We study the existence of homoclinic orbits for the second order Hamiltonian systemqis T-periodic in t. A map K satisfies the "pinching" condition b 1 |q| 2 K(t, q) b 2 |q| 2 , W is superlinear at the infinity and f is sufficiently small in L 2 (R, R n ). A homoclinic orbit is obtained as a limit of 2kT -periodic solutions of a certain sequence of the second order differential equations.

We shall be concerned with the existence of homoclinic solutions for the second order Hamiltonian systemq − V q (t, q) = f (t), where t ∈ R and q ∈ R n . A potential V ∈ C 1 (R × R n , R) is T -periodic in t, coercive in q and the integral of V (·, 0) over [0, T ] is equal to 0. A function f : R → R n is continuous, bounded, square integrable and f = 0. We will show that there exists a solution q 0 such that q 0 (t) → 0 anḋ q 0 (t) → 0, as t → ±∞. Although q ≡ 0 is not a solution of our system, we are to call q 0 a homoclinic solution. It is obtained as a limit of 2kT -periodic orbits of a sequence of the second order differential equations.

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system q + V q (t, q) = 0, where q ∈ R n and V ∈ C 1 (R × R n , R), V 0. We will assume that V and a certain subset M ⊂ R n satisfy the following conditions. M is a set of isolated points and #M 2. For every sufficiently small ε > 0 there exists δ > 0 such that for all (t, z) ∈ R×R n , if d(z, M) ε then −V (t, z) δ. The integrals ∞ −∞ −V (t, z) dt, z ∈ M, are equi-bounded and −V (t, z) → ∞, as |t| → ∞, uniformly on compact subsets of R n \ M. Our result states that each point in M is joined to another point in M by a solution of our system.

The paper is devoted to the study of bifurcation in the von Kármán equations with two parameters α, β ∈ R+ that describe the behaviour of a thin round elastic plate lying on an elastic base under the action of a compressing force. The problem appears in the mechanics of elastic constructions. We prove the necessary and sufficient condition for bifurcation at points of the set of trivial solutions. Our proof is based on reducing the von Kármán equations to an operator equation in Banach spaces with a nonlinear Fredholm map of index 0 and applying the Crandall-Rabinowitz theorem on simple bifurcation points or a finite-dimensional reduction and degree theory.2000 Mathematics Subject Classification: 35Q72, 46T99.

We prove the existence of two almost homoclinic solutions for a class of second-order perturbed Hamiltonian systems [Formula: see text], where L : ℝ → ℝn2is a continuous matrix-valued function such that L(t) is symmetric and positive definite for each t ∈ ℝ, the smallest eigenvalue of L(t) → ∞ as |t| → ∞, W : ℝ × ℝn→ ℝ is a C1-map satisfying the Ambrosetti–Rabinowitz condition and f : ℝ → ℝnis continuous and sufficiently small in L2(ℝ, ℝn). We improve a result by Rabinowitz and Tanaka [Math. Z. 206 (1991) 473–499].