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Subharmonic solutions for convex nonautonomous Hamiltonian systems. (English) Zbl 0879.34043

Nonautonomous Hamiltonian systems of the form

Ix ˙(t)=H x x ( t ) , t(1)

where I is the standard symplectic matrix

I=0I N -I N 0

x=(x 1 ,,x 2N ) 2N and the Hamiltonian H is a T-periodic (in the second variable) function, i.e. H(x,t+T)=H(x,t), for all (x,t) is considered. Precisely, for any given integer p1 the existence of multiple periodic (subharmonical) solutions of Eq. (1) x(0)=x(pT) is investigated.

It is known, that under rather general assumptions on H for any integer p1 the subharmonic problem (1) was shown to have a distinct solution X (p) and if H has subquadratic or superquadratic growth at the origin and at infinity then the minimal period of X (p) tends towards infinity as p+.

Some new multiplicity results are obtained for H convex with subquadratic growth at the origin and at infinity. Roughly speaking, it is shown that there exist many subharmonics with exact minimal period pT provided S p (the smallest prime factor of p) is sufficiently large. Using a dual variational method the author answers the question whether the number of solutions of (1) with minimal period pT tends towards infinity as p+.

MSC:
34C25Periodic solutions of ODE
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
34C11Qualitative theory of solutions of ODE: growth, boundedness