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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES FOR FOURTH-ORDER BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS COEFFICIENTS AND TRANSMISSION
CONDITIONS
MUSTAFA KANDEM ·IR
Abstract. We investigate a fourth-order boundary value problem with dis-continuous coe¢ cients, functional many points and transmission conditions. In this problem, boundary conditions contain not only endpoints of the con-sidered interval, but also a point of discontinuity, a …nite number internal points and abstract linear functionals. We discuss asymptotic distribution of its eigenvalues. Finally, we obtain asymptotic formulas for the eigenvalues of the problem in sectors of the complex plane.
1. Introduction
In classical theory, boundary-value problems for ordinary di¤erential equations are usually considered for equations with continuous coe¢ cients and for boundary conditions which contain only end-points of the considered interval. However, this paper deals with one nonclassical boundary-value problem for ordinary di¤erential equation with discontinuous coe¢ cients and boundary conditions containing not only end-points of the considered interval, but also a point of discontinuity and internal points. This type problems are connected with di¤erent applied problems which include various transfer problems such as heat transfer in heterogeneous media. Naturally, transmission problems arise in various physical …elds as the theory of di¤raction, elasticity, heat and mass transfer [10], [16], [17], [18].
The investigation of boundary value problem for which the eigenvalue parameter appears both in the equation and boundary conditions originates from the works of G. D. Birkho¤ [4], [5]. There are many papers and books that the spectral properties of such problem are investigated; see[2], [3], [6]. Some spectral properties of such problems with discontinuous coe¢ cients and the eigenvalue parameter both in the di¤erential equation and boundary conditions have been studied by O. Sh.
Received by the editors: March 30, 2016, Accepted: Aug. 25, 2016. 2010 Mathematics Subject Classi…cation. 34A36; 34B09; 34L20.
Key words and phrases. Fourth order problem, eigenvalue parameter, asymptotic distribution, internal points, transmission conditions.
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Mukhtarov, M. Kandemir and some others [7], [8], [9], [11], [12], [13]. In this study, we shall consider fourth-order di¤erential equation
p(x)u(4)+ q(x)u = 4u; x 2 I; (1.1)
with the functional-transmission boundary conditions Lk(u) = 3 X s=0 4 s [ ksu(s)( 1) + ksu(s)( 0) + ksu(s)(+0) + ksu(s)(1) + Z 0 1 u(s)(x) ks(x)dx + Z 1 0 u(s)(x) ks(x)dx + 2 X i=1 Ni ks X j=1 ij ksu (s)(ai ksj)] = 0; k = 1; 2; :::; 8; (1.2)
where I = I1[ I2 = [ 1; 0) [ (0; 1]; p(x) and q(x) are complex valued functions;
p(x) = pj(x) and q(x) = qj(x) for x 2 Ij; j = 1; 2; ks; ks; ks; ks; ksare
com-plex coe¢ cients; aiksj2 Ii internal points and u(mk)( 0) denotes lim x! 0u (mk)(x): Denote: F1ku := 3 X s=0 4 sZ 0 1 u(s)(x) ks(x)dx and F2ku := 3 X s=0 4 sZ 1 0 u(s)(x) ks(x)dx:
F1k and F2k are abstract linear functionals. F1k + F2k acts from Wpk( 1; 0) +
Wk
p(0; 1) into complex plane C continuously. In virtue of the general representation
of the continuous linear functionals in the Lq( 1; 1) spaces and using the well-known
methods of real analysis it may be shown that there exists a function ks(x) 2 Wpk( 1; 0) + Wpk(0; 1) such that for every u 2 Wqk( 1; 0) + Wqk(0; 1); (1p+1q = 1). Wpq( 1; 0; 1) := Wpq( 1; 0) + Wpq(0; 1); 1 < p < 1; q = 0; 1; 2; :::; denotes the
Banach spaces of complex valued functions u = u(x) de…ned on [ 1; 0)[(0; 1]; which belongs to Wq
p( 1; 0) and Wpq(0; 1) on intervals ( 1; 0) and (0; 1); respectively, with
the norm kukWpq( 1;0;1)= kuk p Wpq( 1;0)+ kuk p Wpq(0;1) 1 p where Wq
Note that, without loss of generality we consider the equation (1.1) instead of more general equation
p(x)u(4)+ p3(x)u000+ p2(x)u00+ p1(x)u0+ p0(x)u = 4u; x 2 I: (1.3)
If p36= 0; by using the substitution
u =eue (x); (x) = 8 < : 1 4p1 Rx 1p3(t)dt; x 2 [ 1; 0) 1 4p2 Rx 1p3(t)dt; x 2 (0; 1] ; we can …nd that equation (1.3) takes the form
p(x)eu(4)+pe2(x)ue00+pe1(x)eu0+pe0(x)u =e 4u;e
where pe2; pe1; ep0 are continuous in I and is the same eigenvalue parameter.
Therefore, we can write equation (1.1) instead of equation (1.3) from [14]. Also, it is easy to verify that under this substitution the form of boundary conditions (1.3) has not changed.
2. Eigenvalues of the problem
Let u1j and u2j; j = 1; 2; 3; 4; denote some fundamental systems of solutions of
the di¤erential equation (1.1) on I1 and I1, respectively. By de…ning
u1j(x; ) = 0; x 2 I2
u2j(x; ) = 0; x 2 I1 j = 1; 2; 3; 4;
the general solution of the equation (1.1) can be written in the form u(x; ) = 2 X =1 4 X j=1 c ju j(x; ); (2.1)
where c j are arbitrary constant numbers. Substituting (2.1) into boundary
condi-tions (2.1) yields a system of linear homogeneous equacondi-tions Lk(u(x; )) = 2 X =1 4 X j=1 c jLk(u j) = 0; k = 1; 2; :::; 8 (2.2)
for the determination of the constants c j; = 1; 2; j = 1; 2; 3; 4: Consequently,
the eigenvalues of the boundary value problem (1.1)-(1.2) consist of zeros of the characteristic determinant
( ) = det (Lk(u j))8 8; = 1; 2;
j = 1; 2; 3; 4; k = 1; 2; :::; 8: (2.3)
First, according to considered problem, we shall divide the complex -plane into speci…c sectors, in which we shall …nd the asymptotic expression for solutions of the di¤erential equation, for boundary functionals and boundary value forms
with transmission conditions. Then, by substituting these obtained asymptotic expression into the equation ( ) = 0 we shall …nd the corresponding asymptotic formulas for the eigenvalues of the problem. Note that, such formulas are not only of interest in themselves, but also they may be used for establishing the completeness and basis properties of the system of eigen-and associated functions of considered problem. In this study, we shall investigate the cases of both arg p1 6= arg p2 and
arg p1= arg p2:
3. Asymptotic distribution of eigenvalues for the case arg p16= arg p2
3.1. Separation of the complex plane into speci…c sectors. Throughout the paper we employ the notation
!j1 = (pj) 1 4; ! j2= (pj) 1 4 !j3 = i (pj) 1 4; ! j4= i (pj) 1 4; j = 1; 2 where z14 := jzj e i(arg z)
4 ; < arg z < : Divide the complex plane into eight sectors Sk; k = 1; 2; : : : ; 8; by the rays
lk = 2 Cj Re ! j= 0; ( 1)kIm ! j 0
= 1; 2; j = 1; 2; 3; 4 g :
On all of these sectors each of the real valued functions Re ! j is of a single
sign, since these functions can vanish only on boundaries Sk: Let us consider one
of the sectors (Sk) with …xed index k: Using the same considerations as in [14]
it is easy to verify that for equation (1.1) there exists a fundamental system of particular solutions u1j(x; ) on I1, j = 1; 2; 3; 4; and u2j(x; ) on I2, j = 1; 2; 3; 4;
respectively, which are analytic functions of 2 Sk and for su¢ ciently large j j ;
and which with derivatives, can be expressed in the asymptotic form u j(x; ) = e ! jx(1 + O( 1 )) u(s)j(x; ) = ( ! j)se ! jx(1 + O( 1 )); = 1; 2; j = 1; 2; 3; 4: (3.1)
Here, as usual, the expression O(1) denotes any function of the form f (x; ); where jf(x; )j for x 2 Ij; j = 1; 2; and su¢ ciently large j j always remain less than a
constant. Now let l0
k; k = 1; 2; : : : ; 8; be arbitrary rays, which originate from the point
= 0, distinct from the rays l and situated so as to from the sequence l1; l01; l2; l02; l3; l03; l4; l04; :::; l8; l80:
The rays l0
k divide each sector Sk into two subsectors. Therefore, we have sixteen
sectors which we shall denote as i; i = 1; 2; :::; 16: As it seems from the
construc-tion, the sectors = f 1; 2; :::; 16g can be distributed into two groups of (i)=n (i) 1 ; (i) 2 ; :::; (i) 8 o ; i = 1; 2 such that, the group (k); k = 1; 2; includes those sectors
i; i = 1; 2; :::; 16; in
which
Re ! j! 1; = 1; 2; j = 1; 2; 3; 4; as ! 1:
3.2. Asymptotic expressions for the characteristic determinant ( ) in the sectors. Each of the real valued functions Re !j does not change sign
also in each sector i, since each of them is a subsector of certain sector Sk:
Let u j = u j(x; ); x 2 I ; = 1; 2; j = 1; 2; :::; 8, are functions de…ning as
for the fundamental system in I ; for which satis…ed asymptotic expressions (3.1). Only in one of the sectors of the groups (1) the conditions
Re !11 ! +1; Re !21 0;
Re !13 ! +1; Re !23 0
and only in one of the sectors of the groups (2) the conditions
Re !21 ! +1; Re !11 0;
Re !23 ! +1; Re !13 0
are holds for ! 1. We shall denote these sectors as (1)0 and (2)0 ; respectively. Besides, we shall denote by [A]; A 2 C; any sum of the from A + f( ) when f ( ) ! 0 as ! 1:
First, let vary in (1)0 : Substituting (3.1) into (1.2), remembering that !11 = !12; !13= !14;
!21 = !22; !23= !24
and applying well-known Rieamann-Lebesgue Lemma [14, p. 117, Lemma 7), we have Lk(u11) = 3 X s=0 4 s ( ! 11)s kse !11[1] + ks[1] + ( !11)s Z 0 1 e !11x(1 + O(1)) ks(x)dx + Nks1 X j=1 1j ks( !11)se !11a 1 ksj[1])
= 3 X s=0 4 s( ! 11)s kse !11[1] + ks[1] + Z 1 0 e !11x(1 + O(1)) ks( x)dx + [0]) = 3 X s=0 4 s( ! 11)s( ks[1] + [0]) = 4 k0+ !11 k1+ !211 k2+ !311 k3 ; (3.2) Lk(u12) = 4e !12 k0+ !12 k1+ !212 k2+ !312 k3 ; (3.3) Lk(u13) = 4 k0+ !13 k1+ !213 k2+ !313 k3 ; (3.4) Lk(u14) = 4e !14 k0+ !14 k1+ !214 k2+ !314 k3 ; (3.5) Lk(u21) = 3 X s=0 4 s ( !21)s ks[1] + kse !21[1] + ( !21)s Z 1 0 e !21x(1 + O(1)) ks(x)dx + Nks2 X j=1 2j ks( !21)se !21a 2 ksj[1]) = 4 3 X s=0 !s21 ks[1] + kse !21[1] +!s21e !21 Z 1 0 e !21(1 x)(1 + O(1)) ks(1 x)dx + Nks2 X j=1 2j ks(!21)se !21a 2 ksj[1])
= 4 k0+ !21 k1+ !221 k2+ !321 k3 +e !21 k0+ !21 k1+ !221 k2+ !321 k3 + N2 ks X j=1 e !21a2ksj[!s 21 2j ks]); (3.6) Lk(u22) = 4 k0+ !22 k1+ !222 k2+ !322 k3 +e !22 k0+ !22 k1+ !222 k2+ !322 k3 + N2 ks X j=1 e !22a2ksj[!s 22 2jks]); (3.7) Lk(u23) = 4 k0+ !23 k1+ !223 k2+ !323 k3 +e !23 k0+ !23 k1+ !223 k2+ !323 k3 + N2 ks X j=1 e !23a2ksj[!s 23 2j ks]); (3.8) Lk(u24) = 4 k0+ !24 k1+ !224 k2+ !324 k3 +e !24 k0+ !24 k1+ !224 k2+ !324 k3 + N2 ks X j=1 e !24a2ksj[!s 24 2j ks]): (3.9)
From the system that is obtained by using (3.2)-(3.9), we have the characteristic determinant in (1)0 as asymptotic quasi-polynomial form
1( ) = 32e (!11+!13) [A1] e 11 !21+ + [A ] e 1 !21 + [B1] e 21 !23+ + [B ] e 2 !23 (3.10) where 1 = j1< j2< < j = 1; j = 1; 2; and A1 = A11+ A12; :::; A = A 1+ A 2; B1 = B11+ B12; :::; B = B 1+ B 2
some complex numbers. Let us denote 1 21( ) := 32e (!11+!13) [A1] e 11 !21 + [A2] e 12 !21+ + [A ] e 1 !21 ; (3.11) 1 23( ) := 32 e (!11+!13) [B 1] e 21 !23 + [B2] e 22 !23+ + [B ] e 2 !23 ; (3.12) and 1( ) = 121( ) + 123( ) :
Now, let the sector (1)0 divide two sectors as (1)01 and (1)02: We assume that one of the expressions 1
21( ) and 123( ) vanish in one of the sectors (1) 01 and
(1) 02:
Therefore, let the characteristic determinant 1( ) has the asymptotic
representa-tion in the form (3.11) in (1)01 and in the form (3.12) in (1)01: Here, all determinants are di¤erent from each other. Also, it is easy to see that A11and A12determinants
for …rst coe¢ cient of (3.11)
A11= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; A12= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 :
We can obtain that the other determinants of (3.11) in the same way. B11and B12
determinants for …rst coe¢ cient of (3.12)
B11= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83
10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; B12= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 :
The other determinants of (3.12) can be obtained in the same way. It can be shown analogically that, the characteristic determinant 2( ) in the sector (2)0 has the
next asymptotic quasi-polynomial representation
2( ) = 32e (!21+!23) [M1] e 11 !11+ + [M'] e 1' !11 + [N1] e 21 !13+ + [N'] e 2' !13 (3.13) where 1 = j1< j2< < j'= 1; j = 1; 2; M1 = M11+ M12; :::; M'= M'1+ M'2; N1 = N11+ N12; :::; N'= N'1+ N'2:
Now, let us denote
2 11( ) := 32e (!21+!23) [M1] e 11 !11 + [M2] e 12 !11+ + [M'] e 1' !11 ; (3.14) 2 13( ) := 32e (!21+!23) [N1] e 21 !13 + [N2] e 22 !13+ + [N'] e 2' !13 ; (3.15) and 2( ) = 211( ) + 213( ) :
Let the sector (2)0 divide two sectors as (2)01 and (2)02: We assume that one of the expressions 2
11( ) and 213( ) vanish in one of the sectors (2) 01 and
(2)
02:
There-fore, let the characteristic determinant 2( ) has the asymptotic representation in
di¤erent from each other and some of them in the form. M11and M12determinants
for …rst coe¢ cient of (3.14)
M11= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; M12= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 :
We can obtain that the other determinants of (3.14) in the same way. N11and N12
determinants for …rst coe¢ cient of ((3.15)
N11= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; N12= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83
10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 :
The other determinants of (3.15) can be obtained in the same way.
3.3. Asymptotic distribution of eigenvalues for arg p16= arg p2. Now we can
obtain the asymptotic formulas for the eigenvalues of the boundary value problem for arg p16= arg p2:
Theorem 1. We assume that the following conditions be satis…ed 1) arg p16= arg p2:
2) q(x) 2 Lp( 1; 1); p > 1:
3) Ai; Bi6= 0; i = 1 and i = ; Mi; Ni; 6= 0; i = 1 and i = ':
4) The linear functionals F1k + F2k in the spaces Wpk( 1; 0) + Wpk(0; 1) are
continuous.
Then, the boundary value problem (1.1)-(1.2) has in each sector Sk an precisely
numerable number eigenvalues, whose asymptotic distribution may be expressed by the following formulas.
j n = p 1 4 j ni(1 + O( 1 n)); j = 1; 2; (3.16) j+2 n = p 1 4 j ni(1 + O( 1 n)); j = 1; 2; (3.17) j+4 n = p 1 4 j n(1 + O( 1 n)); j = 1; 2; (3.18) j+6 n = p 1 4 j n(1 + O( 1 n)); j = 1; 2: (3.19)
Proof. By the rays lj0, the complex -plane is divided into eight sectors Dj; j =
1; 2; :::; 8: Let Dj be that sector which contains the rays lj: We shall distribute these
sectors into two groups D(i)=
n
D(i)1 ; D(i)2 ; :::; D(i)8 o
; i = 1; 2:
Obviously that sector of the group D(k) contains two sectors of the group (k) by D0(k) denote that sectors of the group D(k) which contain (k)0 ; k = 1; 2: As seems from the consideration in subsection 3.1 and 3.2 the asymptotic expressing (3.10) and (3.13) hold also in the sectors D0(1) and D(2)0 ; respectively. Let D(1)1 and D(2)1 are the other sectors of the groups D(1) and D(2) respectively. Only in one of the
sectors of the groups D(1) the conditions
Re !12 ! +1; Re !22 0;
and only in one of the sectors of the groups D(2) the conditions
Re !22 ! +1; Re !12 0;
Re !24 ! +1; Re !14 0 :
hold for ! 1: By the similar way as in subsection 3.1 and 3.2, one can prove that the characteristic determinants have the asymptotic quasi-polynomial repre-sentation given by 3( ) = 32e (!11+!13) [K1] e 11 !21 + + [Kr] e 1r !21 + [T1] e 21 !23+ + [Tr] e 2r !23 (3.20) and 4( ) = 32e (!21+!23) [U1] e 11 !11 + + [U%] e1% !11 + [V1] e21 !13+ + [V%] e2% !13 (3.21)
in the sectors D(1)0 and D(2)0 ; respectively, where
1 = j1< j2< < jr = 1; j = 1; 2; K1 = K11+ K12; :::; Kr= Kr1+ Kr2; T1 = T11+ T12; :::; Tr= Tr1+ Tr2 and 1 = j1< j2< < j%= 1; j = 1; 2; U1 = U11+ U12; :::; U% = U%1+ U%2; V1 = V11+ V12; :::; V%= V%1+ V%2: Let us denote 3 21( ) := 32 e (!11+!13) [K 1] e 11 !21 + [K2] e 12 !21+ + [Kr] e 1r !21 ; (3.22) 3 23( ) := 32e (!11+!13) [T1] e 21 !23 + [T2] e 22 !23+ + [Tr] e 2r !23 ; (3.23) and 3( ) = 321( ) + 323( ) :
Let the sector D(1)0 is divided into two sectors as D(1)01 and D(1)02: We assume that one of the expressions 321( ) and 323( ) vanish in one of the sectors D(1)01 and D02(1): Therefore, let the characteristic determinant 3( ) has the asymptotic
way for the sector D(2)0 the characteristic determinant 4( ) has the asymptotic
quasi-polynomial representation in the form in D(2)01
4 11( ) := 32e (!21+!23) [U1] e 11 !11 + [U2] e 12 !11+ + [U%] e1% !11 ; (3.24) and in D(2)02 4 13( ) := 32e (!21+!23) [V1] e21 !13 + [V2] e22 !13+ + [V%] e 2% !13 (3.25) and 4( ) = 411( ) + 413( ) :
Hence, let the characteristic determinant 4( ) has the asymptotic representation
in the form (3.24) in D(2)01 and in the form (3.25) in D(2)02: Here, all determinants are di¤erent from each other and some determinants are in the following form
K11= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; K12= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 :
The other determinants can be obtained in the same way. According to the condi-tion 3 of the theorem, principal term of …rst and last coe¢ cients of the asymptotic quasipolynomials (3.10), (3.13), (3.20) and (3.23) are di¤erent from zero, that is Ai; Bi 6= 0; i = 1 and i = ; Mi; Ni 6= 0; i = 1 and i = '; Ki; Ti 6= 0; i = 1 and
i = r; Ui; Vi 6= 0; i = 1 and i = %:
Since ( ) = j( ) when vary in sector D(i)j and all quasi-polynomials j( )
we shall investigate the equation ( ) = 0 only in the sector (1)0 : We know that
(1)
0 consists of the sectors (1) 01 and
(1)
02: Therefore, from (3.11), we can write the
equation
[A1] e 11 !21+ [A2] e 12 !21+ + [A ] e 1 !21 = 0 (3.26)
in (1)01 and from (3.12), the equation
[B1] e 21 !23+ [B2] e 22 !23+ + [B ] e 2 !23 = 0 (3.27)
in (1)02. By virtue of the [15, p. 100, Lemma 1] the equations (3.26) and (3.27) have an in…nite number of roots n which contain in strips
E01= 2 Cj jRe !21j < h1 2 and E02= 2 Cj jRe !23j < h2 2
in the sectors (1)01 and (1)02; respectively, of …nite width h1; h2> 0 and have the
asymptotic expressions 2 n!21 = 2 n 1 11 (1 + O(1 n)) = n(1 + O(1 n)) (3.28) and 6 n!23 = 2 n 2 21 (1 + O(1 n)) = n(1 + O(1 n)) : (3.29)
Taking into account 2n2 E01, 6n2 E02and 2n2 (1) 01 , 6 n 2 (1) 02 from (3.26) and (3.27) 2 n = (!21) 1 ni(1 + O( 1 n)) = p 1 4 2 ni(1 + O( 1 n)); n = 1; 2; ::: and 6 n = (!23) 1 ni(1 + O( 1 n)) = p 1 4 2 n(1 + O( 1 n)); n = 1; 2; :::;
where there is only one possible choice for the sign of the integer n: Similarly, from (3.14) and (3.15), we can write the following asymptotic expression in (2)01 and
(2) 02; respectively, 1 n = p 1 4 1 ni(1 + O( 1 n)); n = 1; 2; :::; and 5 n= p 1 4 1 n(1 + O( 1 n)); n = 1; 2; ::::
The other formulas in (3.16)-(3.19) can be obtained by the same procedure, which we used in proving above asymptotic formulas.
4. Asymptotic distribution of eigenvalues for the case arg p1= arg p2
4.1. Separation of the complex plane into speci…c sectors. In the case arg p1= arg p2; the lines
l1 = f 2 Cj Re !11= 0g ;
l3 = f 2 Cj Re !21= 0g
and the lines
l2 = f 2 Cj Re !13= 0g ;
l4 = f 2 Cj Re !23= 0g
coincide, then the lines d1 = l1 = l3 and d2 = l2 = l4 divide the complex plane
into four sectors S0
j; j = 1; 2; 3; 4. On all of these sectors each of the real valued
functions Re ! j is a single sign, since these functions can vanish only on
bound-aries S0
j: Now let d0k; k = 1; 2; 3; 4; be arbitrary rays, which originate from the point
= 0, distinct from the rays d and situated so as to from the sequence d1; d01; d2; d02; d3; d03; d4; d04:
The rays d0
k divide each sector Sj0 into two subsectors. Therefore, we have eight
sectors which we shall denote as Gi; i = 1; 2; :::; 8: As it seems from the construction,
the sectors G = fG1; G2; :::; G8g can be distributed into two groups of
G(i)= n
G(i)1 ; G(i)2 ; G(i)3 ; G(i)4 o
; i = 1; 2;
such that the group G(k)4 ; k = 1; 2; includes those sectors G(i); i = 1; 2; :::; 8; in
which
Re ! j! 1; = 1; 2; j = 1; 2; 3; 4; as ! 1:
Only in one of the sectors of the groups G(1) the conditions
Re !11(Re !21) ! +1; Re !13(Re !23) 0;
and only in one of the sectors of the groups G(2) the conditions
hold for ! 1. These sectors denote as G(1)0 and G(2)0 accordingly.
4.2. Asymptotic expressions for the characteristic determinant ( ) in the G sectors. First, we shall consider vary in G(1)0 : Let us substitute (3.1) into (1.2). Therefore, we have the characteristic determinant as asymptotic quasi-polynomial form 5( ) := 32e (!11+!21) [Q11] e 11 !14+ + [Q1l] e 1l !14 + [Q21] e 21 !24+ + [Q2l] e 2l !24 where 1 = j1< j2< < jl= 1; j = 1; 2: Let us denote 51( ) := 32e (!11+!21) [Q11] e 11 !14+ + [Q1l] e 1l !14 ; (4.1) 52( ) := 32e (!11+!21) [Q21] e 21 !24+ + [Q2l] e 2l !24 (4.2) and 5( ) = 51( ) + 51( ) :
Let divide the sector G(1)0 into two sectors as G(1)01 and G(1)02: We assume that one of the expressions 51( ) and 52( ) vanish in one of the sectors G(1)01 and G
(1) 02:
Hence, let the characteristic determinant 5( ) has the asymptotic representation
in the form (4.1) in G(1)01 and in the form ((4.2) in G(1)02 where
Q11= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; Q1l= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83
10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; Q21= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 ; Q2l= 10+ !11 11+ !211 12+ !311 13 20+ !11 21+ !211 22+ !311 23 .. . 80+ !11 81+ !211 82+ !311 83 10+ !24 11+ !224 12+ !324 13 20+ !24 21+ !224 22+ !324 23 .. . ... 80+ !24 81+ !224 82+ !324 83 :
By the same procedure in the sector G(2)0 ; we have the characteristic determinant as asymptotic representation 6( ) = 32e (!13+!23) [R11] et11 !12+ + [R1m] et1m !12 + [R21] et21 !22+ + [R2m] et2m !22 where 1 = tj1< tj2< < tjm= 1; j = 1; 2:
Considering the above idea, we can write the following equalities in sectors G(2)01 and G(2)02
61( ) := 32e (!13+!23)
[R11] et11 !12+ + [R1m] et1m !12 ; (4.3) 62( ) := 32e (!13+!23)
respectively, and
6( ) = 61( ) + 6( ) :
The numbers R j can be seen by the same procedure in sectors G(2)01 and G (2) 02:
4.3. Asymptotic distribution of eigenvalues for arg p1= arg p2. Now we can
prove the next theorem for the problem (1.1)-(1.2).
Theorem 2. We assume that the following conditions be satis…ed 1) arg p1= arg p2:
2) q(x) 2 Lp( 1; 1); p > 1:
3) Qj1; Qjl; Rj1; Rjm; 6= 0; j = 1; 2:
4) The linear functionals F1k + F2k in the spaces Wpk( 1; 0) + Wpk(0; 1) are
continuous.
Then, the boundary value problem (1.1)-(1.2) has an precisely number of eigen-values whose asymptotic distribution may be expressed by the following formulas
1 n = p 1 4 1 n(1 + O( 1 n)); 2 n = p 1 4 2 n(1 + O( 1 n)); 3 n= p 1 4 1 ni(1 + O( 1 n)); 4 n= p 1 4 2 ni(1 + O( 1 n)): in each sector S0 j:
Proof. According to condition (3) of the Theorem, the principal terms of the …rst and last coe¢ cients of the asymptotic quasi-polynomials (4.1), (4.2), (4.3) and (4.2) are di¤erent from zero. These quasi-polynomials in sectors G(1)01, G(1)02; G(2)01 and G(2)02 have an in…nite number of roots 1n , 2n , 3n and 4n , respectively,
and they are contained in strips
E1j = 2 Cj jRe !j4j < h1j 2 ; j = 1; 2; E2j = 2 Cj jRe !j2j < h2j 2 ; j = 1; 2;
respectively, where hij > 0: Again, in view of the [15, p. 100, Lemma 1] eigenvalues
of the problem have the asymptotic representation
j n!j4 = 2 n jl j1 (1 + O(1 n)) = n(1 + O(1 n)) ; j = 1; 2;
j+2 n !j2 = 2 n tjm tj1 (1 + O(1 n)) = n(1 + O(1 n)) ; j = 1; 2: Therefore, we have the sought asymptotic formulas
j n = (!j4) 1 ni(1 + O( 1 n)) = p 1 4 j n(1 + O( 1 n)); j = 1; 2; n = 1; 2; :::; j+2 n = (!j2) 1 ni(1 + O( 1 n)) = p 1 4 j ni(1 + O( 1 n)); j = 1; 2; n = 1; 2; :::: for eigenvalues of the problem (1.1)-(1.2).
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Current address : Mustafa KANDEM ·IR: Department of Mathematics, Faculty of Education, Amasya University, Amasya, 05100 Turkey.
