Order Nonlinear Vector Differential Equations

Transcription

1 It. Joural of Math. Aalyss Vol. 3 9 o Coverget Power Seres Solutos of Hgher Order Nolear Vector Dfferetal Equatos I. E. Kougas Departet of Telecoucato Systes ad Networs Techologcal Educatoal Isttute of Messologh Nafpatos Greece ougas@tees.gr Abstract I ths artcle power seres solutos of hgher order olear vector dfferetal equatos are studed ad suffcet coos are obtaed so that these solutos alog wth ther dervatves are absolutely coverget. The exstece of coplex-valued perodc solutos for a certa type of hgher order olear vector dfferetal equatos s also proved ad cosequetly leads to the exstece of real-valued perodc solutos to a larger class of hgher order dfferetal systes. The ethod used also provdes forato about the for of the exstg solutos. Matheatcs Subject Classfcato: 3A5 3A 34A34 34K3 46E5 46E Keywords: Abstract equato Baach space Frechét dfferetable Hlbert space holoorphc fucto coverget perodc soluto vector dfferetal equato. Itroducto The exstece of holoorphc solutos of the olear Brot-Bouquet dfferetal equato df ( z) z = G( f ( z) ) (.) dz where G( f ( z )) s α holoorphc fucto a eghborhood of zero proved useful whe vestgatg the asyptotc behavor of the solutos to certa types of olear secod order dfferetal equatos of partcular terest ad

2 4 I. E. Kougas ay applcatos such as the Ede equato astrophyscs ad the Fer- Thoas equatos atoc physcs (see [8-] ad the refereces there). I [3] the study of the exstece of holoorphc solutos to equato (.) where f ( z ) s a coplex fucto ad G( f ( z )) s holoorphc f ( z ) satsfyg the tal coos G ( ) = ad G ( ) G ( ) beg the dervatve of the fucto G( w ) at w = led to the exstece of fales of coplex-valued solutos of the secod order olear dfferetal equato dg( z) = G( g t ). G = τ p ad of expoetal type whe These solutos are perodc whe G ( ) = τ f. Moreover the solutos obtaed ply the exstece of fte fales of perodc ad o-perodc real-valued solutos to a larger class of secod order dfferetal systes. Exstece ad uqueess coos for a class of probles of the for zg z + J z G z + F G z = G = G where J s holoorphc soe eghborhood about zero ad F s holoorphc soe eghborhood of G are studed []. The covergece of foral power seres solutos for olear dfferetal systes x p+ du = E( x y u) dx where E s a holoorphc fucto ts arguets ear zero s studed [7]. These results are geeralzed [6] for olear dfferetal systes of the for where A( x ) s a ( ) du dx = E( x y u) A x atrx whose etres are holoorphc ear zero. I a recet paper [5] by cosderg a frst order lear Brot-Bouquet vector dfferetal equato we obtaed suffcet coos o the exstece ad uqueess of coverget power seres solutos to certa types of ohoogeous vector dfferetal equatos. The a purpose of ths artcle s to exted those results to hgher order olear vector dfferetal equatos. We beg by studyg the olear Brot- Bouquet vector dfferetal equato df z z = A( z) f ( z) + G( f ( z) ) (.) dz

3 Coverget power seres solutos 4 where f ( z ) s a vector-valued fucto j square atrx cosstg of eleets aj zero ad G f ( z) G f z G f z G f z = s a olear vectorvalued fucto. (... ) ( ) ( ) ( ) A z = a z j =... s a z holoorphc a eghborhood of We follow the fuctoal aalytc ethod fully descrbed [-4] ad [5] that coverts the questo of the exstece of holoorphc solutos of a dfferetal equato to the study of a operator equato a abstract Baach space ebedded a separable Hlbert space. Moreover the costructve ature of the ethod used eables us to obta forato about the for of the exstg solutos. For equato (.) t s proved that there exsts a faly of absolutely coverget power seres solutos uder certa coo o the atrx A( z ). The ethodology used aes t possble to readly exted these results to hgher order olear vector dfferetal equatos of Euler type. I secto 4 of the paper t s proved that there exst coplex-valued perodc th solutos for the -order olear vector dfferetal equato dgt (... ) = Ag t + G g t g t = g t g t g t ad as a cosequece we obta the exstece of fales of absolutely coverget real-valued perodc solutos to real dfferetal systes dxt dyt where u= ( x y ) v= ( ) ( ) (... ) = Ax By+ u x y x= x x x ( ) (... ) = Bx + Ay + v x y y = y y y x y are vector-valued cotuously dfferetable fuctos whose copoets satsfy the Cauchy-Rea coos ad A costat atrces. are The results preseted here also geeralze those of ref. [3] cocerg sglevalued olear dfferetal equatos. B. Prelary results ad cocepts I order to ae ths wor ore self-cotaed we preset bellow the fuctoal aalytc ethod that we follow. Let H be a abstract separable Hlbert space wth a orthooral bass { e } ad V ts ulateral shft operator such that Ve = e... =. +

4 4 I. E. Kougas The operator V s the adjot of V defed by V : V e = e Ve =. Every fucto f ( z) = a z H ( Δ ) ad H = Δ.e. the Hardy- Lebesgue spaces of holoorphc fuctos the ut dsc { z: z } satsfy the coos a p ad = ( ) where ( ) s the scalar product H ad egeeleets of the operator V. z Δ= p whch a p respectvely s represeted by = f z = f f (.) fz z e z = = p are the Relato (.) costtutes a soorphs betwee the spaces H well as betwee H ( Δ ) ad the Baach space eleets f = Δ ad H as H whch cossts of those = ae H that satsfy the coo f = ( f e ) p where a s the coplex cojugate of a ad deotes the or H. It s clear that the space H Δ cossts of holoorphc fuctos that have absolutely coverget power seres expasos the closed ut dsc Δ= { z: z }. The space H s ebedded H the sese that f f H the f H ad f f wth f = ( f e ) p beg the or H. = Let C be the dagoal operator actg o the eleets e =... of H as follows: Ce = e =... the the operator B: Be = e =... s the copact self-adjot dagoal verse of C ad Bf = ( f e) e. [] * If a( z) H ( Δ) the = = a V s a bouded operator o H ad ado * a ( V) a( z) a H( Δ) = = =. (.) Moreover the followg propertes follow easly (see [ 3]) H s varat uder the operators V V ad V = V =. If the operator V s restrcted to ay subspace { } Ve + =. H e e... e + the

5 Coverget power seres solutos 43 H s varat uder every dagoal operator De = de =... o H ad D = D = sup d. (v) For every f ( z) = a z H = Δ the ufor lt l av exsts = ad defes a bouded operator o f V a V H : = wth f ( V) = f ad V = I =detty operator. (v) Sce VV= I ad ( C I) e = we have that VCV = C I ad C I = VC V. Now cosder a vector valued fucto f ( z) f( z) f( z)... f ( z) copoets belog to the space H ( Δ ) or to H ( Δ ) the product space ( Δ) = ( Δ) ( Δ) H H H tes = = whose f z belogs to the or to H ( Δ) = H ( Δ) H ( Δ) tes By the above the Hlbert space H ( Δ ) ad the Baach space soorphc to the abstract product spaces The ebers f g of ( ) H ad H Δ are H respectvely. f = f f... f ad H H are deoted by = where each f g H( H )... g g g... g product s defed by = ad ther scalar f g = ( f g ) =. We shall use the sae sybol for the ors both spaces H ad H deoted by. Also the or of a eleet f H deoted by the sae sybol as H s gve by f = f. = The Baach space H s ebedded H the sese that f f H the f H ad f f. Defe the operator C H by the relato Cf = ( Cf Cf... Cf ) the ts egevalues are the values =... where each of whch has ultplcty. Also the egeeleets of C correspodg to a egevalue are:

6 44 I. E. Kougas (...) (...)... (... ) e = e e = e e = e. The set e =... =... cosstg of all the egeeleets of the operator C s a orthooral bass for the space H. I fact f e f = for soe f H e f = e f =. Ths ples that f = for =... ad thus f =. The verse operator of C s the copact self-adjot operator B defed by Bf = Bf Bf... Bf. the Now the lear vector Brot-Bouquet dfferetal equato df z z dz where square atrx whose etres aj ( z ) belog to H = A z f z (.3) f z s a vector-valued fucto ad operator equato [5] ( ) = C I f A V f or A z a z j =... s a = j Δ s equvalet to the abstract ( I B) f = BA( V) f (.4) where C ad B are the operators defed above ad A( V ) s gve by the relato A( V) f g =.... = where g ( g g g )... = wth g = a V f j j j= Aother oto that wll be used what follows s that of Fréchet dfferetablty. Defto Let X ad Y be Baach spaces ad U X a ope subset of X a fucto f : U W s called Fréchet dfferetable at the pot x f a bouded lear operator Ax : U Y such that l h ( + ) f x h f x A h h X x Y =. Or alteratvely Defto A fucto f s called Fréchet dfferetable at the pot α If the lt

7 Coverget power seres solutos 45 exsts. l x α f ( α ) x α f x Ths s equvalet to sayg that there exsts a fuctoϕ cotuous at the pot α such that f x f α = ϕ x x α. I ths case f s called Carathéodory dfferetable. I other words the two otos Fréchet ad Carathéodory dfferetablty are equvalet. Let w ( w w w ) the copoets of the fucto =... seres expasos the suppose that the seres j =... r f. G w possess power w varables that start by ters of degree at least two ad = G w s absolutely coverget for w p r I such a case the operators G f ( z ) are defed for f z p r whch j H ( Δ) eas that they are defed fro a ope sphere S H ( Δ) r H ( Δ). The fuctos G f ( z ) have cotuous Fréchet dervatves for every the sphere S r (see the corollary to theore 4.4 of [3]). If N G ( f z ) the the abstract fors of the operators ( ) j to f z f deotes G f z = f N f =.... (.5) z (... ) N f = N f N f N f ca be detered exactly as t s show secto 4 of [3]. The for of the abstract operator Now by (.) ad for f( z) f( z) = fz f ( V) f where = ( z ) f ( z) = ( f f ) we have z f z f f ad f z f z = f V f f V f = f f. H Δ Thus H ( Δ) H ( Δ) H ( Δ) f z f z f z f z. (.6) H s a Baach Algebra ad t ca alteratvely be proved by cosderg the covoluto algebra for sequeces [4]. Relato (.6) eas that ( Δ)

8 46 I. E. Kougas Now for H f z r p = Δ r... we have that H ( Δ). (.7) f z f z f z r N N N... N we get the = Hece by the results [3] for the operator followg. Lea. The abstract for of the operator G the operator N above s a fucto.e. t s cotuously dfferetable defed fro a ope sphere the space H to H ad f N ( ) s ts Fréchet dervatve at the pot f = the As [5] we cosder the subspace M ( λ ) of ( λ + ) eleets = ( λ + ) = ts orthogoal copleet. Let ( λ ) C N = N =. (.8) H whch s spaed by the c e ad deote by M ( λ ) M be the tersecto c ( λ) ( λ) M = M H the ths s a Baach space whch s varat uder the operators B ad A( V ). Lea. Let ( λ ) y M ad f = ξ f + y where ξ C (C=set of coplex ubers) f s the eleet defed Lea 3. of [5] belog to the defto doa of the operator N. The ( ξ + ) M. space ( λ ) N f y s also a eleet of the Proof The copoets of f ( z) f( z) f( z) f ( z) G( f ( z )) are of the for =... the fucto f z = ξ z + b z + b z + =. λ λ+ λ+... The copoets of G have power seres expasos f ( z ) that start by ters of degree at least two whch eas that the copoets of the fucto N ξ f + y are such that ( ) ( ( ξ ) + ) N f + y e = =... =... λ. (.9) M. The Lea ow follows by the defto of the space ( λ )

9 Coverget power seres solutos Exstece of holoorphc solutos to olear vector dfferetal equatos Let the atrx A( z) = aj ( z ) by A( z) A( ) +za ( z ) where ( ) = j =... of equato (.) be gve A s the costat atrx A a j =... the ts abstact equvalet s the operator [5] ( ) Theore 3. Cosder the abstract equato = ( j ) A V = A + VA V. (3.) ( ) = + C I f A V f N f (3.) where A( V ) s the operator (3.) ad N s the operator of the above Leas satsfyg (.8) ad (.9). If at least oe of 3... s a egevalue of the costat atrx A ( ) the equato (3.) has a faly of solutos belogg to the space H ad of the for ( ξ ) f = ξe + b e + =.... (3.3) λ+ λ+ Proof For the operator equato (3.) we see solutos of the for f = ξ f + B y (3.4) wth f ad y beg the eleets defed above. A ope sphere cetered at the org of the product space C M ( λ ) ca be foud such that the eleet (3.4) belogs to the defto doa of the operator N ad sertg t to equato (3.) by Lea 3. of [5] we have ( I B) y ξ A( V) Vf A( V) B y N( ξ f B y) = (3.5) Now the space M ( λ ) s varat uder the operators B A ( V ) ad A( V ) [5]. Moreover Vf s a eleet of that space ad by Lea. the operator ( ) ξ + s a appg fro M ( λ ) to M ( λ ) N f B y. Hece F( ξ y) ( I B) y ξ A( V) Vf A( V) B y N( ξ f B y) C fucto defed fro a ope sphere of C ( λ ) = + (3.6) s a satsfyg the followg coos M to ( λ ) M

10 48 I. E. Kougas F = F = ( I B) A( V) B y. (3.7) I vew of Lea 3.4 of [5] ad by the Iplct Fucto Theore there exsts a eleet y M ( λ ) for every ξ suffcetly sall. Hece the copoets of the solutos of equato (3.) are of the for (3.3) due to the for of the eleet (3.4). We ca ow cosder the olear vector dfferetal equato where df z z = A( z) f ( z) + G( f ( z) ) (3.8) dz = f z f z f z... f z. Theore 3. Let the etres aj ( z) j =... of the atrx A( ) + za ( z) ad the fucto G( w) w ( w w w ) A z = =... be holoorphc soe eghborhood of zero satsfyg the coo G = G = (3.9) G ( ) beg the Fréchet dervatve of G( w) at the pot w = the f the costat atrx A ( ) possesses at least oe postve teger egevalue there exsts a oe paraeter faly of solutos of equato (3.8) of the for ξ ( ξ) ( ξ) f z = z + b z + b z + = (3.) λ λ+ λ+... where λ s the greatest postve teger egevalue of A ( ). Moreover these solutos alog wth the frst dervatves coverge absolutely for every z the Δ= z: z ad for suffcetly sall ξ. closed ut dsc { } a z j =... belog to Δ the equato (3.8) s equvalet to the abstract Proof Assue wth out loss of geeralty that j the Baach space H equato ( ) = + C I f A V f N f (3.) whch s a spler case of equato (3.).

11 Coverget power seres solutos 49 Leas. ad. esure that the operator N fulfls the coos of Theore 3. also by (3.3) we coclude that the copoets of the exstg solutos of equato (3.8) have the for (3.) ad sce they belog to the space H ( Δ ) they coverge absolutely for every z Δ. Now for = we ote that the eleet (3.4) belogs to the defto doa of the operator C the Baach space H ad thus the dervatves of (3.) coverge absolutely for every z Δ. Theore 3.3 Cosder the olear vector dfferetal equato df ( z) d f z ( ) z + z = A z f z + G f z (3.) dz dz where the abstract fors of A( z ) ad G satsfy the coos of Theore 3.. If the costat atrx A ( ) possesses at least oe egevalue of the for λ where λ =± ±... the equato (3.) has a oe paraeter faly of soluto of the for (3.). Moreover these solutos alog wth the dervatves up to order two coverge absolutely for every z Δ ad for suffcetly sall ξ. Proof The dfferetal equato (3.) s equvalet to the operator equato ( C I) f A( V) f N( f) = + (3.3) for whch by Theore 3. there exsts a faly of solutos of the for where ( λ ) f = ξ f + B y (3.4) y M ad ξ suffcetly sall. The coeffcets of the vectorvalued fucto (3.4) belog to the rage of the operator B whch eas that they are eleets of the defto doa of the operator C. Thus the solutos of the olear vector dfferetal equato (3.) are the equvalet eleets to the abstract oes (3.4) ad are vector-valued holoorphc fuctos whch together wth the frst two dervatves belog to the space H ( Δ ). It s obvous at ths pot that vew of Theore 3. we ca extet our results to hgher order olear vector dfferetal equatos of the for + + = z f z z f z zf z A z f z G f z ( 4 ) = + z 4 f z z 3 f z z f z zf z A z f z G f z ad so forth. Furtherore the results preseted above hold true for ore geeral olear vector dfferetal equatos of Euler type.

12 5 I. E. Kougas As Theore 6.4 of [3] f we set z t = ρe ϑ ad f ( z) g( t) = the the copoets of the vector-valued fucto g ( t ) are coplex-valued perodc fuctos ad t s a soluto of a olear vector dfferetal equato of the for dgt = Ag t + G ( g t ) =... (3.5) where A s a costat atrx. Next secto s devoted to the exstece of coplex-valued perodc solutos of olear vector dfferetal equatos of type (3.5). 4. Exstece of perodc solutos Lea 4. Let the abstract equato ( ϑ ) ( C I) f A( V) f N( f) = + (4.) where ϑ s real have a vector-valued soluto f ( f f f ) =... whch belogs to the Baach space H ad t s of the for f = ξ f + B y wth = ( ) ad ( ) f z fz f G f z = f N z f =.... The the coplexvalued perodc fucto ϑt λ ϑλt λ+ ϑλ ( + ) ( ρ ) ξρ ( ξ) ρ g t = f e g t = e + b e + (4.) s a soluto of the olear vector dfferetal equato t dgt ϑt ( ρ ) = A e g t + G( g t ) (4.3) wth the seres (4.) beg absolutely coverget for every real t ad ρ (. Proof By assupto equato (4.) has a soluto of the for f = ξ f + B y where y H ad thus the fuctos f( z) = ( fz f) =... alog wth the frst dervatves are eleets of the Baach space H ( Δ ). Hece the fuctos g t are tes dfferetable the varable t ρ. Now for = we have

13 Coverget power seres solutos 5 The abstract for of So (4.4) becoes dg z ϑ dg t df z = ϑz z= ρe t. dz df t ( z) dz where the eleets s ( fz VCV f) or ( fz ( C I) f) ( ) ϑ z ( z ) = ϑ f C I f = f u []. u = C I f =... belog to the space H. (4.4) (4.5) Slarly for = we have t dg = ( ϑ) ( f ( C I) u ) = ( ϑ) f ( C I ) f z z ad by cotug the sae aer we obta Sce ( ϑ ) dg dg t t ( z ) ( ϑ ) = f C I f or ( f ( ϑ ) ( C I z ) f) = C I f H relato (4.6) s vald for z = ρ =.. (4.6) Furtherore by the represetato (.) the followg relatos hold true ad aj ( z) f ( z) = fz aj ( V) f (4.7) j= j= ( ) z G f z = f N f. (4.8) Hece fro equato (4.) we get equato (4.3). Wrte (4.) the for = (4.9) ϑ ( C I) f A( V) f h( f) where the fucto h( f ) fulfls the sae coos as those of N( f ) Theore 3..

14 5 I. E. Kougas Sce =... t ca be wrtte the followg fors = 4μ 3 = 4μ = 4μ = 4 μ μ = 3... ad so ca tae o the values respectvely. Theore 4. Assue that s a postve odd teger ad the costat atrx A ( ) possesses a o-zero pure agary egevalue. If ν s the largest postve uber for whch ± ν s a egevalue of A ( ) the for every ρ ( ad for every λ = 3... the vector dfferetal equato dgt τν t λ = A ρe g t + G( g t ) (4.) where τ =± for = 4μ ad τ = for = 4μ 3 has a faly of perodc solutos g t = ( g t g t... g ( t) ) that have the for λ+ λ+ τν t τν t λ τνt λ+ λ λ+ λ ξρ ξ ρ ξ ρ g t = e + b e + b e +. (4.) The solutos (4.) alog wth ther frst dervatves coverge absolutely for ξ suffcetly sall. Proof a) Assue that ν s a egevalue of the costat atrx A ( ) ad set ν ϑ = (4.9) wheever = 4μ μ = λ ν b) Assue that ν s a egevalue of the costat atrx A ( ) ad set ϑ = λ (4.9) wheever = 4μ 3 μ = I ether case ϑ A. λ s the greatest egevalue of the atrx Hece by Theore 3. equato (4.) has a faly of solutos of the for (3.3) ad oreover by Lea (4.) equato (4.) has a faly of solutos of the for (4.) whch alog wth the frst dervatves coverge absolutely sce the seres (3.) s absolutely coverget. Theore 4. Assue = 4μ ( = 4μ ) s a eve teger ad the costat atrx A ( ) has a postve (egatve) egevalue. Moreover let ν be the largest postve uber for whch ν ( ν ) ( ] s a egevalue of ρ ad every λ = 3... the vector dfferetal equato A the for every

15 Coverget power seres solutos 53 dgt τν t λ = A ρe g t + G( g t ) τ =± (4.) has a faly of perodc solutos of the for (4.) above. ν Proof Let ϑ λ ad the proof follows the sae aer as that of Theore 4.. Let fuctos are of =± the ν s the greatest egevalue of the atrx ϑ A ( ) A B be ( ) costat atrces ad assue that the vector-valued ( ) ( ) = ( ) ( )... ( ) ( ) = ( ) ( )... ( ) uxy u xy u xy u xy vxy v xy v xy v xy C class whose copoets satsfy the Cauchy-Rea coos ( ) ( ) u x y v x y = x y j ( ) ( ) u x y v x y = y x j j j j =.... (4.3) Hece the fucto G g t = u x y + v x y s holoorphc the varable g = x+ y. u v Furtherore f = = at the pot ( ) the G( ) = G ( ) =. x y j j Now the desoal syste of dfferetal equatos dxt dyt ( ) (... ) = Ax By + u x y x = x x x ( ) (... ) = Bx + Ay + v x y y = y y y wth the above assuptos ca be reduced to the for (4.4) dgt ( A B) g t G( g t ) = + +. (4.5)

16 54 I. E. Kougas By the results of Theore 4. there exst real-valued perodc solutos to (4.4) o the coo that the atrx A + B possesses at least oe pure agary ozero egevalue. Furtherore Theore 4. predcts the exstece of real-valued perodc solutos to (4.4) provded that the atrx A + B possesses at least oe postve egevalue case = 4μ ad a egatve egevalue case = 4μ μ = Thus we have show the followg result. Theore 4.3 For every postve teger λ the syste of dfferetal equatos (4.4) possesses a faly of real-valued perodc solutos of perod πλ o the ν coo that s a odd teger ad the atrx A + B has at least oe pure agary egevalue ± ν ν f. s a eve teger of the for = 4 μ μ = 3... ad the atrx A + B has at least oe postve egevalue ν. s a eve teger of the for = 4μ μ = 3... ad the atrx A + B has at least oe egatve egevalue ν. Lea 4. Let the abstract equato ( ϑ) ( C I) f A( V) f N( f) = + ϑ f (4.6) = of the for f = ξ f + B y. The the = s a soluto of the vector dfferetal equato (4.5) posses a soluto f ( f f... f ) H fucto g t f ( ρe ϑt ) ρ ad for every [ ) for every ( ] t. Proof The proof s otted sce t s slar to that of Lea 4.. A edate cosequece of Lea 4. s the followg result cocerg the th exstece of real-valued solutos of the order vector dfferetal equato dgt t ( ρ ϑ ) ( ) = A e g t + G g t. (4.7) Theore 4.4 Let the costat atrx A ( ) posses at least oe postve ν (egatve ν ) egevalue wth beg a eve (odd) teger the for every postve teger λ equato (4.7) has a faly of real-valued solutos of the for λ λ ν + t ν + t λ νt λ+ λ λ+ λ ξρ ξ ρ ξ ρ g t = e + b e + b e +.

17 Coverget power seres solutos 55 These solutos are absolutely coverget for every ρ ( ] t [ ) suffcetly sall. ad ξ Coclusos The purpose of ths wor was to preset suffcet coos o the exstece of coverget power seres solutos to hgher order olear vector dfferetal equatos cludg equatos of Euler type. Usg a well-ow fuctoal aalytc ethod the dfferetal equatos uder cosderato are coverted to operator equatos a abstract Baach space whch s soorphc to the Hardy space of absolutely coverget vector-valued fuctos. The exstece of fales of coplex-valued perodc solutos to hgher order olear vector dfferetal equatos was also proved whch cosequetly pled the exstece of realvalued perodc solutos to a larger class of dfferetal systes. All the predcted solutos alog wth ther dervatves coverge absolutely wth the ut dsc ad furtherore the costructve ature of the ethod used provdes forato about the for of these solutos. Refereces [] D.C. Bles M.P. Robso ad J.S. Spraer Aalytc solutos for a class of olear ordary dfferetal equatos Coplex Varables Vol. 48 o. (3) pp [] E.K. Ifats A exstece theory for fuctoal-dfferetal equatos ad fuctoal-dfferetal systes J. Dfferetal Equatos 9 (978) pp [3] E.K. Ifats Aalytc solutos for olear dfferetal equatos J. Math. Aal. Appl. 4 o. (987) pp [4] E.K. Ifats Global aalytc solutos of the radal olear wave equato J. Math. Aal. Appl. 4 o. (987) pp [5] I.E. Kougas O holoorphc solutos of vector dfferetal equatos It. J. Cotep. Math. Sceces Vol. 3 o. (8) pp [6] M.-S. Rezaou Covergece of foral solutos of soe olear dfferetal systes at a rregular sgularty Asyptotc Aalyss 46 (6) pp [7] Y. Sbuya Covergece of foral power seres solutos of a syste of olear dfferetal equatos at a rregular sgular pot Lecture Notes Math. Vol. 8 Sprger (98) [8] I. Tsuaoto O the geeralzed Thoas-Fer dfferetal equatos ad applcablty of Sato s trasforato Toyo J. Math. (997) pp. 7-.

18 56 I. E. Kougas [9] I. Tsuaoto Asyptotc behavor of solutos of pα p Osaa J. Math. 4 (3) pp [] I. Tsuaoto O solutos of 3 (5) pp. -4. = αλt + α x e x where α p αλt + α = where x e x [] I. Tsuaoto O asyptotc behavor of postve solutos of where α p Hrosha Math. J. 37 (7) pp Japa. J. Math. = αλt + α x e x Receved: Deceber 8