Miskolc Mathematical Notes
Vol. 17 (2016), No. 2, pp. 911–923
HU e-ISSN 1787-2413
DOI: 10.18514/MMN.2016.1098
STURM-LIOUVILLE PROBLEMS WITH FINITELY MANY POINT
ı INTERACTIONS AND EIGEN-PARAMETER IN BOUNDARY
CONDITION
ABDULLAH KABLAN AND MANAF DZH. MANAFOV
Received 15 January, 2014
Abstract. This paper deals with the Sturm-Liouville equation with a finite number of point
ı interactions and eigenvalue parameter contained in the boundary condition. Sturm–Liouville
problem with discontinuities at one or two points and its different variants have already been
investigated. In this study we extend these results to a finite number of point ı interactions case.
The crucial part of this study is the using graph demonstration to obtain asymptotic representation
of solutions.
2010 Mathematics Subject Classification: 34B05; 34L20
Keywords: discontinuous Sturm-Liouville problems, transmission conditions, ı interactions
1. I NTRODUCTION
We consider the boundary problem (BVP) for the differential equation
`y WD y 00 C q.x/y D y
on Œa; x1 / [ .x1 ; x2 / [ [ .xn
1 ; xn / [ .xn ; b
(1.1)
and boundary condition at x D a
L1 .y/ WD ˛1 y.a/ C ˛2 y 0 .a/ D 0;
(1.2)
with transmission conditions at discontinuous points xi ; i D 1; n
Ui .y/ WD y.xi
0/ D y.xi C 0/ D y.xi /;
Vi .y/ WD y 0 .xi C 0/
y 0 .xi
0/ D i y.xi /
(1.3)
(1.4)
and the eigenparameter-dependent boundary condition at x D b
L2 .y/ WD Œˇ10 y.b/ C ˇ20 y 0 .b/ C Œˇ1 y.b/
ˇ2 y 0 .b/ D 0;
(1.5)
where q.x/ is real-valued function and continuous in L1 Œa; b. We assume that ˛i , i ,
ˇi ; ˇ 0i ; i D 1; 2 are real numbers, satisfying j˛1 j C j˛2 j ¤ 0; is a complex spectral
parameter. Throughout this paper, we assume that
r WD ˇ10 ˇ2
ˇ1 ˇ20 > 0:
(1.6)
c 2016 Miskolc University Press
912
ABDULLAH KABLAN AND MANAF DZH. MANAFOV
Notice that, we can understand problem (1.1), (1.3) and (1.4) as one of the treatments of the equation
y 00 C .p.x/ C q.x// y D y;
x 2 .a; b/;
(1.7)
Pn
when p.x/ D i D1 i ı.x xi /, where ı.x/ is the Dirac function, (see [2]).
Sturmian theory is one of the most extensively developing fields in theoretical and
applied mathematics. Particularly, there has been increasing interest in the spectral
analysis of BVPs with eigenvalue-dependent boundary conditions. There are quite
substantial literatures on such problems. Here we mention the results of [5], [6], [7],
[8], [10], [14], [18] and the corresponding references cited therein.
BVPs with discontinuities inside the interval and eigenvalue contained in the boundary conditions often appear in many branches of natural sciences. We note that SturmLiouville problems with eigen-dependent boundary conditions and with transmission conditions have been investigated in [1], [3], [9], [13], [11], [12]. Furthermore
Green’s formula for impulsive differential equation has been studied in [16] and [17].
In this paper we consider Sturm-Liouville problem with eigenparameter depend
boundary condition and ı interactions at finite number of interior points. The aim of
this article is to carry results in [9] to the case of finitely many ı interactions and the
main difference of this study is the using graph theory (see [4]) for the complicated
asymptotic formulas.
2. T HE H ILBERT SPACE CONSTRUCTION AND SOME PROPERTIES OF THE
SPECTRUM
In the Hilbert space H WD L2 Œa; b ˚ C of two component vectors, we define an
inner product by
Zb
1
(2.1)
hf; giH WD f1 .x/gN 1 .x/dx C f2 g 2
r
a
for
f D
f1 .x/
f2
; gD
g 1 .x/
g2
where f1 .x/; g 1 .x/ 2 L2 .a; b/; f2 ; g2 2 C, the constant r is defined in (1.6). For
convenience we introduce
Rb .y/ WD ˇ1 y.b/
ˇ2 y 0 .b/;
Rb0 .y/ WD ˇ10 y.b/
ˇ20 y 0 .b/:
In the Hilbert space H we define the operator L
LWH !H
STURM-LIOUVILLE PROBLEMS WITH FINITELY MANY POINT ı INTERACTIONS
913
with domain
8
ˇ
nS1
ˇ
ˆ
0
ˆ
ˇ
.xi ; xi C1 / [ .xn ; b ;
<
ˇ f1 ; f1 2 AC Œa; x1 / [
i D1
D.L/ W f 2 H ˇˇ
ˆ
`f1 2 L2 Œa; b; L1 .f1 / D 0;
ˆ
ˇ
:
ˇ U .f / D V .f / D 0 f or i D 1; n; f D R0 .f /
i 1
i 1
2
1
b
9
>
>
=
>
>
;
and operator rule
`f 1
:
Rb .f 1 /
Here AC .:/ denotes the set of all absolutely continuous functions on related interval. In particular, those functions will have limits at the boundary points xi :
It is clear that the eigenvalues of the operator L and the boundary value problem (1.1)-(1.5) are same and the eigenfunctions of (1.1)-(1.5) coincide with the first
components of corresponding eigenelements of the operator L.
L.f / WD
Theorem 1. The operator L is symmetric.
Proof. Let f; g 2 D.L/. From the inner product defined in (2.1), we obtain
hLf; giH
hf; LgiH D ŒW .f; gI x1
C ŒW .f; gI x2
C
n
X1
0/
W .f; gI a/
0/
W .f; gI xpC1
(2.2)
W .f; gI x1 C 0/
0/
W .f; gI xp C 0/
pD2
C ŒW .f; gI b/ W .f; gI xn C 0/
1
C Rb0 .f /Rb .g/ Rb .f /Rb0 .g/
r
where W .f; gI x/ D f .x/g 0 .x/ f 0 .x/g.x/ is the wronkskian of the functions f and
g. Since f and g satisfy the same boundary condition (1.2) and from the transmission
conditions (1.3) and (1.4), we get
W .f; gI a/ D 0;
(2.3)
W .f; gI xi 0/ D W .f; gI xi C 0/; .i D 1; n/;
(2.4)
0
0
Rb .f /Rb .g/ Rb .f /Rb .g/ D rW .f; gI b/:
(2.5)
Substituting (2.3)-(2.5) in (2.2) we obtain hLf; giH D hf; LgiH for f; g 2 D.L/. So
L is symmetric:
Corollary 1. All eigenvalues of the problem (1.1)-(1.5) are real and if 1 and
2 are two different eigenvalues of the problem (1.1)-(1.5) , then the corresponding
eigenfunctions y1 and y2 are orthogonal in the sense of
Z b
1
y1 .x/y2 .x/dx C Rb0 .y1 /Rb0 .y2 / D 0:
r
a
914
ABDULLAH KABLAN AND MANAF DZH. MANAFOV
To define a solution of (1.1)-(1.5), we need the following lemma. The proof of this
assertion reproduces that of Theorem 1.1 in [15, p. 14] or [12].
Lemma 1. Let q.x/ be a real-valued, continuous function and let f ./ and g./
are given entire functions. Then for any 2 C, the equation
y 00 C q.x/y D y;
x 2 Œa; b
has a unique solution y D y.x; / such that
y.a/ D f ./; y 0 .a/ D g./
(or y.b/ D f ./; y 0 .b/ D g./)
and for each x 2 Œa; b, y.x; / is an entire function of .
Now we define two solutions of the equation (1.1) as follows:
8
8
1 .x/;
x 2 Œa; x1 /
.x/;
x 2 Œa; x1 /
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ 1
ˆ
ˆ
ˆ
ˆ
x 2 .x1 ; x2 /
x 2 .x1 ; x2 /
ˆ
ˆ
<2 .x/;
<2 .x/;
::
::
::
::
.x/ D
:
:
; .x/ D ˆ
:
:
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
n .x/;
x 2 .xn 1 ; xn /
n .x/;
x 2 .xn 1 ; xn /
ˆ
ˆ
ˆ
ˆ
:
:
.nC1/ .x/; x 2 .xn ; b
.nC1/ .x/; x 2 .xn ; b
Let 1 .x/ D 1 .x; / be a solution of the equation (1.1) on Œa; x1 / which satisfies
the initial conditions at the point a
y.a/ D ˛ 2 ;
y 0 .a/ D
˛1
(2.6)
From Lemma 1, we can define a solution i C1 .x; / of the equation (1.1) on Œxi ; xi C1 ,
(i D 1; n 1) by means of the solution i .x; / by the nonstandard initial conditions
y.x i C 0/ D i .xi
0/;
y 0 .x i C 0/ Di0 .xi
0/ C i i .xi /:
(2.7)
In the same manner, we define a solution nC1 .x; / of the equation (1.1) on Œxn ; b
by the nonstandard initial conditions
y.x n C 0/ D n .xn
0/;
0
y 0 .x n C 0/ Dn
.xn
0/ C i n .xn /:
(2.8)
Therefore .x; / satisfies the equation (1.1) on the interval Œa; x1 / [ .x1 ; x2 / [ [
.xn 1 ; xn / [ .xn ; b and the boundary condition (1.2) and the transmission conditions
(1.3), (1.4).
Similarly let .nC1/ .x/ D nC1 .x; / be a solution of (1.1) on .xn ; b which
satisfies the initial conditions
y.b/ Dˇ20 C ˇ2 ;
y 0 .b/ Dˇ10 C ˇ1 :
Likewise let define the solution i .x; /, on Œxi
tions
y.x i
0/ D.iC1/ .xi C0/; y 0 .x i
1 ; xi ,
(2.9)
( i D n; 2) satisfies the condi-
0/ D0.i C1/ .xi C0/ i .i C1/ .xi /: (2.10)
STURM-LIOUVILLE PROBLEMS WITH FINITELY MANY POINT ı INTERACTIONS
915
Finally we define a solution 1 .x; / on Œa; x1 / satisfies the conditions
y.x 1
0/ D2 .x1 C 0/;
y 0 .x 1
0/ D02 .x1 C 0/
i 2 .x1 /:
(2.11)
Hence .x; / satisfies the equation (1.1) on Œa; x1 / [ .x1 ; x2 / [ [ .xn 1 ; xn / [
.xn ; b, the boundary condition (1.5) and the transmission conditions (1.3), (1.4).
Let define the wronskians of i and i for x in the intervals Œa; x1 /; .x1 ; x2 /; :::;
.xn 1 ; xn /; .xn ; b respectively:
wi ./ D W .i ; i ; x/
D i .x; /0i .x; /
(2.12)
i0 .x; /i .x; /; i D 1; n C 1:
Lemma 2. For each 2 C w./ WD w1 ./ D w2 ./ D w3 ./ D D wnC1 ./:
Proof. From (2.7), (2.10), (1.3) and (1.4)
W .1 ; 1 I x1
0/ D 1 .x1
D
0; /01 .x1
10 .x1
0; /1 .x1
0
2 .x1 C 0; /Œ2 .x1 C 0; / 1 2 .x1 ; /
Œ20 .x1 C 0; / 1 2 .x1 ; /2 .x1 C 0; /
D W .2 ; 2 I x1 C 0/
0; /
0; /
1 Œ2 .x1 C 0; /2 .x1 ; /
2 .x1 ; /2 .x1 C 0; /
D W .2 ; 2 I x1 C 0/
Since the wronskians are independent of x (see [15, pp.7]), we get
W .2 ; 2 I x1 C 0/ D W .2 ; 2 I x2
0/
and similar calculation gives
W .2 ; 2 I x2
0/ D W .3 ; 3 I x2 C 0/:
Consequently we arrive at
W .1 ; 1 I x1
0/ D W .2 ; 2 I x1 C 0/ D W .3 ; 3 I x2 C 0/ D D W .nC1 ; nC1 I xn C 0/
Theorem 2. The eigenvalues of the problem (1.1)-(1.5) consist the zeros of the
function w./:
Proof. For the proof, we will follow the technique in [9]. Suppose that is the
zero of w./. Then the wronskian of 1 .x; / and 1 .x; / is zero, so that 1 .x; /
is a constant multiple of 1 .x; /, say
1 .x; / D k1 .x; /; x 2 Œa; x1 
for some k ¤ 0: Therefore 1 .x; / satisfies the boundary condition (1.2) and this
means that 1 .x; / is an eigenfunction for the eigenvalue :
916
ABDULLAH KABLAN AND MANAF DZH. MANAFOV
For the converse, let ˚.x/ be any eigenfunction corresponding to eigenvalue .
Then the function ˚.x/ may be represented in the form
8
k1 1 .x; / C k2 1 .x; /;
x 2 Œa; x1 /
ˆ
ˆ
ˆ
ˆ
ˆ
x 2 .x1 ; x2 /
ˆk3 2 .x; / C k4 2 .x; /;
<
::
::
˚.x/ D
:
:
ˆ
ˆ
ˆ
ˆ
k2n 1 n .x; / C k2n n .x; /;
x 2 .xn 1 ; xn /
ˆ
ˆ
:
k2nC1 nC1 .x; / C k2nC2 nC1 .x; /; x 2 .xn ; b
where at least one of the constants kj; .j D 1; 2n C 2/ is not zero. By using the initial
conditions (2.6)-(2.11), the equations
L1 .˚ / D 0I Ui .˚ / D 0; Vi .˚ / D 0; i D 1; nI L2 .˚ / D 0
(2.13)
give a system of linear equations in the variables kj; .j D 1; 2n C 2/ and the coefficient matrix of this system is
2
0 w1 ./
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
N1
0
0
::
:
0
0
0
0
0
0
0
0
0
0
M1
N2
0
0
0
M2
N3
::
:
0
0
0
::
:
::
:
::
: Mn
0
0
1
Nn
0
::
:
0
0
Mn
0
wnC1 ./ 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
.2nC2/.2nC2/
(2.14)
where Mi and Ni are 2 2 matrices defined as follows
.iC1/ .xi C 0/
.i C1/ .xi C 0/
Mi D
0
.x C 0/
0
.x C 0/ ; i D 1; n;
.iC1/
i
.i C1/
i
0/ i .xi 0/
Ni D
; i D 1; n:
0/ 0i .xi 0/
Qn
nC1 ./ which must be zero
The determinant of the matrix (2.14) is
i D1 wi ./:w
in order to the system (2.13) has a nontrivial solution and hence w./ D 0.
i .xi
0
i
.xi
STURM-LIOUVILLE PROBLEMS WITH FINITELY MANY POINT ı INTERACTIONS
917
3. A SYMPTOTIC FORM OF SOLUTIONS AND GRAPH REPRESENTATION
In this section, we shall derive the asymptotic formulas for the characteristic function w./ of (1.1), (1.5) in four different cases. The main difficulty is derivation of
asymptotic formulas for the solutions of (1.1), (1.5). Because these formulas fastly
lead to a very complicated equations for large values of n. A convenient way to manage the resulting asymptotic representation of solutions is by a graph analogy. For
further reading for graph based theory, see, for example, [4]. We start with some
lemmas.
Lemma 3. Let D 2 and .x; / be solution of the equation (1.1). Then the
following integral equations hold for k D 0; 1
dk
dk
1 dk
.x/
D
˛
.cos
.x
a//
˛
.sin .x a//
2
1
1
dx k
dx k
dx k
Z
1 x1 d k
C
.sin .x t //q.t /1 .t /dt;
a dx k
(3.1)
dk
dk
.x/
D
.x
0/
.cos .x xi //
(3.2)
.iC1/
i i
dx k
dx k
1
dk
C i0 .xi 0/ k .sin .x xi //
dx
Z x k
1
d
C
.sin .x t //q.t /.i C1/ .t /dt; i D 1; n:
xi dx k
Proof. The last terms in (3.1) and (3.2) are equal to
Z
1 x1 d k
00
.sin .x t //f2 1 .t / C 1
.t /gdt;
a dx k
Z
1 x dk
.sin .x t //f2 .i C1/ .t / C .i00 C1/ .t /gdt; i D 1; n
xi dx k
by the equation (1.1), respectively. On integrating by parts twice we obtain (3.1) and
(3.2).
918
ABDULLAH KABLAN AND MANAF DZH. MANAFOV
Lemma 4. Let WD I m and ˛2 ¤ 0: As jj ! 1, the asymptotic formula for
1 .x/ is
dk
dk
k 1 j j.x a/
.x/
D
˛
.cos
.x
a//
C
O
e
(3.3)
jj
2
1
dx k
dx k
and the asymptotic formula for .iC1/ .x/; i D 1; n , is obtained from the following
tree:
dk
s
dx k
dk
c
dx k
˛2
.&
c
s
.& .&
c
ss
c
:: ::
:: ::
: :
: :
.& .& .& .&
c ss cc
ss c
# ## ## ## #
c sc sc sc
s
.&
s
c
.& .&
c
ss
c
:: ::
:: ::
: :
: :
.& .& .& .&
c ss cc
ss c
# ## ## ## #
c sc sc sc
s
.x
.xn
.xn
xn /
xn
1
1/
xn
2/
.x2
x1 /
.x1
a/
(3.4)
where c and s denote cosine and sine functions respectively. The roots consist of two
dk
dk
s. The children of roots are written by repeating the processing
nodes dx
k c and
dx k
s s s c. The last children are written by repeating the processing c s. After condk
structing trees for any dx
k i .x/, i D 2; n C 1 using rules above to get its formula,
first we write each terms by multiplying all notes on the branch from root to least
children and then sum these terms. Then, while if ˛2 D 0
˛1 d k
dk
k 2 j j.x a/
.x/
D
.sin
.x
a//
C
O
e
(3.5)
jj
1
dx k
dx k
and
˛1
dk
c
dx k
dk
s
dx k
.&
c
s
.& .&
c
ss
c
:: ::
:: ::
: :
: :
.& .& .& .&
c ss c c s s
c
# ## ## ## #
c sc sc sc
s
.&
s
c
.& .&
c
ss
c
:: ::
:: ::
: :
: :
.& .& .& .&
c ss c c s s
c
# ## ## ## #
c sc sc sc
s
.x
.xn
.xn
xn /
xn
1
1/
xn
.x2
x1 /
.x1
a/
2/
(3.6)
Each result obtained from trees holds uniformly for the intervals a x x1 ; xi x xi C1 ; i D 1; n 1 and xn x b respectively.
STURM-LIOUVILLE PROBLEMS WITH FINITELY MANY POINT ı INTERACTIONS
919
Proof. The proof for i D 1; 2 and 3 without graph representation were given in
[9] and [15]. Here, we propose an alternative proof in general sense by using graph
demonstration. As can be seen in studies [9], [15] the continuation of this process,
although theoretically clear, soon leads to very complicated formulas. It is therefore advantageous to use a graphical representation. Indeed, we can construct a link
about the array of the functions. Below, we exhibit the trees corresponding to each
dk
dk
.x/; i D 1; n C 1 and give a systematic pattern. We will write dx
k i .x/ for
dx k i i D 1; 2; 3 from [9, Lemma 3.2]. For ˛2 ¤ 0 and i D 1 we have
dk
dk
k 1 j j.x a/
.x/
D
˛
.cos
.x
a//
C
O
e
:
jj
2
1
dx k
dx k
The tree for this formula will consist of only one node will be named by root:
dk
c
dx k
˛2
.x
a/
In order to work with trees we introduce the following notations: ˛2 is common
k
d
factor of all terms, dx
k shows the derivative of terms on root and .x
values of functions which are on the same line with it. For i D 2;
(
dk
dk
.x/
D
˛
.cos .x x1 // cos .x1 a/
2
2
dx k
dx k
dk
.sin .x
dxk
CO jjk
a/ is at
(3.7)
x1 // sin .x1
1 j jŒ.x x1 /C.x1
e
a/
o
a/
:
For this formula the tree will consist of two roots and their one child:
˛1
dk
c
dx k
#
c
dk
s
dx k
#
s
.x
x1 /
.x1
a/
In this tree, the multiplication of terms on the vertical branch from root to its child
gives the first term of (3.7) and similarly, second branch gives the second term of
(3.7). Then the sum of these branches gives the formula (3.7). Finally, for i D 3
(
dk
dk
.x/
D
˛
.cos .x x2 // cos .x2 x1 / cos .x1 a/
2
3
dx k
dx k
dk
.cos .x x2 // sin .x2 x1 / sin .x1
dx k
dk
.sin .x x2 // sin .x2 x1 / cos .x1
dx k
a/
a/
920
ABDULLAH KABLAN AND MANAF DZH. MANAFOV
dk
.sin .x
dxk
CO jjk
x2 // cos .x2
x1 / sin .x1
o
a/
1 j jŒ.x x2 /C.x2 x1 /C.x1 a/
e
and corresponds to:
˛2
dk
s
dx k
dk
c
dx k
.&
c
s
#
#
c
s
.&
s
c
#
#
c
s
.x
x2 /
.x2
x1 /
.x1
a/
The above trees help in giving a procedure to construct trees for i 4. From using
dk
the rules above we generalize a tree for dx
k .nC1/ .x/ as in (3.4) and by using the
same technique we obtain a tree for
dk
.x/
dx k .nC1/
when ˛2 D 0 as in (3.6).
Theorem 3. Let D 2 and WD I m. Then the estimates obtained from the
following four trees for w./ are valid.
For ˇ20 ¤ 0; ˛2 ¤ 0, the estimate obtained from a tree which is the same as in (3.4)
except common factor and root changed by
ˇ20 ˛2 3
s
c
.b
xn /
For ˇ20 ¤ 0; ˛2 D 0, the estimate obtained from a tree which is the same as in (3.6)
except common factor and root changed by
ˇ20 ˛1 2
s
c
.b
xn /
For ˇ20 D 0; ˛2 ¤ 0, the estimate obtained from a tree which is the same as in (3.4)
except common factor and root changed by
ˇ10 ˛2 2
c
s
.b
xn /
For ˇ20 D 0; ˛2 D 0, the estimate obtained from a tree which is the same as in (3.6)
except common factor and root changed by
ˇ10 ˛1 c
s
.b
xn /
Proof. The case ˇ20 ¤ 0; ˛2 ¤ 0 will be considered; similar proof works for other
three cases. Since wnC1 ./ in (2.12) is independent of x 2 Œa; b, from (2.9) we get
wnC1 ./ D .ˇ10 C ˇ1 /.nC1/ .b/
0
.ˇ20 C ˇ2 /.nC1/
.b/
(3.8)
Putting x D b in estimates obtained from trees (3.4) and (3.6) for the asymptotic
dk
behavior of dx
k .nC1/ .x/ and then substituting in (3.8) we obtain four different
STURM-LIOUVILLE PROBLEMS WITH FINITELY MANY POINT ı INTERACTIONS
921
cases for the asymptotic behavior of wnC1 ./ and their trees as jj ! 1. For ˇ20 ¤
0; ˛2 ¤ 0, we briefly write
wnC1 ./ D ˇ20 ˛2 3 fsin .b
C O jj2 e j jŒ.b
xn / Œ: : :1 C cos .b
xn /C.xn xn
1 /CC.x2
xn / Œ: : :2 g
x1 /C.x1 a/
(3.9)
where the terms in square brackets Œ: : :1 and Œ: : :2 are the same as the terms in estimate obtained from (3.4). Therefore the tree for (3.9) is similar to the tree (3.4) except
common factor and root. The common factor and root are replaced by
ˇ20 ˛2 3
s
c
.b
xn /
Putting D i . > 0/ in these formula it follows that w./ ¤ 0 for negative
and sufficiently large. The eigenvalues of the problem (1.1)-(1.5) are bounded below
in all cases.
4. A SYMPTOTIC BEHAVIOR OF THE EIGENVALUES AND EIGENFUNCTIONS
Our object in this section is to obtain asymptotic estimates for eigenvalues of problem (1.1)-(1.5). We know from Lemma 2 and Theorem 2 that the eigenvalues coincide with the zeros of the entire function wnC1 ./. Since the eigenvalues are real and
bounded below we can denote the eigenvalues by n .n D 0; 1; : : :/, where
0 1 2 Let ˇ20 ¤ 0; ˛2 ¤ 0: In the formula (3.9) and so in the corresponding tree, after
successive trigonometric operations from last children to the root we arrive at
n
Y
2 j j.b a/
0
3
wnC1 ./ D ˇ2 ˛2 sin .b a/ C O jj e
:
(4.1)
kD1
By applying Rouché’s theorem on a sufficiently large contour it follows that wnC1 ./
has the same number of zeros inside the contour as
n
Y
ˇ20 ˛2 3
sin .b a/:
kD1
2 D , we have for
Hence, if 0 1 2 , are the zeros of wnC1 ./ and m
sufficiently large m
1
.m 1/
CO
:
(4.2)
m D
b a
m
With the same idea, we obtain: For ˇ20 ¤ 0; ˛2 D 0 and for ˇ20 D 0; ˛2 ¤ 0,
m D
.m
1=2/
1
CO
:
b a
m
(4.3)
922
ABDULLAH KABLAN AND MANAF DZH. MANAFOV
For ˇ20 D 0; ˛2 D 0;
m D
m
1
CO
:
b a
m
(4.4)
Let .x; m / is an eigenfunction corresponding to eigenvalue m and ˇ20 ¤ 0; ˛2 ¤
0. After inserting (4.2) into (3.3) and (3.4) and taking into account the definition of
solutions defined in 2 we get
1
.m 1/
.x a/ C O
; x 2 Œa; b;
.x; m / D ˛2 cos
b a
m
and for ˇ20 ¤ 0; ˛2 D 0
.b
.x; m / D ˛1
.m
a/
.m 1=2/
sin
.x
1=2/
b a
1
; x 2 Œa; b;
a/ C O
m2
for ˇ20 D 0; ˛2 ¤ 0
.x; m / D ˛2 cos
.m
1=2/
.x
b a
1
; x 2 Œa; b;
a/ C O
m
for ˇ20 D 0; ˛2 D 0
.x; m / D ˛1
.b a/ m
sin
.x
m
b a
1
; x 2 Œa; b:
a/ C O
m2
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Authors’ addresses
Abdullah Kablan
Gaziantep University, Faculty of Arts and Sciences, Department of Mathematics, 27310 Gaziantep,
Turkey
E-mail address: [email protected]
Manaf Dzh. Manafov
Adiyaman University, Faculty of Arts and Sciences, Department of Mathematics, 02040 Adiyaman,
Turkey
E-mail address: [email protected]