Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems

INFINITELY MANY HOMOCLINIC ORBITS OF SECOND-ORDER p-LAPLACIAN SYSTEMS

In this paper, we give several new sufficient conditions for the existence of infinitely many homoclinic orbits of the second-order ordinary p-Laplacian system d dt (|u̇(t)|p−2u̇(t)) − a(t)|u(t)|p−2u(t) +∇W (t, u(t)) = 0, where p > 1, t ∈ R, u ∈ R , a ∈ C(R,R) and W ∈ C(R × R ,R) are no periodic in t, which greatly improve the known results due to Rabinowitz and Willem.

Existence of Homoclinic Orbits for Hamiltonian Systems with Superquadratic Potentials

and Applied Analysis 3 We make the following assumptions. A1 W t, z ∈ C1 R × R2N,R is 1-periodic in t. W t, 0 0 for all t ∈ R. There exist constants c1 > 0 and μ > 2 such that Wz t, z z ≥ c1|z| for t, z ∈ R × R2N. A2 there exist c2, r > 0 such that |Wz t, z | ≤ c2|z|μ−1 for t ∈ R and |z| ≤ r. A3 there exist c3, R ≥ r and p ≥ μ such that |Wz t, z | ≤ c3|z|p−1 for t ∈ R and |z| ≥ R. A4 there exis...

Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Existence of Infinitely Many Elliptic Periodic Orbits in Four-Dimensional Symplectic Maps with a Homoclinic Tangency

We study the problem of coexistence of a countable number of periodic orbits of different topological types (saddles, saddle–centers, and elliptic) in the case of four-dimensional symplectic diffeomorphisms with a homoclinic trajectory to a saddle–focus fixed point.

Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential

where n ∈ Z, u ∈ RN , u(n) = u(n + ) – u(n) is the forward difference operator, p,L : Z→ RN×N and W : Z× RN → R. As usual, we say that a solution u(n) of system (.) is homoclinic (to ) if u(n)→  as n→±∞. In addition, if u(n) ≡ , then u(n) is called a nontrivial homoclinic solution. In general, system (.) may be regarded as a discrete analogue of the following second order Hamiltonian sy...