# On calculation of eigenvalues and eigenfunctions of a Sturm-Liouville type problem with retarded argument which contains a spectral parameter in the boundary condition

## Abstract

In this study, a discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition and with transmission conditions at the point of discontinuity is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions.

MSC (2010): 34L20; 35R10.

## 1 Introduction

Boundary-value problems for differential equations of the second order with retarded argument were studied in , and various physical applications of such problems can be found in .

The asymptotic formulas for the eigenvalues and eigenfunctions of boundary problem of Sturm-Liouville type for second order differential equation with retarded argument were obtained in .

The asymptotic formulas for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the spectral parameter in the boundary condition were obtained in .

In the articles , the asymptotic formulas for the eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions and with the boundary conditions which include spectral parameter were obtained.

In this article, we study the eigenvalues and eigenfunctions of discontinuous boundary-value problem with retarded argument and a spectral parameter in the boundary condition. Namely, we consider the boundary-value problem for the differential equation

$p\left(x\right){y}^{″}\left(x\right)+q\left(x\right)y\left(x-\Delta \left(x\right)\right)+\lambda y\left(x\right)=0$
(1)

on $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$, with boundary conditions

$y\left(0\right)=0,$
(2)
${y}^{\prime }\left(\pi \right)+\lambda y\left(\pi \right)=0,$
(3)

and transmission conditions

${\gamma }_{1}y\left(\frac{\pi }{2}-0\right)={\delta }_{1}y\left(\frac{\pi }{2}+0\right),$
(4)
${\gamma }_{2}{y}^{\prime }\left(\frac{\pi }{2}-0\right)={\delta }_{2}{y}^{\prime }\left(\frac{\pi }{2}+0\right),$
(5)

where $p\left(x\right)={p}_{1}^{2}$ if $x\in \left[0,\frac{\pi }{2}\right)$ and $p\left(x\right)={p}_{2}^{2}$ if $x\in \left(\frac{\pi }{2},\pi \right]$, the real-valued function q(x) is continuous in $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$ and has a finite limit $q\left(\frac{\pi }{2}±0\right)=\underset{x\to \frac{\pi }{2}±0}{lim}q\left(x\right)$, the real-valued function Δ(x) ≥ 0 continuous in $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$ and has a finite limit $\Delta \left(\frac{\pi }{2}±0\right)=\underset{x\to \frac{\pi }{2}±0}{lim}\Delta \left(x\right),x-\Delta \left(x\right)\ge 0$, if $x\in \left[0,\frac{\pi }{2}\right)$; $x-\Delta \left(x\right)\ge \frac{\pi }{2}$ if $x\in \left(\frac{\pi }{2},\pi \right]$; λ is a real spectral parameter; p1, p2, γ1, γ2, δ1, δ2 are arbitrary real numbers and |γ i | + |δi| ≠ 0 for i = 1, 2. Also, γ1δ2p1 = γ2δ1p2 holds.

It must be noted that some problems with transmission conditions which arise in mechanics (thermal condition problem for a thin laminated plate) were studied in .

Let w1(x, λ) be a solution of Equation 1 on $\left[0,\frac{\pi }{2}\right]$, satisfying the initial conditions

${w}_{1}\left(0,\lambda \right)=0,{w}_{1}^{\prime }\left(0,\lambda \right)=-1.$
(6)

The conditions (6) define a unique solution of Equation 1 on $\left[0,\frac{\pi }{2}\right]$ [2, p. 12].

After defining above solution, we shall define the solution w2 (x, λ) of Equation 1 on $\left[\frac{\pi }{2},\pi \right]$ by means of the solution w1(x, λ) by the initial conditions

${w}_{2}\left(\frac{\pi }{2},\lambda \right)={\gamma }_{1}{\delta }_{1}^{-1}{w}_{1}\left(\frac{\pi }{2},\lambda \right),\phantom{\rule{1em}{0ex}}{\omega }_{2}^{\prime }\left(\frac{\pi }{2},\lambda \right)={\gamma }_{2}{\delta }_{2}^{-1}{\omega }_{1}^{\prime }\left(\frac{\pi }{2},\lambda \right).$
(7)

The conditions (7) are defined as a unique solution of Equation 1 on $\left[\frac{\pi }{2},\pi \right]$.

Consequently, the function w (x, λ) is defined on $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$ by the equality

$w\left(x,\lambda \right)=\left\{\begin{array}{cc}\hfill {\omega }_{1}\left(x,\lambda \right),\hfill & \hfill x\in \left[0,\frac{\pi }{2}\right),\hfill \\ \hfill {\omega }_{2}\left(x,\lambda \right),\hfill & \hfill x\in \left(\frac{\pi }{2},\pi \right]\hfill \end{array}\right\$

is a such solution of Equation 1 on $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$; which satisfies one of the boundary conditions and both transmission conditions.

Lemma 1. Let w (x, λ) be a solution of Equation 1 and λ > 0. Then, the following integral equations hold:

$\begin{array}{ll}\hfill {w}_{1}\left(x,\lambda \right)& =-\frac{{p}_{1}}{s}sin\frac{s}{{p}_{1}}x\phantom{\rule{2em}{0ex}}\\ -\frac{1}{s}\underset{0}{\overset{x}{\int }}\frac{q\left(\tau \right)}{{p}_{1}}sin\frac{s}{{p}_{1}}\left(x-\tau \right){w}_{1}\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \phantom{\rule{1em}{0ex}}\left(s=\sqrt{\lambda },\lambda >0\right),\phantom{\rule{2em}{0ex}}\end{array}$
(8)
$\begin{array}{ll}\hfill {w}_{2}\left(x,\lambda \right)& =\frac{{\gamma }_{1}}{{\delta }_{1}}{w}_{1}\left(\frac{\pi }{2},\lambda \right)cos\frac{s}{{p}_{2}}\left(x-\frac{\pi }{2}\right)+\frac{{\gamma }_{2}{p}_{2}{{w}^{\prime }}_{1}\left(\frac{\pi }{2},\lambda \right)}{s{\delta }_{2}}sin\frac{s}{{p}_{2}}\left(x-\frac{\pi }{2}\right)\phantom{\rule{2em}{0ex}}\\ -\frac{1}{s}\underset{\pi ∕2}{\overset{x}{\int }}\frac{q\left(\tau \right)}{{p}_{2}}sin\frac{s}{{p}_{2}}\left(x-\tau \right){w}_{2}\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \phantom{\rule{1em}{0ex}}\left(s=\sqrt{\lambda },\lambda >0\right).\phantom{\rule{2em}{0ex}}\end{array}$
(9)

Proof. To prove this, it is enough to substitute $-\frac{{s}^{2}}{{p}_{1}^{2}}{\omega }_{1}\left(\tau ,\lambda \right)-{\omega }_{1}^{″}\left(\tau ,\lambda \right)$ and $-\frac{{s}^{2}}{{p}_{2}^{2}}{\omega }_{2}\left(\tau ,\lambda \right)-{\omega }_{2}^{″}\left(\tau ,\lambda \right)$ instead of $-\frac{q\left(\tau \right)}{{p}_{1}^{2}}{\omega }_{1}\left(\tau -\Delta \left(\tau \right),\lambda \right)$ and $-\frac{q\left(\tau \right)}{{p}_{2}^{2}}{\omega }_{2}\left(\tau -\Delta \left(\tau \right),\lambda \right)$ in the integrals in (8) and (9), respectively, and integrate by parts twice.

Theorem 1. The problem (1)-(5) can have only simple eigenvalues.

Proof. Let $\stackrel{̃}{\lambda }$ be an eigenvalue of the problem (1)-(5) and

$ũ\left(x,\stackrel{̃}{\lambda }\right)=\left\{\begin{array}{cc}\hfill {ũ}_{1}\left(x,\stackrel{̃}{\lambda }\right),\hfill & \hfill x\in \left[0,\frac{\pi }{2}\right),\hfill \\ \hfill {ũ}_{2}\left(x,\stackrel{̃}{\lambda }\right),\hfill & \hfill x\in \left(\frac{\pi }{2},\pi \right]\hfill \end{array}\right\$

be a corresponding eigenfunction. Then, from (2) and (6), it follows that the determinant

$W\left[{ũ}_{1}\left(0,\stackrel{̃}{\lambda }\right),{w}_{1}\left(0,\stackrel{̃}{\lambda }\right)\right]=\left|\begin{array}{cc}\hfill {ũ}_{1}\left(0,\stackrel{̃}{\lambda }\right)\hfill & \hfill 0\hfill \\ \hfill {{ũ}^{\prime }}_{1}\left(0,\stackrel{̃}{\lambda }\right)\hfill & \hfill -1\hfill \end{array}\right|=0,$

and by Theorem 2.2.2 in , the functions ${ũ}_{1}\left(x,\stackrel{̃}{\lambda }\right)$ and ${w}_{1}\left(x,\stackrel{̃}{\lambda }\right)$ are linearly dependent on $\left[0,\frac{\pi }{2}\right]$. We can also prove that the functions ${ũ}_{2}\left(x,\stackrel{̃}{\lambda }\right)$ and ${w}_{2}\left(x,\stackrel{̃}{\lambda }\right)$ are linearly dependent on $\left[\frac{\pi }{2},\pi \right]$. Hence,

${ũ}_{1}\left(x,\stackrel{̃}{\lambda }\right)={K}_{i}{w}_{i}\left(x,\stackrel{̃}{\lambda }\right)\phantom{\rule{1em}{0ex}}\left(i=1,2\right)$
(10)

for some K1 ≠ 0 and K2 ≠ 0. We must show that K1 = K2. Suppose that K1K2. From the equalities (4) and (10), we have

$\begin{array}{ll}\hfill {\gamma }_{1}ũ\left(\frac{\pi }{2}-0,\stackrel{̃}{\lambda }\right)-{\delta }_{1}ũ\left(\frac{\pi }{2}+0,\stackrel{̃}{\lambda }\right)& ={\gamma }_{1}{ũ}_{1}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)-{\delta }_{1}{ũ}_{2}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)\phantom{\rule{2em}{0ex}}\\ ={\gamma }_{1}{K}_{1}{w}_{1}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)-{\delta }_{1}{K}_{2}{w}_{2}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)\phantom{\rule{2em}{0ex}}\\ ={\gamma }_{1}{K}_{1}{\delta }_{1}{\gamma }_{1}^{-1}{w}_{2}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)-{\delta }_{1}{K}_{2}{w}_{2}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)\phantom{\rule{2em}{0ex}}\\ ={\delta }_{1}\left({K}_{1}-{K}_{2}\right){w}_{2}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)=0.\phantom{\rule{2em}{0ex}}\end{array}$

Since δ1 (K1 - K2) ≠ 0, it follows that

${w}_{2}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)=0.$
(11)

By the same procedure from equality (5), we can derive that

${w}_{2}^{\prime }\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)=0.$
(12)

From the fact that ${w}_{2}\left(x,\stackrel{̃}{\lambda }\right)$ is a solution of the differential equation (1) on $\left[\frac{\pi }{2},\pi \right]$ and satisfies the initial conditions (11) and (12) it follows that ${w}_{1}\left(x,\stackrel{̃}{\lambda }\right)=0$ identically on $\left[\frac{\pi }{2},\pi \right]$ (cf. [2, p. 12, Theorem 1.2.1]).

By using we may also find

${w}_{1}\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)={w}_{1}^{\prime }\left(\frac{\pi }{2},\stackrel{̃}{\lambda }\right)=0.$

From the latter discussions of ${w}_{2}\left(x,\stackrel{̃}{\lambda }\right)$, it follows that ${w}_{1}\left(x,\stackrel{̃}{\lambda }\right)=0$ identically on $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$. But this contradicts (6), thus completing the proof.

## 2 An existance theorem

The function ω(x, λ) defined in Section 1 is a nontrivial solution of Equation 1 satisfying conditions (2), (4) and (5). Putting ω(x, λ) into (3), we get the characteristic equation

$F\left(\lambda \right)\equiv {w}^{\prime }\left(\pi ,\lambda \right)+\lambda \omega \left(\pi ,\lambda \right)=0.$
(13)

By Theorem 1.1, the set of eigenvalues of boundary-value problem (1)-(5) coincides with the set of real roots of Equation 13. Let ${q}_{1}=\frac{1}{{p}_{1}}{\int }_{0}^{\pi ∕2}|q\left(\tau \right)|d\tau$ and ${q}_{2}=\frac{1}{{p}_{2}}{\int }_{\pi ∕2}^{\pi }q\left(\tau \right)d\tau$.

Lemma 2. (1) Let $\lambda \ge 4{q}_{1}^{2}$. Then, for the solution w1(x, λ) of Equation 8, the following inequality holds:

$\left|{w}_{1}\left(x,\lambda \right)\right|\le \left|\frac{{p}_{1}}{{q}_{1}}\right|,\phantom{\rule{1em}{0ex}}x\in \left[0,\frac{\pi }{2}\right].$
(14)
1. (2)

Let $\lambda \ge max\left\{4\underset{1}{\overset{2}{q}},4{q}_{2}^{2}\right\}$. Then, for the solution w2 (x, λ) of Equation 9, the following inequality holds:

$\left|{w}_{2}\left(x,\lambda \right)\right|\le \frac{2{p}_{1}}{{q}_{1}}\left\{\left|\frac{{\gamma }_{1}}{{\delta }_{1}}\right|+\left|\frac{{p}_{2}{\gamma }_{2}}{{p}_{1}{\delta }_{2}}\right|\right\},\phantom{\rule{1em}{0ex}}x\in \left[\frac{\pi }{2},\pi \right].$
(15)

Proof. Let ${B}_{1\lambda }=\underset{\left[0,\frac{\pi }{2}\right]}{max}\left|{w}_{1}\left(x,\lambda \right)\right|$. Then, from (8), it follows that, for every λ > 0, the following inequality holds:

${B}_{1\lambda }\le \left|\frac{{p}_{1}}{s}\right|+\frac{1}{s}{B}_{1\lambda }{q}_{1}.$

If s ≥ 2q1, we get (14). Differentiating (8) with respect to x, we have

${w}_{1}^{\prime }\left(x,\lambda \right)=-cos\frac{s}{{p}_{1}}x-\frac{1}{{p}_{1}^{2}}\underset{0}{\overset{x}{\int }}q\left(\tau \right)cos\frac{s}{{p}_{1}}\left(x-\tau \right){w}_{1}\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau .$
(16)

From (16) and (14), it follows that, for s ≥ 2q1, the following inequality holds:

$\left|{{w}^{\prime }}_{1}\left(x,\lambda \right)\right|\le \sqrt{\frac{{s}^{2}}{{p}_{1}^{2}}+1}+1.$

Hence,

$\frac{\left|{{w}^{\prime }}_{1}\left(x,\lambda \right)\right|}{s}\le \frac{1}{{q}_{1}}.$
(17)

Let ${B}_{2\lambda }=\underset{\left[\frac{\pi }{2},\pi \right]}{max}\left|{w}_{2}\left(x,\lambda \right)\right|$. Then, from (9), (14) and (17), it follows that, for s ≥ 2q1, the following inequalities holds:

$\begin{array}{c}{B}_{2\lambda }\le \frac{\left|{p}_{1}\right|}{{q}_{1}}\left|\frac{{\gamma }_{1}}{{\delta }_{1}}\right|+\left|{p}_{2}\right|\left|\frac{{\gamma }_{2}}{{\delta }_{2}}\right|\frac{1}{\left|{q}_{1}\right|}+\frac{1}{2{q}_{2}}{B}_{2\lambda }{q}_{2},\\ {B}_{2\lambda }\le \frac{2\left|{p}_{1}\right|}{{q}_{1}}\left\{\left|\frac{{\gamma }_{1}}{{\delta }_{1}}\right|+\left|\frac{{p}_{2}{\gamma }_{2}}{{p}_{1}{\delta }_{2}}\right|\right\}.\end{array}$

Hence, if $\lambda \ge max\left\{4\underset{1}{\overset{2}{q}},4{q}_{2}^{2}\right\}$, we get (15).

Theorem 2. The problem (1)-(5) has an infinite set of positive eigenvalues.

Proof. Differentiating (9) with respect to x, we get

$\begin{array}{ll}\hfill {{w}^{\prime }}_{2}\left(x,\lambda \right)& =-\frac{s{\gamma }_{1}}{{p}_{2}{\delta }_{1}}{w}_{1}\left(\frac{\pi }{2},\lambda \right)sin\frac{s}{{p}_{2}}\left(x-\frac{\pi }{2}\right)+\frac{{\gamma }_{2}{{w}^{\prime }}_{1}\left(\frac{\pi }{2},\lambda \right)}{{\delta }_{2}}cos\frac{s}{{p}_{2}}\left(x-\frac{\pi }{2}\right)\phantom{\rule{2em}{0ex}}\\ -\frac{1}{{p}_{2}^{2}}\underset{\pi ∕2}{\overset{x}{\int }}q\left(\tau \right)cos\frac{s}{{p}_{2}}\left(x-\tau \right){w}_{2}\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau .\phantom{\rule{2em}{0ex}}\end{array}$
(18)

From (8), (9), (13), (16) and (18), we get

(19)

Let λ be sufficiently large. Then, by (14) and (15), Equation 19 may be rewritten in the form

$ssins\pi \frac{{p}_{1}+{p}_{2}}{2{p}_{1}{p}_{2}}+O\left(1\right)=0.$
(20)

Obviously, for large s, Equation 20 has an infinite set of roots. Thus, the theorem is proved.

## 3 Asymptotic formulas for eigenvalues and eigenfunctions

Now, we begin to study asymptotic properties of eigenvalues and eigenfunctions. In the following, we shall assume that s is sufficiently large. From (8) and (14), we get

${\omega }_{1}\left(x,\lambda \right)=O\left(1\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{on}}\phantom{\rule{1em}{0ex}}\left[0,\frac{\pi }{2}\right].$
(21)

From (9) and (15), we get

${\omega }_{2}\left(x,\lambda \right)=O\left(1\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{on}}\phantom{\rule{1em}{0ex}}\left[\frac{\pi }{2},\pi \right].$
(22)

The existence and continuity of the derivatives ${\omega }_{1s}^{\prime }\left(x,\lambda \right)$ for $0\le x\le \frac{\pi }{2},\left|\lambda \right|<\infty$, and ${\omega }_{2s}^{\prime }\left(x,\lambda \right)$ for $\frac{\pi }{2}\le x\le \pi ,\left|\lambda \right|<\infty$, follows from Theorem 1.4.1 in [?].

${\omega }_{1s}^{\prime }\left(x,\lambda \right)=O\left(1\right),\phantom{\rule{1em}{0ex}}x\in \left[0,\frac{\pi }{2}\right]\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}{\omega }_{2s}^{\prime }\left(x,\lambda \right)=O\left(1\right),\phantom{\rule{1em}{0ex}}x\in \left[\frac{\pi }{2},\pi \right].$
(23)

Theorem 3. Let n be a natural number. For each sufficiently large n, there is exactly one eigenvalue of the problem (1)-(5) near $\frac{{p}_{1}^{2}{p}_{2}^{2}}{{\left({p}_{1}+{p}_{2}\right)}^{2}}{\left(2n+1\right)}^{2}$.

Proof. We consider the expression which is denoted by O(1) in Equation 20. If formulas (21)-(23) are taken into consideration, it can be shown by differentiation with respect to s that for large s this expression has bounded derivative. It is obvious that for large s the roots of Equation 20 are situated close to entire numbers. We shall show that, for large n, only one root (20) lies near to each $\frac{4{n}^{2}{p}_{1}^{2}{p}_{2}^{2}}{{\left({p}_{1}+{p}_{2}\right)}^{2}}$. We consider the function $\varphi \left(s\right)=sins\pi \frac{{p}_{1}+{p}_{2}}{2{p}_{1}{p}_{2}}+O\left(1\right)$. Its derivative, which has the form ${\varphi }^{\prime }\left(s\right)=sins\pi \frac{{p}_{1}+{p}_{2}}{2{p}_{1}{p}_{2}}+s\pi \frac{{p}_{1}+{p}_{2}}{2{p}_{1}{p}_{2}}coss\pi \frac{{p}_{1}+{p}_{2}}{2{p}_{1}{p}_{2}}+O\left(1\right)$, does not vanish for s close to n for sufficiently large n. Thus, our assertion follows by Rolle's Theorem.

Let n be sufficiently large. In what follows, we shall denote by ${\lambda }_{n}={s}_{n}^{2}$ the eigenvalue of the problem (1)-(5) situated near $\frac{4{n}^{2}{p}_{1}^{2}{p}_{2}^{2}}{{\left({p}_{1}+{p}_{2}\right)}^{2}}$. We set ${s}_{n}=\frac{2n{p}_{1}{p}_{2}}{{p}_{1}+{p}_{2}}+{\delta }_{n}$. From (20), it follows that ${\delta }_{n}=O\left(\frac{1}{n}\right)$. Consequently

${s}_{n}=\frac{2n{p}_{1}{p}_{2}}{{p}_{1}+{p}_{2}}+O\left(\frac{1}{n}\right).$
(24)

The formula (24) makes it possible to obtain asymptotic expressions for eigenfunction of the problem (1)-(5). From (8), (16) and (21), we get

${\omega }_{1}\left(x,\lambda \right)=O\left(\frac{1}{s}\right),$
(25)
${\omega }_{1}^{\prime }\left(x,\lambda \right)=O\left(1\right).$
(26)

From (9), (22), (25) and (26), we get

${\omega }_{2}\left(x,\lambda \right)=O\left(\frac{1}{s}\right).$
(27)

By putting (24) in (25) and (27), we derive that

$\begin{array}{c}{u}_{1n}={w}_{1}\left(x,{\lambda }_{n}\right)=O\left(\frac{1}{n}\right),\\ {u}_{2n}={w}_{2}\left(x,{\lambda }_{n}\right)=O\left(\frac{1}{n}\right).\end{array}$

Hence, the eigenfunctions u n (x) have the following asymptotic representation:

${u}_{n}\left(x\right)=O\left(\frac{1}{n}\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right].$

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:

1. (a)

The derivatives q'(x) and Δ(x) exist and are bounded in $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$ and have finite limits ${q}^{\prime }\left(\frac{\pi }{2}±0\right)=\underset{x\to \frac{\pi }{2}±0}{lim}{q}^{\prime }\left(x\right)$ and ${\Delta }^{″}\left(\frac{\pi }{2}±0\right)=\underset{x\to \frac{\pi }{2}±0}{lim}{\Delta }^{″}\left(x\right)$, respectively.

2. (b)

Δ'(x) ≤ 1 in $\left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right]$, Δ(0) = 0 and $\underset{x\to \frac{\pi }{2}+0}{lim}\Delta \left(x\right)=0$.

Using (b), we have

$x-\Delta \left(x\right)\ge 0\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,\frac{\pi }{2}\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}x-\Delta \left(x\right)\ge \frac{\pi }{2}\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left(\frac{\pi }{2},\pi \right].$
(28)

From (25), (27) and (28), we have

${w}_{1}\left(\tau -\Delta \left(\tau \right),\lambda \right)=O\left(\frac{1}{s}\right),$
(29)
${w}_{2}\left(\tau -\Delta \left(\tau \right),\lambda \right)=O\left(\frac{1}{s}\right).$
(30)

Under the conditions (a) and (b), the following formulas

$\begin{array}{c}\underset{0}{\overset{\frac{\pi }{2}}{\int }}q\left(\tau \right)sin\frac{s}{{p}_{1}}\left(\frac{\pi }{2}-\tau \right)d\tau =O\left(\frac{1}{s}\right),\\ \underset{0}{\overset{\frac{\pi }{2}}{\int }}q\left(\tau \right)cos\frac{s}{{p}_{1}}\left(\frac{\pi }{2}-\tau \right)d\tau =O\left(\frac{1}{s}\right)\end{array}$
(31)

can be proved by the same technique in Lemma 3.3.3 in [?]. Putting these expressions into (19), we have

$\begin{array}{c}0=\frac{{\gamma }_{1}{p}_{1}}{{p}_{2}{\delta }_{1}}sin\frac{s\pi }{2{p}_{1}}sin\frac{s\pi }{2{p}_{2}}-\frac{{\gamma }_{2}}{{\delta }_{2}}cos\frac{s\pi }{2{p}_{2}}-s{p}_{1}sin\frac{s\pi }{2{p}_{1}}cos\frac{2\pi }{2{p}_{2}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-\frac{s{\gamma }_{2}{p}_{2}}{{\delta }_{2}}cos\frac{s\pi }{2{p}_{1}}sin\frac{s\pi }{2{p}_{2}}+O\left(\frac{1}{s}\right),\end{array}$

and using γ1δ2p1 = γ2δ1p2 we get

$0=\frac{{\gamma }_{2}}{{\delta }_{2}}coss\pi \frac{{p}_{1}+{p}_{2}}{2{p}_{1}{p}_{2}}-s{p}_{1}sins\pi \frac{{p}_{1}+{p}_{2}}{2{p}_{1}{p}_{2}}+O\left(\frac{1}{s}\right).$

Dividing by s and using ${s}_{n}=\frac{2n{p}_{1}{p}_{2}}{{p}_{1}+{p}_{2}}+{\delta }_{n}$, we have

$sin\left(n\pi +\frac{\pi \left({p}_{1}+{p}_{2}\right){\delta }_{n}}{2{p}_{1}{p}_{2}}\right)=O\left(\frac{1}{{n}_{2}}\right).$

Hence,

${\delta }_{n}=O\left(\frac{1}{{n}^{2}}\right),$

and finally

${s}_{n}=\frac{2n{p}_{1}{p}_{2}}{{p}_{1}+{p}_{2}}+O\left(\frac{1}{{n}^{2}}\right).$
(32)

Thus, we have proven the following theorem.

Theorem 4. If conditions (a) and (b) are satisfied, then the positive eigenvalues ${\lambda }_{n}={s}_{n}^{2}$ of the problem (1)-(5) have the (32) asymptotic representation for n → ∞.

We now may obtain a sharper asymptotic formula for the eigenfunctions. From (8) and (29),

${w}_{1}\left(x,\lambda \right)=-\frac{{p}_{1}}{s}sin\frac{s}{{p}_{1}}x+O\left(\frac{1}{{s}^{2}}\right).$
(33)

Replacing s by s n and using (32), we have

${u}_{1n}\left(x\right)=\frac{{p}_{1}+{p}_{2}}{2{p}_{2}n}sin\frac{2{p}_{2}n}{{p}_{1}+{p}_{2}}x+O\left(\frac{1}{{n}^{2}}\right).$
(34)

From (16) and (29), we have

$\frac{{{w}^{\prime }}_{1}\left(x,\lambda \right)}{s}=-\frac{cos\frac{s}{{p}_{1}}x}{s}+O\left(\frac{1}{{s}^{2}}\right),\phantom{\rule{1em}{0ex}}x\in \left(0,\frac{\pi }{2}\right].$
(35)

From (9), (30), (31), (33) and (35), we have

$\begin{array}{l}{w}_{2}\left(x,\lambda \right)=\left\{-\frac{{\gamma }_{1}{p}_{1}\mathrm{sin}\frac{s\pi }{2{p}_{1}}}{s{\delta }_{1}}+O\left(\frac{1}{{s}^{2}}\right)\right\}\mathrm{cos}\frac{2}{{p}_{2}}\left(x-\frac{\pi }{2}\right)\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}-\left\{\frac{{\gamma }_{2}{p}_{2}\mathrm{cos}\frac{s\pi }{2{p}_{1}}}{s{\delta }_{2}}+O\left(\frac{1}{{s}^{2}}\right)\right\}\mathrm{sin}\frac{s}{{p}_{2}}\left(x-\frac{\pi }{2}\right)+O\left(\frac{1}{{s}^{2}}\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}{w}_{2}\left(x,\lambda \right)=-\frac{{\gamma }_{2}{p}_{2}}{s{\delta }_{2}}\mathrm{sin}s\left(\frac{\pi \left({p}_{2}-{p}_{1}}{2{p}_{1}{p}_{2}}+\frac{x}{2{p}_{2}}\right)+O\left(\frac{1}{{s}^{2}}\right).\end{array}$

Now, replacing s by s n and using (32), we have

${u}_{2n}\left(x\right)=-\frac{{\gamma }_{2}\left({p}_{1}+{p}_{2}\right)}{2n{p}_{1}{\delta }_{2}}sinn\left(\frac{\pi \left({p}_{2}-{p}_{1}\right)}{{p}_{1}+{p}_{2}}+\frac{{p}_{1}x}{{p}_{1}+{p}_{2}}\right)+O\left(\frac{1}{{n}^{2}}\right).$
(36)

Thus, we have proven the following theorem.

Theorem 5. If conditions (a) and (b) are satisfied, then the eigenfunctions u n (x) of the problem (1)-(5) have the following asymptotic representation for n → ∞:

${u}_{n}\left(x\right)=\left\{\begin{array}{c}\hfill {u}_{1n}\left(x\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left[0,\frac{\pi }{2}\right),\hfill \\ \hfill {u}_{2n}\left(x\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}x\in \left(\frac{\pi }{2},\pi \right],\hfill \end{array}\right\$

where u1n(x) and u2n(x) defined as in (34) and (36), respectively.

## 4 Conclusion

In this study, first, we obtain asymptotic formulas for eigenvalues and eigenfunctions for discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary condition. Then, under additional conditions (a) and (b) the more exact asymptotic formulas, which depend upon the retardation obtained.

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## Author information

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Correspondence to Erdoğan Şen.

### 5 Competing interests

The authors declare that they have no completing interests.

### 6 Authors' contributions

Establishment of the problem belongs to AB (advisor). ES obtained the asymptotic formulas for eigenvalues and eigenfunctions. All authors read and approved the final manuscript.

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Şen, E., Bayramov, A. On calculation of eigenvalues and eigenfunctions of a Sturm-Liouville type problem with retarded argument which contains a spectral parameter in the boundary condition. J Inequal Appl 2011, 113 (2011). https://doi.org/10.1186/1029-242X-2011-113 