asymptotic formulae of liouville-green type for higher-order

ASYMPTOTIC FORMULAE OF LIOUVILLE-GREEN TYPEFOR HIGHER-ORDER DIFFERENTIAL EQUATIONS

M. S. P. EASTHAM

1. Introduction

We investigate the asymptotic form of n linearly independent solutions of the n-thorder differential equation

qy = 0 (1.1)

as x -^ oo, where x is the independent variable and the prime denotes d/dx. Thefunctions q, rx,..., rn_x are defined on an interval [X, oo) and are all nowhere zerothere.We shall write (1.1) in terms of a first-order system, and consequently werequire no more differentiability on q and the rk than that they are C{2\X, oo). Thereis no assumption that these functions are necessarily real-valued.

The familiar equation(ry')'-qy = O (1.2)

is the case n = 2 of (1.1) and, subject to certain conditions on r and q, (1.2)has solutions with the so-called Liouville-Green asymptotic form

y~(r*r1/4exp[± I (q/r)l'2dtj (1.3)x

as x -> oo. The usual proof of (1.3) is based on the transformation

X

y(x) = {r(x)q(x)}-ll*z(t), t = J {\q(u)\/r(u)}l'2du (1.4)x

and, if differential equations in the complex domain are to be avoided, (1.4) carrieswith it the condition that

I r and q are real-valued (with r > 0 and q nowhere zero),

as well as the conditions

II (q/r)112 is not L(X, oo),

III r(rq)-5l2{(rq)'}2 and (rq)-3'2{r(rq)'y are L(X, oo).

Received 17 November, 1982.

J. London Math. Soc. (2), 28 (1983), 507-518

508 M. S. P. EASTHAM

As references for (1.3), we cite for example [2, pp. 118-122; 5; 7, pp. 319-320; 10,pp. 190-202]. With the coefficients

r(x) = xa, q{x) = ±xp, (1.5)

II and III are satisfied whena-p<2. (1.6)

A higher-order generalization of (1.2) which is again a particular case of (1.1) isthe equation

{ry{m)){l)-qy = 0 (1.7)

of order n = m + l ^ 2. Hinton [8] showed that, subject again to I and to conditionscorresponding to II and III, (1.7) has solutions

{q/r)ll"dt j (1.8)

as x -> oo, where a>k (I ^ k ^ n) are the n-th roots of unity. With the coefficients(1.5), Hinton's conditions are satisfied when

a-P<n. (1.9)

Here (1.9) does of course reduce to (1.6) when n = 2, and (1.8) reduces to (1.3)when n = 2 and m = 1.

In a recent paper [4] it was shown that there is an alternative and moresystematic approach to the asymptotic theory of (1.2) than that based on (1.4). Inthis approach there is no change of independent variable, such as we have in (1.4),and consequently no restriction to real-valued r and q. More important, the theoryin [4] covers the two following complementary cases, only one of which isrepresented by (1.3) and (1.6).

Case A. As x -> oo, let

(qr)'/qr = o{(q/r)^2} . (1.10)

Case B. Let (qr)' be nowhere zero in [X, oo) and, as x -> oo, let

(q/rY'2 = o{(qr)'/qr} . (1.11)

It is Case A that leads to (1.3) and reduces to (1.6) when the coefficients are (1.5).Case B leads to solutions

y, ~ 1, y2 = o{(qr)-1'2} (x - oo) (1.12)

and it reduces to the condition opposite to (1.6),

a - j 5 > 2 , (1.13)when the coefficients are (1.5).

HIGHER-ORDER DIFFERENTIAL EQUATIONS 509

In this paper we show that the approach developed in [4] for (1.2) can beextended to deal with (1.1). We introduce the appropriate generalizations of Cases Aand B at the end of §2 and, as we have already indicated, complex-valued coefficientsare allowed. The asymptotic forms of n solutions of (1.1) in the two cases are given inTheorem 1 (§3) and Theorem 2 (§4). For the particular equation (1.7), we obtain(1.8) and (1.9) as instances of the result and conditions of Theorem 1, and we arealso able to cover the opposite condition a — /? > n in Theorem 2. Finally wemention that other applications of the general method of this paper to higher-orderdifferential equations may be found in [3] and papers cited in [3] .

2. A transformation of the differential equation

We write (1.1) as the first-order system

Y'= AY (2.1)with

0 rf1

/A =

0q 0 ... 0

and the entries in A are zero where not indicated otherwise. As in other recent paperson the asymptotic theory of differential equations [3, 4], the aim is to transform (2.1)into the standard first-order system

U' = {A + R)U (2.2)

where A is diagonal and R is L{X, oo), and the method of achieving this aim startswith the diagonalization of A. We therefore require the eigenvalues fxk andeigenvectors vk of A.

The characteristic equation of A is easily seen to be ^"~"<?/(ri •••rn-i) — 0-Hence, defining

...rn-1)}11" (2.3)

with any fixed determination of the n-th root, we obtain

fik = cokQ ( 1 ^ / c ^ n ) , (2.4)

where cofc = exp {2(/c— \)n/n}. The nk are of course distinct throughout [X, oo) sinceQ is nowhere zero in [X, oo). An eigenvector vk corresponding to ptk is

vk = (l,Hkri,tfr1r2,...,n"k-xrl ...rn_J (2.5)

where the superscript t denotes the transpose. We then have the diagonalizationof A,

T-'AT = dg(Co1Q,...,con(2) = A, , (2.6)


say, where T = {vl... vn). Let us also note that, by (2.4) and (2.5),

7 = MQ, (2.7)where

M = dg(\,Q1,QlQ2,...,Ql...Qn.l), (2.8)

Qk = Qrk, (2.9)

and Q is the constant matrix whose entry in the (/, m) position is indicated by

Q = ((olm-1) ( l ^ / . m ^ n ) . (2.10)

By (2.6), the transformation

Y = TZ (2.11)takes (2.1) into

Z' = {Ky-T-XT)Z = AlZ1 (2.12)

say. From (2.7) and (2.8) we have

Ax = A 1 -Q~ 1 A 2 Q, (2.13)where

( . Q n - 1 ) - (2.14)

In order to arrive at a transformed system of the form (2.2), we have to repeat theabove diagonalization process, this time for the matrix Ax in (2.13). There are twocases to consider depending on whether Ax dominates A2 as x -> oo or vice versaand, stated more precisely, the two cases are as follows.

Case A. As x -• oo, let

(Qrk)'/Qrk = o(Q) (2.15)

for 1 ^ k ^ n— 1, where Q is given by (2.3).

Case B. Let {Qrk)' be nowhere zero in [X, GO) and, as x -» oo, let

Q = o{{Qrky/Qrk} (2.16)

for 1 ^ k ^ n-\.

By (2.6), (2.9) and (2.14), we can say that Aj dominates A2 as x -> oo in Case Aand similarly that A2 (apart from its zero diagonal entry) dominates Ax in Case B.We note that (2.15) and (2.16) are immediate extensions to (1.1) of (1.10) and(1.11).

In §3 we consider Case A and first state the asymptotic result (Theorem 1) for thesolutions of (1.1) in this case. The proof of Theorem 1 starts with the workingalready carried out up to (2.14) and is completed in §3 with the details of thediagonalization of Ax in Case A. We deal with Case B similarly in §4.

HIGHER-ORDER DIFFERENTIAL EQUATIONS 5 1 1

3. Case A: Ai dominates A2

THEOREM 1. Let q, rl5..., rn_1 be C(2)[X, oo) and nowhere zero on [X, oo). Fork ^n-\,let

(0 (Qrk)'/Qrk = o(Q) ( x - o o ) , (3.1)

(ii) {{Qrk)YlQhlbeL{X,a,), (3.2)

(Hi) {(Qrky/Q2rk}'beL(X,<v), (3.3)

where Q = {qKr^ ... rn_l)}11". Also, let Re{(coj — cok)Q} have one sign in \_X, oo) for

each j and k. Then (1.1) has solutions yk (1 ^ k ^ n) with the asymptotic forms, asx -> oo,

eXP

Proof. We note first that (3.1) is just a repeat of (2.15) and that (3.2) and (3.3)are further conditions which are required in the proof. As explained at the end of §2,we have to diagonalize the matrix A^ in (2.12) and (2.13), and we begin by findingits eigenvalues kk. By (2.13), we can write the characteristic equation of Ax,det(^!-A/) = 0, in the form

xKx-vl)-Q-l\2QL) = 0, (3.5)where

v = XIQ. (3.6)Let us also write

Q~1A2 =dg(0 ,p 1 , . . . , p n _ 1 ) . (3.7)

Then, by (2.9) and (2.14), conditions (3.1)—(3.3) can be expressed in terms of the pk

as

pk = o(l) ( x - o o ) , (3.8)

Qp2k\sL(X,oD), (3.9)

p'k is L{X, oo) . (3.10)

By (2.6), (2.10) and (3.7), (3.5) is

detKT1(o>I11-v-p,_1)) = 0 (p0 = 0), (3.11)

where again we display the entry in the (/, m) position, and it follows from (3.8) thatthis equation for v has solutions

\ = <Pk~nk {\^k^n) (3.12)

with r\k = o{\) as x -> oo. In order to determine the nk more exactly, we substitute


(3.12) into (3.11) and expand the resulting determinant by the /c-th column to obtain

r]kDuk + (ok{rik-pl)D2,k + ... + (Dnk-\r]k-pn-l)Dn,k = 0 , (3.13)

where the Djk are the obvious co-factors. Now,

where Aj k is the determinant obtained from de tn on deleting the j-th row and /c-thcolumn. Hence (3.13) gives

M

(3.14)2) + ... + O(pn

2_1).

To simplify (3.14) we first express the Ajk in terms of Ank. To do this, wemultiply the columns of Ajtk in turn by a)n

m~j with m = 1, 2,.. . , n but omitting, ofcourse, m = k. Then, using col, = I and rearranging the rows of Aik, we obtain

AJfjk = ( - l y W I! ®«YI"\* = (-l)""M-jAM, (3.15)\m±k J

since f]com = ( -1)"" 1 . (Actually, (3.15) is a special case of a more general resultinvolving determinants where the com are replaced by arbitrary numbers [9, p. 36,Problem 18].) On substituting (3.15) into (3.14), we obtain

(3.16)

Now detO and An k are determinants of Vandermonde type whose values are

1 < / < m $ n

and

1 < / < m ^ n

with cok omitted from Y\' [9, p. 17]. Hence

( d e t Q ) / A n k = { — l)"~i{ — l)k~l{cok — cjl){...){cok — a)n)

= (-lynwr1 •

Hence, dividing out by de tn in (3.16), we obtain

rjk = n~l{px +.. . + pn_1) + O(p2) + ... + O(p2_1). (3.17)


Now, by (3.7) and (2.14), we have p t + . . . + £„_! = Q'1^'/^, where

0) _ f\n-lf\n-2 O2 (1 — /) i(n-l) /2rn-l , .n-2 Jl r fl 1 Q\

by (2.9). Hence, by (3.17), (3.12), (3.6) and (3.9), we can write

Xk = (DkQ-n-'2!l£ + hk, (3.19)

where hk is L(X, oo).For reference shortly, we note that on differentiating (3.13) with respect to x it

follows that rj'k = O{p\) + ... + O{p'n-x). Hence, by (3.10),

ri'k is L{X, co). (3.20)

To complete the diagonalization of AY, we require the eigenvectors wk

corresponding to the Xk. These eigenvectors then form the columns of a matrix 7"isuch that

T-'AJ, = d g a i , . . . , A J = A, (3.21)

say. We show that Ti can be made to satisfy

Tt -»• / (x -> oo) (3.22)where

T\ is L(X, oo) (3.23)

and, to do this, we writewk = ek + uk (3.24)

where ek has /c-th component unity and other components zero while uk has /c-thcomponent zero. Then the eigenvalue equation Ax wk = Xkwk can by (2.13) and (3.6)be written

(Q-lAl-Q-lCl-lA2Q-vkI)uk = - ( Q " 1 A i - Q " 1 ^ ~ 1 A 2 O - v f c / ) e f c . (3.25)

In this vector equation we ignore the /c-th component and we denote by u the(n— l)-component vector obtained from uk on deleting the zero /c-th component.Then, in terms of u, (3.25) has the form

(A3 + 5)u = s, (3.26)

where A3 = dg(co1 — vk,..., con — vk) (with cok — vk omitted) and, by (3.7), the entriesin S and the components of s are all linear combinations of p l s . . . , pn_j . It followsfrom (3.26) that

u = {A3 + Syls = O(p1) + ... + O(pn_1)

and this, together with (3.8) and (3.24), establishes (3.22). Further, by (3.12),differentiation of (3.26) gives

= s'


and this, together with (3.10) and (3.20) gives (3.23).We are now in a position to complete the proof of (3.4). By (3.21), the

transformationZ = T1U (3.27)

takes (2.12)intoU' = (A-T;lT\)U (3.28)

which, by (3.22) and (3.23), has the required form (2.2). By hypothesisin Theorem 1, Re{(a)J — cok)Q} has one sign in [X, oo) and it then follows from(3.19) that the system (3.28) satisfies the conditions of the standard asymptotictheorem of [1, p. 92; 6, p. 6]. Hence (3.28) has solution Uk (1 ^ k ^ n) such that

X

Uk = {ek + o(l)}exp([xk(t)dt\ (3.29)x

where ek is as in (3.24). Since hk is L{X, oo) in (3.19), we can by (3.19) write

.V

Uk = 5-1/"{eJk + o( l )}expK I 6 ( 0 * ) (3.30)

after adjusting a constant multiple in Uk. We now transform back to (1.1) via (3.27),(2.11) and (2.1), bearing in mind that the first component of Y is y. Then (3.4)follows from (3.30) when we recall (3.22), (3.18) and (2.5).

In proving (3.4) we have of course selected the first component y of Y. As usualin the asymptotic theory of higher-order differential equations, there are alsoasymptotic formulae for the derivatives of yk. These formulae, which are obtained byselecting other components of Y, are in essence what one would expect on formallydifferentiating each side of (3.4).

For the particular equation (1.7), the asymptotic formula (3.4) agrees withHinton's result [8, Theorem 1]. Also, apart from allowing complex-valuedcoefficients, our conditions in Theorem 1 above agree in essence with those ofHinton, although there is of course some difference of form between our condition(3.1) and Hinton's {q/r)11" £ L(X, oo). Conditions (3.1)—(3.3) do however reduce to(1.9) when the coefficients are (1.5). We also note that the condition in the theoremconcerning Re{(<Wj — cok)Q} is certainly satisfied when q and the rk are real-valued.

4. Case B: A2 dominates At

We continue the asymptotic analysis of the solutions of (1.1) by considering thediagonalization of Ax when (2.16) holds. To prove the theorem which follows, werequire the strengthened form of (2.16) which is contained in (4.1), (4.2) and (4.4).

THEOREM 2. Let there be a C{1)[X, oo) function p, nowhere zero in [X, oo), suchthat

(i) (Qrk)'/pQrk^ck ( x - c o , 1 < f c < n - l ) , (4.1)


where the ck are finite limits with the property that

Cj + ... + ch±0 (4.2)

for all j and k with I ^ j ^ k ^ n — 1 ;

(ii) {(Qrk)'/pQrk}f is L(X, oo) (1 ^ k ^ n - \ ) ; (4.3)

(Hi) Q = o(p) ( x - » « ) ; (4.4)

(iv) (Q/p)'isL(X,oo); (4.5)

(v) Q2/pisL(X,oo). (4.6)

Further, for each j and k with 1 ^ j ^ k ^ n-l, let the derivative

\Qk-j+lrj...rk\' (4.7)not change sign in [X, oo).

Then (1.1) has solutions yk such that, as x -*• oo,

yx - i (4.8)and

yk = oiiQ'-'r, . . . r , . , ) - 1 } (2 ̂ k ^ n). (4.9)

Proof. As in the proof of Theorem 1 we start by considering the characteristicequation d e t ^ — XI) = 0. By (2.13), we can write this equation as

det((p-1A2-v/)Q-p"1QA1) = 0, (4.10)where

v = - A / p , (4.11)

and (4.10) and (4.11) now replace (3.5) and (3.6). We also write

p~lA2 = dg(ao,a1,...,(Tn_1) (4.12)and

S = Q/p (4.13)

where o0 = 0 by (2.14) and 3 = o(l) by (4.4). In place of (3.12), we seek solutions of(4.10) in the form

\ = ok-i~U (l^k^n), (4.14)

where £k = o(l). The method of determining £k more exactly is similar to that usedfor rjk in Theorem 1. We substitute (4.14) into (4.10) and expand the resultingdeterminant by the /c-th row to obtain

(Ck-d(D1)(oki~lDk,l+... + (Ck-Sajn)cokn~1Dk,n = 0, (4.15)

where the Dkj are the obvious co-factors. Then, in place of (3.14), we obtain

where we have used (4.1) and (4.2) to say that, because of (2.14), (4.12) and (4.14),


Oj — vk is bounded away from zero as x -> oo wheny ^ k — l. On using (3.15) andproceeding as before, we obtain, in place of (3.17),

The first term on the right-hand side is zero, and hence (k = O{32). Then, by (2.14),(4.11), (4.13) and (4.14), we obtain

(4.16)

K = -«2i... Qk-J/(Qi... & 7 J

for the eigenvalues of Ax.To determine suitable eigenvectors vvk of Ax, we write the eigenvalue equation

Axwk = Xkwkas

(p-'A.-p-'QA.Q-^Qw, = vkQwk, (4.17)

where we have used (2.13) and (4.11). If we now write

Qwk = ek + uk (4.18)

in place of (3.24), then (4.17) has a form similar to (3.25). Arguing as before, we seethat (4.18) implies that the diagonalizing matrix Tt can be chosen to satisfy

0 7 ; - + / ( x ^ o o ) (4.19)

in place of (3.22). Further, on differentiating (4.17) and using (4.3) and (4.5), we findthat

T\ is L(X, oo) (4.20)as in the case of (3.23).

We now make the transformation

Z = 71t/ , (4.21)

as in (3.27), to obtain (3.28) again, where A is given by (3.21) and (4.16). By (4.19)and (4.20), (3.28) has the form (2.2). In order to apply Levinson's standardasymptotic theorem [1, pp. 92-95; 6, p. 6] once again, we have to check that theusual condition on Re(Ak —1}) is satisfied. By (4.16), we have, for) ^ k— 1,

where

By (4.6) and (4.7), ^2' does not change sign in [X, oo) and Q2/p is L{X, oo), andtherefore the conditions of Levinson's theorem are satisfied. It follows that (3.29)holds again but now with the lk given by (4.16).

HIGHER-ORDER DIFFERENTIAL EQUATIONS 5 1 7A,

On substituting (4.16) into (3.29) and adjusting a constant multiple in Uk, weobtain

l / 1 = e 1 + o ( l ) , )(4.22)

Then, by (2.7), (2.11), (4.19) and (4.21), we obtain corresponding solutions Yk of(2.1) in the form

Yk = MQT, Uk = M{I + o(\)}Uk. (4.23)

On choosing the first component in this vector equation and using (4.22), (2.8) and(2.9), we obtain (4.8) and (4.9). The proof of Theorem 2 is now complete.

5. Concluding remarks

(i) First we consider Theorems 1 and 2 as applied to the coefficients

rk(x) = (const.)*"*, q(x) = (const.)x*.

In this case (2.3) gives Q{x) = (const.)xv, where

y = {p-(a1+... + ocn.l)}/n, (5.1)

and conditions (3.1)-(3.3) in Theorem 1 are all satisfied if

a 1 + . . . + aB_1-j8 < n. (5.2)

Here, (5.2) reduces to Hinton's condition (1.9) when all but one of the ak are zero.In Theorem 2, conditions (4.1) and (4.3)-(4.6) are satisfied if

when the choice p{x) = x" 1 is made. In (4.1) we have ck = y + cck, where y is as in(5.1), and then (4.2) can be thought of as ruling out certain values of /? in terms ofthe ak.

If there is only one non-zero ak, say a, then (4.2) gives (k— j+l)y + a ± 0, that is,y =£ -ct/K(K = 1, 2, . . . , n - l ) . Substituting y = (/?-a)/n from (5.1), we can write thecondition in terms of /? as

(ii) Here we consider the condition of Re(/lfc+1— X}) which is required inLevinson's asymptotic theorem and which we have already expressed in terms of(4.7). Another way of expressing this condition is in terms of the constants ck in (4.1).By (4.16) we have

518 HIGHER-ORDER DIFFERENTIAL EQUATIONS

Hence, if p is real-valued and if Re(cJ + ... + cfc) ^ 0, we have

and the conditions of Levinson's theorem are then satisfied.

(iii) Although (4.9) is only an o-formula for yk (2 ^ k ^ n), it is possible toobtain asymptotic formulae for certain derivatives of these yk in analogy with (4.8).Let us take, not the first component, but the fe-th component in (4.23). Denoting thiscomponent on the left-hand side by (Yk)k, we obtain

(Yk)k = 1+0(1) (5.3)

from (2.8) and (4.22). Further, from (2.1) we have

(5.4)

and then (5.3) and (5.4) give the asymptotic formula involving yk (2 ^ k ^ n)which corresponds to (4.8). In particular, for k = 2 we have

\ (x->co).

References

1. E. A. CODDINGTON and N. LEVINSON, Theory of ordinary differential equations (McGraw-Hill, NewYork, 1955).

2. W. A. COPPEL, Stability and asymptotic behaviour of differential equations (Heath, Boston, 1965).3. M. S. P. EASTHAM, 'Asymptotic theory for a critical class of fourth-order differential equations', Proc.

Roy. Soc. London Sect. A, 383 (1982), 465-476.4. M. S. P. EASTHAM, 'The Liouville-Green asymptotic theory for second-order differential equations: a

new approach and some extensions', Ordinary differential equations and operators: a tribute to F. V.Atkinson, Lecture Notes in Mathematics 1032 (Springer, Berlin, 1983).

5. W. N. EVERITT, 'On the transformation theory of ordinary second-order linear symmetric differentialequations', Czechoslovak Math. J., 32 (1982), 275-306.

6. W. A. HARRIS and D. A. LUTZ, 'On the asymptotic integration of linear differential systems', J. Math.Anal. Appl.,4$ (1974), 1-16.

7. P. HARTMAN, Ordinary differential equations (Wiley, New York, 1964).8. D. B. HINTON, 'Asymptotic behaviour of solutions of (ryim)){k)±qv = 0', J. Differential Equations, 4

(1968), 590-596.9. L. MIRSKY, An introduction to linear algebra (Oxford University Press, Oxford, 1955).

10. F. W. J. OLVER, Asymptotics and special functions (Academic Press, New York, 1974).

Department of Mathematics,Chelsea College,

University of London,552 King's Road,

London SW10 0UA.

asymptotic formulae of liouville-green type for higher-order

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