Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 53, pp. 1–13.

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

OSCILLATION FOR SECOND ORDER NONLINEAR

DIFFERENTIAL EQUATIONS WITH A

SUB-LINEAR NEUTRAL TERM

YINGZHU WU, YUANHONG YU, JINSEN XIAO

Abstract. We study the oscillatory behavior of solution to the second order
nonlinear differential equations with a sub-linear neutral term(

a(t)[(x(t) + p(t)xα(τ(t)))′]γ
)′

+ q(t)xβ(σ(t)) = 0, t ≥ t0 > 0.

A new criterion is established that improves related results reported in the

literature. Moreover, some examples are provided to illustrate the main results.

1. Introduction

This article concerns the second order nonlinear differential equation with a sub-
linear neutral term(

a(t)[(x(t) + p(t)xα(τ(t)))′]γ
)′

+ q(t)xβ(σ(t)) = 0, t ≥ t0 > 0, (1.1)

under the following assumptions:

(A1) α, β and γ are the ratios of odd positive integers with α ∈ (0, 1];
(A2) a ∈ C1([t0,∞), (0,∞)), a′(t) > 0, p, q ∈ C([t0,∞), (0,∞)), limt→∞ p(t) = 0

and q is not eventually zero on [t∗,∞) for t∗ ≥ t0;
(A3) τ ∈ C([t0,∞), R), σ ∈ C1([t0,∞), R), τ(t) ≤ t, σ(t) ≤ t, σ′(t) > 0, and

limt→∞ τ(t) = limt→∞ σ(t) =∞.

A function x(t) ∈ C1([Tx,∞), R), Tx ≥ t0 is called a solution of (1.1) if a(t)[(x(t) +
p(t)xα(τ(t)))′]γ ∈ C1([Tx,∞), R) and satisfies (1.1) on an interval [Tx,∞). We only
consider the nontrivial solutions of (1.1), which means sup{|x(t)| : t ≥ T} > 0 for
all T ≥ Tx. A solution of(1.1) is said to be oscillatory if it has an arbitrarily large
zeros on [Tx,∞); otherwise, it is called non-oscillatory. Equation (1.1) is said to be
oscillatory if all its solutions are oscillatory.

Oscillation phenomena arise in various models from real world applications; see,
e.g., [7, 13, 16, 20, 22]. In particular, we refer the reader to the papers [16, 22]
for models from mathematical biology and physics where oscillation and/or delay
actions may be formulated by means of cross-diffusion terms. The increasing in-
terest in oscillatory criteria for the second order nonlinear differential equations is
motivated by their applications in the natural sciences and engineering, see, for

2020 Mathematics Subject Classification. 34C10, 34K11.

Key words and phrases. Neutral differential equation; sub-linear neutral term; oscillation.
©2022. This work is licensed under a CC BY 4.0 license.
Submitted March 15, 2022. Published July 19, 2022.

1



2 Y. WU, Y. YU, J. XIAO EJDE-2022/53

example, [2, 3, 4, 8, 9, 10, 15, 17, 18, 21, 23, 26] and the references cited therein.
For (1.1), one important special case is when γ = 1,(

a(t)(x(t) + p(t)xα(τ(t)))′
)′

+ q(t)xβ(σ(t)) = 0. (1.2)

Note that if p(t) ≡ 0, this equation is the well-known Emden-Fowler equation which
has been widely applied in mathematics and theoretical physics (see [19, 25]). When
α = 1, Li et al. [17, 18, 21] obtained oscillation criteria under canonical and non-
canonical conditions. In 2014, Agarwal et al. [1] discussed the oscillatory behavior
of (1.2) with β = 1. Later on, Grace and Graef [12] and Tamilvanan et al. [24]
considered the sub-linear (0 < β < 1) and super-linear (β > 1) cases under the
non-canonical condition ∫ ∞

t0

1

a(t)
dt <∞ (1.3)

and obtained the oscillation criteria of all solutions of (1.2). Tamilvanan et al. [24]
studied the sub-linear and super-linear cases under the canonical condition∫ ∞

t0

1

a(t)
dt =∞ (1.4)

and obtained the oscillation of all solutions of (1.2).
Another important special case of (1.1) is (when β = γ)(

a(t)[(x(t) + p(t)xα(τ(t)))′]γ
)′

+ q(t)xγ(σ(t)) = 0. (1.5)

If p(t) ≡ 0, this equation is called the half-linear differential equation which is
first studied by Hungarian mathematicians Bihari and Elbert in the 1970s, and has
attracted considerable attentions in recent years, see, for example, [2, 4, 9, 10, 20,
11, 23, 25]. However, all of those results on the oscillation of (1.5) are for α = 1
and few for 0 < α < 1.

Now in this article we shall use the Riccati transformation technique to study the
oscillation behaviors of (1.1) for different positive constants α, β and γ. In Section
2, we establish a new oscillation criterion for (1.1). In Section 3, we present some
examples to illustrate our results.

2. Main results

Without loss of generality, we only deal with the positive solution of (1.1) in the
proofs of our results. In what follows, all functional inequalities are assumed to be
satisfied for all t sufficiently large. We define the functions

z(t) := x(t) + p(t)xα(τ(t)), π(t) :=

∫ ∞
t

1

a1/γ(s)
ds,

ϕ(t) :=
p(t)

π1−α(τ(t))
, ψ(t) :=

p(t)πα(τ(t))

π2−α(t)

and assume that

(A4) π(t0) <∞;
(A5) max{ϕ(σ(t)), ψ(σ(t))} < 1.

Before giving the proof of our main theorem, we need the following lemma derived
by Zhang and Wang [27].



EJDE-2022/53 OSCILLATION FOR SECOND ORDER NONLINEAR DES 3

Lemma 2.1. Let ξ be a ratio of odd positive integers, A,B > 0 and w ≥ 0. Then

Aw −Bw1+ 1
ξ ≤ ξξ

(ξ + 1)ξ+1
Aξ+1B−ξ.

Theorem 2.2. Let (A1)–(A5) hold. If there exist a positive nondecreasing function
ρ ∈ C1([t0,∞), (0,∞)) and two positive constants K,M such that

lim
t→∞

∫ t

T

[
ρ(s)q(s)(1− ϕ(σ(s)))β − a(θ(s))(ρ′(s))ξ+1

(ξ + 1)ξ+1(Kρ(s)σ′(s))ξ

]
ds =∞, (2.1)

lim
t→∞

∫ t

T

[
πη(s)q(s)(1− ψ(σ(s)))β − (

η

η + 1
)η+1 (η/M)η

π(s)a1/γ(s)

]
ds =∞ (2.2)

hold for sufficiently large T ≥ t0, where ξ = min{β, γ}, η = max{β, γ} and

θ(t) =

{
t, γ > β,

σ(t), γ ≤ β,

then every solution of (1.1) is oscillatory.
Especially, when β = γ, equation (1.1) is oscillatory if assumptions (2.1) and

(2.2) hold for K = 1 and M = η = β.

Proof. Suppose to the contrary that (1.1) has an eventually positive solution x(t),
i.e., there exists a t1 ≥ t0 such that x(t) > 0, x(τ(t)) > 0, and x(σ(t)) > 0 for all
t ≥ t1. It follows that z(t) > 0 for t ≥ t1. From (1.1), we see that

(a(t)(z′(t))γ)′ = −q(t)xβ(σ(t)) < 0, (2.3)

which implies that a(t)(z′(t))γ is decreasing and thus z′(t) does not change sign
eventually. Therefore, there exists a t2 ≥ t1 such that either z′(t) > 0 or z′(t) < 0
for all t ≥ t2.

Case I. First we assume that z′(t) > 0 for all t ≥ t2. Recall that z(t) = x(t) +
p(t)xα(τ(t)), hence we have z(t) ≥ x(t) and

x(t) ≥ z(t)− p(t)zα(τ(t)), t ≥ t2. (2.4)

Since z(t) is a positive increasing function, π(t) is a positive decreasing function
and π(t)→ 0 as t→∞, then there exists a t3 ≥ t2 such that

z(t) ≥ π(t), t ≥ t3. (2.5)

Hence we obtain

x(t) ≥
(

1− p(t)

π1−α(τ(t))

)
z(t). (2.6)

This, (1.1) and the definition of ϕ(t), imply that

(a(t)(z′(t))γ)′ + q(t)[1− ϕ(σ(t))]βzβ(σ(t)) ≤ 0, t ≥ t3. (2.7)

Now define a function w(t) by

w(t) := ρ(t)
a(t)(z′(t))γ

zβ(σ(t))
, t ≥ t3. (2.8)

It follows that w(t) > 0 for t ≥ t3, and

w′(t) = ρ′(t)
a(t)(z′(t))γ

zβ(σ(t))
+ ρ(t)

(a(t)(z′(t))γ)′

zβ(σ(t))
− βρ(t)a(t)(z′(t))γσ′(t)z′(σ(t))

zβ+1(σ(t))
.



4 Y. WU, Y. YU, J. XIAO EJDE-2022/53

By (2.7) and (2.8), we obtain

w′(t) ≤ −ρ(t)q(t)(1−ϕ(σ(t)))β +
ρ′(t)

ρ(t)
w(t)− βρ(t)a(t)(z′(t))γσ′(t)z′(σ(t))

zβ+1(σ(t))
. (2.9)

We discuss this inequality in three cases. If γ < β, notice that a1/γ(t)z′(t) ≤
a1/γ(σ(t))z′(σ(t)), then

w′(t) ≤ −ρ(t)q(t)(1−ϕ(σ(t)))β +
ρ′(t)

ρ(t)
w(t)− βσ′(t)

(ρ(t)a(σ(t)))1/γ
[z(σ(t))]

β−γ
γ w

γ+1
γ (t).

Since z(σ(t)) is increasing, then there exist constants k1 > 0 and t4 ≥ t3 such that

[z(σ(t))]
β−γ
γ ≥ k1 for t ≥ t4. It follows that

w′(t) ≤ −ρ(t)q(t)(1− ϕ(σ(t)))β +
ρ′(t)

ρ(t)
w(t)− γk1σ

′(t)

(ρ(t)a(σ(t)))1/γ
w
γ+1
γ (t). (2.10)

An easy computation shows that k1 = 1 as γ = β. Now if γ > β, we claim that

[z′(t)]
β−γ
β is increasing. Observing that (a(t)(z′(t))γ)′ ≤ 0 and a′(t) > 0, then

z′′(t) ≤ 0, which implies that z′(t) is decreasing and hence [z′(t)]
β−γ
β is increasing.

Therefore, there exist constants k2 > 0, t5 ≥ t4 such that [z′(t)]
β−γ
β ≥ k2 for t ≥ t5.

It follows from (2.9) that

w′(t) ≤ −ρ(t)q(t)(1− ϕ(σ(t)))β +
ρ′(t)

ρ(t)
w(t)− βσ′(t)

(ρ(t)a(t))
1
β

[z′(t)]
β−γ
β w

β+1
β (t)

≤ −ρ(t)q(t)(1− ϕ(σ(t)))β +
ρ′(t)

ρ(t)
w(t)− βk2σ

′(t)

(ρ(t)a(t))
1
β

w
β+1
β (t), t ≥ t5.

Combining this inequality and (2.10), we have

w′(t) ≤ −ρ(t)q(t)(1− ϕ(σ(t)))β +
ρ′(t)

ρ(t)
w(t)− ξKσ′(t)

(ρ(t)a(θ(t)))
1
ξ

w
ξ+1
ξ (t) (2.11)

for t ≥ t3, where ξ = min{γ, β}, K = min{k1, k2}, and

θ(t) =

{
t, γ > β,

σ(t), γ ≤ β.

Obviously, K = 1 if β = γ and K is a positive constant if β 6= γ.

For the inequality (2.11), applying Lemma 2.1 with A = ρ′(t)
ρ(t) and

B =
ξKσ′(t)

(ρ(t)a(θ(t)))
1
ξ

,

we obtain

w′(t) ≤ −ρ(t)q(t)(1− ϕ(σ(t)))β +
(ρ′(t))ξ+1a(θ(t))

(ξ + 1)ξ+1(Kρ(t)σ′(t))ξ
. (2.12)

Integrating this inequality from T > t5 to t ≥ T , we have∫ t

T

[ρ(s)q(s)(1− ϕ(σ(s)))β − (ρ′(s))ξ+1a(θ(s))

(ξ + 1)ξ+1(Kρ(s)σ′(s))ξ
]ds

≤ w(T )− w(t) ≤ w(T ),

which contradicts (2.1).



EJDE-2022/53 OSCILLATION FOR SECOND ORDER NONLINEAR DES 5

Case II. Assume that z′(t) < 0 for t > t2. In view of (1.1) we have that
(a(t)(−z′(t))γ)′ ≥ 0, t ≥ t2. Then a1/γ(t)(−z′(t)) is an increasing function and
thus

z′(s) ≤ (
a(t)

a(s)
)1/γz′(t), s ≥ t ≥ t2. (2.13)

Integrating this inequality from t to u, we obtain

z(u)− z(t) ≤ a1/γ(t)z′(t)

∫ u

t

a−1/γ(s)ds, t ≥ t2.

Letting u→∞ we then obtain

z(t) ≥ π(t)a1/γ(t)(−z′(t)), t ≥ t2. (2.14)

We define the function

V (t) :=
a(t)(−z′(t))γ

zβ(t)
, t ≥ t2. (2.15)

It follows that V (t) > 0 for all t ≥ t2. By (2.14) we have

zγ(t) ≥ πγ(t)(a(t)(−z′(t)))γ . (2.16)

For this inequality, we first observe that if γ ≥ β, then zγ−β(t) is a decreasing
function and thus there exist constants l1 > 0 and t3 ≥ t2 such that zγ−β(t) ≤ l1
for t ≥ t3. By (2.16) we obtain

l1 ≥ zγ−β(t) ≥ πγ(t)V (t), γ ≥ β, t ≥ t3. (2.17)

Now from (2.14) we see that

zβ(t) ≥ πβ(t)
(
a1/γ(t)(−z′(t))

)β−γ+γ
. (2.18)

If γ < β, then (a1/α(t)(−z′(t)))β−γ is an increasing function. Hence there exist
constants l2 > 0 and t4 ≥ t3, such that

l2 ≥
(
r1/γ(t)(−z′(t))

)γ−β
≥ πβ(t)V (t), t ≥ t4. (2.19)

Combining (2.17) and (2.19), we have

0 < πη(t)V (t) ≤ l, t ≥ t4, (2.20)

where η = max{γ, β} and l = max{l1, l2}.
Observing (2.14) again we obtain( z(t)

π(t)

)′ ≥ 0, t ≥ t4. (2.21)

It follows that z(t)
π(t) is a positive increasing function. From assumption (A3) we have

τ(t) ≤ t and thus
z(τ(t))

π(τ(t))
≤ z(t)

π(t)
.

By this inequality and (2.4) we deduce that

x(t) ≥ z(t)− p(t)π
α(τ(t))

πα(t)
zα(t). (2.22)

Since z(t)
π(t) is a positive increasing function and π(t) is a positive decreasing function

tends to 0 as t→∞, then there exists t5 ≥ t4 such that

z(t) ≥ π2(t), t ≥ t5. (2.23)



6 Y. WU, Y. YU, J. XIAO EJDE-2022/53

In view of the above two inequalities, for t ≥ t5, we have

x(t) ≥
(

1− p(t)πα(τ(t))

πα(t)z1−α(t)

)
z(t)

≥
(

1− p(t)πα(τ(t))

π2−α(t)

)
z(t)

= (1− ψ(t))z(t).

By (1.1) and the fact z′(t) < 0, we have(
a(t)(−z′(t))γ

)′ ≥ q(t)(1− ψ(σ(t)))βzβ(t). (2.24)

Taking derivative of the function V (t) defined by (2.15) and then using (2.24) we
obtain

V ′(t) ≥ q(t)(1− ψ(σ(t)))β +
βa(t)

(
− z′(t)

)γ+1

zβ+1(t)
, t ≥ t5. (2.25)

For this inequality, we first treat the case γ > β. Since [z(t)]
β−γ
γ is an increasing

function, then there exist constants m1 > 0 and t6 ≥ t5, such that [z(t)]
β−γ
γ ≥ m1

for t ≥ t6. Hence,

V ′(t) ≥ q(t)(1− ψ(σ(t)))β +
β

a1/γ(t)
[z(t)]

β−γ
γ V

γ+1
γ (t)

≥ q(t)(1− ψ(σ(t)))β +
βm1

a1/γ(t)
V
γ+1
γ (t), t ≥ t6.

(2.26)

If γ < β, we find that [a1/γ(t)(−z′(t))]
β−γ
β is an increasing function. Thus there

exist constants m2 > 0 and t7 ≥ t6 such that [a1/γ(t)(−z′(t))]
β−γ
β ≥ m2 for t ≥ t7.

In view of (2.25), we then have

V ′(t) ≥ q(t)(1− ψ(σ(t)))β +
β

a1/γ(t)
[a1/γ(t)(−z′(t))]

β−γ
β V

β+1
β (t)

≥ q(t)(1− ψ(σ(t)))β +
βm2

a1/γ(t)
V
β+1
β (t), t ≥ t7.

(2.27)

Now if γ = β, it is easy to see that m1 = m2 = 1. Hence (2.26) and (2.27) still
hold.

Combining (2.26) and (2.27) we obtain

V ′(t) ≥ q(t)(1− ψ(σ(t)))β +
M

a1/γ(t)
V
η+1
η (t), t ≥ t7, (2.28)

where η = max{γ, β} and M =

{
β, γ = β,

const > 0, γ 6= β.
Multiplying (2.28) by πη(t)

and integrating the resulting inequality from T ≥ t7 to t, we have∫ t

T

πη(s)q(s)(1− ψ(σ(s)))βds

≤
∫ t

T

πη−1(s)a−1/γ(s)[ηV (s)−Mπ(s)V
η+1
η (s)]ds+ πη(t)V (t)− πη(T )V (T ).

Again using Lemma 2.1 we obtain

ηV (s)−Mπ(s)V
η+1
η (s) ≤

( η

η + 1

)η+1( η
M

)η
π−η(s).



EJDE-2022/53 OSCILLATION FOR SECOND ORDER NONLINEAR DES 7

In view of the above two inequalities and (2.3), we have∫ t

T

[
πη(s)q(s)(1− ψ(σ(s)))β − (

η

η + 1
)η+1 (η/M)η

π(s)a1/γ(s)

]
ds ≤ l. (2.29)

Letting t → ∞ in the above inequality we then get a contradiction to (2.2). The
proof is complete. �

Recently, Jadlovská and Džurina [14] used the characteristic equation method
to establish a variant of Kneser oscillation theorem for the second order differential
equation (

r(t)|y′(t)|α−1y′(t)
)′

+ q(t)|y(τ(t))|α−1y(τ(t)) = 0 (2.30)

under non-canonical conditions. This result is extended by Chatzarakis et al. [6]
for the equation(

r(t)|y′(t)|α−1y′(t)
)′

+

m∑
i=1

qi(t)|y(τi(t))|α−1y(τi(t)) = 0. (2.31)

Another extension is due to Bohner et al. [5], who studied the half-linear neutral
differential equation (

a(t)(z′(t))α
)′

+ q(t)yα(σ(t)) = 0 (2.32)

where z(t) = y(t)+p(t)y(τ(t)), and obtained a generalization of the Kneser theorem.

Theorem 2.3 ([5, Theorem 2]). Assume

λ∗ := lim inf
t→∞

π(σ(t))

π(t)
<∞.

If

lim inf
t→∞

a1/α(t)πα+1(t)q(t) >
αmax{Kα(1−K)λ−αK∗ : 0 < K < 1}

(1− p0)α
,

then equation (2.32) is oscillatory.

Obviously, (2.32) is a special case of equation (1.1) (when α = 1 and β = γ).
Now by Theorem 2.2 we derive the following Kneser oscillation criterion for (1.1)
under non-canonical conditions.

Corollary 2.4. Theorem 2.2 still holds if the conditions (2.1) and (2.2) are replaced
by

lim sup
t→∞

∫ t

T

q(s)(1− ϕ(σ(s)))βds =∞, (2.33)

lim inf
t→∞

πη+1(t)a1/γ(t)q(t)(1− ψ(σ(t)))β > (
η

η + 1
)η+1(

η

M
)η, (2.34)

respectively.

Proof. Equality (2.33) follows by substituting ρ(t) ≡ 1 into (2.1). Now suppose
(2.34) holds and denote µ = ( η

η+1 )η+1( ηM )η. Then, for any ε > 0, there exists a

sufficiently large T , such that

πη(t)q(t)(1− ψ(σ(t)))β >
µ− ε

π(t)a1/γ(t)
, t ≥ T.



8 Y. WU, Y. YU, J. XIAO EJDE-2022/53

Integrating the above inequality from T to t we obtain∫ t

T

[
πη(s)q(s)(1− ψ(σ(s)))β − µ

π(s)a1/γ(s)

]
ds

>

∫ T

t

ε

π(s)a1/γ(s)
ds

= −ε
∫ t

T

dπ(s)

π(s)
= ε
(

ln
1

π(T )
− ln

1

π(t)

)
.

(2.35)

Then (2.2) holds as t→∞. This completes the proof. �

The following theorem is for (1.2), the special case of (1.1) with γ = 1, and the
theorem is Kneser oscillation criteria for Emden-Fowler neutral (1.2) and sub-linear
neutral (1.5).

Theorem 2.5. Suppose γ = 1. Then Corollary 2.4 still holds if (2.2) is replaced
by any one of the following conditions:

(i) β > 1, lim inft→∞ πη+1(t)a(t)q(t)(1− ψ(σ(t)))β > ( β
β+1 )β+1( βM )β;

(ii) β < 1, lim inft→∞ π2(t)a(t)q(t)(1− ψ(σ(t)))β > 1
4M ;

(iii) β = 1, lim inft→∞ π2(t)a(t)q(t)(1− ψ(σ(t))) > 1/4.

The following theorem is for (1.5).

Theorem 2.6. Suppose β = γ. Then Corollary 2.4 still holds if (2.2) is replaced
by

lim inf
t→∞

πγ+1(t)a1/γ(t)q(t)(1− ψ(σ(t)))γ >
( γ

γ + 1

)γ+1

.

Note that Theorem 2.5 improves [1, Theorem 2.2] and [12, Theorems 1-4]. The-
orem 2.6 improves [2, Theorem 2.2], [4, Theorems 2.2-2.4], [10, Theorem 3 and
Corollary 1].

3. Examples

In this section, we provide some examples to illustrate our results.

Example 3.1. Consider the Emden-Fowler neutral differential equation(
t3/2

(
x(t) + p0x(

t

2
)
)′)′

+ q0t
δxβ(t) = 0, t ≥ 1, (3.1)

where β is a ratio of odd positive integers, p0 ∈ [0,
√
2
2 ), q0 > 0 and δ ≥ −1.

We are going to use Theorem 2.5 to show that (3.1) is oscillatory. Note that
a(t) = t3/2, τ(t) = t/2, σ(t) = t, p(t) = p0, q(t) = q0t

δ and α = γ = 1. Thus we
have

π(t) =

∫ ∞
t

1

a1/γ(s)
ds =

∫ ∞
t

s−
3
2 ds = 2t−1/2,

ϕ(t) =
p(t)

π1−α(τ(t))
= p0, ψ(t) =

p(t)πα(τ(t))

π2−α(t)
=
√

2p0.

Therefore, (A1)–(A5) and (2.33) hold.
Now we see that if β > 1, then η = max{β, 1} = β and

lim inf
t→∞

πη+1(t)a(t)q(t)(1− ψ(σ(t)))β



EJDE-2022/53 OSCILLATION FOR SECOND ORDER NONLINEAR DES 9

= lim inf
t→∞

2β+1(t)q0(1−
√

2p0)βtδ−
β
2 +1 =∞

holds for δ > β
2 − 1. Hence Theorem 2.5-(i) is satisfied as β > 1 and δ > β

2 − 1.
If β < 1, we have η = 1 and thus

lim inf
t→∞

π2(t)a(t)q(t)(1− ψ(σ(t)))β = lim inf
t→∞

4q0
(
1−
√

2p0
)β
tδ+

1
2 =∞

holds for δ > −1/2. Then Theorem 2.5-(ii) is satisfied as β < 1 and δ > −1/2.
If β = 1, we have η = 1 and thus

lim inf
t→∞

π2(t)a(t)q(t)(1− ψ(σ(t)))β = lim inf
t→∞

4q0(1−
√

2p0)tδ+
1
2 >

1

4

holds for δ = −1/2. Now Theorem 2.5-(iii) is satisfied as β = 1 and δ = −1/2.
We, therefore, can make clear from the above discussion that (3.1) is oscillatory

if (i) β > 1 and δ > β
2 − 1, or (ii) β < 1 and δ > −1/2, or (iii) β = 1 and δ = −1/2.

Note that Li et al. [15] considered the equation(
t3/2y′(t)

)′
+ y(t) = 0, (3.2)

which is a special case of (3.1) with p0 = 0, q0 = 1, δ = 0, β = 1.
The following example illustrates Theorem 2.6.

Example 3.2. Consider the half-linear neutral differential equation(
tγ+1

[(
x(t) + p0x(

t

2
)
)′]γ

)′ + q0x
γ(λt) = 0, t ≥ 1, (3.3)

where γ is a ratio of odd positive integers, q0 ∈ (0,∞), p0 ∈ [0, γ
√

0.5), λ ∈ (0, 1].

Observing that a(t) = tγ+1, τ(t) = t/2, σ(t) = λt, p(t) = p0, and q(t) = q0, we
have

π(t) =

∫ ∞
t

1

a1/γ(s)
ds =

γ

t1/γ
,

ϕ(t) =
p(t)

π1−γ(τ(t))
= p0, ψ(t) =

p(t)π(τ(t))

π(t)
=

γ
√

2p0.

Hence, the conditions (A1)–(A5) and (2.33) hold.
An easy calculation shows that

lim inf
t→∞

πγ+1(t)a1/γq(t)(1− ψ(σ(t)))γ = γγ+1(t)q0(1− γ
√

2p0)γ .

Then by Theorem 2.6, we conclude that (3.3) is oscillatory if

q0(1− γ
√

2p0)γ >
( 1

γ + 1

)γ+1

. (3.4)

Remark 3.3. This example is also studied by Bohner et al. [4]. According to [4,
Theorem 2.1], equation (3.3) is oscillatory if

q0(1− γ
√

2p0)γ >
( 1

γ

)γ
. (3.5)

Obviously, this restriction is contained in (3.4).
In [2], the authors considered the special case of (3.3) when γ = 1, λ = 1, i.e.,(

t2
(
x(t) + p0x

( t
2

))′)′
+ q0x(t) = 0. (3.6)



10 Y. WU, Y. YU, J. XIAO EJDE-2022/53

By [2, Theorem 2.1], equation (3.6) is oscillatory if

q0(1− 2p0) >
1

4
, (3.7)

which is just a special case of (3.4) when γ = 1. Note that if p0 = 0 and γ = 1,
then (3.4) or (3.7) reduces to q0 > 1/4, which is sharp for oscillation of the Euler
differential equation (see [2, 4])(

t2x(t)
)′

+ q0x(t) = 0.

Another special case of (3.3) is studied by Džurina and Jadlovská [10], which
reads (

tγ+1(x′(t))γ
)′

+ q0x
γ(λt) = 0, (3.8)

where q0 > 0, γ ≥ 1, λ ∈ (0, 1]. Due to [10, Theorem 3], one can conclude that (3.8)
is oscillatory if q0 > 1. However, by Theorem 2.6 we can obtain a sharp condition

q0 >
( 1

γ + 1

)γ+1
. (3.9)

Example 3.4. Consider the sub-linear neutral Emden-Fowler equation(
t2
(
x(t) +

1

t2
xα
( t

2

))′)′
+ λtbxβ

( t
2

)
= 0, t ≥ 4, (3.10)

where α, β are ratios of odd positive integers with α ∈ (0, 1].

For this example we set a(t) = t2, p(t) = 1/t2, q(t) = λtb, b > −1, λ > 0,
τ(t) = σ(t) = t/2. Hence we obtain

π(t) =
1

t
, ϕ(σ(t)) =

4α

t1+α
, ψ(σ(t)) =

8α

t2α
.

Therefore, (A1)–(A5) and (2.33) hold.
Now if b > η − 1, we have

lim inf
t→∞

πη+1(t)a(t)q(t)(1− ψ(σ(t)))β = lim inf
t→∞

λ(1− 8α

t2α
)βtb−η+1 =∞.

Then (3.10) is oscillatory because of Corollary 2.4.
If b = η − 1, we see that when η = β = 1,

lim inf
t→∞

πη+1(t)a(t)q(t)(1− ψ(σ(t)))β = lim inf
t→∞

π2(t)a(t)q(t)(1− ψ(σ(t))) = λ.

By Theorem 2.5(iii), we conclude that (3.10) is oscillatory for λ > 1/4.

Remark 3.5. Grace and Graef [12] considered (3.10) when α = 1/3, λ = 1, and
used Theorems 1–4 to obtain oscillatory results for (1) β = 3 and b = 9, (2) β = 3
andb > 4, (3) β = 1 and b > 0, and (4) β = 1/3 and b > 0. Another result for
(3.10) when β = 1 and b = 0 from Agarwal et al. [1], showed that this equation is
oscillatory if λ > 1/4.

Example 3.6 ([8, Example 3.10]). Consider the equation(
tγ+1

[(
x(t) + p(t)xα

( t
2

))′]γ)
+ λtδxβ

( t
2

)
= 0, (3.11)

where a(t) = tγ+1, π(t) = γ
t1/γ

, q(t) = λtδ, δ ≥ 0, ψ(t) = p(t)πα(τ(t))
π2−α(t) , and p(t) is is

a positive function such that

lim
t→∞

p(t)πα(τ(t))

π2−α(t)
= 0.



EJDE-2022/53 OSCILLATION FOR SECOND ORDER NONLINEAR DES 11

When β > γ, equation (3.11) is a super-linear equation. From Theorem 2.2 we
see that if

g(β, γ) := lim inf
t→∞

πβ+1(t)a1/γ(t)q(t)(1− ψ(σ(t)))β >
( β

β + 1

)β+1( β
M

)β
, (3.12)

then (3.11) is oscillatory. A direct computation shows that

g(β, γ) = lim inf
t→∞

πβ+1(t)a1/γ(t)q(t)(1− ψ(σ(t)))β = lim inf
t→∞

t1+
1
γ−

β
γ−

1
γ+δ. (3.13)

Setting δ + 1 > β
γ then gives g = ∞ and hence the inequality (3.12) holds. Note

that from [8, Theorem 3.1], one can also deduce that (3.11) is oscillatory for δ ≥ β
γ .

Now if β = γ, equation (3.11) is a half-linear differential equation. Again using
Theorem 2.2 we conclude that (3.11) is oscillatory if

g(β, β) = lim inf
t→∞

πβ+1(t)a
1
β (t)q(t)(1− ψ(σ(t)))β >

( β

β + 1

)β+1
. (3.14)

Letting δ = 0, we obtain

g(β, β) = lim inf
t→∞

λ(
β

t
1
β

)β+1t
β+1
β . (3.15)

It follows that if λ > 1
(β+1)β+1 , then (3.14) holds. We also mention here that if

β = γ and δ = 0, from [8, Theorem 3.2], one can deduce that (3.11) is oscillatory
for λ > 1/ββ . Therefore, Theorem 2.2 improves both [8, Theorems 3.1 and 3.2].

Remark 3.7. The theorems of [8] only hold for 0 < α < 1. However, Theorem 2.2
in this paper can also be applied to the case of α = 1. In addition, the conditions
of the theorem only require to find the limit.

4. Conclusion

In this article, we use Riccati transformation and some inequality techniques to
establish some new Kneser oscillation criteria for second order nonlinear differential
equations with sublinear term. Our method is different from those reported in
[5, 6, 14] and the results can be used to deal with the half-linear, the sublinear, the
linear and the Emden-Fowler neutral differential equations. Those examples given
in the last section show that our results improve some well-known results published
recently in the literature.

Acknowledgements. This research was supported by the National Natural Sci-
ence Foundation of China (No. 11501131), by the Guangdong Basic and Applied
basic Research Foundation (No. 2019A1515010955), and by the Science and Tech-
nology Project of Maoming, China (No. 2017286). The authors want to thank the
editor and the anonymous referee for their constructive comments and suggestions,
which greatly improved this this article.

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Yingzhu Wu

Department of Mathematics, Guangdong University of Petrochemical Technology,

Maoming 525000, China
Email address: wuyingzhu1978@gdupt.edu.cn



EJDE-2022/53 OSCILLATION FOR SECOND ORDER NONLINEAR DES 13

Yuanhong Yu

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing

100190, China
Email address: yu84845366@126.com

Jinsen Xiao (corresponding author)
Department of Mathematics, Guangdong University of Petrochemical Technology,

Maoming 525000, China

Email address: jinsenxiao@gdupt.edu.cn


	1. Introduction
	2. Main results
	3. Examples
	4. Conclusion
	Acknowledgements

	References