HETEROCLINIC MOTIONS JOINING ALMOST PERIODIC SOLUTIONS FOR A CLASS OF LAGRANGIAN SYSTEMS

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

Volume 5, Number 3, July 1999

HETEROCLINIC MOTIONS JOINING ALMOST PERIODIC SOLUTIONS FOR A CLASS

OF LAGRANGIAN SYSTEMS

FRANCESCA ALESSIO

, CARLO CARMINATI

AND PIERO MONTECCHIARI

Dipartimento di Matematica “R. Caccioppoli”, Universit` a di Napoli “Federico II” Via Cintia, I–80126 Napoli, Dipartimento di Matematica, Universit` a di Pisa, Via Buonarroti 2, I–56127 Pisa, e-mail carminat@mail.dm.unipi.it

Dipartimento di Matematica, Universit` a di Ancona, Via Brecce Bianche, I–60131 Ancona,

(Communicated by Antonio Ambrosetti)

Abstract. We regard second order systems of the form ¨q = ∇

W (q, t), t ∈ R, q ∈ R

, where W (q, t) is Z

periodic in q and almost periodic in t. Variational arguments are used to prove the existence of heteroclinic solutions joining almost periodic solutions to the system.

1. Introduction. In this paper we are concerned with Lagrangian systems of the form

q = ∇ ¨

W (q, t), t ∈ R, q ∈ R

(L) where W satisfies

(W 1 ) W ∈ C ² (R

× R, R) is almost periodic in t, uniformly with respect to q ∈ R

and Z

-periodic in q;

(W 2 ) there exist ρ ∈ (0, ¹ ₄ ) and ω, σ, δ ∈ (0, +∞) such that for all t ∈ R (i) ω|y| ² ≥ ∇ ²

W (x, t)y · y ≥ σ|y| ² for all x ∈ B

(0), y ∈ R

; (ii) W (x, t) − W (0, t) ≥ δ for all x ∈ R

\ ∪

B

(η).

Since the system (L) does not change if we add to the potential a purely time dependent function, without loss of generality, we assume that W (0, t) = 0 for any t ∈ R. We also set

µ = sup

t∈R

|∇

W (0, t)|.

The value µ plays the role of a perturbative parameter. When µ = 0 one can easily recognize that our assumptions implies that W (x, t) ≥ 0 and W (η, t) = 0 for any x ∈ R

, η ∈ Z

, t ∈ R. Then in this limit case system (L) admits as trivial solutions the equilibria q

( t) = η, η ∈ Z

, t ∈ R. The solution q

is in fact the unique solution of ( L) which remains inside the ρ-neighborhood of η in the configuration space for any time.

1991 Mathematics Subject Classification. 34C37, 58Exx, 58Fxx.

Key words and phrases. Almost periodic Lagrangian systems, almost periodic solutions, heteroclinic, variational methods.

1

Supported by Consiglio Nazionale delle Ricerche (CNR), Italy.

1,2

Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lin- eari”.

3

Partially supported by Consiglio Nazionale delle Ricerche (CNR), Italy.

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