On the Asymptotic Behavior of Solutions of Differential Systems
There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let $ϕ (t)$ be a scalar continuous monotonically increasing positive function tend...
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509696132120576 |
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| author | Pham, Van Viet Vu, Tuan Фам, Ван В'єт Ву, Туан |
| author_facet | Pham, Van Viet Vu, Tuan Фам, Ван В'єт Ву, Туан |
| author_sort | Pham, Van Viet |
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| datestamp_date | 2020-03-18T19:59:02Z |
| description | There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions.
Let $ϕ (t)$ be a scalar continuous monotonically increasing positive function tending to ∞ as $t → ∞$. It is established that if all solutions of a differential system satisfy the inequality
$$\left\| {x(t;t_0 ,\;x_0 )} \right\| \leqslant M\frac{{\varphi (t_0 )}}{{\varphi (t)}}\quad \operatorname{for} \;all\quad t \geqslant t_0 ,\quad x_0 \in \left\{ {x:\left\| x \right\| \leqslant \alpha } \right\},$$
then the solution $x(t; t_0, x_0)$ of this differential system tends to 0 faster than $M\frac{{\varphi (t_0 )}}{{\varphi (t)}}$. |
| first_indexed | 2026-03-24T02:45:12Z |
| format | Article |
| fulltext |
UDC 517.9
Vu Tuan, Pham Van Viet (Hanoi Univ. Education, Vietnam)
ON THE ASYMPTOTIC BEHAVIOR
OF SOLUTIONS OF DIFFERENTIAL SYSTEMS
PRO ASYMPTOTYÇNU POVEDINKU
ROZV’QZKIV DYFERENCIAL|NYX SYSTEM
There are many studies on the asymptotic behavior of solutions of differential equations. In the present
paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of
solutions.
Let ϕ( )t be a scalar continuous monotonically increasing positive function tending to ∞ as t →
→ ∞. It is established that if all solutions of a differential system saisfy the inequality:
x t t x
t
t
M( ; , )
( )
( )
0 0
0≤
ϕ
ϕ
for all t t≥ 0 , x x x0 ∈ ≤{ }: α ,
then the solution x t t x( ; , )0 0 of this differential system tends to 0 faster that M
t
t
ϕ
ϕ
( )
( )
0 .
Asymptotyçnij povedinci rozv’qzkiv dyferencial\nyx rivnqn\ prysvqçeno çymalo doslidΩen\.
U danij roboti problemu rozhlqnuto z inßoho boku, a same, z toçky zoru ßvydkosti asymp-
totyçno] zbiΩnosti rozv’qzkiv.
Nexaj ϕ( )t skalqrna neperervna monotonno zrostagça dodatna funkciq, wo prqmu[ do ∞
pry t → ∞. Vstanovleno, wo qkwo vsi rozv’qzky dyferencial\no] systemy zadovol\nqgt\ ne-
rivnist\
x t t x
t
t
M( ; , )
( )
( )
0 0
0≤
ϕ
ϕ
dlq vsix t t≥ 0 , x x x0 ∈ ≤{ }: α ,
to rozv’qzok x t t x( ; , )0 0 ci[] dyferencial\no] systemy prqmu[ do 0 ßvydße, niΩ M
t
t
ϕ
ϕ
( )
( )
0 .
1. Introduction and preliminaries. Let I denote the interval a ≤ t < ∞ , a ≥ 0, and
R
n denote Euclidean n-space. For x n∈R , let x be the Euclidean norm of x.
We shall denote by Sα the set of x such that x ≤ α .
Consider a system of differential equations [1 – 9]
dx
dt
X t x= ( , ), X t( , )0 0≡ , (1)
where X t x( , ) is defined on a region in I n× R and continuous in ( , )t x .
Moreover, suppose that X t x( , ) satisfies uniqueness condition of solution.
Throughout this paper, a solution passing through a point ( , )t x0 0 in I n× R will be
denoted by such form as x t t x( ; , )0 0 . We denote by C x0( ) the family of functions
which satisfy locally Lipschitz condition with respect to x, and assume that ϕ( )t is a
scalar continuous, monotonically increasing function in I, ϕ( )a ≥ 1, ϕ( )t → +∞ as
t → +∞ .
We have the following definitions:
Definition 1. The solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable if given
any ε > 0 and any t I0 ∈ , there exist δ δ ε= >( , )t0 0 such that if x0 < δ ,
then x t t x( ; , )0 0 < ε ϕ
ϕ
( )
( )
t
t
0 for all t t≥ 0 .
© VU TUAN, PHAM VAN VIET, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 137
138 VU TUAN, PHAM VAN VIET
Definition 2. The solution x t( ) ≡ 0 of (1) is ϕ-uniform asymptotically stable
if δ in Definition 1 is independent of t0 .
Definition 3. The solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable in the
large if for any α > 0, there exist K( )α > 0 such that if x S0 ∈ α , then
x t t x( ; , )0 0 < K
t
t
x( )
( )
( )
α ϕ
ϕ
0
0 for all t t≥ 0 .
Let V t x( , ) be a continuous scalar function defined on an open set S and let
V t x( , ) ∈ C x0( ). This function is called Liapunov function [9]. We also define the
function:
′ = + +( ) −[ ]
→ +
V t x
h
V t h x hX t x V t x
h
( )( , ) lim , ( , ) ( , )1
0
1 .
Let x t( ) be a solution of (1) that stays in S. Denote by ′( )V t x t, ( ) the upper right-
hand derivative of V t x t, ( )( ) , i.e.,
′V t x( )( , )1 = lim , ( ) ( , )
h h
V t h x t h V t x
→ +
+ +( ) −[ ]
0
1 .
We have [9]
′ = ′V t x V t x( )( , ) ( , )1 .
In the case where V t x( , ) has continuous partial derivatives of the first order, it is
evident that
′ = ∂
∂
+ ∂
∂
⋅V t x V
t
V
t
X t x( )( , ) ( , )1 ,
where “.” denotes the scalar product.
2. Sufficient conditions.
Theorem 1. Suppose that there exists a Liapunov function V t x( , ) defined on I,
x H< , which satisfies the following conditions:
(i) x V t x≤ ( , ) and V t( , )0 0≡ ;
(ii) ′V t x( )( , )1 ≤ −λ( ) ( , )t V t x , where λ is a scalar continuous positive function
in I and
a
t dt
+∞
∫ = +∞λ( ) .
Then the solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable.
Proof. For any ε > 0 (ε < H ), t I0 ∈ , we can choose δ δ ε= ( , )t0 such that
x0 < δ implies V t x( , )0 0 < ε . Let x t t x( ; , )0 0 be a solution of (1) such that
x0 < δ . Applying Theorem 4.1 in [9], by (ii) we have
V t x t t x V t x d d
t
t
t
t
, ( ; , ) ( , ) exp ( ) exp ( )0 0 0 0
0 0
( ) ≤ −
< −
∫ ∫λ ξ ξ ε λ ξ ξ .
Denote ϕ( )t = exp ( )−( )∫a
t
dλ ξ ξ . Because of the feature of the function λ, we can
see that ϕ is continuous monotonically increasing function on I, ϕ( )a = 1, ϕ( )t →
→ +∞ as t → +∞ and the above estimate leads to
V t x t t x V t x
t
t
t
t
, ( ; , ) ( , )
( )
( )
( )
( )0 0 0 0
0 0( ) ≤ <ϕ
ϕ
ε ϕ
ϕ
.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL SYSTEMS 139
Thus, by (ii) we obtain
x t t x V t x t t x
t
t
( ; , ) , ( ; , )
( )
( )0 0 0 0
0≤ ( ) < ε ϕ
ϕ
for all t t≥ 0 if x0 < δ .
That is, the solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable.
The theorem is proved.
Theorem 2. Suppose that there exists a Liapunov function V t x( , ) defined on I,
x H< , which satisfies the following conditions:
(i) x V t x b x≤ ≤ ( )( , ) , where b r CIP( ) ∈ [9, 7];
(ii) ′V t x( )( , )1 ≤ −λ( ) ( , )t V t x , where λ is the function defined in Theorem 1.
Then the solution x t( ) ≡ 0 of (1) is ϕ-uniform asymptotically stable.
Proof. For a given ε > 0, we can choose δ ε( ) > 0 so that δ ε ε( ) ( )< −b 1 and the
remainder of the proof can be verified by the same argument as in Theorem 1.
Corollary 1. If λ( )t c≡ ( )c > 0 , then x t( ) ≡ 0 of (1) is exponential-
asymptotically stable, that is x t t x( ; , )0 0 ≤ εe c t t− −( )0 for all t t≥ 0 (see
Definition 7.8 in [9]).
Theorem 3. Suppose that there exists a Liapunov function V t x( , ) defined on
I n× R satisfying the following conditions:
(i) x V t x≤ ( , ) and V t( , )0 0≡ ;
(ii) ′ ( ) ≤V t t x( ) , ( )1 0ϕ , where ϕ( )t is a scalar monotonically increasing,
differentiable function on I, ϕ( )a ≥ 1 and ϕ( )t → +∞ as t → +∞ .
Then the solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable.
Proof. For any ε > 0 and a fixed t I0 ∈ , we can find a number δ δ ε= ( , )t0
such that x0 < δ implies V t t x0 0 0, ( )ϕ( ) < ε. Under assumption that x t t x( ; , )0 0 is
a solution of (1) satisfying x0 < δ , we have x t t x( ; , )0 0 < ε for all t t≥ 0 .
Indeed, if there exists t t1 0> such that x t t x( ; , )1 0 0 ≥ ε, by (i) and (ii) we obtain
ε ≤ x t t x( ; , )1 0 0 ≤ ϕ( ) ( ; , )t x t t x1 1 0 0 ≤ V t t x t t x1 1 1 0 0, ( ) ( ; , )ϕ( ) ≤ V t t x0 0 0, ( )ϕ( ) < ε.
This is a contradiction.
On the other hand, conditions (i), (ii) imply:
ϕ( ) ( ; , )t x t t x0 0 ≤ V t t x t t x, ( ) ( ; , )ϕ 0 0( ) ≤ V t t x0 0 0, ( )ϕ( ) < ε.
Thus, we have x t t x( ; , )0 0 < ε
ϕ( )t
≤ ε ϕ
ϕ
( )
( )
t
t
0 for all t t≥ 0 if x0 < δ .
This shows that the solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable.
Theorem 4. Suppose that there exists a Liapunov function V t x( , ) defined on
I n× R which satisfies the following conditions:
(i) V t( , )0 0≡ ;
(ii) ϕ( )t x ≤ V t x( , ) , where ϕ( )t is a continuous monotonically increasing
function on I, ϕ( )a ≥ 1 and ϕ( )t → +∞ as t → +∞ ;
(iii) ′ ( ) ≤V t x( ) ,1 0 .
Then the solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable.
Proof. The proof can be given by the same idea as in the proof of Theorem 3.
Theorem 5. Suppose that there exists a Liapunov function V t x( , ) defined on
I n× R which satisfies the following conditions:
(i) x V t x K x≤ ≤( , ) ( )α for x S∈ α , K( )α is a positive number;
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
140 VU TUAN, PHAM VAN VIET
(ii) ′V t x( )( , )1 ≤ −λ( ) ( , )t V t x , where λ is the function defined in Theorem 1.
Then the solution x t( ) ≡ 0 of (1) is ϕ-asymptotically stable in the large.
The proof can be given by the same idea as in the proof of Theorem 11.6 in [9].
Example. Consider the equation
′ = −x
t
x1
2
,
(2)
′ = − −y
t
y e
t
x y
t1
2
2
2 , t ≥ 1.
Let V t x y( , , ) = x y2 2
1
2+( ) , then V t x y( , , ) is a Liapunov function defined on
1 2≤ < ∞{ } ×t R , which satisfies condition (i) of Theorem 5; condition (ii) is also
satisfied because:
′V t x y( )( , , )2 = 1
2
2 2 2 2
1
2xx yy x y′ + ′( ) +( )− =
= − − −
+( )−1
2
1
2
12 2
2
2 2 2 2
1
2
t
x
t
y
t
e
t
x y x y
t
≤
≤ − +( ) +( )−1
2
2 2 2 2
1
2
t
x y x y = − +( )1
2
2 2
1
2
t
x y = − 1
2 t
V t x y( , , ).
Then, for a given α > 0 we have z t t z( ; , )0 0 ≤ e
e
z
t
t
0
0 for all t t≥ ≥0 1,
z S0 ∈ α , where z t t z( ; , )0 0 = colon x t t x y t t y( ; , ), ( ; , )0 0 0 0( ) , and z0 = colon x y0 0,( ) .
Thus, the solution x t( ) ≡ 0 of equation (2) is e t -asymptotically stable in the
large.
3. Converse theorems on ϕϕϕϕ-asymptotic stability. Let us begin with converse
theorems on ϕ-asymptotic stability of linear systems. Consider the system
dx
dt
A t x= ( ) , (3)
where A t( ) is continuous n n× matrix on I.
Theorem 6. Suppose that there exists K ≥ 1 satisfying the following condition:
x t t x K
t
t
x( ; , )
( )
( )0 0
0
0≤ ϕ
ϕ
, (4)
where x t t x( ; , )0 0 is a solution of (3), ϕ( )t is a function defined as in the
Theorem 3 and ′ >ϕ ( )t 0 on I.
Then there exists a function V t x( , ) defined on I n× R which satisfies the
following conditions:
(i) x V t x K x≤ ≤( , ) ;
(ii) V t x V t x K x x( , ) ( , )
* *
− ≤ − ;
(iii) ′ ≤ −V t x t V t x( )( , ) ( ) ( , )3 λ , λ ϕ
ϕ
( )
( )
( )
t
t
t
=
′
for all t I∈ ;
(iv) ′ ( ) ≤V t t x( ) , ( )3 0ϕ .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL SYSTEMS 141
Proof. Put V t x x t t x
t
t
( , ) sup ; ,
( )
( )
= +( ) +
≥τ
τ ϕ τ
ϕ0
.
Due to (4), we can see that condition (i) will be held. In fact,
x ≤ sup ; ,
( )
( )τ
τ ϕ τ
ϕ≥
+( ) +
0
x t t x
t
t
≤ sup
( )
( )
( )
( )τ
ϕ
ϕ τ
ϕ τ
ϕ≥ +
+
0
K
t
t
x
t
t
= K x .
Moreover,
V t x V t x( , ) ( , )
*
− = sup ( ; , )
( )
( )
sup ( ; , )
( )
( )*
τ τ
τ ϕ τ
ϕ
τ ϕ τ
ϕ≥ ≥
+ + − + +
0 0
x t t x
t
t
x t t x
t
t
≤
≤ sup ( ; , ) ( ; , )
( )
( )*
τ
τ τ ϕ τ
ϕ≥
+ − + +
0
x t t x x t t x
t
t
= sup ( ; , )
( )
( )*
τ
τ ϕ τ
ϕ≥
+ − +
0
x t t x x
t
t
≤
≤ sup
( )
( )
( )
( )*
τ
ϕ
ϕ τ
ϕ τ
ϕ≥ +
− +
0
K
t
t
x x
t
t
= K x x− * .
Thus, condition (i) is satisfied.
The proof of the continuity of V t x( , ) can be performed by the same method used
in the proof of Theorem19.1 in [9].
Now, we shall prove (iii). Let x x t h t x* ; ,= +( ), h > 0. Then we have
V t h x( , )
*
+ = sup ( ; , )
( )
( )*
τ
τ ϕ τ
ϕ≥
+ + + + +
+0
x t h t h x
t h
t h
=
= sup ( ; , )
( )
( )
( )
( )τ
τ ϕ τ
ϕ
ϕ
ϕ≥
+ + + +
+0
x t h t x
t h
t
t
t h
=
= sup ( ; , )
( )
( )
( )
( )τ
τ ϕ τ
ϕ
ϕ
ϕ≥
+ +
+h
x t t x
t
t
t
t h
≤
≤ sup ( ; , )
( )
( )
( )
( )τ
τ ϕ τ
ϕ
ϕ
ϕ≥
+ +
+0
x t t x
t
t
t
t h
=
ϕ
ϕ
( )
( )
( , )
t
t h
V t x
+
,
which implies
1 1 1
h
V t h x V t x
h
t
t h
V t x( , ) ( , )
( )
( )
( , )*+ −[ ] ≤
+
−
ϕ
ϕ
.
Since the function ϕ( )t is differentiable, the above inequality implies
′ ≤ −V t x t V t x( )( , ) ( ) ( , )3 λ ,
where λ ϕ
ϕ
( )
( )
( )
t
t
t
= ′
, t I∈ . Condition (iii ) is proved.
Finally, we shall establish (iv). Since system (3) is linear, we have the relation
x t t t x; , ( )0 0 0ϕ( ) = ϕ( ) ; ,t x t t x0 0 0( ), whence we obtain
V t h t h x( , )*+ +( )ϕ = sup ( ; , ( ) )
( )
( )*τ
τ ϕ ϕ τ
ϕ≥
+ + + + + +
+0
x t h t h t h x
t h
t h
=
= sup ( ; , ( , , )) ( )
( )
( )τ
τ ϕ ϕ τ
ϕ≥
+ + + + + + +
+0
x t h t h x t h t x t h
t h
t h
=
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
142 VU TUAN, PHAM VAN VIET
= sup ( ; , ) ( )
τ
τ ϕ τ
≥
+ + + +
0
x t h t x t h = sup ( ; , ) ( )
τ
τ ϕ τ
≥
+ +
h
x t t x t =
= sup ( ; , ( ) )
( )
( )τ
τ ϕ ϕ τ
ϕ≥
+ +
h
x t t t x
t
t
≤
≤ sup ( ; , ( ) )
( )
( )τ
τ ϕ ϕ τ
ϕ≥
+ +
0
x t t t x
t
t
= V t t x, ( )ϕ( ) ,
which implies V t h t h x( , )*+ +( )ϕ – V t t x, ( )ϕ( ) ≤ 0 and then ′ ( )V t t x( ) , ( )7 ϕ ≤ 0.
Theorem 7. Suppose that there exists K ≥ 1 such that x t t x( ; , )0 0 ≤
≤ K
t
t
x
ϕ
ϕ
( )
( )
0
0 , where x t t x( ; , )0 0 is a solution of (3), ϕ( )t is a function defined
as in the Theorem 4.
Then there exists a function V t x( , ) defined on I n× R which satisfies the
following conditions:
V t( , )0 0≡ , ϕ( ) ( , )t x V t x≤ , V t x V t x K t x x( , ) ( , ) ( )* *− ≤ −ϕ
and
′ ≤V t x( )( , )7 0.
Proof. By the same idea used in the proof of Theorem 6, this theorem can be
proved by choosing V t x( , ) = sup ( ; , ) ( )
τ
τ ϕ τ
≥
+ +
0
x t t x t .
1. Kurzweil J., Vrkoc I. The converse theorems of Liapunov and Persidskii concerning the stability of
motion // Ibid. – 1957. – 7(82). – P. 254 – 274.
2. Levinson N. The asymptotic behavior of a system of linear differential equation // Amer. J. Math. –
1946. – 68. – P. 1 – 6.
3. Malkin I. G. Theory of stability of the motion. – Moscow: Gos. Izdat. Tekn.-Theoret. Lit., 1952.
(Engl. transl.: AEC tr-3352, 1958).
4. Martynyuk A. A. Stability analysis: nonlinear mechanic equations (stability and control: theory,
methods and applications). – London; New York, 1995. – Vol. 2.
5. Martynyuk A. A., Lakshmikantham V., Lecla S. Stability of motion: method of integral inequalities.
– Kiev: Naukova Dumka, 1989.
6. Martynyuk A. A. A survey of some classical and modern development of stability theory //
Nonlinear Analysis. – 2000. – 40. – P. 483 – 496.
7. Rouche N., Habets P., Laloy M. Stability theory by Liapunov’s direct method. – New York etc.:
Springer, 1977.
8. Rumyantsev V. V., Oziraner A. S. Partial stability and stabilization of motion [in Russian]. –
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Received 25.10.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
|
| id | umjimathkievua-article-3581 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:45:12Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c5/dd04b311416eb77824fefe0e716947c5.pdf |
| spelling | umjimathkievua-article-35812020-03-18T19:59:02Z On the Asymptotic Behavior of Solutions of Differential Systems Про асимптотичну поведінку розв'язків диференціальних систем Pham, Van Viet Vu, Tuan Фам, Ван В'єт Ву, Туан There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let $ϕ (t)$ be a scalar continuous monotonically increasing positive function tending to ∞ as $t → ∞$. It is established that if all solutions of a differential system satisfy the inequality $$\left\| {x(t;t_0 ,\;x_0 )} \right\| \leqslant M\frac{{\varphi (t_0 )}}{{\varphi (t)}}\quad \operatorname{for} \;all\quad t \geqslant t_0 ,\quad x_0 \in \left\{ {x:\left\| x \right\| \leqslant \alpha } \right\},$$ then the solution $x(t; t_0, x_0)$ of this differential system tends to 0 faster than $M\frac{{\varphi (t_0 )}}{{\varphi (t)}}$. Асимптогичній поведінці розв'язків диференціальних рівнянь присвячено чимало досліджень. У даній робогі проблему розглянуто з іншого боку, а саме, з точки зору швидкості асимптотичної збіжності розв'язків. Нехай $ϕ (t)$ скалярна неперервна монотонно зростаюча додатна функція, що прямує до ∞ при $t → ∞$. Встановлено, що якщо всі розв'язки диференціальної системи задовольняюсь нерівнісгь $$\left\| {x(t;t_0 ,\;x_0 )} \right\| \leqslant M\frac{{\varphi (t_0 )}}{{\varphi (t)}}\quad \operatorname{for} \;all\quad t \geqslant t_0 ,\quad x_0 \in \left\{ {x:\left\| x \right\| \leqslant \alpha } \right\},$$ то розв'язок $x(t; t_0, x_0)$ цієї диференціальної системи прямує до 0 швидше, ніж $M\frac{{\varphi (t_0 )}}{{\varphi (t)}}$. Institute of Mathematics, NAS of Ukraine 2005-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3581 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 1 (2005); 137–142 Український математичний журнал; Том 57 № 1 (2005); 137–142 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3581/3895 https://umj.imath.kiev.ua/index.php/umj/article/view/3581/3896 Copyright (c) 2005 Pham Van Viet; Vu Tuan |
| spellingShingle | Pham, Van Viet Vu, Tuan Фам, Ван В'єт Ву, Туан On the Asymptotic Behavior of Solutions of Differential Systems |
| title | On the Asymptotic Behavior of Solutions of Differential Systems |
| title_alt | Про асимптотичну поведінку розв'язків диференціальних систем |
| title_full | On the Asymptotic Behavior of Solutions of Differential Systems |
| title_fullStr | On the Asymptotic Behavior of Solutions of Differential Systems |
| title_full_unstemmed | On the Asymptotic Behavior of Solutions of Differential Systems |
| title_short | On the Asymptotic Behavior of Solutions of Differential Systems |
| title_sort | on the asymptotic behavior of solutions of differential systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3581 |
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