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

Advances in Difference Equations, Feb 2017

In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales T of the form { ( p ( t ) u Δ ( t ) ) Δ + q σ ( t ) u σ ( t ) = f ( σ ( t ) , u σ ( t ) ) , △ -a.e.  t ∈ T , u ( ± ∞ ) = u Δ ( ± ∞ ) = 0 . 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. MSC: 34B15, 34C25, 34N05.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.advancesindifferenceequations.com/content/pdf/s13662-017-1098-1.pdf

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

Su et al. Advances in Difference Equations Homoclinic orbits for a class of second order dynamic equations on time scales via variational methods You-Hui Su 0 Xingjie Yan 2 Daihong Jiang 1 Fenghua Liu 0 0 School of Mathematics and Physics, Xuzhou University of Technology , Xuzhou, Jiangsu 221111 , China 1 School of Electrical Engineering, Xuzhou University of Technology , Xuzhou, Jiangsu 221111 , China 2 College of Sciences, China University of Mining and Technology , Xuzhou, Jiangsu 221008 , China In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales T of the form time scales; variational structure; homoclinic orbits; critical point theorem - (p(t)u (t)) + qσ (t)uσ (t) = f (σ (t), uσ (t)), u(±∞) = u (±∞) = 0. -a.e. t ∈ T, 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. MSC: 34B15; 34C25; 34N05 1 Introduction In 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 [–]) 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. Although considerable attention has been dedicated to the existence of homoclinic and heteroclinic solutions for continuous or discrete ordinary differential equations, see [– ] and the references therein, to the best of our knowledge, there is little work on homoclinic orbits for differential equations on time scales []. One of interesting and open problems on dynamic equations on time scales is to investigate discrete or continuous differential equations on time scales with one goal being the unified treatment of differential equations (the continuous case) and difference equations (the discrete case). In particular, not much work has been seen on the existence of solutions or homoclinic orbits to © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. (p(t)u (t)) + qσ (t)uσ (t) = f (σ (t), uσ (t)), u(±∞) = u (±∞) = , where p(t) : T → R is nonzero and is -differential, q : T → R is Lebesgue integrable and f : T × R → R is Lebesgue integrable with respect to t for -a.e. t ∈ T. Providing that f (t, x) grows superlinearly both at origin and at infinity or is an odd function with respect to x ∈ R, we explore the existence of a nontrivial homoclinic orbit of the dynamic equation () by means of the mountain pass lemma and the existence of an unbounded sequence of nontrivial homoclinic orbits by using the symmetric mountain pass lemma. 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. We say that a property holds for -a.e. t ∈ A ⊂ T or -a.e. on A ⊂ T whenever there exists a set E ⊂ A with the null Lebesgue -measure such that this property holds for every t ∈ A \ E. Definition  We say that a solution u of equation () is homoclinic to zero if it satisfies u(t) →  as t → ±∞, where t ∈ T. In addition, if u = , then u is called a nontrivial homoclinic solution. Throughout this paper, we make the following assumptions: (H) limx→ f (tx,x) =  uniformly for -a.e. t ∈ T; (H) there exists a constant β >  such that for -a.e. t ∈ T and |x| ≥ . It follows from () and () that |xl|i→m∞ f (tx, x) = –∞ T uniformly for -a.e. t ∈ . Hence, we have the following remark. Remark  () u(t) ≡  is a trivial homoclinic solution of equation (). () f (t, x) grows superlinearly both at infinity and at origin. The paper is structured as follows. In Section , we introduce two technical lemmas which will be used in the proofs of our main results. In Section , the variational structure of the dynamic equation () is presented. In Section , we summarize our main results on the existence homoclinic solution of the dynamic equation () on time scales and present two examples. We demonstrate the proofs in Section . (...truncated)


This is a preview of a remote PDF: http://www.advancesindifferenceequations.com/content/pdf/s13662-017-1098-1.pdf

You-Hui Su, Xingjie Yan, Daihong Jiang, Fenghua Liu. Homoclinic orbits for a class of second order dynamic equations on time scales via variational methods, Advances in Difference Equations, 2017, pp. 47, 2017, DOI: 10.1186/s13662-017-1098-1