Integral sets of quasilinear pulse systems
We obtain sufficient conditions for the existence of integral sets of a weakly linear system of differential equations with impulse action on surfaces. We study the asymptotic behavior of the solutions beginning on the integral sets and in a neighborhood of these sets.
Збережено в:
| Дата: | 1992 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1992
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2314 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508240598532096 |
|---|---|
| author | Akhmetov , M. U. Perestyuk, N. A. Ахметов , М. У. Перестюк , Н. А. |
| author_facet | Akhmetov , M. U. Perestyuk, N. A. Ахметов , М. У. Перестюк , Н. А. |
| author_sort | Akhmetov , M. U. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | We obtain sufficient conditions for the existence of integral sets of a weakly linear system of differential equations with impulse action on surfaces. We study the asymptotic behavior of the solutions beginning on the integral sets and in a neighborhood of these sets. |
| first_indexed | 2026-03-24T02:22:04Z |
| format | Article |
| fulltext |
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ISSN 0041-6053. Y,cp, MaT. ~ypH., 1992, T. 44, J.& l .J. '-
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Kafl, •trno 1/x ({;)II > h.
n ycTh Tenepb IlOCT051HHbie Q H [ .uono.TJHHTeJJb!-10 K YK333HHblM BbIWe ycJio
BHHM Y.ll.OBJieTBOpHIOT HepaBeI-ICTBaM
a>l, (1 + 2alk(l))lk(l)a3 < I (13)
H 0003H3'-!HM
<l )= (1 + 2alk(l))( l + lk(l)a) _
p ' a I - ( I + 2alk (/)) lll (l) a3 '
q (l , a, y) = y-a2 (I + a2 p(l, a)) l- 0-1 In (1 + k(l) la (a2p (i, a)+ 1)).
By.ueM Cl.JHT3Tb TaK.tKe , l.JTO
q (l, a, y) > 0. (14)
T eo p e Ma 2. Ilycmb cucme.Ma (9), (10) yooe11emeopH.em yKasaHH.btM yc-
11oeu. flAt u , Kpo,ue mo20 , 6btnOAHR1omcfl yc11oaufl (13), (14) u z0 =I= 0 11z0/I ~ H.
1111011 > IIGoll- To2oa cyU{ecrneyern mo•tKa ~. s < + 00 maKafl , 'Imo: 1) I z m11 ~
¾ H; 2) npu ecex t , t0 ~ t < s, cnpaeeo11uebt HepaeeHcrnea II z (t)/1 ~ H u
II (t)I '::::;;. 11 ri o II e g(/, a ,y)(l-lo). (15)
11 ! =- a(l+(a2p(l , a)+ I) afl (l) '
3) cyU{ecmayem oeiicm eumeAbHoe <1uc110 8 > 0 maKoe, ,1mo OAR acex t E (~, ~ +
+ 6) cn paeeo11ueo coomHoiueHue llz (t)II > H .
JI. 0 I< a 3 a T e JI h CT B 0 . TTpeJJ.nOJIOl!mM, lJTO BhIIIOJIH5110TC51 Bee YCJIOBH H
TeopeMhI 1-1 nm1 scex t > t0 cnpaseJJ.j JHBO He pase1-1CTBO l!z (t) II ,s:;:; H . n oKa)KeM,
LJTO B 3TOM cJJyqae }I.JI51 ncex t > to BblIIOJI H 51eTCH 1-1 e p aBeHCTBO (15). Il.TJ51 3Toro
y6enHMCH , n peH{ ll.e ncero, '-ITO ecJI H \l£oll ~ llri ol l, TO np11 scex t ;;), t0 cnpanenJIH
no Hep aae11c-rao
II£ (t)II ~ a2p u. a)II ri (t)II- (1 6)
ECJIH npeJJ.nO,TJO)l{HTh IIpOTHBHoe, TO cyw.ecTBY lOT TOl.JKH t1, t2, to ~ ti < t2 ,
TaKHe, '-ITO B TOtJKe f1 BnepBbi e , Hi.ll.JHH35l C MOMeHTa t =to, BhinOJIH51eTCfl Hepa-
8 ISSN 0041-6053. YKp. MaT, xypn., 1992, T, 44, M I
BeHCTBO
II 11 (t1)ll ~ II s (t1)II ~ a2p (l , a)fl ri (t1)'1, (17)
a t2 - TO'!Ka, B KOTOpOH BrrepBb!e BblIJOJIHHeTCH COOTHOllleHHe
( 18)
,lleiicTBHTeJJbHO, 3TO HeB03MO)KHO JlHlllb B CJiyqae, Kor.na cyl.I.I,eCTayeT TOl!Ka
p a 3pb!Ba t;, .!l,Jl5l KOTOp Otl crrp aBe.LIJIHBbl HepaBeHCTBa
lls(t;)ll<ll11 (t; )i1 , lls(t; +)ll>a2p(t, a)l111(t; +)II- ( 19)
IloKa:>KeM, l!TO 3TO rrpe.nrro.TJO)Ke1rne He rrpHae,n.eT K rrpoTHBopetimo. 1'fa pa
BeHCTB
H (19) Hatt~M
11
s (t; +) = s (t;) + X1 (t; + , t1) Jf (z (t;)),
11 (t; +) = !1 (t ; ) + x2 (t; + , t;) J; (z (t;))
II s (t , +)II ~ 1111 (t; )II + 2alll (l)II 11 (t,)II,
II n (t, +)II > 1111 (tJII - 2alli (!)II 11 (t; )II
II 11 (t ;) 11 (1 + 2alk (!)) ~ a2p (l, a) ( l - 2al/.: (l)) II 11 (t;)IJ.
IlocJie.D,Hee HepaBeHCTBO np1rne.neT K COOTHOllleHHIO a ~ 1, KOTOpoe npOTH
Bopel!IIT Bb!6opy a > 1.
T e nepb, IJOJlb3YHCb HI-ITerpaJibl-IblMH rrpe,n.cTaBJieI-IJH!MH .LIJIH s (t) H 11 (t)
[2], aHa.TJoroM JieMMbI r po11yoJIJia-6eJIJIMaHa H Hepase 1-1cTBaMH ( 17), ( 18), no
JIYllHM
II s (t2)II ~ a (I + 2alli (l)) II s (t1)II exp (-('y - 2al -
- 0-1 In ( l + 2/l (l) la)) (t2 - ti)),
II !1 (t1)II ~ a (1 + a2 (1 + all(!) l (l + a2 p(l, a)))II 11 (t2)II X
x exp(-q(l, a, y) (t2 -t1))
H OTCIOll.a lls (t2)II < a2p (l, a) 1111 (t2) II, l!TO npornaopel! IIT Bb16opy TO'IKH f2 .
CJie,n.oBaTeJibHO, COOTHOUieHHe ( 16) c npaBe,UJIHBO .
HcnoJih3YH HHTerpaJibHOe rrpe,UCTaBJieHHe peUiemrn 11 (t) Cl-iCTeMbl (9), (10),
H aHaJior JieMMbl fpoHyOJIJia-6eJIJIMaHa, HaXO.D,HM
II 11 Uo)U ~ eq(t ,a.y)(t.-tl II 11 (t)II a (I + (a2p (l, a)+ 1) ak (l)
H, cJie.nosaTeJibHO, HepaseHCTBO (15) cnpase.ummo .UJIH t > t0 rrpH llz (t)I] ~ H.
TaKIIM o6pa30M, ITOJIYl!eHO npOTHBopetme H, CJie,UOBaTeJihHO, MHO)KeCTBO TOl!eK
t, y,n.osJiernopmou.i:Hx ycJIOBHHM t > t0 H II z (t)II > H Henycrn.
JlerKO npoaepHTh, 4TO IH!)KH5151 T04H 3 5l rpa!-!b 3Toro MHO)KeCTBa HBJIHeTCH
HCKOMOH TOl!KOH cp. T eoper..rn .uoKa3aHa.
Yrsep,1<;i e1-rne , a1-raJion-1<iHoe reopeMe 2, Mon<Ho A0Ka3aTb 1-1 AJifl cJJy'-laH
t ~ to, ilsoll > !1110II-
nycTb Tenepb 2 = (s, 11), llzoll ~ H,- pellleHHe, H3l!HH3 IOUI.eecH Bl-le rr o
BepXHOCTH (D+, Ii BblI10JlH51IOTC51 ycJIOBHH TeopeMbl 2. EcJIH il'lloll > ll sol l, TO crrpa
Be)lJilIBO He paBe HCTBO ( 15). vlcnOJib3YH paccy )KJleHHH II3 [3], MO>l<I-IO IJOK333Tb ,
'!TO H B npoTHBHOM cJiyqae ll soll > 1111011, llzojl ~ H pewe1rne 2 (t) rrpmmM aeT
rrpH HeKOTOpblX t, t ;;;,, to, 3HatJ e im51 B 06JiaCTl1 1/zj j > H H3 -3 a p ocra no MOJlYJIIO
3Hal!eHHH COCT3BJ!HIOULeH 11 (t) npH yseJIHllemm t.
4. 1l 11 cp cp e p e 1-1 u 11 a JI b 1-1 bl e C B O H CT B a H H Te r p a JI b I-I bl X
no Be p x 1-1 o c T e 11. T eo p e M a 3. Dycmb cucmeMa (1) yiJoBAemBopRem
ycAOBUH,,H nn. 2, 3 u, KpoMe mow, cpyl-lKU,UU l (t, x), I; (x), t; (x) 1-lenpepblBl-lO iJu¢
q1epeH4upyeMbt no x u cnpaBeaAuBo coom/-lotuel-ltte (5). ToziJa pewel-lue x (t, t0,
c), 1-la'lul-laiou.1,eecfl Ha noBepxHocmu qr+• onpeiJeAel-ll-lOU ypaBl-le/-luflAtu (11), uMeem
B-npou38oal-lble no c;, j = l, m, Ha npo.MeJ1C ymKe R+ = [t0 , +=).
ISSN 0041-6053. :iKp. MaT. ~ypn., 1992, T. 44, M 1 9·
.il o Ka 3 a T e JI b c TB o. MmKHO npoBepHTb, l.!T0 npeo6pa3oBaHHe JIHnyHo
sa C MaTpHueii L (t) coxpam1eT HenpepbIBHYI0 .umpcpepeHUHpyeMOCTb cpyHKUHH
Ii, J{, j = I , 2. Kpo\.fe Toro , 011eBH.UH0, l.!T0 3TH cpy HKUHH y ,uoBJJeTBop H!OT yc-
JJ0Bl1 IO JlHnwima . Mo)KHO c1111TaTb, l.!T0
II of i~!· z) II ¾ l , I/ oJtz) II ¾ k (!) l ,
ec.rrn II z \I ¾ Mh 11 l )J.0CTaT0 Y.H0 MaJJo, r .ue M = sup II L (t)! I- IToKa)KeM, •n o
I
p em e~rne z (t, 10 , c) y p aBHeH11H (9), (10), r )J.e c orrpe,ueJieHo 113 (11), HMeeT B-rrpo -
H3B0)J.Hbie no Cj, j = 1, m . T aKHM o 6 p a30M, Ha 0 CH0BaHl111 JieMMbl l 6y)J.eT )J.0Ka-
3aHa Teop eMa.
CornaCHO (51, Z (l, t0 , c) = (S (l), T) (t)) 51BJIHeTC51 T3K)Ke p ellleHHeM Cl1CTeMbl
t
s(l) = X1(t,to) c+S X1(t,u)f1(u, z (u))du + L X1(t,t;)J}(z(ti)), (20)
t0 t0 ,,;;_ t, <t
00
11(1) = - S X 2 (l, u)f2 (u , z (u))du - L X 2 (l, t ;)J7(z(ti)).
t t ,;;t,
ITycTb z (t, to C + µ ej) - perneHHe ypaBHeHHH (20), B K0T0p0M BeKT0p c
3aMeHeH BeKT0p0M C + µej. O6o3H3'-I.HM qepe3 (I) (t, lo, c) = (u, v) perneHHe CH
CTeMbl
t
U (t) = X 1 (l, t0) ei + S X 1 (t , -r) f l z (-t) W (-r) d,: + L X 1 (t, ti) J}zw (t,),
10 t0 ,s;;_ti< t
00
v(t)= - S X 2 (l, -r)f2z('t)ffi(-r)d1:- L X 2 (l, l;)J~2w(t1), (21)
t t<t,
B K0T0p0H
j = 1, 2.
Paccy)K)J.a51 aHaJJ0rH'-1.H0 T0MY, KaK 3T0 )J.eJiaJI0Cb .UJJH CHCTeMbl (13), npoBe
BepHM, '-I.TO y p aBHeHHH (21) HMelOT e)J.HHCTBeHH0e orpatt1111em10e rrpH t >= to
pememre w0 (t).
IToKa}!{eM, l.!T0 cpyHKUHH Wo (t) eCTb B-npOH3B0)J.Ha51 peweHHH Z (t, to, c)
no c1. )lJIH 3Toro .n.ocTaT0Y.H0 y6e)J.HTbCH, l.!T0 npH Bcex t E R+ crrpaBe.D.JIHB0 co
-0rnornettHe v (t) = o (µ), r)J.e
v (t) = z (t, t0 , c + µei) - z (t , t0 , c) - µw 0 (l).
HcrrOJib3YH YCJI0BHH TeopeMbl, HaX0)J.11M, l.!T0 cpyHKUHH V (l) = (<p, ,P}. ecTb
-0rpaHHl.!eHH0e peweHHe CHCTeMbl
I
<p (t) = S X1 (t, u) [f1z (u) v (u) + o1 (v (u) + µw 0 (u))l du +
lo
+ L X1 (t, t,)[J izv (t1) + 0 2 (v (/1) + µw 0 (/1))],
t 0~ t, <t
00
,i, (t) = - S X2 (t, u) [f 22 (u) v (u) + o3 (v (u) + µffi0 (u))J du -
t
- L X 2 (t, t1) [J7,v(t1) + o4 (v(t1) + µffi0 (t 1))l.
t<t,
B 3THX ypaBHeHHHX Bbipa)KeHl151 o,._ ( •. • ), k = T,4, y.uoBJieTB0pH!OT oueHKaM
II o,._ ( ... )II~ e (µ)I µ I e-o(l-lo)' e (µ) = 0 (µ) .
10 . ISSN 0041-6053. ":ltcp. MaT. xypn., 1992, r. 44, M I
IT03TOMY
t
11 v (t)\\ ~ S ae-v(l-u) [l \Iv (u)]I + 0 (µ)\ µ \ e-a<u-t.)] du+
lo
00
+ S aev(l- u) ll 11 v (u)II + 0 (µ)\µI e-a(u-t•>1 du +
t
+ L ae-vU-ti> [k (/) l II v (t;)II + 0 (~t) Iµ I e-au,-to>J +
t0~t;< t
+ L ae vu-t ,> [/< (l) l \I '\' (t;)II + 0 (~t)I µ I e-a(l,-to>].
t<t;
OTc10,na, 06031-ia4HB B = su p ii v (t)lt, noJ1ytmM
i
2a(J_ +----) y 1 - e- ve
f3 ~ ---'----(-1 - --1,-z-'--- e ( ~l) I µ 1.
1 - 2al y + 1 _:_ (e~vo )
TeopeMa nm<a3aHa.
3aMeHHB B npoBe.D,eHHb!X BbIW e paccy:H<}leHJHIX t Ha -t, y6e,!IJlMC51' l!T0 no
BepxH0CTh ,y _ o6JJa,naeT .nmpcjiepenu1-1a.1bHb!MH CBO}lCTBaMH, aI-IaJJOrH11HbIMH .IlJJH
TI0BepXH0CTH '\P+·
I. Eo2011106oe H. H., MumponOAbCKUU FO. A. AcnMTITOTli'IeCKHe MeTOllbl B TeOpH!! HeJJHHeii1IblX
KOJJe6a1mf1. - l\'1 . : H avKa , 1974. - 502 c.
2. Ca,uot1,1e1-tKo A. M., riepecm,oK H. A. D.mjxj>epe11uuaJ11,111,1e ypam1emrn c HMI1YJ1bC11bIM no3-
nef1crn11eM. - KueB: B111ua WK. , 1987. -288 c.
3. rI AUCC B . A. I11nerpa.nhllb!C Ml!()}l,ecrna nepHOJ1f14eCKHX CHCTCM ll1Hjxpepe1mua.%Hb!X ypaB-
11e H11 {1.- M.: H ayKa, 1977 .- 304 c.
4. Tporpu.At<lUK C. H. D.oCTaT04Hble ycJJOBI-1 51 OTCYTCTBHH «611e1111fl» B HMTIYJJbCHblX CHCTer.rnx / /
D.mjxjJepeHUH3JibHbie ypan11e1-111H c napaMeTpOM.- Kues : I-111-T MaTeMaTHKH AH YCCP,
1989.- ' C. 13:-l- 139.
5. Ca.1wt1Ae1-tKO A. M., riepecmmK H. A., Ax,1re11wo M. ,Y. M eToll cpan 11e11ttH llJJH cucTeM nmji
cjJepe 1rnuaJJ1,1-1b1 x ypaBHCHHi-i C H1'!11YJ1bC ll blM 803Jl ef1cnrneM.- KneB, 1989.- 41 c.- (TTpe
npmrr I AH :i1CCP. 11H-T MaTeMaTHKH; 89.:3).
6. AxAtemoo M. Y. , riepecm,nK /J. A . n epHOllH'lec1me pewen:rn JlHcjxpepemni aJJbHbIX ypas
ne1mii C HMTIYJJbCHblM B03;J.eiicrnHeM 113 nonepXIIOCTHX / / TTpll\teHe1111e acHM OTOTH4eCKHX MeTo
}lOB B TeopHH HeJIHHeii111,1x m1<jxjlepe1mHaJibHblX ypaB!JeHHH.- KHell : I-111-T MaTeMaTHKH AH
YCCP , 1987.- C. 11-14.
7. AxMemoo M. Y., fl epecm10K fl . A. 0 llBHlKemm C HMTIYJJbCHblM B03l!eflcTBHeM Ha nosepxHOC·
rnx I! l13B. AH Ka3CCP. Cep. cjJH 3.-MaT .- 1988.- C. 11-1 4.
8. Ax,11emoo M. Y., fl epecm,oK /-!. A. 0 n11cjxjJepe11u11pyeMofl 3aBHCHMOcrn pewe1111i'i HMn VJ1bc11b1X
CHCTeM OT H3ll3.1bllb!X lIMIHbl X // YKp. MaT. lKypH.- 1989.- 41 , N2 8. - C. 1028-1033 .
9. Ax,uemoe M . Y . . n PprcmwK ff. A. Y cToii'IHBOCTb nep11onw1ec1rnx pernetrn fl cnCTeM JlHcjxjJe
penuuaJJbHbIX ypas11e 1111 i-i C HMl1YJlbCHLIM B03)1.eflcTBHeM Ha TIOBepXHOCTHX // TaM lKe.
N2 12.- C. 1596-1601.
DoJiy<Jeuo 09.11.89
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| id | umjimathkievua-article-2314 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T02:22:04Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2e/5a5fc4181e0d7aa23cf81150a325822e.pdf |
| spelling | umjimathkievua-article-23142023-09-14T12:03:56Z Integral sets of quasilinear pulse systems Интегральные множества квазилинейных импульсных систем Akhmetov , M. U. Perestyuk, N. A. Ахметов , М. У. Перестюк , Н. А. Effect Pulse System Differential Equation Asymptotic Behavior Nonlinear System Pulse Диференціальне рівняння Асимптотична поведінка Імпульсна система Нелінійна система We obtain sufficient conditions for the existence of integral sets of a weakly linear system of differential equations with impulse action on surfaces. We study the asymptotic behavior of the solutions beginning on the integral sets and in a neighborhood of these sets. Указаны достаточные условия существования интегральных множеств слабонелинейной системы дифференциальных уравнений с импульсным воздействием на поверхностях. Исследовано асимптотическое поведение решений, начинающихся на интегральных множествах и в окрестности этих множеств. Вказані достатні умови існування інтегральних множин слабонелінійних систем диференціальних рівнянь з імпульсним збуренням на поверхнях. Досліджено асимптотичну поведінку розв’язків, які починаються на інтегральних множинах і в околі цих множин. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2314 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 5-11 Український математичний журнал; Том 44 № 1 (1992); 5-11 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/2314/1535 |
| spellingShingle | Akhmetov , M. U. Perestyuk, N. A. Ахметов , М. У. Перестюк , Н. А. Integral sets of quasilinear pulse systems |
| title | Integral sets of quasilinear pulse systems |
| title_alt | Интегральные множества квазилинейных импульсных систем |
| title_full | Integral sets of quasilinear pulse systems |
| title_fullStr | Integral sets of quasilinear pulse systems |
| title_full_unstemmed | Integral sets of quasilinear pulse systems |
| title_short | Integral sets of quasilinear pulse systems |
| title_sort | integral sets of quasilinear pulse systems |
| topic_facet | Effect Pulse System Differential Equation Asymptotic Behavior Nonlinear System Pulse Диференціальне рівняння Асимптотична поведінка Імпульсна система Нелінійна система |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2314 |
| work_keys_str_mv | AT akhmetovmu integralsetsofquasilinearpulsesystems AT perestyukna integralsetsofquasilinearpulsesystems AT ahmetovmu integralsetsofquasilinearpulsesystems AT perestûkna integralsetsofquasilinearpulsesystems AT akhmetovmu integralʹnyemnožestvakvazilinejnyhimpulʹsnyhsistem AT perestyukna integralʹnyemnožestvakvazilinejnyhimpulʹsnyhsistem AT ahmetovmu integralʹnyemnožestvakvazilinejnyhimpulʹsnyhsistem AT perestûkna integralʹnyemnožestvakvazilinejnyhimpulʹsnyhsistem |