Periodic solutions for Duffing type p-Laplacian equations
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Periodic solutions for Duffing type p-Laplacian equations
Yuanhong Wei
1
, Shaoyun Shi
1,2∗
1
College of Mathematics, Jilin University, Changchun 130012, P. R. China
2
Key Laboratory of Symbolic Computation and
Knowledge Engineering of Ministry of Education,Changchun 130012, P. R. China
E-mail: yuanhongwei@email.jlu.edu.cn, shisy@mail.jlu.edu.cn,
Abstract
In this paper, we study periodic solutions for a class of Duffing type p-Laplacian equations.
By using the Man´asevich-Mawhin continuation theorem, some new results on the existence of
p eriodic solutions are obtained.
Keywords: Periodic solution; Duffing type p -Laplacian euqations; Man´asevich-Mawhin con-
tinuation theorem
Mathematics Subject Classification: 34C25, 54H25
1 Introduction
In recent years, many works have focused on the investigation of existence and
uniqueness of periodic solutions for Duffing equations, see, for instance, [3, 4, 5, 7,
8, 9, 10] and references therein.
In [1], Zhang and Li considered the one-dimensional Duffing type p-Laplacian
equation
(ϕ
p
(x
0
(t)))
0
+ Cx
0
(t) + g(t, x(t)) = e(t), (1)
where t, x ∈ R, p > 1, ϕ
p
: R → R is given by ϕ
p
(s) = |s|
p−2
s for s 6= 0 and
ϕ
p
(0) = 0, C is a constant, g(t, x) is continuous and g(t, ·) = g(t + T, ·), e(t) is a
continuous function, e(t) = e(t + T ),
R
T
0
e(t)dt = 0. They showed that if
(A1) (g(t, u
1
) − g(t, u
2
))(u
1
− u
2
) < 0 for u
1
6= u
2
, t ∈ R;
(A2) xg(t, x) < 0 for |x| > 0, t ∈ R;
(A3) There exist constants K > 0 and M > 0, such that
2
2−p
MT
p
< 1, g(t, x) > −M|X|
p−1
− K, for x > 0, t ∈ R,
∗
Supported by Program for New Century Excellent Talents in University, NSFC grant (10771083), SRFDP grant
(20040183030).
1
http://www.paper.edu.cn
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