Conditions for the stability of an impulsive linear equation with pure delay
We establish necessary and sufficient conditions for the stability of one class of impulsive linear differential equations with delay.
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| Date: | 2009 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2009
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3092 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509122762375168 |
|---|---|
| author | Ivanov, I. L. Slyn'ko, V. I. Іванов, I. Л. Слинько, В. І. |
| author_facet | Ivanov, I. L. Slyn'ko, V. I. Іванов, I. Л. Слинько, В. І. |
| author_sort | Ivanov, I. L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:45:12Z |
| description | We establish necessary and sufficient conditions for the stability of one class of impulsive linear differential equations with delay. |
| first_indexed | 2026-03-24T02:36:05Z |
| format | Article |
| fulltext |
UDK 517. 9
I. L. Ivanov (Ky]v. nac. un-t im. T. Íevçenka),
V. I. Slyn\ko (In-t mexaniky NAN Ukra]ny, Ky]v)
UMOVY STIJKOSTI LINIJNOHO RIVNQNNQ
Z ÇYSTYM ZAPIZNENNQM TA IMPUL|SNOG DI{G
Necessary and sufficient stability conditions are established for the class of linear differential equations
with delay and pulse action.
Ustanovlen¥ neobxodym¥e y dostatoçn¥e uslovyq ustojçyvosty dlq klassa lynejn¥x dyffe-
rencyal\n¥x uravnenyj s zapazd¥vanyem y ympul\sn¥m vozdejstvyem.
Stijkist\ rozv’qzkiv dyferencial\nyx rivnqn\ z impul\snog di[g, vklgçagçy
periodyçni systemy, bula predmetom vyvçennq v rqdi robit (dyv., napryklad, [1 –
3]). U robotax [4 – 6] bulo doslidΩeno systemy dyferencial\nyx rivnqn\ z im-
pul\snog di[g i zapiznennqm. V osnovu cyx doslidΩen\ [4, 5] pokladeno prqmyj
metod Lqpunova v po[dnanni z koncepci[g B. S. Razumixina. Aktual\nog zada-
çeg [ pobudova analoha teori] Floke dlq c\oho klasu dyferencial\nyx rivnqn\.
U danij roboti dlq skalqrnoho rivnqnnq z çystym zapiznennqm, velyçyna qkoho
zbiha[t\sq z periodom impul\sno] di], pry deqkyx dodatkovyx prypuwennqx pobu-
dovano analoh operatora monodromi] u funkcional\nomu prostori i vstanovleno
neobxidni ta dostatni umovy stijkosti linijnoho rivnqnnq. V osnovu metodu do-
slidΩennq pokladeno pryncyp porivnqnnq dlq dyskretnyx vidobraΩen\ [ 7 ] .
DoslidΩennq stijkosti zvedeno do znaxodΩennq dijsnyx koreniv deqkoho trans-
cendentnoho rivnqnnq.
Rozhlqnemo pytannq pro stijkist\ dyferencial\noho rivnqnnq vyhlqdu
�x bx t= −( )θ , t k≠ θ ,
(1)
x t cx t t k k( ) ( ),+ = = ∀ ∈θ N0 ,
de bc ≥ 0, θ > 0, u prostori funkcij X = C 0, θ[ ) ∩ C k k
k
1
1
1θ θ, ( )+( )
=
∞
∪ . Roz-
hlqduvane prypuwennq bc ≥ 0 vvedeno z metog harantuvannq rozv’qznosti na
dijsnij osi transcendentnoho rivnqnnq, qke bude otrymano dali.
ZauvaΩymo, wo vypadok c\oho rivnqnnq pry b = 0 [ tryvial\nym. Joho my
rozhlqnemo potim, a zaraz prypustymo, wo b ≠ 0.
Oskil\ky bc ≥ 0, to moΩlyvi 2 vypadky:
1) b > 0, c ≥ 0;
2) b < 0, c ≤ 0.
Dali obmeΩymosq detal\nym rozhlqdom lyße perßoho vypadku, zvertagçy
uvahu na druhyj vypadok lyße ßlqxom remarok.
Poznaçymo Ω = ( , )0 θ i sformulg[mo dlq (1) poçatkovi umovy
x t f t( ) ( )= , t ∈Ω , (2)
de f (t) — neperervna funkciq.
Viz\memo poslidovnist\ ϕn n{ } ∈N0
funkcij ϕn : Ω → R i rozhlqnemo za-
daçu
d t
dt
b tn
n
ϕ
ϕ
( )
( )= −1 , t ∈Ω , n ∈N , (3)
© I. L. IVANOV, V. I. SLYN|KO, 2009
1200 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
UMOVY STIJKOSTI LINIJNOHO RIVNQNNQ Z ÇYSTYM ZAPIZNENNQM … 1201
ϕ θ ϕn nc( ) ( )− = −1 0 ,
ϕ0( ) ( )t f t= , t ∈Ω , (4)
de funkci] ϕn , n ∈N , neperervno dyferencijovni v oblasti vyznaçennq (ϕ0 [
neperervnog).
Oznaçennq 1. Systema (3) nazyva[t\sq stijkog, qkwo dlq dovil\noho ε >
> 0 isnu[ δ > 0 take, wo koly ϕ δ0( ) ( )t C Ω < , to ϕ εn Ct( ) ( )Ω < rivno-
mirno po n.
Oznaçennq 2. Systema (3) nazyva[t\sq asymptotyçno stijkog, qkwo
vona stijka i ϕn Ct( ) ( )Ω → 0 pry n → ∞.
Lehko baçyty, wo miΩ rozv’qzkamy zadaç (1), (2) ta (3), (4) isnu[ zv’qzok
ϕ θn t x n t( ) ( )= + , t ∈( ]0, θ , (5)
a tomu umovy stijkosti ta asymptotyçno] stijkosti systemy (1) rivnosyl\ni umo-
vam vidpovidno stijkosti ta asymptotyçno] stijkosti systemy (3).
Oznaçennq 3. Nexaj ϕn{ } — rozv’qzok (3). Todi operator T : C( )Ω →
→ C( )Ω , oznaçenyj rivnistg T n nϕ ϕ= +1 dlq dovil\noho n ∈N , nazyva-
[t\sq operatorom monodromi] dlq (3).
Oçevydno, wo dlq c\oho operatora ma[ misce zobraΩennq
T t c b s dsn n n
t
ϕ ϕ ϕ
θ
( ) ( ) ( )= +
−
∫0 . (6)
MoΩna pokazaty linijnist\ operatora T. Beruçy do uvahy teoremu Banaxa –
Ítejnhauza, a takoΩ oznaçennq operatora monodromi] ta stijkostej, lehko
baçyty, wo stijkist\ (3) ekvivalentna obmeΩenosti poslidovnosti T n
C( )Ω
vzqta( dlq operatora norma [ zvyçajnog operatornog normog, porodΩenog
normog ⋅ )C( )Ω , a asymptotyçna stijkist\ ekvivalentna spivvidnoßenng
lim
( )n
n
C
T
→∞
=
Ω
0 .
Vvedenyj operator monodromi] ma[ vyhlqd
T t c b s dsn n n
t
ϕ ϕ θ ϕ( ) ( ) ( )= + ∫
0
. (7)
Rozhlqnemo pytannq pro vidßukannq zahal\noho vyhlqdu vyrazu T n1.
Doslidymo T n1, vzqvßy dekil\ka poçatkovyx znaçen\ n:
T 01 1= na Ω ,
T c b ds c bt
t
1 1 1
0
= ⋅ + = +∫ na Ω ,
T T T c c b b c bt ds
t
2
0
1 1= ( ) = + + +∫( ) ( ) ( )θ =
= c cb b ct bt2 21
2
+ + +
θ na Ω .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1202 I. L. IVANOV, V. I. SLYN|KO
Zvidsy vydno, wo T n1 ma[ zahal\nyj vyhlqd T P tn
n1 = ( ) , de P tn ( ) — mnoho-
çlen stepenq n.
MoΩna pokazaty, wo Tt c
k
btk k k= +
+
+θ
1
1
1
, a tomu, poznaçyvßy dlq do-
vil\noho n ∈N0 çerez bn vil\nyj çlen polinoma P tn ( ) , dlq mnohoçlena
P tn ( ) otryma[mo zobraΩennq
P t b
b
n
t b
b
n
t b
b
n
n
n
n
n
n
n
( )
! ( )! ( )
= +
−
+
−
−
−
−
0 1
1
1
2
2
1 2 !!
t b bt bn
n n
−
−+ … + +2
1 .
(8)
Ale todi
P t TP t b
b
n
t b
b
n
t bn n
n
n
n
n
+
+
+= =
+
+ +1 0
1
1
1 2
1
( ) ( )
( )! !
bb
n
t
n
n
−
−
−
+
1
1
1( )!
…
…+ +b t cPn n ( )θ ,
zvidky
b cP c b b b
n
bn n n n n
n
+ − −= = + + + … +
1 1
2
2 0
2
( )
! !
θ β
β β
, (9)
de β θ= b .
Rivnist\ (9) [ rekurentnym spivvidnoßennqm, qke da[ zmohu znajty bn + 1 , ko-
ly vidomi b1 , b2 , … , bn . Paralel\no z cym spivvidnoßennqm budemo rozhlqda-
ty takoΩ spivvidnoßennq
� �b c
k
bn
k
n k
k
+
=
∞
−= ∑1
0
1
!
β (10)
dlq deqko] poslidovnosti
�bn{ } . Oskil\ky, qk lehko baçyty, b0 1= , to pokla-
demo
�b0 1= .
Budemo ßukaty rozv’qzok (10) u vyhlqdi
�b qn
n= (pryçyny, çerez qki roz-
v’qzok ßuka[t\sq u takomu specyfiçnomu vyhlqdi, polqhagt\ u tomu, wo
dovil\nyj rozv’qzok systemy (10) zavΩdy moΩe maΩoruvatys\ pry n → + ∞ roz-
v’qzkom u zaproponovanomu vyhlqdi, pomnoΩenomu na deqku konstantu; nas tut
cikavyt\ poslidovnist\ iz najßvydßym zrostannqm). Todi spivvidnoßennq (10)
nabere vyhlqdu
q c q
k
n n k
k
k
+ −
=
∞
= ∑1
0
β
!
,
abo, pislq sprowennq ta vidßukannq sumy rqdu,
q ce q= β/
. (11)
MoΩna pokazaty, wo pry b > 0, c > 0 (vypadok c = 0 vidpovida[ rivnqnng z
tryvial\no stijkym nul\ovym rozv’qzkom) transcendentne rivnqnnq (11) ma[
[dynyj dijsnyj korin\, do toho Ω dodatnyj. Ce vydno z toho, wo funkciq pravo]
çastyny rivnqnnq nabuva[ lyße dodatnyx znaçen\, a na pravij pivosi monotonno
spada[ vid + ∞ do odynyci. Podibnymy mirkuvannqmy moΩna vstanovyty, wo u
vypadku, koly b < 0 i c < 0, ce rivnqnnq matyme [dynyj dijsnyj rozv’qzok, do
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
UMOVY STIJKOSTI LINIJNOHO RIVNQNNQ Z ÇYSTYM ZAPIZNENNQM … 1203
toho Ω vid’[mnyj. OtΩe, nexaj q zadovol\nq[ (11), todi dlq
�bn = qn
, n ∈Z ,
bude vykonuvatys\ spivvidnoßennq (10).
Dlq dovil\noho n ∈N0 poznaçymo
θn
n
n
b
b
= � . (12)
MoΩna perekonatys\, wo θn zadovol\nq[ spivvidnoßennq
θ θ
β
θ
β
θ
β
β
n
q
n n ne
q q n q
+
−
− −= + +
+ … +1 1
2
2
1
2
1
! !
n
θ0 . (13)
Qkwo poznaçyty β
β
1 =
q
, to otryma[mo
θ θ β θ
β
θ
β
θβ
n n n n
n
e
n
+
−
− −= + + + … +
1 1 1
1
2
2
1
0
1
2! ! .
Vykorystavßy zaminu A ek
kk
= −β β1( / !)
, budemo maty
θ θn k n k
k
n
A+ −
=
= ∑1
0
. (14)
Poznaçymo S An kk
n=
=∑ 0
ta r Sn n= −1 .
Lema 1. Nexaj poslidovnist\ θn oznaçena rivnistg (12). Todi isnu[ θ∗
take, wo rivnomirno po n vykonu[t\sq spivvidnoßennq θ∗ ≤ θn ≤ 1.
Dovedennq. OtΩe, θ νn n nr+ = −1 1( ) , de νn — „zvaΩene seredn[”, νn =
= A Ak n kk
n
kk
nθ −= =∑ ∑0 0
.
PokaΩemo spoçatku, wo ma[ misce prava nerivnist\ tverdΩennq lemy. Dove-
demo ]] metodom matematyçno] indukci].
Pry n = 0 nerivnist\ vykonu[t\sq, θ0 1≤ . Prypustymo ]] vykonannq dlq
dovil\noho k ≤ n (θk ≤ 1 ) ta vstanovymo ]] dlq n + 1. Dijsno,
θ ν ν θn n n n
k
kr+ = − ≤ ≤ { } ≤1 1 1( ) max ,
wo j potribno bulo dovesty.
Perejdemo do livo] nerivnosti. Rozhlqnemo we odnu poslidovnist\, zadanu
rekurentno:
� �θ θn l nr+ = −1 1( ) ,
de l take, wo dlq dovil\noho k ≤ n, k ≠ l bude
� �θ θl k< , a
�θ0 1= . Z dopomo-
hog metodu matematyçno] indukci] lehko dovesty, wo l = n, tobto
�θn{ } mo-
notonno spada[ (bo 1 1− <�rn ).
PokaΩemo, wo dlq dovil\noho n θ θn n≥ �
(metodom matematyçno] induk-
ci]).
Pry n = 0 θ θ0 0≥ �
. Prypustymo vykonannq nerivnosti pry l ≤ n (tobto
θ θl l≥ �
) ta dovedemo, wo θ θn n+ +≥1 1
�
. Dijsno,
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1204 I. L. IVANOV, V. I. SLYN|KO
θ ν θ θn n n
i
i n
i
i nr r r+ = − ≥ { } − ≥ { } −1 1 1 1( ) min ( ) min (� )) =
= � �θ θn n nr( )1 1− = + ,
wo j potribno bulo dovesty.
PokaΩemo, wo
�θn obmeΩena znyzu dodatnym çyslom.
Dijsno,
� � � �θ θ θ θn n n n n nr r r+ − −= − = − − = … =1 1 1 01 1 1 1( ) ( ) ( ) ( −−
=
∑ rk
k
n
)
0
.
OtΩe, pytannq pro obmeΩenist\
�θn ekvivalentne pytanng pro zbiΩnist\ (do
nenul\ovoho znaçennq) dobutku ( )1
0
−=
∞∏ rnn
. Ostannij zbiha[t\sq todi i lyße
todi, koly isnu[ suma rqdu rnn =
∞∑ 0
[8]. Ale
r
k
n
k
k n
=
= +
∞
∑ β1
1
1
!
.
Viz\memo minimal\ne l ≥ n take, wo l > β1 , i prodovΩymo:
β β β1
1
1
1
1
1
1 1 1k
k n
k
k n
l
k
k lk k k! ! != +
∞
= + = +
∞
∑ ∑ ∑= + =
= β
β β
1
1
1 1
1
1
1 2
k
k n
l l k l
k lk l l l k! ! ( ) ( )= +
−
= +
∞
∑ ∑+
+ + …
<
< β
β β β
1
1
1 1
1
11
1
k
k n
l l k l
k l
k l
k
k l l! ! ( )= +
−
−
= +
∞
∑ ∑+
+
=
kk l l
l
k n
l l
! != +
∑ +
+ −
+
1
1 1
11
1
1
1
β β
β =
=
β β
β
1
1
1
1
1
1
1
k
k n
l l
k l l! != +
+
∑ +
+ −
.
Vidkynemo ti rn , u qkyx l > n. Ce ne vplyne na zbiΩnist\ rqdu. Nexaj teper
n = l, todi
r
n n n
n
n
e
n
n n n
n
<
+ −
< =
+β
β
β
β
β
β
π
1
1
1
1
1
1
11
1
2
! !
ee nδ
,
de δn
n
∈
0
1
12
, [9], i prodovΩymo
β
β
π
β
π
β
β
δ δ
1
1
1
1
2
1
2
n
n n
n
n
e
e n
n
e
en n
=
< 11
1
11
2n
e
n n
β
β
< ,
qkwo vzqty n take, wob
n
eβ1
2> . Ale rqd
β
β1
1 1
2nn =
∞∑ = , tomu zbiΩnym [
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
UMOVY STIJKOSTI LINIJNOHO RIVNQNNQ Z ÇYSTYM ZAPIZNENNQM … 1205
rqd rnn =
∞∑ 0
, dobutok ( )1
0
−=
∞∏ rnn
, a tomu isnu[ take θ∗ , wo θ θn > ∗
dlq
vsix n ∈N .
Lemu 1 dovedeno.
Z lemy vyplyva[ oçevydnyj naslidok.
Naslidok 1. Nexaj bn — poslidovnist\ vil\nyx çleniv P t Tn
n( ) ( )= 1 , de
T — operator monodromi] dlq (4) pry b > 0, c ≥ 0, q — rozv’qzok rivnqnnq
q ce q= β/
. Todi:
1) qkwo q < 1, to bn → 0 , n → ∞;
2) qkwo q = 1, to isnugt\ n∗ , c1 , c2 , 0 < c1 < c2 , taki, wo dlq vsix
n n> ∗
c1 < bn < c2 ;
3) qkwo q > 1, to bn → ∞ , n → ∞.
Lema 2. Nexaj P t Tn
n( ) ( )= 1 , de T — operator monodromi] dlq (3), b > 0,
c ≥ 0, a bn > 0 — vil\ni çleny P tn ( ) . Todi:
1) qkwo bn → ∞ , n → ∞, to Pn C( )Ω → ∞ , n → ∞;
2) qkwo isnu[ n∗ ∈N take, wo dlq vsix n n> ∗
0 < c1 < bn < c2 , to is-
nu[ n∗∗ take, wo dlq vsix n n> ∗∗
γ γ1 2< <Pn C( )Ω ;
3) qkwo bn → 0 , n → ∞, to Pn C( )Ω → 0 , n → ∞.
Dovedennq. 1. Nexaj bn → ∞ , ale
Pn C( )Ω = b
b
n k
tk
n k
n k
k
n
C
−
−
= −∑
( )!
( )0 Ω
≥ Pn ( )0 = bn ,
tomu Pn C( )Ω → ∞ , n → ∞.
2. Nexaj isnu[ n∗ ∈N take, wo dlq vsix n n> ∗
0 < c1 < bn < c2 . Todi
P b
b
n k
t P bn C k
n k
n k
k
n
C
n n( )
( )
( )!
( )Ω
Ω
1
1
0
0=
−
≥ =
−
−
=
∑ >> c1 .
Z inßoho boku,
P b
b
n k
t b
b
n
n C k
n k
n k
k
n
C
k
n k
( )
( )
( )! (Ω
Ω
1
1
0
=
−
=
−
−
=
−
∑ −−
−
=
∑
k
n k
k
n
)!
θ
0
=
= b
b
k c
b
c
cn k
k
k
n
n−
=
+∑ = <
( )
!
θ
0
1 2
1 1
.
OtΩe, pry n n> ∗
c P
c
c
n C1
2
1
< <( )Ω .
3. Nexaj bn → 0 , n → ∞, todi Pn C( )Ω1
=
b
c
n + 1
→ 0, n → ∞.
Lemu 2 dovedeno.
Lema 3. Nexaj T — operator monodromi] dlq (3), b > 0, c ≥ 0. Todi dlq
dovil\noho n T fn
C( )Ω
≤ f TC
n
C( ) ( )
( )Ω Ω
1 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
1206 I. L. IVANOV, V. I. SLYN|KO
Dovedennq. Dostatn\o pokazaty, wo T f tn ( ) ≤ f T tC
n
( ) ( ) ( )Ω 1 .
PokaΩemo ce metodom matematyçno] indukci]. Pry n = 0 tverdΩennq [ pra-
vyl\nym, oskil\ky T f t0 ( ) = f t( ) ≤ f C( )Ω = f TC( ) ( )Ω
0 1 . Prypustymo,
wo T f tn ( ) ≤ f T tC
n
( ) ( ) ( )Ω 1 , i rozhlqnemo
T f t T T f t bT f c T f s dsn n n n
t
+ = = + ∫1
0
( ) ( ) ( ) ( )θ ≤
≤ b T f c T f s dsn n
t
( ) ( )θ + ∫
0
≤
≤ b T f c T s ds fn
C
n
t
C( ) ( ) ( ) ( )( ) ( )1 1
0
θ Ω Ω+ ∫ =
= f bT c T s ds f TC
n n
t
C
n
( ) ( )( ) ( ) ( ) ( ) ( ) (Ω Ω1 1 1
0
1θ + =∫ + tt) .
Lemu dovedeno.
OtΩe, bulo vstanovleno, wo qkwo q — rozv’qzok rivnqnnq q ce q= βθ/
, to
povedinka poslidovnosti bn vyznaça[t\sq roztaßuvannqm q po vidnoßenng do
odynyci (naslidok 1), povedinka T n
C
1
( )Ω
— povedinkog poslidovnosti bn
(lema 2), a povedinka T fn
C( )Ω
— povedinkog T n
C
1
( )Ω
. Tomu moΩna sfor-
mulgvaty takyj naslidok.
Naslidok 2. Nexaj q — rozv’qzok rivnqnnq q ce q= βθ/
, a systema (3)
taka, wo v nij b > 0, c ≥ 0. Todi:
1) qkwo q < 1, to rivnqnnq (1) asymptotyçno stijke;
2) qkwo q = 1, to rivnqnnq (1) stijke;
3) qkwo q > 1, to rivnqnnq (1) nestijke.
Zaznaçymo, wo qkwo b = 0, to stijkist\ (3) vyznaça[t\sq roztaßuvannqm mo-
dulq parametra c po vidnoßenng do odynyci, oskil\ky rozv’qzok ci[] systemy
dopuska[ analityçne zobraΩennq
ϕ θn
nt c f( ) ( )= , t ∈Ω , n ∈N .
U vypadku, koly b < 0 ta c ≤ 0, rozv’qzok zadaçi (3), (4) moΩna podaty u
vyhlqdi ϕ ϕn
n
n= −( )1 � , de �ϕn — rozv’qzok zadaçi (3), (4), v qkij koefici[nty b
ta c zamineno ]x modulqmy. Tomu pytannq pro stijkist\ ϕn ta �ϕn [ ekviva-
lentnymy.
U zv’qzku z cym, vraxovugçy, wo rozv’qzok systemy (1) ma[ zv’qzok z rozv’qz-
kom systemy (3), wo vyraΩa[t\sq rivnistg (5), moΩna sformulgvaty taku teo-
remu.
Teorema. Nexaj q — rozv’qzok rivnqnnq q ce q= βθ/
, a systema (1) taka,
wo bc ≥ 0. Todi:
1) qkwo q < 1 , to systema (1) asymptotyçno stijka;
2) qkwo q = 1 , to systema (1) stijka;
3) qkwo q > 1 , to systema (1) nestijka.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
UMOVY STIJKOSTI LINIJNOHO RIVNQNNQ Z ÇYSTYM ZAPIZNENNQM … 1207
Takym çynom, pytannq pro stijkist\ rozv’qzkiv rozhlqduvanoho rivnqnnq zvo-
dyt\sq do vyznaçennq roztaßuvannq po vidnoßenng do odynyci rozv’qzku trans-
cendentnoho rivnqnnq (11), vzqtoho za modulem.
1. Samojlenko A. M., Perestgk N. A. Dyfferencyal\n¥e uravnenyq s ympul\sn¥m vozdejst-
vyem. – Kyev: Vywa ßk., 1987. – 282 s.
2. Perestgk M. O., Çernikova O. S. Deqki suçasni aspekty teori] dyferencial\nyx rivnqn\ z
impul\snog di[g // Ukr. mat. Ωurn. – 2008. – 60, # 1. – S. 81 – 94.
3. Bojçuk A. A., Perestgk N. A., Samojlenko A. M. Peryodyçeskye reßenyq ympul\sn¥x dyf-
ferencyal\n¥x system v krytyçeskyx sluçaqx // Dyfferenc. uravnenyq. – 1991. – 27, # 9.
– S. 1516 – 1521.
4. Martynyuk A. A., Shen J. H., Stavroulakis I. P. Stability theorems in impulsive functional differen-
tial equations with infinite delay // Adv. Stabil. Theory. Stability and Control: Theory, Meth. and
Appl. – London: Taylor & Francis, 2003. – 13. – P. 153 – 174.
5. Sl¥n\ko V. Y. Ob uslovyqx ustojçyvosty dvyΩenyq lynejn¥x ympul\sn¥x system s zapaz-
d¥vanyem // Prykl. mexanyka. – 2005. – 41, # 6. – S. 130 – 138.
6. Perestgk M. O., Slgsarçuk V. G. Umovy isnuvannq nekolyvnyx rozv’qzkiv nelinijnyx dy-
ferencial\nyx rivnqn\ iz zapiznennqm ta impul\snym zburennqm u banaxovomu prostori //
Ukr. mat. Ωurn. – 2003. – 55, # 6. – S. 790 – 798.
7. Lakßmykantam V., Lyla S., Mart¥ngk A. A. Ustojçyvost\ dvyΩenyq: metod sravnenyq. –
Kyev: Nauk. dumka, 1991. – 248 s.
8. Hursa ∏. Kurs matematyçeskoho analyza. T. 1, çast\ 2. RazloΩenye v rqd¥. Heometryçeskye
pryloΩenyq. – M., L.: Hostexteoryzdat, 1933. – 235 s.
9. Qnke E., ∏mde F., Leß F. Specyal\n¥e funkcyy. – M.: Nauka, 1977. – 344 s.
OderΩano 11.09.08,
pislq doopracgvannq — 06.04.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
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| id | umjimathkievua-article-3092 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:36:05Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a7/d635e1e8ee4a4530721bed47cdb068a7.pdf |
| spelling | umjimathkievua-article-30922020-03-18T19:45:12Z Conditions for the stability of an impulsive linear equation with pure delay Умови стійкості лінійного рівняння з чистим запізненням та імпульсною дією Ivanov, I. L. Slyn'ko, V. I. Іванов, I. Л. Слинько, В. І. We establish necessary and sufficient conditions for the stability of one class of impulsive linear differential equations with delay. Установлены необходимые и достаточные условия устойчивости для класса линейных дифференциальных уравнений с запаздыванием и импульсным воздействием. Institute of Mathematics, NAS of Ukraine 2009-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3092 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 9 (2009); 1200-1207 Український математичний журнал; Том 61 № 9 (2009); 1200-1207 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3092/2932 https://umj.imath.kiev.ua/index.php/umj/article/view/3092/2933 Copyright (c) 2009 Ivanov I. L.; Slyn'ko V. I. |
| spellingShingle | Ivanov, I. L. Slyn'ko, V. I. Іванов, I. Л. Слинько, В. І. Conditions for the stability of an impulsive linear equation with pure delay |
| title | Conditions for the stability of an impulsive linear equation with pure delay |
| title_alt | Умови стійкості лінійного рівняння з чистим запізненням та імпульсною дією |
| title_full | Conditions for the stability of an impulsive linear equation with pure delay |
| title_fullStr | Conditions for the stability of an impulsive linear equation with pure delay |
| title_full_unstemmed | Conditions for the stability of an impulsive linear equation with pure delay |
| title_short | Conditions for the stability of an impulsive linear equation with pure delay |
| title_sort | conditions for the stability of an impulsive linear equation with pure delay |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3092 |
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