y+kf(y)y+g(y)=bkp( ), ON NON-LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER: IV. THE GENERAL EQUATION

ON NON-LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER: IV. THE GENERAL EQUATION

y+kf(y)y+g(y)=bkp( ),

B Y

J. E. LITTLEWOOD

in Cambridge

w t . W e e n t e r n o w on o u r c o m p l e t e a c c o u n t of t h e m o r e general e q u a t i o n ij+kit(y)~+g(y)=bkp(q~), ~ = t + ~ .

T h e f u n c t i o n s i t, g, p are fixed, b is n o n - n e g a t i v e , a n d k is large a n d positive. W e proceed t o s t a t e t h e long list of a s s u m p t i o n s a b o u t it, g, p. I t m a y h e l p t o w a r d s easier r e a d i n g t o i m a g i n e t h a t it a n d g are p o l y n o m i a l s a n d p a t r i g o n o m e t r i c a l p o l y n o m i a l : in so f a r as h y p o t h e s e s a b o u t t h e s m o o t h n e s s of f, g, p a r e concerned o u r a r g u m e n t s are n o t essentially different f r o m w h a t t h e y w o u l d t h e n be, a n d the r e a d e r m a y t r u s t us to h a v e t a k e n care of t h e details. H e m a y similarly t a k e on t r u s t details a b o u t t h e c o n s t a n t s c o n n e c t e d w i t h these functions, a n d t h e v a r i o u s a p p e a l s t o t h e ], g, p d i c t i o n a r y (w 3) t h a t occur in t h e a r g u m e n t s .

p h a s continuous p " , is periodic w i t h period n o r m a l i z e d to 2 zt, has m e a n v a l u e 0, a n d is s k e w - s y m m e t r i c , i.e. p (z~ + ~ ) = - p (~). A n y integral f p dq~ is periodic; we define Pl (~) be t h a t one for which t h e m e a n v a l u e is 0. I t also is s k e w - s y m m e t r i c . I t is n o w a n essential a s s u m p t i o n t h a t Pl attains its upper (and consequently also its lower) bound once only in a period. W e n o r m a l i z e p t o m a k e 1 t h e u p p e r b o u n d of Pl, to be a t t a i n e d a t ~ g. So Pl (~ zt) = - pl ( - ~ ze) = l, p ( + ~ 7t) = 0. p ' ( - ~ 7t) is n o n - n e g a t i v e ; we suppose it a positive c o n s t a n t a 2.

it(y) is even, w i t h continuous it". I t has a single p a i r of zeros, n o r m a l i z e d t o +__ 1; /' (1) is a positive c o n s t a n t al, a n d it has a positive lower b o u n d in (say) y_>2.

y

W e define F ( y ) = f i t ( y ) d y ; F is odd. W e n o r m a l i z e ] to m a k e F ( - 1 ) ( c e r t a i n l y

0

positive) ~. This will m a k e ~ t h e critical v a l u e of b, as for v a n der P o l ' s e q u a t i o n ; t h e b e h a v i o u r for b > ~ is crude, a n d we suppose for s i m p l i c i t y t h a t 0 < b < 2 as before.

1 -- 573805, Acta mathematica. 98. I m p r i m 6 le 19 n o v e m b r e 1957.

2 J . E . LrL-rLEWOOD

W e define H , a c o n s t a n t > 1 , b y F ( H ) = F ( - I ) = ~ . 1

T h e final a s s u m p t i o n s a r e a b o u t g: we suppose it odd, w i t h c o n t i n u o u s g". W e should in a n y case suppose t h a t g" h a s a positive lower bound. I n o r d e r to a v o i d c e r t a i n complications we s u p p o s e g'>_ 1. 2

w 2. C o n s t a n t s L are t h r o u g h o u t positive c o n s t a n t s d e p e n d i n g o n l y on t h e func- tions [, g, p, a n d t h e c o n s t a n t s implied in O's are a l w a y s of t y p e L.

Before going on we s t a t e t h e essential

L E ~ M A 1. Suppose (as always) that O<_b <_2. T h e n every trajectory F ultimately satisfies

lYl<_L ',

where we m a y suppose L~' > 20, say. a I f it satisfies these at t = t o, then it will satisfy

lyI-<L *, I l<_L*k re, t>_to.

I / b > O, F (strictly) crosses y = 0 infinitely often.

This is p r o v e d (for still m o r e general [, g, p) e l s e w h e r e )

w 3. I n t h e light of L e m m a 1 we define, slightly e x t e n d i n g t h e n a t u r a l m e a n i n g of the adjective, a n " e v e n t u a l " F t o be one satisfying I Y l <_ L~, I Y l <_ L~ k a t t h e ( a r b i t r a r y ) origin of t i m e t = 0. I t t h e n satisfies I Y [ <_ L*, I Y l <- L* k for t >_ 0.

W e m a y o b s e r v e t h a t this u l t i m a t e b e h a v i o u r holds (for a suitable L~ a n d L*) s u b j e c t to v e r y general conditions on [, g, p.4 Once g r a n t e d this, it is e n o u g h for o u r f u r t h e r p u r p o s e s t h a t t h e m o r e s t r i n g e n t conditions we i m p o s e on f, g, p should hold in t h e r e s t r i c t e d r a n g e l Y I -< L*.

W e give for convenience of reference a " d i c t i o n a r y " of f, g, p.

L EMMA 2. p ( ~ ) s has continuous p " . I t has period 2 rc and m e a n value O, and p (q) + ~z) = - p (q~). Pl (q)), the integral of p with mean value O, has Pl (~ § ~z) = - Pl ((P).

= 1 and nowhere else, and attains its pl attaius its upper bound, which is 1, at q ~ - - ~ ,

lower bound - 1 at ~ ~ - - 1 ~ and nowhere else.

p ( - F ~ : ~ ) = O , p , ( _ _ _ l ~ ) = ~ a s , a z > O ; p l ( • _ 1 . (1)

i In v a n der Pol's equation this critical height H has the value 2. We choose to normalize the critical b to -~ rather than H to 2.

s Since the period is normalized to 2 :r this is a real restriction on one parameter of the equa- tion a n d m a y be unnecessary. W e could alternatively a s s u m e that ]' > 0 in 1 < y < H .

3 Constants with *'s (and with or without suffixes) are p e r m a n e n t (see w 4 below).

4 See M. L. CARTWRIOHT and J. E. LITTLEWOOD, Ann. o/ Math., 48 (1948).

5 ~ is the phase, and is of the form t +r162 since the period is 2x~. We work sometimes in ~, sometimes in t. We have of course

p' = d p/dg~ = d p/dt = p.

THE GENERAL EQUATION ~ "~- ~ / (y) y § g (y) = b k p (q~), (p = t § ~r 3

- I

Fig. 1. Graph of: P (y).

F o r ] ~ ] < ~ we have

Ip(-~+~)l<_Ll~l,

(2)

l + p l ( - ~-:~ + yJ) = 1 a2~p~ + 0(y~a ), (3) Lye2_< 1 + P l ( - ~ r~ + YJ)-< Lv2 ~, (4) with corresponding results /or Pl (~ g + v2) = - Pl ( - 89 rc + v2).

/ is even, with continuous /". g is odd, with continuous g".

/(+1)--o, ~ ( T 1 ) = +~, / ' ( + 1 ) = + a . a~>O. (5)

F ( H ) = F ( - D -~ H > I . (6)

y = l + ~ ; / ( y ) = a ~ + O ( ~ ~) ([y]_<L*); ~ ( y ) / ~ L (O<_y<_L*). (7)

$ ' ( y ) - F ( 1 ) = ~ a ~ y ~ + O ( ~ 3) (lyl_<L*);

$ ' ( y ) - F(1) >_0 (y_> - H ) ; (8)

L ~ 2 < _ $ ' ( y ) - F ( 1 ) < _ L ~ ~ ( - ~ ( I + H)<_y<_L*);

g" is continuous, and 1 <_ g' <_ L*, /or I Y l <- L*. (9) (1) is agreed, and (2), (3) follow from (1) and the continuity of p". The second inequality in (4) is a trivial corollary of (3); the first, however, depends on the fact that the lower bound P l ( - ~ ) is attained at ~ - - - ~ only.

The results about / and F are either agreed, or simple consequences of ( 5 ) a n d the continuity of /". (9) is agreed.

w ~. N o t a t i o n f o r u p p e r b o u n d s . W e use L (as we said above) for positive constants depending only on the functions /, g, p; and we use A (x, y . . . . ) for positive

4 J . E . LITTLEWOOD

constants depending only on these functions and the x, y . . . . I n the rather rare cases when A is used as an index [as log Ak, or /c -A] it means a positive absolute con- stant. We use D for constants A (~) depending on a J whose rSle is similar to t h a t of the ~ in the Introduction 1 (w 12). This ~ is to be t h o u g h t of as "small": it has in the end to be less t h a n some definite L; we suppose always, a n d tacitly, t h a t ~ satisfies a n y inequality J < L called for b y the run of the argument. Each L, D, A (), as it occurs will in general depend on previously occurring ones; the chain, e.g., of D ' s could be made one of explicitly defined constants.

Many L ' s and /)'s do n o t need identification. Where t h e y need identification throughout a single argument we use temporary suffixes, restarting the suffixes again a t 1 on the next occasion. We sometimes use dashes in the same w a y (when suffixes are too thick on the ground). Where constants need permanent identification we use stars (as well as suffixes): thus D~ (w 20) is always the same D. Suffixes to things other than constants L, D are used in m a n y distinct senses; we hope t h a t these are sufficiently disparate not to be confused; our notational problems are very dif- ficult.

The upper bounds implied in O's are always L's.

We have to employ Lemmas with undetermined non-negative or positive con- stants d, d'; these are blank cheques, constants chosen in different ways on different occasions; when chosen, t h e y m a y be 0, or L, or D, b u t are always one of these.

The assertions of the Lemmas, which involve such things as A (d, d'), k o (d, d'), con- sequently involve D's at worst, when t h e y actually come to be applied. (Indeter- minate constants other t h a n d, d' are sometimes used, b u t only temporarily and with ad hoc explanations.)

The constant b requires some discussion. I t is always (as explained in w 1) subject to 0_< b_<2, and for some results no restriction other than this is necessary. B u t both b = 0 and b = ~ are generally critical, and bounds of various things depend on the nearness of b to 0 or ~. Behaviour when b = 0 has considerable interest of its own, and our first intention was to introduce a second "5", ~', and a hypothesis b ~> J ' in the relevant contexts. B y leaving the orders of J and ~' independent we should arrive at results which were at least pointers to the case of small b (the real answers probably require such b to be a function of k). The complications of having more than one ~, however, proved almost prohibitive, and we adopted the simplifica- tion of making all 5 the same. I t turned out in the end, however, t h a t even the 1 See Paper I I I Acta Math. Vol. 97 (1957). This paper will be referred to in future as the In- troduction. Both papers ~re based on joint work with M. L. CAItTW~IGHT.

THe. G~.~.RAL ~.QUATIO~ ~ + k f (y) ?~ + g (y) = b k p (q), q = t +

a s s u m p t i o n b > (~ ( ~ ' = (~) led t o a v e r y g r e a t increase of c o m p l i c a t i o n ; a n d our final h y p o t h e s i s (where t h e critical values 0, ~ are relevant) is b E B, where B is t h e range

1-~0_<b < - 2 - ~ - a 1 0 o . ( 1 )

W e r e g r e t this m a s k i n g of b e h a v i o u r for small b, b u t it seems t h e lesser evil.

W h e n b E B a n A (b) or A (b, ~), if c o n t i n u o u s in b, as it a l w a y s is in practice, lies between t w o L ' s or D ' s respectively.

T h e dependence of ko=ko(x, y . . . . ) on c o n s t a n t s (of. I n t r o d u c t i o n w167 5, 9 ) r e q u i r e s o n l y a s h o r t explanation, k 0 (x, y . . . . ) is a l w a y s an A (x, y . . . . ) a n d depends o n l y on /, g, p a n d t h e x, y . . . W h e r e we h a v e Lemmas containing ( u n d e t e r m i n e d ) d ' s t h e k 0 n a t u r a l l y depends on thes d's. T h e k0's of Theorems generally d e p e n d on (~, b u t never on u n d e t e r m i n e d p a r a m e t e r s (which T h e o r e m s never contain).

A k o in a L e m m a or T h e o r e m is " s u f f i c i e n t l y large". I t has to be continually re-chosen as t h e argument, proceeds. Suppose, for example, we h a v e p r o v e d X < D 1 k -89 where k > k o. W e t h e n have, e.g., X < k -~ for k > k0, where k 0 = m a x (ko, D~), a n d s a y

" X < D l k - i < k -89 b y re-choice of k0". I t would be intolerable to m e n t i o n all t h e re- choices, and, once h a v i n g directed a t t e n t i o n s t r o n g l y t o t h e point, we shall more a n d m o r e f r e q u e n t l y suppose t a c i t l y t h a t a n y n e c e s s a r y rechoice is being made.

w 5. W e n o w seriously begin our long a n d intricate story, which, after t h e literary excursions of t h e I n t r o d u c t i o n , we shall n o t t r y to lighten. W h a t we h a v e a i m e d a t is to m a k e things as e a s y as m a y be for a reader who omits t h e proo/s of t h e L e m m a s (or m e r e l y skims t h e m for t h e general idea) a n d c o n c e n t r a t e s o n their s t a t e m e n t s (and of course t h e connecting explanations). W e have t a k e n pains t o m a k e t h e chain of s t a t e m e n t s as lucid a n d efficient as we can. E a c h L e m m a of t h e chain, further, has a l m o s t a l w a y s a self-contained p r o o f ; clumsinesses t h a t h a p p e n inside these proofs do n o t c a r r y over outside. P a r t of t h e plan is to collect all n e e d e d results of a similar k i n d into one L e m m a a t a time, a n d some of t h e L e m m a s are long " d i c t i o n a r i e s " .

~ 6 . L E M M A 3. (i) Let 0 _ < b < 2 , and let d be a non-negativeand d' a positive constant. Then there is a ]co(d, d') such that when k>_ko, the /ollowing properties hold.

Suppose that an eventual trajectory has a piece X Y Z lying entirely in y >_ 1 - d k-89 suppose also that (a) X Y has time length at least d', (b) Y Z contains a point at which

~ _ _ 1 ~ , (c) Y Z has time length at least k-89 log k. Then /or any Q o/ Y Z,

J. E. L I T T L E W O O D

I ~ l < A ( d , d ' ) ,

I n the identity

l i j l < A ( d , d ' ) k t , I i j l < A ( d , d ' ) k ;

~] = b p (q)) + 0 (A (d, d') k-t);

~)f=bp(qJ) + O(A (d,d') k - l [ y - l [-1).

(1) (2) (3)

t

E _ F ( 1 ) = C + b ( l + p l ( q ) ) _ k - l f g d t _ ~ k -1 (4)

0

we may substitute ~= 0 (A (d, d')) in the stretch Y Z .

(ii) Let 0 <b <_ 2, and suppose that d is a positive constant, and that k > k o, where k o is a certain k o (d). Then at a Q that has been preceded by a piece o/ an eventual trajectory lying in y > 1 +d, and lasting a time k -1 log~k at least, we have

l Y I < A ( d ) , ] ? ) l < A ( d ) , I ~ ] l < A ( d ) ;

with various consequences, e.g. (2) is valid with error term improved to O ( A (d) k-l), or

I Y/-- b p @) I < A1 (d) ]r (5)

A]l.~

y= 1 - d k - ~

F i g . 2. Mr, M2, Ma c o r r e s p o n d t o c a s e s (i), (ii), (iii) r e s p e c t i v e l y .

W e a b b r e v i a t e c o n s t a n t s A (d, d') to A.

W e begin b y p r o v i n g t h e result I~)1 < A in (i). On t h e t r a j e c t o r y we h a v e a l w a y s I Y l -< L r I n a n y piece of t h e t r a j e c t o r y of t i m e i n t e r v a l d' I~)l c a n n o t everywhere exceed 2 L~/d'. H e n c e (see fig. 2) t h e r e is a n R of X Y w i t h l YR l < L I d ' . L e t l Yl a t t a i n its m a x i m u m v for R Z a t M . W e m a y in w h a t follows s u p p o s e t h a t v is g r e a t e r t h a n a n y p a r t i c u l a r A t h a t arises, since otherwise we h a v e w h a t we w a n t . (In p a r t i c u l a r we s y s t e m a t i c a l l y reject a l t e r n a t i v e s v < A as t h e y p r e s e n t themselves.) W e suppose a l w a y s v > 1.

W e m a y suppose M n o t a t R ( M = R would give w h a t we want). B y t h e h y p o - thesis a b o u t Y Z, R Z c o n t a i n s a point, S within 2 ~ on one side or t h e o t h e r of M, for which ~s~-- - ~ ~- L e t ~VM~ -- 1 ~ + V2 ' where [~v] _< 2 ~.

THE GENERAL EQUATION y § k / (y) ~j + g (y) = b k p (~), ~ = t + ~ 7 w 7. W e m a y suppose, b y prolonging Y Z t o t h e n e x t intersection 1 w i t h t h e line y = l - d k - 8 9 t h a t Z lies on this line (the h y p o t h e s e s being satisfied a / o r t i o r i in t h e n e w case). W e h a v e n o w t o distinguish t h r e e cases:

(i) M identical w i t h Z (when YM is n e g a t i v e b y t h e g e o m e t r y a n d YM = --V);

(if) M is n o t Z, y M = - - v ; (iii) M is n o t Z, YM= + v -

I n cases (i) a n d (if) ~M = --V. F r o m t h e ~-identity, writing gl for f g d t , we h a v e

~ls - YM = -- k ( F (Yz) -- F (1)) + k ( F (YM) ^{-- ~} (1)) + b k (Pl (~s) - Pl (~M)) -- [gl] s . T h e l e f t - h a n d side is Ys + v _> 0. On t h e r i g h t t h e first t e r m is n o n - p o s i t i v e b y L e m m a 2 (8); t h e second is n o t g r e a t e r t h a n L k ~ b y L e m m a 2 (8); t h e f o u r t h is less t h a n L since { t s - tMI ~ 2 z . W e have, therefore,

O<-Lk~2M+ bk(P~ (q~s) - P ~ (q~M)) + L.

N O W -- P l (~0S) -~- P l (~M) = 1 § P l ( -- 1:7~ § ~)) ~__ L ~2, b y L e m m a 2 (4). H e n c e

by~<_L~eM+ L k -1 in cases (i) a n d (if). (1) N e x t , in either of t h e cases (if), (iii), M is s t r i c t l y interior to R Z , a n d conse- q u e n t l y ijM= 0; whence b y s u b s t i t u t i o n in t h e differential e q u a t i o n

y M / ( y M ) = b p ( ~ M ) - - g ( y M ) ] ~ -1 i n cases (if) a n d (iii). (2) B y (7) a n d (2) of L e m m a 2, a n d since ~M = - - ~ : ~ + ~ , we h a v e

V l ~ M { < b L { y J [ § L k -~ in cases (if) and (iii). (3) I n case (if) we h a v e (1) a n d (3), a n d so also

v ~ ~ < b2L 2 ~p2 + L 2 k-2 < LbyJ2 + L k - ~ < L (L~2M+ L k -1) + L k -1 < L~2M + L k -~.

Since we m a y suppose v~> 2 L1, this gives

{~M[</k -~,

a n d a /ortiori I~M{ < A k-89 This last inequality, j u s t p r o v e d for case (if), is t r u e also in case (i) (when ~ M = ~ Z = - d E - 8 9 I n either case we n o w h a v e byJ~<A]c -1.

1 This need not happen immediately.

8 J . E . L I T T L E W O O D

Hence, summing u p :

~ M = - - V,

In cases

(i)

and

(ii) b ~ < A k -1, I ~ l < A k -89

(4)

We continue to t a k e cases (i) a n d (ii) together, and now consider t h e reversed motion (r.m.) from M ; if v is its t i m e variable we h a v e t=~pM--T. The ~i-identity for this r.m. is

dy

dv

v - k ( F (y) - F (1)) + k ( F ( y . ) - F (1)) +

T

D

with y ( 0 ) - - I + ~ M ,

(dy/dv)o=v.

We write

y = y - l = k - t z , v=k- 89 P = P ( z ) =

k ( F (y) - P (1)), Po = P (Zo)- Then P, Po >- 0, z 0 = k 89 ~M > -- A. The differential equation becomes

dz - - = v + P - P o + b k p ( - ~ + v 2 } ~ +O(k~)+O(v).

dx

Now b y (4) a n d L e m m a 2 (2),

bkp(-lz~+v2)=O(k~)=O(k 89

a n d when we sub- stitute from this and for T the last differential equation becomes

dz

d-x = v + P - Po + 0 (Ax) + 0 (A x~). (5) L e t

X=log 89

t h e n

X<Llog 89

a n d (since the r.m. lasts a t i m e k- 89 which corresponds to a range y-1 log k of x, without y reaching L*) (5) has a solution in 0_<x<_X t h a t is bounded b y

L k89

We show n e x t t h a t either v < a certain A, as desired, or else

dz/dx,

initially positive, remains positive t h r o u g h o u t

O<_x<_X.

I f

dz/dx

ever vanishes, let it vanish

[irst

a t x = ~ _ < X ; t h e n in (0,~) z_>z 0 > - A . Now i f z o < 0 , t h e n P - P o > - - P 0 > - A ;

z

a n d if Zo>_0, t h e n

P - P o = f(positive) dz>_O;

in either case

Ze

dz ~ x > V - A - A ~ - A ~ 2 > v - A - A X * > v - A - A l o g

(v + 2),

which is positive a t x = ~ (contrary to hypothesis) unless v < A.

We m a y suppose, then, t h a t

dz/dx>O

and

z>_z o

in (0, X). I n this range we have certainly - 1 < y < L * , and so, b y L e m m a 2 (8),

Lz~ < p < Lz ~.

THE OE~ERAL v, QUATm~ ?~ + k i t (y) # + g (y) = b k p (~), q = t + 9

I t follows n o w t h a t { L z * - A (z•

P - P 0 > L ( z - Z o ) 2 ( % > 0 ) .

F o r if Z o < 0 , t h e n z o = O ( A ) , P o < A , a n d P - P o > L z 2 - A . I f %>_0, t h e n

(6)

t ' - Po= k f/d,__ k f L,7 e,7,

~M ~M

b y L e m m a 2 (7), a n d so

P - P o >- L k (rl 2 - ~ ) = L (z ~ - z~) > L (z - Zo) 2,

since Z>Zo; a n d this completes t h e proof of (6).

F r o m (5) a n d (6) we h a v e in (0, X) for t h e case z o < 0 d z

- - > v - A - A X - A X 2 + L z 2 > v - A - A log ( v + 2 ) + L z 2,

d x ⁽⁷⁾

a n d a similar inequality with L ( z - z0) 2 in place of t h e last t e r m for t h e case z 0 >_ 0.

N o w either v is less t h a n a certain A, as desired, or else (7) gives, in, e.g., t h e

c a s e Z 0 < 0 ,

d Z > ~ v + Lz~,

z f j

a n d t h e n log89 (v + 2) = X = dx <_ _{v ~} < + < 2 = L v- ~- _'

0 zo 0

a n d v < A. I n t h e case z 0 > 0, t h e a l t e r n a t i v e t o v < A is d z

~ x > ~ v + L (z - zo) 2,

a n d t h e rest is m u c h as before.

W e h a v e n o w p r o v e d l Yl < A in eases (i) a n d (ii).

w I t remains to consider case (iii), in which yM=V: here we h a v e to p a y close a t t e n t i o n t o signs (of ~a a n d ~/M).

W e recall t h e i d e n t i t y (2) of w 7 [valid for case (iii)].

YM/(YM) = b p (qJM) -- g (YM) k-l"

This gives, b y L e m m a 2 (7) (whatever t h e sign of

~M)

b p (q)M) < LV~M-~ L k -1, ⁽¹⁾

(algebraically, note) a n d also

10 J. E. LITTLEWOOD

V I~MI < Lb ]P(q)M)I A-Lk-1,

(2)

< L b l v ] + L k -1, (3)

b y L e m m a 2 (2) (since ~M = -- ~ ~ + v2).

I n t h e ~-identity between M a n d S of w 7 we h a v e now for t h e l e f t - h a n d side

?]s-?/M t h e lower b o u n d - 2 v in place of t h e original 0; t h e conelusiom:(il) of w c o n s e q u e n t l y modified to

b~)2 < i~2M+ L v k -1.

(4)

Combining this with (3) (and using b < L , v > l ) we h a v e v~ ~7~M < L l ~ + L v k -1, a n d unless v 2 < 2 LI, which we c a n reject, we h a v e

v2~2M<Lvk -1, ]~M[ < L ( v k ) -89 (5)

We prove next that either v is less than a certain A (which we reject), or else

~1> 0 /or a time interval k -89 be]ore M . Suppose t h e second a l t e r n a t i v e false; t h e n t h e r e is a s t a t i o n a r y point Z, with ?~=0, a t time v < k -89 before M, a n d we m a y suppose it t h e nearest such p o i n t t o M . T h e p o i n t 5: is in X M (since X M has t i m e - l e n g t h a t least d ' > k - 8 9 hence y ~ > l - d k -~, -dk-89 a n d so ~ _ _ _ ~ + d 2 k -1. Con- s e q u e n t l y

0 - v = ?~= - ~)M = -- k (F (y~) - F (1)) + k (F (YM) -- F (1)) + b k (Pl (~M -- 3) -- Pl (~M)) + [gl] M

> - L k ~ l ~ + O + b k ( - ~ p ( q j M ) - L ~ 2 ) + O ,

b kT p(q)M) > V - L (k~?~M+ A ) + O - b k L k - I + O > v - L v - I - A ,

b y (5). Unless v is less t h a n a certain A, which we reject, this is greater t h a n 89 a n d t h e n

bp (~M) > ~ vk-1 T -~ >-- ~ vk-89 (6)

On t h e other h a n d (1) a n d (5) give

bp (q~M) < Lv89 k-~ + L k -~

which contradicts (6) unless v < L , which we can reject. T h e n ~ > 0 for an interval k -89 before M, as stated.

B y (5) we n o w have, for tM--k-89 <_t <--tM,

- d k - 8 9 < ~ < ~ M < L v - 8 9 k-89 (7)

b y (1), b y (5), unless v < A.

THE (]ENERAL EQUATION y + k / (y) ?) -{- g (y) = b k p (~0), ~ = t + a 11 T h e ?)-identity b e t w e e n t a n d tM gives

?) = v + k (F (YM) -- F (y) ) + b k (Pl (q~) - Pl (qJM) ) + [gl]~ M

>_ V + k (F (YM) -- F (y)) + b k (t - tM) p (q~M) -- L k (t - tM) ~ + O. (8) W e n o w distinguish t h e cases (i) r / > 0, (ii) ~1 _<0, a n d p r o v e in e a c h case t h a t

?) > ~v a t t h e p o i n t in question (of t h e k -89 interval), or else v < A . YM

I n (i) y~>_y>_l a n d F ( y M ) - : F ( y ) = f / d y ? _ O . Y

I n (ii) ]~l<_dk- 89 a n d so

F ( y M ) - - F ( y ) = (F(yM)-- F ( 1 ) ) - ( F ( y ) - F ( 1 ) ) > - O - L r / ~ , b y L e m m a 2 (8), > - A k -1.

This last i n e q u a l i t y is t h e r e f o r e t r u e in either case, a n d t h e n (8) gives

?) > _ _ v - A - L k . k-89 M a x (0, bp(gM))-- L + O

> _ v - A - L k ~ v ] ~ M I ,

> _ v - A - L y e ,

~lv~

I g n o r i n g t h e v < A alternatives, then, we h a v e ?)> ~ v t h r o u g h o u t t h e t i m e i n t e r v a l k- 89 before M . B u t t h e n a t t i m e t M - k -89 we h a v e

*l <- ~IM-- i r k - 8 9 <_Lv-~ k-~ - ~ vk- 89

T h e l e f t - h a n d side being a t least - d k - 8 9 this leads t o v < A , which is therefore established.

W e t a k e n e x t t h e (easier) proof t h a t l Yl < A k89 on Y Z . L e t X1 be t h e p o i n t of X Y of t i m e h a l f w a y b e t w e e n tx a n d tr. W e h a v e I?)] < A I (say) ^{o n}

X1Z

(by t h e ?) result). T h e n we c a n n o t h a v e 1?~[>2hl/(-~d') on t h e whole of X 1 Y (or ? ) w o u l d s o m e w h e r e exceed A1) ; t h e r e is t h e r e f o r e a p o i n t R of X I Y a t which I~al_<A. L e t t h e m a x i m u m of I?Jl for R Z occur a t M . We m a y suppose M n o t a t R, which would give w h a t we w a n t . This t i m e we distinguish t w o cases:

(~) [~/M[_<dk -89 (this includes t h e case M = Z ) , (fl) ~]M>dk -89

I n ease (~) we use t h e fact t h a t t h e r e is a n S of R Z , within 2 ~ on one side o r t h e o t h e r of M , with ~ s ~ - ~ g , a n d then, b y t h e ?)-identity,

12 J . r . LrrrLv.WOOD b k (1 + Pl (~M)) = b k (Pl (qM) -- Pl (qs))

= ?)M-- ?)S + k (F (YM) -- $' (1)) -- k ( F (Ys) -- F (1)) + [~l]S M

<_A + A + L k ~ M - O + 2 7 e L < A .

B y L e m m a 2 (2) the left-hand side is at least

bkLv22,

where ~M ~ - 1 ~ + v 2 , 1~01_<2~; hence b ~ < A k -1, a n d

b lp(q~M) l < b L l ~l < L(by~2)89 < h k -89

Since

]/(YM)I < L lrl~l < i k -t,

we have

~M=l--k/~-g+bkplM<_lcAk- 89189189

which proves w h a t we want.

I n case (fl) M is strictly interior to

R Z;

consequently y m = 0, or

0 = - - k iyM ] (YM) -- k / ' (YM) ~)~M -- 9' (YM) YI~ +

b k p" (qM).

Since

]/(yM)]>LI~M[>Ak- 89

b y L e m m a 2 (7),

[ {]M I <- A k89 I - ]' (YM) ~l~ -- k-1 g, (y~) ~JM + b p' (~fM) l

< A k 8 9

89

which completes the proof t h a t I Y I < A k 89 for Y Z.

The proof of l Y] < A k is m u c h like t h a t of the ~ result, b u t simpler, since t h e t e r m

bkp"

is crudely

O(k).

We differentiate once more a n d use y~V=0 in one half of the a r g u m e n t (as for ~)): it is this t h a t requires us to assume the existence of continuous second derivatives of /, g, p.X

We h a v e now established (1) of the L e m m a : (2) a n d (4) are immediate conse- quences. F o r (3) we have

1~] = b p ' -

g'y]c -1 - / ' y ~ ^- y k - 1 = O ( i ) , a n d so

ij=O(A

l Y - 1 I-~), and we have only to substitute this in

? ) / - b p = - g k - l - ~ ] k -1.

This completes the proof of p a r t (i).

w 9. I n p a r t (ii) let T = l o g S

k/k,

a n d consider the r.m. from the point concerned

t

as time origin, over the time 0 < t < ~ . L e t

T = f / d t .

Since

y>_l+d,

we h a v e

] > L d

0

( L e m m a 2 (7)),

T>_tLd.

The r.m. is

1 W e n e e d t h e i n e q u a l i t y for y : it is n o t a l u x u r y .

or

F r o m (1) we h a v e

and so

ij= k / (y) ~ ) - g + b kp, ~ d-t (Ye-kr) = (b k p - if) e - ~ r . d

y = k / i j § ?)2-g" y §

d t (?/e-~r) = (/c]' Y2 - g ' Y § bIcp) e -kr. d F r o m (2)

t t

~e - ~ - ~)o = f (b k V - g) e - ~ dt = f 0 (k) e - ~ ~t

o o

t

13 (1)

(2)

(3)

= o (k) f e - k ~ d t = o (k) f e -~'L~ d t = o (d-l),

o o

Hence, either ~ o = O ( d - 1 ) , or else

I ~ e k T l > h l y 0 1 > l .

T h e last a l t e r n a t i v e m a k e s [?)1 > c x p (Ld log s k) a t t i m e T, c o n t r a r y t o ?)= 0 (Lk).

T h u s Yo= O(d-1), as desired.

F o r ?)0 we a r g u e similarly f r o m (3), s u b s t i t u t i n g ~ = O ( L d -1) o n t h e right- h a n d side. F o r ~) t h e a r g u m e n t is similar.

This c o m p l e t e s t h e p r o o f of L e m m a 3.

w t 0 . W e t a k e n e x t t h e k e y L e m m a B of t h e I n t r o d u c t i o n , ( L e m m a 5, below) prefacing it b y L e m m a C ( L e m m a 4, below); we r e s t a t e t h e m for convenience ( t h e y are u n a l t e r e d in form, e x c e p t for a n a d d i t i o n to L e m m a B).

L EMMA 4. Suppose Yl, Y~ are respectively solutions o/

~ = O ( y , t ) + Rl.~,

where ~P is continuous in (y, t), RI, u continuous and R I > R ~ /or t>_t 0.

(i) I / now ?/1 (Q) >- Y2 (to), then Yl > Y~ /or t > t o.

(ii) The conclusion is true i/ R 1 > R~ /or t > t o only, provided we know independ- ently that Yl > Y2 /or small positive t - t o.

F o r t h e proof see w 14 of t h e I n t r o d u c t i o n .

1 The argument of p involves - t and a constant, but neither detail affects the reasoning.

1 4 J , E. LITTLEWOOD

L E M M A 5. Consider the (Riceati) equation, /or x>_O, d z

--=z2-x~+l+~-28x,

z ( 0 ) = 0 , d x

where ~ > _ - 1, and fl [urther satis/ies fl < 0 when a = - 1, so that z is positive/or small positive x.

There is a .B o = flo (~r such that

(i) i/ fl>flo [or O > f l > f l o when r 1 6 2 then z changes sign t o negative at an

x=xo(~, fl)>0;

(ii) i/ fl<flo, then z-->+ oo at an asymptote x = x o ( : c , f l ) > 0 ; (iii) i/ fl =rio, there is a solution in (0, ~ ) for which z > 0 and

z = x + flo + F (x, ~),

where F ( x , ~) is continuous in (x,~)* and F = O ( 1 / x ) as x--->oo.

Further flo (or and ~'o (~) = :r § fl~ (cr are continuous and (strictly) increasing. 8o (~) is large with large positive o:.

Finally 8io(a) has the sign o/ ~ (and 8 0 ( 0 ) = 0).

(iv) I f 8 = 8 o ( ~ ) , 0 < / l < - l + ~ - < l z , then d z / d x > A (l,,l~)>O.

z is positive for small positive x, since z' (0) > 0 if ~ > - 1, a n d z' (0) = O, z" (0) = - 2 8 > 0 if ~ = - 1 .

L e t z = u + x + 8, ? = ~ + 82 (and 70 = ~ + 802) ; t h e e q u a t i o n b e c o m e s d u 2

~ x = U + 2 ( x + 8 ) u + ? = u ( u + 2 x + 2 8 ) + ? , u ( 0 ) = - 8 . (1) L e t Cz, Cu be t h e curves z = z (x), u = u (x) (both for x_> 0), d e t e r m i n e d b y t h e e q u a - tions a n d t h e i r initial conditions, a n d let Fu be t h e h y p e r b o l a

u ( u + 2 x + 2 f l ) + 7 = O . T h e slope of Cu can v a n i s h o n l y a t a p o i n t of Fu.

F o r given ~ > _ - 1 t h e r e are 3 m u t u a l l y exclusive possibilities in r e s p e c t of fl:

(A) 2 Cz has a vertical a s y m p t o t e w h e r e z - - > + co; (B) C~ crosses O x (from positive t o n e g a t i v e z) (C) n e i t h e r (A) n o r (B) h a p p e n s ; we s a y in t h e respective eases t h a t fl 6 (A), (B), (C) (the classes v a r y w i t h a).

1 We include this obvious fact because it is explicitly used later.

2 Initial of "asymptote".

THE GENERAL EQ~ATIO~ ~ + k f (y) ?~ + g (y) = b k p (q), ~0 = t +

- 2

u

/

/

(i)

u

/

/

- f l /

(ii)

Fig. 3. (i) r fl > 0. The region

du/dx

> 0 is shaded. (ii) r fl < 0. The region

du/dx

> 0 is shaded.

(Erratum:

in Fig. 3 (i) - f l should be placed between - 2 fl and origo.)

15

i n t h e first place we h a v e b y c o n t i n u i t y :

T h e classes (A), (B) are open, a n d v a r y c o n t i n u o u s l y with ~, a n d f r o m L e m m a 4 we h a v e :

I f

fiE(A)

so does

fl'<fl;

if

fiE(B)

so does

fl'>fl.

(2)

(3)

T h u s (A) a n d (B), unless one of t h e m is null, are infinite o p e n intervals, s e p a r a t e d b y t h e c o m p l e m e n t a r y (C), which is either a closed interval or a single point.

W e aim first at p r o v i n g t h e following results:

(a)+ I f ~ > 0 , a small positive f i e (A), a n d e v e r y large positive tiff (B).

(a)_ If - 1 < ~ < 0, t h e n fl = - I ~ 189 E (A), a n d a n y small negative fl E (B).

(b) F o r ~=~0, if f i e (C), t h e n z>_O,

z=x+fl+F(x,~), $'=O(1/x),

a n d (C) contains e x a c t l y one ft.

(c)+ This fl is large w i t h large positive ~.

Suppose these results established. F r o m (a)+, (a)_, (b)+, (c)+ a n d (2) (continuity) it follows t h a t a unique continuous fl0 (a) exists for all ~ (including a = 0), t h a t fl0 (~) E (C), t h a t rio(a) has t h e sign of ~, a n d t h a t rio(s) is large w i t h large positive ~. F u r t h e r

16 J . E . L I ' I " r l J E W O O D

(i), (ii), (iii) of t h e L e m m a hold. I t r e m a i n s o n l y t o p r o v e (iv) a n d t h a t fl0 (~) a n d a + f l 0 ~ (cr are m o n o t o n i c increasing, a n d t h e s e we postpone, going on n o w t o p r o v e (a)+ t o (c)+. W e can divide (b) into t h e t w o cases (b)+ a n d (b)_ c o r r e s p o n d i n g t o

~ > 0 a n d ~ < 0 .

Begin w i t h t h e results involving ~ > 0 , n a m e l y (a)+, (b)+, (c)+.

I n (a)+ a n d (b)+ we m a y suppose ~ > 0 , f l > 0 . So ~ > 0 , I ~ is as in fig. 3 ( i ) a n d does n o t c u t O u (the e q u a t i o n w i t h x = 0 has no real roots).

C~ 1 s t a r t s a t P ( 0 , - f l ) w i t h positive slope ~. I f C~ cuts Ox before c u t t i n g F~, t h e slope c a n n o t v a n i s h t h e r e a f t e r , we h a v e d u / d x > u 2 + ~ ,, a n d Ca, a n d so also Cz, h a s a vertical a s y m p t o t e a t a n x=xo(~,;3); fl belongs to Class (A).

I f C~ cuts Fu before c u t t i n g O x, t h e slope of C~ becomes n e g a t i v e a n d r e m a i n s n e g a t i v e t h e r e a f t e r ; u is n e g a t i v e a t t h e crossing, u = - 2 say, a n d s u b s e q u e n t l y decreases, so t h a t l ul___ 2. N e x t , we m u s t h a v e u < - ~ ( x + f l ) for some large x, since t h e c o n t r a r y i n e q u a l i t y for all large x, c o m b i n e d w i t h (1), would i m p l y

u + 2 x § 2fl>_~(x § fl) d u

a n d ~ x < - l lu ] (x + fl) § ~<_ - ~ 2 (x + fl) + ~,

so t h a t u would go t o - c o a t least as f a s t as - 8 8 ~, a contradiction. Since u < - ~ (x § fl) implies z < - ~ (x + fl), this last is t r u e for some large x ; c o n s e q u e n t l y Cz m u s t cross O x a t a positive x = x o(~, fl), a n d fl belongs to class (B).

T o s u m u p : fl belongs to class (A) if Cu cuts Ox before Fu, a n d to class (B) if it cuts Fu before O x.

F o r a small positive fl, C~, h a v i n g slope ~ > cr a t P, clearly cuts O x first; a n d we h a v e t h e first p a r t of (a)+.

Suppose n e x t t h a t fl is large a n d positive. T h e n d z / d x is large a n d n e g a t i v e for some positive x. F o r suppose not, t h e n Cz does n o t go to - ~ o for finite x.

F u r t h e r , b y L e m m a 4, Cz is below t h e c u r v e

d--~=~+l+~, $(0)=0,

d x

which has a n a s y m p t o t e a t x = l : ~ ( l + ~ ) - ~ = c , say, a n d satisfies ~<c', say, in 0 _-_~2v.< ~ ( ! ~ I n t h e r a n g e { c _< x ~ ~ c we have, on t h e one h a n d z < c', a n d on t h e other, b y hypothesis, d z / d x > - K, where K is i n d e p e n d e n t of fl, a n d so z > _ 1 c K. H e n c e at x = l c ,

1 Cu is an auxiliary curve for proving facts about C z.

~ . O~.NER~a~ V, QUATm~T 9 + k ] (y) ~ + g (y) = b k p (~0), ~0 = t + 17 dZ__<z2 + l + ~ _ 2/5. 88 + icK)2 + l +o~_[ /sc,

dx

a n d as this is l a r g e a n d n e g a t i v e we h a v e a contradiction.

So d z / d x a n d a ]ortiori d u / d x , is large a n d n e g a t i v e for s o m e positive x. B u t for such x Cu m u s t h a v e a l r e a d y crossed Fu (since d u / d x is positive until Fu is crossed). Cu c a n n o t h a v e first crossed O x, since it w o u l d t h e n continue to m o v e u p w a r d s . This establishes t h e second half of (a)+.

W h e n /5 is of class (C), z r e m a i n s positive. F u r t h e r , Cu cuts n e i t h e r Ox n o r F~, a n d c o n s e q u e n t l y a p p r o a c h e s 0 x b e t w e e n O x a n d t h e a s y m p t o t i c b r a n c h of F~; henc u = 0 ( l / x ) , a n d z = x + fl + 0 ( l / x ) . I f this h a p p e n e d for t w o distinct /5's, /51 a n d /52>/51, we should h a v e z 2 = z l + ( / 5 , - / 5 1 ) + O ( 1 / x ) > z 1 for large x, w h e r e a s z2<z 1 b y L e m m a 4. W e h a v e accordingly p r o v e d (b)+.

T h a t fl0(~) is large w i t h large positive ~ is e v i d e n t ; if ~ is large a n d /5 is not, t h e large initial slope of Cu will t a k e it across Ox, a n d /5 will n o t be of class (C).

T h i s is (c)+, a n d we h a v e p r o v e d all t h e ~ > 0 results.

w W e n o w t a k e u p t h e ~ < 0 results, n a m e l y (a)_, (b)_. I n (a)_ we h a v e

~ < 0 , /5<0.

W h e n v = O , or / 5 = - I~189 t h e u e q u a t i o n is

d u

g%=u2+2(x+/5)u, u(0)=l/51.

This is soluble in finite t e r m s , a n d t h e solution h a s a n u p w a r d a s y m p t o t e : this p r o v e s (a)_.

F o r small n e g a t i v e /5 we h a v e 7 < 0 , a n d Fu is as in fig. 3 (ii). Since d u / d x vanishes on a n d o n l y on Fu, C~ c e r t a i n l y c a n n o t cross t h e lower b r a n c h of Fu. H e n c e if C~ crosses O x (as it clearly does for a small n e g a t i v e /5), u t a k e s a n e g a t i v e v a l u e - ) t a n d t h e r e a f t e r decreases further. I f we n o w h a d u > _ - ~ (x +/5) for all large x, we should h a v e

d u

u + 2 ( x + / 5 ) > 1 _~(x+/5), - - < - ~ ( x + / ~ ) + 7 d x -

for large x, a n d u would go to - oo like - 88 ~ x ~ a t least, t h e r e b y crossing t h e lower b r a n c h of Fu, which is impossible. H e n c e for s o m e large x u < - ~ (x +fl), a n d so z < - ~ ( x + f l ) , Cz crosses O x, a n d /5 belongs t o class (B). (a)_ is n o w proved.

F o r a fl of class (C) Cu m u s t go t o oo b e t w e e n Ox a n d t h e u p p e r b r a n c h of F~, since if it crosses t h e F~, u would s u b s e q u e n t l y increase; for large x we should h a v e

2 - 573805. Acta mathematica. 98. I m p r i m 6 le 19 n o v e m b r e 1957.

18 a.. ~. ia-rrLEWOOD

du/dx>u2+xu

w i t h a v e r t i c a l a s y m p t o t e . H e n c e

u=O(1/x),

a n d t h e r e s t of t h e p r o o f of (b)_ is t h e s a m e as for (b)+.

I t r e m a i n s t o p r o v e /50 (~) a n d y0(~) ( s t r i c t l y ) i n c r e a s i n g , a n d f i n a l l y (iv).

Consider C1, C2, t h e Cz for (~1, 80 (%)), (~2,/~0 (~2)) r e s p e c t i v e l y , a n d l e t $ = z 1 - z v W e h a v e

~ ' = P ~ - 2 b x + a ,

~ ( 0 ) = 0 ,

P=zl+z 2,

w h e r e a = ~1 - ~2, b =/~0 (0~1) -- 80 ((X2)' a n d t h i s g i v e s

X X

~=ee'f (-2bx+a)e-Pidx, Pl= f Pdx.

0 0

A s x - > o o ,

P=2x+O(1), Pl=x~+O(x).

I f n o w % > ~ a n d so a > 0 , ~ will t e n d t o oo l i k e e e' u n l e s s b > 0 . Since

~ = O ( 1 ) , b > 0 a n d /~0(~) is i n c r e a s i n g .

N e x t , a g a i n w i t h ~1 > ~ , l e t ~ = z I (x) - z 2 (x +/~1 -/~2) w h e r e we write/~1.2 =/~0 (~1, ~),

7 1 , 2 = 7 0 ((Xl, 2)" W e f i n d

~'=P~1+(71-72), P=Zl(X)+Z2(X+fll-fl2),

if:

0

W e h a v e a g a i n P = 2 x + 0 (1), P1 = x2 + 0 (x). A l s o ~ (0) < 0 a n d we shall h a v e 7 - + - oo, w h i c h is false, u n l e s s 7 1 - 7 2 > 0 ; 70(x) is incresing.

I n (iv) C~ c a n n o t cross t h e h y p e r b o l a z 2 - x 2 - 2 80 x + 1 + ~ = 0, since i t s

dz/dx

w o u l d t h e r e a f t e r b e n e g a t i v e , a n d

dz/dx>O

for x>_0; also

dz/dx-->l

a s

x-->oo, dz/dx

h a s

i

a p o s i t i v e m i n i m u m A (a), c o n t i n u o u s in a, a n d t h e d e s i r e d r e s u l t s follows.

w 12. " L i n k a g e of

v, co, V

a t 'U f o r a s e t t l e d t r a j e c t o r y " . 1 This, in full d e t a i l , a n d for g e n e r a l ], g, p, is o u r n e x t t a s k .

T h e r e a r e in p o i n t of f a c t t w o d i s t i n c t s e t s of c i r c u m s t a n c e s in w h i c h we n e e d t o e s t a b l i s h " l i n k a g e " a t U on y = 1 b e t w e e n v a n d o~ ( a n d V, w h i c h is a c o m b i n a t i o n of v a n d co). One, d i s c u s s e d a t l e n g t h in t h e I n t r o d u c t i o n , is t h e case of a r r i v a l a t U a f t e r a " l o n g d e s c e n t " t o y = 1, w i t h p o s s i b l e dips. H e r e we e s t a b l i s h n o t o n l y t h e linkage, b u t (from L e m m a 3) also a n u p p e r b o u n d for I wl (one of o r d e r k - t ) . T h e o t h e r b e c o m e s i m p o r t a n t o n l y m u c h l a t e r . I n this, o n t h e one h a n d n o t h i n g is a s s u m e d a b o u t t h e p r e v i o u s h i s t o r y of t h e t r a j e c t o r y e a r l i e r t h a n a t i m e k -89 log k b e f o r e U; o n t h e o t h e r h a n d we a r e

given

t h a t eo is of o r d e r / c - 8 9 W e g i v e a s e p a r a t e

1 Cp. Introduction w 11, 12.

THE GENERAL EQUATION y + ]C l (Y) Y + g (Y) = b k p (~), ~ = t + r 19 L e m m a for each case; when we come to proo/s, tiowever, it is n a t u r a l to establish first the restriction on eo in the first case, after which everything reduces to proving the second case (where the co-restriction is a hypothesis).

I n dealing with linkage, we n a t u r a l l y transform our variables v, to (as in the Introduction) to p a r a m e t e r s m, fl appropriate for the application of L e m m a 5. The s t a t e m e n t of the two L e m m a s is further complicated b y (i) the necessity of working with undetermined d, d'; (ii) the need for a specific error.term in the "linkage".

We set out first some p e r m a n e n t notation. F o r a trajectory (in the first instance otherwise unrestricted) arriving a t U on y = 1 from above, let

~t1= _ 1 ~ _ ~ , ]o~[~<az; --~lv=v; V = v + b l c ( l + p i ( ' l ~ - t - o ) ) ; (1}

and let the change of variables to ~, fl be defined b y

fl=2- 89189 ( - P ( - l g - ~ ~ / (2}

l + o t = v / v * , V*=v*=al 89189 j rio(m) is the function of L e m m a 5.

We h a v e now

1 < <_ 2, and let d be a non.negative, and d ' a positive constant.

L E M M A 6. Let ir

Suppose that an eventual trajectory F satislies the two /ollowing sets o I conditions (A) and (B):

(A) it ends with a piece W U (U on y = l ) lying in y>_l and of time-length at least k-~ log k.

(B) W U is preceded by a piece X Y W ; the whole o/ X W (and so o/ X U) is in y >_ 1 - d k - 8 9 X Y has time-length at least d'; and Y U contains a point at which q~=_ _ i n .

I / now /urther k>_ko(d,d'), then we have upper bounds (tot v , w , V , ~ , f l ) as /ollows, in which A is an abbreviation /or A (d, d'):

(a) Io~l<A~-%

(b) 0 _ < v < V_<A;

(e) I ~ [ < i

(d) 0 _ < I + ~ < A .

1 We de]ine V* by V* = v*.

20 z. •. ~vrLEWOOD And we have linkage given (in terms o/ ~, fl) by

(e) fl =rio(a) + 0 (k-89 log A k). 1

We note /or convenience the asymptotic relations 2 (/or k large, eo small)

(f)

V = V * ( l + a + f l o ~ ( ~ ) ) + O ( h k - t l o g A b). I

Finally we have (/or re/erence)

(g) V* = v*, L < V* < L.

L ~ M M A 7. The conclusions o/ Lemma 6, with d absent from k o and A's, are valid (in form s) when (B) is replaced by

(B')

Io~l<_d' k-~.

W e prove first (a) of L e m m a 6. I n L e m m a 6 L e m m a 3 (4) is valid for Y U, so t h a t for points Y U

F (y) - F (1) = C + b (1 + Pl (~0)) + 0 (A It-l). (3) NOW Y U contains a p o i n t S where ~0-- - ~ , a n d so 1 +p(~0)=0; also F ( y s ) - F ( 1 ) >_ 0;

hence C > - A / c - i , a n d t a k i n g y = 1, ~0=~0v in (3), we h a v e b (1 + Pl (~0v)) < - C + 0 (A k -1) < 0 (A k-l), a n d so f r o m L e m m a 2 (4) 4

o~ 2 = 0 ( A

k-l),

as desired.

w t 3 . E v e r y t h i n g n o w reduces t o p r o v i n g L e m m a 7; for in L e m m a 6 we h a v e p r o v e d (a), i.e. condition (B') is fulfilled with A (d, d') for d', a n d this leads t o t h e s a m e final results. Our a r g u m e n t s are n o w based on (B') a n d t h e f a c t t h a t t h e r.m.

f r o m U does n o t go outside 1 _< y_< L* within a time k- 89 log k.

I n t h e O-identity for t h e direct m o t i o n (d.m.) f r o m U, viz.

t

= - v - k ( F (y) - F (1)) § b k (p~ (q:) - p, (q)v)) - / g dr,

t u

1 W e do n o t a i m a t b e s t p o s s i b l e p o w e r s of l o g k i n t h e e r r o r t e r m , t h e m o r e so t h a t w e c a n a b s o r b a f a c t o r A b y c h a n g i n g t h e A.

2 T h e s e a r e s t r a i g h t f o r w a r d c a l c u l a t i o n s f r o m (a) . . . (e), a n d t h e p r o p e r t i e s of t h e f u n c - t i o n s p, Pl.

a d ' h a s a n e w m e a n i n g i n L e m m a 7, a n d d d o e s n o t o c c u r . 4 T h e s p e c i a l a s s u m p t i o n a b o u t Pl is i n v o l v e d .

THE GENERAL V.QUATIO~

i]+kf(y)?)+g(y)=bkp(cp), q~=t+(x

21 we write t = Tv--T to obtain the r.m. with t i m e variable zero a t U. This gives

d ~ = ~ + k (F(y) - ~ ( 1 ) ) - _{b k} (Pl ( _ 1 ~ _ ~ _ 3 ) - pl ( - ~ - ~ ) ) + ₀ (3)

=v+ k (F(y)- F (1)) + blcp ( - ~ r - o ~ ) ~ - l b kp' ( - l ~-o~) ~ + O (k~s) + O(v),

with y ( 0 ) = 1, or ~ ( 0 ) = 0. I n this we write

~ = y - l = c k - 8 9 ~;=Tk-89

where c, F are given in t e r m s of the f u n d a m e n t a l constants b y

l c T a t = 1 ,

~Tac-tba~=l,

a n d t h e n write

~ + 1 = 7 c - l v ,

f l = 8 9 1 8 9 -1,

which yield the values of w 12 for ~, /~.

The result of t h e substitutions is

dz p , ( _ l ~ _

dx=l+~+,p(z) p , ( _ ~ : r ; ) x ~ - 2 ~ x + O ( k - ~ x 3 ) + O ( k - ~ x ) ,

where

~p(z)=~p(z,k)=~,c-lk(F(y)-F(1)).

Since

w=O(Ak- 89

the coefficient of x 2 is - l + 0 ( A k - 8 9 Thus t h e r.m. f r o m U, in (z,x) form, is

dz d--~=l+~+~p(z)-x~-2flx+xe(x),

z(0) = 0 , (1)

where, over the range 0 _ x _ ~ -1 log k,

(x) = 0 (A k-89 (1 + x + x~), (2)

a n d /3 = 0 (k~ ~o) = 0 (A). (3)

9 The solution z is finite and non-negative in the range, y satisfies 1 _<y_< L*, and so, b y L e m m a 2 (8), ~o satisfies

a n d

F r o m this state of things results of the L e m m a .

We begin b y proving (which is (d) of the L e m m a ) .

~0 = z 2 + 0 ( k - ~ za), (4)

L1 z2 ~ ~p (z) ~ .L2z 2.

(5)

[and for suitable

ko(d ,

d')] we have to deduce the

I + ~ < A (6)

22 J. E. LITTLEWOOD F o r x_< 1 (and suitable k e (d, d')) we h a v e

] - x ~ - 2 f l x + x e ( x ) ] < I + A + A k - ~ < A 1 . Suppose n o w t h a t 1 + a > A l + l ; t h e n f r o m (1)

d z > l + v 2 (z) > l + L2z2;

d x

b y L e m m a 4 z is a b o v e t h e solution of d z / d x = LszS+ l, which has a n a s y m p t o t e a t x= 89189 This n u m b e r is less t h a n 1 if l is a suitable chosen L, a n d we h a v e t h e n a c o n t r a d i c t i o n w i t h "z < oo (0 < x < 1)". H e n c e 1 + ~ > Ax + 1 implies l < L, a n d this p r o v e s (6).

F o r O < x < ? - l l o g k (and suitable /co) we h a v e

[1 + c r s - 2 f l x + x e (x)l < A + ?-Slog 2 k + A log k + A k - 8 9 log a k < 2 7 -s log s k.

d z

H e n c e d x = ~ + 2 v~ ?-s log~ k (0 _<_ x _< ? - s log k), (7) where Iv ~ ] _< 1.

W e p r o v e n e x t t h a t in t h e s h o r t e r r a n g e 0_<x_<~ ? - s log k d z

d-x < l~ k. (8)

F o r suppose not, so t h a t d z / d x = l o g a k for t h e first t i m e a t a n x = ~ satisfying

_ ~ ? log k. Consider n o w t h e r a n g e f r o m ~ t o ~1, where ~1 is either ? - 1 log k, or else t h a t x > ~ a t which first d z / d x = O , w h i c h e v e r is least. I n (~, ~l) Z is non-decreasing

a n d s o

d~>_ ~ - 2 ? - 2 logs k >_ L~ - 2 ? Z s ~ 2 log s k >_ L s z ~ (~) - 2 ? - 2 log s k = ( L j L ~ ) (L lz 2 (~) - L log s k) > 0, since LlZS(~)>~;0(~) = ~ - 2 0 ? - S l o g S k = l o g 3 k - 2 v q e ? - 2 1 o g S k > ~ l o g a k . (9) H e n c e t h e a l t e r n a t i v e d z / d x = O does n o t h a p p e n first, so t h a t ~ 1 = ? -s log k. T h u s in (~, ? - t log ^{k )}

z > z ( ~ ) > L l o g ~l' k, d ~ > L s - 2 ? - S l o g S k ^{Z s} > 1 L - - ~ . ~ S z-* ~

lo~ [ d z L

~ ] o g k < _ l o g k - ~ = j d x < t ~ < - - < l

J ~Lsz z(~)

z ( D

b y (9). This being false, we h a v e e s t a b l i s h e d (8).

THE GENERAL E Q U A T I O ~ ff -4- k f (Y) Y + g (Y) = b k p ((p), (p = t + r162 23 For t h e range 0 < x_< 89 ^{~ - 1} log ]~ ^{w e n o w} h a v e I z I < t 7 -1 l~ 4 k, b y (8); also

[el(X)l<Ak- 89

b y (2). F r o m these a n d (1), (4) t h e z, x e q u a t i o n now becomes

d z

d ~ = l + ~ q - 2:2 - - X2 - - 2 / ~ X - l - Z 81 ( X ) , Z ( 0 ) = 0 ,

[ e 1 (x) I < A k - t log 12 k < k - 89 lo g A k.

L e t 0 = f l - fl0 (~), let ~ = ~ (x, ~) be the solution in 0_< X ~ 89 7 -1 log k of

d $ ~z

d ~ = l + c r -- x~-- 2 flo (:r x, ~(0) = 0 ,

and let

u = z - ~ .

We shall prove t h a t ]0]_<2 k-89 logA'k, t h e r e b y establishing t h e re- maining result (e) of t h e L e m m a . Suppose t h a t , on t h e c o n t r a r y , 101> 2 k - t log A'

k,

a n d suppose first t h a t 0 is positive. T h e n 2 0 - el ( x ) > 0. N o w u satisfies

d u

d-x = u (~ + z) - (2 O--e 1 (x)) x, u (0) = O, a n d b y L e m m a

4 u< w,

where

d w

d-~=w(r +z)-Ox,

^w(0)-=0,

X X X

and so

w=-Oexp(f(r fxexp(-f(r

⁽¹⁰⁾

0 o 0

Now, b y L e m m a 5 (iii), I $ - x l = l f l 0 ( a ) + F ( x , a ) l < A , since - I < ~ < A , a n d b y (8) we h a v e 0 < z_< x log a k. H e n c e

x x x x

f x e x p

( - - f ( : + z ) d x ) d x > y x e x p

( - f ( x + i + x l o g 3 k ) d x ) d x

0 0 0 0

x

= f x exp ( -- A x - ~ (1 + log S

k) x 2) dx

0

> A log -S k (11)

for x = 1 a n d therefore for x > 1. So for x > l we h a v e from (10) ] w [ =

--w>O

exp ( f ~ d x ) . A l o g - 3 k,

0

]w] _> (2 k -89 log a' k). exp ( i x ~, _ A x ) - A log -3 k (1 < x < ~ ~- t-1 log k). (12)

~ 4 J . E . LITTLEWOOD

On the other hand,

I w l = - w _ < - u = < x + A + x log a k. (13)

(12) and (13) are incompatible (for a suitable k0) when x = 89 7-1 log k, and the assumed inequality for 0 is false.

In the case of negative 0, assuming 0 < - 2 k -89 loga'k, we have w non-negative, u>~w, and so w _ < z - $ _ < z + l ~ ] , and the rest of the argument is the same.

This completes the proof of L e m m a 7 (and L e m m a 6).

w tt~. L ~ M ~ A 8. ("Dip or shoot-through at a U".) Let ~ < _ b < 2 . Let the piece W U o/ F satis/y the conditions (A), (B) o/ Lemma 6. Abbreviate constants A (d, d', 6) to A. t

(i) Suppose V-> V* + ~; then /or k-> k o (d, d', ~}) the d.m. /tom U 1 shoots through and reaches 2 y = - l ( l + H ) in time at most A k -89 Up to this moment we have

- 3 - > V* > L, and

-~) = v + k (F ( y ) - $' (1)) + 0 ( A ) ;

and /inally the velocity o~ arrival at y = - ~ (1 + H) satisfies - ? ) > L k.

(ii) Suppose V ~ V * - ~ ; then /or k->ko(d,d',~), (a) the d.m. /rom U makes a dip o/ depth Ak- 89 at most below y = l , emerging at time Ak- 89 at most later. I t then {b) pursues approximately the curve C1, the branch o/

F ( y ) - F ( 1 ) = b (1 + Pl (~))

lying in y_>l, and (c) i/ F has been above y = l - d k -89 /or a time 3rl be~ore u it arrives at y = 1 again at a time approximately 2 ~ later.

In either case F satisfies, up to its arrival at U, the hyl0otheses, and therefore the conclusions, of L e m m a 6.

The d.m. from U, taking t = 0 at U, is

- ?) = v + k ( F (y) - F (1)) - b k (Pl (q~v + t) - pl (~v)) + g~

= ( v + k ( F ( y ) - F ( 1 ) ) ~ - b l c { p ( - ~ - o J ) t + ~ p ' ( - l r e - m ) t 2 ~ + O ( k t a ) + O ( t ) . (1)

= (v + k ( F (y) - F (1))} + 0 (k o~ t) + 0 (k t 2) + 0 (k t a) + O (t). (2)

1 I n a p p l i c a t i o n s A b e c o m e D ' s . T h e b l a n k c h e q u e s d, d ' a r e s t i l l i n v o l v e d , v i a t h e h y p o t h e s e s

(A), (B).

W h a t w e do ( w h i l e w e a r e a b o u t it), is t o f o l l o w t h e s h o o t - t h r o u g h u p t o a p o i n t a d i s t a n c e L below y = - l : t h i s is a m o r e c o n v e n i e n t p l a c e t h a n y = - 1 f o r t h e n e x t s t a r t i n g p o i n t .

THE GENERAL EQUATION y § k f (y) ~ + g (y) = b k p (~0), ~0 = t + 25 L e t c, ~, ~, 8 be t h e n u m b e r s a n d ~(z) t h e f u n c t i o n of w 13, a n d write y = 1 - c k - 8 9 t=~,k- 89 (1) t h e n gives (with a n e(x) different f r o m t h a t of w

d x - l + ~ + ~ ( - ~ ) - x S + 2 8 x + x e ( x ) , ~ ( 0 ) = 0 , ( (3) e (x) = O ( h k -89 (1 + x + xS). ]

(that is, formally, (1) of w 13 with - $ for z a n d - 8 for 8).

Case (i). V_> V* + d. B y L e m m a 6 (f) we h a v e a + 8~ (cr > L ~, a n d so, b y L e m m a 5,

> A (~) a n d 8o (~) > A1 (5). Since

18- o( )1 < log a k < (6), b y L e m m a 6 (e), we h a v e f l > A ((~).

Consider n o w (3) for t h e r a n g e of x after 0 to t h e value for which (for t h e first

1 1

time) ~ = 0, or I E I = k~, or x - / c ~ , whichever h a p p e n s first. I n this range 2 fl + e (x) > 0 _ ! + 3

a n d y J ( - ~ ) = ~ 2 + 0 ( k 2 10), a n d so

d S>l+89

dx B y L e m m a 4 $ ~> w, where

dw d - - x = l + ~ + w S - x 2 , w ( 0 ) = 0.

B y L e m m a 5 w>_0 a n d w has an a s y m p t o t e t o + ~ a t x = x o ( a ) < A2(5). H e n c e t w o 1

of t h e a l t e r n a t i v e s fail, a n d ~ reaches t h e value + k~ before x = A 2 (5) a t most, which 1

corresponds to t = A k- 89 a t most, a n d t h e n - ~ = (c/y) d ~/d x > L ~2 _ A > L kg. F u r t h e r

~_>x, since ~ is n o t less, b y L e m m a 4, t h a n t h e solution of d u

d-x= l + u 2 - x 2, u ( 0 ) = 0 ,

which is u = x ; hence d $ / d x > 1 + ~ > 1 throughout, equivalent to -~)> V*.

R e t u r n n o w t o (1). We h a v e - ~ > V * u p t o a time t l < A k - - ~ , a n d at t=tl,

1 1

y - 1 = - c ki~-~. Consider t h e range f r o m t = t 1 until either - y = V*, or y = - 1 (1 + H), or t - t l = k - ~ , whichever h a p p e n s first. I n this range (2) gives

( - ~) - {v + k ( F (y) - F (1))} = O (h), (4)

since w = 0 (Ak- 89 I n particular

26 J. E. LITTLEWOOD

- y > k ( F ( y ) - F ( 1 ) ) - A I > L ~ k (1 _ y ) 2 A1

b y L e m m a 2 (8). F u r t h e r - # > V * > 0 a n d k ( 1 - y ) 2 > _ k ( 1 - y ) ~ = t = L k ~, a n d so

- y > i L2k (1-y)2. (5)

1 1

N o w this m o t i o n , i/ uninterrupted, m a k e s y go t o - o o in t i m e O ( k - ~ - ~ ) with - y > V* t h r o u g h o u t . W e infer t h a t of t h e t h r e e a l t e r n a t i v e s it is y = - -~ (1 + H ) t h a t h a p p e n s first, a n d in t i m e a t m o s t ( A + l ) k - 8 9 a f t e r U, a n d then, b y (5), - # > L k . This completes t h e proof of case (i).

w t5. Case (ii). V < V * - ( ~ . Much of this is parallel t o case (i). B y L e m m a 6 (f) we h a v e ~ < - A ( ( ~ ) , so t h a t , b y L e m m a 5, r i 0 ( a ) < - A 1 ( 5 ) ; also 1fl-flo(~)[< 89 a n d so fl < - ~ A 1 ((~). Consider t h e ~, x e q u a t i o n [(3) of w 14] for t h e r a n g e of x a f t e r

1 1

0 t o t h e value for which (for t h e first time) ~ = 0 , or [ r or x = k~, w h i c h e v e r _1+__8

h a p p e n s first. I n this r a n g e 2 fl + ~ (x) < 0, a n d yJ ( - ~) = ~2 + 0 (k ~ 10), a n d so d ~ < 1 - 4 - 8 9 1 6 2 2.

B y L e m m a 4 ~ < w, where

dw T x = l + 8 9 w(0) = 0 .

B y L e m m a 5 (since a < 0) w, initially positive, b e c o m e s n e g a t i v e a t x = A (~) a n d is b o u n d e d b y a n A (~r before this point. W e infer t h a t obvious a l t e r n a t i v e s fail, a n d t h a t t h e d.m. f r o m U ' m a k e s a dip, as described in (ii).

w L e t t h e dip e m e r g e a t U', w i t h yv,=v'>-O. 1 i W e t a k e t = 0 a t U ' , a n d w e h a v e n o w to discuss t h e d.m. f r o m U', for which

N o w for t < k - ~

9 = v' - k ( F (y) - F (1)) + b k (pl (~v" + t) - P l (~gu')) - gl"

b k (pl (q~u. + t) - p l (q~v.) ) = b kt p ( - 89 ze + O),

where 0 = - ~o+ (q~u,-qpu)+v~t, which is (a) small, a n d (b) g r e a t e r t h a n - c o , which is positive with - f l ( L e m m a 6). Since p ' ( - - ~ z t ) is positive,

b k p ( - ~ze + O ) > _ b k p ( - ~ z t - e o ) = L b ~ ]fll k89 Ak89

1 The dashes in U', v' are t e m p o r a r y n o t a t i o n only, inside the proofs, a n d while we are dealing w i t h dips.