On the smoothness of conjugation of circle diffeomorphisms with rigid rotations
We consider the operator which is a variable hysteron that describes, according to the Krasnosel'skii -Pokrovskii scheme, a nonstationary hysteresis nonlinearity with characteristics varying under external influences. We obtain sufficient conditions under which this operator is defined for...
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| Date: | 2008 |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509199971123200 |
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| author | Borzdyko, V. I. Борздыко, В. И. Борздыко, В. И. |
| author_facet | Borzdyko, V. I. Борздыко, В. И. Борздыко, В. И. |
| author_sort | Borzdyko, V. I. |
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| datestamp_date | 2020-03-18T19:47:10Z |
| description | We consider the operator which is a variable hysteron that describes, according to the Krasnosel'skii -Pokrovskii scheme, a nonstationary hysteresis nonlinearity with characteristics varying under external influences.
We obtain sufficient conditions under which this operator is defined for inputs from the class of functions H1[t0, T]
that satisfy the Lipschitz condition on the interval [t0, T]. |
| first_indexed | 2026-03-24T02:37:19Z |
| format | Article |
| fulltext |
UDK 517.9 + 539.214
V. Y. Borzd¥ko (Yn-t matematyky AN Respublyky TadΩykystan, Dußanbe)
HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY
We consider the operator which is a variable hysteron that describes, according to the Krasnosel’skii –
Pokrovskii scheme, a nonstationary hysteresis nonlinearity with characteristics varying under external
influences. We obtain sufficient conditions under which this operator is defined for inputs from the class
of functions H t T1
0 ,[ ] that satisfy the Lipschitz condition on the interval t T0 ,[ ].
Otrymano dostatni umovy, za qkyx operator – zminnyj histeron, wo opysu[ za sxemog Krasno-
sel\s\koho – Pokrovs\koho nestacionarnu histerezysnu nelinijnist\, xarakterystyky qko] zmi-
nggt\sq pid vplyvom zovnißnix syl, [ vyznaçenym dlq vxodiv iz klasu funkcij H t T1
0 ,[ ] , wo
zadovol\nqgt\ na vidrizku t T0 ,[ ] umovu Lipßycq.
Mnohye zadaçy v estestvenn¥x naukax — fyzyke, mexanyke, πkolohyy — pryvo-
dqt k rassmotrenyg dyfferencyal\n¥x uravnenyj s osoboho roda sloΩn¥my
nelynejnostqmy, naz¥vaem¥my hysterezysn¥my. V [1, c. 99] predloΩena sxema
dlq opysanyq nelynejn¥x system s hysterezysom, xarakterystyky kotor¥x me-
nqgtsq so vremenem, y postavlena problema v¥delenyq klassov vxodov, dlq ko-
tor¥x πta sxema realyzuetsq. V dannoj stat\e formulyrugtsq uslovyq, pry
kotor¥x πta sxema Krasnosel\skoho – Pokrovskoho osuwestvyma dlq lgboho
vxoda, udovletvorqgweho uslovyg Lypßyca. Matematyçeskaq teoryq system s
hysterezysom, zaloΩennaq v [1], osnovana na obwej ydeolohyy teoryy system
[2]. V nej kaΩdaq hysterezysnaq nelynejnost\ traktuetsq so svoym prostran-
stvom sostoqnyj, operatoramy-hysteronamy „vxod-v¥xod”. Ona oxvat¥vaet raz-
rabotann¥e ranee fenomenolohyçeskye modely. V p.B1 pryveden¥ neobxodym¥e
v dal\nejßem ponqtyq [1].
1. Statyçeskyj hysteron. Dlq opysanyq konkretnoho hysterona W opre-
delqgt oblast\ eho vozmoΩn¥x sostoqnyj Ω( )W , raspoloΩennug na ploskos-
ty Π = { }u x, , y odnoznaçn¥e operator¥
x t( ) = W t u x u t( , , ) ( )0 0 0 , t t≥ 0 , (1.1)
sopostavlqgwye dopustym¥m vxodam u t( ) yz nekotoroho klassa funkcyj v¥-
xod¥ x t( ), esly preobrazovatel\-hysteron naxodylsq v naçal\nom sostoqnyy
u x0 0,{ } ∈B Ω( )W . Ymeet mesto poluhruppovoe toΩdestvo
W t u x u t( , , ) ( )0 0 0 = W t u t W t u x u t u t1 1 0 0 0 1, ( ), ( , , ) ( ) ( )[ ] . (1.2)
Oblast\ vozmoΩn¥x sostoqnyj Ω( )W hysterona xarakteryzuetsq rqdom
svojstv [1, c. 25]. V Ω( )W v¥delen¥ dve kryv¥e: Φl y Φr . Oblast\ Ω0( )W =
= Ω Φ Φ( )W l r∪( ) rassloena v systemu neperesekagwyxsq hrafykov neprer¥v-
n¥x funkcyj Π( , )u M , M W∈Ω0( ) — nekotoraq toçka na Π( , )u M . Po kryv¥m
Π( , )u M y Φl , Φr po opredelennomu pravylu vvodytsq systema oprede-
lqgwyx kryv¥x T u M( , ), – ∞ < u < + ∞, M ∈ B Ω( )W — proyzvol\naq toçka [1,
c. 27]. Zatem daetsq opysanye operatorov (1.1) dlq hysterona W. Pust\ naçal\-
noe sostoqnye hysterona zadano toçkoj M0 = u x0 0,{ } = u t x( ),0 0{ } ∈ B Ω( )W y
vxod u t( ) monotonen y neprer¥ven. Tohda vxod v (1.1) opredelqetsq ravenstvom
x t( ) = W t u x u t( , , ) ( )0 0 0 = T u t M( ), 0[ ], t ≥ t0. (1.3)
© V. Y. BORZDÁKO, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3 295
296 V. Y. BORZDÁKO
Operator-hysteron moΩno opredelyt\ na lgbom kusoçno-monotonnom nepre-
r¥vnom vxode u t( ) , razbyv oblast\ opredelenyq funkcyy u t( ) na promeΩutky
ee monotonnosty y vospol\zovavßys\ poluhruppov¥m toΩdestvom (1.2). Zatem
okaz¥vaetsq vozmoΩn¥m opredelenye hysterona W na lgbom neprer¥vnom vxo-
de u t( ) s pomow\g predel\noho perexoda na osnovanyy svojstva vybrokor-
rektnosty hysterona [1, c. 28]. Vybrokorrektnost\ oznaçaet, çto yz u x0 0,{ } ∈
∈B Ω( )W y ravnomernoj na kaΩdom koneçnom promeΩutke t t0 1,[ ] sxodymosty
neprer¥vn¥x vxodov u tn( ), t ≥ t0, k vxodu u t∗( ), t ≥ t0, hde u tn( )0 = u t∗( )0 = u0,
sleduet ravnomernaq na kaΩdom promeΩutke sxodymost\ neprer¥vn¥x v¥xodov
x tn( ) = W t u0 0,( , x u tn0) ( ), t ≥ t0, n = 1, 2,B…B, k neprer¥vnomu v¥xodu x t∗( ) =
= W t u x u t( , , ) ( )0 0 0
∗
, t ≥ t0.
2. Peremenn¥j hysteron. 2.1. Matematyçeskaq sxema dlq konstrukcyy
peremennoho hysterona. Pust\ zadano odnoparametryçeskoe semejstvo Wt
,
t0 ≤ t ≤ T, hysteronov, pry t0 ≤ t1 ≤ t2 ≤ T dlq oblastej vozmoΩn¥x sostoqnyj
Ω( )Wt
v¥polnqetsq uslovye Ω( )Wt1 � Ω( )Wt 2
, na t T0,[ ] zadan neprer¥vn¥j
vxod u t( ) y τ1 0∈[ )t T, . Tohda dlq lgboho τ τ2 1∈[ ), T y proyzvol\noho na-
çal\noho sostoqnyq u x( ),τ2 0{ } ∈B Ω( )W τ 1
odnoznaçno opredelen v¥xod x t( ) =
= W uτ τ τ1
2 2, ( )[ , x u t0] ( ), τ2 ≤ t ≤ T. V [1, c. 99] predloΩena sledugwaq „sxema
m” dlq opredelenyq peremennoho hysterona. Razob\em otrezok t T0,[ ] na
koneçnoe çyslo promeΩutkov ∆i , i = 1,B…B, n, toçkamy t0 = τ0 < τ1< …B < τn = T .
Melkost\g πtoho razbyenyq S qvlqetsq çyslo δ( )S = max –
, ,
–
i n
i i= …1
1τ τ . Razbye-
nyg S sopostavlqetsq peremenn¥j operator-hysteron W S( ) , oblast\g na-
çal\n¥x sostoqnyj kotoroho qvlqetsq Ω( )W
τ 0
. V¥xod x t( ) pry prymenenyy
W S( ) k neprer¥vnomu vxodu u t( ) , t ≤ t0 ≤ T, sootvetstvugwyj proyzvol\nomu
naçal\nomu sostoqnyg u t x( ),0 0{ } ∈B Ω( )W
τ 0
, opredelqetsq formulamy
x t( ) =
W u t x u t t t
W u x u t t
W u xi
i i i
τ
τ
τ
τ τ
τ τ τ τ τ
τ τ τ
0
1
1
0 0 0 1 0 1
1 1 1 2 1 2
1 1 1
, ( ), ( ) , ,
, ( ), ( ) ( ) , ,
, ( ), ( )–
– – –
[ ] ∈ = [ )
[ ] ∈ = [ )
……………………………… …………………………
[ ]
pry
pry
∆
∆
uu t t
W u x u t t T
i i i
n n n n n
n
( ) , ,
, ( ), ( ) ( ) , .
–
– – – –
–
pry
pry
∈ = [ )
……………………………… …………………………
[ ] ∈ = [ ]
∆
∆
τ τ
τ τ τ ττ
1
1 1 1 1
1
(2.1)
V¥xod x t( ) neprer¥ven na t T0,[ ]. Esly pry rassmotrenyy razlyçn¥x razbye-
nyj S otrezka t T0,[ ] y pry stremlenyy melkosty razbyenyj δ( )S k nulg v¥-
xod¥ x tS( ), opredelqem¥e formulamy (2.1), sootvetstvugwymy S, sxodqtsq v
kakom-lybo sm¥sle po metryke nekotoroho prostranstva neprer¥vn¥x na t T0,[ ]
funkcyj k funkcyy y t( ), to πtu funkcyg budem oboznaçat\ çerez y t( ) =
= W t u t0 0, ( )[ , x u t0] ( ), t0 ≤ t ≤ T, y sçytat\ v¥xodom peremennoho hysterona W ,
sootvetstvugwym vxodu u t( ) y naçal\nomu sostoqnyg u t x( ),0 0{ } ∈B Ω( )W τ 0
. V
dal\nejßem budem rassmatryvat\ ravnomernug sxodymost\ v¥xodov x tS( ) na
otrezke t T0,[ ] pry δ( )S → 0, t.Be. sxodymost\ po metryke prostranstva
C t T0,[ ]. V πtom sluçae funkcyq y t( ) neprer¥vna na t T0,[ ]. Oblast\g naçal\-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY 297
n¥x sostoqnyj peremennoho hysterona W qvlqetsq Ω( )W τ 0
, t.Be. Ω( )W =
= Ω( )W τ 0
. V [1, c. 101] postavlena problema naxoΩdenyq uslovyj, pry kotor¥x
opysannaq „sxema m” opredelenyq peremennoho hysterona realyzuetsq dlq
lgboho vxoda u t( ) yz kakoho-lybo klassa L � C t T0,[ ]. V dannoj stat\e pred-
lahagtsq nekotor¥e dostatoçn¥e uslovyq, obespeçyvagwye realyzacyg „sxem¥
m” dlq lgboho vxoda u t( ) , prynadleΩaweho klassu funkcyj H t1
0[ , T], udov-
letvorqgwyx uslovyg Lypßyca.
Zameçanye 2.1. V [1, c. 88 – 106] rassmotren podxod k opysanyg peremenno-
ho hysterona, yspol\zugwyj vybrokorrektn¥e dyfferencyal\n¥e uravnenyq s
ohranyçytelqmy (sm. takΩe [3]). Klass peremenn¥x hysteronov, opysann¥j v
pp.B2.1, vklgçaet v sebq peremenn¥e hysteron¥, opys¥vaem¥e vybrokorrektn¥-
my dyfferencyal\n¥my uravnenyqmy.
Rassmotrennaq v¥ße sxema yspol\zuetsq dlq opysanyq nelynejn¥x system s
hysterezysom, xarakterystyky kotor¥x menqgtsq v sylu yzmenenyq so vremenem
parametrov vneßnej sred¥. V [4, 5] rassmatryvaetsq nestacyonarnost\ system¥
s hysterezysom, kohda ona obæqsnqetsq vnutrennymy svojstvamy funkcyony-
rovanyq πtoj system¥.
2.2. Uslovyq realyzacyy „sxem¥ m ” dlq lgboho vxoda u t( ) ∈∈∈∈
∈∈∈∈$ H t T1
0,[[ ]]. Pust\ funkcyq f t( ) ∈ B H t T1
0,[ ], t.Be. udovletvorqet na t T0,[ ]
uslovyg Lypßyca. Rassmotrym mynymal\nug postoqnnug Lypßyca funkcyy
f t( ) na t T0,[ ], opredelqemug ravenstvom
l = sup ( ) – ( ) –
, , ,
–
t t t T t t
f t f t t t
1 2 0 1 2
1 2 1 2
1
∈[ ] ≠
, (2.2)
y budem hovoryt\, çto f t( ) ∈B H t Tl
1
0,[ ].
Lemma 2.1. Pust\ f t( ) ∈ B H t Tl
1
0,[ ]. Tohda suwestvuet posledovatel\-
nost\ mnohoçlenov P tk ( ) ∈ B H t Tlk
1
0,[ ], k = 1, 2,B…B, kotoraq ravnomerno sxo-
dytsq na t T0,[ ] k funkcyy f t( ) y
l lk → pry k → ∞. (2.3)
Zameçanye 2.2. Ydeq dokazatel\stva πtoj lemm¥ prynadleΩyt ∏. M. Mu-
xamadyevu. Pervonaçal\no teoremuB2.1 dokazal avtor, yspol\zovav teoremu
V.BK. Dzqd¥ka [6, c. 271].
Lemma 2.2. Pust\ g x( ) = 1 + cx, hde c > 0. Tohda dlq lgb¥x neotry-
catel\n¥x çysel xi, i = 1,B… B, n, verna ocenka 1 ≤ g x( )1 g x( )2 B…B g xn( ) ≤
≤B exp ( )cM , hde xii
n
=∑ 1
≤ M.
Lemma 2.3. Pust\ zi , i = 1,B…B, n, — proyzvol\n¥e neotrycatel\n¥e çys-
la, udovletvorqgwye uslovyg zii
n
=∑ 1
≤ d. Tohda ymeet mesto ocenka F z1( ,
z2,B… B, zn) = zn + ( ) –1 1+ cz zn n B+ ( )( )– –1 1 1 2+ +cz cz zn n n +B… B+ ( )1 + czn ×
× ( )–1 1+ czn ( )1 2 1+ cz z ≤ d exp ( )cd . Zdes\ d > 0 y c > 0 — nekotor¥e po-
stoqnn¥e.
Rassmotrym nekotor¥j hysteron W s oblast\g vozmoΩn¥x sostoqnyj
Ω( )W . Pust\ u t( ) ∈ B H t T1
0,[ ] — nekotor¥j dopustym¥j dlq πtoho hysterona
vxod. Oboznaçym çerez Φu = ϕ( )t{ }, t t T∈[ ]0, , mnoΩestvo v¥xodov, sootvet-
stvugwyx πtomu vxodu pry razlyçn¥x naçal\n¥x sostoqnyqx yz oblasty Ω( )W .
Pust\ 0 ≤ Br < + ∞ — nekotoroe çyslo y dann¥j vxod udovletvorqet uslovyg
u t( ) ≤ r, t t T∈[ ]0, . Oboznaçym çerez Φu
r
mnoΩestvo v¥xodov yz Φu , udov-
letvorqgwyx uslovyg
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
298 V. Y. BORZDÁKO
ϕ( )t ≤ r, t t T∈[ ]0, . (2.4)
Lemma 2.4. Pust\ vse kryv¥e, opredelqgwye hysteron W , udovletvorq-
gt obwemu uslovyg Lypßyca s odnoj y toj Ωe postoqnnoj m. Pust\ u t( ) ∈
∈ H t T1
0,[ ] — nekotor¥j dopustym¥j dlq hysterona W vxod, ymegwyj po-
stoqnnug Lypßyca l. Tohda vse v¥xod¥ yz semejstva Φu prynadleΩat
klassu H t T1
0,[ ] s odnoj y toj Ωe postoqnnoj Lypßyca ml.
Lemma 2.5. Pust\ vse kryv¥e, vxodqwye v opredelenye hysteronov yz se-
mejstva W t
, t0 ≤ t ≤ T, udovletvorqgt obwemu uslovyg Lypßyca s odnoj y
toj Ωe postoqnnoj m. Pust\ suwestvugt takye poloΩytel\n¥e konstant¥
γ, ε, µ y neotrycatel\naq B, çto dlq proyzvol\noj toçky M = u z0 0,{ } ∈
∈B Ω( )Wt0 ymeet mesto neravenstvo
T u h M T u h Mt t1 2
0 0( , ) – ( , )+ + ≤ B h t t2
1 2– µ
(2.5)
pry h ≤ γ, t t1 2– ≤ ε, t1, t2 ∈ B t T0,[ ]. Tohda dlq lgboho vxoda u t( ) ∈
∈B H t T1
0,[ ] y lgboho naçal\noho sostoqnyq u t x( ),0 0{ } ∈ B Ω( )Wt0 suwestvuet
takoe poloΩytel\noe çyslo r, çto pry lgbom razbyenyy S s δS ≤ ε soot-
vetstvugwyj v¥xod udovletvorqet uslovyg
x t rS( ) ≤ (2.6)
pry t t T∈[ ]0, .
Podrobn¥e dokazatel\stva lemmB2.1 – 2.5 pryveden¥ v [7].
Teorema 2.1. Pust\ vse kryv¥e, opredelqgwye hysteron¥, vxodqwye v se-
mejstvo Wt
, t0 ≤ t ≤ T, udovletvorqgt obwemu uslovyg Lypßyca s odnoj y
toj Ωe postoqnnoj. Pust\ v¥polnen¥ uslovyq:
1) suwestvugt takye poloΩytel\n¥e konstant¥ γ , ε0, µ y neotryca-
tel\naq B, çto dlq proyzvol\noj toçky M = u z0 0,{ } ∈ B Ω( )Wt0 ymeet mesto
neravenstvo
T u h M T u h Mt t1 2
0 0( , ) – ( , )+ + ≤ B h t t2
1 2– µ
(2.7)
pry h ≤ γ, t t1 2– ≤ ε0, t1, t2 ∈B t T0,[ ];
2) pust\ dlq lgboho neotrycatel\noho çysla r suwestvugt takye po-
stoqnn¥e çysla 0 ≤ Kr < + ∞ y 0 < α r< + ∞, çto esly vxod u t( ) ∈ B H t T1
0,[ ]
s postoqnnoj Lypßyca l udovletvorqet uslovyg u t( ) ≤ r, t0 ≤ t ≤ T, to
dlq lgb¥x v¥xodov ϕ( )t , ψ( )t yz semejstva Φu
r ( )τ (sm. (2.4)), postroenn¥x
po hysteronu W τ
yz semejstva Wt
, t0 ≤ t ≤ T, ymeet mesto neravenstvo
ϕ ψ( ) – ( )t t ≤ 1 +[ ]∗ ∗ ∗c t t t t( – ) ( ) – ( )ϕ ψ (2.8)
pry t t t r∈ +[ ]∗ ∗, δ , hde
c = lKr + ε , (2.9)
δr = min – , ( )–T b lr1
1α εε+ +{ }, α εr + = min ,γ α εr +{ }, (2.10)
b T1 ∈( , )τ — proyzvol\noe fyksyrovannoe çyslo, t∗
— lgboe çyslo yz [ τ, b1],
ε > 0 — nekotoroe çyslo, M = u t( )∗{ , ϕ( )t∗ } ∈ B Ω( )Wt0 , N = u t( )∗{ , ψ( )t∗ } ∈
∈B Ω( )Wt0 .
Tohda „sxema m” realyzuetsq dlq lgboho vxoda u t( ) yz klassa H t T1
0,[ ].
Dokazatel\stvo teorem¥ razob\em na dva πtapa. Na pervom πtape dokaz¥-
vagtsq neravenstva, kotor¥e yspol\zugtsq v dal\nejßem. Pust\ zadan vxod
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY 299
u t( ) ∈B H t Tl
1
0,[ ], l > 0, udovletvorqgwyj uslovyg
u t( ) ≤ r, t t T∈[ ]0, , (2.11)
hde r > 0 — nekotoraq postoqnnaq. Pust\ τ ∈ B t T0,[ ), t∗ ∈ B τ, b1[ ], b1 < T,
u t( )∗{ , z0} ∈B Ω( )Wt0 . Oboznaçym çerez x z t( , , )τ 0 v¥xod hysterona W τ
pry t ≥
≥ t∗
: x τ( , z t0, ) = W t u tτ ∗ ∗[ , ( ) , z u t0] ( ). Pust\ u t z( ),∗{ }0 , u t y( ),∗{ }0 ∈ B Ω( )Wt0 y
v¥xod¥
x z t( , , )τ 0 ≤ r, x y t( , , )τ 0 ≤ r, t t T∈[ ]∗, . (2.12)
Tohda v sylu uslovyq (2.8) pry lgbom t ∈B t t r
∗ ∗ +[ ], δ v¥polnqetsq neravenstvo
x z t x y t( , , ) – ( , , )τ τ0 0 ≤ 1 0 0+ ( )[ ]∗c t t z y– – . (2.13)
DokaΩem, çto esly τ τ1 2– ≤ ε0, t0 ≤ τ1 ≤ τ2 ≤ t∗ < b1, y
x z t( , , )τ1 0 ≤ r, x z t( , , )τ2 0 ≤ r, t t T∈[ ]∗, , (2.14)
to pry t ∈B t t r
∗ ∗
++[ ], δ ε ymeet mesto neravenstvo
x z t x z t( , , ) – ( , , )τ τ1 0 2 0 ≤ λ τ τ µ
u t t1 2– ( – )∗
, (2.15)
hde µ > 0 — postoqnnaq yz uslovyq (2.7), a neotrycatel\naq postoqnnaq λu ne
zavysyt ot t∗
, z0 y opredelqetsq zadann¥m vxodom u t( ) . DokaΩem neravenstvo
(2.15) snaçala dlq kusoçno-monotonnoho vxoda u t( ) , udovletvorqgweho na
t T0,[ ] uslovyg Lypßyca s postoqnnoj l. Vvedem oboznaçenyq u0 = u t( )∗
y
M = u z0 0,{ }. V sylu (2.10) pry t ∈B t t r
∗ ∗ +[ ], δ ymeem neravenstvo
u t u( ) – 0 = u t u t( ) – ( )∗ ≤ l t t( – )∗ ≤ α εr + .
Razob\em otrezok t t r
∗ ∗ +[ ], δ na promeΩutky monotonnosty funkcyy u t( ) B:
t i( – )1[ , t i( )], i = 1,B… B, k, hde t( )0 = t∗
, t k( ) = t∗ + δr . Oboznaçym ∆ i t = t i( ) –
– t i( – )1
, ∆ i u = u t i( )( ) – u t i( – )1( ). Tohda pry t ∈B t i( – )1[ , t i( )]
u t u ti i( ) ( – )–( ) ( )1 ≤ l t t i– ( – )1( ) ≤ l ti∆ , i = 1,B…B, k. (2.16)
V sylu (1.3), (2.7), (2.16) pry t ∈B t t∗[ ], ( )1
x z t x z t( , , ) – ( , , )τ τ1 0 2 0 = T u t M T u t Mτ τ1 2( ), – ( ),( ) ( ) ≤
≤ B u t u t( ) – ( ) –∗ 2
1 2τ τ µ ≤ Bl t t r
2
1 2– –∗( ) τ τ δµ . (2.17)
Vvedem oboznaçenyq x1 = T uτ1
0( + ∆1u M, ) = T u tτ1 1( )( )( , M) , y1 = T u uτ2
0 1( + ∆ ,
M) B= T u tτ2 1( )( )( , M) , M1 = u t x( ) ,1
1( ){ } , N1 = u t y( ) ,1
1( ){ }, xi = T u t iτ1 1( – )( )[ + ∆ i u ,
Mi –1] = T u t iτ 1 ( )( )( , Mi –1) , y i = T u t iτ2 1( – )( )[ + ∆ i iu N, –1] = T u t iτ2 ( )( )( , Ni –1),
Mi –1 = u t i( – )1( ){ , xi –1}, Ni –1 = u t i( – )1( ){ , yi –1}, i = 2,B…B, k. Yz (1.2), (1.3), (2.7),
(2.8), (2.12) – (2.14) y (2.17) pry t ∈B t t( ) ( ),1 2[ ] poluçaem
x z t x z t( , , ) – ( , , )τ τ1 0 2 0 = T u t M T u t Nτ τ1 2
1 1( ), – ( ),[ ] [ ] ≤
≤ T u t M T u t Nτ τ1 1
1 1( ), – ( ),[ ] [ ] B+B T u t N T u t Nτ τ1 2
1 1( ), – ( ),[ ] [ ] ≤
≤ 1 1
1 1+ ( )[ ]c t t x y– –( ) B+B Bl t t2 1 2
1 2– –( )( ) τ τ µ ≤
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300 V. Y. BORZDÁKO
≤ 1 1 2 1 2
1 2+ ( )[ ] ∗c t t Bl t t– – –( ) ( ) τ τ µ B+B Bl t t2 1 2
1 2– –( ) τ τ µ ≤
≤ Bl t c t t t t2
1 2 2 2 11τ τ µ– –∆ ∆ ∆+ +[ ]{ }( )∗ .
Analohyçno, pry t ∈B t t( ) ( ),2 3[ ]
x z t x z t( , , ) – ( , , )τ τ1 0 2 0 ≤ 1 2
2 2
2 2 2
1 2+ ( )[ ] + ( )c t t x y Bl t t– – – –( ) ( ) τ τ µ ≤
≤ Bl t t t c t t c t2
1 2 2 2 1 31 1– –∗( ) + +[ ]{ } +[ ]τ τ µ ∆ ∆ ∆ ∆ +
+ Bl t t t2 2
3 1 2– –( )( )∆ τ τ µ ≤
≤ Bl t t t c t t c t c t t2
1 2 3 3 2 3 2 11 1 1– –∗( ) + +[ ] + +[ ] +[ ]{ }τ τ µ ∆ ∆ ∆ ∆ ∆ ∆ .
Dalee, yspol\zuq metod polnoj matematyçeskoj yndukcyy, pry t ∈B t ti i( – ) ( ),1[ ]
poluçaem neravenstvo
x z t x z t( , , ) – ( , , )τ τ1 0 2 0 ≤ Bl t t t c t ti i i
2
1 2 11– – –
∗( ) + +[ ]{τ τ µ ∆ ∆ ∆ +
+ 1 1 1 1 11 2 1 2 1+[ ] +[ ] + … + +[ ] +[ ]… +[ ] }c t c t t c t c t c t ti i i i i∆ ∆ ∆ ∆ ∆ ∆ ∆– – – .
(2.18)
Poskol\ku ∆ss
i
t=∑ 1
≤ δr y ∆s t > 0, s = 1,B…B, i, yspol\zuq v neravenstve
(2.18) lemmuB2.3, v sylu (2.9) pry t ∈B t ti i( – ) ( ),1[ ] ymeem sledugwug ocenku:
x z t x z t( , , ) – ( , , )τ τ1 0 2 0 ≤ λ τ τ µ
u t t1 2– – ∗( ) , (2.19)
hde
λu = Bl lkr r r
2δ δεexp( )+ . (2.20)
Yz (2.19), (2.20) v¥tekaet neravenstvo (2.15) na otrezke t t r
∗ ∗ +[ ], δ v sluçae ku-
soçno-monotonnoho vxoda u t( ) ∈ B H t T1
0,[ ], ymegweho postoqnnug Lypßyca
l > 0. PredpoloΩym teper\, çto vxod u t( ) qvlqetsq proyzvol\noj neprer¥vnoj
na t T0,[ ] funkcyej yz H t Tl
1
0,[ ], udovletvorqgwej uslovyg (2.11). Sohlasno
lemmeB2.1 suwestvuet posledovatel\nost\ vxodov-mnohoçlenov, t.Be. kusoçno-
monotonn¥x funkcyj p tn( ) ∈B H t Tl n
1
0,[ ], n = 1, 2,B…B, takaq, çto
p tn( ) → u t( ) ravnomerno na t T0,[ ], ln → l pry n → ∞. ( 2 3. ′ )
V sylu svojstv oblasty vozmoΩn¥x sostoqnyj hysterona [1, c. 25] moΩno sçy-
tat\, çto pry fyksyrovannom t∗
toçky Ln = p tn( )∗{ , z0} ∈ B Ω( )Wt0 pry dosta-
toçno bol\ßom n. Predpolahaetsq, çto v¥xod x z t( , , )τ 0 , sootvetstvugwyj
vxodu u t( ) , pry lgb¥x τ1, τ2 0∈[ )t T, y t ≥ t∗
udovletvorqet neravenstvu
(2.12). Oboznaçym sootvetstvenno
x z tn( , , )τ1 0 = W t p t z p tn n
τ1
0
∗ ∗[ ], ( ), ( ),
(2.21)
x z tn( , , )τ2 0 = W t p t z p tn n
τ2
0
∗ ∗[ ], ( ), ( ) .
V sylu vybrokorrektnosty (p.B1) hysteronov W τ1 , W τ2 yz (2 3. ′ ) v¥tekaet,
çto
x z tn( , , )τ1 0 → x z t( , , )τ1 0 ,
(2.22)
x z tn( , , )τ2 0 → x z t( , , )τ2 0 pry n → ∞
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HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY 301
ravnomerno na otrezke t T∗[ ], .
Vsledstvye (2.11), (2.12), ( 2 3. ′ ) y (2.22) moΩno sçytat\, çto pry zadannom
ε > 0 dlq dostatoçno bol\ßoho n ymegt mesto neravenstva
p tn( ) ≤ r B+B ε,
x z tn( , , )τ1 0 ≤ r B+B ε, (2.23)
x z tn( , , )τ2 0 ≤ r B+B ε, t t T∈[ ]0, ,
l ln –[ ] ≤ ε. (2.24)
V sylu kusoçnoj monotonnosty vxodov p tn( ) , n = 1, 2,B…B, uslovyj (2.23), (2.24),
sootnoßenyj (2.19), (2.20), dokazann¥x pry πtyx uslovyqx, ymeet mesto nera-
venstvo
x z t x z tn n( , , ) – ( , , )τ τ1 0 2 0 ≤ Bl l kn r
n
n r r
2
2δ δε ε ε+ + +
( ) exp( ) , (2.25)
hde δ εr
n
+
( ) = min –T b1{ , α εr + 2 ( )–ln + }2 1ε > 0, α εr + 2 > 0, k r + 2ε ≥ 0 opredelq-
gtsq po r + ε sohlasno uslovyg 2 teorem¥. Perexodq v (2.25) k predelu pry
n → ∞ , yz (2 3. ′ ), (2.22) poluçaem, çto dlq proyzvol\noho neprer¥vnoho vxoda
(2.11) pry uslovyy na v¥xod¥ (2.12) na otrezke t t r
∗ ∗
++[ ], δ ε ymeet mesto nera-
venstvo (2.15), v kotorom postoqnn¥e pereoboznaçen¥ tak, kak ony yspol\zugt-
sq v dal\nejßem:
c = lkr + 2ε , λu = Bl lkr r r
2
2δ δε ε ε+ + +exp( ) , (2.26)
hde postoqnnaq B ≥ 0 yz (2.7), b1 ∈B( , )τ2 T , ε > 0, r + ε > 0, l + ε > 0, a δ εr + >
> 0 — opredelqemoe po nym y α εr + 2 çyslo v sootvetstvyy s formuloj (2.10).
Otmetym, çto v sylu uslovyq 2 teorem¥ δr ≥ δ εr + > 0.
Pryvedem ewe odno neravenstvo, yspol\zuemoe nyΩe. Pust\, kak y ranee,
u t( )∗{ , z0}, u t y( ),∗{ }0 ∈B Ω( )Wt0 , t0 ≤ τ1 ≤ τ2 ≤ t∗ < t∗ + δ εr + , pryçem v¥xod¥,
sootvetstvugwye vxodu (2.11), udovletvorqgt ocenke (2.12) (ε > 0 — zadannoe
çyslo). Tohda, kak sleduet yz neravenstva (2.15) y uslovyq 2 teorem¥, pry t ∈
∈B t t r
∗ ∗
++[ ], δ ε ymeet mesto neravenstvo
x z t x y t( , , ) – ( , , )τ τ1 0 2 0 ≤
≤ x z t x z t( , , ) – ( , , )τ τ1 0 2 0 + x z t x y t( , , ) – ( , , )τ τ2 0 2 0 ≤
≤ λ τ τ µ
u t t1 2– – ∗( ) + 1 0 0+ ( )[ ]∗c t t z y– – , (2.27)
hde c, λu opredelen¥ formuloj (2.26).
Perejdem teper\ ko vtoromu πtapu dokazatel\stva teorem¥. Pust\ ymegtsq
dva razlyçn¥x razbyenyq otrezka t T0,[ ]:
S t k m1 0 0
1
1
1 1 1
1
: ( ) ( ) ( ) ( )≤ < < … < < … <τ τ τ τ = T,
S t i m2 0 0
2
1
2 2 2
2
: ( ) ( ) ( ) ( )≤ < < … < < … <τ τ τ τ = T
s melkost\g razbyenyj δ( )S1 , δ( )S2 sootvetstvenno. Oboznaçym çerez S sme-
ßannoe razbyenye otrezka t T0,[ ], v kotorom uçastvugt toçky delenyq oboyx
razbyenyj S1, S2 . Pust\ zadan vxod u t( ) ∈ B H t Tl
1
0,[ ], udovletvorqgwyj uslo-
vyg (2.11). Oboznaçym çerez x t1( ), x t2( ) y x t( ) v¥xod¥, sootvetstvugwye
πtomu vxodu, naçal\nomu sostoqnyg u t x( ),0 0{ } ∈ B Ω( )Wt0 y razbyenyqm soot-
vetstvenno S1, S2 po pravylu (2.1). Vvedem oboznaçenye δ0 = max ( )δ S1{ , δ( )S2 }.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
302 V. Y. BORZDÁKO
Tohda melkost\ smeßannoho razbyenyq δ = δ( )S ≤ δ0. Zafyksyruem nekotoroe
b1 ∈ B( , )t T0 y opredelym po nemu y ε > 0, l + ε, r + ε çyslo δ εr + sohlasno
(2.10) yz uslovyq 2 teorem¥. Pust\
δ0 ≤ min ,δ εεr +{ }0 . (2.28)
Ocenym raznost\ x t x t1( ) – ( ) na otrezke t b r0 1, +[ ]+δ ε . Dlq πtoho pryvedem
formulu, sootvetstvugwug smeßannomu razbyenyg S : t0 = τ0
1( ) < τ1
1( ) <B…B
…B<B τk 1
1( ) < τ1
2( ) < τ2
2( ) <B… B< τi1
2( ) < τk 1 1
1
+
( ) < τk 1 2
1
+
( ) <…B< τk s1
1
+
( ) < τk s1 1
1
+ +
( ) < B…B
… < τk 2
1( ) < τi1 1
2
+
( ) < τi1 2
2
+
( ) <B…B< τi2
2( ) < τk 2 1
1
+
( ) :
x t( ) – x t1( ) =
=
= ≤ ≤
( ) ( )[ ] ( ) ( )[ ]
≤ ≤
0 0 0
1
1
2
1
2
1
2
1 1
2
1
2
1
2
1 1
2
1
2
2
2
2
1
2
1
1
2
2
, ,
, , ( ) – , , ( ),
,
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
(
( )
( )
( )
t t
W u x u t W u x u t
t
W
k
τ τ
τ τ τ τ τ τ
τ τ
τ
τ τ
τ 22
2
2
2
2
2
2
2
2
1 2
2
2
2
3
2
1
1
1
2
) ( ) ( ) ( ) ( ) ( )
( ) ( )
, , ( ) – , , ( ),
,
( )
( )
u x u t W u x u t
t
W
k
i
τ τ τ τ τ
τ τ
τ
τ
τ
( ) ( )[ ] ( ) ( )[ ]
≤ ≤
………………………………………………………………………………
ii i i i i i
i k
k k
u x u t W u x u t
t
W u
k
k
1 1 1
1
1
1 1 1
1 1
1 1
1
1 1
2 2 2 2 2
1
2
2
1
1
1
1
1
1
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) (
, , ( ) – , , ( ),
,
,
( )
( )
τ τ τ τ τ
τ τ
τ τ
τ
τ
( ) ( )[ ] ( ) ( )[ ]
≤ ≤ +
+ +
+ )) ( ) ( ) ( ) ( )
( ) ( )
, ( ) – , , ( ),
,
( )( ) ( )[ ] ( ) ( )[ ]
≤ ≤
……………………………………………………………………
+ + + +
+ +
+
x u t W u x u t
t
k k k k
k k
kτ τ τ τ
τ τ
τ
1
1 1
1
1 1 1
1 1
1
1
1
1
1
1
1 1
1
1
1
2
1
……………
( ) ( )[ ] ( ) ( )[ ]
≤ ≤
+ +
+ + + + + +
+ +
W u x u t W u x u t
t
k s k s
k s k s k s k s k s k s
k s k s
τ τ
τ τ τ τ τ τ
τ τ
1
1
1 1 1
1
1
1 1 1
1 1
1 1 1 1 1
1
1
1
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
, , ( ) – , , ( ),
++
………………………………………………………………………………
( ) ( )[ ] ( ) ( )[ ]
1
1
1 1 1 1 1
1
12
1
2 2 2
2
1
2 2 2
( )
( ) ( ) ( ) ( ) ( ) ( )
,
, , ( ) – , , ( ),
( ) ( )
W u x u t W u x u t
k k
k k k k k k
τ τ
τ τ τ τ τ τ
ττ τ
τ τ τ τ τ τ
τ
τ τ
k i
i i i i i i
i
t
W u x u t W u x u t
i k
2 1
1 1
2
1 1 1
2
1
1 1 1
1
1
1
2
1
2
1
2
1
2
1
2
1
2
1 1
2
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
,
, , ( ) – , , ( ),
( ) ( )
≤ ≤
( ) ( )[ ] ( ) ( )[ ]
+
+ + + + + +
+
+
11
2
2
2
1
( ) ( ) ,≤ ≤
………………………………………………………………………………
+t iτ
(2.29)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY 303
Teper\, yspol\zovav (2.29), ocenym x t x t( ) – ( )1 . Otmetym, çto v sylu uslovyj
(2.7), (2.28) y lemm¥B2.5 moΩno sçytat\, çto pry vsex trex razbyenyqx S1, S2 y
S sootvetstvugwye ym v¥xod¥ udovletvorqgt uslovyqm x ti( ) ≤ r, i = 1, 2;
x t( ) ≤ r, t t T∈[ ]0, . Poπtomu dlq ocenok moΩno prymenqt\ neravenstva (2.15),
(2.26) y (2.27). Pry t t∈[ ]0 1
2, ( )τ ymeem
x t x t( ) – ( )1 = 0, (2.30)
pry t ∈[ ]τ τ1
2
2
2( ) ( ), —
x t x t( ) – ( )1 ≤ λ τ τ τ
µ
u k t1
2 1
1
2
1
( ) ( ) ( ), – ≤ λ τ τ τ
µ
u t1
2
0
2
1
2( ) ( ) ( ), – . (2.31)
Yz (2.29), (2.31) v sylu lemm¥B2.3 sleduet, çto pry t ∈[ ]τ τ2
2
3
2( ) ( ),
x t x t( ) – ( )1 = W u x u t W u x u tkτ τ
τ τ τ τ τ τ2
2
1
1
2
2
2
2
2
2
2
2
2
2
1 2
2( ) ( )
( ) ( ) ( ) ( ) ( ) ( ), , ( ) – , , ( )( ) ( )[ ] ( ) ( )[ ] ≤
≤ 1 2
2
2
2
1 2
2+ ( )[ ] ( ) ( )c t x x– –( ) ( ) ( )τ τ τ + λ τ τ τ
µ
u k t2
2 1
2
2
1
( ) ( ) ( )– – ≤
≤ 1 3
2
2
2
1
2 1
2
2
1
2
1
+ ( )[ ]c u kτ τ λ τ τ τ τ
µ( ) ( ) ( ) ( ) ( ) ( )– – – +
+ λ τ τ τ τ
µ
u k2
2 1
3
2
2
2
1
( ) ( ) ( ) ( )– – ≤
≤ λ τ τ τ τ τ τ τ τ
µ
u k c2
2 1
3
2
2
2
3
2
2
2
2
2
1
2
1
1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )– – – –( ) + + ( )[ ]( ){ } ≤
≤ λ τ τ τ τ τ τ
µ
u k c2
2 1
3
2
1
2
3
2
1
2
1
( ) ( ) ( ) ( ) ( ) ( )– – exp –( )[ ] (2.32)
y tak dalee do promeΩutka τi1
2( ) ≤ t ≤ τk1 1
1
+
( )
. Pry t i k∈[ ]+τ τ
1 1
2
1
1( ) ( ), po yndukcyy
poluçaem
x t x t( ) – ( )1 ≤ λ τ τ
µ
u i k1 1
2 1( ) ( )– ×
× τ τ τ τ τ τi i i i i ic
1 1 1 1 1 1
2
1
2 2
1
2
1
2
2
21( )
–
( ) ( )
–
( )
–
( )
–
( )– – –( ){ + + ( )[ ]( ) +B…
…B+ 1 1 1
1 1 1 1
2
1
2
1
2
2
2
3
2
2
2+ ( )[ ] + ( )[ ]… + ( )[ ]c c ci i i iτ τ τ τ τ τ( )
–
( )
–
( )
–
( ) ( ) ( )– – – ×
× τ τ τ τ λ τ τ τ
µ
2
2
1
2
1
1 2
1
1 2 21
1 1 1 1 1
( ) ( ) ( ) ( ) ( ) ( ) ( )– – – –( )} + ( )[ ] + ( )+ +c tk i u k i i ≤
≤ λ τ τ τ τ
µ
u k k k i1 1 1 11
1 1
1
1 2
+ +( ){( ) ( ) ( ) ( )– – +
+ 1
1 1 1 11
1 2 2
1
2+ ( )[ ] ( )+c k i i iτ τ τ τ( ) ( ) ( )
–
( )– – +
+ 1 1
1 1 1 1 1 11
1 2 2
1
2
1
2
2
2+ ( )[ ] + ( )[ ] ( )+c ck i i i i iτ τ τ τ τ τ( ) ( ) ( )
–
( )
–
( )
–
( )– – – +B…
…B+ 1 1 1
1 1 1 11
1 2 2
1
2
3
2
2
2
2
2
1
2+ ( )[ ] + ( )[ ]… + ( )[ ] ( )}+c c ck i i iτ τ τ τ τ τ τ τ( ) ( ) ( )
–
( ) ( ) ( ) ( ) ( )– – – – .
Ocenyvaq pravug çast\ πtoho neravenstva sohlasno lemmeB2.3 pry d = τk1 1
1
+
( ) –
–B τk 1
1( ) ≥ τk1 1
1
+
( ) – τ1
2( )
y yspol\zuq takΩe ocenky (2.30) – (2.32), pry t ∈ B τk1
1( )[ ,
τk1 1
1
+ ]( ) poluçaem neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
304 V. Y. BORZDÁKO
x t x t( ) – ( )1 ≤ λ τ τ τ τ
µ
u k k k kc
1 1 1 11
1 1
1
1
1 1
+
+
+( ) ( )[ ]( ) ( ) ( ) ( )– exp – ≤
≤ λ δ τ τµ
u k k c T t
1 11
1 1
0+( ) ( )[ ]( ) ( )– exp – . (2.33)
Rassmotrym sledugwyj vremennoj ynterval τk1 1
1
+
( ) ≤ t ≤ τi1 1
2
+
( )
. Kak sleduet
yz (2.27), (2.29) y (2.33), ymegt mesto neravenstva:
pry τk1 1
1
+
( ) ≤ t ≤ τk1 2
1
+
( )
x t x t( ) – ( )1 ≤ 1
1 1 1 1 11
1
1
1 1 1
1
1 1+ ( )[ ] ( ) ( )[ ]+ +
+
+c t ck u k k k k– – exp –( ) ( ) ( ) ( ) ( )τ λ τ τ τ τ
µ
, (2.34)
pry τk1 2
1
+
( ) ≤ t ≤ τk1 3
1
+
( )
x t x t( ) – ( )1 ≤ 1 1
1 1 12
1
2
1
1
1+ ( )[ ] + ( )[ ]+ + +c t ck k k– –( ) ( ) ( )τ τ τ ×
× λ τ τ τ τ
µ
u k k k kc
1 1 1 11
1 1 1
1
1 1
+
+
+( ) ( )[ ]( ) ( ) ( ) ( )– exp – (2.35)
y t.Bd. Nakonec, pry τk 2
1( ) ≤ t ≤ τi1 1
2
+
( ) ymeem
x t x t( ) – ( )1 ≤ 1 1
2 2 2
1 1
1
1+ ( )[ ] + ( )[ ]c t ck k k– –( ) ( )
–
( )τ τ τ …
… 1 1
1 1 1 13
1
2
1
2
1
1
1+ ( )[ ] + ( )[ ]+ + + +c ck k k kτ τ τ τ( ) ( ) ( ) ( )– – ×
× λ τ τ τ τ
µ
u k k k kc
1 1 1 11
1 1
1
1
1 1
+
+
+( ) ( )[ ]( ) ( ) ( ) ( )– exp – . (2.36)
Yz lemm¥B2.2 y neravenstv (2.34) – (2.36) sleduet, çto ymeet mesto ocenka:
pry τk1 1
1
+
( ) ≤ t ≤ τi1 1
2
+
( )
x t x t( ) – ( )1 ≤ λ τ τ τ
µ
u i k it
1 2 11
2 1
1
2
+ +−( )( ) ( ) ( )– +
+ λ τ τ τ τ τ
µ
u i k k i kc t c1
1 1 1 1 11
2
1
1 1
1
1
2 1+ ( )[ ]( ) ( )[ ]+ +
+
+– – exp –( ) ( ) ( ) ( ) ( ) , (2.37)
pry τi1 2
2
+
( ) ≤ t ≤ τi1 3
2
+
( )
x t x t( ) – ( )1 =
= W u x u t
k
i i i
τ
τ τ τ2
1
1 1 12
2
2
2
1 2
2
( )
( ) ( ) ( ), , ( )+ + +( ) ( )( ) – W u x u ti
i i i
τ
τ τ τ1 2
2
1 1 12
2
2
2
2
2+
+ + +( ) ( )( )( )
( ) ( ) ( ), , ( ) ≤
≤ λ τ τ τ
µ
u i k it
1 2 12
2 1
2
2
+ +
( )( ) ( ) ( )– – + 1
1 1 2 1 12
2
1
2 1
2
2
1
2+ ( )[ ] ( ) ( )+ + + +c t i i k i i– – –( ) ( ) ( ) ( ) ( )τ τ τ τ τ
µ
+
+ 1 1
1 1 12
2
2
2
1
2+ ( )[ ] + ( )[ ]+ + +c t ci i i– –( ) ( ) ( )τ τ τ τ τ τ τ
µ
k k i kc
1 1 1 11
1 1
1
1
2 1
+
+
+( ) ( )[ ]
( ) ( ) ( ) ( )– exp –
(2.38)
y t.Bd. Prymenqq lemm¥B2.2 y 2.3, pry τi 2
1( ) ≤ t ≤ τk 2 1
1
+
( )
poluçaem ocenku
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY 305
x t x t( ) – ( )1 =
= W u x u tk
i i i
τ
τ τ τ2
1
2 2 2
2 2
1
2
( )
( ) ( ) ( ), , ( )( ) ( )[ ] – W u x u t
i
i i i
τ
τ τ τ2
2
2 2 2
2 2 2
( )
( ) ( ) ( ), , ( )( ) ( )[ ] ≤
≤ λ τ τ τ
µ
u i k it
2 2 2
2 1 2( ) ( ) ( )– –( ) ( ) +
+ 1
2 2 2 2 2
2
1
2 1 2
1
2+ ( )[ ]( ) ( )c t i i k i i– – –( )
–
( ) ( ) ( )
–
( )τ τ τ τ τ
µ
+ 1
2
2+ ( )[ ]c t i– ( )τ ×
× 1
2 2 2 2 2 2
2
1
2
2
2 1
1
2
2
2+ ( )[ ] ( ) ( )c i i i k i iτ τ τ τ τ τ
µ( )
–
( )
–
( ) ( )
–
( )
–
( )– – – + …
… + 1 1
2 2 2
2 2
1
2+ ( )[ ] + ( )[ ]c t ci i i– –( ) ( )
–
( )τ τ τ … 1
1 13
2
2
2+ ( )[ ]+ +c i iτ τ( ) ( )– ×
× τ τ τ τ
µ
i k i i1 2 1 11
2 1
2
2
1
2
+ + +( ) ( )( ) ( ) ( ) ( )– – + λ τ τ τu i ic t c1 12
2 2
1
2
2 2
+ ( )[ ] + ( )[ ]−– –( ) ( ) ( ) …
… 1
1 1 1 1 1 12
2
1
2
1
1 1 1
1
2 1+ ( )[ ] ( ) ( )[ ]+ + +
+
+c ci i k k i kτ τ τ τ τ τ
µ( ) ( ) ( ) ( ) ( ) ( )– – exp – ≤
≤ λ τ τ τ τ
µ
u k k i it c t
2 2 1 11
1 1
1
2
1
2
+ + +( ) ( ) ( )[ ]( ) ( ) ( ) ( )– – exp – +
+ λ τ τ τ
µ
u k k kc t
1 1 11
1 1 1 1
+
+( ) ( )[ ]( ) ( ) ( )– exp – ≤
≤ λ δ τ τ τ τµ
u k k k k2 2 1 11
1 1
1
1 1
+ +( ) + ( )[ ]( ) ( ) ( ) ( )– – ≤ λ δ µ
u T t c T t( – ) exp ( – )0 0[ ]. (2.39)
Yz lemm¥B2.3 y neravenstv (2.37) – (2.39) sleduet, çto ocenka (2.39) ymeet
mesto pry lgbom t k k∈[ ]+τ τ
2 2
1
1
1( ) ( ), . Poskol\ku ynterval¥ vydov τ τk k1 1
1
1
1( ) ( ), +[ ],
τ τk i1 11
1
1
2
+ +[ ]( ) ( ), y τk 2
1( )[ , τk 2 1
1
+ ]( )
çeredugtsq, to x t x t( ) – ( )1 ≤ λ δ µ
u T t( – )0 ×
× exp ( – )c T t0[ ] pry lgbom t ∈B t b0 1,[ + δ εr + ].
Oçevydno, çto na tom Ωe promeΩutke ymeet mesto takaq Ωe ocenka dlq
x t( ) – x t2( ) . Otsgda sleduet, çto pry t0 ≤ t ≤ b1 + δ εr + ≤ T ymeet mesto nera-
venstvo
x t x t1 2( ) – ( ) ≤ x t x t1( ) – ( ) + x t x t( ) – ( )2 ≤ 2 0 0λ δ µ
u T t c T t( – ) exp ( – )[ ].
(2.40)
V sylu polnot¥ prostranstva neprer¥vn¥x funkcyj C t b r0 1, +[ ]+δ ε yz
(2.40) sleduet, çto esly melkost\ δ( )S proyzvol\noho razbyenyq S otrezka
t T0,[ ] stremytsq k nulg, to v¥xod x ts( ) , sootvetstvugwyj vxodu u t( ) ∈
∈B H t Tl
1
0,[ ], udovletvorqgwemu uslovyg (2.11) y dannomu razbyenyg S, po-
stroenn¥j sohlasno formulam (2.1), ravnomerno stremytsq na otrezke t b0 1,[ +
+ δ εr + ] k nekotoroj neprer¥vnoj funkcyy y t1( ) . Esly b1 + δ εr + = T, to πto
dokaz¥vaet teoremu; postroennaq funkcyq y t1( ) sluΩyt v¥xodom peremennoho
hysterona W y „sxema m” realyzuetsq.
PredpoloΩym, çto b1 + δ εr + < T. Tohda vvedem oboznaçenyq dlq b1
1( ) = b1 +
+ δ εr + y δ εr +
( )1 > 0, çysla, postroennoho sohlasno formule (2.10) po b1
1( )
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
306 V. Y. BORZDÁKO
α εr + 2 B> 0 y l + 2ε > 0. Povtorqq pryvedennug v¥ße proceduru y sravnyvaq
δ εr +
( )1 c δ εr + , pryxodym k v¥vodu, analohyçnomu pryvedennomu v¥ße: v¥xod
x ts( ) pry δ( )s → 0 na otrezke t b r0 1
1 1, ( ) ( )+[ ]+δ ε ravnomerno stremytsq k nepre-
r¥vnoj funkcyy y t2( ). Na t b0 1
1, ( )[ ] funkcyy y t2( ) y y t1( ) sovpadagt. Pry
b1
2( ) = b1
1( ) + δ εr +
( )1
= T procedura zakonçena. Esly Ωe b1
2( ) < T , to ee moΩno
prodolΩyt\ dalee. Poskol\ku α εεr l+ +2
12( )– > 0 ne zavysyt ot v¥bora toçky
b1 ∈ ( , )t T0 , procedura zakonçytsq çerez koneçnoe çyslo p ßahov, hde p ravno
lybo n = E T b l r( – )( )1 2
12+[ ]+
−ε α ε — celoj çasty droby, lybo n + 1.
Postroennaq na k-m ßahe neprer¥vnaq funkcyq y tk ( ) sluΩyt prodolΩe-
nyem postroennoj na pred¥duwem ßahe funkcyy y tk – ( )1 . Postroennaq na p-m
ßahe funkcyq y tp( ) qvlqetsq v¥xodom peremennoho hysterona, sootvetstvug-
wym zadannomu vxodu u t( ) ∈ H t Tl
1
0,[ ]. Ytak, „sxema m” realyzuetsq.
Teorema dokazana.
Prymer 2.1. Rassmotrym semejstvo hysteronov W τ
, udovletvorqgwyx
uslovyqm teorem¥B2.1. Pry kaΩdom fyksyrovannom τ ∈[ ]t T0, , τ0 > 0, hysteron
W τ
predstavlqet soboj systemu, analohyçnug obobwennomu lgftu [1, c. 13]
na ploskosty P = u z,{ } s opredelqgwymy kryv¥my: Φe = { ( , )u z ; z = u + H,
H > 0}, Φr = ( , )u z{ ; z = u – d0, 0 < d0 < 1
4T };
Π( , )u M =
z z H u z
u u z z u u z
d
0 0 0
0
2
0 0 0
01 1 4
2
pry
pry
– ,
( – )
–
,
≤ ≤
+ ≤ ≤ = + +
∗τ τ
ττ
hde M = u z0 0,{ } ∈B Ω( )W τ
pry u0 = z0, uτ
∗
— abscyssa pervoj toçky pereseçe-
nyq parabol¥ z = ( – )u z0
2 τ + z0 s prqmoj z = u – d0. Kak sleduet yz [1, c. 27],
T u M( , ) =
u H u z H
u M z H u u
u d u u
+ ≤
≤ ≤
≤
∗
∗
pry
pry
pry
0
0
0
– ,
( , ) – ,
– .
Π τ
τ
Poπtomu pry lgb¥x τ1, τ2 ∈ τ0, T[ ), M ∈B Ω( )W τ
T u h M T u h Mτ τ1 2
0 0( , ) – ( , )+ + =
=
0 0 0
2
1 2 0 0
2
1 2 0
1 2
1 2
pry
pry
pry
u h z
h z u h u u
h u u u h
+ ≤
≤ + ≤ { }
≤ { } ≤ +
∗ ∗
∗ ∗
,
– min , ,
– min , .
τ τ
τ τ
τ τ
τ τ
(2.41)
Otsgda sleduet v¥polnenye dlq semejstva hysteronov W τ
uslovyq (2.7) pry B
= 1, µ = 1. Pust\ τ τ∈[ )0, T , M = u z0 0,{ } ∈ B Ω( )W τ
, N = u y0 0,{ } ∈ B Ω( )W τ
;
u t( ) , u t( )∗ = u0 , t T∗ ∈[ )τ0, , — monotonn¥j neprer¥vn¥j na τ0, T[ ) vxod.
Tohda v sylu (2.41) ymeet mesto neravenstvo
W t u z u t W t u y u tτ τ( , , ) ( ) – ( , , ) ( )∗ ∗
0 0 0 0 ≤ z y0 0– . (2.42)
Yspol\zuq (1.2) y vybrokorrektnost\ hysteronov W τ
, πto neravenstvo moΩno
perenesty na kusoçno-monotonn¥e vxod¥, a zatem na proyzvol\n¥e neprer¥vn¥e
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY 307
na τ0, T[ ) vxod¥ u t( ) . Yz (2.42) sleduet, çto semejstvo hysteronov W τ
udov-
letvorqet uslovyg (2.8). Poπtomu po nemu moΩno postroyt\ peremenn¥j hyste-
ron W , mnoΩestvo dopustym¥x vxodov kotoroho soderΩyt klass funkcyj
H T1
0τ ,[ ]. Peremenn¥j hysteron W u( , , )0 0 0ω = W , u0 ≤ ω0, opredelen dlq
monotonnoho vxoda u t( ) ∈B H T1 0,[ ], u( )0 = u0, u( )τ0 = ω0, formuloj
Wu t( ) =
ω τ τ ω
τ ω τ
0 0 0 0
0
2
0 0
0, , ( ) ,
( ) – ( ) , .
≤ ≤ =
[ ] + ≤ ≤
t u
t u t u t T
V sledugwej lemme formulyrugtsq dostatoçn¥e uslovyq, pry kotor¥x
hysteron¥ yz semejstva W τ
udovletvorqgt uslovyg (2.8).
Lemma 2.6. Pust\ kryv¥e, opredelqgwye hysteron W, udovletvorqgt ob-
wemu uslovyg Lypßyca s odnoj y toj Ωe postoqnnoj. Pust\ dlq lgboho ne-
otrycatel\noho çysla r suwestvugt takye postoqnn¥e 0 ≤ kr < + ∞ y 0 <
< αr < + ∞, çto
T u h M T u h N T u M T u N z y h( , ) – ( , ) – ( , ) – ( , ) ( – )– –
0 0 0 0 0 0
1 1+ +( ) ( )[ ] ≤ kr
(2.43)
pry lgbom h h r( )0 ≤ ≤ α , dlq lgb¥x z0, y0 ( )z y0 0≠ , u0 takyx, çto z0 ≤
≤ r, y0 ≤ r, u0 ≤ r, M = u z0 0,{ } ∈ B Ω( )W , N = u y0 0,{ } ∈ B Ω( )W , y pust\
b1 ∈ ( , )t T0 . Tohda dlq lgb¥x v¥xodov ϕ( )t , ψ( )t ∈ B Φu
r( )
ymeet mesto nera-
venstvo (2.8) pry uslovyqx (2.9), (2.10).
Dokazatel\stvo lemm¥ soderΩytsq v [7] (lemmaB2.2).
Zameçanye 2.2. I. Yz lemm¥B2.6 sleduet, çto uslovye 2 v teoremeB2.1 moΩno
zamenyt\ sledugwym:
′2 ) pust\ dlq kaΩdoho t t T∈[ ]0, systema opredelqgwyx kryv¥x T u M( , )
hysterona Wt
udovletvorqet neravenstvu (2.43).
II. Oboznaçym T u h M( , )0 + = z h u zh( , , )0 0 . Pust\ funkcyq z h u zh( , , )0 0
udovletvorqet sledugwym uslovyqm: esly u0 ≤ r, u = u h0 + ≤ r, z0 ≤ r,
r > 0, u z, 0{ } ∈ Ω( )W , to funkcyq z h u zh( , , )0 0 neprer¥vna, ymeet nepre-
r¥vnug pervug çastnug proyzvodnug ∂ ∂z zh 0 , a smeßannaq proyzvodnaq
∂
∂
∂
∂h
z
z
h
0
neprer¥vna vezde v Ω( )W , za ysklgçenyem koneçnoho mnoΩestva
toçek M z( )0 na kaΩdom otrezke Lz0
= ( , )u h z0 0+{ ∈ Ω( )W ; 0 ≤ h ≤ hz0 } , pry-
çem ( ,u h z0 0+{ } ∈ Ω( )W , 0 < h < hz0
; toçky ( , )u z0 0 , ( , )u h zz0 00
+ , v kotor¥x
predpolahaetsq suwestvovanye sootvetstvenno proyzvodn¥x
∂
∂
∂
∂h
z
z
h
h0 0
→ +
, ∂
∂
∂
∂h
z
z
h
h hz0 0
0
→ –
, (2.44)
prynadleΩat hranyce mnoΩestva Ω( )W , y pry πtom proyzvodn¥e
∂
∂
∂
∂h
z
z
h
0
y
(2.44) ohranyçen¥ po modulg postoqnnoj c, 0 ≤ c < + ∞, ne zavysqwej ot v¥bo-
ra toçky ( , )u z0 0 ∈ Ω( )W . Predpolahaetsq takΩe, çto dlq lgboho yntervala
F = ( , ) ( )u h z W0 0+{ ∈ Ω ; h0 < h < h ; ( , )u h z0 0 0+ , ( , )u h z0 0+ ∈ Ω( )W } ymeet
mesto neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
308 V. Y. BORZDÁKO
∂
∂
∂
∂
z h u z
z
z
z
h u z h hh h( , , )
– ( , , ) ( – )–0 0
0 0
0 0 0 0
1
≤ c,
hde c — konstanta, opredelennaq v¥ße. Tohda dlq funkcyj T u h M( , )0 + =
= z h u zh( , , )0 0 ymeet mesto neravenstvo (2.43).
UtverΩdenye II moΩno dokazat\, yspol\zovav tot fakt, çto levaq çast\ v
(2.43) ravna [8, c. 459 – 461]
∂
∂
∂ θ θ
∂h
z h h u y z y
z
h z0 1 0 0 2 0 0
0
0
+ +[ ]
, , ( – )
, 0 < θ
1
, θ
2
< 1,
y svojstva oblasty Ω( )W .
Prymer 2.2. Na osnove zameçanyq 2.2 (II) moΩno postroyt\ konkretn¥j
prymer hysterona W, dlq kotoroho ymeet mesto uslovye lemm¥ 2.6, obespe-
çyvagwej v¥polnenye neravenstva (2.8). Pust\ W — hysterezysnaq systema,
analohyçnaq obobwennomu lgftu [1, c. 13] s opredelqgwymy kryv¥my Φe =
= ( , )u z{ ; z = 3u H+ , H > }0 , Φr = ( , )u z{ ; z = 3u},
Π( , )u M =
=
z z H u z
H
z r u u z
H
u z
H
z r u u z
H
u
c c
c c
0 0 0
1 2 1 2
0 0
2 2 2 2
0
1
3
5
6
1
3 6
1
3 6
1
3
1 3
12
1
3 12
13 3
pry
pry
pry
– – ,
– – – – ,
– – –
( ) ( )
( ) ( )
( ) ≤ ≤ ( )
( ) ( ) ≤ ≤ +
+( )
+ ( ) +( )
≤ ≤≤ ( )
1
3
5
60z H– ,
hde r = H
6
2 3( )+ , zc
( )1 = z0 + 3
2
r , uc
( )1 = 1
3 60z H–
– r
2
, zc
( )2 = z0 – 3
2
r ,
uc
( )2 = 1
3
5
60z H–
+ r
2
, M = ( , )u z0 0 ∈ Ω( )W pry z0 = 3 0u + H
2
. Netrudno
proveryt\, çto funkcyy z h u zh( , , )0 0 = T u M( , ) dlq hysterona W obladagt tem
svojstvom, çto proyzvodn¥e
∂
∂
z
z
h
0
,
∂
∂
∂
∂h
z
z
h
0
neprer¥vn¥, a potomu ohranyçen¥
na kaΩdom zamknutom ohranyçennom mnoΩestve
( , ) ( ); ; ,u z W u u h A z A A0 0 0 0∈ = + ≤ ≤ >{ }Ω .
Sledovatel\no, hysteron W udovletvorqet uslovyqm zameçanyq 22 (II).
Dostatoçn¥e uslovyq realyzacyy „sxem¥ m”, soderΩawyesq v teoremeB2.1 y
lemmeB2.6, ne b¥ly pryveden¥ v takom vyde ny v odnoj yz pred¥duwyx rabot av-
tora, posvqwenn¥x πtoj teme [9 – 11].
Zameçanye 2.3. Peremenn¥j hysteron W , opredelenn¥j v p.B2.1, obladaet
rqdom svojstv, prysuwyx statyçeskomu hysteronu:
1) esly pry naçal\nom sostoqnyy ( , )u x0 0 ∈ Ω( )Wt0 dopustym¥j vxod u t( )
postoqnen, u t( ) ≡ u0, t0 ≤ t ≤ T, to W t u t x u t0 0 0, ( ), ( )[ ] ≡ x0, t0 ≤ t ≤ T;
2) dlq hysterona W spravedlyvo poluhruppovoe toΩdestvo (sm. sootvetst-
venno (1.2));
3) pust\ vse opredelqgwye kryv¥e vsex hysteronov Wt
, t0 ≤ t ≤ T, udovlet-
vorqgt uslovyg Lypßyca s odnoj y toj Ωe postoqnnoj m y „sxema m” realy-
zuetsq dlq nekotoroho vxoda u t H t Tl( ) ,∈ [ ]1
0 y naçal\noho sostoqnyq u t( )0{ ,
x0} ∈ Wt0 ; tohda sootvetstvugwyj v¥xod y t( ) peremennoho hysterona W
prynadleΩyt klassu H t T1
0,[ ] s postoqnnoj Lypßyca ml;
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HYSTEREZYSNÁE NESTACYONARNÁE NELYNEJNOSTY 309
4) esly vse funkcyy, hrafyky kotor¥x uçastvugt v opredelenyy hystero-
nov, vxodqwyx v semejstvo Wt
, t0 ≤ t ≤ T, udovletvorqgt obwemu uslovyg
Lypßyca, to hysteron W udovletvorqet uslovyg Lypßyca vyda
W t u t x u t W t t z t0 0 0 0 0 0, ( ), ( ) – , ( ), ( )[ ] [ ]v v ≤
≤ λ max – , max ( ) – ( )x z u s s
t s t
0 0
0 ≤ ≤
v ,
hde λ — nekotoraq postoqnnaq, u t( ) y v( )t — proyzvol\n¥e neprer¥vn¥e vxo-
d¥, dopustym¥e sootvetstvenno pry naçal\n¥x sostoqnyqx u t x( ),0 0{ }, v( )t0{ ,
z W t
0
0} ∈Ω( ) (sm. sootvetstvugwee svojstvo statyçeskoho hysterona [1, c. 35]).
∏ty svojstva v¥tekagt yz svojstv statyçeskoho hysterona, konstrukcyy
peremennoho hysterona (p.B2.1) y lemm¥B2.4 dlq dokazatel\stva svojstva 3.
1. Krasnosel\skyj M. A., Pokrovskyj A. V. System¥ s hysterezysom. – M.: Nauka, 1983. – 272 s.
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3. Krasnosel\skyj M. A., Pokrovskyj A. V. Vybroustojçyvost\ reßenyj dyfferencyal\n¥x
uravnenyj // Dokl. AN SSSR. – 1970. – 195, # 3. – S.544 – 547.
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Tam Ωe. – 1989. – 305, # 5. – S. 1065 – 1069.
5. Chernorutskii V. V., Krasnosel’skii M. A. Histeresis systems with variable characteristics // Non-
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8. Fyxtenhol\c H. M. Kurs dyfferencyal\noho y yntehral\noho ysçyslenyq: V 2 t. – M.:
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9. Borzd¥ko V. Y. Peremenn¥j hysteron // Dokl. RAN. – 1992. – 324, # 2. – S. 269 – 272.
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Poluçeno 09.07.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
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| id | umjimathkievua-article-3157 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:37:19Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/59/a84b473244ccff0fc4c87f1862994d59.pdf |
| spelling | umjimathkievua-article-31572020-03-18T19:47:10Z On the smoothness of conjugation of circle diffeomorphisms with rigid rotations Гистерезисные нестационарные нелинейности Borzdyko, V. I. Борздыко, В. И. Борздыко, В. И. We consider the operator which is a variable hysteron that describes, according to the Krasnosel'skii -Pokrovskii scheme, a nonstationary hysteresis nonlinearity with characteristics varying under external influences. We obtain sufficient conditions under which this operator is defined for inputs from the class of functions H1[t0, T] that satisfy the Lipschitz condition on the interval [t0, T]. Отримано достатні умови, за яких оператор - змінний гістерон, що описує за схемою Красносельського - Покровського нестаціонарну гістерезисну нелінійність, характеристики якої змінюються під впливом зовнішніх сил, є визначеним для входів із класу функцій H1[t0, T], що задовольняють на відрізку [t0, T] умову Ліпшиця. Institute of Mathematics, NAS of Ukraine 2008-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3157 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 3 (2008); 283–292 Український математичний журнал; Том 60 № 3 (2008); 283–292 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3157/3062 https://umj.imath.kiev.ua/index.php/umj/article/view/3157/3063 Copyright (c) 2008 Borzdyko V. I. |
| spellingShingle | Borzdyko, V. I. Борздыко, В. И. Борздыко, В. И. On the smoothness of conjugation of circle diffeomorphisms with rigid rotations |
| title | On the smoothness of conjugation of circle diffeomorphisms with rigid rotations |
| title_alt | Гистерезисные нестационарные нелинейности |
| title_full | On the smoothness of conjugation of circle diffeomorphisms with rigid rotations |
| title_fullStr | On the smoothness of conjugation of circle diffeomorphisms with rigid rotations |
| title_full_unstemmed | On the smoothness of conjugation of circle diffeomorphisms with rigid rotations |
| title_short | On the smoothness of conjugation of circle diffeomorphisms with rigid rotations |
| title_sort | on the smoothness of conjugation of circle diffeomorphisms with rigid rotations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3157 |
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