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 [1–5], and various physical applications of such problems can be found in [2].

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 [5].

The asymptotic formulas for the eigenvalues and eigenfunctions of Sturm-Liouville problem with the spectral parameter in the boundary condition were obtained in [6].

In the articles [7–9], 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 $q\left(\frac{\pi }{2}\pm 0\right)=\underset{x\to \frac{\pi }{2}\pm 0}{lim}q\left(x\right)$, 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 $\Delta \left(\frac{\pi }{2}\pm 0\right)=\underset{x\to \frac{\pi }{2}\pm 0}{lim}\Delta \left(x\right),x-\Delta \left(x\right)\ge 0$, 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 [10].

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

$$w_2\left(\frac{\pi }{2},\lambda \right)={\gamma }_1{\delta }_1^{-1}{w}_1\left(\frac{\pi }{2},\lambda \right),{\omega }_2^{\prime }\left(\frac{\pi }{2},\lambda \right)={\gamma }_2{\delta }_2^{-1}{\omega }_1^{\prime }\left(\frac{\pi }{2},\lambda \right).$$

(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:

$$\begin{array}{ll}\hfill w_1\left(x,\lambda \right)& =-\frac{p_1}{s}sin\frac{s}{p_1}x\\ -\frac{1}{s}\underset{0}{\overset{x}{\int }}\frac{q\left(\tau \right)}{p_1}sin\frac{s}{p_1}\left(x-\tau \right)w_1\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \left(s=\sqrt{\lambda },\lambda >0\right),\end{array}$$

(8)

$$\begin{array}{ll}\hfill w_2\left(x,\lambda \right)& =\frac{{\gamma }_1}{{\delta }_1}{w}_1\left(\frac{\pi }{2},\lambda \right)cos\frac{s}{p_2}\left(x-\frac{\pi }{2}\right)+\frac{{\gamma }_2{p}_2{w^{\prime }}_1\left(\frac{\pi }{2},\lambda \right)}{s{\delta }_2}sin\frac{s}{p_2}\left(x-\frac{\pi }{2}\right)\\ -\frac{1}{s}\underset{\pi ∕2}{\overset{x}{\int }}\frac{q\left(\tau \right)}{p_2}sin\frac{s}{p_2}\left(x-\tau \right)w_2\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \left(s=\sqrt{\lambda },\lambda >0\right).\end{array}$$

(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 $\tilde{\lambda }$ be an eigenvalue of the problem (1)-(5) and

$$ũ\left(x,\tilde{\lambda }\right)=\left\{\begin{array}{cc}\hfill {ũ}_1\left(x,\tilde{\lambda }\right),\hfill & \hfill x\in \left[0,\frac{\pi }{2}\right),\hfill \\ \hfill {ũ}_2\left(x,\tilde{\lambda }\right),\hfill & \hfill x\in \left(\frac{\pi }{2},\pi \right]\hfill \end{array}\right.$$

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

$$W\left[{ũ}_1\left(0,\tilde{\lambda }\right),w_1\left(0,\tilde{\lambda }\right)\right]=\left|\begin{array}{cc}\hfill {ũ}_1\left(0,\tilde{\lambda }\right)\hfill & \hfill 0\hfill \\ \hfill {{ũ}^{\prime }}_1\left(0,\tilde{\lambda }\right)\hfill & \hfill -1\hfill \end{array}\right|=0,$$

and by Theorem 2.2.2 in [2], the functions ${ũ}_1\left(x,\tilde{\lambda }\right)$ and $w_1\left(x,\tilde{\lambda }\right)$ are linearly dependent on $\left[0,\frac{\pi }{2}\right]$. We can also prove that the functions ${ũ}_2\left(x,\tilde{\lambda }\right)$ and $w_2\left(x,\tilde{\lambda }\right)$ are linearly dependent on $\left[\frac{\pi }{2},\pi \right]$. Hence,

$${ũ}_1\left(x,\tilde{\lambda }\right)=K_i{w}_i\left(x,\tilde{\lambda }\right)\left(i=1,2\right)$$

(10)

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

$$\begin{array}{ll}\hfill {\gamma }_1ũ\left(\frac{\pi }{2}-0,\tilde{\lambda }\right)-{\delta }_1ũ\left(\frac{\pi }{2}+0,\tilde{\lambda }\right)& ={\gamma }_1{ũ}_1\left(\frac{\pi }{2},\tilde{\lambda }\right)-{\delta }_1{ũ}_2\left(\frac{\pi }{2},\tilde{\lambda }\right)\\ ={\gamma }_1{K}_1{w}_1\left(\frac{\pi }{2},\tilde{\lambda }\right)-{\delta }_1{K}_2{w}_2\left(\frac{\pi }{2},\tilde{\lambda }\right)\\ ={\gamma }_1{K}_1{\delta }_1{\gamma }_1^{-1}{w}_2\left(\frac{\pi }{2},\tilde{\lambda }\right)-{\delta }_1{K}_2{w}_2\left(\frac{\pi }{2},\tilde{\lambda }\right)\\ ={\delta }_1\left(K_1-K_2\right)w_2\left(\frac{\pi }{2},\tilde{\lambda }\right)=0.\end{array}$$

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

$$w_2\left(\frac{\pi }{2},\tilde{\lambda }\right)=0.$$

(11)

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

$$w_2^{\prime }\left(\frac{\pi }{2},\tilde{\lambda }\right)=0.$$

(12)

From the fact that $w_2\left(x,\tilde{\lambda }\right)$ 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 $w_1\left(x,\tilde{\lambda }\right)=0$ identically on $\left[\frac{\pi }{2},\pi \right]$ (cf. [2, p. 12, Theorem 1.2.1]).

By using we may also find

$$w_1\left(\frac{\pi }{2},\tilde{\lambda }\right)=w_1^{\prime }\left(\frac{\pi }{2},\tilde{\lambda }\right)=0.$$

From the latter discussions of $w_2\left(x,\tilde{\lambda }\right)$, it follows that $w_1\left(x,\tilde{\lambda }\right)=0$ 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 $q_1=\frac{1}{p_1}{\int }_0^{\pi ∕2}\left|q\left(\tau \right)\right|d\tau$ and $q_2=\frac{1}{p_2}{\int }_{\pi ∕2}^{\pi }q\left(\tau \right)d\tau$.

Lemma 2. (1) Let $\lambda \ge 4q_1^2$. Then, for the solution w₁(x, λ) of Equation 8, the following inequality holds:

$$\left|w_1\left(x,\lambda \right)\right|\le \left|\frac{p_1}{q_1}\right|,x\in \left[0,\frac{\pi }{2}\right].$$

(14)

• (2) Let $\lambda \ge max\left\{4\underset{1}{\overset{2}{q}},4q_2^2\right\}$. Then, for the solution w₂ (x, λ) of Equation 9, the following inequality holds:

  $$\left|w_2\left(x,\lambda \right)\right|\le \frac{2p_1}{q_1}\left\{\left|\frac{{\gamma }_1}{{\delta }_1}\right|+\left|\frac{p_2{\gamma }_2}{p_1{\delta }_2}\right|\right\},x\in \left[\frac{\pi }{2},\pi \right].$$

  (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:

$$\begin{array}{c}{B}_{2\lambda }\le \frac{\left|p_1\right|}{q_1}\left|\frac{{\gamma }_1}{{\delta }_1}\right|+\left|p_2\right|\left|\frac{{\gamma }_2}{{\delta }_2}\right|\frac{1}{\left|q_1\right|}+\frac{1}{2q_2}{B}_{2\lambda }{q}_2,\\ B_{2\lambda }\le \frac{2\left|p_1\right|}{q_1}\left\{\left|\frac{{\gamma }_1}{{\delta }_1}\right|+\left|\frac{p_2{\gamma }_2}{p_1{\delta }_2}\right|\right\}.\end{array}$$

Hence, if $\lambda \ge max\left\{4\underset{1}{\overset{2}{q}},4q_2^2\right\}$, 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

$$\begin{array}{ll}\hfill {w^{\prime }}_2\left(x,\lambda \right)& =-\frac{s{\gamma }_1}{p_2{\delta }_1}{w}_1\left(\frac{\pi }{2},\lambda \right)sin\frac{s}{p_2}\left(x-\frac{\pi }{2}\right)+\frac{{\gamma }_2{w^{\prime }}_1\left(\frac{\pi }{2},\lambda \right)}{{\delta }_2}cos\frac{s}{p_2}\left(x-\frac{\pi }{2}\right)\\ -\frac{1}{p_2^2}\underset{\pi ∕2}{\overset{x}{\int }}q\left(\tau \right)cos\frac{s}{p_2}\left(x-\tau \right)w_2\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau .\end{array}$$

(18)

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

$$\begin{array}{c}-\frac{s{\gamma }_1}{p_2{\delta }_1}\left(-\frac{p_1}{s}sin\frac{s\pi }{2p_1}-\frac{1}{s{p}_1}\underset{0}{\overset{\frac{\pi }{2}}{\int }}q\left(\tau \right)sin\frac{s}{p_1}\left(\frac{\pi }{2}-\tau \right){\omega }_1\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \right)\\ \times sin\frac{s\pi }{2p_2}\\ +\frac{{\gamma }_2}{{\delta }_2}\left(-cos\frac{s\pi }{2p_1}-\frac{1}{p_1^2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}q\left(\tau \right)cos\frac{s}{p_1}\left(\frac{\pi }{2}-\tau \right){\omega }_1\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \right)\\ \times cos\frac{s\pi }{2p_2}-\frac{1}{p_2^2}\underset{\pi ∕2}{\overset{\pi }{\int }}q\left(\tau \right)cos\frac{s}{p_2}\left(\pi -\tau \right){\omega }_2\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \\ +\lambda \left(\frac{{\gamma }_1}{{\delta }_1}\left[-\frac{p_1}{s}sin\frac{s\pi }{2p_1}-\frac{1}{s{p}_1}\underset{0}{\overset{\frac{\pi }{2}}{\int }}q\left(\tau \right)sin\frac{s}{p_1}\left(\frac{\pi }{2}-\tau \right){\omega }_1\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \right]\right.\\ \times cos\frac{s\pi }{2p_2}\\ +\frac{{\gamma }_2{p}_2}{{\delta }_{2}s}\left[-cos\frac{s\pi }{2p_1}-\frac{1}{p_1^2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}q\left(\tau \right)cos\frac{s}{p_1}\left(\frac{\pi }{2}-\tau \right){\omega }_1\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \right]\\ \left.\times sin\frac{s\pi }{2p_2}-\frac{1}{s{p}_2}\underset{\frac{\pi }{2}}{\overset{\pi }{\int }}q\left(\tau \right)sin\frac{s}{p_2}\left(\pi -\tau \right){\omega }_2\left(\tau -\Delta \left(\tau \right),\lambda \right)d\tau \right)=0.\end{array}$$

(19)

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

$$ssins\pi \frac{p_1+p_2}{2p_1{p}_2}+O\left(1\right)=0.$$

(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

$${\omega }_1\left(x,\lambda \right)=O\left(1\right)\mathsf{\text{on}}\left[0,\frac{\pi }{2}\right].$$

(21)

From (9) and (15), we get

$${\omega }_2\left(x,\lambda \right)=O\left(1\right)\mathsf{\text{on}}\left[\frac{\pi }{2},\pi \right].$$

(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 [?].

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

(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 $\frac{4n^2{p}_1^2{p}_2^2}{{\left(p_1+p_2\right)}^2}$. We consider the function $\varphi \left(s\right)=sins\pi \frac{p_1+p_2}{2p_1{p}_2}+O\left(1\right)$. Its derivative, which has the form ${\varphi }^{\prime }\left(s\right)=sins\pi \frac{p_1+p_2}{2p_1{p}_2}+s\pi \frac{p_1+p_2}{2p_1{p}_2}coss\pi \frac{p_1+p_2}{2p_1{p}_2}+O\left(1\right)$, 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 $\frac{4n^2{p}_1^2{p}_2^2}{{\left(p_1+p_2\right)}^2}$. 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:

$$u_n\left(x\right)=O\left(\frac{1}{n}\right)\mathsf{\text{for}}x\in \left[0,\frac{\pi }{2}\right)\cup \left(\frac{\pi }{2},\pi \right].$$

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:

• (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 $q^{\prime }\left(\frac{\pi }{2}\pm 0\right)=\underset{x\to \frac{\pi }{2}\pm 0}{lim}{q}^{\prime }\left(x\right)$ and ${\Delta }^{″}\left(\frac{\pi }{2}\pm 0\right)=\underset{x\to \frac{\pi }{2}\pm 0}{lim}{\Delta }^{″}\left(x\right)$, respectively.

• (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

$$x-\Delta \left(x\right)\ge 0\mathsf{\text{for}}x\in \left[0,\frac{\pi }{2}\right)\mathsf{\text{and}}x-\Delta \left(x\right)\ge \frac{\pi }{2}\mathsf{\text{for}}x\in \left(\frac{\pi }{2},\pi \right].$$

(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

$$\begin{array}{c}0=\frac{{\gamma }_1{p}_1}{p_2{\delta }_1}sin\frac{s\pi }{2p_1}sin\frac{s\pi }{2p_2}-\frac{{\gamma }_2}{{\delta }_2}cos\frac{s\pi }{2p_2}-s{p}_{1}sin\frac{s\pi }{2p_1}cos\frac{2\pi }{2p_2}\\ -\frac{s{\gamma }_2{p}_2}{{\delta }_2}cos\frac{s\pi }{2p_1}sin\frac{s\pi }{2p_2}+O\left(\frac{1}{s}\right),\end{array}$$

and using γ₁δ₂p₁ = γ₂δ₁p₂ we get

$$0=\frac{{\gamma }_2}{{\delta }_2}coss\pi \frac{p_1+p_2}{2p_1{p}_2}-s{p}_{1}sins\pi \frac{p_1+p_2}{2p_1{p}_2}+O\left(\frac{1}{s}\right).$$

Dividing by s and using $s_n=\frac{2n{p}_1{p}_2}{p_1+p_2}+{\delta }_n$, we have

$$sin\left(n\pi +\frac{\pi \left(p_1+p_2\right){\delta }_n}{2p_1{p}_2}\right)=O\left(\frac{1}{n_2}\right).$$

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

$$u_{1n}\left(x\right)=\frac{p_1+p_2}{2p_{2}n}sin\frac{2p_{2}n}{p_1+p_2}x+O\left(\frac{1}{n^2}\right).$$

(34)

From (16) and (29), we have

$$\frac{{w^{\prime }}_1\left(x,\lambda \right)}{s}=-\frac{cos\frac{s}{p_1}x}{s}+O\left(\frac{1}{s^2}\right),x\in \left(0,\frac{\pi }{2}\right].$$

(35)

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

$$\begin{array}{l}{w}_2(x,\lambda )=\left\{-\frac{{\gamma }_1{p}_1\sin \frac{s\pi }{2p_1}}{s{\delta }_1}+O\left(\frac{1}{s^2}\right)\right\}\cos \frac{2}{p_2}\left(x-\frac{\pi }{2}\right)\\ -\left\{\frac{{\gamma }_2{p}_2\cos \frac{s\pi }{2p_1}}{s{\delta }_2}+O\left(\frac{1}{s^2}\right)\right\}\sin \frac{s}{p_2}\left(x-\frac{\pi }{2}\right)+O\left(\frac{1}{s^2}\right),\\ w_2(x,\lambda )=-\frac{{\gamma }_2{p}_2}{s{\delta }_2}\sin s\left(\frac{\pi (p_2-p_1}{2p_1{p}_2}+\frac{x}{2p_2}\right)+O\left(\frac{1}{s^2}\right).\end{array}$$

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 → ∞:

$$u_n\left(x\right)=\left\{\begin{array}{c}\hfill u_{1n}\left(x\right)\mathsf{\text{for}}x\in \left[0,\frac{\pi }{2}\right),\hfill \\ \hfill u_{2n}\left(x\right)\mathsf{\text{for}}x\in \left(\frac{\pi }{2},\pi \right],\hfill \end{array}\right.$$

where u_1n(x) and u_2n(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.