A twist condition and periodic solutions of Hamiltonian systems

Abstract

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian systemUnder a general twist condition for the Hamiltonian function in terms of the difference of the ConleyZehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions. Compared with the existing work in the literature, our results do not require the Hamiltonian function to have linearization at infinity. Our results allow interactions at infinity between the Hamiltonian and the linear spectra. The general twist condition raised here seems to resemble more the spirit of Poincaré's last geometric theorem.

Some abstract critical point theorems for self-adjoint operator equations and applications

“…Up to the authors known, some similar conditions as (1.5) in (P ∞ ) were introduced in [14,18]. [6,Theorem 3.1.7], the functional Φ behaves as a quadratic functional at infinity but in our Theorem 1.2 it is only required that the functional Φ is estimated from below by a quadratic functional.…”

“…We can use the abstract critical point Theorem1.2 and Theorem1.3 to deal with the existence and multiplicity problems of solutions of nonlinear elliptic equations as in [19], the periodic solutions of asymptotically linear Hamiltonian systems as in [18] and the Lagrangian boundary value problems of asymptotically linear Hamiltonian systems as in [14,17]. To avoid tedious, in the following, we only show an application of the abstract critical point Theorem 1.2 and Theorem 1.3 to the problem of nonlinear Hamiltonian systems with P -periodic boundary conditions.…”

Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions

“…In order to proof Theorem1.3 and Theorem1.4, we need the following lemma which is similar to Lemma 3.4 in [22] and Lemma 3.3 in [23]. (1) There exists an increasing sequence of real numbers M k → ∞(k → ∞) such that…”

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

“…In order to prove Theorem 1.4, inspired by [19] and [20], consider a sequence of modified problems −(Jż + L(t)z) = H m (t, z).…”

“…Condition (H ∞ ) is a two-pinching condition near the infinity, learning from the idea of [18,19], we can relax (H ∞ ) to conditions (H ± ∞ ) as follows. (H ± ∞ ) There exist some R 0 > 0 and a continuous symmetric matrix function…”