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Многопараметрическое семейство решений интегрального уравнения Вольтерра с особенностью в банаховом пространстве.

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???????? ?????. ??????????
2011, ? 1, c. 59?71
http://www.ksu.ru/journals/izv_vuz/
???. ????? ?????? ?? ??? "?????????????" 0421100123 \0006
?.?. ????????
???????????????????? ????????? ???????
????????????? ????????? ????????? ? ????????????
? ????????? ????????????
?????????. ????????? ???????????? ????????? ????????? I ???? ? ???????????? ? ??????????
??????? ????? ? ???????????? ??????????? ??????? ?? ?????????? ? ????????? ????????????, ??????? ?????????? ????????????????????? ????????? ???????.
???????? ?????: ???????????? ?????????, ???????? ????????????, ??????????? ?????, ??????.
???: 517.968
Abstract. We construct a multiparametric set of solutions to a singular Volterra integral equation
of the ?rst kind with a su?ciently smooth kernel in the space of integrable functions whose values
belong to a Banach space.
Keywords: integral equation, Banach space, operator pencil, spectrum.
????????
???????? ?? ??????? ????? ??????????, ??????????? ???????? ????????? ?????????,
?????? ???????? ???????? ?????????? ? ???????????? ????????? ???????. ????????????
???????? ??? ?????? ???????? ? ????? ????? [1]?[3], ??? ???????? ?????? ?????? ????????
???????????? ????????? I ? II ????, ????????? ??????? ?????? ? ????????? ?????????????
? ?????? ???????????? ????. ? ????????? ???? ????????????? ????????? ? ?????????????
??????????????, ??????????? ???????? ??????????. ??? ???? ????????? ??????????????? ??? ? ?????????????, ??? ? ? ???????????????? ????????? ????????????? [4]?[9].
? ?????? [10] ????????? ???????????? ????????? ????????? I ???? ? ???????????? ? ?????????? ??????? ????? ? ???????????? ??????????? ?? [0, ?] ??????? ?? ?????????? ?
????????? ???????????? E. ?????? ?????? ???????? ???????????? [10].
1. ?????????? ??????. ???????????? ????????? ??????????
? ???????????? ????????? ???????????? E ??????????? ????? и E . ??? ????? ?????????? ? ???????????? L(E) ???? ???????? ???????????? ?????????? ?? E ???????????
?????
AL(E) = sup AxE .
xE =1
????????? 02.06.2009
59
60
?.?. ????????
? ???????????? C([0, ?], E) ??????????? ? ????? и E ?? [0, ?] ??????? ?? ??????????
? E ?????, ??? ??????, ???????????? ?? ???????
?C([0,?],E) = max ?(x)E .
0?x??
???????, ? ???????????? C, ????????? ?? ???? ??????????? ? ????? иL(E) ?? ???????????? 0 ? t ? x ? ? ??????? ?? ?????????? ? L(E), ???????? ?????
|Q| = max Q(x, t)L(E) .
C
0?t?x??
??????????????? ???????????? ????????? ????????? I ???? ????
x
K(x, t)u(t)dt = 0 (0 ? x ? ?)
(1)
0
? L1 ([0, ?], E), ??? K(x, t) ? ???????? ??????? ?? ?????????? ? L(E), ??????? ???????????
??????? ??????????? ?? ??????? N + m + 1 (N , m ? ??????????? ?????) ????????????,
?????? ??? ??????? ??????????? ?? ??????? m ? 1 ????? ???? ? ????? (0, 0), ? ???????
??????????? m-?? ??????? ?? ??? ????? ???? ? ????? (0, 0), u(x) ? ??????? ???????????
??????? ?? [0, ?] ?? ?????????? ? E.
?????????? K(x, t) ?? ??????? ???????
K(x, t) =
?+?=N
+m
K ?? x? t? +
?+?=m
?? (x, t)x? t? ,
K
(2)
?+?=N +m+1
???
?K ?+? (0, 0) 1 1
(? + ? = m, . . . , N + m).
?x? ?t?
?! ?!
?????? ??????????? ?????
1
??
K ?
s?+??1 ds + Am ,
Q? =
K ?? =
??? Am =
?+?=m
?>0
(3)
0
K ?? .
?+?=m
???????. ????? ????????? ????????? ???????:
1) ????? (3) ????? ?????????????????? ????? ? = ? + iх (? > 0), ???????? ????????????? ??????????? ?????? e0 = e01 + ie02 ? ??????? ?????????????? ???????? ek =
ek1 + iek2 (k = 1, 2, . . . , q1 );
2) ?????????? ??????????? ????? k ?????, ??? ? = ? + k = (? + k) + iх ????? ???????? ?????????????????? ?????? ????? (3), ???????? ????????????? ????????????
??????????? ?????? f0 = f10 + if20 ? ??????? ?????????????? ???????? fk = f1k + if2k
(k = 1, 2, . . . , q2 );
3) ??????????? ????? N ?????, ???
1
K(x, x) ?1 1?m | C и max s?+N ds < 1;
(4)
2 |Kx (x, t)x
0?x??
xm
L(E) 0
4) ???????? Am = lim
x??0
K(x,x)
xm
????? ???????????? ????????.
????? ??? ????????? (1) ?????????? (q1 + 1)-??????????????? ????????? ??????? ????
????????? ?????????, ???????????????????? ????????? ???????
??1
ul (x) = x
l N
r=0
i
arl
i x
+
N +1
arl
N +1 (x)x
sin(х ln x) +
N
i=0
61
i
brl
i x
+
N +1
brl
N +1 (x)x
О
i=0
q2 N
r
?+k?1 l+1
rl i
rl
N +1
sin(х ln x)+
ln (x)
ai x + aN +1 (x)x
О cos(х ln x) ln x + x
r=0
+
N
i=0
rl i
bi x +
N +1
brl
N +1 (x)x
r
cos(х ln x) ln x , (5)
i=0
rl
rl
rl (l = 0, 1, . . . , q ; i =
??? arl
1
i , bi (l = 0, 1, . . . , q1 ; i = 0, 1, . . . , N ; r = 0, 1, . . . , l), ai , bi
rl
(x),
b
0, 1, . . . , N ; r = 0, 1, . . . , q2 ) ? ??????? ???????????? ? E, ??????? arl
N +1
N +1 (x),
rl
rl
aN +1 (x), bN +1 (x) ???????? ???????????? ?? [0, ?] ?? ?????????? ? ????????? ???????????? E.
???????, ??? ? ?????? [10] ???????? ??????, ????? ????? (? + k) + iх (k = 1, 2, . . . , N ) ??
???????? ??????????????????? ??? ????? (3).
2. ?????????????? ??????? (?????????? ???????)
????????? (1) ?????????????????? ???????? ? ????????????? ????????? ?????????
II ????
x
Kx (x, t)u(t)dt,
K(x, x)u(x) = ?
0
???????, ????????? (2), ??????? ? ????
N
+m
k
N +m+1
Ak x + AN +m+1 (x)x
u(x) = ?
+m
x N
0
k=m
?+?=m
?>0
+
K ?? ?x??1 t? +
R
??
(x, t)x t u(t)dt, (6)
? ?
?+?=m+N
???
Ak =
?+?=k
K ?? , AN +m+1 (x) =
R
??
? ?
(x, t)x t =
?+?=m+N
?? (x, x),
K
?+?=N +m+1
?? (x, t)x? t?
K
.
x
?+?=N +m+1
?????????? (5) ? (6) ? ??????????? ???????????? ??? x?+k+m?1 sin(х ln x) lnl+1+? x ?
x?+k+m?1 cos(х ln x) lnl+1+? x (? = q2 , q2 ? 1, . . . , 0), ????????
1
q2
rl
?l
l+1+? r??
??
?+?+k?1
K ?
s
(arl
s ds,
Am a0 = ?
0 cos(х ln s) ? b0 sin(х ln s))Cl+r+1 ln
0
?+?=m
?>0
?l
Am b0
=?
?+?=m
?>0
r=?
1
K ?? ?
s?+?+k?1
0
q2
r=?
(7)
(arl
0 sin(х ln s) +
rl
b0
l+1+? r??
cos(х ln s))Cl+r+1
ln
s ds.
62
?.?. ????????
?? ??????????? (7) ??? ? = q2 [10] ? ???????????????? ???????? ???????????? E ???????
?????????
q2 l
Q(?+k)+iх (aq02 l + ib0 ) = 0,
??????? ????????, ??? ?????? aq02 l + ibq02 l ???????? ??????????? ???????? ??? ????????????
????? Q? , ?????????? ??????????????????? ????? (? + k) + iх. ???????? ????????, ???
???? a0 + ib0 ???????? ??????????? ???????? ???????????? ????? (3), ?????????? ??????????????????? ????? (? + k) + iх, ?? aq02 l = a0 , bq02 l = b0 ????? ???????? ??????? ?????????
(7) ??? ? = q2 . ??? ????????, ??? aq02 l = f10 , bq02 l = f20 ???????? ???????? ???? ???????.
???????? ???????? [10], ??? ??????? ????????? (7) ??? ? = q2 ?1 ? ? = q2 ?z (z = 2, . . . , q2 )
???????? ???????????? E:
???????????? ?????????? ? ???????????????? E
Q(?+k)+iх (a0 q2 ?1,l + ibq02 ?1,l ) + (q2 + l + 1)Q(?+k)+iх f0 = 0,
Q(?+k)+iх (aq02 ?z,l + ibq02 ?z,l ) + (q2 ? z + l + 2)Q(?+k)+iх (aq02 ?z+1,l + ibq02 ?z+1,l )+
1 q2 ?z+2,l
Q
(aq2 ?z+2,l + ib0
) + иии+
2! (?+k)+iх 0
1 (z)
+ (q2 ? z + l + 2) О и и и О (q2 + l + 1) Q(?+k)+iх f0 = 0.
z!
+ (q2 ? z + l + 2)(q2 ? z + l + 3)
? b?l
???
???????? ????????? ??????????? ??? ??????????? ????????????? a?l
0
0
? = q2 ? 1, . . . , 0:
Q? (aq02 ?1,l + ibq02 ?1,l ) + Q ? (q2 + l + 1)f0 = 0,
Q? (aq02 ?2,l + ibq02 ?2,l ) + Q ? (q2 + l)(aq02 ?1,l + ibq02 ?1,l ) +
Q?
(q2 + l)(q2 + l + 1)f0 = 0,
2!
.........................................................
Q? (a0l
0
+
ib0l
0)
+
+
Q
?
3!
Q ? (l
+
2)(a1l
0
+
ib1l
0 )+
Q?
2!
2l
(l + 2)(l + 3)(a2l
0 + ib0 )+
(q ?1)
(l + 2)(l + 3)(l +
4)(a3l
0
+
3l
ib0 )
+ иии +
Q? 2
(q2 ? 1)!
(l + 2)(l + 3) О и и и
(q )
О (q +
l)(aq02 ?1,l
+
ibq02 ?1,l ) +
Q? 2
q2 !
(l + 2)(l + 3) О и и и О (q2 + l + 1)f0 = 0. (8)
????? 1. ????? Q? ? ??????????? ?????, ? ? ??? ?????????????????? ?????. ?????
??????? ?????????
?
?
Q? e0 = 0,
?
?
?
?
?
?Q? e1 + Q? e0 = 0,
(9)
........................
?
?
(q2 )
?
?
Q
?
?
?Q? eq2 + Q? eq2 ?1 + и и и + ? e0 = 0
q2 !
????? ??????? e0 = c0 f0 , ek = c0 fk (k = 1, 2, . . . , q2 ), c0 ? ???????????? ??????????.
????????? ?????????, ???????????????????? ????????? ???????
63
????? 2. ???? e0 , e1 , . . . , eq2 ? ??????? ??????? (9), ?? e0 = e0 , e1 = ?1 e1 , e2 = ?1 ?2 e2 , . . . ,
e
q2 = ?1 и ?2 О и и и О ?q2 eq2 (?i ? ???????????? ??????????) ????? ???????? ???????
?
?
Q? e0 = 0,
?
?
?
?
?
e0 = 0,
?Q? e1 + Q ? ?1 (10)
........................
?
?
(q
)
?
?
Q 2
?
?
?Q? eq2 + Q ? ?q2 eq2 ?1 + и и и + ? ?1 и ?2 О . . . О ?q2 e0 = 0.
q2 !
?????????????? ????? 2 ????? ???????? ???????????????? ???????????? ej (j = 0, . . . , q2 )
? ??????? (10).
?? ????????? ???? 1 ? 2 ????? ???????? ??????? ??????? (8)
jl
ajl
0 + ib0 =
(q2 + l + 1)!
c0 fq2 ?j (j = 0, 1, 2, . . . , q2 ).
(j + l + 1)!
(q2 +l+1)!
q2 ?j
(q2 +l+1)!
q2 ?j
, bjl
(j = 0, 1, . . . , q2 ) ????? ????????
?????????????, ajl
0 = (j+l+1)! c0 f1
0 = (j+l+1)! c0 f2
??????? ????????? (7).
???????? ????????, ??? ???????? (8) ????? ?????
ajl
0
+ ibjl
0
q2 ?j
(q2 + l + 1)! =
ci fq2 ?j?i ,
(j + l + 1)!
(11)
i=0
??? ci ? ???????????? ??????????.
??????????? ???????????? ??? x?+k+m?1+p sin(х ln x) lnl+1+? x ? x?+k+m?1+p cos(х ln x)О
lnl+1+? x (p = 1, 2, . . . , N ) ? ????? ? ?????? ?????? ?????????, ??????????? ????? ??????????? (5) ? (6), ???????? ???????, ????????????? ?????????
?l
Q?+p (a?l
p +ibp )
=?
Ak a?l
i ?
k+i=m+p
k>m
?
?b?l
i
brl
i
l+?+1 r??
sin(х ln s))Cl+r+1
ln
sin(х ln s))ds?i
+
0
l+?+1 r??
cos(х ln s))Cl+r+1
ln
1
0
K
??
?s
?+k+?+i?1
q2
1
0
K
K
??
?+?+i=m+p
?>0
1
s ds +
0
??
?s
?+k+?+i?1
?+?+i=m+p
?+?>m
?>0
(arl
i cos(х ln s)?
r=?+1
?+?+i=m+p
?>0
s ds ?
Ak b?l
i +
k+i=m+p
k>m
brl
i
1
?s
?+k+?+i?1
(a?l
i cos(х ln s)?
q2
(arl
i sin(х ln s)+
r=?+1
K
??
?s
?+k+?+i?1
?+?+i=m+p
?+?>m
?>0
(a?l
i sin(х ln s)+
+
b?l
i
cos(х ln s))ds . (12)
????????????
??????? A: ????? ? + p (p = 1, 2, . . . , N ) ?? ???????? ??????????????????? ??? ????? Q? .
64
?.?. ????????
?l
????? ????????? (12) ?????????? ????????? ?, ?????????????, ????????? a?l
p ? bp ???
? = q2 , q2 ? 1, . . . , 0; p = 1, 2, . . . , N .
??????????? ???????????? ??? x?+m?1 lnl?d x sin(х ln x), x?+m?1 lnl?d x cos(х ln x) (d =
0, 1, . . . , l), ???????? ???????
Am al?d,l
0
=?
K
??
=?
s?+??1
0
?+?=m
?>0
Am bl?d,l
0
1
?
K
??
rl
l?d r?l+d
[arl
ln
s ds,
0 cos(х ln s) ? b0 sin(х ln s)]Cr
r=l?d
1
s?+??1
?
0
?+?=m
?>0
l
l
rl
l?d r?l+d
[arl
ln
s ds,
0 sin(х ln s) + b0 cos(х ln s)]Cr
r=l?d
(13)
?? ??????? [10] ??????? ????????? ? E
Q?+iх (all0 + ibll0 ) = 0,
+ ibl?1,l
) + lQ?+iх (all0 + ibll0 ) = 0,
Q?+iх (al?1,l
0
0
..........................................
+ ibl?d,l
) + (l ? d + 1)Q?+iх (al?d+1,l
+ ibl?d+1,l
) + (l ? d + 1)(l ? d + 2)
Q?+iх (al?d,l
0
0
0
0
+ ibl?d+2,l
) + . . . + (l ? d + 1)(l ? d + 2) О и и и О l
О Q?+iх (al?d+2,l
0
0
1
О
2!
1 (d)
Q
(all + ibll0 ) = 0.
d! ?+iх 0
(14)
?? ???????? ???????? ????????, ??? ???????? ??????? (14) ?????
+ ibl?1,l
= le1 , al?2,l
+ ibl?2,l
= l(l ? 1)e2 , . . . , al?d+1,l
+
all0 + ibll0 = e0 , al?1,l
0
0
0
0
0
= l(l ? 1) О и и и О (l ? d + 2)ed?1 , al?d,l
+ ibl?d,l
= l(l ? 1) О и и и О (l ? d + 1)ed .
+ ibl?d+1,l
0
0
0
=
?????????????, al?d,l
0
l!
d
(l?d)! e1 ,
bl?d,l
=
0
l!
d
(l?d)! e2 ????? ???????? ??????? (13).
x?+m+p?1 lnl?d x sin(х ln x), x?+m+p?1 lnl?d x cos(х ln x)
??????????? ???????????? ???
(d = 0, 1, . . . , l; p = 1, 2, . . . , k ? 1), ???????? ???????, ????????????? ????????? ? E
Q(?+p)+iх (al?d,l
p
+
ibl?d,l
)
p
=?
k+i=m+p
k>m
О
l
?
0
cos(х ln s)
l
(arl
p
r=l?d+1
? brl
i
sin(х ln s))Crl?d lnr?l+d s ds
1
?
0
cos(х ln s) ?
brl
p
K
??
? s?+?+i?1 О
?+?+i=m+p
?+?>m
?>0
(arl
i
r=l?d
О
1
Ak al?d,l
i
sin(х ln s))Crl?d lnr?l+ds ds
K
??
? s?+?+p?1 О
?+?=m
?>0
?i
k+i=m+p
k>m
Ak bl?d,l
+
i
????????? ?????????, ???????????????????? ????????? ???????
1
+
0
+
brl
i
??
K
?+?+i=m+p
?+?>0
?>0
cos(х ln s))Crl?d lnr?l+d s ds
65
l
? s?+?+i?1
(arl
i sin(х ln s)+
r=l?d
1
+
0
K
??
? s?+?+p?1
?+?=m
?>0
l
(arl
p sin(х ln s)+
r=l?d+1
+
brl
p
cos(х ln s))Crl?d lnr?l+d s ds
. (15)
????????????
??????? B: ????? ?+p = (? +p)+iх (p = 1, 2, . . . , k ?1) ?? ???????? ???????????????????
??? ????? Q? .
? bl?d,l
(d =
????? ????????? (15) ?????????? ????????? ?, ?????????????, ????????? al?d,l
p
p
0, 1, . . . , l; p = 1, 2, . . . , k ? 1).
??????????? ???????????? ??? x?+m+k?1 lnl?d x sin(х ln x), x?+m+k?1 lnl?d x cos(х ln x)
(d = 0, 1, . . . , l), ???????? ??????? ?????????
Ak al?d,l
i
=?
0
k+i=m+k
Оln
r?l+d
1
1
s ds?
0
Ak bl?d,l
i
k+i=m+k
K
=?
0
??
l
rl
l?d
? s?+?+i?1
[arl
i cos(х ln s)?bi sin(х ln s)]Cr О
r=l?d
?+?+i=m+k
?>0
??
q2
l?d
?+?+k?1
rl
r+1+d
? s
(arl
s ds,
0 cos(х ln s)?b0 sin(х ln s))Cl+r+1 ln
r=0
1
K
??
l
rl
? s?+?+i?1
[arl
i sin(х ln s) + bi cos(х ln s)]О
r=l?d
?+?+i=m+k
?>0
1
О
K
?+?=m
?>0
Crl?d lnr?l+d s ds
?
0
K
??
q2
?+?+k?1
? s
[arl
0 sin(х ln s)+
r=0
?+?=m
?>0
l?d
r+1+d
s ds. (16)
+ brl
0 cos(х ln s)]Cl+r+1 ln
????????? ???????? (11), ???????? ????????, ??? ??? d = 0 ??????? ????????? (16)
???????????? ?????????
Q? allk
+
ibllk
q
q2
?i+1
q2
2 +1
(q2 + l + 1)!
(k) 1
(k) 1
c0
fq ?k+1 +
fq ?k?i+1 ?
Q?
ci
Q?
=?
l!
k! 2
k! 2
k=1
i=1
k=1
?
k
p=1
p
Q? allk?p + ibllk?p . (17)
66
?.?. ????????
????-???????????? ???????????? ????????? Q? ????? ????????? ??????????????? ? E.
?
?????? ????????????? ??????????? ? ??????? ???????? ????????? ????????? Q? . ????????? ???????? (z, f ) ???????? ??????????? f ?? ???????? z, ????????????? ???????????????? ???????? ???????????? E. ???????????, ??? ???????? Q? ????? ??????, ??????
????. ????? ??? ???????????? ????????? (17) ????? ??????????? ?????????? ???????
q
k
2 +1
(q2 + l + 1)
?
?
(k) 1
c0
fq2?k+1 , f
Q?
Qp? allk?p + ibllk?p , f ,
(18)
=
?
l!
k!
p=1
k=1
?
Q?? f
= 0.
???
???????? ????????, ??? ?? ??????????? ? ????????????? ?????????????? ???????? ?
???????????? ??????? ???????????? ????? Q? ???????
q2
?i+1
q2
?
(k) 1
fq2?k?i+1 , f
ci
Q?
= 0.
k!
i=1
?????????
k=1
q
2 +1
(k)
Q?
k=1
1
?
fq ?k+1 , f
k! 2
= 0,
??? ??? ? ????????? ?????? ??????????? ?????? f0 ???? ?? q2 +1 ?????????????? ????????,
??? ???????????? ??????? ??????? ???????. ?????????????, ?? ????????? (18) ??????????
???????????? c0 . ????? ????????? (17) ????????? ? ??? ??????? ????? ???????? ? ????
allk
+
ibllk
=
Allk
q2 +1
(q2 + l + 1) +
ci fq2?i+1 ,
l!
i=1
???
Allk
???????? ???????? ?????????
Q? Allk = ?
q2 +1
k
(q2 + l + 1)! (k) 1
p
c0
fq2?k+1 ?
Q?
Q? allk?p + ibllk?p .
l!
k!
p=1
k=1
?????????????,
q2 +1
q2 +1
(q2 + l + 1)! (q2 + l + 1)! ll
ll
ll
ll
ci fq2 ?i+1 , bk = Im Ak +
ci fq2 ?i+1 .
ak = Re Ak +
l!
l!
i=1
i=1
???????? ? ??????? ??????? ????????? (16) ??? d = 1. ??? ??????? ???????????? ????
????? ? E
Q? al?1,l
k
+
ibl?1,l
k
=
?Q? l
q2
ll
(r+2) l?1 rl
ll
ak + ibk ?
Q?
cl+r+1 a0 + ibrl
0 ?
r=0
?
k
p
Q? al?1,l
k?p
+ ibl?1,l
k?p
p=1
?l
k
p (Q? ) allk?p + ibllk?p , (19)
p=1
??????? ????? ????????? ?????????????? ????? ???????? ? ????
+ibl?1,l
Q? al?1,l
k
k
=
(q2 + l + 1)!
c0
?Q? lAllk ?
(l ? 1)!
q
2 +2
k=2
(k)
q2 +1 (k)
(q2 + l + 1)! Q?
fq ?i+2 ?
c1
fq ?k+1 ?
k! 2
(l ? 1)!
k! 2
Q?
k=1
????????? ?????????, ???????????????????? ????????? ???????
67
q2 +1 q2
?i+2 (k)
k
Q?
p l?1,l
(q2 + l + 1)! p ll
ll
fq2?k+2?i ?
?
ci
Q? ak?p + ibl?1,l
)
+
ib
+
l(Q
a
.
?
k?p
k?p
k?p
(l ? 1)!
k!
i=2
p=1
k=1
??? ???????????? ????? ????????? ????? ??????????? ?????????? ???????
(q2 + l + 1)!
c1
?
(l ? 1)!
q
2 +1
(k)
Q?
fq2 ?k+1 , f
k!
k=1
?
=
Q? lAllk
q2 +2 (k)
(q2 + l + 1)! Q?
c0
fq ?i+2 +
+
(l ? 1)!
k! 2
k=2
k
p l?1,l
?
p ll
l?1,l ll
Q? ak?p + ibk?p + l(Q? ) ak?p + ibk?p , f ,
+
p=1
?? ???????? ?????????? ???????????? c1 . ?????????????, ????????? (19) ????????? ? ???
??????? ????? ???????? ? ????
al?1,l
k
+
ibl?1,l
k
=
Al?1,l
k
+
q
2 +2
i=2
???
Al?1,l
k
(q2 + l + 1)!
ci fq2 +2?i ,
(l ? 1)!
????????????? ?????????
Q? Al?1,l
k
=
?Q? lAllk
+2?i (k)
q2
1
k
Q?
p l?1,l
(q2 + l + 1)! fq2 ?k+2?i ?
?
ci
Q? ak?p + ibl?1,l
+
k?p
(l ? 1)!
k!
p=1
i=0
k=2?i
p + l(Q? ) allk?p + ibllk?p .
?????
al?1,l
k
bl?1,l
k
q
2 +2
(q2 + l + 1)!
l?1,l
ci fq2 +2?i ,
= Re Ak
+
(l ? 1)!
i=2
q
2 +2
(q2 + l + 1)!
l?1,l
ci fq2+2?i .
= Im Ak
+
(l ? 1)!
i=2
???????? ????? ?????????????? ????????, ???????? ????????, ??? ??????? ?????????
(16) ??? d = l ? ? (2 ? ? ? l) ???????????? ????????? ? E
Q? l??+2,l
l??,l l??+1,l
A
+
ib
[(l
?
?
+
1)A
]
+
(l
?
?
+
1)(l
?
?
+
2)
+ иии+
=
?
Q
Q? al??,l
?
k
k
k
2! k
??1 q2 +?+1?i
(?)
Q(k)
Q
(q2 + l + 1)! ?
fq +?+1?i?k +
ci
+ (l ? ? + 1)(l ? ? + 2) О и и и О l ? Allk ?
?!
(l ? ?)!
k! 2
i=0
+ c?
q
2 +1
k=1
+
(k)
Q?
k!
?
i=1
fq2 ?k+1 +
q
2 +? q2 +?+1?i
i=?+1
p
(Q? )(i) i!
k=1
(k)
Q?
k!
fq2 +?+1?k?i ?
(l ? ? + 1)(l ? ? + 2) О . . . О (l ? ? +
???? ?????????? c? ?? ?????????
k=?+1?i
k p
Q? al??,l
+ ibl??,l
+
k?p
k?p
p=1
i)[al??+i,l
k?p
+
ibl??+i,l
]
k?p
. (20)
68
?.?. ????????
(q2 + l + 1)!
c?
(l ? ?)!
О (l ? ? + 2)
О
??1
ci
i=0
q
2 +1
(k)
Q?
Q?
2!
fq2 ?k+1 , f
k!
k=1
?
+ (l ? ? + 1)О
=?
Q? (l ? ? + 1)Al??+1,l
k
(?)
Al??+2,l
k
q2 ?i+?+1
+ и и и + (l ? ? + 1)(l ? ? + 2) О и и и О l
?!
Allk
+
(q2 + l + 1)!
О
(l ? ?)!
p
k ?
(Q? )(i) p l??,l
l??,l fq +?+1?i?k +
(l ? ? + 1)О
Q? ak?p + ibk?p +
k! 2
i!
p=1
i=1
?
l??+i,l
l??+i,l + ibk?p ]
,f
= 0,
О (l ? ? + 2) О . . . О (l ? ? + i)[ak?p
(k)
Q?
k=?+1?i
Q?
?? ????????? (20) ?????????:
al??,l
k
+ ibl??,l
k
=
Al??,l
k
q2 +?+1
(q2 + l + 1)! +
ci fq2 +?+1?i ,
(l ? ?)!
i=?+1
???????? ???????? ?????????
??? Al??,l
k
Q? l??+2,l
l??+1,l A
=
?
Q
+ и и и + (l ? ? + 1)О
(l
?
?
+
1)A
+
(l
?
?
+
1)(l
?
?
+
2)
Q? Al??,l
k
?
k
2! k
??1 q2 +?+1?i
(?)
k Q(k)
Q?
(q2 + l + 1)! p
ll
?
Ak ?
fq2 +?+1?i?k ?
ci
Q? al??,l
+
О(l??+2)Ои и иОl
k?p
?!
(l ? ?)!
k!
p=1
i=0
k=?+1?i
p
?
(Q? )(i) l??,l l??+i,l
l??+i,l (l ? ? + 1)(l ? ? + 2) О . . . О (l ? ? + i)[ak?p
+ ibk?p ] .
+ ibk?p +
i!
i=1
?????????????,
al??,l
k
q2 +?+1
(q2 + l + 1)! l??,l
= Re Ak
+
ci fq2+?+1?i ,
(l ? ?)!
bl??,l
k
q2 +?+1
(q2 + l + 1)! l??,l
= Im Ak
+
ci fq2 +?+1?i .
(l ? ?)!
i=w+1
i=w+1
?
????? ???????, ?? ?????? ?????????????? ???????? ???????????? ???????????? a?l
k
(? = l, l ? 1, . . . , 0).
b?l
k
??????????? ???????????? ??? x?+m+p?1 ln? x sin(х ln x), x?+m+p?1 ln? x cos(х ln x) (? =
0, 1, . . . , l; p = k + j (j = 1, . . . , N ? k)) ? ????? ? ?????? ?????? ?????????, ??????????? ?????
??????????? (5) ? (6), ???????? ??????? ?????????, ????????????? ????????? ? E
1 ?l
?l ?l
?l
??
Ak ai ?
Ak ai ?
K ? s?+k+j+??1 О
Q?+j ak+j + ibk+j = ?
k+i=m+k+j
k>m
О
l
r=?+1
arl
k+j
cos(х ln s)
? brl
k+j
0
k+i=m+j
sin(х ln s) Cr? lnr?? s ds ?
1
0
?+?=m
?>0
?+?+i=m+k+j
?+?>m
?>0
K
??
? s?+i+??1 О
????????? ?????????, ???????????????????? ????????? ???????
О
l
arl
i
cos(х ln s) ?
brl
i
sin(х ln s) Cr? lnr?? s ds ?
0
r=?
О
q2
arl
i
cos(х ln s)?brl
i
+
0
1
0
K
??
Ak b?l
i +
Ak b?l
i +
k+i=m+j
l
? r??
rl
?+i+??1
? s
s ds+
ai sin(х ln s) + brl
i cos(х ln s) Cr ln
r=?
?+?+i=m+k+j
?+?>m
?>0
+
r=?+1
+
? s?+k+i+??1 О
l
rl
? s?+k+j+??1
cos(х ln s) Cr? lnr?? s ds+
ak+j sin(х ln s) + brl
k+j
?+?=m
?>0
1
K
??
K
??
?+?+i=m+j
?>0
k+i=m+k+j
k>m
1
0
?
l+r+1??
sin(х ln s) Cl+r+1 ln s
s ds?i
r=0
1
69
K
??
q2
?
rl
rl
?+k+i+??1
l+r+1??
? s
ai sin(х ln s)+bi cos(х ln s) Cl+r+1 ln
s ds .
r=0
?+?+i=m+j
?>0
(21)
? ???? ??????? A ????????? (21) ?????????? ????????? ?, ?????????????, ????????????
? b?l
(? = l, l ? 1, . . . , 0; j = 1, 2, . . . , N ? k) ????????????.
a?l
k+j
k+j
??? ????, ????? ??????? (5) ???? ???????? ????????? (1), ??????? ??????? arl
N +1 (x),
(r = 0, . . . , q2 ) [10] ?????? ????????????? ??????? ?????????
brl
N +1 (x)
K(x, x) ?l
aN +1 (x) =
xm
K(x, x) ?l
bN +1 (x) =
xm
1
0
[Kx (x, xs)x1?m ]s?+k+N [?a?l
N +1 (xs) cos(х ln s)+
?l
+ b?l
N +1 (xs) sin(х ln s)]ds + f 1 (x),
1
0
[Kx (x, xs)x1?m ]s?+k+N [?a?l
N +1 (xs) sin(х ln s)?
?l
? b?l
N +1 (xs) cos(х ln s)]ds + f 2 (x),
(22)
?l
??? f ?l
1 (x) ? f 2 (x) (? = q2 , . . . , 0) ? ????????? ??????????? ?? [0, ?] ??????? ?? ??????????
? ????????? ???????????? E.
?????????? ?? ????? ? ???? ? ????? ???????????? ?????????????-??????? K(x,x)
xm
???, ????? ??? 0 ? x ? ? ?????????? ????? ?????????? ???????????? ???????? ????????
K(x,x) ?1
, ??????? ?????????? ?? ????? ?? [0, ?]. ?????????????, ??????? ????????? (22)
xm
70
?.?. ????????
??? x ? [0, ?] ????? ???? ???????????? ? ????
K(x, x) ?1 1 ?l
aN +1 (x) =
[Kx (x, xs)x1?m ]s?+k+N ? a?l
N +1 (xs)О
m
x
0
K(x, x) ?1 ?l
?l
f 1 (x),
О cos(х ln s) + bN +1 (xs) sin(х ln s) ds +
xm
K(x, x) ?1 1 ?l
bN +1 (x) =
[Kx (x, xs)x1?m ]s?+k+N ? a?l
N +1 (xs)О
m
x
0
K(x, x) ?1 ?l
?l
f 2 (x).
О sin(х ln s) ? bN +1 (xs) cos(х ln s) ds +
xm
(23)
?????????? ???????? ???????????? E О E = {z : z = (u, v) , u ? E, v ? E; zEОE =
uE + vE }. ????????? ????? C([0, ?], E О E) ???????????? ??????????? ?? [0, ?] ? ?????
???????? ???????????? E О E ??????? ? ?????? z(x)C([0,?], EОE) = max z(x)EОE .
0?x??
?????
?l
?l
?l
z ?l (x) = (z ?l
1 (x), z 2 (x)) = (aN +1 (x), bN +1 (x)) ? C([0, ?], E О E),
K(x, x) ?1 ?l
K(x, x) ?1 ?l
?l
f (x) =
f 1 (x),
f 2 (x)
? C([0, ?], E О E).
xm
xm
????? ??????? ????????? (23) ????? ?????????? ? ??????????? ?????
z ?l (x) = (Bz ?l )(x) + f ?l (x),
(24)
??? ???????? B ????????? ?? ??????????? ?? [0, ?] ? ????? ???????? ???????????? E О E
??????? z ?l (x) ?? ?????????? ? E О E ?? ???????
1
?l
s?+k+N K(x, s)z ?l (xs)ds,
(Bz )(x) =
0
???
?
? K(x, x) ?1 Kx (x, xs)
K(x, x) ?1 Kx (x, xs)
cos(х ln s)
sin(х ln s) ?
??
xm xm?1
xm xm?1
?.
K(x, s) = ?
?
? K(x, x) ?1 K (x, xs)
K(x, x) ?1 Kx (x, xs)
x
sin(х
ln
s)
?
cos(х
ln
s)
?
xm
xm?1
xm
xm?1
???????? ????? ????????? B ? ???????????? C([0, ?], E О E) [10], ? ???? (4) ????????
1
K(x, x) ?1 1?m
|C О max s?+N ds < 1.
BC([0,?],EОE)?C([0,?],EОE) ? 2|Kx (x, t)x
0?x?? xm
L(E) 0
??? ????????, ??? ???????? ?????? ????????? ? (24) ??? ? = q2 , . . . , 0 ?????????? ?????????
??????? ???????????????? ???????????.
?l
??????? ??????? a?l
N +1 (x) ? bN +1 (x) (? = l, l ? 1, . . . , 0) ?????? ????????????? ?????????
(25)
z ?l (x) = (Bz ?l )(x) + f ?l (x),
???
?l
?l
?l
?l
z ?l (x) = (a?l
N +1 (x), bN +1 (x)) ? C([0, ?], E ОE), f (x) = (f1 (x), f2 (x)) ? C([0, ?], E ОE).
????????? ?????????, ???????????????????? ????????? ???????
71
???????? B ????????? ?? ??????????? ?? [0, ?] ? ????? ???????? ???????????? E О E
??????? z ?l (x) ?? ?????????? ? ???? ?? ???????????? ?? ???????
1
s?+N K(x, s)z ?l (xs)ds.
(Bz ?l )(x) =
0
??? ????????? B ????? ??????????? ???????????
BC([0,?],EОE)?C([0,?], EОE) < 1.
?????????????, ?????? ????????? ? (25) ??? ? = l, . . . , 0 ?????????? ????????? ???????
?l
???????????????? ???????????. ????? ???????, ???????????? a?l
N +1 (x) ? bN +1 (x), ???????
??????????? ???????????? C([0, ?], E).
??????????
[1] ????????? ?.?. ? ????????????? ???????????????????? ???????? ??????? ????????????? ????????? ????????? 1-?? ????, ??? ???? 235 (4), 772?774 (1977).
[2] ????????? ?.?. ???????????????????? ????????? ??????? ???????????? ????????? ?????????,
??? ???? 240 (2), 268?271 (1978).
[3] ????????? ?.?. ???????? ???????????? ????????? ????????? I ? III ????, ????. ??????. ?????. ?
?????. ???. 19 (4), 970?988 (1979).
[4] ????? ?.?., ???????? ?.?. ? ??????? ??????? ??????? ????????????? ????????? ????????? ? ????????????, ????. ??? 355 (4), 450?452 (1997).
[5] ????? ?.?., ???????? ?.?. ?? ???????????? ?????????? ????????? ? ?????????????, ??? 50 (4),
140 (1995).
[6] Krein S.G. Singular integral Volterra equations, Abstracts of Internat. Congress of Math. (Zurich, August
3?11, 1994), p. 125.
[7] Krein S.G., Sapronov I.V. One class of solutions of Volterra equations with regular singularity, ???. ?????.
????. 49 (3), 424?432 (1997).
[8] ???????? ?.?. ?? ????? ?????? ??????? ????????? ????????? II ???? ? ?????????? ???????????? ?
????????? ????????????, ???. ?????. ??????????, ? 6, 48?58 (2004).
[9] ???????? ?.?. ???????????????????? ????????? ??????? ????????????? ????????? ????????? ? ???????????? ? ????????? ????????????, ???. ?????. ??????????, ? 2, 81?83 (2005).
[10] ???????? ?.?. ????????? ????????? ? ???????????? ? ????????? ????????????, ???. ?????. ??????????, ? 11, 45?55 (2007).
?.?. ????????
??????, ?????????? ???????? ??????????,
??????????? ??????????????? ??????????????? ???????? (?????),
??. ??????????, ?. 8, ?. ???????, 394613,
e-mail: vglta311@mail.ru
I.V. Sapronov
Associate Professor, Head of the Chair of Mathematics,
Voronezh State Forestry Academy,
8 Timiryazev str., Voronezh, 394613 Russia
e-mail: vglta311@mail.ru
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