See discussions, stats, and author profiles for this publication at: https://www.researchgate.

net/publication/222010168

Existence result for periodic solutions of a class of

Hamiltonian systems with super quadratic potential

Article in Nonlinear Analysis · November 2005

DOI: 10.1016/j.na.2005.05.018

CITATIONS READS

11 15

1 author:

Leonard Shilgba
Admiralty University of Nigeria
18 PUBLICATIONS 32 CITATIONS

SEE PROFILE

All content following this page was uploaded by Leonard Shilgba on 22 April 2025.

The user has requested enhancement of the downloaded file.

 Nonlinear Analysis 63 (2005) 565 – 574
www.elsevier.com/locate/na

Existence result for periodic solutions of a class of

Hamiltonian systems with super quadratic potential
Leonard Karshima Shilgba∗
Hirano Laboratory, Division of Information Media and Environment Sciences, Graduate School of Environment
and Information Sciences, Yokohama National University, 79-5, Tokiwadai, Hodogaya-Ku, Yokohama-shi,
Kanagawa 240-8501, Japan
Received 27 May 2003; accepted 18 May 2005

Abstract
We have presented a variant existence result for periodic solutions to a class of Hamiltonian systems
with a potential which is both super quadratic and sign-indefinite.
䉷 2005 Elsevier Ltd. All rights reserved.
MSC: 34B15

Keywords: Periodic solutions; Hamiltonian systems; Potential; Linking

1. Introduction

Our objective is to investigate the existence of periodic solutions to the class of Hamil-
tonian systems:

ẍ + A(t)x + b(t)V  (x) = 0 on [0, T ],

x(0) = x(T ),

ẋ(0) = ẋ(T ), (P)

∗ Tel.: +81 45 50 24 115; fax: +81 45 33 94 207.

E-mail address: shilgba@yahoo.com.

0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2005.05.018
566 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574

where

1.1 A ∈ C 0 (R, RN×N ) is a symmetric T -periodic matrix function that is not sign-definite
on [0, T ];
1.2 b ∈ C 0 (R, R) is T -periodic and hanges sign on [0, T ];
1.3 V ∈ C 2 (RN , R) with V (x)
V (0) = 0 for all x ∈ RN , and has super quadratic
behaviour.

The above problem has been studied by several authors:

Case A: A ≡ 0.
While Lassoued [14] obtained existence of T -periodic solutions when V is strictly convex
and homogeneous, Ben Naoum et al. [7] provided existence results by relaxing condition
on V to only homogeneity.
Girardi and Matzeu [13] used the Alama–Tarantello condition in [1] and proved some
existence and multiplicity results for T -periodic and subharmonic solutions. The condition
is given by:
There exist c > 0,
> 2 : |V  (x)x −
V (x)|  c|x|2 for all x ∈ RN . (1.4)
Case B: A  = 0.
In this case, when b(.) > 0, refer to [3,2,10] (with the references contained therein),
severally for existence of periodic, homoclinic, and subharmonic solutions. Besides, when
we assume b changes sign and A is a negative definite matrix, there are some existence and
multiplicity results of periodic, subharmonic, and homoclinic solutions (refer to [12,8,4]).
When b(.) changes sign and A is sign-indefinite, the author is aware of only the results of
Antonacci [5], Xu and Guo [19], and the author in [18].
Antonacci proved the existence of non-trivial periodic solutions under the assumption
that

T
(A(t), ) dt > 0 for all  ∈ RN , || = 1. (1.5)
0
Antonacci observed the vital role played by assumption (1.5) as he posed two open problems:
Given that A(.) is not negative-definite, to

1. study the existence of T -periodic solutions of (P) in the case (1.5) holds and A(.) is
sign-indefinite in any interval [0, T ];
2. find some existence results in the case that A does not satisfy (1.5).

As mentioned above, Antonnaci attempted an elegant solution of problem 1 in [5]. Xu and

Guo, in attempting the solution of open problem 2 in [18], relied, among other assumptions,
on the glooseh or weaker version of (1.5), that:
T
There exist  > 0,
> 2,  > 0,  : 0 <  < ( 0 b(t) dt)
such that

T
(A(t), ) dt
−  for all  ∈ RN , || = 1, (1.6)
0

where V (x)
|x|
−  for all sufficiently large x ∈ RN .

 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574 567

T
It is clear from (1.6) that 0 (A(t), ) dt,  ∈ RN , || = 1, is neither assumed to be
definitely positive nor negative, which is a relaxation on condition (1.5.); however, it is only
T
valid when 0 b(t) dt > 0. In [18] we resolved the open problem 2 posed by Antonnaci by
assuming some super-convex quadratic condition on the potential V :
There exist real numbers, R > 0, ,  :  >  > 1 such that



 b(t) dt + b(t) dt > 0,
I I¯

and for all z1 , z2 ∈ RN , z1  = z2 , and |z1 + z2 | > R, we have

V (z1 + z2 )
(V (z1 ) + V (z2 )) − |z1 − z2 |2 . (V∗ )

In this paper we drop the super-convex quadratic condition (V∗ ).

2. Existence results

We shall investigate the periodic solutions of (P) in the Sobolev space HT1 = H 1 ([0, T ],
RN ) = {u : [0, T ] → RN is absolutely continuous, u(0) = u(T ), u̇ ∈ L2 (0, T ; RN )} with
the norm

T
T 1/2
u = |u̇| +
2
|u| 2
for all u ∈ HT1 ,
0 0

assuming conditions 1.1–1.3 are verified in addition to the following assumptions:

a.1 0 < l = max |A(t)| < (1+442 )T 2 : (A(t)x, x)  l|x|2 , for all x ∈ RN , t ∈ [0, T ]
2

t∈[0,T ]
T
b.1 0 b(t) dt = 0.
m.1 There exist  > 0,  : 0 <  < l, t0 ∈ [0, T ], R1 > 0:
(i) b(t) > 0 for all t ∈ I , I = [t0 − , t0 + ] ⊂ [0, T ].
(ii) I (A(t)x, x) dt
 I |x|2 dt for all x ∈ HT1 .
 
(iii)  I¯ (A(t), ) dt  T for all  ∈ RN , || = 1, (I¯ = [0, T ]\I ).
V .1 There exist
> 2, a1 > 0, R2 > 0:
(i)
V (x)  V  (x)x for all x ∈ RN , |x|
R2 ,
(ii) V (x)
a1 |x|
for all x ∈ RN , |x|
R2 .
V .2 There exist 

, R3 > 0 : V (x)
|V  (x)||x| for all x ∈ RN , |x|
R3 (xy denotes
the inner product of the pair of vectors x, y ∈ RN ).
V .3 There exists R4 > 0 : V (x) − (V  (x), x)  c|x|2 for all x ∈ RN , |x|
R4 where
 
−2 4 2
c< − l , m = max b(t).
2m (1 + 42 )T 2 t∈[0,T ]

V (x)
V .4 lim = 0.
|x|→0 |x|
2
568 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574

Remark. Assumption m.1 indicates that

T
(A(t), ) dt
(2 − T ) for all  ∈ RN , || = 1; 2 − T < 0.
0

This is an improvement on condition (0.2) in [5].

Hamiltonian action: We reduce problem (P) to finding critical points of the functional:

T
T 
T
1
I (x) = |ẋ|2 − A(t)x.x − b(t)V (x) x ∈ HT1 (∗ )
2 0 0 0

associated with (P). We note that the critical points of this functional correspond to periodic
solutions of (P).

Definition 2.1 (Cerami–Palais–Smale condition). Given a real Banach space X, we say

that I ∈ C 1 (X, R) satisfies the Cerami–Palais–Smale condition at level d ∈ R (i.e
condition (CP S)d for short) if for any sequence (xn ) ∈ X such that I (xn ) → d and
(1 + xn )I  (xn ) → 0 it implies that (xn ) contains a strongly convergent subsequence
in X.

Remark. The Cerami–Palais–Smale condition was introduced by Cerami [9] as a weaker

variant of the classical Palais–Smale condition. We shall rely on the following linking
theorem of Benci–Rabinowitz (see [17,6]).

Theorem 2.2 (Linking theorem). Let E be a Banach space and I ∈ C 1 (E, R) such that

(a) E = E− ⊕ E+ , dim E− < ∞ and E+ is closed in E.

(b) There exist r, R : 0 < r < R, and ∈ E+ with  =1 : sup{I (x) : x ∈ jQ}  inf{I (x) :
x ∈ D} where
Q = {u + :
0, u ∈ E− , u +   R} ⊂ E− ⊕ R ;
D = {x ∈ E+ : x = r}.

(c) I satisfies the Cerami–Palais–Smale condition (CP S)d ,

d = inf sup I (g(y)), = {g ∈ C(Q, E) : g(y) = y for all y ∈ jQ}.

g∈ y∈Q

Then d
inf D I and d is a critical value of I. Moreover, if d = inf D I , there is a critical

point xd ∈ D : I (xd ) = d.

Lemma 2.3. Let conditions 1.1–1.3, a.1, b.1, V .1–V .3 be verified. Then I ∈ C 1 (HT1 , R)
given as in (∗) satisfies the Cerami–Palais–Smale condition for all real d.

Proof. We only need to show that given any sequence (xn ) ∈ HT1 such that I (xn ) is bounded
and (1 + xn )I  (xn ) → 0 as n → ∞ i.e sup{(1 + xn )I  (xn ) : ∈ HT1 ,   = 1} → 0
 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574 569

as n → ∞), then (xn ) contains a strongly convergent subsequence in HT1 . Boundedness of

I (xn ) implies that there exists k > 0 such that



T
1 T 1 T
|ẋn |2  (A(t)xn , xn ) + b(t)V (xn ) + k. (2.1)
2 0 2 0 0

Since (1 + xn )I  (xn ) is linear, it follows from Riesz’s representation theorem that there
exists a sequence (zn ) ∈ HT1 such that
(1 + xn )I  (xn )H −1 = zn  = n →0 as n → +∞ and
T

(1 + xn )I  (xn ) = zn , H 1 for all  ∈ HT1 .

T

Thus, we have from Cauchy–Schwartz inequality that

T
T
T
|ẋn |2
(A(t)xn , xn ) + b+ (t)(V  (xn ), xn )
0 0 0

T
− 
− b (t)(V (xn ), xn ) − n ,
0

where b+ (t) = max{0, b(t)} and b− (t) = − min{0, b(t)}, t ∈ [0, T ].

Clearly, b(t) = b+ (t) − b− (t) for all t ∈ [0, T ].
Hence,

T
T
T
T
|ẋn |

2
(A(t)xn , xn ) +  b(t)V (xn ) − mc |xn |2 − n.
0 0 0 0
Therefore,

T
T
T
T
1 1 mc n
− |ẋn |2  − (A(t)xn , xn ) − b(t)V (xn ) + |xn |2 + .
 0  0 0  0 
(2.2)
Combining (2.1) and (2.2) results



−2 T ( − 2)l + 2mc T n
|ẋn |2  |xn |2 + + k.
2 0 2 0 
That is,

T
T
( − 2) |ẋn |2  [( − 2)l + 2mc] |xn |2 + k̄, (0 < k̄ < ∞).
0 0
Or, according to Wirtinger’s inequality, we have that

T
T2
( − 2) − [( − 2)l + 2mc] 2 |ẋn |2  T [( − 2)l + 2mc]|x̄n |2 + k̄,
4 0

where

T
1
x̄n = xn (t) dt, n ∈ N.
T 0
570 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574

Clearly from condition V .3,

T2
d̄ = ( − 2) − [( − 2)l + 2mc] > 0.
42
Thus, setting

T [( − 2)l + 2mc] k̄

d1 = (> 0) and d2 = (> 0),
d̄ d̄
we obtain

T
|ẋn |2  d1 |x̄n |2 + d2 . (2.3)
0

Moreover, applying Sobolev inequality and from (2.3), it is not difficult to verify that

min |xn (t)|
d3 |x̄n | − d4 , (d3 = 1 − (T d 1 )1/2 , d4 = (T d 2 )1/2 ).

t∈[0,T ]

We claim that 1 − T d 1 > 0. Suppose the contrary, that 1 − T d 1  0, i.e if and only if
 
42
2mc
( − 2) −l .
(1 + 42 )T 2
This is a contradiction. Hence, we have,

min |xn (t)|
d3 |x̄n | − d4 , (d3 , d4 > 0). (2.4)

t∈[0,T ]

Clearly, if the sequence (|x̄n |) of real numbers is bounded, then so is (xn ) in HT1 . Suppose
|x̄n | → +∞ as n → +∞. Then V (xn )
a1 |xn |
→ +∞. So, we can define a sequence
in HT1 :

xn (t)
n (t) = , t ∈ [0, T ]
(1 + xn )V (xn (t))
which is well defined at infinity. Furthermore,
V (xn )ẋn − xn [V  (xn )ẋn ]
˙ n = .
(1 + xn )(V (xn ))2
So,

( + 1)2 |ẋn |2
|˙n |2 
[(1 + xn )V (xn )]2
for sufficiently large n, and

[{T 2 /42 + ( + 1)2 }d1 + T ]|x̄n |2 + const.

n 2  →0 as n → +∞.
a12 (1 + xn )2 (d3 |x̄n | − d4 )2

 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574 571

Hence,
(1 + xn )I  (xn )n → 0 as n → +∞.
It is clear that
 
 T ˙  → 0 as n → +∞.
(i) (1 + xn )  0 ẋn n
 
 T 
(ii) (1 + xn )  0 A(t)xn n  → 0 as n → +∞.

Now,

T
T
T
 |xn |2
(1 + xn ) b(t)V (xn )n
 b(t) dt − b+ (t) dt.
0 0 0 V (xn )
Thus,

T
T
(1 + xn ) b(t)V  (xn )n →  b(t) dt  = 0 as n → +∞
0 0
since


 T |xn |2  m(T 2 /42 )d1 + T )|x̄n |2 + const.
 +
b (t) dt < →0 as n → +∞.
 V (xn ) 
0 a1 (d3 |x̄n | − d4 )

This contradicts our supposition. Therefore, (xn ) is bounded, and so by a similar proof in
[4], admits a strongly convergent subsequence. 

Theorem 2.4 (Main result). Given that assumptions 1.1–1.3 and a.1–V .4 hold, then (P )
has at least one periodic solution.
T
Proof. HT1 = H ⊕ RN , H = {x ∈ HT1 : 0 x(t) dt = 0}.
Let x ∈ H . Then,



T
1 T 2 1 T
I (x) = |ẋ| − (A(t)x, x) − b(t)V (x)
2 0 2 0 0
 
T
T
1 4 2

− l |x| 2
− m V (x).
2 T2 0 0

According to V .4, we can choose

1
,0< < (42 /T 2 − l) such that there exists r  ( ) > 0 (sufficiently small):
2m
 
T
1 4 2
I (x)
− l − m |x|2 > 0 for all x ∈ H , x∞ = r  . (1)
2 T2 0

Clearly, there exists c1 , c2 > 0 such that

T
T
I (x)
c1 |ẋ|2 and I (x)
c2 |x|2 for all x ∈ H , x∞ = r  .
0 0
572 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574

So,
I (x)
cm x2 , cm = min{c1 , c2 } for all x ∈ H , x∞ = r  .
Thus, we can find r(r  ) > 0 : I (x) > 0 for all x ∈ H , x = r.
Furthermore, choose a constant vector 0 ∈ RN : |0 |2 = /(2 + 2 ).
Define,

0 cos (t +  − t0 ) t ∈ I ,
(t) = 
0 t ∈ I¯ = [0, T ]\I .

So, we have (t) ∈ HT1 ,   = 1.(Particularly, (t) ∈ H .) Besides, Supp{ } ⊂ I .

Let H̄ = + z :
0, z ∈ RN . Thus, for every + z ∈ H̄ , we have

T


2
|z|2 
I( + z)  |˙ |2 − (A(t), ) − | + z|2
2 0 2 I¯ 2 I



− b(t)V ( + z) − V (z) b(t)
I I¯



2
|z|2 
 |˙ |2 + T − | + z|2
2 I 2 2 I



− a1
˜ | + z|
− V (z) b(t),
I I¯

where
˜ = minI b(t) > 0 and  ∈ RN ,  = z/|z|(|| = 1).

Now, defining some two norms on H̄ :

1/2 
1/

 + z1 = | + z| 2
,  + z2 = | + z| , 
2,
I I

it is clear that they are equivalent since dim H̄ < ∞. Therefore, for our fixed
> 2, we
have that there exists k1 () > 0 such that  + z2
k1  + z1 for all
0, z ∈ RN .
Besides, we can find some constant k2 > 0 (independent of z ∈ RN ,
0):





I ( + z)  k2 −  + z21 − k1 a1
˜  + z − V (z) b(t). (2.5)
2 I¯
1

We have two cases, namely,


T 
Case 1 : b(t) > 0 i.e b(t) > 0 .
I¯ 0



Case 2 : b(t) < 0 if and only if − b(t) > 0 .
I¯ I¯

Case 1: We have from (2.5) that




I( + z)  k2  + z21 −  + z21 − k1 a1
˜  + z .
2 1
 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574 573

Clearly, we can find R > r (sufficiently large) such that

I( + z)  0 for all + z ∈ H̄ :  + z = R.

Therefore,

sup{I ( + z) :
0 z ∈ RN ,  + z = R}  inf{I (x) : x ∈ H , x = r}.

Hence the conditions of the linking theorem are satisfied. Consequently, I has a non-constant
critical point in HT1 which is a solution of (P).
Case 2: Without loss of generality, we assume that:
 > r, a2 > a1 : V (z)  a1 |z|
+ a2 |z|2 for all z ∈ RN , |z|
R.
1. There exists R 
 2

¯ 2 /(2)

˜ .
2
2. There exists ¯ > 0 : −
¯ I¯ b(t) < 0 and k1 > 
Then, we have


2  T 2
I( + z)  2
− | + z|2 +
|z|
 2
+ 2 2 I 2



− a1
˜ | + z|
+ V (z) − b(t)
I I¯

2 2 
 2
− 2
+ (T − 2)|z|2
2 + 2 2(2 + 2 ) 2




− k1 a1
˜

+ (2)
/2 |z|
+ 
¯ a1 |z|
+ 
¯ a2 |z|2
( 2
+ 2 )
/2

22 − 2 1
 2
+ (T − 2 + 2
¯ a2 )|z|2
2( 2
+ 2 ) 2


˜

k1


− a1

− a1 [k1
˜ (2)
/2 − 
¯ ]|z|

(2 + 2 )
/2

 D1 2
+ D2 |z|2 − D3
− D4 |z|
,

where D1 , D2 , D3 , D4 ∈ R+ .

 + z2 = 2
+ T |z|2 for all
0, z ∈ RN .

Thus, as  + z → +∞, I ( + z) → −∞.

Therefore, we can find R > r (sufficiently large) such that

I( + z) < 0 for all + z ∈ H̄ :  + z = R.

This verifies the Linking theorem and hence completes proof of Theorem 2.4. 
 574 L.K. Shilgba / Nonlinear Analysis 63 (2005) 565 – 574

References

[1] S. Alama, G. Tarantello, On semilinear elliptica equations with indefinite non linearities, Calc. Var. Partial
Diff. Eq. 93 (1991) 1–18.
[2] A. Ambrosetti, V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Semin.
Mat. Univ. Padova. 89 (1993) 177–194.
[3] A. Ambrosetti, G. Mancini, Solutions of minimal period to a class of convex Hamiltonian systems, Math.
Ann. 255 (1981) 405–421.
[4] F. Antonacci, Periodic and Homoclinic solutions to a class of Hamiltonian systems with potential indefinite
in sign, Boll. Unione. Mat. Ital. 10-B (1996) 303–324.
[5] F. Antonacci, Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign,
Nonlinear Anal. TMA 29 (1997) 1353–1364.
[6] V. Benci, P.H. Rabinowitz, Critical point theorem for indefinite functionals, Invent. Math. 52 (1979)
241–273.
[7] A.K. Ben Naoum, C. Trostler, M. Willem, Existence and multiplicity results for homogeneous second order
differential equations, J. Diff. Equations 112 (1994) 239–249.
[8] P. Caldiroli, P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing
sign, Comm. Appl. Anal. 1 (1994) 97–129.
[9] G. Cerami, Un criterio di esistenza per in punti critici su varieta illimit ate, Istit. Lombardo Accad. Sci. Lett.
Rend. A 112 (1978) 332–336.
[10] V. Coti Zelati, P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing super
quadratic potential, J. Ann. Math. Soc. 4 (1991) 693–727.
[12] Y.H. Ding, M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potential
changing sign, Dynamical systems Appl. 2 (1993) 131–145.
[13] M. Girardi, M. Matzeu, Existence and multiplicity results for periodic solutions for super quadratic
Hamiltonian systems where the potential changes sign, Nonlinear Diff. Equations Appl. 2 (1995) 35–61.
[14] L. Lassoued, Periodic solutions to a second order super quadratic system with a change of sign in the potential,
J. Diff. Equations 93 (1991) 1–18.
[17] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS
Regional Conference Series in Mathematics, American Mathematical Society, vol. 65, Providence, RI, 1986.
[18] L.K. Shilgba, A variant existence result for periodic solutions of a class of Hamiltonian systems with indefinte
potential, Journal of Nonlinear and convex analysis JNCA, to appear.
[19] Y.-T. Xu, Z.-M. Guo, Existence of periodic solutions to second order Hamiltonian systems with potential
indefinite in sign, Nonlinear Anal. TMA 51 (7) (2002) 1273–1283.

Further reading

[11] M. Degiovani, Basic tools of critical point theory, 2002.

[15] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
[16] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math. 31 (1978) 157–184.

View publication stats