Periodic solutions for a class of second order delay differential systems

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R E S E A R C H

Open Access

Periodic solutions for a class of second

order delay differential systems

Qiong Meng

*

_{and Juntang Ding}

*_{Correspondence:}

[email protected] School of Mathematical Science, Shanxi University, Taiyuan, Shanxi 030006, P.R. China

Abstract

Consider the periodic boundary value problem

¨

u(t) +ru(t–

τ

) +∇F(t,u(t–

τ

)) = 0, a.e.t∈[0, 2

τ

],

u(0) –u(2

τ

) =u˙(0) –u˙(2

τ

) = 0,

where

τ

> 0 is a given constant,r∈Ris a parameter. Periodic solutions are obtained by using linking theorems.

MSC: 34K13; 34K18; 58E50

Keywords: periodic boundary value problem; delay diﬀerential systems; linking theorems

1 Introduction

Consider the periodic boundary value problem

¨

u(t) +ru(t–τ) +∇F(t,u(t–τ)) = , a.e.t∈[, τ],

u() –u(τ) =u˙() –u˙(τ) = , (.) whereτ >  is a given constant, r∈Ris a parameter.F: [, τ]×RN _→_R_{satisﬁes the}

following assumptions:

(A)F(t,x) is measurable with respect tot, for allx∈RN, continuously diﬀerentiable inx, for a.e.t∈[, τ], and there exista∈C(R+_{; R}+_{) and}_b_∈_L_{([, }_τ_{]; R}+_{) such that}

F(t,x),∇F(t,x)≤a|x|b(t)

for allx∈RN_{and a.e.}_t_∈_{[, }_τ_].

Variational methods are very powerful techniques in nonlinear analysis and are exten-sively used in many disciplines of pure and applied mathematics, including ordinary and partial diﬀerential equations, mathematical physics, and geometrical analysis. The exis-tence and multiplicity of solutions for Hamilton systems and Schrödinger equations have been studied extensively via critical point theory; see [–].

In the past several years, some results on the existence of periodic solutions for the func-tional diﬀerential equation by the critical point theory have been obtained (see [–]). In

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[], the authors obtained the multiplicity results for periodic solutions to (.) by using critical point theory. Our proof method in this paper is diﬀerent from the literature [].

Motivated by the above observation, in this paper, we study the existence of periodic solutions to the system (.). The following theorems are the main results of our paper.

(H) There exists a constantR> such that

(i) F(t,x)≥,∀|x| ≤R,t∈[, τ]; or

(ii) F(t,x)≤,∀|x| ≤R,t∈[, τ].

(H) lim|x|→∞infF_|(_xt_|,x) > uniformly fort∈[, τ].

(H) lim_|x|→|∇F_|x(t|,x)|= uniformly fort∈[, τ].

(H) There existα> anda> such that

_∇F(t,x)≤a|x|α_{+ }_, _∀₍_t_,_x₎_∈_{[, }_τ_]_×_RN_.

(H) There existμ> , <β< ,R> , and a functionb(t)∈L([, τ]; R+)such that

μF(t,x)≤∇F(t,x),x+b(t)|x|β, ∀|x| ≥R,t∈[, τ].

(H) There existβ>α≥andb> ,R> such that

∇F(t,x),x– F(t,x)≥b|x|β_, _∀|_x_{| ≥}_R_,_t_∈_{[, }_τ_].

Theorem . Assume that(H)-(H)hold,ifis an eigenvalue of L(L is deﬁned in Sec-tion),BVP(.)has at least one nontrivial solution.

Theorem . Assume that(H)-(H)and(H)hold,ifis an eigenvalue of L(L is deﬁned in Section),BVP(.)has at least one nontrivial solution.

Remark . The condition (H) and the condition (H) are appeared, respectively, in [] and [].

2 Preliminaries

In order to seek τ-periodic orbits of (.), let us transform (.) to

¨

u(t) +rλ_u₍_t_–_π_{) +}_λ_∇_F₍_λ_t_,_u₍_t_–_π_{)) = ,} _a.e._t_∈_{[, }_π_],

u() –u(π) =u˙() –u˙(π) = , (.) by making the change of variablet→π_τt=λ–t, which implies that a π-periodic solution of (.) corresponds to a τ-periodic solution of (.). Hence we will only look for the π -periodic solutions of (.) in the sequel.

LetL₍_S_{, R}N_{) be the space of square integrable }_π_{-periodic vector-valued functions}

with dimension N, andC∞(S_{, R}N_{) be the space of }_π_-periodic_C∞ _{vector-valued}

func-tions with dimension N. For anyu∈C∞(S, RN), it has the following Fourier expansion in the sense that it is convergent in the spaceL₍_S_{, R}N_):

u(t) =√a π +



√ π

+∞

j=

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where a,aj,bj∈RN. Let H(S, RN) be the closure ofC∞(S, RN) with respect to the

Hilbert norm

u=

|a|+

+∞

j=

 +j|aj|+|bj| /

.

More speciﬁcally,H₍_S_{, R}N_{) =}_{_u_∈_L₍_S_{, R}N_{) :}_u_{< +}_∞}_{with the inner product}

u,v=

π



u(t),v(t)+u˙(t),v˙(t)dt

for anyu,v∈H₍_S_{, R}N_{), where (}_·_,_·_{) denotes the usual inner product in R}N_{. The norm on}

H₍_S_{, R}N_{) is deﬁned by}

u=

π



u(t)+u˙(t)dt

/

. (.)

It is well known thatH₍_S_{, R}N_{) is compactly embedded in}_C₍_S_{, R}N_{) (see Proposition .}

of []). Deﬁne two operatorsLandLfromH(S, RN) intoH∗(S, RN) as follows: for any

u∈H₍_S_{, R}N_{), which are given by}

(Lu)(v) =

π



˙

u(t+π),v˙(t)dt,

(Lu)(v) = –rλ

π



u(t),v(t)dt,

for allv∈H∗(S_{, R}N_{), where}_u_˙₍_t_{) and}_H∗₍_S_{, R}N_{) denote the weak derivative of}_u_{and the}

dual space ofH₍_S_{, R}N_{), respectively. By the Riesz representation theorem, we can}

iden-tifyH∗(S_{, R}N_{) with}_H₍_S_{, R}N_{). Thus,}_L

iucan also be viewed as an element belonging to

H₍_S_{, R}N_{) such that}_L

iu,v= (Liu)(v) for allu,v∈H(S, RN), wherei=  or . It is easy

to check thatLandLare bounded linear operators onH(S, RN). Note thatH(S, RN)

is compactly embedded inC(S_{, R}N_{), thus}_L

is compact onH(S, RN).

Lemma .[] The operator L:=L+L,that is,

Lu,v= (Lu)(v) =

π



˙

u(t+π),v˙(t)dt–rλ π



u(t),v(t)dt

is self-adjoint on H₍_S_{, R}N_)._{The operators L}

and Lare also self-adjoint on H(S, RN).

Lemma .[] The essential spectrum of Lon H(S, RN)is just{–, }.

In view of the above facts,  is not in the essential spectrum ofLas it is a compact perturbation of the self-adjoint operatorL. This implies that  is at most an eigenvalue

of ﬁnite multiplicity ofL.

Remark . H₍_S_{, R}N_{) has an orthogonal decomposition}

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whereH₌_Ker_L_{is ﬁnite dimensional, and}_H+_,_H–_are_L_{-invariant subspaces such that}

for someσ> ,

Lu,u ≥σu (.)

for allu∈H+_,

Lu,u ≤–σu (.)

for allu∈H–_.

Remark . By [], we can choose the properrto make that the set{j≥|(–)_jjj_+–rλ =

} =∅, which meansH₌_∅_{, where}_j_∈_Z_.

Lemma .[] There exists C> such that

uLp≤Cu, u∞≤Cu, (.)

for p=  +α, ,u∞=maxt∈[,π]|u(t)|,∀u∈H(S, RN).

Now consider the functionalϕdeﬁned onH(S, RN) given by

ϕ(u) = 

Lu,u–λ



π



Fλt,u(t)dt. (.)

SinceFsatisﬁes the assumption (A), a standard argument shows the following.

Lemma .[] The functionalϕis continuously diﬀerentiable on H₍_S_{, R}N₎_and_ϕ _is

deﬁned by

ϕ(u),v=Lu,v–λ π



∇Fλt,u(t),v(t)dt (.)

for all v∈H₍_S_{, R}N_).

Lemma . [] A critical point u∈H₍_S_{, R}N₎_{of functional} _ϕ _{is equivalent to a} __π

-periodic solution of system(.).

In this paper, we will use the following local linking Theorem A to prove our theorems. The following concepts appeared in [, , ].

LetX be a real Banach space with direct decompositionX=X_⊕_X_{. Consider two}

sequences of subspaces:

X_j⊂X_j⊂ · · · ⊂Xj

such thatXj₌ n∈NX

j

n,j= , . For every multi-indexα= (α,α)∈N, we denote byXα

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Deﬁnition .[] A sequence{αn} ⊂Nis said to be admissible if, for everyα∈Nthere

ism∈Nsuch thatn≥m⇒αn>α.

Deﬁnition .[] Letf ∈C₍_X_,_R_{). Then the function}_f _{satisﬁes the (}_PS₎∗ _{condition if}

every sequence{uαn}such that{αn}is admissible and

uαn∈Xαn, sup n

f(uαn) <∞, fαn(uαn)→

contains a subsequence which converges to a critical point off, wherefα=f|Xα.

Deﬁnition .[] LetXbe a Banach space with a direct sum decompositionX=X_⊕_X_.

The functionf ∈C₍_X_,_R_{) has a local linking at , with respect to (}_X_,_X_{), if, for some}_r_{> ,}

f(x)≥, ∀x∈X,x ≤r,

f(x)≤, ∀x∈X,x ≤r.

Theorem A[] Suppose that f ∈C(X, R)satisﬁes the following assumptions:

(f) f has a local linking atandX={}; (f) f satisﬁes(PS)∗condition;

(f) f maps bounded sets into bounded sets;

(f) for everym∈N,f(x)→–∞asx → ∞onX

m⊕X.

Then f has at least one nonzero critical point.

3 Proofs of theorems

In this section,cistand for diﬀerent positive constants fori∈Z+, Z+is the set of all positive

integers. Let

X=X⊕X, (.)

whereX₌_H+_,_X₌_H–_⊕_H_.

Lemma . Under assumption(H), (H)-(H),the functionalϕsatisﬁes the(PS)∗ condi-tion.

Proof Let{uαn}be a sequence such that{αn}is admissible and

ϕ(uαn) <∞, ϕαn(uαn)→.

For the sake of notational simplicity, setun=uαn.

Claim.{un}is bounded inX.

If not, passing to a subsequence if necessary, we assume thatun → ∞asn→ ∞. Set

vn=u_un_n, then{vn}is bounded inX. Hence, there exists a subsequence, still denoted by {vn}. Writevn=v+n+v–n+vnandv=v++v–+v, then

vn v, vn→v inX, vn→v inC

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In view of (H), we have

F(t,x)≤c

|x|+α+|x|, ∀x∈RN andt∈[, τ]. (.)

For|x| ≤Randt∈[, τ], one has

F(t,x)≤c

|x|+α+|x|≤c

R+α+|R|:=A,

together with (H), one has

μF(t,x)≤∇F(t,x),x+b(t)|x|β+μA, ∀x∈RN andt∈[, τ]. (.)

Hence, by (.) and (.), we have

μ

 – 

Lun,un=μϕ(un) –

ϕ(un),un

+λ π



μF(λt,un) –

∇F(λt,un),un

dt

≤c+cun+λ π



b(t)|un|β+μA

dt

≤c+cun+cunβ+c.

This implies that

Lun,un ≤c+cun+cunβ (.)

sinceμ> . Together with (.), we have

Lvn,vn=

Lun,un un →

, asn→ ∞,

since  <β< . Hence,

Lv,v= .

That means that

Lv+,v++Lv–,v–= .

By using (.) and (.), we discuss two cases:

Ifv+= , we havev–= , then we obtainv=v=vn= .

Ifv+ _{= , we have}_v– _{= , then we obtain}_v _{= .}

So we getv(t)≡ fort∈[, π]. Set

E=

t∈[, π] :v(t)= ,

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From (H), we get

lim

|un|→∞ inf

π

 F(λt,un)dt

un

≥ π

 [lim|un|→∞infF(λt,un)]dt un

≥

E

lim

|un|→∞

infF(λt,un) |un| |

vn| dt≥ E lim

|un|→∞

infF(λt,un) |un| |

v|

dt> . (.)

By the boundedness ofϕ(un) and (.), we have ϕ(un)

un

=Lun,un un

–λ

π

 F(λt,un)dt

un

≤c+cun+cunβ

un

–λ

π

 F(λt,un)dt

un

,

which together with  <β<  implies that

lim

|un|→∞ inf

π

 F(λt,un)dt

un

= .

This contradicts (.). Therefore,{un}is bounded inX.

Claim.{un}possesses a strong convergent subsequence inX.

Writeun=u+n+u–n+unandu=u++u–+u, then

u±_n u±, u_n→u inX, u±_n →u± inCS, RN.

In view of (.) andu–

n→u–inC(S, RN), it is easy to verify π



∇F(λt,un) –∇F(λt,u),u–n–u–

dt→, asn→ ∞.

Note that

ϕ(un) –ϕ(u),u–n–u–

→, asn→ ∞.

Thus,

–ϕ(un) –ϕ(u),u–n–u–

= –Lu–_n–w–,u–_n–u–+λ π



dt

≥σu–_n–u–+λ π



dt.

This yieldsu–

n→u–inX. Similarly,u+n→u+inX. Henceun→uinX. Hence,{un}

pos-sesses a strong convergent subsequence inX. The proof of Lemma . is complete.

Lemma . Under assumption(H), (H),and(H),the functionalϕsatisﬁes the(PS)∗

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Proof Let{uαn}be a sequence such that{αn}is admissible and

ϕ(uαn) <∞, ϕαn(uαn)→.

For the sake of notational simplicity, setun=uαn.

Claim.{un}is bounded inX.

If not, passing to a subsequence if necessary, we assume thatun → ∞asn→ ∞. In

view of (H), there existsc>  such that

∇F(t,x),x– F(t,x)≥b|x|β–c, ∀(t,x)∈[, τ]×RN. (.)

Hence, we have

ϕ(un) –

ϕ(un),un

=λ π



∇F(λt,un),un

– F(λt,un)

dt

≥λ π



b|un|β–c

dt

=bλ π



|un|βdt– πcλ.

This implies that

T

 |un| β_dt

un →

, asn→ ∞. (.)

Letun=u+n+u–n+un∈H+⊕H–⊕H. By (H), we have

ϕ(un),u+n

=Lu+_n,u+_n–λ π



∇F(λt,un),u+n

dt

≥σu+_n–aλ π



|un|αun++u+ndt

≥σu+_n–aλ(π)β–βα

π



|un|α×

β α_dt

α

β

u+_n_∞– πaλu+_n_∞

≥σu+_n–aλ(π)β–βα_C_u

nα_Lβu+n– πaλCu+n

for alln, where we use the Hölder inequality and (.). We have

u+

n un→

, asn→ ∞. (.)

Similarly foru–

n, we also get u–

n un →

, asn→ ∞. (.)

Again, by (H), we have

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sinceβ>α≥. Since dimH_<_∞_,

ϕ(un) –

ϕ(un),un

=λ π



∇F(λt,un),un

– F(λt,un)

dt

≥λ π



b|un|–c

dt

≥λ π



bu

n–bu+n–bu–n–c

dt

≥cun–cun+–cu–n–c.

So we get

u_n

un

→, asn→ ∞. (.)

Hence by (.)-(.), one has

 =un

un≤ u

n+u+n+u–n

un →

, asn→ ∞,

a contradiction. Therefore,{un}is bounded inX.

Using similar arguments to the proof of Claim  in Lemma .,{un}possesses a strong

convergent subsequence inX. The proof of Lemma . is complete.

Lemma . Under assumption(H)-(H),the functionϕsatisﬁes the conditions(f), (f),

and(f)of TheoremA.

Proof We only consider the case (i) in (H). The other case is similar. () We claim thatϕhas a local linking at  with respect to (X,X). By (H), for anyε> , there existsδ>  such that

F(t,x)≤ε|x|, ∀t∈[, τ],|x| ≤δ. (.)

By (.) and (.), there existsc>  such that

F(t,x)≤ε|x|+c|x|+α, ∀t∈[, τ],x∈RN. (.)

Hence, foru∈X_{, we have}

ϕ(u)≥ 

Lu,u–λ



π



ε|u|+c|u|+α

dt≥σ

u

_–_ελ_C_u_–_c

λCu+α.

Takingε=σ/(λC) and noting thatα>  in (H), we can ﬁnd a constantρ>  such

that

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On the other hand, letu=u₊_u–_∈_X_satisfying_u_≤_ρ

=R/(C) and let

=

t∈[, π]|u–≤R/

, =

t∈[, π]|u–>R/

. Then we have

_u_≤_u

∞≤Cu≤R/

for allt∈[, π]. Consequently,

|u| ≤u+u–≤R, for allt∈.

Hence, from (H)(i), we obtain



F(t,u)dt≥.

On the one hand, one obtains

|u| ≤u+u–≤R/ +u–≤u–, for allt∈.

It follows from (.) that

F(λt,u)≤ε|u|+c|u|+α≤εu– 

+ +αcu– +α

, for allt∈, which implies that



F(λt,u)dt≤ε



u–dx+ +αc



u–+αdt

≤ε π



u–dx+ +αc

π



u–+αdt

≤εCu–+ (C)+αcu– +α

. Settingε=σ/(C_{), we have}

ϕ(u)≤–σ 

u–,u––λ π



F(λt,u)dt≤–σ u

–_{+ (}_C₎+α_c u–

+α

.

Consequently,

ϕ(u)≤, ∀u∈Xwithu ≤ρ,

whereρ<ρis small enough.

() We claim that for everym∈N,ϕ(u)→–∞asu → ∞onX

m⊕X.

SinceX_m andHare ﬁnite dimensional, we can chooseσm> ,Cm>  such that

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u ≤CmuL, for allu∈X_m⊕H.

By (H), there existsc>  such that

F(t,x)≥σm

Cm|

x|–c, ∀t∈[, τ],∀x∈RN.

So foru∈X

m⊕X, ϕ(u) = 



Lu+,u++ 

Lu–,u––λ π



Fλt,u(t)dt

≤σm

 u

+_–σ

u

–_–σm

Cm π



u(t)dt+c

= σm  u

+

–σ u

–

–σm

Cm

u+_L+u–_L+u_L

+c

≤–σm  u

+

–σ u

–

–σmu+c→–∞,

foru → ∞.

() By (.), we see thatϕmaps bounded sets into bounded sets.

The proof of Lemma . is complete.

Proof of Theorem. By Lemmas . and ., all conditions of Theorem A are satisﬁed. Thus, problem (.) has at least one nonzero critical point.

Proof of Theorem. By Lemmas . and ., all conditions of Theorem A are satisﬁed. Thus, problem (.) has at least one nonzero critical point.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors would like to express their sincere thanks to the referees for their helpful comments. Supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-1) of China and by the National Natural Science Foundation of China (No. 61473180).

Received: 28 August 2015 Accepted: 25 January 2016

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