International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 1 (2017), 69-76
http://www.etamaths.com

SUFFICIENT CONDITIONS FOR THE OSCILLATION OF ODD-ORDER

NEUTRAL DYNAMIC EQUATIONS

BAŞAK KARPUZ∗

Abstract. In this paper, we study oscillation and asymptotic behaviour of odd-order delay dynamic

equations. We first state an oscillation test for odd-order nonneutral equations, then by comparison

we provide sufficient conditions for all solutions of neutral equations to be oscillatory or tend to zero
depending on two main ranges of the neutral coefficient.

1. Introduction

In this paper, we will study the oscillation of solutions to the higher-order delay dynamic equations
of the form [

x(t) + A(t)x
(
α(t)

)]∆n
+ B(t)x

(
β(t)

)
= 0 for t ∈ [t0,∞)T, (1.1)

where n ∈ N, T is a time scale unbounded above, t0 ∈ T, A ∈ Crd([t0,∞)T,R), B ∈ Crd([t0,∞)T, [0,∞)R),
and α,β ∈ Crd([t0,∞)T,T) are unbounded nondecreasing functions such that α(t),β(t) ≤ t for
all t ∈ [t0,∞)T. We also assume when necessary that α ∈ C([t0,∞)T,T) has an inverse α−1 ∈
C([α(t0),∞)T, [t0,∞)T). We will confine our attention to the following ranges of the coefficient A.
(R1) A ∈ Crd([t0,∞)T, [0, 1]R) with lim supt→∞A(t) < 1.
(R2) A ∈ Crd([t0,∞)T, [−1, 0]R) with lim inft→∞A(t) > −1.

The oscillation problem for dynamic equations on time scales has attracted a lot of attention im-
mediately after the discovery of time scale calculus. Although there are several such works in the
literature, the majority is focuses on second-order equations. An important reason for this is due to
lack of technical inequalities which combines higher-order derivatives with lower-order ones. In this
paper, we will generalize and improve the technique by Das [7] employed for differential equations. For
some works on higher-order dynamic equations we refer the readers to [8–14, 17, 18, 20, 22, 23]. We will
be giving comparison tests for the oscillation of higher-order delay dynamic equations with first-order
delay dynamic equations. As we will be making comparison with first-order delay dynamic equations,
we find useful to cite the following references [2, 3, 6, 16, 19, 24], where the authors study oscillation of
all solutions of first-order delay dynamic equations.

To give an exact definition of a solution of delay dynamic equation (1.1), we need to define t−1 :=
min{α(t0),β(t0)}.

Definition 1.1 (Solution). A function x : [t−1,∞)T → R, which is rd-continuous on [t−1, t0]T and
x+A·x◦α is n-times rd-continuously ∆-differentiable on [t0,∞), is called a solution of (1.1) provided
that it satisfies the functional delay equation (1.1) identically on [t0,∞).

It can be shown as in [15] that (1.1) admits a unique solution, which exists on the entire interval
[t−1,∞)T, when an initial function ϕ : [t−1, t0]T → R, which is n-times rd-continuously ∆-differentiable,
is provided. More precisely, we mean in the equation that x∆

j

(t) = ϕ∆
j

(t) for t ∈ [t−1, t0]T and
j = 0, 1, · · · ,n.

Definition 1.2 (Oscillation). A solution x of (1.1) is called nonoscillatory if there exists s ∈ [t0,∞)T
such that x is either positive or negative on [s,∞). Otherwise, the solution is said to oscillate (or is
called oscillatory).

Received 12th January, 2017; accepted 22nd March, 2017; published 2nd May, 2017.

2010 Mathematics Subject Classification. Primary: 34N05; Secondary: 34K11, 39A12, 39A21.
Key words and phrases. oscillation; odd-order; delay dynamic equations.

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.

69



70 KARPUZ

This paper is constructed in the following setting. § 2 includes some fundamental results on time
scale polynomials and nonoscillatory functions (§ 2.1). In this section (§ 2.2), we also quote some
results from the recent paper [17]. In § 3.1, we provide some comparison theorems on the qualitative
behaviour of higher-order delay dynamic equations when the neutral term is absent (i.e., A(t) ≡ 0 for
t ∈ [t0,∞)T), and in § 3.2, we extend these results to higher-order neutral delay dynamic equations
together with some illustrative general examples.

2. Auxiliary lemmas

In this section, we will form the background for the proof of our main result.

2.1. Time scales. This section is dedicated to some general results on time scales.
In the sequel, we introduce the definition of the generalized polynomials on time scales (see [1,

Lemma 5] and/or [4, § 1.6]) hk ∈ C(T×T,R) as follows:

hk(t,s) :=




1, k = 0∫ t
s

hk−1(η,s)∆η, k ∈ N
for s,t ∈ T. (2.1)

Note that, for all s,t ∈ T and all k ∈ N0, the function hk satisfies

h∆1k (t,s) =

{
0, k = 0

hk−1(t,s), k ∈ N
for s,t ∈ T. (2.2)

In particular, for T = Z, we have hk(t,s) = (t− s)(k)/k! for all s,t ∈ Z and k ∈ N0, where (·) is the
usual factorial function, and for T = R, we have hk(t,s) = (t−s)k/k! for all s,t ∈ R and k ∈ N0.

Property 2.1 ([14, Property 1]). By using induction and (2.1), it is easy to see for all k ∈ N0 that
hk(·,s) ≥ 0 on [s,∞)T and (−1)khk(·,s) ≥ 0 on (−∞,s]T. In view of (2.2), for all k ∈ N, hk(·,s) is
increasing on [s,∞)T, and (−1)khk(·,s) is decreasing on (−∞,s]T.

Below, we give two lemmas related to the generalized polynomials on time scales.

Lemma 2.1 (Taylor’s formula [4, Theorem 1.113]). If n ∈ N, s ∈ T and f ∈ Cnrd(T,R), then

f(t) =

n−1∑
k=0

hk(t,s)f
∆k (s) +

∫ t
s

hn−1
(
t,σ(η)

)
f∆

n

(η)∆η for t ∈ T.

Lemma 2.2 ([17, Lemma 2]). If k ∈ N, ` ∈ N0 and s ∈ T, then

hk+`(t,s) =

∫ t
s

hk−1
(
t,σ(η)

)
h`(η,s)∆η for t ∈ T.

As an immediate consequence of Lemma 2.2, we can give the following alternative definition of the
generalized polynomials:

hk(t,s) :=




1, k = 0∫ t
s

hk−1
(
t,σ(η)

)
∆η, k ∈ N

for s,t ∈ T. (2.3)

The following is the main tool for studying qualitative properties of higher-order dynamic equations.

Lemma 2.3 (Kiguradze’s lemma [1, Theorem 5]). Let sup T = ∞, n ∈ N and f ∈ Cnrd([t0,∞)T,R
+
0 ).

Suppose that either f∆
n

≥ 0(6≡ 0) or f∆
n

≤ 0( 6≡ 0) on [t0,∞)T. Then, there exist s ∈ [t0,∞)T and
m ∈ [0,n)Z such that (−1)n−mf∆

n

(t) ≥ 0 for all t ∈ [s,∞)T. Moreover, the following assertions hold.
(i) f∆

k

(t) > 0 holds for all t ∈ [s,∞)T and all k ∈ [0,m)Z.
(ii) (−1)m+kf∆

k

(t) > 0 holds for all t ∈ [s,∞)T and all k ∈ [m,n)Z.

A basic result on higher-order derivatives of a function is quoted below.

Lemma 2.4 ( [1, Lemma 7]). If sup T = ∞, n ∈ N and f ∈ Cnrd([t0,∞)T,R), then the following
conditions are true.

(i) lim inft→∞f
∆n (t) > 0 implies limt→∞f

∆k (t) = ∞ for all k ∈ [0,n)Z.



OSCILLATION OF ODD-ORDER NEUTRAL DYNAMIC EQUATIONS 71

(ii) lim supt→∞f
∆n (t) < 0 implies limt→∞f

∆k (t) = −∞ for all k ∈ [0,n)Z.

The following can be easily obtained from the previous result.

Corollary 2.1 ([10, Corollary 2.10]). If sup T = ∞, n ∈ N and f ∈ Cnrd([t0,∞)T,R
+
0 ), then

lim
t→∞

f∆
k

(t) = 0, ∀k ∈ (m,n)Z,

where m ∈ [0,n)Z is the key number in Kiguradze’s lemma.

The following result is the key tool of this paper.

Lemma 2.5 (Cf. [7, Lemma 1]). Assume that sup T = ∞, n ∈ N be odd and f ∈ Cnrd([t0,∞)T,R
+
0 )

with f∆
n

≤ 0( 6≡ 0) on [t0,∞)T. If Kiguradze’s lemma holds with m ∈ [0,n)Z, then

f(r) ≥ hm(r,s)hn−m−1(r,t)f∆
n−1

(t) for all t ∈ [r,∞)T, (2.4)

where r ∈ [s,∞)T.

Proof. By Lemma 2.3 (i), r ∈ [s,∞)T implies f∆
j

(r) > 0 for all j ∈ [0,m)Z and f∆
m+1

(r) ≤ 0, where
m ∈ [0,n)Z is even. Using Property 2.1, Lemma 2.2 and Taylor’s formula, we get

f(r) =

m−1∑
k=0

hk(r,s)f
∆k (s) +

∫ r
s

hm−1
(
r,σ(η)

)
f∆

m

(η)∆η

≥
(∫ r

s

hm−1
(
r,σ(η)

)
∆η

)
f∆

m

(t) = hm(r,s)f
∆m (r) (2.5)

for all r ∈ [s,∞)T. By Lemma 2.3 (i), t ∈ [r,∞)T implies (−1)m+kf∆
k

(t) > 0 for all k ∈ [m,n)Z and
f∆

n

(t) ≤ 0. Using Property 2.1, (2.3) and Taylor’s formula, we get

f∆
m

(r) =

n−m−1∑
k=0

hk(r,t)f
∆m+k (t) +

∫ r
t

hn−m−2
(
r,σ(η)

)
f∆

n−1
(η)∆η

=

n−m−1∑
k=0

(−1)khk(r,t)(−1)m+kf∆
m+k

(t) +

∫ t
r

(−1)n−m−2hn−m−2
(
r,σ(η)

)
f∆

n−1
(η)∆η

≥
(∫ t

r

(−1)n−m−2hn−m−2
(
r,σ(η)

)
∆η

)
f∆

n−1
(t)

=

(∫ r
t

hn−m−2
(
r,σ(η)

)
∆η

)
f∆

n−1
(t) = hn−m−1(r,t)f

∆n−1 (t) (2.6)

for all t ∈ [r,∞)T. Using (2.5) and (2.6), we arrive at (2.4), which completes the proof. �

2.2. Recent results. Below, we quote two fundamental results from [17], which will be required in
the sequel.

Theorem 2.2 ( [17, Theorem 2 (ii)]). Assume that (R1) holds and n ∈ N is odd. If (1.1) has a
nonoscillatory solution x with lim supt→∞ |x(t)| > 0, then so does

x∆
n

(t) +
[
1 −A

(
α(t)

)]
B(t)x

(
β(t)

)
= 0 for t ∈ [t0,∞)T.

Theorem 2.3 ( [17, Theorem 3]). Assume that (R2) holds and n ∈ N. If (1.1) has a nonoscillatory
solution x with lim supt→∞ |x(t)| > 0, then so does

x∆
n

(t) + B(t)x
(
β(t)

)
= 0 for t ∈ [t0,∞)T.

3. Main results

Now, we can study oscillation and asymptotic behaviour of higher-order delay dynamic equations.



72 KARPUZ

3.1. Nonneutral equations. In this section, we will focus on the higher-order delay dynamic equation

x∆
n

(t) + B(t)x
(
β(t)

)
= 0 for t ∈ [t0,∞)T. (3.1)

Our first comparison result is the following, which presents results on oscillation.

Theorem 3.1. Assume that n ∈ N is odd with n ≥ 3, and there exist two functions δ,γ ∈ Crd([t0,∞)T,T)
satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)T such that

x∆(t) + B(t) min
k∈[0,n)Z
k=even

{
hk
(
β(t),δ(t)

)
hn−k−1

(
β(t),γ(t)

)}
x
(
γ(t)

)
= 0 for t ∈ [t0,∞)T (3.2)

is oscillatory. Then, every solution of (3.1) is oscillatory.

Proof. Assume the contrary that x is an eventually positive solution of (3.1). Then, there exists
t1 ∈ [t0,∞)T such that x(t), x

(
β(t)

)
> 0 for all t ∈ [t1,∞)T. By Kiguradze’s lemma, there exist

t2 ∈ [t1,∞)T and an even integer m ∈ [0,n)Z such that for all t ∈ [t2,∞)T, we have x∆
k

(t) > 0 for all

k ∈ [0,m)Z and (−1)m+kx∆
k

(t) > 0 for all k ∈ [m,n)Z. In particular, we have x∆
n−1

> 0 on [t2,∞)T.
It follows from Lemma 2.5 that

x
(
β(t)

)
≥ hm

(
β(t),δ(t)

)
hn−m−1

(
β(t),γ(t)

)
x∆

n−1(
γ(t)

)
for all t ∈ [t3,∞)T, (3.3)

where t3 ∈ [t2,∞)T satisfies δ(t) ≥ t2 for all t ∈ [t3,∞)T. Substituting (3.3) into (3.2), we see that
x∆

n−1
is an eventually positive solution of

y∆(t) + B(t)hm
(
β(t),δ(t)

)
hn−m−1

(
β(t),γ(t)

)}
y
(
γ(t)

)
≤ 0 for all t ∈ [t3,∞)T

or

y∆(t) + B(t) min
k∈[0,n)Z
k=even

{
hk
(
β(t),δ(t)

)
hn−k−1

(
β(t),γ(t)

)}
y
(
γ(t)

)
≤ 0 for all t ∈ [t3,∞)T. (3.4)

By [5, Theorem 3.1 and Corollary 4.2], (3.2) admits an eventually positive solution too. This is a
contradiction and the proof is complete. �

Depending on the oscillation test to be applied for the first-order delay equation, the functions δ
and γ can be chosen appropriately as it is illustrated by an example below. Before we proceed, let
us recall a well-known result due to Myškis [21], which ensures that the first-order delay differential
equation

x′(t) + B(t)x
(
β(t)

)
= 0 for t ∈ [t0,∞)R

is oscillatory if

lim sup
t→∞

(
t−β(t)

)
< ∞ and lim inf

t→∞

(
t−β(t)

)
lim inf
t→∞

B(t) >
1

e
.

Now, we can give a simple example to explain how to choose appropriately the functions δ and γ.

Example 3.1. Let T = R, and consider the following odd-order delay differential equation

x(n)(t) + B(t)x(t−β0) = 0 for t ∈ [t0,∞)R, (3.5)

where n ≥ 3 is an odd integer, β0 ∈ R+ and B ∈ C([t0,∞)R,R+0 ). Choosing δ(t) = t − δ0 and
γ(t) = t−γ0, where δ0 ≥ β0 ≥ γ0, we get

min
k∈[0,n)Z
k=even

{(
(t−β0) − (t−δ0)

)k
k!

(
(t−β0) − (t−γ0)

)n−k−1
(n−k − 1)!

}
= min
k∈[0,n)Z
k=even

{
(δ0 −β0)k(β0 −γ0)n−k−1

k!(n−k − 1)!

}

=

(
min{δ0 −β0,β0 −γ0}

)n−1
(n− 1)!

.

The last expression is maximized if δ0 − β0 = β0 − γ0. So, by letting δ0 = β0 + α and γ0 = β0 − α,
where β0 ≥ α ≥ 0, we obtain the dynamic equation

x′(t) +
B(t)αn−1

(n− 1)!
x
(
t− (β0 −α)

)
= 0 for t ∈ [t0,∞)T. (3.6)



OSCILLATION OF ODD-ORDER NEUTRAL DYNAMIC EQUATIONS 73

Due to the result by Myškis (quoted above), every solution of ( (3.6) and hence) (3.5) is oscillatory if

lim inf
t→∞

B(t)αn−1(β0 −α)
(n− 1)!

>
1

e
. (3.7)

Now, define the function

f(λ) = λn−1(β0 −λ) for β0 ≥ λ ≥ 0.
We compute that f′(λ) = λn−2

(
β0(n− 1) −nλ

)
. Then,

f′
(n− 1

n
β0

)
= 0 and f′′

(n− 1
n

β0

)
= −n

(
n− 1
n

β0

)n−2
< 0,

which yields

max
0≤λ≤β0

{
f(λ)

}
= f

(n− 1
n

β0

)
=

(n− 1)n−1

nn
βn0 .

Therefore, the oscillation condition (3.7) is optimized as

lim inf
t→∞

B(t) >
n!

eβn0

( n
n− 1

)n−1
.

3.2. Neutral equations. In the previous section, we have stated some oscillation conditions for
nonneutral equations, however, for neutral equations this phrase replaces with the so-called “almost
oscillation”, i.e., every solution oscillates or tends to zero at infinity. Our main tools in this section
will be the comparison tests Theorem 2.2 and Theorem 2.3, and the oscillation test Theorem 3.1.

Our first result for neutral equations investigates (1.1) when the neutral coefficient is in the range
(R1).

Theorem 3.2. Assume that n ∈ N is odd with n ≥ 3 and (R1) holds. Further, assume that there exist
two functions δ,γ ∈ Crd([t0,∞)T,T) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)T such that

x∆(t) + [1 −A
(
β(t)

)
]B(t) min

k∈[0,n)Z
k=even

{
hk
(
β(t),δ(t)

)
hn−k−1

(
β(t),γ(t)

)}
x
(
γ(t)

)
= 0 for [t0,∞)T (3.8)

is oscillatory. Then every solution of (1.1) is oscillatory or tends to zero asymptotically.

Proof. The proof follows from Theorem 2.2 and Theorem 3.1. �

As an immediate consequence of the theorem above, we can give the following corollary by combining
Theorem 3.2 by [2, Theorem 1] and [19, Theorem 2].

Corollary 3.1. Assume that n ∈ N is odd with n ≥ 3 and (R1) holds. If there exist two functions
δ,γ ∈ Crd([t0,∞)T,T) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)T such that

lim inf
t→∞

inf
−λϕ∈R+([γ(t),t)T)

λ>0

{
1

λe−λϕ
(
t,γ(t)

)} > 1
or

lim inf
t→∞

∫ t
γ(t)

ϕ(η)∆η > M and lim sup
t→∞

∫ σ(t)
γ(t)

ϕ(η)∆η > 1 −
(

1 −
√

1 −M
)2
,

where

ϕ(t) := [1 −A
(
β(t)

)
]B(t) min

k∈[0,n)Z
k=even

{
hk
(
β(t),δ(t)

)
hn−k−1

(
β(t),γ(t)

)}
for t ∈ [t0,∞)T,

then every solution of (1.1) oscillates or tends to zero asymptotically.

Our next result treats (1.1) when the neutral coefficient is in the range (R2).

Theorem 3.3. Assume that n ∈ N is odd with n ≥ 3 and (R2) holds. Further, assume that there exist
two functions δ,γ ∈ Crd([t0,∞)T,T) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)T such that
(3.2) is oscillatory. Then, every solution of (1.1) is oscillatory or tends to zero asymptotically.

Proof. The proof follows from Theorem 2.3 and Theorem 3.1. �



74 KARPUZ

Corollary 3.2. Assume that n ∈ N is odd with n ≥ 3 and (R2) holds. If there exist two functions
δ,γ ∈ Crd([t0,∞)T,T) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)T such that

lim inf
t→∞

inf
−λψ∈R+([γ(t),t)T)

λ>0

{
1

λe−λψ
(
t,γ(t)

)} > 1
or

lim inf
t→∞

∫ t
γ(t)

ψ(η)∆η > M and lim sup
t→∞

∫ σ(t)
γ(t)

ψ(η)∆η > 1 −
(

1 −
√

1 −M
)2
,

where

ψ(t) := B(t) min
k∈[0,n)Z
k=even

{
hk
(
β(t),δ(t)

)
hn−k−1

(
β(t),γ(t)

)}
for t ∈ [t0,∞)T,

then every solution of (1.1) oscillates or tends to zero asymptotically.

Consider now the following additional assumptions∫ ∞
t0

B(η)∆η = ∞ (3.9)

and

lim sup
t→∞

A(t) < 0 (3.10)

under which we will state an oscillation test for (1.1).

Theorem 3.4. Assume that n ∈ N is odd with n ≥ 3, (3.9) and (R2) hold with (3.10). Further,
assume that there exist k0 ∈ N and a function γ ∈ Crd([t0,∞)T,T) satisfying β(t) ≤ γ(t) ≤ t for all
t ∈ [t0,∞)T such that

x∆(t) +

k0−1∑
`=0

[
`−1∏
ν=0

A
(
αν(t)

)]
B(t)hn−1

(
β(t),γ(t)

)
x
(
γ(t)

)
= 0 for t ∈ [t0,∞)T (3.11)

is oscillatory. Then, every solution of (1.1) is oscillatory.

Proof. Assume the contrary that x is an eventually positive solution of (1.1). Then, there exists
t1 ∈ [t0,∞)T such that x(t), x

(
α(t)

)
, x
(
β(t)

)
> 0 for all t ∈ [t1,∞)T. Set

y(t) := x(t) + A(t)x
(
α(t)

)
for t ∈ [t1,∞)T. (3.12)

Therefore, we have

y∆
n

(t) + B(t)x
(
β(t)

)
= 0 for t ∈ [t1,∞)T (3.13)

showing that for j ∈ [0,n)Z the functions y∆
j

are monotonic on [t2,∞)T for some t2 ∈ [t1,∞)T.
(C1) Let y > 0 on [t2,∞)T. By Kiguradze’s lemma, we learn that there exist t2 ∈ [t1,∞)T and

an even integer m ∈ [0,n)Z such that t ∈ [t2,∞)T implies y∆
k

(t) > 0 for all k ∈ [0,m)Z and
(−1)m+ky∆

k

(t) > 0 for all k ∈ [m,n)Z.
(a) Let m ∈ [2,n)Z, i.e., y∆ > 0 on [t2,∞)T. Therefore, limt→∞y(t) > 0. We also have x ≥ y

on [t2,∞)T, i.e., lim inft→∞x(t) > 0. This implies by (3.9) that limt→∞y∆
n−1

(t) = −∞,
which yields limt→∞y(t) = −∞.

(b) Let m = 0, i.e., y∆ < 0 on [t2,∞)T. Then, by recursively substituting x into (3.12), we get
for k ∈ N and all t ∈ [t3,∞)T that

x(t) =

k−1∑
`=0

[
`−1∏
ν=0

A
(
αν(t)

)]
y
(
α`(t)

)
+

[
k−1∏
ν=0

A
(
αν(t)

)]
x
(
αk(t)

)
≥
k−1∑
`=0

[
`−1∏
ν=0

A
(
αν(t)

)]
y(t),

where αk(t) ≥ t2 for all t ∈ [t3,∞)T. The rest of the proof follows from the proofs of [17,
Theorem 3] and Theorem 3.1.



OSCILLATION OF ODD-ORDER NEUTRAL DYNAMIC EQUATIONS 75

(C2) Let y < 0 on [t2,∞)T. In this case, we see that x(t) ≤ A(t)x
(
α(t)

)
for all t ∈ [t2,∞)T, which

implies boundedness of the function x. Therefore, y is bounded too. Applying Kiguradze’s lemma
for the function (−y), we learn that there exist t2 ∈ [t1,∞)T and an odd integer m ∈ [0,n)Z
such that t ∈ [t2,∞)T implies y∆

k

(t) < 0 for all k ∈ [0,m)Z and (−1)m+ky∆
k

(t) < 0 for all
k ∈ [m,n)Z. In particular, we have y∆ < 0 on [t2,∞)T, i.e., limt→∞y(t) < 0. Then, we obtain

y(t) ≥−A(t)x
(
α(t)

)
for all t ∈ [t2,∞)T,

which yields

x(t) ≥−
y
(
α−1(t)

)
A
(
α−1(t)

) for all t ∈ [t2,∞)T.
Thus, we have lim inft→∞x(t) > 0 by (R2). Proceeding as in the case (C1a), we obtain
limt→∞y(t) = −∞, which contradicts the boundedness of y.

Thus, the proof is complete. �

Corollary 3.3. Assume that n ∈ N is odd with n ≥ 3, (3.9) and (R2) hold. If there exist k0 ∈ N and
a function γ ∈ Crd([t0,∞)T,T) satisfying β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)T such that

lim inf
t→∞

inf
−λψk0∈R

+([γ(t),t)T)
λ>0

{
1

λe−λψk0
(
t,γ(t)

)} > 1
or

lim inf
t→∞

∫ t
γ(t)

ψk0 (η)∆η > M and lim sup
t→∞

∫ σ(t)
γ(t)

ψk0 (η)∆η > 1 −
(

1 −
√

1 −M
)2
,

where

ψk(t) :=

k−1∑
`=0

[
`−1∏
ν=0

A
(
αν(t)

)]
B(t)hn−1

(
β(t),γ(t)

)
for t ∈ [t0,∞)T and k ∈ N,

then every solution of (1.1) oscillates.

Remark 3.1. If A is a constant function, i.e., A(t) ≡ −a0, where a0 ∈ (0, 1)R, then ψk0 in Corol-
lary 3.3 can be replaced by

B(t)

1 −a0
hn−1

(
β(t),γ(t)

)
for t ∈ [t0,∞)T.

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Dokuz Eylül University, Tınaztepe Campus, Faculty of Science, Department of Mathematics, Buca, 35160

İzmir, Turkey.

∗Corresponding author: bkarpuz@gmail.com


	1. Introduction
	2. Auxiliary lemmas
	2.1. Time scales
	2.2. Recent results

	3. Main results
	3.1. Nonneutral equations
	3.2. Neutral equations

	References