a Xi :0911.5263 1 [ma h.FA] 27 No 2009
On bes p oximi y poin s in me ic and Banach spaces
Ra a Esp´ınola & Au o a Fe n´andez-Le´on∗
Abs ac
In his pape we s udy he exis ence and uniqueness o bes p oximi y poin s o cyclic con-
ac ions as well as he con e gence o i e a es o such p oximi y poin s. We do i om wo
di e en app oaches, leading each one o hem o di e en esul s which comple e, i no im-
p o e, o he simila esul s in he heo y. Resul s in his pape s and o Banach spaces, geodesic
me ic spaces and me ic spaces. We also include an appendix on CAT(0) spaces whe e we s udy
he pa icula beha io o hese spaces ega ding he p oblems we a e conce ned wi h.
MS Classi ica ion: 54H25, 47H09
1 In oduc ion
Le Aand Bbe wo nonemp y closed subse s o a comple e me ic space X. Conside a mapping
T:A∪B→A∪Bsuch ha
T(A)⊆Band T(B)⊆A
wi h he addi ional condi ion ha he e exis s k∈(0,1) such ha
d(Tx, Ty)≤kd(x, y) o all x∈Aand y∈B,
hen A∩B6=∅and Thas a unique ixed poin in A∩B.
In [3, 4, 5, 17] a gene aliza ion o his si ua ion was s udied unde he assump ion o A∩B=∅.
Mo e p ecisely, in [3, 17] i was assumed ha he e exis s k∈(0,1) such ha
d(Tx, Ty)≤kd(x, y) + (1 −k) dis (A, B) (1.1)
o all x∈Aand y∈B o ob ain exis ence, uniqueness and con e gence o i e a es o he so-called
bes p oximi y poin s; ha is, a poin xei he in Ao Bsuch ha d(x, Tx) = dis (A, B). This
was i s s udied in [3] o uni o mly con ex Banach spaces, see also [11] o mo e on ela ed opics.
Then, in [17], he p ope y UC (see Sec ion 3 o de ini ion) was in oduced o a pai (A, B) o
subse s o a me ic space so a esul on exis ence, uniqueness and con e gence o i e a es s ands
(Theo em 2.6 in Sec ion 2) in gene al me ic spaces. Since, as i is also p o ed in [17], p ope y
UC happens o a la ge collec ion o pai s o subse s o uni o mly con ex Banach spaces, Theo em
2.6 ac ually con ains he main heo em o [3] (Theo em 3.10 in [3]) as a pa icula case. P ope y
UC was e en p o ed, in [17], o happen ou side he se ing o uni o mly con ex Banach spaces. In
ac , his was ob ained o UCED (uni o mly con ex in e e y di ec ion) Banach spaces and s ic ly
con ex Banach spaces bu in bo h cases unde he e y s ong condi ion (see Theo em 3.7 in Sec ion
3) o one o he se s o be o compac closu e. In his wo k we i s in oduce a new p ope y, he
so-called p ope y WUC, which is p o ed o happen unde a less es ic i e condi ions han whe e
∗Bo h au ho s we e pa ially suppo ed by he Minis e y o Science and Technology o Spain, G an BFM 2000-
0344-CO2-01 and La Jun a de An aluc´ıa p ojec FQM-127.
1
p ope y UC seems o each, and hen an exis ence, uniqueness and con e gence heo em is p o ed
o pai s o se s e i ying p ope y WUC. Second, we ocus he same p oblem om he new app oach
sugges ed by one o he au ho s in [5] o ob ain s ill new esul s on he same p oblem. As a esul ,
a pa ial answe in he posi i e is gi en o a ques ion aised in [3].
The wo k is o ganized as ollows: in Sec ion 2 we in oduce mos o he de ini ions, no a ions
and p e ious esul s we will need. In Sec ion 3 we look o weake condi ions han p ope y UC.
We in oduce p ope ies WUC and W-WUC and show ha simila esul s o hose in [17] hold
unde condi ions which a e easie o e i y. In Sec ion 4, we app oach he same p oblem by he
in oduc ion o a semime ic. This is applied in a success ul way by showing ha he mappings
e i ying he con ac i e condi ion (1.1), unde sui able assump ions, a e con ac ions wi h espec
o a ce ain semime ic. We inish his wo k wi h a ema k on CAT(0) spaces. In [4] i was shown
ha when he ambien space is a Hilbe space hen he kind o mappings we a e dealing wi h
ac ually beha e as nonexpansi e ones. In [5] i is shown ha he semime ic he e de ined coincides
wi h he me ic o he ambien space when his is a Hilbe space. Ou ema k on CAT(0) spaces, in
a ce ain sense he nonlinea coun e pa s o Hilbe spaces, s a es ha some hing simila happens
in hese spaces.
2 P elimina ies
In his sec ion we compile he main concep s and esul s we will wo k wi h along his pape . We
begin wi h some basic de ini ions and no a ions ha a e needed. Le (X, d) be a me ic space and
le Aand Bbe wo subse s o X. De ine
dis (x, A) = in {d(x, y) : y∈A};
PA(x) ={y∈A:d(x, y) = dis (x, A)};
dis (A, B) = in {d(x, y) : x∈A, y ∈B};
diam(A) = sup{d(x, y) : x, y ∈D}.
Recall ha he se Ais said o be a Chebyshe se wi h espec o Bi PA(x) is a single on o any
x∈B.
A me ic space (X, d) is said o be a geodesic space (D-geodesic space, espec i ely) i e e y wo
poin s xand yo X(wi h d(x, y)≤D) a e joined by a geodesic, i.e, a map c: [0, l]⊆R→X
such ha c(0) = x,c(l) = y, and d(c( ), c( ′)) = | − ′| o all , ′∈[0, l]. Mo eo e , (X, d) is
called uniquely geodesic (D-uniquely geodesic) i he e is exac ly one geodesic joining xand y o
each x, y ∈X(wi h d(x, y)≤D). When he geodesic be ween wo poin s is unique, i s image
(called geodesic segmen ) is deno ed by [x, y]. The midpoin min be ween wo poin s xand yin a
uniquely geodesic me ic space is he only poin in [x, y] such ha d(x, m) = d(y, m). Any Banach
space is a geodesic space wi h usual segmen s as geodesic segmen s. Th oughou his wo k we will
jus use geodesic me ic space o e e o a uniquely geodesic space since all ou geodesic spaces
will be uniquely geodesic.
A e y impo an class o geodesic me ic spaces a e he CAT(k) spaces, ha is, me ic spaces
o cu a u e uni o mly bounded abo e by k. These spaces ha e been he objec o a lo o in e es
by many esea ches and we will ge back o hem, especially o CAT(0) spaces, a ce ain momen s
o ou exposi ion. Fo a e y ho ough ea men on CAT(k)-spaces he eade can check [2].
A subse Ao a geodesic me ic space (X, d) is said o be con ex i he geodesic joining each
pai o poin s xand yo Ais con ained in A.
We will need he no ion o uni o mly con ex geodesic me ic space (see also [8, pg. 107]).
2
De ini ion 2.1 A geodesic me ic space (X, d)is said o be uni o mly con ex i o any > 0and
any ε∈(0,2] he e exis s δ∈(0,1] such ha o all a, x, y ∈Xwi h d(x, a)≤ ,d(y, a)≤ and
d(x, y)≥ε i is he case ha
d(m, a)≤(1 −δ)
whe e ms ands o a midpoin o he geodesic segmen [x, y]. A mapping δ: (0,+∞)×(0,2] →(0,1]
p o iding such a δ=δ( , ε) o a gi en > 0and ε∈(0,2] is called a modulus o uni o m con exi y.
I mo eo e δdec eases wi h ( o a ixed ε) we say ha δis a mono one modulus o uni o m
con exi y o X.
The no ion o mono one modulus o uni o m con exi y seems o ha e been s udied o i s ime
in [13]. O cou se, he usual modulus o con exi y o a uni o mly con ex Banach space is mono one
in his sense. Fo mo e on geome y o Banach spaces he eade can check [1, 7, 12].
Rema k 2.2 I in he abo e de ini ion we d op he uni o mi y condi ions hen we ind he no ion
o s ic con exi y. Mo e p ecisely, i Xis a Banach space and such a δexis s o each a, x and
yas abo e wi h d(x, y)>0, hen we will say ha Xis a s ic ly con ex Banach space. I he
same condi ion is imposed on a geodesic me ic space X, hen we can ind he spaces o nonposi i e
cu a u e in he sense o Busemann, see [15] o a de ailed s udy on hem.
Cyclic con ac ions and bes p oximi y poin s a e de ined nex .
De ini ion 2.3 Le Aand Bbe wo nonemp y subse s o a me ic space X. A map T:A∪B→
A∪Bis a cyclic con ac ion map i i sa is ies:
(1) T(A)⊆Band T(B)⊆A.
(2) The e is some k∈(0,1) such ha d(T x, Ty)≤kd(x, y) + (1 −k) dis (A, B), o all x∈A
and y∈B.
Rema k 2.4 No ice ha condi ion (2) implies ha Tis a ela i ely nonexpansi e mapping, i.e.,
Tsa is ies ha d(Tx, T y)≤d(x, y) o all x∈Aand y∈B, which we e he main objec o s udy
in [4, 5].
Nex we de ine he no ion o bes p oximi y poin .
De ini ion 2.5 Le Aand Bbe wo nonemp y subse s o a me ic space X. Le T:A∪B→A∪B
such ha T(A)⊆Band T(B)⊆A. A poin x∈A∪Bis said o be a bes p oximi y poin o T
i d(x, Tx) = dis (A, B).
Exis ence, uniqueness and con e gence o i e a es o a bes p oximi y poin o cyclic con ac-
ions ha e ecen ly been s udied in [3, 17]. The goal o his wo k is o ind imp o emen s o main
esul s in hese wo ks. Nex we s a e he main esul om [17] ( he de ini ion o p ope y UC is in
Sec ion 3).
Theo em 2.6 Le (X, d)be a me ic space and le Aand Bbe nonemp y subse s o Xsuch ha
(A, B)sa is ies he p ope y UC. Assume ha Ais comple e. Le Tbe a cyclic con ac ion on
A∪B. Then Thas a unique bes p oximi y poin zin Aand {T2nx}con e ges o z o e e y
x∈A.
3
Main esul in [3] s a es basically he same bu wi h Aand Bnonemp y closed and con ex, X
a uni o mly con ex Banach space and no men ion o p ope y UC.
In [5] a new app oach o ela i ely nonexpansi e mappings lead o he ac ha such mappings,
unde sui able condi ions, a e ac ually nonexpansi e wi h espec o an adequa e semime ic. In
Sec ion 4 we apply his new app oach o cyclic con ac ions. Nex we in oduce he main no ions
and esul s on semime ic spaces ha we will need.
De ini ion 2.7 Le Mbe a nonemp y se . A unc ion d:M×M→[0,∞)is said o be a
semime ic on Mi
(1) d(x, y) = 0 i , and only i , x=y.
(2) d(x, y) = d(y, x) o any x, y ∈X.
In his case, (M, d)is said o be a semime ic space.
Con ac ions wi h espec o semime ics a e de ined in a simila way o con ac ions wi h
espec o me ics.
De ini ion 2.8 Le (X, d)be a semime ic space. A mapping T:X→Xis said o be a con ac ion
i he e is a cons an k∈(0,1) such ha o all x, y ∈X
d(Tx, Ty)≤kd(x, y).
The nex de ini ion will make easie o s a e some o ou esul s.
De ini ion 2.9 Le Xbe a nonemp y se . Le dand d1be a me ic and a semime ic on X
espec i ely. We say ha dand d1a e compa ible on Xi o e e y ε > 0and x∈X he e exis
x(ε)>0and gx(ε)>0such ha
Bd(x, x(ε)) ⊆Bd1(x, ε)and Bd1(x, gx(ε)) ⊆Bd(x, ε),
whe e Bd(x, )and Bd1(x, )s and, espec i ely, o he closed balls o cen e xand adius wi h
espec o he me ic and he semime ic.
In [9], di e en coun e pa s o Banach’s con ac ion heo em a e gi en o semime ic spaces.
We s a e nex a pa icula case o hose esul s mo e adequa e o ou con ex (see Theo em 1 in
[9]).
Theo em 2.10 Le X,dand d1be as in he de ini ion abo e wi h dand d1compa ible. Le Tbe
a con ac ion on X o he semime ic d1, hen Thas a unique ixed poin x0. Mo eo e , o any
x∈X he sequence {Tnx}∞
n=1 con e ges o x0.
We inish his sec ion in oducing wo geome ical p ope ies o Banach spaces. We begin
desc ibing p ope y (H).
De ini ion 2.11 Le Xbe a Banach space. Xis said o ha e he p ope y (H)i o any sequence
on he uni sphe e o X, weak and no m con e gence coincide.
Rema k 2.12 This p ope y has been e y ex ensi ely s udied in he li e a u e and i is closely
ela ed o he so-called Kadec-Klee p ope y (KK-p ope y, o sho ). Fo mo e on his opic, see
[1, 7, 12, 14].
4
We will also need he ollowing uni o m e sion o he KK p ope y.
De ini ion 2.13 Le Xbe a Banach space. Xis said o ha e he p ope y UKK (uni o m Kadec-
Klee p ope y) i o any ε > 0 he numbe
η(ε) = in {1− kxk} >0,
whe e he in imum is aken o e all poin s xsuch ha xis a weak limi o some sequence {xn}in
he uni ball o Xwi h kxn−xk ≥ ε o all n.
Di e en p ope ies o UKK Banach spaces as well as connec ion among all hese geome ical
no ions can be ound in he abo e-men ioned e e ences. Le us jus no e he e, as a ma e o ac ,
ha uni o mly con ex Banach spaces a e UKK spaces and so hey also ha e p ope y (H). Bo h
no ions, uni o m con exi y and p ope y UKK, ha e o do wi h a ce ain o oundi y o he balls o
he space. This is ob ious o uni o m con exi y and a less ob ious o p ope y UKK as he e
exis Banach spaces which a e UKK and no e en s ic ly con ex.
3 The UC and WUC p ope ies
P ope y UC was de ined in [17] in he ollowing way.
De ini ion 3.1 Le Aand Bbe nonemp y subse s o a me ic space (X, d). Then (A, B)is said
o sa is y he p ope y UC i o {xn}and {x′
n}sequences in Aand {yn}a sequence in Bsuch ha
limnd(xn, yn) = limnd(x′
n, yn) = dis (A, B), hen limnd(xn, x′
n) = 0.
The ollowing p oposi ion shows he uni o m na u e o p ope y UC.
P oposi ion 3.2 Fo Aand Bnonemp y subse s o a me ic space X, he ollowing a e equi alen :
(i) (A, B)has p ope y UC.
(ii) Fo any ε > 0 he e exis s δ > 0such ha diam(A∩B(y, dis (A, B) + δ)) ≤ε o any y∈B.
P oo . Fi s we see (i)⇒(ii). Supposing he con a y implies ha he e is ε0>0 such
ha o e e y δ= 1/n he e exis yn∈Band xn, x′
n∈Asa is ying d(yn, xn)≤dis (A, B) + 1
n,
d(yn, x′
n)≤dis (A, B) + 1
nand d(xn, x′
n)> ε0, which ob iously con adic s p ope y UC.
Now we p o e (ii)⇒(i). Le xn, x′
n∈Aand yn∈Bsuch ha d(yn, xn) and d(yn, x′
n) bo h con-
e ge o dis (A, B) as n→ ∞. Then gi en ε > 0, he e is δ > 0 such ha diam(A∩B(y, dis (A, B)+
δ)) ≤ε o any y∈B. Now, i is enough o ake n0∈Nsuch ha d(xn, yn), d(x′
n, yn)≤
dis (A, B) + δ o any n≥n0 o deduce ha d(xn, x′
n)≤ε o n≥n0.2
In [17] i was shown ha any pai o nonemp y subse s (A, B) o uni o mly con ex Banach
spaces wi h Acon ex enjoy he p ope y UC. Nex we show ha some hing simila can be said o
uni o mly con ex geodesic spaces unde adequa e condi ions on he modulus o con exi y.
P oposi ion 3.3 Le (X, d)be a uni o mly con ex geodesic me ic space wi h a mono one modulus
o con exi y δ( , ε). Le Aand Bbe wo nonemp y subse s o Xwi h Acon ex. Then he pai
(A, B)has p ope y UC.
5

P oo . Suppose on he con a y ha he e exis {xn}and {x′
n}sequences in A,{yn}in B
and ε0>0 such ha o e e y k∈N, he e exis nk≥k o which d(xnk, x′
nk)≥ε0while
limn→∞ d(xn, yn) = limn→∞ d(x′
n, yn) = dis (A, B).
The e is no loss o gene ali y in assuming ha δ( , ε)<1 o , ε > 0 and ha dis (A, B)>0
since o he wise he esul ollows in a i ial way. Fo γ > dis (A, B) and ε1=ε0/γ, choose ε > 0
such ha
ε < min γ−dis (A, B),dis (A, B)δ(γ, ε1)
1−δ(γ, ε1).
Then he e exis s N0∈Nsuch ha i nk≥N0, hen d(xnk, ynk)≤dis (A, B) + εand d(x′
nk, ynk)≤
dis (A, B) + ε. Le mnkbe he mid-poin o he geodesic segmen [xnk, x′
nk]. Using he uni o m
con exi y o X, we ha e ha
d(ynk, mnk)≤1−δ(dis (A, B) + ε, ε1)(dis (A, B) + ε)≤
≤1−δ(γ, ε1)(dis (A, B) + ε)<dis (A, B).
Then, o nk≥N0
d(ynk, mnk)<dis (A, B),
which con adic s he ac ha mnk∈Aby con exi y o A.2
Rema k 3.4 No ice ha he same esul emains ue i he condi ion on he mono onici y o he
modulus o con exi y is eplaced by he condi ion o being lowe semi-con inuous om he igh .
As i was poin ed in he In oduc ion, p ope y UC was also shown in [17] o happen in UCED
Banach spaces and s ic ly con ex Banach spaces bu eques ing Ais ela i ely compac . Rega ding
he assump ion on he compac ness o A he ollowing esul om [3] is ele an .
Theo em 3.5 Le Aand Bbe nonemp y closed subse s o a me ic space (X, d)and le T:A∪B→
A∪Bbe a cyclic con ac ion. I ei he Ao Bis boundedly compac , hen he e exis s xin A∪B
wi h d(x, Tx) = dis (A, B).
No ice ha wha we miss om Theo em 2.6 in his heo em is uniqueness and con e gence o
i e a es. We see nex ha his is easy o ob ain by adding he e y mild condi ion (see he ema k
below o suppo his idea) o being Aa Chebyshe se o p oximinal poin s wi h espec o B.
De ini ion 3.6 Gi en Aand B wo nonemp y subse s o a me ic space, we say ha Ais a
Chebyshe se o p oximinal poin s wi h espec o Bi o any x∈Bsuch ha dis (x, A) =
dis (A, B)we ha e ha PA(x)is a single on.
Then we can p o e he ollowing.
Theo em 3.7 I in he abo e heo em, Ais supposed o be boundedly compac and a Chebyshe
se o p oximinal poin s wi h espec o B, hen he bes p oximi y poin z∈Ais unique and he
sequence {T2nx}con e ges o z o any x∈A.
P oo . We i s show i is unique. Suppose zand z′a e wo bes p oximi y poin s in A
wi h z6=z′. Then he Chebyshe condi ion on Aimplies ha Tz 6=Tz′. Now, he ela i e
nonexpansi i y o Timplies ha
d(T2z, Tz)≤d(z, T z) = dis (A, B)
6
and so, he Chebyshe condi ion on Aalso implies ha zand z′a e ixed poin s o T2. I we w i e
d∗(x, y) = d(x, y)−dis (A, B) hen
d∗(z, Tz′) = d∗(T2z, Tz′)
≤kd∗(z′, Tz) = kd∗(T2z′, Tz)≤k2d∗(z, T z′).
Hence d∗(z, T z′) = 0 and so z=z′.
Finally he con e ges o he i e a es ollows di ec ly om he ac s ha Ais boundedly compac ,
he sequences {T2nx}a e bounded o any x∈Aand ha lim d(T2nx, Tz) = dis (A, B) o any
x∈A.2
Rema k 3.8 No ice ha he condi ion o being Chebyshe is a e y na u al one in his kind o
p oblems. Think o he wise on he se s A={(x, 0) : x∈[0,1]}and B={(x, 1) : x∈[0,1]}
as subse s o he plane wi h he maximum no m. Then any mapping T:A∪B→A∪Bwi h
T(A)⊆Band T(B)⊆Ais a cyclic con ac ion.
We sugges o eplace p ope y UC wi h he weake one WUC which we de ine nex .
De ini ion 3.9 Le Aand Bbe nonemp y subse s o a me ic space (X, d). Then (A, B)is said
o sa is y he p ope y WUC i o any {xm} ⊆ Asuch ha o e e y ε > 0 he e exis s y∈B
sa is ying ha d(xm, y)≤dis (A, B) + ε o m≥m0, hen i is he case ha {xm}is con e gen .
Rema k 3.10 Ano he al e na i e o he abo e de ini ion is o ask he sequence {xm} o be Cauchy
ins ead o con e gen . I is wo hwhile o no e he e ha his is qui e a de ail o a o mal na u e
since in all ou main esul s we always assume A o be comple e.
Nex p oposi ion gi es he ela ion be ween he wo men ioned p ope ies.
P oposi ion 3.11 Le Aand Bbe nonemp y subse s o a me ic space (X, d)such ha Ais
comple e. Suppose he pai (A, B)has p ope y UC. Then (A, B)has p ope y WUC.
P oo . Le {xm} ⊆ Abe such ha o e e y δ > 0 he e exis s y∈Bsa is ying ha d(xm, y)≤
dis (A, B)+δ o m≥m0. I su ices o show ha {xm}is a Cauchy sequence. This ollows di ec ly
om (ii) o P oposi ion 3.2. 2
Nex we show ha p ope y WUC implies a nonuni o m e sion o he equi alence gi en by
P oposi ion 3.2 o p ope y UC. We omi i s p oo .
P oposi ion 3.12 Le Aand Bbe nonemp y subse s o a me ic space (X, d). Suppose (A, B)has
p ope y WUC hen
lim
ε→0diam(A∩B(y, dis (A, B) + ε)) = 0
o any y∈B.
The nex p oposi ions show ha p ope y WUC is likely o happen in mo e si ua ions han
p ope y UC. We i s weaken he no ion o uni o m con ex geodesic space wi h a mono one modulus
o con exi y.
7
De ini ion 3.13 A geodesic me ic space (X, d)is said o be poin wise uni o mly con ex i o any
a∈X, > 0and ε∈(0,2] he e exis s δ=δ(a, , ε)∈(0,1] such ha o all x, y ∈Xwi h
d(x, a)≤ ,d(y, a)≤ and d(x, y)≥ε i is he case ha
d(m, a)≤(1 −δ)
whe e ms ands o he midpoin o he geodesic segmen [x, y]. A mapping δ:X×(0,+∞)×(0,2] →
(0,1] p o iding such a δ o a gi en a∈X, > 0and ε∈(0,2] is called a modulus o poin wise
uni o mly con exi y (o jus modulus o con exi y when con usion canno a ise). I mo eo e δ
dec eases wi h ( o εand a) we say ha δis a mono one modulus o poin wise uni o mly con exi y
o X.
Rema k 3.14 No ice ha bo h no ions o uni o m con exi y coincide o Banach spaces.
P oposi ion 3.15 Le (X, d)be a comple e poin wise uni o mly con ex geodesic me ic space wi h
mono one modulus o con exi y. Le Aand Bbe wo nonemp y subse s o Xwi h Acon ex. Then
he pai (A, B)has p ope y WUC.
P oo . Le {xm}be a sequence in Asuch ha o e e y ε > 0 he e exis y∈Band m0∈N
sa is ying ha d(xm, y)≤dis (A, B) + ε o m≥m0. Suppose {xm}is no con e gen , hen he e
exis s ε0>0 such ha o each k∈N he e a e nk, mk≥k o which d(xnk, xmk)≥ε0. Thus, o
k=m0, we ind nm0, mm0≥m0such ha
d(xnm0, y)≤dis (A, B) + ε, d(xmm0, y)≤dis (A, B) + ε
and
d(xnm0, xmm0)≥ε0.
Le zm0be he mid-poin in he segmen [xnm0, xmm0]. Since Xis poin wise uni o mly con ex, we
ob ain ha
d(zm0, y)≤(dis (A, B) + ε)(1 −δ),
o some δ=δ(y, dis (A, B) + ε, ε1)∈(0,1] as in he p oo o P oposi ion 3.3. The con adic ion
ollows om he ac ha we can epea his easoning o any ε > 0 wi h yand ε0 ixed. 2
Rema k 3.16 This kind o modulus has been p e iously used o hype bolic spaces in [16].
P oposi ion 3.17 Le Xbe a UKK e lexi e and s ic ly con ex Banach space. Then, o A, B ⊆
Xnonemp y and con ex, i is he case ha (A, B)has he p ope y WUC.
P oo . Le {xn} ⊆ Abe as in he abo e p oo . Suppose {xn}is no con e gen . Fi s we show
ha his sequence needs o ha e a sepa a ed subsequence. Conside wo con e gen subsequences
{xnk}and {xnl}o xnwi h espec i e limi s xand x′in he closu e o A. Fo each n∈Nchoose
yn∈Bsuch ha he ales o bo h subsequences a e in B(yn,dis (A, B) + 1/n). Then i is clea
ha
x, x′∈
n∈N
B(yn,dis (A, B) + 1/n).
Since {yn}is bounded we can assume i is weakly con e gen o a poin yin he closu e o B. Then
i mus be he case ha x, x′∈B(y, dis (A,B)) om whe e, since Xis s ic ly con ex, x=x′.
The e o e we can assume ha {xn}does no ha e any con e gen subsequence and so i is a
sepa a ed sequence. Le ε > 0 such ha d(xn, xm)≥ε o e e y n6=m. Since his sequence
8
is bounded and Xis e lexi e, we can also assume {xn}is weakly con e gen o a poin x. Now
we only ha e o apply he UKK p ope y in a simila way as he uni o m con exi y was applied
in he p e ious p oposi ion o deduce ha xis in he closu e o Abu dis (x, B)<dis (A, B),
con adic ing he de ini ion o dis (A, B). 2
The UKK p ope y o ∆-con e gen sequences has been ecen ly s udied in [6, 10] o CAT(k)
spaces. I we assume ha Xis a geodesic space such ha bounded sequences ha e a unique
asymp o ic cen e which belongs o he con ex hull o he sequence (see any o [6, 10] o de ini ions),
he p e ious p oposi ion inds a me ic coun e pa ha we s a e nex and which can be p o ed
exac ly he same.
P oposi ion 3.18 Le Xbe a geodesic me ic space wi h he UKK p ope y o ∆-con e gen
sequences and he abo e-men ioned p ope y o bounded sequences. Suppose also ha he unc ion
µ( , ε)gi en by he UKK p ope y dec eases wi h espec o he adius, hen, o A, B ⊆Xnonemp y
and con ex, i is he case ha (A, B)has he p ope y WUC.
Rema k 3.19 Al hough he si ua ion o Banach spaces is clea in he sense ha uni o m con exi y
implies p ope y UKK, he same seems a o be he case o geodesic spaces as de ined o ∆-
con e gen sequences. Ac ually he UKK p ope y o CAT(k)spaces as shown in [6, 10] seems o
be mo e connec ed wi h he so-called Opial condi ion han wi h he uni o m con exi y. I is wo h
o ecall a his poin ha only Hilbe spaces and he spaces o sequences ℓpa e known o enjoy he
Opial p ope y. Fo mo e on his and ela ed opics he in e es ed eade can consul Chap e s 3,
4, 5 and 16 in [12] o [1, p. 102].
Nex we show ha WUC is enough o lead o a bes p oximi y poin o a cyclic con ac ion.
Due o no a ion pu poses, we will deno e as he con ac i e cons an in he de ini ion o cyclic
con ac ion o he emainde o his sec ion.
Theo em 3.20 Le (X, d)be a me ic space and Aand B wo nonemp y subse s o Xsuch ha
(A, B)sa is ies he p ope y WUC. Assume ha Ais comple e. Le Tbe a cyclic con ac ion on
A∪B. Then Thas a unique bes p oximi y poin zin Aand he sequence {T2nx}con e ges o z
o e e y x∈A.
P oo . As in [17] we conside d∗(x, y) = d(x, y)−dis (A, B). Then d∗(Tx, Ty)≤ d∗(x, y) o
x∈Aand y∈B. In consequence d∗(T2x, T x)≤ d∗(x, Tx) and d∗(Ty, T 2y)≤ d∗(Ty, y) o any
x∈Aand y∈B.
Fix x∈A,n∈Nand le m=n+kwi h k∈N. Then
d∗(T2mx, T2n+1x)≤ 2nd∗(T2kx, Tx)
≤ 2nsup{d(Tx, T 2kx) : k∈N}= 2nM(x).
P oposi ion 3.3 in [3] gua an ees ha M(x) is ini e o each x. Hence, gi en ε > 0 and aking
nsuch ha 2nM(x)< ε we ha e ha
T2mx∈B(T2n+1x, dis (A, B) + ε) (3.2)
o m≥nand so, by he p ope y WUC, {T2nx}is con e gen . Now he p oo ollows he same
pa e ns han he p oo o Theo em 3 in [17]. Le z∈Abe he limi o {T2nx}, hen
d∗(z, Tz) = lim
n→∞ d∗(T2nx, Tz)≤lim
n→∞ d∗(z, T2n−1x)
9
P oposi ion 4.14 The pai (A0, B0)is a nonemp y, closed and con ex pai in X. Fu he mo e,
each poin b∈B0can be joined h ough a geodesic segmen o leng h d= dis (A, B) o i s p oximinal
poin b−hin A0and ice e sa.
P oo . Tha hey a e closed ollows in a s aigh o wa d way om hei de ini ion and he ac
ha Aand Ba e bo h closed. The ac ha A0and B0a e nonemp y also ollows in a simila way
o he linea case unde he assump ion o e lexi i y, since i is a e y well-known ac (see [6, 10])
ha dec easing sequences o nonemp y bounded closed and con ex subse s o a CAT(0) space ha e
nonemp y in e sec ion. Finally, he con exi y o he se s A0and B0 ollows om he con exi y o
he me ic o CAT(0) spaces (see P oposi ion 2.2 in Chap e II.2 o [2]). 2
The nex hing we need o do is o de ine he semime ic d1on B0. We will de ine i in such a
way ha a hi d se C0is no needed.
De ini ion 4.15 We de ine he unc ion d1:B0×B0→[0,∞)by
d1(x, y) = in { > 0 : y∈B(x−h, d + )and y−h∈B(x, d + )},
whe e d=dis (A, B).
Rema k 4.16 No ice ha De ini ion 4.5 and De ini ion 4.15 coincide in linea spaces.
Theo em 4.17 The semime ic d1coincides wi h he me ic dinduced by Xon B0.
P oo . This esul ollows as an easy applica ion o The Fla Quad ila e al Theo em ([2, p.181]).
Indeed, conside he ou poin x, y, x −hand y−h. Then x( espec i ely, y) is he p oximinal
poin o x−h( ep., y−h) in B0, and ice e sa. In consequence, he angles ∠x(x−h, y), ∠y(x, y −
h), ∠y−h(x−h, y) and ∠x−h(y−h, x) a e all g ea e han o equal o π/2. The e o e he Fla
Quad ila e al Theo em implies ha he con ex hull o he poin s x, y, x −hand y−his isome ic
o a ec angle in he 2-dimensional Euclidean space. Now, by he Py hago ean heo em, i is
immedia e o deduce ha
B1(x, pd2+ 2−d) = B0∩B(x−h, pd2+ 2) = B0∩B(x, ),
as we wan ed o p oo . 2
We close his appendix by obse ing ha i is also possible o show ha he mapping T′(b) =
Tb +h o b∈B0is ac ually a con ac ion. To see his we jus need o p oceed as in he p oo o
Theo em 4.10 and ecall, a he p ope momen , ha he con ex hull o he poin s x, y, x −hand
y−his ac ually a ec angle.
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Depa amen o de An´alisis Ma em´a ico
Facul ad de Ma em´a icas
Uni e sidad de Se illa
P.O.Box: 1160
41080-Se illa
emails: [email protected], au o a [email protected]
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