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
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(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)