Skip to main content

Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian

Abstract

By means of the generalized Mawhin’s continuation theorem, we present some sufficient conditions which guarantee the existence of at least one T-periodic solution for a generalized Liénard neutral functional differential system with p-Laplacian.

MSC:34B15, 34L30.

1 Introduction

This paper is devoted to investigating the following p-Laplacian Liénard neutral differential system:

( φ p ( ( x ( t ) B x ( t τ ) ) ) ) +f ( x ( t ) ) x (t)+g ( x ( t γ ( t ) ) ) =e(t),
(1.1)

where x(t)= ( x 1 ( t ) , x 2 ( t ) , , x n ( t ) ) ;

eC(R, R n ) with e(t+T)=e(t); γC(R,R) with γ(t+T)=γ(t); τ is a given constant; B= [ b i j ] n × n is a real matrix with |B|= ( i = 1 n j = 1 n | b i j | 2 ) 1 / 2 .

When the matrix B is a constant, Zhang [1] studied the properties of a difference operator A and obtained the following results: define the operator A on C T

A: C T C T ,[Ax](t)=x(t)cx(tτ),tR,
(1.2)

where C T ={x:xC(R,R),x(t+T)x(t)}, c is a constant. If |c|1, then A has a unique continuous bounded inverse A 1 satisfying

[ A 1 f ] (t)={ j 0 c j f ( t j τ ) , if  | c | < 1 , f C T , j 1 c j f ( t + j τ ) , if  | c | > 1 , f C T .

On the basis of Zhang’s work, Lu [2] further studied the properties of the difference operator A and gave the following inequality properties for A:

  1. (1)

    A 1 1 | 1 | k | | ;

  2. (2)

    0 T |[ A 1 f](t)|dt 1 | 1 | k | | 0 T |f(t)|dt, f C T ;

  3. (3)

    0 T | [ A 1 f ] ( t ) | 2 dt 1 | 1 | k | | 0 T | f ( t ) | 2 dt, f C T .

After that, by using the above results, many researchers studied the existence of periodic solutions for some kinds of differential equations; see [37]. In a recent paper [8], when the constant c of (1.2) is a variable c(t), we generalized the results of [1] and obtained the following results. If |c(t)|1, then the operator A has continuous inverse A 1 on C T , satisfying

  1. (1)
    [ A 1 f ] (t)={ f ( t ) + j = 1 i = 1 j c ( t ( i 1 ) τ ) f ( t j τ ) , c 0 < 1 , f C T , f ( t + τ ) c ( t + τ ) j = 1 i = 1 j + 1 1 c ( t + i τ ) f ( t + j τ + τ ) , σ > 1 , f C T .
  2. (2)
    0 T | [ A 1 f ] ( t ) | dt{ 1 1 c 0 0 T | f ( t ) | d t , c 0 < 1 , f C T , 1 σ 1 0 T | f ( t ) | d t , σ > 1 , f C T .

Using the above results, we have obtained some existence results of periodic solutions for first-order, second-order and p-Laplacian neutral equations with a variable parameter; see [911].

However, when B of (1.1) is a matrix, there are few existence results of periodic solutions for neutral differential systems. In [12], when B is a symmetric matrix, the authors studied a second-order p-Laplacian neutral functional differential system and obtained the existence of periodic solutions. In [13], when B is a general matrix, the authors studied a second-order neutral differential system. But for p-Laplacian functional differential system, to the best our knowledge, there are no results on the existence of periodic solutions. Hence, in this paper, we will study system (1.1) and obtain the existence of periodic solutions by using the generalization of Mawhin’s continuation theorem.

2 Main lemmas

In this section, we give some notations and lemmas which will be used in this paper. Let

X= C T 1 with the norm x=max{ | x | 0 , | x | 0 }, Z= C T with the norm

| x | 0 = max 0 t T | x ( t ) | , | x ( t ) | = ( i = 1 n x i 2 ) 1 / 2 ,

U is a complex such that

UB U 1 = E λ =diag( J 1 , J 2 ,, J n )
(2.1)

is a Jordan’s normal matrix, where

J i = ( λ i 1 0 0 0 0 λ i 1 0 0 0 0 0 λ i 1 0 0 0 0 λ i ) n i × n i

with i = 1 l n i =n, { λ i :i=1,2,,l} is the set of eigenvalues of matrix B. Let

A 1 : C T C T ,[ A 1 x](t)=x(t)Bx(tτ).
(2.2)

Furthermore, we suppose that γ(t) C 1 (R,R) with γ (t)<1, tR. It is obvious that the function tγ(t) has a unique inverse denoted by μ(t).

Lemma 2.1 ([13])

Suppose that the matrix U and the operator A 1 are defined by (2.1) and (2.2), respectively, and for all i=1,2,,l, | λ i |1. Then A 1 has its inverse A 1 1 : C T C T with the following properties:

  1. (1)

    A 1 1 | U 1 ||U| σ 0 , σ 0 = i = 1 l j = 1 n i k = 1 j 1 | 1 λ i | k .

  2. (2)

    For all f C T , 0 T | [ A 1 1 f ] ( s ) | p ds | U 1 | p | U | p σ 1 0 T | f ( s ) | p ds, p[1,+), where

    σ 1 ={ i = 1 l j = 1 n i ( k = 1 j 1 | 1 λ i | k ) 2 , p = 2 , n 2 p 2 [ i = 1 l j = 1 n i ( k = 1 j 1 | 1 λ i | k ) q ] p q , p [ 1 , 2 ) , [ i = 1 l j = 1 n i ( k = 1 j 1 | 1 λ i | k ) q ] p q , p [ 2 , + )

and q>0 is a constant with 1/p+1/q=1.

  1. (3)

    A 1 1 f C T 1 , [ A 1 1 f ] (t)=[ A 1 1 f ](t), for all f C T 1 , tR.

Definition 2.1 ([14])

Let X and Z be two Banach spaces with norms X , Z , respectively. A continuous operator

M:XdomMZ

is said to be quasi-linear if

  1. (i)

    ImM:=M(XdomM) is a closed subset of Z;

  2. (ii)

    KerM:={xXdomM:Mx=0} is linearly homeomorphic to R n , n<.

Definition 2.2 ([14])

Let ΩX be an open and bounded set with the origin θΩ, N λ : Ω ¯ Z, λ[0,1] is said to be M-compact in Ω ¯ if there exists a subset Z 1 of Z satisfying dim Z 1 =dimKerM and an operator R: Ω ¯ ×[0,1] X 2 being continuous and compact such that for λ[0,1],

  1. (a)

    (IQ) N λ ( Ω ¯ )ImM(IQ)Z,

  2. (b)

    Q N λ x=0, λ(0,1)QNx=0, xΩ,

  3. (c)

    R(,0)0 and R(,λ) | λ =(IP) | λ ,

  4. (d)

    M[P+R(,λ)]=(IQ) N λ , λ[0,1],

where X 2 is the complement space of KerM in X, i.e., X=KerM X 2 ; P, Q are two projectors satisfying ImP=KerM, ImQ= Z 1 , N= N 1 , λ ={x Ω ¯ :Mx= N λ x}.

Lemma 2.2 ([14])

Let X and Z be two Banach spaces with norms X , Z , respectively and ΩX be an open and bounded nonempty set. Suppose

M:XdomMZ

is quasi-linear and N λ : Ω ¯ Z, λ[0,1] is M-compact in Ω ¯ . In addition, if the following conditions hold:

(A1) Mx N λ x, (x,λ)Ω×(0,1);

(A2) QNx0, xKerMΩ;

(A3) deg{JQN,ΩKerM,0}0, J:ImQKerM is a homeomorphism.

Then the abstract equation Mx=Nx has at least one solution in domM Ω ¯ .

Lemma 2.3 ([15])

Let s,σC(R,R) with s(t+T)s(t) and σ(t+T)σ(t). Suppose that the function tσ(t) has a unique inverse μ(t), tR. Then s(μ(t+T))s(μ(t)).

For fixed lZ and a R n , define

G l (a)= 1 T 0 T φ p 1 ( a + l ( t ) ) dt.

Lemma 2.4 ([16])

The function G l has the following properties:

  1. (1)

    For any fixed lZ, there must be a unique a ˜ = a ˜ (l) such that the equation

    G l (a)=0.
  2. (2)

    The function a ˜ :Z R n defined as above is continuous and sends bounded sets into bounded sets.

Lemma 2.5 ([17])

Let p(1,+) be a constant, sC(R,R) such that s(t)s(t+T), uX. Then

0 T | u ( t ) u ( t s ( t ) ) | p dt2 ( max t [ 0 , T ] | s ( t ) | ) p 0 T | u ( t ) | p dt.

3 Main results

For convenience of applying Lemma 2.2, the operators A, M, N λ are defined by

(3.1)
(3.2)
(3.3)

where domM={xX: φ p [ ( A x ) ] C T 1 }. For convenience of the proof, let

F(t,x)=f ( x ( t ) ) x (t)g ( x ( t γ ( t ) ) ) +e(t),
(3.4)

then ( N λ x)(t)=λF. By (3.1)-(3.3), Eq. (1.1) is equivalent to the operator equation Nx=Mx, where N 1 =N. Then we have

Since KerM R n , ImM is a closed set in Z, then we have the following.

Lemma 3.1 Let M be as defined by (3.2), then M is a quasi-linear operator.

Let

Lemma 3.2 If f, g, e, γ satisfy the above conditions, then N λ is M-compact.

Proof Let Z 1 =ImQ. For any bounded set Ω ¯ X, define R: Ω ¯ ×[0,1]KerP,

R(x,λ)(t)= A 1 { 0 t φ q [ a x + 0 s λ ( F ( r , x ( r ) ) ( Q F ) ( r ) ) d r ] d s } ,t[0,T],

where F is defined by (3.4) and a x is a constant vector in R n which depends on x. By Lemma 2.4, we know that a x exists uniquely. Hence, R(x,λ)(t) is well defined.

We first show that R(,λ) is completely continuous on Ω ¯ ×[0,1]. Let

G λ (t)= 0 t φ q [ a x + 0 s λ ( F ( r , x ( r ) ) ( Q F ) ( r ) ) d r ] ds,t[0,T],

we have

R(x,λ)(t)= [ A 1 G λ ] (t).

From the properties of f, g, e, γ, obviously, x Ω ¯ , G λ (t) C T . Then by Lemma 2.1 R(x,λ) is uniformly bounded on Ω ¯ ×[0,1]. Now, we show R(x,λ) is equicontinuous. t 1 , t 2 [0,T], ε>0 is sufficiently small, then there exists δ>0, for | t 1 t 2 |<δ, by G λ , A 1 G λ C T we have

| [ A 1 G λ ] ( t 1 ) [ A 1 G λ ] ( t 2 ) | <ε.

Hence, R(x,λ) is equicontinuous on Ω ¯ ×[0,1]. By using the Arzelà-Ascoli theorem, we have R(x,λ) is completely continuous on Ω ¯ ×[0,1].

Secondly, we show that N λ is M-compact in four steps, i.e., the conditions of Definition 2.2 are all satisfied.

Step 1. By Q 2 =Q, we have Q(IQ) N λ ( Ω ¯ )=θ, so (IQ) N λ ( Ω ¯ )KerQ=ImM, here θ is an n-dimension zero vector. On the other hand, zImM. Clearly, Qz=θ, so z=zQz=(IQ)z, then z(IQ)Z. So, we have

(IQ) N λ ( Ω ¯ )ImM(IQ)Z.

Step 2. We show that Q N λ x=θ, λ(0,1)QNx=θ, xΩ. Because Q N λ x= 1 T λFdr=θ, we get 1 T Fdr=θ, i.e., QNx=θ. The inverse is true.

Step 3. When λ=0, from the above proof, we have a x =θ. So, we get R(,0)=θ. x λ ={x Ω ¯ :Mx= N λ x}, we have ( φ p [ ( A x ) ] ) =λF and QF=θ. In this case, when a x = φ p [ ( A x ) (0)], we have

G λ ( T ) = 0 T φ q [ a x + 0 s λ ( F ( r , x ( r ) ) ( Q F ) ( r ) ) d r ] d s = 0 T φ q [ φ p [ ( A x ) ( 0 ) ] + 0 s λ F ( r , x ( r ) ) d r ] d s = 0 T φ q [ φ p [ ( A x ) ( 0 ) ] + 0 s ( φ p [ ( A x ) ( r ) ] ) d r ] d s = 0 T ( A x ) ( s ) d s = ( A x ) ( T ) ( A x ) ( 0 ) = θ .

Hence,

R ( x , λ ) ( t ) = A 1 { 0 t φ q [ φ p [ ( A x ) ( 0 ) ] + 0 s λ ( F ( r , x ( r ) ) ( Q F ) ( r ) ) d r ] d s } = A 1 { 0 t φ q [ φ p [ ( A x ) ( 0 ) ] + 0 s λ F ( r , x ( r ) ) d r ] d s } = A 1 { 0 t φ q [ φ p [ ( A x ) ( 0 ) ] + 0 s ( φ p [ ( A x ) ( r ) ] ) d r ] d s } = A 1 { 0 t ( A x ) ( s ) d s } = A 1 [ ( A x ) ( t ) ( A x ) ( 0 ) ] = [ ( I P ) x ] ( t ) .

Step 4. x Ω ¯ , we have

Hence, N λ is M-compact in Ω ¯ . □

Theorem 3.3 Suppose that 0 T e(s)ds=θ, λ 1 , λ 2 ,, λ l are eigenvalues of the matrix B with | λ i |1, i=1,2,l, and there exist positive constants D>0, l>0 and σ>0 such that

(H1) x i g i ( x i )>0, x i R, | x i |>D, for each i=1,2,,n,

(H2) | g i ( u 1 ) g i ( u 2 )|l| u 1 u 2 |, u 1 , u 2 R, for each i=1,2,,n,

(H3) | f i ( x i )|σ, x i R, for each i=1,2,,n.

Then Eq. (1.1) has at least one T-periodic solution if one of the following two conditions is satisfied:

where f R 2 = max | x | R 2 |f(x)|, R 2 is defined by (3.14).

Proof We complete the proof in three steps.

Step 1. Let Ω 1 ={xdomM:Mx= N λ x,λ(0,1)}. We show that Ω 1 is a bounded set. If x Ω 1 , then Mx= N λ x, i.e.,

( φ p [ ( A x ) ] ) =λf ( x ( t ) ) x (t)λg ( x ( t γ ( t ) ) ) +λe(t).
(3.5)

Integrating both sides of (3.5) over [0,T], we have

0 T g ( x ( t γ ( t ) ) ) dt=θ,

which together with assumption (H1) leads to the fact that there exists a point ξ i R such that

| x i ( ξ i γ ( ξ i ) ) | D,for each i=1,2,,n.

Let ξ i γ( ξ i )=kT+ η i , kZ, η i [0,T]. Then

| x i ( η i ) | D,for each i=1,2,,n.

Thus,

| x i |D+ 0 T | x i ( s ) | ds,for each i=1,2,,n.
(3.6)

By (3.6), we have

| x | = ( x 1 2 + x 2 2 + + x n 2 ) 1 2 n ( D + 0 T | x ( s ) | d s )
(3.7)

and

| x | 0 n ( D + 0 T | x ( s ) | d s ) .
(3.8)

On the other hand, multiplying the two sides of Eq. (3.5) by [ x ( t ) ] from the left side and integrating them over [0,T], we have

0 T [ x ( t ) ] ( φ p [ ( A x ) ] ) d t = λ 0 T [ x ( t ) ] f ( x ( t ) ) x ( t ) d t λ 0 T [ x ( t ) ] g ( x ( t γ ( t ) ) ) d t + λ 0 T [ x ( t ) ] e ( t ) d t .
(3.9)

Let ω(t)= φ p [ ( A x ) (t)], then

0 T [ x ( t ) ] ( φ p [ ( A x ) ] ) dt= 0 T { A 1 ( φ q ( ω ( t ) ) ) } dω(t)=0.

By (3.9), we have

(3.10)

By assumption (H3), we have

σ i = 1 n 0 T | x i ( t ) | 2 d t i = 1 n 0 T | f i ( x i ( t ) ) | [ x i ( t ) ] 2 d t = | i = 1 n 0 T f i ( x i ( t ) ) [ x i ( t ) ] 2 d t | .
(3.11)

From (3.10) and (3.11), we have

σ i = 1 n 0 T | x i ( t ) | 2 dt 0 T | [ x ( t ) ] g ( x ( t γ ( t ) ) ) | dt+ 0 T | [ x ( t ) ] e ( t ) | dt.
(3.12)

From 0 T [ x ( t ) ] g(x(t))dt=0, assumption (H2), Lemma 2.5 and (3.12), we have

(3.13)

Since σ> 2 l max t [ 0 , T ] |γ(t)|, by (3.13), there exists a positive constant R 1 such that

0 T | x ( s ) | ds R 1 .

Then by (3.8),

| x | 0 n (D+ R 1 ):= R 2 .
(3.14)

By (3.5), we have

| ( φ p [ ( A x ) ] ) | f R 2 | x ( t ) | + g R 2 + | e | 0 ,

where f R 2 = max | x | R 2 |f(x)|, g R 2 = max | x | R 2 |g(x)|. Take φ p [ ( A x ) (t)]=y(t), then

| y | 0 f R 2 | x ( t ) | + g R 2 + | e | 0
(3.15)

and ( A x ) (t)= φ q (y(t)). Because there exists a t i [0,T] such that y( t i )=0, i=1,2,,n, so by (3.15), we get

| y ( t ) | n T | y | 0 n T f R 2 | x ( t ) | + n T g R 2 + n T | e | 0

and

| ( A x ) ( t ) | ( n T f R 2 | x ( t ) | + n T g R 2 + n T | e | 0 ) q 1 .
(3.16)

By (3.16) and Lemma 2.1, we have

| x ( t ) | = | [ A 1 A x ] ( t ) | | U 1 | | U | σ 0 | ( A x ) ( t ) | | U 1 | | U | σ 0 ( n T f R 2 | x ( t ) | + n T g R 2 + n T | e | 0 ) q 1 .
(3.17)

Now, we consider ( n T f R 2 | x ( t ) | + n T g R 2 + n T | e | 0 ) q 1 . In the formal case, we get

(3.18)

By classical elementary inequalities, we see that there is a constant h(p)>0, which is dependent on p only, such that

( 1 + x ) p <1+(1+p)x,x(0,h(p)].
(3.19)

Case 2.1. If n T g R 2 + n T | e | 0 n T f R 2 | x ( t ) | >h, then

| x ( t ) | < n T g R 2 + n T | e | 0 n T f R 2 h := M 1 .
(3.20)

Case 2.2. If n T g R 2 + n T | e | 0 n T f R 2 | x ( t ) | h, by (3.18) and (3.19), we have

(3.21)

From (3.17) and (3.21), we have

| x ( t ) | | U 1 | | U | σ 0 ( n T f R 2 ) q 1 | x ( t ) | q 1 + | U 1 | | U | σ 0 q ( n T g R 2 + T | e | 0 ) ( n T f R 2 ) q 2 | x ( t ) | q 2 .
(3.22)

When q=2, from | U 1 ||U| σ 0 n T f R 2 <1, we know that there exists a constant M 2 >0 such that

| x ( t ) | M 2 .
(3.23)

When 1<q<2, there must be a constant M 3 >0 such that

| x ( t ) | M 3 .
(3.24)

Hence, from (3.14), (3.20), (3.23) and (3.24), we have

x<max{ R 2 , M 1 , M 2 , M 3 }+1:=L.

Step 2. Let Ω 2 ={xKerM:QNx=θ}, we shall prove that Ω 2 is a bounded set. x Ω 2 , then x=a, a R n , we have g i ( a i )=0 for each i=1,2,,n. By assumption (H1) we have | a i |D and |a| n D. So, Ω 2 is a bounded set.

Step 3. Let Ω={xX:x<L}, then Ω 1 Ω 2 Ω, (x,λ)Ω×(0,1). From the above proof, Mx N λ x is satisfied. Obviously, condition (A2) of Lemma 2.2 is also satisfied. Now, we prove that condition (A3) of Lemma 2.2 is satisfied. Take the homotopy

H(x,μ)=μx(1μ)JQNx,x Ω ¯ KerM,μ[0,1],

where J:ImQKerM is a homeomorphism with Ja=a, a R n . xΩKerM, we have x= a 1 R n , | a 1 |=L>D, then

H ( x , μ ) = a 1 μ ( 1 μ ) 1 T 0 T ( g ( a 1 ) + e ( t ) ) d t = a 1 μ + ( 1 μ ) g ( a 1 ) ,

then we have

a 1 H(x,μ)= a 1 a 1 μ+(1μ) a 1 g( a 1 ).

By using assumption (H1), we have H(x,μ)0. And then, by the degree theory,

deg { J Q N , Ω Ker M , 0 } = deg { H ( , 0 ) , Ω Ker M , 0 } = deg { H ( , 1 ) , Ω Ker M , 0 } = deg { I , Ω Ker M , 0 } 0 .

Applying Lemma 2.2, we complete the proof. □

Remark Assumption (H1) guarantees that condition (A2) of Lemma 2.2 is satisfied. Furthermore, using assumptions (H1)-(H3), we can easily estimate prior boud of the solution to Eq. (1.1).

As an application, we consider the following example:

( φ p [ ( x ( t ) B x ( t π ) ) ] ) +f ( x ( t ) ) x (t)+g ( x ( t π ) ) =e(t),
(3.25)

where

x(t)= ( x 1 ( t ) x 2 ( t ) ) R 3 ,g(x)= ( 1 100 x 1 1 100 x 2 ) ,B= ( 1 3 4 0 ) ,

e(t)= ( sin t , cos t ) , τ=γ=π, p=1.5, T=2π, f(x)=(5+sin x 1 ,10+cos x 2 ).

Obviously, λ 1 =3±1, λ 2 =4±1,

0 2 π e(t)dt= ( 0 2 π cos t d t 0 2 π sin t d t ) = ( 0 0 ) .

Since

x 1 g 1 ( x 1 )= 1 100 x 1 2 >0for| x 1 |>D>0, x 2 g 2 ( x 2 )= 1 100 x 2 2 >0for | x 2 |>D>0,

so assumption (H1) is satisfied. Take l= 1 100 , then

| g i ( u 1 ) g i ( u 2 ) | 1 100 | u 1 u 2 |, u 1 , u 2 R, for each i=1,2,

and assumption (H2) is satisfied. Take σ=4, then

| f 1 ( x 1 ) | =|5+sin x 1 |4, | f 2 ( x 2 ) | =|10+cos x 2 |4,

and assumption (H3) is satisfied. Hence, assumptions (H1)-(H3) are all satisfied. Take

U= ( 1 1 4 3 ) , U 1 = ( 3 7 1 7 4 7 1 7 )

such that

UB U 1 = ( 3 0 0 4 ) .

Take γ(t)=π, then

σ> 2 l max t [ 0 , T ] | γ ( t ) | .

By using Theorem 3.3, we know that Eq. (3.25) has at least one 2π-periodic solution.

References

  1. Zhang M: Periodic solutions of linear and quasilinear neutral functional differential equations. J. Math. Anal. Appl. 1995, 189: 378–392. 10.1006/jmaa.1995.1025

    MathSciNet  Article  MATH  Google Scholar 

  2. Lu S, Ge W, Zheng Z: Periodic solutions to neutral differential equation with deviating arguments. Appl. Math. Comput. 2004, 152: 17–27. 10.1016/S0096-3003(03)00530-7

    MathSciNet  Article  MATH  Google Scholar 

  3. Kaufmann ER, Raffoul YN: Periodic solutions for a neutral nonlinear dynamical equation on a time scale. J. Math. Anal. Appl. 2006, 319: 315–325. 10.1016/j.jmaa.2006.01.063

    MathSciNet  Article  MATH  Google Scholar 

  4. Li Y: Periodic solutions for delay Lotka-Volterra competition systems. J. Math. Anal. Appl. 2000, 246: 230–244. 10.1006/jmaa.2000.6784

    MathSciNet  Article  MATH  Google Scholar 

  5. Fan M, Wang K: Global periodic solutions of a generalized n -species Gilpin-Ayala competition model. Comput. Math. Appl. 2000, 40: 1141–1151. 10.1016/S0898-1221(00)00228-5

    MathSciNet  Article  MATH  Google Scholar 

  6. Chen F, Lin F, Chen X: Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control. Appl. Math. Comput. 2004, 158: 45–68. 10.1016/j.amc.2003.08.063

    MathSciNet  Article  MATH  Google Scholar 

  7. Lu S, Ge W: On the existence of periodic solutions for neutral functional differential equation. Nonlinear Anal. TMA 2003, 54: 1285–1306. 10.1016/S0362-546X(03)00187-1

    MathSciNet  Article  MATH  Google Scholar 

  8. Du B, Guo L, Ge W, Lu S: Periodic solutions for generalized Liénard neutral equation with variable parameter. Nonlinear Anal. TMA 2008, 70: 2387–2394.

    MathSciNet  Article  MATH  Google Scholar 

  9. Du B, Wang X: Periodic solutions for a second-order neutral differential equation with variable parameter and multiple deviating arguments. Electron. J. Differ. Equ. 2010, 2010: 1–10.

    MathSciNet  MATH  Google Scholar 

  10. Du B, Zhao J, Ge W: Periodic solutions for a neutral differential equation with variable parameter. Topol. Methods Nonlinear Anal. 2009, 33: 275–283.

    MathSciNet  MATH  Google Scholar 

  11. Du B, Sun B: Periodic solutions to a p -Laplacian neutral Duffing equation with variable parameter. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: Article ID 55

    Google Scholar 

  12. Lu S: Periodic solutions to a second order p -Laplacian neutral functional differential system. Nonlinear Anal. TMA 2008, 69: 4215–4229. 10.1016/j.na.2007.10.049

    Article  MathSciNet  MATH  Google Scholar 

  13. Lu S, Xu Y, Xia D: New properties of the D -operator and its applications on the problem of periodic solutions to neutral functional differential system. Nonlinear Anal. TMA 2011, 74: 3011–3021. 10.1016/j.na.2011.01.023

    MathSciNet  Article  MATH  Google Scholar 

  14. Ge W, Ren J: An extension of Mawhin’s continuation theorem and its application to boundary value problems with a p -Laplacian. Nonlinear Anal. TMA 2004, 58: 477–488. 10.1016/j.na.2004.01.007

    MathSciNet  Article  MATH  Google Scholar 

  15. Lu S, Ge W: Existence of positive periodic solutions for neutral population model with multiple delays. Appl. Math. Comput. 2004, 153: 885–902. 10.1016/S0096-3003(03)00685-4

    MathSciNet  Article  MATH  Google Scholar 

  16. Manásevich R, Mawhin J: Periodic solutions for nonlinear systems with p -Laplacian-like operators. J. Differ. Equ. 1998, 145: 367–393. 10.1006/jdeq.1998.3425

    Article  MathSciNet  MATH  Google Scholar 

  17. Lu S, Ge W: Periodic solutions for a kind of Liénard equation with a deviating argument. J. Math. Anal. Appl. 2004, 289: 231–243. 10.1016/j.jmaa.2003.09.047

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by NSF of Jiangsu education office (11KJB110002), Postdoctoral Fundation of Jiangsu (1102096C), Postdoctoral Fundation of China (2012M511296) and Jiangsu province fund (BK2011407).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bo Du.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The first author QY gave an example for verifying the paper’s results. The corresponding author BD gave the proof for all the theorems. QY and BD read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Yang, Q., Du, B. Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian. J Inequal Appl 2012, 270 (2012). https://doi.org/10.1186/1029-242X-2012-270

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2012-270

Keywords

  • periodic solutions
  • neutral equations
  • generalized Mawhin’s continuation theorem