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

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales ...... of the form ...... 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 ......, ...... and ...... are not required.

Su et al. Advances in Dierence Equations (2017) 2017:47 DOI 10.1186/s13662-017-1098-1

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

http://crossmark.crossref.org/dialog/?doi=10.1186/s13662-017-1098-1&domain=pdf

Web End = You-Hui Su1*, Xingjie Yan2, Daihong Jiang3 and Fenghua Liu1

*Correspondence: mailto:[email protected]

Web End [email protected]

1School of Mathematics and Physics, Xuzhou University of Technology, Xuzhou, Jiangsu 221111, ChinaFull list of author information is available at the end of the article

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 xed 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 dierential equations.

Although considerable attention has been dedicated to the existence of homoclinic and heteroclinic solutions for continuous or discrete ordinary dierential equations, see [ ] and the references therein, to the best of our knowledge, there is little work on homoclinic orbits for dierential 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 unied treatment of dierential equations (the continuous case) and dierence 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.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 2 of 16

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 equation on time scales T of the form

(p(t)u (t)) + q (t)u (t) = f ((t), u (t)), -a.e. t

T,

u() = u () = ,

()

where p(t) : T

R is nonzero and is -dierential, q : T

R is Lebesgue integrable

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 innity or is an odd function with respect to x

and f : T

R

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.

Denition We say that a solution u of equation () is homoclinic to zero if it satises u(t) as t , where t

T. In addition, if u = , then u is called a nontrivial ho-

moclinic solution.

Throughout this paper, we make the following assumptions:

(H) limx f(t,x)x = uniformly for -a.e. t

T;

(H) there exists a constant > such that

xf (t, x)

x

f (t, s) ds < for -a.e. t

T and for all x

R

\ {}; ()

T and

(,)T p(t) t < +; (H) q (t) < for -a.e. t

T, lim|t| q (t) = and

(,)T |q (t)| t < +.

(H) p(t) > for -a.e. t

Let F(t, x) =

x f (t, s) ds, it follows from () that

dF

F

x

f (t, s) ds

(t)|x| for -a.e. t

T and |x| . ()

It follows from () and () that

lim

|x|

f (t, x)x = uniformly for -a.e. t

T. ()

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 3 of 16

Hence, we have the following remark.

Remark

() u(t) is a trivial homoclinic solution of equation (). () f (t, x) grows superlinearly both at innity 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 Sobolevs 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 and : X

Suppose that satises the following conditions:

(i) () = ;(ii) every sequence {uj}jN in X such that {(uj)}jN is bounded in

R and (uj) in

X as j + contains a convergent subsequence as j + (the PS condition);

(iii) there exist constants and > such that |B () ;(iv) there exists e X \ B () such that (e) , where B () is an open ball in X of radius centered at .

Then possesses a critical value c given by

c = inf

g

=

Suppose that satises the following conditions:

(i) () = ;(ii) satises the PS condition;(iii) there exist constants and > such that |B () ;(iv) for each nite-dimensional subspace

E E, there is = (

E \ .

Then possesses an unbounded sequence of critical values.

R be a C-smooth functional.

max

s[,]

g(s)

,

where

g C [,], E : g() = , g() = e

.

Lemma ([]) Let X be a real Banach space and : X

R be a C-smooth functional.

E) such that on

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 4 of 16

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.

For p

R and p , we let the space

Lp (, )T, R

= f : (, )T

R :

(,)T f

(t)

p t

< +

be equipped with the norm

f L

p =

(,)T f

p

.

(s)

p s

Then Lp ((, )T,

R) is a Banach space together with the inner product given by

f , g L

p = (,)T

f (t)g(t) t,

where (f , g) Lp ((, )T,

R) Lp ((, )T,

R).

Let

H, = H,

(, )T, R

= u : (, )T

u is absolutely continuous and bounded measurable functional, u L ((, )T,

R)

R

.

It is a Hilbert space with the norm dened by

u = u H

,

= (,)T |

u| t + (,)T u t

for u H, .Dene

E =

u H,

(,)T [p(t)(u ) q (t)(u )] t < +, and there exist , a (, )T are real such that

(,a)T u(t) t =

.

Then E is a Hilbert space with the norm dened by

u E = (,)T p(t) u

q (t)

u

t for u E,

and the inner product is

u, v = (,)T

p(t)u v q (t)

u

v

t for any u, v E.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 5 of 16

Let

L

(, +)T, R

= u : (, +)T

R

u is bounded measurable function a.e. on (, +)T

,

and L ((, +)T,

R) is called the essentially bounded space on time scales, which is equipped with the norm

u L

:= ess sup

u(t)

: t (, +)T

= inf

(E)=,EE

sup

t(,+)T\E

u(t) ,

where u(t) is bounded on (, +)T \ E, and E is 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:

u L

C u . ()

Moreover, there exist , a (, )T are real such that

(,a)T u(t) t = , then

u L

L u L

, ()

where t (, +)T, holds.

Proof Going to the components of u(t), we can assume that n = , and there exist , a (, +)T are real. If u(t) H, , then there exists [, a]T such that u() = inft[,a]T u(t), it follows that

a

(,a)T u(t)

t

a

(,a)T u(

) t = u().

Thus, there exists constant c > such that |u()| c|

(,a)T u(t) t|. Hence, for t

(, )T, one can get

u(t)

=

u(

) + (,t)T u (t)

t

u(

) + (,t)T u (t)

t

c

(,a)T u(t)

t

+ |t |

(,t)T u (t) t

ca

(,)T u(t) t

+ |t |

(,)T u (t) t

,

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 6 of 16

then

u L

= inf

(E)=,EE

sup

t(,+)T\E

u(t)

max ca

, inf

(E)=,EE

sup

t(,+)T\E |

t |

(,)T u(s) t

+

(,)T u (s) t

C u .

If

(,a)T u(t) t = , then

u(t)

=

u(

) + (,t)T u (t)

t

u(

) + (,t)T u (t)

t

c

(,a)T u(t)

t

+ |t |

(,t)T u (t) t

,

which implies () holds.

Lemma Assume that the sequence {un} E such that un u in E, then the sequence un satises un u in L ((, )T,

R).

Proof Without loss of generality, assume that un in E for any > . It follows from (H) that there exists negative T

T such that

for -a.e. t (, T)T. ()

Similarly, we also have there exists positive T

T such that

q (t)

for -a.e. t (T, )T. ()

From (H) and (H), we have un u in EI, where

EI =

u H,

q (t)

t < + .

Hence, {un} is bounded in EI, which implies that {un} is bounded in L ((T, T)T,

(T,T)T p(t) u (t)

q (t)

u (t)

R). Due

to the uniqueness of the weak limit in L ((T, T)T, R), one obtains un on (T, T)T, then there is n such that

(T,T)T un(t) t

for all n n ()

since

sup

n

(,)T

p(t)

u n(t)

q (t)

un(t)

t < +.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 7 of 16

Let

A = max

(,T)T

q (t)

u

n(t)

t,

(,T)T

q (t)

u

n (t)

t

,

then < A < +.According to (), we have

(,T)T u

n(t)

t

max (,T)T

q (t)

u

n(t)

t,

(,T)T

q (t)

u

n (t)

t

A. ()

Let

A = max

(T,)T

q (t)

u

n(t)

t,

(T,)T

q (t)

u

n (t)

t

,

then < A < +.In view of (), we have

(T,)T u

n(t)

t

max (T,)T

q (t)

u

n(t)

t,

(T,)T

q (t)

u

n (t)

t

A. ()

Since is arbitrary, combining (), () and (), one has

un u in L (,

)T

.

In the following, we dene and prove the variational framework of the dynamic equation ().

Dene the functional E

R by

, R

(u) =

(,)T p(t) u (t)

q (t)

u (t)

t +

(,)T

F

(t), u (t)

t

=

u E + (,)T

F

(t), u (t)

t, ()

where F(t, ) =

f (t, s) ds.

Lemma The functional is continuously dierentiable on E, and

(u)v =

(,)T p(t)u v

q (t)u v

t +

(,)T

f

(t), u

v t for u, v E.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 8 of 16

Proof Let us rst consider the existence of the Gteaux derivative.For any v E and

R ( < || < ), we have

(u + v) (u)

=

(,)T

p(t)u v + p(t)

v

q (t)u (t)v (t) + q (t)

v (t)

+

(,)T

F((t), u + v ) F((t), u ) t.

Given u

R, the mean value theorem indicates that there exists (, ) such that

|| F

(t), u + v

F

(t), u

=

||

F

v

=

f

(t), u + v

v .

((t),u +v )

Note that

f

(t), u + v

v

L (,

)T

, R

.

It follows from Lebesgues dominated convergence theorem on time scales that

(u)v = lim

(u + v) (u)

= (,)T p(t)u v

q (t)u v

t +

(,)T

f

(t), u

v t.

Next, we show the continuity of the Gteaux derivative.

Assume that the sequence {un} E satises un u as n in E. Using Lebesgues dominated convergence theorem on time scales and (H) yields

(,)T f

(t), un

as n . ()

It follows from Theorem . in [] that E L ((, )T,

R) is compact, then un u

as n in L ((, )T,

f

(t), u

t

R). For arbitrary v E, there holds

(un)v (u)v

=

(,)T

p(t)

u n u

v t

(,)T

q (t)

un u

v t +

(,)T f

(t), un

f

(t), u v t.

Hlders inequality on time scales and Lemma reduce to

(u

n)v (u)v

(,)T p(t)

u n u

v t

+ (,)T q

(t)

un u

v t

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 9 of 16

+ (,)T f

(t), un

f

(t)u

v

t

v L

(,)T u

n u t

(,)T p(t) t

+

v

L

(,)T u

n u t

(,)T q

(t)

t

+ (,)T f

(t), un

f

(t), u

v

t

C v u

n u

L

(,)T p(t) t

v u

n v

L

+ C

(,)T q

(t)

t

+ C

v

(,)T f

(t), un

f

(t), u

t.

Thus, from the above discussion, (), (H) and (H), we have

(u

n) (u)

C u

n u

L

(,)T p(t) t

+ C v

v u

n v

L

(,)T q

(t)

t

+ C v

v (,)T f

(t), un

f

(t), u

t

as n ,

which implies (un) (u) as n .

For any v E, the dynamic equation () gives

(,)T p(t)u (t) v

t +

(,)T

q (t)u (t)v t

(,)T

f

(t), u (t)

v t

= (,)T p(t)u (t)v

+ q (t)u (t)v

t

(,)T

f

(t), u

v t

= .

So, nding the homoclinic solutions to the zero of dynamic equation () is equivalent to nding the critical points of the associated functional dened in ().

4 Main results

In this section, we state the results of the existence of nontrivial homoclinic orbits of the dynamic equation () on time scales. As an elementary illustration, two examples are given to show the usefulness of these criteria.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 10 of 16

Theorem If conditions (H), (H), (H) and (H) are satised, then the dynamic equation () has one nontrivial homoclinic orbit to such that

< (,)T

p(t)

u (t)

q (t)

u (t)

+ F

(t), u t < +.

Example Let

T ={, , , , , , , , , } [., +) (, .).

Consider the following second order boundary value problem on time scales T of the form

(tu (t)) (t )u = (t)(u (t)), -a.e. t

T,

u() = u () = .

()

x f (t, s) ds = tx, one can check that all conditions of Theorem are fullled. It follows from Theorem that the dynamic equation () has one nontrivial homoclinic orbit to .

Since

Theorem If conditions (H), (H), (H), (H) and the following condition are satised

(H) f (t, x) = f (t, x) for all x

R and -a.e. t

T,

then the dynamic equation () has an unbounded sequence in E of a homoclinic orbit to .

Example Let a, b > be real numbers,

P =

k(a + b), k(a + b) + a

,

and

P =

k(a + b) a, k(a + b)

.

Consider the following second order boundary value problem on time scales P P of the form

(tu (t)) |t |u = (t)(u (t)), -a.e. t P P, u() = u () = .

()

x f (t, s) ds = tx, one can check that all conditions of Theorem are fullled. It follows from Theorem that the dynamic equation () has an unbounded sequence in E of a homoclinic orbit to .

5 Proof of theorems

In this section, we show our main results on the existence of nontrivial homoclinic orbits of the dynamic equation () on time scales.

Since

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 11 of 16

Proof of Theorem Since we have already known that C(E,

R) and () = , in the following we prove that all the other conditions of Lemma are fullled with respect to the functional .

Firstly, we claim that satises the PS condition.

Assume that there exist a sequence {un} E and a constant c such that

(un) as n and (un) c, n = , , . . . , ()

we show that {un} has a convergent subsequence in E.It follows from () and (H) that there is a constant d such that

d + un E (un)

(un)un

=

u E + (,)T F

(t), u

f

(t), u

v

t

u E,

which implies that {un} is bounded in E. Hence, there is a subsequence (still denoted by {un}, un u in E). It follows from Lemma that un u in L ((, )T,

R). Now,

according to (H), un, u E, for any > , we have that there exist constants > , > and L

T such that

|un| < , |u| < and un u L

< for -a.e. |t| > L, ()

which implies that

f

(t), un

u

n

and

f

(t), u

u

for -a.e. |t| > L. ()

Since

(,)T f

(t), un

f

(t), u (un u) t

= [L,L]T f

(t), un

f

(t), u (un u) t

+ (,L)T f

(t), un

f

(t), u (un u) t

+ (L,)T f

(t), un

f

(t), u (un u) t, ()

let

L ,loc(T, R) =

: T

R| for arbitrary compact interval K

T, IK L (T, R)

,

where IK is an indicator function of interval K and

IK =

(x), x K,

, x /

K.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 12 of 16

It follows from the uniform continuity of f (t, x) in x and un u in L ,loc(

T, Rn) that

[L,L]T f

(t), un

(t), u (un u) t as n .

Combining Hlders inequality on time scales, () and () leads to

f

(,L)T f

(t), un

f

(t), u (un u) t

(,L)T f

(,L)T

(t), un

f

(t), u

t

(un u) t

(,L)T u

n

+

u

t

M.

By using the same technique, we obtain

(L,)T f

M,

where M, M depend on the bounds for un and u in E. Then

(,)T f

(t), un

f

(t), u (un u) t

(t), un

f

(t), u (un u) t as n ()

since

(un) (u)

(uk u)

= un u E (,)T f

(t), u (un u) t. ()

Equations () and () imply that un u in E. Consequently, satises the PS condition.

Secondly, we prove that there exist constants and > such that satises the assumption (iii) of Lemma .

It follows from Lemma that there exists > such that

u L

u E for u E.

On the other hand, according to (H) and (H), we have that there exists > such that

u u E,

where

u = max

t(,)T

u(t) .

(t), un

f

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 13 of 16

(H) implies that there is > such that

F(t,

x)

|x| for |x| .

Let =

and u E , we have u

= , then

F t,

u

u

for

u

and -a.e. t

T,

which implies that

(,)T

F

t, u

t u L u E.

Hence, if u E = , we have

(u) =

u E + (,)T

F

(t), u

t

u E

u E =

.

Choosing = , we have

(u)

= > .

Thirdly, we claim that there exists e X \ B() such that satises the assumption (iv) of Lemma .

Let u E be such that |u(t)| , for any , it follows from () that

(u) =

u E + (,)T

F

(t), u

t

u E (,)T

u

(t) t

= u E |

| (,)T u

(t) t,

which implies that there exists such that u > and (u) = ().

Hence, all the conditions of Lemma are satised, the desired results follow.

Proof of Theorem It follows from (H) that is even. In addition, we have already proved that C(E,

T), () = and satises the Palais-Smale condition. We prove that all the other conditions of the symmetric mountain pass theorem are satised with respect to the functional . We have already showed that satises condition (iii) of the symmetric mountain pass theorem in the proof of Theorem .

In the following, we claim that satises condition (iv) of the symmetric mountain pass theorem.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 14 of 16

Let

E E be a nite-dimensional subspace. Consider u

E E with u = . It follows

from () that

(,)T

F

t, u

t (,)T (t)

u(t) t,

and

(,)T

F

t, u

t (,)T (t)

u(t) t.

We also have

u E c u for u

E,

where c = c(

E).

Dene m = inf u =(

(,)T (t)|u(t)| t +

(,)T (t)|u(t)| t), if m = , we have

u = for -a.e. t {t | |u(t)| > }, which contradicts u = , then m > , and we have

(u)

c u + (,)T

F

(t), u

t

+

(,)T

F

(t), u

t +

[,]T F

(t), u

t

c u + (,)T

F

(t), u

t +

(,)T

F

(t), u

t

c u (,)T

(t)

u(t) t

(,)T

(t)

u(t) t

=

c u

u (,)T

(t)

|u(t)|

u

t

+

(,)T

(t)

|u(t)|

u

t

c u

m

u .

Since > , there exists a constant C such that (u) if u C.

Consequently, it follows from Lemma that the functional possesses an unbounded sequence of critical values {cj} with cj = (uj), where uj satises

= (uj)uj = uj E + (,)T

f

(t), uj

uj t,

which implies that

uj E = (,)T

f

(t), uj

uj t.

Su et al. Advances in Dierence Equations (2017) 2017:47 Page 15 of 16

(H) implies that

cj =

(,)T

(,)T

f

(t), uj

(t), uj

uj t +

F

t

(,)T

(t), uj

uj t = uj E.

Then {uj} is unbounded in E because of cj as j . The proof is completed.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors read and approved the nal 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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