Page 1

Journal of Applied Analysis

Vol. 8, No. 2 (2002), pp. 261–278

OSCILLATION OF SOLUTIONS TO

NONLINEAR NEUTRAL DELAY

DIFFERENTIAL EQUATIONS

S. H. SAKER and I. KUBIACZYK

Received August 28, 2001 and, in revised form, May 15, 2002

Abstract. In this paper we shall consider the nonlinear neutral delay

differential equations with variable coefficients. Some new sufficient con-

ditions for oscillation of all solutions are obtained. Our results extend

and improve some of the well known results in the literature. Some ex-

amples are considered to illustrate our main results. The neutral logistic

equation with variable coefficients is considered to give some new suffi-

cient conditions for oscillation of all positive solutions about its positive

steady state.

1. Introduction

In recent years the literature on the oscillation theory of neutral delay

differential equations is growing very fast. It is relatively a new field with

interesting applications in real world life problems. In fact, the neutral

delay differential equations appear in modelling of the networks containing

lossless transmission lines (as in high-speed computers where the lossless

transmission lines are used to interconnect switching circuits), in the study

2000 Mathematics Subject Classification. 34K11, 34K40.

Key words and phrases. Oscillation, nonlinear neutral delay differential equations, neu-

tral delay logistic equation.

ISSN 1425-6908

c ? Heldermann Verlag.

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262 S. H. SAKER and I. KUBIACZYK

of vibrating masses attached to an elastic bar, as the Euler equation in some

variational problems, theory of automatic control and in neuromechanical

systems in which inertia plays an important role. see Hale [16], Driver

[8], Brayton and Willoughby [5], Popov [23] and Boe and Chang [4] and

reference cited therein).

Recently some papers [13, 34] have appeared which are concerned with

the oscillation and nonoscillation behavior of the neutral delay differential

equation with variable coefficients,

d

dt[a(t)x(t) − P(t)x(t − τ)] + Q(t)f(x(t − σ)) = 0, t ≥ t0

where

a,P, Q ∈ C([t0,∞),?+),

and f satisfies

f ∈ C([t0,∞),?), uf(u) > 0 for u ?= 0

and lim

u→0

Let ρ = max{σ,τ} and let t1≥ t0. By a solution of equation (1.1) on [t1,∞)

we mean a function x ∈ C([t1−ρ,∞),?), such that (a(t)x(t)−P(t)x(t−τ)) is

continuously differentiable on [t1,∞) and such that equation (1.1) is satisfied

for t ≥ t1.

Let t1≥ t0be a given point, let φ ∈ C([t1− ρ,t1],?) be a given initial

function. By using the method of steps one can see that equation (1.1) has

a unique solution x ∈ C([t1− ρ,∞),?) such that

x(t) = φ(t) for t ∈ [t1− ρ,t1].

As usual, we say that the equation (1.1) is oscillatory if every solution

of (1.1) is oscillatory, i.e., for every initial point t1 ≥ t0 and for every

initial function φ ∈ C([t1− ρ,t1],?) the unique solution of equations (1.1)

and (1.4) has arbitrarily large zeros. Otherwise the solution is called non-

oscillatory. The oscillation of various functional differential equations has

been investigated by several authors. For some contributions we refer to the

monographs [1, 2, 3, 9, 15, 21].

The first systematic work about oscillation of neutral delay differential

equations is given by Zahariev and Bainov [32]. For the oscillation of equa-

tion (1.1) when P(t) and Q(t) are constants, a(t) = 1 and f(x) = x, we refer

to the articles by Ladas and Sficas [20], Grammatikopoulos et al. [10] and

Zhang [33] and the references cited therein. For P(t) equal to a constant,

a(t) = 1 and f(x) = x we refer to the articles by Grammatikopoulos, Grove

and Ladas [12] Zhang [33] and Saker and Elabbasy [24]. Grammatikopouo-

los et al. [11] considered the neutral delay differential equation (1.1) when

a(t) = 1 and f(x) = x and presented some finite sufficient conditions for

(1.1)

σ,τ ∈ [0,∞),

(1.2)

u

f(u)= β exists.

(1.3)

(1.4)

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OSCILLATION OF SOLUTIONS263

oscillation of all solutions when P(t) takes some values in the interval [0,1].

Chuanxi and Ladas [7] and Kubiaczyk and Saker [19] considered the neutral

delay differential equation (1.1) when a(t) = 1 and f(x) = x and established

new sufficient conditions for oscillation of all solutions under less restrictive

hypotheses on P(t). All the above mentioned papers except [24] given the

oscillation conditions for equation (1.1) when P(t) ≤ 1 under the condition

?∞

In the case when (1.5) does not hold (in this case equation (1.1) is called has

integrally small coefficients) Yu, Wang and Chuanxi [28] considered equation

(1.1) when P(t) ≡ 1 and relaxed the condition (1.5) to the condition

?∞

In fact, Chen, Yu and Huang [6] observed that for the equation (1.1), it is

sufficient to have a point t∗≥ t0so that

P(t∗+ kτ) ≤ 1

without the assumptions (1.5) and (1.6) and proved a comparison theorem

for oscillation of equation (1.1) with the absence of positive solutions of

the corresponding delay differential inequality. They succeeded in getting

oscillation theorem which involve joint behavior of P and Q (for example

see [6, Theorems 3 and 5]) and using the condition

P(t − σ)Q(t) ≤ Q(t − τ)

to transfer the equation (1.1) to the inequality

d

dt[y(t) − y(t − τ)] + Q(t)y(t − σ) ≤ 0, t ≥ t0,

and using the results in Yu, Wang and Chuanxi [28] with the condition

(1.6) in the proofs of the main results, and obtained some finite sufficient

conditions for oscillation of all solutions. However, most of the results in

the literature involve conditions placed separately on Q mimicking the con-

ditions on Q sufficient for oscillation of (1.1) and placing conditions on P

which allow extension of arguments used in the case where P(t) ≡ 1 as

in Yu, Wang and Chuanxi [28]. Recently Li and Saker [22] considered the

equation (1.1) when a(t) = 1 and given some finite sufficient conditions for

oscillation of all solutions and applied these results to the logistic neutral

delay differential equations. For further oscillation results when (1.5) does

not hold we refer the reader to the articles by Yu [26, 27] and Yu et al. [29,

30, 31].

Our aim in this paper in Section 2 is to give some new integral sufficient

conditions for oscillation of all solutions of equation (1.1) and show that

t0

Q(s)ds = ∞.

(1.5)

t0

sQ(s)

?∞

s

Q(u)duds = ∞.

(1.6)

k = 0,1,2,...

(1.7)

(1.8)

(1.9)

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264S. H. SAKERand I. KUBIACZYK

the combined growth of P and Q without the condition (1.8) in the linear

case can give oscillation even when (1.5) and (1.6) fail. Our results extend

and improve some well known results for the oscillation of (1.1), and im-

prove some theorems about oscillation of the linear neutral delay differential

equations. In Section 3 some examples are considered to illustrate our main

results and in Section 4 we apply our results to the neutral logistic equa-

tion with variable coefficients to give some oscillation criteria for all positive

solutions about its positive steady state.

In the sequel, when we write a functional inequality we will assume that

it holds for all sufficient large values of t.

Before stating our main results we need the following lemma.

Lemma 1.1. Assume that (1.2), (1.3) hold, and there exist t∗≥ t0 such

that

P(t∗+ iτ)

a(t∗+ (i − 1)τ)≤ 1

Let x(t) be an eventually positive solution of equation (1.1), and set

y(t) = a(t)x(t) − P(t)x(t − τ).

Then we have eventually

y(t) > 0.

for i = 0,1,2,... .

(1.10)

(1.11)

(1.12)

Proof. Let t1≥ t0be such that x(t)>0, x(t − σ) > 0 and x(t − τ) > 0 for

t ≥ t1. Then by (1.1) and (1.11) we have

y?(t) = −Q(t)f(x(t − σ)) < 0

which implies that y(t) is nonincreasing on [t1,∞) and does not equal a

constant eventually. Hence if (1.12) does not hold, then eventually

y(t) < 0,

Therefore there exist t2≥ t1and M>0 such that

y(t) < −M,

Set z(t) = a(t)x(t)>0, then

P(t)

a(t − τ)z(t − τ),

Now we chose a positive integer i to be such that t∗+ iτ > t2. Then by

(1.10) and (1.13) we get

z(t∗+ iτ) ≤ −M(i + 1) + z(t∗) → −∞ as i → ∞

which contradicts the fact that z(t) is eventually positive. Then (1.12) holds.

The proof is complete.

for t ≥ t1

t ≥ t1.

t ≥ t2.

z(t) < −M +

t ≥ t2.

(1.13)

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OSCILLATION OF SOLUTIONS265

2. Main results

In this section we will establish some sufficient conditions for oscillation

of all solutions of equation (1.1).

Theorem 2.1. Assume that (1.2), (1.3) and (1.10) hold,

0 < d≤liminf

t→∞

?t+σ

t

Q1(s)

a(s − σ)ds

for t ≥ t0,

(2.1)

and

?∞

t0

Q1(t)

a(t − σ)exp

??t+σ

t

Q1(s)

a(s − σ)ds

?

dt = ∞

(2.2)

where Q1(t) = Q(t)/(β + ε) for some small positive constant ε. Then every

solution of equation (1.1) oscillates.

Proof. Assume, by the way of contradiction, that equation (1.1) has an

eventually nonoscillatory solution. Without loss of generality we assume

that equation (1.1) has an eventually positive solution x(t) (the case that

x(t) is negative is similar and will be omitted). Set x(t) > 0 and x(t−σ) > 0,

for t ≥ t0. From (1.3) since the limit exists, we can assume that there exists

Tεsufficiently large such that for t ≥ Tε, 0<x(t−σ) and x(t − σ)/(β + ε) ≤

f(x(t − σ)) ≤ x(t − σ)/(β − ε). Set y(t) as in (1.11), then from equation

(1.1) and Lemma 1.1, y(t) is a positive function and satisfies the inequalities:

y?(t) + Q(t)x(t − σ)

β + ε

≤ 0, y?(t) + Q(t)x(t − σ)

β − ε

≥ 0, t ≥ Tε.

(2.3)

Then from (1.11) and (2.3) we have

y?(t) ≤ −

Q1(t)

a(t − σ)y(t − σ) −Q1(t)P(t − σ)

Q1(t)

a(t − σ)y(t − σ) +

Hence y(t) is positive and satisfies the inequality

a(t − σ)

Q1(t)

Q1(t − τ)

x(t − τ − σ)

P(t − σ)

a(t − σ)y?(t − τ), t ≥ Tε.

≤ −

y?(t) −

Q1(t)

Q1(t − τ)

P(t − σ)

a(t − σ)y?(t − τ) +

Q1(t)

a(t − σ)y(t − σ) ≤ 0.

(2.4)

Set

λ(t) = −y?(t)

y(t).

(2.5)

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266 S. H. SAKERand I. KUBIACZYK

Then (2.4) reduces to

λ(t) ≥λ(t − τ)

Q1(t)

a(t − σ)

Q1(t)

a(t − σ)exp

P(t − σ)

Q1(t − τ)exp

??t

??t

t−τ

λ(s)ds

?

+

t−σ

λ(s)ds

?

.

(2.6)

It is obvious that λ(t) > 0 for t ≥ t0. From (2.6) it is clear that

λ(t) ≥ Q(t)exp

??t

t−σ

λ(s)ds

?

with Q(t) = Q1(t)/[a(t − σ)]. Let A(t) = exp

??t+σ

t

Q(s)ds

?

?

, then

λ(t) ≥ Q(t)exp

?

1

A(t)A(t)

?t

t−σ

λ(s)ds.

(2.7)

By using the inequality

ex/r≥ 1 +x

r2

for x ≥ 0, r ≥ 1(2.8)

we have from the inequality (2.7) that

A(t)λ(t) − Q(t)

?t

t−σ

λ(s)ds ≥ Q(t)A(t).

Then for N > T,

?N

By interchanging the order of integration, we find that

?N

Hence

?N

?N

From (2.9) and (2.10), it follows that

?N

On the other hand from the definition of λ(t), then from (2.6) y(t) is positive

function and satisfies the delay differential inequality

y?(t) + Q(t)y(t − σ) ≤ 0.

T

λ(t)A(t)dt −

?N

T

Q(t)

?t

t−σ

λ(s)dsdt

≥

?N

T

Q(t)A(t)dt.

(2.9)

T

Q(t)

??t

t−σ

λ(s)ds

?

dt ≥

?N−σ

T

λ(t)

??t+σ

t

Q(s)ds

?

dt

T

λ(t)A(t)dt −

?N−σ

?N

T

λ(t)

??t+σ

?t

t

Q(s)ds

?

dt

≥

T

λ(t)A(t)dt −

T

Q(t)

t−σ

λ(s)dsdt.

(2.10)

T

λ(t)A(t)dt +

?T

N−σ

λ(t)

??t+σ

t

Q(s)ds

?

dt ≥

?N

T

Q(t)A(t)dt.

(2.11)

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OSCILLATION OF SOLUTIONS267

Integrating the last inequality from t to t + σ we have

y(t + σ) − y(t) +

?t+σ

t

Q(s)y(s − σ)ds ≤ 0.

Then

y(t) >

?t+σ

t

Q(s)y(s − σ1)ds > y(t)

?t+σ

t

Q(s)ds

which implies that

d ≤

?t+σ

t

Q(s)ds < 1 and ed≤ A(t) < e.

(2.12)

Therefore

A(t) >

?t+σ

t

Q(s)ds.

Then

?N

Then from (2.12) and (2.13) we have

T

λ(t)A(t)dt +

?T

N−σ

λ(t)A(t)dt ≥

?N

T

Q(t)A(t)dt.

(2.13)

?N

N−σ

λ(t)dt ≥1

e

?N

?N

T

Q(t)A(t)dt

or

logy(N − σ)

y(N)

≥1

e

T

Q(t)A(t)dt.

(2.14)

In view of (2.2) we have

lim

t→∞

y(t − σ)

y(t)

= ∞.

(2.15)

Because of d≤liminft→∞

as k → ∞ and there exist ζk∈ (tk,tk+ σ) for every k such that

?ζk

Integrating both sides of the inequality (2.11) over the intervals [tk,ζk] and

[ζk,tk+ σ], we have

?ζk

and

?tk+σ

?t+σ

t

Q(s)ds there exists a sequence {tk}, tk→ ∞

tk

Q(s)ds ≥d

2

and

?tk+σ

ζk

Q(s)ds ≥d

2.

(2.16)

y(ζk) − y(tk) +

tk

Q(s)y(s − σ)ds ≤ 0(2.17)

y(tk+ σ) − y(ζk) +

ζk

Q(s)y(s − σ)ds ≤ 0.

(2.18)

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268S. H. SAKERandI. KUBIACZYK

From (2.16), (2.17) and (2.18), we have

−y(tk) +d

2y(ζk− σ) ≤ 0and

− y(ζk) +d

2y(tk) ≤ 0.

Then

y(ζk− σ)

y(ζk)

≤

?2

d

?2

which contradicts (2.15). Therefore, every solution of equation (1.1) oscil-

lates.

The following theorems are improved Theorem 2.1 which indicate that

the oscillation conditions of all solutions of equation (1.1) depend on P and

Q.

Theorem 2.2. Assume that (1.2), (1.3) and (1.10) hold,

0 < d≤liminf

t→∞

?t+σ

??t+σ

t

Q1(s)ds for t ≥ t0,

(2.19)

and

?∞

t0

Q1(t)exp

t

Q1(s)ds

?

dt = ∞.

(2.20)

Then every solution of equation (1.1) oscillates, where

Q1(t) =

Q1(t)P(t − σ)

a(t − σ)a(t − τ − σ).

Proof. Without loss of generality, we assume that equation (1.1) has an

eventually positive solution x(t). Then from Theorem 2.1 then y(t)>0 and

its generalized equation is given by (2.6). From (2.6) one can see that λ(t) ≥

Q1(t)/[a(t − σ)], then λ(t − τ) ≥ Q1(t − τ)/[a(t − τ − σ)], substituting in

(2.6) we have

Q1(t)P(t − σ)

a(t − σ)a(t − τ − σ)exp

+

a(t − σ)exp

It is obvious that λ(t) > 0 for t ≥ t0and then

λ(t) ≥

The remainder of the proof is similar to the proof of Theorem 2.1 and will

be omitted.

λ(t) ≥

??t

t−τ

?

λ(s)ds)

?

Q1(t)

??t

t−σ

λ(s)ds.

(2.21)

Q1(t)P(t − σ)

a(t − σ)a(t − τ − σ)exp

??t

t−τ

λ(s)ds

?

.

(2.22)

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OSCILLATION OF SOLUTIONS269

Theorem 2.3. Assume that (1.2), (1.3) and (1.10) hold, τ ≥ σ,

d≤liminf

t→∞

?t+σ

t

Q2(s)ds,

?∞

t0

Q2(t)exp

??t+σ

t

Q2(s)ds

?

dt = ∞,

(2.23)

then every solution of equation (1.1) oscillates, where

Q2(t) =

1

a(t − σ)

?Q1(t)P(t − σ)

a(t − τ − σ)

+ Q1(t)

?

.

Proof. The proof is similar to the proof of Theorem 2.1 from the inequality

(2.6) and will be omitted.

Theorem 2.4. Assume that (1.2), (1.3) and (1.10) hold and τ ≥ σ,

d≤liminf

t→∞

?t+σ

t

Q3(s)ds,

?∞

t0

Q3(t)exp

??t+σ

t

Q3(s)ds

?

dt = ∞.

(2.24)

then every solution of equation (1.1) oscillates, where

Q3(t) =Q1(t)P(t − σ)P(t − τ − σ)

a(t − τ − σ)a(t − 2τ − σ)

+

Q1(t)

a(t − σ).

Proof. Without loss of generality, we assume that equation (1.1) has an

eventually positive solution x(t). As in Theorem 2.1 from equation (2.6)

it is obvious that λ(t) > 0 for t ≥ t0, and λ(t) ≥ Q1(t)/[a(t − σ)]. Hence

λ(t − τ) ≥ Q1(t − τ)/[a(t − τ − σ)] and

Q1(t)P(t − σ)

a(t − σ)a(t − τ − σ)exp

+

a(t − σ)exp

which guarantees that

Q1(t)P(t − σ)

a(t − σ)a(t − τ − σ)

and then

λ(t − τ) ≥

λ(t) ≥

??t

t−τ

?

λ(s)ds

?

Q1(t)

??t

t−σ

λ(s)ds

(2.25)

λ(t) ≥

Q1(t − τ)P(t − τ − σ)

a(t − τ − σ)a(t − 2τ − σ).

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270S. H. SAKER andI. KUBIACZYK

From (2.6) we have

λ(t) ≥Q1(t)P(t − σ)P(t − τ − σ)

Q1(t)

a(t − σ)exp

as τ ≥ σ we obtain

λ(t) ≥

a(t − τ − σ)a(t − 2τ − σ)exp

+

??t

t−τ

λ(s)ds

?

??t

t−σ

λ(s)ds

?

(2.26)

?Q1(t)P(t − σ)P(t − τ − σ)

+

a(t − σ)

a(t − τ − σ)a(t − 2τ − σ)

Q1(t)

exp

???t

t−σ

λ(s)ds

?

.

(2.27)

The remainder of the proof is similar to the proof of Theorem 2.1 and will

be omitted.

Theorem 2.5. Assume that (1.2), (1.3) and (1.10) hold,

1

e<

?t+σ1

?t+σ1

t

Q(s)ds

and

?∞

t0

Q(t)log

?

e

t

Q(s)ds

?

dt = ∞.

Then every solution of equation (1.1) oscillates.

Proof. The proof is similar to the proof of Theorem 2.1 by choosing

?

and will be omitted.

A(t) = log

e

?t+σ1

t

Q(s)ds

?

,

Theorem 2.6. Assume that (1.2), (1.3) and (1.10) hold,

1

e≤

?t+σ1

t

Q(s)ds

and

?∞

t0

Q(t)

?

exp

??t+σ1

t

Q(s)ds −1

e

?

ds − 1

?

dt = ∞.

Then every solution of equation (1.1) oscillates.

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OSCILLATION OF SOLUTIONS 271

Proof. The proof is similar to the proof of Theorem 2.1 by choosing

?

and will be omitted.

A(t) =exp

??t+σ1

t

Q(s)ds −1

e

?

ds − 1

?

In fact, if we take

Q1(t) =

Q1(t)P(t − σ)

a(t − σ)a(t − τ − σ),

1

a(t − σ)[Q1(t)P(t − σ)

Q3(t) =Q1(t)P(t − σ)P(t − τ − σ)

a(t − τ − σ)a(t − 2τ − σ)

then respectively we have the following new sufficient conditions for oscilla-

tion of all solutions of equation (1.1).

Q2(t) =

a(t − τ − σ)

+ Q1(t)],

+

Q1(t)

a(t − σ),

Theorem 2.7. Assume that (1.2), (1.3) and (1.10) hold,

1

e<

and

?∞

Then every solution of equation (1.1) oscillates.

?t+τ

?t+τ

t

Q1(s)ds

t0

Q1(t)log

?

e

t

Q1(s)ds

?

dt = ∞.

Theorem 2.8. Assume that (1.2), (1.3) and (1.10) hold,

1

e≤

and

?∞

Then every solution of equation (1.1) oscillates.

?t+σ

t

Q1(s)ds

t0

Q1(t)

?

exp

??t+σ

t

Q1(s)ds −1

e

?

ds − 1

?

dt = ∞.

Theorem 2.9. Assume that (1.2), (1.3) and (1.10) hold,

1

e<

and

?∞

Then every solution of equation (1.1) oscillates.

?t+σ

?t+σ

t

Q2(s)ds

t0

Q2(t)log

?

e

t

Q2(s)ds

?

dt = ∞.

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272S. H. SAKER andI. KUBIACZYK

Theorem 2.10. Assume that (1.2), (1.3) and (1.10) hold,

1

e≤

?t+σ

t

Q2(s)ds

and

?∞

t0

Q2(t)

?

exp

??t+σ

t

Q2(s)ds −1

e

?

ds − 1

?

dt = ∞.

Then every solution of equation (1.1) oscillates.

Theorem 2.11. Assume that (1.2), (1.3) and (1.10) hold,

1

e<

?t+σ

t

Q3(s)ds

and

?∞

t0

Q3(t)log

?

e

?t+σ

t

Q3(s)ds

?

dt = ∞.

Then every solution of equation (1.1) oscillates.

Theorem 2.12. Assume that (1.2), (1.3) and (1.10) hold,

1

e≤

?t+σ

t

Q3(s)ds

and

?∞

t0

Q3(t)

?

exp

??t+σ

t

Q3(s)ds −1

e

?

ds − 1

?

dt = ∞.

Then every solution of equation (1.1) oscillates.

Remark 2.1. Our results can be extended to the more the general equation

j=1

dn

dtn

a(t)x(t) −

m

?

Pj(t)x(t − τj)

+

n

?

i=1

Qi(t)f(x(t − σi)) = 0, t ≥ t0.

Due to limited space, their statements are omitted here.

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OSCILLATION OF SOLUTIONS 273

3. Examples

In this section we introduce some examples to illustrate our results.

Example 3.1. Consider the neutral delay differential equation

?

x(t) − (3

2+ sint)x(t − π)

??

+3

2

?

(√2 +1

e)2

π+ cost

?

(ex(t−π/2)−1) = 0,

t ≥ 0(3.1)

Here σ = π/2 and

f(u) = eu− 1

with β = 1 and ε = 1/2, then

Q1(t) = (√2 +1

e)2

π+ cost > 0for t ≥ 0

and

?t

Hence

t−π/2

Q1(s)ds =

?t

t−π/2

?

(√2 +1

e)2

π+ coss

?

ds =

√2 +1

e+ sint + cost.

liminf

t→∞

?t

t−π/2

Q1(s)ds =1

e.

Then according to Theorem 3.3 in [22] equation (3.1) cannot has an oscil-

latory solution, but one can prove by Theorem 2.1 that every solution of

equation (3.1) oscillates.

Example 3.2. Consider the neutral delay differential equation

?

x(t) − (3

2+ sint)x(t − π)

??

+3

2

0.6

απ +√2(2α + cost)(ex(t−π/2)− 1) = 0,

t ≥ 0(3.2)

withσ = π/2,

f(u) = eu− 1

with β = 1, ε = 1/2, a(t) = 1, and α =√2(0.6e + 1)/[π(0.6e − 1)]

Q1(t) =

απ +√2(2α + cost) > 0

and

?t

Hence

liminf

t→∞

t−π/2

0.6

for t ≥ 0

t−π/2

Q1(s)ds =

?t

?t

t−π/2

0.6

απ +√2(2α + coss)ds.

Q1(s)ds =1

e

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274 S. H. SAKERandI. KUBIACZYK

and

limsup

t→∞

?t

t−π/2

Q1(s)ds = 0.6.

Then Theorem 3.3 in [22] is failed to apply on the equation (3.2), but one

can see by Theorem 2.1 that every solution of equation (3.2) oscillates.

Example 3.3. Consider the neutral delay differential equation

?

x(t) − (3

2+ sint)x(t − π)

??

+3

2

?1

e+

1

t + 1

?

(1 − e−x(t−1)) = 0,

t ≥ 0 (3.3)

with σ = 1,

f(u) = 1 − e−u

?1

with β = 1, ε = 1/2, and a(t) = 1,

Q1(t) =

e+

1

t + 1

?

for t ≥ 0 and

?t

t−1

Q1(s)ds =

?t

t−1

?1

?t

e+

1

s + 1

?

ds = logt + 1

t

+1

e.

Hence

liminf

t→∞

t−1

Q1(s)ds =1

e.

For T>1 we have

?T

?T

as T → ∞ where ex≥ ex for all real x. Then by Theorem 2.1 every solution

of equation (3.3) oscillates.

1

Q1(t)exp

?1

??t+1

e+

t + 1

t

Q1(s)ds

??

?

dt=

?T

?

1

?1

e+

1

t + 1

?

exp

?

logt + 2

t + 1+ 1

?

dt

≥ e

1

1

logt + 2

t + 1+ 1

dt → ∞

4. Oscillation in non-autonomous neutral delay logistic equation

The scalar autonomous ordinary differential equation

N?(t) = rN(t)

?

1 −N(t)

K

?

is known as the logistic equation in mathematical ecology and it is a proto-

type in the modelling in the dynamics of single-species population systems

whose biomass or density is denoted by a differentiable function N(t). The

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OSCILLATION OF SOLUTIONS 275

constant r, is called the growth rate and K, is called the carrying capacity

of the habitat. Hutchinson [17] suggested the following modification

?

Equation (4.1) is commonly known as the “delay equation” and has been

extensively investigated by numerous authors (see for example Wright [25],

Kakutani and Markus [18]). Gyori [14] considered the neutral delay logistic

equation with constant coefficients of the form,

?

and established oscillation criteria for all positive solutions.

The effects of varying environment are often important in dynamical na-

ture of populations, then we consider the non-autonomous neutral delay

equation

?

where

N?(t) = rN(t)1 −N(t − τ)

K

?

.

(4.1)

N?(t) = N(t)

r(1 −N(t − σ)

K

) + cN?(t − τ)

?

(4.2)

N?(t) = N(t)

r(t)(1 −N(t − σ)

K

) + cN?(t − τ)

?

(4.3)

r ∈ C[[t0,∞),?+],K,τ,σ,c ∈ (0,∞)(4.4)

and r(t) is the growth rate function, K is the carrying capacity of the

environment and c is the growth rate associated with the growth rate at

time t − τ.

With equation (4.3) one associate an initial condition of the form,

N(t) = φ(t)for − γ ≤ t ≤ 0, φ ∈ C[[−τ,0],?+] and φ(0) > 0

where γ = max{τ,σ}, then by the method of steps, the initial value problem

(4.3) and (4.5) has a unique solution N(t) which is valid for t ≥ 0. We

will only consider those solutions N(t) which are positive. Note that such

solution exist because if φ(0) > 0, then N(t)>0 for t ≥ 0.

and Saker presented some finite sufficient conditions for oscillation of all

positive solutions of equation (4.3) about K when 0<c(t)<1. In this section

we introduce some new infinite sufficient conditions for oscillation of all

positive solution of equation (4.3) when c(t) = c is a constant by applying

one of the above theorems.

(4.5)

In [22] Li

Theorem 4.1. Assume that (4.4) holds, 0 < c < 1,

0 < liminf

t→∞

?t+σ

t

r(s)ds