Periodic solutions for p Laplacian neutral differential equation with multiple delay and variable coefficients

R E S E A R C H

Open Access

Periodic solutions for

p

-Laplacian neutral

differential equation with multiple delay and

variable coefficients

Zhonghua Bi

_{, Zhibo Cheng}

_{and Shaowen Yao}

*_{Correspondence:}

[email protected]

1_{School of Mathematics and} Information Science, Henan Polytechnic University, Jiaozuo, China

Full list of author information is available at the end of the article

Abstract

In this paper, we first discuss some properties of the neutral operator with multiple delays and variable coefficients (Ax)(t) :=x(t) –n_i=1ci(t)x(t–

δ

an extension of Mawhin’s continuation theorem, a second orderp-Laplacian neutral differential equation

φ

x(t) –

n

i=1

ci(t)x(t–

δ

=˜f

(

is studied. Some new results on the existence of a periodic solution are obtained. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from those known in the literature.

MSC: 34C25; 34K14

Keywords: Neutral operator;p-Laplacian; Periodic solution; Extension of Mawhin’s continuation theorem; Singularity

1 Introduction

In this paper, we consider a second orderp-Laplacian neutral differential equation

φp

x(t) –

n

i=1

ci(t)x(t–δi)

=f˜t,x(t),x(t) , (1.1)

whereφp:R→Ris given byφp(s) =|s|p–2s, herep> 1 is a constant,ci(t)∈C1(R,R) and

ci(t+T) =ci(t) andδi are constants in [0,T) fori= 1, 2, . . . ,n;f˜: [0,T]×R×R→R is anL2_{-Carathéodory function, i.e., it is measurable in the first variable and} continu-ous in the second variable, and for every 0 <r<sthere existshr,s∈L2[0,T] such that

|˜f(t,x(t),x(t))| ≤hr,sfor allx∈[r,s] and a.e.t∈[0,T].

The study of the properties of the neutral operator (A1x)(t) :=x(t) –cx(t–δ) began with the paper of Zhang [2]. In 2004, Lu and Ge [14] investigated an extension ofA1, namely the neutral operator (A2x)(t) :=x(t) –

n

i=1cix(t–δi). Afterwards, Du [6] discussed the neu-tral operator (A3x)(t) :=x(t) –c(t)x(t–δ), herec(t) is aT-periodic function. And by using

Mawhin’s continuation theorem and the properties ofA3, they obtained sufficient condi-tions for the existence of periodic solucondi-tions to the following Liénard neutral differential equation:

x(t) –c(t)x(t–τ) +fx(t) x(t) +gxt–γ(t) =e(t).

In recent years, many works have been published on the existence of periodic solutions of second-order neutral differential equations (see [1,3–5,7,9,11–13,16–19]). In 2007, Zhu and Lu [19] discussed the existence of periodic solutions for ap-Laplacian neutral differential equation

φp

x(t) –cx(t–τ) +gt,xt–δ(t) =p(t).

Since (φp(x(t)))is nonlinear (i.e., quasilinear), Mawhin’s continuation theorem [8] can-not be applied directly. In order to get around this difficulty, Zhu and Lu translated the

p-Laplacian neutral differential equation into a two-dimensional system

⎧ ⎨ ⎩

(x1(t) –cx1(t–τ))(t) =φq(x2(t)) =|x2(t)|q–2x2(t),

x₂(t) = –g(t,x1(t–δ(t))) +p(t),

where 1_p+1_q = 1, for which Mawhin’s continuation theorem can be applied. Afterwards, Du [5] discussed the existence of a periodic solution for ap-Laplacian neutral differential equation

φp

x(t) –c(t)x(t–τ) +fx(t)x(t) +gxt–γ(t) =e(t),

by applying Mawhin’s continuation theorem.

However, the existence of a periodic solution forp-Laplacian neutral differential equa-tion (1.1) has not been studied until now. The obvious difficulty lies in the following two respects. First, although (Ax)(t) =x(t) –n_i=1ci(t)x(t–δi) is a natural generalization of the operators A1,A2andA3, the class of neutral differential equations withAtypically possesses a more complicated nonlinearity than neutral differential equations withA1,A2 andA3. Second, we do not get (Ax)(t) = (Ax)(t), meanwhile a priori bounds of periodic solutions are not easy to estimate.

2 Analysis of the generalized neutral operator

Let

ci:= max t∈[0,T]

ci(t), i= 1, 2, . . .n; ck:=max

c1,c2, . . . ,cn

.

SetCT:={x∈C(R,R) :x(t+T) =x(t),t∈R}, then (CT, · ) is a Banach space. Define operatorsA,B:CT→CT, by

(Ax)(t) =x(t) – n

i=1

ci(t)x(t–δi), (Bx)(t) = n

i=1

ci(t)x(t–δi).

Lemma 2.1 Ifn_i=1ci = 1,then operator A has a continuous inverse A–1on CT,

satisfy-ing (1)

A–1x (t)≤

⎧ ⎪ ⎨ ⎪ ⎩

x

1–n_i=1ci, for

n

i=1ci< 1; 1

_ckx

1–_ck1–n_i=1,i=kckci, for

n

i=1ci> 1;

(2)

T

0

A–1x (t)dt≤ ⎧ ⎪ ⎨ ⎪ ⎩

1 1–n_i=1ci

T

0 |x(t)|dt, for

n

i=1ci< 1; 1

ck

1–_ck1–ni=1,i=k_ckci

T

0 |x(t)|dt, for

n

i=1ci> 1.

Proof Case 1:

n

i=1

ci< 1.

(Bx)(t) = n

i=1

ci(t)x(t–δi);

B2x (t) = n

l1=1

cl1(t) n

l2=1

cl2(t–δl1)x(t–δl1–δl2);

B3x (t) = n

l1=1

cl1(t) n

l2=1

cl2(t–δl1) n

l3=1

cl2(t–δl1–δl2)x(t–δl1–δl2–δl3).

Therefore, we have

Bjx (t) = n

l1=1

cl1(t) n

l2=1

cl2(t–δl1)· · · n

lj=1

The operator (Ax)(t) =x(t) –n_i=1ci(t)x(t–δi) can be converted to

Meanwhile, we obtain

T

3 Periodic solutions for equation (1.1)

In order to use an extension of Mawhin’s continuation theorem [10], we recall it firstly. LetXandZbe Banach spaces with norms · Xand · Z, respectively. A continuous operatorM:X∩domM→Zis said to be quasilinear if

(1) ImM:=M(X∩domM)is a closed subset ofZ;

LetX1=kerMandX2be the complement space ofX1inX, thenX=X1⊕X2. Further-more,Z1 is a subspace ofZandZ2is the complement space ofZ1inZ, soZ=Z1⊕Z2. Suppose thatP:X→X1andQ:Z→Z1are two projections andΩ⊂Xis an open and bounded set with the originθ∈Ω.

LetNλ:Ω¯ →Z,λ∈[0, 1] be a continuous operator. DenoteN1byN, and let

λ={x∈

¯

Ω:Mx=Nλx}. ThenNλis said to beM-compact inΩ¯ if

(3) there is a vector subspaceZ1ofZwithdimZ1=dimX1and an operatorR:Ω¯ ×X2

being continuous and compact such that forλ∈[0, 1],

(I–Q)Nλ(Ω¯)⊂ImM⊂(I–Q)Z, (3.1)

QNλx= 0, λ∈(0, 1)⇔QNx= 0, (3.2)

R(·, 0)is the zero operator andR(·,λ)|_λ= (I–P)|_λ, (3.3)

and

MP+R(·,λ)= (I–Q)Nλ. (3.4)

LetJ:Z1→X1be a homeomorphism withJ(θ) =θ.

Next, we investigate existence of periodic solutions for Eq. (1.1) by applying the exten-sion of Mawhin’s continuation theorem.

Lemma 3.1([10]) Let X and Z be Banach spaces with norm · Xand · Z,respectively,

andΩ⊂X be an open and bounded set withθ∈Ω.Suppose that M:X∩domM→Z is a quasilinear operator and

Nλ:Ω¯ →Z, λ∈(0, 1)

is an M-compact mapping.In addition,if (a) Mx=Nλx,λ∈(0, 1),x∈∂Ω,

(b) deg{JQN,Ω∩kerM, 0} = 0,

where N=N1,then the abstract equation Mx=Nx has at least one solution inΩ¯. Theorem 3.2 Assumen_i=1ci = 1,Ωis an open bounded set in C1T.Suppose the following

conditions hold:

(i) For eachλ∈(0, 1),the equation

φp(Ax)(t) =λf˜

t,x(t),x(t) (3.5)

has no solution on∂Ω. (ii) The equation

F(a) := 1

T

T

0

˜

f(t,a, 0)dt= 0

(iii) The Brouwer degree

deg{F,Ω∩R, 0} = 0.

Then Eq. (1.1)has at least oneT-periodic solution onΩ¯.

Proof In order to use Lemma3.1we study the existence of periodic solutions to Eq. (1.1). We setX:={x∈C[0,T] :x(0) =x(T)}andZ:=C[0,T],

M:X∩domM→Z, (Mx)(t) =φp(Ax)(t), (3.6) wheredomM:={u∈X:φp(Au)∈C1(R,R)}. ThenkerM=R. In fact,

kerM=x∈X:φp(Ax)(t) = 0

=x∈X:φp(Ax)≡c

=x∈X: (Ax)≡φq(c) :=c1

=x∈X: (Ax)(t)≡c1t+c2

,

whereq> 1 is a constant with 1_p +_q1= 1 andc,c1,c2 are constants inR. Since (Ax)(0) = (Ax)(T), then we getkerM={x∈X: (Ax)(t)≡c2}. In addition,

ImM=

y∈Z, forx(t)∈X∩domM,φp(Ax) =y(t),

T

0

y(t)dt=

T

0 (φp

(Ax) dt= 0

.

SoMis quasilinear. Let

X1=kerM, X2=

x∈X:x(0) =x(T) = 0,

Z1=R, Z2=ImM.

Clearly,dimX1=dimZ1= 1, andX=X1⊕X2,P:X→X1,Q:Z→Z1, are defined by

Px=x(0), Qy= 1

T

T

0

y(s)ds.

For∀ ¯Ω⊂X, defineNλ:Ω¯ →Zby

(Nλx)(t) =λ˜f

t,x(t),x(t) .

We claim that (I–Q)Nλ(Ω¯)⊂ImM= (I–Q)Zholds. In fact, forx∈ ¯Ω, we observe that

T

0

(I–Q)Nλx(t)dt

=

T

0

=

T

0

λf˜t,x(t),x(t) dt–

T

0

λ T

T

0

˜

fs,x(s),x(s) ds dt

= 0.

Therefore, we have (I–Q)Nλ(Ω¯)⊂ImM.

Moreover, for anyx∈Z, it is obvious that

T

0

(I–Q)x(t)dt=

T

0

x(t) –

T

0 1

T

T

0

x(t)dt

dt= 0.

So, we have (I–Q)Z⊂ImM. On the other hand,x∈ImMand₀Tx(t)dt= 0, so we have

x(t) =x(t) –₀Tx(t)dt. Hence, we getx(t)∈(I–Q)Z. Therefore,ImM= (I–Q)Z. FromQNλx= 0, we get_Tλ

T

0 f˜(t,x(t),x(t))dt= 0. Sinceλ∈(0, 1), we have 1 T

T

0 f˜(t,x(t),

x(t))dt= 0. Therefore,QNx= 0, and so Eq. (3.4) also holds. LetJ:Z1→X1,J(x) =x, thenJ(0) = 0. DefineR:Ω¯×[0, 1]→X2,

R(x,λ)(t) =A–1

t

0

φ_p–1

a+

s

0

λf˜u,x(u),x(u) du

–λs

T

T

0

˜

fu,x(u),x(u) du

ds, (3.7)

wherea∈Ris a constant such that

R(x,λ)(T) =A–1

T

0

φ_p–1

a+

s

0

λf˜u,x(u),x(u) du–λs

T

T

0

˜

fu,x(t),x(u) du

ds

= 0. (3.8)

From Lemma 2.3 of [15], we know thatais uniquely defined by

a=a˜(x,λ),

wherea˜(x,λ) is continuous onΩ¯×[0, 1] and maps bounded sets ofΩ¯×[0, 1] into bounded sets ofR.

From Eq. (3.4), one can find that

R :Ω¯ ×[0, 1]→X2.

Now, for anyx∈λ={x∈ ¯Ω:Mx=Nλx}={x∈ ¯Ω: (φp(Ax)(t))=λf˜(t,x(t),x(t))}, we have₀Tf˜(t,x(t),x(t))dt= 0, which, together with Eq. (3.7), gives

R(x,λ)(t) =A–1

t

0

φ_p–1(a+

s

0

λf˜u,x(u),x(u)du ds

=A–1

t

0

φ_p–1

a+

s

0

φp(Ax)(u) du

ds

=A–1

t

0

Takinga=φp(Ax)(0), we then have

R(x,λ)(T) =A–1

T

0

φ_p–1φp(Ax)(s)

ds

=A–1

T

0

(Ax)(t)ds

=A–1(Ax)(T) – (Ax)(0)

=x(T) –x(0)

= 0,

whereais unique, and we see that

a=a˜(x,λ) =φp(Ax)(0), ∀λ∈[0, 1].

Thus, we derive

R(x,λ)(t)|x∈λ=A

–1

t

0

φ–1_p

φp(Ax)(0) +

s

0

λf˜t,u,x(u),x(u) du

ds

=A–1

t

0

φ–1_p φp(Ax)(s)

ds

=A–1

t

0

(Ax)(s)ds

=x(t) –x(0)

= (I–P)x(t),

which yields the second part of Eq. (3.3). Meanwhile, ifλ= 0, then

λ

={x∈ ¯Ω:Mx=Nλx}={x∈ ¯Ω: (φp(Ax)(t))=λf˜(t,x(t),x(t))}=c3,

wherec3∈Ris a constant, so by the continuity ofa˜(x,λ) with respect to (x,λ),a=a˜(x, 0) =

φp(Ac)(0) = 0. Hence,

R(x, 0)(t) =A–1

t

0

φ_p–1(0)ds= 0, ∀x∈ ¯Ω,

which yields the first part of Eq. (3.3). Furthermore, we consider

M(P+R) = (I–Q)Nλ,

and, in fact,

d dtφp

Integrating both sides of (3.9) over [0,s], we have

Therefore, we arrive at

φp

Integrating both sides of (3.10) over [0,t], we derive

t

Hence, we have thatNλisM-compact onΩ¯. Obviously, the equation

φp(Ax)(t) =λf˜

t,x(t),x(t)

can be converted to

whereMandNλare defined by Eqs. (3.6) and (3.7), respectively. As proved above,

Nλ:Ω¯ →Z, λ∈(0, 1),

is anM-compact mapping. From assumption (i), one finds

Mx=Nλx, λ∈(0, 1),x∈∂Ω,

and assumptions (ii) and (iii) imply thatdeg{JQN,Ω∩kerM,θ}is valid and

deg{JQN,Ω∩kerM,θ} = 0.

So by applications of Lemma3.1, we see that Eq. (1.1) has aT-periodic solution.

4 Application of Theorem3.2:p-Laplacian equation

As an application, we consider the followingp-Laplacian neutral Liénard equation:

φp

x(t) –

n

i=1

ci(t)x(t–δi)

+fx(t) x(t) +gt,x(t) =e(t), (4.1)

whereφp:R→Ris given byφp(s) =|s|p–2s, herep> 1 is a constant,gis a continuous func-tion defined onR2_{and periodic intwithg(t,·) =g(t+T,·),f∈C(R,R),eis a continuous} periodic function defined onRwith periodTand₀Te(t)dt= 0. Next, by applications of Theorem3.2, we will investigate the existence of periodic solution for Eq. (4.1) in the case thatn_i=1ci = 1.

Define

σ:=

⎧ ⎪ ⎨ ⎪ ⎩

1

1–ni=1ci, for

n

i=1ci< 1; 1

ck

1–_ck1–_ckci, for

n

i=1ci> 1.

Theorem 4.1 Supposen_i=1ci = 1holds.Assume the following conditions hold: (H1) There exists a constantD> 0such that

xg(t,x) > 0, ∀(t,x)∈[0,T]×R, with|x|>D. (H2) There exist positive constantsm,n˜such that

f(x)≤m|x|p–2+n˜, x∈R.

(H3) There exist positive constantsa,b,Bsuch that

g(t,x)≤a|x|p–1+b, for|x|>Bandt∈[0,T].

Then Eq. (4.1)has at least oneT-periodic solution,if

σT1q

mn_i=1ci 2p–1 +

aT(1 +n_i=1ci) 2p

1 p

+σT

n

Proof Consider the homotopic equation

φp

x(t) –

n

i=1

ci(t)x(t–δi)

+λfx(t) x(t) +λgt,x(t) =λe(t). (4.2)

Firstly, we claim that the set of allT-periodic solutions of Eq. (4.2) is bounded. Letx(t)∈

CT be an arbitraryT-periodic solution of Eq. (4.2). Integrating both sides of (4.2) over [0,T], we have

T

0

gt,x(t) dt= 0. (4.3)

From the mean-value theorem for integrals, there is a constantξ∈[0,T] such that

gξ,x(ξ) = 0.

In view of condition (H1), we obtain

x(ξ)≤D.

Then, we have

x= max

t∈[0,T]

x(t)= max

t∈[ξ,ξ+T]

x(t)

=1 2t∈max[ξ,ξ+T]

_x(t)+x(t–T)

=1 2t∈max[ξ,ξ+T]

x(ξ) +

T

ξ

x(s)ds+x(ξ) –

ξ

t–T

x(s)ds

≤D+1 2

t

ξ

x(s)ds+

ξ

t–T

x(s)ds

≤D+1 2

T

0

x(s)ds. (4.4)

Multiplying both sides of Eq. (4.2) by (Ax)(t) and integrating over the interval [0,T], we get

T

0

φp(Ax)(t) (Ax)(t)dt+λ

T

0

fx(t)x(t)(Ax)(t)dt+λ

T

0

gt,x(t)(Ax)(t)dt

=λ

T

0

e(t)(Ax)(t)dt. (4.5)

Substituting ₀T(φp(Ax)(t))(Ax)(t)dt= –

T

0 |(Ax)(t)| p_dt, T

0 f(x(t))x(t)x(t)dt= 0 into Eq. (4.5), we see that

–

T

0

(Ax)(t)pdt=λ

T

0

fx(t) x(t)

_n

i=1

ci(t)x(t–δi)

–λ

Thus, we have

wheree:=maxt∈[0,T]|e(t)|,gB:=max|x(t)|≤B|g(t,x(t))|andN1:= (1+

Next, we introduce a classical inequality: there exists aκ(p) > 0, which is depends onp

only, such that

(1 +x)p≤1 + (1 +p)x, forx∈0,κ(p). (4.8) Then, we consider the following two cases:

Case 1: If D

From Eq. (4.4), it is clear that

+n˜

By applying Lemma2.1and Hölder inequality, we get

+σT1q

aT(1 +n_i=1ci)(1 +p)D

2p–1 +

mn_i=1cipD 2p–2

1 p T

0

x(t)dt

p

p–1

+σT1q

_˜

nn_i=1ci 2

1 p T

0

x(t)dt 2

p

+σT1q

˜

n

n

i=1

ciD+

N1 2

1 p _T

0

x(t)dt 1

p

+σT1q_(N 1D)

1 p₊σT

n

i=1ci 2

T

0

x(t)dt+σT

n

i=1

c_iD. (4.12)

SinceσT1q₍m

n

i=1ci 2p–1 +

aT(1+n_i=1ci) 2p )

1 p ₊σT

n

i=1ci

2 < 1, it is easily see that there exists a constantM₁> 0 (independent ofλ) such that

T

0

x(t)dt≤M₁. (4.13)

From Eq. (4.4), we obtain

x ≤D+1 2

T

0

x(s)ds≤D+1 2M

1:=M12. (4.14)

LetM1=

M₁₁2 +M₁₂2 + 1. As (Ax)(0) = (Ax)(T), there exists a pointt0∈(0,T) such that (Ax)(t0) = 0. Moreover, sinceφp(0) = 0, due to Eq. (4.14), it is obvious that

φp(Ax)(t)=

_t0tφp(Ax)(s) ds

≤λ

T

0

fx(t) x(t)dt+λ

T

0

gt,x(t) dt+λ

T

0

e(t)dt

≤ fM1

T

0

x(t)dt+TgM1+Te

≤ fM1M1+TgM1+Te:=M2,

wherefM1:=max|x(t)|≤M1|f(x(t))|andgM1:=max|x(t)|≤M1|g(t,x(t))|. Next we claim that there exists a positive constantM∗₂>M₂+ 1, such that, for allt∈R,

(Ax)≤M∗₂. (4.15)

In fact, if (Ax)is not bounded, there exists a positive constantM₂such that(Ax)>M₂

contra-diction. Hence, Eq. (4.15) holds. From Lemma 2.1 and Eq. (4.15), we have

x=A–1Ax

=A–1Ax (t)

≤σ

(Ax)(t) +

n

i=1

c_i(t)x(t–δi)

≤σ(Ax)+σ

_n

i=1

c_ix

≤σM∗₂+σ

n

i=1

c_iM1:=M2. (4.16)

SettingM=M2

1+M22+ 1, we get

Ω=x∈C_T1(R,R)|x ≤M+ 1,x≤M+ 1,

and we know that Eq. (4.1) has no solution on∂Ω asλ∈(0, 1). When x(t)∈∂Ω∩R,

x(t) =M+ 1 orx(t) = –M– 1, and from Eq. (4.4) we know thatM+ 1 >D. Thus, from condition (H1), we see that

1

T

T

0

g(t,M+ 1)dt> 0,

1

T

T

0

g(t, –M– 1)dt< 0,

since₀Te(t)dt= 0. So condition (ii) of Theorem3.2is also satisfied. Set

H(x,μ) =μx+ (1 –μ)1

T

T

0

g(t,x)dt, x∈∂Ω∩R,μ∈[0, 1].

Obviously, from condition (H1), we can getxH(x,μ) > 0 and thusH(x,μ) is a homotopic transformation, as well as

deg{F,Ω∩R, 0}=deg

1

T

T

0

g(t,x)dt,Ω∩R, 0

=deg{x,Ω∩R, 0} = 0.

So condition (iii) of Theorem3.2is satisfied. In view of Theorem3.2, there exists at least

oneT-periodic solution.

5 Application of Theorem3.2:p-Laplacian equation with singularity

In this section, we consider Eq. (4.1) with a singularity. Hereg(t,x(t)) =g0(x) +g1(t,x(t)),

g0∈C((0,∞);R) andg1is anL2-Carathéodory function, andg0has a singularity atx= 0, i.e.,

1

0

Next, we consider the existence of periodic solutions for Eq. (4.1) with singularity by ap-plying Theorem3.2.

Theorem 5.1 Supposen_i=1ci = 1and condition(H2)hold.Assume that the following

conditions hold:

(H4) There exist positive constants 0 <D1 <D2 such that x is a positive

continu-ous T-periodic function satisfying ₀Tg(t,x(t))dt < 0, for some x∈ (0,D1) and

T

0 g(t,x(t))dt> 0,for somex∈(D2,∞). (H5) There exist positive constantsαandβsuch that

g(t,x)≤αxp–1+β, fort∈[0,T], andx> 0. (5.2)

Then Eq. (4.1)has at least one T -periodic solution if

σT1q

mn_i=1ci 2p–1 +

αT(1 +n_i=1ci) 2p–1

1 p

+σT

n

i=1ci 2 < 1.

Proof Consider the homotopic equation

φp(Ax)(t) +λf

x(t) x(t) +λgt,x(t) =λe(t). (5.3)

We follow the same strategy and notation as in the proof of Theorem4.1. From condition (H4), we know that there exists a constantD2> 0 such that

x(t)≤D2+ 1 2

T

0

x(t)dt. (5.4)

From Eq. (4.5), we have

T

0

_(Ax)(t)p

dt≤

n

i=1

cix

T

0

_f

x(t) x(t)dt

+

1 + n

i=1

ci

x

T

0

gt,x(t) dt

+

1 + n

i=1

ci

x

T

0

e(t)dt.

From Eq. (4.3) and condition (H5), we get

T

0

gt,x(t) dt=

g(t,x(t))≥0

g+t,x(t) dt–

g(t,x(t))<0

g–t,x(t) dt

= 2

g(t,x(t))≥0

g+t,x(t) dt

≤2

T

0

αxp–1+β dt

whereg+_{:=max{g(t,x), 0}. Using condition (H2) and Eq. (5.5), we derive}

Following the same strategy and notation as in the proof of Theorem4.1, we can obtain, sinceσT1q₍m (independent ofλ) such that

T

From Eqs. (4.15), (4.16) and (5.8), we get that there exists a constantM₃∗, such that, for all

From Eq. (5.3), we have

t

τ

φp(Ax)(t) x(t)dt

=x

t

τ

φp(Ax)(t) dt

≤x

λ

T

0

fx(t) x(t)dt+λ

T

0

gt,x(t) dt+λ

T

0

e(t)dt

≤λM∗₃

fM3

T

0

x(t)dt+ 2αTxp–1+ 2βT+Te

≤λM∗₃fM3M3+ 2αT(M3)p–1+ 2βT+Te ,

wherefM3:=max|x(t)|≤M3|f(x(t))|. From Eqs. (5.7) and (5.8), we obtain

λ

t

τ

fx(t) x(t)x(t)dt≤λ

T

0

fx(t) x(t)x(t)dt

≤λfM3

T

0

x(t)2dt

≤λfM3

M∗₃ 2T,

λ

t

τ

g1(t,x)x(t)dt≤λ

T

0

g1(t,x)x(t)dt

≤λg1M3M3, whereg1M3:=max|x(t)|≤M3|g1(t,x)|,

λ

t

τ

e(t)x(t)dt≤λ

T

0

e(t)x(t)dt

≤λeM₃. From these inequalities, we get

_x(x_τ(₎t)g0(u)du≤M∗₃fM3M3+ 2αT(M3)p–1+ 2βT+Te

+fM3

M∗₃ 2T+g1M3M3+eM3.

In view of Eq. (5.1), we know that there exists a constantM4> 0 such that

x(t)≥M4, ∀t∈[τ,T]. (5.11)

The caset∈[0,τ] can be treated similarly. From Eqs. (5.8), (5.9) and (5.11), we have

Ω=x∈C_T1(R,R)|E1≤x≤E2,x≤M∗3,∀t∈[0,T]

,

where 0 <E1<min(M4,D1),E2>max(M3,D2). This proves the claim, and the rest of the

6 Examples see that there exists a constant D= 1 such that condition (H1) holds. Obviously, we get |f(x)|=|₂₀1x| ≤ ₂₀1|x|+ 3, herem= ₂₀1, n˜ = 3, and condition (H2) holds. Consider

1

Therefore, by Theorem4.1, we know that Eq. (6.2) has at least one positive π₄-periodic solution.

Example6.3 Consider the followingp-Laplacian Liénard equation with singularity

6. Next, we consider the condition

σT1q

Therefore, by Theorem5.1, we know that Eq. (6.3) has at least one positive π

4-periodic solution.

Example6.4 Consider the followingp-Laplacian Liénard equation with singularity

Comparing Eq. (6.4) with Eq. (4.1), it is easy to see thatc1(t) = 1₆cos(6t+π₅) +11₆,c2(t) =

Therefore, by Theorem5.1, we know that Eq. (6.4) has at least one positive π₃-periodic solution.

7 Conclusions

In this paper, we first investigated some properties of the neutral operator with multiple delays and variable coefficients (Ax)(t) :=x(t) –n_i=1ci(t)x(t–δi). Afterwards, by using an extension of Mawhin’s continuation theorem due to Ge and Ren, properties of the neutral operatorA, we studied the existence of a periodic solution for equation (1.1). At last, by ap-plying Theorem3.2, we discussed the existence of a periodic solution for twop-Laplacian neutral differential equations. In comparison to [5] and [19], we avoided translating the equation into a two-dimensional system.

Acknowledgements

ZHB, ZBC and SWY are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.

Funding

This work was supported by National Natural Science Foundation of China (11501170, 71601072), China Postdoctoral Science Foundation funded project (2016M590886), Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302).

Abbreviations

Not applicable.

Availability of data and materials

Not applicable.

Ethics approval and consent to participate

ZHB, ZBC and SWY contributed to each part of this study equally and declare that they have no competing interests.

Competing interests

ZHB, ZBC and SWY declare that they have no competing interests.

Consent for publication

ZHB, ZBC and SWY read and approved the final version of the manuscript.

Authors’ contributions

ZHB, ZBC and SWY contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

Author details

Publisher’s Note

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Received: 10 August 2018 Accepted: 27 December 2018

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