ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

2002,22B(3): 335-345

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONALTRANSMISSION CONDITIONS 1 O.Sh.Mukhtarov Department of Mathematics, Science and Arts Faculty, Gaziosmanpasa University, Tokat, Turkey

Mustafa Kandemir Department of Mathematics, Faculty of Amasya Education, Ondokuz Mayis University, Amasya, Turkey

Abstract In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and abstract linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found. Key words Asymptotic behaviour of eigenvalues, boundary-value problems, functionalconditions, discontinuous coefficients 2000 MR Subject Classification

1

34L20

Introduction

The investigation of-boundary-value problems for which the eigenvalue parameter appears in both the equation and boundary conditions originates from the works of G.D. Birkhofl1 1 ,2]. There are many papers and books, where the spectral properties of such problems are investigated (see, for example [5, 8, 10, 11, 12, 13] and corresponding bibliography). Usually, in many monographs and papers, the theory of boundary- value problems for ordinary differential equations is considered for equations with a constant coefficient at the highest derivative and for boundary conditions which contain only endpoints of the considered interval. But this paper deals with one nonstandard boundary-value problem for second order ordinary differential equation with discontinuous coefficients and boundary conditions containing not only endpoints of the considered interval, but also point of discontinuity, and abstract linear functionals. Moreover, the eigenvalue parameter appear both in differential equation and boundary conditions. 1 Received

February 26, 2001. The research was supported by TUBITAK.

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Some spectral properties of such problems and its applications to the corresponding initialboundary-value problems for parabolic equations was investigated by 0.Sh.Mukhtarov[5] and O.S H. Mukhtarov, H.Demir[6]. The similar problems for differential equations with discontinuous coefficients, but without the abstract linear functionals in boundary conditions was investigated by M.L. Rasulov in monographs [8, 9]. One must note that in the series of S.Y.Yakubov works, appearing in the recent years, have been construct an abstract theory of boundary-value problems with a parameter in boundary conditions (see [11, 12, 13] and corresponding bibliography). In these works, in particular, the main spectral properties of the more general problems, but with continuous coefficients are investigated.

2

Statement of the Problem Let us consider a differential equation.

a(x)y" + c(x)y == >..2y,

x E [-1,0) U (0,1]

(1)

with the functional-transmission conditions

L 1

Ll(y) ==

,\l-k (aiky(k) (-1) +Jiky(k) (-0) +riky(k) (+0) + ,Biky(k) (1) +Tiky) == 0, i == 1,2,3,4,

k=O

(2)

where a(x) and c(x) are complex-valued functions; a(x) == a1 at x E [-1,0), a(x) == a2 at x E (0,1]' aik, Jik, rik, ,Bik are complex coefficients; Tvk are abstract linear functionals; y(k) (±o) denotes limx~±o y(k) (x). Below by W;(-I,O) + W;(O, 1), q E (1,00), k == 0,1,2,··· we denote Banach space of complex-valued functions y == y(x) defining on [-1,0) U (0, 1]' which belongs to W;(-l,O) and (0,1) on intervals (-1,0) and (0,1) respectively, with the norm

W;

W;

W;

. where (-1,0) and (0,1) are usual Sobelev spaces. Note that, without loss of generality we consider the equation (1) instead of more general equation

a(x)y" Since, by using the substitution y ==

+ b(x)y' + c(x)y == ,\2y.

yeCP(x) ,

where

z

__1_ j b(t)dt at 2al

cp(x) ==

__l_jb(t)dt 2a2

x E [-1,0);

-1 x

at

XE(O,l],

o

we find that this equation takes the form (1) with the same eigenvalue parameter A. Also, it is easy to verify that under this substitution the form of boundary conditions (2) has not changed.

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Besides, for simplicity we replace the general domain [a, c) U (C, b], a < for we can return to the general case by making the substitution

C

< b by

337

[-1, 0) U (0, 1],

x E [-1,0);

x=={ c+(c-a)t at c + (b - c)t at

x E (0,1].

Some special cases of the problem (1)-(2) arise in a varied physical transfer problems, in particular, in the heat and mass transfer problemslsl.

3

Eigenvalues of the Problem

As usual, these values of parameter A for which the considered boundary-value problem (1)-(2) has a nontrivial solutions, we called the eigenvalues of the problem (1)-(2). Let YI0(X, A), Y20(X, A) and Y30(X, A), Y40(X, A) denote a some fundamental systems of solutions of the differential equation (1) on [-1,0) and (0,1] consequently. Defining .(

YJ

') _ { YjO(X, A) , x E In;

X,A

-

o

,

x ~

In.

Where II == 12 == [-1,0),13 == 14 == (0,1], the general solution of the equation (1) can be represented by the form 4

y(x, A)

==

L CjYj(x, A).

(4)

j=1

Substituting (4) into the boundary conditions (2) we obtain a system of linear homogeneous equations 4

LcjLV(Yj) == 0,

v == 1,2,3,4,

(5)

j==1

for the determination the constants Cj, j ==~. Consequently eigenvalues of the problem (1)-(2) consists of the zeros of characteristic determinant

(6) At first, according to the considered problem, we shall divide the complex A-plane into specific sectors, in which by turns we shall find the asymptotic expressions for solutions of the differential equation, for boundary functionals and boundary-value forms. Then by substi tuting these obtained asymptotic expressions into the equation ~ (A) == 0 we shall find the corresponding asymptotic formulas for the eigenvalues. Note that, such formulas not only of interest in themselves, but they also may be used for the establishing the completeness and basis properties of the system of eigen-and associated functions of the considered problem The cases argc-

=1=

arga2 and argc, == arga2 shall be investigated separately.

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i-

Asymptotic Behaviour of Eigenvalues for the Case argal

argOQ

4.1

Separation of the complex A-plane into specific sectors Throughout the paper we used the notations: WI == (yffil) -1, W2 == - (yffil)-1, W3 == (y!a2)-1, W4 == _(y!a2)-I, where Vi:== Izlei~, - 7r < argz ~ tt, Divide the complex A-plane into four sectors 3 v , v == ~, by the rays lk == {A E CI ReAwk == 0, (-l)k ImAwk ~ O}. On each of these sectors each of the real-valued functions ReAwk is of a single sign since this functions can vanish only on boundaries of the sectors 3 v . Let us consider one of the sectors (3 v ) with fixed index u. Using the same considerations as in [7] it is easy to verify that corresponding to the equation (1) there exists a fundamental system of particular solutions Y10(X, A), Y20(X, A) on [-1,0) and Y30(X, A), Y40(X, A) on (0,1] respectively, which are regular analytic functions of A E S; and for sufficiently large IAI, and which with their derivatives, can be expressed in the asymptotic form

(7) Here, as usual, the expression 0(*) denotes any function of the form f(X,A)/A, where I k and sufficiently large IAI always remains less than a constant. Now let lj (j == ~) are arbitrary rays, which originated from the point A == 0, distinct from the rays lj and situated so as to the from the sequence

If(x, A)I for x E

(8) The rays

lj divide each sector 3 v into two subsectors. Thus we have the eight sector which we

shall signify as OJ, j == U. As it seems from the construction, the sectors 0 == {01, O2 , ' ' . ,Os} can be distributed into two groups of 0(1) == {0~1), ... , 0~1)} and 0(2) == {0~2), ... , 0~2)} such that, the group O(k) includes those sectors OJ in which

4.2

Asymptotic expressions for the characteristic determinant

~(A)

in the

n sec-

tors for large values of IAI Each of the real-valued functions ReAwk is not changed sign also in each of the sectors OJ, since each of they is a subsector of certain sector 3 v .

Let Yk == Yk(X, A), k == ~ are functions, defining as for the solutions YkO(X, A) of the equation in I k , for which satisfied the asymptotic expressions (7). First we need asymptotically estimate the expressions TvkYn (., A) as A --+ 00 in the sectors OJ. Since the linear functionals T vk acts from W;(-l,O) + W;(O, l ) into complex plane C continuously, in virtue of general representation from of the continuous linear functionals in the Lq(a, b) spaces and using the well-known methods of real analysis it may be shown that there exists a functions Uvks E W;( -1,0) the equations k

TVk(Y) =

+ W;(O, 1) such

0

'2)/ y(s)(x)Uvks(x)dx + / s=o -1

0

that for every Y E W;( -1,0)

1

y(s)(x)Uvks(x)dx),

v

=~,

k

+ W;(O, 1)

= 0,1

(9)

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Mukhtarov & Kandemir: ASYMPTOTIC BEHAVIOUR OF EIGENVALUES

are holds, where 1:.p

+ 1.q == 1[4].

Only in one of the sectors of the group 0(1) the conditions Re"\w3 ~ 0 and Re"\wl -t +00 as ,.\ -t 00 are holds, as well only in one of the sectors of the group 0(2), the conditions Re"\wl ~ 0 and Re"\w3 -t +00 as ,.\ -t 00, are holds. These sectors denote as 0~1) and 0~2) accordingly. We shall calculate asymptotic approximation expressions for TvkYj only for these sectors of the groups 0(1) and 0(2), since calculations for others sectors can be made analogically. For convenience throughout below by [M], M E C, we shall denote any sum of the from M + f("\), when f(,.\) -t 0 as ,.\ -t 00. W4 ==

First let ,.\ vary in 0~1). Substituting (7) into (9), having in view the equations -W3 and applying the well-known.

W2

==

-wI,

Riemann-Lebesque Lemma [cf. 8, p.117, Lemma 7] we get

! y~~)

k

k

0

Tvdyd = L

(x, A)Uvks(x)dx = L

8=0_ 1

k

= (AWl)k '2)AWl)S-k 8=0

!

1

! 0

(AwdSeAWIX (1 + O( ~ ))Uvks(x)dx

8=0_ 1

(10)

e-AWIX{Uvks( -x)(l + O( ~))}dX

0

== ,.\k[O],

k

T vk(Y2) = L

! y~~)(x,

k

0

8=0_ 1

k

!

1

= (Aw2le-AW2 L(AW2)s-k 8=0

!

0

A)Uvks(x)dx = L(Aw2)Se-AW2

e AW2{1+X) (1 + O( ~))Uvks(X)dX

8 = 0 _1

e-AW2X{Uvks(X - 1)(1 + O(~))}dX

0

== ,.\ k e- A W 2 [0], (11) k

T vk(Y3) = L

! y~~)(x, 1

8=0 0

= (Aw3)keAW3

! 1

k

L(AW3)s-k 8=0

!

1

k

A)Uvks(x)dx = L(Aw3)SeAW3 8=0

e- Aw 3 (1-

x) ( 1

+ O( ~))UVkS(x)dX

0

e- AW3X{Uvks(1- x)(l

+ O( ~ ))}dx

0

== Ak e A W 3 [0], (12) k

T vk(Y4) = L

! yi~)

k

1

(x, A)Uvks(x)dx

8=0 0

k

=

(AW4)k L( AW4)s-k 8=0

== ,.\k[O].

!

1

0

=L

!

1

(Aw4)SeAW4X(1 + O( ~))UVkS(X)dX

8=0 0

e-AW3X{Uvks(x)(1 + O( ~))}dX

(13)

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Using this formulas we find the next asymptotic expressions for LvYj in the n~l) , as A -+

00,

1

L i (Y1) ==

L: A1-k{aik(Aw1)ke-AWl [1] + Jik( AW1)k[1] + (AW1)k [0]}

k=O == A[JiO+W1 Ji1],

(14)

1

L i (Y2) ==

L: A1-k{aik(AW2)ke-AW2[1] + Jik( Aw2)k[1] + (AW2)ke-AW2[0]} k=O

== Ae- AW2

1

L:{[aikW~] +

(15) [Jikw~]eAW2} == Ae- AW2[aiO

+ W2 ai1],

k=O

1

L i (Y3) ==

L: A1- k{')'ik( AW3)k[1] + fJik(Aw3)keAW3[1] + (AW3)keAW3[0]} k=O

(16)

== A{[riO + W3ri1] + e AW3[{3iO + W3{3i1]} and analogically (17) All calculations for the sector n~2) are carried out analogically. After needed calculations we get the next asymptotic expressions for T v k (Yi) and L, (Yj) as well

and

By substituting the asymptotic expressions (14)-(17) for L i (Yj) into the determinant ~(A), taking the common factors A of each column and the common factor e- AW2 of second column outside the determinant and taking into account W2 == -WI, W4 == -W3 we may representate this determinant as the next asymptotic form;

where -1 == ml

< m2 < ... < m

U1

== 1, and A;k) some complex numbers.

Furthermore, it is easy to see that

+ W1 J11 J2 0 + W1 J21 J3 0 + WIJ31 J4 0 + W1 641

J1 0 A(l) 1

-

+ W2 a11 a20 + W2 a21 a30 + W2 a31 a40 + W2 a41

a10

rIO + W3')'11

+ W3')'21 ')'30 + W3')'31 ')'40 + W3')'41

')'20

+ W4fJ11 fJ20 + W4fJ21 fJ30 + W4fJ31 fJ40 + W4fJ41

fJ10

Mukhtarov & Kandemir: ASYMPTOTIC BEHAVIOUR OF EIGENVALUES

No.3

+ W1 611 620 + W1 621 630 + W1 631 640 + W1 641 610

A(l) 0"1

==

+ W2 a11 a20 + W2 a21 a30 + W2 a31 a40 + W2 a41 a10

+ W3f311 (320 + W3(321 (330 + W3(331 (340 + W3(341 f310

341

+ W4r11 r20 + W4r21 r30 + W4r31 r40 + W4r41 rIO

It can be shown analogically that, the characteristic determinant ~(A) in the sector n~2) has the next asymptotic representation;

(21) where -1 ==

n1

< n2 < ... < n0"2

== 1 and

+ W1 a11 a20 + W1 a21 a30 + W1 a31 a40 + W1 a41

a10 A(2) -

1 -

610 + W1 611

A(2) 0"2

==

610 + W2 611

+ W3f311 620 + W2 621 (320 + W3 (321 630 + W2 631 (330 + W3(331 640 + W2 641 (340 + W3(341

+ W4r21 r30 + W4r31 r40 + W4r41

+ W3(311 (320 + W3(321 (330 + W3(331 (340 + W3(341

+ W4r11 r20 + W4r21 r30 + w4r31 r40 + W4r41

+ W2 a11 620 + W1 621 a20 + W2 a21 630 + W1 631 a30 + W2 a3l 640 + W1 641 a40 + W2 a41 a10

f310

(310

rIO +W4r11 r20

rIO

4.3

Asymptotic behaviour of the eigenvalues Now we can find the needed asymptotic formulas for the eigenvalues of the considered problem for the case arga l i= arga2. Theorem 1 Let the following conditions be satisfied. 1) arg a1

i= arg a2' q

+ (-1)i a 11 a20vaI + (-I)i a 2 1 a30vaI + (-1)i a 3 1 a40vaI + (-I)i a 4 1

810 vaI +- (-1)i+ 1611

a10vaI

i= 0,

> 1.

2) c(·) E L q(-I, 1), 3) ()ij

i == 1, 2,

(310 Viii + (-l)j f311

620 vaI + (-1)i+1621 (320 Viii + (-I)j (321 830 vaI + (-1)i+ 1631 640 vaI + (-1)i+ 1641

+ (-l)j f331 f340Viii + (-1)jf341 f330Viii

+ (-1)j+1 r 11 r20Viii + (-I)j+1 r 2 1 r30Viii + (-1)j+1 r 3 1 r40Viii + (-1)j+1 r 4 1 rIO Viii

j == 1, 2.

4) The linear functionals Tvk in the spaces W;(-l,O) + Wqk(O, 1) are continuous. Then the boundary-value problem (1)-(2), has in each sector Sv an precisely numerable number eigenvalues, whose asymptotic behaviour may be expressed by the following formulas.

(22)

A~2) =

-J(il1rni(1 +

O(~)),

(23)

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Ah3 ) = An(4)

Vii21rni(1

Vo1.22 Ser.B

+ O(.! )),

(24)

n

= - Y!£i21fni(l + O(.!)), n

(25)

U:l

n == 1,2, .... Proof By the rays lj divided the complex A-plane into four sectors R j , j == 1,4. Let R, that sector, which contains the ray lj. This sector we shall distribute into two groups R(l) == {R 1,R2} and R(2) == {R 3,R4}. Obviously that each sector of the group R(k) consists of two sectors of the groups n(k) by R~k) denote that sector of the group R(k) which contain n~k) (k == 1,2). As seems from the consideration in Subsections 4.1 and 4.2 the asymptotic expressing (20) and (21) are holds also in the sectors R~l) and R~2) respectively. Let Ri 1) and Ri 2) are the other sectors of the groups R(l) and R(2) respectively. By the similar way as in Subsections 4.1. and 4.2, one can prove that the characteristic determinant ~(A) has the asymptotic representations given by

(26) and ~(A) == A4 e- Aw 3 ([B1 2 )]eh 1)

AWl

+ [B~2)]et2AW1 + ... + [B~;)]et7"2 AWl),

2)

in the sectors Ri and Ri respectively, where -1 == -1 == t 1 < t2 < ... < t-, == 1,

+ w1 Q11 Q20 + W1 Q21 Q30 + W1 Q31 Q40 + W1 Q41

810

+ W1 Q11 Q20 + W1 Q21 Q30 + W1 Q31 Q40 + w1 Q41

810

+ W1 Q11 Q20 + W1 Q21 Q30 + W1 Q31 Q40 + W1 Q41

810

Q10

B(l) 1 -

Q10

B(l) '1

==

Q10

B(2) 1 -

+ W1 611 820 + W1 821 630 + W1 631 640 + W1 641 610

B(2) '2

==

+ W2 811 820 + W2 821 830 + W2 831 840 + W2 841 + W2 811 820 + W2 821 630 + W2 631 840 + W2 841 + W2 811 620 + W2 821 830 + w 2 831 840 + W2 841 + W2 Q11 Q20 + W2 Q21 Q30 + W2 Q31 Q40 + W2 Q41 Q10

81

< 82 < ... < 8'1

+ w3')'11 ')'20 + W3')'21 ')'30 + W3')'31 ')'40 + W3')'41 ')'10

(310 (320 (330 (340

+ W3(311 + W3(321 + W3 (331 + W3(341

+ W3')'11 ')'20 + W3')'21 ')'30 + W3')'31 ')'40 + W3')'41 ')'10

+ W3')'11 ')'20 + W3')'21 ')'30 + w3')'31 ')'40 + W3')'41 ')'10

(310 (320 (330 (340

== 1,

+ w4(311 + W4(321 + W4(331 + W4(341

+ W4')'11 ')'20 + W4')'21 ')'30 + W4')'31 ')'40 + W4')'41

')'10

(310 (320 (330 (340

(310 (320 (330 (340

+ W4(311 + W4(321 + W4(331 + w4(341 + W4(311 + W4(321 + W4(331 + W4(341

(27)

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According to the condition 3 of the Theorem, the principal terms of first and last coefficients 2), of the asymptotic quasipolinoms (20), (21), (26) and (27), i.e. the numbers Ail), A~11)' Ai A~2} ( l ) B(l) B(2) B(2) Bl ' ri » l' T2 are diff itterent f rom zero. Since Ll(A) == Ll~j) (A) when A vary in sector R~j) and all quasipolynoms Ll~j) (A) has the same form, it is enough to investigate only one of them. Namely, we shall investigate the equation Ll(A) == a only in the sector R~l) i.e. the equation

[Ail)]emlAW3

+ ... + [A~~)]emCTl

AW3

== O.

(28)

By virtue of the [8, P.100, lemma 1] the equation (28) has in the sector number of roots An which are contained in a strip

ITo

(1)

of finite width h

R6

1

)

an infinite

h

= {A E CI IR eAw3 I < 2}

> a and have the asymptotic expression 1 IAn W 31== l7rn(1+0(-))I· n

(29)

Taking into account An E TI~l) and An E R~l) from (29) we get the needed asymptotic formula

An = va27fni(l

+ O( !)), n

n

= ±l, ±2, ... ,

(30)

where there is only one choice possible for the sign of the integer n. 1 ) it can be shown that the asymptotic By the same consideration as used for sector 1 behaviour of the eigenvalues, which are contain in the sector ) also has the same form as (30), but the integer n has the opposite sign. The other formulas (22) and (23) can be obtained by the same procedure, which we used in proving the formula (30).

R6

5 5.1

Ri

Asymptotic Behaviour of Eigenvalues for the Case argtz, ==

arg~

Separation of the complex A-plane into half-planes

Since, in the case argu, == arga2 the lines II == {A E CI ReAwl == a} and l2 == {A E CI ReAw3 == a} coincided, then the line l == II == l2 is divide the complex A-plane into two half-plane. In each of these half-planes the real-valued functions Re(Aw1) and Re(Aw3) is not changed their signs and moreover, have the same signs. These half-planes we notice as 2:1 ==

<

{A E CI ReAw1 2:: 0, ReAw3 2:: a} and 2:2 == {A E CI ReAw1 ::; 0, ReAw3 a}. 5.2 Asymptotic expressions for the characteristic determinant Ll(A) in the halfplanes L1 and L2 First we find a suitable asymptotic representation of the TvkYn, when A

-t 00

in the

half-planes 2:1 and L2' where Yn == Yn(x, A) are the same solutions as in Section 4.2. By the similar considerations as used in the case arga1 i= arga2 it can be shown that for the T i k (Yj) takes place the asymptotic representations

when A E 2:1' A -t

00

and

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when A E 2::2' A -+ 00. Taking advantage of this expressions for Tik(Yj), we obtain the asymptotic representations

which are valid both in determinant ~(A)

==

L1

and in

L2.

Substituting these formulas into the characteristic

det(LiYj) we obtain, after some simple rearrangements the asymptotic

representation

== k1 < k2 < ... < ki == 1,

where -1

P1==

Pi==

(}:10

+ W1(}:11

J 10 + w2 J11

/10

+ W3/11

/310

+ W4/311

(}:20

+ W1(}:21

J 20 + W2 J21

/20

+ W3/21

/320

+ W4/321

(}:30

+ W1(}:31

J30 + W2 J31

/30

+ W3/31

/330

+ W4/331

(}:40

+ W1(}:41

J40 + W2 J41

/40

+ W3/41

/340

+ W4/341

J 10 + W1 J11

(}:10

+ W2(}:11

f310

+ W3f311

/10 +W4/11

J20 + W1 J21

(}:20

+ W2(}:21

f320

+ W3/321

/20

+ W4/21

J 30 + w1 J31

(}:30

+ W2(}:31

/330

+ W3/331

/30

+ W4/31

J40 + WI J41

(}:40

+ W2(}:41

/340

+ W3/34I

/40

+ W4/4I

5.3 Asymptotic behaviour of the eigenvalues

Now we can prove the next theorem Theorem 2 Let the following conditions

1) argai

== arge-,

2) c(·) E L q(-1,1), 3)

()12

(()12

1= 0,

and

()21

~e

satisfied.

s > 1,

()2I 1= 0, are the same determinant as in Theorem 1)

4) The linear functionals Ti k in the spaces W;(-1,0)

(i=="G,

+ W;(0,1)

are continuous.

k==0,1) Then the boundary value problem (1)-(2) has in each half-plane L1 and 2:2 an precisely numerable number eigenvalues, whose asymptotic behaviour may be expressed by the following formulas,

A~l)

=

A~2) = -

..;al..;a:2 7fni(l + O( ~)), n = 1,2, ... ..;al+..;a:2 n ..;al..;a:2 7fni(l + O( ~ )), n ..;al+..;a:2 n

= 1,2,··· .

Proof According to the condition 3 of this Theorem, the principal terms of the first and last coefficients of the asymptotic quasipolynom (33), i.e. the numbers PI and Pi, are different

from zero. Again, by virtue of the [8, P.IOO, Lemmal], the quasipolynom (33) in the half-planes

Mukhtarov & Kandemir: ASYMPTOTIC BEHAVIOUR OF EIGENVALUES

No.3

2:1

and

2:2 has

an infinite number of roots {A~l)} and {A~2)} respectively, which are contained

in a strip

of finite width h

345

II = {-~ E q

ReA(wl

+ W3) < ~ }

> 0 and have the asymptotic representations (34)

Taking into account that the eigenvalues ~~1) and A~2) belongs to the strip we get the needed asymptotic formulas. '(8) An

== r;:;: valvfiii, r;:;: 7rnl·(1 + 0(1)) - , n .== ±1 , ±2 ... , ya1

+ ya2

n

S

TI

from (34)

== 1, 2,

where there is only one choice possible for the sign of integer n, but opposite signs for the half-planes 2:1 and 2:2 which completes the proof. References 1 Birkhoff G D. On the asymptotic character of the solution of the certain linear differential equations containing parameter. Trans Amer Mat Soc, 1908, 9: 219-231 2 Birkhoff G D. Boundary value and expantion problems of ordinary linear differential equations. Trans Amer Mat Soc, 1908, 9: 373-395 3 Likov A V, Mikhailov Yu A. The Theory of Heat and Mass Transfer. Qosenergoizdat, 1963 (Russian) 4 Maz'ja V G. Sobolev Space. Berlin: Springer-Verlag, 1985 5 Mukhtarov 0 She Discontinuous Boundary value problem with spectral parameter in boundary conditions. TUrkish J Math, 1994, 18(2): 183-192 6 Mukhtarov 0 Sh, Demir H. Coerciveness of the discontinuous intial-boundary value problem for parabolic equations. Israel J Math, 1999, 114: 239-252 7 Naimark M A. Linear Differential Operators. New York: Ungar, 1967 8 Rasulov M L. Methods of contour integration. Amsterdam: North Holland Publishing Company, 1967 9 Rasulov M L. Applications of the Contour Integral Method. Moskow: Nauka, 1975 (Russion) 10 Shkalikov A A. Boundary Value Problems for Ordinary Differential Equations With a Parameter in Boundary Condition. Trudy Sem Imeny I G Petrowsgo, 1983, 9: 190-229 11 Yakubov S. Completeness of Root Functions of Regular Differential Operators. New York: Longman, Scientific, Technical, 1994 12 Yakubov S, Yakubov Y. Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman and Hall/CRC, Boca Raton, 2000. 568 13 Yakubov S. Solution of irreguler problems by the asymptotic method. Asymptotic Analysis, 2000, 22: 129-148