VNU JOURNAL OF SCIENCE, Nat Sci., t.x v . - 1999
O N L I N E A R M U L T I P O I N 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 I N D E X - 2 D I F F E R E N T I A B L E - A L G E B R A I C E Q U A T I O N S
N g u y e n V a n N g h i Fnciiky o f Mcìthcỉiiỉìtics College of Natiirai Sciences - VNU
A b s t r a c t . T h i s p a p e r deals w i t h j n u l f i 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 s f e r a 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 n d e x ~ 2 s y s t e m s .
I. IN T R O D U C T IO N
Consider th e following m ultipoint b o u n d a r y - v a lu e problem (B V P) for linear (liffoi- en tial-a lg e b ra ic equ atio ns (DAEs):
L.r A{ f ) x ^ + B { t ) x - ry(/), / e J := [/(),T] (1.1)
I t
Ị
d i ] { t ) . r { t ) = -i, ( 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 b ounded variations, (]{f) ^ c := C ( ( J ,R ^ ') and 7 E K ” aio given function and vector respe^ctively.
By the Riesz th e o m ii, the left han d side of (1.2) represents a goneial form of liiK'Hi bounded op erato rs from c to R “ .
In w h a t f ol lo ws , Wf* russumo t h a t D A E ( 1 . 1) w i t h tli(' pair { A , B \ is tractal)ì(' w i t h index 2 , i.e., (sec [1, 2]):
1) T h e re exists a continuously differontiablc p r o je c to r " function Q € c ’ (.7, ). i Q'^(t) — Q{t), such t h a t Koi A{f ) for evory f e J.
2) T h e m atrix Ai { t ) = ^ o ( 0 + ^ o ( 0 Ọ ( 0 ' Ao := -4, i ?0 ~ B ~ A P \ is singular and the m a trix yl2(0 ( 0 + ( 0 (0> whore Q\ {f ) tloiiotcs a projoction o nto the nnllspace Koi (Bo - Aq[ P PxỴ ) P . is nonsiiigular for all t e [/(),T . D enote by p an d P\ the oporatoi'S I — Q and Ỉ - Q\ rosportivel\'. O h \’iously, p and Pi aro also p ro jecto r functions satisfying lolations: p e c * (J, P Ọ = Q P - P\Q\ — Q \P \ — 0 . 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 e y words and phrases. D A Es, index 2, m u ltip o in t BVP, N oether o p erato r.
30
w ith the norni X :r 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 = 0. 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,)' - P Pi A^ ^B ]Y -, Y{ s , s ) = I.
D enote 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 th e homogeneous I V P :
A ( t ) X ' + B ( t ) X = 0 ; P{ s ) P, { s ) [ X{ s . s ) - I ] = 0 .
It has been proved t h a t Ker X { t , s ) = Kor P{s ) Py{s} for all t , s e {fo,T]. Moroover, the Ỉ V P :
-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
h( f ) = P P . A ^ ' q + [ P P y Y P Q . A ^ ' q , ( 1,3 ) W'hcie
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 ap- plied. Since proofs of most s tatem e n ts in this article can be can io d out in similar ways as m 1]. they will bo om itted.
ĨĨ U K C Ì U . A T Ì \ Í I I Ĩ T Ĩ P 0 Ĩ N T Ĩ W P
We cleiiotí' bv D t h r shootiiig m atrix í/a/(/) vY(^/()) and by 7^0 i'll** following suhs('t o f K";
7^() ;= { I <l n{t ). r[ t ) : .r e C ' Y
T h e o r e m 2.1. PiuiAcni (1.1), ( Ỉ. 2) is Iiniquelv sulvalilc un Ả' Ĩ U I a n v q G
c
Hiid 7 e TZq if aiid oaJy if tiic shooting m a tr ix D satisfies cuiiditions:Kci D = K c i A ự o ) 0 K c i A ị ( t o ) = Kei Pựị )) Pi ị to),
ỉ I U D = TZo-
In particular, W P can consider tlip followiníỉ, m ultipoint condition:
Ì l )
r . r : = ^ ơ , . r ( ^ ) = 7 ,
7 = 1
(2.1) (2.2)
(2.3) whero fo < 11 < t'2 < ■ ■ ■ < < T and D, e (/: = l,? n ) arc given constant matricos.
32 N g u y e n Van Nghi C o r o l l a r y 2.1. Piulììciii (1.1), (2.3) is uniquely suluihlc on ,v for iiiiy Í/ G c iiiid Ị 6
Ĩ Ì Ì
D-2...D , „ ) i f ỈÌIICỈ o n l y i f t h e s h v u t i n g n i H t r i x D = y " D , S Ht i s t e s
conditions:
1.=. 1
Kei D — K crA (fo) 0 Kci A i(fa) = K c r P (f o ) Pi(/()).
7h jD - I w ( D ị , , D,„).
III. IR R E G U L A R M U L T IP O IN T BVP
In this section, W(> suppose th a t condition (2.1) a iu i/o r (2.2) a n ' not satisfif'd.
Consider a linear bounded o p erato r £ acting fioiii Af to y := CỊ.,^X R ” (lofined by:
/ L.r Vt
Ct : =
where •= {'/ e
c
; Q i A .2 ^q e c * } and \\q ;= q oo + IVi Ker P(^o)-Pi(^o) c Kei D, for the sake of simplicity wo can suppose th a tdim A'er p ự o ) Pị ị t o) = u < dim K e r D = p.
Lot be an o ith o n o n n a l basis of Ke i P { t o ) P ị { t o ) . — ^I.i- 'vhoiT th(' superscript T denotes th e transposition. Let be an extension of to an OI- thonorm al basis of Ker D. Define a colum n m a trix ỉ>(/) := (ip^+i{f)----, ỌpỰ)) with (/?,(#) = X{ t , t o ) i J ° {i = u + l , p ) and p u t M := r/^ Ir is easy to prove th a t M is nonsingular and -Y can be decomposed into a direct sum of clospd suhspac os:
Af = K e r C e K c i M , w h e r e ( U . T ) ( t ) : = ^ ( t ) { s ) r { s ) d s . a nd K ei £ = {.r =
$ ( f ) a : a 6 = Span F u rther, (leaoto by bp an o rth o no n iial basis
+ IKQi SiliCO
i > ( ‘ I I X { p ~ I ' ) / / -A n I i i t i l 1
/ respectively.
T h e o r e m 3 .1 . The folloxving stateinents hold:
i) c : A! y is a bounded linear Noether operator, inci c dim Kor c ~ codiin Iin £ = —
= - á ì m ị K e i A{fo) & K e v Ai {t o) } = - d i m Kor P (/o) Pi (^i).
it) Problem (1.1), (1.2) is solvable on X t f and only i f the given data {r/,7 } •sa/z.s/y condition:
v r ( 7 - f
‘Ito
where f { t , t o ) ■= X ( t , t o ) f ' X{ t o , T ) h ị r ) dT + q{t) a n d h ( t ) , q{t) are difijied by (1.3), (1.4) respectively.
i n ) A general solution of ( l . l ) j (1-2) is o f the forrri:
x { t ) = X { t . t o ) { x o + W o a ) + / ( / , t o) + m o .
whem .7-0 = a = - A / - > { t ) { X ( t . J o) x o+f { f , f o) } df, a e is mi arbitrary vector and D denote the restriction of D onto h n D ^ '.
4. E x a m p l e s
Consider s y s t P i n (1.1) with the following data:
/ 1 0 0 \
0 1 0
V o 0 0 /
B =
/
0 - 1t 0
\ 1 - ^ 1
- n - t
0 /
q e C i J ^ R ^ ) - J : = 0, 1
A simple co m p u ta tio n shows that:
/ 0 0 0 \ / 1 0 0 \ / 1 0 - 1 \
Q = 0 0 0 ; ip — 0 1 0 ; A 1 — 0 1 - f
M ) 0 l )
Vo 0 oy
u 0 0 // \ - f 1 0 ^ / 1 0 - 1 -- f \
Q i = f 0 1 Q i Q = 0; /h = 0 1 - f 2
\ l - f 1 o y u 0 1 Ì
/ t -1 0 ^
/ 0 - e ' 0 )
Px = f ^ - t 1 - / 0 ; X { i ) = 0 ( l - t ) e ' 0
-1 l y \ 0 fe^ o j
i / Suppose tliat:
^ 0 0 0 \
0 1 0
\ 0 0 0
/
111 ttiis case the shooting m atrix is of the form:
1 / 0 0
( h l ( f ) X { f ) ^ D =
■>0
0 r - 2 0
VO 0 0 /
Sinc('
(4.1)
(4.2)
Ker D = Kvv P(0) Pi (0) = Span { ( 1 ,0 , (1, 0, 0)^};
I n i D = TZo = S p a n { ( 0 , 1, 0 )^ },
it follows from Theorem 2.1 t h a t Problem ( 1.1), (1.2) w ith d a t a (4.1), (4.2) is uniquely
s o lv a b le fo r o v e i v (] G C ( J , K ^ ) a n d 72 G K .
ii/ Now let
/ 1 0 0
^
0 1 0
\ 0 0 0 /
(if; 7 : = (7 , 72, 0)'^. (4.3)
T he shooting m a trix D is defined as:
34 N g u y e n Van Nghi
...
D = / đr ì { t ) X{ t ) = 0 e - 2 0
\0 0 0/
Thus, K e v D = K e r P ( 0 ) P i ( 0 ) = S p a n { ( l, 0, 0 ^ ; (0,0,1)'^'}, but Im D = Span { ( 1 - e , e - 2 ,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 T h eo rem 3.1, p art (ii), we comp to tho following necessary and sufficient condition for th e existence of solutions of (1.1), (1-2) with d a t a (4.1), (4.3):
( e - 2 ) / e ' l [ { I - T ) e ^ [ { T - l ) q i { T ) + {t^ - ì ) q 3 Ì r ) ] d T \ d t + Jo '■ - 'o
+ { l - e )
Ị
( l - / ) e ‘ | ^ (1 - r ) e ^ [ ( r - l ) ợ i ( r ) + ( r ^ - l ) ( 7 3 ( r ) ] r f r | f i f ++ { 2 - e ) f {(1 - t )qi {t ) + Qiit) + q3{ t ) } d t + Jo
+ ( l - e ) [ { { I + f q 2{f ) Jo
= (2 - e)7i + (1 - e)72.
A c k n o w l e d g e m e n t . T h e a u th o r th a n k s DSc, p. K. Anh for suggesting the considered topic and several helpful discussions.
R E F E R E N C E S
1] R, Maz. O n 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 l i u o V i i i g I i i c t l i o d f o i f u l l v u i i p l i c i t i i u l i ' x 2 D A E o . S I A M J. S e t .
Comput. 1(1997), 94-114
3] P.K.Arih. M ultipoint B V Ps for transferable DAEs. I - Linear case. Viei. ,/.
Math.,2b 4(1997) 347 - 358.
TAP CHI KHOA HOC ĐHQGHN, KHTN, t.x v , n ° l - 1999
V Ê BÀI T O Á N BIÈN N H IEU DIEM
ĐỐI V Ớ I P H Ư Ơ N G T R ÌN H VI P H À N ĐẠI
số
CHỈsố
2N g u y ễ n V ă n N g h i
Khoa toán Đại học Khoa học T ự nhiên - DH Q G Hà Nội
Bài báo đề cập đ ế n bài to án biên nhiều đ iểm đối với p h ư a n g trình vi p h ân đ ạ i số chì số 2. K ết q u ả chính củ a bài báo là chì ra rằ n g kết quả nhận đư ợc bới [3] đối với cỉii số 1 có thể m ờ rộng lên cho p h ư ơ n g trìn h chỉ số 2 .
