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. [1].

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

Theorem 2.1

[1]

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

$${\lambda }_n^{2/p}=n{\pi }_p+\frac{1}{p{(n{\pi }_p)}^{p-1}}{\int }_0^{1}q(t)dt+\frac{2}{p{(n{\pi }_p)}^{\frac{p-2}{2}}}{\int }_0^{1}r(t)dt+O\left(\frac{1}{n^{\frac{p}{2}}}\right).$$

(2.4)

Theorem 2.2

[1]

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

$$\begin{array}{rcl}{x}_j^n& =& \frac{j}{n}+\frac{j}{p{n}^{p+1}{({\pi }_p)}^p}{\int }_0^{1}q(t)dt+\frac{2j}{p{n}^{\frac{p}{2}+1}{({\pi }_p)}^{\frac{p}{2}}}{\int }_0^{1}r(t)dt+\frac{2}{{(n{\pi }_p)}^{{}^{\frac{p}{2}}}}{\int }_0^{x_j^n}r(x)S_p^{p}dx\\ +\frac{1}{{(n{\pi }_p)}^{{}^p}}{\int }_0^{x_j^n}q(x)S_p^{p}dx+O\left(\frac{1}{n^{\frac{p}{2}+2}}\right).\end{array}$$

Proof

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

$$\frac{j\cdot {\pi }_p}{{\lambda }_n^{2/p}}=x_j^n-{\int }_0^{x_j^n}\frac{2r(x)}{{\lambda }_n}{S}_p^{p}dx-{\int }_0^{x_j^n}\frac{q(x)}{{\lambda }_n^2}{S}_p^{p}dx.$$

By using the estimates of eigenvalues as

$$\frac{1}{{\lambda }_n^{2/p}}=\frac{1}{n{\pi }_p}+\frac{1}{p{(n{\pi }_p)}^{p+1}}{\int }_0^{1}q(t)dt+\frac{2}{p{(n{\pi }_p)}^{\frac{p}{2}+1}}{\int }_0^{1}r(t)dt+O\left(\frac{1}{n^{\frac{p}{2}+2}}\right),$$

we obtain the result. □