Homoclinic orbits for a class of second order dynamic equations on time scales via variational methods

Advances in Difference Equations

Xingjie Yan¹  Daihong Jiang²  You-Hui Su³  Fenghua Liu³ 

[1] College of Sciences, China University of Mining and Technology, Xuzhou, China;School of Electrical Engineering, Xuzhou University of Technology, Xuzhou, China;School of Mathematics and Physics, Xuzhou University of Technology, Xuzhou, China

关键词: time scales;    variational structure;    homoclinic orbits;    critical point theorem;    34B15;    34C25;    34N05;   

DOI  :  10.1186/s13662-017-1098-1

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scalesT$\mathbb{T}$of the form{(p(t)uΔ(t))Δ+qσ(t)uσ(t)=f(σ(t),uσ(t)),△-a.e. t∈T,u(±∞)=uΔ(±∞)=0.$$ \left \{ \textstyle\begin{array}{l} ( p(t)u^{\Delta}(t) ) ^{\Delta}+q^{\sigma}(t)u^{\sigma}(t)= f(\sigma(t),u^{\sigma}(t)),\quad \triangle\text{-a.e. } t\in\mathbb{T}, \\ u(\pm\infty)=u^{\Delta}(\pm\infty)=0. \end{array}\displaystyle \right . $$We construct a variational framework of the above-mentioned problem, and some new results on the existence of a homoclinic orbit or an unbounded sequence of homoclinic orbits are obtained by using the mountain pass lemma and the symmetric mountain pass lemma, respectively. The interesting thing is that the variational method and the critical point theory are used in this paper. It is notable that in our study any periodicity assumptions onp(t)$p(t)$,q(t)$q(t)$andf(t,u)$f(t,u)$are not required.

【 授权许可】

CC BY