DSpace at VNU: On linear multipoint boundary - value problems for index-2 differentiable - algebraic equations

VNU JOURNAL OF SCIENCE, Nat Sci., t.x v - 1999 O N L IN E A R M U L T IP O IN T B O U N D A R Y - V A L U E P R O B L E M S F O R IN D E X -2 D IF F E R E N T IA B L E - A L G E B R A IC E Q U A T IO N S N g u y en V an N ghi Fnciiky o f Mcìthcỉiiỉìtics College of Natiirai Sciences - VNU A b stra c t T h i s p a p e r deals w it h j n u l fi p o rn t B V P i i f o r l i n e a r iride.r~2 D A E s It fios been s h o w n t h a t t he 7'esulfs o b l a n ỉ e d by ỊSj f o r f r a n sf e b l e DAPJs cmi be (’.rlciidrd to l i n e a r t i m e v a i ' y m g i ndex~ s y s t e m s I IN T R O D U C T IO N Consider the following m ultipoint b o u n d a ry -v a lu e problem (BVP) for linear (liffoien tial-algebraic equations (DAEs): L.r A{f ) x^ + B { t ) x - ry(/), / e J := [/(),T] I t Ị d i ] { t ) r{ t ) = -i, (1.1) (1-2) ■'to where A, B e C ( J , are continuous m a trix -v a lu e d functions, // E D V ( J W ^ ' ‘) is a m atrix - valued function of bounded variations, (]{f) ^ c := C ((J,R ^ ') and E K ” aio given function and vector respe^ctively By the Riesz th e o m ii, the left hand side of (1.2) represents a goneial form of liiK'Hi bounded operators from c to R “ In wh at follows , Wf* russumo that D A E (1.1) w i t h tli(' pair { A , B \ is tractal)ì(' w i t h index , i.e., (sec [1, 2]): 1) There exists a continuously differontiablc p ro je c to r" function Q € c ’ (.7, Q'^(t) — Q{t), such th a t 2) ) i Koi A{f ) for evory f e J T h e m atrix Ai { t ) = ^ o ( + ^ o ( Ọ ( ' Ao := -4, i ?0 ~ B ~ A P \ is singular and the m atrix yl2(0 (0 + (0 (0> whore Q\ {f ) tloiiotcs a projoction onto the nnllspace Koi (Bo - A q[ P P xỴ ) P is nonsiiigular for all t e [/(),T Denote by p and P\ the oporatoi'S I — Q and Ỉ - Q\ rosportivel\' O h \’iously, p and Pi aro also projector functions satisfying lolations: p e c * (J, PỌ = Q P P\Q\ — Q\P \ — Since ( 1.1) can be refornmlatecl as Ấ [(P.r)' - P'x] +z?.r = q, wo sliould look for solutions belonging to the Banach space X : = { x e C { I R n : P-r K ey words and phrases DAEs, index 2, m u ltipo int BVP, Noether o perator 30 O n L in e a r M u l ti p o i n t B o u n d a r y - Value P r o b l e m s f o r with the norni 31 :r X oc Let Q\ G C ’ ( J , K " ’*” ) and without loss of generality, we can suppose that Qi is a canonical projpction satisfying Q i Q = It follows from the last relation th a t P P ị X = FP] P r e C ‘ ( J ,R " ) Lot Y{ t ) be a fundam ental solution of the following O D E : Y ' = [(PP,)' - PPiA^^B]Y-, Y{ s , s ) = I Denote by X ( t , s ) tlio m atrix M( f ) V ( t , s ) P{s) Pi{s), where M{ t ) := I + Q ị Q Q i i P Q i Ỵ - A A ; ' Z ? ] P P , thou X( f s ) is a solution of the homogeneous I V P : A ( t ) X ' + B ( t ) X = ; P{ s ) P, { s ) [ X{ s s ) - I ] = It has been proved th a t Ker X { t , s ) = Kor P{s) Py{s} for all t , s e {fo,T] Moroover, the ỈVP: -4(0.r' + B{t).r = q{t)- P{to)Pi ito){x{fo) - T o) = 0, has a Iiniquo solution of the form (cf [2]): r{f ) = X { t J o ) r o + X { f J o ) [ X Ựo t ) h( r ) d r q { f ) , ■ho W'hcie h(f ) = P P A ^ ' q + [ P P y Y P Q A ^ ' q , ( 1,3 ) and W ) ■= {PQi + Q P i ) A ^ \ ] + Q Q , { P Q i A ^ ^ q Y - Q Q , ( P Q y Y P Q i A ^ ' q (1.4) For investiftatiug imiltipoiiit B V P ( 1.1), (1.2), the technique described in [3] can he applied Since proofs of most statem ents in this article can be caniod out in similar ways as m 1] they will bo om itted ĨĨ UKCÌU.ATÌ \ÍIIĨT ĨP ĨN T ĨW P We cleiiotí' bv D t h r shootiiig m atrix í/a/(/) vY(^/()) and by 7^0 i'll** following suhs('t of K"; 7^() ;= { I 0 VO r - 0 0/ Sinc(' Ker D = Kvv P(0) Pi (0) = Span { (1 ,0 , In i D = TZo ( 1, 0, 0)^}; = S p a n { ( , 1, )^ }, it follows from Theorem 2.1 th a t Problem ( 1.1), (1.2) with d a ta (4.1), (4.2) is uniquely so lva b le fo r o vei v (] G C ( J , K ^ ) a n d 72 G K ii/ Now let /1 0 0^ (if; \0 0/ : = (7 , 72, 0)'^ (4.3) N g u y e n Van Nghi 34 The shooting m atrix D is defined as: D= / đr ì {t ) X{t ) = \0 e-2 0 0/ Thus, K e v D = K e r P ( ) P i ( ) = S p a n { (l, 0, ^ ; (0,0,1)'^'}, but Im D = Span { (1 - e , e ,0 )^} 7^ Teo = S p a n { ( l , 0, 0)"^; (0, 1, 0)'^} Therefore, condition (2.2) of Theorem 2.1 is not valid Using Theorem 3.1, part (ii), we comp to tho following necessary and sufficient condition for the existence of solutions of (1.1), (1-2) with d a ta (4.1), (4.3): ( e - ) / e ' l [ { I - T ) e ^ [ { T - l ) q i { T ) + {t ^ - ì ) q Ì r ) ] d T\ d t + Jo '■-'o Ị + { l - e ) + {2-e) f (l-/)e ‘| ^ (1 - r ) e ^ [ ( r - l) ợ i( r ) + (r^ - l) ( (r )]rfr|fif+ {(1 - t)qi{t) + Qiit) + q3 { t ) }dt + Jo + (l-e) [ { { I + f Jo = (2 - e)7i + (1 - e)72 q {f) A c k n o w l e d g e m e n t T h e a u th o r th an ks DSc, p K Anh for suggesting the considered topic and several helpful discussions REFERENCES 1] R, Maz On linear differential - algebr aic equations and linearizations J Appl Num Math 18(1995) 267-292 Ị2 11 L a u i o u i A b li u o V i i i g I i i c t l i o d f oi fullv uiiplicit iiuli'x D A E o Comput 1(1997), 94-114 3] P.K.Arih M ultipoint B V Ps for transferable DAEs S I A M J I - Linear case Viei Se t ,/ Math.,2b 4(1997) 347 - 358 TAP CHI KHOA HOC ĐHQGHN, KHTN, t.x v , n ° l - 1999 VÊ BÀI T O Á N BIÈN N HIEU DIEM ĐỐI VỚI P H Ư Ơ N G T R ÌN H VI PH À N ĐẠI số CHỈ số N gu yễn V ăn Nghi Khoa toán Đại học Khoa học Tự nhiên - DH QG Hà Nội Bài báo đề cập đ ến toán biên nhiều điểm p h an g trình vi ph ân đ ại số chì số Kết báo chì rằn g kết nhận bới [3] cỉii số mờ rộng lên cho phư ơng trình số ... considered topic and several helpful discussions REFERENCES 1] R, Maz On linear differential - algebr aic equations and linearizations J Appl Num Math 18(1995) 26 7-2 92 Ị2 11 L a u i o u i A b li u o... (4.3): ( e - ) / e ' l [ { I - T ) e ^ [ { T - l ) q i { T ) + {t ^ - ì ) q Ì r ) ] d T d t + Jo ' -' o Ị + { l - e ) + {2-e) f (l-/)e ‘| ^ (1 - r ) e ^ [ ( r - l) ợ i( r ) + (r^ - l) ( (r )]rfr|fif+... df, = -A /-> is mi arbitrary vector and D denote the restriction of D onto h n D ^ ' E x a m p l e s Consider s y s t P i n (1.1) with the following data: /1 0 Vo 0 / / B = -1 t - n -t - ^ qeCiJ^R^)-