VNU, JOURNAL OF SCIENCE, Nat Sci., t.xv, n ° l - 1999 O N SO L V A B IL IT Y IN A C L O S E D F O R M OF A C L A SS OF S IN G U L A R IN T E G R A L E Q U A T IO N S W IT H R O T A T IO N N guyen Tan H oa Gia Lai Teacher’s tra.iiiing college A b s t r a c t In this paper we shall give some algebraic charactenzations of the oper ator s „ k of the form (2) and study solvability in a closed form of singular integral equation of the form (1) By algebraic method we reduce the equation (1) to the system of singular inte gral equations and then obtain all Solutions in a closed form Suppose t h a t r = {/ : |/| = 1},D + = {z : l^l < = {z : |2 | > 1}, arc respectively the boundary, interior and exterior of th e unit disk on th e com plex plan Consider the singular integral equation of the form where ự>{f), f { f ) , àự) Ễ H^{T) (0 < /i < 1) Define ( „,a ^ ) ( = — 7Ĩ / / ^ /p T ^ i 1: » t /V (2 ) ‘T T ^1 == e.r.p( — ), £ị = e l (U» ( = It is easy to chock th at s w - w s 5„.a-VV - W Sn ^- 5„,a-S - 55„.A (3) Denote p = i ( / + S ) < ? = ị ( / - S ) , = (4) u=\ Thon 16 O n S o lv a b ility i n a Closed F o rm of 17 = p, Q'^ = Cl P Q = Q P = (5) P ,P ,= 6,jP, (6) Ỉ — P\ + Po + ■■■ + F-2„+ + e ^L ' P 2n i r " = 54 ÍpP , + + e4^P P 22 + (7) A' = Y - A ' - = A ' , /-1 where - ỌA' x , - P j X A'+ - FA ', L e m m a Let defined lie HS [j = IS the Kronecker symbol in (2) Then Sr,.k- = 5Pa- - SP„+k (Ẳ- = ri - 1), (8 ) u-iiere we p v t Pq = P-2 „ Pr oof : Fioiii t h e i de nt it y ^ n — [ — k ỷk- 7-n ^ fri ^ ‘2 n ~ l — kỷk- ^ n ỵ2rt _ f ‘2n — ì — k ỷ ĩ ì + k ^2n _ fin ■ Wo obtain /■ « I' ì i - - ị ~ k ỷ k /' ^^ \ 2n I' ( ị Ị ~ ì - k ị ĩ ì + k /■ ^ Ĩ n-l-A-.r)+A ị- - „ l ~ í ,í = { S F , ^ ) { t ) - ( S F „ ^ ,^ ) ịt) (svv L e in n ia E v e i y u p e n ì tu i S'„,A i.s ÍÌÍJ íìỉgelníìic opciHtui with the cỉiHiHctciintic pulynoIiiial A'^ - À for ĩì > 1, A'“ - for it — Proof L('t n = 1, from (3) (5), (6 ) and (8 ), we got S l o = { S P o - S P , f = Po + P, = / , It is easy to check th a t Pỵ Let IÌ (A) = À* - > from (3), (5) and (8 ), \VP got = S' i , , Sn, , = {P, + P „ + , ) ( S P , - S P n +, ) N g u y e n Tan Hoa = SPk - SPn +k = s„,k- To finish the proof it suffices to show th a t for every polynomial Q(A) = a \ ^ + such th a t Q{Sn fc) = we can follow Q = /3 = = {a, ị3,~f e C) Indepd, from (5 vr have r = PkQ{Sn,k) = (a + l)Pk + 0SPk, 10 = P„ +kQ{Sn,k) = (7 - a)Pr +k - í3SPn +kFrom the last equalities, we get a = /3 = = L e m m a [2j I f th e function K { t J ) can be extended to such a manner ti a t A'(r, is analytic in both variables in D+ an d is continuoiis ill D + Then I K ( t , f)ự>{T)dt e for every if ^ X \ f I K{T,t)ip'^{T)dT = for every e In the following, for every function a{f) € X , we shall write Ự){f) = f {f ) (9) L e m m a T he equation (9) is equivalent to the following system ị (P,ự:>){t) + b { t ) { s p , ^ m - k ( t ) { s P n + k ^ m = {Pkf)(f), < {Pn +k ^ m + ỉ>,{t){SPK-ự^){t) - b{f){SPn +, < p m = (Pn+A-/)(/), , ^{ t) = f i t ) - b { f ) { S P ,^ ) { t ) + b{t){SPn +, ự ^ m where m = l/= l Proof: According to Lem m a 3, we have = E l - ' ) - ' ’!"-')- u =i ( 10) On S o l v a b i l i t y in a Closed F o rm of 19 I \ K , F , = K ,,, p, = p„ t A-A),p,, u = ỉ(t)(SP,^)Ự) - h,{f){SP„^,.^){t) = i P, f ){ f) {P„^k-^){f) + ỉ> ịụ) {Sỉ\ự ^)(t ) ~ b i t ) ị S P „ u - ^ ) ự ) = ( p „ + A / ) ( Mon'ov('i , lias hpon prove th at tho last system is equivaleiit to ( 10) H('ncc in onlf'r to St)l\'c tli(’ ('C|natioii (9) it suffices to solvf' th(’ following aystf'iii r ^, { f ) + b{t)(S'^,]it) - ĩ>ị{f)is^„+,){t) = I in th(> s p a c e x \ L e m m a + / ; , ( ( ^ J ( = (P„+A-/)(0- X x „ + f_ li ip„ ịj,.) is ii sohitiuii o f S y s t e m (11) in A' X A' then P„+/.-ự!„ is ÍÌ s u h i t i u i i o f S y s t viii ( 1 ) i n A'/, X Proof S n p p o s v that y-!„ is a soli irion o f S y s t e m (11) in A' X A' Actin'^ t o b o t h sid e of system ( 11) by oporatois P/,, p„ rrspertivcly by virtue of Lf'iniiia \V(‘ g('t (Ay-aOíO + h{f){SP,yO,){t) - h A f ) { S P „ , , ^ „ , , ) { t ) = (P,VP)(/), I ỉ^ u r „n ) ( t ) h, (t){.^P, llciicc (ì \ ^ị , ^ + j A,tA)(/) /,.) is a so lu tio n oi' s>-st('iu (11) in \ \ X A'„ /, ( / ’„, / ) ( / ) ■ □ D u e t o K'siilts ()1 L( 'nin ia a n d Li 'imiia \V(' o b t a i n t!i(' followiiij’ K'snlt L e i i i i n a I'iic cqiinliun (9) is sulvnl)!c in X i f n n d on ly if the s y s t e m (I I ) is solvỉìhlc in A X A ^ MunH nv r e v c i y subitivii ol (9) CHII l>c (ỉctcnninctì i>y the furiuuhi ^o(/) = / ( n _ b { t ) { S P , ^ ) ( t ) + h { t ) { s p „ , ^ , ^ ) { t ) , wlieiv ỷ{t ) = ( ỉ \ ph){t) + {P,r-ị ^-‘p„ ịi,-){f ) [Ọk 'Pu +k) )■*>'a solution oi'systein ( I I ) ill X X X T h e o r e m Suppose tlint (l[f)(i{T) is a continuoiis fmictiou in (r, f) e r X r which admits fill lUUilytic pnAongHtiuii in i>uth vHiifihlcs uiitu D * , where a{f ) = 'b{f) + b^{i), d{t) = b { t ) (12) Then th e equ ati on (9) adiiiits Ỉìlỉ suììitioii in H closed foini Proof: Diu' to the rosults of Lemm a 7, it suffices to show th a t the systrai ( 11) adm its all solution in a closod form T he systom (11) is oquivalf'nt to the following system N g u y e n Tan Hoa 20 I ự>kự) + i ) {t ) =92Ìf ) 4>i {t ) + { K a S r J > ) W = i { t ) , (14) o V.-'2(0 - [ K a S K a H W ) = { t ) - d { t ) ( S g , ) { t ) In order to solve th e System (14), we have only to solve the equation ĩj>2{t) - KdSKaSĩJ^2 (15 [ K d S K a { ĩ l > ĩ + rj’ĩ ) ] i f ) = ( , (16) where g s ự ) = g { t ) - d { t ) { S g ) { t ) Rewrite (15) as th e following v (0 - V’2 ^ ( - where ĩ l >ỉ { t ) = {Pi'2)if), = -(< V '2 )(0 -(V '2 e ^ ' V-2 )• By our assum ption for ( r - f ) “ 'rf(f)a(r) , by the Lem m a 3, wc have { K u S K a t ỉ m = (17 v (f) - { Kl SKaĩ ỉ ’ĩ ) { t ) - v V (0 = ĩ){t) e x \ ệ (f) := V’2 (0 e X 19 Hence, the equation (18) is just a Riem ann boun dary problem ệ+ự)-c(>-{f)=93{t) 20 T he equation (20) has the solution { = 2.93 (^) + ^ { S g s ) ^ ) , (í>~'ự) = =Y93{f) + ụ s g ) { t ) (21 U n S o lv a b ility i n a Closed F o r m of., 21 From (19) and (21), we obtain V-2(0 = - */V(0 = ộ ^ { t ) - ộ - ụ ) + ( KaSKaậ~) { f ) = fi3Ìf) - — {K,iSKag3)(t) + -{K\iSKaSg-.ị){f) The thooiom is proved by a similar argunuMit as above, we prove a dual statem en t, namely WP ha\'e T h e o r e m Suppose that (r - 1}-^ r/(f }a(r) is n continuous fiiiictioij in ( r j ) e r x r which Hcỉiniĩs HỈ1 analytic prolongation in i)oth variahỉes on to D ~ , where ^ ( , ^ ( defined by (12) Then the equation (9) adinits ail solution in H closed fonn A c k n o w l e d g m e n t The au th or is greatly indo'bted to professor Nguyen Van Mail for valuable advice an d various suggestions th a t lod to iniprovpnient of this work REFERENCES F.D Gakliov Bouiidary value pvohleins, Oxford 1966 (3i‘(l Russian coniploinpiitod and collected edition, Moscow, 1977) X g V M a u Goiieralized algebraic olonionts a n d linear singu lar integral e q u a tio n with transform ed arguineuts, W PW , Warszawa 1989 X g V M a i i N g M T u a i i O i l s o l u t i o n s o f i n t o g r a l f q u a ti o MS w i t l i a i i a l y i i r k i ' in el a n d rotations Annales Polomci Maflieinatici L X l l L 3, 1996 D Pizeworska-Rolovvicz Ịìiations with transformed arguments, A n algebraic appioiH.ii Aiiiriinildiii - Wai^fiWft 197.] D Pr z ('\ vo is ka -R ol ('\ vi c z s R()l(‘\\'icz l u Ị Ui ì i ÌOTÌS lĩì L i ĩ ì a i r Sj i ace A nistordam - Waizawa 1968 TAP CHI KHOA HOC ĐHQGHN KHTN, t XV, v 'e t í n p ỉ g iả i đ ợ c d x g - 1999 đ ó n g c ủ a m ọ t l p p h x g t r ìn h TÍC H PHÂN KỲ DI VỚI PFỈÉP QUAY N guyễn T ắn H òa Cao âầng Sìi p h m CÌH Líìi Bài báo đồ cập đốn vài đặc tinrng đại số toán tử 5„ A- dạng (2) nghiên i-thitính giải đ ợ c dạng đóng cùa phươĩig trình tích phản kỳ dị (lạng ( 1), Bằng p h n g pháp (lại số đ a pliương trình (1) hệ phương tiìiih tích phân kỳ (iỊ sau th u đưực tấ t rác ngliiộm dạng đóng ... collected edition, Moscow, 1977) X g V M a u Goiieralized algebraic olonionts a n d linear singu lar integral e q u a tio n with transform ed arguineuts, W PW , Warszawa 1989 X g V M a i i N... as above, we prove a dual statem en t, namely WP ha'e T h e o r e m Suppose that (r - 1}-^ r/(f }a( r) is n continuous fiiiictioij in ( r j ) e r x r which Hcỉiniĩs HỈ1 analytic prolongation in. .. i)oth variahỉes on to D ~ , where ^ ( , ^ ( defined by (12) Then the equation (9) adinits ail solution in H closed fonn A c k n o w l e d g m e n t The au th or is greatly indo'bted to professor