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

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

Consider the problem (1.3), (1.4) in .

## Theorem 2.1



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



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. 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. 