R E S E A R C H
Open Access
Homoclinic orbits for a class of second
order dynamic equations on time scales via
variational methods
You-Hui Su
1*, Xingjie Yan
2, Daihong Jiang
3and Fenghua Liu
1*Correspondence: [email protected] 1School of Mathematics and
Physics, Xuzhou University of Technology, Xuzhou, Jiangsu 221111, China
Full list of author information is available at the end of the article
Abstract
In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scalesTof the form
(p(t)u(t))+qσ(t)uσ(t) =f(
σ
(t),uσ(t)), -a.e.t∈T, u(±∞) =u(±∞) = 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 onp(t),q(t) andf(t,u) are not required.
MSC: 34B15; 34C25; 34N05
Keywords: time scales; variational structure; homoclinic orbits; critical point theorem
1 Introduction
In the past decades, there has been an increasing interest in the study of dynamic equa-tions on time scales, employing and developing a variety of methods (such as the varia-tional method, the fixed point theory, the method of upper and lower solutions, the coin-cidence 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 ho-moclinic 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 dif-ferential equations on time scales with one goal being the unified treatment of differential equations (the continuous case) and difference equations (the discrete case). In particu-lar, not much work has been seen on the existence of solutions or homoclinic orbits to
dynamic equations on time scales through the variational method and the critical point theory [–].
In this paper, we consider the existence of nontrivial homoclinic orbits to zero of equa-tion on time scalesTof the form
(p(t)u(t))+qσ(t)uσ(t) =f(σ(t),uσ(t)), -a.e.t∈T,
u(±∞) =u(±∞) = , ()
where p(t) :T→Ris nonzero and is-differential, q:T→Ris Lebesgue integrable andf :T×R→Ris Lebesgue integrable with respect totfor-a.e.t∈T. Providing thatf(t,x) grows superlinearly both at origin and at infinity or is an odd function with respect tox∈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 onp(t),q(t) and f(t,u) are not required.
We say that a property holds for-a.e.t∈A⊂Tor-a.e. onA⊂Twhenever there exists a setE⊂Awith the null Lebesgue-measure such that this property holds for everyt∈A\E.
Definition We say that a solutionuof equation () is homoclinic to zero if it satisfies u(t)→ ast→ ±∞, wheret∈T. In addition, ifu= , thenuis called a nontrivial ho-moclinic solution.
Throughout this paper, we make the following assumptions:
(H) limx→f(tx,x)= uniformly for-a.e.t∈T;
(H) there exists a constantβ> such that
xf(t,x)≤β x
f(t,s)ds< for-a.e.t∈Tand for allx∈R\ {}; ()
(H) p(t) > for-a.e.t∈Tand
(–∞,∞)Tp
(t)t< +∞;
(H) qσ(t) < for-a.e.t∈T,lim|t|→∞qσ(t) = –∞and(–∞,∞)T|q
σ(t)|t< +∞.
LetF(t,x) =xf(t,s)ds, it follows from () that
dF F ≥
β
xdx for|x| ≥,
which implies that there is a real functionα(t) > such that
x
f(t,s)ds≤–α(t)|x|β for-a.e.t∈Tand|x| ≥. ()
It follows from () and () that
lim
|x|→∞ f(t,x)
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 .
2 Preliminaries
In this section, we present two lemmas which can help us to better understand our main results and proofs. For the basic terminologies such as measure, absolute continuity, the Lebesgue integral and Sobolev’s spaces on time scales, we refer the reader to references [–].
Let us recall the mountain pass theorem [] and the symmetric mountain pass theorem [], respectively.
Lemma ([]) Let X be a real Banach space andϕ:X→Rbe a C-smooth functional.
Suppose thatϕsatisfies the following conditions: (i) ϕ() = ;
(ii) every sequence{uj}j∈NinXsuch that{ϕ(uj)}j∈Nis bounded inRandϕ(uj)→in
X∗asj→+∞contains a convergent subsequence asj→+∞(the PS condition); (iii) there exist constantsandα> such thatϕ|∂B()≥α;
(iv) there existse∈X\ ¯B()such thatϕ(e)≤,whereB()is an open ball inXof radiuscentered at.
Thenϕpossesses a critical value c≥αgiven by
c=inf
g∈ s∈max[,]ϕ
g(s),
where
=g∈C[, ],E:g() = ,g() =e .
Lemma ([]) Let X be a real Banach space andϕ:X→Rbe a C-smooth functional.
Suppose thatϕsatisfies the following conditions: (i) ϕ() = ;
(ii) ϕsatisfies the PS condition;
(iii) there exist constantsandα> such thatϕ|∂B()≥α;
(iv) for each finite-dimensional subspaceE⊂E,there isγ =γ(E)such thatϕ≤on
E\βγ.
3 Variational framework
In this section, we state some basic notations, some lemmas which are closely related to our main results, and construct a variational framework of our problem.
Forp∈Randp≥, we let the space
be equipped with the norm
fLp
uis absolutely continuous and bounded measurable functional,
It is a Hilbert space with the norm defined by
u=uH,
ThenEis a Hilbert space with the norm defined by
uE=
(–∞,∞)T
p(t)u–qσ(t)uσt foru∈E,
and the inner product is
u,v=
(–∞,∞)T
Let
L∞
(–∞, +∞)T,R=
u: (–∞, +∞)T→Ruis bounded measurable
function a.e. on (–∞, +∞)T
,
andL∞((–∞, +∞)T,R) is called the essentially bounded space on time scales, which is
equipped with the norm
uL∞ :=ess supu(t):t∈(–∞, +∞)T = inf
μ(E)=,E⊂E
sup
t∈(–∞,+∞)T\E
u(t),
whereu(t) is bounded on (–∞, +∞)T\E, andEis a set of measure zero in the space
(–∞, +∞)T.
Now, we list three technical lemmas which will be used in the proofs of our main results in the next section.
We have the following lemma.
Lemma There exist positive constants C∗and L such that the following inequality holds:
uL∞ ≤C∗u. ()
Moreover,there exist,a∈(–∞,∞)Tare real such that(,a)Tu(t)t= ,then
uL∞ ≤LuL
, ()
where t∈(–∞, +∞)T,holds.
Proof Going to the components ofu(t), we can assume thatn= , and there exist ,a∈ (, +∞)Tare real. Ifu(t)∈H,, then there existsτ∈[,a]Tsuch thatu(τ) =inft∈[,a]Tu(t), it follows that
a
(,a)T
u(t)t≥ a
(,a)T
u(τ)t=u(τ).
Thus, there exists constant c > such that |u(τ)| ≤c|
(,a)Tu(t)t|. Hence, for t∈
(–∞,∞)T, one can get
u(t)=u(τ) +
(τ,t)T
u(t)t≤u(τ)+
(τ,t)T
u(t)t
≤c
(,a)
T
u(t)t+|t–τ|
(τ,t)T
u(t)t
≤ca
(–∞,∞)T
u(t)t
+|t–τ|
(–∞,∞)T
u(t)t
then
uL∞ = inf
μ(E)=,E⊂E
sup
t∈(–∞,+∞)T\E
u(t)
≤maxca , inf
μ(E)=,E⊂E
sup
t∈(–∞,+∞)T\E
|t–τ|
×
(–∞,∞)T
u(s)t
+
(–∞,∞)T
u(s)t
≤C∗u.
If(,a)Tu(t)t= , then
u(t)=u(τ) +
(τ,t)T
u(t)t≤u(τ)+
(τ,t)T
u(t)t
≤c
(,a)T
u(t)t+|t–τ|
(τ,t)T u
(t)t
,
which implies () holds.
Lemma Assume that the sequence{un} ⊂E such that unu in E,then the sequence un satisfies un→u in L((–∞,∞)T,R).
Proof Without loss of generality, assume thatun inEfor anyε> . It follows from (H) that there exists negativeT∈Tsuch that
–
qσ(t)≤ε for-a.e.t∈(–∞,T)T. ()
Similarly, we also have there exists positiveT∈Tsuch that
–
qσ(t)≤ε for-a.e.t∈(T,∞)T. ()
From (H) and (H), we haveunuinEI, where
EI=
u∈H,
(T,T)T
p(t)u(t)–qσ(t)uσ(t)t< +∞
.
Hence,{un}is bounded inEI, which implies that{un}is bounded inL((T,T)T,R). Due
to the uniqueness of the weak limit inL
((T,T)T,R), one obtainsun→ on (T,T)T,
then there isnsuch that
(T,T)T
un(t)t≤ε for alln≥n ()
since
sup
n
(–∞,∞)T
Let
According to (), we have
Sinceεis arbitrary, combining (), () and (), one has
un→u inL
(–∞,∞)T,R.
In the following, we define and prove the variational framework of the dynamic equa-tion ().
Define the functionalE→Rby
ϕ(u) =
Lemma The functionalϕis continuously differentiable on E,and
Proof Let us first consider the existence of the Gâteaux derivative. For anyv∈Eandε∈R( <|ε|< ), we have
ε
ϕ(u+εv) –ϕ(u)
=
(–∞,∞)T
ε
p(t)εuv+p(t)εv– εqσ(t)uσ(t)vσ(t) +εqσ(t)vσ(t)
+
(–∞,∞)T
F(σ(t),uσ+εvσ) –F(σ(t),uσ)
ε t.
Givenu∈R, the mean value theorem indicates that there existsλ∈(, ) such that
|ε|F
σ(t),uσ+εvσ–Fσ(t),uσ
=
|ε| ∂F
∂ξ
(σ(t),uσ+λ
εvσ)
εvσ=fσ(t),uσ+λεvσvσ.
Note that f
σ(t),uσ+λεvσvσ∈L
(–∞,∞)T,R.
It follows from Lebesgue’s dominated convergence theorem on time scales that
ϕ(u)v=lim
ε→
ε
ϕ(u+εv) –ϕ(u)
=
(–∞,∞)T
p(t)uv–qσ(t)uσvσt+
(–∞,∞)T
fσ(t),uσvσt.
Next, we show the continuity of the Gâteaux derivative.
Assume that the sequence{un} ⊂Esatisfiesun→uasn→ ∞inE. Using Lebesgue’s dominated convergence theorem on time scales and (H) yields
(–∞,∞)T
fσ(t),uσn–fσ(t),uσt→ asn→ ∞. ()
It follows from Theorem . in [] thatE→L((–∞,∞)T,R) is compact, thenun→u asn→ ∞inL((–∞,∞)T,R). For arbitraryv∈E, there holds
ϕ(un)v–ϕ(u)v
=
(–∞,∞)T
p(t)un–uvt
–
(–∞,∞)T
qσ(t)uσn–uσvσt+
(–∞,∞)T
fσ(t),uσn–fσ(t),uσvσt.
Hölder’s inequality on time scales and Lemma reduce to
ϕ(un)v–ϕ(u)v
≤
(–∞,∞)T
p(t)un –uvt+
(–∞,∞)T
+
So, finding the homoclinic solutions to the zero of dynamic equation () is equivalent to finding the critical points of the associated functionalϕdefined in ().
4 Main results
Theorem If conditions(H), (H), (H)and(H)are satisfied,then the dynamic equation
()has one nontrivial homoclinic orbit tosuch that
<
(–∞,∞)T
p(t)u(t)–qσ(t)uσ(t)+Fσ(t),uσt< +∞.
Example Let
T={, , , , , , , , , } ∪[., +∞)∪(–∞, –.).
Consider the following second order boundary value problem on time scalesTof the form
(tu(t))– (tσ)uσ= – σ(t)(u
σ(t)), -a.e.t∈T,
u(±∞) =u(±∞) = . ()
Sincexf(t,s)ds= –tx, one can check that all conditions of Theorem are fulfilled. It
follows from Theorem that the dynamic equation () has one nontrivial homoclinic orbit to .
Theorem If conditions(H), (H), (H), (H)and the following condition are satisfied
(H) f(t, –x) = –f(t,x)for allx∈Rand-a.e.t∈T,
then the dynamic equation()has an unbounded sequence in E of a homoclinic orbit to.
Example Leta,b> be real numbers,
P=
∞
k=
k(a+b),k(a+b) +a,
and
P=
∞
k=
–k(a+b) –a, –k(a+b).
Consider the following second order boundary value problem on time scalesP∪Pof
the form
(tu(t))–|tσ|uσ = – σ(t)(u
σ(t)), -a.e.t∈P ∪P,
u(±∞) =u(±∞) = . ()
Sincexf(t,s)ds= –tx, one can check that all conditions of Theorem are fulfilled. It follows from Theorem that the dynamic equation () has an unbounded sequence inE of a homoclinic orbit to .
5 Proof of theorems
Proof of Theorem Since we have already known thatϕ∈C(E,R) andϕ() = , in the
following we prove that all the other conditions of Lemma are fulfilled with respect to the functionalϕ.
Firstly, we claim thatϕsatisfies the PS condition.
Assume that there exist a sequence{un} ⊂Eand a constantcsuch that
ϕ(un)→ asn→ ∞ and ϕ(un)≤c, n= , , . . . , ()
we show that{un}has a convergent subsequence inE.
It follows from () and (H) that there is a constantd≥ such that
d+unE≥ϕ(un) – βϕ
(u n)un
=
–
β
uE+
(–∞,∞)T
Fσ(t),uσ–fσ(t),uσvσt
≥
–
β
uE,
which implies that{un}is bounded inE. Hence, there is a subsequence (still denoted by
{un},unu inE). It follows from Lemma that un→uinL((–∞,∞)T,R). Now,
according to (H),un,u∈E, for anyε> , we have that there exist constantsδ> ,δ>
andL∈Tsuch that
|un|<δ, |u|<δ and un–uL
<ε for-a.e.|t|>L, ()
which implies that
fσ(t),uσ
n≤εuσn and f
σ(t),uσ
≤εuσ for-a.e.|t|>L. ()
Since
(–∞,∞)T
fσ(t),uσn–fσ(t),uσ(un–u)t
=
[–L,L]T
fσ(t),uσ
n
–fσ(t),uσ
(un–u)t
+
(–∞,–L)T
fσ(t),uσn–fσ(t),uσ(un–u)t
+
(L,∞)T
fσ(t),uσn–fσ(t),uσ(un–u)t, ()
let
L,loc(T,R) = :T→R|for arbitrary compact intervalK⊂T,IK∈L (T,R) ,
whereIKis an indicator function of intervalKand
IK=
It follows from the uniform continuity off(t,x) inxandun→uinL,loc(T,Rn) that
[–L,L]T
fσ(t),unσ–fσ(t),uσ(un–u)t→ asn→ ∞.
Combining Hölder’s inequality on time scales, () and () leads to
(–∞,–L) T
fσ(t),uσn–fσ(t),uσ(un–u)t
≤
(–∞,–L)T
fσ(t),uσn–fσ(t),uσt
(–∞,–L)T
(un–u)t
≤ε
(–∞,–L)T uσ
n+u
σ
t
≤εM.
By using the same technique, we obtain
(L,∞) T
fσ(t),uσn–fσ(t),uσ(un–u)t
≤εM,
whereM,Mdepend on the bounds forunanduinE. Then
(–∞,∞)T
fσ(t),unσ–fσ(t),uσ(un–u)t→ asn→ ∞ ()
since
ϕ(un) –ϕ(u)
(uk–u)
=un–uE–
(–∞,∞)T
fσ(t),uσn–fσ(t),uσ(un–u)t. ()
Equations () and () imply thatun→uinE. Consequently,ϕsatisfies the PS
condi-tion.
Secondly, we prove that there exist constantsandα> such thatϕsatisfies the as-sumption (iii) of Lemma .
It follows from Lemma that there existsα> such that
uL
≤αuE foru∈E.
On the other hand, according to (H) and (H), we have that there existsα> such that
u∞≤αuE,
where
u∞= max t∈(–∞,∞)T
(H) implies that there isδ> such that
F(t,x)≤ε|x| for|x| ≤δ.
Letρ= δ
α anduE≤ρ, we haveu∞≤ δ
αα=δ, then
Ft,uσ≤εuσ foruσ≤δand-a.e.t∈T,
which implies that
(–∞,∞)T
Ft,uσt≥–εuL
≥–εα
uE.
Hence, ifuE=ρ, we have
ϕ(u) = u
E+
(–∞,∞)T
Fσ(t),uσt
≥
u
E–εαuE=
–εα
ρ.
Choosingε=α
, we have
ϕ(u)≥ ρ
=α> .
Thirdly, we claim that there existse∈X\ ¯Bρ() such thatϕsatisfies the assumption (iv)
of Lemma .
Letu∈Ebe such that|u(t)| ≥, for anyσ≥, it follows from () that
ϕ(σu) = σ
u
E+
(–∞,∞)T
Fσ(t),σuσt
≤ σ
u
E–
(–∞,∞)T
σuσβα(t)t
= σ
u
E–|σ|
β
(–∞,∞)T
uσβα(t)t,
which implies that there existsσ≥ such thatσu>ρandϕ(σu)≤ =ϕ().
Hence, all the conditions of Lemma are satisfied, the desired results follow.
Proof of Theorem It follows from (H) thatϕis even. In addition, we have already proved
thatϕ∈C(E,T),ϕ() = andϕsatisfies the Palais-Smale condition. We prove that all the
other conditions of the symmetric mountain pass theorem are satisfied with respect to the functionalϕ. We have already showed thatϕsatisfies condition (iii) of the symmetric mountain pass theorem in the proof of Theorem .
LetE⊂Ebe a finite-dimensional subspace. Consideru∈E⊂Ewithu= . It follows
(H) implies that
cj= –
(–∞,∞)T
fσ(t),uσjujt+
(–∞,∞)T
Fσ(t),uσjt
≤–
(–∞,∞)T
fσ(t),uσjujt= uj
E.
Then{uj}is unbounded inEbecause ofcj→ ∞asj→ ∞. The proof is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1School of Mathematics and Physics, Xuzhou University of Technology, Xuzhou, Jiangsu 221111, China.2College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China.3School of Electrical Engineering, Xuzhou University of Technology, Xuzhou, Jiangsu 221111, China.
Acknowledgements
This work is partially supported by the Natural Science Foundation of China (Nos. 11361047, 11501560), the Natural Science Foundation of JiangSu Province (No. BK20151160), the Six Talent Peaks Project of Jiangsu Province (2013-JY-003) and 333 High-Level Talents Training Program of Jiangsu Province (BRA2016275).
Received: 4 January 2017 Accepted: 30 January 2017
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