Two Almost Homoclinic Solutions for a Class of Perturbed Hamiltonian Systems Without Coercive Conditions - Zhang, Ziheng; Yuan, Rong (Mathematical Modelling and Analysis, 2015)
In this paper we consider the existence of almost homoclinic solutions for the following second order perturbed Hamiltonian systems ü- L(t)u + ∇W (t, u) = f (t), (PHS) where is a symmetric and positive definite matrix for all t ∈ R, W ∈ C1(R×Rn, R) and ∇W (t, u) is the gradient of W (t, u) at u, f ∈ C(R, Rn) and belongs to L2(R, Rn). The novelty of this paper is that, assuming L(t) is bounded in the sense that there are two constants 0 < τ1 < τ2 < ∞ such that τ1 ∣u∣2 ≤ (L(t)u, u) ≤ τ2 ∣u∣2 for all (t, u) ∈ R × Rn, W(t, u) satisfies Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, f (t) is sufficiently small in L2(R, Rn), we obtain some new criterion to guarantee that (PHS) has at least two nontrivial almost homoclinic solutions. Recent results in the literature are generalized and significantly improved.