Fixed poin s, selec ions and common ixed poin s o
nonexpansi e- ype mappings
Ra a Esp´ınola1, Pepa Lo enzo1, Ad iana Nicolae2
1Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa, P.O. Box 1160, 41080 Se illa, Spain
2Depa men o Ma hema ics, Babe¸s-Bolyai Uni e si y, Kog˘alniceanu 1, 400084, Cluj-Napoca, Romania
Manusc ip co espondence:
Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa,
P.O.Box 1160, 41080 Se illa, Spain
email: [email p o ec ed]
Phone: 0034 954 559 560
Fax: 0034 954 557 972
Email add esses: [email p o ec ed] (R. Esp´ınola), [email p o ec ed] (P. Lo enzo), [email p o ec ed]cluj. o
(A. Nicolae)
1
Abs ac
We s udy he exis ence o ixed poin s in he con ex o uni o mly con ex geodesic
me ic spaces, hype con ex spaces and Banach spaces o single and mul i alued
mappings sa is ying condi ions ha gene alize he concep o nonexpansi i y. Be-
sides, we use he ixed poin heo ems p o ed he e o gi e common ixed poin esul s
o commu ing mappings.
Key-wo ds: ixed poin , selec ion o mul i unc ions, gene alized nonexpansi e map-
pings, commu ing mappings, me ic space, Banach space.
1 In oduc ion
In [26], T. Suzuki ex ends he concep o single alued nonexpansi e mapping in he ol-
lowing way: a mapping de ined on a subse Ko a Banach space is said o sa is y con-
di ion (C) i o x, y ∈Kwi h (1/2)kx− (x)k≤kx−yk, hen k (x)− (y)k≤kx−yk.
T. Suzuki [26] p o es some basic p ope ies and gi es ixed poin heo ems and con-
e gence esul s o mappings sa is ying condi ion (C). Following [26], A. Razani and
H. Salahi a d [23] s a e pa o T. Suzuki’s [26] esul s in he con ex o a comple e
CAT(0) space and gene alize condi ion (C) o he mul i alued case: a mul i alued map-
ping Tde ined on subse o a CAT(0) space is said o sa is y condi ion (C) i o each
x, y ∈Kand ux∈T(x) wi h (1/2)d(x, ux)≤d(x, y) he e exis s uy∈T(y) such ha
d(ux, uy)≤d(x, y). This condi ion is used in [23] o p o e a ixed poin heo em o
mul i alued mappings and some common ixed poin esul s. Mo i a ed by he esul s
in [26], J. Ga c´ıa-False , E. Llo ens-Fus e and T. Suzuki conside in [7] wo gene aliza-
ions in he single alued case o condi ion (C) gi ing examples and es ablishing ixed
poin esul s.
The pu pose o his pape is o s udy condi ion (C) o mul i alued mappings in he
con ex o geodesic me ic spaces (wi h special a en ion o he case o R- ees) and Ba-
nach spaces, and condi ion (C) o single alued mappings in he con ex o hype con ex
spaces. A e some p elimina y con en s in Sec ion 2, we begin Sec ion 3 by s udying he
mul i alued case in geodesic spaces. We assume condi ion (C) o mul i alued mappings
as in [23] whe e di e en esul s in his di ec ion we e ob ained o CAT(0) spaces. In
ou wo k, we de i e a echnical lemma (Lemma 3.2) which is a mul i alued e sion o
he key ac which is behind he main esul s in [7, 26]. Ou esul s a e i s ob ained
o as gene al as comple e uni o mly con ex geodesic spaces and hen pa icula ized o
mo e p ecise geome ies. Since CAT(0) spaces a e a pa icula class o uni o mly con ex
geodesic spaces, we ob ain mo e gene al esul s han hose om [23]. Mo eo e , hanks
mainly o Lemma 3.2, we ill in a gap in he p oo o he main mul i alued esul in [23].
We con inue Sec ion 3 by in oducing a new condi ion o mul i alued mappings in he
spi i o (C). We gi e examples showing ha his condi ion is ac ually weake han con-
di ion (C) and p o e a selec ion heo em in R- ees o mappings sa is ying his newly
in oduced condi ion om whe e a s onge ixed poin esul o mul i alued mappings
ollows. This selec ion esul esembles a e y impo an one, see o ins ance [12, 25], o
hype con ex spaces (no ice, see [14], ha comple e R- ees a e hype con ex) al hough
he app oach he e is comple ely di e en as he p oo elies on e y pa icula p ope ies
o R- ees a he han on hype con exi y. I is wo hwhile o poin ou ha R- ees ind
2
a lo o applica ions in di e en a eas as, o ins ance, he indexing o in o ma ion o
phylogene ics. We close Sec ion 3 wi h an appendix whe e we s udy he exis ence o
ixed poin s o single alued mappings wi h p ope y (C) in hype con ex me ic spaces.
I is e y well-known (see [17, Chap e 13]) ha nonexpansi e sel -mappings de ined on
nonemp y bounded and closed hype con ex spaces ha e ixed poin s. The e o e i is
na u al o wonde abou his p oblem o mappings wi h condi ion (C). We i s s udy
he compac case p o iding a posi i e answe . Fo he mo e gene al case we need o
in oduce a new condi ion on he mapping unde conside a ion. In pa icula i is shown
ha a 2-lipschi zian sel -mapping wi h condi ion (C) de ined on a nonemp y closed and
bounded hype con ex space has a ixed poin . This esul is signi ican among he class
o known esul s o mappings wi h condi ion (C) since i is he i s one wi hou com-
pac ness condi ions o which nei he he uniqueness o asymp o ic cen e s no any hing
simila o he Opial p ope y is equi ed (see Sec ions 2 and 4 o de ini ions). The e o e,
his esul ollows h ough a comple ely new app oach compa ed o hose in [7, 23, 26]
and implies new esul s e en, o ins ance, in injec i e Banach spaces.
In Sec ion 4 we e isi he classical heo y o nonexpansi e mul i alued mappings on
Banach spaces o s udy i unde condi ion (C). We show he exis ence o ixed poin s o
such a mapping in a Banach space wi h he Opial p ope y. The me hod o asymp o ic
cen e s allows us o es ablish he same esul in a uni o mly con ex in e e y di ec ion
(UCED) Banach space. Mo eo e , i we also assume he con inui y o he mapping we
can p o e he exis ence o ixed poin s in a Banach space o which he asymp o ic cen e
o a bounded sequence wi h espec o a bounded closed con ex subse is nonemp y and
compac , ha is, a coun e pa o he Ki k-Massa heo em. Finally, in Sec ion 5, we
appeal o he ixed poin heo ems p o ed in his pape in o de o gi e some common
ixed poin esul s o commu ing mappings.
2 P elimina ies
Le (X, d) be a me ic space. A geodesic pa h om x o yis a mapping c: [0, l]⊆R→X
wi h c(0) = x, c(l) = yand d(c( ), c( 0)) = | − 0| o e e y , 0∈[0, l]. The image
c([0, l]) o c o ms a geodesic segmen which joins xand yand is no necessa ily unique.
I no con usion a ises, we will use [x, y] o deno e a geodesic segmen joining xand y.
(X, d) is a (uniquely) geodesic space i e e y wo poin s x, y ∈Xcan be joined by a
(unique) geodesic pa h. A poin z∈Xbelongs o he geodesic segmen [x, y] i and only
i he e exis s ∈[0,1] such ha d(z, x) = d(x, y) and d(z, y) = (1 − )d(x, y), and we
will w i e z= (1 − )x+ y o simplici y. A subse Ko Xis con ex i i con ains any
geodesic segmen ha joins e e y wo poin s o i .
In a geodesic space (X, d), he me ic d:X×X→Ris con ex i o any x, y, z ∈X
one has
d(x, (1 − )y+ z)≤(1 − )d(x, y) + d(x, z) o all ∈[0,1].
A geodesic space which me ic is con ex will be e e ed as a space wi h con ex me ic.
A i ial example o a uniquely geodesic space wi h con ex me ic is a s ic ly con ex
Banach space. Fo mo e de ails abou geodesic me ic spaces one may check [2].
A geodesic space (X, d) is uni o mly con ex i o any > 0 and ∈(0,2] he e exis s
3
δ∈(0,1] such ha i a, x, y ∈Xwi h d(x, a)≤ ,d(y, a)≤ and d(x, y)≥ hen
d(1
2x+1
2y, a)≤(1 −δ) .
F om he de ini ion, i is easy o see ha uni o mly con ex me ic spaces a e uniquely
geodesic.
A mapping δ: (0,∞)×(0,2] →(0,1] p o iding such a δ=δ( , ) o a gi en > 0 and
∈(0,2] is called a modulus o uni o m con exi y. The mapping δis mono one ( esp.
lowe semi-con inuous om he igh ) i o e e y ixed i dec eases ( esp. is lowe
semi-con inuous om he igh ) wi h espec o (see also [5], [18]). CAT(0) spaces in
he sense o G omo (see [2]) a e uni o mly con ex me ic spaces wi h con ex me ic.
Le (X, d) be a me ic space and le (xn)n∈Nbe a bounded sequence in X. Fo x∈X,
de ine (x, (xn)) = lim supn→∞ d(x, xn). The asymp o ic adius o (xn)n∈Nis gi en by
((xn)) = in { (x, (xn)) : x∈X},
and he asymp o ic cen e o (xn)n∈Nis he se
A((xn)) = {x∈X: (x, (xn)) = ((xn))}.
Th oughou his pape we will deno e a uni o mly con ex me ic space wi h mono one
(o lowe semi-con inuous om he igh ) modulus o uni o m con exi y as a UC space.
In [5], he au ho s p o e ha e e y bounded sequence in a UC space has a unique
asymp o ic cen e .
A bounded sequence (xn)n∈Nin a comple e UC space is egula i ((xn)) = ((xnk))
o e e y subsequence (xnk)k∈No (xn)n∈N. I is known ha in a Banach space e e y
bounded sequence con ains a egula subsequence (see, o ins ance, [17], Chap e 2,
Lemma 5.2). Since he p oo has a me ic na u e we can conclude ha e e y bounded
sequence (xn)n∈Nin a comple e UC space has a egula subsequence (xnk)k∈Nand hus
e e y subsequence o (xnk)k∈Nhas he same asymp o ic cen e as (xnk)k∈N.
Le (X, d) be a me ic space. Taking z∈Xand > 0 we deno e he closed ball
cen e ed a zwi h adius by e
B(z, ). Gi en Ya nonemp y subse o X, we de ine he
dis ance o a poin z∈X o Yby dis (z, Y ) = in y∈Yd(z, y).The me ic p ojec ion (o
nea es poin mapping)PYon o Yis he mapping
PY(z) = {y∈Y:d(z, y) = dis (z, Y )}, o e e y z∈X.
I Yis addi ionally bounded, he diame e o Yis gi en by diamY= supx,y∈Yd(x, y).
In his pape we also conside he ollowing amilies o se s:
P(X) = {Y⊆X:Yis nonemp y},
Pb(X) = {Y⊆X:Yis nonemp y and bounded},
Pb,c (X) = {Y⊆X:Yis nonemp y, bounded and con ex},
Pcl,c (X) = {Y⊆X:Yis nonemp y, closed and con ex},
Pb,cl,c (X) = {Y⊆X:Yis nonemp y, bounded, closed and con ex},
4
Pcp(X) = {Y⊆X:Yis nonemp y and compac },
Pcp,c (X) = {Y⊆X:Yis nonemp y, compac and con ex}.
A me ic space (X, d) is me ically con ex i o any wo dis inc poin s x, y ∈X
and any α, β > 0 such ha d(x, y) = α+β he e exis s z∈Xwi h d(x, z) = αand
d(y, z) = β.Xhas he bina y in e sec ion p ope y i Ti∈Ie
Bi6=∅ o e e y collec ion o
balls ( e
Bi)i∈Isuch ha any wo o hese balls in e sec .
A me ic space (X, d) is hype con ex i Ti∈Ie
B(xi, i)6=∅ o e e y collec ion o poin s
(xi)i∈Iin Xand posi i e numbe s ( i)i∈Isuch ha d(xi, xj)≤ i+ j o any i, j ∈
I. Hype con exi y is equi alen o he bina y in e sec ion p ope y and he me ic
con exi y. Mo e abou hype con ex spaces can be ound in [1, 12, 25] o in Chap e 13
o [17].
Gi en (X, d) a me ic space and A⊆X, he numbe x(A) = supy∈Ad(x, y) is called
he adius o A ela i e o x∈X. The adius o Ais (A) = in x∈X x(A), he cen e
o Ais he se C(A) = {x∈X: x(A) = (A)}and he admissible co e o Ais de ined
by co (A) = T{e
B:e
Bis a closed ball and A⊆e
B}. The se Ais said o be admissible i
A= co (A). Fo Xa hype con ex space and A⊆X, co (A) = Tx∈Xe
B(x, x(A)) and
diam(A)=2 (A) ( o de ails see Chap e 13 o [17]).
An R- ee is a uniquely geodesic me ic space Xsuch ha i [y, x]∩[x, z] = {x} hen
[y, x]∪[x, z]=[y, z] o each x, y, z ∈X. F om he de ini ion i immedia ely ollows ha
i x, y, z ∈X, hen [x, y]∩[x, z]=[x, w] o some w∈X. Likewise, i Kis a closed
and con ex subse o an R- ee X, hen o e e y x∈X,PK(x) is a single on and o
any y∈K,d(x, y) = d(x, PK(x)) + d(PK(x), y). A s anda d example o an R- ee is R2
endowed wi h he so-called i e me ic. Fo x= (x1, x2), y = (y1, y2)∈R2, he i e
me ic (deno ed by ρ) is de ined by
ρ(x, y) = |x2−y2|i x1=y1,
|x2|+|y2|+|x1−y1|o he wise.
I is known ha R- ees a e CAT(0) spaces and ha a me ic space is a comple e R- ee
i and only i i is hype con ex and has unique geodesic segmen s (see [14]). Mo e abou
he ixed poin heo y in R- ees can be ound in [4, 15, 21, 22].
In [26], T. Suzuki conside ed he ollowing gene alized amily o nonexpansi e map-
pings in he se ing o a Banach space. We will use in he sequel he no m no a ion, bu
he same de ini ions also hold when wo king in he me ic se ing (na u ally, he no m
will be eplaced by he dis ance).
De ini ion 2.1. Le Xbe a Banach space, K∈P(X)and :K→X. Then sa is ies
condi ion (C)i
1
2kx− (x)k ≤ kx−yk=⇒ k (x)− (y)k ≤ kx−yk,
o all x, y ∈K.
Ob iously, e e y nonexpansi e mapping mee s condi ion (C). We nex summa ize
some o he basic p ope ies p o ed in [26] in ela ion o hese mappings. The p oo s
o hese esul s a e me ic in na u e so he p ope ies also apply in he me ic case.
Th oughou his pape we deno e he se o ixed poin s o a mapping by Fix( ).
5

Lemma 2.2. Le Xbe a Banach space and K∈P(X). Assume ha he mapping
:K→Xsa is ies condi ion (C). Then o each x, y ∈K,
(i) i z∈Fix( ), hen kz− (x)k≤kz−xk, ha is, is quasinonexpansi e;
(ii) k (x)− (y)k ≤ kx−yko k 2(x)− (y)k≤k (x)−yk;
(iii) kx− (y)k ≤ 3k (x)−xk+kx−yk.
Using hese p ope ies, T. Suzuki [26] p o es ixed poin heo ems o mappings
sa is ying condi ion (C).
In [7], he au ho s s udy wo gene aliza ions o condi ion (C) gi ing examples and
es ablishing ixed poin esul s. One o hese condi ions is he ollowing.
De ini ion 2.3. Le Xbe a Banach space, K∈P(X), :K→Xand µ≥1. The
mapping sa is ies condi ion (Eµ)i o all x, y ∈K,
kx− (y)k ≤ µk (x)−xk+kx−yk.
Lemma 2.2, (iii) yields ha condi ion (C) implies (E3), bu Example 3 o [7] shows
ha (E3) does no imply (C). O he examples o di e en alues o µa e s udied in
[7].
In he nex sec ions we will make use o he lemma below which is a special case o
P oposi ion 2 in [9].
Lemma 2.4. Le Xbe a geodesic me ic space wi h con ex me ic, α∈(0,1) and
(xn)n∈Nand (yn)n∈Nbounded sequences in Xsuch ha xn+1 = (1 −α)xn+αynand
d(yn+1, yn)≤d(xn+1, xn) o e e y n∈N. Then limn→∞ d(xn, yn)=0.
The ollowing wo heo ems we e p o ed in [23], bu in he se ing o a comple e
CAT(0) space. I is easy o see ha hese esul s hold in mo e gene al con ex s. We will
o mula e he i s esul in he amewo k o a uniquely geodesic me ic space.
Theo em 2.5. Le Xbe a uniquely geodesic me ic space and K∈Pcl,c (X). Suppose
:K→Ksa is ies condi ion (C)and Fix( )6=∅. Then Fix( )is closed and con ex.
The p oo o he second heo em only equi es he uniqueness o he asymp o ic cen e
and he con exi y o he me ic. This is why we s a e his esul unde he hypo hesis
o a comple e UC space wi h con ex me ic.
Theo em 2.6. Le Xbe a comple e UC space wi h con ex me ic and suppose K∈
Pb,cl,c (X). I :K→Ksa is ies condi ion (C) hen Fix( )is nonemp y, closed and
con ex.
In [23], he au ho s also ex end Suzuki’s [26] condi ion (C) o he mul i alued case
in he ollowing way.
De ini ion 2.7. Le Xbe a me ic space and K∈P(X). A mapping T:K→P(X)is
said o sa is y condi ion (C)i o each x, y ∈Kand ux∈T(x)such ha
1
2d(x, ux)≤d(x, y),
he e exis s uy∈T(y)such ha
d(ux, uy)≤d(x, y).
6
The abo e condi ion is used in [23] o gi e a ixed poin heo em o mul i alued
mappings and some common ixed poin esul s.
In he es o his pape we use condi ion (C) o bo h single and mul i alued map-
pings wi h he con ex dis inguishing be ween he wo cases. The same also holds o
o he condi ions we make use o .
3 Fixed poin s and selec ions in geodesic spaces
In his sec ion we s udy he mul i alued e sion o mappings wi h condi ion (C) in
geodesic me ic spaces. Following he single alued case, we in oduce he nex condi ion
and p o e ha o µ= 3 i is a gene aliza ion o condi ion (C).
De ini ion 3.1. Le Xbe a me ic space, K∈P(X),T:K→P(X)and µ≥1.
The mapping Tsa is ies condi ion (Eµ)i o each x, y ∈Kand ux∈T(x) he e exis s
uy∈T(y)such ha
d(x, uy)≤µd(x, ux) + d(x, y).
We p o e nex ha a mul i alued mapping which sa is ies condi ion (C) also sa is ies
(E3). This p ope y will cons i u e a key ool in p o ing ou esul s.
Lemma 3.2. Le Xbe a me ic space, K∈P(X)and le T:K→P(K)sa is y
condi ion (C). Then Tsa is ies condi ion (E3).
P oo . Le x, y ∈Kand ux∈T(x). Because (1/2)d(x, ux)≤d(x, ux) he e exis s
x∈T(ux) such ha
d(ux, x)≤d(x, ux).(1)
We p o e ha ei he 1
2d(x, ux)≤d(x, y) (2)
o 1
2d(ux, x)≤d(ux, y) (3)
holds. Suppose (1/2)d(x, ux)> d(x, y) and (1/2)d(ux, x)> d(ux, y). Then, using (1)
we ob ain he ollowing con adic ion
d(x, ux)≤d(x, y) + d(y, ux)<1
2d(x, ux) + 1
2d(ux, x)≤d(x, ux).
Hence, i (2) holds, hen he e exis s uy∈T(y) such ha d(ux, uy)≤d(x, y), so
d(x, uy)≤d(x, ux) + d(ux, uy)≤d(x, ux) + d(x, y).
I (3) holds, hen he e exis s uy∈T(y) such ha d( x, uy)≤d(ux, y). Using again (1)
we ha e ha
d(x, uy)≤d(x, ux) + d(ux, x) + d( x, uy)≤2d(x, ux) + d(ux, y)≤3d(x, ux) + d(x, y).
Thus, he inequali y holds in each o he wo cases and we a e done.
7
De ini ion 3.3. Le Xbe a me ic space, K∈P(X)and T:K→P(X). We say ha
(xn)n∈N⊆Kis an app oxima e ixed poin sequence o he mapping Ti o each n∈N
he e exis s yn∈T(xn)such ha limn→∞ d(xn, yn)=0.
The nex esul p o ides an app oxima e ixed poin sequence o a mul i alued
mapping sa is ying condi ion (C). We use his esul in he es o he pape because
many o ou p oo s ely on i .
P oposi ion 3.4. Le Xbe geodesic me ic space wi h con ex me ic, K∈Pb,c (X)
and T:K→P(K). I Tsa is ies condi ion (C), hen Thas an app oxima e ixed poin
sequence.
P oo . Le x1∈K,y1∈T(x1) and ake x2= (1/2)x1+ (1/2)y1. Then (1/2)d(x1, y1) =
d(x1, x2) so, by condi ion (C), he e exis s y2∈T(x2) such ha d(y1, y2)≤d(x1, x2).
Con inuing in his ein, we can build he sequences (xn)n∈Nand (yn)n∈Nsuch ha
yn∈T(xn), xn+1 = (1/2)xn+ (1/2)ynand d(yn+1, yn)≤d(xn+1, xn) o e e y n∈N.
Using Lemma 2.4 we ob ain ha limn→∞ d(xn, yn) = 0.
Ou i s ixed poin esul o mul i alued mappings is gi en o sel -mappings on a
compac se .
Theo em 3.5. Le Xbe a geodesic space wi h con ex me ic and K∈Pcp,c (X). Sup-
pose T:K→Pcl(K)sa is ies condi ion (C). Then Fix(T)6=∅.
P oo . By P oposi ion 3.4, he e exis wo sequences (xn)n∈Nand (yn)n∈Nin Ksuch ha
yn∈T(xn) and limn→∞ d(xn, yn) = 0. Since Kis compac , we can ind a subsequence
(xnk)k∈No (xn)n∈Nsuch ha (xnk)k∈Ncon e ges o some x∈K.
Using Lemma 3.2, we ha e ha o all k∈N
dis (xnk, T(x)) ≤3d(xnk, ynk) + d(xnk, x).
Taking he limi as k→ ∞ we ob ain ha dis (x, T (x)) = 0. Since T(x) is closed i
ollows ha x∈T(x).
In he ollowing heo em we mo e he compac ness condi ion om he domain o he
images o he mapping. This heo em is ac ually an ex ension o Theo em 3.2 o [23] in
he con ex o a comple e UC space wi h con ex me ic. We also emo e he con exi y
condi ion on he image se s o he mapping. Mo eo e , we ob ain ou esul s in a simple
way as a consequence o Lemma 3.2 which a oids o go h ough a delica e poin in he
p oo o Theo em 3.2 o [23].
Theo em 3.6. Le Xbe a comple e UC space wi h con ex me ic and K∈Pb,cl,c (X).
Suppose T:K→Pcp(K)sa is ies condi ion (C). Then Fix(T)6=∅.
P oo . By P oposi ion 3.4, we can ind he sequences (xn)n∈Nand (yn)n∈Nin Ksuch
ha yn∈T(xn) and limn→∞ d(xn, yn) = 0. As explained in Sec ion 2, we may suppose
ha (xn)n∈Nis egula (o he wise choose a egula subsequence o i ). Deno e he unique
asymp o ic cen e o (xn)n∈Nby x. Le n∈N. Applying Lemma 3.2 o xn, x and yn
espec i ely i ollows ha he e exis s zn∈T(x) such ha
d(xn, zn)≤3d(xn, yn) + d(xn, x).
8
Le (znk)k∈Nbe a subsequence o (zn)n∈N ha con e ges o some z∈T(x). Then, o
each k∈N,
d(xnk, z)≤d(xnk, znk) + d(znk, z)≤3d(xnk, ynk) + d(xnk, x) + d(znk, z).
Taking he supe io limi as k→ ∞ and knowing ha he asymp o ic cen e o (xnk)k∈N
is p ecisely xwe ob ain ha x=z∈T(x). Hence, he p oo is comple e.
Rema k 3.7. F om he abo e p oo i is immedia e ha in Theo em 3.6 we can d op
he con exi y o he me ic and assume ins ead ha he mapping admi s an app oxima e
ixed poin sequence.
In he nex esul we will conside he ollowing new condi ion o mul i alued map-
pings which will be shown o be weake han condi ion (C).
De ini ion 3.8. Le Xbe a me ic space, K∈P(X)and T:K→P(X). The mapping
Tsa is ies condi ion (C0)i o each x, y ∈Kand ux∈T(x)wi h
d(x, ux) = dis (x, T(x)) and 1
2d(x, ux)≤d(x, y),
he e exis s uy∈T(y)such ha
d(ux, uy)≤d(x, y).
We p o e nex a selec ion heo em in R- ees o mul i alued mappings sa is ying
condi ion (C0) and analyze a e wa ds he ela ion o (C0) o (C) and (E3) espec i ely.
Theo em 3.9. Le Xbe an R- ee, K∈P(X)and T:K→Pcl,c (X)a mapping which
sa is ies (C0). Then he mapping :K→Xde ined by (x) = PT(x)(x) o each x∈K
is a selec ion o T ha sa is ies condi ion (C).
P oo . No ice ha he p ope ies o R- ees (see Sec ion 2) gua an ee ha is well-
de ined. Le x, y ∈Ksuch ha (x)6= (y) and (1/2)d(x, (x)) ≤d(x, y). Conside
p(x) = PT(y)( (x)) and p(y) = PT(x)( (y)).
Fi s , suppose p(x)6= (y) and p(y)6= (x). Since p(x) is he p ojec ion o (x) on o
T(y) i ollows ha
d( (x), (y)) = d( (x), p(x)) + d(p(x), (y)),
i.e., p(x)∈[ (x), (y)]. Since T(y) is con ex, [p(x), (y)] ⊆T(y). This implies
[ (x), (y)] ∩[ (y), y] = { (y)}because o he wise he minimali y o (y) would be
con adic ed. Thus, (y)∈[ (x), y]. Simila ly, (x)∈[ (y), x]. Then (x), (y)∈[x, y]
(o he wise supposing o example ha z∈[x, (y)]∩[ (y), y] wi h z6= (y) we ha e ha
(x)∈[z, (y)] and (y)∈[z, (x)] which is alse). The e o e, d( (x), (y)) ≤d(x, y).
In ac , d( (x), (y)) = d(x, y)−dis (x, T(x)) −dis (y, T(y)).
Now assume p(x) = (y). Then d( (x), (y)) = dis ( (x), T(y)) and so, by condi ion
(C0),
d( (x), (y)) = dis ( (x), T(y)) ≤d(x, y).
9
Recall ha Xis said o be uni o mly con ex in e e y di ec ion (UCED, in sho ) i
δz()>0 o all  > 0 and z∈Xwi h kzk= 1, whe e δz() is he modulus o con exi y
o Xin he di ec ion zde ined by
δz() = in 1−1
2kx+yk:kxk ≤ 1,kyk ≤ 1, x −y=z.
Ob iously, uni o mly con ex Banach spaces a e UCED. I is known ha in a UCED
Banach space, he asymp o ic cen e o a sequence wi h espec o a weakly compac
con ex se is a single on. Hence, e e y egula sequence wi h espec o such a se is
asymp o ically uni o m.
Theo em 4.6. Le Kbe a weakly compac and con ex subse o a UCED Banach space
X. Suppose T:K→Pcp(K)is a mapping sa is ying condi ion (C). Then Fix(T)6=∅.
P oo . Le (xn)n∈Nand (yn)n∈Nbe wo sequences in Ksuch ha yn∈T(xn) and
limn→∞ kxn−ynk= 0. Wi hou loss o gene ali y, me may assume ha (xn)n∈Nis
egula wi h espec o K. Le zbe he unique poin in he asymp o ic cen e o (xn)n∈N
in K. By Lemma 3.2, o each n∈N he e exis s n∈T(z) such ha
kxn− nk ≤ 3kxn−ynk+kxn−zk.
F om he compac ness o T(z) we can assume ha ( n)n∈Ncon e ges o a poin ∈T(z).
I ollows ha
lim sup
n→∞
kxn− k ≤ lim sup
n→∞
kxn−zk.
Since (xn)n∈Nis egula we conclude ha =z∈T(z).
Dhompongsa e al. [3] ha e ecen ly p o ed he Tin a iance o he asymp o ic cen e
in Ko an app oxima e ixed poin sequence o T, when Tis a single alued mapping
sa is ying condi ion (C). We now s a e a esul which can be seen as an adap a ion o
his ac o he mul i alued case.
P oposi ion 4.7. Le Kbe a weakly compac subse o a Banach space X. Suppose
T:K→Pcp(K)sa is ies condi ion (C)and (xn)n∈Nis an app oxima e ixed poin
sequence o T. Then, he e exis s a subsequence (zn)n∈No (xn)n∈Nsuch ha
T(x)∩A6=∅, o all x∈A:= A(K, (zn)).
P oo . Since Tis a sel -mapping we can build a subsequence (zn)n∈No (xn)n∈Nwhich is
egula and asymp o ically uni o m wi h espec o K. Deno e (K, (zn)) by . Taking
any x∈Aand ollowing he same a gumen as in he p oo o he abo e heo em we
ob ain a sequence ( n)n∈N⊆T(x) no m con e gen o a poin ∈T(x) such ha
lim sup
n→∞
kxn− k ≤ lim sup
n→∞
kxn−xk= .
This shows ha ∈A, and so T(x)∩A6=∅.
Now we a e eady o p o e an analogous esul o he Ki k-Massa heo em [16] o
mappings sa is ying condi ion (C).
16

Theo em 4.8. Le Kbe a bounded, closed and con ex subse o a Banach space X
and T:K→Pcp,c (K)be a con inuous mapping wi h espec o he Pompeiu-Hausdo
dis ance sa is ying condi ion (C). Suppose ha each sequence in Khas a nonemp y and
compac asymp o ic cen e ela i e o K. Then Fix(T)6=∅.
P oo . Acco ding o he p e ious p oposi ion we can ake a sequence (xn)n∈Nin Ksuch
ha
T(x)∩A6=∅, o all x∈A:= A(K, (xn)).
Now we de ine he mapping ˜
T:A→Pcp,c (A) by ˜
T(x) = T(x)∩A. Since Tis con inuous,
om P oposi ion 2.45 in [11] we know ha he mapping ˜
Tis uppe semi-con inuous.
Since T(x)∩Ais a compac con ex se we can apply he Kaku ani-Bohnenblus -Ka lin
heo em (see [10]) o ob ain a ixed poin o ˜
Tand hence o T.
Rema k 4.9. Recall ha a mul i alued mapping T:K→Pb(X) is said o be nonex-
pansi e i
H(T(x), T(y)) ≤ kx−yk o all x, y ∈K,
whe e Hdeno es he Pompeiu-Hausdo dis ance. I is wo h poin ing ou ha ano he
na u al ex ension o he Suzuki’s condi ion (C) o a mul i alued mapping T:K→P(X)
is he ollowing: o all x, y ∈K
1
2dis (x, T(x)) ≤ kx−yk=⇒H(T(x), T(y)) ≤ kx−yk.
Ob iously, a nonexpansi e mapping mee s he abo e condi ion. Howe e , i is no
clea i a mapping sa is ying he abo e condi ion also sa is ies (C). S ill, i T akes
compac alues is easy o see ha his new condi ion implies condi ion (C). Since in
ou heo ems Tis assumed o be compac alued, such esul s gene alize classical ixed
poin heo ems o mul i alued mappings (see [16],[19],[20]).
5 Common ixed poin s
In ou las sec ion we will apply some o he ixed poin heo ems s a ed in p e ious
sec ions o ob ain esul s on he exis ence o common ixed poin .
De ini ion 5.1. Le Xbe a me ic space and K∈P(X). Suppose :K→Kand
T:K→P(K). Then and Ta e commu ing mappings i (y)∈T( (x)) o all x∈K
and y∈T(x).
We s a by gi ing a lemma ha will cons i u e a main ool in p o ing ou esul s.
Lemma 5.2. Le Xbe a me ic space, K∈P(X), :K→Ksa is ying condi ion (C)
and wi h Fix( )6=∅. Supppose T:K→P(K)is such ha o e e y x, y ∈Fix( ),
he se PT(y)(x)is a single on. I and Tcommu e, hen PT(y)(x)∈Fix( ) o all
x, y ∈Fix( ).
P oo . Le x, y ∈Fix( ) and deno e PT(y)(x) by u. Because mee s condi ion (C) and
0 = (1/2)d(x, (x)) ≤d(x, u) we ob ain ha d(x, (u)) = d( (x), (u)) ≤d(x, u) =
dis (x, T(y)). Bu (u)∈T(y) because and Tcommu e, y∈Fix( ) and u∈T(y).
Hence, (u) = uand he conclusion ollows.
17
The ollowing heo em is an ex ension o Theo em 4.2 o [23] in he se ing o a UC
space wi h con ex me ic. No ice ha ou app oach is di e en in he second hal o he
p oo om ha o [23]. In pa icula , ou s ills a gap in he p oo o [23]. No ice also
ha his heo em ex ends some o he esul s in he heo y, see, o ins ance, [6, 24].
Theo em 5.3. Le Xbe a comple e UC space wi h con ex me ic and K∈Pb,cl,c (X).
Suppose :K→Kand T:K→Pcp,c (K)sa is y condi ion (C). I and Tcommu e,
hen he e exis s z∈Ksuch ha z= (z)∈T(z).
P oo . Using Theo em 2.6, i ollows ha Fix( ) is nonemp y, closed and con ex. Since
he se ing we wo k in is a UC space, he p ojec ion on o each compac and con ex se is
a single on. By Lemma 5.2, PT(x)(x)∈T(x)∩Fix( ) o each x∈Fix( ) and so we can
conside he mapping T(·)∩Fix( ) : Fix( )→Pcp(Fix( )). We show ha his mapping
sa is ies condi ion (C). Le x, y ∈Fix( ), ux∈T(x)∩Fix( ) such ha (1/2)d(x, ux)≤
d(x, y). Since T ul ills (C), he e exis s y∈T(y) such ha d(ux, y)≤d(x, y). Le uy
s and o PT(y)(ux). Acco ding o Lemma 5.2, uy∈T(y)∩Fix( ). I is also clea ha
d(ux, uy)≤d(ux, y)≤d(x, y). Thus, he mapping T(·)∩Fix( ) : Fix( )→Pcp(Fix( ))
sa is ies (C) which means, using Theo em 3.6, ha he e exis s z∈Ksuch ha z=
(z)∈T(z).
Likewise, one can p o e he ollowing esul in he amewo k o R- ees.
Theo em 5.4. Le Xbe a bounded comple e R- ee. Suppose :X→Xand T:X→
Pcl,c (X)sa is y condi ions (C)and (C0) espec i ely. I and Tcommu e, hen he e
exis s z∈Ksuch ha z= (z)∈T(z).
P oo . Acco ding o Theo em 2.6, i ollows ha Fix( ) is nonemp y, closed and con ex
(and so also hype con ex). This means ha Fix( ) is in i s own u n a comple e R- ee.
Since in an R- ee he p ojec ion on o each closed and con ex se is a single on we can
apply Lemma 5.2 and so T(x)∩Fix( )6=∅ o each x∈Fix( ). Now conside he
mapping T(·)∩Fix( ) : Fix( )→Pcl,c (Fix( )). We show ha his mapping sa is ies
condi ion (C0). Le x, y ∈Fix( ), ux∈T(x)∩Fix( ) such ha d(x, ux) = dis (x, T(x)∩
Fix( )) and (1/2)d(x, ux)≤d(x, y). Applying Lemma 5.2, PT(x)(x)∈T(x)∩Fix( )
which implies ha dis (x, T (x)) = dis (x, T(x)∩Fix( )), so d(x, ux) = dis (x, T(x)).
Because Tsa is ies (C0), he e exis s y∈T(y) such ha d(ux, y)≤d(x, y). Le uy
s and o PT(y)(ux). Acco ding o Lemma 5.2, uy∈T(y)∩Fix( ). I is also clea
ha d(ux, uy)≤d(ux, y)≤d(x, y). Thus, he mapping T(·)∩Fix( ) : Fix( )→
Pcl,c (Fix( )) sa is ies (C0) which means, using Co olla y 3.11, ha he e exis s z∈K
such ha z= (z)∈T(z).
6 Acknowledgmen s
We would like o hank A apol Kaewkhao o no icing and le ing us know abou a
mis ake in a p e ious e sion o his pape .
The esea ch o he i s wo au ho s was pa ially suppo ed by DGES, G an
MTM2009-10696-C02-01 and Jun a de Andaluc´ıa, G an FQM-127. The hi d au ho
was suppo ed by p og ams co- inanced by he Sec o al Ope a ional P og amme Human
Resou ces De elopmen , Con ac POS DRU 6/1.5/S/3 - “Doc o al s udies: h ough
18
science owa ds socie y”. She would also like o exp ess he app ecia ion o he Depa -
men o Ma hema ical Analysis and o he Ins i u e o Ma hema ics o he Uni e si y o
Se ille (IMUS) o hei suppo .
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