J. Appl. Math. & Computing Vol. 21(2006), No. 1 - 2, pp. 99 - 118

Website: http://jamc.net

OSCILLATION OF NONLINEAR NEUTRAL DELAYDIFFERENTIAL EQUATIONS

ELMETWALLY M. ELABBASY, TAHER S. HASSAN∗ AND SAMIR H. SAKER

Abstract. In this paper, we study the oscillatory behavior of first ordernonlinear neutral delay differential equations. Several new sufficient condi-tions which ensure that all solutions are oscillatory are given. The obtained

results extend and improve several known results in the literature. Someexamples are considered to illustrate the main results.

AMS Mathematics Subject Classification : 34K15, 34C10.Key words and phrases : Oscillation, nonoscillation, neutral delay of differ-ential equations.

1. Introduction

In this paper, we shall consider the first order nonlinear neutral delay differ-ential equation

(x(t) − q(t)x(t − σ))′ + f(t, x(τ(t))) = 0, t ≥ t0, (1)

where

q, τ ∈ C([t0,∞),R+), σ ∈ (0,∞) τ(t) < t, limt→∞

τ(t) = ∞, (2)

andn∑

i=1

i∏

j=1

1q(tj)

−→ ∞ as n −→ ∞, (3)

f ∈ C([t0,∞) ×R,R), uf(t,u) ≥ 0, q(t) 6= 1. (4)

In connection with the nonlinear function f(t, u) in (1) we suppose that thefollowing assumption (H) holds :

Received October 10, 2005. Revised January 16, 2006. ∗Corresponding author.

c© 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

99

100 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

(H) There are a piecewise continuous functions

p : [t0,∞) → R+ = [0,∞) , g ∈ C(R,R+

)

and a number ε0 > 0 such that(i) g is nondecreasing on R+

(ii) g (−u) = g (u) , limu→0

g (u) = 0,

(iii)

∞∫

0

g(e−u

)du < ∞,

(iv) 1|u| |f(t, u) − p (t) u| ≤ p (t) g (u) for t ≥ t0 and 0 < |u| < ε0,

(v) For each ϕ ∈ C ([t0,∞) ,R) with limt→∞

ϕ (t) > 0,

∞∫

t0

f (t, ϕ (τ (t))) dt = ∞,

∞∫

t0

f (t,−ϕ (τ (t))) dt = −∞.

A solution x (t) of equation (1) is said to be oscillatory if it has arbitrarily largezeros on [t0,∞) . Otherwise it is nonoscillatory and the equation (1) is calledoscillatory if every solution of this equation is oscillatory.

When q (t) = 0, Eq.(1) reduces to the equation

x′(t) + f (t, x (τ (t))) = 0,

which has been studied recently by Tang and Shen [32]. The oscillatory be-havior of various other neutral delay differential equations have been investi-gated by many authors. For contributions we refer the reader to the papers,[1 − 3, 6 − 12, 14− 27, 29− 36], and references cited therein.Also, in a recent papers Elabbasy and Saker [6], Kubiaczyk and Saker [17] andtheir references obtained an infinite integral conditions for oscillation of the linearneutral delay differential equation

(x (t) − q (t) x (t − σ))′+ p (t) x (t − τ) = 0,

Our aim in this paper is to extend the results in [7, 18] and provide some newfinite and infinite integral sufficient conditions for the oscillation of all solutionsof (1). Some examples that are illustrating our main results are given.

The following notations will be used throughout this paper,δ (t) = max {τ (t) : t0 ≤ s ≤ t} and δ−1 (t) = min {s ≥ t0 : δ (s) = t} .

Clearly, δ and δ−1 are nondecreasing and satisfy(A) δ (t) < t and δ−1 (t) > t,(B) δ

(δ−1 (t)

)= t and δ−1 (δ (t)) ≤ t.

Let δ−k (t) be defined on [t0,∞) by

δ−(k+1) (t) = δ−1(δ−k (t)

), k = 1, 2, ... (5)

Oscillation of solutions 101

Also, we use the sequence {pk} of functions defined as follows:

p1 (t) =

δ−1(t)∫

t

p (s) ds, t ≥ t0,

pk+1 (t) =

δ−1(t)∫

t

p (s) pk (s) ds, t ≥ t0, k = 1, 2, ...

In what follows, when we write a functional inequality we will mean that itholds for all sufficiently large values of t.

2. Main results

To prove our main results we shall need the following Lemmas.

Lemma 1. Assume that (2) , (3) and (4) hold, let x (t) be an eventually positivesolution of (1) and set

z (t) = x (t) − q (t) x (t − σ) . (6)

Then z (t) is eventually nonincreasing positive function.

Proof. From (1) , (4) , we have z′(t) = −f (t, x (τ (t))) ≤ 0 eventually. We prove

that z (t) is a positive function. If not, then there exist T ≥ t0 and α < 0 suchthat z (t) < α for t ≥ T. Then from (6) , we have

x (t) < α + q (t) x (t − σ) ,

which implies thatx (t + σ) < α + q (t + σ) x (t) .

Now we choose k such that tk = t∗+kσ > T. Then x (tk+1) < α+q (tk+1) x (tk) .Applying this inequality by induction it gives

x (tn) < α

1 +

n∑

i=k+2

n−i∏

j=0

q (tn−j)

+

n∏

i=k+1

q (ti) x (tk) .

Now define qn and dn by

qn = 1 +n∑

i=k+2

n−i∏

j=0

q (tn−j) , dn =n∏

i=k+1

q (ti) ,

and let

sn =n∑

i=1

i∏

j=1

1q (tj)

.

102 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

Then

s∗n =qn

dn=

sn −

k+1∑

i=1

i∏

j=1

1q (tj)

q (tk+1) ...q (t1) → ∞ as n → ∞,

by condition (3). Using the last inequality, one can see that

x (tn) <

[s∗n +

x (tk)α

]αdn → −∞ as n → ∞,

and this contradicts the assumption that x (t) > 0. Then z (t) must be positivefunction. The proof is complete. �

Note that the proof of Lemma 1 is similar to that of Lemma 1 in [4] and westate it here for the sake of completeness.

Lemma 2. Assume that (2) , (3), (4) and (H) hold, let x (t) be an eventuallypositive solution of 1. Then x (t) and z (t) are convergent to zero monotonicallyas t → ∞.

Proof. By Lemma 1 z (t) is a nonincreasing positive function and satisfy theequation

z′(t) = −f (t, x (τ (t))) .

Choose a t1 ≥ t0 such that x (t) > 0, z (t) > 0 for t ≥ t1.It follows from equations (2) , (4) and (H) that there exists t2 > t1 such that

τ (t) ≥ t1 and z′(t) ≤ 0 for t > t2.

Hence

limt→∞

x (t) ≥ limt→∞

z (t) = α ≥ 0 exists.

If α > 0, then from (1) , we have

z (t) − z (t0) = −t∫

t0

f (t, x(τ (s))) ds.

It follows from the assumption (H) (v) that limt→∞

z (t) = −∞, which contradicts

that z(t) being positive. Then limt→∞

z (t) = 0. Since q (t) 6= 1, we have also

limt→∞

x (t) = 0. Then the proof is complete. �

Oscillation of solutions 103

Lemma 3. Assume that (2) , (3), (4) and (H) hold. If x (t) is a nonoscillatorysolution of equation (1), there exist A > 0, ε > 0 and T ∈ (0,∞) such that fort ≥ T

|x (t)| ≤ A exp

−1

2

t∫

T

p (s) ds

+ ε. (7)

Proof. We shall assume x (t) to be eventually positive [if x(t) is eventuallynegative the proof is similar]. By Lemma 2, there exists t1 > 0 such that

0 < x

(τ (t)

)< ε0 for t ≥ t1.

From (H), we find that for t ≥ t1

f (t, x(τ (t))) ≥ p (t) [1 − g (x (τ (t)))]x (τ (t)) ,

and limt→∞

x (t) = 0. By assumption (H), there exists T > t1 such that for t ≥ T

f (t, x(τ (t))) ≥ 12p (t) x (τ (t)) ≥ 1

2p (t) x (t) ,

and it follows from (1) that for t ≥ T

(x (t) − q (t) x (t − σ))′+

12p (t) x (t) ≤ 0,

and

z′(t) +

12p (t) z (t) ≤ 0,

where z (t) = x (t) − q (t) x (t − σ) . This yields for t ≥ T

z (t) ≤ A exp

−1

2

t∫

T

p (s) ds

,

and

|x (t)| ≤ A exp

−1

2

t∫

T

p (s) ds

+ ε,

where A = x (T )− q (T )x (T − σ) . The proof is complete. �

Lemma 4. Assume that (2) , (3), (4) and (H) hold. If equation (1) has anonoscillatory solution, then eventually

t∫

τ(t)

p (s) ds ≤ 2 and pk (t) ≤ 2k, k = 1, 2, ... (8)

104 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

Proof. Let us suppose that x(t) is a nonoscillatory solution of equation (1)which we shall assume to be eventually positive [if x(t) is eventually negativethe proof is similar]. By Lemma 2, there exists T ≥ 0 such that

x (τ (t)) ≥ x (t) > 0 for t ≥ T,

(x (t) − q (t) x (t − r))′+

12p (t) x (τ (t)) ≤ 0,

and

z′(t) +

12p (t) z (τ (t)) ≤ 0 for t ≥ T. (9)

Integrating both sides from τ (t) to t yields that

z (t) − z (τ (t)) +12

t∫

τ(t)

p (s) z (τ (s)) ds ≤ 0 for t ≥ T.

By the decreasing nature of z (t) for large t and the increasing nature of τ (t) ,there exists T1 ≥ T such that

z (t) − z (τ (t)) +12z (τ (t))

t∫

τ(t)

p (s) ds ≤ 0 for t ≥ T1.

Then, we havet∫

τ(t)

p (s) ds ≤ 2.

Also, Integrating both sides of equation (9) from t to δ−1 (t) yields

z(δ−1 (t)

)− z (t) +

12

δ−1(t)∫

t

p (s) z (τ (s)) ds ≤ 0 for t ≥ T.

By the decreasing nature of z (t) for large t and the increasing nature of τ (t) ,there exists T1 ≥ T such that

z(δ−1 (t)

)− z (t) +

12

δ−1(t)∫

t

p (s) ds

z

(τ

(δ−1 (t)

))≤ 0 for t ≥ T1.

or

z(δ−1 (t)

)− z (t) +

12

δ−1(t)∫

t

p (s) ds

z (t) ≤ 0 for t ≥ T1.

Oscillation of solutions 105

Then, we have

p1 (t) =

δ−1(t)∫

t

p (s) ds ≤ 2.

By iteration we deduce from this that

pk (t) ≤ 2k,

which shows that (8) holds for t ≥ T1. The proof of Lemma 4 is complete. �

Lemma 5. Assume that (2) , (3), (4) and (H) hold, and that

limt→∞

inf

t∫

τ(t)

p(s)ds > 0. (10)

If x (t) is a nonoscillatory solution of equation (1) , thenz (τ (t))

z (t), which is well

defined for large t, is bounded.

Proof. Let us suppose that x(t) is a nonoscillatory solution of equation (1) whichwe shall assume to be eventually positive [if x(t) is eventually negative the proofis similar]. By the same argument as in the proof of Lemma 3, there existsT > 0, such that

x (τ (t)) ≥ x (t) > 0 for t ≥ T,

(x (t) − q (t) x (t − σ))′+

12p (t) x (τ (t)) ≤ 0,

and

z′(t) +

12p (t) z (τ (t)) ≤ 0 for t ≥ T.

The rest of the proof is similar to that of Lemma 5 in [24] respectively, and henceis omitted. �

Theorem 1. In addition to the assumptions (2) , (3), (4) and (H)assume that

limt→∞

inf

t∫

τ(t)

P (s) ds >1e, (11)

or

limt→∞

sup

t∫

τ(t)

P (s) ds > 1, (12)

106 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

where P (t) = (1 − ε) p (t) . Then every solution of Eq. (1) oscillates.

Proof. Let us suppose that x(t) is a nonoscillatory solution of equation (1)which we shall assume to be eventually positive [if x(t) is eventually negativethe proof is similar]. By Lemma 2, there exists t1 > 0 such that

0 < x (τ (t)) < ε0 for t ≥ t1.

From (H), we find that for t ≥ t1

f (t, x(τ (t))) ≥ p (t) [1 − g (x (τ (t)))]x (τ (t)) ,

and limt→∞

x (t) = 0. By assumption (H), there exists T > t1 such that for t ≥ T

f (t, x(τ (t))) ≥ (1 − ε) p (t) x (τ (t)) ,

and it follows from (1) that for t ≥ T

(x (t) − q (t) x (t − σ))′+ (1 − ε) p (t) x (τ (t)) ≤ 0,

and

z′(t) + P (t) z (τ (t)) ≤ 0. (13)

But, then by Corollary 3.2.2 [11] the delay differential equation

z′(t) + P (t) z (τ (t)) = 0, (14)

has an eventually positive solution as well. It is also well known that (11) or(12) implies (14) has no eventually positive solution (see, [11] Theorem 3.4.3).This contradiction completes the proof. �

Remark 1. It is clear that every solution of (1) oscillates if (14) has no even-tually positive solutions.

It is clear that there is a gap between (11) and (12) for the oscillation of allsolutions of (1) . The problem how to fill this gap for the equation (1) when thelimit

limt→∞

t∫

τ(t)

P (s) ds,

does not exist needs to be considered. This problem has been cleared for thelinear Eq. (14) . Let the numbers k and l be defined by

k = limt→∞

inf

t∫

τ(t)

P (s)ds, l = limt→∞

sup

t∫

τ(t)

P (s)ds, .

0 < k ≤ 1e, l < 1,

Oscillation of solutions 107

and λ is the smallest root of the equation λ = ekλ. Then Eq. (14) will beoscillatory if either of the following conditions is satisfied:

(C1) l >ln λ + 1

λ, [18]

(C2) l > 1 − 1 − k −√

1 − 2k − k2

2, [34]

(C3) l >1 + ln λ

λ−

1 − k −√

1− 2k − k2

2, [14]

(C4) l > 2k +2λ− 1, [19]

(C5) l >ln λ − 1 +

√5 − 2λ + 2kλ

λ, [30]

(C6) l >e − 1e − 2

(k +

1λ1

)− 1

e − 2, [7]

Remark 2. Theorem 1 implies that Eq. (1) will also be oscillatory if either ofthe conditions (C1) − (C6) is satisfied.

In the following theorems we present new infinite integral sufficient conditionsfor the oscillation of all solutions of (1)

Theorem 2. Assume that (2) , (3), (4) , (10) and (H) hold, and suppose thatthere exists a positive integer n such that

∞∫

t0

p (t) ln (pn (t) + 1) dt = ∞. (15)

Then every solution of (1) oscillates.

Proof. Assume that (1) has a nonoscillatory solution x (t) which will be assumedto be eventually positive (if x (t) is eventually negative the proof is similar). ByLemma 2 and assumption (H), there exists t∗0 ≥ t0 such that

0 < x (t) ≤ x (δ (t)) ≤ x (τ (t)) < ε0, g (x (τ (t))) < 1, t ≥ t∗0, (16)

where ε0 is given by assumption (H). (16) and (H) yield that for t ≥ t∗0,

f (t, x (τ (t))) ≥ p (t) [1 − g (x (τ (t)))] x (τ (t)) ≥ p (t) [1 − g (x (τ (t)))] z (δ (t)) ,(17)

and it follows from (1) that

z′ (t)z (t)

+ p (t)z (δ (t))

z (t)[1 − g (x (τ (t)))] ≤ 0, t ≥ t∗0. (18)

108 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

By Lemmas 1 − 5, there exist T > t2, A > 0, ε > 0 and M > 0 such that fort ≥ T,

x (τ (t)) ≤ A exp

−1

2

τ(t)∫

T

p (s) ds

+ ε, (19)

t∫

δ(t)

p (s) ds ≤t∫

τ(t)

p (s) ds ≤ 2, pk (t) ≤ 2k, k = 1, 2, ..., (20)

z (δ (t))z (t)

≤ z (τ (t))z (t)

≤ M. (21)

Let tk = δ−k (T ) , k = 1, 2, ...Clearly tk → ∞ as k → ∞. Set

λ (t) =−z

′(t)

z (t), t ≥ T.

Then

z (δ (t))z (t)

= exp

t∫

δ(t)

λ (s) ds, t ≥ t1,

and from (18) we have for t ≥ t1

λ (t) ≥ p (t) exp

t∫

δ(t)

λ (s) ds − p (t) g (x (τ (t)))z (δ (t))

z (t). (22)

It follows from (19) − (22) that for t ≥ t1,

λ (t) ≥ p (t) exp

t∫

δ(t)

λ (s) ds − Mp (t) g

A exp

−1

2

τ(t)∫

T

p (s) ds

+ ε

≥ p (t) exp

t∫

δ(t)

λ (s) ds − Mp (t) g

A1 exp

−1

2

t∫

T

p (s) ds

+ ε

,

(23)

where A1 = eA. By the inequality ec ≥ c for c ≥ 0, we have for t ≥ t1

λ (t) ≥ p (t)

t∫

δ(t)

λ (s) ds − Mp (t) g

A1 exp

−1

2

t∫

T

p (s) ds

+ ε

.

(24)

Oscillation of solutions 109

Set

α (t) =12

t∫

T

p (s) ds, t ≥ T, (25)

and

λ0 (t) = λ (t) , t ≥ T,

λk (t) = p (t)

t∫

δ(t)

λk−1 (s) ds, t ≥ tk, k = 1, 2, ..., n, , (26)

and

G0 (t) = 0, t ≥ T,

Gk (t) = p (t)

t∫

δ(t)

Gk−1 (s) ds

+ Mp (t) g (A1 exp (−α (t)) + ε) , t ≥ tk, k = 1, 2, ..., n,

.(27)

Clearly (10) implies that α (t) is nondecreasing on [T,∞) and α (t) → ∞ ast → ∞. By iteration we deduce from (24) that

λ (t) ≥ λk (t) − Gk (t) , t ≥ tk, k = 1, 2, ...n− 1, (28)

and so by (23) ,

λ (t) ≥ p (t) exp

t∫

δ(t)

λn−1 (s) ds

exp

−

t∫

δ(t)

Gn−1 (s) ds

− G1 (t) , t ≥ tn.

(29)

From (27) , one can easily obtain

Gk+1 (t) − Gk (t)

= p (t)

t∫

δ(t)

[Gk (s) − Gk−1 (s)] ds, t ≥ tk+1, k = 1, 2, ..., n− 1. (30)

By (20) , (25) and (27) , for t ≥ t2 we have

t∫

δ(t)

G1 (s) ds = M

t∫

δ(t)

p (s) g (A1 exp (−α (s)) + ε) ds

= 2M

α(t)∫

α(δ(t))

g(A1e

−u + ε)du (31)

110 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

≤ 2M

α(t)∫

α(t)−1

g(A1e

−u + ε)du.

Thus, from (30) , we get

G2 (t) − G1 (t) = p (t)

t∫

δ(t)

G1 (s) ds

≤ 2Mp (t)

α(t)∫

α(t)−1

g(A1e

−u + ε)du, t ≥ t2,

G3 (t) − G2 (t) = p (t)

t∫

δ(t)

[G2 (s) − G1 (s)] ds

≤ 2Mp (t)

t∫

δ(t)

p (s)

α(s)∫

α(s)−1

g(A1e

−u + ε)duds

= 4Mp (t)

α(t)∫

α(δ(t))

v∫

v−1

g(A1e

−u + ε)dudv

≤ 4Mp (t)

α(t)∫

α(t)−1

v∫

v−1

g(A1e

−u + ε)dudv

≤ 4Mp (t)

α(t)∫

α(t)−2

g(A1e

−u + ε)du, t ≥ t3.

By induction, one can prove in general that for k = 2, 3, ..., n− 1,

Gk (t) − Gk−1 (t) ≤ (2)k−1 (k − 2)!Mp (t)

α(t)∫

α(t)−(k−1)

g(A1e

−u + ε)du, t ≥ tk,

and so

Gn−1 (t) =n−1∑

k=1

[Gk (t) − Gk−1 (t)]

≤ G1 (t) + Mp (t)n−1∑

k=2

(2e)k−1 (k − 2)! (32)

Oscillation of solutions 111

×α(t)∫

α(t)−(k−1)

g(A1e

−u + ε)du, t ≥ tn−1.

By (20) , (21) and (26) , we obtain

λ1 (t) = p (t)

t∫

δ(t)

λ (s) ds = p (t) ln[z (δ (t))

z (t)

]

≤ p (t) ln M, t ≥ t1,

λ2 (t) = p (t)

t∫

δ(t)

λ1 (s) ds ≤ p (t) ln M

t∫

δ(t)

p (s) ds

≤ 2p (t) ln M, t ≥ t2,........λn−1 (t) ≤ 2n−2p (t) ln M, t ≥ tn−1.

. (33)

Set

D (t) = p (t) exp

t∫

δ(t)

λn−1 (s) ds

1 − exp

−

t∫

δ(t)

Gn−1 (s) ds

+G1 (t) , t ≥ tn.

One can easily see that

0 ≤ 1 − e−c ≤ c, c ≥ 0. (34)

From, (20) , (32) , (33) , (33) and (34) we have

D (t) ≤ p (t) exp

t∫

δ(t)

λn−1 (s) ds

t∫

δ(t)

Gn−1 (s) ds + G1 (t)

≤ G1 (t) + p (t) exp

2n−2 ln M

t∫

δ(t)

p (s) ds

×t∫

δ(t)

[G1 (s) + Mp (s)

n−1∑

k=2

(2)k−1 (k − 2)!

α(s)∫

α(s)−(k−1)

g(A1e

−u + ε)du

]ds

≤ G1 (t) + 2Mp (t) exp((2)n−1 ln M

) α(t)∫

α(t)−1

g(A1e

−u + ε)du

112 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

+Mp (t) exp((2)n−1 ln M

)

×n−1∑

k=2

(2)k−1 (k − 2)!

t∫

δ(t)

p (s)

α(s)∫

α(s)−(k−1)

g(A1e

−u + ε)duds (35)

≤ G1 (t) + M1p (t)n−1∑

k=1

(2)k−1 (k − 1)!

α(t)∫

α(t)−k

g(A1e

−u + ε)du, t ≥ tn,

where M1 = 2M exp((2)n−1 ln M

). Let T ∗ > tn be such that α (T ∗) > n +

ln A1. It follows from (36) and (H) that

∞∫

T∗

D (t) dt ≤∞∫

T∗

G1 (t) dt + M1

n−1∑

k=1

(2)k−1 (k − 1)!

∞∫

T∗

p (t)

α(t)∫

α(t)−k

g(A1e

−u)dudt

≤ 2M

∞∫

α(T∗)

g(A1e

−u)du

+2M1

n−1∑

k=1

(2)k−1 (k − 1)!

∞∫

α(T∗)

v∫

v−k

g(A1e

−u)dudv

≤ 2M

∞∫

α(T∗)

g(A1e

−u)du + 2M1

n−1∑

k=1

(2)k−1k!

∞∫

α(T∗)−(k+1)

g(A1e

−u)du

∞∫

T∗

D (t) dt ≤ 2M

∞∫

0

g(e−u

)du + 2M1

n−1∑

k=1

(2)k−1k!

∞∫

0

g(e−u

)du < ∞.

(36)

Since

p (t) exp

t∫

δ(t)

λn−1 (s) ds

exp

−

t∫

δ(t)

Gn−1 (s) ds

− G1 (t)

= p (t) exp

t∫

δ(t)

λn−1 (s) ds

− D (t) , t ≥ tn,

Oscillation of solutions 113

it follows from (29) that

λ (t) ≥ p (t) exp

t∫

δ(t)

λn−1 (s) ds

− D (t) , t ≥ tn, (37)

or

λ (t) ≥ p (t) exp

1

pn (t)pn (t)

t∫

δ(t)

λn−1 (s) ds

− D (t) , t ≥ tn.

One can easily show that eγx ≥ x +ln (γ + 1)

γfor all x ≥ 0 and γ > 0, and so

for t ≥ tn,

pn (t) λ (t) − p (t)

t∫

δ(t)

λn−1 (s) ds ≥ p (t) ln (pn (t) + 1) − pn (t) D (t) .(38)

For N > δ−n (T ∗) , we have

N∫

T∗

pn (t) λ (t) dt −N∫

T∗

p (t)

t∫

δ(t)

λn−1 (s) dsdt

≥N∫

T∗

p (t) ln (pn (t) + 1) dt −N∫

T∗

pn (t) D (t) dt. (39)

Let

δ1 (t) = δ (t) , δk+1 (t) = δ(δk (t)

), k = 1, 2, , ..., n.

Then by interchanging the order of integration, we have

N∫

T∗

p (t)

t∫

δ(t)

λn−1 (s) dsdt ≥δ(N)∫

T∗

λn−1 (t)

δ−1(t)∫

t

p (s) dsdt

=

δ(N)∫

T∗

p (t) p1 (t)

t∫

δ(t)

λn−2 (s) dsdt

≥δ2(N)∫

T∗

λn−2 (t)

δ−1(t)∫

t

p (s) p1 (s) dsdt

114 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

=

δ2(N)∫

T∗

p (t) p2 (t)

t∫

δ(t)

λn−3 (s) dsdt

..........

≥δn(N)∫

T∗

λ (t) pn (t) dt.

From this and (39) we have

N∫

δn(N)

pn (t) λ (t) dt ≥N∫

T∗

p (t) ln (pn (t) + 1) dt −N∫

T∗

pn (t) D (t) dt, (40)

which together with (20) yields

2n

N∫

δn(N)

λ (t) dt ≥N∫

T∗

p (t) ln (pn (t) + 1) dt − 2n

N∫

T∗

D (t) dt,

or

lnx (δn (N))

x (N)≥ 2−n

N∫

T∗

p (t) ln (pn (t) + 1) dt −N∫

T∗

D (t) dt. (41)

In view of (15) and (36) , we have

limN→∞

x (δn (N))x (N)

= ∞. (42)

On the other hand, (21) implies that

x (δn (N))x (N)

=x

(δ1 (N)

)

x (N).x

(δ2 (N)

)

x (δ1 (N))...

x (δn (N))x (δn−1 (N))

≤ Mn.

This contradicts (42) and completes the proof. �

Remark 3. From Lemma 3 we have

limt→∞

inf pk (t) ≤ (2)k−1 limt→∞

inf

δ−1(t)∫

t

p (s) ds ≤ (2)k−1 limt→∞

inf

t∫

τ(t)

p (s) ds.

As a result, by Theorem 2 we have

Oscillation of solutions 115

Corollary 1. Assume that (2) , (3), (4) , and (H) hold, and that there exists apositive integer n such that

limt→∞

inf pn (t) > 0.

Then every solution of (1) oscillates.

Example 1. Consider the neutral delay differential equation(

x (t) −(

52

+ sin t

)x (t − π)

)′

+ f (t, x (τ (t))) = 0, t ≥ 3, (43)

where

τ (t) = t − 1 and f (t, u) = [exp 3 (sin t − 1) + |u|]13 u.

Let

p (t) = exp (sin t) − 0.1 and g (u) = e2 |u|13 .

It is easy to see that assumption (H) holds. Clearly

limt→∞

inf

t∫

t−1

p (s) ds <1e,

∞∫

0

p (t) ln (p1 (t) + 1) dt

≥∞∫

0

exp (sin t − 1) ln

t+1∫

t

exp (sin s) ds

dt

By Jensen’s inequality,

∞∫

0

p (t) ln

t+1∫

t

p (s) ds + 1

dt ≥

∞∫

0

exp (sin t − 1)

t+1∫

t

sin sdsdt

=2 sin 2−1

e

∞∫

0

exp (sin t) sin(

t +12

)dt.

On the other hand, it is easy to see thatt∫

0

exp (sin s) cos sds

116 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

is bounded and2π∫

0

exp (sin t) sin tdt > 0.

Thus∞∫

0

p (t) ln (p1 (t) + 1) dt = ∞.

By Theorem 2 every solution of (43) oscillates.

Note that most of the results that has been given in [3, 11] required thecondition that 0 ≤ q (t) ≤ 1, which cannot be applied on (43) since q (t) =(

52 + sin t

)> 1.

References

1. R. P. Agarwal, S. R. Grace and D. O,Regan, Oscillation theory for difference and Func-tional differential equations, Kluwer, Dordrecht (2000) .

2. R. P. Agarwal, X. H. Tang and Z. C. Wang, The existence of positive solutions to neutraldifferential equations, J. Math. Anal. Appl. 240 (1999), 446-467.

3. D. D. Bainov and D. P. Mishev, Oscillation Theory for Neutral Differential with Delay,Adam Hilger, NewYork (1991) .

4. K. Dib and R. M. Mathsen, Oscillation of solutions of neutral delay differential equations,Math. Comp. Model. 32 (2000), 609-619.

5. R. D. Driver, A mixed neutral system, Nonlinear Analysis 8 (1976), 182-189.6. E. M. Elabbasy and S. H. Saker, Oscillation of first order neutral delay differential equa-

tions, Kyungpook. Math. J. 41 (2001), 311-321.7. E. M. Elabbasy and T. S. Hassan, Oscillation criteria for first order delay differential

equations, Serdica Math. J. 30 (2004), 71-86.8. A. Elbert and I. P. Stavroulakis, Oscillation and non-oscillation criteria for delay differ-

ential equations, Proc. of Amer. Math. Soc. 124 (1995), 1503-1511.9. M. K. Grammatikopoulos, E. A. Grove and G. Ladas, Oscillation of first order neutral

delay differential equations, J. Math. Anal. Appl. 120 (1986), 510-520.10. M. K. Grammatikopoulos, Y. G. Sficas and G. Ladas, Oscillation and asymptotic behavior

of neutral equations with variable coefficients, Radovi Mathematicki 2 (1986), 279-303.

11. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applica-tions, Oxford Mathematical Monographs (1991).

12. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York (1977).13. J. Jaros and I. P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J.

Math. 28 (1999), 197-207.14. M. Kon, Y. G. Sficas and I. P. Stavroulakis, Oscillation criteria for delay equations, Proc.

Amer. Math. Soc. 128 (2000), 2989-2997.15. M. R. S. Kulenovic, G. Ladas and A. Meimaridou, Necessary and sufficient condition for

oscillations of neutral differential equations, J. Austral. Math. Soc. Ser. B. 28 (1987),362-375.

Oscillation of solutions 117

16. I. Kubiaczyk, S. H. Saker and J. Morchalo, New oscillation criteria for first order nonlinearneutral deelay differential equations, Applied Mathematics and Computation 142 (2003),225-242.

17. I. Kubiaczyk and S. H. Saker, Oscillation of solutions of neutral delay differential equa-tions, Math. Slovaca, to appear.

18. M. K. Kwong, Oscillation of first order delay equations, J. Math. Anal. Appl. 159 (1991),469-484.

19. M. Kon, Y. G. Sficas and I. P. Stavroulakis, Oscillation criteria for delay equations, Proc.Amer. Math. Soc. 128 (2000), 2989-2997.

20. G. Ladas and Y. G. Sficas, Oscillation of neutral delay differential equations, Canad.Math. Bull. 29 (1986), 438-445.

21. B. S. Lalli and B. G. Zhang, Oscillation of first order neutral differential equations, Ap-plicable Analysis 39 (1990), 265-274.

22. B. Li, Oscillations of delay differential equations with variable coefficients, J. Math. Anal.Appl. 192 (1995), 312-321.

23. B. Li, Oscillations of first order delay differential equations, Proc. Amer. Math. Soc. 124(1996), 3729-3737.

24. B. Li and Y. Kuang, Sharp conditions for oscillations in some nonlinear nonautonomousdelay differential equations, Nonlinear Anal. Appl. 29 (1997), 1265-1276.

25. W. T. Li, H. Quan and J. Wu, Oscillation of first order neutral differential equations withvariable coefficients, Commun. Appl. Anal. 3 (1999), 1-13.

26. W. T. Li and and J. Yan, Oscillation of first order neutral differential equations withpositive and negative coefficients, Collect. Math. 50 (1999), 199-209.

27. Z. Luo and J. Shen, Oscillation and nonoscillation of neutral differential equations withpositive and negative coefficients, Czechoslovak Math. J. 129 (2004), 79-93.

28. E. P. Popove, Automatic Regulation and Control, Nauka, Moscow (1966) , In Russian.29. C. Qian, M. R. S. Kulenovic and G. Ladas, Oscillation of neutral equations with variable

coefficients, Radovi Mathematicki, 5 (1989), 321-331.30. Y. G. Sficas and I. P. Stavroulakis, Oscillation criteria for first order delay equations,

Bull. London Math. Soc. 35 (2003), 239-246.31. X. Tang and J. Shen, Oscillations of delay differential equations with variable coefficients,

J. Math. Anal. Appl. 217 (1998), 32-42.32. X. Tang and J. Shen, New oscillation criteria for first order nonlinear delay differential

equations, Colloquim Mathematicum 83 (2000), 21-41.33. J. S. Yu, Neutral differential equations with positive and negative coefficients, Acta Math.

Sinica 34 (1992), 517-523.34. J. S. Yu and Z. C. Wang, Some futher results on oscillation of neutral equations, Bull.

Austral. Math. Soc. 46 (1992), 149-157.35. B. G. Zhang, Oscillation of first order neutral functional differential equations, J. Math.

Anal. Appl. 139 (1989), 311-318.36. B. G. Zhang, On the positive solution of kind of neutral equations, Acta Math. Appl.

Sinica 19 (1996), 222-230.

Elmetwally M. Elabbasy received his BS (1973) from Mansoura University and MSC(1977) from Ain Shams University Egypt. He got his Ph. D. (1980) from Wales, UK.His subject interest is oscillation theory of differential and difference equations and theirapplications and also the nonlinear dynamical systems. Now he is teaching in Departmentof Mathematics, Mansoura University.

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516,Egypte-mail: emelabbasy@mans.edu.eg

118 Elmetwally M. Elabbasy, Taher S. Hassan and Samir H. Saker

Taher S. Hassan received his BS (1997) and MSC (2003) from Mansoura University,Egypt. His subject interest is oscillation theory of functional differential equations. Nowhe is teaching in Department of Mathematics, Mansoura University.

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516,Egypte-mail: tshassan@mans.edu.eg

Samir H. Saker received his BS (1993) and MSC (1997) from Mansoura University, Egyptand he got his Ph. D. from Adam Mickiewiz University, Poland (2002). His research interestfocus on the oscillation theory of differential, difference equations and their applicationsand oscillation of dynamic equations on time scales which unify the oscillation of differentialand difference equations. Now he is teaching in Department of Department of Mathematics,Mansoura University.

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516,Egypte-mail: shsaker@mans.edu.eg