OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 12, December 1996, Pages 3729–3737 S 0002-9939(96)03674-X

OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS BINGTUAN LI (Communicated by Hal L. Smith)

Abstract. We introduce a new technique to analyze the generalized characteristic equations to obtain some infinite integral conditions for oscillation of the nonautonomous delay differential equations.

1. Introduction Consider the first order delay differential equation x0 (t) + p(t)x(t − τ ) = 0

(1)

where p(t) ≥ 0 is a continuous function and τ is a positive constant, or the more general one (2)

0

x (t) +

n X

pi (t)x(t − τi ) = 0

i=1

where pi (t) ≥ 0 are continuous and τi are positive constants. By a solution of equation (1) (or (2)) we mean a function x ∈ C([t − ρ, ∞), R) for some t, where ρ = τ (or ρ = max1≤i≤n {τi }) satisfies equation (1) (or (2)) for all t ≥ t. As is customary, a solution of equation (1) (or (2)) is said to oscillate if it has arbitrarily large zeros. Ladas [2] and Koplatadze and Chanturia [3] obtained the well-known oscillation criterion for Eq. (1) Z t 1 (3) lim inf p(s) ds > . t→∞ e t−τ Ladas and Stavroulakis [4] and Arino and Gy¨ori [5] established sufficient conditions for oscillation of Eq. (2) which are in some sense extensions of (3) in the case of several delays. These authors based their techniques on the study of functions of the form x(t)/x(t − τi ) or the generalized characteristic equations to obtain finite integral conditions for oscillation of equation (1) or (2). Integral conditions like (3) have been employed by many authors in the study of the oscillatory properties of various functional differential equations. For example, see papers by Grammatikopoulos, Grove and Ladas [7], by Ladas and Qian [11], and by Zhang and Gopalsamy [10]. Received by the editors May 12, 1995. 1991 Mathematics Subject Classification. Primary 34K15; Secondary 34C10. Key words and phrases. Oscillation, nonoscillation, delay differential equations. c

1996 American Mathematical Society

3729

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3730

BINGTUAN LI

For other relevant results, the reader is referred to Hunt and Yorke [6], Gy¨ori [8], Cheng [12], Kwong [14] and Tramov [1]. In a recent paper [16], this author showed that if Z t 1 (4) p(s) ds ≥ for some t0 > 0 e t−τ and

Z

(5)

 1 dt = ∞, p(s) ds − e t−τ

Z

∞

p(t) t0

t

then every solution of Eq. (1) oscillates. This result improves condition (3). In this paper, we make use of the generalized characteristic equations and a new technique to examine equations (1) and (2). In section 3, we present an infinite integral condition for oscillation of Eq. (1) which indicates that condition (3) or (4) or even the condition Z t lim inf p(s) ds > 0 t→∞

t−τ

is no longer necessary. In section 4, we establish sufficient conditions for oscillation of Eq. (2). 2. Lemmas We need the following lemmas for the proofs of our main results. Lemma 1. If

Z

t+τi

lim sup t→∞

pi (s) ds > 0, t

for some i, and x(t) is an eventually positive solution of Eq. (2), then for the same i, (6)

lim inf t→∞

x(t − τi ) < ∞. x(t)

Proof. In view of the assumption there exist a constant d > 0 and a sequence {tk } such that tk → ∞ as k → ∞ and Z tk+τ i pi (s) ds ≥ d, k = 1, 2, . . . . tk

Then there exists ξk ∈ (tk , tk + τi ) for every k such that Z ξk Z tk +τi d d (7) pi (s) ds ≥ and pi (s) ds ≥ . 2 2 tk ξk On the other hand, Eq. (2) implies (8)

x0 (t) + pi (t)x(t − τi ) ≤ 0

eventually. By integrating (8) over the intervals [tk , ξk ] and [ξk , tk + τi ], we find Z ξk (9) x(ξk ) − x(tk ) + pi (s)x(s − τi ) ds ≤ 0 tk

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OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

and (10)

Z

tk +τi

x(tk + τi ) − x(ξk ) +

3731

pi (s)x(s − τi ) ds ≤ 0.

ξk

By omitting the first terms in (9) and (10) and by using the decreasing nature of x(t) and (7), we find d −x(tk ) + (ξk − τi ) ≤ 0 2

d − x(ξk ) + x(tk ) ≤ 0 2

and

or x(ξk − τi ) ≤ x(ξk )

 2 2 . d

This completes the proof. Lemma 2. If Eq. (2) has an eventually positive solution, then Z t+τi (11) pi (s) ds ≤ 1, i = 1, 2, . . . , n, t

eventually. Proof. See the proof of Theorem 2.1.3 in [9]. 3. Equations with a single delay Our objective in this section is to establish the following result. R t+τ Theorem 1. Suppose that t p(s) ds > 0 for t ≥ t0 for some t0 > 0 and  Z t+τ  Z ∞ (12) p(t) ln e p(s) ds dt = ∞. t0

t

Then every solution of Eq. (1) oscillates. Proof. Assume the contrary. Then we may have an eventually positive solution x(t) of Eq. (1). Obviously x(t) is eventually monotonically decreasing. Let λ(t) = −x0 (t)/x(t). Clearly for large t, function λ(t) is nonnegative and continuous, and Rt x(t) = x(t1 ) exp(− t1 λ(s) ds), where x(t1 ) > 0 for some t1 ≥ t0 . Furthermore, λ(t) satisfies the generalized characteristic equation Z t  λ(t) = p(t) exp (13) λ(s) ds . t−τ

One can easily show that (14) and thus

erx ≥ x +

ln(er) r

 λ(t) = p(t) exp A(t) · 

1 ≥ p(t) A(t)

Z

for r > 0

1 A(t)

Z

t

 λ(s) ds

t−τ

t

ln(eA(t)) λ(s) ds + A(t) t−τ

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3732

BINGTUAN LI

where A(t) =

R t+τ

p(s) ds. It follows that  Z Z t t+τ p(s) − p(t) λ(s) ds ≥ p(t) ln e

t

Z

(15)

λ(t) t

t−τ

Then, for N > T , Z

Z

N

T

(16)

Z

Z ≥

 Z p(t) ln e

T

Z

N

p(s) ds dt − t N

p(s) ds .

t

t+τ

λ(t)



t+τ

t

p(t) T

λ(s) ds dt



t+τ

p(s) ds

t−τ

dt.

t

By interchanging the order of integration, we find  Z N Z t Z N −T Z s+τ p(t) λ(s) ds dt ≥ p(t)λ(s) dt ds T

t−τ

T N −τ

s

Z

(17)

Z

T Z N −τ

=

p(t) dt ds s Z t+τ

λ(t) T

From this and (16), it follows that Z N Z t+τ Z (18) λ(t) p(s) ds dt ≥ N −τ

t

p(s) ds dt. t

N

 Z p(t) ln e

T

By Lemma 2, we have

Z

s+τ

λ(s)

=



t+τ

p(s) ds

dt.

t

t+τ

p(s) ds ≤ 1

(19) t

eventually. Then by using (18) and (19), we have  Z t+τ  Z N Z N λ(t)dt ≥ p(t) ln e p(s) ds dt N −τ

T

or (20)

ln

x(N − τ ) ≥ x(N )

Z

t

N

 Z p(t) ln e

T



t+τ

p(s) ds

dt.

t

In view of (12), (21)

lim

t→∞

x(t − τ ) = ∞. x(t)

On the other hand, (12) implies that there exists a sequence {tn } with tn → ∞ as n → ∞ such that Z tn +τ 1 p(s) ds ≥ for all n. e tn Hence by Lemma 1, we obtain lim inf t→∞

x(t − τ ) < ∞. x(t)

This contradicts (21) and completes the proof.

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OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

3733

Theorem 1 substantially improves condition (3). In fact, if (3) holds, then Z ∞ (22) p(t) dt = ∞ t0

and there exists c > 0 such that for large t,   Z t+τ (23) p(s) ds ≥ c. ln e t

R t+τ (22) and (23) imply (12). Condition (12) is an evaluation of p(t) and t p(s) ds R t+τ in an infinite interval. Obviously, condition t p(s) ds > 0 is necessary for (12). Example 1. Consider the delay differential equation x0 (t) + exp(k sin t − 1)x(t − 1) = 0

(24)

where p(t) = exp(k sin t − 1) and k is a positive constant. Clearly, Z t lim inf p(s) ds < 1/e. t→∞

t−1

So condition (3) is not satisfied. By Jensen’s inequality,  Z t+1  Z ∞ Z ∞ Z t+1 p(t) ln e p(s) ds dt ≥ p(t) k sin s ds dt t

0

0

t

  Z 2k sin 12 ∞ 1 = exp(k sin t) sin t + dt. e 2 0 Rt On the other hand, it is easy to see that 0 exp(k sin t) cos t dt is bounded and Z 2π exp(k sin t) sin t dt > 0. R∞

It follows that 0 p(t) ln(e Eq. (24) oscillates.

0

R t+1 t

p(s) ds) dt = ∞. By Theorem 1, every solution of

We would like to point out that Theorem 1 in [15] is not always true. We use the following example to illustrate this. Example 2. Consider the delay differential equation   1 1 0 x (t) + (25) 1+ x(t − 1) = 0. e t Let p(t) = 1e (1 + 1t ). Clearly,  Z t+1     Z ∞ Z 1 ∞ 1 p(t) ln e p(s) ds dt ≥ ln 1 + ln 1 + dt = ∞. e 1 t 1 t By Theorem 1, every solution of (25) oscillates. On the other hand, the delay equation 1 (26) x0 (t) + x(t − 1) = 0 e has a positive solution, e−t . Therefore (25) and (26) possess different oscillatory behaviors though limt→∞ 1e (1 + 1t )/ 1e = 1. In the proof of Lemma 2 in [15], the chosen neighborhood δ(µ0 ) ⊂ M of µ0 actually depends on the given large constant m. As m → +∞, δ(µ0 ) may not exist. Therefore, the proof is incorrect.

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3734

BINGTUAN LI

4. Equations with several delays In this section we obtain sufficient conditions for the oscillation of all solutions of Eq. (2). The main result is the following theorem. R t+τ P Theorem 2. Let τn = max{τ1 , τ2 , . . . , τn }. Suppose that ni=1 t i pi (s) ds > 0 for t ≥ t0 for some t0 > 0 and that Z t+τn lim sup (27) pn (s) ds > 0. t→∞

If, in addition,

Z

n X

∞

(28) t0

t

! pi (t) ln e

i=1

n Z X

!

t+τi

dt = ∞,

pi (s) ds

t

i=1

then every solution of Eq. (2) oscillates. Proof. Assume the contrary. Then Eq. (2) may have an eventually positive and decreasing solution x(t). Let λ(t) = −x0 (t)/x(t). Then λ(t) is nonnegative and continRt uous, and there exists t1 ≥ t0 with x(t1 ) > 0 such that x(t) = x(t1 ) exp(− t1 λ(s) ds). Furthermore, λ(t) satisfies the generalized characteristic equation Z t  n X λ(t) = pi (t) exp λ(s) ds . Let B(t) =

t−τi

i=1

R t+τi

Pn

pi (s) ds. By using (14), we find   Z t n X 1 λ(t) = pi (t) exp B(t) · λ(s) ds B(t) t−τi i=1   Z t n X ln(eB(t)) 1 pi (t) λ(s) ds + ≥ B(t) t−τi B(t) i=1

i=1 t

or

n Z X

t+τi

pi (s) ds λ(t) −

t

i=1

n X

≥

(29)

!

T

! pi (t) ln e

N

≥

(30)

T

n Z X

pi (s) ds λ(t) dt −

n X

! pi (t) ln e

i=1

n Z X i=1

t

λ(s) ds t−τi

t+τi

pi (s) ds .

n Z X i=1

Z

N

T

i=1

t−τi

i=1

=

T

λ(s) ds dt

!

t+τi

pi (s) ds

t−τi

dt.

t

T

n Z X i=1

t

pi (t)

By interchanging the order of integration, we find Z t Z NX n n Z N −τi Z X pi (t) λ(s) ds dt ≥ (31)

!

t

i=1

!

t

i=1

Z

t+τi

Z pi (t)

i=1

i=1

Then for N > T Z N X n Z

n X

s

N −τi

s+τi

Z

t+τi

λ(t)

T

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 pi (t)λ(s) dt ds pi (s) ds dt.

t

OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

3735

From (30) and (31) it follows that (32) n Z X i=1

Z

N

t+τi

λ(t)

N −τi

Z

N

pi (s) ds dt ≥

n X

T

t

! pi (t) ln e

i=1

On the other hand, by Lemma 2, we have Z t+τi (33) pi (s) ds ≤ 1,

n Z X i=1

!

t+τi

pi (s) ds

dt.

t

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

t

eventually. Then by (32) and (33), we find ! Z N X n Z N n n Z X X λ(t) dt ≥ pi (t) ln e i=1

N −τi

T

i=1

(34)

n X

x(N − τi ) ln ≥ x(N ) i=1

Z

!

n X

N

T

pi (t) ln e

i=1

n Z X i=1

pi (s) ds

dt

t

i=1

or

!

t+τi

t+τi

! pi (s) ds

dt.

t

In view of (28), (35)

lim

t→∞

n Y x(t − τi )

x(t)

i=1

= ∞.

This implies (36)

lim

t→∞

x(t − τn ) = ∞. x(t)

However, by Lemma 1, we have lim inf t→∞

x(t − τn ) < ∞. x(t)

This contradicts (36) and completes the proof. Corollary 1. If (37)

lim inf t→∞

n Z X

t+τi

pi (s) ds >

t

i=1

1 , e

then every solution of Eq. (2) oscillates. Proof. Let τ1 < τ2 < · · · < τn . Then it follows from (37) that there is an m with 1 ≤ m ≤ n such that Z t+τm (38) pm (s) ds > 0 lim sup t→∞

and (39)

lim inf t→∞

t

m Z X i=1

t

t+τi

pi (s) ds >

1 . e

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3736

BINGTUAN LI

Now assume, for the sake of contradiction, that Eq. (2) has an eventually positive solution x(t). Then x(t) is also an eventually positive solution of the inequality x0 (t) +

(40)

m X

pi (t)x(t − τi ) ≤ 0.

i=1

So, by Corollary 3.2.2 in [13], we know that the equation 0

(41)

y (t) +

m X

pi (t)y(t − τi ) = 0

i=1

has an eventually positive solution as well. On the other hand, from (39) we see that for some t0 > 0, ! ! Z ∞ X m m Z t+τi X (42) pi (t) ln e pi (s) ds dt = ∞. t0

i=1

i=1

t

Then by Theorem 2, every solution of Eq. (41) oscillates. This is a contradiction and the proof is complete. Acknowledgment I would like to thank Professor Y. Kuang for his advice. Also, I want to thank the referee for making several helpful suggestions. References 1.

2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

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15. G. Ladas, C. Qian and J. Yan, A comparison result for the oscillation of delay differential equations, Proc. Amer. Math. Soc. 114 (1992), 939–946. MR 92g:34097 16. B. Li, Oscillations of delay differential equations with variable coefficients, J. Math. Anal. Appl. 192 (1995), 312–321. CMP 95:12 17. L. H. Erbe, Qingkai Kong and B. G. Zhang, Oscillation theory for functional differential equations, Marcel Dekker, New York, 1995. CMP 95:6 Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804 E-mail address: [email protected]

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