Integral inequalities and stability of an equilibrium state on a time scale
We present some integral inequalities on a time scale and establish sufficient conditions for the uniform stability of an equilibrium state of a nonlinear system on a time scale.
Saved in:
| Date: | 2010 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2010
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2973 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508979137871872 |
|---|---|
| author | Luk’yanova, T. A. Martynyuk, A. A. Лукьянова, Т. А. Мартынюк, А. А. Лукьянова, Т. А. Мартынюк, А. А. |
| author_facet | Luk’yanova, T. A. Martynyuk, A. A. Лукьянова, Т. А. Мартынюк, А. А. Лукьянова, Т. А. Мартынюк, А. А. |
| author_sort | Luk’yanova, T. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:41:38Z |
| description | We present some integral inequalities on a time scale and establish sufficient conditions for the uniform stability of an equilibrium state of a nonlinear system on a time scale. |
| first_indexed | 2026-03-24T02:33:48Z |
| format | Article |
| fulltext |
UDK 517.929
T. A. Luk\qnova, A. A. Mart¥ngk (Yn-t mexanyky NAN Ukrayn¥, Kyev)
YNTEHRAL|NÁE NERAVENSTVA
Y USTOJÇYVOST| SOSTOQNYQ RAVNOVESYQ
NA VREMENNOJ ÍKALE
Some integral inequalities on the time scale are presented and sufficient conditions of the uniform
stability of equilibrium of a nonlinear system on the time scale are obtained.
Navedeno deqki intehral\ni nerivnosti na çasovij ßkali ta otrymano dostatni umovy rivnomirno]
stijkosti stanu rivnovahy nelinijno] systemy na çasovij ßkali.
1. Vvedenye. Yntehral\n¥e neravenstva qvlqgtsq mown¥m y ßyroko pry-
menqem¥m sredstvom dlq kaçestvennoho yssledovanyq yntehral\n¥x y dyf-
ferencyal\n¥x uravnenyj. V çastnosty, v knyhe [1] neravenstvo Hronuolla
y1eho obobwenyq yspol\zugtsq dlq yzuçenyq yntehral\n¥x uravnenyj s de-
heneratyvn¥m qdrom, system dyfferencyal\n¥x uravnenyj obweho vyda y
kvazylynejn¥x system. V dannoj rabote podxod, predloΩenn¥j v [1], ras-
prostranqetsq na sluçaj yntehral\n¥x y dyfferencyal\n¥x uravnenyj na vre-
mennoj ßkale.
Yssledovanye dynamyçeskyx uravnenyj na vremennoj ßkale qvlqetsq ak-
tual\n¥m, poskol\ku pozvolqet kak odnovremenno opysat\ dynamyku system v
neprer¥vnom y dyskretnom sluçaqx, tak y yssledovat\ dynamyku system¥ vo
vremennoj oblasty „meΩdu” πtymy sostoqnyqmy. Krome toho, suΩenye rezul\-
tatov, poluçenn¥x dlq obwej vremennoj ßkal¥, daet vozmoΩnost\ poluçat\
nov¥e rezul\tat¥ dlq dyskretn¥x system (naprymer, sledstvyq 3 y 4), analo-
hyçn¥e yzvestn¥m dlq neprer¥vnoho sluçaq.
V dannoj rabote na osnove nov¥x yntehral\n¥x neravenstv poluçen¥ ocenky
reßenyj system yntehral\n¥x y dyfferencyal\n¥x uravnenyj na proyzvol\-
noj vremennoj ßkale. ∏ty ocenky prymenen¥ dlq yssledovanyq ustojçyvosty
sostoqnyq ravnovesyq system¥ dyfferencyal\n¥x uravnenyj obweho vyda.
∏ffektyvnost\ poluçenn¥x rezul\tatov prodemonstryrovana na konkretnom
prymere.
2. Osnovn¥e oboznaçenyq y neobxodym¥e teorem¥. Vremennoj ßkaloj T
naz¥vaetsq proyzvol\noe nepustoe zamknutoe podmnoΩestvo mnoΩestva ve-
westvenn¥x çysel R . Osnovn¥e ponqtyq y teorem¥ matematyçeskoho analyza
na vremennoj ßkale, takye kak opredelenyq proyzvodnoj y yntehrala, pravyla
dyfferencyrovanyq y yntehryrovanyq, opredelenye πksponencyal\noj, rehres-
syvnoj y rd-neprer¥vnoj funkcyj, podrobno yzloΩen¥ v rabotax [2, 3]. Pry-
vedem tol\ko nekotor¥e neobxodym¥e ponqtyq y opredelenyq.
Oboznaçym çerez Crd
n( , )T R mnoΩestvo vsex rd-neprer¥vn¥x funkcyj
g : T → Rn
. Funkcyq f : T × Rn → Rn
naz¥vaetsq rd-neprer¥vnoj, esly po-
roΩdaem¥j eg operator superpozycyy ( )fx perevodyt mnoΩestvo Crd
n( , )T R
v sebq.
Funkcyq f : T → R naz¥vaetsq rehressyvnoj, esly 1 + µ( ) ( )t f t ≠ 0 pry vsex
t k∈T , y poloΩytel\no rehressyvnoj, esly 1 + µ( ) ( )t f t > 0 pry vsex t k∈T .
MnoΩestvo vsex r d-neprer¥vn¥x y poloΩytel\no rehressyvn¥x funkcyj f :
T → R oboznaçym çerez R+
.
Funkcyq f : T × Rn → Rn
naz¥vaetsq rehressyvnoj, esly pry lgbom
© T. A. LUK|QNOVA, A. A. MARTÁNGK, 2010
1490 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
YNTEHRAL|NÁE NERAVENSTVA Y USTOJÇYVOST| SOSTOQNYQ RAVNOVESYQ … 1491
t k∈T operator F : Rn → Rn , dejstvugwyj po formule Fx = x + µ( ) ( , )t f t x ,
obratym.
Krome toho, oboznaçym çerez R+ = u ∈{ R : u ≥ }0 , x = xii
n 2
1
1 2
=∑( ) /
nor-
mu vektora x n∈R , a, +∞[ )T = t ∈{ T : a ≤ t < +∞} , a ∈T .
Dalee nam potrebugtsq sledugwye lemm¥.
Lemma 1 ([4], teorema 3.5). Pust\ funkcyy u, p : T → R+ r d-neprer¥vn¥
na T, funkcyq f : T → R+ qvlqetsq ∆-dyfferencyruemoj na T y f t∆ ( ) ≥
≥ 0. Esly
u t( ) ≤ f t( ) + p s u s s
a
t
( ) ( ) ∆∫
dlq vsex t ∈T , to
u t( ) ≤ f a e t ap( ) ( , ) + f s e t s s
a
t
p
∆ ∆( ) , ( )∫ ( )σ
dlq vsex t ∈T , hde funkcyq e t ap( , ) , a ∈T , qvlqetsq reßenyem naçal\noj
zadaçy
x t∆ ( ) = p t x t( ) ( ) , x a( ) = 1.
Lemma 2 ([5], zameçanye 2). Esly funkcyq p : T → R+ qvlqetsq r d-ne-
prer¥vnoj na T, to
e t ap( , ) ≤ exp ( )p s s
a
t
∫
∆
pry vsex a ∈T , t ∈ a, +∞[ )T .
V dal\nejßem budut neobxodym¥ takye svojstva πksponencyal\noj funk-
cyy:
1) e a ap( , ) = 1, e t ap( , ) > 0 pry p ∈ +R , a ∈T y t ∈ a, +∞[ )T ;
2) p e s a s
a
t
p∫ ( , ) ∆ = e t ap( , ) – 1 pry p ∈ +R , a ∈T y t ∈ a, +∞[ )T ;
3) esly λ ∈ +R , to lim ( , )t e t a→+∞ �λ = 0, hde �λ = −
+
∈ +λ
µ λ1 ( )t
R
(sm.1[6]).
PredpoloΩym, çto na vremennoj ßkale T opredelena systema dynamyçe-
skyx uravnenyj
x t∆ ( ) = f t x t, ( )( ) , t I∈ , (1)
x t t x( ; , )0 0 0 = x0 , t I0 ∈ , x n
0 ∈R , (2)
hde x n∈R , I = α, +∞[ )T , α ∈T , f : I × Rn → Rn
, f t( , )0 ≡ 0, sup T = + ∞.
Krome toho, predpoloΩym, çto dlq zadaçy (1), (2) v¥polnqgtsq uslovyq su-
westvovanyq edynstvennoho reßenyq na t0, +∞[ )T pry lgb¥x naçal\n¥x dan-
n¥x ( , )t x0 0 ∈ I × Rn
. ∏to reßenye budem oboznaçat\ x t( ) = x t t x( ; , )0 0 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1492 T. A. LUK|QNOVA, A. A. MARTÁNGK
Opredelenye 1. Sostoqnye ravnovesyq x t( ) ≡ 0 system¥ (1) naz¥vaetsq
ravnomerno ustojçyv¥m, esly dlq lgboho ε > 0 suwestvuet postoqnnaq δ =
= δ ε( ) > 0 takaq, çto yz uslovyq x0 < δ sleduet ocenka x t t x( ; , )0 0 < ε
pry vsex t ∈ t0, +∞[ )T y t I0 ∈ .
V rabote [3] dlq proyzvol\noj vremennoj ßkal¥ dokazana sledugwaq teo-
rema suwestvovanyq y edynstvennosty.
Lemma 3 ([3], teorema 8.24). PredpoloΩym, çto dlq lgb¥x znaçenyj t ∈T
y x n∈R opredelen¥ okrestnosty Ic = ( , )t c t c− + ∩ T y S b( ) = y n∈{ R :
y x− < b} , hde c > 0, inf T ≤ t – c , sup T ≥ t + c, takye, çto vektor-
funkcyq f : Ic × S b( ) → Rn
qvlqetsq r d-neprer¥vnoj, ohranyçennoj na
Ic × S b( ) y udovletvorqet uslovyg
f t x f t x( , ) ( , )1 2− ≤ L t x x x( , ) 1 2− , L t x( , ) > 0,
pry vsex ( , )t x1 , ( , )t x2 ∈ Ic × S b( ) . Krome toho, predpoloΩym, çto su-
westvugt poloΩytel\n¥e neprer¥vn¥e funkcyy p, q : T → R+ takye, çto
f t x( , ) ≤ p t x( ) + q t( )
pry vsex ( , )t x ∈ T × Rn
.
Tohda naçal\naq zadaça
x t∆ ( ) = f t x t, ( )( ) , t ∈T ,
x t t x( ; , )0 0 0 = x0 , t0 ∈T , x n
0 ∈R ,
ymeet toçno odno reßenye na T.
3. Ocenky reßenyj system yntehral\n¥x y dyfferencyal\n¥x urav-
nenyj. Rassmotrym yntehral\noe uravnenye vyda
x t( ) = g t( ) + B t U s x s s
a
t
( ) , ( )( )∫ ∆ , t ∈ a, +∞[ )T , (3)
hde x, g : a, +∞[ )T → Rn
, B : a, +∞[ )T → R, U : a, +∞[ )T × Rn → Rn
— rd-ne-
prer¥vn¥e funkcyy, a I∈ . Pust\ suwestvuet reßenye uravnenyq (3) na
a, +∞[ )T .
Lemma 4. PredpoloΩym, çto suwestvugt rd-neprer¥vn¥e funkcyy L, M :
I × R+ → R+ takye, çto:
1) U t x( , ) ≤ L t x,( ) pry vsex t I∈ , x n∈R ;
2) 0 ≤ L t u( , ) – L t( , )v ≤ M t u( , ) ( )v v− pry vsex t I∈ , u ≥ v ≥ 0.
Tohda dlq reßenyq x t( ) yntehral\noho uravnenyq (3) pry vsex t ∈ a[ ,
+∞)T ymeet mesto ocenka
x t g t( ) ( )− ≤ B t L s g s e t s s
a
t
p( ) , ( ) , ( )( ) ( )∫ σ ∆ ,
hde p t( ) = B t M t g t( ) , ( )( ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
YNTEHRAL|NÁE NERAVENSTVA Y USTOJÇYVOST| SOSTOQNYQ RAVNOVESYQ … 1493
Dokazatel\stvo. Pust\ funkcyq x t( ) qvlqetsq reßenye uravnenyq (3).
Oboznaçym
y t( ) = U s x s s
a
t
, ( )( )∫ ∆ (4)
y yz (3) poluçym
x t( ) = g t( ) + B t y t( ) ( ) . (5)
Dyfferencyruq (4), s uçetom (5) ymeem
y t∆ ( ) = U t g t B t y t, ( ) ( ) ( )+( ) , y a( ) = 0 .
Tohda
y t∆ ( ) = U t g t B t y t, ( ) ( ) ( )+( ) ≤ L t g t B t y t, ( ) ( ) ( )+( ) ≤
≤ L t g t B t y t, ( ) ( ) ( )+( ) = L t g t B t y t, ( ) ( ) ( )+( ) – L t g t, ( )( ) +
+ L t g t, ( )( ) ≤ L t g t, ( )( ) + M t g t B t y t, ( ) ( ) ( )( ) (6)
y, poskol\ku y t( ) = y s s
a
t ∆ ∆( )∫ ≤ y s s
a
t ∆ ∆( )∫ , yz (6) sleduet ocenka
y t( ) ≤ y s s
a
t
∆ ∆( )∫ ≤ L s g s s
a
t
, ( )( )∫ ∆ +
+ B s M s g s y s s
a
t
( ) , ( ) ( )∫ ( ) ∆ . (7)
Oboznaçaq f t( ) = L s g s s
a
t
, ( ) ,( )∫ ∆ p t( ) = B t M t g t( ) , ( )( ) , neravenstvo (7)
zapys¥vaem v vyde
y t( ) ≤ f t( ) + p s y s s
a
t
( ) ( )∫ ∆ ,
pry πtom f t∆ ( ) = L t g t, ( )( ) ≥ 0, f a( ) = 0. S uçetom lemm¥11 pry vsex
t ∈ a, +∞[ )T poluçaem neravenstvo
y t( ) ≤ f a e t ap( ) ( , ) + L s g s e t s s
a
t
p, ( ) , ( )( ) ( )∫ σ ∆ ,
otkuda ymeem ocenku
x t g t( ) ( )− ≤ B t y t( ) ( ) ≤ B t L s g s e t s s
a
t
p( ) , ( ) , ( )( ) ( )∫ σ ∆ .
Lemma 4 dokazana.
Lemma 5. PredpoloΩym, çto suwestvuet rd-neprer¥vnaq funkcyq S : a[ ,
+∞)T × R+ → R+ takaq, çto pry vsex t ∈ a, +∞[ )T y x , y n∈R v¥polnq-
etsq neravenstvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1494 T. A. LUK|QNOVA, A. A. MARTÁNGK
U t x y U t x( , ) ( , )+ − ≤ S t x y,( ) .
Tohda dlq reßenyq x t( ) yntehral\noho uravnenyq (3) pry vsex t ∈ a[ ,
+∞)T ymeet mesto ocenka
x t g t( ) ( )− ≤ B t U s g s e t s s
a
t
p( ) , ( ) , ( )( ) ( )∫ σ ∆ , (8)
hde p t( ) = B t S t g t( ) , ( )( ) .
Dokazatel\stvo. Kak y pry dokazatel\stve lemm¥14, dlq
y t( ) = U s x s s
a
t
, ( )( )∫ ∆
poluçaem ocenku
y t∆ ( ) = U t g t B t y t( , ( ) ( ) ( ))+ ≤ U t g t( , ( )) + S t g t B t y t, ( ) ( ) ( )( ) ,
otkuda sleduet, çto
y t( ) ≤ y s s
a
t
∆ ∆( )∫ ≤ U s g s s
a
t
, ( )( )∫ ∆ +
+ B s S s g s y s s
a
t
( ) , ( ) ( )∫ ( ) ∆ . (9)
Esly poloΩyt\ f t( ) = U s g s s
a
t
, ( )( )∫ ∆ , p t( ) = B s S t g t( ) , ( )( ) , to nera-
venstvo (9) prymet vyd
y t( ) ≤ f t( ) + p s y s s
a
t
( ) ( )∫ ∆ .
Tohda s uçetom lemm¥11 poluçaem
y t( ) ≤ U s g s e t s s
a
t
p, ( ) , ( )( ) ( )∫ σ ∆ ,
otkuda ymeem ocenku
x t g t( ) ( )− ≤ B t y t( ) ( ) ≤ B t U s g s e t s s
a
t
p( ) , ( ) , ( )( ) ( )∫ σ ∆ .
Lemma 5 dokazana.
Sledstvye 1 . PredpoloΩym , çto suwestvugt rd-neprer¥vn¥e funkcyy
L, M : I × R+ → R+ takye, çto:
1) f t x( , ) ≤ L t x,( ) pry vsex t I∈ , x n∈R ;
2) 0 ≤ L t u( , ) – L t( , )v ≤ M t u( , ) ( )v v− pry vsex t I∈ , u ≥ v ≥ 0.
Tohda dlq reßenyq x t( ) zadaçy (1), (2) pry vsex t ∈ t0, +∞[ )T ymeet mes-
to ocenka
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
YNTEHRAL|NÁE NERAVENSTVA Y USTOJÇYVOST| SOSTOQNYQ RAVNOVESYQ … 1495
x t x( ) − 0 ≤ L s x e t s s
t
t
p, , ( )0
0
( ) ( )∫ σ ∆ , (10)
hde p t( ) = M t x, 0( ) .
Dokazatel\stvo. Perepyßem uravnenyq (1) v vyde
x t( ) = x0 + f s x s s
t
t
, ( )( )∫
0
∆ .
Yspol\zuq lemmu14 pry g t( ) ≡ x0 , B t( ) ≡ 1, a = t0 , poluçaem ocenku (10).
Sledstvye 2. PredpoloΩym, çto suwestvuet rd -neprer¥vnaq funkcyq
S : a, +∞[ )T × R+ → R+ takaq, çto pry vsex t ∈ a, +∞[ )T y x , y n∈R v¥-
polnqetsq neravenstvo
U t x y U t x( , ) ( , )+ − ≤ S t x y,( ) .
Tohda dlq reßenyq x t( ) zadaçy (1), (2) pry vsex t ∈ t0, +∞[ )T ymeet mesto
ocenka
x t x( ) − 0 ≤ f s x e t s s
t
t
p( , ) , ( )0
0
∫ ( )σ ∆ , (11)
hde p t( ) = S t x, 0( ) .
Dokazatel\stvo. Kak y pry dokazatel\stve sledstvyq11, yspol\zuq lem-
mu15 pry g t( ) ≡ x0 , B t( ) ≡ 1, a = t0 , poluçaem ocenku (11).
4. Osnovnoj rezul\tat. Teper\ m¥ moΩem dokazat\ sledugwye utverΩ-
denyq.
Teorema 1. PredpoloΩym, çto suwestvugt rd -neprer¥vn¥e funkcyy L,
M : I × R+ → R+ takye, çto L t( , )0 ≡ 0 y v¥polnqgtsq neravenstva:
1) f t x( , ) ≤ L t x,( ) pry vsex t I∈ , x n∈R ;
2) 0 ≤ L t u( , ) – L t( , )v ≤ M t u( , ) ( )v v− pry vsex t I∈ , u ≥ v ≥ 0.
Tohda esly suwestvugt postoqnn¥e K > 0, δ0 > 0 takye, çto
M s s( , )δ
α
+∞
∫ ∆ ≤ K
dlq lgboho 0 ≤ δ ≤ δ0 , to sostoqnye ravnovesyq x t( ) ≡ 0 system¥ (1) rav-
nomerno ustojçyvo.
Dokazatel\stvo. Yspol\zuq sledstvye11 y lemmu12, dlq reßenyq x t( )
system¥ (1) poluçaem ocenku
x t( ) ≤ x0 + x t x( ) − 0 ≤ x0 + L s x e t s s
t
t
p, , ( )0
0
( ) ( )∫ σ ∆ ≤
≤ x0 + L s x p s
t
t
s
t
, exp ( )
( )
0
0
( )
∫ ∫ τ τ
σ
∆ ∆ =
= x0 + L s x M x s
t
t
s
t
, exp ,
( )
0 0
0
( ) ( )
∫ ∫ τ τ
σ
∆ ∆ . (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1496 T. A. LUK|QNOVA, A. A. MARTÁNGK
V¥berem proyzvol\noe ε > 0 y poloΩym δ = δ ε( ) = min ε/2{ , ε/( )2KeK ,
δ0} . V sylu uslovyq12 teorem¥ L t u( , ) ≤ M t u( , )0 pry vsex t ∈ t0, +∞[ )T y
u ≥ 0, otkuda sleduet neravenstvo
L s u s
t
t
( , )
0
∫ ∆ ≤ u M s s
t
t
( , )0
0
∫ ∆ ≤ u M s s( , )0
α
+∞
∫ ∆ ≤ Ku .
ProdolΩaq ocenku (12), pry vsex x0 ≤ δ y t ∈ t0, +∞[ )T poluçaem
x t( ) ≤
ε
2
+ L s x M x s
t
t
s
t
, exp ,
( )
0 0
0
( ) ( )
∫ ∫ τ τ
σ
∆ ∆ ≤
≤
ε
2
+ L s x M x s
t
t
, exp ,0 0
0
( ) ( )
∫ ∫
+∞
τ τ
α
∆ ∆ ≤
≤
ε
2
+ e L s x sK
t
t
, 0
0
( )∫ ∆ ≤
ε
2
+ e K xK
0 ≤ ε,
otkuda y sleduet ravnomernaq ustojçyvost\ nulevoho sostoqnyq ravnovesyq
system¥ (1).
Teorema 1 dokazana.
Teorema 2. PredpoloΩym, çto suwestvuet rd -neprer¥vnaq funkcyq S :
α, +∞[ )T × R+ → R+ takaq, çto pry vsex t ∈ α, +∞[ )T y x , y n∈R v¥pol-
nqetsq neravenstvo
f t x y f t x( , ) ( , )+ − ≤ S t x y,( ) .
Tohda esly suwestvugt postoqnn¥e K > 0 y δ0 > 0 takye, çto
S u u, δ
α
( )
+∞
∫ ∆ ≤ K
dlq lgboho 0 ≤ δ ≤ δ0 , to sostoqnye ravnovesyq x t( ) ≡ 0 system¥1(1) rav-
nomerno ustojçyvo.
Dokazatel\stvo. Polahaq y = – x pry vsex t I∈ y x n∈R , poluçaem ne-
ravenstvo
f t x( , ) ≤ S t x x,( ) ,
otkuda sohlasno lemme15 ymeem
x t( ) ≤ x0 + x t x( ) − 0 ≤ x0 + f s x e t s s
t
t
p( , ) , ( )0
0
∫ ( )σ ∆ ≤
≤ x0 + x S s x p s
t
t
s
t
0 0
0
, exp ( )
( )
( )
∫ ∫ τ τ
σ
∆ ∆ =
= x0 + x S s x S x s
t
t
s
t
0 0 0
0
, exp ,
( )
( ) ( )
∫ ∫ τ τ
σ
∆ ∆ , (13)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
YNTEHRAL|NÁE NERAVENSTVA Y USTOJÇYVOST| SOSTOQNYQ RAVNOVESYQ … 1497
hde p t( ) = S t x, 0( ) , t ∈ t0, +∞[ )T . V¥berem proyzvol\noe ε > 0 y poloΩym
δ = δ ε( ) = min ε/2{ , ε/( )2KeK , δ0} . ProdolΩaq ocenku (13), pry vsex x0 ≤ δ
y t ∈ t0, +∞[ )T poluçaem
x t( ) ≤
ε
2
+ x S s x S x s0 0 0, exp ,( ) ( )
+∞ +∞
∫ ∫
α α
τ τ∆ ∆ ≤
≤
ε
2
+ x e S s x sK
0 0,( )
+∞
∫
α
∆ ≤
≤
ε
2
+ e K xK
0 ≤ ε,
otkuda y sleduet ravnomernaq ustojçyvost\ nulevoho sostoqnyq ravnovesyq
system¥ (1).
Teorema 2 dokazana.
Zameçanye 1. V sluçae, kohda T = R, yntehral y ∆-proyzvodnaq na T
sovpadagt s yntehralom Rymana y πjlerovoj proyzvodnoj. Poπtomu teore-
m¥13.5.1 y 3.5.7 yz [1] avtomatyçesky poluçagtsq kak sledstvyq yz teorem 1 y 2.
Pust\ teper\ T = Z. V πtom sluçae naçal\naq zadaça (1), (2) prynymaet vyd
∆x( )τ = f x tτ, ( )( ) , τ ∈ I , (14)
x x( ; , )τ τ0 0 0 = x0 , τ0 ∈ I , x n
0 ∈R , (15)
hde x n∈R , ∆x( )τ = x( )τ + 1 – x( )τ , I = α{ , α + 1, α + 2, …} , α ∈Z , f : I ×
× Rn → Rn
, f ( , )τ 0 ≡ 0, y dlq zadaçy (14), (15) v¥polnqgtsq uslovyq suwest-
vovanyq y edynstvennosty reßenyq na τ0, +∞[ )T pry lgb¥x naçal\n¥x dann¥x
( , )τ0 0x ∈ I × Rn
.
Sledstvye 3. PredpoloΩym, çto suwestvugt funkcyy L , M : I × R+ →
→ R+ takye, çto L( , )τ 0 ≡ 0 y v¥polnqgtsq neravenstva:
1) f x( , )τ ≤ L xτ,( ) pry vsex τ ∈ I , x n∈R ,
2) 0 ≤ L u( , )τ – L( , )τ v ≤ M u( , ) ( )τ v v− pry vsex τ ∈ I , u ≥ v ≥ 0.
Esly suwestvugt postoqnn¥e K > 0, δ0 > 0 takye, çto
M( , )τ δ
τ α=
+∞
∑ ≤ K
dlq lgboho 0 ≤ δ ≤ δ0 , to sostoqnye ravnovesyq x t( ) ≡ 0 system¥1(14) rav-
nomerno ustojçyvo.
Sledstvye 4. PredpoloΩym, çto suwestvuet funkcyq S : I × R+ → R+
takaq, çto pry vsex t I∈ y x, y n∈R v¥polnqetsq neravenstvo
f t x y f t x( , ) ( , )+ − ≤ S t x y,( ) .
Tohda esly suwestvugt postoqnn¥e K > 0 y δ0 > 0 takye, çto
S( , )τ δ
τ α=
+∞
∑ ≤ K
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
1498 T. A. LUK|QNOVA, A. A. MARTÁNGK
dlq lgboho 0 ≤ δ ≤ δ0 , to sostoqnye ravnovesyq x t( ) ≡ 0 system¥ (14) rav-
nomerno ustojçyvo.
5. Prymer. Rassmotrym systemu dynamyçeskyx uravnenyj na vremennoj
ßkale T vyda
x t A t x t∆ ( ) ( ) ( )= , (16)
hde x = ( , )x x1 2
T
, x1 , x2 : T → R, A t( ) =
0
0
a t
a t
( )
( )
, I = α, +∞[ )T . Predpo-
loΩym, çto funkcyq a : T → R qvlqetsq neprer¥vnoj, rehressyvnoj, a t( ) ≠ 0
pry vsex t I∈ y
a s s( )
α
+∞
∫ ∆ ≤ M .
Qsno, çto funkcyq f t x( , ) = A t x( ) qvlqetsq rd-neprer¥vnoj. PokaΩem,
çto ona rehressyvna. Dlq πtoho dostatoçno pokazat\, çto pry lgbom fyksyro-
vannom t I∈ operator F : R2 → R2
, dejstvugwyj po formule F x( ) = x +
+ µ( ) ( )t A t x , obratym. Dlq lgboho z ∈R2
uravnenye F( )ξ = z ymeet edynst-
vennoe reßenye ξ = z t a t2 1( +/( ( ) ( ))µ , z t a t1 1/( ( ) ( ))+ )µ T
, çto y oznaçaet obra-
tymost\ operatora F y, sootvetstvenno, rehressyvnost\ funkcyy f t x( , ) .
Poskol\ku f t x( , ) = a t x( ) , v¥polnen¥ uslovyq lemm¥13, t.1e. su-
westvuet edynstvennoe reßenye system¥ (16) na t0, +∞[ )T pry lgb¥x naçal\-
n¥x dann¥x ( , )t x0 0 ∈ I × R2
.
Proverym v¥polnenye uslovyj teorem¥11. Lehko vydet\, çto L t u( , ) =
= a t u( ) ≥ 0 pry u ≥ 0 y, krome toho, 0 ≤ L t u( , ) – L t( , )v = a t u( ) ( )− v pry
u ≥ v ≥ 0 y M t( , )v = a t( ) . Funkcyy L t u( , ) , M t( , )v rd-neprer¥vn¥, po-
πtomu uslovyq teorem¥11 v¥polnen¥ y sostoqnye ravnovesyq x t( ) ≡ 0 syste-
m¥1(16) ravnomerno ustojçyvo.
V çastnosty, funkcyq
a t( ) = � �λ αλe t( , ) =
λ
µ λ
αλ
1 + ( )
( , )
t
e t�
pry neprer¥vnoj µ( )t qvlqetsq neprer¥vnoj y rehressyvnoj pry λ > 0. Po-
skol\ku
λ
µ λ
α
α
λ
1 +
+∞
∫ ( )
( , )
s
e s s� ∆ = −
+∞
∫ � �λ α
α
λe s s( , ) ∆ = − +∞e s�λ αα( , ) =
= − −
→+∞
lim ( , ) ( , )
s
e s e� �λ λα α α = 1,
sostoqnye ravnovesyq x t( ) ≡ 0 system¥ (16) ravnomerno ustojçyvo.
6. Zaklgçytel\n¥e zameçanyq. Za redkym ysklgçenyem (sm. [4]) osnov-
n¥m yntehral\n¥m neravenstvom, prymenqem¥m na vremennoj ßkale, qvlqetsq
neravenstvo Hronuolla (sm. [3]) y nekotor¥e eho modyfykacyy. Lemm¥ 4 y 5,
pryvedenn¥e v πtoj stat\e, pozvolqgt rasßyryt\ hranyc¥ prymenymosty yn-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
YNTEHRAL|NÁE NERAVENSTVA Y USTOJÇYVOST| SOSTOQNYQ RAVNOVESYQ … 1499
tehral\n¥x neravenstv na vremennoj ßkale v processe analyza reßenyj dyna-
myçeskyx uravnenyj. Yx prymenenye moΩet okazat\sq perspektyvn¥m v soçeta-
nyy s metodom funkcyy Lqpunova dlq dynamyçeskyx uravnenyj (sm. [7]).
1. Dragomir S. S. The Gronwall type lemmas and applications. – Timisoara: Tipografia Univ. Timi-
soara, 1987. – 90 p.
2. Boxner M., Mart¥ngk A. A. ∏lement¥ teoryy ustojçyvosty A. M. Lqpunova dlq dynamy-
çeskyx uravnenyj na vremennoj ßkale // Prykl. mexanyka. – 2007. – 43, # 9. – S. 3 – 27.
3. Bohner M., Peterson A. Dynamic equations on time scales: An introduction with applications. –
Boston: Birkhäuser, 2001. – 358 p.
4. Pachpatte D. B. Explicit estimates on integral inequalities with time scale // J. Inequalities in Pure
and Appl. Math. – 2006. – 7, # 4.
5. Bohner M. Some oscillation criteria for first order delay dynamic equations // Far East J. Appl.
Math. – 2005. – 18, # 3. – P. 289 – 304.
6. Peterson A. C., Raffoul Y. N. Exponential stability of dynamic equations on time scales // Adv.
Difference Equat. – 2005. – 2005, # 2. – P. 133 – 144.
7. Mart¥ngk-Çernyenko G. A. K teoryy ustojçyvosty dvyΩenyq nelynejnoj system¥ na vre-
mennoj ßkale // Ukr. mat. Ωurn. – 2008. – 60, # 6. – S. 776 – 782.
Poluçeno 30.11.09,
posle dorabotky — 02.07.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 11
|
| id | umjimathkievua-article-2973 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:33:48Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/20/0c23ab975e4ca57d46d69def7e3ad620.pdf |
| spelling | umjimathkievua-article-29732020-03-18T19:41:38Z Integral inequalities and stability of an equilibrium state on a time scale Интегральные неравенства и устойчивость состояния равновесия на временной шкале Luk’yanova, T. A. Martynyuk, A. A. Лукьянова, Т. А. Мартынюк, А. А. Лукьянова, Т. А. Мартынюк, А. А. We present some integral inequalities on a time scale and establish sufficient conditions for the uniform stability of an equilibrium state of a nonlinear system on a time scale. Наведено деякі інтегральні нерівності на часовій шкалі та отримано достатні умови рівномірної стійкості стану рівноваги нелінійної системи на часовій шкалі. Institute of Mathematics, NAS of Ukraine 2010-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2973 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 11 (2010); 1490–1499 Український математичний журнал; Том 62 № 11 (2010); 1490–1499 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2973/2696 https://umj.imath.kiev.ua/index.php/umj/article/view/2973/2697 Copyright (c) 2010 Luk’yanova T. A.; Martynyuk A. A. |
| spellingShingle | Luk’yanova, T. A. Martynyuk, A. A. Лукьянова, Т. А. Мартынюк, А. А. Лукьянова, Т. А. Мартынюк, А. А. Integral inequalities and stability of an equilibrium state on a time scale |
| title | Integral inequalities and stability of an equilibrium state on a time scale |
| title_alt | Интегральные неравенства и устойчивость состояния равновесия на временной шкале |
| title_full | Integral inequalities and stability of an equilibrium state on a time scale |
| title_fullStr | Integral inequalities and stability of an equilibrium state on a time scale |
| title_full_unstemmed | Integral inequalities and stability of an equilibrium state on a time scale |
| title_short | Integral inequalities and stability of an equilibrium state on a time scale |
| title_sort | integral inequalities and stability of an equilibrium state on a time scale |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2973 |
| work_keys_str_mv | AT lukyanovata integralinequalitiesandstabilityofanequilibriumstateonatimescale AT martynyukaa integralinequalitiesandstabilityofanequilibriumstateonatimescale AT lukʹânovata integralinequalitiesandstabilityofanequilibriumstateonatimescale AT martynûkaa integralinequalitiesandstabilityofanequilibriumstateonatimescale AT lukʹânovata integralinequalitiesandstabilityofanequilibriumstateonatimescale AT martynûkaa integralinequalitiesandstabilityofanequilibriumstateonatimescale AT lukyanovata integralʹnyeneravenstvaiustojčivostʹsostoâniâravnovesiânavremennojškale AT martynyukaa integralʹnyeneravenstvaiustojčivostʹsostoâniâravnovesiânavremennojškale AT lukʹânovata integralʹnyeneravenstvaiustojčivostʹsostoâniâravnovesiânavremennojškale AT martynûkaa integralʹnyeneravenstvaiustojčivostʹsostoâniâravnovesiânavremennojškale AT lukʹânovata integralʹnyeneravenstvaiustojčivostʹsostoâniâravnovesiânavremennojškale AT martynûkaa integralʹnyeneravenstvaiustojčivostʹsostoâniâravnovesiânavremennojškale |