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Erratum to: Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations
Boundary Value Problems volume 2014, Article number: 204 (2014)
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:
where is T-periodic, and 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: where depends on and which is only defined for and cannot be continuously extended; therefore, should not be defined on since can occur, where and .
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:
where . In [1], the authors embed (2.1) into a family of equations with one parameter ,
They have proved that there exists a constant such that
and there exists such that .
In fact, to use the continuation theorem, it suffices to prove that there exists a positive constant such that, for any possible solution () of (2.2), the following condition holds:
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): , for all and ;
(H2): there exists such that
(H3): there exists β, γ such that
(H4): for all ,
Under the conditions mentioned above, we prove that (2.4) holds.
Since and g, e are continuous, we find that there exists such that
By (H3), there exists a positive constant such that
Next, we shall prove that
Assume by contradiction that there exist such that
and
Noticing that , we have, ,
which is a contradiction.
By (H3), there exists a positive constant such that
By using a similar argument, we can prove that
Therefore, we get from the continuity of , for any solution () of (2.2),
Consequently, (2.4) holds.
Putting
we can use Mawhin’s continuation theorem on Ω.
References
Li J, Wang Z: Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations. Bound. Value Probl. 2012., 2012: 10.1186/1687-2770-2012-109
Acknowledgements
The authors would like to thank Professor J Webb for pointing out the errors of the paper [1].
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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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DOI: https://doi.org/10.1186/s13661-014-0204-5
Keywords
- complementary proof
- prescribed mean curvature Rayleigh equations