- Email: [email protected]
Subharmonic oscillations of convex Hamiltonian systems
Nonhnear Anolwx Theo?. Prmred m Great Briram.
Merhods
& Apphcormnr.
SUBHARMONIC
Vol
9. No. 11. pp. 1303-1311.
0362-546X 85 $3 00 - .oo G 1985 Pergamon Press Ltd
1985.
OSCILLATIONS OF CONVEX HAMILTONIAN SYSTEMS M. WILLEM
Institut
de Mathkmatique
Pure et AppliquCe. Universitk Catholique de Louvain. Louvain-la-Heuve. Belgium (Received
Key words and phrases:
Chemin
du Cyclotron
1. B-1348
5 March 1983: received for publication 5 June 1985)
Subharmonic
oscillations.
Hamiltonian
systems.
nonconvex
duality
INTRODUCTION
IN [l] BREZIS and Coron DH(o) = 0 and
proved
that,
if HE
H(u)*
for every
T sufficiently
large.
there
Hamiltonian
such that
Iu/ + x,
x,
H(u)/lul’+ then,
C’(R*“, R) is a convex
/U/+ x,
0,
is a solution
Jic + DH(u)
of
= 0
with minimal period T.When H is not conservative and T-periodic with respect to the time t, only kT-periodic solutions are expected. We prove that if H(t, u) is convex with respect to U, H(r, U) *
/U/---, x,
=.
lUl--, 5,
H(t, U)/jUl~ --;, 0. then,
for every
k E N*,
there
exists a kT-periodic
solution
Us of
JU + D,,H(t. u) = 0 such that ~~~~~~ x, k-, x, and the minimal period of uk tends to infinity. This theorem generalizes the results of [6] on long period subharmonics of subquadratic Hamiltonian systems by a completely different method. Before treating first order systems we consider the case of classical mechanics ii + D,,V(t. u) = 0. The proofs are similar and depends of Clarke and Ekeland [3]. 1. SUBHARMONICS
For
on a variational
OF SECOND
T > 0 and n E N” let us denote
characterization
ORDER
by H the Hilbert 1303
involving
HAMILTONIAN
space L’(O,
the “dual action”
SYSTEMS
T;BP).Let L be defined
1304
M. WILLEM
bY D(L)
=
{u: [O, T]+ T-periodic
R” : u and U are absolutely and ii E H}
continuous
and
Lu = ii. Then L is a self-adjoint operator, Ker L is the space of constant functions and R(L) = Ker L-. Let V : R X R” + R : (t, u) += V(t, u) be a continuous function. convex and differentiable with respect to u and such that D,V(t, u) is continuous on R x W”. Let us write K = L-’ v*(t,
: Z?(L)*
Z?(L) f7 D(L), cl E R8”. fEiR
u) = ,“E”wp” [(u. u) - V(t. u)],
and Q?(U) =
r [~(Ku, u) + V*(t. u)] dt.
u E R(L)
where (. , .) is the usual inner product on R”. The function V* is the Fenchel transform of V with respect to u and p7 is the dual action. Using Fourier expansions, it is easy to verify the following (equivalent) inequalities
IT
(Ku. u) a - -$jT i
0
where
1.1 is the Euclidean
LEMMA
1. If there
vu E R(L).
I$. 0
norm
is a. /3 > 0 and y E IO. 4r?/T’[
such that
PIUI - 1Y~V(r,U)~y~+a’ then there
exist a solution
- ii minimizses
(3)
u of Lu + D,V(t,
such that
Iu!Z
u) = 0
Q; on R(L).
Proof. We only outline the argument which follows the proof conservative systems. For E E IO, (A - y)/2[, where A = 4rr’/T’, let us define El@ V,(t, v) = V(t, v) + 2
of Brezis
and Coron
for
Subharmonic oscillations of convex Hamiltonian systems
130.5
and Q)E(U)= /r E(K u, v) + VT (t, u)l df,
u E R(L),
0
where VT is the Fenchel transform of V, with respect to u. Assumption (3) and the definition of V,*imply that
By (1) there is 6 > 0, independent
of E, such that, for every u E R(L), q,(u)
2 6
I0
r Jr_/’dt - CUT.
(4)
Thus there exists u, minimizing qE on R(L). By the first order necessary condition Ku, + D,V: (t, u,) = W, E Ker L. If u, = w, - KuE, then Lu, = -u, and, by duality, Lu, + D,V(t,
Moreover,
u,) +
EU,
(5)
0.
=
for every h E R(L), ve(uc) s vdh)
c v(h).
(6)
Since POE
c q,(O) s /r V”(t, 0) dt 0
;
deduce from (4) that Jilii12 = Jilui’ < c ‘. By the convexity of V and (3) we obtain, as in
S(u) - as V(t, u) c (D,V(t, u), u) +
V(t, 0)
c (DuV(t, u), 2.4)+ a.
(7)
Using (5) we then have B~oTl~~~~20tT+~~(-U.-~~~,u~) 0
~2aT+
I0
T IL.12=Gc”.
Using those estimates it suffices to pass to the limit in (5) and (6).
LEMMA 2. If there is LY,y > 0 such that -cYs
/u12 V(t, u) s y2 +
(Y,
4
1306
M. WILLEM
then 6
Proof.
Assuming
/V,(t, cl)\2s (V(t, u). u) + 2n.
f = V,(t, u) # 0, let us write g = f/If].
We have,
for every r E R,
(f, rg - U) + V(t, u) s V(t, rg) or 0 =GV(r, rg) - llfi + (f, u) - V(G u). By assumption 0 s rz ; + 2ff - rifl + (f. U). Then
Remarks 1. The proof follows Brezis and Nirenberg 2. The constant 1/2y is the best possible (see [7]). LEMMA
3. If there
is CY,/? > 0 and ‘J E IO, 2&/T’[ CYS V(t.u)
Blulthen every
solution
[2. p. 3121
such that
I4
=S y2
+ (Y.
of Lu + D,V(r.
u) = 0
(8)
is such that (9) and
Proof. Lemma
2 implies
that,
for every u E H.
T(1),V(r.u).ii)dr~~~TlD,.V(r,il)~’dr-2aT. I0 0 It follows
from (2), (8) and (10) that 0 =
J
[(ii. U) + (D,,V(t.
u), u)] dt
(10)
Subharmonic oscillations of convex Hamiltonian systems
This is equivalent
1307
to (9). We deduce from (7) and (8) that
Remark.
The estimation in the proof of lemma 1 uses the construction of the solution. In contrast to this u posteriori estimation (Dolph [5]), the estimates of lemma 3 are a priori.
THEOREM
1. If V is T-periodic with respect to t, IUI+ =,
V(t, u) + =c,
(II)
and V(r,
IUI+ =,
4ll4’ * 0,
(12)
uniformly in t, then for every k E N*, there exists a kT-periodic
solution uk of
ii + D,V(t, U) = 0
(13)
such that
IUkL= yEy bk(Ol--,=, and the minimal period of uk tends to 3~as k+
k-,x,
(14)
cc.
Proof. Without loss of generality, we can assume that T = 2~r. Assumption convexity of V imply the existence of a, b > 0 such that blul - a s V(t. u).
(11) and the
(15)
It follows then from assumption (12) and lemma 1 applied to T = 2kn that there is a 2knperiodic solution uk of (13) such that -ii, minimizes Zkx qk(")
=
[(Ku.
I0
u)
+
V*(t, u)] dt
on R(K). (Clearly K depends on k). Let us estimate ck = q(-iik) from above. Using (15) and the definition of V*, it is easy to verify that IuI s b + V*(t, u) c a. Let e E R” be such that lel = 1. Since hk(t) = b(sin(t/k))e
(16)
E R(K), it follows from (16) that
Zk.7 ck
c
qk@k)
=S =
i0 -nk’
[
-k’(sin(t/k))’ + 2unk.
+ a] dt (17)
1308
M. WILLEM
If, for some subsequence (k,), luknlr is bounded, bounded. Thus there is c’ > 0 such that
it follows from (13) that (iik, Ix is also
l%nl”ltiknlf cc’.
(18)
v*(t, u) 1 -V(t, 0) 3 -c”
(19)
The definition of V* implies that
where c” = mEa: V(t, 0). By using (18) and (19) we obtain ‘kn = q( -ii/,,)
= /Q’k”X[(iikn, uk,) + v*(t, 2 -2kJr(C
-iikn)]
dt
+ c”),
contrary to (17). Thus luklx+ =, k-, s. It remains only to prove that the minimal period of uk tends to = when k-, x. Suppose that, for a subsequence (k,). the minimal period of Ukn is less than a constant T > 0. By assumption there is LY,/3 > 0 and y E ]0,2n’/r’[ such that
It follows then from lemma 3 that
and
It is easy to verify that those estimates and T, S T imply that luknjx is bounded, (14). n
contrary to
Remarks 1. A similar proof shows that the amplitude of the oscillations in [l] tends to infinity
with the minimal period. 2. Theorem COROLLARY
1. If V E C’(W.
1 generalizes corollary 3.2 of [4].
58) is convex, f E C(R. W) is T-periodic and
then, for every k E N*, there exists a kT-periodic
solution uk of
ii + DV(U) = f(t)
Subharmonic oscillations of convex Hamiltonian systems
1309
such that IL& --, x, k --, =, and the minimal period of uk tends to = as k ---, =. Proof. It suffices to apply theorem
2 with V(t, u) = V(u) - (f(t), u).
Example 1. Let g : R + IR be g(--3c) < g < g(x). If equation
a
continuous
nondecreasing
n
function
such
ii + g(q) = f(t) where f : R+
R is continuous
that (20)
and T-periodic, has a T-periodic solution, it is clear that g(-r-)
< -fyf
< g(x).
(21)
0
Let us write
fi = y*f.f*
=f -f1,
and u = q - h.
h = Kf2
0
Then equation (20) is equivalent
to ii+g(u+h)=f,.
Let us define V(t, u) = G(u + h(t)) -flu
where G(r) = _&g(s) d.s. Since (21) implies (ll), theorem 1 is applicable under this condition. Assumption (21) is a Landesman-Lazer type condition ([2]). Example 2. Let v E]-l,O[ applies to system
and let
f: R+ R”be continuous and T-periodic. Corollary 1 ii + I+4
= f(t)
describing forced oscillations of a weak spring. The existence of at least one T-periodic solution proved in [4] by duality. When f is odd one can use [8]. If v = 0, there is no subharmonic oscillation.
is
2. SUBHARMONIC
OF FIRST ORDER
HAMILTONIAN
SYSTEMS
For T > 0 and n E N* let us denote by X the Hilbert space L2(0, T; R*‘). Let L be defined by D(L) = {u : [0, T] ---, lR*” : u is absolutely continuous and T-periodic and ri E EX} Lu=JU
where
J=
0
-1,
1,
0
1
J
M. WILLEM
1310
is the symplectic matrix. Then L is a self-adjoint operator, Ker L is the space of constant functions and R(L) = Ker L-. The eigenvalues of L are 2kx/T, k E Z. and the corresponding eigenfunctions are 2nkt u(t) = e sin -++ecosT where
T
e E R2’Y{O}.If K = L-l
it
2rrkt
is easy to verify
: R(L)+
R(L) n D(L),
that
vu IT(KV,“)9-~j-ilU12,
E R(L).
0
T i
(Lu, u) 3 - L/T
0
JLUI’,
vu E R(L).
By using (22), (23) instead THEOREM
(23)
0
LetH:RXlR2”+R:(t,u) + H(t, u) be a continuous with respect to u and such that D,H(t, u) is continuous given by q(v)
(22)
0
=
’ [(Ku. v) + H*(t, v)] dt, I0
of (l),
2. If H is T-periodic
function, convex and differentiable on [w+ R’“. The dual action is now
(2) one proves
with respect
v E R(L).
the following
result.
to t.
H(t, u) + x,
/uI---, =
and qt. uniformly
in t, then,
for every
u)/lLq
k E N”. there
-
0,
/ul--, x
exists a kT-periodic
solution
uk of
J; + D,H(t. u) = 0 such that (ukll+
x, k+
x, and the minimal
Remark. Theorem 2 generalizes the results subquadratic Hamiltonian systems.
period
of u,! tends
to x.
of [6] on long period
subharmonics
of convex
REFERENCES 1. BREZIS H. & CORON J. M., Periodic
and hamiltonian systems. Am. J. Mafh. (1980). 2. BREZIS H. & NIRENBERG L.. Chardcterrzations of the ranges of some non linear operators and applications to boundary value problems, Annali Scu. norm. sup. Pisa 5. 225-325 (1978). 3. CLARKE F. & EKELAND I., Hamiltonian tralectorles having prescribed minimal period, Communs pure appl. Math. 33, 103-115 (1980). 4. CLARKE F. & EKELAND I., Nonlinear oscillations and boundary value problems for hamiltonian systems, Arch ration. Mech. Analysis 78, 315-333 (1982). 103. 559-570
solutions
of non linear wave equations
Subharmonic oscillations of convex Hamiltonian systems
1311
DOLPH C. L., Nonlinear integral equations of the Hammerstein type, Trans. Am. math. Sec. 66. 289-307 (1949). RABINOWITZ P., On subharmonic solutions of hamiltonian systems, Communspure uppl. Marh. 33.609433 (1980). 7. WILLEM M., Remarks on the dual least action principle, Z. Analysis Anwend. 1, 85-90 (1982). 8. WILLEM M., Periodic oscillations of odd second order hamiltonian systems, Boll. Un. Mar. Ital. 3B. 23F304 (1984).
5. 6.
Merhods
& Apphcormnr.
SUBHARMONIC
Vol
9. No. 11. pp. 1303-1311.
0362-546X 85 $3 00 - .oo G 1985 Pergamon Press Ltd
1985.
OSCILLATIONS OF CONVEX HAMILTONIAN SYSTEMS M. WILLEM
Institut
de Mathkmatique
Pure et AppliquCe. Universitk Catholique de Louvain. Louvain-la-Heuve. Belgium (Received
Key words and phrases:
Chemin
du Cyclotron
1. B-1348
5 March 1983: received for publication 5 June 1985)
Subharmonic
oscillations.
Hamiltonian
systems.
nonconvex
duality
INTRODUCTION
IN [l] BREZIS and Coron DH(o) = 0 and
proved
that,
if HE
H(u)*
for every
T sufficiently
large.
there
Hamiltonian
such that
Iu/ + x,
x,
H(u)/lul’+ then,
C’(R*“, R) is a convex
/U/+ x,
0,
is a solution
Jic + DH(u)
of
= 0
with minimal period T.When H is not conservative and T-periodic with respect to the time t, only kT-periodic solutions are expected. We prove that if H(t, u) is convex with respect to U, H(r, U) *
/U/---, x,
=.
lUl--, 5,
H(t, U)/jUl~ --;, 0. then,
for every
k E N*,
there
exists a kT-periodic
solution
Us of
JU + D,,H(t. u) = 0 such that ~~~~~~ x, k-, x, and the minimal period of uk tends to infinity. This theorem generalizes the results of [6] on long period subharmonics of subquadratic Hamiltonian systems by a completely different method. Before treating first order systems we consider the case of classical mechanics ii + D,,V(t. u) = 0. The proofs are similar and depends of Clarke and Ekeland [3]. 1. SUBHARMONICS
For
on a variational
OF SECOND
T > 0 and n E N” let us denote
characterization
ORDER
by H the Hilbert 1303
involving
HAMILTONIAN
space L’(O,
the “dual action”
SYSTEMS
T;BP).Let L be defined
1304
M. WILLEM
bY D(L)
=
{u: [O, T]+ T-periodic
R” : u and U are absolutely and ii E H}
continuous
and
Lu = ii. Then L is a self-adjoint operator, Ker L is the space of constant functions and R(L) = Ker L-. Let V : R X R” + R : (t, u) += V(t, u) be a continuous function. convex and differentiable with respect to u and such that D,V(t, u) is continuous on R x W”. Let us write K = L-’ v*(t,
: Z?(L)*
Z?(L) f7 D(L), cl E R8”. fEiR
u) = ,“E”wp” [(u. u) - V(t. u)],
and Q?(U) =
r [~(Ku, u) + V*(t. u)] dt.
u E R(L)
where (. , .) is the usual inner product on R”. The function V* is the Fenchel transform of V with respect to u and p7 is the dual action. Using Fourier expansions, it is easy to verify the following (equivalent) inequalities
IT
(Ku. u) a - -$jT i
0
where
1.1 is the Euclidean
LEMMA
1. If there
vu E R(L).
I$. 0
norm
is a. /3 > 0 and y E IO. 4r?/T’[
such that
PIUI - 1Y~V(r,U)~y~+a’ then there
exist a solution
- ii minimizses
(3)
u of Lu + D,V(t,
such that
Iu!Z
u) = 0
Q; on R(L).
Proof. We only outline the argument which follows the proof conservative systems. For E E IO, (A - y)/2[, where A = 4rr’/T’, let us define El@ V,(t, v) = V(t, v) + 2
of Brezis
and Coron
for
Subharmonic oscillations of convex Hamiltonian systems
130.5
and Q)E(U)= /r E(K u, v) + VT (t, u)l df,
u E R(L),
0
where VT is the Fenchel transform of V, with respect to u. Assumption (3) and the definition of V,*imply that
By (1) there is 6 > 0, independent
of E, such that, for every u E R(L), q,(u)
2 6
I0
r Jr_/’dt - CUT.
(4)
Thus there exists u, minimizing qE on R(L). By the first order necessary condition Ku, + D,V: (t, u,) = W, E Ker L. If u, = w, - KuE, then Lu, = -u, and, by duality, Lu, + D,V(t,
Moreover,
u,) +
EU,
(5)
0.
=
for every h E R(L), ve(uc) s vdh)
c v(h).
(6)
Since POE
c q,(O) s /r V”(t, 0) dt 0
;
deduce from (4) that Jilii12 = Jilui’ < c ‘. By the convexity of V and (3) we obtain, as in
S(u) - as V(t, u) c (D,V(t, u), u) +
V(t, 0)
c (DuV(t, u), 2.4)+ a.
(7)
Using (5) we then have B~oTl~~~~20tT+~~(-U.-~~~,u~) 0
~2aT+
I0
T IL.12=Gc”.
Using those estimates it suffices to pass to the limit in (5) and (6).
LEMMA 2. If there is LY,y > 0 such that -cYs
/u12 V(t, u) s y2 +
(Y,
4
1306
M. WILLEM
then 6
Proof.
Assuming
/V,(t, cl)\2s (V(t, u). u) + 2n.
f = V,(t, u) # 0, let us write g = f/If].
We have,
for every r E R,
(f, rg - U) + V(t, u) s V(t, rg) or 0 =GV(r, rg) - llfi + (f, u) - V(G u). By assumption 0 s rz ; + 2ff - rifl + (f. U). Then
Remarks 1. The proof follows Brezis and Nirenberg 2. The constant 1/2y is the best possible (see [7]). LEMMA
3. If there
is CY,/? > 0 and ‘J E IO, 2&/T’[ CYS V(t.u)
Blulthen every
solution
[2. p. 3121
such that
I4
=S y2
+ (Y.
of Lu + D,V(r.
u) = 0
(8)
is such that (9) and
Proof. Lemma
2 implies
that,
for every u E H.
T(1),V(r.u).ii)dr~~~TlD,.V(r,il)~’dr-2aT. I0 0 It follows
from (2), (8) and (10) that 0 =
J
[(ii. U) + (D,,V(t.
u), u)] dt
(10)
Subharmonic oscillations of convex Hamiltonian systems
This is equivalent
1307
to (9). We deduce from (7) and (8) that
Remark.
The estimation in the proof of lemma 1 uses the construction of the solution. In contrast to this u posteriori estimation (Dolph [5]), the estimates of lemma 3 are a priori.
THEOREM
1. If V is T-periodic with respect to t, IUI+ =,
V(t, u) + =c,
(II)
and V(r,
IUI+ =,
4ll4’ * 0,
(12)
uniformly in t, then for every k E N*, there exists a kT-periodic
solution uk of
ii + D,V(t, U) = 0
(13)
such that
IUkL= yEy bk(Ol--,=, and the minimal period of uk tends to 3~as k+
k-,x,
(14)
cc.
Proof. Without loss of generality, we can assume that T = 2~r. Assumption convexity of V imply the existence of a, b > 0 such that blul - a s V(t. u).
(11) and the
(15)
It follows then from assumption (12) and lemma 1 applied to T = 2kn that there is a 2knperiodic solution uk of (13) such that -ii, minimizes Zkx qk(")
=
[(Ku.
I0
u)
+
V*(t, u)] dt
on R(K). (Clearly K depends on k). Let us estimate ck = q(-iik) from above. Using (15) and the definition of V*, it is easy to verify that IuI s b + V*(t, u) c a. Let e E R” be such that lel = 1. Since hk(t) = b(sin(t/k))e
(16)
E R(K), it follows from (16) that
Zk.7 ck
c
qk@k)
=S =
i0 -nk’
[
-k’(sin(t/k))’ + 2unk.
+ a] dt (17)
1308
M. WILLEM
If, for some subsequence (k,), luknlr is bounded, bounded. Thus there is c’ > 0 such that
it follows from (13) that (iik, Ix is also
l%nl”ltiknlf cc’.
(18)
v*(t, u) 1 -V(t, 0) 3 -c”
(19)
The definition of V* implies that
where c” = mEa: V(t, 0). By using (18) and (19) we obtain ‘kn = q( -ii/,,)
= /Q’k”X[(iikn, uk,) + v*(t, 2 -2kJr(C
-iikn)]
dt
+ c”),
contrary to (17). Thus luklx+ =, k-, s. It remains only to prove that the minimal period of uk tends to = when k-, x. Suppose that, for a subsequence (k,). the minimal period of Ukn is less than a constant T > 0. By assumption there is LY,/3 > 0 and y E ]0,2n’/r’[ such that
It follows then from lemma 3 that
and
It is easy to verify that those estimates and T, S T imply that luknjx is bounded, (14). n
contrary to
Remarks 1. A similar proof shows that the amplitude of the oscillations in [l] tends to infinity
with the minimal period. 2. Theorem COROLLARY
1. If V E C’(W.
1 generalizes corollary 3.2 of [4].
58) is convex, f E C(R. W) is T-periodic and
then, for every k E N*, there exists a kT-periodic
solution uk of
ii + DV(U) = f(t)
Subharmonic oscillations of convex Hamiltonian systems
1309
such that IL& --, x, k --, =, and the minimal period of uk tends to = as k ---, =. Proof. It suffices to apply theorem
2 with V(t, u) = V(u) - (f(t), u).
Example 1. Let g : R + IR be g(--3c) < g < g(x). If equation
a
continuous
nondecreasing
n
function
such
ii + g(q) = f(t) where f : R+
R is continuous
that (20)
and T-periodic, has a T-periodic solution, it is clear that g(-r-)
< -fyf
< g(x).
(21)
0
Let us write
fi = y*f.f*
=f -f1,
and u = q - h.
h = Kf2
0
Then equation (20) is equivalent
to ii+g(u+h)=f,.
Let us define V(t, u) = G(u + h(t)) -flu
where G(r) = _&g(s) d.s. Since (21) implies (ll), theorem 1 is applicable under this condition. Assumption (21) is a Landesman-Lazer type condition ([2]). Example 2. Let v E]-l,O[ applies to system
and let
f: R+ R”be continuous and T-periodic. Corollary 1 ii + I+4
= f(t)
describing forced oscillations of a weak spring. The existence of at least one T-periodic solution proved in [4] by duality. When f is odd one can use [8]. If v = 0, there is no subharmonic oscillation.
is
2. SUBHARMONIC
OF FIRST ORDER
HAMILTONIAN
SYSTEMS
For T > 0 and n E N* let us denote by X the Hilbert space L2(0, T; R*‘). Let L be defined by D(L) = {u : [0, T] ---, lR*” : u is absolutely continuous and T-periodic and ri E EX} Lu=JU
where
J=
0
-1,
1,
0
1
J
M. WILLEM
1310
is the symplectic matrix. Then L is a self-adjoint operator, Ker L is the space of constant functions and R(L) = Ker L-. The eigenvalues of L are 2kx/T, k E Z. and the corresponding eigenfunctions are 2nkt u(t) = e sin -++ecosT where
T
e E R2’Y{O}.If K = L-l
it
2rrkt
is easy to verify
: R(L)+
R(L) n D(L),
that
vu IT(KV,“)9-~j-ilU12,
E R(L).
0
T i
(Lu, u) 3 - L/T
0
JLUI’,
vu E R(L).
By using (22), (23) instead THEOREM
(23)
0
LetH:RXlR2”+R:(t,u) + H(t, u) be a continuous with respect to u and such that D,H(t, u) is continuous given by q(v)
(22)
0
=
’ [(Ku. v) + H*(t, v)] dt, I0
of (l),
2. If H is T-periodic
function, convex and differentiable on [w+ R’“. The dual action is now
(2) one proves
with respect
v E R(L).
the following
result.
to t.
H(t, u) + x,
/uI---, =
and qt. uniformly
in t, then,
for every
u)/lLq
k E N”. there
-
0,
/ul--, x
exists a kT-periodic
solution
uk of
J; + D,H(t. u) = 0 such that (ukll+
x, k+
x, and the minimal
Remark. Theorem 2 generalizes the results subquadratic Hamiltonian systems.
period
of u,! tends
to x.
of [6] on long period
subharmonics
of convex
REFERENCES 1. BREZIS H. & CORON J. M., Periodic
and hamiltonian systems. Am. J. Mafh. (1980). 2. BREZIS H. & NIRENBERG L.. Chardcterrzations of the ranges of some non linear operators and applications to boundary value problems, Annali Scu. norm. sup. Pisa 5. 225-325 (1978). 3. CLARKE F. & EKELAND I., Hamiltonian tralectorles having prescribed minimal period, Communs pure appl. Math. 33, 103-115 (1980). 4. CLARKE F. & EKELAND I., Nonlinear oscillations and boundary value problems for hamiltonian systems, Arch ration. Mech. Analysis 78, 315-333 (1982). 103. 559-570
solutions
of non linear wave equations
Subharmonic oscillations of convex Hamiltonian systems
1311
DOLPH C. L., Nonlinear integral equations of the Hammerstein type, Trans. Am. math. Sec. 66. 289-307 (1949). RABINOWITZ P., On subharmonic solutions of hamiltonian systems, Communspure uppl. Marh. 33.609433 (1980). 7. WILLEM M., Remarks on the dual least action principle, Z. Analysis Anwend. 1, 85-90 (1982). 8. WILLEM M., Periodic oscillations of odd second order hamiltonian systems, Boll. Un. Mar. Ital. 3B. 23F304 (1984).
5. 6.