Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 306917, 8 pages
http://dx.doi.org/10.1155/2013/306917
Research Article
Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a
Sturm-Liouville Problem with Retarded Argument
ErdoLan Fen,1,2 Jong Jin Seo,3 and Serkan Araci4
1
Department of Mathematics, Faculty of Arts and Science, Namik Kemal University, 59030 Tekirdağ, Turkey
Department of Mathematics Engineering, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey
3
Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea
4
Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey
2
Correspondence should be addressed to Jong Jin Seo; seo2011@pknu.ac.kr
Received 18 November 2012; Accepted 1 March 2013
Academic Editor: Suh-Yuh Yang
Copyright © 2013 Erdoğan Şen 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
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) cos 𝛼 + 𝑦󸀠 (0) sin 𝛼 = 0,
(2)
𝑦 (𝜋) cos 𝛽 + 𝑦󸀠 (𝜋) sin 𝛽 = 0,
(3)
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Journal of Applied Mathematics
and transmission conditions
Lemma 1. Let 𝑤(𝑥, 𝜆) be a solution of (1) and 𝜆 > 0. Then the
following integral equations hold:
𝑦 (ℎ1 − 0) − 𝛿𝑦 (ℎ1 + 0) = 0,
(4)
𝑦󸀠 (ℎ1 − 0) − 𝛿𝑦󸀠 (ℎ1 + 0) = 0,
(5)
𝑤1 (𝑥, 𝜆) = sin 𝛼 cos 𝑠𝑥 −
𝑦 (ℎ2 − 0) − 𝛾𝑦 (ℎ2 + 0) = 0,
(6)
−
𝑦󸀠 (ℎ2 − 0) − 𝛾𝑦󸀠 (ℎ2 + 0) = 0,
(7)
cos 𝛼
sin 𝑠𝑥
𝑠
1 𝑥
∫ 𝑞 (𝜏) sin 𝑠 (𝑥 − 𝜏) 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏
𝑠 0
(𝑠 = √𝜆, 𝜆 > 0) ,
(14)
where the real-valued function 𝑞(𝑥) is continuous in [0, ℎ1 ) ∪
(ℎ1 , ℎ2 ) ∪ (ℎ2 , 𝜋] and has finite limits
𝑞 (ℎ1 ± 0) = lim 𝑞 (𝑥) ,
𝑥 → ℎ1 ±0
𝑞 (ℎ2 ± 0) = lim 𝑞 (𝑥) ,
𝑥 → ℎ2 ±0
(8)
𝑤2 (𝑥, 𝜆) =
the real-valued function Δ(𝑥) ≥ 0 continuous in [0, ℎ1 ) ∪
(ℎ1 , ℎ2 ) ∪ (ℎ2 , 𝜋] and has finite limits
Δ (ℎ1 ± 0) = lim Δ (𝑥) ,
𝑥 → ℎ1 ±0
Δ (ℎ2 ± 0) = lim Δ (𝑥) ;
𝑥 → ℎ2 ±0
1
𝑤 (ℎ , 𝜆) cos 𝑠 (𝑥 − ℎ1 )
𝛿 1 1
+
𝑤1󸀠 (ℎ1 , 𝜆)
sin 𝑠 (𝑥 − ℎ1 )
𝑠𝛿
−
1 𝑥
∫ 𝑞 (𝜏) sin 𝑠 (𝑥 − 𝜏) 𝑤2 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏
𝑠 ℎ1
(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 sin 𝛼 sin 𝛽 ≠ 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) = sin 𝛼,
𝑤1󸀠 (0, 𝜆) = − cos 𝛼.
𝑤2󸀠 (ℎ1 , 𝜆) = 𝛿−1 𝑤1󸀠 (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
solution 𝑤3 (𝑥, 𝜆) of (1) on [ℎ2 , 𝜋] by means of the solution
𝑤2 (𝑥, 𝜆) using the initial conditions
𝑤3 (ℎ2 , 𝜆) = 𝛾−1 𝑤2 (ℎ2 , 𝜆) ,
𝑤3󸀠 (ℎ2 , 𝜆) = 𝛾−1 𝑤2󸀠 (ℎ2 , 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 (𝑥, 𝜆) , 𝑥 ∈ [0, ℎ1 ) ,
{
{ 1
𝑤 (𝑥, 𝜆) = {𝑤2 (𝑥, 𝜆) , 𝑥 ∈ (ℎ1 , ℎ2 ) ,
{
{𝑤3 (𝑥, 𝜆) , 𝑥 ∈ (ℎ2 , 𝜋] ,
𝑤3 (𝑥, 𝜆) =
1
𝑤 (ℎ , 𝜆) cos 𝑠 (𝑥 − ℎ2 )
𝛾 2 2
+
𝑤2󸀠 (ℎ2 , 𝜆)
sin 𝑠 (𝑥 − ℎ2 )
𝑠𝛾
−
1 𝑥
∫ 𝑞 (𝜏) sin 𝑠 (𝑥 − 𝜏) 𝑤3 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏
𝑠 ℎ2
(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
𝑤2 (ℎ1 , 𝜆) = 𝛿−1 𝑤1 (ℎ1 , 𝜆) ,
(𝑠 = √𝜆, 𝜆 > 0) ,
(15)
(13)
is a solution of (1) on [0, ℎ1 ) ∪ (ℎ1 , ℎ2 ) ∪ (ℎ2 , 𝜋], which satisfies one of the boundary conditions and four transmission
conditions.
(𝑠 = √𝜆, 𝜆 > 0) .
(16)
Proof. To prove this lemma, it is enough to substitute −𝑠2 𝑤1
(𝜏, 𝜆) − 𝑤1󸀠󸀠 (𝜏, 𝜆), −𝑠2 𝑤2 (𝜏, 𝜆) − 𝑤2󸀠󸀠 (𝜏, 𝜆), and −𝑠2 𝑤3 (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.
̃ be an eigenvalue of the problem (1)–(7) and
Proof. Let 𝜆
̃ ,
𝑦̃ (𝑥, 𝜆)
{
{ 1
̃
̃ ,
𝑦̃ (𝑥, 𝜆) = {𝑦̃2 (𝑥, 𝜆)
{
̃
{𝑦̃3 (𝑥, 𝜆) ,
𝑥 ∈ [0, ℎ1 ) ,
𝑥 ∈ (ℎ1 , ℎ2 ) ,
𝑥 ∈ (ℎ2 , 𝜋] ,
(17)
be a corresponding eigenfunction. Then, from (2) and (10), it
follows that the determinant
󵄨󵄨
󵄨󵄨
̃
󵄨
󵄨
̃ , 𝑤 (0, 𝜆)]
̃ = 󵄨󵄨󵄨𝑦̃1 (0, 𝜆) sin 𝛼 󵄨󵄨󵄨 = 0, (18)
𝑊 [𝑦̃1 (0, 𝜆)
1
󵄨󵄨𝑦̃󸀠 (0, 𝜆)
̃
− cos 𝛼󵄨󵄨󵄨󵄨
󵄨󵄨 1
̃ and 𝑤
and, by Theorem 2.2.2, in [5] the functions 𝑦̃1 (𝑥, 𝜆)
1
̃ are linearly dependent on [0, ℎ ]. We can also prove that
(𝑥, 𝜆)
1
̃ and 𝑤 (𝑥, 𝜆)
̃ are linearly dependent on
the functions 𝑦̃ (𝑥, 𝜆)
2
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Journal of Applied Mathematics
3
̃ and 𝑤 (𝑥, 𝜆)
̃ are linearly dependent on
[ℎ1 , ℎ2 ] and 𝑦̃3 (𝑥, 𝜆)
3
[ℎ2 , 𝜋]. Hence
̃ = 𝐾 𝑤 (𝑥, 𝜆)
̃
𝑦̃𝑖 (𝑥, 𝜆)
𝑖 𝑖
(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
̃ − 𝛾̃
̃
̃2 (ℎ2 , 𝜆)
=𝑦
𝑦3 (ℎ2 , 𝜆)
̃ − 𝛾𝐾 𝑤 (ℎ , 𝜆)
̃
= 𝐾2 𝑤2 (ℎ2 , 𝜆)
3 3
2
(20)
̃ = 0.
= 𝛾 (𝐾2 − 𝐾3 ) 𝑤3 (ℎ2 , 𝜆)
̃ = 0.
𝑤3 (ℎ2 , 𝜆)
̃ = 0.
𝑤3󸀠 (ℎ2 , 𝜆)
(22)
̃ is a solution of the differential (1)
From the fact that 𝑤3 (𝑥, 𝜆)
on [ℎ2 , 𝜋] and satisfies the initial conditions (21) and (22), it
̃ = 0 identically on [ℎ , 𝜋].
follows that 𝑤3 (𝑥, 𝜆)
2
By using this method, we may also find
̃ = 𝑤󸀠 (ℎ , 𝜆)
̃ = 0,
𝑤2 (ℎ2 , 𝜆)
2
2
̃ = 𝑤󸀠 (ℎ , 𝜆)
̃ = 0.
𝑤1 (ℎ1 , 𝜆)
1
1
(23)
̃ it follows that 𝑤
From the latter discussions of 𝑤3 (𝑥, 𝜆),
2
̃
̃
(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) cos 𝛽 + 𝑤󸀠 (𝜋, 𝜆) sin 𝛽 = 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
𝑥 ∈ [ℎ1 , ℎ2 ] .
(27)
(3) Let 𝜆 ≥ max{4𝑞12 , 4𝑞22 , 4𝑞32 }. Then for the solution
𝑤3 (𝑥, 𝜆) of (16), the following inequality holds:
𝑥 ∈ [ℎ2 , 𝜋] .
(28)
1
𝐵1𝜆 ≤ √ sin2 𝛼 +
cos2 𝛼 1
+ 𝐵1𝜆 𝑞1 .
𝑠2
𝑠
(29)
If 𝑠 ≥ 2𝑞1 we get (26). Differentiating (14) with respect to 𝑥,
we have
𝑤1󸀠 (𝑥, 𝜆) = − 𝑠 sin 𝛼 sin 𝑠𝑥 − cos 𝛼 cos 𝑠𝑥
𝑥
− ∫ 𝑞 (𝜏) cos 𝑠 (𝑥 − 𝜏) 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏.
0
(30)
From expressions of (30) and (26), it follows that, for 𝑠 ≥ 2𝑞1 ,
the following inequality holds:
󵄨
󵄨󵄨 󸀠
󵄨󵄨𝑤1 (𝑥, 𝜆)󵄨󵄨󵄨
󵄨 ≤ 1 √4𝑞2 sin2 𝛼 + cos2 𝛼.
󵄨
1
𝑠
𝑞1
(31)
Let 𝐵2𝜆 = 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:
𝐵2𝜆 ≤
1
2
√4𝑞12 sin2 𝛼 + cos2 𝛼 +
𝐵 𝑞.
𝑞1 |𝛿|
2𝑞2 2𝜆 2
(32)
Hence, if 𝜆 ≥ max{4𝑞12 , 4𝑞22 }, it reduces to (27). Differentiating
(15) with respect to, we get
𝑠
𝑤2󸀠 (𝑥, 𝜆) = − 𝑤1 (ℎ1 , 𝜆) sin 𝑠 (𝑥 − ℎ1 )
𝛿
+
𝑤1󸀠 (ℎ1 , 𝜆)
cos 𝑠 (𝑥 − ℎ1 )
𝛿
𝑥
ℎ
2
󵄨
󵄨
𝑞2 = ∫ 󵄨󵄨󵄨𝑞 (𝜏)󵄨󵄨󵄨 𝑑𝜏,
ℎ
2
(2) Let 𝜆 ≥ max{4𝑞12 , 4𝑞22 }. Then for the solution 𝑤2 (𝑥, 𝜆)
of (15), the following inequality holds:
(21)
By the same procedure from equality (7) we can derive that
𝜋
(26)
Proof. Let 𝐵1𝜆 = max[0,ℎ1 ] |𝑤1 (𝑥, 𝜆)|. Then from (14), it follows
that, for every 𝜆 > 0, the following inequality holds:
Since 𝛾(𝐾2 − 𝐾3 ) ≠ 0 it follows that
󵄨
󵄨
𝑞3 = ∫ 󵄨󵄨󵄨𝑞 (𝜏)󵄨󵄨󵄨 𝑑𝜏.
ℎ
𝑥 ∈ [0, ℎ1 ] .
16
󵄨
󵄨󵄨
󵄨󵄨𝑤3 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ 󵄨󵄨 󵄨󵄨 √4𝑞12 sin2 𝛼 + cos2 𝛼,
𝑞1 󵄨󵄨𝛿𝛾󵄨󵄨
̃ − 𝐾 𝛾𝑤 (ℎ , 𝜆)
̃
= 𝐾2 𝛾𝑤3 (ℎ2 , 𝜆)
3
3
2
ℎ
󵄨 1
󵄨󵄨
󵄨󵄨w1 (𝑥, 𝜆)󵄨󵄨󵄨 ≤ √4𝑞12 sin2 𝛼 + cos2 𝛼,
𝑞1
4
󵄨
󵄨󵄨
√4𝑞12 sin2 𝛼 + cos2 𝛼,
󵄨󵄨𝑤2 (𝑥, 𝜆)󵄨󵄨󵄨 ≤
𝑞1 |𝛿|
̃ − 𝛾𝑦̃ (ℎ + 0, 𝜆)
̃
𝑦̃ (ℎ2 − 0, 𝜆)
2
1
󵄨
󵄨
𝑞1 = ∫ 󵄨󵄨󵄨𝑞 (𝜏)󵄨󵄨󵄨 𝑑𝜏,
0
Lemma 3. (1) Let 𝜆 ≥ 4𝑞12 . Then for the solution 𝑤1 (𝑥, 𝜆) of
(14), the following inequality holds:
− ∫ 𝑞 (𝜏) cos 𝑠 (𝑥 − 𝜏) 𝑤2 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏.
ℎ1
(25)
(𝑠 = √𝜆, 𝜆 > 0) .
(33)
4
Journal of Applied Mathematics
From (26) and (33), it follows that, for 𝑠 ≥ 2𝑞1 and 𝑠 ≥ 2𝑞2 ,
the following inequality holds:
󵄨
󵄨󵄨 󸀠
󵄨󵄨𝑤2 (𝑥, 𝜆)󵄨󵄨󵄨
󵄨 ≤ 4 √4𝑞2 sin2 𝛼 + cos2 𝛼.
󵄨
1
𝑠
|𝛿| 𝑞1
× cos 𝑠 (𝜋 − ℎ2 )
+
1
cos 𝛼
𝑠
{− (sin 𝛼 cos 𝑠ℎ1 −
sin 𝑠ℎ1
𝑠𝛾
𝛿
𝑠
(34)
−
Let 𝐵3𝜆 = 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:
𝐵3𝜆
8
1
≤ 󵄨󵄨 󵄨󵄨 √4𝑞12 sin2 𝛼 + cos2 𝛼 + 𝐵3𝜆 𝑞3 .
𝑠
𝑞1 󵄨󵄨𝛿𝛾󵄨󵄨
× 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
× sin 𝑠 (ℎ2 − ℎ1 )
(35)
+
1
(− 𝑠 sin 𝛼 sin 𝑠ℎ1 − cos 𝛼 cos 𝑠ℎ1
𝛿
Hence, if 𝜆 ≥ max{4𝑞12 , 4𝑞22 , 4𝑞32 } we procure (28).
ℎ1
− ∫ 𝑞 (𝜏) cos 𝑠 (ℎ1 − 𝜏)
0
Theorem 4. The problem (1)–(7) has an infinite set of positive
eigenvalues.
Proof. Differentiating (16) with respect to 𝑥, we readily see
that
𝑠
𝑤3󸀠 (𝑥, 𝜆) = − 𝑤2 (ℎ2 , 𝜆) sin 𝑠 (𝑥 − ℎ2 )
𝛾
+
𝑤2󸀠
1 ℎ1
∫ 𝑞 (𝜏) sin 𝑠 (ℎ1 − 𝜏)
𝑠 0
(ℎ2 , 𝜆)
cos 𝑠 (𝑥 − ℎ2 )
𝛾
×𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
× cos 𝑠 (ℎ2 − ℎ1 )
ℎ2
− ∫ 𝑞 (𝜏) cos 𝑠 (ℎ2 − 𝜏) 𝑤2 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏}
ℎ1
× sin 𝑠 (𝜋 − ℎ2 )
𝑥
− ∫ 𝑞 (𝜏) cos 𝑠 (𝑥 − 𝜏) 𝑤3 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏.
ℎ2
(𝑠 = √𝜆, 𝜆 > 0) .
(36)
1 𝜋
− ∫ 𝑞 (𝜏) sin 𝑠 (𝜋 − 𝜏) 𝑤3 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏] cos 𝛽
𝑠 ℎ2
𝑠 1
cos 𝛼
+ [− { ( sin 𝛼 cos 𝑠ℎ1 −
sin 𝑠ℎ1
𝛾 𝛿
𝑠
−
With the helps of (14), (15), (16), (24), (30), and (36), we have
the following:
1 ℎ1
∫ 𝑞 (𝜏) sin 𝑠 (ℎ1 − 𝜏)
𝑠 0
×𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
cos 𝛼
1 1
[ { (sin 𝛼 cos 𝑠ℎ1 −
sin 𝑠ℎ1
𝛾 𝛿
𝑠
−
1 ℎ1
∫ 𝑞 (𝜏) sin 𝑠 (ℎ1 − 𝜏) 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
𝑠 0
× cos 𝑠 (ℎ2 − ℎ1 )
−
1
(𝑠 sin 𝛼 sin 𝑠ℎ1 + cos 𝛼 cos 𝑠ℎ1
𝑠𝛿
ℎ1
× cos 𝑠 (ℎ2 − ℎ1 )
+ ∫ 𝑞 (𝜏) cos 𝑠 (ℎ1 − 𝜏)
0
1
−
(𝑠 sin 𝛼 sin 𝑠ℎ1 + cos 𝛼 cos 𝑠ℎ1
𝑠𝛿
× 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
ℎ1
+ ∫ 𝑞 (𝜏) cos 𝑠 (ℎ1 − 𝜏) 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
0
× sin 𝑠 (ℎ2 − ℎ1 )
1 ℎ2
− ∫ 𝑞 (𝜏) sin 𝑠 (ℎ2 − 𝜏) 𝑤2 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏}
𝑠 ℎ1
× sin 𝑠 (ℎ2 − ℎ1 )
1 ℎ2
− ∫ 𝑞 (𝜏) sin 𝑠 (ℎ2 − 𝜏) 𝑤2 (𝜏 − Δ (𝜏) , 𝜆) }
𝑠 ℎ1
× sin 𝑠 (𝜋 − ℎ2 )
Journal of Applied Mathematics
+
5
cos 𝛼
1
𝑠
{− × (sin 𝛼 cos 𝑠ℎ1 −
sin 𝑠ℎ1
𝛾
𝛿
𝑠
Lemma 5. The following holds true:
󸀠
𝑤1𝑠
(𝑥, 𝜆) = 𝑂 (1) ,
ℎ1
1
− ∫ 𝑞 (𝜏) sin 𝑠 (ℎ1 − 𝜏)
𝑠 0
× 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
(43)
󸀠
𝑤2𝑠
(𝑥, 𝜆) = 𝑂 (1) ,
𝑥 ∈ [ℎ1 , ℎ2 ] ,
(44)
󸀠
𝑤3𝑠
(𝑥, 𝜆) = 𝑂 (1) ,
𝑥 ∈ [ℎ2 , 𝜋] .
(45)
Proof. By differentiating (16) with respect to 𝑠, we get, by (43)
and (44) the following:
× sin 𝑠 (ℎ2 − ℎ1 )
+
𝑥 ∈ [0, ℎ1 ] ,
1
(− 𝑠 sin 𝛼 sin 𝑠ℎ1 − cos 𝛼 cos 𝑠ℎ1
𝛿
󸀠
𝑤3𝑠
(𝑥, 𝜆)
1 𝑥
󸀠
= − ∫ 𝑞 (𝜏) sin 𝑠 (𝑥 − 𝜏) 𝑤3𝑠
(𝜏 − Δ (𝜏) , 𝜆) + 𝑍 (𝑥, 𝜆) ,
𝑠 ℎ2
ℎ1
− ∫ 𝑞 (𝜏) cos 𝑠 (ℎ1 − 𝜏)
0
(|𝑍 (𝑥, 𝜆)| ≤ 𝑍0 ) .
(46)
× 𝑤1 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏)
× cos 𝑠 (ℎ2 − ℎ1 )
󸀠
Let 𝐷𝜆 = max[ℎ2 ,𝜋] |𝑤3𝑠
(𝑥, 𝜆)|. Then the existence of 𝐷𝜆 follows from continuity of derivation for 𝑥 ∈ [ℎ2 , 𝜋]. From (46)
ℎ2
− ∫ 𝑞 (𝜏) cos 𝑠 (ℎ2 − 𝜏) 𝑤2 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏}
1
𝐷𝜆 ≤ 𝑞3 𝐷𝜆 + 𝑍0 .
𝑠
ℎ1
× cos 𝑠 (𝜋 − ℎ2 )
𝜋
− ∫ 𝑞 (𝜏) cos 𝑠 (𝜋 − 𝜏) 𝑤3 (𝜏 − Δ (𝜏) , 𝜆) 𝑑𝜏] sin 𝛽 = 0.
ℎ2
(37)
Let 𝜆 be sufficiently big. Then, by (26), (27), and (28), (37)
may be rewritten in the following form:
−
𝑠 sin 𝛼 sin 𝛽
{cos 𝑠ℎ2 sin 𝑠 (𝜋 − ℎ2 )
𝛾𝛿
(38)
+ sin 𝑠ℎ2 cos 𝑠 (𝜋 − ℎ2 )} + 𝑂 (1) = 0,
𝑠 sin 𝑠𝜋 + 𝑂 (1) = 0.
(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)
on [0, ℎ1 ] .
(40)
on [ℎ1 , ℎ2 ] .
(41)
By (15) and (27), this leads to
𝑤2 (𝑥, 𝜆) = 𝑂 (1)
By (16) and (28), this leads to
𝑤3 (𝑥, 𝜆) = 𝑂 (1)
on [ℎ2 , 𝜋] .
(42)
󸀠
(𝑥, 𝜆)
𝑤1𝑠
for
The existence and continuity of the derivatives
󸀠
0 ≤ 𝑥 ≤ ℎ1 , |𝜆| < ∞, 𝑤2𝑠 (𝑥, 𝜆) for ℎ1 ≤ 𝑥 ≤ ℎ2 , |𝜆| < ∞ and
󸀠
𝑤3𝑠
(𝑥, 𝜆) for ℎ2 ≤ 𝑥 ≤ 𝜋, |𝜆| < ∞ follows from Theorem 1.4.1
in [5].
(47)
Now let 𝑠 ≥ 2𝑞3 . Then 𝐷𝜆 ≤ 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) = 𝑠 sin 𝑠𝜋 + 𝑂(1). Its derivative, which has
the form 𝜙󸀠 (𝑠) = sin 𝑠𝜋 + 𝑠𝜋 cos 𝑠𝜋 + 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 = 𝑠𝑛2 the eigenvalue of the problem (1)–(7) situated near 𝑛2 .
We set 𝑠𝑛 = 𝑛+𝛿𝑛 . Then from (39) it follows that 𝛿𝑛 = 𝑂(1/𝑛).
Consequently
1
𝑠𝑛 = 𝑛 + 𝑂 ( ) ,
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
1
𝑤1 (𝑥, 𝜆) = sin 𝛼 cos 𝑠𝑥 + 𝑂 ( ) ,
𝑠
(50)
𝑤1󸀠 (𝑥, 𝜆) = −𝑠 sin 𝛼 sin 𝑠𝑥 + 𝑂 (1) .
(51)
6
Journal of Applied Mathematics
From expressions of (15), (31), (39), and (41), we easily see that
𝑤2 (𝑥, 𝜆) =
sin 𝛼
1
cos 𝑠𝑥 + 𝑂 ( ) ,
𝛿
𝑠
1
sin 𝛼
𝑤3 (𝑥, 𝜆) =
cos 𝑠𝑥 + 𝑂 ( ) .
𝛿𝛾
𝑠
𝑢3𝑛
sin 𝛼
1
= 𝑤3 (𝑥, 𝜆 𝑛 ) =
cos 𝑛𝑥 + 𝑂 ( ) .
𝛿𝛾
𝑛
(53)
for 𝑥 ∈ [0, ℎ1 ) ,
𝑥 − Δ (𝑥) ≥ ℎ1 ,
𝑥 ∈ (ℎ1 , ℎ2 ) ,
1 sin (𝛼 − 𝛽)
(
𝑠 sin 𝛼 sin 𝛽
1 𝜋
− ∫ 𝑞 (𝜏) cos 𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏)
2 0
+ 𝑂(
(55)
𝑥 ∈ (ℎ2 , 𝜋]
1
𝑤1 (𝜏 − Δ (𝜏) , 𝜆) = sin 𝛼 cos 𝑠 (𝜏 − Δ (𝜏)) + 𝑂 ( ) ,
𝑠
on [0, ℎ1 ), (ℎ1 , ℎ2 ) and (ℎ2 , 𝜋], respectively.
1
).
𝑠2
Again, if we take 𝑠𝑛 = 𝑛 + 𝛿𝑛 , then from (48)
tan ((𝑛 + 𝛿𝑛 ) 𝜋) = tan 𝛿𝑛 𝜋
=
1 sin (𝛼 − 𝛽)
(
𝑛 sin 𝛼 sin 𝛽
1 𝜋
− ∫ 𝑞 (𝜏) cos 𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏)
2 0
+ 𝑂(
𝛿𝑛 =
1
).
𝑛2
(62)
1 sin (𝛼 − 𝛽) 1 𝜋
(
− ∫ 𝑞 (𝜏) cos 𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏)
𝑛𝜋 sin 𝛼 sin 𝛽 2 0
+ 𝑂(
(56)
sin 𝛼
1
cos 𝑠 (𝜏 − Δ (𝜏)) + 𝑂 ( ) , (57)
𝛿
𝑠
sin 𝛼
1
𝑤3 (𝜏 − Δ (𝜏) , 𝜆) =
cos 𝑠 (𝜏 − Δ (𝜏)) + 𝑂 ( )
𝛿𝛾
𝑠
(61)
Hence for big 𝑛,
are obtained.
By (50), (52), and (55), we have
𝑤2 (𝜏 − Δ (𝜏) , 𝜆) =
Hence
for 𝑥 ∈ (ℎ2 , 𝜋] .
𝑥 ∈ [0, ℎ1 ) ,
(60)
1
+ 𝑂 ( ) = 0.
𝑠
tan 𝑠𝜋 =
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) =
lim𝑥 → ℎ1 ±0 𝑞󸀠 (𝑥), 𝑞󸀠 (ℎ2 ±0) = lim𝑥 → ℎ2 ±0 𝑞󸀠 (𝑥) and Δ󸀠󸀠 (ℎ1 ±0) =
lim𝑥 → ℎ1 ±0 Δ󸀠󸀠 (𝑥), Δ󸀠󸀠 (ℎ2 ±0) = lim𝑥 → ℎ2 ±0 Δ󸀠󸀠 (𝑥), respectively.
(b) Δ󸀠 (𝑥) ≤ 1 in [0, ℎ1 ) ∪ (ℎ1 , ℎ2 ) ∪ (ℎ2 , 𝜋], Δ(0) = 0,
lim𝑥 → ℎ1 +0 Δ(𝑥) = 0, and lim𝑥 → ℎ2 +0 Δ(𝑥) = 0.
It is easy to see that, using (b)
𝑥 − Δ (𝑥) ≥ ℎ2 ,
sin 𝛼 sin 𝛽 𝜋
∫ 𝑞 (𝜏) cos 𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏
2𝛿𝛾
0
for 𝑥 ∈ (ℎ1 , ℎ2 ) , (54)
𝑥 − Δ (𝑥) ≥ 0,
(59)
cos 𝑠𝜋 sin (𝛼 − 𝛽) − 𝑠 sin 𝛼 sin 𝛽 sin 𝑠𝜋
𝛿𝛾
−
Hence the eigenfunctions 𝑢𝑛 (𝑥) have the following asymptotic representation:
1
sin 𝛼 cos 𝑛𝑥 + 𝑂 ( ) ,
{
{
{
𝑛
{
{
{
{
{
{
{
{ sin 𝛼
1
cos 𝑛𝑥 + 𝑂 ( ) ,
𝑢𝑛 (𝑥) = {
𝛿
𝑛
{
{
{
{
{
{
{
{
{
{ sin 𝛼 cos 𝑛𝑥 + 𝑂 ( 1 ) ,
𝑛
{ 𝛿𝛾
𝑥
1
∫ 𝑞 (𝜏) sin 𝑠 (2𝜏 − Δ (𝜏)) 𝑑𝜏 = 𝑂 ( )
𝑠
0
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
1
𝑢1𝑛 = 𝑤1 (𝑥, 𝜆 𝑛 ) = sin 𝛼 cos 𝑛𝑥 + 𝑂 ( ) ,
𝑛
𝑢2𝑛
𝑥
1
∫ 𝑞 (𝜏) cos 𝑠 (2𝜏 − Δ (𝜏)) 𝑑𝜏 = 𝑂 ( ) ,
𝑠
0
(52)
By substituting (48) into (50), (52), we find that
sin 𝛼
1
= 𝑤2 (𝑥, 𝜆 𝑛 ) =
cos 𝑛𝑥 + 𝑂 ( ) ,
𝛿
𝑛
Under the conditions (a) and (b) the following formulas:
(58)
1
)
𝑛2
(63)
and finally
𝑠𝑛 = 𝑛 +
1 sin (𝛼 − 𝛽) 1 𝜋
(
− ∫ 𝑞 (𝜏) cos 𝑠 (𝜋 − Δ (𝜏)) 𝑑𝜏)
𝑛𝜋 sin 𝛼 sin 𝛽 2 0
+ 𝑂(
1
).
𝑛2
Thus, we have proven the following theorem.
(64)
Journal of Applied Mathematics
7
Theorem 7. If conditions (a) and (b) are satisfied, then the
eigenvalues 𝜆 𝑛 = 𝑠𝑛2 of the problem (1)–(7) have the (64)
asymptotic formula for 𝑛 → ∞.
−
sin (𝛼 − 𝛽)
sin 𝑛𝑥
[(
𝑛𝜋
sin 𝛼 sin 𝛽
1 𝜋
− ∫ 𝑞(𝜏) cos(𝑛(𝜋−Δ(𝜏)))𝑑𝜏) 𝑥
2 0
Now, we may obtain sharper asymptotic formulas for the
eigenfunctions. From (14), (56), and (59), we have
𝑤1 (𝑥, 𝜆) = sin 𝛼 cos 𝑠𝑥 [1 +
1 𝑥
+(cot𝛼+ ∫ 𝑞(𝜏)cos(𝑛Δ (𝜏)) 𝑑𝜏) 𝜋]}
2 0
𝑥
1
∫ 𝑞 (𝜏) sin 𝑠Δ (𝜏) 𝑑𝜏]
2𝑠 0
+ 𝑂(
sin 𝑠𝑥
sin 𝛼 𝑥
−
[cos 𝛼 +
∫ 𝑞 (𝜏) cos 𝑠Δ (𝜏) 𝑑𝜏]
𝑠
2 0
+ 𝑂(
1
),
𝑠2
𝑥 ∈ [0, ℎ1 ) .
(65)
(68)
From (16), (58), (59), (65), and (67) and after long operations, we have
𝑤3 (𝑥, 𝜆) =
Now, replacing 𝑠 by 𝑠𝑛 and using (64), we have
1 𝑥
sin 𝛼 cos 𝑠𝑥
[1 + ∫ 𝑞 (𝜏) sin 𝑠Δ (𝜏) 𝑑𝜏]
𝛿𝛾
2𝑠 0
−
𝑢1𝑛 (𝑥)
= 𝑤1 (𝑥, 𝜆 𝑛 )
1 𝑥
+(cot𝛼+ ∫ 𝑞 (𝜏) cos (𝑛Δ (𝜏)) 𝑑𝜏)𝜋]}
2 0
=
sin 𝛼
1 𝑥
{cos 𝑛𝑥 [1 +
∫ 𝑞 (𝜏) sin (𝑛Δ (𝜏)) 𝑑𝜏]
𝛿𝛾
2𝑛 0
From (15), (57), (59), and (65) we have
1 𝑥
+(cot𝛼+ ∫ 𝑞 (𝜏)cos(𝑛Δ(𝜏)) 𝑑𝜏) 𝜋]}
2 0
sin 𝛼 cos 𝑠𝑥
1 𝑥
[1 + ∫ 𝑞 (𝜏) sin 𝑠Δ (𝜏) 𝑑𝜏]
𝛿
2𝑠 0
𝑥
sin 𝑠𝑥
sin 𝛼
[cos 𝛼 +
∫ 𝑞 (𝜏) cos 𝑠Δ (𝜏) 𝑑𝜏]
𝑠𝛿
2 0
1
),
𝑠2
𝑢2𝑛 (𝑥)
= 𝑤2 (𝑥, 𝜆 𝑛 )
sin 𝛼
1 𝑥
{cos 𝑛𝑥 [1 +
∫ 𝑞 (𝜏) sin (𝑛Δ (𝜏)) 𝑑𝜏]
𝛿
2𝑛 0
+ 𝑂(
1
).
𝑛2
(70)
Thus, we have proven the following theorem.
𝑥 ∈ (ℎ1 , ℎ2 ) .
Now, replacing 𝑠 by 𝑠𝑛 and using (64), we have
sin (𝛼 − 𝛽)
sin 𝑛𝑥
[(
𝑛𝜋
sin 𝛼 sin 𝛽
1 𝜋
− ∫ 𝑞(𝜏)cos(𝑛(𝜋−Δ(𝜏)))𝑑𝜏)𝑥
2 0
(66)
+ 𝑂(
(69)
= 𝑤3 (𝑥, 𝜆 𝑛 )
−
1
).
𝑛2
−
𝑥 ∈ (ℎ2 , 𝜋] .
𝑢3𝑛 (𝑥)
1 𝜋
− ∫ 𝑞 (𝜏) cos (𝑛 (𝜋 − Δ (𝜏))) 𝑑𝜏) 𝑥
2 0
𝑤2 (𝑥, 𝜆) =
1
),
𝑠2
Now replacing 𝑠 by 𝑠𝑛 and using (64), we have
sin (𝛼 − 𝛽)
sin 𝑛𝑥
−
[(
𝑛𝜋
sin 𝛼 sin 𝛽
+ 𝑂(
sin 𝑠𝑥
sin 𝛼 𝑥
[cos 𝛼 +
∫ 𝑞 (𝜏) cos 𝑠Δ (𝜏) 𝑑𝜏]
𝑠𝛿𝛾
2 0
+ 𝑂(
1 𝑥
= sin 𝛼 {cos 𝑛𝑥 [1 +
∫ 𝑞 (𝜏) sin (𝑛Δ (𝜏)) 𝑑𝜏]
2𝑛 0
=
1
).
𝑛2
(67)
Theorem 8. If conditions (a) and (b) are satisfied, then the
eigenfunctions 𝑢𝑛 (𝑥) of the problem (1)–(7) have the following
asymptotic formula for 𝑛 → ∞:
𝑢 (𝑥) , 𝑥 ∈ [0, ℎ1 ) ,
{
{ 1𝑛
𝑢𝑛 (𝑥) = {𝑢2𝑛 (𝑥) , 𝑥 ∈ (ℎ1 , ℎ2 ) ,
{
{𝑢3𝑛 (𝑥) , 𝑥 ∈ (ℎ2 , 𝜋] ,
(71)
where 𝑢1𝑛 (𝑥), 𝑢2𝑛 (𝑥), and 𝑢3𝑛 (𝑥) are determined as in (49),
(68), and (70), respectively.
8
References
[1] G. Teschl, Ordinary Differential Equations and Dynamical Systems, vol. 140, American Mathematical Society, Providence, RI,
USA, 2012.
[2] M. Shahriari, A. J. Akbarfam, and G. Teschl, “Uniqueness for
inverse Sturm-Liouville problems with a finite number of transmission conditions,” Journal of Mathematical Analysis and
Applications, vol. 395, no. 1, pp. 19–29, 2012.
[3] T. Kim, “Identities involving Frobenius-Euler polynomials arising from non-linear differential equations,” Journal of Number
Theory, vol. 132, no. 12, pp. 2854–2865, 2012.
[4] S. B. Norkin, “On a boundary problem of Sturm-Liouville type
for a second-order differential equation with a retarded argument,” Izvestija Vysših Učebnyh Zavedeniı̆ Matematika, vol. 6,
no. 7, pp. 203–214, 1958 (Russian).
[5] S. B. Norkin, Differential Equations of the Second Order with Retarded Argument, vol. 31 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA,
1972.
[6] R. Bellman and K. L. Cooke, Differential-Difference Equations,
Academic Press, London, UK, 1963.
[7] G. V. Demidenko and V. A. Likhoshvaı̆, “On differential equations with retarded argument,” Siberian Mathematical Journal,
vol. 46, no. 3, pp. 417–430, 2005.
[8] G. V. Demidenko and I. I. Matveeva, “On numerical study of
asymptotic stability of solutions to linear periodic differential
equations with a parameter,” Journal of Computational Mathematics and Optimization, vol. 5, no. 3, pp. 163–173, 2009.
[9] A. Bayramov, S. Öztürk Uslu, and S. Aliskan, “Computation of
eigenvalues and eigenfunctions of a discontinuous boundary
value problem with retarded argument,” Applied Mathematics
and Computation, vol. 191, no. 2, pp. 592–600, 2007.
[10] E. Şen, S. Araci, and M. Acikgoz, “Asymptotic behavior of eigenvalues and fundamental solutions of one discontinuous fourthorder boundary value problem,” Research Journal of Mathematical and Statistical Sciences, vol. 1, no. 1, pp. 23–32, 2013.
[11] E. Şen and S. Araci, “Computation of Eigenvalues and fundamental solutions of a fourth-order boundary value,” Proceedings
of the Jangjeon Mathematical Society, vol. 15, no. 4, pp. 455–464,
2012.
[12] E. Şen and A. Bayramov, “On calculation of eigenvalues and
eigenfunctions of a Sturm-Liouville type problem with retarded
argument which contains a spectral parameter in the boundary
condition,” Journal of Inequalities and Applications, vol. 2011,
article 113, 2011.
[13] E. Şen and A. Bayramov, “On a discontinuous Sturm-Liouville
type problem with retarded argument,” American Institute of
Physics Conference Proceedings, vol. 1389, pp. 1172–1175, 2011.
[14] E. Şen and A. Bayramov, “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,
vol. 54, no. 11-12, pp. 3090–3097, 2011.
[15] A. Bayramov and E. Şen, “On a Sturm-Liouville type problem
with retarded argument,” Mathematical Methods in the Applied
Sciences, vol. 36, no. 1, pp. 39–48, 2013.
[16] C. T. Fulton, “Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions,” Proceedings of the Royal Society of Edinburgh A, vol. 77, no. 3-4, pp.
293–308, 1977.
Journal of Applied Mathematics
[17] P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville
problems with boundary conditions rationally dependent on
the eigenparameter II,” Journal of Computational and Applied
Mathematics, vol. 148, no. 1, pp. 147–168, 2002.
[18] N. Altinisik, M. Kadakal, and O. Sh. Mukhtarov, “Eigenvalues
and eigenfunctions of discontinuous Sturm-Liouville problems
with eigenparameter-dependent boundary conditions,” Acta
Mathematica Hungarica, vol. 102, no. 1-2, pp. 159–175, 2004.
[19] N. B. Kerimov and Kh. R. Mamedov, “On a boundary value
problem with a spectral parameter in the boundary conditions,”
Sibirskiı̆ Matematicheskiı̆ Zhurnal, vol. 40, no. 2, pp. 325–335,
1999.
[20] I. Titeux and Y. Yakubov, “Completeness of root functions for
thermal conduction in a strip with piecewise continuous coefficients,” Mathematical Models & Methods in Applied Sciences,
vol. 7, no. 7, pp. 1035–1050, 1997.
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