Critical case for periodic solutions of a class of neutral equations with a small parameter

Pub . Ma . UAB Nó 13, Juliol 1979 CRITICAL CASE FOR PERIODIC SOLUTIONS OF A CLASSOF NEUTRAL EQUATIONSWITH A SMALL PARAMETER Ped oMa ínez Amo es Secciónde Ma emá icas - Uni e sidad de G anada In oduc ion In his palie we s udy a class o neu al unc io nal di e en ialequa ionswhich a ises om a coupled sys em o di e en ial-di e ence and o dina y di e ence equa ions ha occu in a ious applica ions, as elec ical ci cui swi hlossless ansmision lines, [11 . In [101, [151 is s udied he s abili yo such sys em and in [131 a e gi en - condi ions o he exis ence o pe iodic solu ions . In his pape , we conside a nonlinea sys em wi h a small pa ame-- e and we s udy he exis en e o pe iodic solu ionswhen - he co esponding linea sys em can ha e pe iodic solu ions (c i ical case) . This is done by applying he me hod o Ha- le [5,61 andi s ex ension o delay di e en ial equa ions - (see [141) o he neu al equa ions ob ained om he coupled sys em a e aking in o accoun he esul so [101 . l . No a ion and summa y o known esul s Le E n be a complex n-dimensional linea ec o space wi h no m 1 .1 andle be a ixed posi i e numbe . C = C  ( [ - ,  o 1, E n) is he spaceo con inuous unc ions 1 : [- , 0 1 --~ E n wi h no m I~¡= sup~  ~(9)1 :9 , o1 ~~ Suppose D, L a e bounded linea ope a o s om C o E n ,  0 = H (0) - [dl+ (A)~~ (9) - 0 L ( 1 ) = - [d 1 (9)] (8) whe e  li  is an n x n ma ix, de li =~ 0, , ~ a e n xn ma ix Func ions o . bounded a ia ion on [- , 0] wi h M nona omic a ze o . We assume N has no singula pa . I .  x  is a con inuous unc ion mapping [a'- ,  ) in o E n , hen o any E[0,w)we de ine x in C by -  - x (9) = x ( + A) ,9E[- ,  0] . An au onomouslinea homoge- - neousneu al un ionaldi e en ial equa ion (N F D E ) is - de ined o be d d D(x ) =  L (x ) A solu ion x= x (¢) o  (1 .] .) h ough a poin JE C  a  = 0 is a con inuous unc ion aking [- , A),  - A> 0, in o E n such ha  xo = , D(x ) is con inuously di-- e en iable on [0, A) and equa ion (1 .1) is sa i,s ied on - his in e al . I is p o ed in [2,4] ha he e is a unique solu ion  x (~) h ough  and x (1)( )  is con inuous in  ( , b I . he ans o ma ion T( ) : C -+C is de ined by - T ( ) ~ = x (1), hen i is shown in [11] ha 1 T ( ), >,0 } - is a s onglycon inuoussemig oup o linea ope a o s wi h he in ini esimal gene a o A : -9(A) -  C,  A  ( 9) _  ( 9 ),  - and he spec um  0'(A) o A consis o hose X which sa is y de 62 0  0 O %) = 0,  a(ñ) -_ ñH _~J  e e d~(9) _ J  é6dj(9 _  - The undamen al ma ixsolu ion o (1 .1) is de i- ned o be he n x n ma ix solu ion o he equa ion D(X ) = I+J  L ( X) ds, > 0 0 s 0 , - c9<0 Xo (0)  0 (1 .6)  x -X o G( ) = T( - )[ -X oG(C)]+  +JT T( -s)X o F(s)ds- J e . (d sT( -s)X o IG(s) o > , E C, whe e i is always unde s ood ha he in e g alsin (1 .6) a e ac uallyan in eg als in E n . Fo mula (1 .6) sugge he changeo a iables x - Xo G( ) = z ,  -X o G (T) = om C - P C .  I his is done,  equa ion (1 .6 ) becomes  (1 .7) z =T ( -T) +  T ( -s)X F(s)ds- [d s T( -s),XJG(s) .  T  o De ini ion 1 .1 . The ope a o D is said o be s able i he- e is a -4 > 0  such ha all oo so he equa ions de D (e x . I) = 0 s is y ReÁ<-a . I D(+) = H j(0) - M«- ), hen D is s able i he oo so he polynomialequa ion de (H -PM) = 0 sá is y  - An impo an p ope yo equa ion (1 .1) when D is s ableis he ollowing (see [3]) : I D is s able, hen - he e is a cons an  aD< 0 such ha o any a > a D , he e - a e only a ini enimbe o oo so de (j(ñ) = 0 wi h - - Reñ) a . Le D be s able .  I A = ~ : de á (X) = 0,  Re X> .0 }, hen A is a ini e se and i is ollows om [11] ha he space C can be descomposed as C=P 49 Q, whe e P, Q a e - subspaces o C in a ian unde T( ), he space P is ini e dimensional and co esponda o he ini ial alues o all - hose solu ions o (1 .1) which a e o he o m p( )e x ,  - whe e  p( ) is a polynomial in and ÁEA . I í  is a basis o P hen o e e y +E P he eexis a a ec o aE E  such ha +=í a .  En his case, we can de ine T( )+= l e B a,  - 6 3 I  F, G : [0, ~)  --~ En a e con inuous, a nonhoimo- : geneous linea N FD E  is de ined as (1 .3)  d  ID (x )-G( »=  L(x )+ F( ) . A solu ion h ough a  = T  o (1 .3) is de i- ned as be o e and is known o exis on [o'- , c o) . The a ia ion o cons an s o mula o (1 .3) (see [9]) s a es ha he solu ion o (1 .3) h ough ( , ) is gi- en by  + (1 .4) x( ) = T( -( - ) 1(0) + ue X( -s)F(s)ds- .. [d s X( -s)]G(s)-G( ), o > l T, whe e X is he undamen al ma ix solu ion gi en by (1 .2) . Equa ion (1 .4) can be w i en as (1 .5)  x( ) -X(o)G( )=T( -Q»)~(0)-X( -T)G(T)  o >,(' . Now, le PC be he space o unc ions aking  - [- ,0] in o E n which a e uni o mly con inuouson E - .0) and may be discon inuous a ze o . Wi h he ma ixX as de ined 0 be o e,  i is clea ha PC= C+ ( X0 ), whe e (X 0 ) is he - - span o X 0 ; ha is any1,EP C  is gi en as ^+ = 1+ X 0 b,  1 E C, n bEE .  We make P Cano med ec o space by de ining he no m 1+1= max{/41, b) . Le us de ine x (~ ) = T ( ) ~ , whe e iEP C  and  - x(^ )  is he solu ion o (1 .1) h ough + . The ope a o -  - T( ) : PC - a ( unc ions on [- , 0] ) is linea , bu T( ) does 64 no ake P C --j PC . I is an ex ensiono he o iginal semi g oup T .( ) on C . I we use his no a ion, hen he a ia ion o cons an s o mula (1 .5) can be w i en as whe e  B  is an n x n g a ix de ined by  Aí = B .  The spec um o B is A . I  C  is decomposed by n as  C = P G Q hen equa-- ion (1 .6) is equi alen o x - Xp G( ) = T( -T)C,~P- XpG(Q')]+ . . T( -s)XpF(s)ds - - [ds T( -s)XP]G(s) 0 whe e he supe sc i s P and Q designa e he p ojec ions o - he co espondine unc ions on o he subspaces P and Q, es- pec i ely, and hey can be de e minedby meanso adjoin di- e en ial equa ion o (1 .1), see (111 . 2 . The li nea p oblem x - XQ G( ) = T( -~)[¢Q- XQ G(c)J + T T( -s) XQ F(s)ds - (ds T( -s)XQG(S) 0 1 In his sec ion, we conside he sys em a) z( ) = A1x( )+A2y( - ) b) y( )-A 3 x( )+A 4 y( - )=0 whe e x, y a e n- ec o andall ma ices a e cons an s . Fo any aEE n,-- EC, onecan de ine a solu ion o (2 .1) wi h  - ini ial alue x (o) = a,  y  = ^~ .  I we de ine,  C = C ([- , 0], E n ) 0 (2 .2) D 1 L 1 D, L :  Enx  C ----> E nx n'  D= [D  L - 0 2 7 D 1 (a, - )= a D 2 (a,^ ) =' (0) - A 3 a - A4 L 1 (a,°%) = A l a+A 2 +(- )  65 henequa ion (2,1) is a specialcase o he NF D E (2 .3) d d andone ob ain he sys em (2,1) by equi ing ha is a s cngly con inuous semig oupo The in ini esimal gene- _es o e o T (+) í_- 0 0 D (x ( ),  y )  =  L(x ( ), y ) D 2 (a,^%)= 0 Equá ion (2-3) de inesa semig oup T( ) on Enx C , I we de- ine (E n xC) 0 = « a, )EE n xC : D2(a,^ )=0} hen (E n xC) 0 can be conside ed as a Banach spacee Fu he mose, o any -  - (a, -+)E(E n xC) 0 , he solu ion o  (2,3) h ough (a,- )  will - be in (E nx C) 0  since i co esponds o he solu ion o  (2 .1) h ough(a,- )o Consequen ly, T  ( )  de  T ( ) (  ,  (E n. C)  -~  (En . C ) 0  0  0 n  1 (E x C) 0 ni esimal gene a o o T( )¿One shows ha T(A 0)=Ii1E4 :: Obse e ha = A l  whe e A is he in i ¡(E nx C) 0 ~ I-A 1 de ~(ñ)=0},  o (ñ) =  -A 3 a  I 0 a 0 0 a D( a ,^~ . ~(O) -A 3 a - A4 Y(- )  -A 3 I  ^Y(0 )  0  A4 ^¡'(- ) de =  H ~(O) -M 4(- ) a  A1 0 a ío A L(a,~) -- - Y(O ) -A3a-A4  ~,(- )  0  0  .~((0) + 0  0 66 d-  Ni (0) + P ~(- )o -A 2 e  Á I-A,e X duli less han 1, hen D is s able . ix solu ion X( ) o (2 .3) as I  X = ( X 11  X 121 ~ _whe e  X i .,  i, j = 1 9 2 1 a e nxn ma i 21 22  j  _ ces, hen X mus be a solu ion o (2 .3) wi h he ini ial - da a speci ied abo e . The e o e, he ma ices X, . mus sa-- 1sJ is y (2 .4) No ice ha  (2 .4a)  impliesX 11 ,  X 21 a e solu ions o  (2 .1) . The unc ions X12, X22 do no sa is y (2 .1 b) . This implies ha he a ia ion o X( ) sa is ies he sys em (2 .1) . UsingLaplace ans o m o he same ype o a gu-- men s as in Hale () pag . 303, onecan p o e he ollowing Lemma 2 .1 .  I he eigen alues o A 4 ha e moduliless han 1 and all oo s o de á (ñ) = 0 sa is y Reñ<-d<0, hen - he e a e posi i econs an s K, o( such ha Ix 11 ( )1,  1x21( )ls  I X ij ( )j_4 Ke C( s a.e  >,0,  i,j=1,2 . Now we conside henonhomogeneous sys em (2 .5) Thus, i he eigen alues o he ma ix A 4 ha e mo- Al so, om (1 .2) . we can de ine he undamen al ma I 0 X(9) = H 1 o  (A 3Ie = 0, x0 (e) = o,  - i6 9 <0 . a) D2 (x 11 ( ), x 21 . ) = o , > " o b) D2 (x 12 ( ), x 22 . ) =  , >, o . X( )=A 1 x( )+A 2 y( - )+ ( ) y( ) -A 3 x( ) -A 4 y( - ) - g( ) =0 n whe e , g a e con inuous unc ions om [0, -0) o E . Wi h D, L de ined as in (2 .2), sys em (2 .5) is a specialcase o he N F D E (2 .6)  c {D (w )-G( )}- L(w )+F( ) whe e  w  = col (x ( ),  y  ),  w  _ ¢  col (a, ^~ )E E nxC =  , -  o G = col (0,g)EEnxn$ F = col( , O)EE nxn and one ob ains he sys em (2 .5) om (2 .6) by equi ing ha D 2 (a,y)= g(0) . As in Sec ion 1, i we ex end he de ini ion o - T( ) o E n x (C + ( X_ ) ) de Y,  hen he gene alsolu ion o .  - (2 .6) is gi enby he a ia ion o cons an s o mula (2 .7) w -Xo G( ) = T( «- X 0 G(0),+  T( -s)X 0 F(s)ds - 0 - [d s T( -s) X0 ]G(s) . o This las o mula sugges s he change o a iables w - X0 G( ) = z s  J- X0 G(O) =5,  1 om'E nx C  o  Y . I his is done, o mula (24) becomes  L (2 .8)  z T( ) + T( -s) X0 F(s)ds- [ds T( -s)X 0 ]G(s~ >,0 . Onecan gi e an explici decomposi ion o (2 .8) by using he adjoin equa ion o (2 .3) . Fo his, we w i e - - (2 .3) in he o m d á ~H w ( ) - Mw( - ) )= Nw( ) +P w( - ), w0 =JEE nx C which is equi alen o (2 .9) d {w( )-Mw ( - )}= Ñw( )+Pw( - ) lince H- 1 M=M and whe e H 1 N=N, H 1 P=P . We de ine he adjoin equa ion o (2 .9) as (2 .10) d j ( )- .( + ) M}= - ( )Ñ- ( + ) P, O = <GE n XC~ In he lame way, we may w i e (2 .6) in he o m 68 (2 .11)  d {w( )-Mw( - )-H 1 G( )}= Nw( )+Pw( - )+H 1 F( ) Using he samea gumen s as in [7,13), i is - -- easily shown ha i he eigen alues o A4 ha e moduli less han 1 and EnxC  is -decomposed by / = I% : de ,&(a) =0 ,  Rein> .0 as P eQ henequa ion (2 .8) is equi alen o (2 .12)  - a)  z = T( )  P+ O  T( -s) Xó H 1 F(s) ds  - - [d  T( -s) X P ] H 1 G(s) 0 s  0 b) zQ=T( )3 Q + T( -s) XóH 1 F(s) ds - [d s T( -s) Xo]H _1 G(s) 0 whe e  JS  = 4 -XoG (0),  4  = o  i  z = íu ( ),  whe e  is a basis o P and T( ) ~_  eB ,  - he spec um o B is 11 ., hen  u  sa is ¡es he equa ion (2 .13)  ü( )=Bu ( )+^1(0)H 1 F( )+B i(0) H 1 G( ), -oo4 coo whe e IP is a basis o he ini ial alues o hose solu- - ions o (2 .10) o he o m p( )e - ~ , p a polynomial,IEA . I 9- is he Banachspaceo con inuous and  - T-pe iodic unc ions wi h no m 1  =  sup 1 ( ( )j , E[0, T]}, - hen onecan s a e he heo em on he F edholm al e na i e o pe iodic solu ions as : Theo em 2 .1,03] I he eigen alues o A 4 ha e moduliless han 1 and , gE -9, hensys em (2 .5) has a solu ion in i and only i (2 .14)  0 ( )H . 1 F(s)ds- jd ( )]H 1 G(s)=0 . I he unc ion z * (a, £ ) in Lemma 3.1  is di e en iable  - wi h espec o a, we can apply he implici unc ion heo em o (3 .9) o ha e a = a(6 ) . g = £ g, whe e  ( ,  g( ,  ) ble in ~ and we de ine In pa icula , i£ = F -F ., a e con inuoslydi e en ia (3 . .10)  F 1 (a, E ) = T X11 (- ) H 1 ( 9 w (a, F .) )  d + o TX 12 Fi 1 g ( ,  w* (a, E ) ) d =  0 0 F 2(a, E . )  = , jT X 21 (- ) H  1 ( , w (a, E)  )  d  + o T 1 + 0  X 22 H  g ( , w ( a, b) d = 0 1 hen, we ha e he ollowing heo em o he i s app oxi- ma ion Theo em 3 .2 . Le , g sa is y he aboyé condi ions . I - he e is an  ao , I í eB ao 1 4 T ,  such ha Ó(F,F) l (3 .11)  F1(a0, 0)=0, F 2 (a o , 0)=0,  de  1  2 (a 0 , 0) J 9E 0 áa hen he e is an E .7 0 such ha sys em'(3 .1) has a T-pe io o  - dic solu ionw (a 0 , E ) , 01IE) < E 0 , con inuous in 6 : and B w (a 0 , 0 ) _  e  a 0 . P oo .  The hypo hesis (3 .11) and he implici unc ion heo em imply he e is an  i o> 0, such ha equa ions (3 .10) ha - e a solu ion  a ' (E),1 a(6)1-<a,  0-IEI<¿ 0 . Theo em 3.1 .  - implies he esse ions o he heo em . REFE RENCES B ay on, R . Small - signal s abili y c i e ion o elec-- icalna wo kscon aining lossles ansmision lines . I . B .M . J . Res . De elop . 12(1968), 431-440- [2] C uz, M . and J . Hale . Exis ence, uniqueness and con i- - nuousdependence o he edi a ysys ems . Annaly Ma h . Pu a Appl . 85 (1970), 63-82 . C uz, 11 . and J . Hale . Exponen ial es ima es and he - saddle poin o neu al unc ional di e en ialequa- ions . J . Ma h . Anal . Appl . 34 (1971), 267-288 . Hale, J . 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