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VNU JOURNAL OF SCIENCE, Sci., tXVI, Nat 2000 n93- ON A CLASS OF EQUATIONS INDUCED BY A NILPOTENT OPERATOR AND ITS APPLICATIONS Nguyen Tan Hoa Gia Lai Education and Training Department Abstract./n deal with this paper we a class of equations of the form n—1 = k=0 Ay S*x where Ay,S € Lo(X)(k = 0,n—1),S is a nilpotent operator We show that under sor assumptions on operators Ay(k = 0,n —1), all solutions of these equations can be obtaine We shall give some applications to study some classes of singular integral and differenti equations A particular case of these equations with Ao, Ay, : , An— being operators of mul plication by complex numbers ap, a1,°-+ ,@n—1(ao # 0) respectively was solved by Przewor Rolewicz in [4) Let X be a linear space over the field of complex numbers We denote by L(X) tl set of all linear operatiors with the domains and the ranges contained in the space X at Lo(X) = {A € L(X): domA = X} An algebraic operator S € Lo(X) is said to be nilpotent operator of order n(n > if its characteristic polynomial is of the form S"=0,5* 201 Theorem Let S, Ao, Aj,-++ n(n > 2) Suppost , An—1 P(t) = ¡=2 ` V248 aed: there is then in X More precisely if y(t) © Xi (8) and such that Bis > By, ,u)(0), "(By Ftp €VT1 petit where N ~ max(n.m), By ~ Ax D*(k = ToT) Proof By our assumption we AuXu = Ấy: =1,2, It is easy to check that AX, Applying Theorem Acknowledgment V Mau have Ao, A;.-++.An—1 CX, we obtain € Lo(X), Ao is invertible an #=1.n=l,P=1,2,/ - the required result The author would like to express his deep fratitude to professor Ng for helpful suggestions in the preparation of this article REFERENCE P.D Gakhov Boundary value problems and corrected edition, Moscow, 1977.) Oxford, 1996 (3rd Russian complemente: Ng V Mau Generalized algebraic elements and linear singular integral equation with transformed arguments, Wydawnictwa Politechniki Warszawskiej], Warszaw« 1989 D Przeworska D Przeworska - Rolewicz Algebraic - Rolewiez analysis Equations with Amsterdam, transformed approach Amsterdam, Warszawa 1973 D Przeworska - Rolewiez, Equations in linear space TAP CHI KHOA HOC ĐHQGHN, KHTN, t XVI, n3 Warszawa, arguments, Amsterdam, 1988 An algebrai Warszawa, 196% - 2000 VE MỘT LỚP PHƯƠNG TRÌNH SINH BỞI TỐN TỬ LÙY LINH VÀ ÁP DỰNG Nguyễn Tấn Hoà Trường Củo đẳng Sư phạm Gia Lai Bai báo ApS & Lo X)k thiết Az(k — dong đề cập đến lớp phương trình nà] dạng Se kỦ Ay Sha y d 0n — T),5 tốn tử lũy lính Chúng ta với vai gi 0,1—n) tất nghiệm phương trình thu dạn San ta dua mot vai ap dung phương, trình vi phản tích phải ... AI AoƠ By, Ag Byy ô++ Ag By AY De kale? Ap Bi By As! By, Ag! Bj, «+ Ag Bj, Aa tụ +n-15 , Mapahand From (1), (2) and (3) we obtain By Ag! B;, Ag | Bj ++ Ag !By Ap! = Oif k+ a1 + beer dt find alk,...‘na class of equations induced by Bj, Ag Bin: Ag) By: 40 By Ag'' By, Ag" By, + Ag By, Ag‘ t3, 14213744 +2y.jSn-l Nay 2(-19 11-7 Bp AQ'' By Ag! Bj, Ag! Bi Ag''Y Hạ Áo: Hy Ag''`Đy „+5 0D") xe n-lon... space of all real valued defined on a closed interval {a, 6] and aving continuous derivatives of an arbitrary order in (a, b) Let d (Dat) = Sad), (Agr) (t) / t?2apj(Tr)(D?z)(r)dr- (k = Tn—T.n Ao =