# Oscillation Properties for Systems of Higher-Order Partial Differential Equations with Distributed Deviating Arguments

Authors: Youjun Liu; Jianwen Zhang; Jurang Yan
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

New sufficient conditions are obtained for oscillation for the solutions of systems of a class of higher-order quasilinear partial functional differential equations with distributed deviating arguments. The obtained results are illustrated by example.


---

## Body

## 1. Introduction and Preliminary

In this paper, we consider systems of higher-order quasilinear partial functional differential equations with distributed deviating arguments:(1)
∂
n
∂
t
n
r
t
u
i
x
,
t
+
∫
α
β
‍
p
t
,
τ
u
i
x
,
t
-
τ
d
τ
=
a
i
(
t
)
Δ
u
i
(
x
,
t
)
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
b
i
j
k
(
t
)
Δ
u
j
(
x
,
ρ
k
(
t
)
)
-
q
i
(
x
,
t
)
u
i
(
x
,
t
)
-
∑
j
=
1
m
‍
∑
γ
=
1
l
‍
∫
c
γ
d
γ
‍
f
i
j
γ
(
x
,
t
,
u
j
(
x
,
t
-
σ
γ
)
)
d
σ
γ
,
d
d
(
x
,
t
)
∈
Ω
×
[
0
,
∞
)
≡
G
,
i
=
1,2
,
…
,
m
,
where n is even, Ω is a bounded domain in R
N with a piecewise smooth boundary ∂
Ω, and
(2)
Δ
u
i
x
,
t
=
∑
r
=
1
N
‍
∂
2
u
i
x
,
t
∂
x
r
2
,
i
=
1,2
,
…
,
m
.Throughout this paper, we assume that the following conditions hold:(H1)
β
>
α
>
0, d
γ
>
c
γ
>
0, r
∈
C
n
(
[
0
,
∞
)
,
[
0
,
∞
)
), lim
⁡
t
→
∞
∫
0
t
‍
1
/
r
(
s
)
d
s
=
∞; p
∈
C
n
(
[
0
,
∞
)
×
[
α
,
β
]
,
[
0
,
∞
)
), γ
∈
I
l
=
{
1,2
,
…
,
l
};
(H2)
q
i
∈
C
(
G
¯
,
[
0
,
∞
)
), q
i
(
t
)
=
min
⁡
x
∈
G
¯
q
i
(
x
,
t
), q
(
t
)
=
min
⁡
1
≤
i
≤
m
q
i
(
t
), i
∈
I
m
=
{
1,2
,
…
,
m
};
(H3)
f
∈
C
(
G
¯
×
R
,
R
), f
i
j
γ
(
t
,
x
,
u
j
)
/
u
j
≥
h
i
j
γ
(
x
,
t
) if u
j
≠
0, where h
i
j
γ
∈
C
(
G
¯
,
R
), h
i
i
γ
(
x
,
t
)
>
0, h
i
i
γ
(
t
)
=
min
⁡
x
∈
G
¯
h
i
i
γ
(
x
,
t
), h
¯
i
j
γ
(
t
)
=
sup
⁡
x
∈
G
¯
|
h
i
j
γ
(
x
,
t
)
|, and Q
γ
(
t
)
=
min
⁡
1
≤
i
≤
m
{
h
i
i
γ
(
t
)
-
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
(
t
)
}
≥
0, i
,
j
∈
I
m
,
γ
∈
I
l;
(H4)
a
i
∈
C
(
[
0
,
∞
)
,
[
0
,
∞
)
), b
i
j
k
∈
C
(
[
0
,
∞
)
,
(
0
,
∞
)
), B
k
(
t
)
=
min
⁡
1
≤
i
≤
m
{
b
i
i
k
(
t
)
-
∑
j
=
1
,
j
≠
i
m
‍
|
b
i
j
k
(
t
)
|
}
≥
0, i
,
j
∈
I
m, k
∈
I
s
=
{
1,2
,
…
,
s
};
(H5)
ρ
k
∈
C
(
[
0
,
∞
)
,
R
)
,
ρ
k
(
t
)
≤
t, ρ
k; is nondecreasing functions and lim
⁡
t
→
∞
ρ
k
(
t
)
=
∞
,
k
∈
I
s.We consider two kinds of boundary conditions:(3)
∂
u
i
(
x
,
t
)
∂
v
+
g
i
x
,
t
u
i
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
∞
,
i
∈
I
m
,
where v is the unit exterior normal vector to ∂
Ω
,
g
i
(
x
,
t
) is a nonnegative continuous function on ∂
Ω
×
[
0
,
∞
), and
(4)
u
i
(
x
,
t
)
=
0
,
(
x
,
t
)
∈
∂
Ω
×
[
0
,
∞
)
,
i
∈
I
m
.During the past three decades, the investigation of oscillatory theory for partial functional differential equations has attracted attention of numerous researchers due to their significance in theory and applications. In 2000, Li [1] has investigated systems of second-order partial differential equations:
(5)
∂
2
∂
t
2
p
t
u
i
x
,
t
+
∑
r
=
1
d
‍
λ
r
t
u
i
x
,
t
-
τ
r
=
a
i
(
t
)
Δ
u
i
(
x
,
t
)
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
a
i
j
k
(
t
)
Δ
u
j
(
x
,
ρ
k
(
t
)
)
-
q
i
(
x
,
t
)
u
i
(
x
,
t
)
-
∑
j
=
1
m
‍
∑
h
=
1
l
‍
q
i
j
h
(
x
,
t
)
u
j
(
x
,
σ
h
(
t
)
)
,
(
x
,
t
)
∈
Ω
×
[
0
,
∞
)
≡
G
,
i
=
1,2
,
…
,
m
,
and obtained sufficient conditions for the oscillation of solutions. In 2004, Lin [2] has studied higher-order nonlinear differential equation:
(6)
∂
n
∂
t
n
p
t
u
i
x
,
t
+
∑
r
=
1
d
‍
λ
r
t
u
i
x
,
t
-
τ
i
t
=
a
i
(
t
)
Δ
u
i
(
x
,
t
)
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
a
i
j
k
(
t
)
Δ
u
j
(
x
,
ρ
k
(
t
)
)
-
q
i
(
x
,
t
)
u
i
(
x
,
t
)
-
∑
j
=
1
m
‍
∑
h
=
1
l
‍
q
i
j
h
(
x
,
t
)
u
j
(
x
,
σ
h
(
t
)
)
,
(
x
,
t
)
∈
Ω
×
[
0
,
∞
)
≡
G
,
i
=
1,2
,
…
,
m
,
and obtained sufficient conditions for the oscillation of solutions. In 2004, Li et al. [3] have investigated a class of vector hyperbolic differential equations with deviating arguments:
(7)
∂
2
∂
t
2
U
x
,
t
=
a
(
t
)
Δ
U
(
x
,
t
)
+
∑
k
=
1
m
‍
b
k
(
t
)
Δ
U
(
x
,
τ
k
(
t
)
)
-
p
(
x
,
t
)
U
(
x
,
t
)
-
∑
h
=
1
l
‍
∫
a
b
‍
q
h
(
x
,
t
,
ζ
)
U
(
x
,
g
h
(
t
,
ζ
)
)
d
σ
(
ζ
)
+
F
(
x
,
t
)
,
(
x
,
t
)
∈
Ω
×
[
0
,
∞
)
≡
G
,
and obtained sufficient conditions for H
-oscillation of solutions. In 2009, Gui and Xu [4] have investigated even-order neutral type partial differential equations with distributed deviating arguments:
(8)
∂
∂
t
p
t
∂
m
-
1
∂
t
m
-
1
u
x
,
t
+
c
t
u
i
x
,
t
-
τ
=
a
0
(
t
)
Δ
u
(
x
,
t
)
+
a
1
(
t
)
Δ
u
(
x
,
t
-
ν
)
-
∫
a
b
‍
F
t
,
ξ
,
u
x
,
g
t
,
ξ
d
μ
(
ξ
)
,
dddddssssdd
(
x
,
t
)
∈
Ω
×
[
0
,
∞
)
≡
G
.For related work, we refer the reader to [5–10]. However, to the best of our knowledge, few investigitions have been reported on the oscillation theory of systems of partial functional differential equations with distributed deviating arguments. Thus, the research presents its significance.Definition 1.
The vector functionu
(
x
,
t
)
=
{
u
1
(
x
,
t
)
,
u
2
(
x
,
t
)
,
…
,
u
m
(
x
,
t
)
}
⊤ is said to be a solution of problem (1), (3) (or (1), (4)) if it satisfies (1) in G
=
Ω
×
[
0
,
∞
) and boundary condition (3) (or (4)).Definition 2.
A nontrivial componentu
i
(
x
,
t
) of the vector function u
(
x
,
t
)
=
{
u
1
(
x
,
t
)
,
u
2
(
x
,
t
)
,
…
,
u
m
(
x
,
t
)
}
⊤ is said to oscillate in Ω
×
[
μ
0
,
∞
) if, for each μ
>
μ
0, there is a point (
x
0
,
t
0
)
∈
Ω
×
[
μ
,
∞
) such that u
i
(
x
0
,
t
0
)
=
0.Definition 3.
The vector solutionu
(
x
,
t
)
=
{
u
1
(
x
,
t
)
,
u
2
(
x
,
t
)
,
…
,
u
m
(
x
,
t
)
}
⊤ of problem (1), (3) (or (1), (4)) is said to be oscillatory in G
=
Ω
×
[
0
,
∞
) if at least one of its nontrivial components is oscillatory in G. Otherwise, the vector solution u
(
x
,
t
) is said to be nonoscillatory.


## 2. The Main Results

Theorem 4.
Assume thatr
(
t
)
≥
1 is a monotone decreasing function, and
(9)
∫
α
β
‍
p
(
t
,
τ
)
d
τ
<
1
.
If there exist some γ
0
∈
I
l such that
(10)
∫
t
1
∞
‍
Q
γ
0
t
∫
c
γ
0
d
γ
0
‍
1
-
∫
α
β
‍
p
t
-
σ
γ
0
,
τ
d
τ
d
σ
γ
0
d
t
=
∞
,
t
0
>
0
,
then every solution of boundary value system (1), (3) is oscillatory.Proof.
Suppose (1) and (3) have a nonoscillatory solution {
u
1
,
u
2
,
…
,
u
m
}. We assume that u
i
x
,
t
>
0 for t
≥
t
0
≥
0, i
∈
I
m. Let δ
i
=
sgn
⁡
u
i
(
x
,
t
), Z
i
(
x
,
t
)
=
δ
i
u
i
(
x
,
t
); then Z
i
(
x
,
t
)
>
0, (
x
,
t
)
∈
Ω
×
[
t
0
,
∞
), i
∈
I
m. From (H1) and (H5), there exists a number t
1
≥
t
0 such that Z
i
(
x
,
t
)
>
0
,
Z
i
(
x
,
t
-
τ
)
>
0
,
Z
i
(
x
,
ρ
k
(
t
)
)
>
0
,
Z
i
(
x
,
t
-
σ
γ
)
>
0, in Ω
×
[
t
0
,
∞
)
,
i
∈
I
m
,
k
∈
I
s
,
γ
∈
I
l. Multiplying both sides of (1) by δ
i and integrating with respect to x over the domain Ω, we get
(11)
d
n
d
t
n
r
t

∫
Ω
‍
δ
i
u
i
x
,
t
d
x

+

∫
α
β
‍

p
t
,
τ
∫
Ω
‍

δ
i
u
i
x
,
t
-
τ
d
x
d
τ

=
a
i
(
t
)
∫
Ω
‍
δ
i
Δ
u
i
(
x
,
t
)
d
x
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
b
i
j
k
(
t
)
∫
Ω
‍
δ
i
Δ
u
j
(
x
,
ρ
k
(
t
)
)
d
x
-
∫
Ω
‍
q
i
(
x
,
t
)
δ
i
u
i
(
x
,
t
)
d
x
-
∑
j
=
1
m
‍
∑
γ
=
1
l
‍
∫
Ω
‍
δ
i
∫
c
γ
d
γ
‍
f
i
j
γ
(
x
,
t
,
u
j
(
x
,
t
-
σ
γ
)
)
d
σ
γ
d
x
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d
d
s
s
s
s
s
s
s
t
≥
t
1
,
i
∈
I
m
.
Therefore,
(12)
d
n
d
t
n
r
t
∫
Ω
‍
Z
i
x
,
t
d
x

+

∫
α
β
‍

p
t
,
τ
∫
Ω
‍

Z
i
x
,
t
-
τ
d
x

d
τ
≤
a
i
(
t
)
∫
Ω
‍
Δ
Z
i
(
x
,
t
)
d
x
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
b
i
j
k
(
t
)
∫
Ω
‍
δ
j
δ
i
Δ
Z
j
(
x
,
ρ
k
(
t
)
)
d
x
-
∫
Ω
‍
q
i
(
x
,
t
)
Z
i
(
x
,
t
)
d
x
-
∑
j
=
1
m
‍
∑
γ
=
1
l
‍
δ
j
δ
i
∫
c
γ
d
γ
‍
∫
Ω
‍
h
i
j
γ
x
,
t
Z
j
x
,
t
-
σ
γ
d
x
d
σ
γ
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d
d
d
s
s
s
s
s
s
s
s
s
s
t
≥
t
1
,
i
∈
I
m
.
From Green’s formula and boundary condition (3), it follows that
(13)
∫
Ω
‍
Δ
Z
i
x
,
t
d
x
=
∫
∂
Ω
‍
∂
Z
i
(
x
,
t
)
∂
v
d
S
=
-
∫
∂
Ω
‍
g
i
(
x
,
t
)
Z
i
(
x
,
t
)
d
S
≤
0
,
∫
Ω
‍
Δ
Z
i
x
,
ρ
k
t
d
x
=
∫
∂
Ω
‍
∂
Z
i
(
x
,
ρ
k
(
t
)
)
∂
v
d
S
=
-
∫
∂
Ω
‍
g
i
(
x
,
ρ
k
(
t
)
)
Z
i
(
x
,
ρ
k
(
t
)
)
d
S
≤
0
,
where d
S is the surface element on ∂
Ω. Combining (12)-(13), we find
(14)
d
n
d
t
n
r
t
∫
Ω
‍
Z
i
x
,
t
d
x

+

∫
α
β
‍
p
t
,
τ
∫
Ω
‍

Z
i
x
,
t
-
τ
d
x

d
τ
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
b
i
j
k
(
t
)
δ
j
δ
i
∫
∂
Ω
‍
g
i
(
x
,
ρ
k
(
t
)
)
Z
i
(
x
,
ρ
k
(
t
)
)
d
S
+
q
i
(
t
)
∫
Ω
‍
Z
i
(
x
,
t
)
d
x
+
∑
γ
=
1
l
‍
h
i
i
γ
(
t
)
∫
c
γ
d
γ
‍
∫
Ω
‍
Z
i
(
x
,
t
-
σ
γ
)
d
x
d
σ
γ
-
∑
γ
=
1
l
‍
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
t
∫
c
γ
d
γ
‍
∫
Ω
‍
Z
j
x
,
t
-
σ
γ
d
x
d
σ
γ
≤
0
.
Therefore,
(15)
d
n
d
t
n
r
t
∫
Ω
‍
Z
i
x
,
t
d
x

+

∫
α
β
‍

p
t
,
τ
∫
Ω
‍

Z
i
x
,
t
-
τ
d
τ

d
x
+
∑
k
=
1
s
‍
b
i
i
k
t
∫
∂
Ω
‍
g
i
x
,
ρ
k
t
Z
i
x
,
ρ
k
t
d
S
-
∑
j
=
1
,
j
≠
i
m
‍
∑
k
=
1
s
‍
b
i
i
k
t
∫
∂
Ω
‍
g
i
(
x
,
ρ
k
(
t
)
)
Z
i
(
x
,
ρ
k
(
t
)
)
d
S
+
q
i
(
t
)
∫
Ω
‍
Z
i
(
x
,
t
)
d
x
+
∑
γ
=
1
l
‍
h
i
i
γ
(
t
)
∫
Ω
‍
∫
c
γ
d
γ
‍
Z
i
(
x
,
t
-
σ
γ
)
d
σ
γ
d
x
-
∑
γ
=
1
l
‍
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
(
t
)
∫
Ω
‍
∫
c
γ
d
γ
‍
Z
j
(
x
,
t
-
σ
γ
)
d
σ
γ
d
x
≤
0
.
Set
(16)
V
i
=
∫
Ω
‍
Z
i
x
,
t
d
x
,
W
i
=
∫
∂
Ω
‍
g
i
x
,
ρ
k
t
Z
i
x
,
ρ
k
t
d
S
,
t
≥
t
1
,
i
∈
I
m
.
Then (15) yields
(17)
d
n
d
t
n
r
t
V
i
t
+
∫
α
β
‍
p
t
,
τ
V
i
t
-
τ
d
τ
+
∑
k
=
1
s
‍
b
i
i
k
t
W
i
ρ
k
t
b
i
j
k
t
W
j
ρ
k
t
b
i
j
k
t
W
j
ρ
k
t
sssssssfsss
-
∑
j
=
1
,
j
≠
i
m
‍
b
i
j
k
t
W
j
ρ
k
t
+
q
i
(
t
)
V
i
(
t
)
+
∑
γ
=
1
l
‍
h
i
i
γ
(
t
)
∫
c
γ
d
γ
‍
V
i
(
t
-
σ
γ
)
d
σ
γ
∑
j
=
1
,
j
≠
i
m
‍
d
d
d
f
d
d
-
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
(
t
)
∫
c
γ
d
γ
‍
V
j
(
t
-
σ
γ
)
d
σ
γ
≤
0
.
Now, let V
(
t
)
=
∑
i
=
1
m
‍
V
i
(
t
), W
(
t
)
=
∑
i
=
1
m
‍
W
i
(
t
), t
≥
t
1; from (17) we have
(18)
d
n
d
t
n
r
t
V
t
+
∫
α
β
‍
p
t
,
τ
V
t
-
τ
d
τ
+
∑
k
=
1
s
‍
∑
i
=
1
m
‍
b
i
i
k
t
W
i
ρ
k
t
-
∑
j
=
1
,
j
≠
i
m
‍
b
i
j
k
t
W
j
ρ
k
t
+
q
(
t
)
V
(
t
)
+
∑
γ
=
1
l
‍
∑
i
=
1
m
‍
h
i
i
γ
(
t
)
∫
c
γ
d
γ
‍
V
i
(
t
-
σ
γ
)
d
σ
γ
∑
j
=
1
,
j
≠
i
m
‍
s
s
s
s
s
s
s
s
s
s
s
s
s
-
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
(
t
)
∑
γ
=
1
l
‍
∫
c
γ
d
γ
‍
V
j
(
t
-
σ
γ
)
d
σ
γ
≤
0
.
Note that
(19)
∑
i
=
1
m
‍
∑
j
=
1
,
j
≠
i
m
‍
h
i
i
γ
t
∫
c
γ
d
γ
‍
V
i
t
-
σ
γ
d
σ
γ
s
s
s
s
s
s
s
-
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
t
∫
c
γ
d
γ
‍
V
j
t
-
σ
γ
d
σ
γ
=
h
11
γ
(
t
)
∫
c
γ
d
γ
‍
V
1
(
t
-
σ
γ
)
d
σ
γ
∑
j
=
1
,
j
≠
1
m
‍
sssss
-
∑
j
=
1
,
j
≠
1
m
‍
h
¯
1
j
γ
(
t
)
∫
c
γ
d
γ
‍
V
j
(
t
-
σ
γ
)
d
σ
γ
+
h
22
γ
(
t
)
∫
c
γ
d
γ
‍
V
2
(
t
-
σ
γ
)
d
σ
γ
∑
j
=
1
,
j
≠
2
m
‍
s
s
s
s
s
-
∑
j
=
1
,
j
≠
2
m
‍
h
¯
2
j
γ
(
t
)
∫
c
γ
d
γ
‍
V
j
(
t
-
σ
γ
)
d
σ
γ
+
⋯
+
h
m
m
γ
(
t
)
∫
c
γ
d
γ
‍
V
m
(
t
-
σ
γ
)
d
σ
γ
∑
j
=
1
,
j
≠
m
m
‍
s
s
s
s
s
-
∑
j
=
1
,
j
≠
m
m
‍
h
¯
m
j
γ
(
t
)
∫
c
γ
d
γ
‍
V
j
(
t
-
σ
γ
)
d
σ
γ
=
[
h
11
γ
(
t
)
-
∑
j
=
1
,
j
≠
1
m
‍
h
¯
j
1
γ
(
t
)
]
∫
c
γ
d
γ
‍
V
1
(
t
-
σ
γ
)
d
σ
γ
+
[
h
22
γ
(
t
)
-
∑
j
=
1
,
j
≠
1
m
‍
h
¯
j
2
γ
(
t
)
]
∫
c
γ
d
γ
‍
V
2
(
t
-
σ
γ
)
d
σ
γ
+
⋯
+
[
h
m
m
γ
(
t
)
-
∑
j
=
1
,
j
≠
1
m
‍
h
¯
j
m
γ
(
t
)
]
∫
c
γ
d
γ
‍
V
m
(
t
-
σ
γ
)
d
σ
γ
≥
min
⁡
1
≤
i
≤
m
{
h
i
i
γ
(
t
)
-
∑
j
=
1
,
j
≠
i
m
‍
h
¯
j
i
γ
(
t
)
}
∑
i
=
1
m
‍
∫
c
γ
d
γ
‍
V
i
(
t
-
σ
γ
)
d
σ
γ
=
Q
γ
(
t
)
∫
c
γ
d
γ
‍
V
(
t
-
σ
γ
)
d
σ
γ
.
Similarly, we have
(20)
∑
k
=
1
s
‍
∑
i
=
1
m
‍
b
i
i
k
t
W
i
ρ
k
t
-
∑
j
=
1
,
j
≠
i
m
‍
b
i
j
k
t
W
j
ρ
k
t
≥
min
⁡
1
≤
i
≤
m
b
i
i
k
t
-
∑
j
=
1
,
j
≠
i
m
‍
b
j
i
k
t
∑
i
=
1
m
‍
W
i
ρ
k
t
=
B
k
t
W
ρ
k
t
.
Thus, from (18) for t
≥
t
1, we find
(21)
r
t
V
t
+
∫
α
β
‍
p
t
,
τ
V
t
-
τ
d
τ
(
n
)
+
∑
k
=
1
s
‍
B
k
(
t
)
W
(
ρ
k
(
t
)
)
+
q
(
t
)
V
(
t
)
+
∑
γ
=
1
l
‍
Q
γ
(
t
)
∫
c
γ
d
γ
‍
V
(
t
-
σ
γ
)
d
σ
γ
≤
0
,
t
≥
t
1
.
It is clear that
(22)
W
ρ
k
t
=
∑
i
=
1
m
‍
W
i
ρ
k
t
≥
0
,
t
≥
t
1
,
k
∈
I
s
.
Therefore, it follows that
(23)
r
t
V
t
+
∫
α
β
‍
p
t
,
τ
V
t
-
τ
d
τ
(
n
)
+
q
t
V
t
+
∑
γ
=
1
l
‍
Q
γ
t
∫
c
γ
d
γ
‍
V
t
-
σ
γ
d
σ
γ
≤
0
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
t
≥
t
1
.
Let
(24)
Y
(
t
)
=
r
(
t
)
V
(
t
)
+
∫
α
β
‍
p
(
t
,
τ
)
V
(
t
-
τ
)
d
τ
.
We have Y
(
t
)
>
0
,
Y
(
n
)
(
t
)
<
0, for t
≥
t
1. Hence, Y
(
n
-
1
)
(
t
) is a monotone decreasing function in the interval [
t
1
,
∞
). We can claim that Y
(
n
-
1
)
(
t
)
>
0, for t
≥
t
1. In fact, if Y
(
n
-
1
)
(
t
)
<
0, for t
≥
t
1, then there exists a T
>
t
1, such that r
(
T
)
Y
(
n
-
1
)
(
T
)
<
0. This implies that
(25)
Y
k
t
<
0
,
k
=
0,1
,
2
,
…
,
n
-
2
,
which contradicts the fact that Y
(
t
)
>
0. Now, we claim that Y
′
(
t
)
≥
0
,
t
≥
t
1. Otherwise Y
(
t
)
<
0. But from the well-known lemma of Kiguradze [5] and the fact that n is even, the inequalities imply that higher derivatives of that function are also negative. This contradicts the fact that Y
(
n
-
1
)
(
T
)
≥
0. Thus we have proved that Y
(
t
) is the increasing function of t.
From (23), we find that there exists a γ
0
∈
I
l such that
(26)
Y
(
n
)
(
t
)
+
Q
γ
0
(
t
)
∫
c
γ
0
d
γ
0
‍
V
(
t
-
σ
γ
0
)
d
σ
γ
0
≤
0
,
t
≥
t
1
.
Thus we obtain
(27)
Y
(
n
)
(
t
)
+
Q
γ
0
(
t
)
×
∫
c
γ
0
d
γ
0
‍
Y
t
-
σ
γ
0
∫
α
β
‍
r
t
-
σ
γ
0
-
1
-
∫
α
β
‍
p
t
-
σ
γ
0
,
τ
V
t
-
σ
γ
0
-
τ
d
τ
×
∫
α
β
‍
r
t
-
σ
γ
0
-
1
d
σ
γ
0
≤
0
,
t
≥
t
1
.
Since Y
(
t
)
≥
r
(
t
)
V
(
t
)
≥
V
(
t
)
,
Y
′
(
t
)
≥
0, from (27), we have
(28)
Y
(
n
)
(
t
)
+
Q
γ
0
(
t
)
×
∫
c
γ
0
d
γ
0
‍
1
-
∫
α
β
‍
p
t
-
σ
γ
0
,
τ
d
τ
Y
(
t
-
σ
γ
0
)
r
(
t
-
σ
γ
0
)
d
σ
γ
0
≤
0
,
s
s
s
s
s
s
s
s
s
s
s
s
d
d
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
t
≥
t
1
.
Integrating inequality (28) from t
1 to t,
(29)
Y
(
n
-
1
)
(
t
)
-
Y
(
n
-
1
)
(
t
1
)
+
∫
t
1
t
‍
Q
γ
0
(
s
)
∫
c
γ
0
d
γ
0
‍
[
1
-
∫
α
β
‍
p
(
s
-
σ
γ
0
,
τ
)
d
τ
]
Y
(
t
-
σ
γ
0
)
r
(
t
-
σ
γ
0
)
s
s
s
s
s
s
s
s
s
s
s
s
d
d
s
s
s
s
s
×
Y
(
t
-
σ
γ
0
)
r
(
t
-
σ
γ
0
)
d
σ
γ
0
d
s
≤
0
,
t
≥
t
1
.
Furthermore, we have
(30)
∫
t
1
t
‍
Q
γ
0
s
∫
c
γ
0
d
γ
0
‍
1
-
∫
α
β
‍
p
s
-
σ
γ
0
,
τ
d
τ
d
σ
γ
0
d
s
≤
r
(
t
1
-
d
γ
0
)
Y
(
t
1
-
d
γ
0
)
(
-
Y
(
n
-
1
)
(
t
)
+
Y
(
n
-
1
)
(
t
1
)
)
,
≤
r
(
t
1
-
d
γ
0
)
Y
(
n
-
1
)
(
t
1
)
Y
(
t
1
-
d
γ
0
)
,
t
≥
t
1
,
which contradicts condition (10). This proof is complete.Theorem 5.
Assume thatr
(
t
)
≥
1 is a monotone decreasing function and (9) holds. And
(31)
∫
t
0
∞
‍
q
s
1
-
∫
α
β
‍
p
s
,
τ
d
τ
d
s
=
∞
,
t
0
>
0
.
Then every solution of boundary value system (1), (3) is oscillatory.Proof.
As in the proof of Theorem4, we obtain (23). Therefore,
(32)
Y
(
n
)
(
t
)
+
q
(
t
)
V
(
t
)
≤
0
,
t
≥
t
0
.
The remaining part of the proof is similar to that of Theorem 4; therefore it is omitted. The proof is complete.Corollary 6.
If inequality (23) has no eventually positive solution, then every solution u
(
x
,
t
) of system (1), (3) is oscillatory in G.Theorem 7.
Suppose that there exists a positive constantM such that 0
<
M
≤
r
(
t
)
<
1, and
(33)
∫
α
β
‍
p
t
,
τ
d
τ
<
M
.
If there exist some γ
0
∈
I
l such that
(34)
∫
t
1
∞
‍
Q
γ
0
t
∫
c
γ
0
d
γ
0
‍
1
-
M
-
1
∫
α
β
‍
p
t
-
σ
γ
0
,
τ
d
τ
d
σ
γ
0
d
t
=
∞
,
t
0
>
0
,
then every solution of boundary value system (1), (3) is oscillatory.Proof.
As in the proof of Theorem4, we obtain (27). Using the fact that r
(
t
)
<
1, from (27), we have
(35)
Y
n
t
+
Q
γ
0
(
t
)
×
∫
c
γ
0
d
γ
0
‍
Y
t
-
σ
γ
0
∫
α
β
‍
s
s
s
s
s
f
f
s
s
s
s
s
-
∫
α
β
‍
p
t
-
σ
γ
0
,
τ
V
t
-
σ
γ
0
-
τ
d
τ
d
σ
γ
0
≤
0
,
t
≥
t
1
.
Since Y
(
t
)
≥
r
(
t
)
V
(
t
)
≥
M
V
(
t
)
,
Y
′
t
≥
0, from (35) we have
(36)
Y
(
n
)
(
t
)
+
Q
γ
0
(
t
)
×
∫
c
γ
0
d
γ
0
‍
[
1
-
M
-
1
∫
α
β
‍
p
(
t
-
σ
γ
0
,
τ
)
d
τ
]
Y
(
t
-
σ
γ
0
)
d
σ
γ
0
≤
0
,
t
≥
t
1
.
Integrating inequality (36) from t
1 to t,
(37)
Y
(
n
-
1
)
(
t
)
-
Y
(
n
-
1
)
(
t
1
)
+
∫
t
1
t
‍
Q
γ
0
(
s
)
∫
c
γ
0
d
γ
0
‍
1
-
M
-
1
∫
α
β
‍
p
(
s
-
σ
γ
0
,
τ
)
d
τ
∫
α
β
‍
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
×
∫
c
γ
0
d
γ
0
‍
Y
(
t
-
σ
γ
0
)
d
σ
γ
0
d
s
≤
0
,
sssssssssssssssssssssssssssssssssssssssssssss
t
≥
t
1
.
Furthermore, we have
(38)
∫
t
1
t
‍
Q
γ
0
s
∫
c
γ
0
d
γ
0
‍
1
-
M
-
1
∫
α
β
‍
p
s
-
σ
γ
0
,
τ
d
τ
d
σ
γ
0
d
s
≤
(
-
Y
(
n
-
1
)
(
t
)
+
Y
(
n
-
1
)
(
t
1
)
)
Y
(
t
1
-
d
γ
0
)
,
≤
Y
(
n
-
1
)
(
t
1
)
Y
(
t
1
-
d
γ
0
)
,
t
≥
t
1
,
which contradicts condition (10). This proof is complete.Theorem 8.
Suppose that there exists a positive constantM such that 0
<
M
≤
r
(
t
)
<
1, and (33) holds. If
(39)
∫
t
0
∞
‍
{
q
(
s
)
[
1
-
M
-
1
∫
α
β
‍
p
(
s
,
τ
)
d
τ
]
}
d
s
=
∞
,
t
0
>
0
,
then every solution of boundary value system (1), (3) is oscillatory.Proof.
As in the proof of Theorem5, we obtain (32). The remaining part of the proof is similar to that of Theorem 7 and we omit it.
Consider the smallest eigenvalueλ
0 of the Dirichlet problem:
(40)
Δ
ω
(
x
)
+
λ
ω
(
x
)
=
0
,
x
∈
Ω
,
ω
(
x
)
=
0
,
x
∈
∂
Ω
,
where λ is a positive constant and the corresponding eigenfunction ϕ
(
x
) is positive in Ω.Theorem 9.
Assume all conditions of Theorem4 hold. Then every solution u
(
x
,
t
) of system (1), (4) is oscillatory in G.Proof.
Suppose (1) and (4) have a nonoscillatory solution {
u
1
,
u
2
,
…
,
u
m
}
T. We assume that |
u
i
(
x
,
t
)
|
>
0 for t
≥
t
0
≥
0
,
i
∈
I
m. Let δ
i
=
sgn
⁡
u
i
(
x
,
t
)
,
Z
i
(
x
,
t
)
=
δ
i
u
i
(
x
,
t
); then Z
i
(
x
,
t
)
>
0
,
(
x
,
t
)
∈
Ω
×
[
t
0
,
∞
)
,
i
∈
I
m. From (H
1) and (H
5), there exists a number t
1
≥
t
0 such that Z
i
(
x
,
t
)
>
0
,
Z
i
(
x
,
t
-
τ
)
>
0
,
Z
i
(
x
,
ρ
k
(
t
)
)
>
0
,
Z
i
(
x
,
t
-
σ
γ
)
>
0, in Ω
×
[
t
0
,
∞
)
,
i
∈
I
m
,
k
∈
I
s
,
γ
∈
I
l. Multiplying both sides of (1) by δ
i
ϕ
(
x
), and integrating with respect to x over the domain Ω, we have
(41)
d
n
d
t
n
r
(
t
)
∫
Ω
‍
δ
i
u
i
(
x
,
t
)
ϕ
(
x
)
d
x
∫
α
β
‍
s
s
s
s
s
s
+
∫
α
β
‍
p
t
,
τ
∫
Ω
‍
δ
i
u
i
x
,
t
-
τ
ϕ
x
d
x
d
τ
=
a
i
(
t
)
∫
Ω
‍
δ
i
ϕ
(
x
)
Δ
u
i
(
x
,
t
)
d
x
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
b
i
j
k
(
t
)
∫
Ω
‍
δ
i
ϕ
(
x
)
Δ
u
j
(
x
,
ρ
k
(
t
)
)
d
x
-
∫
Ω
‍
q
i
(
x
,
t
)
δ
i
u
i
(
x
,
t
)
ϕ
(
x
)
d
x
-
∑
γ
=
1
l
‍
∫
Ω
‍
δ
i
ϕ
(
x
)
∫
c
γ
d
γ
‍
f
i
j
γ
(
x
,
t
,
u
j
(
x
,
t
-
σ
γ
)
)
d
σ
γ
d
x
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
f
s
s
f
f
f
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
t
≥
t
1
,
i
∈
I
m
.
Therefore,
(42)
d
n
d
t
n
r
(
t
)
∫
Ω
‍
Z
i
(
x
,
t
)
ϕ
(
x
)
d
x
∫
α
β
‍
d
d
d
d
+
∫
α
β
‍
p
(
t
,
τ
)
∫
Ω
‍
Z
i
(
x
,
t
-
τ
)
ϕ
(
x
)
d
x
d
τ
≤
a
i
(
t
)
∫
Ω
‍
Δ
Z
i
(
x
,
t
)
ϕ
(
x
)
d
x
+
∑
j
=
1
m
‍
∑
k
=
1
s
‍
b
i
j
k
(
t
)
∫
Ω
‍
δ
j
δ
i
Δ
Z
j
(
x
,
ρ
k
(
t
)
)
ϕ
(
x
)
d
x
-
∫
Ω
‍
q
i
(
x
,
t
)
Z
i
(
x
,
t
)
ϕ
(
x
)
d
x
-
∑
j
=
1
m
‍
∑
γ
=
1
l
‍
δ
j
δ
i
∫
Ω
‍
∫
c
γ
d
γ
‍
h
i
j
γ
(
x
,
t
)
Z
j
(
x
,
t
-
σ
γ
)
d
σ
γ
ϕ
(
x
)
d
x
,
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
d
d
d
s
s
s
s
s
s
s
s
s
s
s
s
s
t
≥
t
1
,
i
∈
I
m
.
Green’s formula and boundary condition (4) yield
(43)
∫
Ω
‍
Δ
Z
i
x
,
t
ϕ
x
d
x
=
∫
Ω
‍
Z
i
(
x
,
t
)
Δ
ϕ
(
x
)
d
x
∫
Ω
‍
Δ
Z
i
x
,
t
ϕ
x
d
i
x
=
-
λ
0
∫
Ω
‍
Z
i
(
x
,
t
)
ϕ
(
x
)
d
x
≤
0
,
∫
Ω
‍
Δ
Z
i
x
,
ρ
k
t
ϕ
x
d
x
=
∫
Ω
‍
Z
i
(
x
,
ρ
k
(
t
)
)
Δ
ϕ
(
x
)
d
x
=
-
λ
0
∫
Ω
‍
Z
i
(
x
,
ρ
k
(
t
)
)
Δ
ϕ
(
x
)
d
x
≤
0
.
Combining (42)-(43), we get
(44)
d
n
d
t
n
r
t
∫
Ω
‍
Z
i
x
,
t
ϕ
x
d
x
∫
α
β
‍
s
s
s
s
g
s
+
∫
α
β
‍
p
t
,
τ
∫
Ω
‍
Z
i
x
,
t
-
τ
ϕ
x
d
x
d
τ
≤
-
λ
0
∑
k
=
1
s
‍
b
i
i
k
t
∫
Ω
‍
Δ
Z
i
x
,
ρ
k
t
ϕ
(
x
)
d
x
+
λ
0
∑
j
=
1
,
j
≠
i
m
‍
∑
k
=
1
s
‍
b
i
j
k
t
∫
Ω
‍
Δ
Z
j
x
,
ρ
k
t
ϕ
x
d
x
-
q
i
(
t
)
∫
Ω
‍
Z
i
(
x
,
t
)
ϕ
(
x
)
d
x
-
∑
γ
=
1
l
‍
h
i
i
γ
(
t
)
∫
c
γ
d
γ
‍
∫
Ω
‍
Z
i
(
x
,
t
-
σ
γ
)
d
σ
γ
ϕ
(
x
)
d
x
+
∑
j
=
1
,
j
≠
i
m
‍
∑
γ
=
1
l
‍
h
¯
i
j
γ
(
t
)
∫
c
γ
d
γ
‍
∫
Ω
‍
Z
j
(
x
,
t
-
σ
γ
)
ϕ
(
x
)
d
x
d
σ
γ
≤
0
.
Set
(45)
V
i
=
∫
Ω
‍
Z
i
(
x
,
t
)
ϕ
(
x
)
d
x
,
t
≥
t
1
,
i
∈
I
m
.
Then (44) yields
(46)
r
t
V
i
t
+
∫
α
β
‍
p
t
,
τ
V
i
t
-
τ
d
τ
(
n
)
+
λ
0
∑
k
=
1
s
‍
∑
k
=
1
s
‍
b
i
i
k
t
Δ
V
i
x
,
ρ
k
t
b
i
j
k
t
Δ
V
j
x
,
ρ
k
t
b
i
j
k
t
sssssssgsssss
-
∑
j
=
1
,
j
≠
i
m
‍
b
i
j
k
t
Δ
V
j
x
,
ρ
k
t
∑
k
=
1
s
‍
+
q
i
(
t
)
V
i
(
x
,
t
)
+
∑
γ
=
1
l
‍
h
i
i
γ
(
t
)
∫
c
γ
d
γ
‍
V
i
(
x
,
t
-
σ
γ
)
d
σ
γ
∑
j
=
1
,
j
≠
i
m
‍
s
s
s
s
s
s
s
s
s
s
s
-
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
(
t
)
∫
c
γ
d
γ
‍
V
j
(
x
,
t
-
σ
γ
)
d
σ
γ
≤
0
.
Let V
t
=
∑
i
=
1
m
‍
V
i
t
,
t
≥
t
1; from (46) we have
(47)
r
t
V
t
+
∫
α
β
‍
p
t
,
τ
V
t
-
τ
d
τ
(
n
)
+
λ
0
∑
k
=
1
s
‍
∑
i
=
1
m
‍
b
i
i
k
t
Δ
V
i
x
,
ρ
k
t
∑
j
=
1
,
j
≠
i
m
‍
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
-
∑
j
=
1
,
j
≠
i
m
‍
b
i
j
k
t
Δ
V
j
(
x
,
ρ
k
(
t
)
)
+
q
i
(
t
)
V
(
x
,
t
)
+
∑
γ
=
1
l
‍
∑
i
=
1
m
‍
h
i
i
γ
(
t
)
∫
c
γ
d
γ
‍
V
i
(
x
,
t
-
σ
γ
)
d
σ
γ
∑
j
=
1
,
j
≠
i
m
‍
s
s
s
s
s
s
s
s
s
s
s
s
s
s
-
∑
j
=
1
,
j
≠
i
m
‍
h
¯
i
j
γ
(
t
)
∫
c
γ
d
γ
‍
V
j
(
x
,
t
-
σ
γ
)
d
σ
γ
≤
0
.
Now, as in the proof of Theorem 4, from (47) we have
(48)
r
t
V
t
+
∫
α
β
‍
p
t
,
τ
V
t
-
τ
d
τ
(
n
)
+
λ
0
∑
k
=
1
s
‍
B
k
(
t
)
V
(
ρ
k
(
t
)
)
+
q
(
t
)
V
(
t
)
+
∑
γ
=
1
l
‍
Q
γ
(
t
)
∫
c
γ
d
γ
‍
V
(
t
-
σ
γ
)
d
σ
γ
≤
0
,
t
≥
t
1
.
The remaining part of the proof is similar to that of Theorem 4; therefore it is omitted. The proof is complete.Corollary 10.
If inequality (48) has no eventually positive solution, then every solution u
(
x
,
t
) of system (1), (4) is oscillatory in G.It is not difficult to see that the following theorems are true.Theorem 11.
Assume thatr
(
t
)
≥
1 is a monotone decreasing function, and (9) holds. If there exists some k
0
∈
I
s such that
(49)
∫
t
0
∞
‍
λ
0
B
k
0
t
1
-
∫
α
β
‍
p
ρ
0
t
,
τ
d
τ
d
t
=
∞
,
t
0
>
0
,
then every solution of boundary value system (1), (4) is oscillatory in G.Theorem 12.
Suppose that there exists a positive constantM such that 0
<
M
≤
r
(
t
)
<
1, and (33) holds. If there exists some k
0
∈
I
s such that
(50)
∫
t
0
∞
‍
λ
0
B
k
0
t
1
-
M
-
1
∫
α
β
‍
p
ρ
0
t
,
τ
d
τ
d
t
=
∞
,
t
0
>
0
,
then every solution of boundary value system (1), (4) is oscillatory in G.Theorem 13.
Assume all conditions of Theorem5 hold. Then every solution u
(
x
,
t
) of boundary value system (1), (4) is oscillatory in G.Theorem 14.
Assume all conditions of Theorem7 hold. Then every solution u
(
x
,
t
) of boundary value system (1), (4) is oscillatory in G.Theorem 15.
Assume all conditions of Theorem8 hold. Then every solution u
(
x
,
t
) of boundary value system (1), (4) is oscillatory in G.


## 3. Example

Example 1.
Consider the systems of second-order partial differential equations(51)
∂
2
∂
t
2
(
(
1
+
e
-
t
)
u
1
(
x
,
t
)
+
1
2
∫
π
/
2
π
‍
u
1
(
x
,
t
-
τ
)
d
τ
)
=
Δ
u
1
(
x
,
t
)
+
3
2
+
3
e
-
t
Δ
u
1
x
,
t
-
3
π
2
-
3
2
u
1
(
x
,
t
)
-
e
-
t
+
4
2
∫
π
/
2
π
‍
u
1
x
,
t
-
σ
1
d
σ
1
-
e
-
t
2
∫
π
/
2
π
‍
u
2
x
,
t
-
σ
1
d
σ
1
(52)
∂
2
∂
t
2
1
+
e
-
t
u
2
x
,
t
+
1
2
∫
π
/
2
π
‍
u
2
x
,
t
-
τ
d
τ
=
e
-
t
Δ
u
2
(
x
,
t
)
+
3
2
+
2
e
-
t
Δ
u
2
x
,
t
-
3
π
2
-
5
2
u
2
(
x
,
t
)
-
e
-
t
2
∫
π
/
2
π
‍
u
1
x
,
t
-
σ
1
d
σ
1
-
e
-
t
+
4
2
∫
π
/
2
π
‍
u
2
x
,
t
-
σ
1
d
σ
1
sssssssssssssss
(
x
,
t
)
∈
(
0
,
π
)
×
[
0
,
∞
)
with boundary condition
(53)
∂
∂
x
u
i
(
0
,
t
)
=
∂
∂
x
u
i
(
π
,
t
)
=
0
,
t
≥
0
,
i
=
1,2
.
Here,N
=
1, n
=
2, m
=
2, α
=
π
/
2, β
=
π, k
=
1, c
1
=
π
/
2, d
1
=
π, i
=
2, a
1
(
t
)
=
1, a
2
(
t
)
=
e
-
t, q
1
(
x
,
t
)
=
3
/
2, q
2
(
x
,
t
)
=
5
/
2, q
111
(
t
)
=
e
-
t
+
4
/
2, q
121
(
t
)
=
e
-
t
/
2, q
212
(
t
)
=
e
-
t
/
2, q
221
(
t
)
=
e
-
t
+
4
/
2, f
i
j
γ
(
x
,
t
,
u
j
)
=
u
j. It is easy to see that all conditions of Theorem 4 are fulfilled. Then every solution of system (52), (53) oscillates in (
0
,
π
)
×
[
0
,
∞
). In fact, u
1
(
x
,
t
)
=
cos
⁡
x
sin
t
,
u
2
(
x
,
t
)
=
cos
⁡
x
cos
⁡
t. is such a solution.



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*Source: 739636-2015-01-01.xml*