Fixed points, selections and common fixed points for nonexpansive-type mappings

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. 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