Homoclinic orbits of sub-linear Hamiltonian systems with perturbed terms

Haiyan Lv,Guanwei Chen

doi:10.1186/s13661-021-01510-z

Abstract

By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.

Highlights

1 Introduction and the main result In this paper, we study the existence of homoclinic orbits for the following second order Hamiltonian systems with perturbed terms:
We say that u(t) is a homoclinic orbit of (1.1) if u(t) is a solution of (1.1) and u(t) ∈ C2(R, RN ) such that u(t) → 0 as |t| → ∞
Since homoclinic orbits play a key role in the research of fluid mechanics and gas dynamics

Summary

Introduction

1 Introduction and the main result In this paper, we study the existence of homoclinic orbits for the following second order Hamiltonian systems with perturbed terms: Homoclinic orbits of Hamiltonian systems have been studied by many authors [1–12, 14–21]. If the matrix A(t) is positive definite uniformly in t, the authors [7–9, 18, 19] have obtained the existence of homoclinic orbits for (1.2). Theorem 1.1 (1) If F(0) = 0 and h(t) ≡ 0, (1.1) has at least one homoclinic orbit.

Results

Conclusion