classescncn

Khanetal.
FixedPointTheoryandApplications2012,2012:149http://www.
xedpointtheoryandapplications.
com/content/2012/1/149RESEARCHOpenAccessAthree-stepiterativeschemeforsolvingnonlinearφ-stronglyaccretiveoperatorequationsinBanachspacesSafeerHussainKhan1*,ArifRaq2andNawabHussain3*Correspondence:safeer@qu.
edu.
qa1DepartmentofMathematics,StatisticsandPhysics,QatarUniversity,Doha,2713,QatarFulllistofauthorinformationisavailableattheendofthearticleAbstractInthispaper,westudyathree-stepiterativeschemewitherrortermsforsolvingnonlinearφ-stronglyaccretiveoperatorequationsinarbitraryrealBanachspaces.
Keywords:three-stepiterativescheme;φ-stronglyaccretiveoperator;φ-hemicontractiveoperator1IntroductionLetKbeanonemptysubsetofanarbitraryBanachspaceXandXbeitsdualspace.
ThesymbolsD(T),R(T)andF(T)standforthedomain,therangeandthesetofxedpointsofTrespectively(forasingle-valuedmapT:X→X,x∈XiscalledaxedpointofTiT(x)=x).
WedenotebyJthenormalizeddualitymappingfromEtoEdenedbyJ(x)=f∈X:x,f=x=f.
LetT:D(T)X→Xbeanoperator.
Thefollowingdenitionscanbefoundin[–]forexample.
DenitionTiscalledLipshitzianifthereexistsL>suchthatTx–Ty≤Lx–y,forallx,y∈K.
IfL=,thenTiscallednonexpansive,andifsuchthatforeachx,y∈D(T),thereexistsj(x–y)∈J(x–y)satisfyingReTx–Ty,j(x–y)≤tx–y.
2012Khanetal.
;licenseeSpringer.
ThisisanOpenAccessarticledistributedunderthetermsoftheCreativeCommonsAttribu-tionLicense(http://creativecommons.
org/licenses/by/2.
0),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Khanetal.
FixedPointTheoryandApplications2012,2012:149Page2of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149(ii)TissaidtobestrictlyhemicontractiveifF(T)isnonemptyandifthereexistsat>suchthatforeachx∈D(T)andq∈F(T),thereexistsj(x–y)∈J(x–y)satisfyingReTx–q,j(x–q)≤tx–q.
(iii)Tissaidtobeφ-stronglypseudocontractiveifthereexistsastrictlyincreasingfunctionφ:[,∞)→[,∞)withφ()=suchthatforeachx,y∈D(T),thereexistsj(x–y)∈J(x–y)satisfyingReTx–Ty,j(x–y)≤x–y–φx–yx–y.
(iv)Tissaidtobeφ-hemicontractiveifF(T)isnonemptyandifthereexistsastrictlyincreasingfunctionφ:[,∞)→[,∞)withφ()=suchthatforeachx∈D(T)andq∈F(T),thereexistsj(x–y)∈J(x–y)satisfyingReTx–q,j(x–q)≤x–q–φx–qx–q.
Clearly,eachstrictlyhemicontractiveoperatorisφ-hemicontractive.
Denition(i)Tiscalledaccretiveiftheinequalityx–y≤x–y+s(Tx–Ty)holdsforeveryx,y∈D(T)andforalls>.
(ii)Tiscalledstronglyaccretiveif,forallx,y∈D(T),thereexistsaconstantk>andj(x–y)∈J(x–y)suchthatTx–Ty,j(x–y)≥kx–y.
(iii)Tiscalledφ-stronglyaccretiveifthereexistsj(x–y)∈J(x–y)andastrictlyincreasingfunctionφ:[,∞)→[,∞)withφ()=suchthatforeachx,y∈X,Tx–Ty,j(x–y)≥φx–yx–y.
RemarkIthasbeenshownin[,]thattheclassofstronglyaccretiveoperatorsisapropersubclassoftheclassofφ-stronglyaccretiveoperators.
IfIdenotestheidentityoperator,thenTiscalledstronglypseudocontractive(respectively,φ-stronglypseudocon-tractive)ifandonlyif(I–T)isstronglyaccretive(respectively,φ-stronglyaccretive).
Chidume[]showedthattheManniterativemethodcanbeusedtoapproximatexedpointsofLipschitzstronglypseudocontractiveoperatorsinLp(orlp)spacesforp∈[,∞).
ChidumeandOsilike[]provedthateachstronglypseudocontractiveoperatorwithaxedpointisstrictlyhemicontractive,buttheconversedoesnotholdingeneral.
Theyalsoprovedthattheclassofstronglypseudocontractiveoperatorsisapropersubclassoftheclassofφ-stronglypseudocontractiveoperatorsandpointedoutthattheclassofφ-stronglypseudocontractiveoperatorswithaxedpointisapropersubclassoftheclassKhanetal.
FixedPointTheoryandApplications2012,2012:149Page3of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149ofφ-hemicontractiveoperators.
Theseclassesofnonlinearoperatorshavebeenstudiedbyvariousresearchers(see,forexample,[–]).
Liuetal.
[]provedthat,undercer-tainconditions,athree-stepiterationschemewitherrortermsconvergesstronglytotheuniquexedpointofφ-hemicontractivemappings.
Inthispaper,westudyathree-stepiterativeschemewitherrortermsfornonlinearφ-stronglyaccretiveoperatorequationsinarbitraryrealBanachspaces.
2PreliminariesWeneedthefollowingresults.
Lemma[]Let{an},{bn}and{cn}bethreesequencesofnonnegativerealnumberswith∞n=bnifandonlyifthereisf∈J(x)suchthatRey,f≥.
Lemma[]SupposethatXisanarbitraryBanachspaceandA:E→Eisacontinuousφ-stronglyaccretiveoperator.
ThentheequationAx=fhasauniquesolutionforanyf∈E.
3Strongconvergenceofathree-stepiterativeschemetoasolutionofthesystemofnonlinearoperatorequationsFortherestofthissection,LdenotestheLipschitzconstantofT,T,T:X→X,L=(+L)andR(T),R(T)andR(T)denotetherangesofT,TandTrespectively.
Wenowproveourmainresults.
TheoremLetXbeanarbitraryrealBanachspaceandT,T,T:X→XLipschitzφ-stronglyaccretiveoperators.
Letf∈R(T)∩R(T)∩R(T)andgenerate{xn}fromanarbitraryx∈Xbyxn+=anxn+bnf+(I–T)yn+cnvn,yn=anxn+bnf+(I–T)zn+cnun,zn=anxn+bnf+(I–T)xn+cnwn,n≥,(.
)where{vn}∞n=,{un}∞n=and{wn}∞n=areboundedsequencesinXand{an},{cn},{an},{bn},{cn},{an},{bn},{cn}aresequencesin[,]and{bn}in(,)satisfyingthefollowingcondi-tions:(i)an+bn+cn==an+bn+cn=an+bn+cn,(ii)∞n=bn=∞,(iii)∞n=bncom/content/2012/1/149eachSiisdemicontinuousandxistheuniquexedpointofSi;i=,,,andforallx,y∈X,wehave(I–Si)x–(I–Si)y,j(x–y)≥φix–yx–y≥φi(x–y)(+φi(x–y)+x–y)x–y=θi(x,y)x–y,whereθi(x,y)=φi(x–y)(+φi(x–y)+x–y)∈[,)forallx,y∈X;i=,,.
Letx∈i=F(Si)bethexedpointsetofSi,andletθ(x,y)=infmini{θi(x,y)}∈[,].
Thus(I–Si)x–(I–Si)y,j(x–y)≥θ(x,y)x–y;i=,,.
(.
)ItfollowsfromLemmaandinequality(.
)thatx–y≤x–y+λ(I–Si)x–θ(x,y)x–(I–Si)y–θ(x,y)y,(.
)forallx,y∈Xandforallλ>;i=,,.
Setαn=bn+cn,βn=bn+cnandγn=bn+cn,then(.
)becomesxn+=(–αn)xn+αnSyn+cn(vn–Syn),yn=(–βn)xn+βnSzn+cn(un–Szn),zn=(–γn)xn+γnSxn+cn(wn–Sxn),n≥.
(.
)Wehavexn=(+αn)xn++αn(I–S)xn+–θxn+,xxn+––θxn+,xαnxn+–θxn+,xαn(xn–Syn)+αn(Sxn+–Syn)–+–θxn+,xαncn(vn–Syn).
Furthermore,x=(+αn)x+αn(I–S)x–θxn+,xx––θxn+,xαnx,sothatxn–x=(+αn)xn+–x+αn(I–S)xn+–θxn+,xxn+–(I–S)x–θxn+,xx––θxn+,xαnxn–x+–θxn+,xαn(xn–Syn)+αn(Sxn+–Syn)–+–θxn+,xαncn(vn–Syn).
Khanetal.
FixedPointTheoryandApplications2012,2012:149Page5of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149Hence,xn–x≥(+αn)xn+–x+αn(+αn)(I–S)xn+–θxn+,xxn+–(I–S)x–θxn+,xx––θxn+,xαnxn–x––θxn+,xαnxn–Syn–αnSxn+–Syn–+–θxn+,xαncnvn–Syn≥(+αn)xn+–x––θxn+,xαnxn–x––θxn+,xαnxn–Syn–αnSxn+–Syn–+–θxn+,xαncnvn–Syn.
Hence,xn+–x≤[+(–θ(xn+,x))αn](+αn)xn–x+αnxn–Syn+αnSxn+–Syn++–θxn+,xαncnvn–Syn≤+–θxn+,xαn–αn+αnxn–x+αnxn–Syn+αnSxn+–Syn+cnvn–Syn≤–θxn+,xαn+αnxn–x+αnxn–Syn+αnSxn+–Syn+cnvn–Syn.
(.
)Furthermore,wehavethefollowingestimates:zn–x=(–γn)xn–x+γnSxn–x+cn(wn–Sxn)≤(–γn)xn–x+γnSxn–x+cnwn–Sxn≤(–γn)xn–x+Lγnxn–x+cnwn–x+Sxn–x≤+(L–)γn+Lcnxn–x+cnwn–x≤(L–)xn–x+cnwn–x,(.
)yn–x=(–βn)xn–x+βnSzn–x+cn(un–Szn)≤(–βn)xn–x+βnSzn–x+cnun–Szn≤(–βn)xn–x+Lβnzn–x+cnun–x+Lzn–x≤–βn+L(L–)βn+L(L–)cnxn–x+Lβncn+Lcncnwn–x+cnun–x≤L(L–)–xn–x+Lcnwn–x+cnun–x,(.
)Khanetal.
FixedPointTheoryandApplications2012,2012:149Page6of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149xn–Syn≤xn–x+Lyn–x≤+LL(L–)–xn–x+Lcnwn–x+Lcnun–x,(.
)Sxn+–Syn≤Lxn+–yn=L(–αn)(xn–yn)+αn(Syn–yn)+cn(vn–Syn)≤L(–αn)xn–yn+αnSyn–yn+cnvn–Syn≤Lxn–yn+αnSyn–yn+cnvn–Syn.
(.
)Using(.
)and(.
),xn–yn=βn(xn–Szn)–cn(un–Szn)≤βnxn–Szn+cnun–Szn≤+L(L–)βn+L(L–)cnxn–x+Lβn+cncnwn–x+cnun–x≤+L(L–)βn+L(L–)cnxn–x+Lcnwn–x+cnun–x.
(.
)Using(.
),Syn–yn≤Syn–x+yn–x≤(+L)yn–x≤(+L)[L(L–)–]xn–x+L(+L)cnwn–x+(+L)cnun–x.
(.
)Again,using(.
),vn–Syn≤vn–x+Lyn–x≤LL(L–)–xn–x+vn–x+Lcnwn–x+Lcnun–x.
(.
)Substituting(.
)-(.
)in(.
),weobtainSxn+–Syn≤L+L(L–)βn+L(L–)cn+L(L–)–(+L)αn+Lcnxn–x+LLcn+(+L)αn+Lcncnwn–x+Lcn+(+L)αn+Lcncnun–x+Lcnvn–x.
(.
)Khanetal.
FixedPointTheoryandApplications2012,2012:149Page7of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149Substituting(.
),(.
)and(.
)in(.
),weobtainxn+–x≤++L(+L)L(L–)–αn+LL(L–)–αnβn+L(L–)αncn+LL(L–)–αncn+LL(L–)–cnxn–x–θxn+,xαnxn–x+L(+L)αncn+Lαncn+Lαncncn+Lcncnwn–x+L(+L)αncn+Lαncn+Lαncncn+Lcncnun–x+(L+)cnvn–x.
(.
)Since{vn},{un}and{wn}arebounded,wesetM=supn≥vn–x+supn≥un–x+supn≥wn–xcncn+LcncnM+L(+L)αncn+Lαncn+Lαncncn+LcncnM+(L+)cnM=(+δn)xn–x–θxn+,xαnxn–x+σn≤(+δn)xn–x+σn,(.
)whereδn=+L(+L)L(L–)–αn+LL(L–)–αnβn+L(L–)αncn+LL(L–)–αncn+LL(L–)–cn,σn=ML(+L)αncn+Lαncn+Lαncncn+LcncnL(+L)αncn+Lαncn+Lαncncn+Lcncn+(L+)cn.
Sincebn∈(,),theconditions(iii)and(iv)implythat∞n=δn.
ThenthereexistsapositiveintegerNsuchthatxn–x≥δforalln≥N.
Sinceθxn+,xxn–x=φ(xn+–x)+φ(xn+–x)+xn+–xxn–x≥φ(δ)δ(+φ(D)+D),Khanetal.
FixedPointTheoryandApplications2012,2012:149Page8of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149foralln≥N,itfollowsfrom(.
)thatxn+–x≤xn–x–φ(δ)δ(+φ(D)+D)αn+λnforalln≥N.
Hence,φ(δ)δ(+φ(D)+D)αn≤xn–x–xn+–x+λnforalln≥N.
Thisimpliesthatφ(δ)δ(+φ(D)+D)nj=Nαj≤xN–x+nj=Nλj.
Sincebn≤αn,φ(δ)δ(+φ(D)+D)nj=Nbj≤xN–x+nj=Nλjyields∞n=bnorTix→∞asx→∞;i=,,.
Let{an},{bn},{cn},{an},{bn},{cn},{an},{bn},{cn},{wn},{un},{vn},{yn}and{xn}beasinTheorem.
Then,foranygivenf∈X,thesequence{xn}convergesstronglytothesolutionofthesystemTix=f;i=,,.
ProofTheexistenceofauniquesolutiontothesystemTix=f;i=,,followsfrom[]andtheresultfollowsfromTheorem.
TheoremLetXbearealBanachspaceandKbeanonemptyclosedconvexsubsetofX.
LetT,T,T:K→KbethreeLipschitzφ-strongpseudocontractionswithanonemptyxedpointset.
Let{an},{bn},{cn},{an},{bn},{cn},{an},{bn},{cn},{wn},{un}and{vn}beasinTheorem.
Let{xn}bethesequencegeneratediterativelyfromanarbitraryx∈Kbyxn+=anxn+bnTyn+cnvn,yn=anxn+bnTzn+cnun,zn=anxn+bnTxn+cnwn,n≥.
Then{xn}convergesstronglytothecommonxedpointofT,T,T.
ProofAsintheproofofTheorem,setαn=bn+cn,βn=bn+cn,γn=bn+cntoobtainxn+=(–αn)xn+αnTyn+cn(vn–Tyn),yn=(–βn)xn+βnTzn+cn(un–Tzn),Khanetal.
FixedPointTheoryandApplications2012,2012:149Page9of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149zn=(–γn)xn+γnTxn+cn(wn–Txn),n≥.
SinceeachTi;i=,,isaφ-strongpseudocontraction,(I–Ti)isφ-stronglyaccretivesothatforallx,y∈X,thereexistj(x–y)∈J(x–y)andastrictlyincreasingfunctionφ:(,∞)→(,∞)withφ()=suchthat(I–Ti)x–(I–Ti)y,j(x–y)≥φx–yx–y≥θ(x,y)x–y;i=,,.
TherestoftheargumentnowfollowsasintheproofofTheorem.
RemarkTheexamplein[]showsthattheclassofφ-stronglypseudocontractiveoper-atorswithnonemptyxedpointsetsisapropersubclassoftheclassofφ-hemicontractiveoperators.
ItiseasytoseethatTheoremeasilyextendstotheclassofφ-hemicontractiveoperators.
Remark(i)Ifwesetbn==cnforalln≥inourresults,weobtainthecorrespondingresultsfortheIshikawaiterationschemewitherrortermsinthesenseofXu[].
(ii)Ifwesetbn==cn=bn==cnforalln≥inourresults,weobtainthecorrespondingresultsfortheManniterationschemewitherrortermsinthesenseofXu[].
RemarkLet{αn}and{βn}berealsequencessatisfyingthefollowingconditions:(i)≤αn,βn≤,n≥,(ii)limn→∞αn=limn→∞βn=,(iii)∞n=αn=∞,(iv)∞n=βncommonsolutionforanitefamilyofφ-stronglyaccretiveoperatorequationsinareexiveBanachspacewithweaklycontinuousdualitymapping.
Someremarksontheirworkcanbeseenin[].
(ii)Alltheaboveresultscanbeextendedtoanitefamilyofφ-stronglyaccretiveoperators.
CompetinginterestsTheauthorsdeclarethattheyhavenocompetinginterests.
Authors'contributionsAlltheauthorsstudiedandapprovedthemanuscript.
Authordetails1DepartmentofMathematics,StatisticsandPhysics,QatarUniversity,Doha,2713,Qatar.
2HajveryUniversity,43-52IndustrialArea,Gulberg-III,Lahore,Pakistan.
3DepartmentofMathematics,KingAbdulazizUniversity,P.
O.
Box80203,Jeddah,21589,SaudiArabia.
Khanetal.
FixedPointTheoryandApplications2012,2012:149Page10of10http://www.
xedpointtheoryandapplications.
com/content/2012/1/149AcknowledgementsThelastauthorgratefullyacknowledgesthesupportfromtheDeanshipofScienticResearch(DSR)atKingAbdulazizUniversity(KAU)duringthisresearch.
Received:30June2012Accepted:29August2012Published:12September2012References1.
Chidume,CE:IterativeapproximationofxedpointsofLipschitzstrictlypseudo-contractivemappings.
Proc.
Am.
Math.
Soc.
99,283-288(1987)2.
Chidume,CE:Iterativesolutionofnonlinearequationswithstronglyaccretiveoperators.
J.
Math.
Anal.
Appl.
192,502-518(1995)3.
Chidume,CE:IterativesolutionsofnonlinearequationsinsmoothBanachspaces.
NonlinearAnal.
TMA26,1823-1834(1996)4.
Chidume,CE,Osilike,MO:FixedpointiterationsforstrictlyhemicontractivemapsinuniformlysmoothBanachspaces.
Numer.
Funct.
Anal.
Optim.
15,779-790(1994)5.
Chidume,CE,Osilike,MO:IshikawaiterationprocessfornonlinearLipschitzstronglyaccretivemappings.
J.
Math.
Anal.
Appl.
192,727-741(1995)6.
Deimling,K:Zerosofaccretiveoperators.
Manuscr.
Math.
13,365-374(1974)7.
Liu,LW:Approximationofxedpointsofastrictlypseudocontractivemapping.
Proc.
Am.
Math.
Soc.
125,1363-1366(1997)8.
Liu,Z,Kang,SM:ConvergenceandstabilityoftheIshikawaiterationprocedureswitherrorsfornonlinearequationsoftheφ-stronglyaccretivetype.
NeuralParallelSci.
Comput.
9,103-118(2001)9.
Liu,Z,Bouxias,M,Kang,SM:Iterativeapproximationofsolutiontononlinearequationsofφ-stronglyaccretiveoperatorsinBanachspaces.
RockyMt.
J.
Math.
32,981-997(2002)10.
Kim,JK,Liu,Z,Kang,SM:AlmoststabilityofIshikawaiterativeschemeswitherrorsforφ-stronglyquasi-accretiveandφ-hemicontractiveoperators.
Commun.
KoreanMath.
Soc.
19(2),267-281(2004)11.
Osilike,MO:Iterativesolutionofnonlinearequationsoftheφ-stronglyaccretivetype.
J.
Math.
Anal.
Appl.
200,259-271(1996)12.
Osilike,MO:Iterativesolutionofnonlinearφ-stronglyaccretiveoperatorequationsinarbitraryBanachspaces.
NonlinearAnal.
36,1-9(1999)13.
Raq,A:Iterativesolutionofnonlinearequationsinvolvinggeneralizedφ-hemicontractivemappings.
IndianJ.
Math.
50(2),365-380(2008)14.
Tan,KK,Xu,HK:IterativesolutionstononlinearequationsofstronglyaccretiveoperatorsinBanachspaces.
J.
Math.
Anal.
Appl.
178,9-21(1993)15.
Xu,Y:IshikawaandManniterativeprocesseswitherrorsfornonlinearstronglyaccretiveoperatorequations.
J.
Math.
Anal.
Appl.
224,91-101(1998)16.
Chidume,CE,Osilike,MO:Nonlinearaccretiveandpseudo-contractiveoperatorequationsinBanachspaces.
NonlinearAnal.
TMA31,779-789(1998)17.
Ding,XP:Iterativeprocesswitherrorstononlinearφ-stronglyaccretiveoperatorequationsinarbitraryBanachspaces.
Comput.
Math.
Appl.
33,75-82(1997)18.
Ishikawa,S:Fixedpointbyanewiterationmethod.
Proc.
Am.
Math.
Soc.
44,147-150(1974)19.
Kamimura,S,Khan,SH,Takahashi,W:IterativeschemesforapproximatingsolutionsofrelationsinvolvingaccretiveoperatorsinBanachspaces.
FixedPointTheoryAppl.
5,41-52(2003)20.
Khan,SH,Hussain,N:Convergencetheoremsfornonself-asymptoticallynonexpansivemappings.
Comput.
Math.
Appl.
55,2544-2553(2008)21.
Khan,SH,Yildirim,I,Ozdemir,M:ConvergenceofageneralizediterationprocessfortwonitefamiliesofLipschitzianpseudocontractivemappings.
Math.
Comput.
Model.
53,707-715(2011).
doi:10.
1016/j.
mcm.
2010.
10.
00722.
Miao,Y,Khan,SH:StrongconvergenceofanimplicititerativealgorithminHilbertspaces.
Commun.
Math.
Anal.
4(2),54-60(2008)23.
To-MingLau,A:SemigroupofnonexpansivemappingsonaHilbertspace.
J.
Math.
Anal.
Appl.
105,514-522(1985)24.
To-MingLau,A:InvariantmeansandsemigroupsofnonexpansivemappingsonuniformlyconvexBanachspaces.
J.
Math.
Anal.
Appl.
153,497-505(1990)25.
To-MingLau,A:Fixedpointpropertiesofsemigroupsofnonexpansivemappings.
J.
Funct.
Anal.
254,2534-2554(2008)26.
Liu,Z,An,Z,Li,Y,Kang,SM:Iterativeapproximationofxedpointsforφ-hemicontractiveoperatorsinBanachspaces.
Commun.
KoreanMath.
Soc.
19(1),63-74(2004)27.
Tan,KK,Xu,HK:ApproximatingxedpointsofnonexpansivemappingsbytheIshikawaiterationprocess.
J.
Math.
Anal.
Appl.
178,301-308(1993)28.
Kato,T:Nonlinearsemigroupsandevolutionequations.
J.
Math.
Soc.
Jpn.
19,508-520(1967)29.
Gurudwan,N,Sharma,BK:Approximatingsolutionsforthesystemofφ-stronglyaccretiveoperatorequationsinreexiveBanachspace.
Bull.
Math.
Anal.
Appl.
2(3),32-39(2010)30.
Mann,WR:Meanvaluemethodsiniteration.
Proc.
Am.
Math.
Soc.
4,506-510(1953)31.
Raq,A:Oniterationsforfamiliesofasymptoticallypseudocontractivemappings.
Appl.
Math.
Lett.
24(1),33-38(2011)doi:10.
1186/1687-1812-2012-149Citethisarticleas:Khanetal.
:Athree-stepiterativeschemeforsolvingnonlinearφ-stronglyaccretiveoperatorequationsinBanachspaces.
FixedPointTheoryandApplications20122012:149.