# On the Oscillation of Even-Order Half-Linear Functional Difference Equations with Damping Term

Authors: Yaşar Bolat; Jehad Alzabut
Journal: International Journal of Differential Equations
 (2014)

---

## Abstract

We investigate the oscillatory behavior of solutions of themth order half-linear functional difference equations with damping term of the form Δ
[p
n
Q
(
Δ
m
-
1
y
n
)
]
+
q
n
Q
(
Δ
m
-
1
y
n
)
+
r
n
Q
(
y
τ
n
)
=
0, n
≥
n
0, where m is even and Q
(
s
)
=
s
α
-
2
s, α
>
1 is a fixed real number. Our main results are obtained via employing the generalized Riccati transformation. We provide two examples to illustrate the effectiveness of the proposed results.


---

## Body

## 1. Introduction

Consider the second order half-linear difference equation:(1)
Δ
[
p
n
|
Δ
y
n
|
α
-
2
Δ
y
n
]
+
r
n
|
y
n
+
1
|
α
-
2
y
n
+
1
=
0
,
n
≥
n
0
,
α
>
1
,
where Δ is the forward difference operator and {
p
n
}, {
r
n
} are sequences of nonnegative real numbers with {
p
n
}
>
0. The study of (1) has been initiated by Rehák in [1]. It is well known that there is a close similarity between (1) and the linear second order difference equation. Indeed, if {
y
n
} is a solution of (1), then so is {
c
y
n
} for any constant c. Thus, (1) has one half of linearity properties [2].In the presence of damping, (1) has been extended further to the second order half-linear difference equation with damping term of the form
(2)
Δ
[
p
n
|
Δ
y
n
|
α
-
2
Δ
y
n
]
+
q
n
|
Δ
y
n
|
α
-
2
Δ
y
n
+
r
n
|
y
n
+
1
|
α
-
2
y
n
+
1
=
0
,
00
0000000000000000000000000000000
000000
n
≥
n
0
∈
N
,
where {
q
n
} is a sequence of nonnegative real numbers. It is to be noted that neither (1) nor (2) has involved a delaying term. There are numerous numbers of oscillation criteria established in the literature for the solutions of (1) and (2). Most of these results were obtained by using certain efficient tools among them we name the Riccati transformation, variational principle, and some inequality techniques; see, for instance, the monograph [3] in which many contributions have been cited therein and to the recent papers [4–9].LetQ
:
R
→
R be defined by Q
(
s
)
=
|
s
|
α
-
2
s; α
>
1 is a fixed real number and N
n
0
=
{
n
0
,
n
0
+
1
,
…
}. Consider the mth order half-linear functional difference equation with damping term of the form
(3)
Δ
[
p
n
Q
(
Δ
m
-
1
y
n
)
]
+
q
n
Q
(
Δ
m
-
1
y
n
)
+
r
n
Q
(
y
τ
n
)
=
0
,
00000000000000000000000000000000000
n
∈
N
n
0
,
where m is even number, and (H1)
{
p
n
}
:
N
n
0
→
R
+ with Δ
p
n
≥
0 for all n
≥
n
0;
(H2)
{
q
n
} and {
r
n
}
:
N
n
0
→
R with q
n
≥
0 and r
n
>
0;
(H3)
{
τ
n
}
:
N
n
0
→
Z with τ
n
<
n and lim
⁡
n
→
∞
⁡
τ
n
=
∞.For close results regarding the continuous counterparts of (1), (2), and (3), the reader is suggested to consult [10–14].A primary purpose of this paper is to establish sufficient conditions that guarantee the oscillation of solutions of (3). Our main results are obtained via employing the generalized Riccati transformation. In view of (3), one can easily figure out that it is formulated in more general form so that it includes some particular cases which have been studied in the literature; see [15–23] for more details. To the best of authors’ observation, however, no published result has been concerned with the investigation of oscillatory behavior of solutions of (3) or its continuous counterpart. Therefore, our paper is new and presents a new approach.


## 2. Main Results

We start by recalling the following standard definitions.Definition 1.
A nontrivial sequencey
n is called a solution of (3) if it is defined for all n
≥
σ where n
∈
Z, σ
=
min
⁡
i
≥
n
0
{
τ
i
}, and p
n
Q
(
Δ
m
-
1
y
n
) is differenceable on N
n
0 and satisfies (3) for all n
∈
N
n
0.Definition 2.
A nontrivial solutiony
n of (3) is said to be oscillatory if the terms of the sequence y
n are not eventually positive or not eventually negative. Otherwise, the solution is called nonoscillatory. A difference equation is called oscillatory if all its solutions oscillate.To obtain our main results, we need the following essential lemmas. The first of these is the discrete analogue of the well-known Kiguradze’s lemma.Lemma 3 (see [24]).
Lety
n be defined for n
≥
n
0
∈
N and y
(
n
)
>
0 with Δ
m
y
n of constant sign for n
≥
n
0 and not identically zero. Then, there exists an integer l, 0
≤
l
≤
m with (
m
+
l
) odd for Δ
m
y
n
≤
0 and (
m
+
l
) even for Δ
m
y
(
n
)
≥
0 such that(i)
l
≤
m
-
1 implies (
-
1
)
l
+
i
Δ
i
y
n
>
0 for all n
≥
n
0, l
≤
i
≤
m
-
1,
(ii)
l
≥
1 implies Δ
i
y
n
>
0 for all large n
≥
n
0, 1
≤
i
≤
l
-
1.Lemma 4 (see [25]).
Lety
n be defined for n
≥
n
0 and y
n
>
0 with Δ
m
y
n
≤
0 for n
≥
n
0 and not identically zero. Then, there exists a large integer n
1
≥
n
0 such that
(4)
y
n
≥
1
(
m
-
1
)
!
(
n
-
n
1
)
m
-
1
Δ
m
-
1
y
2
m
-
l
-
1
n
,
n
≥
n
1
,
where l is defined as in Lemma 3. Further, if y
n is increasing, then
(5)
y
n
≥
1
(
m
-
1
)
!
(
n
2
m
-
1
)
m
-
1
Δ
m
-
1
y
n
,
n
≥
2
m
-
1
n
1
.Lemma 5.
Lety
n satisfy conditions of Lemmas 3 and 4 and Δ
m
-
1
y
n
Δ
m
y
n
≤
0 for n
≥
n
1
≥
n
0. Further, if y
n is increasing, then
(6)
Δ
y
n
-
k
≥
M
n
m
-
2
Δ
m
-
1
y
n
,
n
≥
n
1
,
where M
=
(
1
/
(
(
m
-
1
)
!
2
(
m
-
1
)
2
)
)
>
0.The proof of Lemma5 is straightforward and it can be achieved by using the last inequality of Lemma 4.Lemma 6.
Lety
n be an eventually positive solution of (3). If
(7)
(
Λ
1
)
lim
⁡
n
→
∞
⁡
∑
s
=
n
1
n
-
1
‍
(
1
p
s
(
1
-
∏
k
=
n
1
s
-
1
(
1
-
q
k
p
k
)
)
)
1
/
(
α
-
1
)
=
∞
,
then Δ
m
-
1
y
n
>
0, Δ
m
y
n
≤
0, and Δ
y
n
>
0 for all n
≥
n
1
≥
n
0.Proof.
The fact thaty
n is eventually positive solution of (3) implies y
n
>
0 and y
τ
n
>
0 for all n
≥
n
1
≥
n
0. In view of (3), we get
(8)
Δ
[
p
n
Q
(
Δ
m
-
1
y
n
)
]
+
q
n
Q
(
Δ
m
-
1
y
n
)
<
0
,
which leads to
(9)
Δ
[
p
n
Q
(
Δ
m
-
1
y
n
)
-
∏
k
=
n
1
n
-
1
‍
(
1
-
q
k
p
k
)
p
n
1
Q
(
Δ
m
-
1
y
n
1
)
]
<
0
.
Hence,
(10)
p
n
Q
(
Δ
m
-
1
y
n
)
-
∏
k
=
n
1
n
-
1
‍
(
1
-
q
k
p
k
)
p
n
1
Q
(
Δ
m
-
1
y
n
1
)
is decreasing and Δ
m
-
1
y
n is eventually positive or eventually negative.
We claim that(11)
Δ
m
-
1
y
n
>
0
,
n
≥
n
1
.
Assume, on the contrary, that Δ
m
-
1
y
n
<
0, n
≥
n
1. Then, from (10), we obtain
(12)
p
n
|
Δ
m
-
1
y
n
|
α
-
2
Δ
m
-
1
y
n
-
∏
k
=
n
1
n
-
1
‍
(
1
-
q
k
p
k
)
p
n
1
|
Δ
m
-
1
y
n
1
|
α
-
2
Δ
m
-
1
y
n
1
≤
p
n
1
|
Δ
m
-
1
y
n
1
|
α
-
2
Δ
m
-
1
y
n
1
-
∏
k
=
n
1
n
1
-
1
‍
(
1
-
q
k
p
k
)
p
n
1
|
Δ
m
-
1
y
n
1
|
α
-
2
Δ
m
-
1
y
n
1
,
where ∏
k
=
n
1
n
1
-
1
‍
(
1
-
(
q
k
/
p
k
)
)
p
n
1
|
Δ
m
-
1
y
n
1
|
α
-
2
Δ
m
-
1
y
n
1
=
0. Therefore, from (12), we have
(13)
p
n
|
Δ
m
-
1
y
n
|
α
-
2
Δ
m
-
1
y
n
-
∏
k
=
n
1
n
-
1
‍
(
1
-
q
k
p
k
)
p
n
1
|
Δ
m
-
1
y
n
1
|
α
-
2
Δ
m
-
1
y
n
1
≤
p
n
1
|
Δ
m
-
1
y
n
1
|
α
-
2
Δ
m
-
1
y
n
1
-
∏
k
=
n
1
n
1
-
1
‍
(
1
-
q
k
p
k
)
p
n
1
|
Δ
m
-
1
y
n
1
|
α
-
2
Δ
m
-
1
y
n
1
≡
-
M
1
α
-
1
,
where M
1
=
p
n
1
1
/
(
α
-
1
)
|
Δ
m
-
1
y
n
1
|
>
0. It follows that
(14)
(
-
Δ
m
-
1
y
n
)
α
-
1
≥
M
1
α
-
1
p
n
(
1
-
∏
k
=
n
1
n
-
1
‍
(
1
-
q
k
p
k
)
)
or
(15)
Δ
m
-
1
y
n
≤
-
M
1
(
1
p
n
(
1
-
∏
k
=
n
1
n
-
1
‍
(
1
-
q
k
p
k
)
)
)
1
/
(
α
-
1
)
.
Consequently, we obtain
(16)
Δ
m
-
2
y
n
≤
Δ
m
-
2
y
n
1
-
M
1
∑
s
=
n
1
n
-
1
‍
(
1
p
s
(
1
-
∏
k
=
n
1
s
-
1
‍
(
1
-
q
k
p
k
)
)
)
1
/
(
α
-
1
)
.
Letting n
→
∞ in the above inequality, one gets lim
⁡
n
→
∞
⁡
Δ
m
-
2
y
n
=
-
∞. Hence, y
n is an eventually negative function which contradicts that y
n
>
0. Therefore, inequality (11) holds.
From (3), we get
(17)
Δ
[
p
n
Q
(
Δ
m
-
1
y
n
)
]
=
p
n
+
1
Δ
(
Δ
m
-
1
y
n
)
α
-
1
+
(
Δ
m
-
1
y
n
)
α
-
1
Δ
p
n
≤
0
from which it follows that
(18)
Δ
(
Δ
m
-
1
y
n
)
α
-
1
≤
0
,
n
≥
n
1
≥
n
0
.
The above inequality implies that (
Δ
m
-
1
y
n
)
α
-
1 is nonincreasing. Therefore, we can write
(19)
Δ
(
Δ
m
-
1
y
n
)
α
-
1
=
(
Δ
m
-
1
y
n
+
1
)
α
-
1
-
(
Δ
m
-
1
y
n
)
α
-
1
=
[
Δ
m
-
1
y
n
+
1
-
Δ
m
-
1
y
n
]
×
[
(
Δ
m
-
1
y
n
+
1
)
α
-
2
+
(
Δ
m
-
1
y
n
+
1
)
α
-
3
(
Δ
m
-
1
y
n
)
+
(
Δ
m
-
1
y
n
+
1
)
α
-
4
(
Δ
m
-
1
y
n
)
2
+
⋯
+
(
Δ
m
-
1
y
n
)
α
-
2
]
≤
0
.
Since (
Δ
m
-
1
y
n
)
α
-
1 is nonincreasing and positive, then from the above inequality, we have
(20)
Δ
(
Δ
m
-
1
y
n
)
α
-
1
≤
[
Δ
m
-
1
y
n
+
1
-
Δ
m
-
1
y
n
]
(
α
-
1
)
(
Δ
m
-
1
y
n
)
α
-
2
=
(
α
-
1
)
Δ
(
Δ
m
-
1
y
n
)
(
Δ
m
-
1
y
n
)
α
-
2
≤
(
α
-
1
)
Δ
m
y
n
(
Δ
m
-
1
y
n
)
α
-
2
≤
0
by which we have
(21)
Δ
m
y
n
≤
0
.
In virtue of (21) and Lemma 3, we deduce that since m is even then l is odd. Hence Δ
y
n
>
0 for n
≥
n
1
≥
n
0. The proof is complete.Theorem 7.
Let condition(
Λ
1
) hold. Further, assume that there exists a constant λ
>
α
-
1 such that
(22)
(
Λ
2
)
lim
⁡
sup
⁡
n
→
∞
⁡
1
(
n
-
n
1
)
λ
∑
k
=
n
1
n
-
1
‍
[
(
α
-
1
)
α
-
1
α
α
X
n
k
α
Y
n
k
1
-
α
(
n
-
k
)
λ
r
k
00000000000000000000000000
-
(
α
-
1
)
α
-
1
α
α
X
n
k
α
Y
n
k
1
-
α
]
=
∞
,
where
(23)
X
n
k
=
(
(
n
+
1
-
k
)
λ
-
(
n
-
k
)
λ
(
1
+
q
k
p
k
+
1
)
)
,
Y
n
k
=
(
α
-
1
)
M
τ
k
m
-
2
(
n
-
k
)
λ
p
k
p
k
+
1
>
0
,
and M is as in Lemma 5. Then, (3) is oscillatory.Proof.
For the sake of contradiction, assume that (1) has a nonoscillatory solution y
n. Without loss of generality, we assume that y
n is eventually positive (the proof is similar when y
n is eventually negative). That is, y
n
>
0,y
τ
n
>
0 and y
τ
n
-
k
>
0 for all n
≥
n
1
≥
n
0. By Lemma 6, we have Δ
m
-
1
y
n
>
0, Δ
m
y
n
≤
0, and Δ
y
n
>
0 for n
≥
n
1. Consider the function
(24)
w
n
=
p
n
Q
(
Δ
m
-
1
y
n
)
Q
(
y
τ
n
-
k
)
=
p
n
(
Δ
m
-
1
y
n
y
τ
n
-
k
)
α
-
1
>
0
,
n
≥
n
1
.
Taking into account that Δ
y
n
>
0 and y
n is increasing and τ
n
-
k
<
τ
n, we deduce that Δ
m
y
n
≤
0 and Δ
m
-
1
y
n is nonincreasing. Lemmas 3 and 4, (1), and (24) yield
(25)
Δ
w
n
=
-
q
n
(
Δ
m
-
1
y
n
)
α
-
1
-
r
n
(
y
τ
n
)
α
-
1
(
y
τ
n
+
1
-
k
)
α
-
1
-
p
n
(
Δ
m
-
1
y
n
)
α
-
1
Δ
(
y
τ
n
-
k
)
α
-
1
(
y
τ
n
-
k
)
α
-
1
(
y
τ
n
+
1
-
k
)
α
-
1
≤
-
r
n
-
q
n
p
n
+
1
w
n
+
1
-
(
α
-
1
)
p
n
(
Δ
m
-
1
y
n
)
α
-
1
(
y
τ
n
-
k
)
α
-
2
Δ
y
τ
n
-
k
(
y
τ
n
-
k
)
α
-
1
(
y
τ
n
+
1
-
k
)
α
-
1
≤
-
r
n
-
q
n
p
n
+
1
w
n
+
1
-
(
α
-
1
)
p
n
(
Δ
m
-
1
y
n
)
α
-
1
(
y
τ
n
-
k
)
α
-
2
M
τ
n
m
-
2
Δ
m
-
1
y
τ
n
(
y
τ
n
-
k
)
α
-
1
(
y
τ
n
+
1
-
k
)
α
-
1
≤
-
r
n
-
q
n
p
n
+
1
w
n
+
1
-
(
α
-
1
)
p
n
(
Δ
m
-
1
y
n
)
α
M
τ
n
m
-
2
(
y
τ
n
+
1
-
k
)
α
≤
-
r
n
-
q
n
p
n
+
1
w
n
+
1
-
(
α
-
1
)
M
τ
n
m
-
2
p
n
(
Δ
m
-
1
y
n
+
1
)
α
(
y
τ
n
+
1
-
k
)
α
=
-
r
n
-
q
n
p
n
+
1
w
n
+
1
-
p
n
(
α
-
1
)
M
τ
n
m
-
2
p
n
+
1
(
Δ
m
-
1
y
n
+
1
)
α
p
n
+
1
(
y
τ
n
+
1
-
k
)
α
=
-
r
n
-
q
n
p
n
+
1
w
n
+
1
-
(
α
-
1
)
M
τ
n
m
-
2
p
n
p
n
+
1
w
n
+
1
α
/
(
α
-
1
)
.
Multiplying by (
n
-
k
)
λ and summing up from n
1 to n
-
1, we obtain
(26)
∑
k
=
n
1
n
-
1
‍
(
n
-
k
)
λ
r
k
≤
-
∑
k
=
n
1
n
-
1
‍
(
n
-
k
)
λ
Δ
w
k
-
∑
k
=
n
1
n
-
1
‍
(
n
-
k
)
λ
q
k
p
k
+
1
w
k
+
1
-
∑
k
=
n
1
n
-
1
‍
(
α
-
1
)
M
τ
k
m
-
2
(
n
-
k
)
λ
p
k
p
k
+
1
w
k
+
1
α
/
(
α
-
1
)
≤
(
n
-
n
1
)
λ
w
n
1
-
w
n
+
∑
k
=
n
1
n
-
1
‍
(

(
n
-
k
)
λ
(
1
+
q
k
p
k
+
1
)
(
n
+
1
-
k
)
λ
-
(
n
-
k
)
λ
(
1
+
q
k
p
k
+
1
)
)
w
k
+
1
-
∑
k
=
n
1
n
-
1
‍
(
α
-
1
)
M
τ
k
m
-
2
(
n
-
k
)
λ
p
k
p
k
+
1
w
k
+
1
α
/
(
α
-
1
)
≤
(
n
-
n
1
)
λ
w
n
1
+
∑
k
=
n
1
n
-
1
‍
(

(
n
-
k
)
λ
(
1
+
q
k
p
k
+
1
)
(
n
+
1
-
k
)
λ
-
(
n
-
k
)
λ
(
1
+
q
k
p
k
+
1
)
)
w
k
+
1
-
∑
k
=
n
1
n
-
1
‍
(
α
-
1
)
M
τ
k
m
-
2
(
n
-
k
)
λ
p
k
p
k
+
1
w
k
+
1
α
/
(
α
-
1
)
or
(27)
1
(
n
-
n
1
)
λ
[
∑
k
=
n
1
n
-
1
‍
(
n
-
k
)
λ
r
k
-
∑
k
=
n
1
n
-
1
‍
(
X
n
k
w
k
+
1
-
Y
n
k
w
k
+
1
α
/
(
α
-
1
)
)
]
≤
w
n
1
,
where
(28)
X
n
k
=
(
(
n
+
1
-
k
)
λ
-
(
n
-
k
)
λ
(
1
+
q
k
p
k
+
1
)
)
,
Y
n
k
=
(
α
-
1
)
M
τ
k
m
-
2
(
n
-
k
)
λ
p
k
p
k
+
1
>
0
.
Let
(29)
F
(
w
k
+
1
)
=
X
n
k
w
k
+
1
-
Y
n
k
w
k
+
1
α
/
(
α
-
1
)
.
Then, F has maximum value at w
k
+
1
=
(
(
α
-
1
)
/
α
)
α
-
1
X
n
k
α
-
1
Y
n
k
1
-
α. That is,
(30)
F
max
⁡
=
(
α
-
1
)
α
-
1
α
α
X
n
k
α
Y
n
k
1
-
α
.
Therefore, (27) can be rewritten as
(31)
1
(
n
-
n
1
)
λ
∑
k
=
n
1
n
-
1
‍
[
(
n
-
k
)
λ
r
k
-
(
α
-
1
)
α
-
1
α
α
X
n
k
α
Y
n
k
1
-
α
]
≤
w
n
1
.
Hence, we have
(32)
lim
⁡
sup
⁡
n
→
∞
1
(
n
-
n
1
)
λ
∑
k
=
n
1
n
-
1
‍
[
(
n
-
k
)
λ
r
k
-
(
α
-
1
)
α
-
1
α
α
X
n
k
α
Y
n
k
1
-
α
]
≤
w
n
1
which contradicts condition (
Λ
2
). The proof is complete.Theorem 8.
Let condition(
Λ
1
) hold. Further, assume that there exists a function δ
n
:
N
→
R
+ such that
(33)
(
Λ
3
)
lim
⁡
sup
⁡
n
→
∞
⁡
∑
k
=
n
1
n
-
1
‍
[
δ
k
+
1
r
k
-
1
α
α
(
Δ
δ
k
δ
k
)
α
(
M
τ
k
m
-
2
)
1
-
α
]
=
∞
,
n
1
≥
n
0
,
where M is as in Lemma 5. Then, (3) is oscillatory.Proof.
For the sake of contradiction, assume that (3) has a nonoscillatory solution y
n. Without loss of generality, we assume that y
n is eventually positive (the proof is similar when y
n is eventually negative). That is, y
n
>
0, y
τ
n
>
0 and y
τ
n
-
k
>
0 for all n
≥
n
1
≥
n
0. By Lemma 6, we have Δ
m
-
1
y
n
>
0, Δ
m
y
n
≤
0, and Δ
y
n
>
0 for n
≥
n
1. Consider the function
(34)
w
n
=
δ
n
p
n
(
Δ
m
-
1
y
n
y
τ
n
-
k
)
α
-
1
>
0
,
n
≥
n
1
.
By utilizing the same approach as in the proof of Theorem 7, we arrive at
(35)
Δ
w
n
≤
-
δ
n
+
1
r
n
+
1
α
α
(
Δ
δ
n
δ
n
)
α
(
M
τ
n
m
-
2
)
1
-
α
.
Summing up (35) from n
1 to n
-
1, we have
(36)
∑
k
=
n
1
n
-
1
‍
[
δ
k
+
1
r
k
-
1
α
α
(
Δ
δ
k
δ
k
)
α
(
M
τ
k
m
-
2
)
1
-
α
]
≤
w
n
1
.
Letting n
→
∞ in the above inequality and taking the upper limit, we get a contradiction to (
Λ
3
). The proof is complete.Remark 9.
In view of the statements of Theorems7 and 8, one can easily deduce that condition (
Λ
3
) is a generalization of (
Λ
2
).Example 10.
Consider the fourth order half-linear functional difference equation with damping(37)
Δ
[
n
(
Δ
3
y
n
)
2
]
+
n
(
Δ
3
y
n
)
2
+
1
n
y
n
-
1
2
=
0
,
n
≥
2
,
where p
n
=
n, q
n
=
n, r
n
=
1
/
n, τ
n
=
n
-
1, m
=
4, and α
=
3. It is easy to see that conditions (H1)–(H3) are satisfied. It remains to check the validity of conditions Λ
1 and Λ
2.
Forn
≥
2, we have
(38)
Γ
1
:
=
∑
s
=
n
1
n
-
1
‍
(
1
p
s
(
1
-
∏
ν
=
n
1
s
-
1
‍
(
1
-
q
ν
p
ν
)
)
)
1
/
(
α
-
1
)
=
∑
s
=
2
n
-
1
(
1
s
)
1
/
2
.
It is clear that Γ
1
→
∞ as n
→
∞. Therfore, condition (
Λ
1
) holds. For n
≥
2 and λ
=
3
>
α
-
1
=
2, we have
(39)
Γ
2
:
=
1
(
n
-
n
1
)
λ
∑
k
=
n
1
n
-
1
‍
[
(
n
-
k
)
λ
r
k
-
(
α
-
1
)
α
-
1
α
α
X
n
k
α
Y
n
k
1
-
α
]
=
1
(
n
-
2
)
3
∑
k
=
2
n
-
1
‍
[
(
n
-
k
)
3
k
-
4
27
X
n
k
3
Y
n
k
2
]
,
where
(40)
X
n
k
3
=
[
(
k
+
1
)
(
n
+
1
-
k
)
3
-
(
2
k
+
1
)
(
n
-
k
)
3
k
+
1
]
3
,
Y
n
k
2
=
(
k
+
1
)
2
4
M
2
k
2
(
k
-
2
)
8
(
n
-
k
)
6
.
It is clear that Γ
2
→
∞ as n
→
∞. Then, condition (
Λ
2
) holds. Thus, by the conclusion of Theorem 7, (37) is oscillatory.Example 11.
Consider the sixth order half-linear functional difference equation with damping(41)
Δ
[
n
(
Δ
5
y
n
)
2
]
+
n
(
Δ
5
y
n
)
2
+
n
2
y
n
-
1
2
=
0
,
n
≥
2
,
where p
n
=
n, q
n
=
n, r
n
=
n
2, τ
n
=
n
-
1, m
=
6, and α
=
3. It is easy to see that conditions (H1)–(H3) are satisfied. In Example 10, we have seen that (
Λ
1
) is satisfied. It remains to check the validity of condition (
Λ
3
).
Forn
≥
2 and δ
n
=
n, we have
(42)
Γ
3
:
=
∑
k
=
n
1
n
-
1
‍
[
δ
k
+
1
r
k
-
1
α
α
(
Δ
δ
k
δ
k
)
α
(
M
τ
k
m
-
2
)
1
-
α
]
=
∑
k
=
2
n
-
1
‍
[
k
2
(
k
+
1
)
-
1
27
M
2
k
3
(
k
-
1
)
8
]
=
∑
k
=
2
n
-
1
‍
[
27
M
2
k
5
(
k
+
1
)
(
k
-
1
)
8
-
1
27
M
2
k
3
(
k
-
1
)
8
]
.
It is clear that Γ
3
→
∞ as n
→
∞. Then, condition (
Λ
3
) holds. Thus, by the conclusion of Theorem 8, (41) is oscillatory.Remark 12.
It is not possible to decide the oscillatory behavior of solutions of (37) and (41) by using any of the results reported in [12, 13]. This implies that the results of our paper extend and generalize some known theorems.Remark 13.
The main results of this paper remain valid for nondelay difference equations of the form(43)
Δ
[
p
n
Q
(
Δ
m
-
1
y
n
)
]
+
q
n
Q
(
Δ
m
-
1
y
n
)
+
r
n
Q
(
y
n
)
=
0
,
0000000000000000000000000000000000
n
∈
N
n
0
.



---

*Source: 791631-2014-05-19.xml*