Homoclinic solutions for Hamiltonian system with impulsive effects

Jian Liu,Mingyong Lai,Fei Xu,Lizhao Yan

doi:10.1186/s13662-018-1774-9

Abstract

In this article, we investigate a class of impulsive Hamiltonian systems with a p-Laplacian operator. By establishing a series of new sufficient conditions, the existence of homoclinic solutions to such type of systems is revealed. We show the existence of homoclinic orbit induced by impulses by introducing some conditions. To illustrate the applications of the main results in this article, we create an example.

Highlights

The aim of the article is to investigate the impulsive Hamiltonian systems with p-Laplacian operator of the form p u (t) – ∇F t, u(t) = 0, t = ti, t ∈ R, (1)– p u = gi u(ti), i ∈ Z. (2)Here, we are interested in the existence of homoclinic orbits for such systems
The existence and multiplicity of homoclinic orbits attracted the attention of researchers from all over the world and as such have been extensively investigated in the literature [1,2,3,4,5,6,7,8]
Mathematical techniques such as the dual variational method [9], concentration compactness method and Ekeland variational principle [10, 11], and the approximation method [12] have been used in evaluating the existence of homoclinic orbits for Hamiltonian systems

Summary

Introduction

The existence and multiplicity of homoclinic orbits attracted the attention of researchers from all over the world and as such have been extensively investigated in the literature [1,2,3,4,5,6,7,8]. M, gj is continuous and m-periodic in j, and F and gj satisfy the following conditions:. (H1) F : R × RN → R is continuously differentiable and T -periodic, and there exist positive constants r1, r2 > 0 such that r1|u|2 ≤ F(t, u) ≤ r2|u|2, ∀(t, u) ∈ [0, T] × Rn;. For the general case of p = 2, due to the complex structure of problem (1) and (2), it is challenging to construct an appropriate functional such that the existence of its critical point implies a homoclinic orbit of the system. Theoerem 2 Assume that F, gi satisfies the following conditions:.

Ek δ C

By b

It is easy to see that