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

Abstract

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 coefficients 𝛿=𝛾=1 and retarded argument Δ≡0 in 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 (taking 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)𝑦(𝜋)sin𝛼=0,cos𝛽+𝑦^′(𝜋)sin𝛽=0, (2)(3) and transmission conditions 𝑦ℎ₁−0−𝛿𝑦ℎ₁𝑦+0=0,_′ℎ₁−0−𝛿𝑦^′ℎ₁𝑦ℎ+0=0,₂−0−𝛾𝑦ℎ₂𝑦+0=0,_′ℎ₂−0−𝛾𝑦^′ℎ₂+0=0, (4)(5)(6)(7) where the real-valued function 𝑞(𝑥) is continuous in [0,ℎ₁)∪(ℎ₁,ℎ₂)∪(ℎ₂,𝜋] and has finite limits 𝑞ℎ₁±0=lim_{𝑥→ℎ1±0}𝑞(𝑥),𝑞ℎ₂±0=lim_{𝑥→ℎ2±0}𝑞(𝑥), (8) the real-valued function Δ(𝑥)≥0 continuous in [0,ℎ₁)∪(ℎ₁,ℎ₂)∪(ℎ₂,𝜋] and has finite limits Δℎ₁±0=lim_{𝑥→ℎ1±0}Δ(𝑥),Δℎ₂±0=lim_{𝑥→ℎ2±0}Δ(𝑥); (9) if 𝑥∈[0,ℎ₁) then 𝑥−Δ(𝑥)≥0; if 𝑥∈(ℎ₁,ℎ₂) then 𝑥−Δ(𝑥)≥ℎ₁; if 𝑥∈(ℎ₂,𝜋) then 𝑥−Δ(𝑥)≥ℎ₂; 𝜆 is a real spectral parameter; ℎ₁, ℎ₂, 𝛼, 𝛽, 𝛿, 𝛾≠0 are arbitrary real numbers such that 0<ℎ₁<ℎ₂<𝜋 and 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].

Let 𝑤₁(𝑥,𝜆) be a solution of (1) on [0,ℎ₁], satisfying the initial conditions 𝑤₁(0,𝜆)=sin𝛼,𝑤^′₁(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 solution 𝑤₂(𝑥,𝜆) of (1) on [ℎ₁,ℎ₂] by means of the solution 𝑤₁(𝑥,𝜆) using the initial conditions 𝑤₂ℎ₁,𝜆=𝛿⁻¹𝑤₁ℎ₁,𝜆,𝑤^′₂ℎ₁,𝜆=𝛿⁻¹𝑤^′₁ℎ₁,𝜆. (11) The conditions (11) define a unique solution of (1) on [ℎ₁,ℎ₂].

After describing the above solution, then we will give the solution 𝑤₃(𝑥,𝜆) of (1) on [ℎ₂,𝜋] by means of the solution 𝑤₂(𝑥,𝜆) using the initial conditions 𝑤₃ℎ₂,𝜆=𝛾⁻¹𝑤₂ℎ₂,𝜆,𝑤^′₃ℎ₂,𝜆=𝛾⁻¹𝑤^′₂ℎ₂,𝜆. (12) The conditions (12) define a unique solution of (1) on [ℎ₂,𝜋].

Consequently, the function 𝑤(𝑥,𝜆) is defined on [0,ℎ₁)∪(ℎ₁,ℎ₂)∪(ℎ₂,𝜋] by the equality 𝑤()=𝑤𝑥,𝜆¹(𝑥,𝜆),𝑥∈0,ℎ₁,𝑤₂(𝑥,𝜆),𝑥∈ℎ₁,ℎ₂,𝑤₃(𝑥,𝜆),𝑥∈ℎ₂,𝜋, (13) 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. Let 𝑤(𝑥,𝜆) be a solution of (1) and 𝜆>0. Then the following integral equations hold: 𝑤₁(𝑥,𝜆)=sin𝛼cos𝑠𝑥−cos𝛼𝑠−1sin𝑠𝑥𝑠^𝑥₀𝑞(𝜏)sin𝑠(𝑥−𝜏)𝑤₁((𝜏))√𝜏−Δ,𝜆𝑑𝜏𝑠=,𝑤𝜆,𝜆>0₂()=1𝑥,𝜆𝛿𝑤₁ℎ₁,𝜆cos𝑠𝑥−ℎ₁+𝑤^′₁ℎ₁,𝜆𝑠𝛿sin𝑠𝑥−ℎ₁−1𝑠^𝑥_ℎ1𝑞(𝜏)sin𝑠(𝑥−𝜏)𝑤₂((𝜏))√𝜏−Δ,𝜆𝑑𝜏𝑠=,𝑤𝜆,𝜆>0₃()=1𝑥,𝜆𝛾𝑤₂ℎ₂,𝜆cos𝑠𝑥−ℎ₂+𝑤^′₂ℎ₂,𝜆𝑠𝛾sin𝑠𝑥−ℎ₂−1𝑠^𝑥_ℎ2𝑞(𝜏)sin𝑠(𝑥−𝜏)𝑤₃((𝜏))√𝜏−Δ,𝜆𝑑𝜏𝑠=𝜆,𝜆>0. (14)(15)(16)

Proof. To prove this lemma, it is enough to substitute −𝑠²𝑤₁(𝜏,𝜆)−𝑤₁^′′(𝜏,𝜆), −𝑠²𝑤₂(𝜏,𝜆)−𝑤₂^′′(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 eigenvalues.

Proof. Let 𝜆 be an eigenvalue of the problem (1)–(7) and 𝑦𝜆=𝑦𝑥,¹𝑥,𝜆,𝑥∈0,ℎ₁,𝑦₂𝑥,𝜆,𝑥∈ℎ₁,ℎ₂,𝑦₃𝑥,𝜆,𝑥∈ℎ₂,𝜋, (17) be a corresponding eigenfunction. Then, from (2) and (10), it follows that the determinant 𝑊𝑦₁0,𝜆,𝑤₁𝜆=𝑦0,¹𝜆𝑦0,sin𝛼_′10,𝜆−cos𝛼=0, (18) and, by Theorem  2.2.2, in [5] the functions 𝑦₁(𝑥,𝜆) and 𝑤₁(𝑥,𝜆) are linearly dependent on [0,ℎ₁]. We can also prove that the functions 𝑦₂(𝑥,𝜆) and 𝑤₂(𝑥,𝜆) are linearly dependent on [ℎ₁,ℎ₂] and 𝑦₃(𝑥,𝜆) and 𝑤₃(𝑥,𝜆) are linearly dependent on [ℎ₂,𝜋]. Hence 𝑦_𝑖𝑥,𝜆=𝐾_𝑖𝑤_𝑖𝑥,𝜆(𝑖=1,2,3), (19) for some 𝐾₁≠0,  𝐾₂≠0, and 𝐾₃≠0. We must show that 𝐾₁=𝐾₂ and 𝐾₂=𝐾₃. Suppose that 𝐾₂≠𝐾₃. From the equalities (6) and (19), we have 𝑦ℎ₂−0,𝜆−𝛾𝑦ℎ₂𝜆=𝑦+0,₂ℎ₂,𝜆−𝛾𝑦₃ℎ₂,𝜆=𝐾₂𝑤₂ℎ₂,𝜆−𝛾𝐾₃𝑤₃ℎ₂,𝜆=𝐾₂𝛾𝑤₃ℎ₂,𝜆−𝐾₃𝛾𝑤₃ℎ₂,𝜆=𝛾𝐾₂−𝐾₃𝑤₃ℎ₂,𝜆=0. (20) Since 𝛾(𝐾₂−𝐾₃)≠0 it follows that 𝑤₃ℎ₂,𝜆=0. (21) By the same procedure from equality (7) we can derive that 𝑤^′₃ℎ₂,𝜆=0. (22) From the fact that 𝑤₃(𝑥,𝜆) is a solution of the differential (1) on [ℎ₂,𝜋] and satisfies the initial conditions (21) and (22), it follows that 𝑤₃(𝑥,𝜆)=0 identically on [ℎ₂,𝜋].
By using this method, we may also find 𝑤₂ℎ₂,𝜆=𝑤^′₂ℎ₂,𝜆𝑤=0,₁ℎ₁,𝜆=𝑤^′₁ℎ₁,𝜆=0. (23) From the latter discussions of 𝑤₃(𝑥,𝜆), it follows that 𝑤₂(𝑥,𝜆)=0 and 𝑤₁(𝑥,𝜆)=0 identically 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)

By Theorem 2 the set of eigenvalues of boundary-value problem (1)–(7) coincides with the set of real roots of (24). Let 𝑞₁=^ℎ1₀𝑞(𝜏)𝑑𝜏,𝑞₂=^ℎ2_ℎ1𝑞(𝜏𝑞)𝑑𝜏,₃=^𝜋_ℎ2𝑞(𝜏)𝑑𝜏. (25)

Lemma 3. (1) Let 𝜆≥4𝑞²₁. Then for the solution 𝑤₁(𝑥,𝜆) of (14), the following inequality holds: w₁(≤1𝑥,𝜆)𝑞₁4𝑞²₁sin²𝛼+cos²𝛼,𝑥∈0,ℎ₁. (26)
(2) Let 𝜆≥max{4𝑞²₁,4𝑞²₂}. Then for the solution 𝑤₂(𝑥,𝜆) of (15), the following inequality holds: 𝑤₂(≤4𝑥,𝜆)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)≤16𝑞₁𝛿𝛾4𝑞²₁sin²𝛼+cos²𝛼,𝑥∈ℎ₂,𝜋. (28)

Proof. Let 𝐵_1𝜆=max_[0,ℎ1]|𝑤₁(𝑥,𝜆)|. Then from (14), it follows that, for every 𝜆>0, the following inequality holds: 𝐵_1𝜆≤sin²𝛼+cos²𝛼𝑠²+1𝑠𝐵_1𝜆𝑞₁. (29) If 𝑠≥2𝑞₁ we get (26). Differentiating (14) with respect to 𝑥, we have 𝑤^′₁()−𝑥,𝜆=−𝑠sin𝛼sin𝑠𝑥−cos𝛼cos𝑠𝑥_𝑥0𝑞(𝜏)cos𝑠(𝑥−𝜏)𝑤₁(𝜏−Δ(𝜏),𝜆)𝑑𝜏. (30) From expressions of (30) and (26), it follows that, for 𝑠≥2𝑞₁, the following inequality holds: 𝑤^′₁(𝑥,𝜆)𝑠≤1𝑞₁4𝑞²₁sin²𝛼+cos²𝛼. (31) Let 𝐵_2𝜆=max_[ℎ1,ℎ2]|𝑤₂(𝑥,𝜆)|. Then from (11), (26), and (31) it follows that, for 𝑠≥2𝑞₁ and 𝑠≥2𝑞₂, the following inequality holds: 𝐵_2𝜆≤2𝑞₁|𝛿|4𝑞²₁sin²𝛼+cos²1𝛼+2𝑞₂𝐵_2𝜆𝑞₂. (32) Hence, if 𝜆≥max{4𝑞²₁,4𝑞²₂}, it reduces to (27). Differentiating (15) with respect to, we get 𝑤^′₂()𝑠𝑥,𝜆=−𝛿𝑤₁ℎ₁,𝜆sin𝑠𝑥−ℎ₁+𝑤^′₁ℎ₁,𝜆𝛿cos𝑠𝑥−ℎ₁−^𝑥_ℎ1𝑞(𝜏)cos𝑠(𝑥−𝜏)𝑤₂((𝜏))√𝜏−Δ,𝜆𝑑𝜏.𝑠=𝜆,𝜆>0. (33) From (26) and (33), it follows that, for 𝑠≥2𝑞₁ and 𝑠≥2𝑞₂, the following inequality holds: 𝑤^′₂(𝑥,𝜆)𝑠≤4|𝛿|𝑞₁4𝑞²₁sin²𝛼+cos²𝛼. (34) Let 𝐵_3𝜆=max_[ℎ2,𝜋]|𝑤₃(𝑥,𝜆)|. Then from (16), (27), and (34) it follows that, for 𝑠≥2𝑞₁,  𝑠≥2𝑞₂ and 𝑠≥2𝑞₃, the following inequality holds: 𝐵_3𝜆≤8𝑞₁𝛿𝛾4𝑞²₁sin²𝛼+cos²1𝛼+𝑠𝐵_3𝜆𝑞₃. (35) Hence, if 𝜆≥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 to 𝑥, we readily see that 𝑤^′₃()𝑠𝑥,𝜆=−𝛾𝑤₂ℎ₂,𝜆sin𝑠𝑥−ℎ₂+𝑤^′₂ℎ₂,𝜆𝛾cos𝑠𝑥−ℎ₂−^𝑥_ℎ2𝑞(𝜏)cos𝑠(𝑥−𝜏)𝑤₃((𝜏))√𝜏−Δ,𝜆𝑑𝜏.𝑠=𝜆,𝜆>0. (36) With the helps of (14), (15), (16), (24), (30), and (36), we have the following: 1𝛾1𝛿sin𝛼cos𝑠ℎ₁−cos𝛼𝑠sin𝑠ℎ₁−1𝑠^ℎ1₀𝑞(𝜏)sin𝑠ℎ₁−𝜏𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×cos𝑠₂−ℎ₁−1𝑠𝛿𝑠sin𝛼sin𝑠ℎ₁+cos𝛼cos𝑠ℎ₁+^ℎ1₀𝑞(𝜏)cos𝑠ℎ₁−𝜏𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×sin𝑠₂−ℎ₁−1𝑠^ℎ2_ℎ1𝑞(𝜏)sin𝑠ℎ₂−𝜏𝑤₂(𝜏−Δ(𝜏),𝜆)𝑑𝜏×cos𝑠𝜋−ℎ₂+1−𝑠𝑠𝛾𝛿sin𝛼cos𝑠ℎ₁−cos𝛼𝑠sin𝑠ℎ₁−1𝑠^ℎ1₀𝑞(𝜏)sin𝑠ℎ₁−𝜏×𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×sin𝑠₂−ℎ₁+1𝛿−𝑠sin𝛼sin𝑠ℎ₁−cos𝛼cos𝑠ℎ₁−^ℎ1₀𝑞(𝜏)cos𝑠ℎ₁−𝜏×𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×cos𝑠₂−ℎ₁−^ℎ2_ℎ1𝑞(𝜏)cos𝑠ℎ₂−𝜏𝑤₂(𝜏−Δ(𝜏),𝜆)𝑑𝜏×sin𝑠𝜋−ℎ₂−1𝑠^𝜋_ℎ2𝑞(𝜏)sin𝑠(𝜋−𝜏)𝑤₃((𝜏))+−𝑠𝜏−Δ,𝜆𝑑𝜏cos𝛽𝛾1𝛿sin𝛼cos𝑠ℎ₁−cos𝛼𝑠sin𝑠ℎ₁−1𝑠^ℎ1₀𝑞(𝜏)sin𝑠ℎ₁−𝜏×𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×cos𝑠₂−ℎ₁−1𝑠𝛿𝑠sin𝛼sin𝑠ℎ₁+cos𝛼cos𝑠ℎ₁+^ℎ1₀𝑞(𝜏)cos𝑠ℎ₁−𝜏×𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×sin𝑠₂−ℎ₁−1𝑠^ℎ2_ℎ1𝑞(𝜏)sin𝑠ℎ₂−𝜏𝑤₂(𝜏−Δ(𝜏),𝜆)×sin𝑠𝜋−ℎ₂+1𝛾−𝑠𝛿×sin𝛼cos𝑠ℎ₁−cos𝛼𝑠sin𝑠ℎ₁−1𝑠^ℎ1₀𝑞(𝜏)sin𝑠ℎ₁−𝜏×𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×sin𝑠₂−ℎ₁+1𝛿−𝑠sin𝛼sin𝑠ℎ₁−cos𝛼cos𝑠ℎ₁−^ℎ1₀𝑞(𝜏)cos𝑠ℎ₁−𝜏×𝑤₁((𝜏))ℎ𝜏−Δ,𝜆𝑑𝜏×cos𝑠₂−ℎ₁−^ℎ2_ℎ1𝑞(𝜏)cos𝑠ℎ₂−𝜏𝑤₂(𝜏−Δ(𝜏),𝜆)𝑑𝜏×cos𝑠𝜋−ℎ₂−^𝜋_ℎ2𝑞(𝜏)cos𝑠(𝜋−𝜏)𝑤₃(𝜏−Δ(𝜏),𝜆)𝑑𝜏sin𝛽=0. (37) Let 𝜆 be sufficiently big. Then, by (26), (27), and (28), (37) may be rewritten in the following form: −𝑠sin𝛼sin𝛽𝛾𝛿cos𝑠ℎ₂sin𝑠𝜋−ℎ₂+sin𝑠ℎ₂cos𝑠𝜋−ℎ₂(1)(1)+𝑂=0,𝑠sin𝑠𝜋+𝑂=0. (38)(39) Obviously, for big 𝑠 (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)on0,ℎ₁. (40) By (15) and (27), this leads to 𝑤₂(𝑥,𝜆)=𝑂(1)onℎ₁,ℎ₂. (41) By (16) and (28), this leads to 𝑤₃(𝑥,𝜆)=𝑂(1)onℎ₂,𝜋. (42) The existence and continuity of the derivatives 𝑤^′_1𝑠(𝑥,𝜆) for 0≤𝑥≤ℎ₁, |𝜆|<∞, 𝑤^′_2𝑠(𝑥,𝜆) for ℎ₁≤𝑥≤ℎ₂, |𝜆|<∞ and 𝑤^′_3𝑠(𝑥,𝜆) for ℎ₂≤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,𝑤^′_2𝑠(𝑥,𝜆)=𝑂(1),𝑥∈ℎ₁,ℎ₂,𝑤^′_3𝑠(𝑥,𝜆)=𝑂(1),𝑥∈ℎ₂,𝜋. (43)(44)(45)

Proof. By differentiating (16) with respect to 𝑠, we get, by (43) and (44) the following: 𝑤^′_3𝑠()1𝑥,𝜆=−𝑠^𝑥_ℎ2𝑞(𝜏)sin𝑠(𝑥−𝜏)𝑤^′_3𝑠((𝜏))(),|𝑍(𝜏−Δ,𝜆+𝑍𝑥,𝜆𝑥,𝜆)|≤𝑍₀. (46) Let 𝐷_𝜆=max_[ℎ2,𝜋]|𝑤^′_3𝑠(𝑥,𝜆)|. Then the existence of 𝐷_𝜆 follows from continuity of derivation for 𝑥∈[ℎ₂,𝜋]. From (46) 𝐷_𝜆≤1𝑠𝑞₃𝐷_𝜆+𝑍₀. (47) Now let 𝑠≥2𝑞₃. Then 𝐷_𝜆≤2𝑍₀ and the validity of the asymptotic formula (45) follows. Formulas (43) and (44) may be proved analogically.

Theorem 6. Let 𝑛 be a natural number. For each sufficiently big 𝑛 there is exactly one eigenvalue of the problem (1)–(7) near 𝑛².

Proof. We consider the expression which is denoted by 𝑂(1) in (39). If formulas (40)–(45) are taken into consideration, it can be shown by differentiation with respect to 𝑠 that for big 𝑠 this expression has bounded derivative. We will show that, for big 𝑛, only one root (39) lies near to each 𝑛. We consider the function 𝜙(𝑠)=𝑠sin𝑠𝜋+𝑂(1). Its derivative, which has the form 𝜙^′(𝑠)=sin𝑠𝜋+𝑠𝜋cos𝑠𝜋+𝑂(1), does not vanish for 𝑠 close to 𝑛 for sufficiently big 𝑛. Thus our assertion follows by Rolle’s Theorem.

Let 𝑛 be sufficiently big. In what follows we will denote by 𝜆_𝑛=𝑠²_𝑛 the eigenvalue of the problem (1)–(7) situated near 𝑛². We set 𝑠_𝑛=𝑛+𝛿_𝑛. Then from (39) it follows that 𝛿_𝑛=𝑂(1/𝑛). Consequently 𝑠_𝑛1=𝑛+𝑂𝑛,𝜆_𝑛=𝑛²+𝑂(1). (48)(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()1𝑥,𝜆=sin𝛼cos𝑠𝑥+𝑂𝑠,𝑤^′₁(𝑥,𝜆)=−𝑠sin𝛼sin𝑠𝑥+𝑂(1). (50)(51) From expressions of (15), (31), (39), and (41), we easily see that 𝑤₂(𝑥,𝜆)=sin𝛼𝛿1cos𝑠𝑥+𝑂𝑠,𝑤₃(𝑥,𝜆)=sin𝛼1𝛿𝛾cos𝑠𝑥+𝑂𝑠. (52) By substituting (48) into (50), (52), we find that 𝑢_1𝑛=𝑤₁𝑥,𝜆_𝑛1=sin𝛼cos𝑛𝑥+𝑂𝑛,𝑢_2𝑛=𝑤₂𝑥,𝜆_𝑛=sin𝛼𝛿1cos𝑛𝑥+𝑂𝑛,𝑢_3𝑛=𝑤₃𝑥,𝜆_𝑛=sin𝛼1𝛿𝛾cos𝑛𝑥+𝑂𝑛. (53) Hence the eigenfunctions 𝑢_𝑛(𝑥) have the following asymptotic representation: 𝑢_𝑛(𝑥)=1sin𝛼cos𝑛𝑥+𝑂𝑛,for𝑥∈0,ℎ₁,sin𝛼𝛿1cos𝑛𝑥+𝑂𝑛,for𝑥∈ℎ₁,ℎ₂,sin𝛼1𝛿𝛾cos𝑛𝑥+𝑂𝑛,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 fulfilled.

(a) The derivatives 𝑞^′(𝑥) and Δ^′′(𝑥) exist and are bounded in [0,ℎ₁)∪(ℎ₁,ℎ₂)∪(ℎ₂,𝜋] and have finite limits 𝑞^′(ℎ₁±0)=lim_{𝑥→ℎ1±0}𝑞^′(𝑥), 𝑞^′(ℎ₂±0)=lim_{𝑥→ℎ2±0}𝑞^′(𝑥) and Δ^′′(ℎ₁±0)=lim_{𝑥→ℎ1±0}Δ^′′(𝑥),  Δ^′′(ℎ₂±0)=lim_{𝑥→ℎ2±0}Δ^′′(𝑥), respectively.

(b) Δ^′(𝑥)≤1 in [0,ℎ₁)∪(ℎ₁,ℎ₂)∪(ℎ₂,𝜋], Δ(0)=0, lim_{𝑥→ℎ1+0}Δ(𝑥)=0, and lim_{𝑥→ℎ2+0}Δ(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))((𝜏1𝜏−Δ,𝜆=sin𝛼cos𝑠𝜏−Δ))+𝑂𝑠,𝑤₂(𝜏−Δ(𝜏),𝜆)=sin𝛼𝛿((𝜏1cos𝑠𝜏−Δ))+𝑂𝑠,𝑤₃(𝜏−Δ(𝜏),𝜆)=sin𝛼((𝜏1𝛿𝛾cos𝑠𝜏−Δ))+𝑂𝑠 (56)(57)(58) on [0,ℎ₁),  (ℎ₁,ℎ₂) and (ℎ₂,𝜋], respectively.

Under the conditions (a) and (b) the following formulas: ^𝑥₀𝑞(𝜏)((𝜏1cos𝑠2𝜏−Δ))𝑑𝜏=𝑂𝑠_𝑥0,𝑞(𝜏)((𝜏1sin𝑠2𝜏−Δ))𝑑𝜏=𝑂𝑠 (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 cos𝑠𝜋sin𝛼−𝛽−𝑠sin𝛼sin𝛽sin𝑠𝜋−𝛿𝛾sin𝛼sin𝛽2𝛿𝛾^𝜋₀𝑞(𝜏)((𝜏1cos𝑠𝜋−Δ))𝑑𝜏+𝑂𝑠=0. (60) Hence 1tan𝑠𝜋=𝑠sin𝛼−𝛽−1sin𝛼sin𝛽2^𝜋₀𝑞(𝜏)((𝜏1cos𝑠𝜋−Δ))𝑑𝜏+𝑂𝑠². (61) Again, if we take 𝑠_𝑛=𝑛+𝛿_𝑛, then from (48) tan𝑛+𝛿_𝑛𝜋=tan𝛿_𝑛𝜋=1𝑛sin𝛼−𝛽−1sin𝛼sin𝛽2^𝜋₀𝑞(𝜏)((𝜏1cos𝑠𝜋−Δ))𝑑𝜏+𝑂𝑛². (62) Hence for big 𝑛, 𝛿_𝑛=1𝑛𝜋sin𝛼−𝛽−1sin𝛼sin𝛽2^𝜋₀𝑞(𝜏)((𝜏1cos𝑠𝜋−Δ))𝑑𝜏+𝑂𝑛² (63) and finally 𝑠_𝑛1=𝑛+𝑛𝜋sin𝛼−𝛽−1sin𝛼sin𝛽2^𝜋₀𝑞(𝜏)((𝜏1cos𝑠𝜋−Δ))𝑑𝜏+𝑂𝑛². (64) Thus, we have proven the following theorem.

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 for 𝑛→∞.

Now, we may obtain sharper asymptotic formulas for the eigenfunctions. From (14), (56), and (59), we have 𝑤₁()1𝑥,𝜆=sin𝛼cos𝑠𝑥1+2𝑠^𝑥₀𝑞(𝜏)(𝜏)−sin𝑠Δ𝑑𝜏sin𝑠𝑥𝑠cos𝛼+sin𝛼2^𝑥₀𝑞(𝜏)(𝜏)1cos𝑠Δ𝑑𝜏+𝑂𝑠²,𝑥∈0,ℎ₁. (65) Now, replacing 𝑠 by 𝑠_𝑛 and using (64), we have 𝑢_1𝑛(𝑥)=𝑤₁𝑥,𝜆_𝑛1=sin𝛼cos𝑛𝑥1+2𝑛^𝑥₀𝑞(𝜏)((𝜏−sin𝑛Δ))𝑑𝜏sin𝑛𝑥𝑛𝜋sin𝛼−𝛽−1sin𝛼sin𝛽2^𝜋₀𝑞(𝜏)(𝑛((𝜏𝑥+1cos𝜋−Δ)))𝑑𝜏cot𝛼+2^𝑥₀𝑞(𝜏)((𝜏𝜋1cos𝑛Δ))𝑑𝜏+𝑂𝑛². (66) From (15), (57), (59), and (65) we have 𝑤₂(𝑥,𝜆)=sin𝛼cos𝑠𝑥𝛿11+2𝑠^𝑥₀𝑞(𝜏)(𝜏)−sin𝑠Δ𝑑𝜏sin𝑠𝑥𝑠𝛿cos𝛼+sin𝛼2^𝑥₀𝑞(𝜏)(𝜏)1cos𝑠Δ𝑑𝜏+𝑂𝑠²,𝑥∈ℎ₁,ℎ₂. (67) Now, replacing 𝑠 by 𝑠_𝑛 and using (64), we have 𝑢_2𝑛(𝑥)=𝑤₂𝑥,𝜆_𝑛=sin𝛼𝛿1cos𝑛𝑥1+2𝑛^𝑥₀𝑞(𝜏)((𝜏−sin𝑛Δ))𝑑𝜏sin𝑛𝑥𝑛𝜋sin𝛼−𝛽−1sin𝛼sin𝛽2^𝜋₀𝑞(𝜏)(𝑛((𝜏𝑥+1cos𝜋−Δ)))𝑑𝜏cot𝛼+2^𝑥₀𝑞(𝜏)((𝜏𝜋1cos𝑛Δ))𝑑𝜏+𝑂𝑛². (68)

From (16), (58), (59), (65), and (67) and after long operations, we have 𝑤₃(𝑥,𝜆)=sin𝛼cos𝑠𝑥1𝛿𝛾1+2𝑠^𝑥₀𝑞(𝜏)(𝜏)−sin𝑠Δ𝑑𝜏sin𝑠𝑥𝑠𝛿𝛾cos𝛼+sin𝛼2^𝑥₀𝑞(𝜏)(𝜏)1cos𝑠Δ𝑑𝜏+𝑂𝑠²,𝑥∈ℎ₂,𝜋. (69) Now replacing 𝑠 by 𝑠_𝑛 and using (64), we have 𝑢_3𝑛(𝑥)=𝑤₃𝑥,𝜆_𝑛=sin𝛼1𝛿𝛾cos𝑛𝑥1+2𝑛^𝑥₀𝑞(𝜏)((𝜏−sin𝑛Δ))𝑑𝜏sin𝑛𝑥𝑛𝜋sin𝛼−𝛽−1sin𝛼sin𝛽2^𝜋₀𝑞(𝜏)(𝑛((𝜏𝑥+1cos𝜋−Δ)))𝑑𝜏cot𝛼+2^𝑥₀𝑞(𝜏)((𝜏𝜋1cos𝑛Δ))𝑑𝜏+𝑂𝑛². (70)

Thus, we have proven the following theorem.

Theorem 8. If conditions (a) and (b) are satisfied, then the eigenfunctions 𝑢_𝑛(𝑥) of the problem (1)–(7) have the following asymptotic formula for 𝑛→∞: 𝑢_𝑛(𝑥)=𝑢_1𝑛(𝑥),𝑥∈0,ℎ₁,𝑢_2𝑛(𝑥),𝑥∈ℎ₁,ℎ₂,𝑢_3𝑛(𝑥),𝑥∈ℎ₂,𝜋, (71) where 𝑢_1𝑛(𝑥),𝑢_2𝑛(𝑥), and 𝑢_3𝑛(𝑥) are determined as in (49), (68), and (70), respectively.