Electronic Journal of Differential Equations, Vol. 2004(2004), No. 80, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
OSCILLATION OF SOLUTIONS FOR SYSTEMS OF
HYPERBOLIC EQUATIONS OF NEUTRAL TYPE
PEIGUANG WANG, YONGHONG WU
Abstract. In this paper, we obtain sufficient conditions for the oscillation of
solutions to systems of hyperbolic differential equations of neutral type. We
consider such systems subject to two kinds of boundary conditions.
1. Introduction
We consider the following system of hyperbolic differential equations of neutral
type
m
i
X
∂2 h
u
(x,
t)
−
cij uj (x, t − τ )
i
2
∂t
j=1
= ai (t)∆ui (x, t) + bi (t)∆ui (x, t − ρ) − pi (x, t)ui (x, t)
m Z b
X
−
qik (x, t, ξ)uk [x, g(t, ξ)]dµ(ξ), (x, t) ∈ Ω × R+ ≡ G,
k=1
(1.1)
a
subject to either of the following boundary conditions
∂ui
+ νi (x, t)ui = 0, (x, t) ∈ ∂Ω × R+
∂n
ui (x, t) = 0, (x, t) ∈ ∂Ω × R+ ,
(1.2)
(1.3)
n
where Ω is a bounded domain in R with a piecewise smooth boundary ∂Ω, R+ =
[0, ∞), ∆ is the Laplacian operator in Rn , cij , τ > 0 and ρ > 0 are constants, n is
the unit outward normal vector of ∂Ω.
There has been considerable interest in obtaining sufficient conditions for oscillatory solutions of partial functional differential equations, as this type of equations
arise frequently in many application fields (see for example the monograph [7]).
Recently, several papers concerning systems of hyperbolic functional differential
equations have appeared in literatures [1, 2, 3]. It is noted that previous work
focused only on the cases where the neutral coefficient number lies between −1 and
0, that is −1 ≤ c(t) ≤ 0. To the best of our knowledge, very little work has been
done for other cases.
2000 Mathematics Subject Classification. 35B05, 35R10.
Key words and phrases. Oscillation, systems of hyperbolic equations,
boundary value problem, distributed deviating arguments.
c
2004
Texas State University - San Marcos.
Submitted December 19, 2003. Published June 7, 2004.
Supported by the Natural Science Foundation of Hebei Province of China (A2004000089).
1
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PEIGUANG WANG, YONGHONG WU
EJDE-2004/80
The aim of this paper is to study the oscillation problem for the above system of
hyperbolic differential equations of neutral type, and give some oscillatory criteria
for such systems. Throughout the paper, we assume that the following conditions
hold:
(H1) ai (t), bi (t) ∈ C(R+ , R+ );
(H2) pi (x, t) ∈ C(G, R+ ), qik (x, t, ξ) ∈ C(G × [a, b], R);
(H3) νi (x, t) ∈ C(∂Ω × R+ , R+ );
(H4) g(t, ξ) ∈ C(R+ × [a, b], R), g(t, ξ) ≤ t for ξ ∈ [a, b], and
lim inf t→∞,ξ∈[a,b] {g(t, ξ)} = ∞;
(H5) µ(ξ) ∈ ([a, b], R) is nondecreasing, and the integral in (1.1) is a Stieltjes
integral.
Definition. A vector function u(x, t) = {u1 (x, t), . . . , un (x, t)}T is said to be a
solution of (1.1) and (1.2) or (1.1) and (1.3) if it satisfies equation (1.1) in G ≡
Ω × R+ and the associated boundary condition on the boundary of G.
Definition. A vector solution u(x, t) of the boundary value problem is said to be
oscillatory in the domain G if at least one of its nontrivial components is oscillatory.
Otherwise, the vector solution u(x, t) is said to be nonoscillatory.
For the following neutral differential inequality
Z b
d2
[y(t)
−
λ(t)y(t
−
τ
)]
+
p(t)y(t)
+
q(t, ξ)y[g(t, ξ)]dµ(ξ) ≤ 0, t ≥ 0, (1.4)
dt2
a
where λ(t) ∈ C 0 (R+ , R+ ), p(t) ∈ C(R+ , R+ ), q(t, ξ) ∈ C(R+ × [a, b], R+ ), we assume that the following conditions hold.
(A1) There exists a function h(t, ξ) ∈ C(R+ × [a, b], R+ ), such that h(h(t, ξ), ξ) =
g(t, ξ), and h(t, ξ) is nondecreasing with respect to t and ξ, and also t ≥
h(t, ξ) ≥ g(t, ξ);
Rt
Rb
(A2) lim inf t→∞ g(t,b) a q(s, ξ)dµ(ξ)ds > 1/e;
Rt
Rb
(A3) lim inf t→∞ h(t,b) a q(s, ξ)dµ(ξ)ds > 0.
We remark here that the existence of the function h(t, ξ) has been proved in [4].
The following two Lemmas are derived from the known literatures and are useful
to the proof of the main results of this paper.
Lemma 1.1 ([4]). Under assumptions (A1)–(A3), the first-order differential inequality
Z b
x0 (t) +
q(t, ξ)x[g(t, ξ)]dµ(ξ) ≤ 0
(1.5)
a
has no eventually positive solution.
Lemma 1.2 ([6]). Assume that λ(t) ≤ 1, and (A1) holds. If for some 0 ≤ ε0 ≤ 1
and ξ0 ∈ [a, b],
Z t Z b
ε0 q(s, ξ)g(s, ξ)dµ(ξ)ds > 0,
(1.6)
lim inf
t→∞
Z
t
Z
lim inf
t→∞
h(t,b)
a
b
ε0 q(s, ξ)g(s, ξ)dµ(ξ)ds >
g(t,ξ0 )
a
h
1
exp − lim inf
t→∞
e
Z
t
i
ε0 sp(s)ds ,
g(t,ξ0 )
(1.7)
then inequality (1.4) has no eventually unbounded positive solution.
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OSCILLATION OF SOLUTIONS
3
2. Main Results
We introduce the following notation:
pi (t) = min{pi (x, t)},
P (t) = min pi (t);
1≤i≤n
x∈Ω
Q∗ik (t, ξ)
Qii (t, ξ) = min{qii (x, t, ξ)},
x∈Ω
(2.1)
= max{qik (x, t, ξ)}.
x∈Ω
Theorem 2.1. Suppose that 0 ≤ C ≤ 1 and (A1) holds. If for some constants
0 ≤ ε0 ≤ 1 and ξ0 ∈ [a, b],
Z
t
Z
b
lim inf
t→∞
Z
t
Z
lim inf
t→∞
ε0 Q(s, ξ)g(s, ξ)dµ(ξ)ds > 0,
h(t,b)
b
ε0 Q(s, ξ)g(s, ξ)dµ(ξ)ds >
g(t,ξ0 )
(2.2)
a
a
h
1
exp −lim inf
t→∞
e
Z
t
i
ε0 sP (s)ds ,
g(t,ξ0 )
(2.3)
then each unbounded solution u(x, t) of (1.1)–(1.2) is oscillatory in the domain G,
where
m
n
o
X
C = max cii +
|cji | ,
1≤i≤n
m
n
o
X
Q(t, ξ) = min Qii (t, ξ) −
Q∗ji (t, ξ) .
i≤i≤n
j=1,j6=i
j=1,j6=i
Proof. Suppose to the contrary that there is a nonoscillatory solution u(x, t) =
(u1 (x, t), . . . , un (x, t))T of the problem (1.1) and (1.2), and that |ui (x, t)| > 0 for
t ≥ 0, i = 1, 2, . . . , n. From (H4), there exists a t1 ≥ 0, for j = 1, 2, . . . , m,
k = 1, 2, . . . , m, such that
|uj (x, t − τ )| > 0,
|ui (x, t − ρ)| > 0,
|uk [x, g(t, ξ)]| > 0,
t ≥ t1 , ξ ∈ [a, b].
Let
δi = sgn ui (x, t),
Yi (x, t) = δi ui (x, t),
δk = sgn uk [x, g(t, ξ)];
Yk [x, g(t, ξ)] = δk uk [x, g(t, ξ)],
(2.4)
then Yi (x, t) > 0, Yj (x, t − τ ) > 0, Yi (x, t − ρ) > 0 and Yk [x, g(t, ξ)] > 0 for t ≥ t1 ,
ξ ∈ [a, b], i = 1, 2, . . . , n, j = 1, 2, . . . , m and k = 1, 2, . . . , m.
Integrating equation (1.1) with respect to x over the domain Ω, for t ≥ t1 , we
obtain
Z
m Z
i
X
d2 h
u
(x,
t)dx
−
cij uj (x, t − τ )dx
i
2
dt
Ω
j=1 Ω
Z
m Z Z b
X
+
pi (x, t)ui (x, t)dx +
qik (x, t, ξ)uk [x, g(t, ξ)]dµ(ξ)dx
Ω
k=1
Z
= ai (t)
Ω
Z
∆ui (x, t − ρ)dx.
∆ui (x, t)dx + bi (t)
Ω
a
Ω
(2.5)
4
PEIGUANG WANG, YONGHONG WU
EJDE-2004/80
Furthermore, it follows from (2.4) that
Z
m Z
i
X
d2 h
δi
Y
(x,
t)dx
−
c
Y
(x,
t
−
τ
)dx
i
ij
j
dt2 Ω
δj
j=1 Ω
Z
Z
m Z
X
δi b
+
pi (x, t)Yi (x, t)dx +
qik (x, t, ξ)Yk [x, g(t, ξ)]dµ(ξ)dx
δk a
Ω
k=1 Ω
Z
Z
= ai (t)
∆Yi (x, t)dx + bi (t)
∆Yi (x, t − ρ)dx .
Ω
Ω
It is clear that
Z Z b
Z
qik (x, t, ξ)Yk [x, g(t, ξ)]dµ(ξ)dx =
Ω
(2.6)
a
b
Z
qik (x, t, ξ)Yk [x, g(t, ξ)]dxdµ(ξ).
a
Ω
(2.7)
From the Green’s formula and the boundary condition, we have
Z
Z
Z
∂Yi (x, t)
∆Yi (x, t)dx =
dω = −
νi (x, t)Yi (x, t)dω ≤ 0,
∂n
Ω
∂Ω
∂Ω
Z
Z
∆Yi (x, t − ρ)dx = −
νi (x, t − ρ)Yi (x, t − ρ)dω ≤ 0,
Ω
(2.8)
(2.9)
∂Ω
where dω is the surface integral element on ∂Ω. Moreover, it follows from (2.1)
that
Z
Z
pi (x, t)Yi (x, t)dx ≥ pi (t)
Yi (x, t)dx.
(2.10)
Ω
Ω
Combining (2.7)–(2.10) and noting (2.1), we obtain
d2 h
dt2
Z
m Z
X
i
δi
Yj (x, t − τ )dx
δj
Ω
j=1 Ω
Z
Z b
hZ
i
+ pi (t)
Yi (x, t)dx +
Qii (t, ξ)
Yi [x, g(t, ξ)]dx dµ(ξ)
Yi (x, t)dx −
Ω
−
a
Z
m
X
k=1,k6=i
cij
b
(2.11)
Ω
hZ
i
Q∗ik (t, ξ)
Yk [x, g(t, ξ)]dx dµ(ξ) ≤ 0.
a
Ω
Let
Z
Vi (t) =
Yi (x, t)dx,
i = 1, 2 . . . , n,
(2.12)
Ω
then Vi (t) > 0, and it follows from (2.11) that
m
i
X
d2 h
δi
V
(t)
−
c
V
(t
−
τ
)
+ pi (t)Vi (t)
j
i
ij
dt2
δj
j=1
Z b
Z b
m
X
+
Qii (t, ξ)Vi [g(t, ξ)]dµ(ξ) −
Q∗ik (t, ξ)Vk [g(t, ξ)]dµ(ξ) ≤ 0 .
a
k=1,k6=i
a
(2.13)
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OSCILLATION OF SOLUTIONS
5
Pm
Furthermore, let V (t) = i=1 Vi (t), then V (t) > 0, and it follows from (2.1) that
m
m
io
X
d2 n X h
δi
V
(t)dx
−
cij Vj (t − τ ) + P (t)V (t)
i
2
dt i=1
δj
j=1
(2.14)
Z bX
m n
m
o
X
∗
+
Qii (t, ξ)Vi [g(t, ξ)] −
Qik (t, ξ)Vk [g(t, ξ)] dµ(ξ) ≤ 0.
a i=1
k=1,k6=i
Noting that
m h
X
Vi (t) −
i=1
=
m
X
cij
j=1
m h
X
i
δi
Vj (t − τ )
δj
i=1
≥
m h
X
m
X
Vi (t) − cii Vi (t − τ ) −
Vi (t) − cii Vi (t − τ ) −
i=1
cij
j=1,j6=i
m
X
i
δj
Vj (t − τ )
δi
i
|cij |Vj (t − τ )
j=1,j6=i
m
X
h
= V1 (t) − c11 V1 (t − τ ) −
i
|c1j |Vj (t − τ )
j=1,j6=1
m
X
h
+ · · · + Vn (t) − cnn Vn (t − τ ) −
i
|cnj |Vj (t − τ )
j=1,j6=n
=
m
X
m
X
Vi (t) − c11 +
i=1
|cj1 | V1 (t − τ )
j=1,j6=1
m
X
− · · · − cnn +
|cjn | Vn (t − τ )
j=1,j6=n
≥
m
X
Vi (t) − max
1≤i≤n
i=1
m
m
n
oX
X
cii +
|cji |
Vi (t − τ )
j=1,j6=i
i=1
= V (t) − CV (t − τ ),
and
m n
m
o
X
X
Qii (t, ξ)Vi [g(t, ξ)] −
Q∗ij (t, ξ)Vj [g(t, ξ)]
i=1
h
= Q11 (t, ξ)V1 [g(t, ξ)] −
j=1,j6=i
m
X
i
Q∗1j (t, ξ)Vj [g(t, ξ)]
j=1,j6=1
h
+ · · · + Qnn (t, ξ)Vn [g(t, ξ)] −
m
X
i
Q∗nj (t, ξ)Vj [g(t, ξ)]
j=1,j6=n
h
= Q11 (t, ξ) −
m
X
i
Q∗j1 (t, ξ) V1 [g(t, ξ)]
j=1,j6=1
h
+ · · · + Qnn (t, ξ) −
m
X
j=1,j6=n
i
Q∗jn (t, ξ) Vn [g(t, ξ)]
6
PEIGUANG WANG, YONGHONG WU
≥ min
1≤i≤n
EJDE-2004/80
m
m
oX
n
X
Vi [g(t, ξ)]
Qii (t, ξ) −
Q∗ji (t, ξ)
i=1
j=1,j6=i
= Q(t, ξ)V [g(t, ξ)],
we have from (2.14) that
d2
[V (t) − CV (t − τ )] + P (t)V (t) +
dt2
Z
b
Q(t, ξ)V [g(t, ξ)]dµ(ξ) ≤ 0.
(2.15)
a
From Lemma 1.2, inequality (2.15) has no eventually positive solutions, which is in
contradiction with V (t) > 0. This completes the proof of Theorem 2.1.
To investigate the boundary-value problem (1.1), (1.3), we consider the following
Dirichlet problem
∆u + αu = 0, (x, t) ∈ Ω × R+
(2.16)
u = 0, (x, t) ∈ ∂Ω × R+ ,
where α is a constant. It is well-known [5] that the least eigenvalue α0 of problem
(23) is positive and the corresponding eigenfunction ϕ(x) is positive for x ∈ Ω.
We further introduce the following notation
A(t) = min {ai (t)},
B(t) = min {bi (t)}
1≤i≤n
1≤i≤n
(2.17)
Theorem 2.2. Suppose that 0 ≤ C ≤ 1, (A1) and (2.2) hold. If for some constants
0 ≤ ε0 ≤ 1 and ξ0 ∈ [a, b],
Z t
Z b
lim inf
ε0 Q(s, ξ)g(s, ξ)dµ(ξ)ds
t→∞
g(t,ξ0 )
a
1
> exp − lim inf
t→∞
e
h
Z
(2.18)
t
i
ε0 s[α0 A(s) + P (s)]ds ,
g(t,ξ0 )
then each unbounded solution u(x, t) of the boundary-value problem (1.1) and (1.3)
is oscillatory in the domain G.
Proof. Suppose to the contrary that there is a non-oscillatory solution u(x, t) =
(u1 (x, t), . . . , un (x, t))T of the problem (1.1) and (1.3), and |ui (x, t)| > 0 for t ≥ 0,
i = 1, 2, . . . , n. Proceeding as the proof of Theorem 2.1, there exists a t1 ≥ 0 such
that Yi (x, t) > 0, Yj (x, t − τ ) > 0, Yi (x, t − ρ) > 0 and Yk [x, g(t, ξ)] > 0 for t ≥ t1 ,
ξ ∈ [a, b] and j = 1, 2, . . . , m, k = 1, 2, . . . , m.
Multiplying both sides of equation (1.1) by the eigenfunction ϕ(x) and then
integrating the equation with respect to x over the domain Ω, for t ≥ t1 , we obtain
Z
m Z
i
X
d2 h
u
(x,
t)ϕ(x)dx
−
cij uj (x, t − τ )ϕ(x)dx
i
2
dt
Ω
j=1 Ω
Z
m Z Z b
X
+
pi (x, t)ui (x, t)ϕ(x)dx +
qik (x, t, ξ)uk [x, g(t, ξ)]ϕ(x)dµ(ξ)dx
Ω
k=1
Z
= ai (t)
Ω
∆ui (x, t − ρ)ϕ(x)dx.
∆ui (x, t)ϕ(x)dx + bi (t)
Ω
a
Z
Ω
(2.19)
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OSCILLATION OF SOLUTIONS
7
Furthermore, we obtain
Z
m Z
i
X
d2 h
δi
Y
(x,
t)ϕ(x)dx
−
c
Y
(x,
t
−
τ
)ϕ(x)dx
i
ij
j
dt2 Ω
δj
j=1 Ω
Z
Z
Z
m
X
δi b
+
pi (x, t)Yi (x, t)ϕ(x)dx +
qik (x, t, ξ)Yk [x, g(t, ξ)]ϕ(x)dµ(ξ)dx
δk a
Ω
k=1 Ω
Z
Z
= ai (t)
∆Yi (x, t)ϕ(x)dx + bi (t)
∆Yi (x, t − ρ)ϕ(x)dx.
Ω
Ω
(2.20)
From the Green’s formula and the boundary condition (1.3), we have
Z
Z
Z
∆Yi (x, t)ϕ(x)dx =
Yi (x, t)∆ϕ(x)dx = −α0
Yi (x, t)ϕ(x)dx,
Ω
Ω
Ω
Z
Z
∆Yi (x, t − τ )ϕ(x)dx = −α0
Yi (x, t − τ )ϕ(x)dx.
Ω
(2.21)
(2.22)
Ω
Combining (2.20)–(2.22), we obtain
Z
m Z
i
X
δi
d2 h
Y
(x,
t)ϕ(x)dx
−
c
Y
(x,
t
−
τ
)ϕ(x)dx
i
ij
j
dt2 Ω
δj
j=1 Ω
Z
Z b
Z
+ pi (t)
Yi (x, t)ϕ(x)dx +
Qii (t, ξ)
Yi [x, g(t, ξ)]ϕ(x)dxdµ(ξ)
Ω
−
a
Z
m
X
k=1,k6=i
b
Q∗ik (t, ξ)
a
Ω
Z
Yk [x, g(t, ξ)]ϕ(x)dxdµ(ξ)
Ω
Z
= −α0 ai (t)
Z
Yi (x, t)ϕ(x)dx − α0 bi (t)
Ω
Yi (x, t − τ )ϕ(x)dx .
Ω
Let
Z
Ui (t) =
Yi (x, t)ϕ(x)dx,
i = 1, 2 . . . , n,
(2.23)
Ω
then the remainder of the proof is the same as that for Theorem 2.1 and thus is
omitted here. This completes the proof.
References
[1] D. Bainov, D. Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Bristol, Adam Hilger, 1991.
[2] Y. K. Li,Oscillation of solutions to systems of hyperbolic differential equations with deviating
arguments, Acta Math. Sin., 40(1997),100-105.
[3] W. N. Li, L. Debnath,Oscillation of a system of delay hyperbolic differential equations, Int. J.
Appl. Math., 4(2000),417-431.
[4] Y. Ruan, A class of differential inequalities with continuous deviating argument, Acta Math.
Sin., 30(1987),661-670.
[5] V. S. Vladimirov, Equations of Mathematical Physics Moscow, Nauka. 1981.
[6] P. G. Wang, Y. H. Wu, Oscillation property of hyperbolic equations of neutral type, Int. Pure
Appl. Math., 2,2(2002).
[7] J. W. Wu, Theory and applications of partial functional differential equations, SpringerVerlag, New York, 1996.
8
PEIGUANG WANG, YONGHONG WU
EJDE-2004/80
Peiguang Wang
College of Electronic and Information Engineering, Hebei University, Baoding, 071002,
China
E-mail address: pgwang@mail.hbu.edu.cn
Yonghong Wu
Department of Mathematics and Statistics, Curtin University of Technology, GPO
BOX U1987, Perth, WA6845, Australia
E-mail address: yhwu@maths.curtin.edu.au