# 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. .

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

## Theorem 2.1



The eigenvalues${\lambda }_{n}$of the Dirichlet problem (1.3), (1.4) are

${\lambda }_{n}^{2/p}=n{\pi }_{p}+\frac{1}{p{\left(n{\pi }_{p}\right)}^{p-1}}{\int }_{0}^{1}q\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+\frac{2}{p{\left(n{\pi }_{p}\right)}^{\frac{p-2}{2}}}{\int }_{0}^{1}r\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+O\left(\frac{1}{{n}^{\frac{p}{2}}}\right).$
(2.4)

## Theorem 2.2



For the problem (1.3), (1.4), the nodal point expansion satisfies

$\begin{array}{rcl}{x}_{j}^{n}& =& \frac{j}{n}+\frac{j}{p{n}^{p+1}{\left({\pi }_{p}\right)}^{p}}{\int }_{0}^{1}q\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+\frac{2j}{p{n}^{\frac{p}{2}+1}{\left({\pi }_{p}\right)}^{\frac{p}{2}}}{\int }_{0}^{1}r\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+\frac{2}{{\left(n{\pi }_{p}\right)}^{{}^{\frac{p}{2}}}}{\int }_{0}^{{x}_{j}^{n}}r\left(x\right){S}_{p}^{p}\phantom{\rule{0.2em}{0ex}}dx\\ +\frac{1}{{\left(n{\pi }_{p}\right)}^{{}^{p}}}{\int }_{0}^{{x}_{j}^{n}}q\left(x\right){S}_{p}^{p}\phantom{\rule{0.2em}{0ex}}dx+O\left(\frac{1}{{n}^{\frac{p}{2}+2}}\right).\end{array}$

## Proof

Let $\lambda ={\lambda }_{n}$; integrating (2.3) from 0 to ${x}_{j}^{n}$, we have

$\frac{j\cdot {\pi }_{p}}{{\lambda }_{n}^{2/p}}={x}_{j}^{n}-{\int }_{0}^{{x}_{j}^{n}}\frac{2r\left(x\right)}{{\lambda }_{n}}{S}_{p}^{p}\phantom{\rule{0.2em}{0ex}}dx-{\int }_{0}^{{x}_{j}^{n}}\frac{q\left(x\right)}{{\lambda }_{n}^{2}}{S}_{p}^{p}\phantom{\rule{0.2em}{0ex}}dx.$

By using the estimates of eigenvalues as

$\frac{1}{{\lambda }_{n}^{2/p}}=\frac{1}{n{\pi }_{p}}+\frac{1}{p{\left(n{\pi }_{p}\right)}^{p+1}}{\int }_{0}^{1}q\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+\frac{2}{p{\left(n{\pi }_{p}\right)}^{\frac{p}{2}+1}}{\int }_{0}^{1}r\left(t\right)\phantom{\rule{0.2em}{0ex}}dt+O\left(\frac{1}{{n}^{\frac{p}{2}+2}}\right),$

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. 