Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

RESEARCH Open Access Composi ions and con ex combina ions o asymp o ically egula i mly nonexpansi e mappings a e also asymp o ically egula Heinz H Bauschke 1* , Vic o ia Ma ín-Má quez 2 , Sa ah M Mo a 1 and Xian u Wang 1 * Co espondence: heinz. [email p o ec ed] 1 Ma hema ics, Uni e si y o B i ish Columbia, Kelowna, BC V1V 1V7, Canada Full lis o au ho in o ma ion is a ailable a he end o he a icle Abs ac Because o Min y’s classical co espondence be ween i mly nonexpansi e mappings and maximally mono one ope a o s, he no ion o a i mly nonexpansi e mapping has p o en o be o basic impo ance in ixed poin heo y, mono one ope a o heo y, and con ex op imiza ion. In his no e, we show ha i ini ely many i mly nonexpansi e mappings de ined on a eal Hilbe space a e gi en and each o hese mappings is asymp o ically egula , which is equi alen o saying ha hey ha e o “almos ha e” ixed poin s, hen he same is ue o hei composi ion. This signi ican ly gene alizes he esul by Bauschke om 2003 o he case o p ojec o s (nea es poin mappings). The p oo esides in a Hilbe p oduc space and i elies upon he B ezis-Ha aux ange app oxima ion esul . By wo king in a sui ably scaled Hilbe p oduc space, we also es ablish he asymp o ic egula i y o con ex combina ions. 2010 Ma hema ics Subjec Classi ica ion: P ima y 47H05, 47H09; Seconda y 47H10, 90C25. Keywo ds: asymp o ic egula i y, i mly nonexpansi e mapping, Hilbe space, maxi- mally mono one ope a o , nonexpansi e mapping, esol en , s ongly nonexpansi e mapping 1 In oduc ion and s anding assump ions Th oughou his a icle, Xis a eal Hilbe s p ace wi h inne p oduc ·,·  (1) and induced no m || ⋅||. We assume ha m∈ { 2, 3, 4, ... } and I:= { 1, 2, ...,m }. (2) Recall ha an ope a o T: X ®Xis i mly nonexpansi e (see, e.g., [1-3] o u he in o ma ion) i (∀xÎX)(∀yÎX)||Tx -Ty|| 2 ≤〈x-y, Tx -Ty〉and ha ase - alued ope a o A: X ⇉Xis maximally mono one i i is mono one,i.e., o all(x, x*)and(y, y*) in he g aph o A, we ha e 〈x - y, x* - y*〉≥0 and i he g aph o Acanno be p op- e ly enla ged wi hou des oying mono onici y (We shall w i e dom A={xÎX|Ax≠ Ø} o he domain o A, anA=A(X)=∪ xÎX Ax o he ange o A,andg A o he g aph o A.) These no ions a e equi alen (see [4,5]) in he sense ha i Ais maximally mono one, hen i s esol en J A : = (Id + A) -1 is i mly nonexpansi e, and i Tis i mly Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 © 2012 Bauschke e al; licensee Sp inge . This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (h p://c ea i ecommons.o g/licenses/by/2.0), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. nonexpansi e, hen T -1 - Id is maximally mono one. (He e and elsewhe e, Id deno es he iden i y ope a o on X.) The Min y pa ame iza ion (see [4] and also [[1], Rema k 23.22(ii)]) s a es ha i Ais maximally mono one, hen g A={ ( JAx,x−JAx ) |x∈X}.(3) In op imiza ion, one main p oblem is o ind ze os o maximally mono one ope a- o s– hese ze os may co espond o c i ical poin s o solu ions o op imiza ion p o- blems. In e ms o esol en s, he co esponding p oblem is ha o inding ixed poin s. Fo backg ound ma e ial in ixed poin heo y and mono one ope a o heo y, we e e he eade o [1-3,6-16]. The aim o his no e is o p o ide app oxima e ixed poin esul s o composi ions and con ex combina ions o ini ely many i mly nonexpansi e ope a o s. The i s main esul (Theo em 4.6) subs an ially ex ends a esul by Bauschke [17] on he composi ions o p ojec o s o he composi ion o i mly nonexpansi e map- pings. The second main esul (Theo em 5.5) ex ends a esul by Bauschke, Mo a and Wang [18] on he con ex combina ion o i mly nonexpansi e ope a o s om Eucli- dean o Hilbe space. The emainde o his sec ion p o ides he s anding assump ions used h oughou he a icle. E en hough he main esul s a e o mula ed in he gi en Hilbe space X, i will u n ou ha he key space o wo k in is he p oduc space Xm:= {x= ( xi ) i∈I| ( ∀i∈I ) xi∈X} . (4) This p oduc space con ains an embedding o he o iginal space X ia he diagonal subspace := {x= ( x ) i∈I|x∈X} . (5) We also assume ha we a e gi en m i mly nonexpansi e ope a o s T 1 ,..., T m ;equi a- len ly, m esol en s o maximally mono one ope a o s A 1 ,..., A m : (∀i∈I)Ti=JA i =(Id+Ai)−1is i mly nonexpansi e . (6) We now de ine a ious pe inen ope a o s ac ing on X m . We s a wi h he Ca e- sian p oduc ope a o s T:Xm→Xm: ( xi ) i∈I→ ( Tixi ) i∈ I (7) and A:Xm⇒Xm: ( xi ) i∈I→ ( Aixi ) i∈I . (8) Deno ing he iden i y on X m by Id, we obse e ha J A= ( Id +A ) −1=T1×···×Tm=T . (9) O cen al impo ance will be he cyclic igh -shi ope a o R :Xm→Xm: ( x1,x2...,xm ) → ( xm,x1,...,xm−1 ) (10) and o con enience we se M=Id −R . (11) Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 2 o 11 We also ix s ic ly posi i e con ex coe icien s (o weigh s) (l i ) iÎI , i.e., (∀i∈I)λi∈]0, 1[ and  i∈ I λi=1 . (12) Le us make X m in o he Hilbe p oduc space X:=Xm,wi hx,y= i∈I xi,yi.(13) The o hogonal complemen o Δwi h espec o his s anda d inne p oduc is known (see, e.g., [[1], P oposi ion 25.4(i)]) o be ⊥=  x=(xi)i∈I| i∈I xi=0 . (14) Finally, gi en a nonemp y closed con ex subse Co X, hep ojec o (nea es poin mapping) on o Cis deno ed by P C . I is well known o be i mly nonexpansi e. 2 P ope ies o he ope a o M In his sec ion, we collec se e al use ul p ope ies o he ope a o M, including i s Moo e-Pen ose in e se (see [19] and e.g., [[1], Sec ion 3.2] o u he in o ma ion.). To ha end, he ollowing esul –which is p obably pa o he olklo e–will u n ou o be use ul. P oposi ion 2.1 Le Y be a eal Hilbe space and le B be a con inuous linea ope a- o om X o Y wi h adjoin B*and such ha an B is closed. Then he Moo e-Pen ose in e se o B sa is ies B†=P a nB∗◦B−1◦P a nB . (15) P oo Take yÎY. De ine he co esponding se o leas squa es solu ions (see, e.g., [[1], P oposi ion 3.25]) by C:=B -1 (P an B y). Since an Bis closed, so is an B* (see, e. g., [[1], Co olla y 15.34]); hence, a U:= ( Ke B ) ⊥= an B∗= anB ∗ . Thus, C=B † y+ ke B=B † y+U ┴ . The e o e, since an B † = anB* (see, e.g., [[1], P oposi ion 3.28( )]), P U (C)=P U B † y=B † y, as claimed. Be o e we p esen a ious use ul p ope ies o M, le us ecall he no ion o a ec angula (which is also known as s a o 3* mono one, see [20]) ope a o . A mono one ope a o B: X⇉Xis ec angula i (∀(x,y∗)∈dom B× an B)sup ( z,z∗ ) ∈g Bx−z,z∗−y<+ ∞ . Theo em 2.2 De ine b L :⊥→X:y→ m−1  i =1 m−i mRi−1y . (16) Then he ollowing hold. (i) Mis con inuous, linea , and maximally mono one wi h dom M=X. (ii) Mis ec angula . (iii) ke M= ke M*=Δ. (i ) an M= an M*=Δ ┴ is closed. ( ) an L=Δ ┴ . ( i) M◦L=Id| ⊥ . Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 3 o 11 ( ii) M−1:X⇒X:y→  Ly +,i y∈⊥; ∅,o he wise . ( iii) M†=P⊥◦L◦P⊥=L◦P ⊥ . (ix) M†= m  k =1 m−(2k−1) 2mRk− 1 . P oo .(i):Clea ly,domM=Xand (∀xÎX)||Rx|| = ||x||. Thus, Ris nonexpansi e and he e o e M=Id -Ris maximally mono one (see, e.g., [[1], Example 20.27]). (ii): See [[1], Example 24.14] and [[17], S ep 3 in he p oo o Theo em 3.1] o wo di e en p oo s o he ec angula i y o M. (iii): The de ini ions o Mand Rand he ac ha R* is he cyclic le shi ope a o eadily imply ha ke M= ke M*=Δ. (i ), ( i), and ( ii): Le y=(y 1 , ..., y m )ÎX. Assume i s ha yÎ an M. Then he e exis s x=(x 1 ,...,x m ) such ha y 1 =x 1 -x m ,y 2 =x 2 -x 1 ,...,andy m =x m -x m-1 .I ollows ha ∑ iÎI y i = 0, i.e, yÎΔ ┴ by [[1], P oposi ion 25.4(i)]. Thus, an M ⊆ ⊥ . (17) Con e sely, assume now ha yÎΔ ┴ . Now se x:= Ly = m− 1  i =1 m−i mRi−1y . (18) I will be no a ionally con enien o w ap indices a ound, i.e., y m+1 =y 1 ,y 0 =y m and likewise. We hen ge (∀i∈I)xi=m−1 m yi+m−2 m yi−1+···+1 m yi+2 . (19) The e o e,  i ∈ I xi=m−1 m i ∈ I yi+m−2 m i ∈ I yi+···+1 m i ∈ I yi=m−1 2 i ∈ I yi=0 . (20) Thus xÎΔ ┴ and an L ⊆ ⊥.(21) Fu he mo e, (∀i∈I)xi−xi−1=m−1 m yi−1 m yi−1−1 m yi−2−···− 1 m yi+ 1 (22a) =yi−1 m  j ∈I yj=yi . (22b) Hence Mx =x-Rx =yand hus yÎ an M. Mo eo e , in iew o (iii), M−1 y =x+ke M=x+ . (23) We hus ha e shown ⊥⊆ an M . (24) Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 4 o 11 Combining (17) and (24), we ob ain an M=Δ ┴ . We hus ha e e i ied ( i), and ( ii). Since an Mis closed, so is an M* (by, e.g., [[1], Co olla y 15.34]). Thus (i ) holds. ( iii)&( ): We ha e seen in P oposi ion 2.1 ha M†=P a nM∗◦M−1◦P a nM . (25) Now le zÎX.Then,by(i ),y:= P a nMZ=P⊥Z∈ ⊥ .By( ii),M -1 y=Ly +Δ.So M † z=P an M∗ M −1 P an M z=P an M* M −1 y=P  ⊥ (Ly +)=P  ⊥ Ly =Ly =(L◦P  ⊥ ) z because an L⊆Δ ┴ by (21). Hence ( iii) holds. Fu he mo e, by (i ) and e.g., [[1], P oposi ion 3.28( )], an L= anL◦P  ⊥= anM†= anM∗= ⊥ and so ( ) holds. (ix): No e ha P  ⊥=Id −P  and ha P Δ =m -1 ∑ jÎI R j . Hence P⊥=Id −1 m j ∈I Rj . (26) Thus, by ( iii) and (16), M†=L◦P⊥=1 m m−1  i=1 (m−i)Ri−1◦⎛ ⎝ Id −1 m j∈I Rj ⎞ ⎠ (27) =1 m m−1  i=1 (m−i)Ri−1−1 m2 m−1  i=1 (m−i) j ∈I Ri+j−1 . (28) Re-a anging his exp ession in e ms o powe s o Rand simpli ying leads o M†=(Id −R)†= m  k =1 m−(2k−1) 2mRk−1 . (29) Rema k 2.3 Suppose ha ˜ L: ⊥→ X sa is ies M◦˜ L=Id| ⊥ Then M−1:X⇒X:y→   Ly+,i y∈⊥; ∅,o he wise . (30) One may show ha M†=P  ⊥◦˜ L◦P ⊥ and ha P  ⊥◦ ˜ L= L (see (16)). Conc e e choices o ˜ L and La e ⊥→X: ( y1,y2,...,ym ) → ( y1,y1+y2,...,y1+y2+y3+...+ym ); (31) howe e , he ange o he la e ope a o is no equal Δ ┴ whene e X≠{0}. Rema k 2.4 Deno ing he symme ic pa o Mby M+=1 2 M+1 2 M∗and de ining he quad a ic o m associa ed wi h Mby qM:x→1 2 x, Mx  , we no e ha [[17], P opo- si ion 2.3] implies ha c an M+= domq∗ M = ⊥ . Fac 2.5 (B ezis-Ha aux) (See [20] and also, e.g., [[1], Theo em 24.20].) Suppose A and B a e mono one ope a o s on X such ha A + B is maximally mono one, dom A⊆ dom B, and B is ec angula . Then in an(A+B)=in ( an A+ an B)and an(A+B)= an A+ anB. Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 5 o 11 Applying he B ezis-Ha aux esul o ou gi en ope a o s Aand M, we ob ain he ollowing. Co olla y 2.6 The ope a o A+Mis maximally mono one and an ( A+M ) =⊥+ an A P oo . Since each A i is maximally mono one and ecalling Theo em 2.2(i), we see ha Aand Ma e maximally mono one. On he o he hand, dom M=X. Thus, by he well known sum heo em o maximally mono one ope a o s (see, e.g., [[1], Co olla y 24.4 (i)]), A+Mis maximally mono one. Fu he mo e, by Theo em 2.2(ii) and (i ), Mis ec angula and an M=Δ ┴ . The esul he e o e ollows om Fac 2.5. 3 Composi ion We now use Co olla y 2.6 o s udy he composi ion. When m= 2, hen Theo em 3.1 ( ) also ollows om [[21], p. 124]. Theo em 3.1 Suppose ha ( ∀i∈I ) 0∈ an ( Id −Ti ) .Then he ollowing hold. (i) 0∈ an ( A+M ) . (ii) (∀ε>0)(∃(b, x)ÎX×X)||b|| ≤εand x=T(b+Rx). (iii) (∀ε>0)(∃(c, x)ÎX×X)||c|| ≤εand x=c+T(Rx). (i ) (∀ε>0)(∃xÎX)(∀iÎI)||T i-1 ···T 1 x m -T i T i-1 ···T 1 x m -x i-1 +x i || ≤(2i- 1)ε, whe e x 0 =x m . ( ) (∀ε>0)(∃xÎX)||x-T m T m-1 ···T 1 x|| ≤m 2 ε. P oo . (i): The assump ions and (3) imply ha ( ∀i∈I ) 0∈ an Ai. Hence, 0 ∈ an A . Ob iously, 0ÎΔ ┴ .I ollows ha 0 ∈ ⊥+ an A .Thus,byCo olla y2.6, 0∈ an ( A+M ) . (ii): Fix ε>0. In iew o (i), he e exis s xÎXand bÎXsuch ha ||b|| ≤εand b ÎAx +Mx. Hence b+Rx Î(Id +A)xand hus x=J A (b +Rx) =T(b +Rx). (iii): Le ε>0. By (ii), he e exis s (b, x)ÎX×Xsuch ha ||b|| ≤εand x=T(b + Rx). Se c=x-T(Rx) =T(b +Rx) -T(Rx). Then, since Tis nonexpansi e, ||c|| = ||T(b+Rx)-T(Rx)|| ≤||b|| ≤ε. (i ): Take ε>0. Then, by (iii), he e exis s xÎXand cÎXsuch ha ||c|| ≤εand x=c+T(Rx). Le iÎI. Then x i =c i +T i x i-1 . Since ||c i || ≤||c|| ≤εand T i is non- expansi e, we ha e  TiTi−1···T1x0−xi  ≤  TiTi−1···T1x0−Tixi−1  +  Tixi−1−xi  (32a) ≤  TiTi−1···T1x0−Tixi−1  +ε . (32b) We hus ob ain induc i ely TiTi−1···T1x0−xi≤iε.(33) Hence, Ti−1···T1x0−xi−1≤ ( i−1 ) ε . (34) Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 6 o 11 The conclusion now ollows om adding (33) and (34), and ecalling he iangle inequali y ( ): Le ε>0. In iew o (i ), he e exis s xÎXsuch ha ( ∀i∈I ) Ti−1···T1xm−TiTi−1···T1xm−xi−1+xi≤ ( 2i−1 )ε (35) whe e x 0 =x m .Nowse (∀iÎI)e i =T i-1 ···T 1 x m -T i T i-1 ···T 1 x m -x i-1 +x i .Then (∀iÎI)||e i || ≤(2i-1)ε. Se x=x m . Then m  i =1 ei= m  i =1 Ti−1...T1xm−TiTi−1...T1xm−xi−1+x i (36) =x− TmTm −1... T 1x . (37) This, (35), and he iangle inequali y imply ha x−TmTm−1···T1x≤ m  i =1 ei≤ m  i =1 (2i−1)ε=m2ε . (38) This comple es he p oo . Co olla y 3.2 Suppose ha ( ∀i∈I ) 0∈ an ( Id −Ti ) .Then 0∈ an ( Id −TmTm−1···T1 ) . P oo . This ollows om Theo em 3.1( ). Rema k 3.3 The con e se implica ion in Co olla y 3.2 ails in gene al: indeed, con- side he case when X≠{0}, m=2,and ÎX {0}. Now se T 1 X®X: x ↦x+ and se T 2 X®X: x ↦x- . Then 0/∈ an ( Id −T1 ) ={− } and 0/∈ an ( Id −T2 ) ={ } how- e e , T 2 T 1 = Id and an ( Id −T2T1 ) ={0 } . Rema k 3.4 Co olla y 3.2 is op imal in he sense ha e en i (∀iÎI)weha e0Î an(Id - T i ), we canno deduce ha 0 Î an(Id - T m T m-1 ···T 1 ): indeed, suppose ha X = R2 and m=2.Se C 1 : = epi exp and C2:= R× { 0 } . Suppose u he ha T1=PC 1 and T2=PC 2 .Then(∀iÎI)0Î an(Id - T i ); howe e , 0∈ an ( Id −T2T1 ) an ( Id −T2T1 ) . 4 Asymp o ic egula i y The ollowing no ions ( aken om B uck and Reich’s seminal a icle [22]) will be e y use ul o ob ain s onge esul s. De ini ion 4.1 ((s ong) nonexpansi eness and asymp o ic egula i y) Le S: X ® X. Then: (i) Sisnonexpansi e i (∀xÎX)(∀yÎX)||Sx - Sy|| ≤||x - y||. (ii) Siss ongly nonexpansi e i S is nonexpansi e and whene e (x n ) nÎN and (y n ) nÎN a e sequences in X such ha (x n -y n ) nÎN is bounded and ||x n -y n ||- ||Sx n - Sy n || ®0, i ollows ha (x n -y n )-(Sx n -Sy n )®0. (iii) S is asymp o ically egula i (∀xÎX)S n x-S n+1 x®0. The ollowing esul illus a es ha s ongly nonexpansi e mappings gene alize he no ion o a i mly nonexpansi e mapping.Inaddi ion, heclasso s ongly Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 7 o 11 nonexpansi e mappings is closed unde composi ions. (In con as , he composi ion o wo (necessa ily i mly nonexpansi e) p ojec o s may ail o be i mly nonexpansi e.) Fac 4.2 (B uck and Reich) The ollowing hold. (i) E e y i mly nonexpansi e mapping is s ongly nonexpansi e. (ii) The composi ion o ini ely many s ongly nonexpansi e mappings is also s ongly nonexpansi e. P oo . (i): See [[22], P oposi ion 2.1]. (ii): See [[22], P oposi ion 1.1]. The sequences o i e a es and o di e ences o i e a es ha e s iking con e gence p ope ies as we shall see now. In passing, we no e ha Fac 4.3(i) also appea s in [[21], Theo em 3.7(b)] e en in ce ain Banach spaces. Fac 4.3 (B uck and Reich) Le S: X ®X be s ongly nonexpansi e and le x ÎX. Then he ollowing hold. (i) The sequence (S n x-S n+1 x) nÎN con e ges s ongly o he unique elemen o leas no m in an ( Id −S ) . (ii) I Fix S =Ø, hen ||S n x||®+∞. (iii) I Fix S≠Ø, hen (S n x) nÎN con e ges weakly o a ixed poin o S. P oo (i): See [[22], Co olla y 1.5]. (ii): See [[22], Co olla y 1.4]. (iii): See [[22], Co ol- la y 1.3]. Suppose S: X ®Xis asymp o ically egula . Then, o e e y xÎX,0¬S n x-S n+1 x =(Id-S)S n xÎ an(Id - S) and hence 0∈ an ( Id −S ) . The opposi e implica ion ails in gene al (conside S= - Id), bu i is ue o s ongly nonexpansi e mappings. Unde he assump ion ha Sis i mly nonexpansi e, he ollowing esul also ollows om [[23], Co olla y 2]. Co olla y 4.4 Le S: X ®X be s ongly nonexpansi e. Then S is asymp o ically egu- la i and only i 0∈ an ( Id −S ) . P oo .“⇒": Clea . “⇐": Fac 4.3(i). Co olla y 4.5 Se S =T m T m-1 ···T 1 .Then S is asymp o ically egula i and only i 0∈ an ( Id −S ) . P oo Since each T i is i mly nonexpansi e, i is also s ongly nonexpansi eby Fac 4.2 (i). By Fac 4.2(ii), Sis s ongly nonexpansi e. Now apply Co olla y 4.4. Al e na i ely, 0∈ an ( Id −S ) by Co olla y 3.2 and again Co olla y 4.4 applies. We a e now eady o ou i s main esul . When m=2, hen heconclusionalso ollows om [[21], p. 124]. Theo em 4.6 Suppose ha each T i is asymp o ically egula . Then T m T m-1 ···T 1 is asymp o ically egula as well. P oo . Theo em 3.1( ) implies ha 0∈ an ( Id −TmTm−1···T1 ) . The conclusion hus ollows om Co olla y 4.5. As an applica ion o Theo em 4.6, we ob ain he main esul o [17]. Example 4.7 Le C 1 , ..., C m be nonemp y closed con ex subse s o X. Then he com- posi ion o he co esponding p ojec o s, PC m PC m −1...PC1is asymp o ically egula . Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 8 o 11 P oo . Fo e e y iÎI, he p ojec o PC i is i mly nonexpansi e, hence s ongly nonex- pansi e, and Fix PC i =Ci= ∅ . Suppose ha (∀i∈I)Ti=PC i , which is hus asymp o i- cally egula by Co olla y 4.4. Now apply Theo em 4.6. 5 Con ex combina ion In his sec ion, we use ou ixed weigh s (l i ) iÎI (see (12)) o u n X m in o a Hilbe p o- duc space di e en om Xconside ed in he p e ious sec ions. Speci ically, we se Y:=Xmwi h x,y=  i∈ I λixi,yi  (39) so ha ||x|| 2 =∑ iÎI l i ||x i || 2 . We also se Q:Xm→Xm:x→ (¯ x)i∈I,whe e ¯ x:=  i∈ I λixi . (40) Fac 5.1 (See [[1], P oposi ion 28.13].) In he Hilbe p oduc space Ywe ha e P Δ = Q. Co olla y 5.2 In he Hilbe p oduc space Y he ope a o Qis i mly nonexpansi e and s ongly nonexpansi e. Fu he mo e,FixQ=Δ≠Ø, 0Î an(Id -Q), and Qis asymp o ically egula . P oo By Fac 5.1, he ope a o Qis equal o he p ojec o P Δ and hence i mly non- expansi e. Now apply Fac 4.2(i) o deduce ha Qis s ongly nonexpansi e. I is clea ha Fix Q=Δand ha 0Î an(Id -Q). Finally, ecall Co olla y 4.4 o see ha Qis asymp o ically egula . P oposi ion 5.3 In he Hilbe p oduc space Y he ope a o Tis i mly nonexpansi e. P oo . Since each T i is i mly nonexpansi e, we ha e (∀x=(x i ) iÎI ÎY)(∀y=(y i ) iÎI Î Y)||T i x i -T i y i || 2 ≤〈x i -y i ,T i x i -T i y i 〉⇒||Tx -Ty|| 2 =∑ iÎI l i ||T i x i -T i y i || 2 ≤∑ iÎI l i 〈x i -y i ,T i x i -T i y i 〉=〈x-y, Tx -Ty〉. Theo em 5.4 Suppose ha ( ∀i∈I ) 0∈ an ( Id −Ti ) .Then he ollowing hold in he Hilbe p oduc space Y. (i) 0∈ an ( Id −T ) . (ii) Tis asymp o ically egula . (iii) Q○Tis asymp o ically egula . P oo . (i): This ollows because (∀x=(x i ) iÎI )||x-Tx|| 2 =∑ iÎI l i ||x i -T i x i || 2 . (ii): Combine Fac 4.2(i) wi h Co olla y 4.4. (iii): On he one hand, Qis i mly nonexpansi e and asymp o ically egula by Co - olla y 5.2. On he o he hand, Tis i mly nonexpansi e and asymp o ically egula by P oposi ion 5.3 and Theo em 5.4(ii). Al oge he , he esul ollows om Theo- em 4.6. We a e now eady o ou second main esul , which conce ns con ex combina ions o i mly nonexpansi e mappings. Fo u he esul s in his di ec ion-namely con ex Bauschke e al.Fixed Poin Theo y and Applica ions 2012, 2012:53 h p://www. ixedpoin heo yandapplica ions.com/con en /2012/1/53 Page 9 o 11