continuability of solutions of perturbed differential equations

Nonlinear Analysis. Theory, Methods & Applications, Vol. 8. No. 8. pp. 963-975. 1984. 0362-546X/84 S3.00+ .00 Printed in Great Britain. ~ 198a Pergamon Press Ltd.

C O N T I N U A B I L I T Y O F S O L U T I O N S O F P E R T U R B E D

D I F F E R E N T I A L E Q U A T I O N S

T. I-tARA, T. YONEYAMA and J. SUGIE Department of Mathematical Science, University of Osaka Prefecture, Sakai, 591, Japan

(Received 25 August 1983)

Key words and phrases: Continuability of solutions, global existence, perturbations, ordinary differential equations, delay-differential equations, differential inequality.

1. I N T R O D U C T I O N

CONSIDER the following systems of differential equations;

(E) x' = f(t, x),

(P) x' = f(t, x) + g(t, x),

(Ph) X' = f(t , X) + h(t),

where x, f, g, h are n-vectors, f ( t , x ) ,g ( t , x ) are continuous on [0, ~) x R n, and h(t) is continuous on [0, ~).

Strauss-Yorke [8, 9] have studied the perturbation problem of stability of solutions from (E) to (P) and (Ph), and Hara-Yoneyama-Okazaki [5] have studied that of boundedness of solutions. Bernfeld [1] has studied the continuability of solutions of perturbed scalar differential equations.

The purpose of this paper is to prove theorems on the perturbation of the continuability of solutions from (E) to (P) and (ph).

For a scalar equation r' = ¢ffr) where @: [0, ~)---> (0, ~) is continuous, it is well-known that the solutions are defined in the future if and only iff~ d r/~p(r) = oo~ We study the continuability of solutions of nonautonomous scalar equations

(ES) r' = A (t) @ (r),

(PS) r' = ,~(t)c)(r) + l~(t)~/(r),

where A, .u, ¢~, ~V: [0, oo)---, [0, o0) are continuous, and ¢~(r) > 0 and 1p(r) > 0 for all r -> 6 => 0. Throughout this paper we assume that A, ~, ¢p and ~V satisfy the above conditions.

We introduce several classes of functions ¢~, for which we investigate the continuability of solutions of (ES) and (PS) (Sections 2 and 3). In particular we show that the solutions of (ES) are defined in the future if and only if f~ dr/C~(r)= oo when 2(0 is not identically zero (theorem 3.1). We also give necessary and sufficient conditions that the solutions of (PS) are defined in the future under certain conditions (theorems 3.2 and 3.3).

In Section 4, we study the continuability of solutions of the system (E) and its perturbed systems (P) and (Ph). For several classes off( t , x), for which the solutions of (E) are defined

963

964 T. HA~,A, T. YONEYAMA and J. Suom

in the future, we determine the admissible classes of g(t, x) which preserve that the solutions of (P) and (Ph) are defined in the future (theorems 4.1 and 4.2).

In Section 5, we study the continuability of solutions of perturbed delay-differential equations (theorems 5.1 and 5.2).

2. DEFINITION AND PRELIMINARIES

Let R ~ denote Euclidean n-space and let I1" II denote any n-dimensional norm. We now present the definition of the continuability of solutions that we shall use later. The

following definition is stated for (E) but of course it applies to (P). We denote the solutions of (E) through (to, x0) by x(t; to, x0), where to ->- 0 and x0 ~ R ~.

Defini t ion 2.1. The solutions of (E) are of global existence (GE) if all solutions of (E) are defined in the future.

By the comparison theorem, the following lemma can be easily seen.

LEMMA 2.1. Suppose that Ill(t, x)l[--- ~.(t)~(llxl[) and Jig(t, x)ll <= ,u(t)~(llx)l) on [0, ~) x R", then

(i) the solutions of (ES) are GE implies those of (E) are GE, (ii) the solutions of (PS) are GE implies those of (P) are GE.

We shall first consider a class of continuous scalar functions q~(r);

)__ "~°= ¢P: ~(r) z¢ .

The following result is well-known as the Wintner-Conti theorem [3], [10], [11].

LEMMA 2.2. Suppose that [If(t, x)ll --< ~.(t)q~(I)xll) on [0, ~) × R ". If q~ ~ ~0, then the solutions of (E) are GE.

We consider a special case of lemma 2.2.

LEMMA 2.3. The solutions of r' = q~(r) are GE if and only if ~ ~ 90. The solutions of r' = O(r) are GE implies the solutions of (ES) are GE.

Proof . It is trivial that the solutions of r' = @(r) are GE is equivalent to ~ ~ -~0. We show the second assertion. Suppose that q~E ~0 and there exist (to, r0) E [0, ~) x [0, :¢), T > to and a solution r(t) = r(t; to, ro) of (ES) such that

r(t; to, to) --~ oQ as t ~ T-.

Then we may assume without loss of generality that r0 -> 6. Let L = max k.(t), then to ~-t~_ T

r' = ),(t)~p(r) <= Ldp(r) on[t0, T).

Continuability of solutions of perturbed differential equations 965

We separate variables in the above inequality, integrate from to to t < T, and obtain

ff(0 < L ( T - t0). d

0

The left-hand side would tend to + ~ as t---, T- , which is a contradiction. Thus the proof is completed.

We next consider the scalar differential equation

r' = ). (t) q~ (r) + ,u (t).

We notice that q~ ~ ~0 does not imply that the solutions of (2.1) are GE.

(2.1)

Example 2.1. In [1], the following example is given• For each integer n > 0, let I, = {r: n + 1In 2 <= r <= n + 1 - 1 / (n + 1)2}. Define ~(r) as follows;

1 ( i ) q , ( n ) = n ,

1 1 (ii) q~(r) r 2 o n l , ,

1 (iii) q~(r) Is linear elsewhere.

Let ).(t) = p(t) = 1 for all t => 0. Then q~ ~ ~0 but q~ + 1 ~ 90. Therefore, by lemma 2.3, we see that there exists/fit) such that even if ¢ E ~0 the solutions of (2.1) are not necessarily GE.

Here we consider another class of continuous scalar functions @(r);

{ f" cp(~)d~ _>_ q~(lr) for all } ~K = (p E ~o: there exist N > O and k > O such that k )~ r >- N .

For the class ~£, we obtain the following

LEMMA 2.4. Let $ ~ ~ , then the solutions of (2.1) are GE.

To prove lemma 2.4 we use lemma 2.5. Conti [4] and Strauss [7] gave sufficient conditions for continuability of solutions of (E).

The authors [6] investigated more flexible results• We now present the Conti-Strauss theorem for reference. For K > 0, let Sx = {x E R": Ilxll --< K}.

LEMMA 2.5. Let V: [0, ~ ) × S~c-'-, R 1 be continuous and locally Lipschitzian in x, where S~c is the complement of Sk in R". Let

v ( t , x ) - - , ~ asllxlf---, ~ for each fixed t, (2 .2)

l?(e)(t, x) -- lim sup h - l { V ( t + h, x + h f ( t , x ) ) - V( t , x ) } h~O~

<- co(t, V( t , x)) . (2.3)

966 T. HAP, A, T. YONEYAMA and J. SUGIE

Suppose co: [0, ~ ) × R I---, R I is continuous and the solutions of u' = co(t, u) are GE. Then the solutions of (E) are GE.

Proof of lemma 2.4. Let

f0' f' d~ V(t. r) = - ).(s) ds + @(~),

then (2.2) is satisfied, and we obtain

1 l)'(2.u(t, r) = ~(t) q~(r-"--)

<- k.u(t)V(t, r) + kl~(t) fo' X(s) ds for all r => N,

therefore (2.3) holds. By lemma 2.5 the solutions of (2.1) are GE.

3. NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUABILITY OF SOLUTIONS OF SCALAR EQUATIONS

In this section we shall consider necessary and sufficient conditions on cp(r) so that the solutions of (ES) or (PS) are GE. The following result is a generalization of lemma 2.3.

THEOREM 3.1. Suppose that ~.(t) is not identically zero. Then the solutions of (ES) are G E if and only if q~ ~ ~0.

Proof. The sufficiency is immediate from lemma 2.3. So we prove the necessity. Suppose that q~ ~ ~;0 and the solutions of (ES) are GE. Since ~.(t) is not identically zero, there exist to ->- 0, r > 0 and e > 0 such that ~.(t) - e on [to, to + r]. Therefore we obtain

r' >-- edp(r) on [to, to + r]. (3.1)

Since ¢ ~ ~o, there exists ro > 6 such that

f ~ < e r . (3.2) dr

o ~ ( r )

We consider the solution r( t) = r(t; to, ro). It follows from (3.1) that

fS 0 >= e(t - to) on [to, to + r]. d

0 ~,(~) This is a contradiction to (3.2) at t = to + r. Thus the theorem is proved.

Before we state results for (PS), we consider two classes of continuous scalar functions ¢(r ) ;


and

{ f f dr = } .~,= t~: r + q ) ( r ) ~ '

We notice that ~;r ~ ~1. For the proof, see lemma 4.1. The following lemma shows the property of .~ and ~, , and the proof is omitted.

LEMMA 3.1. For each L > 0 and M > 0,

f ~ dr _ M + L ~ (r)

if and only if tp ~ ~1 (resp. tp ~ ~r).

resp. Mr +-Lop(r) = ~

Here we define B(K) and A(K) as follows;

B(K) = (r >- 6: qJ(r) <= K} and A(K) = {r >- 6: V2(r) <= Kr}.

Let m(B~(K)) and m(A¢(K)) denote the Lebesgue measure of the complements B~(K) of B(K) and A~(K) of A(K) in [6, ~) , respectively. We now state a result for (PS).

THEOREM 3.2. Suppose that Z(t)l~(t) is not identically zero, and suppose that

lira inf ~p(r) > 0, r ~ z

there exists K > 0 such that m(B~(K)) < ~.

Then the solutions of (PS) are GE if and only if q~ ~ ~1.

(3.3)

(3.4)

Proof. We first prove the sufficiency. By lemma 2.3 it is enough to show that q~ + ~ E ~ ~0. It follows from (3.4) that

fB dr fs dr

From q~ ~ ,~ and lemma 3.1

Hence,

f dr ca $ ( ; ) - + ¢ ( r ) + v o + K "

f~ dr fs dr fs dr cp(r) + ~p(r) = (~ cp(r) + ~p(r) + cOO ~(r) + ~p(r)

~s dr (~ ¢(r) + g

We next prove the necessity. Suppose that ¢ ~ ~;1 and the solutions of (PS) are GE. Since ~.(t)~(t) is not identically zero, there exist t0_>O, v > O and e > O such that ~.(t)~ e and

968 T. HAaA, T. YONEYAMA and J. Suom

/~(t) ->_ e on [to, to + r]. Therefore, we have

r'--- ~(q~(r) + ~(r)) on[t0, t0+ r].

If q~ + ~p ~ ~0, the theorem is proved as in the proof of theorem 3.1. In order to show that cp + ~p ~ ~0, one can proceed as follows. From (3.3), there exist r /> 0 and N > 0 such that V/(r) --- r /for all r => N. Let I = {r: 0 --< r - N} and J = {r: r >_- N}. Hence, we obtain

f~ dr <- fs dr f dr ¢p(r) + lp(r) (h3 cp(r) + Vd(r ) + - - ~(h3 K

<--fa dr ~-fB dr ft~ --dr (tOn: ~p(r) + r 1 (K)~I cp(r) + ~p(r) + ¢(x3 K"

(P ~ ~t and (3.4) imply that the right hand side in the last inequality converges, so that + u 2 fi5 ~0. Thus the proof of theorem 3.2 is now complete.

Remark 3.1. The above result is a generalization of theorem 4.1 and corollary 4.2 in [1].

The following result for (2.1) is immediate from theorem 3.2.

COROLLARY 3.1. Suppose that A(t)/~(t) is not identically zero. Then the solutions of (2.1) are GE if and only if q~ ~ ~ l .

We next give another result for (PS).

THEOREM 3.3. Suppose that A(t)u(t) is not identically zero, and suppose that

lim inf V/(r) > 0, t ~ ~ r

there exists K > 0 such that m(AC(K)) < ~.

Then the solutions of (PS) are GE if and only if q~ ~ ~ , .

(3.5)

(3.6)

The proof is very similar to that of theorem 3.2 and omitted.

4. P E R T U R B A T I O N T H E O R E M S

In this section we investigate the continuability of solutions of (E) and its perturbed system (P). We first consider the following classes of continuous scalar functions q~(r);

~0 = {q~ ~ ,~0: q~(r) is nondecreasing},

~0 = { q~: there exist q~ E ~0 and N > 6 such that q~ (r) =< ~ (r) for all r --- N},

= { qX there exist K > 0 and N > 6 such that @(r) =< Kr for all r - N},

= {q~: q~(r) is bounded for all r ->__ 6}.

For ~0. ~ , ~ , and the above classes, we give the following lemmas.


LEMMA 4.1.

Proof. ~ ~ ~ ~ ~o and 9;, C 9;t C 9;0 are trivial. Example 2.1 shows that 9;1 ~: ~0.

The following example shows that 9;, :~ 9;t. Let I, = {r: n + 1In <= r <= n + 1 - 1/(n + 1)}. Define q~(r) as follows:

1 (i) ¢ (n )

1 (ii) q~(r)

1 (iii) •(r)

Thus 9;, ~ 9;1 ~ 9;0 is proved.

We next prove that

To prove (4.1), we need to show

= n ,

1 = m on I . ,

r

- - is linear elsewhere.

~0 ~ 9;,-

~o C 9;,.

(4.1)

(4.2)

Let

~t = {q~: there exist K > 0and N > ~ such that ¢(r) --- Kr for all r_- > N}.

If we can show the following assertions (4.3)-(4.5), then (4.2) is proved.

C 9;,, (4.3)

~t D ~o C 9;,, (4.4)

(~e u ~ ) c n 9;0 c 9;,. (4.5)

The proof of (4.3) and (4.4) are trivial. Let ~ E (Se U ~ )c n ~0. Then we may assume without loss of generality that there exist sequences {a,}, {b,}, {rz,} and {r~÷t} such that a,---~ oo, b,---~ 0 as n---~ oo, rz,- i =< r2., @( r ~ ) = a,r2, and q~(r2,÷l)= b,r~+l . Therefore, we have

ff ~-i d r > r2n + 1 - - r 2 .

z~ r + @(r) (1 + b,)r2,+t

Notice that rz~ = (1/a ,) ~P(r2,) =< (1/a,) q~(r2, + 1) = (b,/a,)r2,+ 1. Then we have

fnf -I dr > 1 - b J a , ~ = --~I as n---~ ~. r + q~(r) 1 + b,

Thus q~ E 9;,, which proves (4.5).

970 T. HARA, T. YONEYAMA and J. SCOrn

NOW we are ready to show (4.1). Let ~ ~ ~0. Then it follows from the definition of ~0 and (4.2) that there exist • ~ o ~ , and N > 0 such that $(r) _-< q~(r) for all r -> N. Therefore, we have q, E o~r. Let I~ = {r: n + ¼ -< r -~ n + ~}. Define q~(r) as in example 2.1. Then ~ ~ ~ , but 4 ~ ~ ~;0, and so #0 :# ~ . The proof of lemma 4.1 is now complete.

For a class ~ C ~0, let o ~n be the class of continuous functions f: [0, ~ ) x R ~---, R" such that ]lf(t,x)ll <-~.(t)~(l[x[I ) on [0, w) x R", where ;.(t) is a continuous function for all t -> 0 and q~(r) belongs to ~.

The following lemma is immediate from lemma 4.1.

LEMMA 4.2.

For a class ~;" C 5;~, define the perturbation classes ~3(~") and ~ ( ~ " ) by

wa(~) = {g(t, x): f ( t , x) E 9 ;~ implies that the solutions of (P) are GE },

~ ( ~ ) = {h(t): f(t, x) ~ ~" implies that the solutions of (Ph) are GE}.

It is easy to show the following property of ~3 and ~ .

LEMMA 4.3. Let ~ ' - ~ o-~,_.,r~. Then ~3(~,~) D~d(~) and ~ ( ~ ) D ~ ( ~ ) .

Let H~ be the set of all continuous functions h:[0, =)--~ R". We now state a result for (1.1).

THEOREM 4.1. For the classes ~;~, ~ , Y~, ~ , 5g" and ~" , we have

Ho = ~ ( ~ " ) = ~e(~e") = ~ ( ~ ) = ~ e ( ~ ) = ~ ( ~ ) ~ ~ ( ~ ) .

Proof. By lemmas 4.2 and 4.3, we have

nc D ~ ( ~ ) D ~(2E ~) D ~ ( ~ ) D ~ ( ~ ) D ~($;~) D ~ ( ~ ) .

We shall prove that

nc C ~(o~).

Let f ~ ~ f and h ~ H, , then there exist continuous scalar functions & and q~ ~ ~1. By lemma 2.1, it is sufficient to show that the solutions of

r' = ~.(t) ~(r) + Hh(t)II (4.6)

are GE. Suppose that there exist (to, ro) ~ [0, ~) x [0, =), T > to and a solution r ( . ) of (4.6) such that

r(t;to, ro)--, ~ ast--~ T-.

Let L = max 3.(0 and M = max []h(t)H, then there exist i ~-< T and a solut ion/( t ; to, ro) of to <~t<= T to~_t<= r


r' = L ¢p(r) + M such that

?(t;to, r o ) ~ ~ a s t ~ 7 ~-. (4.7)

By lemma 3.1, however, we obtain L~p + M ~ g0. Then it follows from lemma 2.3 that the solutions of r' = L ~O(r) + M are GE. This contradicts (4.7).

By example 2.1, it is clear that there exists h(t) ~ ~ ( ~ ) . Thus the theorem is proved.

Remark 4.1. Let (i) and (ii) in example 2.1 be replaced by 1/~p(n) = n: and 1/¢p(r) = 1 on In, respectively. Then we see that <p E ~t (in fact q~ ~ ~ ) but q~ ~ ~ .

Conversely, let (i) and the definition of I, in example 2.1 be replaced by 1/q~(n) = 1 + log n and I, = {r: n + 1/n(1 + log n) =< r -< n + 1 - 1/(n + 1)(1 + log(n + 1))}, respectively. Then we see that q~ ~ ~K but ~ ~ ~ . Thus ~f and ~ , and so ~ , and ~7 are not related by inclusion. But by lemmas 2.1 and 2.4, we have

Hc = X(~K") = ~(~).

We next give a result for (P).

THEOREM 4.2. For the classes ~7, ,~,", ~£~ and ~" , we have

~ 3 ( ~ " ) D ~ , ~ ( ~ " ) D , ~ , ~3(~;7) D27 ~ and ~(~;7) D ~ n.

Remark 4.2. It is immediate from lemmas 4.2 and 4.3 that

~ ( ~ " ) D ~(:e") D ~ ( ~ ) D ~(~,") D ~ ( ~ ? ) D ~ ( ~ ) .

What class of functions is contained in ~($;~) is an open question.

Proof. In each case the proof is similar and hence we only show ~3(~ n) D ~;~. Let f ~ ~" and g ~ ; ~ , then there exist e E ~ and V E ~;i such that [If(t,x)ll <=

A(0~(llxll) and IIg(t,x)ll <= ~(/)~(llxll), where ~.(t) and #(t) are continuous functions for all t ~ 0. By lemmas 2.1 and 2.3, it suffices to show that ~p + ap ~ g0.

Since ¢p ~ ~ , there exists M > 0 such that

~ O ( r ) + V ( r ) = < M + V ( r ) f o r r ~ 6 ,

and it follows from V E $;t that

I/ dr > =oo ¢(r) + ~(r) M + V (r) '

which shows that ¢ + V E $;0.

Remark 4.3. Bernfeld [1, Section 3] studied the continuability of solutions of (PS). Our assumptions are weaker than those of Bernfeld in [1, corollaries 3.3, 3.5 and theorem 3.7].

Finally we give extensions of theorem 4.2. Let ~ , (resp. ~7,) be the class of continuous functions g(t,x): [0, o0)x R" such that

[Ig(t,x)ll <-~(0~(llxll) on [0, ~) × R n, where /a(t) is a continuous function for all t = > 0 and

972 T. HARA, T. YONEYAMA and J. Surom

~p(r) is a positive and continuous function oo (resp. re (At (K) ) < ~ ) for some K > 0. Then theorems 3.2, 3.3 and their proofs.

for all r=> 6 such that m(B~(K))< we obtain the following corollary from

COROLLARY 4.1. For the classes ~ and ~;7, we have

~ ( ~ ? ) D ~ - ~ ~ and ~ J ( ~ I D ~ , ~ . Y ".

5. P E R T U R B A T I O N T H E O R E M S FOR D E L A Y - D I F F E R E N T I A L E Q U A T I O N S

In this section we present results for continuability of solutions of the perturbed delay- differential equations.

We first consider

x' = f ( t , x) + g(t, x ( t - z'(t))) (5.1)

where f , g: [0, ~ ) × R n---, R n, r:. [0, ~)---, [0, ~) , and all functions are continuous. For to >- 0, the initial interval at to is given by

El0 = {to} tA {s: s = t - r(t) < to fort => to}.

We denote the solutions of (5.1) with a continuous initial function Z: E,0---~ R n by x(t; to, ;4). For a class ~;n C ~ , define the perturbation class ~3"($ ;~) by

~* (~ ~) = {g(t, x): f ( t , x) ~ ~ implies the solutions of (5.1) are GE }.

THEOREM 5.1. For the class ~n, we have

~*(~e") ~ ~ .

In order to prove theorem 5.1, we give some lemmas. We consider two classes of continuous scalar functions @(r);

~ i = {@ E ~1: @(r) is nondecreasing}

and

,~r = {@ E ~ : @(r) is nondecreasing}.

Then the following lemma is easy to see by (4.2).

LEMMA 5.1.

~ 0 = 9 1 = <Tr.

Here we consider a scalar delay-differential equation

r = ~.(t)dp(r) + bt( t )~V(r( t - v(t))) (5.2)

where all functions are continuous and nonnegative on [0, o,), and @(r) > 0 and ~v(r) > 0 for all r = > 6 = > 0.

Notice that each solution of (5.2) is nondecreasing for t => to as long as the solution is defined.


LEMMA 5.2. Suppose q~E FI- If there exists a noncontinuable solution r(t) of (5.2) on [to, T) with T < ~, then r(T) = 0.

Proof . If r(t) is not continuable to T, then

r(t) ---, oe as t---~ T-. (5.3)

Suppose r (T) > 0. Then there exists 7 ~ [to, T) such that t - r(t) _~ 7~for all t E [to, T]. Since r(. ) is bounded on Eto tJ [to, 7~], r(t - r(t)) is bounded for all t ~ [to, T]. Let L = max Z(t)

to<=t~T

and M = max , u ( t ) l p ( r ( t - r(t))), then t o m t i t

r'(t) = )t(t)4p(r(t)) + u(t)~p(r(t - "t'(t))) _--< Ldp(r( t ) ) + M on[t0, T).

By lemmas 2.3 and 3.1, however, the solutions of r' = Lop(r) + M are GE. This is a contradiction to (5.3).

LEMMA 5.3. Let ¢, ~ 2? and ~v E `90. Then the solutions of (5.2) are GE.

Proof . Suppose that there exist to->-0, T > to, a continuous initial function p: E~---, [0, ~ ) and a solution r(t) = r(t; to, p) of (5.2) such that

r(t) --> ~ as t---> T-. (5.4)

Then we may assume without loss of generality that p(to) >- 6. Since 27 C `gt, by lemma 5.2, r ( T ) = 0 . Thus, there exists t t~ [ t0 , T) such that t - r ( t ) - > t 0 on [ t l ,T] . Let M = max [A(t),/~(t)], then for tt -< t < T, we have tl~t~T

r'(t) = ;~ (t) dp (r(t)) + #(t) V2 (r(t - r(t)))

<-- A(t) ~p(r(t)) + #( t )~p(r( t ) )

<- M ( cp(r(t)) + lp(r(t))) .

By theorem 4.2 and lemma 5.1, we obtain 4, + q, ~ ,90. Then it follows from lemma 2.3 that the solutions of r' = M(q,(r) + V.(r)) are GE. This contradicts (5.4).

Now we give the proof of theorem 5.1.

P r o o f o f theorem 5.1. Let f ~ 2?n and g ~ fig, then there exist ¢, ~ 27 and qp E if0 such that Ill(t, x)fl--< ~.(t)q'(llxl[) and IIg(t, x)II -< .u(t)~'(llxll), where ;t(t) and .u(t) are continuous functions for all t _-> 0.

Suppose that there exists a noncontinuable solution x(t) of (5.1). Let r(t) = [Ix(t) H, then we have

r'(t) <-IIx'(t)ll

<= ~.(t)dp(r(t)) + I~(t)lp(r(t - r(t))).

On the other hand, by lemma 5.3 the solutions of r' = ) .( t) tp(r) + #( t )~p(r( t - r(t))) are GE. This is a contradiction.

97~ T. H.~A, T. YONEYAMA and J. SUGtE

Next we consider

x ' = f ( t , x) + g( t , x,) (5.5)

where f : [0, ~ ) x Rn---~ R n, g: [0, ~ ) x Cq---~ R" are continuous, C q = C ( [ - q , 0], R ~) for q 0, and x,(s) = x ( t + s) for s ~ I - q , 0].

Assume that there exist ;t, #, ~ and 9: [0, ~ ) ~ [0, ~) such that Ilf(t, x)ll-- z(t)@(llxll), [Ig(t, z ) l l - (/)W(lllxlll) where Z ~ Cq and lllzlll = sup IIz(s)ll. Then we have the following.

s~{-q,0]

THEOREM 5.2. Suppose that q~ ~ ~ and W ~ ~0, then the solutions of (5.5) are GE.

Proo f . We first show that the solutions of

r' = A(t)@(r) + ~(0W(lllr,lll) (5.6) are GE.

Suppose that there exist t 0 ~ 0 , a continuous initial function p : [ - q , 0] ---, [0, ~ ) and a noncontinuable solution r(t) = r(t; to, p ) of (5.6). Then r(t) ~ oo as t---~ T- for some T > to, since r(t) is nondecreasing for t _>-- to. Choose tl _-> to so that r ( t l ) ~ sup p(s), then for tl t < T, Mr, Ill = r(t) and hence ~et-q,01

r'(t) = ,~(t)~p(r(t)) + ~(t)~p(r(0 ).

It follows from q~ ~ ~e and W E if0 that r(t) is bounded on [to, T], which is a contradiction. Let x( t ) be any solution of (5.5) and let r(t) = IIx(011, then we have

r'(t) <- A(t)@(r(0) ÷ ~(0~'(lllr,lll).

Since the solutions of (5.6) are GE, the comparison theorem shows that x( t ) is continuable in the future.

R e m a r k 5.1. Burton and Haddock [2] gave the following result; Suppose that there are continuous functions )., /~, q~ and ~p such that IJf(t,x)tl

-< x(t) (llxl[) and [[g(t,x)[I (t)W(llxll), where @~ ~0, ~P~ ~0 and q~+ lpE g0. Then the solutions of (5.1) are GE.

Finally we remark that the above results can be extended without difficulty to the following result.

THEOREM 5.3. Suppose that ~p ~ if0 and q~ + ~ ~ 90, then the solutions of (5.1) and (5.5) are GE.

REFERENCES 1. BERNFELD S. R., The extendability of solutions of perturbed scalar differential equations, Pacific J. Math. 42,

277-288 (1972). 2. BL~TON T. A. & HADDOCK J. R., On the delay-differential equations x'(0 + a( t ) f ( x ( t - r ( t ) ) )= 0 and

x"(t) + a(t)f(x(t - r(t))) = O, J. math. Analysis Applic. 54, 37--48 (1976). 3. CorcrI R., Limitazioni "in ampiezza" delle soluzioni di un sistema di equazioni differenziali e applicazioni, Boll.

Un. mat. ital. (3) 11,344--349 (1956). MR 18, 309. 4. CoyI-I R., Sulla prolungabilith deUe soluzioni di un sistema di equazioni differenziali ordinarie, Boll. Un. mat.

ital. (3) 11,510--514 (1956). 5. H.M~, T., YONEY~A T., & OKAZAKI Y., On the boundedness of solutions of perturbed linear systems, J. math.

Analysis Applic. 83, 188--208 (1981).


6. HAgA T., YOrCEYAMA T. & SUGm J., Continuation results for differential equations by two Liapunov functions, Annali Mat. pura appL (IV) CXXXIII, 79-92 (1983).

7. S'rRAUSS A., A note on a global existence result of R. Conti, Boll. Un. mat. ital. 22,434-441 (1967). 8. STgAUSS A. & YOP.Y~ J. A., Perturbing uniform asymptotically stable nonlinear systems, J. diff. Eqns 6, 452-483

(1969). 9. S~gA~SS A. & Yogr~ J. A., Perturbing uniform stable linear systems with and without attraction, SIAMJ. appl.

Math. 17, 725-738 (1969). 10. W~rrl~R A., The non-local existence problem of ordinary differential equations, Am. J. Math. 67, 27%284

(1945). 11. WlN'r~R A., The infinities in the non-local existence problem of ordinary differential equations, Am. J. Math.

68, 173-178 (1946).

continuability of solutions of perturbed differential equations

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