Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 70, pp. 1–12.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.70

OSCILLATION CRITERIA OF FOURTH-ORDER NONLINEAR

SEMI-NONCANONICAL NEUTRAL DIFFERENTIAL

EQUATIONS VIA A CANONICAL TRANSFORM

GANESH PURUSHOTHAMAN, KANNAN SURESH,
ERCAN TUNÇ, ETHIRAJU THANDAPANI

Abstract. In this work first we transform the semi-noncanonical fourth or-

der neutral delay differential equations into canonical type. This simplifies the
investigations of finding the relationships between the solution and its compan-

ion function which plays an important role in the oscillation theory of neutral

differential equations. Moreover, we improve these relationships based on the
monotonic properties of positive solutions. We present new conditions for the

oscillation of all solutions of the corresponding equation which improve the
oscillation results already reported in the literature. Examples are provided

to illustrate the importance of our main results.

1. Introduction

In recent years, the oscillation theory has expanded and developed greatly since
this phenomena take part in different models from real world applications, see,
e.g., the papers [7, 8, 21] dealing with biological mechanisms (for models from
mathematical biology where oscillation and/or delay actions may be formulated by
means of cross-diffusion terms). Moreover, the study of neutral functional differ-
ential equations has attracted considerable/significant attention because it arise in
many fields such as control theory, communication, mechanical engineering, biody-
namics, physics, economics and so on, see [10, 29, 30] and the references therein. In
particular, Emden–Fowler differential equations have many applications in mathe-
matical, theoretical and chemical physics; we refer the reader to the papers [18, 19]
for more details. In view of the above observations, one can see that the investiga-
tion of oscillatory and asymptotic behavior of solutions of delay and neutral type
fourth order functional differential equations has received immense interest in re-
cent times; for example, see [1, 2, 3, 4, 5, 6, 12, 14, 20, 22, 23, 24, 26, 27, 28, 31, 32]
and the references cited therein. The aim of this study is to establish new oscillation
conditions for all solutions of the neutral delay differential equation

L4z(t) + q(t)xα(σ(t)) = 0, t ≥ t0 > 0, (1.1)

2020 Mathematics Subject Classification. 34C10, 34K11, 34K40.
Key words and phrases. Oscillation; neutral differential equations; semi-noncanonical;

fourth-order equation.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted September 8, 2023. Published October 16, 2023.

1



2 G. PURUSHOTHAMAN, K. SURESH, E. TUNÇ, E. THANDAPANI EJDE-2023/70

where z(t) = x(t) + a(t)x(τ(t)), α is a ratio of odd positive integers, and L4 is an
iterated operator defined as follows:

L0z = z, Liz = pi(Li−1z))
′ for i = 1, 2, 3, and L4z = (L3z))

′.

During this study, we assume the following assumptions:

(A1) pi ∈ C(4−i)([t0,∞), (0,∞)) for i = 1, 2, 3;
(A2) a, q ∈ C([t0,∞), [0,∞)) with 0 ≤ a(t) < 1 and q does not vanish eventually;
(A3) τ ∈ C1([t0,∞),R) with τ ′(t) > 0, σ ∈ C([t0,∞),R) is nondecreasing,

τ(t) ≤ t, σ(t) ≤ t, and limt→∞ τ(t) = limt→∞ σ(t) =∞.

We define

Ωi(t) =

∫ ∞
t

1

pi(s)
ds for i = 1, 2, 3,

and introduce the classification as in [28]. The equation (1.1) is in semi-noncanonical
form if either one of the 3 conditions hold:

Ω1(t0) <∞, Ω2(t0) =∞, Ω3(t0) <∞, (1.2)

Ω1(t0) <∞, Ω2(t0) <∞, Ω3(t0) =∞, (1.3)

Ω1(t0) =∞, Ω2(t0) <∞, Ω3(t0) <∞. (1.4)

By a solution of (1.1), we mean a function x ∈ C([t∗,∞),R) for t∗ ≥ t0, which
has the property Liz ∈ C1([t0,∞),R) for i = 1, 2, 3 and sup{|x(t)| : t ≥ tx} > 0
for tx ≥ t∗ and x satisfies (1.1) on [t∗,∞). Such a solution x of (1.1) is said to be
oscillatory if it is neither eventually positive nor eventually negative. Otherwise,
it is said to be nonoscillatory. The equation itself is called oscillatory if all its
solutions oscillate.

Recently in [1, 3, 4, 5, 14, 31], the authors studied the oscillatory properties of
solutions of (1.1) in each one of following cases:

Ωi(t0) =∞, for i = 1, 2, 3, i.e., equation (1.1) is in canonical form;

Ω1(t0) = Ω2(t0) =∞, Ω3(t0) <∞;

Ω1(t0) <∞, Ω2(t0) <∞, Ω3(t0) =∞;

without changing the form of the equation. In [26, 27, 28] the authors studied
equation (1.1) when

Ωi(t0) <∞, i = 1, 2, 3, i.e., equation (1.1) is in noncanonical form,

or (1.3) or (1.4) hold, by transforming the equations into canonical form. The main
advantage of studying (1.1) in canonical form is that using famous Kiguradze’s
Lemma (see [13]) to classify the behavior of nonoscillatory solutions results in there
exisitng only two types of solutions where as six types for semi-noncanonical equa-
tions. Suppose, we keep the equation (1.1) as it is and if x is a positive solution
of (1.1), then the companion function z must satisfy six possible cases and it is
very difficult to get a relationship between z and x and this is certainly essential to
obtain oscillation criteria for the equation (1.1). Further note that if the studied
fourth order neutral differential equation is not in canonical form, then the authors
proved only that every solution is either oscillatory or tends to zero asymptotically,
see [5, 11, 17]. To overcome these difficulties first we transform (1.1) into canonical
type, which reduce the classification into two cases and from these one can easily
obtain the relation between x and z (see also the paper [9] for more interesting
details). Thus, our method not only reduces the number of classification types of



EJDE-2023/70 OSCILLATION CRITERIA FOR FOURTH-ORDER NEUTRAL EQUATIONS 3

non-oscillatory solutions but it is also very helpful in finding a relation between
z and x. Hence, the authors believe that the results obtained here form a signif-
icant contribution to the oscillation theory of fourth order functional differential
equations.

2. Main results

Throughout, and withouth further mention, we assume that (1.2) holds. For this
case, we use the notation

a3(t) = p3(t)Ω2
3(t), a2(t) =

p2(t)

Ω3(t)Ω1(t)
, a1(t) = p1(t)Ω2

1(t), y(t) =
z(t)

Ω1(t)
.

Theorem 2.1. Let ∫ ∞
t0

1

a2(t)
dt =∞. (2.1)

Then the semi-noncanonical operator L4z has the canonical form

L4z(t) =
1

Ω3(t)
(a3(t)(a2(t)(a1(t)y′(t))′)′)′. (2.2)

Proof. With a simple calculation we observe that(
p1(t)Ω2

1(t)
( z(t)

Ω1(t)

)′)′
=
(
Ω1(t)p1(t)z′(t) + z(t)

)′
= Ω1(t)(p1(t)z′(t))′.

Now, (
p3(t)Ω2

3(t)
( p2(t)

Ω3(t)Ω1(t)

(
p1(t)Ω2

1(t)
( z(t)

Ω1(t)

)′)′)′)′
=
(
p3(t)Ω2

3(t)
( p2(t)

Ω3(t)

(
p1(t)z′(t)

)′)′)′
= [p3(t)Ω3(t)(p2(t)(p1(t)z′(t))′)′ + p2(t)(p1(t)z′(t))′]′

= Ω3(t)(p3(t)(p2(t)(p1(t)z′(t))′)′)′.

Therefore,

L4z(t) =
1

Ω3(t)
(a3(t)(a2(t)(a1(t)y′(t))′)′)′.

To see that (2.2) is in canonical form, note that∫ ∞
t0

1

a3(t)
dt =

∫ ∞
t0

1

p3(t)Ω2
3(t)

dt = lim
t→∞

1

Ω3(t)
− 1

Ω3(t0)
=∞,∫ ∞

t0

1

a1(t)
dt =

∫ ∞
t0

1

p1(t)Ω2
1(t)

dt = lim
t→∞

1

Ω1(t)
− 1

Ω1(t0)
=∞,∫ ∞

t0

1

a2(t)
dt =∞

by (2.2). This completes the proof. �

From Theorem 2.1, we see that under condition (2.1), equation (1.1) can be
written in the equivalent canonical form

L4y(t) + Ω3(t)q(t)xα(σ(t)) = 0,

where L0y = y, Liy = ai(Li−1y)′ for i = 1, 2, 3, and L4y = (L3y)′.



4 G. PURUSHOTHAMAN, K. SURESH, E. TUNÇ, E. THANDAPANI EJDE-2023/70

Corollary 2.2. The semi-noncanonical equation (1.1) is oscillatory if and only if
the canonical equation

L4y(t) + Ω3(t)q(t)xα(σ(t)) = 0 (EC1)

is oscillatory.

Lemma 2.3. Assume that (2.1) holds. If x(t) is an eventually positive solution of
(EC1), then the companion function y(t) is positive and satisfies either

y(t) ∈ S1 ⇔ L1y(t) > 0, L2y(t) < 0, L3y(t) > 0, L4y(t) ≤ 0,

or
y(t) ∈ S3 ⇔ L1y(t) > 0, L2y(t) > 0, L3y(t) > 0, L4y(t) ≤ 0.

Hence the set S of all positive solutions of (EC1) has the decomposition S = S1∪S3.

For convenience, we denote:

f [0](t) = t, f [j](t) = f(f [j−1](t)) for j = 1, 2, . . . ,

Aj(t) =

∫ t

t1

1

aj(s)
ds, j = 1, 2, 3,

Q2(t) =

∫ t

t1

1

a2(s)
A3(s)ds, Q3(t) =

∫ t

t1

1

a1(s)
Q2(s)ds,

D1(t) = Ω3(t)q(t)Bα1 (σ(t);m), D2(t) = Ω3(t)q(t)Bα2 (σ(t);m),

R1(t) =
( 1

a2(t)

∫ ∞
t

1

a3(s)

∫ ∞
s

D1(v)dvds
)(∫ σ(t)

t1

1

a1(s)
ds
)α
,

R2(t) = D2(t)
(∫ σ(t)

t1

1

a1(s)

∫ s

t1

1

a2(v)

∫ v

t1

1

a3(s1)
ds1dvds

)α
,

and we assume without further mention that

a(τ [2r](t))
Ω1(τ [2r+1](t))

Ω1(τ [2r](t))
< 1

for every integer r ≥ 0 and t ≥ t1 for some t1 ≥ t0.

Lemma 2.4. Suppose that x is an eventually positive solution of (EC1). Then,
eventually,

x(t) ≥
m∑
r=0

( 2r∏
l=0

a
(
τ [l](t)

))[Ω1(τ [2r](t))y(τ [2r](t))

a(τ [2r](t))
− Ω1(τ [2r+1](t))y(τ [2r+1](t))

]
(2.3)

for each integer m ≥ 0.

Proof. From the definition of x and z, we have

x(t) = z(t)− a(t)x(τ(t))

= z(t)− a(t)z(τ(t)) + a(t)a(τ(t))x(τ [2](t))

= z(t)− a(t)z(τ(t)) + a(t)a(τ(t))z(τ [2](t))− a(t)a(τ(t))a(τ [2](t))x(τ [3](t))

and so on. Thus,

x(t) ≥
m∑
r=0

(−1)r
( r∏
l=0

a
(
τ [l](t)

))z(τ [r](t))

a(τ [r](t))



EJDE-2023/70 OSCILLATION CRITERIA FOR FOURTH-ORDER NEUTRAL EQUATIONS 5

for each odd integer m ≥ 0, or

x(t) ≥
m∑
r=0

( 2r∏
l=0

a
(
τ [l](t)

))[z(τ [2r](t))

a(τ [2r](t))
− z(τ [2r+1](t))

]
for each integer m ≥ 0. Now using z(t) = Ω1(t)y(t) we obtain the desired result. �

Lemma 2.5. Assume that x is an eventually positive solution of (1.1) and suppose
that (2.1) holds. Then

(i) if y(t) ∈ S1, then y(t)
A1(t) is decreasing for t ≥ t1 for some t1 ≥ t0;

(ii) if y(t) ∈ S3, then y(t)
Q3(t) is decreasing and L1y(t) ≥ Q2(t)L3y(t) for t ≥ t1

for some t1 ≥ t0.

Proof. Let x(t) be an eventually positive solution of (1.1). Then, x(t) is also an
eventually positive solutions of (EC1). Thus, by Lemma 2.3, the companion func-
tion y(t) is positive and satisfies either y(t) ∈ S1 or y(t) ∈ S3. The remainder of
the proof is similar to that of [6, Theorem 3.1] and so the details are omitted. �

Lemma 2.6. Assume that x is an eventually positive solution of (1.1) and suppose
(2.1) holds. If the companion function y(t) ∈ S1, then

x(t) ≥ B1(t;m)y(t), (2.4)

and if y(t) ∈ S3, then

x(t) ≥ B2(t;m)y(t), (2.5)

where

B1(t;m) =

m∑
r=0

( 2r∏
l=0

a
(
τ [l](t)

))
Ω1(τ [2r](t))

[ 1

a(τ [2r](t))
−Ω1(τ [2r+1](t))

Ω1(τ [2r](t))

]A1(τ [2r](t))

A1(t)
,

and

B2(t;m) =

m∑
r=0

( 2r∏
l=0

a
(
τ [l](t)

))
Ω1(τ [2r](t))

[ 1

a(τ [2r](t))
−Ω1(τ [2r+1](t))

Ω1(τ [2r](t))

]Q3(τ [2r](t))

Q3(t)

for all positive integer m ≥ 0.

Proof. From Lemma 2.4, we have (2.3) holds. Based on the monotonic properties
of y(t) ∈ S1, we see that y(τ [2r+1](t)) ≤ y(τ [2r](t)) for r = 0, 1, 2, . . . , is obtained.
Thus, (2.3) becomes

x(t) ≥
m∑
r=0

( 2r∏
l=0

a
(
τ [l](t)

))[Ω1(τ [2r](t))

a(τ [2r](t))
− Ω1(τ [2r+1](t))

]
y(τ [2r](t)). (2.6)

From Lemma 2.5 (i), we see that

y(τ [2r](t)) ≥
(A1(τ [2r](t))

A1(t)

)
y(t). (2.7)

Using (2.7) in (2.6), one can obtain (2.4). Again based on the monotonic properties
of y(t) ∈ S3, we see that

y(τ [2r+1](t)) ≤ y(τ [2r](t)), for r = 0, 1, 2, . . . .



6 G. PURUSHOTHAMAN, K. SURESH, E. TUNÇ, E. THANDAPANI EJDE-2023/70

Thus again (2.6) holds. Now from Lemma 2.5 (ii), we see that

y(τ [2r](t)) ≥
(Q3(τ [2r](t))

Q3(t)

)
y(t). (2.8)

Substituting (2.8) in (2.6), we obtain (2.5). The proof of lemma is complete. �

Remark 2.7. It is easy to verify that for m = 0, we have

B1(t; 0) = B2(t; 0) = Ω1(t)
(

1− a(t)
Ω1(τ(t))

Ω1(t)

)
.

Thus, the relation (2.4) and (2.5) reduce to

x(t) ≥ Ω1(t)
(

1− a(t)
Ω1(τ(t))

Ω1(t)

)
y(t).

Theorem 2.8. Let (2.1) hold. Suppose that both first-order delay differential equa-
tions

w′(t) +R1(t)wα(σ(t)) = 0, (2.9)

u′(t) +R2(t)uα(σ(t)) = 0 (2.10)

are oscillatory. Then equation (1.1) is oscillatory.

Proof. Let x(t) be an eventually positive solution of (1.1), say x(t) > 0, x(τ(t)) > 0
and x(σ(t)) > 0 for t ≥ t1 for some t1 ≥ t0. Then, x(t) is also an eventually
positive solutions of (EC1). Thus, it follows from Lemma 2.3 that either y(t) ∈ S1

or y(t) ∈ S3 for t ≥ t1. First we assume that y(t) ∈ S1. From (EC1) and (2.4), we
have

L4y(t) +D1(t)yα(σ(t)) ≤ 0. (2.11)

Since a1(t)y′(t) is decreasing, we see that

y(t) ≥
∫ t

t1

a1(s)
y′(s)

a1(s)
ds ≥ a1(t)y′(t)

∫ t

t1

1

a1(s)
ds. (2.12)

Integrating (2.11) from t to ∞, we obtain

(a2(t)(a1(t)y′(t))′)′ ≥ yα(σ(t))

a3(t)

∫ ∞
t

D1(s)ds. (2.13)

Integrating (2.13) from t to ∞, we obtain

− (a1(t)y′(t))′ ≥ yα(σ(t))

a2(t)

∫ ∞
t

1

a3(v)

∫ ∞
v

D1(s) ds dv. (2.14)

From (2.12) and (2.14), we observe that

− (a1(t)y′(t))′ ≥ R1(t)(a1(σ(t))y′(σ(t)))α. (2.15)

Letting w(t) = a1(t)y′(t) in (2.15), it follows from (2.15) that w is a positive solution
of the differential inequality

w′(t) +R1(t)wα(σ(t)) ≤ 0.

Therefore, by [25, Theorem 1], the associated delay differential equation (2.9) also
has a positive solution. This contradiction implies that S1 is empty.

Next, we shall assume that y(t) ∈ S3. From (EC1) and (2.5), we have

L4y(t) +D2(t)yα(σ(t)) ≤ 0. (2.16)



EJDE-2023/70 OSCILLATION CRITERIA FOR FOURTH-ORDER NEUTRAL EQUATIONS 7

Noting that a3(t)(a2(t)(a1(t)y′(t))′)′ is decreasing, we see that

a2(t)(a1(t)y′(t))′ ≥
∫ t

t1

1

a3(s)
a3(s)(a2(s)(a1(s)y′(s))′)′ds

≥ a3(t)(a2(t)(a1(t)y′(t))′)′
∫ t

t1

1

a3(s)
ds.

Integrating the last inequality, we obtain

y′(t) ≥ a3(t)(a2(t)(a1(t)y′(t))′)′
1

a1(t)

∫ t

t1

1

a2(v)

∫ v

t1

1

a3(s)
ds dv.

Integrating once more, we see that u(t) = a3(t)(a2(t)(a1(t)y′(t))′)′ satisfies

y(t) ≥ u(t)

∫ t

t1

1

a1(s)

∫ s

t1

1

a2(v)

∫ v

t1

1

a3(s1)
ds1dvds.

Using the last estimate in (2.16), we see that u is a positive solution of the differ-
ential inequality

u′(t) +R2(t)uα(σ(t) ≤ 0,

which, in view of Philos [25, Theorem 1], implies that the corresponding differential
equation (2.10) also has a positive solution. This is again a contradiction and so
S3 is empty. The proof of the theorem is complete. �

Applying suitable criteria for the oscillation of (2.9) and (2.10) with α ∈ (0, 1],
we obtain immediately the following conditions for the oscillation of (1.1). The first
one is due to [16, Theorem 1], whereas the second one is due to [15, Theorem 2].

Corollary 2.9. Let α = 1 and let (2.1) hold. If

lim inf
t→∞

∫ t

σ(t)

H(s)ds >
1

e
, (2.17)

where H(t) = min{R1(t), R2(t)}, then (1.1) is oscillatory.

Corollary 2.10. Let (2.1) hold and α ∈ (0, 1). If∫ ∞
t0

H(t)dt =∞, (2.18)

then (1.1) is oscillatory.

Lemma 2.11. Let x(t) be an eventually positive solution of (EC1). Then

(i) if y(t) ∈ S1, then yα−1(t) ≥ φ1(t), where

φ1(t) =


1, if α = 1,

ε1, if α > 1,

ε2A
α−1
1 (t), if α < 1,

and ε1 and ε2 are positive constants for all t ≥ t1 ≥ t0;
(ii) if y(t) ∈ S3, then yα−1(t) ≥ φ2(t), where φ2(t) is given by

φ2(t) =


1, if α = 1,

ε3, if α > 1,

ε4Q
α−1
3 (t), if α < 1,

and ε3 and ε4 are positive constants for all t ≥ t1 ≥ t0.



8 G. PURUSHOTHAMAN, K. SURESH, E. TUNÇ, E. THANDAPANI EJDE-2023/70

The proof of the above lemma is similar to taht of [27, Lemma 2.10] and it is
are omitted here. By using Riccati transformation method we obtain the following
result.

Theorem 2.12. Let (2.1) hold. If there are positive functions ρ1, ρ2 ∈ C1([t0,∞),R)
such that

lim sup
t→∞

∫ t

t1

(ρ1(v)

a2(v)

∫ ∞
v

1

a3(s)

∫ ∞
s

F1(s1)ds1ds−
a1(v)(ρ′1(v))2

4ρ1(v)

)
dv =∞, (2.19)

lim sup
t→∞

∫ t

t1

(ρ2(s)F2(s)− a1(s)(ρ′2(s))2

4ρ2(s)Q2(s)
))ds =∞, (2.20)

where

F1(t) =
D1(t)Aα1 (σ(t))

Aα1 (t)
φ1(t) and F2(t) =

D2(t)Qα3 (σ(t))

Qα3 (t)
φ2(t)

for all t ≥ t1 ≥ t0, then equation (1.1) is oscillatory.

Proof. Let x(t) be an eventually positive solution of (1.1), say x(t) > 0, x(τ(t)) > 0
and x(σ(t)) > 0 for t ≥ t1 for some t1 ≥ t0. Then, x(t) is also an eventually
positive solutions of (EC1). Thus, it follows from Lemma 2.3 that either y(t) ∈ S1

or y(t) ∈ S3 for t ≥ t1.
First we assume that y(t) ∈ S1. In this case, from Lemma 2.5 (i) and Lemma

2.11 (i), we observe that

yα(σ(t)) ≥ Aα1 (σ(t))

Aα1 (t)
φ1(t)y(t).

Using the above estimate in (2.11), we see that

L4y(t) + F1(t)y(t) ≤ 0.

An integration of the latter expression from t to ∞ yields

L3y(t) ≥
∫ ∞
t

F1(s)y(s)ds ≥ y(t)

∫ ∞
t

F1(s)ds. (2.21)

Now integrating (2.21) from t to ∞, we have

L2y(t) +
(∫ ∞

t

1

a3(v)

∫ ∞
v

F1(s) ds dv
)
y(t) ≤ 0. (2.22)

Let us define

µ1(t) = ρ1(t)
L1y(t)

y(t)
, t ≥ t1. (2.23)

From (2.22) and (2.23), we observe that

µ′1(t) = ρ′1(t)
L1y(t)

y(t)
+
ρ1(t)

a2(t)

L2y(t)

y(t)
− ρ1(t)y′(t)L1y(t)

y2(t)

≤ −ρ1(t)

a2(t)

∫ ∞
t

1

a3(v)

∫ ∞
v

F1(s) ds dv +
a1(t)(ρ′1(t))2

4ρ1(t)
.

(2.24)

Integrating (2.24) from t1 to t yields∫ t

t1

(ρ1(v)

a2(v)

∫ ∞
v

1

a3(s)

∫ ∞
s

F1(s1)ds1ds−
a1(v)(ρ′1(v))2

4ρ1(v)

)
dv ≤ µ1(t1),

which contradicts (2.19) as t→∞.



EJDE-2023/70 OSCILLATION CRITERIA FOR FOURTH-ORDER NEUTRAL EQUATIONS 9

Next assume that y(t) ∈ S3. Then from Lemma 2.5 (ii) and Lemma 2.11 (ii), we
see that

yα(σ(t)) ≥ Qα3 (σ(t))

Qα3 (t)
φ2(t)y(t), (2.25)

L1y(t) ≥ Q2(t)L3y(t). (2.26)

Using (2.25) in (2.16), we obtain

L4y(t) + F2(t)y(t) ≤ 0. (2.27)

We define

µ2(t) = ρ2(t)
L3y(t)

y(t)
, t ≥ t2. (2.28)

From (2.27)-(2.28), we obtain, for t ≥ t2,

µ′2(t) = ρ′2(t)
L3y(t)

y(t)
+
ρ2(t)L4y(t)

y(t)
− ρ2(t)L3y(t)y′(t)

y2(t)

≤ −ρ2(t)F2(t) +
ρ′2(t)

ρ2(t)
µ2(t)− Q2(t)µ2

2(t)

ρ2(t)a1(t)

≤ −ρ2(t)F2(t) +
a1(t)(ρ′2(t))2

4ρ2(t)Q2(t)
.

(2.29)

Integrating (2.29) from t2 to t yields∫ t

t2

(ρ2(s)F2(s)− a1(s)(ρ′2(s))2

4ρ2(s)Q2(s)
))ds ≤ µ2(t2),

which contradicts (2.20) as t→∞. The proof is complete. �

Letting ρ1(t) = A1(t), ρ2(t) = Q3(t) and α = 1, one can immediately get the
following result.

Corollary 2.13. Let α = 1. If

lim sup
t→∞

∫ t

t1

(A1(v)

a2(v)

∫ ∞
v

1

a3(s)

∫ ∞
s

D1(s1)
A1(σ(s1))

A1(s1)
ds1ds−

1

4a1(v)A1(v)

)
dv =∞

and

lim sup
t→∞

∫ t

t1

(
Q3(σ(s))D2(s)− Q2(s)

4a1(s)Q3(s)

)
ds =∞ (2.30)

for all t1 ≥ t0, then (1.1) is oscillatory.

3. Examples

In this section, we provide two examples to show the importance of our results.

Example 3.1. Consider the semi-noncanonical neutral delay differential equation(
t2
( 1

t2
(t2z′(t))′

)′)′
+
h

t2
x(λt) = 0, t ≥ 1, (3.1)

where z(t) = x(t)+ 1
4x( t2 ), h > 0 is a constant, and λ ∈ (0, 1). A simple computation

shows that

Ω3(t) = Ω1(t) =
1

t
, a1(t) = a2(t) = a3(t) = 1 and y(t) = t

(
x(t) +

1

4
x(
t

2
)
)
.



10 G. PURUSHOTHAMAN, K. SURESH, E. TUNÇ, E. THANDAPANI EJDE-2023/70

The transformed equation is

y(4)(t) +
h

t3
x(λt) = 0, t ≥ 1,

which is clearly in canonical form. Now we see that, for m = 0,

B1(t; 0) = B2(t; 0) =
1

2t
,

D1(t) = D2(t) =
h

2λt4
, R1(t) ≈ h

12t
,

R2(t) ≈ hλ2

12t
, H(t) =

hλ2

12t
.

Clearly (2.1) holds. Condition (2.17) becomes

lim inf
t→∞

∫ t

λt

hλ2

12s
ds =

hλ2

12
ln

1

λ
>

1

e
.

Hence by Corollary 2.9, equation (3.1) is oscillatory if h > 12
λ2e ln 1

λ

.

Note that using Corollary 2.13, we see that equation (3.1) is oscillatory if h >
9/λ2 and therefore Corollary 2.9 gives better condition than Corollary 2.13.

Example 3.2. Consider the semi-noncanonical nonlinear neutral differential equa-
tion (

t2
( 1

t2
(t2z′(t))′

)′)′
+ htx3(λt) = 0, t ≥ 1, (3.2)

where z(t) = x(t) + 1
4x( t2 ), h > 0 is a constant, and λ ∈ (0, 1). The transformed

equation is

y(4)(t) + hx3(λt) = 0

and it is clearly of canonical type. A simple calculation shows that

B1(t; 0) = B2(t; 0) =
1

2t
, D1(t) = D2(t) =

h

8λ3t3
,

A1(t) ≈ t, A2(t) ≈ t, A3(t) ≈ t, Q2(t) ≈ t2

2
,

Q3(t) ≈ t3

6
, φ1(t) = ε1, φ2(t) = ε3, F1(t) ≈ hε1

8t3
, F2(t) ≈ hλ6ε3

8t3
.

Choose ρ1(t) = 1 and ρ2(t) = t2, we see that conditions (2.19) and (2.20) become

lim sup
t→∞

∫ t

1

hε1
16s

ds = lim
t→∞

hε1
16

ln t =∞,

lim sup
t→∞

∫ t

1

(hλ6ε3
8s

− 2

s2

)
ds =∞.

That is, conditions (2.19) and (2.20) are satisfied. Hence, by Theorem 2.12, equa-
tion (3.2) is oscillatory.

Remark 3.3. Note that none of the results reported in the literature [4, 5, 6, 14,
19, 22, 23, 27] applied to (3.1) and (3.2) to get any conclusion.



EJDE-2023/70 OSCILLATION CRITERIA FOR FOURTH-ORDER NEUTRAL EQUATIONS 11

4. Conclusions

In this paper, by transforming the semi-noncanonical equation to canonical type
equation, we establish oscillation criteria using comparison and Riccati transfor-
mation methods. The oscillation criteria presented in this paper are new in the
sense that it gives all solutions are oscillatory instead of every solution is either
oscillatory or tends to zero asymptotically.

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Ganesh Purushothaman

Department of Mathematics, St. Joseph’s College of Engineering, Chennai-600119, In-
dia

Email address: gpmanphd@gmail.com

Kannan Suresh

Department of Mathematics, St. Joseph’s College of Engineering, Chennai-600119, In-

dia
Email address: dhivasuresh@gmail.com

Ercan Tunç
Department of Mathematics, Faculty of Arts and Sciences, Tokat Gaziosmanpaşa Uni-

versity, 60240, Tokat, Turkey

Email address: ercantunc72@yahoo.com

Ethiraju Thandapani

Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chen-
nai - 600 005, India

Email address: ethandapani@yahoo.co.in


	1. Introduction
	2. Main results
	3. Examples
	4. Conclusions
	References