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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 310, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/310\hfil Sturm-Liouville problems]
{Sturm-Liouville problems with retarded argument and  a finite number of \\
 transmission conditions}

\author[E. \c{S}en \hfil EJDE-2017/310\hfilneg]
{Erdo\u{g}an \c{S}en}

\address{Erdo\u{g}an \c{S}en \newline
Department of Mathematics,
Faculty of Arts and Science,
Namik Kemal University, 59030,
Tekirda\u{g}, Turkey}
\email{erdogan.math@gmail.com}

\dedicatory{Communicated by Ludmila S. Pulkina}

\thanks{Submitted July 26, 2017. Published December 29, 2017.}
\subjclass[2010]{34L20, 35R10}
\keywords{Retarded argument; eigenparameter; transmission conditions; 
\hfill\break\indent asymptotics of eigenvalues and eigenfunctions}

\begin{abstract}
The main goal of the present paper is to study the asymptotic behaviour of
eigenvalues and eigenfunctions of a discontinuous boundary-value problem
with retarded argument with a finite number of transmission conditions.
\end{abstract}

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

\section{Introduction}

Spectral properties of boundary-value problems with retarded argument and
with discontinuities inside the interval are studied by many authors
\cite{a1,b1,b2,c1,h1,s1,s2,s3,y1}.
Following these studies, in this work, we consider the boundary-value problem
for the differential equation
\begin{equation}
y''(x)+q(x)y(x-\Delta (x))+\mu ^2y(x)=0  \label{e1}
\end{equation}
on $[ 0,r_1) \cup ( r_1,r_2) \cup \dots \cup (r_m,\pi ]$, with boundary conditions
\begin{gather}
d_1y(0)+d_2y'(0)=0,  \label{e2} \\
y'(\pi )+\mu ^2y(\pi )=0,  \label{e3}
\end{gather}
and transmission conditions
\begin{gather}
y(r_i-0)-\delta _iy(r_i+0)=0,\quad i=\overline{1,m}, \label{e4} \\
y'(r_i-0)-\delta _iy'(r_i+0)=0,\quad i=\overline{1,m}  \label{e5}
\end{gather}
where the real-valued function $q(x)$ is continuous in
 $[0,r_1) \cup ( r_1,r_2) \cup \dots \cup ( r_m,\pi]$ and has finite limits
\begin{equation*}
q(r_i\pm 0)=\lim_{x\to r_i\pm 0}q(x),
\end{equation*}
the real valued function $\Delta (x)\geq 0$ continuous in $[
0,r_1) \cup ( r_1,r_2) \cup \dots \cup ( r_m,\pi
] $ and has finite limits
\begin{equation*}
\Delta (r_i\pm 0)=\lim_{x\to r_i\pm 0}\Delta (x),
\end{equation*}
$x-\Delta (x)\geq 0$ if $x\in [ 0,r_1)$;
$x-\Delta(x)\geq r_1$, if $x\in ( r_1,r_2) $;\dots ,
$x-\Delta(x)\geq r_{m-1}$, if $x\in ( r_m,\pi ) $;
$\mu $ is a real positive eigenparameter;
$r_i,\delta _i\neq 0$ are arbitrary real
numbers such that $0<r_1<r_2<\dots <r_m<\pi $ and $d_1d_2\neq 0$.

The goal of this article is to obtain asymptotic formulas for eigenvalues of
eigenfunctions for  problem \eqref{e1}--\eqref{e5}.
To this aim, first, the principal
term of asymptotic distribution of eigenvalues and eigenfunctions of
\eqref{e1}--\eqref{e5} was obtained up to $O(1/N) $, but,
afterwards under some additional conditions we improve these formulas up to
$O(1/N^2) $. Thus, when the number of points of
discontinuity is more than one, we see how the asymptotic behaviour of
eigenvalues and eigenfunctions of a boundary-value problem with retarded
argument which contains a spectral parameter in the boundary conditions
change. We point out that our results are extension and/or generalization to
those in \cite{a3,f1,k1,l1,l2,m1,m2,s3,s4,t1}.
 For example, 
if the retardation function $\Delta \equiv 0$ in \eqref{e1} 
and $\delta _i=1$ $( i=\overline{1,m}) $;
or $\delta _1\neq 1$ and $\delta _i=1$ $(i=\overline{2,m}) $;
or $\delta _{1,2}\neq 1$ and $\delta _i=1$ $( i=\overline{3,m}) $;
or $\delta _i\neq 1$ $( i=\overline{1,m}) $ results obtained in
this paper coincide with the results of \cite{f1,k1,m1,s4},
respectively.

Differential equations with deviating argument, in particular differential
equations with retarded argument, describe processes with aftereffect; they
find many applications, particularly in the theory of automatic control, in
the theory of self-oscillatory systems, in the study of problems connected
with combustion in rocket engines (see \cite{n1} and the references therein).

Boundary value problems containing a spectral parameter in the boundary
conditions have many interesting applications, especially in mathematical
physics (e.g. \cite[pp. 146-152]{t2}). It must be also noted that recently
boundary-value problems with transmission conditions attracted much attention
in connection with the inverse acoustic scattering problem (see, e.g.,
\cite{a2,b3,c3} and the references therein).

Let $w_1(x,\lambda )$ be a solution of  \eqref{e1} on $[0,h_1] $, 
satisfying the initial conditions
\begin{equation}
w_1( 0,\mu ) =d_2\quad\text{and} \quad
w_1'(0,\mu ) =-d_1.  \label{e6}
\end{equation}
The conditions \eqref{e6} define a unique solution of \eqref{e1} on 
$[ 0,h_1] $  \cite[p. 12]{n1}.

After defining the above solution, then we shall define the solution $
w_{i+1}( x,\mu ) $ of  \eqref{e1} on $[r_i,r_{i+1}] $ by means of 
the solution $w_i( x,\mu ) $
using the initial conditions
\begin{equation}
w_{i+1}( r_i,\mu ) =\delta _i^{-1}w_i( r_i,\mu
) \text{and}\quad w_{i+1}'( r_i,\mu )
=\delta _i^{-1}w_i'( r_i,\mu ) ,\text{ }i=
\overline{2,m-1}  \label{e7}
\end{equation}
The conditions \eqref{e7} define a unique solution of \eqref{e1} on
 $[ r_i,r_{i+1}]$.

Continuing in this manner we may define the solution 
$w_{m+1}( x,\mu) $ of \eqref{e1} on $[ r_m,\pi ] $ by means
of the solution $w_m( x,\mu ) $ using the initial conditions
\begin{equation}
w_{m+1}( r_m,\mu ) 
=\delta _m^{-1}w_m( r_m,\mu) \text{and}\quad w_{m+1}'( r_m,\mu )
=\delta _m^{-1}w_m'( r_m,\mu ) .
\label{e8}
\end{equation}
The conditions \eqref{e8} define a unique solution of \eqref{e1} 
on $[ r_m,\pi ] $. 

Consequently, the function $w( x,\mu ) $ is defined on 
$[0,r_1) \cup ( r_1,r_2) \cup \dots \cup ( r_m,\pi] $ by the equality
\begin{equation*}
w(x,\lambda )=\begin{cases}
w_1(x,\mu ), & x\in [ 0,r_1), \\
w_i( x,\mu ) , & x\in ( r_i,r_{i+1}) ,\;i= \overline{2,m-1}, \\
w_{m+1}( x,\mu ) , & x\in ( r_m,\pi ]
\end{cases}
\end{equation*}
is a solution of  \eqref{e1} on $[ 0,r_1) \cup
( r_1,r_2) \cup \dots \cup ( r_m,\pi ] $; which
satisfies one of the boundary conditions and transmission conditions.

\begin{lemma} \label{lem1.1}
Let $w( x,\mu ) $ be a solution of \eqref{e1}. Then
the following integral equations hold:
\begin{gather}
\begin{aligned}
w_1(x,\mu ) 
& =d_2\cos \mu x-\frac{d_1}{\mu }\sin \mu x  \\
&\quad -\frac{1}{\mu }\int_{0}^{{x}}q( \tau ) 
\sin \mu (x-\tau ) w_1( \tau -\Delta ( \tau ) ,\mu )\, \,d\tau ,
\end{aligned} \label{e9}\\
\begin{aligned}
w_{i+1}(x,\mu )
& =\frac{1}{\delta _i}w_i( r_i,\mu ) \cos
\mu ( x-r_i) +\frac{w_i'( r_i,\lambda )
}{\mu \delta _i}\sin \mu ( x-r_i)   \\
&\quad  -\frac{1}{\mu }\int_{r_i}^{{x}}q( \tau ) \sin s(
x-\tau ) w_{i+1}( \tau -\Delta ( \tau ) ,\mu )\,d\tau ,
\end{aligned}  \label{e10}
\end{gather}
\end{lemma}

\begin{proof}
To prove this lemma, it suffices to substitute
 $-\mu ^2w_1(\tau ,\mu)-w_1''(\tau ,\mu )$ and
$-\mu ^2w_{i+1}(\tau ,\mu )-w_{i+1}''(\tau ,\mu )$ by
$-q(\tau )w_1(\tau -\Delta (\tau ),\mu )$ and 
$-q(\tau )w_{i+1}(\tau -\Delta (\tau ),\mu )$
in the integrals in \eqref{e9}, \eqref{e10} respectively, and then
integrate by parts twice.
\end{proof}

\section{An existence theorem}

In this chapter, we show that the characteristic function of the problem
\eqref{e1}--\eqref{e5} has an infinite set of roots.

\begin{theorem} \label{thm2.1}
Problem \eqref{e1}-\eqref{e5} can have only simple
eigenvalues.
\end{theorem}

\begin{proof}
Let $\widetilde{\mu }$ be an eigenvalue of  \eqref{e1}-\eqref{e5} and
\begin{equation*}
\widetilde{y}(x,\widetilde{\mu })
=\begin{cases}
\widetilde{y}_1(x,\widetilde{\mu }), & x\in [ 0,r_1), \\
\dots  \\
\widetilde{y}_{m+1}(x,\widetilde{\mu }), & x\in ( r_m,\pi ]
\end{cases}
\end{equation*}
be a corresponding eigenfunction. Then, from \eqref{e2} and 
\eqref{e6}, it follows that the determinant
\begin{equation*}
W[ \widetilde{y}_1(0,\widetilde{\mu }),w_1(0,\widetilde{\mu })] 
=\begin{vmatrix}
\widetilde{y}_1(0,\widetilde{\mu }) &d_2 \\
\widetilde{y}_1'(0,\widetilde{\mu }) &-d_1
\end{vmatrix}
 =0,
\end{equation*}
and the functions $\widetilde{y}_1(x,\widetilde{\mu })$ and 
$w_1(x,\widetilde{\mu })$ are linearly dependent on $[ 0,r_1] $. 
We can also prove that the functions $\widetilde{y}_{i+1}(x,\widetilde{\mu })$
and $w_{i+1}(x,\widetilde{\mu })$ are linearly dependent on 
$[r_i,r_{i+1}] $, $i=\overline{2,m-1}$ and 
$\widetilde{y}_{m+1}(x,\widetilde{\mu })$ and 
$w_{m+1}(x,\widetilde{\mu })$ are linearly dependent on $[ r_m,\pi ] $. Hence
\begin{equation}
\widetilde{y}_i(x,\widetilde{\mu })=K_iw_i(x,\widetilde{\mu })\quad
( i=\overline{1,m+1})  \label{e11}
\end{equation}
for some $K_i\neq 0$. We must show that $K_i=K_{i+1}$. From the
equalities \eqref{e4} and \eqref{e11}, we have
\begin{align*}
\widetilde{y}(r_i-0,\widetilde{\mu })-\delta _i\widetilde{y}(r_i+0,
\widetilde{\mu })& =\widetilde{y_i}(r_i,\widetilde{\mu })-\delta _i
\widetilde{y_{i+1}}(r_i,\widetilde{\mu }) \\
& =K_iw_i(r_i,\widetilde{\mu })-\delta _iK_{i+1}w_{i+1}(r_i,
\widetilde{\mu }) \\
& =K_i\delta _iw_{i+1}(h_i,\widetilde{\mu })-K_{i+1}\delta
_iw_{i+1}(h_i,\widetilde{\mu }) \\
& =\delta _i( K_i-K_{i+1}) w_{i+1}(h_i,\widetilde{\mu })=0.
\end{align*}
Since $\delta _i( K_i-K_{i+1}) \neq 0$ it follows that
\begin{equation}
w_{i+1}( r_i,\widetilde{\mu }) =0.  \label{e12}
\end{equation}
By the same procedure from equality \eqref{e5} we can derive that
\begin{equation}
w_{i+1}'( r_i,\widetilde{\mu }) =0.
\label{e13}
\end{equation}
From the fact that $w_i(x,\widetilde{\mu })$ is a solution of the
differential \eqref{e1} on $[ r_i,r_{i+1}] $ and
satisfies the initial conditions \eqref{e12} and \eqref{e13}
it follows that $w_{i+1}(x,\widetilde{\mu })=0$ identically on 
$[r_i,\pi ] $.

By using this method, we may also find
\begin{equation*}
w_{m+1}( r_i,\widetilde{\mu }) =w_{m+1}'( r_i,\widetilde{\mu }) =0.
\end{equation*}
From the latter discussions of $w_{m+1}(x,\widetilde{\mu })$ it follows that
$w_m(x,\widetilde{\mu })=0$, $w_i(x,\widetilde{\mu })=0$, 
$w_1(x,\widetilde{\mu })=0$ identically on $( r_{m-1},r_m) $, 
$(r_{i-1},r_i) $ and $[ 0,r_1) $. But this contradicts 
\eqref{e6}, thus completing the proof.
\end{proof}

The function $w(x, \mu ) $ is defined in introduction is a nontrivial
solution of  \eqref{e1} satisfying conditions \eqref{e2}
and \eqref{e4}-\eqref{e5}. Putting $w(x,\mu ) $ into \eqref{e3},
we get the characteristic equation
\begin{equation}
H(\mu )\equiv w'(\pi ,\mu )+\mu ^2w(\pi ,\mu )=0. \label{e14}
\end{equation}

By Theorem \ref{thm2.1} the set of eigenvalues of boundary-value problem 
\eqref{e1}-\eqref{e5} coincides with the set of real roots of 
\eqref{e17}. Let
\begin{equation*}
 q_1=\int_{0}^{r_1}|q(\tau )|\,d\tau, \quad
q_i=\int_{r_{i-1}}^{ri}| q(\tau )| \,d\tau, \quad
q_{m+1}=\int_{r_m}^{{\pi }}| q(\tau )|\,d\tau ,  i=\overline{2,m}
\end{equation*}

\begin{lemma} \label{lem2.2}
(1) Let $\mu \geq 2q_1$. Then for the solution $w_1( x,\mu ) $
of  \eqref{e11}, the following inequality holds:
\begin{equation}
| w_1( x,\mu ) | \leq \frac{1}{q_1}\sqrt{4q_1^2d_2^2+d_1^2},\quad
x\in [ 0,r_1].  \label{e15}
\end{equation}
(2) Let $\mu \geq \max \{ 2q_1,2q_2,\dots ,2q_{m+1}\} $. 
Then for the solution $w_{i+1}( x,\mu ) $ $(i=\overline{1,m})$ of  
\eqref{e12}, the following inequality holds:
\begin{equation}
| w_{i+1}( x,\mu ) | \leq \frac{4^{i}}{q_1\prod_{j=1}^{i}| \delta _{j}| }
\sqrt{4q_1^2d_2^2+d_1^2},\quad x\in [ r_1,r_2].  \label{e16}
\end{equation}
\end{lemma}

The proof of the above lemma  is similar to that of \cite[Lemma 2]{s3}.

\begin{theorem} \label{thm2.3}
Problem \eqref{e1}-\eqref{e5} has an infinite set of
positive eigenvalues.
\end{theorem}

\begin{proof}
We readily see that
\begin{equation}
\begin{aligned}
\frac{\partial }{\partial x}w_{i+1}(x,\mu )
&=-\frac{\mu }{\delta _i}
w_i( r_i,\mu ) \sin \mu ( x-r_i) +\frac{\frac{
\partial }{\partial x}w_{i+1}( r_i,\mu ) }{\delta _i}\cos \mu
( x-r_i)    \\
&\quad -\int_{r_i}^{{x}}q(\tau )\cos \mu ( x-\tau )
w_{i+1}(\tau -\Delta ( \tau ) ,\mu )\,d\tau .
\end{aligned}\label{e17}
\end{equation}
Let $ \mu  $ be sufficiently big. With the helps of \eqref{e8},
\eqref{e9}), \eqref{e16}, \eqref{e17}, \eqref{e14} and \eqref{e15},
Equation \eqref{e17} can be reduced to the form
\begin{equation}
\mu \cos \mu \pi +O(1)=0.  \label{e18}
\end{equation}
Obviously, for big $ \mu $, \eqref{e18} has an infinite set
of roots. Thus, the proof of theorem is complete.
\end{proof}

\section{Asymptotic formulas for eigenvalues and eigenfunctions}

Now we begin to study asymptotic properties of eigenvalues and
eigenfunctions. In the following we shall assume that $\mu$ is
sufficiently big. From \eqref{e9} and \eqref{e15}, we obtain
\begin{equation}
w_1(x, \mu )=O(1)\quad \text{on}\quad [ 0, r_1].
\label{e19}
\end{equation}
Equations \eqref{e10} and \eqref{e16}, lead to
\begin{gather}
w_{i+1}(x, \mu ) =O(1),\quad (i=\overline{1,m-1})\quad \text{on }
[ r_i, r_{i+1}].  \label{e20} \\
w_{m+1}(x, \mu ) =O(1)\quad \text{on } [ r_m, \pi ].
\label{e21}
\end{gather}
The existence and continuity of the derivatives
 $\frac{\partial }{\partial \mu }w_1(x, \mu ) $ for
 $ 0\leq x\leq r_1, |\mu |<\infty $,
$\frac{\partial }{\partial \mu }w_{i+1}(x, \mu ) $ for 
$ r_i\leq x\leq r_{i+1}$ $(i=\overline{1,m-1}), |\mu |<\infty $ and 
$\frac{\partial }{\partial \mu }w_{m+1}(x, \mu ) $ for 
$ r_m\leq x\leq \pi$, $|\mu|<\infty $ follows from 
\cite[Theorem 1.4.1]{n1}.

\begin{lemma} \label{lem3.1}
The following statements hold:
\begin{gather}
\frac{\partial }{\partial \mu }w_1(x,\mu ) =O(1),\quad x\in [0,r_1],  \label{e22} \\
\frac{\partial }{\partial \mu }w_{i+1}(x,\mu ) =O(1),\quad 
(i=\overline{1,m-1})\; x\in [ r_i,r_{i+1}],  \label{e23} \\
\frac{\partial }{\partial \mu }w_{m+1}(x,\mu ) =O(1),\quad x\in [ r_m,\pi ].  \label{e24}
\end{gather}
\end{lemma}

\begin{proof}
By differentiating \eqref{e9} with respect to $\mu$, we get, by 
\eqref{e19}-\eqref{e21}
\begin{equation}
\begin{aligned}
\frac{\partial }{\partial \mu }w_{m+1}(x,\mu )
&=-\frac{1}{\mu }\int_{r_m}^{x}q(\tau )
 \sin \mu (x-\tau )\frac{\partial }{\partial
\mu }w_{m+1}( \tau -\Delta ( \tau ) ,\mu ) \\
&\quad +R(x,\mu ),\quad (| R(x,\mu )| \leq &R_{0}).
\end{aligned} \label{e25}
\end{equation}
Let $D_{\mu }=\max_{[r_m,\pi ]}| \frac{\partial }{\partial \mu }
w_{m+1}(x,\mu )| $.
Then the existence of $D_{\mu }$ follows from
continuity of derivation for $x\in [ r_m,\pi ]$. From
\eqref{e25}
\begin{equation*}
D_{\mu }\leq \frac{1}{\mu }q_{m+1}D_{\mu }+R_{0}.
\end{equation*}
Now let $\mu \geq 2q_{m+1}$. Then $D_{\mu }\leq 2R_{0}$ and the
validity of the asymptotic formula \eqref{e24} follows.
 Formulas \eqref{e22} and \eqref{e23} may be proved analogically.
\end{proof}

\begin{theorem} \label{thm3.2}
Let $N$ be a natural number. For each sufficiently big $N$ there is exactly
one eigenvalue of the problem \eqref{e1}-\eqref{e5} near $N^2$.
\end{theorem}

\begin{proof}
We consider the expression which is denoted by $ O(1)$ in \eqref{e18}.
 If formulas \eqref{e19}-\eqref{e24} are
taken into consideration, it can be shown by differentiation with respect to
$ \mu $ that for big $\mu $ this expression has bounded derivative.
We shall show that, for big $ N$, only one root \eqref{e18} lies
near to each $N$. We consider the function 
$ \phi (\mu )=\mu \cos \mu \pi +O(1)$. Its derivative, which has the form 
$ \frac{\partial }{\partial \mu } \phi (\mu )
=\cos \mu \pi -\mu \pi \sin \mu \pi +O(1)$, does not vanish for 
$ \mu  $ close to$ N $for sufficiently big$ $ $N$. Thus our assertion
follows by Rolle's Theorem.
\end{proof}

Let $ N $ be sufficiently big. In what follows we shall denote by 
$\mu_n^2$ the eigenvalue of the problem \eqref{e1}-\eqref{e5}
 situated near $N^2$. We set $\mu_N=N+\frac{1}{2}+\delta_N$. Then
from \eqref{e18} it follows that $\delta_N=O( \frac{1}{N}) $. Consequently
\begin{equation}
\mu_N=N+\frac{1}{2}+O\bigl ({\frac{1}{N}}\bigr ),  \label{e26}
\end{equation}
Formula \eqref{e26} make it possible to obtain asymptotic
expressions for eigenfunction of the problem \eqref{e1}-\eqref{e5}. 
From \eqref{e9}, \eqref{e19}, we get
\begin{equation}
w_1(x, \mu )=d_2\cos \mu x+O\bigl ({\frac{1}{\mu }}\bigr ).
\label{e27}
\end{equation}
From expressions of \eqref{e10}, \eqref{e23}, \eqref{e27}, we easily see that
\begin{equation}
w_{i+1}(x, \mu )={\frac{d_2}{\prod_{j=1}^{i}\delta _{j}}}\cos \mu
x+O\bigl ({\frac{1}{\mu }}\bigr ),\quad ( i=\overline{1,m}) .
\label{e28}
\end{equation}
By substituting \eqref{e26} in \eqref{e27} and \eqref{e28}, we find that
\begin{gather*}
U_{1N} =w_1( x,\mu_N) =d_2\cos \big( ( N+\frac{1}{2}
) x\big) +O\bigl({\frac{1}{N}}\bigr), \\
U_{( i+1) N} =w_{i+1}( x,\mu_N) 
={\frac{{d}_2}{\prod_{j=1}^{i}\delta _{j}}}\cos \big( ( N+\frac{1}{2})
x\big) +O\bigl ({\frac{1}{N}}\bigr ),\text{ }( i=\overline{1,m}) .
\end{gather*}
Under some additional conditions the more exact asymptotic formulas which
depend upon the retardation may be obtained. Let us assume that the
following conditions are fulfilled:

(a) The derivatives $q'(x)$ and $\Delta ''(x)$ exist
and are bounded in $[ 0,r_1) \cup ( r_1,r_2)
\cup \dots \cup ( r_m,\pi ] $ and have finite limits 
$q'( r_i\pm 0) =\lim_{x\to r_i\pm 0}q'(x)$,
and $\Delta ''( r_i\pm 0) =\lim_{x\to r_i\pm 0}\Delta ''(x)$ 
$( i=\overline{1,m}) $.

(b) $\Delta '( x) \leq 1$ in $[ 0,r_1) \cup( r_1,r_2) \cup \dots 
\cup ( r_m,\pi ] $, $\Delta ( 0) =0$, $\lim_{x\to h_1+0}\Delta ( x) =0$
and $\lim_{x\to r_i+0}\Delta ( x) =0$ $( i=\overline{1,m}) $.

It is easy to see that, using (b)
\begin{gather}
x-\Delta ( x)  \geq 0,\text{ }x\in [ 0,r_1) ,\label{e29} \\
x-\Delta ( x)  \geq r_i,\text{ }x\in (r_i,r_{i+1}) \quad (i=\overline{1,m-1}) ,
\label{e30}\\
x-\Delta ( x) \geq r_m,\quad x\in ( r_m,\pi ]\label{e31}
\end{gather}
are obtained.
By \eqref{e27}-\eqref{e31}, we have
\begin{gather}
w_1( \tau -\Delta ( \tau ) ,\mu ) 
=d_2\cos \mu ( \tau -\Delta ( \tau ) ) +O( \frac{1}{\mu }) ,  \label{e32} \\
w_{i+1}( \tau -\Delta ( \tau ) ,\mu ) 
=\frac{d_2}{\prod_{j=1}^{i}\delta _{j}}\cos \mu ( \tau -\Delta ( \tau
) ) +O( \frac{1}{\mu })  \label{e33}
\end{gather}
on $[ 0,r_1)$, $( r_i,r_{i+1}) $ 
$( i=\overline{1,m-1}) $ and $( r_m,\pi ] $ respectively.

Under conditions (a) and (b) the following two formulas
\begin{equation}
\begin{gathered}
\int_{0}^{x}q( \tau ) \cos \mu ( 2\tau -\Delta ( \tau
) ) \,d\tau =O(1\mu) , \\
\int_{0}^{x}q( \tau ) \sin \mu ( 2\tau -\Delta ( \tau
) ) \,d\tau =O(1/\mu )
\end{gathered} \label{e34}
\end{equation}
can be proved by the same technique in \cite[Lemma 3.3.3]{n1}.

Using \eqref{e32}, \eqref{e33} and \eqref{e34},
after long operations we have
\begin{align*}
&-\frac{d_1+d_2}{\prod_{j=1}^{m}\delta _{j}}\sin \mu \pi +\frac{
\mu d_2}{\prod_{j=1}^{m}\delta _{j}}\cos \mu \pi -\frac{d_2\sin
\mu \pi }{2\prod_{j=1}^{m}\delta _{j}}\int_{0}^{\pi }q( \tau
) \cos \mu \Delta ( \tau ) \,d\tau \\
&+\frac{d_2\cos \mu \pi }{2\prod_{j=1}^{m}\delta _{j}}\int_{0}^{\pi
}q( \tau ) \sin \mu \Delta ( \tau ) \,d\tau +O(
\frac{1}{\mu }) =0.
\end{align*}
Again, if we take $\mu_N=N+\frac{1}{2}+\delta_N$,  for
sufficiently big $N$, we obtain
\begin{equation*}
\delta_N=\frac{1}{( N+\frac{1}{2}) \pi }\Big( \frac{d_1}{
d_2}-1-\frac{1}{2}\int_{0}^{\pi }q( \tau ) \cos \big( ( N+
\frac{1}{2}) \Delta ( \tau ) \big) \,d\tau \Big) +O(1/N^2)
\end{equation*}
and finally
\begin{equation}
\mu_N=N+\frac{1}{2}+\frac{1}{( N+\frac{1}{2}) \pi }(
\frac{d_1}{d_2}-1-\frac{1}{2}\int_{0}^{\pi }q( \tau ) \cos
\big( ( N+\frac{1}{2}) \Delta ( \tau ) \big) \,d\tau
) +O(1/N^2) .  \label{e35}
\end{equation}
Thus, we have proven the following theorem.

\begin{theorem} \label{thm3.3}
If conditions (a) and (b) are satisfied then, the eigenvalues $\mu_N$ of
the problem \eqref{e1}-\eqref{e5} have the \eqref{e35}
 asymptotic formula for $N\to \infty $.
\end{theorem}

Now, we may obtain sharper asymptotic formulas for the eigenfunctions. 
From \eqref{e9}), \eqref{e32}, \eqref{e34} and replacing 
$\mu $ by $\mu_N$ we have
\begin{align*}
u_{1N}(x)
&=d_2\Big\{ \frac{\sin ( ( N+\frac{1}{2})
x) }{N\pi }\Big[ \Big( \frac{d_1}{d_2}+\frac{1}{2}
\int_{0}^{x}q( \tau ) \cos ( ( N+\frac{1}{2})
\Delta ( \tau ) ) \,d\tau \Big) \pi \\
&\quad +\Big( \frac{d_1}{d_2}-1-\frac{1}{2}\int_{0}^{\pi }
 q( \tau) \cos \Big( ( N+\frac{1}{2}) \Delta ( \tau )
\Big) \,d\tau \Big) x\Big] \\
&\quad  +\cos \big( ( N+\frac{1}{2}) x\big) \Big[ 1+\frac{1}{2N
}\int_{0}^{x}q( \tau ) \sin \big( ( N+\frac{1}{2})
\Delta ( \tau ) \big) \,d\tau \Big] \Big\} +O(1/N^2) .
\end{align*}
From \eqref{e10}, \eqref{e33} and \eqref{e34}, and
replacing $\mu $ by $\mu_N$ we have
\begin{align*}
u_{( i+1) N}(x)
&=\frac{d_2}{\prod_{j=1}^{i}\delta _{j}}
\Big\{ \cos \big( ( N+\frac{1}{2}) x\big) 
\Big[ 1+\frac{1}{2N} \int_{0}^{x}q( \tau ) \sin \big( ( N+\frac{1}{2})
\Delta ( \tau ) \big) \,d\tau \Big]  \\
&\quad 
+\frac{\sin \big( ( N+\frac{1}{2}) x\big) }{N\pi }
\Big[\Big( \frac{d_1}{d_2}-1-\frac{1}{2}\int_{0}^{\pi }q( \tau )
\cos \big( ( N+\frac{1}{2}) \Delta ( \tau ) \big)\,d\tau \Big) x \\
&\quad  -\Big( \frac{d_1}{d_2}+\frac{1}{2}\int_{0}^{x}q(
\tau ) \cos \big( ( N+\frac{1}{2}) \Delta ( \tau
) \big) \,d\tau \Big) \pi \Big] \Big\} +O(1/N^2) .
\end{align*}

\begin{thebibliography}{00}

\bibitem{a1} F. A. Akg\"{u}n, A. Bayramov, M. Bayramoglu;
\emph{Discontinuous boundary value problems with retarded argument and
eigenparameter dependent boundary conditions}, Mediterr. J. Math., 10 (1) (2013),
 277-288.

\bibitem{a2} T. Aktosun, D. Gintides, V. G. Papanicolaou;
\emph{The uniqueness in the inverse problem for transmission eigenvalues
for the spherically symmetric variable-speed wave equation},
Inverse Probl., 27 (115004) (2011), 1-17.

\bibitem{a3} K. Aydemir, O. S. Mukhtarov;
\emph{Asymptotic distribution of eigenvalues and eigenfunctions for a
 multi-point discontinuous Sturm-Liouville problem},
Electron. J. Differential Equations (EJDE), 2016
(131) (2016), 1-14.

\bibitem{b1} M. Bayramoglu, A. Bayramov, E. \c{S}en;
\emph{A regularized trace formula for a discontinuous Sturm-Liouville
operator with delayed argument},
Electron. J. Differential Equations (EJDE), 2017 (104) (2017). 1-12.

\bibitem{b2} A. Bayramov, S. \d{C}al\i\d{s}kan, S. Uslu;
 \emph{Computation of eigenvalues and eigenfunctions of a discontinuous
 boundary value problem with retarded argument}, Appl. Math. Comput., 
191 (2007), 592-600.

\bibitem{b3} S. A. Buterin, C.-F. Yang;
\emph{On an inverse transmission problem from complex eigenvalues},
Results Math., 71 (3-4) (2017), 859-866.

\bibitem{c1} F. A. Cetinkaya, K. R. Mamedov;
\emph{A boundary value problem with retarded argument and discontinuous
 coefficient in the differential equation}, Azerb. J. Math., 7 (1) (2017), 130-140.

\bibitem{c3} D. Colton, Y-J. Leung, S. Meng;
\emph{Distribution of complex transmission eigenvalues for spherically
symmetric stratified media}, Inverse Probl., 31 (035006) (2015), 1-19.

\bibitem{f1} C. T. Fulton;
\emph{Two-point boundary value problems with
eigenvalue parameter contained in the boundary conditions}, Proc. Roy. Soc.
Edinburgh, A 77 (1977), 293-308.

\bibitem{h1} F. Hira;
\emph{A trace formula for the Sturm-Liouville type
equation with retarded argument}, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math.
Stat., 66 (1) (2017), 124-132.

\bibitem{k1} M. Kadakal, O. S. Mukhtarov;
\emph{Sturm-Liouville problems with discontinuities at two points},
 Computers and Mathematics with Applications 54 (2007), 1367-1379.

\bibitem{l1} B. M. Levitan;
 Expansion in characteristic functions of
differential equations of the second order, GITTL, Moscow, 1950. (Russian)

\bibitem{l2} B. M. Levitan, I. S. Sargsyan;
Sturm-Liouville and Dirac operators, Springer-Verlag, New-York, 1991.

\bibitem{m1} A. S. Makin;
\emph{Asymptotics of the spectrum of the Sturm-Liouville operator with
regular boundary conditions}, Differential Equations, 44 (5) (2008), 645--658.

\bibitem{m2} O. S. Mukhtarov, M. Kadakal, F. S. Muhtarov;
 \emph{On discontinuous Sturm-Liouville problems with transmission conditions},
J. Math. Kyoto Univ., 44 (4) (2004), 779-798.

\bibitem{n1} S. B. Norkin;
 Differential equations of the second order with retarded argument,
Translations of Mathematical Monographs, AMS, Providence, RI, 1972.

\bibitem{s1} E. \c{S}en, M. Acikgoz, S. Araci;
\emph{Spectral problem for Sturm-Liouville operator with retarded argument
which contains a spectral parameter in the boundary condition},
Ukrainian Mathematical Journal, 68 (8) (2017), 1263-1277.

\bibitem{s2} E. \c{S}en, A. Bayramov;
\emph{Spectral analysis of boundary value problems with retarded argument},
 Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66 (2) (2017), 175-194.

\bibitem{s3} E. \c{S}en, A. Bayramov;
\emph{Calculation of eigenvalues and eigenfunctions of a discontinuous boundary 
value problem with retarded argument which contains a spectral parameter 
in the boundary condition},
Mathematical and Computer Modelling, 54 (11-12) (2011), 3090-3097.

\bibitem{s4} E. \c{S}en, O. S. Mukhtarov;
\emph{Spectral properties of discontinuous Sturm--Liouville problems with a
 finite number of transmission conditions}, Mediterr. J. Math., 13 (1) (2016),
 153-170.

\bibitem{t1} E. C. Titchmarsh;
Eigenfunctions expansion associated with second order differential equation
1, 2nd ed., Oxford University Press, London, 1962.

\bibitem{t2} A. N. Tikhonov, A. A. Samarskii;
Equations of Mathematical Physics, Pergamon press, London, 1963.

\bibitem{y1} C.-F. Yang;
\emph{Trace and inverse problem of a discontinuous
Sturm-Liouville operator with retarded argument}, 
J. Math. Anal. Appl., 395 (2012), 30-41.

\end{thebibliography}

\end{document}