Subharmonics of nonconvex Hamiltonian systems

Abstract

In this note we prove the existence of subharmonic solutions of the hamiltonian system u JrHtY u where H is a nonconvex unbounded hamiltonian with a bounded gradient.

Subharmonic solutions for nonautonomous sublinear first order Hamiltonian systems

“…In section 4, we will study the minimality of periods of the subharmonic solutions. We will give examples in order to show the originality of our results which improve many previous results among them [1,14,16]. For the proofs, we will apply a Generalized Saddle Point Theorem to the Least Action Integral and use a Generalized Egoroff's Lemma.…”

“…|x|−→∞ H(t, x) = ±∞, unif ormly in t ∈ [0, T ], [16] has shown that the system (H) admitted a sequence of subharmonic solutions. After that, [1] generalized this result to the sublinear case.…”

Subharmonics of Not Uniformly Partially Coercive Hamiltonian Systems

“…or -00 as |x| -• +00 uniformly for almost every t G [0, T], the author proved the existence of such solutions (see [8]). In this work, we replace the uniform coercivity by the local partial coercivity; that is, replacing the above coercivity condition by the following local partial coercivity: there exist a decomposition M. 2N = A © B of and some non empty open subset C of [0, T] …”

“…Then f~l([ (3,7]) contains at least cuplength (V) + 1 critical points of f. E + = lx € E / x(t) -^^ ex P ( a.e.l. m<-1 V / J Then E = E° © E + © E~.…”

“…distinct kT-periodic solutions. Many solvability conditions are given, such as the coercivity condition (see [2,4,5,6,8]), the convexity condition (see [1,7,9]), the boundedness condition (see [6]). However, most of the results proving the existence of subharmonics have made use of a uniformly coercivity or a uniformly resonance assumption in all the variables of the Hamiltonian H.…”