Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space
We obtain necessary and sufficient conditions for the oscillation of solutions of nonlinear second-order differential equations with pulse influence in a Banach space.
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| Datum: | 1999 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4586 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510734022082560 |
|---|---|
| author | Slyusarchuk, V. E. Слюсарчук, В. Е. Слюсарчук, В. Е. |
| author_facet | Slyusarchuk, V. E. Слюсарчук, В. Е. Слюсарчук, В. Е. |
| author_sort | Slyusarchuk, V. E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We obtain necessary and sufficient conditions for the oscillation of solutions of nonlinear second-order differential equations with pulse influence in a Banach space. |
| first_indexed | 2026-03-24T03:01:42Z |
| format | Article |
| fulltext |
Y]IK 517.9
B. E. CJnocapqyK (l'Hn.~., Te~. yH-T)
HEOBXOJIHMBIE H ~OCTATOqI-~IE YCJIOBHH
O C H H J I S L q I I H H P E I I I E H H ~ H E ~ ~ I X
J I H | 1 7 4 Y P A B H E H I 4 B C I 4 M I I W d l b C H H M
B O 3 j I E t l C T B t l E M B B A H A X O B O M I I P O C T P A H C T B E
We obtain necessary and sufficient conditions of oscillation of solutions of second order nonlinear
differential equations with pulse influence in the Banach space.
O/xep~aHo Heo6xiRni It ~loc'raTHi y~on~t oct~t, matdi po~'~s~n lteJ'liHiRHttX ~3~q~CpCHRhUlbHHX pinHgHb
~apyroro nop~zy s iMnyJn,cHo~)/li~o s 6anaxono~y npocropi.
IIycT~ E - - ~teltermrreYn, Hoe 6aHaXOBO IIpOCTDaHCTBO, E 1 -- no~npocTpax-iCTSO npo-
cTpaHersa E, ~tJ~a KOTOpOro codimE l = 1, r b JmHel~matt HenpeptaBHtat~ dpyH~-
ttaoHa~ Ha E c aRpo~ Kercp = E l , ~ + = [0, +~,), T - - npoHzBOJn,HOe caeTHoe
MHO3KCCTBO BCR~eCTBeHHblX qHCeJ~ fn, n e ~ , ]~YlJ~ KOTOptax 0 < t I < t2 < "'" < tn <
... n l i m t n=+o*.
n - ~ e o
PaccMOTpHM r~tny~XbCHy~O CHCTeMy, OnHChlBaeMylO cHCTeMOII ypaBHeHHIt
~ + f(t, x(t)) = O, t e ]R+ T, \
dx(t+O) dx(t-O) I-g(t,x(t-O)) = O, teT, (1)
dt dt
x( t+O) = x ( t - O ) = x(t), t e T ,
r~e f : ( R + \ T ) x E --->E H g : T x E - - ->E- - npoH3BO,m, mae Henpepr~Bmae
OTO6pa3KeHH~I.
PemeHae x(t) CHcTeMr~ (1) 6y/IeM HastamaTb octm,a~tHpy~ottmM oTnocrrrezbHO
noRnpocTpaHcr~a E t , ece, H ~ ma~aoro aHcJm a > 0 Hatt~zyrca m~cJaa x l ' X2 e
e (a, +-0), aJaz KOTOpblX
'o(xf~0)'o(xCxe)) < o.
B ~a~HO~ pa6OTr npH ~OrIOJIHHTeJIbHHX 01"paHHqeI-IHnX Ha OTO6pa3KeHtl.q f H g
yKSOKCM HCO~XO~I, IMI~Ie H ~OCTaTOqHHe yc.rIOBH~ O C I ~ 4 H OTHOCHTC$1~HO E | pC-
mCHH~ CHCTCMH (1), IIpO~OJDKHB TOM CaMHM HCCJIC~OBaHH~/ 0 KOJIe6JIeMOCTH
I~mCHH~ CHCTCMI:~ (1), HaqaTl~e aBTOpOM B [I - 3]. 3aMCTrIM, qTO BOrIpOCaM KOJIe6-
~r TpaCKTOpHi~ HMnyJIbCHHX CHCTCM K HaCTO~HI~CMy BpCMeHH He yRcJIgJI0Cb
~03I~HOF0 BHHMaHHR, XOTH TcopHH HMny.rlI, CHHX cIlc'rcM [4 - 8] H TCOpH~I K0-
ae6mM0CTH TpaeK't-op,~ raa~m~x nrmaMwaec~Hx cHerer~ [9 - 20] n o s t y ~ m r~j6o-
K0C pa3BHTl, le.
1. OcrtoBmae Tpe60"aHHa K cHcreMe (I). ByneM c~wrarb, ~ro/Xaa rrpo~3-
BOm,H~X ,.mc~.a t o �9 ~ .+\T H ~4HOXeCTBa {X l, X2} C E CHCTeMa (1) HMCCT r
C'rBVHHOC 011pcRr Ha ~ + I~tlICHHC X(f), ynos~eT~opmomee yC~IOBH$1M
x ( t o ) = x t H x'(to)=X 2. ~ro pemeHHe 0 6 o z H a ~ ~epe~ x(t, to, x l , x 2 ) .
Pacc~toTpm~ r, tHO~eCr~
~) B.E. C,/'[IOCAPqYK, 1999
98 ISSN 0041.6033. Y~p. ~lam. ~.'vpn., 1999, m. 51,1~ l
HEOBXO./~HMBIE H s YCJIOBHfl OCI.[H/IJIJ:ILIHH ... 99
E 2 = {x E E ; t p ( x ) > 0 } , E 3 = {X E E : t p (x )<0} .
KpoMe yCJIOBH~ HenpepuBHOCTH qbyHKUHtl f ( t, X) H g ( t, X) COOTBeTCTBeHHO Ha
( R +\T) x E H T x E, 6y/IeM Tpe6oaaTl,, qTO6~a aTH qbynKl~tt Ilpr B
BH~C
f ( t , x ) = ~.~ p t ( t ) f t ( x ) ,
k = l
m
g( t , x ) = E qt(t)gt (x ) '
i=1
rz~e n, m e I~I, P k ( t ), k = 1, n, ~ H e n p e p ~ a l a e Ha ~ + \ T orpamlqeHarae t@yHKI.U'IH
co 3Ha,aeHHxrari B ]R +, q l : T ---> ~ +, l = 1, m, ~ nportaBO:n, Hue OTo6pa~em4x H
f k : E ---> E , k = 1, n, g t: E ---> E , l = 1, m , ~ HenpepuBHue OTo6pa~emix, a ~ x KO-
TOpbtx f k E i c E i, y l E i C E i /X~a Bcex k = l,----n, l = L m n i = 1,-'-3.
2. YC.rlOBH~i octluci.rlaIlnn pemenHt i CHCTeMIbl (1). 0603rzatlnM qepea tP k MHO-
Y,(eCTBO HenpepuaHUX Ha ~ + H /~Hd~cl0epeHllHpyeMrax Ha ] R + \ T qbyHKl-lHtt
Z = z k ( t ) CO aHaqenna~H a E k, k = 2,"--3, ~.na Ica .~o l t Ha KoTOpUX I'p(zk(t)) I--
MOHO'rOHHaX Hey6uamomax Ha ]R+ qbynKtma, a qepea A y ( t n ) ~ paaHoc-n, nepBoro
nop~tara OT Y ( t n ) , T.e. y(tn+ 1 ) - y ( t ~ , ) . 3~ecb t h e T = { t 1, t 2 . . . . }.
OTo6pa~eHne h : E ---->E HaaoBeM noKa~n_~HO ~_I4nmHtIeBraM, eC~H / ~ npOHa-
aO~mHUX b ~ E n r e (0, + ~ ) Ha~aeTcx noc'roarIHaa M > 0 , ~tax KOTOpO~
IIh(x)-h(y)ll <- Mllx-yll
z ~ a ~cex x , y ~ a ( b , r) = { x ~ E : IIx - bll--- r } .
C n p a ~ e ~ r m o c~Ie/I3no~ee y T ~ e p ~ e H H e .
T e o p e ~ a 1. Flycmb:
1) e~monttn~omca mpe6oeanua nynrma 1;
2) inf (p(f~(z(s))) > 0, k = 1,'--~, u inf 9(g~(z(s))) > O, 1= 1, m, On~
, ~ , ~ o ~o(A(z( t ) ) ) , ~ , ~ o ~o(gt(z(t)))
acex z e 0 2 [.J 0 3 ;
3) neco6cmoenntae unmezpa/tta
u nucnoaw.e p.cugbz
0 tP(fk(z(t))) ' k = 1, n,
n • = 2 'Atp(z( tn- l ) ) l = 1, m,
~(gdZ(tn)))'
cxo&,anca a~a acex z ~ ~P2 (.J ~P3.
T o g a a ~ oc~uana~uu a c e x p e u , enua x ( t , O, x I, x2), (x l, x 2) ~ (E x E ) \ ( E l x
x E l ) , cucme~t~a ypaanenut i (1) omnocumeabnO noanpocmpancmaa E l aocma-
mo,mo, a a c.ay~ae .aoranbno aunumtleew.X unu ro~marmn~x omoapa.,'renua f k : E -.->
-.-> E, k = 1, n, g t : E..-> E , l = 1, m , u neogxoau~to a~no.anenue coomnoutenu.~
ISSN 0041-6053. Y~p. ~tam. :aypa., 1999 , m. 51, N e I
100 B.E. CAIIOCAPHYK
+ ' ~ ,~
f t pi(t)dt + E tEqt(t) = +...
0 k-~i t e T 1=1
(2)
]~oxa.~men~emao. ~ocmamonnocmb. rlycr~ BblrlOJ'lI-IJleTCJl cooa~omerme (2).
r l p e ~ o a o ~ a ~ t , wro cacre~,a ypanaema / (1 ) m~eeT ueoctum.rmpy~omee OTnOca'rear~-
rio E l pemem~e z ( t ) , Z a s ~0Toporo (Z(0), Z ' (0)) e E l • E I. He orpaHmmBaa
o 6 / / J ~ o c r r l / ~ o K a . ~ T e ~ a , nona rae~
r > 0 r~ tp(z(t)) >_ 0 V t > a (3)
rleKOTOpOrO ~ c ~ a a e (0, +oo)\To rlOCKOJrl, Ky Ha ocrloBarmH (1)
dz(s) dz(t) rs
ds d t
t ueCt, s)f~T
~Jl~ I'IpOH3BOJIbHI~IX t > a n s > t, t, s ~ T, r~, c.rle/~OBaTeJlbHO,
+(ez(~)h (az(t)~ 8 n
+ ~ ~ t,~(u)~(A(z(u)))au +
t k = [
m
+ E ~_~ ql (u) r (gl(ZCu))) = O, s > t >_ a, (4)
ue( t , s )AT l=l
TO cor~acHo (3) H yCJIOBIIIO 1 TeOpeMhl llO.rlyqlln
$ n
J ~ p~c.)q~(ACzC.)))d. +
t k = l
m
+ E Eql(u)q~(gt (z(u))) >- 0
ue( t , s )NT l=l
_(dz(t+O)~
g a s scex t rl s, a a s KoTop~x S > t _> a. rlo~voMy qbyrmrarfg lp t - ~ .)
~n,qe-rca Hesoapacrammefl qbyHKur~e~ Ha [a, +o , ) . Czle~toBarezbno, qbyHKUHS
~(z ( t ) ) ~ u t e r c g n o r H y T o l t r m [a, +**) [21, c. 1 7 ] r a c y m e c ' m y e r n 0 e z e z
lim tp( dz(t +0>~ =
t~+*, k ~" ') c>_O
(C He MO~eT 6blTl, OTprluaTe~bHblM Ha ocrlosaHrlrI (3) rl BOFHyTOCTH q b y H ~ r l
~(z(t)) Ha [a, +**)). Orcroza a r m (4) c.~e~yer, ~rro ~(z(t)) ~ MOHOTOrma~ aey-
6bll~lloma~ Ha [a, +oo) C~yIIKIIR.q H
+ ~
(dz(t +O)~ n
:
t k,~l
i n
+ ~ ~qt(u)+CsjCzCu))) vt>_a. (5)
u~(t,§ l ~ l
ISSN 0041-6053. Ytcp. ~tam. ~h'y. p~., ! 999, m. 51, N ~ 1
HEOBXO/2HMME H/2OCTATOqHME YC/IOBH.q OCI2HJUIflI2HH ... 101
H
0~03HaqHM qcpc3 8 HaHMCHbmee H3 qHCe31
inf q~(fk(z(s)))
s > t ~ a r ( fk (Z(t))) '
k = 1, n,
inf ffJ(gl(z(s))) 1 = 1, m ,
s >, ~ a ~p (gt (z( t ) ) ) '
KOTOpOe Ha OCHOBaHHH BTOpOro ycalOBrlJl TeopeMra nono:~KrlTeamHoe.
BK~IOqeHHe z ( t ) e �9 2, CXO;aHMOCTb HeCO6CTBeHHblX HHTeI'pSdlOB
YqHTblBa~I
+ : dcp(z(t))
9 ( f k ( z ( t ) ) ) ' k = 1, n,
a
H HHCMOB~X p~OB
A • ~ l = 1, m ,
~=, '~(gt(z(t,,)))'
s cHny yCnOBH~ 3 TeopeMrJ 1 (3~ecb k m HaHMeHbmee H3 qHceR MHO~'~ecTBa 1~,
/~J~ KOTOpr~x tk_ 1 > a ) H TO, qTO
Aq~(z( t , ,_ l ) ) = Iq~ d t =
tn-I
tn (+.n ,, "~
= + I / I z,',<u>+<:,<:<u,,:/<" +
tn_i \ t k = l ]
>_ + 1 1 z t-I
! n
> qt(ti)cP(gt(z(ti t - l ,
i = n =
Ha OCHOSaHrm (5), npH•eM K SHBo~ty, qTO ~nZ K a w ~ o r o k = 1,---n H l = i ' m
_ ( d z ( t +O)'~
+7" + ? < " t - - m - ) d :
+** > ~ q, (A fzft))) = J,, q, CAfz(O)) t
= Ja q~(fk(z(t))) c + , i=t~ pi(u)ffJ(f i(z(u)))du d t +
t))) ~,:+r ./
ISSN 0041-6053. YKp. ~ m . :~. pe.. 1999, m. 51, N e 1
102 B.E.C.YlIOCAPqFK
a \ ~ v~J~(z(r))) ,}
!('r I ": > 8 pk(U)du d t = 8 ( t - a ) p t ( t ) d t >-0,
t a
d-oo > nYkfP(gl(Z(tn ))) >_
-> ~kq~(gl i=~, ~ 1 l l(z(fi))) At,,-I
>
n--~ \ i=n ~P(gt(z(tn))) qt(ti )j Atn-I
>
CJI~OBaTCJIbHO, HCCO6CTBeHH~Ir HHTCrpaJIbI
(t - a)pk(t) dt , k = 1,'-'n,
a
(t n - t~_l)qt(tn), l = 1,-'-~,
n - - k
~tmcaoB~e p ~ a
cxolIffrcyt. I'[O~TOMy cxo/Iffrc.,.q HeCO6C'I'BeHI-I~e aHTeFpa..rl~
r~ ,-mc.noBrae p a ~
t Pk (t) d t , k = 1,"-~,
a
~. t~ q~(t~), t = 1,--~,
n=k
wro npOTHBOpCqwr COOTHOmem4~ (2).
TaKHM o6paaoM, npc]InoJ~o~euHe o cymCCTBoBam4H HCOCRHJ~J~HpyU)mero OTHO-
cwrcJ~HO no~npocTpascTsa E L pcmemta z ( t ) cHCrCM~ ypaBHetml~ (1), ~Jzx KOTO-
poro (z(0), z'(0)) e E l x El, ~oz~o.
]I0cTaTo~moc~ COOTH0meHHa (2) ~n~ 0Ctm.qJ~anHH cooTseTcrBymm~x l~meHa~
CHc-reMm ypa~ueHHR (1) ~oKasaHa.
Heo6xoOu~tocmb. l'lyca~ sco pemeHH~ x (t, 0, x l, x :z), ( xl, x = ) e (E x
X E)~(E I X El) , CHCTeMH ypanHCI-IHl~l (I) JIB.IIJIIOTC.q 0c~p~OIHJHMI, I OTHOCH-
TC/~HO E i.
PaccMoTpm~ c.~qaR aoKam, HO m m m H a e s ~ x 0T06pa~r f k , k = 1, n , H g t ,
l = I ,m. rlpe~tno~o~T,a~, wro coo'r~omomar (2) He sbmosm~erca, T. e.
ISSN 0041.6053. Yr, p. ~,am. J~ypn., 1999, m. 51, N ~ 1
HEOBXO~41V~E H ~OCTATOqHHE YC_JIOBHfl OCIg4JlJIglg4H ... 103
+ ~ n m
0 < f t E p~(t)dt + E t ~ ql(,) < +**. (6)
0 k = l t e T 1=1
BOaI, MeM rlpoH3so.rl~rll~t BeXTOp y e E 2 [.J E 3, 3a-~KHy'r~ map B(y, r), r > O,
/I~-a Koroporo E l [7 B (y, r) = ~ , H paccMorpm~ ypanHerme
+oo
n
Z(t) = y - f ( s - t ) ~ pt(s)f~(z(s))ds -
t k = l
m
(s - t ) ~ qt(s)gl(Z(S)), t >. a, (7)
sr 1=1
r/~e a ~ TaKOe llO.rlO:h'Cd4TeJlbHOe HHCJIO, qTO a ~ T,
H
.~oe
n
f ( s - a ) ~ pt(s) sup IlA(x)lld~ +
a k = I x ~ B(y, r)
m
+ E ( s - a ) E ql(s) sup Ilgt(x)ll -< r (8)
sr 1 - - 1 xcB(y , r )
I -t- c~,~ gl m
l" ! (s-t,~--I P'(s)f'(zl(S))ds+ ..(t,+**)OTZ (s-t'Z,=l qt(s)gt(zl(s))-
-1.r
t!
- ~ (s-t)E pk(s)fk(z2(s))ds -
t kffil " H
~-a ( s - t ) ~ qtCs)gt(z2Cs)) <-
s r lffil
1 sup Ilzt(0- z2(t) ll (9)
< 2 t ~ a
R~a scex aerxpep~m~x a orparmqemlux Ha [a, +~*) E-zaaamax dpyI~KUntt zi(t),
i = 1,"~, ~.rI~l KOTOpblX s u p ~ z i ( t ) - - y l l -< r, i = 1,--'L
t ~ a
Coo 'momemlz (8) H (9) BOaMO~Hbl Ha OCHOBamtn (6) a ZOICanmHOa .riHnmmte-
BOCTH OTo6paacerml~ f k, k = 1, n, H g l, l = 1, m.
]Ianee paccMoTpHM 6auaxo~o npocTpaHCTBO X Henpep~aBmax rl orparmqem1~x
Ha [a, +o.) E-3naynux qbyHKma~ x = x(t) c HopMoIt
II x IIx = sup II x(,)I1,
t k a
orpana~eimoe aar~ayToe H B~anyK~oe MHO~eCTnO D Bcex dpyaKt~Hlt X = X(t) r X,
~sI~ KOTOpUX x(t) eB(y , r) Vt >_. a, H onepaTop ~ : X ---->X, onpe~eaeamal l
paBeHCT~M
+ . a
(9~x)(t) = y - ~ ( s - Of(s, x(s))ds - ~ ( s - t)g(s, x(s)), t > a. (10)
t ae(t,+**)flr
Ha onpe~e~erma onepaTopa 9.1 H COOT~omemna (8) H (9) BUTeKaeT, wro ~ID c D rl
< 2 !ll~-zllx V,~zG o.
ISSN 0041-6053. Yxp. ~tam. ~.'vpn.. 1999, m. 51 ,1~ I
104 B.E. C2IIOCAPqYK
HO~TOMy Ha OCHOBaHHH npHmmna cmaT~x OTo6pa)KeHHfl [22, c. 72] HaflgeTca
dpyHKl~Yl Z -- z ( t ) E D, KOTOpa~ 6yg~r pemeHHe~ ypasHeHHa (7). YlerKo y 6 e ~ r b -
C~ S TOM, WTO ~'s cI~yHKI.~q Tax.~e 6y~eT pemetmeM C~CTem~ ypaBHemd~ (I ) ~ n a t >
>_ a. ~ro pemeaHe He 6y~CT OCRH.rI21.pyIoIRI4M OTHOCHTeJIbHO E l , KaK 9JICMeHT
MH0~KCCTBa D. B cHny Tpe6OBaHHfl nyHKTa I Ha~CTC$I peIIICHH~ y ( t ) CHCTCMI~I
ypaBHem4fl (I), coBna~momcc c Z(t) Ha [a, +oo). ~ L q ~TOrO petUeHHa 6y/IeT
nwanomcaTbCa BK.mo'~eHHe ( y ( 0 ) , / ( 0 ) ) a (E • l x E l ) cor~acHo e~.HCTBeH-
.OC-m pe m e a a a x ( t ) CHCTe~ (1), mIa KOTOpOro x(O) = y(O) . x ' ( O ) = y ' (O)
(ecna (x(0) , x ' (0 ) ) a E 1 • E 1 , TO C.CTeMa (1) .MeeT pemeHHe x( t ) , :~.ns KOTOpOro
( X( t ), X" ( t )) ~ E t • E t V t > O, wro cne~yeT .~ BK:IIOqeH~I~ f kE ~ C E 1' g t E ~
c E 1, k = 1, n.
HTax. B cnyqae HeBunOnHeH.S COOT~OmeH.S (2) CaCTeMa (1) . M e e t Heocu~X-
a~Ip3nomee OTHOC.Te~Ho E I pemeHHe x = x ( t ) , m m KOTOpOro (X(0), X'(0))
(E x E ) \ ( E 1 x E 1 ), wro IIpOTaBOpeq.T yc.rlOBltlO -- BCe pemeH.a x( t , O, x 1 , x 2)
CHCTeMI:,I ypaBHeaalt (1), /Llia KOTOpI:,IX (Xl, X2) ~ ~ x E ) \ ( E ~ X E l ) , HBJIJIIOTCJt
octm.a.rmpylonmm,I OTaOcrrre.rmrlO rto~npocTpaUcT~a E 1.
TaK.M 06pa3OM.. HeO6XO~.MOCTB COOTHOIIIeHH~I (2) ~na ocu:annzu:a~ COOTBeTCF-
BylOtti.X pcmeHrrl~ C.CTCM~ ypaBHeH.~ (I) B cnyqae 2IOKa.HbHO .m, lnllI.iieB~x OTO-
6pa~eH.fl f~, k = 1,'n, H g l, I = I, m, ~oKaaaHa.
PaccMoTp.M c.nyqa.q KOMnaKTah~X o'ro6pa~elm.q fk, k = I, n, . g t, I = I, m.
IlycTb COOT~omea, e (2) He BMHOJIHXeTC$1, T. e. BblIIOJIH$1eTCJt COOTHOIIIeHHe (6).
BOabMeM npon3~o:mmai~ BeKTOp y ~ E 2 [3 E 3 "~mcno r > 0, aTO6ra E l ~ B ( y , r) =
= 0 , H paccMoTpma ypamte.He (7), r~ae a ~ TaKoe aHCnO n3 ~ +\T, aTO nMeeT
MCCTO COOTnomeHHe (8). B~a6op TaKOrO ancna a BO3MOX~CH Ha OCHonamm COOT-
HomeHaa (6) a Ko~naxTaOCam OTO6pameHaia f i , k = 1, n, n g l, l = 1, m.
J~anee, KaK. B cnyqae noKa.m.Ho nnnmm~eBhtx OT06pa~emat f t , k = 1, n, a g t,
l = 1, m, paccMoap-M 6aHaxo~o npocTpaacrno X, orpaH.aeHHoe 3aMKHyToe ~r~nyK-
hoe MnomecT~O D ~cex qbyHKtmia X = X ( t ) ~ X, ~nX KOTOpraX x ( t ) ~ B ( y , r)
V t>_.a, n onepaTop ~ : X - . X , onpe~eneHmW.t pa~eacamoM (10). Ha ( 8 ) . (10)
BbITCKaCT, qTO ~[ D c D.
PaCCMOTpHM ~ y t m m . o
5 ( 0 = L ( s - t ) pk(s )ds +
,, k=l
r~e
m)
s ~ (t, +.o)rlr l=l
f
L = sup IIAf )L sup ~gt(x)l[~'.
k= l,n;l= l,m [xe B(y. r) xeB(y.r) J
~YI~ Ka)K/1oro y e D
<
sup ,, �9 <
,G[a.+.-)\r ~ dt l[
f. E +
a k=l
I$SN 0041-6053. YKp. ~tam. ~ p n . . 1999, m. 51, N ~ !
HEOBXO/],HMHE H ~OCTATOHHblE YC.FIOBH~ OCI.[HJ'I,/'LqHHH ... 105
+
m
~., qt(s)llgl(y(s))ll <
se(a,+**)~T /=1
f+o~ oo 1
< L ~ ! ~__lPt~(s'ds+ Z t q l (s, <0..
s~(a,+**)NT 1=1
YIOaTOMy MHO~S:eCTBO BCeX d,byHKl.I~[t Z = Z (t) e ~ D paBHOCTeneHHO HenpepusHO Ha
[a, + ~ ) . 1-IocKo.~Ky/XJ~a K a ~ O ~ dpyHK~XIH Z = z(t) e 9~D HMeeT MecTo OUeHKa
I lz( t ) -y l i <- ~(t) V t > a
lim ~( t ) = 0,
t-.~ +oo
To Ha OCHOBaHHH yC~OBH~ 2 TeOpeM~ 1 H o6o6meaHo~ TeOpeM~ Apue.~a [22, c. 110]
MHO~eCTBO ~(D OTHOClITe31bHO KOMIIaKTHO.
HTaK, OTO6pa)KeHrle ~ : D --> D KOMrlaKTHO. Cor~acHo TeopeMe IIIay/~epa o
neno/IBH~Oil TOqKe [23, C. 37] OTO6pa,'KeHlle ~ HMeeT a D Henoz~Ba~Hym TOqKy.
C.,qC./~OaaTe.JlbHO, ypaBHeHHe (7) HMeeT pelJ_ICHHe Z = Z (t) e D, KOTOpOe 6y~eT pettte-
rmeM n cacTeMr~ (1) nprx t > a.
Paccy~z~a.a ~a~ee TaK, KaK H B c~yqae JlOKaJIbHO .rlHI'IIIIHIAeBHX OTo6pa~enrI~ fk,
k = 1,---'n, n gt , i = 1, m, TaK~e npaxo~r~M K npoTaBope,~mo.
TaKaM 06pa3oM, HeO6XO~HMOC'n, cooTHomemla (2) ~ z z o c t m . q ~ a H COOTBeTCr-
By~omax petttearat CHCTeM~a ypaBHeHn~ (1) B c.ny,~ae KOMrmKTmaX OTo6pa.~erm~ f ~ ,
k= 1, n, H gt, l = 1, m, TaK:~e O6OCHOBaHa.
TeopeMa 1 ~oi<aaaHa.
3. KOMMeHTapHH KO nT0poMy H Tpe'rbeMy yC.rlOBH~iM TeOpeMl,i 1. Ha OFpa-
HHqeHH~ Ha E l BbrTeKaeT, qTO 6aHaXOBO npOCTpaHCT~O E MO~HO Ilpe~CTaBHTt, B
BrIRe E = El + Y, r~e Y ~ ~ae~cr~aTe~snoe O~HOMepHoe no~tnpocrpaHcrso npocT-
pancrea E. 1-IycT~, P l - npoeKTop Ha El napa~ae~sHO Y n P2 ~ n p o e z T o p Ha Y
napa~ze~u, no E~.
YKa)KeM yc.rIOBrt~l, o6ecHeqi4Balol.~He BblIIOYlHeHHe yc~qOSH~t 2 r~ 3 TeopeMH 1.
l ' Ipe~no~mcr~, aTO P2ft(x) = P2fk(P2x) H P2gt(x) = P2gt(P2 x) 1~.nz Bcex
X ~ E ; k = 1, n , l = 1, m . BO3~MeM npOa3BOJn, H ~ t Heny~eBo~i ~ e ~ e H T b e Y
paccMorprlM qbyaKtma
k(s) = IIP2A(sb)l l , k(s) = IIPeA(-sb) l l , k= 1, n,
Tt(s) = I IP2gt (sb) l l , ~ l ( s ) = IIP2gt(-sb)ll, l= 1, m ,
r/Ie s r ~-, +.
YC~OBHe 2 re0peMu 1 BunOy[HaeTca, ec.n_H, HanpHMep, paccMoTpeHnue ~yHK-
t~m at(S ), [~k(S) , y l ( s ) a S t ( s ) , k - 1,-"n, l - 1, m , ~IBYI21OTC~I B03paCTaIOI.ILtlMH
Ha ~ + .
l'IpH 3THX ~Ke IIp~[IIOJIO~K~HHSIX BHHO31H~/C ~c$I I! yC.rlOBHe 3 TP.op0MI:~ 1, 0CJIH
~OIIOJIHHTOJI~HO CXO,/~.g'l~$l H~CO~2TBOHHI~0 HH'I't~I padlta
ISSN 0041-6053. Yxp. ~am. ~..'vpu., 1999, m. 51, bl e I
106 B.E.C.rlIOCAPqYK
+*o +o~ +~a +o*
J dx f dx k=l,-"-n ~,dx ! dx l=l, rn"
ak(X)" ' ' 1 1
3a~e~M, qTO CX0~HM0CTIa HeC06CTBeHHOrO HHTerpa~Ia +~;0 ~* d x V(x)' r~e v ( x ) - -
Henpeph-~naJ~ ~oapacTazomaa Ha [X 0, + '~ ) qbyHKl.IH~l CO 3HaqeHH~IMH B (0 , +~'),
o6ecnewasaeT CXOZm~OCT~ qHCa'IOSOFO p ~ a
" • AXn-I
n = 1 V ( X n )
/~JI~I Ka~g~Oi~ Hey6r~BmOttlcll ffOCJIe/~OBaTe~H0CTH X0, X 1, X2, " " , TaK KaK
Xn X n
v(x) = v (x . )
Xn_l X - I
(sTO aa~e~amle y : q m m a e r nprmnax A6e.na H ~HHH [24, C. 290]).
4. IIprlYlOh'~eHHe TeOpeMl,! 1 K /1H~qbepeHtttla,rll,lll~ir,! ypal~HeHHaM. E c ~ 4
q l ( t ) = 0 , l = 1, m, TO ~ C~4CTeMC ypaBHSHHI~ (1) g(t, X)~ O. FIoaTo~4y a c~yqac
HenpepblBHOfl Ha ]R + X E tx~yHKI/2IH f ( t , X), KaK qacTrlht~ cJlyqalt TeOpeMhl 1, ~O~-
H0 no~yqnT~ yTaep~K~eH~e o6 0 C t t a ~ a t m a 0THOCHTe~O E l peuaean~ ~nqb-
(~ep~Hl.~Ia./l.bHOrO ypaBHeHa$1
TeopeMa 2. Flycmb:
n
d2x(t) + f ( t , x(t)) = O,
d t 2
t>O. (11)
1) f ( t , x ) = ~ pk( t ) fk(x) , eae n E 1~I, pie(t) , k = ~ ,n , - -nenpepb tomaena
k=l
IR § otpanu,~enn~e dpynmcuu co 3na~enu~lu o IR +, fk: E --> E , k = 1, n, ~ ro~i-
naKmnbte u~u noKanbno nunututleebte omo@a~enu~, an~ romop~x f k E i ~ E i aria
ocex k = 1, n u i = 1,3;
' 2) a.a.~ npouzoonbn~x ~ucea t o >_. 0 u ~mo~ecmoa { x l, x2} c E ypaonenue
(11) u~teem eauncm~ennoe onpeOenennoe na ~ + peuwnue x( t , t o, x ], x 2 ) , ~na
romopozo x ( t o, t o, x l, x2) = x 1, x ' ( to ' to' x l, x2) = x2;
3) inf eP(fk(z(s))) > O, k = I,"'~, Oa~acex z ~ O 2 U ~ 3 ;
s ~ t > 0 r
4) neco6cmoennbte unmezpanbl
f dq~(z(t))
o k= 1 , . ,
cxoS~mc~ O/t~ ocex z r 0 2 U @3.
1Ins moeo t~m05~ c9~ npouz~nbnbtx ( x l, x2) ~ (E x E ) \ ( E 1 x E ! ) pevaenue
x ( t, O, x I , x 2 ) ypaonenu.~ (11)5btnO OCgUnnupy10u~u~t omnocumenbno noanpocm-
pancmoa E ~ , neodxoOu~to u cgocmamo~no obmoanenue coomnomenua
ISSN 0041.6053. Y~p. ~am. ~.'vpu., 1999, m. 5 !, N ~ 1
HEOBXO~HMblE H s YCJIOBH$I OCHH,FIII$1UMH ... 107
ypaBneHnlt (1) T = 1~I
nprtHriMaeT Bri/~
n
t ~ pk( t )dt = + oo.
0 kffii
3 a ~ e ~ a . u e 1. B ycaoBrlJtX 3 . 4 TeOpeMH 2 blHOXeCTBa ~ k , k = 2, 3, MO.,'KHO
aaMeHHTS MHO>KeCTaaMH Wk, k = 2,'--~, r~e W k --MHOXeCTSO C I-oTo6pa~eHa~t
Z k : ]R + --~ E t , k = 2, 3, asia Kamaoro Ha KOTOpUX I 9 ( Z t ( t)) [ - - MOHOTOHHaZ He-
y6uBammaa Ha IR + dpynKttna, qacTnhat cayqa l t TeopeMra 2, Korzta n = 1 H OTO-
6pa~enHe f aOKa.nbHo armtuHUeBo, npHm~eH aBTOpOM a [25].
3 a ~ e ~ a n u e 2. I IycTs E = R n f ( t , X) = p ( t ) x 2m + l, r/Ie p ( t ) - - HenpepTaBHaa
neoTprmaTeasHaz Ha ]R + qbyHKana H m e l~I. TeopeMa 2 o6o6maeT H ycHanBaeT
TeopeMy ATKHHCOHa [10]: Bce p e t u e a n a ypaaHeHHa (11). rpoMe Hy~eBoro, OCttH~-
znpy~oT TOr/la H TOZb~0 Tor~a, Korea
+.e
~ tp(t)dt = +oo.
o
5. I I p u ~ o z ~ e n n e Teopenbi 1 X p a a n o c T n U n y p a a n e u n u n . I Iycx~ B CHCTeMe
rl p k ( t ) - - 0 , k = 1, n. To r~a f ( t , x ) = - 0 H CHCTeMa (1)
1-IO~TOMy
r~e
d2x(t) = O, tH ]R+\I~
dt 2
dx(n + O) _ dx (n - O) + g(n, x(n - 0)) = 0, n H l ~ , (12)
dt d t
x(n+O) = x(n-O) = x(.), .r I~I.
OqeBH~HO, qTO Ka.~K.~O0 13eIIIeHHe 3TOI~ CHCTOMM ypaBHeHHI~I KyCOqHO-J'IHHeI~HO,
dx(n+O) = x(n + I) - x(n) = Ax(n)
dt
d x ( n - O ) = x ( n ) - x ( , - l ) = A x ( n - 1), n r H .
dt
A2x(n- I) + g(n,x(n)) = O, nE I,I, (13)
A2x(n - I) = Ax(n) -Ax(n- I) = x(n+ I) - 2x(n) + x(n- 1).
CHCTeMa ypaBHeHHfl (1) TCCHO CB~I3aHa He TOJIbKO C RHqbqbepeHIBIaJII, HHMH ypaB-
HeHH.qMH, KaK BH,/~HO H3 rlyHKTa 4, HO H C pa3HOCTHHMH ypaBHeHH..qMH. Y q H T ~ a ~
CB.q3b MC~X~j' CHCTCMO~ ypaBHCHHI~ (12) H p a 3 H O ~ ypaBHCHHOM (13) H Teopes~, 1,
nonyqaeM yC~OBH.q OCH.HJIJLqld~HH OTHOCHTe, JIIaHO E l peUICHHfl ypaBHOHn..q (13).
O6OaHaqHM qCpr V k MHOXCCTBO OTO6pa,~r Zk: H - - ) E k, k = 2, 3, ~ s
Ka~/IOr'O H3 KoTopux A ltp(Z k (n)) ] -> 0 V n r N , H tlcpea x(n, x 1, x2) - - pemgHHe
ypaBHCHHH (13), y~oa~eTaopmomee ycs[OBHm, l
ISSN 0041-6053. Yrp. ~tam. u,.'vpH., 1999, m. 51, PO 1
108 B.E.C.rlIOCAPqYK
x(l,xl,x2) = x I, x(2, Xl,X 2) = x 2.
Teope~a 3. llycmb:
1) g(n,x) = ~ ql(n)gt(n), eDe m ~I, qt: l~I-->~+,l=I,m,--npouz-
I=I
aO~lbt.tble omo6paz, cenu~ c omnocumenbnO roamarmnb~tu ~mo~ecm~a~tu znat~enua,
g t : E --> E , l = 1, m , - - ro~marraabte unu no~a.abno nunutut4ev~e omoSpa.~enu.a, 8 ~
romopbvc g l E i ~ E i On.a~cex l = 1, m u i = 1,3;
2) i n f 9(g~(z(p))) > O, l = l"-,m, On~ecex z ~ V2[.JV3;
3) ~tuc.~oabte p.a~bt
,~= Acp(z(n-1)),
cxoOJ~rnca Ona ocex z ~ V2 [J e3.
] lna moeo ~mo6t, t Ona npouzoont, nbV: ( x 1, x2) ~ (E x E ) \ ( E 1 x E 1 ) peu tenue
x ( n, x 1 , x 2 ) ypa~uenu~ (13) 6btno oct~unnupytoulust omnocumentmo no~npocmpan-
cmsa E 1, neo6xoOu~to u Docmamotmo e~tno.~nenu~ coomnoutettua
~ n q l ( n ) = + ~ .
n = l I = l
3aTae~auue 3, A H a ~ o r TeopeMl~ 3 ~ J ~ pa3HOCTHhlX ypaBHeHHlt n 6aHaXOBOM
npocTpa~cTBe npHBe~teH B [26].
3 a ~ e ~ a n u e 4. B T e o p e ~ e 30TCyTCTByeT a l ia .nor yC~OBHa 2 T e o p e ~ , x 2. 3TOT
aHaJIOF, OqeBH~HO, HMeeT MeCTO, qTO BtJTeKaeT H3 rlpoCTOTbl CHCTeMrJ ypaBHe-
Hma (12) ( f ( t , x ) - - 0).
1. Cntocap~yr B. I0. OcttxJlattia poan'JlaKiB /IHd/~pellttiaJlbllHX piBH~ltlb 3 i~myzU,CUOIO /tielo n
6aHaxoBoMy npocTopi// KoHC'rpyKTaBni MeTO/m/toc.aitt~emta ~aHtl.~epemfia.at, uax pimlaw.: 36.
nayK. n p a ~ . -Kain : ht-T MaTeMaTRKa HAH YKpaimt, 1993. --C. 174 - 178.
Cntocap~yx B./O. ~[ocTa'mi yMonn OCl~HJImtii "lpaeK'ropitt iMny.qbctmx cttcTeM a ueqbiKCO~mmMH
MO~eHTaMa iMny•bcaoi ~tii//IHTer'pa.qbni nepeTaopemla Ta ix 3aerocyBaHna ~to xpaltonrlx ~ t a q :
36. HayK. IIpaub. -- KHiB: IH'T MaTeMaTHKrl HAH YKpaIHH, 1994. --C. 192 - 197.
C,atocap~tyr B. I0. Heo6xi~Hi i /IOCTaTHi yMOBH OCttHJIJlUi] poza'~13Kin tleJIiHiitHHX iMHyJIbCHHX
er~erern 3 My~bTan~iKa'rHmm po3~iJleHOlO npanolo qaCTHHOIO/1 YKp. MaT. ~Kypu. -- 1995. -- 47,
N ~ 3. - C. 381 - 389.
Xa,~anafi A., BeKcaep IL KaqecTBeHHaa 'reopua H~my~bcHrax erleTeM. - M.: Mnp, 1971. - 311 e.
bl~nxuu .~. 3 , Honro6 !0. C. TeopHa HeJmHetlutax uMny~bcHt~X CHCTeM. -- M.: HayKa, 1973. -
415 c.
3aeanumun C. T., CeceKun A. H., ]lposaen~o C. E. ~HHaMHqeCKI4e CHC'I'eMbl C tl~nyRbCnOtl
eTpy~crypott. - Ceep/l.qoaeK: Cpe/Jt.-Ypa.rt. tl~/~-~o, 1983. - 112 e.
r A. O. ~qbqbepeHvtrta~IbH~r ypaaHen~a c paapmBnoll npanoll qaca~,~o. - M.: HayKa,
1985. - 224 C.
Ca~o~laetnr A. M.. Flepecm~or H.A. j[JJ, I c ~ p e t l l U , la,,qbHld~ ypaBllC~HH$1 C HMIIyJII~IIMM BO3/J, ef/CT-
eHea~.- K~eB: Bama rex., 1987. - 2 8 8 e.
Sturm C. Sur les &luations diff~rentielles line, ares du second ordre//J . Math. Pura Appl. - 1836. -
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P. 643 - 647.
11. B /had L Oscillation and monotonicity theorems concerning non-linear differential equations of the
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2.
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5.
6.
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12. Belohorec 3. Oscilatorick6 riegnia istej neline~rnej diferencialnej rowniee druhtho radu // Mat.-
fys. 6asop. - 1961. - 11. - S. 250 - 255.
13. Kon~pam~o B. A. 0 Ko.~e6aeMoca~ pemenn~l ypanneaaa yen) + p(x)y = 0 /I Tp. MocK. MaT.
o-ma. - 1961. - 1 0 . - C . 419 -436 .
14. KuzypaDze 14. T. HeKoToptae caary~lapat~e Kpaenme 3a~aqH /~Ra OtblKIIOBeHHIdlX /]HCI~c~pOH-
uHa~bnt~X ypaBneHHtl. -T6H~MCM: T6H~HC. yH-T, 1975. -- 352 C.
15~ lllene,ao B.H. OcuMJi~alma petuen~tl /mqbqbepentma~bmax ypanHenHtl c OTKROHS~mtHMCa
apryMerrroM.- KHen: HayK./lyMKa, 1978.- 155 c.
16. lllapKotcKufi A. H., Maticmpen~o !0..11., Po~tanenKo E. I0. Pa3HOCTHr~e ypaBHenHa rt HX npH~O-
:~enHa. - KrteB: HayK. ~tyMga, 1986. - 280 e.
17. KueypaOze 14. T., qanmypu~ T. A. Ac~Mn'rOTHqeCKne CBOItCTBa pemeHn~l HeaaTOHOMH~X
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18. Cn~ocap~yx B. E. YcH.r]enHe reopeMu Kue~epa o ~y.nax petueHt~lt ypanneHHa y" + p(x)y = 0 //
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22. Kon~toeopoo A. H., ~o~tuu C. 8. :9.neMenT~a Teopm~ qbyHKtl~ rt qbyHKUr~OHa.qbHoro armnHaa. -
M.: HayKa, 1968. - 496 c.
23. HupenSepe.ll. JIeK~tHH no He~mHeltHoMy qbyHKU~O~m.m,~oMy aHa.naay. - M.: Map, 1977. - 2 3 2 e.
24. <Du.t-meneonbt4 F. M. Kypc l~xqbqbepem~rm.m, Horo a HHTerpa.r~,Horo Hctme.neHaa. T. 2. - M.:
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25. Cntocap~yx B. I0. OcttH.na~iia po3n'aaKia ,/1Hd/~peiltliadlblll4X piBHmm B 6aHaxoBoMy n p o e r o p i / /
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C. 269 - 275.
2 m
Cntocapt~yr B. !0. OcUHJl$1Ui.q poan'aaKin piaHmteaoro piBHmnl.q A x(n)+~tffi~pt(n)gt~(x(n))=O 26.
n 6aHaxoBoMy npoeropi//CncTeMn eno~uotii~hmx piBHaHb a nic~a]lie~: 36. HayK. npat~. - KHia:
IH-T MaTeMaTHKH HAH YKpahm, 1995. - C . 98 - 102.
I1o~yqeno 21.03.96,
noc.qe nopa6o~H -- 20.07.98
ISSN 0041-6053. Ys,'p. ~tam. ~.'ypn., 1999, m. 51, N g I
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| id | umjimathkievua-article-4586 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:01:42Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/66/c6f8deb865929c90386aa29565a0e866.pdf |
| spelling | umjimathkievua-article-45862020-03-18T21:09:14Z Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space Необходимые и достаточные условия осцилляции решений нелинейных дифференциальных уравнений с импульсным воздействием в банаховом пространстве Slyusarchuk, V. E. Слюсарчук, В. Е. Слюсарчук, В. Е. We obtain necessary and sufficient conditions for the oscillation of solutions of nonlinear second-order differential equations with pulse influence in a Banach space. Одержано необхідні й достатні умови осциляції розв'язків нелінійних диференціальних рівнянь другого порядку з імпульсною дією в банаховому просторі. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4586 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 98–109 Український математичний журнал; Том 51 № 1 (1999); 98–109 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4586/5876 https://umj.imath.kiev.ua/index.php/umj/article/view/4586/5877 Copyright (c) 1999 Slyusarchuk V. E. |
| spellingShingle | Slyusarchuk, V. E. Слюсарчук, В. Е. Слюсарчук, В. Е. Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space |
| title | Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space |
| title_alt | Необходимые и достаточные условия осцилляции решений нелинейных дифференциальных уравнений с импульсным воздействием в банаховом пространстве |
| title_full | Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space |
| title_fullStr | Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space |
| title_full_unstemmed | Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space |
| title_short | Necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space |
| title_sort | necessary and sufficient conditions for the oscillation of solutions of nonlinear differential equations with pulse influence in a banach space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4586 |
| work_keys_str_mv | AT slyusarchukve necessaryandsufficientconditionsfortheoscillationofsolutionsofnonlineardifferentialequationswithpulseinfluenceinabanachspace AT slûsarčukve necessaryandsufficientconditionsfortheoscillationofsolutionsofnonlineardifferentialequationswithpulseinfluenceinabanachspace AT slûsarčukve necessaryandsufficientconditionsfortheoscillationofsolutionsofnonlineardifferentialequationswithpulseinfluenceinabanachspace AT slyusarchukve neobhodimyeidostatočnyeusloviâoscillâciirešenijnelinejnyhdifferencialʹnyhuravnenijsimpulʹsnymvozdejstviemvbanahovomprostranstve AT slûsarčukve neobhodimyeidostatočnyeusloviâoscillâciirešenijnelinejnyhdifferencialʹnyhuravnenijsimpulʹsnymvozdejstviemvbanahovomprostranstve AT slûsarčukve neobhodimyeidostatočnyeusloviâoscillâciirešenijnelinejnyhdifferencialʹnyhuravnenijsimpulʹsnymvozdejstviemvbanahovomprostranstve |