Perturbation theorems for nonlinear systems of ordinary differential equations

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

APPLICATIONS

86. 194-207 (1982)

Perturbation Theorems for Nonlinear Systems of Ordinary Differential Equations ZHIVKO

S. ATHANASSOV*

Itlstitute of Mathemarics, Polish Academy of Sciences. 00-950 Warsaw. Sniadeckich 8, Poland Submirted bv J. P. LaSalle

As is traditional in a perturbation theory of nonlinear differential equations, the behavior of solutions of a perturbed equation is determined in terms of the behavior of solutions of an unperturbed equation. There are two useful methods for studying the qualitative behavior of the solutions of perturbed nonlinear system of differential equations: the second method of Liapunov and the use of variation of constants formula. Recently. Brauer 13-5 1, Strauss [ 15 1. Marlin and Struble [ 131, Brauer and Strauss [6]. Fennel1 and Proctor [8]. Harbertson and Struble [9]. Harbertson [IO], May [ 141, Basti and Lalli 121 and possibly others have obtained results on qualitative behavior of solutions of perturbed nonlinear systems, using the nonliner variation of constants formula of Alekseev [ 11. Brauer and Strauss [6 ] defined a new type of stability, so-called uniform stability in variation, which is less general than integral stability. In this paper, a different approach, based on Liapunov’s second method and a comparison principle, is used to obtain some results on the asymptotic behavior and growth properties of solutions of perturbed nonlinear systems. Our approach is inspired by Kato’s work [ 111. The paper is organized as follows. In Section 1 we introduce some preliminary. rather standard, definitions and notations, which will be used throughout the paper. In Section 2 we prove Massera type converse theorems; in other words we motivate the use of Liapunov’s second method. Liapunov functions, the existence of which is proved in this section, are used later on. The main results are presented in Section 3. The different types of stability and growth properties of perturbed nonlinear systems are discussed and several theorems are proved. In the final section we study simple examples to illustrate some of our results. * Ths paper is dedicated to the memory of my mother. 194 0022~247X~82.;030194-14302.OO~O CopyrIght All rights

‘C 1982 by Academic Press. Inc of reproduction in an) form reserved

195

PERTURBED DIFFFRENTIAL EQUATIONS

1 We are interested in the relations between the solutions of the unperturbed system x’ =f(t,

x)

(1.1)

and the solutions of the perturbed system

1” =f(r. 4’) + g(t. 1’).

(1.2)

Here x, y,f and g are elements of R”, an n-dimensional real Euclidean space. Let R+ denote the interval [0, co) and C[X, Y] denote the space of continuous functions from X to Y, where X and Y are any convenient spaces. We shall always assume that fig E C[R+ x R”, R”], and that f is continuously differentiable with respect to the components of .Y on R’xR”,f(t,O)=O for all tER+. The symbol 1.1 will be used to denote arbitrary vector norm in R”. Throughout this work x(t, t,, x,,) and ~(t, fO,J,) will denote the unique solutions of (1.1) and (1.2), satisfying the initial conditions x(t, , t,, x0) = x,, and j(to. f,, , yO) = y,-,, respectively. We shall denote by @(t. t,, ?cO)the fundamental matrix solution of the variational system z’ =J;(r. x(t, for x(J) z

(1.3)

of (1.1) with respect to the solution ~(t, r,,, x0) of (1.1) which is the identity matrix for t = t,. Here,f,(f, X) is the matrix whose element in the ith row, jth column is the partial derivative of the ith component off with respect to the jth component of x. It is known that

DEFINITION

1.1. The type number of a vector-valued

function

c(t) is

defined by x(v) = lim $up + log 1u(t)l. DEFINITION 1.2. The function u(t) is said to be slow@ growing if x in the preceding definition is nonpositive. DEFINITION 1.3. The solution x = 0 of (1.1) is said to be global/y uniformly stable in variation if there exists a constant M such that

I WV hl, %)I ,< M

for all t > t,

and

lx,,l < co.

(1.6)

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ZHlVKo S. ATHANASSOL

DEFINITION 1.4. The solution x = 0 of (1.1) is said to be globall! uniformly slowly growing in variation if for every E > 0 there exists a constant K, possibly depending on E, such that

I @(t, t,, x0)1 < Ke”‘-‘o)

for all t>to>O

and

]xO] < co.

(1.7)

DEFINITION 1.5. The solution x = 0 of (1.1) is said to be global&~ exponentially stable in variation if there exist two positive constants N and u which are independent of the initial values, such that

1@(t, t,, x,)1 < Ne-ncf-fo’

for all tat,,>0

and

]x,]
(1.8)

Definition 1.1 is well known in stability theory and Definitions 1.2, 1.3 and 1.4 are due to Brauer and Strauss [6]. Definition 1.5 appears to be new. As indicated in these definitions, we shall be considering all of R”. Moreover, there is in general no difficulty in restricting the discussion to some open subset of R”. Consider a continuous function V: R + x R” + R. V is said to be globally Lipschitz if I V(t,x) - V(t, y)I ,< L /x - y/ for some L > 0 and for all (t, X, y) E R ’ X R” x R”. Corresponding to V we define the “total derivative” I” with respect to system (1.1) by Vi ,.,, (t, x) = IiF ;up $ (V(t + h. x + hf(t, x)) - V(t, x)), + + and if x(t) is a solution of (1.1) we denote by V’(t. x(t)) the upper right-hand derivative of V(t, x(t)), i.e., V’(t, x(t)) = Ii? ;up + (V(t + h, x(t + h)) - V(t. x)). + +

(1.10)

If V is Lipschitz with respect to x, Yoshizawa ] 17, p. 3 ] has proved that

y,.,,(t, x) = V’(t, x(t)).

(1.11)

In this section we prove Massera type converse theorems for the kinds of stability from Definitions 1.3, 1.4 and 1.5. Liapunov functions which are constructed in these theorems will be used in the next section. The techniques and results of this section are similar to those in [ 171. Let D denote a region in R”. We state the following known lemma, due to Alekseev [ 11, but given here in a slightly different form [5].

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LEMMA 2.1. Let E be a convex subset of D, let x,,, y,, E E, and let t be such that x(t, t,, x0) E D and x(t, t,, y,) E D. Then (2.1)

1X(& t, 7x0> - x(t, to 1Ydl < Ix0 - YoI s,tpDI @k to 9VI.

THEOREM2.2. Let the trivial solution of (1.1) be globally uniformly stable in variation. Then there exists a function U(t, x) with the following properties : (a) CJ(t,x) is defined and continuous on R + x R”. (b) Ixl
IU(t,x)-U(t,y)l
(d)

U{,.,,(t,x)
Proof

(t,x)ER+

(t,x),(t,y)ER+

xR”.

xR”.

Define the “Liapunov function” U(t. x) = stt Ix@ + r, t, xl,

(2.2)

where x(t + r, t, x) is a solution of (1.1) for (t, x) E R + x R”. From x(t, to, x0) =x0 j’ @(t, to. sxo) ds .o

(2.3)

and the assumption of the theorem we have

l~~~~~o~~,~l~~l~,l,t~~,>,~~ Further,

1x01=cco*

SUPr>OIx0 + 5, tvXI > Ix0, I, 4 = lx I,

and

(2.4) by

(2.4)

U(t, x) < M (xl. Therefore (b) is satisfied. From (2.4) and the uniqueness of solutions of (1.1) it follows that U(t, x) is defined on R + X R”. Let (t, x), t, jr) E R + x R”. Then

I U(t, x) - qt, y)l < wg Ix(t + r, t, x) - x(t + r, t,y,)I. Applying Lemma 2.1 and using the hypothesis of the theorem we have, l~(t,“u)--(t,Y)l~Mlx-Yl. Now we shall prove the continuity of U(t, x). Take a 6 > 0 and (t, x), (t, y) E R+ x R”. Then,

I U(t + 6, y) - U(t, x)1 < I w + 4 Y) - w + 6, x)1 + 1U(t + 6, x) - U(t + 6, x(t + 6, t, x))I + 1cJ(t + 6, x(t + 6, t, x)) - U(t, x)1.

198

ZHIVKO

S. ATHANASSOV

The first two terms of the preceding inequality are small when /.Y-!‘I and 6 are small, since CTis Lipschitz in s and x(t + 6.1. x) is continuous in 6. For the third term we have I U(t + 6, x(t + 6. t. x.)) - U(f, .K)l = I ;;T /X(l + r, f, x)1 - mg lx(t + 5, tx)l I. Let a(6) = SUP,>~ lx(f + r. t. x)1. Then a(S) -+ a(O) as 6 + 0 and therefore I U(f + 6, x(t + 6, f. x) - qt. x)1 + 0 as 6 --) 0. Hence, the continuity of U(t, x) is verified. With the help of (2.2). (1.10) and the uniqueness of solutions of (1.1) we have U’(f, x(f)) = liy+;up + (St! Ix(f + 5, f, x)1 - ;y{ lx(f + r, f. .~)I) < 0. Thus, since U is Lipschitz in x, from the above relation and (I. 11) it follows that U’,,, ,,(t, x) < 0, proving (d). The theorem is proved completely. THEOREM 2.3. Let rhe friuial solution of (1.1) be globally uniforml& slowly growing in Ljariafion. Then fhere exisfs a function V(t, x) with the following proper ties :

(a)

V(r. x) is defined and continuous on R + X R”.

(b)

Ixl~V(t,.~)~Klxlfor(t,?c)ER+XR”.

(c)

I V(f.x) - V(t.y)( < K lx -1’1 for (t, x). (f.y) E Rt x R”.

(d)

Vi ,,,, (t. x) < EV(t, x) for (I. x) E R + x R”.

Proof:

Define the “Liapunov

function”

V(f. x) = sup Ix(t + r. f, x)1 em”. (f, x) E Rf X R”. r>O

(2.5)

Relations (a), (b) and (c) can be verified as in the proof of Theorem 2.2. If h > 0, x* = x(t + h, t, x). and r* is such that V(r + h, x*) = ;,LII~ Ix(t + h + r*. I + h, x(t + h, f, x))l eetT”, and if r = r* + h, then by the uniqueness of solutions of ( 1.1) V(t + h, x*) = ~22 jx(t + r. f, x)1 e-“eCh < V(f. x) efh,

PERTURBED

DIFFERENTIAL

EQUATIONS

19’)

whence we obtain

qt + h, x*) - V(t, x) h

-1

< qt. x) G.

From this, we deduce V’,,., ,(t. x) < sV(t. x). The proof is complete. THEOREM 2.4. Let the trivial solution of (1.1) be globally exponentially stable in variation. Then there exists a function W(t, x) with the following properties:

(a) (b)

W(t, x) is defined and continuous on R + x R”. ]x < W(t,x)
(c) I W(t, x) - W(t,y)I < N Ix - .Y1for (t, x), (t. .I,)E R + x R”. W;,.,, (t, x) < --u W(t, x) for (t, x) E R + x R”.

(d)

Proof: Define the “Liapunov

function”

W(t, x) = sup Ix(t + r, t. x)1 e”‘, (t, x) E T>O

R + x R”.

(2.6)

Thus, this theorem can be proved by following the proof of Theorem 2.3.

3 In this section we give some results on asymptotic behavior and growth properties of the solutions of (1.2) in relation with properties of the solutions of (l.l), under suitable restrictions on perturbation. The following basic comparison lemma, due to Conti [7]. underlies all results in this section. LEMMA 3.1. Let x(t) = x(t, to, x0) be a solution of (1.1) existing for t > to. Suppose F E C[R X R”. Rt 1,F(t, x) is Lipschitzian in x and F’(t, x) satisfies an inequality of the form

F’(t, x) < w(t. F(t, x))

for

(t,.r)ERxR”,

where o E C[R X R ‘, R]. Let r(t) = r(t. to, uo) be the maximal solution of the scalar dtgerential equation u’ = w(t. u),

u(to) = uo > 0,

(3.1)

existing for t > to. Then F(t, x(t)) < r(t) for t > to, whenever F(t, , x0) < u. , Note that, following definitions for (3.1). 409’86!1-I4

Definitions

1.3, 1.4 and 1.5 we an formulate similar

200

ZHIVKO

S. ATHANASSO\

THEOREM 3.2. Let the triuial solution of (1.1) be globally unrfirmf> stable in variation. Suppose that the perturbation term in (1.2) satisfies

Idt.4’11 < WI I?‘I. t>t,>o, I?‘1< 00,

(3.2)

where 1 E C[R +, R ’ ] satisfies the inequal+ WI < PR’(t)lW)

(3.3)

for t > t, and R(t) is a polynomial of degree n > 1. Then, all solutions of (1.2) do not grow more rapidly than a polynomial of degree MPn as t + 00. Proof. By Theorem 2.2 there exists a function properties from this theorem. We have

U(t,x)having

the four

~;,.,,k xl ,< cl:,. ,,k -xl + M I g(t. xl. With the help of properties (b) and (d) of U(t, X) and (3.2) this implies U;,.&

s)
/xl.

We wish to apply Lemma 3.1. where w(t, u) = MA(t) u. Let y(t) = (t. t,,y,) be a solution of (1.2) such that U(t,. .rO)< Mu,, u, > 0. The maximal solution of the differential equation u’ = MA(t) u,

44,) = Mu,,

is

1

r(t, t,, u,) = Mu, exp M \.f A(s) ds . . 10 Using Lemma 3.1, we get U(t, y(t, t, yo)) ,< Mu,exp

M (( A(s) ds . I . r,l

(3.4)

Hence, relations (3.3) and (3.4) and property (b) of U(t. s) lead us to the inequality

I v(f, t,,I’,,)l < M+W,)]

-“‘1R(t)l’“‘~

which proves the stated result. Define ,4(t) = (t - to)-’ J‘:, A(s) ds. If A(t) is bounded it follows that A(t) is bounded. If A(t)+ 0 as t + co or if A(t) is diminishing [ 161, i.e., if lim _i:+’ A(s) ds = 0 as t --) co, then A(t) + 0 in both cases. Following the proof of Theorem 3.2 it is not difficult to prove the following theorem due to Brauer and Strauss [6].

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201

EQUATIONS

THEOREM 3.3. Let the trivial solution of (1.1) be globally uniformly stable variation. Suppose that the perturbation term in (1.2) satisfies I g(t. Y)l < W)

for

t>to>O,

]y/ < co,

where 1E C[R+, R+]nL,[t,, 00). Let A(t)=(t-tt,))‘liO1(s)ds. each solution of (1.2) satisfies:

Then

(a)

y(t, t,,, yO) = O(t) as t + co, tf A(t) is bounded for t > to > 0;

(b)

y(t, t,,y,)=o(t)

as t-, 03, tfA(t)+O

as t+ 03.

THEOREM 3.4. Let the trivial solution of (1.1) be globally uniformly slowly growing in variation. Suppose that the perturbation term in (1.2) satisfies t>t,>o, 14’1
u(t,) = u, > 0

(3.6)

is globally uniformly slowly growing, where E and K are the constants from Definition 1.4. Then, the solutions of (1.2) are globa& untform& slowly growing. Proof By Theorem 2.3 there exists a function V(t, X) having the four properties from this theorem. Using property (d), (3.5) and hypotheses on w, we get V;,.,,(t, x> < EV, x) + Kyl(t, W, ~1).

t>t()>o.

IxI<

co.

Let r(t, t,u,) be the maximal solution of (3.6) existing for t > t, > 0, with U, > K ( ~‘~1. I ~‘~1< co, and y(t, t, .v,) be any solution of (1.2) existing for t > t, > 0. Then V(t,, y,) Q K I y,,l < u0 and by Lemma 3.1, (3.6) and (3.7) it follows that V(t, y(t, t,, y,J) < r(t, t,, uO). From this and property (b) of V(t, x), we obtain I rk to h)l < 4, toy 4J,

t > t, > 0.

(3.7)

From the assumption that r(t, t,, uO) is globally uniformly slowly growing. the desired result follows. From Definitions 1.1 and 1.2 it follows that a function v(t) is slowly growing if and only if for every E > 0 there exists a constant K, which may depend on E, such that 1v(t)] < Ke”, t > 0.

202

LHIVKO S. ATHANASSO\'

COROLLARY 3.5. Let the trivial solution of (1.1) be globally uniforml~~ slowly growing in variation and let the perturbation term in (1.2) satisjj

t>t,>o,

I g(t, Y)I < l,(t) + Lz(t) I Yl.

I?‘( < 00.

(3.8)

where A, is do wly growing, &E C[Rf,R+]nL,[t,, co) and (t-t,)-‘J’fOAz(s)ds+O as t-+0. Then the conclusion of Theorem 3.4 remains true. ProoJ Fllowing the proof equation (3.6) takes the form

of Theorem 3.4. the scalar differential

u’ = (E + Kl,) u + KL,(t),

u(to) = u(’ > 0.

(3.9)

Let r(t, t,, uO) be the maximal solution of (3.9) existing for t > t, > 0 with u,, > K j~~j. I ~‘~1< 00, and y(t.t, .rO) be any solution of (1.2) existing for t > t, > 0. Let A(t) = (t - to)-’ j f, I?(s) ds. Then by the hypothesis on A, we have r(t, t,, u,,) < (uO + K’(t - to)) e”fm’08 ““(‘~‘l)‘.

(3.10)

Since A(t)+ 0 as t+ og, we can choose K, = K,(u,, E) and T so that u, + K2(t - tO) < K, ee(rPro)for t 1 T. Therefore. r(t. t,u,) < K, e’r’r-rO’

for

t > T.

which shows that the maximal solution of (3.9) is globally uniformly slowly growing. By Theorem 3.4 the conclusion of the corollary follows. From Definition 1.5 and Eq. (2.3) we conclude that if the solution x = 0 of ( 1.1) is exponentially stable in variation, then it is exponentially stable in the usual sense with the same constant a as the exponent. Brauer [5 1 has shown that if the trivial solution of (1.1) is exponentially stable and if the perturbation g(r, .r) in (1.2) satisfies an estimate g(t,y) = O((~~1)as / .v( a 0, uniformly in t, then the solution J = 0 of (1.2) is also exponentially stable. Now we consider the perturbed system (1.2) under a condition on g(t. y) more general than that of Brauer. THEOREM 3.6. Let the trivial solution of ( 1.1) be globally exponentiall) stable in variation and let the perturbation g(t, ~1)satisfJ1 the inequality

I gv, J-11< W) I 4’13

t>t,>o,

14’1
where ilEC[R+,R+JnL,(t,,cg) and (t-t,))‘!‘:,,A(s)ds+O Then all solutions of (1.2) approach zero as t--t 00.

as t+oO.

PERTURBED DIFFERENTIAL EQUATIONS

203

This theorem can be proved in the same manner as that for Theorem 3.4, so we omit the details. COROLLARY 3.1. Let the trivial solution of (1.1) be globally exponentially stable in variation and

Then the solution y = 0 of (1.2) is global!~~ exponentially stable if u - NB > 0. where the constants a and N are given in Definition 1.5. The following lemma which appers to be very useful in a stability theory, is essentially due to Levinson [ 121.

LEMMA 3.8. If ,u E C[R +, R ’ ] is diminishing. then iAaze Ofl’f eUsp(s)ds = 0

for all (T > 0.

.”

Observe that either of the conditions ,u(t) -+ co or ,UE L,[O. 00) for some ,u(s) ds -+ 0 as t + co. For a discussion of this condition (and examples illustrating that the above are sufficient conditions only), see Strauss and Yorke [ 161.

p > 1, is sufficient (but not necessary) for j:”

THEOREM3.9. Let the trivial solution of (1.1) be globally exponentially stable in variation and

I .a Y)IGPO)

for all t > t, > 0 and I y 1< 00,

where ,uE C[R+, R+] is diminishing. Then the solution y(t, t, JI,,) of (1.2) satisfies the condition / y(t, to y,)] + 0 as t + co. In particular, if g(t, 0) = 0, then y = 0 is asymptoticalbpstable for (1.2). Proof By Theorem 2.4 there exists a function W(t.x) with properties (at(d) from this theorem. In view of the hypothesis on g(t, ~7) it is easy to derive the inequality T ,&

x> < --a W, x) + NW,

where the constants a and N are the same as those in Theorem 2.4. The solution of (3.1) with w(t, u) = au + Np(t) is given by

u(t, t,u,) = u,,e-a’t-‘o’ + Ne-“’ ]‘A,u(s) ensds. _10

(3.11)

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ZHl\'liO S. ATHANASSOC

Let u0 > N ] y0 1 for ] JJ~1< co, and let y(t, t,,, yO) be any solution of (1.2) existing for t > to 2 0. Then W(to ,yO) < N 1y,,] < U, for to > 0 and 1yO] < co. With the help of Lemma 3.1 and property (b) of W(t. x) we get I J’(tqto3Yol < a- fouo)

for

t> to.

From (3.1 l), by application of Lemma 3.8, it follows that u(t, to, no) + 0 as t -+ co. Hence I y(t, to, yo)] + 0 as I + co and the proof is complete.

4 In this section we study simple examples which illustrate some results from the previous section. EXAMPLE

1. Consider the scalar differential equation x’ = -e’x’,

x(t,) = -Yo,

(4.1)

and its corresponding perturbed equation y’ = -efx3

+ g(t, y),

Ato) = 4’0*

(4.2)

The solution of (4.1) is given by x(t, to, x0) = xo[

1 + 2xi(e’ - el”)] Ii*.

t > to > 0,

so that @(t, to, x0) = [ 1 + 2xi(e’ - efo)]- 3:2,

and ] @(t, to, x0)] < 1 for all t > to > 0 and all real x0. Hence, the zero solution of (4.1) is globally uniformly stable in variation. Suppose that the perturbation g(t,y) satisfies (3.2) and (3.3). Then, by Theorem 3.2 it follows that the solutions of (4.2) do not grow more rapidly than polynomials as t+a3. EXAMPLE

2. Consider the scalar differential equation X’=(tCOSr2-

1)x,

x(&J = x0 9

(4.3)

and its corresponding perturbed equation y’ = (t cos t* - 1) x + g(t, y),

y(t,) = 4’0.

(4.4)

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205

EQUATIONS

The solution of (4.3) is given by x(t, t,, x0) =x0 exp[+(sin t* - sin Ii)] exp[-(t - t,)].

f 2 f, > 0,

and @(4 to, x0) = exp [f t, > 0 and all real x,, and the zero solution of (4.3) is globally exponentially stable in variation. EXAMPLE 3. Consider the scalar differential equation (borrowed from [151) x’ = ee2’x3,

x(t,)

= x0

(4.5)

and its corresponding perturbed equation y’ = eC2’y3 + g(f, y),

(4.6)

Ye0 ) = Yo *

The solution of (4.5) is given by x(f, to, x0) =x,[

1 + xi(e-”

- e-2’0)]-“2,

and @(f,f,, x0) = [ 1 +

xi(emzf - e-2ro)]-3/2,

f > f. > 0.

There are solutions of (4.5) which are not bounded: x(t, 0, 1) = exp f and x(f, 0, -1) = -exp f. All solutions of (4.5) are bounded in the future if and only if lx01 < expf,. We have 1@(f, fox,) to > 0 and 1x0(< a, a = exp to. Hence, if [x0/ < exp to, the zero solution of (4.5) is uniformly, but not globally uniformly, stable in variation. Let te perturbation term in (4.6) satisfy g E C[R+ x R, R] and for

(f, y) E R + x R,

(4.7)

where v, is integrable on every finite sub-interval of R +. Define V(f, x) = x2. We obtain V;4.6)(fX) < 2v2u, 4 + 2df) m x) and (3.1) is now u’ = w(t, u) = 2u2 + 2rp(f) u,

u(t,) = u, > 0

(4.8)

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ZHIVKO

S. 4THANASSOI

The above equation has the solution I-r u(t, t,, tin) = un exp 2 .f v(s) ds 1 - 2U, .r,,exp (2J)W4 ( .111) I

df’). (4.9)

Let u0 > 0 and 8-l = 2 .I‘:, exp(2 j:, q(s) ds) dt. We claim the following: (1)

If U, > p there exists a unique finite time t, > 0 such that (4.10)

and lim u(t) = co as f + r,. (2) Proof. monotone such that lim u(t) =

If U, < /3. then lim u(t) = 0 as t + 00. (1) Since the integral Ji, exp(2 j:,p(s) ds ds is a continuous, function of t, follows that there is a unique value of time, say, f,, (4.10) is satisfied and the dominator of (4.9) vanishes. Obviously, 00 as t + t, .

(2) In order that u0 < ,8 be possible it must be that j3 > 0, which is to say that integral j:, exp(2 jr, p(s) ds) dr must tend to a finite limit as t + co. Therefore, it must be that lim exp(j;,p(s) ds) = 0 as t + co and it follows from (4.9) that lim u(t) = 0 as t + co. Let u0 =x: and XI < 00. Then. from the above claim and by Conti’s lemma it follows that all solutions of (4.6) tend to zero as t + 00. We see that the perturbation g(t, J), which satisfies (4.7), “improves” the behavior of the solutions of (4.5). ACKNOWLEDGMENTS The author wishes to express his sincere gratitude helpful discussions.

to Professor C. Olech for numerous

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