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Periodic solutions for the p-Laplacian neutral functional differential system

Abstract

By using the generalized Borsuk theorem in coincidence degree theory, we prove the existence of periodic solutions for the p-Laplacian neutral functional differential system.

MSC:34C25.

1 Introduction

In recent years, the existence of periodic solutions for the Rayleigh equation and the Liénard equation has been studied (see [19]). By using topological degree theory, some results on the existence of periodic solutions are obtained.

Motivated by the works in [19], we consider the existence of periodic solutions of the following system:

d d t ϕ p [ ( x ( t ) C x ( t τ ) ) ] + d d t gradF ( x ( t ) ) +gradG ( x ( t ) ) =e(t),
(1.1)

where F C 2 ( R n ,R), G C 1 ( R n ,R), eC(R, R n ) are periodic functions with period T; C= [ c i j ] n × n is an n×n symmetric matrix of constants, τR is a constant. ϕ p : R n R n is given by

ϕ p (u)= ϕ p ( u 1 ,, u n ):= ( | u 1 | p 2 u 1 , , | u n | p 2 u n ) T ,1<p<.

The ϕ p is a homeomorphism of R n with the inverse ϕ q . By using the theory of coincidence degree, we obtain some results to guarantee the existence of periodic solutions. Even for p=2, the results in this paper are also new.

In what follows, we use , to denote the Euclidean inner product in R n and | | p to denote the l p -norm in R n , i.e., | x | p = ( i = 1 n | x i | p ) 1 / p .

The norm in R n × n is defined by A p = sup | x | ρ = 1 , x R n | A x | p .

The corresponding L p -norm in L p ([0,T], R n ) is defined by

x p = ( i = 1 n 0 T | x i ( t ) | p d t ) 1 p = ( 0 T | x ( t ) | p p d t ) 1 p ,

and the L -norm in L ([0,T], R n ) is

x = max 1 i n x i ,

where x i = sup t [ 0 , ω ] | x i (t)| (i=1,,n).

Let W= W 1 , p ([0,T],R] be the Sobolev space.

Lemma 1.1 (See [8])

Suppose uW and u(0)=u(T)=0, then

u p ( T π p ) u p ,

where

π p =2 0 ( p 1 ) 1 / p d s ( 1 s p p 1 ) 1 / p = 2 π ( p 1 ) 1 / p p sin ( π p ) .

In order to use coincidence degree theory to study the existence of T-periodic solutions for (1.1), we rewrite (1.1) in the following form:

{ ( x ( t ) C x ( t τ ) ) ( t ) = ϕ q ( y ( t ) ) , y ( t ) = d d t grad F ( x ( t ) ) grad G ( x ( t ) ) + e ( t ) .
(1.2)

If z(t)= ( x ( t ) y ( t ) ) is a T-periodic solution of (1.2), x(t) must be a T-periodic solution of (1.1). Thus, the problem of finding a T-periodic solution for (1.1) reduces to finding one for (1.2).

Let C T ={xC(R, R n ):x(t+T)x(t)} with the norm x = max 1 i n x i , X=Z={z= ( x ( ) y ( ) ) C(R, R 2 n ):z(t+T)z(t)} with the norm z=max{ x , y }. Clearly, X and Z are Banach spaces.

Denote the operator A by

A: C T C T ,(Ax)(t)=x(t)Cx(tτ).

Meanwhile, let

L : Dom L X Z , ( L z ) ( t ) = z ( t ) = ( ( A x ) ( t ) y ( t ) ) , N : X Z , ( N z ) ( t ) = ( ϕ q ( y ( t ) ) d d t grad F ( x ( t ) ) grad G ( x ( t ) ) + e ( t ) ) : = H ( z , t ) .

It is easy to see that KerL= R 2 n , ImL={zZ: 0 T z(s)ds=0}. So, L is a Fredholm operator with index zero. Let P:XKerL and Q:ZImQ be defined by

P u = 1 T 0 T u ( s ) d s , u X ; Q v = 1 T 0 T v ( s ) d s , v Z ,

and let K p denote the inverse of L | Ker P Dom L .

Obviously, KerL=ImQ= R 2 n and

( K p z)(t)= ( ( A 1 F x ) ( t ) ( F y ) ( t ) ) ,
(1.3)

where z= ( x T ( ) , y T ( ) ) T Z, (Fh)(t)= 0 t h(s)ds 1 T 0 T 0 t h(s)dsdt, h C T .

From (1.3), one can easily see that N is L-compact on Ω ¯ , where Ω is an open bounded subset of X.

Lemma 1.2 (See [9])

Suppose that λ 1 , λ 2 ,, λ n are eigenvalues of the matrix C. If | λ i |1, i{1,2,,n}, then A has a continuous bounded inverse A 1 with the following relationships:

  1. (1)

    A 1 u ( i = 1 n 1 | 1 | λ i | | )u, u C T ;

  2. (2)

    A 1 u p p dtσ u p p dt, u C T , p1, where

    σ= { max i { 1 , 2 , , n } { 1 | 1 | λ i | | 2 } , p = 2 , ( i = 1 n 1 | 1 | λ i | | 2 p 2 p ) ( 2 p ) / 2 , p [ 1 , 2 ) , ( i = 1 n 1 | 1 | λ i | | q ) p / q , p ( 2 , + ) ,

    where q is a constant with 1/p+1/q=1;

  3. (3)

    A x = ( A x ) , x C T .

In the proof of our results on the existence of periodic solutions, we use the following generalized Borsuk theorem in coincidence degree theory of Gaines and Mawhin [10].

Lemma 1.3 Let X and Z be real normed vector spaces. Let L be a Fredholm mapping of index zero. Ω is an open bounded subset of X and Ω is symmetric with respect to the origin and contains it. Let N ˜ : Ω ¯ ×[0,1]Z be L-compact and such that

  1. (a)

    N ˜ (x,0)= N ˜ (x,0), x Ω ¯ ,

  2. (b)

    Lx N ˜ (x,λ), xDomLΩ.

Then, for every λ[0,1], the equation Lx= N ˜ (x,λ) has at least one solution in Ω.

2 Main results

Theorem 2.1 Suppose that the matrix C satisfies the conditions of Lemma  1.2 and that there exist constants a>0, b>0, c0 and α>1 such that

  • (H1) y T 2 F ( x ) x 2 ya | y | 2 2 or y T 2 F ( x ) x 2 ya | y | 2 2 , x,y R n ;

  • (H2) y,gradG(x)b | y | α α c, x,y R n .

    Then equation (1.1) has at least one T-periodic solution for 1<p2.

Proof For any λ[0,1], let

N ˜ (z,λ)(t)= 1 + λ 2 H(z,t) 1 λ 2 H(z,t).

Consider the following parameter equation:

(Lz)(t)= N ˜ (z,λ)(t),λ[0,1].
(2.1)

Let z(t)= ( x ( t ) y ( t ) ) be a possible T-periodic solution of (2.3) for some λ[0,1], then x=x(t) is a T-periodic solution of the following system:

( ϕ p ( ( A x ) ( t ) ) ) + 1 + λ 2 d d t grad F ( x ( t ) ) 1 λ 2 d d t grad F ( x ( t ) ) + 1 + λ 2 grad G ( x ( t ) ) 1 λ 2 grad G ( x ( t ) ) = λ e ( t ) .
(2.2)

Noticing that x(t) is a T-periodic solution, we have

A x p p = 0 T A x , ( ϕ p ( A x ) ) dt.
(2.3)

Multiplying the two sides of (2.2) by (Ax)(t) and integrating them on the interval [0,T], by (2.3) and (H1)-(H2), we obtain

A x p p +a A x 2 2 +b A x α α cT e β A x α ,where  1 α + 1 β =1.
(2.4)

On the other hand,

0 T A x , [ ϕ p ( A x ) ] dt=0.

So, multiplying the two sides of (2.2) by (A x )(t) and integrating them on the interval [0,T], by (H1)-(H2), we get

a A x 2 2 cTa A x 2 2 +b A x α α cT e 2 A x 2 .

Furthermore, we have

A x 2 c T a + e 2 2 4 a 2 + e 2 2 a := R 1 .

It is obvious that there exist c 1 >0 and c 2 >0 such that

c 1 | x | 2 | x | p c 2 | x | 2 ,x R n .

Thus,

A x p p = 0 T | ( A x ) ( t ) | p p d t c 2 p ( 0 T | A x ( t ) | 2 2 d t ) p / 2 T ( 2 p ) / 2 ( c 2 R 1 ) p T ( 2 p ) / 2 : = R 2 ,
(2.5)

where 1<p2.

From (2.4) and (2.5), we can see

b A x α α e β A x α cT R 2 a A x 2 2 R 2 ,

from which it follows that there exists a positive number R 3 such that

A x α R 3 .

By using Lemma 1.2, we get

x α = ( 0 T | x ( t ) | α α d t ) 1 / α = ( 0 T | A 1 ( A x ) ( t ) | α α d t ) 1 / α σ 1 / α ( 0 T | ( A x ) ( t ) | α α d t ) 1 / α σ 1 / α R 3 : = R 4 .
(2.6)

From (2.6), there exists t 0 [0,T) such that | x ( t 0 ) | α R 4 T 1 / α , and

| x i ( t ) | = | x i ( t 0 ) + t 0 t x i ( s ) d s | R 4 T 1 / α + T ( 0 T ( x i ( s ) ) 2 d s ) 1 / 2 R 4 T 1 / α + T R 1 : = R 5 .

Therefore x R 5 and | x ( t ) | p n 1 / p R 5 .

Since F C 2 ( R n ,R), G C 1 ( R n ,R), there exist R 6 and R 7 such that

2 F ( x ) x 2 p R 6 , | grad G ( x ) | p R 7 for  | x | p n 1 / p R 5 .

From (2.4), we have

0 T | ( ϕ p ( A x ) ) | p d t R 6 0 T | x | p d t + R 7 T + 0 T | e ( t ) | p d t R 6 T 1 / q x p + R 7 T + 0 T | e ( t ) | p d t R 6 T 1 / q R 2 1 / p + R 7 T + 0 T | e ( t ) | p d t : = R 8 .

Clearly, for each i=1,,n, there exists t i (0,T) such that x i ( t i )=0. Thus, for any t[0,T], we have

| y i ( t ) | = | ϕ p ( ( A x i ) ( t ) ) | = | ϕ p ( ( A x i ) ( t ) ) ϕ p ( ( A x i ) ( t i ) ) | = | t i t ( ϕ p ( ( A x i ) ( s ) ) ) d s | R 8 .

Therefore y R 8 .

Choose a number R 9 >max( R 5 , R 8 ), and let Ω={zX:z< R 9 }, then Lz N ˜ (z,λ) for any zDomLΩ, λ[0,1]. It is easy to see that N ˜ is L-compact on Ω ¯ ×[0,1], Lz= N ˜ (z,1) is (2.1) and N ˜ (z,0)= N ˜ (z,0). From Lemma 1.3, (2.1) has at least one T-periodic solution z ˜ = ( x ˜ ( t ) y ˜ ( t ) ) , x ˜ (t) is a T-periodic solution of (1.1). □

Theorem 2.2 Let λ =max{| λ 1 |,| λ 2 |,,| λ n |}, where λ 1 , λ 2 ,, λ n are eigenvalues of the matrix C with | λ i |1, i{1,2,,n}. Suppose that there exist constants b0, c0 and d>0 such that

  • (H3) there is a constant r0 such that lim | x | + | grad F ( x ) | | x | p 1 r;

  • (H4) y,gradG(x)b | y | p p +c, x,y R n ;

  • (H5) i{1,,n}, either x i [ G ( x ) x i e ¯ i ]>0 or x i [ G ( x ) x i e ¯ i ]<0 for | x i |>d, where e ¯ i = 1 T 0 T e i (t)dt.

    Then (1.1) has at least one T-periodic solution for ( λ r+b) T π p <σ.

Proof Let z(t)= ( x ( t ) y ( t ) ) be a possible T-periodic solution of (2.1). From assumption (H3), there exists a constant ρ>d such that

| grad F ( x ) | <r | x | p 1 ,x R n  with | x i |>ρ for i=1,2,,n.

From (H3) and (2.2), we have

A x p p + 0 T A x ( t ) , 1 + λ 2 d d t grad F ( x ( t ) ) 1 λ 2 d d t grad F ( x ( t ) ) d t + b x p p + c T λ 0 T x ( t ) , e ( t ) d t e q x p ,

i.e.,

A x p p 0 T C x ( t τ ) , 1 + λ 2 d d t grad F ( x ( t ) ) 1 λ 2 d d t grad F ( x ( t ) ) d t + b x p p + e q x p + c T C x p grad F ( x ( t ) ) p p 1 + b x p p + e q x p + c T λ x p ( r x p p 1 + θ ) + b x p p + e q x p + c T ,
(2.7)

where θ= max | u | n p |gradF(u)| T ( p 1 ) / p .

Integrating both sides of (2.2) over [0,T], we get

1 + λ 2 0 T [ G ( x ( t ) ) x i e ¯ i ] dt 1 λ 2 0 T [ G ( x ( t ) ) x i e ¯ i ] dt=0,i=1,,n.

So, there exist t ˜ i [0,T] such that

1 + λ 2 0 T [ G ( x ( t ˜ i ) ) x i e ¯ i ] dt 1 λ 2 0 T [ G ( x ( t ˜ i ) ) x i e ¯ i ] dt=0,i=1,,n.

From (H4), one can see | x i ( t ˜ i )|d. Let χ i (t)= x i (t+ t ˜ i ) x i ( t ˜ i ), χ(t)= ( χ 1 ( t ) , , χ n ( t ) ) T , then χ(0)=χ(T)=0. By Lemma 1.1, one can obtain

χ p T π p χ p.

Noticing the periodicity of x(t), we have

x i p p = 0 T | x i ( t ) | p d t = 0 T | x i ( t + t ˜ i ) | p d t 0 T ( | χ i ( t ) | + d ) p d t ( χ i p + T 1 / p d ) p .

From Minkovski’s inequality, we have

x p = ( i = 1 n x i p p ) 1 / p ( i = 1 n χ i ( t ) + T 1 / p d ) p χ p + ( n T ) 1 / p d T π p χ p + ( n T ) 1 / p d = T π p x p + ( n T ) 1 / p d .

In view of (2.7) and Lemma 1.2, we get

σ x p p λ x p ( r ( T π p x p + ( n T ) 1 / p d ) p 1 + θ ) + b ( T π p x p + ( n T ) 1 / p d ) p + e q ( T π p x p + ( n T ) 1 / p d ) + c T .
(2.8)

Since ( λ r+b) T π p <σ, from (2.8), there exists a constant R 9 >0 such that

x p R 9 .
(2.9)

Therefore,

x p T π p R 9 + ( n T ) 1 / p d:= R 10 .
(2.10)

From (2.9) and (2.10), we know that the rest of the proof of the theorem is similar to that of Theorem 2.1. □

Remark 2.1 If C 0 n × n , system (1.1) can be reduced to the system in [2].

If C 0 n × n and p=2, system (1.1) can be reduced to the system in [3].

Example 2.1 Consider the following system:

d d t ϕ p [ ( x ( t ) C x ( t τ ) ) ] + d d t gradF ( x ( t ) ) +gradG ( x ( t ) ) =e(t),
(2.11)

where F C 2 ( R 2 ,R), G C 1 ( R 2 ,R), eC(R, R 2 ) are periodic functions with period T; C= ( 1 1 1 0 ) . Clearly, λ 1 , 2 = 1 ± 5 2 ±1.

Let

x= ( x 1 , x 2 ) T ,F( x 1 , x 2 )= x 1 2 + x 2 2 x 1 x 2 2 ,G( x 1 , x 2 )= x 1 4 + x 1 3 1 4 x 1 2 x 2 2 + x 2 4 ,

then, by Theorem 2.1, (2.11) has at least one T-periodic solution for 1<p2.

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Acknowledgements

The authors thank the referees for helpful suggestions. This work was supported by Grants Nos. 10871052 and 109010600 from NNSF of China, and by Grant No. S2011010005029 from NSF of Guangdong.

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Correspondence to Changxiu Song.

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The first author carried out the studies, the second author participated in the studies and drafted the manuscript. All authors read and approved the final manuscript.

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Wang, Z., Song, C. Periodic solutions for the p-Laplacian neutral functional differential system. Adv Differ Equ 2013, 367 (2013). https://doi.org/10.1186/1687-1847-2013-367

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Keywords

  • p-Laplacian
  • periodic solutions
  • coincidence degree