\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 14, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/14\hfil Inverse nodal problem for $p$-Laplacian]
{Inverse nodal problem for a $p$-Laplacian Sturm-Liouville equation
with polynomially boundary condition}

\author[H. Koyunbakan, T. Gulsen, E. Yilmaz \hfil EJDE-2018/14\hfilneg]
{Hikmet Koyunbakan, Tuba Gulsen, Emrah Yilmaz}

\address{Hikmet Koyunbakan \newline
Firat University,
Department of Mathematics,
23119, Elaz\i{g},  Turkey}
\email{hkoyunbakan@gmail.com}

\address{Tuba Gulsen \newline
Firat University,
Department of Mathematics,
23119, Elaz\i{g},  Turkey}
\email{tubagulsen87@hotmail.com}

\address{Emrah Yilmaz \newline
Firat University,
Department of Mathematics,
23119, Elaz\i{g},  Turkey}
\email{emrah231983@gmail.com}

\dedicatory{Communicated by Ira Herbst}

\thanks{Submitted February 20, 2017. Published January 10, 2018.}
\subjclass[2010]{34A55, 34L05, 34L20}
\keywords{Inverse nodal problem; Pr\"ufer substitution;
 Sturm-Liouville equation}

\begin{abstract}
 In this article, we extend solution of inverse nodal problem for
 one-dimensional  $p$-Laplacian equation to the case when the
 boundary condition is polynomially eigenparameter.
 To find the spectral data as eigenvalues  and nodal parameters, a
 Pr\"ufer substitution is used.
 Then, we give a reconstruction formula of the potential function
 by using nodal lengths. This method is similar to used in \cite{yang1},
 and our results are more general.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Consider  $p$-Laplacian Sturm-Liouville eigenvalue problem
\begin{equation}
-\big(  y^{'(p-1)}\big)  '=(  p-1)  \big(\lambda-q_m(x)\big)
 y^{(p-1)},\quad 0\leq x\leq1,\label{e1.1}
\end{equation}
with the boundary conditions
\begin{equation}
\begin{gathered}
y(0)=0,\quad y'(0)=1, \\
y'(1,\lambda)+f(\lambda)y(1,\lambda)=0,
\end{gathered}\label{e1.2}
\end{equation}
where $p>1$,
\begin{equation}
f(\lambda)=a_1\sqrt{\lambda}+a_2(  \sqrt{\lambda})
^{2}+\dots+a_m(  \sqrt{\lambda})  ^{m},\quad
a_i\in \mathbb{R},\; a_m\neq0,\;  m\in \mathbb{Z}^{+}, \label{e1.3}
\end{equation}
$\lambda$ is a spectral parameter and 
$y^{(p-1)}=|y| ^{(p-2)}y$. Throughout this study, we suppose that $q_m(x)$ is
a real-valued $C[  0,1]$-function defined on the interval $0\leq
x\leq1$ for each $m\in \mathbb{Z} ^{+}$ and
 $y(x,\lambda)$ denotes the solution of the problem \eqref{e1.1}-\eqref{e1.2}.
When $p=2$, Equation \eqref{e1.1} becomes the well-known Sturm-Liouville equation. 
The idea of inverse eigenvalue problems with an eigenparameter together with 
the boundary conditions is of great interest to many problems of mathematical 
physics and mechanics. 
These type problems have many physical applications. For instance,
Sturm-Liouville equation including spectral parameter with the boundary
conditions arises in heat and one-dimensional wave equation by seperation of
variables. There are many literatures on these type of problems (see
\cite{brownesleeman,bttrn,elbrt,esin,freiling,glyev,sad,wltr,wangshieh,yangxp}).

Inverse spectral problem involves recovering differential equation from its
spectral parameters like eigenvalues, norming constants and nodal points
(zeros of eigenfunctions). These type of problems have been divided into two
parts; inverse eigenvalue problem and inverse nodal problem. They play an
important role and also have many applications in applied mathematics. Inverse
nodal problem was firstly studied by McLaughlin in 1988. She showed that the
knowledge of a dense subset of nodal points is sufficient to determine the
potential function of Sturm-Liouville problem up to a constant \cite{mccc}.
Also, some numerical results about this problem were given in \cite{haldd}.
Nowadays, many authors have given some interesting results about inverse nodal
problems for different type of operators 
(see \cite{buttshieh,eti,koyunbb,lawyang,zkan,wangcheng,yurko1}).

In this study, we devote our effort with the inverse nodal problem for
$p$-Laplacian Sturm-Liouville equation with boundary condition polynomially
dependent on spectral parameter. Essentially, we give asymptotics of
eigenparameters and reconstruction formula for potential function. Note that
inverse eigenvalue problems for different $p$-Laplacian operators have been
studied by several authors 
(see \cite{bndngdrabk,chenn,koyunbakan1,laww,wangg,wangcheng}).

The zero set $X_n=\{  x_{j,m}^n\}  _{j=1}^{n-1}$ of the
eigenfunction $y_{n,m}(x)$ corresponding to $\lambda_{n,m}$ is called the set
of nodal points. And, $l_{j,m}^n=x_{j+1,m}^n-x_{j,m}^n$ is referred as
the nodal length of $y_{n,m}$. The eigenfunction $y_{n,m}(x)$ has exactly
$n-1$ nodal points in $(0,1)$, say 
$0=x_{0,m}^{(n)}<x_{1,m}^{(n)}<\dots<x_{n-1,m}^{(n)}<x_{n,m}^{(n)}=1$.

Let us now recall some important results. Firstly, we need to introduce the
generalized sine function $S_p$ which is the solution of the initial value
problem
\begin{equation} \label{e1.4}
\begin{gathered}
-\big(  S_p^{'(p-1)}\big)  '=(p-1)S_p^{(p-1)}, \\
S_p(0)=0,\text{ }S_p'(0)=1.
\end{gathered}
\end{equation}
$S_p$ and $S_p'$ are periodic functions which satisfy the identity
\[
| S_p(x)| ^p+| S_p'(x)|^p=1,
\]
for any $x\in \mathbb{R}$. These functions are $p$-analogues of classical 
sine and cosine functions.
It is well known that
\[
\hat{\pi}=\int_0^1\frac{2}{(  1-t^p)  ^{\frac{1}{p}}}dt
=\frac{2\pi}{p\sin (  \frac{\pi}{p})  },
\]
is the first zero of $S_p$ in positive axis \cite{chenn}.

\begin{lemma}[\cite{chenn}]\label{lem1.1}
(a) For $S_p'\neq0$,
\[
(  S_p')  '=-| \frac{S_p}
{S_p'}| ^{p-2}S_p.
\]
(b)
\[
(  S_pS_p^{'(p-1)})  '=| S_p
'| ^p-(p-1)S_p^p=1-p| S_p|
^p=(1-p)+p| S_p'| ^p.
\]
\end{lemma}

Using $S_p(x)$ and $S_p'(x)$, the generalized tangent function
$T_p(x)$ can be defined as follows \cite{chenn}
\[
T_p(x)=\frac{S_p(x)}{S_p'(x)},\quad \text{for }x\neq \big(
k+\frac{1}{2}\big)  \hat{\pi}.
\]

The remaining part of this study is organized as follows; 
In section 2, we give some asymptotic formulas for eigenvalues and nodal 
parameters for $p$-Laplacian Sturm-Liouville eigenvalue problem 
\eqref{e1.1}-\eqref{e1.2} with boundary
condition polynomially dependent on spectral parameter by using modified
Pr\"ufer substitution. In section 3, we give a reconstruction for the
potential function of the problem \eqref{e1.1}-\eqref{e1.2}.


\section{Asymptotic behavior of some eigenparameters}

In this section, we present some results on \eqref{e1.1}-\eqref{e1.2}. 
One of them is the Pr\"ufer's transformation which is one of the most powerful 
method for solving inverse problem. Recall that the Pr\"ufer's transformation for a
nonzero solution $y$ of \eqref{e1.1} takes the form
\begin{equation} \label{e2.1}
\begin{gathered}
y(x)  =R(x)S_p(  \lambda^{1/p} \theta(x,\lambda)),\\
y'(x)  =\lambda^{1/p}R(x)S_p'(  \lambda^{1/p} \theta(x,\lambda))  ,
\end{gathered}
\end{equation}
or
\begin{equation}
\frac{y'(x)}{y(x)}=\lambda^{1/p}\frac{S_p'(
\lambda^{1/p} \theta(x,\lambda))  }{S_p(  \lambda
^{1/p} \theta(x,\lambda))  },\label{e2.2}
\end{equation}
where $R(x)$ is amplitude and $\theta(x)$ is the Pr\"ufer variable
\cite{yantr}. Standard manipulations \cite{wangcheng} yield
\begin{equation}
\theta'(x,\lambda)=1-\frac{q_m(x)}{\lambda}S_p^p\big(
\lambda^{1/p} \theta(x,\lambda)\big)  .\label{e2.3}
\end{equation}


\begin{lemma}[\cite{wangcheng}]\label{lem2.1}
Define $\theta (x,\lambda_n)$ as in \eqref{e2.1} and
 $\phi_n(x)=S_p^p(\lambda_n^{1/p} \theta(x,\lambda_n))  -\frac{1}{p}$.
Then, for any $g\in L^1(0,1)$,
\[
\int_0^1\phi_n(x)g(x)dx=0.
\]
\end{lemma}

\begin{theorem}\label{thm2.2}
The eigenvalues $\lambda_{n,m}$ of the $p$-Laplacian Sturm-Liouville eigenvalue
problem given in problem \eqref{e1.1}-\eqref{e1.2} have the form
\begin{gather}
\lambda_{n,1}^{1/p}=n\hat{\pi}-\frac{1}{a_1(  n\hat{\pi})
^{\frac{p-2}{2}}}+\frac{1}{p(  n\hat{\pi})  ^{p-1}}
 \int_0^1 q_1(x)dx+O(  \frac{1}{n^{p-2}}) ,\quad
\text{for }m=1,\label{e2.4} \\
\begin{aligned}
\lambda_{n,2}^{1/p}
&=n\hat{\pi}-\frac{1}{a_1(  n\hat{\pi})
^{\frac{p-2}{2}}+a_2(  n\hat{\pi})  ^{p-1}}
+\frac{1}{p(n\hat{\pi})  ^{p-1}}\int_0^1q_2(x)dx\\
&\quad +O\big( \frac{1}{n^{2p-1}}\big) ,\quad \text{for }m=2,
\end{aligned} \label{e2.5} \\
\begin{aligned}
\lambda_{n,m}^{1/p}
&=n\hat{\pi}-\frac{1}{a_1(  n\hat{\pi})
^{\frac{p-2}{2}}+\dots+a_m(  n\hat{\pi})  ^{\frac{mp-2}{2}}} \\
&\quad +\frac{1}{p(  n\hat{\pi})  ^{p-1}}
 \int_0^1  q_m(x)dx+O\big(  \frac{1}{n^{2p-1}}\big) ,\quad
\text{for }m\geq3, 
\end{aligned} \label{e2.6}
\end{gather}
as $n\to \infty$.
\end{theorem}

\begin{proof} Let $\theta(0,\lambda)=0$ for 
\eqref{e1.1}-\eqref{e1.2}. Integrating both sides of \eqref{e2.3} with respect 
to $x$ from $0$ to $1$, we obtain
\[
\theta(1,\lambda)=1-\frac{1}{\lambda}\int_0^1q_m(x)S_p^p
(\lambda^{1/p} \theta(x,\lambda))  dx.
\]
By Lemma \ref{lem2.1},
\[
\int_0^1q_m(x)\{  S_p^p(  \lambda^{1/p} \theta(x,\lambda))  
-\frac{1}{p}\}  dx=o(1),\quad \text{as }n\to \infty.
\]
Hence, we obtain
\begin{equation}
\theta(1,\lambda)=1-\frac{1}{p\lambda}\int_0^1q_m(x)dx
+O\big(\frac{1}{\lambda^{2}}\big)  .\label{e2.7}
\end{equation}
Let $\lambda_{n,m}$ be an eigenvalue of the problem \eqref{e1.1}-\eqref{e1.2}. 
For $m=1$, by
\eqref{e1.2}, we have
\[
\lambda_{n,1}^{1/p}R(1)S_p'(  \lambda_{n,1}^{1/p}\text{ }
\theta(1,\lambda_{n,1}))  +a_1\sqrt{\lambda_{n,1}}R(1)S_p\big(
\lambda_{n,1}^{1/p} \theta(1,\lambda_{n,1})\big)  =0,
\]
or
\[
-\frac{\lambda_{n,1}^{\frac{1}{p}-\frac{1}{2}}}{a_1}=\frac{S_p(
\lambda_{n,1}^{1/p} \theta(1,\lambda_{n,1}))  }{S_p^{\prime
}(  \lambda_{n,1}^{1/p} \theta(1,\lambda_{n,1}))  }
=T_p\big(  \lambda_{n,1}^{1/p} \theta(1,\lambda_{n,1})\big)  .
\]
As $n$ is sufficiently large, it follows that
\begin{equation}
\lambda_{n,1}^{1/p} \theta(1,\lambda_{n,1})=T_p^{-1}\Big(
-\frac{\lambda_{n,1}^{\frac{1}{p}-\frac{1}{2}}}{a_1}\Big)
 =n\hat{\pi}-\frac{\lambda_{n,1}^{\frac{1}{p}-\frac{1}{2}}}{a_1}
+o(  \lambda _{n,1}^{\frac{2}{p}-1})  .\label{e2.8}
\end{equation}
By considering \eqref{e2.7} and \eqref{e2.8} together, we obtain
\[
\lambda_{n,1}^{1/p}=n\hat{\pi}-\frac{1}{a_1(  n\hat{\pi})
^{\frac{p-2}{2}}}+\frac{1}{p(  n\hat{\pi})  ^{p-1}}
\int_0^1q_1(x)dx+O\big(  \frac{1}{n^{p-2}}\big)  .
\]
For $m=2$, by \eqref{e1.2}, using the same process as in $m=1$, 
we can easily obtain 
\[
\lambda_{n,2}^{1/p}R(1)S_p'(  \lambda_{n,2}^{1/p}\text{ }
\theta(1,\lambda_{n,2}))  +(  a_1\sqrt{\lambda_{n,2}}
+a_2(  \sqrt{\lambda_{n,2}})  ^{2})  R(1)S_p(
\lambda_{n,2}^{1/p} \theta(1,\lambda_{n,2}))  =0,
\]
or
\begin{equation}
-\frac{\lambda_{n,2}^{\frac{1}{p}}}{a_1\sqrt{\lambda_{n,2}}+a_2(
\sqrt{\lambda_{n,2}})  ^{2}}=\frac{S_p(  \lambda_{n,2}
^{1/p} \theta(1,\lambda_{n,2}))  }{S_p'(
\lambda_{n,2}^{1/p} \theta(1,\lambda_{n,2}))  }=T_p(
\lambda_{n,2}^{1/p} \theta(1,\lambda_{n,2}))  .\label{e2.9}
\end{equation}
Therefore, 
\[
\lambda_{n,2}^{1/p}=n\hat{\pi}-\frac{1}{a_1(  n\hat{\pi})
^{\frac{p-2}{2}}+a_2(  n\hat{\pi})  ^{p-1}}+\frac{1}{p(
n\hat{\pi})  ^{p-1}}
\int_0^1 q_2(x)dx+O\big(\frac{1}{n^{2p-1}}\big)  .
\]
Finally, by \eqref{e1.2}, we have
\begin{align*}
&\lambda_{n,m}^{1/p}R(1)S_p'(  \lambda_{n,m}^{1/p}
\theta(1,\lambda_{n,m}))  +(  a_1\sqrt{\lambda_{n,m}}
+\dots\\
&+a_m(  \sqrt{\lambda_{n,m}})  ^{m})  R(1)S_p(
\lambda_{n,m}^{1/p} \theta(1,\lambda_{n,m}))  =0,
\end{align*}
or
\begin{equation}
\begin{aligned}
-\frac{\lambda_{n,m}^{\frac{1}{p}}}{a_1\sqrt{\lambda_{n,m}}+\dots+a_m(
\sqrt{\lambda_{n,m}})  ^{m}}
&=\frac{S_p(  \lambda_{n,m}
^{1/p} \theta(1,\lambda_{n,m}))  }{S_p'(
\lambda_{n,m}^{1/p} \theta(1,\lambda_{n,m}))  }\\
&=T_p\big(\lambda_n^{1/p} \theta(1,\lambda_{n,m})\big)  ,
\end{aligned} \label{e2.10}
\end{equation}
for $m\geq3$, by considering \eqref{e2.7} and \eqref{e2.10} together, 
we deduce that
\begin{align*}
&\lambda_{n,m}^{1/p}=n\hat{\pi}-\frac{1}{a_1(  n\hat{\pi})
^{\frac{p-2}{2}}+\dots+a_m(  n\hat{\pi})  ^{\frac{mp-2}{2}}} \\
&+\frac{1}{p(  n\hat{\pi})  ^{p-1}}\int_0^1q_m(x)dx+O(  \frac{1}{n^{2p-1}})  .
\end{align*}
\end{proof}


\begin{theorem}\label{thm2.3}
The nodal points for problem \eqref{e1.1}-\eqref{e1.2} satisfy the following
asymptotic estimates:
\begin{gather}
\begin{aligned}
x_{j,1}^n  & =\frac{j}{n}-\frac{j}{a_1n^{\frac{p+2}{2}}\hat{\pi}^p
}+\frac{j}{pn^{p+1}\hat{\pi}^p}
\int_0^1 q_1(t)dt
+ \int_0^{x_{j,1}^n} \frac{q_1(t)}{(  n\hat{\pi})  ^p}S_p^pdt
+O\big(  \frac{j}{n^p}\big) ,\\
&\quad \text{for }m=1,
\end{aligned}\label{e2.11}\\
\begin{aligned}
x_{j,2}^n  
& =\frac{j}{n}-\frac{j}{a_1n^{\frac{p+2}{2}}\hat{\pi}^{\frac
{p}{2}}+a_2n^{p+1}\hat{\pi}^p}+\frac{j}{pn^{p+1}\hat{\pi}^p}
\int_0^1  q_2(t)dt \\
&\quad + \int_0^{x_{j,2}^n} \frac{q_2(t)}{(  n\hat{\pi})  ^p}S_p^pdt
+O\big(  \frac
{j}{n^{2p+1}}\big)  ,\quad \text{for }m=2,
\end{aligned} \label{e2.12} \\
\begin{aligned}
x_{j,m}^n  
& =\frac{j}{n}-\frac{j}{a_1n^{\frac{p+2}{2}}\hat{\pi}^{\frac
{p}{2}}+\dots+a_mn^{\frac{mp+2}{2}}\hat{\pi}^{\frac{mp}{2}}}+\frac{j}
{pn^{p+1}\hat{\pi}^p}\int_0^1q_m(t)dt\label{e2.13}\\
&\quad + \int_0^{x_{j,m}^n} \frac{q_m(t)}{(  n\hat{\pi})  ^p}S_p^pdt+O\big(  \frac
{j}{n^{2p+1}}\big)  ,\quad \text{for }  m\geq3,
\end{aligned}
\end{gather}
as $n\to \infty$.
\end{theorem}


\begin{proof} Integrating  \eqref{e2.3} from $0$ to $x_{j,m}^n$~and
letting $\theta(x_{j,m}^n,\lambda)=\frac{j\hat{\pi}}{\lambda_{n,m}^{1/p}}$,
we have
\begin{equation}
x_{j,m}^n=\frac{j\hat{\pi}}{\lambda_{n,m}^{1/p}}+
 \int_0^{x_{j,m}^n}
\frac{q_m(t)}{\lambda_{n,m}}S_p^pdt.\label{e2.14}
\end{equation}
For $m=1$, from  \eqref{e2.4}, we deduce that
\begin{equation}
\frac{1}{\lambda_{n,1}^{1/p}}=\frac{1}{n\hat{\pi}}-\frac{1}{a_1(
n\hat{\pi})  ^{\frac{p+2}{2}}}+\frac{1}{p(  n\hat{\pi})
^{p+1}}
\int_0^1 q_1(x)dx+O\big(  \frac{1}{n^p}\big)  ,\label{e2.15}
\end{equation}
and therefore, we obtain formula \eqref{e2.11} by using \eqref{e2.14} 
and \eqref{e2.15}.

For $m=2$, from formula \eqref{e2.5}, the asymptotic estimate of
eigenvalues $1/\lambda_{n,2}^{1/p}$ is considered as
\begin{equation}
\frac{1}{\lambda_{n,2}^{1/p}}
=\frac{1}{n\hat{\pi}}-\frac{1}{a_1(
n\hat{\pi})  ^{\frac{p+2}{2}}+a_2(  n\hat{\pi})  ^{p+1}
}+\frac{1}{p(  n\hat{\pi})  ^{p+1}}
\int_0^1 q_2(x)dx +O\big(  \frac{1}{n^{2p+1}}\big)  ,\label{e2.16}
\end{equation}
and, we conclude formula \eqref{e2.12} by using \eqref{e2.14} 
and \eqref{e2.16}.

For $m\geq3$, from the formula \eqref{e2.6}, it can easily be shown that
\begin{equation}
\begin{aligned}
\frac{1}{\lambda_{n,m}^{1/p}}
&=\frac{1}{n\hat{\pi}}-\frac{1}{a_1(
n\hat{\pi})  ^{\frac{p+2}{2}}+\dots+a_m(  n\hat{\pi})
^{\frac{mp+2}{2}}}+\frac{1}{p(  n\hat{\pi})  ^{p+1}}
\int_0^1 q_m(x)dx \\
&\quad +O\big(  \frac{1}{n^{2p+1}}\big)  ,
\end{aligned} \label{e2.17}
\end{equation}
and, we obtain formula \eqref{e2.13} by using \eqref{e2.14} and \eqref{e2.17}.
\end{proof}

\begin{theorem}\label{thm2.4}
Asymptotic estimate of the nodal lengths for the problem \eqref{e1.1}-\eqref{e1.2}
satisfies
\begin{gather} \label{e2.18}
\begin{aligned}
l_{j,1}^n  & =\frac{1}{n}-\frac{1}{a_1n^{\frac{p+2}{2}}\hat{\pi}^{\frac
{p}{2}}}+\frac{1}{pn^{p+1}\hat{\pi}^p}
\int_0^1 q_1(t)dt\\
&\quad +\frac{1}{( n\hat{\pi})^p}
\int _{x_{j,1}^n}^{x_{j+1,1}^n} 
q_1(t)S_p^pdt+O\big(  \frac{1}{n^p}\big),\quad 
\text{for } m=1,
\end{aligned} \\
\label{e2.19}
\begin{aligned}
l_{j,2}^n  
& =\frac{1}{n}-\frac{1}{a_1n^{\frac{p+2}{2}}\hat{\pi}^{\frac{p}{2}}
+a_2n^{p+1}\hat{\pi}^p}+\frac{1}{pn^{p+1}\hat{\pi}^p}
\int_0^1 q_2(t)dt\\
& +\frac{1}{(  n\hat{\pi})  ^p}
\int _{x_{j,2}^n}^{x_{j+1,2}^n}
q_2(t)S_p^pdt+O\big(  \frac{1}{n^{2p+1}}\big),\quad 
 \text{for } m=2,
\end{aligned} \\
\label{e2.20}
\begin{aligned}
l_{j,m}^n  
& =\frac{1}{n}-\frac{1}{a_1n^{\frac{p+2}{2}}\hat{\pi}^{\frac
{p}{2}}+\dots
+a_mn^{\frac{mp+2}{2}}\hat{\pi}^{\frac{mp}{2}}}+\frac{1}{pn^{p+1}\hat{\pi}^p}
\int_0^1 q_m(t)dt\\
& +\frac{1}{(  n\hat{\pi})  ^p}
\int _{x_{j,m}^n}^{x_{j+1,m}^n}
q_m(t)S_p^pdt+O\big(  \frac{1}{n^{2p+1}}\big),\quad \text{for } m\geq3.
\end{aligned}
\end{gather}
\end{theorem}

\begin{proof}
For a large $n\in \mathbb{N}$, integrating  \eqref{e2.3} on 
$[x_{j,m}^n,x_{j+1,m}^n]$ and using the
definition of nodal lengths, we have
\begin{equation}
\begin{aligned}
\frac{\hat{\pi}}{\lambda_{n,m}^{1/p}}
&=x_{j+1,m}^n-x_{j,m}^n-\frac
{1}{p\lambda_{n,m}}
\int _{x_{j,m}^n}^{x_{j+1,m}^n} q_m(t)S_p^pdt \\
&\quad -\frac{1}{\lambda_{n,m}}
\int _{x_{j,m}^n}^{x_{j+1,m}^n}
q_m(t)\big(  S_p^p-\frac{1}{p}\big)  dt,
\end{aligned}\label{e2.21}
\end{equation}
or
\[
l_{j,m}^n=\frac{\hat{\pi}}{\lambda_{n,m}^{1/p}}+\frac{1}{p\lambda_{n,m}}
 \int _{x_{j,m}^n}^{x_{j+1,m}^n}
q_m(t)S_p^pdt+O\big(  \frac{1}{\lambda_{n,m}}\big)  .
\]
\end{proof}

For  $m=1$, $m=2$ and $m\geq3$, we can easily obtain \eqref{e2.18},
\eqref{e2.19} and \eqref{e2.20} by using the formulas 
\eqref{e2.15}, \eqref{e2.16}, \eqref{e2.17} and \eqref{e2.21}, respectively.

\section{Reconstruction of the potential function}

In this section, we give an explicit formula for the potential function by
using the nodal lengths. The method used in the proof of the theorem is
similar to classical problems; $p$-Laplacian Sturm-Liouville eigenvalue
problem and $p$-Laplacian energy-dependent Sturm-Liouville eigenvalue problem
(see \cite{koyunbakan1,laww,wangg,wangcheng}).

\begin{theorem}\label{thm3.1}  Let $q_m(x)$ 
be a real-valued $C[  0,1]$-function on the interval
 $0\leq x\leq1$. Then
\begin{equation}
q_m(x)=\lim_{n\to \infty}~p\lambda_{n,m}\Big(  \frac{\lambda
_{n,m}^{1/p}l_{j,m}^n}{\hat{\pi}}-1\Big)  ,\label{e3.1}
\end{equation}
for $j=j_n(x)=\max \{  j:x_{j,m}^n<x\}  $ and 
$m\in\mathbb{Z}^{+}$.
\end{theorem}

\begin{proof} 
We need to consider Theorem \ref{thm2.3} for the proof. From
\eqref{e2.21}, we have
\[
\frac{p\lambda_{n,m}^{1/p+1}}{\hat{\pi}}l_{j,m}^n=p\lambda_{n,m}
+\frac{\lambda_{n,m}^{1/p}}{\hat{\pi}}
 \int _{x_{j,m}^n}^{x_{j+1,m}^n}
q_m(t)dt+\frac{p\lambda_{n,m}^{1/p}}{\hat{\pi}}
\int _{x_{j,m}^n}^{x_{j+1,m}^n}
q_m(t)(  S_p^p-\frac{1}{p})  dt.
\]
Then, we can use similar procedure as those in \cite{laww} for 
$j=j_n(x)=\max \{  j:x_{j,m}^n<x\}  $ to show
\[
\frac{\lambda_{n,m}^{1/p}}{\hat{\pi}}
 \int _{x_{j,m}^n}^{x_{j+1,m}^n}
q_m(t)dt\to q_m(x),
\]
and
\[
\frac{p\lambda_{n,m}^{1/p}}{\hat{\pi}}
\int _{x_{j,m}^n}^{x_{j+1,m}^n}
q_m(t)\big(  S_p^p-\frac{1}{p}\big)  dt\to0,
\]
pointwise almost everywhere. Hence, we obtain
\[
q_m(x)=\lim_{n\to \infty}~p\lambda_{n,m}\Big(  \frac{\lambda
_{n,m}^{1/p}l_{j,m}^n}{\hat{\pi}}-1\Big)  .
\]
\end{proof}

\begin{theorem}\label{thm3.2} 
Let $\{  l_{j,m}^{(n)}:j=1,2,\dots,n-1\}  _{n=2}^{\infty}$ 
be a set of the nodal lengths of problem \eqref{e1.1}-\eqref{e1.2}, 
where $q_m$ is a real-valued $C[0,1]$-function. Let us define
\begin{gather}
F_{n,1}(x)
=p(  n\hat{\pi})  ^p(  nl_{j,1}^{(n)}-1)
-\frac{p}{a_1}\big( n\hat{\pi}\big)  ^{p/2}+\int_0^1
q_1(t)dt,\quad \text{for }m=1.\label{e3.2} \\
F_{n,2}(x)=p(  n\hat{\pi})  ^p\big(  nl_{j,2}^{(n)}-1\big)
-\frac{p(  n\hat{\pi})  ^{p/2}}{a_1+a_2(  n\hat{\pi})  ^{p/2}}
+\int_0^1q_2(t)dt,\quad \text{for } m=2.\label{e3.3} 
\\
\begin{aligned}
F_{n,m}(x) & =p(  n\hat{\pi})  ^p\big(  nl_{j,m}^{(n)}-1\big)
-\frac{p(  n\hat{\pi})  ^{p/2}}{a_1+\dots+a_m(  n\hat{\pi
})  ^{\frac{mp-p}{2}}} 
+\int_0^1q_m (t)dt,\\
&\quad  \text{for } m\geq3.
\end{aligned}\label{e3.4}
\end{gather}
Then $\{F_{n,m}(x)\}$  converges to $q_m$  pointwise almost everywhere in 
$L^1(0,1)$, for all cases.
\end{theorem}

\begin{proof} We prove this theorem only for $m=1$. Other cases can
be shown similarly. For $m=1$, by the asymptotic formulas of 
eigenvalues \eqref{e2.4}
and nodal lengths \eqref{e2.18}, we obtain
\[
p\lambda_{n,1}\Big(  \frac{\lambda_{n,1}^{1/p}l_{j,1}^n}{\hat{\pi}
}-1\Big)  
=p\lambda_{n,1}\big(  nl_{j,1}^{(n)}-1\big)  
-\frac{p}{a_1 \pi}(  n\hat{\pi})  ^{p/2+1}l_{j,1}^{(n)}+nl_{j,1}^{(n)}
\int_0^1q_1(t)dt+o(1).
\]
Considering $nl_{j,1}^{(n)}=1+o(1)$, as $n\to \infty$, we have
\[
p(  n\hat{\pi})  ^p(  nl_{j,1}^{(n)}-1)  -\frac
{p}{a_1}(  n\hat{\pi})  ^{p/2}\to q_1(x)-\int
_0^1q_1(t)dt,
\]
pointwise almost everywhere in $L^1(0,1)$.
\end{proof}

\subsection*{Conclusion}
In this study, we give some asymptotic estimates for eigenvalues, nodal
parameters and potential function of the $p$-Laplacian Sturm-Liouville
eigenvalue problem \eqref{e1.1}-\eqref{e1.2}. We show that the obtained 
results are the generalizations of the classical problem.

\subsection*{Acknowledgements}
The authors is deeply indebted to the reviewer, who made remarks 
which contributed to the improvements in the text and in the
transparency of the results.

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\end{document}