Open Access

Existence of periodic solutions for a prescribed mean curvature Liénard p-Laplacian equation with two delays

Advances in Difference Equations20142014:290

https://doi.org/10.1186/1687-1847-2014-290

Received: 2 September 2014

Accepted: 13 November 2014

Published: 25 November 2014

Abstract

This paper is concerned with the prescribed mean curvature Liénard type p-Laplacian equation with two arguments. By employing Mawhin’s coincidence degree theorem and the analysis techniques, some new existence results of periodic solutions are obtained. We also give an example to illustrate the application of our main results.

Keywords

periodic solution Mawhin’s continuation theorem Liénard type p-Laplacian equation

1 Introduction

Recently, the study of the periodic solutions of prescribed mean curvature equations has become very active; see [15] and the references therein. Meanwhile, the Liénard equation, Liénard system, and p-Laplacian equations are also studied by many people; see for example [68]. These kinds of equations have wide applications in many fields, such as physics, mechanics, and engineering.

In [9] Wang studied the following prescribed mean curvature Rayleigh equation with one deviating argument:
( φ p ( x ( t ) 1 + x 2 ( t ) ) ) + f ( t , x ( t ) ) + g ( t , x ( t τ ( t ) ) ) = e ( t )
(1.1)
under the assumptions:
f ( t , u ) a | u | r , ( t , u ) R 2 and g ( t , u ) e ( t ) m 1 | u | m 2 , t R , x d .
(1.2)

By using the theory of coincidence degree, the author obtained some sufficient conditions for the existence and uniqueness of periodic solution in the case of τ ( t ) = 0 .

Stimulated by [9], we study the periodic solutions for a kind of prescribed mean curvature Liénard p-Laplacian equation with two deviating arguments in the form
( φ p ( x ( t ) 1 + x 2 ( t ) ) ) + f ( x ( t ) ) x ( t ) + g ( x ( t τ ( t ) ) ) + h ( x ( t γ ( t ) ) ) = e ( t ) ,
(1.3)

where φ p ( s ) = | s | p 2 s , p 2 , τ, γ are continuous functions with period T, τ ( t ) < 1 , γ ( t ) < 1 , and 0 T e ( t ) d t = 0 , f , g , h C 1 ( R , R ) .

To the best of our knowledge, there are few results on this topic. The purpose of this paper is to establish a criterion to guarantee the existence of T-periodic solution. Our methods and results are different from the corresponding ones of [9]. So our results are essentially new.

2 Preliminaries

Let X and Y be real Banach spaces and L : D ( L ) X Y be a Fredholm operator with index zero, that is, X = Ker L X 1 and Y = Im L Y 1 . Furthermore, let P : X Ker L and Q : Y Y 1 be the continuous projectors. Clearly, Ker L ( D ( L ) X 1 ) = { 0 } , thus the restriction L P = L | D ( L ) X 1 is invertible. Denote by K the inverse of L P .

Let Ω be an open bounded subset of X with D ( L ) Ω Φ . A map N : Ω ¯ Y is called L-compact in Ω ¯ if Q N ( Ω ¯ ) is bounded and the operator K ( I Q ) N : Ω ¯ X is compact. The following, Mawhin’s continuation theorem, is well known.

Lemma 2.1 (Gaines and Mawhin [10])

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 subset and N : Ω ¯ Y is L-compact in 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 { J Q N , Ω Ker L , 0 } 0 , where J : Im Q Ker L is an isomorphism.

     

Then the equation L x = N x has a solution in Ω ¯ D ( L ) .

To use Mawhin’s continuation theorem to study (1.3), we firstly transform (1.3) into the following form:
{ x 1 ( t ) = φ q ( x 2 ( t ) ) 1 φ q 2 ( x 2 ( t ) ) , x 2 ( t ) = f ( x 1 ( t ) ) φ q ( x 2 ( t ) ) 1 φ q 2 ( x 2 ( t ) ) g ( x 1 ( t τ ( t ) ) ) h ( x 1 ( t γ ( t ) ) ) + e ( t ) .
(2.1)

Denote by ϕ ( x ) = φ q ( x ) 1 φ q 2 ( x ) , φ q ( s ) = | s | q 2 s , 1 / p + 1 / q = 1 , then prescribed mean curvature p-Laplacian operator φ p ( x 1 + x 2 ) is marked by ϕ 1 ( x ) . Clearly, if x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) is a T-periodic solution to (2.1), then x 1 ( t ) must be a T-periodic solution to (1.3).

Set C T 1 = { x : x C ( R , R 2 ) , x ( t + T ) x ( t ) } with the norm | x | 0 = max t [ 0 , T ] | x ( t ) | , X = Y = { x ( t ) = ( x 1 ( t ) , x 2 ( t ) ) C ( R , R 2 ) : x ( t + T ) x ( t ) } with the norm x = max { | x 1 | 0 , | x 2 | 0 } , | x | 2 = ( 0 T x 2 ( t ) d t ) 1 / 2 . Then X and Y are Banach spaces. Let
L : D ( L ) X Y , L x = x = ( x 1 x 2 ) ,
(2.2)
N : X Y , where  N x = ( ϕ ( x 2 ( t ) ) f ( x 1 ( t ) ) ϕ ( x 2 ( t ) ) g ( x 1 ( t τ ( t ) ) ) h ( x 1 ( t γ ( t ) ) ) + e ( t ) ) .
(2.3)
It is easy to see that Ker L = R 2 , Im L = { x : x Y , 0 T x ( s ) d s = 0 } . So L is a Fredholm operator with index zero. Define projectors P : X Ker L and Q : Y Im Q by
P x = 1 T 0 T x ( s ) d s , Q y = 1 T 0 T y ( s ) d s ,
and let K be the inverse of L | Ker P D ( L ) . Obviously Ker L = Im Q = R 2 ,
[ K y ] 1 ( t ) = 0 T k ( t , s ) y ( s ) d s ,
(2.4)
where
k ( t , s ) = { s T , 0 s < t T , s T T , 0 t s T .

From (2.3) and (2.4), one can easily see that N is L-compact on Ω, where Ω is an open bounded subset of X. The following lemma is useful to estimate a priori bounds of periodic solutions of (2.1).

Lemma 2.2 (Liu et al. [11])

Suppose that x ( t ) C T 1 , and that there is a point t R such that x ( t ) = 0 , then
0 T | x ( t ) | 2 d t T 2 π 2 0 T | x ( t ) | 2 d t .

Lemma 2.3 ([12])

Let τ C T 1 , | τ | 0 < 1 . If x ( t ) C T 1 then
0 T | x ( t τ ( t ) ) | 2 d t τ 0 0 T | x ( t ) | 2 d t ,

where τ 0 = 1 1 | τ | 0 , γ ( t ) satisfies the above inequality and γ 0 = 1 1 | γ | 0 .

At the end of this section, we list the basic assumptions which will be used in Section 3.

(H1) ( g ( x 1 ) g ( x 2 ) ) ( x 1 x 2 ) > 0 , x 1 , x 2 R and x 1 x 2 , ( h ( x 1 ) h ( x 2 ) ) ( x 1 x 2 ) > 0 , x 1 , x 2 R and x 1 x 2 .

(H2) There is a constant d > 0 such that x ( g ( x ) + h ( x ) ) > 0 if and only if | x | > d .

(H3) There exist nonnegative constants k 1 , k 2 , c 1 , c 2 such that
| g ( x ) | k 1 | x | + c 1 , | x | > d ; | h ( x ) | k 2 | x | + c 2 , | x | > d .

(H4) There is a constant σ > 0 such that | f ( s ) | σ for all s R .

3 Main results and the proof

In this section we state the main results and give its proof.

Theorem 3.1 Suppose the assumptions (H1)-(H4) hold. Then (1.3) has at least one T-periodic solution provided T π ( k 1 τ 0 + k 2 γ 0 ) < σ .

Proof Let Ω 1 = { x X : L x = λ N x , λ ( 0 , 1 ) } , suppose x ( t ) Ω 1 , then
{ x 1 ( t ) = λ φ q ( x 2 ( t ) ) 1 φ q 2 ( x 2 ( t ) ) λ ϕ ( x 2 ( t ) ) , x 2 ( t ) = λ f ( x 1 ( t ) ) φ q ( x 2 ( t ) ) 1 φ q 2 ( x 2 ( t ) ) λ g ( x 1 ( t τ ( t ) ) ) λ h ( x 1 ( t γ ( t ) ) ) + λ e ( t ) .
(3.1)
Integrating both sides of the second equation of (3.1) from 0 to T, we have
0 T ( g ( x 1 ( t τ ( t ) ) ) + h ( x 1 ( t γ ( t ) ) ) ) d t = 0 ,
which implies that there exists a point ξ [ 0 , T ] such that
g ( x 1 ( ξ τ ( ξ ) ) ) + h ( x 1 ( ξ γ ( ξ ) ) ) = 0 .
(3.2)
Now we claim that there must exist a point t 0 R such that
| x 1 ( t 0 ) | d .
(3.3)

In fact, let ξ τ ( ξ ) = t 1 , ξ γ ( ξ ) = t 2 , then from (3.2), g ( x 1 ( t 1 ) ) + h ( x 1 ( t 2 ) ) = 0 . If x 1 ( t 1 ) = x 1 ( t 2 ) holds, (3.3) is clearly true. Now assume x 1 ( t 1 ) < x 1 ( t 2 ) .

Set F ( x ) g ( x ) + h ( x ) , according to (H1), one has
F ( x 1 ( t 1 ) ) = g ( x 1 ( t 1 ) ) + h ( x 1 ( t 1 ) ) < g ( x 1 ( t 1 ) ) + h ( x 1 ( t 2 ) ) < g ( x 1 ( t 2 ) ) + h ( x 1 ( t 2 ) ) = F ( x 1 ( t 2 ) ) ,

namely F ( x 1 ( t 1 ) ) < 0 < F ( x 1 ( t 2 ) ) , then there must exist a t 0 such that F ( x 1 ( t 0 ) ) = 0 . That is, g ( x 1 ( t 0 ) ) + h ( x 1 ( t 0 ) ) = 0 . According to (H2), (3.3) holds. If x 1 ( t 1 ) > x 1 ( t 2 ) holds, by a similar method, we can show that (3.3) is also true.

Let t ˜ = n T + t 0 , where t ˜ [ 0 , T ] and n is an integer. Then
| x 1 ( t ) | = | x 1 ( t ˜ ) + t ˜ t x 1 ( s ) d s | d + 0 T | x 1 ( s ) | d s for all  s [ 0 , T ] .
Thus we get
| x 1 | 0 = max t [ 0 , T ] | x 1 ( t ) | d + T | x 1 | 2 .
(3.4)
On the other hand, multiplying both sides of the second equation of (3.1) by x 1 ( t ) and integrating over [ 0 , T ] , we have
0 T ( ϕ 1 ( x 1 ( t ) λ ) ) x 1 ( t ) d t + 0 T f ( x 1 ( t ) ) ( x 1 ( t ) ) 2 d t + λ 0 T g ( x 1 ( t τ ( t ) ) ) x 1 ( t ) d t + λ 0 T h ( x 1 ( t γ ( t ) ) ) x 1 ( t ) d t = λ 0 T e ( t ) x 1 ( t ) d t ,
(3.5)
where ϕ 1 ( x 1 ( t ) λ ) = φ p ( x 1 ( t ) λ 1 + ( x 1 ( t ) λ ) 2 ) . Furthermore, in view of (H4), we have
| 0 T f ( x 1 ( t ) ) ( x 1 ( t ) ) 2 d t | = 0 T | f ( x 1 ( t ) ) | | x 1 ( t ) | 2 d t | 0 T ( ϕ 1 ( x 1 ( t ) λ ) ) x 1 ( t ) d t | + λ | 0 T g ( x 1 ( t τ ( t ) ) ) x 1 ( t ) d t | + λ | 0 T h ( x 1 ( t γ ( t ) ) ) x 1 ( t ) d t | + λ | 0 T e ( t ) x 1 ( t ) d t | .
(3.6)
Denote w ( t ) = ϕ 1 ( x 1 ( t ) / λ ) , then
0 T ( ϕ 1 ( x 1 ( t ) / λ ) ) x 1 ( t ) d t = λ 0 T ϕ ( ω ( t ) ) d ω ( t ) = 0 .
Together with (3.6) and the fact λ ( 0 , 1 ) yields
σ 0 T | x 1 ( t ) | 2 d t | 0 T g ( x 1 ( t τ ( t ) ) ) x 1 ( t ) d t | + | 0 T h ( x 1 ( t γ ( t ) ) ) x 1 ( t ) d t | + | 0 T e ( t ) x 1 ( t ) d t | .
Set
E 1 = { t [ 0 , 1 ] , | x 1 ( t τ ( t ) ) | d } , E 2 = { t [ 0 , 1 ] , | x 1 ( t τ ( t ) ) | > d } , E 3 = { t [ 0 , 1 ] , | x 1 ( t γ ( t ) ) | d } , E 4 = { t [ 0 , 1 ] , | x 1 ( t γ ( t ) ) | > d } .
In view of Lemma 2.3, we obtain
σ | x 1 | 2 2 E 1 | g ( x 1 ( t τ ( t ) ) ) | | x 1 ( t ) | d t + E 2 | g ( x 1 ( t τ ( t ) ) ) | | x 1 ( t ) | d t + E 3 | h ( x 1 ( t γ ( t ) ) ) | | x 1 ( t ) | d t + E 4 | h ( x 1 ( t γ ( t ) ) ) | | x 1 ( t ) | d t + 0 T | e ( t ) | | x 1 ( t ) | d t g d 0 T | x 1 ( t ) | d t + h d 0 T | x 1 ( t ) | d t + k 1 0 T | x 1 ( t τ ( t ) ) | | x 1 ( t ) | d t + k 2 0 T | x 1 ( t γ ( t ) ) | | x 1 ( t ) | d t + c 1 0 T | x 1 ( t ) | d t + c 2 0 T | x 1 ( t ) | d t + | e | 0 T | x 1 ( t ) | d t ( g d + h d + c 1 + c 2 + | e | ) T | x 1 | 2 + k 1 τ 0 | x 1 | 2 | x 1 | 2 + k 2 γ 0 | x 1 | 2 | x 1 | 2 ,
(3.7)
where τ 0 , γ 0 are defined by Lemma 2.3, and
g d = max | x | d | g ( x ) | , h d = max | x | d | h ( x ) | , | e | = max t [ 0 , T ] | e ( t ) | .
From (3.3), there exists a point t 0 R such that | x 1 ( t 0 ) | d . Let ϖ ( t ) = x 1 ( t ) x 1 ( t 0 ) , we have ϖ ( t + T ) = ϖ ( t ) , ϖ ( t ) = x 1 ( t ) , and ϖ ( t 0 ) = 0 ; then, by Lemma 2.2,
| ϖ | 2 T π | ϖ | 2 = T π | x 1 | 2 .
By the Minkowski inequality, we get
| x 1 | 2 = ( 0 T | x 1 ( t ) | 2 d t ) 1 2 = ( 0 T | ϖ ( t ) + x 1 ( t 0 ) | 2 d t ) 1 2 ( 0 T | x 1 ( t 0 ) | 2 d t ) 1 2 + ( 0 T | ϖ ( t ) | 2 d t ) 1 2 .
(3.8)
So
| x 1 | 2 T d + | ϖ | 2 T d + T π | x 1 | 2 .
Combining (3.7) and (3.8), we obtain
σ | x 1 | 2 2 ( g d + h d + c 1 + c 2 + | e | ) T | x 1 | 2 + ( k 1 τ 0 + k 2 γ 0 ) | x 1 | 2 ( T d + T π | x 1 | 2 ) = ( g d + h d + c 1 + c 2 + | e | ) T | x 1 | 2 + ( k 1 τ 0 + k 2 γ 0 ) T d | x 1 | 2 + T π ( k 1 τ 0 + k 2 γ 0 ) | x 1 | 2 2 .
(3.9)

Since T π ( k 1 τ 0 + k 2 γ 0 ) < σ , there is a constant M 0 > 0 independent of λ such that | x 1 | 2 M 0 , i.e., | x 1 | 0 d + T | x 1 | 2 d + T M 0 : = M 1 .

By the first equation of (3.1), we have
0 T λ φ q ( x 2 ( t ) ) 1 φ q 2 ( x 2 ( t ) ) d t = 0 ,
together with φ q ( 0 ) = 0 , which implies that there is a constant η [ 0 , T ] such that x 2 ( η ) = 0 . Hence
| x 2 | 0 0 η | x 2 ( s ) | d s 0 T | x 2 ( s ) | d s .
(3.10)
By the two equations of (3.1) and Hölder’s inequality, we obtain
0 T | x 2 ( s ) | d s 0 T | f ( x 1 ( t ) ) x 1 ( t ) | d t + λ 0 T | g ( x 1 ( t τ ( t ) ) ) | d t + λ 0 T | h ( x 1 ( t γ ( t ) ) ) | d t + λ 0 T | e ( t ) | d t 0 T | f ( x 1 ( t ) ) | | x 1 ( t ) | d t + g M 1 T + h M 1 T + | e | T < | f | 0 M 0 T + ( g M 1 + h M 1 + | e | ) T ,
(3.11)

where | f | 0 = max | x 1 | M 1 | f ( x 1 ) | , g M 1 = max | x 1 | M 1 | g ( x 1 ) | , h M 1 = max | x 1 | M 1 | h ( x 1 ) | . By (3.10), (3.11), | x 2 | 0 ( g M 1 + h M 1 + | e | ) T + | f | 0 M 0 T + d = : M 2 .

Let Ω 2 = { x Ker L : N x Im L } . If x Ω 2 , then Q N x = 0 . In view of 0 T e ( t ) d t = 0 , we have
{ φ q ( x 2 ) 1 φ q 2 ( x 2 ) = 0 , 0 T g ( x 1 ) d t + 0 T h ( x 1 ) d t = 0 .
(3.12)
So x 2 = 0 M 2 . Together with (H2) yields
| x 1 | d M 2 .

Let Ω = { x : x = ( x 1 , x 2 ) X , | x 1 | 0 < M 1 + 1 , | x 2 | 0 < M 2 + 1 } . Then Ω ( Ω 1 Ω 2 ) and Ω is a bounded open set of X. So (1) and (2) of Lemma 2.1 are satisfied.

In the next step we show that condition (3) of Lemma 2.1 holds. Define a linear isomorphism J : Im Q Ker L by J ( x 1 , x 2 ) = ( x 2 , x 1 ) , and let
H ( v , μ ) : = μ v + 1 μ T J Q N v , ( v , μ ) Ω × [ 0 , 1 ] .
The direct computation and (H2) show that for ( x , μ ) ( Ω Ker L ) × [ 0 , 1 ] ,
x H ( x , μ ) = μ ( x 1 2 + x 2 2 ) + 1 μ T ( g ( x 1 ) + h ( x 1 ) ) x 1 + 1 μ T | x 2 | q 2 x 2 2 1 φ q 2 ( x 2 ) > 0 .
Thus, x H ( x , μ ) 0 for ( x , μ ) Ω Ker L × [ 0 , 1 ] , which implies
deg { J Q N , Ω Ker L , 0 } = deg { H ( x , 0 ) , Ω Ker L , 0 } = deg { H ( x , 1 ) , Ω Ker L , 0 } = deg { I , Ω Ker L , 0 } 0 .

Condition (3) of Lemma 2.1 holds. By Lemma 2.1, the equation L x = N x has a solution. This completes the proof of Theorem 3.1. □

We can use a similar method to conclude the following result, the details are omitted.

Theorem 3.2 Suppose (H3), (H4), and the following assumptions hold:

( H 1 ) ( g ( x 1 ) g ( x 2 ) ) ( x 1 x 2 ) < 0 , x 1 , x 2 R , and x 1 x 2 , ( h ( x 1 ) h ( x 2 ) ) ( x 1 x 2 ) < 0 , x 1 , x 2 R , and x 1 x 2 .

( H 2 ) There is a constant d > 0 such that x ( g ( x ) + h ( x ) ) < 0 if and only if | x | > d .

Then (1.3) has at least one T-periodic solution if T π ( k 1 τ 0 + k 2 γ 0 ) < σ .

4 Uniqueness results

It is difficult to establish the uniqueness of the T-periodic solution of (1.3), but in the special case we obtain the uniqueness of (1.3).

Theorem 4.1 Assume ( H 1 ) holds, and τ ( t ) = ε 1 , γ ( t ) = ε 2 ( ε 1 , ε 2 are sufficiently small constants). Then (1.3) has at most one T-periodic solution.

Proof Denote
I ( x ) = 0 x f ( w ) d w , y ( t ) = φ p ( x ( t ) 1 + x 2 ( t ) ) + I ( x ( t ) ) .
Then (1.3) can be converted into the following form:
{ x ( t ) = φ q ( y ( t ) I ( x ( t ) ) ) 1 φ q 2 ( y ( t ) I ( x ( t ) ) ) , y ( t ) = g ( x ( t ε 1 ) ) h ( x ( t ε 2 ) ) + e ( t ) .
(4.1)
Suppose that x i ( t ) ( i = 1 , 2 ) are two T-periodic solutions of (1.3). Then from (4.1), we have
{ x i ( t ) = φ q ( y i ( t ) I ( x i ( t ) ) ) 1 φ q 2 ( y i ( t ) I ( x i ( t ) ) ) , y i ( t ) = g ( x i ( t ε 1 ) ) h ( x i ( t ε 2 ) ) + e ( t ) , i = 1 , 2 .
(4.2)
Let u ( t ) = x 1 ( t ) x 2 ( t ) , v ( t ) = y 1 ( t ) y 2 ( t ) , which together with (4.2) yields
{ u ( t ) = φ q ( y 1 ( t ) I ( x 1 ( t ) ) ) 1 φ q 2 ( y 1 ( t ) I ( x 1 ( t ) ) ) φ q ( y 2 ( t ) I ( x 2 ( t ) ) ) 1 φ q 2 ( y 2 ( t ) I ( x 2 ( t ) ) ) , v ( t ) = ( g ( x 1 ( t ε 1 ) ) g ( x 2 ( t ε 1 ) ) ) v ( t ) = ( h ( x 1 ( t ε 2 ) ) h ( x 2 ( t ε 2 ) ) ) .
(4.3)
Now, we claim that
v ( t ) 0 , t R .

We argue by contradiction, in view of v C 2 [ 0 , T ] and v ( t + T ) = v ( t ) , t R , we obtain max t R v ( t ) > 0 .

Then there must exist t R such that
{ v ( t ) = max t [ 0 , T ] v ( t ) = max t R v ( t ) > 0 , v ( t ) = ( g ( x 1 ( t ε 1 ) ) g ( x 2 ( t ε 1 ) ) ) v ( t ) = ( h ( x 1 ( t ε 2 ) ) h ( x 2 ( t ε 2 ) ) ) = 0 , v ( t ) = ( g ( x 1 ( t ε 1 ) ) x 1 ( t ε 1 ) g ( x 2 ( t ε 1 ) ) x 2 ( t ε 1 ) ) v ( t ) = ( h ( x 1 ( t ε 2 ) ) x 1 ( t ε 2 ) h ( x 2 ( t ε 2 ) ) x 2 ( t ε 2 ) ) 0 .
(4.4)
Noticing ( H 1 ), that is, g ( x ) < 0 , h ( x ) < 0 , ε 1 , ε 2 sufficiently small, it follows from the second equation of (4.4) that
x 1 ( t ε 1 ) x 2 ( t ε 1 ) , x 1 ( t ε 2 ) x 2 ( t ε 2 ) .
Then from the third equation of (4.4), we get
v ( t ) = g ( x 1 ( t ε 1 ) ) ( x 1 ( t ε 1 ) x 2 ( t ε 1 ) ) h ( x 1 ( t ε 2 ) ) ( x 1 ( t ε 2 ) x 2 ( t ε 2 ) ) = g ( x 1 ( t ε 1 ) ) ( φ q ( y 1 ( t ε 1 ) I ( x 1 ( t ε 1 ) ) ) 1 φ q 2 ( y 1 ( t ε 1 ) I ( x 1 ( t ε 1 ) ) ) φ q ( y 2 ( t ε 1 ) I ( x 2 ( t ε 1 ) ) ) 1 φ q 2 ( y 2 ( t ε 1 ) I ( x 2 ( t ε 1 ) ) ) ) h ( x 1 ( t ε 2 ) ) ( φ q ( y 1 ( t ε 2 ) I ( x 1 ( t ε 1 ) ) ) 1 φ q 2 ( y 1 ( t ε 2 ) I ( x 1 ( t ε 2 ) ) ) φ q ( y 2 ( t ε 2 ) I ( x 2 ( t ε 2 ) ) ) 1 φ q 2 ( y 2 ( t ε 2 ) I ( x 2 ( t ε 2 ) ) ) ) .
In view of g ( x 1 ( t ε 1 ) ) < 0 , h ( x 1 ( t ε 2 ) ) < 0 , v ( t ) = y 1 ( t ) y 2 ( t ) > 0 , x 1 x 2 , φ q ( x ) are increasing on x, and ε 1 , ε 2 are sufficiently small, thus we have
v ( t ) > 0 .
This contradicts the third equation of (4.4), which implies that
v ( t ) = y 1 ( t ) y 2 ( t ) 0 , t R .

By using a similar argument, we can also show that y 2 ( t ) y 1 ( t ) 0 .

Hence, we obtain
y 1 ( t ) y 2 ( t ) , t R .
Then from (4.3) and g ( x ) < 0 , we have x 1 ( t ε 1 ) x 2 ( t ε 1 ) , x 1 ( t ε 2 ) x 2 ( t ε 2 ) , t R . That is
x 1 ( t ) x 2 ( t ) .

Therefore, (1.3) has at most one T-periodic solution. The proof of Theorem 4.1 is now complete. □

From Theorem 3.2 and Theorem 4.1, we see that (1.3) has an unique T-periodic solution when τ ( t ) = ε 1 , γ ( t ) = ε 2 .

5 An example

In this section we give an example to illustrate the application of Theorem 3.1.

Example 5.1 Consider the prescribed mean curvature Liénard equation with two deviating arguments:
( φ 4 ( x ( t ) 1 + x 2 ( t ) ) ) + f ( x ( t ) ) x ( t ) + g ( x ( t cos 3 π t 4 π ) ) + h ( x ( t sin 3 π t 4 π ) ) = sin 3 π t ,
(5.1)

where f ( x ) = σ + x 2 , σ > 5 , τ ( t ) = cos 3 π t 4 π , γ ( t ) = sin 3 π t 4 π , e ( t ) = sin 3 π t , g ( x ) = 85 16 x + 6 , h ( x ) = 5 x + 2 . So τ 0 = γ 0 = 4 . Choosing T = 2 3 , d = 10 , k 1 = 85 16 , k 2 = 5 , then ( k 1 τ 0 + k 2 γ 0 ) T π 4.38 < 5 < σ . It is obvious that (H1), (H2), (H3), and (H4) hold. Then it follows from Theorem 3.1 that (5.1) has at least one 2 3 -periodic solution.

Declarations

Acknowledgements

This work was sponsored by the NSFC (11471073) and the Fundamental Research Funds for the Central Universities (2014B11414). The authors are very grateful to the referee for his/her careful reading of the manuscript and for his/her valuable suggestions on improving this article.

Authors’ Affiliations

(1)
College of Science, Hohai University
(2)
Department of Mathematics and Physics, Hohai University
(3)
Department of Mathematics, Beijing Institute of Technology

References

  1. Li W, Liu Z: Exact number of solutions of a prescribed mean curvature equation. J. Math. Anal. Appl. 2010, 367: 486-498. 10.1016/j.jmaa.2010.01.055MathSciNetView ArticleGoogle Scholar
  2. Zhang X, Feng M: Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities. J. Math. Anal. Appl. 2012, 395: 393-402. 10.1016/j.jmaa.2012.05.053MathSciNetView ArticleGoogle Scholar
  3. Pan H, Xing R: Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations II. Nonlinear Anal. TMA 2011, 74: 3751-3768. 10.1016/j.na.2011.03.020MathSciNetView ArticleGoogle Scholar
  4. Feng M: Periodic solutions for prescribed mean curvature Liénard equation with a deviating argument. Nonlinear Anal., Real World Appl. 2012, 13: 1216-1223. 10.1016/j.nonrwa.2011.09.015MathSciNetView ArticleGoogle Scholar
  5. Lu S, Lu M: Periodic solutions for a prescribed mean curvature equation with multiple delays. J. Appl. Math. 2014., 2014: Article ID 909252 10.1155/2014/909252Google Scholar
  6. Cheung W, Ren J: Periodic solutions for p -Laplacian Liénard equation with a deviating argument. Nonlinear Anal. 2004, 59: 107-120.MathSciNetGoogle Scholar
  7. Cheung W, Ren J: On the existence of periodic solutions for p -Laplacian generalized Liénard equation. Nonlinear Anal. 2005, 60: 65-75.MathSciNetGoogle Scholar
  8. Lu S, Ge W: Periodic solutions for a kind of Liénard equations with deviating arguments. J. Math. Anal. Appl. 2004, 249: 231-243.MathSciNetView ArticleGoogle Scholar
  9. Wang D: Existence and uniqueness of periodic solution for prescribed mean curvature Rayleigh type p -Laplacian equation. J. Appl. Math. Comput. 2014, 46: 181-200. 10.1007/s12190-013-0745-2MathSciNetView ArticleGoogle Scholar
  10. Gaines RE, Mawhin J Lecture Notes in Mathematics 568. In Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin; 1977.Google Scholar
  11. Liu X, Jia M, Ren R: On the existence and uniqueness of periodic solutions to a type of Duffing equation with complex arguments. Acta Math. Sci. Ser. A 2007, 27: 37-44. (in Chinese)MathSciNetGoogle Scholar
  12. Wang Z, Qian L, Lu S, Cao J: The existence and uniqueness of periodic solutions for a kind of Duffing-type equation with two deviating arguments. Nonlinear Anal. 2010, 73: 3034-3043. 10.1016/j.na.2010.06.071MathSciNetView ArticleGoogle Scholar

Copyright

© Li et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. 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.