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Two Almost Homoclinic Solutions for a Class of Perturbed Hamiltonian Systems Without Coercive Conditions

    Ziheng Zhang Affiliation
    ; Rong Yuan Affiliation

Abstract

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.

Keyword : homoclinic solutions, critical point, variational methods, mountain pass theorem

How to Cite
Zhang, Z., & Yuan, R. (2015). Two Almost Homoclinic Solutions for a Class of Perturbed Hamiltonian Systems Without Coercive Conditions. Mathematical Modelling and Analysis, 20(1), 112-123. https://doi.org/10.3846/13926292.2015.1003619
Published in Issue
Feb 3, 2015
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This work is licensed under a Creative Commons Attribution 4.0 International License.