Implicit Mann approximation with perturbations for nonexpansive semigroups in CAT(0) spaces

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R E S E A R C H

Open Access

Implicit Mann approximation with

perturbations for nonexpansive semigroups

in CAT(0) spaces

Xue-song Li

1*

_{and Tsz-leung Yip}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Sichuan University, Chengdu, Sichuan 610064, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, we study the convergence of implicit Mann iteration processes with bounded perturbations for approximating a common ﬁxed point of nonexpansive semigroup in CAT(0) spaces. We obtain the-convergence results of implicit Mann iteration schemes with bounded perturbations for a family of nonexpansive mappings in CAT(0) spaces. Under certain and diﬀerent conditions, we also get the strong convergence theorems of implicit Mann iteration schemes with bounded perturbations for nonexpansive semigroups in CAT(0) spaces. The results presented in this paper extend and enrich the existing literature.

MSC: 47H05; 47H10; 47J25

Keywords: common ﬁxed point;-convergence; nonexpansive semigroup; CAT(0) space; implicit approximation

1 Introduction

Let (X,d) be a metric space andK be a subset ofX. A mappingT :K→Xis said to be nonexpansive ifd(Tx,Ty)≤d(x,y) for allx,y∈K. We denote the set of all nonnegative real numbers byR+_{and the set of all ﬁxed points of}_T_by_F₍_T_),_i.e._,

F(T) ={x∈K:Tx=x}.

For eachn∈N, letTn:K→Kbe nonexpansive mappings and denote the common ﬁxed points set of the family{Tn}by

∞

n=F(Tn). A family of mappings{Tn}is said to be uni-formly asymptotically regular if, for any bounded subsetBofK,

lim n→∞sup_z_∈_Bd

Tnz,Ti(Tnz)

= 

for alli∈N.

A nonexpansive semigroup is a family

:=T(t) :t≥

of mappingsT(t) onKsuch that

() T(s+t)x=T(s)(T(t)x)for allx∈Kands,t≥;

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() T(t) :K→Kis nonexpansive for eacht≥;

() for eachx∈K, the mappingT(·)xfromR+_to_K_{is continuous.}

We denote byF() the common ﬁxed points set of nonexpansive semigroup,i.e.,

F() = t∈R+

FT(t)=x∈X:T(t)x=xfor eacht≥.

Note that, ifKis a nonempty, compact and convex subset of a Banach space, thenF() is nonempty (see [–]).

A geodesic fromxtoyinXis a mappingfrom a closed interval [,l]⊂RtoXsuch that() =x,(l) =yandd((t),(t)) =|t–t|for allt,t ∈[,l]. In particular,is an isometry andd(x,y) =l. The imageofis called a geodesic (or metric) segment joining

xandy. The space (X,d) is said to be a geodesic space if any two points ofXare joined by a geodesic segment, andXis said to be uniquely geodesic if there is exactly one geodesic joiningxandyfor anyx,y∈X, which is denoted by [x,y] and is called the segment joining

xandy. A subsetKof a geodesic spaceXis said to be convex if for anyx,y∈K, [x,y]⊂K. A geodesic triangle(x,x,x) in a geodesic metric space (X,d) consists of three points

x,x,xinX(the vertices of) and a geodesic segment between each pair of vertices

(the edges of). A comparison triangle for the geodesic triangle(x,x,x) in (X,d) is a

triangle ¯(x,x,x) =(x¯,x¯,x¯) inRsuch thatdR(x¯_i,x¯_j) =d(x_i,x_j) for alli,j∈ {, , }.

It is known that such a triangle always exists (see []). A geodesic space is said to be a CAT() space if all geodesic triangles of appropriate size satisfy the following comparison axiom (CA):

(CA) Letbe a geodesic triangle in(X,d)and let ¯ ⊂R_{be a comparison triangle for}

. Thenis said to satisfy the CAT() inequality if, for allx,y∈ and all comparison pointsx¯,y¯∈ ¯,

d(x,y)≤d(¯x,y¯).

The complete CAT() spaces are often called Hadamard spaces (see []). For anyx,y∈X, we denote byαx⊕( –α)ythe unique pointz∈[x,y] which satisﬁes

dx,αx⊕( –α)y= ( –α)d(x,y) and dy,αx⊕( –α)y=αd(x,y).

It is known that if (X,d) is a CAT() space andx,y∈X, then for anyβ∈[, ], there exists a unique pointβx⊕( –β)y∈[x,y]. For anyz∈X, the following inequality holds:

dz,βx⊕( –β)y≤βd(z,x) + ( –β)d(z,y),

whereβx⊕( –β)y∈[x,y] (for metric spaces of hyperbolic type, see []).

Recently, Choet al.[] studied the strong convergence of an explicit Mann iteration sequence{zn}for approximating a common ﬁxed point ofin a CAT() space, where{zn} is generated by the following iterative scheme for a nonexpansive semigroup={T(t) :

t∈R+_}_:

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whereα∈(, ) and{tn} ⊂R+. The existence of ﬁxed points, an invariant approximation and convergence theorems for several mappings in CAT() spaces have been studied by many authors (see [–]).

On the other hand, Thong [] considered an implicit Mann iteration process for a non-expansive semigroup={T(t) :t∈R+_}_{on a closed convex subset}_C_{of a Banach space, as}

follows:

x∈C, xn=αnxn–+ ( –αn)T(tn)xn, n≥. (.) Under diﬀerent conditions, Thong [] proved the weak convergence and strong conver-gence results of implicit Mann iteration scheme (.) for nonexpansive semigroups in cer-tain Banach spaces. In the last twenty years, many authors have studied the convergence of implicit iteration sequences for nonexpansive mappings, nonexpansive semigroups and pseudocontractive semigroups in Banach spaces (see [–] and the references therein). Readers may consult [, , ] for the convergence of Ishikawa iteration sequences for nonexpansive mappings and nonexpansive semigroups in certain Banach spaces. In the literature of approximating convergence for nonexpansive mappings and nonexpansive semigroups in CAT() spaces,explicititeration schemes are very abundant butimplicit

iteration process remains unaddressed. Therefore, it is of interest to investigate the con-vergence ofimplicitMann type process with perturbations for nonexpansive semigroups in CAT() spaces.

Motivated and inspired by the work mentioned above, we consider the following implicit Mann iteration scheme with perturbations{un}for a family of nonexpansive mappings in a CAT() space:

x∈K, xn=αnxn–⊕( –αn)

( –θn)Tnxn⊕θnun

, ∀n≥, (.)

where{αn} ⊂(, ] and{θn} ⊂[, ] are given sequences of real numbers,{un}is a bounded sequence inK. We prove that{xn}generated by (.) is-convergent to some point in

∞

n=F(Tn) under appropriate conditions. We also consider the following implicit Mann iteration process with perturbations{un}for a nonexpansive semigroup={T(t) :t≥} in a CAT() space:

x∈K, xn=αnxn–⊕( –αn)

( –θn)T(tn)xn⊕θnun

, ∀n≥, (.)

where{αn} ⊂(, ] and{θn} ⊂[, ] are given sequences of real numbers,{un}is a bounded sequence inK. Under various and appropriate conditions, we obtain that{xn}generated by (.) converges strongly to a common ﬁxed point of. We extend the strong convergence result in [] and establish the-convergence results of implicit Mann type approxima-tion for nonexpansive semigroups in CAT() spaces.

2 Deﬁnitions and lemmas

Let{xn}be a bounded sequence in a CAT() space (X,d). For anyx∈X, denote

rx,{xn}

=lim sup n→∞

d(x,xn).

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(ii) rK({xn}) =inf{r(x,xn) :x∈K}is called the asymptotic radius of{xn}with respect

toK;

(iii) the setA({xn}) ={x∈X:r(x,{xn}) =r({xn})}is called the asymptotic center of{xn};

(iv) the setAK({xn}) ={x∈K:r(x,{xn}) =rK({xn})}is called the asymptotic center of {xn}with respect toK.

Deﬁnition .[, ] A sequence{xn}in a CAT() spaceXis said to be-convergent to

a pointxinX, ifxis the unique asymptotic center of{xnj}for all subsequences{xnj} ⊆ {xn}.

In this case, we write-limn→∞xn=xandxis called the-limit of{xn}.

For the sake of convenience, we restate the following lemmas that shall be used.

Lemma .[] Let(X,d)be a CAT() space. Then,

d( –t)x⊕ty,z≤( –t)d(x,z) +td(y,z)

for all x,y,z∈X and t∈[, ].

Lemma .[] Let(X,d)be a CAT() space. Then,

d( –t)x⊕ty,z ≤( –t)d(x,z)+td(y,z)–t( –t)d(x,y)

for all x,y,z∈X and t∈[, ].

Lemma .[] Let K be a closed convex subset of a complete CAT() space and T :

K →K be a nonexpansive mapping. Suppose that {xn} is a bounded sequence in K

such thatlimn→∞d(xn,Txn) = and{d(xn,p)}converges for all p∈F(T). Then,ωw(xn) =

A({xnj})⊂F(T), where the union is taken over all subsequences{xnj}of{xn}. Moreover, ωw(xn)consists of exactly one point.

Lemma .[] Let{zn}and{wn}be bounded sequences in a CAT() space X. Let{αn}be

a sequence in[, ]such that <lim infn→∞αn≤lim supn→∞αn< . Deﬁne zn=αnzn–⊕

( –αn)wnfor all n∈Nand suppose that

lim sup n→∞

d(wn+,wn) –d(zn+,zn) ≤.

Thenlimn→∞d(wn,zn) = .

Lemma .[] Let{an},{αn}and{βn}be sequences of nonnegative real numbers such

that

an+≤( +αn)an+βn, n≥n,

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3 Main results

To focus on the convergence results of this present paper, it is necessary to show that the sequences generated by implicit Mann iteration processes (.) and (.) are well deﬁned.

Lemma . Let K be a nonempty, closed and convex subset of a complete CAT() space

X and Tn:K→K be nonexpansive mappings. Suppose that{un}is a bounded sequence

in K ,{αn} ⊂(, ]and{θn} ⊂[, ]are given parameter sequences. Then the sequence{xn}

generated by implicit Mann iteration process (.) is well deﬁned.

Proof For eachn∈Nand any givenu,v∈K, deﬁne a mappingSn:K→Kby

Snx:=αnu⊕( –αn)

( –θn)Tnx⊕θnv

, ∀n≥.

It can be veriﬁed that for each ﬁxedn∈N,Snis a contractive mapping. Indeed, if setting

pn= ( –θn)Tnx⊕θnvandqn= ( –θn)Tny⊕θnv, then we haveSnx=αnu⊕( –αn)pnand

Sny=αnu⊕( –αn)qn. It follows from Lemmas . and . that

d(Snx,Sny)



=dSnx,αnu⊕( –αn)qn



≤( –αn)

d(Snx,qn)



+αn

d(Snx,u)



–αn( –αn)

d(u,qn)



≤( –αn)

( –αn)

d(pn,qn)



+αn

d(u,qn)



–αn( –αn)

d(u,pn)



+αn( –αn)

d(pn,u)–αn( –αn)

d(u,qn) = ( –αn)

d(pn,qn) .

Consequently, we have thatd(Snx,Sny)≤( –αn)d(pn,qn) andd(pn,qn)≤( –θn)d(Tnx,

Tny)≤( –θn)d(x,y). Thus,

d(Snx,Sny)≤( –αn)( –θn)d(x,y),

which shows that for eachn∈N,Snis a contractive mapping. By induction, Banach’s ﬁxed theorem yields that the sequence{xn}generated by (.) is well deﬁned. This completes

the proof.

We need the following lemma for our main results. The analogs of [, Lemma .] and [, Lemma .] are given below. We sketch the proof here for the convenience of the reader.

Lemma . Let K be a nonempty, closed and convex subset of a complete CAT() space X,

{un}be a bounded sequence in K and Tn:K→K be nonexpansive mappings. Let{αn} ⊂ (, ]and{θn} ⊂[, ]be given sequences such that <lim infn→∞αn≤lim supn→∞αn< .

Suppose that{xn}generated by (.) is bounded (or, equivalently,{Tnxn}is bounded) and

either

lim

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Proof First, we show the equivalence between the boundedness of{xn}and the bounded-ness of{Tnxn}. If{xn}is bounded, then set

M=sup

d(xn,x) :n∈N

< +∞, M=sup

d(un,x) :n∈N

< +∞

for some given pointx∈Xandα=lim infn→∞αn> ,β=lim supn→∞αn< . Setting  <

a<α≤β<b< , we know that there existsn∈Nsuch that for alln≥n,

θn≤/,  <a<αn<b< . (.)

It follows from Lemma . that

d(Tnxn,x)≤d(Tnxn,xn) +d(xn,x)

=dTnxn,αnxn–⊕( –αn)

+d(xn,x)

≤αnd(Tnxn,xn–) + ( –αn)d

Tnxn, ( –θn)Tnxn⊕θnun

+d(xn,x)

≤αnd(Tnxn,xn–) + ( –αn)θnd(Tnxn,un) +d(xn,x) ≤αnd(Tnxn,x) +αnd(xn–,x) + ( –αn)θn

d(Tnxn,x) +d(un,x) +d(xn,x)

≤αn+ ( –αn)θn d(Tnxn,x) +d(xn–,x) +d(un,x) +d(xn,x).

Hence, for alln≥n, from (.) we have

d(Tnxn,x)≤

 ( –αn)( –θn)

d(xn–,x) +d(un,x) +d(xn,x) ≤(M+ M)/( –b),

which means that{Tnxn}is bounded. Conversely, if{Tnxn}is bounded, then set

Q=sup

d(Tnxn,x) :n∈N

< +∞ and M=sup

d(un,x) :n∈N

< +∞

for some given pointx∈X. DenoteQ=max{Q,M}. From Lemma ., we have

d(xn,x) =d

αnxn–⊕( –αn)

,x

≤αnd(xn–,x) + ( –αn)( –θn)d(Tnxn,x) + ( –αn)θnd(un,x) ≤αnd(xn–,x) + ( –αn)Q

≤maxd(xn–,x),Q

.

By induction, we know thatd(xn,x)≤max{d(x,x),Q}, which shows that{xn}is bounded. Then, we prove Lemma .. Iflim_n_→∞d(Tn+xn,Tnxn) = , then we have

lim sup n→∞

d( –θn+)Tn+xn+⊕θn+un+, ( –θn)Tnxn⊕θnun

–d(xn+,xn) ≤lim sup

n→∞

d( –θn+)Tn+xn+⊕θn+un+,Tn+xn+

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+d(Tn+xn,Tnxn) +d

Tnxn, ( –θn)Tnxn⊕θnTnxn+

n→∞

θn+d(un+,Tn+xn+) +θnd(Tnxn,Tnxn+)

≤lim sup n→∞

θn+d(un+,x) +θn+d(Tn+xn+,x) +θnd(xn,x) +θnd(xn+,x)

≤lim sup

n→∞ (θn++θn)

d(un+,x) +d(Tn+xn+,x) +d(xn+,x) +d(xn,x) ≤lim sup

n→∞

(M+Q+ M)(θn++θn) = . (.)

Similarly, iflimn→∞d(Tn+xn+,Tnxn+) = , then from inequality (.) we have

lim sup n→∞

d( –θn+)Tn+xn+⊕θn+un+, ( –θn)Tnxn⊕θnun

n→∞

d( –θn+)Tn+xn+⊕θn+un+,Tn+xn+

+d(Tn+xn+,Tnxn+)

+d(Tnxn+,Tnxn) +d

Tnxn, ( –θn)Tnxn⊕θnTnxn+

n→∞

θn+d(un+,Tn+xn+) +θnd(Tnxn,Tnxn+) = .

It follows from Lemma . thatlimn→∞d(( –θn)Tnxn⊕θnun,xn) = . Note that

lim

n→∞d(xn,Tnxn)≤nlim→∞

dxn, ( –θn)Tnxn⊕θnun

+d( –θn)Tnxn⊕θnun,Tnxn

≤ lim

n→∞

d( –θn)Tnxn⊕θnun,xn

+θnd(un,Tnxn)

≤ lim

n→∞

+θn

d(un,x) +d(Tnxn,x)

≤ lim

n→∞

+ (M+Q)θn = .

We have thatlimn→∞d(Tnxn,xn) = . This completes the proof.

As a direct consequence of Lemma ., the following lemma is immediate.

Lemma . Let K be a nonempty, closed and convex subset of a complete CAT() space X,

{un}be a bounded sequence in K and Tn:K→K be nonexpansive mappings. Let{αn} ⊂ (, ]and{θn} ⊂[, ]be given sequences such that <lim infn→∞αn≤lim supn→∞αn< .

Suppose that{xn}generated by (.) is bounded (or, equivalently,{Tnxn}is bounded) and

lim

n→∞sup_x_∈_Kd(Tn+x,Tnx) = 

holds. Iflimn→∞θn= , thenlimn→∞d(Tnxn,xn) = .

We now present our main results. The following theorem discusses the-convergence of implicit Mann iteration sequence (.) with perturbations for a family of nonexpansive mappings in CAT() spaces.

Theorem . Let K be a nonempty, closed and convex subset of a complete CAT() space

X, and Tn:K→K be uniformly asymptotically regular and nonexpansive mappings such

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given sequences such that <lim infn→∞αn≤lim supn→∞αn< . Then the sequence{xn}

generated by (.) is well deﬁned. Suppose that∞_n_=θn< +∞and either lim

n→∞d(Tn+xn,Tnxn) =  or nlim→∞d(Tn+xn+,Tnxn+) = 

holds. Then{xn} -converges to some point in

∞

n=F(Tn).

Proof By Lemma ., we know that the sequence{xn}generated by (.) is well deﬁned. For anyp∈∞_n_=F(Tn), from (.) and Lemma ., we have

d(xn,p) =d

αnxn–⊕( –αn)

,p

≤αnd(xn–,p) + ( –αn)

( –θn)d(Tnxn,p) +θnd(un,p)

≤αnd(xn–,p) + ( –αn)( –θn)d(xn,p) + ( –αn)θnd(un,p).

Sinceαn≤ – ( –αn)( –θn), it follows from (.) that for alln≥n,

d(xn,p)≤d(xn–,p) +

( –αn)θn  – ( –αn)( –θn)

d(un,p)≤d(xn–,p) +



aθnd(un,p). (.)

Since∞_n_=θn< +∞and{un}is bounded, Lemma . yields that{d(xn,p)}converges and thus{xn}is bounded.

Applying Lemma ., we havelimn→∞d(xn,Tnxn) = . We prove that for eachi∈N,

lim

n→∞d(xn,Tixn) = .

Because the family of nonexpansive mappings{Ti}is uniformly asymptotically regular, we have

d(xn,Tixn)≤d(xn,Tnxn) +d

Tnxn,Ti(Tnxn)

+dTi(Tnxn),Tixn

≤d(xn,Tnxn) + sup z∈{xn}

dTnz,Ti(Tnz)

→.

Since{d(xn,p)}converges for anyp∈

∞

n=F(Tn), an application of Lemma . yields that

ωw(xn) consists of exactly one point and is contained inF(Ti), for alli∈N. This shows that {xn} -converges to some point in

∞

n=F(Tn). This completes the proof. By Lemma . and Theorem ., the following theorem holds trivially.

Theorem . Let K be a nonempty, closed and convex subset of a complete CAT() space

X, and Tn:K→K be uniformly asymptotically regular and nonexpansive mappings such

that∞_n_=F(Tn)=∅. Let{un}be a bounded sequence in K ,{αn} ⊂(, ]and{θn} ⊂[, ]be

given sequences such that <lim infn→∞αn≤lim supn→∞αn< . Then the sequence{xn}

generated by (.) is well deﬁned. Suppose that∞_n_=θn< +∞and lim

n→∞sup_x_∈_Kd(Tn+x,Tnx) = 

holds. Then{xn} -converges to some point in

∞

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Finally, we study the strong convergence of implicit Mann iteration sequence (.) with perturbations for a nonexpansive semigroup in CAT() spaces, under various and appro-priate conditions.

Theorem . Let K be a compact and convex subset of a complete CAT() space X, and

={T(t) :t∈R+_}_{be a nonexpansive semigroup on K . Let} _{_u

n}be a bounded sequence

in K , {αn} ⊂(, ] and{θn} ⊂[, ] be given sequences such that  <lim infn→∞αn≤ lim sup_n_→∞αn< . Then the sequence{xn}generated by implicit scheme (.) is well

de-ﬁned. Suppose that{tn}is a sequence inR+such that lim inf

n→∞ tn<lim sup_n_→∞ tn and nlim→∞(tn+–tn) = .

If∞_n_=θn< +∞, then{xn}converges strongly to some point in F().

Proof Following the proof details of the main result of [], we know thatF() in complete CAT() spaces is nonempty (see [, ]). From Lemma ., it is easy to see that the sequence {xn}generated by (.) is well deﬁned. We show that

lim n→∞sup_x_∈_Kd

T(tn)x,T(tn–)x

= . (.)

Assume for the contrary that (.) does not hold. There exist a subsequence{tnk} ⊂ {tn},

a sequence{yk} ⊂Kand anη>  such that for allk∈N,

dT(tnk)yk,T(tnk–)yk

≥η.

SinceKis compact, there exists a convergent subsequence of{yk}. Without loss of gener-ality, we assume thatyk→yask→ ∞. Obviously,y∈Kand so

 < η

≤lim sup k→∞ d

T(tnk)yk,T(tnk–)yk

T|tnk–tnk–|

yk,T()yk

≤lim sup k→∞

dT|tnk–tnk–|

yk,T

|tnk–tnk–|

y

+dT|tnk–tnk–|

y,T()y+dT()y,T()yk

≤lim sup k→∞

d(yk,y) +d

T|tnk–tnk–|

y,T()y = ,

which is a contradiction. Formula (.) follows readily. By Lemma ., we have

lim n→∞d

T(tn)xn,xn

= .

Following the proof of [, Theorem .], we can show that there exists a subsequence {xnj} convergent tox∗, where x∗ is a common ﬁxed point of {T(t) :t∈R

+_}_{. Since} _x∗

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Remark . The proofs of Theorems . and . are respectively similar to [,

Theo-rems . and .]. As we know, the existing literature of approximating convergence for several mappings in CAT() spaces restricts to explicit iteration schemes. Theorems .-. extend and develop some existing results such as [, Theorems . and .] and [, Theorems .-.] from explicit Mann iteration schemes to implicit Mann iteration pro-cesses with perturbations.

We prove another strong convergence theorem which is diﬀerent from Theorem ..

Theorem . Let K be a compact and convex subset of a complete CAT() space X, and

={T(t) :t∈R+_}_{be a nonexpansive semigroup on K . Let}_{_u

n}be a bounded sequence in

K ,{αn} ⊂(, ]and{θn} ⊂[, ]be given sequences. Then the sequence{xn}generated by

scheme (.) is well deﬁned. If

lim

n→∞tn=nlim→∞

αn+θn

tn = 

and∞_n_=θn< +∞, then{xn}converges strongly to a common ﬁxed point x∗of.

Proof Following the proof details of the main result of [], we know thatF() in complete CAT() spaces is nonempty (see [, ]). From Lemma ., we know that{xn}generated by (.) is well deﬁned.

Claim : If{rn}is a sequence of nonnegative real numbers such thatlimn→∞rn= , then

lim n→∞sup_x_∈_Kd

T(rn)x,T()x

= . (.)

Assume for the contrary that (.) does not hold. There exist a subsequence{rnk} ⊂ {rn},

a sequence{yk} ⊂Kand anη>  such that for allk∈N,

dT(rnk)yk,T()yk

≥η.

SinceKis compact, there exists a convergent subsequence of{yk}. Without loss of gener-ality, we assume thatyk→yask→ ∞. Obviously,y∈Kand so

 < η

T(rnk)yk,T()yk

≤lim sup k→∞

dT(rnk)yk,T(rnk)y

+dT(rnk)y,T()y

+dT()y,T()yk

≤lim sup k→∞

d(yk,y) +d

T(rnk)y,T()y = ,

which is a contradiction. Formula (.) follows readily.

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Lemma . that

dxn,T(tn)xn

=dαnxn–⊕( –αn)

( –θn)T(tn)xn⊕θnun

,T(tn)xn

≤αnd

xn–,T(tn)xn

+ ( –αn)θnd

un,T(tn)xn

≤αn+ ( –αn)θn d

xn,T(tn)xn

+αnd(xn–,xn) + ( –αn)θnd(un,xn).

Thus, we have

dxn,T(tn)xn

≤ αn+θn

( –αn)( –θn)

d(xn–,xn) +d(un,xn).

For any givenx∈X, letM=sup{d(xn,x) :n∈N}andM=sup{d(un,x) :n∈N}. SinceK is compact and{un}is bounded, it follows thatM< +∞andM< +∞. Also, we know

that αn≤/ and θn≤ / for suﬃciently large n∈N, because limn→∞αn =  and lim_n_→∞θn= . Consequently,

dxn,T(tn)xn

≤(αn+θn)

d(xn–,x) + d(xn,x) +d(un,x)

≤(M+M)(αn+θn)→. (.)

Hence, it follows from (.) that

lim sup n→∞

dT()xn,xn

≤lim sup n→∞

dT()xn,T(tn)xn

+dT(tn)xn,xn

≤ lim

n→∞sup_x_∈_Kd

T()x,T(tn)x

+lim sup n→∞ d

xn,T(tn)xn

= . (.)

For any givent> , from (.) we know that

dT()xn,T(t)xn

≤

[t/tn]–

k=

dT(ktn)xn,T

(k+ )tn

xn

+dT[t/tn]tn

xn,T(t)xn

≤[t/tn]d

xn,T(tn)xn

+dTt– [t/tn]tn

xn,xn

≤(M+M)t

αn+θn

tn

+maxdT(s)xn,xn

: ≤s≤tn

= (M+M)t

αn+θn

tn

+dT(sn)xn,xn saysn∈[,tn]

≤(M+M)t

αn+θn

tn

+dT(sn)xn,T()xn

+dT()xn,xn

≤(M+M)t

αn+θn

tn

+dT()xn,xn

+sup x∈K

dT(sn)x,T()x

,

where [t/tn] is the integer part oft/tn. Sincelimn→∞αnt+nθn = , it follows from (.) and

(.) that

lim n→∞d

xn,T(t)xn

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Therefore, from the above equality, we know thatx∗ ∈F(). From (.), Lemma . yields thatlimn→∞d(xn,x∗) exists and thus{xn}converges strongly tox∗asn→ ∞. This

completes the proof.

Remark . The main results presented in this paper can be immediately applied to any

CAT(k) space withk≤, because any CAT(k) space is a CAT(k) space for anyk >k(see [, ]).

IfXis a Banach space, from Theorem ., we have the following corollary.

Corollary . Let K be a compact and convex subset of a Banach space X, and={T(t) :

t∈R+_}_{be a nonexpansive semigroup on K . Let}_{_u

n}be a bounded sequence in K ,{αn} ⊂ (, ]and{θn} ⊂[, ]be given sequences. Then the sequence{xn}generated by

x∈K, xn=αnxn–+ ( –αn)

( –θn)T(tn)xn+θnun

, ∀n≥

is well deﬁned. If

lim

n→∞tn=nlim→∞

αn+θn

tn = 

and∞_n_=θn< +∞, then{xn}converges strongly to a common ﬁxed point x∗of.

Remark . Whenθn≡, Corollary . reduces to [, Theorem .]. Therefore,

The-orem . and Corollary . extend [, TheThe-orem .] from implicit Mann iteration pro-cesses to implicit Mann iteration propro-cesses with bounded perturbations.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China.}2_{Department of Logistics and}

Maritime Studies, Faculty of Business, Hong Kong Polytechnic University, Hong Kong, P.R. China.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11026063, 11171237, 11101069). This work was also supported by a Hong Kong Polytechnic University Postdoctoral Fellowship (Number G-YX4N) and an Internal Competitive Research Grant (Number A-PL05).

Received: 27 March 2012 Accepted: 22 August 2012 Published: 11 September 2012 References

1. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA54, 1041-1044 (1965) 2. DeMarr, R: Common fixed points for commuting contraction mappings. Pac. J. Math.13, 1139-1141 (1963) 3. Lim, TC: A fixed point theorem for families of nonexpansive mappings. Pac. J. Math.53, 487-493 (1974) 4. Bridson, M, Haefliger, A: Metric Spaces of Non-Positive Curvature. Springer, New York (1999)

5. Kirk, WA: A fixed point theorem in CAT(0) spaces andR-trees. Fixed Point Theory Appl.4, 309-316 (2004) 6. Ćirić, L: Common fixed point theorems for a family of non-self mappings in convex metric spaces. Nonlinear Anal.

71(5-6), 1662-1669 (2009)

7. Cho, YJ, ´Ciri´c, L, Wang, SH: Convergence theorems for nonexpansive semigroups in CAT(0) spaces. Nonlinear Anal.74, 6050-6059 (2011)

(13)

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

10. Dhompongsa, S, Kirk, WA, Panyanak, B: Nonexpansive set-valued mappings in metric and Banach spaces. J. Nonlinear Convex Anal.8, 35-45 (2007)

11. Dhompongsa, S, Panyanak, B: On-convergence theorems in CAT(0) spaces. Comput. Math. Appl.56, 2572-2579 (2008)

12. Khan, AR, Khamsi, MA, Fukhar-ud-din, H: Strong convergence of a general iteration scheme in CAT(0) spaces. Nonlinear Anal.74, 783-791 (2011)

13. Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal.68, 3689-3696 (2008) 14. Laowang, W, Panyanak, B: Strong andconvergence theorems for multivalued mappings in CAT(0) spaces. Fixed

Point Theory Appl.2009, Art. ID 730132 (2009)

15. Laowang, W, Panyanak, B: Approximation ﬁxed points of nonexpansive nonself mappings in CAT(0) spaces. Fixed Point Theory Appl.2010, Art. ID 367274 (2010)

16. Leustean, L: A quadratic rage of asymptotic regularity for CAT(0) spaces. J. Math. Anal. Appl.325, 386-399 (2007) 17. Nanjaras, B, Panyanaka, B, Phuengrattana, W: Fixed point theorems and convergence theorems for

Suzuki-generalized nonexpansive mappings in CAT(0) spaces. Nonlinear Anal. Hybrid Syst.4, 25-31 (2010) 18. Saejung, S: Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl.2010, Art. ID 471781 (2010) 19. Razani, A, Salahifard, H: Invariant approximation for CAT(0) spaces. Nonlinear Anal.72, 2421-2425 (2010) 20. Thong, DV: An implicit iteration process for nonexpansive semigroups. Nonlinear Anal.74, 6116-6120 (2011) 21. Chidume, CE, Shahzad, N: Strong convergence of an implicit iteration process for a ﬁnite family of nonexpansive

mappings. Nonlinear Anal.62, 1149-1156 (2005)

22. Li, XS, Huang, NJ, Kim, JK: General viscosity approximation methods for common ﬁxed points of nonexpansive semigroups in Hilbert spaces. Fixed Point Theory Appl.2011, Art. ID 783502 (2011)

23. Qin, XL, Cho, SY: Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions. Appl. Math. Lett.23, 1252-1255 (2010)

24. Xu, HK, Ori, RG: An implicit iteration process for nonexpansive mappings. Numer. Funct. Anal. Optim.22(5-6), 767-773 (2001)

25. Buong, N: Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space. Comput. Math. Appl.61, 2546-2554 (2011)

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

27. Lim, TC: Remarks on some ﬁxed point theorems. Proc. Am. Math. Soc.60, 179-182 (1976)

28. Suzuki, T: Strong convergence theorems for inﬁnite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl.1, 103-123 (2005)

doi:10.1186/1687-1812-2012-145