞

d(xn_kl,u)

≤lim sup

k→∞

d(xnk,u)

≤lim sup

n→∞

d(xn,u).

Since

lim

n→∞d(xn,z)≤lim sup_n→∞ d(xn,u) for allu∈X,

we havez=x. Since

lim

k→∞d(xnk,z)≤lim sup_k→∞ d(xnk,u) for allu∈X,

we havez=y. This implies thaty=x.

Proposition . Let X be a completeCAT()space and{Tn}:X→X be a countable family

of quasi-nonexpansive mappings withF :=∞_n=F(Tn)=∅.Let{xn}be a sequence in X such

that d(x, F) <π/and

xn+=tnxn⊕( –tn)Tnxn for all n∈N,

(i) {xn}is well defined andr({xn}) <π/.In particular,ω({xn})=∅and

d(xn+,p)≤d(xn,p)for alln∈Nwheneverd(x,p) <π/andp∈F.

(ii) Ifω({xn})⊂F,then{xn}-converges to an element ofF.

Proof (i) Letp∈F be such thatd(x,p) <π/. Sinced(Tx,p)≤d(x,p) <π/, we have

thatd(x,Tx) <π. This implies thatxis well defined. It follows from Lemma . that

cosd(x,p) =cosd

tx⊕( –t)Tx,p

≥tcosd(x,p) + ( –t)cosd(Tx,p)

≥tcosd(x,p) + ( –t)cosd(x,p)

=cosd(x,p),

and we have

d(x,p)≤min

d(x,x) +d(x,p),d(Tx,x) +d(Tx,p)

=min( –t)d(x,Tx) +d(x,p),td(x,Tx) +d(Tx,p)

≤min{ –t,t}d(x,Tx) +d(x,p)

≤ 

d(x,Tx) +d(x,p) <π

 +

π

 =π.

Therefore,d(x,p)≤d(x,p) <π/. Using mathematical induction, we can conclude that

the sequence{xn}is well defined and

d(xn+,p)≤d(xn,p)≤d(x,p) <π/ for alln∈N.

Thenlimn→∞d(xn,p) exists which is less thanπ/, and sor({xn}) <π/. This implies that ω({xn})=∅.

(ii) Letu∈ω({xn})⊂F. Then there exists a subsequence{xnk}of{xn}such that{xnk}

-converges tou. Notice thatr(u,{xnk}) =r({xnk})≤r({xn}) <π/. Thus there isN∈N

such thatd(xN,u) <π/. Similar to the first step, we have thatd(xn+,u)≤d(xn,u) for

alln≥N. This implies thatlimn→∞d(xn,u) exists. It follows immediately from

Proposi-tion . that{xn}-converges to an element of F and the proof is finished.

3.1 Countable nonexpansive mappings

The following concept is introduced by Aoyamaet al.[]. LetXbe a complete metric space and{Tn}be a countable family of mappings fromXinto itself with F :=

∞

n=F(Tn)=

∅. We say that ({Tn},T) satisfiesAKTT-conditionif

• ∞_n=sup{d(Tn+y,Tny) :y∈Y}<∞for each bounded subsetYofX; • Tx:=limn→∞Tnxfor allx∈XandF(T) = F.

Remark . Assume that ({Tn},T) satisfies AKTT-condition.

() For eachx∈X, we have{Tnx}is a Cauchy sequence and hence the mappingT above is well defined.

() If{xn}is bounded, then

∞

Theorem . Let X be a completeCAT()space and{Tn}be a countable family of

map-pings from X into itself.Let{xn}be a sequence in X defined by x∈X and

xn+=tnxn⊕( –tn)Tnxn for all n∈N,

where{tn}is a sequence in[, ].Suppose that

(Cn) Tnis nonexpansive for alln∈NandF :=

∞

n=F(Tn)=∅; (C) d(x, F) <π/;

(C) ∞_n=tn( –tn) =∞;

(C) ({Tn},T)satisfies AKTT-condition.

Then the sequence{xn}-converges to a common fixed point of{Tn}.

Proof We first show thatlim_n→∞d(xn,Tnxn) exists. Using the nonexpansiveness ofTnand

the definition of{xn}, we obtain that

d(xn+,Tn+xn+)

≤d(xn+,Tnxn) +d(Tnxn,Tnxn+) +d(Tnxn+,Tn+xn+)

≤d(xn+,Tnxn) +d(xn,xn+) +d(Tnxn+,Tn+xn+)

=d(xn,Tnxn) +d(Tnxn+,Tn+xn+)

for alln∈N. It follows from Lemma . and∞_n=d(Tnxn+,Tn+xn+) <∞that

lim

n→∞d(xn,Tnxn)

exists.

Next, we show thatlim_n→∞d(xn,Tnxn) = . Assume thatlimn→∞d(xn,Tnxn) > . Thus,

without loss of generality, there is a positive real numberAsuch that

A≤d(xn,Tnxn) <π for alln∈N.

To get the right inequality of the preceding expression, letp∈F be such thatd(x,p) <π/.

By Proposition .(i), we have

d(xn,Tnxn)≤d(xn,p) +d(Tnxn,p)≤d(xn,p) < d(x,p) <π.

PutAn:=d(xn,Tnxn) for alln∈N. By elementary trigonometry and Lemma ., we get

that

cosd(xn+,p)sinAn

=cosdtnxn⊕( –tn)Tnxn,p

sinAn

≥cosd(xn,p)sin(tnAn) +cosd(Tnxn,p)sin

( –tn)An

≥cosd(xn,p)

sin(tnAn) +sin

( –tn)An

and it follows that

cosd(xn+,p) –cosd(xn,p) ≥cosd(xn,p)

sin(tnAn) +sin(( –tn)An)

sinAn

– 

=cosd(xn,p)sin(tnAn/)sin(( –tn)An/) cos(An/)

≥cosd(x,p)sin(tnA/)sin(( –tn)A/)

cos(A/)

≥tn( –tn)cosd(x,p)sin(A/)

cos(A/)

for alln∈N. Notice thatcosd(x,p),cos(A/), andsin(A/) are positive. Consequently,

∞

n=

tn( –tn)≤

cos(A/) cosd(x,p)sin(A/)

∞

n=

cosd(xn+,p) –cosd(xn,p)

<∞,

which is a contradiction. Then we get thatlimn→∞d(xn,Tnxn) =  and hence

d(xn,Txn)≤d(xn,Tnxn) +d(Tnxn,Txn)

≤d(xn,Tnxn) +sup

d(Tny,Ty) :y∈Y

→,

whereY:={xn}.

Finally, we show that {xn} -converges to an element of F(T). To apply

Proposi-tion .(ii), we show thatω({xn})⊂F(T). Letu∈ω({xn}). Then there exists a

subse-quence{xnk}of{xn}such that{xnk}-converges tou. Clearly,uis the unique asymptotic

center of{xnk}. Using the nonexpansiveness ofTandd(xn,Txn)→, we get that

lim sup

k→∞ d(xnk,Tu)≤lim supk→∞ d(xnk,Txnk) +lim supk→∞ d(Txnk,Tu)

≤lim sup

k→∞

d(xnk,u).

This implies thatTu=u, that is,ω({xn})⊂F(T). This completes the proof.

As an immediate consequence of Theorem ., we obtain the following result.

Corollary . Let X be a completeCAT()space and T:X→X be a mapping.Let{xn}be

the Mann algorithm(.)in X.Suppose that

(C) Tis nonexpansive withF(T)=∅; (C) d(x, F(T)) <π/;

(C) ∞_n=tn( –tn) =∞.

Then the sequence{xn}-converges to a fixed point of T.

sense that ifd(x, F(T)) =π/, then we may construct the Mann algorithm for a

nonex-pansive mapping which is not-convergent.

Example . LetS_{be the unit sphere of the Euclidean spaceR}_{with the geodesic metric.}

LetT:S_→S_{be defined by}

T(x,y,z) = (–y,x,z) for all (x,y,z)∈S.

ThenT is nonexpansive and F(T) ={(, , ), (, , –)}. Let{xn}be a sequence inS

de-fined byx= (, , ) and

xn+=

 xn⊕



Txn for alln∈N.

Thend(x, F(T)) =π/ and

xn+=

cosnπ  ,sin

nπ

 ,  for alln∈N.

It is easy to see that{xn+}has the unique asymptotic center which is{(, , )}and{xn+}

has the unique asymptotic center which is{(, , )}. Hence,{xn}is not-convergent.

3.2 Countable quasi-nonexpansive mappings

In this subsection, we give a supplement result to Theorem .. Obviously, every nonex-pansive mapping with a fixed point is quasi-nonexnonex-pansive. Moreover, ifTis nonexpansive, thenTis-demiclosed[], that is, if for any-convergent sequence{xn}inX, its-limit

belongs to F(T) wheneverlimn→∞d(xn,Txn) = .

In the following theorem, we deal with quasi-nonexpansive mappings satisfying -demiclosedness. This interesting class of mappings includes the metric projections []. However, there are many metric projections such that they are not nonexpansive.

Theorem . Let X be a completeCAT()space and{Tn}be a countable family of

map-pings from X into itself.Let{xn}be a sequence in X defined by x∈X and

xn+=tnxn⊕( –tn)Tnxn for all n∈N,

where{tn}is a sequence in(, ).Suppose that

(Cn) Tnis quasi-nonexpansive for alln∈NandF =

∞

n=F(Tn)=∅; (C) d(x, F) <π/;

(C) lim infn→∞tn( –tn) > ;

(C) there exists a mappingT:X→Xsuch that

• {Tn}converges uniformly toT on each bounded subset ofX; • F(T) = F;

(C) Tis-demiclosed.

Then the sequence{xn}-converges to a common fixed point of{Tn}.

() (Cn)⇒(Cn); () (C)⇒(C);

() (C) and (C)⇒(C); () (C)⇒(C).

Proof of Theorem. We first show thatlimn→∞d(xn,Tnxn) = . Letp∈F(T) be such that

d(x,p) <π/. We have thatA:=limn→∞d(xn,p) exists which is less thanπ/ andr({xn}) < π/ by Proposition .(i). Put B:= lim sup_n→∞d(xn,Tnxn). Notice that B< π. Since

lim infn→∞tn( –tn) > , we may assume that there exists a subsequence{nk}ofNsuch

thatlimk→∞d(xnk,Tnkxnk) =Bandtnk→t∈(, ). Using the quasi-nonexpansiveness of

Tnand Lemma ., we get that

cosd(xnk+,p)sind(xnk,Tnkxnk)

=cosdtnkxnk⊕( –tnk)Tnkxnk,p

sind(xnk,Tnkxnk)

≥cosd(xnk,p)sin

tnkd(xnk,Tnkxnk)

+cosd(Tnkxnk,p)sin

( –tnk)d(xnk,Tnkxnk)

≥cosd(xnk,p)

sintnkd(xnk,Tnkxnk)

+sin( –tnk)d(xnk,Tnkxnk)

.

Lettingk→ ∞yields

cosAsinB≥cosAsintB+sin( –t)B.

Using elementary trigonometry, we get thatB= . Hence it follows that

lim

n→∞d(xn,Tnxn) = .

By condition (C), we get that

d(xn,Txn)≤d(xn,Tnxn) +d(Tnxn,Txn)

≤d(xn,Tnxn) +sup

d(Tny,Ty) :y∈Y

→,

whereY:={xn}.

Finally, we show thatω({xn})⊂F(T). Letu∈ω({xn}). Then there exists a subsequence {xnk}of{xn}such that{xnk}-converges tou. It follows from the-demiclosedness ofT

andd(xn,Txn)→ thatTu=u, that is,ω({xn})⊂F(T). Hence the result follows from

Proposition .(ii). The proof is now finished.

As an immediate consequence of Theorem ., we obtain the following result.

Corollary . Let X be a completeCAT()space and T :X→X be a mapping with F(T)=∅.Let{xn}be the Mann algorithm(.)in X.Suppose that

(C) Tis quasi-nonexpansive and-demiclosed; (C) d(x, F(T)) <π/;

(C) lim infn→∞tn( –tn) > .

Then the sequence{xn}-converges to a fixed point of T.

Let {Tn}: X→X be a countable family of quasi-nonexpansive mappings with F :=

∞

n=F(Tn)=∅. We next show how to generate a family{Wn} and a mappingW

satis-fying (C) and (C), and hence Theorem . is applicable.

Theorem . Let X be a completeCAT()space such that d(u,v) <π/for all u,v∈X, and let{Tn}:X→X be a countable family of quasi-nonexpansive mappings withF :=

∞

n=F(Tn)=∅.Then there exist a family of quasi-nonexpansive mappings{Wn}:X→X

and a quasi-nonexpansive mapping W :X→X such that

(i) ({Wn},W)satisfies AKTT-condition andF(W) =

∞

n=F(Wn) = F; (ii) W is-demiclosed wheneverTnis-demiclosed for alln∈N.

To prove Theorem ., we need the following lemmas.

Lemma .([]) Let X be a completeCAT()space such that d(u,v) <π/for all u,v∈X, and let S,T:X→X be quasi-nonexpansive mappings withF(S)∩F(T)=∅.Then,for each  <t< , F(S)∩F(T) = F(tS⊕( –t)T)and the mapping tS⊕( –t)T is quasi-nonexpansive.

The following lemma is essentially proved in []. For the sake of completeness, we show the proof.

Lemma . Let X be a completeCAT()space such that d(u,v) <π/for all u,v∈X, and let S,T:X→X be quasi-nonexpansive mappings withF(S)∩F(T)=∅.Let{xn}be a

sequence of X.If d(xn,_Sxn⊕_Txn)→,then d(xn,Sxn)→and d(xn,Txn)→.

Proof PutW := _S⊕_T. By Lemma ., we have thatW is quasi-nonexpansive and F(W) = F(S)∩F(T). Letp∈F(W). By Lemma . and the quasi-nonexpansiveness ofS andT, we get that

cosd(Wxn,p)sin

d(Sxn,Txn)

 cos

d(Sxn,Txn)



=cosd

 Sxn⊕



Txn,p sind(Sxn,Txn)

≥cosd(Sxn,p)sin

d(Sxn,Txn)

 +cosd(Txn,p)sin

d(Sxn,Txn)



≥cosd(xn,p)sin

d(Sxn,Txn)

 .

This implies that

d(Sxn,Txn) =  or cos

d(Sxn,Txn)

 ≥

cosd(xn,p)

cosd(Wxn,p)

.

It follows fromd(xn,Wxn)→ that _cosdcosd_(Wx(xn_n,p_,p)₎→, that is,d(Sxn,Txn)→. Thus

d(xn,Sxn)≤d

xn,

 Sxn⊕



Txn +d

 Sxn⊕



Txn,Sxn

=d

xn,

 Sxn⊕

 Txn +

d(Sxn,Txn)

 →.

We are now ready to prove Theorem ..

Proof of Theorem. PutSn:=_I⊕_Tnfor alln∈N. We define a family of mappings {Wn}:X→Xby

Wx=Sx;

Wx=

 Sx⊕

 Sx; Wx=

 Sx⊕

 

 Sx⊕

 Sx ; ..

.

Wnx=

 Sx⊕

 

 Sx⊕

  · · · ⊕   Sn–x⊕



Snx · · ·

;

.. .

It follows from Lemma . thatWnis quasi-nonexpansive for alln∈Nand

∞

n=F(Wn) =

∞

n=F(Tn).

We first show that∞_n=sup{d(Wn+x,Wnx) :x∈X}<∞. Fork∈N, putVk(k):=Skand

V_n(k):= Sk⊕

 

 Sk+⊕

  · · · ⊕   Sn–⊕

 Sn · · ·

for alln>k. By Lemma ., we have that

d(Wn+x,Wnx) =d

 Sx⊕

 V

()

n+x,

 Sx⊕

 V () n x ≤ √   d  Sx⊕

 V

()

n+x,

 Sx⊕

 V () n x ≤ √    d  Sx⊕

 V

()

n+x,

 Sx⊕

 V () n x ≤ · · · ≤ √   n– d  Snx⊕



Sn+x,Snx

≤ √   n– ·π 

for allx∈Xandn∈N. Then

supd(Wn+x,Wnx) :x∈X

≤ √   n– ·π 

and the result follows. In particular,{Wnx}is a Cauchy sequence for eachx∈X. We now

define the mappingW:X→Xby

Wx:= lim

Next, we show that F(W) =∞_n=F(Tn). It is easy to see that

∞

n=F(Tn)⊂F(W). On the

other hand, letq∈∞_n=F(Tn) =

∞

n=F(Sn) andp∈F(W). We prove thatp∈F(Tk) for all

k∈N. Letk∈Nbe given. For anyn>k, it follows from Lemma . that

cosd(q,Wnp)

=cosd

q, Sp⊕

 

 Sp⊕

  · · · ⊕   Sn–p⊕



Snp · · ·

≥

cosd(q,Sp) +  cosd

q, Sp⊕

  · · · ⊕   Sn–p⊕



Snp · · ·

≥

cosd(q,Sp) +· · ·+ 

kcosd(q,Skp) +· · ·+



n–cosd(q,Sn–p)

+ 

n–cosd(q,Snp)

≥

 +· · ·+

 k–+



k+ +· · ·+

 n–+



n– cosd(q,p) +



kcosd(q,Skp)

=

 – 

k cosd(q,p) +



kcosd(q,Skp).

Lettingn→ ∞yieldsWnp→Wp=pand

cosd(q,p) =cosd(q,Wp)≥

 – 

k cosd(q,p) +



kcosd(q,Skp).

This implies thatcosd(q,p) =cosd(q,Skp). It follows from Lemma . that

cosd(q,p)sind(p,Tkp) =cosd(q,Skp)sind(p,Tkp)

=cosd

q, p⊕



Tkp sind(p,Tkp)

≥cosd(q,p)sind(p,Tkp)

 +cosd(q,Tkp)sin

d(p,Tkp)



≥cosd(q,p)sind(p,Tkp)  .

Using elementary trigonometry, we get thatd(p,Tkp) = . Sincekis arbitrary, we have

p∈∞_n=F(Tn). Hence (i) is proved.

Finally, we prove (ii). We assume thatTnis-demiclosed for alln∈N. We show thatW

is-demiclosed. Let{xn} ⊂Xbe such thatlimn→∞d(xn,Wxn) =  and{xn}-converges

tox∈X. It follows from the definitions of{Wn}and{Vn()}that

Wn=

 S⊕

 V

()

n for alln≥.

Similar to the proof of the first and the second steps, we can define the quasi-nonexpansive mappingV:X→Xby

Vx:= lim

n→∞V

()

and F(V) =∞_n=F(Tn). This implies that

W= S⊕



V and F(W) = F(S)∩F(V).

Then, by Lemma ., we obtain thatd(xn,Sxn)→ andd(xn,Vxn)→. Thus

d(xn,Txn) = d

xn,

 xn⊕



Txn = d(xn,Sxn)→.

Since T is -demiclosed, we have x ∈F(T). Continuing this procedure gives x ∈

∞

n=F(Tn) = F(W). This completes the proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1_{Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan.}2_{Department of} Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand. 3_{Centre of Excellence in} Mathematics, CHE, Sriayudthaya Road, Bangkok, 10400, Thailand.

Acknowledgements

The authors thank the referee for their suggestions which enhanced the presentation of the paper. The first author is supported by Grant-in-Aid for Scientific Research No. 22540175 from Japan Society for the Promotion of Science. The second author is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education of Thailand. The last author is supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0188/2552) and Khon Kaen University under the RGJ-Ph.D. scholarship.

Received: 28 June 2013 Accepted: 15 November 2013 Published:13 Dec 2013

References

1. Bridson, MR, Haefliger, A: Metric Spaces of Non-Positive Curvature. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 319. Springer, Berlin (1999)

2. Kirk, WA: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003). Colecc. Abierta, vol. 64, pp. 195-225. Univ. Sevilla Secr. Publ., Seville (2003)

3. Kirk, WA: Geodesic geometry and fixed point theory II. In: International Conference on Fixed Point Theory and Applications, pp. 113-142. Yokohama Publ., Yokohama (2004)

4. Espínola, R, Fernández-León, A: CAT(k)-Spaces, weak convergence and fixed points. J. Math. Anal. Appl.353, 410-427 (2009)

5. Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc.4, 506-510 (1953)

6. He, JS, Fang, DH, López, G, Li, C: Mann’s algorithm for nonexpansive mappings in CAT(κ) spaces. Nonlinear Anal.75, 445-452 (2012)

7. Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal.68, 3689-3696 (2008) 8. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28.

Cambridge University Press, Cambridge (1990)

9. Dhompongsa, S, Kirk, WA, Sims, B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal.65, 762-772 (2006)

10. Tan, K-K, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl.178, 301-308 (1993)

11. Kimura, Y, Satô, K: Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one. Fixed Point Theory Appl.2013, 14 (2013)

12. Kimura, Y, Satô, K: Convergence of subsets of a complete geodesic space with curvature bounded above. Nonlinear Anal.75, 5079-5085 (2012)

13. Aoyama, K, Kimura, Y, Takahashi, W, Toyoda, M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal.67, 2350-2360 (2007)

14. Kimura, Y, Satô, K: Image recovery problem on a geodesic space with curvature bounded above by one (submitted)

10.1186/1687-1812-2013-336