Neutral operator with variable parameter and third order neutral differential equation

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R E S E A R C H

Open Access

Neutral operator with variable parameter and

third-order neutral differential equation

Yun Xin

1*

_{and Zhibo Cheng}

2

*_{Correspondence:}

[email protected]

1_{College of Computer Science and}

Technology, Henan Polytechnic University, Jiaozuo, 454000, China Full list of author information is available at the end of the article

Abstract

In this article, we discuss the properties of the neutral operator with variable parameter (Ax)(t) =x(t) –c(t)x(t–

δ

(t)) and by applying Green’s function of a

third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity of positive periodic solutions for a generalized third-order neutral differential equation.

Keywords: neutral operator; variable parameter; positive solutions; third-order; Green’s function

1 Introduction

In [], Zhang discussed the properties of the neutral operator (Ax)(t) =x(t) –cx(t–δ), which became an eﬀective tool for the research on diﬀerential equations with this pre-scribed neutral operator (see,e.g., [–]). Lu and Ge [] investigated an extension ofA, namely the neutral operator (Ax)(t) =x(t) –

n

i=cix(t–δi), and obtained the existence of

periodic solutions for the corresponding neutral differential equation. Afterwards, Duet al.[] studied the neutral operator (Ax)(t) =x(t) –c(t)x(t–δ), herec(t) isω-periodic func-tions. By means of Mawhin’s continuation theorem and the properties ofA, they obtained sufficient conditions for the existence of periodic solutions to a Liénard neutral differen-tial equation. Recently, in [], Renet al.investigated the neutral operator with variable delay (A)x(t) –cx(t–δ(t)). By applying coincidence degree theory, they obtained suffi-cient conditions for the existence of periodic solutions to a Rayleigh neutral differential equation.

Motivated by [, –], in this paper, we consider the neutral operator (Ax)(t) =x(t) –

c(t)x(t–δ(t)), here |c(t)| = , c,δ∈C₍_R_,_R_{) and} _δ _{is an}_{ω-periodic function for some} ω> . Notice that here the neutral operatorAis a natural generalization of the famil-iar operatorAi, i= , , , . But Apossesses a more complicated nonlinearity thanAi,

i= , , , . For example, the neutral operatorAi,i= , , is homogeneous in the following sense (Aix)(t) = (Aix)(t),i= , , whereas the neutral operatorAin general is inhomo-geneous. As a consequence, many of the new results for diﬀerential equations with the neutral operatorAwill not be a direct extension of known theorems for neutral diﬀeren-tial equations.

The paper is organized as follows. In Section , we ﬁrst analyze qualitative properties of the generalized neutral operatorAwhich will be helpful for further studies of diﬀeren-tial equations with this neutral operator; in Section , we consider a third-order neutral

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diﬀerential equation as follows:

x(t) –c(t)xt–δ(t)=a(t)x(t) –λb(t)fxt–τ(t), (.) hereλis a positive parameter;δ(t) is said to be variable delay,c,δ∈C₍_R_,_R_{) and}_δ_{is an} ω-periodic function for someω> ,f∈C(R, [,∞)), andf(x) >  forx> ;a∈C(R, (,∞)) withmax{a(t) :t∈[,ω]}< 

√( π

ω),b∈C(R, (,∞)),τ∈C(R,R),a(t),b(t) andτ(t) are ω-periodic functions. By applying Green’s function of a third-order differential equation and a fixed point theorem in cones, we obtain sufficient conditions for the existence, mul-tiplicity and nonexistence of positive periodic solutions to the third-order neutral differ-ential equation. We will give an example to illustrate our results, and an example is also given in this section. Our results improve and extend the results in [–].

2 Analysis of the generalized neutral operator with variable parameter

Let

c∞= max

t∈[,ω]

c(t), c= min

t∈[,ω]

c(t).

LetX={x∈C(R,R) :x(t+ω) =x(t),t∈R}with the normx=maxt∈[,ω]|x(t)|, and letC+

ω={x∈C(R, (,∞)) :x(t+ω) =x(t)},Cω–={x∈C(R, (–∞, )) :x(t+ω) =x(t)}. Then (X, · ) is a Banach space. A coneKinXis deﬁned byK={x∈X:x(t)≥αx,∀t∈R},

whereαis a ﬁxed positive number withα< . Moreover, deﬁne operatorsA,B:Cω→Cω by

(Ax)(t) =x(t) –c(t)xt–δ(t), (Bx)(t) =c(t)xt–δ(t).

Lemma . If|c(t)| = ,then the operator A has a continuous inverse A–_{on C}_ω,_satisfying

()

A–f(t) =

⎧ ⎪ ⎨ ⎪ ⎩

f(t) +∞_j_=j_i_=c(Di)x(t–j_i_=δ(Di)) for|c(t)|< ,∀f ∈Cω, –f_c(₍t_t₊+_δδ(₍_tt))₎₎–∞_j_=f(t+δ(t)+

j i=δ(Di))

c(t+δ(t))j_i_=c(D_i) for|c(t)|> ,∀f ∈Cω.

()

A–f(t)≤

⎧ ⎨ ⎩

f

–c∞ forc∞< ∀f ∈Cω,

f

c– forc> ∀f ∈Cω.

()

ω



A–f(t)dt≤ ⎧ ⎨ ⎩

 –c∞

ω

 |f(t)|dt forc∞< ∀f ∈Cω, 

c–

ω

 |f(t)|dt forc> ∀f ∈Cω.

Proof Case :|c(t)| ≤c∞< .

Lett=DandDj=t– j

i=δ(Di),j= , , . . . .

(Bx)(t) =c(t)xt–δ(t)=c(D)x

t–δ(D)

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Case :|c(t)|>c> .

LetD_=t,D_j=t+j_i_=δ(D_i),j= , , . . . . And set

E:Cω→Cω, (Ex)(t) =x(t) – 

c(t)x

t+δ(t),

B:Cω→Cω, (Bx)(t) = 

c(t)x

t+δ(t).

By the deﬁnition of the linear operatorB, we have

Bj_f(t) =_j 

i=c(Di) f

t+

j

i= δD_i

,

hereDiis deﬁned as in Case . Summing overjyields

∞

j=

Bj_f(t) =f(t) +

∞

j= 

j i=c(Di)

f

t+

j

i= δD_i

.

SinceB< , we obtain that the operatorEhas a bounded inverseE–,

E–:Cω→Cω, E–= (I–B)–=I+

∞

j=

Bj_,

and∀f ∈Cωwe get

E–f(t) =f(t) +

∞

j=

Bj_f(t).

On the other hand, from (Ax)(t) =x(t) –c(t)x(t–δ(t)), we have

(Ax)(t) =x(t) –c(t)xt–δ(t)= –c(t)

xt–δ(t)– 

c(t)x(t)

,

i.e.,

(Ax)(t) = –c(t)(Ex)t–δ(t).

Letf ∈Cωbe arbitrary. We are looking forxsuch that

(Ax)(t) =f(t),

i.e.,

–c(t)(Ex)t–δ(t)=f(t).

Therefore

(Ex)(t) = –f(t+δ(t))

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and hence

proving thatA–_{exists and satisﬁes}

Statements () and () are proved. From the above proof, () can easily be deduced.

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3 Positive periodic solutions for third-order neutral equations

At ﬁrst, we introduce the following Green’s functions and properties of Green’s functions, which can be found in [].

has a unique solution which is of the form

u(t) =

has a uniqueω-periodic solution

u(t) =

Now we present the properties of the Green’s functions for (.), (.).

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Theorem . _ωG(t,s)ds=_ρ and if √

ρω<_πholds,then <l<G(t,s)≤L for all

t∈[,ω]and s∈[,ω].

Theorem . _ωG(t,s)ds=_ρ and if √

ρω<_πholds,then <l<G(t,s)≤L for all [,ω]and s∈[,ω].

Deﬁne the Banach spaceXas in Section . Denote

M=maxa(t) :t∈[,ω], m=mina(t) :t∈[,ω], ρ=M,

k=l(M+m) +σLM, k=

k–√k_{– σ}_LlMm

σLM , α=

l[m– (M+m)c∞]

LM( –c∞) .

It is easy to see thatM,m,β,k,k> . Now we consider (.). First let

f_=lim x→

f(x)

x , f∞=xlim→∞ f(x)

x , f=_xlim_→_

f(x)

x , f∞=_xlim_→∞

f(x)

x ,

and denote

i= number of ’s in (f,f∞), i= number of ’s in (f_,f_∞);

i∞= number of∞’s in (f,f∞), i∞= number of∞’s in (f_,f_∞).

It is clear thati,i,i∞,i∞∈ {, , }. We will show that (.) hasiori∞positivew-periodic

solutions for suﬃciently large or smallλ, respectively.

In what follows, we discuss (.) in two cases, namely the case wherec(t) <  and –c∞>

–min{k,_Mm₊_m}.

From –c∞> –Mm+m, we have α =

l[m–(M+m)c∞]

LM(–c∞) >

l(m–(M+m)· m M+m)

LM(–c∞) = . So, we getα > . Moreover, we consider the equation

σLMx–kx+lm= .

Then the equation has a solutionx=k=k–

√

k_–_σ_LlMm

σLM . Fromc∞<k, we can get

σLMc_∞–kc∞+lm< . So, we have

σLMc_∞–l(M+m) +σLMc∞+lm< ,

we get

σc∞>l[m– (M+m)c∞]

LM( –c∞) =α.

On the other hand, the case wherec>  andc∞<min{Mm+m, LM–lm

(L–l)M–lm}(note thatc∞< m

M+m impliesα> ;c∞< LM–lm

(L–l)M–lm impliesα< ). Obviously, we havec∞< , which makes

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LetK={x∈X:x(t)≥αx}denote the cone inXas deﬁned in Section , whereαis just as deﬁned above. We also useKr={x∈K:x<r}and∂Kr={x∈K:x=r}.

Lety(t) = (Ax)(t), then from Lemma . we havex(t) = (A–_y₎₍_t_{). Hence (.) can be} trans-formed into

y(t) –a(t)A–y(t) = –λb(t)fA–yt–τ(t), (.) which can be further rewritten as

y(t) –a(t)y(t) +a(t)Hy(t)= –λb(t)fA–yt–τ(t), (.) whereH(y(t)) =y(t) – (A–_y₎₍_t_{) = –}_c₍_t₎₍_A–_y₎₍_t_–_δ(_t_)).

Now we discuss the two cases separately.

3.1 Case I:c(t) < 0 and –c_∞> – min{k1,_Mm₊_m} Now we consider

y(t) –a(t)y(t) +a(t)Hy(t)=h(t), h∈C–ω, (.) and deﬁne the operatorsT,Hˆ :X→Xby

(Th)(t) =

t+ω

t

G(t,s)

–h(s)ds, (Hyˆ )(t) = –M+a(t)y(t) –a(t)Hy(t).

ClearlyT, Hˆ are completely continuous, (Th)(t) >  forh(t) <  and ˆH ≤(M–m+

M_–c∞_c_∞). By Theorem ., the solution of (.) can be written in the form

y(t) = (Th))(t) + (THyˆ )(t). (.)

In view ofc(t) <  and –c∞> –min{k,_Mm₊_m}, we have

THˆ ≤ T ˆH ≤M–m+mc∞

M( –c∞) < , (.)

where we used the fact_tt+ωG(t,s)ds=_M. Hence

y(t) = (I–THˆ)–(Th)(t).

Deﬁne an operatorP:X→Xby

(Ph)(t) = (I–THˆ)–(Th)(t).

Obviously, for anyh∈Cω–, ifmax{a(t) :t∈[,ω]}<_√_( π ω)

_,_y₍_t_{) = (}_Ph₎₍_t_{) is the unique}

positiveω-periodic solution of (.).

Lemma . P is completely continuous and

(Th)(t)≤(Ph)(t)≤ M( –c∞)

m– (M+m)c∞Th for all h∈C

–

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Proof By the Neumann expansion ofP, we have

P= (I–THˆ)–T

=I+THˆ + (THˆ)+· · ·+ (THˆ)n+· · ·T

=T+THTˆ + (THˆ)T+· · ·+ (THˆ)nT+· · ·. (.)

SinceTandHˆ are completely continuous, so isP. Moreover, by (.), and recalling that THˆ ≤ M–m+mc∞

M(–c∞) < , we get

(Th)(t)≤(Ph)(t)≤ M( –c∞)

m– (M+m)c∞Th.

Deﬁne an operatorQ:X→Xby

Qy(t) =Pλb(t)fA–yt–τ(t). (.)

Lemma . Q(K)⊂K.

Proof From the deﬁnition ofQ, it is easy to verify thatQy(t+ω) =Qy(t). Fory∈K, we have from Lemma . that

Qy(t) =Pλb(t)fA–yt–τ(t)

≥Tλb(t)fA–yt–τ(t)

=λ

t+ω

t

G(t,s)b(s)f

A–ys–τ(s)ds

≥λl

ω



b(s)fA–ys–τ(s)ds.

On the other hand,

≤ M( –c∞)

m– (M+m)c∞

Tλb(t)fA–yt–τ(t)

=λ M( –c∞)

m– (M+m)c∞tmax∈[,ω]

t+ω

t

G(t,s)b(s)f

A–ys–τ(s)ds

≤λ M( –c∞)

m– (M+m)c∞L

ω



b(s)fA–ys–τ(s)ds.

Therefore

Qy(t)≥l[m– (M+m)c∞]

LM( –c∞) Qy=αQy,

i.e.,Q(K)⊂K.

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the existence of positive periodic solutions for (.) is equivalent to the existence of ﬁxed points ofQinK. Furthermore, ifQhas a ﬁxed pointyinK, it means that (A–_y₎₍_t_{) is a} positiveω-periodic solution of (.).

Lemma . If there existsη> such that

fA–yt–τ(t)≥A–yt–τ(t)η for t∈[,ω]and y∈K,

then

Qy ≥λlη

α  –c

 – c∞

 –c

∞

ω



b(s)dsy, y∈K.

Proof By Lemma . and Lemma ., we have fory∈Kthat

≥Tλb(t)fA–yt–τ(t)

=λ

t+ω

t

G(t,s)b(s)f

A–ys–τ(s)ds

≥λlη

ω



b(s)A–ys–τ(s)ds

≥λlη

α  –c

 – c∞

 –c

∞

ω



b(s)dsy.

Hence

Qy ≥λlη

α  –c

 – c∞

 –c

∞

ω



b(s)dsy, y∈K.

Lemma . If there existsε> such that

fA–yt–τ(t)≤A–yt–τ(t)ε for t∈[,ω]and y∈K,

then

Qy ≤λε LM

ω  b(s)ds

m– (M+m)c∞y, y∈K.

Proof By Lemma . and Lemma ., we have

Qy(t) ≤λ M( –c∞)

m– (M+m)c∞L

ω



b(s)fA–ys–τ(s)ds

≤λ M( –c∞)

m– (M+m)c∞Lε

ω



b(s)A–ys–τ(s)ds

≤λε LM

ω  b(s)ds

m– (M+m)c∞y.

Deﬁne

F(r) =max

f(t) : ≤t≤ r

 –c∞

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f(r) =min

f(t) :

α  –c

 – c∞

 –c

∞

r≤t≤ r

 –c∞

.

Lemma . If y∈∂Kr,then

Qy ≥λlf(r)

ω



b(s)ds.

Proof By Lemma ., we obtain ( α –c_ –

c∞

–c∞)r≤(A

–_y₎₍_t_–_τ₍_t₎₎_≤ r

–c∞ fory∈∂Kr, which yieldsf((A–_y₎₍_t_–_τ₍_t₎₎₎_≥_f

(r). The lemma now follows analogous to the proof

of Lemma ..

Lemma . If y∈∂Kr,then

Qy ≤λLM( –c∞)F(r)

m– (M+m)c∞

ω



b(s)ds.

Proof By Lemma ., we can have ≤(A–_y₎₍_t_–_τ₍_t₎₎_≤ r

–c∞ fory∈∂Kr, which yields

f((A–y)(t–τ(t)))≤F(r). Similar to the proof of Lemma ., we get the conclusion.

We quote the ﬁxed point theorem which our results will be based on.

Lemma .[] Let X be a Banach space and K be a cone in X.For r> ,deﬁne Kr= {u∈K:u<r}.Assume that T:Kr→K is completely continuous such that Tx=x for x∈∂Kr={u∈K:u=r}.

(i) IfTx ≥ xforx∈∂Kr,theni(T,Kr,K) = ; (ii) IfTx ≤ xforx∈∂Kr,theni(T,Kr,K) = .

Now we give our main results on positive periodic solutions for (.).

Theorem .

(a) Ifi= or,then(.)hasipositiveω-periodic solutions forλ>_f  ()l

ω

b(s)ds> ;

(b) Ifi_∞= or,then(.)hasi_∞positiveω-periodic solutions for

 <λ<_LM_(–m_c–(_∞M₎_F+_()m)cω∞

b(s)ds;

(c) Ifi∞= ori= ,then(.)has no positiveω-periodic solutions for suﬃciently small

or suﬃciently largeλ> ,respectively.

Proof (a) Chooser = . Take λ=_f  (r)l

ω

b(s)ds > , then for allλ>λ, we have from Lemma . that

Qy>y fory∈∂Kr. (.)

Case . Iff_= , we can choose  <¯r<r, so thatf(u)≤εufor ≤u≤ ¯r, where the constantε>  satisﬁes

λε LM

ω  b(s)ds

m– (M+m)c∞ < . (.)

Letr= ( –c∞)¯r, we havef((A–y)(t–τ(t)))≤ε(A–y)(t–τ(t)) fory∈Kr. By Lemma ., we have ≤(A–_y₎₍_t_–_τ₍_t₎₎_≤ y

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have fory∈∂Kr that

Qy ≤λε LM

ω  b(s)ds

m– (M+m)c∞y<y.

It follows from Lemma . and (.) that

i(Q,Kr,K) = , i(Q,Kr,K) = ,

thusi(Q,Kr\ ¯Kr,K) = – andQhas a ﬁxed pointyinKr\ ¯Kr, which means that (A–y)(t) is a positiveω-positive solution of (.) forλ>λ.

Case . Iff_∞= , there exists a constantH˜ >  such thatf(u)≤εuforu≥ ˜H, where the constantε>  satisﬁes

λε LM

ω  b(s)ds

m– (M+m)c∞ < . (.)

Letr=max{r,

˜

H(–c )(–c∞)

α(–c∞)–c∞(–c)}, we havef((A

–_y₎₍_t_–_τ₍_t₎₎₎_≤_ε(_A–_y₎₍_t_–_τ₍_t_{)) for} _y_∈

Kr. By Lemma ., we have (A

–_y₎₍_t_–_τ₍_t₎₎_≥₍ α –c

 – c∞

–c_∞)y ≥ ˜Hfory∈∂Kr. Thus by Lemma . and (.), we have fory∈∂Kr that

Qy ≤λε LM

ω  b(s)ds

m– (M+m)c∞y<y.

Recalling Lemma . and (.) that

i(Q,Kr,K) = , i(Q,Kr,K) = ,

theni(Q,Kr\ ¯Kr,K) =  andQhas a ﬁxed pointyinKr\ ¯Kr, which means that (A–y)(t) is a positiveω-positive solution of (.) forλ>λ.

Case . Iff_=f_∞= , from the above arguments, there exist  <r<r<rsuch thatQ has a ﬁxed pointy(t) inKr\ ¯Krand a ﬁxed pointy(t) inKr\ ¯Kr. Consequently, (A–y)(t) and (A–_y

)(t) are two positiveω-periodic solutions of (.) forλ>λ. (b) Letr= . Takeλ=_LM_(–m_c_∞–(₎M_F+₍_rm)c∞

)ωb(s)ds > , then by Lemma . we know ifλ<λ then

Qy<y, y∈∂Kr. (.)

Case . Iff

=∞, we can choose  <¯r<rso thatf(u)≥ηufor ≤u≤ ¯r, where the constantη>  satisﬁes

λlη

α  –c

 – c∞

 –c

∞

ω



b(s)ds> . (.)

Letr= ( –c∞)r¯, we havef((A–y)(t–τ(t)))≥η(A–y)(t–τ(t)) fory∈Kr. By Lemma ., we have ≤(A–_y₎₍_t_–_τ₍_t₎₎_≤ y

–c∞ ≤ ¯rfory∈∂Kr. Thus by Lemma . and (.),

Qy ≥λlη

α  –c

 – c∞

 –c

∞

ω



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It follows from Lemma . and (.) that

i(Q,Kr,K) = , i(Q,Kr,K) = ,

which impliesi(Q,Kr\ ¯Kr,K) =  andQhas a ﬁxed pointyinKr\ ¯Kr. Therefore (A–y)(t) is a positiveω-periodic solution of (.) for  <λ<λ.

Case . Iff_∞=∞, there exists a constantH˜ >  such thatf(u)≥ηuforu≥ ˜H, where the constantη>  satisﬁes

λlη

α  –c

 – c∞

 –c

∞

ω



b(s)ds> . (.)

Letr=max{r,

˜

H(–c)(–c∞)

α(–c∞)–c∞(–c_)}, we havef((A

–_y₎₍_t_–_τ₍_t₎₎₎_≥_η(_A–_y₎₍_t_–_τ₍_t_{)) for}_y_∈

Kr. By Lemma ., we have (A

–_y₎₍_t_–_τ₍_t₎₎_≥₍ α –c–

c∞

–c∞)y ≥ ˜Hfory∈∂Kr. Thus by Lemma . and (.), we have fory∈∂Kr that

Qy ≥λlη

α  –c

 – c∞

 –c

∞

ω



b(s)dsy>y.

It follows from Lemma . and (.) that

i(Q,Kr,K) = , i(Q,Kr,K) = ,

i.e.,i(Q,Kr\ ¯Kr,K) = – andQhas a ﬁxed pointyinKr\ ¯Kr. That means (A–y)(t) is a positiveω-periodic solution of (.) for  <λ<λ.

Case . Iff

=f∞=∞, from the above arguments,Qhas a ﬁxed pointy inKr\ ¯Kr and a ﬁxed pointy inKr\ ¯Kr. Consequently, (A–y)(t) and (A–y)(t) are two positive ω-periodic solutions of (.) for  <λ<λ.

(c) By Lemma ., ify∈K, then (A–_y₎₍_t_–_τ₍_t₎₎_≥₍ α –c_–

c∞

–c∞)y>  fort∈[,ω]. Case . Ifi_= , we havef

>  andf∞> . Letb=min{

f(u)

u ;u> }> , then we obtain

f(u)≥bu, u∈[, +∞).

Assume that y(t) is a positive ω-periodic solution of (.) for λ > λ, where λ = (–c)(–c∞)

lb[α(–c∞)–c∞(–c)]

ω

b(s)ds> . SinceQy(t) =y(t) fort∈[,ω], then by Lemma . if λ>λ,

we have

y=Qy ≥λlb

α  –c

 – c∞

 –c

∞

ω



b(s)dsy>y,

which is a contradiction.

Case . Ifi∞= , we havef<∞andf∞<∞. Letb=max{f(_uu) :u> }> , then we obtain

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Assume that y(t) is a positive ω-periodic solution of (.) for  <λ<λ, where λ=

m–(M+m)c∞

bLMωb(s)ds. SinceQy(t) =y(t) fort∈[,ω], it follows from Lemma . that

y=Qy ≤λb

LM_ωb(s)ds

m– (M+m)c∞y<y,

which is a contradiction.

Theorem .

(a) If there exists a constantb> such thatf(u)≥buforu∈[, +∞),then(.)has

no positiveω-periodic solution forλ> (–c)(–c∞)

lb[α(–c∞)–c∞(–c)]

ω

b(s)ds.

(b) If there exists a constantb> such thatf(u)≤buforu∈[, +∞),then(.)has

no positiveω-periodic solution for <λ<_bm–(M+m)c∞ LMωb(s)ds.

Proof From the proof of (c) in Theorem ., we obtain this theorem immediately.

Theorem . Assume i_=i=i_∞=i∞= ,and that one of the following conditions holds:

() f_≤f_∞; () f

>f∞;

() f

≤f∞≤f≤f∞;

() f_∞≤f

≤f∞≤f.

If

( –c

)( –c∞)

l[α( –c

∞) –c∞( –c)]

ω

 b(s)dsmax{f,f,f∞,f∞}

<λ< m– (M+m)c∞

LM_ωb(s)dsmin{f

,f,f∞,f∞} ,

then(.)has one positiveω-periodic solution.

Proof Case . Iff_≤f_∞, then

( –c

)( –c∞)

l[α( –c

∞) –c∞( –c)]

ω  b(s)ds

<λ<m– (M+m)c∞

LM_ωb(s)ds .

It is easy to see that there exists  <ε<f∞such that

( –c

)( –c∞)

(f¯∞–ε)l[α( –c

∞) –c∞( –c_)]_ωb(s)ds<λ<

m– (M+m)c∞

(f

+ε)LM

ω  b(s)ds

.

For the aboveε, we choose¯r>  such thatf(u)≤(f_+ε)ufor ≤u≤ ¯r. Letr= (–c∞)¯r, we havef((A–y)(t–τ(t)))≤(f

+ε)(A

–_y₎₍_t_–_τ₍_t_{)) for}_y_∈_Kr

. By Lemma ., we have ≤(A–_y₎₍_t_–_τ₍_t₎₎_≤ y

–c∞ ≤ ¯rforK∈∂Kr. Thus by Lemma . we have fory∈∂Krthat

Qy ≤λ(f

+ε)

LM_ωb(s)ds

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On the other hand, there exists a constantH˜ >  such thatf(u)≥(f_∞–ε)uforu≥ ˜H. Letr=max{r,

˜

H(–c_)(–c_∞)

α(–c∞)–c∞(–c_)}, we havef((A

–_y₎₍_t_–_τ₍_t₎₎₎_≥₍_f

∞–ε)(A–y)(t–τ(t)) for

y∈Kr. By Lemma ., we have (A–y)(t–τ(t))≥( α –c_–

c∞

–c∞)y ≥ ˜Hfory∈∂Kr. Thus by Lemma ., fory∈∂Kr,

Qy ≥λl(f_∞–ε)

α  –c_–

c∞

 –c

∞

ω



b(s)dsy>y.

It follows from Lemma . that

i(Q,Kr,K) = , i(Q,Kr,K) = ,

thusi(Q,Kr\ ¯Kr,K) = – andQhas a ﬁxed pointyinKr\ ¯Kr. So (A

–_y₎₍_t_{) is a positive}

ω-periodic solution of (.). Case . Iff

>f∞, in this case, we have ( –c_)( –c

∞)

¯

fl[α( –c∞) –c∞( –c)]

ω  b(s)ds

<λ< m– (M+m)c∞

f_∞LM_ωb(s)ds.

It is easy to see that there exists  <ε<fsuch that

( –c_)( –c

∞)

(f¯–ε)l[α( –c∞) –c∞( –c)]

ω  b(s)ds

<λ< m– (M+m)c∞ (f_∞+ε)LM_ωb(s)ds.

For the aboveε, we chooser¯>  such thatf(u)≥(f–ε)ufor ≤u≤ ¯r. Letr= (–c∞)¯r, we havef((A–y)(t–τ(t)))≥(f_–ε)(A–y)(t–τ(t)) fory∈Kr. By Lemma ., we have ≤(A–_y₎₍_t_–_τ₍_t₎₎_≤ y

–c∞≤ ¯rfory∈∂Kr. Thus we have by Lemma . that fory∈∂Kr,

Qy ≥λl(f_–ε)

α  –c

 – c∞

 –c

∞

ω



b(s)dsy>y.

On the other hand, there exists a constantH˜ >  such thatf(u)≤(f_∞+ε)uforu≥ ˜H. Letr=max{r,

˜

H(–c_)(–c∞)

α(–c∞)–c∞(–c_)}, we havef((A

–_y₎₍_t_–_τ₍_t₎₎₎_≤₍_f

∞+ε)(A–y)(t–τ(t)) for

y∈Kr. By Lemma . we have (A–y)(t–τ(t))≥( α –c_ –

c∞

–c∞)y ≥ ˜Hfory∈∂Kr. Thus by Lemma ., fory∈∂Kr,

Qy ≤λ(f_∞+ε) LM

ω  b(s)ds

m– (M+m)c∞y.

It follows from Lemma . that

i(Q,Kr,K) = , i(Q,Kr,K) = .

Thusi(Q,Kr\ ¯Kr,K) = – andQhas a ﬁxed pointyinKr\ ¯Kr, proving that (A–y)(t) is a positiveω-periodic solution of (.).

Case .f

≤f∞≤f≤f∞. The proof is the same as in Case . Case .f_∞≤f

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3.2 Case II:c(t) > 0 andc_∞< min{_Mm₊_m,₍_LLM_–_l₎_M–lm_–_lm} Deﬁne

f(r) =min

f(t) : α  –c

r≤t≤ r

 –c∞

.

Similarly as in Section ., we get the following results.

Theorem .

(a) Ifi= or,then(.)hasipositiveω-periodic solutions forλ>_f  ()l

_ω

b(s)ds> .

(b) Ifi_∞= or,then(.)hasi∞positiveω-periodic solutions for

 <λ<_LM_(–m_c–(M+m)c∞ ∞)F()_ωb(s)ds.

(c) Ifi∞= ori_= ,then(.)has no positiveω-periodic solution for suﬃciently small or largeλ> ,respectively.

Theorem .

(a) If there exists a constantb> such thatf(u)≥buforu∈[, +∞),then(.)has no

positiveω-periodic solution forλ> –c

lαbωb(s)ds.

(b) If there exists a constantb> such thatf(u)≤buforu∈[, +∞),then(.)has no

positiveω-periodic solution for <λ<_bm–(M+m)c∞ LM

ω

b(s)ds.

Theorem . Assume that i_=i=i∞=i∞= hold,and that one of the following con-ditions holds:

() f_≤f_∞; () f

>f∞;

() f

≤f∞≤f≤f∞;

() f_∞≤f

≤f∞≤f.

If

 –c

lα_ωb(s)dsmax{f

,f,f∞,f∞}

<λ< m– (M+m)c∞

LM_ωb(s)dsmin{f

,f,f∞,f∞} ,

then(.)has one positiveω-periodic solution.

Remark  In a similar way, one can consider the third-order neutral functional diﬀerential equation (x(t) –c(t)x(t–δ(t)))+a(t)x(t) =λb(t)f(x(t–τ(t))).

We illustrate our results with an example.

Example . Consider the following third-order neutral diﬀerential equation:

u(t) +  

 – sint

ut–cost

– 

 – sin

_t

u(t)

= –λ( –cost)ut–τ(t)au(t–τ(t)), (.)

whereλand  <a<  are two positive parameters,τ(t+π) =τ(t).

Comparing (.) to (.), we see thatδ(t) =cost,c(t) = –_ ( –_sint),a(t) =_( – 

sin

_t_),_b₍_t_{) =  –}_cos__t_,_ω₌_π_,_f₍_u_{) =}_u_au_{. Clearly,}_c

∞= ,c=  ,M=

 ,m=

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and we getρ=_, noticing that

√

π  <

π

 holds.f= ,f∞= ,i= . By Theorem ., we easily get the following conclusion: equation (.) has two positiveπ-periodic solutions forλ> 

πr, wherer=min{f(.),f(  )}. In fact, by simple computations, we have

l= 

ρ(exp(ρω) – )= ., L=

 + exp(–ρω_ ) ρ_{( –}_exp_(–ρω

 )) = .

k= ., k= ., α= .,

c∞=  <min

k,

m M+m

= ., c∞= 

< . =α,

and

f() =min

f(t) : .≈ α  –c_–

c∞

 –c

∞ ≤

t≤



=min

f(.),f

 

=r,



f()l

π  b(s)ds

=  πr

.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YX and ZBC worked together in the derivation of the mathematical results. Both authors read and approved the ﬁnal manuscript.

Author details

1_{College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, 454000, China.}2_{School of}

Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China.

Acknowledgements

YX and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (Nos. 11326124, 11271339) and Education Department of Henan Province project (No. 14A110002).

Received: 16 June 2014 Accepted: 13 October 2014 Published: 24 October 2014

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doi:10.1186/1687-1847-2014-273