Two nontrivial homoclinic orbits for a class of nonperiodic perturbed Hamiltonian systems

Abstract

In this paper, we prove the existence and  multiplicity of nontrivial homoclinic orbits for a class of second order nonperiodic  perturbed Hamiltonian systems:
$$\ddot{x}(t)-\lambda K^{'}(t,x(t))+W^{'}(t,x(t))=f(t),  $$
where $\lambda \geq 1$ is a parameter, $ t\in \mathbb{R},K, W \in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$ and $f\in L^{2}(\mathbb{R},\mathbb{R}^{N})\backslash\left\{0\right\}$ small enough. The proof is based on Ekeland's Variational Principle and the Mountain Pass Theorem. Some  examples are also given to illustrate our main theoretical results.