DSpace at VNU: Linear equations with polyinvolutions

Siipposc tiiHt Coiiditioli 2 is satisfied... Equations in linear spaces.

L I N E A R E Q U A T I O N S W I T H P O L Y I N V O L U T I O N S

T i a u T h i T ao

FHcuity o f MHthciiiHtics Hanoi ưnivcTsity o f Scỉviìce- V N Ư

A b s t r a c t I.et X he (I l i n e a r s pace o v e r c D e n o t e hy L q ( X ) t h e sef o f (ill !nie(ir

o p e r a t o r s A G L ( X —* Y) Wifh d o m A = X m i d by X s u b a ỉ g eh r a o f L ( ) ( A )

L e t S i , 5, „ be i n v o l u i i o n s o f o r d e r s 7Í 1 , , r e s p e c t i v e l y , s a t i s f y i n g S , S j ~

S j S i for any } j = 1 , 2 ??/ ('onsider the equaiwn

k - 1 ,IM

I n t his p a p e r we p r e s e n t a m e t h o d to r ed u c e t h e e q u a t i o n ( 0 ) to t h e s y s t e m o f

e q u a t i o n s w i t h o u t a n y i n v o l u H o n T h e n w e are able to g i v e all soỉutiOTìs o f EquatìOìì (0) in a close.d f o r w

1 S o m e fu n d a m e n ta l p r o p e r tie s o f p o l y i n v o l u t i o n o p e r a to r s

a An o p erato r s G L q { X) is said to bo an involution of onloi Ì) if S'” “ / ami S’^

I = 1, / ; - 1.

Suppose t h a t s is an involution of o rd er 1Ìth e n

27T/

k = \

are callocl projections aysoc’iat(‘cl with s.

T h e projections P i , , p,, satisfy th e following proportios:

1 P j P j = w h e r e is t h e K r o n o c k o r s y m b o l

2 ± r , = / ,

.7=1

3 SPr

Hence, it implies th a t

S = Y , ^ ' P r 5 "

X = 0 X j , w h e i o A', = P j X (j = Ty>).

7=1

42

b Lot Si , , Syji b e c o m m u ta tiv e involution op erators of orders 77.1, , respect i vel y

an d Pkj^Uk = 1,7/Ả:) be p ro je ctio n s asso ciated w ith Sk{ k = l , m ) We denote

— Plji } •'-I Pmj„^ 1 1 ^ s '^'ki

A = { ( j z ,- ,J r n ) |P o ) 7 ^ 0 } ,

27T7

P r o p o s i t i o n VVe h ave the following relations

a) ^ p „ , = Ị

U ) e A

^>) P{ i ) P{ 3 ) — ị -f c \ _ / \ '

0 ) e A

{{i) = U ) i k = jk V Ả := l,m )

Tlfc

Proof: a) From ^ = / VẲ: = 1, m a n d the com m u tativ ity of Pf^j^ we get

m ntf.

fc = 1 rn

b) It n a t u r a l l y h o l d s b y th e c o m m u t a t i v i t y a n d th.e a s s o c i a t i v i t y o f p r o j e c t i o n s

c) It is im plied from a) an d b) □

2 T h e equation (1) can be re w ritte n in th e from

(•)€/

where Ặ) =

We consider th e e q u a tio n (1) u n d er th e assumptions; V P(j), P(^), where (j), (Ả:)

= ^{r)U)ik)P{k) (o r P ( j ) ^ ( ,) = A ^ ) U ) w P i k ) )

'f^0)^(«)0)(fc)'f’(í) = 0 V(/) (k)

Acting on b o t h sides of e q u a tio n (1) by P(j), we ob tain th e system

Pu) E = P(j)y v ơ ) G A {^)e^

(t)er(Ẳ:)€A

(i)er{fc)€A

=>

(r)€r(fc)6A

w h e r e

3-(fc) = P(k)X € X(fc).

L en im a 1 ỉ f the condition (2) is satisfied, then the eqìiãtỉon (1) has a solution T e X i f

and only i f s ys t em (3) has a soiỉiủioíi (^(A:))(A:)eA ^ Ỵ ^ioreover, i f T is a solution

(k)eA

o f (1) then (P{k)'^ĩ'){k)eA ^ soiiitioii o f (3) Hiid conversely, i f (j'(k)}(k)€A ^ ^(A) is a

(*)€A

solution o f (3) then X = ^ j:(A:) is a sohition o f (1).

{k)eA Proof: It is obvious t h a t if th e equation (1) has a solution X th en from (3), {P{k)-^){k)eA

is a solution of (3)

C o n v e r s e l y , s u p p o s e th a t t h e s y s t e m ( 3) h a s a s o l u t i o n (j^(A:))(ye)6A ^ W e

(t)6A

prove th at X = ^ is a solution of the equation (1).

{ k ) e A

Iiidet'd, since •'(it) t ^ { k ) ) t A (Iiicl ^ ( k ) ~

(fc)€A

= -Tịk)- Fur the rrnoiP, i ^( k ) ) { k ) e ^ ^ s o l u t i o n o f ( 3) , which implies th at

{ t ) e r { k ) e A

Thus,

(fc)GA

is a solution of (1) L em m a 1 is proved □

L e m m a 2 Sup pose thnt the condi t i on (2) is satisfied I f the s ys tem (3) has solutions

soliJtion o f (3) in the space

(k)eA

Pr oof : L(*t (-T{k)){k-)eA ^ ^ solution of system (3), i e., v ( j ) 6 A

^(00)(A-)''**'^^’^-'>"(A-) = P u y y

(Oer (Ẳ-)€A

( 0 e r ( ) 6 A (0.A^

(t)er(A-)eA

Y1 12 Y1

( kAc= ,\ ( t ) € A

j i V

(Ắ'}

( O e r ( A ) e A

From C ondition (2), it follows th a t the left side of the last equality belongs to ker Pịj)

ij)

{ 0 } w e get

(0er(A-)6A

( 0 €r (Ẳ ) €A

i.(\ (^^(A-Ị-í^íAOÌíẢìeA ^ solution of systoni (3) □

Com bining two rosults ju st obtained yields ih(* following

T h e o r e m Siipposc tiiHt Coiiditioli (2) is satisfied T h e eqiiatioii (1) ÌÌH.S sohitioiis if and only i f the svstciii (3) Ììrìs soììitioĩi Moreover, if 1 ' is H sobition o f (Ĩ) tiicn {P{k-)-^')(k ) e \

a solution o f (3) and coiiverscly, //" ^ soìĩìtion o f (3) then 1' = ^

{k-)eA

ri sollỉĩioỉi o f ( 1 ).

R e m a r k I f *4(,) ((;) 6 r) are com m utative w ith th e operators Sk- (A' — 1, 777) then the

system (3) becomes the iiKlepondont system

(Oer

3 E x a m p l e s

E x a m p l e 1 Consider th e Volterra - C arlem an integral equation of the form

ự:>(.r,y,t) ~ y y / A',,(.7:./y ^ r ) ^ [ f i , ( r ) , / i , ( y ) , r ]r /r = ! / ( r Ị j , t ) (4)

1) g{ r Ị/ , t ) K, i ( v , Ị j , t , T ) aif' c o n t i n u o u s f u n c t i o n s V/ = 1, n , J = l , ? n

2) a ( r ) , f3{y) ai(' C a r l p m a n t i a n s f o n n a t i o i i s o f o r d o i V a n d ni l e s p p c t i v r l y o n R, i.e.,

a(A + i)(.r) = a[QA.(.T)],QA-{.r) / ,T if 1 < Ả-< n, Q„(:r) = ,r.

â ( k + i ) { y ) = d{ í h- { y) ] f l k- { y) / ỉ/ if 1 < A' < i n, 0 „ , { y ) = y.

3) y , t , T ) a r e i n v a r i a n t u n d e r t h o t r a n s f o r m a t i o n s a { r ) , f3{y).

W e define the o p e r a to r s V , W , A j j E Loi-r) as follows:

{ V i p ) { r , y J ) = ự > a { r ) , y , t

( U » ( r , i / ,0 = ự > \ x j 3 { y ) j ] ,

A , , ^ ) { x , y , f ) = / K ^ j { x , y J , T ) i p { x , y , T ) d T

./0

Then the equation (4) can be rew ritten in th e foini

n 771

We note t h a t v , w are co m m u ta tiv e involution o p e r a to rs of o rd er Vand 7Ì1resp ec­

tively and the o p erato rs A,J also co m m u te w ith V, \V {i — 1, n , j — 1, m ).

Denote by — l , n ) and Qị^{ỊJ- — 1, 77?) th e p ro je c tio n s asso ciated with th e V, w

respectively

From the above o b tain ed results, in o rd er to s tu d y of E q u a tio n (4), we can s tu d y the system of the in d ep en den t equations

- / M ^ f , { T , y , f , T ) i p ^ , , { T , y , T ) d T = g ^ f , { x , y , t ) V ; / = 1, n , / i = 1 , 7 » ,

If)

1 = 1 7=1

f i = e x p -, f 2 = e x p

E x a m p l e 2; Consider th e Fredholm - Carleinari in tegral e q u a tio n of form

ip{x

t = l j = l ‘' - i

K ^ j { x , t , T ) t f [ a ^ { T ) , P j { T ) ] d r = g { x j ) , (5

where

1) g { x , y , t ) an d K i j { x , t , r ) are c o n tinu o u s fu n ctio n s X e R ; t , r e [ - 1 , 1

2) q(t) is C arlem an tra n sfo rm atio n of o rd er n

3 ) / ? ( V - - ^ 0 2 Ì t ) = m t ) ) = t^

We define th e o p erato rs V, w , Ai j as follows:

{Vip){x,t) =

{ Wự > ) { x , f ) = - t ) ,

{A, j i p ) { x , i ) = Ị K , j { x , f , T ) ^ { x , r ) d T , V?: = T7n; j = 1, 2.

T h e n the e q u a tio n (5) can be re w ritte n in the form:

We get

V/" = / , ^ y ị y ^

D enote by p ^ ( u = 1, 7?) th e p ro jectio n s associated w ith V and

Qi = ị n - W ) , Q^ = ụ i + W),

2 n i

e = e x p —

n

We prove th a t the co n d itio n (2) is satisfied Indeed, p u ttin g

] ^ ^

” k = l 1=1

t h e n

f P l ' Q s-^7 j p u Q r — ^ijiysf^rf^^Qr‘>

\ Pi/Qs ^ijusfir PoQi) — 0 i f ^ o r r; / r

v 4 ,;(/ = ^ 1, 2) and V P ^,P ^(/v,/i 1,7?,) Q s Q r { s r = 1,2).

T h ( ' 1 ont IB t o n l i o w t h a t a i v t l i r i i i t c g i u l o p c i a t o i y w \ - h a v e

” k = \ 1=1

P u t t i n g

Ơ = /32_ ; ( r ) r = Pi{ơ),fÍT =

the light side of the last equality can also be w ritten as

2n

P u ttin g

OI r n e lasT e q u a l i t y c a n a ls o be w r i t t e n as

/ K , A a , { x ) , m , P i { c T M x , a ) d a

' A-=l /- 1

We get

Thus, instead of stu d y in g the equation (5) we can s tu d y th e following onv id of stu d y in g the equation (5) we can s tu d y th e following onv

- Ẻ Ề Ê Ẻ i i 1 1

(-2 = 1 J = 1 ^ = 1 r = l A-=l i = l

X K i j { a k { x ) , l3 i ( f ) , P i { T ) ] i p ^ , r { : r , T ) d T

- P ^ Q s g i ^ J ) V i / = l , n ; s = 1, 2

X

( 6 )

E q u a tio n (5) has solutions if and only if th e system (6 ) has solutions Moreover, if

s = \ 2 solutions of (6 ), th en

l/=l S=1

is a solution of (5)

R E F E R E N C E

1] Nguven Van Mail Generalized algebraic elernents and Linear singular infecral equa­ tions with t r a n s f o r m e d arguments W arsaw 1989.

2] D P Rolewicz Equations in linear spaces A m sterd am - W arsaw 1968.

3] D P Rolewicz Algebraic Analysis A m sterd am - W arsaw 1987.

T A P C H Í K H O A H O C D H Q G H N , K H T N , t.xv, n ^l - 1999

P H Ư Ơ N G T R ÌN H T U Y Ế N T ÍN H V Ớ I CÁ C T O Á N T Ử ĐA P H Ố l H Ợ P

Trần T h ị T ạ o

K hoa Toán - Cơ - Tin bọc Dại học Khoã học Tựnb ièn - Đ H Q G HàNội

X l à m ộ t khòrig g i a n t u y ế n t í n h t r ê n t r ư ờ n g c L o { X ) là t ậ p t ấ t cả r ác t o á n t ư

tu v ến tính A trẻ n X với dom i4 = X X Ik đại số con củ a L q { X) G iả sử 5 i , , Sn, là

các to án t ử đối hợ p c ấ p r ?i , tư ơ n g ứng đòi m ột giao hoán với nhau

Xét ph ư ơng trìn h

trong đó

t ^ — 1, »1

fc=(l ,m)

e X { i k = l , T i k , k = l , m ) và x, y € X

Nội dun g của bài báo này là đ ư a ph ư ơng trìn h (*) về hệ p h ư a n g trình không còn

to án t ử đối h ạ p m à tín h giải đ ư ợ c củ a nó khả th i h ơ n nhiều, đồng th ờ i cho mối liên hệ giữa cấu trú c nghiệm của p h ư a n g trìn h (*) vái cấu trú c nghiệm củ a hệ đó