Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation
We present conditions under which a linear homogeneous second-order equation is nonoscillatory on a semiaxis and conditions under which its solutions have infinitely many zeros.
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| Date: | 2010 |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509002365927424 |
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| author | Slyusarchuk, V. Yu. Слюсарчук, В. Ю. |
| author_facet | Slyusarchuk, V. Yu. Слюсарчук, В. Ю. |
| author_sort | Slyusarchuk, V. Yu. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:41:53Z |
| description | We present conditions under which a linear homogeneous second-order equation is nonoscillatory on a semiaxis and conditions under which its solutions have infinitely many zeros. |
| first_indexed | 2026-03-24T02:34:10Z |
| format | Article |
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K O R O T K I P O V I D O M L E N N Q
UDK 517.925.46
V. G. Slgsarçuk (Nac. un-t vodn. hosp-va ta pryrodokorystuvannq, Rivne)
POSYLENNQ TEOREMY KNEZERA PRO NULI ROZV’QZKIV
RIVNQNNQ ′′uu qq tt uu+ ( ) = 0 Z VYKORYSTANNQM ODNOHO
FUNKCIONAL|NOHO RIVNQNNQ
We present conditions under which a linear homogeneous second-order equation is nonoscillatory on the
semiaxis and also conditions under which its solutions have infinitely many zeros.
Pryvede¥ uslovyq, pry kotor¥x lynejnoe odnorodnoe uravnenye vtoroho porqdka qvlqetsq
neoscyllyrugwym na poluosy, a takΩe uslovyq, pry kotor¥x eho reßenyq ymegt beskoneçnoe
çyslo nulej.
1. Postanovka osnovno] zadaçi. Vstanovymo umovy kolyvnosti rozv’qzkiv
linijnoho dyferencial\noho rivnqnnq
′′ +u q t u( ) = 0, (1)
de q : [ , )1 + ∞ → R — neperervna funkciq.
Cq zadaça — ob’[kt doslidΩen\ bahat\ox matematykiv (dyv. [1 – 12]).
Rozhlqnemo odyn pidxid do doslidΩennq rivnqnnq (1), wo dozvolyt\ inßym
sposobom posylyty teoremu Knezera pro nuli rivnqnnq (1) (perßyj variant na-
vedeno avtorom v [10]) ta otrymaty rezul\taty pro odne vaΩlyve dlq (1) funk-
cional\ne rivnqnnq ta joho rozv’qzky.
Spoçatku vykona[mo zaminu zminnyx t ta u v rivnqnni (1).
VvaΩatymemo, wo
t = e es + −1 i u t( ) = ω ( ) ( )s z s , (2)
de ω ( )s i z s( ) — dviçi neperervno dyferencijovni na [ , )1 + ∞ funkci].
Vykorystovugçy pravyla dyferencigvannq funkcij [13], otrymu[mo
du
dt
=
du
ds
ds
dt
=
d
ds
z
dz
ds
e sω
ω+
− ,
d u
dt
2
2 =
d
du
dt
ds
ds
dt
=
d
d
ds
z
dz
ds
e
ds
e
s
s
ω ω+
−
− =
© V. G. SLGSARÇUK, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12 1705
1706 V. G. SLGSARÇUK
= − +
+ + +−d
ds
z
dz
ds
e
d
ds
z
d
ds
dz
ds
dsω
ω
ω ω
ω
2
2
2
2
zz
ds
e es s
2
− − =
= ω
ω ω
ω
ωd z
ds
d
ds
dz
ds
d
ds
z
dz
ds
d
ds
z
2
2
2
22+ +
− +
−e s2 . (3)
Zavdqky (2) i (3) rivnqnnq (1) matyme vyhlqd
e
d z
ds
d
ds
dz
ds
d
ds
d
ds
s− + −
+ −
2
2
2
2
22ω
ω
ω
ω ω
+ + −z q e e zs( )1 ω = 0. (4)
Dali vyberemo funkcig ω ( )s tak, wob 2
d
ds
ω
ω− ≡ 0 i ω ( )1 = 1. Ci umovy,
oçevydno, zadovol\nq[ funkciq
ω = e s( )/−1 2 . (5)
Oskil\ky
d
ds
d
ds
2
2
ω ω
− = –
1
4
1 2e s( )/− ,
to rivnqnnq (4) rivnosyl\ne rivnqnng
d z
ds
e q e e zs s
2
2
2 1
1
4
+ + − −
( ) = 0. (6)
OtΩe, qkwo vykonaty v rivnqnni (1) zaminu zminnyx t ta u zhidno z (2) i (5),
to pryjdemo do rivnqnnq (6), wo analohiçne (1).
Teper utoçnymo osnovnu metu ci[] statti.
U podal\ßomu z’qsu[mo, qkyj vyhlqd povynna maty funkciq q t( ) v rivnqnni
(1), wob pry rozhlqnutyx vywe zaminax zminnyx t ta u dyferencial\ne rivnqn-
nq (6) zbihalosq z vyxidnym rivnqnnqm (1), i doslidymo ce rivnqnnq pry takomu q
na predmet kolyvnosti rozv’qzkiv. Oçevydno, wo funkciq q t( ) , wo nas cika-
vyt\, povynna buty rozv’qzkom funkcional\noho rivnqnnq
x t( ) = e x e et t2 1
1
4
( )+ − − , t ≥ 1. (7)
2. DoslidΩennq funkcional\noho rivnqnnq (7). Rozhlqnemo funkci]
v0( )t =
t e
e
− +1
, vn t( ) =
ln ( ( ))e t e
e
nv − − +1 1
, n ≥ 1,
Q tk ( ) = vn
n
k
t( )
=
∏
0
, k ≥ 0,
wo vyznaçeni i neperervni na [ , )1 + ∞ .
Teorema71. Funkcional\ne rivnqnnq (7) ma[ [dynyj neperervnyj na [ , )1 + ∞
rozv’qzok K K t= ( ) , wo zobraΩu[t\sq u vyhlqdi
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
POSYLENNQ TEOREMY KNEZERA PRO NULI ROZV’QZKIV RIVNQNNQ … 1707
K t( ) =
1
4 2
1
2
1 e Q tn
nn ( ( ))−=
+∞
∑ . (8)
Dovedennq. Vykorysta[mo novu zminnu
τ = e et + −1 .
Todi funkcional\ne rivnqnnq (7) nabere vyhlqdu
x ( )τ =
1
4 1
1
1
12 2( ) ( )
ln ( )( )
τ τ
τ
− +
+
− +
− +
e e
x e , τ ≥ 1. (9)
Oskil\ky dlq vsix τ ≥ 1
1
1 2( )τ − + e
≤
1
2e
< 1,
to v banaxovomu prostori Cb ([ , ), )1 + ∞ R neperervnyx i obmeΩenyx funkcij
x : [ , )1 + ∞ → R z normog x Cb ([ , ), )1 + ∞ R = sup ( )
τ
τ
≥1
x linijnyj operator
( ) ( )Ax τ =
1
4 1
1
1
12 2( ) ( )
ln ( )( )
τ τ
τ
− +
+
− +
− +
e e
x e , τ ≥ 1,
[ styskagçym. Tomu u prostori Cb ([ , ), )1 + ∞ R cej operator ma[ [dynu neruxo-
mu toçku (poznaçymo ]] çerez K K t= ( ) ), a funkcional\ne rivnqnnq (9) — [dy-
nyj rozv’qzok K Cb∈ + ∞([ , ), )1 R . Lehko pereviryty, wo
K t( ) =
1
4 1
1
4 1 1 12 2 2( ) ( ) ln ( )( )t e t e t e e− +
+
− + − + − +
+ …
… +
1
4 2
1
2e Q tn
n( ( ))−
+ … ,
tobto spravdΩu[t\sq rivnist\ (8). Zaznaçymo, wo suma funkcional\noho rqdu v
pravij çastyni poperedn\oho spivvidnoßennq [ neperervnog na [ , )1 + ∞ funk-
ci[g, oskil\ky çleny c\oho rqdu neperervni na [ , )1 + ∞ i cej rqd maΩoru[t\sq
na [ , )1 + ∞ çyslovym rqdom
1
4
1
4
1
42 4 2e e e n
+ + … + + … .
PokaΩemo, wo rivnqnnq (7) ne moΩe maty neobmeΩenyj neperervnyj na
[ , )1 + ∞ rozv’qzok. Poznaçymo çerez v( )t dovil\nyj neperervnyj rozv’qzok
c\oho rivnqnnq i, otΩe, rivnqnnq (9). Oskil\ky
ln ( )τ − +1 e ≤ τ,
1
4 1 2( )τ − + e
≤
1
4 2e
i
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
1708 V. G. SLGSARÇUK
1
1 2( )τ − + e
≤
1
2e
< 1
dlq vsix τ ≥ 1, to zavdqky (9) dlq koΩnoho çysla T ≥ 1 vykonu[t\sq neriv-
nist\
max ( )
1≤ ≤τ
τ
T
v ≤
1
4
1
2 2 1e e T
+
≤ ≤
max ( )
τ
τv .
Zvidsy vyplyva[, wo
max ( )
1≤ ≤τ
τ
T
v ≤
1
4
1
1
2
2
e
e
−
=
1
4 12( )e −
, T ≥ 1.
OtΩe, koΩnyj neperervnyj rozv’qzok rivnqnnq (7) [ elementom prostoru
Cb ([ , ), )1 + ∞ R .
TeoremuI1 dovedeno.
3. Zv’qzok miΩ rozv’qzkamy rivnqn\ (1) i (6) u vypadku q (((( t )))) = K (((( t )))) . Zav-
dqky doslidΩennqm, vykladenym u p. 1, funkciqm (2), (5) ta totoΩnosti
K ( t ) ≡ e K e et t2 1
1
4
( )+ − −
pravyl\nym [ nastupne tverdΩennq.
Teorema72. Qkwo funkciq z = z ( t ) [ rozv’qzkom dyferencial\noho rivnqn-
nq
′′ +y K t y( ) = 0, (10)
to rozv’qzkom c\oho rivnqnnq takoΩ [ funkciq
u ( t ) = v0 1( ) ln ( )( )t z t e− + .
4. Nekolyvnist\ rozv’qzkiv rivnqnnq (10). Spoçatku pokaΩemo, wo
pravyl\nym [ nastupne tverdΩennq.
Teorema73. Isnu[ [dynyj rozv’qzok rivnqnnq (10), dlq qkoho z ( 1 ) = 1 i
z ( t ) = v0 1( ) ln ( )( )t z t e− + , t ≥ 1. (11)
Dovedennq. Nexaj z = z ( t ) — rozv’qzok rivnqnnq (10), wo zadovol\nq[
umovu z ( 1 ) = 1 (takyx rozv’qzkiv [ neskinçenno bahato). Todi za teoremogI2
funkciq
u ( t ) =
t e
e
z t e
− +
− +
1
1( )ln ( )
takoΩ [ rozv’qzkom rivnqnnq (10) i u ( 1 ) = 1. Vyberemo rozv’qzok z c\oho riv-
nqnnq tak, wob
′u ( )1 = ′z ( )1 . (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
POSYLENNQ TEOREMY KNEZERA PRO NULI ROZV’QZKIV RIVNQNNQ … 1709
Oskil\ky
′u t( ) =
1
2 1
1 2 1
e t e
z t e z t e
− +
− + + ′ − +( ( ) ( ))ln ( ) ln ( ) , t ≥ 1,
to
′u ( )1 =
1
2
1 2 1
e
z( )( )+ ′ .
Na pidstavi (12)
′z ( )1 =
1
2
1 2 1
e
z( )( )+ ′ .
Zvidsy otrymu[mo
′z ( )1 =
1
2 1( )e −
.
OtΩe, isnu[ [dynyj rozv’qzok z rivnqnnq (10), dlq qkoho
z ( 1 ) = 1 i ′z ( )1 = ′u ( )1 .
Oskil\ky rivnqnnq (10) ma[ [dynyj rozv’qzok y, wo zadovol\nq[ poçatkovi umo-
vy
y ( 1 ) = 1 i ′y ( )1 =
1
2 1( )e −
,
a rozv’qzky z i u rivnqnnq (10) takoΩ zadovol\nqgt\ ci umovy, to vykonu[t\sq
spivvidnoßennq (11).
TeoremuI3 dovedeno.
Teorema74. Funkciq z = z ( t ) , wo zadovol\nq[ umovy teoremyI3, dodatna
na [ , )1 + ∞ .
Dovedennq. Qkwo toçka t∗ ∈ + ∞[ , )1 [ nulem funkci] z ( t ) , to
ln ( )t e∗ − +1 = t∗
zavdqky (11) i tomu, wo v0( )t > 0 dlq vsix t ∈ + ∞[ , )1 . Oskil\ky ln ( )t e− +1 <
< t dlq vsix t > 1 i ln ( )t e− +1 = t til\ky dlq t = 1, to t∗ = 1. Odnak
z ( )1 = 1. Zvidsy vyplyva[, wo mnoΩyna nuliv funkci] z t( ) [ poroΩn\og.
OtΩe, zavdqky neperervnosti z t( ) na [ , )1 + ∞ ta rivnosti z ( )1 = 1 mno-
Ωyna znaçen\ ci[] funkci] mistyt\sq v ( , )0 + ∞ .
TeoremuI4 dovedeno.
Lehko pokazaty, wo funkciq z ( t ) , qka zadovol\nq[ umovy teoremyI3, ma[ vy-
hlqd
z ( t ) = vk
k
t( )
=
+∞
∏
0
.
Osnovnym u c\omu punkti [ nastupne tverdΩennq.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
1710 V. G. SLGSARÇUK
Teorema75. KoΩnyj nenul\ovyj rozv’qzok dyferencial\noho rivnqnnq (10) na
promiΩku [ , )1 + ∞ ma[ ne bil\ße odnoho nulq.
Ce tverdΩennq — naslidok teoremyI4 ta teoremy Íturma pro vidokremlen-
nq nuliv [1, 5, 9].
5. Umovy kolyvnosti rozv’qzkiv rivnqnnq (1). U c\omu punkti navedemo
osnovne tverdΩennq pro kolyvnist\ rozv’qzkiv rivnqnnq (1), wo posylg[
teoremu Knezera pro nuli rozv’qzkiv c\oho rivnqnnq i pokazu[ vaΩlyvist\
funkci] K t( ) .
Teorema76. Nexaj q : [ , )1 + ∞ → R — neperervna funkciq. Qkwo vykonu-
[t\sq nerivnist\
q t( ) ≤ K t( )
dlq vsix dosyt\ velykyx t ≥ 1, to koΩnyj rozv’qzok dyferencial\noho rivnqn-
nq (1) na promiΩku [ , )1 + ∞ ma[ ne bil\ße skinçennoho çysla nuliv. Qkwo dlq
deqkoho natural\noho çysla n
lim ( ( ) ( )) ( ( ))
t
nq t K t Q t
→+∞
− 2 > 0, (13)
to koΩnyj nenul\ovyj rozv’qzok rivnqnnq (1) ma[ neskinçennu mnoΩynu nuliv u
koΩnomu intervali ( , )t1 + ∞ , t1 1≥ .
Cq teorema navedena avtorom u statti [10]. Povtorennq ]] tut [ pryrodnym i
pidkreslg[ vaΩlyvist\ provedenyx u poperednix punktax doslidΩen\, pov’qza-
nyx iz funkci[g K t( ) . Funkciq K t( ) u pevnomu sensi [ universal\nog (cq funk-
ciq krawa, niΩ funkci], qki rozhlqdaly Xille [3] i Xartman [4] (dyv. takoΩ nas-
tupnyj punkt)). Krim c\oho my navedemo inße dovedennq çastyny tverdΩennq
teoremyI6, wo stosu[t\sq kolyvnosti rozv’qzkiv rivnqnnq (1).
Dovedennq teoremy76. Rozhlqnemo çyslo t0 1∈ + ∞( , ) , dlq qkoho vykonu-
[t\sq spivvidnoßennq
q t( ) ≤ K t( ) , t ≥ t0 . (14)
TakoΩ rozhlqnemo dovil\nyj nenul\ovyj rozv’qzok y = y t( ) rivnqnnq (1).
Zavdqky teoremi porivnqnnq [9, c. 588], teoremiI4 ta spivvidnoßenng (14) rozv’q-
zok y = y t( ) rivnqnnq (1) na promiΩku [ , )t0 + ∞ moΩe maty ne bil\ße odnoho
nulq. Cej rozv’qzok na promiΩku [ , ]1 0t moΩe maty lyße skinçenne çyslo
nuliv (dyv., napryklad, [9, c. 582, 583]). Tomu mnoΩyna nuliv rozv’qzku y na
[ , )1 + ∞ [ skinçennog mnoΩynog.
OtΩe, perßu çastynu teoremy dovedeno.
Dovedemo teper druhu çastynu teoremy.
Poznaçymo nyΩng hranycg u spivvidnoßenni (13) çerez γ. Za dopomohog
ci[] hranyci funkcig q t( ) moΩna podaty u vyhlqdi
q t( ) =
ψ ε+
+ +
=
∑( )
( ( )) ( ( ))( )
t
Q t e Q tn
k
kk
n
2 2 1 2
0
1
4
,
de ε : [ , )1 + ∞ → R — neperervna funkciq, dlq qko] lim ( )
t
t
→+∞
ε = 0. Vyko-
navßy u rivnqnni (1) zaminu zminnyx t ta u za formulamy
t = e es1 1+ − , u ( t ) = v0 1 1 1( ) ( )s z s ,
qk i v perßomu punkti, pryjdemo do dyferencial\noho rivnqnnq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
POSYLENNQ TEOREMY KNEZERA PRO NULI ROZV’QZKIV RIVNQNNQ … 1711
d z s
ds
q s z s
2
1 1
1
2 1 1 1 1
( )
( ) ( )+ = 0, (15)
de
q s1 1( ) =
γ ε+
+
−
+
=
−
1 1
1 1
2 2 1
1
2
0
1
4
( )
( ( )) ( ( ))( )
s
Q s e Q sn
k
kk
n 11
∑
i
ε1 1( )s = ε ( )ln ( )t e− +1 .
Dali, vykonavßy u rivnqnni (15) zaminu zminnyx s1 ta z1 za formulamy
s1 = e es2 1+ − , z s1 1( ) = v0 2 2 2( ) ( )s z s ,
otryma[mo dyferencial\ne rivnqnnq
d z s
ds
q s z s
2
2 2
2
2 2 2 2 2
( )
( ) ( )+ = 0, (16)
de
q s2 2( ) =
γ ε+
+
−
+
=
−
2 2
2 2
2 2 1
1
2
0
1
4
( )
( ( )) ( ( ))( )
s
Q s e Q sn
k
kk
n 22
∑
i
ε2 2( )s = ε1 1 1( )ln ( )s e− + = ε ( )ln (ln ( ) )t e e− + − +1 1 .
Vykonugçy dali poslidovno analohiçni zaminy zminnyx za formulamy
s2 = e es3 1+ − , z s2 2( ) = v0 3 3 3( ) ( )s z s ,
. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . .
sn = e esn+ + −1 1 , z sn n( ) = v0 1 1 1( ) ( )s z sn n n+ + + ,
pryxodymo do dyferencial\noho rivnqnnq
d z s
ds
s z sn n
n
n n n n
2
1 1
1
2 1 1 1 1
+ +
+
+ + + ++ +
( )
( ( )) ( )γ ε = 0, (17)
v qkomu εn+ + ∞ →1 1: [ , ) R — neperervna funkciq i
lim ( )
s
n n
n
s
+ → +∞
+ +
1
1 1ε = 0. (18)
Oçevydno, wo
u t( ) = v v v0 1 0 2 0 1 1 1( ) ( ) ( ) ( )s s s z sn n n… + + + . (19)
Oskil\ky γ > 0 , to koΩnyj nenul\ovyj rozv’qzok rivnqnnq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
1712 V. G. SLGSARÇUK
d w
ds
w
n
2
1
2 2+
+
γ
= 0
ma[ na promiΩku [ , )1 + ∞ neskinçenne çyslo nuliv. Na pidstavi toho, wo dlq
vsix dosyt\ velykyx dodatnyx sn+1
γ
ε
2
1 1+ + +n ns( ) > 0
(zavdqky (18)), ta teoremy porivnqnnq [9, c. 588] analohiçnu vlastyvist\ magt\
usi nenul\ovi rozv’qzky dyferencial\noho rivnqnnq (17). Zavdqky (19) i tomu,
wo
v v v0 1 0 2 0 1( ) ( ) ( )s s sn… + ≥ 1
dlq vsix s1 1≥ , s2 1≥ , … , sn+ ≥1 1, nenul\ovi rozv’qzky rivnqnnq (1) magt\
neskinçenne çyslo nuli na koΩnomu promiΩku [ , )t1 + ∞ , t1 1≥ .
TeoremuI6 dovedeno.
6. Porivnqnnq teoremy76 z teoremog Knezera. Navedeni vywe rezul\taty
tisno pov’qzani z teoremog Knezera pro nuli rozv’qzkiv rivnqnnq (1), tobto z na-
stupnym tverdΩennqm.
Teorema7Knezera [2]. Qkwo v rivnqnni (1) koefici[nt q t( ) zadovol\nq[
umovu
0 < q t( ) ≤
1
4 2t
, t ≥ t0 ,
to joho nenul\ovi rozv’qzky ne moΩut\ maty neskinçenne çyslo nuliv v intervali
( , )t0 + ∞ . Qkwo Ω
q t( ) >
1
4 2
+ α
t
, α > 0, t ≥ t1 ,
to koΩnyj nenul\ovyj rozv’qzok rivnqnnq (1) ma[ neskinçennu mnoΩynu nuliv v
intervali ( , )t1 + ∞ .
Xille [3] i Xartman [4] perekonalysq, wo tverdΩennq teoremy Knezera zaly-
ßagt\sq pravyl\nymy, qkwo v cij teoremi funkci]
p t1 0( , ) ≤
1
4 2t
i p t1( , )α ≤
1
4 2
+ α
t
zaminyty vidpovidno funkciqmy p tn ( , )0 i p tn ( , )α , n ≥ 2, de
p tn ( , )β =
1 1
42 1
t
p tn+
− (ln , )β , n ≥ 2, β α∈{ , }0 .
Oskil\ky dlq koΩnoho n ∈N isnu[ take çyslo α > 0, wo
K t( ) > p tn ( , )0
dlq vsix t > a i
lim ( ( , ) ( )) ( ( ))
t
n np t K t Q t
→+∞ −−α 1
2 = α
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
POSYLENNQ TEOREMY KNEZERA PRO NULI ROZV’QZKIV RIVNQNNQ … 1713
(u c\omu lehko perekonatysq), to teoremaI6 posylg[ ne til\ky teoremu Knezera,
a j vidpovidni rezul\taty Xille [3] i Xartmana [4].
ZauvaΩennq. Vywe zaznaçalosq, wo funkciq K t( ) v pevnomu sensi [ uni-
versal\nog.
Zhidno z doslidΩennqmy Req (dyv.I [8], teoremaI5) dlq dyferencial\noho riv-
nqnnq (1), nenul\ovi rozv’qzky qkoho ne oscylggt\, isnu[ neperervna na
[ , )1 + ∞ funkciq �q t( ) , dlq qko]
q t( ) < �q t( ) , t ≥ 1,
i vsi nenul\ovi rozv’qzky dyferencial\noho rivnqnnq
′′ +u q t u�( ) = 0
takoΩ ne oscylggt\.
U vypadku dyferencial\noho rivnqnnq (10) isnu[ neperervna na [ , )1 + ∞
funkciq
�K t( ) , dlq qko] takoΩ
K t( ) < �K t( ) , t ≥ 1,
i vsi nenul\ovi rozv’qzky dyferencial\noho rivnqnnq
′′ +u K t u� ( ) = 0
ne oscylggt\ (takyx funkcij [ neskinçenno bahato). Odnak znaxodΩennq ta-
kyx funkcij [ skladnog zadaçeg, oskil\ky potribno vykorystovuvaty rozv’qzok
Y = Y t( ) rivnqnnq (10), dlq qkoho nevlasnyj intehral
ds
Y s2
1 ( )
+∞
∫
[ zbiΩnym, funkcig
η( )t = Y t
ds
Y st
( )
( )2
+∞
∫ ,
qki ne [ prostymy vnaslidok hromizdkosti funkci] K t( ) , ta inßi dopomiΩni
funkci] i spivvidnoßennq.
Na zaverßennq zaznaçymo, wo funkci] vn t( ) , n ≥ 0, i vnn
t( )=
+∞∏ 0
vyko-
rystovuvalysq avtorom takoΩ dlq doslidΩennq zbiΩnosti çyslovyx rqdiv [14].
1. Sturm C. Sur les équation différentielles linéaires du second order // J. math. pures et appl. – 1836.
– 1, # 1. – P. 106 – 186.
2. Kneser A. Untersuchung über die reellen Nullstellen der Integrale linearer Differentialgeleichunges
// Math. Ann. – 1893. – 42. – S. 409 – 435.
3. Hille E. Nonoscillation theorems // Trans. Amer. Math. Soc. – 1948. – 64. – P. 234 – 252.
4. Hartman P. On the linear logarithmicoexponential differential equations of the second order //
Amer. J. Math. – 1948. – 70. – P. 768 – 779.
5. Xartman F. Ob¥knovenn¥e dyfferencyal\n¥e uravnenyq. – M.: Myr, 1970. – 720Is.
6. Bellman R. Teoryq ustojçyvosty reßenyj dyfferencyal\n¥x uravnenyj. – M.: Yzd-vo
ynostr. lyt., 1954. – 216Is.
7. Kyhuradze Y. T., Çanturyq T. A. Zameçanye ob asymptotyçeskom povedenyy reßenyj urav-
nenyq ′′ + =u a t u( ) 0 // Dyfferenc. uravnenyq. – 1970. – 6, # 6. – S.I1115 – 1117.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
1714 V. G. SLGSARÇUK
8. Wray S. D. Integral comparison theorems in oscillation theory // J. London Math. Soc. – 1974. – 2,
# 8. – P. 595 – 606.
9. Matveev N. M. Metod¥ yntehryrovanyq ob¥knovenn¥x dyfferencyal\n¥x uravnenyj. –
Mynsk: V¥ßπjß. ßk., 1974. – 768Is.
10. Slgsarçuk V. E. Usylenye teorem¥ Knezera o nulqx reßenyj uravnenyq ′′ + =y p x y( ) 0
// Ukr. mat. Ωurn. – 1996. – 48, # 4. – S.I520 – 524.
11. Slgsarçuk V. E. Uzahal\nennq teoremy Knezera pro nuli rozv’qzkiv rivnqnnq ′′ +y
+I p t y( ) = 0 // Tam Ωe. – 2007. – 59, # 4. – S.I571 – 576.
12. Evtuxov V. M., Vasyl\eva N. S. Uslovyq koleblemosty y nekoleblemosty reßenyj odnoho
klassa polulynejn¥x dyfferencyal\n¥x uravnenyj vtoroho porqdka // Tam Ωe. –
S.I458 – 466.
13. Fyxtenhol\c H. M. Kurs dyfferencyal\noho y yntehral\noho ysçyslenyq. – M.: Nauka,
1966. – T.I1. – 608 s.
14. Slgsarçuk V. G. Zahal\ni teoremy pro zbiΩnist\ çyslovyx rqdiv. – Rivne: Rivnen. derΩ.
texn. un-t, 2001. – 240Is.
OderΩano 21.07.09,
pislq doopracgvannq — 10.08.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 12
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| id | umjimathkievua-article-2993 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:34:10Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3c/47db96afd865aaaec3a2d065cf60523c.pdf |
| spelling | umjimathkievua-article-29932020-03-18T19:41:53Z Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation Посилення теореми Киезера про нулі розв'язків рівняння $u″ + q(t)u = 0$ з використанням одного функціонального рівняння Slyusarchuk, V. Yu. Слюсарчук, В. Ю. We present conditions under which a linear homogeneous second-order equation is nonoscillatory on a semiaxis and conditions under which its solutions have infinitely many zeros. Приведены условия, при которых линейное однородное уравнение второго порядка является неосциллирующим на полуоси, а также условия, при которых его решения имеют бесконечное число нулей. Institute of Mathematics, NAS of Ukraine 2010-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2993 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 12 (2010); 1705–1714 Український математичний журнал; Том 62 № 12 (2010); 1705–1714 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2993/2736 https://umj.imath.kiev.ua/index.php/umj/article/view/2993/2737 Copyright (c) 2010 Slyusarchuk V. Yu. |
| spellingShingle | Slyusarchuk, V. Yu. Слюсарчук, В. Ю. Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation |
| title | Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation |
| title_alt | Посилення теореми Киезера про нулі розв'язків рівняння
$u″ + q(t)u = 0$ з використанням одного функціонального рівняння |
| title_full | Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation |
| title_fullStr | Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation |
| title_full_unstemmed | Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation |
| title_short | Strengthening of the Kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation |
| title_sort | strengthening of the kneser theorem on zeros of solutions of the equation $u″ + q(t)u = 0$ using one functional equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2993 |
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