Open Access

Periodic solutions for a kind of neutral functional differential systems

Boundary Value Problems20142014:151

https://doi.org/10.1186/s13661-014-0151-1

Received: 23 March 2014

Accepted: 3 June 2014

Published: 23 September 2014

Abstract

In this paper, we analyze some properties of the linear difference operator A : C T C T , [ A x ] ( t ) = x ( t ) V ( t ) x ( t τ ) , and then, by using the coincidence degree theory of Mawhin, a kind of neutral differential systems with non-constant matrix is studied. Some new results on the existence of periodicity are obtained. It is worth noting that V ( t ) is no longer a constant matrix, which is different from the corresponding ones of past work.

Keywords

periodic solutionsneutralMawhin’s continuation theorem

1 Introduction

The field of neutral functional equations (in short NFDEs) is making significant breakthroughs in its practice; it is no longer only a specialist’s field. In many practical systems, models of systems are described by NFDEs in which the models depend on the delays of state and state derivatives. Practical examples for neutral systems include population ecology, heat exchanges, mechanics, and economics; see [1]–[4]. In particular, qualitative analysis such as periodicity and stability of solutions of NFDEs has been studied extensively by many authors. We refer to [5]–[12] for some recent work on the subject of periodicity and stability of neutral equations.

In the last few years, the stability of neutral systems of various classes with time delays has received an ever-growing interest from many authors. Many sufficient conditions have been proposed to guarantee the asymptotic stability for neutral time delay systems. We only mention the work of some authors [13]–[15]. It is well known that the existence of periodic solutions of neutral equations and neutral systems is a very basic and important problem, which plays a role similar to stability. Thus, it is reasonable to seek conditions under which the resulting periodic neutral system would have a periodic solution. Much progress has been seen in this direction and many criteria are established based on different approaches. However, there is no paper for investigating the existence of periodic solutions of neutral system with non-constant matrix. In addition, to the best of our knowledge, most of the existing results deal with scalar NFEDs or neutral systems with a constant matrix. For example, in papers [16]–[20], based on Mawhin’s continuation theorem, several types of scalar neutral equations have been studied:
d 2 d t 2 ( u ( t ) k u ( t τ ) ) = f ( u ( t ) ) u ( t ) + α ( t ) g ( u ( t ) ) + j = 1 n β j ( t ) g ( u ( t γ j ( t ) ) ) + p ( t ) , d N d t = N ( t ) [ α ( t ) β ( t ) N ( t ) j = 1 n b j ( t ) N ( t σ j ( t ) ) i = 1 m c i ( t ) N ( t τ i ( t ) ) ] , d N d t = N ( t ) [ r ( t ) j = 1 n α j ( t ) ln N ( t σ j ( t ) ) i = 1 m b i ( t ) d d t ln ( t τ i ( t ) ) ] , x ( t ) + α x ( t τ ) = f ( t , x ( t ) ) , ( u ( t ) + B u ( t τ ) ) = g 1 ( t , u ( t ) ) g 2 ( t , u ( t τ 1 ) ) + p ( t ) .
For a neutral system, we note that Lu and Ge [21] studied the following system:
d 2 d t 2 ( x ( t ) C x ( t τ ˜ ) ) + d d t grad F ( x ( t ) ) + grad G ( x ( t τ ( t ) ) ) = p ( t ) .
But C is a constant symmetric matrix. The purpose of this paper is to investigate the existence of periodic solutions to the nonlinear neutral system with non-constant matrix of the form
d 2 d t 2 ( x ( t ) C ( t ) x ( t τ ) ) + d d t grad F ( x ( t ) ) + grad G ( x ( t γ ( t ) ) ) = p ( t ) ,
(1.1)

where x R n , C ( t ) = diag ( c 1 ( t ) , c 2 ( t ) , , c n ( t ) ) , C ( t + T ) = C ( t ) ; F ( x ) C 2 ( R n , R ) , G ( x ) C 1 ( R n , R ) ; p ( R , R n ) , p ( t + T ) = p ( t ) ; γ C ( R , R ) , γ ( t + T ) = γ ( t ) ; T, and τ are given constants with T > 0 .

Throughout this paper, we use some notation:
  1. (1)

    I n = { 1 , 2 , , n } ; a = ( a 1 , a 2 , , a n ) T R n , | a | = ( i = 1 n | a i | 2 ) 1 2 ;

     
  2. (2)
    C T = { x : x C ( R , R n ) , x ( t + T ) = x ( t ) , t R } with the norm
    | φ | 0 = max t [ 0 , T ] | φ ( t ) | , φ C T ;
     
  1. (3)
    C T 1 = { x : x C 1 ( R , R n ) , x ( t + T ) = x ( t ) , t R } with the norm
    φ = max t [ 0 , T ] { | φ | 0 , | φ | 0 } , φ C T 1 .
     

Clearly, C T and C T 1 are Banach spaces.

2 Main lemmas

Lemma 2.1

[22]

If | c ( t ) | 1 , then operator A 1 has a continuous inverse A 1 1 on C T , satisfying
( 1 ) [ A 1 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 ) 0 T | [ A 1 1 f ] ( t ) | d t { 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 , ( 3 ) A 1 1 f 0 { | f | 0 1 c 0 , c 0 < 1 , f C T , | f | 0 σ 1 , σ > 1 , f C T .
Here
c 0 = max t [ 0 , T ] | c ( t ) | , σ = min t [ 0 , T ] | c ( t ) | .
Let
A : C T C T , [ A ] ( t ) = x ( t ) V ( t ) x ( t τ ) ,

where t R , V ( t ) C T 1 is a real symmetric matrix.

We will give some properties of A .

Lemma 2.2

Suppose that λ 1 ( t ) , λ 2 ( t ) , , λ n ( t ) are eigenvalues of V ( t ) . Then the operator A has continuous inverse A 1 on C T , satisfying
( 1 ) 0 T | [ A 1 f ] ( t ) | d t { ( i = 1 n 1 ( 1 λ i , L ) 2 ) 1 2 0 T | f ( t ) | d t , λ i , L < 1 , f C T , ( i = 1 n 1 ( 1 λ i , l ) 2 ) 1 2 0 T | f ( t ) | d t , λ i , l > 1 , f C T , ( 2 ) | [ A 1 f ] | 0 { ( i = 1 n 1 ( 1 λ i , L ) 2 ) 1 2 | f | 0 , λ i , L < 1 , f C T , ( i = 1 n 1 ( 1 λ i , l ) 2 ) 1 2 | f | 0 , λ i , l > 1 , f C T ,
where
λ i , L = max t [ 0 , T ] | λ i ( t ) | , λ i , l = min t [ 0 , T ] | λ i ( t ) | , i I n .

Proof

  1. (1)
    Since V ( t ) is a real symmetric matrix, there exists an orthogonal matrix U ( t ) such that
    U ( t ) V ( t ) U T ( t ) = E λ ( t ) = diag ( λ 1 ( t ) , λ 2 ( t ) , , λ n ( t ) ) .
     
Consider the system
x ( t ) V ( t ) x ( t τ ) = f ( t ) ,
where we have equivalence to
y ( t ) E λ ( t ) y ( t τ ) = f ˜ ( t ) ,
(2.1)
where f ˜ ( t ) = U ( t ) f ( t ) , y ( t ) = U ( t ) x ( t ) . On the other hand, a component of the vector in system (2.1) is
y i ( t ) λ i ( t ) y i ( t τ ) = f ˜ i ( t ) , i I n .
From Lemma 2.1, we have
y i ( t ) = { f ˜ i ( t ) + j = 1 k = 1 j λ i ( t ( k 1 ) τ ) f ˜ i ( t j τ ) , λ i , L < 1 , f ˜ i ( t + τ ) λ i ( t + τ ) j = 1 k = 1 j + 1 1 λ i ( t + k τ ) f ˜ i ( t + j τ + τ ) , λ i , l > 1 .
(2.2)
Thus, A 1 exists and
A 1 : C T C T , A 1 f ( t ) = x ( t ) = U T ( t ) y ( t ) , t [ 0 , T ] .
(2.3)
When λ i , L < 1 , by (2.2) we get
| y i ( t ) | max t [ 0 , T ] | f ˜ i ( t ) | 1 λ i , L , i I i ,
i.e.,
max t [ 0 , T ] | y i ( t ) | max t [ 0 , T ] | f ˜ i ( t ) | 1 λ i , L , i I i .
Thus, by (2.3) we have
| A 1 f | 0 = max t [ 0 , T ] | U T ( t ) y ( t ) | = max t [ 0 , T ] | y ( t ) | = max t [ 0 , T ] ( i = 1 n y i 2 ( t ) ) 1 2 ( i = 1 n max t [ 0 , T ] y i 2 ( t ) ) 1 2 ( i = 1 n max t [ 0 , T ] | f ˜ i ( t ) | 2 ( 1 λ i , L ) 2 ) 1 2 ( i = 1 n 1 ( 1 λ i , L ) 2 ) 1 2 | f ˜ | 0 = ( i = 1 n 1 ( 1 λ i , L ) 2 ) 1 2 | U f | 0 = ( i = 1 n 1 ( 1 λ i , L ) 2 ) 1 2 | f | 0 .
Obviously,
0 T | A 1 f ( t ) | d t ( i = 1 n 1 ( 1 λ i , L ) 2 ) 1 2 0 T | f ( t ) | d t .
  1. (2)
    Similar to the above proof, when λ i , l > 1 , we get
    | A 1 f | 0 ( i = 1 n 1 ( 1 λ i , l ) 2 ) 1 2 | f | 0 , 0 T | A 1 f ( t ) | d t ( i = 1 n 1 ( 1 λ i , l ) 2 ) 1 2 0 T | f ( t ) | d t .
     

 □

Let X and Y be two Banach spaces and let L : D ( L ) X Y be a linear operator, a Fredholm operator with index zero (meaning that ImL is closed in Y and dim Ker L = codim Im L < + ). If L is a Fredholm operator with index zero, then there exist continuous projectors P : X X , Q : Y Y such that Im P = Ker L , Im L = Ker Q = Im ( I Q ) , and L D ( L ) Ker P : ( I P ) X Im L is invertible. Denote by K p the inverse of L P .

Let Ω be an open bounded subset of X, a map N : Ω ¯ Y is said to be L-compact in Ω ¯ if Q N ( Ω ¯ ) is bounded and the operator K p ( I Q ) N ( Ω ¯ ) is relatively compact. We first give the famous Mawhin continuation theorem.

Lemma 2.3

[23]

Suppose that X and Y are two Banach spaces and L : D ( L ) X Y is a Fredholm operator with index zero. Furthermore, Ω X is an open bounded set and N : Ω ¯ Y is L-compact on Ω ¯ . If all the following conditions hold:
  1. (1)

    L x λ N x , x Ω D ( L ) , λ ( 0 , 1 ) ,

     
  2. (2)

    N x Im L , x Ω Ker L ,

     
  3. (3)

    deg { Q N , Ω Ker L , 0 } 0 ,

     

then the equation L x = N x has a solution on Ω ¯ D ( L ) .

3 Main results

Theorem 3.1

Suppose that 0 T p ( t ) d t = 0 , φ ( t ) is a nonzero periodic solution of (3.1) and there exists a constant M > 0 such that

(H1): i I n , G x i is bounded in the set 1 (or 2 ), where
1 = { x : x = ( x 1 , x 2 , , x n ) R n , x i ( , M ] , x j R , j I n { i } } , 2 = { x : x = ( x 1 , x 2 , , x n ) R n , x i [ M , ) , x j R , j I n { i } } .

(H2): x i G x i > 0 (or <0), whenever | x i | > M , i I n .

(H3):Suppose that μ 1 , μ 2 , , μ n are eigenvalues of 2 F ( v ) x 2 , v R n , and there exists a constant λ F 0 such that
max { | μ 1 | , | μ 2 | , , | μ n | } λ F .
Then system (1.1) has at least one T-periodic solution, if λ 0 , i < 1 2 (or σ 0 , i > 1 ), ( λ 2 , i T n + n λ 1 , i n ) T + λ 0 , i < 1 , and τ = m T , m Z , where
λ 0 , i = max t [ 0 , T ] { | c i ( t ) | , i I n } , λ 1 , i = max t [ 0 , T ] { | c i ( t ) | , i I n } , λ 2 , i = max t [ 0 , T ] { | c i ( t ) | , i I n } , σ 0 , i = min t [ 0 , T ] { | c i ( t ) | , i I n } .

Proof

Define
A : C T C T , [ A x ] ( t ) = x ( t ) C ( t ) x ( t τ ) , t R , N : C T 1 C T , ( N x ) ( t ) = d d t grad F ( x ( t ) ) grad G ( x ( t γ ( t ) ) ) + p ( t ) , L : D ( L ) C T C T 1 , L x = ( A x ) ,
where D ( L ) = { x : x C T 1 , x C ( R , R n ) } . Then system (1.1) obeys the operator equation L x = N x . We have ( x ( t ) C ( t ) x ( t τ ) ) = 0 , x Ker L . Then
x ( t ) C ( t ) x ( t τ ) = c ˜ 1 t + c ˜ 2 ,
where c ˜ 1 , c ˜ 2 R n . Since x ( t ) C ( t ) x ( t τ ) C T , we have c ˜ 1 = 0 . Let φ ( t ) C ( R , R n ) be a nonzero periodic solution of
x ( t ) C ( t ) x ( t τ ) = I ,
(3.1)
then | φ ( t ) | 2 > 0 and 0 T φ 2 ( t ) d t 0 , where I is an unit matrix. We get
Ker L = { a 0 φ ( t ) : a 0 R } , Im L = { y : y C T , 0 T y ( s ) d s = 0 } .
Obviously, ImL is closed in C T and dim Ker L = codim Im L = n . So L is a Fredholm operator with index zero. Define continuous projectors P, Q:
P : C T Ker L , ( P x ) ( t ) = 0 T x ( t ) φ ( t ) d t 0 T φ 2 d t φ ( t )
and
Q : C T C T / Im L , Q y = 1 T 0 T y ( s ) d s .
Let
L P = L | D ( L ) Ker P : D ( L ) Ker P Im L ,
then
L P 1 = K P : Im L D ( L ) Ker P .

Since Im L C T and D ( L ) Ker P C T 1 , K P is an embedding operator. Hence K P is a completely continuous operator in ImL. By the definitions of Q and N, one knows that Q N ( Ω ¯ ) is bounded on Ω ¯ . Hence the nonlinear operator N is L-compact on Ω ¯ . We complete the proof by three steps.

Step 1. Let Ω 1 = { x D ( L ) C T 1 : L x = λ N x , λ ( 0 , 1 ) } . We show that Ω 1 is a bounded set. We have L x = λ N x x Ω 1 , i.e.,
d 2 d t 2 ( x ( t ) C ( t ) x ( t τ ) ) + λ d d t grad F ( x ( t ) ) + λ grad G ( x ( t γ ( t ) ) ) = λ p ( t ) .
(3.2)
Integrating both sides of (3.2) over [ 0 , T ] , we have
0 T grad G ( x ( t γ ( t ) ) ) d t = 0 ,
i.e., i I n ,
0 T G ( x ( t γ ( t ) ) ) x i d t = 0 .
(3.3)
Let G x i be bounded in 1 and
x i G x i > 0 , whenever  | x i | > M .
(3.4)
Let
E 1 = { t : t [ 0 , T ] , x ( t γ ( t ) ) M } , E 2 = { t : t [ 0 , T ] , x ( t γ ( t ) ) > M } .
By assumption (H1), if x i M , there exists a constant M 1 > 0 such that | G x i | M 1 . From (3.3) and (3.4), we get
E 2 | G ( x ( t γ ( t ) ) ) x i | d t = E 2 G ( x ( t γ ( t ) ) ) x i d t E 1 | G ( x ( t γ ( t ) ) ) x i | d t M 1 T .
Thus
0 T | G ( x ( t γ ( t ) ) ) x i | d t = E 1 | G ( x ( t γ ( t ) ) ) x i | d t + E 2 | G ( x ( t γ ( t ) ) ) x i | d t 2 M 1 T ,
i.e.,
0 T | grad G ( x ( t γ ( t ) ) ) | d t = 0 T [ i = 1 n ( G ( x ( t γ ( t ) ) ) x i ) 2 ] 1 2 d t 0 T [ i = 1 n | G ( x ( t γ ( t ) ) ) x i | ] d t 2 n M 1 T .
(3.5)
We claim that there exists a point t 1 R such that
| x i ( t 1 ) | M .
(3.6)
In fact, for t [ 0 , T ] , we have x i ( t γ ( t ) ) > M , and by (3.4), we have 0 T G ( x ( t γ ( t ) ) ) x i d t > 0 , which is a contradiction; see (3.3). So there must be a point ξ [ 0 , T ] such that
x i ( ξ λ ( ξ ) ) M .
(3.7)
Similar to the above proof, there must be a point η [ 0 , T ] such that
x i ( η γ ( η ) ) M .
(3.8)
  1. (1)
    If x i ( ξ γ ( ξ ) ) M , by (3.7) we have
    | x i ( ξ λ ( ξ ) ) | M .
     
Let t 1 = ξ γ ( ξ ) . This proves (3.6).
  1. (2)
    If x i ( ξ γ ( ξ ) ) < M , from (3.8) and the fact that x i ( t ) is continuous on , there is a point t 1 between ξ γ ( ξ ) and η γ ( η ) such that | x i ( t 1 ) | M . This also proves (3.6). Let t 1 = k π + t 2 , k Z , t 2 [ 0 , T ] . Then | x i ( t 1 ) | = | x i ( t 2 ) | M . Hence we get
    | x i ( t ) | = max t [ 0 , T ] | x i ( t 2 ) + t 2 t x i ( s ) d s | | x i ( t 2 ) | + 0 T | x i ( s ) | d s M + 0 T | x ( s ) | d s , | x | 0 n ( M + 0 T | x ( s ) | d s ) n ( M + T 1 2 ( 0 T | x ( s ) | 2 d s ) 1 2 ) .
     
Multiplying the two sides of system (3.2) by x T ( t ) and integrating them over [ 0 , T ] , combining with τ = m T , by (3.9) we have
0 T | x ( t ) | 2 d t + λ 2 , i 0 T | x ( t ) | 2 d t + n λ 1 , i | x | 0 0 T | x ( t ) | d t + λ 0 , i 0 T | x ( t ) | 2 d t + λ 0 T x T ( t ) grad G ( x ( t γ ( t ) ) d t λ 0 T x T ( t ) p ( t ) d t 0 ,
i.e.,
( 1 λ 0 , i ) 0 T | x ( t ) | 2 d t λ 2 , i T n ( M + 0 T | x ( t ) | d t ) 2 + n λ 1 , i n ( M + 0 T | x ( t ) | d t ) 0 T | x ( t ) | d t + ( | p | 0 + 2 n M 1 ) T ( M + 0 T | x ( t ) | d t ) = ( λ 2 , i T n + n λ 1 , i n ) ( 0 T | x ( t ) | d t ) 2 + ( 2 λ 2 , i T n M + n λ 1 , i n M + | p | 0 T + 2 n M 1 T ) 0 T | x ( t ) | d t + λ 2 , i T n M 2 + ( | p | 0 + 2 n M 1 ) T M ( 2 λ 2 , i T n + n λ 1 , i n ) T 0 T | x ( t ) | 2 d t + ( 2 λ 2 , i T n M + n λ 1 , i n M + | p | 0 T + 2 n M 1 T ) T 1 2 × ( 0 T | x ( t ) | 2 d t ) 1 2 + λ 2 , i T n M 2 + ( | p | 0 + 2 n M 1 ) T M .
(3.9)
From (3.9) and ( λ 2 , i T n + n λ 1 , i n ) T + λ 0 , i < 1 , there is a constant M 2 > 0 such that
0 T | x ( t ) | 2 d t M 2 .
(3.10)
In view of (3.9) and (3.10), we get
| x | 0 n ( M + T 1 2 M 2 1 2 ) : = M 3 .
(3.11)
From Lemma 2.2, ( A x ( t ) ) = A x ( t ) 2 C ( t ) x ( t τ ) C ( t ) x ( t τ ) and (3.2), if λ 0 , i < 1 2 , we have
x ( t ) + A 1 [ λ d d t grad F ( x ( t ) ) + λ grad G ( x ( t γ ( t ) ) ) ] = A 1 [ 2 C ( t ) x ( t τ ) + C ( t ) x ( t τ ) + A 1 ( λ p ( t ) ) ] , 0 T | x ( t ) | d t ( i = 1 n 1 ( 1 λ 0 , i ) 2 ) 1 2 0 T | x ( t ) | d t × ( 0 T | 2 F ( x ( t ) x 2 | | x ( t ) | d t + 0 T | grad G ( x ( t γ ( t ) ) ) | d t 0 T | x ( t ) | d t + 2 T λ 1 , i 0 T | x ( t ) | d t + T λ 2 , i | x | 0 + T | p | 0 ) .
(3.12)
From assumption (H3) and (3.10)-(3.12), we get
0 T | x ( t ) | d t ( i = 1 n 1 ( 1 λ 0 , i ) 2 ) 1 2 ( λ F T 1 2 M 2 1 2 + 2 n M 1 T + 2 T λ 1 , i T 1 2 M 2 1 2 + T λ 2 , i M 3 + T | p | 0 ) .
So there exists a constant M 4 > 0 such that
0 T | x ( t ) | d t M 4 .
(3.13)
Since x ( t ) C T 1 , 0 T x ( t ) d t = 0 , there is a constant vector α R n such that x ( α ) = 0 ; then by (3.13) we get
| x ( t ) | 0 T | x ( t ) | d t M 4 .
Thus
| x | 0 M 4 .
Step 2. Let Ω { x Ker L : Q N x = 0 } , we shall prove that Ω 2 is a bounded set. We have x ( t ) = a 0 φ ( t ) , a 0 R x Ω 2 ; then
0 T grad G ( a 0 φ ( t γ ( t ) ) ) d t = 0 T grad G ( a 0 φ ( t ) ) d t = 0 .
(3.14)
When λ 0 , i < 1 2 , i I n , we have
φ i ( t ) = A 1 ( 1 ) = 1 + j = 1 k = 1 j c i ( t ( k 1 ) τ ) 1 j = 1 k = 1 j λ 0 , i = 1 λ 0 , i 1 λ 0 , i = 1 2 λ 0 , i 1 λ 0 , i : = δ > 0 .
Then we have
| φ ( t ) | n δ .
Thus
a 0 M n δ .
Otherwise, if, t [ 0 , T ] , | a 0 φ ( t ) | > M , then from assumption (H2), we have
G ( a 0 φ ( t ) ) x i > 0 ( or  < 0 ) , i I n ,
which is a contradiction; see (3.14). When σ 0 , i > 1 , i I n , we have
φ i ( t ) = A 1 ( 1 ) = 1 c i ( t + τ ) j = 1 k = 1 j 1 c i ( t + k τ ) 1 λ i , l j = 1 k = 1 j + 1 1 λ 0 , i = 1 λ i , l 1 : = γ < 0 .
Then we have
| φ ( t ) | n | γ | .
Thus
a 0 M n | γ | .
Otherwise, if t [ 0 , T ] , | a 0 φ ( t ) | > M , then from assumption (H2), we have
G ( a 0 φ ( t ) ) x i > 0 ( or  < 0 ) , i I n ,

which is a contradiction; see (3.14). Hence Ω 2 is a bounded set.

Step 3. Let Ω = { x C T 1 : | x | 0 < n M 2 + 1 , | x | 0 < n M 4 + 1 } , then Ω 1 Ω 2 Ω , ( x , λ ) Ω × ( 0 , 1 ) , and from the above proof, L x λ N x is satisfied. Obviously, condition (2) of Lemma 2.3 is also satisfied. Now we prove that condition (3) of Lemma 2.3 is satisfied. We have | x 0 | = | a 0 φ | 0 , a 0 R , x 0 Ω Ker L . There at least exists a i I n such that | x i 0 | > M . When x i 0 > M , take the homotopy
H ( x , μ ) = μ x + ( 1 μ ) Q N x , x Ω ¯ Ker L , μ [ 0 , 1 ] .
Then, by using assumption (H2), we have H ( x , μ ) 0 . When x i 0 < M , take the homotopy
H ( x , μ ) = μ x ( 1 μ ) Q N x , x Ω ¯ Ker L , μ [ 0 , 1 ] .
We also have H ( x , μ ) 0 . Then by degree theory,
deg { Q N , Ω Ker L , 0 } = deg { H ( , 0 ) , Ω ker L , 0 } = deg { H ( , 1 ) , Ω ker L , 0 } = deg { I , Ω ker L , 0 } 0 .

Applying Lemma 2.3, we reach the conclusion. □

Remark 3.1

When 1 2 λ 0 , i < 1 or σ 0 , i < 1 , there are no existence results for periodic solutions for system (1.1). We hope that there is interest in doing further research on this issue.

As an application, we consider the following system:
d 2 d t 2 ( x ( t ) C ( t ) x ( t 4 π ) ) + d d t grad F ( x ( t ) ) + grad G ( x ( t 5 cos t ) ) = p ( t ) ,
(3.15)
where
x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) T , τ = 4 π , γ ( t ) = 5 cos t , C ( t ) = diag ( sin t 1 , 000 , cos t 1 , 000 ) , F ( x ) = 1 2 π ( x 1 2 + 2 x 1 x 2 + x 2 2 + 2 x 1 + 3 x 2 + 1 ) , G ( x ) = 1 2 π ( x 1 + x 2 ) , p ( t ) = ( sin t , cos t ) T .
Clearly, system (3.15) is a particular case of system (1.1). Obviously,
grad G ( x ) = 1 2 π ( x 1 , x 2 ) T , 2 F ( v ) x 2 = ( 1 π 1 π 1 π 1 π ) .
Here assumptions (H1)-(H2) are all satisfied. In addition,
T = 2 π , λ 0 , i = λ 1 , i = λ 2 , i = 1 1 , 000 , n = 2 , ( λ 2 , i T n + n λ 1 , i n ) T + λ 0 , i 0.0976 < 1 .

By using Theorem 3.1, when λ 0 , i < 1 2 , we know that system (3.15) has at least one 2π-periodic solution.

Declarations

Acknowledgements

The authors wish to thank the anonymous referee for his/her valuable suggestions to this paper. This work is supported by the NSFC of China (11171085).

Authors’ Affiliations

(1)
Tianmu College, Zhejiang A and F University, Hangzhou, China
(2)
Department of Mathematics, Huaiyin Normal University, Huaian, China

References

  1. Kuang Y: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York; 1993.Google Scholar
  2. Hale J: Theory of Functional Differential Equations. Springer, New York; 1997.Google Scholar
  3. Kolmannovskii V, Myshkis A: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic, London; 1999.View ArticleGoogle Scholar
  4. Xu S, Lam J, Zou Y: Further results on delay-dependent robust stability conditions of uncertain neutral systems. Int. J. Robust Nonlinear Control 2005, 15: 233-246. 10.1002/rnc.983MathSciNetView ArticleGoogle Scholar
  5. Babram MA, Ezzinbi K: Periodic solutions of functional differential equations of neutral type. J. Math. Anal. Appl. 1996, 204: 898-909. 10.1006/jmaa.1996.0475MathSciNetView ArticleGoogle Scholar
  6. Corduneanu C: Existence of solutions for neutral functional differential equations with causal operators. J. Differ. Equ. 2000, 168: 93-101. 10.1006/jdeq.2000.3879MathSciNetView ArticleGoogle Scholar
  7. Weng P, Liang M: The existence and behavior of periodic solution of Hematopoiesis model. Math. Appl. 1995, 8: 434-439.MathSciNetGoogle Scholar
  8. Chen FD: Positive periodic solutions of neutral Lotka-Volterra system with feedback control. Appl. Math. Comput. 2005, 162: 1279-1302. 10.1016/j.amc.2004.03.009MathSciNetView ArticleGoogle Scholar
  9. Beretta E, Solimano F, Takeuchi Y: A mathematical model for drug administration by using the phagocytosis of red blood cell. J. Math. Biol. 1996, 35: 1-19. 10.1007/s002850050039MathSciNetView ArticleGoogle Scholar
  10. Rabah R, Sklyar GM, Rezounenko AV: Stability analysis of neutral type systems in Hilbert space. J. Differ. Equ. 2005, 214: 391-428. 10.1016/j.jde.2004.08.001MathSciNetView ArticleGoogle Scholar
  11. Shen J, Liu Y, Li J: Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses. J. Math. Anal. Appl. 2007, 332: 179-189. 10.1016/j.jmaa.2006.09.078MathSciNetView ArticleGoogle Scholar
  12. Wang K: Global attractivity of periodic solution for neutral functional differential system with multiple deviating arguments. Math. Methods Appl. Sci. 2011, 34: 1308-1316. 10.1002/mma.1437MathSciNetView ArticleGoogle Scholar
  13. Fridman E: New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Syst. Control Lett. 2004, 40: 1087-1092.Google Scholar
  14. Han Q: On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty. Automatica 2001, 42: 313-322.Google Scholar
  15. He Y, She J, Liu G: Delay-dependent robust stability criteria for uncertain neutral systems with mixed delay. Syst. Control Lett. 2004, 51: 57-65. 10.1016/S0167-6911(03)00207-XMathSciNetView ArticleGoogle Scholar
  16. Lu S, Ren J, Ge W: Problems of periodic solutions for a kind of second order neutral functional differential equation. Appl. Anal. 2003, 82: 411-426. 10.1080/0003681031000103013MathSciNetView ArticleGoogle Scholar
  17. Lu S: On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments. J. Math. Anal. Appl. 2003, 280: 321-333. 10.1016/S0022-247X(03)00049-0MathSciNetView ArticleGoogle Scholar
  18. Lu S, Ge W: Existence of positive periodic solutions for neutral logarithmic population model with multiple delays. J. Comput. Appl. Math. 2004, 166: 371-383. 10.1016/j.cam.2003.08.033MathSciNetView ArticleGoogle Scholar
  19. Serra E: Periodic solutions for some nonlinear differential equations of neutral type. Nonlinear Anal. 1991, 17: 139-151. 10.1016/0362-546X(91)90217-OMathSciNetView ArticleGoogle Scholar
  20. Liu B, Huang L: Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equation. J. Math. Anal. Appl. 2006, 322: 121-132. 10.1016/j.jmaa.2005.08.069MathSciNetView ArticleGoogle Scholar
  21. Lu S, Ge W: On the existence of periodic solutions for a kind of second order n -dimensional neutral functional differential systems. Acta Math. Sin. Engl. Ser. 2003, 46: 601-610.MathSciNetGoogle Scholar
  22. Du B, Ge W, Lu S: Periodic solutions for generalized Liénard neutral equation with variable parameter. Nonlinear Anal. 2009, 70: 2387-2394. 10.1016/j.na.2008.03.021MathSciNetView ArticleGoogle Scholar
  23. Gaines R, Mawhin J: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1977.Google Scholar

Copyright

© He and Du; licensee Springer 2014

This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.