ON H-TRICHOTOMY IN BANACH SPACES

Transcription

1 CODRUTA STOICA IHAIL EGA O H-TRICHOTOY I BAACH SPACES Absrac: I his papr w mphasiz h oio of skw-oluio smiflows cosidrd a gralizaio of smigroups oluio opraors ad skw-produc smiflows which aris i h sabiliy hory. W dfi ad characriz a paricular cas of richoomy calld h H-richoomy which is usful i dscribig h bhaiors of h soluio of oluio quaios. W mphasiz h fac ha h richomy iroducd i fii dimsios i [] ad [5] is a aural gralizaio of dichoomy. A similar cocp for sabiliy was sudid for oluio opraors i []. This papr cosidrs also ohr asympoic propris as xpoial growh ad dcay sabiliy ad isabiliy. ahmaics Subjc Classificaio: 4D9 Kywords: oluio quaio skw-oluio smiflow H-richoomy ITRODUCTIO Th cocp of skw-oluio smiflows ariss i h hory of oluio quaios which as wll as h hory of opimal corol is a impora ool i dscribig procsss drid from girig or coomics. Th dyamical sysms ha sudy h ral lif phoma ar complx ad h idificaio of appropriad mahmaical modls is difficul bcaus i h cas of sysms dscribd by parial diffrial quaios h sa spac is of of ifii dimsio. I is irsig o rcosidr h dfiiios of asympoic propris for diffrial quaios by mas of skwoluio smiflows. I wha follows w will cosidr a mor gral cas for asympoic bhaiors ha o iols cssarily xpoials bu isad proprly dfid fucios. L us dfi h s Γ of all coiuous fucios H : R. W will do by Θ h s of all fucios f : R wih h propry ha hr xiss a cosa μ R such ha f = μ wih h subss Θ ad Θ for posii rspcily gai alus of μ. By Ψ is dod h s of coiuous fucios h : R [ dfid such ha for all H Γ hr xis a fucio f Θ ad a cosa k > wih h propris hs kf s H s ad hh s H s s. Rmark.. Th s Ψ is o mpy as w ca cosidr ν ν h = f = ad H = ν >. W will mphasiz h oio of skw-oluio smiflows by mas of oluio smiflows ad oluio cocycls as iroducd by us i [4]. Thy aurally graliz oios as opraors smigroups oluio opraors or skwproduc smiflows. A skw-oluio smiflow dpds o hr ariabils corary o a skwproduc smiflow which dpds oly o wo ad hc h sudy of asympoic bhaiors for skw-oluio smiflows i h ouiform copyrigh FACULTY of EGIEERIG HUEDOARA ROAIA 5

2 ACTA TECHICA CORVIIESIS BULLETI of EGIEERIG sig ariss as aural rlai o h hird ariabil. I his papr w will also cosidr h dfiiios ad characrizaios of som asympoic propris by mas of h s of fucios Θ Γ ad Ψ. SKEW-EVOLUTIO SEIFLOWS L us cosidr X d a mric spac V a ral or complx Baach spac V is opological dual ad BV h family of liar V-alud boudd opraors dfid o V. Th orm of cors ad opraors is. I wha follows w will do Y = X V ad w will cosidr h s T = { R }. By I is dsigd h idiy opraor o V. Dfiiio.. A mappig ϕ : T X X wih h propris: s ϕ x = x x R X ; s ϕ s ϕ s x = ϕ x ss T x X is calld oluio smiflow o X. Dfiiio.. A mappig : T X BV wih h propris: c x = I x R X ; c s ϕs x sx = x ss T x X is calld oluio cocycl or h oluio smiflow ϕ. Dfiiio.. Th mappig C : T Y Y Csx = ϕsx s x whr ϕ is a oluio smiflow o X ad h mappig is a oluio cocycl or ϕ is calld skw-oluio smiflow o Y. Th x xampl mphasizs a skw-oluio smiflow grad by a sysm of diffrial quaios. Exampl.. L us cosidr h sysm of diffrial quaios u = si u w = cos w z = cos z. L us dfi h spacs X = R ad V = R which is dowd wih h orm = whr = V. Th mappig ϕ : T R ϕ s x = - s x is a oluio smiflow o R. Th mappig : T X B V sx Us W s Zs = whr U s=uu - s W s=ww - s Z s=zz - s s T ad u w ad z whr R ar h soluios of h gi sysm of diffrial quaios is a oluio cocycl or h oluio smiflow ϕ o h mric spac R. W obai ha C = ϕ is a skw-oluio. Th followig asympoic bhaiors of skwoluio smiflow ar usful i characrizig h propry of H-richoomy as wll as hir characrizaios. Dfiiio.. A skw-oluio smiflow C = ϕ is said o ha xpoial growh if hr xiss a odcrasig fucio g : R [ wih h propry lim g = such ha: x g s s x ss T x Y. Proposiio.. A skw-oluio smiflow C = ϕ has xpoial growh if ad oly if hr xis som cosas ad ω > such ha: x s s x ω ss T x Y. Proof. cssiy. L s ad b h igr par of h ral umbr - s. W obai succssily x g x... [ g] x ω s s x s x for all ss T ad all x Y whr w ha dod = g > ad ω = l >. Sufficicy. I is obaid immdialy if w ωu cosidr gu = u. ω Dfiiio.. A skw-oluio smiflow C = ϕ is said o b wih xpoial dcay if hr xiss a odcrasig fucio g : R [ wih h propry lim g = such ha: 54 9/Fascicul /Jauary arch/tom II

3 ACTA TECHICA CORVIIESIS BULLETI of EGIEERIG s x g s x ss T x Y. H s s x P x x P x ; Proposiio.. A skw-oluio smiflow C = ϕ has xpoial dcay if ad oly if hr xis som cosas ad ω > such ha: H s s x P x P x x ; ω s s x x ss T x Y. Proof. cssiy. L s. Thr xiss a aural umbr such ha. W ha followig rlaios s x... [ g] g s s x x ω s x x ω for all ss T ad all x Y whr w ha cosidrd h cosas = g > ad ω = l >. Sufficicy. I follows immdialy for ωu gu = u O THE PROPERTY OF H-TRICHOTOY A gral cocp of xpoial richoomy is mphasizd i his scio. Dfiiio.. A mappig P:Y Y gi by Px=x Px whr Px is a projcio o Y x = {x} V ad x X is calld projcor o Y. Dfiiio.. A skw-oluio smiflow C = ϕ is said o b H-richoomic if hr xis som mappigs : projcors familis {P k } k followig codiios hold: R ad hr {} such ha R for ach projcor P k k {} h rlaio P ϕ sx s x = s xpx holds for all s T ad all x X; for all x X h projcios P x P x ad P x saisfy h codiios P xp xp x=i ad P i xp j x= for all i j {} i j; followig iqualiis x P x s H s x P x ad s xp x Hs xp x hold for all ss T for all x Y ad all H Γ. ν Rmark.. I h paricular cas H = ν > h xpoial richoomy for skw-oluio smiflows dfid ad characrizd by us i [] for oluio opraors is obaid i a ouiform sig. Rmark.. i A projcor P o Y wih propry is also calld iaria rlai o h skwoluio smiflow C = ϕ ; ii If hr projcors familis {P k } k {} saisfy rlaios ad of Dfiiio. hy ar usually said o b compaibl wih h skwoluio smiflow C. I wha follows w will do a skw-oluio smiflow C k = ϕ k k {} whr k s x = sx P k x s T x Y. Exampl.. L us cosidr h skw-oluio smiflow gi i Exampl.. W obai for h oluio cocycl : T X BV followig rlaios = s x si s si cos cos s s si si s s = cos s cos s si si s s L us dfi h projcors P x= P x= ad P x=. As followig rlaio holds cos - s cos s - si si s - s - 5s 4 s T w ha ha 9/Fascicul /Jauary arch/tom II 55

4 ACTA TECHICA CORVIIESIS BULLETI of EGIEERIG H s x P x s s x T Y whr w ha dod H = ad 5s 4 s =. Accordig o h iqualiy si s si s cos - cos s - s s - s T i follows ha s x P x H s s x T Y whr w ha cosidrd H = s s =. Also as - si si s - s s s T w ha s xp x sh s x T Y ad as - si si s - s - s - s T w obai H s s xp x ad s x T Y whr i boh cass w ha dod u u H u = ad u =. As a rmark w ca cosidr wihou ay loss of graliy h fucio dod H = mi{h H H }. I follows ha h skw-oluio smiflow C = ϕ is H-richoomic. Th x mai rsul of his papr ca b cosidrd as a igral characrizaio for h cocp of H-richoomy. Thorm.. L H Γ ad h Ψ. A skwoluio smiflow C = ϕ is H-richoomic if ad oly if hr xis som mappigs : R som fucios f f Θ ad hr projcors familis {P k } k {} compaibl wih C such ha h skw-oluio smiflow C has xpoial growh ad h skw-oluio smiflow C has xpoial dcay ad such ha followig codiios hold: i ii H h τ τx dτ P x ; h τ x dτ x ; H τ iii f τ s τ x dτ s x ; s i f τ τ x dτ x s for all s s T ad all x Y V wih. Proof. cssiy. As h skw-oluio smiflow C is H-richoomic i implis ha h rlaios of Dfiiio. hold. i Thr xis a fucio f Θ ad a cosa k > wih h propry hs kf sh s. L us do f = ν >. W obai s x s s H s s x x for all ss T ad for all x Y whr w ha cosidrd h fucio : R u u = k. hu W obai furhr H k h τ τ P x τ x τ x d τ d τ whr w ha dod u = kν u u. ii Thr xis a fucio f Θ ad a cosa k > wih h propry kfs Hs s. L us h cosidr f = ν >. W ha s x x H s s k x h k h ν s s s x x 56 9/Fascicul /Jauary arch/tom II

5 ACTA TECHICA CORVIIESIS BULLETI of EGIEERIG for all ss T ad for all x Y whr w ha dod h νu fucio : R u = k u. iii ad i ar obaid by a similar argumaio accordig o Proposiio. ad Proposiio.. Sufficicy. i L ad s [ ]. As H Γ ad h Ψ hr xiss a cosa α > such ha α s hs H for all s T. Th as h skw-oluio smiflow C has xpoial growh accordig o Proposiio. hr xis som cosas ad ω > such ha followig rlaios hold α ω α ω α τ x x = α τ x dτ α τ ω τ τ ϕ τ x τ x dτ α τ P x τ ϕ τ P x x dτ By akig suprmum rlai o w ha α x P x for all ad all x Y whr α ω u = u u. O h ohr had for [ ] ad x Y w obai ω α x ˆ whr w ha dod ˆ follows ha α ω =. Hc i α x [ ˆ ] for all T ad for all x Y. Furhr if w cosidr H u = fu ad u = [ u ˆ ]fu νu whr fu = Θ ad u w obai rlaio. ii W ha cosidrd H Γ ad h Ψ hc hr xiss a cosa β > such ha β s hs H for all s T. As h skwoluio smiflow C has xpoial growh accordig o Dfiiio. hr xiss a odcrasig fucio g : [ wih R h propry lim g = such ha s x g s x ss T x Y. W will do βτ K = g τdτ. W obai succssily followig rlaios β τ K P x = g τ x dτ β τ β τ x dτ x for all T ad for all x Y. Accordig o Dfiiio. his rlaio is quial wih β s s x x K for all ss T ad for all x Y. If w ak H u = fu ad u ufu = u for fu = Θ ad u rlaio is obaid. iii ad i ca simmilarly b prod. COCLUSIO I h las dcads a gra progrss cocrig h sudy of asympoic bhaiors for oluio quaios ca b obsrd. Th possibiliy of rducig h oauoomous cas i h sudy of oluioary familis or skw-produc flows o h auoomous cas of oluio smigroups o Baach spacs is cosidrd a impora way oward irsig applicaios. Th sudy of h asympoic bhaior of liar skw-produc smiflows has b usd i h hory of oluio quaios i ifii dimsioal spacs. Th approach from h poi of iw of asympoic propris for h oluio 9/Fascicul /Jauary arch/tom II 57

6 ACTA TECHICA CORVIIESIS BULLETI of EGIEERIG smigroup associad o h liar skw-produc smiflows was ssial. Isad i our sudy w ha cosidrd mor gral characrizaios for h asympoic propris of h soluios of oluio quaios dscribd by mas of skw-oluio smiflows which graliz h abo oios. Also h approach was o rsraid by cosidrig i h dfiiios xpoials. As a rmark i Dfiiio. w ha h dfiiios for H-sabiliy H-isabiliy H-growh ad H-dcay characrizd rspcily by Thorm. which xds oward applicaios i girig ad coomics h sudy of oluio quaios. AUTHORS & AFFILIATIO CODRUTA STOICA IHAIL EGA DEPARTET OF ATHEATICS AD COPUTER SCIECE UIVERSITY AUREL VLAICU OF ARAD ROAIA DEPARTET OF ATHEATICS WEST UIVERSITY OF TIISOARA ROAIA ACKOWLEDGET This work is fiacially suppord by h Rsarch Gra CCSIS P II ID 8 of h Romaia iisry of Educaio Rsarch ad Ioaio. REFERECES [] S. ELAYDI O. HAJEK Expoial richoomy of diffrial sysms J. ah. Aal. Appl []. EGA O H-sabiliy of oluio opraors Prpri Sris i ahmaics Ws Uirsiy of Timisoara []. EGA C. STOICA O uiform xpoial richoomy of oluio opraors i Baach spacs Igral Equaios Opraor Thory 6 o [4]. EGA C. STOICA Expoial isabiliy of skw-oluio smiflows i Baach spacs Sudia Ui. Babs-Bolyai ah. LIII o [5] R.J. SACKER G.R. SELL Exisc of dichoomis ad iaria spliigs for liar diffrial sysms III J. Diffrial Equaios /Fascicul /Jauary arch/tom II