Convergence theorems of common fixed points for some semigroups of nonexpansive mappings in complete CAT(0) spaces

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R E S E A R C H

Open Access

Convergence theorems of common ﬁxed

points for some semigroups of nonexpansive

mappings in complete CAT(0) spaces

Lai-Jiu Lin

1*

_{, Chih-Sheng Chuang}

1

_{and Zenn-Tsun Yu}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

National Changhua University of Education, Changhua, 50058, Taiwan

Full list of author information is available at the end of the article

Abstract

In this paper, we consider some iteration processes for one-parameter continuous semigroups of nonexpansive mappings in a nonempty compact convex subsetCof a complete CAT(0) spaceXand prove that the proposed sequence converges to a common ﬁxed point for these semigroups of nonexpansive mappings. Note that our results generalize Choet al.result (Nonlinear Anal. 74:6050-6059, 2011) and related results.

Keywords: common ﬁxed point; CAT(0) space; nonexpansive semigroup; implicit iteration process

1 Introduction

Fixed point theory in CAT() spaces was first studied by Kirk [, ]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT() space always has a fixed point. Since then, the fixed point theory for single-valued and multivalued mappings in CAT() spaces has been rapidly developed, and many papers have appeared; for example, one can see [–] and related references.

Let (X,d) be a metric space. A geodesic path joiningx∈Xtoy∈X(or, more brieﬂy, a geodesic fromxtoy) is a mapcfrom a closed interval [,]⊆RtoXsuch thatc() =x, c() =y, andd(c(t),c(t)) =|t–t|for allt,t∈[,]. In particular,cis an isometry and d(x,y) =. The imageαofcis called a geodesic (or metric) segment joiningxandy. 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 ofXare joined by a geodesic, andXis said to be uniquely geodesic if there is exactly one geodesic joiningxandyfor eachx,y∈X. A subsetY⊆Xis said to be convex ifYincludes every geodesic segment joining any two of its points.

A geodesic triangle (x,x,x) in a geodesic space (X,d) consists of three pointsx,

x, andxinX(the vertices ofand a geodesic segment between each pair of vertices

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

triangle(x,x,x) :=(x¯,x¯,x¯) in the Euclidean planeEsuch thatdE(x¯i,x¯j) =d(xi,xj) fori,j∈ {, , }.

A geodesic space is said to be a CAT() space if all geodesic triangles of appropriate size satisfy the following comparison axiom.

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¯

x,y¯∈,d(x,y)≤dE(x,¯ ¯y). It is well known that any complete, simply connected Rieman-nian manifold having nonpositive sectional curvature is a CAT() space. Other examples include pre-Hilbert spaces [], R-trees [], the complex Hilbert ball with a hyperbolic met-ric [], and many others.

Ifx,y,yare points in a CAT() space, and ifyis the midpoint of the segment [y,y],

then the CAT() inequality implies

d(x,y)≤

 d

_(x,_y ) +

 d

_(x,_y ) –

 d

_(y ,y).

This is the (CN) inequality of Bruhat and Tits []. In fact, a geodesic space is a CAT() space if and only if it satisﬁes the (CN) inequality [, p. ].

In , Dhompongsa and Panyanak [] gave the following result, and the proof is sim-ilar to the proof of the remark in [, p. ].

Lemma .[] Let X 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∈[, ].

By the above lemma, we know that CAT() space is a convex metric space. Indeed, it is a metric spaceXwith a convex structure if there exists a mappingW:X×X×[, ]→X such that

dW(x,y,t),z≤td(x,z) + ( –t)d(y,z)

for allx,y,z∈Xandt∈[, ] and this spaceXis called a convex metric space []. Fur-thermore, Takahashi [] has proved that

d(x,y) =tdx,W(x,y,t)+ ( –t)dy,W(x,y,t)

for allx,y,z∈Xandt∈[, ] whenXis a convex metric space with a convex structure. So, we also get the following result, and it is proved in [].

Lemma .[] Let X be a CAT()space and x,y∈X.For each t∈[, ],there exists a unique point z∈[x,y]such that d(x,z) =td(x,y)and d(y,z) = ( –t)d(x,y).

For convenience, from now on we will use the notationz= ( –t)x⊕ty. Therefore, we have

z= ( –t)x⊕ty ⇔ z∈[x,y],

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

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positive integers,Rbe the set of real numbers, and letR+be the set of nonnegative real

numbers.

A family{T(t) :t≥}of mappings is called a one-parameter continuous semigroup of nonexpansive mappings on a nonempty closed convex subsetCof a CAT() spaceXif the following conditions hold:

(SG) for eacht∈R+,T(t)is a nonexpansive mapping onC;

(SG) T(s+t) =T(t)◦T(s)for allt,s∈R+;

(SG) for eachx∈X, the mappingT(·)xfromR+intoCis continuous.

A family{T(t) :t≥}of mappings is called a one-parameter strongly continuous semi-group of nonexpansive mappings on a nonempty closed convex subsetCof a CAT() space Xif conditions (SG)i,i= , , , and the following condition are satisﬁed:

(SG) T()x=xfor allx∈C.

Note that ifCis a nonempty compact subset of a Banach space and{T(t) :t≥}is a semi-group of nonexpansive mappings, then_t_≥_F(T(t))=∅[]; see also [–] and others. For the example of a one-parameter continuous semigroup of nonexpansive mappings, one can see [].

Construction of common ﬁxed points of a nonexpansive semigroup is an important sub-ject in the theory of nonexpansive semigroup mappings and its applications. Fox example, one can refer to [, ]. In [], Shioji and Takahashi introduced the implicit iteration

(A) un=αnu+ ( –αn)_t_n tn

 T(s)unds,n≥,

whereCis a nonempty closed convex subset of a real Hilbert spaceH,u∈C,{αn}is a se-quence in (, ),{tn}is a sequence of positive real numbers divergent to∞. Under suitable conditions, Shioji and Takahashi [] proved strong convergence of{xn}to a member of

t≥F(T(t)). Note that their iteratexnat stepnis constructed through the average of a semigroup over the interval (,∞).

In , Suzuki [] introduced the following implicit iteration process in a Hilbert space:

(B) xn=αnu+ ( –αn)T(tn)xn,n≥

for a nonexpansive semigroup, where C is a nonempty closed convex subset of a real Hilbert spaceH,u∈C,{αn}is a sequence in (, ),{tn}is a sequence of positive real num-bers. Note thatxnis constructed directly from the T(tn). So, Zegeye and Shahzad [] viewed Suzuki’s iteration process (B) as an extension of the implicit process (A) to nonex-pansive semigroups.

In , Suzuki [] considered an iterative process{xn}for a one-parameter continu-ous semigroup of nonexpansive mappings onC, whereCis a nonempty compact convex subset of a Banach spaceE, deﬁned by

⎧ ⎨ ⎩

x∈C chosen arbitrary,

xn+:=λT(tn)xn+ ( –λ)xn,

(.)

whereλ∈(, ) and{tn} ⊆[,∞). Then Suzuki [] proved that{xn}converges strongly to a common ﬁxed point of{T(t) :t≥}if

lim inf

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In , Dhompongsaet al.[] gave the following important result for a strongly con-tinuous semigroup of nonexpansive mappings.

Theorem .[] Let C be a nonempty bounded closed convex subset of a complete CAT() space X,and let{T(t) :t≥}be a strongly continuous semigroup of nonexpansive map-pings on C.Let{αn}and{tn}be sequences of real numbers satisfying <αn< ,tn> ,and limn→∞tn=limn→∞αntn = .Let x∈C,and let{xn}be a sequence in C with

xn=αnx⊕( –αn)T(tn)xn, ∀n∈N.

Then_t_≥_F(T(t))=∅and{xn}converges to the element of

t≥F(T(t))nearest to x.

In , Choet al.[] gave the following result for a continuous semigroup of non-expansive mappings on a nonempty compact convex subset C of a complete CAT() spaceX.

Theorem .[] Let C be a nonempty compact convex subset of a complete CAT()space X,and{T(t) :t≥}be a one-parameter continuous semigroup of nonexpansive mappings on C.Let{tn}be a sequence in[,∞)satisfying

lim inf

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

For anyλ∈(, ),deﬁne a sequence{xn}in C by ⎧

⎨ ⎩

x∈C chosen arbitrary,

xn+:=λT(tn)xn⊕( –λ)xn.

(.)

Then{xn}converges to a common ﬁxed point of the semigroup{T(t) :t≥}.

Remark . By Theorems . and ., we know that

(a) ifCis a nonempty bounded closed convex subset of a complete CAT() spaceX, and{T(t) :t≥}is a strongly continuous semigroup of nonexpansive mappings on

C, then_t_≥_F(T(t))=∅;

(b) ifCis a compact convex subset of a complete CAT() spaceX, and{T(t) :t≥}is a one-parameter continuous semigroup of nonexpansive mappings onC, then

t≥F(T(t))=∅.

Motivated by the above works and related results, we study the following iteration pro-cesses for some families of one-parameter continuous semigroups of nonexpansive map-pings on a nonempty compact convex subsetCof a complete CAT() spaceX.

Let{T(t) :t≥}and{Q(q) :q≥}be one-parameter continuous semigroups of nonex-pansive mappings on a nonempty compact convex subsetCof a complete CAT() spaceX. We consider the following iteration processes (.), (.), and (.):

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn:= ( –βn)xn⊕βnT(tn)xn, xn+:= ( –αn)xn⊕αnQ(qn)yn,

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⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn:= ( –βn)xn⊕βnT(tn)xn, xn+:= ( –αn)yn⊕αnQ(qn)yn,

(.)

where {αn} and{βn}are sequences in (, ), {tn}and{qn} are sequences in [,∞). We prove that the proposed sequences converge to a common ﬁxed point of these families of mappings. Note that our results generalize Theorem . (i.e., Theorem . in []).

Besides, Thong [] considered an implicit iteration for nonexpansive semigroups

{T(t) :t≥}on a nonempty compact convex subsetCof a real Banach spaceE as fol-lows:

⎧ ⎨ ⎩

xn:=αn–xn+ ( –αn)T(tn)xn,

(.)

where{αn}is a sequence in (, ), and{tn}is a sequence in [,∞).

In this paper, motivated by [], we also consider the following implicit iteration process for some families of one-parameter continuous semigroups of nonexpansive mappings on a nonempty compact convex subsetCof a complete CAT() spaceX:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn:= ( –βn)xn⊕βnT(tn)xn,

xn+:= ( –αn+)Q(qn+)xn+⊕αn+yn,

(.)

where{αn}and{βn}are sequences in (, ),{tn}and{qn}are sequences in [,∞). For a special case of the iteration process (.), we have the following types:

⎧ ⎨ ⎩

xn+= ( –αn+)Q(qn+)xn+⊕αn+xn,

(.)

⎧ ⎨ ⎩

xn+:= ( –βn)xn⊕βnT(tn)xn.

(.)

We prove the proposed sequences converge to a common ﬁxed point of three families of mappings. Our result for the iteration process (.) is similar to (.) on complete CAT() spaces. Our results for the iteration processes (.) and (.) generalize Theorem .. Note that the iteration process (.) is also a special case of the iteration processes (.) and (.).

2 Preliminaries

In , Suzuki [] gave the following result, and it is an important tool in this paper.

Lemma . [] Let {tn} be a real sequence and τ be a real number satisfying

lim infn→∞tn≤τ≤lim supn→∞tn.Suppose that either of the following holds:

(i) lim sup_n_→∞(tn+–tn)≤or

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Thenτ is a cluster point of{tn}.Moreover,for anyε> and k,m∈N,there exists m≥m

such that|tj–τ|<εfor all j∈Nwith m≤j≤m+k.

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

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

for all t∈[, ]and x,y,z∈X.

Deﬁnition . Let{xn}be a bounded sequence in a CAT() spaceX, and letCbe a subset ofX. Now, we use the following notations:

(i) r(x,{xn}) :=lim supn→∞d(x,xn).

(ii) r({xn}) :=infx∈Xr(x,{xn}).

(iii) rC({xn}) :=infx∈Cr(x,{xn}).

(iv) A({xn}) :={x∈X:r(x,{xn}) =r({xn})}.

(v) AC({xn}) :={x∈C:r(x,{xn}) =rC({xn})}.

Note thatx∈Xis called an asymptotic center of{xn}ifx∈A({xn}). It is known that in a CAT() space,A({xn}) consists of exactly one point [].

Deﬁnition .[] Let (X,d) be a CAT() space. A sequence{xn}inX is said to be -convergent tox∈Xifxis the unique asymptotic center of{un}for every subsequence

{un} of{xn}. That is,A({un}) ={x}for every subsequence{un} of{xn}. In this case, we write-limnxn=xand callxthe-limit of{xn}.

Lemma .[] Let(X,d)be a CAT()space.Then every bounded sequence in X has a

-convergent subsequence.

Lemma .[] Let C be a nonempty closed convex subset of a CAT()space X.If{xn}is a bounded sequence in C,then the asymptotic center A({xn})of{xn}is in C.

Lemma .[] Let C be a nonempty closed convex subset of a complete CAT()space X, and let T:C→C be a nonexpansive mapping.Let{xn}be a bounded sequence in C with

-limnxn=x andlimn→∞d(xn,Txn) = .Then x∈C and Tx=x.

Lemma .[] Let X be a CAT()space.Let{xn}and{yn}be two bounded sequences in X withlimn→∞d(xn,yn) = .If-limnxn=x,then-limnyn=x.

Next, we give the following results and these results show that the intersection of the ﬁxed point sets for a continuous semigroup of nonexpansive mappings is nonempty. Note that Theorem . is diﬀerent from Theorem ..

A: Common ﬁxed point for a strongly nonexpansive semigroup

Lemma . Let C be a nonempty bounded closed convex subset of a complete CAT()space X.Let{T(t) :t∈R}be a continuous semigroup of nonexpansive mappings on C.Let{tn}be a sequence inR+.Let{αn}be a sequence in(, ),and let{xn}be deﬁned as

⎧ ⎨ ⎩

x∈C chosen arbitrary,

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Assume that limn→∞tn = limn→∞αntn = . Then

t≥F(T(t)) = ∅ if and only if

limn→∞d(T()xn,xn) = .

Proof Suppose thatlim_n_→∞d(T()xn,xn) = . Since Cis bounded, there exist a subse-quence{un}of{xn}andx¯∈Csuch that-limn→∞un=x. By Lemma .,¯ T()x¯=x.¯

Take anyt>  and lettbe ﬁxed. Forn>  with [_tt

n] > , we have

dxn,T(t)x¯

≤dxn,T()xn

+

[_tnt]–

k=

dT(k+ )tn

xn,T(ktn)xn +d T t tn tn xn,T

t tn tn ¯ x +d T t tn tn ¯

x,T(t)x¯

≤dxn,T()xn + t tn

dT(tn)xn,xn

+d(xn,x) +¯ d T t– t tn tn ¯

x,T()x¯

≤dxn,T()xn

+tαn tn

dT(tn)xn,xn–

+d(xn,x)¯

+maxdT(s)x,¯ T()x¯: ≤s≤tn

≤dxn,T()xn

+tαn tn

M+d(xn,x) +¯ max

dT(s)x,¯ T()x¯: ≤s≤tn

for someM>  (note thatCis bounded). Hence, we know that

lim sup n→∞ d

un,T(t)x¯

≤lim sup n→∞ d(un,x)¯

for each t > . Since -limn→∞un = x,¯ T(t)x¯ = x¯ for each t > . Therefore,

¯

x∈_t_≥_F(T(t))=∅.

Conversely, suppose that_t_≥_F(T(t))=∅. Now, take anyw∈_t_≥_F(T(t)), and letw be ﬁxed. Then we have

d(xn,w)

≤αnd(xn–,w)+ ( –αn)d

T(tn)xn,w 

–αn( –αn)d

xn–,T(tn)xn

≤αnd(xn–,w)+ ( –αn)d(xn,w)–αn( –αn)d

xn–,T(tn)xn

≤αnd(xn–,w)+ ( –αn)d(xn,w).

And this implies thatd(xn,w)≤d(xn–,w) andlimn→∞d(xn,w) exists. Hence,

dxn–,T(tn)xn

≤d(xn–,w)–d(xn,w)+αnd

xn–,T(tn)xn

.

By assumption,limn→∞d(xn–,T(tn)xn) = . Next, we get

d(xn,xn–) =d

αnxn–⊕( –αn)T(tn)xn,xn–

= ( –αn)d

T(tn)xn,xn–

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This implies thatlimn→∞d(xn,xn+) =  andlimn→∞d(xn,T(tn)xn) = . Furthermore, it follows that

lim sup n→∞ d

T()xn,xn

≤lim sup n→∞

dT()xn,T( +tn)xn

+dT(tn)xn,xn

≤lim sup n→∞

dT()xn,T()◦T(tn)xn

+dT(tn)xn,xn

≤lim sup n→∞ d

xn,T(tn)xn

= .

Therefore, the proof is completed.

By Lemma ., we get the following theorem, and it is diﬀerent from Theorem ..

Theorem . Let C be a nonempty bounded closed convex subset of a complete CAT()

space X.Let{T(t) :t∈R}be a strongly continuous semigroup of nonexpansive mappings on C. Let{tn} be a sequence inR+, and{αn} be a sequence in(, ) with limn→∞tn= limn→∞αntn = .Let{xn}be deﬁned as

⎧ ⎨ ⎩

x∈C chosen arbitrary,

xn=αnxn–⊕( –αn)T(tn)xn, n≥.

Then _t_≥_F(T(t))=∅.Furthermore, if C is a compact set,thenlimn→∞d(xn,x) = ¯ for somex¯∈ .

Proof Since{T(t) :t∈R}is a strongly continuous semigroup of nonexpansive mappings onC, it is easy to see thatlim_n_→∞d(T()xn,xn) = . By Lemma .,

t≥F(T(t))=∅, and

the proof is completed.

B: Common ﬁxed point for a nonexpansive semigroup

Lemma . Let C be a nonempty bounded closed convex subset of a complete CAT()space X,and{T(t) :t≥}be a one-parameter continuous semigroup of nonexpansive mappings on C.Let{tn}be a sequence in[,∞)satisfying

lim inf

⎨ ⎩

xn+:=λT(tn)xn⊕( –λ)xn.

Then_t_≥_F(T(t))=∅if and only iflimn→∞d(xn,T()xn) = .

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By Lemma ., we get the following theorem. Notice also that it is a consequence of Theorem ..

Theorem . Let C be a nonempty compact convex subset of a complete CAT()space

X,and{T(t) :t≥}be a one-parameter continuous semigroup of nonexpansive mappings on C.Let{tn}be a sequence in[,∞)satisfying

lim inf

⎨ ⎩

xn+:=λT(tn)xn⊕( –λ)xn. Then_t_≥_F(T(t))=∅.

Proof Following the same argument as in the proof of Theorem . in [], we can prove thatlimn→∞d(xn,T()xn) = . By Lemma .,

t≥F(T(t))=∅.

3 Main results

Theorem . Let C be a nonempty compact convex subset of a complete CAT()space X.

Let{T(t) :t∈R+}and{Q(q) :q∈R+}be continuous semigroups of nonexpansive mappings

on C.Let{tn}and{qn}be sequences inR+.Let{αn}and{βn}be sequences in[, ].Suppose that

:= t≥

FT(t)∩ q≥

FQ(q)=∅.

Let{xn}be deﬁned as

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn:= ( –βn)xn⊕βnT(tn)xn, xn+:= ( –αn)xn⊕αnQ(qn)yn. Assume that

(i)  <a≤αn≤b< ,lim infn→∞βn( –βn) > ;

(ii) max{lim inf_n_→∞tn,lim infn→∞qn}<min{lim supn→∞tn,lim supn→∞qn};

(iii) eitherlim sup_n_→∞(tn+–tn)≤orlim infn→∞(tn+–tn)≥holds;

(iv) eitherlim sup_n_→∞(qn+–qn)≤orlim infn→∞(qn+–qn)≥holds. Thenlim_n_→∞xn=x for some¯ x¯∈ .

Proof Take anyw∈ , and letwbe ﬁxed. Then for eachn∈N,

d(yn,w)

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≤( –βn)d(xn,w)+βnd

T(tn)xn,w 

–βn( –βn)d

xn,T(tn)xn 

≤( –βn)d(xn,w)+βnd(xn,w)–βn( –βn)d

xn,T(tn)xn 

≤d(xn,w), (.)

and

d(xn+,w)

=d( –αn)xn⊕αnQ(qn)yn,w 

≤( –αn)d(xn,w)+αnd

Q(qn)yn,w 

–αn( –αn)d

xn,Q(qn)yn 

≤( –αn)d(xn,w)+αnd

Q(qn)yn,w 

≤( –αn)d(xn,w)+αnd(yn,w)

≤d(xn,w). (.)

Hence,limn→∞d(xn,w) exists, and{xn}and{yn}are bounded sequences. By (.), we get

d(xn,w)+

d(xn+,w)–d(xn,w) a

≤d(xn,w)+

d(xn+,w)–d(xn,w)

αn =d(xn+,w)

_{– ( –}_α

n)d(xn,w)

αn

≤dQ(qn)yn,w 

≤d(yn,w)

≤d(xn,w). (.)

And this implies that

lim

n→∞d(xn,w) =nlim→∞d(yn,w) =nlim→∞d

Q(qn)yn,w

. (.)

By (.) and (.), we have

lim

n→∞βn( –βn)d

xn,T(tn)xn 

= . (.)

By assumption and (.),

lim n→∞d

xn,T(tn)xn

= . (.)

By (.), we also have

αn( –αn)d

xn,Q(qn)yn 

≤d(xn,w)–d(xn+,w),

and this inequality and assumption imply that

lim n→∞d

xn,Q(qn)yn

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By (.), we have

lim sup n→∞

d(xn,yn) =lim sup n→∞

d( –βn)xn⊕βnT(tn)xn,xn

≤lim sup n→∞ d

xn,T(tn)xn

= . (.)

By (.), it is easy to see that

lim

n→∞d(xn+,xn) = . (.)

By (.), (.), and (.), it is easy to see that

lim n→∞d

yn,Q(qn)yn

= lim n→∞d

xn+,Q(qn)yn

= . (.)

Furthermore, by (.), it follows that

lim sup n→∞ d

T()xn,xn

≤lim sup n→∞

dT()xn,T( +tn)xn

+dT(tn)xn,xn

≤lim sup n→∞

dT()xn,T()◦T(tn)xn

+dT(tn)xn,xn

≤lim sup n→∞ d

xn,T(tn)xn

= . (.)

By (.), it follows that

lim sup n→∞ d

Q()yn,yn

= . (.)

Next, ﬁxτ,δ∈Rwith max

lim inf

n→∞ tn,lim infn→∞ qn

<τ<τ+δ<min

lim sup

n→∞ tn,lim supn→∞ qn

.

Following the same argument as in the proof of Theorem  in [], we choose a subse-quence{ni}of{n}such that

lim n→∞

d(T(tni)xni,xni)

tni–τ

= lim n→∞

d(Q(qni)yni,yni)

qni–τ

= ,

lim

i→∞tni=lim

i→∞qni=τ.

(.)

For completeness, we give the following proof. By (.) and (.), there existsm∈Nsuch

that

dxn,T(tn)xn

< 

, and d

yn,Q(qn)yn

<  

for alln≥m. By Lemma ., we note thatτ+δ_ is a cluster point of{tn}and{qn}. Hence, there existsn>msuch that

τ+δ

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By (.) and (.) again, there existsm>nsuch that

dxn,T(tn)xn

< 

, and d

yn,Q(qn)yn

<  

for alln≥m. By Lemma . again, we note thatτ+_δ is a cluster point of{tn}and{qn}. Hence, there existsn>msuch that

τ+δ

<tn,qn<τ+

δ

.

Continuing this argument, we can deﬁne a subsequence{ni}of{n}satisfying

maxdxni,T(tni)xni

,dyni,Q(qni)yni

< 

(i+ ), and

τ+ δ

i+ <tni,qni<τ+

δ

i

for alli∈N. Then it is obvious thatτ<min{tni,qni}for alli∈N,

lim

i→∞tni=ilim→∞qni=τ, and

lim i→∞d

T(tni)xni,xni

=lim

i→∞d

Q(qni)yni,yni

= .

We also have

≤lim sup i→∞

d(T(tni)xni,xni)

tni–τ

≤lim sup i→∞

/(i+ )

δ/(i+ ) =lim sup

i→∞ 

δ(i+ )= . Similarly,

lim i→∞

d(Q(qni)yni,yni)

qni–τ

= .

Since{xn}is a bounded sequence, there exist a subsequence{xn_ik} of{xni}andx¯∈C such that-limk→∞xn_ik =x. Let¯ uk=xn_ik,vk=yn_ik,ak=tn_ik, andbk=sn_ik. Then we get

lim sup k→∞

duk,T(τ)x¯

≤lim sup k→∞

duk,T(ak)uk

+dT(ak)uk,T(ak)x¯

+dT(ak)x,¯ T(τ)x¯

≤lim sup k→∞

duk,T(ak)uk

+lim sup k→∞

d(uk,x) +¯ lim sup k→∞

dT(ak–τ)x,¯ T()x¯

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and

lim sup k→∞

dvk,Q(τ)x¯

≤lim sup k→∞

dvk,Q(bk)vk

+dQ(bk)vk,Q(bk)x¯

+dQ(bk)x,¯ Q(τ)x¯

≤lim sup k→∞ d

vk,Q(bk)vk

+lim sup

k→∞ d(vk,x) +¯ lim sup

k→∞ d

Q(bk–τ)x,¯ Q()x¯

≤lim sup k→∞

d(vk,x).¯

Since-limk→∞uk=x, we know that¯ T(τ)x¯=x. By (.),¯ -limk→∞uk=x, and Lem-¯ ma ., we know that-limk→∞vk=x, and this implies that¯ Q(τ)x¯=x. Also, we have¯

T()x¯=T()◦T(τ)x¯=T(τ+ )x¯=T(τ)x¯=x.¯

Similarly,Q()x¯=x.¯

Take anyt> , and lettbe ﬁxed. Then fork∈Nwithak–τ<t, we have

duk,T(t)x¯

=duk,T(t)◦T(τ)[t/(ak–τ)]x¯

=duk,T

t/(ak–τ)

τ+tx¯

≤duk,T()uk

+

[t/(ak–τ)]–

m=

dT(m+ )ak

uk,T(mak)uk

+dTt/(ak–τ)

ak

uk,T

t/(ak–τ)

ak

¯

x

+dTt/(ak–τ)

ak

¯

x,Tt/(ak–τ)

τ+tx¯

≤duk,T()uk

+t/(ak–τ)

·dT(ak)uk,uk

+d(uk,x)¯

+dTt–t/(ak–τ)

(ak–τ)

¯

x,T()x¯

≤duk,T()uk

+t·d(T(ak)uk,uk) ak–τ

+d(uk,x)¯

+maxdT(s)x,¯ T()x¯: ≤s≤ak–τ

.

By (.) and (.), we get

lim sup k→∞

duk,T(t)x¯

≤lim sup k→∞

d(uk,x).¯

Since-limk→∞uk=x, we get¯ T(t)x¯=x¯for eacht≥. Therefore,x¯is a common ﬁxed point of{T(t) :t≥}. Furthermore, we know thatx¯∈ by following the same argument. In fact, sinceC is a compact set, we may assume thatlimk→∞d(xn_ik,x) = . So,¯ x¯ is a cluster point of{xn}andlim infn→∞d(xn,x) = . Since¯ limn→∞d(xn,w) exists for eachw∈ ,limn→∞d(xn,x) = . Therefore, the proof is completed.¯

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Theorem . Let C be a nonempty compact convex subset of a complete CAT()space X. Let{T(t) :t∈R+}and{Q(q) :q∈R+}be continuous semigroups of nonexpansive mappings

on C.Let{tn}and{qn}be sequences inR+.Let{αn}and{βn}be sequences in[, ].Suppose that

:= t≥

FT(t)∩ q≥

FQ(q)=∅.

Let{xn}be deﬁned as ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn:= ( –βn)xn⊕βnT(tn)xn, xn+:= ( –αn)yn⊕αnQ(qn)yn. Assume that

(i) lim infn→∞αn( –αn) > ,lim infn→∞βn( –βn) > ;

(ii) max{lim infn→∞tn,lim infn→∞qn}<min{lim supn→∞tn,lim supn→∞qn};

(iii) eitherlim sup_n_→∞(tn+–tn)≤orlim infn→∞(tn+–tn)≥holds;

(iv) eitherlim sup_n_→∞(qn+–qn)≤orlim infn→∞(qn+–qn)≥holds. Thenlimn→∞xn=x for some¯ x¯∈ .

Proof Take anyw∈ , and letwbe ﬁxed. Then for eachn∈N,

d(yn,w)

=d( –βn)xn⊕βnT(tn)xn,w 

= ( –βn)d(xn,w)+βnd

T(tn)xn,w 

–βn( –βn)d

xn,T(tn)xn 

≤d(xn,w), (.)

and

d(xn+,w)

=d( –αn)yn⊕αnQ(qn)yn,w 

= ( –αn)d(yn,w)+αnd

Q(qn)yn,w 

–αn( –αn)d

yn,Q(qn)yn 

≤d(yn,w)–αn( –αn)d

yn,Q(qn)yn 

≤d(xn,w)–αn( –αn)d

yn,Q(qn)yn 

. (.)

Hence,limn→∞d(xn,w) exists,limn→∞d(xn,w) =limn→∞d(yn,w), and{xn}and{yn}are bounded sequences. By (.) andlimn→∞d(xn,w) =limn→∞d(yn,w), we have

lim

n→∞βn( –βn)d

xn,T(tn)xn 

= . (.)

By assumption and (.),

lim n→∞d

xn,T(tn)xn

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By (.) andlimn→∞d(xn,w) exists,

lim n→∞d

yn,Q(qn)yn

= . (.)

By (.), we have

lim sup

n→∞ d(xn,yn) =lim supn→∞ d

xn, ( –βn)xn⊕βnT(tn)xn

≤lim sup n→∞

dxn,T(tn)xn

= . (.)

Next, following the same argument as in the proof of Theorem ., we get the proof of

Theorem ..

Remark . Theorem . is also a special case of Theorem ..

Theorem . Let C be a nonempty compact convex subset of a complete CAT()space X.

Let{T(t) :t∈R}and{Q(q) :q∈R}be continuous semigroups of nonexpansive mappings on C.Let{tn}and{qn}be sequences inR+.Let{αn}and{βn}be sequences in[, ]with <

αn< ,lim infn→∞βn( –βn) > .Suppose that := (

t≥F(T(t)))∩(

q≥F(Q(q)))=∅.

Let{xn}be deﬁned as

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

yn= ( –βn)xn⊕βnT(tn)xn,

xn+= ( –αn+)Q(qn+)xn+⊕αn+yn. Assume that

(i) lim infn→∞tn<lim supn→∞tn;

(ii) eitherlim sup_n_→∞(tn+–tn)≤orlim infn→∞(tn+–tn)≥holds;

(iii) limn→∞qn=limn→∞αn_q_n= . Thenlimn→∞xn=x for some¯ x¯∈ .

Proof Take anyw∈ and letwbe ﬁxed. Then for eachn∈N,

d(yn,w)

=d( –βn)xn⊕βnT(tn)xn,w 

≤( –βn)d(xn,w)+βnd

T(tn)xn,w 

–βn( –βn)d

xn,T(tn)xn 

xn,T(tn)xn 

≤d(xn,w), (.)

and

d(xn+,w)

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≤( –αn+)d

Q(qn+)xn+,w



+αn+d(yn,w)–αn+( –αn+)d

Q(qn+)xn+,yn 

≤( –αn+)d(xn+,w)+αn+d(yn,w). (.)

By (.) and (.),

d(xn+,w)≤d(yn,w)≤d(xn,w). (.)

Hence, lim_n_→∞d(xn,w) exists, limn→∞d(xn,w) = limn→∞d(yn,w), {xn} and {yn} are bounded sequences. By (.) and assumptions,

lim n→∞d

xn,T(tn)xn

= . (.)

By (.) and assumptions,

lim n→∞d

Q(qn+)xn+,yn

= . (.)

By (.) and (.), we have

lim

n→∞d(xn,yn) =  and nlim→∞d

Q(qn+)xn+,xn+

= . (.)

Furthermore, by (.) and (.), it follows that

lim n→∞d

T()xn,xn

=  and lim n→∞d

Q()xn,xn

= . (.)

Following the same argument as in the proof of Theorem ., there exist a subsequence

{un} of{xn}andx¯∈Csuch thatlimn→∞d(un,x) = , and¯ x¯is a common ﬁxed point of

{T(t) :t≥}. Hence,lim infn→∞d(xn,x) = .¯

Forq= , by (.), we getlimn→∞d(Q()un,un) = , and it is easy to see thatQ()x¯=x.¯ Next, for eachq> ,

dxn,Q(q)x¯

≤dxn,Q()xn

+

[_qnq]–

k=

dQ(k+ )qn

xn,Q(kqn)xn +d Q q qn qn xn,Q

q qn qn ¯ x +d Q q qn qn ¯

x,Q(q)x¯

≤dxn,Q()xn + q qn

dQ(qn)xn,xn

+d(xn,x) +¯ d Q q– q qn qn ¯

x,Q()x¯

≤dxn,Q()xn

+qαn qn

dQ(qn)xn,yn

+d(xn,x) +¯ max

dQ(s)x,¯ Q()x¯: ≤s≤qn

≤dxn,Q()xn

+qαn qn

M+d(xn,x) +¯ max

dQ(s)x,¯ Q()x¯: ≤s≤qn

for someM> .

By assumptions and (.), we know that

lim sup n→∞

dun,Q(q)x¯

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for eachq> , and this implies thatx¯is also a common ﬁxed point of{Q(q) :q≥}. Now,

¯

x∈ and this implies thatlimn→∞d(xn,x) exists. Since¯ lim infn→∞d(xn,x) = , we know¯ thatlimn→∞d(xn,x) = , and the proof is completed.¯

Remark . Theorem . is also a generalization of Theorem ..

The following result is similar to Theorem . in [].

Corollary . Let C be a nonempty compact convex subset of a complete CAT()space X. Let{Q(q) :q∈R}be a continuous semigroup of nonexpansive mappings on C.Let{qn}be a sequence inR+.Let{αn}be a sequence in[, ]with <αn< .Let{xn}be deﬁned as

⎧ ⎨ ⎩

xn+= ( –αn+)Q(qn+)xn+⊕αn+xn.

Assume thatlim_n_→∞qn=limn→∞α_q_nn= .Thenlimn→∞xn=x for some¯ x¯∈ .

Proof For eacht≥, letT(t) :C→Cbe deﬁned byT(t)x:=xfor eachx∈C. Clearly,

{T(t) :t≥}is a continuous semigroup of nonexpansive mappings onC. SinceC is a compact set, :=_q_≥_F(Q(q))=∅. By Theorem ., we get the conclusion.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

LJL: Problem resign, coordinator, discussion, revise the important part, and submit. CSC: Responsible for the important results of this paper, discuss, and draft. ZTY: Responsible for the results of this paper, discuss, and draft.

Author details

1_{Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan.}2_{Department of}

Electronic Engineering, Nan Kai University of Technology, Nantour, 54243, Taiwan.

Acknowledgements

This research was supported by the National Science Council of Republic of China.

Received: 15 February 2012 Accepted: 29 August 2012 Published: 18 September 2012

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