Xin and Cheng Advances in Difference Equations (2016) 2016:41 DOI 10.1186/s13662-015-0721-2

RESEARCH

Open Access

Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument Yun Xin1* and Zhibo Cheng2 *

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 p-Laplacian Liénard type differential equation with singularity and deviating argument:

(ϕp (x (t))) + 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 solution; p-Laplacian; Liénard equation; singularity; deviating argument

1 Introduction In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument:          ϕp x (t) + f x(t) x (t) + g t, x(t – σ ) = e(t),

(.)

where ϕp : R → R is given by ϕp (s) = |s|p– s, here p >  is a constant, f is continuous function; g is a continuous function defined on R and periodic in t with g(t, ·) = g(t + T, ·), g has a singularity at x = ; σ is a constant and  ≤ σ < T; e : R → R are continuous periodic T functions with e(t + T) ≡ e(t) and  e(t) dt = . As is well known, the existence of periodic solutions for Liénard type differential equations was extensively studied (see [–] and the references therein). In recent years, there also appeared some results on a Liénard type differential equation with singularity; see [, ]. In , using coincidence degree theory, Zhang considered the existence of Tperiodic solutions for the scalar Liénard equation     x (t) + f x(t) x (t) + g t, x(t) = , when g becomes unbounded as x → + . The main emphasis was on the repulsive case, i.e. when g(t, x) → +∞, as x → + . Afterwards, Wang [] studied the existence of periodic © 2016 Xin and Cheng. 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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solutions of the Liénard equation with a singularity and a deviating argument,     x (t) + f x(t) x (t) + g t, x(t – σ ) = , where σ is a constant. When g has a strong singularity at x =  and satisfies a new small force condition at x = ∞, the author proved that the given equation has at least one positive T-periodic solution. However, the Liénard type differential equation (.), in which there is a p-Laplacian Liénard type differential equation, has not attracted much attention in the literature. There are not so many existence results for (.) even as regards the p-Laplacian Liénard type differential equation with singularity and deviating argument. In this paper, we try to fill this 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 [, ].

2 Preparation 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 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). For the sake of convenience, throughout this paper we will adopt the following notation:   |u|∞ = max u(t),

  |u| = min u(t), t∈[,T]

t∈[,T]

 |u|p =

T

|u|p dt

 p

 h¯ = T

,





T

h(t) dt. 

Lemma . ([]) If ω ∈ C  (R, R) and ω() = ω(T) = , then  

T

 ω(t)p dt ≤



T πp

where  ≤ p < ∞, πp = 

p 

T

 ω (t)p dt,



 (p–)/p 

ds sp )/p (– p–

=

π (p–)/p . p sin(π /p)

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Lemma . If x ∈ C  (R, R) with x(t + T) = x(t), and t ∈ [, T] such that |x(t )| < d, then 

T

 x(t)p dt

 p

 ≤



T πp



T

 x (t)p dt

 p



+ dT p .



Proof Let ω(t) = x(t + t ) – x(t ), and then ω() = ω(T) = . By Lemma . and Minkowski’s inequality, we have 

T

 x(t)p dt

 p



T

 ω(t) + x(t )p dt

=



 p





T

 ω(t)p dt

≤

 p





 ≤  =

T πp

T πp

 x(t )p dt

T

+

 p





T

 ω (t)p dt

 p



+ dT p





T

 x (t)p dt

 p



+ dT p .



This completes the proof of Lemma ..



In order to apply the topological degree theorem to study the existence of a positive periodic solution for (.), we rewrite (.) in the form ⎧ ⎨x (t) = ϕ (x (t)), q   ⎩x (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 an T-periodic solution to (.), then

x (t) must be an T-periodic solution to (.). Thus, the problem of finding an T-periodic solution for (.) reduces to finding one for (.). Now, set 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  R, R : x(t + T) = x(t), t ∈ R ⊂ X → Y by  x (t) (Lx)(t) =  x (t) 

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

(.)

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Then (.) can be converted to 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)

then Im P = Ker L, Ker Q = Im L. Let K denote the inverse of L|Ker p∩D(L) . It is easy to see that Ker L = Im Q = R and 

T

[Ky](t) =

G(t, s)y(s) ds, 

where G(t, s) =

⎧ ⎨s,

T ⎩ s–t , T

 ≤ s < t ≤ T;  ≤ t ≤ s ≤ T.

(.)

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 Main results Assume that ψ(t) = lim sup x→+∞

g(t, x) xp–

(.)

exists uniformly a.e. t ∈ [, T], i.e., for any ε >  there is gε ∈ L (, T) such that   g(t, x) ≤ ψ(t) + ε x + gε (t),

(.)

for all x >  and a.e. t ∈ [, T]. Moreover, ψ ∈ C(R, R) and ψ(t + T) = ψ(t). For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel: (H ) (Balance condition) There exist constants  < D < D such that if x is a positive continuous T-periodic function satisfying 

T

  g t, x(t) dt = ,



then D ≤ x(τ ) ≤ D , for some τ ∈ [, T].

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(H ) (Degree condition) g¯ (x) <  for all x ∈ (, D ), and g¯ (x) >  for all x > D . (H ) (Decomposition condition) g(t, x) = g (x) + g (t, x), where g ∈ C((, ∞); R) and 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 any b >  there is hb ∈ L (, T; R+ ) such that   g (t, x) ≤ hb (t),

a.e. t ∈ [, T], ∀ ≤ x ≤ b.

(H ) (Strong force condition at x = )

 

g (x) dx = –∞.

Theorem . Assume that conditions (H )-(H ) hold. Suppose the following condition is satisfied: (H ) ( πTp )p |ψ|∞ < . 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 ⎧ ⎨x (t) = λϕ (x (t)), q    ⎩x (t) = –λf (x (t))x (t) – λg(t, x (t – σ )) + λe(t). 

(.)



Substituting x (t) =

 ϕ (x (t)) λp– p 

into the second equation of (.)

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

(.)

Integrating both sides of (.) over [, T], we have 

T

  g t, x (t – σ ) dt = .

(.)



From (H ), there exist positive constants D , D , and ξ ∈ [, T] such that D ≤ x (ξ ) ≤ D .

(.)

Then we have    t  t         ≤ D + x (s) ds, x (t) = x (ξ ) + 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

t–T

   x (s) ds, 

t ∈ [ξ , ξ + T].

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Combining 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 +   

(.)

Multiplying both sides of (.) by x (t) and integrating over the interval [, T], we get 

T 

  ϕp x (t) x (t) dt + λp 



T 

  f x (t) x (t)x (t) dt + λp



T

  g t, x (t – σ ) x (t) dt



T

= λp

e(t)x (t) dt.

(.)



Substituting we have 

T 

(ϕp (x (t))) x (t) dt = –

 x (t)p d = λp 

T





T

T 

|x (t)|p dt,



 g t, x (t – σ ) x (t) dt – λp



T

f (x (t))x (t)x (t) dt =  into (.),





T

e(t)x (t) dt.

(.)



For any ε > , there exists a function gε ∈ L (, T) such that (.) holds. Since x (t) > , t ∈ [, T], it follows from (.) that     p– g t, x (t – σ ) x (t) ≤ ψ(t) + ε x (t – σ )x (t) + gε (t)x (t).

(.)

We infer from (.) and (.) 

 x (t)p dt 

T 



T

≤ λp 



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



    ψ(t) + ε xp– (t – σ )x (t) dt + 

T 

≤ 

  ≤ |ψ|∞ + ε



T

 x (t – σ )p dt





T

 gε (t) dt +

+ |x |∞ 

  ≤ |ψ|∞ + ε



 e(t) dt



T

 gε (t) + e(t) x (t) dt



    gε (t) + e(t) x (t) dt

T 



  p– p

T



T

 x (t)p dt

 p





   x (t)p dt + |x |∞



 gε (t) dt +

T 

T





  e(t) dt .

T

(.)



From Lemma . and (.), we have 

T

 x (t)p



 p

 ≤

T πp

 

T

 x (t)p dt 

 p



+ D T p .

(.)

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Substituting (.), (.) into (.), we get 

T 

 x (t)p dt  

≤ |ψ|∞ + ε





T πp



 x (t)p dt 

T 

 p + D T

 p

p

   T   T      T        x (t) dt gε (t) dt + e(t) dt + D +       p  T   p   T x (t) dt ≤ |ψ|∞ + ε  πp   p–   p–  T p   p  T x (t) dt p + · · · + Dp T D T +p    πp   p   T    p      x (t) dt T  |gε | + |e| + D + T q     p  T   p  T  x (t) dt = |ψ|∞ + ε  πp     T  p– p   p  T p–     x (t) dt D T p + · · · + |ψ|∞ + ε p πp   p  T   p      x (t) dt |gε | + |e| + T q+       p   + |ψ|∞ + ε D T + T  D |gε | + |e| , where |gε | = (

T 

(.)



|gε (t)| dt)  . Since ε is sufficiently small, from (H ) we know that

( πTp )p |ψ|∞ < . So, it is easy to see that there exists a positive constant M such that  

T

 x (t)p dt ≤ M .  

From (.), we have  |x |∞ ≤ D + 



 x (t) dt 

 

Tq ≤ D +  ≤ D +

T



T 

 x (t)p dt 

 p

 q

T    p M := M . 

(.)

Write  

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

 

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

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Then we get from (.) and (.) 

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 + 



√   p– ≤  |ψ|∞ + ε TM +  T|gε | .



T

 gε (t) dt



(.)

By the second equations of (.) and (.), we obtain 

 x (t) dt 

T 



T

≤λ

   f x (t) x (t) dt + λ 







≤ λ|f |M T q ≤ λ|f |M T

 q





 x (t)p dt 

T

M

 p



 p

T

  g t, x (t – σ )  dt + λ





T

 e(t) dt



√ √     p– + λ  |ψ|∞ + ε TM +  T|gε | + λ T|e|

√ √     p– + λ  |ψ|∞ + ε TM +  T|gε | + λ T|e|

:= λM ,

(.)

where |f |M = max
T

 x (s)q– x (s) ds = ,



which implies that there is a constant t ∈ [, T] such that x (t ) = , so |x |∞ ≤

 

 

 x (s) ds ≤   

t 

 

 x (s) ds ≤ λ M := λM .   

T

(.)

On the other hand, it follows from (.) that          ϕp x (t + σ ) + λp f x (t + σ ) x (t + σ ) + g t + σ , x (t) = λp e(t + σ ).

(.)

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

(.)

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

(.)

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

x (t)



t

g (u) du = λp

x (τ )

τ



t

=– τ

–λ

  g x (s) x (s) ds   ϕp x (s + σ ) x (s) ds – λp



p τ

t



 τ

 g s + σ , x (s) x (s) ds + λp

t

  f x (s + σ ) x (s + σ )x (s) ds



t

τ

e(s + σ )x (s) ds.

(.)

By (.), (.), (.), (.), and (.), we have  t           ϕ x (t + σ ) x (s) ds p     τ  t        ϕp x (t + σ )  x (s) ds ≤   τ

  ≤ x ∞



 

T 



≤ λp x ∞

    ϕp x (t + σ )   dt 



   f x (t) x (t) dt + 

T 



T

  g t, x (t – σ )  dt +





T

 e(t) dt





      p   p–  p– ≤ λp M |f |M M T q +  |ψ|∞ + ε TM + T  gε+  + T  |e| . We have  t        f x (s + σ ) x (s + σ )x (s) ds ≤ |f |M     

T



τ





≤ |f |M T q

τ

 x (t)p dt 

T 

  p

 ≤ |f |M T M p ,   t  √    T    g t, x(t – σ )  dt ≤ Mp– T|gM | , g s + σ , x (s) x (s) ds ≤ x     q

   

 x (s) ds 





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

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

x (t) x (τ )

  g (u) du ≤ M ,

(.)

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.

(.)

Xin and Cheng Advances in Difference Equations (2016) 2016:41

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From (.), (.), and (.), we let

 = x = (x , x ) : E ≤ |x |∞ ≤ E , |x |∞ ≤ E , ∀t ∈ [, T] , where  < E < min(M , D ), E > max(M , D ), 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 |p– 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 |p– 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 the ¯ ∩ D(L), i.e., (.) has an T-periodic equation Lx = Nx has a solution x = (x , x ) on   solution x (t). Finally, we present an example to illustrate our result.

Xin and Cheng Advances in Difference Equations (2016) 2016:41

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Example . Consider the p-Laplacian Liénard type differential equation with singularity and deviating argument:          ϕp x (t) + f x(t) x (t) + (cos t + )x (t – σ ) – κ = sin t,  x (t – σ )

(.)

where κ ≥  and p = , f is a continuous function, σ is a constant, and  ≤ σ < T.  , ψ(t) =  (cos t + ). It is It is clear that T = π , g(t, x) =  (cos t + )x (t – σ ) – xκ (t–σ ) obvious that (H )-(H ) hold. Now we consider the assumption (H ). Since |ψ|∞ ≤  , we have 

T πp

p

 |ψ|∞ =

T π (p–)/p p sin(π /p)

p

 |ψ|∞ ≤

π π (–)/  sin 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 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 (No. 11501170), Fundamental Research Funds for the Universities of Henan Province (NSFRF140142), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055). Received: 22 May 2015 Accepted: 8 December 2015 References 1. Atslega, S, Sadyrbaev, F: On periodic solutions of Liénard type equations. Math. Model. Anal. 18, 708-716 (2013) 2. Cheung, WS, Ren, JL: On the existence of periodic solutions for p-Laplacian generalized Liénard equation. Nonlinear Anal. TMA 60, 65-75 (2005) 3. Feng, MQ: Periodic solutions for prescribed mean curvature Liénard equation with a deviating argument. Nonlinear Anal., Real World Appl. 13, 1216-1223 (2012) 4. Gao, FB, Lu, SP: Existence of periodic solutions for a Liénard type p-Laplacian differential equation with a deviating argument. Nonlinear Anal. 69, 4754-4763 (2008) 5. Gutiérrez, A, Torres, P: Periodic solutions of Liénard equation with one or two weak singularities. Differ. Equ. Appl. 3, 375-384 (2011) 6. Liu, WB: Existence and uniqueness of periodic solutions for a kind of Liénard type p-Laplacian equation. Nonlinear Anal. TMA 69, 724-729 (2008) 7. Liu, WB, Liu, JY, Zhang, HX, Hu, ZG, Wu, YQ: Existence of periodic solutions for Liénard-type p-Laplacian systems with variable coefficients. Ann. Pol. Math. 109, 109-119 (2013) 8. Ma, TT, Wang, ZH: Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete Contin. Dyn. Syst. 33, 1563-1581 (2013) 9. Meng, H, Long, F: Periodic solutions for a Liénard type p-Laplacian differential equation. J. Comput. Appl. Math. 224, 696-701 (2009) 10. Tiryaki, A, Zafer, A: Global existence and boundedness for a class of second-order nonlinear differential equations. Appl. Math. Lett. 26, 939-944 (2013) 11. Zhang, MR: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203, 254-269 (1996) 12. Wang, ZH: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal., Real World Appl. 16, 227-234 (2014) 13. Gaines, R, Mawhin, J: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977) 14. Zhang, MR: Nonuniform nonresonance at the first eigenvalue of the p-Laplacian. Nonlinear Anal. TMA 29, 41-51 (1996)