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

The Original Article was published on 09 October 2012

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 e∈C(R,R) is T-periodic, and f,g∈C(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 ¯ ={x∈X:∥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 t∈R;

(H4): for all t,x∈R,

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 ,∀t∈R.
(2.5)

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

f(t,x)≥γx− M 4 ,∀x>0 and âˆ€t∈R.
(2.6)

Next, we shall prove that

x(t)≤ M 3 + M 4 ( M 3 + M 4 ) 2 + γ 2 ,∀t∈R.

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 âˆ€t∈R.

By using a similar argument, we can prove that

x 2 (t)≥− M 3 + M 5 ( M 3 + M 5 ) 2 + β 2 ,for t∈R.

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 ,∀t∈R.

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. 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].

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Correspondence to Jin Li.

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The authors declare that they have no competing interests.

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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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Li, J., Wang, Z. Erratum to: Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations. Bound Value Probl 2014, 204 (2014). https://doi.org/10.1186/s13661-014-0204-5

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