On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points
We establish new properties of $C^1 (0, + ∞)$-solutions of the linear functional differential equation $\dot{x}(t) = ax(t) + bx(qt) + cx(qt)$ in the neighborhoods of the singular points $t = 0$$ and t = + ∞$.
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| Datum: | 2005 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2005
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3717 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509850512916480 |
|---|---|
| author | Bel’skii, D. V. Pelyukh, G. P. Бельский, Д. В. Пелюх, Г. П. Бельский, Д. В. Пелюх, Г. П. |
| author_facet | Bel’skii, D. V. Pelyukh, G. P. Бельский, Д. В. Пелюх, Г. П. Бельский, Д. В. Пелюх, Г. П. |
| author_sort | Bel’skii, D. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:02:57Z |
| description | We establish new properties of $C^1 (0, + ∞)$-solutions of the linear functional differential equation
$\dot{x}(t) = ax(t) + bx(qt) + cx(qt)$ in the neighborhoods of the singular points $t = 0$$ and t = + ∞$. |
| first_indexed | 2026-03-24T02:47:39Z |
| format | Article |
| fulltext |
UDK 517.9
H. P. Pelgx, D. V. Bel\skyj (Yn-t matematyky NAN Ukrayn¥, Kyev)
O POVEDENYY REÍENYJ LYNEJNÁX
DYFFERENCYAL|NO-FUNKCYONAL|NÁX URAVNENYJ
S POSTOQNNÁMY KO∏FFYCYENTAMY Y LYNEJNO
PREOBRAZOVANNÁM ARHUMENTOM V OKRESTNOSTY
OSOBÁX TOÇEK
*
We establish new properties of C
1
( 0, + ∞ )-solutions of the functional-differential equation ˙( )x t =
= a x ( t ) + b x ( q t ) + cx qt˙( ) in neighborhoods of the singular points t = 0 and t = + ∞.
Vstanovleno novi vlastyvosti C
1
( 0, + ∞ )-rozv’qzkiv linijnoho dyferencial\no-funkcional\-
noho rivnqnnq ˙( )x t = a x ( t ) + b x ( q t ) + cx qt˙( ) v okoli osoblyvyx toçok t = 0 i t = + ∞.
V dannoj rabote rassmatryvaetsq lynejnoe dyfferencyal\no-funkcyonal\noe
uravnenye
˙( )x t = a x ( t ) + b x ( q t ) + cx qt˙( ), (1)
hde { a, b, c } ⊂ R , 0 < q < 1, t ∈ ( 0, + ∞ ). Razlyçn¥e çastn¥e sluçay takyx
uravnenyj yzuçalys\ mnohymy matematykamy, y v nastoqwee vremq ymeetsq rqd
ynteresn¥x rezul\tatov, kasagwyxsq yzuçenyq svojstv eho reßenyj. Tak, v [1]
dostatoçno polno yssledovan¥ asymptotyçeskye svojstva reßenyj uravnenyq
(1) pry c = 0, v [2] ustanovlen¥ nov¥e svojstva reßenyj πtoho uravnenyq pry
a = 0, c = 0, v [3] poluçen¥ uslovyq suwestvovanyq analytyçeskyx poçty pery-
odyçeskyx reßenyj uravnenyq (1) pry c = 0, v [4] postroeno predstavlenye ob-
weho reßenyq uravnenyq (1) pry | c | > 1, v [5] poluçen rqd nov¥x rezul\tatov o
suwestvovanyy ohranyçenn¥x y fynytn¥x reßenyj uravnenyj s lynejno preob-
razovann¥m arhumentom, v [6] opredelen¥ maΩorant¥ dlq reßenyj uravnenyq
(1). Nesmotrq na obylye rezul\tatov, posvqwenn¥x yssledovanyg asymptoty-
çeskyx svojstv reßenyj ßyrokyx klassov dyfferencyal\no-funkcyonal\n¥x
uravnenyj y yx vaΩn¥e pryloΩenyq [7], mnohye vopros¥ teoryy dyfferency-
al\no-funkcyonal\n¥x uravnenyj vyda (1) yzuçen¥ nedostatoçno. Osobenno
πto kasaetsq svojstv reßenyj uravnenyq (1) v okrestnostqx osob¥x toçek t = 0
y t = + ∞, yssledovanye kotor¥x qvlqetsq osnovnoj cel\g nastoqwej rabot¥.
Poluçenn¥e v nej rezul\tat¥ dopolnqgt y razvyvagt yzvestn¥e rezul\tat¥
mnohyx matematykov (sm. [1, 8 – 10] y pryvedennug v nyx byblyohrafyg), posvq-
wenn¥e yssledovanyg asymptotyçeskyx svojstv reßenyj lynejn¥x dyfferen-
cyal\no-funkcyonal\n¥x uravnenyj nejtral\noho typa s peremenn¥my y pos-
toqnn¥my otklonenyqmy arhumenta.
Rassmotrym snaçala sluçaj, kohda t ∈ ( 0, 1 ].
Dlq yssledovanyq povedenyq reßenyj uravnenyq (1) v okrestnosty nulq v¥-
polnym zamenu peremenn¥x
x ( t ) = z
t
1
, x ( q t ) = z
qt
1
,
d
dt
x t( ) = ′
−
z
t t
1 1
2 ,
x ′ ( q t ) = ′
−
z
qt q t
1 1
2 2 , t =
1
qτ
.
V rezul\tate poluçym uravnenye
z ′ ( τ ) =
1
2c
az q bz
τ
τ τ( ) ( )+( ) +
q
c
z q
2
′( )τ , (2)
*
Vykonano pry finansovij pidtrymci DerΩavnoho fondu fundamental\nyx doslidΩen\ pry Mi-
nisterstvi Ukra]ny z pytan\ nauky i texnolohij.
© H. P. PELGX, D. V. BEL|SKYJ, 2005
1668 ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
O POVEDENYY REÍENYJ LYNEJNÁX DYFFERENCYAL|NO- … 1669
kotoroe moΩno zapysat\ v vyde
z ( τ ) =
q
c
z q( )τ + z
q
c
z q e
b
c( ) ( )1
1 1
−
−( )/τ
+
ac bq
c
e e
z qs
s
ds
b
c
b
c
s+ −( ) − −( )/ /
∫2
1 1
1
1 1
2
τ τ
( )
.
Oboznaçaq
f ( τ ) =df z
q
c
z q e
b
c( ) ( )1
1 1
−
−( )/τ
+
ac bq
c
e e
z qs
s
ds
b
c
b
c
s+ −( ) − −( )/ /
∫2
1 1
1
1 1
2
τ τ
( )
, (3)
poluçaem uravnenye
z ( τ ) =
q
c
z q( )τ + f ( τ ). (4)
Ymeet mesto sledugwaq teorema.
Teorema 1. Pust\ { a, b, c } ⊂ R, 0 < q < 1. Tohda:
1) pry 0 < | c | < q reßenye uravnenyq (1) ymeet pravostoronnyj predel v
nule tohda y tol\ko tohda, kohda ono ohranyçeno v okrestnosty nulq;
2) pry | c | > q vse reßenyq uravnenyq (1) ymegt pravostoronnyj predel v
nule.
DokaΩem pervoe utverΩdenye teorem¥. Suwestvovanye pravostoronneho
predela v nule u reßenyq x ( t ) oçevydn¥m obrazom podrazumevaet eho ohrany-
çennost\ v okrestnosty nulq.
Obratno, pust\ x ( t ), a znaçyt, y z ( τ ) ohranyçen¥ na otrezkax ( 0, 1 ] y
[ q
–
1, + ∞ ) sootvetstvenno. Tohda lehko pokazat\, çto yntehral v (3) sxodytsq
absolgtno pry τ → + ∞, t. e. suwestvuet koneçn¥j predel lim ( )τ τ→+ ∞ f =df M.
Poskol\ku koπffycyent¥ uravnenyq (1) qvlqgtsq dejstvytel\n¥my çyslamy,
bez ohranyçenyq obwnosty sçytaem, çto x ( t ) ∈ R dlq lgboho t > 0 y, sledova-
tel\no, M ∈ R.
Yz uravnenyq (4) sleduet
z ( q
–
j
τ ) =
q
c
z q
j
+1
( )τ +
q
c
f
j
( )τ +
q
c
f q
j
−
−
1
1( )τ + …
… +
q
c
f q j( )− +1τ + f q j( )− τ , j ≥ 1.
PredpoloΩym, çto q > c > 0. Tohda yz opredelenyq predela lim ( )τ τ→+ ∞ f =
= M ymeem
∀ ε > 0 ∃ L : τ > L ⇒ M + ε > f ( τ ) > M – ε
y
z ( q
–
j
τ ) =
q
c
z q
j
+1
( )τ +
q
c
f
j
( )τ +
q
c
f q
j
−
−
1
1( )τ + …
… +
q
c
f q j( )− +1τ + f q j( )− τ >
q
c
z q
j
+1
( )τ +
+
q
c
q
c
q
c
M
j j
+
+…+ +
−
−1
1 ( )ε =
=
q
c
z q
j
+1
( )τ +
q c
q c
M
j/
/
( ) −
−
−
+1 1
1
( )ε =
q
c
z q
M
q c
j
+ −
−
+
/
1
1
( )τ ε
–
M
q c
−
−/
ε
1
.
Ytak,
z ( q
–
j
τ ) >
q
c
z q
M
q c
j
+ −
−
+
/
1
1
( )τ ε
–
M
q c
−
−/
ε
1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1670 H. P. PELGX, D. V. BEL|SKYJ
Uçyt¥vaq, çto z ( τ ) — ohranyçennaq na otrezke [ 1, + ∞ ) funkcyq, poluçaem
z ( q τ ) +
M
q c
−
−/
ε
1
≤ 0,
t. e.
z ( q τ ) ≤
−
−/
M
q c 1
+
ε
q c/ −1
pry τ > L. V protyvnom sluçae z ( τ ) — neohranyçennaq na [ 1, + ∞ ) funkcyq,
çto protyvoreçyt predpoloΩenyg. Analohyçno ymeem
z ( q
–
j
τ ) <
q
c
z q
j
+1
( )τ +
q c
q c
M
j/
/
( ) −
−
+
+1 1
1
( )ε =
=
q
c
z q
M
q c
j
+ +
−
+
/
1
1
( )τ ε
–
M
q c
+
−/
ε
1
,
otkuda (v sylu ohranyçennosty z ( τ ))
z ( q τ ) +
M
q c
+
−/
ε
1
≥ 0
yly
z ( q τ ) ≥
−
−/
M
q c 1
–
ε
q c/ −1
pry τ > L. Summyruq rezul\tat¥, poluçaem
–
M
q c/ − 1
–
ε
q c/ −1
≤ z ( q τ ) ≤ –
M
q c/ −1
+
ε
q c/ −1
pry τ > L.
Rassmotrym sluçaj – q < c < 0. Tohda
z ( τ ) =
q
c
z q( )τ + f ( τ ) =
=
q
c
z q
2
2( )τ +
q
c
f q( )τ + f ( τ ) =
q
c
z q
2
2( )τ + f1 ( τ ),
hde
f1 ( τ ) =df
q
c
f q( )τ + f ( τ ) →
q
c
M+
1
pry τ → + ∞. Sledovatel\no, m¥ svely zadaçu k tol\ko çto rassmotrennomu
sluçag q > c > 0. Pervaq çast\ teorem¥ dokazana.
DokaΩem vtoroe utverΩdenye teorem¥. Yz uravnenyq (2) neposredstvenno
sleduet
′z ( )τ ≤
1
2τ
τ τ
c
b z a z q( ) ( )+( ) +
q
c
z q
2
′( )τ .
Yntehryruq poslednee neravenstvo na otrezke [ 1, t] , naxodym
1
t
z d∫ ′( )τ τ ≤
b
c
z
d
t
1
2∫ ( )τ
τ
τ +
a
c
z q
d
t
1
2∫ ( )τ
τ
τ +
q
c
z d
q
qt
∫ ′( )τ τ ≤
≤
b
c
z
d
t
1
2∫ ( )τ
τ
τ +
a
c
z q
d
t
1
2∫ ( )τ
τ
τ +
q
c
z d
q
1
∫ ′( )τ τ +
q
c
z d
qt
1
∫ ′( )τ τ . (5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
O POVEDENYY REÍENYJ LYNEJNÁX DYFFERENCYAL|NO- … 1671
Oboznaçym
1
u
z d∫ ′( )τ τ =df s ( u ),
hde u > 0. Tohda pry τ > 0 ymeem
| z ( τ ) | ≤ s ( τ ) + | z ( 1 ) |.
Esly 1 ≤ t ≤ q
–
1
, to
s ( q t ) =
1
qt
z d∫ ′( )τ τ ≤
q
qt
z d∫ ′( )τ τ +
qt
z d
1
∫ ′( )τ τ =
=
q
z d
1
∫ ′( )τ τ =df N ≤ N + s ( t ).
Pry t ≥ q
–
1
naxodym
s ( q t ) =
1
qt
z d∫ ′( )τ τ ≤
1
t
z d∫ ′( )τ τ = s ( t ) ≤ s ( t ) + N.
Takym obrazom, pry t ≥ 1 ymeem s ( q t ) ≤ s ( t ) + N.
Yz (5) sleduet
s ( t ) ≤
b
c
z s
d
t
1
2
1∫ +( ) ( )τ
τ
τ +
a
c
z N s
d
t
1
2
1∫ + +( ) ( )τ
τ
τ +
+
q
c
N +
q
c
N s t+( )( ) ≤ K +
b a
c
s
d
t+ ∫
1
2
( )τ
τ
τ +
q
c
s t( ),
hde
K =
b z a z a N
c
d
( ) ( )1 1 1
1
2
+ + + ∞
∫ τ
τ +
2q
c
N ,
yly (v sylu toho, çto q < | c | )
s ( t ) ≤ 1
1
−
−
q
c
K + 1
1
1
2−
+−
∫q
c
b a
c
s
d
t
( )τ
τ
τ = F + M
s
d
t
1
2∫ ( )τ
τ
τ ,
hde
F = 1
1
−
−
q
c
K , M = 1
1
−
+−
q
c
b a
c
.
Otsgda y yz lemm¥ Hronuolla – Bellmana sleduet
s ( t ) ≤ Fe
M
s
ds
t
1
2
1
∫
≤ Fe
M
s
ds
1
2
1+∞
∫
,
t. e.
1
+∞
∫ ′z d( )τ τ =
0
1
∫ ˙( )x t dt ≤ Fe
M
s
ds
1
2
1+∞
∫
.
Teorema dokazana.
Perejdem k yssledovanyg ustojçyvosty uravnenyq (1).
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1672 H. P. PELGX, D. V. BEL|SKYJ
Teorema 2. Esly { a, b, c } ⊂ R , 0 < q < 1, a < 0 y
c
q
+
b
a
c
q
+ < 1, to
uravnenye (1) asymptotyçesky ustojçyvo.
Dokazatel\stvo. Zapyßem uravnenye (1) v πkvyvalentnoj (v klasse C
1
( 0,
+ ∞ )-reßenyj) yntehral\noj forme
x ( t ) =
c
q
x qt( ) + x
c
q
x q ea t( ) ( ) ( )1 1−
− +
bq ac
q
e x qs ds
t
a t s+ ∫ −
1
( ) ( ) . (6)
PokaΩem, çto nulevoe reßenye uravnenyq (6) asymptotyçesky ustojçyvo v
klasse C [ q, + ∞ )-reßenyj. S πtoj cel\g reßym zadaçu
x ( t ) =
c
q
x qt g
c
q
g q e
bq ac
q
e x qs ds t
g t t q
a t
t
a t s( ) ( ) ( ) ( ) , ,
( ), , ,
( ) ( )+ −
+ + >
∈[ ]
− −∫1 1
1
1
1
(7)
hde g ( t ) — nekotoraq funkcyq yz klassa C [ q, 1 ], s pomow\g metoda posledo-
vatel\n¥x pryblyΩenyj, kotor¥e opredelym sootnoßenyqmy
xm ( t ) =
=
c
q
x qt g
c
q
g q e
bq ac
q
e x qs ds t
g t t q
m
a t
t
a t s
m−
− −
−+ −
+ + >
∈[ ]
∫1
1
1
11 1
1
( ) ( ) ( ) ( ) , ,
( ), , ,
( ) ( )
(8)
m ≥ 1,
x0 ( t ) =
0 2
1 2 1 2
1
, ,
( )( ), ,
( ), , .
t
g t t
g t t q
≥
− < <
∈[ ]
V sylu (8) pry t ≥ 2 q
–
1
ymeem
| x1 ( t ) – x0 ( t ) | = | x1 ( t ) | =
= g
c
q
g q e
bq ac
q
e x qs dsa t
q
a t s( ) ( ) ( )( ) ( )1 1
1
2
0
1
−
+ +− −
−
∫ = K eat
1 ,
hde
K1 = g
c
q
g q e
bq ac
q
e x qs dsa
q
as( ) ( ) ( )1
1
2
0
1
−
+ +− −
−
∫ .
Tohda, oçevydno, | x1 ( t ) – x0 ( t ) | ≤ Keat
, t ≥ q, hde K — nekotoraq konstanta.
Poskol\ku a < 0, to pry vsex t > 1 naxodym
| x2 ( t ) – x1 ( t ) | ≤
c
q
| x1 ( q t ) – x0 ( q t ) | +
bq ac
q
e x qs x qs ds
t
a t s+ −∫ −
1
1 0
( ) ( ) ( ) ≤
≤
c
q
Keaqt +
bq ac
q
e Ke ds
t
a t s aqs+ ∫ −
1
( ) =
=
c
q
bq ac
q
e
a q
Ke
a q t
aqt+ + −
−
− −1
1
1 1( )( )
( )
≤
c
q
bq ac
q a q
Keaqt+ +
−
1
1( )
.
RassuΩdaq po yndukcyy, poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
O POVEDENYY REÍENYJ LYNEJNÁX DYFFERENCYAL|NO- … 1673
| xm + 1 ( t ) – xm ( t ) | ≤ K
c
q
bq ac
q a q
+ +
−
1
1( )
…
…
c
q
bq ac
q a q
em
aq tm
+ +
−
1
1( )
=df K em
aq tm
, m ≥ 1. (9)
Poskol\ku a < 0 ,
c
q
+
bq ac
q
+
< 1, yz (9) neposredstvenno sleduet, çto po-
sledovatel\nost\ neprer¥vn¥x funkcyj { xm ( t ) } ravnomerno sxodytsq pry t ∈
∈ [ q, + ∞ ) k neprer¥vnomu reßenyg x* ( t ) uravnenyq (7), dlq kotoroho spraved-
lyva ocenka
| x* ( t ) | ≤
m
m mx t x t
=
+∞
+∑ −
0
1( ) ( ) ≤ Keat +
m
m
aq tK e
m
=
+∞
∑
1
, t ≥ 2.
Lehko pokazat\, çto funkcyq x* ( t ) qvlqetsq edynstvenn¥m reßenyem zada-
çy (7). Sledovatel\no, ona neprer¥vna pry t ∈ [ q, + ∞ ) y stremytsq k nulg,
kohda t → + ∞.
MnoΩestvo C [ q, + ∞ )-reßenyj uravnenyq (6) qvlqetsq mnoΩestvom reße-
nyj zadaçy (7) pry razlyçn¥x „naçal\n¥x” funkcyqx g ( t ) yz klassa C [ q, 1 ].
Sledovatel\no, reßenyq uravnenyq (6) stremqtsq k nulg, kohda t → + ∞. Po-
skol\ku konstanta K v yzloΩenn¥x v¥ße rassuΩdenyqx zavysyt lyß\ ot ve-
lyçyn¥ sup ( ),t q g t∈[ ]1 , ymeet mesto asymptotyçeskaq ustojçyvost\ nulevoho
reßenyq.
Teorema dokazana.
Yssleduem svojstva ohranyçenn¥x na otrezke [ 1, + ∞ ) reßenyj uravnenyq
(1) pry t → + ∞.
Teorema 3. Esly { a, b, c } ⊂ R, 0 < q < 1, b / c > 0 y
q
c
+ 2
a
b
q
c
+ < 1, to
ohranyçenn¥e na otrezke [ 1, + ∞ ) reßenyq (1) udovletvorqgt sootnoßenyg
| x ( t ) | ≤ 1 1
1
1
1 1
1
2
− − +
−
−
−
− + − − +
+
−
−
q
c
a
b
q
c
x
q
c
x q e
b
c
q
c
a
b
q
c
a
c
bq
c
t
( ) ( )
( )
pry t ≥ 1, hde
–
b
c
+ 1
1
2− − +
+
−q
c
a
b
q
c
a
c
bq
c
< 0.
Dokazatel\stvo. Rassmotrym uravnenye
x ( t ) =
q
c
x q t( )−1 + x
c
q
x q e
b
c
t
( ) ( )
( )
1 1
1
−
− − −
–
–
a
c
bq
c
e x q s ds
t b
c
t s
+
∫
− − −
2
1
1
( )
( ) , (10)
πkvyvalentnoe uravnenyg (1) v klasse C
1
( 0, + ∞ )-reßenyj. PredpoloΩym, çto
x ( t ) ohranyçeno na otrezke [ 1, + ∞ ). Yz (10) poluçaem
| x ( t ) | ≤
q
c
x q t( )−1 + x
q
c
x q e
b
c
t
( ) ( )
( )
1 1
1
− − − −
+
+
a
c
bq
c
e x q s ds
t b
c
t s
+ ∫
− − −
2
1
1
( )
( ) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1674 H. P. PELGX, D. V. BEL|SKYJ
Opredelym ν ( t ) =df sup ( )s t x s≥ (v sylu ohranyçennosty x ( t ) funkcyq ν ( t ) op-
redelena).
Poskol\ku b / c > 0, dlq t1 ≥ t ≥ 1 naxodym
| x ( t1 ) | ≤
q
c
ν ( t ) + x
q
c
x q e
b
c
t
( ) ( )
( )
1 1
11
− − − −
+
+
a
c
bq
c
e x q s ds
t b
c
t s
+ ∫
− − −
2
1
11( )
( ) +
a
c
bq
c
e x q s ds
t b
c
t s
+ ∫
− − −
2
1
1
1
1( )
( ) ≤
≤
q
c
ν ( t ) + x
q
c
x q e
b
c
t
( ) ( )
( )
1 1
1
− − − −
+
+
a
c
bq
c
e s ds
t b
c
t s
+ ∫
− −
2
1
( )
( )ν +
a
c
bq
c
c
b
t+ 2 ν( ) .
Otsgda sleduet, çto | x ( t1 ) | moΩno zamenyt\ na ν ( t ). Uçyt¥vaq, çto
L =df
q
c
+
a
b
q
c
+ < 1,
ymeem
ν ( t )e
b
c
t( )−1
≤ ( ) ( ) ( )1 11 1− −− −L x
q
c
x q + ( ) ( )
( )
1 1
2
1
1
− +− −
∫L
a
c
bq
c
e s ds
t b
c
s
ν ,
y v sylu lemm¥ Hronuolla – Bellmana poluçaem
ν ( t ) ≤ ( ) ( ) ( )
( ) ( )
1 11 1
1 11
2− −− −
− + − +
−−
L x
q
c
x q e
b
c
L
a
c
bq
c
t
.
PokaΩem, çto yz uslovyj teorem¥ sleduet
–
b
c
+ ( )1 1
2− +−L
a
c
bq
c
< 0.
Dejstvytel\no,
–
b
c
+ ( )1 1
2− +−L
a
c
bq
c
= –
b
c
+ ( )1 1− −
−L L
q
c
b
c
< 0 ⇔
⇔ 2L –
q
c
< 1 ⇔
q
c
+ 2
a
b
q
c
+ < 1.
Teorema dokazana.
Teorema 4. Esly { a, b, c } ⊂ R, 0 < q < 1, a > 0 y
c
q
+
b
a
c
q
+ < 1, to re-
ßenyq uravnenyq (1), ohranyçenn¥e na otrezke [ 1, + ∞ ), udovletvorqgt uslo-
vyg x ( t ) = O ( t
u
), t → + ∞, hde
u =
1
1ln
ln
q
c
q
b
a
c
q− + +
.
Dokazatel\stvo. Zapyßem uravnenye (1) v vyde
˙( )x t s+ = a x ( t + s ) + b x ( q ( t + s ) ) + cx q t s˙ ( )+( ) , (11)
hde s ≥ 0. Pust\ reßenye x ( t ) ohranyçeno na otrezke [ 1, + ∞ ) y a > 0.
UmnoΩaq uravnenye (11) na e
–
a
s
y yntehryruq eho po s na otrezke [ 0, + ∞ ),
poluçaem
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
O POVEDENYY REÍENYJ LYNEJNÁX DYFFERENCYAL|NO- … 1675
0
+∞
−∫ +e x t s dsas ˙( ) = a e x t s dsas
0
+∞
−∫ +( ) +
+ b e x q t s dsas
0
+∞
−∫ +( )( ) + c e x q t s dsas
0
+∞
−∫ +( )˙ ( ) ,
x ( t ) =
c
q
x qt( ) – b
ca
q
e x q t s dsas+
+( )
+∞
−∫
0
( ) . (12)
Opredelym ν ( t ) =df sup ( )s t x s≥ (v sylu ohranyçennosty x ( t ) funkcyq ν ( t ) op-
redelena). Dlq lgboho t1 ≥ t ≥ 1 yz (12) naxodym
| x ( t1 ) | ≤
c
q
| x ( q t1 ) | + b
ca
q
e x q t s dsas+
+( )
+∞
−∫
0
1( ) ≤
c
q
b
a
c
q
qt+ +
ν( ) .
Sledovatel\no,
ν ( t ) ≤
c
q
b
a
c
q
qt+ +
ν( ) .
Poskol\ku
c
q
+
b
a
c
q
+ < 1,
ν ( t ) → 0 pry t → + ∞. Ocenym „skorost\”, s kotoroj ν ( t ) → 0 pry t → + ∞.
Dlq ocenky funkcyy ν ( t ) v¥polnym zamenu peremennoj ν ( t ) = t
u
y ( t ), hde
u =
1
1ln
ln
q
c
q
b
a
c
q− + +
.
V rezul\tate poluçym neravenstvo
t
u
y ( t ) ≤
c
q
b
a
c
q
+ +
q
u
t
u
y ( q t ) ⇔ y ( t ) ≤ y ( q t ).
Sledovatel\no, y ( t ) ohranyçena na [ 1, + ∞ ).
Teorema dokazana.
Najdem dostatoçnoe uslovye edynstvennosty ohranyçennoho na otrezke [ 1,
+ ∞ ) nenulevoho reßenyq uravnenyq (1).
Teorema 5. Esly { a, b, c } ⊂ R, 0 < q < 1, b / c > 0 y 2
q
c
+
a
b
q
c
+ < 1, to
x ( t ) =
k
k
b
c
q t
x e
k
=
+∞ −
∑
−
0
,
hde
xk =
ac bq
bc q
x
k
k k
+
−
− +
− −
1
11( )
, k ≥ 1,
x0 — proyzvol\no, budet edynstvenn¥m s toçnost\g do x0 nenulev¥m ohrany-
çenn¥m na otrezke [ 1, + ∞ ) reßenyem uravnenyq (1).
Dokazatel\stvo. Pust\ ( H, ρ ) — polnoe metryçeskoe prostranstvo ohra-
nyçenn¥x funkcyj yz klassa C [ 1, + ∞ ) s metrykoj ρ ( x, y ) = sup ( ) ( )1≤ −t x t y t .
Opredelym operator F na osnovanyy uravnenyq (10), πkvyvalentnoho uravne-
nyg (1) (v klasse C
1
[ 1, + ∞ )-reßenyj):
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
1676 H. P. PELGX, D. V. BEL|SKYJ
( Fx )( t ) =
q
c
x q t( )−1 + g
q
c
x q e
b
c
t
1
1
1
−
− − −
( )
( )
–
–
a
c
bq
c
e x q s ds
t b
c
t s
+
∫
− − −
2
1
1
( )
( ) , t ≥ 1,
hde g1 — proyzvol\noe kompleksnoe çyslo. Poskol\ku b / c > 0, to F : H → H
y
ρ ( F x, F y ) ≤ 2
q
c
a
b
q
c
x y+ +
ρ( , ) .
Sohlasno predpoloΩenyg
2
q
c
+
a
b
q
c
+ < 1,
sledovatel\no, F — operator sΩatyq, dlq kotoroho v H suwestvuet edyn-
stvennaq nepodvyΩnaq toçka.
Teorema dokazana.
1. Kato T., McLeod J. B. The functional-differential equation y ′ ( x ) = a y ( λ x ) + b y ( x ) // Bull. Amer.
Math. Soc. – 1971. – 77. – P. 891 – 937.
2. De Bruijn N. G. The difference-differential equation F ′ ( x ) = e
α
x
+
β
F ( x – 1 ). I, II // Ned. Akad.
Wetensch. Proc. Ser. A. Math. – 1953. – 15. – P. 449 – 464.
3. Frederickson P. O. Series solutions for certain functional-differential equations // Lect. Notes
Math. – 1971. – 243. – P. 249 – 254.
4. Pelgx H. P., Íarkovskyj A. N. Vvedenye v teoryg funkcyonal\n¥x uravnenyj. – Kyev:
Nauk. dumka, 1974. – 119 s.
5. Derfel\ H. A. Veroqtnostn¥j metod yssledovanyq odnoho klassa dyfferencyal\no-
funkcyonal\n¥x uravnenyj // Ukr. mat. Ωurn. – 1989. – 41, # 10. – S. 12 – 16.
6. Bel\skyj D. V. Ob asymptotyçeskyx svojstvax reßenyj lynejn¥x dyfferencyal\no-
funkcyonal\n¥x uravnenyj s postoqnn¥my koπffycyentamy y lynejno preobrazovann¥m
arhumentom // Nelinijni kolyvannq. – 2004. – 7, # 1. – S. 48 – 52.
7. Gumovski I., Mira C. Recurrences and discrete dynamic systems // Lect. Notes Math. – 1980. –
809. – 267 p.
8. Xejl DΩ. Teoryq funkcyonal\no-dyfferencyal\n¥x uravnenyj. – M.: Myr, 1984. – 421Vs.
9. Kurbatov V. H. Lynejn¥e dyfferencyal\no-raznostn¥e uravnenyq. – VoroneΩ: VoroneΩ.
un-t, 1990. – 168 s.
10. Slgsarçuk V. G. Absolgtna stijkist\ dynamiçnyx system iz pislqdi[g. – Rivne: Vyd-vo
UDUVHP, 2003. – 288 s.
Poluçeno 10.10.2003,
posle dorabotky — 28.01.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 12
|
| id | umjimathkievua-article-3717 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:47:39Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/16f7b9ee0ca5db0552e1d1288e83e70d.pdf |
| spelling | umjimathkievua-article-37172020-03-18T20:02:57Z On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points O поведении решений линейных дифференциально-функциональных уравнений с постоянными коэффициентами и линейно преобразованным аргументом в окрестности особых точек Bel’skii, D. V. Pelyukh, G. P. Бельский, Д. В. Пелюх, Г. П. Бельский, Д. В. Пелюх, Г. П. We establish new properties of $C^1 (0, + ∞)$-solutions of the linear functional differential equation $\dot{x}(t) = ax(t) + bx(qt) + cx(qt)$ in the neighborhoods of the singular points $t = 0$$ and t = + ∞$. Встановлено нові властивості $C^1(0, +\infty)$-розв'язків лінійного диференціально-функціонального рівняння $\dot{x}(t) = ax(t) + bx(qt) + c\dot{x}(qt)$ в околі особливих точок $t = 0$ і $t = +\infty$. Institute of Mathematics, NAS of Ukraine 2005-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3717 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 12 (2005); 1668–1676 Український математичний журнал; Том 57 № 12 (2005); 1668–1676 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3717/4159 https://umj.imath.kiev.ua/index.php/umj/article/view/3717/4160 Copyright (c) 2005 Bel’skii D. V.; Pelyukh G. P. |
| spellingShingle | Bel’skii, D. V. Pelyukh, G. P. Бельский, Д. В. Пелюх, Г. П. Бельский, Д. В. Пелюх, Г. П. On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points |
| title | On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points |
| title_alt | O поведении решений линейных дифференциально-функциональных уравнений с постоянными коэффициентами и линейно преобразованным аргументом в окрестности особых точек |
| title_full | On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points |
| title_fullStr | On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points |
| title_full_unstemmed | On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points |
| title_short | On the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points |
| title_sort | on the behavior of solutions of linear functional differential equations with constant coefficients and linearly transformed argument in neighborhoods of singular points |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3717 |
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