Xin et al. Boundary Value Problems (2016) 2016:33 DOI 10.1186/s13661-016-0545-3

RESEARCH

Open Access

Positive periodic solution for high-order p-Laplacian neutral differential equation with singularity Yun Xin1* , Xuefeng Han2 and Zhibo Cheng2,3 *

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 paper, we consider the following high-order p-Laplacian neutral differential equation with singularity:

(ϕp (x(t) – cx(t – τ ))(n) )(m) + f (x(t))x (t) + g(t, x(t – σ )) = e(t). By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established. MSC: 34C25; 34K13; 34K40 Keywords: positive periodic solution; p-Laplacian; high-order; neutral operator; singularity

1 Introduction In this paper, we consider the following high-order p-Laplacian neutral differential equation with singularity:   (n) (m)     ϕp x(t) – cx(t – τ ) + f x(t) x (t) + g t, x(t – σ ) = e(t),

(.)

where p ≥ , ϕp (x) = |x|p– x for x =  and ϕp () = ; g : [, T] × (, ∞) → R is an L Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for every  < r < s there exists hr,s ∈ L [, T] such that |f (t, x(t))| ≤ hr,s for all x ∈ [r, s] and a.e. t ∈ [, T]. g(t, x) being singular at  means that g(t, x) becomes unbounded when x → + . τ and σ are constants and  ≤ τ , σ < T; e : R → R is a continuous T periodic function with e(t + T) ≡ e(t) and  e(t) dt = . T is a positive constant, c is a constant and |c| = ; n, m are positive integers. Generally speaking, differential equations with singularities have been considered from the very beginning of the discipline. The main reason is that singular forces are ubiquitous in applications, gravitational and electromagnetic forces being the most obvious examples. In , Lazer and Solimini [] discussed the second-order singular equation u +

 = h(t), uα

(.)

© 2016 Xin et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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and they showed that, if h(t) was continuous and T-periodic, then for all α >  a positive T-periodic solution existed if and only if h(t) had a positive mean value. Afterwards, they studied the singular equation u –

 = h(t), uα

(.)

and they found that if α ≥ , a positive T-periodic solution existed if and only if h(t) had a negative mean value. This last result was best possible in that for any α,  < α < , h can be chosen so that h had a negative mean value and the equation had no T-periodic solution. Lazer and Solimini’s work has attracted the attention of many specialists in differential equations. More recently, the method of lower and upper solutions [–], the PoincaréBirkhoff twist theorem [–], topological degree theory [, ], the Schauder fixed point theorem [–], the Leray-Schauder alternative principle [–], the Krasnoselskii fixed point theorem in a cone [, ], and the fixed point index theory [] have been employed to investigate the existence of positive periodic solutions of singular second-order, thirdorder, and fourth-order differential equations. However, the singular differential equation (.), in which there are p-Laplacian and high-order cases, has not attracted much attention in the literature. There are not so many results concerning the existence of a positive periodic solution for (.) even when we have a neutral operator. In this paper, we try to fill gap and establish the existence of a positive periodic solution of (.) using coincidence degree theory. Our new results generalize in several aspects some recent results contained in []. In what follows, we need the notations:   |u|∞ = max u(t), t∈[,T]

  |u| = min u(t), t∈[,T]



T

|u|p = 

|u|p dt

 p ,

 h¯ = T



T

h(t) dt. 

2 Preparation Let CT = {φ ∈ C(R, R) : φ(t + T) ≡ φ(t)} with the norm |φ|∞ = maxt∈[,T] |φ(t)|. Define operators A as follows: A : CT → CT ,

(Ax)(t) = x(t) – cx(t – τ ).

Lemma . (see []) If |c| = , then the operator A has a continuous inverse A– on CT , satisfying: () ⎧ ⎨f (t) + ∞ cj f (t – jτ ), –

for |c| < , ∀f ∈ CT , j= A f (t) =  f (t+τ ) ∞  ⎩– – j= cj+ f (t + (j + )τ ), for |c| > , ∀f ∈ CT . c |f |∞ () |[A– f ](t)| ≤ |–|c|| , ∀f ∈ CT . T T  – ()  |[A f ](t)| dt ≤ |–|c||  |f (t)| dt, ∀f ∈ CT .

Let X and Y be real Banach spaces and L : D(L) ⊂ X → Y be a Fredholm operator with index zero, here D(L) denotes the domain of L. This means that Im L is closed in Y and dim Ker L = dim(Y / Im L) < +∞. Consider supplementary subspaces X , Y of X, Y , respectively, such that X = Ker L⊕X , Y = Im L⊕Y . Let P : X → Ker L and Q : Y → Y denote the

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natural projections. Clearly, Ker L ∩ (D(L) ∩ X ) = {} and so the restriction LP := L|D(L)∩X is invertible. Let K denote the inverse of LP . Let  be an open bounded subset of X with D(L) ∩  = ∅. A map N :  → Y is said to be L-compact in  if QN() is bounded and the operator K(I – Q)N :  → X is compact. Lemma . (Gaines and Mawhin []) Suppose that X and Y are two Banach spaces, and L : D(L) ⊂ X → Y is a Fredholm operator with index zero. Let  ⊂ X be an open bounded set and N :  → Y be L-compact on . Assume that the following conditions hold: () Lx = λNx, ∀x ∈ ∂ ∩ D(L), λ ∈ (, ); () Nx ∈/ Im L, ∀x ∈ ∂ ∩ Ker L; () deg{JQN,  ∩ Ker L, } = , where J : Im Q → Ker L is an isomorphism. Then the equation Lx = Nx has a solution in  ∩ D(L). In order to apply coincidence degree theorem, we rewrite (.) in the form ⎧ ⎨(Ax )(n) (t) = ϕ (x (t)),  q  ⎩x(m) (t) = –f (x (t))x (t) – g(t, x (t – σ )) + e(t), 

(.)



where p + q = . Clearly, if x(t) = (x (t), x (t)) is a T-periodic solution to (.), then x (t) must be a T-periodic solution to (.). Thus, the problem of finding a T-periodic solution for (.) reduces to finding one for (.). Now, set X = {x = (x (t), x (t)) ∈ C(R, R ) : x(t + T) ≡ x(t)} with the norm |x|∞ = max{|x |∞ , |x |∞ }; Y = {x = (x (t), x (t)) ∈ C  (R, R ) : x(t + T) ≡ x(t)} with the norm x = max{|x|∞ , |x |∞ }. Clearly, X and Y are both Banach spaces. Meanwhile, define     L : D(L) = x ∈ C n+m R, R : x(t + T) = x(t), t ∈ R ⊂ X → Y by 

(Ax )(n) (t) (Lx)(t) = x(m)  (t)



and N : X → Y by  ϕq (x (t)) . (Nx)(t) = –f (x (t))x (t) – g(t, x (t – σ )) + e(t) 

(.)

Then (.) can be converted into the abstract equation Lx = Nx. From the definition of L, one can easily see that  Ker L ∼ =R , 



T

Im L = y ∈ Y :







 

y (s)  ds = y (s) 

.

So L is a Fredholm operator with index zero. Let P : X → Ker L and Q : Y → Im Q ⊂ R be defined by  (Ax )() ; Px = x () 

 Qy = T



T 



 y (s) ds, y (s)

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then Im P = Ker L, Ker Q = Im L. Setting LP = L|D(L)∩Ker P and writing L– P : Im L → D(L) to denote the inverse of LP , then



L– P y

 (A– Gy )(t) , (t) = (Gy )(t) 

[Gy ](t) =

 t n–    (Ax )(i) ()t i + (t – s)n– y (s) ds, i! (n – )!  i=

 t m–    (i) i [Gy ](t) = x ()t + (t – s)m– y (s) ds, i!  (m – )!  i=

(.)

where (Ax )(i) (), i = , , . . . , n –  are defined by the following: ⎛

E Z = B,

 ⎜ c ⎜  ⎜ ⎜ c where E = ⎜ ⎜ ··· ⎜ ⎜ ⎝cn– cn–

  c

  

··· ··· ···

  

cn– cn–

cn– cn–

··· ···

 c

⎞  ⎟ ⎟ ⎟ ⎟ ⎟ . ⎟ ⎟ ⎟ ⎠  (n–)×(n–)

T  i Z = ((Ax )(n–) (), . . . , (Ax ) (), (Ax ) ()) , B = (b , b , . . . , bn– ) , bi = – i!T  (T – s) × j (i) T y (s) ds, and cj = (j+)! , j = , , . . . , n – . x (), i = , , . . . , m – , are determined by the equation ⎛

E W = F,

 ⎜ ⎜ c ⎜ ⎜ c where E = ⎜ ⎜ ··· ⎜ ⎜ ⎝cm– cm–

  c

  

··· ··· ···

  

cm– cm–

cm– cm–

··· ···

 c

⎞  ⎟ ⎟ ⎟ ⎟ ⎟ . ⎟ ⎟ ⎟ ⎠  (m–)×(m–)

T  i W = ((x )(m–) (), . . . , (x ) (), (x ) ()) , F = (d , d , . . . , dn– ) , di = – i!T  (T – s) y (s) ds, j T and cj = (j+)! , j = , , . . . , m – . From (.) and (.), it is clear that QN and K(I –Q)N are continuous, QN() is bounded and then K(I – Q)N() is compact for any open bounded  ⊂ X, which means N is L¯ compact on .

3 Existence of positive periodic solutions for (1.1) For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel: (H ) There exist constants  < D < D such that if x is a positive continuous T-periodic function satisfying  

T

  g t, x(t) dt = ,

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then D ≤ x(τ ) ≤ D for some τ ∈ [, T]. (H ) g¯ (x) <  for all x ∈ (, D ), and g¯ (x) >  for all x > D . (H ) Assume that g(t, x) , xp–

ψ(t) = lim sup x→+∞

exist uniformly a.e. t ∈ [, T], i.e., for any ε >  there is gε ∈ L (, T) such that   g(t, x) ≤ ψ(t) + ε xp– + gε (t) for all x >  and a.e. t ∈ [, T]. Moreover, ψ ∈ C(R, R) and ψ(t + T) = ψ(t). (H ) g(t, x) = g (x) + g (t, x), where g ∈ C((, ∞); R) and g : [, T] × [, ∞) → R is an L -Carathéodory function.  (H )  g (x) dx = –∞. (H ) There exist two positive constants a, b such that    f x(t)  ≤ a|x |p– + b,

∀x ∈ R.

Theorem . Assume that conditions (H )-(H ) hold. Suppose one of the following conditions is satisfied: p ∞ T)T (i) p >  and (a+|ψ| ( T )(n–)(p–)+(m–) < ; p |–|c||p– π bT  ( T )n+m–

p

∞ T)T π ( T )(n–)(p–)+(m–) + |–|c|| < . (ii) p =  and (a+|ψ| p |–|c||p– π Then (.) has at least one positive T-periodic solution.

Proof Consider the equation Lx = λNx,

λ ∈ (, ).

Set  = {x : Lx = λNx, λ ∈ (, )}. If x(t) = (x (t), x (t)) ∈  , then ⎧ ⎨(Ax )(n) (t) = λϕ (x (t)),  q  ⎩x(m) (t) = –λf (x (t))x (t) – λg(t, x (t – σ )) + λe(t). 

(.)



Substituting x (t) = λ–p ϕp [(Ax )(n) (t)] into the second equation of (.) 

ϕp (Ax )(n) (t)

(m)

    + λp f x (t) x (t) + λp g t, x (t – σ ) = λp e(t).

(.)

Integrating of both sides of (.) from  to T, we have  

T

  g t, x (t – σ ) dt = .

(.)

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In view of (H ), there exist positive constants D , D , and ξ ∈ [, T] such that D ≤ x (ξ ) ≤ D . Then we have    t  t         x (t) = x (ξ ) + x (s) ds ≤ D + x (s) ds,  ξ

t ∈ [ξ , ξ + T],

ξ

and        x (t) = x (t – T) = x (ξ ) – 

ξ

t–T

   x (s) ds ≤ D +

ξ

   x (s) ds,

t–T

t ∈ [ξ , ξ + T].



Combing the above two inequalities, we obtain     |x |∞ = max x (t) = max x (t) t∈[ξ ,ξ +T]

t∈[,T]

  t   ξ            x (s) ds + x (s) ds ≤ max D + t∈[ξ ,ξ +T]  ξ  t–T   T    x (s) ds. ≤ D +   

(.)

Since  (n) (n) (n) (Ax )(n) (t) = x (t) – cx (t – σ ) = x(n)  (t) – cx (t – σ ) = Ax (t), from Lemma . and the first equation of (.), we have    (n)  x  = max A– Ax(n) (t)  ∞  t∈[,T]

≤

maxt∈[,T] |(Ax )(n) (t)| | – |c||

≤

ϕq (|x |∞ ) . | – |c||

(.)

() = x(m–) (T), there exists a point t ∈ [, T] such that On the other hand, from x(m–)   (m–) x (t ) = , which together with the integration of the second equation of (.) on the interval [, T] yields    T  (m)    x (t) dt  ≤  x(m–) (t ) +  x(m–) (t)       T       –f x (t) x (t) – g t, x (t – σ ) + e(t) dt ≤λ  



T

≤ 

   f x (t) x (t) dt + 

 

T

  g t, x (t – σ )  dt +

 

T

 e(t) dt.

(.)

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Write     I+ = t ∈ [, T] : g t, x (t – σ ) ≥  ;

    I– = t ∈ [, T] : g t, x (t – σ ) ≤  .

Then we get from (H ) and (.) that 

T

  g t, x (t – σ )  dt =





  g t, x (t – σ ) dt –

I+



  g t, x (t – σ ) dt

I–







g t, x (t – σ ) dt

= I+

 ≤

I+

  p–  ψ(t) + ε x (t – σ ) + gε (t) dt

  ≤  |ψ|∞ + ε



T

 x (t)p– dt + 





T

 gε (t) dt.

(.)



Substituting (.) and (.) into (.), and from (H ), we have   (t) ≤ a x(m–) 



   x (t)p– x (t) dt + b 

T 

  +  |ψ|∞ + ε





 x (t) dt 

T



 x (t)p– dt + 

T 



 gε (t) dt +

T 



T

 e(t) dt



p–  T   T         T    x (t) dt + b x (t) dt x (t) dt ≤ a D +        p–     T     x (t) dt +  |ψ|∞ + ε T D + + T  |gε | + |e|∞ T.   For a given constant δ > , which is only dependent on k > , we have ( + x)k ≤  + ( + k)x for x ∈ [, δ]. From (.), we have p–  p–  T      D  ≤ a  x(m–) x (t) (t) dt +  T    p–   |x (t)| dt  p– (|ψ|∞ + ε)T D + +   T  p–  |x (t)| dt p–  T  T         x (t) dt + T  |gε | + |e|∞ T x (t) dt · +b  



    D (p – ) D p (|ψ|∞ + ε)T a  + + ≤ p–  +  T T    p–  |x (t)| dt  |x (t)| dt p–  T  T       x (t) dt x (t) dt + T  |gε | + |e|∞ T +b ·   

 =



a p–



+

(|ψ|∞ + ε)T p–





T

 x (t) dt 

p–

(.)

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  aD (p – ) + |ψ|∞ + ε TD p + p–  T p–  T        x (t) dt + T  |gε | + |e|∞ T.  x (t) dt · +b  

(.)



From the Wirtinger inequality (see [], Lemma .), we get  

T

 x (s) ds ≤ T  



T 

 x (s) ds 

 

  T    (n)  T n–   ≤T x (s) ds π   n–  (n)  T x  . ≤T  ∞ π  



(.)

Substituting (.) into (.), we have    (m–)  (a + (|ψ|∞ + ε)T)T p– T (n–)(p–)  (n) p–  x    x (t) ≤  ∞ p– π   (aD (p – ) + (|ψ|∞ + ε)TD p)T p– T (n–)(p–)  (n) p– x ∞ + p– π  n–  (n)  T x  + T  |gε | + |e|∞ T. + bT  ∞ π

(.)

Substituting (.) into (.), we have    (a + (|ψ|∞ + ε)T)T p– T (n–)(p–) (ϕq (|x |∞ ))p–  ≤ (t) x(m–)  p– π | – |c||p–   (aD (p – ) + (|ψ|∞ + ε)TD p)T p– T (n–)(p–) (ϕq (|x |∞ ))p– + p– π | – |c||p–  n– ϕq (|x |∞ ) T  + T  |gε | + |e|∞ T + bT π | – |c||   (a + (|ψ|∞ + ε)T)T p– T (n–)(p–) |x |∞ = p– π | – |c||p–   –q (aD (p – ) + (|ψ|∞ + ε)TD p)T p– T (n–)(p–) |x |∞ + p– π | – |c||p–  n– q– |x |∞ T  + T  |gε | + |e|∞ T. + bT (.) π | – |c|| T T Since  (ϕq (x (t))) dt =  (Ax (t))(n) (t) dt = , there exists a point t ∈ [, T] such that x (t ) = . From the Wirtinger inequality, we can easily get   T    x (t) dt |x |∞ ≤    √  T      T x (t) ≤   

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√  m–  T    (m–)  T T   x (t) dt ≤  π    T T m–  (m–)  x ≤ . ∞  π

(.)

Combination of (.) and (.) implies T |x |∞ ≤  ≤



T π

m–

 (m–)  x  

∞

  (a + (|ψ|∞ + ε)T)T p– T (n–)(p–) |x |∞ p– π | – |c||p–   –q (aD (p – ) + (|ψ|∞ + ε)TD p)T p– T (n–)(p–) |x |∞ + p– π | – |c||p–  n–  q– T |x |∞  + T  |gε | + |e|∞ T . + bT π | – |c||

T 



T π

m– 

So, we have |x |∞ ≤

  (a + (|ψ|∞ + ε)T)T p T (n–)(p–)+(m–) |x |∞ p | – |c||p– π     (aD (p – ) + |ψ|∞ + ε TD p)T p– T (n–)(p–)+(m–) + |x |–q ∞ p– | – |c||p– π     q–  bT  T n+m– |x |∞ T T m–   T  |gε | + |e|∞ T . + +  π | – |c||  π

Case (i): If p > , we can get  < q < . Since ε sufficiently small, we know that   (a + |ψ|∞ T)T p T (n–)(p–)+(m–) < , p | – |c||p– π there exists a positive constant M such that |x |∞ ≤ M .

(.)

Case (ii): If p = , we can get q = . Since ε is sufficiently small, we know that   T n+m– ) (a + |ψ|∞ T)T p T (n–)(p–)+(m–) bT  ( π < , + p p–  | – |c|| π | – |c|| there exists a positive constant M such that |x |∞ ≤ M . On the other hand, from (.), we have q–  (n)  x  ≤ ϕq (|x |∞ ) ≤ M := Mn .  ∞ | – |c|| | – |c||

(.)

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Since x () = x (T), there exists a point t ∈ [, T] such that x (t ) = . From the Wirtinger inequality, we can easily get     T      x (t) dt ≤ T x (t) dt        T T (n–)  (n)  x ∞ ≤  π   T T n–  ≤ Mn := M .  π

  x  ≤   ∞ 



T

(.)

Hence, from (.), we have |x |∞ ≤ D +

 



T 

 x (t) dt ≤ D + TM := M .  

(.)

From (.), (.), and (.) we have  T     (m–)  (m)  ≤  max x x (t) dt    ∞    T       λ –f x (t) x (t) – g t, x (t – σ ) + e(t) dt ≤    √    λ p– ≤ |f |M TM +  |ψ|∞ + ε TM +  T|gε | + T|e|∞ := λMm– ,  where |f |M = max


 x (t) dt 

T 

 T      T x (t) dt      T T (m–)  (m–)  x ≤ ∞  π   T T (m–) ≤ λMm– := λM .  π

≤

 

Next, it follows (.) that 

ϕp (Ax )(n) (t + σ )

(m)

     + λp f x (t + σ ) x (t + σ ) + λp g t + σ , x (t)

= λp e(t + σ ).

(.)

Namely, 

ϕp (Ax )(n) (t + σ ) = λp e(t + σ ).

(m)

       + λp f x (t + σ ) x (t + σ ) + λp g x (t) + g t + σ , x (t) (.)

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Multiplying both sides of (.) by x (t), we get 

(m)   ϕp (Ax )(n) (t + σ ) x (t) + λp f x (t + σ ) x (t + σ )x (t)     + λp g x (t) x (t) + λp g t + σ , x (t) x (t) = λp e(t + σ )x (t).

(.)

Let τ ∈ [, T], for any τ ≤ t ≤ T, we integrate (.) on [τ , t] and get  λp

x (t)

g (u) du x (τ )



t

= λp τ



t

=– τ

  g x (s) x (s) ds ϕp (Ax )(n) (s + σ )



(m)

  g s + σ , x (s) x (s) ds + λp

t

– λp τ



x (s) ds – λp 

τ

t τ

t

  f x (s + σ ) x (s + σ )x (s) ds

e(s + σ )x (s) ds.

(.)

By (.), (.), (.), and (.), we have   t    (m)  (n)  ϕp (Ax ) (s + σ ) x (s) ds  τ  t      ϕp (Ax )(n) (s + σ ) (m) x (s) ds ≤  τ

  ≤ x ∞



 

T 

 ϕp (Ax )(n) (t + σ )

(m)   dt



≤ λp x ∞



T 

   f x (t) x (t) dt + 



T

  g t, x (t – σ )  dt +





      p– ≤ λ M |f |M M +  |ψ|∞ + ε TM + T  gε+  + T|e|∞ .

T

 e(t) dt





p

We have  t       f x (s + σ ) x (s + σ )x (s) ds ≤ |f |M M T,       τ  t   √      T    g s + σ , x (s) x (s) ds ≤ x  g(t, x(t – σ ) dt ≤ Mp– T|gM | ,       

τ

where gM = max≤x≤M |g (t, x)| ∈ L (, T) is as in (H ); we have  t     e(t + σ )x (t) dt  ≤ M T|e|∞ .    τ

From these inequalities we can derive from (.) that    

x (t) x (τ )

  g (u) du ≤ M ,

(.)

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for some constant M , which is independent on λ, x, and t. In view of the strong force condition (H ), we know that there exists a constant M >  such that x (t) ≥ M ,

∀t ∈ [τ , T].

(.)

The case t ∈ [, τ ] can be treated similarly. From (.), (.), and (.) and (.), we let     = x = (x , x ) : E ≤ |x |∞ ≤ E , x ∞ ≤ E , |x |∞ ≤ E and    x  ≤ E , ∀t ∈ [, T] ,  ∞ where  < E < min{M , D }, E > max{M , D }, E > M , E > M , and E > M .  = {x : x ∈ ∂ ∩ Ker L} then ∀x ∈ ∂ ∩ Ker L  QNx = T

 

T



 ϕq (x (t)) dt. –f (x (t))x (t) – g(t, x (t – σ )) + e(t)

If QNx = , then x (t) = , x = E or –E . But if x (t) = E , we know 

T

 g(t, E ) – e(t) dt.

= 

From assumption (H ), we have x (t) ≤ D ≤ E , which yields a contradiction. Similarly if x = –E . We also have QNx = , i.e., ∀x ∈ ∂ ∩ Ker L, x ∈/ Im L, so conditions () and () of Lemma . are both satisfied. Define the isomorphism J : Im Q → Ker L as follows: J(x , x ) = (x , –x ) . Let H(μ, x) = –μx + ( – μ)JQNx, (μ, x) ∈ [, ] × , then ∀(μ, x) ∈ (, ) × (∂ ∩ Ker L),  T [g(t, x ) – e(t)] dt –μx – –μ  T . H(μ, x) = –μx – ( – μ)|x |q– x 

We have

T 

e(t) dt = . So, we can get

 T –μx – –μ g(t, x ) dt   T , H(μ, x) = –μx – ( – μ)|x |q– x 

∀(μ, x) ∈ (, ) × (∂ ∩ Ker L).

From (H ), it is obvious that x H(μ, x) < , ∀(μ, x) ∈ (, ) × (∂ ∩ Ker L). Hence   deg{JQN,  ∩ Ker L, } = deg H(, x),  ∩ Ker L,    = deg H(, x),  ∩ Ker L,  = deg{I,  ∩ Ker L, } = . So condition () of Lemma . is satisfied. By applying Lemma ., we conclude that equa¯ ∩ D(L), i.e., (.) has an T-periodic solution tion Lx = Nx has a solution x = (x , x ) on  x (t). 

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Example . Consider the p-Laplacian type high-order neutral differential equation with singularity       ϕp x(t) – x(t – τ ) + x (t)x (t) + (cos t + )x (t – σ ) – κ  x (t – σ ) = sin t,

(.)

where κ ≥  and p = , σ and τ are constants, and  ≤ σ , τ < T.  , ψ(t) = It is clear that T = π , m = n = , c = , g(t, x) =  (cos t + )x (t – σ ) – xκ (t–σ )     (cos t + ), |ψ| = , f (x(t)) = x (t), and |f (x(t))| ≤ |x (t)| + ; here a = , b = . It is ∞   obvious that (H )-(H ) hold. Now we consider the assumption of the condition   (a + |ψ|∞ T)T p T (n–)(p–)+(m–) p | – |c||p– π =

( + π )π   ×     ×  

=

( + π )π  < .  × 

So by Theorem ., we know (.) has at least one positive π -periodic solution.

Competing interests The authors declare that they have no competing interests. Authors’ contributions YX, XFH, and ZBC worked together in the derivation of the mathematical results. All 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. 3 Department of Mathematics, Sichuan University, Chengdu, 610064, China. Acknowledgements YX, XFH, and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11501170), Fundamental Research Funds for the Universities of Henan Provience (NSFRF140142), Education Department of Henan Province project (No. 16B110006), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055). Received: 12 November 2015 Accepted: 26 January 2016 References 1. Lazer, AC, Solimini, S: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109-114 (1987) 2. Rachunková, I, Tvrdý, M, Vrko˘c, I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ. 176, 445-469 (2001) 3. Bonheure, D, De Coster, C: Forced singular oscillators and the method of lower and upper solutions. Topol. Methods Nonlinear Anal. 22, 927-938 (2003) 4. Hakl, R, Torres, P: On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differ. Equ. 248, 111-126 (2010) 5. Fonda, A, Manásevich, R: Subharmonics solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 24, 1294-1311 (1993) 6. Xia, J, Wang, ZH: Existence and multiplicity of periodic solutions for the Duffing equation with singularity. Proc. R. Soc. Edinb., Sect. A 137, 625-645 (2007) 7. Cheng, ZB, Ren, JL: Periodic and subharmonic solutions for Duffing equation with singularity. Discrete Contin. Dyn. Syst., Ser. A 32, 1557-1574 (2012) 8. Wang, ZH: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal., Real World Appl. 16, 227-234 (2014) 9. Cheng, ZB: Existence of positive periodic solutions for third-order differential equation with strong singularity. Adv. Differ. Equ. 2014, 162 (2014)

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