DOI: 10.2478/s11533-006-0030-9
Research article
CEJM 4(4) 2006 656–668
Unsaturated solutions for
partial difference equations with forcing terms
Zhi-Qiang Zhu1 , Sui Sun Cheng2∗
1
Department of Computer Science,
Guangdong Polytechnic Normal University,
Guangzhou 510665, P.R. China
2
Department of Mathematics,
Tsing Hua University, Hsinchu,
Taiwan 30043, R.O. China
Received 7 June 2006; accepted 24 July 2006
Abstract: The concept of unsaturated infinite double sequence is introduced by making use of frequency
measures. Unsaturated solutions are then studied for a partial difference equation. Conditions for all
solutions to be unsaturated are obtained. Since unsaturated solutions are oscillatory, our results yield
oscillation criteria.
c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.
Keywords: Partial difference equation, frequency measure, unsaturated solution, oscillation criteria
MSC (2000): 39A11
1
Introduction
Let N = {0, 1, 2, ...} and Z+ = {1, 2, 3, ...} . In this paper, we are concerned with partial
difference equations of the form
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)
2
for (i, j) ∈ Z+ , where f (i, j), P1 (i, j), P2 (i, j) and P3 (i, j) = −2 are all real functions
defined on Z+2 , Δ1 u(i, j) = u(i + 1, j) − u(i, j) and Δ2 u(i, j) = u(i, j + 1) − u(i, j).
Such an equation may arise from numerical simulations of solutions of partial difference
∗
E-mail: [email protected]
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equations [1, 2]. For example, consider an initial-boundary problem of the form
aux + ut = 0, x > 0, t > 0,
u(x, 0) = g(x), x > 0.
By means of the finite difference method for calculating an approximate solution u(ih, jτ ) =
ui,j of this problem, we are led to the following difference equation
h
a(ui,j − ui−1,j ) + (ui,j − ui,j−1 ) = 0, i, j = 1, 2, 3, ...,
τ
which can be regarded as a special case of (1).
Equation (1) may also arise when stationary heat distribution is sought in the dynamic
heat diffusion of a plane lattice molecules. More specifically, consider the temperature
distribution of a large set of molecules in the plane. Assume that the molecules are bonded
in a regular manner such that each molecule can be labeled by (i, j) where i, j ∈ Z. Let
(t)
the temperature of each molecule in the time period t ∈ N be denoted by uij . In the time
(t)
(t)
period t, if the temperature uij of the (i, j) molecule is lower than the temperature ui+1,j
of the neighboring molecule (i+1, j), then heat will flow from the (i+1, j) molecule to the
(i, j) molecule. The amount of increase of temperature in molecule (i, j) from time t to
(t+1)
(t)
time t + 1 is ui,j − uij , and it is reasonable to postulate that the increase is proportional
(t)
(t)
to the difference ui−1,j − uij , that is,
(t+1)
(t)
(t)
(t)
uij − uij = α ui−1,j − uij ,
where α is a proportionality constant. Similarly, heat may flow from the other three
neighbors. Thus, it is reasonable that the total effect under the superposition principle is
(t+1)
(t)
(t)
(t)
(t)
(t)
(t)
(t)
(t)
(t)
uij − uij = α ui−1,j − uij + β ui,j−1 − uij + γ ui+1,j − uij + δ ui,j+1 − uij .
(2)
If we assume further that each molecule can also allow a control mechanism (such as
heat storing or releasing mechanism) or a perturbation, then a more general evolutionary
equation may result:
(t+1)
(t)
(t)
(t)
(t)
(t)
uij − uij = α ui−1,j − uij + β ui,j−1 − uij
(t)
(t)
(t)
(t)
+ γ ui+1,j − uij + δ ui,j+1 − uij + Q(i, j).
A special solution may arise that is independent of time. Such a solution u(i, j) then
satisfies
0 = α (u(i − 1, j) − u(i, j)) + β (u(i, j − 1) − u(i, j))
+γ (u(i + 1, j) − u(i, j)) + δ (u(i, j + 1, j) − u(i, j)) + Q(i, j).
In particular, when γ = 0 = δ, we end up with the equation
αΔ1 u(i − 1, j) + βΔu2 (i, j − 1) + Q(i, j) = 0,
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which is of the form (1).
A solution of (1) is a double sequence {u(i, j)} defined for (i, j) ∈ N 2 \{(0, 0)} which
renders (1) into an identity for each (i, j) ∈ Z+2 by substitution. In case P3 (i, j) = −2 for
(i, j) ∈ Z+2 , the existence of solutions of (1) is easily established. Indeed, since (1) can be
rewritten as
(2 + P3 (i, j))u(i, j) + (P1 (i, j) − 1)u(i − 1, j) + (P2 (i, j) − 1)u(i, j − 1) = f (i, j),
if boundary distributions of the form
u(0, j) = ϕj , j ∈ Z+ ,
(3)
u(i, 0) = φi , i ∈ Z+
(4)
and
are given, we can calculate from (1)
u(1, 1); u(1, 2), u(2, 1), u(2, 2); u(1, 3), u(2, 3), u(3, 1), u(3, 2), u(3, 3); . . .
successively in an unique manner. We remark that since we will be concerned with
asymptotic properties of solutions of (1), we may, for the sake of convenience, extend
the domain of definition of a solution of (1) by including the point (0, 0). Henceforth, a
solution of (1) is a double sequence u = {u(i, j)} defined for (i, j) ∈ N 2 which renders
(1) into an identity for each (i, j) ∈ Z+2 by substitution.
As usual, a solution of (1) or, in general, a double sequence {u(i, j)} defined on N 2
is said to be eventually positive if there is a positive integer M such that u(i, j) > 0
for all i ≥ M and j ≥ M . Eventually negative, eventually nonpositive and eventually
nonnegative solutions are similarly defined. {u(i, j)} is said to be oscillatory if for any
subset of N 2 of the form {(i, j) ∈ N 2 : i, j ≥ T } , there are (α, β), (s, t) in it such that
u(α, β)u(s, t) ≤ 0.
In this note, we will be concerned with solutions of (1) which have “unsaturated upper
positive parts”. A solution with “unsaturated upper positive part” cannot be eventually
positive nor eventually nonpositive. Thus by finding unsaturated solutions, we are able
to find oscillatory solutions as well.
2
Main results
To define unsaturated solutions, we need to recall some terminologies in [3]. Let S be
a countable set. The notation |S| denotes the number of elements of S. The union and
intersection of two sets S1 and S2 are denoted by S1 + S2 and S1 · S2 respectively. A
lattice point z = (i, j) is a point in the plane with integer coordinates. A neighbor of
lattice point (i, j) means the lattice point (i − 1, j), (i + 1, j), (i, j + 1) or (i, j − 1). Let
Ω be a subset of N 2 . The lattice points z1 , z2 , ..., zn is said to form a path with terminals
z1 and zn if z1 is a neighbor of z2 , z2 is a neighbor of z3 , etc. The subset Ω of N 2 is
said to be connected if any two of its points are terminals of a path in Ω. A nonempty
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connected subset Ω of N 2 is called a domain. Let Ω be a domain of N 2 and z ∈ N 2 ,
z is said to be an exterior boundary point of Ω if it does not belong to Ω but has at
least one neighbor in Ω. The set of all exterior boundary points of Ω will be denoted
by ∂Ω. For any z = (i, j) ∈ N 2 , let zL = (i − 1, j), zR = (i + 1, j), zT = (i, j + 1) and
zD = (i, j − 1). Further, we let ∂L Ω = {z ∈ ∂Ω : zR ∈ Ω}, ∂R Ω = {z ∈ ∂Ω : zL ∈ Ω},
∂T Ω = {z ∈ ∂Ω : zD ∈ Ω} and ∂D Ω = {z ∈ ∂Ω : zT ∈ Ω}.
For k ∈ N, let
Wk = (i, j) ∈ N 2 : max {i, j} = k .
For any n ∈ N and any subset Ω of N 2 , let Ω(n) = {(i, j) ∈ Ω ∩ Wk : k ≤ n}, X n Ω =
{(i + n, j) : (i, j) ∈ Ω} and Y n Ω = {(i, j + n) : (i, j) ∈ Ω}. Let α, β, γ, δ ∈ N with α ≤ β
and γ ≤ δ, the union of δn=γ βm=α X m Y n Ω will be denoted by Xαβ Yγδ Ω.
We can easily see that for any lattice point (i, j) ∈ N 2 , it holds that
(i, j) ∈ N 2 − Xαβ Yγδ Ω ⇔ (i − s, j − t) ∈ N 2 − Ω
(5)
for α ≤ s ≤ β and γ ≤ t ≤ δ, where (i − s, j − t) is null when i < α or j < γ.
The frequency measure on lattice planes has been defined in [3]. A variant is defined
below.
Definition 2.1. Let Ω be a subset of N 2 . If
lim sup
n→∞
|Ω(n) |
(n + 1)2
exists, then this limit, denoted by μ∗ (Ω), will be called the upper frequency measure of
Ω. Similarly, if
|Ω(n) |
lim inf
n→∞ (n + 1)2
exists, this limit, denoted by μ∗ (Ω), will be called the lower frequency measure of Ω. If
μ∗ (Ω) = μ∗ (Ω), then the common limit denoted by μ(Ω) will be called the frequency
measure of Ω.
For any double sequence x = {x(i, j)}(i,j)∈S and any real numbers c1 and c2 , let
(x > c1 ) = {(i, j) ∈ S : x(i, j) > c1 }. The set (x ≥ c1 ), (x < c1 ) etc. have similar
meanings.
Definition 2.2. Let u = {u(i, j)}i,j∈N 2 be any double sequence. If μ∗ (u > 0) = ω ∈
(0, 1), then u is said to have unsaturated upper positive part. If μ∗ (u > 0) = ω ∈ (0, 1),
then u is said to have unsaturated lower positive part. The double sequence u is said to
have unsaturated positive part if μ∗ (u > 0) = μ∗ (u > 0) = ω ∈ (0, 1).
A double sequence u with unsaturated negative part can be defined accordingly. It
is easy to see that if u = {u(i, j)} is eventually positive or eventually nonpositive, then
μ∗ (u > 0) = 1 or μ∗ (u > 0) = 0 respectively. Hence, a sequence u = {u(i, j)} with
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unsaturated positive part cannot be eventually positive nor eventually nonpositive. In
other words, u is oscillatory.
The following properties of frequency measures hold. They are similar to those in [3].
Lemma 2.3. Let Ω and Γ be subsets of N 2 . The following hold:
(i) if Ω ⊆ Γ, then μ∗ (Ω) ≤ μ∗ (Γ) and μ∗ (Ω) ≤ μ∗ (Γ);
(ii) μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ) ≤ μ∗ (Ω + Γ) ≤ μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ);
(iii) μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ) ≤ μ∗ (Ω + Γ) ≤ μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ);
(iv) μ∗ (Ω) + μ∗ (N 2 − Ω) = 1;
(v) if Ω · Γ is finite, then μ∗ (Ω) + μ∗ (Γ) ≤ 1.
Lemma 2.4. Let Ω be a subset of N 2 and α, β, γ and δ be integers such that α ≤ β and
γ ≤ δ. Then
μ∗ (Xαβ Yγδ Ω) ≤ (β − α + 1)(δ − γ + 1)μ∗ (Ω)
and
μ∗ (Xαβ Yλτ Ω) ≤ (β − α + 1)(δ − γ + 1)μ∗ (Ω).
From Lemma 2.3, we can easily obtain the following result.
Lemma 2.5. If Ω1 , ..., Ωn be subsets of N 2 , then
n
n
n
Ωi ≤
μ∗ (Ωi ) − (n − 1)μ∗
Ωi
μ∗
i=1
and
μ∗
n
i=1
Ωi
≤ μ∗ (Ω1 ) +
i=1
(6)
i=1
n
μ∗ (Ωi ) − (n − 1)μ∗
n
i=2
Ωi
.
(7)
i=1
Next, we will make use of the frequency measures defined above to establish some
criteria for solutions of (1) to have unsaturated positive parts. Before doing so, we quote
a result in [3].
Lemma 2.6. Let Ω be a finite domain of N 2 and suppose {u(i, j)} is a bivariate sequence
such that Δ1 u(i − 1, j) and Δ2 u(i, j − 1) are defined on Ω. Then
Δ1 u(i − 1, j) =
u(i − 1, j) −
u(i, j)
as well as
(i,j)∈Ω
(i,j)∈∂R Ω
(i,j)∈∂L Ω
Δ2 u(i, j − 1) =
(i,j)∈Ω
(i,j)∈∂T Ω
u(i, j − 1) −
u(i, j).
(i,j)∈∂D Ω
2.1 Nonhomogenous case
This section is concerned with (1) under the condition that the forcing term is not identically zero.We will denote the set {a, a + 1, · · · , b} of integers by Z[a, b] and the Cartesian
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product of Z[a, b] and Z[c, d] by Z[a, b] × Z[c, d].
Theorem 2.7. Suppose that
P3
P3
∗
∗
= ω1 , μ P2 > 1 +
= ω2 , μ∗ (P3 > −2) = ω3
μ P1 > 1 +
2
2
+
∗
∗
μ (f > 0) = ωf , μ (f < 0) = ωf− .
Suppose further that there exists a constant ω0 ∈ (0, 1) such that
9(ω1 + ω2 + ω3 + ωf + ω0 ) < 1,
(8)
where ωf = max{ωf− , ωf+ }. Then every solution u = {u(i, j)} of (1) has unsaturated upper
positive part.
Proof. We assert that μ∗ (u > 0) ∈ (ω0 , 1). Otherwise, there are two cases, that is
μ∗ (u > 0) ≤ ω0 or μ∗ (u > 0) = 1. In case μ∗ (u > 0) ≤ ω0 , in view of Lemma 2.3(ii),(iv),
Lemma 2.4 and (8), we have
P3
P3
2
1
1
μ∗ N − X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f > 0)
2
2
2
∗
1
1
+μ N − X−1 Y−1 (u > 0)
P3
P3
∗
1
1
= 2 − μ X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f > 0)
2
2
1 1
−μ∗ X−1 Y−1 (u > 0)
P3
P3
∗
∗
∗
∗
≥ 2 − 9 μ (P1 > 1 + ) + μ (P2 > 1 + ) + μ (P3 > −2) + μ (f > 0) + μ∗ (u > 0)
2
2
> 1.
Then, by Lemma 2.3(v), the intersection
P3
P3
2
1
1
N − X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f > 0) ∩
2
2
2
1
1
N − X−1
Y−1
(u > 0)
is infinite. This, together with (5), implies that there exists a lattice point (ac , bc ) ∈ N 2
with (ac − 1, bc − 1) ∈ N 2 such that when (i, j) ∈ Z[ac − 1, ac + 1] × Z[bc − 1, bc + 1],
P3 (i, j) ≤ −2,
(9)
P3 (i, j)
P3 (i, j)
, P2 (i, j) ≤ 1 +
, f (i, j) ≤ 0.
2
2
From (1) and (10), we obtain for (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1] that
u(i, j) ≤ 0, P1 (i, j) ≤ 1 +
(10)
Δ1 u(i − 1, j) + Δ2 u(i, j − 1) ≤ 0, (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1].
(11)
Let us set Ω = {(ac , bc )} and view P1 (ac , bc ) as P1 , P2 (ac , bc ) as P2 and P3 (ac , bc ) as P3 .
According to (11), we need to deal with three cases.
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(i) Δ1 u(ac − 1, bc ) ≤ 0 and Δ2 u(ac , bc − 1) ≤ 0. Then, by Lemma 2.6 and (10), we
have from (1) that
u(i − 1, j) −
u(i, j) +
u(i, j − 1) −
u(i, j)
0≥
(i,j)∈∂R Ω
P1
(i,j)∈∂L Ω
u(i − 1, j) + P2
(i,j)∈Ω
u(i, j) −
(i,j)∈∂L Ω
+P1
(i,j)∈∂L Ω
u(i, j) +
u(i, j) + (2 + P3 )
(i,j)∈∂D Ω
≥−
u(i, j)
(i,j)∈Ω
(i,j)∈∂D Ω
u(i, j) + P2
(i,j)∈∂D Ω
u(i, j − 1) + P3
(i,j)∈Ω
=−
(i,j)∈∂T Ω
u(i, j) + P1
(i,j)∈∂L Ω+∂D Ω
(i,j)∈∂L Ω
(i,j)∈∂L Ω+∂D Ω
(i,j)∈∂L Ω
u(i, j)
(i,j)∈Ω
u(i, j) + P2
u(i, j)
(i,j)∈∂D Ω
P3 P3 +(1 + )
u(i − 1, j) + (1 + )
u(i, j − 1)
2
2
(i,j)∈Ω
(i,j)∈Ω
u(i, j) + P1
u(i, j)
=−
+P2
u(i, j) + (1 +
(i,j)∈∂D Ω
≥−
P3
)
2
u(i, j) + 2(1 +
(i,j)∈∂L Ω+∂D Ω
= (1 + P3 )
u(i, j)
(i,j)∈∂L Ω+∂D Ω
P3
)
2
u(i, j)
(i,j)∈∂L Ω+∂D Ω
u(i, j),
(i,j)∈∂L Ω+∂D Ω
this means that P3 ≥ −1, which is contrary to (9).
(ii) Δ1 u(ac − 1, bc ) ≥ 0 and Δ2 u(ac , bc − 1) ≤ 0. Note from (9)-(10) that (1) implies
0 ≥ Δ2 u(i, j − 1) + P3 (i, j)u(i, j), (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1],
so that by Lemma 2.6 , we have
u(i, j − 1) −
0≥
(i,j)∈∂T Ω
u(i, j) + P3
(i,j)∈∂D Ω
u(i, j)
(i,j)∈Ω
= −u(ac , bc − 1) + (1 + P3 )u(ac , bc )
≥ −u(ac , bc − 1) + (1 + P3 )u(ac , bc − 1) = P3 u(ac , bc − 1).
Now we see that P3 ≥ 0 which is contrary to (9).
(iii) Δ1 u(ac − 1, bc ) ≤ 0 and Δ2 u(ac , bc − 1) ≥ 0. Since (1) implies
0 ≥ Δ1 u(i − 1, j) + P3 (i, j)u(i, j), (i, j) ∈ Z[ac , ac + 1] × Z[bc , bc + 1],
by similar arguments as above, we are led to the same contradiction as P3 ≥ 0.
In case μ∗ (u > 0) = 1, then in view of Lemma 2.3(iv), μ∗ (u ≤ 0) = 0. Note from (8)
that
9(ω1 + ω2 + ω3 + ωf ) < 1.
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By the same reasons as above, we see that
P3
P3
2
1
1
N − X−1 Y−1 (P1 > 1 + ) + (P2 > 1 + ) + (P3 > −2) + (f < 0) ∩
2
2
2
1
1
N − X−1
Y−1
(u ≤ 0)
is infinite. Hence, there exists a lattice point (ac , bc ) ∈ N 2 with (ac − 1, bc − 1) ∈ N 2 such
that
u(i, j) > 0, P1 (i, j) ≤ 1 +
P3 (i, j)
P3 (i, j)
, P2 (i, j) ≤ 1 +
, f (i, j) ≥ 0, P3 (i, j) ≤ −2
2
2
for (i, j) ∈ Z[ac − 1, ac + 1] × Z[bc − 1, bc + 1]. By similar arguments as above, we may
show that μ∗ (u > 0) = 1 does not hold. The proof is complete.
We remark that the conditions P1 (i, j) ≤ 1 + P3 (i, j)/2 and P2 (i, j) ≤ 1 + P3 (i, j)/2 in
(10) can be replaced by P1 (i, j) ≤ 0 and P2 (i, j) ≤ 0. We remark further that the proof
above requires the inequality μ∗ (Ω + Γ) ≤ μ∗ (Ω) + μ∗ (Γ) − μ∗ (Ω · Γ) and the right side
has been magnified.
Theorem 2.8. Suppose that
μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (P3 > −2) = ω3 , μ∗ (f > 0) = ωf+ ,
μ∗ (f < 0) = ωf− , μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] = ω +
and
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f < 0)] = ω − .
Suppose further that there exists a constant ω0 ∈ (0, 1) such that
9(ω1 + ω2 + ω3 + ωf + ω0 − 3ω) < 1,
(12)
where ωf = max{ωf− , ωf+ } and ω = min{ω − , ω + }. Then every solution u = {u(i, j)} of
(1) has unsaturated upper positive part.
Proof. In case μ∗ (u > 0) ≤ ω0 . In view of Lemma 2.3(iv), Lemma 2.4, (6) in Lemma 2.5
and (12), we have
1
1
μ∗ N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0)]
1
1
Y−1
(u > 0)
+μ∗ N 2 − X−1
1 1
Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0)]
= 2 − μ∗ X−1
1 1
Y−1 (u > 0)
−μ∗ X−1
≥ 2 − 9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (P3 > −2) + (f > 0)
− 3μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] + μ∗ (u > 0)}
> 1.
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Then, by Lemma 2.3(v), the intersection
1
1
1
1
N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0)] ∩ N 2 − X−1
Y−1
(u > 0)
is infinite. Similar to those in Theorem 2.7, we will arrive at a contradiction. Consequently, μ∗ (u > 0) ≤ ω0 does not hold. Analogously, we can show that μ∗ (u > 0) = 1.
That is, μ∗ (u > 0) ∈ (ω0 , 1). The proof is complete.
Theorem 2.9. Suppose that
μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (f > 0) = ωf+ , μ∗ (f < 0) = ωf−
and
μ∗ [(P1 > 0) · (P2 > 0) · (f > 0)] = ω + , μ∗ [(P1 > 0) · (P2 > 0) · (f < 0)] = ω − .
Suppose further that there exist a constant ω0 ∈ (0, 1) such that
μ∗ (P3 ≤ −2) > 9(ω1 + ω2 + ωf + ω0 − 2ω),
(13)
where ωf = max{ωf− , ωf+ } and ω = min{ω − , ω + }. Then every solution u = {u(i, j)} of
(1) has unsaturated upper positive part.
Proof. Similar to the proofs to Theorem 2.7-2.8, we need only to prove that μ∗ (u > 0) ≤
ω0 and μ∗ (u > 0) = 1 do not hold. Indeed, by Lemma 2.3(iv), we have
1
1
1 = μ∗ N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)]
1 1
Y−1 [(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)]
+μ∗ X−1
1
1
Y−1
[(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)] +
≤ μ∗ N 2 − X−1
9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (f > 0) + μ∗ (u > 0)
−2μ∗ [(P1 > 0) · (P2 > 0) · (f > 0)]}
1
1
< μ∗ N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)] + μ∗ (P3 ≤ −2)
when μ∗ (u > 0) ≤ ω0 , and
1
1
1 = μ∗ N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)]
1 1
Y−1 [(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)]
+μ∗ X−1
1
1
Y−1
[(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)] + μ∗ (P3 ≤ −2)
< μ∗ N 2 − X−1
when μ∗ (u > 0) = 1 (which implies that μ∗ (u ≤ 0) = 0). Hence, the intersections
and
1
1
N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (f > 0) + (u > 0)] ∩ (P3 ≤ −2)
1
1
N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (f < 0) + (u ≤ 0)] ∩ (P3 ≤ −2)
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are infinite. The rest of the proof is analogous to those as above and we will reach the
fact that μ∗ (u > 0) ∈ (ω0 , 1). The proof is complete.
Note that
1
1
μ∗ N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)]
1 1
Y−1 [(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)]
= 1 − μ∗ X−1
≥ 1 − 9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (P3 > 0) + μ∗ (f > 0) + μ∗ (u > 0)}
+27μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] .
Thus, when
27μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)]
> 9 {μ∗ (P1 > 0) + μ∗ (P2 > 0) + μ∗ (P3 > 0)} − 1,
it follows that
1
1
μ∗ N 2 − X−1
Y−1
[(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)] > 0.
1
1
By Definition 2.1, the set N 2 −X−1
Y−1
[(P1 > 0) + (P2 > 0) + (P3 > −2) + (f > 0) + (u > 0)]
is infinite and we have
P1 (i, j) ≤ 0, P2 (i, j) ≤ 0, P3 (i, j) ≤ −2, f (i, j) ≤ 0, u(i, j) ≤ 0
for some domain. Repeating the proofs as above, we may then obtain the following result.
Theorem 2.10. Suppose that
μ∗ (P1 > 0) = ω1 ,
μ∗ (P3 > −2) = ω3 ,
μ∗ (P2 > 0) = ω2 ,
μ∗ (f > 0) = ωf+ ,
μ∗ (f < 0) = ωf−
and
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] = ω + ,
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f < 0)] = ω − .
Suppose further that there exist a constant ω0 ∈ (0, 1) such that
ω>
ω1 + ω2 + ω3 + ωf + ω0
1
− ,
3
27
(14)
where ωf = max{ωf− , ωf+ } and ω = min{ω − , ω + }. Then every solution u = {u(i, j)} of
(1) has unsaturated upper positive part.
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2.2 Homogeneous case
In this section,we are concerned with (1) under condition that the forcing term is identically zero.The proofs are the same as Theorems 2.8-2.10 and hence omitted.
Theorem 2.11. Suppose that
μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (P3 > −2) = ω3
and
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2)] = ω.
Suppose further that there exists a constant ω0 ∈ (0, 1) such that
9(ω1 + ω2 + ω3 + ω0 − 2ω) < 1.
(15)
Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part.
Theorem 2.12. Suppose that
μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2
and
μ∗ [(P1 > 0) · (P2 > 0)] = ω.
Suppose further that there exist a constant ω0 ∈ (0, 1) such that
μ∗ (P3 ≤ −2) > 9(ω1 + ω2 + ω0 − ω).
(16)
Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part.
Theorem 2.13. Suppose that
μ∗ (P1 > 0) = ω1 , μ∗ (P2 > 0) = ω2 , μ∗ (P3 > −2) = ω3 .
Suppose further that there exist a constant ω0 ∈ (0, 1) such that
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2)] >
ω1 + ω2 + ω3 + ω0
1
− .
2
18
(17)
Then every solution u = {u(i, j)} of (1) has unsaturated upper positive part.
3
Examples
We consider two examples.
Example 3.1. Consider the partial difference equation with forcing term
Δ1 u(i − 1, j) + Δ2 u(i, j − 1) − u(i − 1, j) − u(i, j − 1) − 2u(i, j) = f (i, j),
(18)
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where P1 (i, j) = P2 (i, j) = −1, P3 (i, j) = −3 and
⎧
⎪
⎨ 1 i = 30m or j = 30n, m, n ∈ N
f (i, j) =
.
⎪
⎩ 0 otherwise
Then μ∗ (P1 > 0) = μ∗ (P2 > 0) = μ∗ (P3 > −2) = 0, μ∗ (P3 ≤ −2) = 1, μ∗ (f > 0) =
and μ∗ (f < 0) = 0. Moreover,
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2)] = 0,
1
30
(19)
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f > 0)] = 0
as well as
μ∗ [(P1 > 0) · (P2 > 0) · (P3 > −2) · (f < 0)] = 0.
1
, we see that (8) and (12)-(14) hold. By Theorems 2.7-2.10, every solution
18
u = {u(i, j)} of (18) has unsaturated upper positive part and hence it is oscillatory.
Given ω0 =
Example 3.2. Consider the partial difference equation
Δ1 u(i − 1, j) + Δ2 u(i, j − 1) − u(i − 1, j) − u(i, j − 1) − 6u(i, j) = 0,
(20)
where P1 (i, j) = P2 (i, j) = −1 and P3 (i, j) = −6. Similar to Example 3.1, we have
1
μ∗ [(P1 > 0) · (P2 > 0)] = 0 and (19) holds. On the other hand, when given ω0 =
,
10
(15)-(17) are satisfied. By Theorems 2.11-2.13, every solution u = {u(i, j)} of (20) has
unsaturated upper positive part and hence is oscillatory. In fact, u = {(−1)i+j+1 } is such
a solution of (20).
4
Remarks
We obtained conditions for all solutions of our equations to be unsaturated. This is the
first time an unsaturated solution is introduced, and hence there are no previous results
for comparison. However, we have mentioned the fact that unsaturated solutions are
also oscillatory. Hence our previous results are oscillation theorems. Usually, oscillation
theorems are proved by showing neither eventually positive nor eventually negative solutions can exist. Hence these theorems do not provide any clues to how frequent solutions
oscillate!
Although frequently oscillatory solutions are also introduced in studying the oscillatory behavior of partial difference equations (see e.g. [4, 5]), these solutions can be
different from unsaturated solutions. To see this, let u = {u(i, j)}(i,j)∈N 2 be defined by
⎧
⎪
⎪1 i < j
⎪
⎪
⎨
u(i, j) = 0 i = j ,
⎪
⎪
⎪
⎪
⎩ −1 i > j
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then μ(u ≤ 0) = μ(u ≥ 0) = 1/2 and hence u is not frequently oscillatory of upper degree
1/2. Yet, u has unsaturated positive part because μ(u > 0) = 1/2!
As our final remark, we mention that similar ideas can be used to establish criteria
for solutions of (1) with unsaturated upper negative parts.
References
[1] Y.Z. Lin and S.S. Cheng: “Stability criteria for two partial difference equations”,
Comput. Math. App., Vol. 32(7), (1996), pp. 87–103.
[2] Y.Z. Lin and S.S. Cheng: “Bounds for solutions of a three-point partial difference
equation”, Acta Math. Sci., Vol. 18(1), (1998), pp. 107–112.
[3] S.S. Cheng: Partial Difference Equations, Taylor and Francis, 2003.
[4] C.J. Tian and B.Q. Zhang: “Frequent oscillation of a class of partial difference equations”, J. Anal. Appl., Vol. 18(1), (1999), pp. 111–130.
[5] S.L. Xie and C.J. Tian: “Frequent oscillatory criteria for partial difference equations
with several delays”, Comput. Math. Appl., Vol. 48, (2004), pp. 335–345.
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