A multipoint problem with multiple nodes for linear hyperbolic equations
We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds...
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| Datum: | 1999 |
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| Hauptverfasser: | , , , , , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4729 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510895178776576 |
|---|---|
| author | Beresnevich, V. V. Bernik, V. I. Vasylyshyn, P. B. Ptashnik, B. I. Бересневіч, В. В. Бернік, В. І. Василишин, П. Б. Пташник, Б. Й. |
| author_facet | Beresnevich, V. V. Bernik, V. I. Vasylyshyn, P. B. Ptashnik, B. I. Бересневіч, В. В. Бернік, В. І. Василишин, П. Б. Пташник, Б. Й. |
| author_sort | Beresnevich, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem. |
| first_indexed | 2026-03-24T03:04:16Z |
| format | Article |
| fulltext |
YJ2K 517.944+511
B. L BepniK, B. B. Bepeclteniq (In-T MaTeMarrmH HAH 6Dtopyci. Milmi,K),
1I. B. B a c r m n m r m (l-lptmapna'r. y~H. Inano-~panKil~cl,K).
I~. 1~. IITallIIIHK (IH-T npHK.It, npo6~l. ~aexauiKH i Ma'reMa'nmrl, J'hmit0
B A F A T O T O q K O B A 3A~AtlA 3 KPATHHMH B Y 3 J I A M H
j~dI~l dlII-III~HHX F I I I E P B O d l I H H I 4 X P I B H I I H b
We establish conditions of the univalent solvability of multipoint (in time coordinate) problem with
multiple knots for linear hyperbolic equations with constant coefficients in the class of functions periodic
in spatial variable. We prove metric statements concerning lower bounds of small denominators which
appear when constructing a solution of the problem.
BCTaIIOBJIeIIO yMonH O/tllO31taqllOi po3n'~aiIocTi 6avaTO'VOqKOBO'i (3a qaconolo KoopltHlla'roIo) 3altaqi 3
KpaTIII.IMH ny3.~laMl-I /tJlI, I 31illii.[llnX i'illep60.11iqllllX piBIl,',lllh 3i C'IaJItIMH KOeC[Jillir It K.tlaci
dpylIKHJi,I, nepiolmmmx aa npoc'ropot~olo 3MilmoIo. J2ol~elteHo Melpn,~ui 'rBeplta~emla. mo crocylo'n,ca
OltillKH 311H3y MaJItlX 311aMCIIIIHKilL :,lKi lU.llnIKaIOTb llpl.I no6yJto~i po3B'.',13Ky 3altaqi.
1. 3azIaqi 3 6aFaTOTOqKOBrlMH yMOBaMH 3a BHzIi~qeHOIO 3MiHHOIO/I~q piBH$1Hb i3 qac-
THHHrlMrl llOXi/glHMrl ~, B3ara.rli, yMOBIIO KopeKTHrlMH, a ix pO3B'~I3HiCTb y 6aFaTbOX
Brlna/IKax nOB'~laaHa 3 npo6:aeMo~o ~,m:mx 3naMemmKiB. , / I ~ BHIIa~Ky rlpOCTHX By3-
:lib "raKi 3a~aqi BaBqa:mcb y 6araTr, OX pO6OTaX (/roB., Hanprm,na~, [1 - 12]).
Y/xaaifi cTaa-ri, aKa e poamrmor, t pO6OTn [3], ;~oc~ai~xyr 6araToTO,aKoBa 3a-
/tuna a KpaTmIMn Byz.aaMII ZLna rinep6o:~i'~mlX pimlam, n - t o .nopa~tKy (n > 2) 3i
c'ra.nm, m KoetI~iuien'raMH B K,naci tl)ytmuii:~, 2X-rlepio/m'~mtx 3a rlpOCrOpOBOIO 3Mill-
HOrn. Aiiaamri'~Ha 3a~aqa ~,:la Oll;ttoro K.nacy ~tH~epetmia~am~o-ormpaToprmx pim~aHb
BnBqaamcb y po6o-ri [13].
13yaeM0 BrlKOprlCTOByBaTrl ~aKi noarm,~eHttz: O~n ~ o,aHOBm, tiprma Top, "rO6T0
KOaO oamm,~Horo pa,aiyca; D t = { (t. x) ~ IR 2 : t e [0, T], x ~ f~[n } ; F ~ npocTip
TparoltOMeTpllqltHX MttoroqaCHiB
in
P,,(x) = ~ C, exp(ikx), x ~ [ 0 , 2 x ] , m = 0 . 1 . . . . .
k = - m
3 Ko~rt.neKCmtMrt Koe~itdCHTaMrt, S aKOr, ty 36i:~rtic'r~ Bnana~aerbca "raKau qHaOM:
F ~ P,, --~ P, aKmo cTeneHi Bcix no~tiHoMiB P,(x) He nepemtmytoTb aeaKoro ~iK-
coBan0ro ram,ha N i nptI n ---> ~ P,(x) ---> P ( x ) ; F ' ~ rtpocTip Bcix .aini~nnx
nenepepmtrtx d~yrmttiotta.rtiB naz~ F 3i c~m6Koto 36ia~niCTtO, aKn~t cniBna,aar 3
npocropoM qbopMa.ql, rmx TpHroHOMeTpHttH~IX p~/IiB [14]; Ha( s ), 0t e R ,
r i~6epTiB NpocTi'p Kovtn:teKctto3naaartx 2n-nepio~rt~rmx qbyttKllifl mtr~a~ay y(x )=
= E I , I ~ 0 Yi exp(ikx) 3 nOpMOtO
= X I'-,
Ikl>-0
n~( D t ) , a ~ R, n e 7/+, ~ ri.m,@pTiB npocTip qbyaKuia h (t, x) "tam, x, mo z~aa
Komuoro t E [0, T] 3Sh/Ots~ Hct_s(f~/n), s = 0 , 1 . . . . . n , i
x, ll2 ? :IX d , < - ; [Ih(t, at
Cn([0 , T], F ) ( C n ( [ 0 , T], r ' ) ) - -K.r lac cloyHKtd[.l g(t , X) TaKttX, mo ~ ~toBi.m,-
noro t a [0, T] ~ g / 3 r I ~ F ( F ' ) , j=O, 1 ...... n.
�9 B. I. BEPHIK, B. B. E;EPECHEBlq, FI. 6. BACH$1HLLIHH, B. ft. FITAIIIHHK, 1999
ISSN 0041.6053, Y~:p. .Itam. ~.'Vlm. 1999. m~ 51, N e I0 1311
1312 B.I. BEPHIK, B. B. BEPECHEBIq, H. B. BACHflHIIIHH, Ig. I;I. FITAIIIHHK
2. B 06~acTi D ~ poaFJDIHCMO aa~aqy
~"u(t, x)
L ( u ) - a,. ~t" ~x" - " = O,
,u ffi= O
a s ~ R , a,, = 1, (1)
t m j = l ..... ~), j = l ..... 1, 1
Njmj (U) - u (mj- l ) ( t j , x ) = tPj,nj (X) ~ ri = n, O < tl < t2 < ... < tl < T , (21
i=1
Re 2 -< rj -< n 1 , j = 1, . . . , 1; piBH2HH~t (1) - - CTpOro rirtep6oniqHe aa 1-IeTpOBCb"
KHM, TO6TO Bci KOpeHi ~.j, j = 1 . . . . . n, piBHJtrlH~I
tl
~ a,.7~ ~' = 0
s - - 0
r i piaHnMa. B ~ r ~ z I o6~aer i D I HaK~az~ar yMOBH 2~ -nep!o/lHqHOCTi aa
3Mi~HOm X HaqbyHKtfii u ( t , X ) i r m j = 1 . . . . . rj, j = 1 . . . . . I.
POaB'~aOK aa/Iaqi (1), (2) myKaeMo y mw~a/l i pa/Iy
u ( t , x ) = ~_, u k ( t ) e x p ( i k x ) . ( 3 )
Ikl>O
KO~KHa 3 qbyHKail~ Ult(t), k r 7/, Br/3Haqae'rhc~ J~K pO3B'~I3OK TaKO] 6araTOTOqKOBOi
3a~aqi 1 ~ 3BaqaflHOrO ~ahqbepeHttia~bHOrO piBH~IHHJI:
n
2 a.,'(ik)"-s t'~ s)(t) = O, (4)
s=O
(m j - l ) t
uk (~i) = tPjmj (k) , mj = 1 . . . . . 13, j = 1 . . . . . 1. (5)
1 2~
IXe tPjmj(]C) ---- " ~ Is r ( x ) e x p ( - i k x ) d x ' k r 7/.
PiBH~IHHJt (4) Mac TaKy qbytt/IaMeHTaJlbHy CHCTeMy pO3B'$13KiB:
{exp(ik ~, rot), k ~ 71 \ {0};
Ukm (t) = tin_l, k = O, m = 1 . . . . . n .
~ KO~KHOrO k E 71 pO3B'$I30K 3aRaqi (4) (5).3o6pamaeThCJ~ qbopMy~om
t/
Uk(t) = 2 Ckm Uk'n(t)' k E Z , (6)
mffi|
s axitt KoedpittienTr~ Ctm, m = 1 . . . . . n , Bn3HaqalOTbC~ 3 CtlC'I'eMH piBtlSIHr,
n (taj-l),' ., 2 ttkq [tj) Ckq - r mj = 1 . . . . . ~ , j = 1 . . . . . I. .(7)
qffil
~.rl~[ KO)KHOFO k E �9 BH3HaqHHK A'(k) cHcreMrl (7) Bapa:~aerbc~ qb0pMy.rIaMH
A'(k) = J! 1-I (te - ts)' k = 0, (8)
mffil ~. j=l l< s<p~ l
ISSN 0041,6053. Yrp. ~om. 3,.Ti~l., 1999. m. 51. N e 10
BAFATOTOqKOBA 3AZI.AqA 3 KPATHHMH BY3.rlAMH...
$1KH~ MaC BHFJDIll
exp(ik~.~tt)
~'1 exp (ik3,1 t 1)
~,~1-1 exp (ik~. I tl)
A(~) ..........................
exp(ik3,~tt)
~,! exp (ik~,l it)
xq'-~ exp(/~tt~)
Teope~a 1. Hexa~ c n p a ~ y t o m ~ c ~ yztoou
(VkH 7/\{0})
1313
a A(k), k H Z \ { 0 } , ~ ~HaHaqnnK n - r o nopzllKy,
exp(ik~,2tl) ... exp(ik~,ntl)
~'2 exp(ik~'2tl) ... ~'n exp(ik~'ntl)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . , . . . . .
~r~-I exp(ik~.2tl) ... ~n-I exp(ik~,ntl)
. , . , . . . . . . . . . . . . . . . . . . . . . . o . . . . . . . . . . . . . . . . . . . . . . . . . .
exp(ik~,2tl) ... exp(ikXntl)
~'2 exp(ik~'Etl) ... ~'n exp(ik~ntl)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , , , . , . . o , . , .
~,~-1 exp(ik~,2tl) ... ~,~-I exp(ik~,ntl)
(9)
A(k) # O. (10)
Rru~o tpj,n j H F ( F ' ) , mj = 1 . . . . . rj , j = 1 . . . . . l, mo icnye e3unufl poza',~tzor
n 0 zaaa~i (1), (2) iz npocmopy C ([. , T], F ) ( cn([o , T], F ' ) )
~[o6eaenu~ nposollrm, ca 3a cxeMoIo lloBelleHna TcOpeM 1 i3 [10] Ta 2 i3 [9] i
6aayerbcs Ha qbopMy~i, SKa ao6pa~ae qbopMa.m,HO pOaB'SaOK 3allani (1), (2):
n t rj AJ,,iq(k) tPjmj(k) exp[ ik(x + ~,qt)]
u(t,x) = uo(t)+ Y E E (ik)q-' , (11)
Ikl>0 q--I j=l mjffil
( n - 1) -ro crenena, AJmjq (k) - - a.nre6paiqne llonoauermg lie ( u0 t) MHOFOqJIeH
mi-I exp(ikXqtj) y nrlaHaqHaKy A(k). eJleMeHTa ..q
3ayBa~nMo, mo yMOBH (I0) r TaKo~r ueo6xillHnMa yMoBaivIH r poaB.'J~axy
3a~aqi (i), (2) n Knaci C"([0, T], F ' ) (llnm [I0]).
B iamHx aHnallKaX pall (I I), B3ara~i, poa6i~HHll, OCKi~bKH a i ~ i H n a Bill a y ~ a
BeJIHqHHa A(k) MO~Ke Ha6yBaTH ~IK 3aBrOllHO Ma~HX 3a Molly~eM 3Haqerlb ll.rDl He-
CKiHHeHHOi MHOJKHHH 3HaqeHb k H Z .
TeopeMa 2. Hexaa cnpa~9~yembca y~to6a (10) i icnye cma~a ~ > 0 mara. u4o
(VkH Z/, Ikl>g>0) I~(k)l>lkI-E (12)
.,qt~u~o tPj,n j a H~+a(f21x), my = 1 . . . . . i), j = 1 . . . . . l, mo icnye r poza'~g3or
zaaa~ti (1), (2) iz npocmopy H~(DI) ', ~ru~ nenepepono zanemumb oi0 qb)'nr~i~
q)jmj �9
,~[oee~enns npoBOllrrr~cs ~a CXeMOIO llOBellOHH$1 TeOp~MH 3.4 ia poz~ai~y 2 [7].
3, 3'JlcyHMo, KOJIH BrlKOHylOTbCJI OUiHKH (12). 306pa3HMO A(k) y sHr~a~i
cn 1
qHF ~, j=l
ISSN 0041-6053, Ylcp. ~tam. m'y. pu.. 1999. m. 5 I. 1~ I0
1314 B.I. BEPHIK. B. B. BEPECHEBIH, n . B. BACI, I,flI.1LIJI'IH. B. [/I. FITAII1FIHK
7to ~ = (~'1 . . . . . ~',,), [ = ( t l . . . . , . t !) . F ~ MHO'~mm l~CKropi~ q =.(q~ . . . . . %,)
~ n KOMIIOIIBIITH IIKIlXyTBOplOIOTh Bci.~I~Ki Hcpecral~OBKH ql lccJ! 1 . . . . . l , 2 . . . . . 2,, . . .
r I ~
(Ki.~mKicT~, TaKI.~X .epec'ra~iol~oK pi[~m N = n ! ! ( r I ! r / ! ) ) ; . . . . / . . . . . 1
P,,(x~ . . . . . x,,)= ~ 5,~'i'...~';:
s ~ Gq
blllOFOqJlCllll CTClIClI,,'I I" (]IJIIL (8) ) 6C3 BiJIbllOrO qdll311tl, a Gq ~ MIIO.YK.tlIIa BBK'I'O-
p[B S = (S I . . . . . Sn) E 7/+ "raKtlX, lllO I KO~41l()l|Clrl KO;.KIIOFO 3 l i l t \ ltopiBliIOIOrl,
-- = = max {i;,,}, uy.mo, q e F , 5s. Aopimnoe 1 a 6 o - I: s j < ~o I, j I . . . . . n , P-0 I ~,,,~ 1
I . , 1 = , . l lpl iqoMy sal = sa , = . . . . s(z I - 0 , YlKIII.O qu,, r qcq,, l < m < h < 1. Po3-
I'JI.',IIIeMO Bci , o x b m i 3a 3Millllllblll t I . . . . . t I ~mpa3y (13) l to IIeBIIOFO /l~OCl.l'l~l, I~HCOKO-
r o HopRAKy
QI3 = c")t~' ..3t/~' = (/k)l131 "~-" PI~"/(~')cxp ik~ff, Xit,/~ , k e 7 / \ { 0 } , (14)
�9 q e l / ' \ j = l
lxe ~ = ([Jl . . . . . ~ t ) ~ 7/1+, P~.q(~L) --U0JfiHOMI,~ c'rcncH:4 r + I~1 3 Itia~u,,m KOe-
t|fit~ienTa~UL CePeA uoxi lmHx (14) Bu6cpc~40 N raK~t~t mmOM. too6 Brz3HammK N -
ro .opa l tKy .5(X) = detllPl~.q . e t; uc jtopi~molm~ r o ' r o ~ H o HyJmlfi. ac M
MIlO~lma ~,lyJu,'rl.tilijtcKcir{ [J = (~ t . . . . . f i t) , aKi Bilgloailtalo:rl, m,16paHHl~ IIOXil, b
IIII~[. FIo3HaqHlqO b = ,fni.n II~l, B = )'naxl[~l r lpms , cTm~o, lifo AJla z t eaKoio ./~ e 1~
�9 p e m p r M "
sup max Ia13(~,, 7, r -< Ikl ~', I~ ~ M.
~.~IW' ieto.r ] / "
~ a l l a CflCTOMa i i e p i m m c w c i i piBHOCn.m, Ha c1/c'reMi p i l m m m
a~(~.,f,~,) = 0~(~.,~)1~,1 "~,, f ~ M, (~5)
~4e 1013(~,, 7)1 < 1, 13 e M. Cttc-rcMa pi!maHb (15) r a iu i i iuom am'e6pai,aHOlO cHc're-
�9 " t i t "
IqOIOBIZBIOCIIO 3MiIIIII'IX e x p ( t k ~ j = t ~, j tqj)~yl , ,] ,~ -~ t . . . . . N, KO~H~.I 3 .,qKl'lX 3 a ,
~4o~yaeM AopiBmoe oAmumi. ~eTepMiHanT cHc're~.m (15) 3o6pa;~aeTbCa tbop~4y~mm
A*(~. ,k) ( i k ) r : ~(~.) , "/2"= Y 2 ( M , N ) , / ~ 7 / \ { 0 } . (16)
H a Or .rleMH 2.3 i3 po3, /I iJ ly 1 [7] BCTalIOB.rlIOCTbC$1, Ill, O 1][d-151 Ma|J)Ke B C i X ~.. e ~ n i
~ l a neaKoro ./3 = "/3 (M) cnpaBnxy lo 'n , ca nepi~mocTi
IA*(~.,k)l > C~(~.)l~:l v-', Ci(~) > 0 , k e Z \ { O } . (17)
Ha OCHOai @opMy.a KpaMepa ia (15) o'rprtMycr, m
I -- lypl = I A p ( L 0 , k ) l l X * ( ~ . , k ) l -~, P = 1 . . . . . N , (18)
/Ie ,~p(~., 0, k) ~BI43HaqHHK. ollepacaHn~t ttLaaXO~ 3alVliltrl y al, laaaqHrlKy A*(~., k)
p - to CTOBnUt~ CTOBnues~ npaBrtx qaCTl-llt cricTe~HpiBnanb (15), 0 = { 013 ( ~., t'), 13
M} . I3 (15) i (!6) o~tep~Kye~o
I%(X,0,k)l ~ C2(~)lkl v'+vz=~, P = 1 . . . . , ~ , C2(~.)>0, ke 7/\{0}. (19)
ISSN 0041-6053. YKp. ~tam. .~.'ypa.. 1999. m. 51, N" I0
EAI"ATOTOqKOBA 3A~AqA 3 KPATHHMH BY3.flAMH... 1315
I3 ottiao~: (17) i (19) Bnrt.naBae, tUo ~sla Hata:~e Bcix 7~ ~ IR" i ~t.na I k [ > K > 0 nprt
"YI < b + Y3 - T 2 piBnoc'ri (18) cynepeq.nnBi; TOMy Moacna Baa~a'rra, mo xoqa 6 o~tna
i3 aacrnnmix noxi~amlx nopaaKy ] ]31, 13 ~ M, aa~aoao.nba,ae nepiBaicn,
a j~ a(~ . , ~, ~)
> C3tkl h+'c3-r-'-e:2, C3>0. ~>0, (20)
/~a~a Ha,tixie Bcix ~, ~ R" i ~n~l Bcix ~ ~ [0, T] l.
TeopeMa 3. ]Lax ,~ta~e acix (eiOnocno ~lipu ./le6eza a R 1) 8ermopia t E
[0, T] l i 0,,1~ .~,a~.x,<e ecix (eianocno ,~dpu Jle6em e R " ) ee~cmopi8 ~ nepie.
nicme,
]A(k)] > ]k] -~'4-~, 74 = B+~I2-*t3 - b , 8 > 0 , (21)
cnpaeO,w.yembc,~ O.aa ecix k ~ Z, ] k ] > K > O .
]IoaeOenus~ o~epxiyeMo ia o/liHm~ (20) na oc .oui ~eMn 2.3 ia poa~iaxy 1 [7] Ta
.rleHrl Bope .na- KaHTe.n.ni [5].
TeopeMa 4. Ar.a~ ~ta~iace ~cix (eiOnocno ~tipu Jle6em e N" ) ae~mopie ~ e N n
i ~n,~ Ooeimmux qbikcoemtux t ~ ~t nepieniemb
I ~ ( L T , k ) l > Ikl -In+~ e > 0 , (22)
ourom'em~e~ 8n~ ocix (~pi~t cKi'n,~em,ozo ~tucna) k ~ Z, Oe 5= ~tq-=l I ~ ! .,.=q+~ r.,. rq,
2 t co = ( ~ / ) ~ , , , : _ ~,. (~,. - t) .
,O[oee8ennx npoao~rm,cx 3a cxeMoIO aoBeaen .a "reopeHn 4 ia [16] (arm. TaKOm
[8, 11 .17]) ; npa aboHy m4ananHn~: A ( ~ , t , k) ottimoe-rscz 3HI43y Ha ocaoBi OtliHOK
no6yaoBaHnx satmM ananaqmiKOH qbyHKtti~ g j ( ~, k ) , j = 1 . . . . . It(l), ~t ( l) = 1 +
I-1
+ (r t - 1 ) /+ ~, , ,=2 m r m , Ta iX noxi~n.X aa ~OMnO~enTaMrl aeKTopa ~. a aaKoprlc-
Ta~m~/~eMrt 2 is [6] Ta .neHri Bope.na - KaaTe.a~i.
JIeMa. Hexaa f = (f~ . . . . . f n) : U --~ R " - - nenepepeno Oudigepen~iaoane ~iOo-
@a.z,<ennn, Oe U c R" - - eiOrpuma niO~mo~runa. ,r A c U ~tae nynboey ~tipy
flegeea, mo f (A ) me.~ ~tae nynboay ~,ipy fle6eea.
,~oee~enus 6aaye'r~ca Ha TeopeHi Capaa [ 18], ari~Ho a .aKOm Kpwrraarfi aaa~ea-
HA nepeTaopeHHa f yTaop~o~OTr, HnOaCaHy Mipn ny,m, y rrpocropi N" ; roHy Kpn-
THqrli TO'-IKrl MOaCHa Bi/IKarlyrn i Baa~tcaTrl Bi/Io6pa.agenna f ,noKa.nr~naH ~Inqbeo-
Mopdpia~oM, ~a.na a~coro TBepzlaceHr~a .neon o,-teaH~r|e (daB., nanpnK.naa, .nelly 5 a
[19, c. 146]).
Poar.rl~neMo piBnaHn~l
n n- 1
I "t + an- t I -t + ' " + a lI t + a0 = 0, (23)
/Ie a = (a0, a I . . . . . a , ,_l) ~ R", i noanaqnMo qepea ~ = (I.t! . . . . . It,,) e C n BeK-
Top, CK~la/IeHI41~ 3 KopeHia pianarlHa (23). J;IK atttO~O, Bi//JIoai/IniCTb Mi~K r, mo~rmom
aeKTopia h i ~noacnnOlO BeKTopia !~ e BaaeHno O/tHoanaq~o~o.
TeopeMa 5. Hexaa A c R n - niOzmo.~'una eerraopia ~t ny,~t, oeof ~tipu fIeSe.
za. ToOi ai~noeiOna i'~ ,~mo~una aermopi~ ~t me~ ~tae nym,oey ~dtD' .lle6eea.
ISSN 0041-6053. Ytcp, ~tam. u,w, pn., 1999. m. 51o I~ !0
1316 B.I. BEPHIK, B. B. BEPECHEBIH, FI. B. BACH.fIHIIII4H, 13. ~. I'ITAIIIHHK
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n i t . h a / t o o n p amo n a tma - 12.03.99
ISSN 0041-6053. Yrp. ,~tam. ~.'vpu.. 1999 m, .51. N"- I0
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| id | umjimathkievua-article-4729 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:04:16Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a8/c21629e9023c7e1c42afb8509e54eea8.pdf |
| spelling | umjimathkievua-article-47292020-03-18T21:12:54Z A multipoint problem with multiple nodes for linear hyperbolic equations Багатоточкова задача з кратними вузлами для лінійних гіперболічних рівнянь Beresnevich, V. V. Bernik, V. I. Vasylyshyn, P. B. Ptashnik, B. I. Бересневіч, В. В. Бернік, В. І. Василишин, П. Б. Пташник, Б. Й. We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem. Встановлено умови однозначної розв'язності багатоточкової (за часовою координатою) задачі з кратними вузлами для лінійних гіперболічних рівнянь зі сталими коефіцієнтами в класі функцій, періодичних за просторовою змінною. Доведено метричні твердження, що стосуються оцінки знизу малих знаменників, які виникають при побудові розв'язку задачі. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4729 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1311–1316 Український математичний журнал; Том 51 № 10 (1999); 1311–1316 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4729/6159 https://umj.imath.kiev.ua/index.php/umj/article/view/4729/6160 Copyright (c) 1999 Beresnevich V. V.; Bernik V. I.; Vasylyshyn P. B.; Ptashnik B. I. |
| spellingShingle | Beresnevich, V. V. Bernik, V. I. Vasylyshyn, P. B. Ptashnik, B. I. Бересневіч, В. В. Бернік, В. І. Василишин, П. Б. Пташник, Б. Й. A multipoint problem with multiple nodes for linear hyperbolic equations |
| title | A multipoint problem with multiple nodes for linear hyperbolic equations |
| title_alt | Багатоточкова задача з кратними вузлами для лінійних гіперболічних рівнянь |
| title_full | A multipoint problem with multiple nodes for linear hyperbolic equations |
| title_fullStr | A multipoint problem with multiple nodes for linear hyperbolic equations |
| title_full_unstemmed | A multipoint problem with multiple nodes for linear hyperbolic equations |
| title_short | A multipoint problem with multiple nodes for linear hyperbolic equations |
| title_sort | multipoint problem with multiple nodes for linear hyperbolic equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4729 |
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