Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 32, pp. 1–11.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

OSCILLATORY BEHAVIOR OF SOLUTIONS TO THIRD-ORDER

NONLINEAR DIFFERENTIAL EQUATIONS WITH A

SUPERLINEAR NEUTRAL TERM

ERCAN TUNÇ, SAID R. GRACE

Abstract. This article studies the oscillatory and asymptotic behavior of
solutions to a class of third-order nonlinear differential equations with super-

linear neutral term. The results are obtained by a comparison with first-order

delay differential equations whose oscillatory behavior is known, and by using
integral criteria. Two examples are provided to illustrate the results.

1. Introduction

This article concerns the oscillatory and asymptotic behavior of solutions to
third-order nonlinear differential equation with superlinear neutral term(

x(t) + p(t)xα(τ(t))
)′′′

+ q(t)xβ(σ(t)) = 0, t ≥ t0 > 0 . (1.1)

In this paper we use the following hypotheses:

(H1) α and β are the ratios of odd positive integers with α ≥ 1;
(H2) p, q : [t0,∞) → R are real-valued continuous functions with p(t) ≥ 1,

p(t) 6≡ 1 for large t, q(t) ≥ 0, and q(t) is not identically zero for large t;
(H3) τ, σ : [t0,∞) → R are real-valued continuous functions such that σ(t) ≤

τ(t) ≤ t, τ is strictly increasing, and limt→∞ τ(t) = limt→∞ σ(t) =∞. We
denote by τ−1 the inverse function of τ .

By a solution to (1.1), we mean a function x ∈ C3([tx,∞),R), and which satisfies
(1.1) on [tx,∞). We consider only non-trivial solutions, i.e. those that satisfy

sup
t≥t1
|x(t)| > 0 for every t1 ≥ tx .

Moreover, we tacitly assume that (1.1) possesses solutions, and the functions p, q, τ, σ
are smooth enough for the solutions to be continuous. A solution x(t) of (1.1) is
said to be oscillatory if it has arbitrarily large zeros on its domain [tx,∞); i.e., for
any t1 ∈ [tx,∞) there exists t2 ≥ t1 such that x(t2) = 0; otherwise x is called
nonoscillatory, hence eventually positive or eventually negative. Equation (1.1) is
said to be oscillatory if all its solutions are oscillatory.

2010 Mathematics Subject Classification. 34C10, 34K11, 34K40.
Key words and phrases. Oscillation of solutions; asymptotic behavior;

neutral differential equation.
c©2020 Texas State University.

Submitted March 3, 2020. Published April 13, 2020.

1



2 E. TUNÇ, S. R. GRACE EJDE-2020/32

A differential equation in which the highest order derivative of the unknown
function appears both with and without delays is called a neutral differential equa-
tion. Qualitative properties of solutions such equations have been studied by many
authors utilizing various methods. One reason for this is that neutral delay differen-
tial equations have applications to electric networks containing lossless transmission
lines such as in high speed computers. They also occur in problems dealing with
vibrating masses attached to an elastic bar and as the Euler equation for variational
problems involving delay equations. See [13] for additional applications.

The problem of oscillatory and asymptotic behavior of solutions for third order
neutral differential and dynamic equations has been a very active area of research
over the years; see for example [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 17, 18, 20, 21,
22, 23, 24, 25] and their references. However, the results obtained are for the cases
α = 1 and/or 0 < α < 1, i.e., for linear neutral terms; see [2, 3, 4, 5, 6, 7, 8, 10,
11, 12, 14, 17, 18, 20, 21, 22, 23, 24, 25]. For the sublinear neutral term see [9].
This means that the results obtained in these papers cannot be applied to the case
where α > 1.

Motivated by the above observation, we wish to establish oscillation criteria
for equation (1.1) via a comparison with first-order delay differential equations
whose oscillatory behavior is known, and by using integral criteria. The results in
this paper can be applied when limt→∞ p(t) = ∞ for α > 1, and when p(t) is a
bounded and/or limt→∞ p(t) = ∞ for α = 1. To the best of our knowledge, there
are no results for third-order differential equations with superlinear neutral terms.
So this article fills partially the gap in oscillation theory for third-order neutral
differential equations. We would like to point out that the results presented in this
paper can easily be extended to more general third-order differential equations with
superlinear neutral term (see Remark 2.12 below).

2. Main results

For proving our result we use the additional hypotheses:

(H4) For every set of positive constants c, d, θ with 0 < θ < 1, we have

Ψ(t) :=
1

p(τ−1(t))

[
1− (

τ−1(τ−1(t))

τ−1(t)
)2/αθ c

1
α−1

p1/α(τ−1(τ−1(t)))

]
≥ 0 (2.1)

and

Ω(t) :=
1

p(τ−1(t))

[
1− d

1
α−1

p1/α(τ−1(τ−1(t)))

]
≥ 0 (2.2)

for all sufficiently large t.

Note that if α > 1, these assumptions require limt→∞ p(t) = ∞. The following
lemma will play an important role in establishing our main results.

Lemma 2.1 ([1, Lemma 2.2.3]). Let f ∈ Cn([t0,∞), (0,∞)), f (n)(t)f (n−1)(t) ≤ 0
for t ≥ tx ≥ t0, and assume that limt→∞ f(t) 6= 0. Then for every λ ∈ (0, 1), there
exists a tλ ∈ [tx,∞) such that, for all t ∈ [tλ,∞),

f(t) ≥ λ

(n− 1)!
tn−1

∣∣∣f (n−1)(t)
∣∣∣ .

To abbreviate notation we define

z(t) = x(t) + p(t)xα(τ(t)).



EJDE-2020/32 OSCILLATORY BEHAVIOR OF SOLUTIONS 3

The following lemma follows from Kiguradze [15], so we omit its proof.

Lemma 2.2. Suppose that (H1)-(H3) are satisfied and x is an eventually positive
solution of equation (1.1). Then, there exists t1 ∈ [t0,∞) such that for t ≥ t1, the
corresponding function z satisfies one of the following two cases:

(A) z(t) > 0, z′(t) > 0, z′′(t) > 0, and z′′′(t) ≤ 0,
(B) z(t) > 0, z′(t) < 0, z′′(t) > 0, and z′′′(t) ≤ 0.

Lemma 2.3. Let x(t) be a positive solution of (1.1) with z(t) satisfying case (A)
of Lemma 2.2 for t ≥ t1 ≥ t0. Then, for every θ with 0 < θ < 1, we have

z(t) ≥ θ

2
tz′(t) (2.3)

for all large t.

Proof. Note that in case (A), z′ > 0 and z′′ is decreasing. Then by integration we
have

z′(t) = z′(t1) +

∫ t

t1

z′′(s) ds ≥ (t− t1)z′′(t) for t ≥ t1 ≥ t0 .

Then for t ≥ t2 = t1 + 1, we have( z′(t)
t− t1

)′
=

(t− t1)z′′(t)− z′(t)
(t− t1)2

≤ 0 .

hence z′(t)/(t − t1) is non-increasing for t ≥ t2. Using this monotonicity and
t2 = t1 + 1, we have

z(t) = z(t2) +

∫ t

t2

(s− t1)z′(s)

s− t1
ds ≥ z′(t)

t− t1

∫ t

t2

(s− t1) ds

=
z′(t)

t− t1

[ (t− t1)2 − (t2 − t1)2

2

]
=

z′(t)

t− t1

[ (t− t1 + 1)(t− t2)

2

]
≥ z′(t)

t− t1

[ (t− t1)(t− t2)

2

]
= z′(t)

(t− t2)

2
≥ z′(t)θ

2
t, for t ≥ θ + t2 .

Then (2.3) follows. �

Lemma 2.4. Suppose that (H1)–(H3) and (2.1) hold, and that x is an eventually
positive solution of (1.1) with z(t) satisfying case (A) of Lemma 2.2. Then

z′′′(t) + q(t)Ψβ/α(σ(t))zβ/α(τ−1(σ(t))) ≤ 0, (2.4)

for large t.

Proof. Let x(t) be an eventually positive solution of (1.1) such that x(t) > 0,
x(τ(t)) > 0 and x(σ(t)) > 0 for t ≥ t1 ≥ t0. Then, from the definition of z, we have

xα(τ(t)) =
1

p(t)
(z(t)− x(t)) ≤ z(t)

p(t)
,

from which and the fact that τ(t) ≤ t is strictly increasing, it is easy to see that

x(τ−1(t)) ≤ z1/α(τ−1(τ−1(t)))

p1/α(τ−1(τ−1(t)))
. (2.5)



4 E. TUNÇ, S. R. GRACE EJDE-2020/32

From the definition of z and (2.5), we obtain

xα(t) =
1

p(τ−1(t))

[
z(τ−1(t))− x(τ−1(t))

]
≥ 1

p(τ−1(t))

[
z(τ−1(t))− z1/α(τ−1(τ−1(t)))

p1/α(τ−1(τ−1(t)))

]
.

(2.6)

Since z(t) satisfies case (A), (2.3) holds, and so we obtain(z(t)
t2/θ

)′
=
z′(t)− 2

θtz(t)

t2/θ
≤ 0.

Therefore z(t)/t2/θ is decreasing. Since τ(t) ≤ t and τ is strictly increasing, it
follows that τ−1 is increasing and t ≤ τ−1(t). Thus,

τ−1(t) ≤ τ−1(τ−1(t)). (2.7)

Since z(t)/t2/θ is decreasing, it follows that

(τ−1(τ−1(t)))2/θz(τ−1(t))

(τ−1(t))2/θ
≥ z(τ−1(τ−1(t))).

Using this inequality in (2.6), we obtain

xα(t) ≥ 1

p(τ−1(t))

[
z(τ−1(t))− (τ−1(τ−1(t)))2/αθ

(τ−1(t))2/αθ

z1/α(τ−1(t))

p1/α(τ−1(τ−1(t)))

]
=
z(τ−1(t))

p(τ−1(t))

[
1− (

τ−1(τ−1(t))

τ−1(t)
)2/αθ z

1
α−1(τ−1(t))

p1/α(τ−1(τ−1(t)))

]
.

(2.8)

Since z(t) is positive and increasing for t ≥ t1, there exist a t2 ∈ [t1,∞) and a
constant c > 0 such that

z(t) ≥ c for t ≥ t2. (2.9)

Using this inequality in (2.8) yields

xα(t) ≥ z(τ−1(t))

p(τ−1(t))

[
1− (

τ−1(τ−1(t))

τ−1(t)
)2/αθ c

1
α−1

p1/α(τ−1(τ−1(t)))

]
= Ψ(t)z(τ−1(t)),

with Ψ(t) defined by (2.1). Using this inequality in (1.1) gives

z′′′(t) ≤ −q(t)Ψβ/α(σ(t))zβ/α(τ−1(σ(t))), (2.10)

and (2.4) holds. This completes the proof. �

Lemma 2.5. Suppose that (H1)–(H3) and (2.2) hold, and x is an eventually posi-
tive solution of (1.1) with z(t) satisfying case (B) of Lemma 2.2. Then, z(t) either
satisfies the inequality

z′′′(t) + q(t)Ωβ/α(σ(t))zβ/α(τ−1(σ(t))) ≤ 0, (2.11)

for large t, or limt→∞ x(t) = limt→∞ z(t) = 0.

Proof. Let x(t) be an eventually positive solution of (1.1) such that x(t) > 0,
x(τ(t)) > 0 and x(σ(t)) > 0 for t ≥ t1 ≥ t0. Proceeding as in the proof of Lemma
2.4, we again see that (2.6) and (2.7) hold. Since z′(t) < 0, it follows from (2.7)
that

z(τ−1(t)) ≥ z(τ−1(τ−1(t))).



EJDE-2020/32 OSCILLATORY BEHAVIOR OF SOLUTIONS 5

Substituting this inequality in (2.6) yields

xα(t) ≥ z(τ−1(t))

p(τ−1(t))

[
1− z

1
α−1(τ−1(t))

p1/α(τ−1(τ−1(t)))

]
. (2.12)

Since z(t) satisfies case (B) of Lemma 2.2, there exists a constant κ such that

lim
t→∞

z(t) = κ <∞.

Case (i): κ > 0. Then there exists t2 ≥ t1 such that

z(t) ≥ κ for t ≥ t2. (2.13)

Then

z
1
α−1(t) ≤ κ 1

α−1.

Using this inequality in (2.12) gives

xα(t) ≥ z(τ−1(t))

p(τ−1(t))

[
1− κ

1
α−1

p1/α(τ−1(τ−1(t)))

]
= Ω(t)z(τ−1(t)),

with Ω(t) defined by (2.2). Using this inequality in (1.1) yields

z′′′(t) ≤ −q(t)Ωβ/α(σ(t))zβ/α(τ−1(σ(t))) (2.14)

for t ≥ t3 ≥ t2, hence (2.11) holds.

Case (ii): κ = 0. Then limt→∞ z(t) = 0. Since 0 < x(t) ≤ z(t) on [t1,∞), we
have limt→∞ x(t) = 0. This completes the proof. �

Theorem 2.6. Let (H1)–(H4) hold. If∫ ∞
t0

q(s)Ψβ/α(σ(s))ds =∞ (2.15)

and ∫ ∞
t0

q(s)Ωβ/α(σ(s))ds =∞ , (2.16)

then every solution x(t) of (1.1) is either oscillatory or satisfies limt→∞ x(t) = 0.

Proof. Let x(t) be a nonoscillatory solution of (1.1), say x(t) > 0, x(τ(t)) > 0, and
x(σ(t)) > 0 for t ≥ t1 ≥ t0, and assume (2.1) and (2.2) hold for t ≥ t1. The proof
when x(t) is eventually negative is similar, so we omit it. Then, from Lemma 2.2,
z(t) satisfies either case (A) or case (B) for t ≥ t1.

First, we consider case (A). From Lemma 2.4, we see that inequalities (2.9) and
(2.10) hold for t ≥ t3 ≥ t2. Using (2.9) in (2.10) gives

z′′′(t) ≤ −cβ/αq(t)Ψβ/α(σ(t)) for t ≥ t3. (2.17)

Integrating from t3 to t yields

z′′(t) ≤ z′′(t3)− cβ/α
∫ t

t3

q(s)Ψβ/α(σ(s))ds→ −∞ as t→∞,

which contradicts z′′(t) being positive.

Now we consider case (B). From Lemma 2.5, we again have case (i) or case (ii).
In case (i), we see that (2.13) and (2.14) hold for t ≥ t3. Using (2.13) in (2.14), we
arrive at

z′′′(t) ≤ −κβ/αq(t)Ωβ/α(σ(t)) for t ≥ t3. (2.18)



6 E. TUNÇ, S. R. GRACE EJDE-2020/32

Integrating from t3 to t yields

z′′(t) ≤ z′′(t3)− κβ/α
∫ t

t3

q(s)Ωβ/α(σ(s))ds→ −∞ as t→∞,

which contradicts z′′(t) being positive. In case (ii), as in Lemma 2.5, we see that
x(t)→ 0 as t→∞. This completes the proof. �

Next, we establish a new oscillation criterion for (1.1) via a comparison with
first-order delay differential equations whose oscillatory behavior is known.

Theorem 2.7. Let (H1)–(H3), (2.1) and (2.2) hold. If there exist constants λ1, λ2

in (0, 1) such that the first-order delay differential equations

w′(t) +
λ
β/α
1

2β/α
(τ−1(σ(t)))2β/αq(t)Ψβ/α(σ(t))wβ/α(τ−1(σ(t))) = 0, (2.19)

for some constant θ ∈ (0, 1), and

y′(t) +
λ
β/α
2

2β/α
(τ−1(σ(t)))2β/αq(t)Ωβ/α(σ(t))yβ/α(τ−1(σ(t))) = 0 (2.20)

are oscillatory, then a solution x(t) of (1.1) is either oscillatory, or limt→∞ x(t) =
0.

Proof. Let x(t) be a nonoscillatory solution of (1.1), say x(t) > 0, x(τ(t)) > 0, and
x(σ(t)) > 0 for t ≥ t1 ≥ t0, and assume that (2.1) and (2.2) hold for t ≥ t1. Then,
from Lemma 2.2, z(t) satisfies either case (A) or case (B) for t ≥ t1.

First we consider case (A). Proceeding as in the proof of Lemma 2.4, we again
arrive at (2.10) for t ≥ t3 ≥ t2. Now z(t) > 0 and z′(t) > 0 on [t3,∞) ⊆ [t2,∞), so

lim
t→∞

z(t) 6= 0,

and hence by Lemma 2.1 and case (A), for every λ, 0 < λ < 1, there exists tλ ≥ t3
such that

z(t) ≥ λ

2
t2z′′(t) for t ≥ tλ, (2.21)

from which we see that

z(τ−1(σ(t))) ≥ λ

2
(τ−1(σ(t)))2z′′(τ−1(σ(t))) for t ≥ t5, (2.22)

where τ−1(σ(t)) ≥ tλ for t ≥ t5 ≥ tλ. Using (2.22) in (2.10) gives

z′′′(t) +
λβ/α

2β/α
(τ−1(σ(t)))2β/αq(t)Ψβ/α(σ(t))(z′′(τ−1(σ(t))))β/α ≤ 0,

for every λ with 0 < λ < 1. Letting w(t) = z′′(t) in the above inequality, we see
that w is a positive solution of the first-order delay differential inequality

w′(t) +
λβ/α

2β/α
(τ−1(σ(t)))2β/αq(t)Ψβ/α(σ(t))wβ/α(τ−1(σ(t))) ≤ 0 for t ≥ t5.

(2.23)
Integrating from t ≥ t5 to u and letting u→∞, we obtain

w(t) ≥
∫ ∞
t

λβ/α

2β/α
(τ−1(σ(s)))2β/αq(s)Ψβ/α(σ(s))wβ/α(τ−1(σ(s)))ds



EJDE-2020/32 OSCILLATORY BEHAVIOR OF SOLUTIONS 7

for t ≥ t5. The function w(t) is decreasing on [t5,∞) for every λ ∈ (0, 1), and so by
[19, Theorem 1], there exists a positive solution of equation (2.19). This contradicts
the fact that equation (2.19) is oscillatory.

Now we consider case (B). From Lemma 2.5, we again have case (i) or case (ii).
In case (i), we again have limt→∞ z(t) 6= 0 for t ≥ t2 and (2.14) holds for t ≥ t3.
Since limt→∞ z(t) 6= 0 for t ≥ t3, by Lemma 2.1, for every λ, with 0 < λ < 1, there
exists tλ ≥ t3 such that (2.21) holds for t ≥ tλ. Using (2.21) in (2.14) yields

z′′′(t) +
λβ/α

2β/α
(τ−1(σ(t)))2β/αq(t)Ωβ/α(σ(t))(z′′(τ−1(σ(t))))β/α ≤ 0,

for every λ with 0 < λ < 1 and for t ≥ t5 ≥ tλ. Letting y(t) = z′′(t) in the above
inequality, we see that y is a positive solution of the first-order delay differential
inequality

y′(t) +
λβ/α

2β/α
(τ−1(σ(t)))2β/αq(t)Ωβ/α(σ(t))yβ/α(τ−1(σ(t))) ≤ 0. (2.24)

for t ≥ t5. As in case (A), we see that there exists a positive solution of equation
(2.20), which contradicts that (2.20) is oscillatory.

In case (ii), as in Lemma 2.5, we see that x(t) → 0 as t → ∞. This completes
the proof. �

It is well known from [16] (see also [1, Lemma 2.2.9] that if

lim inf
t→∞

∫ t

ζ(t)

R(s)ds >
1

e
, (2.25)

then the first-order delay differential equation

x′(t) +R(t)x(ζ(t)) = 0 (2.26)

is oscillatory, where R, ζ ∈ C([t0,∞),R) with R(t) ≥ 0, ζ(t) ≤ t, and limt→∞ ζ(t) =
∞. Thus, from Theorem 2.7, we have the following oscillation result.

Corollary 2.8. Let (H1)–(H4) be satisfied and α = β. If

lim inf
t→∞

∫ t

τ−1(σ(t))

(τ−1(σ(s)))2q(s)Ψ(σ(s))ds >
2

e
(2.27)

and

lim inf
t→∞

∫ t

τ−1(σ(t))

(τ−1(σ(s)))2q(s)Ω(σ(s))ds >
2

e
, (2.28)

then a solution x(t) of (1.1) either oscillates, or satisfies limt→∞ x(t) = 0.

Proof. From (2.27), one can choose a positive constant λ1 with 0 < λ1 < 1 such
that

lim inf
t→∞

λ1

∫ t

τ−1(σ(t))

(τ−1(σ(s)))2q(s)Ψ(σ(s))ds >
2

e
. (2.29)

Now, in view of (2.25)–(2.26), inequality (2.29) ensures that (2.19) is oscillatory in
the case when α = β. Again, in view of (2.25)–(2.26), inequalities (2.28) ensures
that (2.20) is oscillatory in the case when α = β. So, by Theorem 2.7, the conclusion
holds. �

From Theorem 2.7, we have the following result.



8 E. TUNÇ, S. R. GRACE EJDE-2020/32

Corollary 2.9. Let (H1)–(H4) hold and β < α. If∫ ∞
t0

(τ−1(σ(s)))2β/αq(s)Ψβ/α(σ(s))ds =∞ (2.30)

and ∫ ∞
t0

(τ−1(σ(s)))2β/αq(s)Ωβ/α(σ(s))ds =∞, (2.31)

then a solution x(t) of equation (1.1) either oscillates, or satisfies limt→∞ x(t) = 0.

Proof. Let x(t) be a nonoscillatory solution of (1.1), say x(t) > 0, x(τ(t)) > 0,
and x(σ(t)) > 0 for t ≥ t1 ≥ t0, and that assume (2.1) and (2.2) hold for t ≥ t1.
Proceeding as in the proof of Theorem 2.7, we again arrive at (2.23) and (2.24)
for t ≥ t5. Using that w(t) := z′′(t) is positive and decreasing, and noting that
τ−1(σ(t)) ≤ t, we have

w(τ−1(σ(t))) ≥ w(t)

and so, (2.23) can be written as

w′(t) +
λβ/α

2β/α
(τ−1(σ(t)))2β/αq(t)Ψβ/α(σ(t))wβ/α(t) ≤ 0,

or
w′(t)

wβ/α(t)
+
λβ/α

2β/α
(τ−1(σ(t)))2β/αq(t)Ψβ/α(σ(t)) ≤ 0 for t ≥ t5. (2.32)

Integration from t5 to ∞ gives∫ ∞
t5

(τ−1(σ(s)))2β/αq(s)Ψβ/α(σ(s))ds ≤ (
2

λ
)β/α

w1− β
α (t5)

1− β
α

<∞,

which contradicts (2.30). Using the similar arguments, the remainder of proof
follows from inequality (2.24) and case (ii) in Theorem 2.7; we omit the details. �

We conclude this paper with the following examples and remarks to illustrate the
above results. Our first example is concerned with the equation with superlinear
neutral term in the case where p(t) → ∞ as t → ∞, and the second example
deals with the equation with linear neutral term in the case where p is a constant
function.

Example 2.10. Consider the third-order differential equation with superlinear
neutral term

z′′′(t) +
t

2
x3(

t

4
) = 0, t ≥ 1, (2.33)

with

z(t) = x(t) + tx3(
t

2
).

Here p(t) = t, q(t) = t/2, τ(t) = t/2, σ(t) = t/4, α = 3, and β = 3. Then, it is easy
to see that conditions (H1)–(H3) hold, and

τ−1(t) = 2t, τ−1(τ−1(t)) = 4t, τ−1(σ(t)) = t/2.

It follows from (2.15) and (2.16) that∫ ∞
t0

q(s)Ψβ/α(σ(s))ds =

∫ ∞
1

(1− 22/3θ

c2/3s1/3
)ds =∞,



EJDE-2020/32 OSCILLATORY BEHAVIOR OF SOLUTIONS 9

and ∫ ∞
t0

q(s)Ωβ/α(σ(s))ds =

∫ ∞
1

(1− 1

d2/3s1/3
)ds =∞;

thus (2.15) and (2.16) hold. Then by Theorem 2.6, a solution x(t) of equation
(2.33) is either oscillatory, or satisfies limt→∞ x(t) = 0.

Example 2.11. Consider the third-order differential equation with linear neutral
term

z′′′(t) + (1 + tµ)x1/5(
t

3
) = 0, t ≥ 1, (2.34)

with

z(t) = x(t) + 20x(
t

2
).

Here p(t) = 20, q(t) = 1 + tµ with µ ≥ 0, τ(t) = t/2, σ(t) = t/3, α = 1, and
β = 1/5. Then, it is easy to see that (H1)–(H3) hold, and

τ−1(t) = 2t, τ−1(τ−1(t)) = 4t, and τ−1(σ(t)) = 2t/3.

Choosing θ = 1/2, it follows from (2.30) and (2.31) that∫ ∞
t0

(
τ−1(σ(s))

)2β/α
q(s)Ψβ/α(σ(s))ds =

(2

3

)2/5( 1

100

)1/5 ∫ ∞
1

s2/5(1 + sµ)ds =∞

and∫ ∞
t0

(
τ−1(σ(s))

)2β/α
q(s)Ωβ/α(σ(s))ds =

(2

3

)2/5( 19

400

)1/5 ∫ ∞
1

s2/5(1 + sµ)ds =∞;

thus (2.30) and (2.31) hold. Then by Corollary 2.9, a solution x(t) of equation
(2.34) either oscillates, or satisfies limt→∞ x(t) = 0.

Remark 2.12. The results of this paper can be easily extended to the third-order
differential equation with superlinear neutral term

(r(t)(z′′(t))γ)′ + q(t)xβ(σ(t)) = 0, t ≥ t0 > 0, (2.35)

under the two conditions∫ ∞
t0

r−1/γ(t)dt =∞,
∫ ∞
t0

r−1/γ(t)dt <∞,

where r ∈ C([t0,∞), (0,∞)), γ is the ratio of odd positive integers, and the other
functions and constant β in the equation are defined as in this paper.

Remark 2.13. It would be of interest to study the oscillatory behavior of all
solutions of (1.1) for p(t) ≤ −1 with p(t) 6≡ −1 for large t.

Acknowledgments. The authors would like to express their gratitude to Prof.
Julio G. Dix and the anonymous referees for the careful reading of the original
manuscript. Their comments helped us improve the presentation of the results,
and accentuate important details.



10 E. TUNÇ, S. R. GRACE EJDE-2020/32

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EJDE-2020/32 OSCILLATORY BEHAVIOR OF SOLUTIONS 11

Ercan Tunç

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpasa University,

60240, Tokat, Turkey
Email address: ercantunc72@yahoo.com

Said R. Grace
Department of Engineering Mathematics, Faculty of Engineering, Cairo University,

Orman, Giza 12221, Egypt

Email address: saidgrace@yahoo.com


	1. Introduction
	2. Main results
	Acknowledgments

	References