DSpace at VNU: On Solvability in a Closed form of a class of singular integral equations with rotation

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