OSCILLATION OF HIGHER ORDER STRONGLY SUPERLINEAR AND STRONGLY SUBLINEAR DIFFERENCE EQUATIONS†

Journal of Applied Mathematics & Informatics.
2014.
Sep,
32(3_4):
455-464

- Received : April 23, 2013
- Accepted : July 22, 2013
- Published : September 28, 2014

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We establish some new criteria for the oscillation of
m
th order nonlinear difference equations. We study the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions. We also present a sufficient condition for every solution to be asymptotic at ∞ to a factorial expression (
t
)
^{(m-1)}
.
AMS Mathematics Subject Classification: 39A10, 39A21.
a
) = {
a
,
a
+1, ...} where
a
∈ ℕ and ℕ(
a, b
) = {
a
,
a
+1, ...,
b
},
b
∈ ℕ(
a
).
Consider the
m
th order nonlinear difference equation
where △ is the forward difference operator defined by △
x
(
t
) =
x
(
t
+ 1) −
x
(
t
),
m
is a positive even integer. We shall assume that
By a solution of equation (1), we mean a nontrivial sequence {
x
(
t
)} satisfying equation (1) for all
t
∈ ℕ(
t
_{0}
), where
t
_{0}
∈ ℕ. A solution {
x
(
t
)} is said to be oscillatory if it is neither eventually positive nor eventually negative and it is nonoscillatory otherwise. An equation is said to be oscillatory if all its solutions are oscillatory.
Equation (1) (or the function
ƒ
) is said to be strongly superlinear if there exists a constant
β
> 1 such that
and it is said to be strongly sublinear if there exists a constant
γ
∈ (0, 1) such that
(3) holds with
β
= 1, then equation (1) is called superlinear and (4) holds with
γ
= 1 is called sublinear.
The literature on oscillation of solutions of difference equantions is almost devoted to study of equation (1) when
m
= 1 and 2 and for recent contribution we refer to Agarwal et.al.
[1
,
2
,
3]
. Only few results are available for the oscillation of equation (1) when
m
> 2, see Agarwal et.al.
[2
,
4
,
5]
.
Therefore, the purpose of this paper is to establish some new results for the oscillation of strongly superlinear and strongly sublinear difference equations. We also provide conditions, which guarantee that every solution defined for all large
t
∈ ℕ(
t
_{0}
) is asymptotic at ∞ to (
t
)
^{(m-1)}
.
The obtained results improve and unify these which have appeared in the recent literature.
Lemma 2.1.
(Discrete Toylors Formula) Let x(t) be defined on
ℕ(
t
_{0}
).
Then for all t
∈ ℕ(
t
_{0}
)
and
0 ≤
n
≤
j
− 1
Further, for all t
∈ ℕ(
t
_{0}
,
z
),
where z
∈
t
∈ ℕ(
t
_{0}
)
and
0 ≤
n
≤
j
− 1
Lemma 2.2.
(Discrete Kneser’s Theorem) Let x(t) be defined on
ℕ(
t
_{0}
),
x
(
t
) > 0
and
Δ
^{m}x
(
t
)
be eventually of one sign on
ℕ(
t
_{0}
).
Then there exists an integer k,
0 ≤
k
≤
m with m
+
k odd for
Δ
^{m}x
(
t
) ≤ 0
and
(
m
+
k
)
even for
Δ
^{m}x
(
t
) ≥ 0
such that
Lemma 2.3.
Let x
(
t
)
be defined on
ℕ(
t
_{0}
)
and x
(
t
) > 0
with
Δ
^{m}x
(
t
) ≤ 0
for t
∈ ℕ(
t
_{0}
)
and not identically zero. Then there exists large t
_{1}
∈ ℕ(
t
_{0}
)
such that
where k is as in Lemma 2.2. Furthermore, if x
(
t
)
is increasing, then
Lemma 2.4.
(Gronwal Inequality) Let for all t
∈ ℕ(
t
_{0}
)
the following inequality be satisfied
where
{
p
(
t
)}, {
q
(
t
)}, {
ƒ
(
t
)} and {
x
(
t
)}
are non-negative real-valued sequence de-fined on
ℕ(
t
_{0}
).
Then for all t
∈ ℕ(
t
_{0}
),
Theorem 3.1.
Suppose that equation
(1)
is strongly superlinear. If
for some c
≠ 0
and k
∈ {1, 3, ...,
m
− 1},
then equation
(1)
is oscillatory
.
Proof
. Let {
x
(
t
)} be a non-oscillatory solution of equation (1), say
x
(
t
) > 0 for
t
≥
t
_{0}
∈ ℕ(
t
_{0}
). By Lemma 2.2, there exists an integer
k
∈ {1, 3, ...,
m
− 1} such that (7) holds for
t
≥
t
_{1}
≥
t
_{0}
.
Clearly, Δ
^{k−1}
x
(
t
) is positive and increasing for
t
≥
t
_{1}
. Thus from (5), we find for
s
≥
t
≥
t
_{0}
On the other hand, there exists a constant
c
> 0 such that
From (6) with
n
=
k
and
j
=
m
, and equation (1), we have
Using the strong superlinearity of
ƒ
we obtain
Using (13) in (16) and the fact that
m
− 1 ≥
k
, we have
for
t
≥
t
_{1}
. Or
where
Now, since Δ
^{k}x
(
t
) is positive and decreasing for
t
≥
t
_{1}
+ 1, we find
Summing this inequality from
t
_{1}
+ 1 to
T
≥
t
_{1}
+ 1 we get
which contradicts condition (12). This completes the proof.
When
k
= 1, condition (12) is reduced to
For the case when
m
= 2, we obtain
Corollary 3.2.
Suppose that equation
(1)
with m
= 2
is strongly superlinear. If
then equation
(1)
with m
= 2
is oscillatory
.
For strongly sublinear equation (1), we have
Theorem 3.3.
Let equation
(1)
be strongly sublinear. If
for some constant c
≠ 0,
then equation
(1)
is oscillatory
.
Proof
. Let {
x
(
t
)} be a non-oscillatory solution of equation (1), say
x
(
t
) > 0 for
t
∈ ℕ(
t
_{0}
). By Lemma 2.2, these exists a
t
_{1}
≥
t
_{0}
and constants
c
_{1}
and
c
_{2}
such that
and by Lemma 2.3, we find
Summing equation (1) from
t
≥
t
_{2}
to
u
≥
t
and letting
u
→ ∞, we get
Using the strong sublinearity of
ƒ
in the above inequality, we obtain
By applying (23) in the (24), we find
Denoting the right-hand side of (25) by
z
(
t
), we find
or
Summing this inequality from
t
_{2}
+ 1 to
T
≥
t
_{2}
+ 1, we get
which contradicts condition (20). This completes the proof.
Remark 3.1
. One can easily see that equation (1) is oscillatory if
for some constants
c
≠ 0.
Remark 3.2.
The results of this section are presented in a form which is essentially new. It extend and improve many of the existing results appeared in the literature, see
[1
,
2
,
3
,
4
,
5]
.
Remark 3.3.
When
m
= 2, the results obtained include many of the known oscillation results for related second order nonlinear difference equations, see
[1
,
2
,
3]
.
Remark 3.4.
The results of this section can be extended to
m
th order nonlinear difference equation with deviating arguments of the form
where
ƒ
is as in equation (1) and
g
∈
{
g
: ℕ(
t
^{∗}
) → ℕ for some
t
^{∗}
∈ ℕ :
g
(
t
) ≤
t
, lim
_{t}
→∞
g
(
t
) = 0}, {
g
(
t
)} is a nondecreasing sequence.
In fact, we may replace s in conditions (12) and (20) by g(s). The details are left to the reader.
x
defined for all large
t
∈ ℕ(
t
_{0}
) of equation (1) to satisfy
where
c
is some real number (depending on solution {
x
(
t
)}.
We assume that
where
γ
∈ (0, 1], {
a
(
t
)} and {
b
(
t
)} are nonnegative real-valued sequences.
Theorem 4.1.
If
then every solution
{
x
(
t
)},
t
∈ ℕ(
t
_{0}
)
of equation
(1)
satisfies
and
where c is some constant (depending on solution
{
x
(
t
)}.
Proof
. Let {
x
(
t
)} be a solution for
t
≥
t
_{0}
∈ ℕ(
t
_{0}
) of equation (1). Then (1) gives
Thus, by (29), we obtain for
t
≥
t
_{0}
,
or
Using the elementary inequality
we find
By the assumption (29), these exists constat
C
> 0 such that
Applying Lemma 2.4 and using condition (29) we can conclude that there exists a positive constant
M
such that
Now, by using (28) and (32), we derive
Thus, because of (29), it follows that
But, (1) gives
Therefore,
i.e. (30) holds. Finally, by L’Hospital rule, we obtain
and consequently the solution {
x
(
t
)} satisfies (31). This completes the proof.
Of course, Theorem 4.1 remains valid for the equations of the form
where
ƒ
satisfies conditions (28) and (29).
We may note that the second part of the condition (29) can be replaced by
and also the conclusion (31) can be replace by
As illustrative example, we consider the equation
where the {
p
(
t
)} and {
q
(
t
)} are non negative real sequence and
γ
∈ (0, 1] is a constant. Now, if
then by Theorem 4.1, we conclude that every solution of equation (36) satisfies
where
C
is some real number depending on the solution {
x
(
t
)} .
while condition (20) takes the form
2. Theorem 4.1 when
m
= 2 is a discrete analog of the results in
[6
,
7
,
8
,
9]
. Moreover, it improve and unify some of them.
Let
m
> 4 is a positive even integer,
we have
so
ƒ
(
t,x
(
t
)) is strong sublinear.
Since
c
≠ 0,
Then,
Because
m
> 4 is a positive even integer,
let
t
_{0}
=
m
− 2, so
and
then (40) is divergence, so
Theorem 3.3, the solution of function (39) is oscillatory.
Said R. Grace
Department of Engineering Mathematics, Faculty of Engineering Cairo University, Orman, Giza 12221, Egypt.
e-mail: saidgrace@yahoo.com
Zhenlai Han
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China.
e-mail: hanzhenlai@163.com
Xinhui Li
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China.
e-mail: lixinhui1011@163.com

1. Introduction

In what follows, we shall denote ℕ = {0, 1, ...}, ℕ(
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2. Preliminaries

We shall need the following lemmas given in
[2]
.
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3. Oscillation Criteria

We shall study the oscillatory behavior of all solutions of equation (1) when it is either strongly superlinear or strongly sublinear.
We begin with strongly superlinear case of equation(1).
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4. Asymptotic Behavior

In this section we give a sufficient condition for every solution
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5. General Remarks

1. We many note that conditions (12) and (18) can be replaced by the stronger condition
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6. Example

Consider the following mth order nonlinear difference equation
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BIO

Agarwal R. P.
,
Bohner M.
,
Grace S. R.
,
O’ Regan D.
2005
Discrete Oscillation Theory
Hindawi publisher
New York

Agarwal R. P.
2000
Difference Equations and Inequalities
Second Edition
Marcel Dekker
New York
Revised and extended

Agarwal R. P.
,
Wong P. J. Y.
1997
Advanced Topices in Difference Equations
Kluwer
Dordrecht

Agarwal R. P.
,
Grace S. R.
(2002)
Oscillation criteria for certain higher order difference equations
Math. Sci. Res. J.
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60 -
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,
O’ Regan D.
(2005)
On the Oscillation of higher order difference equations
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245 -
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Rogovchenko S. P.
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(2001)
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Tong J.
(1982)
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Citing 'OSCILLATION OF HIGHER ORDER STRONGLY SUPERLINEAR AND STRONGLY SUBLINEAR DIFFERENCE EQUATIONS†
'

@article{ E1MCA9_2014_v32n3_4_455}
,title={OSCILLATION OF HIGHER ORDER STRONGLY SUPERLINEAR AND STRONGLY SUBLINEAR DIFFERENCE EQUATIONS†}
,volume={3_4}
, url={http://dx.doi.org/10.14317/jami.2014.455}, DOI={10.14317/jami.2014.455}
, number= {3_4}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={GRACE, SAID R.
and
HAN, ZHENLAI
and
LI, XINHUI}
, year={2014}
, month={Sep}