On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations
We consider a quasilinear system of difference equations with certain conditions. We prove that there exists a formal partial o-solution of this system in the form of functional series of special type. We also prove a theorem on the asymptotic behavior of this solution.
Збережено в:
| Дата: | 1997 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
1997
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| Назва видання: | Український математичний журнал |
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/156977 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations / A.V. Kostin, I.V. Skripnik // Український математичний журнал. — 1997. — Т. 49, № 5. — С. 672–677. — англ. |
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Ukrainian Mathematical Journal, Vol. 49, No. 5, 1997
ON ASYMPTOTIC DECOMPOSITIONS OF o-SOLUTIONS IN THE THEORY
OF QUASILINEAR SYSTEMS OF DIFFERENCE EQUATIONS
A. V. Kos t i n a n d I. V. S k r i p n i k UDC 517.949
We consider a quasilinear system of difference equations with certain conditions. We prove that there
exists a formal partial o-solution of this system in the form of functional series of special type. We also
prove a theorem on the asymptotic behavior of this solution.
Consider the system of difference equations
n
Ayk(t) = qk(t) + Epki ( t )Y i ( t ) +
i=1
t ~ N,
oo
Z P~l-.-h, (t)y~l (t) ... yn k" ( t) ,
h i + . . . + k n = 2
t > t o, k = 1 . . . . . n,
with the condit ions
(i) Ip~l . . .h , ( / ) l < AR -(k'+'''+h"),
k = l . . . . . n, k i + . . . + k n > 2, A , R ~ + ;
(ii) qh(t) = o(1 ) , k = l . . . . . n, t - - ~ + ~ ;
(iii) 3 P 0 = lim P(t) , Po e I~. nxn, P( t ) = (Pla(t))n'n
t - -~+oo
(1)
(2)
Assume that the characteristic numbers Xk, k = 1 . . . . . n, of the matrix P0 possess the proper ty 11 + ~'h ] ~: 0,
k = l . . . . . n.
Inequali ty (2) guarantees the absolute and uniform convergence of the series in system (1) in any domain of the
form
R 0 ~ ~ + .
Furthermore, we assume that the functions qh(t), Phi(t), and pkkx ...h., k, i = 1 . . . . . n, kl +. . . + kn > 2, admit , in
a certain sense (see Definit ions 6 and 7), formal expansions into series of the form
Chth2 ...kpfl kl ( t ) f ~ 2 ( t ) . . . f ; P (t) ,
k I + . . .+kp = 0
%k2...kp ~ (I;, (3)
Odessa University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 672--677, May, 1997. Original article
submitted May 15, 1995.
0041-5995/97/4905-0747 $18.00 �9 1998 Plenum Publishing Corporation 747
748
where f t (t), k = 1 . . . . . p , is a fixed set of functions such that
Aifk(t) = o(1), AOfk(t) de=f fk( t ) ,
In what follows, we denote the set of functions fk( t ) , k = 1 . . . . . p, by ( f ) .
We rewrite system (1) as
AY(t) = Q ( t ) + P ( t ) Y ( t ) + ~F(t, Y( t ) ) .
Consider the series
A. V. KOSTIN AND I. V. SKRIPNIK
k = l . . . . . p, i = 0 , 1 , 2 . . . . .
(4)
s - I s-1 i lp
, (5>
s=0 i=0 i=0
where k = k o . . , ks-i ... lo . . . Is-1 and, for any fixed value s, the exponents ko . . . . . ks_l . . . . . lo . . . . . ls-I can be
integer nonnegative numbers satisfying the condition
ko + 2kl + . . . + sks_l + . . . + lo + 2ll + . . . + sls_l = s.
The coefficients ck are columns of the same dimension n. The numbers s are called the orders of the correspond-
ing terms in (5).
Definition 1. A vector function (p(t), t ~ N, t > to, which is a finite sum o f the type
Z * * n• 1
(p(t) = ckCSk(t ), c k ~ ~ , k = ko ... ks-l ... lo ... ls-1,
S=$ 0
where the terms have the same order so, is called a funct ion o f order so, which is denoted as fo l lows:
H(q) ( t ) ) = so.
Property 1. I f l-I(q0(t)) = so and c ~ (E, then H(cqo( t ) ) = so.
Property2. If H ( q o l ( t ) ) = s o and F l ( q ) 2 ( t ) ) = s o , then
l-I(qOl(t) + qOz(t)) = sO.
Property3. I f l ' I ( g l ( t ) ) = s l and H ( q o z ( t ) ) = s a , then
I I (qo l ( t )qo2( t ) ) = Sl + s2.
Definition 2. The fo l lowing series are called, respectively, the sum, difference, and product o f two formal
series ~ ; = o W s and ~ ; = o V s o f t ype (5 ) :
s=0 s=0 s=0
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nasplib_isofts_kiev_ua-123456789-1569772025-02-09T23:40:57Z On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations Об асимптотических разложениях o-решений в теории квазилинейных систем разностных уравнений Kostin, A.V. Skripnik, I.V. Статті We consider a quasilinear system of difference equations with certain conditions. We prove that there exists a formal partial o-solution of this system in the form of functional series of special type. We also prove a theorem on the asymptotic behavior of this solution. Досліджується квазілінійиа система різницевих рівнянь при певних умовах. Доводиться існування формального частинного о-розв'язку цієї системи у вигляді функціональних рядів спеціального типу. Доводиться також теорема про асимптотичний характер цього розв'язку. 1997 Article On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations / A.V. Kostin, I.V. Skripnik // Український математичний журнал. — 1997. — Т. 49, № 5. — С. 672–677. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/156977 517.949 en Український математичний журнал application/pdf Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Kostin, A.V. Skripnik, I.V. On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations Український математичний журнал |
| description |
We consider a quasilinear system of difference equations with certain conditions. We prove that there exists a formal partial o-solution of this system in the form of functional series of special type. We also prove a theorem on the asymptotic behavior of this solution. |
| format |
Article |
| author |
Kostin, A.V. Skripnik, I.V. |
| author_facet |
Kostin, A.V. Skripnik, I.V. |
| author_sort |
Kostin, A.V. |
| title |
On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations |
| title_short |
On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations |
| title_full |
On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations |
| title_fullStr |
On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations |
| title_full_unstemmed |
On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations |
| title_sort |
on asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations |
| publisher |
Інститут математики НАН України |
| publishDate |
1997 |
| topic_facet |
Статті |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/156977 |
| citation_txt |
On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations / A.V. Kostin, I.V. Skripnik // Український математичний журнал. — 1997. — Т. 49, № 5. — С. 672–677. — англ. |
| series |
Український математичний журнал |
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