On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations
We investigate different measure transformations of the mapping-multiplication type in the cases where the corresponding chains of differential equations can be efficiently found and integrated.
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| Date: | 2005 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2005
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3603 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509720496832512 |
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| author | Mokhonko, A. A. Мохонько, О. А. |
| author_facet | Mokhonko, A. A. Мохонько, О. А. |
| author_sort | Mokhonko, A. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:42Z |
| description | We investigate different measure transformations of the mapping-multiplication type in the cases where the corresponding chains of differential equations can be efficiently found and integrated. |
| first_indexed | 2026-03-24T02:45:35Z |
| format | Article |
| fulltext |
UDK 530.1
O.�A.�Moxon\ko (Ky]v. nac. un-t im.T.�Íevçenka)
DEQKI ROZV’QZNI KLASY NELINIJNYX
NEIZOSPEKTRAL|NYX RIZNYCEVYX RIVNQN|
We investigate different measure transformations of mapping-multiplication type for the cases where
corresponding chains of differential equations may be effectively found and integrated.
DoslidΩugt\sq rizni vypadky peretvorennq miry typu vidobraΩennq-mnoΩennq dlq tyx sytu-
acij, koly vidpovidni lancgΩky dyferencial\nyx rivnqn\ moΩna efektyvno znajty i zinte-
hruvaty.
1. Vstup. Nexaj L — ermitiv linijnyj operator u prostori l2 , qkyj moΩna
zobrazyty qkobi[vog matryceg z çyslamy bn ∈ R na holovnij diahonali i an > 0
na dvox sumiΩnyx iz neg, wo pokoordynatno di[ tak: L ( u )n = an –1 un –1 + bn un +
+ an un +1 , de a–1 = 0, a oblastg joho vyznaçennq [ mnoΩyna D ( L ) = { u : u ∈ l2 ,
L u ∈ l2 }. Cej operator bude obmeΩenym todi i til\ky todi, koly poslidovnosti
an , bn [ obmeΩenymy.
Dyskretnyj analoh zadaçi Íturma – Liuvillq
an –1 Pn –1 ( λ ) + bn Pn ( λ ) + an Pn +1 ( λ ) = λ Pn ( λ ) , P–1 ( λ ) ≡ 0, P0 ( λ ) ≡ 1 ⇒
(1)
⇒ Pn +1 = 1
an
(( λ – bn ) Pn – an –1 Pn –1 )
porodΩu[ systemu polinomiv perßoho rodu { } =
∞Pn n( )λ 0. Dlq operatora L budu-
[t\sq spektral\na boreleva mira d ρ ( λ ) i dovodyt\sq, wo ci polinomy utvorg-
gt\ ortonormovanyj bazys hil\bertovoho prostoru L2 ( R , d ρ ) . Klas usix spek-
tral\nyx mir obmeΩenyx qkobi[vyx matryc\ — ce skinçenni miry na borelevij
σ -alhebri z kompaktnym nosi[m i neskinçennog mnoΩynog toçok rostu. Po
koΩnij takij miri za vidomog klasyçnog procedurog budu[t\sq qkobi[va matry-
cq, dlq qko] cq mira [ spektral\nog (obernena spektral\na zadaça). Detal\nyj
vyklad teori] qkobi[vyx matryc\ moΩna znajty, napryklad, u [1, 2].
U cij statti rozhlqda[t\sq zadaça Koßi:
˙ ( )a tn = Fn ( a ( t ) , b ( t ) ) ,
˙ ( )b tn = Gn ( a ( t ) , b ( t ) ) ,
a tn t( ) =0 = an ( 0 ) , b tn t( ) =0 = bn ( 0 ) , n ∈ N0 = {0, 1, … }, t ∈ [ 0, T ] , T ≤ ∞ .
Tut a ( t ) = ( )( )a tn n=
∞
0, b ( t ) = ( )( )b tn n=
∞
0 — poslidovnosti ßukanyx rozv’qzkiv, a
Fn , Gn — deqki riznycevi nelinijni vyrazy. Rozv’qzky ßukagt\sq sered odyn raz
neperervno dyferencijovnyx funkcij an ( t ) > 0, bn ( t ) ∈ R , wo [ obmeΩenymy
po n pry koΩnomu fiksovanomu t .
U vypadku klasyçnoho lancgΩka Tody na pivosi [3] dlq rozv’qzku ci[] zadaçi
Koßi v [4, 5] bulo zaproponovano pidxid, pov’qzanyj z robotamy [6, 7]. Vin polq-
ha[ u tomu, wo z ßukanym rozv’qzkom ( a ( t ) , b ( t ) ) asocig[t\sq qkobi[va matrycq
(operator) L ( t ) z b ( t ) na holovnij diahonali i a ( t ) na dvox sumiΩnyx. ZaleΩ-
nist\ an ( t ) , bn ( t ) vid t , wo dyktu[t\sq rivnqnnqmy Tody, [ skladnog. Ale
vyqvylosq, wo zaleΩnist\ spektral\no] miry d ρ ( ⋅ , t ) operatora L ( t ) vid po-
çatkovo] miry d ρ ( ⋅ , 0 ) [ prostog: d ρ ( λ , t ) = eλ
t
d ρ ( λ , 0 ) . Ce dalo moΩlyvist\
© O.�A.�MOXON|KO, 2005
356 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
DEQKI ROZV’QZNI KLASY NELINIJNYX NEIZOSPEKTRAL|NYX … 357
zaproponuvaty takyj sposib rozv’qzuvannq zadaçi Koßi: za poçatkovymy danymy
( a ( 0 ) , b ( 0 ) ) budu[mo qkobi[vu matrycg L ( 0 ) i znaxodymo ]] spektral\nu miru
d ρ ( ⋅ , 0 ) . Dali znaxodymo vkazanym vywe sposobom spektral\nu miru d ρ ( ⋅ , t )
dlq t ∈ ( 0, T ] . Rozv’qzugçy obernenu spektral\nu zadaçu, za mirog d ρ ( ⋅ , t )
vidnovlg[mo matrycg L ( t ) , koefici[nty qko] i budut\ ßukanymy rozv’qzkamy
( a ( t ) , b ( t ) ) . Formuly, wo dozvolqgt\ ce zrobyty, moΩna znajty, napryklad, u
[5, 8, 9].
Pizniße u robotax [8, 9] bulo zaproponovano uzahal\nennq c\oho pidxodu do
intehruvannq deqkyx klasiv nelinijnyx riznycevyx rivnqn\, koly d ρ ( ⋅ , t ) zna-
xodyly po d ρ ( ⋅ , 0 ) bil\ß skladnym çynom, aniΩ mnoΩennq na e λ
t ( vono
pov’qzane z toçkog zoru, vykladenog u knyzi [10]). U c\omu uzahal\nenni
supp d ρ ( ⋅ , t ) zming[t\sq z çasom, tobto zadaçi ne [ „izospektral\nymy”. Zakon
zminy d ρ ( λ , t ) po t zada[t\sq pevnym çynom za dopomohog dvox funkcij Φ, Ψ .
U�danij roboti rozhlqdagt\sq vypadky, koly cej zakon moΩna kal\kulqtyvno
realizuvaty: navodqt\sq efektyvni formuly dlq poßuku d ρ ( λ , t ) .
2. (((( ΦΦΦΦ, ΨΨΨΨ )))) -peretvorennq miry. Qk bulo zaznaçeno u vstupi, za poçatkovu
umovu viz\memo miru d ρ ( ⋅ , 0 ) . Pobudu[mo po nij miru d ρ ( ⋅ , t ) u dva kroky:
çerez vidobraΩennq i mnoΩennq miry.
VidobraΩennq miry budu[t\sq z vykorystannqm zadaçi Koßi:
d t
dt
λ µ( , )
= Φ ( λ ( t , µ ) , t ) ,
(2)
λ ( 0 , µ ) = µ ; t ∈ [ 0, T ] , µ ∈ M ,
de koefici[nt Φ ( λ , t ) = ϕ λi
i
i
l
t( )
=
∑
0
[ zadanym, M = supp d ρ ( ⋅ , 0 ) — nosij miry.
Dlq dovil\noho fiksovanoho t ∈ ( 0, T ] budu[mo vidobraΩennq: ω t : M → R ,
ωt ( µ ) = λ ( t , µ ) . Vono [ neperervnym, a tomu obmeΩenym na kompakti M × [ 0, T ] .
VidobraΩennq miry budu[t\sq tak:
ˆ( , )ρ ∆ t = ρ ω( )( ),t
−1 0∆ , ∆ ∈ B ( R ) , t ∈ [ 0, T ] .
Oskil\ky obraz ωt ( µ ) [ obmeΩenym, to mira, pobudovana z vykorystannqm pro-
obrazu i miry z kompaktnym nosi[m, sama ma[ supp ˆ( , )ρ ⋅ t -kompakt.
MnoΩennq pobudovano] miry ρ̂ na funkcig zdijsng[t\sq z vykorystannqm
rivnqnnq z çastynnymy poxidnymy. Nexaj Ψ ( λ , t ) = ϕ λi
i
i
m
t( )
=
∑
0
— inßyj fikso-
vanyj koefici[nt. Rozhlqnemo rivnqnnq
∂ λ
λ
λ ∂ λs t
t
s t
t
( , )
( , )
( , )
∂
+
∂
Φ = Ψ ( λ , t ) s ( λ , t ) ,
(3)
s ( λ , 0 ) = 1; t ∈ [ 0, T ] , λ ∈ R .
Nexaj s ( λ , t ) — joho rozv’qzok. Budu[mo ßukanu miru ostatoçno:
ρ ( ∆ , t ) = s t d t( , ) ˆ( , )λ ρ λ
∆
∫ = s t t d
t
( )( , ), ( , )
( )
λ µ ρ µ
ω
0
1−
∫
∆
.
Zahal\nyj vyhlqd lancgΩkiv dlq pobudovano] miry bude takym:
ȧn = {Φ ( L ( t ) , t )}n +1, n +
an
2
( { Ψ ( L ( t ) , t )}n +1, n – {Ψ ( L ( t ) , t )}n , n ) +
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
358 O.�A.�MOXON|KO
+ an +1 {Φ ( L ( t ) , t ) DL ( t ) }n +2, n – an –1 {Φ ( L ( t ) , t ) DL ( t ) }n +1, n –1 +
+ ( bn +1 – bn ) {Φ ( L ( t ) , t ) DL ( t ) }n +1, n ,
(4)
ḃn = {Φ ( L ( t ) , t )}n , n + an { 2 Φ ( L ( t ) , t ) DL ( t ) + Ψ ( L ( t ) , t ) }n +1, n –
– an –1 { 2 Φ ( L ( t ) , t ) DL ( t ) + Ψ ( L ( t ) , t ) }n , n –1 ,
n ∈ N0, a–1 = 0, t ∈ [ 0, T ] .
Tut DL ( t ) — operator dyferencigvannq po λ u bazysi { } =
∞P ti i( , )λ 0 hil\berto-
voho prostoru L2 ( R , d ρ ( λ , t ) ) .
Osnovog usix podal\ßyx vykladok [ takyj fakt [8, 9]:
Rozhlqnemo zadaçu Koßi dlq rivnqn\ (4): dlq dovil\nyx poçatkovyx danyx
( a ( 0 ) , b ( 0 ) ) , de an ( 0 ) > 0, bn ( 0 ) ∈ R , n ∈ N0 , — obmeΩeni rivnomirno po n ,
znajty ]x rozv’qzok. Todi isnu[ dostatn\o male T > 0 take, wo rozv’qzok isnu[
u klasi
K [ 0, T ] = ( )( ), ( ) ,a t b t Cn n T∈{ [ ]→0
1
R
∀t ∈ [0, T ] , ∀n ∈ N0
vykonu[t\sq an ( t ) > 0, sup max max ( ), ( )
,
( )
n t T n na t b t
∈ ∈[ ]
< + ∞}
N0
0
,
[ [dynym i moΩe buty znajdenyj za opysanog vywe procedurog.
Dali budemo doslidΩuvaty rivnqnnq, wo vynykatymut\ pry riznyx Φ ( λ , t ) ,
Ψ ( λ , t ) . Qk uΩe zaznaçalosq, perevaha opysanoho pidxodu do rozv’qzuvannq lan-
cgΩkiv dyferencial\nyx rivnqn\ polqha[ u tomu, wo skladni zakonomirnosti
dlq ]x rozv’qzkiv budut\ opysuvatys\ vidpovidnymy spivvidnoßennqmy dlq mir,
qki vyqvlqgt\sq znaçno prostißymy. Tut vynykagt\ dvi zadaçi: neobxidno vka-
zaty lancgΩok i zapysaty vidpovidne peretvorennq miry. My rozhlqnemo koΩnu
z nyx okremo.
Znaçni trudnowi pry qvnomu zapysi lancgΩkiv vynykagt\ pry rozkrytti do-
dankiv typu {Φ ( L ( t ) , t ) DL ( t ) }i , j . Tomu poçnemo z doslidΩennq operatora dy-
ferencigvannq DL ( t ) .
3. Operator dyferencigvannq DL (((( t )))) . Operator D L ( t ) dyferencigvannq
za zminnog λ di[ na linijnij pidmnoΩyni neperervno dyferencijovnyx funkcij
prostoru L2 ( R , d ρ ( λ , t ) ) , i joho moΩna podaty u vyhlqdi matryci ( ) ,dij i j=
∞
0 u
bazysi { } =
∞P tn n( , )λ 0 (nadali zminnu t dlq zruçnosti vkazuvaty ne budemo).
Oskil\ky deg Pn ( λ ) = n , to deg ′Pn( )λ = n – 1, a tomu ′Pn( )λ = d Pin i
i
n
( )λ
=
−
∑
0
1
. Tobto
matrycq operatora DL [ stroho verxn\o-trykutnog.
Lehko otrymu[t\sq rekurentnist\, wo da[ moΩlyvist\ buduvaty vsg matry-
cg zliva napravo, stovpçyk za stovpçykom:
d
d
a P b P a Pn n n n n nλ
λ λ λ− − ++ +[ ]1 1 1( ) ( ) ( ) = d
d
Pnλ
λ λ( )[ ] ⇒
⇒ ′+Pn 1( )λ = 1
1 1a
P b P a P P
n
n n n n n n( ) ( ) ( ) ( )λ λ λ λ λ− ′ − ′ + ′[ ]− − .
Operator L (wo parametryçno zaleΩyt\ vid t ) u prostori L 2 ( R , d ρ ( λ , t ) ) [
operatorom mnoΩennq na λ . Cej fakt dozvolq[ rozklasty dodanok λ λ′Pn( ) za
vkazanym bazysom. Ostatoçno ma[mo
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
DEQKI ROZV’QZNI KLASY NELINIJNYX NEIZOSPEKTRAL|NYX … 359
d
d
d
d
d
d
n
n
n n
n n
n n
n n
0 1
1 1
3 1
2 1
1 1
1
0
,
,
,
,
,
,
+
+
− +
− +
− +
+
�
�
= 1
0
0
0
0
0
1
0
0
0
0
1
3
2
1
1
0
a
b
d
d
d
d
d
a
d
n
n
n
n
n n
n n
n n
n
�
�
�
�
−
−
−
−
−
−
,
,
,
,
,
,nn
n
n n
n n
n
n
n n
n n
n n
d
d
d L
d
d
d
d
d
−
−
− −
− −
−
−
−
+
1
1 1
3 1
2 1
0
1
3
2
10
0
0
0
0
,
,
,
,
,
,
,
,
�
�
�
�
.
Poznaçymo d
(
n
) = ( d0, n ; d1, n ; … ; dn –1, n ; 0; 0; … ) . Dva startovi vektory znaxo-
dqt\sq bezposeredn\o: ′P0( )λ = 0 ⇒ d
(
0
) = ( 0 ; 0 ; … ) , ′P1( )λ = 1 / a0 ⇒ d
(
1
) =
= ( 1 / a0 ; 0 ; … ) . Obçyslymo takoΩ
L d
(
n
) = ( b0 d0, n + a0 d1, n ; a0 d0, n + b1 d1, n + a1 d2, n ; … ; an –3 dn –3, n + bn –2 dn –2, n +
+ an –2 dn –1, n ; an –2 dn –2, n + bn –1 dn –1, n ; an –1 dn –1, n ; 0 ; 0 ; … ) .
Nas bude cikavyty ne sama matrycq operatora D L , a qvni (tobto ne�reku-
rentni) vyrazy dlq dn , n +1 , dn –1, n +1 , dn –2, n +1 i t.�d.
Lema 1.
dn , n +1 =
n
an
+ 1
, dn –1, n +1 = 1
1 0
1
a a
b nb
n n
i
i
n
n
− =
−
∑ −
,
dn –2, n +1 = 1 2 1 1
1 2
2
0
2
1
2
1
2
0
2
1
0
2
a a a
a n a n b b b b b b
n n n
i
i
n
n n n i
i
n
n n i
i
n
− − =
−
− −
=
−
−
=
−
∑ ∑ ∑− − + − + − +
( ) ( ) ( ) ,
dn – j , n +1 =
a a
a a
d b b a d a di i j
n n j
i j i i j i i j i j i i i j i
i j
n − −
−
− − − − + − − −
=
…
…
− + −( )∑ 1
1 1 1, , ,( ) , 1 ≤ j ≤ n .
Dovedennq ci[] lemy provodyt\sq za indukci[g. Neobxidno porqdkovo zapy-
saty vidpovidni rivnosti u vektornij rekurentnosti (same tomu ]] bulo navedeno u
stovpçykovomu vyhlqdi).
4. Pobudova lancgΩkiv. Rozhlqnemo spoçatku izospektral\nyj vypadok:
Φ ( λ , t ) ≡ 0. Vid lancgΩkiv (4) zalyßa[t\sq:
ȧn = an
2
( { Ψ ( L ( t ) , t )}n +1, n – {Ψ ( L ( t ) , t )}n , n ) ,
(5)
ḃn = an { Ψ ( L ( t ) , t ) }n +1, n – an –1 { Ψ ( L ( t ) , t ) }n , n –1 .
Tut peretvorennq miry budu[t\sq do kincq, i my otrymu[mo klas zadaç, qki za-
proponovanym metodom rozv’qzugt\sq povnistg. Zokrema, pry Ψ( λ , t ) = λ oder-
Ωymo lancgΩok�Tody [3], a pry Ψ (λ , t ) = λ2
— lancgΩok Kas – van Moerbeke [7].
Dlq vypadku Ψ ( λ , t ) = λ3 vidpovidnyj lancgΩok ma[ vyhlqd
ȧn = a
a b a b b b an
n n n n n n n2
2
1
2
1
2
1 1 2 1
2+ +( ) + +( + + + + + +( ) –
– a b b b a a bn n n n n n n− − −+( ) − + +( ) )1
2
1 1
2 2 2 ,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
360 O.�A.�MOXON|KO
ḃn = a a b b b a a bn n n n n n n n
2
1
2
1
2
1
2
1
2
− + + ++ + + + +( )( ) –
– a a b b b a a bn n n n n n n n− − − − −+ + + + +( )1
2
2
2
1 1 1
2 2 2( ) .
Qkwo Ψ ( λ , t ) = λ4, to
ȧn = a
a a a b a b b a a an
n n n n n n n n n n2 1
2 2 2
1
2
1
2 2
1 2
2
1
2
1
2
2
2
− + + + + + + ++ + +( ) + + +( )( ) –
– a a a b a b b a a an n n n n n n n n n− − − − − ++ + +( ) + + +( )2
2
1
2
1
2 2 2 2
1
2
1
2 2
1
2( ) ,
ḃn = a b b a a b a a b b b a b bn n n n n n n n n n n n n n
2
1 1
2
1
2 2 2
1
2
1
2
1 1
2
1 22( ) ( ) ( )− − − + + + + + ++ + + + + +( ) + + +( ) –
− + + + + + +( ) + + +( )− − − − − − − − +a b b a a b a a b b b a b bn n n n n n n n n n n n n n1
2
2 1 2
2
2
2
1
2
1
2 2 2
1
2
12( ) ( ) ( ) .
Rozhlqnemo bil\ß zahal\nyj neizospektral\nyj vypadok:
Φ ( λ , t ) = ϕ λi
i
i
t( )
=
∑
0
3
, Ψ ( λ , t ) = ψ λi
i
i
t( )
=
∑
0
3
.
Neobxidno poraxuvaty usi dodanky v lancgΩku (4). Poznaçymo
Φi , j = {Φ ( L ( t ) , t )}i , j , Ω i , j = {Φ ( L ( t ) , t ) DL ( t ) }i , j
i otryma[mo
Φn +1, n = a b b a b b b b a an n n n n n n n n nϕ ϕ ϕ1 2 1 3 1
2
1 1
2 2
1
2+ + + + + + + +( )( )( )+ − + + +( ) ( ) ,
Ψn , n = ψ ψ ψ1 2 1
2 2 2
3 1
2 2 2b a a b a a b bn n n n n n n n+ + +( ) + + +( )[− − +
+ ( ) ( )b b a b b an n n n n n+ − −+ + + ] +1
2
1 1
2
0ψ ,
Φn , n = ϕ ϕ ϕ1 2 1
2 2 2
3 1
2 2 2b a a b a a b bn n n n n n n n+ + +( ) + + +( )[− − +
+ ( ) ( )b b a b b an n n n n n+ − −+ + + ] +1
2
1 1
2
0ϕ ,
Ψn +1, n = a b b a b b b b a an n n n n n n n n nψ ψ ψ1 2 1 3 1
2
1 1
2 2
1
2+ + + + + + + +( )( )( )+ − + + +( ) ( ) ,
Ω n +1, n –1 = ϕ3 1 1a a nn n− −( ),
Ω n +1, n = a n b b a bn n n n j
j
n
ϕ ϕ ϕ ϕ2 3 1 3 3 1
1
+ +( ) ++ −
=
∑ .
Pry pidraxunku Ω i , j sutt[vo vykorystovuvalasq lema 1. Ostatoçnyj lancg-
Ωok ma[ vyhlqd
ȧn = 1
2 3 1
3
1 3 1
1
1 2ψ ϕ ϕb a b b a b b b a nn n n n n j
j
n
n n n+ + −
=
++ − + −∑( ) ( ) +
+ ψ ψ ψ ψ ψ3 1
2 2
1
2
2 1
2
3
3
2
2
1 12
a n b b a
a
a b b bn n n n
n
n n n n+ − − +− −( ) − + + −( ) +
+ a b a
a
b a b a b bn n n
n
n n n n n nψ ψ ψ ψ ψ ψ3 1 1
2
3
2
2 1
2
2 1
2
3 1
3
12+ + + + ++ − + + + −( ) +
+ ϕ ψ2 3
2
1
2 2
1
2
1
2
12a b a a a b b a b bn n n n n n n n n n+ + + + + +( )− + + + +
+ a a b b a b a b a nn n n n n n n n n
1
2
1
23 1
2
2 3 1
2
3 1 1
2
2 1 1 1
2
3ψ ψ ψ ϕ ϕ ϕ+ + − − − + +− − + + +
,
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DEQKI ROZV’QZNI KLASY NELINIJNYX NEIZOSPEKTRAL|NYX … 361
ḃn = b a n a b a a b b bn n n n n n n n n− − +( ) + + + +(− − − + +2 1
2
3 3 1
2
1 1
2
3 1
2
3 1
3
3
2
2ψ ψ ϕ ψ ψ ψ ψ +
+ ψ ϕ ϕ2 1 2 3 1 1
2 2
1
2
1
22 3 2 4b n b a a b b a b an n n n n n n n n+ − − + −+ ) + + + +
–
– 2 1
2 2
1
1
1
3 1
2
2
2
1
2 2
2 1
2
1a a b a a b b a b bn n j
j
n
n n n n n n n− −
=
−
− − − − −−( )
− + +( ) − +( )∑ ψ ψ +
+ 2 22
3 3
2
1
2
3 1 1 1
2
3 1
4
1
2a n b a b b a n b a a an n n n n n n n n nϕ ψ ϕ ψ ψ ψ+ + − − ++ + − − +
+ ψ ϕ ϕ ϕ ϕ ϕ ϕ ϕ3
4
1
2
2 1
2
3 1 2
2
3
3
2 1
2
2
2
02 2 3a a n a b n b b a an n n n n n n n− − + + + + +− − − − .
Qk çastynnyj vypadok, zvidsy moΩna oderΩaty lancgΩok dlq Φ ( λ , t ) = ϕ0 +
+ ϕ1 λ + ϕ2 λ2, Ψ ( λ , t ) = ψ0 + ψ1 λ , dlq qkoho u [8, 9] [ bahato prykladiv. Lema�1
da[ moΩlyvist\ u qvnomu vyhlqdi zapysuvaty lancgΩky dlq dovil\nyx Φ, Ψ.
5. Pobudova peretvoren\ mir. Osnovni trudnowi u zastosuvanni zapropono-
vanoho metodu pov’qzani z intehruvannqm zvyçajnyx dyferencial\nyx rivnqn\ i
rivnqn\ z çastynnymy poxidnymy. Navedemo prosti vypadky, koly ce intehruvan-
nq moΩna zdijsnyty povnistg. Proanalizu[mo spoçatku vypadok, koly [ lyße
vidobraΩennq miry, tobto Ψ ( λ , t ) ≡ 0:
1. Φ ( λ , t ) ≡ 0 ⇒ ˆ( , )ρ ∆ t = ρ ( ∆ , 0 ) . Ce [ vypadok, koly poçatkova mira ne
deformu[t\sq zovsim. Ma[mo
d t
dt
λ µ
λ µ µ
( , )
( , )
=
=
0
0
⇒ λ ( t , µ ) = µ ⇒ ωt = ′IM ⇒ ωt
−1 = ′IM ⇒
⇒ ˆ( , )ρ ∆ t = ρ ω( )( ),t
−1 0∆ = ρ ( ∆ , 0 ) .
2. Φ ( λ , t ) ≡ 1 ⇒ ˆ( , )ρ ∆ t = ρ ( ∆ – t , 0 ) . Tut nosij poçatkovo] miry z çasom
zsuva[t\sq. Ma[mo
d t
dt
λ µ
λ µ µ
( , )
( , )
=
=
1
0
⇒ λ ( t , µ ) = t + µ = ωt ( µ ) ⇒ ω λt
−1( ) = λ – t ⇒
⇒ ˆ( , )ρ ∆ t = ρ ω( )( ),t
−1 0∆ = ρ ( ∆ – t , 0 ) .
3. Φ ( λ , t ) = a ( t ) λ + b ( t ) ⇒
⇒ ˆ( , )ρ ∆ t = ρ ω( )( ),t
−1 0∆ = ρ
τ τ
∆ ⋅ ∫ − ∫
− −
∫e b u e du
a u du a d
tt u
( ) ( )
( ) ;0 0 0
0
.
Zadaça Koßi rozv’qzu[t\sq metodom variaci] dovil\no] stalo]:
d t
dt
a t b t
λ µ λ
λ µ µ
( , )
( ) ( )
( , )
= +
=
0
⇒ λ ( t , µ ) = b u e du e
a d
t
a u du
u t
( )
( ) ( )−∫ +
∫∫
τ τ
µ0 0
0
= ωt ( µ ) ⇒
⇒ ω λt
−1( ) = λ
τ τ
e b u e du
a u du a d
tt u− −∫ − ∫∫
( ) ( )
( )0 0
0
.
Napryklad, pry Φ ( λ , t ) = λ oderΩu[mo ˆ( , )ρ ∆ t = ρ( ),∆ ⋅ −e t 0 .
4. Φ ( λ , t ) = a ( t ) λn, n ≥ 2 ⇒ ˆ( , )ρ ∆ t = ρ ∆1
0
1 1
1 0−
−
+ −
∫n
t n
n a u du( ) ( ) ;
/( )
.
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362 O.�A.�MOXON|KO
Dyferencial\ne rivnqnnq rozv’qzu[t\sq metodom vidokremlennq zminnyx:
d t
dt
a t nλ µ λ
λ µ µ
( , )
( )
( , )
=
=
0
⇒ λ ( t , µ ) = µ1
0
1 1
1−
−
+ −
∫n
t n
n a u du( ) ( )
/( )
= ωt ( µ ) ⇒
⇒ ω λt
−1( ) = λ1
0
1 1
1−
−
+ −
∫n
t n
n a u du( ) ( )
/( )
.
Napryklad, pry Φ ( λ , t ) = λ2 oderΩu[mo ˆ( , )ρ ∆ t = ρ ∆−
−
+( )
∫1
0
1
0a u du
t
( ) ; .
Neobxidno zaznaçyty, wo pry velykyx znaçennqx n zapys vidpovidnoho lancgΩ-
ka sta[ duΩe skladnog zadaçeg, xoça peretvorennq miry opysu[t\sq prosto.
Teper rozhlqnemo druhyj etap peretvorennq miry (pry riznyx variantax per-
ßoho): koly [ mnoΩennq.
1. Φ ( λ , t ) ≡ 0 ∀Ψ ( λ , t ) ⇒ d ρ ( λ , t ) = e d
d
t Ψ( , )
( , )
λ τ τ
ρ λ0 0∫ ,
∂ λ
∂λ
∂ λ
∂
λ λ
λ
s t s t
t
t s t
s
( , ) ( , )
( , ) ( , )
( , )
⋅ + =
=
0
0 1
Ψ
⇒ s ( λ , t ) = e
d
t Ψ( , )λ τ τ
0∫ ,
ρ ( ∆ , t ) = s t d t( , ) ˆ( , )λ ρ λ
∆
∫ = e d
d
t Ψ
∆
( , )
( , )
λ τ τ
ρ λ0 0∫∫ .
Napryklad, pry Ψ ( λ , t ) ≡ 0 mira ne zming[t\sq: ρ ( ∆ , t ) = ρ ( ∆ , 0 ) . Pry Ψ ( λ , t ) =
= λ oderΩu[mo peretvorennq miry dlq lancgΩka Tody: d ρ ( λ , t ) = eλ
t
d ρ ( λ , 0 ) .
Vzahali, dlq Ψ ( λ , t ) = λn, n ≥ 0, zokrema, i dlq tyx vypadkiv, dlq qkyx u statti
bulo navedeno vidpovidni lancgΩky (n = 1, 2, 3, 4) , di[ take peretvorennq miry:
d ρ ( λ , t ) = e d
ntλ ρ λ( , )0 .
2. Φ ( λ , t ) ≡ 1 ∀Ψ ( λ , t ) ⇒ d ρ ( λ , t ) = e d t
u t u du
t Ψ( , )
( , )
− +∫ −
λ
ρ λ0 0 ,
∂ λ
∂λ
∂ λ
∂
λ λ
λ
s t s t
t
t s t
s
( , ) ( , )
( , ) ( , )
( , )
⋅ + =
=
1
0 1
Ψ
⇒ s ( λ , t ) = f t e
t d
( )
( , )
− ∫ + −
λ
τ τ λ τλ Ψ
0 ,
1 = f e
d
( )
( , )
−
∫ −
λ
τ τ λ τ
λ
Ψ
0 ⇒ f ( λ ) = e
d− +
−
∫ Ψ( , )τ τ λ τ
λ
0 ⇒
⇒ s ( λ , t ) = e e
t d t d
t
− + − + −
− +
∫ ∫Ψ Ψ( , ) ( , )τ τ λ τ τ τ λ τ
λ λ
0 0 = e
u t u du
t
Ψ( , )− +∫ λ
0 ,
ρ ( ∆ , t ) = s t d t( , ) ˆ( , )λ ρ λ
∆
∫ = e d
u t u du
t
Ψ
∆
( , )
( , )
− +∫
∫
λ
ρ λ0 0 .
U c\omu vypadku moΩna otrymuvaty u pokaznyku eksponenty polinomy po t .
Prypustymo, nam neobxidno otrymaty miru e d ta a t a tn
n
0 1 0( ) ( ) ( ) ( , )λ λ λ ρ λ+ + + −… . Dlq
c\oho neobxidno rozv’qzaty intehral\ne rivnqnnq Ψ( , )u t u du
t
− +∫ λ
0
= a0 ( λ ) +
+ a1 ( λ ) t + … + an ( λ ) t
n vidnosno Ψ (qkwo cej rozv’qzok isnu[). Zvorotna zadaça
[ lehkog: napryklad, Φ ( λ , t ) ≡ 1, Ψ ( λ , t ) = λ – t ⇒ d ρ ( λ , t ) = e d tt t− + −
2
0λ ρ λ( , ).
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DEQKI ROZV’QZNI KLASY NELINIJNYX NEIZOSPEKTRAL|NYX … 363
3. Φ ( λ , t ) = a ( t ) λ + b ( t ) ∀Ψ ( λ , t ) ⇒ d ρ ( λ , t ) = exp
( )
Ψ e
a d
t
t
ξ ξ
η
∫
∫
0
×
× λ τ τ η η ρ λ
ξ ξ
η
τ τ
τ+ ∫
∫ − ∫
∫ ∫
− −
b e d d d e b u e du
a d
t
a u du a d
tt t u
( ) , ( ) ,
( ) ( ) ( )
0 0
0
0 .
Na c\omu vypadku prodemonstru[mo metod intehruvannq rivnqn\ z çastynnymy
poxidnymy, wo zastosovuvavsq dlq otrymannq usix podal\ßyx rezul\tativ.
Dyferencial\ne rivnqnnq ma[ vyhlqd
∂ λ
∂λ
λ ∂ λ
∂
s t
a t b t
s t
t
( , )
( ) ( )
( , )+( ) + = Ψ( λ , t ) s ( λ , t ) ,
s ( λ , 0 ) = 1.
Zastosu[mo metod xarakterystyk. Neobxidno znajty poverxng s = s ( λ , t ) , wo
proxodyt\ çerez kryvu � : s ( λ , 0 ) = 1, λ ∈ R . Kryvu � podamo u parametryzova-
nomu vyhlqdi: � = { ( λ , t , s ) : λ = v , t = 0, s = 1 , v ∈ R }. Rozhlqnemo try xarakte-
rystyçni rivnqnnq:
dt
du
t
=
=
1
0 0( )
⇒ t ( u ) = u ,
d
du
a t u u b t u
λ λ
λ
= +
=
( ) ( )( ) ( ) ( )
( )0 v
⇒ λ ( u ) = e b e d
a d a d
uu
( ) ( )
( )
ξ ξ ξ ξ
τ τ
τ
0 0
0
∫ ∫ +
−
∫ v ,
ds
du
u t u s u
s
=
=
Ψ( )( ), ( ) ( )
( )
λ
0 1
⇒ s ( u ) = e
t d
uΨ( )( ), ( )λ η η η
0∫ .
Iz perßyx dvox vyraΩa[mo u ta v çerez t i λ :
u = t, v = λ τ τ
ξ ξ ξ ξτ
e b e d
a d a d
tt− −∫ − ∫∫
( ) ( )
( )0 0
0
i pidstavlq[mo ]x u vyraz dlq λ ( u ) , zaminggçy u zminnog η :
λ ( η ) = e b e d e b e d
a d a d a d a d
tt
( ) ( ) ( ) ( )
( ) ( )
ξ ξ ξ ξ
η
ξ ξ ξ ξη τ τ
τ τ λ τ τ0 0 0 0
0 0
∫ ∫ + ∫ − ∫
− − −
∫ ∫ .
Teper dlq otrymannq rozv’qzku zalyßylosq pidstavyty otrymani zobraΩennq
λ ( η ) ta t ( η ) = η u s ( λ , t ) = s ( u ( λ , t ) , v ( λ , t ) ) = e
t d
uΨ( )( ), ( )λ η η η
0∫ .
Analohiçno rozhlqda[t\sq nastupnyj vypadok:
4. Φ ( λ , t ) = a ( t ) λn, n ≥ 2, ∀Ψ ( λ , t ) ⇒ d ρ ( λ , t ) = exp ( )Ψ n
t
−
∫ 1
0
×
× a d d d n a u du
t
n
n
n
t n
( ) ; ( ) ( ) ;
/( ) /( )
ξ ξ λ τ τ ρ λ
τ
∫ ∫+
+ −
−
−
−
−
1
1 1
1
0
1 1
1 0 .
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364 O.�A.�MOXON|KO
Pevnyj interes ma[ vypadok, koly koefici[nty Φ ne�zaleΩat\ vid t . MoΩna
zdijsnyty vidokremlennq zminnyx i zintehruvaty rivnqnnq do kincq (tobto ne�vy-
nyka[ sytuaciq typu rivnqnnq Rikkati), ale ce vse odno ne�dozvolq[ znajty qvnu
formulu dlq peretvorennq miry:
Φ ( λ , t ) = ϕ0 + ϕ1 λ + … + ϕn λ
n, deg Φ ≥ 2, ϕi ∈ R , ∀Ψ ( λ , t ) ⇒
⇒
d t
dt
t
λ µ λ λ
λ µ µ
( , )
( , ) ( )
( , )
= =
=
Φ Φ
0
⇒
d
t
ξ
ξ
λ µ
Φ( )
( , )
0
∫ = t +
dξ
ξ
µ
Φ( )0
∫ .
Tak z’qvlqgt\sq pevni problemy z odnoznaçnym vyrazom dlq λ ( t , µ ) . Navedemo
zahal\nyj rozv’qzok dlq mnoΩennq miry:
∂ λ
∂λ
λ ∂ λ
∂
λ λ
λ
s t s t
t
t s t
s
( , )
( )
( , )
( , ) ( , )
( , )
Φ Ψ+ =
=
0 1
⇒
⇒ s ( λ , t ) = f t d e
t d
d
−
∫
∫
∫
+
τ
τ
λ
τ
ξ
ξ
τ
τ
λ
τ
λ
Φ
Ψ
Φ
Φ
( )
,
( )
( )
0
1
0
.
OtΩe, pry sprobi znajty mnoΩennq miry vynykagt\ ti sami trudnowi.
Zapyßemo we peretvorennq miry dlq vypadku Φ ( λ , t ) = λn, Ψ ( λ , t ) = λm:
a) Φ ( λ , t ) = λn, Ψ ( λ , t ) = λm, n ≥ 2, m – n + 1 ≠ 0 ⇒ d ρ ( λ , t ) =
= exp ( ) ( ) ;
/( ) /( )1
1
1 1 01 1 1 1 1 1 1
m n
n t d n tm n n m n n n
− +
− − +( )( )
+ −[ ]( )+ − − − + − −
λ λ ρ λ ;
b) Φ ( λ , t ) = λn, Ψ ( λ , t ) = λm, n ≥ 2, m – n + 1 = 0 ⇒
⇒ d ρ ( λ , t ) = λ
λ
ρ λ
t n
d n t
n n
n n
( )
( ) ;/( )
/( )
− +( )
+ −[ ]( )− −
− −
1
1 0
1 1 1
1 1 1
;
v) Φ ( λ , t ) = λ , Ψ ( λ , t ) = λm, m ≠ 0 ⇒
⇒ d ρ ( λ , t ) = exp ;
( )λ ρ λ
m mt
te
m
d e
1
0
−
( )
−
− ;
h) Φ ( λ , t ) = λ , Ψ ( λ , t ) = λm, m = 0 ⇒ d ρ ( λ , t ) = e d et tρ λ( );− 0 .
6. Dodatkovi zauvaΩennq. 1. U razi, qkwo neobxidno proanalizuvaty sytua-
cig, koly [ lyße odne peretvorennq miry, dostatn\o u vsi poperedni rezul\taty
pidstavyty Ψ ( λ , t ) ≡ 0 dlq vypadku, koly vidbuva[t\sq lyße vidobraΩennq
miry, i Φ ( λ , t ) ≡ 0, koly vidbuva[t\sq lyße mnoΩennq.
2. Vywe bulo rozhlqnuto vypadok Φ ( λ , t ) ≡ 0 ∀Ψ ( λ , t ) i otrymano vidpo-
vidne peretvorennq miry: d ρ ( λ , t ) = e
d
t Ψ( , )λ τ τ
0∫ d ρ ( λ , 0 ) . Tut vidbuva[t\sq lyße
mnoΩennq miry. Oskil\ky nosij miry ne�zming[t\sq z çasom, to ce peretvorennq
[ izospektral\nym.
Qkwo u poçatkovyx danyx b t t( ) =0 = 0, to mira d ρ ( λ , 0 ) [ parnog. Qkwo
poklasty Φ ( λ , t ) = λn, de n — parne, to mira d ρ ( λ , t ) takoΩ bude parnog.
Peretvorennq miry matyme vyhlqd d ρ ( λ , t ) = e
ntλ d ρ ( λ , 0 ) . Dlq lancgΩkiv
vynyka[ we odna zakonomirnist\: qkwo n — parne, to usi bn ( t ) ≡ 0, i lancgΩok
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
DEQKI ROZV’QZNI KLASY NELINIJNYX NEIZOSPEKTRAL|NYX … 365
dlq ˙ ( )b tn znyka[, a dlq ˙ ( )a tn syl\no sprowu[t\sq; qkwo Ω n — neparne, to
mira parnog vΩe ne�bude, i skazaty wos\ pro bn ( t ) vΩe ne�moΩna. Dlq doveden-
nq ci[] vlastyvosti neobxidno akuratno proanalizuvaty�(5). Navedemo dekil\ka
prykladiv. OtΩe, skriz\ Φ ( λ , t ) ≡ 0 :
Ψ ( λ , t ) = λ2 ⇒ ȧn = 1
2 1
2
1
2a a an n n+ −−( ) ,
Ψ ( λ , t ) = λ4 ⇒ ȧn =
=
a
a a a a a a a a a a a an
n n n n n n n n n n n n2 1
2 2 2
1
2
2
2
1
2 2
1
2 2 2 2
1
2
2
2
1
2
+ + + − − + − −+( ) + + − +( ) − −( ).
Cej rezul\tat moΩna we uzahal\nyty: usi poperedni vysnovky zalyßagt\sq
virnymy, koly Ψ ( λ , t ) — bud\-qka parna po λ funkciq. Krim toho, moΩna spe-
cial\no skonstrugvaty Φ ( λ , t ) � 0 tak, wob ω λt
−1( ) bula parnog funkci[g.
Todi ˆ( , )ρ ∆ t = ρ ω( )( ),t
−1 0∆ teΩ bude parnog, i qkwo dodatkovo Ψ ( λ , t ) [
parnog po λ , to d ρ ( λ , t ) — parna mira, i znovu z’qvlq[t\sq moΩlyvist\ vyko-
rystaty navedenyj rezul\tat.
Viz\memo, napryklad, µ = ω λt
−1( ) = λ
λ
2
21+ t
. Todi λ( t , µ) =
µ
µ
2
21− t
, λ(0, µ) =
= µ ,
d t
dt
λ µ( , )
= 1
2
3λ µ( , )t . OtΩe, slid vzqty Φ ( λ , t ) = 1
2
3λ . Poklademo, napryk-
lad, Ψ ( λ , t ) = t2 + t λ2. Mira u c\omu vypadku ne�[�izospektral\nog i [ parnog:
d ρ ( λ , t ) = e d
t
t t3 2 23 2
2
21
0/ / ;+
+
λ ρ λ
λ
. LancgΩok u c\omu vypadku ma[ vyhlqd
ȧn = 1
2
11
2 2
1
2
1
2
1
2
1
2
1
2a a a a a a t a a na a a a nn n n n n n n n n n n n− + + − + −+ +( ) + −( ) + − −[ ]( ) .
1. Axyezer/N./Y. Klassyçeskaq problema momentov y nekotor¥e vopros¥ analyza, svqzann¥e s
neg. – M.: Fyzmathyz, 1961. (Anhl. red.: Akhiezer N. I. The classical moment problem and some
related questions in analysis. – New York: Hafner, 1965.)
2. Berezanskyj/G./M. RazloΩenye po sobstvenn¥m funkcyqm samosoprqΩenn¥x operatorov.
– Kyev: Nauk. dumka, 1965. –�800�s. (Anhl. red.: Berezansky Yu. M. Expansions in eigenfunc-
tions of self-adjoint operators. – Providence: Amer. Math. Soc., 1968.)
3. Toda M. Theory of nonlinear lattices // Springer Ser. Solid-State Sci. – 1981. – # 20. – 205 p.
4. Berezansky Yu. M. Integration of nonlinear difference equations by the inverse spectral problem
method // Sov. Math. Dokl. – 1985. – 21. – P. 264 – 267.
5. Berezansky Yu. M. The integration of semi-infinite Toda chain by means of inverse spectral
problem // Repts Math. Phys. – 1986. – 24. – P. 21 – 47.
6. Moser J. Three integrable Hamiltonian systems connected with isospectral deformations // Adv.
Math. – 1975. – 16, # 2. – P. 197 – 220.
7. Kac M., van Moerbeke P. On an explicitly soluble system of nonlinear differential equations
related to certain Toda lattices // Ibid. – 1975. – 16, # 2. – P. 160 – 169.
8. Berezansky Yu., Shmoish M. Nonisospectral flows on semi-infinite Jacobi matrices // Nonlinear
Math. Phys. – 1994. – 1. – P. 116 – 146.
9. Berezansky Yu. M. Integration of nonlinear nonisospectral difference-differential equations by
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| id | umjimathkievua-article-3603 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:35Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e9/84e9708dcbb51f2a46cb2b7e543584e9.pdf |
| spelling | umjimathkievua-article-36032020-03-18T19:59:42Z On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations Деякі розв'язні класи нелінійних неізоспектральних різницевих рівнянь Mokhonko, A. A. Мохонько, О. А. We investigate different measure transformations of the mapping-multiplication type in the cases where the corresponding chains of differential equations can be efficiently found and integrated. Досліджуються різні випадки перетворення міри типу відображення-множення для тих ситуацій, коли відповідні ланцюжки диференціальних рівнянь можна ефективно знайти і зінтегрувати. Institute of Mathematics, NAS of Ukraine 2005-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3603 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 3 (2005); 356–365 Український математичний журнал; Том 57 № 3 (2005); 356–365 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3603/3937 https://umj.imath.kiev.ua/index.php/umj/article/view/3603/3938 Copyright (c) 2005 Mokhonko A. A. |
| spellingShingle | Mokhonko, A. A. Мохонько, О. А. On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations |
| title | On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations |
| title_alt | Деякі розв'язні класи нелінійних неізоспектральних різницевих рівнянь |
| title_full | On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations |
| title_fullStr | On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations |
| title_full_unstemmed | On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations |
| title_short | On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations |
| title_sort | on some solvable classes of nonlinear nonisospectral difference equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3603 |
| work_keys_str_mv | AT mokhonkoaa onsomesolvableclassesofnonlinearnonisospectraldifferenceequations AT mohonʹkooa onsomesolvableclassesofnonlinearnonisospectraldifferenceequations AT mokhonkoaa deâkírozv039âzníklasinelíníjnihneízospektralʹnihríznicevihrívnânʹ AT mohonʹkooa deâkírozv039âzníklasinelíníjnihneízospektralʹnihríznicevihrívnânʹ |