- Erratum
- Open Access

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

- Jin Li
^{1}Email author and - Zaihong Wang
^{2}

**2014**:204

https://doi.org/10.1186/s13661-014-0204-5

© Li and Wang; licensee Springer 2014

**Received:**18 August 2014**Accepted:**18 August 2014**Published:**25 September 2014

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’.

## Keywords

- complementary proof
- prescribed mean curvature Rayleigh equations

## 1 Introduction

where $e\in C(R,R)$ is *T*-periodic, and $f,g\in C(R\times R,R)$ 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: ${F}_{\mu}(x)=\mu L({Q}_{1}(t,{x}_{1},{x}_{2}))$ where ${Q}_{1}$ depends on $\psi ({x}_{2})$ and $\psi (x)=\frac{x}{\sqrt{1-{x}^{2}}}$ which is only defined for $|x|<1$ and cannot be continuously extended; therefore, ${F}_{\mu}$ should not be defined on $\overline{\mathrm{\Omega}}=\{x\in X:\parallel x\parallel <M\}$ since $|{x}_{2}(t)|>1$ can occur, where $\parallel x\parallel =max\{{\parallel {x}_{1}\parallel}_{\mathrm{\infty}},{\parallel {x}_{2}\parallel}_{\mathrm{\infty}}\}$ and $M=1+max\{{D}_{1},{D}_{2}\}$.

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

## 2 Complementary proof

and there exists $\eta \in [0,T]$ such that ${x}_{2}(\eta )=0$.

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

(H_{1}): $(g(t,{x}_{1})-g(t,{x}_{2}))({x}_{1}-{x}_{2})<0$, for all $t,{x}_{1},{x}_{2}\in R$ and ${x}_{1}\ne {x}_{2}$;

_{2}): there exists $l>0$ such that

_{3}): there exists

*β*,

*γ*such that

_{4}): for all $t,x\in R$,

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

*g*,

*e*are continuous, we find that there exists ${M}_{3}>0$ such that

_{3}), there exists a positive constant ${M}_{4}>0$ such that

which is a contradiction.

_{3}), there exists a positive constant ${M}_{5}>0$ such that

Consequently, (2.4) holds.

we can use Mawhin’s continuation theorem on Ω.

## Notes

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

## References

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

## Copyright

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.