Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order
We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type.
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| Дата: | 2007 |
|---|---|
| Автори: | , , , |
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| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509476570791936 |
|---|---|
| author | Evtukhov, V. M. Stekhun, A. A. Евтухов, В. М. Стехун, А. А. Евтухов, В. М. Стехун, А. А. |
| author_facet | Evtukhov, V. M. Stekhun, A. A. Евтухов, В. М. Стехун, А. А. Евтухов, В. М. Стехун, А. А. |
| author_sort | Evtukhov, V. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:53:10Z |
| description | We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type. |
| first_indexed | 2026-03-24T02:41:43Z |
| format | Article |
| fulltext |
UDK 517.925
V. M. Evtuxov (Odes. nac. un-t),
A. A. Stexun (Odes. nac. mor. un-t)
ASYMPTOTYÇESKYE PREDSTAVLENYQ REÍENYJ
ODNOHO KLASSA NELYNEJNÁX NEAVTONOMNÁX
DYFFERENCYAL|NÁX URAVNENYJ
TRET|EHO PORQDKA
Asymptotic representations are established for unbounded solutions of nonlinear nonautonomous third-
order differential equations that, in a certain sense, are close to equations of the Emden – Fowler type.
Vstanovleno asymptotyçni zobraΩennq dlq neobmeΩenyx rozv’qzkiv nelinijnyx neavtonomnyx
dyferencial\nyx rivnqn\ tret\oho porqdku, wo u deqkomu sensi [ blyz\kymy do rivnqn\ typu
Emdena – Faulera.
Rassmatryvaetsq dyfferencyal\noe uravnenye
′′′ =y p t yα ϕ0 ( ) ( ), (1)
hde α0 1 1∈{− }, , p : a, ω[ [ → 0, + ∞] [ ( – ∞ < a < ω ≤ + ∞)1
— neprer¥vnaq
funkcyq, ϕ : y 0, +∞[ [ → 0, + ∞] [ — dvaΩd¥ neprer¥vno dyfferencyruemaq
funkcyq takaq, çto
lim ( )
y
y
→ +∞
ϕ =
yly
yly
0,
,+ ∞
′ ≠ϕ ( )y 0, lim
( )
( )y
y y
y→ +∞
′′
′
ϕ
ϕ
= σ = const ≠ 0.
(2)
Eho çastn¥m sluçaem qvlqetsq uravnenye typa Emdena – Faulera
′′′ = +y p t y yα σ
0
1( ) sign , σ ≠ 0.
Posle yssledovanyq asymptotyçeskyx svojstv reßenyj πtoho uravnenyq name-
tylys\ nov¥e ydey v dopolnenye k tem, kotor¥e yspol\zovalys\ pry yzuçenyy
dyfferencyal\n¥x uravnenyj vtoroho porqdka, pozvolyvßye v dal\nejßem
(sm. monohrafyg Y.:T.:Kyhuradze y T.:A.:Çanturyq [1], a takΩe rabot¥ [2 – 6])
postroyt\ asymptotyçeskug teoryg nelynejn¥x dyfferencyal\n¥x uravne-
nyj typa ∏mdena – Faulera n-ho porqdka.
Poπtomu dyfferencyal\noe uravnenye (1) sluΩyt vaΩn¥m promeΩutoçn¥m
zvenom pry perexode k nelynejn¥m dyfferencyal\n¥m uravnenyqm n-ho po-
rqdka bolee obweho vyda, çem uravnenyq typa ∏mdena – Faulera, y trebugt
detal\noho yssledovanyq asymptotyçeskyx svojstv vsex eho vozmoΩn¥x typov
reßenyj.
Reßenye y uravnenyq (1) budem naz¥vat\ Pω λ1 0( )-reßenyem, esly ono opre-
deleno v nekotoroj levoj okrestnosty ω y udovletvorqet sledugwym trem
uslovyqm:
lim ( )
t
y t
↑ω
= + ∞, lim ( )( )
t
ky t
↑ω
=
yly
yly
0,
,±∞
k = 1, 2, (3)
lim
( )
( ) ( )t
y t
y t y t↑
′′[ ]
′′′ ′ω
2
= λ0 . (4)
V [7] b¥ly poluçen¥ neobxodym¥e y dostatoçn¥e uslovyq suwestvovanyq, a
1
Pry ω = + ∞ sçytaem, çto a > 0.
© V. M. EVTUXOV, A. A. STEXUN, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10 1363
1364 V. M. EVTUXOV, A. A. STEXUN
takΩe asymptotyçeskye pry t ↑ ω predstavlenyq Pω λ1 0( )-reßenyj uravnenyq
(1), dlq kotor¥x λ0 ∈ R \ 0 1
2
1, ,{ }.
Dannaq stat\q posvqwena Pω λ1 0( )-reßenyqm uravnenyq (1), sootvetstvug-
wym znaçenyqm λ0 = ± ∞ y λ0 = 0.
Vvedem neobxodym¥e dlq dal\nejßeho dopolnytel\n¥e uslovyq. Budem ho-
voryt\, çto funkcyq ϕ( )y udovletvorqet uslovyg Sk , k ∈ {1, 2}, esly dlq
lgboj neprer¥vno dyfferencyruemoj funkcyy L : t0, +∞[ [ → 0, + ∞] [ takoj,
çto
lim
( )
( )t
t L t
L t→ +∞
′
= 0, (5)
funkcyq ψ( )y =
ϕ
σ
( )y
y1+ dopuskaet asymptotyçeskoe predstavlenye vyda
ψ ψt L t t ok k( ) ( ) ( )( ) = +[ ]1 1 pry t → + ∞. (6)
Uslovyqm Sk zavedomo udovletvorqgt funkcyy ϕ, dlq kotor¥x funkcyq
ϕ
σ
( )y
y1+ ymeet koneçn¥j predel pry y → + ∞, a takΩe funkcyy vyda ϕ( )y =
= y y1+ σ µln , ϕ( )y = y y y1+ σ µ νln ln ln , hde µ, ν ≠ 0, y dr.
Teorema 1. Pust\ funkcyq ϕ udovletvorqet uslovyg S1. Tohda dlq su-
westvovanyq Pω1 0( )-reßenyj dyfferencyal\noho uravnenyq (1) neobxodymo,
çtob¥ ω = + ∞, v¥polnqlos\ neravenstvo
α σ0 2 0J t( ) < pry t a∈ + ∞] [, (7)
y ymely mesto predel\n¥e sootnoßenyq
lim ( ) /
t
t J t
→ +∞
−
2
1 σ = + ∞, lim
( )
( ) ( )t
J t
J t J t→ +∞
′[ ]
′′
2
2
2 2
= 0, (8)
hde
J t2( ) =
A
t
J d
2
1∫ ( )τ τ , J t1( ) =
A
t
p s s ds
1
∫ ( ) ( )ϕ ,
A
a p s s ds
p s s ds
a
a
1 =
= +∞
+∞ < +∞
+∞
+∞
∫
∫
, ( ) ( ) ,
, ( ) ( ) ,
esly
esly
ϕ
ϕ
A
a J d
J d
a
a
2
1
1
=
= +∞
+∞ < +∞
+∞
+∞
∫
∫
, ( ) ,
, ( ) ,
esly
esly
τ τ
τ τ
pryçem kaΩdoe takoe reßenye dopuskaet pry t → + ∞ asymptotyçeskye pred-
stavlenyq
y t t J t( ) ( ) /∼ −σ σ
2
1
, ′ ∼ −y t J t( ) ( ) /σ σ
2
1
, ′′ ∼ − +y t J t J t( ) ( ) ( ) ( )/α σ σ σ
0 1 2
1
.
(9)
Bolee toho, uslovyq (7) y (8) qvlqgtsq dostatoçn¥my dlq suwestvovanyq
P+∞1 0( )-reßenyj uravnenyq (1) v sluçae – 2 < σ < 0, a takΩe v sluçae, kohda
suwestvuet koneçn¥j yly ravn¥j ± ∞ predel lim
( )
( )t
t J t
J t→ +∞
′
1
1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
ASYMPTOTYÇESKYE PREDSTAVLENYQ REÍENYJ ODNOHO KLASSA … 1365
Dokazatel\stvo. Neobxodymost\. Pust\ y : t0, ω[ [ → y 0, +∞[ [ — proyz-
vol\noe Pω1 0( )-reßenye dyfferencyal\noho uravnenyq (1). Tohda v sylu (1) y
opredelenyq Pω1 0( )-reßenyq v¥polnqgtsq uslovyq (3), y, y′, y′′, y′′′ otlyçn¥
ot nulq na nekotorom promeΩutke t1, ω[ [ ⊂ t0, ω[ [ y
lim
( ) ( )
( )t
y t y t
y t↑
′′′ ′
′′[ ]ω 2 = ± ∞. (10)
Poskol\ku
′′′ ′
′′[ ]
y t y t
y t
( ) ( )
( )
2 =
′′
′
′
′′
′
+
y t
y t
y t
y t
( )
( )
( )
( )
2 1 pry t ∈ t1, ω[ [, (11)
yz (10) s uçetom toho, çto lim ( )
t
y t
↑
′
ω
raven lybo nulg, lybo + ∞, poluçaem
′′
′
−
y t
y t
( )
( )
1
= − +[ ]∫
C
t
d oγ τ τ( ) ( )1 1 pry t ↑ ω,
hde
lim ( )
t
t
↑ω
γ = ± ∞, C =
t d
d
t
t
1
1
1
, ( ) ,
, ( ) .
esly
esly
ω
ω
γ τ τ
ω γ τ τ
∫
∫
= ±∞
=
const
Otsgda neposredstvenno sleduet, çto
lim
( ) ( )
( )t
t y t
y t↑
′′
′ =
ω
ωπ
0 , hde πω( )t =
t
t
, ,
, ,
esly
esly
ω
ω ω
= +∞
− < +∞
y poπtomu, yspol\zuq pravylo Lopytalq, naxodym
lim
( ) ( )
( )
lim
( ) ( ) ( )
( )t t
t y t
y t
y t t y t
y t↑ ↑
′
=
′ + ′′
′
=
ω
ω
ω
ωπ π
1.
Yz πtoho predel\noho sootnoßenyq v sylu pervoho yz uslovyj (3) sleduet, çto
funkcyq πω qvlqetsq poloΩytel\noj v nekotoroj levoj okruΩnosty ω. Po-
skol\ku πto vozmoΩno lyß\ v sluçae, kohda ω = + ∞, rassmatryvaemoe reßenye
dyfferencyal\noho uravnenyq (1) qvlqetsq P+∞1 0( )-reßenyem. Poskol\ku ω
= + ∞, to dlq dannoho reßenyq v sylu yzloΩennoho v¥ße
lim
( )
( )t
t y t
y t→ +∞
′′
′
= 0, lim
( )
( )t
t y t
y t→ +∞
′
= 1. (12)
Otsgda, v çastnosty, sleduet, çto
lim
( )
( )t
t
y t
t
y t
t
→ +∞
′
= 0,
y poπtomu vsledstvye v¥polnenyq uslovyq S1 ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1366 V. M. EVTUXOV, A. A. STEXUN
ϕ ψ ψ ψσ
y t
y t
y t t
y t
t
t o
( )
( )
( )
( )
( ) ( )
( ) = ( ) = ⋅
= +[ ]+1 1 1 pry t → + ∞. (13)
Krome toho, s uçetom (4) naxodym
′′
′[ ]
′
+
y t
y t
( )
( )
1 σ =
′′′
′[ ]
− +
′′[ ]
′′′ ′
+
y t
y t
y t
y t y t
( )
( )
( )
( )
( ) ( )1
2
1 1σ σ =
=
′′′
′[ ]
+[ ]+
y t
y t
o
( )
( )
( )1 1 1σ pry t → + ∞.
V sylu πtyx dvux asymptotyçeskyx predstavlenyj yz (1) s uçetom vtoroho yz
uslovyj (12) sleduet, çto
′′
′[ ]
′
+
y t
y t
( )
( )
1 σ = α ϕ0 1 1p t t o( ) ( ) ( )+[ ] pry t → + ∞.
Yntehryruq πto sootnoßenye na promeΩutke ot t1 do t, poluçaem
′′
′[ ] +
y t
y t
( )
( )
1 σ = C + α0 1 1 1J t o( ) ( )+[ ] pry t → + ∞,
hde C — nekotoraq postoqnnaq.
PokaΩem, çto otsgda v¥tekaet predstavlenye vyda
′′
′[ ] +
y t
y t
( )
( )
1 σ = α0 1 1 1J t o( ) ( )+[ ] pry t → + ∞. (14)
V samom dele, esly b¥ πto b¥lo ne tak, to predel yntehryrovanyq A1 v J1 b¥l
b¥ raven + ∞ y ymelo b¥ mesto predstavlenye
′′
′[ ] +
y t
y t
( )
( )
1 σ = C + o( )1 pry t →
→ + ∞, hde C ≠ 0. Uçyt¥vaq eho, a takΩe (12) y (13), yz (1) poluçaem
′′′
′′
y t
y t
( )
( )
=
α ϕ0 1 1
C
p t t o( ) ( ) ( )+[ ] pry t → + ∞,
otkuda sleduet, çto
lim ln ( )
t
y t
→ +∞
′′ = const.
Odnako, πtoho b¥t\ ne moΩet, poskol\ku dlq rassmatryvaemoho reßenyq y v¥-
polnqetsq vtoroe yz uslovyj (3). Znaçyt, ymeet mesto (14).
Yntehryruq sootnoßenye (14) na promeΩutke ot t1 do t y prynymaq vo
vnymanye vtoroe yz uslovyj (3), a takΩe uslovye σ ≠ 0, ymeem
′[ ] −
y t( )
σ
= − +[ ]α σ0 2 1 1J t o( ) ( ) pry t → + ∞.
Otsgda qsno, çto v¥polnqetsq neravenstvo (7) y ymeet mesto vtoroe yz asymp-
totyçeskyx predstavlenyj (9). V sylu πtoho predstavlenyq yz vtoroho yz uslo-
vyj (12) y sootnoßenyq (14) v¥tekagt pervoe y tret\e yz asymptotyçeskyx
predstavlenyj (9).
Yspol\zuq teper\ asymptotyçeskye predstavlenyq (9), a takΩe sootnoßenye
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
ASYMPTOTYÇESKYE PREDSTAVLENYQ REÍENYJ ODNOHO KLASSA … 1367
′′′y t( ) = α ϕ σ
0
1
1 1p t t y t o( ) ( ) ( ) ( )′[ ] +[ ]+
pry t → + ∞,
kotoroe v¥tekaet yz (1), (13) y vtoroho yz uslovyj (12), poluçaem na osnovanyy
pervoho yz uslovyj (3) y uslovyq (4), hde ω = + ∞ y λ0 = 0, uslovyq (8).
Dostatoçnost\. Pust\ funkcyq ϕ udovletvorqet uslovyg S1, ω = + ∞ y
v¥polnqgtsq uslovyq (7), (8). Yspol\zuq vtoroe yz uslovyj (8), toçno tak Ωe,
kak yz (10) b¥lo poluçeno pry dokazatel\stve neobxodymosty pervoe yz uslo-
vyj:(12), ustanavlyvaem, çto
lim
( )
( )t
t J t
J t→ +∞
′
2
2
= 0. (15)
Prynymaq vo vnymanye pervoe yz uslovyj (8), podbyraem çyslo t0 ≥ a na-
stol\ko bol\ßym, çtob¥ pry t ≥ t0 v¥polnqlos\ neravenstvo
1
2 2
1t J tσ σ( ) /− ≥
≥ max {0, y0}.
Dyfferencyal\noe uravnenye (1) s pomow\g preobrazovanyq
τ = ln t, y t( ) = t J tσ τσ2
1
11( ) ( )− +[ ]v ,
(16)
′y t( ) = σ τσJ t2
1
21( ) ( )− +[ ]v , ′′y t( ) = α σ τ
σ
σ0 1 2
1
31J t J t( ) ( ) ( )− +
+[ ]v
y s uçetom toho, çto
ϕ σ σt J t2
1
11( ) − +( )
v =
= ϕ σ σt J t2
1
( ) −
+ t J t t J tσ ϕ σσ σ2
1
2
1
1( ) ( )− −′
v +
+ 1
2
12
2
2
2
1
1
2t J t t J tσ ϕ σ ξσ σ( ) ( ) ( )− −′′ +
v ,
hde ξ = ξ( , )t v1 udovletvorqet neravenstvu 0 < ξ < v1 pry t ≥ t0 y v1 ≤ 1
2
,
svedem k systeme dyfferencyal\n¥x uravnenyj
′ = + − +[ ] +v v v1 1 1 1 21q q( ) ( )τ τ ,
′ = −v v v2 1 2 3q ( )( )τ ,
′ = + + − + +
+
v v v v3 2 1
1
2
3 11
1
q f c
q
q
V( ) ( ) ( )
( ) ( )
( )
( , )τ τ τ σ τ
τ
τ , (17)
v kotoroj
q t
t J t
J t1
2
2
τ( )
( )
( )
( ) =
′
, q t
t J t
J t2
1
1
τ( )
( )
( )
( ) =
′
,
f t
t J t
t J t
q t
q t
τ
ϕ σ
ϕ σ
σ τ
τ
σ
σ
σ
( )
( )
( ) ( )
( ) ( )
( )
( ) =
− + + ( )
( )
−
− +
2
1
2
1
1
2
1
1
,
c t
t J t t J t
t J t
τ
σ ϕ σ
ϕ σ
σ σ
σ
σ
( )
( ) ( )
( ) ( )
( ) =
′
− −
− +
2
1
2
1
2
1 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1368 V. M. EVTUXOV, A. A. STEXUN
V t
t J t t J t
t J t
τ
σ ϕ σ ξ
ϕ σ
σ σ
σ
σ
( ),
( ) ( ) ( )
( ) ( )
v v1
2
2
2
2
1
2
1 1
2
1
2
( ) =
′′ +
− −
− + .
Zdes\ v sylu uslovyj (2), S1, (8) y (15)
lim ( )
τ
τ
→ +∞
q1 = 0, lim
( )
( )τ
τ
τ→ +∞
q
q
1
2
= 0, lim ( )
τ
τ
→ +∞
f = 0, lim ( )
τ
τ
→ +∞
c = 1 + σ
(18)
y
lim
( , )
v
v
v1 0
1 1
1→
V τ
= 0 ravnomerno po τ ∈ +∞[ [ln ,t0 . (19)
Krome toho,
ln
( )
( )
( )
ln ( )
t
i
t
i
i
i t
q d
J s ds
J s
J s
0 0
0
3
3
3
+∞ +∞
−
−
−
+∞
∫ ∫=
′
= = ± ∞τ τ , i = 1, 2. (20)
Poπtomu pry – 2 < σ < 0 systema dyfferencyal\n¥x uravnenyj (17) ymeet
sohlasno teoreme 1.3 y zameçanyg 1.4 yz rabot¥ [8] xotq b¥ odno reßenye
( )vi i=1
3
: ln ,t1 +∞[ [ → R
3
, hde t1 ≥ t0 , stremqweesq k nulg pry τ → + ∞. Emu v
sylu zamen (16) sootvetstvuet reßenye y :: t1, +∞[ [ → R, dopuskagwee pry t →
→ + ∞ asymptotyçeskye predstavlenyq (9).
Dopustym teper\, çto suwestvuet (koneçn¥j yly ravn¥j ± ∞ ) predel
lim
( )
( )t
t J t
J t→ +∞
′
1
1
. Tohda, uçyt¥vaq (15) y yspol\zuq pravylo Lopytalq, poluçaem
0 = lim
( )
( )t
t J t
J t→ +∞
′
2
2
= lim
( )
( )t
t J t
J t→ +∞
( )′
′
2
2
= lim
( )
( )t
t J t
J t→ +∞
+
′′
′
1 2
2
= lim
( )
( )t
t J t
J t→ +∞
+
′
1 1
1
.
Sledovatel\no,
lim
( )
( )t
t J t
J t→ +∞
′
1
1
= – 1 y lim ( )
τ
τ
→ +∞
q2 = – 1. (21)
Ustanovyv πtot fakt, systemu dyfferencyal\n¥x uravnenyj (17) s pomow\g
preobrazovanyq
v1 1= z , v2 2 1 1 31= + + −[ ]z q z z( ) ( )τ σ , v3 3= z (22)
pryvedem k vydu
′ = + − + +[ ] + −z q q z z q z1 1 1 1 2 1 31 2( ) ( ) ( ) ( )τ σ τ τ ,
′ = + − + +[ ]z q f c z z q z q V z2 1 1 1 1 2 1 3 2 12( ) ( ) ( ) ( ) ( ) ( , )τ τ τ σ σ τ τ τ , (23)
′ = + + − + +
+
z q f c z
q
q
z V z3 2 1
1
2
3 11
1
( ) ( ) ( )
( ) ( )
( )
( , )τ τ τ σ τ
τ
τ ,
hde
f q f q1 2 11( ) ( ) ( ) ( ) ( )τ τ τ σ τ= − + , c q q q c1 1 2 21( ) ( ) ( ) ( ) ( ) ( )τ σ σ τ τ τ τ= − + +[ ] +
y v sylu uslovyj (18), (21) takov¥, çto
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ASYMPTOTYÇESKYE PREDSTAVLENYQ REÍENYJ ODNOHO KLASSA … 1369
lim ( )
τ
τ
→ +∞
f1 = 0, lim ( )
τ
τ
→ +∞
c1 = 0. (24)
Esly teper\ uçest\, çto σ ≠ 0 y v¥polnqgtsq uslovyq (18) – (21), (24), to,
v¥byraq proyzvol\n¥m obrazom çyslo δ ∈ 0 1,] [ y prymenqq k systeme (23)
novoe dopolnytel\noe preobrazovanye
z w1 1= δ , z w2 2= , z w3 3= , (25)
poluçaem systemu dyfferencyal\n¥x uravnenyj, kotoraq na osnovanyy teore-
m¥ 1.3 y zameçanyq 1.4 yz rabot¥ [8] ymeet xotq b¥ odno reßenye ( )wi i=1
3
:
ln ,t1 +∞[ [ → R
3
, hde t1 ≥ t0 , stremqweesq k nulg pry τ → + ∞. Emu v sylu
zamen (25), (22) y (16) sootvetstvuet reßenye y :: t1, +∞[ [ → y0, +∞[ [ dyffe-
rencyal\noho uravnenyq (1), dopuskagwee pry t → + ∞ asymptotyçeskye pred-
stavlenyq (9). Yspol\zuq πty predstavlenyq, vyd uravnenyq (1), a takΩe uslo-
vyq S1 y (8), lehko ubeΩdaemsq v tom, çto dannoe reßenye qvlqetsq P+∞1 0( )-
reßenyem uravnenyq (1).
Teorema dokazana.
Zameçanye 1. Esly v¥polnen¥ ukazann¥e v dannoj teoreme dostatoçn¥e
uslovyq suwestvovanyq P+∞1 0( )-reßenyj uravnenyq (1), dopuskagwyx pry t →
→ + ∞ asymptotyçeskye predstavlenyq (9), to s yspol\zovanyem zameçanyq 1.1
rabot¥ [8] netrudno ustanovyt\, çto pry α σ0 > 0 suwestvuet odnoparamet-
ryçeskoe semejstvo takyx reßenyj, a pry α σ0 < 0 — dvuparametryçeskoe.
Teorema 2. Pust\ funkcyq ϕ udovletvorqet uslovyg S2 . Tohda dlq
suwestvovanyq Pω1( )±∞ -reßenyj uravnenyq (1) neobxodymo y dostatoçno,
çtob¥ ω = + ∞, v¥polnqlos\ neravenstvo
α σ0 3 0J t( ) < pry t a∈ +∞] [, (26)
y ymely mesto predel\n¥e sootnoßenyq
lim ( )
t
t J t
→ +∞
−2
3
1
σ = + ∞, lim
( )
( )t
t J t
J t→ +∞
′
3
3
= 0, (27)
hde
J t p s s ds
A
t
3
1
21
2
3
( ) ( ) ( )=
+
∫
σ
ϕ ,
A
a p s s ds
p s s ds
a
a
3
2
2
=
= + ∞
+ ∞ < + ∞
+∞
+∞
∫
∫
, ( ) ( ) ,
, ( ) ( ) .
esly
esly
ϕ
ϕ
Bolee toho, kaΩdoe takoe reßenye dopuskaet pry t → + ∞ asymptotyçeskye
predstavlenyq
y t t J t( ) ~ ( )
2
3
1
2
σ σ
−
, ′ = −y t t J t( ) ( )σ σ3
1
, ′′ −y t J t( ) ~ ( )σ σ3
1
. (28)
Dokazatel\stvo. Neobxodymost\. Pust\ y : : t0, ω[ [ → y0, +∞] [ —
Pω1( )±∞ -reßenye uravnenyq (1). Tohda v sylu (1) y opredelenyq Pω1( )±∞ -reße-
nyq y, y′, y′′, y′′′ otlyçn¥ ot nulq na nekotorom promeΩutke t1, ω[ [ ⊂ t0, ω[ [ ,
pryçem y y y′ qvlqgtsq poloΩytel\n¥my na πtom promeΩutke. Krome toho,
sohlasno (11) y uslovyg (4), hde λ0 = ± ∞, ymeem
′′
′
′
′′
′
= − +
y t
y t
y t
y t
o
( )
( )
( )
( )
( )2 1 1 pry t ↑ ω.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1370 V. M. EVTUXOV, A. A. STEXUN
Otsgda s uçetom (3) sleduet, çto
lim
( ) ( )
( )t
t y t
y t↑
′′
′
=
ω
ωπ
1, lim
( ) ( )
( )t
t y t
y t↑
′
=
ω
ωπ
2 ,
hde
πω( )t =
t
t
, ,
, .
esly
esly
ω
ω ω
= + ∞
− < + ∞
V sylu uslovyj y t( ) > 0 y ′y t( ) > 0 pry t ∈ t1, ω[ [ vtoroe yz πtyx predel\n¥x
sootnoßenyj, oçevydno, vozmoΩno lyß\ v sluçae, kohda ω = + ∞. Sledova-
tel\no, rassmatryvaemoe reßenye y uravnenyq (1) qvlqetsq P+∞ ±∞1( )-reßeny-
em y dlq neho
lim
( )
( )t
t y t
y t→ +∞
′′
′
= 1, lim
( )
( )t
t y t
y t→ +∞
′
= 2, lim
( )
( )t
t y t
y t→ +∞
′′′
′′
= 0 . (29)
Poskol\ku funkcyq ϕ udovletvorqet uslovyg S2 y v sylu vtoroho yz
predel\n¥x sootnoßenyj (29)
lim
( )
( )t
t
y t
t
y t
t
→ +∞
′
=
2
2
0 ,
to
ϕ ψ ψ ψσ
y t
y t
y t t
y t
t
t o
( )
( )
( )
( )
( ) ( )
( ) = ( ) = ⋅
= +[ ]+1
2
2
2 1 1 pry t → + ∞. (30)
Otsgda s yspol\zovanyem (29) naxodym
ϕ ψ
σ
σ σ
y t t y t t o( ) ( ) ( ) ( )( ) =
′′[ ] +[ ]
+
+ +1
2
1 1
1
2 2 1 2
pry t → + ∞,
yly
ϕ ϕ
σ σ
y t t y t o( ) ( ) ( ) ( )( ) =
′′[ ] +[ ]
+ +1
2
1 1
1
2 1
pry t → + ∞.
Poπtomu yz (1) ymeem
′′′
′′[ ]
=
+[ ]+
+y t
y t
p t t o
( )
( )
( ) ( ) ( )1 0
1
21
2
1 1σ
σ
α ϕ pry t → + ∞. (31)
Yntehryruq πto sootnoßenye na promeΩutke ot t1 do t y uçyt¥vaq, çto
lim ( )
t
y t
→ +∞
′′ raven lybo nulg, lybo ± ∞, poluçaem
′′[ ] = − +[ ]−
y t J t o( ) ( ) ( )
σ α σ0 3 1 1 pry t → + ∞.
Otsgda sleduet, çto v¥polnqetsq neravenstvo (26) y ymeet mesto tret\e yz
asymptotyçeskyx predstavlenyj (28). Yz πtoho predstavlenyq s uçetom perv¥x
dvux predel\n¥x sootnoßenyj (29) poluçaem pervoe y vtoroe yz asymptotyçes-
kyx predstavlenyj (28). Uslovyq (27) neposredstvenno v¥tekagt yz (28), esly
uçest\ pervoe yz uslovyj (3), tret\e yz predel\n¥x sootnoßenyj (29), a tak-
Ωe:(31).
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ASYMPTOTYÇESKYE PREDSTAVLENYQ REÍENYJ ODNOHO KLASSA … 1371
Dostatoçnost\. Pust\ funkcyq ϕ udovletvorqet uslovyg S2 , ω = + ∞ y
v¥polnqgtsq uslovyq (26), (27). V sylu pervoho yz uslovyj (27) najdetsq çyslo
t0 ∈ a, +∞[ [ takoe, çto pry t ≥ t0 ymeet mesto neravenstvo
1
4
2
3
1
t J tσ σ( ) − ≥
≥ max ,0 0y{ }. V¥byraq takym obrazom çyslo t0 y prymenqq k uravnenyg (1)
preobrazovanye
y t( ) = t J t
2
3
1
12
1σ τσ( ) ( )− +[ ]v , ′y t( ) = t J tσ τσ3
1
21( ) ( )− +[ ]v ,
(32)
′′y t( ) = σ τσJ t3
1
31( ) ( )− +[ ]v , τ = ln t,
s uçetom (26) y toho, çto
ϕ σ σ
t J t
2
3
1
12
1( ) ( )− +
v =
= ϕ σ σ
t J t
2
3
1
2
( ) −
+
t J t t J t
2
3
1 2
3
1
12 2
σ ϕ σσ σ( ) ( )− −′
v +
+
1
8 2
14
3
2 2
3
1
1
2t J t t J tσ ϕ σ ξσ σ( ) ( ) ( )− −′′ +
v ,
hde ξ = ξ( , )t v1 udovletvorqet neravenstvu 0 < ξ < v1 pry t ≥ t0 y v1 ≤ 1
2
,
poluçaem systemu dyfferencyal\n¥x uravnenyj vyda
′ = + −[ ] +v v v1 1 22 2q q( ) ( )τ τ ,
′ = + −[ ] +v v v2 2 31q q( ) ( )τ τ , (33)
′ = + + +[ ]v v v v3 1 3 1q f c V( ) ( ) ( ) ( , )τ τ τ τ ,
hde
q t
t J t
J t
τ
σ
( )
( )
( )
( ) =
′
3
3
, f t
t J t
t J t
τ
ϕ σ
ϕ σ
σ
σ σ
σ
( )
( )
( ) ( )
( ) = −
−
+ − +1
2
1
2
2
3
1
1
2
3
1 ,
c t
t J t t J t
t J t
τ
σ ϕ σ
ϕ σ
σ σ
σ σ
σ
( )
( ) ( )
( ) ( )
( ) = −
′
− −
+ − +
2
3
1 2
3
1
1
2
3
1
2 2
1
2
,
V t
t J t t J t
t J t
τ
σ ϕ σ ξ
ϕ σ
σ σ
σ σ
σ
( ),
( ) ( ) ( )
( ) ( )
v v1
4
3
2 2
3
1
1
2
3
1 1
28 2
1
1
2
( ) = −
′′ +
− −
+ − + .
Zdes\ v sylu uslovyj (27), (2) y S2
lim ( )
τ
τ
→ +∞
=q 0 , lim ( )
τ
τ
→ +∞
=f 0 , lim ( )
τ
τ σ
→ +∞
= − −c 1 , (34)
lim
( , )
v
v
v1 0
1
1
0
→
=V τ
ravnomerno po τ ∈ +∞[ [ln ,t0 . (35)
Krome toho,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1372 V. M. EVTUXOV, A. A. STEXUN
ln
( )
( )
( )
ln ( )
t t t
q s ds
J u du
J u
J u
0 0
0
3
3
3
1
+∞ +∞ +∞
∫ ∫=
′
= = ± ∞
σ σ
. (36)
Teper\, prymenqq k systeme dyfferencyal\n¥x uravnenyj (33) dopolnytel\-
noe preobrazovanye
v1 1= z , v2 2= z , v3 3 1 2 2 1= + +z h z h z( ) ( )τ τ , (37)
hde ( )hi i=1
2
: ln ,t1 +∞[ [ → R
2
( t1 ≥ t0 ) — ysçezagwee pry τ → + ∞ reßenye
system¥ dyfferencyal\n¥x uravnenyj
′ = − −h h h h1 1 2 1
22 ,
′ = + −h q c h h h2 2 1 22( ) ( )τ τ ,
suwestvugwee v sylu uslovyj (34) sohlasno teoreme 1.3 y zameçanyg 1.4 rabot¥
[8], poluçaem systemu dyfferencyal\n¥x uravnenyj
′ = + −[ ] +z q q z z1 1 22 2( ) ( )τ τ ,
′ = + + + −[ ] +z q h z q h z z2 2 1 1 2 31( ) ( ) ( ) ( )τ τ τ τ , (38)
′ = + +
− −( )′
− −
+z q f q c
h h
h h
z q V z3 1 1
1 2
1 2
3 1
1
1
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( , )τ τ τ τ
τ τ
τ τ
τ τ ,
v kotoroj
f f h h1 1 2( ) ( ) ( ) ( )τ τ τ τ= − − , c
c
h h1
1 2
1
1
( )
( )
( ) ( )
τ τ
τ τ
= +
− −
.
Poskol\ku
lim ( )
τ
τ
→ +∞
=hi 0, i = 1, 2, (39)
y v¥polnqgtsq vtoroe y tret\e yz uslovyj (34), to
lim ( )
τ
τ
→ +∞
=f1 0 , lim ( )
τ
τ σ
→ +∞
= − ≠c1 0 . (40)
V sylu uslovyj (34) – (36), (39) y (40) systema dyfferencyal\n¥x uravnenyj
(38) ymeet na osnovanyy teorem¥ 1.3 y zameçanyq 1.4 rabot¥ [8] xotq b¥ odno
reßenye ( )zi i=1
2
: ln ,t2 +∞[ [ → R
2
( t2 ≥ t1), stremqweesq k nulg pry τ → + ∞.
Emu v sylu zamen (37) y (32) sootvetstvuet reßenye y : : t2, +∞[ [ → y0, +∞[ [
dyfferencyal\noho uravnenyq (1), dopuskagwee pry t → + ∞ asymptotyçeskye
predstavlenyq (28). Uçyt¥vaq πty predstavlenyq y uslovyq (27), lehko ubeΩ-
daemsq v tom, çto dannoe reßenye uravnenyq (1) qvlqetsq P+∞ ±∞1( )-reßenyem.
Teorema dokazana.
Zameçanye 2. Esly v¥polnqgtsq uslovyq (26) y (27), to, uçyt¥vaq zameça-
nye 1.1 yz rabot¥ [8], netrudno proveryt\, çto sluçae α0 > 0 uravnenye (1)
ymeet dvuparametryçeskoe semejstvo P+∞ ±∞1( )-reßenyj, dopuskagwyx pry
t → + ∞ asymptotyçeskye predstavlenyq (28), a v sluçae α0 < 0 — trexpara-
metryçeskoe semejstvo takyx reßenyj.
V kaçestve prymera, yllgstryrugweho ustanovlenn¥e rezul\tat¥, rassmot-
rym dyfferencyal\noe uravnenye
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ASYMPTOTYÇESKYE PREDSTAVLENYQ REÍENYJ ODNOHO KLASSA … 1373
′′′ = +y t y yα γ σ µ1 ln , (41)
hde α , σ, γ, µ ∈ R, pryçem α ≠ 0, σ ≠ 0 y (1 + σ)2 + µ2 ≠ 0. Ono qvlqetsq
uravnenyem vyda (1), v kotorom α0 = sign α, p t( ) = α γt , ϕ( )y = y y1+ σ µln .
Zdes\ funkcyq ϕ udovletvorqet uslovyqm (2), a takΩe uslovyqm S1 y S2 .
Krome toho, dlq funkcyj Jk , k = 1, 2, 3, yz teorem 1 y 2 ymegt mesto pry t →
→ + ∞ asymptotyçeskye predstavlenyq
J t1( ) = α τ τ τγ σ µ
A
t
d
1
1∫ + + ln ∼
∼
α
γ σ
γ σ
α
µ
γ σ µ
α γ σ µ
γ σ µ
µ
t t
t
t
2
1
2
2 0
1
2 0 1
2 0 1
+ +
+
+ +
+ + ≠
+
+ + = ≠ −
+ + = = −
ln
, ,
ln
, ,
ln ln , ,
esly
esly y
esly y
J t2( ) =
A
t
J d
2
1∫ ( )τ τ ∼
∼
α
γ σ γ σ
γ σ γ σ
α
µ
γ σ µ
α γ σ µ
α
µ
γ σ µ
α
γ σ µ
µ
µ
t t
t
t
t t
t
3
1
1
2 3
2 3 0
1
3 0 1
3 0 1
1
2 0 1
+ +
+
+
+ + + +
+ + + + ≠
−
+
+ + = ≠ −
− + + = = −
+
+ + = ≠ −
ln
( )( )
, ( )( ) ,
ln
, ,
ln ln , ,
ln
, ,
ln
esly
esly y
esly y
esly y
lnln , ,t esly y2 0 1+ + = = −
γ σ µ
J t3( ) =
α τ τ τσ µ
γ σ µ
21
2 2
3
+ −
+ +∫
A
t
dln ∼
∼
α
γ σ
γ σ
α
µ
γ σ µ
α γ σ µ
γ σ µ
σ µ
µ
σ µ
σ µ
t t
t
t
3 2
1
1
1
1
2 3 2
3 2 0
2 1
3 2 0 1
2
3 2 0 1
+ +
+ −
+
+ −
+ −
+ +
+ + ≠
+
+ + = ≠ −
+ + = = −
ln
( )
, ( ) ,
ln
( )
, ,
ln ln
, .
esly
esly y
esly y
V sylu πtyx predstavlenyj yz teorem 1 y 2 nastoqwej rabot¥ neposredstvenno
v¥tekagt sledugwye utverΩdenyq.
Sledstvye 1. Dlq suwestvovanyq P+∞1 0( )-reßenyj uravnenyq (41) neob-
xodymo y dostatoçno, çtob¥ 3 + γ + σ = 0 y v¥polnqlos\ neravenstvo
α σ µ( )1 0+ > , esly µ ≠ – 1,
ασ > 0, esly µ = – 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
1374 V. M. EVTUXOV, A. A. STEXUN
Bolee toho, esly µ ≠ – 1, to kaΩdoe takoe reßenye dopuskaet pry t → + ∞
asymptotyçeskye predstavlenyq
y t t t( ) (ln )∼
− +
ρ
µ
σ
1
1
, ′ ∼
− +
y t t( ) (ln )ρ
µ
σ
1
1
, ′′ ∼ − +
− + +
y t
t
t
( )
( ) (ln )ρ µ
σ
µ σ
σ
1
1
1
,
hde ρ1 =
ασ
µ
σ
+
−
1
1
, a esly µ = – 1, to — asymptotyçeskye predstavlenyq vyda
y t t t( ) (ln ln )∼
−
ρ σ
2
1
, ′ ∼
−
y t t( ) (ln ln )ρ σ
2
1
, ′′ ∼ −
− +
y t
t
t t
( )
(ln ln )
ln
ρ
σ
σ
σ
2
1
,
hde ρ2 = α σ σ
− 1
.
Sledstvye 2. Dlq suwestvovanyq P+∞ ±∞1( )-reßenyj uravnenyq (41) ne-
obxodymo y dostatoçno, çtob¥ 3 + γ + 2σ = 0 y v¥polnqlos\ neravenstvo
ασ µ( )1 0+ < , esly µ ≠ – 1,
ασ < 0, esly µ = – 1.
Bolee toho, esly µ ≠ – 1, to kaΩdoe takoe reßenye dopuskaet pry t → + ∞
asymptotyçeskye predstavlenyq
y t
t
t( ) (ln )∼
− +
ρ
µ
σ3
2 1
2
, ′ ∼
− +
y t t t( ) (ln )ρ
µ
σ
3
1
, ′′ ∼
− +
y t t( ) (ln )ρ
µ
σ
3
1
,
hde ρ3 =
ασ
µ σ µ
σ
( )+ + −
−
1 21
1
, a esly µ = – 1, to — asymptotyçeskye predstav-
lenyq vyda
y t
t
t( ) (ln ln )∼
−ρ σ4
2 1
2
, ′ ∼
−
y t t t( ) (ln ln )ρ σ
4
1
, ′′ ∼
−
y t t( ) (ln ln )ρ σ
4
1
,
hde ρ4 =
ασ
σ µ
σ
21
1
+ −
−
.
V¥vod¥. V rabotax [5, 6] dlq obobwennoho dyfferencyal\noho uravnenyq
typa ∏mdena – Faulera n-ho porqdka b¥l vveden dostatoçno ßyrokyj klass tak
naz¥vaem¥x Pω -reßenyj, kotor¥j po svoym asymptotyçeskym svojstvam raspa-
daetsq na n + 2 neperesekagwyxsq podmnoΩestva. Dlq Pω -reßenyj kaΩdoho
yz ( n + 2 ) -x vozmoΩn¥x typov b¥ly poluçen¥ neobxodym¥e y dostatoçn¥e
uslovyq suwestvovanyq, a takΩe asymptotyçeskye predstavlenyq pry t ↑ ω.
V nastoqwej stat\e v kaçestve obæekta yssledovanyq v¥brano dvuçlennoe
dyfferencyal\noe uravnenye tret\eho porqdka (1) s nelynejnost\g bolee ob-
weho vyda, çem u uravnenyj typa ∏mdena – Faulera. Dlq πtoho uravnenyq raz-
rabotana metodyka, pozvolqgwaq ustanovyt\ asymptotyçeskye predstavlenyq
pry t ↑ ω neohranyçenn¥x Pω -reßenyj, dlq kotor¥x λ0 = 0 y λ0 = ± ∞. Pry
πtom b¥ly takΩe poluçen¥ neobxodym¥e y dostatoçn¥e uslovyq suwestvova-
nyq takyx reßenyj. Sleduet obratyt\ vnymanye na to, çto zdes\, v otlyçye ot
rabot¥ [7], hde yssledovalys\ neohranyçenn¥e Pω -reßenyq uravnenyq (1) so
znaçenyqmy λ0 ∉ 0 1
2
1, , , ±∞{ }, asymptotyçeskye formul¥ pry t ↑ ω v¥pys¥-
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ASYMPTOTYÇESKYE PREDSTAVLENYQ REÍENYJ ODNOHO KLASSA … 1375
vagtsq v qvnom vyde. Poluçenn¥e rezul\tat¥ proyllgstryrovan¥ na klassy-
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Poluçeno 08.12.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 10
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| id | umjimathkievua-article-3395 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:41:43Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f1/53fa994e938159fbe6043768e4074cf1.pdf |
| spelling | umjimathkievua-article-33952020-03-18T19:53:10Z Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order Асимптотические представления решений одного класса нелинейных неавтономных дифференциальных уравнений третьего порядка Evtukhov, V. M. Stekhun, A. A. Евтухов, В. М. Стехун, А. А. Евтухов, В. М. Стехун, А. А. We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type. Встановлено асимптотичні зображення для необмежених розв'язків нелінійних неавтономних диференціальних рівнянь третього порядку, що у деякому сенсі є близькими до рівнянь типу Емдена - Фаулера. Institute of Mathematics, NAS of Ukraine 2007-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3395 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 10 (2007); 1363–1375 Український математичний журнал; Том 59 № 10 (2007); 1363–1375 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3395/3533 https://umj.imath.kiev.ua/index.php/umj/article/view/3395/3534 Copyright (c) 2007 Evtukhov V. M.; Stekhun A. A. |
| spellingShingle | Evtukhov, V. M. Stekhun, A. A. Евтухов, В. М. Стехун, А. А. Евтухов, В. М. Стехун, А. А. Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order |
| title | Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order |
| title_alt | Асимптотические представления решений одного класса нелинейных неавтономных дифференциальных уравнений третьего порядка |
| title_full | Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order |
| title_fullStr | Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order |
| title_full_unstemmed | Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order |
| title_short | Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order |
| title_sort | asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3395 |
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