Browder's convergence for uniformly asymptotically regular nonexpansive semigroups in Hilbert spaces

Hindawi Publishing Co po a ion Fixed Poin Theo y and Applica ions Volume 2010, A icle ID 418030, 8pages doi:10.1155/2010/418030 Resea ch A icle B owde ’s Con e gence o Uni o mly Asymp o ically Regula Nonexpansi e Semig oups in Hilbe Spaces Gena o L ´ opez Acedo1and Tomona i Suzuki2 1Depa amen o de An´ alisis Ma em´ a ico, Facul ad de Ma em´ a icas, Uni e sidad de Se illa, 41080 Se illa, Spain 2Depa men o Ma hema ics, Kyushu Ins i u e o Technology, Toba a, Ki akyushu 804-8550, Japan Co espondence should be add essed o Gena o L´ opez Acedo, [email p o ec ed] Recei ed 6 Oc obe 2009; Accep ed 14 Oc obe 2009 Academic Edi o : Tomas Dominguez Bena ides Copy igh q2010 G. L´ opez Acedo and T. Suzuki. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, 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. We gi e a sufficien and necessa y condi ion conce ning a B owde ’s con e gence ype heo em o uni o mly asymp o ically egula one-pa ame e nonexpansi e semig oups in Hilbe spaces. 1. In oduc ion Le Cbe a closed con ex subse o a Hilbe space E. A mapping Ton Cis called a nonexpansi e mapping i Tx −Ty≤x−y o all x, y ∈C. We deno e by FT he se o ixed poin s o T. B owde , see 1, p o ed ha FTis nonemp y p o ided ha Cis, in addi ion, bounded. Ki k in a e y celeb a ed pape , see 2, ex ended his esul o he se ing o e lexi e Banach spaces wi h no mal s uc u e. B owde 3ini ia ed he in es iga ion o an implici me hod o app oxima ing ixed poin s o nonexpansi e sel -mappings de ined on a Hilbe space. Fix u∈C, he s udied he implici i e a i e algo i hm z  u 1− Tz .1.1 Namely, z , ∈0,1, is he unique ixed poin o he con ac ion x→ u 1− Tx,x∈C. B owde p o ed ha lim →0z Pu, whe e Puis he elemen o FTnea es o u. Ex ensions o he amewo k o Banach spaces o B owde ’s con e gence esul s ha e been done by many au ho s, including Reich 4, Takahashi and Ueda 5, and O’Ha a e al. 6. 2 Fixed Poin Theo y and Applica ions A amily o mappings {T : ≥0}is called a one-pa ame e s ongly con inuous semig oup o nonexpansi e mappings nonexpansi e semig oup, o sho on Ci he ollowing a e sa is ied. NS1Fo each ≥0, T is a nonexpansi e mapping on C. NS2Ts Ts◦T  o all s, ≥0. NS3Fo each x∈C, he mapping → T x om 0,∞in o Cis s ongly con inuous. The e a e many pape s conce ning he exis ence o common ixed poin s o {T : ≥0}; see, o ins ance, 7–13. As a ma e o ac , B owde 8p o ed ha i Cis bounded, hen  ≥0FT  is nonemp y. B owde ’s ype con e gence heo em o nonexpansi e semig oups is p o ed in 11, 14–18and o he s. Fo example, he ollowing heo em is p o ed in 17. Theo em 1.1 see 17.Le Cbe a closed con ex subse o a Hilbe space E.Le {T : ≥0} be a nonexpansi e semig oup on Csuch ha  ≥0FT  / ∅.Le {αn}and { n}be sequences in R sa is ying C10<α n<1and 0≤ n; C2limn nlimnαn/ n0,whe e1/0∞. Fix u∈Cand de ine a sequence {xn}in Cby xnαnu1−αnT nxn.1.2 Then {xn}con e ges s ongly o he elemen o  ≥0FT  nea es o u. We no e ha C1is needed o de ine {xn}. A nonexpansi e semig oup {T : ≥0}on Cis said o be uni o mly asymp o ically egula u.a. .i o e e y ≥0 and o e e y bounded subse Ko C, lim s→∞sup x∈K Ts x−Tsx01.3 holds. The ollowing is p o ed by Dom´ ınguez Bena ides e al. 16;seealso15. Theo em 1.2 see 16.Le E,C, and {T : ≥0}be as in Theo em 1.1. Assume ha {T : ≥ 0}is u.a. . Le {αn}and { n}be sequences in Rsa is ying (C1) and D2limnαn0and limn n∞. Fix u∈Cand de ine a sequence {xn}in Cby 1.2.Then{xn}con e ges s ongly o he elemen o  ≥0FT  nea es o u. The e is an in e es ing diffe ence be ween Theo ems 1.1 and 1.2, ha is,{ n}in Theo em 1.1 con e ges o 0 and { n}in Theo em 1.2 di e ges o ∞. By he way, e y ecen ly, Akiyama and Suzuki 14gene alized Theo em 1.1. They eplaced C2o Theo em 1.1 by Fixed Poin Theo y and Applica ions 3 he ollowing: C2{ n}is bounded; C3limnαn/ n−τ0 o all τ∈0,∞. They also showed ha he conjunc ion o C2and C3is bes possible; see also 18. In his pape , mo i a ed by he p e ious conside a ions, we gene alize Theo em 1.2 conce ning {αn}and { n}. Also, we will show ha ou new condi ion is bes possible. 2. Main Resul s We deno e by N he se o all posi i e in ege s and by R he se o all eal numbe s. Fo ∈R, we deno e by   he maximum in ege no exceeding . The ollowing p oposi ion plays an impo an ole in his pape . P oposi ion 2.1. Le Cbe a se o a sepa a ed opological ec o space E.Le {T : ≥0}be a amily o mappings on Csuch ha Ts◦T Ts  o all s, ∈0,∞. Assume ha {T : ≥0}is asymp o ic egula , ha is, lim s→∞ T sx−Tsx02.1 o all ∈0,∞and x∈C.Then FT   s≥0 FTs 2.2 holds o all ∈0,∞. P oo . Fix ∈0,∞. I is ob ious ha FT  ⊃sFTs holds. Le z∈Cbe a ixed poin o T . Fo e e y h∈0,∞, we ha e Thz−zlim n→∞ Th◦T nz−T nz lim n→∞ Thn z−Tn z lim s→∞ Thsz−Tsz 0, 2.3 and hence zis a common ixed poin o {T : ≥0}. I is well known ha e e y Hilbe space has he Opial p ope y. P oposi ion 2.2 Opial 19.Le Ebe a Hilbe space. Le {xn}be a sequence in Econ e ging weakly o z0∈H. Then he inequali y lim in nxn−z≤lim in nxn−z0implies zz0. We gene alize Theo em 1.2. 4 Fixed Poin Theo y and Applica ions Theo em 2.3. Le Cbe a closed con ex subse o a Hilbe space E.Le {T : ≥0}be a u.a. . nonexpansi e semig oup on Csuch ha  ≥0FT  / ∅.Le {αn}and { n}be sequences in Rsa is ying (C1) and D2limnαnlimnαn/ n0. Fix u∈Cand de ine a sequence {xn}in Cby 1.2.Then{xn}con e ges s ongly o he elemen o  ≥0FT  nea es o u. P oo . Pu FT ≥0FT .Le be he elemen o FTnea es o u. Since xn− 1−αnT nxnαnu−  ≤1−αnT nxn− αnu−  ≤1−αnxn− αnu− , 2.4 we ha e xn− ≤u− . The e o e {xn}is bounded. Hence {T xn:n∈N, ≥0}is also bounded. We pu M:sup{T xn−u:n∈N, ≥0}<∞.2.5 Le { n}be an a bi a y subsequence o {n}. Then he e exis s a subsequence {gn}o {n} such ha {x ◦gn}con e ges weakly o x. We choose a subsequence {hn}o {n}such ha τ:lim n→∞ ◦g◦hnlim sup n→∞ ◦gn.2.6 Pu yjx ◦g◦hj,βjα ◦g◦hj,and sj ◦g◦hj. We will show x∈FT, di iding he ollowing h ee cases: iτ∞, ii0<τ<∞, iiiτ0. In he i s case, we ix ≥0. Fo sufficien ly la ge j∈N, we ha e T x−yj≤T x−T yjT yj−yj ≤x−yjβjT yj−u1−βjT yj−Tsjyj ≤x−yjβjM1−βjTsj− yj−yj ≤x−yjβjM1−βjβjTsj− yj−u1−βj2Tsj− yj−Tsjyj ≤x−yjβj2−βjM1−βj2Tsj−  yj−Tsj− yj, 2.7 Fixed Poin Theo y and Applica ions 5 and hence lim in j→∞  T x−yj ≤lim in j→∞  x−yj .2.8 By he Opial p ope y, we ob ain T xx.Thusx∈FT. In he second case, we ha e Tτx−yj≤Tτx−TsjxTsjx−TsjyjTsjyj−yj ≤Tτx−Tsjxx−yjβjTsjyj−u ≤Tτ−sjx−T0xx−yjβjM, 2.9 and hence lim in j→∞  Tτx−yj ≤lim in j→∞  x−yj .2.10 By he Opial p ope y, we ob ain Tτxx.ByP oposi ion 2.1,weob ainx∈FT. In he hi d case, we ix ≥0. Fo sufficien ly la ge j∈N, we ha e T x−yj≤T x−T /sjsjxT /sjsjx−T /sjsjyj   /sj−1  k0 Tksjyj−Tk1sjyjT0yj−yj ≤T − /sjsjx−T0xx−yj  /sjTsjyj−yjT0yj−TsjyjTsjyj−yj ≤T − /sjsjx−T0xx−yj  /sjTsjyj−yjyj−TsjyjTsjyj−yj T − /sjsjx−T0xx−yj /sj2Tsjyj−yj T − /sjsjx−T0xx−yj /sj2βjTsjyj−u ≤maxTsx−T0x:0≤s≤sjx−yj βj/sj2βjM. 2.11 Hence 2.8holds. Thus we ob ain x∈FT. We nex p o e ha {yj}con e ges s ongly o . Since βj yj−   21−βjyj−Tsjyj− −Tsj ,y j−  βju− ,yj− , yj−Tsjyj− −Tsj ,y j−  ≥yj− 2−Tsjyj−Tsj yj− ≥0, 2.12 6 Fixed Poin Theo y and Applica ions we ob ain yj− 2≤u− , yj− . Since u− ,x − ≤0, we ha e  yj−   2≤u− , yj−  u− ,yj−xu− ,x −  ≤u− ,yj−x, 2.13 and hence {yj}con e ges s ongly o . Since {x n}is a bi a y, we ob ain ha {xn} con e ges s ongly o . Using 20, Theo em 7, we ob ain he ollowing Mouda i’s ype con e gence heo em; see 21. Co olla y 2.4. Le E,C,{T : ≥0},{αn},and { n}be as in Theo em 2.3.Le Φbe a con ac ion on C; ha is, he e exis s ∈0,1such ha Φx−Φy≤ x−y o x, y ∈C. De ine a sequence {xn}in Cby xnαnΦxn1−αnT nxn.2.14 Then {xn}con e ges s ongly o he unique poin z∈Csa is ying P◦Φzz,whe ePis he me ic p ojec ion om Con o  ≥0FT . We will show ha D2is bes possible. Example 2.5. Pu E2N, ha is,Eis a Hilbe space consis ing o all he unc ions x om Nin o Rsa is ying k∈N|xk|2<∞wi h inne p oduc x, yk∈Nxkyk. De ine a bounded closed con ex subse Co Eby Cx∈E:0≤xk≤pk,2.15 whe e pk2−k/2. De ine a u.a. . nonexpansi e semig oup {T : ≥0}on Cby T xkmaxxk− pk2,0.2.16 Le {ek}be he canonical basis o Eand pu u∞ k1pkek.Le {αn}and { n}be sequences in Rsa is ying C1and de ine {xn}in Cby 1.2. Then {xn}con e ges o a common ixed poin o {T : ≥0}only i limnαnlimnαn/ n0. P oo . Fo α∈0,1and ≥0, we de ine xα, by xα, αu 1−αT xα, .2.17 Fixed Poin Theo y and Applica ions 7 We no e xα, k⎧ ⎪ ⎨ ⎪ ⎩ αpk,i α≤ pk, 1 pk− pk αpk,i α≥ pk. 2.18 So, xα, k≥αpk. I is ob ious ha  ≥0FT   {0}. We assume limnxnlimnxαn, n Pu 0. Then 0lim n→∞ xn1 p1 ≥lim n→∞αn.2.19 A guing by con adic ion, we assume lim supnαn/ n>0. Then he e exis κ∈Nand a subsequence { n}o {n}such ha α n n ≥2pκ.2.20 Since limnx nκ0, we ha e 0lim n→∞ x nκ pκ lim n→∞ 1 npκ− npκ α n ≥lim sup n→∞ 1− npκ α n≥1 2>0, 2.21 which is a con adic ion. The e o e we ob ain limnαn/ n0. By Theo em 2.3 and Example 2.5, we ob ain he ollowing. Theo em 2.6. Le Ebe an in ini e-dimensional Hilbe space. Le {αn}and { n}be sequences in R sa is ying (C1). Then he ollowing a e equi alen : ilimnαnlimnαn/ n0, iii Cis a bounded closed con ex subse Co E,{T : ≥0}is a u.a. . nonexpansi e semig oup on C,u∈C, and {xn}is a sequence in Cde ined by 1.2, hen{xn}con e ges s ongly o he elemen o  ≥0FT  nea es o u. Compa e D2wi h he conjunc ion o C2and C3. We can ell ha he diffe ence be ween bo h condi ions is u.a. . Acknowledgmen s The i s au ho was pa ially suppo ed by DGES, G an MTM2006-13997-C02-01 and Jun a de Andaluc´ ıa, G an FQM-127. The second au ho is suppo ed in pa by G an s-in-Aid o Scien i ic Resea ch om he Japanese Minis y o Educa ion, Cul u e, Spo s, Science and Technology. 8 Fixed Poin Theo y and Applica ions Re e ences 1F. E. B owde , “Fixed-poin heo ems o noncompac mappings in Hilbe space,” P oceedings o he Na ional Academy o Sciences o he Uni ed S a es o Ame ica, ol. 53, pp. 1272–1276, 1965. 2W. A. 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