THE

CONSTRUCTION

GENERALIZED

OF CAUCH

ORDINARY

Y

SOME PROBLEM

DIFFERENTIAL

V.

i. We

G.

SOLUTIONS FOR

OF

THE

NONLINEAR

EQUATIONS

Pisarenko

UDC

517.93

will consider the ordinary autonomous nonlinear differential equation d~x(t) , dt ~ -1- O~oX(t) = g~(g, x(t)),

(1)

w h e r e w0 = 30; f(g, z) is an analytic function of the a r g u m e n t g in s o m e c o m p l e x n e i g h b o r h o o d G = (g [I Ig i < go) of the point g = 0 with polynomial coefficients in z in the series expansion with respect to g: r

nk

gk f (g' Z) = E k=O

where d~, = dkt, max Ida, l --- do < c~, 2~
~. E dkS, 1=2

max nk = n < c~. O-.
We will a s s u m e t h a t f o r a given s m a l l gi > O, a n u m b e r B(gi, %) e x i s t s such t h a t in the r e g i o n of the p h a s e plane

Ro~ = (x,:~lllx I + lxl <~ B(g~,r

(2)

(.00

all p h a s e t r a j e c t o r i e s for (1) f o r - gl -< g -< gl h a v e the f o r m of c y c l e s enveloping the o r i g i n x = i = O. F i r s t we c o n s i d e r the C a u c h y p r o b l e m f o r the i n h o m o g e n e o u s l i n e a r e q u a t i o n s : d2u(t) -3F + a~u ( t ) = F(t), (3)

u (to + O) = uo,

u" (to + O) = u .

w h e r e F PC[0, ~). We extend the solution u(t) of this p r o ~ e m and the function F(t) to z e r o f o r t < t o (t > to) , and denote the e x t e n d e d functions b y t~ and F (u and F), r e s p e c t i v e l y . The functions t~, ~', ~?, and a r e g e n e r a l i z e d functions in the s p a c e D'(R) = D', which is c o n j u g a t e to the s p a c e of alt finite infinitely diff e r e n t i a b l e functions D(R) =-D, and they s a t i s f y the following equations in D' [1]:

d2u dr---r + a~'~ = > (t).+ u06' ( t - - to) + u~6 (t -- t0),

a~u

' a~ = 7"(0-

u0~' (t - - to) - . , ~

(t -

(4)

to)-

(5)

dt.~ ~Definition 1. The g e n e r a l i z e d C a u e h y p r o b l e m to the r i g h t (to the left) of the point to f o r the l i n e a r o r d i n a r y d i f f e r e n t i a l equation (3) with s o u r c e f E D ' and initial (finite) p e r t u r b a t i o n s u 0 and u 1 is the p r o b l e m of finding a g e n e r a l i z e d function ~t E D ' (u E D') which v a n i s h e s for t < t o (t > to) and s a t i s f i e s (4) ((5)). Definition 2. The g e n e r a l i z e d C a u e h y p r o b l e m to the r i g h t (to the left) of the point t o f o r the n o n l i n e a r o r d i n a r y d i f f e r e n t i a l equation (1) with initial (finite) p e r t u r b a t i o n s u 0 and ul is ~he p r o b l e m of finding a g e n e r a l i z e d function ~ fi D' (~E D') for which the g e n e r a l i z e d function f(g, 37) (f(g, ~), defined in D ' , v a n i s h e s f o r t < t o (t > to) and s a t i s f i e s (6) ((7)) in D ' : Institute of Mathematics, Academy of Sciences of the Ukrainian SSR. Matematicheskii Zhttrnal, Vol. 23, No. 4, pp. 555-562, July-August, 1971. February I, 1971.

Translated from Ul~ainskii Original article submitted

9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West ]7th Street, '~'ew York, N. Y. lOOll. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

467

d~tr dt~ § %2t; - - g f ( g , ~ = u05' (t-- t 0) % utS(t-d2g + dt ~

(e,

= -

8' (t-

to) -

.,6 (t -

to),

(6)

to).

(7)

Definition 3. The r e t a r d e d G r e e n ' s function Gret(t) f o r the n o n l i n e a r o r d i n a r y differential equation (1) is a g e n e r a l i z e d function f r o m D v which v a n i s h e s for t < 0 and s a t i s f i e s the following equation in D':

d~z (t) ,

2

~y ~ %z (t)-- gf (g, z(t)) = 6 (t).

(6a)

Definition 4. The a d v a n c e d G r e e n ' s function GadV(t) for the n o n l i n e a r o r d i n a r y differential equation (1) is a g e n e r a l i z e d function f r o m D' which v a n i s h e s for t > 0 and s a t i s f i e s the equation (6a) in D ' . Now we c o n s i d e r (6). In g e n e r a l , the s e t of solutions of (6) in D' does not c o i n c i d e with x(t)0 ( t - t 0 ) , w h e r e x(t) is the r e a l solution of (1) with the initial conditions X ( t 0 ~ - 0 ) = U 0 = ~0,

~ ( t 0 ~'- 0 ) = U 1 =

~1'

{L ~ ~ 0,

(8)

(Sa)

0(~)= 0, ~ < 0 .

Below, we c o n s t r u c t a solution of the g e n e r a l i z e d C a u c h y p r o b l e m to the r i g h t of the point t o f o r the n o n l i n e a r o r d i n a r y d i f f e r e n t i a l equation (1) with the initial p e r t u r b a t i o n s u 0 and u 1 which b e l o n g s to the s e t D~, * and is p e r i o d i c in t for t > t o; such a solution e x i s t s and is unique u n d e r the conditions of T h e o r e m 1. B e f o r e s t a t i n g T h e o r e m 1, we m a k e a c h a n g e of v a r i a b l e in (6) and unknown function: t

t

(9)

~o+ ~gkBk r

~(t)

"%( ~

EgkB h

)

~,=0 c~

k=O

T H E O R E M 1. In (6) let the function f(g; z) h a v e the following p r o p e r t i e s : 1) the function f(g, z) is art a n a l y t i c function of the a r g u m e n t g in s o m e c o m p l e x n e i g h b o r h o o d G = (g II [gl < go) of the point g = 0 with p o l y n o m i a l c o e f f i c i e n t s in z in the s e r i e s expansion with r e s p e c t to g: e~

g~

nk

~ ( g , z ) = ~__~. ~ d k S k~O

where

max 0~k
'

(11)

l='2

[ dkt ] ~_ d o < o:., dkt = dkl' max n k = n < ,-,o O~k
2....~l.~n k

2) the function f(g, z) is s u c h that f o r a n y given gl < g2 < go, a r e a l c o n s t a n t B(g I, w0) e x i s t s such that in the r e g i o n of the p h a s e plane

*D ~ d e n o t e s the set of all g e n e r a l i z e d functions in D' which have bounded s u p p o r t on the left: D~ = fill supple (cr r

cf > --o~, fED').

D~ d e n o t e s the s e t of all g e n e r a l i z e d functions in D' which have bounded support on the right: D~ = (/]lsupp/~(--co, cf), c f ,~ co, [ ~ D').

468

t

\

all phase t r a j e c t o r i e s of (1) for g e [ - g t , g~] have the f o r m of cycles enveloping the origin x = ~: = O. Moreover, let the functions u0(g) and ut(g) be analytic funetions of the variable g in the r e g i o n Igl < g~ such that the s e r i e s

ldO(g)~--- ~jglZak, /./l(e)~- E ~bh' ah=ah, Oh=bh, h~0

k=0

(12)

converge, and also let

(Uo(g), ul(g))gRg,,

gE[--g, gI].

Then t h e r e exists a solution of (6) with the initial p e r t u r b a t i o n s (12), which belong~ to the set ~ and is periodic in t for t > to, for which all p r o d u c t s of the coefficients of the s e r i e s (10) (~pt)~l... (ypr) r r = 1, 2, 3 . . . . ; p ~ = 0 , 1, 2 . . . . ; 1 - < j - < r , l 1 + . . . + l t - < n , 0 - < - l j a r e i n t e g e r s , are defined as g e n e r a l i z e d functions in i~'; this solution is unique in D~ and can be obtained in the f o r m of the s e r i e s (10) as the sum of solutions f r o m D~ which a r e periodic in t for t > to, and a r e the solutions of sequences of generalized Cauchy p r o b l e m s to the right of the point t o for equations obtained f o r m a l l y by substituting the e x p r e s s i o n s (9), (10), (1I), and (12) into (6), and equating t e r m s for identical powers of g. The solution so c o n s t r u c t e d depends continuously on the initial p e r t u r b a t i o n s u 0 and ul in the sense of the weak topology in D T. In the p r o o f of T h e o r e m 1 we use a t h e o r e m on the uniqueness of the solution of (4) in the set D~ (see the t h e o r e m in [1, w Section 3]). R e m a r k 1. The generalized solution c o n s t r u c t e d in T h e o r e m 1 for t > t o coincides with the solution of (1) with the initial conditions (8) and (12), and is unique under the conditions of T h e o r e m 1. Using the method of s u c c e s s i v e approximations we will establish that this solution of (i) with the initial conditions (8) and (12) is analytically continued with r e s p e c t to g to the whole c i r c l e Igi < gl in the plane of the complex variable g for sufficiently small gl. COROLLARY t. F o r u 0 = 0 and uj = 1 T h e o r e m 1 gives a r e t a r d e d Green's function for the nonJinear o r d i n a r y differential equation (1). For 0 < t = coast this r e t a r d e d G r e e n ' s function is analytically continued with r e s p e c t to the variable g to the whole c i r c l e ]gl < gl for sufficiently small gt. The following t h e o r e m , which is s i m i l a r to T h e o r e m 1, is proved for (7). THEOREM 2.

Let all conditions of T h e o r e m 1 hold.

Then t h e r e exists a solution of (7) with the initial p e r t u r b a t i o n s (12) which belongs to the=set D~, and is periodic in t for t > t 0, for which all products of coefficients of the s e r i e s (10a): ( y p / L . . (ypr)/r, r = 1, 2, 3 . . . . ; p j = 0 , 1, 2 . . . . ; 1 -
~ ( t o - o) = .~ (g) =

~ b ~ '~

(13)

k=O

and with the solution of (i) extended to the whole axis t 6 (-~o, ~ ) with initial conditions (8) and (12), and is unique under the conditions of Theorem 2. In this connection we conclude from Remark 1 that this solution admits an analytic continuation in g to the whole circle ]g] < g~ in the plane of the complex variable g for sufficiently small gl. COROLLARY 2. For u 0 = 0 and u~ =-i, linear ordinary differential equation (i).

Theorem2

gives an advanced

Green's

function for the non-

469

F o r 0 > t = const this advanced G r e e n ' s function a d m i t s an analytic continuation in the v a r i a b l e g to the whole c i r c l e Igl < gl f o r s u f f i c i e n t l y s m a l l g~. C O R O L L A R Y 3. T h e r e is an e n t i r e f a m i l y of p e r i o d i c solutions of (1) with the p e r i o d s Ti(g ) = 2~r/w 0 (1 + gfli(g)), which, f o r g = 0, b e c o m e the solution x0(t ) = c o c o s wot +(d0/co 0) sinw0t of the g e n e r a t e d equation d2z/dt2O+ c02z(t) = 0; each solution of this f a m i l y is a s o l u t i o n of the C a u c h y p r o b l e m for (1) with initial c o n ditions of the f o r m (12) : r

r

x (t o + O) = U(oi)(g) = co @ 2 g~a~),

(14)

x (l o -b O) = u~i)(g) = d o @ ~, gkb(~i).

k=l

k~l

2. E x a m p l e . We will c o n s t r u c t the r e t a r d e d and a d v a n c e d G r e e n ' s function f o r the n o n l i n e a r o r d i n a r y d i f f e r e n t i a l equation:

d2x (l) + co2ox (t) - - gx a (t) = O. dt ~

(15)

The g e n e r a l i z e d C a u c h y p r o b l e m to the right of the point t o = 0 for (15) with the initial p e r t u r b a t i o n s u 0 = 0 and u 1 = 0 r e d u c e s to the following equation in D': d2y

(t)

dt ~ + co~y(t)--g~(t) = 6(t).

(16)

It is obvious that this e q u a t i o n s a t i s f i e s all of the conditions of T h e o r e m 1 if we let 0)4

(17)

O
coo B (gv coo) = V~-~l 9 be the c o n s t a n t s gl and B(g l, r

(18)

in the conditions (2).

We obtain the r e t a r d e d G r e e n ' s function Gret(t) f o r (15) f r o m T h e o r e m 1 and C o r o l l a r y 1. t e r m s of o r d e r 0(g2), this r e t a r d e d G r e e n ' s function h a s the f o r m :

Gr~t(t) = O(t) [l where %(0=[%

sin , l (t)

3~

9

sm 3,1(t) + ~

9g

]

sin , l (t) + O (g~) .

To within

(19)

8co 3g +O(g2)] t 9

The whole s e r i e s with r e s p e c t to g f o r the function Gret(t), b y T h e o r e m 1 and C o r o l l a r y 1, gives an a n a l y t i c function of g in the r e g i o n [g[ < gl < r f o r 0 < t = const f o r sufficiently s m a l l gl. The g e n e r a l i z e d C a u c h y u 0 = 0 and u 1 = - 1, r e d u c e s to t h e inequalities (17) and (18), C o r o l l a r y 2. To within t e r m s

p r o b l e m to the left of the point t o = 0 f o r (15), with the initial p e r t u r b a t i o n s (16) in D'. C h o o s i n g the c o n s t a n t s gl and B(g~, COo) in c o r r e s p o n d e n c e with we obtain the a d v a n c e d G r e e n ' s function Gadv(t) f o r (15) f r o m T h e o r e m 2 and of o r d e r 0(g2), this a d v a n c e d Green' s function has the f o r m :

O~dv(t)=--O(--t)[-~osin,~(t)-t-3~sin3,1(t

) + ~9 g sin,~(t) ~-.

]

L e t t i n g X[u0, ul ] (t) denote the u s u a l solution of the C a u c h y p r o b l e m for the o r d i n a r y differential e q u a t i o n (15) with the initial conditions (8), we obtain, to within 0(g2), O~v (t) = - - X~o,q (t) + ~et (t) + 0 (g2)

(21)

as a c o n s e q u e n c e of the fact that x[0,q (t) = x[0,_~l ( - l) + 0 (g2).

(22)

In g e n e r a l , the equations (21) and (22) do not hold f o r the g e n e r a l f o r m of the function gf(g, z) which s a t i s f i e s the condition of T h e o r e m s 1 and 2.

470

3. It is convenient to u s e the g e n e r a l i z e d solutions f r o m T h e o r e m 1 f o r the c o n s t r u c t i o n of sointions of the Cauehy p r o b l e m f o r n o n l i n e a r i n h o m o g e n e o u s o r d i n a r y d i f f e r e n t i a l equations with a d i s c o n t i n u i t y on the r i g h t - h a n d side. Thus, we c o n s i d e r the p r o b l e m of finding a s o l u t i o n f o r t > t o of the equation

d2x(t) dt~ + with the initial conditions

'4x(~

gf(g' x ( t ) ) = F(t )

(23)

(8): x G + 0) = Uo,

x(t0 + 0) = u .

w h e r e f(g, z) s a t i s f i e s all eonditions of T h e o r e m 1, and the "function" F(t) equals z e r o f o r all t > t 0, except at a finite n u m b e r of points t = tj > t 0, j = 1, 2, 3 . . . . p; at the m o m e n t s t = tj the "function" F(t) t a k e s v a l u e s which g u a r a n t e e a given finite jump of the d e r i v a t i v e of the solution x(t) of (23) :

x(t + O)-- x(t --O) = O, x(t+O)--x(t--O)=O,

t~to,

if

x(ti+0)--x(tj--0)=

t > to;

if

t@ti;

(24)

ai4=0, ] = 1,2,3 . . . . . p.

It is obvious t h a t the "function" F(t) so defined is not a function in the u s u a l s e n s e . But F(t) c a n be defined as a g e n e r a l i z e d function o v e r the s p a c e D of all finite infinitely d i f f e r e n t i a l functions: p

F (l) = E 6 (t -- t) % E D'. ]=1

F o r the eonstrueLion of the solution of the p r o b l e m (23), (8), and (24), we divide the h a l f - a x i s t > t 0 b y the points t = t t . . . . . tp into the s e g m e n t s At, A 2. . . . . Ap, w h e r e Aj = it II t;_, < t < t ) , ] = 1, 2 . . . . . p,

(25)

A,+, = (t II t > O .

In the p h a s e plane we c o n s i d e r the c l o s e d (with r e s p e c t to the n o r m of t w o - d i m e n s i o n a l E u c l i d e a n space) c o n n e c t e d r e g i o n Hg 1 which c o n t a i n s the o r i g i n {0}, and the c y c l e l y i n g in the r e g i o n R g 1 defined b y (2):

{o} r G, =Hg, ~R~.

(2s)

Let e(x0, k0) d e n o t e the c y c l e p a s s i n g t h r o u g h t h e point (x o, ~:0) of the p h a s e plane. In the p h a s e plane we define the c o n n e c t e d r e g i o n s Kj(g~) and Kj(g~), j = 0, 1, 2 . . . . . of the f o r m u l a s : K~.(gx) = (x, x t l l x - a l

<1%l,(x,~)r

~j(g? =

]=

p, by memos

1, 2 . . . . . p,

U c (x, x), (z,x)EKi(m)

Furthermore,

we obtain

Hg, _~ Ko (gl) ~ t(o (g,) c= K.~ (gl) ~ ~(~ (g~) c . . . ~ K; (gO c ~(p (gO ~ Ra.

(28)

T H E O R E M 3. Let all of the conditions of T h e o r e m 1 hold f o r the flmction fig, z) in (23), and let p r e a l e o n s t a n t s a 1. . . . . a p be given s u c h tha~ (28) holds f o r all fixed gt < g2 < go f o r the r e g i o n s Kj(g 3 and Kj(gl) defined b y (27). Then there exists a unique solution of (23) with the initial conditions (8) and (12), and with the righthand side F(t), which guarantees the condition (24), on the interval t E (tj_ t, tj) = Aj, j = i, 2, 3 ..... p + i, and the solution coincides on this interval with the solution from the set D~of the equation

-d~(v) - dt 2

~_offJ(O-gt(g,

~(t))

= .o,-~

dS(t--ti-l) dt

+ ( u l i_ ~+c~f_ 1)6(t.--ff _ 1)

(29)

"

471

which is unique in D~ and is given by T h e o r e m 1; f u r t h e r m o r e , u0, j _ 1 and ul, j _ ~ a r e the values of the solution f r o m D~ of the equation

d~'Y(~) ~- 0)2~(t) -- gf (e, ~'~(t)) = UO,]._2 dt 2

d5 (t ~ tr__~.)

.3V (Ul,]_ 2 jZ_ C~]_2)~ ( t - - ~]--2)

dt

(30)

and its f i r s t d e r i v a t i v e with r e s p e c t to t at the point t = t j _ 1 - 1 , and u0, 0 = u0; u0,1 = ul, t p + 1 -- ~, a0 = 0. We note that the method which r e s u l t s f r o m T h e o r e m s the Cauchy p r o b l e m for the o r d i n a r y differential equation (1) the solution of the g e n e r a l i z e d Cauchy p r o b l e m is n e a r to the p r e s e n t e d , for example, in the m o n o g r a p h [2], but it gives a solution.

1 and 2 for the construction of the solution of with the initial conditions (8) and (12) using method of u n d e t e r m i n e d coefficients which is s i m p l e r a l g o r i t h m for the construction of the

We r e c a l l that the question of the b a s i s of the method of a v e r a g i n g for nonlinear equations of the type (23) with a discontinuous r i g h t - h a n d side of this specific f o r m has been c o n s i d e r e d in a s e r i e s of works([13], s e e also the bibliography in [3]). LITERATURE

1. 2. 3.

472

CITED

V. S. Vladimirov, Equations of M a t h e m a t i c a l P h y s i c s [in Russian], Nauka, Moscow (1967). N. N. Bogolyubov and Yu. A. Mitropol' skii, A s y m p t o t i c Methods in the T h e o r y of Nonlinear Vibrations [in Russian], Fizmatgiz, Moscow (1963). A. M. Samoilenko, "On the p r o b l e m of the b a s i s of the method of a v e r a g i n g for the investigation of v i b r a t i o n s in s y s t e m s , subjected to an impulse action, " Ukrainsk. Matem. Zh., 19, No. 5 {1967).