VNU JOURNAL OF SCIENCE Nat, Sci, t XV - 1999 L IN E A R E Q U A T IO N S W IT H P O L Y IN V O L U T IO N S T ia u T h i Tao FHcuity o f MHthciiiHtics Hanoi ưnivcTsity o f Scỉviìce- V N Ư 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 ) A bstract I.et X operators A G L (X —* Y) Wifh d o m A = X m i d by X 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 7Í1 , , t h e sef o f (ill !nie(ir s u b a ỉ g eh r a o f L()( A) respectively, s a tisfy in g S , S j ~ S j S i for any } j = , ??/ ('onsider the equaiwn ; ; r r = ,v A{S).r= (0) k - 1,IM e w h er e i/ ^ XỤị, — 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 uc e t h e e q u a t i o n ( ) to t h e s y s t e m o f equations without a n y involuHon 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 S o m e fu n d a m en tal p rop erties o f p o ly in v o lu t io n o p e r a to r s a An op erato r s G Lq { X) is said to bo an involution of onloi Ì) if S'” “ / ami S’^ I = 1, / ; - Suppose th a t s is an involution of order Ì then P — _ y It 27T/ ^ f = erp — n k= \ are callocl projections aysoc’iat(‘cl with s The projections P i , , p,, satisfy th e following proportios: P j P j = where is the Kron ock or s y m b o l ± r , = /, 7=1 SPr Hence, it implies th at S = Y,^'Pr 5" ĩ=\ X = Xj, 1=1 w h eio A', = P j X 7=1 42 (j = Ty>) L i n e a r E q u a t i o n s w i th P o l y i n v o l u t i o n s 43 b Lot Si , , Syji be co m m u tativ e involution operators of orders 77.1, , respectively and Pkj^Uk = 1,7/Ả:) be projections associated with Sk{k = l , m ) We denote k = l,m } , r ^ {(?■) = (il, cim oc(») _ CM , — Plji } •'-I Pmj„^ 1 ^ s '^'ki A = { ( jz ,- ,J r n )|P o ) ^ } , 27T7 e P ro p o sitio n a) ^ — eJ VVe have the following relations p „ , = I U)eA ^>) P{ i ) P{ 3) — ị -f c \ _ I A'(J) cj ^ = / F(J) {{i)\ =' U ) i k = jk V Ả := l,m ) i f (?:) = ); where X(j) = 0)eA Tlfc Proof: a) From ^ = / VẲ: = 1, m an d the com m utativity of Pf^j^ we get m ntf k = i Jk = ĩ j f c = T ^ (j)eA fc= rn b) It n a t u r a l l y h o l d s b y the 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 ec t io n s c) It is im plied from a) and b) □ T he equation (1) can be re w ritte n in th e from = A{S)x= (1) (•)€/ where A(.) = We consider th e eq u atio n (1) under th e assumptions; V P(j), P(^), G A, ( € r, where (j), (Ả:) € X satisfying = ^{r)U)ik)P{k) (o rP (j)^ (,) = A^)U)wPik)) (fc)€A 'f^0)^(«)0)(fc)'f’(í) = V(/) (k) Acting on b o th sides of equ ation (1) by P(j), we obtain the system (2) TYan Thi Tao 44 Pu) E {^)e^ = P(j)y vơ) G A Y1 ^ P{])y V(j) e A (t)er(Ẳ:)€A E E V0 ' ) € A = (i)er{fc)€A => E E ■4(.)o k * ) '''‘" ' ’’ (‘ ) = n , ) ! / V(j) e A, (3) (r)€r(fc)6A w h ere 3-(fc) = P(k)X € X(fc) L enim a ỉ f the condition (2) is satisfied, then the eqìiãtỉon (1) has a solution T e X if and only i f sys tem (3) has a soiỉiủioíi (^(A:))(A:)eA ^ of (1) then (P{k)'^ĩ'){k)eA Ỵ (k)eA ^ioreover, if T is a solution ^ soiiitioii of (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 the equation (1) has a solution is a solution of (3) X then from (3), {P{k)-^){k)eA Con ver sely, s u p p o s e th a t t h e s y s t e m (3) has a s o lu ti o n (j^(A:))(ye)6A ^ We (t)6A prove that X = ^ is a solution of the equation (1) {k)eA Iiidet'd, since •'(it) t ^{k) ) t A (Iiicl ^ ( k ) ~ (fc)€A = -Tịk)- FurtherrnoiP, i ^( k) ) { k) e^ V(J) € A, 5; ^ s o lu ti o n o f (3), which implies th at = Pu)y 5; {t)er{k)eA (r)er(fc)€A ^ J2 (*)er(fc)€A Y ^ i ’){j)wP{k)S^' ^T = F(j)ị/ (i)e r(/c )6 A ^ = -^0 )^ (t)e r ^0) Z ! (j)eA => Y (i)er ^ )2^ (j)eA Thus, (fc)GA is a solution of (1) Lem m a is proved Y (Oer □ = y- L i n e a r E q u a ti o n s w ith P o l y i n v o l u t i o n s 45 L e m m a Suppose thnt the condi t i on (2) is satisfied I f the system (3) has solutions ãỉKÌ ( /■(ả-))(a-)€A ^ one its solution in the space soliJtion of (3) in the space then (Pạ-)-'ĩ'ik))(k)eA ^ (k)eA Proof : L(*t (-T{k)){k-)eA ^ ^ solution of system (3), i e., v ( j ) A ^(00)(A-)''**'^^’^-'>"(A-) = P u y y (Oer (Ẳ-)€A ^ ^ X] (i)er(A :)eA (0er(.)6A = Puyy { i ) eA (0.A^ jiV (t)er(A-)eA Y1 12 ( kAc= ,\ Y1 (Ắ'} ( t ) €A (Oer(A)eA From Condition (2), it follows th at the left side of the last equality belongs to ker Pịj) and Its light side belongs to Note th a t Kor Pị^j) = Xị,) ^ => kerf(^) n.Y(^) ^ ij) { } we get ^\ỉ)y ~ ^ X! (0er(A-)6A ^ u (0€r(Ẳ)€A i.(\ (^^(A-Ị-í^íAíẢìeA ^ solution of systoni (3) □ Combining two rosults just obtained yields ih(* following T h e o r e m Siipposc tiiHt Coiiditioli (2) is satisfied The eqiiatioii (1) ÌÌH.S sohitioiis if and only if the svstciii (3) Ììrìs sìitioĩi 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 ( ) R e m a r k I f *4(,) ((;) r) are com m utative with the operators Sk- (A' — 1, 777) then the system (3) becomes the iiKlepondont system E (Oer = Puyy VU) G A, E x a m p l e s E x a m p l e Consider the Volterra - C arlem an integral equation of the form IVtzn Thi Tao 16 ự:>(.r,y,t) ~ y y / A',,(.7:./y ^ r ) ^ [ f i , ( r ) , / i , ( y ) , r]r/r = !/(.r.Ịj,t) (4) \vh(‘i 1) g{.r Ị/ , t ) K, i ( v , Ịj,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, 2) a ( r ) , f3{y) ai(' Ca r lp m a n t i a n s f o n n a t i o i i s o f or doi a(A + i)(.r) = a[QA.(.T)],QA-{.r) / â ( k + i ) { y ) = d{ í h- { y) ] f l k- { y) / 3) y,t,T) ,T ỉ/ if if V J = l, ? n a n d ni l e s p p c t i v r l y on < Ả-< n, Q„(:r) = ,r < A' < i n, „ , { y ) = y R, i.e., a r e in 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 erators V , W , A j j E Loi-r) as follows: {Vip){.r,yJ) = ự>a{.r),y,t ( U » ( r , i/,0 = ự > \ x j { y ) j ] , A,,^){x,y,f) = / K^j{x,yJ,T)ip{x,y,T)dT ./0 Then the equation (4) can be rew ritten in the foini n v ,w We note th a t 771 are com m utative involution o p e to rs of order V and 7Ì1 respec tively and the o perators A,J also com m ute w ith V, \V {i — 1, n , j — 1, m) Denote by — l , n ) and Qị^{ỊJ- — 1, 77?) th e pro jection s associated with the V, w respectively From the above obtained results, in order to s tu d y of E q u a tio n (4), we can s tu d y the system of the independent equations - / M^f,{T,y,f,T)ip^,,{T,y,T)dT = g^f,{x,y,t) V ;/= 1, n , / i = ,7 » , If) n wheiT rfi 1= 7=1 27T? f i = e x p , n 27T/ f2 = exp rn E x a m p l e 2; Consider th e Fredholm - Carleinari integral e q u atio n of form n ip{x K^j { x, t , T) t f [ a^{ T) , Pj{T)]dr = g{xj), t = l j = l ‘' - i where 1) g { x , y , t ) and K i j { x , t , r ) are continuous functions X e R ; t , r e [ - , 2) q (t ) is Carlem an transform ation of order n )/? (V ^ 02Ìt) = m t ) ) = t^ We define th e operators V, w , Ai j as follows: (5 L inear Equations w ith P o lyin vo lu tio n s 47 {Vip){x,t) = { Wự>) { x , f ) = -t), Ị {A, j i p ) { x , i ) = V?: = T7n; j = K,j{x,f,T)^{x,r)dT, 1, T h en the e q u atio n (5) can be re w ritte n in the form: We get V/" = / , ^ yịy ^ Denote by p ^ ( u = 1, 7?) the projections associated with V and Qi = ị n - W ) , e = exp Q^ = ụ i + W), 2ni — n We prove th at the condition (2) is satisfied Indeed, p uttin g ] ^ ^ ” k = l 1=1 then f P l' Q s-^7 j p u Q r — ^ijiysf^rf^^Qr‘> \ Pi/Qs ^ijusfir PoQi) v.4,;(/ = ^ 1, 2) and V P^,P^(/v,/i T h ( ' ont IB t o n l i o w t h a t ^ " —0 if ^ or r; / r 1,7?,) Q s Q r { s r = 1,2) a iv tlir iiitcgiul o p c ia to iy w\- have “ rl ^ / K , , [ n f , { : T ) j 3i { t ) , T ] < f ị r , - l Ì T ) ] d T ” k = \ 1=1 Putting Ơ = /32_ ;( r ) r = Pi{ơ),fÍT = the light side OI of rn thee lasT last eequality q u a l ity ccan a n aalso lso be w rritten i t t e n as 2n ' P u tting / A-=l /-1 K,Aa,{x),m ,Pi{cTM x,a)da Tran Jhi Tao 48 We get Thus, instead id of studying the equation (5) we can stu d y the following onv - Ẻ Ề Ê Ẻ i i 1 (- 2= J = ^ = r = l A-=l i = l X (6) X K i j { a k { x ) , l3 i ( f ) , P i { T ) ] i p ^ , r { : r , T ) d T - P^Qsgi^J) V i/= l , n ; s = 1, Equation (5) has solutions if and only if the system (6 ) has solutions Moreover, if s=\ solutions of (6 ), then TỈ l/=l S=1 is a solution of (5) REFERENCE 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 Warsaw 1989 2] D P Rolewicz Equations in linear spaces A m sterdam - Warsaw 1968 3] D P Rolewicz Algebraic Analysis A m sterdam - Warsaw 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 TÍN H V Ớ I CÁC T O Á N T Ử ĐA P H Ố l HỢP Trần T h ị T ạo Khoa Toán - Cơ - Tin bọc Dại học Khoã học Tựnbièn - Đ H Q G HàNội X m ộ t khòrig g i a n t u y ế n t ín h t r n g c L o { X) t ậ p tất rác t o n tư tuvến tính A trẻn X với dom i4 = X X Ik đại số Lq{ X) Giả sử i , , Sn, toán tử đối hợp cấp r ?i , tư n g ứng đòi giao hốn với Xét phương trình t ^ —1, »1 fc=(l ,m) e X{ i k = l,Tik,k = l , m) x, y € X Nội dung báo đ a phương trìn h (*) hệ p h an g trình khơng tốn tử đối h p m tính giải đ ợ c khả thi h n nhiều, đồng thời cho mối liên hệ cấu trúc nghiệm p h a n g trìn h (*) vái cấu trú c nghiệm hệ ... Mail Generalized algebraic elernents and Linear singular infecral equa tions with t r a n s f o r m e d arguments Warsaw 1989 2] D P Rolewicz Equations in linear spaces A m sterdam - Warsaw 1968... pro jection s associated with the V, w respectively From the above obtained results, in order to s tu d y of E q u a tio n (4), we can s tu d y the system of the independent equations - / M^f,{T,y,f,T)ip^,,{T,y,T)dT... e [ - , 2) q (t ) is Carlem an transform ation of order n )/? (V ^ 02Ìt) = m t ) ) = t^ We define th e operators V, w , Ai j as follows: (5 L inear Equations w ith P o lyin vo lu tio n s 47 {Vip){x,t)