Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 78, pp. 1–15.

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

HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA FOR

HALF-LINEAR DYNAMIC EQUATIONS WITH MIXED

DERIVATIVES ON A TIME SCALE

KAZUKI ISHIBASHI

Dedicated to Professor Jitsuro Sugie on his 65th birthday

Abstract. This article deals with half-linear dynamic equations that have

two types of derivatives, and obtains sufficient conditions for all solutions to

be non-oscillatory. The obtained results extend a previous Hille-Nehari type
theorems for problems of dynamic equations. To prove our main result, we

use a generalized Riccati inequality. As an application, we apply the main

result to self-adjoint Euler type linear differential and difference equations
with a changing sign coefficient. The equation selected for this application is

of Mathieu type.

1. Introduction

Equations of continuous type are often used for modeling natural science phe-
nomena in physics and chemistry. While discrete models are used for approximating
the continuous models. However, since the beginning of this century, with the de-
velopment of basic theories of difference equations, many phenomena have been
modeled directly with discrete type models and excellent reports have been ob-
tained. On the other hand, the idea of a theory that can unify continuous type
models (differential equations) and discrete type model (difference equations) was
initiated by Stefan Hilger [12], and it is known as the theory of time scales [6, 7].
Here, a time scale T is defined as a nonempty closed subset of the real numbers.
In time scales, operators such as σ, ρ, µ, and ν are often used, and are defined in
Section 5.

This article concerns the non-oscillation of solutions to the half-linear dynamic
equation with mixed derivatives(

r(t)Φp(x
∆)
)∇

+ c(t)Φp(x) = 0, t ∈ [t0,∞)T, (1.1)

where t0 ∈ T; the function r:T → R is continuous and r(t) > 0 for all t ∈ T; the
function c:T → R is real and left-dense continuous; p is a parameter greater than

2010 Mathematics Subject Classification. 39A21, 34C10.
Key words and phrases. Half-linear dynamic equations; nonoscillation; time scale;

Riccati dynamic inequality; linear differential equation; linear difference equation.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted June 16, 2021. Published September 15, 2021.

1



2 K. ISHIBASHI EJDE-2021/78

1; Φp is the so-called one dimensional p-Laplacian defined for s ∈ R by

Φp(s) =

{
|s|p−2s s 6= 0

0 s = 0 .

For simplicity, let q be the conjugate number of p; namely, 1/p+1/q = 1. Then, Φq is
the inverse function of Φp. Here, the term mixed derivatives indicates the use of ∆-
derivative and∇-derivative (see Section 5 for details). Note that if T = R, then (1.1)
becomes a half-linear differential equation (see [9]). If T = Z, then (1.1) becomes a
half-linear difference equation. Some results from half-linear differential equations
have bee generalized to elliptic partial differential equations, see for example [11].

When p = 2, equation (1.1) becomes the Sturm-Liouville linear dynamic equation(
r(t)x∆

)∇
+ c(t)x = 0. (1.2)

Many papers have been devoted to finding conditions which guarantee that all non-
trivial solutions of (1.2) (and more general nonlinear equations) are oscillatory, and
non-oscillatory. See for example [1, 2, 3, 4, 10] and the references cited therein. In
particular, the definitions of oscillatory and non-oscillatory for dynamic equations
with mixed derivatives are given by Messer [16].

Definition 1.1. A non-trivial solution x of (1.1) is said to have a generalized zero
at t if x(t) = 0, or if t is left-scattered and x(ρ(t))x(t) < 0. Here ρ(t) = sup{s ∈ T :
s < t} which is the backward jump operator.

Definition 1.2. Let t∗ = supT and a ∈ T. When t∗ < ∞, assume ρ(t∗) = t∗. A
non-trivial solution x of (1.1) is said to be oscillatory on [a, t∗) if every non-trivial
solution has infinitely many generalized zeros in [a, t∗). Otherwise, it is said to be
non-oscillatory on [a, t∗).

Looking back on the history, the use of mix derivatives as in (1.2) was considered
by Messer [16] for oscillation problems (see also [7, Chap. IV]). In extension, Došlý
and Marek [8] studied the half-linear dynamic equation (1.1) and its oscillatory
properties (for example, Sturm’s comparison theorem and Hille-Nehari type oscil-
lation). In this article, we prove the following the Hille-Nehari type non-oscillation
theorem of the type studied by Došlý and Marek [8, Theorem 4.5].

Theorem 1.3. Let

Ap(ρ(t)) =
(∫ ρ(t)

t0

(
r(ρ(s))

)1−q∇s)p−1(∫ ∞
ρ(t)

c(s)∇s
)
.

Assume that
∫∞
t0

(
r(ρ(t))

)1−q∇t =∞,
∫∞
t0
c(t)∇t <∞, and

lim
t→∞

ν(t)
(
r(ρ(t))

)1−q∫ ρ(t)
t0

(
r(ρ(s))

)1−q∇s = 0. (1.3)

If there exists Ap(ρ) such that

lim inf
t→∞

Ap(ρ(t)) > −2p− 1

p

(p− 1

p

)p−1

and

lim sup
t→∞

Ap(ρ(t)) <
1

p

(p− 1

p

)p−1

,

then all non-trivial solutions of (1.1) are non-oscillatory.



EJDE-2021/78 HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA 3

In Theorem 1.3, Došlý and Marek [8, Theorem 4.2] established the Hille-Nehari
type nonoscillation criterion by considering the lower limit condition

lim inf
t→∞

Ap(ρ(t)) > −2p− 1

p

(p− 1

p

)p−1

and some other conditions. For the case p = 2, the lower limit condition is

lim inf
t→∞

(∫ ρ(t)

t0

1

r(ρ(s))
∇s
)(∫ ∞

ρ(t)

c(s)∇s
)
> −3

4
.

The purpose of this article is to extended result of Theorem 1.3, by finding con-
ditions on the lower limit value. The new Hille-Nehari type nonoscillation criterion
for (1.1) is as follows.

Theorem 1.4. Assume that
∫∞
t0

(
r(ρ(t))

)1−q∇t = ∞,
∫∞
t0
c(t)∇t < ∞, and (1.3)

holds. Let h:T→ R be a positive function such that h∇(t) ≤ 0 for large t. If there
exists h(ρ) such that

lim inf
t→∞

Ap(ρ(t)) > −(h(ρ(t)))1/q − h(ρ(t)), and (1.4)

lim sup
t→∞

Ap(ρ(t)) < (h(ρ(t)))1/q − h(ρ(t)), (1.5)

then all non-trivial solutions of (1.1) are non-oscillatory, where Ap(ρ) is the func-
tion given by Theorem 1.3.

In Section 2 we shall show that Theorem 1.4 includes Theorem 1.3. Note that
both Theorem 1.3 and Theorem 1.4 assume the integral condition∫ ∞

t0

(
r(ρ(t))

)1−q∇t =∞ .

Therefore, Theorems 1.3 and 1.4 cannot be applied when∫ ∞
t0

(
r(ρ(t))

)1−q∇t <∞ . (1.6)

Under this condition, the Hille-Nehari type non-oscillation theorem of (1.1) is not
given. Under assumption (1.6), this article shows a non-oscillation condition cor-
responding to Theorem 1.4. A new Hille-Nehari type non-oscillation criterion for
(1.1) reads as follows.

Theorem 1.5. Let

Bp(ρ(t)) =
(∫ ∞

ρ(t)

(
r(ρ(s))

)1−q∇s)p−1(∫ ρ(t)

t0

c(s)∇s
)
.

Assume (1.6) holds, and

lim
t→∞

ν(t)
(
r(ρ(t))

)1−q∫∞
ρ(t)

(
r(ρ(s))

)1−q∇s = 0. (1.7)

Let h:T → R be a positive function such that h∇(t) ≥ 0 for large t. If there exists
h(ρ) such that

lim inf
t→∞

Bp(ρ(t)) > −(h(ρ(t)))1/q − h(ρ(t)), and (1.8)

lim sup
t→∞

Bp(ρ(t)) < (h(ρ(t)))1/q − h(ρ(t)), (1.9)



4 K. ISHIBASHI EJDE-2021/78

then all non-trivial solutions of (1.1) are non-oscillatory.

In Theorem 1.5 we investigated the same bound with distinct conditions. Under
these conditions, all non-trivial solutions of (1.1) are also non-oscillatory. However,
since it is not the same conditions as in Theorem 1.3, Theorem 1.4 and Theorem 1.5
can be considered as new results. Moreover, the main results extend Moore’s results
[17] (see Section 2). In Section 4, as an application of Theorem 1.5, we give a non-
oscillation theorem for the linear differential and difference equation with a changing
sign coefficient. For these equations we cannot use Theorem 1.4 directly. By using
Theorem 1.5, we show that the linear differential and the difference equations have
similar non-oscillation results.

2. Remarks about Theorems 1.4 and 1.5

Let us compare Theorem 1.4 with Theorem 1.3. In the case that h(ρ) ≡ ((p −
1)/p)p, by using p/q = p− 1, we have the upper limit value of

(h(ρ(t)))1/q − h(ρ(t)) =
(p− 1

p

)p−1(
1− p− 1

p

)
=

1

p

(p− 1

p

)p−1

and the lower limit value of

−(h(ρ(t)))1/q − h(ρ(t)) = −
(p− 1

p

)p−1(
1 +

p− 1

p

)
= −2p− 1

p

(p− 1

p

)p−1

.

Hence, the condition of Theorem 1.4 becomes the one of Theorem 1.3. For the case
p = 2, from Theorem 1.3, we have

lim inf
t→∞

Ap(ρ(t)) > −3

4
= −0.75 and lim sup

t→∞
Ap(ρ(t)) <

1

4
= 0.25.

In the case p = 2, from Theorem 1.4 ((1.4) and (1.5)), we assume that there exists
h(ρ) ≡ k (positive constant) such that

lim inf
t→∞

Ap(ρ(t)) > −
√
k − k and lim sup

t→∞
Ap(ρ(t)) <

√
k − k ≤ 1

4
.

Notice that the parameter k gives us opportunity to obtain the desired values. If
k = 1/4, then we have the same result as the one from Došlý and Marek. If we set
k = 1/3, then

lim inf
t→∞

Ap(ρ(t)) > − 1√
3
− 1

3
≈ −0.91068 · · · ,

lim sup
t→∞

Ap(ρ(t)) <
1√
3
− 1

3
≈ 0.24401 · · · .

Thus we decreased the the lower limit from −0.75 to −0.91068 · · · . Therefore,
we can conclude that by setting the parameter k, Theorem 1.4 can extend the
lower limit. Under these conditions, all non-trivial solutions of (1.1) are also non-
oscillatory. However, since it is not the same conditions with Theorem 1.3, Theorem
1.4 can be considered as a new result.

Let T = R and p = 2. Then (1.1) becomes the linear differential equation

(r(t)x′)′ + c(t)x = 0. (2.1)

Decreasing the lower limit −3/4 for the Hille-Nehari type non-oscillation result has
been actively studied by Moore [17], Wray [22] and Wu and Sugie [23]. In fact,
Moore [17] gave the following two theorems.



EJDE-2021/78 HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA 5

Theorem 2.1. Suppose that
∫∞
t0

(r(t))−1dt =∞ and
∫∞
t0
c(t)dt <∞. If there exists

a constant k > 0 such that(
1 +

∫ t

t0

1

r(s)
ds
)(∫ ∞

t

c(s)ds
)
≥ −
√
k − k,(

1 +

∫ t

t0

1

r(s)
ds
)(∫ ∞

t

c(s)ds
)
≤
√
k − k ≤ 1

4
,

then all nontrivial solutions of (2.1) are non-oscillatory.

Theorem 2.2. Suppose that
∫∞
t0

(r(t))−1dt < ∞. If there exists a constant k > 0

such that (
1 +

∫ ∞
t

1

r(s)
ds
)(∫ t

t0

c(s)ds
)
≥ −
√
k − k,(

1 +

∫ ∞
t

1

r(s)
ds
)(∫ t

t0

c(s)ds
)
≤
√
k − k ≤ 1

4
,

then all nontrivial solutions of (2.1) are non-oscillatory.

Theorems 1.4 and 1.5 are generalization to Theorems 2.1 and 2.2. Indeed, we
assume that T = R, p = 2 and h(ρ(t)) ≡ k (positive constant) for Theorems 1.4

and 1.5. Then, we have the upper bound
√
k − k and the lower bound −

√
k − k.

Recently, Wu, She and Ishibashi [24] gave the Moore-type nonoscillation theorem
for half-linear difference equations. Theorems 1.4 and 1.5 also extend their results,
when T = N.

3. Proof of Theorems 1.4 and 1.5

First we show some preliminary results that are used for proving the main results.
The readers can find more preliminaries that support the proof in [8].

Lemma 3.1. Let f :R → R be a differentiable function and let g:T → R be ∇-
differentiable function. Then we have

[f(g(t))]∇ = f ′(ξ)g∇(t),

where g(ρ(t)) ≤ ξ(t) ≤ g(t).

Lemma 3.2. Equation (1.1) is non-oscillatory if and only if there exists a ∇-
differentiable function w satisfying (3.1) such that R[w] ≤ 0, where

R[w] :=


w∇(t) + c(t) + (p− 1) |w(t)|q

Φq(r(t)) if ρ(t) = t,

w∇(t) + c(t) + w(ρ(t))
ν(t)

(
1− r(ρ(t))

Φp

(
Φq(r(ρ(t)))+ν(t)Φq(w(ρ(t)))

)) if ρ(t) < t

for large t.

Remark 3.3. Let x be a non-oscillatory solution of (1.1). Then, in R[w], we see
that

Φq(r(ρ(t))) + ν(t)Φq(w(ρ(t))) > 0 (3.1)

for large t. In other words, we need only one function w and establish R[w] ≤ 0 for
each case (left scattered case, and left dense case) to prove our main theorems.



6 K. ISHIBASHI EJDE-2021/78

Proof of Theorem 1.4. For simplicity, let

r̂(t) := r(ρ(t)), ŵ(t) := w(ρ(t)),

Ap(t) :=
(∫ t

t0

(r̂(s))1−q∇s
)p−1(∫ ∞

t

c(s)∇s
)
.

Also, put

w(t) = h(t)
(∫ t

t0

(r̂(s))1−q∇s
)1−p

+

∫ ∞
t

c(s)∇s.

Using Lemma 3.1, we can calculate[( ∫ t

t0

(r̂(s))1−q∇s
)1−p]∇

= (1− p)(r̂(s))1−q (θ(t))−p,

where ∫ ρ(t)

t0

(r̂(s))1−q∇s ≤ θ(t) ≤
∫ t

t0

(r̂(s))1−q∇s.

Also, using the Lagrange mean value, we have

ŵ(t)

ν(t)

(
1− r̂(t)

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

))
=
ŵ(t)

ν(t)

(Φp(Φq(r̂(t)) + νΦq(ŵ(t))
)
− Φp(Φq(r̂(t)))

Φp
(
Φq(r̂(t)) + νΦq(ŵ(t))

) )
= (p− 1)

|η(t)|p−2|ŵ(t)|q

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

) ,
where

Φq(r̂(t)) ≤ η(t) ≤ Φq(r̂(t)) + νΦq(ŵ(t)).

From (1.4) and (1.5), there exists ε > 0 such that

|Ap(ρ(t)) + h(ρ(t))|q(1 + ε) < h(ρ(t)).

We also need to calculate

|ŵ(t)|q =
(∫ ρ(t)

t0

(r̂(s))1−q∇s
)−p
|Ap(ρ(t)) + h(ρ(t))|q.

Next we consider two cases: t > ρ(t) and t = ρ(t).

Case (i): t > ρ(t). Since h∇(t) ≤ 0 for large t, we have

R[w] = w∇(t) + c(t) +
ŵ(t)

ν(t)

(
1− r̂(t)

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

))
= −(p− 1)h(ρ(t))(θ(t))−p(r̂(t))1−q + h∇(t)

(∫ t

t0

(r̂(s))1−q∇s
)1−p

− c(t)

+ c(t) + (p− 1)
|η(t)|p−2|ŵ(t)|q

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

)
≤ (p− 1)(r̂(t))1−q

[
− h(ρ(t))

(∫ t

t0

(r̂(s))1−q∇s
)−p

+
(∫ ρ(t)

t0

(r̂(s))1−q∇s
)−p |η(t)|p−2(r̂(t))q−1

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

) |Ap(ρ(t)) + h(ρ(t))|q
]



EJDE-2021/78 HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA 7

=
(p− 1)(r̂(t))1−q( ∫ t
t0

(r̂(s))1−q∇s
)p [−h(ρ(t)) + Z1(t)|Ap(ρ(t)) + h(ρ(t))|q],

where

Z1(t) :=
( ∫ t

t0
(r̂(s))1−q∇s∫ ρ(t)

t0
(r̂(s))1−q∇s

)p |η(t)|p−2(r̂(t))1−q

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

) .
We can see that

ν(t)
∣∣ ŵ(t)

r̂(t)

∣∣q−1

= ν(t)

∣∣h(ρ(t))
( ∫ ρ(t)

t0
(r̂(s))1−q∇s

)1−p
+
∫∞
ρ(t)

c(s)∇s
∣∣q−1

(r̂(t))q−1

=
ν(t)(r̂(t))1−q∫ ρ(t)
t0

(r̂(s))1−q∇s

∣∣∣h(ρ(t)) +
(∫ ρ(t)

t0

(r̂(s))1−q∇s
)p−1(∫ ∞

ρ(t)

c(s)∇s
)∣∣∣q−1

→ 0

as t→∞ because of (1.3). Therefore, we can estimate

|Z1(t)| =
(∫ ρ(t)

t0
(r̂(s))1−q∇s+ ν(t)(r̂(t))1−q∫ ρ(t)

t0
(r̂(s))1−q∇s

)p |Φq(r̂(t)) + νΦq(ŵ(t))|p−2(r̂(t))1−q

Φp(Φq(r̂(t)) + ν(t)Φq(ŵ(t)))

=
(

1 +
ν(t)(r̂(t))1−q∫ ρ(t)
t0

(r̂(s))1−q∇s

)p (r̂(t))(q−1)(p−1)|1 + ν(t)Φq(ŵ(t)/r̂(t))|p−2

r̂(t)Φp(1 + ν(t)Φq(ŵ(t)/r̂(t)))

=
(

1 +
ν(t)(r̂(t))1−q∫ ρ(t)
t0

(r̂(s))1−q∇s

)p 1

1 + ν(t)Φq(ŵ(t)/r̂(t))
→ 1

as t→∞. Summarizing all estimates, we see that

R[w] ≤ (p− 1)(r̂(t))1−q( ∫ t
t0

(r̂(s))1−q∇s
)p [−h(ρ(t)) + Z1(t)|Ap(ρ(t)) + h(ρ(t))|q(1 + ε)] < 0

for large t.

Case (ii): t = ρ(t). In this case r̂ = r and ŵ = w. Hence, the Riccati-type
expression is

R[w] = w∇(t) + c(t) + (p− 1)
|w(t)|q

Φq(r(t))

= −(p− 1)h(ρ(t))
(∫ t

t0

(r(s))1−q∇s
)−p

(r(t))1−q

+ h∇(t)
(∫ t

t0

(r(s))1−q∇s
)1−p

− c(t) + c(t)

+ (p− 1)

( ∫ t
t0

(r(s))1−q∇s
)−p|Ap(ρ(t)) + h(ρ(t))|q

Φq(r(t))

= (p− 1)
(∫ t

t0

(r(s))1−q∇s
)−p

(r(t))1−q[−h(ρ(t)) + |Ap(ρ(t)) + h(ρ(t))|q] < 0

for large t. From Lemma 3.2, we can complete the proof. �



8 K. ISHIBASHI EJDE-2021/78

Proof of Theorem 1.5. From (1.6), we define

w(t) = −h(t)
(∫ ∞

t

(r̂(s))1−q∇s
)1−p

−
∫ t

t0

c(s)∇s

= −
(∫ ∞

t

(r̂(s))1−q∇s
)1−p

[Bp(t) + h(t)],

where r̂(t) = r(ρ(t)) and

Bp(t) =
(∫ ∞

t

(r̂(s))1−q∇s
)p−1(∫ t

t0

c(s)∇s
)
.

By using Lemma 3.1, we need to calculate[( ∫ ∞
t

(r̂(s))1−q∇s
)1−p]∇

=
[( ∫ ∞

T

(r̂(s))1−q∇s−
∫ t

T

(r̂(s))1−q∇s
)1−p]∇

= (p− 1)(r̂(t))1−q(θ(t))−p,

where ∫ ∞
t

(r̂(s))1−q∇s ≤ θ(t) ≤
∫ ∞
ρ(t)

(r̂(s))1−q∇s

for sufficiently large T . Then, by using the product rule, we obtain

w∇(t) = −h(ρ(t))(p− 1)(r̂(s))1−q(θ(t))−p − k∇(t)
(∫ ∞

t

(r̂(s))1−q∇s
)1−p

− c(t).

While

|ŵ(t)|q =
(∫ ∞

ρ(t)

(r̂(s))−p∇s
)−p

[Bp(ρ(t)) + h(ρ(t))]q,

where ŵ(t) = w(ρ(t)). Here, we consider two cases: t > ρ(t) and t = ρ(t).

Case (i): t > ρ(t). Since h∇(t) ≥ 0 for large t, we have

R[w] = w∇(t) + c(t) +
ŵ(t)

ν(t)

(
1− r̂(t)

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

))
= −(p− 1)h(ρ(t))(θ(t))−p(r̂(t))1−q − h∇(t)

(∫ ∞
t

(r̂(s))1−q∇s
)1−p

− c(t)

+ c(t) + (p− 1)
|η(t)|p−2|ŵ(t)|q

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

)
≤ (p− 1)(r̂(t))1−q

[
− h(ρ(t))

(∫ ∞
ρ(t)

(r̂(s))1−q∇s
)−p

+
(∫ ∞

ρ(t)

(r̂(s))1−q∇s
)−p

Z2(t)|Bp(ρ(t)) + h(ρ(t))|q
]

=
(p− 1)(r̂(t))1−q( ∫∞
ρ(t)

(r̂(s))1−q∇s
)p [− h(ρ(t)) + Z2(t)|Bp(ρ(t)) + h(ρ(t))|q

]
,

where

Z2(t) :=
|η(t)|p−2(r̂(t))q−1

Φp
(
Φq(r̂(t)) + ν(t)Φq(ŵ(t))

) .



EJDE-2021/78 HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA 9

In a similar way to the proof of Theorem 1.4, by using (1.7) one can show that
Z2(t)→ 1 as t→∞. Therefore, from (1.8) and (1.9), we have

R[w] ≤ (p− 1)(r̂(t))1−q( ∫∞
ρ(t)

(r̂(s))1−q∇s
)p [− h(ρ(t)) + |Bp(ρ(t)) + h(ρ(t))|q

]
< 0

for large t. Thus, the proof of Case (i) is complete.

Case (ii): t = ρ(t). We follow the proof of Theorem 1.4 Csae (ii) to have

R[w] = w∇(t) + c(t) + (p− 1)
|w(t)|q

Φq(r(t))
< 0

for large t. �

4. Linear differential and difference equations with a changing sign
coefficient

As a special case of (1.1), we consider the linear dynamic equation with a chang-
ing sign coefficient(

σ(t)σ(σ(t))x∆
)∇

+
(
− α+ β cos

(
log t− π

4

))
x = 0, t ∈ [1,∞)T, (4.1)

where α and β are real numbers. Since the coefficient is −α + β cos(log t − π/4),
equation (4.1) is a non-periodic Mathieu type dynamic equation; see [13, 14, 15, 19,
21] for the Mathieu type differential equations. In the case T = R, equation (4.1)
becomes the new self-adjoint Euler type linear differential equation with a changing
sign coefficient(

t2x′
)′

+
(
− α+ β cos

(
log t− π

4

))
x = 0, t ≥ t0 = 1. (4.2)

See [18, 20] for the oscillation of the usual self-adjoint Euler type differential equa-
tions.

On the other hand, in the case T = N, equation (4.1) becomes the linear differ-
ence equation

∆
(
t(t+ 1)∆x(t− 1)

)
+
(
− α+ β cos

(
log t− π

4

))
x(t) = 0, t ≥ t0 = 1. (4.3)

Note that equation (4.3) for T = N and ∇(r(t)∆x(t)) = ∆(r(t − 1)∆x(t − 1)), we
see that ∇((t+ 1)(t+ 2)∆x(t)) = ∆(t(t+ 1)∆x(t− 1)).

Note that Theorems 1.3 and (1.4) cannot be applied to equations (4.2) and (4.3).
In this section, we present an example of which all non-trivial solutions of (4.2) and
(4.3) are non-oscillatory.

In Theorem 1.5, we assume that T = R (or T = N), p = 2 and h(ρ(t)) ≡ k, a
positive constant. Then, we have the following corollaries.

Corollary 4.1. Assume that
∫∞
t0

(
r(t)

)−1
dt <∞. If there exists a constant k > 0

such that

lim inf
t→∞

B2(t) > −
√
k − k, and (4.4)

lim sup
t→∞

B2(t) <
√
k − k ≤ 1

4
, (4.5)



10 K. ISHIBASHI EJDE-2021/78

then all non-trivial solutions of second-order linear differential equation (2.1) are
non-oscillatory, where

B2(t) =

∫ ∞
t

1

r(s)
ds

∫ t

t0

c(s)ds.

Corollary 4.2. Assume that

∞∑
t=1

1

r(t− 1)
<∞ and lim

t→∞

1
r(t−1)∑∞
j=t

1
r(j−1)

= 0.

If there exists a constant k > 0 such that

lim inf
t→∞

B2(t− 1) > −
√
k − k, and (4.6)

lim sup
t→∞

B2(t− 1) <
√
k − k ≤ 1

4
, (4.7)

then all non-trivial solutions of second-order difference equation

∆(r(t− 1)∆x(t− 1)) + c(t)x(t) = 0 (4.8)

are non-oscillatory, where

B2(t− 1) =

∞∑
j=t

1

r(j − 1)

t−1∑
j=1

c(j).

We expand Corollary 4.1 (or Corollary 4.2) in order to apply it to the equation
(4.2) (or equation (4.3)). This result is obtained as follows.

Theorem 4.3. If there exists a constant k > 0 such that

|β| <
√

2(
√
k + k − α), (4.9)

|β| <
√

2(
√
k − k + α), (4.10)

then all non-trivial solutions of (4.2) and (4.3) are non-oscillatory.

All non-trivial solutions of (4.2) and (4.3) are non-oscillatory if a pair of coordi-
nates (α, β) is contained in the grey part and the dark part of Figure 1. The grey
part is the region

R1 := {(α, β) : k −
√
k < α ≤ k, |β| <

√
2(
√
k − k + α)}.

On the other hand, the dark part is the region

R2 := {(α, β) : k ≤ α <
√
k + k, |β| <

√
2(
√
k + k − α)}.

Thus, the union of areas R1 and R2 is represent the when conditions (4.9) or (4.10)
are satisfied. As an example, let k = 1.5. Then, from numerical calculations

k −
√
k = 1.5−

√
1.5 ≈ 0.275255 · · · ,

√
k + k =

√
1.5 + 1.5 ≈ 2.72474 · · · ,

√
2(
√
k − k + α) =

√
2(
√

1.5− 1.5) +
√

2α ≈ −0.38927 · · ·+
√

2α,
√

2(
√
k + k − α) =

√
2(
√

1.5 + 1.5)−
√

2α ≈ 3.85337 · · ·+
√

2α,

we see that the nonoscillation regions

R̃1 := {(α, β) : 0.275255 · · · < α ≤ 1.5, |β| < −0.38927 · · ·+
√

2α},



EJDE-2021/78 HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA 11

α

β

k- k k+ kk

2 k

- 2 k

R1 R2

Figure 1. Nonoscillation region given by conditions (4.9) and (4.10)

R̃2 := {(α, β) : 1.5 ≤ α < 2.72474 · · · , |β| < 3.85337 · · · −
√

2α}.

For example, if α = 2 and β = 1, then a point (α, β) = (2, 1) ∈ R̃1 ∪ R̃2. In
fact, from numerical simulation, we draw a solution curve of (4.2) and (4.3) for
(α, β) = (2, 1) (see, Figures 2 and 3). In Figures 2 and 3, the non-oscillation curve
of (4.2) and (4.3) starting at the point (0, 1). The solution curves for (4.2) and
(4.3) are very similar. Hence, from numerical simulations, we see that the linear
dynamic equation (4.1) unifies linear differential equation (4.2) and linear difference
equation (4.3).

5 10 15 20
t

1
2
3
4

x

Figure 2. A non-oscillatory solution of (4.2) when (α, β) = (2, 1)

5 10 15 20
t

2

4

6

x

Figure 3. A non-oscillatory solution of (4.3) when (α, β) = (2, 1)



12 K. ISHIBASHI EJDE-2021/78

Finally, by using Corollaries 4.1 and 4.2, we prove Theorem 4.3.

Proof of Theorem 4.3. We split the proof into two cases: Linear differential equa-
tion (4.2), and Linear difference equation (4.3).

Case (i): Equation (4.2). Let t0 = 1. To show that (4.2) is non-oscillatory, we
utilize Corollary 4.1. Comparing equation (4.2) with equation (2.1), we see that
r(t) = t2 and

c(t) = −α+ β cos
(

log t− π

4

)
= −α+

β
√

2

2

(
cos(log t) + sin(log t)

)
.

Since

lim
t→∞

∫ t

1

1

r(s)
ds = lim

t→∞

∫ t

1

1

s2
ds = lim

t→∞

[
− 1

s

]t
1

= lim
t→∞

(
− 1

t
+ 1
)

= 1,

we see that
∫∞

1

(
r(t)

)−1
dt <∞ is satisfied. In addition, we obtain

B2(t) =

∫ ∞
t

1

r(s)
ds

∫ t

t0

c(s)ds

=
[
− 1

s

]∞
t

[
− αs+

β
√

2

2
s sin(log s)

]t
1

=
1

t

(
− αt+

β
√

2

2
t sin(log t) + α

)
= −α+

β
√

2

2
sin(log t) +

α

t
.

Hence, from conditions (4.4) and (4.5), we see that conditions (4.9) and (4.10) hold.

Case (ii): Equation (4.3). To show that (4.3) is non-oscillatory, we utilize Corol-
lary 4.2. Comparing equation (4.3) with equation (4.8), we see that r(t−1) = t(t+1)
and

c(t) = −α+ β cos
(

log t− π

4

)
= −α+

β
√

2

2

(
cos(log t) + sin(log t)

)
.

From r(t− 1), it is easy to check that
∞∑
t=1

1

r(t− 1)
=

∞∑
t=1

( 1

t(t+ 1)

)
=

∞∑
t=1

(1

t
− 1

t+ 1

)
= 1 <∞,

∞∑
j=t

1

r(j − 1)
=

∞∑
j=t

( 1

j(j + 1)

)
=

∞∑
j=t

(1

j
− 1

j + 1

)
=

1

t
,

lim
t→∞

1
r(t−1)∑∞
j=t

1
r(j−1)

= lim
t→∞

1
t(t+1)

1
t

= lim
t→∞

1

t+ 1
= 0.

Hence, conditions (4.2) are satisfied. By a straightforward calculation, it follows
that
t−1∑
j=1

c(j) = −
t−1∑
j=1

α+

t−1∑
j=1

β
√

2

2

(
cos(log j) + sin(log j)

)
= −α(t− 1) +

β
√

2

2
t

t∑
j=1

1

t

[
cos
(

log(
j

t
t)
)

+ sin
(

log(
j

t
t)
)]



EJDE-2021/78 HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA 13

− β
√

2

2
(cos(log t) + sin(log t))

=− α(t− 1) +
β
√

2

2
t

t∑
j=1

1

t

[
cos
(

log
(j
t

)
+ log t

)
+ sin

(
log
(j
t

)
+ log t

)]
− β
√

2

2
(cos(log t) + sin(log t)) .

By using addition theorem of trigonometric functions, we have

t−1∑
j=1

c(j) = −α(t− 1) +
β
√

2

2
t

t∑
j=1

1

t

[
cos(log t)

{
sin
(

log
(j
t

))
+ cos

(
log
(j
t

))}]

+
β
√

2

2
t

t∑
j=1

1

t

[
sin(log t)

{
cos
(

log
(j
t

))
− sin

(
log
(j
t

))}]
− β
√

2

2
(cos(log t) + sin(log t)) .

Hence, we see that

lim
t→∞

B2(t− 1) = lim
t→∞

∞∑
j=t

1

r(j − 1)

t−1∑
j=1

c(j)

= − lim
t→∞

α(t− 1)

t

+ lim
t→∞

β
√

2

2
cos(log t)

t∑
j=1

1

t

[
sin
(

log
(j
t

))
+ cos

(
log
(j
t

))]

+ lim
t→∞

β
√

2

2
sin(log t)

t∑
j=1

1

t

[
cos
(

log
(j
t

))
− sin

(
log
(j
t

))]
− lim
t→∞

β
√

2

2t
(cos(log t) + sin(log t)) .

Taking into account that

lim
t→∞

t∑
j=1

1

t

[
sin
(

log
(j
t

))
+ cos

(
log
(j
t

))]
=

∫ 1

0

(sin(log δ) + cos(log δ))dδ

= lim
ε→0+

[
δ sin(log δ)

]1
ε

= 0

and

lim
t→∞

t∑
j=1

1

t

[
cos
(

log
(j
t

))
− sin

(
log
(j
t

))]
=

∫ 1

0

(cos(log δ)− sin(log δ))dδ

= lim
ε→0+

[
δ cos(log δ)

]1
ε

= 1,



14 K. ISHIBASHI EJDE-2021/78

we can check that

lim inf
t→∞

B2(t− 1) = −α− |β|
√

2

2
and lim sup

t→∞
B2(t− 1) = −α+

|β|
√

2

2
.

Hence, from Corollary 4.2, conditions (4.6) and (4.7) hold. �

5. Basic definitions on a time scales

The ∆-derivative is defined as

x∆(t) := lim
s→t

x(σ(t))− x(s)

σ(t)− s
which was introduced by Bohner and Peterson [6]. This is one of the mixed deriva-
tives of (1.1). The ∇-derivative, is defined as

x∇(t) := lim
s→t

x(ρ(t))− x(s)

ρ(t)− s
which was introduced by Atici and Guseinov[5]. This the another mixed derivative
of (1.1). Here, σ(t) := inf{s ∈ T : s > t} is the forward jump operator, ρ(t) =
sup{s ∈ T : s < t} is the backward jump operator. The functions µ, ν:T → [0,∞)
are called forward graininess and backward graininess respectively, and are defined
by

µ(t) = σ(t)− t and ν(t) = t− ρ(t).

A point t ∈ T is said to be right-dense if µ(t) = 0, and it is said to be right-scattered
if µ(t) > 0. Similarly, a point t ∈ T is said to be left-dense if ν(t) = 0, and it is
said to be left-scattered if ν(t) > 0. We will use abbreviations rd, rs, ld and ls
respectively. When T = R, we have

x∆(t) = x′(t) = x∇(t) .

When T = Z, we have

x∆(t) = ∆x(t) = x(t+ 1)− x(t) and x∇(t) = ∇x(t) = x(t)− x(t− 1) .

A function u:T → R is said to be rd-continuous if it is right continuous at all rd
points and the left limit at ld points exists. If u is rd-continuous, then there exists
a ∆-differentiable function U such that U∆(t) = u(t). While a function v:T→ R is
said to be ld-continuous if it is left continuous at all ld points and the right limit at
rd points exists. If v is ld-continuous, then there exists a ∇-differentiable function
V such that V ∇(t) = v(t). The ∆-integral and the ∇-integral are defined by∫ b

a

u(t)∆t = U(b)− U(a) and

∫ b

a

v(t)∇t = V (b)− V (a)

for a < b. In particular, if T = R, then∫ b

a

u(t)∆t =

∫ b

a

u(t)dt and

∫ b

a

v(t)∇t =

∫ b

a

v(t)dt,

while if T = Z, we have∫ b

a

u(t)∆t =

b−1∑
t=a

u(t) and

∫ b

a

v(t)∇t =

b∑
t=a+1

v(t).



EJDE-2021/78 HILLE-NEHARI TYPE NON-OSCILLATION CRITERIA 15

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Kazuki Ishibashi
Department of Electronic Control Engineering, National institute of Technology

(KOSEN), Hiroshima College, Toyota-gun 725-023, Japan

Email address: ishibashi kazuaoi@yahoo.co.jp


	1. Introduction
	2. Remarks about Theorems ?? and ??
	3. Proof of Theorems ?? and ??
	4. Linear differential and difference equations with a changing sign coefficient
	5. Basic definitions on a time scales
	References