Advances in Pure Mathematics, 2011, 1, 90-94
doi:10.4236/apm.2011.13020 Published Online May 2011 (http://www.scirp.org/journal/apm)
Periodic Solutions to Non-Autonomous
Second-Order Dynamical Systems
An-Min Mao, Miao-Miao Yang
School of Mathemat i cal Sci ences, Qufu Normal University, Qufu, China
Received January 17, 2011; revised March 7, 2011; accepted March 12, 2011
Abstract
We study the multiple existence of periodic solutions for a second-order non-autonomous dynamical systems

,=0
u
uVtu
 (1). Using the method of invariant sets of descending flow and chain of rings theorem, we
obtain the existence of seven 2π-periodic solutions.
Keywords: Critical Point, Invariant Sets of Descending Flow, (PS) Condition
1. Introduction
In this paper, we consider the existence of seven non-
trivial solutions for the second order non-autonomous
systems
 

,=0. 0,2π,.
N
u
uVtu tuR 
 (1)
For each
N
uR the function

,Vtu is periodic in t
with period 2π.
Problem (1) has been studied by many authors and
there is a large literature, see, for example, [1-11] (cf. the
references quoted in them).
Using the method of invariant sets of descending flow,
Z. Liu and J. Sun [1] got at least four periodic solutions
of (1). Via the variational method, which has been most-
ly used to prove the existence of solutions of (1), M.
Willem, J. Mawhin, S. Li, M. Schechter, C. Tang and
others proved existence under various conditions (cf. the
reference given in these publications). Also, the fixed
point theorems in cones can be chosen to establish the
existence of solutions for (1), see [12].
The goal of this paper is to find more periodic solu-
tions for problem (1). We get at least seven periodic so-
lutions of (1) by using the method of invariant sets of
descending flow and Chain of rings Theorem, which is
obtained in [7].
Let us give some notations. For two functions u and
v defined on


0, 2π and taking their values in
N
R,
we define a partial order by uv if and only if
 
ii
ut vt for all


0, 2πt and =1,2, ,iN. If
 
<
ii
ut vt for all


0, 2πt and =1,2, ,iN, the
relationship between u and v will be denoted by
uv.
(H1) There are two couples of functions ,,,




2,
N
CRR with period 2π such that:
,,,,


 



and

,,
uVt

 


,,
uVt

 


,,
uVt

 


,,
uVt

 

for any =1,2, ,iN , there is


0, 2π
i
t such that
  
<,,
iii i
i
tVtt
u




for any =1,2, ,iN, there is


0, 2π
i
t
 such that
  
>,,
iii i
i
tVtt
u





for any =1,2, ,iN, there is


0, 2π
i
t such that
  
<,,
iii i
i
tVtt
u



  
for any =1,2, ,iN , there is


0, 2π
i
t such that
  
>,.
iii i
i
tVtt
u



 

(H2) There exists a constant >0K such that every
entry of the matrix


2
,
uu tu KVI is nonnegative if
u is a function satisfying u

 or u

 or u


or u


, where


,
uu tuV is the Hessian matrix of

A. M. MAO ET AL. 91
V and I is the NN unit matrix.
(H3) There exists >2

and >0R such that, for
uR,
 
0<,, ,
u
Vtuu Vtu  
where u means the Euclidean norm of u in
N
R.
Theorem 1. If (H1), (H2) and (H3) are satisfied, then
problem (1) has at least seven periodic solutions.
Remark 1.1. Conditions in Theorem 1 can be satisfied
by generic functions. As examples, it can be proved that
4
2
=1
2
=2
1
N
iij
iij
N
Vu uu u




satisfies all the conditions in Theorem 1. One should take
0u

 and 0,v

 in which12 ===,
N
uu u
12
===
N
vv v, u and v are sufficiently small.

and

can be chosen in the same way.
Remark 1.2. Our work is based on the results in [1,7].
2. Preliminary and Lemmas
Let
H
be the Hilbert space of vector functions


ut
having period 2π and belonging to 1
H
on


0, 2π,
with the following inner product


22
0
,= d,uvuv Kuvt



where
K
is a fixed number satisfying (H2). The corre-
sponding norm in
H
is denoted by
H
 and



2
22
22
0
=, =d.
H
uuu uKut



Let
X
be the Banach space of N-vector functions

ut having period 2π and belonging to 1 C on


0, 2π and
X
is continuously imbedded in
H
.
Define a functional :
J
HR as
 
2π2
0
1
=,d.
2
J
uuVtut





Then the critical points of
J
correspond to the solu-
tions of problem (1). Here


:== 0
K
uHJu X


.
We have
 

1
2
22
2
d
=,
du
J
uuK VtuKu
t


 

 (2)
here
1
2
2
2
d
dK t





is the inverse operator of
2
2
2
d
d
K
t
 with the periodic condition of period 2π.
Denote


1
2
22
2
d
=,.
du
A
uKVtuKu
t


 


Now we will explain that (2) holds: Noting that
  
2π
0
,=, d,
u
J
uvuvVtuvt






and



2π2
0
,= d,
A
uvAuvKAuvt

then
 


2π22
0
2
222
2
0
,,= d
=d.
u vAu vuvKuvAuvKAuvt
d
uvKAuK uv t
dt




 
 





 

For


1
2
22
2
d
=,
du
A
uKVtuKu
t


 

 , we have





 
,=, ,.
J
uvuv Auv

For 0
uX

, consider the initial value problem

 
0
d=,0=
d
uuAuuu 

 (3)
both in
H
and in
X
. Let

0
,uu  and


0
,uu 
 be
the unique solution of (3) in
H
and in
X
respectively,
with maximal right existence interval



0
0, u 

 and



0
0, u 

.
Lemma 1. (Lemma 5.1 in [1])



00
=uu 
 and




00
,=,uu uu 
 for all

0
0<u  . Moreover, if



00
,=
lim uuu u



 in the
H
topology for some
uK


, the critical set of
J
, then the limit is also valid
in the
X
topology.
Definition 1. (Chain of rings) (Definition 5 in [7])
Assume that

12
,,, 2
n
DDD n all are nonempty
path connected sets. When 4n, 1 D intersects only
with 2 D and n D, 1=
nn
DDA

. i D intersects only
with 1i D

, 1i D



=2,3, ,1in, and 1=
ii i
DD A




=1,2, ,1in

, we say that 12 ,,,
n
DD D form a
chain of rings. For the case of =3n, 121 =DDA,
232
=DDA

 , 313 =DDA

, and 12 DD

3=D

, we say that 123 ,,DDD form a chain of rings.
For the case of =2n, 1212 =DDAA, 12
A
A

=

, we say that 1 D and 2 D form a chain of rings.
Lemma 2. (Theorem 4 and Remark 5 in [7]) Assume
that
H
is a Hilbert space,

11
,
J
CHR,


J
u

=uAu

, uH

, J satisfies the (PS) condition on
H
. 12 ,,,
n
DD D


3n are open convex subsets of
X
, and form a chain of rings.

Xi i
A
DD


=1,2, ,in,
1) when n is even, if


22
221
=1 =1
>or >,
inf inf
nn
XX
uA uA
jj
jj
Ju Ju
 



A. M. MAO ET AL. 92
then
J
has at least 31
2n critical points;
2) when n is odd, if
 
11
22
212 1
=1 =1
>or >,
inf inf
nn
XXX
uA uAA
jj
jj
Ju Ju

 


then
J
has at least

313
2n
critical points.
3. Proof of Theorem 1
We now give the proof of Theorem 1.
Proof. Step 1. First we will prove that
J
satisfies
(PS) condition.
(H3) implies the existence of constants 1>0C and
2>0C such that

12
,,.
N
VtuCuCu R

 (4)
Actually let

=,
g
xVtxu,
  
 
d1
=,= ,
d
,= ,
uu
gx Vtxuu Vtxuxu
xx
Vtxu gx
x
x



i.e.


dd gx
x
g
xx

. For 1
x
, we get


11
dd,
xx
gs
s
gs s



 
lnln 1ln0,gx gx


 
1,
g
xg x


 
,,.VtxuVtu x


thus,


1
[0,2π]
,=,|| ,
,1 :=.
min
t
uu
Vtu VtuVtu
uu
Vtu Cu










For <1x, one has


1
1
dd.
x
x
gs
s
g
ss



Similarly,
 
1,
g
xg x


 
,,.VtxuVtu x


then



2
0, 2π
,=|,||,|
,1:= .
max
t
uu
Vtu VtuVtu
uu
Vtu C









If <uR , by the continuousness of V, we can take
proper 2
C such that (4) holds.
For uR, by (H3) and (4), it follows that, for
uH

,
  
 
 

2π2π
22
00
2π
0
2π2π
13
00
1,
2
11
=,d ,d
22
=, ,d
2
1,d 1d.
22
u
u
JuJ uu
uVtutu Vtuut
u
VtuVtu t
VtutCutC


 










 
 
 
 




For <uR, by the continuousness of


,Vtu

,
2u
uVtu , we can take proper 3
C such that
 

2π
13
0
1,1d,.
22
J
uJuu CutCuH








(5)
Similarly, we can obtain that
 

 
 
 
2π2π
22
00
2π2π
2
00
2π2π
22 2
22
00
2π
0
2
1,
11
=,d ,d
2
11
=d,,d
2
11 11
=dd
22
,,d
11 11
22
u
u
u
H
JuJuu
uVtutuVtuut
u
utVtuVtu t
uKut Kut
u
VtuVtut
u














 

 
 
 

 
 

 



 









2π2
2
0d.Kut





(6)
From these inequalities, we see that, if

1
n
uH

such that


<
n
J
uC and

0
n
Ju  as n ,
then by (5), one has
2π
13
0
1d1,
2nn
H
CutCC u



 



2π
45
0d
nn
H
utCCu


 (7)

A. M. MAO ET AL. 93
By (6) and (7), we obtain
2π
22
2
0
11 11d1 ,
22
nnn
H
H
uKutCu

 

 
 

2π
22
67
0d
nnn
H
H
uCCutu 
 (8)
and




22
2π2π2π
22
22
00 0
22
2π
8845
0
2
84 584 5
d1d d
=d
11
nn
nn
H
nn
HH
uttu t
CutCCCu
CC CuCC Cu








 


 

 
 
 (9)
It follows from (8) and (9) that
2
910, nn
HH
uCCu (10)
and here 410
CC are positive constants.
(10) implies that

n
u is bounded in
H
and there-
fore it has a subsequence, relabeled

n
u, weakly con-
vergent in
H
and strongly convergent in
p
L

*
1<2.p Thus



,0as,
nn
JuJuu un


and
   
 
2π
0,,d
,, 0as
unun
p
q unun
L
L
VtuVtuuut
VtuVtuu un
 
 







   
2π
2
20
,
=,,d,
nn
nunun
L
JuJuu u
uuVtu Vtuuut


 


then we have
2
20as .
nL
uu n  
by
22 2
22
=
nnn
L
L
uu uuuu

and
20as ,
nL
uu n 
we get
0as .
n
uu n 
Step 2. Let

12
=,=,Du XuDu Xu

 

34
=,=,DuXuDu Xu

 
It is easy to see that 14 DD are all open convex
subsets of
X
, and
121 232
=, =,DDA DDA


343 414
=, =.DDA DDA

 
Since

,

, we have 13 =DD

,
24
=DD

.
Hence

4
1
i
D form a chain of rings.
If 1X uD

, then u


. Condition (H2) implies that








 


22
12
0
,,
=, 0.
uu
uu
VtKVtu Ku
Vtu suKIuds




 

Note that

1
22π
2
20
=, d
d
K
uksuss
dt


 


, where





 

cos π
,0 2π;
2sin π
,=
cos ,0 2π.
2sin π
hts Kst
KhK
kts hstKts
KhK







Since


,>0kts , for any 0,2πts, we have
 

1
2
2
2
22
=
,,0.
uu
d
AAu K
dt
VtKVtu Ku









And by (H1)

1
2
2
2
2
22
2
=
,0.
u
d
AK
dt
dKVtK
dt
















The maximum principle shows that
A

 . There-
fore Au

 for all 1X uD

, that is 1
A
uD for all
1X
uD

. Hence


11X
A
DD. In a similar way,


Xi i
A
DD, for =2,3,4i.
Since

112
=,
XXX
ADDuX u



334
=,
XXX
ADDuX u

  
and
J
is bounded on a bounded set, we get


12 34
>,and >.
inf inf
XX XX
DD DD
Ju Ju



From Lemma 5 and Remark 1, we know that
J
has
at least 341=7
2 critical points.
Remark 3.1. If (H1) (H2) and the following condition

A. M. MAO ET AL. 94
are satisfied.
(H4) There exists 1>0R and positively definite con-
stant matrixes
A
and B with =
A
BBA such that

1
,,, .
N
uuVtuu RuR AB
Since =
A
BBA, there is an orthogonal matrix T such
that =

TATTBT are simultaneously diagonal ma-
trixes. Let

1
=,,
N
diag


TAT and =

TBT

1,,
N
diag

  and assume also that >0
i

for
=1,2, ,iN and that



2
=1 ,=0,1, 2,=.
N
iii nn
  
Then (1) has at least seven periodic solutions.
[1] shows


J
u satisfies the (PS) condition under
(H4). From the proof of Theorem 1, we can get this con-
clusion.
4. Acknowledgements
We should like to express our appreciation to the referees
for suggesting how to improve our paper.
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