An approximation method for common fixed points of a finite family of asymptotic pointwise nonexpansive mappings

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R E S E A R C H

Open Access

An approximation method for common ﬁxed

points of a ﬁnite family of asymptotic

pointwise nonexpansive mappings

Bancha Nanjaras

1

_{and Bancha Panyanak}

1,2*

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Faculty of Science, Chaing Mai University, Chiang Mai, 50200, Thailand

2_{Centre of Excellence in}

Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400, Thailand

Abstract

In this article, we consider an iterative scheme to approximate a common ﬁxed point for a ﬁnite family of asymptotic pointwise nonexpansive mappings. We obtain weak and strong convergence theorems of the proposed iteration in uniformly convex Banach spaces. The related results for complete CAT(0) spaces are also included.

MSC: 47H09; 47H10

Keywords: common ﬁxed point; asymptotic pointwise nonexpansive mapping;

weak convergence; strong convergence; Banach space; CAT(0) space

1 Introduction

It is well known that many of the most important nonlinear problems of applied math-ematics reduce to solving a given equation which in turn may be reduced to finding the fixed points of a certain operator. It is important not only to know the fixed points exist, but also to be able to construct that fixed points. Lau is a great mathematician who has published many good papers concerning to the existence and the approximation of fixed points for various types of mappings (see,e.g., [–]).

The existence of fixed points for nonexpansive mappings was studied independently by three authors in  (see Browder [], Göhde [], and Kirk []). Since then the iteration methods for approximating fixed points of nonexpansive mappings has rapidly been developed and many of papers have appeared (see,e.g., [–]). One of the popular classes of generalized nonexpansive mappings is the class of asymptotically nonexpansive mappings which was introduced by Goebel and Kirk [] in . Later on, Kirk and Xu [] introduced the concept of asymptotic pointwise nonexpansive mappings which gen-eralizes the concept of asymptotically nonexpansive mappings and proved the existence of fixed points for such maps in a uniformly convex Banach space. In , Kozlowski [] defined an iterative sequence for an asymptotic pointwise nonexpansive mappingT on a convex subsetCof a Banach spaceXbyx∈Cand

xk+= ( –tk)xk+tkTnkyk, ()

yk= ( –sk)xk+skTnkxk, k∈N,

where{tk}and{sk}are sequences in [, ] and{nk}is an increasing sequence of natural

numbers. He proved, under some suitable assumptions, that the sequence{xk}deﬁned by

() converges weakly to a ﬁxed point ofT whereXis a uniformly convex Banach space

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which satisﬁes the Opial condition and{xk}converges strongly to a ﬁxed point ofT

pro-videdTr_{is a compact mapping for some}_r_∈_N_{. Recently, Pasom and Panyanak []}

ex-tended Kozlowski’s results to a ﬁnite family of asymptotic pointwise nonexpansive map-pingsT, . . . ,Tm. Precisely, they proved weak and strong convergence theorems of the

it-erative process deﬁned by

xk+= ( –tmk)xk+tmkTmnky(m–)k, ()

y(m–)k= ( –t(m–)k)xk+t(m–)kTmnk–y(m–)k,

y(m–)k= ( –t(m–)k)xk+t(m–)kTmnk–y(m–)k,

.. .

yk= ( –tk)xk+tkTnkyk,

yk= ( –tk)xk+tkTnkyk,

yk=xk, k∈N,

where {tik}∞_k_= are sequences in [, ] for alli= , , . . . ,m, and{nk} be an increasing

se-quence of natural numbers. On the other hand, Kettapunet al.[] studied the iterative process deﬁned by

xk+= ( –tmk)y(m–)k+tmkTmny(m–)k, ()

y(m–)k= ( –t(m–)k)y(m–)k+t(m–)kTmn–y(m–)k,

y(m–)k= ( –t(m–)k)y(m–)k+t(m–)kTmn–y(m–)k,

.. .

yk= ( –tk)yk+tkTnyk,

yk= ( –tk)yk+tkTnyk,

yk=xk, k∈N,

whereT, . . . ,Tmare asymptotically quasi-nonexpansive mappings onC.

In this article, motivated by the results mentioned above, we obtain weak and strong convergence theorems of the iterative process deﬁned by

xk+= ( –tmk)y(m–)k+tmkTmnky(m–)k, ()

y(m–)k= ( –t(m–)k)y(m–)k+t(m–)kTmnk–y(m–)k,

y(m–)k= ( –t(m–)k)y(m–)k+t(m–)kTmnk–y(m–)k,

.. .

yk= ( –tk)yk+tkTnkyk,

yk= ( –tk)yk+tkTnkyk,

yk=xk, k∈N,

whereT, . . . ,Tmare asymptotic pointwise nonexpansive mappings onC,{tik}∞k=are

se-quences in [, ] for alli= , , . . . ,m, and{nk}be an increasing sequence of natural

num-bers.

2 Preliminaries and lemmas

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(i) nonexpansiveifd(Tx,Ty)≤(x,y)for allx,y∈C,

(ii) asymptotically nonexpansiveif there is a sequence{kn}of positive numbers with

the propertylim_n→∞kn= and such that

dTnx,Tny≤knd(x,y), for allx,y∈Candn≥,

(iii) asymptotically quasi-nonexpansiveif there is a sequence{kn}of positive numbers

with the propertylimn→∞kn= and such that

dTnx,p≤knd(x,p), for allx∈C,p∈F(T)andn≥,

(iv) asymptotic pointwise nonexpansiveif there exists a sequence of functions

αn:C→[,∞)such thatlim supn→∞αn(x)≤and

dTnx,Tny≤αn(x)d(x,y), for allx,y∈Candn≥.

The following implications hold.

Tis nonexpansive⇒Tis asymptotically nonexpansive ⇒Tis asymptotically quasi-nonexpansive

⇓

Tis asymptotic pointwise nonexpansive

The existence of ﬁxed points for asymptotic pointwise nonexpansive mappings in uni-formly convex Banach spaces was proved by Kirk and Xu [] as the following result.

Theorem . Let C be a nonempty bounded closed and convex subset of a uniformly con-vex Banach space X. Then every asymptotic pointwise nonexpansive mapping T:C→C has a ﬁxed point. Moreover, F(T)is closed and convex.

For common ﬁxed points of a family of commuting mappings, Pasom and Panyanak [] obtained the following result.

Theorem . Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Then every commuting familyS of asymptotic pointwise nonexpansive mappings on C has a nonempty closed convex common ﬁxed point set.

LetCbe a nonempty subset of a metric space (X,d). We shall denote byT(C) the class of all asymptotic pointwise nonexpansive mappings fromCintoC. LetT, . . . ,Tm∈T(C),

without loss of generality, we can assume that there exists a sequence of mappingsαn:

C→[,∞) such that for allx,y∈C,i= , . . . ,m, andn∈N,

dT_inx,T_iny≤αn(x)d(x,y) and lim sup

n→∞ αn(x)≤. ()

Letan(x) = max{αn(x), }. Again, without loss of generality, we can assume that

dT_inx,T_iny≤an(x)d(x,y), _n→∞liman(x) =  and an(x)≥, ()

for allx,y∈C,i= , . . . ,m, andn∈N. We deﬁnebn(x) =an(x) – , then for eachx∈Cwe

havelimn→∞bn(x) = .

Deﬁnition .[] DeﬁneTr(C) as a class of allT∈T(C) such that ∞

n=

bn(x) <∞ for everyx∈C, and ()

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LetCbe a nonempty subset of a Banach spaceXandT, . . . ,Tm∈Tr(C). Let{tik}∞k=⊂

(, ) be bounded away from  and  for alli= , , . . . ,mand{nk}be an increasing sequence

of natural numbers. Letx∈Cand deﬁne a sequence{xk}inCas

xk+= ( –tmk)y(m–)k+tmkTmnky(m–)k, ()

y(m–)k= ( –t(m–)k)y(m–)k+t(m–)kTmnk–y(m–)k,

y(m–)k= ( –t(m–)k)y(m–)k+t(m–)kTmnk–y(m–)k,

.. .

yk= ( –tk)yk+tkTnkyk,

yk= ( –tk)yk+tkTnkyk,

yk=xk, k∈N.

We say that the sequence{xk}in () is well deﬁned iflim supk→∞ank(xk) = . As in [], we

observe thatlim_k→∞ak(x) =  for everyx∈C. Hence, we can always choose a subsequence {ank}which makes{xk}well deﬁned.

Deﬁnition . A strictly increasing sequence{nk} ⊂Nis calledquasi-periodicif the

se-quence{nk+–nk}is bounded, or equivalently if there exists a numberp∈Nsuch that any

block ofpconsecutive natural numbers must contain a term of the sequence{nk}. The

smallest of such numberspwill be called a quasi-period of{nk}.

Recall that a mappingT:C→Cis calledsemi-compactif for any sequence{xn}inC

such that

lim

n→∞d(xn,Txn) = ,

there exists a subsequence{xnj}of{xn}andq∈C such thatlimj→∞xnj=q. A family of

mapping{Ti:i= , , . . . ,m}onC is said to satisfyCondition(A) if there exists a

non-decreasing functionf : [,∞)→[,∞) withf() =  andf(r) >  for allr>  such that d(x,Tjx)≥f(dist(x,F)), for somej= , . . . ,mfor allx∈C, where dist(x,F) =inf{d(x,p) :p∈

F=m_i_=F(Ti)}.

Lemma .[, Lemma .] Let{sn}and{un}be sequences of nonnegative real numbers

satisfy:

sn+≤( +un)sn, for all n∈N, and ∞

n=

un<∞.

Then (i)limnsnexists (ii) iflim infnsn= , thenlimnsn= .

Lemma .[, Lemma ] Suppose{rk}is a bounded sequence of real numbers and{dk,l}

is a doubly index sequence of real numbers which satisfy

lim sup

k→∞

lim sup

l→∞ dk,l≤, and rk+l≤rk+dk,l

for each k,l∈N. Thenlim_k→∞rk=a for some a∈R.

Lemma .[, ] Let X be a uniformly convex Banach space and let{tn}be a sequence

in[a,b]for some a,b∈(, ). Suppose that{un}and{vn}are sequences in X such that

lim sup

n→∞ un ≤r, lim supn→∞ vn ≤r, and n→∞limtnun+ ( –tn)vn=r,

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Lemma .[, Lemma .] Let C be a nonempty closed convex subset of a uniformly convex Banach space X and let T ∈Tr(C). Iflimn→∞xn–Txn= then for any m∈N,

lim_n→∞xn–Tmxn= .

Lemma .[, Theorem .] Let X be a uniformly convex Banach space with the Opial property and let C be a nonempty closed convex subset of X. Let T ∈Tr(C) and let ω∈X, {xn} ⊂X be such that weak-limn→∞xn=ω and limn→∞xn–Txn= . Then

ω∈F(T).

3 Results in Banach spaces

3.1 Results for bounded domains

Recall that a subsetCof a metric space (X,d) is said to beboundedif

diam(C) :=supd(x,y) :x,y∈C<∞.

Lemma . Let C be a nonempty closed convex subset of a Banach space X and T, . . . ,Tm∈

Tr(C). Let{tik}∞k=⊂[, ]and{nk} ⊂Nbe such that{xk}in () is well deﬁned. Assume that

F:=m_i_=F(Ti)=∅. Then for each p∈F, there are sequences of nonnegative real numbers {γk}and{δk}(depending on p) such that ∞k=γk<∞, ∞k=δk<∞and the following

state-ments hold: (i) Tnk

i y(i–)k–p ≤( +γk)y(i–)k–p, for alli= , . . . ,m;

(ii) yik–p ≤( +γk)ixk–p, for alli= , . . . ,m– ;

(iii) xk+–p ≤( +δk)xk–p;

(iv) ifCis bounded, thenlim_k→∞xk–pexists.

Proof Letp∈Fandγk=bnk(p) for allk∈N. Then

∞

k=γk<∞.

(i) Fori= , , . . . ,m, we have Tnk

i y(i–)k–p≤( +γk)y(i–)k–p.

(ii) By (), we obtain

yk–p=( –tk)(xk–p) +tk

Tnk

 xk–p ≤( –tk)xk–p+tkTnkxk–p ≤( –tk)xk–p+tk( +γk)xk–p ≤( +γk)xk–p.

We assume thatyjk–p ≤( +γk)jxk–pholds for somej= , . . . ,m– . From part (i),

we have

y(j+)k–p=( –t(j+)k)(yjk–p) +t(j+)k

Tnk

j+yjk–p ≤( –t(j+)k)yjk–p+t(j+)kTjn+kyjk–p ≤( –t(j+)k)yjk–p+t(j+)k( +γk)yjk–p ≤( +γk)yjk–p

≤( +γk)( +γk)jxk–p

= ( +γk)j+xk–p.

By mathematical induction, we obtain

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(iii) By part (ii), we get

xk+–p=( –tmk)(y(m–)k–p) +tmk

Tnk

my(m–)k–p ≤( –tmk)y(m–)k–p+tmkTmnky(m–)k–p ≤( –tmk)y(m–)k–p+tmk( +γk)y(m–)k–p ≤( +γk)y(m–)k–p

≤( +γk)( +γk)m–xk–p ≤( +γk)mxk–p

≤( +δk)xk–p,

whereδk=

_m



γk+

_m



γ k +· · ·+

_m

m

γm k . Since

∞

k=γk<∞, then ∞k=δk<∞.

(iv) By part (iii), we havexk+–p ≤ xk–p+ diam(C)δkfor allk∈N. Thus, for each

l∈N,

xk+l–p ≤ xk–p+ diam(C) k+l–

i=k δi.

Since ∞_i_=δi< ∞, lim supk→∞lim supl→∞ ki=+kl–δi = . The conclusion follows from

Lemma . by lettingrk=xk–panddk,l= diam(C) ki=+kl–δi.

Lemma . Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X and T, . . . ,Tm∈Tr(C). Let{tik}_k∞_=⊂[a,b]⊂(, )and{nk} ⊂Nbe such

that{xk}in () is well deﬁned. Assume that F:=

m

i=F(Ti)=∅. Then

(i) limk→∞y(i–)k–Tinky(i–)k= , for alli= , , . . . ,m;

(ii) lim_k→∞xk–Tinky(i–)k= , for alli= , , . . . ,m;

(iii) If the setJ ={k∈N:nk+=  +nk}is quasi-periodic, thenlimk→∞xk–Tixk= ,

for alli= , , . . . ,m.

Proof (i) Letp∈F, then by Lemma .(iv) we havelimk→∞xk–pexists. Let

lim

k→∞xk–p=c. ()

By () and Lemma .(ii), we get that

lim sup

k→∞ yik–p ≤c, for alli= , . . . ,m– . ()

Note that

xk+–p ≤( –tmk)y(m–)k–p+tmkTmnky(m–)k–p ≤( –tmk)y(m–)k–p+tmk( +γk)y(m–)k–p ≤( +γk)y(m–)k–p

= ( +γk)( –t(m–)k)(y(m–)k–p) +t(m–)k

Tnk

m–y(m–)k–p ≤( +γk)

( –t(m–)k)y(m–)k–p+t(m–)k( +rk)y(m–)k–p

≤( +γk)y(m–)k–p

.. .

≤( +γk)m–iyik–p,

for alli= , . . . ,m– . So that

c≤lim inf

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From () and (), we have

lim

k→∞yik–p=c, for alli= , , . . . ,m– . ()

That is,

lim

k→∞( –tik)(y(i–)k–p) +tik

Tnk

i y(i–)k–p=c, for alli= , , . . . ,m– . ()

By Lemma .(i) and (), we get that

lim sup

k→∞

Tnk

i y(i–)k–p≤c, for allj= , , . . . ,m– . ()

By (), (), (), and Lemma ., we obtain

lim

k→∞y(i–)k–T nk

i y(i–)k= , for alli= , , . . . ,m– . ()

For the casei=m, by Lemma .(i), we have Tnk

my(m–)k–p≤( +γk)y(m–)k–p.

This implies by () that

lim sup

k→∞

Tnk

my(m–)k–p≤c. ()

Moreover,

lim

k→∞( –tmk)(y(m–)k–p) +tmk

Tnk

my(m–)k–p= lim

k→∞xk+–p=c.

Again, by Lemma ., we get that

lim

k→∞y(m–)k–T nk

my(m–)k= . ()

Thus, () and () imply that

lim

k→∞y(i–)k–T nk

i y(i–)k= , for alli= , . . . ,m. ()

(ii) From (), we have

yik–y(i–)k=tikT nk

i y(i–)k–y(i–)k, for alli= , . . . ,m– .

By (), we obtain

lim

k→∞yik–y(i–)k= , fori= , . . . ,m– . ()

From

xk–yik ≤ xk–yk+yk–yk+· · ·+y(i–)k–yik, for alli= , . . . ,m– ,

it follows by () that

lim

k→∞xk–yik= , for alli= , . . . ,m– . ()

From

xk–Tinky(i–)k≤ xk–y(i–)k+y(i–)k–Tinky(i–)k,

it implies by () and () that

lim

k→∞xk–T nk

i y(i–)k= , for alli= , , . . . ,m. ()

(iii) Fori= , from (ii) we have

lim

k→∞T nk

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Ifi= , , . . . ,m, then Tnk

i xk–xk≤Tinkxk–Tinky(i–)k+Tinky(i–)k–xk ≤ank(xk)xk–y(i–)k+T

nk

i y(i–)k–xk.

By (), (), andlim sup_k→∞ank(xk) = , we get

lim sup

k→∞

Tnk

i xk–xk=  for alli= , , . . . ,m. ()

By () and (), we have

lim

k→∞T nk

i xk–xk=  for alli= , , . . . ,m. ()

From (), we have

xk+–xk ≤( –tmk)y(m–)k–xk+tmkTmnky(m–)k–xk

≤( –tmk)y(m–)k–xk+tmkTmnky(m–)k–y(m–)k+y(m–)k–xk

=y(m–)k–xk+tmkTmnky(m–)k–y(m–)k.

From () and (),

lim

k→∞xk+–xk= . ()

The proof of the remaining part is identical to the proof of [, Lemma .(iii)] upon

replacingd(·,·) with · .

By using Lemma . and the argument in the proof of [, Theorem .], we can obtain the following result.

Lemma . Let C be a nonempty bounded closed convex subset of a Banach space X and T, . . . ,Tm∈Tr(C). Let{tik}∞k=⊂[, ]and{nk} ⊂Nbe such that{xk}in () is well deﬁned.

Assume that F:=m_i_=F(Ti)=∅. Then{xk}converges strongly to a point in F if and only if

lim inf_k→∞dist(xk,F) = .

Theorem . Let X be a uniformly convex Banach space with the Opial property and C be a nonempty bounded closed convex subset of X. Let T, . . . ,Tm∈Tr(C)be such that F:=

m

i=F(Ti)=∅.{tik}∞_k_=⊂[a,b]⊂(, )and{nk} ⊂Nbe such that{xk}in () is well deﬁned.

If the set J ={k∈N:nk+ =  +nk}is quasi-periodic, then the sequence {xk} converges

weakly to a common ﬁxed point of the family{Ti:i= , . . . ,m}.

Proof We have by Lemma . thatlim_n→∞xk–pexists for everyp∈F. We shall prove

that{xk}has a unique weak subsequential limit inF. For this, we suppose that there are

subsequences{xkl}and{xkj}of{xk}which converge weakly touandv, respectively. By

Lemma .(iii),limk→∞Tixk–xk=  for alli= , . . . ,m. It follows from Lemma . that

u,v∈F(Ti) for alli= , . . . ,m. That isu,v∈F. Finally, we prove thatu=v. Suppose not,

then by the Opial property we get that

lim

k→∞xk–u=l→∞limxkl–u

< lim

l→∞xkl–v

= lim

k→∞xk–v

=lim

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< lim

j→∞xkj–u

= lim

k→∞xk–u.

This is a contradiction. Therefore, the proof is complete.

Theorem . Let X be a uniformly convex Banach space and C be a nonempty bounded closed convex subset of X. Let T, . . . ,Tm∈Tr(C)be such that Tilis semi-compact for some

i∈ {, . . . ,m}and l∈N.{tik}∞_k_=⊂[a,b]⊂(, )and{nk} ⊂Nbe such that{xk} in () is

well deﬁned. Suppose that F:=m_i_=F(Ti)=∅and the setJ ={k∈N:nk+=  +nk}is

quasi-periodic. Then{xk}converges strongly to a common ﬁxed point of the family{Ti:i=

, , . . . ,m}.

Proof By Lemma ., we have

lim

k→∞xk–Tixk= , for alli= , . . . ,m. ()

Leti∈ {, . . . ,m}be such thatTl

i is semi-compact. Thus, by Lemma .,

lim

k→∞xk–T l ixk= .

We can also ﬁnd a subsequence{xnj}of{xk} such thatlimj→∞xkj =q∈C. Hence, from

(), we have

q–Tiq=_j→∞limxkj–Tixkj= , for alli= , . . . ,m.

Thusq∈F. Therefore,{xkj}converges strongly toq∈F. But sincelimk→∞xk–qexists,

{xk}must itself converges toq. This completes the proof.

Theorem . Let X be a uniformly convex Banach space and C be a nonempty bounded closed convex subset of X. Let{T, . . . ,Tm} ⊂Tr(C)be satisfy Condition (A). Let{tik}∞k=⊂

[a,b]⊂(, ) and {nk} ⊂Nbe such that {xk} in () is well deﬁned. Suppose that F :=

m

i=F(Ti)=∅and the setJ ={k∈N:nk+=  +nk}is quasi-periodic. Then{xk}converges

strongly to a common ﬁxed point of the family{Ti:i= , , . . . ,m}.

Proof By Lemma .,lim_k→∞xk–Tixk= , for alli= , , . . . ,m. By using Condition (A),

there exists a nondecreasing functionf : [,∞)→[,∞) withf() = ,f(r) >  forr∈ (,∞) such that

lim

k→∞f

dist(xk,F)

≤ lim

k→∞xk–Tjxk=  for somej= , . . . ,m.

This implies thatlim_k→∞dist(xk,F) = . The conclusion follows from Lemma ..

3.2 Results for unbounded domains

To relax the boundedness of the domains we have to add some condition on the sequence

{bnk}.

Lemma . Let C be a nonempty closed convex subset of a Banach space X and T, . . . ,Tm∈

Tr(C)be such that F :=

m

i=F(Ti)=∅. Let {tik}∞_k_= ⊂[, ] and{nk} ⊂N be such that {xk}in () is well deﬁned. Assume that ∞k=supx∈Cbnk(x) <∞. Then for p∈F, we have

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Proof Similar to the proof of Lemma ., we can show thatxk+–p ≤( +ηk)xk–p

for allk∈N, whereηk=

_m



sk+

_m



s

k+· · ·+

_m

m

sm

k andsk=supx∈Cbnk(x). By assumption,

we have ∞_k_=si

k<∞for alli= , . . . ,m. It follows that ∞

k=ηk<∞. By Lemma ., we get

thatlimk→∞xk–pexists.

By using Lemma . and the argument in Section . we can obtain the following results.

Lemma . Let C be a nonempty closed convex subset of a Banach space X and T, . . . ,Tm∈

Tr(C)be such that F:=

m

i=F(Ti)=∅. Let{tik}k∞=⊂[, ]and{nk} ⊂Nbe such that{xk}

in () is well deﬁned. Assume that ∞_k_=sup_x∈Cbnk(x) <∞. Then

(i) lim_k→∞y(i–)k–T nk

i y(i–)k= , for alli= , , . . . ,m;

(ii) limk→∞xk–Tinky(i–)k= , for alli= , , . . . ,m;

(iii) If the setJ ={k∈N:nk+=  +nk}is quasi-periodic, thenlimk→∞xk–Tixk= ,

for alli= , , . . . ,m.

Lemma . Let C be a nonempty closed convex subset of a Banach space X and T, . . . ,Tm∈

Tr(C)be such that F:=mi=F(Ti)=∅. Let{tik}∞_k_=⊂[, ]and{nk} ⊂Nbe such that{xk}

in () is well deﬁned. Assume that ∞_k_=sup_x∈Cbnk(x) <∞. Then{xk}converges strongly to

a point in F if and only iflim infk→∞dist(xk,F) = .

Theorem . Let X be a uniformly convex Banach space with the Opial property and C be a nonempty closed convex subset of X. Let T, . . . ,Tm ∈ Tr(C) be such that F :=

m

i=F(Ti)=∅.{tik}∞_k_=⊂[a,b]⊂(, ) and{nk} ⊂Nbe such that{xk} in () is well

de-ﬁned. Assume that ∞_k_=sup_x∈Cbnk(x) <∞and the setJ ={k∈N:nk+=  +nk}is

quasi-periodic. Then the sequence{xk}converges weakly to a common ﬁxed point of the family {Ti:i= , . . . ,m}.

Theorem . Let X be a uniformly convex Banach space and C be a nonempty closed convex subset of X. Let T, . . . ,Tm ∈Tr(C)be such that Til is semi-compact for some i∈ {, . . . ,m}and l∈N,{tik}∞_k_=⊂[a,b]⊂(, )and{nk} ⊂Nbe such that{xk}in () is well

deﬁned. Suppose that ∞_k_=sup_x∈Cbnk(x) <∞, F:=

m

i=F(Ti)=∅and the setJ ={k∈N:

nk+=  +nk}is quasi-periodic. Then{xk}converges strongly to a common ﬁxed point of the

family{Ti:i= , , . . . ,m}.

Theorem . Let X be a uniformly convex Banach space and C be a nonempty closed convex subset of X. Let{T, . . . ,Tm} ⊂Tr(C)be satisfy Condition (A). Let{tik}∞_k_=⊂[a,b]⊂

(, )and{nk} ⊂Nbe such that{xk}in () is well deﬁned. Suppose that ∞k=supx∈Cbnk(x) <

∞, F:=m_i_=F(Ti)=∅and the setJ ={k∈N:nk+=  +nk}is quasi-periodic. Then{xk}

converges strongly to a common ﬁxed point of the family{Ti:i= , , . . . ,m}.

4 Results in CAT(0) spaces

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the fundamental role they play in geometry, we refer the reader to Bridson and Haeﬂiger [].

Letx,y∈X, by Lemma .(iv) of [] for eacht∈[, ], there exists a unique pointz∈ [x,y] such that

d(x,z) =td(x,y) and d(y,z) = ( –t)d(x,y). ()

From now on, we will use the notation ( –t)x⊕tyfor the unique pointzsatisfying (). Let{xn}be a bounded sequence in a metric space (X,d). Forx∈X, we set

rx,{xn}

=lim sup

n→∞ d(x,xn).

Theasymptotic radius r({xn}) of{xn}is given by

r{xn}

=infrx,{xn}

:x∈X,

and theasymptotic center A({xn}) of{xn}is the set

A{xn}

=x∈X:rx,{xn}

=r{xn}

.

It is known from Proposition  of [] that in a CAT() space,A({xn}) consists of exactly

one point. We now give the deﬁnition of-convergence.

Deﬁnition .[, ] A sequence{xn}in a metric spaceXis said to-convergetox∈X

ifxis the unique asymptotic center of{un}for every subsequence{un}of{xn}. In this case

we write–limnxn=xand callxthe-limitof{xn}.

LetCbe a nonempty closed convex subset of a CAT() spaceXand ﬁxx∈C. Deﬁne a

sequence{xk}inCas

xk+= ( –tmk)y(m–)k⊕tmkTmnky(m–)k, ()

y(m–)k= ( –t(m–)k)y(m–)k⊕t(m–)kTmnk–y(m–)k,

y(m–)k= ( –t(m–)k)y(m–)k⊕t(m–)kTmnk–y(m–)k,

.. .

yk= ( –tk)yk⊕tkTnkyk,

yk= ( –tk)yk⊕tkTnkyk,

yk=xk, k∈N,

whereT, . . . ,Tm∈T(C),{tik}∞k=are sequences in [, ] for alli= , , . . . ,m, and{nk}be an

increasing sequence of natural numbers.

By using the argument in Section  together with the results in [, , , ], we can also obtain the analogous results for CAT() spaces.

Theorem . Let C be a nonempty closed convex subset of a complete CAT() space X. Let T, . . . ,Tm ∈Tr(C)be such that F :=

m

i=F(Ti)=∅,{tik}∞k= ⊂[a,b]⊂(, ) and {nk} ⊂Nbe such that{xk} in () is well deﬁned. Suppose that either C is bounded or

∞

k=supx∈Cbnk(x) <∞. If the setJ ={k∈N:nk+=  +nk}is quasi-periodic, then the

sequence{xk}-converges to a common ﬁxed point of the family{Ti:i= , . . . ,m}.

Theorem . Let C be a nonempty closed convex subset of a complete CAT() space X. Let T, . . . ,Tm∈Tr(C)be such that F:=

m

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i∈ {, . . . ,m}and l∈N. Let{tik}∞k=⊂[a,b]⊂(, )and{nk} ⊂Nbe such that{xk}in ()

is well deﬁned. Suppose that either C is bounded or ∞_k_=sup_x∈Cbnk(x) <∞. If the setJ =

{k∈N:nk+=  +nk}is quasi-periodic, then{xk} converges strongly to a common ﬁxed

point of the family{Ti:i= , , . . . ,m}.

Theorem . Let C be a nonempty closed convex subset of a complete CAT() space X. Let{T, . . . ,Tm} ⊂Tr(C)be satisfy Condition (A) and F:=im=F(Ti)=∅. Let{tik}∞_k_=⊂

[a,b]⊂(, )and{nk} ⊂Nbe such that{xk}in () is well deﬁned. Suppose that either C is

bounded or ∞_k_=sup_x∈Cbnk(x) <∞. If the setJ ={k∈N:nk+=  +nk}is quasi-periodic,

then{xk}converges strongly to a common ﬁxed point of the family{Ti:i= , , . . . ,m}.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

This article is dedicated to Professor Anthony To-Ming Lau for celebrating his great achievements in the development of ﬁxed point theory and applications. It was supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. Bancha Nanjaras also thanks the Graduate School of Chiang Mai University, Thailand.

Received: 17 March 2012 Accepted: 20 June 2012 Published: 2 July 2012 References

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doi:10.1186/1687-1812-2012-108