Dichotomies and asymptotic integration of linear differential equations of order n

Nonlinear

Analysis,

Theory,

Mefhods

&Applications, Vol. 30, NO. 2. pp. 1125-l 132, 1997 Proc. 2nd World Congress of Nonlinear Analysts 0 1997 Elsevier Science Ltd printed in Great Britain. All rights reserved 0362-546X197 $17.00 + 0.00

Pergamon

PII: SO362-546X(97)00298-8

DICHOTOMIES AND ASYMPTOTIC INTEGRATION LINEAR DIFFERENTIAL EQUATIONS OF ORDER Ra61 Naulin ’ Departamento

Key

1

words

and

phrases:

Scalar

t 11

de Matemiticas, Cumani 61OlA-285.

equations,

scalar

OF N

Universidad Venezuela

dichotomies,

de Oriente

Ghizzetti

asymptotic

integration.

Int reduction

The theory of asymptotic integration of linear systems [2, 4, 5, 7] constitutes an important field of research in ordinary differential equations. The aim of this theory is the description of the solutions of the perturbed system Y’ = (A(t) + B(t)) Y, (1.1) where we assume to have some information

on the solutions

of the equation

z’ = A(t)z.

(14

These studies started with the research of Levinson [7] for system (1.1) for a diagonal function Lately, this theory has been developed in [2, 4, 9, 101 under different dichotomic behaviors of System (1.2). Although the development of the theory of dichotomies of system (1.2) and the corresponding applications to the asymptotic integration of system (1.1) has achieved a respectful degree of perfection, the main ideas of the theory of dichotomies have not found explicit applications in the study of the general ordinary differential equation

A(t).

y(") + a,-1(t)y("-')

+ . . . t ao(t)y = &$)y”’

t . . . + bo(t)y,

0 < p 5 72 - 1,

(1.3)

except in those cases, when these equations can be reduced to the linear System (l.l), and the respective linear System (1.2) has some of the known dichotomic properties. This procedure has several inconvenients. For example, if we reduce the escalar equation

dn) + a,~&)z("-') t ... + ao(t)z = 0,

(1.4)

to System (1.2), it is not clear what dichotomic behavior this system may have. On the other hand, in the study of Eq. (1.4), frequently, properties of the solutions of this equation are known up to a order p < n - 1 and therefore the fundamental matrix of System (1.2) is not known. In this paper we wish to show that it is possible a dichotomic theory for the escalar Eq. (1.4) similar to the Levinson dichotomy for the diagonal system (1.2) [5]. Thus, for the equation (1.3), we will obtain n linearly independent solutions yi of Eq. (1.3) h aving the asymptotic formulas

y!” I llResearch

supported

by Proyecto

UDO

= (1 + $)z!‘) , Y #

CI-S-025-00730/95.

1125

iEN,

T E J%

1126

Second World

Congress

of Nonlinear

Analysts

where N= and the functions We will apply y(")

{1,2 ,...)

)...) p},

Oip
pl are small at t = co. this theory to the analysis of the asymptotic

behavior

+ c,-1y("-')

n},

Np={O,l

t c"-zy("-2)

+ . . . t coy = b,-l(t)y(n-l)

where c; are constants. We will show that under certain teristic polynomial the equation x(n) t c,-ldn-l)

t c,-2z("-2)

of the solutions + . . . + bo(t)Y,

conditions

of (1.5)

on the the roots of the charac-

t .. . t coz = 0

(l-6)

has a scalar dichotomy. Henceforthn, by mean of the notion of scalar dichotomy it will be possible to obtain not only the known result of Ghizzetti regarding the asymptotic integration of the equation y(“) = b”-lz+l) but we will obtain

2

Scalar

also a Ghizzetti

+ b,-2z(“-2)

like theorem

+ . . . f boz,

for a more general

Eq. (1.5).

dichotomies

In order to motivate the introduction of the scalar dichotomies, dichotomic conditions. Let us consider the diagonal matrix A(t) = diadh(t), where the functions &(t) are assumed to be complex In the study of the perturbed linear system [7] Y’ = (Nt) Levinson

(1.7)

assumed that the coefficients

h(t),

we recall the definition

. . . , h(t)),

valued and continuous

+ B(t))

of the linear

of the Levinson

on the interval

J = [0, 00).

(2.8)

Y1

system

s’(t) = A(t)z(t),

(2.9)

have the following dichotomic properties (under which one says that System (2.9) has a Levinson dichotomy): For each pair of indexes (i,j) and some positive constant K, the functions Xi, Xj satisfy either (2.10) s
t P o (Xi(U) J

or R

- Xj(U))dU

J I

’ (Xi(U) -

A,(U))

--t -CO a.5 i! + 00

dU 2 X-l,

s5t.

Let us denote

Mj = {i E N : (i,j) Nj = {i E N: (i,j)

satisfies (2.10)-(2.11)}, satisfies

(2.12)},

1 5 j < n, 1 5 j 5 n.

(2.11) (2.12)

SecondWorld Congress of NonlinearAnalysts By e; we will denote the canonical basic vector in the i-direction basis of the vector space of solutions of Eq. (2.9) is given by

1127

of the n-dimensional space C”. A (2.13)

B = {Xl,%...,GJ, where q(t)

= Jo’ X’(T)drei.

Regarding the asymptotic integration of the System (2.8), Levinson [7] considered the set of n integral equations y(t)

=

-

xdt) + iz

CJ’

ieN,

3

lt eS: Xdr)dr 2 bipypy(s)ds p=l

(2.14)

e-f: X’(r)dT 5 b,,y,y(s)ds. a

p=l

He proved that under condition B(t) E L’, this integral equation has a unique solution yi that is a solution of Eq. (2.8) [3], with the asymptotic formula Yi(t)

= (1 +

O(l))Xi.

The solutions yi are linearly independent and therefore they constitute a basis a of the space of solutions of Eq. (2.8). Using basis (2.13), we could rewrite conditions (2.10))-(2.12) to the form (2.15) and (2.16)

or (2.17)

The Levinson dichotomic conditions, written in the form (2.15)-(2.17), suggestthe following construction for Eq. (1.4). Let us denote by x = {x1,22,... ,%a)

(2.18)

a setof linear independentsolutionsof Eq. (1.4). By W(t) we denote the wronskian of thesesolutitins, W,(t) is the determinant of the matrix obtained from the Wronskii matrix by replacing the i-column by the n-basis vector e, = col(0,. . . ,O, 1). We will require a set of positive continuous functions m;(t), i E N, T E Jy,, satisfying the conditions lx~‘l 5 rni.

(2.19)

In order to have (2.19) we have to estimate the derivatives of the solutions of Eq. (1.6) up to order p, where p 5 n - 1. Definition El:

1 We shall say that Eq. (1.4) has a Levinson

there exist a basis (2.18)

and a set of continuous

scalar dichotomy

functions

(LSD)

rni satisfying

ifl

(2.1 g),

1128 E2:

Second World Congress of Nonlinear Analysts for any pair of integers

(i, j),

0 5 i, j < n, one satisfies either

?!!.@sp, my(s) m;(s)’

-

,

and (2.21)

(2.22) where M is a constant. The reader may observe the similarity between definitions (2.20)(2.22) and (2.15)(2.17). ,\ large class of scalar Eq. (1.4) has the dichotomic properties described by (2.20)-(2.22), among them a large class of scalar equations with constant coefficients (1.6). Partially, we will show this situation in the forthcoming Examples 1,2,3. For a positive continuous function h, we denote by L’(h) the space of h-integrable functions, that is f E L’(h) iff h-‘f is integrable. In [8] is proven the following Theorem 1 Let us assume that Eq. (1.4) has the LSD (2.20)-(2.22) of a set of positive and continuous functions L=

and let us assume the existence

{ho,hl,...rhp),

such that mr(t) Iwittl , ,wtt),

I KM%

Vi E N,

where K is a constant. Under these circumtances, if 6, E L’(h;‘), linearly independent solutions yl, y2,. . . , y,, , such that y!”t = xi” + o(my), The proof of this theorem 7(y)(t)

=

-

can be accomplished zj(t)

C icF,

Jm zi(t)$#

then Eq.

(1.3) has a set of n

T E Np.

by considering

+ C J* zip iEI, lo

(2.23)

r E 4,

(b,(s)y”‘(s)

(bp(s)y(‘)(s)

the operator

[6]

+ . . . + bo(s)y(s))

+ . . . + bo(s)y(s))

ds

ds,

'

where for each j E N, we have denoted 4 = {i E N; (i, j) satisfies

(2.20)-(2.21)

},

Fj = {i E N;(i,

j) satisfies (2.22)).

Second World Congress of Nonlinear Analysts

3

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Examples

Let us consider Example

1.

some examples. Let us consider

the asymptotic

integration

y" + (1-l - h(t))y'

of

- (t-2 t b,(t))y

= 0,

(3.24)

where the Euler equation 2” $ t-lx’

-t-22

= 0,

(3.25)

has the basis zl(t) = t-l, z2(t) = t. We will accomplish the asymptotic integration of Eq. (3.24) by means of Theorem 1. In this example my(t) = lzl(l)) = t-l, m;(t) = 122(1)1 = 1, m:(1) = )z;(t)l = tw2, and m:(t) = (z’,(t)1 = 1. The condition (2.23) is obtained for ho(t) = t, h,(t) = 1. In this simple example it is easy to verify that the dichotomic conditions (2.20)-(2.22) are satisfied. Thus Eq. (3.25) has a LSD . Theorem 2.1 implies that Eq. (3.24) has two linearly independent solutions satisfying y!‘)(t) I if b1 E L’(1)

= (1 + o(l))z!“) I 2 i = 1 72 7k = 0 11 .

and b. E L*(t-‘)

Example 2 The equation y” It (1 + 4(t) + f(t))y = 0 gives interesting examples of scalar dichotomies can be applied. We restrict our attention to the intensively 2” - (1 + 4(t) t f(t))% Where Theorem

we will rely on the following A Zf the continuous

known

where the method studied equation (3.26)

= 0,

result [l].

real function

4(t) satisfies

the conditions

(a) 4(t) ---) 0 as t --* 00,

(b) .f” 4(tj2dt < 00, then the equation 2” - (1 + c$(t))x = 0 has two linear

independent

solutions

(3.27)

x1 and x2 with the asymptotic

formulas

x1(1)= exp(tt i l' 4(T)dT t o(l)), x2(t) = exp(-t

- i 1’ 4(r)dT

+ o(l)).

Under conditions of this theorem we do not obtain the asymptotic x:,x;. But the given information is sufficient to obtain the following Theorem

2 Zf in addition

to the conditions

of Theorem

integration

of the derivatives

A, we assume:

(c) The continuous function f(t) is absolulely integrable. Then Eq. (3..!?6) has two linearly independent solutions y1 and y2 satisfying

yl(t) = (1 t o(l))exp(t y2(t) = (I+

o(l))exp(-l

t i 1’ +(T)dr - f 1’ $(7-P

+ o(l)),

(3.28)

+ 41))

(3.29)

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Second World Congress of Nonlinear Analysts

Proof

The solutions m:(t)

of the basis X = {x1, ~2) of Eq. (3.27) have the estimates m;(t)

= exp(t + i /,’ 4(r)d~),

= exp(-t

(2.23) if

- f Jo’ 4(r)&).

With these functions it is easy to prove that Eq. (3.27) has a SLD. Since W(t) = 1, Wi(t) = mi, W, = rnf, then condition (2.23) is accomplished with h, = 1. The application of Theorem 1 gives the formulas (3.28) and (3.29). Example

3.

The following

result is due to Ghizzetti

Theorem

B Let us assume that

m t”-‘-kpkldt < co, J independent solutions yi having the form

then Eq. (1.7) h as a set of linear

y,(‘)(t) = (1 + o(l))ti-‘,

This result can be obtained

[4, 51

0 5 r 5 i, y,!‘)(t) = o(P),

from Theorem

i + 1 5 T 5 12- 1.

(3.30)

1. The equation .(n) = l-j

(3.31)

has the basis Si(t) = tivl, 1 < i 5 n. Let m;(t) = timr, 0 < r 5 n - 1. These functions satisfy (2.19). Further, the pair of indexes (i,j) satisfies (2.20)-(2.21) if i < j, otherwise satisfies (2.22) if j < i. Thus Eq. (3.31) has a LSD. The condition (2.23) of Eq. (3.31) reduces to mr(t)

IwiCt)l

’

-

Now, the result (3.30) follows from Theorem using canonical forms of linear algebra.

4

A generalization

=

ti-l--rtn-i

p-1-T

IWt)l (1). Note, that this new proof has been obtained

of Ghizzetti

without

theorem

The notion of a scalar dichotomy allows to extend the result of Ghizzetti in many directions [8]. In this section we will analyze one of them. We will assume that P(X), the characteristic polynomial of Eq. (1.6) can be decomposed into the form P(A)

= &(A)X”l,

15 I+ 5 n,

Q(0) # 0.

A basis (2.18) is given by Xi(t)

=

(4.32)

i = 1,. . . , VI,

tie’,

and Zi(t) = tjexk*, where uk denotes

the multiplicity ml(t)

v1tl

5

e 5

n,

0 <

j

5

Vk,

Xt

#

0,

(4.33)

of the root Xk. Let us define = t+,

1 5 i 5 pl,

0 5 r 5 n - 1,

(4.34)

and my(t) = tjeSLXkr,

v1 + 1 5 i 5 72, 0 <

T

5 n - 1.

(4.35)

Second World

Lemma

1

If the roots of polynomial

Congress

Q(X)

of Nonlinear

Analysts

1131

satisfy WA # 0, then Eq. (1.6) has a LSD.

Proof Let us consider the basis of the Eq. (1.6) defined by (4.32)(4.33). Then the estimates L e t us consider two solutions of the Eq. (1.6) are accomplished by (4.34)(4.35). s;,(t)

= tjLeAklt,

Zi,(t)

= tjkxk~t,

where X1;, # 0 and XC, # 0. On the other hand, if !RAL, < RAk,, then the pair (2.20)-(2.21). If IliXk, = ?fzXb, and ji < j, then the pair (ii,iz) satisfies (2.20)(2.21). Xi,(t)

= t”,

Xi # O},a(r)

Theorem 3 Let us assume the conditions (1.5) satisfy c, E L’(h;‘), where then Eq. (1.5) has a set of linearly

independent

(1-t o(l))P, y!“(t) I

satisfies (2.22).

If ?ii&

< 0 then

If

the cmficients

of polynomial

= t”(‘), solutions

(4.36) yi satisfying

the asymptotic

formulas

0 5 T < i,

=

( 1

satisfies

= max{P - 1,vi - 1 - r}.

of Lemma 4.1 be fulfilled. h,(t)

(ii,&) Let

Zi2(t) = tj2exkst.

If %AI;, > 0, then the pair (ii, iz) satisfies (2.20)-(2.21), and (iZ,il) the pair (ii,iZ) satisfies (2.22) and (iz,ii) satisfies (2.20)-(2.21). In order to establish the forthcoming theorem, let us define V = max{vi;

(2.18)

i + 15

O(t’-‘),

T

i = 1,2 )...)

vi,

5 12- 1,

and y!”I = (1 + o(l))(tieAk’)(‘), Proof

Let us consider

the basis (4.32)-(4.33).

IW ,w(t),tt)l 5 KtY’-i, we obtain

vi + 1 <- i -< n 7 As # 0. Then

1 < i 5 vi, IwiCt>l - p-j-le-Ski Iwt)l

,u,+lIi5n,

for At # 0 the bound mT(t) Iwitt>l ’ IWWI

< up-1 -

(4.37)

’

and for Xi = 0

mT(t)IwiCt)l < ~~~‘-1-r ’ IWQI . The estimates (4.34)-(4.35) and (4.37)(4.38) imply that (2.23) is obtained Now the proof of the theorem follows from Lemma 1 and Theorem 1.

(4.38) with the function

(4.36).

Remark 1 Theorem 3 generalizes the known Theorem B. Under conditions of Theorem 3, the existence of simple roots of polynomial Q(A) satisfying RXh = 0 can be considered. This case is analysed in [8].

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Remark

Second World

2 The equation

with

constant

Congress

of Nonlinear

Analysts

coefficients

@) + 2zc4)+ .@) = 0, with respect to the basis Zl(t)

= 1, X2(t) = t, 23(t) = ei*, z4(t) = te”,

z5(t) = e-“,

z6(t)

= te-“,

has not a LSD. This is due to the existence of imaginary roots of the characteristic polynomial with multiplicity greater than one. Equations with constant coefficients with such a property have not a LSD [8].

REFERENCES 1. BELLMAN, R., Stability Theory of Diflerential Equations, Dover Publications, New York (1953). 2. BRAUER, F., WONG, J.S., On the asymptotic relationships between solutions of two systems of ordinary differential equations, J. Dig Eqm., 9, 527-543 (1969). 3. CODDINGTON E.A., LEVINSON N., Theory of Ordinary Di'emntial Equations, McGraw-Hill, New York (1955). 4. COPPEL W.A., Stability and Asymptotic Behavior of Differential Equations, D.C.Heath and Company, Boston (1965). 5. EASTHAM, M. S. P., The Asymptotic Solution of Linear Differential Systems (Applications of the Levimon Theorem), Clarendon Press, Oxford (1989). 6. KUSANO T., Asymptotic relationships between two higher order ordinary differential equations, Internat. J. Moth. and Moth. SC;. Vol. 6 No 3, 559- 566 (1983). 7. LEVINSON, N., The asymptotic nature of solutions of linear systems of differential equations, Duke Moth. J., 15 (1948), 129-139. 8. NAULIN R., Asymptotic integration of linear ordinary differential equations of order n, preprint, Departamento de Matemdticss, Universidad de Oriente, Nticleo de Sucre (1995). 9. NAULIN R., PINTO, M., Roughess of (h,k)-dichotomies, J. Ditf. Eqns., Vol. 118, No 1, 20-35 (1995). 10. NAULIN R., PINTO M., Dichotomies and asymptotic solutions of nonlinear differential systems, Nonlinear Analysis and Applications, T.M.A., 23, 7, 871-882 (1994).