Multiple periodic solutions to a class of nonautonomous second-order delay differential equation

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R E S E A R C H

Open Access

Multiple periodic solutions to a class of

nonautonomous second-order delay

differential equation

Qiong Meng

1*

_{and Xiaosheng Zhang}

2

*_{Correspondence:}

[email protected]

1_{School of Mathematical Science,}

Shanxi University, Wucheng Road, Taiyuan, 030006, P.R. China Full list of author information is available at the end of the article

Abstract

The existence of the nontrivial periodic solutions for nonautonomous second-order delay diﬀerential equation

x(t) +

λ

x(t) = –f

(

t,x(t),x(t–

τ

))

is investigated, where

λ

>

π

2_/τ2_,

_τ

_{> 0,}_f_∈_C1₍_R_×_R2_,_R_{). Multiple periodic solutions}

are obtained by some recent critical point theorems.

MSC: 34K13; 34K18; 58E50

Keywords: second-order delay diﬀerential equation; nonautonomous; periodic solution; critical point theorems

1 Introduction

It is well known that the critical point theory is a powerful tool to deal with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equa-tions (see [–]). In , Li and He [] first applied the critical point theory to study the multiplicity of periodic solutions for delay differential equations. Especially, in , Guo and Yu [] established a variational framework for delay differential autonomous systems. In the past several years, some results on the existence of periodic solutions for the func-tional differential equation have been obtained by the critical point theory (see [–]). However, most of these functional differential equations are autonomous, the results on the non-autonomous functional differential equations are relatively few (see [, ]).

Motivated by the work of [, ], we consider a class of nonautonomous second-order delay diﬀerential equation

x(t) +λx(t) = –ft,x(t),x(t–τ), (.)

whereλ>π/τ,τ> ,f∈C₍_R_×_R_,_R_).

In this paper, we have the following conditions onf.

(f) f(t+τ,x) =f(t,x),f(t, –x, –y) = –f(t,x,y)and

∂f(t,x,y)

∂y =

∂f(t,y,x)

∂x

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for allx,y∈R,t∈[,τ];

(f) there exist fourτ-periodic and continuous functionsa(t),b(t),c(t)andd(t)such that

f(t,x,y) =a(t)x+b(t)y+◦(r) asr=x₊_y→_

and

f(t,x,y) =c(t)x+d(t)y+◦(r) asr=x₊_y→ ∞

uniformly fort∈[,τ];

(f±) |∇F(t,z) –B(t)z|is bounded for allz= (x,y)∈R_,_t_∈_[,_τ_]_and

F(t,z) –  

B(t)z,z→ ±∞ as|z| → ∞

uniformly fort∈[,τ], where

F(t,x,y) = x



f(t,s,y)ds+ y



f(t,s, )ds,

A(t) =

a(t) b(t) b(t) a(t)

, B(t) =

c(t) d(t) d(t) c(t)

.

Theorem . Assume that f satisﬁes (f)-(f–) with (A(t)z,z) ≤ , (B(t)z,z) > (λ –

π_/_τ₎_|_z_|_{> }_{for z}_∈_R_\{__}_,_t_∈_[,_τ_]._Then_(.)_{possesses at least}__{m pairs}__τ_-periodic solutions,where

m=max

j∈Z+:B(t)z,z> λ–π 

τj 

|z|> for z∈R\{},t∈[,τ]

,

Z+_{is the set of all positive integers}_.

Theorem . Assume that f satisﬁes (f)-(f+₎ _with ₍_A₍_t₎_z_,_z₎_≥ _{, (}_B₍_t₎_z_,_z_{) < (}_λ _–

πm_/τ)|z|< for z∈R\{},m∈Z+,t∈[,τ].Then(.)possesses at least[m– m+ ]pairsτ-periodic solutions,where

m=max

j∈Z+:B(t)z,z< λ–π 

τj 

|z|< for z∈R\{},t∈[,τ]

.

In this paper, the main purpose is to study the multiplicity of periodic solutions for sys-tems (.) via some recent critical point theorems for strongly indefinite functionals. In order to achieve this, some preliminaries are necessary. Let XandY be Banach spaces withXbeing separable and reflexive, and setE=X⊕Y. LetS⊂X*_{be a dense subset. For} eachs∈S, there is a semi-norm onEdefined by

ps:E→R, ps(u) =s(x)+y foru=x+y∈X⊕Y.

We denote byTS the topology onEinduced by the semi-norm family{ps}, and letωand

ω*_{denote the weak-topology and weak*-topology, respectively. Clearly, the}_T

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contains the product topology onE=X⊕Yproduced by the weak topology onXand the strong topology onY.

For a functional∈C₍_E_,_R_{), we write}

a={u∈E:(u)≥a}. Recall thatis weakly

sequentially continuous ifukuinE, one haslimk→∞(uk)v→(uk)vfor eachv∈E,

i.e.,: (E,ω)→(E*_,_ω*_{) is sequentially continuous. For}_c_∈_R_{, we say that}_{satisﬁes the} (C)ccondition if any sequence{uk} ⊂E, such that(uk)→cand ( +uk)(uk)→ as

k→ ∞, contains a convergent subsequence. Suppose that

() ∈C(E,R),cisTS-closed for everyc∈Rand: (c,TS)→(E*,ω*)is

continu-ous;

() there existsρ> such thatκ:=inf(Bρ∩Y) >  =(), whereBρ={u∈E:u= ρ};

() there exist a ﬁnite dimensional subspaceY⊂YandR>ρsuch thatc:=sup(E) <

∞andsup(E\S) <inf(Bρ∩Y), whereE:=X⊕Y, andS={u∈E:u ≤R}.

The following critical point theorem will be used later (see [, ]).

Theorem A Assume thatis even and()-()are satisﬁed.Thenhas at least m= dimYpairs of critical points with critical values less than or equal to c providedsatisﬁes the(C)ccondition for all c∈[κ,c].

2 Preliminaries

In this section, we establish a variational structure which enables us to reduce the exis-tence of τ periodic solutions of (.) to a classic Hamiltonian system. First, we have the following lemma by using similar arguments [].

Lemma . Suppose that f ∈C₍_R_×_R_,_R₎_satisﬁes₍_f_),_{then the function}

H(t,x,y) = x



f(t,s,y)ds+ y



f(t,s, )ds (.)

satisﬁes

∂H(t,x,y)

∂x =f(t,x,y),

∂H(t,x,y)

∂y =f(t,y,x)

and H(t,x,y)∈C₍_R_×_R_,_R_).

Proof It is obvious that ∂H

∂x =f(t,x,y). By (f),

∂f(t,x,y)

∂y =

∂f(t,y,x)

∂x . Then we have

∂H(t,x,y)

∂y = x



∂f(t,y,s)

∂s ds+f(t,y, ) =f(t,y,x).

The proof of Lemma . is complete.

Suppose thatxis a periodic solution of (.) with τ, lety(t) =x(t–τ), then

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We denotez= (x,y). Then

z(t) +λz(t) = –∇Ht,z(t), (.)

whereH(t,z) is deﬁned in (.) and∇H(t,z) denotes the gradient ofH(t,z) with respect to thezvariable. It is easy to obtain the following lemma.

Lemma . For anyτ periodic solution of(.),let y(t) =x(t–τ),then z= (x,y)is aτ

periodic solution of(.).Conversely,for anyτperiodic solution z= (x,y)of(.)satisfying y(t) =x(t–τ),x is aτ periodic solution of(.).

ForS₌_R_/(_τ_Z_{), let}_C∞₍_S_,_R_{) denote the space of }_τ_periodic_C∞_{functions on}_R_with

values inR_{. For any}_z_∈_C∞₍_S_,_R_{), it has the following Fourier expansion in the sense} that it is convergent in the spaceL₍_S_,_R_),

z(t) =√a τ +



√ τ

∞

j=

ajcos

π τjt

+bjsin

π τjt

,

wherea,aj,bj∈R,j= , , . . . .

Let z∈L₍_S_,_R_{). If there exists a function} _y_∈ _L₍_S_,_R_{) such that, for every} _x _∈ C∞(S_,_R_),

τ



z(t),x(t)dt= – τ



y(t),x(t)dt,

thenyis called a weak derivative ofzdenoted byy=z(t). Let

HS,R=

z∈LS,R τ



_z(t)+z(t)dt< +∞

.

For anyz,y∈H(S,R),·,·and · can be explicitly expressed by

z,y= τ



z(t),y(t)+z(t),y(t)dt

and

z=

|a|+

∞

j=

 +j|aj|+|bj|

/ .

We deﬁne an operatorL:H₍_S_,_R₎_→_H₍_S_,_R_{) by the Riesz representation theorem}

Lz,y= τ



˙

z(t),y˙(t)dt–λ

τ



z(t),y(t)dt.

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By (f) and (.) in Section , there exist two continuous functionsα(t) >  andβ(t)≥ such that

H(t,z)≤α(t)|z|+β(t), ∀z∈R.

By using similar arguments as in [], we have that

ϕ(z) = 

Lz,z+ψ(z),

where

ψ(z) = τ



Ht,z(t)dt.

Thus the critical points ofϕ(z) inH₍_S_,_R_{) are classical solutions of (.).}

DenoteE={z= (x,y)∈H₍_S_,_R_{) :}_y₍_t_{) =}_x₍_t_–_τ₎_}_{. By a direct computation, we have}

E=

z∈HS,R|z(t) =√ τa

 

+√

τ ∞

j= ajcos

π τjt

+bjsin

π τjt

 (–)j

⎫_⎬

⎭,

wherea,aj,bj∈R,j= , , . . . .

Note that for anyz= (x,y)∈E,y(t) =x(t–τ) and∇H(t,z) = (f(t,x,y),f(t,y,x)). Due to the fact thatf(t,x(t–τ),y(t–τ)) =f(t,y(t),x(t)), we know that∇H(t,z)∈Ewhateverz∈E. Therefore, we have the following lemma.

Lemma . If z(t)is a critical point ofϕin E,then z(t)is a critical point ofϕin H(S,R).

Moreover, we also denote byM+₍_·_),_M–₍_·_{) and}_M₍_·_{) the positive definite, negative} defi-nite and null subspaces of the self-adjoint linear operator defining it, respectively.

ThenEhas an orthogonal decomposition

E=M+(L)⊕M–(L)⊕M(L).

Let

zj(t) =



√

τ ajcos π τjt

+bjsin

π τjt

 (–)j

, j= , , . . . ,

whereaj,bj∈R. We have

Lzj,zj=

  +j

π τj

_–_λ

zj, j= , , . . . .

There existsσ>  such that

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Lzj,zj ≤–σzj forj∈J–,

whereJ+₌_{_j_≥_{ :}  +j(π



τj–λ) > },J–={j≥ :_+_j(π 

τj–λ) < },J={j≥ :_+_j(π 

τj–

λ) = }. Therefore, we get

Lz,z ≥σz forz∈M+(L), (.)

Lz,z ≤–σz forz∈M–(L). (.)

Remark . The conditionλ>π/τis to ensureJ–₌_∅_.

3 Proofs of theorems

In this section,cistand for diﬀerent positive constants fori∈Z+.

By a direct computation, (f) and (f) imply thatH(t,z) is even and satisﬁes

∇H(t,z) =A(t)z+◦|z| as|z| →, (.)

∇H(t,z) =B(t)z+◦|z| as|z| → ∞ (.)

uniformly fort∈[,τ]. (f±) implies that

_∇H(t,z) –B(t)z is bounded and

H(t,z) – 

B(t)z,z→ ±∞ as|z| → ∞

(.±)

uniformly fort∈[,τ].

Lemma . Suppose that f satisﬁes(f)-(f±).Then the functionϕsatisﬁes the(C)c

con-dition for any c∈R.

Proof First we deﬁne an operator

Bz,y= τ



B(t)z(t),y(t)dt

for anyz,y∈E. By a direct computation,Bis a bounded self-adjoint linear operator onE. ThusL+Bis also a self-adjoint linear operator onE. ThenEhas an orthogonal decompo-sition

E=M+(L+B)⊕M–(L+B)⊕M(L+B).

Also there existsσ>  such that

(L+B)z,z≥σz forz∈M+(L+B), (.)

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So, set

(z) = –ϕ(z) = – 

(L+B)z,z– τ



H(t,z) –  

B(t)z,zdt

and

(z),y= –(L+B)z,y– τ



∇H(t,z) –B(t)z,ydt

for anyz,y∈E,ϕ(z) is deﬁned in Section . Let{zk} ⊂Ebe any sequence such that

(zk)→c,

 +zk

(zk)→ ask→ ∞.

We ﬁrst prove that{zk}is bounded. For anyε> , (.) implies that there exists a constant

Cε>  such that

_∇H(t,z) –B(t)z<ε|z|+Cε (.)

for allz∈R_and_t_∈_{[, }_τ_{]. Since}_z

k∈E, we have

zk=zk++z–k+zk,

wherez+

k∈M+(L+B),z–k ∈M–(L+B),zk∈M(L+B). By (.±), there exists a constant

d>  such that

_∇H(t,z) –B(t)z≤d.

Therefore we have

( +c)z+_k≥–(zk),z+k

=(L+B)z+_k,z+_k+ τ



∇H(t,zk) –B(t)zk,z+k

dt

≥σz+k



–dz+_k.

Thus we have that{z+

k}is bounded. Using similar arguments, we can prove that{z–k}

is bounded. Consider{z_k}. Arguing indirectly, we suppose{z_k}is unbounded, then we havez_k → ∞. According to the deﬁnition ofM₍_L₊_B_{), this implies that there are} constantsd,d>  such that

dzk≤zk≤dzk. (.)

By (.), we have

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Then

c= lim

k→∞

(zk) –(zk)zk

= lim

k→∞

τ



∇H(t,zk) –B(t)zk,zk

dt

–  τ



H(t,zk) –

 

B(t)zk,zk

dt

. (.)

By (.) and (.±), (.) is a contradiction. Hence{z

k}is bounded. Therefore there

exists a constantd>  such that

zk=z+k+z–k+zk≤d.

So, we get{zk}is bounded, and going if necessary to a subsequence, we can assume that

zkzinE andzk→zinC(S,RN). Write zk=z+k +z–k +zk andz=z++z–+z, then

z±_k z±inE,z

k→zinEandz±k →z±inC(S,RN).

In view of (.) andz–

k →z–inC(S,R), it is easy to verify

τ



∇H(t,zk) –B(t)zk,z–k–z–

dt→

and τ



∇H(t,z) –B(t)z,z_k––z–dt→.

But then(zk) –(z),z–k–z– → ask→ ∞, and

(zk) –(z),z–k–z–

= –(L+B)z–_k–z–,z–_k–z–

– τ



dt

+ τ



∇H(t,z) –B(t)z,z–_k–z–dt

≥σz–k–z––

τ



dt

+ τ



∇H(t,z) –B(t)z,z–_k–z–dt.

This yieldsz–

k →z– inE. Similarly,z+k →z+inEand hencezk→zinE, that is,

sat-isﬁes the (C)ccondition. Thusϕ satisﬁes the (C)ccondition. The proof of Lemma . is

complete.

Proof of Theorem. Forz∈E, letX=M+₍_L₎_⊕_M₍_L_),_Y₌_M–₍_L_),_E₌_X_⊕_Y_{, and}

(z) = –ϕ(z), (z) =ψ(z), (.)

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In order to obtain this theorem, we apply Theorem A to the functional(z). The proof of this theorem is divided into the following three steps.

Step .satisﬁes ().

We ﬁrst check that c is TS -closed for any c∈R. Let {zk} be any sequence TS

-converging to somez∈E. Writezk=z+_k+z_k+z–_k andz=z++z+z–, thenz–_k →u– in

Eand hence{z–}is bounded in the norm topology. Note that (B(t)z,z) >λ–π/τ>  for z∈R_\{__}_{and for}_t_∈_{[, }_τ_{], then for any}_z_∈_E_{and small}_ε_{> , we have by (.) and (.)}

(z) = τ



H(t,z)dt

= τ   

∇H(t,sz),zds dt

≥ τ   

B(t)sz,zds dt– τ

 



_∇H(t,z) –B(t)sz,z|z|ds dt

≥ 

 τ



B(t)z,zdt– τ

 



ε|sz|+Cε

|z|ds dt

≥ 

 λ–

π τ –ε

z_L–CεzL≥cz

L–czL,

which implies thatis bounded from below onE. Consequently, combiningzk∈cand

 Lz

+

k,zk+= –Lz –

k,z–k–(zk) –(zk) shows that{zk+}is bounded inEby (.) and hence

(zk) = –

 

Lz+_k,z+_k– 

Lz–_k,z_k––(zk)

≤– 

Lz+_k,z+_k– 

Lz–_k,u–_k–c≤c. (.)

Moreover, sincezk_L =z+kL+zkL+z–kL, we have

(zk)≥czk_L–czkL≥ z_k_L–c≥z_k

L–c.

It follows from (.) and the above inequality that{z_k}is also bounded inE since all norms are equivalent in a ﬁnite dimensional space. Thenzk(=z+k+zk+z–k)

is bounded, and hence we can assume that{zk}converges weakly toz=z++z+z–inE.

Thus we have(zk)→(z). Note thatLy,y/is an equivalent norm onH+. By the lower

semi-continuity of the norm, we get

c≤lim sup

k→∞

(zk) =lim sup k→∞

– 

Lz_k+,z+_k– 

Lz–_k,z–_k–(zk)

= –lim inf

k→∞

 

Lz_k+,z+_k– 

Lz–,z––(z)≤(z),

that is,z∈cand hencecisTS-closed.

Next, we prove that: (c,TS)→(E*,ω*) is continuous. To achieve this, it is suﬃcient

to demonstrate thathas the same property. SupposezkuinE, then{zk}converges

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con-verges to (∇H(t,z(t)),y(t)) in measure on [, τ]. Moreover, by (.), one has

_∇Ht,zk(t)

,y(t))≤εzk∞+b¯zk∞+Cε

y∞≤c

for allkandt∈[, τ], whereb¯=maxt∈[,τ]{|B(t)|}and · ∞denotes the natural norm of C(S_,_R_{). Thus, the Vitali theorem is applicable and}

(zk),y

=

τ



∇H(t,zk),y

dt→

τ



∇H(t,z),ydt=(z),y

for anyy∈E. So,satisﬁes (). Step .satisﬁes ().

By (.), there exist  <ε<σ/,ρ>  such that

H(t,z)≤ε|z|+ 

A(t)z,z for all|z|<ρ, t∈[, τ]. (.)

By Proposition . in [], there is a positive constantξ such thatz∞≤ξz. Set small

ρ<ρ/ξ, then for eachz∈Ywithz ≤ρ, one hasz∞≤ρ, and hence by (.)

(z) = –

Lz,z– τ



H(t,z)dt

≥ σ

z _–

τ



ε|z|+ 

A(t)z,zdt≥ σ  –ε

z.

Therefore,

κ:=inf(Bρ∩Y)≥ σ

 –ε

ρ> ,

and hence () holds. Step .satisﬁes (). Let

Y=span

cos π

τjt

 (–)j

,sin π

τjt

 (–)j

:j∈Z+,j≤m

.

Obviously,Y⊂YanddimY= m. In order to obtain the desired conclusion, it is suﬃ-cient to prove that(z)→–∞asz → ∞onE:=X⊕Y.

Let=λ–π_τ. By the deﬁnition ofm, there exists a constant  <δ<σ such that

B(t)≥+δ (.)

fort∈[,τ]. Clearly, for anyy∈Y,

Ly,y ≥–y_L. (.)

LetF(t,z) =H(t,z) –_(B(t)z,z). We claim that

z– τ



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Indeed, for =z∈E, by (.), one has

z– τ



F(t,z)dt

=z– τ

 



∇H(t,sz) –B(t)sz,zds dt

≤ z– τ



ε|z|+Cε

|z|dt

≤ z–εz_L+CεzL

≤ε+ Cε

z,

which implies that (.) is true by the arbitrariness ofε. Then, forz=z+₊_z₊_z–_∈_E , by (.), (.) and (.), one has

(z) = – 

Lz–,z–– 

Lz+,z+– τ



F(t,z)dt

≤

z –

L–

δ

z +_– 

 τ



B(t)z,zdt– τ



F(t,z)dt

≤

z –

L–

δ

z +

– 

(+δ)z 

L– τ



F(t,z)dt

≤

z –

L–

δ

z +

– 

(+δ)z –

L+z  L – τ 

F(t,z)dt

≤–δ z

–

L+z+ 

+z

L

– τ



F(t,z)dt.

SinceM_and_Y_{are ﬁnitely dimensional, (.) and the above estimate imply that}₍_z₎_→ –∞asz → ∞,z∈E:=X⊕Y. Hence () holds.

The proof of Theorem . is complete.

Proof of Theorem. Forz∈E, letX=M–₍_L₎_⊕_M₍_L_),_Y₌_M+₍_L_),_E₌_X_⊕_Y_{, and}

(z) =ϕ(z), (z) = –ψ(z)

and

Y=span cos π τjt  (–)j ,sin π

τjt

 (–)j

:j=m,m+ , . . . ,m

.

Then the conclusion is obtained by the same argument as in the proof of Theorem .. The

proof of Theorem .. is complete.

Example . Consider the following equation:

x(t) +λx(t) = –α(t)x(t) –β(t)x(t–π) – x(t)

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wherex(t)≥. Let

f(t,x,y) =α(t)x+β(t)y+ x x₊_y₊_γ₍_t₎.

Then

∂f(t,x,y)

∂y =β(t) –

xy

(x₊_y₊_γ₍_t₎₎ =

∂f(t,y,x)

∂x .

Let

H(t,x,y) =  α(t)x

₊_β₍_t₎_xy₊ α(t)y

₊ ln

x₊_y₊_γ₍_t₎

γ₍_t₎ .

Then

∂H

∂x =f(t,x,y),

∂H

∂y =f(t,y,x).

By a straightforward computation, we have

f(t,x,y) = 

γ₍_t₎+α(t)

x+β(t)y+◦(r) asr=x₊_y→_

and

f(t,x,y) =α(t)x+β(t)y+◦(r) asr=x₊_y_{→ ∞}

uniformly fort∈[,π].

Takeλ= ,α(t) = – +sinπt,β(t) =  –sinπt,γ₍_t_{) =} 

. By Theorem ., we havea(t) = 

γ₍_t₎+α(t)≥,b(t) =d(t) =  –sinπt≥,c(t) = – +sinπtfort∈[,π],z= (x,y),x≥, y≥,

A(t)z,z=

a(t) b(t) b(t) a(t)

x y

,

x y

≥,

B(t)z,z=

c(t) d(t) d(t) c(t)

x y

,

x y

=c(t)x+y+ d(t)xy≤c(t) +d(t)x+y

= –x+y< – x+y< .

Then we havem=m= . By Theorem ., (.) possesses at least two pairs π-periodic solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{School of Mathematical Science, Shanxi University, Wucheng Road, Taiyuan, 030006, P.R. China.}2_{School of Mathematical}

Sciences, Capital Normal University, Beijing, 100048, P.R. China.

Acknowledgements

The authors are grateful for the referees’ careful reviewing and their valuable suggestions. The work is partially supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-1) of China.

Received: 18 May 2013 Accepted: 12 September 2013 Published:19 Nov 2013 References

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