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Erratum to: Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations

The original article was published in Boundary Value Problems 2012 2012:109

Abstract

In this paper, we give a complementary proof on the paper ‘Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations’.

1 Introduction

In [1], the authors were concerned with the existence and uniqueness of anti-periodic solutions of the following prescribed mean curvature Rayleigh equation:

( x 1 + x 2 ) +f ( t , x ( t ) ) +g ( t , x ( t ) ) =e(t),
(1.1)

where eC(R,R) is T-periodic, and f,gC(R×R,R) are T-periodic in the first argument, T is a constant.

The paper mentioned above obtained the main result by using Mawhin’s continuation theorem in the coincidence degree theory. Unfortunately, the proof of main result Theorem 3.1 (see [1]) has a serious problem: F μ (x)=μL( Q 1 (t, x 1 , x 2 )) where Q 1 depends on ψ( x 2 ) and ψ(x)= x 1 x 2 which is only defined for |x|<1 and cannot be continuously extended; therefore, F μ should not be defined on Ω ¯ ={xX:x<M} since | x 2 (t)|>1 can occur, where x=max{ x 1 , x 2 } and M=1+max{ D 1 , D 2 }.

In this paper, we shall give a complementary proof to correct the errors.

2 Complementary proof

Rewrite (1.1) in the equivalent form as follows:

{ x 1 ( t ) = ψ ( x 2 ( t ) ) = x 2 ( t ) 1 x 2 2 ( t ) , x 2 ( t ) = f ( t , ψ ( x 2 ( t ) ) ) g ( t , x 1 ( t ) ) + e ( t ) ,
(2.1)

where ψ(x)= x 1 x 2 . In [1], the authors embed (2.1) into a family of equations with one parameter λ(0,1],

{ x 1 ( t ) = λ x 2 ( t ) 1 x 2 2 ( t ) = λ ψ ( x 2 ( t ) ) , x 2 ( t ) = λ f ( t , ψ ( x 2 ( t ) ) ) λ g ( t , x 1 ( t ) ) + λ e ( t ) .
(2.2)

They have proved that there exists a constant D 1 >0 such that

| x 1 | 2 D 1 ,and| x 1 | D 1 ,
(2.3)

and there exists η[0,T] such that x 2 (η)=0.

In fact, to use the continuation theorem, it suffices to prove that there exists a positive constant 0< ε 0 1 such that, for any possible solution ( x 1 (t), x 2 (t)) of (2.2), the following condition holds:

| x 2 (t)|<1 ε 0 .
(2.4)

In what follows, we shall give a complementary proof for the main result in [1] by giving a proof of (2.4).

In [1], the authors assume that

(H1): (g(t, x 1 )g(t, x 2 ))( x 1 x 2 )<0, for all t, x 1 , x 2 R and x 1 x 2 ;

(H2): there exists l>0 such that

|g(t, x 1 )g(t, x 2 )|l| x 1 x 2 |for all t, x 1 , x 2 R;

(H3): there exists β, γ such that

γ lim inf | x | f ( t , x ) x lim sup | x | f ( t , x ) x β,uniformly in tR;

(H4): for all t,xR,

f ( t + T 2 , x ) =f(t,x),g ( t + T 2 , x ) =g(t,x),e ( t + T 2 ) =e(t).

Under the conditions mentioned above, we prove that (2.4) holds.

Since | x 1 | < D 1 and g, e are continuous, we find that there exists M 3 >0 such that

M 3 <g ( t , x 1 ( t ) ) +e(t)< M 3 ,tR.
(2.5)

By (H3), there exists a positive constant M 4 >0 such that

f(t,x)γx M 4 ,x>0 and tR.
(2.6)

Next, we shall prove that

x(t) M 3 + M 4 ( M 3 + M 4 ) 2 + γ 2 ,tR.

Assume by contradiction that there exist t 2 > t 1 >η such that

x 2 ( t 1 ) = M 3 + M 4 ( M 3 + M 4 ) 2 + γ 2 , x 2 ( t 2 ) > M 3 + M 4 ( M 3 + M 4 ) 2 + γ 2 ,

and

x 2 (t)> M 3 + M 4 ( M 3 + M 4 ) 2 + γ 2 ,for t ( t 1 , t 2 ) .

Noticing that λ(0,1], we have, t( t 1 , t 2 ),

x 2 (t)=λ ( f ( t , ψ ( x 2 ( t ) ) ) g ( t , x 1 ( t ) ) + e ( t ) ) <0,

which is a contradiction.

By (H3), there exists a positive constant M 5 >0 such that

f(t,x)βx+ M 5 ,x<0 and tR.

By using a similar argument, we can prove that

x 2 (t) M 3 + M 5 ( M 3 + M 5 ) 2 + β 2 ,for tR.

Therefore, we get from the continuity of x 2 (t), for any solution ( x 1 (t), x 2 (t)) of (2.2),

M 3 + M 5 ( M 3 + M 5 ) 2 + β 2 x 2 (t) M 3 + M 4 ( M 3 + M 4 ) 2 + γ 2 ,tR.

Consequently, (2.4) holds.

Putting

Ω= { x = ( x , x ) C T 0 , 1 2 ( R , R 2 ) = X : x < M , | x 2 ( t ) | < 1 ε 0 } ,

we can use Mawhin’s continuation theorem on Ω.

References

  1. 1.

    Li J, Wang Z: Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations. Bound. Value Probl. 2012., 2012: 10.1186/1687-2770-2012-109

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Acknowledgements

The authors would like to thank Professor J Webb for pointing out the errors of the paper [1].

Author information

Correspondence to Jin Li.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

The online version of the original article can be found at 10.1186/1687-2770-2012-109

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Keywords

  • complementary proof
  • prescribed mean curvature Rayleigh equations