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

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• 作者：You-Hui Su ; Xingjie Yan ; Daihong Jiang ; Fenghua Liu
• 关键词：time scales ; variational structure ; homoclinic orbits ; critical point theorem
• 刊名：Advances in Difference Equations
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1630KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847
• 卷排序：2017

文摘

In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales \(\mathbb{T}\) of the form $$ \left \{ \textstyle\begin{array}{l} ( p(t)u^{\Delta}(t) ) ^{\Delta}+q^{\sigma}(t)u^{\sigma}(t)= f(\sigma(t),u^{\sigma}(t)),\quad \triangle\text{-a.e. } t\in\mathbb{T}, \\ u(\pm\infty)=u^{\Delta}(\pm\infty)=0. \end{array}\displaystyle \right . $$ 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 on \(p(t)\), \(q(t)\) and \(f(t,u)\) are not required.Keywordstime scalesvariational structurehomoclinic orbitscritical point theoremMSC34B1534C2534N051 IntroductionIn the past decades, there has been an increasing interest in the study of dynamic equations on time scales, employing and developing a variety of methods (such as the variational method, the fixed point theory, the method of upper and lower solutions, the coincidence degree theory, and the topological degree arguments [1–13]) motivated, at least in part, by the fact that the existence of homoclinic and heteroclinic solutions is of utmost importance in the study of ordinary differential equations.