Structure of a general solution of systems of nonlinear difference equations
We investigate the structure of a general solution of systems of nonlinear difference equations with continuous argument in a neighborhood of the state of equilibrium.
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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/4736 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510899289194496 |
|---|---|
| author | Pelyukh, G. P. Пелюх, Г. П. Пелюх, Г. П. |
| author_facet | Pelyukh, G. P. Пелюх, Г. П. Пелюх, Г. П. |
| author_sort | Pelyukh, G. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We investigate the structure of a general solution of systems of nonlinear difference equations with continuous argument in a neighborhood of the state of equilibrium. |
| first_indexed | 2026-03-24T03:04:19Z |
| format | Article |
| fulltext |
YIIK 517.962.2
rs IL/IedllOX (I/[iI-T Ma'r~MaTHKH HAH YKpaHIIbl, KrleB)
O C T P Y K T Y P E O B I I J ~ F O P E I I I E H H I I
C H C T E M H E J I H H E H H H X P A 3 H O C T H b I X Y P A B H E H H ~
We investigate the structure of general solution of systems of nonlinear difference equations with
continuous argument in a neighborhood of equilibrium.
~oc~lilt~KeuO erpyx'lTpy 3araJmnoro po3B'aaxy cHcl'e~ ue~filfifllmX pi311mteBaX piBnam, 3 nenepep-
BIIHM apryMelrroM B OKO.Ui CT'alIy pinuonal'rl.
PaccMo'rpHM CrICTeMy nenHHeltmax paaHOCTHbXX ypaBHenrx~
x(t + 1) = Ax(t) + f ( t , x(t)) , (1)
r~ae t ~ R = (_0% + 00), A ~ BetUecTBeHHaa, nOCTOZnnaa (n • n)-MepHaa HaTprxua,
f : R • Cn---> C n, x ( t ) ~ HeH3BeCTHaa KoMn~eKcHo3HaqHaZ BeKTOp-qbynKUna pa3-
Mepaocm n. By~eM rtccne/zoBar~, c'rpyKTypy MHO:~eCTBa HenpepbxBH~aX petueHaia
BTOI~I CHCTeMbI, Haxo~$II!IHXCJ~ B oKpeCTHOCTH ee TpHBHaJIbHOFO petueHr~a
( f ( t , 0) - 0). HpH paaJm,~max ripe~nono~KeHHaX STa 3a~a,aa aay~anacb MHOrrXMH Ma-
TeHaTriKa~rL B '~aCTHOCTn, /Z-ha tUrXpOKHX Ka~aCCOB TaKrXX ypaBneHnil nocTpoeHo
npe~cTanaenHe o6tuero nenpepraBHoro pemenHa [1 - 6]. Ylpo/zoa~Kaa ManaTee B [5,
6] Hccne/~OBaHrla, B HacTo~meit pa6oTe y~anocb nony 'mTb aHa.rlorHqHble pe3yn~,Ta-
T~ npa MeHee o6peMCHX4Te~qbHbXX npeRnoa~o:KeHHSX OTHOCHTeJ'IBHO BeKTop-c13yHKI.IHH
f ( t , X).
1. O 6 m e e pemeHHe caerem,~ He,anHeflm,[x pa~noeTH~X y p a n n e n n f l (1)
oKpeCTHOeTH ee TpHaaam,HOrO p e m e n H a x ( t ) m O. Pacc~oTpa~ CaCTeMy ypa~-
HeHH[t (1) npri c.ue~y~otuaX npe~anono:~eaHax:
1) co6cT~eHHrae ,mc.aa ~,i, i = 1 . . . . . n, MaTprxur,, A ~emecT~e~nbt a y/~o-
~eT~opz~OT COOTHOmeHHaM
~'i ~: ~'j, i ~:j, 0 < ]ki] < 1, i , j = 1 . . . . . n;
.l+r
2) Z,;~TC < 1, rzte k , = min{]~,il, i = 1 .... . n}, Z,* =max'{l~,il, i=l ..... n};
f ( t , x) = (A(t , x) ..... f~(t, x)) an:laeTca Henpep~Bnofi w or-
aenpep~anHO ~aqbdpepeHuHpye~o~ no x n p a t e R ,
3) BeKTOp-C~yHKIIH~
paHHqeHHO~ rio t,
I x l= max I x i l A b
l < i S n
f ( t , O) -- O,
Of(t, x)
4)
Of(t ,x) l Of( t ,x) :0fi(t , x)~;
Of<t' Y)l LIx- yl a, rae
ay I
Iof<t x)l of ct, x)
[ O . g [ l<_i<n
L = const>0, 0 < o~ < 1, (t,x),(t,y)~ D=R•
IIOCKO~n,Ky n eMny ycnonaa 1 c y m e e m y e T ueoco6aa aaMerm nepeMeHHHX x ( t ) =
= Cy(t ) , nprmo~ama.a crlereMy ypanx~eautt (1) K BH/Lv
y ( t + 1) = C- lACy(t ) + C-If(t, Cy(t)),
�9 F. rl. IIF~IOX. 1999
1368 I$5N 0041-605 3, YKp. ,uam. ,~. pu. . 1999, m. 51, N~- I O
O CTPYKTYPE OBIIIEFO PEILIEHHJt CHCTEM HEJIHHE}tHHX PA3HOCTHHX ... 1369
npl~,~eM C - I A C = diag(~L 1 . . . . . ~'n), TO B ~tanbHetameM 6y~ae~ c,mTaT~, qTO ca~a MaT-
pnaa A aMeeT Ta~O~ aUa, T. e. A = diag(~q ..... ~'n).
TeopeMa 1. I lycmb obmonna~omca ycnoou~ 1 - 4 . TozOa cyu4ecmoyem za~ena
nepe~tem~btx
y(t) = X(t) + y(t, x(t)), (2)
zOe oe~mop-dpynK~Cua T(t, x) = (7~(t, x) ..... 7n(t, x)) aonaemca nenpep~eno~ u
ozpanu~enno~ no t, nenpepbtono cgudpdpepen~upye~to~ no x o nexomopo~ o6nacmu
D . = R • [ - b . , b , ] , b , < b, u max.o~, ~mo 8tgnonnmomca coomnotueuua
r x)
7(t,O) --- 0, T x=0 - 0,
(3)
( t , x ) , ( t , y ) e D , ; [ r M l x - y l ~, M = c o n s t > 0 ,
npuooOaucaa cucmezty ypaonenua (1) ~ nune~no~t)' eucgy
y ( t + 1) = Ay(t). (4)
~yl~l/~OKa3aTe~IBCTBa TeopeM~a ROCTaTOqHO, OqeBH/~HO, rlOKa3aTb, qTO c y ~ e c T -
ByeT p e m e m m CtlCTeMBI ypaBneHHfl
~(t + 1, Ax + f ( t , x)) = AT(t, x) - f ( t , x), (5)
y~os~eTBopaIoIaee yKaaarm~aM B TeopeMe yc~or~riaM.
C no~0tubm COOTnOmenma
~'0(t, x) = 0,
(6)
Tm(t ,x) = A - I T m _ l ( t + l , A . x + f ( t , x ) ) + A - I f ( t , x ) , m = 1 ,2 . . . . .
onpe~aenrxM nocm~touaTenmmCTb BeKXOp-~yHKtmtt { ~/m(t, X)} n ~tOKa~eM, wro B
neKoTopO~ o6nacTa D . c D oHa paBno~epao CXO~anTCX K BeKTOp-qbyrmtlaH
$(t, X), KOTOpaa y/lonneTBopzeT yKa3aHmaM B weopeMe yc~aosrtma r~ Jm~aeTca
pememmM C~CTerar~ ypa~Henalt (5).
Crm,~a.rla rio~a~:eM, qTO npr~ ~ROCTaTO'~HO ~a~orn b. < b H Bcex m = 1, 2 . . . .
BBIIIOJIH~IOTC.q OReHKI4
17m(t,x) - T , , _ , ( t , x ) I <- M 0 0 " - * l x ] *+=,
(7) (xt, x) ~ m - I (t'x) < MoOm-l lx l" '
M0, (Mo, > X: X < 0 < 1,
( t , x ) , ( t , y ) e D . .
B ca rom acne, nocKom, zy
~/~(t, x) - T0(t, x) = A- t f ( t , x),
TO B Casly ycsmBa~ 3, 4 ouemcH (7) m~nosmxmrcx n p ~ m = 1. Paccyw,4xa~ no
anayKimn, npennosmxcaM cnpaBe~Ismuoc'r~ oueHOK (7) ns~X HeKOTOpOrO m _> 1 H
~OKa~KeM, qTO OHH c0xpaH,qIOTC~[ np~ nepexone O T m K m 4- I . ~[~O~CTBHTC./I/aH0,
I"IOCKOJIbKy IIpa t H R, [ x ] < b , h'MeeM (BblTeKaCT Ha ycJIOBl~fl 3, 4)
ISSN 0041-6053. Yxp. ~mm. ~'ypn, 1999. m, M, N e /0
1370 F. FI. rlEIIIOX
af(t, x) I Ax+f( t , x ) I s (X* +/5)lxl, A + ~ _< X* + 8 ,
r a e ~ = 8 ( b , ) ---> 0 npH b. -o O, TO, IIpHHHMa$1BO B H ~ C COOTnOmearta
Tm + 1( t, x) - Tin(t, x) = A-I[ I'm (t + 1, Ax + f(t, x))'-
- V . - 1( t + 1, A x + f(t, x)) ],
C~Ym(t,x) = ~ X A--l[~xm(t"~ 1' Ax dl" f(t, X)) --
af(t, x)~,
+ - - i f - x )
~Ym+l(t,x)
bx
i)Ym-l(t'x) (t + l'Ax + f(t'x)) ](
~axo~H~
[Ym + t( t, x) - Tm (t, x) I < ~.-.llT m (t + 1, Ax + f(t, x)) -
- 7,n_t(t + l, Ax + f(t ,x))] < 9~.lMoOm-llAx + f(t ,x)l l+= <
< MoOre - 1~,.1 (~*.+ 5)1 + ct[ x I l + =,
[ ~Tm+l(t'x)~x ~Tm(t'x)l<)(-*ll-~(t+'l'A'x+f(t'x))-~x -
~T'~x-I'(t't'I'Ax+ f(t'x))]IA + Tl-of(t'x)] <
< ~,-,1goOm-llAx+f(t, x)laO ~* +6) < MoOm-Ix-,I(~, * +8)1+=1x1%
[~[m+l(t,x) ~Ym+l(t,Y) < ~-,ll_.~(t+l, Ax+ f(t,x))( A Of(t,x)] .
a x - +
OTm ( Of(t, y , ) [ + <. I a:(,, y) I
Oy (t+l, Ay+ f(t,y)) A + Oy I Ox OY I -
< ~,.ll~x(t+l, Ax+f( t ,x))_~_~(t+l , Ay+f(t,y))ll A + Of(t,x__._..~)Ox I +
+ ~: '1~:"+1 '~" + ~'-*ll~f(t'x']ox Of(t,Oy y,<_
< ~ . - . t M l l A x + f ( t , x ) - A y - f ( t , y ) l a A + ~ + 7~. t
x L l x - y ~ t + k- .ILIx-y~t. < MlX-.t()~*+~)~+alx-yla + M~ LX;lO~*+
1 - 0
-1 * +cs)l+a +~)ba, l x - Y ~ + X-,ILIx-y~ t = M l )(,, (~, +
~l(~. ~ + 8) MoLM?Iba * + ~,-,tLMi I ]Ix-y[ a.
+ 1 . 0
Tax KaK ~:l~, *~+~ < 0, TO npn ROCTaTOqHO Ma.rlOM b. H B[OCTaTOqHO 60.rlblIIOM M1
IqblCA~M
ISSN 0041-6053. Yrp. J~am. ~.Tp,.. 1999. m. 51. N e 10
O CTPY.KTYPE OBIIIEFO PEUIEHH~I CHCTEM HEJIHHEFIHBIX PA3HOCTHHX ... 1371
~-~1(~* § * + X: tLM11 < 1, +
14, cJIe~OBaTea'IbHO, BIc,/I'IOJIH$IIOTCR COOTHOIIIeHH$1
IVm+l(t ,X) - - Vm(t,X)[ <- Moemlxl ~+~,
~7 m § ~ (t,. x )
~ m + l ( t , x ) .
Tx
~tm(~f'x).] <_ MOOm[x[ ct ,
Y) I <- M~lx - yl ~.
3y I
TeM caM~,M ~oKa3aHO, qTO oRenKa (7) BblIIOJIH~IIOTC~I npn ( t, x) , (t, y ) e D . 'H Bcex
m > l .
H3 (7) Henocpe~cTBeaHo BBITeKaeT, HTO IIOCJIe/~OBaTedlbHOC'rb BeKTOp~-c13Y~I~HI~I
7m(t, X), m = 0, 1 . . . . . onpe~e.neHHUX COOTHOtaeH14ZMn (6), paBnOMCpno CXOZmTCa
nprl (t, x) e D �9 K aeKTop-dpyHKt~an T(t, X), KOTOpaa B o6~acTa / ) . - Jt'B~aeTc ~
HenpepblBHOl~ I4 orpaH14qeHHOfi I10 t, HeHpepblBHO ~14qbqbepcHtI14pyeMol~t no x 14
y/~onneT~opaeT yCnOBHZM (3). Hepexoz~a B (6) K npejaeny rip14 m ---> ~ , Moacao
y6e/I14TbCZ, aTO BeKTop-qbyHKtma T(t, X) = lim,n~**'[m(t, X) aanaeTcs pemermeM
C14CTeMrJ ypaBHeHma (5). TeopeMa 1/loKasaHa.
3 a ~ e , a u u e 1. TeopeMa 1 cnpaBe/in14r~a 14 B cnyqae. KOr/~a cpe/I14 CO6CTBeHmaX
qHce.rl ~.i, i.= 1 , ; . . , n, HMelOTC~I KOMrl.rleKCHHe.
Hcnonb3yz TeOpeMy 1, MOmHO nony'a14Tb npencTaBneHHe nm6oro Henpepr.rBHOrO
petueH14J~ c14cTerm~ ~aBHeH14i~ (1) B oKpeCTHOCTH Tp14BrlaYmaOrO perneHrla. ~e~CTB14-
TenbHO, nOCKO.rlbKy o6mee nenpep~aBHoe pemeH14e C14CTeMr~ ypaBHeH14It (4) HMeeT
BHK
yi(t) = l~il%i(t), i = 1,. ,n , (8)
r/le coi(t) ~ nponaBOnbn~e 14enpepun14r~e qbyrrmlmk y~loBneT~opatom14e ycnoB14a~
oJi(t + 1)= sign~,itOi(t), i = 1 . . . . . n, TO (B~a-reKaeT 14a (2) 14 (3)) ar ia npo14aBOm,Horo
HenpepbmHoro petueH14a C14CTeM~a ypa~He1414~ (1), y~loaneTBopa~otuero np14 t > 0
yC~OBmO IX(t) I < b,, no~yqaeM
x(t) = y(t) + T-l(t, y(t)), (9)
r ze y( t )=(lXt l tO~l( t ) . . . . . [xn]t0}n(t)) 14 T- ' ( t , y ) - - HeKOTOpaa Henpep~Bnaa 14
orpaH14qenaarl no t, HenpepblB140 ~HqbqbepeauapyeMaa no y a HeKOTOpOIt o6nacT~
R • ], /~ < b,, BeKTOp-qbyHKR14~t, yZloBneT~opam~aa ycnoB14mn (3), a (Oi(t),
i = I . . . . . n ~ HeKOTOp~ae aenpep~ar~Hr~e qbynKnrm, y/IoaneTnopamttme ycnonnaM
~i ( t + 1)= sign ~,ir i = 1 . . . . . n.
2. HHnapHaHTH~ae MHOrOO6paaHa cHereM HedlHHe~HIAX paanocrHh[x ypan-
HeHHI~ H HX CBOI~CTBa. PaCCMOTp14M Tencpb cTlyqa~, Kor/~a HapylIIalOTC~l yCnOBa~l 1,
2. I/IMCHHO, Ilpe/~IOJI0~HM, qTO C 0 ~ I ' B C H H M e ~a~cza ~'i, i = 1 . . . . . n , MaTpHII/=I A
y~OBJIC'rBop~IOT yCJIOB14~IM:
1') X i ~ X j , i ~ j , i , j = 1 . . . . . n, 0 < I~,i[ < 1 < [~'Jl'
i = 1 . . . . . p , j = p + l . . . . . n;
2 ' ) X, IX*~+a< 1, r/~e ~,, = min{[~,i[ i = l ..... p} , ~* - max{IXil ,
i = l . . . . . p } .
ISSN 0041-6053. Yrp. ~tam. ~t'y. pu.o 1999, m. 51, IW I0
1372 F. 1"I. IIEd'llOX
~su /y / Io6c~a CHcTe~Iy ypaBHeHrltt (1) sanatue~ B BHRe
xCt + 1) = fix(t) + ~f(t, xCt), y(t)),
(10)
y(t + 1) = Ax(t) + f ( t , x(t), y(t)),
r~e ~ .=d iag{X 1 ..... X,}, A = d i a g { X . + 1 ..... Xn} , x = ( x I ..... xp). Y=(Yp+I .. . .
.... Yn), J? = (3] ..... fp), f = (fp+ 1 ..... fn), a rtpeartoaoa~nM, wro neKxop-qbynKtmn
.f(t, x, y), f ( t , x, y) y/xoBaeT~opamx yc~onnar~ 3, 4. KaK n npez~ne, nameia
KOHeqHOI~I llea-lblO ..qB.t'I~eTC~ rlOCTpOeHHe rlpe/ICTaBJleHrI~ o6mero nenpep~mnoro
pettteana cnc're~ra (10) B o~pec'rnocTn xprmaasn,Horo pemenna x( t ) = 0. y ( t ) = O.
I'IOCKO.rlt,Ky B ~TObl C.rlyqac TeopeMa 1, BOO6tlle FoBop$1, He HMCeT MeCTa, TO rlOSlyHHTI,
npe~tera~s~erme o6tttero Henpepra~noro pemerma mt/Ia (9) He npe/ic'raBsx~ieTcsi BO3-
~O~r O~HaKO B ~TO~ caly,aae cnpa~eRsmBa cs~e/xy~otua~ TeopeMa, KOWOpa~/IaeT
~O3MOX<HOCT~ cytuec'I~eHno yrIpoernT~ rtccsm/iosarme cr~ereM~a ypanrieHrxlt (1 0).
Teope/~a 2. Hycm~ ot, monnsuomc~ yc/toou~ 1', 2' , 3, 4. Tozcga cyu4ecmoyem
3a~tena nepe~tennbtx
x(t) = 2(t), y(t) = y(t) + V(t, .~(t)) (1 1)
malcaai, ~mo (n - p)-~tepna.~ oelcmop-~yntcl~U.~ ~(t, ~(t)) ~on~emca~ nenpept, tonoa u
oepanu~enno~ no t, nenpep~ono Oudpqbepen,upye~to~ no ~ o netcomopoa o6nac-
mu D , = R • [ - b , , b, ], b. < b, yOo~nemoop~em ycnoou~zt
0 ~(t, 0) --- 0,
(12)
M = const > 0,
cucmezta ypaonenua (10) uzteem
1 ~3v(t'~) 3~t3(-~-_'Y)l < M I ~ - Y l
3~
(t, ~), (t, y) ~ IX, u 8 noobzx nepe~lemttax ~, y
ouO
~(t + 1) = . ~ ( t ) + f ( t , ~(t), y(t)),
z
y(t + 1) = Ay(t) + a~(t, ~(t), y(t)),
(13)
](t, 3, y), :(t, y) toe oerrnop-cllynxt4uu ycgoonemoop~1om ycnoouatzt 3 . 4 u
=o.
3a~e,~allue 2. YCSlOBrle a~(t, ~, 0) =-- 0 o31taqaex, qTO MHoroo6pa3rle y = lit(t, x)
~IBM_~TCH JIOK831bH0 HHBapHaHTHHM OTHOCHTeMbHO OTO6pa~eHH~
+ ?(t,x,y),
�9 -
y---> Ay + f ( t , x, y),
t - - > t + l .
/ff(otca.~meau~cmoo m e o p e ~ 2. Bbmo~naa s (10) aaMeHy nepeMeHH~x (11),
noay,4acM (13), r.~e
] ( t , .~(t), y c t ) ) = ?(t, gCt), YCt) + ~(t , ~(0)) ,
ISSN 0041-6053. Yxp. ~utm. ~. pn., 1999, m. 51, N e 10
O CTPYKTYPE OBI.L[EFO PEILIEHH,q CHCTEM HEJIHHEI~HblX PA3HOCTHHX ... 1373
a~(t, g(t), y(t)) = ) ( t , ,~(t), y(t) + V(t, ~(t))) + AV(t, ~(t)) -
- V(t + I, ~ ( t ) + jT(t, ~(t), yet) + V(t, ~(t)))).
OTclo/Ia HelIOCpe/~CTBeHHO BSITeKaeT, qTO/~JI.q/~OKa3aTCJlbCq-Ba TeOpeMhI ~OCTaTOqHO
/~oKa3aT,o CytlIeCTBOBaHIte pemenna CHCTeMhl OpyHKI..U, IOHa.rlbHHX ypaBHeHnl~l
V ( / + 1, . /~ + f ( t , .~, V(t, .~))) = AV(t, ~) § : ( t , ~, V(t,-~)) (14)
c yKa3aI-IHblMH B ToopoMo CBO]~C"rBaMH.
C rlOMOIIlblO COOTHOIIIeHHIt
V0(t, .~) = 0,
(15)
Vm(/,~') = A - l ~ l m _ l ( t ~ r l , ~ + } ( t , X , Vm_l( t ,~ ' ) ) ) -
-- ~ - l j~ ( t , X, ~/m_,(t, X')), m = 1,2 . . . . .
orIpe~e~ma nocne~ouaTesU, HOCT~ BeKTop-qbynKmIlt { ~m(t, ~)} H/~oKa~eM, wro
aeKOTOpO~ o6.nacTa D , orIa paunoraepno CXO~aTCJ~ K r~eKTop-dpyHKImH g ( t , .~),
KOTOpa.q y/IoB.rleTBOp~qeT yKaaaHma~ B TeopeMe yCJ!0BH.qM H JtB.rlJteTc.q petueH~IeM
C~CTema ypaunenmt (14).
Cgaqa.rla nOKa2KeM, qTO IlpH /~OCTaTONHO Ma./IhlX b. < b ri gcex m = 0, 1, ...
BbIHOJIH~IIOTC$I OHeHKH
]Vm(t,.~) -~ /Fm_l(t, yc ) A N0e~'-'[~l '+=, (16)
~/m(t ,x) 3Vm-l(t,x)[ < g,o~'-'l~l = (17) ~ /9,~ - '
~]lm(t,.~" ) ~]lm(t,.~" ) < NI~'-~�9 ~,
~.~ ~.~ -
r~te
(18)
(t, ~) . (t, .~'). (t, ~") ~ D . . N 0, N 1, N - - neKoTopr~e aono~rrre.qbH~e nOCTO-
~*=max{l~l, i = l . . . . . p } , HHHI~Ie, ~-~l ~,.I + ct < 0 o < 1 , 0 ~ < 0 1 < 1 , npH~ieta
X, = min{[ki] , i = p + l ..... n}.
B caMoM/lcdle, IIOCKOJIbKy
$1(t, ~) = - A - l : ( t , ~, 0),,
TO B cany yCJIOBHit 3, 40I.[eHKH (16) - (18) BSIIIO.rlH.qlOTC.q l'IpH /rl = 1. Hpe~tnono-
2r, nM, qTO ottemcH (16) - (18) aoKaaaH~ yme ~na HeKOrOporo m > 1, a no~a>KeM,
wro OHa coxpaHa~OTCa np.n nepexo/Ie OTm K m + 1.
jIe~tCTBaTeJU, HO, 8 criny yCaOBZJl 1", 3, 4 rt (16) - (18) npn ~XOCTaTOqHO Man~x
]x[<<.b.<b ( b , < ! ) I~MeeM
I:(t,x',y') - f ( t , x " , y " ) I <- Lb~,(Ix " - x " l + l y ' - y ' l ) ,
I: - 1 (t, x' , y ' ) - " " - f ( t , x , y " ) < Lba,(Ix" - x " l + l y ' - Y I ) ,
r~e ( t , x , y ' ) , " " D , ; �9 ( t .x ,y ) r
No xl,+~
]Wm(t,,~)]'< I 0 , O I < b,
ISSN 0041.6053. YKp. ~wm. ~. pu, 1999. m. 51, N'-' 10
1374 F. rl. HEJIIOX
I ~ l m ( t , ~ ' ) - W m ( t , , " ) l < Nba, i ~ ' - ~"l ,
I~1.1,'1.1,,I-< b.; (19)
!x~ + :(,,~,v.(,,~))l-< (~'+~)l~l-< b.
8 = 8(b.)--+0 npn b , - - + 0 .
Tortla Henocpe~crnemm Ha (15), (19) nbrrexaer
IYm+,(t . -~) - ~m(t.-~)l < I~-l[l~m(t+l.~c+]~(t.'.u
-- l[Im(t + 1,.7~ + .f(t, ~, ~/m_ l(t, ")))[ + [~-1 [[~lm( t + 1 ,~ + f(t,~, Wm_t(t,~))) -
- ~l/m_ l(t-I-1,/~-I- j~(t, ' , ~[Im_ l(t, .~)))1-b IA -111~, ~l/m(f' ")) --
-- } ( t , ' , Vm_i(t, 2))1 < ~. . 'Nl]( t ,~ , Vm(t, ~)) - f( t , E, ~l/m_l(t, '))[ +
4" ~,,IN0o~":II.7~ 4- ?(,.,. I[Im_l(t.,))ll*lz q - ~...l[:(t.,. IVm(,. ' ) ) -
- } ( , , , , v.,_ic,, ~)1] -- ~.: 'NL~NoOa'-'I~I '+~ +
+ ~..'NoO~-'(X* +5)'+~'1,1,+~, + ~.;' Lb,'XNoO~'-'l,I '+c, <
lIoc,o.,'n,Xy ~ ' r ' § TO 0 0 = ~ ' ( r + 6)'+a+~'LNb~, +~'Lb~, <1 npH
/I;OCTaTOqHO M ,aarloM b. ~ I4, caleaosaTealbHo, otte14~:a (16) coxpaHaeTCa npa nepexo~te
o r m ~ m + l .
IIprmm, taa BO BH14MaHHe (19) H COOTHOmeH14e
g
- ~ '
HaXO/Iilbt
-- ~;.( , '}" 1 . /~ ~ ? ( t . , . , r a . l ( t . ~ ) ) ) l + 1 - ~ (t ae 1 . /~ q " ](t, ' . ,re_l(/ . '))) --
ISSN 0041.6053. Yxp. Jnam, .r, ff. pn,, 1999. m. 51, bl e 10
O CTPYKTYPE OBIIIEFO PEIIIEHH,q CHCTEM HF_J'IHHEI~HbIX PA3HOCTHbIX ... 1375
• (t, ~, ~/m(t, ~)) - ~ , ,
~u .~) ~/m 3~ (t, ~) + []0Wm . - l(t, .~) [ 0"-~0~ (t, ~, • ~lm(t, ~) ) -
>1 E( _~f(t,~.,lgm_lCt,~. ) _< ~1 Nl].(t,~,Vm(t.~)).](t,~,~,m_,Ct,~))l +
+ NI o?- 'IL~ + ?(, ,z Vm_,(', ~))l ~ )(~* + L(I~I + [ ~gm(t, .~)l)a +
+ L([~] + Ill/me/, ~)[)~N[xl ~ ) + N I ~ + fi(t, ~, Vm_,(t, ~))l~ •
x (L l~ /m( t .~ ) -u " + L(I~I + lYre(t, ~)I)"N~0~'-'I~I" +
+ gl~l~'LlVm(t,~)-Vm_,(t,~)l=)] + ~'-,l~gm(t,.~)--~Itm_l(t,.~)la+
+ ~.%l(z([~l + IVm(t, ~)l)=g~0?-~l~l= + gl~ZlVm(t,~) -
-Vm_l(t,~)l=) < ~.E~I(NL~NoO'g-'I~II+'~+ N, 0~'-'(~.* +8)=1~1=)•
+ g(~.* + 8)"1~1" L.a,r ~/~ ~0 .~'ct(l+a) + L I~1 + 1- 0 o ) x
1 ] ~- ' r m~I:l~(m- 1) ~ r162 x g~o~n-~l~l" + NLl,~l"N~Ocd(m't)l~l "(~+") + '~, '-,,o,,o - ~ +
ISSN 0041-60&L Y~g. ~am. s~y, gn., 1999. m. 5! . N ~ 10
1376 F, rl. I'IEJIIOX
~o,j A,
+ b. f -0o-" )
( No hi*o: t,o.) b ~ ' + L b . + l _ 0 o . ) +
"l" tN~SN?i(O~]m-lhcx(l+oO~Ol) - , ][.~[cto
z ,- , l+a
TaKma o6pa3oM, n0CKO.abKy ~.'~ Z < 00 < 0~ t < 0 l < I, TO a3 n o c n e ~ n e r o C00"r-
HotUeHH.a rtpH )IOCTaT0qBO Ma~oM b , nosxyqaeM
1 3~ltm+l(t, X) 3~m(t, 2)t < Nl0]nl2,cx '
32 32 -
T. r otleHKa (17) crrpaeea.nnBa npn In § 1.
AHanOFHqHO HaXO~HM
1 3v , .~ t , 2") 3v , .+! t ,2") I -
- - - - - '" + " ~ 2 ( t , . ~ ' , ~/m(t, 2")) +
+ ~---~[ (t, 2", Vm(t, 2'))3V~2")l +
--t~ 3 7 --t
~x 2", Vm(t, 2")) + 32 (t, 2", Vm(t, 2")) -
+~'Oatt%(~' ) (t, 2",m(t,~'))-~(t, 2"',Vra(t. 2"'))§
I$$N 0041-6053. Y~p. ~am. ~.'~pu., 1999. m. 51. N e 10
O CTPYKTYPE OBLUEFO PEULIEHH~I CHCTEM HF.3"IHHE~tHblX PA3HOCTHHX ... 1377
I- )
-- ~pp ~sp ~r ~pp
< k ' N ( f l 2 ' - 2"1 + Lba,(I 2 " - 2"1 + Nba, 12 " - 2 " 1 ) ) = ( f +
+ Z(12'l + gb,~12'l) = + Z(12"l + gb~ '~ ) +
+ + L . 02"l + Nb. 12"0) ('(12 " - 2"1 +
+ gb'~,l 2 ' - 2"'1) = + gl2'l~Z(I 2' - 2"1 + gb'~,l 2 ' - 2"1) = +
§ L(I2 , [ § Nba. 12"'t)aNl2 " - 2"[ a ) + ~ . 'L ( [2 ' - 2"1 +
+ gb'~,12" 2"l) '~ + ~.-,'( U12"l'~Z(12 " - 2"1 + gb~
+ L(12"I+ Nb, lx II Hl~ - 2 ' I ~ I -~ HI ,2"- 2" ' I~Z ! ~.* +
+ Lb~(1 + Nba,))a(~, + L(b, + Nbl,+a)a (1 + N.b:l )
+ (~,*b, + Lb~,(b, + Ob', + (x))•(L(1 + Obj,) a + LNb,a(l +Nb,a) = §
OTCIO~a c.,'le~yeT, qTO lIpH ]IOCTaTOHHO MaJIOM b , H /~0CTaTOqttO 6OJII)IIIOM N
HM0eM
T. e. ot~enKa (18) mano.anaeTca npH aaMene m , Ha m + I. TamxM o6paaoM, npa
~OCTaTOqHO MaJIOM b , • ]~0CTaTOtlHO 60JlbIIIOM N OHeHKH (16) - (18) cIIpaBe,/~J'IHBH
npn Bcex m > 0.
Henocpe~tcrsenno Ha (16) - (18) ~bIweKaer, 'aT0 nocne/~osare.q~,rIOCT~ BeKT0p-
cl~yHKIMIt { Vm(t, 2) }, onpezenenH~x C00TH0mOHHJtMH (15), paBHOMepno cxo/Ixrrca
npn 121 < b , K BeKrop-qbynK~rlrl ~(t , 2), KOT0paJt y~aosJ~eTBopaer yCJIOBaSt~,
yzaaanmaM S TeopeMe 2. l-lepexo~a ~ (15) K npe~ezy npn m --> **, n e r z o y6e-
anTiCS, wro SeKT0p-qbynKtma ~(t, 2) = limm~**~m(t, 2) aBn~ZeTCZ petuerme~
CaCTe~a ypanaermlt (14). TeopeMaaoKaaaHa.
B cn.rt~' Teope~ta 2 nccneaosam~e oKpecT~OCTn TpuBaa.rmnoro pememt,a CHCTeMta
ypaBHCHHI~I (10) C~O~HTC,~ K HCC3Ie/~OBaHHIO 0KpeCTH0CTH TpH~na~IbHOrO pemeaH~
CnCTema (13). HOKa~Ke~ cria~iana, ,aTO ecnH (2(0, y(t)) ~ neKoTopoe nenpep~amIoe
npn t _> 0 pemerme CHCTeM~ ypa~Henalt (13), Haxo/xameeca ,~ oKpeCTnOem ee TprI-
sHa.m,aoro pemem~a (0, 0), TO y(t) ~ 0. Heilcr~rrre,rmr~o, nycT~ ~rro ne raK, T. e.
r~eeTcz r~eKoTopoe nenpepraBnoe npr~ t -> 0 petucm~e (2(0, y(t)) cr~cTema (13),
Haxo~[M/~r162 B HCKOTOpO~ OKpeCTHOCTH Cr TpHBHa31bHOFO p~LIIeHHYI H TaKo(~, HTO
y(t) ~ 0. Tor/Ia s cnny ycnosata 1', 3, 4 r~ ~(t, 2, 0) = 0 redeem
ISSN 0041-6053. YKp. ,~vzm. .a~. pttL 1999, m. 51, N ~ 10
1378 r . H . HEJIIOX
[~(t,~, Y)I <- elYl,
r a e e ~ 0 n p a I~1, lYl ~ 0 , a r ~ a ( 1 3 ) n o a y ~ a e ~ t
lY(t+l)l > - (~ . . - e )Y( t ) l .
I IocKoJmKy ~,, > I, TO ~v, -- I~ > I n p a ~OCTaTOqHO M~LHOM
COOTHOIIICHH~! BlxrreKaeT
H H3 nocne~nero
lY(t+m)l >- , - e ly(t)l, m >_. ~,
r . e . I y ( t ) [ --* ** n p n t ~ 0-. I loay,~caI~oe n p o r r m o p c , m e ~oKazuaaeT , a r o I y ( t ) [ --
- 0. C n e ~ t o s a v e n b a o , ~ z n o c T p o e m i a Bcex l t e n p e p u s m a x n p n t > 0 p e m e m i f l
CHCTCMKI ypaBHCl-II41~ (13) , naxo / I zmnxcJ~ a OKpeBTHOCTH ee TprlBHa.rlbHOr'O p e m e r m a ,
a o e r a r o ' m o nOCTpOWI~ a c e nenpep~asa r~e r lpn t > 0 p e m e m t a c r t c r e ~ m ypaBHemtl~
z(t + ~) = ~(t) +/(t, z(t), 0), (20)
naxoAattracca s 0KpCCTHOCTrl CC Tprmrlani, Horo pcmerma ~(t) = 0. TaK KaK An~
cHcrcma (20) BunonnS~OTCa ace yc.uomia TC0pCMU I, TO ec odmee ncnpepusHoe nprl
t >_ 0 pemexrm rlMeeT Blt~ (9), r~ae n = p. HprlHrlMaJl m) aHrIMallI4e (I I), nonyqaeM,
qTO ecm~ (~(t), y(t)) ~ nexovopoe rlenpeplasrloe npn t -> 0 pemeHrte cncveMm
ypasHeHHfl (I0), Haxo~amceca a ]~OCTaTOqH0 MaJl01~ 0KpCCTHOCTH CC TpHBHa.rlBHOFO
petuemm (0, 0), TO X(t) -- z(t)+y-t(t,z(t)), y(t) = V(t,z(t)+y-l(t,z(t))), r~e
0 II' ) z(t) = Xt I%l(t) ..... Xp mp(t ) ~ mi( t ) , i = 1 . . . . . p , ~ H e K o T o p ~ e H e n p e p ~ B -
U u e qbyHztmH, y ~ I o s ~ e T ~ p ~ m H e y r ~ i ( t + 1) - s ign ~.io~i(t), i = 1 . . . . . p.
1. Birkhoff G,D., Trjitzinsky W.Z Analitic theory of singular difference equa t ions / /Ac ta math. -
1932.- 6 0 . - P. 1 - 8 9 .
2. Harris Jr., W.A,, Sibuya Y. General solution of nonlinear difference equa t ions / /Trans . Amer.
Math. Soc. - 1965. - 115. - P. 62 - 75.
3. Takano By. K. General solution of a nonlinear difference equations of Briot - Bouquet type / /
Funkc. ekvacioj.- 1971.- 13, N o 3 . - P . 179 - 198.
'4. Takano By. K. Solution containing arbitrary periodic functions of systems of nonlinear difference
equations//Ibid. - 1973. -13, N 9 2. - P. 137 - 164.
5. Ilemox I".17. 0 c'rpytcrype nenpep~mn~tx pemen~fl olutovo zJtacca ne.~tmie~t~z~-ax paanocTnt,lx
y p a a n e n m ; / / J ~ p e t m , ypaanem, m. - 1994. - 30, N ~ 6. - C. 1083 - 1085.
6. llemox F.fl. I'IpeltcTa~mtme pemenvtfl pa311OCTlndx ypammtml~ c IlenpepbiBIlblM apryMetn'om //
TaM ,,Ke. - - 1 9 9 6 . - - 3 2 , I ~ 2 . - C . 3 0 4 - 3 1 2 .
Flo.rtyqetlo 28.01.99
ISSN 0041-6053. Yrp. ~mm. ~.'vpu., 1999. m. 51, N ~ 10
|
| id | umjimathkievua-article-4736 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:19Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/05/d7d253feeafde1278f910c2087d58e05.pdf |
| spelling | umjimathkievua-article-47362020-03-18T21:12:54Z Structure of a general solution of systems of nonlinear difference equations О структуре общего решения систем нелинейных разностных уравнений Pelyukh, G. P. Пелюх, Г. П. Пелюх, Г. П. We investigate the structure of a general solution of systems of nonlinear difference equations with continuous argument in a neighborhood of the state of equilibrium. Досліджено структуру загального розв'язку систем нелінійних різницевих рівнянь з неперервним аргументом в околі стану рівноваги. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4736 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1368–1378 Український математичний журнал; Том 51 № 10 (1999); 1368–1378 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4736/6173 https://umj.imath.kiev.ua/index.php/umj/article/view/4736/6174 Copyright (c) 1999 Pelyukh G. P. |
| spellingShingle | Pelyukh, G. P. Пелюх, Г. П. Пелюх, Г. П. Structure of a general solution of systems of nonlinear difference equations |
| title | Structure of a general solution of systems of nonlinear difference equations |
| title_alt | О структуре общего решения систем нелинейных разностных уравнений |
| title_full | Structure of a general solution of systems of nonlinear difference equations |
| title_fullStr | Structure of a general solution of systems of nonlinear difference equations |
| title_full_unstemmed | Structure of a general solution of systems of nonlinear difference equations |
| title_short | Structure of a general solution of systems of nonlinear difference equations |
| title_sort | structure of a general solution of systems of nonlinear difference equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4736 |
| work_keys_str_mv | AT pelyukhgp structureofageneralsolutionofsystemsofnonlineardifferenceequations AT pelûhgp structureofageneralsolutionofsystemsofnonlineardifferenceequations AT pelûhgp structureofageneralsolutionofsystemsofnonlineardifferenceequations AT pelyukhgp ostruktureobŝegorešeniâsistemnelinejnyhraznostnyhuravnenij AT pelûhgp ostruktureobŝegorešeniâsistemnelinejnyhraznostnyhuravnenij AT pelûhgp ostruktureobŝegorešeniâsistemnelinejnyhraznostnyhuravnenij |