On periodic solutions of a certain nonlinear equation

ON PERIODIC SOLUTIONS OF A CERTAIN NONLINEAR EQUATION (0 PERIODICAESKOM

ODNOQO NELINEINOGO

RESAENII URAVNENIIA)

PMM Vo1.27,

1963,

No.6, I.D.

pp.f107-iii0

KILL’

(Yoscor) (Received

The Question dlz x2 is

This

existence

dz ecostdt+

of

periodic

is

I and -z,

be odd in t.

[d

Cesari’s

invariant

with

(1)

Consider val

[O,

2~1,

the

space

which

respect

equation

[21

(O
(1)

to

the

its

simultaneous

solution

inter-

may supposed

may be rewritten = 4e sin 1

p

S of

are

of

method.

gd- i -S_c~~~s~sin 1 + Tcosl z +

solutions

and hence

and t and -t,

Equation

1963)

4e sin t z = 1 + ecost

1 +:cosrein

by means of

equation of

the

23,

2e sin t

1+

considered

change to

of

May

functions

of

odd and periodic

(,z = 1 + 2 e cos t )

integrable with

sauare

period

2v,

on the

(2)

inter-

endowed with

the

norm v (a$ = [$G

(I) dl]lif

(3)

0 For x in S,

we have 2-

2

v (z) = i;;

b, sin kt,

k=l

Consider,

in the

space

&I’*

(4)

k-l

S,

the

linear

Pz = b, sin t + b, sin 21,

operator la

Hz = -

1699

P and H such

x ke2b, sin lit h’_=l

that (5)

I.D.

1700

If

A

Kill’

then

E 0,

cu

2

2

(t)-

b, siu kt,

(6)

Ii-n Now consider Qz._?._!?c-

Let

1 f

k=n

the

e cos t

operators

z -

p sin -L-y--+

1 $-

X* be a function Px*

Q and F

e co5

4e sin t,

i

Fz

H(Qs--PQz)

=

(7)

such that

=

x* 7

t + b, sin 2t

b, sin

(x* ES)

(8)

Then v (x*)

Consider s*=

the

2-‘I’

=

subset

(b,2 +

b,2)liz < c

(c =

const)

(9)

S*

[Z,2ES,Px=z*,

Y(Z)


v(x-

x*) <‘a!

(d, 6 =

For arbitrary values of the constants d and 6, the subset since each X* belongs to S*. The space s is complete,

empty, closed;

consequently,

Following

S* is

hl . let

us

a complete

introduce

const~

S* is not and S* is

space.

the

operator

T

y=Tx==Px+Fx-Px+H(Qx-PQx)

If

x is

in S*,

PFx

employing

(6), v (y -

Employing

the

obvious

readily

If(Qx.

3-’ v (Qr -

P Qx)

PQ4

-

(12)

we obtain Py)

d

ZJZ)< v (z),

(13)

2 E “7

e

ecos (-j-.qcost

cos t

l+ecostP’-P (z -

-

y-Py-m

that

v(Qx--PQx)
px x*,

inequality v (2 -

one obtains

(11)

then

Py = PPT +

From this,

(W

Pz)]

-I- pv

[sin j:~&~]

Ti2 (b,, b,. e) -I- 7s * (b,, b,, 4 “’ I

2

p.,]

__+

< [F (b,, bz, e) es +,_,+p

(14)

Here F (b,, b,, e) = 2; \ @* (t) dt iJ

(Q,

(1) =

e (1

.+

cos t e

cos

t)

x* (4 )

Periodic

solutrons

a certain

of

nonltnear

where y1 and y2 are the coefficients of sin t and of sin 2t Fourier expansion of the function a(t). Now suppose that the coefficients the inequal it ies

lhl


IhI

and b,

bl


of equation

(al, a,

1701

equation

in the

(8)

satisfy

= const)

(15)

Then we may put c

From (14)

we obtain

2-‘/Q(al”

=

(16)

that -r12f

v(Qr-PQz)
a,“)‘/’

+

the closed

rectangle

2

1

722 ‘1s

in the bl,

(17)

space which is defined

b,

by

(15). In order

that y belong

to SO, it

v(Y----Y)

The first

of relations

is obviously


will

sufficient

that

d==c+-S

(18)

be fulfilled,

provided

that

(1% If

(19) holds

for

a certain

6 = 6,.

then,

upon setting

d=c+a,

(20)

we obtain T (S*) E S*

(21)

Let us prove that, in a certain range of values of the parameters u and e, the operator T is a contracting operator on S*. Writing y1 = Txl y2 = TX,, with XI and x2 in S, we obtain yl-yp=

v

Hence,

(~1 -

Qq)---P(Qq-

HI(Qq-

YA < 3-% [(Qrl -

Q4

- P (Qq -

Q41

Qzz)l Q 39 v (Qq -

44

if

(22)

I.D.

1702

then

is a contracting

T

Thus,

if

relations

in view of point x*,

(19)

Banach’s It

y in S*. and,

operator to

of

The right-hand

the

side

the

twice,

(2) +

P

6, +

and bp.

We have

of

(23)

is,

almost

t,

in view

has this

everywhere, of

a twice

hypothesis

Equations

(25)

coefficients

solution

of

solutions ities

bl

(15).

con-

H;

property.

we obtain

QY

-

PQY

=

4e -

of

L’

0,

[3-

9~ + J’

(2).

when P[d2y/dt2

46, +

: -

(24)

QY] -

(35)

(%a(b,, b,) = 0

sin ~1 dl

the

T,

- PQy)

is a periodic solution the same thing, when j31 (h,, 11,)-

the operator

[31, has a unique fixed a continuous function of

y is h,

(23)

also

then

theorem

that

function

(23)

The function y(t) F 0 or, which is 11 =: --

[d,

equation side

flY dt2

fulfilled,

PY -1 II (Q!/

left-hand

Differentiating

Qyl

of

are

mapping

coefficients

differentiable

sequently,

(22)

may be proved

!I

continuously

on S*.

contraction

therefore,

Kill’

(72

1,2)

may be regarded as equations for the determination of of the existence of a periodic and b2. The question to the question of the existence of reduces, therefore, b,

(2)

and b, Let

of

equations

us substitute

(25)

which

equations

also

(25)

satisfy

by the

the

inequal-

approximate

equa-

tions

?r:

2n

ecost

1 +ecost

Let plane

z* (t)

sin nt dt +

us map, by means of into

Let plane. system following

a domain

in the

sin

_k!-

formulas

5* 1 +

(26),

(t) ecost

the

sin nt df

domain D of

1, 2)

(n =

the

b,,

b,

l/V plane.

co be the boundary of the corresponding domain A0 in the UV If the origin of coordinates of the UV plane belongs to Ao, then (26)

has a solution inequality

satisfying

(15).

If,

in particular,

the

holds:

nlaxo v(CJ -

U,)* $- (V -

VJ2
[CO,01

(27)

Periodic

min [C,. 01 is

where to

a point

of

the mapping

(0,

solutions

of

the

for

O
the

parameters.

(19)

has the

a

distance

nonlinear

(25)

the
also

has a solution

question

of

the

and e = 0.6. Let

us choose

a1 = 3.5

A, also

coordinates as a result

contains

satisfies

of

periodic

(22)

of

obtained

which

existence

Condition

1703

equation

from the origin

c,,,

holds

for

and a2 = 0.5,

the

origin

(15). solutions these

then

values

of of

inequality

form 3-2 IO.415 +

and is

certain

then the domain D by means of formulas (25),

of

Now consider (2)

least

boundary

and hence

6).

the

of

fulfilled

for

d = 2.556

+ 0.134

hand side

of

8 = 0.056

p_ Let

1.58 $p]

+ 0.134

us estimate

< 6

(28)

CI. Equation the

quantity

(20)

then

appearing

gives on the

left-

(27)

1U -

U, ( =

] fll -

file 1 =

1f \ $$$;

(y -

Py) sin t dt +

il 273 +

+

5

2sin

0 Employing

the

Y + PY Y - PY cos sin t dt 2 (1 + e cos t) 2 (1 +- e cos 1) I

Cauchy-Buniakovskii

inequality

for

integrals,

we obtain

2rr $- k

(1

(y -

Pylz

dt)“’ (p

d

(~ ,“;;t

t)2 dt)“’ <

0 dt ‘I* )

Analogously,

it

follows

that sin2 2t

(1 Choosing and (39),

c = 0.6,

(2%

1

and calculating

the

integrals

(1 + e cos t)? appearing

dt

‘/I )I

in

(30)

(29)

we obtain

For p = 1,

from

(31)

it

follows

mas o d{uxT+

that (V -

V,)2 < 0.683

(32)

1704

I.D.

Let into

us map the

the

rectangle

W plane,

lb11

by means of

Kill’

3.5,

<

lb,1

formulas the

domain

curve

0.5,

<

(26),

for

(see

c,, (l),

of the bl, bp plane, CI = 1. Then we obtain

Figure),

bounded

by the

where

mirl IQ (11,01 > 0.780 From (32) and (33) equality (27) holds. If

we map the

cry plane,

again

we obtain

the

is

bounded

follows

that

same rectangle using

domain

by the

From the

it

(X,

(26)

Figure

it

Figure)

cc is

the

but with

(see

curve

into

in-

~1 = 0,

which

(0).

readily

seen

that

min [CO(I), 0,] < miri [co [O), 0] uO, V0 are

The quantities the

following

inequality

linear

functions

of

~1; hence,

for

(34) 0 <:cl%l,

holds: (35)

min [cO (I), 01 < min IQ (p), 01 On the

other

hand,

the

quantity

(31)

is

an increasing

function

(32)

is

also

true

for

all

that

(27)

holds

easily the

hence

equation

The mapping carried

out

(2) of

(27)

the

the

CI
and thus

11.From

b,,

b,

side

satisfying

of

inequality

this

and 0 < CI< 1. For these

has solutions

boundary

on the right-hand

0<

interval

e = 0.6

has periodic

by means of

~1 for

p in the for

system

parameters,

appearing

of

it

follows

values (15),

of and

solutions. of

the

computing

rectangle

into

machine

“Strela”.

the

UV plane

was

BIBLIOGRAPHY 1.

Cesari, L., Periodic Symposium on Active Brooklyn,

2.

Beletskii.

N. Y., V.V.,

Zskusstvennye

3.

Kantorovich, rovannykh

Fizmatgiz,

solutions

of

Networks

differential

and

Feedback

systems. Systems.

Proc.

New York,

1960;

1961. 0 libratsii sputnik;

L.V.

sputnika Zemli,

and Akilov,

prostranstvakh

(On librations

No. 3, G.P.,

(Functional

of

satellites).

1959.

Funktsional

Analysis

‘nyi

anal

in

Vormed

iz

v normi.spaces).

1959. Translated

by J.B.D.