PERTURBATIONS OF DISCONTINUOUS SOLUTIONS OF NON-LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS,

PERTURBATIONS OF DISCONTINUOUS SOLUTIONS OF NON-LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS,

By

N O R M A N L E V I N S O N I) of CAMBRIDGE, U. S. A.

1. T h e s t u d y of t h e solutions of t h e s y s t e m

( 1 . 0 ) dxj~ = X ~ ( x l . . . x n, t, ~) i = 1, 2 . . . . n ,

d t

w h e r e t h e X i are r e g u l a r f u n c t i o n s of e for small e is classical. More r e c e n t l y n o n - linear s y s t e m s like (l.0) h a v e b e e n s t u d i e d w h e n one or m o r e of t h e X i h a s a pole a t e --~ 0, or w h a t is e q u i v a l e n t , w h e r e e or s o m e p o w e r of e occurs as t h e coefficient of t h e left m e m b e r of one or m o r e of t h e e q u a t i o n s (1.0), [1, 2, 3, 4, 5, 6]. I n this case t h e s y s t e m w h e n e ~ 0 i s of lower order t h a n w h e n e 4=- 0. I n t h e s t u d i e s [2, 4, 6] it is a s s u m e d t h a t t h e s y s t e m h a s a s o l u t i o n w i t h a c o n t i n u o u s d e r i v a t i v e in case e --- 0 a n d conditions a r e g i v e n for this t o be t h e case w h e n e 4 0.

I n a p p l i e d m a t h e m a t i c s t h e r e are cases w h e r e t h e s y s t e m h a s o n l y d i s c o n t i n u o u s

" s o l u t i o n s " w h e n e ~ 0 a n d y e t is k n o w n e m p i r i c a l l y t h a t w h e n e > 0 t h e s y s t e m h a s a c o n t i n u o u s s o l u t i o n w h i c h a p p r o a c h e s t h e d i s c o n t i n u o u s one as e - ~ 0. T h i s f a c t h a s b e e n e x p l o i t e d b y t h e R u s s i a n school of n o n - l i n e a r m e c h a n i c s . H e r e a r i g o r o u s t r e a t m e n t will b e g i v e n for a case w h e r e t h e s y s t e m h a s a d i s c o n t i n u o u s " s o l u t i o n "

w h e n e ~ 0. T h e m a i n r e s u l t h a s a l r e a d y b e e n a n n o u n c e d , w i t h o u t proof, [3]. Since [3] h a s a p p e a r e d , a s y s t e m w i t h a d i s c o n t i n u o u s s o l u t i o n h a s b e e n t r e a t e d [5] b y T i h o n o v . I n T i h o n o v ' s t r e a t m e n t t h e " j u m p a r c s " i n s t e a d of b e i n g solutions of {2.2) m u s t b e s t r a i g h t lines. Also t h e e x i s t e n c e of d e r i v a t i v e s w i t h r e s p e c t to initial v a l u e s is n o t considered.

T h e specific s y s t e m we shall consider h e r e is

1 J o h n Simon G u g g e n h e i m Memorial Fellow on leave from the M a s s a c h u s e t t s Irlstitute of Tochnology.

dx~ d u d2d d u

(1.1) d~ ~- f i ~d~+q~i , i : 1, 2 . . . n , S - d ~ + g - ~ + h -~ O.

H e r e fi, ~i, g a n d h are f u n c t i o n s of xl, x~ . . . xn, u, t a n d e. T h e y are c o n t i n u o u s in e for small e ~ 0. I t will be c o n v e n i e n t to use a v e c t o r n o t a t i o n a n d to d e n o t e t h e v e c t o r with c o m p o n e n t s x i b y x, t h e v e c t o r with c o m p o n e n t s f l b y f , a n d ?~ b y q~.

T h u s we can write (1.1) as

d x d u d2u d u

(1.2) dt = f d i +q~' ~ [ ; + g ~ [ + h = 0

w h e r e the first e q u a t i o n of (1.2) is a v e c t o r e q u a t i o n . T h e v e c t o r f i s f ( x , u, t, e) a n d similarly for t h e v e c t o r ~ a n d t h e scalars g a n d h.

B y t h e d e g e n e r a t e s y s t e m we shall m e a n (1.2) with ~ --- 0. W e o b s e r v e t h a t t h e d e g e n e r a t e s y s t e m is of lower o r d e r t h a n (1.2). W e shall write t h e d e g e n e r a t e s y s t e m as

d y d v dv

(1.3) ~ = 1 ~ + ~ , g d i + h = 0

where y is a v e c t o r with c o m p o n e n t s Yl . . . Yn a n d v is a scalar. I n ( 1 . 3 ) f is f ( y , v, t, O) a n d similarly for ~, g a n d h. L e t us n o w consider a solution of (1.3) as a c u r v e in t h e n + 2 d i m e n s i o n a l space (y, v, t). W e assume t h a t such a solution s t a r t s a t a point, A. W e observe t h a t when t h e solution r e a c h e s a p o i n t on the h y p e r s u r f a c e g ( y , v , t, O)

0, a singular s i t u a t i o n m a y prevail with r e g a r d to dv/dt.

This s i t u a t i o n can be seen clearly b y t a k i n g a v e r y simple special case. W e t a k e t h e v a n der Pol e q u a t i o n with a change of t i m e scale which m a y be w r i t t e n as

d2u d u

(1.4) ~ - : = + ( u ~ - l ) - : + u = o

dt~ - - dt a n d we consider t h e r e l a t e d d e g e n e r a t e e q u a t i o n

(1.5)

(v~-l)hi+v= 0.

dv

T h e solution v(t) o f (1.5) which a t t = 0 satisfies v(0) > 1 is r e a d i l y o b t a i n a b l e . H o w e v e r all we n e e d observe is t h a t since

dV dt v is decreasing. T h u s for v ~ 1

V 2 - 1 < 0

Perturbations of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 73

dv 1

dt v~(O)-- 1

a n d a f t e r a f i n i t e e l a p s e of t we h a v e f o r s o m e t - - tl, v(t~) ~ 1. As t -~ t ~ - - 0 , ~ -~ - - e ~ . dv T h a t is f o r v ~ l d - 0 , dv/dt ~ - - o o . A n y a t t e m p t t o c o n t i n u e t h e s o l u t i o n , as a c o n t i n u o u s f u n c t i o n of t, b e y o n d t ---- t l , fails since f o r v ---- 1 - - 0 , d v / d t ~ d - o o . T h u s t h e s o l u t i o n c a n n o t p a s s c o n t i n u o u s l y f r o m a b o v e v ~ 1 t o b e l o w v ~ 1. M o r e o v e r since v ~ I is o b v i o u s l y n o t a s o l u t i o n of (1.5) t h e s o l u t i o n c a n n o t be c o n t i n u e d as v ~ 1. W e n o t e t h a t v ---- 1 h e r e c o r r e s p o n d s t o g ~ 0 in (1.3).

I f w e t u r n t o (1.4) w i t h s > 0 w e see t h a t t h e line u = 1 offers n o s p e c i a l d i f f i c u l t y . T h u s a s o l u t i o n u(t) of (1.4) o b v i o u s l y c a n b e c o n t i n u e d b e y o n d a p o i n t w h e r e u = l-t-0. L e t u = 1~-0 w h e n t ~ t l - - O a n d let us i n t e g r a t e (1.4) f r o m t l - - ~ t o t1~-5 w h e r e 6 > 0 is small. W e f i n d

dU -] t1+6 [ U 3 ~ ] tl~-(~ ( 'tl~r

d u (tl +_ ~)

N o w let us p r o c e e d h e u r i s t i c a l l y . I f as s -+ 0 we a s s u m e dt a p p r o a c h f i n i t e l i m i t i n g v a l u e s a n d if w e a s s u m e t h a t lu[ r e m a i n s u n i f o r m l y b o u n d e d in t h e r a n g e ( t l - - 8 , tld-5) t h e n we g e t

]t~+~ 0(~)

N o w l e t t i n g 6 ~ 0 a n d r e c a l l i n g t h a t u ( t t - - O ) = 1, w e g e t

89167 = O .

S o l v i n g t h i s l a s t e q u a t i o n we g e t e i t h e r u ( t l ~ - O ) ~- 1 o r u ( t l ~ - O ) = - - 2 . T h e v a l u e 1 we d i s c a r d o n t h e basis of o u r e x p e r i e n c e w i t h (1.5) a n d w e a r e led t o i n v e s t i g a t e f u r t h e r t h e p o s s i b i l i t y u ( t l ~- 0) = - - 2. A c t u a l l y (1.4) c a n b e i n v e s t i g a t e d d i r e c t l y [ 1]

a n d it is i n d e e d f o u n d t h a t as s ~ 0, s o l u t i o n s of (1.4) o n r e a c h i n g u = 1 + 0 t e n d t o j u m p t o u = - - 2 . W e shall n o t p u r s u e t h i s i n t u i t i v e d i s c u s s i o n f u r t h e r b u t r a t h e r p r o c e e d in w 2 t o g i v e a d e f i n i t i o n of a s o l u t i o n of (1.3) w h i c h m a y b e d i s c o n t i n u o u s . T h e d e f i n i t i o n will b e j u s t i f i e d b e c a u s e w e shall s h o w t h a t as ~ -~ 0 s o l u t i o n s of (1.2) t e n d t o s o l u t i o n s , as we d e f i n e t h e m h e r e , of (1.3).

As w a s o b s e r v e d in [3] t h e s y s t e m (1.2) i n c l u d e s as a special case t h e s y s t e m

d x d w

(1.6) d-[ = H ( z , w , t, s) , s--zT. = a ( x , w , t, e) ott

where x a n d H are v e c t o r s a n d H a n d G are c o n t i n u o u s in e for small ~ ~ 0. G a n d w are scalars. W i t h e n o t a p p e a l i n g on t h e r i g h t this is t h e s y s t e m t r e a t e d b y T i h o n o v [5]. I n case t h e r i g h t m e m b e r s of (1.6) are n o t linear in w t h e s y s t e m can b e b r o u g h t to t h e f o r m (1.2) ( w i t h f ~ 0) s i m p l y b y d i f f e r e n t i a t i n g t h e last e q u a t i o n w i t h r e s p e c t to t. On t h e o t h e r h a n d if t h e r i g h t m e m b e r s of (1.6) are linear in w t h e n , if we

d u

i n t r o d u c e t h e v a r i a b l e u g i v e n b y d~ = w, t h e s y s t e m (1.6) assumes t h e f o r m

d x du d2u du

= f ( x , t, ~ ) ~ + q ~ ( x , t, ~) ~ + g ( x , t, ~ ) w + h ( x , t, ~)

d t ^g~ dt 2 ^a~

which is a special case of (1.2).

~ 0

2. W e shall now give t h e definition of a solution of (1.3). W e consider t h e solution as a curve, S0, in t h e n + 2 dimensional space (y, v, t). A t t h e p o i n t A let t -~ a a n d g > 0. F o r a ~ t < 31, let (y(t), v(t)) be a solution of (1.3) a n d let

(2.0) g(y(t), v(t); t, O) > O, a ~ t < vl .

As t -~ r l - - 0 let g -+ 0. T h e p o i n t ( y ( ~ - - 0 ) , v ( v ~ , 0 ) , v,) we d e n o t e b y B 1. W e shall d e n o t e v(v~--O) b y v B a n d y ( ~ l - - 0 ) b y YB" A B I is an arc of So. W e assume t h a t . a t B 1

n ~ g . ~g

(2.l) I = ~'~L-~.J/+ 4= 0

u y i ~ v "

i ~ l

H e r e I ~ I B. T h e n e x t arc of S O is B~C1 w h e r e B~C 1 is a c u r v e y(v) in t h e h y p e r p l a n e t : Vl which satisfies t h e v e c t o r e q u a t i o n s

(2.2) d y

d v = f ( y ' v, v~, 0 ) .

(The s y s t e m (2.2) is (1.3) with t held c o n s t a n t a n d with t h e last e q u a t i o n of (1.3) o m i t t e d ) . T h e solution of (2.2) s t a r t s a t B~. W e t h e n consider (2.2) for increasing or decreasing v a c c o r d i n g as h a t B is r e s p e c t i v e l y n e g a t i v e or positive. W e a s s u m e t h a t t h e solution of (2.2) can be c o n t i n u e d for increasing or decreasing v, d e p e n d i n g as we h a v e seen on t h e sign of h a t B, until we r e a c h t h e first v a l u e of v 4= v B for which

v

(2.3) I a(y(v), v, ~,, o) dv = o.

vB

This v a l u e of v we call v c a n d t h e p o i n t C 1 is g i v e n b y (y(Vc), vc,, 31). A t C 1 we a s s u m e (2.5) g(y(vc), we, ~,, o) = g~ > o .

On t h e basis of r e m a r k s a l r e a d y m a d e we h a v e

Perturbations of Discontinuous Soluti:(~r~s of Non-Linear Syst. of Diff. Equations. 75

(2.5)

( v c - - v ~ ) h B < 0

w h e r e h B = h(yB, VB, T1, 0). ( T h e i n t e g r a l c o r r e s p o n d i n g to (2.3) in t h e case of t h e v a n der P o l e q u a t i o n we c o n s i d e r e d a b o v e is s i m p l y

i

v ( v~--1) dv =. 0

1

a n d since h a t B 1 (where v -~ ]) is p o s i t i v e we m u s t t a k e v decreasing. T h u s we see

~ a t h e r e v c = - - - 2 . )

A t C1 we r e t u r n t o t h e s y s t e m (1.3) a n d consider t h e s o l u t i o n w i t h initial v a l u e s a t C1 a n d w i t h t increasing. W e a s s u m e t h a t t h i s s o l u t i o n c a n be c o n t i n u e d w i t h g > 0 u n t i l t ~ 32--0 w h e r e g ~+ 0. F r o m t h i s p o i n t w h i c h we d e n o t e b y B 2 we a s s u m e we c a n p r o c e e d in t h e m a n n e r a l r e a d y i n d i c a t e d a t B 1 w i t h v i n c r e a s i n g or d e c r e a s i n g d e p e n d i n g on t h e sign of h a t B2- P r o c e e d i n g in t h i s w a y t h e s o l u t i o n So is d e f i n e d g e o m e t r i c a l l y as A B 1 C 1 B 2 C ~ . . . B N C ~ A " w h e r e A ' is a n o r d i n a r y p o i n t (i. e. o n e w h e r e g > 0). W e a s s u m e t h a t (1.3) h a s a s o l u t i o n So as j u s t d e f i n e d f o r a g t _~ ft.

W e a s s u m e f u r t h e r t h a t t h e r e exists a n o p e n set R in t h e n ~ - 2 d i m e n s i o n a l s p a c e (y, v, t) c o n t a i n i n g t h e c u r v e S 0 such t h a t f ( y , v, t, e), % g a n d h a n d t h e i r f i r s t o r d e r p a r t i a l d e r i v a t i v e s w i t h r e s p e c t t o Yi, v a n d t a r e u n i f o r m l y c o n t i n u o u s a n d b o u n d e d as f u n c t i o n s of y, v, t a n d e w h e n (y, v, t) is in R a n d ~ ~ 0 is small. W e shall also a s s m n e t h a t ff a n d h h a v e second o r d e r p a r t i a l d e r i v a t i v e s c o n t i n u o u s in R a n d for s m a l l e a l t h o u g h t h i s a s s u m p t i o n c a n b e a v o i d e d .

W e see t h a t S0 c o n s i d e r e d as a c u r v e ( b u t n o t as a f u n c t i o n of t) is c o n t i n u o u s b u t t h a t a t t h e p o i n t s C1, C2, etc. it h a s a d i s c o n t i n u o u s t a n g e n t . W e see f u r t h e r t h a t So is t h e s u m of t w o k i n d s of arcs, t h e a r c s AB1, C1B2, C~B3, etc. w h i c h a r e s o l u t i o n s of (1.3) a n d m i g h t b e called r e g u l a r arcs, a n d t h e arcs B1CI, B2C2, etc.

w h i c h a r e solutions of (2.2) lying in p l a n e s t --~ ~i a n d w h i c h m i g h t b e called j u m p arcs since t h e arcs a r e t r a v e r s e d in a zero elapse of t. As a f u n c t i o n of t t h e s o l u t i o n is d i s c o n t i n u o u s a n d j u m p s f r o m Bj to Cj a t t - - ~j, j = 1, 2 . . . N .

T h e c o n d i t i o n (2.1) c a n b e w e a k e n e d b y allowing I to v a n i s h On g = 0 b u t r e q u i r i n g t h a t I b e d i f f e r e n t f r o m zero a n d of t h e s a m e sign off g ---- 0 in t h e neigh- b o r h o o d of B. A s o m e w h a t s i m p l e r s i t u a t i o n w h e r e t h e r e will b e no j u m p a t all arises w h e n I c h a n g e s sign in p a s s i n g t h r o u g h g =- 0. T h e s e cases will n o t be p u r s u e d f u r t h e r .

I n w h a t follows t h e n o r m of a v e c t o r is d e f i n e d as Ix[ = X[xil.

T h e basic r e s u l t s f o r (1.2) are g i v e n in t h e following t h e o r e m s ,

T h e o r e m 1. Let the degenerate system (1.3) have a solution S o in the sense defined above, for ~ ~ t ~ ft. Let ~ > O. I f ~,

01

and 62 are small enough there is a solution x(t), u(t) of (1.2) over (~, fl) for any set of initial values satisfying

I x ( ~ ) - y ( ~ ) ] + [ u ( ~ ) - v ( ~ ) J ~ 01 du (~) dv(o,) < 02

dt dt = ~"

Moreover as ~, 01 and 02 tend to zero, the curve representing the solution x(t), u(t) i n (x, u, t) tends to S o. I n particular for a n y f i x e d (~ > 0

( 2 . 6 ) ]x(t)--y(t)[-F[u(t)-:v(t)[

tends uniformly to zero over the intervals, ~ ~_ t ~_ Vl--(~, Vl-~5 ~ t ~_ T 2 - - 0 , . . . , vN~-(~ ~ t ~ fl, as e, 0~ and (~2 -~ O. Also

du dv

(2.7) dt dt

tends uniformly to zero over the intervals ~-~0 ~_ t ~ Vl--~, ~ 1 ~ ~-- t ~_ ~ - - ~ . . . d~u d~v

~ - 0 ~ t ~ fl as e, 0~ a n d 0~ --> O. The same is true for

dt ~- dt ~

T h e o r e m 2. I t is also the case that i f S o is a solution of (1.3) for ~ ~ t ~ fl then corresponding to any set of initial values sufficiently near y(~), v(~) there is a solution of (1.3) which tends to S o as the initial values tend to those of S o. Moreover the convergence is u n i f o r m i n t i f the portions of S o between r j ~ O are omitted.

Theorem 2 is in a sense T h e o r e m 1 for t h e case e = ~-0.

In w h a t follows let us d e n o t e b y ~a differentiation w i t h respect to one of t h e n ~ - I initial coordinates (x(~), u(a)) or w i t h respect to t h e corresponding initial coordinate of (y(~), v(~)). T h e n we h a v e

T h e o r e m 3. Subject to the same hypothesis as Theorem 1 we have

~x(t) ~y(t) ~u(t) ~ v ( t ) - ~ 0

(2.8) ~a ~a ~- ~a aa '

(2.9) I ~ du ~ dv

~ a d t ~ a ~ - + 0

for ~, 01, and 02 -+ O, the convergence being uniform for (2.8) and (2.9) over the same

Perturbations of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 77 intervals of t as for (2.6) and (2.7) respectively. Moreover denoting the initial value of du/dt at a by b we have

~x(t) ~u(t) o and

~--,o

uniformly over the same sets of intervals as for (2.6) and (2.7) respectively as s, 6~ and 6~-+ O.

T h e o r e m 4. The functions 3y(t) and --~a are uniformly continuous with respect ~v(t) to changes in the initial values of y and v at a over the same set of intervals of t as in Theorem 2.

T h e o r e m 4 is in a sense T h e o r e m 3 for t h e case e = -t-0.

As a n a p p l i c a t i o n of t h e s e t h e o r e m s in case t h e r i g h t m e m b e r s do n o t c o n t a i n t or in case t h e y are periodic in t we h a v e t h e f a c t t h a t if t h e d e g e n e r a t e s y s t e m (1.3) has a periodic solution a n d if t h e J a c o b i a n associated w i t h t h e d e t e r m i n a t i o n of this solution b y v a r y i n g initial c o o r d i n a t e s is d i f f e r e n t f r o m zero, t h e n it follows b y T h e o r e m 3 t h a t (1.2) will also h a v e a periodic solution. W e shall show this in w 8.

3. W e shall r e q u i r e several lemmass T h e first is well k n o w n . L e m m a 1. Let ~(t) be a vector with an integrable derivative and let

d~ < a(t)-t-b(t)l~l , to < t < t l ,

(3.0) dt l = ~- ~-

where b(t) ~ O. Then for t o ~ t • tl ,

(3.1) [$(t)--r l ~ [~(to)l(e ftob(s)ds--1)-~- oa(T)e Itb(s)&dr . Clearly (3.1) implies

(3.2) I~(t)[ ~ I~(to)left'ob(s)ds ~ - Iioa(~)ef~b(~)dSd~ .

Correspo~zling results hold i f tl < to.

P r o o f . L e t

i

^{t d~}

z(t) = ~ dr.

tO T h e n I~(t)--~(to)l < Z(t) a n d (3.0) b e c o m e s

d Z

_ _ < a(t)+b(t)Z+b(t)l~(to) 1 . dt - -

T h u s

dt /

T h u s i n t e g r a t i n g we find

e- I~o b(~jd~ <= (a(t) + b(t) l~(to)l)e- I~o b(s)d~ .

z(t)e- f~o "~)~ <= Ito(a( v)-4-b(v)l~(to)])e- ftobtS)dS dv '

f r o m which (3.1) follows.

T h e n e x t l e m m a is v e r y similar t o one F r i e d r i c h s a n d W a s o w [2]. H e r e z is a v e c t o r a n d w is a scalar a n d L(z, w, t) a n d M(z, w, t) are v e c t o r s while H(z, w, t) a n d J(z, w, t) are scalars.

L e m m a 2. Let z(t), w(t) satisfy

(3.3) dz L dw M , d~w H dw

a-t= ~ + ~ b + ~ [ = g

for a <~ t -<- V where L, M , H and J are continuous in the region given by ~ <~ t <_

and [z]-4-lwl <= 2 for some 2 > o. I n this region let

(3.4) ILl < k, ]M] < k ( [ z l + l w ] ) + e l , [J] < k(lzl-~-]wl)-~-t~l, H ~> m > 0 . Moreover at t : ~x let

(3.5) [~(~,)J+lw(~)l =< 6~,

Let lq = k ( k + m + l ) / m and let

4(k-4-1)2 (m-4-1) A =

m k

i dw(~) I 6~ --~1 <=-

(61"~- 62-~ El) 9

T h e n for a <~ t _<_ V, and i f el, 61, and 62 are small enough

Iz(t)J + Iw(t)l <= Ae k'(~-~)

and

Perturbations of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 7 9

- ~ d w ~ 5~ e_m(t_~):e + _ _ e ~( -") kA k t II't P r o o f of L e m m a 2.

(3.6) Or

d w F r o m ( 3 . 3 ) w e h a v e if we set - - = O,

dt

dz d w dO

dt L O + M , dt O, e dt + H O J .

+ -d-i{ <= (k+l)lOl+~(Izl+lwl)+~l.

If ~ denotes the vector (z, w) t h e n L e m m a 1 yields

(3.7) Lz(t)[+lw(t)l < +51 e k ( t - ~ ) + ( k + l ) [O(~)[ek(t-~)d~.

O~

F r o m t h e last e q u a t i o n of (3.6)

e e~' = O(a) + J e e~' d~ . Or since H > m > 0

e_m(t_a)/e_~ 1 I t

IO(t)l <~ ^{IO(cr -} IJle-m(t-v,/edv.

E

< 5~ e_m,t_~,)i~+ k (t 10(t)l

T h u s

(3.s)

F r o m (3.7)

F r o m this

k(k+e 1) e_m(t_T)/ed7: iolO(a) [ek(~_a)da "

,39)

⁼ - - e k(t-~') +51 e k(t-~') IO(a)le-k(~-~')da.

f

e m m ,~x

t

Applying L e m m a 1 w i t h ~ ~ ta[O(a)Ie-k((~-~)da we get

I n (3.7) a n d (3.9) this proves t h e l e m m a providing sl, 51 a n d 5~ are small e n o u g h so t h a t A ek~(Y-~)< )..

B e f o r e t u r n i n g t o t h e p r o o f of T h e o r e m 1 we m a k e t h e following o b s e r v a t i o n which is valid for T h e o r e m 1, 2, 3 a n d 4. I t suffices to t a k e t h e case w h e r e t h e r e is o n l y one j u m p arc on S o since b y r e p e a t e d use of t h e t h e o r e m for t h e case of one j u m p arc, t h e t h e o r e m for N j u m p arcs follows a t once. T h e r e f o r e we shall a s s u m e t h a t S o is of t h e f o r m

AB~CiA'.

Clearly t h e r e will be no confusion if we call

So, A B C A ' .

T h e p r o o f of T h e o r e m 1 is d i v i d e d into f o u r parts. I n t h e first p a r t we p r o c e e d f r o m A t o a p o i n t s h o r t of B ; t h e second p a r t involves t h e i m m e d i a t e v i c i n i t y of B ; t h e t h i r d p a r t t a k e s t h e arc

B C

w i t h t h e t w o small p o r t i o n s a t ends B a n d C o m i t t e d ; t h e f o u r t h p a r t t a k e s t h e rest of S o to A'. W e shall use K t h r o u g h o u t t h e p a p e r to r e p r e s e n t finite c o n s t a n t s which d e p e n d on t h e b o u n d s of [fl, IFI, Igl, Ihi a n d t h e i r first o r d e r p a r t i a l d e r i v a t i v e s in R or p a r t of R for small

e,

on t h e distance f r o m S o t o t h e n e a r e s t p o i n t on t h e b o u n d a r y of R, a n d on the l e n g t h of So. I n p a r t i c u l a r t h e c o n s t a n t s K will r e m a i n finite as ~ + ~ + 5 2 -~ + 0 . A n y d e v i a t i o n f r o m this use of K will be n o t e d .

P r o o f o f T h e o r e m 1, P a r t I.

H e r e we p r o v e T h e o r e m 1 for the i n t e r v a l a ~ t ~ 7 w h e r e }, < T~. I f we d e n o t e

x - - y

b y z a n d

u - - v

b y w we h a v e f r o m (1.2) a n d (1.3)

dz dw

(3.10) d-t = f(x, u, t, ~ ) = + ~

a s

(3.11) where F 1

a n d

d2w dw

~ +~(x, u, t, ~)-~. = F~

ct~

is a v e c t o r a n d

h(y,

v~

t, O) FI = - - [ f ( x , u, t, s)--f(y, v, t,

0)]

g (y, v, t, O)

t-q~(x, u, t, s)-q~(U, v, t, o) F~ = [g(x, u, t, ~)--g(y, v, t,

0)] h(U, v, t, O)

g(y, v, t, O) h(x, u, t, s)

~ [ O dhat h

O) dg]

+h(U, v, t, 0)+-= [g(y, v, t, )==-- (U, v, t, g~

_{d t J}

L e t t h e m i n i m u m of

g(y(t), v(t),

t, 0) o v e r (~, V) be d e n o t e d b y 2m unless this m i n i m u m exceeds 1 in which case we t a k e m = 1. T h e n so long as

(x, u, t)

is in R we h a v e using t h e m e a n v a l u e t h e o r e m ,

IFll ~ m ([Z{+lwI§

K

v, t, s ) - f ( y , v, t,

0 ) [ §

v, t, E)-cf(y, v, t,

0 ) [ ) .

P e r t u r b a t i o n s of D i s c o n t i n u o u s S o l u t i o n s of N o n - L i n e a r S y s t . of Diff. E q u a t i o n s . 81

Since f a n d ~ are u n i f o r m l y c o n t i n u o u s for small e t h e r e m u s t be a c o n t i n u o u s f u n c t i o n

~(e) such t h a t ~(0) = 0 a n d such t h a t

If(y, v, t, e)--f(y, v, t, 0)[ ~ ~(e) with similar results for ~, g a n d h so long as (y, v, t) is in R.

Clearly we c a n choose W(e) > e. W e get

(3.12) IFll

K

m

dh dg

a n d similarly c o m p u t i n g ~ a n d ~ we find

(3:13) ]P~] _<-- ~ ( I z l + Iw] + ~ ( ~ ) ) . K

W e h a v e f(x, u, t, ~) -~ f ( y ( t ) + z , v(t)q-w, t, e) a n d similarly for ~, g, a n d h. I f w e now a p p l y L e m m a 2 t o (3.10) a n d (3.11) a n d m a k e use of (3.12) a n d (3.13) we see t h a t if e, 6~, a n d ~ as d e f i n e d in t h e s t a t e m e n t of T h e o r e m 1 are small enough, (x, u, t) is in R for a ~ t ~ 7, a n d i n d e e d for a ~ t g ~, a t least so long as g(x, u, t, e) ~ m,

(3.14) Ix(t)--y(t)l-[-lu(t)-v(t)l g e K/m* (~1~-5~-~(~))

a n d

du dv

(3.15) - ~ . - - - ~ ~ e-mC'-a)/~+eK/m'((~x-~(~+V2(~)).

F r o m (3.14) a n d t h e c o n t i n u i t y of g we see t h a t we will i n d e e d h a v e g(x, u, t, e) ~ m if e, (~1 a n d ~2 are small enough. W e see f r o m (3.14) a n d (3.15) t h a t for t < 31, T h e o - r e m 1 is established e x c e p t for t h e difference of t h e second d e r i v a t i v e s of u a n d v.

This we shall show in L e m m a 4. T h e d i s c o n t i n u i t y a t 31 is precisely t h e p o i n t of i n t e r e s t here a n d we begin t o h a n d l e it in w 4.

4. I n t h e n e x t p a r t of t h e p r o o f of T h e o r e m 1 we shall show t h a t for small e, 5~

a n d ~ t h e solution of (1.2) i n t e r s e c t s t h e h y p e r s u r f a c e g = 0 a t a p o i n t w h i c h t e n d s du - - -+ ~ or - - c ~ a t t h e p o i n t of to B as e, ~1 a n d ~ -~ 0. M o r e o v e r as e, 61, (~ -~ 0, dt

intersection.

P r o o f of T h e o r e m 1, P a r t 2.

W e shall consider h e r e t h e case w h e r e h(y, v, t, 0) < 0 at B. T h e case w h e r e h > 0 is t r e a t e d in e x a c t l y t h e same m a n n e r . A t B we h a v e t ~ 31 a n d we shall designate y a t B b y YB a n d v b y v B. As t - + 31--0 we h a v e , since h < 0 , ~ - ~ dv

6. Acta mathematica, 82, Imprim~ le 18 decerabre 1949.

h q-~x~. L e t v 1 < v B be n e a r e n o u g h t o v B so t h a t as t increases f r o m a t o w a r d r 1 g

t h e r e is a v a l u e of t ~ t 1 n e a r to v 1 such t h a t v ( t l ) ~ Vl a n d d v / d t is large for tl ~ t < rl.

W e shall d e n o t e t h e p o i n t ( y ( t l ) , v(t I), tl) b y t h e letter Q. I f we choose y so t h a t tl < y < rl, a n d a p p l y t h e results (3.14) a n d (3.15) we see t h a t if e, 51 a n d 52 are small e n o u g h t h e n for some t (which t e n d s to t 1 as s, ~1, 52 -+ 0) we h a v e u(t) -~ v 1. L e t us d e n o t e this p o i n t b y P . T h e n a t P we h a v e t = tp, x ( t p ) ~ Xp a n d u = u t, --- u ( t e ) = v I.

Also as s, 6x, 53 -+ 0, P -~ Q. Clearly we c a n choose Q as n e a r to B as we wish.

W e n o w c h a n g e f r o m t to v (and u) as t h e i n d e p e n d e n t variable. Since w h e n v = u t h e values of t for t h e points on S o a n d t h e solution of (1.2) are n o t in general equal we will reserve t for t h e s y s t e m (1.2) a n d in this section d e s i g n a t e t h e v a r i a b l e t for (1.3) b y t h e letter s. W e h a v e t h e n t h a t (1.2) c a n be w r i t t e n as

d x dt d p

(4.0) d u = f +q~P' p = duu' s-d-UU = P 2 g + p ~ h

where f = f ( X , u , t, e), etc. while (1.3) b e c o m e s

(4.1) ely d s

d v = f q-cpq, q = -d-v' 0 =- g q - q h .

where f -~ f ( y , v, s, 0), etc. Since u a n d v are t h e i n d e p e n d e n t variables here we c a n with no c o n f u s i o n use t h e m i n t e r c h a n g e a b l y . W e consider (4.0) f o r u ~ v 1. T h e s o l u t i o n of (1.2) c a n n o w be r e g a r d e d a t least in c e r t a i n r a n g e of u as a s o l u t i o n

of (4.0).

S u p p o s e v2 > VB- L e t us choose

(4.2) K 1 > 2(Ifl~-IqIq-1 )

for all (x, u, t) in R. (Clearly K 1 is a K). L e t R1 d e n o t e t h e region of (x, u, t) b o u n d e d b y t h e planes u = vl, a n d u = v 2 a n d b y

I X - - X p l q - I t - t e l <=

gl(v2--Vl).

Clearly if v 1 a n d vz are chosen n e a r e n o u g h to vB we h a v e R 1 c o n t a i n e d in R for small e.

I f we consider t h e c h a n g e in g as we follow-a s o l u t i o n of (4.0) we h a v e

Or

Perturbations of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations.

where

dg

du J~(x, u, t, e)~-pJ~(x, u, t, s)

~g ~ 3g ~ ~g

83

W e recall t h e definition of I in (2.1) a n d o u r a s s u m p t i o n t h a t I ~ 0. Since g(y(v), v, s(v), 0) > 0 f o r v < v B a n d zero a t v = v B we see t h a t dg/dv < O. This fact, t h e f a c t t h a t I # 0 a n d t h e f a c t t h a t dv/dt is large implies t h a t I < 0. L e t K s be chosen so t h a t in R a n d for small ~, IJ~l < Ks. If R1 is small e n o u g h , t h a t is if v~

a n d v2 are n e a r e n o u g h t o vB, we c e r t a i n l y h a v e J1 < 89 a n d h < 89 B for (x, u, t) in R1 a n d for small e.

L e t v 1 be n e a r e n o u g h t o v B so t h a t

q(vl) = -g-Q < - - - - I

hQ 10K~"

This is possible since gQ -+ 0 as vl -+ v B. L e t e, (~ a n d ~ be so small t h a t P is n e a r e n o u g h t o Q a n d p n e a r e n o u g h t o q a t u ---- vl so t h a t

2gp I

0 < v ( v , ) < - <

N o w let us s u p p o s e t h a t for o u r solution of (4.0) t h e r e is a va, vl < va < v2 s u c h t h a t for vl ~= u < v a, p ( u ) < - - 2 g p / h B b u t t h a t for u ~ - v a we h a v e either p---- --2gp/h B or (x, u, t) reaches t h e b o u n d a r y of R~. W e shall show t h a t this is impossible.

B y i n t e g r a t i n g

(4.3) s - - = dp ( g + p h ) d u

~9 2

f r o m v~ t o v 3 we see t h a t p > 0 since if p = 0 t h e l e f t side di,derges. 'For vz g u < va, since 0 < p < - - 2 g p / h B, we c a n a s s u m e p < 1 since we c a n t a k e Q a n d t h e r e f o r e P n e a r B w h e r e g vanishes. F r o m (4.0)

V3

I x - : x p ] ~ ] t - - t p [ g ~ Iv (]f]-~]q~l~- l )du < Kx(vz--v~) ^o

1

T h u s (x(u), t(u)) is in R 1 for v~ _< u _< va a n d t h e r e f o r e we m u s t h a v e p = - - 2 g p / h B a t u = v s. Since p < --2gp/h B < - - I / 4 K ~ for Vl < u < va, we h a v e

(4.4) dg ^-- J~A-pJ~ < 1I-4-PK2 < ~ I - - ~ I 88 < O.

du

T h u s g is d e c r e a s i n g a s u i n c r e a s e s u p t o va. S i n c e g is d e c r e a s i n g w e h a v e a t va g + p h < g p @ l p h B ~ g p - - g p = 0 .

T h u s b y (4.3), d p / d u < 0 a t v a. T h a t is p is d e c r e a s i n g a n d t h e r e f o r e w e c a n n o t h a v e p = - - 2 g p / h B f o r t h e f i r s t t i m e a t va.

W e see t h e n t h a t o u r s o l u t i o n of (4.0) r e m a i n s i n R 1 a n d c a n b e e x t e n d e d t o u = v2 a n d t h a t 0 < p < - - 2 g p / h B f o r v 1 =~ u =~ v 2. W e c a n t a k e vl a s c l o s e t o v B a s w e w i s h . T h u s Q c a n b e a s n e a r B a s w e w i s h . B y t a k i n g e, 51 a n d 5~ s m a l l e n o u g h w e c a n b r i n g P a s c l o s e t o Q, a n d t h e r e f o r e t o B , a s w e w i s h . T h e n e a r e r w e t a k e P t o B t h e s m a l l e r is gp a n d t h e r e f o r e t h e s m a l l e r is p f o r v I ~ u ~ v 2. S i n c e q = - - g / h u n t i l g = 0 a f t e r w h i c h q is i d e n t i c a l l y z e r o o n So, u n t i l v = v C, w e see t h a t q f o r Vl ~ v ~ v 2 a l s o g e t s s m a l l e r a s w e t a k e v I n e a r e r t o v B. S i n c e u a n d v a r e i n d e p e n d e n t v a r i a b l e s f o r t h e r e s p e c t i v e s y s t e m s w e c a n i d e n t i f y u w i t h v. T h u s w e c a n w r i t e o u r d i f f e r e n t i a l e q u a t i o n s a s

d x dt

d v - - f ( x , v , t, 0)-~(D 1 , d v P ' (4.5)

d y ds

d v f ( y , v, s, O ) + w 2 , dv q '

w h e r e ~Ol = f ( x , v, t, s ) - - f ( x , v, t, O)+pq~(x, v, t, ~) a n d o~2 = qq~(y, v, s, 0). U s i n g t h e f a c t s j u s t e n u m e r a t e d w e h a v e , f o r vl ~-- v _< v2

[(Di]-71-[O)2] ~ K(gp@gQ@~f(~))

w h e r e ~(c) h a s t h e s a m e p r o p e r t i e s a s i n w 3. A p p l y i n g a s t a n d a r d t h e o r e m t o (4.5) r e l a t i n g t w o a p p r o x i m a t e s o l u t i o n s of a s y s t e m o r e l s e s e t t i n g ~ ---- ( x - - y , t - - s ) a n d m a k i n g u s e of L e m m a 1 w e h a v e

[ x ( v l - - y(v)[ + lt(v)--s(v)] <= K[~p(e )-~-gp-]-gQ-~- I X ( V l ) - y ( v l ) t +

]t(Vl)--8(Vl)l]

f o r v 1 g v _~ v2. S i n c e t h e r i g h t m e m b e r c a n b e m a d e a s s m a l l a s w e w i s h w e h a v e d e m o n s t r a t e d T h e o r e m 1 u p t o t h e i n t e r s e c t i o n of t h e h y p e r - p l a n e v ---- v2 w i t h S 0.

W e o b s e r v e t h a t v 2 c a n b e k e p t f i x e d i n t h e l a t t e r p a r t of o u r a r g u m e n t a s e, 51, a n d 52 ~ 0 w h i l e vl m u s t a p p r o a c h v B.

P r o o f o f T h e o r e m 1, P a r t 3. H e r e w e d e m o n s t r a t e T h e o r e m 1 o v e r t h e p a r t of t h e j u m p a r c B C b e g i n n i n g w i t h v = v2 a n d e n d i n g s h o r t of t h e p o i n t C.

Perturbations of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 85

Since I < 0 a n d gp > 0 a n d is small, we see t h a t (4.4) implies t h a t t h e solution of (4.0) crosses t h e h y p e r s u r f a c e g = 0 in e x a c t l y one p o i n t w h i c h is n e a r B. L e t us designate t h e p o i n t where t h e solution crosses g = 0 b y t h e subscript 4. W e h a v e b y i n t e g r a t i n g (4.3)

= ( g + p h ) d u .

Or for u >='u 4 (and so long as x a n d p r e m a i n s finite)

(4.6) (u) - - - p h d u .

P /74 4

I n (4.6) g is g(x(u), u, t(u), s) a n d similarly for h.

W e h a v e a l r e a d y s~en t h a t p(u) < - - 2 g p / h B for vl --~ v ~ v~. T h u s g i v e n a n y 8a > 0, b y t a k i n g P n e a r e n o u g h to B, we can m a k e p(u) ~ 53, v 1 ~ u ~ v2. G i v e n a n y 84 > 0 we can choose s, 51, a n d 82 small e n o u g h so t h a t

(4.7) [x(v~)--y(v2)[ + [t(v2)-- s(v~)] ~ 84.

N o w so long as p(u) ~ 8a we h a v e f r o m c o m p a r i n g t h e t w o s y s t e m s

dx dt

dv - - f ( x , v, t, ~ ) ~ - p ~ ( x , v, t, ~) , dv - - p

d y _ f ( Y , ds

v,s, 0), d v = 0

for v2 ~ v < v c, just as in t h e a r g u m e n t following (4.5), (4.8) I x - - y i + [t--s I ~ K ( S a + 8 4 + ~ ( e ) ) .

W e recali i n c i d e n t a l l y t h a t on B C , s(v) ~-- ~:1. W e shall show t h a t p(u) ~ 83 a l m o s t up t o u --- v c. W e choose 55 > 0 as small as we wish a n d t h e n choose 56 > 0 small e n o u g h so t h a t

V

(4.9) I g(y(v), v, s(v), 0) v < -480, v2 <_- v __< vc-8 .

V B

This is possible since g < 0 for v B < v ~ v 2 a n d since t h e c o n t i n u o u s f u n c t i o n of v

V

I g dv < 0 v B < v < v C 9 V B

T h e logical p r o c e d u r e in this section is to first choose 85 a n d t h e n 8~. W e t h e n o b s e r v e

t h a t we can r e q u i r e 6 a a n d 64 to be as small as we wish" if we m a k e e, 61, a n d 02 small enough. B y t a k i n g 6al 64 a n d e small e n o u g h we h a v e f r o m (4.8) t h a t for v c > v ~ v2 a n d so long as p ~ 63

(4.1o) fg(x(v), v, t(v), ~)-g(y(,,), v,

8(v), 0)l <

6 J ( v c - v ~ + l) .

W e can also s a t i s f y (4.10) f o r vl ~ v g v~ on t h e basis of P a r t 2 s i m p l y b y t a k i n g e, 01, a n d 63 small e n o u g h . T u r n i n g t o (4.6) we h a v e

(4.11)

p(u) = E

f,(

g x(v), v, t(v), e)dv--

f

phdv

P 4 u u 4

B u t b y (4.10) followed b y (4.9) we h a v e f o r v~ < u < v c - - 0 5

~ u

u ~

1

Since we can b r i n g u 4 as close to v B as we wish b y t a k i n g ~, 61, a n d ~ small enough, we can m a k e

dv < 6 , .

I

T h u s

U

- f g(x(v), v, t(v), ~)dv > 26o.

e. U 4

Also so long as p ~ 0a,

I S " p h d v ~ K 6 3 3 "

9 u 1

I f we choose 6 a small e n o u g h so Ka6 a < 08 t h e n c e r t a i n l y t h e last t w o inequalities used in (4.11) yield

(4.12) p(u) < ~/6,

u p to t h e p o i n t w h e r e we first h a v e e i t h e r p(u) -~ 6a or u ~ vc--65. B u t if e is small e n o u g h so t h a t E/66 < 6a we c e r t a i n l y c a n n o t h a v e p(u) ~ 6a in u 4 ~ u ~ vc--65.

T h u s (4.8) holds in this r a n g e a n d we h a v e established T h e o r e m 1 on So u p t o a n y p o i n t j u s t s h o r t of C.

T h e o r e m 1, P a r t 4. H e r e we p r o v e T h e o r e m 1 in t h e n e i g h b o r h o o d of C a n d b e y o n d . G i v e n a n y 65 > 0 we saw t h a t for v B ~_ v ~_ Vc--65 we h a v e f x - - y [ ~ f t - - ~ ' l l ~

P e r t u r b a t i o n s of Discontinyous Solutions of Non-Linear Syst. of Diff. Equations. 87 - - --> 0 a s s, 0~, 02 ~ 0. W e n o w s h o w t h a t i n t h e n e i g h b o r h o o d of C, p g r o w s q u i c k l y .

dt du

A s w e h a v e a l r e a d y s e e n f r o m (4.3) p > 0 s o l o n g a s p r e m a i n s s m a l l . W e s h a l l s h o w t h a t p m u s t e x c e e d m i n

[gc/(2K),

1] i n t h e n e i g h b o r h o o d of C w h e r e n o w K is t h e m a x of Ih] i n R .

L e t u s d e n o t e a v a l u e of v < v c b y v 5 a n d a v a l u e v >

v C

b y v s. W e c h o o s e v s a n d v6 n e a r v c . S o l o n g a s p < I w e c a n c o n s t r u c t R2, m u c h l i k e R 1, s o t h a t if v 5 a n d v ~ a r e c h o s e n c l o s e e n o u g h t o g e t h e r t h e s o l u t i o n o f (4.0) c a n b e c o a t i n u e d u p t o v 6 a n d w i l l lie i n R 2 w h i c h i n t u r n w i l l lie i n R . T h i s is c e r t a i n l y t h e c a s e t h e n if p is s m a l l . W e c a n a l s o c h o o s e v~ a n d v G n e a r e n o u g h t o vr so t h a t i n R2, g > ~gc > 0.

I n t e g r a t i n g (4.3) b e t w e e n v I a n d u4 w e h a v e

s f (g§

U4

- - o

P 4 P ! v l

S i n c e w e c a n c h o o s e vl a s c l o s e t o

v B

a s w e w a n t a n d s i n c e a s e, 01, a n d 03 -~ 0, u 4 ~ v B w e s e e t h a t w e c a n r e q u i r e t h a t

f o r a n y 07 > 0. T h u s

Iv~i4(g+ph)du]<(~7

E - - < - - - ~ - 0 7 9

P4 P l

N o w s i n c e w e c a n c h o o s e ~, 01, a n d ~ a s s m a l l a s w e w i s h a f t e r h a v i n g s e l e c t e d v 1 w e c a n m a k e

s/ql

a n d t h e r e f o r e a l s o

e/p 1

a s s m a l l a s w e w a n t . T h u s w e c a n r e q u i r e

T h u s (4.6) g i v e s

8

- - < 2 ( $ 7 . P 4

p(u)>

207- !:4gdu-- I~4 phdu"

F r o m t h e r e s u I t of p a r t 3 i t f o l l o w s t h a t if w e t a k e v~ c l o s o e n o u g h t o v c a n d e, 01, a n d 62 s m a l l e n o u g h w e h a v e

. Iu(g+ph)du <0~.

V5

W e see t h a t t h e c h o i c e o f 07 a f f e c t s t h e c h o i c e of v 5 b u t n o t v~. W e h a v e

(4.13) p(u) >

N o w let us a s s u m e t h a t

(4.14) g + p h ~ - - - - , 587 v5 ~-- u ~ v6.

V 6 - - V 5

T h e n we f i n d f r o m (4.13) t h a t p(u) -+ c~, c o n t r a r y to o u r a s s u m p t i o n t h a t p < 1.

T h u s (4.14) c a n n o t h o l d a n d we h a v e

5~ 7 5~7

ph < g < g

V 6 - - 72 5 V 6 - - V C

f o r s o m e u, v5 < u < v6. I f ~7 is s m a l l e n o u g h t h i s c a n be r e p l a c e d b y ph < -- ~gv 9

Since gc > 0 we m u s t h a v e h =~ 0. I f h > 0 we h a v e p < 0 w h i c h is impossible 9 I f h < 0 a n d [h I < K we h a v e p > gc/(2K). T h u s if p < 1 we c e r t a i n l y h a v e p > gv/(2K) for s o m e u, v5 < u < vs.

Since we c a n choose v 6 as close to C as we wish we see t h a t i n d e e d we c a n enclose C in a s p h e r e in (x, u, t) w i t h c e n t e r a t C a n d of r a d i u s a r b i t r a r i l y small a n d t h a t h a v i n g chosen t h e s p h e r e we can, b y t a k i n g s, ~1, a n d ~2 small e n o u g h be sure t h a t R 2 lies in t h e s p h e r e a n d t h a t t h e s o l u t i o n of (4.0) e n t e r s t h e s p h e r e w i t h p v e r y small b u t a t s o m e p o i n t in t h e s p h e r e p ~- m i n (gc/(2K), 1). L e t us d e n o t e this p o i n t b y D.

A t D we t r a n s f o r m b a c k to t h e original i n d e p e n d e n t v a r i a b l e s . T o s h o w t h a t t h e s o l u t i o n (1.3) c h a n g e s b u t little f r o m vl t o t D, i r r e s p e c t i v e of w h e t h e r ~ - - t D is p o s i t i v e or n e g a t i v e we h a v e o n l y t o n o t e t h a t tD--V 1 --> 0 as 8, (~1, (~2 --> 0. W e c a n n o w a p p l y L e m m a 2 a t D as we d i d a t t h e p o i n t A a n d we g e t T h e o r e m 1 for t h e r a n g e of t g i v e n b y m a x (31, tO) --~ t --~ /~.

T h e p r o o f of T h e o r e m 2 is v e r y m u c h s i m p l e r a n d s h o r t e r t h a n t h a t of T h e o r e m 1.

I t is quite d i r e c t e x c e p t n e a r B w h e r e we m u s t use a n a r g u m e n t s i m i l a r to t h a t used in t h e case of T h e o r e m 1, P a r t . 2.

5. T h e p r o o f s of T h e o r e m s 3 a n d 4 p r o c e e d a l o n g s o m e w h a t d i f f e r e n t lines. W e shall first p r o v e T h e o r e m 4. W e use t h e l e t t e r a t o d e s i g n a t e a n initial v a l u e of x i du(~) w h i c h h a s no c o u n t e r p a r t in t h e a n d Yl or of u a n d v. T h e initial v a l u e of ~ -

d e g e n e r a t e s y s t e m we shall d e n o t e b y b. W e o b s e r v e t h a t a t t ~ ~, ~yJ~a is zero

Perturbations of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 89 unless a is t h e intial v a l u e of Yi in w h i c h case

Oyi/Oa

~ 1 a t t ~ ~. Similarly w i t h

Ov/Oa.

W e h a v e 0y 0-a = 0--a 0x a n d 0v 0-a = 0--a a t t -~ ~ since a r e p r e s e n t s t h e s a m e initial 0u

c o o r d i n a t e in (y, v) as in (x, u). W e a l w a y s h a v e ~a d-/ = 0 a t t = ~.

( 5 . o )

P r o o f o f T h e o r e m 4. W e h a v e o n d i f f e r e n t i a t i n g w i t h r e s p e c t t o a

dOy ~d Ov Of dr acp

dt Oa =

h Og ~h

dt Oa g2

H e r e

f = f ( y , v, t,

0), etc. a n d

Of n Of Oyi Of Ov

= 2 . ' ov

a n d similarly for % g a n d h. T h e s y s t e m (5.0) is linear in

Oy/Oa

^{a n d}

Ov/Oa

a n d t h e coefficients are c o n t i n u o u s for ~ < t < T1. T h e initial v a l u e s are k n o w n a n d t h u s

~y/Oa

^{a n d}

Ov/~aare

d e t e r m i n e d for ~ < t < vl. M o r e o v e r for cr ~< t _< ? < T~ t h e f u n c t i o n s

Oy/Oa

a n d

Ov/~a

are u n i f o r m l y c o n t i n u o u s w i t h r e s p e c t t o t a n d w i t h respect to changes in t h e initial values of y a n d v a t t = cr

Os/Oa

a t v - = v 0 § by.

v(s, a) = Vo,

A t v ~ - v 0 < v

B

b u t w i t h v0 n e a r v B we c h a n g e f r o m t t o v as t h e inde- p e n d e n t variable. W e observe t h a t v ~ v 0 d e t e r m i n e s a p o i n t o n So n e a r B.

As before it is c o n v e n i e n t t o replace t b y s a n d reserve t for t h e s y s t e m (1.2).

L e t s a t v ~ - v 0 be so w h e r e clearly s 0 < T 1. W e d e n o t e

Oy/~a

a n d

Ov/Oa

a t

Oy (so) Ov (So)

s -~ so, or m o r e precisely a t s = so--O, b y ~ a n d - ~ - - . W e d e n o t e

Oy/Oa

^{a n d}

Oy(vo) Os(vo)

a n d - - . Clearly since

v(so)= vo

we h a v e f r o m

~a ~a

T h u s

(5.1)

Also

dv OS(Vo) OV(8o)

ds Oa Oa

~s (%) Ov (8o) ds (%)

~a ~a dv

(5.~) ~y(Vo) dy ~s~vo) ~y(so) dy(vo) ~v(so) ~y(so)

~a ~a dv ~a ~a

~a ds

F o r vo ~ v ~ v B w e h a v e

d ~ y _ ~ f ~cfds d ~s d O s _

(5.3) dv ~a ~a ~-~a dv-~q~dv ~a' dv ~a h ~

w h e r e t h e i a t i t i a l v a l u e s a r e t a k e n f r o m (5.1) a n d (5.2). F o r v B ~ v ~ v c, w e h a v e

(5.4) d ~ y ~f - - - ~ d ~s ~ 0 .

dv ~a ~a ' dv ~a

T h u s ~y(v)/~a a n d ~s(v)/~a a r e d e t e r m i n e d u p t o t h e p o i n t C.

W e h a v e g(y~, v~, v~, 0) ~ 0. T h u s

Since

~g ~YBi ~g ~v B ~g ~v~

)--"~ ~y~ ~a t-~v-~a-~ ~t ~a - o .

~y(VB)

~YB dy ~v B

~a dv ~a 3a

w e h a v e , r e c a l l i n g t h e d e f i n i t i o n of I -~ I B ,

I ~vB ~g ~Yi(VB) ~: ~g

~T1

- - 0 .

Since 1 ~= 0 we see t h a t ~vB/~a is d e t e r m i n e d in t e r m s of ~v~/~a.

(5.5)

W e h a v e

T h e c o o r d i n a t e v c is g i v e n b y

V C

g ( y ( v ) , v, O)dv = o .

D i f f e r e n t i a t i n g w i t h r e s p e c t t o a a n d r e c a l l i n g t h a t g ~ 0 a t B we h a v e 8v c ~e( 9r ~Yi ~g ~vl\ ,

g(Yc'Vc'V~' 0 ) ~ a q- I ^.B

Va+

^{~ a ) av -~ O .}

~ 1 -- ~s(vB) + ds(vB) ~v B

~a ~a dv ~a

Since ds(vB)/dv ~_ 0 we see 1 t h a t ~ ' l / ~ a -~- ~s(v~)/~a. T h u s (5.5) d e t e r m i n e s ~vv/~a. Also 1 Strictly speaking we find DvB/~a in terms of Dvl/aa and then ~rl/c~a in terms of ~vB/~a which is not rigorous. Actually we should proceed with finite differences corresponding to an incre- ment in a, Aa, and then take limits as Aa --> 0 to show the existence of ~VB/COa from a single formula obtained by eliminating the term Avl/z]a from the equation involving I and the equation following (5.5).

P e r t u r b a t i o n s of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 91

(5.6) ~Yc _ dy ~v v ~y(vc)

Oa dv ~a ~ ~a

where ~y(vc)/~a is ~y/~a at v ~-- ve--0. As we approach C from s > s c we have

(5.7) ~v v __ ~v(sc) ~ dv ~s v

~a Oa ds ~a

~Yc ~Y(Sv) dy Os v

(5.S)

~a ~a dt ~a

where ~y(se)/~a is ~y(s)/~a at s ~ sc~-0 and similary for ~v(sc)/Oa. However ~sc/~a ~--

~'~l/~a. Using (5.5) and (5.7) we have

- - _

T h a t is ~v(sc)/~a is determined. Likewise ~y(se)/Oa is determined. I n d e e d from (5.8) and (5.6)

3y(sc) dy(vv) ~vv ~y(vc) dy(sv) ~

~ a - - - dv Oa ~ ~a ds ~a"

F r o m (5.7) and (1.3) therefore

(5,10) ~Y(Sc) ~v(sc)' ~Y(Vc) ~vl

~a -- fo ~ - ~ Oa q~v Oa"

We now use (5.0) again from s c ~ s ~ fl and thereby determine ~y(fl)/~a and

~v(fl)/~a. The functions ~y(s)/~a and ~v(s)/~a are clearly uniformly continuous with respect to the initial values of y and v at s : ~ and with respect to s for a ~ S ~ V l - - ~ , 31~-~ ~ t ~ f l for a n y given 5 > 0.

This completes the proof of Theorem 4.

6, We proceed now to the proof of Theorem 3. I n the course of the proof we shall make use of Theorem 4. We recall the remarks made at the beginning of w 5 concerning the meaning of a and b. We require the following resnlt which is a slightly modified form of L e m m a 2 and is proved in the same way.

L e m m a 3. I n Lemma 2 let el in (3.4) be replaced by

e V - ~- - - e - m ( t - ~ ) / e ~2

and let ~1 and ~2 be replaced by ~3 and ~4 respectively. Then the conclusion of Lemma 2 becomes

Iz(t)l-~-[w(t)l

~

Z~l eki(t-~)

I d w I <

64 e_mtt_a)/s_~ s2(t--~) e_m(t_a)/s4 - ]czJl ckl(t_a) for a <~ t <~ ~ where A~ replaces A in Lemma 2 and

4 ( k + 1)2(mA- 1) 2

zJ 1 ~-~ (~1---}-~'2-~-~3-"~-~4) ,

km ~ and ]c~ is the same as in Lemma 2.

W e b e g i n b y c o n s i d e r i n g ~ ~ t _~ ~ < T 1.

(6.0)

P r o o f o f T h e o r e m 3 , P a r t 1. H e r e we h a v e

dOx dOu Ofdu Oq~

dt Oa : ~ J dt ~ a + ~ a - ~ ~ Oa d 2 Ou d Ou ~9du Oh

Oa = 0

w h e r e

f -~ f(x(t), u(t), t, ~),

etc. a n d w h e r e

Of Of Oxi Of ~u

~a -- )--'/ ~ ~a -~ Ou ~a

etc. T h e s y s t e m is linear in

~x/~a

a n d

Ou/Oa.

T h e coefficients a r e a l l f u n c t i o n of t ( a n d of s a n d t h e initial v a l u e s of

x, u, du/dt

a t t -~ ~).

F o r t h e d e g e n e r a t e s y s t e m we h a v e (5.0) w h e r e t h e l a s t e q u a t i o n c a n be w r i t t e n as

d ~v 09dv ~h

= o .

0x 0y 0u 0v

N o w let z - - a n d w - - T h e initial v a l u e s of z a n d w are zero. F r o m

0a 0a 0a 0a

(6.0) a n d (5.0) we h a v e

dz dw Of Of \ d v ~ Oq~ ~q~

(6.1) -~ = f ( x ( t ) , u ( t ) , t , ~ ) y i + ( 2 ~ z ~ + ~ w ) ~ i + 2 ' ~ z ~ + ~ w + F .

d2w , ,dw [ ~ Og

,

09 \ d v Oh Oh

(6.2) e d t ~

~-g(x(t), u(t), t, e)~-[+ ~ 2 , ~x i zi-t-~u W) d~-~ ~ x i zi+~-vu w + G -~ 0

Perturbations of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 93

w h e r e

F =

+ ^{,_,[ ~f} d ~f\~y~ ~ f v [ ~f\~v]

8r / 3 r ~ r d ~v

~y~y~)-~a4-[~u ~v)~a4-2~ ~a [f(x, u, t, s)--f(y, v, t,

0 ) ] . As e, 6~, a n d 6~ -> 0 we h a v e for ~ ~ t ~ y < vt f r o m T h e o r e m 1, p a r t 1, t h a t

~f(x, u, t, ~) ~f(y, v, t, 0) 1 0

u n i f o r m l y w h e r e x is

x(t), y

is

y(t)

etc. S i m i l a r r e s u l t s h o l d for t h e o t h e r differences

du dv

t h a t o c c u r in F e x c e p t for t h e difference - - - . F o r t h e l a t t e r we h a v e b y (3.15)

dt dt

dvl _<

dt dt l

d ~v ~ dv dv h

T h e t e r m d-t a-a = ~aa di- is r e p l a c e d b y use of -d-t- ~- - - - " W e h a v e easily f r o m (5.0) a n d g

L e m m a 1 t h a t t h e t e r m s

~yJ~a

a n d

~v/~a

are b o u n d e d o v e r a ~-- t ~ ~, b y a b o u n d t h a t d e p e n d s o n l y on K a'nd

m.

T h u s so long as

]zl+lw I

~ 1 t h e r e is a n

E,

a f u n c t i o n of K a n d

m,

w h i c h b e c o m e s large w h e n m g e t s small, s u c h t h a t

(6.3)

IFI <= E[~p(~, (~, 62)4-~2e-m(t-~

L J

w h e r e ~(e, 61, 62) is a c o n t i n u o u s f u n c t i o n of E, 81, 62, a n d m w h i c h t e n d s t o zero w h e n e, 6t a n d 62 -~ 0. A s i m i l a r r e s u l t holds for G w h i c h h a s t e r m s like those of F a n d t h e a d d i t i o n a l t e r m S~ad- ~ . I t is h e r e t h a t we use t h e e x i s t e n c e of t h e s e c o n d

d2v

o r d e r p a r t i a l d e r i v a t i v e s of g a n d h. W e find t h a t G satisfies a n i n e q u a l i t y of t h e s a m e f o r m as (6.3). F r o m t h e l a s t e q u a t i o n of ( 5 . 0 ) ~ a ~ / a t t = a c a n be c o m p u t e d .

dv

du dw

W e h a v e - - 0 a t t = ~. T h u s - - a t t = a is bounde~t a n d t h e r e f o r e 64 is of t h e

3a dt dt

f o r m

Ke.

W i t h this we see t h a t (6.1) a n d (6.2) s a t i s f y t h e h y p o t h e s i s of L e m m a 3 w h e r e st = E ~ a n d e2 = ES~. U s i n g L e m m a 3 we see t h a t T h e o r e m 3 is v a l i d f o r

~ t ~ I~ < ~t in so f a r as t h e d e r i v a t i v e s w i t h r e s p e c t to a a r e concerned.

As r e g a r d s t h e d e r i v a t i v e s w i t h r e s p e c t to b we use (6.0) w i t h o u t (5.0). W e s e t

z ~- ~x/~b

a n d

w -~ ~u/~b

a n d a p p l y L e m m a 3. Since a t t ~ a, z = w ~- 0 we h a v e h e r e t h a t ~3 --= 0. Since b

du dt

1 a t t ~-- ~ we h a v e 5~ ~ ~. C l e a r l y f o r

IzlA-Iw[ < 1

we h a v e

I g ~ + g < g I ~ - ~ / I + ~ t ~/ + g

~ E I ~(e' ~' 62)-~ ~e-m(t-c~)/e-~-Izl-~-Iw[

w h e r e E is t h e s a m e k i n d of f u n c t i o n as a p p e a r e d in (6.3). A s i m i l a r r e s u l t h o l d s f o r

~g du ~h I

I f w e n o w use L e m m a 3 we f i n d t h a t I

ox(t) i I Ou(t)

~ .

~ - I + f - ~ ~+ 0 u n i f o r m l y

f

~b

dt

I

o v e r a _< t ~ 7 < vl as e, ^{d 1} a n d d2 -~ 0 while i ~ - ~ --> 0 u n i f o r m l y o v e r ~ - ~ ~< t ~<

< ~ f o r a n y f i x e d ~ > 0. T h i s c o m p l e t e s t h e p r o o f of T h e o r e m 3 o v e r

T h e o r e m 3, Part 2.

A t v = v0 < vB we c h a n g e f r o m t t o u (or v) as t h e i n d e p e n d e n t v a r i a b l e . W e h a v e f r o m (4.0)

d ~x ~f ~r ~p d ~t ~p

- ~ a p + ~ a , -

du ~a ~a du ~a ~a

(6.4)

d ~p

du ~a = 2p~pgA_ p2~aA-3p2~ah+p3~a. Og ~p ~h

W e also h a v e (5.3) f o r t h e d e g e n e r a t e s y s t e m . T h e initial v a l u e s a t u -~ v = v o of

~X(Vo)/~a

a n d

~t(Vo)/~a

m a y be f o u n d in t h e s a m e w a y as

~y(Vo)/Oa

a n d

~s(vo)/~a

in (5.1) a n d (5.2). H e r e

~x(vo)/~a

m e a n s

~X(Vo-f-O)/~a

etc. F r o m P a r t 1 we k n o w t h a t

~x(t0) ~y(s0) ~u(t0) ~v(s0)

a n d c a n b e m a d e as s m a l l as we w a n t b y t a k i n g e, ~1, a n d

~a ~a ~a ~a

~x(%) ~y(vo) ~t(vo) ~s(vo)

52 s m a l l e n o u g h . T h u s a n d c a n b e m a d e as s m a l l as we

~a ~a ~a ~a

wish.

W e h a v e also t o consider

~p(vo)/~a.

W e h a v e since p ~ 1 / - -

dt

Perturbatious of Discontinuous Solutions of Non-Linear Syst. of Diff. Equations. 95

(6.5)

3p(vo) - 1 [d~u3t(v~) d ~u_(to) )

~a (du~ ~ -d~ ~a ~ dt ~a J"

\ d t ]

A similar f o r m u l a is v a l i d for

~q(vo)/~a.

T h e t e r m s in (6.5) t e n d t o t h e c o r r e s p o n d i n g t e r m s in

~q(%)/~a,

w i t h t h e possible e x c e p t i o n of d~u

dt--~, as s, ~1,

a n d 53 -~ 0 b y T h e o r e m 1 a n d P a r t 1 of T h e o r e m 3. H e r e we h a v e t h e following l e m m a .

d2u d2v

L e m m a 4.

For a~-(~ ~ t ~ ~ < vl, where

~ > 0,

dt ~ dt -~OuniJbrmlyass,6l, and ~ -~ 0 under the hypothesis of Theorem 1. The result also holds for v1+8 ~ t ~ ft.

W e shall p r o v e t h i s l e m m a a t t h e e n d of t h e section. W e see n o w t h a t i n d e e d

~p(vo) ~q(vo) ~p(v) ~q(v)

~a ~a - ~ 0 as s, 81, a n d 8 ~ - ~ 0 . I t is e a s y t o see t h a t ~a ~a ~ 0 u n i -

ds g

f o r m l y f o r

v o ~ v ~ VB--~

f o r a n y (~ > 0 as s, 51 , a n d ~ - ~ 0. Since q - -

d v - - h

we also see t h a t ~ e x i s t s a n d is a c o n t i n u o u s f u n c t i o n of v f o r v <

v B

a n d m o r e o v e r

Oq(VB--O)

exists. I f we n o w let z r e p r e s e n t \ ~ a ~a ' 0a

~a a n d (6.4)

fd~l y ( x , v , t , ~ ) ~f(y,v,s,O)l+ K ~p ~ql

I~vt < = ~a ~a I ~a---~l

P ~ ( x , v, t, e) ~ ( y , v, s, 0)] ~q

~a q ~a ! -~ -~a I~(x, v, t, s)---~(y, v, s, 0 t .

So long as Ipl~-Iql ~ 1 a n d as Izt ~ 1 ,

z

l ~ l =tzl~l+ ~x~ ~y, Val + ~ i - ~ ~al

+ ' ~ i ~ x i t~-aa T { ~ [ - - ~ s { t ~ a l + K I p - - q l + t~a--~at + K I q ~ ( x ' v ' t ' s ) - ~ ( y ' v ' t ' O)["

L e t va < v v. W e o b s e r v e t h a t v3 is u n r e l a t e d to v 3 of T h e o r e m 1. I f ~(s, 81, 62, va) r e p r e s e n t s a f u n c t i o n w h i c h t e n d s t o zero as s, ~i, a n d ~, --> 0 t h e n f o r v0 ~ v ~ v 3

t d~ . K { ~P aq I

Idv ~ KIzI+KJp-qIT I~a-~a +~(s'81'5~'va)"