VNU Journal o f Science, Mathematics - Physics 25 (2009) 169-177 On the set o f periods for periodic solusions o f some linear differential equations on the multidimensional sphere 5" Dang Khanh Hoi* H o a B in h U n iv e r s ity , Ỉ N g u v e n Trai, T h a n h X u a n , H a n o i, V ie tn a m R e c e iv e d A u g u s t 0 A b s t r a c i T h e p r o b le m about p eriod ic s o lu tio n s for the f a m ily o f linear d ifferen tial eq u a tio n La = ^ d \id t ~ aA ^ J Ii(x, t) — G {u “ / ) !S c o n s id e re d on th e m u itid im e n sio n a l sp h e re X E '* u n d e r the p e r io d ic ity c o n d itio n ĩ i \ t = o — uịt^b and ỉ/(.r, t ) d x — Here a is g iv e n real, and A is a fix ed c o m p le x num ber, G u ( x , / ) is a linear integral operator, It is s h o w n that the set o f param eters { , b ) for w h i c h is th e L a p la c c operator on th e a b o v e p r o b le m a d m its a u n iq u e solutio n is a m e a s u b le set o f full m e a su re in c X riiis work further develops part o f the authors’ result in [1, ], on the problem on the periodic solution to the equation (L - X)ii — G { a - / ) Here L is Schrồdinger operator on sphere " and A bclonựs 10 ihe spectrum o f L Particularly, the authors consider the case that A is an eigenvalue o f L ( the case which can be always converted to the case A ) It is shown that the main results are all right (but) on the c o m p le m e n t o f eigcnspace o f A in the d o m a in o f L W e c o n s i i i c r llie p r o h lc ii i o n perinrli c s ol ut ion' ; for t h e n o n l o c a l Schrỉ Sdi ng cr t;ypc e q u a t i o n (lẳ ^ with these conditions : u\t^o ^ Mere ỉt Ì J \t ) - is a com plex function on / Ii{x, t ) d x ^ (2) Js^ X [0,6], 5^^ - is the multidimensional sphere, n > 2; a ệ- 0, V - arc given com plcx numbers, / ( x , /) - is a given function T he change o f variables t = br reduces our problem to a problem with a fixed period, but with a new equation in which the coefficient o f the r ~ derivative is equal to - : h Ỡ ib d r ~aA u ( x , br) = G {ĩi{x, b r) - f { x , b r )) / Thus, problem (1), (2) turns into the problem on periodic solution o f the equation ĨAI = n_d_ -T -r a A y I bdt x i{x ,t) = G { u { x , t ) - f { x , t ) ) , l-m ail dangk.hanhhoi@>ahcx).com 169 (3) D K ỉỉo i / VNU Journal ofSciencc, Mathematics - Physics 25 (2008) ỊỐ9-Ỉ77 170 with the following conditions: w|i=:0 == íí|í=:ỉ, u{x,t)dx-^Q / (4) Js^ Here G u [ x , t) — ị (j{x, y ) u { y , t ) d y ( dy is the L eb e sg u e -llau sd o ríĩ m easure on the sphere 5^*) is an Js^ inteirral o p e r a t o r o n th e s p a c e / ^ ( " X [0 1]) w ith s m o o t h k e r n e l d e f in e d on ” X " s u c h that [ g{x,y)dx= (5) for all y in S'" Ỡ -a A is assumed lo be defined for the functions u ( x , t ) G i bot and with the condition I i { x , t ) d x = Let L - denote The difFereritial operation Qocị^gn ^ ịq^ Ỡ (lA in w — />2( '^ X [0, ll) So an element w t 7Í belongs to the closure o f this operation - - - — ib o i the domain V { L ) o f L — — a A , if and only if there is a sequence {Wj} c c ^ ( " X [0,1]) Uj d x — 0, such that liniiZj ~ u, l i m L u j ~ L u in H Let Ho is a subspacc and o f s p a c e L ( ' ^ X [0, I ) , Hự) ^ {ỉi(x, /) e /^2 ( " X [0 1]) I i u {j\ f )dx ()} It is well known that the eigenvalues o f the Laplace operator A on the sphere 5'^ are o f the form —k { k + n — 1), /c G z , /l' > and that A admits the corresponding orthonormal basis o f eigenfunctio ■ Wk{x) € c ^ ( ^ ( s e e , e.g [3]) L e m m a The fu n c tio n s k , m e z , k > are eig cn fim ciio m o f the /) ~ operator L in the space H q that correspond to the eigenvalues >^krn ^ in 7ĩ , , mix ^ -I- ri - 1) + Xk (G, These fu n c tio n s fo r m an ortìĩoìĩormal basis in Ho- The dom ain o f L is ịịiven by the f o n m d a X^(Z/) — { u “ ^^km^krn I ^ " I 7n,k£ ^ Co} ,k> The spectrum { L ) in the closure o f the set {A^:rn}Lem m a \ G \ f < MỒ = [ [ \g{x, y ) \ ‘^ ( l x d y Js^ Proof We have G u (x ,0 p = | [ g{x,y)n{ij,t)dyf < [ \g{x,y)\\kj Ị Js^ |G u ( x ,i) ||^ = Js^ i Ị \G x i { x ,i ) \^ d x d t < Jo J s ^ f / [ ( Í _ \3{x,y)\% \ u{ y , t ) \ ' ^ dy ) d x d t Jo Js" KJS’^ J S ’' / \uiyj)\^dy D K ỉỉo i / VNU Journal o f Science, híaihcmaĩics ~ Physics 25 (2009) Ĩ69~Ỉ77 G'u(.r, t) 171 Ị ^< / [ \g{x,y)\^dxdy [ \ u { y , t ) f d y d t - M ^ \\u Js" Js" J s " Jo | | G | | < Mo- I'he Ic'mma is proved \ \ c note that the Laplace operator is formally selfadjoint relative to the scalar product { u , v ) / u ( x ) v { x ) ( Lr on the space C '^ { S '^ ) , The product A x o G = coincides with the integral operator with the kernel A r ( ; ( x , y ) Let the function Ay~g{ x v) be continuous on " X A / - m a x { | | A , G ’i |, | |G ’l|} We put " L e m m a Lcl (' - C u ,k>o^'kynekm, then (7) i k i k 71 w l w e (H,n = ei,,n), and Y , Proof Since the Laplacc operator is selfadjoint, for ^' > we have ^ h n ~ {‘^ x G u ^ ^km) ~ (Gzi, akm ^ t)) — (Gli, ~ki^k + n — t)) + n - l ) ( u , Ckmi x, t)) ^ - k { k + n - ll follows that 0(h ^ i k{k + n - l ) ) ^ ' By the Parseval identity, we have Y1 \f^krn\'^ — ||^ x G i i || ^ < < Vh whence AÍ { k { k - h n - 1) ) - The lemma is proved Wc assum e that a IS a real number I hen by Lem m a 1, the spcctrum ( L ) lies on the real axis Most typical and intercslintĩ is the ease where the num ber ab/{27T) is irrational In this case, / A;^.,nVA:, r/ỉ e z,k the niimbers is cver>'vvhcrc dense on R and { L ) — M Then in the subspace Ho the inverse > and llie li.Wcyl theorem (sec, e.g., [4]), says that, in this case, the set o f operator L ~^ is well defined , but unbounded T he expression for this inverse operator involves small denominators r - l L / X _ v{x,i) - > ^^krn (8) ^ekrn '^km where Vkm is tlie F ourier coefllcient o f the series l/(x, t) — F^or positive num bers Ơand c, let Aa{C) ^ ^ ^Arn^fcm* A:,mG ,fc>0 denote the set o f all positive ^hn > b such that c (9) for all in, k e z , k > This definition sho w s that the set Acj[C) extends as c reduces and as Ơ grows Therefore, in wha! follows, to prove that such a set or its part is nonempty, w e require that c > be sufficiently D K ỉỉo i / VNU Journal o f Science Mathematics - Physics 25 (2008) Ĩ69-Ỉ77 172 small and Ơsufliciently larae I.et Arr denote the union o f l h c sets (9) is fulfilled for some h and all over all c > If inequality k\ then it is fulfilled for m = 0; this provides a conditioii necessary for the nonem ptiness o f Acr{C): c < k^+^^ị nki k + n - 1)1 V A- > ( 10 ) We put d = 7rỉrn*.ỆZ + n - 1)1 > T h e o re m The seis /lcr(C), A a are Borel The set A a has fu ll measure, i.e its com plcm eut to ĩhe h a lf4ine is of zero measure oc Proof Obviously, the sets A( j { C) are closed in The set A a ~ A a { \ / r ) - is Borel beinu a r= I countable union o f closed sets Wc show that Acr has full measure in Suppose Ị, / > Í) c < 7; vve consider the com plem ent (0, i ) \ A a { C ) This set consists o f all positive numbers h, for which there exist 777, k, such that 11 Solving this inequality for h, we see that, for rn, k fixed, the num ber b form an interval Ik.rn — mRi \ vvhf»rf» 77) =: tt tt — n k { k + 71 - 1)1 + \ ak { k + n - l)i U l+ a The length o f Ikrn is rnỗi^, with 4nCk Since c < - by assum ption, vvc have IGttC ỏk < 3Ả-'+'^|aA-(Ả- + n - 1)|2' 12 f-'or Ả' fixed and m var> iiig, there are only finitely many o f intervals Ikm tliat intersect the given segment ( ,/) Such intervals arise for the values o f 7n = ,2 , satisfying m c ik < /, i.e., < m < : ^ i \ a k i k + n - 1)1 -f Since < ị \ a k { k + ĨI - 1)1, we can write simpler restrictions on ÌÌI : I < m < ^ ^ \ak{k + 27T I - 1)1 < ^ \ a k { k + n - 1) 7T '13 The measure o f the intervals indicated ( for k fixed ) is d o m inated by SkSk, where S k = Sk{l) is the sum o f all integers m satisfying (13) Summing an arithmetic progression, we obtain S k < ~ \ a k { k + n - l ) \ { l \ a k { k + n - )1 + 7t} (M) D.K ỉ/oi / i'NU Journal of Science Maihematics - Physics 25 (2009) 169-177 173 l’assinv» to the union o f the intervals in question over k and m , and Iising ( 12), we sec that k =0 where 8/{/| aẢ:(Ả: + n - l ) | + 7r} ^ 37rk^'^^\ak(k + n “ 1) k =0 O bserve that the qnantitv I (l k( k + 71 — 1) -j- 7T 7T is dom inated a k { k + n ~ 1) a constant D: therefore, A:=l We have /i((0 , /) \ A , ) < /i( ( , /) \ / U ( C ) ) < CS { ) It follows that ; i ( ( , 1} \ A a ) = VC > V/ > T hus,/i((0, oo) \ A ^ ) — and A a - has full measure The ihcorcin is proved T h e o re m Su p p o se (]{x, y) is a fu n c tio n defined on X " such that the fu n c tio n A j:g {x , y) is i'oniifiiious OỈÌ S ' ' X S ' ' a n d g { x , y ) d x ^ My e " Lei < Ơ < 1, a n d let b e Aa { C) Then in the space Hi) the inverse oper ator L ~ ^ is well defined, a n d the op erato r o G is compact Proof Since h e /lcr(C ), we have V k , r n e z , k > so that is well defined and j^,24-2cr looks like the expression in ( 8) O bserv e that liiri—— -^ = as /t' -+ oo Therefore, given { k[ k + n - 1)) 7^ s > we can find an inlcíĩcr ko > 0, such that L [k(k ^ I It ^ 1))'^ ‘ i’(x, t) = Q u , V -f Qko '^i, A /‘ for all k > k() We write V = G’ii, where ()k(j ))2 * c ^ Consequently, IKA'02 ° G | | < £ Since G is com pact and I A*.-,'Til is bounded, ^ - o G is compact Next, w e have l i ^ " ‘ o G -C > ;i-o, o G | | = | | Q , „ , o G | | < f Thus, we see that the operator L ~ ^ o G is the limit o f sequence o f com pact operators Therefore, it is compact itself The theorem is proved We denote K — Kị, — L ~ ^ o G T h e o re m Suppose b € A a { C ) Then problem (I ) , (2) adm its a u iiiq u e p e r io d ic solution wiih p e r io d h f o r all e c , except, possibly, an at m ost couutahle discrete sei o f values o f y Proof Equation (1) reduces to _ r-l We write L ^ o G - - ~ /v V Ư Since K — o G is a com pact operator, its spectrum o { K ) is at most countable, and th( limit point o f cr(A') ( if any ) can only be zero Therefore, the set “ I - G { K ) } is at most connfable anH fiiscrete, and for all Ư ^ , ỷ s the operator ( K — —) is invertible, i.e., equation ( 1) is uniquely solvable T h e theorem is proved Wc pass to the q u estion about the solvability o f problem (1), (2) for fixed Ư We need to study the structure o f the set i? c c X K \ that consists o f all pairs (i/, 6), such that // 7^ and - ị ni Kf , ) Ư where Kị, — L~^ o G T h e o re m 4, E is a m e a s u b le s e t o f fu ll measure in c X For the prooi', w c need several auxiliar\ statements L e m m a F o r a?iy e > (here exists an integer k() such that ||/\fo — Kh\\ < s f o r all b G < O' < 1, where r ~ , r' „ r —1 , h t u ^ Lb ^ ^ ^'Ẳ:m yr ỉ^bU ^ '^^krn ^ ("km ■ k'Q, < ( { k { k + n - 1))2 ^ V A / < Ơ < I W e have (/Ú - K ,)u = ^krn k>ko for all D.K ỉỉo i / VNIJ Journal o f Science, Mathematics - Physics 25 (2009) ỈỐQ-Ị77 175 u k>ko I'hus l | / ú - ĩ By Lem m a there exists an integer /I'o { independent ĨhJ)-\ ^ ù^b) such that ||/v^6 - / ũ l l ^ \\ỉ (} such (liaf f o r all compact ( a m i even h o u n d e d ) operators D with \\D — /v II < Ỗ H'e have ( B ) c a { K ) + K,(0), (16 a[K) c a{B) + Here Ve(0) — {A € c I |A| < £} is the e-neighborhood o f the p o in t in c Proof Let K be a com pact operator; we fix £ > The structure o f the spectrum o f a compact operator shows that there exists < e / such that E\ set o f all spectrum points À with |A| > |A| for all A o { K ) Let and let V” ^ [ J s { À Ị , Aa:} be the Vei(A) T hen V is neighborhood o f A65 u {0} { K ) and V' c ( K ) I r ( ) By the w ell-known property o f spectra ( see, e.g.,[5], Theorem 10.20) there existi Ồ > such that a { B ) c V for any bounded operator D with \\D — /v II < s Moreover (see, e.g., [5], p.293, Exercise 20), the num ber Ổ > can be cho sen so that { B ) n 7^ ^ VA E u {()} T hen for all bounded operators D with Hi? — / \ II < Ò the required inclusions { K ) c o { D) -f V'2, , ( ) c [ D ) 4- v , { ) and ự ) c V c a [ K ) -f v , { ) are fulfilled T he lemma is proved It is casv to deduce the following statement from I x m m a P ropo sition The funclion p(A, A') - d i s t { X , { K ) ) is continuous on c X Coinp{7i{)) Proof Suppose A E c , K G C o ĩn p{H ị)) and £: > Bv Lem ma , there exists Ỗ > such that for any operator / / lying in the (5-neighborhood o f K , \\H - A'll < Ỗ, the inclusions (16) are fulfilled; these inclusions directly imply the estimate |/)(À, K ) - p(A, / / ) | < e T h e n for all /Í c with I// - A| < £• and all / / with \\H - A'll < Ỏ we have p(/i, A') - p(A, / / ) | < A') - p(A, A')| -f |p(A, / \ ) - p(A, / / ) | < \fi - x\ e < 2e, Since £ > is arbitrary; the function p(A, K ) is continuous The proposition is proved Cornbinirm Proposition and Lem ma vve obtain the follovvinii fact C o ro lla ry TỈÌC funciion p{ XJ) ) ~ d i s t { \ ^ { K ị ) ) is continuous on (À,fe) G c X ^ ( - ) Now we arc readv to prove Theorem Proof o f Theorem By Corollar\' 1, the function f ) { \ / u , b ) is continuo us with respect to the variable {u, 6) € (C \ {0}) X ^ ( ~ ) Consequently, the set Dr {{I'M I p( l / / ^, 6) ỹ Ế0, hEA„{-)) ỉỉo i / VNiỉ Journal o f Science, Mathematics - Physics 25 (2009) ì 69-177 IS mca.suiablc, and so is the set B = U r/ir- Clearly, /Í Obviouisly, I3{) lies in the set c X full m easure in c E and / Í — z? u Ữ0 , 177 where Bq ~ E \ B \ Acj) o f zero measure ( recall that, by Theorem Ị, Afj has ) Sincc the Lebesgue measure is complete, Do is measurable Thus, the set E is m easurable, beitm the union o f tw o measurable sets Next, by Theorem 3, for b G Afj the scction - -Ị G c I (ư h) G F } has full measure, because its complement { l / i ^ \ E ơ{Kị , ) } is at most countable Therefore, the set / Ĩ is o f full plane Lebesguc measure The Theorem is proved The following im portanl statem ent is a conscqucnce o f Theorem C o ro ll j r \ ’ F o r a.e G c , prohlevi (I), (2) has a unique p erio d ic solution wilh alm ost every' p e rio d I) Proof Since the set E is m easu rab le and has full measure, for a.e z/ G c the section E y — [b ^ Ị (-a/;) E E } : - {h e IX^ Ị ì / Ệ a(A'/j)} has full measure, and for such / / s problem (1), (2) has unique pe riodic solution w ith period h 'I'he Corollar} is proved Rcfere nces | I | A.n.AntoncNich Dang Ivhanh Hoi On the SCI o f pcritxls for poritKÌic solulions o f mtxlcỉ quasilinear dilĩcreruial equations D iffe r U n iv n T N o (2006) 1041 12| D ang Khanh Hoi, On ihc slruciurc oi'the set o f periods tor periodic solutions o f some linear integro-din'crcniial cqutioans on the iTiultidimcntional sphere A lg e b r a a n d A n a ly s is , Tom 18, N o (2006) 83 (Russian) | | M.A Subin, P se u d o d ĩJ J e r e n tĩa ỉ o p e r a to r s a n d sp e c tra l th e o ry , " Nauka." Moscow 1978 | ị Ỉ.P, Konicld Ya.Ci Sinai, s v 1'oỉĩiin K rg o d ĩc theo i'y, " Nauka,” Moscow, 1980 | | \v Rudin F u n c h o n a l (m a b jszs 2nd ed McGraw-Hill, Inc., New York, 1991  ... solulions o f mtxlcỉ quasilinear dilĩcreruial equations D iffe r U n iv n T N o (2006) 1041 12| D ang Khanh Hoi, On ihc slruciurc oi 'the set o f periods tor periodic solutions o f some linear. .. satisfy the condition |A 6|2 |6(6 -f A 6)|2 Then II/Ú+AÒ M' ^r h- o^ C {k o) < e < 3c This shows that the operator-valued function b Kh is continuous on ylcr(-) The Lem m a is proved L e m m a The. .. Arr denote the union o f l h c sets (9) is fulfilled for some h and all over all c > If inequality k then it is fulfilled for m = 0; this provides a conditioii necessary for the nonem ptiness