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Asymptotic behaviour of the eigenfunctions and eigenvalues of integral operators with δ-type kernels
ASYMPTOTIC BEHAVIOUR OF THE EIGENFUNCTIONS AND EIGENVALUES OF INTEGRAL OPERATORS WITH 6 -TYPE KERNELS* YU. G. MALINOVSKII Chelyabinsk (Received 12 December 1972; revised 5 March 1973)
THE asymptotic behaviour of the eigenvalues and eigenfunctions of a class of integral operators, dependent on a parameter e > 0 and such that, as E+ 0 the operator kernel behaves as a 6 -function, is investigated. In [ I] we discussed the asymptotic behaviour of the solutions of integral equations of the second kind, dependent on a small parameter E > 0 in such a way that, as E * 0 the kernel of the integral operator behaves like a 6 -function. Our present aim is to discover the asymptotic behaviour of the eigenfunctions and eigenvalues of self-conjugate integral operators with kernels of the same type. We consider the integral operator lx-
yl
U(Y)dY
e
with a kernel k (x, y, g) , which is symmetric with respect to x and y and satisfies the following conditions. Condition 1. Constants h and OD
c> 0
am+n
SC
cl
forallxandyinthesquare require. Condition 2.
l
exist such that
k (5, Y, 8
axmayn
y=Gl
1
ehE ‘dE
andforthemand
+m * k(x,s, --CO
*Z/L vjkhisl Mat. put. Fiz., 13,5, 1124-1133, 1973.
32
ItI)@,=
1.
n>O,
thatweshall
Behaviour of integral operators
33
Condition 3. += k(s,z,l~/)gZdpJ=O f --cc Condition 4.
npR
+=f eizEk(~,r,I~l)d~=#=l -.s a
-IGxGf,
for3:=*1
andanyrealz#o.
1. Preliminary Remarks We shall consider the eigenvalue problem
K&B = haus,
(1.1)
where us (2) E .L[ --I, 11. It can easily be seen that the eigenfunctions u8 (x) of the operator K,, which belong to L, [ -1, 13 and correspond to non-zero 3,,, are continuous with respect to x in L-1, 11. * The most natural approach to our problem is to seek h, and us (5) as series in powers of e:
(1.2)
h c=
c
hkEk,
(1.3)
k=O
According to [l], given any f(s)
E C”‘+i t-1,
11
where
n
f(l) (x)
Kfl(xJ)=C j!(n-Z)!
+= P-%(z,x, _m J
iso
1-g)
i?yn-’
and
2, (z, E) = 0 (&m+l)(
U.S.S.R.
Vol.
13. No.
5-C
Isl
mf2
~cllnf3l.
S” a,
(Here and below, cp(5, a) = 0 (a’) for 5 E S, provided numbers c and e. >O exist such that 1cp(x, E) 1 G d for all IZ:E S and 0 < F,< co.) We introduce the notation
where
Let N be a positive integer. Assu~g that the functions uk (5) are sufficiently smooth in [-I, 1] , we can formally substitute the series (I .2) and (1.3) in (I. 1), and on using the expansion (1.4), equate the coefficients of like powers of G up to sN+‘: ho = I,
Al = 0,
L(u0) -Ahzua -0,
(1.5)
L(&) - h&A = FA(u, h) + ?bk+ZUO,
where
k-i k = 1,2, L, . ) N,
F,(r2,h)=C~~i?U”i(Z)i: I=i
&+-l(s,zh). I=!
A recurrence system of differential equations, dependent on A has been obtained for the function uk (z).Notice that we do not know the boundary conditions on the nk (5) and furthermore, we cannot expect that the partial sum of the series (1.3), composed of solutions of the system (IS), will describe the eigenfunctions of the operator % in the nei~bourhood of boundary points of the segment [-1, 1J . To determine the behaviour of the eigenfunctions in the neighbourhood of z = --I we introduce new variables sl = (5 i- 1) / a, ti = (.v f I) / a. The operator K. takes the form =/e K& =
f
k(ESi - 1, at* - 1, I.%- tiI)n(&ti - I)&.
u
Following [l], we expand the function k(&.s%- 1, E& - 1, 5) in a series in powers of a:
k (e-s1 - 1,Et1 - 1, a
(1.6)
Behaviour of integral operators
35
where
Using the expansion (1.6) and taking any h,, of the form (1.2), we construct the function N
vi, (Si) =
lilUik(ki)eh
(1.7)
k=o
in such a way that the c~fficients of like powers of E up to aN are equal on both sides of the equation OD
~uis(si)=
J FC( ‘=I - f, Et, -
1, 1si - tr I) via (t,) &,
0
Recalling that ho = 1, and hi = 0, we obtain for the functions uik ( si) a recurrence system of integral equations on the semi-axis with a kernel dependent on 1s1 -tl 1 : u&J-j
k(-l,
-1,
is,-t~l)l)ik(ti)dti=Qih(ui,$1,3L),
Cl.%
where k=O,l,...,i,qto-0;
k -
I!l
hnvk-n
(si) .
?I=*
To determine the behaviour of the eigenfunctions of the operator in the neighbourhood of the point x = 1, we use the expansion mapping s2 = (1 - ~)/a, tz = (1- y ) le, and construct the function
U2eC%)= 2
U2k(S2)
Eh,
(1.9)
k=O
where the vuc(s2) satisfy a system of integral equations similar to the system (1.8). On repeating the proof of Lemma 4.3 of [l] , we find that the solutions of these systems are sums of polynomials and functions of the boundary layer type. (A function rp(s) is of the boundary layer type if there exist numbers c and y > 0 such that ( ci, (s) 1 G ce+* for s > 0. We shall denote functions of the boundary layer type by p (s) .) In fact,
36
Yu. G. Mdinovskii
UiO(Si)
=AiO(si
ai + piO(Si)),
+
(1.10) vik(Si)
=Qih(Si)
+
pik(Si)
+Ai~UiO(Si)
“P*(Si)
+Pjh(S+),
where k = 1, 2, . . . , N, the ai are certain numbers, the Ati are arbitrary constants, the Qti (&>are polynomials of degrees si, and Pik(si) = QiA(si) + A, (si + ai). Here and below, i = 1 or 2. Note. The polynom~s eIA(Q) and the functions ~tk (si) are fully defined by the values of hz+zand Ail C%t
in Taylor series in powers of x + 1 with remainder term of order N - k, then substituting 2 -I- 1 = ES, and changing the order of summation with respect to the powers of e. We get N N r&k(Z)Ek= ek(Si)Ek f&(&, E), c c A=0
(1.11)
h=O
Similarly, N
N
c
Uk(5)Ek=
lil
WZA(4
&k+ Rz, (sz,E),
k=O
k=O
We now try to select numbers hk, boundary conditions z&k (d) constants A, in such a way that the system (1.5) is solvable and
Pik(Si)
=
WO (Si)
9
OSk,cN.
2. The Iterative Process Our problem is simpered by the following
and arbitrary
(1.12)
31
Behaviour of integral operators
Lemma I
If the system (1.5) is solvable, conditions (1.12) will follow, for the polynomials Pik (si) and OS (si) described above, from the conditions
P&(O)= Oik(“)
(2.1)
Pip.‘(O) = oikl(O).
(2.2)
I
The proof is virtually a word for word repetition of the proof of Lemma 5.1 of [l] and will therefore be omitted. Let us show that the systems (1.5) and (1.8), and conditions (2.1) and (2.2), generate an iterative process, whereby we can find successively the functions uk (5) and the values of &+a, and &, 0 < k < N. With k = 0, it follows from condition (2.2) that Aio = 0, i.e., u~O (si) s 0 in the expansions (1.7) and (1.9). Substituting this value of the Aio in (2.1) with k = 0, we get uo(-1)
=o,
(2.3)
Uo(1) =o.
Consider the first equation of the system (1.5): L(u0) -h&o
=
0.
(2.4)
We know (see e.g., [2]) that an unbounded monotonic sequence of real numbers PI, CL%* * - exists, such that the problem (2.3), (2.4) has a non-trivial solution if and only if hz = p,, for some n, this solution $n(z) being unique, apart from a constant factor. We will fur n. After numbering the 9,,(z) in such a way that llQ,Jl = 1 (here and below we employ the usual notation for the norm and scalar product in L,[ -1, 4 ] ) , we set u. (z) = $,, (5) , hz = p,,. The recurrence process will then determine uniquely Aikforl
= !T
dQir (0)
d&--I (-1) dx
-
ds,
A
’
=
2r
_
~Qz? (0)
dur-i(l) dx
-
dsz
’
Yu. G. Malinovskii
38
Substituting the values obtained for the Ai, in (2.1) with k = r, we get u7(-I)
=Qir(O) +.A&&,
u,(l)
=&(O)
+A276
(2.5)
We consider the corresponding equation of system (1 S): L
(2.6)
(u,) - hzu, = F, (u, A) + hr+zuo.
By Green’s formula, the problem (2.9, (2.6) is solvable if and only if %i. r+2 = - (F,, uo) - PWO’~ -,‘. Solving (2.6) subject to the boundary conditions (2.5), with the value found for h,+z, we can define the function u,(z),after subjecting it to an auxiliary condition, e.g., (UI, UO)= 0. The iterative process is thus a possibility. In fact, if we choose a positive integer n, and hence AZ= pn and ~0 (5) = I#,,(2) , then the numbers J,A+~,and A, and functions uk (2) , 1 G k < N, such that conditions (1 .12) are satisfied, will be uniquely determined.
3. Estimation of the Error For every positive integer n we consider N
h
nNe
=
1+
hkEk,
Unh?
(z>
=
&NE
(5)
+
PnN
b,
8))
c k=,
where
and the numbers itk and the functions uk (r), and the iterative process described in Section 2.
f&k (Si)
are determined by means of
Consider the error ZMve(2) = 3LnNeUnNc - K&nNs.
Behaviour of integral operators
39
Lemma 2 Given any 6 > 0 we have znNe(5) ==0 (E~+*--~)for II:ES [--I,
1 f,
Proof. Positive constants cl, cz, c3 exist such that pnN(2, E) =O(&N+l)
for
1x1G I-
c,e]ln a(
(3.1)
(these functions are of the boundary-payer type by de~ni~on), (3.2) (by virtue of (3.1) and condition 1 on the kernel k (5, y, g) ) , and (3.3) (by virtue of the choice of the functions uk fs) and the expansion (1.4)). We set e = max (ct, cz, cI) , divide the interval [-1, I] into three parts tz=l-CCE ) lna ) bymeansofthepoints A,= [--I, z,], x,=--1+ce 1 In&], = [x2, I] and estimate the error z,& (t) in each of these subintervals. Ahz= [x,, 4, As With z E Az,we have, by virtue of (3.1)-(3.3) z,N,(~)
=
(hnN&,Ne
-
and the choice of the constant c,
~ei&,N~)+ hnNe&tN- K&V = O(eN+‘).
Let II:E Ai (the case 2 E A, can be considered in the same way) and let S be an arbitrary positive number. In view of (1.7), (1.1 I), and (1.12), we have, even in an interval longer than A$, (e.g.,withJ:~[--1,-1+~elu~e]) UnNe
(z) = vi, (si) f o(&“+‘-“),
Then, by condition 1 on the kernel k(z, y, g) with J: E Ai,
zn~e (5) = (hnnevie- Kevi,) + O(e"+'-a) , But, by definition of the function ut, (s,) and condition 1, with x E A,, &,V,vie - K&e = 0 (gN+‘--O).
(3.4)
Yu G. Malinovskii
40
Hence, recalhng (3.4), znNe(5) = 0 (&N+*-‘)for x E Ai. The lemma is proved. Theorem I Let conditions l-4 be satisfied, let n and N be positive integers, let CL,,and I$,,(z) be the n-th eigenvalue and corresponding eigenfunction of the differential operator L under homogeneous boundary conditions, and let 6 be a positive number. Then the eigenvalue h,, and corresponding eigenfunction uns (2) of the operator K,, exist which can be written as N
a
nc =
1+ pne2+
lchhd + 0 (IF+‘-“),
(3.5)
k=3
u..(+b.(x)+fp[ ukb)+pla(q) +@A k=,
I
(3.6)
where the numbers &, and the functions uk (x) and pti (si) are defined by means of the interative process described in Section 2, while d,(e) is the distance from I,,,. to the nearest different eigenvalue of K,. Proof: Let the hypotheses be satisfied. By Lemma 12 of [3], an eigenvalue h,, of the operator K, exists such that
(3.7)
Obviously, JIrz,J from (3.7).
= 1+ 0 (6). H ence, by Lemma 2, the expansion (3.5) follows
By Lemma 13 of [3], the eigenfunction u,,~(2) of the operator K., corresponding to the eigenvalue 5 na, and a constant C > 0 exist such that
nN8
-
ne
A &l(E)
’
Behaviour of integraloperators
41
where d,(e) is the distance from h,, to the nearest different eigenvalue of K,.
weput x E [-I,
rnNe
(5) = unNs(z) - u,,~(z). By virtue of (3.7) and Lemma 2, with
I] we have hnsu,,m - Ksunm=
Hence, using (3.8), ]]rnNa]]=
0 (eN+-).
(3.9)
O(eN+"'ldn(e)).
To obtain an estimate for 1r’nNe(5) 1with J: E [ -1, hnsrnNa- &‘,,Ns = 0 (EN+'-'). Hence
+hn,11r’nNa 1< IlKelIbnNLIi
+
0
I] notice that, from Eq. (3.9)
Obviously, IlKJl = 0 (IT'/') and h, = I+ 0 (8). Hence, from (3.10), O(~~+“‘-~ld,,(a)) when ZE [-1, I].
4. Operators with Kernel I+
(3.10)
bN+‘-?.
-
r’nNa
(z)
=
‘y I/e)
Consider the integral operator
K..u=$jk.(12ey')n(y)dy. -1 Condition Ia An h > 0 exists such that k, ( 1g I ) = 0 ( e-hur).
We shall prove in this section Theorem 2
Let the function k. (5) satisfy conditions la and 2-4, let x,, arranged in decreasing order, let N and M be positive integers, and let 6 be a positive number. Then, given sufficiently small e and any integer n < M the eigenvalue X,, of K,,, is simple and has the asymptotic form (4.1)
while the corresponding eigenfunction Ti,. (s)can be written as E,,(s)
= sinn+(z
+ 1)
(4.2)
Yu. G. Malinovskii
42
N-Z
Ek uk(z) + l!.l[
+
piA
(+)
fP2k
(+)I
+O(&N--“‘z-b),
k=i
where the numbers AR, the functions ~~(2) E C” [ -1, by the iterative process described in Section 2, and Po=j
11 and the pti(s) are obtained
kot8E2@. 0
Pr005 Let the hypotheses of the theorem be satisfied. Then, by Theorem 1, given any integer n an eigenvalue A,, and corresponding eigenfunction u,~ (s)of the operator Ko,, exist which can be written as
La,,,
(2) = sin ff (z + 1)
.Z[
+
d+
Uk(t)
+
Plk
($L)
(4.3)
+
@k
(-+)I
+
O(~“+“-vd,(E)),
k=,
where d,(e) is the distance from h,, to the nearest different eigenvalue of Ko,. In view of conditions 3 and 4, p. > 0, so that, for sufficiently small a , A,, >
h2E
>
. .
. >
a,,.
The next lemmas form part of the proof of Theorem 2. We take the Fourier transform of the kernel k. ( 1g I) with real z:
Lemma 3 Under the conditions of Theorem 2, constants a > 0 and b -C 1 exist such that, first K, (z) increases monotonically in the interval [ 0, a] and second, K,(z) < /3 for any 2 Z a. Proo$ We have
Behaviour of integral operators
Obviously, for z > 0 small, the sign of&‘(z) that K,,(z) is monotonic.
43
is opposite to the sign of pO,which proves
The second assertion follows from the obvious statements: K, (z) + 0 as z + 00; K. (z) is continuous for z E (- 00, i- -) ; by virtue of condition 4, K. (z) < 1 for real z Z 0. Lemma 3 is proved. Let Y be an arbitrary positive number. We set a = 1 i- y. In the square we consider the function
(the series is obviously convergent by virtue of condition la). The function g (5, y, e) is symmetric with respect to z and y, while in the square - 1 < 5, y < 1
(’x-“‘I+
g(x,Y,&)=ko
(4.4)
0 (e+‘e) .
c
We define in L, [ ---a, a] the self-conjugate integral operator a
cu=$Jg(x,y,e)u(y)dy. --(I
Lemma 4
All the terms of the sequence v,,~ = K, (nnel2u), eigenvalues of the operator G,
and only these terms, are
ProoJ Given any positive integer n,
G,sin~(.cfa)=-2a
1
+-
J ( k,
& A>
lx-yl &
1
sinnz(y
+ a)dy
Each of the functions sin [ nn (x -I- a) /2u] is thus an eigenfunction of the operator G, with corresponding eigenvalue TInE. Since the system {sin[nn(z + a)/2u]}, n = 1, 2,. . . , is complete and orthogonal, there are no other eigenvalues of G,. The lemma is proved.
To compfete the proof of Theorem 2, let qls, &, G, arranged in decreasing order, Since E-1, 1 I forms part of the interval [ --(t, by Courant’s minimax theory for n = 1, 2, . . .
. . . be positive eigenvalu~ of
u f and Eq. (4.4) holds, we have
(4.5) By Lemma 4, given suf~~~ently small E we haye ij,* = qtnefor 1< n < M f i. It then fohows from (4.5) that, for suf~c~ent~y small E not more than M eigenvalues ;Z,, exist such that
In view of the expansion (1.4), Y)~.+~, P= 1 -&&+W+ $)A?(1 +y))z&z+ufeJ). But, for y < l/M and sufficiently small E all the eigenvalues A,,, 1 4 n < M (see (4.3)) satisfy the inequality
Compar~g (4.6) and (4.7), we see that Xne = A,,, f < n G iw, for su~ciently small a, Hence all the x,,, are simple and a number c > Oexists such that, for any 1, kg.Mm, j+k, WehaVe /$ij8-X,,/ >Ct3’. The expansions (4.1) and (4.2) have thus been proved. Theorem 2 is pmved. Translated by D, E. Brown REFERENCES 1.
~~IN~VSKII, Yu G., Asymptotic expansion of the solutions of integral equations with B-form kernels, Zh. ~j%isl H&t. mat. F&T.,l&6, 1554-1564, 1912.
2,
HARTMAN, P., ~~i~~$~ diffE?en~~~e4u~~o~~, Wiley, 1964.
3.
VISHIK, M. I., and LYUSTERNIK, L. A., Regular degeneration and the boundary layer for linear differential equations with a small parameter, Crsp mat, Nut&, l&.5, 3-122, 1957.
THE asymptotic behaviour of the eigenvalues and eigenfunctions of a class of integral operators, dependent on a parameter e > 0 and such that, as E+ 0 the operator kernel behaves as a 6 -function, is investigated. In [ I] we discussed the asymptotic behaviour of the solutions of integral equations of the second kind, dependent on a small parameter E > 0 in such a way that, as E * 0 the kernel of the integral operator behaves like a 6 -function. Our present aim is to discover the asymptotic behaviour of the eigenfunctions and eigenvalues of self-conjugate integral operators with kernels of the same type. We consider the integral operator lx-
yl
U(Y)dY
e
with a kernel k (x, y, g) , which is symmetric with respect to x and y and satisfies the following conditions. Condition 1. Constants h and OD
c> 0
am+n
SC
cl
forallxandyinthesquare require. Condition 2.
l
exist such that
k (5, Y, 8
axmayn
y=Gl
1
ehE ‘dE
andforthemand
+m * k(x,s, --CO
*Z/L vjkhisl Mat. put. Fiz., 13,5, 1124-1133, 1973.
32
ItI)@,=
1.
n>O,
thatweshall
Behaviour of integral operators
33
Condition 3. += k(s,z,l~/)gZdpJ=O f --cc Condition 4.
npR
+=f eizEk(~,r,I~l)d~=#=l -.s a
-IGxGf,
for3:=*1
andanyrealz#o.
1. Preliminary Remarks We shall consider the eigenvalue problem
K&B = haus,
(1.1)
where us (2) E .L[ --I, 11. It can easily be seen that the eigenfunctions u8 (x) of the operator K,, which belong to L, [ -1, 13 and correspond to non-zero 3,,, are continuous with respect to x in L-1, 11. * The most natural approach to our problem is to seek h, and us (5) as series in powers of e:
(1.2)
h c=
c
hkEk,
(1.3)
k=O
According to [l], given any f(s)
E C”‘+i t-1,
11
where
n
f(l) (x)
Kfl(xJ)=C j!(n-Z)!
+= P-%(z,x, _m J
iso
1-g)
i?yn-’
and
2, (z, E) = 0 (&m+l)(
U.S.S.R.
Vol.
13. No.
5-C
Isl
mf2
~cllnf3l.
S” a,
(Here and below, cp(5, a) = 0 (a’) for 5 E S, provided numbers c and e. >O exist such that 1cp(x, E) 1 G d for all IZ:E S and 0 < F,< co.) We introduce the notation
where
Let N be a positive integer. Assu~g that the functions uk (5) are sufficiently smooth in [-I, 1] , we can formally substitute the series (I .2) and (1.3) in (I. 1), and on using the expansion (1.4), equate the coefficients of like powers of G up to sN+‘: ho = I,
Al = 0,
L(u0) -Ahzua -0,
(1.5)
L(&) - h&A = FA(u, h) + ?bk+ZUO,
where
k-i k = 1,2, L, . ) N,
F,(r2,h)=C~~i?U”i(Z)i: I=i
&+-l(s,zh). I=!
A recurrence system of differential equations, dependent on A has been obtained for the function uk (z).Notice that we do not know the boundary conditions on the nk (5) and furthermore, we cannot expect that the partial sum of the series (1.3), composed of solutions of the system (IS), will describe the eigenfunctions of the operator % in the nei~bourhood of boundary points of the segment [-1, 1J . To determine the behaviour of the eigenfunctions in the neighbourhood of z = --I we introduce new variables sl = (5 i- 1) / a, ti = (.v f I) / a. The operator K. takes the form =/e K& =
f
k(ESi - 1, at* - 1, I.%- tiI)n(&ti - I)&.
u
Following [l], we expand the function k(&.s%- 1, E& - 1, 5) in a series in powers of a:
k (e-s1 - 1,Et1 - 1, a
(1.6)
Behaviour of integral operators
35
where
Using the expansion (1.6) and taking any h,, of the form (1.2), we construct the function N
vi, (Si) =
lilUik(ki)eh
(1.7)
k=o
in such a way that the c~fficients of like powers of E up to aN are equal on both sides of the equation OD
~uis(si)=
J FC( ‘=I - f, Et, -
1, 1si - tr I) via (t,) &,
0
Recalling that ho = 1, and hi = 0, we obtain for the functions uik ( si) a recurrence system of integral equations on the semi-axis with a kernel dependent on 1s1 -tl 1 : u&J-j
k(-l,
-1,
is,-t~l)l)ik(ti)dti=Qih(ui,$1,3L),
Cl.%
where k=O,l,...,i,qto-0;
k -
I!l
hnvk-n
(si) .
?I=*
To determine the behaviour of the eigenfunctions of the operator in the neighbourhood of the point x = 1, we use the expansion mapping s2 = (1 - ~)/a, tz = (1- y ) le, and construct the function
U2eC%)= 2
U2k(S2)
Eh,
(1.9)
k=O
where the vuc(s2) satisfy a system of integral equations similar to the system (1.8). On repeating the proof of Lemma 4.3 of [l] , we find that the solutions of these systems are sums of polynomials and functions of the boundary layer type. (A function rp(s) is of the boundary layer type if there exist numbers c and y > 0 such that ( ci, (s) 1 G ce+* for s > 0. We shall denote functions of the boundary layer type by p (s) .) In fact,
36
Yu. G. Mdinovskii
UiO(Si)
=AiO(si
ai + piO(Si)),
+
(1.10) vik(Si)
=Qih(Si)
+
pik(Si)
+Ai~UiO(Si)
“P*(Si)
+Pjh(S+),
where k = 1, 2, . . . , N, the ai are certain numbers, the Ati are arbitrary constants, the Qti (&>are polynomials of degrees si, and Pik(si) = QiA(si) + A, (si + ai). Here and below, i = 1 or 2. Note. The polynom~s eIA(Q) and the functions ~tk (si) are fully defined by the values of hz+zand Ail C%t
in Taylor series in powers of x + 1 with remainder term of order N - k, then substituting 2 -I- 1 = ES, and changing the order of summation with respect to the powers of e. We get N N r&k(Z)Ek= ek(Si)Ek f&(&, E), c c A=0
(1.11)
h=O
Similarly, N
N
c
Uk(5)Ek=
lil
WZA(4
&k+ Rz, (sz,E),
k=O
k=O
We now try to select numbers hk, boundary conditions z&k (d) constants A, in such a way that the system (1.5) is solvable and
Pik(Si)
=
WO (Si)
9
OSk,cN.
2. The Iterative Process Our problem is simpered by the following
and arbitrary
(1.12)
31
Behaviour of integral operators
Lemma I
If the system (1.5) is solvable, conditions (1.12) will follow, for the polynomials Pik (si) and OS (si) described above, from the conditions
P&(O)= Oik(“)
(2.1)
Pip.‘(O) = oikl(O).
(2.2)
I
The proof is virtually a word for word repetition of the proof of Lemma 5.1 of [l] and will therefore be omitted. Let us show that the systems (1.5) and (1.8), and conditions (2.1) and (2.2), generate an iterative process, whereby we can find successively the functions uk (5) and the values of &+a, and &, 0 < k < N. With k = 0, it follows from condition (2.2) that Aio = 0, i.e., u~O (si) s 0 in the expansions (1.7) and (1.9). Substituting this value of the Aio in (2.1) with k = 0, we get uo(-1)
=o,
(2.3)
Uo(1) =o.
Consider the first equation of the system (1.5): L(u0) -h&o
=
0.
(2.4)
We know (see e.g., [2]) that an unbounded monotonic sequence of real numbers PI, CL%* * - exists, such that the problem (2.3), (2.4) has a non-trivial solution if and only if hz = p,, for some n, this solution $n(z) being unique, apart from a constant factor. We will fur n. After numbering the 9,,(z) in such a way that llQ,Jl = 1 (here and below we employ the usual notation for the norm and scalar product in L,[ -1, 4 ] ) , we set u. (z) = $,, (5) , hz = p,,. The recurrence process will then determine uniquely Aikforl
= !T
dQir (0)
d&--I (-1) dx
-
ds,
A
’
=
2r
_
~Qz? (0)
dur-i(l) dx
-
dsz
’
Yu. G. Malinovskii
38
Substituting the values obtained for the Ai, in (2.1) with k = r, we get u7(-I)
=Qir(O) +.A&&,
u,(l)
=&(O)
+A276
(2.5)
We consider the corresponding equation of system (1 S): L
(2.6)
(u,) - hzu, = F, (u, A) + hr+zuo.
By Green’s formula, the problem (2.9, (2.6) is solvable if and only if %i. r+2 = - (F,, uo) - PWO’~ -,‘. Solving (2.6) subject to the boundary conditions (2.5), with the value found for h,+z, we can define the function u,(z),after subjecting it to an auxiliary condition, e.g., (UI, UO)= 0. The iterative process is thus a possibility. In fact, if we choose a positive integer n, and hence AZ= pn and ~0 (5) = I#,,(2) , then the numbers J,A+~,and A, and functions uk (2) , 1 G k < N, such that conditions (1 .12) are satisfied, will be uniquely determined.
3. Estimation of the Error For every positive integer n we consider N
h
nNe
=
1+
hkEk,
Unh?
(z>
=
&NE
(5)
+
PnN
b,
8))
c k=,
where
and the numbers itk and the functions uk (r), and the iterative process described in Section 2.
f&k (Si)
are determined by means of
Consider the error ZMve(2) = 3LnNeUnNc - K&nNs.
Behaviour of integral operators
39
Lemma 2 Given any 6 > 0 we have znNe(5) ==0 (E~+*--~)for II:ES [--I,
1 f,
Proof. Positive constants cl, cz, c3 exist such that pnN(2, E) =O(&N+l)
for
1x1G I-
c,e]ln a(
(3.1)
(these functions are of the boundary-payer type by de~ni~on), (3.2) (by virtue of (3.1) and condition 1 on the kernel k (5, y, g) ) , and (3.3) (by virtue of the choice of the functions uk fs) and the expansion (1.4)). We set e = max (ct, cz, cI) , divide the interval [-1, I] into three parts tz=l-CCE ) lna ) bymeansofthepoints A,= [--I, z,], x,=--1+ce 1 In&], = [x2, I] and estimate the error z,& (t) in each of these subintervals. Ahz= [x,, 4, As With z E Az,we have, by virtue of (3.1)-(3.3) z,N,(~)
=
(hnN&,Ne
-
and the choice of the constant c,
~ei&,N~)+ hnNe&tN- K&V = O(eN+‘).
Let II:E Ai (the case 2 E A, can be considered in the same way) and let S be an arbitrary positive number. In view of (1.7), (1.1 I), and (1.12), we have, even in an interval longer than A$, (e.g.,withJ:~[--1,-1+~elu~e]) UnNe
(z) = vi, (si) f o(&“+‘-“),
Then, by condition 1 on the kernel k(z, y, g) with J: E Ai,
zn~e (5) = (hnnevie- Kevi,) + O(e"+'-a) , But, by definition of the function ut, (s,) and condition 1, with x E A,, &,V,vie - K&e = 0 (gN+‘--O).
(3.4)
Yu G. Malinovskii
40
Hence, recalhng (3.4), znNe(5) = 0 (&N+*-‘)for x E Ai. The lemma is proved. Theorem I Let conditions l-4 be satisfied, let n and N be positive integers, let CL,,and I$,,(z) be the n-th eigenvalue and corresponding eigenfunction of the differential operator L under homogeneous boundary conditions, and let 6 be a positive number. Then the eigenvalue h,, and corresponding eigenfunction uns (2) of the operator K,, exist which can be written as N
a
nc =
1+ pne2+
lchhd + 0 (IF+‘-“),
(3.5)
k=3
u..(+b.(x)+fp[ ukb)+pla(q) +@A k=,
I
(3.6)
where the numbers &, and the functions uk (x) and pti (si) are defined by means of the interative process described in Section 2, while d,(e) is the distance from I,,,. to the nearest different eigenvalue of K,. Proof: Let the hypotheses be satisfied. By Lemma 12 of [3], an eigenvalue h,, of the operator K, exists such that
(3.7)
Obviously, JIrz,J from (3.7).
= 1+ 0 (6). H ence, by Lemma 2, the expansion (3.5) follows
By Lemma 13 of [3], the eigenfunction u,,~(2) of the operator K., corresponding to the eigenvalue 5 na, and a constant C > 0 exist such that
nN8
-
ne
A &l(E)
’
Behaviour of integraloperators
41
where d,(e) is the distance from h,, to the nearest different eigenvalue of K,.
weput x E [-I,
rnNe
(5) = unNs(z) - u,,~(z). By virtue of (3.7) and Lemma 2, with
I] we have hnsu,,m - Ksunm=
Hence, using (3.8), ]]rnNa]]=
0 (eN+-).
(3.9)
O(eN+"'ldn(e)).
To obtain an estimate for 1r’nNe(5) 1with J: E [ -1, hnsrnNa- &‘,,Ns = 0 (EN+'-'). Hence
+hn,11r’nNa 1< IlKelIbnNLIi
+
0
I] notice that, from Eq. (3.9)
Obviously, IlKJl = 0 (IT'/') and h, = I+ 0 (8). Hence, from (3.10), O(~~+“‘-~ld,,(a)) when ZE [-1, I].
4. Operators with Kernel I+
(3.10)
bN+‘-?.
-
r’nNa
(z)
=
‘y I/e)
Consider the integral operator
K..u=$jk.(12ey')n(y)dy. -1 Condition Ia An h > 0 exists such that k, ( 1g I ) = 0 ( e-hur).
We shall prove in this section Theorem 2
Let the function k. (5) satisfy conditions la and 2-4, let x,, arranged in decreasing order, let N and M be positive integers, and let 6 be a positive number. Then, given sufficiently small e and any integer n < M the eigenvalue X,, of K,,, is simple and has the asymptotic form (4.1)
while the corresponding eigenfunction Ti,. (s)can be written as E,,(s)
= sinn+(z
+ 1)
(4.2)
Yu. G. Malinovskii
42
N-Z
Ek uk(z) + l!.l[
+
piA
(+)
fP2k
(+)I
+O(&N--“‘z-b),
k=i
where the numbers AR, the functions ~~(2) E C” [ -1, by the iterative process described in Section 2, and Po=j
11 and the pti(s) are obtained
kot8E2@. 0
Pr005 Let the hypotheses of the theorem be satisfied. Then, by Theorem 1, given any integer n an eigenvalue A,, and corresponding eigenfunction u,~ (s)of the operator Ko,, exist which can be written as
La,,,
(2) = sin ff (z + 1)
.Z[
+
d+
Uk(t)
+
Plk
($L)
(4.3)
+
@k
(-+)I
+
O(~“+“-vd,(E)),
k=,
where d,(e) is the distance from h,, to the nearest different eigenvalue of Ko,. In view of conditions 3 and 4, p. > 0, so that, for sufficiently small a , A,, >
h2E
>
. .
. >
a,,.
The next lemmas form part of the proof of Theorem 2. We take the Fourier transform of the kernel k. ( 1g I) with real z:
Lemma 3 Under the conditions of Theorem 2, constants a > 0 and b -C 1 exist such that, first K, (z) increases monotonically in the interval [ 0, a] and second, K,(z) < /3 for any 2 Z a. Proo$ We have
Behaviour of integral operators
Obviously, for z > 0 small, the sign of&‘(z) that K,,(z) is monotonic.
43
is opposite to the sign of pO,which proves
The second assertion follows from the obvious statements: K, (z) + 0 as z + 00; K. (z) is continuous for z E (- 00, i- -) ; by virtue of condition 4, K. (z) < 1 for real z Z 0. Lemma 3 is proved. Let Y be an arbitrary positive number. We set a = 1 i- y. In the square we consider the function
(the series is obviously convergent by virtue of condition la). The function g (5, y, e) is symmetric with respect to z and y, while in the square - 1 < 5, y < 1
(’x-“‘I+
g(x,Y,&)=ko
(4.4)
0 (e+‘e) .
c
We define in L, [ ---a, a] the self-conjugate integral operator a
cu=$Jg(x,y,e)u(y)dy. --(I
Lemma 4
All the terms of the sequence v,,~ = K, (nnel2u), eigenvalues of the operator G,
and only these terms, are
ProoJ Given any positive integer n,
G,sin~(.cfa)=-2a
1
+-
J ( k,
& A>
lx-yl &
1
sinnz(y
+ a)dy
Each of the functions sin [ nn (x -I- a) /2u] is thus an eigenfunction of the operator G, with corresponding eigenvalue TInE. Since the system {sin[nn(z + a)/2u]}, n = 1, 2,. . . , is complete and orthogonal, there are no other eigenvalues of G,. The lemma is proved.
To compfete the proof of Theorem 2, let qls, &, G, arranged in decreasing order, Since E-1, 1 I forms part of the interval [ --(t, by Courant’s minimax theory for n = 1, 2, . . .
. . . be positive eigenvalu~ of
u f and Eq. (4.4) holds, we have
(4.5) By Lemma 4, given suf~~~ently small E we haye ij,* = qtnefor 1< n < M f i. It then fohows from (4.5) that, for suf~c~ent~y small E not more than M eigenvalues ;Z,, exist such that
In view of the expansion (1.4), Y)~.+~, P= 1 -&&+W+ $)A?(1 +y))z&z+ufeJ). But, for y < l/M and sufficiently small E all the eigenvalues A,,, 1 4 n < M (see (4.3)) satisfy the inequality
Compar~g (4.6) and (4.7), we see that Xne = A,,, f < n G iw, for su~ciently small a, Hence all the x,,, are simple and a number c > Oexists such that, for any 1, kg.Mm, j+k, WehaVe /$ij8-X,,/ >Ct3’. The expansions (4.1) and (4.2) have thus been proved. Theorem 2 is pmved. Translated by D, E. Brown REFERENCES 1.
~~IN~VSKII, Yu G., Asymptotic expansion of the solutions of integral equations with B-form kernels, Zh. ~j%isl H&t. mat. F&T.,l&6, 1554-1564, 1912.
2,
HARTMAN, P., ~~i~~$~ diffE?en~~~e4u~~o~~, Wiley, 1964.
3.
VISHIK, M. I., and LYUSTERNIK, L. A., Regular degeneration and the boundary layer for linear differential equations with a small parameter, Crsp mat, Nut&, l&.5, 3-122, 1957.