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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

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