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 , 2 Ỉ 6 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 3 A u g u s t 2 0 0 9

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) — ư 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 5 '* 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 — 0.

Here a is g iv e n real,
and A



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 5 " X
A / - m a x { | | A , G ’i |, | |G ’l|}.

We put

"

L e m m a 3. Lcl (' - C u

AÍ
{ k { k - h n - 1) ) 2 -

The lemma is proved.
Wc assum e that a IS a real number.

I hen by Lem m a 1, the spcctrum ơ ( L ) lies on the real

axis. Most typical and intercslintĩ is the ease where the num ber ab/{27T) is irrational.

In this case,

0 / A;^.,nVA:, r/ỉ e

z,k

the niimbers

is cver>'vvhcrc dense on R and ơ { L ) — M. Then in the subspace Ho the inverse

> 0 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

/


b such

that

c
(9)

for all in, k e z , k > 0 .
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 > 0 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

r= I

countable union o f closed sets. Wc show that Acr has full measure in

. Suppose Ò, / > 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) =: 1 9
2 tt

2 tt

—

n k { k + 71 - 1)1 +

\ ak { k + n - l)i

U l+ a

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 )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
0. we can find an inlcíĩcr ko > 0, such that
L

[k(k

^

I It

^

1))'^

‘ i’(x, t) = Q u , V -f Qko 2 '^i,

A /‘

for all k > k(). We write

V = G’ii,

where


/ 1 \2j,2+2a ^

''^ L n

, / {k{k + n ^>ko

I ^

V "

I

„

2

^

^
k>k(j

1 ))2 * c ^

Consequently, IKA'02 ° G | | < £.
Since G is com pact and

I A*.-,'Til

is bounded,
;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 3. 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
1

_ 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 5 “


^'Ẳ:m

yr

ỉ^bU ^

'^^krn

^
("km ■
0 k'Q,

< (
{ k { k + n - 1))2 ^ V A /

0 < 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



I

N

e

x

t

,

+ ỉ^^kob),

w h e n c e wc obtain

I|A'6+A6 - / h | | < ||i v 6+A6 - A'fcll +

+ ||/U-o6

Consider the operators Kb+Sb^ I^b- We have

0
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 6 . The s p e d m m a { K ) o f the com pact operator K depends c o n tin u o u sly on K in the space
Comp[ H{ ) ) o f com paci operators on Ho, in the sense that f o r a n y e there exists Ỗ > (} 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
ơ ( 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 0 in c .
Proof. Let K be a com pact operator; we fix £ > 0. The structure o f the spectrum o f a compact operator
shows that there exists

< e / 2 such that E\

set o f all spectrum points À with |A| >

|A| for all A 6 o { K ) . Let

and let V” ^ [ J



A') - p(A, A')| -f |p(A, / \ ) - p(A, / / ) | < \fi - x\

e < 2e,

Since £ > 0 is arbitrary; the function p(A, K ) is continuous. The proposition is proved.
Cornbinirm Proposition 1 and Lem ma 5 vve obtain the follovvinii fact.
C o ro lla ry 1. TỈÌC funciion p{ XJ) ) ~ d i s t { \ ^ ơ { K ị ) ) is continuous on (À,fe) G c X . 4 ^ ( - ) .
Now we arc readv to prove Theorem 4.
Proof o f Theorem 4. By Corollar\' 1, the function f ) { \ / u , b ) is continuo us with respect to the variable
{u, 6) € (C \ {0}) X .4 ^ ( ~ ) . 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


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 / ư 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 4 2 . N o 8 (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 4 (2006) 83. (Russian)
| 3 | 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.
| 4 ị Ỉ.P, Konicld. Ya.Ci. Sinai, s . v . 1'oỉĩiin. K rg o d ĩc theo i'y, " Nauka,” Moscow, 1980.
| 5 | \v. Rudin. F u n c h o n a l (m a b jszs. 2nd ed.. McGraw-Hill, Inc., New York, 1991.