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

The Original Article was published on 12 December 2013

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)dt+ 2 p ( n π p ) p 2 2 0 1 r(t)dt+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 dx 0 x j n q ( x ) λ n 2 S p p dx.

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)dt+ 2 p ( n π p ) p 2 + 1 0 1 r(t)dt+O ( 1 n p 2 + 2 ) ,

we obtain the result. □

References

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

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Correspondence to Hikmet Koyunbakan.

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The online version of the original article can be found at 10.1186/1687-2770-2013-272

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Koyunbakan, H. Erratum: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation. Bound Value Probl 2014, 222 (2014). https://doi.org/10.1186/s13661-014-0222-3

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