Subharmonic solutions of a nonconvex noncoercive hamiltonian system

RAIRO Oper. Res. 38(2004) 27-37 DOI: 10.1051/ro:2004010

SUBHARMONIC SOLUTIONS OF A NONCONVEX NONCOERCIVE HAMILTONIAN SYSTEM

Najeh Kallel

and Mohsen Timoumi

Communicated by Jean-Pierre Crouzeix

Abstract. In this paper we study the existence of subharmonic solu- tions of the Hamiltonian system

Jx˙+u^∗∇G(t, u(x)) =e(t)

where u is a linear map, G is a C¹-function and e is a continuous function.

1. Introduction

In this paper we are interested in the existence of periodic solutions, of the noncoercive Hamiltonian system

Jx˙+u^∗∇G(t, u(x)) =e(t)

whereu:R²ⁿ−→R^m(1≤m≤2nis a linear map not identically null with adjoint u^∗;G:R×R^m−→R,(t, y)−→G(t, y) is a continuous function,T-periodic in the first variable (T > 0), differentiable with respect to the second variable and its derivative∇Gis continuous; e: R−→ R²ⁿ is a continuous, T-periodic function with mean value zero, and

J =

0 −I_n I_n 0

is the standard symplectic matrix.

Received December, 2002.

1 Institut pr´eparatoire des ´Etudes d’ing´enieurs de Sfax, D´epartement de Math´ematiques, BP 805, CP 3018, Tunisia.

2 Facult´e des sciences de Monastir, D´epartement de Math´ematiques, CP 5000, Tunisia;

e-mail:m−timoumi@yahoo.com

c EDP Sciences 2004

There have been many papers studying the multiplicity of periodic solutions of the Hamiltonian system ˙x=JH(t, x). They have been obtained by using many different techniques, for example Morse theory, Minimax methods, ... However, most of the results proving the existence of subharmonic solutions have made use of a convexity and a coercivity assumptions on H, see [1, 2, 7, 8].

Our first result is the following:

Theorem 1.1. Assume that Gsatisfies:

(G₁) ∃M >0 :∀t∈R,∀x∈R^m,||∇G(t, x)|| ≤M; (G₂)either

(i) lim

|x|−→+∞G(t, x) = +∞,uniformly int∈[0, T], or

(ii) lim

|x|−→+∞G(t, x) =−∞,uniformly in t∈[0, T], then the Hamiltonian system

Jx˙+u^∗∇G(t, u(x)) =e(t) (He) has at least oneT-periodic solution.

In the case where the forcing termeis null, we obtain:

Theorem 1.2. Assume that rgu > n.

Under the assumptions (G₁, G₂), the Hamiltonian system (H0) possesses for all integerk≥1, a kT-periodic solution x_k satisfying

k−→+∞lim ||x_k||∞= +∞, where||x||∞= sup{|x(t)|, t∈R}·

Concerning the minimality of the period, we have the following result:

Theorem 1.3. Assume that rgu > n.

If the Hamiltonian Gsatisfies(G₁) and the assumption(G₂)either (i) lim

|x|−→+∞ ∇G(t, x), x= +∞,uniformly int∈[0, T], or

(ii) lim

|x|−→+∞ ∇G(t, x), x=−∞,uniformly int∈[0, T],

then for any sufficiently large prime number k, the system (H₀) possesses a kT- periodic solution with minimal periodkT.

2. Preliminaries [5]

LetX =W⊕Z be a Banach space andX_n=W_n⊕Z_n be a sequence of closed subspaces with Z₀ ⊂Z₁ ⊂... ⊂ Z, W₀ ⊂ W₁ ⊂... ⊂W,1 ≤dimW_n <∞. For everyϕ:X−→Rwe denote byϕ_n the functionϕrestricted toX_n. Let us recall that, for anyA⊂X, A_n =A∩X_n.

Definition 2.1. Letc∈Randϕ∈C¹(X,R). The functionalϕsatisfies the (P S)^∗_c condition if every sequence (x_nj) ⊂ X satisfying: ϕ_nj(x_nj) −→ c, ϕ_n

j(x_nj)−→0, possesses a subsequence which converges inX to a critical point ofϕ.

Theorem 2.1 (Generalized Saddle Point theorem). Let ϕ∈ C¹(X,R). Assume that there exists r >0 such that, withY ={w∈W;|w|=r}:

a) sup_Y ϕ≤inf_Zϕ;

b) ϕis bounded from above onA={x∈W;||x|| ≤r};

c) ϕsatisfies the(P S)^∗_c condition, wherec= inf_A∈Asup_x∈Aϕ(x), with

A=

A⊂X;A is closed, Y ⊂A,catX,Y^∞ (A) = 1

· Then c is a critical value ofϕandc≥inf_Zϕ.

Remark 2.1. In a) we may replaceZ byq+Z, q∈W.

3. Proof of Theorem 1.1

We will prove here the case whereG satisfies (G₂)(i), the case (G₂)(ii) is the same.

Before giving a variational formulation of (H_e), some preliminary materials on function spaces and norms are needed.

Let L²(S¹,R²ⁿ) be the space of square integrable functions defined on S¹ = R/TZ, with value inR²ⁿ. Each functionx∈L²(S¹,R²ⁿ) has a Fourier expansion

x(t) =

m∈Z

exp 2π

T mtJ

xˆ_m

where ˆx_m∈R²ⁿ and

m∈Z|ˆx_m|²<∞.

Set

H^1/2(S¹,R²ⁿ) =

x∈L²(S¹,R²ⁿ);||x||_H^1/2 <∞ , where

||x||_H^1/2 =

m∈Z

(1 +|m|)|ˆx_m|² _1/2

. Consider the subspace

X =

x∈H^1/2(S¹,R²ⁿ);x₀∈keru^⊥

·

It is easy to check that the quadratic formQdefined onX by Q(x) = 1/2

_T

0

Jx, x˙ dt satisfies, for a smoothx∈X

Q(x) =−π

m∈Z

m|xˆ_m|². (1)

Set

X⁰= (keru)^⊥, X⁺=



x∈X;x(t) =

m≤−1

exp 2π

T mtJ

xˆ_m a.e.



 X⁻=



x∈X;x(t) =

m≥1

exp 2π

T mtJ

xˆ_m a.e.



,

thenX =X⁺⊕X⁻⊕X⁰, andX⁺, X⁻, X⁰,is respectively the subspace ofX on whichQis positive definite, negative definite, and null. X⁺, X⁻, X⁰are mutually orthogonal with respect to the associated inner product and mutually orthogonal inL²(S¹,R²ⁿ).

These remarks show that ifx=x⁺+x⁻+x⁰∈X,

||x||²=|x⁰|²+Q(x⁺)−Q(x⁻) (2) serves as an equivalent norm onX. Henceforth we use the norm defined in (2) as the norm for X. For each p∈[1,∞[, X is compactly embedded inL^p(S¹,R²ⁿ).

In particular there is anα_p>0 such that

∀x∈X,||x||_L^p≤α_p||x||. (3) Now, consider the functional

f(x) = _T

0 [1/2 Jx, x˙ +G(t, u(x))− e(t), x] dt

defined on the spaceX. The functionalf is continuously differentiable and verifies

∀x, y ∈X, f(x).y = _T

0

Jx˙+u^∗∇G(t, u(x))−e(t), ydt. (4) We claim that the critical points of the functionalf correspond to theT-periodic solutions of the system (H_e). Indeed, let xbe a critical point of f on X, then by (4) there exists a constantξ∈kerusuch that

Jx˙ +u^∗∇G(t, u(x))−e(t) =ξ a.e. (5)

Integrating (5) we obtain u^∗

_T

0 ∇G(t, u(x)) dt=T ξ,

so ξ∈(keru)^⊥, which proves that ξ = 0 andxis aT-periodic solution of (He).

Inversely, it is clear that every solution of (He) is a critical point off onX.

To find critical points off we shall apply the Generalized Saddle Point theorem to the functional f onX with these subspacesW =X⁻, Z=X⁺⊕X⁰ and the sequence of closed subspaces

X_n=



x∈X;x(t) =

|m|≤n

exp 2π

T mtJ

xˆ_m a.e.,



·

First, from (1), the assumption (G₁) and the mean value theorem applied to the functionG(t, .), we have for x=x⁰+x⁺∈Z,

f(x)≥π

2||x⁺||²−c||x⁺||+ _T

0

G(t, u(x⁰))dt, (6) wherecis a constant. Since the functionuis invertible onX⁰, we deduce from (6) and the assumption (G₂) thatf(x) goes to infinity asxgoes to infinity inZ.

Secondly as in the lastly, there exist two constantsa, b∈Rsuch that for every x∈W

f(x)≤ −π

2||x||²+a||x||+b.

Consequently there existsr >0 such that supY f ≤inf

Z f

where Y = {x ∈ W;||x|| = r}. Furthermore, f is bounded from above onD_r, whereD_ris the closed disc in W centered in zero, with radiusr.

Finally we will show that for all c ∈ R, the functional f satisfies the Palais Smale condition (P S)^∗_c with respect to (X_n). Let (x_nj) be a sequence such that

n_j−→ ∞, x_nj ∈X_nj, f(x_nj)−→c, f_nj(x_nj)−→0, (7) we setx_nj =x⁰_nj+˜x_nj, wherex⁰_nj is the projection ofx_nj ontoX⁰. We have from (1) and (4):

f_n

j

x_nj x⁺_n

j −x⁻_n

j

= 2π

1≤|m|≤nj

|m||ˆx_j,m|²

+ _T

0

u^∗∇G(t, u(x))−e(t), x⁺_n

j −x⁻_n

jdt.

It follows from (7) and the assumption (G₁) that (˜x_nj) is bounded inX. Elsewhere, we have

f(x_nj) = 1/2 _T

0

Jx˜_nj,x˜_njdt+ _T

0

G(t, u(x_nj))dt− _T

0

e,x˜_njdt,

so by (7), (_T

0 G(t, u(x_nj))dt) is bounded. By applying the mean value theorem to the function G(t, .), there existsy_nj ∈X such that

_T

0

G(t, u(x_nj))dt= _T

0

G(t, u(x⁰_nj))dt+ _T

0

u^∗∇G(t, u(y_nj)), u(˜x_nj)dt and we deduce from the assumption (G₁) that (_T

0 G(t, u(x⁰_nj))dt) is bounded, so by the assumption (G₂)(i),(x⁰_nj) is also bounded inX. Up to a subsequence, we can assume that x_nj xand x⁰_nj −→x⁰in X. Notice that

Q(x⁺_nj−x⁺) = f_nj(x_nj)−f(x), x⁺_nj −x⁺

− _T

0 ∇G(t, u(x_nj))− ∇G(t, u(x)), u(x⁺_nj −x⁺)dt.

This implies thatx⁺_nj −→x⁺ in X. Similarly x⁻_nj −→ x⁻ in X. It follows then thatx_nj −→xinX andf(x) = 0.

The function f satisfies all the assumptions of the Generalized Saddle Point theorem sof has at least one critical point. The proof of Theorem 1.1 is complete.

4. Proof of Theorem 1.2

As in the proof of Theorem 1.1, we will prove here the case where Gsatisfies (G₁) and (G₂)(i), the case (G₂)(ii) is the same. By making the change of variable t−→ _k^t, the system (H₀) transforms to:

Jx˙+ku^∗∇G(kt, u(x)) = 0. (Hk) Hence to findkT-periodic solutions of (H₀), it suffices to findT-periodic solutions of (H_k), which are the critical points of the continuously differentiable functional

f_k(x) = 1 2

_T

0

Jx, x˙ dt+k _T

0

G(kt, u(x))dt

defined on the space X introduced above. By applying the Theorem 2.1 to the functional f_k on the space X, we prove as in Theorem 1.1, that for all integer k≥1 the system (H_k) possesses aT-periodic solutionx_k such that

f_k(x_k)≥ inf

x∈Zf_k(x). (8)

Now we will prove that the sequence (x_k) obtained above has the following property

k−→∞lim 1

kf_k(x_k) =∞. (9)

This will be done by the following lemma:

Lemma 4.1. Assume that Gsatisfies(G₂), then:

k−→∞lim inf

x∈Z

f_k

k¹²(ϕ+x)

k =∞ (10)

whereϕ= _π1/2¹ exp_2π

T tJ

e₀ ande₀ is a particular element inX⁰,(Je₀∈keru).

Proof of Lemma 4.1. Assume by contradiction that there exist sequencesk_j−→ ∞, x_j=x⁺_j +x⁰_j ∈Z and a constantc∈Rsuch that

∀j∈N, f_kj

k_j¹²(ϕ+x_j)

≤k_jc. (11)

Takingx_j =x_j⁺+x_j^o, x_jⁱ∈Xⁱ, i= +,0, by an easy calculation, we obtain f_kj

k_j¹² (ϕ+x_j)

=k_j ||x⁺_j||²− ||e₀||²+ _T

0

G k_jt, u

k_j¹²(ϕ+x_j) dt

; (12) so by (G₂) there exists a constantc₁>0 such that

f_kj

k_j¹²(ϕ+x_j)

≥k_j

||x⁺_j||²−c₁

and we conclude, from the inequality (11), that (x⁺_j) is a bounded sequence inX. Taking a subsequence if necessary, we findx⁺∈X⁺ such that

x⁺_j(t)−→x⁺(t) as j−→ ∞ for a.e. t∈[0, T]. (13) We claim that (x⁰_j) is also bounded in X⁰. If we suppose otherwise, we easily deduce from (13) and the fact thatuis invertible onX⁰ that

u k_j¹²

ϕ(t) +x⁺_j(t) +x⁰_j−→ ∞ as j−→ ∞ for a.e. t∈[0, T].

Consequently by (G₂) and Fatou’s lemma, we obtain _T

0

G k_jt, u

k_j¹²

ϕ(t) +x⁺_j(t) +x⁰_j

dt−→ ∞ as j−→ ∞, (14)

and we deduce from (12) that

f_kj(k_j¹²(ϕ+x⁺_j +x⁰_j))

k_j −→ ∞ as j−→ ∞, (15)

which contradicts (11) and proves our claim. Going if necessary to a subsequence, we can assume that there existsx⁰∈X⁰ such that

ϕ(t)+x⁺_j(t)+x⁰_j −→x(t) =ϕ(t)+x⁺(t)+x⁰ as j−→ ∞ for a.e. t∈[0, T].

By Fourier analysis, we havex(t)∈kerufor almost everyt∈[0, T]. Therefore u

k_j¹²

ϕ(t) +x⁺_j(t) +x⁰_j−→ ∞ as j−→ ∞,

and by (G₂) and Fatou’s lemma, we obtain (15), which contradicts (11). Thus (10)

must hold. The proof of Lemma 4.1 is complete.

Hence Theorem 2.1 (Rem. 2.1) implies that for allk∈N, we have f_k(x_k) =b_k≥ inf

x∈Zf_k

k¹²(ϕ+x) .

So we have by Lemma 4.1:

b_k

k −→ ∞ as k−→ ∞.

We claim that||x_k||_∞−→ ∞as k−→ ∞. Indeed, if we suppose otherwise, (x_k) possesses a bounded subsequence (x_kp). Since

f_k(x_k) k =−1

2 _T

0

u^∗∇G(kt, u(x_k)), x_kdt+ _T

0

G(kt, u(x_k))dt

the sequence _b

kkpp

is bounded, contrary to (9). Consequently, we have

k−→∞lim ||x_k||_∞=∞,

which implies thatv_k(t) =x_k(_k^t) is akT-periodic solution of the system (H₀) and verifies:

k−→∞lim ||v_k||∞= lim

k−→∞||x_k||∞=∞. That concludes the proof of Theorem 1.2.

5. Proof of Theorem 1.3.

We begin the proof by the following lemma:

Lemma 5.1. The assumptions(G₁),(G₂)imply the assumption(G₂).

Proof of Lemma 5.1: Assume for example (G₁) and (G₂)(i) hold and let us prove (G₂)(i). LetR >0 be such that

|x| ≥R=⇒ ∇G(t, x), x ≥1,∀t∈R. Then, for|x| ≥R, we have

G(t, x) =G(t,0) + ^R

|x|

0 ∇G(t, sx), xds+ ₁

|x|R

∇G(t, sx), xds

≥G(t,0)−MR+ ₁

|x|R

ds s

≥inf

t∈RG(t,0)−MR+ Log |x|

R

and (G₂)(i) follows.

Letkbe an integer≥1, we consider the functionalϕ_k ϕ_k(x) = 1

2 _kT

0

Jx, x˙ dt+ _kT

0

G(t, u(x))dt defined on the space

X^k =

x∈H¹²(S_k¹,R²ⁿ); ˆx₀∈(keru)^⊥ , whereS_k¹=R/kTZ.

It is clear thatx∈X is a critical point off_k if and only ifv(t) =x(_k^t) belongs toX^k and is a critical point ofϕ_k. By the Lemma 5.1, the assumptions (G₁, G₂) are satisfied, so we use the Theorem 1.2.

Let (v_k) ⊂ X^k be the sequence of critical points of ϕ_k associated to the se- quence (x_k)⊂X of critical points off_k obtained in the proof of Theorem 1.2, the property (9) is written

k−→∞lim 1

kϕ_k(v_k) =∞. (16)

On the other hand, let us denote by S_T, the set of T-periodic solutions of (H0) which are in X. We claim thatS_T is bounded inX. Indeed, assume by contra- diction that there exists a sequence (v_n) in S_T such that ||v_n||∞ −→ ∞. Let us write ˜v_n =v⁺_n +v_n⁻ and ¯v_n the mean value ofv_n, thenv_n= ˜v_n+ ¯v_n. Multiplying both sides of the identity

Jv˙_n+u^∗∇G(t, u(v_n)) = 0 (17)

byJv˙_n and integrating, we obtain:

||v˙_n||²L²+ _T

0 ∇G(t, u(v_n(t))), u(Jv˙_n)dt= 0.

Using assumption (G₁), we easily deduce that ( ˙v_n) is bounded in L². So (˜v_n) is bounded inL^∞([0, T]) and inX, and then (u(˜v_n)) is bounded inL^∞([0, T]). The identity

||v_n||²=|¯v_n|²+||˜v_n||² shows then that

n−→∞lim |¯v_n|=∞, and then

n−→∞lim |u(¯v_n)|=∞. Consequently, we obtain

n−→∞lim min

t∈[0,T]|u(v_n(t))|=∞. (18) Multiplying (17) byv_n and integrating, we get

_T

0

Jv˙_n, v_ndt+ _T

0 ∇G(t, u(v_n)), u(v_n)dt= 0. (19) Now, since (˜v_n) is bounded inX, we deduce from (19) that

_T

0 ∇G(t, u(v_n)), u(v_n)dt

is bounded.

But this is in contradiction with (18) and (G₂). The claim follows immediately.

As a consequence, ϕ₁(S_T) is bounded, and since for anyv ∈S_T one hasϕ_k(u) = kϕ₁(u) , we have

∃c >0 :∀v∈S_T,∀k≥1,1

k|ϕ_k(v)| ≤c. (20) Consequently, (16) and (20) show that, fork sufficiently large,v_k ∈S_T, and ifk is chosen to be a prime number, the minimal period ofv_k has to bekT and the

proof is complete.

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