Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 606149, 15 pages
doi:10.1155/2010/606149
Research Article
Frequent Oscillatory Behavior of Delay Partial
Difference Equations with Positive and Negative
Coefficients
Li Hua Xu1 and Jun Yang1, 2
1
2
College of Science, Yanshan University, Qinhuangdao, Hebei 066004, China
Mathematics Research Center in Hebei Province, Shijiazhuang 050000, China
Correspondence should be addressed to Li Hua Xu, xlh2007.love@163.com
Received 12 November 2009; Accepted 11 February 2010
Academic Editor: Binggen Zhang
Copyright q 2010 L. H. Xu and J. Yang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
This paper is concerned with a class of nonlinear delay partial difference equations with positive
and negative coefficients, which also contains forcing terms. By making use of frequency measures,
some new oscillatory criteria are established.
1. Introduction
Partial difference equations are difference equations that involve functions with two or more
independent integer variables. Such equations arise from considerations of random walk
problems, molecular structure problems, and numerical difference approximation problems.
Recently, there have been a large number of papers devoted to partial difference equations,
and the problem of oscillatory of solutions and frequent oscillatory solutions for partial
difference equations is receiving much attention.
In 1, authors considered oscillatory behavior of the partial difference equations with
positive and negative coefficients of the form
Am1,n Am,n1 − Am,n pm, nAm−k,n−l − qm, nAm−k ,n−l 0,
but they have not discussed frequent oscillations of this equation.
1.1
2
Advances in Difference Equations
In 2, authors considered oscillatory behavior for nonlinear partial difference
equations with positive and negative coefficients of the form
ωfm, n, Am−σ,n−τ , Am−u,n−v Am−1,n Am,n−1 − Am,n pAmk,nl − qAmk ,nl 0,
ωm,n fm, n, Am−σ,n−τ , Am−u,n−v Am−1,n Am,n−1 − Am,n pmn Amk,nl − qmn Amk ,nl 0
1.2
In 3, authors considered frequent oscillation in the nonlinear partial difference
equation
um,n um1,n um,n1 pm,n |um−k1 ,n−l1 |α sgn um−k1 ,n−l1
qm,n |um−k2 ,n−l2 |β sgn um−k2 ,n−l2 0.
1.3
In 4, authors considered oscillations of the partial difference equations with several
nonlinear terms of the form,
um1,n um,n1 − um,n h
pi m, n|um−ki ,n−li |αi sgn um−ki ,n−li 0,
1.4
i1
and in 5 authors considered frequent oscillations of these equations.
In 6, authors considered unsaturated solutions for partial difference equations with
forcing terms
Δ1 u i − 1, j Δ2 u i, j − 1 P1 i, j u i − 1, j P2 i, j u i, j − 1 P3 i, j u i, j f i, j .
1.5
Let Z be the set of integers, Zk, l {i ∈ Z | i k, k 1, . . . , l}, and Zk, ∞ {i ∈ Z |
i k, k 1, . . .}.
In this paper, we will consider the equation of the following form:
um1,n um,n1 − um,n h
i1
pi m, num−ki ,n−li −
g
qj m, num−sj ,n−tj fm, n,
1.6
j1
where m,n ∈ Z0, ∞, pi m, n ≥ 0 i 1, 2, . . . , h, qj m, n ≥ 0 j 1, 2, . . . , g and
H1 ki , li i 1, 2, . . . , h; sj , tj j 1, 2, . . . , g are nonnegative integers;
H2 pi {pi m, n}m,n∈Z0,∞ i 1, 2, . . . , h, qj {qj m, n}m,n∈Z0,∞ j 1, 2, . . . , g,
fm, n are real double sequences.
The usual concepts of oscillation or stability of steady state solutions do not catch all
their fine details, and it is necessary to use the concept of frequency measures introduced in
7 to provide better descriptions. In this paper, by employing frequency measures, some new
oscillatory criteria of 1.6 are established.
Advances in Difference Equations
3
Let
k max ki , sj > 0,
1≤i≤h
1≤j≤g
l max li , tj > 0,
1≤i≤h
1≤j≤g
ρ min{um−ki ,n−li },
θ max um−sj ,n−tj ,
σ max{um−ki ,n−li },
τ min um−sj ,n−tj .
1.7
In addition to H1 and H2 , we also assume
H3 ρ ≥ θ, σ ≤ τ;
g
H4 hi1 pi m, n − j1 qj m, n ≥ 0, i 1, 2, . . . , h; j 1, 2, . . . , g.
For the sake of convenience, Z−k, ∞ × Z−l, ∞ will be denoted by Ω in the sequel.
Given a double sequence {um,n }, the partial differences um1,n − um,n and um,n1 − um,n will be
denoted by Δ1 um,n and Δ2 um,n , respectively.
To the best of our knowledge, nothing is known regarding the qualitative behaviour
of the solutions of 1.6, because these equations contain positive and negative coefficients,
and also contain forcing terms.
Our plan is as follows. In the next section, we will recall some of the basic results
related to frequency measures. Then we obtain several criteria for all solutions of 1.6 to be
frequently oscillatory and unsaturated. In the final section, we give one example to illustrate
our results.
2. Preliminary
The union, intersection, and difference of two sets A and B will be denoted by A B, A · B,
and A \ B, respectively. The number of elements of a set S will be denoted by |S|. Let Φ be a
subset of Ω, then
i m, j ∈ Ω | i, j ∈ Φ ,
Y m Φ i, j m ∈ Ω | i, j ∈ Φ
XmΦ 2.1
are the translations of Φ. Let α, β, λ, and δ be integers satisfying α ≤ β and λ ≤ δ. The union
β δ
β δ
i j
jλ X Y Φ will be denoted by Xα Yλ Φ. Clearly,
iα
β
i, j ∈ Ω \ Xα Yλδ Φ ⇐⇒ i − d, j − e ∈ Ω \ Φ
2.2
for α ≤ d ≤ β and λ ≤ e ≤ δ.
For any m, n ∈ Z0, ∞, we set
Φm,n i, j ∈ Φ | −k1 ≤ i ≤ m, −l1 ≤ j ≤ n .
2.3
4
Advances in Difference Equations
If
lim sup
m,n → ∞
m,n Φ
mn
2.4
exists, then the superior limit, denoted by μ∗ Φ, will be called the upper frequency measure
of Φ. Similarly, if
lim inf
m,n → ∞
m,n Φ
mn
2.5
exists, then the inferior limit, denoted by μ∗ Φ, will be called the lower frequency measure
of Φ. If μ∗ Φ μ∗ Φ, then the common limit is denoted by μΦ and is called the frequency
measure of Φ.
Clearly, μ∅ 0, μΩ 1, and 0 ≤ μ∗ Φ ≤ μ∗ Φ ≤ 1 for any subset Φ of Ω;
furthermore, if Φ is finite, then μΦ 0.
The following results are concerned with the frequency measures and their proofs are
similar to those in 8.
Lemma 2.1. Let Φ and Γ be subsets of Ω. Then μ∗ Φ Γ ≤ μ∗ Φ μ∗ Γ. Furthermore, if Φ and
Γ are disjoint, then
μ∗ Φ μ∗ Γ ≤ μ∗ Φ Γ ≤ μ∗ Φ μ∗ Γ ≤ μ∗ Φ Γ ≤ μ∗ Φ μ∗ Γ,
2.6
μ∗ Φ μ∗ Ω \ Φ 1.
2.7
so that
Lemma 2.2. Let Φ be a subset of Ω, and let α, β, λ, and δ be integers such that α ≤ β and λ ≤ δ.
Then
β
μ∗ Xα Yλδ Φ ≤ β − α 1 δ − λ 1μ∗ Φ,
β
μ∗ Xα Yλδ Φ
≤ β − α 1 δ − λ 1μ∗ Φ.
2.8
Advances in Difference Equations
5
Lemma 2.3. Let Φ1 , . . . , Φn be subsets of Ω. Then
μ∗
μ∗
n
n
Φi
i1
Φi
i1
≤
n
μ∗ Φi − n − 1μ∗
i1
≤ μ∗ Φ1 μ
n
∗
i2
Φi
n
Φi ,
i1
− n − 1μ∗
2.9
n
Φi .
i1
Lemma 2.4. Let Φ and Γ be subsets of Ω. If μ∗ Φ μ∗ Γ > 1, then the intersection Φ · Γ is infinite.
For any real double sequence {vi,j } defined on a subset of Ω, the level set {i, j ∈ Ω |
vi,j > c} is denoted by v > c. The notations v ≥ c, v < c, and v ≤ c are similarly defined.
Let u {ui,j }i,j∈Ω be a real double sequence. If μ∗ u ≤ 0 0, then u is said to be frequently
positive, and if μ∗ u ≥ 0 0, then u is said to be frequently negative.
u is said to be frequently oscillatory if it is neither frequently positive nor frequently
negative. If μ∗ u > 0 ω ∈ 0, 1, then u is said to have unsaturated upper positive part, and
if μ∗ u > 0 ω ∈ 0, 1, then u is said to have unsaturated lower positive part. u is said to
have unsaturated positive part if μ∗ u > 0 μ∗ u > 0 ω ∈ 0, 1.
The concepts of frequently oscillatory and unsaturated double sequences were
introduced in 5–11. It was also observed that if a double sequence u {ui,j }i,j∈Ω is
frequently oscillatory or has unsaturated positive part, then it is oscillatory; that is, u is not
positive for all large m and n, nor negative for all large m and n. Thus if we can show that
every solution of 1.6 is frequently oscillatory or has unsaturated positive part, then every
solution of 1.6 is oscillatory.
3. Frequently Oscillatory Solutions
Lemma 3.1. Suppose there exist m0 ≥ 2k and n0 ≥ 2l such that
pi m, n ≥ 0
i 1, 2, . . . , h,
qj m, n ≥ 0
j 1, 2, . . . , g
3.1
for m, n ∈ Zm0 − 2k , m0 1 × Zn0 − 2l, n0 1. Let {um,n } be a solution of 1.6. If um,n ≥ 0,
fm, n ≤ 0 for m, n ∈ Zm0 − 2k, m0 1 × Zn0 − 2l, n0 1, then
Δ1 um,n ≤ 0,
Δ2 um,n ≤ 0 for m, n ∈ Z m0 − k, m0 × Z n0 − l, n0 ,
3.2
and if um,n ≤ 0, fm, n ≥ 0 for m, n ∈ Zm0 − 2k, m0 1 × Zn0 − 2l, n0 1, then
Δ1 um,n ≥ 0,
Δ2 um,n ≥ 0 for m, n ∈ Z m0 − k, m0 × Z n0 − l, n0 .
3.3
6
Advances in Difference Equations
Proof. If um,n ≥ 0, fm, n ≤ 0 for m, n ∈ Zm0 − 2k, m0 1 × Zn0 − 2l, n0 1, it follows
from 1.6 and H3 that
um,n um1,n um,n1 h
pi m, num−ki ,n−li −
i1
g
qj m, num−sj ,n−tj − fm, n
j1
⎤
⎡
g
h
≥ um1,n um,n1 θ⎣ pi m, n − qj m, n⎦ − fm, n
i1
3.4
j1
≥ um1,n um,n1 .
Hence Δ1 um,n ≤ 0, Δ2 um,n ≤ 0 for m, n ∈ Zm0 − k, m0 × Zn0 − l, n0 .
Similarly, we also have Δ1 um,n ≥ 0, Δ2 um,n ≥ 0 for m, n ∈ Zm0 − k, m0 × Zn0 −
l, n0 .
Theorem 3.2. Suppose that
μ∗ pi < 0 ωi ,
i 1, 2, . . . , h,
μ∗ f < 0 ωf− ,
μ∗ qj < 0 ωj ,
j 1, 2, . . . , g ,
μ∗ f > 0 ωf ,
⎞
⎛
g
h
μ∗ ⎝
pi < 0
qj < 0 · f > 0 ⎠ ω ,
i1
j1
⎞
⎛
g
h
μ∗ ⎝
pi < 0
qj < 0 · f < 0 ⎠ ω− ,
i1
j1
⎞
⎞
⎛⎛
⎞
⎛
g
g
h
h
μ∗ ⎝⎝ pi − qj ⎠ > 1⎠ > 4 k 1 l 1 ⎝ ωi ωj ωf − h g ω⎠,
i1
j1
i1
j1
3.5
where ωf max{ωf , ωf− } and ω min{ω , ω− }. Then every nontrivial solution of 1.6 is
frequently oscillatory.
Proof. Suppose to the contrary that u {um,n } is a frequently positive solution of 1.6. Then
μ∗ u ≤ 0 0. By Lemmas 2.1–2.3, we have
⎧
⎨
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
1 μ∗ Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎩
⎭
i1
j1
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
μ∗ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
⎩
i1
j1
⎧
⎨
Advances in Difference Equations
⎧
⎤⎫
⎡
g
h
⎬
⎨
2k 2l ⎣
≤ μ∗ Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
⎩
i1
j1
7
4 k1 l1
⎫
⎧ ⎛
⎞
g
h
⎬
⎨
pi < 0 qj < 0 f > 0 ⎠ μ∗ u ≤ 0
× μ∗ ⎝
⎭
⎩
i1
j1
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
Y−1
≤ μ∗ Ω \ X−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
⎩
i1
j1
⎧
⎨
⎞
⎛
g
h
4 k 1 l 1 ⎝ ωi ωj ωf − h g ω⎠
i1
j1
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
< μ∗ Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
⎩
i1
j1
⎧
⎨
⎛⎛
⎞
⎞
g
h
μ∗ ⎝⎝ pi − qj ⎠ > 1⎠.
i1
j1
3.6
Therefore, by Lemma 2.4, the intersection
⎞
⎞
⎤⎫ ⎛⎛
⎡
g
g
h
h
⎬
2k 2l ⎣
Ω \ X−1 Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦ · ⎝⎝ pi − qj ⎠ > 1⎠
⎭
⎩
i1
j1
i1
j1
⎧
⎨
3.7
is infinite. This implies that there exist m0 ≥ 2k and n0 ≥ 2l such that
!
h
pi m0 , n0 −
i1
pi m, n ≥ 0,
g
"
qj m0 , n0 3.8
> 1,
j1
i 1, 2, . . . , h,
qj m, n ≥ 0,
j 1, 2, . . . , g ,
3.9
fm, n ≤ 0,
um,n > 0
8
Advances in Difference Equations
hold for m, n ∈ Zm0 − 2k, m0 1 × Zn0 − 2l, n0 1. In view of 3.9 and Lemma 3.1, we
may see that Δ1 um,n ≤ 0 and Δ2 um,n ≤ 0 for m, n ∈ Zm0 − k, m0 × Zn0 − l, n0 , and hence
θ ≥ um0 ,n0 , so by 3.9 and H3 , we have that
0 ≥ um0 1,n0 um0 ,n0 1 − um0 ,n0 ρ
g
h
pi m0 , n0 − θ qj m0 , n0 − fm0 , n0 i1
⎛
≥ um0 1,n0 um0 ,n0 1 − um0 ,n0 θ⎝
j1
h
g
pi m0 , n0 −
i1
⎞
qj m0 , n0 ⎠ − fm0 , n0 j1
⎞
g
h
um0 ,n0 ⎝ pi m0 , n0 − qj m0 , n0 ⎠ − fm0 , n0 ⎛
≥ um0 1,n0 um0 ,n0 1 − um0 ,n0
i1
⎛
≥ um0 ,n0 ⎝
3.10
j1
⎞
g
h
pi m0 , n0 − qj m0 , n0 − 1⎠ > 0,
i1
j1
which is a contradiction.
In a similar manner, if u {um,n } is a frequently negative solution of 1.6 such that
μ∗ u ≥ 0 0, then we may show that
⎤⎫ ⎛⎛
⎡
⎞
⎞
g
g
h
h
⎬
2k 2l ⎣
Ω \ X−1
pi < 0 qj < 0 f < 0 u ≥ 0⎦ · ⎝⎝ pi − qj ⎠ > 1⎠
Y−1
⎭
⎩
i1
j1
i1
j1
⎧
⎨
3.11
is infinite. Again we may arrive at a contradiction as above. The proof is complete.
Theorem 3.3. Suppose that
μ∗ pi < 0 ωi ,
i 1, 2, . . . , h,
μ∗ f < 0 ωf− ,
μ∗ qj < 0 ωj ,
j 1, 2, . . . , g , μ∗ f > 0 ωf ,
⎛⎛
⎞
⎞
g
h
μ∗ ⎝⎝ pi − qj ⎠ ≤ 1⎠ ω ,
i1
⎛
μ∗ ⎝
⎞⎞
⎞
g
g
h
pi < 0
qj < 0 · f > 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠ ω ,
h
i1
⎛
μ∗ ⎝
j1
⎛⎛
i1
j1
j1
⎞⎞
⎞
g
g
h
pi < 0
qj < 0 · f < 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠ ω− ,
⎛⎛
h
i1
h
ω>
i1
j1
i1
ωi g
j1
ωj ωf ω
hg1
j1
1
− ,
4 hg1 k1 l1
3.12
Advances in Difference Equations
9
where ωf max{ωf , ωf− }, and ω min{ω , ω− }. Then every nontrivial solution of 1.6 is
frequently oscillatory.
Proof. Suppose to the contrary that u {um,n } is frequently positive solution of 1.6. Then
μ∗ u ≤ 0 0. By Lemmas 2.1–2.3, we know
μ
∗
⎧
⎨
Ω\
⎩
⎞
⎞
⎛⎛
⎤⎫
g
g
h
⎬
pi < 0 qj < 0 f > 0 ⎝⎝ pi − qj ⎠ ≤ 1⎠ u ≤ 0⎦
⎭
j1
i1
j1
⎡
h
2k 2l ⎣
X−1
Y−1
i1
⎞ ⎞
⎛⎛
⎤⎫
⎡
g
g
h
h
⎬
2k 2l ⎣
1 − μ∗ X−1
Y−1
pi < 0 qj < 0 f > 0 ⎝⎝ pi − qj ⎠ ≤ 1⎠ u ≤ 0⎦
⎭
⎩
i1
j1
i1
j1
⎧
⎨
≥1−4 k1 l1
⎫
⎧ ⎡
⎞
⎞⎤
⎛⎛
g
g
h
h
⎬
⎨
pi < 0 qj < 0 f > 0 ⎝⎝ pi − qj ⎠ ≤ 1⎠⎦ μ∗ u ≤ 0
× μ∗ ⎣
⎭
⎩
i1
j1
i1
j1
⎡
g
h
∗
⎣
≥1−4 k1 l1
μ pi < 0 μ∗ qj < 0
i1
j1
⎛⎛
⎞
⎞
g
h
∗ ⎝⎝
μ f >0 μ
pi − qj ⎠ ≤ 1⎠ − h g 1
∗
i1
j1
⎞⎞⎤
⎛⎛
⎛
⎞
g
g
h
h
·μ∗ ⎝
pi < 0
qj < 0 · f > 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠⎦
i1
i1
j1
j1
> 0.
3.13
Therefore, by Lemma 2.4, we know that
⎧
⎨
⎞
⎛⎛
⎞
⎤⎫
⎡
g
g
h
h
⎬
2k 2l ⎣
Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 ⎝⎝ pi − qj ⎠ ≤ 1⎠ u ≤ 0⎦
⎩
⎭
i1
j1
i1
j1
3.14
is infinite. This implies that there exist m0 ≥ 2k and n0 ≥ 2l such that 3.8 and
pi m, n ≥ 0,
i 1, 2, . . . , h,
fm, n ≤ 0,
qj m, n ≥ 0,
um,n > 0
j 1, 2, . . . , g ,
3.15
hold for m, n ∈ Zm0 − 2k, m0 1 × Zn0 − 2l, n0 1. By similar discussions as in the proof
of Theorem 3.2, we may arrive at a contradiction against 3.8.
10
Advances in Difference Equations
In case u {um,n } is a frequently negative solution of 1.6, then μ∗ u ≥ 0 0. In an
analogous manner, we may see that
⎞
⎛⎛
⎞
⎤⎫
⎡
g
g
h
h
⎬
2k 2l ⎣
Ω \ X−1
Y−1
pi < 0 qj < 0 f < 0 ⎝⎝ pi − qj ⎠ ≤ 1⎠ u ≥ 0⎦
⎭
⎩
i1
j1
i1
j1
⎧
⎨
3.16
is infinite. This can lead to a contradiction again. The proof is complete.
4. Unsaturated Solutions
The methods used in the above proofs can be modified to obtain the following results for
unsaturated solutions.
Theorem 4.1. Suppose there exists constant ω0 ∈ 0, 1 such that
μ∗ pi < 0 ωi ,
i 1, 2, . . . , h,
μ∗ f < 0 ωf− ,
μ∗ qj < 0 ωj ,
j 1, 2, . . . , g ,
μ∗ f > 0 ωf ,
⎞
⎛
g
h
μ∗ ⎝
pi < 0
qj < 0 · f > 0 ⎠ ω ,
i1
j1
⎞
⎛
g
h
μ∗ ⎝
pi < 0
qj < 0 · f < 0 ⎠ ω− ,
i1
j1
⎞
⎞
⎞
⎛
g
g
h
h
μ∗ ⎝⎝ pi − qj ⎠ > 1⎠ > 4 k 1 l 1 ⎝ ωi ωj ωf ω0 − h g ω⎠,
⎛⎛
i1
j1
i1
j1
4.1
where ωf max{ωf , ωf− }, and ω max{ω , ω− }. Then every nontrivial solution of 1.6 has
unsaturated upper positive part.
Proof. Let u {um,n } be a nontrivial solution of 1.6. We assert that μ∗ u > 0 ∈ ω0 , 1.
Otherwise, then μ∗ u > 0 ≤ ω0 or μ∗ u > 0 1. In the former case, applying arguments
similar to the proof of Theorem 3.2, we may then arrive at the fact that
⎞
⎞
⎤⎫ ⎛⎛
⎡
g
g
h
h
⎬
2k 2l ⎣
Y−1
Ω \ X−1
pi < 0 qj < 0 f < 0 u > 0⎦ · ⎝⎝ pi − qj ⎠ > 1⎠
⎭
⎩
i1
j1
i1
j1
⎧
⎨
4.2
is infinite and a subsequent contradiction.
Advances in Difference Equations
11
In the latter case, we have μ∗ u ≤ 0 0. By Lemmas 2.1–2.3, we have
⎧
⎨
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
1 μ∗ Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎩
⎭
i1
j1
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
μ∗ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
⎩
i1
j1
⎧
⎨
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
≤ μ∗ Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
⎩
i1
j1
⎧
⎨
⎧ ⎛
⎫
⎞
g
h
⎬
⎨
4 k1 l1
μ∗ ⎝
pi < 0 qj < 0 f > 0 ⎠ μ∗ u ≤ 0
⎩
⎭
i1
j1
⎤⎫
⎡
g
h
⎬
2k 2l ⎣
≤ μ∗ Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
⎩
i1
j1
⎧
⎨
4.3
⎛
⎞
g
h
4 k 1 l 1 ⎝ ωi ωj ωf ω0 − h g ω⎠
i1
<μ
∗
⎧
⎨
Ω\
⎩
j1
⎤⎫
g
⎬
pi < 0 qj < 0 f > 0 u ≤ 0⎦
⎭
j1
⎡
h
2k 2l ⎣
X−1
Y−1
i1
⎛⎛
⎞
⎞
g
h
μ∗ ⎝⎝ pi − qj ⎠ > 1⎠.
i1
j1
Therefore, by Lemma 2.4, we know that the set
⎞
⎞
⎤⎫ ⎛⎛
⎡
g
g
h
h
⎬
2k 2l ⎣
Ω \ X−1 Y−1
pi < 0 qj < 0 f > 0 u ≤ 0⎦ · ⎝⎝ pi − qj ⎠ > 1⎠
⎭
⎩
i1
j1
i1
j1
⎧
⎨
4.4
is infinite. Then by discussions similar to these in the proof of Theorem 3.2 again, we may
arrive at a contradiction. The proof is complete.
Combining Theorems 3.3 and 4.1, we have the following Theorem 4.2 and the proof of
this theorem is omitted.
12
Advances in Difference Equations
Theorem 4.2. Suppose there exists constant ω0 ∈ 0, 1 such that
μ∗ pi < 0 ωi ,
μ∗ qj < 0 ωj ,
j 1, 2, . . . , g ,
⎞
⎛⎛
⎞
g
h
∗ ⎝⎝
μ
pi − qj ⎠ ≤ 1⎠ ω ,
i 1, 2, . . . , h,
μ∗ f < 0 ωf− ,
i1
μ∗ f > 0 ωf ,
j1
⎞⎞
⎛⎛
⎛
⎞
g
g
h
h
μ∗ ⎝
pi < 0
qj < 0 · f > 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠ ω ,
i1
i1
j1
j1
⎛
⎞
⎞⎞
⎛⎛
g
g
h
h
μ∗ ⎝
pi < 0
qj < 0 · f < 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠ ω− ,
i1
i1
j1
h
i1
ω>
ωi g
j1
ωj ωf ω ω0
hg1
j1
1
− ,
4 hg1 k1 l1
4.5
where ωf max{ωf , ωf− }, and ω min{ω , ω− }. Then every nontrivial solution of 1.6 has
unsaturated upper positive part.
Theorem 4.3. Suppose there exists constant ω0 ∈ 0, 1 such that
μ∗ pi < 0 ωi ,
i 1, 2, . . . , h,
∗
μ f < 0 ωf− ,
μ∗ qj < 0 ωj ,
μ∗ ⎝
μ∗ ⎝
μ∗ f > 0 ωf ,
⎞
⎞
g
h
μ∗ ⎝⎝ pi − qj ⎠ ≤ 1⎠ ω ,
j1
⎞
⎞⎞
⎛⎛
g
g
h
pi < 0
qj < 0 · f > 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠ ω ,
h
i1
⎛
j 1, 2, . . . , g ,
⎛⎛
i1
⎛
i1
j1
j1
⎞⎞
⎛⎛
⎞
g
g
h
pi < 0
qj < 0 · f < 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠ ω− ,
h
i1
i1
j1
j1
⎛
⎞
g
h
4 k 1 l 1 ⎝ ωi ωj ωf ω ω0 − h g 1 ω⎠ < 1,
i1
j1
4.6
where ωf max{ωf , ωf− }, and ω min{ω , ω− }. Then every nontrivial solution of 1.6 has an
unsaturated upper positive part.
Advances in Difference Equations
13
Proof. We claim that μ∗ u > 0 ∈ ω0 , 1. First, we prove that μ∗ u > 0 > ω0 . Otherwise, if
μ∗ u > 0 ≤ ω0 , by Lemmas 2.1–2.3, we have
⎧
⎨
μ∗
⎡
g
h
2k 2l ⎣
Ω \ X−1
Y−1
pi < 0 qj < 0 f < 0
⎩
i1
j1
⎛⎛
⎞
⎞⎤⎫
g
h
⎬
2k 2l
⎝⎝ pi − qj ⎠ ≤ 1⎠⎦ μ∗ Ω \ X−1
Y−1 u > 0
⎭
i1
j1
2 − μ∗
⎧
⎨
⎡
g
h
2k 2l ⎣
X−1
Y−1
pi < 0 qj < 0 f < 0
⎩
i1
j1
⎛⎛
⎞⎤⎫
⎞
g
h
⎬
2k 2l
⎝⎝ pi − qj ⎠ ≤ 1⎠⎦ − μ∗ X−1
Y−1 u > 0
⎭
i1
j1
⎧
g
h
⎨
≥2−4 k1 l1
μ∗ pi < 0 μ∗ qj < 0
⎩ i1
j1
4.7
⎛⎛
⎞
⎞
g
h
μ f < 0 μ∗ ⎝⎝ pi − qj ⎠ ≤ 1⎠ μ∗ u > 0
∗
i1
j1
⎛
g
h
− h g 1 μ∗ ⎝
pi < 0
qj < 0
i1
j1
⎞⎞⎫
g
h
⎬
· f < 0 · ⎝⎝ pi − qj ⎠ ≤ 1⎠⎠ > 1.
⎭
i1
j1
⎛⎛
⎞
Hence, by Lemma 2.4, we see that
⎞⎤⎫
⎛⎛
⎞
⎡
g
g
h
h
⎬
2k 2l ⎣
Ω \ X−1
Y−1
pi < 0 qj < 0 f < 0 ⎝⎝ pi − qj ⎠ ≤ 1⎠⎦
⎭
⎩
i1
j1
i1
j1
⎧
⎨
4.8
2k 2l
· Ω \ X−1
Y−1 u > 0
is infinite. Then there exist m0 ≥ 2k and n0 ≥ 2l such that 3.8 and
pi m, n ≥ 0,
i 1, 2, . . . , h,
fm, n ≤ 0,
qj m, n ≥ 0,
um,n ≤ 0
j 1, 2, . . . , g ,
4.9
14
Advances in Difference Equations
hold for m, n ∈ Zm0 − 2k, m0 1 × Zn0 − 2l, n0 1. Applying similar discussions as in
the proof of Theorem 3.2, we can get a contradiction. Next, we prove that μ∗ u > 0 < 1.
Otherwise, μ∗ u ≤ 0 0. Analogously, we see that
⎞
⎞⎤ ⎫
⎛⎛
⎡
g
g
h
h
⎬
2k 2l ⎣
Ω \ X−1
Y−1
pi < 0 qj < 0 f > 0 ⎝⎝ pi − qj ⎠ ≤ 1⎠⎦
⎭
⎩
i1
j1
i1
j1
⎧
⎨
4.10
2k 2l
· Ω \ X−1
Y−1 u ≤ 0
is infinite. Then, we can also lead to a contradiction. The proof is complete.
We remark that every nontrivial solution of 1.6 has an unsaturated lower positive
part under the same conditions as Theorems 4.1, 4.2, or 4.3. So we can obtain that every
nontrivial solution of 1.6 has an unsaturated positive part.
5. Examples
We give one example to illustrate our previous results.
Example 5.1. Consider the partial difference equation
um1,n um,n1 − um,n p1 m, num−4,n−3 p2 m, num−3,n−2 − q1 m, num−1,n−1 fm, n,
5.1
where
fm, n ⎧
⎨1,
m 30η,
⎩0,
otherwise
n 10ξ,
η, ξ ∈ N
p1 m, n 1,
,
p2 m, n 3,
5.2
and q1 m, n 2.
It is clear that μ∗ p1 < 0 0, μ∗ p2 < 0 0, μ∗ q1 < 0 0, μ∗ f > 0 1/300,
∗
μ f < 0 0, and μ∗ p1 p2 q1 > 1 1.
Moreover,
μ∗
p1 < 0 p2 < 0 q1 < 0 f < 0 0,
μ∗
μ∗
p1 < 0 p2 < 0 q1 < 0 f > 0 0,
1
4 k 1 l 1 ωf .
p1 p2 q1 > 1 1 > 4 × 4 1 × 3 1 ×
300
5.3
Then according to Theorems 3.2 or 3.3, we know that every nontrivial solution of 5.1 is
frequently oscillatory. If ω0 ∈ 0, 11/1200, we see that all conditions in Theorems 4.1, 4.2, or
4.3 are satisfied. Thus, every nontrivial solution of 5.1 has an unsaturated upper positive
part.
Advances in Difference Equations
15
Acknowledgments
This project was supported by the NNSF of P.R.China 60604004 and by Natural Science
Foundation of Hebei province no. 07M005.
References
1 S. T. Liu and B. G. Zhang, “Oscillatory behavior of delay partial difference equations with positive
and negative coefficients,” Computers & Mathematics with Applications, vol. 43, no. 8-9, pp. 951–964,
2002.
2 S. T. Liu, Y. Q. Liu, and F. Q. Deng, “Oscillation for nonlinear delay partial difference equations with
positive and negative coefficients,” Computers & Mathematics with Applications, vol. 43, no. 10-11, pp.
1219–1230, 2002.
3 J. Yang, Y. J. Zhang, and S. S. Cheng, “Frequent oscillation in a nonlinear partial difference equation,”
Central European Journal of Mathematics, vol. 5, no. 3, pp. 607–618, 2007.
4 B. G. Zhang and Q. Xing, “Oscillation of certain partial difference equations,” Journal of Mathematical
Analysis and Applications, vol. 329, no. 1, pp. 567–580, 2007.
5 J. Yang and Y. J. Zhang, “Frequent oscillatory solutions of a nonlinear partial difference equation,”
Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 492–499, 2009.
6 Z. Q. Zhu and S. S. Cheng, “Unsaturated solutions for partial difference equations with forcing
terms,” Central European Journal of Mathematics, vol. 4, no. 4, pp. 656–668, 2006.
7 C. J. Tian, S.-L. Xie, and S. S. Cheng, “Measures for oscillatory sequences,” Computers & Mathematics
with Applications, vol. 36, no. 10–12, pp. 149–161, 1998.
8 S. S. Cheng, Partial Difference Equations, vol. 3 of Advances in Discrete Mathematics and Applications,
Taylor & Francis, London, UK, 2003.
9 Z. Q. Zhu and S. S. Cheng, “Frequently oscillatory solutions of neutral difference equations,” Southeast
Asian Bulletin of Mathematics, vol. 29, no. 3, pp. 627–634, 2005.
10 Z. Q. Zhu and S. S. Cheng, “Frequently oscillatory solutions for multi-level partial difference
equations,” International Mathematical Forum, vol. 1, no. 29–32, pp. 1497–1509, 2006.
11 B. Zhang and Y. Zhou, Qualitative Analysis of Delay Partial Difference Equations, vol. 4 of Contemporary
Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2007.