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

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 418030, 8pages doi:10.1155/2010/418030 Research Article Browder’s Convergence for Uniformly Asymptotically Regular Nonexpansive Semigroups in Hilbert Spaces Genaro L ´ opez Acedo1and Tomonari Suzuki2 1Departamento de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, 41080 Sevilla, Spain 2Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan Correspondence should be addressed to Genaro L´ opez Acedo, [email protected] Received 6 October 2009; Accepted 14 October 2009 Academic Editor: Tomas Dominguez Benavides Copyright q2010 G. L´ opez Acedo and T. Suzuki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give a sufficient and necessary condition concerning a Browder’s convergence type theorem for uniformly asymptotically regular one-parameter nonexpansive semigroups in Hilbert spaces. 1. Introduction Let Cbe a closed convex subset of a Hilbert space E. A mapping Ton Cis called a nonexpansive mapping if Tx −Ty≤x−yfor all x, y ∈C. We denote by FTthe set of fixed points of T. Browder, see 1, proved that FTis nonempty provided that Cis, in addition, bounded. Kirk in a very celebrated paper, see 2, extended this result to the setting of reflexive Banach spaces with normal structure. Browder 3initiated the investigation of an implicit method for approximating fixed points of nonexpansive self-mappings defined on a Hilbert space. Fix u∈C, he studied the implicit iterative algorithm zttu 1−tTzt.1.1 Namely, zt,t∈0,1, is the unique fixed point of the contraction x→ tu 1−tTx,x∈C. Browder proved that limt→0ztPu, where Puis the element of FTnearest to u. Extensions to the framework of Banach spaces of Browder’s convergence results have been done by many authors, including Reich 4, Takahashi and Ueda 5, and O’Hara et al. 6. 2 Fixed Point Theory and Applications A family of mappings {Tt:t≥0}is called a one-parameter strongly continuous semigroup of nonexpansive mappings nonexpansive semigroup, for shorton Cif the following are satisfied. NS1For each t≥0, Ttis a nonexpansive mapping on C. NS2TstTs◦Ttfor all s, t ≥0. NS3For each x∈C, the mapping t→ Ttxfrom 0,∞into Cis strongly continuous. There are many papers concerning the existence of common fixed points of {Tt:t≥0}; see, for instance, 7–13. As a matter of fact, Browder 8proved that if Cis bounded, then t≥0FTt is nonempty. Browder’s type convergence theorem for nonexpansive semigroups is proved in 11, 14–18and others. For example, the following theorem is proved in 17. Theorem 1.1 see 17.Let Cbe a closed convex subset of a Hilbert space E.Let{Tt:t≥0} be a nonexpansive semigroup on Csuch that t≥0FTt / ∅.Let{αn}and {tn}be sequences in R satisfying C10<α n<1and 0≤tn; C2limntnlimnαn/tn0,where1/0∞. Fix u∈Cand define a sequence {xn}in Cby xnαnu1−αnTtnxn.1.2 Then {xn}converges strongly to the element of t≥0FTt nearest to u. We note that C1is needed to define {xn}. A nonexpansive semigroup {Tt:t≥0}on Cis said to be uniformly asymptotically regular u.a.r.if for every t≥0 and for every bounded subset Kof C, lim s→∞sup x∈K Tstx−Tsx01.3 holds. The following is proved by Dom´ ınguez Benavides et al. 16;seealso15. Theorem 1.2 see 16.Let E,C, and {Tt:t≥0}be as in Theorem 1.1. Assume that {Tt:t≥ 0}is u.a.r. Let {αn}and {tn}be sequences in Rsatisfying (C1) and D2limnαn0and limntn∞. Fix u∈Cand define a sequence {xn}in Cby 1.2.Then{xn}converges strongly to the element of t≥0FTt nearest to u. There is an interesting difference between Theorems 1.1 and 1.2,thatis,{tn}in Theorem 1.1 converges to 0 and {tn}in Theorem 1.2 diverges to ∞. By the way, very recently, Akiyama and Suzuki 14generalized Theorem 1.1. They replaced C2of Theorem 1.1 by Fixed Point Theory and Applications 3 the following: C2{tn}is bounded; C3limnαn/tn−τ0 for all τ∈0,∞. They also showed that the conjunction of C2and C3is best possible; see also 18. In this paper, motivated by the previous considerations, we generalize Theorem 1.2 concerning {αn}and {tn}. Also, we will show that our new condition is best possible. 2. Main Results We denote by Nthe set of all positive integers and by Rthe set of all real numbers. For t∈R, we denote by tthe maximum integer not exceeding t. The following proposition plays an important role in this paper. Proposition 2.1. Let Cbe a set of a separated topological vector space E.Let{Tt:t≥0}be a family of mappings on Csuch that Ts◦TtTstfor all s, t ∈0,∞. Assume that {Tt:t≥0}is asymptotic regular, that is, lim s→∞ Ttsx−Tsx02.1 for all t∈0,∞and x∈C.Then FTt  s≥0 FTs 2.2 holds for all t∈0,∞. Proof. Fix t∈0,∞. It is obvious that FTt ⊃sFTs holds. Let z∈Cbe a fixed point of Tt. For every h∈0,∞, we have Thz−zlim n→∞ Th◦Ttnz−Ttnz lim n→∞ Thntz−Tntz lim s→∞ Thsz−Tsz 0, 2.3 and hence zis a common fixed point of {Tt:t≥0}. It is well known that every Hilbert space has the Opial property. Proposition 2.2 Opial 19.Let Ebe a Hilbert space. Let {xn}be a sequence in Econverging weakly to z0∈H. Then the inequality lim infnxn−z≤lim infnxn−z0implies zz0. We generalize Theorem 1.2. 4 Fixed Point Theory and Applications Theorem 2.3. Let Cbe a closed convex subset of a Hilbert space E.Let{Tt:t≥0}be a u.a.r. nonexpansive semigroup on Csuch that t≥0FTt / ∅.Let{αn}and {tn}be sequences in Rsatisfying (C1) and D2limnαnlimnαn/tn0. Fix u∈Cand define a sequence {xn}in Cby 1.2.Then{xn}converges strongly to the element of t≥0FTt nearest to u. Proof. Put FTt≥0FTt.Letvbe the element of FTnearest to u. Since xn−v1−αnTtnxnαnu−v ≤1−αnTtnxn−vαnu−v ≤1−αnxn−vαnu−v, 2.4 we have xn−v≤u−v. Therefore {xn}is bounded. Hence {Ttxn:n∈N,t≥0}is also bounded. We put M:sup{Ttxn−u:n∈N,t≥0}<∞.2.5 Let {fn}be an arbitrary subsequence of {n}. Then there exists a subsequence {gn}of {n} such that {xf◦gn}converges weakly to x. We choose a subsequence {hn}of {n}such that τ:lim n→∞tf◦g◦hnlim sup n→∞ tf◦gn.2.6 Put yjxf◦g◦hj,βjαf◦g◦hj,and sjtf◦g◦hj. We will show x∈FT, dividing the following three cases: iτ∞, ii0<τ<∞, iiiτ0. In the first case, we fix t≥0. For sufficiently large j∈N, we have Ttx−yj≤Ttx−TtyjTtyj−yj ≤x−yjβjTtyj−u1−βjTtyj−Tsjyj ≤x−yjβjM1−βjTsj−tyj−yj ≤x−yjβjM1−βjβjTsj−tyj−u1−βj2Tsj−tyj−Tsjyj ≤x−yjβj2−βjM1−βj2Tsj−ttyj−Tsj−tyj, 2.7 Fixed Point Theory and Applications 5 and hence lim inf j→∞  Ttx−yj ≤lim inf j→∞  x−yj .2.8 By the Opial property, we obtain Ttxx.Thusx∈FT. In the second case, we have 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 inf j→∞  Tτx−yj ≤lim inf j→∞  x−yj .2.10 By the Opial property, we obtain Tτxx.ByProposition 2.1,weobtainx∈FT. In the third case, we fix t≥0. For sufficiently large j∈N, we have Ttx−yj≤Ttx−Tt/sjsjxTt/sjsjx−Tt/sjsjyj  t/sj−1  k0 Tksjyj−Tk1sjyjT0yj−yj ≤Tt−t/sjsjx−T0xx−yj t/sjTsjyj−yjT0yj−TsjyjTsjyj−yj ≤Tt−t/sjsjx−T0xx−yj t/sjTsjyj−yjyj−TsjyjTsjyj−yj Tt−t/sjsjx−T0xx−yjt/sj2Tsjyj−yj Tt−t/sjsjx−T0xx−yjt/sj2βjTsjyj−u ≤maxTsx−T0x:0≤s≤sjx−yjtβj/sj2βjM. 2.11 Hence 2.8holds. Thus we obtain x∈FT. We next prove that {yj}converges strongly to v. Since βj yj−v  21−βjyj−Tsjyj−v−Tsjv,y j−v βju−v,yj−v, yj−Tsjyj−v−Tsjv,y j−v ≥yj−v2−Tsjyj−Tsjvyj−v≥0, 2.12 6 Fixed Point Theory and Applications we obtain yj−v2≤u−v, yj−v. Since u−v,x −v≤0, we have  yj−v  2≤u−v, yj−v u−v,yj−xu−v,x −v ≤u−v,yj−x, 2.13 and hence {yj}converges strongly to v. Since {xfn}is arbitrary, we obtain that {xn} converges strongly to v. Using 20, Theorem 7, we obtain the following Moudafi’s type convergence theorem; see 21. Corollary 2.4. Let E,C,{Tt:t≥0},{αn},and {tn}be as in Theorem 2.3.LetΦbe a contraction on C; that is, there exists r∈0,1such that Φx−Φy≤rx−yfor x, y ∈C. Define a sequence {xn}in Cby xnαnΦxn1−αnTtnxn.2.14 Then {xn}converges strongly to the unique point z∈Csatisfying P◦Φzz,wherePis the metric projection from Conto t≥0FTt. We will show that D2is best possible. Example 2.5. Put E2N,thatis,Eis a Hilbert space consisting of all the functions xfrom Ninto Rsatisfying k∈N|xk|2<∞with inner product x, yk∈Nxkyk. Define a bounded closed convex subset Cof Eby Cx∈E:0≤xk≤pk,2.15 where pk2−k/2. Define a u.a.r. nonexpansive semigroup {Tt:t≥0}on Cby Ttxkmaxxk−tpk2,0.2.16 Let {ek}be the canonical basis of Eand put u∞ k1pkek.Let{αn}and {tn}be sequences in Rsatisfying C1and define {xn}in Cby 1.2. Then {xn}converges to a common fixed point of {Tt:t≥0}only if limnαnlimnαn/tn0. Proof. For α∈0,1and t≥0, we define xα, tby xα, tαu 1−αTtxα, t.2.17 Fixed Point Theory and Applications 7 We note xα, tk⎧ ⎪ ⎨ ⎪ ⎩ αpk,if α≤tpk, 1tpk−tpk αpk,if α≥tpk. 2.18 So, xα, tk≥αpk. It is obvious that t≥0FTt  {0}. We assume limnxnlimnxαn,t n Pu 0. Then 0lim n→∞ xn1 p1 ≥lim n→∞αn.2.19 Arguing by contradiction, we assume lim supnαn/tn>0. Then there exist κ∈Nand a subsequence {fn}of {n}such that αfn tfn ≥2pκ.2.20 Since limnxfnκ0, we have 0lim n→∞ xfnκ pκ lim n→∞ 1tfnpκ−tfnpκ αfn ≥lim sup n→∞ 1−tfnpκ αfn≥1 2>0, 2.21 which is a contradiction. Therefore we obtain limnαn/tn0. By Theorem 2.3 and Example 2.5, we obtain the following. Theorem 2.6. Let Ebe an infinite-dimensional Hilbert space. Let {αn}and {tn}be sequences in R satisfying (C1). Then the following are equivalent: ilimnαnlimnαn/tn0, iiif Cis a bounded closed convex subset Cof E,{Tt:t≥0}is a u.a.r. nonexpansive semigroup on C,u∈C, and {xn}is a sequence in Cdefined by 1.2,then{xn}converges strongly to the element of t≥0FTt nearest to u. Compare D2with the conjunction of C2and C3. We can tell that the difference between both conditions is u.a.r. Acknowledgments The first author was partially supported by DGES, Grant MTM2006-13997-C02-01 and Junta de Andaluc´ ıa, Grant FQM-127. The second author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology. 8 Fixed Point Theory and Applications References 1F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965. 2W. A. 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