On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces
We present some conditions for the asymptotic equilibrium of nonlinear differential equations and, in particular, a linear inhomogeneous equation in Banach spaces. We also discuss analogous problems for a linear equation with unbounded operator. Some obtained results are applied to problems of asymp...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509227376705536 |
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| author | Nguyen, The Hoan Nguyen, Minh Man Nguyen, Sinh Bay Нгуєн, Зе Хоан Нгуєн, Мінь Ман Нгуєн, Сінь-Бей |
| author_facet | Nguyen, The Hoan Nguyen, Minh Man Nguyen, Sinh Bay Нгуєн, Зе Хоан Нгуєн, Мінь Ман Нгуєн, Сінь-Бей |
| author_sort | Nguyen, The Hoan |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:47:45Z |
| description | We present some conditions for the asymptotic equilibrium of nonlinear differential equations and, in particular, a linear inhomogeneous equation in Banach spaces. We also discuss analogous problems for a linear equation with unbounded operator. Some obtained results are applied to problems of asymptotic equivalence. |
| first_indexed | 2026-03-24T02:37:45Z |
| format | Article |
| fulltext |
UDC 517.9
Nguyen Sinh Bay, Nguyen The Hoan, Nguyen Minh Man (Hanoi, Vietnam)
ON THE ASYMPTOTIC EQUILIBRIUM
AND ASYMPTOTIC EQUIVALENCE
OF DIFFERENTIAL EQUATIONS IN BANACH SPACES
PRO ASYMPTOTYÇNU RIVNOVAHU
TA ASYMPTOTYÇNU EKVIVALENTNIST|
DYFERENCIAL|NYX RIVNQN| U BANAXOVYX PROSTORAX
We present some conditions for the asymptotic equilibrium of nonlinear differential equations in Banach
spaces, in particular, of the linear nonhomogenous equation. We also discuss analogous problems for the
linear equation with a nonbounded operator. Some obtained results are applied to problems of
asymptotic equivalence.
Navedeno deqki umovy asymptotyçno] rivnovahy nelinijnyx dyferencial\nyx rivnqn\ u banaxo-
vyx prostorax i, zokrema, linijnoho neodnoridnoho rivnqnnq. TakoΩ rozhlqnuto analohiçni py-
tannq dlq linijnoho rivnqnnq iz neobmeΩenym operatorom. Deqki otrymani rezul\taty zastoso-
vano do zadaç asymptotyçno] ekvivalentnosti.
1. Introduction. Asymptotic equilibrium and asymptotic equivalence of differential
equation systems in Rn were investigated in papers [1 – 4]. Some extensions for the
case of linear differential equations in Banach spaces were given in [5]. This paper
studies the same problem for nonlinear differential equations and, particularly, for the
nonhomogenous linear equation in Banach spaces E. We also discuss analogous
problems for the linear equation with a nonbounded operator. At last, we apply some
obtained results to problems of asymptotic equivalence.
2. Asymptotic equilibrium for nonhomogenous linear equations.
Definition 1. We say that the equation
ẋ f t x= ( , ) (1)
has an asymptotic equilibrium if every its solution has a finite limit at the infinity and
for each u E0 ∈ there exists a solution x t( ) of (1) such that x t( ) → u0 as t →
→ + ∞.
Here and in the following, E denotes a Banach space. I, L E( ), L a b E1 [ , ],( ) ,
C T E[ , ],0( ), … are well-known notations. For the linear equation
ẋ A t x= ( ) , (2)
where A t( ) is a linear bounded operator strongly continuous on [ , )0 ∞ , the following
statement was proved in [5].
Theorem 1. Equation (2) has a linear asymptotic equilibrium if and only if the
equation
dU
dt
A t U= ( ) (2′ )
considered in the space of all linear bounded operators L E( ) has a solution V t( )
which strongly tends to I as t → + ∞ and which has V t−1( ) ∈ L E( ) for t ≥
≥ t0 ≥ 0.
We consider now the nonhomogenous linear equation
˙ ( ) ( )x A t x f t= + , (3)
where f t( ) is a function continuous on 0, ∞[ ) . Suppose that equation (2) has a linear
asymptotic equilibrium and let V t( ) be the operator mentioned in Theorem 1. It is
easy to verify that
© NGUYEN SINH BAY, NGUYEN THE HOAN, NGUYEN MINH MAN, 2008
626 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ON THE ASYMPTOTIC EQUILIBRIUM AND ASYMPTOTIC EQUIVALENCE … 627
x t V t V t x V t V f d
t
t
( ) ( ) ( ) ( )= + ( ) ( )− −∫1
0 0
1
0
τ τ τ , t0 ≥ 0, (4)
is a solution of equation (3) which satisfies condition x t( )0 = x0 . Let f t( ) be such
that integral
0
1+∞ −∫ V f d( )τ τ τ( ) converges. By virtue of properties of V t( ) and from
the formula (4), we can state that there exists lim ( )
t
x t
→ +∞
: = x( )+∞ . We show now
that, for u E0 ∈ , the solution x t( ) of (3) satisfying condition x t( )0 = x0 with
x0 = V t u V t V f d
t
( ) ( )0 0 0
1
0
−
+∞
−∫ ( ) ( )τ τ τ (5)
tends to u0 as t → + ∞.
In fact, replacing the expression of x0 from (5) into (4), we obtain
x t V t u V t V f d
t
( ) ( ) ( )= ( ) ( )0
1−
+∞
−∫ τ τ τ .
Now, our statement is implied from the property of V t( ) and the Banach – Steinhass
theorem. Thus, we have prove the following statement.
Theorem 2. Let equation (2) have a linear asymptotic equilibrium and let
continuous function f t( ) be such that integral
0
1+∞ −∫ V t f t dt( ) ( ) converges.
Then equation (3) has an asymptotic equilibrium.
We note that
0
1+∞ −∫ V t f t dt( ) ( ) converges if V t−1( ) ≤ M ∀ t ≥ 0 (for some
M > 0) and f ∈ L E1 0, ,∞[ )( ). In particular, if the operator function A t( ) satisfies the
condition of Theorem 3 in [5], then equation (3) has an asymptotic equilibrium if
f ∈ L E1 0, ,∞[ )( ). In fact, in this case there exists a solution V t( ) of equation (2 ′ )
which tends to I by norm of the space L E( ) as t → + ∞. Consequently, it is easy to
verify that V t−1( ) ≤ M for t ≥ 0.
3. The case of nonlinear differential equations. We consider now the equation
˙ ( , )x f t x= , (6)
where f : 0, +∞[ ) × E → E. Further, we need the following statement (see [6]).
Proposition 1. If f : [ ]0, T × E → E is a compact operator, then the operator F :
[ ]0, T × D → C T E[ ]( )0, , , defined by the formula
( )( ) : , ( )Fx t x f x d
t
= + ( )∫0
0
τ τ τ , t T∈[ ]0, , x D∈ ,
is also a compact operator, where D is a set of continuous functions x : [ ]0, T → E.
Theorem 3. Let the compact operator f t x( , ) satisfy the following conditions:
f t x g t h x( , ) ( )≤ ( ), ( , ) ,t x E∈ ∞[ ) ×0 ,
where
0
+∞
∫ g t dt( ) < + ∞; h u( ) is a positive continuous nondecreasing function such
that
u
du
h u
0
+∞
∫ = + ∞
( )
, u0 0> .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
628 NGUYEN SINH BAY, NGUYEN THE HOAN, NGUYEN MINH MAN
Then equation (6) has an asymptotic equilibrium.
Proof. Let x t( ) be an arbitrary solution of (6) satisfying the condition x t( )0 =
= x0 . Then
x t( ) = x0 +
t
t
f x d
0
∫ ( )τ τ τ, ( ) . (7)
Hence,
x t x g h x d
t
t
( ) ( ) ( )≤ + ( )∫0
0
τ τ τ .
According to the theorem about the integral inequality, we have then x t( ) ≤ y t( ),
where y t( ) is a solution of the problem
˙ ( ) ( )y g t h y= ,
(8)
y t x( )0 0= .
From (8) we have
x
y t
t
t
t
du
h u
g d g d
0 0 0
( )
( )
( ) ( )∫ ∫ ∫= ≤ < + ∞
+ ∞
τ τ τ τ .
This shows that y t( ) is upper bounded. Hence, x t( ) ≤ M for t ≥ t0 . Let now t1,
t2 > t0 satisfy the inequality
t
t
g d
h M
1
2
∫ <( )
( )
τ τ ε .
Then
x t x t( ) ( )1 2− =
t
t
f x d
1
2
∫ ( )τ τ τ, ( ) ≤
t
t
g h x d
1
2
∫ ( )( ) ( )τ τ τ ≤
≤ h M g d
t
t
( ) ( )
1
2
∫ τ τ < ε.
This means that there exists lim ( )
t
x t
→ +∞
. Let now u E0 ∈ be an arbitrary element of
E. Let x t( ) be a solution of (6) which tends to u0 as t → +∞ . Then
u x f x d
t
0 0
0
= + ( )
+∞
∫ τ τ τ, ( ) . (9)
From (7), (9) we obtain
x t u f x d
t
( ) , ( )= − ( )
+∞
∫0 τ τ τ . (10)
Thus, x t( ) is a solution of integral equation (10). Consequently, it remains only to
prove the existence of solutions for integral equation (10). For this purpose, we denote
by Ω the set of continuous functions x t( ) satisfying inequality x t( ) ≤ R for t ≥
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ON THE ASYMPTOTIC EQUILIBRIUM AND ASYMPTOTIC EQUIVALENCE … 629
≥ t0 ≥ 0, where R is large enough. Clearly, Ω is closed, bounded, and convex.
Define now a map F by
( )( ) : , ( )Fx t u f x d
t
= − ( )
+∞
∫0 τ τ τ, x ∈Ω , t t≥ 0 ,
t0 is large enough,
( )( ) , ( ) ( ) ( )Fx t u f x d u h R g d
t t
≤ + ( ) ≤ +
+∞ +∞
∫ ∫0 0
0 0
τ τ τ τ τ .
We choose R > 2 u0 and t0 be large enough such that
t
g t dt
0
+∞
∫ ( ) < R
h R2 ( )
.
Then ( )( )Fx t ≤ R. This shows that F : Ω → Ω .
We prove now that F is a compact operator. In fact,
( )( ) , ( ) , ( ) ( )( ) ( )( )Fx t u f x d f x d Gx t Hx t
t
T
T
= − ( ) − ( ) = +∫ ∫
+∞
0 τ τ τ τ τ τ , (11)
where
( )( ) : , ( )Gx t u f x d
t
T
= − ( )∫0 τ τ τ , t ≥ t0 ,
( )( ) : , ( )Hx t f x d
T
= − ( )
+∞
∫ τ τ τ .
Choosing T > t0 such that
T
g d
h R
+∞
∫ <( )
( )
τ τ ε
4
we get
( )( ) ( ) ( ) ( ) ( )Hx t h x g d h R g d
T T
≤ ( ) < <
+∞ +∞
∫ ∫τ τ τ τ τ ε
4
.
By proposition mentioned above, operator G is compact. Consequently, sequence
( )( )Gx tn{ } contains a subsequence ( )( )Gx tnj{ } which converges. This means that
there exists a number K > 0 such that
Gx t Gx tn nj j p( ) − ( ) <
+
( ) ( ) ε
2
∀ j > K, p ∈ N.
From (11), we obtain
Fx t Fx tn nj j p( ) − ( ) < + =
+
( ) ( ) ε ε ε
2 2
∀ t ≥ t0 .
This shows that F : Ω → Ω is a compact operator. According to the Schauder
theorem, there exists an element x ∈ Ω such that x = F x( ) or
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
630 NGUYEN SINH BAY, NGUYEN THE HOAN, NGUYEN MINH MAN
x t u f x d
t
( ) ( , ( ))= −
+∞
∫0 τ τ τ .
It is easy to verify that x t( ) is a solution of (6) which tends to u0 as t → + ∞.
Theorem is completely proved.
Corollary 1. If the compact operator f t x( , ) satisfies conditions
f t x g t x( , ) ( )≤ α , 0 < α ≤ 1,
t
g t dt
0
+∞
∫ < + ∞( ) , t0 ≥ 0,
then equation (6) have an asymptotic equilibrium.
Theorem 4. Let the compact operator f t x( , ) satisfy the condition
f t x f t y g t h x y( , ) ( , ) ( )− ≤ −( ), x, y ∈ E, t ≥ 0,
where
0
+∞
∫ < + ∞g t dt( )
and the positive continuous and nondecreasing function h u( ) satisfies the condition
u
du
h u
0
+∞
∫ = + ∞
( )
, u0 > 0.
Then equation (6) has an asymptotic equilibrium.
The proof of this theorem is analogous to that of Theorem 3.
4. The case of linear equations with nonbounded linear operator. In this
section, we consider the equation
˙ ( )x A t x= (12)
in the Hilbert space H. A t( ) is a linear operator defined in D A( ) ⊆ H . We suppose
that D A( ) does not depend on t ∈ 0, +∞[ ) and that D A( ) is everywhere dense in
H. Moreover, we suppose that the Cauchy problem x( )0 = x0 , x0 ∈ D A( ) , has a
solution defined on 0, +∞[ ).
Theorem 5. Let, for each h ∈ D A( ) , A t h( ) ∈ L1 0, +∞[ ) and let the operator
A t( ) be self-adjoint. Then every bounded solution of equation (12) has a weak finite
limit at the infinity. Moreover, if the inclusion A t h( ) ∈ L1 0, +∞[ ) is uniform for
h ∈ S( , )0 1 ∩ D A( ) (see [5]), then every bounded solution of (12) has a strong finite
limit at the infinity.
Proof. Let x t( ) be any bounded solution of (12), i.e., there is M > 0 such that
x t( ) ≤ M ∀ t ≥ 0. Then, for any h ∈ D A( ) , we have
x t h( ), = 〈 〉x h0, +
t
t
A x h d
0
∫ ( ) ( ),τ τ τ = 〈 〉x h0, +
t
t
x A h d
0
∫ ( ), ( )τ τ τ , (13)
where x0 = x t( )0 . Hence,
x t x t h x A h d M A h d
t
t
t
t
( ) ( ), ( ), ( ) ( )1 2
1
2
1
2
− = ≤ <∫ ∫τ τ τ τ τ ε
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ON THE ASYMPTOTIC EQUILIBRIUM AND ASYMPTOTIC EQUIVALENCE … 631
if t1, t2 > T, where T is large enough. This shows that there exists lim ( ),
t
x t h
→ +∞
for all h ∈ D A( ) . Because of the denseness of D A( ) and the boundedness of x t( ),
we easily prove that this limit exists for all h ∈ H. Thus, the first statement is proved.
Since H is weakly complete, there exists h0 ∈ H such that
lim ( ),
t
x t h
→ +∞
= 〈 〉h h0, , h ∈ H.
By virtue of (13), we have
〈 〉h h0, = 〈 〉x h0, +
t
x A h d
0
+∞
∫ ( ), ( )τ τ τ. (14)
From (13), (14) we obtain
〈 〉x t h( ), = 〈 〉h h0, –
t
x A h d
+∞
∫ ( ), ( )τ τ τ, h ∈ D A( ) . (15)
Hence,
〈 〉x t h( ), ≤ 〈 〉h h0, + M A h d
t0
+∞
∫ ( )τ τ < 〈 〉h h0, + ε (16)
if t0 large enough. By virtue of (16) and the denseness of D A( ) , we have
x t h( ) ≤ +0 ε (17)
for t large enough. On the other hand, by theorem about the weak convergence,
h x t0 ≤ +( ) ε (18)
for t large enough. Inequalities (17), (18) show that lim ( )
t
x t
→ +∞
= h0 . Since x t( )
weakly tends to h0 , we obtain that lim ( )
t
x t
→ +∞
= h0 .
Theorem is proved.
We extend now the notion “solution”.
Definition 2. Let A t( ) = A t*( ), t ≥ t0 ≥ 0, x t( ) is said to be an extended
solution of the equation (12) if it satisfies the relation
d
dt
x t y x t A t y( ), ( ), ( )= ∀ ∈y D A( ), t ≥ t0 ≥ 0.
This definition of solution is given in [7].
Theorem 6. Let A t h( ) ∈ L1 0, +∞[ ) uniformly for h ∈ S D A( , ) ( )0 1 ∩ ; A t( ) =
= A t*( ). Then for each h D A0 ∈ ( ) there exists an extended solution x t( ) of
equation (16) such that
lim ( )
t
x t
→ +∞
= h0 . (19)
Proof. Consider the functional
ζ τ τ τ1 0 0( , ) , ( ) ( ),t h h h A x h d
t
= 〈 〉 −
+∞
∫ ,
where t ≥ t0 , h D A∈ ( ) , x t0( ) ≡ h0 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
632 NGUYEN SINH BAY, NGUYEN THE HOAN, NGUYEN MINH MAN
ζ1( , )t h ≤ h0 h +
t
x A h d
+∞
∫ 0( ) ( )τ τ τ ≤ h0 h q+( ) , (20)
where q =
t
A h d
0
+∞
∫ ( )τ τ . We choose t0 be large enough such that 0 < q < 1.
Inequality (20) shows that ζ1( , )t h is a linear continuous functional defined in D A( ) .
Because of the denseness of D A( ) on H, we can extend continuously this functional
in H with the norm preserving. We denote the extended functional also by ζ1( , )t h .
According to the Riesz theorem, there exists an element x t1( ) in H such that
ζ1( , )t h = x t h1( ), .
Clearly, x t1( ) ≤ ( )1 0+ q h . Consider now the functional
ζ2( , )t h : = 〈 〉h h0, –
t
x A h d
+∞
∫ 1( ), ( )τ τ τ , h ∈ D A( ) .
By the analogous proof, we obtain that ζ2( , )t h is a linear continuous functional
defined in H. Consequently,
ζ2( , )t h = x t h2( ), ,
where x t2( ) ≤ ( )1 2
0+ +q q h . Continuing this process, we have the linear
continuous functional
ζn t h( , ) : = 〈 〉h h0, –
t
nx A h d
+∞
−∫ 1( ), ( )τ τ τ , (21)
defined in D A( ) . The continuous extension of this functional has a form
ζn t h( , ) = x t hn( ), , (22)
x t q q h
h
qn
n( ) ≤ + + … +( ) ≤
−
1
10
0 . (23)
We show now that the sequence x tn( ){ } uniformly converges on t0, +∞[ ) . To prove
this statement, it suffices to show that
x t x t h qn n
n( ) ( )− ≤−1 0 . (24)
In fact, for n = 1 we have
x t x t1 0( ) ( )− ≤ sup ( ) ( ),
h
x t x t h
≤
−
1
1 0 =
sup ( ) ( ),
( , ) ( )h S D A
x t x t h
∈
−
0 1
1 0
∩
≤
≤ sup ( ) ( )
( , ) ( )h S D A t
A h x d h q
∈
+∞
∫ ≤
0 1
0 0
∩
τ τ τ ,
i.e., formula (24) is true for n = 1. Let us now assume that (24) is true for n. Then
x t x tn n+ −1( ) ( ) = sup ( ) ( ),
h
n nx t x t h
≤
+ −
1
1 =
=
sup ( ) ( ), ( )
( , ) ( )h S D A t
n nx x A h d
∈
+∞
−∫ −
0 1
1
0
∩
τ τ τ τ ≤
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ON THE ASYMPTOTIC EQUILIBRIUM AND ASYMPTOTIC EQUIVALENCE … 633
≤
t
n nx x A h d
0
1
+∞
−∫ −( ) ( ) ( )τ τ τ τ ≤ h qn
0
1+ ,
i.e., formula (15) is valid for n + 1. Since 0 < q < 1, inequality (24) shows that
sequence x tn( ){ } uniformly converges on t0, +∞[ ) .
Setting x t( ) = lim ( )
n
nx t
→ +∞
and tending n → + ∞ in (21), (22), we obtain
x t h h h x A h d
t
( ), , ( ), ( )= 〈 〉 −
+∞
∫0 τ τ τ , h ∈ D A( ) . (25)
This show that x t( ) is an extended solution of (12) and that x t( ) weakly tends to h0
as t → + ∞. We prove now that x t( ) strongly tends to h0 as t → + ∞. By virtue of
the uniform convergence of x tn( ){ }, it suffices to show that x tn( ) → h0 as t → + ∞.
In fact, we have
x t h h x A h d
h
q
A h dn
t
n
t
( ) , ( ) ( ) ( )− < ≤
−
+∞
−
+∞
∫ ∫0 1
0
0 0
1
τ τ τ τ τ .
Hence,
x t h
h q
qn( ) − ≤
−0
0
1
.
Since q → 0 as t → + ∞, our statement is proved.
5. Asymptotic equivalence. In this section, we consider equations
˙ ( )y A t y= , (26)
˙ ( ) ( , )x A t x f t x= + . (27)
Definition 3. Equations (26), (27) are said to be asymptotically equivalent if to
each solution x t( ) of (27) there exists a solution y t( ) of (26) such that
lim ( ) ( )
t
x t y t
→ +∞
− = 0 (28)
and conversely, to each solution y t( ) of (26) there exists a solution x t( ) of (27)
satisfying (28).
We assume throughout that A t( ) ∈ L E( ) for t ≥ 0 and A t( ) is strongly
continuous on 0, +∞[ ); f : 0, +∞[ ) × E → E is a continuous operator. We denote by
U t( ) the Cauchy operator of (26) satisfying U( )0 = I. Consider the equation
˙ ( ) , ( )z U t f t U t z= [ ]−1 . (29)
Theorem 7. Let equation (26) be stable and consequently U t( ) ≤ M.
Moreover, we suppose that equation (29) has an asymptotic equilibrium. Then
equations (26), (27) are asymptotically equivalent.
Proof. I.et x t( ) be an arbitrary solution of (27). It is easy to verify that z t( ) =
= U t x t−1( ) ( ) is a solution of (29). By virtue of the assumptions, there exists z+∞ =
= lim ( )
t
z t
→ +∞
. Setting y t( ) = U t z( ) +∞, we easily verify that y t( ) is a solution of (26)
which satisfies relation (28). Conversely, let now y t( ) be an arbitrary solution of (26)
satisfying condition y( )0 = y0. Then y t( ) = U t y( ) 0 . According to the assumption,
there exists a solution z t( ) of (29) such that lim ( )
t
z t
→ +∞
= y0. Let x t( ) = U t z t( ) ( ) . It
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
634 NGUYEN SINH BAY, NGUYEN THE HOAN, NGUYEN MINH MAN
is easy to verify that x t( ) is a solution of (27) and
lim ( ) ( )
t
x t y t
→ +∞
− ≤ M z t y
t
lim ( )
→ +∞
− 0 = 0.
Theorem is proved.
Remark. We have proved that, in the condition of stability of equation (26), the
asymptotic equilibrium of equation (29) is a sufficient condition for the asymptotic
equivalence of equations (26), (27). In general, this condition is not necessary.
Example. Consider the following example:
ẋ Ax B t x= + ( ) ,
ẏ Ay= ,
where
A =
−
−
1 0
0 2
, B t
e
e
t
t
( ) =
−
−
0
0
.
In this case,
U t
e
e
t
t
( ) =
−
−
0
0 2
, U t
e
e
t
t
− =
1
2
0
0
( ) , U t B t U t
e t
−
−
=
1
20
1 0
( ) ( ) ( ) .
By the Levison theorem (see [8, p. 159]), above equations are asymptotically
equivalent. However, equation
ż U t B t U t z= ( ) ( ) ( )−1
has not an asymptotic equilibrium. In fact, this equation can be written in the form
ż e zt
1
2
2= − ,
ż z2 1= .
Suppose that this system has an asymptotic equilibrium. Then for h0 = (1, 1), there
exists a solution z t z t1 2( ), ( )( ) such that z t1( ) → 1; z t2( ) → 1 as t → + ∞. Hence,
˙ ( )z t2 → 1 as t → + ∞. Therefore,
1 12− < < +ε ε˙ ( )z t ∀ ≥ >t T 0.
Consequently,
z t z T t T2 2 1( ) ( ) ( )( )> + − −ε .
Tending t → + ∞, we obtain a contracdition.
However, we have the following theorem.
Theorem 8. Let equation (26) be bistable (see [9, p. 165]). Then the
asymptotic equilibrium of equation (29) is a necessary and sufficient conditions for
the asymptotic equivalence of (26), (27).
Proof. According to assumptions, we have
U t M( ) ≤ , U t M− ≤1( ) ∀ ≥t 0.
Obviously, we remain to prove the necessary condition. Let y E0 ∈ and y t( ) =
= U t y( ) 0 be a solution of (26). According to our assumption, there exists a solution
x t( ) of (27) such that x t y t( ) ( )− → 0 as t → + ∞. Consider z t( ) = U t x t−1( ) ( ). It
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ON THE ASYMPTOTIC EQUILIBRIUM AND ASYMPTOTIC EQUIVALENCE … 635
is a solution of (29) and
z t y U t x t y t M x t y t( ) ( ) ( ) ( ) ( ) ( )− ≤ − ≤ −−
0
1 .
Therefore, z t( ) → y0 as t → + ∞. Let now z t( ) be an arbitrary solution of (29).
Then x t( ) = U t z t( ) ( ) is a solution of (27). According to our assumption, there exists
a solution y t( ) = U t y( ) 0 y y0 0=( )( ) of (26) such that x t y t( ) ( )− → 0 as t → + ∞.
Consequently, we have that z t y( ) − 0 ≤ U t−1( ) x t y t( ) ( )− ≤ M x t y t( ) ( )− → 0
as t → + ∞. This shows that z t( ) → y0 as t → + ∞. Thus, equation (29) has an
asymptotic equilibrium.
Theorem 9. Let equation (26) be bistable. The compact operator f t x( , )
satisfies conditions of Theorem 3 or Theorem 4. Then equations (26), (27) are
asymptotically equivalent.
In fact, in this case conditions of Theorem 3 or Theorem 4 are satisfied for
equation (29). Hence, it has an asymptotic equilibrium. By virtue of Theorem 6, we
obtain the assertion of this theorem.
Acknowledgement. The authors wish to thank the referees for their helpful
remarks.
1. Nguyen The Hoan. Some asymptotic behaviours of solutions to nonlinear system of differential
equation // Differents. Uravnenija. – 1981. – 12, # 4.
2. Voskresenski E. V. On Cezari problem // Ibid. – 1989. – 25, # 9.
3. Seah S. W. Existence of solutions and asymptotic equilibrium of multivalued differential system //
J. Math. Anal. and Appl. – 1982. – 89. – P. 648 – 663.
4. Seah S. W. Asymptotic equivalence of multivalued differential system // Boll. Unione math. ital. B.
– 1980. – 17. – P. 1124 – 1145.
5. Nguyen Minh Man, Nguyen The Hoan. On some asymptotic behaviour for solutions of linear
differential equations // Ukr. Math. J. – 2003. – 55, # 4.
6. Krasnoselski M. A., Krein C. G. On the theory of differential equations in Banach spaces // Trudy
Semin. Functional. Anal. – 1956. – 2.
7. Balakrishnan A. V. Introduction to theory of optimization in Hilbert spaces. – Moscow: Mir, 1974
(in Russian).
8. Demidovich B. P. Lectures on mathematical theory of stability. – Moscow: Nauka, 1967 (in
Russian).
9. Daletskii J. L., Krein M. G. Stability of solutions for differential equations in Banach spaces. –
Moscow: Nauka, 1970 (in Russian).
Received 23.08.05,
after revision — 22.01.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
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| id | umjimathkievua-article-3181 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:45Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/78/0d14a9ee613d76f1befc7ce23908a978.pdf |
| spelling | umjimathkievua-article-31812020-03-18T19:47:45Z On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces Про асимптотичну рівновагу та асимптотичну еквівалентність диференціальних рівнянь у банахових просторах Nguyen, The Hoan Nguyen, Minh Man Nguyen, Sinh Bay Нгуєн, Зе Хоан Нгуєн, Мінь Ман Нгуєн, Сінь-Бей We present some conditions for the asymptotic equilibrium of nonlinear differential equations and, in particular, a linear inhomogeneous equation in Banach spaces. We also discuss analogous problems for a linear equation with unbounded operator. Some obtained results are applied to problems of asymptotic equivalence. Наведено деякі умови асимптотичної рівноваги нелінійних диференціальних рівнянь у банахо-вих просторах і, зокрема, лінійного неоднорідного рівняння. Також розглянуто аналогічні питання для лінійного рівняння із необмеженим оператором. Деякі отримані результати застосовано до задач асимптотичної еквівалентності. Institute of Mathematics, NAS of Ukraine 2008-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3181 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 5 (2008); 626–635 Український математичний журнал; Том 60 № 5 (2008); 626–635 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3181/3109 https://umj.imath.kiev.ua/index.php/umj/article/view/3181/3110 Copyright (c) 2008 Nguyen The Hoan; Nguyen Minh Man; Nguyen Sinh Bay |
| spellingShingle | Nguyen, The Hoan Nguyen, Minh Man Nguyen, Sinh Bay Нгуєн, Зе Хоан Нгуєн, Мінь Ман Нгуєн, Сінь-Бей On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces |
| title | On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces |
| title_alt | Про асимптотичну рівновагу та асимптотичну еквівалентність диференціальних рівнянь у банахових просторах |
| title_full | On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces |
| title_fullStr | On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces |
| title_full_unstemmed | On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces |
| title_short | On the asymptotic equilibrium and asymptotic equivalence of differential equations in Banach spaces |
| title_sort | on the asymptotic equilibrium and asymptotic equivalence of differential equations in banach spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3181 |
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