Available at
http://pvamu.edu/pages/398/asp
Applications and Applied
Mathematics
(AAM):
An International Journal
Vol. 1, Issue 2 (December, 2006), pp. 83 – 95
(Previously, Vol. 1, No. 2)
Oscillations of Hyperbolic Systems with Functional Arguments*
Yutaka Shoukaku
Faculty of Engineering
Kanazawa University
Ishikawa 920-1192, Japan
E-mail: shoukaku@t.kanazawa-u.ac.jp
Norio Yoshida
Department of Mathematics
University of Toyama
Toyama 930-8555, Japan
E-mail: nori@sci.u-toyama.ac.jp
Received May 25, 2006; accepted October 18, 2006
Abstract
Hyperbolic systems with functional arguments are studied, and sufficient conditions are
obtained for every solution of boundary value problems to be weakly oscillatory (that is, at
least one of its components is oscillatory) in a cylindrical domain. Robin-type boundary
condition is considered. The approach used is to reduce the multi-dimensional oscillation
problems to one-dimensional oscillation problems by using some integral means of solutions.
Keywords: Oscillation; hyperbolic systems; functional arguments
AMS MSC No.: 35B05; 35R10
1. Introduction
We are concerned with the oscillation of the system of hyperbolic equations with functional
arguments

∂2 

2 U ( x, t ) + ∑ H i (t ) U ( x, ρ i (t ))  − A(t )∆U ( x, t )
∂t 
i =1

K
m
i =1
i =1
− ∑ Bi (t )∆U ( x, τ i (t )) + ∑ Pi ( x, t )ϕ i (U ( x, σ i (t )) )
(1)
( x, t ) ∈ Ω ≡ G × (0, ∞),
= F ( x, t ), where G is a bounded domain in R n with piecewise smooth boundary ∂G , ∆ is the
Laplacian in R n , and
83
84
Y. Shoukaku and N. Yoshida
H i (t ) = (hijk (t ) ) j ,k =1 , A(t ) = (a jk (t ) ) j ,k =1 , Bi (t ) = (bijk (t ) ) j ,k =1 ,
M
M
M
Pi ( x, t ) = ( pijk ( x, t ) ) j ,k =1 ,
M
U ( x, t ) = (u1 ( x, t ), , u M ( x, t ) ) ,
T
U ( x, ρ i (t )) = (u1 ( x, ρ i (t )), , u M ( x, ρ i (t )) ) ,
T
U ( x, τ i (t )) = (u1 ( x, τ i1 (t )), , u M ( x,τ iM (t )) ) ,
T
ϕ i (U ( x, σ i (t )) ) = (ϕ i1 (u1 ( x, σ i (t ))), , ϕ iM (u M ( x, σ i (t ))))T ,
F ( x, t ) = ( f1 ( x, t ), , f M ( x, t ) ) ,
T
the superscript T denoting the transpose.
It is easy to see that (1) can be written in the following system:
 M
∂2 

(
,
)
u
x
t
hijk (t )u k ( x, ρ i (t )) 
+

∑∑
j
2
∂t 
i =1 k =1

M
K
k =1
i =1 k =1
M
− ∑ a jk (t )∆u k ( x, t ) − ∑∑ bijk (t )∆u k ( x,τ ik (t ))
m
(2)
M
( j = 1, 2,  , M )
+ ∑∑ pijk ( x, t )ϕ ik (u k ( x, σ i (t )) ) = f j ( x, t ) i =1 k =1
for ( x, t ) ∈ Ω ≡ G × (0, ∞) .
The boundary condition to be considered is the following:
α (xˆ )
∂u j
∂ν
+ µ (xˆ ) u j = α ( xˆ )ψ~ j + µ (xˆ )ψ j on　∂G × (0, ∞) ( j = 1, 2,  , M ), (BC)
2
2
where ψ j ,ψ~ j ∈ C (∂G × (0, ∞); R) , α (xˆ ), µ (xˆ ) ∈ C (∂G; [0, ∞)) , α ( xˆ ) + µ (xˆ ) ≠ 0 , and ν
denotes the unit exterior normal vector to ∂G .
In case α ( xˆ ) ≡ 0 on ∂G , then µ ( xˆ ) ≠ 0 on ∂G , and hence the boundary condition (BC)
reduces to
u j = ψ j on　∂G × (0, ∞) ( j = 1, 2,  , M ).
If α ( xˆ ) = 1 on ∂G , then (BC) can be written in the form
∂u j
∂ν
+ µ (xˆ ) u j = ψ~ j + µ (xˆ )ψ j on ∂G × (0, ∞) ( j = 1, 2,  , M ),
moreover, if µ ( xˆ ) ≡ 0 on ∂G , then (BC) can be written as
∂u j
∂ν
= ψ~ j on　∂G × (0, ∞) ( j = 1, 2,  , M ).
AAM: Intern. J., Vol. 1, Issue 2 (2006) [Previously Vol. 1, No. 2]
85
We assume that:
(H1 ) hijk (t ) ∈ C 2 ([0, ∞);[0, ∞)) (i = 1, 2, ,  ; j, k = 1, 2, , M ),
a jk (t ) ∈ C ([0, ∞); [0, ∞)) ( j , k = 1, 2, , M ),
bijk (t ) ∈ C ([0, ∞); [0, ∞)) (i = 1, 2, , K ; j , k = 1, 2, , M ),
p ijk ( x, t ) ∈ C (G × [0, ∞); [0, ∞) )　(i = 1, 2, , m ; j , k = 1, 2, , M ),
f j ( x, t ) ∈ C (G × [0, ∞); R )　( j = 1, 2,  M ) ;
(H 2 )　ρ i (t ) ∈ C 2 ([0, ∞); R), lim t →∞ ρ i (t ) = ∞　(i = 1, 2, , ),
τ ik (t ) ∈ C ([0, ∞); R), lim t →∞ τ ik (t ) = ∞　(i = 1, 2,  , K ; k = 1, 2, , M ),
σ i (t ) ∈ C ([0, ∞); R), lim t →∞ σ i (t ) = ∞　(i = 1, 2,  , m) ;
(H3 )　ϕ ik (ξ ) ∈ C (R; R),
ϕ ik (ξ ) ≥ 0 for ξ ≥ 0, ϕ ik (−ξ ) = −ϕ ik (ξ ) for　ξ > 0 and
ϕ ik (ξ ) are　convex　in (0, ∞) (i = 1, 2,  , m ; k = 1, 2,  , M ) ;
M
(H 4 )　hikk (t ) − ∑ hijk (t ) ≥ 0　(i = 1, 2, ,  ; k = 1, 2, , M ) ;
j=1
j≠ k
M
(H5 )　a kk (t ) − ∑ a jk (t ) ≥ 0　(k = 1, 2 ,, M ) ;
j =1
j≠k
M
(H 6 )　bikk (t ) − ∑ bijk (t ) ≥ 0　(i = 1, 2, , K ; k = 1, 2, , M ) ;
j =1
j≠k
M
(H7 )　pikk ( x, t ) − ∑ pijk ( x, t ) ≥ 0　(i = 1, 2, ,
j =1
j≠k
m ; k = 1, 2,  , M ) ;
(H 8 )　ϕˆ i (ξ ) ≡ 1min
ϕ ik (ξ ) is nondecreasing　and
≤k ≤M
(0, ∞)
convex　in (i = 1, 2,  , m).
Definition 1: By a solution of system (2) we mean a vector function (u1 ( x, t ), , u M ( x, t ) )
such that u j ( x, t ) ∈ C 2 (G × [t −1 , ∞); R ) ∩ C (G × [~
t−1 , ∞); R ) , and u j ( x, t ) ( j = 1, 2,  , M )
satisfy (2) in Ω , where


t −1 = min 0, min inf τ ik (t ) , min inf ρ i (t ) ,
1≤i ≤  t ≥ 0

 11≤≤ik≤≤KM t ≥0
{
{
} {
{
}}
}
~
t−1 = min 0, min inf σ i (t ) .
1≤i ≤ m
t ≥0
Definition 2: A solution (u1 ( x, t ), , u M ( x, t ) ) of system (2) is said to be weakly oscillatory
in Ω if at least one of its components is oscillatory in Ω (cf. Ladde, Lakshmikantham and
Zhang (1987, Definition 6.2.1)).
86
Y. Shoukaku and N. Yoshida
In 1984 the oscillations of delay hyperbolic equations have been first investigated by Mishev
and Bainov (1984) (cf. Mishev 1989, Mishev and Bainov, 1986). Parhi and Kirane (1994)
investigated the oscillatory properties of solutions of coupled hyperbolic equations.
Oscillation of hyperbolic systems with deviating arguments was studied by Li (1997), and
then oscillation results have been established by several authors, see, e.g., Li (2000),
Agarwal, Meng and Li (2002) and the references cited therein. However, all of them pertain
to the case where the matrices H i (t ) are the diagonal matrices or H i (t ) ≡ 0 .
The purpose of this paper is to derive sufficient conditions for every solution of the boundary
value problem (2), (BC) to be weakly oscillatory in a cylindrical domain G × (0, ∞) . We note
that the matrices H i (t ) are not necessarily the diagonal matrices.
2. Oscillation results
In this section we establish a lemma and two oscillation theorems for the boundary value
problem (2), (BC). Two examples are also given in this section to illustrate oscillation results.
− ∆w = λ w　in　G,
∂w
α (xˆ ) + µ (xˆ ) w = 0　on　∂G
∂ν
is nonnegative, and the corresponding eigenfunction Φ (x) can be chosen so that Φ ( x) > 0 in
G (see Ye and Li, 1990, Theorem 3.3.22). In case µ ( xˆ ) ≡ 0 on ∂G , then we can choose
λ1 = 0 and Φ ( x) = 1 . If µ (xˆ ) ≡/ 0 on ∂G , then there exist λ1 > 0 and the eigenfunction
Φ ( x) > 0 in G .
We use the notation :
Γ1 = {xˆ ∈ ∂G ; α ( xˆ ) = 0},
Γ2 = {xˆ ∈ ∂G ; α ( xˆ ) ≠ 0}.
Lemma: If u j satisfy the boundary condition (BC) and Φ (x) is the eigenfunction
corresponding to the smallest eigenvalue λ1 ≥ 0 , then we obtain
 ∂u j
∂Φ ( x) 
Φ ( x) − u j
KΦ ∫ 
 dS = Ψ j (t ),
∂G ∂ν
∂ν 

where
KΦ =
(∫
G
Φ ( x) dx
)
−1
,



∂Φ ( x)
µ (xˆ ) 
Ψ j (t ) = K Φ  − ∫ ψ j
dS + ∫ ψ~ j +
ψ j Φ ( x) dS  .
Γ1
Γ2
∂ν
α (xˆ ) 



(3)
AAM: Intern. J., Vol. 1, Issue 2 (2006) [Previously Vol. 1, No. 2]
87
Proof: It is evident that
 ∂u j
 ∂u j
∂Φ ( x) 
∂Φ ( x) 
Φ
−
=
Φ
−
(
x
)
u
dS
(
x
)
u



 dS
j
j
∫∂G  ∂ν
∫Γ1  ∂ν
∂ν 
∂ν 
 ∂u j
∂Φ ( x) 
Φ ( x) − u j
+∫ 
 dS .
Γ2 ∂ν
∂ν 

(4)
Since u j = ψ j on Γ1 and Φ ( x) = 0 on Γ1 , we obtain
 ∂u j
∂Φ ( x) 
∂Φ ( x)
Φ
−
(
x
)
u
dS .

 dS = − ∫Γ1 ψ j
j
∫Γ1  ∂ν
∂ν 
∂ν
(5)
From the boundary condition (BC) we see that
µ (xˆ )
µ (xˆ )
ψj −
= ψ~ j +
u j on　Γ2 ,
α (xˆ )
α (xˆ )
∂ν
∂Φ ( x)
µ (xˆ )
=−
Φ ( x) on　Γ2 .
∂ν
α (xˆ )
∂u j
Hence, we observe that
∫
 ∂u j

µ (xˆ ) 
∂Φ ( x) 
dS = ∫ ψ~ j +
ψ j Φ ( x) dS .
Φ ( x) − u j


Γ2 ∂ν
Γ2
α (xˆ ) 
∂ν 


Combining (4)－(6) yields the desired identity (3).
We note that if α ( xˆ ) = 0 on ∂G , then Γ2 = 0/ and
 ∂u j
∂Φ ( x) 
∂Φ ( x)
KΦ ∫ 
dS = − K Φ ∫ ψ j
dS .
Φ ( x) − u j

∂G ∂ν
∂G
∂ν 
∂ν

If α ( xˆ ) = 1　( xˆ ∈ ∂G ) and µ ( xˆ ) = 0 ( xˆ ∈ ∂G ) , then Γ1 = 0/ and
 ∂u j
∂Φ ( x) 
1
dS =
KΦ ∫ 
Φ ( x) − u j

∂G ∂ν
∂ν 
G

where G = ∫ dx denotes the volume of G .
G
We use the notation :
∫
∂G
ψ~ j dS ,
(6)
88
Y. Shoukaku and N. Yoshida
F j (t ) = K Φ ∫ f j ( x, t )Φ ( x) dx　( j = 1, 2 , , M ),
G
[Θ(t )]± = max{ ± Θ(t ) , 0},
Γ=
{ ((−1)
α1
)
}
, (−1) α 2 ,  , (−1) α M ; α j = 0, 1　( j = 1, 2 ,  , M ) .
{
}
~
~
We note that # Γ = 2 M and − γ ∈ Γ for γ ∈ Γ , and hence Γ = ± γ ; γ ∈ Γ for some Γ ⊂ Γ
~
with # Γ = 2 M −1 . For example, we let M = 2 . Then we observe that
Γ = {(1, 1), (1, − 1), (−1, 1), (−1, − 1)}
and
{
}
~
Γ = { ± (1, 1), ± (1, − 1)} = ± γ ; γ ∈ Γ ,
where
~
Γ = {(1, 1), (1, − 1)}.
Theorem 1: Assume that the hypotheses (H1)－(H8) hold. If the following conditions are
satisfied :
(H 9 ) ρ i (t ) ≤ t　(i = 1, 2, , ) ;
 M
~
(H10 ) [t 0 , ∞) for some　t 0 > 0 ;
∑∑ hij (t ) ≤ 1　on　and
i =1 j =1
such that Θ γ (t ) is os (H11 ) there exist functions Θ γ (t ) ∈ C 2 ([t 0 , ∞); R) (γ ∈ Γ) M
cillatry at t = ∞ and　Θ′γ′ = γ ⋅ (G j (t ) ) j =1 ( ⋅ denotes the scalar product) ;
if for some j 0 ∈ {1, 2,  , m} and for any c > 0
 M
 
 

~
 1 −
ˆ (σ (t ))  dt = ∞,

ˆ
ˆ
ϕ
σ
p
t
h
t
c
(
)
(
(
))
(7)
+
Θ
∑∑
j
j
ij
γ
j
j
∫ t0 0
0
0
0


 
i =1 j =1




+ 

then every solution (u1 ( x, t ), , u M ( x, t ) ) of the boundary value problem (2), (BC) is
weakly oscillatory in Ω , where
∞
~
hij (t ) = max hijk (t ) ,
1≤ k ≤ M
M
K M
1 

 F j (t ) + ∑ a jk (t )Ψk (t ) + ∑∑ bijk (t )Ψk (τ ik (t ))  ,
M
k =1
i =1 k =1





M



pˆ i (t ) = min  min  pikk ( x, t ) − ∑ pijk ( x, t )  ,
x∈G 1≤ k ≤ M
j =1



j ≠k



 M
~
ˆ (t ) = Θ (t ) − ∑∑ h (t )Θ ( ρ (t )) .
Θ
ij
i
γ
γ
γ
G j (t ) =
i =1 j =1
Proof: Suppose that there exists a solution (u1 ( x, t ), , u M ( x, t ) ) of the problem (2), (BC)
which is not weakly oscillatory in Ω . Then, each component u j ( x, t ) is nonoscillatory in Ω .
AAM: Intern. J., Vol. 1, Issue 2 (2006) [Previously Vol. 1, No. 2]
89
We easily see that there is a number t1 ≥ t 0 such that
t ≥ t1 ( j = 1, 2,  , M ) . Letting
u j ( x, t ) > 0 for x ∈ G ,
w j ( x, t ) = δ j u j ( x, t ),
where δ j = sgn u j ( x, t ) , we see that w j ( x, t ) = u j ( x, t ) > 0 in G × [t1 , ∞) . There exists a
number t 2 ≥ t1 such that w j ( x, t ) > 0 , w j ( x, ρ i (t )) > 0 , w j ( x, τ ij (t )) > 0 , w j ( x, σ i (t )) > 0 in
G × [t 2 , ∞) . Proceeding as in the proof of Theoem 1 of Shoukaku and Yoshida (2005), we
observe that the following identity holds :

1
V (t ) +

M

d2
dt 2

M

m

i =1
∑ ∑ δ j δ k hijk (t )Wk ( ρ i (t ))  + ∑ pˆ i (t )ϕˆ i (V (σ i (t )))
i =1 j , k =1
M
(8)
≤ ∑ δ j G j (t ), t ≥ t 2 ,
j =1
where
∑
V (t ) =
There is a γ ∈ Γ such that
∑
M
j =1
M
k =1
W k (t )
M
.
δ j G j (t ) = γ ⋅ (G j (t ) ) jM=1 . Setting
1
M
Y (t ) = V (t ) +

M
∑ ∑δ
i =1 j , k =1
j
δ k hijk (t )Wk ( ρ i (t )) − Θ γ (t ) ,
we see that
m
Y ′′(t ) ≤ −∑ pˆ i (t )ϕˆ i (V (σ i (t )) ) ≤ 0 , t ≥ t 2 .
(9)
i =1
Therefore, Y (t ) > 0 or Y (t ) ≤ 0 on [t 3 , ∞) for some t 3 ≥ t 2 . If Y (t ) ≤ 0 on [t 3 , ∞) , then
V (t ) +
1
M

M
∑ ∑δ
i =1 j , k =1
j
δ k hijk (t )Wk ( ρ i (t )) ≤ Θ γ (t ) , t ≥ t 3 ,
and therefore
1
V (t ) +
M


M


∑∑
 hikk (t ) − ∑ hijk (t )  Wk ( ρ i (t )) ≤ Θ γ (t ) , t ≥ t 3 .
i =1 k =1 
j =1

j ≠k



M
(10)
The left hand side of (10) is positive in view of the hypothesis (H4), whereas the right hand
side of (10) is oscillatory at t = ∞ . This is a contradiction. Hence, we conclude that
Y (t ) > 0 on [t 3 , ∞) . Since Y ′′(t ) ≤ 0 , Y (t ) > 0 on [t 3 , ∞) , we obtain Y ′(t ) ≥ 0 on [t 4 , ∞) for
some t 4 ≥ t 3 . Hence, Y (t ) ≥ Y (t 4 )
for t ≥ t 4 . In view of the fact that V (t ) ≤ Y (t ) + Θ γ (t ) and Y (t ) is nondecreasing, we obtain
90
Y. Shoukaku and N. Yoshida
V (t ) = Y (t ) −
1
M

M
∑ ∑δ
i =1 j , k =1
M
j
δ k hijk (t )Wk ( ρ i (t )) + Θ γ (t )

~
≥ Y (t ) − ∑∑ hij (t )V ( ρ i (t )) + Θ γ (t )
i =1 j =1
 M
~
≥ Y (t ) − ∑∑ hij (t )(Y ( ρ i (t )) + Θ γ ( ρ i (t )) ) + Θ γ (t )
(11)
i =1 j =1
 M

~ 
ˆ (t ) , t ≥ t .
≥ 1 − ∑∑ hij (t )  Y (t ) + Θ
γ
4
i
=
j
=
1
1


Since Y (t ) ≥ Y (t 4 ) and V (t ) > 0 for t ≥ t 4 , from (11) we have
 M


~ 
ˆ (t ) , t ≥ t
V (t ) ≥ 1 − ∑∑ hij (t )  Y (t 4 ) + Θ
γ
4
i =1 j =1

 +

and therefore
 M



~
ˆ (σ (t )) , t ≥ T
V (σ j0 (t )) ≥ 1 − ∑∑ hij (σ j0 (t ))  Y (t 4 ) + Θ
j0
γ
i =1 j =1

 +

(12)
for some T ≥ t 4 . Since ϕˆ j0 (t ) is nondecreasing, from (9) and (12) we obtain
 M
 
 

~
ˆ (σ (t )) 
pˆ j0 (t )ϕˆ j0  1 − ∑∑ hij (σ j0 (t ))  Y (t 4 ) + Θ
j0
γ
 
i =1 j =1
 + 


≤ −Y ′′(t ) , t ≥ T .
(13)
Integrating (13) over [T , t ] yields
 M
 
 

~

ˆ (σ ( s ))  ds


ˆ
ˆ
(
)
1
(
(
))
(
)
p
s
h
s
Y
t
ϕ
σ
+
Θ
−

∑∑
4
j
j
ij
j
j
γ
∫T 0
0
0
0


 
i =1 j =1

 + 

≤ −Y ′(t ) + Y ′(T ) ≤ Y ′(T ), t ≥ T .
t
This contradicts the hypothesis (7) and completes the proof.
Theorem 2: Assume that the hypotheses (H1)－(H8) hold. Every solution
(u1 ( x, t ),, u M ( x, t ) ) of the boundary value problem (2), (BC) is weakly oscillatory in
Ω if for any γ ∈ Γ
(
)
t
s
M
lim inf ∫ 1 −  γ ⋅ (G j ( s ) ) j =1 ds = −∞
T
t →∞
 t
for all large T .
Proof: Suppose that there exists a solution (u1 ( x, t ), , u M ( x, t ) ) of the problem (2), (BC)
which is not weakly oscillatory in Ω . Arguing as in the proof of Theorem 1, we observe that
AAM: Intern. J., Vol. 1, Issue 2 (2006) [Previously Vol. 1, No. 2]
91
(8) holds, and hence
d2
dt 2

1
V (t ) +

M


M

M

j =1
∑ ∑ δ j δ k hijk (t )Wk ( ρ i (t ))  ≤ ∑ δ j G j (t ), t ≥ t 2
i =1 j , k =1
(14)
for some t 2 > 0 . Letting
1
~
V (t ) = V (t ) +
M

M
∑ ∑δ
i =1 j , k =1
j
δ k hijk (t )Wk ( ρ i (t )) ,
~
we note that V (t ) > 0 (t > t 3 ) for some t 3 ≥ t 2 (cf. the proof of Theorem 1). Integrating
(14) twice over [t 3 , t ] , we obtain
~
t 

s  M
V (t ) c1
 t 
≤ + c 2 1 − 3  + ∫ 1 −  ∑ δ j G j ( s )  ds
t
t
t  t3  t  j =1


for some constants c1 and c 2 . The left hand side of the above inequality is positive, whereas
the right hand side is not bounded from below by the hypothesis. This is a contradiction.
{
}
~
~
~
Remark 1: We find that Γ = ± γ ; γ ∈ Γ for some Γ ⊂ Γ with # Γ = 2 M −1 . Hence, Theorem
1 holds true if the hypothesis (H11) and the condition (7) are replaced by
~
~
(H11 ) there exist functions Θ γ (t ) ∈ C 2 ([t 0 , ∞) ; R) (γ ∈ Γ) such that Θ γ (t ) is
oscillatory at
( ⋅ denotes the scalar and
t = ∞ and　Θ′γ′ (t ) = γ ⋅ (G j (t ) ) j =1 M
product)
∫
∞
t0
 M
 
 

~
ˆ (σ (t ))  dt = ∞,
pˆ j0 (t ) ϕˆ j0  1 − ∑∑ hij (σ j0 (t ))  c ± Θ
γ
j0
 
i =1 j =1
 + 


respectively.
Remark 2: Under the same hypotheses of Theorem 2, the conclusion of Theorem 2 holds
~
true if for any γ ∈ Γ
t
s
M
lim inf ∫ 1 −  γ ⋅ (G j ( s ) ) j =1 ds = −∞,
T
t →∞
 t
t
s
M
lim sup ∫ 1 −  γ ⋅ (G j ( s ) ) j =1 ds = ∞
T
t →∞
 t
for all large T .
(
(
Example 1: We consider the hyperbolic system
)
)
92
Y. Shoukaku and N. Yoshida
 ∂2 
1
1

 2  u1 ( x, t ) + u1 ( x, t − 2π ) + u 2 ( x, t − 2π ) 
4
2

 ∂t  2
2
2
u
u
u
∂
∂
∂
1
1 ∂ 2u 2
　2π
1
2
1
x
t
e
x
t
x
t
−
−
−
−
−
π
3
(
,
)
(
,
)
(
,
)
( x, t − 2π )

8
8 ∂x 2
∂x 2
∂x 2
∂x 2

1
1
+ u1 ( x, t − π ) + e π u 2 ( x, t − π ) = e 2π (sin x ) e −t ,
　2
 2 2
1
1
∂ 

 ∂t 2  u 2 ( x, t ) + 4 u1 ( x, t − 2π ) + 4 u 2 ( x, t − 2π ) 



2
2
2
∂ u1
1 ∂ u1
1 ∂ 2u2
2π ∂ u 2
　( x, t ) −
( x, t − π ) −
( x, t − 2π )
− 2 2 ( x, t ) − e

2 ∂x 2
4 ∂x 2
∂x
∂x 2

1
3
+ u1 ( x, t − π ) + e π u 2 ( x, t − π ) = (sin x )sin t + 1 + 3e 2π (sin x ) e −t ,
　4
2

(
x
,
t
)
(
0
,
π
)
(
0
,
)
∈
×
∞

(
with the boundary condition
(15)
)
u j (0, t ) = u j (π , t ) = 0, t > 0 ( j = 1, 2) .
(16)
G = (0, π ) , n = 1 , M = 2 ,  = K = m = 1 , h111 (t ) = 1 / 2 , h112 (t ) = h121 (t ) =
h122 (t ) = 1 / 4 , ρ1 (t ) = t − 2π , a11 (t ) = 3 , a12 (t ) = (1 / 8) e 2π , a21 (t ) = 2 , a 22 (t ) = e 2π ,
b111 (t ) = 1 , b112 (t ) = 1 / 8 , b121 (t ) = 1 / 2 , b122 (t ) = 1 / 4 , τ 11 (t) = t − π , τ 12 (t )=t −2π ,
p111 ( x,t )=1/ 2 , p112 ( x,t )=(1/ 2)e π , p121 ( x, t ) = 1 / 4 , p122 (t ) = (3 / 2) e π , σ 1 (t) = t − π ,
Here
ϕ 11 (ξ ) = ϕ 12 (ξ ) = ξ ,
f1 ( x, t ) = e 2π (sin x ) e − t ,
f 2 ( x, t ) = (sin x )sin t + (1 + 3e 2π ) (sin x ) e − t ,
( j = 1, 2) , α ( xˆ ) = 0 and Γ2 = 0/ . We observe that
ψj =0　~
~
1
1
1
h11 (t ) = , h12 (t ) = , pˆ 1 (t ) = .
2
4
4
It is easy to see that
2π
(π / 8)e e
−t
( j = 1, 2) , G1 (t ) = (1 / 2) F1 (t ) =
λ1 = 1 , Φ ( x) = sin x , Ψ j (t ) = 0　(
)
and G2 (t ) = (1 / 2) F2 (t ) = (π / 8) sin t + (π / 8) 1 + 3e 2π e − t . Since we can choose
Θ (1, −1)
π
(− sin t + (1 + 4e ) e ) ,
8
π
(t ) = (sin t − (1 + 2e ) e ) ,
8
Θ (1,1) (t ) =
2π
2π
−t
−t
we obtain
ˆ (t ) = π  − 1 sin t + 1 + 4e 2π 1 − 3 e 2π e −t  ,
Θ
(1,1)

8  4
 
 4
π 1
 
 3
ˆ
 sin t − 1 + 2e 2π 1 − e 2π e −t  ,
Θ
(1, −1) (t ) =
84
 
 4
(
(
and therefore
)
)
AAM: Intern. J., Vol. 1, Issue 2 (2006) [Previously Vol. 1, No. 2]
93

1  3 
π 1

 3
1 −  c ±  − sin(t − π ) + 1 + 4e 2π 1 − e 2π  e −( t −π )  dt


t0 4
8 4

 4
 +
 4 
∞
1

π

 3
sin t + 1 + 4e 2π 1 − e 2π e π e −t  dt
≥

∫
t
32 0  4

 4
±
=∞
∫
(
∞
(
)
)
and
1  3 
3 2π  −(t −π ) 
π 1
2π 
 dt
1 −  c ±  sin(t − π ) − 1 + 2e 1 − e  e
4  4 
84

 4
 +
∞


π
1

 3
≥
− sin t − 1 + 2e 2π 1 − e 2π e π e −t  dt

∫
t
0
32

 4
 4
±
=∞
∫
(
∞
t0
(
)
)
for any c > 0 . Hence, it follows from Theorem 1 and Remark 1 that every solution
(u1 ( x, t ), u 2 ( x, t ) ) of the problem (15), (16) is weakly oscillatory in (0, π ) × (0, ∞) . One such
solution is
 (sin x )sin t 
.
U ( x, t ) = 
−t 
 (sin x ) e 
Example 2: We consider the hyperbolic system
 ∂2 
1
1

 2  u1 ( x , t ) + u1 ( x , t − π ) + u 2 ( x , t − π ) 
2
2

 ∂t  2
2
2
u
u
u
∂
∂
∂
∂ 2u 2
π
　( 3 / 2 )π
1
2
1
x
t
x
t
e
x
t
−
−
−
+
−
(
,
)
(
,
)
(
,
)
( x, t + π )

2
∂x 2
∂x 2
∂x 2
∂x 2

3
1
+ u1 ( x, t − π ) + u 2 ( x, t − π ) = 2 + 3e π (cos x ) e −t ,
　2
2

 ∂2 
1
1

 2  u 2 ( x, t ) + u1 ( x, t − π ) + u 2 ( x, t − π ) 
3
2

 ∂t 
2
2
2

∂ u1
∂ u2
2 (3 / 2 )π ∂ u1
π
3 ∂ 2u2
− 2 ( x, t ) −
( x, t ) − e
( x, t + ) −
( x, t + π )
　3
2
2 ∂x 2
∂x
∂x 2
∂x 2

+ u1 ( x, t − π ) + u 2 ( x, t − π ) = −2(cos x )sin t + 2e π + 1 (cos x ) e −t ,
　( x, t ) ∈ (0, π ) × (0, ∞)
2

(
)
(
(17)
)
with the boundary condition
 ∂u1
− ∂x (0, t ) = 0,
 ∂u
− 2 (0, t ) = 0,
 ∂x
Here G = (0, (π / 2)) , n = 1 , M = 2 ,
h122 (t ) = 1 / 2 ,
ρ1 (t) = t − π ,
∂u1  π 
−t
 , t  = −e ,
∂x  2 
∂u 2  π 
 , t  = − sin t , t > 0.
∂x  2 
(18)
 = K = m = 1 , h111 (t ) = h112 (t ) = 1 / 2 , h121 (t ) = 1 / 3 ,
a11 (t ) = a12 (t ) = a 21 (t ) = a 22 (t ) = 1 , b111 (t ) = e (3 / 2 )π ,
94
Y. Shoukaku and N. Yoshida
b112 (t ) = 1 , b121 (t ) = (2 / 3)e (3 / 2 )π , b122 (t ) = 3 / 2 , τ 11 (t ) = t + (π / 2) , τ 12 (t) = t + π ,
p111 ( x, t ) = 3 / 2 , p112 ( x, t ) = 1 / 2 , p121 ( x, t ) = p122 ( x, t ) = 1 , σ 1 (t)=t −π , ϕ 11 (ξ )=ϕ 12 (ξ )=ξ ,
α (xˆ ) = 1 ,
f1 ( x, t ) = (2 + 3e π ) (cos x ) e − t ,
f 2 ( x, t ) = −2(cos x )sin t + (2e π + 1) (cos x )e − t ,
µ (xˆ ) = 0 and Γ1 = 0/ . We find that
~
~
1
1
h11 (t ) = h12 (t ) = , pˆ 1 (t ) = .
2
2
It is easy to check that λ1 = 0 , Φ ( x) = 1 and that
1
G1 (t ) =
2 + 4e π e − t ,
2π
1 
8 π −t 
G2 (t ) =
 − 3 sin t + e e  .
2π 
3

Choosing
1 
 
 20
 3 sin t +  e π + 2 e −t  ,
Θ (1,1) (t ) =
2π 
 
 3
1 
 
4
 − 3 sin t +  e π + 2 e −t  ,
Θ (1, −1) (t ) =
2π 
 
3
we see that
ˆ (t ) = 1  6 sin t +  20 e π + 2  1 − e π e −t  ,
Θ
(1,1)

2π 

 3


1 

4
ˆ
 − 6 sin t +  e π + 2  1 − e π e −t  .
Θ
(1, −1) (t ) =
2π 

3

Since
∞1
∫t0 2 ± Θˆ (1,1) (t − π ) + dt

1 ∞

 20
=
− 6 sin t +  e π + 2  e π − e 2π e −t  dt

∫
4π t0 

 3
±
=∞
and
∞1
∫t0 2 ± Θˆ (1,−1) (t − π ) + dt

1 ∞

4
6 sin t +  e π + 2  e π − e 2π e −t  dt
=

∫
4π t0 

3
±
= ∞,
Theorem 1 and Remark 1 imply that every solution (u1 ( x, t ), u 2 ( x, t ) ) of (17), (18) is weakly
oscillatory in (0, π2 ) × (0, ∞) . In fact,
(
)
(
)
(
[
)
]
(
[
]
(
 (cos x ) e − t 

U ( x, t ) = 

 (cos x )sin t 
is such a solution.
)
)
AAM: Intern. J., Vol. 1, Issue 2 (2006) [Previously Vol. 1, No. 2]
95
References
Agarwal, R. P., F. W. Meng, and W. N. Li, (2002), Oscillation of solutions of systems of
neutral type partial functional differential equations, Comput. Math. Appl., 44, pp. 777–
786.
Ladde, G. S., V. Lakshmikantham, and B. G. Zhang, (1987), Oscillation Theory of Differential
Equations with Deviating Arguments, Marcel Dekker, Inc., New York.
Li, Y. K., (1997), Oscillation of systems of hyperbolic differential equations with deviating
arguments, Acta Math. Sinica, 40, pp. 100–105, (in Chinese).
Li, W. N., (2000), Oscillation properties for systems of hyperbolic differential equations of
neutral type, J. Math. Anal. Appl., 248, pp. 369–384.
Li, W. N., (2003), Forced oscillation properties for certain systems of partial functional
differential equations, Appl. Math. Comput., 143, pp. 223–232.
Mishev, D. P., (1989), Oscillation of the solutions of hyperbolic differential equations of
neutral type with “ maxima”, Godishnik Vissh. Uchebn. Zaved. Prilozhna Mat., 25, pp. 9
–18.
Mishev, D. P. and D. D. Bainov, (1984), Oscillation properties of the solutions of hyperbolic
equations of neutral type, Proceedings of the Colloquium on Qualitative Theory of
Differential Equations, Szeged, pp. 771 –780.
Mishev, D. P. and D. D. Bainov, (1986), Oscillation properties of the solutions of a class of
hyperbolic equations of neutral type, Funkcial. Ekvac., 29, pp. 213–218.
Parhi, N. and M. Kirane, (1994), Oscillatory behaviour of solutions of coupled hyperbolic
differential equations, Analysis, 14, pp. 43–56.
Shoukaku, Y. and N. Yoshida, (2005), Oscillations of parabolic systems with functional
arguments, Toyama Math. J., 28, pp. 105–131.
Ye, Q. X. and Z. Y. Li, (1990). Introduction to Reaction-Diffusion Equations, Beijing Science
Press.