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

${\left(\frac{{x}^{\prime }}{\sqrt{1+{x}^{\prime 2}}}\right)}^{\prime }+f\left(t,{x}^{\prime }\left(t\right)\right)+g\left(t,x\left(t\right)\right)=e\left(t\right),$
(1.1)

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

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:

$\left\{\begin{array}{c}{x}_{1}^{\prime }\left(t\right)=\psi \left({x}_{2}\left(t\right)\right)=\frac{{x}_{2}\left(t\right)}{\sqrt{1-{x}_{2}^{2}\left(t\right)}},\hfill \\ {x}_{2}^{\prime }\left(t\right)=-f\left(t,\psi \left({x}_{2}\left(t\right)\right)\right)-g\left(t,{x}_{1}\left(t\right)\right)+e\left(t\right),\hfill \end{array}$
(2.1)

where $\psi \left(x\right)=\frac{x}{\sqrt{1-{x}^{2}}}$. In , the authors embed (2.1) into a family of equations with one parameter $\lambda \in \left(0,1\right]$,

$\left\{\begin{array}{c}{x}_{1}^{\prime }\left(t\right)=\lambda \frac{{x}_{2}\left(t\right)}{\sqrt{1-{x}_{2}^{2}\left(t\right)}}=\lambda \psi \left({x}_{2}\left(t\right)\right),\hfill \\ {x}_{2}^{\prime }\left(t\right)=-\lambda f\left(t,\psi \left({x}_{2}\left(t\right)\right)\right)-\lambda g\left(t,{x}_{1}\left(t\right)\right)+\lambda e\left(t\right).\hfill \end{array}$
(2.2)

They have proved that there exists a constant ${D}_{1}>0$ such that

$|{x}_{1}^{\prime }{|}_{2}\le {D}_{1},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}|{x}_{1}{|}_{\mathrm{\infty }}\le {D}_{1},$
(2.3)

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

In fact, to use the continuation theorem, it suffices to prove that there exists a positive constant $0<{\epsilon }_{0}\ll 1$ such that, for any possible solution (${x}_{1}\left(t\right),{x}_{2}\left(t\right)$) of (2.2), the following condition holds:

$|{x}_{2}\left(t\right)|<1-{\epsilon }_{0}.$
(2.4)

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

In , the authors assume that

(H1): $\left(g\left(t,{x}_{1}\right)-g\left(t,{x}_{2}\right)\right)\left({x}_{1}-{x}_{2}\right)<0$, for all $t,{x}_{1},{x}_{2}\in R$ and ${x}_{1}\ne {x}_{2}$;

(H2): there exists $l>0$ such that

(H3): there exists β, γ such that

(H4): for all $t,x\in R$,

$f\left(t+\frac{T}{2},-x\right)=-f\left(t,x\right),\phantom{\rule{2em}{0ex}}g\left(t+\frac{T}{2},-x\right)=-g\left(t,x\right),\phantom{\rule{2em}{0ex}}e\left(t+\frac{T}{2}\right)=-e\left(t\right).$

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

Since ${|{x}_{1}|}_{\mathrm{\infty }}<{D}_{1}$ and g, e are continuous, we find that there exists ${M}_{3}>0$ such that

$-{M}_{3}<-g\left(t,{x}_{1}\left(t\right)\right)+e\left(t\right)<{M}_{3},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in R.$
(2.5)

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

(2.6)

Next, we shall prove that

$x\left(t\right)\le \frac{{M}_{3}+{M}_{4}}{\sqrt{{\left({M}_{3}+{M}_{4}\right)}^{2}+{\gamma }^{2}}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in R.$

Assume by contradiction that there exist ${t}_{2}^{\ast }>{t}_{1}^{\ast }>\eta$ such that

${x}_{2}\left({t}_{1}^{\ast }\right)=\frac{{M}_{3}+{M}_{4}}{\sqrt{{\left({M}_{3}+{M}_{4}\right)}^{2}+{\gamma }^{2}}},\phantom{\rule{2em}{0ex}}{x}_{2}\left({t}_{2}^{\ast }\right)>\frac{{M}_{3}+{M}_{4}}{\sqrt{{\left({M}_{3}+{M}_{4}\right)}^{2}+{\gamma }^{2}}},$

and

Noticing that $\lambda \in \left(0,1\right]$, we have, $\mathrm{\forall }t\in \left({t}_{1}^{\ast },{t}_{2}^{\ast }\right)$,

${x}_{2}^{\prime }\left(t\right)=\lambda \left(-f\left(t,\psi \left({x}_{2}\left(t\right)\right)\right)-g\left(t,{x}_{1}\left(t\right)\right)+e\left(t\right)\right)<0,$

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

By using a similar argument, we can prove that

Therefore, we get from the continuity of ${x}_{2}\left(t\right)$, for any solution (${x}_{1}\left(t\right),{x}_{2}\left(t\right)$) of (2.2),

$-\frac{{M}_{3}+{M}_{5}}{\sqrt{{\left({M}_{3}+{M}_{5}\right)}^{2}+{\beta }^{2}}}\le {x}_{2}\left(t\right)\le \frac{{M}_{3}+{M}_{4}}{\sqrt{{\left({M}_{3}+{M}_{4}\right)}^{2}+{\gamma }^{2}}},\phantom{\rule{1em}{0ex}}\mathrm{\forall }t\in R.$

Consequently, (2.4) holds.

Putting

$\mathrm{\Omega }=\left\{x=\left(x,x\right)\in {C}_{T}^{0,\frac{1}{2}}\left(R,{R}^{2}\right)=X:\parallel x\parallel

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

## Acknowledgements

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

## Author information

Authors

### Corresponding author

Correspondence to Jin Li.

### 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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