International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 2 (2017), 147-154
http://www.etamaths.com

OSCILLATION OF NONLINEAR DELAY DIFFERENTIAL EQUATION WITH

NON-MONOTONE ARGUMENTS

ÖZKAN ÖCALAN1,∗, NURTEN KILIÇ2, SERMIN ŞAHİN3 AND UMUT MUTLU ÖZKAN3

Abstract. Consider the first-order nonlinear retarded differential equation

x′(t) + p(t)f (x (τ(t))) = 0, t ≥ t0
where p(t) and τ(t) are function of positive real numbers such that τ(t) ≤ t for t ≥ t0, and limt→∞ τ(t) =
∞. Under the assumption that the retarded argument is non-monotone, new oscillation results are
given. An example illustrating the result is also given.

Keywords: delay differential equation; non-monotone argument; oscillatory solutions; nonoscil-
latory solutions.

2010 Mathematics Subject Classification: 34K11, 34K06.

1. Introduction

Consider the nonlinear retarded differential equation

x′(t) + p(t)f (x (τ(t))) = 0, t ≥ t0 (1.1)
where p(t) and τ(t) are functions of nonnegative real numbers, and τ(t) is non-monotone or nonde-
creasing such that

τ(t) ≤ t for t ≥ t0, and lim
t→∞

τ(t) = ∞, (1.2)

and
f ∈ C(R,R) and xf(x) > 0 for x 6= 0. (1.3)

By a solution of (1.1) we mean a continuously differentiable function defined on [τ(T0),∞] for some
T0 ≥ t0 and such that (1.1) is satisfied for t ≥ T0. Such a solution is called oscillatory if it has
arbitrarily large zeros. Otherwise, it is called nonoscillatory.

Recently there has been an increasing interest in the study of the oscillatory behavior of the following
special form of (1.1)

x′(t) + p(t)x (τ(t)) = 0, t ≥ t0. (1.4)
See, for example, [1−19] and the references cited therein. The first systematic study for the oscilla-

tion of all solutions of equation (1.4) was made by Myshkis. In 1950 [17] he proved that every solution
of (1.4) oscillates if

lim sup
t→∞

[t− τ(t)] < ∞ and lim inf
t→∞

[t− τ(t)] lim inf
t→∞

p(t) >
1

e
.

In 1972, Ladas, Lakshmikantham and Papadakis [16] proved that the same conclusion holds if, in
addition, τ is a non-decreasing function and

lim sup
t→∞

t∫
τ(t)

p(s)ds > 1. (1.5)

In 1982, Koplatadze and Canturija [14] established the following result.

Received 12th April, 2017; accepted 5th June, 2017; published 3rd July, 2017.
2010 Mathematics Subject Classification. 34K11, 34K06.
Key words and phrases. delay differential equation; non-monotone argument; oscillatory solutions; nonoscillatory

solutions.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

147



148 ÖCALAN, KILIÇ, ŞAHİN AND ÖZKAN

If τ(t) is a non-monotone or nondecreasing and

lim inf
t→∞

t∫
τ(t)

p(s)ds >
1

e
, (1.6)

then all solutions of Eq.(1.4) oscillate, while if

lim sup
t→∞

t∫
τ(t)

p(s)ds <
1

e
, (1.7)

then the equation (1.4) has a nonoscillatory solution.
To the best of our knowledge, there are few papers dealing with the oscillatory behavior of solutions

of (1.1), see, for example, [9, 17]. The following theorem was given by Ladde et al. in [17].
THEOREM A. Assume that the f, p and τ in Eq.(1.1) satisfy the following conditions:
i) The condition (1.2) holds and let τ(t) be strictly increasing on R+,
ii) p(t) is locally integrable and p(t) ≥ 0, a.e.;
iii) The condition (1.3) holds and let f be nondecreasing, and

lim
x→0

x

f(x)
= N < +∞.

Assume further that

lim sup
t→∞

t∫
τ(t)

p(s)ds > N,

or

lim inf
t→∞

t∫
τ(t)

p(s)ds >
N

e
.

Then every solution of Eq.(1.1) is oscillatory.
The following theorem was given by Fukagai and Kusano in [9].
THEOREM B. Suppose that the conditions (1.2) and (1.3) hold. Suppose moreover that

lim sup
x→0

|x|
|f(x)|

= λ < ∞.

If

lim inf
t→∞

t∫
τ(t)

p(s)ds >
λ

e
,

then every solution of Eq.(1.1) is oscillatory.
Thus, in this paper, our aim is to obtain some oscillation criteria for all solutions of Eq.(1.1) under

the assumption that τ(t) is non-monotone.

2. Main Results

In this section, we present a new sufficient conditions for the oscillation of all solutions of Eq.(1.1),
under the assumption that the argument τ(t) is non-monotone or nondecreasing. Set

h(t) := sup
s≤t

τ(s), t ≥ 0. (2.1)

Clearly, h(t) is nondecreasing, and τ(t) ≤ h(t) for all t ≥ 0.
Assume that the f in Eq.(1.1) satisfy the following condition:

lim sup
x→0

x

f(x)
= M, 0 ≤ M < ∞. (2.2)



OSCILLATION OF NONLINEAR DELAY DIFFERENTIAL 149

Theorem 2.1. Assume that (1.2), (1.3) and (2.2) holds. If τ(t) is non-monotone or nondecreasing,
and

lim inf
t→∞

t∫
τ(t)

p(s)ds >
M

e
, (2.3)

then all solutions of Eq.(1.1) oscillate.

Proof. Assume, for the sake of contradiction, that there exists a nonoscillatory solution x(t) of (1.1).
Since −x(t) is also a solution of (1.1), we can confine our discussion only to the case where the solution
x(t) is eventually positive. Then there exists t1 > t0 such that x(t), x (τ(t)) > 0, for all t ≥ t1. Thus,
from (1.1) we have

x′(t) = −p(t)f (x (τ(t))) ≤ 0, for all t ≥ t1.
Thus x(t) is nonincreasing and has a limit l ≥ 0 as t →∞.

Now, we claim that l = 0. Condition (2.3) implies that

∞∫
a

p(t)dt = ∞. (2.4)

In view of (2.4) and by the Theorem 3.1.5 in [17] that limt→∞x (t) = 0. Suppose M > 0. Then, in
view of (2.2) we can choose t2 > t1 so large that

f(x (t)) ≥
1

2M
x(t) for t ≥ t2. (2.5)

On the other hand, we know from Lemma 2.1.1 [7] that

lim inf
t→∞

t∫
h(t)

p(s)ds = lim inf
t→∞

t∫
τ(t)

p(s)ds. (2.6)

Since h(t) ≥ τ(t) and x(t) is nonincreasing , by (1.1) and (2.5) we have

x′(t) +
1

2M
p(t)x(h(t)) ≤ 0, t ≥ t3. (2.7)

Also, from (2.3) and (2.6) it follows that there exists a constant c > 0 such that

t∫
h(t)

p(s)ds ≥ c >
M

e
, t ≥ t3 ≥ t2. (2.8)

So, from (2.8), there exists a real number t∗ ∈ (h(t), t), for all t ≥ t3 such that
t∗∫

h(t)

p(s)ds >
M

2e
and

t∫
t∗

p(s)ds >
M

2e
. (2.9)

Integrating (2.7) from h(t) to t∗ and using x(t) is nonincreasing then we have

x(t∗) −x (h(t)) +
1

2M

t∗∫
h(t)

p(s)x (h(s)) ds ≤ 0,

or

x(t∗) −x (h(t)) +
1

2M
x (h(t∗))

t∗∫
h(t)

p(s)ds ≤ 0.

Thus, by (2.9), we have

−x (h(t)) +
1

2M
x (h(t∗))

M

2e
< 0. (2.10)



150 ÖCALAN, KILIÇ, ŞAHİN AND ÖZKAN

Integrating (2.7) from t∗ to t and using the same facts , we get

x(t) −x (t∗) +
1

2M

t∫
t∗

p(s)x (h(s)) ds ≤ 0.

Thus, by (2.9), we have

−x (t∗) +
1

2M
x (h(t))

M

2e
< 0. (2.11)

Combining the inequalities (2.10) and (2.11), we obtain

x(t∗) > x (h(t))
1

4e
> x (h(t∗))

(
1

4e

)2
,

and hence we have
x (h(t∗))

x(t∗)
< (4e)

2
for t ≥ t4.

Let

w =
x(h(t∗))

x(t∗)
≥ 1,

and because of 1 ≤ w < (4e)2 , w is finite.
Now dividing (1.1) with x(t) and then integrating from h(t) to t we obtain

t∫
h(t)

x′(s)

x(s)
ds +

t∫
h(t)

p(s)
f(x(τ(s)))

x(s)
ds = 0

and

ln
x(t)

x(h(t))
+

t∫
h(t)

p(s)
f(x(τ(s)))

x(τ(s))

x(τ(s))

x(s)
ds = 0

Since x(t) is nonincreasing, we get

ln
x(t)

x(h(t))
+

t∫
h(t)

p(s)
f(x(τ(s)))

x(τ(s))

x(h(s))

x(s)
ds ≤ 0

and

ln
x(h(t))

x(t)
≥
f(x(τ(ξ)))

x(τ(ξ))

x(h(ξ))

x(ξ)

t∫
h(t)

p(s)ds, (2.12)

where ξ is defined with h(t) < ξ < t, while t −→ ∞, ξ −→ ∞ and because of this h(t) −→ ∞. Then
taking lower limit on both side of (2.12), we obtain ln w ≥ w

e
. But this is impossible since ln x ≤ x

e
for

all x > 0. The case where M = 0 can be discussed similarly. The proof of the theorem is completed. �

Theorem 2.2. Assume that (1.2), (1.3), (2.2) and (2.4) holds. If τ(t) is non-monotone, and

lim sup
t→∞

t∫
h(t)

p(s)ds > 2M (2.13)

where h(t) is defined by (2.1), then all solutions of Eq.(1.1) oscillate.



OSCILLATION OF NONLINEAR DELAY DIFFERENTIAL 151

Proof. Assume, for the sake of contradiction, that there exist a nonoscillatory solution x(t) of (1.1).
In view of (2.4), we know from Theorem 2.1 that lim

t→∞
x(t) = 0, for t ≥ t1.

Considering equation (1.1)

x′(t) + p(t)f(x(τ(t))) = 0

by (2.5) we get

x′(t) +
1

2M
p(t)x(τ(t)) ≤ 0

Since h(t) ≥ τ(t) and x(t) is nonincreasing

x′(t) +
1

2M
p(t)x(h(t)) ≤ 0 (2.14)

Integrating (2.14) from h(t) to t, and using the fact that the function x(t) is nonincreasing and the
function h(t) is nondecreasing

x(t) −x(h(t)) +
1

2M

t∫
h(t)

p(s)x(h(s))ds ≤ 0

or

x(t) −x(h(t)) +
1

2M
x(h(t))

t∫
h(t)

p(s)ds ≤ 0

This implies

x(t) −x(h(t)) +


1 − 1

2M

t∫
h(t)

p(s)ds


 ≤ 0

and hence

t∫
h(t)

p(s)ds < 2M

for sufficiently t. Therefore,

lim sup
t→∞

t∫
h(t)

p(s)ds ≤ 2M

This is a contradiction to (2.13). The proof is completed. �

Now, assume that f is nondecreasing function then we have the following result.

Theorem 2.3. Assume that (1.2), (1.3), (2.2) and (2.4) hold. If τ(t) is non-monotone, f is nonde-
creasing function and

lim sup
t→∞

t∫
τ(t)

p(s)ds > M (2.15)

where h(t) is defined by (2.1), then all solutions of Eq.(1.1) oscillate.



152 ÖCALAN, KILIÇ, ŞAHİN AND ÖZKAN

Proof. Assume, for the sake of contradiction, that there exist a nonoscillatory solution x(t) of (1.1).
In view of (2.4), we know from Theorem 2.1 that lim

t→∞
x(t) = 0, for t ≥ t1.

Considering equation (1.1)

x′(t) + p(t)f(x(τ(t))) = 0

Since τ(t) ≤ h(t), x(t) is nonincreasing and f is nondecreasing we have

x′(t) + p(t)f(x(h(t))) ≤ 0 (2.16)
Integrating (2.16) from h(t) to t and using the fact that x(t) is nonincreasing and f, h(t) are

nondecreasing

x(t) −x(h(t)) +
t∫

h(t)

p(s)f(x(h(s)))ds ≤ 0

or

x(t) −x(h(t)) + f(x(h(t)))
t∫

h(t)

p(s)ds ≤ 0

and so

x(t) −x(h(t))


1 − f(x(h(t)))

x(h(t))

t∫
h(t)

p(s)ds


 ≤ 0

Therefore

1 >
f(x(h(t)))

x(h(t))

t∫
h(t)

p(s)ds

≥
1

M
lim sup
t→∞

t∫
h(t)

p(s)ds

That is a contradiction. The proof is completed. �

We remark that if τ(t) is nondecreasing, then we have τ(t) = h(t) for all t, and the condition (2.13)
and (2.15), respectively, reduce to

lim sup
t→∞

t∫
τ(t)

p(s)ds > 2M (2.16)

and

lim sup
t→∞

t∫
τ(t)

p(s)ds > M (2.17)

Now, we have the following example.

Example 2.1. Consider the nonlinear delay differential equation

x′(t) +
1

e
x (τ(t)) ln (10 + |x (τ(t))|) = 0, t > 0, (2.18)



OSCILLATION OF NONLINEAR DELAY DIFFERENTIAL 153

where

τ(t) =




t− 1, if t ∈ [3k, 3k + 1]
−3t + 12k + 3, if t ∈ [3k + 1, 3k + 2]
5t− 12k − 13, if t ∈ [3k + 2, 3k + 3]

, k ∈ N0.

By (2.1), we see that

h(t) := sup
s≤t

τ(s) =




t− 1, if t ∈ [3k, 3k + 1]
3k, if t ∈ [3k + 1, 3k + 2.6]
5t− 12k − 13, if t ∈ [3k + 2.6, 3k + 3]

, k ∈ N0.

If we put p(t) = 1
e

and f(x) = x ln(10 + |x|). Then, we have

M = lim sup
x→0

x

f(x)
= lim sup

x→0

x

x ln(10 + |x|)
=

1

ln 10

and

lim inf
t→∞

∫ t
τ(t)

p(s)ds =
1

e
>
M

e
=

1

e ln 10

that is, all conditions of Theorem 2.1 are satisfied and therefore all solutions of (2.18) oscillate.

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154 ÖCALAN, KILIÇ, ŞAHİN AND ÖZKAN

1Akdeniz University, Faculty of Science, Department of Mathematics, 07058, Antalya, Turkey

2Dumlupınar University, Faculty of Science and Arts, Department of Mathematics, 43000, Kütahya,

Turkey

3Afyon Kocatepe University, Faculty of Science and Arts, Department of Mathematics, ANS Campus,

03200, Afyon, Turkey

∗Corresponding author: ozkanocalan@akdeniz.edu.tr


	1. Introduction
	2. Main Results
	References