On sharp conditions for the global stability of a difference equation satisfying the Yorke condition
Continuing our previous investigations, we give simple sufficient conditions for global stability of the zero solution of the difference equation xn+1 = qxn + fn (xn ,..., xn-k ), n ∈ Z, where nonlinear functions fn satisfy the Yorke condition. For every positive integer k, we rep...
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| Дата: | 2008 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3138 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509177196052480 |
|---|---|
| author | Nenya, O. I. Tkachenko, V. I. Trofimchuk, S. I. Неня, О. І. Ткаченко, В. І. Трофімчук, С. І. |
| author_facet | Nenya, O. I. Tkachenko, V. I. Trofimchuk, S. I. Неня, О. І. Ткаченко, В. І. Трофімчук, С. І. |
| author_sort | Nenya, O. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:46:36Z |
| description | Continuing our previous investigations, we give simple sufficient conditions for global stability
of the zero solution of the difference equation
xn+1 = qxn + fn (xn ,..., xn-k ), n ∈ Z,
where nonlinear functions fn satisfy the Yorke condition.
For every positive integer k, we represent the interval (0, 1]
as the union of [(2k + 2) /3] disjoint subintervals, and, for q from each subinterval, we present a
global-stability condition in explicit form. The conditions obtained are sharp for the class of equations satisfying the Yorke condition. |
| first_indexed | 2026-03-24T02:36:57Z |
| format | Article |
| fulltext |
UDK 517.9
O. I. Nenq (Nac. ekonom. un-t, Ky]v),
V. I. Tkaçenko (In-t matematyky NAN Ukra]ny, Ky]v),
S. I. Trofymçuk (In-t matematyky ta fizyky un-tu Tal\ky, Çyli)
PRO TOÇNI UMOVY HLOBAL|NO} STIJKOSTI
RIZNYCEVOHO RIVNQNNQ, QKE ZADOVOL|NQ{
UMOVU JORKA*
Continuing our previous investigations, we give simple sufficient conditions for global stability of the
zero solution of the difference equation xn +1 = qxn + f x xn n n k( , , )–… , n ∈Z , where nonlinear
functions fn satisfy the Yorke condition. For every positive integer k, we represent the interval 0 1,( ]
as the union of ( ) /2 2 3k +[ ] disjoint subintervals, and, for q from each subinterval, we present a
global-stability condition in explicit form. The conditions obtained are sharp for the class of equations
satisfying the Yorke condition.
V prodolΩenye pred¥duwyx yssledovanyj avtorov pryveden¥ prost¥e dostatoçn¥e uslovyq
hlobal\noj ustojçyvosty nulevoho reßenyq raznostnoho uravnenyq xn +1 = qxn + f xn n( , …
… , xn k– ) , n ∈Z , hde nelynejn¥e funkcyy fn udovletvorqgt uslovyg Jorka. Dlq kaΩdoho
natural\noho k ynterval 0 1,( ] predstavlen kak obæedynenye ( ) /2 2 3k +[ ] pod¥ntervalov, y
dlq q s kaΩdoho pod¥ntervala v qvnom vyde pryvedeno uslovye hlobal\noj ustojçyvosty.
Poluçenn¥e uslovyq qvlqgtsq toçn¥my dlq klassa uravnenyj, udovletvorqgwyx uslovyg
Jorka.
Rozhlqnemo riznyceve rivnqnnq
xn +1 = qxn + f x xn n n k( , , )–… , x n ∈R , n ∈Z , (1)
de q ∈ 0 1,( ], k ≥ 1, i nelinijni funkci] fn : Rk +1 → R zadovol\nqgt\ nastupnu
umovu Jorka: isnu[ a < 0 take, wo
aM ( )φ ≤ fn( )φ ≤ − −aM ( )φ , n ∈Z , (2)
dlq vsix φ ∈ Rk +1
. Funkcional M : R
k +1 → R + oznaça[t\sq qk M ( )φ =
= max ,i i0 φ{ } .
Rozv’qzok xn{ } rivnqnnq?(1) z poçatkovog umovog xi = ϕi , i = – k, … , 0, is-
nu[ dlq vsix n ≥ 0 i moΩe buty odnoznaçno pobudovanyj poslidovno.
Qk vyplyva[ z (2), toçka x = 0 [ [dynog neruxomog toçkog rivnqnnq?(1).
Rivnqnnq (1) z riznymy vydamy nelinijnostej magt\ çyslenni zastosuvannq v
matematyçnij biolohi] qk dyskretni modeli evolgci] biolohiçnyx populqcij
(dyv., napryklad, [1 – 7] ). VaΩlyvym [ doslidΩennq umov stijkosti neruxomo]
toçky matematyçno] modeli j ocinka oblasti prytqhannq ci[] toçky, zokrema
osoblyvo cikavym ta zruçnym dlq zastosuvan\ [ vypadok hlobal\no] stijkosti
neruxomo] toçky. Cij zadaçi prysvqçeno bahato doslidΩen\ (dyv., napryklad,
[8 – 15] ).
U danij roboti my pokaΩemo, wo dlq koΩnoho natural\noho k interval
0 1,( ] zminy parametra q moΩna rozbyty na dekil\ka pidintervaliv i dlq znaçen\
q z koΩnoho pidintervalu vkazaty prostu dostatng umovu hlobal\no] stijkosti
nul\ovoho rozv’qzku rivnqnnq (1), pryçomu ci umovy [ toçnymy dlq klasu riv-
nqn\ (1), qki zadovol\nqgt\ umovu (2).
U robotax avtoriv [16 – 18] otrymano nastupni umovy hlobal\no] stijkosti
nul\ovoho rozv’qzku rivnqnnq (1).
Teorema 1. Prypustymo, wo q q qk k+ + … +1 ( ) < 1 i funkci] fn zadovol\-
nqgt\ umovu (2). Todi rivnqnnq (1) [ hlobal\no asymptotyçno stijkym dlq
koΩno] trijky parametriv ( , , )a q k , qki zadovol\nqgt\ umovu
*
Çastkovo pidtrymano Fondom fundamental\nyx doslidΩen\ Ukra]ny (hrant 14.1/007 (V. I Tka-
çenko)) ta Fondecyt, Çyli (hranty 1071053 (S. I Trofymçuk) ta 7070091 (V. I Tkaçenko)).
© O. I NENQ, V. I. TKAÇENKO, S. I. TROFYMÇUK, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1 73
74 O. I NENQ, V. I. TKAÇENKO, S. I. TROFYMÇUK
a
q1 –
> –
–
1
1
1
1
+ +
+
q
q
k
k . (3)
Ce oznaça[, wo isnugt\ λ ∈ (0, 1) i H > 0 taki, wo
max
– ,j n k n
jx
=
≤ Hλn s– max
– ,j s k s
jx
=
, n s≥ , (4)
dlq koΩnoho rozv’qzku xn{ } rivnqnnq (1). Umova (3) [ toçnog dlq klasu riv-
nqn\ (1), qki zadovol\nqgt\ nerivnist\ q q qk k+ + … +1 ( ) < 1 ta umovu (2).
Teorema 2. Nexaj q ∈ (0, 1) i funkci] fn zadovol\nqgt\ umovu (2). Todi
koΩnyj rozv’qzok xn{ } rivnqnnq?(1) prqmu[ do 0 i zadovol\nq[ nerivnist\?(4),
qkwo
Θ( , , )k q a = max ( , , , )
, ,s k
s k q a
= …0
Ω > – 1, (5)
?
de
Ω( , , , )s k q a = qk s+ +1 + aq s
q
q
s
k
+
+1
1
1–
–
+ a
q sq s
q
s
2
2
1 1
1
+ ( – – )
( – )
.
Umova (5) [ toçnog dlq klasu rivnqn\ (1), qki zadovol\nqgt\ umovu Jorka
(2): dlq koΩnyx znaçen\ parametriv a, k i q, qki ne zadovol\nqgt\ umovu (5),
isnu[ rivnqnnq (1), qke ne [ hlobal\no stijkym.
Teorema 3. Nexaj q = 1, funkci] fn zadovol\nqgt\ umovu (2) i
f z zj j j k
j
( , , )–…∑ = ∞ (6)
dlq koΩno] poslidovnosti zm{ }, qka ma[ nenul\ovu hranycg na neskinçennosti.
Todi koΩen rozv’qzok xn{ } rivnqnnq (1) prqmu[ do nulq, qkwo
a ≥ max ( , )
,j
jg k
=1 2
τ = max
( ) – ( ),j
j j j jk k=
−
+ + + + + +1 2 2
4
1 1 4 1τ τ τ τ
, (7)
de τ1 — cila çastyna çysla
1
3
1 12k k k− + + +( ), a τ2 = τ1 + 1.
Umova (5) xoç i konstruktyvna, ale hromizdka i ne da[ uqvlennq pro heomet-
ryçnu strukturu oblasti parametriv stijkosti. BaΩano otrymaty umovu typu (3)
ta (7) ne til\ky dlq malyx q çy q = 1, a i dlq vsix q ∈ 0 1,( ]. V danij roboti my
rozv’qzu[mo cg zadaçu.
Spoçatku vvedemo deqki poznaçennq.
Dlq q ∈ (0, 1), natural\noho k ta ciloho s, 0 ≤ s ≤ k, oznaçymo funkcig
ω ( , , )s k q =
q q q q s
s q
k k s k s2 2 2 3 2
2 2
1 1
1 1
+ + + + + +
+
– – ( – )( )
( ) ( – )
. (8)
Pry q = 1 poklademo
ω ( , , )s k 1 =
s k
s
–
( )
2
2 1+
qk hranycg pravo] çastyny (8) pry q → 1.
Lehko pereviryty, wo
ω ( , , )s k q = Ω s k q
q
s
k
, , , −
+
+1
1
= Ω s k q
q
s
k
+ −
+
+
1
1
1
, , , .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
PRO TOÇNI UMOVY HLOBAL|NO} STIJKOSTI RIZNYCEVOHO RIVNQNNQ … 75
Za dovedenog nyΩçe lemog?1 pry 0 ≤ s ≤ sk = 2 2 3k –( )[ ] ( ⋅[ ] — cila
çastyna çysla) rivnqnnq ω ( , , )s k q = – 1 ma[ [dynyj rozv’qzok q = q ks
∗( ) ∈ 0 1,( ].
Pry s > 2 2 3k –( )[ ] zavΩdy ω ( , , )s k q > – 1.
Poznaçymo çerez a k qj ( , ) , j = 1, … , k, pravyj korin\ kvadratnoho vidnosno a
rivnqnnq Ω( , , , )j k q a = – 1. Vidpovidno a k q0 ( , ) — [dynyj rozv’qzok linijnoho
rivnqnnq Ω( , , , )0 k q a = – 1.
Teper my moΩemo sformulgvaty osnovnyj rezul\tat roboty.
Teorema 4. Nexaj q ∈ (0, 1) i funkci] fn zadovol\nqgt\ umovu Jorka (2).
Todi dlq prqmuvannq koΩnoho rozv’qzku xn{ } rivnqnnq (1) do 0 ta vykonannq
nerivnosti (4) dostatnimy [ nastupni umovy:
0) a ∈ a k q0 0( , ),( ) , qkwo q ∈ 0 0, ( )q k∗( ];
j) a ∈ a k qj ( , ), 0( ), qkwo q ∈ q k q kj j−
∗ ∗[ ]1( ), ( ) , 1 ≤ j ≤ sk ;
sk + )1 a ∈ ask +( 1 ( , ),k q 0), qkwo q ∈ q ksk
∗[ )( ), 1 .
Qkwo q = 1 i vykonugt\sq umovy (2) ta (6), to dlq prqmuvannq do nulq
koΩnoho rozv’qzku rivnqnnq (1) dostatn\o vykonannq nastupno] umovy: a ?∈
∈ ( , )ak 0 , de ak — pravyj korin\ kvadratnoho rivnqnnq
a s sk k
2 1 2( )( )+ + + 2 2a s kk( )+ + + 4 = 0.
Navedeni umovy [ toçnymy pry vidpovidnyx znaçennqx q dlq klasu rivnqn\ (1),
qki zadovol\nqgt\ umovy (2).
ZauvaΩennq 1. Lehko pereviryty, wo q k0
∗( ) — [dynyj dodatnyj korin\ riv-
nqnnq q q qk k+ + … +1( ) = 1, a Ω( , , , )0 k q a = qk +1 + a qk( – )1 1+
/ (1 – q), tomu
umova a?∈ a k q0 0( , ),( ) nabuva[ vyhlqdu (3). OtΩe, umova 0) teoremy?4 ekviva-
lentna umovi (3) teoremy?1.
ZauvaΩennq 2. V kinci statti my dovedemo, wo umova a?∈ ( , )ak 0 hlobal\-
no] stijkosti rivnqnnq (1) pry q = 1 ekvivalentna umovi (7) teoremy?3.
Vykorystovugçy qvnyj vyhlqd vyraziv ω ( , , )s k q ta Ω( , , , )s k q a , moΩna ob-
çyslyty konkretni znaçennq q ks
∗( ) i dlq fiksovanyx k ta q vkazaty optymal\-
ni umovy hlobal\no] stijkosti rivnqnnq (1). Navedemo dlq prykladu znaçennq
q ks
∗( ) dlq k = 4 : q0 4∗( ) = 0,8288, q1 4∗( ) = 0,9344, q2 4∗( ) = 1.
Tomu dostatnimy umovamy hlobal\no] stijkosti rivnqnnq (1) pry k = 4 [ taki:
a?∈ a q0 4 0( , ),( ), qkwo q?∈ 0 0 8288; ,( ], de a q0 4( , ) = – ( )1 5+ q / 1 2+ +( +q q
+ q q3 4+ );
a?∈ a q1 4 0( , ),( ), qkwo q?∈ 0 8288 0 9344, ; ,[ ], de a q1 4( , ) — pravyj korin\ kvad-
ratnoho rivnqnnq a2
?+? a q q q q q( )2 2 3 4 5+ + + + + q6
?+ ?1 = 0;
a?∈ a q2 4 0( , ),( ) , qkwo q?∈ 0 9344 1, ;[ ], de a q2 4( , ) — pravyj korin\ kvadrat-
noho rivnqnnq a q2 2 1( )+ + a q q q q q( )3 2 3 4 5 6+ + + + + q7
?+ ?1 = 0;
a?∈ ( / , )−1 3 0 , qkwo q = 1. Çyslo −1 3/ [ pravym korenem kvadratnoho riv-
nqnnq 3 2a + 4 a + 2 = 0. Vidmitymo takoΩ, wo a2 4 1( , ) = −1 3/ .
Na rysunku v koordynatax ( , )q a sucil\nog kryvog pokazano funkci]
a q0 4( , ) , a q1 4( , ) ta a q2 4( , ) , a toçky nad ci[g lini[g zadagt\ oblast\ hlobal\-
no] stijkosti rivnqnnq xn +1 = qxn + f x x xn n n n( , , , )– –1 4… . Dlq porivnqnnq ob-
last\ nad ßtryxovog kryvog [ oblastg lokal\no] stijkosti nul\ovoho roz-
v’qzku rivnqnnq.
U roboti [18] umovu 0) teoremy?4 bulo dovedeno dlq q?∈ 0 0, ( )q k( ] z deqkym
q k0( ) < q k0
∗( ) dlq rivnqn\ (1) z syl\no nelinijnymy funkcionalamy fn , qki za-
dovol\nqgt\ uzahal\nenu umovu Jorka
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
76 O. I NENQ, V. I. TKAÇENKO, S. I. TROFYMÇUK
Oblasti hlobal\no] ta lokal\no] stijkosti v koordynatax (q, a).
a
b
M
M
( )
( )
φ
φ1 +
≤ fn( )φ ≤
– ( )
– ( )
a
b
M
M
−
−
φ
φ1
, b > 0, n ∈Z , (9)
dlq vsix φ ∈ Rk +1
takyx, wo min j jφ > − −b 1
?∈ ( , )−∞ 0 .
Hipoteza. Teorema?4 [ spravedlyvog dlq syl\no nelinijnyx rivnqn\ (1), qki
zadovol\nqgt\ umovu (9).
Lema 1. Funkciq ω ( , , )s k q [ stroho monotonno spadnog po q ta stroho
monotonno zrostagçog po s.
Dovedennq. Vykona[mo peretvorennq
ω ( , , )s k q =
q q q q s
s q
k k s k s2 2 2 3 2
2 2
1 1
1 1
+ + + + + +
+
– – ( – )( )
( ) ( – )
=
=
q
s q
q q q q s
k s
k s k s k k
+ +
+ −
+
+ + … + + +( )
2
2
2 3 1
1 1
1
( ) ( – )
– ( )– – =
= –
( )
– –
q
s
q
k s
j
j
k i
i
s+ +
==+ ∑∑
2
2
0
1
01
. (10)
Z (10) vydno, wo ∂ω ( , , )s k q / ∂q < 0, q > 0.
Teper pokaΩemo, wo funkciq ω ( , , )s k q [ monotonno zrostagçog po s:
ω ( , , )s k q+ 1 > ω ( , , )s k q , s = 0, … , k – 1. (11)
Dijsno, rozhlqdagçy s qk neperervnyj arhument ta vraxovugçy nerivnist\
−ln q / ( )1 − q > 1, dlq q?∈ (0, 1) otrymu[mo
∂ω
∂
( , , )s k q
s
=
−
+
+
+
+ + + +ln
( – ) ( ) ( – )
q
q
q
s
q
s q
k s k s
1 1 1 1
2 2 3
2 +
+
q
s q
k s+ +
+
2
21 1( ) ( – )
–
2 1
1 1
2 2 1
3 2
q q
s q
k s+ +
+
( – )
( ) ( – )
>
q
s q
k +
+
2
21 1( ) ( – )
×
× q s q q q q
q
s q
s s k s
s
( )( – ) –
( – )
( )( – )
+ + +
+
+
+
1 1
2 1
1 1
1
1
≥
≥
q
s q
k +
+
2
21 1( ) ( – )
q s q q q q
q
s q
s s s s
s
( )( – ) –
( – )
( )( – )
+ + +
+
+
+
1 1
2 1
1 1
1
1
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
PRO TOÇNI UMOVY HLOBAL|NO} STIJKOSTI RIZNYCEVOHO RIVNQNNQ … 77
=
q
s q
k s+ +
+
2
21 1( ) ( – )
( )( – ) –
( – )
( )( – )
s q q
q
s q
s
s
+ + +
+
+
+
1 1 1
2 1
1 1
1
1
≥
≥
q
s q
k s+ +
+
2
21 1( ) ( – )
( )( – ) –s q qs+ +( )+1 1 11 ≥ 0
dlq s ≥ 0, q?∈ (0, 1). My vykorystaly nerivnosti
q s qs + +1 1 1( )( – ) < 1 1– qs + < ( )( – )s q+ 1 1
dlq s > 0, q?∈ (0, 1).
Lemu dovedeno.
Dovedennq teoremy 4. Pry k = 1 formuly v teoremi nabyragt\ prostoho
vyhlqdu s1 = 0, ω ( , , )0 1 q = –q3
, q0 1∗( ) = 1, Ω( , , , )0 1 q a = q2 + a q( )1 + ,
a q0 1( , ) = – ( )q2 1+ / ( )q + 1 . Umova hlobal\no] stijkosti a?∈ a q0 1 0( , ),( ) pry
q?∈ ∈ 0 1,( ] [ bezposerednim naslidkom teoremy?1.
Dali rozhlqda[mo vypadok k ≥ 2.
Rozib’[mo dovedennq na kil\ka krokiv.
1. Doslidymo funkcig Ω( , , , )t k q a . Rozhlqnemo riznycg
R t k q a( , , , )+ 1 = Ω( , , , )t k q a+ 1 – Ω( , , , )t k q a =
= q q a q t q a t qk k t+ ++ + + + +( )1 1 21 1 1 1( – ) ( ( )( – )) ( ) .
Lehko pereviryty, wo R t k q q tk( , , , / ( ))+ − ++1 11 = 0 i
∂
∂
R t k q a
a a q tk
( , , , )
/ ( )
+
= − ++
1
1 1
< 0.
Tomu Ω( , , , )t k q a+ 1 > Ω( , , , )t k q a , qkwo a < − +qk 1 / ( t + 1 ) , i Ω( , , , )t k q a+ 1 <
< Ω( , , , )t k q a , qkwo a?∈ − +( )+q tk 1 1 0/ ( ), .
Nexaj − +qk 1
/ s < a < − +qk 1
/ (s + 1), 1 ≤ s ≤ k – 1. Todi
Ω( , , , )k k q a > … > Ω( , , , )s k q a+ 1 > Ω( , , , )s k q a ,
(12)
Ω( , , , )s k q a < Ω( – , , , )s k q a1 < … < Ω( , , , )0 k q a .
OtΩe, pry − +qk 1
/ s < a < − +qk 1
/ (s + 1) minimum u (5) dosqha[t\sq na
Ω( , , , )s k q a , tobto Θ( , , )k q a = Ω( , , , )s k q a .
Pry a = − +qk 1
/ (s + 1) otrymu[mo
… > Ω( , , , )s k q a+ 2 > Ω( , , , )s k q a+ 1 = Ω( , , , )s k q a < Ω( – , , , )s k q a1 < … .
Vidpovidno, pry a < − +qk 1
Ω( , , , )k k q a > … > Ω( , , , )1 k q a > Ω( , , , )0 k q a , (13)
tobto minimum u (5) dosqha[t\sq na Ω( , , , )0 k q a . Qkwo Ω a > − +qk 1
/ k, to
Ω( , , , )k k q a < … < Ω( , , , )1 k q a < Ω( , , , )0 k q a . (14)
2. Dovedemo, wo ∂Ω( , , , )s k q a / ∂a > 0 pry a ≥ − +qk 1
/ s, s ≥ 1, q?∈ (0, 1):
∂
∂
Ω( , , , )s k q a
a
= q s
q
q
s
k
+
+1
1
1–
–
+ 2
1 1
1 2a
q sq q
q
s s– ( – )
( – )
+
≥
≥ q s
q
q
s
k
+
+1
1
1–
–
–
2 1 1
1
1
2
q q sq q
s q
k s s+ +( )– ( – )
( – )
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
78 O. I NENQ, V. I. TKAÇENKO, S. I. TROFYMÇUK
= q s qs kq( )+ + + … +1 –
2
1
1
1
1q
s q
q q q q q
k
s s s s
+
+ + … +
( – )
( – – – )– >
> q s q qs k( )+ + + … +1 – q sk + +1 1( ) > 0.
Tomu pry − +qk 1
/ s < a < − +qk 1
/ (s + 1)
ω ( – , , )s k q1 = Ω( , , , – )s k q q sk +1 < Ω( , , , )s k q a <
< Ω s k q q sk, , , – ( )+ +( )1 1 = ω ( , , )s k q . (15)
3. Nexaj q < q k0
∗( ). Todi Ω 0 1, , , –k q qk +( ) = ω ( , , )0 k q > – 1. Qkwo a?∈
∈? −[ +qk 1
/ t q tk, /( )− + )+1 1 , 1 ≤ t ≤ k – 1, to za formulamy (12), (15) otrymu[mo
Ω( , , , )j k q a ≥ Ω( , , , )t k q a ≥ Ω t k q q tk, , , – +( )1 = ω ( – , , )t k q1 ≥
≥ ω ( , , )0 k q > ω 0 0, , ( )k q k∗( ) = – 1, j = 0, … , k.
Qkwo a ≥ –qk +1
/ k, to za nerivnostqmy (14) dlq j = 0, 1, … , k vykonu[t\sq
Ω( , , , )j k q a ≥ Ω( , , , )k k q a ≥ Ω k k q q kk, , , – /+( )1 = ω ( – , , )k k q1 > – 1.
OtΩe, pry a ≥ –qk +1
vykonu[t\sq umova (5), a pry a < –qk +1
z uraxuvannqm
(13) vykonannq umovy (5) ekvivalentne vykonanng nerivnosti Ω( , , , )0 k q a > – 1.
4. Prypustymo, wo q ks
∗( ) ≤ q < q ks +
∗
1( ), s ≥ 0.
Qkwo a?∈? − − +[ )+ +q q sk k1 1 1, ( ) , a same a?∈? − − +[ )+ +q j q jk k1 1 1, ( ) , 1 ≤ j ≤
≤ s, na pidstavi (15) ta lemy?1
Ω( , , , )j k q a < Ω j k q
q
j
k
, , ,
– +
+
1
1
= ω ( , , )j k q ≤ ω ( , , )s k q ≤
≤ ω s k q ks, , ( )∗( ) = – 1.
Qkwo a < –qk +1
, to
Ω( , , , )0 k q a = –qk +1 + a q qk( – ) ( – )1 11+ < ω ( , , )0 k q < ω 0 0, , ( )k q k∗( ) = – 1.
Tomu v c\omu vypadku Θ( , , )k q a < – 1 , tobto a ne moΩe buty menßym za
–qk +1
/ (s + 1).
Qkwo − +qk 1
/ (s + 1) ≤ a < –qk +1
/ (s + 2), to za nerivnostqmy (12)
… > Ω( – , , , )s k q a1 > Ω( , , , )s k q a ≥ Ω( , , , )s k q a+ 1 ,
Ω( , , , )s k q a+ 1 < Ω( , , , )s k q a+ 2 < … .
Najmenße znaçennq nabuva[t\sq na Ω( , , , )s k q a+ 1 , tobto Θ( , , )k q a =
= Ω( , , , )s k q a+ 1 . Vidmitymo, wo
Ω s k q
q
s
k
+
+
+
1
1
1
, , ,
–
= ω ( , , )s k q ≤ ω s k q ks, , ( )∗( ) = – 1,
Ω s k q
q
s
k
+
+
+
1
2
1
, , ,
–
= ω ( , , )s k q+ 1 > ω s k q ks+( )+
∗1 1, , ( ) = – 1
i ∂Ω( , , , )s k q a+ 1 / ∂a > 0 pry a ≥ − +qk 1
/ (s + 1). Tomu kvadratne vidnosno
a rivnqnnq Ω( , , , )s k q a+ 1 = – 1 ma[ toçno odyn korin\ na intervali
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
PRO TOÇNI UMOVY HLOBAL|NO} STIJKOSTI RIZNYCEVOHO RIVNQNNQ … 79
− +[ +q sk 1 1( ) ,? − + )+q sk 1 2( ) . Druhyj korin\ c\oho rivnqnnq leΩyt\ liviße toç-
ky − +qk 1
/ (s+ 1).
Nareßti, qkwo − +qk 1
/ (s + 2) ≤ a < − +qk 1
/ k, a same − +qk 1
/ j ≤ a < − +qk 1 / (j +
+ 1) dlq deqkoho k – 1 ≥ j ≥ s + 2, to za spivvidnoßennqmy (15) ta lemog?1
Ω( , , , )j k q a ≥ Ω( , , , / )j k q q jk− +1 = ω ( – , , )j k q1 > ω ( , , )s k q+ 1 ≥
≥ ω s k q ks+( )+
∗1 1, , ( ) = – 1.
Vraxovugçy (12), otrymu[mo Θ( , , )k q a > – 1.
Qkwo Ω a ≥ − +qk 1
/ k, to dlq vsix j = 0, 1, … , k
Ω( , , , )j k q a ≥ Ω( , , , )k k q a ≥ Ω k k q
q
k
k
, , ,
– +
1
= ω ( – , , )k k q1 > – 1.
OtΩe, pry a?∈? − +[ )+q sk 1 1 0( , zavΩdy Θ( , , )k q a > – 1.
Teoremu dovedeno.
Dovedennq zauvaΩennq 2. Poklademo Ω( , , , )s k a1 = a s s2 1( )+ / 2 + ( k + s +
+ 1 ) a + 1 qk hranycg lim ( , , , )q s k q a→1 Ω . Lehko pereviryty, wo nerivnist\
Ω( , , , )s k a1 > – 1 ekvivalentna nerivnosti a > g k s( , ) , de g k s( , ) oznaçeno v (7).
Wob pereviryty, wo umova a?∈?( , )ak 0 teoremy?4 ekvivalentna umovi (7) teo-
remy?3, dosyt\ pokazaty, wo max ( , ),j jg k=1 2 τ = g k sk( , )+ 1 . Dlq oznaçenyx ra-
niße çysel
sk = 2
3
1( – )k
, τ1 = τ1( )k = 1
3
1 12k k k– + + +( )
,
τ2 = τ2( )k = τ1 + 1
rozhlqnemo try vypadky.
1. Viz\memo vsi çysla k = 3 l + 1, l = 0, 1, … , todi sk = 2
3
1( – )k
= 2 l, a
τ1 = 1
3
3 9 9 32l l l+ + +( )
= l l l+ + +
2 1
3
= l + l l2 1
3
+ +
= 2 l.
Vidpovidno τ2 = 2 l + 1 i g ( 3 l + 1, τ1
) = g ( 3 l + 1, τ2
). Tomu umova a?∈ ?( , )ak 0 ek-
vivalentna nerivnosti a > max ( , )j jg l3 1+ τ = g l sk( , )3 1 1+ + .
2. Viz\memo çysla k = 3 l + 2, l = 0, 1, … , todi sk = 2 2 3l +[ ] = 2 l,
τ1 = l + 1
3
15
9
7
9
2+ + +
l l = 2 l + 1,
oskil\ky
l + 1 ≤ 1
3
+ l l2 15
9
7
9
+ + < l + 2.
U c\omu vypadku sk + 1 = τ1( )k = 2 l + 1 i
g k( , )τ1 =
−
+ + + +
4
5 4 9 16 82l l l
> g k( , )τ2 =
−
+ + + +
4
5 5 9 10 12l l l
.
Tomu umova a?∈?( , )ak 0 ekvivalentna nerivnosti a > max ( , )j jg l3 2+ τ = g l(3 2+ ,
sk + 1) .
3. Viz\memo vsi çysla k = 3 l + 3, todi sk = 2 4 3l +[ ] = 2 l + 1,
τ1 = l + 2
3
7
3
13
9
2+ + +
l l = 2 l + 1,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
80 O. I NENQ, V. I. TKAÇENKO, S. I. TROFYMÇUK
oskil\ky
l + 1 ≤ 2
3
+ l l2 7
3
13
9
+ + < l + 2.
U c\omu vypadku sk + 1 = τ2( )k = 2 l + 2 i
g k( , )τ1 =
−
+ + + +
4
5 5 9 26 172l l l
< g k( , )τ2 =
−
+ + + +
4
5 6 9 20 122l l l
.
Tomu umova a?∈?( , )ak 0 ekvivalentna nerivnosti a > max ( , )j jg l3 3+ τ = g l3( +
+ 3 1, sk + ).
TverdΩennq dovedeno.
1. Botsford L. W. Further analysis of Clark’s delayed recruitment model // Bull. Math. Biol. – 1992. –
54. – P. 275 – 293.
2. Clark C. W. A delayed recruitment model of population dynamics with an application to baleen
whale populations // J. Math. Biol. – 1976. – 3. – P. 381 – 391.
3. El-Morshedy H. A., Liz E. Globally attracting fixed points in higher order discrete population mo-
dels // Ibid. – 2006. – 53. – P. 365 – 384.
4. Gurney W. S. C., Nisbet R. M. Ecological dynamics. – Oxford Univ. Press, 1998. – 352 p.
5. Kocic V. L., Ladas G. Global asymptotic behavior of nonlinear difference equations of higher order
with applications. – Dordrecht: Kluwer Acad., 1993. – 228 p.
6. Liz E., Tkachenko V., Trofimchuk S. Global stability in discrete population models with delayed-
density dependence // Math. Biosci. – 2006. – 199. – P. 26 – 37.
7. Thieme H. R. Mathematics in population biology. – Princeton: Princeton Univ. Press, 2003. –
543 p.
8. Berezansky L., Braverman E., Liz E. Sufficient conditions for the global stability of nonautono-
mous higher order difference equations // J. Different. Equat. and Appl. – 2005. – 11. – P. 785 – 798.
9. Ivanov A. F. On global stability in a nonlinear discrete model // Nonlinear Anal. – 1994. – 23. –
P. 1383 – 1389.
10. Li X. Global attractivity in a genotype selection model // Int. J. Math. and Math. Sci. – 2002. – 29,
# 9. – P. 537 – 544.
11. Matsunaga H., Hara T., Sakata S. Global attractivity for a nonlinear difference equation with va-
riable delay // Comput. Math. and Appl. – 2001. – 41. – P. 543 – 551.
12. Graef J. R., Qian C. Global attractivity of the equilibrium of a nonlinear difference equation //
Czech. Math. J. – 2002. – 52. – P. 757 – 769.
13. Cull P., Chaffee J. Stability in discrete population models // Amer. Inst. Phys. Conf. Proc. – 2000.
– 517. – P. 263 – 276.
14. Levin S. A., May R. M. A note on difference-delay equations // Theor. Populat. Biol. – 1976. – 9. –
P. 178 – 187.
15. Muroya Y., Ishiwata E., Guglielmi N. Global stability for nonlinear difference equations with va-
riable coefficients // J. Math. Anal. and Appl. – 2007. – 334. – P. 232 – 247.
16. Nenya O., Tkachenko V., Trofimchuk S. On the global stability of the nonlinear difference equation
// Nonlinear Oscillations. – 2004. – 7, # 4. – P. 473 – 480.
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kolyvannq. – 2006. – 9, # 4. – S. 525 – 534.
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OderΩano 02.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
|
| id | umjimathkievua-article-3138 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:36:57Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7a/3021d35568de203ba03b5064e125987a.pdf |
| spelling | umjimathkievua-article-31382020-03-18T19:46:36Z On sharp conditions for the global stability of a difference equation satisfying the Yorke condition Про точні умови глобальної стійкості різницевого рівняння, яке задовольняє умову Йорка Nenya, O. I. Tkachenko, V. I. Trofimchuk, S. I. Неня, О. І. Ткаченко, В. І. Трофімчук, С. І. Continuing our previous investigations, we give simple sufficient conditions for global stability of the zero solution of the difference equation xn+1 = qxn + fn (xn ,..., xn-k ), n &isin; Z, where nonlinear functions fn satisfy the Yorke condition. For every positive integer k, we represent the interval (0, 1] as the union of [(2k + 2) /3] disjoint subintervals, and, for q from each subinterval, we present a global-stability condition in explicit form. The conditions obtained are sharp for the class of equations satisfying the Yorke condition. В продолжение предыдущих исследований авторов приведены простые достаточные условия глобальной устойчивости нулевого решения разностного уравнения xn+1 = qxn + fn (xn ,..., xn-k ), n &isin; Z, где нелинейные функции fn удовлетворяют условию Йорка. Для каждого натурального k интервал (0, 1] представлен как объединение [(2k + 2)/3] подынтервалов, и для q с каждого подынтервала в явном виде приведено условие глобальной устойчивости. Полученные условия являются точными для класса уравнений, удовлетворяющих условию Йорка. Institute of Mathematics, NAS of Ukraine 2008-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3138 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 1 (2008); 73–80 Український математичний журнал; Том 60 № 1 (2008); 73–80 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3138/3024 https://umj.imath.kiev.ua/index.php/umj/article/view/3138/3025 Copyright (c) 2008 Nenya O. I.; Tkachenko V. I.; Trofimchuk S. I. |
| spellingShingle | Nenya, O. I. Tkachenko, V. I. Trofimchuk, S. I. Неня, О. І. Ткаченко, В. І. Трофімчук, С. І. On sharp conditions for the global stability of a difference equation satisfying the Yorke condition |
| title | On sharp conditions for the global stability of a difference equation satisfying the Yorke condition |
| title_alt | Про точні умови глобальної стійкості різницевого рівняння, яке задовольняє умову Йорка |
| title_full | On sharp conditions for the global stability of a difference equation satisfying the Yorke condition |
| title_fullStr | On sharp conditions for the global stability of a difference equation satisfying the Yorke condition |
| title_full_unstemmed | On sharp conditions for the global stability of a difference equation satisfying the Yorke condition |
| title_short | On sharp conditions for the global stability of a difference equation satisfying the Yorke condition |
| title_sort | on sharp conditions for the global stability of a difference equation satisfying the yorke condition |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3138 |
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