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  • Open Access

Erratum: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation

Boundary Value Problems20142014:222

https://doi.org/10.1186/s13661-014-0222-3

  • Received: 19 September 2014
  • Accepted: 19 September 2014
  • Published:

The original article was published in Boundary Value Problems 2013 2013:272

In this note, we correct some mistakes in Theorem 2.1 and Theorem 2.2 which are given in Ref. [1].

Consider the problem (1.3), (1.4) in [1].

Theorem 2.1

[1]

The eigenvalues λ n of the Dirichlet problem (1.3), (1.4) are
λ n 2 / p = n π p + 1 p ( n π p ) p 1 0 1 q ( t ) d t + 2 p ( n π p ) p 2 2 0 1 r ( t ) d t + O ( 1 n p 2 ) .
(2.4)

Theorem 2.2

[1]

For the problem (1.3), (1.4), the nodal point expansion satisfies
x j n = j n + j p n p + 1 ( π p ) p 0 1 q ( t ) d t + 2 j p n p 2 + 1 ( π p ) p 2 0 1 r ( t ) d t + 2 ( n π p ) p 2 0 x j n r ( x ) S p p d x + 1 ( n π p ) p 0 x j n q ( x ) S p p d x + O ( 1 n p 2 + 2 ) .

Proof

Let λ = λ n ; integrating (2.3) from 0 to x j n , we have
j π p λ n 2 / p = x j n 0 x j n 2 r ( x ) λ n S p p d x 0 x j n q ( x ) λ n 2 S p p d x .
By using the estimates of eigenvalues as
1 λ n 2 / p = 1 n π p + 1 p ( n π p ) p + 1 0 1 q ( t ) d t + 2 p ( n π p ) p 2 + 1 0 1 r ( t ) d t + O ( 1 n p 2 + 2 ) ,

we obtain the result. □

Notes

Authors’ Affiliations

(1)
Department of Mathematics, Firat University, Elazig, 23119, Turkey

References

  1. Koyunbakan H: Inverse nodal problem for p -Laplacian energy-dependent Sturm-Liouville equation. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-272Google Scholar

Copyright

© Koyunbakan; licensee Springer 2014

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.

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