Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces
In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations.
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| author | Dang, Dinh Chau Vu, Tuan Данг, Дінь Чау Ву, Туан |
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| description | In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations. |
| first_indexed | 2026-03-24T02:45:33Z |
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UDC 517.9
Dang Dinh Chau (Hanoi Univ. Sci., Vietnam),
Vu Tuan (Hanoi Pedagog. Univ., Vietnam)
ASYMPTOTICAL EQUIVALENCE OF TRIANGULAR
DIFFERENTIAL EQUATION IN HILBERT SPACES
ASYMPTOTYÇNA EKVIVALENTNIST|
DYFERENCIAL|NOHO RIVNQNNQ TRYKUTNO} FORMY
V�HIL|BERTOVYX PROSTORAX
In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert
spaces. Besides, we discuss the relation between properties of solutions of differential equations of
triangular form and those of truncated differential equations.
Vyvçeno umovy asymptotyçno] ekvivalentnosti dyferencial\nyx rivnqn\ u hil\bertovyx prosto-
rax. Rozhlqnuto takoΩ zv’qzok miΩ vlastyvostqmy rozv’qzkiv dyferencial\nyx rivnqn\ try-
kutno] formy ta nepovnyx dyferencial\nyx rivnqn\.
In a separable Hilbert space H, let us consider the differential equations
dx
dt
f t x= ( , ), (1)
dy
dt
g t y= ( , ) , (2)
where f : R+ × H → H and g : R+ × H → H are operators satisfying the conditions
f ( t, 0) = 0, g( t, 0) = 0 ∀ t ∈ R+ and all conditions of global theorem of existence and
uniqueness of solution (see [1, p. 187 – 189]).
Definition 1 [2 – 4]. Differential equations (1) and (2) are said to be
asymptotically equivalent if there exists a bijection between a set of solutions {x( t )}
of (1) and the one of {y( t )} of (2) such that
lim ( ) ( )
t
x t y t
→∞
− = 0 .
Let ei{ }∞1 be a basis of the separable Hilbert space H and let x x ei ii
= =
∞∑ 1
be
an element of H. Then the operator Pn : H → H defined as
P x x en i i
i
n
=
=
∑
1
is a projection on H. We introduce the notation Hn = Im Pn .
Suppose that J = {n1 , n2 , … , nj , …} is a strictly increasing sequence of natural
numbers (nj → ∞ as j → + ∞). Together with system (1), (2), we consider the
following systems of differential equations:
dx
dt
P f t P xm m= ( , ),
(3)
( I – Pm ) x = 0, m ∈ J,
dy
dt
P g t P ym m= ( , ),
(4)
( I – Pm ) y = 0, m ∈ J.
© DANG DINH CHAU, VU TUAN, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3 329
330 DANG DINH CHAU, VU TUAN
The stability of the solutions of differential equations (1) (or (2)) with the right-hand
side satisfying the conditions
f ( t, Pm x ) ≡ Pm f ( t, Pm x ) , (5)
g( t, Pm x ) ≡ Pm g( t, Pm x ) (6)
(∀ t ∈ R+ ∀ m ∈ J ∀ x ∈ H)
was already studied in [5, 6]. In the present paper, we give some new definitions of
asymptotical equivalence for these classes of differential equations and the
corresponding results.
Definition 2. Differential equations (1) a n d (2) are called asymptotically
equivalent by part with respect to the set J (or J -asymptotically equivalent) if
systems (3) and (4) are asymptotically equivalent for all m ∈ J.
Since (5) we have the following lemma.
Lemma 1. For any solution x ( t ) = x( t, t0 , Pm x0 ) , x 0 ∈ H, of differential
equation (1) the following relation:
x( t, t0 , Pm x0 ) = Pm x( t, t0 , Pm x0 )
will be held for all t ∈ R+, m ∈ J, x0 ∈ H.
Proof. For given m ∈ J, let us consider the differential equation
du
dt
f t P um= ( , ), u ∈ H, t ∈ R+. (7)
For ξ0 ∈ Pm H, the solution u( t ) = x( t, t0 , ξ0) of (7) is also a solution of the equation
u t f P u dm
t
t
( ) ( , ( ))= + ∫ξ τ τ τ0
0
. (8)
By virtue of (5) and the equation Pm ξ 0 = ξ0, we have
u t P P f P u dm m m
t
t
( ) ( , ( ))= + ∫ξ τ τ τ0
0
or
u t P f P u dm m
t
t
( ) ( , ( ))= +
∫ξ τ τ τ0
0
.
Hence,
u( t ) = Pm u( t ) ∀ t ∈ R+.
Consequently, we can rewrite (8) as follows:
u t f u d
t
t
( ) ( , ( ))= + ∫ξ τ τ τ0
0
.
This shows that u( t ) = x( t, t0 , ξ0) is a solution of (1), too.
Denoting by x( t ) = x( t, t0 , ξ0) the solution of differential equation (1) satisfying
the condition x( t0 ) = ξ0 , by uniqueness of solution, we have
x( t ) = u( t ) .
Hence, for x0 ∈ H, any solution x( t ) = x( t, t0 , Pm x0 ) , m ∈ J, of differential
equation (1) will satisfy the relation
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
ASYMPTOTICAL EQUIVALENCE OF TRIANGULAR DIFFERENTIAL EQUATION … 331
x( t, t0 , Pm x0 ) = Pm x ( t, t0 , Pm x0 ) (∀ t ∈ R+) .
The lemma is proved.
Remark 1. By virtue of Lemma 1, we can see that if conditions (5) and (6) are
satisfied, then all solutions of equations (3), (4) are solutions of equations (1), (2),
respectively. Therefore, from the asymptotical equivalence of system (1), (2), we can
deduce their J-asymptotical equivalent.
Now we consider the following linear differential equations:
dx
dt
Ax= , (9)
dy
dt
A B t y= +[ ]( ) , (10)
where A ∈ L ( H ) , B( t ) ∈ L ( H ) ∀ t ∈ [0, ∞) and
B d( )τ τ < ∞
∞
∫
0
. (11)
We assume that conditions (5), (6) are satisfied for these equations, that is
( A – Pm A ) Pm x = 0, (12)
(B( t ) – Pm B( t )) Pm x = 0 (13)
∀ m ∈ J ∀ x ∈ H.
Together with (9), (10), we consider also the sequences of truncated differential
equations
dx
dt
AP xm= ,
(14)
( I – Pm ) x = 0, m ∈ J,
dy
dt
A B t P ym= +[ ]( ) ,
(15)
( I – Pm ) y = 0, m ∈ J.
We denote by Xm ( t ) the Cauchy operator of (14) satisfying Xm (0) = Em and by
Ym ( t ) the Cauchy operator of (15) satisfying Ym ( t0 ) = Em , where Em is a unit
operator in Hm .
Lemma 2. If all solutions of equation (14) are bounded, then:
1. The Cauchy operator Xm ( t ) of (14) can be written in the form:
Xm ( t ) = Um ( t ) + Vm ( t ) ,
where Um ( t ) and Vm ( t ) : Hm → H m , so that there exist positive constants a m ,
bm , cm satisfying
U t a em m
b tm( ) ≤ −
∀ t ∈ R+, (16)
V t cm m( ) ≤ ∀ t ∈ R. (17)
2. The operators Fm : H → H defined by
F V t B Y P dm m m m
t
ξ τ τ τ ξ τ= −
∞
∫ ( ) ( ) ( )0
0
, ξ ∈ H,
are bounded and moreover the following inequality is valid:
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
332 DANG DINH CHAU, VU TUAN
|| Fm || ≤ αm < 1, t0 ≥ ∆ > 0.
Proof. Using condition (12), we can write the Cauchy operator of (14) in the form
X t em
AP tm( ) = . Since dim Im Pm < ∞ and Xm ( t ) is bounded uniformly in t for
every fixed m, using the same method in [2, p.160], we can prove Conclusion 1 of
Lemma 2.
Denote by Ym ( t ) the Cauchy operator of equation (14) satisfying Ym ( t0 ) = E. We
see that Ym ( t ) satisfies the equations
Y t X t t X t B Y dm m m m
t
t
( ) ( ) ( ) ( ) ( )= − + −∫0
0
τ τ τ τ ⇒
⇒ Y t X t t X t B Y dm m m m
t
t
( ) ( ) ( ) ( ) ( )≤ − + −∫0
0
τ τ τ τ.
By virtue of (16), (17), we have
Y t a a B Y dm m
t
t
( ) ( ) ( )≤ + ∫1 1
0
τ τ τ ,
where a1 = 2max( am , cm ) .
Due to the Gronwall – Bellman lemma and condition (11), we have
Y t a e a em
B d B dt
t
( )
( ) ( )
≤ ∫ ≤ ∫
∞
1 1
0 0
τ τ τ τ
.
Hence, there exists a number Km independent of t0 so that
|| Ym ( t ) || ≤ Km ∀ t ∈ R+. (18)
Moreover, for any αm ≤ 1, we can find a number ∆ > 0 so that
B d
c K
t
m
m m
( )τ τ α
0
+∞
∫ ≤ ∀ t0 > ∆.
This implies that
F V t B Y dm m m
t
≤ −
∞
∫ ( ) ( ) ( )0
0
τ τ τ τ ≤
≤ c K B dm m
t
m( )τ τ α
0
1
∞
∫ ≤ ≤ ∀ t0 > ∆.
The lemma is proved.
Theorem 1. Assume that, for any m ∈ J, the solutions of (14) are bounded.
Then differential equations (9) and (10) are J-asymptotically equivalent.
Proof. For each m ∈ J, we put
Qm x = ( I + Fm ) Pm x, x ∈ H.
Due to Lemma 2, the inequality || Fm || < 1 holds for t0 > ∆. Therefore, the operator
Qm is invertible.
Denoting η ξ0
1
0= −Qm , ξ0 ∈ Hm , m ∈ J, we consider the solutions x( t ) = x( t,
t0 , ξ0) of (14) and y( t ) = y( t, t0 , η0) of (15).
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
ASYMPTOTICAL EQUIVALENCE OF TRIANGULAR DIFFERENTIAL EQUATION … 333
It is clear that
x( t ) = Xm ( t – t0 )ξ0
and
y t X t t X t B y dm m
t
t
( ) ( ) ( ) ( ) ( )= − + −∫0 0
0
η τ τ τ τ .
By analogy with Lemma 2, since Hm is a finite dimensional subspace of H and
Xm ( t ) is bounded uniformly in t for every fixed m, by using the same method in [2,
p. 160], we can prove that
Xm ( t – t0 ) = Um ( t – t0 ) + Vm ( t – t0 ) ,
Vm ( t – τ ) = Xm ( t – t0 )Vm( t0 – τ ) .
From the definition of Qm , we have
ξ η η τ τ τ η τ0 0 0 0 0
0
= = + −
∞
∫Q V t B Y dm m
t
m( ) ( ) ( ) .
Hence,
x t X t t X t t V t B Y dm m m
t
m( ) ( ) ( ) ( ) ( ) ( )= − + − −
∞
∫0 0 0 0 0
0
η τ τ τ η τ =
= X t t V t B Y dm m
t
m( ) ( ) ( ) ( )− + −
∞
∫0 0 0
0
η τ τ τ η τ .
Consequently,
y t x t( ) ( )− =
= − − + −
∞
∫ ∫V t B Y d X t B y dm m
t
m
t
t
( ) ( ) ( ) ( ) ( ) ( )τ τ τ η τ τ τ τ τ0
0 0
⇔
⇔ y t x t( ) ( )− =
⇔ − − + −
∞
∫ ∫V t B Y t d U t B y dm m
t
m
t
t
( ) ( ) ( ) ( ) ( ) ( )τ τ η τ τ τ τ τ0
0 0
+
+ V t B y dm
t
t
( ) ( ) ( )−∫ τ τ τ τ
0
.
Since y( t ) = Ym ( t )η0 , we have
y t x t( ) ( )− =
= − − + −
∞
∫ ∫V t B y d U t B y dm
t
m
t
t
( ) ( ) ( ) ( ) ( ) ( )τ τ τ τ τ τ τ τ
0 0
+
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
334 DANG DINH CHAU, VU TUAN
+ V t B y dm
t
t
( ) ( ) ( )−∫ τ τ τ τ
0
⇔
⇔ y t x t( ) ( )− =
= − − + −
∞
∫ ∫V t B y d U t B y dm
t
m
t
t
( ) ( ) ( ) ( ) ( ) ( )τ τ τ τ τ τ τ τ
0
.
Using (16) – (18) and taking into account y( t ) = Ym ( t )η0 , we have
y t x t( ) ( )− ≤ a K e B d c K B dm m
b t
t
t
m m
t
mη τ τ η τ ττ
0 0
0
− −
∞
∫ ∫+( ) ( ) ( )
or
y t x t( ) ( )− ≤ M e B d M B db t
t
t
t
m
1 2
0
− −
∞
∫ ∫+( ) ( ) ( )τ τ τ τ τ ∀ t ≥ t0 ,
where
M1 = amKm|| η0 || , M2 = cmKm|| η0 || .
Then, for every positive number ε > 0, there exists a sufficiently large number t
and t > 2t0 such that the following inequalities are valid:
e B d e B d
M
b t
t
t b t
t
m
m
− − − ∞
∫ ∫≤ <( )
/
( ) ( )τ τ τ τ τ ε
0
2
2
13
,
B d
M
t
t
( )
/
τ τ ε
2 13∫ < , B d
M
t
( )τ τ ε∞
∫ <
3 2
.
Hence,
y t x t( ) ( )− ≤ M e B d e B db t
t
t
b t
t
t
m m
1
2
20
− − − −∫ ∫+
( )
/
( )
/
( ) ( )τ ττ τ τ τ +
+ M B d
t
2 3 3 3
( )τ τ ε ε ε ε
∞
∫ < + + = .
This means that
lim ( ) ( )
t
y t x t
→∞
− = 0.
By the uniqueness of solutions of differential equations (14) and (15), the map Qm is
bijective between two sets of solutions of equations (14) and (15).
The theorem is proved.
Lemma 3. If all solutions of the differential equations (9) are bounded, then:
1) there exists a positive number ∆ = ∆( α ) such that
|| Fm || ≤ α < 1 ∀ t0 ≥ ∆ ∀ m ∈ J;
2) {Fm} and {Qm} are convergent sequences of operators as m → ∞.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
ASYMPTOTICAL EQUIVALENCE OF TRIANGULAR DIFFERENTIAL EQUATION … 335
Proof. Due to the boundedness of all solutions of (9), there is a number β1 > 0
such that the Cauchy operator X( t ) of (9) satisfies the relation
|| X( t ) || ≤ β1 ∀ t ∈ R+.
If we denote by Y( t ) the Cauchy operator of (10) satisfying Y( t0 ) = E, we see
that Y( t ) satisfies the equation
Y t X t t X t B Y d
t
t
( ) ( ) ( ) ( ) ( )= − + −∫0
0
τ τ τ τ ⇒
⇒ Y t X t t X t B Y d
t
t
( ) ( ) ( ) ( ) ( )≤ − + −∫0
0
τ τ τ τ ⇒
⇒ Y t B Y d
t
t
( ) ( ) ( )≤ + ∫β β τ τ τ1 1
0
.
Due to the Gronwall – Bellman lemma and condition (11), there exists a number β2
independent of t0 and of m such that
|| Y( t ) || ≤ β2 ∀ t ∈ R+.
Consequently,
|| Xm ( t ) || ≤ β1 , || Ym ( t ) || ≤ β2 ∀ t ∈ R+ ∀ m ∈ J.
On the other hand, for any 0 < α < 1, we can find a number ∆ = ∆( α ) > 0
such that
B t d
t
( ) τ α
β β
0
1 2
∞
∫ ≤ < + ∞ ∀ t0 ≥ ∆.
Analogously, as in the proof of Lemma 2, we have
F V t B Y dm m m
t
≤ −
∞
∫ ( ) ( ) ( )0
0
τ τ τ τ ≤
≤ β β τ τ α1 2
0
1B d
t
( )
∞
∫ ≤ < ∀ m ∈ J ∀ t0 ≥ ∆.
By definition,
F V t B Y P dm m
t
mξ τ τ τ ξ τ= −
+∞
∫ ( ) ( ) ( )0
0
.
From (12) and (13) we can show that for all m, m + p ∈ J, p > 0,
X t t P X t t Pm p m m m+ − = −( ) ( )0 0ξ ξ ∀ ξ ∈ H,
Y t P Y t Pm p m m m+ =( ) ( )ξ ξ ∀ ξ ∈ H.
Hence,
F P F Pm p m m m+ =ξ ξ ∀ m, m + p ∈ J, p > 0.
We now prove the convergence of {Fm} . In fact, for all m, m + p ∈ J, p > 0, we
have
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
336 DANG DINH CHAU, VU TUAN
F F F P F Pm p m m p m p m m+ + +− = − = F P P F F Pm p m p m m p m m+ + +− + −( ) ( ) =
= F P P F P Pm p m p m m p m p m+ + + +− ≤ −( ) .
By definition, limm mP I→∞ = . Hence, by the boundedness of Fm , the above
yields that {Fm} is a Cauchy sequence, so {Fm} is convergent. This implies the
convergence of {Qm} .
Theorem 2. If all solutions of the differential equation (9) are bounded, then
equations (9) and (10) are asymptotically equivalent.
Proof. By virtue of Lemma 3, we can put
F F
m
m=
→∞
lim and Q Q
m
m=
→∞
lim .
Hence, Q = I + F. Since || Fm || ≤ α < 1 ∀ m ∈ J ∀ t0 ≥ ∆, we have
|| F || ≤ α < 1 ∀ t0 ≥ ∆.
Therefore, Q : H → H is an invertible operator.
By uniqueness of solutions of equations (9) and (10), we deduce that the map Q is
also bijective between two sets of solutions {x( t )} of (9) and {y( t )} of (10).
Let y0 = Q–1x0 and x( t ) = X( t – t0 )x0 , y( t ) = Y( t )y0 . Since
lim
m
mP y y
→∞
=0 0, lim
m
mQ y Qy x
→∞
= =0 0 0 ,
we can deduce that, for any arbitrarily given, ε < 0 there exists sufficiently large
m1 ∈ J such that, for all m ≥ m1 , we have
y t t y y t t P ym( ; , ) ( ; , )0 0 0 0 3
− < ε ,
x t t y x t t Q ym( ; , ) ( ; , )0 0 0 0 3
− < ε
for all t ≥ t0 .
By virtue of Theorem 1 and boundedness of all solutions of (9), we deduce that
differential equations (9) and (10) are J-asymptotically equivalent. Consequently,
there exists τ0 ∈ ( t0 , ∞) such that, for all t ≥ τ0 ,
x t t Q y y t t P ym m( ; , ) ( ; , )0 0 0 01 1 3
− < ε ,
where t0 is choosen sufficiently large such that
|| Fm || ≤ α < 1 ∀ m ∈ J.
Therefore,
y t t y x t t x( ; , ) ( ; , )0 0 0 0− ≤
≤ y t t y y t t P ym( ; , ) ( ; , )0 0 0 01
− + y t t P y x t t Q ym m( ; , ) ( ; , )0 0 0 01 1
− +
+ x t t Q y x t t xm( ; , ) ( ; , )0 0 0 01
− ≤
≤ ε ε ε ε
3 3 3
+ + = ∀ t ≥ τ0 .
This implies that
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
ASYMPTOTICAL EQUIVALENCE OF TRIANGULAR DIFFERENTIAL EQUATION … 337
lim ( ; , ) ( ; , )
t
x t t x y t t y
→∞
− =0 0 0 0 0.
The theorem is proved.
By virtue of Lemma 1, we can immediately obtain the following corollaries:
Corollary 1. Assume that all solutions of the differential equation (9) are
bounded. Then the differential equations (9) and (10) are asymptotically equivalent
if and only if they are J-asymptotically equivalent.
Corollary 2. If all solutions of differential equations (14) are uniformly bounded
for all m ∈ J, then differential equations (9) a n d (10) are asymptotically
equivalent.
Remark 2. In the case where the supposition of the boundedness of solutions of
differential equation (9) is not satisfied, by similar way as in [4], we can consider ψ-
asymptotical equivalence. Therefore, it is clear that, by choosing suitable in the Hilbert
space H, we can get the broadeness of the Levisions theorem for linear differential
equation with the operator A on the right-hand side of (9) being compact self-adjoint.
1. Barbashin E. A. Introduction to the stability theory. – Moscow: Nauka, 1967 (in Russian).
2. Demidivitch B. P. Lectures on the mathematical theory of stability. – Moscow: Nauka, 1967 (in
Russian).
3. Levinson N. The asymptotic behavior of systems of linear differential equations // Amer. J. Math. –
1946. – 63. – P. 1 – 6.
4. Nguyen The Hoan. Asymptotic equivalence of systems of differential equations // Izv. Akad. Nauk
AzSSR. – 1975. – # 2. – P. 35 – 40 (in Russian).
5. Dang Dinh Chau. Studying the instability of the infinite systems of differential equations by
general characteristic number // Sci. Bull. (Vestnik) Nat. Univ. Belarus. Ser. 1. Phys., Math. and
Mech. – 1983. – # 1. – P. 48 – 51 (in Russian).
6. Vu Tuan, Dang Dinh Chau. On the Lyapunov stability of a class of differential equations in Hilbert
spaces // Sci. Bull. Univ. Math. Ser. – Vietnam, 1996.
Received 12.12.2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
|
| id | umjimathkievua-article-3601 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:45:33Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/71/849e819e7f17639c9d56b326319dba71.pdf |
| spelling | umjimathkievua-article-36012020-03-18T19:59:42Z Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces Асимптотична еквівалентність диференціального рівняння трикутної форми в гільбертових просторах Dang, Dinh Chau Vu, Tuan Данг, Дінь Чау Ву, Туан In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations. Вивчено умови асимптотичної еквівалентності диференціальних рівнянь у гільбертових просторах. Розглянуто також зв'язок між властивостями розв'язків диференціальних рівнянь трикутної форми та неповних диференціальних рівнянь. Institute of Mathematics, NAS of Ukraine 2005-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3601 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 3 (2005); 329–337 Український математичний журнал; Том 57 № 3 (2005); 329–337 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3601/3933 https://umj.imath.kiev.ua/index.php/umj/article/view/3601/3934 Copyright (c) 2005 Dang Dinh Chau; Vu Tuan |
| spellingShingle | Dang, Dinh Chau Vu, Tuan Данг, Дінь Чау Ву, Туан Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces |
| title | Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces |
| title_alt | Асимптотична еквівалентність диференціального рівняння трикутної форми в гільбертових просторах |
| title_full | Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces |
| title_fullStr | Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces |
| title_full_unstemmed | Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces |
| title_short | Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces |
| title_sort | asymptotic equivalence of triangular differential equations in hilbert spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3601 |
| work_keys_str_mv | AT dangdinhchau asymptoticequivalenceoftriangulardifferentialequationsinhilbertspaces AT vutuan asymptoticequivalenceoftriangulardifferentialequationsinhilbertspaces AT dangdínʹčau asymptoticequivalenceoftriangulardifferentialequationsinhilbertspaces AT vutuan asymptoticequivalenceoftriangulardifferentialequationsinhilbertspaces AT dangdinhchau asimptotičnaekvívalentnístʹdiferencíalʹnogorívnânnâtrikutnoíformivgílʹbertovihprostorah AT vutuan asimptotičnaekvívalentnístʹdiferencíalʹnogorívnânnâtrikutnoíformivgílʹbertovihprostorah AT dangdínʹčau asimptotičnaekvívalentnístʹdiferencíalʹnogorívnânnâtrikutnoíformivgílʹbertovihprostorah AT vutuan asimptotičnaekvívalentnístʹdiferencíalʹnogorívnânnâtrikutnoíformivgílʹbertovihprostorah |