Strongly convergent approximations to fixed points of total asymptotically nonexpansive mappings

S ongly Con e gen App oxima ions o Fixed Poin s o To al Asymp o ically Nonexpansi e Mappings Yako Albe Depa men o Ma hema ics The Technion-Is ael Ins i u e o Technology 32000 Hai a, Is ael. Email: albe ya@ echunix. echnion.ac.il Ra a Esp´ınola∗and Pepa Lo enzo Depa amen o de An´alisis Ma em´a ico Uni e sidad de Se illa, P.O. Box 1160 41080-Se ille, Spain. Emails: [email p o ec ed], [email p o ec ed] ∗Co esponding au ho Abs ac In his wo k we p o e a new s ong con e gence esul o he egula ized successi e app oxima ion me hod gi en by yn+1 =qnz0+ (1 −qn)Tnyn, n = 1,2, ..., whe e lim n→∞ qn= 0 and ∞ X n=1 qn=∞, o Ta o al asymp o ically nonexpansi e mapping, i.e., Tis such ha kTnx−Tnyk ≤ kx−yk+k(1) nφ(kx−yk) + k(2) n, whe e k1 nand k2 na e eal null con e gen sequences and φ:R+→R+is con inuous and such ha φ(0) = 0 and lim →∞ φ( ) ≤C o a ce ain cons an C > 0. Among o he ea u es, ou esul s essen ially gene alize exis ing esul s on s ong con e gence o Tnonexpansi e and asymp o ically nonexpansi e. The con e gence and s abili y analysis is gi en o bo h sel - and nonsel -mappings. Running i le: S ong con e gence o ixed poin s Key wo ds: asymp o ically nonexpansi e mappings, bes app oxima ion, ixed poin , duali y map, i e a ion schemes. Ma hema ics Subjec Classi ica ion 2000: 47A58, 47H10. 1 2 1 In oduc ion I e a i e p ocedu es o nonlinea ope a o s ha e been la gely s udied by many au ho s in he las decades. One o he i s esul s o his na u e was ob ained by B owde [5] o nonexpansi e sel -mappings de ined on Hilbe spaces. He e B owde s udied he i e a i e me hod: xω=ωz0+ (1 −ω)T xω.(1.1) o Ω a closed and con ex subse o H,z0∈Ω an a bi a y poin and T: Ω →Ω a nonexpansi e mapping wi h nonemp y ixed poin se N(T) := {x∈Ω : T x =x}. In [5], B owde p o ed ha lim ω→0xωexis s and is a ixed poin o T. This esul was ex- ended by Reich [17] o he case when Xis a uni o mly smoo h Banach space. Fu he mo e, he showed ha he ixed poin se o Tis a sunny nonexpansi e e ac o Ω. The ecu si e o mula (explici scheme) y1∈Ω, yn+1 =qnz0+ (1 −qn)Tyn, n = 1,2, ..., (1.2) was in oduced by Halpe n [12] who discussed i s con e gence in he amewo k o Hilbe spaces. La e i has been in es iga ed in [12, 18, 19, 20] wi h di e en addi ional p ope ies on he sequence {qn}, he ope a o Tand he space X. B owde ’s and Halpe n’s i e a i e p ocedu es ha e mo i a ed di e en schemes o ind ixed poin s o asymp o ically nonexpansi e mappings (see Rema k 1.2 o de ini ion). In his way, T.C. Lim and H.K. Xu [15] s udied he algo i hm o Tasymp o ically nonexpan- si e which gene a es he sequence (implici scheme) xn=qnz0+ (1 −qn)Tnxn.(1.3) They showed ha he sequence {xn}con e ges s ongly o a ixed poin o Tin he ame- wo k o a uni o mly smoo h Banach space, unde sui able condi ions on he coe icien s. Ve y ecen ly, in [6], he s ong con e gence o he explici scheme gi en by y1∈Ωyn+1 =qnz0+ (1 −qn)Tnyn, n = 1,2, ..., (1.4) whe e z0∈Ω,and lim n→∞ qn= 0 and ∞ X n=1 qn=∞,(1.5) has been s udied in uni o mly smoo h spaces. I is wo hwhile o poin ou ha he con- e gence o he implici scheme gi en by (1.3) is an impo an ool in o de o p o e he s ong con e gence o explici schemes as (1.2). In his pape we will conside he class o he o al asymp o ically nonexpansi e map- pings which ha e been in oduced e y ecen ly in [2]. De ini ion 1.1 (c . [2]) A mapping T: Ω →Ωis called o al asymp o ically nonexpansi e i he e exis nonnega i e eal sequences {k(1) n}and {k(2) n}wi h k(1) n, k(2) n→0as n→ ∞, and a con inuous unc ion φ:R+→R+wi h φ(0) = 0 such ha kTnx−Tnyk ≤ kx−yk+k(1) nφ(kx−yk) + k(2) n.(1.6) 3 Rema k 1.2 I φ(λ)≡0 hen (1.6) akes he o m kTnx−Tnyk ≤ kx−yk+k(2) n. Hence, i Ωis a bounded se and TNis con inuous o some in ege N≥1 he mapping T is o asymp o ically nonexpansi e ype. I φ(λ) = λ hen we can w i e kTnx−Tnyk ≤ (1 + k(1) n)kx−yk+k(2) n. In addi ion, i k(2) n= 0 o all n≥1 hen we ob ain he de ini ion o asymp o ically nonex- pansi e mapping: kTnx−Tnyk ≤ knkx−yk, kn→1. I k(1) n= 0 and k(2) n= 0 o all n≥1 hen we ob ain he class o nonexpansi e mappings: kTx −Tyk ≤ kx−yk. I k(2) 1= 0 hen i ollows om (1.6) ha Tis uni o mly con inuous, howe e , i can be uni o mly con inuous e en i k(2) 16= 0. To cons uc he s ong con e gen app oxima ions o solu ions o he equa ion Tx =x(1.7) wi h a o al asymp o ically nonexpansi e mapping T, we apply he i e a i e scheme gi en by (1.4). Fo nonexpansi e ope a o s T, he algo i hm (1.4) is w i en down in he ollowing o m: y1∈Ω, yn+1 =qnz0+ (1 −qn)Tyn, n = 1,2, ..., (1.8) whe e lim n→∞ qn= 0. We show nex ha (1.8) is he egula ized successi e app oxima ion me hod o (1.7). As i is known, equa ion (1.7) is equi alen o Ax = 0 (1.9) wi h he acc e i e ope a o A=I−T: Ω →Ω.Tha is, in his case hAx −Ay, J(x−y)i ≥ 0,(1.10) whe e Js ands o he no malized duali y map. I x∗is a solu ion o (1.7) hen Ax∗= 0. In he sequel, we assume ha he ixed poin se N(T) o Tis no emp y. We emphasize ha he p oblem (1.7) belongs o he class o ill-posed p oblems ( o mo e on ill-pos ed p oblems see [4]). S ongly con e gen app oxima ions o x∗can be ob ained only by using some egula iza ion p ocedu e. Le ωbe a pa ame e such ha 0 < ω < 1 and ω→1.Ob iously, i x∗is a solu ion o (1.9) hen i is solu ion o he equa ion ωAx = 0 (1.11) 4 o any ixed ω > 0.Using he gene al heo y (see o example [4], Sec ion 2.7), cons uc o (1.11) he ope a o egula iza ion me hod wi h egula iza ion pa ame e α= 1−ω→0, namely, ωAx + (1 −ω)(x−z0) = 0,(1.12) whe e z0∈Ω.I is easy o see ha (1.12) is equi alen o x= (1 −ω)z0+ωT x. (1.13) Deno e Tωx= (1 −ω)z0+ωT x. Since Ω is con ex and closed, we ha e ha Tω: Ω →Ω,and (1.13) can be ew i en as x=Tωx. (1.14) Consequen ly, by Banach Con ac ion P inciple, equa ion (1.12) has a unique solu ion xωand he successi e app oxima ion me hod x1∈Ωxn+1 = (1 −ω)z0+ωT xn con e ges s ongly o xω.Le Xbe uni o mly smoo h and ωk→1 as k→ ∞.Conside now he egula ized equa ion ωkAx + (1 −ωk)(x−z0) = 0 (1.15) wi h k ixed and deno e by xki s unique solu ion. Then he e exis s ¯x∗∈ N(T) such ha xk→¯x∗as k→ ∞.Mo eo e (see [4]), ¯x∗sa is ies he inequali y h¯x∗−z0, J(¯x∗−x∗)i ≥ 0∀x∗∈ N(T). I can be shown in he same way ha (1.8) wi h Tnin place o Tis he egula ized successi e app oxima ion me hod o he equa ion (1.7) wi h o al asymp o ically nonex- pansi e mapping. Indeed, i o al asymp o ically nonexpansi e mappings a e conside ed in place o nonexpansi e mappings, hen hAnx−Any, J(x−y)i ≥ −k(1) nφ(kx−yk)kx−yk − k(2) nkx−yk,(1.16) whe e An=I−Tn.I is clea ha he analysis o s ong con e gence is mo e di icul in his si ua ion, mo eo e , e y li le is known abou he s uc u e o he solu ion se . In pa icula he same holds o asymp o ically nonexpansi e mappings o which (1.16) is hAnx−Any, J(x−y)i ≥ −k(1) nkx−yk2.(1.17) The main esul o his pape , Theo em 3.1 in Sec ion 3, s a es a s ong con e gence esul o he i e a i e scheme (1.4) in e lexi e Banach spaces wi h a weakly con inuous duali y map on unbounded domains. No ice ha i is an open ques ion we he a e lexi e Banach space admi ing a weakly sequen ially con inuous duali y mapping is uni o mly smoo h. An implici scheme con e gence esul is also p o ed. This esul is used o gua an ee he exis ence o sunny nonexpansi e e ac ions. In Sec ion 4 we s udy he same i e a i e scheme o o al asymp o ically nonexpansi e nonsel -mapping. Finally, in Sec ion 5, ou las sec ion we in es iga e he s abili y p ob- lem o i e a i e schemes wi h espec o pe u ba ions o cons ain se s o nonexpansi e nonsel -mappings. 5 2 P elimina ies Le Xbe a eal Banach space wi h no m k·k,le X∗be i s dual space wi h he no m k · k∗ and, as usual, deno e he duali y pai ing o Xand X∗by hϕ, xi,whe e x∈Xand ϕ∈X∗ (in o he wo ds, hϕ, xiis he alue o ϕa x). I is said ha Xis uni o mly smoo h i o any gi en ε > 0, he e exis s δ > 0 such ha o all x, y ∈Xwi h kxk= 1 and kyk ≤ δ, he inequali y 2−1(kx+yk+kx−yk)−1≤εkyk holds. The unc ion ρX(τ) = sup{2−1(kx+yk+kx−yk)−1 : kxk= 1,kyk=τ} is called he modulus o smoo hness o he space X. This unc ion is inc easing and app oaches o ze o as τ→0.Deno e hX(τ) = ρX(τ) τ. Obse e ha he space Xis uni o mly smoo h i and only i lim τ→0hX(τ) = 0. Le ψ: [0,∞)→[0,∞) be a con inuous s ic ly inc easing unc ion such ha ψ( )→ ∞ as → ∞ and ψ(0) = 0. The gene alized duali y mapping Jψ:X→2X∗associa ed o a gauge unc ion ψis de ined as Jψ(x) = {x∗∈X∗:hx, x∗i=ψ(kxk)kxk,kx∗k=ψ(kxk)}, x ∈X. In he case ha ψ( ) = hen Jψ=Jwhich is he no malized duali y map. We say ha a Banach space Xhas a weakly con inuous duali y map ([5]) i he e exis s a gauge unc ion ψ o which he gene alized duali y map Jψis single- alued and weak- o- weak* sequen ially con inuous. I is well-known ha Jψis he subdi e en ial, in he sense o con ex analysis, o he con ex unc ion Φ( ) = Z 0 ψ(τ)dτ, o τ≥0, and ha Jψis single- alued i and only i Xis smoo h. We will need he ollowing subdi - e en ial inequali y which is known o hold in smoo h spaces: Φ(kx+yk)≤Φ(kxk) + hy, Jψ(x+y)i o any x, y ∈X. Nex we in oduce some de ini ions and auxilia y esul s ha will be needed in he sequel. De ini ion 2.1 Le Xbe a Banach space and Ca nonemp y closed con ex subse o X. An ope a o T:C→Xis demiclosed (a y) i T(x) = ywhene e {xn} ⊆ Cis a sequence weakly con e gen o xand T(xn)→yas n→ ∞. 6 De ini ion 2.2 A Banach space Xsa is ies he Opial’s condi ion i o each sequence {xn} in X, he ela ion xn* x implies ha lim sup n→∞ kxn−xk<lim sup n→∞ kxn−yk o all y∈Xwi h x6=y. De ini ion 2.3 A Banach space Xsa is ies he Gene alized Gossez-Lami Dozo p ope y (GGLD-p ope y) i lim in n→∞ kxnk<lim sup m→∞ lim sup n→∞ kxm−xnk whene e {xn}is a weak null sequence which is no no m con e gen . The ollowing demiclosedness p inciple can be ound in [11]. Theo em 2.4 Le Xbe a Banach space wi h GGLD-p ope y and Opial’s condi ion. Le Cbe a weakly compac con ex subse o Xand T:C→Ca uni o mly con inuous mapping o asymp o ically nonexpansi e ype. Then I−Tis demiclosed a ze o. The ollowing esul is well-known (see [13] and [14]). P oposi ion 2.5 I in a e lexi e Banach space X he duali y mapping Jis weakly con in- uous hen Xsa is ies GGLD-p ope y and Opial’s condi ion. The nex co olla y ollows as a consequence o his p oposi ion and a ca e ul eading o he o iginal p oo o Theo em 2.4. Co olla y 2.6 Le Xbe a e lexi e Banach space wi h a weakly con inuous duali y mapping J. Le Cbe a closed con ex subse o Xand T:C→Ca uni o mly con inuous mapping and o al asymp o ically nonexpansi e wi h bounded o bi s. Then I−Tis demiclosed a ze o. We will also use he concep o a sunny nonexpansi e e ac ion [10] and, in pa icula , i s cha ac e iza ion by means o he duali y map in a smoo h Banach space. De ini ion 2.7 Le Cbe a non-emp y subse o a Banach space Xand Da subse o C. A mapping Q:C→Dis said o be (i) a e ac ion on o Di Q2=Q; (ii) a nonexpansi e e ac ion i i also sa is ies he inequali y kQx −Qyk ≤ kx−yk, o all x, y ∈C; (iii) a sunny e ac ion i o all x∈Cand o all 0≤ < ∞, Q(Qx + (x−Qx)) = Qx, whene e Qx + (x−Qx)∈C. 7 P oposi ion 2.8 Assume ha Cis a non-emp y closed con ex subse o a smoo h Banach space Xand Dis a subse o C. Then a nonexpansi e mapping Q:C→Dis a sunny e ac ion i and only i o all x∈Cand o all ξ∈D, hx−Qx, J(ξ−Qx)i ≤ 0. In pa icula , he e is a mos one sunny nonexpansi e e ac ion on D. Rema k 2.9 P oposi ion 2.5 and Co olla y 2.6 emain s ill alid i he no malized duali y mapping Jis eplaced by he duali y mapping Jψwi h he gauge unc ion ψ( ).Mo eo e , we can use Jψ o cha ac e ize sunny nonexpansi e e ac ions in a smoo h Banach space gi en by P oposi ion 2.8. Le G1and G2be nonemp y closed subse s o X. The Hausdo dis ance be ween G1 and G2is de ined by he ollowing o mula: H(G1, G2) = max{sup z1∈G1 in z2∈G2 kz1−z2k,sup z1∈G2 in z2∈G1 kz1−z2k}. We need he ollowing lemma [3] in o de o p o e he main esul o Sec ion 5. Lemma 2.10 I Xis a uni o mly smoo h Banach space, Ω1and Ω2a e closed con ex subse s o Xsuch ha he Hausdo dis ance H(Ω1,Ω2)≤σand QΩ1and QΩ2a e he (unique) sunny nonexpansi e e ac ions on o he subse s Ω1and Ω2, espec i ely, hen kQΩ1x−QΩ2xk2≤16R(2 +d)hX(16LR−1σ),(2.1) whe e =kxk, d = max{d1, d2}, R = 2(2 +d) + σand 1<L<1.7is he Figiel cons an [1, 2, 9]. He e di=dis (θ, Ωi), i = 1,2,and θis he o igin o he space X. We will o en apply he ollowing lemma on nume ical ecu en inequali ies. Lemma 2.11 Le {λn}and {γn}be nonnega i e, {αn}be posi i e eal numbe s such ha λn+1 ≤λn−αnλn+γn,∀n≥1. Le o all n > 1γn αn ≤c1and αn≤α. (2.2) Then λn≤max{λ1, K∗},whe e K∗= (1 + α)c1. In addi ion, i ∞ X 1 αn=∞and γn αn →0 hen λn→0as n→ ∞. 8 3 Con e gence Analysis o Successi e App oxima ion Me hod The goal o his sec ion is o p o e s ong con e gence o he egula ized successi e app ox- ima ion me hod (1.4). Le us conside he explici scheme (1.4) gi en by y0∈Ω, yn+1 =qnz0+ (1 −qn)Tnyn, n = 1,2, ..., wi h lim n→∞ qn= 0 and ∞ X n=1 qn=∞.(3.1) Theo em 3.1 Le Ωbe a nonemp y closed and con ex subse o a smoo h e lexi e Banach space Xwi h a weakly sequen ially con inuous duali y map Jψ,T: Ω →Ωa uni o mly con inuous mapping which is o al asymp o ically nonexpansi e wi h nonemp y ixed poin se N(T).Le N(T)be such ha he e exis s a sunny nonexpansi e e ac ion Q: Ω → N (T). Le z0∈Ωand {qn} ⊂ (0,1] a sequence sa is ying (3.1). Le he sequence {yn}be gene a ed by (1.4). Assume ha lim n→∞ k(1) n+k(2) n qn = 0,(3.2) and ha he e exis posi i e cons an s M0and M1such ha φ(λ)≤M0λ o λ≥M1. Suppose ha lim nkyn−Tynk= 0, hen {yn}con e ges s ongly o he ixed poin ¯x∗=Qz0 o T. P oo . Fi s ly we obse e ha he sequence {yn} ⊂ Ω because Ω is con ex. Take x∗∈ N(T).I ollows om (1.4) ha kyn+1 −x∗k ≤ qnkz0−x∗k+ (1 −qn)kTnyn−Tnx∗k ≤qnkz0−x∗k+ (1 −qn)kyn−x∗k+k(1) nφ(kyn−x∗k) + k(2) n. Deno ing λn=kyn−x∗kwe ha e λn+1 ≤(1 −qn)λn+qnkz0−x∗k+ (1 −qn)k(1) nφ(λn) + k(2) n.(3.3) Since φis con inuous i a ains i s maximum Mon [0, M1]. Then i is easy o e i y ha o all λ∈[0,∞) φ(λ)≤M+M0λ. The inequali y (3.3) is ew i en as λn+1 ≤λn−qn−(1 −qn)k(1) nM0λn+γn, whe e γn= (1 −qn)k(1) nM+k(2) n+qnkz0−x∗k. 9 Wi hou loss o gene ali y, in iew o (3.2), we assume ha he e exis cons an s α∈(0,1) and M2>0 such ha o all n≥1 k(1) n qn ≤M0(1 −α) 1−qn ,(3.4) and γn qn ≤αM2. Then λn+1 ≤λn−αqnλn+γn. By Lemma 2.11, we conclude ha λn≤max{λ1,(1 + α)M2}. Thus, he sequence {yn−x∗}is bounded, which, clea ly, implies ha {yn}is a bounded sequence. Applying he subdi e en ial inequali y o yn+1 −Qz0= (1 −qn)(Tnyn−Qz0) + qn(z0−Qz0) we deduce ha Φ(kyn+1 −Qz0k)≤Φ((1 −qn)kTnyn−Qz0k) + qnhz0−Qz0, Jψ(yn+1 −Qz0)i. Since Tis o al asymp o ically nonexpansi e and Qz0∈ N(T), we ha e kTnyn−Qz0k ≤ kyn−Qz0k+νn, whe e νn=Mk(1) n+M0k(1) nkyn−Qz0k+k(2) nis bounded and anishes as n→ ∞. Now, since Φ is a con ex and nondec easing, o nla ge enough we ha e Φ(kTnyn−Qz0k)≤(1 −νn)Φ(kyn−Qz0k) + νnΦ(kyn−Qz0k+ 1) ≤Φ(kyn−Qz0k) + νnM3 o M3a sui able cons an . Consequen ly Φ(kyn+1 −Qz0k)≤(1 −qn)Φ(kyn−Qz0k) + (1 −qn)νnM3+ (3.5) +qnhz0−Qz0, Jψ(yn+1 −Qz0)i. We claim ha lim sup n→∞ hz0−Qz0, Jψ(yn−Qz0)i ≤ 0. Indeed, since he sequence {yn}is bounded and he space X e lexi e he e exis s a sub- sequence {ynk}which is weakly con e gen in Ω. Le ¯ybe i s weak limi . We can ix his subsequence so ha lim sup n→∞ hz0−Qz0, Jψ(yn−Qz0)i= lim k→∞hz0−Qz0, Jψ(ynk−Qz0)i. 16 P oo . Fi s o all, we no e ha N(QΩT) is non-emp y because N(T)⊆ N (QΩT). We show nex ha {yn} ⊆ Ω is bounded. Take x∗∈ N(T). In iew o he inequali ies kyn+1 −x∗k ≤ qnkz0−x∗k+ (1 −qn)k(QΩT)nyn−(QΩT)nx∗k ≤qnkz0−x∗k+ (1 −qn)kQΩT(QΩT)n−1yn−QΩT(QΩT)n−1x∗k ≤qnkz0−x∗k+ (1 −qn)kT(QΩT)n−1yn−T(QΩT)n−1x∗k ≤qnkz0−x∗k+ (1 −qn)kyn−x∗k+k(1) nφ(kyn−x∗k) + k(2) n, we conclude, by analogy wi h Theo em 3.1, ha he e exis posi i e cons an s C, C1and ¯ Msuch ha kynk ≤ C, kyn−x∗k ≤ C1and φ(kyn−x∗k)≤¯ M. Since QΩis a nonex- pansi e mapping, i is no di icul o e i y ha he e exis s a cons an ¯ C > 0 such ha k(QΩT)nynk ≤ ¯ C. I ollows om (4.1) ha lim n→∞ yn+1 −(QΩT)nyn= 0.(4.3) On he o he hand, we can show ha lim n→∞ yn+1 −(QΩT)nyn+1= 0.(4.4) Indeed, kyn+1 −(QΩT)nyn+1k≤kyn+1 −(QΩT)nynk+k(QΩT)nyn−(QΩT)nyn+1k. Due o he o al asymp o ical nonexpansi eness o T, we ob ain k(QΩT)nyn−(QΩT)nyn+1k ≤ kT(QΩT)n−1yn−T(QΩT)n−1yn+1k ≤ kyn−yn+1k+k(1) nφ(kyn−yn+1k) + k(2) n. Now he boundedness o {(QΩT)nyn}, he ac ha kyn−yn+1k → 0 and (4.3) p o e (4.4). Fu he , we ha e kyn−QΩTynk ≤ kyn−yn+1k+kyn+1 −(QΩT)nynk+k(QΩT)nyn−QΩT ynk. F om he uni o m con inui y o T, we es ima e he las e m in he o m: kQΩT(QΩT)n−1yn−QΩTynk ≤ kT(QΩT)n−1yn−Tynk ≤ ω(k(QΩT)n−1yn−ynk). By (4.3) and (4.4), one ge s lim(yn−QΩTyn) = 0. 17 No ice ha QΩTis a mapping as equi ed in Theo em 3.1 and hen we can apply Theo em 3.1 o he sequence {yn}. Indeed, we w i e yn+1 −Qz0= (1 −qn)((QΩT)nyn−Qz0) + qn(z0−Qz0). we use he subdi e en ial inequali y as in Theo em 3.1, and he es o he p oo ollows he pa e n wi h he only di e ence ha we ob ain ha {yn}s ongly con e ges o Qz0= ¯x∗ which is a ixed poin o QΩT. 5 S abili y Analysis o Nonexpansi e Nonsel -mappings Nex we s udy he s abili y p oblem o i e a i e p ocesses wi h espec o pe u ba ions o cons ain se s. Speci ically, we conside a p ocess which in ol es nonexpansi e nonsel - mappings in he ollowing o m: y1∈Ω1, yn+1 =qnz0+ (1 −qn)QΩn+1 Tyn, n = 1,2, ..., (5.1) whe e QΩn:X→Ωnis a sunny nonexpansi e e ac ion, and he p oximi y be ween he o iginal se Ω and Ωnwi h n= 1,2, ... is gi en by he Hausdo dis ance: H(Ωn,Ω) ≤σn.(5.2) Le G=∩nΩnand ¯ G= Ω ∩G6=∅. Theo em 5.1 Le Xbe a uni o mly smoo h Banach space which has a weakly sequen ially con inuous duali y map Jψ. Assume ha D⊂Xis a closed con ex se , Ω⊂Dand Ωn⊂D, n = 1,2, ... a e closed con ex subse s o Xwi h p ope y (5.2). Le T:D→Xbe a nonexpansi e mapping wi h ixed poin se N(T)such ha N(T)∩Ω6=∅.Take z0some poin in ¯ G, x∗∈ N(T)∩Ω,{qn}a sequence in (0,1) sa is ying (3.1) and a non-inc easing sequence σn≤σ, such ha σn→0as n→ ∞.Le he sequence {yn}be gene a ed by (5.1). We suppose ha he sequence {T yn}is bounded, lim n→∞ |qn−qn−1| qn = 0 (5.3) and lim n→∞ phX(σn) qn = 0.(5.4) Then {yn}con e ges s ongly o he ixed poin ¯x∗=Qz0o QΩT, whe e Q: Ω → N (QΩT) is he unique sunny nonexpansi e e ac ion on o N(QΩT). P oo . We show i s ha {yn}is bounded. I is no di icul o see ha kyn+1 −x∗k=kqnz0+ (1 −qn)QΩTyn−QΩTx∗+ (1 −qn)(QΩn+1 Tyn−QΩTyn)k ≤qnkz0−x∗k+ (1 −qn)kQΩTyn−QΩTx∗k+ (1 −qn)kQΩn+1 Tyn−QΩTynk. 18 Since {Tyn}is bounded and due o Lemma 2.10, he e exis posi i e cons an s M5and M6 such ha kQΩn+1 Tyn−QΩTynk ≤ M5qhX(M6σn). This implies kyn+1 −x∗k ≤ qnkz0−x∗k+ (1 −qn)kyn−x∗k+M5qhX(M6σn). Deno ing λn=kyn−x∗kwe ob ain λn+1 ≤(1 −qn)λn+qnkz0−x∗k+M5qhX(M6σn). F om Lemma 2.11 i ollows ha {yn−x∗}is bounded. Le kyn−x∗k ≤ C1 o all n. Nex we e alua e he ollowing di e ence: yn+1 −yn= (1 −qn)(QΩn+1 T yn−QΩnT yn−1)+(qn−qn−1)(z0−x∗) +(qn−1−qn)(QΩnTyn−1−QΩTx∗) = (1 −qn)(QΩn+1 T yn−QΩn+1 T yn−1) + (1 −qn)(QΩn+1 T yn−1−QΩnT yn−1) +(qn−qn−1)(z0−x∗)+(qn−1−qn)(QΩnTyn−1−QΩnTx∗) +(qn−1−qn)(QΩnTx∗−QΩT x∗). Using he ollowing es ima es kQΩn+1 Tyn−QΩn+1 T yn−1k ≤ kyn−yn−1k, kQΩnTyn−1−QΩnT x∗k ≤ kyn−1−x∗k, kQΩnTx∗−QΩT x∗k ≤ M7qhX(M8σn), and kQΩn+1 Tyn−1−QΩnT yn−1k ≤ M9qhX(M10σn), o sui able cons an s M7, ..., M10 (see Lemma 2.10), we ob ain kyn+1 −ynk ≤ (1 −qn)kyn−yn−1k+ (1 −qn)M9qhX(M10σn) +|qn−qn−1|kz0−x∗k+C1) + M7qhX(M8σn). Deno ing λn=kyn−yn−1kone has λn+1 ≤(1 −qn)λn+γn, whe e γn=|qn−qn−1|(kz0−x∗k+C1) + |qn−qn−1|M7qhX(M8σn) + (1 −qn)M9qhX(M10σn). 19 Now, by (5.3), (5.4) and Lemma 2.11, we conclude ha kyn−yn−1k → 0. Nex we show ha lim n→∞ kyn−QΩTynk= 0.(5.5) Indeed, kyn−QΩTynk≤kyn−QΩnT yn−1k+kQΩnTyn−1−QΩT yn−1k +kQΩTyn−1−QΩT ynk ≤ kyn−QΩnTyn−1k+M7qhX(M8σn) + kyn−1−ynk, which s a es ou claim since kyn−yn−1k → 0 and phX(M8σn)→0 as n→ ∞, and kyn−QΩnTyn−1k=qn(z0−QΩnT yn−1) which, om he boundedness o {QΩnT yn−1}, also ends o 0. Now we w i e yn+1 −Qz0= (1 −qn)(QΩn+1 T yn−Qz0) + qn(z0−Qz0), and apply he subdi e en ial inequali y o Jψas in Theo em 3.1 o deduce ha φ(kyn+1 −Qz0k)≤φ((1 −qn)kQΩn+1 Tyn−Qz0k) + qnhz0−Qz0, Jψ(yn+1 −Qz0)i. Since Qz0∈ N(QΩT)⊆Ω, kQΩn+1 Tyn−Qz0k ≤ kQΩn+1 T yn−QΩn+1 T Qz0k+kQΩn+1 T Qz0−QΩT Qz0k ≤ kyn−Qz0k+νn, whe e νn=M7phX(M8σn+1) is bounded and anishes as n→ ∞. F om (5.4) we can ollow he same easoning as in Theo em 3.1 o ob ain ( o nla ge enough) φ(kyn+1 −Qz0k)≤(1 −qn)φ(kyn−Qz0k) + (1 −qn)νnM11+ (5.6) +qnhz0−Qz0, Jψ(yn+1 −Qz0)i. o M11 a sui able cons an . We claim ha lim sup n→∞ hz0−Qz0, Jψ(yn−Qz0)i ≤ 0.(5.7) Since {yn}is bounded he e exis s a subsequence {ynk}, ynk∈Ωnk o each k, which weakly con e ges o some poin ¯yand lim sup n→∞ hz0−Qz0, Jψ(yn−Qz0)i= lim k→∞hz0−Qz0, Jψ(ynk−Qz0)i. Now we apply he ollowing esul : 20 Lemma 5.2 [16] I he se Ωis con ex and closed se in e lexi e Banach space Xand he sequence o se s Ωn⊆Xsa is y he limi ela ion H(Ωn,Ω) →0as n→ ∞ hen e e y weak limi poin uo any sequence {un}, un∈Ωn,belongs o he subse Ω. To deduce ha ¯y∈Ω.The es o he p oo ollows he pa e n o Theo em 3.1 once we p o e ha ¯yis a ixed poin o QΩT. To p o e his we make use o Opial’s condi ion used in he ollowing o m: lim in l→∞ kynl−¯yk<lim in l→∞ kynl−QΩT¯yk ≤lim in l→∞ (kynl−QΩTynlk+kQΩT ynl−QΩT¯yk) ≤lim in l→∞ kQΩTynl−QΩT¯yk ≤ lim in l→∞ kynl−¯yk. Rema k 5.3 No ice ha in his heo em we do no ob ain con e gence o a ixed poin o T. Con e gence o a ixed poin o Tcan be ob ained i we impose ce ain bounda y condi ions on Tas, o ins ance, ha T(∂Ω) ⊆Ω. I is no ha d o see ha in his case N(QΩT) = N(T). Rema k 5.4 The sunny nonexpansi e e ac ion on o N(QΩT)always exis s in his case since QΩTis a nonexpansi e sel -mapping [8]. Rema k 5.5 I he condi ion ¯ G= Ω∩G6=∅does no hold we p o e he p e ious heo em in he ollowing way. Ins ead o Ωnwe conside he collec ion o se s Ω0 n=co(Ω ∪Ωn), whe e co(A)s ands o he closed con ex closu e o a se A. I is easy o see ha H(Ω,Ω0 n)→0 as n→ ∞. Now i su ices o ollow he same p oo . Rema k 5.6 The sequence {T yn}is bounded i , o ins ance, {yn}is bounded. This ob i- ously holds i , o ins ance, Ωis bounded. Re e ences [1] Albe , Ya.I.: Gene alized p ojec ion ope a o s in Banach spaces: p ope ies and appli- ca ions. Func ional Di e en ial Equa ions 1, 1-21 (1994). [2] Albe , Ya.I., Chidume C.E., Zegeye H.: App oxima ing ixed poin s o o al asymp o - ically nonexpansi e mapings , Fixed Poin Theo y and Applica ions 2006, a icle ID 10673, 1-20. 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