Proceedings of the Third International Colloquium on Differential Equations, Plovdiv, Bulgaria, 18–22 August 1992 9783112318928, 9783112307656

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Proceedings of the Third International Colloquium on Differential Equations

Proceedings of the

Third International Colloquium on Differential Equations Plovdiv, Bulgaria, 18-22 August 1992

Editors: D. Bainov and V. Covachev

///VSP///

Utrecht, The Netherlands, 1993

VSPBV P.O. Box 346 3700 AH Zeist The Netherlands

© VSP BV 1993 First published in 1993 ISBN 90-6764-153-7

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed, in Great Britain by Dookcraft (Bath) Ltd., Midsomer

Norton.

CONTENTS

Preface Time delay models of population growth with stage structure W. Aiello

vii

1

Sufficient conditions for point dissipative quadratic nonlinear dynamical systems A. K. Bose, A.S. Cover and J A. Reneke

13

Oscillatory properties for damped hyperbolic equations with deviating arguments D.D. Bainov and C. Baotong

23

Initial value problem for a singularly perturbed nonlinear impulsive system in the critical case V. Covachev

31

Recent developments on numerical integrators for curves on prescribed surfaces J.M. Ferrdndiz and M.T. Perez

45

On the contact angle in capillarity R. Finn

55

Kernel asymptotic of exotic second-order operators S.A. Fulling

63

Self adjoint differential equations and Karamata functions L.J. Grimm and L.M. Hall

11

Ordinary differential equations in Frechet spaces G. Herzog and R. Lemmert

89

Entire solutions of nonlinear parabolic equations G.N. Hile and C.P. Mawata A class of weighted-mean Banach spaces T.L. Gill and W.W. Zachary Global existence of holomorphic solutions of differential equations with complex parameters J. Kajiwara

95 105 119

On the Riemann-Hilbert problem for the one-dimensional Schrödinger equation M. Klaus

133

Optimal control for nonlinear systems of partial differential equations related to ecology A.W. Leung

147

Stokes multipliers for a certain n-th order differential equation T.K. Puttaswamy

157

Supersymmetric quantum mechanics and multi-soliton solutions of higher order K-dV equations C.V. Sukumar

169

On integral and differential equations arising from probability distributions K. Takano

177

Singularities for Monge-Ampere equation M. Tsuji

193

Oscillation properties of solutions of arbitrary order neutral differential equations A. Zafer and R.S. Dahiya

205

Behaviour at infinity of solutions of systems of differential equations with delay J.Diblik 219

Preface The Third International Colloquium on Differential Equations was organized by UNESCO and Plovdiv Technical University with the help of the Austrian Mathematical Union, the Canadian Mathematical Society, Hamburg University, Institute of Mathematics of the Bulgarian Academy of Science, Kyushu University, the London Mathematical Society, Plovdiv University 'Paissii Hilendarski', Technical University-Berlin, Union of the Bulgarian Mathematicians-Plovdiv Section, and sponsored by Bulgarian Mails and Telecommunications Ltd., and the firms 'Eureka' and 'Microtechnika Ltd.' It was held on August 18-22, 1992 in Plovdiv, Bulgaria. Over 200 participants from 43 countries took part in its work. Round tables were organized on 'Impulse Differential Equations', Non-linear Differential Equations' and 'Differential Equations with Maxima', resulting in fruitful discussions among their participants. The subsequent colloquia will take place each year from 18-22 August in Plovdiv, Bulgaria. The address of the organising committee is: Stoyan Zlatev, Mathematical Faculty of the Plovdiv University, Tsar Assen Str. 24, Plovdiv 4000, Bulgaria. The Editors

Third International Colloquium on Differentia! Equations pp. 1-12 (1993) D. Bainov and V. Covachev (Eds) © 1993

TIME DELAY MODELS OF POPULATION WITH STAGE STRUCTURE

GROWTH

WALTER AIELLO

Department of Mathematics, University of Alberta Edmonton, Alberta, Canada T6G 2G1 ABSTRACT

Models of population growth incorporating immature and mature stages of growth are investigated. In particular, a comparison is made between single species models of populations where the time to maturity is a fixed constant, and where the time to maturity is a function of population concentration.

INTRODUCTION

Modelling single species growth has a long history in the area of mathematical ecology, where modern developments can be said to have started with the work of Malthus in 1798. The logistic equation, i ( t ) = rx(t) plays an important role, where x(t) tion,

r

t> 0

is the concentration of individuals in the popula-

is an instrinsic growth rate, and

k

is the environmental carrying capacity.

Solutions of the logistic equation are monotone functions of time decreasing to the limit

x(t) = k

t,

increasing or

if x(0) > 0. The literature on this subject is quite

extensive, for example see Freedman [1], The existence of features such as gestation in natural systems lead investigators to introduce time delays into the logistic equation. One form of the logistic equation with time delays, z(t) = rx(t) ( 1 ~ * ( * ~ r ) ) is known as Hutchinson's equation.

Here

r, k,

0 and

r

are positive constants.

Hutchinson's equation has been investigated by May [2], Kakutani and Markus [3]

2

Waller Aiello

and by Zhang and Gopalsamy [4]. In our investigations, we consider a population that is divided into immature and mature stages of growth. The equation for this model thus appears in two parts, one for the immature stage and one for the mature stage of growth. CONSTANT TIME DELAY MODEL - DEVELOPMENT When the time to maturity remains constant, the model takes on the form, ¿,( 0

(1)

px2M(T).

is the intrinsic birth rate of the population,

7

is the intrinsic death rate

of immatures, and ¡3 is the logistic death rate of matures. For 0 < t < r, xm(t — r) takes on the value of an initial function {t) that is defined on the interval

[—r, 0].

To assure continuity of initial conditions we assume that,

Thus we see that introduction of new members into the population is by birth of new immatures, and that those immatures born at time

t—r

who survive to time t

leave the immature population and pass into the mature population. Assuming the initial function to be continuous (for mathematical reasons) and positive (for biological reasons) we prove that for system (1) the following are true: 1) Solutions

(xi(t),

xm(t))

exist and axe uniquely continuable,

2) Solutions

(xi(t),

xm(t))

are positive for all

3) Solutions

(xj(t), xm(t))

t,

axe bounded.

The proof of 1) follows from the theory of existence and uniqueness of delay differential systems as outlined in Driver [5]. Properties 2) and 3) are proven in Aiello and Freedman [6]. EQUILIBRIA A N D STABILITY FOR CONSTANT DELAY CASE There axe two possible equilibria, one at the origin, E(xi,xm).

In Aiello and Freedman [6] it is shown that

-EQ(0; E

0),

and in the interior,

exists, and that

x,

and

Models of Popular

xm

3

Growth

satisfy, Si = a20-1 xm

=

The variational system about (A + 7)(A —

= 0.

J'16-^(1

-

« r ' e - ^ .

EQ gives us a characteristic equation This equation has at least one real, positive eigenvalue

and at least one real, negative eigenvalue. Thus Concerning the interior equilibrium

EQ is a "saddle" point. in Aiello and Freedman [6], it is

E(xi,xm),

proven that this equilibrium is globally asymptotically stable. S U M M A R Y OF C O N S T A N T D E L A Y C A S E We have shown that the model for a population with two growth stages and fixed time delay has a globally asymptotically stable interior equilibrium. The results of a computer simulation of several solutions with different initial conditions is illustrated in figure number 1. STATE D E P E N D E N T T I M E D E L A Y M O D E L - D E V E L O P M E N T Observation of the population dynamics of certain mammalian species such as seals and whales in the Antarctic Ocean appears to indicate that the time to maturity is shortened with decreased populations and the resulting increase in food supply that becomes available to each member of the species. Hence we consider a model of single species growth where the time to maturity is a monotone nondecreasing function of members of the species present. Thus for our model we apply the system, ii(t) x

where 0,

xm(t)

RM < T(Z)

m

( t )

= axm(t) =

A E - r t * )

— tpm(t) > 0 < TM.

Here

-

for z{t)

7®i(0 - a e " 7 r ( j ; ) x m (t X m

(

t

-

T(Z))

-

—rm < t < 0,

= XI{t)

+

r(z))

(2)

0x2m(t)

and where

a > 0, fi > 0, 7 >

xm{t).

The variable delay system is similar to the previous constant delay model except that the time delay is now a function of z(t)

rather than a constant. The time delay

Walter

4

Figure 1.

Aiello

Globally Asymptotically Stable Equilibrium

Models of Popular

function

r(z)

lim We further assume that m

5

is assumed to be nondecreasing, and it is further assumed to satisfy, lim t(z) i-»0+

t

Growth

dr/dz

= rm > 0

t(z)

= tm


0 , 0 < r m < t(z)
8m

for all

such that

Thus if the initial function

xm(t)

> 0

t > 0.

for all

Then there exists a Sm = 6m(tpm)

—r m < t < 0.

< Am

xm(t)

Then

>

t > 0.

> O for

ipm(t)

0.

-rM 0. 0 < Sm( — rm

of the state

dependent

system.

Then,

ti < t < t\ + TM,

xm(t)

< a/3~1e~'yTm

for

xm(t)

> a/3_1e_TrM

for

for all such that

> a/3~1e~'r™

for all

t2 < t < t2 +

tm,

t > tx.

This immediately leads to the following corollary: COROLLARY. If the initial xm(t)

< otP~1e— 0.

L]J

-0J "1"

1

1]A

0

0

If u is in Z f then either u =

L -0-

_ 0

or u = _

U

-

U

. Thus

2

3 -

0 Zr n B

1

< = ]

0 _0 _

"U2 -

U

u,

0, u 2

0

2

0 NOW[U 2

0

0]A

= - u * < 0 a n d [ 0 u2

u2]A -ii.

if u is a nontrivial element of Zf n B± then u T A u < 0.

=

- 3u* + Ug = -

i s in B 1 and

Since L

1 1 -1

is in B

and (2

-1

1]A

< 0, i.e.,

1J

= 1 > 0 T h e o r e m 2 does not apply.

Again

Point dissipativi

0

1 L

non-linear

dynamical

19

systems

0 is in Z f a n d [0

1

1 JA 1

5 > 0 and so T h e o r e m 1 d o e s not apply. If we c h o o s e K

Li-

1 -

•1 -2BT then

A + B K = A - 2BB

1

3'

2 - 1 0

=

L

3

-1

N o t e t h a t A - 2BB

T.

is n o t n e g a t i v e

-1. 0

definite, b u t

0

0](A-2BB

= -u* < 0. T h e r e f o r e if U i s in Z

f

)

= - u* < 0 and [0

-u

u ] ( A - 2BB )

-u„

t h e n u T ( A - 2 B B )w < 0. H e n c e x' = (A - 2BBT)X + f ( x ) is

point dissipative. In figure 2 two trjectories are given. O n e w h e n the s y s t e m is u n c o n t r o l e d , x = 0, and o n e w h e n the system is controled, x = - 2. For both trajectories (- 2, - 2, 2) is the initial point.

[ - 1 1 dx dt

3

2 1 2

"0" +X

1

0 [0

1

1 ] X + - xz

[ 3 1 1

XY 4

Figure 2

20 4.

Anil K. Bose el al.

Extension of previous results

T h e sufficient condition for a quadratic dynamical system to be point dissipative discussed above uses a relation between the quadratic and linear parts of the system when f is conservative. T h e following l e m m a allows us to extend the condition to the case where there exists a positive definite matrix S such that S M = MS and Sf is conservative. See I 4 ]. L e m m a 6. Let M x ' = N + Ax + f(x) (2.1) be a quadratic dynamical system f o r which there exists a positive definite matrix S such that S M = M S and Sf is conservative. T h e n there exists matrix H for which the c h a n g e of variables y = Hx transforms the dynamical system (2.1) into ( H M H _ 1 ) y ' = H N + By + g(y) which has a conservative quadratic term. Furthermore, x T S A x = y T B y . When Sf is conservative then we say that a vector a is a d m i s s i b l e for the dynamical system M x ' = N + Ax + f(x) if - x T S A x + a T H f(x) is a positive definite function where S = H T H . When f is conservative then the condition for an admissible a reduces to - x T A x + T a f ( x ) is positive definite. T h e proofs of the theorems when the quadratic part of the system is conservative entail demonstrating the existence of an admissible a for the system. T h e s e results can be restated as follows: T h e o r e m 4. A quadratic dynamical system M x ' = N + Ax + f(x) is point dissipative when there exists a positive definite matrix S such that SM = MS and Sf is conservative and there exists an admissible a . Another direction of generalizing the past results is to consider nonlinear d y n a m i c a l systems which have nonquadratic nonlinear terms as well as quadratic terms. Relative to the nonlinear terms there again must exit a positive definite matrix S such that the nonlinear terms premultiplied by S are conservative. See [ 4 J. T h e o r e m 5. W h e n there exists a positive definite matrix S such that S M = MS, Sg and the quadratic function Sf are conservative, and Condition (A) For some admissible a for M x ' = N + Ax + f(x) there exists an ordered triple of numbers (e, C, M) such that - a T H g ( x ) < C llxll2"£ for all x with llxll > M. then M x ' = N + Ax + f(x) + g(x) is point dissipative. Note that condition (A) can be replaced by either of the stronger conditions Condition (B) There is an admissible a II g II = o II x II 2 . Condition (C) There is an admissible a

(B) or

(C).

for Mx' = N + Ax + f(x) and

for M x ' = N + Ax + f(x) and g is

bounded.

Point dissipative

Theorem quadratic (1) (2) and (3)

non-linear

dynamical

systems

21

6. A quadratic dynamical system Mx' = N + Ax + f(x), A a matrix and f a function, is point dissipative when there exists a positive definite matrix S such that SM = MS and Sf is conservative, the zeros of f contains an (n - l)-dimensional hyperplane, Z T SAZ < 0 for any z which is a nontrivial zero of f.

This result can be generalized by adding to the differential equation any conservative function g(x) whose growth is restricted. The corollary states this condition. Corollary. Let g and the quadratic function f be conservative. If Z(f) contains an (n - 1)hyperplane and x * 0 in Z(f) implies x^A x < 0 and Condition (A) For some admissible a there exists an ordered triple of numbers (e, C, M) such that - a T g(x) < llxll2"6 for all x with llxll > M. then Mx 1 = N + Ax + f(x) + g(x) is point dissipative. Note that condition (A) can be replaced by either of the stronger conditions (B) or (C). Condition (B) There is an admissible a for Mx1 = N + Ax + f(x) and II g II = O II X II 2 . Condition (C) There is an admissible a for Mx1 = N + Ax + f(x) and g is bounded. 5. Conclusions Quadratic and "almost" quadratic nonlinear systems can exhibit a wide range of qualitative behavior. Even the subclass of such systems with compact attractors contains systems with point attractors, limit cycles and strange attractors [ 6 j. Linear feedback control problems with system objectives of steering to desired limit points or of minimizing the diameter of a compact attractor have yet to be formulated and solved. This paper represents only a first step, i.e., using linear feedback to produce a controlled system with a compact attractor. References 1. A. K. Bose and J. A. Reneke, Sufficient conditions for two-dimensional point dissipative systems, International Journal of Mathematics and Mathematical Sciences, 12(1989), 693696. 2. A. K. Bose, A. S. Cover, and J. A. Reneke, On Point-dissipative Systems of Differential Equations with Quadratic Nonlinearity, International Journal of Mathematics and Mathematical Sciences, 14(1989), 99-110. 3. A. K. Bose, A. S. Cover, and J. A. Reneke, A class of point dissipative n-dimensional nonlinear dynamical systems, Proceedings of the Twenty-Fourth Southeastern Symposium on System Theory, 1992, 2-6.

22

Anil K. Bose el al.

4. A. K. Bose, A. S. Cover, and J. A. Reneke, On point dissipative n-dimensional systems of differential equations with quadratic nonlinearity, International Journal of Mathematics and Mathematical Sciences, accepted. 5. E. Lorenz, Deterministic non-periodic flow, Journal of Atmospheric Science, 20(1963), 13014. 6. C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, SpringerVerlag, New York, 1982.

Third International Colloquium on Differential Equations pp. 23-30 (1993) D. Bainov and V. Covachev (Eds) © 1993

OSCILLATORY PROPERTIES F O R DAMPED HYPERBOLIC EQUATIONS W I T H D E V I A T I N G ARGUMENTS

D.D. B A I N O V 1 and CUI B A O T O N G 2 ^Academy of Medicine, P. 0. Box 45, Sofia 1504, BULGARIA Dept. of Math., B i n z h o u Normal College, Shandong 256604, P.R.

CHINA

Abstract. Sufficient conditions for oscillation of the solutions of damped hyperbolic equations w i t h deviating arguments are found.

Keywords: oscillatory, damped hyperbolic equations, argument.

1.

deviating

INTRODUCTION

Nowadays one can observe a n expanding interest toward the study of oscillations for partial differential equations w i t h deviating a r g u ments. We refer the reader to Mishev and Bainov Georgiou & K r e i t h [4] and Cui

[1, 2], Yoshida

[3],

[5, 6]. However, the oscillations of

solutions of damped partial differential equations w i t h deviating arguments have not b e e n studied till now. This paper gives some sufficient conditions for oscillation of solutions of the damped hyperbolic equation w i t h deviating arguments k m (1) u. . — b ( t ) A u - Y b .(t)Au(x,t-r .) + p(t)u + V q.(x,t)u(x,t-x.) tt j=l ^ J i=l 1 + a ( t ) u t = 0,

(x, t) € [î x (0, co),

where fi is a bounded domain in IRn w i t h piecewise smooth boundary u = u(x, t) and A is the Laplacian in the Euclidean n-space R n . We assume throughout this paper that (i) p(t) 6 C ( [0, oo) ; [0,co)), q^e C(G;

[0,®)), i = 1, 2

G = £2 x (0, oo) and b . , b, a e C([0,o>);

[0,co)), j = 1, 2

m, k.

24

D.D. Bainov and C. Baotong

(11) T. > 0 and r. > 0 are constants and q.(t) = rain q.(x, t) l l = l Jj xsfi (i = 1, 2 m) are not identically zero on any ray [t, oo). Consider two kinds of boundary conditions (B.) 1

3N

+ r(x,t)u = 0

on

an x [0, ®),

where N is the unit exterior normal vector to an and r(x,t) is a nonnegative continuous function on an x [0, oo), and (B 2 )

u = 0

on

an x [0, oo).

Our objective is to present conditions which imply that every solution u(x, t) of (1) satisfying a certain boundary condition is oscillatory in n x [0, oo) in the sense that u has a zero in n x [t, a>) for any t > 0.

2. OSCILLATION FOR THE PROBLEM (1), (B )

Theorem 1. If 00

a2it)

(C 1 )

f

(C 2 )

there exist a € (1, oo) and ¡3 e

[pit) -

]dt = 00, [0, 1) such that

m 1 1 n R lim sup i - J- {(t - y) y Ip(y) + £ q.(y)] t > 00 t tg ' i=l

- I t (t-y)ya(y) + ay -p(t-y) ] 2 ( t - y ) a " V " 2 > d y = oo, then every solution of the problem (1), (B^) oscillates in G. Proof. Suppose to the contrary that there is a solution u(x, t) of the problem (1), (B^) which has no zero in n x [t^.oo) for some t^> 0. Let u > 0 in n x [t^ , co). Then there exists t^ a t^ such that u(x, t - T. ) > 0 and u(x, t - r.) > 0 (i = 1, 2 1 J j = 1, 2, ..., k) for any (x, t) € £2 x [t , co).

m,

and

Integrating (1) over n, we have that (2)

dZ dt

(

J" udx v. n

)

k

- b (t) J"Audx - £ b . (t) J"Au(x, t-r . )dx + p(t)Judx J J n j=i J n n

+ £ J" q. (x,t)u(x,t-T. )dx + a(t)^r [ f udx ) = 0, t a t 1 i=l n 1 £1 ' It follows from Green's formula that, for t £ t^ , (3)

J~Audx = S n an

dS = - J" r(x,t)u(x,t)dS s 0, and an

.

Oscillations for damped

equations

J"Au(x, t-r . )dx = - JY (x,t-r . )u(x,t-r . )dS < 0, j = 1,2 J J J n an

k,

(4)

J"q. ( x , t ) u ( x , t - T . ) dx i q . ( t ) J u ( x , t - T . ) d x , i = 1, 2, . . . , m. „ l l l „ l n q Thus, b y (2) ~ (4) w e o b t a i n that m (5) v (t) + p ( t ) v ( t ) + £ q . ( t ) v ( t - T . ) + a(t)v (t) s 0, t £ t , 1 1=1 1 w h e r e v ( t ) = J\i(x,t)dx. n /

Set w(t) = v , (6)

w

Then by

(t)/v(t) for t 2: t

, then from

(5) w e h a v e

that

m v(t-x. ) 7 (t) + a ( t ) w ( t ) + w (t) + p(t) + Y. q t ( t ) v ( t ) 1 s 0, t £ ^ (6)

w'(t) +

]2 -

[w(t) +

.

2 +

p(t)

y i t^ ,

J ( t - y ) a A ' (y)dy + J(t-y) Va(y)w(y)dy + J-(t-y)Vw 2 (y)dy y

y

y

+ J"(t-y)Vfp(y) y

y

v

+ I

i=l

q, (y)ldy s 0.

J

Since X(t-y) a y' 3 w y

we have that

(y)dy = aj"(t-y) a

1

y' 3 w(y)dy,

y

J ( t - y ) V [ p ( y ) + I q^ylldy s w ( y ) ( t - y ) V - J(t-y)Vw 2 (y)dy y

i=i

- S [ (t-y)a(y)y + cty - p(t-y) ] (t-y) and hence

y

y a_1

3 1

y' ~ w(y)dy>

D.D. Bainov

26

and C.

Baotong

t p m ~ / ( t - y J V [ p ( y ) + I q. (y) ]dy £ w(y)(t-y) a y P y 1=1 1 - J" { ( t - y ) a / 2 y 0 / 2 w ( y ) + H(y,t)} 2 dy + J" H 2 (y,t)dy, y

y

where H = | [(t-y)ya(y) + ay - |3(t-y) ] U-y)a/2~1yl3/2~1.

Thus, we

obtain that t „/ J*( J * ( t - y ) V P(y) y ^

(9)

+

m I q,(y) dy i=l >

- J1 H 2 (y,t)dy s w(y)(t-y) a /,

t > y a t2 .

y

Divide (9) by t a and take the upper limit as t

> co. Using (C^),

we can obtain a contradiction. If u(x,t) < 0 for (x,t) e n x [t^ , oo), then the proof follows from the fact that -u(x,t) is a positive solution of the problem (l),(Bj). This completes the proof of the theorem.

Corollary 2. Assume that (C^) holds and there exist a e (1, co) and 13 e [0, 1) such that lim sup t t > co

t m g f (t-y) y' [p(y) + £ q.(y)]dy = co, t 1=1 1

lim sup t" a f [ (t-y)ya(y) + ay - |3(t-y)] 2 (t-y) a ~ 2 y P_2 dy < co, t >« tQ then every solution of the problem (1), (B^) oscillates in G.

From the proof of Theorem 1 it is easy to obtain the following theorem.

Theorem 3. If the inequality m v (t) + p(t)v(t) + Z q.(t)v(t-T.) + a(t)v (t) £ 0 1 i=l 1 has no ultimately positive solutions, then every solution of the problem (1), (Bj) oscillates in G. Theorem 4. If (a) t

lim J" exp | - J f " a(y)dy|ds a(y)dy c = oo, for every t^ £ 0, > co t I t„ >

Oscillations for damped equations

(b) there exist a e (1, w), |3 e [0, 1) and e € (0, 1) such that

)

1 lim sup — S • (t-y)V [ p ( y ) 4. 4.« I t » dx J £2 > £2 j=l J £2

^ dt

+ p (t)JuOdx + £ J~q. (x,t)u(x,t-T. )$dx + a(t)^r ( Ju$dx 1 1 £2 i=l n 1 ^ £2 ' From the divergence theorem and (ii) it follows that (17)

/Auidx = J"! an^ a

$ + aN

=

u aN

0.

|dS + JuA$dx > a

J"uA$dx = - a n J"u$dx; Q £2

J"q. (x, t)u(x, t-T. )$dx £ q. (t)Ju(x, t-T. )$dx, 1 = 1 , 2 , ...,m, 1 1 £21 n , 9u(x,t-r ) , J~Au(x, t-r . )$dx = J- ™ — * " u ( x , t - r . ) ^ dS + Ju(x,t-r.)A$dx J J aN J n an I a N J n (18)

= Ju(x,t-r,)A$dx = - a Ju(x, t-r .)$dx, j = 1, 2 k. J J £2 £2 Combining (6), (17) with (18), we obtain „ k m (19) v (t) + [a n b(t)+p(t)]v(t)+ a. £ b .(t)v(t-r .)+ £ q.(t)v(t-T.) J J 1 0 j=l i=l + a(t)v (t) s o ,

t a tQ .

Thus v(t) = J"u(x, t)$(x)dx > 0 for t £ t Q is a positive solution of the inequality

(19), which contradicts the condition of the theorem.

Similarly to Theorems 1-5, using the inequality (19),we obtain the following

theorems.

Theorem 7. If J" [a b (t) + pit) -

]dt = m and there exist

t

as

0 (1, to) and (3 e [0, 1) such that . t ( , k m lim sup i - S j ( t - y ) V k , b ( y ) + p(y) + a Q £ b (y) + £ q t > to t t [ ^ j=l J i=l - I

(y) '

t (t-y)ya(y) + ay - 0(t-y) ] 2 (t-y) < X _ 2 y' 3 _ 2 | dy = co,

then every solution of the problem (1), (B^) oscillates in G. Theorem 8. If

t

lim S exp > co t 0

I

- S t

and there exist a € (1, oo),

0

a(y)dy ds = oo >

for every

t^ t. 0,

[0, 1) and c € (0, 1) such that

30

D.D. Bainov and C.

lim s u p — f tt 4t > oo 4t- It^ i

Baolong

k y-r . C t - y l V f « , b(y)+p(y)+a e £ — - ^ b j=l

[(t-y)ya(y) + a y - ß ( t - y ) ] 2 ( t - y ) a

then e v e r y s o l u t i o n of the p r o b l e m

(1),

2 ß

y

m y-T

,

(y)+e

2

d y = oo,

(I^) o s c i l l a t e s in G.

T h e o r e m 9. If (a) (in T h e o r e m 4) holds, a n d there e x i s t s e e (0, 1) s u c h that the H

inequality

(t) + [et(a b ( t ) + p ( t ) ) + a(t)]H(t) + ane T bJ . (t)(t-r . )H(t-f .) 0 j=l J J m + e T q.(t)(t-T.)H(t-x.} £ 0 1=1 1

h a s no u l t i m a t e l y p o s i t i v e solution, blem

(1),

then e v e r y s o l u t i o n of the p r o -

( B 2 ) o s c i l l a t e s in G.

References

1. D.P. M i s h e v a n d D.D. Bainov, Appi. Math. Comput., 28,

97-111

(1988). 2.

, Funk. Ekvac., 29, 2 1 3 - 2 1 8

(1986).

3. N. Y o s h i d a , Bull. Austral. Math. Soc., 36, 2 8 9 - 2 9 4

(1987).

4. D. G e o r g i o u a n d K. K r e i t h , J. Math. Anal. Appi., 107,

414-424

(1985). 5. Cui B a o t a n g , Math. J. T o y a m a Univ., 6.

14, 113-123

(1991).

, O s c i l l a t i o n P r o p e r t i e s for P a r a b o l i c E q u a t i o n s of N e u t r a l

Type, Comment. Math. Univ. Carolinae, Vol. 33, No. 4 (in press). 7. A.K. K a r t s a t o s , S t a b i l i t y of Dynamical Systems: T h e o r y a n d A p p l i cations, L e c t u r e N o t e s in P u r e a n d Applied Mathematics, 28, New York

(1977)

pp.17-72.

8. J u r a n g Yan, J. Math. Anal. Appi.,

122, 380-384

9. Wei Junjie, Ann. Diff. Eqs., 4, 473-478

(1988).

(1987).

Springer,

Third International Colloquium on Differential Equations pp. 31-44 (1993) D. Bainov and V. Covachev (Eds) © 1993

INITIAL V A L U E PROBLEM F O R A SINGULARLY PERTURBED NONLINEAR IMPULSIVE SYSTEM IN THE CRITICAL CASE V. COVACHEV Institute of Mathematics, Bulgarian Academy of Science, Sofia, BULGARIA

Abstract. A n asymptotic method for solving the initial value problem is justified for a nonlinear singularly perturbed impulsive system of differential equations in the "critical" case.

Keywords: critical case,initial value,impulsive,singularly

1.

perturbed.

INTRODUCTION

The interest in differential equations w i t h impulse effect steadily increases in relation to the investigation of mathematical models of real processes and phenomena which at certain moments of their evolution undergo rapid changes. The initial value problem in the so called noncritical case was investigated in [1], In the present paper an asymptotic method for solving the initial value problem is justified for a nonlinear

singu-

larly perturbed impulsive system of differential equations in the so called critical case, that is w h e n the linearized degenerate

system

has an eigenvalue zero of constant multiplicity. The p l a n of the paper is as follows.

In § 2 the necessary asser-

tions from [2] concerning the existence of a stable manifold in the critical case are given.

In § 3 the problem is formulated.

In § 4 the

construction of an asymptotic solution of this problem is carried out, and in § 5 the asymptotic method is justified, i.e. the existence of a n exact solution of the problem considered is proved and an estimate of the difference between the exact and the asymptotic solution is obtained.

V. Covachev

32

2. A U X I L I A R Y

ASSERTIONS

C o n s i d e r the

equation

(1)

z = F(z, a), m

w h e r e z e R , a is a p a r a m e t e r , F is c o n t i n u o u s l y d i f f e r e n t i a b l e r e s p e c t to z, F ( 0 , a ) = 0, F ^ t O . a ) has a n e i g e n v a l u e 0 of k

(independent o f a) a n d

Suppose, moreover,

with

multiplicity

m - k e i g e n v a l u e s w i t h n e g a t i v e real

parts.

that for a n y v a l u e of the p a r a m e t e r a there are k

l i n e a r l y i n d e p e n d e n t n u l l e i g e n v e c t o r s , a n d d e n o t e b y ip(a) a n (m x k ) m a t r i x of rank k eigenvectors)

(whose c o l u m n s are the linearly i n d e p e n d e n t

null

s u c h that F

z

(0, ot)i>(a) = 0.

H e n c e f o r t h w e shall o f t e n s u p p r e s s the d e p e n d e n c e o n the p a r a m e t e r a, a n d d e n o t e b y A the m a t r i x F

z

(0, a).

The s t a t i o n a r y p o i n t z = 0 is in general not a s y m p t o t i c a l l y i.e. a n a r b i t r a r y s o l u t i o n of

(1) w i t h initial c o n d i t i o n

c l o s e to 0 d o e s n o t h a v e to tend to 0 as t

> oo. H o w e v e r ,

initial c o n d i t i o n is c h o s e n in a n a p p r o p r i a t e way, t h e n the w i l l e x p o n e n t i a l l y t e n d to 0 as t

> to. M o r e p r e c i s e l y ,

stable,

arbitrarily if

the

solution

the

follow-

ing a s s e r t i o n is valid. L e m m a 1. In a s u f f i c i e n t l y small n e i g h b o u r h o o d of the p o i n t z = 0 there e x i s t s a n ( m - k ) - d i m e n s i o n a l m a n i f o l d u and p o s i t i v e c o n s t a n t s y a n d o- s u c h that if z(0) e to, t h e n for t i 0 the s o l u t i o n z(t) s f i e s the

inequality |z(t) | i 3-e -trt .

(2)

The m a n i f o l d w is c o n s t r u c t e d b y the m e t h o d of s u c c e s s i v e mations,

sati-

r e p r e s e n t i n g the r i g h t - h a n d side of

approxi-

(1) a s a sum of a

linear

term A z a n d a n o n l i n e a r i t y G(z) = F(z) - Az: (3)

z = A z + G(z).

The f u n c t i o n G ( z ) o b v i o u s l y e n j o y s the f o l l o w i n g two p r o p e r t i e s : 1. G ( 0 ) = 0. 2. For a n y e >0 there e x i s t s S >0 s u c h that for we have

|z^| ^ 6,

|z^| s S

Critical case

|G( Zl ) - G(Z 2 )| * C| Z L - z 2 |. The matrix A has a k-tuple eigenvalue 0 and m - k eigenvalues with negative real parts. Then there exists a matrix B, as smooth with respect to the parameter a as A is, which reduces A to a block-diagonal form (4)

B

2

AB

C 0

0 0

where the (m-k) x (m-k) -matrix C has the above mentioned eigenvalues 1

m-k ' If for (1) we consider the linearized system, i.e. we set in (3)

B„

rB

ll

B

12

B

21

B

22

„.m-k _k x € IR , y e IR , B =

G(z) = 0, and denote z =

where

, i, j = 1, 2, are blocks of appropriate dimensions, in a linear

approximation the manifold w can be represented in the form (5)

y

= B21Bi!X

provided that the matrix B ^

is nonsingular.

By Lemma 1 the manifold u is constructed locally (in a neighbourhood of 0). Extending to the direction of t < 0 the trajectories starting on u, we obtain an extended manifold £2 enjoying the same properties as w, i.e. any trajectory z(t) starting on fi for t = 0 remains on fl for all t > 0 and exponentially tends to 0 as t

> ro.

Formula (5) suggests us to assume that in some domain Z)(x, a) the manifold fi = Q(a) can be represented in the form (6)

y = P(x, a),

where P(x, a) is a sufficiently smooth function on D(x, a). (The parameter a will be again suppressed henceforth). F : (x, y) Denote H(x) F2

E

IR

F

11 =

P (x), F(z) = F(x, y) x 3

xFl

12

3 F F y 1 ' 21

F 2 (X, y) 3 F x 2

F

22 =

m-k F 1 e IR' a

yF2

Now consider a nonhomogeneous system of equations whose corresponding homogeneous system is the system in variations for (1): (7)

A = F (t)A + 0(t) z where F (t) = F (z(t)), z(t) e £2, é{t) is a vector-valued function. As z z above, we denote A

.

=

V.

34

Covachev

Lemma 2. The change of variables (8)

A x = 5 X , A 2 = H(t)61 + S 2 ,

where H(t) = H(x(t)), reduces system (7) to the form *1

(9)

S

= a

ll 5 l

2 = a 22 6 2

+ a +

12 6 2 %

+

~

+1 ' H

V '

where a

(10)

ll =

F

u

+ F

12 H '

a

12 = F 1 2 ' a 22 = F 22 "

HF

12 "

In the above formulae the dependence on t is suppressed too. Now suppose that in system (7) the nonhomogeneous part ifi(t) is the matrix F (t) IRm,

i

i = 1, p, £2 i s a domain in IR , A z ( t ^ ) = zit^+O) - z ( t ^ ) , m

the jump

f u n c t i o n s I ^ ( z ) and the i n i t i a l c o n d i t i o n z^ a r e assumed independent of c , f o r the sake of

simplicity.

Assume t h a t the following c o n d i t i o n s hold: HI. For some n £ 2 the v e c t o r - v a l u e d f u n c t i o n f € C n + 2 ( G , IRm), where G = [0, T] x fi x [0, H2. I . e C

n+2

(fi,

eQ).

R ). ra

The f u r t h e r assumptions w i l l be made in the course of the e x p o s i tion. F i r s t c o n s i d e r the degenerate system (15)

f(t,

z, 0 ) = 0.

H3. For each t e [0, T] system ( 1 5 ) has a family of s o l u t i o n s of the form where a = (a^

z = 0. They are taken into account in the proof

of the convergence of the method, a ^ z (t. ), u = 0, v-1. M

ilij J

are expressed in terms of

Critical

Now p r o b l e m

case

(12)-(14) takes the form

e £ IV (t)cV n V>=0

+ V n L>=0

(T. )e" = f ( t , 1

V

+0 ) I izV (t. 1 V = rvl 0V

+

e)

+ f C i ) ( T1 .

,

e),

^(Oile"

= z n ( t . ) + I . ( z n ( t . ) ) + e[E+a I. (z.(t. ))]z. (t. ) O i 1 O 1 ^ z i O i J l i V (e + 3 I . ( x n ( t . ) ) ì z (t.) + a. I , E E- [ z l 0 l J v l ii>[ v=2

+

(0) + , < ° > ( 0 ) ) e " = z n V J 0 " viol " In fact, the above e x p a n s i o n s have sense only up to order n+1. W e equate the c o e f f i c i e n t s at the like powers of e, s e p a r a t e l y depending on t and on

those

.

To d e t e r m i n e the a p p r o x i m a t i o n of order zero z^(t) a n d i = 0, p, w e o b t a i n the systems (25)

f(t, z Q ( t ) , 0) = 0,

(26)

zQ(t) +

(27)

ûï0(t.) = I.(i0(t.)) -

(28)

^

^

V

(0) = z q ,

=

f(t

i

' W

0

^U(0), )

+

¿ ^ V '

0)

'

F r o m (25) in v i e w of c o n d i t i o n H 3 w e find z

(t) = ¥>(t, a

(t)),

w h e r e a 0 (t i )))- V l (t i ( a 0 (t i + 0)) + ¥ > 1 (t., a 0 (t i )),a 0 (t i )).

H8. For i = l,p from equations (32^) 0^(^+0) can be expressed as functions of a.(t.) e D with values in D: 0 l (33)

a 0 (t.+0) = x.ta^t.)).

Thus we determine the jump conditions for the unknown function 0 , c Q e (0, c^] such that for any w^ , w 2 satisfying for O s t s T ,

0 < c £ c^ the inequalities | W j (t, e)| a CjG 2 , j = 1, 2,

for O ^ t s T ,

0 < G S

we have o |G(t, G, w ^ t . c ) ) - G(t, G, w (t,e))| e

2 £ c c max |w (t,c) - w (t,c)|. 0 y ( - S o ) and < .,. > stands for the dot product. The new point is at Euclidean distance h from y(s) and we can guarantee that the error in the tangential direction to y(so) is small relative to the transversal error. Based on these ideas, we have designed a family of multistep methods to integrate normalized differential equations in Rn y' = f(y),

||f|| = l,

in the case that the solutions are spherical curves. The algorithm can be written as Yj+m = y,+m-l + hF{yj,. . . , y ; + m ) ,

(2)

where the function F has an expression of the form F[ Yj, • . . ,y j+m) = --yy+m-l z

+ \l 1

Til Jc+m 11| » ll i+m-l||

c J + m _i being the component in the hyperplane tangent to y J + m _i of the vector that plays the role of the previous approximation quoted above. Usually we select the coefficients in such a way that the substitution of F by it in (2), with m u = E « . [ f ( y J + . ) - f(y 3 '+.-i)] + f(y,+m-i), t=0 would produce a standard code, such as the Adams type ones. Thus we can envisage the algorithm (2) as the result of a certain projection of the former one along the normal and tangent manifolds in a point of the sphere. These schemes leave the surface invariant, a property that we have called spherical exactness. Besides two consecutive points axe always at Euclidean distance h. Their numerical behaviour is also good: they axe consistent, their stability can be proved using the main theorem of Grigorieff [26], and thus they converge. The principal term

Numerical

integrators for curves on

surfaces

49

of the local truncation error has a component neither in the tangential direction nor in the normal one to the spherical surface to y(sj+ m -i), provided that the values of the parameter are chosen to make the points equispaced —the so-called chord criterium. Moreover, the methods are exact along geodesies, and the local error is factorized by the geodesic curvature and its derivatives. So the codes are suitable for integrating orbits near geodesies, and they allow us to gain accuracy with respect to the usual algorithms. 3 A MODIFICATION OF MULTISTEP METHODS Despite the good behaviour that those methods described in the former section show, they have some limitations. The most important one comes from the fact that they can only be applied with success to a rather reduced kind of problems: those which have solutions on a sphere of a Euclidean space of arbitrary dimension. Secondly, they are not advisable when the exact correspondence between an independent varible, fixed forehand, and the calculated points of the solution are to be known. In this section we present a different alternative to modify the classic multistep methods, that turned out to be useful not only in the same problems as the spherically exact algorithms, but also to integrate curves on surfaces near spheres, that is to say, surfaces obtained from spheres by means of small deformations. In order to simplify the algorithms, it is enough to parametrize the solutions with their arc length, then the computed points of the solution correspond to values of this variable. As we will see later in the section dedicated to numerical examples, in many cases these algorithms produced an improvement in accuracy similar to the spherically exact ones —naturally, with no additional noticeable computational cost- a reason that makes them a good choice. In a first approach, we introduce these new methods in an intuitive way. For this purpose we start by considering an initial IVP in Rs x'=f(i,x),

te[a,b],

x(a) = w, where f is a field tangent to the sphere S2, and normalized in such a way that ||f(i,x)|| = 1, Vi € [a, 6], the solution of this problem being a curve on S2 parametrized by its arc length. Let us denote by { t , N , T } the Darboux-Ribaucour trihedrom in any point of the solution curve, where t = f(i, x) is the tangent vector, N the inwards normal in x and T completes the orthonormal frame. Let us recall the following equalities: i ' 0 kn t A N = 1 ~kn 0 < —kg I T J

(

kg\ 0

f

I

t

N T

where kn = kn(t) is the normal curvature of the surface, kg = k„(t) the geodesic curvature and Tg = Tg(t) the geodesic torsion. For a curve on S2 we have kn(t) = 1 and rg(t) = 0, Vi € [a, 6], and the former formulae reduce to t' = N + kgT, N' = - t , T' = ~kg t. If the curve is a geodesic then kt = 0 and the successive derivatives of x can be easily calculated. This can be done in terms of f and x itself because the outwards normal

50

J.M. Ferrandiz and M.T. Perez

field to the sphere is given by x. More precisely, we have x'). See Figure 2. Denoting by i l a horizontal section of Z, we then obtain the equation div

Tu = ku +

A;

Tu-

^ \ 1 + | Dif

(4)

in Q, with v-

Tu

= cos Y

(5)

on L = dCl, v being unit exterior normal on L.

Figure 2: Free surface S in capillaiy tube

The equation (4) is precisely the one originally given by Laplace. It follows indirectly from observations of Young that the divergence expression in (4) is twice the mean curvature H of S\ thus (4) achieves an independent geometrical interest, apart from its physical motivation. The correctness of (4) as a description of reality has to our knowledge never been challenged, although its scientific utility was ridiculed by Young |5], in the context of his general disdain for the mathematical method (and more particularly for those who practiced it).

Coniaci angle in

57

capillarity

In fact, to our knowledge the only attempt that has been made to examine the applicability of (4) experimentally appears in the paper of Bashforth and Adams [6]; that investigation was limited to a relatively simple rotationally symmetric configuration and combines the test of the theory with that of a new numerical method. In the context considered, the test did appear to yield a definitively positive result, on both counts. The boundary condition (5) has also a clear geometrical interest; however its correctness as a description of reality has been - - and still remains - - a matter of continuing controversy, see e.g. the survey by Dussan V [7]. (For a derivation of (5) based on molecular considerations see, e.g. [8] and the references cited there.) The controversy can be traced directly to the difficulty in obtaining repeatable measurements, and different investigators have obtained notably disparate values for presumably identical materials. Certainly, some of the difficulty is attributable to "hysteresis" forces, arising from resistance to motion of the fluid along a solid surface. This view of the matter has led to introduction of the notions of "advancing" and "receding" contact angles, and these in turn are assumed to depend on the speeds of relative motion (and perhaps also on other factors). From this physical point of view the theory becomes exceedingly complicated, and clear explicit workable principles have yet to emerge. In the limited context of equilibrium configurations, a possible unifying principle has been proposed in [9], [10]; apparently, the matter has not been pursued further along such lines. The work described in the present article represents an attempt to address the original question, as to the existence of an intrinsic physical angle, by introducing situations in which the adhesion forces that are the basis for the classical theory are overwhelmingly large in respect to the resistance forces. We consider the "capillary tube" configuration described above, in a gravity-free environment (e.g., in space flight). Then (4) becomes div Tu = 2H* const.

(6)

in Q, while (5) remains unchanged. In (6), the choice of H is not arbitrary. If we integrate (6) over ii, we find by the divergence theorem 2/i|i)| = J> v- Tuds - |e| cos y

(7)

by (5), yielding a value independent of the volume of fluid, so long as il is covered. Let us now choose a subdomain il* c c Q, bounded in part by T c Q and by a subarc L*

c

L. The divergence theorem now yields 2H|il*| = |£*|cos y + j

v Tu ds .

(8)

The essential observation that underlies everything thatfollows is that |v- Tu\ < 1 for any function u. Replacing v- Tu in (8) by its lower bound - 1, we find that the functional

58

R. Finn

[i)*;y] » | r | - |l1*|cos Y + 2W|oH > 0

(9)

for any non-null choice of CI* c c Q. Consider the particular situation in which Q contains a corner of opening angle 2 a . Data cannot be prescribed at the vertex, but we can exclude that point by a small arc A, which is then allowed to shrink to the vertex. Choosing T as in Figure 3, we obtain from (9) after a limiting procedure H /2sin a cos a + / sin a > / cos y •

It can be shown that this holds regardless of behavior at the corner, provided only that (5) holds on the {open) sides inL\no growth hypotheses need be introduced. Letting / -» 0, we find sin a > cos y. We have proved:

Theorem 1: IfQ has a corner point of opening angle 2a, and if a < j®- - y|, then there is no solution to the problem (4), (5) in Q.

Figure 3: Corner configuration

Figure 4: Regular polygon; circumscribed equatorial circle

To put the theorem into context, consider the case in which O is a regular polygon n (for example, the equilateral triangle of Figure 4). A lower hemisphere whose equatorial circle circumscribes F1 provides an explicit solution of (4), (5), for which y = ^ - a . Any y in the range

-y| < a can be obtained by increasing the radius of the hemisphere and/or

replacing the lower hemisphere by an upper one. In fact, for the indicated geometry, the solution u is uniquely determined by the prescribed volume, and is always a spherical cap when it exists. Thus, the result shows a

discontinuous dependence of the solutions on the data y. As y is decreased (CM- increased) from 7t/2, the solution continues to be determined as a spherical cap up to and including a critical value Ycr. But with any change beyond yCT the solution ceases to exist. Experiments conducted by W. Masica have shown that the theorem reflects physical reality. He found the predicted spherical cap when j®- -y| < a , while for

• - y | > a the

fluid climbed up the walls at the comers and partly wet the top of the cylinder (or the reverse). This explains the seeming paradox of the theorem; in fact, a surface S does exist as

Coniaci angle in

59

capillarity

solution of (2), (3) but it cannot be expressed as solution of (4), for the reason that S bends over itself at the corners (Figure 5) and does not cover all of il.

2

Figure 5: Fluid section at corner

Figure 6: Two corner domain; spherical caps, with concentric circles as equator, meet cylinder over L in constant angle

In any event, there is a discontinuity in behavior as the critical contact angle Ya is crossed, with S changing its character dramatically at the critical value. Clearly the surface forces are the overwhelmingly dominant ones in any such configuration, and it is proposed to use the phenomenon in a space experiment to measure contact angle.

Consider the

case 0 < y < 2 • A*1 initial guess y for ya is made, and two angles yi, Y2 are chosen so that 3 > y, > y > y2 > o . A cylinder is to be constructed with section i l as in Figure 6, with aB(Ai). j 2. (For

example, 0.2(H) contains both t r ( E 2 ) and ( t r i ? ) 2 (not to mention terms bilinear in E and in curvature), and these are not separated by the identity operator.) Moreover, formulas for the global quantities an(H)

do not give complete information about the local quantities

an(x).

For one thing, any exact divergence, such as Ao_R, intégrâtes to zero. For another, [a\ (x)\ap may contain both terms proportional to [E(x)]ap

and tëfms proportional to tr E(x)

these are no longer independent after the trace is taken in (8) and (14). coefficient tensors of H do not appear in the untraced an(x)

, and

(Traces of the

for ordinary operators H, but if

H is exotic, they do.) A third way to calculate an(x)

is based on the calculus of pseudodifferential operators

[31, 22, 18, 34, 30, 16, 23, 24]. It has broader validity than the two traditional methods; it offers in principle a complete solution of the problem for operators of the class (12).

(For

more general exotic (3) or higher-order operators, integrals may be encountered that cannot be evaluated in closed form.) Unfortunately, it is much less efficient than the other methods. An attack upon its computational complexity is the main subject of the present work. T h e s y m b o l . In the intrinsic pseudodifferential

calculus [5, 34, 11, 16] each covariant deriva-

tive, V, in an operator such as (3) is represented by a Fourier (cotangent bundle) variable, Thus the operator on C o c ( A 1 ) H0 = - A - I ( V ^ V " + V a V/j)

(A = g^V^y)

(16)

has the intrinsic symbol Sy(F0) = |e|2+ce®e. By definition, an exotic

operator is one with nonscalar

principal

(17) symbol-, that is, the terms

in the symbol of highest degree in £ are not just a multiple of the identity matrix. We may think of Ho as operating either on vector fields or on one-forms; to match up with physics literature, I shall consider vector fields. Then the coefficient tensor in (16) and (17) is [ A " T « - - g ^ t - § {g^&l

+ g"6$.

(18)

Here all the indices axe Greek, since they refer to the same vector bundle (or its dual); their place of origin in the symbol is momentarily preserved by the distinction between the beginning and middle of the alphabet. We consider the class of operators H = ft2 i/o +

V(x).

(19)

SA Fulling

68

(Here b2 and c are constants, obeying sign constraints to be discussed presently, and V is a C°° endomorphism-valued function, called the potential

in analogy with quantum mechanics.)

Equivalently, H is of the form (12) with -E"a

= Vf>a - b2 (l + 0

(20)

R"a

and a2 = b2(c + 1),

(21)

The operator can be parametrized by a and b, to exploit the Hodge decomposition, or by b and c, so that c is the magnitude of the exotic term and b is merely a scale factor. (In [24], V is called X and c is called —a. The terms in (20) proportional to the Ricci tensor

are

the Weitzenboc.k operator for k = 1 and a similar contribution from the desymmetrization of the exotic term in (16).) From the second-order derivative terms one reads off the principal symbol of H as +b

2

m

2

+ c t ® o = b2 lei2 + ( ^ - 6 2 ) i ® e 2 • ?2= a ext^intf + 6 mt^ext^ ,

'

'

where int and ext are the operators of interior and exterior multiplication on forms. (Here £ (A £ has the matrix tPta in a conventional local basis.) Under the assumption that the metric is positive definite and b2 > 0, the condition that both eigenvalues be nonnegative, so that the heat kernel exists, is c > -1,

or

a 2 > 0.

(23)

By convention, b > 0 and a > 0. In the calculation a major role is played by the eigenvalues and eigenprojections of the matrix (22). They are A, =a 2 |e| 2 = 6 2 (c + l)k| 2 , A2 = 62|£|2,

(«WO.

P2 = J i ! l = | ® i

( ± to i ) .

(24a) (24b)

In the notation of [16] we have MA) = [Sy(b2H0)

- A] - 1 = -

2

i= i

P,

•

(25)

3

The first step in calculating the heat kernel of H is to find the symbol of the resolvent operator, ( H — A ) - 1 , following [34] and [16]. For background reading on the resolvent symbol and the heat kernel, I highly recommend the expositions of Gilkey [17, 18, 20]. (They, however, deal with the conventional

pseudodifferential calculus, where £ is a literal Fourier variable

corresponding to conventional partial differentiation. The basic formulas of that approach are simpler than those of the intrinsic formalism, but extra work is needed at the end to express the results in coordinate-independent geometrical form.)

The series of conference

reports [14-15] provides a brief introduction to and summary of [16].

Asymlotics of exotic

THEOREM 3. The intrinsic symbol of the resolvent operator

has an asymptotic

69

operators

parametrix

of an elliptic linear

differential

expansion Sy[(H-

A)" 1 ] ~ ! > ( * > £ > * ) ,

(26)

s> 0

where b0 , as in (25), is the local resolvent by an explicit formula over multiindic.es. the present

of the principal

[15, 16] involving very complicated

The formula

for 62 contains

symbol,

and the higher bs are given

index contractions

40 terms for a generic

and

operator,

summations but only 5 in

case (19): b2 = -b0

Vb0

-AboAanoA^(V^VvI)boiai0 - 2b0A^b0A^(V„V1/Vp^)b0

(27)

2b0At"/boAap(Vti'Vv'Vp$P)bo

-

- SboA^boA^boA^V^V^V^bo

.

[Here a matrix multiplication is implied, and hence the fiber-bundle indices, including the last two indices in (18), are suppressed. I is the parallel transport operator in the tangent bundle, called TE in [16]. See (A.11-13) of [24]; their W will be 0 for us. $(x,y)

is the

tangent vector to the geodesic joining x and y (see [16], Remark 2.5). The important thing is that in the present case all the covariant derivatives of I and

evaluated at y = x as

tacitly implied in (27), can be recursively calculated in terms of the Riemann tensor [9, 10, 12].] More generally,

a term in bs has the (coefficient)

where the Tu are tensors indices

absorbing

U0 = -[-5/2]

form

b0Tub0Tu^

built out of A, V,

the £ factors

by contraction.

(28)

••• boT^bo ^{2U-S), and I and their covariant (Each T is linear in (A, V).)

derivatives, U ranges

with from

= \s/2\ to 2s.

For our special operator, the covariant derivatives of A vanish, and 1 < U < 4 for s

=2.

To obtain the diagonal value of the heat kernel we follow Widom [34], pp. 59-61. THEOREM 4. In an orthonormal

frame at x we have 00

K(t,x,x)

(29)

~ Y^ K,{x), s=0

Ks(x)

= (27r) - m J ^ dm(

( ^ j

J

dX e~txb3(x,

A),

(30)

70

SA Fulling

where T surrounds case, K2n

Ai and X2 in the positive

equals ( 4 7 r < ) - m / 2 t nan

sense.

Ks

will be 0 for s odd.

in our earlier notation (5).]

[In the contrary

Thus the contribution

of the

term (28) is 1

t {s~ m)l 2{coefficient){-l) u(2n)~ m

J

2

^ E " " ¿0=1

® ^

^ " t n L ^

^

E iv=\ ^ ~ ^

PivTv-TiPt,

^

(31) '

~

M2)_M2

where Mj = cardinality H

of {¿t: it = j} = a

2

M 2,

{Mi + M2 = U + 1),

»2=b 2\r,\ 2.

(32) (33)

T o get to (31) one m a k e s the substitutions r? = i 1 / 2 £ ,

n =

t\

a n d renames t r as I \ Whenever Tu equals A, another s u m from 1 to 2 can be introduced into (31), corresponding to the two terms in (18); furthermore, P2 splits into two terms (24b), so the i s u m s are effectively over three values. This implies that each term in t>2n gives rise to a rather large number of terms in an — namely,

where v is the number of

occurrences of the potential V in the term (28). T h u s from (27) for 62 we will get (in a i ) 9 terms involving V a n d 108 + 108 + 108 + 648 = 972 other terms, which are all linear in the Ricci tensor. T h e number of terms in a2 is over a million! T h e s e terms are highly r e d u n d a n t , in the senses that m a n y of them are manifestly proportional, quite a few turn out to b e zero, a n d they are all linear combinations of a small number of linearly independent objects. For example, ai m u s t simplify to a s u m of 5 terms, proportional to V, its transpose, its trace (times the identity), the Ricci tensor, and its trace (the curvature scalar R). In what follows I shall d e m o n s t r a t e that all the integrations in (31) can b e performed in closed form, so that in principle an(x)

has been expressed in terms of elementary functions

( a n d local, polynomial func.tionals of the coefficient tensors). Furthermore, I shall show that m o d e r n computer technology m a k e s practical the actual calculation of a,\ a n d p r o b a b l y a2 by this m e t h o d . I n t e g r a l s . In evaluating formula (31), one encounters three kinds of integrals: 1. Cauchy integrals 2. Gaussian 3. Angular

integrals integrals

over the spectral p a r a m e t e r , fi. over the radial coordinate in Fourier space, over the unit sphere in Fourier space.

THEOREM 5. Define FMi

,fi2) = ^ - J

r

e " A (A - /x, ) - M ' (A -

to)-"'

d\,

(34)

Asymtotics of exotic

71

operators

where a2 - b2 Then FM m (U

v '

U ) ~

^

(~1)M'~1(^2 + J - l ) !

(,2C]-M2-J\

R2(M2+J)p-a^

( - 1 ) m 2 - m ' ~ j ( M 1 + J - 1)! (M, - 1 ) ! ( M / -i

2

-J-1)!J!

2 W | + J ) 1

j

(35)

t,|B|a

1/1

—1

^I^Ais—1

=

W ,

H .

=

T h e proof is a straightforward application of Cauchy's integral formula.

(36)

Contrary to

appearance, the functions in (35) are guaranteed to be nonsingular as r; —»• 0. Henceforth I write .Fmi M2(a> &) instead of F m i M j I / ' i i ^ )

an< ^ r

p l a c e of

W e shall dispose of the angular integrals quickly, since a similar problem has been discussed in depth in the appendix of [8]. THEOREM 6. Consider an integral of the form XF= where a is a multiindex, but not necessarily

corresponding

distinct).

(37)

[ dmr,F(\V\)V- g(x)

=

X

a

then, at infinity, g(x) has index a and is said to be i) slowly varying if a = 0,

If, for each X > 1,

L.J. Grim and L.M. Hall

78

ii) regularly varying if a

0, -»> < a
0, where c and e are measurable, limc(x) = c and lime(ac) = 0. jr—»«»

i - » »

If c(x) = c, then L(x) is called normalized slowly varying. Notation. Slowly varying functions (svf) will be denoted by L(x), and normalized slowly varying functions (nsvf) will be denoted by Lo(x). Lemma 2. Lo(x) is an nsvf if and only if Loi*)

Lemma 3. Let L(x) be an svf. Then, for each T| > 0, lim*' L(x) = °° and

X—»«

limJc"" L(x) = Q.

Jt—

Lemma 4. Let L(x) be an svf. If L is differentiable and convex, that is, if L'(x) is increasing, then -n hr m — — - =U. L(x)

Lemma 5. Suppose that

f(x) = J*y(t)dt = x" a L(x), X

-xf'(x) where w is nonincreasing, a > 0, and L is an svf. Then lim — = a. f(x) Lemma 6. Suppose g(x) > 0, g'(x) is increasing and limx_>„g(x) = 0. -xg'(x) Then g is rapidly varying if and only if lim — — — = g(x)

00

.

QUALITATIVE BEHAVIOR OF SOLUTIONS We first consider the equation {p(x)y')

+ f ( x ) y= 0

(2.o)

where f is continuous, and p is positive and differentiable on [xo, ). If (2.0) is nonoscillatory at x = oo, then a nonzero solution y(x) for which

Self adjoint differential

1 )

79

equations

dx _ p(x)y2(x)~°°

is called a principal solution of (2.0), and a solution independent of y(x) is called a nonprincipal solution. For instance, e 2 t and e 2 t are principal and nonprincipal solutions for the equation y " - y = 0. The above integral will converge with a nonprincipal solution in place of y(x). If, in addition, (2.0) is disconjugate on [xo,

and yo(x) is a principal solution, then

yo (or -yo) will satisfy yo(x) > 0, yo'00 < 0 on [xo,

We show how a result of Marie and

Tomic [5] can extend to characterize the asymptotic behavior of the principal solution in the case where f(x) < 0. In this case, Sturm's theorem ensures the disconjugacy of (2.0), and so the existence of a positive decreasing solution is guaranteed. Hence, for the remainder of this paper, we consider the equation (p(x)y')

- f(x)y

= 0

(2.1)

where f is positive and continuous, and p is positive and differentiable on [xo, Theorem 2.1

Let y be a positive decreasing solution of (2.1) on [xo,

nondecreasing and bounded on [xo,

and let p be

Then y is a slowly varying function if and only if

l i m x j / ( i ) d i = 0.

(2.2)

X

Proof:

Suppose that y is slowly varying. Then, since y is convex, Lemmas 2 and 4

imply that y is an nsvf and so ——>0 y

as

Equation (2.1) can be written in the form

or

p'y'

Integrate and multiply by x to obtain

-+ P

+

y_ \y J

+ p

= /,

y_ \y ;

80

L J. Grim and L.M. Hall

x j ^ - d t + xjpl^dt + xj^\t^dt = x j f d t > 0. \ y \ \y) \t \ y j Since y is slowly varying and p is bounded, the second and third integrals on the left tend to zero as x —> oo; the first is nonpositive. Hence as

ï J / ¿r -> 0

x —> .

Conversely, suppose

0 as

i j f dt

Then integration of (2.1) from x to yields py'\~ =

J/ydt. X

Since y is decreasing, and y' —» 0 as x —> -xp(x) ^

y(x)

we have

- — ] f ( t ) y m * * J f i t ) dt y(x)

and so there is a ^ > x such that -xp(x)y

which tends to zero as x —»

Thus

^ = x Jf / ( t ) d t == s(x) y(x)

= k exp

r c ( 0 dt [tp(t)

and by Lemma 1, y is an nsvf. This completes the proof of Theorem 2.1. Theorem 2.2 Let y be a solution of (2.1) on [xo,°°) such that y is positive and decreasing, and let p be a positive nondecreasing nsvf on [xo,«•=), with lim x _» «,p(x) = 1 . Then y will be a regularly varying function of index a = (1 - V( 1 + 4c))/2 if and only if lim x j f ( t ) d t = c.

^.3)

x

Remark. The restriction limx

oop(x) = 1 is a convenient normalization and can be

replaced by the requirement that p be bounded. Proof : As in the proof of Theorem 2.1, integration of (2.1) leads to / ^ 4—rff+4P\ — 1^+441'— Idt=x\fdt • 1 u J M j J t { y

(2.4)

Suppose y is regularly varying of index a. Then y = x a L(x), where a < 0 (by Lemma 3)

Self adjoinI differential equations

81

and L is an svf. Using Lemma 5, we can show that xL'/L —> °° as x —>

so that L is an

nsvf. Integrating the second integral in (2.4) by parts and simplifying gives +

=

fit)

dt,

and since y'/y = L'/L + a/x a n d p ( x ) —> 1 as x —»

we get xjf(t)dt

a2

- a

as x —» «>. Now assume : J f i t ) dt —» c

as

and let a be the negative root o f a 2 - a = c. For k > 0, let y(x) - k exp

r J

w oi - a m dt t

\

so that rj(x) = ^ -

+ a

and T) satisfies +

y ( x )

J +

j

= f(x)y(x)

(2.5)

Now integrating (2.1) and using the definition of T) leads to 0 < (t7(jc) - a)p(x)

a

2

- a

x

and so T] is bounded. If we now divide (2.5) by py, integrate, and multiply by x, we get

where the right side can be expressed as c + 8(x), 8(x) —> 0 as x — » I n t e g r a t i n g the term containing T)' by parts, and using the fact that a 2 - a = c, (2.6) becomes

x]

(2.7)

nm^M^su.

It is easy to show that the first integral tends to 0 as x tends to

and so it can be absorbed into

the 8(x) term. Thus the other integral converges. Since we know that ri(t) - a > 0, T|(t) - 2 a > 0 as well, and so the sign o f the last two terms on the left side o f (2.7) depends only on the sign o f T|. Clearly, if T) is ultimately of constant sign, T)(x)—> 0 as x—> 0.

82

L.J. Grim and L.M. Hall

Suppose T|(x) has infinitely many zeros Xj°. Let x; be the nearest point where T| has a local maximum to the left of x,0. From (2.5) we have, after some calculations, T](t) -t

J

a

-t

i-v'f/W -Jxr

J

I

dt

pit)

As before, for x; large, the first integral tends to zero and the right side tends to c, so x

p(t)-2ay

) +

a -a

d t +

^(x)

= c

1

x,

+Y(Xi)

VXi

X

\Xi /

i7

where y(x)—» 0 as x—»0. Further simplification and the use of the intermediate value theorem yields f

+ T 1 (x i )-y(x i )-Y(x°)

n(4)(n(g-2a)fi

h.

\

V

The right side —»0 as xj —>

and the left side is the sum of positive terms, so Tl(xj) —> 0 as

x; This, along with Lemma 1, means that y(x) = x«Lo(x), and so y is regularly varying with index a . This completes the proof of Theorem 2.2. Theorem 2.3. Let y be a solution of (2.1) on [xo,) such that y is positive and decreasing, and let p be a positive nondecreasing nsvf on [xo,°°), with limx _>^p(x) = 1. Then y will be rapidly varying if and only if for each k > 1,

J

kx

lim*

(2.8)

f ( t ) d t = °°

X

Proof: Suppose that y is rapidly varying, and let k > 1 be given. Since y is decreasing, integration of (2.1) from x to kx yields: r

-xy'(x) y(x)

.

' p(kx)V

P(x) 1 V v p(x)

y'(kx)^ y'(x)

/

°° provided that y'(kx)/y'(x) does not —> 1 as x—» 0 and so we only need to consider the cases where the

Self adjoint differential

equations

83

lim sup is infinite and finite, respectively. Suppose lim supi_»«())(xi) = °° . Then for 8 > 0, there is a

x;
x, and let y be a positive decreasing solution of (5.1) on (x 0 ,~). Then y is a svf if and only if lim x _ iM x Jx°° f(t) dt =

lim x ^ o o x ix°° g(t) dt = 0.

86

L.J. Grim and L.M.

Hall

Proof. Suppose that y is slowly varying. Since y is convex, y is an nsvf and hence xy'/y - 4 0 as x -> Write (5.1) in the form y"(x)/y(x) = f(x) + g(x)y(h(x))/y(x), integrate and multiply by x to obtain f

v \2

1 ( ty'(t)

y(0 )

dt = x j f ( t ) dt + x j g ( t ) ^ » d t .

Since y is slowly varying, the integrals on the left —» zero as x —>

Since both of the

integrals on the right are nonnegative, each of these must also - » 0. Conversely, suppose that both x Jx°° f(t) dt and x Jx°° g(t) dt —> 0 as x —» Integration of (5.1) from x to °° yields y'(°°) - y ' W = i x ~ f(t)y(t) dt + Jx°° g(t)y(h(t)) dt Since h(t) > t and y is decreasing with y' —> 0 as x —» xy'(x) y(x) which

we have

< x j f ( t ) dt + x j g ( t ) dt = x j ( f ( t ) + g(t)) dt S e(x),

0 as x -> ">o. The proof of the first part of Theorem 5.1 also yields the following result.

Theorem 5.2. Let h(x) > xo for x e (xo, 0} with the topology induced by the system {p n } of seminorms given by p„(x) = max |x^ n ^(a)|. If we define —i < j < l A : E —> E by (yl(x))(i) = x'(a), A is continuous, and (P) with f(t,x) = Ax takes the form ut{s,t) = u.(s,t), - 1 < a < 1, 0 < < < — — tn, as —' re! n=0 n is seen by choosing x0 such that x ^ O ) = re . E x a m p l e 4. Let E = IRW and A be the left shift, i.e., (Ax)„ = x n + 1 . Each x 0 € u C 0 0 (IR + ) with x ^ = 0, n > 0, produces a solution u = (x0,®o>zo'i • • •) ' — -A-U, x(0) = 0. For later purposes, let us consider E = in more detail, cf. [6], Here, a linear continuous operator A (we write A £ L(E)) can be thought of as a row-finite matrix (a*})ij=i> where in each row an at most finite number of elements is different from zero. The topological dual can be indentified with Cn = {y = (yn) : yn € C, yn / 0 for an at most finite number of n's}. If Cjpj is equipped with the weak* topology cr( 0 such t h a t f o r a l l a(x,tK

(N2)

- a(y,s)|

write following :

i n R n + 1 and £ i n Rn, •

I l2 i i |x-y| + It-s |

< K 1

(x,t)

and

+

ia/2 '

| x | 2 + I y 12 + | t |

+ |s| J

,n+l T h e r e e x i s t A, 7 , w i t h A > 0, 7 > 0, s u c h t h a t f o r ( x , t ) e R Ia(x,t) and f o r a l l

For a l l

- 11 < A p ( x , t ) - 7

( x , t ) 6 R n + 1 , u , v e R, z , f e R n ,

|W(x,t,u,z)

- J f ( x , t , u , f ) | < A I z-f I ^ ( x . t ) " 1 " 7

|«(x,t,u,z)

-

tf(x,t,v,z)|

< A |u-v|

- M(x,t,v,z)

For each ( u , z ) e R x R , » ( • , * , u , z ) (x,t)

on

p(x,t)"2"7

( x , t ) £ R n + 1 , z € R n , and u , v e R w i t h u * v , ?f(x,t,u,z)

(N5)

(7)

Rn+1,

in

|a(x,t)

(N4)

' ? > A U|

2

T h e r e e x i s t a , K w i t h 0 < a < 1, K > 0, s u c h t h a t f o r a l l (y,s)

(N3)

(x,t)

.

moreover,

i s m e a s u r a b l e a s a f u n c t i o n of

t h e r e e x i s t numbers p a n d a s u c h

n+2 < p < « \\P°+2

< 0

«(.,*,0,0)|

that

n+2 < a < P < 0, e "jk ej3)(x) =

(3.1)

7

(x), and Jk

e(4)(x) = (Jf *

(3-2)

where

Xl

"jk

,

A

jk

Xl

)(x), £ > 0,

are the characteristic functions of B., and I.,, Jk Jk

respectively, and J £ denotes the mollifier J f (x) = £~n J(|), e > 0, where J is a nonnegative element of Cq(Kn) with supp J contained in the closed unit ball centered at the origin and ||J|| 1 = 1 . Similarly, L^IR") n x. J denotes the mollifier J (x) - t II j ( — ) , e > 0, with j a e f £ i=l nonnegative element of Cq(IR) with supp j contained in the closed interval [-1,1], and ||j||Ll^ = 1.

The corresponding t^ are chosen to be

independent of k and the summation over 7 = (j,k) e T = N * IN is replaced by summation over j £ IN. It follows from known results concerning mollifiers that e ^

£ C®(ft) for i = 2 (i = 4) if Ej k c ft (Ijk C ft) and

£ < dist(Fjk,0ft) (f < dist(Ijk,dft)). The choices (3.1), (3.2) are more convenient for discussion of the relationship between Fp spaces and Denjoy-type integrals in dimensions n > 2 due to the fact that many higher dimensional nonabsolute integrals are only defined on products of intervals. Other possibilities for o {e^, 7 £ T} are L -orthogonal bases of eigenfunctions of unbounded linear

110

T.L.Gill and W.W. Zachary

partial differential operators with purely discrete spectra.

A different 2 n type of choice consists of taking orthonormal wavelet bases for L (R ) [9]. PROPOSITION 3.1.

Let ft denote

in IRn.

an open subset

{ej^; i = 1,3, j,k e IN} are total families (1.3) for all p e [l,a>], (b) there

exists

Then (a) the sets

for LP(ft) in the sense of «Q > 0 such

{ef*); i = 2,4, j ,k £ IN} are total families J (1.3) for all p í (l,a)] if 0 < e < £0.

that the sets

for ^(ft) in the sense

of

Ve do not obtain a result in part (b) for p = 1 because the proof employs a property of the mollifiers that is valid in L^(ft) when 1 < q < DO but not when q = OD. Ve now discuss our results that functions integrable in the general Denjoy sense belong to Fp spaces.

Since the requirements for a function

to be integrable in these nonabsolute integration theories are more stringent in higher dimensions, we discuss the cases n = 1 and n > 2 separately. PROPOSITION 3.2.

Consider

with n = 1 and denote a ^

the sets { e ^ , 7 e T, i = 1,2 (or i = 3,4)}

by F ^

< 00, i = 1,..., 4.

the corresponding

If f is integrable

Fp spaces.

in the general

Suppose Denjoy

that sense,

then f e F^ 1 ' 3 ) for all p £ [1,®] and f £ F^ 2 ' 4 ) for all p £ (1,®]. In the proofs one sets ft = [a,b], where the interval may be of either finite or infinite length.

The proofs rely on the facts that the

following seminorm can be defined on the set of general Denjoy integrable functions on [a,b] [1,10,11]:

(3.3)

||f||D = U 1

X

sup E [a,b]

I / X f(y)dy|, •'a

where the integral is interpreted in the general Denjoy sense; and that

(3.4)

I/bf(x)g(x)dx| < ||f||D (|g(b)|

+

V (g; [a,b])),

111

A class of weighted-mean branch spaces

where the integral is defined as in (3.3) and g is of bounded variation on [a,b], V(g; [a,b]) denoting its total variation over [a,b]. When n > 2 there are many nonequivalent integrals that are more general than the Riemann and Lebesgue integrals [1,2,12].

For our

purposes, the most useful is the one defined in [1] (which we have generalized from dimension n = 2 to n > 2).

In the sequel we will refer

to these integrals as CD-

integrals.

PROPOSITION 3.3.

the sets { e ^ , 7 e T, i = 3,4} in

n > 2 and denote CD-integrable,

Consider

the corresponding

Fp spaces

by F ^ ,

i = 3,4.

dimensions If f is

then f e f£ 3 )(II) for all p e [l,oo] and f e f( 4 )(«) for all

P f (!,«>] • n n [ a , b ] a finite product of 1 1 i=l The seminorms

In the proofs one denotes by ft = compact intervals in IRn, n > 2. ||f||jj = sup IF(X| n (xj,..., x n ) £ I(x)

x ) I, which generalize the seminorms

(3.3), are finite for CD-integrable functions provided that the primitive function F satisfies a certain Holder-type condition on each perfect subinterval of each "component interval" where I(x) =

n II [a.,x.] eft. i=l 1 1

(i = 1,..., n) of ft

If f is CD-integrable (with F satisfying

the condition mentioned above) and g^ is absolutely continuous on [a^,bj], i = l,...,n, then one obtains an estimate which generalizes (3.4), (3-5)

IjT f g l g 2 ... g n d X l ... d x j

< \\%

for each "subinterval" I =

! (l8i(/>i)l

+

V(g ± ; [«i»^])),

n II [a-,/?-] c ft, where V denotes a i=l 1 1

one-dimensional total variation. Of the many Denjoy-type integrals in dimensions n > 2, we have chosen generalizations of the definition in [1] because for them it is

112

T.L.Gill and W.W. Zachary

y

possible to obtain the estimates (3.5) which are analogous to the one-dimensional results (3.4).

For a comparison of a number of such

integrals in the case n = 2, we refer to [12]. Ve have the following connection between the Fp(il) spaces and the Schwartz distributions, D'(ft). PROPOSITION 3.4.

Consider

D'(ll) c f J ^ O O f o r

al1

P

the sets { e ^ , 7 e T} for i = 2,4. £

Then

(1»®]-

Ve note that the implications in the preceding proposition go in the opposite direction compared to the function spaces usually employed in analysis.

Thus, for example, the elements of Sobolev spaces are

distributions with some additional restrictions.

Ve now discuss in more

detail the relationship between the F^ spaces and Schwartz distributions. Several researchers have investigated the relationship between nonabsolutely integrable functions and Schwartz distributions in the one-dimensional case [13-16].

The following theorem generalizes these

results to the case of general Denjoy integrable functions when n = 1 and to CD-integrable functions when n > 2. THEOREM 3.5. Let (3.6)

f(x)fj(x)dx, f e D(Rn),

Tf(p) = / Rn

where f is locally

integrable

in the general

in the CD sense

when n > 2, and the integrals

interpretations

in these

distribution;

respective

cases.

i.e., it is a continuous

Denjoy

sense

(3.6) have Then,

when n = 1 and the

indicated

in each case, T^ is a

linear functional

on D(Rn).

Using the results of Theorem 3.5, we can define similar generalized derivatives to the well-known Schwartz distributional derivatives by interpreting the defining integrals in the general Denjoy sense for n = 1 and in the CD sense for n > 2.

For convenience, we will refer to these

derivatives as GD-distributional derivatives.

Then we can define norms

A class of weighted-mean

branch

113

spaces

II- IpI m (m t IN, 1 < p < a>) formally obtained by replacing the L^ norms in P the usual definitions of the Sobolev norms by F norms and by ' p interpreting the derivatives as GD distributional derivatives.

If we

then attempt to define analogous spaces to the Sobolev spaces by using the norms ||-||

, it is not necessary to make the separate assumptions

F

P that f e Fp because this follows from Propositions 3.2 and 3.3 (for the choices of {e^, 7 c T} therein indicated).

However, there is a

complication in this procedure which is not present in the definition of Sobolev spaces due to the fact that elements of Fp need not be GD-integrable, as the following example shows. EXAMPLE 3.6. It is well-known in integration theory that the series 00 sin mx f(x) = E log m i s a w e l l " d e f i n e d 2x- periodic function on R, but is not GD-integrable.

However, it can be shown that f £ Fp(0,2ir) for all

p £ [l,m) by choosing {e m } as the following complete orthonormal set on L 2 (0,2t): e m (x) = (2T)" 1 / 2 exp(imx), m £ I,

(3.7) and choosing {t p } by

0, (3.8)

tr -

r = 0 r" 2 , r t 0.

T

If 1 < p < 00, there is a method by which Sobolev spaces can be embedded in Fp-type spaces similarly to the way that L^ spaces are embedded in Fp spaces which bypasses the difficulties mentioned above concerning the GD distributional derivatives.

The idea is based on the

fact that, if 1 < p < od, the Sobolev spaces V™(K n ) coincide with

Xj(R n ) = {gfl*u = (-A) a / 2 u; u £ LP(K n ),

0 < 0 < m, 0
1) be open and suppose o

unbounded symmetric extension multiplicity

operator

i which has a purely with corresponding

p e [1,oj), the spaces

on Cq(H) c L (ft) with a discrete

spectrum

eigenfunctions

Bp(ft) with norm

that A is an selfadjoint

Oj}^.,,,

{ e j}j = .(„•

finite Then, for

each

A class of weighted-mean branch

115

spaces

1 1-lln ,Wn) = ( Ê t. |(i •.e.)!?) /? J P j="OD J

and S (ft) with norm

CD / P^/P p l-lls m = ( s 1.1(• ,e.))| ) J P*' > j=-OD J are equivalent in the sense that the sequences of positive •ft-:}?

/

j {tj}^

can be chosen such that

CD

E t- =

numbers

GO /

E t. = 1 and

J =- OD J J =- CD 1/P Pil/P E t. |JA.|P < co so that ll^lljv = ( S Jt-M|P) Nilsv in) f°r al1 J j=-oo P ' j=-oo P > i

£ Kp(ft) n S p ( 0 ) .

The class of operators considered in ([18], Theorem 14.6) satisfy the conditions of this proposition. Proposition 4.1 shows that differential operators may be "removed" from the definition of the norms of some Fp-type spaces. More information can be obtained about the equivalence of Fj spaces if one considers the case in which the functions {e^, ) £ T} form complete orthonormal sets of basis functions for the o corresponding L (ft) spaces. PROPOSITION 4.2. Consider two spaces F(a'b)(ft) with norms of the form (1.4) (with p = 2) with corresponding numbers {t(a'b^, y e p(a'b)} satisfying (1.2) and functions {e(a,b), complete orthonormal

7 e r(a'b)}

2

which are each a b

bases for L (ft). Then, the F^ ' )(ll) norms are

equivalent if there exists a constant C > 0 such that

(4.1)

|E

tja)(eja), e(b))(e^), eja)) | < C2t(b) S^,

for all k e r(b) and a constant B > 0 such that

(4.2)

IS

for all j e T(a)

tib>(4b>, e(a))(eja), e(b))| < B2 tja) i

T.L.Gill and W.W. Zachary

116

Me note t h a t the summation inside the absolute value sign in [ ( 4 . 2 ) ] reduces t o 5 j j / [ > k k / ] when t j ) = a

j e rW

[k

l[t£b)

(4.1)

= 1] f o r a l l

However, when ( t f a ) , j e T ^ } [ { Kt , k £ I ' M } ] J s a t i s f i e s ( 1 . 2 ) , one does not expect t h a t the orthogonality of the £

r(fe)].

q u a n t i t i e s on the l e f t - h a n d sides of ( 4 . 1 ) and ( 4 . 2 ) w i l l be preserved. The following example i l l u s t r a t e s t h i s

EXAMPLE 4 . 3 .

statement.

2 Consider the space L ( - 1 , 1 ) and the two complete

orthonormal s e t s : fal 1 e^ ; ( x ) = — exp(iirmx), m £ 2,

ft

m

• ^ P . ( x ) ,

eM

» = 0,1,...,

where { P m ( x ) ; m = 0 , 1 , . . . } are the Legendre polynomials.

Then ( [ 1 9 ] ,

p.

231), — - r1 ajk E / 1

where

a

e

j

( a ) , 1 (b), x. W ek W d x

=

12k + 1 - k T~ 1

Jk+l/2^Tj)

® e s s e l function of the f i r s t kind.

(a)

0, =

4

U/2

'

Then, i f we choose

j = 0

ijr2>

J * o,

we f i n d , f o r example, t h a t

s t(a) irz j=-„ J J2

a

J4

= - _L * 7V5

0.

ACKNOWLEDGMENTS This research was supported under NSF grant DMS-8813313, AF0SR c o n t r a c t F49620-89-C-0079 and ARO c o n t r a c t DAAL03-89-C-0038.

A class of weighted-mean branch

117

spaces

REFERENCES 1.

V.G. Celidze and A.G. Dzvarseisvili, The Theory Integral

2.

and Some Applications,

R. Henstock, The General

of the Benjoy

Vorld Scientific, Singapore (1989).

Theory

of Integration,

Clarendon Press,

Oxford (1991). 3.

S. Saks, Theory

of the Integral,

4.

L.C. Evans, Veak Convergence

Monografie Matematyezne, Varsaw

(1937). Differential

Equations,

Methods

for Honlinear

Partial

Conference Board of the Mathematical

Sciences, no. 74, Amer. Math. Soc., Providence, R.I. (1990). 5.

T.L. Gill, Trans,

Amer.

Math.

Soc., 279, 617-634 (1983).

6.

T.L. Gill and V.V. Zachary, J. Mathematical

Phvs..

28, 1459-1470

(1987). 7.

J. Kuelbs, J. Functional

8.

V.Y. Steadman, Theory

Analysis. of Operators

Howard University (1988). 9.

Y. Meyer, Ondelettes

5, 354-367 (1970). on Banach

Spaces,

Ph.D. thesis,

t

et Operateurs

I. Ondelettes,

Hermann, Paris

(1990). 10.

A. Alexiewicz, Colloa.

11.

V.L.C. Sargent, J. London

12.

K.M. Ostaszewski, Henstock Math.

Math..

1, 289-293 (1948).

Math.

Soc., 28, 438-451 (1953).

Integration

in the Plane,

Memoirs

Amer.

Soc.. 63, No. 353 (1986).

13.

S. Foglio, Proc.

London

Math.

Soc.. (3), 18, 337-348 (1968).

14.

S.F.L. de Foglio, J. London

15.

S.F.L. de Foglio and R. Henstock, J. London

16.

P.Y. Lee, Amer.

17.

V.P. Ziemer, Veakly

Math.

Soc.. (2), 2, 14-18 (1970). Math.

Soc.. (2), 6,

693-700 (1973). Math.

Monthly.

77, 984-987 (1970).

Differentiable

Functions,

Springer-Verlag, New

York (1989). 18.

S. Agmon, Lectures

on Elliptic

Boundary

Value

Problems,

19.

V. Magnus, F. Oberhettinger, and R.P. Soni, Formulas

D. van Nostrand, Princeton, New Jersey (1965). for the Special New York (1966).

Functions

of Mathematical

Physics,

and

Theorems

Springer-Verlag,

Third International Colloquium on Differential D. Bainov and V. Covachev (Eds) © 1993

Equations

pp. 119-132 (1993)

Global existence of holomorphic solutions differential equations with complex

of

parameters

Joji KAJIWARA Department of Mathematics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan Abstract. Stein

manifold and

concerning (z,r)eDx that

Let D variables

M.

We

be a cylindrical domain in C n , M be a

T be a zeC"

linear partial differential operator with

coefficients

holomorphic

give a neceessary and sufficient condition

, for any holomorphic function g in D x M,

77= g has

a

global

in

holomorphic

solution

f

condition concerns with topological properties of

in

the

equation

Dx

M. T h e

D and analytic

properties of T. Keywords: Global existence,

Holomorphic solutions

Introduction Early in 1956 L. Ehrenpreis [1] discoursed on an application of the sheaf theory to differential equations and gave a criterion for the existence of global solutions of differential equations 77 = g in a domain D when the local existence of solutions for g are assured. J. Kajiwara [2] applied this Ehrenpreis' method to linear ordinary differential equations with meromorphic coefficients and gave a necessry and sufficient condition for the global existence in the meromorphic category. In the holomorphic category the condition is that D is either a simply connected domain or a doubly connected domain without non trivial global single-valued holomorphic homogeneous solutions in D.

J. Kajiwara

120

H. Suzuki [11] stated that the golbal existence of differential eqution du/dx^ = f with complex parameter depends on the Steinness of the sets of cuts over the parameter space. J . Kajiwara-Y. Mori [7] connected [2] with [11] and discussed the global existence of more general ordinary differential equations with complex parameters. J.Kajiwara-K.H.Shon[8] generalized the results to the case of parameter spaces Stein. Concerning partial differential equation, J. Kajiwara [3]-[6] discussed several concrete cases. In the paper [10] K. H. Shon reported promptly the main theorem and showed sheaftheoretically a route of its proof. The aim of the lecture in this third Colloquium is to give a necessary and sufficient condition for the global existence stated in the last paragraph in case of polycylindrical domains. This is a preparation to the fourth Colloquium where we will discuss the case that the cuts are cylindrical but vary with the parameter r e M I . Global existence and local existence In this report, a connected and open set in a topological space is called a domain. Let m be a positive integer and, for each integer j with 1 i / ' i m , Dj be a domain in the complex plane C. We put D = Di x D2 x ••• x DM (1.1). The set D is a domain of holomorphy in the complex m-space C m . Let M be a Stein manifold. We use the manifold M as a parameter space, denote by r a point of M and regard it as a complex parameter. We consider the product manifold Dx M. For each i with 1 i / ' l m , let a' = (a)k{z, r))

be a square matrix of degree m

whose each (J, k) element djk(z, r) is holomorphic function in Dx M. For each / with 1 i / ' i m , let operator defined by

be a differential

T i - ^ + a 1 (1.2). azi Let ODXM be the sheaf of germs of all holomorphic functions on Dx M. Then each differential operator T, defines the sheaf homomorphism Ti:

(OD x M)m

^ (OD x MT

(1.3).

Global

121

existence-2

(0DXM ) m.

of

We put S =

T1T2

(1.4)

•••Tm

and, for any positive integer / with 1 i / ' i m ,

S

i

= T

l

T

2

-

T

we also put

(1.5).

i

Let H(D x M) = H°(Dx M, (OD x Ai)m)

(1-6) 1

be the set of global sections of ( O d x m ) " over DxM. H°(Dx M, Odxm ) is the C-module of all holomorphic functions on D x M . H(D x M) \s the H°(DXM,0DxM )-module of all m-column vector valued holomorphic functions on DxM. Let Ker S and Im S be, respectively, the kernel and image of the homomorphism S: H(D x M)

— > H (DxM)

(1.7).

For any positive integer i with m>i >1, let Ker 5/ and Im 5/ be , respectively, the kernel and image of the homomorphism Si:H(DxM)

—>

(1.8).

H ( D x M )

Their elements are m-column vector valued global holomorphic functions on D x M . We obtain the following lemma by induction with on / (

^

( 0

D x M

)

m

-Sp->

Im

Sp

—>0

(ODXM) 1 " is the canonical injection.

(1.10) The above

short exact sequence of sheaves induces the following long exact sequence of cohomologies over D x M\

J. Kajiwara

122

0 — > H ° ( D X M, Ker Sp) —> H ° ( D x M , l m Sp)

—> H x

H(DxM) - V >

M, Ker Sp)

—> (1.11)

—>•••

U1(DxM,(0DxM)m)

Since D x M is a domain of holomorphy from the theorem of Weierstrass, w e have H1(DxM,(0/,xjii)m) = 0 (1.12) from the t h e o r e m of O k a - C a r t a n - S e r r e . From Proposition 1, w e have Im Sp = (Odxm)™H e n c e w e have the following Proposition: P R O P O S I T I O N 2. For H ! ( D x M, Ker Sp) = H (D i.e., H ' ( D x M , Ker Sp) if and only if H (DxM) = SP(H(D

the above D, M and Sp, there x M) /SP(H(D x M))

holds (1.13)

=0

(1.14)

x M)

(1.15).

Especially, in c a s e that p= proposition: PROPOSITION operator S given H ' ( D x M, Ker S) i.e., H \ D x M, if and only if H ( D x M) =

m, w e have the following

3. For the domain DxM and the in ( 1 . 1 ) - ( 1 . 7 ) , there holds = H ( D x M) / S(H(D x M))

differential

Ker S) = 0

(1.17)

S ( H ( 0 x M))

(1.18).

(1.16),

By induction with respect to p, w e have the following Proposition: P R O P O S I T I O N A. holds Hl(Dx then w e have

If, for an integer

M, Ker Sp) = 0

H l ( D x M , Ker T\)

=0

p with

1< p< m,

there (1.19), (1.20).

Global

PROPOSITION 5. the

domain

D,

,D2

, . . . and

Dp

the are

Let Stein simply

p

be

a

existence-2

positive

manifold

M

connected,

123

integer and then

the

with

1 i(-k,x)

S (*)

k e IR.

i>T{-k,x)

=

e ' h x f r ( k , x ) . m(k,x)

=

(1.5) mi(k,

x)

nir(k,

x)

and J

obeys the vector equation + 2ikJ

m'(k,x)

=

V'(j)

m(k,x).

(1.6)

k G IR,

(1.7)

we see t h a t (1.5) is equivalent to m(—k,x)

= G(k.x)qm(k,x),

where T(k)

G(k,

-R(k)e2

-L{k)e-2ikx

(l.S)

T{k)

If V G L\, then t h e functions mi(k.x) and mr(k,x) are continuous for k G IR and can be extended analytically in k to C + . Moreover, mi(k,x) —> 1 and mr(k,x) —* 1 as & —> oo in C + . Here C + denotes the complex upper half-plane and C+ denotes its closure. Similarly, C~ and C~ denote t h e complex lower half-plane and its closure, respectively. T h u s mi(—k,x) and mT(—k,x) can be extended analytically in k to C - . Hence, when S ( k ) is given, (1.7) represents a Riemann-Hilbert problem for the vector function m(k. x). Once (1.7) is solved, the potential can be obtained from (1.6) using m"(k,

V(x)

x) + 2ikm'i(k, rrii(k,

x)

m"(k.

x) — 2ikm'r(k.

x)

mr(k,

x)

x)

(1.9)

provided, of course, mi(k,x) and mT{k.x) both yield the same potential. In this sense, solving the Riemann-Hilbert problem (1.7) amounts to solving the inverse scattering problem for (1.1). In addition to t h e scattering matrix S ( k ) we introduce the m a t r i x J S(k)

J =

T(k) — L(k)

-R(k)

(1.10)

T(k)

and consider, in analogy to (1.7), the vector Riemann-Hilbert problem n( — k, x) = J G(k.

x) J qn(k,

x).

k G IR.

(1.11)

It turns out t h a t we must allow for n(k. x) to have a singularity at k = 0. T h u s we seek a solution vector n(fc, x) =

j such that, for each fixed j; G IR, ni(k. x) and nr(k,

x) are

continuous in k in I R \ {0} and have analytic continuations to C + such that ni(k,x) — 1 and rir(k.x) —> 1 as A' —> so in C + . The exact behavior of n(k.x) near = 0 depend? >> 'n

Riemann-Hilbert the scattering matrix S ( k ) .

problem

135

A s s u m e for t h e m o m e n t t h a t t h e r e e x i s t s a p o t e n t i a l

c o r r e s p o n d i n g t o t h e s c a t t e r i n g m a t r i x J S ( f c ) J . Let gi(k,x) s o l u t i o n s a s s o c i a t e d with U{x)

a n d put r>i(k,x)

=

a n d gT(k.x) x) a n d nr(k.x)

gi{k,

U(x)

d e n o t e t h e .Jost = e'kx gT(k,

x).

S i m i l a r l y to ( 1 . 6 ) , we have t h a t x) + '2ik J n'(k,

n"(k.

x) = i'(x)

n(k,

x).

(1.12;

D e f i n e t h e ' 2 x 2 m a t r i x M ( f c . x) by

M (k.x)

Let 1 =

and e =

=

1 -1

rri[(k. x) + ni(k,

-

mT(k.

x)

x) — nr(k,

x)

m/(k,

x) — n/(k,

mr(k,

x ) + nr(k.

x)

(1.13)

x)

Then

m(k.x) n(k.x)

= M(ifc,x)i,

(1.14)

JM(k,x)e.

(1.15)

-

W e can c o m b i n e ( 1 . 7 ) and ( 1 . 1 1 ) into t h e m a t r i x R i e m a n n - H i l b e r t p r o b l e m M ( - b )

= G ( l M ) q M ( M ) q ,

fc

€ IR,

(1.16)

w h e r e M ( f c , a ; ) is c o n t i n u o u s for k € 1R \ { 0 } a n d has an a n a l y t i c e x t e n s i o n in k to C + , a n d M(A'. x) —» I as k —• oo in C + for each x. In t h i s p a p e r we only c o n s i d e r t r a n s m i s s i o n coefficients for w h i c h t h e following d i c h o t o m y holds: (i) T(k) c is a real n o n z e r o c o n s t a n t , or ( i i ) T ( k ) —» T ( 0 )

= ick + o ( l ) as k —» 0 where

0. T h e n /?(0) = 1 ( 0 ) = —1 in case (i)

a n d 0 < |fl(0)| = |i(0)| < 1 in c a s e (ii). F o r t h e class L\ t h i s d i c h o t o m y is known to b e valid [ 2 ] . [3], W e will call a p o t e n t i a l V(x) a c c o r d i n g to ( i ) ( ( i i ) ) . N o t e t h a t V(x)

or U(x)

and U(x)

g e n e r i c ( e x c e p t i o n a l ) if T(k)

I t will be shown t h a t t h e r e always e x i s t s a p o t e n t i a l U(x) J S ( £ ) J . but U(x)

behaves

have t h e s a m e t r a n s m i s s i o n coefficients.

will g e n e r i c a l l y be n o n u n i q u e .

whose s c a t t e r i n g m a t r i x is

In t h e g e n e r i c c a s e t h e p o t e n t i a l

c a n n o t lie in L\ s i n c e at k = 0 t h e reflection coefficients for U(x)

U(x)

have t h e ' w r o n g ' value

+ 1 i n s t e a d of —1. N o t e t h a t ( 1 . 1 6 ) is a g e n e r a l i z a t i o n of t h e s t a n d a r d

Riemann-Hilbert

p r o b l e m in t h e s e n s e t h a t M ( & , x ) is not r e q u i r e d to be c o n t i n u o u s at k = 0. F o r m u l a t i o n s o f t h e inverse p r o b l e m as a m a t r i x R i e m a n n - H i l b e r t p r o b l e m in t h e f o r m ( 1 . 1 6 ) . or in different b u t e q u i v a l e n t forms, h a v e a p p e a r e d previously in t h e l i t e r a t u r e [4],[5], [6j. In [7], D e g a s p e r i s a n d S a b a t i e r use a D a r b o u x t r a n s f o r m a t i o n to c o n s t r u c t a o n e - p a r a m e t e r f a m i l y of p o t e n t i a l s a s s o c i a t e d w i t h a s c a t t e r i n g m a t r i x of t h e f o r m J S ( f c ) J . In s o m e e x a m p l e s such f a m i l i e s have also b e e n o b t a i n e d b y different m e t h o d s [5], [8]. [9]. T h i s raises t h e q u e s t i o n of w h e t h e r t h e p o t e n t i a l s o b t a i n e d b y t h e s e m e t h o d s r e p r e s e n t all possible p o t e n t i a l s t h a t c a n b e a s s o c i a t e d with a given s c a t t e r i n g m a t r i x of t h e f o r m JS(k)J.

E v e n w i t h r e s p e c t t o t h e s c a t t e r i n g m a t r i x S ( k ) o n e c a n ask w h e t h e r t h e r e are

a n y p o t e n t i a l s o u t s i d e L\ t h a t have S ( k ) as t h e i r s c a t t e r i n g m a t r i x . W e a r e n o t a b l e to answer t h e s e q u e s t i o n s in full g e n e r a l i t y y e t , b u t we have r e s u l t s if t h e s i n g u l a r i t y t h a t M ( & , : r ) can h a v e at k = 0 is s u i t a b l y r e s t r i c t e d . t h e r e are no p o t e n t i a l s a s s o c i a t e d with JS(k)J

Under t h e s e r e s t r i c t i o n s , we find t h a t

o t h e r t h a n t h o s e given by a o n e - p a r a m e t e r

f a m i l y , and t h a t t h e r e are no p o t e n t i a l s o u t s i d e L\ a s s o c i a t e d with S(A').

M. Klaus

136 2. E X I S T E N C E OF SOLUTIONS

In order to establish the existence of solutions of the Riemann-Hilbert problem (1.16) we need to consider the zero-energy solutions of (1.1), i.e. the solutions of rj}" + V{x)i> = 0.

(2.1)

We divide our analysis into two parts according as V has bound states or not. First suppose t h a t V has no bound states. Two solutions of (2.1) are given by mi(0.x) and m r ( 0 , x ) . It is well known t h a t if V is generic, then mi(0,x) and m T ( 0 , x ) are linearly independent while if V is exceptional, then m ; ( 0 . x ) and rar(0,x) are linearly dependent. Moreover, m / ( 0 , x ) = 1 + o ( l ) as x —> + o o , m ; ( 0 , x ) " — c; x + o(x) as x —> — oo, mT(0, x) = cT x + o(x) as x —> + o o and mr(0, x) = 1 + o ( x ) as x —• — oo, where c r = c; > 0 on account of the Wronskian and the fact that in the absence of bound states m / ( 0 , x ) and m r ( 0 , x ) are strictly positive [10], [1]. In the exceptional case. c r = c; = 0 and m ( (0.x) and r n r ( 0 , x ) are bounded. We refer the reader to [1] for more details about the solutions of (2.1). In the generic case, we introduce a one-parameter family of solutions of (2.1) by

*(x;a)H

f m/fO, x) + a m r ( 0 . x) in x)\ ' (. m (0, r

if •(it

0 < a < oo, a = oo.

(2 2)

"

Let x'(*;a) Then p{x\ a) satisfies the Riccati

,„ „s

equation V(x) = p'{x;a) + p{x;a)2.

If ip(k,x)

is a solution of (1.1),

then

(k, x; a) = ip'(k, x) — p(x; a) i/j(k, x) is a solution

(2.4)

of (1.1) with the new

(2.5)

potential

U(x;a)

= p(x;a)2

— p'{x\a).

(2-6)

In the exceptional case we define x ( x ) = mi{0, and p(x) = m|(0, x ) / m ; ( 0 . x) for, since ni/(0, x) and m r ( 0 , x ) are linearly dependent, the parameter a becomes superfluous. Applying t h e transformation (2.5) to the Jost solutions fi(k,x) and fT(k,x) associated with V(x), we find for the Jost solutions gi(k.x;a) and gr(k,x:a) associated with U(x;a) that gi(k, x; a) = — [}{{k, x) - p(x;a) fi(k,x)}, gT{k, x; a) = ~ and hence

IK

[.f'T{k. x) - p{x\ a) fT(k, x)].

ni(k, x; a) = — [m'[(k, x) + ikmt(k, ik

x) — p(x\ a) mi(k, x)],

nr(k, x; a) = — [ m ' T ( k . x) — ik mT(k, x) — p(x\ a) mT(k, x)]. ?k

(2.7) (2.8) (2.9) (2.10)

137

Riemann-Hilbert problem

We will refer to t h e t r a n s f o r m a t i o n s f r o m ip to given by (2.5) a n d from V to U given by (2.6) as D a r b o u x t r a n s f o r m a t i o n s . It is straightforward to verify t h a t the s c a t t e r i n g m a t r i x corresponding to U(x;a) is J S ( i ; ) J . T h e potentials U t h a t arise as the D a r b o u x t r a n s f o r m a t i o n of potentials in L\ can be completely characterized. For similar results, see [7] and [11]. Let U be t h e family of potentials U t h a t satisfy t h e following t h r e e conditions (i) U is of the f o r m

x2 + 1

H{x)

+ ti

x2 + 1

H{-x)

+

W{x),

(2.11)

where W £ L\ a n d ti = 0 , 1 ( j = 1,2). (ii) T h e Schrodinger equation w i t h p o t e n t i a l U has no b o u n d states, (iii) k = 0 is either a b o u n d s t a t e or a half-bound s t a t e for t h e p o t e n t i a l U. We recall t h a t k — 0 is a half-bound s t a t e when t h e Schrodinger equation has a zeroenergy solution which is b o u n d e d b u t not in L2. For given ej a n d e2 w e d e n o t e by W tli£2 t h e class of p o t e n t i a l s of t h e f o r m UCUC2. Concerning condition (iii) we r e m a r k t h a t k = 0 is a b o u n d s t a t e if a n d only if = 62 = 1 [!]• T h e connection between t h e class L\ and t h e class U is m a d e precise in t h e next t h e o r e m which is a c o m p a c t version of T h e o r e m 2.7 of [1]. Let Vg (V e ) denote t h e class of generic (exceptional) p o t e n t i a l s in L\ without b o u n d states. Since in t h e generic case t h e p a r a m e t e r a enters into t h e D a r b o u x t r a n s f o r m a t i o n , we also introduce t h e set Vg x [0, oo] whose elements we d e n o t e by [V, a], r S Vg, 0 < a < oo. Consider the m a p V : {Vg x [0, oo]) U V e ^ M defined by V{[V, a]) = U{-; a) if V S Vg a n d V(V) = [/(•) if V 6 Ve. T h e n we have T h e o r e m 2.1 Suppose V has no bound states. V is one-to-one and onto. Specifically, V{Vg x {0}) = Wo.i, V{Vg x {oo}) = W li0 , V{Vg x (0, oo)) = Ux>i and D(V e ) = U0fi. Associated with a p o t e n t i a l U{x\a) there is a solution n(k,x-,a) of (1.12) a n d hence a solution M ( A ; , x ; a ) of (1.16). T h i s establishes, in t h e absence of b o u n d s t a t e s , t h e existence of solutions for t h e R i e m a n n - H i l b e r t p r o b l e m (1.16). In t h e generic case we o b t a i n a o n e - p a r a m e t e r family of solutions, whereas in t h e exceptional case we o b t a i n just one solution. We e m p h a s i z e t h a t at this point we do not yet know w h e t h e r t h e r e are any solutions of (1.16) o t h e r t h a n t h e ones just described. We will refer to t h e solutions of (1.16) obtained so far as ' s t a n d a r d solutions'. We r e m a r k t h a t in the generic case t h e m a t r i x M ( A \ ; r ; a ) has a 1/¿-singularity at k = 0. Explicitly, we can show t h a t as k —• 0 inC+

lim k M ( k , x\ a) = k-t o

icT

a

2 x{x;a)

-1

1

1 0

-1 0

ICr

2x(z;oo)

0 < a < oo,

(2.12)

T h u s we see t h a t t h e p a r a m e t e r a can be recovered f r o m t h e small-k behavior of M(A:, x; a). If 0 < a < oo, t h e n t]{x-,a) = l / x ( x ; o ) is the zero-energy L 2 -solution. Moreover, a is related to the n o r m i n g constant corresponding to r](x;a) by

Tj(x: a)2 dx

(2.13)

138

M.

Klaus

In t h e e x c e p t i o n a l c a s e M ( f c . . r ) is c o n t i n u o u s a t k = 0. W e also m e n t i o n t h a t in b o t h t h e e x c e p t i o n a l a n d g e n e r i c case d e t M ( £ , x ) = l/T(k), which implies t h a t M ( k . x ) ~ 1 exists in C + \ {0} a n d is c o n t i n u o u s a t k = 0. For t h e a b o v e details, see [1]. N e x t w e c o n s i d e r t h e case w h e n t h e r e a r e b o u n d s t a t e s T(k) h a s s i m p l e p o l e s a t k = i3j (Pj > 0). j = 1 , . . . , 7 V . Define

< ...
I as k —> + o o in C + for every x, a n d M ( - i . x ) = M ( f c , a : ) for k 6 IR. (iii) M ( & , x ) = 0(k~2)

as k —• 0 in C + for e v e r y x.

Theorem 3.1 S u p p o s e t h a t M ( f c , x ) is a s o l u t i o n of (1.16) w h i c h satisfies (i), (ii) a n d (iii) a n d w h e r e t h e s c a t t e r i n g m a t r i x comes f r o m a n L ] - p o t e n t i a l w i t h o u t b o u n d s t a t e s . T h e n t h e o n l y s o l u t i o n s of (1.16) t h a t can b e a s s o c i a t e d w i t h a S c h r o d i n g e r e q u a t i o n a r e of t h e s t a n d a r d t y p e . In [1], T h e o r e m 3.1, t h e s t r o n g e r a s s u m p t i o n M ( A : , x ) = 0(k-1) was m a d e in p l a c e of iiiil. In t h e proof of T h e o r e m 3.1 we will c o n c e n t r a t e on how to deal w i t h c o n d i t i o n iiiiV

Riemann-Hilbert

problem

139

T h e other aspects of the proof are similar as in the proof of [1], Referring back to (1.9), we also remark that it follows as in [1] that if only one component of m(k,x) (n(k,x)) is associated with a potential V'(x) (U(x)) then the other component is automatically associated with the same potential. Proof: In this proof we denote the potential whose scattering matrix enters into (1.16) by \'o(x) and we use a subscript zero to denote other quantities related to Vo(x). Assume that Vo(z) is generic. Everything goes through in essentially the same way if V(x) is exceptional. We will comment on the exceptional case only where necessary. Now fix b € (0. oo). Let Mo(fc, x; b) denote the corresponding standard solution of (1.16). where b is the parameter in I o(x] b). T h e solution M0(A". x\ b) provides us with a matrix factorization of G ( k , x ) , n a m e l y G(k,x)

-

M0(~k,x\b)

(3.1)

[qM0(/k,i;6)~1q]

where the first factor has an analytic continuation to C~ \ { 0 } and the second factor (in brackets) has an analytic extension to C + \ { 0 } . Recall that the factor M 0 ( — k,x; b) has an 0 ( 1 / ^ - s i n g u l a r i t y at k = 0 while the second factor is continuous at k = 0. Let M ( k . x ) be an arbitrary solution of (1.16) which satisfies (i).(ii) and (iii). Then we can write (1.16) as M 0 ( - f c , x; 6 ) - 1 M ( - f c , x) = q M 0 { k , x; 6 ) " 1 M ( i , x) q. Both sides of (3.2) are 0{k~2) sides must be equal to

as k

(3.2)

0. Hence, by a variant of Liouville's theorem both iAx(x)

A2(X) k2

ï

(3.3)

k

for some matrix functions A i ( x ) and A2(x). B y (3.2) and assumption (ii), we have that q A j ( x ) q = — A j ( x ) , A i ( x ) = A i ( z ) , q A 2 ( x ) q = A 2 ( x ) and A2(x) = A2(x). Hence A i ( x ) and A 2 ( : r ) are of the form Ai(x)

ai(x)

A2(X) where aj(x) and bj(x) (j (1.15) it follows that m(k,

bi(x)

-M-0

-ai(x)

a2(x)

b2(x)

b2(x)

a2(x)

(3.4)

_ '

(3.5)

5

1,2) are real functions. From (3.2), (3.3), (1.13), (1.14) and

x) = m0(fc,

n(k, x ) = n0(k,

x)

x; b)

1 -

M

1 -

+ -7i(a;) J

F Oi(x) k'2

+

n0{k,x;b),

-6i(x)Jm0(k,x),

(3.6)

(3.7)

where h(x) = - « i ( x ) - 6 i ( s ) , 7 2 ( x ) = a2(x) + 6 2 ( x ) , 9-i(x) = bi(x) - a : ( x ) and 02(x) = a2(x) — b2(x). We see from (3.7) that in order to satisfy condition (iii) in the generic case, we must have that 0 2 ( x ) = 0 [1]. We ignore this restriction for the moment. Now we insert (3.6) and (3.7) into (1.6) and (1.12) respectively, where the potentials V(x) and U(x) are viewed as unknowns. By using (2.9) and (2.10) and looking at the magnitudes of the terms as k — 30 (k £ IR). we obtain the following equations (p0 = p0(x: b))

M. Klaus

140

V(x) U{x)

= V0(.r)-21[(x),

(3.8)

= U0(x:b)-29\{x),

- 2 i 2 + il + 2 7 ;

(3.9)

- 2 ( 7 l poy = 0.

(3.10)

72 + 2 7Î 72 - - 7 2 Po = 0,

(3.11)

- 2 9'2 + 9'l + 2 e\ 9, + 2 (flipo)' = 0.

(3.12)

9^ + 29[ 92 - 29'2po

= 0.

(3.13)

We emphasize that the equations (3.8)—(3.131 are both necessary and sufficient for rn(k, x) and n(k,x) to be solutions of a Schrodinger equation. Since m(k1x) and n(k.x) are related to the Jost solutions of V(x) and l~\x) as described in Section 1, we must have that

x ) a n d ni(k,x)

—> 1 a s x —»• + o c . a n d mT(k,x)

a n d nr(k,x)

—+ 1 a s x —> — o o .

Consequently, we require t h a t -ij{x),9j{x)

0

as

,r->±oo,

(; = 1,2).

(3.14)

By integrating (3.10) and (3.12) once, we see that (3.14) implies '/'3(x) —• 0 and 9'j[x) —> 0 as x —» ± o o , ( j = 1.2). We first consider (3.10) and (3.11). The solutions 7i(i') and 72(1) can be expressed in the form Tl(*) = ¿ y ,

=

z

^

r

(3.15)

( 3

y

"

1 6 )

where w(x) and u(x) obey w' — 2 po(x: b) w = 1, u' + 2p0(x)u

(3-17)

= 1,

(3.18)

and where po(x) is related to p0(x; b) by p0(x)

v h\ + -I - ^ . = p0(x-.b)

(3.19)

Solving (3.17) yields w{x)

= xo{x~,

b)2

\o(y,

b)

2

dy + Ci

where C\ is an arbitrary constant. To solve (3.18), let

£o(z)

= Xo(^;

b)

(

J0

\o(y; by 'dy + c j

(3.21)

and note t h a t £o(-r) is a solution of (2.1) with potential V0(x) which is linearly independent of \0(x;b).

Furthermore,

=

(3-22)

Riemann-Hilbert

problem

141

Then t h e solution of (3.IS) is given by u{x)

=

¿¡oGt)-'

/

Jo

(3.23)

(0 ( y f dy + 02

where c 2 is arbitrary. Since in t h e generic case Xo(j/i b) grows linearly as y —» ± 0 0 , we see that t h e integral inside the brackets in (3.21) has finite limits as x —> ±00. Let d± = ± r ° ° Xo(y,b)-'2dy.

(3.24)

Jo

If C\ > or c\ < — d+. then w(x) and £o(x) have no zero but the term in brackets in (3.23) is zero for some unique value x = .r 0 . To see this, note t h a t £o(x) grows linearly as x —> + 0 0 if Cj > or C\ < — d+ and approaches a nonzero limit if Cj = — d+. Similarly £0(2:) grows linearly as 1 ^ —00 if < —d+ or C] > and approaches a nonzero limit if C\ = d_. Hence u ( x 0 ) = 0 and so 71 (x) becomes singular as x —> x0. This is unacceptable. If Ci 6 (—cL. cL), then there is a unique Xj such t h a t the bracket in (3.20) vanishes. Hence ty(xi) = foU'i) = 0. So either x0 ^ Xi in which case u(x0) = 0 immediately, or x0 = x-i in which case u(x) = c(x — xo) 2 + o(x — x0)2 (c ^ 0) as x —» x 0 by expanding u(x) near x = x0. T h u s in all cases 71 (x) is singular at x0. One can also see t h a t 7 i ( x ) is singular at ,ro- T h e cases where one of 71 (x) or 72(1) is identically zero and t h e other is nonzero (formally C\ = oc or c 2 = 00) can also easily be ruled out. Hence we have that 7i(x) = 72(J") — 0 f ° r a l l 1 i s the only acceptable solution. Hence by (3.6), V(x) = Vo(x). In the exceptional case X o ( x ; b) = X o ( x ) is independent of b and has nonzero limits as x —• ± 0 0 . 3 as k —> 0, or if we do not make any such assumption at all, in which case (3.3) would have to be replaced by a full Laurent expansion. < ••• < —/32 < 0. In addition

Now we turn to t h e case when V has „V bound states to (1.16) we consider the Riemann-Hilbert problem M(-fc,x) = G ( M ) q M ( M ) q , where G (Lx) =

(3.25)

-R{k)e2ikx

T(k) -L(k)e-'2ikx

(3.26)

f{k)

with f ( k ) . R(k) and L(k) given by are given by (2.14)-(2.16). Put rhi(k,x) mr(k,

s u c h t h a t m(k.x)

+ ht(k.x) x ) — hr(k,

— M ( À ' . x ) î a n d n(k.x)

rhi(k,x) x)

rhT(k,

= JM(k.x)ê.

—

fi[(k,x)

x) + nT(k,

(3.27

x)

T h e n m(k,x)

a n d n{k,x)

are

solutions of ' 1.6) and (1.121 Also >(.r1 and p(x) are the corresponding quantities given bv

M. Klaus

142

(2.2) and (2.3) with potentials V{x) and U(x), respectively. Also detM{k,x) = l/T{k) and hence M(A;,x) is invertible for k € C+ \ {0}. Similarly as in the proof of Theorem 2.2 the solution of the Riemann-Hilbert problem (3.25) provides us with a factorization of G ( k , x). namely G{k,x)

[qM{k,x)~lq],

= M(-k.x)

k 6 IR.

(3.2S)

In the generic case, we can assume without loss that the parameter a which specifies U(x) satisfies 0 < a < oo. We will suppress a from our notation. Moreover, note t h a t G(k,x) a n d G(k.x)

a r e r e l a t e d by

g

(3-29)

(*> *)=(n i S f ) * )

It is convenient to define N(fc.x) = J ^ M t M ) J ^ -

(3-30)

Inserting (3.29) in (1.16) and using (3.28) and (3.30), we obtain for k € IR

-

1 N ( - A , X ) " 1 M ( - k . x) = ^ j j ( k + i f t ) j qN{k,x)~ M{k,x)q.

(3.31)

Here we impose t h e condition t h a t M(k,x) = 0(l/k)

as

k 0

in

C+,

(3.32)

which is the same condition as in [1], We have not yet studied weaker conditions like M(fc, x) = 0(k~2). in connection with bound states. Since M(A;, x ) _ 1 and hence N ( k , x ) _ 1 are continuous at k = 0, an appeal to Liouville's theorem yields that both sides of (3.31) must be equal to a matrix function in k of the form ,V+l

iX

(3.33)

I+£(»/*)>• A^I) 3=1

Thus, by (3.31), k«

N(/fc,x)

jV+1 I + J i W q A ^ q i=i

(3.34)

Since N ( - k , x ) = N ( L x ) and M ( - f c , x ) = M(fc,x) for k 6 IR, we conclude that the matrices A_,(x) are real and t h a t A ; ( x ) = ( — l ) J q Aj(x) q. In other words, A.3(x) is of the form A,-(z) =

fij(x)

bj(x)

b}{x)

a/(x)

j even,

(3.35)

and Aj(x)

=

aj(x)

bj(x)

— bj(x)

— 00 leads to the following equations (see [1] for more details)

V(x)

=

U(x)

=

V(x)

— 27,(1),

Af

even,

U{x) - 2 7 i ( a ; ) ,

Af

odd,

U{x)-20[(x),

Af

even,

V{x)-20\{x),

Af

odd.

(3.39)

(3.40)

In the e q u a t i o n s (3.41)—(3.44) t h e u p p e r (lower) sign refers to Af even (odd). - H , + 72,-1 + H l v - I T 2 ( 7 2 i - l p ) ' = 0, -Hj+1

-202; +

(3.41) (3.42)

+ 7-2; + H l * j ± 2 7 2 J p = 0,

+ 20'!02,-1 ± 2 (02,-1 p)' = 0,

(3.43) (3.44)

-20-2,+1 + 02, + W J i i T 26' 2j p = 0.

In (3.41)-(3.44) we can let j range f r o m 1 to + 0 0 if we assume t h a t 7 j ( x ) = 0 for j > Af+1 ( j > Af t h e generic case) and 9j(x) = 0 for j > Af + 1. Since mi(k,x), ni(k,x) - + 1 as x —• + 0 0 a n d mT(k,x): nT(k,x) —> 1 as x —> —00, it follows t h a t 7 3 ( x ) a n d 0,(x) must satisfy t h e following b o u n d a r y conditions

0,(+°°) = 7,'(+°°) =

J2

Äi-fty.

(3.45)

tl 0

are respectively upper and lower solutions for (1.1). See [5] for details. Since there are upper and lower bounds for all solutions of (1.1) uniformly for all f £ Cs, we can find a maximizing sequence fn £ Cs and prove in the usual manner that there exist a subsequence which converges weakly in

L2(il)

to an optimal control /* £ CsT h e o r e m 1.2 Let 6 and a(x) trol does exist.

satisfy hypothesis [HI],

then an optimal

con-

Control of partial differential

149

equations

In order to characterize the optimal control, we next find a slightly stronger condition than [HI] to obtain differentiability of u(f)

with respect

to / as described in the following lemma. L e m m a 1.1 Suppose 6 and a(x)

satisfy

0 < 6 < |{2infn a(x) — supfi a(a;)}.

[H2J Then, the

mapping c

is differentiate

3 f ~ «(/) e

s

in the following u ( f + m

-

u{f)

sense: ^

^

w

as ¡3 -> 0, for any f £ Cs and f e L°°(i}) is the unique solution (1.6)

e

M

y

m

w

h

2

{

Q

)

such that f + ¡3f € Cs. Further, £

of

A( + (a - f - 2bu(f))£

fi,

= f u ( f ) in

= 0 on dSl.

We now obtain a characterization of the optimal control as follows. T h e o r e m 1.3 Suppose a(x) and 6 satisfy hypothesis K, M have the

[HZ]; and the

constant

property:

(1.7)

M > [I\ supa]/(26 0 small enough, we have (1-10)

J(f)>J(f

+

Pf).

Dividing by ¡3, and letting ¡3 —> 0, we use Lemma 1.1 to obtain (1.11)

/

Jn

Kft

+

< 0.

Ku(f)f-2Mffdx

Now, define p to be the solution of (1.12)

A p + [ a - f - 2bu(f)]p = - K f

in

ft,

^

= 0 on

dû.

Note that from the lower bound (1.5) for u(f), we have a — f — 2bu(f) < supQ a — 2inffj a + 26 < —36 + 26 = —6. Hence, the problem (1.12) has a unique solution p, and moreover (1.13)

0 < p < K.

Combining (1.11) and (1.12), we integrate by parts and use Lemma 1.1 to obtain / f M f ) ( K - p ) ~ 2Mf]dx Jn

(1.14)

< 0.

Letting € —• 0, (1.14) leads to (1.15)

/

g[u(f)(K

- p) -

2 Mf]dx


^-(K-

)mSln{f 0, we set

f = { '

g

if

1 0 elsewhere.

Controì of partial differential

equations

151

We deduce in the same way as above that (1.14) holds for such /e. Passing to the limit as e —» 0, we obtain an inequality analogous to (1.15) and deduce that (1.17)

in

ftn{/>0}.

Combining (1.16) and (1.17), and using (1.7), we can finally obtain (1-18)

in fi.

From (1.1), (1.12), and (1.18) we easily derive (1.9).

2

Optimal Control of Steady-State P r e y - P r e d a t o r Diffusive Volterra-Lotka Systems

This section considers the optimal harvesting control of two interacting populations. The species concentrations satisfy a predator-prey Volterra-Lotka system under diffusion.

They are in steady-state situation with no-flux

boundary conditions as follows:

(2.1)

A w + w [ ( a i ( x ) — fi(x))

— biu — Cju]

=

0 in ft

Av + w [ ( o 2 ( x ) - f2(x))

+ c2u - b2v]

=

0 in fi

du

dv

dv

dv

0 on dfl.

The functions u(x), v(x) respectively describe prey, predator population concentrations with intrinsic growth rates a1(x),a2(x). f2(x)

The functions

fi(x),

respectively denote distribution of control harvesting effect on the bi-

ological species. The parameters

i = 1,2 are positive constants desig-

nating the crowding and interaction among the species. The optimal control criteria is to maximize economic return expressed by the payoff functional (2.2)

J ( f

u

f2)

=

as in (1.2). Here Ki,K2

f {K.nfi Jn

+ K2vf2

- M J f -

M2tf}dx

are constants describing the price of the prey and

predator species, and M i , M 2 are constants describing the costs of the controls fi,

f2.

152

A.W.

Leung

As in Section 1, we assume di(x) > 0, /¡(x) > 0 a.e. in ft, and a,-6 I ~ ( f i ) ,

/¡€lw(fl),

» = 1,2.

For 0, « = 1,2, we denote C(ii,i2) = {(/i,/2)|0[l + £? 1 ( s )]

(2.11)

u a («) = ( ± ) V ' [ l + £ a («)]

(2.12)

tll(s)

where r, =

P+

^P22

r2 =

+ 4 M

(2.12)

(2.13)

with p = nwx + 1

(2.14)

q = pr2 + 1

(2.15)

u\ (s) and u¡(s) are analytic at every point in some right half plane and in this right half plane E¡ (s) —• 0 as s —• oo for j = 1,2 By rendering precise meaning through certain conventions and by analytic continuation, we find that ui(s) and u 2 (s) are single valued and analytic throughout the s-plane except for poles at the points. s = atj—k, t = 0,1,2,... (2.16) w h e r e a j ( j = 1 , 2 , . : . , n ) are the roots of Po(s) = 0 Next , we determine the constant a and b such that u(s) = aui(s) + bu2{s) (2.17) The constants a and b are evidently to be determined from the initial conditions u(0) = 1 and u(—1) = 0 However, in order to obtain such a function •u(s), we shall assume that none of the poles (2.16) is an integer, the result of removing this restriction, we shall examine at a later point, but it may be observed that in part this restriction was made at the outset, when we assumed that none of the roots PQ(S — 2) = 0 is zero or a negative integer.

T.K.

162

Puttaswamy

If we put ui(s) ui(s + l )

D(s) =

u 2 (s) u 2 (s + l )

(2.18)

We find that «,,(-1)

Ml

(-1)

(2.19)

Our next objective is to find the functional value of the Casorati's determinant D(s). By Heyman's theorem,D(s) satisfies p ( s + i )

Po(s)

(2.20)

where /3j,j = 1 , 2 , . . . , n axe the roots of P2(s — 2) = 0 We observe that, in view of (1.3), 0 is a root of P2(s — 2) = 0 let /?„ be this root. Then, Z?( S )

+ 2) FIfc=i r ( « -Pk

+ 2)

(2.21)

where k ' = i - » V p 2 + * 2 , we conclude that relation (3.8) for y(z) in which ci(A,/i,/ 2 ,... t /„) is no longer to be determined by (3.9) but by the following: Cl(Ml,/2,...,/n) =

n ^ i r ( l - / ? t ) [ / o ( 2 m - / 1 ) ^ + 2) + / i ( m - / 1 ) ^ + l ) ] -1) n L 2 C * - h) u]=2 -1)

j =

J7

(3.17) while Cj(h, li, I2, •. •, ln) for j = 2,3,... ,n are to be determined as before by merely interchanging h and lj in (3.9). In summary, we arrive at the following main theorem: Theorem: If the indicial exponents l j ( j ~ l , 2 , . . . , n ) of the regular singular point z = 00 of the differential equation (1.1) are such that the difference of no two of them is congruent to zero modulo m, then the solution y j ( z ) , j = 1,2 n about the regular singular point z = 0 as defined in (1-4), when considered for values of z of large modulus and for which arg zm ^ arg/j may be developed asymptotically in the form Vji2) ~ J2k=i.Ck(h,li,l2,---,ln)yk(z) where yk(z),k = 1 , 2 , . . . , n are the solutions (1.5) (a) In case, none of the quantities k = 1,2,... ,n is an integer or in case i i l l i s a n integer < 2 , the constants ci(hj, , l 2 , . . . , /„) is determined by (3.9) in which hj is used for h , while the remaining constants c j ( h j , I1J2,... , l n ) ar determined by interchanging ¡1 and lj. (b) In case, = N = an integer > 2, then constant ci(hj, li, / 2 , . . . , /„) is determined by the use of h = hj in (3.17), while the constants Cj(hj, k\, k2, • • •, kn) continues to be determined from (3.9) as in case (a). In arriving at the above theorem, it must be understood that we have assumed fi ^ 0 in (1.1) and the indicial exponents hj(j = 1,2,..., n) are such that the difference of no two of them is congruent to zero modulo m. Moreover, in as much as the series is yj(z), j = 1 , 2 , . . . , n are known in the present instance to be convergent for the indicated values of z, the symbol ~ as used above may be changed to = and the resulting equation may be regarded as those which join the fundeamental solutions (1.4) with the fundamental solutions (1.5) upon a

168

TK. Puttaswamy

suitably constructed Riemann surface having cut the line drawn from z — p to z = oo

Due to the limitation in space, the case when the difference of the indicial exponants about z — oo is congruent to zero modulo m will not be discussed here and will be published elsewhere as a sequel to this paper. This paper extends and generalizes [3] and [4].

REFERENCES [1] W.J.A.Culmer and W.A.Harris Jr, Convergent Solutions of ordinary Linear Homogeneous Difference Equations,Pac.Jour.of Math. 13 (1963),11111138. [2] W.B.Ford, Asymptotic Developments of Functions Defined by Maclaurin Series, Chelsa publiching company (1960). [3] T.K.Puttaswamy, Solution in the Large of a Certain nth Order Differential Equation, Proceedings of Ramanujan Birth Centenary Year International symposium on Analysis, Macmillan India Limited, 1989, 451-465. [4] T.K.Puttaswamy, A connection problem for a certain n—th order Homogeneous Differential Equation, Proceeding of the International Conference on Theory and Applications of Differential Equations, Ohio University Press, 1989, 317-321. [5] T.K.Puttaswamy, A generalization of a theorem of W.B.Ford, Proceedings of the International Conference on New Trends in Geometric Function Theory and Applications, World Scientific, 1991, 90-95.

Third International Colloquium on Differential D. Bainov and V. Covachev (Eds) © 1993

Equations pp. 169-175 (1993)

S U P E R S Y M M E T R I C QUANTUM MECHANICS AND MULTI-SOLITON SOLUTIONS OF HIGHER ORDER K-dV EQUATIONS

C. V. SUKUMAR Theoretical physics, University of Oxford 1 Keble road, Oxford 0 X 1 3NP, England

A B S T R A C T : It is shown that it is possible to construct higher order equations of the K-dV type for which explicit expressions for the ra-soliton solution may be given. For these equations an n-soliton solution and an (n + 1 ) - soliton solution may be paired as the fermionic and bosonic components of a supersymmetric system.

Keywords:

Supersymmetry, Quantum Mechanics, Solitons, K-dV Equation.

In field theory, supersymmetry is a symmetry that generates transformations between bosons and fermions. Starting from a supersymmetric field theory, it is possible to constuct supersymmetric Quantum Mechanics (Witten 1981).

A supersymmetric

Hamiltonian may be defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory (Freedman and Cooper 1983). It has been shown (Andrianov et al 1984, Sukumar 1985) that every one dimensional Hamiltonian Hi can have a partner H2 such that H\ and H2 taken together may be viewed as the components of a supersymmetric Hamiltonian. The consequences of supersymmetry for the spectra of Hi and H2 have been extensively discussed. The supersymmetric pairing may be used to generate a H2 by eliminating the groundstate of H1 or adding a state below the groundstate of Hi or maintaining the spectrum to be the same as that of H i . The standard inverse scattering method based on the Gelfand-Levitan procedure for finding a new potential by (1) eliminating the groundstate of a given potential, (2) adding a state below the groundstate of a given potential and (3) generating the phase equivalent family of a given potential may each be shown to consist of the successive application of two appropriately chosen supersymmetric transformations (Sukumar 1985).

170

C.V.

Sukumar

The connection between the algebra of supersymmetry and the inverse scattering method may be used to construct one dimensional potentials with any number of boundstates at any chosen energies. The reflection coefficient of the potential so constructed is related to the reflection coefficient of a reference potential which supports no boundstates.

All potentials connected by supersymmetry to a reference potential that is

reflectionless therefore have vanishing reflection coefficient . It has been shown that using this idea a reflectionless potential with any number n of non-degenerate boundstates at arbitrarily chosen energies may be constructed (Sukumar 1986). Choosing the reduced mass fi = 1/2, the potential with boundstates at energies Ej ,j = 1,2,...n may be represented in the form

Vn

Bij

=

d2

- 2 ^

=

j =

(1/2)

In Det

B

,

j(x,Tj)

, i,j =

[ eT'x+9'

+

( ~ y

1,2,...n + 1

,

e-T'x~e>

^ ]

,

E j = - r ? , r n > r „ _ i > .... > T i , where $j are arbitrary phase factors. It must be noted that for odd values of j , 4>j is a cosh

function and hence nodeless while for even values of

a single node. Such a structure for

j

ensures that

j , j

is a

sinh

function with

is free of singularities. In terms

Vn

of the normalised eigenstates of Vn given by = { B - ^ j n , j = 1,2, ...n

(2)

the potential Vn may be written as n Vn

=

-2

Y ,

3) in equation (13) for

C.V. Sukumar

174

which the time dependence of the phase is given by 6j{t)

=

-

2m_1 r f t .

(21)

For this class of non-linear equations, the n-soliton solution is given by equations (1) and (23) or by equation (6) with yj suitably modified to produce the appropriate time dependence of the phase. Caudrey et al (1976) derived a hierarchy of K-dV equations using the inverse scattering method. They showed that the non-linear equation of order (2m + 1) with n solitons satisfies

f

+

T? = °

where the operator Lm is to be found by solving the system of equations defined by dLj dx

dx3

4Y

i~i WM dx

dL

_

2

^

dx

W fl*

L

!

(23)

for j — l , 2 . . . m with the starting condition Lo =

Vn .

(24)

It is easy to show that j = 1 leads to the K-dV equation and j = 2 and j = 3 lead to the non-linear equations given by equations (18) and (20) respectively. Further iterations of equation (23) lead to the higher members of the K-dV hierarchy. The n-soliton solution of this hierarchy given by Caudrey et al (1976) can be shown to be identical to the solutions discussed in this paper. An explicit solution of equation (23) for the operator Lk for arbitrary k remains to be discovered. Such an explicit solution for ¿j. would provide an explicit structure for non-linear equations of the K-dV type for arbitrary order. In this paper we have constructed a hierarchy of higher order non-linear equations of the K-dV type generated from the set of equations first considered by Caudrey et al (1976). The non-linear equations which axe members of this hierarchy possess the attractive feature that explicit expressions for n-soliton solutions may be given. Furthermore it can be shown that for all the members of this hierarchy the n and the n + 1 soliton solutions are supersymmetric partners.

Quantum mechanics and multi-soliton solutions

REFERENCES A. A. Andrianov, N. V. Borisov and M. V. Ioffe, Phys. Lett. 105A 19 (1984). P. J . Caudrey, R. K. Dodd and J. D. Gibbon, Proc. R. Soc. Lond. A351 407 (1976). B. Freedman and F. Cooper, Ann. Phys. NY 146 262 (1983). C. S. Gardner, J . M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett. 19 1095 (1967). I. Kay and H. E. Moses, J . Appl. Phys. 27 1503 (1956). W. Kwong and J. L. Rossner, Prog. Theor. Phys. Suppl. 86 366 (1985). K. Sawada and T. Kotera, Prog. Theor. Phys. 51 1355 (1974). A. C. Scott, F. Y. F. Chu and D. W.McLaughlin, Proc. I.E.E.E. 61 1443 (1973). C. V. Sukumar, J . Phys. A 18 2917 (1985). C. V. Sukumar, J . Phys. A 18 2937 (1985). C. V. Sukumar, J . Phys. A 19 2297 (1986). E. Witten, Nucl. Phys. B 188 513 (1981).

Third International Colloquium on Differential Equations pp. 177-192 (1993) D. Bainov and V. Covachev (Eds) © 1993

On integral and d i f f e r e n t i a l equations arising from probability distributions Katsuo Takano College of General Education, Ibaraki University, Mito, Ibaraki 310, Japan 1991 Mathematics

Subject

Classification.

Primary 45E10,

Secondary 34A25, 60E07. 1. Introduction T h e probability density function of variance ratio distribution or t h e F - distribution with m and n d e g r e e s of f r e e d o m is as follows:

_

s

m,n

(x)

.

f((m+n)/2)

r(m/2)r(n/2) ,

x(m/2)'1

Am+n)/2'

x >

0

(1+*)

c f l l . p.946. 26.6.1], T h e variance ratio distribution is an important distribution in statistics.

(1.1)

g

We consider the density function

(x) = J ^ f r

X > 0,

where a and (5 are any positive numbers. It is known that the Laplace t r a n s f o r m of t h e infinitely divisible probability distribution G(x) (1.2)

on [0,oo) is r e p r e s e n t e d in t h e form

J™«"sxdG(x) = e x p U ^ V * * -

D^dK(x)),

(cf.[5l). If a probability distribution Gix)

on [0,oo) is infinitely

divisible and has a probability density g(x), its density satisfies an integral equation

gix)

K.

178

(1.3)

xgix) = J0g(x- t)dK(t), x > 0

with conditions that ( c l ) K(x) (c3) J

l'akano

l/x dKix)

< oo

is nondecreasing, (c2) /£(-0) = 0,

(cflSl).

Conversely, if a density function gix)

satisfies the integral equation

(1.3) with the conditions ( c l ) , (c2) and the probability distribution function Gix) divisible. is (1.4)

on [0,oo) with the density function gix)

If dKix)

is infinitely

is absolutely continuous and its density function

then (1.2) and (1.3) are written as

jJVsxdGix) =

ex?{Qe~sx

)^kix)dxl

-1

rX

(1.5)

xgix) = J0 g(x - t)kit)dt, x > 0.

For the case « = 1, the density function

(1.1) is called the density

function of the Pareto distribution and 0. Thorin [10] first obtained the function Kix)

for the case where 0 is a non-integer.

For the

case where a is any positive number and C is a non-integer, M. J. Goovaerts, L. D'Hooge, N. De Pril [3] obtained the function /£(x). The purpose of this paper is to get the function Kix)

for the case

where ex is any positive number and 0 is any positive integer. Now we get the function kit)

(1.6)

-

Jq ix-tf

of the integral equation

x>0

l+ekit)dt,

(1+x) (1+%-f) for seen any positive 0. As from thea , integral equation (1.5), we can ignore the normalizing constant number of the probability density function and hence in general we can consider the solution or the function kix)

of the integral equation (1.5).

The author would like to emphasize that the idea obtained here can also apply to other probability distributions.

Differential

2.

equations

from probability

179

distributions

On the Laplace transform of the density function of the variance

ratio distribution We make use of the confluent hypergeometric function (aL 2 (a) n (21) Mia b z) = 1 + - z + — Z - + ••• + — a 2 - +••• 2 where

n

(a) Q = 1, (fc)Q = 1, (a)

= a(a + l)(a + 2)---(a + n - l ) , ' (b) = 6(i» + l)(f> + 2)---(i» + n - l ) , b * 0 , - 1 , - 2 , n ' ' ' ' n

The series M(a,b,z)

converges uniformly in z in any compact set.

From U. (3.1.19), (1.3.1)] r(a)U(a,b,s)

we have = j j J Y ' V ^ d + t f ^ d t

and (2.2)

U(a,b,s)

=

r(l-b)

r(l+a-b)

M

(

a b s

)

+

iit^Ls\-bM{l

+

r(a)

a

.

b 2

-b,s).

The gamma function r(z) is defined on the whole complex plane except z - 0 , - 1f, - 2 , is) = L Let (2.3) e~sxg

(x)dx,

Re s > 0.

Suppose 6 = / + h, 1 = 0,1,2,..., 0 < h < 1.

Then tx = a > 0, 0 = 1 - i> > 0

and we have /

U)=

M

(2.4)

Ck±gly(

r(e)

= M(«X,1 - 0. ¿

m

/ l o * * >

We take the principal branch of logs such as 0

for 5 > 0.

If s = -t + ip, t > 0, p > 0, then U (cc,l - 0 , - f ) = l i m ^ + o y ( a , l - B,-f + ip) (2.5)

=

f V " 6 M(«x + 0 , 1 +

- p,-f) + ^

r(a+p)

r(o 0, P > 0, then C7 (oc, 1 — 0 , - f ) = lim (2.6)

+ O i/(oc,l

= -4-^-rAf(oc,l - 0 , - i ) +

- 6 , - i - ip) M(«

+

0,l

+

0,-r).

K. Takano

180 Let

/ m ^ f ^ i M « , ! - e,-r) = r ( « + g ) r ( - g ) ( _ 1 ) s i n n g . f r(p) r(i)r(«) (2.7)

M i a t + 9 l + 9

t)

= u(a,i;r).

Then /mi^£/ r(p)

+

( a , l - e,-t)

= rtcc^M-g) r(s)r(«)

. feM(a

siniTg

+

1 + t

t)

= -u(a,0;i). Let y+(K j _p

f) =

! _0

f)

r(e) +r( 0.

Then

u(oc,0;f) and y(tx,0;i) are linearly independent solutions of the differential equation

( p i e ; t ) y ' ( t ) ) ' + q{ 0.

T h e Wronskian of u(oc,0;f) and y ( a , 0 ; f ) is as follows: (3.2)

Proof.

/»(e;f)(u'((x,0;i)y(oc,0;i) - u(cx,0;f)y'((x,0;i)) = c( 0.

(3.3)

W e can make use of the fact that the functions

M i a . + 0,1 + 0,i) and M ( o c , l - 0 , f ) are solutions of Kummer's equations, t y ' i t ) + ( l + 6 - t ) y ' i t ) - ( a + &)yit)

t y ' i t ) + (1 - 0 - t ) y i t ) - ayit)

= 0 and

= 0, respectively.

With the Wronskian, W i t ) =

tf'( 0 if 0 < a < 1.

If 1 < tx < 2, it holds that Mil

M'i 1 -

,

- oc,l + 0,0) = 1 and

« , 1 + 0,f) = ^ - M ( 2 - oc,2 + M )
0.

Hence A f ( l - K , l + 0,i) has one positive zero. For 2 < a < 3, we make use of the relation such as

6(1 - 6 + t)Mia,b,t) + bib -1 )Mia -1,6 - l,f) - atMia +1,6 + l,i) = 0. cf. [l. 13.4.7].

If we let a = 2 - a ,

6 = 2 + 8, we have

(2 + 0X1 + e)M( 1 - oc,i + 6,f) = 6(6 - DMia -1,6 - i,r) = -6(1 - 6 + t)Mia,b,t) + atMia +1,6 + l,f) = (2 + e)(l + 0 - i ) M ( 2 - oc,2 + 0,f) - (ot - 2 ) i M ( 3 - a , 3 + 0,r). If we denote the zero of Mi2 - k , 2 + 0,i) by i^, we have (2 + ff)(l + e)Mil

-

0), y(tx,0;f)

and so ¿(tx,P;ii) is a finite valued function. By the fact that as r - > + 0, -1 u(a,e;i) / > ft x tan , \ ~e(oc,g)r yloc,0;f) we see that f°°l u J,

1

£k((x,e)x)dx x

1 r°°l

= ns -J J n

0 t

-t tan

-1

dt

u(cKfi]t) , , \ dt

y(

0

2

pi/;f)(u (a,/;i) + y ((x,/;i))

for the density function (1.1) with the positive number k and the positive integer I. Proof.

Suppose that & = I + h, I is a positive integer, 0 < h < 1.

Let us show that Let us put

q( + 0.

= y(a,/;f)/u(o:,/;f).

By Kummer's transformation, we have - 0(,1 + cf. [4. (1.4.1)].

lyt).

If m is a positive integer and m < oc < m + 1,

the confluent hypergeometric function Mil - cx,l + real positive zeros.

Since the functions

u(cx,/;t)

has m simple

and y(tx,/;i)

are the solutions of the differential eqution (5.1), the Sturm separation theorem tells that the zeros of

u(tan

-1

,

u(a,i,t)

for any t such that y ( a , l , t ) / 0 and by the dominated convergence theorem we obtain (x > 0)

as h~* + 0.

The function k((x,l-,x) satisfies the condition (c3).

As A-> + 0, {

(*)-» {

.is),

{'

(*)-* / ' M a,/

0,

Re

s>

fo

* > o,

hold and we have -f

(s) = f .(*)«/ tx,I

e-sxk(*&x)dx

is) L tx,i

L 0

e~sxk^,l,x)dx

Differential

for Re s > 0.

equations

from probability

distributions

By the inversion formula of the

Laplace transform we obtain tx

xg

= J* ?

- t)k( 0, the

In this talk, we will treat the equations of

More precisely, we assume that

equations have two independent first intergrals. has been well studied especially by E. Goursat

A ^ 0, and that the This class of equations

[3].

unknown parameters (r,s,t) from the equation (3).

Let us eliminate the Denote by

X^ and

Monge-Amp£re equations

195

the solutions of second order polynomial: X 2 + B X+ (AC+DE) = 0.

Then the

characteristic strip satisfies the following equations:

dz - pdx - qdy = 0

dz - pdx - qdy = 0

Ddp + Cdx + X ^ y = 0

(I)

or

(II)

Ddp + Cdx + X 2 dy = 0

Ddq + X 2 dx + Ady = 0

Definition 2.

A function

integral" of (I)

(or (II))

V=V(x,y,z,p,q)

is called the "first

if it is constant on any solution of (I)

(or (II) respectively).

Proposition 1. (Darboux [2], Goursat [3], [4]) sufficient condition so that is that

V

V=V(x,y,z,p,q)

The necessary and

be the first integral of (I)

is a solution of the following system of linear first order

equations :

L

(4)

1V

L2V

def

ÌV 3x

def

3y

D

Definition 3.

+

p D

.3V_ 3z

c

3z

Let

u

and

are independent each other, and gradient is not zero.

Then

3V_ 3p

3V 1 3q

=

0

— - A — 2 3p 3q

=

0

v

A

be the first integrals of (I) which

if> be any function of two variables whose

(u,v)

is called an "intermediate integral"

of (1).

Using the above classical notions, we try to solve the Cauchy problem for (1) explicitely.

(5)

where

The initial strip is given by

Su : (x,y,z,p,q) = (x (a), y (a), z (a), p (a), q (a)) 0 0 0 0 0 z Q (a)=p 0 (a) x 0 (a) + q0(ot) y Q (a).

(1) defined in a domain set

U.

{(x,y,z(x,y),3z/3x(x,y),3z/3y(x,y));

U

z=z(x,y) be a solution of

If the initial strip

satisfies the initial condition (5). an open domain

Let

SQ

is contained in a

(x.y) e U}, we say that z=z(x,y)

The Cauchy problem (l)-(5) is to find

where there exists a function

z=z(x,y) satisfying the

A/. Tsuji

196 equation

F=0

in

U

and the initial condition (5).

Now suppose the hypotheses: (HI)

The system of the equations (I)

(or (II))

first integrals, and denote them by (H2)

The initial strip

Remark.

SQ

u

has two independent

and

v.

is not characteristic.

Let us write

L =L,L„-L„L.. [L,,L„1. J 11 1 i del 1 1

sufficient condition for (HI) is that any bracket

The necessary and

[L^L^]

(lSi,jS3) is

a linear combination of L^, L^ and L^. u 0 ( a ) = u | s ^ and

Denote

v

0

=v

|sq'

and

T={(u,v); u=u Q (a),

v=vQ(a)}.

t

If (Up(a) ,VQ(a))? (0,0), it is easy to get locally a function satisfying

(uQ(a),vQ(o) )=0

and

(u,v)

(grad > (u Q (a) ,v 0 (a) )?i(0,0) .

would like to develop the global theory.

Therefore

defined in the large with the above two properties.

(u,v)

But we

should be

To prove the exist-

ence of such a function, we assume the following condition:

(H3)

The curve T is simple and (u^(a) >v^(a) )?K0,0) .

The sets {a;u^(a)

=0} and {a;vQ(a)=0} have not any point of accumulation.

Here we define g(x,y,z,p,q)

-

(u(x,y,z,p,q) ,v(x,y,z,p,q)) , and

consider the Cauchy problem for g(x,y,z,p,q) as follows:

(6)

g(x,y,z,3z/3x,3z/3y)

(z

=

0

,

' 8 z / 3 x ' 3 z / 3 y ) | ( x , y ) = (x 0 (a),y 0 (a))

=

(z

0 ( a ) ' P 0 ( a ) *' q 0 (a) ^ ) '

Then G. Darboux [2] and E. Goursat [3,4] have proved the following Theorem 2.

The solution of the Cauchy problem (l)-(5) is obtained

as a solution of the Cauchy problem (6).

§3. Examples.

Before stating our results in general form, we will give

some examples to explain concretely what we have done. examples first, then we will solve them.

We will give two

Monge-Amptre

equations

197

Example 1. rt - s

2

=

0

z(0,y) = (y)> p(0,y) = 4»' (y)2, q(0,y) = ' (y) 00 1 ip(y) e C (R ).

where

The first integral of the equation

, 2 rt-s =0

are

{p, q, z-px-qy}.

Therefore an intermediate integral corresponding to the 2 initial data is given by g(x,y,z,p,q) = p - q . Hence the solution of (7) is obtained as a solution of the following Cauchy problem:

(8)

If - ( !f )2

=0;

z(o,y) =

*(y) •

As this is just Hamilton-Jacobi equation, we can apply the ideas of [9, 10] to construct the singularities of solutions. Example 2. {

q

2 2 r - pqs + z(rt - s ) = 0 z (0,y) = (y),

co i (y) .

As this is the conservation law, the singularities of shock type may appear. Remark.

The above examples mean that, changing the initial data, we can

get intermediate integrals of various types.

For example, if we give the

initial data in Example 1 by

and

p (0,y) =—()>' (y)

intermediate integral is given by

g=p+q

q(0,y) =()>* (y) , then the

which is a linear equation.

First let us solve Example 1 by the characteristic method.

A system

M. Tsuji

198

of characteristic differential equations is written as dx _ d&

(11)

d^ de

1

'

x(0)=0 ,

dz = "I 2 . ^ = ^ dB de dB

-2q .

y(0)=a , z(0)=^(a) , p C O W i a )

, q(0)=' (a)

We can easily get the solutions of (11) as follows (12)

x=B ,

z=(a)-' (a)2g ,

y=a-2$'(a)B ,

Here we define a smooth mapping H: R 2

3 (a, B)

H

•

p=

0

and

We use the same notations as in Example 1. £={ (a, B) ; D(x,y)/D(a,B)=B' (a)-l=0},

^(a ) ^ 0 . u For example, we write

H(Z)=C, etc.

Though the behavior

of the solution surface of Example 2 is similar to Example 1, there exist several differences.

The principal one is that the solution of Example 2

is continuous up to the boundary C, but that it is not of C^ in a neighbourhood of C.

Moreover Proposition 3 must be modified a little.

The problem in Example 2 which is corresponding to (14) in Example 1 is expressed as follows:

(15)

^

dx

-

z

dy

= 0

Proposition 5.

in fi ; r

z=u

r

on

C D U. r

The Cauchy problem (15) has not a solution of class

C 1 for any open neighbourhood

U

of the point R.

Therefore we can not extend the smooth solution of (9) beyond the curve C.

Summing up these results, we have the following

Theorem 6.

Consider the Cauchy problem (9) under the condition (A.2),

then it holds (i)

CO 1 The Cauchy problem (7) has uniquely a C -solution in {0£x R which satisfies (2) for sufficiently large t, and s u p { | i ( i ) | : t > T} > 0 for any T > Tx. A proper solution of (2) is called oscillatory if it has arbitrarily large zeros; otherwise it is called nonoscillatory. Thus a nonoscillatory solution is necessarily eventually positive or eventually negative. We make the standing hypothesis that (2) posses proper solutions. For questions on existence, uniqueness, and continuous dependence, see Driver [6], Bellman and Cooke [2], and Hale [17]. Throughout unless we state otherwise, the following conditions are always assumed to hold: (a) a g C ' ( [ i 0 , o c ) , i 2 ) , a(t) > 0, and

(b)

p e C([t0,oc),R),

(c)

q 6 C([t0,oo),

(d)

/ 6 C(R,R)

and xf{x)

> 0 for x ± 0;

(e)

r(t)

and a(t)

—> oo as t —> oo.

2

< t, r(t)

0 < p(t)

< 1;

R\ q(t) > 0;

Lemmas

Here we present some lemmas which we will rely on in the next section.

Solutions of arbitrary order neutral differential equations

207

L e m m a 2 . 1 ( K i g u r a d z e [18]) If u(t) is an n-times differentiable function on R+ of constant sign, is of constant sign and not identically zero in any interval [i^oc), and u(t)uik]{t) < 0, then (i) there exists a t2 > ii such that the functions k = 1 , 2 , n — 1, are of sign on [ij, oo), (ii) there is an integer I, 0 < I < n — 1 with n — I is odd, such that for t > i2 w(t)u lfc) (i)

>0

& = 0,1,...,/

(-l)n-fc-1ii.(i)w"c)(i)

>0

k = / , . . . , n - 1,

constant

(4)

and if I < / < n — 1, k'-*-1^)!

>

fork

= 0,1,....,1-1.

(5)

If u(t) satisfies (4), then it is said to be a nonoscillatory function of degree I [9]. L e m m a 2 . 2 ( K i g u r a d z e [18]) Let u(t) be an n-times differentiable function on [ij,oo) withu^k\t), k = 0,..., n — 1, absolutely continuous and of constant sign on[ti,oo), and let u(t)u^k\t) > 0 for every t in [i l5 oo), then either u{t)uik](t)

> 0 k = 0,1,2,...,n - 1

(6)

or there is an integer I, 0 < I < n — 2 with n — I is even, such that for t > £x u(t)u{k]{t) (-lr-^i)«1*^) and if 1 < I < n — 1 then inequality

(5)

>0

A: = 0 , 1 , . . . , *

>0

k = l,...,n-

1,

(7)

holds.

L e m m a 2 . 3 (Q. C h u a n x i and G. L a d a s [4]) Let :,x,p : [i 0 ,oo) —+ R and c G R be such that z(t) = x(t) + p(t)x{t — c), t > i j = 0, then there exists an integer I, I £ { 0 , 1 , . . . , « } with ( —l)n~'_1(5 that for t > T z{t)z{k){t) k

(-l) ~'z(t)z^(t)

>0

k = 0,1,...,/

>0

k = /,.... n,

= 1, such

(8)

and if 1 < I < n — 1 then |.(i-*-D(i)| >

for

k = 0,1,....,/ - 1

(9)

Proof: We may assume that x(t), x(r(t)), and x(a(t)) are positive for all t > t1 > t0. The proof when x(t) is eventually negative can be constructed similarly. Clearly z(t) > 0 for t > tu

(10)

and from (1.1) 8(a(t)z^(t))' Since ¿a(*)~

(n_1)

= -q(t)f(x(a(t)))

t 1 , we can have 8a(t)z ( n ~ 1 > (t) > 0 for t > h

(11)

or Sa(t)z{n-1](t)

< 0 for t > T, > tx.

(12)

Suppose that (12) holds. Then 8a{t)z{n~l\t)

< SaiT^-^Tx)

Tx.

Integrating the above inequality divided by a(t) from Tx to t and using (3) we have 8z(n~2](t)

— - o c as t — oo.

Since this implies that 8z(t) —> —oo as t —• oo, in view of (10) we easily conclude that 5 = —1. This completes the proof of part (i) of the lemma.

Solutions

of arbitrary

order

neutral

differential

equations

209

To prove (ii) we first notice t h a t , since = ia(0=|n,(0

S[a(t)z^(t)}'

+ Sa'{t)z^\t)

=

-q(t)f(x\a(t))),

we have Sz^(t)

=

- S^lf a(t)

a(t)

{ x { l 7 m

.

(13)

Suppose that limi—«, ; ( n _ 2 ' ( i ) / oo when 6 = —1. Then it follows from (i) and (13) t h a t ¿z(n( T,

k

z(t)

>

0, fc = 0 , 1 , 2 , . . . , /

ik

>

0, k = /,/+

( - 1 ) -'z \t)

l,...,n,

(15)

and (9) holds when 1 < I < n - 1. Suppose that l i m ^ , ^ ; ' n _ 2 ) ( i ) = oo and 8 = —1. Then, ; ' n _ 1 ' ( i ) is eventually positive and so from (13) z(n)(t) is also eventually positive. It follows that zfi>, i = 0 , 1 , . . . , n are all eventually positive. But, this case is included in (15).

3

M a i n results

Eq. (2) is said to be of retarted type if cr(t) < t, mixed type if cr(t) > t. In the following theorems we do not impose any condition on the argument 0. (i) 7/5=1, then every solution x(t) of (2) is oscillatory solution x(t) of (2) is either oscillatory or satisfies

when n is even, and

liminf |x(i)| = 0 t—*oo

when n is odd . (ii) If S = —1, then every solution x(t) of (2) is either oscillatory

or else

lim \x(t)\ = oo or l i m i n f i - . ^ |a;(i)| = 0

t—too

when n is even, and every solution x(t) of (2) is either oscillatory lim |x(£)l = i—'OO

when n is odd.

oo

or else

every

A. Zafer and R.S. Dahiya

210

P r o o f : Let x(t) be a nonoscillatory solution of (2). We m a y assume t h a t eventually x(t) > 0. T h e case x(t) < 0 can be t r e a t e d similarly. By the part (i) of L e m m a 2.4, we see t h a t there exist a ti > 0 such t h a t for 0,

it is not difficult to see from (23) and (??) that liminf f(x(t)) f—.oo

= 0 or liminf t _ 0 0 x(t) = 0.

This clearly completes the proof of Theorem 3.1.

Corollary 3.1 Let ( —l)n 0 and z'(t) < 0 for t sufficiently large, limf.,^ z(t) exists. Applying lemma 2.3, we obtain that lim z(t) = 0. t —• oo On the other hand, 0 < x(t) < z(t). Therefore lim x(t) = 0 t—*oo as desired. E x a m p l e 1. Consider the equation [e - ! [x(t) + 2elr~ix(t

- tt)]'"]' + 2V2eh,/\l

+ e" f )z(i - 7tt/4) = 0

(27)

so that S = 1, n = 4, a(t) = e~\ p(t) = 2e3lr'2~t, r(t) = t - tt/2, 12, ( - l ) f c _ 1 i z ( f c , ( i ) > 0 for Jfe = 1,2,

- 1

if n is even

(32)

and (-l)k8z{k](t)

> 0 for k = 1,2, ...,77. - 1

if n is odd.

(33)

In both cases, we have lim z{k\t)

t — OO

= 0 for k = 1,2,..., n - 1.

(34)

Using (34), we now integrate (2) n-times between t and oo to obtain ( - l ) " * [ = ( o o ) - ;(i)] = - ^ r r y (n — 1 ! Jt

- Wq(s)f(x(a(s)))ds,

(35)

Solutions of arbitrary order neutral differential equations where ; ( o o ) = lim^oc

213

z(t).

If ( — = 1, then we see from (32) and (33) that z(t) z(t) is also positive, we have as in T h e o r e m 3.1, f(x( f((l

is increasing for t large.

Since

(36)

- p(a(t))c)

for t > t2 for some t2 > ¿1 and c > 0. Thus, from (35) and (36) we get z{oc)

- z(t2)

>

1 r°° — / (s-t2r-1q(s)f{(l-p(a(s))c))ds, (n — 1): Jt2

(37)

which, because of (31), implies that ; ( o o ) = oo and so contradicts to boundedness of

z(t).

If ( —1)" t2. So, from (35) it follows that -(**)>;

1 z"00 TTi / ( « - « ^ " " ^ ( ^ / ( ( l - p ^ a J J c ) ) ^ , (tj - 1)! Jt2

(38)

On the other hand, since = oo,

j™ tn~lq{t)dt it follows from (38) that l i m i n f f(x(t)) t — oo

= 0 or liminff^oo x(t) = 0.

This completes the proof of Theorem 3.2 when x(t) is eventually positive. T h e proof when x(t) is eventually negative is similar. If a{t) / 1, then the previous theorem can be generalized provided a'(t) > 0. Specifically, we shall prove T h e o r e m 3 . 3 Let a'(t) > 0 for t > t0, and let f be increasing.

r J for

every

-7rJn-lqWf( a{t)

c > 0, then the conclusion

1 - P( 0 and an integer I G { 0 , 1 } with (-l)"-'-^ = 1, such that for t > T z{k](t) (-1

)n-k~lztk](t)

>

0 fc = 1 , 2 , ...I

>

0 k = l,...n-

1.

(40)

B y Taylor's formula for r > t > T ,

--

- ty-'-^-z^isïïds

for T

(42)

(n~

Clearly if (-1)" £i such t h a t for t > i 2 , z(t) > 0, w(t) > 0, and v(t) > 0. For t > i 2 , since (a(i)t'(n_1)(i))'

p(a(t)iuin~l){t

- T))'

=

(aWw^-'^T))'+

=

( a i D z ^ W y + p i a W z ^ i t - r ) ) ' +p[(a(t)z^-1\t

=

(a(i)z'

n_1

- r))'

+ p(a(t)z^~l\t

-

2r))']

'(i))' + 2p(a(i)r'"_1'(i — r))'

+p2(a(i)i(B-1,(i-2r))' =

-Sq(t)f(x(t

- 0

= A0(t)St(t

and these inequalities may be re-proved on interval [i0 + kr,t0 + (k + 1 )r) for each k = 1 , 2 , . . . . We will suppose according of the method of induction that (7) is valid for s = 0 , 1 , . . . , / — 1 < r. Then we obtain, on [t0, t0 + r ) by (6) and Leibniz formula, (-!)'*('>(

¿ ( - D ' K w l ^ i - D ' - ^ C - ^ i=l

> r

/ !

fc=0

¿(-i^Kioj^i-iy-vr'^-j=i

= ( - 1 ) ' [A0{t)4>(t - r)] ( , ) > ( - 1 ) ' V ( ° ( 0 ajid these inequalities may be re-proved on interval [io + kr, io + (k + 1)t) for each k — 1,2, Therefore the left sides of inequalities (10) hold. By analogy we can prove that the right sides of (10) hold in the case if (82) and (92) hold. The inequalities between S 4 ( 0 and 4> v (0 follow from Lemma 2.

MAIN RESULTS Consider the system of functional differential equations of the retarded type

(ii)

y(0 = f(*,y«)

where y € R n , yt is an element from the space of continuous function K = K([—r, 0], R"), yt(6) = y(i + 0) where 6 € [—t-,0], f : ft -> R", ft = (i 0 - t - £,+00) x K, 0 < e,

Behaviour at infinity....

223

e =const and f is a continuous mapping such that each element (S, it) € fl determines a unique solution y(6,7r) on its maximal interval Ds,„ = [¿, a), 8 < a < oo. The elements ( t , i ) e R x i are assumed to be such that (t,ir) £ fi. Let w0 = {(t,y) e R x R n,t £ A} and wi = {(i,y) € u;0,/,-(*, y) < 0, i = 1 , 2 , . . . ,p, m , ( i , y ) < 0, j = 1, 2 , . . . , q,p2 + q2 > 0} where the functions /,-(t, y), i = 1 , 2 , . . . ,p,

rrij(t, y ) , j — 1, 2 , . . . , q are defined as

/.•(t,y) = ( y i - f t ( i ) ) ( y i - i i ( i ) ) , m

i(t,y)

= (yp+j - Sp+:(t))(yp+j -

functions gk(t),f>k{t), k = 1 , 2 , p + q are continuous on countinuously differentiable on I and gk(t) < 6k(t) on / i . The following theorem, which we will use in our investigations, can be proved by analogy with the proof of Theorem 2 from DIBLIK [6] which is based on the topological method of T. Wazewski in the form of RYBAKOWSKI [9]. Therefore its proof is omitted. T h e o r e m 1. Let a) for each i £ { 1 , 2 , . . , , p } ; (t,y) £ dwi if h(t,y) (< + 0,tt(0)) € wj, 0 € [ - r , O ) hold on I: (12)

61 < fi(t,ir)

= 0 and ir £ K, 7r(0) = y ,

if yi = Si(t)

and (13)

¿(/ fp+itt,*)

if Vp+j = 6P+j(t)-

y,

and Then there is a noncountable set of solutions of (ll) y = y(i) such that on I\ the values of corresponding initial functions) (14)

(including

Qi(t) < y

[K(i))(s)]2

0, E

>

0 and

3=1

< o o , s = 0,1;

a,ij(t)dt

of 1-th

0 < (-l)V

class, K

< t )

=

—

£ a;j(i) Then

there

/ t

t € I, i,j = (s)

(i)_,

s ds,

1 , 2 , . . . , n . Lei,

= 0,1 1

and

< K

a.ij(t)

=const,

moreover,

( p ( t ) be an initial

+ (aj((t),

€

C

1

225

( / i ) f o r which

the

inequalities

(81),

(82)

e

Let,

n hold

if 3

=

0;

0


i(i) = Ni exp

t d s

, Wi(t) = Mi exp

-/ «0

d s l i (

s

)

i = 1,2, . . . , n where 0 < Ni < Afi; Ni,Mi =const. Then the inequalities (18), (19) hold. The functions /3,(i), 7 , i ( t ) , i = 1 , 2 , . . . , n may be frequently taken in the form

Pi{t) =

- N i

In ( * £ > * ( * ) )

,7,-(*) =

-Mi In ( K £ a i j ( t ) )

3=1

where N,, Mi and K are positive constants and Nt < Mi.

3=1

226

J.

Diblik

EXAMPLES E x a m p l e 1. Consider the scalar equation x(t) = — [exp(—i)] x(t - 1). If we put (in correspondence with Remark 2) N = 0.5, M = \,K = 1,0 = 0.5t _ 1 , 7(i) = 2 r V o = 10, r = 1 ,V>(t) = 0.5exp(—i 2 + 100), $(