R E S E A R C H
Open Access
Nonexpansive semigroups in
CAT(
κ
)
spaces
Shuechin Huang
*Dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday
*Correspondence:
[email protected] Department of Applied Mathematics, National Dong Hwa University, Hualien, 97401, Taiwan
Abstract
The main purpose of this paper is to study Browder type convergence theorems for a nonexpansive semigroup with geometric approaches in a CAT(
κ
) space. Besides, we determine a necessary and sufficient condition for convergence of a Browder type iteration associated to a uniformly asymptotically regular nonexpansive semigroup on the unit sphere in an infinite-dimensional Hilbert space.MSC: 47H20; 47H10
Keywords: nonexpansive mapping; asymptotically regular; uniformly asymptotically regular;
-convergence; CAT(
κ
) space1 Introduction
Let (X,d) be a metric space,Ca closed convex subset ofX andT :C→Ca mapping. Recall thatT is nonexpansive onCifd(Tx,Ty)≤d(x,y), for allx,y∈C. We denote by
F(T) the fixed point set of the mappingT. A one-parameter familyS={T(t) :t≥}of self-mappings ofC is called astrongly continuous nonexpansive semigrouponC if the following conditions are satisfied:
(i) for eacht≥,T(t)is a nonexpansive mapping onC; (ii) T()x=x, for allx∈C;
(iii) T(s+t) =T(s)◦T(t), for alls,t≥;
(iv) for eachx∈C, the mappingt→T(t)xfrom[,∞)intoCis continuous.
LetF(S) denote the common fixed point set of all mappings inS.
There have been considerably many interesting results of iterative methods for approxi-mating fixed points of nonexpansive mappings, nonexpansive semigroups, and their gen-eralizations which solve some variational inequalities problems due to their various ap-plications in several physical problems, such as in operations research, economics, and engineering design; see,e.g., [–] and the references therein.
Suppose thatXis a real Hilbert space anduis an arbitrary point ofX. IfTis nonexpan-sive onC, then for eachα∈(, ) there exists a uniquexα∈Csuch thatxα=αu+ ( –α)Txα
because the mappingx→αu+ ( –α)Txis a contraction. In , Browder [] was the pio-neer to consider an implicit scheme and prove the following strong convergence theorem of this algorithm in a Hilbert space.
Theorem . Let C be a bounded closed convex subset of a Hilbert space and T a nonex-pansive mapping on C.Let u be an arbitrary point of C and define xα∈C by
xα=αu+ ( –α)Txα, forα∈(, ).
Then asα→,{xα}converges strongly to a point ofF(T)nearest to u.
The extension work of Browder’s type convergence theorems has been tremendously studied for not only one single nonexpansive mapping, but, most significantly, a semigroup of nonexpansive mappings; see,e.g., [, ] and the references therein.
This paper is devoted to studying Browder’s type iterations for a nonexpansive semi-group in aCAT(κ) space, whereκ∈R, which is a specific type of metric space. Intuitively, triangles in aCAT(κ) space are ‘slimmer’ than corresponding ‘model triangles’ in a stan-dard space of constant curvature κ (see Section ). Complete CAT() spaces are often calledHadamard spaces. A few recent new convergence results of classical iterations on
CAT(κ) spaces withκ> are obtained; see,e.g., [–] and the references therein. In [], Dhompongsaet al.extended Suzuki’s result [, Theorem ] on common fixed points of a nonexpansive semigroup in a Hilbert space to a completeCAT() space.
Theorem .([, Theorem .]) Let C be a bounded closed convex subset of aCAT() space,S={T(t) :t≥} a strongly continuous nonexpansive semigroup on C,and two sequences {αn} ⊂(, ),{tn} ⊂(,∞) such that limn→∞tn=limn→∞αn/tn= . Choose
arbitrarily a point x∈C and for each n∈Nlet xnbe the fixed point of the mapping
x→αnx⊕( –αn)T(tn)x.ThenF(S)=∅and{xn}converges strongly to a point ofF(S)
nearest to x.
In , Acedo and Suzuki [] proved a Browder type convergence theorem for uni-formly asymptotically regular(UARfor short) nonexpansive semigroups (see Section ) in Hilbert spaces under a weaker condition on{αn}and{tn}than that in Theorem ..
Theorem . ([, Theorem .]) Let C be a closed convex subset of a Hilbert space,
S={T(t) :t≥} a UAR and strongly continuous nonexpansive semigroup on C such that F(S)=∅, and two sequences {αn} ⊂(, ), {tn} ⊂[,∞) such that limn→∞αn=
limn→∞αn/tn= .Fix x∈C and define a sequence{xn}in C by
xn=αnx+ ( –αn)T(tn)xn.
Then{xn}converges strongly to a point ofF(S)nearest to x.
The preceding two theorems lead naturally to the question of whether or not they can be extended to aCAT(κ) space withκ> . The purpose of this article is to investigate this question with the geometric approaches inCAT(κ) spaces.
This paper is organized as follows. In Section we recall the definition of geodesic met-ric spaces and summarize some useful lemmas and the main properties ofCAT(κ) spaces. In Section we present some technical results about-convergence of a sequence in a completeCAT() space. In Section we establish our main results (Theorems ., ., .) of Browder’s iterations for nonexpansive semigroups inCAT(κ) spaces and conclude that Theorems . and . can be generalized toCAT(κ) spaces under the same respective con-ditions on the coefficients{αn}and{tn}. It is noteworthy that, however, without theUAR
to the nearest point ofF(S) toxiflimn→∞tn=tˆ∈(,∞] even whenlimn→∞αn= (and
thereforelimn→∞αn/tn= ); see Example .. Furthermore, we determine a necessary and
sufficient condition for a Browder type convergence theorem associated to a nonexpansive semigroup on the unit sphere in an infinite-dimensional Hilbert space. We also propose an open problem in Section .
2 Preliminaries
Let (X,d) be a metric space. For any subsetEofXandx∈X, thediameterofEand the distance from x to Eare defined, respectively, by
diamE=supd(x,y) :x,y∈E,
d(x,E) =infd(x,y) :y∈E.
We always denote the open ball and the closed ball centered atxwith radiusr> byB(x,r) andB(x,r), respectively.
Let C be a closed convex subset ofX and letS={T(t) :t≥} be a family of self-mappings ofC. ThenSis called
(i) asymptotically regularonCif for anyh≥and anyx∈C,
lim
t→∞d
T(h)T(t)x,T(t)x= ;
(ii) uniformly asymptotically regular(in shortUAR) onCif for anyh≥and any bounded subsetDofC,
lim
t→∞supx∈Dd
T(h)T(t)x,T(t)x= .
Forx,y∈X, ageodesic pathjoiningxtoy(or ageodesicfromxto y) is an isometric mappingγ : [a,b]⊂R→Xsuch thatγ(a) =x,γ(b) =y,i.e.,d(γ(t),γ(t)) =|t–t|, for all t,t∈[a,b]. Therefored(x,y) =b–a. The image ofγ is called ageodesic(segment) from xtoyand we shall denote a definite choice of this geodesic segment by [x,y]. We remark that composingγ with a translation (this is still an isometry), one can always choose the interval [a,b] to be [,], where=b–a. A pointz=γ(t) in the geodesic [x,y] will be written asz= ( –λ)x⊕λy, whereλ= (t–a)/(b–a), and sod(z,x) =λd(x,y) andd(z,y) = ( –λ)d(x,y).
Letr> . The metric space (X,d) is said to be
(i) ageodesic(metric)spaceif any two points inXare joined by a geodesic; (ii) uniquely geodesicif there is exactly one geodesic joiningxtoyfor allx,y∈X; (iii) r-geodesic spaceif any two pointsx,y∈Xwithd(x,y) <rare joined by a geodesic; (iv) r-uniquely geodesicif any two pointsx,y∈Xwithd(x,y) <rare joined by a unique
geodesic inX.
A subsetCofXisconvexif every pair of pointsx,y∈Ccan be joined by a geodesic in Xand the image of every such geodesic is contained inC. If this condition holds for all pointsx,y∈Cwithd(x,y) <r, thenCis said to ber-convex.
Then-dimensional sphereSnis the set{x= (x
, . . . ,xn+)∈Rn+: (x|x) = }, where (· | ·)
Sn×Sn→Rbe the function that assigns to each pair (A,B)∈Sn×Snthe unique number
d(A,B)∈[,π] such that
cosdSn(A,B) = (A|B).
ThendSnis a metric [, I..].
Definition . Given a real numberκ, we denote byMn
κ the following metric spaces: (i) ifκ= , thenMn
is the Euclidean spaceRn; (ii) ifκ> , thenMn
κ is obtained from the sphereSnby multiplying the distance function by/√κ;
(iii) ifκ< , thenMn
κ is obtained from the sphereHnby multiplying the distance function by/√–κ, whereHnis the hyperbolicn-space.
It is well known that Mn
κ is a geodesic metric space. Ifκ ≤, then Mnκ is uniquely
geodesic. Ifκ> , thenMn
κ isπ/
√
κ-uniquely geodesic, and any open ball (respectively, closed) ball of radius≤π/(√κ) (respectively, <π/(√κ)) inXis convex [, I..]. The diameter ofMnκwill be denotedDκ and thusDκ isπ/
√
κifκ> , and∞otherwise. Given two distinct pointsA,B∈Snwithd(A,B) =<πthere is a natural way to parame-terize a unique geodesic joiningAtoB: consider the pathc(t) = (cost)A+ (sint)u,t∈[,], where the initial vectoru∈Rn+is the unit vector in the direction ofB– (A|B)A. We shall
refer to the image ofcas a minimal great arc joiningAtoB.
Thespherical anglebetween two minimal great arcs issuing from a point ofSn, with the
initial vectorsuandv, say, is the unique numberα∈[,π] such thatcosα= (u|v). A spher-ical triangleinSnconsists of a choice of three distinct points (its vertices)A,B,C∈Sn,
and three minimal great arcs (its sides), one joining each other of vertices. Thevertex angle atCis defined to be the spherical angle between the sides ofjoiningCtoAandCtoB.
Proposition .(The Spherical Law of Cosines [, I..]) Let a geodesic triangle in Mn
κ
(κ> )have sides a,b,c and anglesα,β,γ at the vertices opposite to the sides of length a, b,c,respectively.Then
cos(√κc) =cos(√κa)cos(√κb) +sin(√κa)sin(√κb)cosγ.
In particular,fixing a,b andκ,c is a strictly increasing function ofγ,varying from|a–b| to a+b asγ varies fromtoπ.
Ageodesic trianglein a metric spaceXconsists of three pointsp,q,r∈X, itsvertices, and a choice of three geodesic segments [p,q], [q,r], [r,p] joining them, itssides. Such a geodesic triangle will be denoted([p,q], [q,r], [r,p]) or (less accurately ifXis not uniquely geodesic)(p,q,r). If a pointx∈Xlies in the union of [p,q], [q,r] and [r,p], then we write x∈ .
A triangle=(p¯,q¯,¯r) inM
κis called acomparison trianglefor=([p,q], [q,r], [r,p])
inXifdM
κ(p¯,q¯) =d(p,q),dMκ(q¯,r¯) =d(q,r) anddMκ(r¯,p¯) =d(r,p). Such a triangle ⊂M κ
always exists if the perimeterd(p,q) +d(q,r) +d(r,p) ofis less than Dκ; it is unique up
to an isometry ofMκ[, I..]. A pointx¯∈[¯q,r¯] is called acomparison pointforx∈[q,r]
ifdM
A geodesic triangleinXis said to satisfy theCAT(κ)inequalityif, given a comparison triangle ⊂M
κ for, for allx,y∈ ,
d(x,y)≤dM
κ(¯x,y¯),
wherex¯,¯y∈ are respective comparison points ofx,y.
Definition . Ifκ≤, thenXis called a CAT(κ) space ifX is a geodesic space all of whose geodesic triangles satisfy theCAT(κ) inequality.
Ifκ> , thenXis called aCAT(κ) space ifXisDκ-geodesic and all geodesic triangles in
Xof perimeter less than Dκsatisfy theCAT(κ) inequality.
In particular, Hilbert spaces areCAT(). ACAT(κ) space is aCAT(κ) space for every
κ≥κ. ACAT(κ) spaceXis (Dκ-)uniquely geodesic (ifκ> ) and any open (respectively,
closed) ball of radius≤Dκ/ (respectively, <Dκ/) inXis convex [, II..].
Lemma . Let(X,d)be aCAT(κ)space and letα,β∈[, ].Then
(i) [, II.. Exercise .()]forp∈Xandx,y∈B(p,Dκ/),we have
dαx⊕( –α)y,p≤αd(x,p) + ( –α)d(y,p);
(ii) forx,y∈X,we have
dαx⊕( –α)y,βx⊕( –β)y=|α–β|d(x,y);
(iii) [, Lemma .]ifκ> ,forx,y,z∈Xwithmax{d(x,y),d(y,z),d(x,z)}<M≤Dκ/, we have
dαx⊕( –α)y,αx⊕( –α)z≤sin[( –α)M]
sinM d(y,z).
Let p,q, rbe three distinct points of X withd(p,q) +d(q,r) +d(r,p) < Dκ. Theκ
-comparison anglebetweenqandratp, denoted∠(κ)p (q,r), is the angle atp¯in a comparison
triangle(p¯,q¯,r¯)⊂M
κfor(p,q,r).
Letγ : [,a]→Xand letγ: [,a]→Xbe two geodesic paths withγ() =γ(). Given t∈(,a] andt∈(,a], we consider the comparison triangle(γ(),γ(t),γ(t)) and the
κ-comparison angle∠(κ)γ()(γ(t),γ(t)). The (Alexandrov)angleor theupper anglebetween the geodesic pathsγ andγis the number∠(γ,γ)∈[,π] defined by
∠γ,γ=lim sup
t,t→ ∠(κ)
γ()
γ(t),γt=lim →<supt,t<∠
(κ) γ()
γ(t),γt.
IfXis uniquely geodesic,p=qandp=r, theangleof(p,q,r) inXatpis the (Alexandrov) angle between the geodesic segments [p,q] and [p,r] issuing frompand is denoted∠p(q,r).
Proposition .([, I.., . and II..]) Let X be a metric space and letγ,γ,γbe
three geodesics issuing from a common point.Then
(ii) ∠(γ,γ)≤∠(γ,γ) +∠(γ,γ);
(iii) ifγ : [–a,a]→Xis a geodesic witha> ,and ifγ,γ: [,a]→Xare defined by
γ(t) =γ(–t)andγ(t) =γ(t),then∠(γ,γ) =π.
3 Basic properties and
-convergence
This section contains a number of primary results in [] which are crucial to the study of our problem. The following proposition states very useful properties of the metric projec-tion in a completeCAT() space.
Proposition .([, Proposition .]) Let X be a completeCAT()space and let C⊂X be nonempty closed andπ-convex.Suppose that x∈X such that d(x,C) <π/.Then the following are satisfied:
(i) There exists a unique pointPCx∈Csuch thatd(x,PCx) =d(x,C).
(ii) Ifx∈/Candy∈Cwithy=PCx,then∠PCx(x,y)≥π/. (iii) Ifdiam(X)≤π,then for anyy∈C,
d(PCx,PCy) =d(PCx,y)≤d(x,y).
The mappingPCofXontoCin Proposition . is called the metric projection.
The next result shows the existence property of fixed points for a nonexpansive mapping.
Proposition .([, Theorem .]) Let X be a completeCAT()space such thatdiamX<
π/.Then every nonexpansive mapping T:X→X has at least one fixed point.
The rest of this section is devoted to presenting several closely related characterizations of-convergence. For this purpose, we start with some basic definitions of an asymptotic radius and an asymptotic center. Let{xn}be a bounded sequence in a completeCAT()
spaceX. Forx∈XandC⊂X, letr(x,{xn}) =lim supn→∞d(x,xn). Theasymptotic radius
r({xn}) of{xn}is given by
r{xn}
=infrx,{xn}
:x∈X;
theasymptotic radius rC({xn})with respective to Cof{xn}is given by
rC
{xn}
=infrx,{xn}
:x∈C;
theasymptotic center A({xn}) of{xn}is given by the set
A{xn}
=x∈X:rx,{xn}
=r{xn}
;
theasymptotic center AC({xn})with respective to Cof{xn}is given by the set
AC
{xn}
=x∈C:rx,{xn}
=rC
{xn}
.
Proposition . ([, Proposition .]) Let X be a complete CAT() space and C⊂X nonempty closed andπ-convex.If{xn}be a sequence in X such that rC({xn}) <π/,then
Definition . A sequence{xn}inX is said to-converge tox∈Xifx is the unique
asymptotic center of every subsequence of{xn}. In this case, we write-limn→∞xn=x
andxis called the-limit of{xn}.
The next result is an immediate consequence of the preceding proposition.
Proposition .([, Corollary .]) Let X be a completeCAT()space and{xn}a sequence
in X.If r({xn}) <π/,then{xn}has a-convergent subsequence.
4 Main results
Since the validity of all our results, including the proofs as well, on CAT() spaces can be restored on anyCAT(κ) space withκ> by rescaling without major changes, we will pay our attention toCAT() spaces. In addition, when we deal with aCAT(κ) space, the hypothesisd(x,F(T)) <π/ for each theorem in this section is replaced byd(x,F(T)) <
Dκ/ and so can be dropped ifκ ≤ (in this case, Dκ =∞; refer to Section for the
definition).
To verify our result, the following basic property of asymptotical regularity is required; alsocf.[, Proposition .] in a topological vector space.
Lemma . Let C be a subset of a metric space(X,d)and letS={T(t) :t≥}be a family
of self-mappings of C such that T(s+t) =T(s)◦T(t),for all s,t≥.IfSis asymptotically regular,then
FT(t)=F(S), for all t> .
Proof Fixt> . ThenF(S)⊂F(T(t)). To proveF(T(t))⊂F(S), letxbe a fixed point of T(t). Forh≥, we obtain
dT(h)x,x= lim
k→∞d
T(h)◦T(t)kx,T(t)kx
= lim
k→∞d
T(h+tk)x,T(tk)x
= lim
s→∞d
T(h+s)x,T(s)x
=
and thereforex∈F(S).
Let (X,d) be a completeCAT() space,Ca closedπ-convex subset ofXandT:C→Ca nonexpansive mapping. ThenF(T) is closed. Also it is seen thatF(T) isπ-convex. Indeed, forx,y∈F(T) withd(x,y) <π, we have
dx,Tλx⊕( –λ)y≤dx,λx⊕( –λ)y= ( –λ)d(x,y)
and similarlyd(y,T(λx⊕( –λ)y))≤λd(x,y). Thus both equalities must hold and hence
Now consider a familyS={T(t) :t≥}of nonexpansive self-mappings ofCsuch that
F(S)=∅. From the previous remark, each fixed point setF(T(t)) is closed and convex, hence so isF(S). Fixx∈Cwith <d(x,F(S)) <π/. Letp=PF(S)x(the uniqueness
follows from Proposition .(i)) and=d(x,p). Then for eacht≥,T(t) maps the closed π-convex setC∩B(p,) into itself. For anyα∈(, ] andt≥, define the mappingSα,t:
C∩B(p,)→C∩B(p,) bySα,tx=αx⊕( –α)T(t)x. Observe thatSα,tis a contraction.
In fact, forx,y∈C∩B(p,), we apply Lemma .(iii) to get
d(Sα,tx,Sα,ty)≤sin
( –α)π
dT(t)x,T(t)y≤sin
( –α)π
d(x,y).
ThenSα,thas a unique fixed point inC∩B(p,).
Let [t] denote the maximum integer no greater thant. We now extend Theorem . in Hilbert spaces toCAT(κ) spaces as the following result.
Theorem . Let X be a completeCAT()space,C a closedπ-convex subset of X,S= {T(t) :t≥}a UAR and strongly continuous nonexpansive semigroup on C withF(S)=∅, and two sequences {αn} ⊂(, ],{tn} ⊂(,∞)such thatlimn→∞αn=limn→∞αn/tn= .
Choose arbitrarily a point x∈C such that d(x,F(S)) <π/.Let p=PF(S)xand=
d(x,p).Define a sequence{xn}in C∩B(p,)by the implicit iteration
xn=αnx⊕( –αn)T(tn)xn. (.)
Then{xn}converges strongly to the point p ofF(S)nearest to x.
Proof Ifx∈F(S), then{xn}reduces to the constant sequence{x,x, . . .}. Suppose that
x∈/F(S). We shall prove that any subsequence of{xn}contains a subsequence
converg-ing strongly topfrom which it follows that{xn}also converges strongly to the pointp. Let
{yn}be any subsequence of{xn}. Proposition . asserts that{yn}has a-convergent
sub-sequence. By passing to a subsequence we may assume that{yn}-converges to a pointy.
Let{βn}and{sn}be the respective corresponding subsequences of{αn}and{tn}.
Observe thaty∈F(S). To see this, takeˆs=lim supn→∞sn. There are three cases:
(i) ˆs= , (ii) <ˆs<∞, (iii) ˆs=∞.
By passing to a subsequence if necessary it may be assumed thatˆs=limn→∞sn. We discuss
each case as follows.
Suppose thatˆs= . Fixt≥ and we obtain
dT(t)y,yn
≤d
T(t)y,T
t sn
sn y +d
T t sn sn y,T
t sn
sn yn
+
[t/sn]
j=
dT(j– )sn
yn,T(jsn)yn
≤d T t– t sn
sn y,y +d(y,yn) +
t sn
dyn,T(sn)yn
≤ max ≤s≤sn
dT(s)y,y+d(y,yn) +
tβn
sn
dx,T(sn)yn
This implies that
lim sup
n→∞
dT(t)y,yn
≤lim sup
n→∞
d(y,yn);
henceT(t)y=yfor allt≥, that is,y∈F(S). If <sˆ<∞, then
dT(ˆs)y,yn
≤dT(ˆs)y,T(sn)y
+dT(sn)y,T(sn)yn
+dT(sn)yn,yn
≤dy,T|sn–ˆs|
y+d(y,yn) +d
T(sn)yn,yn
≤dy,T|sn–ˆs|
y+d(y,yn) +βnd
x,T(sn)yn
and thus
lim sup
n→∞
dT(ˆs)y,yn
≤lim sup
n→∞
d(y,yn).
HenceT(ˆs)y=yfrom which it follows thaty∈F(S) by Lemma .. Ifˆs=∞, fixt≥. For all sufficiently largenwe get
dT(t)y,yn
≤dT(t)y,T(t)yn
+dT(t)yn,yn
≤d(y,yn) +βnd
T(t)yn,x
+ ( –βn)d
T(t)yn,T(sn)yn
≤d(y,yn) +βnd
T(t)yn,x
+ ( –βn)d
yn,T(sn–t)yn
≤d(y,yn) +βnd
T(t)yn,x
+βn( –βn)d
x,T(sn–t)yn
+ ( –βn)d
T(sn)yn,T(sn–t)yn
≤d(y,yn) +βnd
T(t)yn,x
+βn( –βn)d
x,T(sn–t)yn
+ ( –βn)d
T(sn–t+t)yn,T(sn–t)yn
which yields
lim sup
n→∞ d
T(t)y,yn
≤lim sup
n→∞ d(y,yn)
since
lim sup
n→∞ d
T(sn–t+t)yn,T(sn–t)yn
≤ lim
n→∞x∈Csup∩B(p,)d
T(hn+t)x,T(hn)x
= ,
wherehn=sn–t. It follows thatT(t)y=yand thereforey∈F(S). This finishes the proof
thatyis a common fixed point of all mappings inS.
Next, we claim that{yn}converges strongly toy. We suppose on the contrary that
r{yn}
=σ> .
Fromy=T(sn)y, we obtain
d(y,yn)≤βnd(y,x) + ( –βn)d
y,T(sn)yn
and hence taking the limit superior asn→ ∞yields
lim sup
n→∞
dy,T(sn)yn
=lim sup
n→∞
d(y,yn) =σ, (.)
sincelimn→∞βn= . Recall that
d(x,yn) = ( –βn)d
x,T(sn)yn
<dx,T(sn)yn
, for alln.
Then
dyn,T(sn)yn
=dx,T(sn)yn
–d(x,yn) > , for alln.
Recall thatx∈/F(S). We havey=xand hence
lim sup
n→∞ d(x,yn) >σ. (.)
According to (.) and (.), by passing to a subsequence again we may assume that
d(x,yn) > , d(y,yn) > , d
y,T(sn)yn
> , for alln.
Let(¯x,y¯,T(sn)yn) be a comparison triangle for(x,y,T(sn)yn) inS. Observe that
∠¯yn(¯x,y¯)≥π/. (.)
For, contrarily, if∠¯yn(x¯,y¯) <π/, since∠y¯n(x¯,T(sn)yn) =π, then∠y¯n(y¯,T(sn)yn) >π/;
see Proposition .. By the law of cosines inS(Proposition .), sinced(yn,T(sn)yn) > ,
it follows that
cosdS
¯
y,T(sn)yn
<cosdS(¯y,y¯n),
which is a contradiction becauseT(sn) is nonexpansive. Notice that
<d(y,yn)≤dS(¯y,y¯n).
Hence (.) implies that
cosdS(x¯,y¯)≤cosdS(x¯,y¯n)cosdS(y¯,y¯n) <cosdS(x¯,¯yn),
or equivalently,
dS(¯x,¯yn) <dS(¯x,¯y).
Similarly,
The previous two inequalities, together with theCAT(κ) inequality, yield
d(x,yn) <d(x,y), d(y,yn) <d(x,y), for alln. (.)
Let(¯x,¯y,y¯n) be a comparison triangle for(x,y,yn) inS. SincedS(¯yn, [¯x,y¯]) <π/, Proposition .(i) assures that there is a unique pointu¯n∈[¯x,¯y] nearest to¯yn. Letun∈
[x,y] such thatd(x,un) =dS(¯x,u¯n) andd(un,y) =dS(¯un,y¯). By passing to a subsequence again it may be assumed that{¯un}and{un}converge, respectively, tou¯∈[¯x,y¯] andu∈
[x,y]. Since
σ=lim sup
n→∞
d(y,yn)≤lim sup n→∞
d(u,yn)
≤lim sup
n→∞
dS(¯u,u¯n) +lim sup
n→∞
dS(u¯n,y¯n)
=lim sup
n→∞
dS(u¯n,y¯n)
≤lim sup
n→∞ dS (¯y,y¯n)
=lim sup
n→∞ d(y,yn), (.)
this guarantees thatu=y.
On the other hand, according to (.), by passing to a subsequence we may suppose that
dS(¯un,¯yn) >
σ
, for alln.
Letan=dS(¯x,u¯n),bn=dS(¯un,¯yn),cn=dS(¯x,¯yn) and γn=∠u¯n(¯x,y¯n). Then by (.), bn>σ/ and cn< dS(¯x,y¯). Observe that un=x. Otherwise, since y=x, we have
∠x¯(¯y,y¯n) =∠u¯n(¯y,¯yn)≥π/ which implies that
d(y,yn) =dS(y¯,y¯n)≥dS(x¯,y¯) =d(x,y).
But this contradicts to (.). Thereforeun=xand soγn≥π/ by Proposition .(ii). The
law of cosines inSyields
coscn=cosancosbn+sinansinbncosγn
≤cosancosbn.
Sinceσ/ <bn≤cn<dS(¯x,y¯) <π/, this implies that
cosan≥
coscn
cosbn
>cosdS(x¯,y¯)
cos(σ/) >cosdS(¯x,y¯).
Hence
dS(x¯,u¯n) =an<cos–
cosdS(¯x,y¯)
It follows that
d(y,un) =dS(¯y,u¯n) =dS(¯x,y¯) –dS(¯x,u¯n) >dS(¯x,y¯) –δ> ,
whereδ=cos–[cosd
S(x¯,y¯)/cos(σ/)]. Taking the limit asn→ ∞yieldsu=y, which is a contradiction. Thusσ= and so{yn}converges strongly toy, as claimed.
It remains to prove thaty=PF(S)x. Letq∈F(S) and consider a comparison triangle
(¯x,q¯,T(sn)yn) for(x,q,T(sn)yn). From
dS
¯
q,T(sn)yn
≤dS(¯q,¯yn),
it is seen that∠y¯n(¯q,T(sn)yn)≤π/. Hence∠y¯n(¯x,q¯)≥π/ and so
d(x,yn) =dS(x¯,y¯n)≤dS(x¯,q¯) =d(x,q).
We then take the limit asn→ ∞and obtain
d(x,y)≤d(x,q),
which shows thatyis the nearest point inF(S) tox. Consequently, we conclude that{xn}
converges strongly toPF(S)x, which completes the proof.
It is worthy emphasizing that a uniformly asymptotical regularity hypothesis of The-orem . is superfluous whenlimn→∞tn=limn→∞αn/tn= ; see case (i) in the proof of
Theorem .. Thus, we state the result as follows.
Theorem . Let X be a completeCAT()space,C a closedπ-convex subset of X,S= {T(t) :t≥}a strongly continuous nonexpansive semigroup on C withF(S)=∅,and two sequences{αn} ⊂(, ],{tn} ⊂(,∞)such thatlimn→∞tn=limn→∞αn/tn= .Choose
ar-bitrarily a point x∈C such that d(x,F(S)) <π/.Let p=PF(S)xand=d(x,p).Then
the sequence{xn}in C∩B(p,)defined by(.)converges strongly to the point p.
Furthermore, ifXis aCAT() space (recall thatD=∞) in Theorem ., then the
corre-sponding assumptiond(x,F(S)) <π/ isd(x,F(S)) <∞, which can be dropped.
More-over, the sequence{xn}defined by (.) is bounded. In fact, since
d(p,xn)≤αnd(p,x) + ( –αn)d
p,T(tn)xn
≤αnd(p,x) + ( –αn)d(p,xn),
this shows thatd(p,xn)≤d(p,x). Then{xn}is bounded and so is{T(t)xn:t≥,n∈N}.
We restate Theorem . without assuming boundedness ofCas follows.
Theorem . Let X be a completeCAT()space,C a closed convex subset of X,S={T(t) : t≥} a strongly continuous nonexpansive semigroup on C withF(S)=∅, and two se-quences{αn} ⊂(, ],{tn} ⊂(,∞)such thatlimn→∞tn=limn→∞αn/tn= .Choose
arbi-trarily a point x∈C.Then the sequence{xn}in C defined by(.)converges strongly to a
Nonetheless, Theorem . has a uniformly asymptotical regularity hypothesis that can-not in general be removed whenlimn→∞tn=ˆt∈(,∞] as illustrated in the following
ex-ample.
Example . Consider the complex plane (C,| · |), where | · |is the absolute value, so that it is a completeCAT() space. Fort≥, define a self-mappingT(t) ofCbyT(t)(z) = zeit,z∈C. ThenS={T(t) :t≥}is a strongly continuous nonexpansive semigroup with
F(S) ={}. The familySis notUARonCbecause
lim
t→∞sup|z|=T(π)T(t)(z) –T(t)(z)=tlim→∞sup|z|=ze
i(t+π)–zeit= .
Letαn= /n,tn=nπ,n∈N, so thatlimn→∞tn=∞andlimn→∞αn/tn= . Choosez= .
Define a sequence{zn}inCby (.), that is,
zn=
n–, nis odd,
, nis even.
Hence the sequence{zn}is divergent.
Next if we takeαn= /n,tn= (/n) + π,n∈N, thenlimn→∞tn= π andlimn→∞αn/
tn= . Choosez= and define a sequence{zn}inCby (.) to get
zn=
/n – [ – (/n)]ei/n.
However, the sequence{zn}converges to ( +i)/ /∈F(S).
These two examples explain that strong convergence of the sequence defined by (.) in Theorem . ceases to be true without theUARassumption iflimn→∞tn=ˆt∈(,∞].
The next example shows that the conditionlimn→∞αn=limn→∞αn/tn= in
Theo-rem . is also necessary whenXis the unit sphere in, the infinite-dimensional Hilbert
space of square-summable sequences.
Example . LetSbe the unit sphere inendowed with the following intrinsic metric d:S×S→R: forx,y∈S,d(x,y)∈[,π] such that
cosd(x,y) = (x|y) = ∞
k=
x(k)y(k).
This space (S,d) is aCAT() space.
Take one element v= (, , . . . , , . . .) of the canonical basis ofS. Let C={x∈S : d(x,v)≤r} be a closed ball centered atvinS, where <r<π/. Fort≥, define a mappingT(t) :C→Cby
T(t)(x) = –e–tv⊕e–tx,
that is,
and fork= , , . . . ,
T(t)(x)(k) = ⎧ ⎨ ⎩
, ifx() = ,
x(k)sin√[e–tcos–x()]
–x() , ifx()= .
(.)
Forx,y∈C, from Lemma .(iii), we obtain
dT(t)x,T(t)y=d –e–tv⊕e–tx, –e–tv⊕e–ty
≤sin(e–tπ/)
sinπ/ d(x,y)
≤d(x,y).
ThenT(t) is nonexpansive. In fact,vis the unique fixed point ofT(t) fort> . Further-more, it is seen thatS={T(t) :t≥}is aUARand strongly continuous nonexpansive semigroup onC.
Chooseu= (uk)∈Cwith <u< so that <cos–u=d(u,v)≤r. Forα∈(, ) and
t∈(,∞), definex(α,t) by
x(α,t) =αu⊕( –α)T(t)x(α,t)
=cos( –α)du,T(t)x(α,t)u+sin( –α)du,T(t)x(α,t) A(α,t)
A(α,t), (.)
where
A(α,t) =T(t)x(α,t) –u|T(t)x(α,t)u.
We remark thatA(α,t)= – (u|T(t)x(α,t)). Let{α
n} ⊂(, ],{tn} ⊂(,∞) be two
sequences and define a sequence{xn}inB(v,cos–u)⊂Cby
xn=x(αn,tn) =αnu⊕( –αn)T(tn)xn.
We shall prove that if the sequence{xn}converges to the common fixed pointvofS, then
limn→∞αn=limn→∞αn/tn= .
First, assume thatβ=lim supn→∞αn. Sincelimn→∞xn() = and|xn(k)| ≤
–xn()
fork≥, we obtain from (.) and (.) that
lim
n→∞
T(tn)(xn)
() = , lim
n→∞
T(tn)(xn)
(k) = .
Thereforelimn→∞T(tn)(xn) =v, which implies that
lim
n→∞
u|T(tn)(xn)
=u, lim
n→∞d
u,T(tn)xn
=cos–u; (.)
hence
lim
n→∞A(αn,tn)() = –u
, nlim→∞A(αn,tn)=
Letθ=d(u,v) =cos–u
so thatsinθ=
–u
. Using (.), (.), and (.), we have
= lim
n→∞xn()
=lim inf
n→∞
ucos
( –αn)d
u,T(tn)xn
+A(αn,tn)() A(αn,tn)
sin( –αn)d
u,T(tn)xn
≤lim sup
n→∞
ucos
( –αn)d
u,T(tn)xn
+lim inf
n→∞
A(αn,tn)()
A(αn,tn)
sin( –αn)d
u,T(tn)xn
=cosθcos( –β)θ+sinθsin( –β)θ
=cos(βθ).
This shows thatβ= becauseθ> and solimn→∞αn= .
To prove thatlimn→∞αn/tn= , we recall the inequalityt≥ –e–tfor allt≥. Since
tnd(v,xn)≥
–e–tnd(v,x
n) =d
xn,T(tn)(xn)
=αnd
u,T(tn)(xn)
,
it follows that
= lim
n→∞d(v,xn)
≥lim sup
n→∞
αn
tn
du,T(tn)(xn)
=θlim sup
n→∞
αn
tn
.
Consequently,lim supn→∞αn/tn= and thereforelimn→∞αn/tn= as desired.
The following result is an immediate consequence of Theorem . and Example ..
Theorem . Let X be the unit sphere in an infinite-dimensional Hilbert space,and let {αn} ⊂(, ],{tn} ⊂(,∞)be two sequences.Then the following conditions are equivalent:
(i) limn→∞αn=limn→∞αn/tn= .
(ii) LetCbe a closed convex subset ofXwithdiamC<π/,S={T(t) :t≥}aUAR
and strongly continuous nonexpansive semigroup onCwithF(S)=∅,x∈Csuch thatd(x,F(S)) <π/,p=PF(S)x,=d(x,p)and{xn}is a sequence inC∩B(p,)
defined by(.).
Then{xn}converges strongly to p.
5 Remark
all left reversible semitopological semigroups includes all groups, all commuting semi-groups, and all normal left amenable semitopological semisemi-groups,i.e., the spaceCB(S) of bounded continuous functions onShas a left invariant mean; see [, , –]. IfSis left (respectively, right) reversible, (S,) is a directed system when the binary relationon Sis defined byabif and only if{a} ∪aS⊃ {b} ∪bS(respectively,{a} ∪Sa⊃ {b} ∪Sb), a,b∈S.
LetSbe a semitopological semigroup andCa closed convex subset of a metric space (X,d). A familyS={T(s) :s∈S}is called arepresentationofSonCif for eachs∈S,T(s) is a mapping fromCintoCandT(st) =T(s)◦T(t), for alls,t∈S. The representation is called astrongly continuous nonexpansive semigrouponC(or acontinuous representation ofS as nonexpansive mappingsonC) if the following conditions are satisfied (see []):
(i) for eacht∈S,T(t)is a nonexpansive mapping onC;
(ii) for eachx∈C, the mappingt→T(t)xfromSintoCis continuous.
The representationSofSis called
(i) asymptotically regularonCif for anyh∈Sand anyx∈C,
lim
t∈Sd
T(h)T(t)x,T(t)x= ;
(ii) uniformly asymptotically regular(in shortUAR) onCif for anyh∈Sand any bounded subsetDofC,
lim
t∈Ssupx∈Dd
T(h)T(t)x,T(t)x= .
Problem . LetX be a completeCAT() space,Ca closedπ-convex subset ofX,Sa commutative (or left reversible) semitopological semigroup, andS={T(t) :t∈S}a con-tinuous representation ofS as nonexpansive mappings onC. Can we obtain the result analogous to Theorem . corresponding to a continuous representationS={T(t) :t∈S} ofSas nonexpansive mappings onC?
To answer this problem, the following conjecture is required and hence needs to be verified.
Conjecture . Let S be a commutative(or left reversible)semitopological semigroup,C a subset of a metric space(X,d)andS={T(t) :t∈S}a representation of S on C.IfSis asymptotically regular,then
FT(t)=F(S), for all t∈S.
Competing interests
The author declares that she has no competing interests.
Acknowledgements
This research was supported by a grant NSC 102-2115-M-259-004 from the National Science Council of Taiwan. The author, therefore, thanks the NSC financial support. The author would like to express the most sincere thanks to the referees for their careful reading of the manuscript, important comments and the citation of Ref. [13]. The author is also very grateful to Professor Anthony Lau for giving valuable remarks and Refs. [14] and [3], and for suggesting presenting Problem 5.1 for future studies.
References
1. Lau, AT-M, Miyake, H, Takahashi, W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal.67, 1211-1225 (2007)
2. Lau, AT-M: Normal structure and common fixed point properties for semigroups of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl.2010, Article ID 580956 (2010)
3. Lau, AT-M, Zhang, Y: Fixed point properties for semigroups of nonlinear mappings and amenability. J. Funct. Anal. 263(10), 2949-2977 (2012)
4. Suzuki, T: Strong convergence of Krasnoselskii and Mann’s type sequences for one parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl.305, 227-239 (2005)
5. Browder, FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal.24, 82-90 (1967)
6. Acedo, GL, Suzuki, T: Browder’s convergence for uniformly asymptotically regular nonexpansive semigroups in Hilbert spaces. Fixed Point Theory Appl.2010, Article ID 418030 (2010)
7. Suzuki, T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc.131(7), 2133-2136 (2003)
8. Espínola, R, Fernández-León, A: CAT(κ)-spaces, weak convergence and fixed points. J. Math. Anal. Appl.353, 410-427 (2009)
9. Huang, S: The-convergence of iterations for nonexpansive mappings in CAT(κ) spaces. J. Nonlinear Convex Anal. 13(3), 541-554 (2012)
10. Pia¸tek, B: Halpern iteration in CAT(κ) spaces. Acta Math. Sin. Engl. Ser.27, 635-646 (2011)
11. Dhompongsa, S, Fupinwong, W, Kaewkhaob, A: Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces. Nonlinear Anal.70, 4268-4273 (2009) 12. Bridson, M, Haefliger, A: Metric Spaces of Non-Positive Curvature. Springer, Berlin (1999)
13. Holmes, RD, Lau, AT-M: Asymptotically non-expansive actions of topological semigroups and fixed points. Bull. Lond. Math. Soc.3, 343-347 (1971)
14. Lau, AT-M: Semigroup of nonexpansive mappings on a Hilbert space. J. Math. Anal. Appl.105(2), 514-522 (1985) 15. Lau, AT-M, Takahashi, W: Weak convergence and nonlinear ergodic theorems for reversible semigroups of
nonexpansive mappings. Pac. J. Math.126, 277-294 (1987)
10.1186/1687-1812-2014-44
