1.Introduction Erdo L an F en, JongJinSeo, andSerkanAraci AsymptoticBehaviourofEigenvaluesandEigenfunctionsofaSturm-LiouvilleProblemwithRetardedArgument ResearchArticle

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Volume 2013, Article ID 306917,8pages http://dx.doi.org/10.1155/2013/306917

Research Article

Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

Erdo L an F en,

^1,2

Jong Jin Seo,

³

and Serkan Araci

⁴

1Department of Mathematics, Faculty of Arts and Science, Namik Kemal University, 59030 Tekirda˘g, Turkey

2Department of Mathematics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey

3Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

4Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey

Correspondence should be addressed to Jong Jin Seo; [email protected] Received 18 November 2012; Accepted 1 March 2013

Academic Editor: Suh-Yuh Yang

Copyright © 2013 Erdo˘gan S¸en et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients𝛿 = 𝛾 = 1and retarded argumentΔ ≡ 0in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

1. Introduction

Delay differential equations arise in many areas of mathematical modelling, for example, population dynamics (tak- ing into account the gestation times), infectious diseases (accounting for the incubation periods), physiological and pharmaceutical kinetics (modelling, for example, the body’s reaction to CO₂, and so forth, in circulating blood) and chemical kinetics (such as mixing reactants), the navigational control of ships and aircraft, and more general control problems. Also, differential equations and nonlinear differential equations have been studied by many mathematicians in several ways for a long time cf. [1–20].

Boundary value problems for differential equations of the second order with retarded argument were studied in [5–

10,13–16], and various physical applications of such problems can be found in [6].

In the papers [13–16], the asymptotic formulas for the eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument and a spectral parameter in the boundary conditions were derived. In spite of their being already a long years, these subjects are still today

enveloped in an aura of mystery within scientific community although they have penetrated numerous mathematical field.

The asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville problem with the spectral parameter in the boundary condition were obtained in [17–

20].

In this paper we study the eigenvalues and eigenfunctions of a discontinuous boundary value problem with retarded argument. Namely, we consider the boundary value problem for the differential equation

𝑦^󸀠󸀠(𝑥) + 𝑞 (𝑥) 𝑦 (𝑥 − Δ (𝑥)) + 𝜆𝑦 (𝑥) = 0, (1)

on[0, ℎ₁) ∪ (ℎ₁, ℎ₂) ∪ (ℎ₂, 𝜋], with boundary conditions

𝑦 (0)cos𝛼 + 𝑦^󸀠(0)sin𝛼 = 0, (2) 𝑦 (𝜋)cos𝛽 + 𝑦^󸀠(𝜋)sin𝛽 = 0, (3)

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and transmission conditions

𝑦 (ℎ₁− 0) − 𝛿𝑦 (ℎ₁+ 0) = 0, (4) 𝑦^󸀠(ℎ₁− 0) − 𝛿𝑦^󸀠(ℎ₁+ 0) = 0, (5) 𝑦 (ℎ₂− 0) − 𝛾𝑦 (ℎ₂+ 0) = 0, (6) 𝑦^󸀠(ℎ₂− 0) − 𝛾𝑦^󸀠(ℎ₂+ 0) = 0, (7) where the real-valued function𝑞(𝑥)is continuous in[0, ℎ₁) ∪ (ℎ₁, ℎ₂) ∪ (ℎ₂, 𝜋]and has finite limits

𝑞 (ℎ₁± 0) = lim

𝑥 → ℎ1±0𝑞 (𝑥) , 𝑞 (ℎ₂± 0) = lim

𝑥 → ℎ2±0𝑞 (𝑥) , (8) the real-valued functionΔ(𝑥) ≥ 0continuous in[0, ℎ₁) ∪ (ℎ₁, ℎ₂) ∪ (ℎ₂, 𝜋]and has finite limits

Δ (ℎ₁± 0) = lim

𝑥 → ℎ1±0Δ (𝑥) , Δ (ℎ₂± 0) = lim

𝑥 → ℎ2±0Δ (𝑥) ; (9) if 𝑥 ∈ [0, ℎ₁) then 𝑥 − Δ(𝑥) ≥ 0; if 𝑥 ∈ (ℎ₁, ℎ₂) then 𝑥 − Δ(𝑥) ≥ ℎ₁; if𝑥 ∈ (ℎ₂, 𝜋)then𝑥 − Δ(𝑥) ≥ ℎ₂; 𝜆is a real spectral parameter;ℎ₁,ℎ₂,𝛼,𝛽,𝛿,𝛾 ̸= 0are arbitrary real numbers such that0 < ℎ₁< ℎ₂< 𝜋and sin𝛼sin𝛽 ̸= 0.

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 [20].

Let𝑤₁(𝑥, 𝜆)be a solution of (1) on[0, ℎ₁], satisfying the initial conditions

𝑤₁(0, 𝜆) =sin𝛼, 𝑤₁^󸀠(0, 𝜆) = −cos𝛼. (10) The conditions (10) define a unique solution of (1) on[0, ℎ₁] ([5], page 12).

After defining the above solution, then we will define the solution𝑤₂(𝑥, 𝜆)of (1) on[ℎ₁, ℎ₂]by means of the solution 𝑤₁(𝑥, 𝜆)using the initial conditions

𝑤₂(ℎ₁, 𝜆) = 𝛿⁻¹𝑤₁(ℎ₁, 𝜆) , 𝑤^󸀠₂(ℎ₁, 𝜆) = 𝛿⁻¹𝑤₁^󸀠(ℎ₁, 𝜆) . (11) The conditions (11) define a unique solution of (1) on[ℎ₁, ℎ₂].

After describing the above solution, then we will give the solution𝑤₃(𝑥, 𝜆)of (1) on[ℎ₂, 𝜋]by means of the solution 𝑤₂(𝑥, 𝜆)using the initial conditions

𝑤₃(ℎ₂, 𝜆) = 𝛾⁻¹𝑤₂(ℎ₂, 𝜆) , 𝑤^󸀠₃(ℎ₂, 𝜆) = 𝛾⁻¹𝑤₂^󸀠(ℎ₂, 𝜆) . (12) The conditions (12) define a unique solution of (1) on[ℎ₂, 𝜋].

Consequently, the function𝑤(𝑥, 𝜆)is defined on[0, ℎ₁) ∪ (ℎ₁, ℎ₂) ∪ (ℎ₂, 𝜋]by the equality

𝑤 (𝑥, 𝜆) ={{ {{ {

𝑤₁(𝑥, 𝜆) , 𝑥 ∈ [0, ℎ₁) , 𝑤₂(𝑥, 𝜆) , 𝑥 ∈ (ℎ₁, ℎ₂) , 𝑤₃(𝑥, 𝜆) , 𝑥 ∈ (ℎ₂, 𝜋] ,

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is a solution of (1) on[0, ℎ₁) ∪ (ℎ₁, ℎ₂) ∪ (ℎ₂, 𝜋], which satisfies one of the boundary conditions and four transmission conditions.

Lemma 1. Let𝑤(𝑥, 𝜆)be a solution of (1)and𝜆 > 0. Then the following integral equations hold:

𝑤₁(𝑥, 𝜆) = sin𝛼 cos𝑠𝑥 −cos𝛼 𝑠 sin𝑠𝑥

−1 𝑠∫^𝑥

0 𝑞 (𝜏)sin𝑠 (𝑥 − 𝜏) 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏 (𝑠 = √𝜆, 𝜆 > 0) ,

(14) 𝑤₂(𝑥, 𝜆) = 1

𝛿𝑤₁(ℎ₁, 𝜆)cos𝑠 (𝑥 − ℎ₁) +𝑤^󸀠₁(ℎ₁, 𝜆)

𝑠𝛿 sin𝑠 (𝑥 − ℎ₁)

−1 𝑠∫^𝑥

ℎ1

𝑞 (𝜏)sin𝑠 (𝑥 − 𝜏) 𝑤₂(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏 (𝑠 = √𝜆, 𝜆 > 0) , (15) 𝑤₃(𝑥, 𝜆) = 1

𝛾𝑤₂(ℎ₂, 𝜆)cos𝑠 (𝑥 − ℎ₂) +𝑤^󸀠₂(ℎ₂, 𝜆)

𝑠𝛾 sin𝑠 (𝑥 − ℎ₂)

−1 𝑠∫^𝑥

ℎ2

𝑞 (𝜏)sin𝑠 (𝑥 − 𝜏) 𝑤₃(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏 (𝑠 = √𝜆, 𝜆 > 0) . (16) Proof. To prove this lemma, it is enough to substitute−𝑠²𝑤₁ (𝜏, 𝜆) − 𝑤^󸀠󸀠₁(𝜏, 𝜆),−𝑠²𝑤₂(𝜏, 𝜆) − 𝑤^󸀠󸀠₂(𝜏, 𝜆), and−𝑠²𝑤₃(𝜏, 𝜆) − 𝑤₃^󸀠󸀠(𝜏, 𝜆)instead of−𝑞(𝜏)𝑤₁(𝜏−Δ(𝜏), 𝜆),−𝑞(𝜏)𝑤₂(𝜏−Δ(𝜏), 𝜆), and−𝑞(𝜏)𝑤₃(𝜏−Δ(𝜏), 𝜆)in the integrals in (14), (15), and (16), respectively, and integrate by parts twice.

Theorem 2. The problem(1)–(7)can have only simple eigen- values.

Proof. Let̃𝜆be an eigenvalue of the problem (1)–(7) and

̃𝑦 (𝑥, ̃𝜆) ={{ {{ {

̃𝑦₁(𝑥, ̃𝜆) , 𝑥 ∈ [0, ℎ₁) ,

̃𝑦₂(𝑥, ̃𝜆) , 𝑥 ∈ (ℎ₁, ℎ₂) ,

̃𝑦₃(𝑥, ̃𝜆) , 𝑥 ∈ (ℎ₂, 𝜋] ,

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be a corresponding eigenfunction. Then, from (2) and (10), it follows that the determinant

𝑊 [ ̃𝑦₁(0, ̃𝜆) , 𝑤₁(0, ̃𝜆)] =󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

̃𝑦₁(0, ̃𝜆) sin𝛼

̃𝑦₁^󸀠(0, ̃𝜆) −cos𝛼󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨= 0, (18) and, by Theorem 2.2.2, in [5] the functions ̃𝑦₁(𝑥, ̃𝜆)and𝑤₁ (𝑥, ̃𝜆)are linearly dependent on[0, ℎ₁]. We can also prove that the functions̃𝑦₂(𝑥, ̃𝜆)and𝑤₂(𝑥, ̃𝜆)are linearly dependent on

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[ℎ₁, ℎ₂]and ̃𝑦₃(𝑥, ̃𝜆)and𝑤₃(𝑥, ̃𝜆)are linearly dependent on [ℎ₂, 𝜋]. Hence

̃𝑦_𝑖(𝑥, ̃𝜆) = 𝐾_𝑖𝑤_𝑖(𝑥, ̃𝜆) (𝑖 = 1, 2, 3) , (19) for some𝐾₁ ̸= 0, 𝐾₂ ̸= 0, and𝐾₃ ̸= 0. We must show that𝐾₁= 𝐾₂and𝐾₂ = 𝐾₃. Suppose that𝐾₂ ̸= 𝐾₃. From the equalities (6) and (19), we have

̃𝑦 (ℎ₂− 0, ̃𝜆) − 𝛾 ̃𝑦 (ℎ₂+ 0, ̃𝜆)

= ̃𝑦₂(ℎ₂, ̃𝜆) − 𝛾̃𝑦₃(ℎ₂, ̃𝜆)

= 𝐾₂𝑤₂(ℎ₂, ̃𝜆) − 𝛾𝐾₃𝑤₃(ℎ₂, ̃𝜆)

= 𝐾₂𝛾𝑤₃(ℎ₂, ̃𝜆) − 𝐾₃𝛾𝑤₃(ℎ₂, ̃𝜆)

= 𝛾 (𝐾₂− 𝐾₃) 𝑤₃(ℎ₂, ̃𝜆) = 0.

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Since𝛾(𝐾₂− 𝐾₃) ̸= 0it follows that

𝑤₃(ℎ₂, ̃𝜆) = 0. (21) By the same procedure from equality (7) we can derive that

𝑤^󸀠₃(ℎ₂, ̃𝜆) = 0. (22) From the fact that𝑤₃(𝑥, ̃𝜆)is a solution of the differential (1) on[ℎ₂, 𝜋]and satisfies the initial conditions (21) and (22), it follows that𝑤₃(𝑥, ̃𝜆) = 0identically on[ℎ₂, 𝜋].

By using this method, we may also find 𝑤₂(ℎ₂, ̃𝜆) = 𝑤₂^󸀠(ℎ₂, ̃𝜆) = 0,

𝑤₁(ℎ₁, ̃𝜆) = 𝑤₁^󸀠(ℎ₁, ̃𝜆) = 0. (23) From the latter discussions of 𝑤₃(𝑥, ̃𝜆), it follows that 𝑤₂ (𝑥, ̃𝜆) = 0and𝑤₁(𝑥, ̃𝜆) = 0identically on(ℎ₁, ℎ₂)and[0, ℎ₁), but this contradicts (10), thus completing the proof.

2. An Existence Theorem

The function 𝑤(𝑥, 𝜆) defined in Section 1 is a nontrivial solution of (1) satisfying conditions (2) and (4)–(7). Putting 𝑤(𝑥, 𝜆)into (3), we get the characteristic equation

𝐹 (𝜆) ≡ 𝑤 (𝜋, 𝜆)cos𝛽 + 𝑤^󸀠(𝜋, 𝜆)sin𝛽 = 0. (24) ByTheorem 2the set of eigenvalues of boundary-value problem (1)–(7) coincides with the set of real roots of (24).

Let

𝑞₁= ∫^ℎ¹

0 󵄨󵄨󵄨󵄨𝑞(𝜏)󵄨󵄨󵄨󵄨𝑑𝜏, 𝑞₂= ∫^ℎ²

ℎ1 󵄨󵄨󵄨󵄨𝑞(𝜏)󵄨󵄨󵄨󵄨𝑑𝜏, 𝑞₃= ∫^𝜋

ℎ₂󵄨󵄨󵄨󵄨𝑞(𝜏)󵄨󵄨󵄨󵄨𝑑𝜏. (25)

Lemma 3. (1) Let𝜆 ≥ 4𝑞²₁. Then for the solution𝑤₁(𝑥, 𝜆)of (14), the following inequality holds:

󵄨󵄨󵄨󵄨w₁(𝑥, 𝜆)󵄨󵄨󵄨󵄨 ≤ 1

𝑞₁√4𝑞²₁sin²𝛼 +cos²𝛼, 𝑥 ∈ [0, ℎ₁] . (26) (2) Let𝜆 ≥max{4𝑞²₁, 4𝑞²₂}. Then for the solution𝑤₂(𝑥, 𝜆) of (15), the following inequality holds:

󵄨󵄨󵄨󵄨𝑤²(𝑥, 𝜆)󵄨󵄨󵄨󵄨 ≤ 4

𝑞₁|𝛿|√4𝑞²₁sin²𝛼 +cos²𝛼, 𝑥 ∈ [ℎ₁, ℎ₂] . (27) (3) Let 𝜆 ≥ max{4𝑞²₁, 4𝑞²₂, 4𝑞²₃}. Then for the solution 𝑤₃(𝑥, 𝜆)of (16), the following inequality holds:

󵄨󵄨󵄨󵄨𝑤³(𝑥, 𝜆)󵄨󵄨󵄨󵄨 ≤ 16

𝑞₁󵄨󵄨󵄨󵄨𝛿𝛾󵄨󵄨󵄨󵄨√4𝑞²₁sin²𝛼 +cos²𝛼, 𝑥 ∈ [ℎ₂, 𝜋] . (28) Proof. Let𝐵_1𝜆=max_[0,ℎ₁_]|𝑤₁(𝑥, 𝜆)|. Then from (14), it follows that, for every𝜆 > 0, the following inequality holds:

𝐵_1𝜆≤ √sin²𝛼 +cos²𝛼 𝑠² +1

𝑠𝐵_1𝜆𝑞₁. (29) If𝑠 ≥ 2𝑞₁we get (26). Differentiating (14) with respect to𝑥, we have

𝑤^󸀠₁(𝑥, 𝜆) = − 𝑠sin𝛼sin𝑠𝑥 −cos𝛼 cos𝑠𝑥

− ∫^𝑥

0 𝑞 (𝜏)cos𝑠 (𝑥 − 𝜏) 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏.

(30) From expressions of (30) and (26), it follows that, for𝑠 ≥ 2𝑞₁, the following inequality holds:

󵄨󵄨󵄨󵄨󵄨𝑤^󸀠¹(𝑥, 𝜆)󵄨󵄨󵄨󵄨󵄨

𝑠 ≤ 1

𝑞₁√4𝑞²₁sin²𝛼 +cos²𝛼. (31) Let𝐵_2𝜆=max_[ℎ₁_,ℎ₂_]|𝑤₂(𝑥, 𝜆)|. Then from (11), (26), and (31) it follows that, for𝑠 ≥ 2𝑞₁and𝑠 ≥ 2𝑞₂, the following inequality holds:

𝐵_2𝜆≤ 2

𝑞₁|𝛿|√4𝑞²₁sin²𝛼 +cos²𝛼 + 1

2𝑞₂𝐵_2𝜆𝑞₂. (32) Hence, if𝜆 ≥max{4𝑞²₁, 4𝑞²₂}, it reduces to (27). Differentiating (15) with respect to, we get

𝑤₂^󸀠(𝑥, 𝜆) = − 𝑠

𝛿𝑤₁(ℎ₁, 𝜆)sin𝑠 (𝑥 − ℎ₁) +𝑤^󸀠₁(ℎ₁, 𝜆)

𝛿 cos𝑠 (𝑥 − ℎ₁)

− ∫^𝑥

ℎ1

𝑞 (𝜏)cos𝑠 (𝑥 − 𝜏) 𝑤₂(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏.

(𝑠 = √𝜆, 𝜆 > 0) . (33)

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From (26) and (33), it follows that, for𝑠 ≥ 2𝑞₁and𝑠 ≥ 2𝑞₂, the following inequality holds:

󵄨󵄨󵄨󵄨󵄨𝑤²^󸀠(𝑥, 𝜆)󵄨󵄨󵄨󵄨󵄨

𝑠 ≤ 4

|𝛿| 𝑞₁√4𝑞²₁sin²𝛼 +cos²𝛼. (34) Let𝐵_3𝜆 =max_[ℎ₂_,𝜋]|𝑤₃(𝑥, 𝜆)|. Then from (16), (27), and (34) it follows that, for𝑠 ≥ 2𝑞₁, 𝑠 ≥ 2𝑞₂and𝑠 ≥ 2𝑞₃, the following inequality holds:

𝐵_3𝜆≤ 8

𝑞₁󵄨󵄨󵄨󵄨𝛿𝛾󵄨󵄨󵄨󵄨√4𝑞²₁sin²𝛼 +cos²𝛼 +1

𝑠𝐵_3𝜆𝑞₃. (35)

Hence, if𝜆 ≥max{4𝑞²₁, 4𝑞²₂, 4𝑞²₃}we procure (28).

Theorem 4. The problem(1)–(7)has an infinite set of positive eigenvalues.

Proof. Differentiating (16) with respect to𝑥, we readily see that

𝑤^󸀠₃(𝑥, 𝜆) = − 𝑠

𝛾𝑤₂(ℎ₂, 𝜆)sin𝑠 (𝑥 − ℎ₂) +𝑤₂^󸀠(ℎ₂, 𝜆)

𝛾 cos𝑠 (𝑥 − ℎ₂)

− ∫^𝑥

ℎ2

𝑞 (𝜏)cos𝑠 (𝑥 − 𝜏) 𝑤₃(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏.

(𝑠 = √𝜆, 𝜆 > 0) . (36) With the helps of (14), (15), (16), (24), (30), and (36), we have the following:

[1 𝛾{1

𝛿(sin𝛼cos𝑠ℎ₁−cos𝛼 𝑠 sin𝑠ℎ₁

−1 𝑠∫^ℎ¹

0 𝑞 (𝜏)sin𝑠 (ℎ₁− 𝜏) 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

×cos𝑠 (ℎ₂− ℎ₁)

− 1

𝑠𝛿(𝑠sin𝛼sin𝑠ℎ₁+cos𝛼cos𝑠ℎ₁ + ∫^ℎ¹

0 𝑞 (𝜏)cos𝑠 (ℎ₁− 𝜏) 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

×sin𝑠 (ℎ₂− ℎ₁)

−1 𝑠∫^ℎ²

ℎ1

𝑞 (𝜏)sin𝑠 (ℎ₂− 𝜏) 𝑤₂(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏}

×cos𝑠 (𝜋 − ℎ₂) + 1

𝑠𝛾{−𝑠

−1 𝑠∫^ℎ¹

0 𝑞 (𝜏)sin𝑠 (ℎ₁− 𝜏)

× 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

×sin𝑠 (ℎ₂− ℎ₁) +1

𝛿(− 𝑠sin𝛼sin𝑠ℎ₁−cos𝛼cos𝑠ℎ₁

− ∫^ℎ¹

0 𝑞 (𝜏)cos𝑠 (ℎ₁− 𝜏)

×𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

− ∫^ℎ²

ℎ1

𝑞 (𝜏)cos𝑠 (ℎ₂− 𝜏) 𝑤₂(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏}

×sin𝑠 (𝜋 − ℎ₂)

−1 𝑠∫^𝜋

ℎ₂𝑞 (𝜏)sin𝑠 (𝜋 − 𝜏) 𝑤₃(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏]cos𝛽 + [−𝑠

𝛾{1

−1 𝑠∫^ℎ¹

×𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

− 1

𝑠𝛿(𝑠sin𝛼sin𝑠ℎ₁+cos𝛼cos𝑠ℎ₁ + ∫^ℎ¹

× 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

×sin𝑠 (ℎ₂− ℎ₁)

−1 𝑠∫^ℎ²

ℎ₁ 𝑞 (𝜏)sin𝑠 (ℎ₂− 𝜏) 𝑤₂(𝜏 − Δ (𝜏) , 𝜆) }

×sin𝑠 (𝜋 − ℎ₂)

(5)

+1 𝛾{−𝑠

𝛿× (sin𝛼cos𝑠ℎ₁−cos𝛼 𝑠 sin𝑠ℎ₁

−1 𝑠∫^ℎ¹

× 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

×sin𝑠 (ℎ₂− ℎ₁) +1

𝛿(− 𝑠sin𝛼sin𝑠ℎ₁−cos𝛼cos𝑠ℎ₁

− ∫^ℎ¹

× 𝑤₁(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)

− ∫^ℎ²

ℎ1

𝑞 (𝜏)cos𝑠 (ℎ₂− 𝜏) 𝑤₂(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏}

×cos𝑠 (𝜋 − ℎ₂)

− ∫^𝜋

ℎ2

𝑞 (𝜏)cos𝑠 (𝜋 − 𝜏) 𝑤3(𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏]sin𝛽 = 0.

(37) Let𝜆be sufficiently big. Then, by (26), (27), and (28), (37) may be rewritten in the following form:

−𝑠sin𝛼sin𝛽

𝛾𝛿 {cos𝑠ℎ₂sin𝑠 (𝜋 − ℎ₂)

+sin𝑠ℎ₂cos𝑠 (𝜋 − ℎ₂)} + 𝑂 (1) = 0, (38)

𝑠sin𝑠𝜋 + 𝑂 (1) = 0. (39) Obviously, for big𝑠(39) has an infinite set of roots. Thus, the proof of the theorem is completed.

3. Asymptotic Formulas for

Eigenvalues and Eigenfunctions

Now we begin to study asymptotic properties of eigenvalues and eigenfunctions. In the following we will assume that is sufficiently big. From (14) and (26), we obtain

𝑤₁(𝑥, 𝜆) = 𝑂 (1) on [0, ℎ₁] . (40) By (15) and (27), this leads to

𝑤₂(𝑥, 𝜆) = 𝑂 (1) on [ℎ₁, ℎ₂] . (41) By (16) and (28), this leads to

𝑤₃(𝑥, 𝜆) = 𝑂 (1) on [ℎ₂, 𝜋] . (42) The existence and continuity of the derivatives𝑤^󸀠_1𝑠(𝑥, 𝜆)for 0 ≤ 𝑥 ≤ ℎ₁,|𝜆| < ∞,𝑤^󸀠_2𝑠(𝑥, 𝜆)forℎ₁≤ 𝑥 ≤ ℎ₂,|𝜆| < ∞and 𝑤_3𝑠^󸀠(𝑥, 𝜆)forℎ₂≤ 𝑥 ≤ 𝜋,|𝜆| < ∞follows from Theorem 1.4.1 in [5].

Lemma 5. The following holds true:

𝑤^󸀠_1𝑠(𝑥, 𝜆) = 𝑂 (1) , 𝑥 ∈ [0, ℎ₁] , (43) 𝑤^󸀠_2𝑠(𝑥, 𝜆) = 𝑂 (1) , 𝑥 ∈ [ℎ₁, ℎ₂] , (44) 𝑤_3𝑠^󸀠 (𝑥, 𝜆) = 𝑂 (1) , 𝑥 ∈ [ℎ₂, 𝜋] . (45) Proof. By differentiating (16) with respect to𝑠, we get, by (43) and (44) the following:

𝑤_3𝑠^󸀠 (𝑥, 𝜆)

= −1 𝑠∫^𝑥

ℎ₂𝑞 (𝜏)sin𝑠 (𝑥 − 𝜏) 𝑤_3𝑠^󸀠 (𝜏 − Δ (𝜏) , 𝜆) + 𝑍 (𝑥, 𝜆) , (|𝑍 (𝑥, 𝜆)| ≤ 𝑍₀) . (46) Let𝐷_𝜆 = max_[ℎ₂_,𝜋]|𝑤^󸀠_3𝑠(𝑥, 𝜆)|. Then the existence of𝐷_𝜆fol- lows from continuity of derivation for𝑥 ∈ [ℎ₂, 𝜋]. From (46)

𝐷_𝜆≤ 1

𝑠𝑞₃𝐷_𝜆+ 𝑍₀. (47) Now let𝑠 ≥ 2𝑞₃. Then 𝐷_𝜆 ≤ 2𝑍₀ and the validity of the asymptotic formula (45) follows. Formulas (43) and (44) may be proved analogically.

Theorem 6. Let𝑛be a natural number. For each sufficiently big𝑛there is exactly one eigenvalue of the problem(1)–(7)near 𝑛².

Proof. We consider the expression which is denoted by𝑂(1) in (39). If formulas (40)–(45) are taken into consideration, it can be shown by differentiation with respect to𝑠that for big 𝑠this expression has bounded derivative. We will show that, for big𝑛, only one root (39) lies near to each𝑛. We consider the function𝜙(𝑠) = 𝑠sin𝑠𝜋 + 𝑂(1). Its derivative, which has the form𝜙^󸀠(𝑠) =sin𝑠𝜋+𝑠𝜋cos𝑠𝜋+𝑂(1), does not vanish for 𝑠close to𝑛for sufficiently big𝑛. Thus our assertion follows by Rolle’s Theorem.

Let𝑛be sufficiently big. In what follows we will denote by 𝜆_𝑛= 𝑠²_𝑛the eigenvalue of the problem (1)–(7) situated near𝑛². We set𝑠_𝑛= 𝑛+𝛿_𝑛. Then from (39) it follows that𝛿_𝑛= 𝑂(1/𝑛).

Consequently

𝑠_𝑛 = 𝑛 + 𝑂 (1

𝑛) , (48)

𝜆_𝑛 = 𝑛²+ 𝑂 (1) . (49)

Formula (48) makes it possible to obtain asymptotic expressions for eigenfunction of the problem (1)–(7). From (14), (30), and (40), we get

𝑤₁(𝑥, 𝜆) =sin𝛼cos𝑠𝑥 + 𝑂 (1

𝑠) , (50)

𝑤₁^󸀠(𝑥, 𝜆) = −𝑠sin𝛼sin𝑠𝑥 + 𝑂 (1) . (51)

(6)

From expressions of (15), (31), (39), and (41), we easily see that 𝑤₂(𝑥, 𝜆) = sin𝛼

𝛿 cos𝑠𝑥 + 𝑂 (1 𝑠) , 𝑤₃(𝑥, 𝜆) = sin𝛼

𝛿𝛾 cos𝑠𝑥 + 𝑂 (1 𝑠) .

(52)

By substituting (48) into (50), (52), we find that 𝑢_1𝑛= 𝑤₁(𝑥, 𝜆_𝑛) =sin𝛼cos𝑛𝑥 + 𝑂 (1

𝑛) , 𝑢_2𝑛 = 𝑤₂(𝑥, 𝜆_𝑛) = sin𝛼

𝛿 cos𝑛𝑥 + 𝑂 (1 𝑛) , 𝑢_3𝑛 = 𝑤₃(𝑥, 𝜆_𝑛) = sin𝛼

𝛿𝛾 cos𝑛𝑥 + 𝑂 (1 𝑛) .

(53)

Hence the eigenfunctions𝑢_𝑛(𝑥)have the following asymptotic representation:

𝑢_𝑛(𝑥) = {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {{ {

sin𝛼cos𝑛𝑥 + 𝑂 (1

𝑛) , for𝑥 ∈ [0, ℎ₁) , sin𝛼

𝛿 cos𝑛𝑥 + 𝑂 (1

𝑛) , for𝑥 ∈ (ℎ₁, ℎ₂) , sin𝛼

𝛿𝛾 cos𝑛𝑥 + 𝑂 (1

𝑛) , for𝑥 ∈ (ℎ₂, 𝜋] . (54)

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 ful- filled.

(a) The derivatives𝑞^󸀠(𝑥)andΔ^󸀠󸀠(𝑥)exist and are bounded in[0, ℎ₁) ∪ (ℎ₁, ℎ₂) ∪ (ℎ₂, 𝜋]and have finite limits𝑞^󸀠(ℎ₁± 0) = lim_{𝑥 → ℎ}₁_±0𝑞^󸀠(𝑥),𝑞^󸀠(ℎ₂±0) =lim_{𝑥 → ℎ}₂_±0𝑞^󸀠(𝑥)andΔ^󸀠󸀠(ℎ₁±0) = lim_{𝑥 → ℎ}₁_±0Δ^󸀠󸀠(𝑥), Δ^󸀠󸀠(ℎ₂±0) =lim_{𝑥 → ℎ}₂_±0Δ^󸀠󸀠(𝑥), respectively.

(b)Δ^󸀠(𝑥) ≤ 1in [0, ℎ₁) ∪ (ℎ₁, ℎ₂) ∪ (ℎ₂, 𝜋],Δ(0) = 0, lim_{𝑥 → ℎ}₁₊₀Δ(𝑥) = 0, and lim_{𝑥 → ℎ}₂₊₀Δ(𝑥) = 0.

It is easy to see that, using (b)

𝑥 − Δ (𝑥) ≥ 0, 𝑥 ∈ [0, ℎ₁) , 𝑥 − Δ (𝑥) ≥ ℎ1, 𝑥 ∈ (ℎ₁, ℎ₂) ,

𝑥 − Δ (𝑥) ≥ ℎ₂, 𝑥 ∈ (ℎ₂, 𝜋]

(55)

are obtained.

By (50), (52), and (55), we have

𝑤₁(𝜏 − Δ (𝜏) , 𝜆) =sin𝛼cos𝑠 (𝜏 − Δ (𝜏)) + 𝑂 (1

𝑠) , (56) 𝑤₂(𝜏 − Δ (𝜏) , 𝜆) = sin𝛼

𝛿 cos𝑠 (𝜏 − Δ (𝜏)) + 𝑂 (1

𝑠) , (57) 𝑤₃(𝜏 − Δ (𝜏) , 𝜆) = sin𝛼

𝛿𝛾 cos𝑠 (𝜏 − Δ (𝜏)) + 𝑂 (1

𝑠) (58) on[0, ℎ₁),(ℎ₁, ℎ₂)and(ℎ₂, 𝜋], respectively.

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

∫^𝑥

0 𝑞 (𝜏)cos𝑠 (2𝜏 − Δ (𝜏)) 𝑑𝜏 = 𝑂 (1 𝑠) ,

∫^𝑥

0 𝑞 (𝜏)sin𝑠 (2𝜏 − Δ (𝜏)) 𝑑𝜏 = 𝑂 (1 𝑠)

(59)

can be proved by the same technique in Lemma 3.3.3 in [5].

Putting the expressions (56), (57), and (58) into (37), and then using (59), after long operations we have

cos𝑠𝜋sin(𝛼 − 𝛽) − 𝑠sin𝛼sin𝛽sin𝑠𝜋 𝛿𝛾

−sin𝛼sin𝛽 2𝛿𝛾 ∫^𝜋

0 𝑞 (𝜏)cos𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏 + 𝑂 (1

𝑠) = 0.

(60)

Hence

tan𝑠𝜋 = 1

𝑠(sin(𝛼 − 𝛽) sin𝛼sin𝛽

−1 2∫^𝜋

0 𝑞 (𝜏)cos𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏) + 𝑂 (1

𝑠²) .

(61)

Again, if we take𝑠_𝑛= 𝑛 + 𝛿_𝑛, then from (48) tan((𝑛 + 𝛿_𝑛) 𝜋) = tan𝛿_𝑛𝜋

= 1

𝑛(sin(𝛼 − 𝛽) sin𝛼sin𝛽

−1 2∫^𝜋

0 𝑞 (𝜏)cos𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏) + 𝑂 (1

𝑛²) .

(62) Hence for big𝑛,

𝛿_𝑛= 1

𝑛𝜋(sin(𝛼 − 𝛽) sin𝛼sin𝛽 −1

2∫^𝜋

0 𝑞 (𝜏)cos𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏) + 𝑂 (1

𝑛²)

(63) and finally

𝑠_𝑛= 𝑛 + 1

𝑛𝜋(sin(𝛼 − 𝛽) sin𝛼sin𝛽 −1

2∫^𝜋

0 𝑞 (𝜏)cos𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏) + 𝑂 (1

𝑛²) .

(64) Thus, we have proven the following theorem.

(7)

Theorem 7. If conditions (a) and (b) are satisfied, then the eigenvalues 𝜆_𝑛 = 𝑠²_𝑛 of the problem (1)–(7) have the (64) asymptotic formula for𝑛 → ∞.

Now, we may obtain sharper asymptotic formulas for the eigenfunctions. From (14), (56), and (59), we have

𝑤₁(𝑥, 𝜆) = sin𝛼cos𝑠𝑥 [1 + 1 2𝑠∫^𝑥

0 𝑞 (𝜏)sin𝑠Δ (𝜏) 𝑑𝜏]

−sin𝑠𝑥

𝑠 [cos𝛼 +sin𝛼 2 ∫^𝑥

0 𝑞 (𝜏)cos𝑠Δ (𝜏) 𝑑𝜏]

+ 𝑂 (1

𝑠²) , 𝑥 ∈ [0, ℎ₁) .

(65) Now, replacing𝑠by𝑠_𝑛and using (64), we have

𝑢_1𝑛(𝑥)

= 𝑤₁(𝑥, 𝜆_𝑛)

=sin𝛼 {cos𝑛𝑥 [1 + 1 2𝑛∫^𝑥

0 𝑞 (𝜏)sin(𝑛Δ (𝜏)) 𝑑𝜏]

−sin𝑛𝑥

𝑛𝜋 [(sin(𝛼 − 𝛽) sin𝛼sin𝛽

−1 2∫^𝜋

0𝑞 (𝜏)cos(𝑛 (𝜋 − Δ (𝜏))) 𝑑𝜏) 𝑥 +(cot𝛼+1

2∫^𝑥

0𝑞 (𝜏)cos(𝑛Δ (𝜏)) 𝑑𝜏)𝜋]}

+ 𝑂 (1 𝑛²) .

(66) From (15), (57), (59), and (65) we have

𝑤₂(𝑥, 𝜆) = sin𝛼cos𝑠𝑥

𝛿 [1 + 1

2𝑠∫^𝑥

−sin𝑠𝑥

𝑠𝛿 [cos𝛼 +sin𝛼 2 ∫^𝑥

+ 𝑂 (1

𝑠²) , 𝑥 ∈ (ℎ₁, ℎ₂) .

(67) Now, replacing𝑠by𝑠_𝑛and using (64), we have

𝑢_2𝑛(𝑥)

= 𝑤₂(𝑥, 𝜆_𝑛)

=sin𝛼

𝛿 {cos𝑛𝑥 [1 + 1 2𝑛∫^𝑥

−sin𝑛𝑥

−1 2∫^𝜋

0𝑞(𝜏)cos(𝑛(𝜋−Δ(𝜏)))𝑑𝜏) 𝑥 +(cot𝛼+1

2∫^𝑥

0𝑞(𝜏)cos(𝑛Δ (𝜏)) 𝑑𝜏) 𝜋]}

+ 𝑂 (1 𝑛²) .

(68) From (16), (58), (59), (65), and (67) and after long operations, we have

𝑤₃(𝑥, 𝜆) = sin𝛼cos𝑠𝑥

𝛿𝛾 [1 + 1

2𝑠∫^𝑥

−sin𝑠𝑥

𝑠𝛿𝛾 [cos𝛼 +sin𝛼 2 ∫^𝑥

+ 𝑂 (1

𝑠²) , 𝑥 ∈ (ℎ₂, 𝜋] .

(69) Now replacing𝑠by𝑠_𝑛and using (64), we have

𝑢_3𝑛(𝑥)

= 𝑤₃(𝑥, 𝜆_𝑛)

= sin𝛼

𝛿𝛾 {cos𝑛𝑥 [1 + 1 2𝑛∫^𝑥

−sin𝑛𝑥

−1 2∫^𝜋

0𝑞(𝜏)cos(𝑛(𝜋−Δ(𝜏)))𝑑𝜏)𝑥 +(cot𝛼+1

2∫^𝑥

0𝑞 (𝜏)cos(𝑛Δ(𝜏)) 𝑑𝜏) 𝜋]}

+ 𝑂 (1 𝑛²) .

(70) Thus, we have proven the following theorem.

Theorem 8. If conditions (a) and (b) are satisfied, then the eigenfunctions𝑢_𝑛(𝑥)of the problem(1)–(7)have the following asymptotic formula for𝑛 → ∞:

𝑢_𝑛(𝑥) ={{ {{ {

𝑢_1𝑛(𝑥) , 𝑥 ∈ [0, ℎ₁) , 𝑢_2𝑛(𝑥) , 𝑥 ∈ (ℎ₁, ℎ₂) , 𝑢_3𝑛(𝑥) , 𝑥 ∈ (ℎ₂, 𝜋] ,

(71)

where𝑢_1𝑛(𝑥), 𝑢_2𝑛(𝑥), and𝑢_3𝑛(𝑥)are determined as in (49), (68), and(70), respectively.

(8)

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1.Introduction Erdo L an F en, JongJinSeo, andSerkanAraci AsymptoticBehaviourofEigenvaluesandEigenfunctionsofaSturm-LiouvilleProblemwithRetardedArgument ResearchArticle

Research Article

Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

Erdo L an F en,

Jong Jin Seo,

and Serkan Araci

1. Introduction

2. An Existence Theorem

3. Asymptotic Formulas for

Eigenvalues and Eigenfunctions

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis