Stabilized Approximations of Strongly Continuous Semigroups

Transcription

1 Sabilized Approxiaios of Srogly Coiuous Seigroups Sarah McAlliser ad Frak Neubrader Sarah McAlliser, IBM T.J. Waso Research Ceer, Yorkow Heighs, NY 0 ad Frak Neubrader, Depare of Maheaics, Louisiaa Sae Uiversiy, Bao Rouge, LA 00 Absrac This paper iroduces sabilizaio echiques for irisically usable, high accuracy raioal approxiaio ehods for srogly coiuous seigroup. The ehods o oly sabilize he approxiaios, bu iprove heir speed of covergece by a agiude of up o /. Key words: srogly coiuous seigroups, Cheroff produc forula, raioal approxiaios of he expoeial, sabilizaio, ie discreizaio 000 MSC: A0, A, A0, L0, M0, M. Preliiaries The Lax-Richyer Equivalece Theore ad he Cheroff Produc Forula play a ipora role i he aalysis of approxiaio ehods for seigroups. Oe of he ai igredies i boh resuls is he sabiliy of he approxiaio ehod uder cosideraio. This paper deals wih varias of hese heores ha cover sabilizaio echiques for irisically usable approxiaio ehods. To pu he resuls io perspecive, recall he saee of he Lax-Richyer Equivalece Theore. Suppose ha A geeraes a srogly coiuous seigroup T ( ) oabaachspacex ad le {V (); [0,τ]} be a approxiaio schee of bouded liear operaors wih V (0) = I ha saisfies he cosisecy codiio V ()x x li = Ax 0 Eail address: sccalli@us.ib.co ad eubrad@ah.lsu.edu (Sarah McAlliser ad Frak Neubrader). Prepri subied o Elsevier March 00

2 for all x i a se D D(A) (he doai of A) haisdeseix. Theiwasshow by Lax ad Richyer i [] (wih a sroger cosisecy codiio) ad i fial for by Cheroff i [] ha he followig saees are equivale (for also [0]): (i) V (0) = I is sable; i.e, here exis ω 0adM,τ > 0 such ha V ( ) Me ω for each [0,τ]ad /τ. (ii) li V ( ) x = T ()x for all 0adx X. If ore iforaio o he approxiaio schees is available, error esiaes ca be give. Based o earlier work by Hersh ad Kao [], his was doe by Breer ad Thoée [] for approxiaio schees V () :=r(a) defied by raioal fucios r = P Q (P, Q polyoials) wih he followig wo properies. (a) The Maclauri series of r coicides wih he expoeial series for he firs ers; i.e., r(z) e z = O( z + )as z 0. (b) The fucio r is A-sable; i.e., r(z) forre(z) 0. The uber is called he approxiaio order of r = P Q.Iisawellkowresulof Padé [] ha deg(p )+deg(q) for all raioal approxiaios of he expoeial fucio. Raioal approxiaios of he expoeial for which =deg(p )+deg(q) are called Padé approxiaios. They are of he for r = P Q,where P (z) = deg(p ) deg(q) ( j)! deg(p )!!j!(deg(p ) j)! zj ad Q(z) = ( j)! deg(q)!!j!(deg(q) j)! ( z)j. As show i [], Padé schees are A-sable if ad oly if deg(q) deg(p ) deg(q). If r is a A-sable approxiaio of he expoeial of approxiaio order ad A geeraes a srogly coiuous seigroup, he he operaors V () := r(a) are cosise (see, for exaple, [], Th.). Exaples are (a) r(z) = z =+z + z + z + (Backward Euler, =, Padé ype), (b) r(z) = +z z = + z z =+z + + z + (Crak-Nicolso, =, Padé ype), (c) r(z) =c + c bz + c ( bz), where b = (+ ), c =,c = c, ad c = (Calaha, =, o Padé ype). If r is a A-sable raioal approxiaio of he expoeial ad V () := r(a) where A geeraes a srogly coiuous seigroup T (), he he error V ( ) x T ()x ca be esiaed (see Theore. below). The esiaes ca be iproved if r saisfies he codiio ( ); i.e., if (i) r(is) < fors R\{0} ad r( ) <, (ii) r(is) =e is+ψ(s) wih ψ(s) =O( s q+ ) for soe posiive ieger q as s 0 (iii) Reψ(s) γs p for s ad soe eve ieger p q +. If r is a A-sable raioal approxiaio of he expoeial ad if eiher (a) A geeraes a bouded aalyic seigroup or (b) A geeraes a srogly coiuous seigroup T ( ) o a Hilber space saisfyig T () e ω, he he approxiaio schee V () := r(a) is sable (see, [] ad []). I geeral, he followig resul is he bes possible. Theore. (Breer-Thoée) Le A geerae a srogly coiuous seigroup T ( ) wih T () Me ω.ifv () :=r(a) for soe A-sable raioal approxiaio r of he expoeial, he V ( ) is cosise bu ay o be sable. However, here are cosas C, κ such ha

3 V ( ) CM e ωκ (.) for all 0. Moreover, if r saisfies codiio ( ), he V ( ) CM (/ (q+)/p) e ωκ, (.) where p ad q are as i he defiiio of codiio ( ). The esiae (.) is sharp for he Crak-Nicolso schee r(z) = +z z ad A = d/dx o C 0 (R) (see []) or o he closure of D(A) il (R)(see []). Exaples of A-sable raioal approxiaios r = P Q saisfyig he codiio ( ) are Padé approxiaios wih deg(p )=deg(q) ordeg(p )=deg(q). For hese cases, he codiio ( ) holdswih q =deg(p )+deg(q) adp =deg(q) (.) (see [], [], []). Le r = P Q be of Padéypewihdeg(P )=deg(q). The i was show by A. Ashyralyev ad P. Sobolevskii ha V () = r(a) is sable if ad oly if A geeraes a srogly coiuous seigroup (see [], Theore II...) Observe ha he codiio deg(p )=deg(q) iplies ha he Padé approxiaio order =deg(p )+deg(q) us be odd; i.e., i geeral, approxiaio schees V () = r(a) ay o sable for Padé approxiaios of eve order (see also Table below). Exaples of raioal Padé approxiaios of odd order are he Backward Euler schee ( = ), he hird-order schee r(z) = +z z+z = C b + C z b wih C = + z.i ad b, =± i; ad he fifh-order Radau IIA schee r(z) = wih 0+z+z 0 z+z z poles a.,. ±.00i. For geeraors A of aalyic seigroups T ( ), he error aalysis of V ( ) x T ()x for raioal approxiaio schees V () = r(a) is well udersood. For such seigroups, he followig key esiaes are due o M. Crouzeix, S. Larsso, S. Piskarev, V. Thoée ad L.B. Wahlbi (see [] ad []). For ay A-sable raioal approxiaio schee r ad V () =r(a) oe has ha V ( ) is sable ad V ( ) x T ()x M A s x for s all x D(A s )(0 s ). Moreover, if deg(p ) < deg(q), he V ( ) x T ()x M x for all x X. These esiaes are based, o a large degree, o he Duford- Riesz fucioal calculus r(a)x := πi Γ r(λ)r(λ, A)xdλ,whereR(λ, A) :=(λi A). I he oaalyic case, he ai idea i he Breer-Thoée ad Hersh-Kao papers is o replace he Duford-Riesz fucioal calculus by he Hille-Phillips fucioal calculus r(a)x := T (s)x dµ(s), where T ( ) is a bouded, srogly coiuous seigroup 0 geeraed by A ad r(z) = ˆ 0 e sz dµ(s) (.) for Re(z) 0 ad soe oralized fucio µ of oal bouded variaio; see also [], []. To see how his fucioal calculus leads o error esiaes for V ( ) x T ()x wih V () := r(a), observe firs ha usig parial fracios, oe obais ha all A- sable raioal fucios r ca be represeed as i (.). Now ake a A-sable raioal approxiaio r of he expoeial of order such ha r(z) e z M z + for z sufficiely sall. If Re(z) 0, he bioial forula yields

4 r( z) e z = r( z) (e z ) = r( z)j (e z ) j r( z) e z M z + 0 as.observehar( z) = e sz dµ 0, (s), where µ, (s) ishe-h covoluio of µ( s ) wih iself ad ha ez = 0 e sz dh (s), where H is he oralized Heaviside fucio wih jup a ie 0. Sice ˆ 0 e sz d[µ, (s) H (s)] = r( z) e z 0 as, i is o surprisig ha µ, H i he L -or (wih precise error esiaes). I has o be poied ou ha he proof of hese facs is o-rivial ad a he hear of he Breer-Thoée ad Hersh-Kao papers; for Laplace rasfor proofs of hese saees, see [] or [0]. By he Hille-Phillips fucioal calculus, V ( ) x T ()x = = ˆ 0 ˆ 0 T (s)xd[µ, (s) H (s)] [µ, (s) H (s)]t (s)ax ds for all x D(A). Therefore, for bouded seigroups, V ( ) x T ()x C µ, H Ax 0 as for all x D(A), Togeher wih error-esiaes for he L -or of µ, H ad is aiderivaives, his is he ai srucure of he proof of he Breer-Thoée resul below (which appears i ore geeral for as Theore.. of [] ad is a refiee of groud-breakig resuls of Hersh ad Kao []; see also [0]). Theore. (Breer-Thoée) Le A be he geeraor of a srogly coiuous seigroup T ( ) wih T () Me ω.ler be a A-sable raioal approxiaio of he expoeial of approxiaio order ad defie V () :=r(a). Ifs =0,,,..., + wih s +, he here are posiive cosas c ad C (depedig oly o r) such ha V ( ) x T ()x CMe cω s ( ) β(s) A s x (.) for every 0, N, ad x D(A s ),where s β(s) := s + If s = +,he V ( ) x T ()x CMe cω + for every 0, N, ad x D(A + ). if 0 s< +, if + <s +. ( ) / l( +) A + x (.)

5 For sable raioal approxiaio schees he facor l( +) ca be reoved fro (.) above (his was show for = i [] ad, for arbirary, i [], Corollary.. or [], Corollary.). Moreover, if r saisfies codiio ( ), he Theore. holds wih β(s) replaced by β (s) :=s ( q q + +i 0, (s ( (q +)) q + )), (.) p where p ad q are as i he defiiio of codiio ( ). The followig able liss soe values of β(s); he sybol idicaes ha β(s) is udefied. Table. Values of he fucio β(s) D(A s ) = = = = = = = = s =0 s = s = s = s = s = s = s = s = For iiial daa x D(A s )wih0 s + ad wihou furher assupios o he A-sable raioal approxiaio r of he expoeial fucio defiig he approxiaio schee V () =r(a), he Baach space X, or o he seigroup T ( ), Table coais he followig egaive essages, predicig for V ( ) x T ()x (a) poeial divergece of order for soe x X (s =0), (b) a axial order of covergece of for x D(A) (s = ) idepede of he approxiaio order of he defiig uerical approxiaio schee r, (c) a axial order of covergece of s+ for x D(A s )(0 s +) o aer how large he approxiaio order. If r = P Q is of Padé ypewihdeg(p )=deg(q) ordeg(p )=deg(q), he he order of covergece of V ( ) x T ()x is β (s),whereβ (s)isasi(.)wihq = ad p =deg(q) (see (.)). Sice deg(p )=deg(q) ordeg(p )=deg(q), i follows ha for each here is exacly oe possible value for deg(q) (ad herefore also for deg(p )); aely, if =or if = +, he deg(q) = +. Thus, if is eve, he q = ad p =deg(q) =+; if is odd, he q = ad p =deg(q) =+. I paricular, if is odd ad r is of Padé ypewihdeg(p )=deg(q), he 0 β (s) =s +. If is eve ad if r is of Padé ypewihdeg(p )=deg(q), he 0 0

6 s( +) β (s) = ( +) s + if 0 s +, if + <s +. Table. Values of β (s) forpadé approxias wih deg(p )=deg(q) if is odd or deg(p )=deg(q) if is eve x D(A s ) = = = = = = = = s = s = s = s = s = s = s = s = 0 0 s = I suary, wihou furher assupios o he seigroup T ( ) geeraed by A (like aalyiciy) or o he space X (like X Hilber space ad T () ), approxiaio schees V ( ) = r( A) defied by A-sable raioal approxiaios r of he expoeial of approxiaio order are (a) guaraeed o be sable oly if r = P Q is of Padé ype, has a odd approxiaio order, ad deg(p )=deg(q) ; (b) covergig o T ()x for x D(A s )(0 s + ) axially wih order s if r saisfies codiio ( ) ad of order s+ if o. The purpose of his paper is o show ha for geeraors A of srogly coiuous seigroups all approxiaio schees V () := r(a) defied by A-sable approxiaios r of he expoeial of approxiaio order ca be sabilized by +power-scaled Backward Euler seps ad ha he sabilizaio ehods o oly sabilize he approxiaios, bu iprove heir speed of covergece by a agiude of up o / ifr does o saisfy codiio ( ). A his poi we recoed a glace a Corollary. o ge a feel for he ypes of resuls o be discussed i he followig secio.. Sabilizaio If A geeraes a aalyic seigroup, he all approxiaio ehods V () = r(a) defied by A-sable raioal approxiaios r are sable ad V ( A) x e A x for all x X. However, if r( ) = (e.g., Crak-Nicolso), he he covergece ca be arbirarily slow for o-sooh x X. R. Raacher [], ad i fial for A. Hasbo [], cobied he high accuracy of he Crak-Nicolso schee wih he soohig properies of he Backward Euler schee o provide a reedy (for exesios, see []). More precisely, le r, r s be A-sable raioal approxiaio schees such ha he high accuracy approxiaio r has order, he sabilizig approxiaio r s order, ad

7 r s ( ) =0.The,ifA geeraes a aalyic seigroup ad R(λ, A) exiss for all λ 0, he sabilized approxiaio schee V (A) :=r( A) r s ( A) saisfies V (A)x T ()x C() x for all x X. I his secio we cosider geeraors A of srogly coiuous seigroups T ( ) oa Baach space X ha are o ecessarily aalyic. As explaied above, a approxiaio schee V () = r(a) defied via a A-sable raioal approxiaio r of he expoeial fucio is always cosise bu ay o be sable. The, by he Lax-Richyer Equivalece Theore, here exiss x X such ha V ( ) x does o coverge o T ()x. However, he followig resul shows ha all such approxiaio schees ca be sabilized by akig firs + odified Backward Euler seps (I α A) for soe appropriaely α chose 0 <α<. Moreover, i will be show ha by sabilizig V () =r(a) defied by a A-sable raioal approxiaio schee r ha does o saisfy he codiio ( ), he speed of covergece ca be iproved fro s+ s o s++ if x D(A s )ad s. Before providig a proof for arbirary s, we sar wih he case s =. Theore. Le A be he geeraor of a srogly coiuous seigroup T ( ) o a Baach space X. Defie V () :=r(a), wherer is a A-sable raioal approxiaio of he expoeial of approxiaio order ad a sabilizig schee W () := α R( α,a)=(i α A), (.) where α = +.The,forallτ>0here exiss a cosa M τ such ha V ( ) W ( ( ) + )+ x T ()x M τ ( x + Ax ) (.) for all [0,τ], allx D(A), ad all sufficiely large +.Furherore, li V ( ) W ( )+ x = T ()x for all x X. Proof. Choose M adω 0 such ha T () Me ω for 0. Sice AR(λ, A) = λr(λ, A) I, i follows ha AW () = ( α R( α,a) I) (a his poi, α > 0is α arbirary). By he Hille-Yosida Theore, λr(λ, A) Mλ λ ω for all λ>ω.moreover,if λ>ω 0 >ωhe λr(λ, A) Mλ λ ω M 0 for all λ>ω 0.Thus, W( ) M 0 ad AW ( α ) (M α 0 + ) for all >ω /α 0.Cosider ( ) V ( ) W ( )+ x T ()x (V ( ) T ())W ( )+ x + T ()(W ( )+ x x), ad observe ha li W ( )x = li α R( α α,a)x = x for all x X sice A α geeraes a srogly coiuous seigroup (see [], Proposiio..). The secod er of ( ) ca be esiaed by usig he bioial forula; i.e., T ()(W ( )+ x x) T () W ( )j W ( )x x.

8 Sice W ( ) M 0 for >ω /α 0 i follows ha T ()(W ( )+ x x) 0as. By (.) wih s = +,hereexisc, c > 0 such ha (V ( ) T ())W ( ( ) )+ x CMe cω (AW ( ))+ x ) ( ) α + (M 0 +) + x CMe cω ( α = M e cω ( ) α(+) x. If 0 <α< + he, for every τ>0, here exiss K τ > 0 such ha (V ( ) T ())W ( )+ x K τ x (.0) α(+) for all [0,τ]. This shows ha li V ( ) W ( )+ x = T ()x for all x X. Now suppose ha x D(A). The, for τ>0hereexissm > 0 such ha for α α T ()(W ( )+ x x) T () W ( )j W ( )x x = T () W ( ) j R( α M,A)Ax T () α ω Ax W ( ) j M e ω α ω Ax (M) j α M α Ax α >ω 0 ad [0,τ]. By ( ) ad (.0), for τ>0hereexissm τ such ha V ( ) W ( ( )+ x T ()x M τ + α(+) α α α ) ( x + Ax ) for [0,τ]adx D(A). Choosig α = + yields + α(+) =. α + The followig able copares he resuls of he Breer-Thoée Theore. o he covergece raes obaied by sabilizig A-sable raioal approxiaio schees. Noice ha for iiial daa x D(A), he rae of covergece + (for a sabilized schee) approaches for large. The leer c idicaes covergece of ukow speed. The colus arked Padé give he covergece raes for he Padé schees r = P Q,where deg(p )= ad deg(q) = + if he approxiaio rae is eve, ad deg(p )= + ad deg(q) = if he approxiaio rae is odd.

9 Table. Copariso of Theore. o Theore. x X, β(s) x X, β (s) x X x D(A),β(s) x D(A),β (s) x D(A) usabilized Padé sabilized usabilized Padé sabilized = c c = c = c c = c = c c = c = c c = 0 c As Theore. shows, approxiaios schees V () = r(a) defied via A-sable approxiaios of he expoeial of approxiaio order ca be sabilized by akig firs + odified Backward Euler seps (I α A). This resuls i a approxiaio α speed of order o D(A) (hereby iprovig he axial usabilized rae of /+ covergece of ), bu ay slow dow he order of approxiaio o D(A k )for / k +. To sabilize such ha he approxiaio order for iiial daa i D(A k )is approxiaely like for sufficiely large, we cosider sabilizers of he for k W () = where 0 <b <b < <b k <b k are arbirarily chose, α := k i= a i 0 α R( b i,a), (.) α +k+,ad a i := ( )k+i b k i k j= b j b i. (.) j i Wih his choice of he coefficies a i ad soe basic algebra, he sabilizer W () cabe represeed i he for W () :=( ) k+ kα A k k i= (b i I α A) + I. (.) Theore. Le A geerae a srogly coiuous seigroup T ( ) o a Baach space X. Defie V () :=r(a), where r is a A-sable raioal approxiaio of he expoeial of approxiaio order ad defie a sabilizig schee W ( ) as i (.). The li V ( ) W ( )+ x = T ()x (.) for all x X. Furherore,forallτ>0here exiss a cosa M τ such ha V ( ) W ( ( ) k +k+ )+ x T ()x M τ ( x + A k x ) (.) for all [0,τ], allx D(A k ), ad all sufficiely large +.

10 Proof. Fix k +adlew ( ) beasi(.). The, for all x X, W ()x x = kα A k k i= = kα A k k i= (b i I α A) x α R( b i α,a)x = k i= AR( b i α,a)x. By he Hille-Yosida Theore, here exiss ω>0 such ha λr(λ, A) M λ ω M for λ > ω. Thus, he operaors λr(λ, A) adar(λ, A) = λr(λ, A) I are uiforly bouded for λ>ω ad λr(λ, A)x x as for all x D(A) ad, herefore, for all x X = D(A). I follows ha li W ( )x = x for all x X. Moreover,if x D(A k )ad bi > ω, he here exiss C α > 0 such ha ( k ) W ()x x R( b i α,a) A k x (.) i= ( k ) (M) k α A k x = C kα A k x. b α i= i λ Also, if bi > ω, he here exiss M α > 0 such ha k a i AW () = α AR( b i α,a) k a i b i α α R( b i,a) I] α M α i= i= We proceed o esiae k a i. i= ( ) V ( ) W ( )+ x T ()x (V ( ) T ())W ( )+ x + T ()(W ( )+ x x). Sice he faily W ( ) is uiforly bouded ad li W ( )x = x for all x X, i follows ha he secod er i ( ) coverges o zero; i.e., T ()(W ( )+ x x) T () W ( )j W ( )x x 0 as for all x X. Moreover,ifx D(A k ), he i follows fro (.) ha here exiss C τ > 0 such ha T ()(W ( )+ x x) T () W ( )j W ( )x x C τ kα Ak x (.) for all 0 τ. To discuss he firs er i ( ), recall fro he Breer-Thoée esiae (.) wih s = + ha here are C, c > 0 such ha 0

11 (V ( ) T ()) W ( ( ) )+ x CMe cω (AW ( ))+ x ( ) ( ) ( CMe cω α + k + α M + a i ) x. Therefore, for all τ>0hereexissk τ > 0 such ha V ( ) W ( )+ x T ()W ( )+ x K τ x (.) α(+) for all [0,τ]. This shows ha li V ( ) W ( )+ x = T ()x for all x X if 0 <α< +. Furherore, cobiig (.) wih (.) yields ha for each τ > 0 here exiss M τ > 0 such ha, for [0,τ]adx D(A k ), V ( ) W ( )+ x T ()x M τ ( + α(+) kα )( x + Ak x ). Choosig α = +k+ yields + =.Thus, α(+) kα +k+ k V ( ) W ( ( ) k k++ )+ x T ()x M τ ( x + A k x ). (.) As Theore. shows, by codiioig raioal approxiaio schees of order by firs applyig k-sabilizers of he for (.) yields (a) covergece for all x X ad (b) k covergece of order k++ for x D(Ak ). If k +k, he k-sabilizig iproves upo he rae of covergece o D(A k ) prediced by he Breer-Thoée Theore.. Ideed, if k + k, he + >kad β(k) =k k k++ k if. Exaple. (-sabilizaio) Le A geerae a srogly coiuous coracio seigroup ad V () :=r(a), wherer is a A-sable raioal approxiaio of he expoeial of approxiaio order. Cosider he -sabilizer W () := b (I α A) + b (I α A), b b b b b b where α = + ad 0 < b < b (see (.) ad (.)). By keepig rack of he cosas, i follows fro he proof of he previous heore for k =ha V ( ) W ( ( τ ) )+ + x T ()x M τ ( x + A x ), for all [0,τ], wherem τ := ax(k τ,n τ ) for N τ = i= b ( +b b b ) + b b b +b b b ad K τ = Cτ + ( b +b b b ) + (wih C depedig oly o r). Therefore he choice of bi affecs he size of M τ. Oe ca show ha M τ + ax(cτ,) for appropriaely chose b i (wih 0 <b <b 0 as ). The sabilizaios of raioal approxiaio schees for srogly coiuous seigroups (Theores. ad.) ad aalyic seigroups (see[], [], []) have a absrac exesio ha ca be viewed as a sabilized versio of he Lax-Richyer Equivalece

12 Theore (i.e., a sabilized Cheroff Produc Forula). The proof of he sabilized Cheroff Produc Forula requires a sabilized versio of he Troer-Kao Theore; he sabilized Troer-Kao ad Lax-Cheroff Theores were firs proved (for he case j =) by Y. Zhuag i []. Proofs of he sabilized versios of he Troer-Kao Theore ad he Cheroff Produc Forula ay be foud i []. Theore. (Sabilized Cheroff Produc Forula.) Le A geerae a srogly coiuous seigroup T ( ) wih T () Me ω for all 0 ad le {V () : [0,τ]} ad {W () : [0,τ]} be srogly coiuous failies of bouded liear operaors o X saisfyig W (0) = V (0) = I. IfV is cosise ad V ( ) j W ( )j Me ω ( j, [0,τ]), he li W ( )j V ( ) j x = T ()x for all x X uiforly o [0,τ]. To see how his resul ca be used here, recall fro he Breer-Thoée Theore. ha if V () := r(a) for a A-sable raioal approxiaio r of he expoeial of approxiaio order, he V ( ) x T ()x Ce c Ax for x D(A). Thus V () x Ce c Ax + Me ω x ad here exi M 0,ω 0 such ha, for all [0,τ], x D(A), ad > V ( ) c x Ce ( ) Ax + Me ω x M 0 e ω0 ( Ax + x ). Defie W () :=(I A). The, as show i he begiig of he proof of Theore., here exiss M>0such ha W ( ) M ad AW ( ) M for all N ad >0. Thus, here exiss M τ > 0 such ha V ( ) W ( ) M τe ω0 for ad [0,τ]. Sice V ad W coue, Theore. yields he followig corollary. Corollary. Le A be he geeraor of a srogly coiuous seigroup T ( ) ad le V () :=r(a), where r is a A-sable raioal approxiaio of he expoeial of approxiaio order. Defie W () := R(,A)=(I A). The li V ( ) W ( )x = T ()x for all x X uiforly o copac iervals. Refereces [] W. Ared, C. Bay, M. Hieber ad F. Neubrader, Vecor-Valued Laplace Trasfors ad Cauchy Probles. Birkhäuser, 00. [] A. Ashyralyev, P. Sobolevskii, Well-posedess of Parabolic Differece Equaios. Birkhäuser,. [] O. Axelsso, A class of A-sable ehods, BIT (), -. [] P. Breer ad V. Thoée, O raioal approxiaios of seigroups. SIAM J. Nuer. Aal. (), -.

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