Abstract
In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems
x
¨
(
t
)
+
B
x
˙
(
t
)
−
A
(
t
)
x
(
t
)
+
H
x
(
t
,
x
(
t
)
)
=
0
,
where
A
(
t
)
∈
(
R
,
R
N
)
is a symmetric matrix for all
t
∈
R
,
B
=
[
b
i
j
]
is an antisymmetric
N
×
N
constant matrix, and
H
(
t
,
x
)
∈
C
1
(
R
×
B
δ
,
R
)
is only locally defined near the origin in
x
for some
δ
>
0
. With the nonlinearity
H
(
t
,
x
)
being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark's theorem.