Nonlinear Analysis 72 (2010) 4575–4586
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Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
The multiplicity of solutions for perturbed second-order Hamiltonian
systems with impulsive effects
✩
Juntao Sun
a,∗
, Haibo Chen
a
, Juan J. Nieto
b
, Mario Otero-Novoa
b
a
Department of Mathematics, Central South University, Changsha, 410075 Hunan, People’s Republic of China
b
Departamento de Análisis Matemático, Universidad de Santiago de Compostela, Santiago de Compostela 15782, Spain
article info
Article history:
Received 5 December 2009
Accepted 17 February 2010
MSC:
34B37
37J45
70H05
70H12
Keywords:
Hamiltonian systems
Perturbed
Impulsive effects
Critical points
Variational method
abstract
In this paper, we study the existence of multiple solutions for a class of second-order
impulsive Hamiltonian systems. We give some new criteria for guaranteeing that the
impulsive Hamiltonian systems with a perturbed term have at least three solutions by
using a variational method and some critical points theorems of B. Ricceri. We extend and
improve on some recent results. Finally, some examples are presented to illustrate our main
results.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
In this paper, we consider the following problem:
−¨ u + A(t )u = λ∇F (t , u) + µ∇G(t , u), a.e. t ∈[0, T ],
∆( ˙ u
i
(t
j
)) =˙ u
i
(t
+
j
) −˙ u
i
(t
−
j
) = I
ij
(u
i
(t
j
)), i = 1, 2,..., N , j = 1, 2,..., l,
u(0) − u(T ) =˙ u(0) −˙ u(T ) = 0,
(1.1)
where A :[0, T ]→ R
N×N
is a continuous map from the interval [0, T ] to the set of N -order symmetric matrices, λ,µ ∈ R, T
is a real positive number, u(t ) = (u
1
(t ), u
2
(t ),..., u
N
(t )), t
j
, j = 1, 2,..., l, are the instants where the impulses occur
and 0 = t
0
< t
1
< t
2
< ··· < t
l
< t
l+1
= T , I
ij
: R → R (i = 1, 2,..., N , j = 1, 2,..., l) are continuous and
F , G :[0, T ]× R
N
→ R are measurable with respect to t , for every u ∈ R
N
, continuously differentiable in u, for almost every
t ∈[0, T ] and satisfy the following standard summability condition:
sup
|u|≤b
(max{|F (·, u)|, |G(·, u)|, |∇F (·, u)|, |∇G(·, u)|}) ∈ L
1
([0, T ]) (1.2)
for all b > 0.
✩
The first author was supported by the Graduate Degree Thesis Innovation Foundation of Central South University (CX2009B023).
∗
Corresponding author.
E-mail addresses: sunjuntao2008@163.com (J. Sun), math_chb@mail.csu.cn (H. Chen), juanjose.nieto.roig@usc.es (J.J. Nieto).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2010.02.034