Nonhomogeneous evolution equations with pulse actions

Для эволюционных уравнений с импульсными воздействиями доказаны основные теоремы теории возмущений. Приведен пример из теории дифференциальных уравнений в частных производных с импульсными воздействиями....

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Date:1992
Main Authors: Samoilenko , A. M., Ilolov , M., Самойленко, А. М., Илолов, М.
Format: Article
Language:Russian
Published: Institute of Mathematics, NAS of Ukraine 1992
Online Access:https://umj.imath.kiev.ua/index.php/umj/article/view/7799
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Journal Title:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
_version_ 1860512761241403392
author Samoilenko , A. M.
Ilolov , M.
Самойленко, А. М.
Илолов, М.
author_facet Samoilenko , A. M.
Ilolov , M.
Самойленко, А. М.
Илолов, М.
author_sort Samoilenko , A. M.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2023-09-14T12:03:56Z
description Для эволюционных уравнений с импульсными воздействиями доказаны основные теоремы теории возмущений. Приведен пример из теории дифференциальных уравнений в частных производных с импульсными воздействиями.
first_indexed 2026-03-24T03:33:55Z
format Article
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B HaCTOHLUeH pa6oTe H3Jiara JOTCH YTBep­ )K,UeHirn O pa3pernHMOCTH a6cTpaKTHOH 3a,uaqH KowH B cJiyqae HeO,UHOpOJJ.HblX no BpeMeHH K03q)(j:JHUHeHTOB. PaccMaTpHBaeTCH npHMep H3 TeopHH .UH<PQlepeH­ UHaJihHb!X ypaBHeHHH B 'laCTHbIX npOH3B0,UHb!X C HMDYJibCaMH. lloKa3aTeJihCT­ BO 0CHOBHbIX pe3 yJibTaTOB conep:»<HT HeKOTOpbie H.UeH H KOHCTPYKUHH, H3JIO­ )KeHHhie B pa6oTaX [3-5]_ TiycTb E - 6attaxoBo npocTpa1-1crno· c 1-10pM011(11" • IIE, [£] - npocTpaHCTBO JIHHettHhIX orpaHHqeHHhIX onepaTopoB, nettcTBYIDLUHX B £, R - ,ue11cTBHTeJih­ Ha51 OCb, N - MHO}KeCTB0 HaTypaJibHbIX qHceJI. TiycTh nocJie,UOBaTeJibH0CTb 'IHCeJI {t;} E [0, s) c: R, i E N,- Q)HI<cttpoBaHHhie MOMeHThI BpeMeHH, npHqeM C A. M. CAM0PIJIEHK0. M HJI0JI0B. 1992 1slN oo4I -6oss. ~IKp . . .uaT. [//Cypn., 1992, T . 44, M 1 ; 93 t, <ti npH i < j . 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OnepaTop A 3aMKHYT H HMeeT nJI0Ttty10 o6JiaeTh onpeneJieHHH D (A (t)) C £ . A 2 • Cyw:ecTsye-r TaKoe w, q-ro Bee 'A> w np1rna1IJie)i(aT pe3oJihBeHTHOMY MHO)Kecrny oneparnpa A (t). A3 . <P ymmm1 (A (t) - 'AE)- 1x(A > w, x E £, / - TO,KneerneHHbIH oneparnp) HHTerpHpyeMa no 5oxttepy. A4 . Cyw:ecTByeT rnKoe M > 0, 4TO l!JIH JII06oro pa36HeHHH s > i: 1 ~ > •2 > ... > •r. > 0 (., =I= t;) 11(11 (., l - 11r1 (A ('t" z) - 11r' ... (A (tn)- 'A)- 1 II ~ M ('A- wt. B1 . O nepaTOpb1 B, 3aMKHYThI c nJIOTHhIMH o6JiaeT51.MH onpeneJieHHH D (B1) B £ . 8 2 • Cyw:eCTBye-r TaKoe w1, 4TO see 'A > w 1 n pHHa.uJiemaT pe30JJbBeHTH0MY MHoMecrny oneparnpa B,, i E N. B a. TTocJienosaTeJi bHOCTh (B, - 'A)-1 x (1. > w1, x E £) exo.nHTC51. no ttop­ Me IJ· IJ£. B4• Cyw:eeTByeT TaKoe M; > 0, i EN, 11ro II (B, - 'A) 111£ 1 ~ M /(1. - ro 1), i EN. B6 • Cyw:eernyJOT orpaHtt4eHHhie oneparnpb1 (I + B;)-1, i E N. CttcTeMe (1), (2) cooTBeTcrnyer onepaTop L E C (E) - E EB L 1 (£) EB EB 11 (£), onpeneneHHhitt paBeHeTBoM Lu= {u (0), duldt-A (t ; u, u (t;) - (I+ B,) u (t1 - O)}, (3) r.ne u (0) E £ . O6nacTb onpe.ueJieHHH oneparnpa L MHO)Keerno D (L) ecTh nepe­ ce11eHHe 06JiacTe11 orrpeJJ.eJieHttH orrepaTopos 9f du/di EC(£)- L 1 (£), A (t) EC(£)- L, (£), I+ B; E l1 (EJ-.11 (£). 1/SSN 0041-6053. ~IKp. MaT. 3/CypH., 1992, T. 44, CcpopMyJrnpyeM YTBep)!rn,e,rne, collep}KaULee oue,m y pem eHH i'.r HMITYJJbCHoro 9BOJIIOUHOHHOro ypaBHeHHH (1 ), (2) C I-JalJaJJbHb!M yeo10mreM ll (0) =~ 11 ., EE. (4) T e o p e M a I. flycmb ce,11euc111Bo onepanwpoe A (t) l/:\teem ceoucnwa A1 - A4 , a on ep a mopbt B; , i E N ,- cuouc11wa B1 - B 0 • T o2aa onepamop L, 3aaaHHblU paBeHCmBO;J,t (3) , U,\teem HenpepbtBHbtll o6pa!1lHblll ll i);z;z peUICHUfl no­ cucme.Me (I ) cnpaeeiJ.,iuea ou,e1-tKa m ax II u (t) II :::;;; M;e"''" {II u (0) II E + S II F (t ) II Edt + L II g; II E}, u)1 > w. 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O603Ha•!l1M ,na,1ee qepes Lo (t): - C(E)-+E EB L1 (£) EB 11 (£) orrepaTOp, ,neiicTByIOW.HH cJJenyIow.aM o6pasoM Ha D ( ! ) c C (£): Lo (t) u (t) = {u (0), duldt - Ao (t) u (t), u (t1) -(/ + Br,;) u (t1 - 0)}, (7) n 110Ka>1<eM, lJT0 Lr, -+ L B CMbTCJJe HopMbI npocTpaHCTBa E EB L 1 (£) EB 11 (£) Ha :1J1e:v1e Inax MIIO>KecTBa D (L) c C (E) . 1lm1 3Toro BbilJTeM 113 paBNICTBa (7} IT0lJJJ eHH0 paBeIICTB0 (3) a BbllJHCJJHM H0PMY II Lr,u - Lu IIE EBL,<E> EBL,<E> = II {O, A11 (t ) u - A (t ) u, B ,s;tt (t1 - 0 ) - s C, - B 1u (t1 - 0)} II = S II f Ao (t) -A (t)J u (t ) II Edt + 1 II [Bo, - B,J X u O<t;<s x u (t ; - O) llr: =,I'> + S6• ISSN 004 1-6053. ¥ Kp. Mar. :HcypH.., 1992, r . 44, IE J. ,,Uo1rnmeM , <rT0 npasaH <raCTb ft, + S0 cTpeMHTCH K O rrpH 6-+ 0. CHaLiaJia paCCMCTpHM HIITerpaJl It,. KaK ycTaH0BJlel-I0 BhllUe, TT0,ILblHTerpaJJbH0e BbipmKel !He ,IL.Tl51 f/~ CTpeMHTC 51 K O npn 8-+ 0. OcTaeTCH .LlOl<a3aTb, <ITO 3T0 Bblp ,rn{eJIHe orpaHH<rel-IO HHTer­ p imyeMoi:\ cl)YHKUHeft. H o A (t) - A o (t) ~ l l - f/0 (t) ] A (t), 11 TaK Ka K JI fl 6 II ~ 2M, To II I - ~7 6 (t) Ii ~ 2M + I, cJienosaTeJJbH0 , II [ 1 - f/ o (t )] .4 (l) u (t) lie ~ (2.M + 1) 11 A (t) u (t) lie- - CTornoJJhKY u (t) ED (L) ED (A (t)) , r.n:e A (l) EC(£) -+ L1 (£ ), TO II A (t) u (t) lie m n erpupyeMa. no TeopeMe Ocry.LLa MmKr-ro yrnep:lK.UaTb , •rro s Io =.\ II [A (t) - Ao(t)J u (t) 11 [-/dt-+ o rr pH O-+ 0. a PacCMOT!)HM cyMMY S6. 3Ta cyMMa CTpeMHTC51 K 0, II0CK0JlbKY Bb!pa}K€1Hle, CT051 - l l.lee non 3HaK0M cyM'vlbl , CTpeMHTC51 !{ HYJIIO npH O-+ 0 H ()110 orpaHH<reno cyM ­ MHpyeMOH cpy1mu11eii . OcraeTC 51 C0CJlaTbC51 Ha cyMMaTopIIblH m1a.1or TeopeMbl Ocr~rna. I1TaK, L r-.-+L npH fi-+0 s EEBL 1 (E) (f)l 1 (E) Ha D(L) c C(E). 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TTycTb tk,l1E{t1} , iEN, rrpH4eM t1t<l1 H (tk, t1)n{t;}= 0 , il Clo(t,'t), tk < 't ~ f < ti; - 3BOJIIOUH0HHblH orrepaTOp nmjxjiepeHUHa.TTbH0fO ypam1emrn dv I dt = (5 fl c (f) V (f). ( 15) .96 fSSN 0041-6053. YKp 11ar. 01CypH ., 19fl2. r. 44 , M 1 Hs orpaHHqeHHH Ha fl 6 (t) BbITeKaeT cooT11Gwe1rne u(t1 -0 , th)= lim u(t,t,,)E[EJ. t➔t1-0 nycTb {t1, t2 , ... 'tm} E [T' sin {t;}, i EN, m > 2, tl < t 2 < ... < tm. vkITOJlb3Y51 pe3yJibT8Tbl pa60Tbl 16/, n oJJy<iaeM CJJe.ay10w,He COOTHOIJJeuHe )lJHI 3BOJIIOUHOH­ uoro onepaTopa S6 (t, 1:): u 6 (t , tm) (/ + B6m) u 6 (tm - 0, tm _ 1) X ... X (/ + B62 ) u 6 (t 2 - 0, ti) X ECJIH X (/ + B6,) ll ii (tl - 0, -r), -r < ti; u6 (t, tm)(J + Bom) 116 (t m - 0, tm_ 1) X .. . X (I+ B6) X X U<, (t2 - 0, f 1), 't = 11 . M!W.>KeCTBO j 't , sj n {t;}, i EN, COCTOHT 113 0./lHOro 3JJeMeHTa t;, TO 50 (t , ·r) = {u0. (t, t;) (I + Bii;) uo (ti- 0, -r), ecmi 1: < ti ; uo (t, ti), ec.rm -r = ti . EcJIH >1,e 11:,s!n{t;}= 0 , TO S e, (t, T) = U/j (t, T). (16) ( 17) (1 8) non:po6r-iee OCTa HOBI-IMCH Ha 3BOJIIOUHOHHOM onepaTope Ue, (t, ,:) ypaBHeHHH (15), r!le 0 < T ~ t ~ ti , flpllM e l!HH Teopmo 3BOJIIOIUIOHHblX oneparnpoB [8] K ypaBHeHHIO (5), noJJy•rneM cpopMyJJy llJJH Uo (t , -r) oo l l l tn-J ll ll (t •• )= 2 6~ sf16(t1) dtd'flt,(t2)dt2 -·· s f16(t,.)dtn- (19) rr.=U T T 't' OnepaTopHbIH pHll ( 19) cxonHTCH B To110JJorn1-1 [ E], rrocKo.11hKY Tenepb, noJib8YHCb coornollleHHHMH ( I 6) - ( 18), HMeeM II s/j (t, 't) 11[£] ~ Me[(a+b)(t-i:J+b]/6(1-(1)0)' (20) r)le a= sup II A (t)(l - 6A (t))-1 ll[EJ' O~t~s , OueHKa (20) Il03BOJIHeT n epeiirn OT CHCTeMbl (12)-(14) K mnerpaJibHO· cyMMaTOpHOMY ypaBHeHI-110 BH.Ua I r;/0 1·/ 6 v(t)=So(t;0)v(0)+ 5so(t,-r)e F(-i-)d-i-+ ~ So(t,t,)e ' g(t;-0). o O<tt<t flp1-1 3TOM <a+b>t+b s (a+b)(l-·t>+b ll V IIE ~ Me iiO-roli> II v0 IIE +MS e 6(1-roOJ 0 (a+b)(t-·tJ+b (a+b )t+b (a+b)'t + 't + ~ e 60-~l e"'o II g, 11£ ~ Me 6(1-roli) {11 Vo 11£ + 5 e- OU-~> • X ll< ti<s 0 (a+b)t; 11 (a+b)t+b X 11F(-t)11Ed-i+ ~ e-O(l-ro6l +bl!g;IIE}~Me 6(1-ro6J {IIVol\s+ O<t;<s ISSN 0041-6053. YKp. Mar. :J1Cyp11... 1992. T. 44. M 1 7-1·812 97 g + -~ 11 F (.) II Ed,:+ L II gdld · 0 O<. t r•, 5 Tenepb noJiy'-IHM Tpe6yeMyIO oueHKY .llJIH u0 (t). HanoMHHM, ttTO pueac,.. , IIO Uo (t) = e-116v (t) MO}KeT 6bITb 3aITHCaHO B BH)le (a+ b )t _..:.,.. +--b- t n Uo (t) 11£ = e-t/0 II V (t ) !IE ~ Me6(1 - wb) " II0-606) {11 u. uE + s II F (t1)" X 0 - {a+b-l } Bo3hMeM w < min 6 , cu • Tor.lla npn Bcex n.ocTaTO'-IHO MaJiblX I - l(a +b ) ro > 6 1 _ w<'> - l . Y'-IHTbIBaH 9TO, MO)KeM 3anncaTh s II uo (t) IIE ~ M1e"''1 {II Uo IIE + S II F (t) IIEdt + ~ II gt IIE} ~ o O< t; <s s ~ M 1e"'•5 {ll Uo IIE + J II F (t) IIEdt + ~ II g I IJE} · o O< ti<s TipaBaH qaCTb IlOCJie)lHer o HepaBeHCTBa He 3 aBHCHT OT t H, CJieD.OBaTeJibHO, Ta }Ke ou,eHKa BepHa H )lJIH max II uo (t) !IE, T. e. O~ t~~ s max II U6 (t) 11£ ~ M1ew, s {II Uo 11£ + J II F (t) II Edt + _ L II g, IIE} · O~t~ s O U< t;< s TioJiy'-leHHOe HepaBeHCTBO 03HatJaeT '-!TO II Loi II ~ M leO)' S. TaKHM o6pa30M, ceMeikTBO onepaTopoB c;;1 orpa HH'-leHO B COBOKYITHOCTH. Tor­ )la B rnJiy TeopeMbI 06 o6paTIIbIX onepaTOpax 13] cymecrnyer onepaTop L- 1 TaKm':i , '-!TO L- 1 = lim L6 1, npH'leM L -t orpaHH'-leH TOH )Ke KOHCTaHTOH 6-+0 M 1 . A 3TO 03HatJaeT, 'ITO .D,JIH pewettmi aa.11,a 'IH (l), (2) , (4) HMee·1 MeCTO ou.e1-1Ka ( * ). TeopeMa 110Ka3aHa. TTycTb ceMettcTBo oneparnpoB A (t), 0 ~ t ~ t0 , H B" i E N, HMeeT CJI~ .llYIOill.He )lOilOJl!-IHTeJibHbJe CBOJ1C'rBa: A11 • Bb!nOJIHHeTCH 1-1epaBeHCTBO (21) A12 . TiocJie.lloBaTeJibHOCTb ceMeHCTB onepaTopoB A,. (t), t =I= t; , y.11osJ1eTBO· pmOil.l,HX ycJIOBHIO (2 1) , HHTerpaJibl-10 CXO,!lHTC51 K ceMeHCTB Y fl ( t ), t =I= t ,, Ha HeKOTOpOM IIJIOTHOM B C (£) MHO)KeCTBe Dt KyCO'-IH0-.!1,HcpcpepeHUHpyeMblX B L1 (£) <j:>yHKUHH 1jJ (t), T. e. I 5 II !An (t) - A (t)J ,~ (t) l!Edt - 0 npu n-. oo. (22) 0 B11 . BElnOJIHHeTCH Hepaae1-1crno l![ l -eB;1- 1 l!lEJ ~ 1 Vi EN, e<e0 . (23) 98 ISSN 0041 -6053. Y,w Mar. {)(Cyp1t., 1992, r . 44, M 1 TiocJienosaTeJihHOCTb ceMeiicTB orrepaTopos B;,.., i E N, y,nosJieTBopmo­ m.ux ycJIOBHIO (23), cyMMaTopHo cxonttTCH K ceMeiicrny B;, i E N, Ha HeKoTOpOM IlJIOTHOM B l1 (£) MHO)KeCTBe, T. e. ~ ll[B;n-B,J'lj)(t;)\ls--o rrpH n--+oo. (24) O<t;<t Terrepb ccpopMyJittpyeM OCHOBHOe yrnep)Knem1e TeopHH B03Mylll,eHHH 3B0- JIIOUHOHHbIX ypaBHe1mii C HMITYJibCHbIMH B03)1.eHCTBHHMH. Ero rJiaBHblM conep­ )K3HHeM HBJIHeTCH yrnep)K1teHHe O TOM, 'ITO MHOMeCTBO TpoeK { u (0), F (t), g,}, rne u (t) - peweHHe CHCTeMbl (1)-(4), 33MKHYTO B E E9 L1 (£) E9 11 (E)..· Teo p e Ma 2. PeLUeHue Un (t) y paeHeHufl dun/di -An (t) Un (t} = F n (t), t =I= ft, C UM.nyAbCl-lbtM.U 803aeucmBUflM.U Un (f;)- (I+ Bi) Un (t; - 0) = gin• yaoeAem eopfl!OU{ee yCAOBU/0 eae Uon --+ Uo 8 £, I s II F n (1) - F (t) ll sdt--+ 0, 0 cxoaumCfl CU/lb flO 8 £ K. peUleHU/0 U (f) 3aaa'I.U (1 )- ( 4). LI. o Ka 3 a Te JI b c TB o. PaccMoTpHM orreparnpbr L,. 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CJJe,uoBaTeJibHO, perne1m e Un (t, x) CXO,UHTC5! B cpe,uHeM no X 11 paBHOMep1-1O no O ~ t ~ 1 K cpyHK­ llJIH r.11.e I -i J C 2 (X,1' ) d ·t I((f,'t, X)=e Im x( l + b1 ,Je -i tm-0 -i s C'(x,..:)d..: Im-I 11-0 _f C'(x , i: )di: 1' F (x, T) = !im Fn (x, 1:), n➔ oo X Poioo<1e1-tKo JO . B. , Tpocfm,1t'l!JK C.H. n ep1wm-1'leCKHe pcwenm1 cJJa6o neJJHHeHHbl X ypaBHe1rn ii D '13CTHblX npo113B0J!Ilb!X napa60JJH'ICCKO ro n rna C J-!.\lllYJJbCHbIM B03J.IeiicTBHCM H HX ycToi:j ­ 'IHB0CTb.- Kuen, 1986.- 44 c. -(TTpenp1111T / AH YCCP . 11H-T MaTe~rnTHKH; 86-65). 2. CaAt0iiAe1-tKo A. M ., HAOA0/3 M. K Teopn1-1 3B0JJI0UH0HHbIX ypaB 11e11m1 c HMHYJJbCHbIM B03JieH­ crnneM // llo,rn. AH CCCP.- 1991.- a 16, N2 3.- C. 82 1-825. 3. MacAO/J 8. fl. AcHMHTOTH'-lecrrne MeTOJibl B TeopHH B03Myme,miL- M. : HayKa, 1988.- 309 C. 4. EaA.aKpUUIHOH A . B. n pHKJJ3JUIOH cjlyHKUHOHaJJbHblil aI-JaJIH3.- M. : HayKa, 1980.- 383 C. 5 . H ocuoa K. <l:>y1IKUH011aJibHbll1 anaJJH3.- M. : MHp, 1967.- 624 c. 6 . CAIOCOfJ'l!fK B. E. 06 3KCHOHeHUH 3Jl bHOii illlX0T0MHH pewermi-i HMHYJlbCHbIX CHCTeM // YKp. MaT. Mypu.- 1989.- 4 1, JI/\! 6.- C. 779-783. 1. JlaAet/KUii IO. JI ., Kpei'm M. r. YcTOH'-IHBOCTb pewennii ,A11qxjlepeuu11aJibHbIX ypanne1111ii B 6a- 11axoB0M npocTpaucTBe.- M. : H ayKa, 1970,- 534 c, Donyqeuo 18,10.91 0085-1 0086 0087 0088 0089 0090 0091 0092-1
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institution Ukrains’kyi Matematychnyi Zhurnal
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publishDate 1992
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spelling umjimathkievua-article-77992023-09-14T12:03:56Z Nonhomogeneous evolution equations with pulse actions Неоднородные эволюционные уравнения с импульсными воздействиями Samoilenko , A. M. Ilolov , M. Самойленко, А. М. Илолов, М. Для эволюционных уравнений с импульсными воздействиями доказаны основные теоремы теории возмущений. Приведен пример из теории дифференциальных уравнений в частных производных с импульсными воздействиями. Для еволюційних рівнянь з імпульсними діями доведено основні теореми теорії збурень. Наведено приклад з теорії диференціальних рівнянь у частинних похідних з імпульсними діями. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7799 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 93-100 Український математичний журнал; Том 44 № 1 (1992); 93-100 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7799/9467 Copyright (c) 1992 A. M. Samoilenko, M. Ilolov
spellingShingle Samoilenko , A. M.
Ilolov , M.
Самойленко, А. М.
Илолов, М.
Nonhomogeneous evolution equations with pulse actions
title Nonhomogeneous evolution equations with pulse actions
title_alt Неоднородные эволюционные уравнения с импульсными воздействиями
title_full Nonhomogeneous evolution equations with pulse actions
title_fullStr Nonhomogeneous evolution equations with pulse actions
title_full_unstemmed Nonhomogeneous evolution equations with pulse actions
title_short Nonhomogeneous evolution equations with pulse actions
title_sort nonhomogeneous evolution equations with pulse actions
url https://umj.imath.kiev.ua/index.php/umj/article/view/7799
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AT samojlenkoam neodnorodnyeévolûcionnyeuravneniâsimpulʹsnymivozdejstviâmi
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