Research

# Positive periodic solutions for a second-order functional differential equation

Yongxiang Li* and Qiang Li

### Author affiliations

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

For all author emails, please log on.

Boundary Value Problems 2012, 2012:140 doi:10.1186/1687-2770-2012-140

 Received: 12 June 2012 Accepted: 12 November 2012 Published: 27 November 2012

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

### Abstract

In this paper, the existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation

u ¨ ( t ) = f ( t , u ( t ) , u ˙ ( t τ 1 ( t ) ) , , u ˙ ( t τ n ( t ) ) ) ,

where f : R × [ 0 , ) × R n R is a continuous function which is ω-periodic in t, τ i C ( R , [ 0 , ) ) is a ω-periodic function, i = 1 , 2 , , n . Our discussion is based on the fixed point index theory in cones.

MSC: 34C25, 47H10.

##### Keywords:
functional differential equation; positive periodic solution; cone; fixed point index

### 1 Introduction

In this paper, we discuss the existence of positive ω-periodic solutions of the second-order functional differential equation with the delay terms of first-order derivative in nonlinearity,

u ¨ ( t ) = f ( t , u ( t ) , u ˙ ( t τ 1 ( t ) ) , , u ˙ ( t τ n ( t ) ) ) , t R , (1)

where f : R × [ 0 , ) × R n R is a continuous function which is ω-periodic in t and τ i C ( R , [ 0 , ) ) is a ω-periodic delay function, i = 1 , 2 , , n .

For the second-order differential equation without delay and the first-order derivative term in nonlinearity,

u ¨ ( t ) = f ( t , u ( t ) ) , t R , (2)

the existence problems of periodic solutions have attracted many authors’ attention and concern. Many theorems and methods of nonlinear functional analysis have been applied to research the periodic problems of Equation (2), such as the upper and lower solutions method and monotone iterative technique [1-4], the continuation method of topological degree [5-7], variational method and critical point theory [8-10], the theory of the fixed point index in cones [11-16], etc.

In recent years, the existence of periodic solutions for the second-order delayed differential equations have also been researched by many authors; see [17-24] and the references therein. In some practice models, only positive periodic solutions are significant. In [20,21,23], the authors obtained the existence of positive periodic solutions for some delayed second-order differential equations as a special form of the following equation:

u ¨ ( t ) + b ( t ) u ˙ ( t ) + a ( t ) u ( t ) = f ( t , u ( t τ 1 ( t ) ) , , u ( t τ n ( t ) ) ) , t R , (3)

by using Krasnoselskii’s fixed point theorem of cone mapping or the theory of the fixed point index in cones. In these works, the positivity of Green’s function of the corresponding linear second-order periodic problems plays an important role. The positivity guarantees that the integral operators of the second-order periodic problems are cone-preserving in the cone

P = { u C [ 0 , ω ] u ( t ) σ u , t [ 0 , ω ] } (4)

in the Banach space C [ 0 , ω ] , where σ > 0 is a constant. Hence, the fixed point theorems of cone mapping can be applied to periodic problems of the second-order delay equation (3) as well as Equation (2) (for Equation (2), see [11-16]). However, few people consider the existence of positive periodic solutions of Equation (1). Since the nonlinearity of Equation (1) explicitly contains the delayed first-order derivative term, the corresponding integral operator has no definition on the cone P. Thus, the argument methods used in [20,21,23] are not applicable to Equation (1).

The purpose of this paper is to discuss the existence of positive periodic solutions of Equation (1). We will use a different method to treat Equation (1). Our main results will be given in Section 3. Some preliminaries to discuss Equation (1) are presented in Section 2.

### 2 Preliminaries

Let C ω ( R ) denote the Banach space of all continuous ω-periodic function u ( t ) with the norm u C = max 0 t ω | u ( t ) | . Let C ω 1 ( R ) be the Banach space of all continuous differentiable ω-periodic function u ( t ) with the norm

u C 1 = u C + u ˙ C .

Generally, C ω n ( R ) denotes the nth-order continuous differentiable ω-periodic function space for n N . Let C ω + ( R ) be the cone of all nonnegative functions in C ω ( R ) .

Let M ( 0 , π 2 ω 2 ) be a constant. For h C ω ( R ) , we consider the linear second-order differential equation

u ¨ ( t ) + M u ( t ) = h ( t ) , t R . (5)

The ω-periodic solutions of Equation (5) are closely related to the linear second-order boundary value problem

{ u ¨ ( t ) + M u ( t ) = 0 , 0 t ω , u ( 0 ) u ( ω ) = 0 , u ˙ ( 0 ) u ˙ ( ω ) = 1 , (6)

see [14]. It is easy to see that problem (6) has a unique solution which is explicitly given by

U ( t ) = cos β ( t ω 2 ) 2 β sin β ω 2 , 0 t ω , (7)

where β = M . By [[14], Lemma 1], we have

Lemma 2.1Let M ( 0 , π 2 ω 2 ) . Then, for every h C ω ( R ) , the linear equation (5) has a uniqueω-periodic solution u ( t ) which is given by

u ( t ) = t ω t U ( t s ) h ( s ) d s : = S h ( t ) , t R . (8)

Moreover, S : C ω ( R ) C ω 1 ( R ) is a completely continuous linear operator.

Since U ( t ) > 0 , for every t [ 0 , ω ] , by (8), if h C ω + ( R ) and h ( t ) 0 , then the ω-periodic solution of Equation (5) u ( t ) > 0 for every t R , and we term it the positive ω-periodic solution. Let

U ¯ = max 0 t ω U ( t ) = 1 2 β sin β ω 2 , U ̲ = min 0 t ω U ( t ) = cos β ω 2 2 β sin β ω 2 , U ¯ 1 = max 0 t ω | U ˙ ( t ) | = max 0 t ω | sin β ( t ω 2 ) | 2 sin β ω 2 = 1 2 , σ = U ̲ U ¯ = cos β ω 2 , C 0 = U ¯ 1 U ̲ = β tan β ω 2 . (9)

Define a set K in C ω 1 ( R ) by

K = { u C ω 1 ( R ) u ( t ) σ u C , | u ˙ ( τ ) | C 0 u ( t ) , τ , t R } . (10)

It is easy to verify that K is a closed convex cone in C ω 1 ( R ) .

Lemma 2.2Let M ( 0 , π 2 ω 2 ) . Then, for every h C ω + ( R ) , the positiveω-periodic solution of Equation (5) u = S h K . Namely, S ( C ω + ( R ) ) K .

Proof Let h C ω + ( R ) , u = S h . For every t R , from (8) it follows that

u ( t ) = t ω t U ( t s ) h ( s ) d s U ¯ t ω t h ( s ) d s = U ¯ 0 ω h ( s ) d s ,

and therefore,

u C U ¯ 0 ω h ( s ) d s .

Using (8), we obtain that

u ( t ) = t ω t U ( t s ) h ( s ) d s U ̲ t ω t h ( s ) d s = U ̲ 0 ω h ( s ) d s σ u C .

For every τ R , since

u ˙ ( τ ) = τ ω τ U ˙ ( τ s ) h ( s ) d s ,

we have

| u ˙ ( τ ) | τ ω τ | U ˙ ( τ s ) | h ( s ) d s U ¯ 1 τ ω τ h ( s ) d s = U ¯ 1 0 ω h ( s ) d s = C 0 U ̲ 0 ω h ( s ) d s C 0 u ( t ) .

Hence, u K . □

Now we consider the nonlinear delay equation (1). Hereafter, we assume that the nonlinearity f satisfies the condition

(F0) There exists M ( 0 , π 2 ω 2 ) such that

f ( t , x , y 1 , , y n ) + M x 0 , x 0 , t R , ( y 1 , , y n ) R n .

Let f 1 ( t , x , y 1 , , y n ) = f ( t , x , y 1 , , y n ) + M x , then f 1 ( t , x , y 1 , , y n ) 0 for x 0 , t R , ( y 1 , , y n ) R n , and Equation (1) is rewritten to

u ¨ ( t ) + M u ( t ) = f 1 ( t , u ( t ) , u ˙ ( t τ 1 ( t ) ) , , u ˙ ( t τ n ( t ) ) ) , t R . (11)

For every u K , set

F ( u ) ( t ) : = f 1 ( t , u ( t ) , u ˙ ( t τ 1 ( t ) ) , , u ˙ ( t τ n ( t ) ) ) , t R . (12)

Then F : K C ω + ( R ) is continuous. We define an integral operator A : K C ω 1 ( R ) by

A u ( t ) = t ω t U ( t s ) F ( u ) ( s ) d s = ( S F ) ( t ) . (13)

By the definition of the operator S, the positive ω-periodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. From assumption (F0), Lemma 2.1 and Lemma 2.2, we easily see that

Lemma 2.3 A ( K ) K and A : K K is completely continuous.

We will find the non-zero fixed point of A by using the fixed point index theory in cones. We recall some concepts and conclusions on the fixed point index in [25,26]. Let E be a Banach space and K E be a closed convex cone in E. Assume Ω is a bounded open subset of E with the boundary Ω, and K Ω . Let A : K Ω ¯ K be a completely continuous mapping. If A u u for any u K Ω , then the fixed point index i ( A , K Ω , K ) has a definition. One important fact is that if i ( A , K Ω , K ) 0 , then A has a fixed point in K Ω . The following two lemmas are needed in our argument.

Lemma 2.4 ([26])

Let Ω be a bounded open subset ofEwith θ Ω and A : K Ω ¯ K be a completely continuous mapping. If λ A u u for every u K Ω and 0 < λ 1 , then i ( A , K Ω , K ) = 1 .

Lemma 2.5 ([26])

Let Ω be a bounded open subset ofEand A : K Ω ¯ K be a completely continuous mapping. If there exists an e K { θ } such that u A u μ e for every u K Ω and μ 0 , then i ( A , K Ω , K ) = 0 .

In the next section, we will use Lemma 2.4 and Lemma 2.5 to discuss the existence of positive ω-periodic solutions of Equation (1).

### 3 Main results

We consider the existence of positive ω-periodic solutions of the functional differential equation (1). Let f C ( R × [ 0 , ) × R n ) satisfy assumption (F0) and f ( t , x , y 1 , , y n ) be ω-periodic in t. Let C 0 be the constant defined by (9) and I = [ 0 , ω ] . For convenience, we introduce the notations

f 0 = lim inf x 0 + min t I , | y i | C 0 | x | , i = 1 , , n ( f ( t , x , y 1 , , y n ) / x ) , f 0 = lim sup x 0 + max t I , | y i | C 0 | x | , i = 1 , , n ( f ( t , x , y 1 , , y n ) / x ) , f = lim inf x + min t I , | y i | C 0 | x | , i = 1 , , n ( f ( t , x , y 1 , , y n ) / x ) , f = lim sup x + max t I , | y i | C 0 | x | , i = 1 , , n ( f ( t , x , y 1 , , y n ) / x ) .

Our main results are as follows.

Theorem 3.1Let f C ( R × [ 0 , ) × R n ) and f ( t , x , y 1 , , y n ) beω-periodic int, τ 1 , , τ n C ω + ( R ) . Iffsatisfies assumption (F0) and the condition

(F1) f 0 < 0 , f > 0 ,

then Equation (1) has at least one positiveω-periodic solution.

Theorem 3.2Let f C ( R × [ 0 , ) × R n ) and f ( t , x , y 1 , , y n ) beω-periodic int, τ 1 , , τ n C ω + ( R ) . Iffsatisfies assumption (F0) and the conditions

(F2) f 0 > 0 , f < 0 ,

then Equation (1) has at least one positiveω-periodic solution.

In Theorem 3.1, the condition (F1) allows f ( t , x , y 1 , , y n ) to be superlinear growth on x and y 1 , , y n . For example,

f ( t , x , y 1 , , y n ) = x 2 + y 1 2 + + y n 2 1 4 π 2 ω 2 ( 2 + sin 2 π t ω ) x

satisfies (F0) with M = 3 4 π 2 ω 2 and (F1) with f 0 = 1 4 π 2 ω 2 and f = + .

In Theorem 3.2, the condition (F2) allows f ( t , x , y 1 , , y n ) to be sublinear growth on x and y 1 , , y n . For example,

f ( t , x , y 1 , , y n ) = x + | y 1 | + + | y n | 1 4 π 2 ω 2 ( 2 + sin 2 π t ω ) x

satisfies (F0) with M = 3 4 π 2 ω 2 and (F2) with f 0 = + and f = 1 4 π 2 ω 2 .

Proof of Theorem 3.1 Choose the working space E = C ω 1 ( R ) . Let K C ω 1 ( R ) be the closed convex cone in C ω 1 ( R ) defined by (10) and A : K K be the operator defined by (13). Then the positive ω-periodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. Let 0 < r < R < + and set

Ω 1 = { u C ω 1 ( R ) u C 1 < r } , Ω 2 = { u C ω 1 ( R ) u C 1 < R } . (14)

We show that the operator A has a fixed point in K ( Ω 2 Ω ¯ 1 ) when r is small enough and R is large enough.

By f 0 < 0 and the definition of f 0 , there exist ε ( 0 , M ) and δ > 0 such that

f ( t , x , y 1 , , y n ) ε x , t I , 0 x δ , | y i | C 0 x , i = 1 , , n . (15)

Let r ( 0 , δ ) . We now prove that A satisfies the condition of Lemma 2.4 in K Ω 1 , namely λ A u u for every u K Ω 1 and 0 < λ 1 . In fact, if there exist u 0 K Ω 1 and 0 < λ 0 1 such that λ 0 A u 0 = u 0 , then by the definition of A and Lemma 2.1, u 0 C ω 2 ( R ) satisfies the delay differential equation

u ¨ 0 ( t ) + M u 0 ( t ) = λ 0 f 1 ( t , u 0 ( t ) , u ˙ 0 ( t τ 1 ( t ) ) , , u ˙ 0 ( t τ n ( t ) ) ) , t R . (16)

Since u 0 K Ω 1 , by the definitions of K and Ω 1 , we have

0 σ u 0 C u 0 ( t ) u 0 C u 0 C 1 = r < δ , | u ˙ 0 ( t τ i ( t ) ) | C 0 u 0 ( t ) , i = 1 , , n , t R . (17)

Hence, from (15) it follows that

f ( t , u 0 ( t ) , u ˙ 0 ( t τ 1 ( t ) ) , , u ˙ 0 ( t τ n ( t ) ) ) ε u 0 ( t ) , t R .

By this, (16) and the definition of f 1 , we have

u ¨ 0 ( t ) + M u 0 ( t ) λ 0 ( M u 0 ( t ) ε u 0 ( t ) ) ( M ε ) u 0 ( t ) , t R .

Integrating both sides of this inequality from 0 to ω and using the periodicity of u 0 , we obtain that

M 0 ω u 0 ( t ) d t ( M ε ) 0 ω u 0 ( t ) d t .

Since 0 ω u 0 ( t ) d t ω σ u 0 C > 0 , it follows that M M ε , which is a contradiction. Hence, A satisfies the condition of Lemma 2.4 in K Ω 1 . By Lemma 2.4, we have

i ( A , K Ω 1 , K ) = 1 . (18)

On the other hand, since f > 0 , by the definition of f , there exist ε 1 > 0 and H > 0 such that

f ( t , x , y 0 , , y n ) ε 1 x , t I , x H , | y i | C 0 x , i = 1 , , n . (19)

Choose R > max { 1 + C 0 σ H , δ } and let e ( t ) 1 . Clearly, e K { θ } . We show that A satisfies the condition of Lemma 2.5 in K Ω 2 , namely u A u μ v for every u K Ω 2 and μ 0 . In fact, if there exist u 1 K Ω 2 and μ 1 0 such that u 1 A u 1 = μ 1 e , since u 1 μ 1 e = A u 1 , by the definition of A and Lemma 2.1, u 1 C ω 2 ( R ) satisfies the differential equation

u ¨ 1 ( t ) + M ( u 1 ( t ) μ 1 ) = f 1 ( t , u 1 ( t ) , u ˙ 1 ( t τ 1 ( t ) ) , , u ˙ 1 ( t τ n ( t ) ) ) , t R . (20)

Since u 1 K Ω 2 , by the definition of K, we have

u 1 ( t ) σ u 1 C , | u ˙ 1 ( τ ) | C 0 u 1 ( t ) , τ , t I . (21)

By the latter inequality of (21), we have that u ˙ 1 C C 0 u 1 C . This implies that u 1 C 1 = u 1 C + u ˙ 1 C ( 1 + C 0 ) u 1 C . Consequently,

u 1 C 1 1 + C 0 u 1 C 1 . (22)

By (22) and the former inequality of (21), we have

u 1 ( t ) σ u 1 C σ 1 + C 0 u 1 C 1 = σ R 1 + C 0 > H , t I .

From this, the latter inequality of (21) and (19), it follows that

f ( t , u 1 ( t ) , u ˙ 1 ( t τ 1 ( t ) ) , , u ˙ 1 ( t τ n ( t ) ) ) ε 1 u 1 ( t ) , t I .

By this inequality, (20) and the definition of f 1 , we have

u 1 ( n ) ( t ) + M ( u 1 ( t ) μ 1 ) ( M + ε 1 ) u 1 ( t ) , t I .

Integrating this inequality on I and using the periodicity of u 1 , we get that

M 0 ω u 1 ( t ) d t ω M μ 1 ( M + ε 1 ) 0 ω u 1 ( t ) d t .

Since 0 ω u 1 ( t ) d t ω σ u 1 C > 0 , from this inequality it follows that M M + ε 1 , which is a contradiction. This means that A satisfies the condition of Lemma 2.5 in K Ω 2 . By Lemma 2.5,

i ( A , K Ω 2 , K ) = 0 . (23)

Now, by the additivity of fixed point index, (18) and (23), we have

i ( A , K ( Ω 2 Ω ¯ 1 ) , K ) = i ( A , K Ω 2 , K ) i ( A , K Ω 1 , K ) = 1 .

Hence, A has a fixed-point in K ( Ω 2 Ω ¯ 1 ) , which is a positive ω-periodic solution of Equation (1). □

Proof of Theorem 3.2 Let Ω 1 , Ω 2 C ω 1 ( R ) be defined by (14). We prove that the operator A defined by (13) has a fixed point in K ( Ω 2 Ω ¯ 1 ) if r is small enough and R is large enough.

By f 0 > 0 and the definition of f 0 , there exist ε > 0 and δ > 0 such that

f ( t , x , y 1 , , y n ) ε x , t I , 0 < x δ , | y i | C 0 x , i = 1 , , n . (24)

Let r ( 0 , δ ) and e ( t ) 1 . We prove that A satisfies the condition of Lemma 2.5 in K Ω 1 , namely u A u μ e for every u K Ω 1 and μ 0 . In fact, if there exist u 0 K Ω 1 and μ 0 0 such that u 0 A u 0 = μ 0 e , since u 0 μ 0 e = A u 0 , by the definition of A and Lemma 2.1, u 0 C ω 2 ( R ) satisfies the delay differential equation

u ¨ 0 ( t ) + M ( u 0 ( t ) μ 0 ) = f 1 ( t , u 0 ( t ) , u ˙ 0 ( t τ 1 ( t ) ) , , u ˙ 0 ( t τ n ( t ) ) ) , t R . (25)

Since u 0 K Ω 1 , by the definitions of K and Ω 1 , u 0 satisfies (17). From (17) and (24) it follows that

f 1 ( t , u 0 ( t ) , u ˙ 0 ( t τ 1 ( t ) ) , , u ˙ 0 ( t τ n ( t ) ) ) ε u 0 ( t ) , t R .

By this, (25) and the definition of f 1 , we have

u 0 ( t ) + M u 0 ( t ) = f 1 ( t , u 0 ( t ) , u ˙ 0 ( t τ 1 ( t ) ) , , u ˙ 0 ( t τ n ( t ) ) ) + M μ 0 ( M + ε ) u 0 ( t ) , t R .

Integrating this inequality on [ 0 , ω ] and using the periodicity of u 0 ( t ) , we obtain that

M 0 ω u 0 ( t ) d t ( M + ε ) 0 ω u 0 ( t ) d t .

Since 0 ω u 0 ( t ) d t ω σ u 0 C > 0 , from this inequality it follows that M M + ε , which is a contradiction. Hence, A satisfies the condition of Lemma 2.5 in K Ω 1 . By Lemma 2.5, we have

i ( A , K Ω 1 , K ) = 0 . (26)

Since f < 0 , by the definition of f , there exist ε 1 ( 0 , M ) and H > 0 such that

f ( t , x , y 1 , , y n ) ε 1 x , t I , x H , | y i | C 0 x , i = 1 , , n . (27)

Choosing R > max { 1 + C 0 σ H , δ } , we show that A satisfies the condition of Lemma 2.4 in K Ω 2 , namely λ A u u for every u K Ω 2 and 0 < λ 1 . In fact, if there exist u 1 K Ω 2 and 0 < λ 1 1 such that λ 1 A u 1 = u 1 , then by the definition of A and Lemma 2.1, u 1 C ω 2 ( R ) satisfies the differential equation

u ¨ 1 ( t ) + M u 1 ( t ) = λ 1 f 1 ( t , u 1 ( t ) , u ˙ 1 ( t τ 1 ( t ) ) , , u ˙ 1 ( t τ n ( t ) ) ) , t R . (28)

Since u 1 K Ω 2 , by the definition of K, u 1 satisfies (21). From the second inequality of (21), it follows that (22) holds. By (22) and the first inequality of (21), we have

u 1 ( t ) σ u 1 C σ ( 1 + C 0 ) u 1 C 1 = σ R ( 1 + C 0 ) > H , t R .

From this, the second inequality of (21) and (27), it follows that

f ( t , u 1 ( t ) , u ˙ 1 ( t τ 1 ( t ) ) , , u ˙ 1 ( t τ n ( t ) ) ) ε 1 u 1 ( t ) , t R .

By this and (28), we have

u ¨ 1 ( t ) + M u 1 ( t ) λ 1 ( M u 1 ( t ) ε 1 u 1 ( t ) ) ( M ε 1 ) u 1 ( t ) , t R .

Integrating this inequality on [ 0 , ω ] and using the periodicity of u 1 ( t ) , we obtain that

M 0 ω u 1 ( t ) d t ( M ε 1 ) 0 ω u 1 ( t ) d t .

Since 0 ω u 1 ( t ) d t ω σ u 1 C > 0 , from this inequality it follows that M M ε 1 , which is a contradiction. This means that A satisfies the condition of Lemma 2.4 in K Ω 2 . By Lemma 2.4,

i ( A , K Ω 2 , K ) = 1 . (29)

Now, from (26) and (29), it follows that

i ( A , K ( Ω 2 Ω ¯ 1 ) , K ) = i ( A , K Ω 2 , K ) i ( A , K Ω 1 , K ) = 1 .

Hence, A has a fixed-point in K ( Ω 2 Ω ¯ 1 ) , which is a positive ω-periodic solution of Equation (1). □

Example 1 Consider the following second-order differential equation with delay:

u ¨ ( t ) = a 1 ( t ) u ( t ) + a 2 ( t ) u 2 ( t ) + a 3 ( t ) u ˙ 2 ( t ω / 2 ) , t R , (30)

where a i ( t ) C ω ( R ) , i = 1 , 2 , 3 . If π 2 ω 2 < a 1 ( t ) < 0 and a 2 ( t ) , a 3 ( t ) > 0 for t [ 0 , ω ] , we can verify that

f ( t , x , y ) = a 1 ( t ) x + a 2 ( t ) x 2 + a 3 ( t ) y 2

satisfies the conditions (F0) and (F1) for n = 1 . By Theorem 3.1, the delay equation (30) has at least one positive ω-periodic solution.

Example 2 Consider the functional differential equation

u ¨ ( t ) = c 1 ( t ) u ( t ) + c 2 ( t ) u 2 ( t ) 3 + c 3 ( t ) u ˙ 2 ( t τ ( t ) ) 3 , t R , (31)

where c i ( t ) C ω ( R ) , i = 1 , 2 , 3 , and τ C ω + ( R ) . If π 2 ω 2 < c 1 ( t ) < 0 and c 2 ( t ) , c 3 ( t ) > 0 for t [ 0 , ω ] . We easily see that

f ( t , x , y ) = c 1 ( t ) x + c 2 ( t ) | x | 2 / 3 + c 3 ( t ) | y | 2 / 3

satisfies the conditions (F0) and (F2) for n = 1 . By Theorem 3.2, the functional differential equation (31) has a positive ω-periodic solution.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

YL carried out the main part of this article. All authors read and approved the final manuscript.

### Acknowledgements

The research was supported by the NNSFs of China (11261053, 11061031).

### References

1. Leela, S: Monotone method for second order periodic boundary value problems. Nonlinear Anal.. 7, 349–355 (1983). Publisher Full Text

2. Nieto, JJ: Nonlinear second-order periodic boundary value problems. J. Math. Anal. Appl.. 130, 22–29 (1988). Publisher Full Text

3. Cabada, A, Nieto, JJ: A generation of the monotone iterative technique for nonlinear second-order periodic boundary value problems. J. Math. Anal. Appl.. 151, 181–189 (1990). Publisher Full Text

4. Cabada, A: The method of lower and upper solutions for second, third, forth, and higher order boundary value problems. J. Math. Anal. Appl.. 185, 302–320 (1994). Publisher Full Text

5. Gossez, JP, Omari, P: Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance. J. Differ. Equ.. 94, 67–82 (1991). Publisher Full Text

6. Omari, P, Villari, G, Zanolin, F: Periodic solutions of Lienard equation with one-sided growth restrictions. J. Differ. Equ.. 67, 278–293 (1987). Publisher Full Text

7. Ge, W: On the existence of harmonic solutions of Lienard system. Nonlinear Anal.. 16, 183–190 (1991). Publisher Full Text

8. Mawhin, J, Willem, M: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differ. Equ.. 52, 264–287 (1984). Publisher Full Text

9. Zelati, VC: Periodic solutions of dynamical systems with bounded potential. J. Differ. Equ.. 67, 400–413 (1987). Publisher Full Text

10. Lassoued, L: Periodic solutions of a second order superquadratic system with a change of sign in potential. J. Differ. Equ.. 93, 1–18 (1991). Publisher Full Text

11. Atici, FM, Guseinov, GS: On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions. J. Comput. Appl. Math.. 132, 341–356 (2001). Publisher Full Text

12. Torres, PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ.. 190, 643–662 (2003). Publisher Full Text

13. Li, Y: Positive periodic solutions of nonlinear second order ordinary differential equations. Acta Math. Sin.. 45, 481–488 (in Chinese) (2002)

14. Li, Y: Positive periodic solutions of first and second order ordinary differential equations. Chin. Ann. Math., Ser. B. 25, 413–420 (2004). Publisher Full Text

15. Li, F, Liang, Z: Existence of positive periodic solutions to nonlinear second order differential equations. Appl. Math. Lett.. 18, 1256–1264 (2005). Publisher Full Text

16. Graef, JR, Kong, L, Wang, H: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differ. Equ.. 245, 1185–1197 (2008). Publisher Full Text

17. Liu, B: Periodic solutions of a nonlinear second-order differential equation with deviating argument. J. Math. Anal. Appl.. 309, 313–321 (2005). Publisher Full Text

18. Li, JW, Cheng, SS: Periodic solutions of a second order forced sublinear differential equation with delay. Appl. Math. Lett.. 18, 1373–1380 (2005). Publisher Full Text

19. Wang, Y, Lian, H, Ge, W: Periodic solutions for a second order nonlinear functional differential equation. Appl. Math. Lett.. 20, 110–115 (2007). Publisher Full Text

20. Wu, J, Wang, Z: Two periodic solutions of second-order neutral functional differential equations. J. Math. Anal. Appl.. 329, 677–689 (2007). Publisher Full Text

21. Wu, YX: Existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter. Nonlinear Anal.. 70, 433–443 (2009). Publisher Full Text

22. Guo, CJ, Guo, ZM: Existence of multiple periodic solutions for a class of second-order delay differential equations. Nonlinear Anal., Real World Appl.. 10, 3285–3297 (2009). Publisher Full Text

23. Cheung, WS, Ren, JL, Han, W: Positive periodic solution of second-order neutral functional differential equations. Nonlinear Anal.. 71, 3948–3955 (2009). Publisher Full Text

24. Lv, X, Yan, P, Liu, D: Anti-periodic solutions for a class of nonlinear second-order Rayleigh equations with delays. Commun. Nonlinear Sci. Numer. Simul.. 15, 3593–3598 (2010). Publisher Full Text

25. Deimling, K: Nonlinear Functional Analysis, Springer, New York (1985)

26. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)