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Bicnux Kuibebxozo yuicepcurnemyCepbz.'q>i9mco-uamemamuuni nayxu
VIII( 517.98
Muxafmo <D. I`opo;lHi17x
Bilcropisx M. POM&H€HKO
llpo o6Meaxeni po3a’n3|cu Jlmbepeu-uianbnoro piBHlllllil 3 celcropianb-mm oneparopmm lcoecbiuiemom.
lIoc.11ioo/ceno numauwz npo icnysaurul iedunicmb 06M€.')IC€H020po3a3131cy du-qbepeuuiwzbnoeo pisrvzuwz
x'(r)= Tx(t)+ f(t), 1 E R,ma anpoxcwuauiro uwzo pose Zvzarcy
pose Gzaxam eidnoeiauux zadau Kowi.Tym T - cexmopianbnuzi onepamop a6aHaxoeoMy npocmopi B, f :R -> B-qbirccoeaua o6meofcena no Rcpynxuix,u;o aaooeofzbww noxa/:buy yuoeyFe/zbdepa.
I<)nouoei cnoea: duipepeuuianbnepienxnwl, cexmopiaabuuu onepamop,anporccwnauin.
IIOCTSIIOBKZ sanaqi :
2004, 2 Bulletin ofthe University ofKievSeries:Phirics& Mathematics
Mykhaylo F. GorodniiVictoria N. RomanenkoOn the bounded solutions of a
differential equation with thesectorial operator coeflicient.
We investigate the question of theexistence and uniqueness of abounded solution of the d#rentialequation
x‘(r)=Tx(r)+ f(t), re R,and approximation of this solution bysolutions of the correspondingCauchy’s problems. Here T is asectorial operator in the Banach spaceB, f .' R ->B is a fxed bounded onR function satiiyfifing localHoldercondition.
Key Words: dWrential equation,sectorial operator, approximation.
Hexaii B- KoMn.ue1ccHu1`»i 6aHaxiB ripocrip 3 HOPMOIO i HYJILOBPIM
SJICMCHTOM 5; T- ce1<TopiaJ11>Hm71 ouepafop, mo nic B B, 3 o6naei'1oBI»13Ha\1eHHsl D( T) [6, c. 26]. 3a¢11iI<cy¢-:Mo Q>yH1<ni1o f :R->B, sixa
33J.IOB0)`II>H}I€ Taxi yMoBn:
on) Hf||w :=sup{||f(t)\| iteR}<+oo;B)11n»1 K0)KHO1"0 to icnyxorr. 'raid aanexcni Bin to CT3J`Ii ,B e (0,l], M > 0,
y>0, mo
`v't,se[t0-y;t0+y]:||f(t)-f(s)|]$M|t-s|‘8. (1)
Oanaqenun. <DyH1<uin x : R->B Hasrrnaernca Bi;1noBiJ1HnM f 06M€7K¢HI#lM
po3n’>13KoM lmqaepeuuiansnoro piBI-IXHHSI ~
x’(r)= Tx(r)+ f(1), r e R , (2)
sncmo xe C (R,B), Hxuw < +oo, an txosinmioro teR x(t)eD(T) i1
B1»1KoHye'rLcx piBHic'rL (2).
0 M.¢.l`opo1;miH,B.M.PouaHeHxo, 200497
Bicuux Kuikzcaxoeo ynisepcumemy 2004,2 Bulle1in ofthe Universigv ofKievC7epz31.'rp¢§wco-Mama/uamuwi nayxu Series:Phisics& Mathematics
Bbryn :
Y nal-1ix71 crarri nocnigmcyernca rnrramm rrpo icnynaurm ra e1mHic1'Lo6Me>x<eHoro po3B’»13Ky 11I»1qJepeHuia.nLHoro piBHsIHH>1 (2) 'ra IIPITHHIPDI upoaupolccumauiro usoro po3a’>131<y po3n’a31<aM1»1 Binnoninnnx 38,1184 Komi.
Binauaqnmo, mo y Bnnamcy o6Me>KeHoro onepaTopa T T8 Henepepsnof i
06MC)K€HOYH8 R cbymcuii f Bigmosiglr, H3 rmrax-mn npo iCHyBaI-IHS! i €,II}IHiCTI>O6M€)l(€H0l`Opo3B’x3Ky pin:-umx-ul (2) Micmn Bi11oMa 'reopema M. l`. Kpeiiua[1, c.1I9]. V po6o'ri [2] UJI 'reopeua yaaransnena H3 BHl'[31IOI( ceIc'ropiam>Horoonepampa T io6Me>l<eHoi` ua R mpyalcuif f snca 3aJ10BiJ'II>H$I€rJ1o6aJ1LHyyMoBy Fensnepa. 1\H3.Tl0X`i'-IHPIITI no Hanenei-xo1‘o B [2] peaynsrar 111151 pinumumarmy (2) Bi11HocHo Heninoxvfoi oI1epaTopHo3Ha\1HoY ¢1>yI-ucnii oglepxcal-IoA.}I. lloporoauemm i T.O. Herponolo [3]. Taxoxc r1o11i6He nmalmasunqaewncn B [4, c. 249]. Ilmanna rrpo aupoxcnmauiro emmoro o6Me>l<eHoropo3s’x3Ky ,cuamlsepex-|uia.m.Horo pinH>IHHs1 (2) y Bunamcy, Konu f aanononbliacrJ1o6am,Hy ymony Fensnepa, nocnimlcysfscx B [5].
V no11aJ1Lu1oMy 6y1I€M0BBa>1<a'm, mo cnelcrp a(T) oneparopa T H6 nepe-THH3€TLC}I 3 ymauoro Biccro iR:= { it {teR}. Hoxnagxemo, xx i B [6, c. 209],0+ a_(T) - qacnmu cnelcrpa cr(T), xxi Micmrbcn ainnoainno y npasiil T8niBiI7I ninnnoumuax C, nplmolvzy at (T) - Henopoucui MHo>xc1»um. Toni a_ (T) -1<oMna1<'ma, 0+ (T ) ~ aammena M1-roxmua; l'I]JOCTiP B poalcnazxacrncsl B npsmycyMy iHBapia.H'rHux Bimxocno oneparopa T ni11r1pocTopiB Bi.; aaymemm Tioneparopa T Ha Bi Maxon, BiJ1l'IOBiJ§[I<I0 cnelcrpn ai(T); T_ - J1iHi1`iHm`»i
o6Me>1<eHuii, T+ - ce1<'ropiaJI1>Hm71 oneparopn. Hkmo Pi - npoexcropn B BBinnonimxo Ha nigmpocwopu Bi, fi(t):= Pif teR , T0 Bi]1l'IOBilIHl/Iifl
c]>yH1<11iY f e11m-1m`fx 06M€7I(CHPII"i po3B’>13o1c x pinrmum (2), 306pEDKy€TI>CXyBHFHHRI
x(t)= x+(t)+x_(t t e R ,
ne xi - Bi,11noBi11Hi cpyulcuism fi ezufmi o6Me>|ceHi poaB’s131<n 11m1>epeH-uiansuux piauam,
xQ(t)=T_x_(t)+f_(t), tell, (3)
xf+(f)=T+x+(t)+f+(f)» tek- (4)ITOKJIHIICMO
x_(t)= .]eT'("s)f_(s)ds, teR, (5)
x+(t)=~+FoeT*(t_`y)f+(s)ds, ren. (6)1
98
Bicuux Kuiiscuxoea ynisepcumemy 200-1,2 Bulletin ofthe University ofKievCepz¥z.'q§i7uxo-Mame/uamu~¢H1` nayxu Series.'Phisics& Mathematics
IH'1`€1"p3.TIPI B (5,6) 36i1“aIoTLc>1 a6comoTHO, 60 CKCHOHCHTH Biz; oneparopin Ti1<ope1<THo B1»13HaqeHi npu t> 0 i 33J10BOJII>H}IIOTB Taxi yMoB1/1:
36>0 E\c>0 Vt>0: eT*[n$ce`6t, i`e`T+'U$ce"5', T+ e'T+!u$§e"St (7)
y npocTopax Bi(T 3riJ11-10 3 Teopemoxo M. F. Kpeima [1, c. 119] Bunnusac, Luo pinnm-mx (3)
Mae cmmnii o6Me>1<eHm71 po3B’>13o1< x_, HKI/IF! 3o6pa>Kye'rLca y Burmmi (5).U_Io11o pinwlnwl (4) cnpanmxyerbcxTeopema 1. Pilansmux (4) Mae enunuil 06MC)K¢HI¢II`/1 po3B’s13o1< x+, HKPIITI
3o6pa>KyeTLc;1 y Bnrnsmi (6).Zloaeneawl. Icaynauml. 3aqJiKcyeMo s>0 i po3m;1HeMo B npocropi B+piB1-LHHHSI
5359 = T+xs(r)+ e'T+Sf+(t), 1 e R. (8)
Jlux CKOPO'-161-IHH sarmc no1<J1a,neMo := e'T+s t , t e R _ Bizlzuafumo, moY s +jg :R -> D( T+), a THKODK 3 ypaxysauuxm (7)
ToMy 1<ope1<THo BI/I3H3“I¢H1 Qlymcnii
MII., S ¢@"”Hf+l|...- ||T+fsl|.. S @""I|f+II., ¢/~<- <9>
»=.<f>= °+1°@'T*(k"')fs(k)dk .
- ’ (10)+00
ws(f)==- I T+@`T+("”’)fs(k)d/f-1
Bi11MiT1»1Mo,u1o BHacI1i11oI< (9) xslloo < +oo, wS[|m < +oo. BI/11<op1»1cToBy1o\m
33.MKH€H1C’[`I> T+ T3 nepexin no iH're1‘pam.H1»1x cym Moxcua nepexouamca, -md
Lum K0)KHOI`0 I e R: ws (t)= T+xs(t).11oBe11eMo, mo mm 11oBim>Horo t e R icnye x; (t) , a 'ra1<o>1< xs (t)
3az1oBim>1-me (8). 3a<1>iI<cyeMo fo e R, B1,1Ip130K [to - y/2,10 +3//2] , ne y
Bi1:moBi,m-Ia to cTa.na 3 (1), Ta no e N , TZKB, mo no > to + y/2. BHaCJ'I1)10l( IISMH
3.2.1 3 [4, c. 57], rrpu n 2 ng zum qmymcuii'
F<f,~)=-'fe'T+("”’)fs(k>d/f, 1600 -M, (11)r
BHKOHYIOTLCH cninnimlomemm`
dF ,-£12 = T+F(f, n)+ 3(1) , (12)
99
Bicuux Kuikcbxozo ymbepcumemy 2004,2 Bulletin ofthe University of KievCepm. rpxéuxo-Juame/uamuuui uayku Series:Phisics& Malhematics
m-< <f>>dk-(1-e~T+<"“'>)fS<f>. (13)I
Tomy
+°° T (k 1)sup |§F(r, n)- xs (r]]= sup I e` + ` ]§(k)dkNsre[r0 -y/2,10 +y/2] ze[r0 »y/2,r0+y/2] n
g¢|],g.Hw +_<|iOe_5Zdz->0, n-»+w, (14)n-(to +y/2)
To6'ro {F(-,n): nzno } a6irae"rx>c>1 pial-Iomipao Ha [to -y/2;t0 +y/2] noqmyuxuii x,(t). I[oBe,ueMo, Luo {T+ F(-,n): n 2 ng } 36iracrl>cn pim-xomipuo Ha[10 - y/2,10 + y/2] 110 cbyH1cui1 ws(t), 3ayBaJ1mMo, mo
ws (1) = -+30 T+@*T+("");;(k)dk 1 +10 T+e`T* ("“’),f,(1)dk =f I'
=-+?°T+e-T+<'f~'><f,<k>-,<,<f>>d/1-11° §(- @~T+‘k-'>,@<f>)dk= <1s>I I
= 51° T19-T+<*~'><fs<k>-mdk-f,<f>1I
Tomy, 3 ypaxyaammm (13,15)
sup !1f+F<f,~>-wS<f>1|f sup f+@“’+("°”><fG[fo~r/2»f0+V/2] f@1fo-1'/2-fo+>'/2] '1
X(/}(k)"fv(f))dkH+ [ _Tip /2]Ue`T+(”")fV(r1' =:/11(n)+A2(n),
npuqomy
A2 (n) 5 ce"5(""° _Y/2)}| fsnm -> 0, n -> +oo, (16)
Taxon:
+00A __ ~6(k+s-tow'/2) C dk :
1(n)< ie k+s-to-y/22"f"°°+10 5 1 uv)
=2||f||w¢ ya (~+2>__dz-»o, n_>0_Y s + z
H-fo '-5
3 (12, 16, 17) nnnnnnae, nocniglonuicrs : n 2 no 36irac'rLc:1
100
Bicnux Kuiacbxozo yuiaepcumemy 200-1,2 Bulletin ofthe University ofKievCep131:d>13u1co-Juamemamuuui uayzcu Series:Phisics& Mathematics
piBHoMipHo H8 [to - y/2,10 + y/2] 110 qaynxnii T+xs (t)+ 1'IepeBipmvxo, mo npn qmilcconauomy n 2 no mbynxuix T+F( t,n) Henepepnna
H3 [t0-y/2;t0+y/2]. 3a<1»iI<cycMo p,te[t0-y/2;t0+y/2], Taxi, mo p<t.BH3C}'IiI[0K (13)
n
u T+F(f»")“ T+F(P'"x| SH ` 1T+e_T+(k“t)(fs(k)' J%(f))dk +
+ r1T+@`T*(k"’)(fs(k)- f.~(p))dk + Il f(f)- f (p)1I+I7
+ U <”-f>fs<f>- <~-p>fS<p>y| 1 E2 + E,3 (1)B1»xm1uBae, mo
E2 < M |t - p|’8 .
3ayBa>1<nMo, mo
E3 = l|e'T+‘"")<f;<f>-/@<p>>-(e°T+1"-P1 -2-f+<"-'>)f,<p>(( S
S e-T+ (n+s-t)‘|" f(t)_ ( e~T+ (t-p) _ I) e-T+ (n+s-r)f(P)u=
=:E3’1 + E3l2.
3 ypaxysalmam (1, 9) crlpaazpxysrscsl onimca
E33 S e`o("+s")cM|t -p IB S cM| t -p |’8.
BHacJ1i11o1< TGOPSM 1.4.3, 1.4.4 3 [4, c. 34]
V/3 > 0 Eicp >0 Vt >0: T+Be"T+'H Sc/3 %e"5';I
7 -T At 1 y\1o<y51,vxeD(T+): (Q + -1)x||s;¢,_yzHTIxN
T3.K0)I(D(T4?, )3D( T + ), a ome (b) Buxonyerbca H3 D( T+ );
Vx eD(T,f'): Tlx S ¢||r3x|]1||x;|1'1_
3 ypaxyBaHH>rM (20, 21)
E12 5L1(f-121,71 e"”1"*‘“'1f(p1{ f1¢<f-pllfnwfg S
$L2|| t-p|/(n+s-to-y/2),11eL1,L2 -,11esu<i c'raJ1i_
Ouirmmo E1 _ Binmimmo, mo
lOl
(18)
(19)
(20)
(21)
(22)
(23)
If unwc Kuikcwcoeo ywiaepcumemy 2004,2 Bulletin ofthe University ofKievCepia.~¢n3uxo-Mamauamuvni nayvcu Series:Phisics& Mathematics
E,,S||3if+@ @~T+<"~%fs<k>-fS<f>>dk
+if-TnfT+ +fT+<**P><mk>-fs<p>>dk
+H-'}T+S‘T+("'P’(f,(k)-f§<p))dk+ 'fT+e”T+("””)<fs<k)-fs(p))dk =I P
=Z EL] + E12 + E13 .
Bpaxonylolm (1 ,9), ;1icTaHeMo
E13 = "}T+e'T+("*f’)<f,<k)-fS<1»)>f1k“ SP
(24)
5 l}e"$(k_p)M(k - py; c/(lc -p)a'k S cMl t- p |‘B;
Pn _T (k+s_ ) ne-5(k+s~_p)
EL25 .[”T+e + pIl!|f+(t)`f+(P)||dk5M(t`P)BI k5t -55 +oo t (25)
S Mn - pf 57 j e“’("'P)dk =¢ M_(f _ p)/“_
PTarcoxc 3 (20, 21) Bunnusac, mo
1| (<ff+"“P> - 1)f+e-f+<’°+‘*'><f+</<>- f+<f>>|| S
S ==.<f - pt( ff# <"+‘-'><f+<k>~ f+(f)1'5
-és5 c.(r - pi' Tfe`T+" e'T+(k't)h2|| f, Hao 5 c.!,(r - p)5?e"5(*"),ne c,, c,,1- nexxi crani. Toni 3Hal`7u1e'rl>c»1 Taxa c1~a.na c,,2 > 0 , mo
Em =H'i(@'T+"'f’) -1)ne-T+<*+~><f+<ff>-f+<f>>dk S
' (26)+<»
S 0,20 _ p) ; e-ff<'f-'Mk S: M_,,(f- p).r
3 (18, 19, 23, 24, 25, 26) Bnmmsae Henepepnuicrs T+F(t,n)
Ha.[t0~y/2,10 +y/2], mo rapamye Henepepsnicrb noximmx. ToMy mu
102
Bicuux Kulkcbxozo ymkepcumemy 2004,2 Bulletin ofthe University ofKievCep13l:¢15*wco-Mameuamuwi uayxu Series:Phisics& Mathematics
;1oBim>Horo te [to - y/2,10 + 71/2] icHye noxilma xg. = T+xs + fs(t), To6To
xs - po31a’;13o1< piBI-IZHHSI (8), mo Bilmoainae fs Orme, }§[0B€}I€H0,Luo mm noainnnoro s>0 piBHsmH>1 (8) Mae
o5Mex<em~n»u`71 po3B’n3oK (10), a T&K0)K
f+@~T+<k-'>fT+S<f+<k>-f,<f>>d/f. anI
3a¢1>i1<cyeMo to e R, Bi1Il`IOBi,l1H€iiomy 73 (1) T8 nepeifizlemo no rpammiH3{I0 -y/2,10 + y/2]np1»1 s -> 0+. I_IOKII8].lCMO
x+(’) “+_t|Pe_T+(k_t)f+(k)dk- <2s>r
1`IepeBipI»1Mo, mo
Sup xs (t)- x+ -> 0, s -> 0. (29)f@[’o -Y/210 +P/2]
3 ypaxysaxmsm (20, 21)
" (e_T+s _ 1) T+e_T+ (k*t)f+ (k) 5 C-»=~5`1/2 T-E/2 e_T+ (k_t)f+ (k) 5
S c..s‘/2 _gig-5(*~'>;1 f pw.
(k-1)3Bi1ICI/I ,
1 "°°( ~T S ) -T (k-r)sup xs(t)-x+(t]| = sup f e + -I T+e + x
’E[’o -J’/2Jo+V/2] ’€[’o~>’/ZJ0+V/2] Y
oo -5(k- )><f(k)dk Hs sup c»,1_;1/2+I
;%dk=few/2,»0+y/21 f (ff-‘Y
= cqsl/2+;O&dz -> 0, s -> 0,. 0 J;
mo ,IIOBOQII/ITB (29).TIOKIIZJICMO
+°° ~T (k-¢)W(t)5=' I T+e + (f+(k)"f+(t))ik» t€R~ (30)
1
Hepeaipumo, mo '
9"(s):= sup £&-w(t)"->0, s->0+. (31)felfo-y/2,f0+v/21 ""
Cnpanni,
1 03
Bicnux Kulbcuxoeo yuiaepcumemy 2004,2 Bulletin ofzhe University of KievCepD|.~¢isuxo-/nam €MdDfll'|Hl'uayxu Series;Phisics& Mathemaxics
1 oo
9”(s) = sup H+) (e'T+s - I)T+e`T+ (k_t)(f+ (k)* f+ (t))dk 5.
'€[fo~V/2»’o+v/2] F
1 +y
S sup ° ; (6-T# _ 1)r+e~T+<'f~'>(f+(k)_ f+(f))dk +re[t0 -y/2,10 +y/2] t
+ sw> MT@”“-Jnf”®”umw-nv»&%f€['o-V/2Jo+v/2] o+>'
=mw+%@>Bpaxosyioqu (20, 21), OTPI/IM8€M0
+CXJ
ms) S sup ( 1 ¢,_s|| r3e~T+<"-'>]!2|; f+ uw dk 5félfo -J'/2.10+w/2] fo+v
+w e~z5(k~1) +<x> e-62S sup I ctlslldké cu s->0, s->0+.
f€[f0-1'/2.10 +>»/2110 +o' (k “f)2 Y/2 Z
3a<1)i1<cycMo 0 < v < B. Toni
fo +V ,,ms); sup j (c*,0svHTl+ e"T+(k")NM(k-t)'6dk]] 5telto -y/2, to +y/2] t
10 +y -6(k-1)< sup I c,,2svf-<-~(k-t)pd/C S
telto-y/2,t0 +y/2] t I (k"t)1+v2y e z
5 c*,2 Ijjkidz sv->0, s-+0+.0 1 V
3 (29, 31) 'ra Herlepepsnocri H3 [to -y/2,10 + y/2] nurmusae,
Luo 11ml noninsuoro te [to - y/2,10 + y/2] icnye x;(t) = w(t). Taxum Lumom,mm 6ym»-xxoro t e R icuye xf,_ (t)= wft).
Sacbilccyemoto e R. 3 (8,l 5) Bmmnnac, modx _ *e T+sf+(fo)= T+xs(fo) ~ (32)
Bpaxosylolm CEKTOpi3JIBHiCTL T+ Ta (31) nepeiulemo B (32) no 1‘pa.H1f1ui npus -> O + _ Onepxcnmo
T+ x.s'(t0)_) x; (t0)" f+(fo), S ") 0+>
104
Iikruux Kuihcr-xoxo yuiaepcumemy 2004,2 Bulletin ofthe University ofKiev(`ep|Jv.'¢f»L9uxo-uamezuamulmi uayxu Series:Ph11vics&Mathematics
a Taxo>1<, 3 ypaxynamvlm (29),xs(t0)-> x_,_(t0), s -> 0+ .
ToMy, Buacninox 3aM1<HeHoc1‘i oneparopa T+, x+(t0)e D(A), a Ta1<o>l<
xl(fo)= T+X+(fo)+f+(fo)~Tal<1»1M lmnom, x+ (t)- po3B’a3oI< piannmm (4). O6Me>1<eHicTL x+ (t) o=1eBu;1Ha,
60+CD +®
l\>f+(f)||S f °@"5("")\If+||°°ff"=C|lf+l|¢, g e"”d2-
€IlIfIHiC'1`L x+ (t) noao/:u»m>ca mm me Meronom, mo li npu ,uoaeneani TCODEMI/I 1
3 [6, c. 65]_
TeopeMy 1 noseneno.
3ac1Ji1<cyeMo tl < 12. Piammmo (3) Binnoainac aanaqa Kolui
vL(r)= T_v_(t)+ f_(r), r 2 fl,_ (33)
V-(f1)= 0»
B npocropi B_, a pisuaumo (4) - sanaqa Komi
V;~(t)=T+V+(’)+f+(f)» ’5f2» (34)
V+ (72 ) = 01
B npocropi B + _
HeBa>x<1<o nepesipum, mo po3B’;131cn 3a11a\1 Korni (33, 34) Maron, Bnrnan
v_ (t) = T7eT' (t's)P_f (s)ds, t ?.t1, (35),
»lv+ (t)= - _[eT+(t_s)P+f(s)ds , t S t2 . (36)
1`Io1<I1a;leMo I v(r) .~= v+(r)+ v_(r), r e [r1,r2], (37)
Toni cnpasmrcyerbcxTeopema 2. H€X&I`;I T - ce1<Topiam.H1»1171 oneparop, a(T)r\ iR = Q. Toni zum
Lloainbuoi rbymcuii f : R -> B , mca sanoninsusle yMoBy (1), Ta Bi)11'IOBiIlHOI`0 Til
po3B’a3Ky x(t) pismmnn (2) BHKOHYCTLCX ouimcaV: e [r1,-z2]:
H x(t)_ v(t) | S e-6(t-rl) + e~5(z2-1) I Nw ) G?)
ne v(t), t e [1],-12] Bu3Ha‘1ae'n,ca aa 1_1or1oMoro1o ¢1>opMyJ1 (35, 36, 37) .
105
Bicnux Kullrcbxoeo yuisepcumemy 2004,2 Bulletin aflhe Universigv of KievCepi.a_'q5L1uxo~./uamauamu~mi uayxu Series:PhLrics& Mathemarics
Iloaenenrm. BH3CJl1lIOK(5,7,35)
vfelfl.-f21 1 llx-(f)-v-(OIISCNIP-ll 1|f(’)|1mt1e_6(s_t)d~°=
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3 (39,40) Bunnunac ouiulca (3 8).Teopemy 2 Lxoneneno.
BIICHOBOKZ
Y ,aaHi171 cTa1'ri Jlocnimxeuo HPITSHHH npo icHyBaHHx Ta ezmuicrbo6Me>|<eHoro po3a’x3Ky maqmepenuianbnoro pixamimvr (2) ra nwrauml npoanpoucumauiro uboro po3n’x31<y po3B’>I31caMu ni11noBi,uHnx sanali Komi.
.JIi'repa1'ypa
1. lIa.r1eul<m`»i IO.II., Kpeiin M. I`. Ycroihmnocrn peulennifl JlumbepeuuuannnbxxYPZIBHCHHH B 6aHaxoB0M npocrpax-xc'rBe. ~ M_,”Hayxa”, 1970. - 534 c.
2. Topozmiii M.<D. Criincicrr. o6Me>1<euux po3B’s131<is nuqyepeuuiansnuxpiBHJIHL 3 MBJIIIIM uapaMe'rpoM y 6anaxosoMy npoc'ropi // Yup. MHT. >1<ypH. -2003. - 55, N9 7 - c. 889 ~ 900.
3. Iloporonuen A.H., Herpona T.A. Orparmqeune u nepeonuqecxne pememmoneparopnoro ypasnemaa Pmucarn c Heorpanmieuum oueparopom //llncbdrep. Ypaaaelma. - 1997. - 33, N2 3. - c. 309 - 315.
4. Xelipn 11. I`eoMe'rpu\1ec1cax Teopwi 11oJ1ym»n-Ie1`7lHI>1x napa6om»mec1cuxypanneuuii. - M.: Mnp, 1985. - 376 c.
5. Pomanem-nco B.M. I-Ia6J1u>1<e1-u-La o6Me>1ceHnx po3s’>131ciB piaunuenux Ta111»1q>epeHuiam.Hux pinmmz, po3B’»I3KaM1»1 sinnosiminx aanaq Komi // B101-II/IKKuiscskom yl-lisepcwrery. - 2002. - Cepin: (bis,-Mar. Haylcu , Ne 2. - c. 142- 147.
6. Iloporonuen AJI. Hepnommecnne u crauuonapnme PBDKI/IMLI Gecxouelnmxne'repM1»1HHpoBaH1>1x H CTOXRCTIF-ICCKHX mmamnqecrcmc cncreu. - K.: Bamaunc., 1992,- 319 c.
Ha11iI7111u1a 110 penalcuii 15.01 .2004p.
106