Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 720235, 21 pages
doi:10.1155/2010/720235
Research Article
Accurate Asymptotic Formulas for
Eigenvalues and Eigenfunctions of
a Boundary-Value Problem of Fourth Order
Hamza Menken
Mathematics Department, Science and Arts Faculty, Mersin University, 33343 Mersin, Turkey
Correspondence should be addressed to Hamza Menken, hmenken@mersin.edu.tr
Received 7 July 2010; Accepted 9 November 2010
Academic Editor: I. T. Kiguradze
Copyright q 2010 Hamza Menken. 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, we consider a nonself-adjoint fourth-order differential operator with
the periodic boundary conditions. We compute new accurate asymptotic expression of the
fundamental solutions of the given equation. Then, we obtain new accurate asymptotic formulas
for eigenvalues and eigenfunctions.
1. Introduction
In the present work, we consider a nonself-adjoint fourth-order operator which is generated
by the periodic boundary conditions:
y4 qxy λy,
yj 0 − yj π 0,
0 ≤ x ≤ π,
j 0, 1, 2, 3,
1.1
1.2
where
qx is a complex-valued function. Without lose of generality, we can assume that
π
qxdx
0.
0
Spectral properties of Sturm-Liouville operator which is generated by the periodic and
antiperiodic boundary conditions have been investigated by many authors, the results on this
direct and references are given details in the monographs 1–5.
In this paper we obtain asymptotic formulas for the eigenvalues and eigenfunctions
of the fourth-order boundary-value problem 1.1, 1.2. For second-order differential
equations, similar asymptotic formulas were obtained in 6–9. We note that in 6, 10, 11,
2
Boundary Value Problems
using the obtained asymptotic formulas for eigenvalues and eigenfunctions, the basis
properties of the root functions of the operators were investigated.
The paper is organized as follows. In Section 2, we compute new asymptotic
expression of the fundamental solutions of 1.1. In Section 3, we obtain new accurate
asymptotic estimates for the eigenvalues. In Section 4, we have asymptotic formulas for
eigenfunctions under the distinct conditions on qx.
2. The Expression of the Fundamental Solutions
It is well known that see 2, page 92 if the complex s-plane s4 λ is divided into eight
sectors S 0, 7, defined by the inequalities
0, 7 ,
π
1π
≤ arg s ≤
,
8
8
2.1
then in each of these sectors 1.1 has four linear independent solutions yk x, s k 1, 2, 3, 4,
which are regular with respect to s in the sector Sl for |s| sufficiently large and which satisfy
the relation
yk x, s e
ωk sx
7
uv,k x
1
,
O 8
v
s
s
υ0
2.2
k 1, 2, 3, 4,
where the numbers ωk are the fourth roots of unity, that is, ω1 −ω4 i and ω2 −ω3 1.
In general, the term Os−N1 at the formula 2.2 depends upon the smoothness of the
function qx. If qx has m continuous derivatives, then one can assert the existence of a
representation 2.2 with N m 3. Here, we assume that qx ∈ C4 0, π. The functions
uυ,k x satisfy the following recursion relations:
4uυ,k x 6ωk3 uυ−1,k x 4ωk2 u
υ−2,k x
4
ωk uυ−3,k x ωk qxuυ−3,k x 0,
uυ,k x ≡ 0, υ < 0.
2.3
Let us put, moreover, u0,k x ≡ 1, uυ,k 0 ≡ 0, for υ ≥ 1. Thus, the functions uυ,k x are
uniquely determined. Thus, we can find from 2.3 that
u0,k x 1,
u1,k x 0,
u4,k x u2,k x 0,
3
qx − q0 ,
8
u6,k x 3ωk2 32
1
u3,k x − 3
4ωk
u5,k x −
q x − q 0 5ωk3 16
x
ωk2
32
3ωk ωk u7,k x −
q x − q 0 −
qx − q0
64
32
0
qtdt,
0
q x − q 0 ,
2.4
2
qtdt
,
0
x
x
3ωk
qtdt −
32
x
0
q2 tdt.
Boundary Value Problems
3
3. The Asymptotic Formulas of Eigenvalues
It follows from the classical investigations see 4, page 65 that the eigenvalues of the
problem 1.1, 1.2 in 0, 1 consist of the pairs of the sequences {λk,1 }, {λk,2 } satisfying
the following asymptotic formula:
1
ξ0
λk,1 λk,2 O k1/2 2kπ4 1 O 3/2
k
k
3.1
for sufficiently large integer k, where ξ0 is a constant.
Theorem 3.1. Assume that qx ∈ C4 0, π. Then, the eigenvalues of the boundary-value problem
1.1, 1.2 form two infinite sequences λk,1 , λk,2 k N, N 1, . . ., where N is a big positive integer
and have the following asymptotic formulas:
λk,1
3
2ki 8π
λk,2
3
2k 8π
π
0
4
4
π
q2 tdt
2ki4
q2 tdt
0
2k4
O
1
,
k8
3.2
1
O 8 .
k
Proof. By derivation of 2.2 up to third order with respect to x, the following relations are
obtained:
m
yk x, s
⎡
7 um x
υ,k
m ωk sx ⎣
ωk s e
sυ
υ0
⎤
1 ⎦
O 8
,
s
3.3
where k 1, 2, 3, 4, m 1, 2, 3 and
m
u0,k x 1,
1
u4,k x m
3
1
qx − q0,
8
8
1
u5,k x m
u1,k x 0,
ωk3
16
m
u2,k x 0,
u3,k x −
3
1
2
u4,k x − qx − q0,
8
8
q x 5ωk3
16
3
q 0,
u5,k x ωk3
16
2
u5,k x q x 5ωk3
16
1
4ωk3
x
qtdt,
0
3
3
3
u4,k x − qx − q0,
8
8
3ωk3
16
q 0,
q x 5ωk3
16
q 0,
4
Boundary Value Problems
1
u6,k x
−
2
u6,k x −
3
u6,k x 1
u7,k x 2
32
3ωk2
32
3ωk2
32
q x q 0 q x −
q x −
5ωk2
32
5ωk2
32
ωk2
ωk2
2
qtdt
32
q 0 q 0 x
32
ωk ωk ωk q x q 0 qx 3q0
64
64
32
,
0
x
32
0
2 x
ωk
9ωk ωk ωk q x q 0 −
qx − 3q0
64
64
32
u7,k x −
3
u7,k x
5ωk2 2
qtdt
,
2
qtdt
,
qtdt −
3ωk
32
0
x
0
x
qtdt −
0
7ωk ωk 3ωk q x q 0 −
qx 3q0
64
64
32
x
0
3ωk
32
x
q2 tdt,
0
x
3ωk
qtdt −
32
q2 tdt,
0
x
q2 tdt.
0
3.4
Now let us substitute all these expressions into the characteristic determinant
U1 y1
U2 y1
Δs U3 y1
U4 y1
U1 y2
U2 y2
U3 y2
U4 y2
U1 y3
U2 y3
U3 y3
U4 y3
U1 y4 U2 y4 ,
U3 y4 U4 y4 3.5
where
j
j
Uj1 yk yk π − yk 0,
j 0, 1, 2, 3 .
3.6
By long computations, for sufficiently large |s|, we obtain that
π
1
O 8
Δs 16is e
s7
s
π 2
3 q0
3 q0
1
3 0 q tdt
1
−sπ
− 2 1−
1−
O 8
O 8
e
7
4
4
2 s
2 s
32
s
s
s
π 2
1
3i 0 q tdt
1
−
2
1
O
× eisπ 1 −
O
32
s7
s8
s8
π 2
3i 0 q tdt
1
−isπ
.
1
O 8
e
32
s7
s
3.7
6
sπ
3 q0
3
1−
−
2 s4
32
0
q2 tdt
Boundary Value Problems
5
Multiplying the last equation by
e
3 q0
3
1
2 s4
32
sπ
π
0
q2 tdt
s7
1
O 8
s
e
isπ
3i
1
32
π
0
q2 tdt
s7
1
O 8
s
,
3.8
it becomes
e
sπ
3
− 1
32
π
0
q2 tdt
s7
1
O 8
s
π 2
2
2 3i 0 q tdt
1
isπ
e − 1
O 8
.
7
32
s
s
3.9
Hence, by Δs 0, for sufficiently large |s|, the following equations hold:
e
sπ
eisπ
π
1
,
s7
s8
π 2
3i 0 q tdt
1
−1
O 8 .
7
32
s
s
3
−1
32
0
q2 tdt
O
3.10
3.11
By Rouche’s theorem, we have asymptotic estimates for the roots sk,1 and sk,2 , k N, N 1, . . . , of 3.10 and 3.11, respectively, where N is a big positive integer
sk,1
3
2ki 32π
sk,2
3
2k 32π
π
0
π
0
q2 tdt
2ki7
q2 tdt
2k7
1
,
k8
3.12
1
O 8 .
k
3.13
O
From the relations 3.12, 3.13 and the relations λk,j s4k,j , j 1, 2, the asymptotic formulas
3.2 are valid for k ≥ N.
4. The Asymptotic Formulas for the Eigenfunctions
Now, we obtain asymptotic formulas for eigenfunctions under the distinct conditions on qx.
Case 1. Assume that qx ∈ C1 0, π and the condition qπ − q0 /
0 holds. Based on the
asymptotic expressions of the fundamental solutions of 1.1 and the asymptotic formulas
for eigenvalues of the boundary-value problem 1.1, 1.2 up to order Os−5 , the following
result is valid.
6
Boundary Value Problems
Theorem 4.1. If the condition qπ − q0 /
0 holds, then eigenfunctions of the boundary-value
problem 1.1, 1.2 corresponding the eigenvalues λk,1 and λk,2 are of the form
1
,
k
1
,
yk,2 x cos2kx cosh2kx O
k
yk,1 x sin2kx − sinh2kx O
4.1
4.2
where k is sufficiently large integer.
Proof. Let us calculate Uj1 yυ x, sk,1 j 0, 1, 2 up to order Os−5
. Since
k,1
e
sk,1 π
3
−1
32
π
0
q2 tdt
s7k,1
O
1
4.3
,
s8k,1
we obtain that
3 qπ − q0
U1 yυ x, sk,1 O
8
s4k,1
U2 yυ x, sk,1 ωυ sk,1
s5k,1
1 qπ − q0
O
8
s4k,1
1
1
s5k,1
1 qπ − q0
U3 yυ x, sk,1 ωυ sk,1 −
O
8
s4k,1
2
,
4.4
,
1
s5k,1
.
Follows from the condition qπ − q0 /
0 that Uj1 yυ x, sk,1 /
0 for j 0, 1, 2. Thus, we
can seek eigenfunction yk,1 x corresponding λk,1 in the form
y1 x, sk,1 y2 x, sk,1 y3 x, sk,1 y4 x, sk,1 U1 y1
U
y
U
y
U
y
1
2
1
3
1
4
yk,1 x .
U2 y1
U
y
U
y
U
y
2
2
2
3
2
4
U3 y1
U3 y2
U3 y3
U3 y4 Then,
U1 y2 U1 y3 U1 y4 yk,1 x y1 x, sk,1 U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 − y2 x, sk,1 U2 y1 U2 y3 U2 y4 U3 y1 U3 y3 U3 y4 4.5
Boundary Value Problems
7
U1 y1 U1 y2 U1 y4 y3 x, sk,1 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4 U1 y1 U1 y2 U1 y3 − y4 x, sk,1 U2 y1 U2 y2 U2 y3 .
U3 y1 U3 y2 U3 y3 4.6
By simple computations, we obtain
U1 y2 U1 y3 U1 y4 3 qπ − q0
1
U2 y2 U2 y3 U3 y4 − 3
1O
,
sk,1
27
s9k,1
U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 3 qπ − q0
1
U2 y1 U2 y3 U2 y4 − 3i
1
O
,
sk,1
27
s9k,1
U3 y1 U3 y3 U3 y4 U1 y1 U1 y2 U1 y4 3 qπ − q0
1
U2 y1 U2 y2 U2 y4 − 3i
1
O
,
sk,1
27
s9k,1
U3 y1 U3 y2 U3 y4 U1 y1 U1 y2 U1 y3 3 qπ − q0
1
U2 y1 U2 y2 U2 y3 − 3
1
O
.
sk,1
27
s9k,1
U3 y1 U3 y2 U3 y3 4.7
Hence, using the formula 2.2, we can write
yk,1 x −
−
3 1
3 qπ − q0
isk,1 x
sk,1 x
−sk,1 x
−isk,1 x
−
ie
ie
−
e
O
e
sk,1
27
s9k,1
3 3 qπ − q0
1
x
−
2i
sinh
s
x
O
2i
sin
s
k,1
k,1
sk,1
27
s9k,1
4.8
3 3i qπ − q0
1
− 6
sin
s
.
x
−
sinh
s
x
O
k,1
k,1
sk,1
2
s9k,1
Therefore, for the normalized eigenfunction, we get
yk,1 x sin sk,1 x − sinh sk,1 x O
1
sk,1
.
4.9
8
Boundary Value Problems
Using the relations 3.3 and 3.12, for sufficiently large integer k, we obtain 4.1
yk,1 x sin2kx − sinh2kx O
Similarly, since Uj1 yυ x, sk,1 / 0 for j
corresponding λk,2 in the form
1
.
k
4.10
1, 2, 3, we can seek eigenfunction yk,2 x
y1 x, sk,2 y2 x, sk,2 y3 x, sk,2 y4 x, sk,2 U2 y1
U
y
U
y
U
y
2
2
2
3
2
4
yk,2 x .
U3 y1
U3 y2
U3 y3
U3 y4 U4 y1 U4 y2
U4 y3
U4 y4 4.11
Then,
U2 y2 U2 y3 U2 y4 yk,2 x y1 x, sk,2 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4 U2 y1 U2 y3 U2 y4 − y2 x, sk,2 U3 y1 U3 y3 U3 y4 U4 y1 U4 y3 U4 y4 U2 y1 U2 y2 U2 y4 y3 x, sk,2 U3 y1 U3 y2 U3 y4 U4 y1 U4 y2 U4 y4 U2 y1 U2 y2 U2 y3 − y4 x, sk,2 U3 y1 U3 y2 U3 y3 .
U4 y1 U4 y2 U4 y3 By similar computations we obtain
U2 y2 U2 y3 U2 y4 3 qπ − q0
1
U3 y2 U3 y3 U3 y4 3i
1O
,
27
sk,2
s6k,2
U4 y2 U4 y3 U4 y4 U2 y1 U2 y3 U2 y4 3 qπ − q0
1
U3 y1 U3 y3 U3 y4 − 3i
1O
,
sk,2
27
s6k,2
U4 y1 U4 y3 U4 y4 4.12
Boundary Value Problems
U2 y1 U2 y2 U2 y4 3 qπ − q0
1
U3 y1 U3 y2 U3 y4 3i
1
O
,
27
sk,2
s6k,2
U4 y1 U4 y2 U4 y4 9
U2 y1 U2 y2 U2 y3 3 qπ − q0
1
U3 y1 U3 y2 U3 y3 − 3i
1
O
.
sk,2
27
s6k,2
U4 y1 U4 y2 U4 y3 4.13
Hence, using the formula 2.2, we can write
yk,2 x 3 1
3i qπ − q0
isk,2 x
sk,2 x
−sk,2 x
−isk,2 x
e
e
e
O
e
sk,2
27
s6k,2
3 3i qπ − q0
1
7
x
2
cosh
s
x
O
2
cos
s
k,2
k,2
sk,2
2
s6k,2
4.14
3 3i qπ − q0
1
6
x
cosh
s
x
O
cos
s
.
k,2
k,2
sk,2
2
s6k,1
Therefore, for the normalized eigenfunction, we get
yk,2 x cos sk,2 x cosh sk,2 x O
1
.
4.15
1
.
k
4.16
sk,2
Hence, for sufficiently large integer k, we obtain 4.2
yk,2 x cos2kx cosh2kx O
Case 2. Assume that qx ∈ C2 0, π and the conditions qπ − q0 0 and q π − q 0 /
0
hold. Based on the asymptotic expressions of the fundamental solutions of 1.1 and the
asymptotic formulas for eigenvalues of the boundary-value problem 1.1, 1.2 up to order
Os−6 , the following result is valid.
Theorem 4.2. If the conditions qπ − q0 0 and q π − q 0 /
0 hold, then eigenfunctions of the
boundary-value problem 1.1, 1.2 corresponding the eigenvalues λk,1 and λk,2 are of the form
1
yk,1 x cos2kx − cosh2kx O
,
k
1
yk,2 x sin2kx − sinh2kx O
,
k
where k is sufficiently large integer.
4.17
4.18
10
Boundary Value Problems
Proof. It is clear that
3 qπ − q0 5ωυ q π − q 0
O
U1 yυ x, sk,1 8
16
s4k,1
s5k,1
U2 yυ x, sk,1 ωυ sk,1
O
1
s6k,1
,
1 qπ − q0 ωυ q π − q 0
−
8
16
s4k,1
s5k,1
2
U3 yυ x, sk,1 ωυ sk,1 −
1
4.19
,
s6k,1
1 qπ − q0 3ωυ q π − q 0
−
8
16
s4k,1
s5k,1
O
1
s6k,1
.
It follows from the conditions qπ − q0 0, q π − q 0 /
0 that Uj1 yυ x, sk,1 /
0 for
j 0, 1, 2. Thus, we can seek eigenfunction yk,1 x corresponding λk,1 in the form
y1 x, sk,1 y2 x, sk,1 y3 x, sk,1 y4 x, sk,1 U1 y1
U
y
U
y
U
y
1
2
1
3
1
4
yk,1 x .
U2 y1
U2 y2
U2 y3
U2 y4 U3 y1 U3 y2
U3 y3
U3 y4 Then,
U1 y2 U1 y3 U1 y4 yk,1 x y1 x, sk,1 U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 − y2 x, sk,1 U2 y1 U2 y3 U2 y4 U3 y1 U3 y3 U3 y4 4.20
Boundary Value Problems
11
U1 y1 U1 y2 U1 y4 y3 x, sk,1 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4 U1 y1 U1 y2 U1 y3 − y4 x, sk,1 U2 y1 U2 y2 U2 y3 .
U3 y1 U3 y2 U3 y3 4.21
By simple computations, we have
U1 y2 U1 y3 U1 y4 U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 U2 y1 U3 y3 U2 y4 U3 y1 U3 y3 U3 y4 3 15i q π − q 0
1
1
O
,
sk,1
210
s12
k,1
3 15i q π − q 0
1
1
O
,
sk,1
210
s12
k,1
U1 y1 U1 y2 U1 y4 3 q π − q 0
15i
1
U2 y1 U2 y2 U2 y4 −
1
O
,
sk,1
210
s12
k,1
U3 y1 U3 y2 U3 y4 U1 y1 U1 y2 U1 y3 3 q π − q 0
1
U2 y1 U2 y2 U2 y3 − 15i
1
O
.
sk,1
210
s12
k,1
U3 y1 U3 y2 U3 y3 4.22
Hence, using the formula 2.2, we can write
3 1
15i q π − q 0
isk,1 x
sk,1 x
−sk,1 x
−isk,1 x
yk,1 x 10
−
e
−
e
e
O
e
sk,1
2
s12
k,1
3 15i q π − q 0
1
9
cos
s
.
x
−
cosh
s
x
O
k,1
k,1
sk,1
2
s12
k,1
4.23
Therefore, for the normalized eigenfunction, we get
yk,1 x cos sk,1 x − cosh sk,1 x O
1
sk,1
.
4.24
12
Boundary Value Problems
Using the relations 3.3 and 3.12, for sufficiently large integer k, we obtain 4.17:
yk,1 x cos2kx − cosh2kx O
1
.
k
4.25
In similar way, we can seek eigenfunction yk,2 x corresponding λk,2 in the form
y1 x, sk,2 y2 x, sk,2 y3 x, sk,2 y4 x, sk,2 U2 y1
U2 y2
U2 y3
U2 y4 yk,2 x .
U3 y1
U3 y2
U3 y3
U3 y4 U4 y1 U4 y2
U4 y3
U4 y4 4.26
Then,
U2 y2 U2 y3 U2 y4 yk,2 x y1 x, sk,2 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4 U2 y1 U2 y3 U2 y4 − y2 x, sk,2 U3 y1 U3 y3 U3 y4 U4 y1 U4 y3 U4 y4 U2 y1 U2 y2 U2 y4 y3 x, sk,2 U3 y1 U3 y2 U3 y4 U4 y1 U4 y2 U4 y4 U2 y1 U2 y2 U2 y3 − y4 x, sk,2 U3 y1 U3 y2 U3 y3 .
U4 y1 U4 y2 U4 y3 By simple computations, we get
U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4 U2 y1 U2 y3 U2 y4 U3 y1 U3 y3 U3 y4 U4 y1 U4 y3 U4 y4 3 3 q π − q 0
1
1O
,
sk,2
210
s9k,2
3 3i q π − q 0
1
1O
,
sk,2
210
s9k,2
4.27
Boundary Value Problems
U2 y1 U2 y2 U2 y4 3 q π − q 0
1
U3 y1 U3 y2 U3 y4 3i
1
O
,
210
sk,2
s9k,2
U4 y1 U4 y2 U4 y4 13
U2 y1 U2 y2 U2 y3 3 q π − q 0
1
U3 y1 U3 y2 U3 y3 3
1
O
.
210
sk,2
s9k,2
U4 y1 U4 y2 U4 y3 4.28
Hence, using the formula 2.2, we can write
3 1
3 q π − q 0
isk,2 x
sk,2 x
−sk,2 x
−isk,2 x
yk,2 x 10
e
− ie
ie
−e
O
sk,2
2
s9k,2
3 3i q π − q 0
1
9
sin sk,2 x − sinh sk,2 x O
.
sk,2
2
s9k,2
4.29
Therefore, for the normalized eigenfunction, we get
yk,2 x sin sk,2 x − sinh sk,2 x O
1
O
sk,2
1
sk,2
.
4.30
Hence, for sufficiently large integer k, we obtain 4.18:
yk,2 x sin2kx − sinh2kx O
1
.
k
4.31
Case 3. Assume that qx ∈ C3 0, π and the conditions qj π − qj 0 0, j 0, 1 and
0 hold. Based on the asymptotic expressions of the fundamental solutions of
q π − q 0 /
1.1 and the asymptotic formulas for eigenvalues of the boundary-value problem 1.1, 1.2
up to order Os−7 , the following result is valid.
Theorem 4.3. If the conditions qj π − qj 0 0, j 0, 1 and q π − q 0 /
0 hold, then
eigenfunctions of the boundary-value problem 1.1, 1.2 corresponding the eigenvalues λk,1 and λk,2
are of the form
1
,
k
1
yk,2 x cos2kx − cosh2kx O
k
yk,1 x sin2kx sinh2kx O
where k is sufficiently large integer.
4.32
4.33
14
Boundary Value Problems
Proof. It is clear that
3 qπ − q0 5ωυ q π − q 0 5ωυ2 q π − q 0
U1 yυ x, sk,1 −
O
8
16
32
s4k,1
s5k,1
s6k,1
1
s7k,1
1 qπ − q0 ωυ q π − q 0 5ωυ2 q π − q 0
−
−
8
16
32
s4k,1
s5k,1
s6k,1
1
,
O
s7k,1
1 qπ − q0 3ωυ q π − q 0 3ωυ2 q π − q 0
2
U3 yυ x, sk,1 ωυ sk,1 −
−
−
8
16
32
s4k,1
s5k,1
s6k,1
1
.
O
s7k,1
,
U2 yυ x, sk,1 ωυ sk,1
4.34
From the conditions qj π − qj 0 0 j 0, 1 and q π − q 0 /
0, we have
0
for
j
0,
1,
2.
Thus,
we
can
seek
eigenfunction
y
x
corresponding
λk,1
Uj1 yυ x, sk,1 /
k,1
in the form
y1 x, sk,1 y2 x, sk,1 y3 x, sk,1 y4 x, sk,1 U1 y1
U1 y2
U1 y3
U1 y4 yk,1 x .
U2 y1
U2 y2
U2 y3
U2 y4 U3 y1 U3 y2
U3 y3
U3 y4 4.35
Then,
U1 y2 U1 y3 U1 y4 yk,1 x y1 x, sk,1 U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 − y2 x, sk,1 U2 y1 U2 y3 U2 y4 U3 y1 U3 y3 U3 y4 U1 y1 U1 y2 U1 y4 y3 x, sk,1 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4 U1 y1 U1 y2 U1 y3 − y4 x, sk,1 U2 y1 U2 y2 U2 y3 .
U3 y1 U3 y2 U3 y3 4.36
Boundary Value Problems
15
By simple calculations, we get
U1 y2 U1 y3 U1 y4 3 q π − q 0
1
U2 y2 U2 y3 U2 y4 − 75
1
O
,
sk,1
213
s15
k,1
U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 3 q π − q 0
1
U2 y1 U2 y3 U2 y4 75i
1
O
,
213
sk,1
s15
k,1
U3 y1 U3 y3 U3 y4 U1 y1 U1 y2 U1 y4 3 q π − q 0
1
U2 y1 U2 y2 U2 y4 75i
1
O
,
213
sk,1
s15
k,1
U3 y1 U3 y2 U3 y4 U1 y1 U1 y2 U1 y3 3 q π − q 0
75
1
U2 y1 U2 y2 U2 y3 −
1
O
.
sk,1
213
s15
k,1
U3 y1 U3 y2 U3 y3 4.37
Hence, using the formula 2.2, we can write
3 1
75 q π − q 0
isk,1 x
sk,1 x
−sk,1 x
−isk,1 x
yk,1 x − 13
ie
− ie
−e
O
e
sk,1
2
s15
k,1
3 75i q π − q 0
1
− 12
sin sk,1 x sinh sk,1 x O
.
sk,1
2
s15
k,1
4.38
Therefore, for the normalized eigenfunction, we get
yk,1 x sin sk,1 x sinh sk,1 x O
1
sk,1
.
4.39
Using the relations 3.3 and 3.12, for sufficiently large integer k, we obtain 4.32
1
yk,1 x sin2kx sinh2kx O
.
k
4.40
In similar way, we can seek eigenfunction yk,2 x corresponding λk,2 in the form
y1 x, sk,2 y2 x, sk,2 y3 x, sk,2 y4 x, sk,2 U2 y1
U
y
U
y
U
y
2
2
2
3
2
4
yk,2 x .
U3 y1
U3 y2
U3 y3
U3 y4 U4 y1 U4 y2
U4 y3
U4 y4 4.41
16
Boundary Value Problems
Then,
U2 y2 U2 y3 U2 y4 yk,2 x y1 x, sk,2 U3 y2 U3 y3 U3 y4 U4 y2 U4 y3 U4 y4 U2 y1 U2 y3 U2 y4 − y2 x, sk,2 U3 y1 U3 y3 U3 y4 U4 y1 U4 y3 U4 y4 4.42
U2 y1 U2 y2 U2 y4 y3 x, sk,2 U3 y1 U3 y2 U3 y4 U4 y1 U4 y2 U4 y4 U2 y1 U2 y2 U2 y3 − y4 x, sk,2 U3 y1 U3 y2 U3 y3 .
U4 y1 U4 y2 U4 y3 By simple computations, we get
U2 y2 U2 y3 U2 y4 3 q π − q 0
1
U3 y2 U3 y3 U3 y4 − 45i
1
O
,
sk,2
213
s12
k,2
U4 y2 U4 y3 U4 y4 U2 y1 U2 y3 U2 y4 3 q π − q 0
1
U3 y1 U3 y3 U3 y4 − 45i
1O
,
sk,2
213
s12
k,2
U4 y1 U4 y3 U4 y4 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4 U4 y1 U4 y2 U4 y4 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4 U4 y1 U4 y2 U4 y4 45i q π − q 0
213
s12
k,2
3 1O
1
sk,2
4.43
,
3 45i q π − q 0
1
1O
.
sk,2
213
s12
k,2
By the formula 2.2, we can write
3 1
45i q π − q 0
isk,2 x
sk,2 x
−sk,2 x
−isk,2 x
yk,2 x − 13
−
e
−
e
e
O
e
sk,2
2
s12
k,2
3 45i q π − q 0
1
12
cos sk,2 x − cosh sk,2 x O
.
sk,2
2
s12
k,2
4.44
Boundary Value Problems
17
Therefore, for the normalized eigenfunction, we get
yk,2 x cos sk,2 x − cosh sk,2 x O
1
sk,2
4.45
.
Hence, for sufficiently large integer k, we obtain the relation 4.33
yk,2 x cos2kx − cosh2kx O
1
.
k
4.46
Case 4. Assume that qx ∈ C4 0, π and the conditions qj π − qj 0 0, j 0, 2 and
0 hold. Based on the asymptotic expressions of the fundamental solutions of
q π − q 0 /
1.1 and the asymptotic formulas for eigenvalues of the boundary-value problem 1.1, 1.2
up to order Os−8 , the following result is valid.
Theorem 4.4. If the conditions qj π − qj 0 0, j 0, 2 and q π − q 0 /
0 hold, then
eigenfunctions of the boundary-value problem 1.1, 1.2 corresponding the eigenvalues λk,1 and λk,2
are of the form
1
,
yk,1 x cos2kx cosh2kx O
k
4.47
1
,
k
4.48
yk,2 x sin2kx sinh2kx O
where k is sufficiently large integer.
Proof. It is clear that
3 qπ − q0 5ωυ q π − q 0
U1 yυ x, sk,1 8
16
s4k,1
s5k,1
5ωυ2 q π − q 0 ωυ q π − q 0
−
−
O
32
64
s7k,1
s6k,1
1
s8k,1
,
1 qπ − q0 ωυ q π − q 0
U2 yυ x, sk,1 ωυ sk,1
−
8
16
s4k,1
s5k,1
5ωυ2 q π − q 0 9ωυ q π − q 0
−
O
32
64
s7k,1
s6k,1
1
s8k,1
,
18
Boundary Value Problems
1 qπ − q0 3ωυ q π − q 0
2
U3 yυ x, sk,1 ωυ sk,1 −
−
8
16
s4k,1
s5k,1
3ωυ2 q π − q 0 ωυ q π − q 0
−
−
O
32
64
s7k,1
s6k,1
1
s8k,1
.
4.49
From the conditions qj π − qj 0 0 j 0, 2 and q π − q 0 /
0, we have
0 for j 0, 1, 2. Thus, we can seek eigenfunction yk,1 x corresponding λk,1
Uj1 yυ x, sk,1 /
in the form
y1 x, sk,1 y2 x, sk,1 y3 x, sk,1 y4 x, sk,1 U1 y1
U
y
U
y
U
y
1
2
1
3
1
4
yk,1 x .
U2 y1
U
y
U
y
U
y
2
2
2
3
2
4
U3 y1
U3 y2
U3 y3
U3 y4 4.50
U1 y2 U1 y3 U1 y4 yk,1 x y1 x, sk,1 U2 y2 U2 y3 U2 y4 U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 − y2 x, sk,1 U2 y1 U2 y3 U2 y4 U3 y1 U3 y3 U3 y4 U1 y1 U1 y2 U1 y4 y3 x, sk,1 U2 y1 U2 y2 U2 y4 U3 y1 U3 y2 U3 y4 U1 y1 U1 y2 U1 y3 − y4 x, sk,1 U2 y1 U2 y2 U2 y3 .
U3 y1 U3 y2 U3 y3 4.51
Then
By simple computations, we get
U1 y2 U1 y3 U1 y4 3 q π − q 0
9i
1
U2 y2 U2 y3 U2 y4 1
O
,
216
sk,1
s18
k,1
U3 y2 U3 y3 U3 y4 U1 y1 U1 y3 U1 y4 3 q π − q 0
1
U2 y1 U2 y3 U2 y4 − 9i
1O
,
sk,1
216
s18
k,1
U3 y1 U3 y3 U3 y4 Boundary Value Problems
U1 y1 U1 y3 U1 y4 3 q π − q 0
1
U2 y1 U2 y3 U2 y4 9i
1
O
,
216
sk,1
s18
k,1
U3 y1 U3 y3 U3 y4 19
U1 y1 U1 y2 U1 y3 3 q π − q 0
1
U2 y1 U2 y2 U2 y3 − 9i
1
O
.
sk,1
216
s18
k,1
U3 y1 U3 y2 U3 y3 4.52
By the formula 2.2, we can write
3 1
9i q π − q 0
isk,1 x
sk,1 x
−sk,1 x
−isk,1 x
yk,1 x 16
e
e
e
O
e
sk,1
2
s18
k,1
3 9i q π − q 0
1
cos sk,1 x cosh sk,1 x O
.
15
sk,1
2
s18
k,1
4.53
Therefore, for the normalized eigenfunction, we get
yk,1 x cos sk,1 x cosh sk,1 x O
1
sk,1
.
4.54
Using the relations 3.3 and 3.12, for sufficiently large integer k, we obtain 4.47:
yk,1 x cos2kx cosh2kx O
1
.
k
4.55
In similar way, we can seek eigenfunction yk,2 x corresponding λk,2 in the form
y1 x, sk,2 y2 x, sk,2 y3 x, sk,2 y4 x, sk,2 U2 y1
U
y
U
y
U
y
2
2
2
3
2
4
yk,2 x .
U3 y1
U3 y2
U3 y3
U3 y4 U4 y1 U4 y2
U4 y3
U4 y4 4.56
20
Boundary Value Problems
By simple computations, we get
U2 y2 U2 y3 U2 y4 3 q π − q 0
63
1
U3 y2 U3 y3 U3 y4 −
1
O
,
sk,2
216
s15
k,2
U4 y2 U4 y3 U4 y4 U2 y1 U2 y3 U2 y4 3 63i q π − q 0
1
U3 y1 U3 y3 U3 y4 1
O
,
216
sk,2
s15
k,2
U4 y1 U4 y3 U4 y4 U2 y1 U2 y2 U2 y4 3 63i q π − q 0
1
U3 y1 U3 y2 U3 y4 1
O
,
216
sk,2
s15
k,2
U4 y1 U4 y2 U4 y4 U2 y1 U2 y2 U2 y3 3 q π − q 0
1
U3 y1 U3 y2 U3 y3 − 63
1
O
.
sk,2
216
s15
k,2
U4 y1 U4 y2 U4 y3 4.57
By the formula 2.2, we can write
3 1
63 q π − q 0
isk,2 x
sk,2 x
−sk,2 x
−isk,2 x
yk,2 x − 16
ie
−
ie
−
e
O
e
sk,2
2
s15
k,2
3 1
63i q π − q 0
x
sinh
s
x
O
− 15
sin
s
.
k,2
k,2
sk,2
2
s15
k,2
4.58
Therefore, for the normalized eigenfunction, we get
yk,2 x sin sk,2 x sinh sk,2 x O
1
sk,2
.
4.59
Hence, for sufficiently large integer k, we obtain the relation 4.48
yk,2 x sin2kx sinh2kx O
1
.
k
4.60
Acknowledgments
This work is supported by The Scientific and Technological Research Council of Turkey
TÜBİTAK. The author would like to thank the referee and the editor for their helpful
comments and suggestions. The author also would like to thank prof. Kh. R. Mamedov for
useful discussions.
Boundary Value Problems
21
References
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