Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities

Abstract

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

In this paper, we look for periodic solutions of planar Hamiltonian systems ...... By using the Poincaré-Birkhoff twist theorem, we prove the existence and multiplicity of periodic solutions of the given system when f satisfies an asymmetric condition and the related time map satisfies an oscillating condition.

Wang and Ma Boundary Value Problems (2017) 2017:46 DOI 10.1186/s13661-017-0780-2

Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities

http://crossmark.crossref.org/dialog/?doi=10.1186/s13661-017-0780-2&domain=pdf

Web End = Zaihong Wang1* and Tiantian Ma2

*Correspondence: mailto:[email protected]

Web End [email protected]

1School of Mathematical Sciences, Capital Normal University, Beijing, 100048, Peoples Republic of China Full list of author information is available at the end of the article

1 Introduction

In this paper, we are concerned with the existence and multiplicity of periodic solutions of planar Hamiltonian systems

x = f (y) + p(t, y),y = g(x) + p(t, x), (.)

where f , g : R R are continuous, pi : R R (i = , ) are continuous and -periodic with the rst variable t.

In the case when f (y) y, p(t, y) and p(t, x) = p(t), system (.) becomes

x = y,

y = g(x) + p(t),

which is equivalent to the dierential equation

x + g(x) = p(t). (.)

The existence and multiplicity of periodic solutions of Eq. (.) have been widely studied in the literature (see [] and the references therein). Recently, the periodic solutions of

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.

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 2 of 16

planar Hamiltonian systems have been studied with an increasing interest (see []). In [], Fonda and Sfecci studied the periodic solutions of the planar Hamiltonian systems of the type

x = g(t, y), y = g(t, x). (.)

Assume that the following conditions hold:

a+ lim inf

y+

g(t, y)

y

lim sup

y+

g(t, y)

y a+,

a lim inf

y

g(t, y)

y

lim sup

y

g(t, y)

y a,

and

b+ lim inf

x+

g(t, x)

x

lim sup

x+

g(t, x)

x b+,

x b,

where a, a, b and b are positive constants. It was proved in [] that system (.) has at least one -periodic solution provided that there exists an integer n > such that

a+ +

b lim inf

x

g(t, x)

x

lim sup

x

g(t, x)

a

b+ +

b <

n (.)

and

a+ +

a

b+ +

b >

n + . (.)

In the present paper, we shall deal with the periodic solutions of system (.) when the non-resonant conditions (.) and (.) do not hold. Assume the following conditions hold:

(h) g satises lim|x|+ sgn(x)g(x) = +;(h) there exists a constant L > such that, for all x, y R, |g(x) g(y)| L|x y|;

(h) the limits lim|y|+ pi(t,y)y = (i = , ) hold uniformly with respect to t [, ]; (h) there are two positive constants a and b such that

lim

y+

f (y)y = a,

f (y)y = b.

It is well known that the time map plays an important role in studying the periodic solutions of Eq. (.) (see [, , ] and the references therein). In this paper, we also use the time map to study the periodic solutions of system (.). Let us set

G(x) =

x

g(s) ds.

lim

y

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 3 of 16

Under condition (h), we can dene the time map

(c) =

d(c)

w(c)

ds (c G(s))

for c > large enough, where w(c) and d(c) satisfy w(c) < < d(c) and G(w(c)) = G(d(c)) = c. Assume that the time map (c) satises the condition:

() There exist a constant > , an integer n > , and two sequences {ak} and {bk} such that limk ak = +, limk bk = +; and moreover

(ak) <

, (bk) >

mn +

,

b and a, b are given in condition (h).

We prove the following theorem.

Theorem . Assume that conditions (hi) (i = , . . . , ) and () hold. Then system (.) has innitely many -periodic solutions {(xk(t), yk(t))}k= which satisfy

lim

k

where m =

a +

min tR

x

k(t)

+

y

k(t)

= +.

Moreover, for each integer k , both xk(t) and yk(t) have exactly n simple zeros in [, ).

From Theorem . we can obtain the following corollary.

Corollary . Assume that a, b are two positive constants, e, p : R R are continuous and conditions (hi) (i = , ) and () are satised. Then the same conclusions of Theorem . still hold for the system

x = ay+ by + e(t), y = g(x) + p(t).

Remark . From condition (h) we know that f can be written in the form

f (y) = ay+ by + h(y),

where h : R R is continuous and satises

lim

|y|+

h(y)y = .

Therefore, it suces for us to prove the main theorem for the system

x = ay+ by + p(t, y),y = g(x) + p(t, x), (.)

where pi (i = , ) satisfy condition (h). In the case a = b = , by introducing a rescaling of the time s = at, u(s) = x( sa), v(s) = y( sa), we nd the equivalent system of (.) (having the

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 4 of 16

classical form)

u = v + q(s, v),

v = g(u) + q(s, u).

Such a rescaling cannot be easily applied in the case a = b because we do not know when the solution will change its sign.

We nally stress the fact that the proofs of the above results will be given under the additional assumptions that f and pi (i = , ) are locally Lipschitz continuous with variables y or x. It is shown in Section that this requirement is not restrictive and that our results are valid for any continuous functions f and pi (i = , ).

2 Basic lemmas

At rst, we consider the auxiliary autonomous system

x = ay+ by, y = g(x). (.)

The orbits of system (.) are curves determined by the equation

c: ay+ +

d(c)

w(c)

by + G(x) = c, (.)

where c is an arbitrary constant. We can easily prove the following lemma.

Lemma . Assume that condition (h) holds. Then there exists a constant c > such that, for any c > c, c is a closed curve which is star-shaped around the origin O.

From Lemma . we know that, for c c, each c intersects with the x-axis at two points (w(c), ) and (d(c), ), where w(c) and d(c) are continuous and satisfy

w(c) < < d(c), G w(c) = G d(c) = c.

Let (xc(t), yc(t)) be the solution of system (.) lying on the curve c with c c. Obviously, (xc(t), yc(t)) is periodic. Let us denote by T(c) the least period of (xc(t), yc(t)). From the rst equation of (.) and (.) we have that

T(c) =

a +

b

ds (c G(s)).

By the denition, T(c) is continuous for c c.

Now we perform some phase-plane analysis for system (.) . Let (x(t), y(t)) = (x(t, x, y), y(t, x, y)) be the solution of system (.) satisfying the initial condition

x() = x, y() = y.

Lemma . Assume that conditions (hi) (i = , , ) hold. Then each solution (x(t), y(t)) of system (.) exists uniquely on the whole t-axis.

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 5 of 16

Proof The proof follows directly from the fact that the nonlinearities are locally Lipschitz continuous and all have at most linear growth.

According to Lemma ., the Poincar map P : R R is well dened by

P : (x, y) (x, y) = x(,

Clearly, the Poincar map P is an area-preserving homeomorphism. The xed points of P correspond to the periodic solutions of system (.) .

Now, we take the polar coordinates transformation x = r cos , y = r sin to system (.) .

Under this transformation, system (.) becomes

Denote by (r(t), (t)) = (r(t, r, ), (t, r, )) the solution of (.) with the initial value

r() = r, () = ,

with x = r cos , y = r sin . Clearly, the Poincar map P can be written in the polar coordinate form P : (r, ) (r, ) with

r = r(, r, ), = (, r, ) + l,

where l is an arbitrary integer.

Applying the polar coordinate transformation x = cos , y = sin to system (.), we get

ddt = (a sin+ b sin ) cos g( cos ) sin ,

ddt = (a sin+ b sin ) sin g( cos ) cos .

x, y), y(, x, y)

.

drdt = r(a sin+ b sin ) cos g(r cos ) sin + p(t, r sin ) cos

+ p(t, r cos ) sin ,

ddt = (a sin+ b sin ) sin rg(r cos ) cos rp(t, r sin ) sin

+ rp(t, r cos ) cos .

(.)

(.)

Denote by ((t), (t)) = ((t, , ), (t, , )) the solution of (.) satisfying the initial value

() = , () = .

Using conditions (hi) (i = , , ), it is not hard to prove the following lemma.

Lemma . Assume that conditions (hi) (i = , , ) hold. Then there exist constants > and R > such that

r

r(t) r, t [, ], r R.

In particular, under conditions (hi) (i = , ), (t) satises the inequality

(t) , t [, ], R.

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 6 of 16

Lemma . Assume that conditions (hi) (i = , , ) hold, and let

(r, ) = (, r, ) ; (r, ) = (, r, ) .

Then, for any suciently small , there exists a positive constant such that

(r

, ) (r, )

for r .

Proof Let (x(t), y(t)) = (x(t, x, y), y(t, x, y)) be the solution of (.) with (x(), y()) = (x, y). It is noted that (x(t), y(t)) = (x(t, x, y), y(t, x, y)) is a solution of system (.) with (x(), y()) = (x, y). Set

u(t) = u(t, x, y) = x(t, x, y) x(t, x, y),

v(t) = v(t, x, y) = y(t, x, y) y(t, x, y).

Then we have

du(t)dt = a y+(t) y+

(t)

b

y(t) y(t) + p

t, y(t)

,

dv(t)dt = g x(t)

g

x(t) + p

t, x(t)

.

Let d(t) =

u(t) + v(t). Then we get

d (t) d(t) + p

with = ( + L), = max{a, b}. From condition (h) we have that, for any suciently small > , there exists c > such that

p

(t, y)

t, y(t)

+

p

t, x(t)

|y| + c, (t, y) R

and

p

(t, x)

|x| + c, (t, x) R.

Therefore, we obtain

d (t) d(t) + x(t)

+

y(t)

+ c.

Solving this inequality, we get

d(t) e

x(t)

+

y(t)

dt + A e

x(t) + y(t) dt + A,

. It follows from Lemma . that, for t [, ],

d(t) r + A,

where A = ce

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 7 of 16

where = e. Write (t) = (t, r, ) = (t, r, )(t, r, ). Clearly, if |(t)| <

, then (t) is just the angle between the vectors (x(t), y(t)) and (x(t), y(t)). Hence, we have

cos (t) = r(t) +

(t) d(t) r(t)(t)

d(t)

r(t)(t).

It follows that

sin (t)

d(t)

r(t)(t).

According to Lemma ., we have that if is suciently small and r is large enough, then

sin (t)

d(t)

r(t)(t)

(r + A)

r

.

Since () = and (t) varies continuously as t increases from to , we have

(t)

sin (t)

.

Consequently, we have that there exists > such that, for r ,

(r

, ) (r, )

, r .

Lemma . Assume that conditions (hi) (i = , ) and () hold. Then there exists a constant > such that, for t R and k large enough,

(t) ,

with ( cos , sin ) ak or ( cos , sin ) bk .

Proof From the denition of T(c) and condition () we know that, for each k N,

T(ak)

n m

.

In what follows, without loss of generality, we assume that the sequence {T(bk)} is bounded. Otherwise, we can replace the sequence {T(bk)} with a bounded one because

T(c) is continuous for c large enough. We shall only deal with the rst case, and the second one can be proved similarly. Let us set

dk = d(ak), wk = w(ak).

Obviously, dk +, wk as k . Next, we prove that there exist two positive constants i (i = , ) such that

lim inf

k

g(dk)

dk =

; T(bk)

n + m

; lim inf

k

g(wk)

wk =

.

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 8 of 16

Assume by contradiction that

lim inf

k

g(dk)

dk = .

Then there exists a subsequence of {dk} (we still denote it by {dk}) such that

lim

k

g(dk)

dk = .

Set

g(dk)

dk =

k.

We have that k as k . From condition (h) we know that, for < x dk,

g(x)x

k

=

g(x)x

g(dk)

dk

|

g(dk) g(x)|

x +

|g(dk)(dk x)|

dkx

L(dk x)

x + L +

|g()|

dk

(dk x)

x .

For simplicity, we assume g() = . Then we get that, for < x dk,

g(x)x

k

L(dk x)

x . (.)

Consequently, we have that, for x dk,

g(x) kx + L(dk x).

It follows that, for x dk,

G(dk) G(x) =

dk

x g(s) ds

dk

x

kx + L(dk x)

dx

dk x + L(dk x).

From the denition of T we have that

T(ak) = m

dk

wk

dx(G(dk) G(x)) m

=

k

dx (G(dk) G(x))

dk

m

dx

k(dk x) + L(dk x)

= m

dk

dt

k( t) + L( t) .

Since

lim inf

k

dt

k( t) + L( t)

L

dt t = +,

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 9 of 16

we have that

lim

k T(ak) = +.

This is a contradiction because T(ak) is a bounded sequence. Therefore, there exists a constant > such that

lim inf

k

g(dk)

dk =

. (.)

Similarly, there exists > such that

lim inf

k

g(wk)

wk =

.

From condition (h) and (.), (.) we know that there exists suciently small > such that, for k large enough and x [( )dk, dk],

g(x)x

g(dk)

dk

L(dk x)

x

L

. (.)

Therefore, if (t) cos (t) [( )dk, dk], then we have

(t) a

sin+ (t) b sin (t)

sin (t)

cos (t) ,

where = min{a, b, }. Next, we deal with the case (t) cos (t) [, ( )dk]. Set xk = ( )dk. Assume that the line x = xk intersects with the curve ak at two points (xk, y+k) and (xk, yk) with yk < < y+k. Then we have

ay+k + G(xk) = ak = G(dk),

byk + G(xk) = ak = G(dk).

Therefore, we get

y+k =

a

G(dk) G

( )dk

, yk =

b

G(dk) G

( )dk

.

From (.) we have

y+k

xk =

a (G(dk) G(( )dk))

( )dk

a( )

and

yk

xk =

b (G(dk) G(( )dk))

( )dk

b( ).

Set

+ = arctan

a( ),

= arctan

b( ).

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 10 of 16

From condition (h) we know that there exists A > such that g(x) for x A. If

(t) cos (t) [A, ( )dk] and (t) sin (t) , then we have

(t) a sin +

a

g( cos ) cos b sin .

If (t) cos (t) [, A] and (t) sin (t) , then we have that, for large enough,

(t) b sin

g( cos ) cos

g( cos ) cos a sin +.

If (t) cos (t) [, A] and (t) sin (t) , then we have that, for large enough,

(t) a sin +

g( cos ) cos

sin +.

Similarly, we have that, if (t) cos (t) [A, ( )dk] and (t) sin (t) , then we have

(t) b sin

sin .

In conclusion, we have proved that there exists > such that

(t) ,

with ( cos , sin ) ak , (t) cos (t) and k large enough. Similarly, we can prove that there exists > such that

(t) ,

with ( cos , sin ) ak , (t) cos (t) and k large enough. Let us set =

min{, }. Then we have that

(t) ,

with ( cos , sin ) ak and k large enough.

Lemma . Assume that conditions (hi) (i = , ) and () hold, and let (, ) = (, , ) . Then there exist two positive constants and such that

(, ) < n , ( cos , sin ) ak, ak ; (, ) > n + , ( cos , sin ) bk, bk .

Proof From Lemma . we have that there exists > such that, for ak or bk ,

(t) , ( cos , sin ) ak or ( cos , sin ) bk.

Write (, ) = l , where l is an integer, < . Let us denote by t the time for (t) to decrease from l to l . If ( cos , sin ) ak , then we have

= lT(ak) + t = lm(ak) + t.

b

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 11 of 16

Since t T(ak) = m(ak), we have

= lm(ak) + t (l + )m(ak) (l + )

n m

.

It follows that l n. If l n + , then we have

(, ) = l (n + ).

If l = n, then we have

t = nm(ak) n

n m

nm.

Therefore, we get

=

nT(ak)+t

nT(ak)

(t) dt nm.

Furthermore,

(, ) n nm.

Set = min{, nm}. Then we have

(, ) n , ( cos , sin ) ak, ak .

Similarly, we can prove

(, ) n + , ( cos , sin ) bk, bk .

3 Proof of the main theorem

At rst, we recall a generalized version of the Poincar-Birkho xed point theorem by Rebelo [].

A generalized form of the Poincar-Birkho xed point theorem. Let A R be an

annular region bounded by two strictly star-shaped curves around the origin, and , int( ), where int( ) denotes the interior domain bounded by . Suppose that F :

int( ) R is an area-preserving homeomorphism and F|A admits a lifting, with the

standard covering projection : (r, ) z = (r cos , r sin ), of the form

F|A : (r, ) w(r,

), + h(r, )

,

where w and h are continuous functions of period in the rst variable. Correspondingly, for

= ( ) and

= ( ), assume the twist condition

h(r, ) > on

; h(r, ) < on

.

Then, F has two xed points z, z in the interior of A such that

h

(z)

= h

(z) = .

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 12 of 16

Remark . The assumption on the star-shaped boundaries of the annulus is a delicate hypothesis. Martins and Urea [] showed that the star-shapedness assumption on the interior boundary is not eliminable. Le Calvez and Wang [] then proved that star-shapedness of the exterior boundary should also be imposed, while this assumption was not made in Dings theorem [].

Proof of Theorem . From Lemmas . and . we know that there exists an integer k > such that, for any k k,

(, r, ) < n, (r cos , r sin ) ak, (.) (, r, ) > n, (r cos , r sin ) bk. (.)

Without loss of generality, we assume that ak < bk < ak+ for k k. Let Dk be an annular region with boundary ak and bk . Consider the Poincar map P : Dk R. Write the

Poincar map in the form

r = r(, r, ), = + (r, ),

where (r, ) = (, r, ) + n. From (.) and (.) we know that, for k k,

(r, ) < , (r cos , r sin ) ak, (r, ) > , (r cos , r sin ) bk.

Therefore, the area-preserving homeomorphism P is twisting on the annulus Dk. On the other hand, by Lemma ., we have that both ak and bk are star-shaped with respect to the origin O for k large enough. Consequently, all assumptions of the generalized form of the Poincar-Birkho xed point theorem are satised. Therefore, the Poincar map P has at least two xed points in annulus Dk. Furthermore, system (.) has innitely many -periodic solutions {(xk(t), yk(t))}k= which satisfy

lim

k

Clearly, each -periodic solution (xk(t), yk(t)) rotates clockwise strictly n turns around the origin in the interval [, ]. It follows that both xk(t) and yk(t) have exactly n simple zeros in [, ). Hence, the conclusion of Theorem . holds.

From Theorem . we know that the existence of periodic solutions of system (.) has tight relation with the asymptotic property of time map (c). In case g is odd, we can easily check condition (). Set

G+ = lim inf

x+

From Theorem . in [], we have the following lemma.

min tR

x

k(t)

k(t)

+

y

= +.

G(x)x , G+ =

lim sup

x+

G(x) x .

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 13 of 16

Lemma . Assume that condition (h) holds, g is odd and G+ < G+. Then

G+ ,

G+

+, +

,

where + = lim infc+ (c), + = lim supc+ (c).

Applying Theorem . and Lemma ., we can obtain the following corollary.

Corollary . Assume that conditions (hi) (i = , . . . , ) hold. Let g(x) be an odd function and G+ < G+. If

n N,

mn

G

+, G+

m =

a +

b ,

then system (.) has innitely many -periodic solutions {(xk(t), yk(t))}k= which satisfy

lim

k

min tR

x

k(t)

+

y

k(t)

= +.

Moreover, for each integer k , both xk(t) and yk(t) have exactly n simple zeros in [, ).

4 Concluding remarks

The restrictions on the local Lipschitz conditions of f and pi (i = , ) made in the proofs of the above sections can be removed. Indeed, Lemmas . and . guarantee the applicability of the following non-uniqueness version of the Poincar-Birkho theorem which was proved by Fonda and Urea in []. We now state this theorem for a general Hamiltonian system in RN. Let us consider the (time-dependent) Hamiltonian system

x = yH(t, x, y), y = xH(t, x, y),

(.)

where the continuous function H : R RN RN R, H = H(t, x, y) is T-periodic in its rst variable t and continuously dierentiable with respect to (x, y), x = (x, . . . , xN), y =

(y, . . . , yN).

We next introduce the denition of rotation number of a continuous path in R. Let w : [t, t] R be a continuous path such that w(t) = (, ) for every t [t, t]. The rotation number of w around the origin is dened as

Rot

w(t); [t, t]

=

(t) (t)

,

where : [t, t] R is a continuous determination of the argument along w, i.e., w(t) = |w(t)|(cos (t), sin (t)).

Assume that for each i = , . . . , N, there are two strictly star-shaped Jordan closed curves around the origin i, i R such that

o D

i

D

i

D

i

.

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 14 of 16

Here we denote by D( ) the open bounded region bounded by the Jordan closed curve .

Consider the generalized annular region

A = D

Theorem . ([]) Under the framework above, denoting zi(t) = (xi(t), yi(t)), assume that every solution z(t) = (z(t), . . . , zN(t)) of (.) departing from z() A is dened on [, T]

and satises

zi(t) = (, ) for every t [, T] and i = , . . . , N.

Assume that there are integer numbers , . . . , N Z such that, for each i = , . . . , N, either

Rot

zi(t); [, T]

or

> i, if zi() i, < i, if zi() i.

Then Hamiltonian system (.) has at least N + distinct T-periodic solutions z(t), . . . , zN(t), with z(), . . . , zN() A, such that

Rot

zki(t); [, T]

U

xi (t, x, y) = pi(t, x, y) (i = , . . . , N).

In this case system (.) is a Hamiltonian system. Simple examples of such functions can be given. For example,

U(t, x, y) = p(t)

i=N

i=

\ D

D N

\ D N

RN.

< i, if zi() i, > i, if zi() i,

Rot

zi(t); [, T]

= i for every k = , . . . , N and i = , . . . , N.

Remark . It is noted that there is no requirement of uniqueness of Cauchy problems associated to system (.) in this higher dimensional Poincar-Birkho theorem for Hamiltonian ows. In [, ], Theorem . is applied to prove the multiplicity of periodic solutions of weakly coupled Hamiltonian systems. Theorem . in the present paper can also be extended to a weakly coupled system of the type

x i = fi(yi) + pi(t, x, y),y i = gi(xi) + pi(t, x, y) (i = , . . . , N), (.)

where x = (x, . . . , xN), y = (y, . . . , yN), fi, gi : R R (i = , . . . , N) are continuous, pji : RN+ R (j = , , i = , . . . , N) are continuous and -periodic with the variable t. Assume that there is a function U : RN+ R such that

U

yi (t, x, y) = pi(t, x, y),

sin xi sin yi.

Assume that the following conditions hold:

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 15 of 16

(h ) gi satises lim|u|+ sgn(u)gi(u) = +, i = , . . . , N;(h ) there exist constants Li > such that, for all u, v R, |gi(u) gi(v)| Li|u v|, i = , . . . , N;

(h ) there are constants M > , M > and < i < such that |pji(t, x, y)| M(|xi| + |yi|)

i + M for all t [, ], (x, y) RN, j = , , i = , . . . , N; (h ) there are positive constants ai and bi such that

lim

v+

fi(v)

v = ai,

lim

v

fi(v)

v = bi.

Set Gi(u) =

u gi(s) ds, i = , . . . , N. Let us dene the time maps

di(c)

wi(c)

ds (c Gi(s))

for c > large enough, where wi(c) and di(c) satisfy wi(c) < < di(c) and Gi(wi(c)) = Gi(di(c)) = c. Assume that the time map i(c) satises the condition:

( ) There exist constants i > , integers ni > , and sequences {aik} and {bik} such that

limk aik = +, limk bik = +; and moreover

i aik < mini i, i bik > mini +i, i = , . . . , N,

where mi =

ai +

i(c) =

bi and ai, bi are given in condition (h ).

With a slight modication of the proof of Theorem . and using the higher dimensional Poincar-Birkho Theorem ., we can prove the result.

Theorem . Assume that conditions (h i) (i = , . . . , ) and ( ) hold. Then Hamiltonian system (.) has innitely many -periodic solutions {(xk(t), yk(t))}k= which satisfy

lim

k

min tR

xk(t)

+

yk(t)

= +,

where |xk(t)| = i=N

i= |xki(t)|, |yk(t)| = i=N

i= |yki(t)|. Moreover, for each index i, both xki(t) and yki(t) have exactly ni simple zeros in the interval [, ).

Competing interests

The authors declare that they have no competing interests.

Authors contributions

ZW proved Lemma 2.5 and gave the concluding remarks. TM proved the other conclusions and helped to draft the manuscript. All authors read and approved the nal manuscript.

Author details

1School of Mathematical Sciences, Capital Normal University, Beijing, 100048, Peoples Republic of China. 2Editorial Department of Journal, Capital Normal University, Beijing, 100048, Peoples Republic of China.

Acknowledgements

Research supported by the National Nature Science Foundation of China, No. 11501381 and the Grant of Beijing Education Committee Key Project, No. KZ201310028031.

The authors are grateful to the referees for many valuable suggestions to make the paper more readable.

Wang and Ma Boundary Value Problems (2017) 2017:46 Page 16 of 16

Publishers Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aliations.

Received: 19 November 2016 Accepted: 23 March 2017

References

1. Ding, T, Iannacci, R, Zanolin, F: Existence and multiplicity results for periodic solutions of semilinear Dung equations.J. Dier. Equ. 105, 364-409 (1993)2. Ding, T: An innite class of periodic solutions of periodically perturbed Dung equations at resonance. Proc. Am. Math. Soc. 86, 47-54 (1982)

3. Hao, D, Ma, S: Semilinear Dung equations crossing resonance points. J. Dier. Equ. 133, 98-116 (1997)4. Qian, D: Time maps and Dung equations across resonant points. Sci. China Ser. A 23, 471-479 (1993)5. Xia, J, Wang, Z: Existence and multiplicity of periodic solutions for the Dung equation with singularity. Proc. R. Soc. Edinb., Sect. A 137, 625-645 (2007)

6. Boscaggin, A: Subharmonic solutions of planar Hamiltonian systems: a rotation number approach. Adv. Nonlinear Stud. 11, 77-103 (2011)

7. Boscaggin, A, Garrione, M: Resonance and rotation numbers for planar Hamiltonian systems: multiplicity results via the Poincar-Birkho theorem. Nonlinear Anal. 74, 4166-4185 (2011)

8. Fabry, C, Fonda, A: Periodic solutions of perturbed isochronous Hamiltonian systems at resonance. J. Dier. Equ. 214, 299-325 (2005)

9. Fonda, A, Ghirardelli, L: Multiple periodic solutions of Hamiltonian systems in the plane. Topol. Methods Nonlinear Anal. 36, 27-38 (2010)

10. Fonda, A, Sabatini, M, Zanolin, F: Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincar-Birkho theorem. Topol. Methods Nonlinear Anal. 40, 29-52 (2012)

11. Fonda, A, Sfecci, A: A general method for the existence of periodic solutions of dierential systems in the plane.J. Dier. Equ. 252, 1369-1391 (2012)12. Rebelo, C: A note on the Poincar-Birkho xed point theorem and periodic solutions of planar systems. Nonlinear Anal. 29, 291-311 (1997)

13. Martins, R, Urea, AJ: The star-shaped condition on Dings version of the Poincar-Birkho theorem. Bull. Lond. Math. Soc. 39, 803-810 (2007)

14. Le Calvez, P, Wang, J: Some remarks on the Poincar-Birkho theorem. Proc. Am. Math. Soc. 138, 703-715 (2010)15. Ding, W: A generalization of the Poincar-Birkho theorem. Proc. Am. Math. Soc. 88, 341-346 (1983)16. Fonda, A, Urea, AJ: A higher dimensional Poincar-Birkho theorem for Hamiltonian ows. Ann. Inst. Henri Poincar, Anal. Non Linaire (2016). doi:http://dx.doi.org/10.1016/j.anihpc.2016.04.002

Web End =10.1016/j.anihpc.2016.04.002

17. Calamai, A, Sfecci, A: Multiplicity of periodic solutions for systems of weakly coupled parametrized second order dierential equations. Nonlinear Dier. Equ. Appl. (2017). doi:10.1007/s00030-016-0427-5

18. Fonda, A, Sfecci, A: Periodic solutions of weakly coupled super-linear systems. J. Dier. Equ. 260, 2150-2162 (2016)