Differential equations with set-valued solutions
Some special space of convex compact sets is considered and notions of a derivative and an integral for multivalued mapping different from already known ones are introduced. The differential equation with multivalued right-hand side satisfying the Caratheodory conditions is also considered and the...
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| Дата: | 2008 |
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| Автори: | , , , , , |
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| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509301967159296 |
|---|---|
| author | Komleva, T. A. Plotnikov, A. V. Skripnik, N. V. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. |
| author_facet | Komleva, T. A. Plotnikov, A. V. Skripnik, N. V. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. |
| author_sort | Komleva, T. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:15Z |
| description | Some special space of convex compact sets is considered and notions of a derivative and an integral for multivalued mapping different from already known ones are introduced.
The differential equation with multivalued right-hand side satisfying the Caratheodory conditions is also considered and the theorems on the existence and uniqueness of its solutions are proved.
In contrast to O. Kaleva's approach, the given approach enables one to consider fuzzy differential equations as usual differential equations with multivalued solutions.
|
| first_indexed | 2026-03-24T02:38:56Z |
| format | Article |
| fulltext |
UDK 517.911.5
T. A. Komleva (Odes. nac. polytexn. un-t),
A. V. Plotnykov (Odes. akad. str-va y arxytektur¥),
N. V. Skrypnyk (Odes. nac. un-t)
DYFFERENCYAL|NÁE URAVNENYQ
S MNOHOZNAÇNÁMY REÍENYQMY
Some special space of convex compact sets is considered and notions of a derivative and an integral for
multivalued mapping different from already known ones are introduced. The differential equation with
multivalued right-hand side satisfying the Caratheodory conditions is also considered and the theorems
on the existence and uniqueness of its solutions are proved. In contrast to O. Kaleva’s approach, the
given approach enables one to consider fuzzy differential equations as usual differential equations with
multivalued solutions.
Rozhlqnuto deqkyj special\nyj prostir opuklyx kompaktnyx mnoΩyn i vvedeno ponqttq poxid-
no] ta intehrala dlq bahatoznaçnoho vidobraΩennq, wo vidriznqgt\sq vid vidomyx raniße. Ta-
koΩ rozhlqnuto dyferencial\ne rivnqnnq z bahatoznaçnog pravog çastynog, qka zadovol\nq[
vymohy Karateodori, i dovedeno teoremy isnuvannq ta [dynosti joho rozv’qzkiv. Cij pidxid da[
moΩlyvist\ rozhlqdaty neçitki dyferencial\ni rivnqnnq qk zvyçajni dyferencial\ni rivnqnnq z
bahatoznaçnymy rozv’qzkamy, wo vidriznq[ joho vid pidxodu O. Kaleva.
Razvytye teoryy mnohoznaçn¥x otobraΩenyj [1, 2] pryvelo k voprosu, çto pony-
mat\ pod proyzvodnoj y yntehralom ot mnohoznaçnoho otobraΩenyq. V 1965 h.
R. J. Aumann [3] vperv¥e vvel ponqtye yntehrala ot mnohoznaçnoho otobraΩe-
nyq, kotoroe b¥lo osnovano na yntehryrovanyy odnoznaçn¥x vetvej. Zatem v
1967 h. M. Hukuhara [4] vvel ponqtye proyzvodnoj y yntehrala dlq mnohoznaç-
n¥x otobraΩenyj, yspol\zovav podxod¥, kotor¥e suwestvovaly dlq odnoznaç-
n¥x otobraΩenyj. V 1969 h. F. S. de Blasi [5] rassmotrel dyfferencyal\n¥e
uravnenyq s proyzvodnoj Xukuxar¥ kak obobwenye ob¥knovenn¥x dyfferen-
cyal\n¥x uravnenyj, a v 1982 h. A. V. Plotnykov [6] — dyfferencyal\n¥e
vklgçenyq s proyzvodnoj Xukuxar¥. V posledugwem dann¥e uravnenyq ras-
smatryvalys\ v rabotax [7 – 17].
V to Ωe vremq s 1965 h. posle rabot¥ L. A. Zadeh [18] naçalos\ razvytye teo-
ryy neçetkyx mnoΩestv, y v 1983 h. M. L. Puri, D. A. Ralescu [19] vvely ponqtyq
H-proyzvodnoj y yntehrala dlq neçetkyx otobraΩenyj, yspol\zovav pry πtom
podxod M. Hukuhara dlq α-srezok neçetkyx otobraΩenyj. V 1985 h. O. Kaleva
[20] rassmotrel neçetkye dyfferencyal\n¥e uravnenyq y dokazal teoremu su-
westvovanyq y edynstvennosty reßenyq zadaçy Koßy dlq sluçaq, kohda pravaq
çast\ udovletvorqet uslovyg Lypßyca. V dal\nejßem neçetkye dyfferency-
al\n¥e uravnenyq rassmatryvalys\ v rabotax [21 – 25].
V dannoj rabote vvodytsq nekotoroe specyal\noe prostranstvo mnoΩestv Ω
y predlahagtsq otlyçn¥e ot yzvestn¥x opredelenyq proyzvodnoj y yntehrala
dlq mnohoznaçnoho otobraΩenyq, πlementamy kotoroho qvlqgtsq mnoΩestva yz
Ω. Dann¥j podxod daet vozmoΩnost\ yspol\zovat\ πty ponqtyq y dlq sluçaq
neçetkyx otobraΩenyj. Pry πtom xotelos\ b¥ otmetyt\, çto v dannom sluçae
ysçezagt te sloΩnosty, kotor¥e voznykagt pry yspol\zovanyy H-proyzvodnoj
y yntehrala yz [19]. TakΩe v fazovom prostranstve Ω rassmotreno dyfferen-
cyal\noe uravnenye y dokazan¥ teorem¥ suwestvovanyq reßenyq, kohda pravaq
çast\ qvlqetsq yzmerymoj po t, neprer¥vnoj po X y ohranyçennoj summyrue-
moj funkcyej, y edynstvennosty, kohda pravaq çast\ dopolnytel\no udovlet-
vorqet uslovyg Lypßyca po X.
Rassmotrym ( n + 1 ) -mernoe evklydovo prostranstvo R
n
+
1
s normoj
© T. A. KOMLEVA, A. V. PLOTNYKOV, N. V. SKRYPNYK, 2008
1326 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
DYFFERENCYAL|NÁE URAVNENYQ S MNOHOZNAÇNÁMY REÍENYQMY 1327
x x xi
i
n
n= +
=
+∑ 2
1
1 .
Oboznaçym çerez Ω prostranstvo mnoΩestv yz R
n × [ 0, 1 ] ⊂ Rn
+
1
takyx,
çto A ∈ Ω tohda y tol\ko tohda, kohda:
1) A ∈ conv ( R
n × [ 0, 1 ] );
2) dlq lgboho α ∈ [ 0, 1 ] suwestvuet a ∈ A takoe, çto a = ( … )a an
T
1, , , α ;
3) esly ( … )a an
T
1, , , α ∈ A, to ( … )a an
T
1 0, , , ∈ A.
Opredelenye 1. Summoj A � B dvux mnoΩestv A , B ∈ Ω naz¥vaetsq
mnoΩestvo
C = A � B =
{ }= = + = = { }+ + +c a b c a b i n c a bi i i n n n� : , , , min ,1 1 1 1 . (1)
Opredelenye 2. Proyzvedenyem λ � A skalqra λ ∈ R na mnoΩestvo A ∈
∈ Ω naz¥vaetsq mnoΩestvo
C = λ � A =
{ }= = = =+ +c a c a i n c ai i n nλ λ� : , , ,1 1 1 . (2)
Lemma 1. Esly A, B ∈ Ω, λ ∈ R, to A � B, λ � A ∈ Ω.
Dokazatel\stvo lemm¥ provodytsq analohyçno [26].
Operacyq summ¥ y umnoΩenyq na skalqr dlq lgb¥x mnoΩestv A, B, C ∈ Ω
y lgb¥x çysel λ, µ ∈ R udovletvorqet sledugwym svojstvam:
1) A � B = B � A;
2) ( A � B ) � C = A � ( B � C );
3) λ � ( µ � A ) = ( λ µ ) � A;
4) λ � ( A � B ) = λ � A � λ � B;
5) v prostranstve Ω suwestvuet nulevoj πlement 0 = { 0 } × [ 0, 1 ];
6) 0 � A = 0;
7) 1 � A = A.
Prostranstvo Ω ne qvlqetsq lynejn¥m prostranstvom s operacyqmy summ¥
mnoΩestv y umnoΩenyq mnoΩestva na skalqr xotq b¥ potomu, çto ne u kaΩdoho
πlementa A ∈ Ω est\ protyvopoloΩn¥j πlement (a lyß\ u mnoΩestv vyda
{ a } × [ 0, 1 ]). Krome toho, ne vsehda v¥polnqetsq neobxodym¥j dlq lynejnosty
zakon dystrybutyvnosty
( λ + µ ) � A = λ � A � µ � A,
a v¥polnqetsq vklgçenye
( λ + µ ) � A ⊂ λ � A � µ � A.
Vvedem v rassmotrenye ßar radyusa r s centrom v 0 sledugwym obrazom:
mnoΩestvo Sr ( 0 ) = Sr ( 0 ) × [ 0, 1 ], 0 — nulevoj πlement prostranstva R
n
,
Sr ( 0 ) = { x ∈ R
n : || x || ≤ r }.
Poskol\ku Sr1
( 0 ) + Sr2
( 0 ) = Sr1
+ r2
( 0 ) y λ Sr ( 0 ) = S| λ | r ( 0 ), to Sr1
( 0 ) �
� Sr2
( 0 ) = Sr1
+ r2
( 0 ) y λ � Sr ( 0 ) = S| λ | r ( 0 ).
Kak y v rabote [4], vvedem raznost\ mnoΩestv v prostranstve Ω y rassmot-
rym nekotor¥e ee svojstva.
Opredelenye 3. Raznost\g A � B dvux mnoΩestv A, B ∈ Ω naz¥vaetsq
mnoΩestvo C ∈ Ω takoe, çto A = B � C.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1328 T. A. KOMLEVA, A. V. PLOTNYKOV, N. V. SKRYPNYK
Lemma 2. Pust\ A, B, C, D ∈ Ω, λ ∈ R. Tohda:
1) ( A � B ) � ( C � D ) = ( A � C ) � ( B � D ), esly raznosty A � C y B �
� D suwestvugt;
2) ( A � B ) � ( C � D ) = ( A � C ) � ( B � D ), esly raznosty ( A � C ) � ( B �
� D ) y C � D suwestvugt;
3) ( A � B ) � ( C � D ) = ( A � C ) � ( D � B ), esly raznosty A � C, D � B
y C � D suwestvugt;
4) λ � ( A � B ) = λ � A � λ � B.
Dokazatel\stvo provodytsq analohyçno [14].
V prostranstve Ω vvedem metryku yly rasstoqnye meΩdu dvumq mnoΩest-
vamy.
Opredelenye 4. Rasstoqnyem po Xausdorfu meΩdu mnoΩestvamy A, B ∈ Ω
nazovem
h ( A, B ) = min
r ≥0
{ A ⊂ B � Sr ( 0 ), B ⊂ A � Sr ( 0 ) }.
V¥polnenye aksyom metryky sleduet yz opredelenyq y svojstv summ¥ mno-
Ωestv.
Lemma 3. Dlq lgb¥x A, B ∈ Ω y lgboho λ ∈ R spravedlyv¥ sledugwye
svojstva rasstoqnyq po Xausdorfu:
1) h ( λ � A, λ � B ) = | λ | h ( A, B ) ;
2) h ( A � B, C � D ) ≤ h ( A, C ) + h ( B, D ) ;
3) h ( A � B, C � B ) = h ( A, C ) ;
4) h ( A, B ) = h ( A � B, { 0 } ), esly raznost\ A � B suwestvuet;
5) h ( A � B, C � D ) ≤ h ( A, C ) + h ( B, D ).
Zameçanye 1. Oçevydno, çto moΩno postroyt\ yzometryçeskoe otobraΩe-
nye γ ( ⋅ ) meΩdu prostranstvom E
n
, kotoroe b¥lo rassmotreno v rabotax [19, 22
– 24], y prostranstvom Ω.
Pust\ I = [ t0 , T ].
Opredelenye 5. Mnohoznaçnoe otobraΩenye F : I → Ω naz¥vaetsq yzme-
rym¥m na I, esly dlq lgboho nepustoho mnoΩestva K ∈ Ω y lgboho ε > 0
mnoΩestvo { t ∈ I : h ( F ( t ), K ) ≤ ε } yzmerymo po Lebehu.
Opredelenye 6. Mnohoznaçnoe otobraΩenye neprer¥vno v toçke t ′ ∈ I, es-
ly dlq lgboho ε > 0 suwestvuet δ ( ε ) > 0 takoe, çto dlq vsex t ∈ I : | t – t ′ | ≤
≤ δ v¥polnqetsq neravenstvo h ( F ( t ), F ( t ′ ) ) ≤ ε.
Opredelenye 7. Mnohoznaçnoe otobraΩenye neprer¥vno na promeΩutke I,
esly ono neprer¥vno v kaΩdoj toçke πtoho promeΩutka.
Opredelenye 8. Yzmerymaq funkcyq f : I → Rn × [ 0, 1 ] naz¥vaetsq odno-
znaçnoj vetv\g mnohoznaçnoho otobraΩenyq F : I → Ω, esly dlq poçty vsex
t ∈ I v¥polnqetsq vklgçenye f ( t ) ∈ F ( t ).
Lemma 4. Esly yzmerymaq funkcyq f : I → Rn × [ 0, 1 ] qvlqetsq odnoznaç-
noj vetv\g mnohoznaçnoho otobraΩenyq F : I → Ω , to funkcyq f̂ (⋅) =
= ( )(⋅) … (⋅)f fn
T
1 0, , , takΩe qvlqetsq odnoznaçnoj yzmerymoj vetv\g mnoho-
znaçnoho otobraΩenyq F : I → Ω.
Dokazatel\stvo sleduet yz opredelenyq yzmerymoj vetvy y svojstv prost-
ranstva Ω.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
DYFFERENCYAL|NÁE URAVNENYQ S MNOHOZNAÇNÁMY REÍENYQMY 1329
Opredelenye 9. Mnohoznaçnoe otobraΩenye X : I → Ω naz¥vaetsq dyf-
ferencyruem¥m v toçke t ∈ I, esly suwestvuet mnoΩestvo D X ( t ) ∈ Ω ta-
koe, çto predel¥
lim
∆ ∆t t↓0
1 � ( X ( t + ∆ t ) � X ( t ) ) y lim
∆ ∆t t↓0
1 � ( X ( t ) � X ( t – ∆ t ) )
suwestvugt y ravn¥ D X ( t ).
Zametym, çto v dannom opredelenyy predpolahaetsq, çto dlq vsex dostatoç-
no mal¥x ∆ t > 0 raznosty X ( t ) � X ( t – ∆ t ) , X ( t + ∆ t ) � X ( t ) suwestvugt.
Esly t — hranyçnaq toçka promeΩutka I, to ymeet sm¥sl hovoryt\ lyß\ o
dyfferencyruemosty sleva (sprava) funkcyy X : I → Ω v toçke t.
Lemma 5. Pust\ mnohoznaçn¥e otobraΩenyq X, Y : I → Ω dyfferencyrue-
m¥ v toçke t ∈ I. Tohda:
1) mnohoznaçnoe otobraΩenye X � Y : I → Ω dyfferencyruemo v toçke t
y pry πtom spravedlyvo ravenstvo
D ( X � Y ) ( t ) = D X ( t ) � D Y ( t );
2) esly v okrestnosty toçky t suwestvuet raznost\ X � Y , to mnoho-
znaçnoe otobraΩenye X � Y : I → Ω dyfferencyruemo v toçke t y pry πtom
spravedlyvo ravenstvo
D ( X � Y ) ( t ) = D X ( t ) � D Y ( t );
3) dlq lgboho λ ∈ R mnohoznaçnoe otobraΩenye λ � X : I → Ω dyfferen-
cyruemo v toçke t y pry πtom spravedlyvo ravenstvo
D ( λ � X ) ( t ) = λ � D X ( t ).
Dokazatel\stvo. 1. V sylu opredelenyq
D X ( t ) = lim
∆ ∆t t↓0
1 � ( X ( t + ∆ t ) � X ( t ) ) = lim
∆ ∆t t↓0
1 � ( X ( t ) � X ( t – ∆ t ) ),
D Y ( t ) = lim
∆ ∆t t↓0
1 � ( Y ( t + ∆ t ) � Y ( t ) ) = lim
∆ ∆t t↓0
1 � ( Y ( t ) � Y ( t – ∆ t ) ).
Tohda dlq lgboho ε > 0 suwestvuet δ ( ε ) > 0 takoe, çto pry 0 < ∆ t ≤ δ v¥-
polnqgtsq neravenstva
h
t
X t t X t DX t1
∆
∆( )( + ) ( ) ( )
� , ≤ ε,
h
t
X t X t t DX t1
∆
∆( )( ) ( − ) ( )
� , ≤ ε,
h
t
Y t t Y t DY t1
∆
∆( )( + ) ( ) ( )
� , ≤ ε,
h
t
Y t Y t t DY t1
∆
∆( )( ) ( − ) ( )
� , ≤ ε.
Pry 0 < ∆ t ≤ δ, yspol\zuq svojstva raznosty y rasstoqnyq po Xausdorfu, po-
luçaem
h
t
X t t Y t t X t Y t DX t DY t1
∆
∆ ∆� � � � �(( ) ( ))( + ) ( + ) ( ) ( ) ( ) ( )
, =
=
h
t
X t t X t Y t t Y t DX t DY t1
∆
∆ ∆� � � � �(( ) ( ))( + ) ( ) ( + ) ( ) ( ) ( )
, =
=
h
t
X t t X t
t
Y t t Y t DX t DY t1 1
∆
∆
∆
∆� �� � � �( ) ( )( + ) ( ) ( + ) ( ) ( ) ( )
, ≤
≤
h
t
X t t X t DX t1
∆
∆� �( )( + ) ( ) ( )
, +
h
t
Y t t Y t DY t1
∆
∆� �( )( + ) ( ) ( )
, ≤
≤ ε + ε = 2ε.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1330 T. A. KOMLEVA, A. V. PLOTNYKOV, N. V. SKRYPNYK
Analohyçno
h
t
X t Y t X t t Y t t DX t DY t1
∆
∆ ∆� � � � �(( ) ( ))( ) ( ) ( − ) ( − ) ( ) ( )
, ≤ 2ε.
Takym obrazom, proyzvodnaq ot mnohoznaçnoho otobraΩenyq ( X � Y ) ( t ) v
toçke t suwestvuet y spravedlyvo ravenstvo
D ( X � Y ) ( t ) = D X ( t ) � D Y ( t ).
Punkt¥ 2 y 3 lemm¥ dokaz¥vagtsq analohyçno.
Opredelenye 10. Mnohoznaçnoe otobraΩenye X : I → Ω naz¥vaetsq dyf-
ferencyruem¥m na I, esly ono dyfferencyruemo vo vsex toçkax t ∈ I.
Lemma 6. Mnohoznaçnoe otobraΩenye X : I → Ω postoqnno tohda y tol\ko
tohda, kohda D X ( t ) ≡ 0.
Dokazatel\stvo provodytsq analohyçno [14].
Opredelenye 11. Obobwenn¥m yntehralom Aumanna ot mnohoznaçnoho
otobraΩenyq F : I → Ω na otrezke I naz¥vaetsq mnoΩestvo
G = F t dt g f t dt i n g f F
t
T
i i
t
T
n
T
( ) = = ( ) =
(⋅) ∈ (⋅)
∫ ∫ +
0 0
1 1, , ; : , (3)
hde f (⋅) = ( )(⋅) … (⋅) (⋅)+f f fn n
T
1 1, , , — odnoznaçnaq vetv\ F(⋅),
g f t f t dt g i n En
E t T t T E
n
t T E
i i+
⊂[ ] [ ]
+
[ ]
= ( ) ( ) = = =
∫1 1
0 0
0
1 0sup inf : , , , mes
, , \
, \
,
yntehral ot fi(⋅), i = 1, n , ponymaetsq v sm¥sle Lebeha.
Lemma 7. Pust\ F : I → Ω — neprer¥vnoe mnohoznaçnoe otobraΩenye.
Tohda F(⋅) yntehryruemo na I y F s ds
t
T
( )∫
0
∈ Ω.
Dokazatel\stvo provodytsq analohyçno [20, 27].
Lemma 8. Pust\ mnohoznaçn¥e otobraΩenyq F, R : I → Ω yntehryruem¥,
λ ∈ R — skalqr. Tohda mnohoznaçn¥e otobraΩenyq F � R, λ � F : I → Ω yn-
tehryruem¥ y
1) ( )( ) = ( ) ( )∫ ∫ ∫F R t dt F t dt R t dt
t
T
t
T
t
T
� �
0 0 0
;
2)
( )( ) = ( )∫ ∫λ λ� �F t dt F t dt
t
T
t
T
0 0
;
3) F s ds F s ds F s ds
t
t
t
T
t
T
( ) ( ) = ( )∫ ∫ ∫
0 0
� , t ∈ ( t0 , T ).
Dokazatel\stvo. UtverΩdenyq lemm¥ sledugt yz svojstv yntehrala Le-
beha, lemm¥ 1 y svojstv supremuma, ynfymuma y mynymuma:
1) dlq lgb¥x selektorov f F(⋅) ∈ (⋅), r R(⋅) ∈ (⋅) y lgboho yzmerymoho pod-
mnoΩestva
( )( ) + ( ) = ( ) + ( )∫ ∫ ∫f t r t dt f t dt r t dti i
J
i
J
i
J
, λ λf t dt f t dti
J
i
J
( ) = ( )∫ ∫ ,
f t dt f t dt f t dti
J
i
J
i
J
( ) + ( ) = ( )∫ ∫ ∫
1 2
, J1 ∪ J2 = J, J1 ∩ J2 = ∅, i = 1, n ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
DYFFERENCYAL|NÁE URAVNENYQ S MNOHOZNAÇNÁMY REÍENYQMY 1331
2) dlq lgb¥x selektorov f F(⋅) ∈ (⋅), r R(⋅) ∈ (⋅)
min sup inf : , , , mes
\
\E I I E
n
I E
i if t f t dt g i n E
⊂
+ ( ) ( ) = ′ = =
∫1 1 0 ,
sup inf : , , , mes
\
\E I I E
n
I E
i ir t r t dt g i n E
⊂
+ ( ) ( ) = ′′ = =
∫1 1 0 =
= sup inf min , :
\
\E I I E
n n
I E
i i if t r t f t r t dt g
⊂
+ +
( ) ( ) ( ) + ( ) ={ } ( )∫1 1 =
= ′ + ′′ = =
g g i n Ei i , , , mes1 0 ;
3) dlq lgboho selektora f F(⋅) ∈ (⋅) y lgboho skalqra λ ∈ R
sup inf : , , , mes
\
\E I I E
n
I E
i if t f t dt g i n E
⊂
+ ( ) ( ) = = =
∫1 1 0λ =
= sup inf : , , , mes
\
\E I I E
n n
I E
i ig t f t f t dt g i n E
⊂
+ +( ) = ( ) ( ) = = =
∫1 1 1 0λ .
Lemma 9. Pust\ mnohoznaçn¥e otobraΩenyq F, G : I → Ω yntehryruem¥.
Tohda:
1) skalqrnaq funkcyq h ( F ( t ), G ( t ) ) yntehryruema;
2) h F s ds G s ds h F s G s ds
t
t
t
t
t
t
( ) ( )
≤ ( ) ( )∫ ∫ ∫ ( )
0 0 0
, , .
Dokazatel\stvo provodytsq analohyçno [20, 27].
Lemma 10. Pust\ mnohoznaçnoe otobraΩenye F : I → Ω neprer¥vno. Tohda
dlq kaΩdoho t ∈ ( t0 , T ) mnohoznaçnoe otobraΩenye G ( t ) = F s ds
t
t
( )∫
0
nepre-
r¥vno dyfferencyruemo y D G ( t ) = F ( t ) vsgdu na I.
Dokazatel\stvo. Voz\mem proyzvol\noe t ∈ ( t0 , T ). Poskol\ku F(⋅) ne-
prer¥vno v toçke t, dlq lgboho ε > 0 suwestvuet δ > 0 takoe, çto dlq vsex
| ∆ t | ≤ δ takyx, çto t + ∆ t ∈ ( t0 , T ), v¥polnqetsq neravenstvo h ( F ( t + ∆ t ),
F ( t ) ) < ε. Tohda dlq vsex ∆ t takyx, çto 0 < ∆ t < δ y t + ∆ t < T, spravedlyvo
h
t
G t t G t F t
t
h G t t G t t F t1 1
∆
∆
∆
∆ ∆� �� �( ) ( )( + ) ( ) ( )
= ( + ) ( ) ( ), , =
=
1 1
0 0
∆
∆
∆
∆
∆ ∆
t
h F s ds F s ds tF t
t
h F s ds tF t
t
t t
t
t
t
t t
( ) ( ) ( )
= ( ) ( )
+ +
∫ ∫ ∫� , , =
=
1 1
∆ ∆
∆ ∆ ∆
t
h F s ds F t ds
t
h F s F t ds
t
t t
t
t t
t
t t
( ) ( )
≤ ( ) ( )
+ + +
∫ ∫ ∫ ( ), , < ε.
Analohyçno, dlq vsex ∆ t takyx, çto 0 < ∆ t < δ y t0 < t – ∆ t,
h
t
G t G t t F t1
∆
∆� �( )( ) ( − ) ( )
, < ε.
Sledovatel\no, lemma dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1332 T. A. KOMLEVA, A. V. PLOTNYKOV, N. V. SKRYPNYK
Lemma 11. Pust\ mnohoznaçnoe otobraΩenye X : I → Ω neprer¥vno dyf-
ferencyruemo, tohda mnohoznaçnoe otobraΩenye D X : I → Ω obobwenno yn-
tehryruemo y
X ( t ) =
X t DX s ds
t
t
( ) ( )∫0
0
� .
Dokazatel\stvo. Sohlasno lemme 7, mnohoznaçnoe otobraΩenye D X ( t )
yntehryruemo na I . Vvedem v rassmotrenye mnohoznaçnoe otobraΩenye Φ ( t ) =
= X t DX s ds
t
t
( ) ( )∫0
0
� . V sylu lemm¥ 10 Φ ( t ) dyfferencyruemo na I kak sum-
ma postoqnnoho mnoΩestva y yntehrala s peremenn¥m verxnym predelom ot ne-
prer¥vnoho mnohoznaçnoho otobraΩenyq, pryçem D Φ ( t ) = D X ( t ). Takym obra-
zom, pry dostatoçno mal¥x ∆ t raznosty Φ ( t + ∆ t ) � Φ ( t ), Φ ( t ) � Φ ( t – ∆ t ),
X ( t + ∆ t ) � X ( t ), X ( t ) � X ( t – ∆ t ) suwestvugt y spravedlyv¥ ravenstva
lim ,
∆ ∆
Φ ∆ Φ
t
h
t
t t t DX t
↓
( )( + ) ( ) ( )
0
1
� =
=
lim ,
∆ ∆
Φ Φ ∆
t
h
t
t t t DX t
↓
( )( ) ( − ) ( )
0
1
� = 0,
lim ,
∆ ∆
∆
t
h
t
X t t X t DX t
↓
( )( + ) ( ) ( )
0
1
� =
= lim ,
∆ ∆
∆
t
h
t
X t X t t DX t
↓
( )( ) ( − ) ( )
0
1
� = 0.
Tohda v sylu svojstv rasstoqnyq po Xausdorfu
h ( Φ ( t + ∆ t ), X ( t + ∆ t ) ) =
= h (( Φ ( t + ∆ t ) � Φ ( t ) ) � Φ ( t ), ( X ( t + ∆ t ) � X ( t ) ) � X ( t ) ) ≤
≤ h (Φ ( t + ∆ t ) � Φ ( t ) , X ( t + ∆ t ) � X ( t ) ) + h ( Φ ( t ), X ( t ) );
h ( Φ ( t ), X ( t ) ) =
= h (Φ ( t + ∆ t ) � ( Φ ( t + ∆ t ) � Φ ( t ) ), X ( t + ∆ t ) � ( X ( t + ∆ t ) � X ( t ) )) ≤
≤ h (Φ ( t + ∆ t ), X ( t + ∆ t ) ) + h ( Φ ( t + ∆ t ) � Φ ( t ), X ( t + ∆ t ) � X ( t ) ).
Sledovatel\no,
| h ( Φ ( t + ∆ t ), X ( t + ∆ t ) ) – h ( Φ ( t ), X ( t ) ) | ≤
≤ h ( Φ ( t + ∆ t ) � Φ ( t ), X ( t + ∆ t ) � X ( t ) ).
Analohyçno
| h ( Φ ( t – ∆ t ), X ( t – ∆ t ) ) – h ( Φ ( t ), X ( t ) ) | ≤
≤ h ( Φ ( t ) � Φ ( t – ∆ t ), X ( t ) � X ( t – ∆ t ) ).
Tohda, yspol\zuq neravenstvo treuhol\nyka, pry ∆ t ↓ 0 poluçaem
1
∆t
| h ( Φ ( t + ∆ t ), X ( t + ∆ t ) ) – h ( Φ ( t ), X ( t ) ) | ≤
≤ 1
∆t
h ( Φ ( t + ∆ t ) � Φ ( t ), X ( t + ∆ t ) � X ( t ) ) ≤
≤
h
t
t t t DX t h
t
X t t X t DX t1 1
∆
Φ ∆ Φ
∆
∆( ) ( )( + ) ( ) ( )
+ ( + ) ( ) ( )
� �, , → 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
DYFFERENCYAL|NÁE URAVNENYQ S MNOHOZNAÇNÁMY REÍENYQMY 1333
Analohyçno
1
∆t
| h ( Φ ( t – ∆ t ) , X ( t – ∆ t ) ) – h ( Φ ( t ), X ( t ) ) | ≤
≤ 1
∆t
h ( Φ ( t ) � Φ ( t – ∆ t ) , X ( t ) � X ( t – ∆ t ) ) ≤
≤
h
t
t t t DX t h
t
X t X t t DX t1 1
∆
Φ Φ ∆
∆
∆( ) ( )( ) ( − ) ( )
+ ( ) ( − ) ( )
� �, , → 0.
Takym obrazom, proyzvodnaq skalqrnoj funkcyy t → h ( Φ ( t ), X ( t ) ) ravna
nulg dlq lgboho t ∈ I, t. e. h ( Φ ( t ), X ( t ) ) postoqnna. Poskol\ku ona ravna 0
pry t = t0 , ona toΩdestvenno ravna nulg, a znaçyt, Φ ( t ) ≡ X ( t ) pry t ∈ I.
Lemma dokazana.
Zameçanye 2. V sluçae, kohda mnohoznaçnoe otobraΩenye F : I → Ω v kaΩ-
d¥j moment vremeny t ∈ I ymeet vyd F ( t ) = F̂ t( ) × [ 0, 1 ] :
1) mnohoznaçnoe otobraΩenye F : I → Ω dyfferencyruemo na I tohda y
tol\ko tohda, kohda F̂ : I → conv ( R
n
) dyfferencyruemo po Xukuxare [4] na I
y spravedlyvo toΩdestvo D F ( t ) = Dh F̂ t( ) × [ 0, 1 ];
2) mnohoznaçnoe otobraΩenye F : I → Ω yntehryruemo na I tohda y tol\ko
tohda, kohda F̂ : I → conv ( R
n
) yntehryruemo po Aumannu [3] na I y spraved-
lyvo toΩdestvo F t dt
t
T
( )∫
0
=
( ) ( )∫A F̂ t dt
t
T
0
× [ 0, 1 ], hde Dh F̂ t( ) — proyzvodnaq
Xukuxar¥ [4], ( ) ( )∫A F̂ t dt
t
T
0
— yntehral Aumanna [3].
Zameçanye 3. MoΩno pokazat\, uçyt¥vaq zameçanye 1, çto:
1) mnohoznaçnoe otobraΩenye F : I → En
dyfferencyruemo na I v sm¥sle
[19, 20] tohda y tol\ko tohda, kohda γ ( F ( ⋅ ) ) : I → Ω dyfferencyruemo na I y
spravedlyvo toΩdestvo D γ ( F ( t ) ) ≡ γ ( DPR
F ( t ) );
2) esly mnohoznaçnoe otobraΩenye F : I → En
yntehryruemo na I v sm¥sle
[19, 20], to γ ( F ( ⋅ ) ) : I → Ω yntehryruemo na I y spravedlyvo toΩdestvo
γ( )( )∫ F t dt
t
T
0
= γ ( ) ( )
∫PR F t dt
t
T
0
, hde DPR
F ( t ), ( ) ( )∫PR F t dt
t
T
0
— proyzvodnaq y
yntehral, opredelenn¥e v [19, 20].
Opredelenye 12. Mnohoznaçnoe otobraΩenye F : I → Ω naz¥vaetsq abso-
lgtno neprer¥vn¥m na promeΩutke I, esly suwestvugt mnoΩestvo X 0 ∈ Ω
y obobwenno yntehryruemoe yzmerymoe mnohoznaçnoe otobraΩenye f : I → Ω
takye, çto
F ( t ) =
X f s ds
t
t
0
0
� ( )∫ .
Zameçanye 4. Kak y v [14], moΩno dokazat\, çto esly mnohoznaçnoe otobra-
Ωenye F : I → Ω yzmerymo y suwestvuet summyruemaq funkcyq k ( t ) takaq, çto
dlq poçty vsex t ∈ I v¥polnqetsq neravenstvo h ( F ( t ), { 0 } ) ≤ k ( t ), to mnoho-
znaçnoe otobraΩenye G ( t ) = F s ds
t
t
( )∫
0
absolgtno neprer¥vno na I y D G ( t ) =
= F ( t ) poçty vsgdu na I.
Rassmotrym dyfferencyal\noe uravnenye
D X = F ( t, X ), X ( t0 ) = X0 , (4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1334 T. A. KOMLEVA, A. V. PLOTNYKOV, N. V. SKRYPNYK
hde F : I × Ω → Ω — mnohoznaçnoe otobraΩenye.
Opredelenye 13. Mnohoznaçnoe otobraΩenye X : J → Ω , J ⊂ I, naz¥vaetsq
reßenyem dyfferencyal\noho uravnenyq (4), esly ono absolgtno neprer¥vno
na J y udovletvorqet πtomu uravnenyg pry poçty vsex t ∈ J.
Narqdu s dyfferencyal\n¥m uravnenyem (4) rassmotrym yntehral\noe
uravnenye
X ( t ) =
X t F s X s ds
t
t
( ) ( )( )∫0
0
� , . (5)
Opredelenye 14. Mnohoznaçnoe otobraΩenye X : J → Ω , J ⊂ I, naz¥vaetsq
reßenyem yntehral\noho uravnenyq (5), esly ono neprer¥vno na J y udovlet-
vorqet πtomu uravnenyg pry vsex t ∈ J.
Tohda v sylu lemm 10, 11 y zameçanyq 4 uravnenyq (4) y (5) qvlqgtsq πkvyva-
lentn¥my, t. e. reßenye uravnenyq (4) qvlqetsq reßenyem (5) y naoborot.
Teorema 1. Pust\ v oblasty Q = { ( t, X ) : t0 ≤ t ≤ t0 + a, h ( X, X0 ) ≤ b } mno-
hoznaçnoe otobraΩenye F ( t, X ) udovletvorqet sledugwym uslovyqm:
a) F ( t, X ) yzmerymo po t dlq lgboho fyksyrovannoho X;
b) F ( t, X ) neprer¥vno po X dlq lgboho fyksyrovannoho t ;
v) h ( F ( t, X ), { 0 } ) ≤ m ( t ), hde m ( t ) summyruema na [ t0 , t0 + a ].
Tohda na otrezke [ t0 , t0 + d ] , hde d > 0, suwestvuet reßenye zadaçy (4)
takoe, çto
d ≤ a, ϕ ( t0 + d ) ≤ b, ϕ ( t ) = m s ds
t
t
( )∫
0
.
Dokazatel\stvo. Dlq lgboho celoho k ≥ 1 voz\mem h = d / k. Posledova-
tel\no na otrezkax
t0 + i h ≤ t ≤ t0 + ( i + 1 ) h, i = 0, 1, … , k – 1,
postroym pryblyΩennoe reßenye, poloΩyv X
k
( t ) = X0 pry t ≤ t0 ,
X
k
( t ) =
X F s X s h dsk
t
t
0
0
� ( )( − )∫ , pry t0 < t ≤ t0 + d. (6)
PokaΩem, çto pry kaΩdom k ≥ 1 mnohoznaçnoe otobraΩenye X
k
( t ) oprede-
leno, neprer¥vno y udovletvorqet neravenstvu
h ( X
k
( t ), X0 ) ≤ b pry t0 ≤ t ≤ t0 + d. (7)
Dlq dokazatel\stva vospol\zuemsq metodom polnoj matematyçeskoj ynduk-
cyy. Pust\ t ∈ [ t0 , t0 + h ], tohda v sylu (6)
X
k
( t ) =
X F s X ds
t
t
0 0
0
� ( )∫ , .
Poskol\ku mnohoznaçnoe otobraΩenye F ( t, X0 ) yzmerymo na [ t0 , t0 + h ], v
sylu zameçanyq 4 X
k
( t ) opredeleno y neprer¥vno. Krome toho, ymeem
h ( X
k
( t ), X0 ) ≤ h F s X ds m s ds t d
t
t
t
t
( )( ) { } ≤ ( ) ≤ ( + )∫ ∫, ,0 0
0 0
0 ϕ ≤ b.
PredpoloΩym, çto mnohoznaçnoe otobraΩenye X
k
( t ) udovletvorqet pere-
çyslenn¥m uslovyqm na otrezke [ t0 , t0 + i h ]. Rassmotrym t ∈ [ t0 , t0 + ( i + 1 ) h ].
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
DYFFERENCYAL|NÁE URAVNENYQ S MNOHOZNAÇNÁMY REÍENYQMY 1335
Tohda mnohoznaçnoe otobraΩenye F ( t, X
k
( t – h ) ) yzmerymo, sledovatel\no,
X
k
( t ) opredeleno y neprer¥vno. Krome toho, ymeem
h ( X
k
( t ), X0 ) ≤ h F s X s h ds m s ds t dk
t
t
t
t
( ) )( ( − ) { } ≤ ( ) ≤ ( + )∫ ∫, , 0
0 0
0ϕ ≤ b.
Takym obrazom, utverΩdenye spravedlyvo.
Yz (7) sleduet, çto posledovatel\nost\ mnohoznaçn¥x otobraΩenyj
{ }( ) =
∞X tk
k 1 ravnomerno ohranyçena:
h ( X0 , { 0 } ) – b ≤ h ( X
k
( t ), { 0 } ) ≤ h ( X0 , { 0 } ) + b.
PokaΩem, çto mnohoznaçn¥e otobraΩenyq X
k
( t ) ravnostepenno neprer¥v-
n¥. Dlq lgb¥x α, β ∈ [ t0 , T ], α ≤ β, y lgboho natural\noho k v¥polnqetsq
neravenstvo
h ( X
k
( β ), X
k
( α ) ) =
h X F s X s h ds X F s X s h dsk
t
k
t
0 0
0 0
� �( ) ( )( − ) ( − )
∫ ∫, , ,
β α
=
= h F s X s h ds h F s X s h dsk k( ) ( )( − ) { }
≤ ( ( − )) { }∫ ∫, , , ,
α
β
α
β
0 0 ≤
≤ m s ds( ) ≤ ( ) − ( )∫
α
β
ϕ β ϕ α . (8)
Funkcyq ϕ ( t ) qvlqetsq absolgtno neprer¥vnoj na [ t0 , t0 + d ] kak ynteh-
ral s peremenn¥m verxnym predelom ot summyruemoj funkcyy. Sledovatel\no,
ϕ ( t ) ravnomerno neprer¥vna, t. e. dlq lgboho ε > 0 suwestvuet δ ( ε ) > 0 ta-
koe, çto dlq vsex α, β ∈ [ t0 , t0 + d ] : 0 ≤ β – α < δ v¥polnqetsq neravenstvo
ϕ ( β ) – ϕ ( α ) < ε. Tohda v sylu (8) h ( X
k
( β ), X
k
( α ) ) < ε pry 0 ≤ β – α < δ, t. e.
posledovatel\nost\ { }( ) =
∞X tk
k 1 ravnostepenno neprer¥vna.
Po teoreme Askoly [28] yz posledovatel\nosty { }( ) =
∞X tk
k 1 v¥berem ravno-
merno sxodqwugsq podposledovatel\nost\. Ee predelom qvlqetsq neprer¥vnoe
mnohoznaçnoe otobraΩenye, kotoroe oboznaçym X ( t ). Poskol\ku
h ( X
k
( s – h ), X ( s ) ) ≤ h ( X
k
( s – h ), X
k
( s ) ) + h ( X
k
( s ), X ( s ) ),
a pervoe slahaemoe pravoj çasty v sylu ravnostepennoj neprer¥vnosty mnoho-
znaçn¥x otobraΩenyj { X
k
( t ) } men\ße ε pry h = d / k < δ ( ε ), po v¥brannoj
podposledovatel\nosty { X
k
( s – h ) } stremytsq k X ( s ). V sylu uslovyj 2 y 3
teorem¥ v (6) moΩno perejty k predelu pod znakom yntehrala. Poluçaem, çto
predel\noe mnohoznaçnoe otobraΩenye X ( t ) udovletvorqet uravnenyg (5) pry
X ( t0 ) = X0 , t. e. qvlqetsq reßenyem zadaçy (4).
Teorema 2. Pust\ v oblasty Q = { ( t, X ) : t0 ≤ t ≤ t0 + a, h ( X, X0 ) ≤ b } mno-
hoznaçnoe otobraΩenye F ( t, X ) udovletvorqet uslovyqm teorem¥ 1 y, krome
toho, suwestvuet postoqnnaq L > 0 takaq, çto dlq lgb¥x toçek ( t, X ),
( t, Y ) ∈ Q v¥polnqetsq neravenstvo
h ( F ( t, X ) , F ( t, Y ) ) ≤ L h ( X, Y ). (9)
Tohda uravnenye (4) ymeet edynstvennoe reßenye.
Dokazatel\stvo. PredpoloΩym protyvnoe, t. e. pust\ uravnenye (4) ymeet
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1336 T. A. KOMLEVA, A. V. PLOTNYKOV, N. V. SKRYPNYK
po krajnej mere dva reßenyq X ( t ) y Y ( t ) takyx, çto θ = max
,t t t h∈[ + ]0 0
h ( X ( t ),
Y ( t ) ) > 0, hde [ t0 , t0 + h ], 0 < h ≤ a — obwyj promeΩutok suwestvovanyq
reßenyj X ( t ) y Y ( t ).
V sylu πkvyvalentnosty uravnenyj (4) y (5)
X ( t ) ≡
X F s X s ds
t
t
0
0
� ( )( )∫ , , Y ( t ) ≡
X F s Y s ds
t
t
0
0
� ( )( )∫ , ,
otkuda, yspol\zuq uslovye (9) y svojstva rasstoqnyq po Xausdorfu, ymeem
h ( X ( t ), Y ( t ) ) ≤
h X F s X s ds X F s Y s ds
t
t
t
t
0 0
0 0
� �( ) ( )( ) ( )
∫ ∫, , , =
= h F s X s ds F s Y s ds h F s X s F s Y s ds
t
t
t
t
t
t
( ) ( ) ( ) ))( ) ( )
≤ ( ( ) ( ( )∫ ∫ ∫, , , , , ,
0 0 0
≤
≤ L h X s Y s ds
t
t
( )( ) ( )∫ ,
0
.
Takym obrazom, poluçaem posledovatel\nost\ ocenok
h ( X ( t ), Y ( t ) ) ≤ L ds L t t L h
t
t
θ θ θ
0
0∫ = ( − ) ≤ ,
h ( X ( t ), Y ( t ) ) ≤ L L s t ds L
t t
L h
t
t
θ θ θ( − ) = ( − ) ≤∫ 0
2 0
2
2
2
0
2 2! !
, … .
Yspol\zuq metod polnoj matematyçeskoj yndukcyy, netrudno pokazat\, çto
dlq lgboho natural\noho n na otrezke [ t0 , t0 + h ] ymeet mesto neravenstvo
h ( X ( t ), Y ( t ) ) ≤ L
h
m
m
m
θ
!
.
Tohda
θ = max
,t t t h∈[ + ]0 0
h ( X ( t ), Y ( t ) ) ≤ L
h
m
m
m
θ
!
,
otkuda v sylu poloΩytel\nosty θ sleduet, çto
1 ≤
( )Lh
m
m
!
(10)
dlq lgboho natural\noho n.
Po pryznaku Dalambera rqd
( )
=
∞∑ Lh
m
m
m !1
sxodytsq y poπtomu v sylu neobxo-
dymoho uslovyq lim
!m
mLh
m→∞
( )
= 0. ∏to oznaçaet, çto dlq ε = 1
2
suwestvuet m ∈ N
takoe, çto
( ) <Lh
m
m
!
1
2
. Tohda v sylu (10) ymeem 1 < 1
2
.
Poluçennoe protyvoreçye voznyklo v rezul\tate nevernoho predpoloΩenyq,
sledovatel\no, uravnenye (4) ymeet edynstvennoe reßenye.
Teorema dokazana.
1. Borysovyç G. H., Hel\man B. D., M¥ßkys A. D., Obuxovskyj V. V. Mnohoznaçn¥e otobraΩe-
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Poluçeno 19.02.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
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| id | umjimathkievua-article-3247 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:38:56Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/c1/60a4202030ec8d5619a42305131543c1.pdf |
| spelling | umjimathkievua-article-32472020-03-18T19:49:15Z Differential equations with set-valued solutions Дифференциальные уравнения с многозначными решениями Komleva, T. A. Plotnikov, A. V. Skripnik, N. V. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. Some special space of convex compact sets is considered and notions of a derivative and an integral for multivalued mapping different from already known ones are introduced. The differential equation with multivalued right-hand side satisfying the Caratheodory conditions is also considered and the theorems on the existence and uniqueness of its solutions are proved. In contrast to O. Kaleva's approach, the given approach enables one to consider fuzzy differential equations as usual differential equations with multivalued solutions. Розглянуто деякий спеціальний простір опуклих компактних множин i введено поняття похідної та інтеграла для багатозначного відображення, що відрізняються від відомих раніше. Також розглянуто диференціальне рівняння з багатозначною правою частиною, яка задовольняє вимоги Каратеодорі, і доведено теореми існування та єдиності його розв'язків. Цій підхід дає можливість розглядати нечіткі диференціальні рівняння як звичайні диференціальні рівняння з багатозначними розв'язками, що відрізняє його від підходу O. Kaleva. Institute of Mathematics, NAS of Ukraine 2008-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3247 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 10 (2008); 1326–1337 Український математичний журнал; Том 60 № 10 (2008); 1326–1337 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3247/3240 https://umj.imath.kiev.ua/index.php/umj/article/view/3247/3241 Copyright (c) 2008 Komleva T. A.; Plotnikov A. V.; Skripnik N. V. |
| spellingShingle | Komleva, T. A. Plotnikov, A. V. Skripnik, N. V. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. Комлева, Т. А Плотников, А. В. Скрипник, Н. В. Differential equations with set-valued solutions |
| title | Differential equations with set-valued solutions |
| title_alt | Дифференциальные уравнения с многозначными решениями |
| title_full | Differential equations with set-valued solutions |
| title_fullStr | Differential equations with set-valued solutions |
| title_full_unstemmed | Differential equations with set-valued solutions |
| title_short | Differential equations with set-valued solutions |
| title_sort | differential equations with set-valued solutions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3247 |
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