The nonoscillatory solutions of delay differential equations with impulses

li NOgI~

- HOLLAND

The Nonoscillatory Solutions of Delay Differential Equations with Impulses J. H. Shen*

Department of Mathematics Normal University of Hunan, Changsha Hunan ~10081, People's Republic of China

Transmitted by Melvin Scott

ABSTRACT We obtain a sufficient condition for the impulsive delay differential equation

x'(t) = - ~ p , ( t ) x ( t - ri(t)) ,

t ¢ tk

i=1

x( t~ ) - x( tk) = I~( x( t~)),

ke

to have a nonoscillatory solution.

1.

INTRODUCTION

Many evolution processes in nature are characterized by the fact that at certain moments of time they experience an abrupt change of state. That was the reason for the development of the theory of impulsive ordinary differential equations (IODEs), and now this theory has been elaborated to a considerable extent [1, 2]. The mathematical modeling of several real-world problems leads to ordinary differential equations with delay (DDEs). With more and more clear indications of the importance of DDEs in the applications and also *This research was partially supported by the NNSF of China.

APPLIED MATHEMATICS AND COMPUTATION 77:153-165 (1996) © Elsevier ScienceInc., 1996 655 Avenue of the Americas, New York, NY 10010

0096-3003/96/$15.00 SSDI 0096-3003(95)00198-Q

154

J.H. SHEN

with the number of interesting mathematical problems involved, it is not surprising that the study of DDEs has undergone a rapid development in the last 20 years [3-5]. It is possible that the first-order DDEs have both nonoscillatory and oscillatory solutions, unlike the first-order ODEs, which have only nonoscillatory solutions. This is one of the reasons why the nonoscillatory and oscillatory theory of DDEs has received extensive attention [4, 6, 7]. It is not difficult to see that when a DDE is subjected to impulsive perturbations (IDDEs), its nonoscillatory solutions may or not may continue to persist. Thus the question as to how one determines the type of DDE for its nonoscillatory solutions to persist under certain impulsive perturbations naturally arises. In this paper we make a contribution to this nonoscillation question by considering the following IDDE with several variable coefficients and variable delays

x ' ( t ) = - ~ p~(t) x ( t - r i ( t ) ) , i=l x ( t [ ) - x(tk) = h ( x ( t k ) ) ,

t¢ tk (1) k ~ N,

where ~ denotes the set of all positive integers. Pi, r~ ~ C([t0,~), R+), t - r~( t) --* oo as t--*o% i = 1 , 2 , . . . , n , I k ( u ) ~ C(R), k ~ ~, and0~< t o to, we define

r°" "~- l<~i<~nmin{ t~rinf( t -

r~( t)}}.

(2)

By a solution of (1) we mean a real-valued function x(t) defined on [ r~, ~) for some (r/> to, which is piecewise left continuous on [ r~, ~), and satisfies (1). Such a solution is said to be nonoscillatory if it is either eventually positive or eventually negative. Otherwise it is called oscillatory. Equation (1) with the special case when n = 1, rl(t) -- r > 0, p1(t) - p > 0, and Ik(u) = bku becomes x'(t) • (t;)

+ px( t - r) = o, -

=

t. k e

(3)

A nonoscillation result of (3) was obtained by Gopalsamy and Zhang [8] by using the Schauder fixed-point theorem, which states that if there exists a

155

Delay Differential Equations

constant c > 0 such that pre ~< 1 - c, and bk > 0, with ~ = l bk < 0% then (3) has a nonoscillatory solution. This result is recently improved and extended by Chen et al. [9] to the IDDE x ' ( t ) + p ( t ) x( t - r) = 0,

t ~ tk

(4)

By using a nice but complicated constructive method, they proved the following general theorem

THEOREM A [9].

Assume that p(t) ~ ([t0, :¢), R+), r > 0, that bk > 0,

for k = 1 , 2 , . . . ,

(5)

and that the delay differential equation

y ' ( t ) + p ( t ) y( t - r) = 0

(6)

has a nonoscillatory solution. Then (4) has also a nonoscillatory solution.

But, as seen in [9], the method used in [9] cannot be applied for Eq. (1). Our aim in this paper is to extend Theorem A to Eq. (1) by using another method. 2.

MAIN RESULTS Throughout this section, we assume that for Eq. (1)

pi, r i ~ C ( [ t o , o o ) , R + ) ,

t-r~(t)---~ooast--~oo,

i= 1,2,...,n.

(7) Ik(u) ~ C ( R ) , k e ~l,

and

0 <~ t o < t l < t 2 < "" < t k ~ ooas k---~ 0%

and denote for any o- >/ to, PC~ = {~b( t ) : [ r~, ar ] -~ R, ~b(t) is piecewise right continuous on [ r~, (r ] with discontinuities of the first kind},

(s)

156

J . H . SHEN

where r~ is defined as in (2). Our main result is the following

Let (7) and (8) hold and assume that

THEOREM 1.

h(

-

u)

=

-

h(u),

>/0,

for u e R ,

k= 1,2,....

(9)

Then the delay differential equation y'(t) = - L p,(t) y(t - r,(t))

(10)

i=1

has a nonosciUatory solution, which implies that (1) also has a nonoscillatory solution.

REMARK 1. W h e n n = 1, rl(t) -~ r > 0, and Ik(u) = bk u, T h e o r e m 1 becomes T h e o r e m A.

REMARK 2. By combining T h e o r e m 1 and various criteria for (10) having a nonoscillatory solution, we can obtain various sufficient conditions for (1) having a nonoscillatory solution. For example, we have the following corollaries.

COROLLARY1.

Consider the impulsive delay equation

x'( t) = - L Pix( t - ri) ,

t ¢ tk

i=l

x( t; ) - x( tk) = b~ x( tk),

k ~ ~.

Assume that pz > 0, r i > 0, i = 1 , . . . , n, bk >1 O, k ~ ~. Then each of the following assumptions implies that this equation has a nonosciUatory solution:

(i) rmax L p , e x p ( r J r m a x ) < 1, i=l

where rma~ = max{ q , . . . ,

rn}

Delay Differential Equations

157

(ii)

A + i piexp( -AT,) z= 1

= 0,

Assume that (7149)

COROLLARY2.

where 7( t> = min 14 zc n(max{t,, solution. To prove the above results,

for some A < 0.

hold and that

t - r1( t))).

Then (1) has a nonoscillatory

we need the following

lemmas.

LEMMA 1. Let u > t,, 4(t) E PC, be given Then (10) has a unique solution y(t): [ rr, m) -+ R satisfying yf t) E 4[a, m), R) and y
PROOF.

For any fixed

A > CT, we define

AI --) R, satisfies

fi = {Y(t>:[r,,

y(t)

E PC,, for r, G t < (T,

and y(t) E C([(+, Define the operator

A], R) for u<

S on R by

4(t), (SY)W

=

t < A}.

d~(u>-

r,
jt‘Ti,l2 pi( s) Y( s - rt( 4)

ds,

a
(11)

158

J . H . SHEN

Clearly, S maps 1~ into itself. Now define the sequence of functions

c~( t ),

r~ < t < er

~o(t) = (12)

yk(t) = ( S y k _ l ) ( t ) ,

r~<~t<~A,k=l,2,....

Set SupT~ ~ t
max {Sup~ ~
1.<
By using (11) and (12) we can obtain for er ~< t ~< A n

[yl(t)

Yo(t)[ < I r E

pi(s)[yo(s

r~(s))l ds <. n a ~ ( t -

-

(r).

rr i = l

And so

l yl( t) - yo( t)l <~ nafl[t - (r[,

r~ <<.t <~ A.

Let us now suppose or(nil)

I Yk(t) -- Yk-l(t)[

<

-

-

k

k!

[ t - ~ l k,

r~< t<~A.

For ¢r ~< t < A we have then n

I y~+

1(t)

-

y~( t)l -<
~(n~) -

-

k~

~

Is - ri( s) - al k ds

a ( nfl ) k+ l

k!

a ( nfl ) k+ l

f: (s-tr)kds

(k+

1)!

(t

-- o') k+1

And so

~,(nt3) k÷i ] yk+ l( t) -- yk( t)] <~

( k + 1)!

]t--(rl k+l,

r~, ~ t <~ A.

Delay Differential Equations

159

By induction we have

.(~g)k I y~(t) - y~_,( t)l < - - i t

- o'l k,

k!

r~ ~< t <~ A, k = 1 , 2 , . . . .

Noting k

y~(t) = y0(t) + E [ yj(t) - yj_ l(t)],

k = 1,2,...

j=l

it follows t h a t L i m k _ ~ yk(t) = y*(t) uniformly on [r~, A]; y*(t) = 4)(t) for r~ ~< t ~< (r; and y*(t) ~ C([g, A], R) for (r < t < A. Therefore, from n

(r 4 t <<.A i=l

we have n

y*( t) = 4)( ~ ) - f j E p,( s) y*( s -

~( ~)) ~,

o'<~t
i=1

and so y*(t) is a solution of(10) on [(r, A] satisfying y*(t) ~ C([(r, A], R) and y*(t) = 4)(t), r~ < t ~ ~. T h e uniqueness of y*(t) can be proved by using the Gronwall's inequality as in the case of O D E s and so we omit it here. Now we choose above A such t h a t t - r~(t) >1 or, i = 1 , . . . , n, for t > l A . T h u s r A >/cr. Next, we consider the new initial condition problem:

y'(t) = - ~ pi(t) y(t-

ri(t)) ,

t>~ A (13)

i=1

y( t) = y*( t),

~A < t < A.

Since y*(t) is continuous on [rA, A], it follows t h a t (13) has a unique solution ~(t) ~ C([ rA, oo), R) satisfying ~(t) -- y*(t), r A ~< t ~< A. Clearly,

y(t) =

y*(t),

r ~ < t~< A

~(t),

A < t <

160

J.H. SHEN

is a unique solution of (10) satisfying y(t) ~ C~[(r, oo), R) and y(t) = ~b(t), r~ ~< t ~< ~. The proof is complete.

LEMMA 2. Let (7) hold and (r >1 t o. Suppose that u( t) and v( t) are two solutions of (10) that satisfy the conditions u, v ~ C ( [ ( r , ~ ) , R), u((r) t> v ( a ) ,

and

v ( t ) > 0for a~< t < oo (14)

v(t)

u(t)

v( a--'-)" >~ ~

>~ 0,

for r~ ~< t < ¢r.

(15)

Then u ( t ) >/ v ( t ) ,

foray< t
(16)

PROOF. The proof is divided into two claims.

CLAIM 1. Show u(t) > 0,

for ~ < t < ~.

(17)

Indeed, from (14) we know u(t) is positive in some right neighborhood of ¢r. Let [ ¢r, T) be the maximal interval in which u(t) is positive. We claim that T = ~. Otherwise, if T < ~, then

u( T ) = O.

(18)

Let for t ~ [(r, T),

Hi(t) = max{(r, t -

So(t) = v'( t)/v( t),

ri( t)},

hi(t) = min{(r, t - ri( t)}

(19)

&( t) = u'( t)/u( t).

(20)

Then s0(t) must satisfy

So(t) + E pi( t) ~=~

v( hi( t)) v(~)

(r~< t < T.

(21)

Delay Differential Equations

161

In fact, from (20) we see t h a t v(t) = v(rr)exp

< t < T.

a o ( S ) ds,

(22)

And so

v(Hi(t))

[

v(t)

t ao(s) ds]

exp - fH,(t)

(r<~t<

'

T,i=l,...,n.

(23)

By dividing both sides of (10) by v(t) and using (20) and (22) we get

ao(t ) + ~ Pi(t) v ( t - ri(t)) ~=,

v(HXt))

q

q=0

o'<~t
T o show (21) it remains to prove t h a t for (r ~< t < T and for every i= 1,2,...,n,

~( t - ~i( t ) ) v( Hi( t ) )

.(hi(t)) v( ,~)

T o this end observe t h a t if t - ri(t) >i (r, then hi(t) = o" and Hi(t) = t ri(t). And so

v ( t - ~(t))

v(H,(t))

v( Hi( t ) )

v( Hi( t ) )

1

~(~) v( o" )

=

v(~,(t)) v( ~r )

On the other hand, if t - ri( t) < cr then hi(t) = t - ri( t) and Hi(t) = o'. And so again

v ( t - rXt)) ~(u~(t))

v(h,(t)) v(¢)

Therefore, (21) holds. Similarly we can prove

~o( t) + Z p,( t) ,=1

u( hi( t) ) ~,( o- )

ex IC o, , q =0

¢~t
(24)

162

J . H . SHEN

We now show that a o ( t ) ~
for tr ~< t < T.

(25)

Let 8(t) ~ O([ or, T), R) be an arbitrary function such that a0(t) ~< 8(t) 0 for o" ~< t < T. Then by (15) and (21) we see that for o" ~< t < T, n

~o( t) = - S. pi( t)

v( hi( t))

i=1

V( Or )

°xp[J:s,, q

t)) exp [ - fi,(t) t s( s) ds] <-..O. - ~ Pi(t) u(hi( u( o'-~

(26)

i=l

For an arbitrary T*: o" < T* < T, we let X be the Banach space of all bounded and continuous functions defined on [or, T*] with the sup-norm. Then a=

{8(t) ~X:a0(t)

~< 8 ( t ) ~<0, o ' < t <

T*}

is a bounded, closed, and convex subset of X. Define a mapping S: ~ ~ X

(sa)(t)

n = - 22 pz(t) i=1

u( hi( t) )

(r~< t~< T*.

?'t( Or )

Then (26) implies that S maps ~ into itself. It is easy to prove, by using the mean value theorem and the Lebesgue-dominated convergence theorem, that S is continuous and the closure of SI~ is compact and so S is completely continuous. Therefore, the Schander's fixed-point theorem [10] implies that S has a fixed point 8*(t) in ~ . T h a t is,

8*(0

= -

p,(t)

u(~)

exp -

i=l

8*(~)ds,

~<

,(t)

From this we can see that the function

u(t), x(t)=

u(~,)exp

f~ ' 8 * ( s ) ds,

r,~< t < o, ~ < t.< T*

t~

T*.

Delay Differential Equations

163

is a solution of (10) on [o', T*] satisfying initial condition X ( t ) = v~t), r~ ~< t ~< o-. In view of the uniqueness of the solution of (10) we get X(t) = u(t) for r,~ <~ t ~< T*, and so

8*(t) = x'( t)/x(

t) = ~'( t ) / u ( t) = rio(t),

~ < t < T*.

Since ao(t) ~ ~*(t) <~ 0, it follows that ao(t) <~ rio(t) for ¢r ~< t ~< T*. In view of the arbitrarity of T*, we see that (25) must hold. Now by using (20) we find

v(t)

=

~(~)exp/~o(~) d~,

~(t) = u ( ~ ) e x p

s2ri0(s) es,

~<

t < T.

Hence (14) and (25) imply that

u(t)>~v(t)

forcr< t<

r.

As v(T) > 0 and in view of the continuity of u(t) we let t ~ T- to obtain ~(T) > 0. This contradicts (18) and so (17) holds.

CLAIM 2. Complete the proof by showing (16). Indeed, from (17) and by changing T by ~ in the proof of Claim 1, we can prove that (16) holds, and so the proof is complete.

PROOF OF THEOREM 1. Let Y0(t) be a nonoscillatory solution of (10). Then by (9) we see that - y o ( t ) is also a nonoscillatory solution of (10). Thus we may assume that y0(t) is eventually positive. Let N be an integer such that y0(t) > 0 for t >~ r t N . Consider the initial function

¢1(t) =

Yo( t),

rtN <<.t < t~

yo( t~) + xN( ~o( t~)),

t = t~.

Clearly, ~bl(t) E P C t . Thus we can apply Lemma 1 to obtain a unique solution yl(t) of (105, which satisfies yl(t) ~ C([tN, o~), R) and yl(t) = ~bl(t) for rtN <~ t <~ t N. In view of (9) we have

yl(tN) = y0(tN) + IN( y0(t~)) >/y0(tN).

164

J.H. SHEN

And so

y0(t)

yl(t)

> o,

for

< t <

By Lemma 2 we have then Yl(t) t> Yo(t) for t >1 t N and so Yl(t) >/ Y0(t),

for t>~ rtN.

(27)

We now consider the second initial function

(~2(t) =

Yl(t),

rtN+l <. t < tN+ 1

Yl( tN+] ) + IN+I( Yl( tN+I) ),

t = tN+ I.

Then ¢2 ~ PC t~÷~- Thus Lemma 1 can again be applied to obtain a unique solution y2(t) of (10) satisfying y2(t) ~ C([tg+l, oo), R) and yz(t) = ~bz(t) for rtu+~ <~ t <~ tN+ 1. Clearly, (9) implies y2(tN+l) >1 y l ( t u + i ) and

y~(t) y,(tN+,)

y2(t) y2(t,,~)

> O,

for rtu+ < t < tN+ 1 1

"

Hence, by using again Lemma 2 we get

y2(t) >1 ~l(t),

for t >~ %+.

(28)

Therefore, by induction we can obtain a sequence of solutions { Ym(t)} of (10) which have the following properties: (i) y,~(t) is a solution of (10) on [tN+m_l,~) satisfying Y m ( t ) ~ C([ tN+ m- 1, ~), R) and the initial condition Ym(t) = 4'm(t)

Y,n- 1(t), =

rtN+ m-1 < t < tN+ m- 1

Ym-~(t~+m-1)

m=

1,2,...,

(ii) y,~(t) >1 y,~_l(t) for t >1 rt~ . . . . , m = 1, 2 . . . . .

Delay Differential Equations

165

Since yo(t) > 0 for t >1 rt~ , it follows that m= 1,2,.... Finally, we define

Y0(1)' x(t)= yl(), ym(t), .

.

.

ym(t) > 0 for t >t rtN. . . . ,

rt~ ~ t ~ t N t N < t ~ tN+ 1 . . .

tN+m_ 1 < t~< tN+m, m = 1 , 2 , . . . .

It is easy to show that x(t) is positive, piecewise left continuous and is a solution of (1) on [ tN~ ~). The proof is complete. REFERENCES 1 V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. 2 D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Wiley, New York, 1989. 3 J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 4 K. Gopalsamy, Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic, Boston, 1992. 5 Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. 6 G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Dekker, New York, 1987. 7 I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon, Oxford, 1991. 8 K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl. 139:110(1989). 9 M-P. Chen, J. S. Yu, and J. H. Shen, The persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations, Comput. Math. Appl. 27:1 (1994). 10 D.R. Smaet, Fixed Point Theorems, Cambridge Univ. Press, Cambridge, UK, 1980.