Memoirs on Differential Equations and Mathematical Physics
Volume 63, 2014, 79–104
Tatiyana Barinova and Alexander Kostin
ON ASYMPTOTIC STABILITY OF SOLUTIONS
OF SECOND ORDER LINEAR NONAUTONOMOUS
DIFFERENTIAL EQUATIONS
Abstract. The sufficient conditions for asymptotic stability of solutions
of second order linear differential equation
y ′′ + p(t)y ′ + q(t)y = 0
with continuously differentiable coefficients p : [0, +∞) → R and q : [0, +∞)
→ R are established in the case where the roots of the characteristic equation
λ2 + p(t)λ + q(t) = 0
satisfy conditions
Re λi (t) < 0 for t ≥ 0,
+∞
∫
Re λi (t) dt = −∞ (i = 1, 2).
t0
2010 Mathematics Subject Classification. 34D05, 34E10.
Key words and phrases. Second order differential equation, linear,
nonautonomous, asymptotic stability.
ÒÄÆÉÖÌÄ. ÌÄÏÒÄ ÒÉÂÉÓ ßÒ×ÉÅÉ ÃÉ×ÄÒÄÍÝÉÀËÖÒÉ ÂÀÍÔÏËÄÁÉÓÀÈÅÉÓ
y ′′ + p(t)y ′ + q(t)y = 0
ÖßÚÅÄÔÀà ßÀÒÌÏÄÁÀÃÉ p : [0, +∞) → R ÃÀ q : [0, +∞) → R ÊÏÄ×ÉÝÉÄÍÔÄÁÉÈ ÃÀÃÂÄÍÉËÉÀ ÀÌÏÍÀáÓÍÄÁÉÓ ÀÓÉÌÐÔÏÔÖÒÉ ÌÃÂÒÀÃÏÁÉÓ ÓÀÊÌÀÒÉÓÉ ÐÉÒÏÁÄÁÉ ÉÌ ÛÄÌÈáÅÄÅÀÛÉ, ÒÏÝÀ ÌÀáÀÓÉÀÈÄÁÄËÉ
λ2 + p(t)λ + q(t) = 0
ÂÀÍÔÏËÄÁÉÓ ×ÄÓÅÄÁÉ ÀÊÌÀÚÏ×ÉËÄÁÄÍ ÐÉÒÏÁÄÁÓ
Re λi (t) < 0, ÒÏÝÀ t ≥ 0,
+∞
∫
Re λi (t) dt = −∞ (i = 1, 2).
t0
On Asymptotic Stability of Solutions. . .
81
1. Introduction
This present paper is a continuation of the article Sufficiency conditions
for asymptotic stability of solutions of a linear homogeneous nonautonomous
differential equation of second-order.
In the theory of stability of linear homogeneous on-line systems (LHS)
of ordinary differential equations
dY
= P (t)Y, t ∈ [t0 ; +∞) = I,
dt
where the matrix P (t) is, in a general case, complex, of great importance
is the study of the LHS stability depending on the roots λi (t) (i = 1, n) of
the characteristic equation
det(P (t) − λE) = 0.
L. Cesáro [1] has considered the system of differential equations of n-th
order
[
]
dY
= A + B(t) + C(t) Y,
dt
where A is a constant matrix, whose roots of the characteristic equation
λi (i = 1, n) are distinct and satisfy the condition Re λi ≤ 0 (i = 1, n);
B(t) → 0 as t → +∞,
+∞
∫
dB(t) dt < +∞,
dt
+∞
∫
∥C(t)∥ dt < +∞,
t0
t0
the roots of the characteristic equation of the matrix A + B(t) have nonpositive material parts.
In his work, C. P. Persidsky [2] considers the case in which elements of
the matrix P (t) are the functions of weak variation, that is, every function
can be represented in the form
f (t) = f1 (t) + f2 (t),
where f1 (t) ∈ CI , and there exists lim f1 (t) ∈ R, and f2 (t) is such that
t→+∞
sup |f2 (t)| < +∞,
t∈I
lim f2′ (t) = 0,
t→+∞
and the condition Re λi (t) ≤ a ∈ R− (i = 1, n) is fulfilled.
N. Y. Lyaschenko [3] has considered the case Re λi (t) < a ∈ R− (i = 1, n),
t ∈ I,
sup ∥A′ (t)∥ ≤ ε.
t∈I
The case n = 2 is thoroughly studied by N. I. Izobov.
I. K. Hale [4] investigated asymptotic behavior of LHS by comparing the
roots of the characteristic equation with exponential functions
Re λi (t) ≤ −gtβ ,
g > 0, β > −1 (i = 1, n).
82
Tatiyana Barinova and Alexander Kostin
Then there are the constants K > 0 and 0 < ρ < 1 such that for solving
the system
dy
= A(t)y
dt
the estimate
ρg 1+β
∥y(t)∥ ≤ Ke− 1+β t ∥y(0)∥
is fulfilled.
In this paper we consider the problem of stability of a real linear homogeneous differential equation (LHDE) of second order
y ′′ + p(t)y ′ + q(t)y = 0
t∈I
(1)
provided the roots λi (t) (i = 1, 2) of the characteristic equation
λ2 + p(t)λ + q(t) = 0
are such that
+∞
∫
Re λi (t) dt = −∞ (i = 1, 2)
Re λi (t) < 0, t ∈ I,
(2)
t0
and there exist finite or infinite limits lim λi (t) (i = 1, 2). We have not
t→+∞
yet encountered with the problems in such a formulation. The case where
at least one of the roots satisfies the condition
+∞
∫
Re λi (t) dt < +∞ (i = 1, 2)
0<
t0
should be considered separately.
Under the term “almost triangular LHS” we agree to understand each
LHS
n
dyi (t) ∑
(3)
=
pik (t)yk (i = 1, n)
dt
k=1
with pik (t) ∈ CI (i, k = 1, n), which differs little from a linear triangular
system
n
dyi∗ (t) ∑
=
pik (t)yk∗ (i = 1, n),
(4)
dt
k=1
and the conditions either of Theorem 0.1 or of Theorem 0.2 due to A. V. Kostin [5] are fulfilled.
Theorem 1. Let the conditions
1) LHS (4) is stable when t ∈ I;
2) for a partial solution σi (t) (i = 1, n) of a linear inhomogeneous
triangular system
i−1
n
∑
dσi (t) ∑
=
|pik (t)| + Re pii (t)σi (t) +
|pik (t)|σk (t) (i = 1, n)
dt
k=1
k=i+1
(5)
On Asymptotic Stability of Solutions. . .
83
with the initial conditions σi (t0 ) = 0 (i = 1, n) the estimate of the
form 0 < σi (t) < 1 − γ (i = 1, n), γ = const, γ ∈ (0, 1) holds for all
t ∈ I.
Then the zero solution of the system (3) is a fortiori stable for t ∈ I.
Theorem 2. Let the system (3) satisfy all the conditions of Theorem 1 and,
moreover,
1) triangular linear system (4) is asymptotically stable for t ∈ I;
2)
lim σi (t) = 0 (i = 1, n).
t→+∞
Then the zero solution of the system (3) is asymptotically stable for t ∈ I.
Theorem 3. Let the system (3) satisfy all the conditions of Theorem 1 and,
moreover,
1) none of the functions
ψi (t) =
i−1
∑
|pik (t)| (i = 2, n) ̸≡ 0 for t ∈ I;
k=1
2)
lim σi (t) = 0 (i = 1, n).
t→+∞
Then the zero solution of the system (3) is stable for t ∈ I.
We will also use the following lemma [5]:
Lemma 1. If the functions p(t), q(t) ∈ CI , Re p(t) < 0, t ∈ I,
+∞
∫
Re p(τ ) dτ = −∞,
q(t)
= 0,
t→+∞ Re p(t)
lim
t0
then
∫t
Re p(τ ) dτ
∫t
t0
e
−
q(τ )e
∫τ
Re p(τ1 ) dτ1
τ0
dτ = o(1), t → +∞.
t0
Further, it will be assumed that all limits and characters o, O are considered as t → +∞.
In case equation (1) has the form
y ′′ + p(t)y = 0,
(6)
√
√
2
where p(t) ∈ CI , p(t) > 0 in I, λ1 (t) = −i p, λ2 (t) = i p, p = p(t), there
is the well-known I. T. Kiguradze’s theorem [6]:
Theorem 4. Let equation (6) be such that
′ − 32
p(+∞) = +∞, p p
= o(1),
(ln p)
−1
∫t
′ − 3 ′
(p p 2 ) dτ = o(1), t → t0 .
a
Then there take place the property of asymptotic stability.
84
Tatiyana Barinova and Alexander Kostin
2. The Main Results
2.1. Reduction of equation (1) to the system of the form (5). Consider the real second order LHDE (1):
y ′′ + p(t)y ′ + q(t)y = 0, t ∈ I,
where p(t), q(t) ∈ CI1 . Let y = y1 , y ′ = y2 . We reduce the equation to an
equivalent system
{
y1′ = 0 · y1 + 1 · y2 ,
(7)
y2′ = −q · y1 − p · y2 .
Consider the characteristic equation of LHS (6):
0 − λ
1 = 0 or λ2 + pλ + q = 0,
−q
−p − λ
and assume that
conjugate roots:
p2
2
(8)
− q < 0 at I. Then this equation has two complexλ1 = α − iβ,
λ2 = α + iβ,
where λi = λi (t) (i = 1, 2), α = α(t) ∈ CI1 , β = β(t) ∈ CI1 . Given (2), we
will consider the case
α(t) < 0,
+∞
∫
α(t) dt = −∞.
(9)
t0
There is the question on the sufficient conditions for stability of the trivial
solution of the system (7). Consider the following transformation for the
system (7):
(
)
(
)
1
1
z1 (t)
Y = C(t)Z, C(t) =
, Z=
,
λ1 (t) λ2 (t)
z2 (t)
where zi (t) are new unknown functions (i = 1, 2).
Z ′ = (C −1 AC − C −1 C ′ )Z,
det C(t) = λ2 (t) − λ1 (1),
(
)
1
λ2 (t) −1
−1
C (t) =
,
λ2 (t) − λ1 (1) −λ1 (t) 1
( ′
(
)
)
1
−λ1 (t) −λ′2 (t)
0
0
′
−1 ′
C (t) =
,
, C C =
λ′2 (t)
λ′1 (t) λ′2 (t)
λ2 (t) − λ1 (1) λ′1 (t)
(
)
λ1 (t)
0
C −1 AC =
.
0
λ2 (t)
On Asymptotic Stability of Solutions. . .
85
The system withe respect to new unknowns zi (t) (i = 1, 2) in a scalar
form is

(
)
λ′1 (t)
λ′2 (t)
′


z
(t)
=
λ
(t)
+
z1 (t) +
z2 (t),
1
 1
λ2 (t) − λ1 (t)
λ2 (t) − λ1 (t)
(10)
(
)

λ′1 (t)
λ′2 (t)

z2′ (t) = −
z1 (t) + λ2 (t) −
z2 (t).
λ2 (t) − λ1 (t)
λ2 (t) − λ1 (t)
It is not difficult to see that
λ′1 (t)
1 β′
λ′2 (t)
1 β′
Re
=−
, Re
=
,
λ2 (t) − λ1 (t)
2 β
λ2 (t) − λ1 (t)
2 β
√
1 ( β ′ )2 ( α ′ )2
λ′1 (t)
λ′2 (t)
h(t) = +
.
=
=
λ2 (t) − λ1 (t)
λ2 (t) − λ1 (t)
2
β
β
In accordance with Theorem 1 we write an auxiliary system of differential
equations:

(
1 β′ )

′

σ1 (t) + h(t)σ2 (t),
σ1 (t) = α −
2 β
(11)
(

1 β′ )

σ2′ (t) = h(t) + α −
σ2 (t).
2 β
Consider a particular solution with initial conditions σi (t0 ) = 0 (i = 1, 2).
This solution has the form

∫τ
∫t
′
′
∫t

− (α− 21 ββ ) dτ1
(α− 21 ββ ) dτ


τ
t
σ
h(τ )e 0
dτ,
e2 (t) = e 0



t0
(12)
∫t
∫τ
′
′
∫t

(α− 21 ββ ) dτ
− (α− 21 ββ ) dτ1


σ

e (t) = et0
h(τ )e
σ2 (τ )e τ0
dτ.

 1
t0
Assume also that there exists a finite or an infinite limit
α
lim
.
t→+∞ β
2.2. Various cases of behavior of the roots λi (t) (i = 1, 2). We consider
the following cases of behavior of the roots of the characteristic equation,
assuming that the condition (9) is fulfilled:
1) α(+∞) ∈ R− , β(+∞) ∈ R;
2) α = o(1), β = o(1),
3) α = o(1), β = o(1),
α
β
α
β
→ const ̸= 0;
→ ∞;
4) α = o(1), β(+∞) ∈ R \ {0};
5) α = o(1), β = o(1),
α
β
→ 0;
6) α(+∞) = −∞, β(+∞) = ∞,
α
β
→ ∞;
7) α(+∞) = −∞, β(+∞) ∈ R \ {0};
86
Tatiyana Barinova and Alexander Kostin
8) α(+∞) = −∞, β = o(1);
9) α(+∞) = −∞, β(+∞) = ∞,
α
β
→ const ̸= 0;
α
β
→ 0.
10) α = o(1), β(+∞) = ∞;
11) α(+∞) ∈ R− , β(+∞) = ∞;
12) α(+∞) = −∞, β(+∞) = ∞,
Theorems 5–16 correspond to the above cases 1)–12).
Theorem 5. Let the condition (9) be fulfilled and
α(+∞) ∈ R− ,
β(+∞) ∈ R.
Then the trivial solution of equation (1) is asymptotically stable.
This case is well known.
Theorem 6. Let the condition (9) and the following conditions
α = o(1),
β = o(1),
α′
= o(1),
α2
α
→ const ̸= 0,
β
β′
= o(1)
β2
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. We consider the system (10), auxiliary system of differential equations (11) and its particular solution (12).
In this case
√ ′
′
( ββ2 )2 + ( βα2 )2
h(t)
1
lim
lim
=
′ =
α
1 β′
t→+∞ α − 1 β
2 t→+∞
2 β
β − 2 β2
α′ β
1
1
|α′ | α
=
lim 2 =
lim
= 0.
2 t→+∞ β α
2 t→+∞ α2 β
Consequently, σ
e2 (t) = o(1), by Lemma 1. Further, we have
lim
t→+∞
h(t)
e2 (t) = 0.
′ σ
α − 12 ββ
Then σ
e1 (t) = o(1). Obviously, ψ(t) = h(t) ̸≡ 0 for t ∈ I. All the conditions
of Theorem 3 are fulfilled and thus Theorem 6 is complete. To obtain the
estimate of solutions yi (t) (i = 1, 2) we make in the system (10) the following
substitution:
δ
zi (t) = e
∫t
t0
α dτ
ηi (t) (i = 1, 2), δ ∈ (0, 1).
(13)
On Asymptotic Stability of Solutions. . .
87
Then the system (10) takes the form

(
)
λ′1 (t)
λ′2 (t)

η1′ (t) = λ1 (t) +
− δα η1 (t) +
η2 (t),

λ2 (t)−λ1 (t)
λ2 (t)−λ1 (t)
(
)

λ′1 (t)
λ′2 (t)

η2′ (t) = −
η1 (t) + λ2 (t) −
− δα η2 (t).
λ2 (t)−λ1 (t)
λ2 (t)−λ1 (t)
(14)
In accordance with Theorem 1, we write an auxiliary system of differential
equations:

(
1 β′ )

σ1′ (t) = (1 − δ)α −
σ1 (t) + h(t)σ2 (t),

2 β
(15)
(
 ′
1 β′ )

σ2 (t) = h(t) + (1 − δ)α −
σ2 (t).
2 β
It’s particular solution with the initial conditions σi (t0 ) = 0 (i = 1, 2) has
the form

) ∫t
)
∫τ (
∫t (
′
′

(1−δ)α− 12 ββ dτ1
−
(1−δ)α− 12 ββ dτ


τ
t
σ
h(τ )e 0
dτ,
e2 (t) = e 0



t
(16)
) ∫0t
)
∫τ (
∫t (
′
′

−
(1−δ)α− 12 ββ dτ1
(1−δ)α− 12 ββ dτ


σ

h(τ )e
σ2 (τ )e τ0
dτ.
e (t) = et0

 1
t0
It is not difficult to see that the replacement (13) does not affect the asymptotic stability. Taking into account the transformation C(t),
{
y1 (t) = z1 (t) + z2 (t),
=⇒
y2 (t) = λ1 (t)z1 (t) + λ2 (t)z2 (t).

t

( δ ∫ α dτ )


y1 (t) = o e t0
,
=⇒
∫t
∫t

(
δ α dτ
δ α dτ )


y2 (t) = o λ1 (t)e t0
+ λ2 (t)e t0
.
∫t
(
t0
y2 (t) = o e
∫t
(
t0
y2 (t) = o e
(
λ′ (t)
δα+ λ1 (t)
1
(
1
α δ+ α
)
∫t
dτ
λ′1 (t)
λ1 (t)
(
+e
)
λ′ (t)
δα+ λ2 (t)
2
t0
∫t
dτ
t0
+e
(
1
α δ+ α
)
dτ )
,
λ′2 (t)
λ2 (t)
)
dτ )
It is easy to see that
λ′1 (t)
λ′ (t)
α′ α + β ′ β
= Re 2
=
,
λ1 (t)
λ2 (t)
α2 + β 2
λ′ (t)
λ′ (t)
α′ β − αβ ′
I(t) = Im 1
= − Im 2
=
.
λ1 (t)
λ2 (t)
α2 + β 2
R(t) = Re
.
88
Tatiyana Barinova and Alexander Kostin
Then
1
λ′ (t)
α′ α + β ′ β
Re 1
= lim
=
t→+∞ α
λ1 (t) t→+∞ α(α2 + β 2 )
)
(
β′
α′
β2
α2
= 0,
= lim
+
α
3
t→+∞ 1 + ( β )2
(α
β) + β
α
1
λ′ (t)
α′ β − αβ ′
lim
Im 1
= lim
=
t→+∞ α
λ1 (t) t→+∞ α(α2 + β 2 )
( α′
)
β′
β2
α2
= lim
− α 2
= 0.
β
α
t→+∞
(β) + 1
β + α
lim
Thus
λ′i (t)
= o(α) (i = 1, 2).
λi (t)
Therefore,
(
δ
yi (t) = o e
∫t
t0
α dτ )
(i = 1, 2), δ ∈ (0, 1).
Theorem 7. Let the condition (9) and the following conditions
α
α = o(1), β = o(1),
→ ∞,
β
α′
β′
= o(β),
= O(1)
α
β2
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replace (13). We obtain the system
(14), an auxiliary system of differential equations (15) and its particular
solution (16).
In this case,
√ ′
′
( ββ )2 + ( αβ )2
h(t)
1
lim
lim
=
′ =
β′
t→+∞ (1 − δ)α − 1 β
2 t→+∞ α(1 − δ − 21 αβ
)
2 β
v
u(
β′
α′
)2 (
)2
u
1
αβ
αβ
t
=
lim
+
=
′
′
β
β
2 t→+∞
1 − δ − 12 αβ
1 − δ − 12 αβ
v
u(
β′ β
α′
)2 (
)2
u
1
β2 α
αβ
t
=
lim
+
= 0.
′
′
β
β
2 t→+∞
1 − δ − 12 ββ2 α
1 − δ − 12 ββ2 α
Consequently, σ
e2 (t) = o(1), by Lemma 1. Further, we have
lim
t→+∞
h(t)
(1 − δ)α −
1 β′
2 β
σ
e2 (t) = 0.
On Asymptotic Stability of Solutions. . .
89
Then σ
e1 (t) = o(1). This implies that Theorem 7 is valid. Moreover,
(
δ
∫t
α dτ )
t0
zi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Next,
β′
(
1
αβ
β2
R(t) = lim
+ α 3
lim
t→+∞ α + β
t→+∞ α
(
)
+
β
β
α
α′
)
α
β
= 0,
β′
)
(
1
αβ
β2
− α 2
= 0.
lim
I(t) = lim
α
2
t→+∞ α
t→+∞ ( ) + 1
(β) + 1
β
α′
Thus
λ′i (t)
= o(α) (i = 1, 2).
λi (t)
Therefore, just as in Theorem 6:
(
yi (t) = o e
δ
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
Theorem 8. Let the condition (9) and the following conditions
α = o(1),
β(+∞) ∈ R \ {0},
α′
= o(1),
α
β′
= o(α)
β
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replacement (13). We obtain the
system (14), an auxiliary system of differential equations (15) and its particular solution (16).
In this case,
√ ′
′
( ββ )2 + ( αβ )2
h(t)
1
lim
lim
=
′ =
β′
t→+∞ α − 1 β
2 t→+∞ α(1 − δ − 12 αβ
)
2 β
√ ′
β 2
α′ 2
α′ ( αβ
) + ( αβ
)
1
1
=
lim
=
lim
= 0.
′
β
2 t→+∞ 1 − δ − 12 αβ
2(1 − δ) t→+∞ αβ
Therefore, σ
e2 (t) = o(1), by Lemma 1. Further, we have
lim
t→+∞
h(t)
(1 − δ)α −
1 β′
2 β
σ
e2 (t) = 0.
Then σ
e1 (t) = o(1). This implies that Theorem 8 is valid. Moreover,
( δ ∫ ατ)
(i = 1, 2), δ ∈ (0, 1).
zi (t) = o e t0
t
90
Tatiyana Barinova and Alexander Kostin
Then
β′
)
( α
1
αβ
α
= 0,
lim
R(t) = lim
+ α 2
β
t→+∞ α +
t→+∞ α
(β) + 1
α β
′
( α
1
α
lim
I(t) = lim
−
t→+∞ α
t→+∞ α α + β
β
β′
αβ
′
α
β
+
)
β
α
= 0.
Thus
λ′i (t)
= o(α) (i = 1, 2).
λi (t)
Therefore, just as in Theorem 6,
(
δ
∫t
α dτ )
(i = 1, 2), δ ∈ (0, 1).
t0
yi (t) = o e
Theorem 9. Let the condition (9) and the following conditions
α = o(1),
β = o(1),
α′
= O(1),
α2
α
→ 0,
β
β′
= o(α)
β
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replacement (13). We obtain the
system (14), an auxiliary system of differential equations (15) and its particular solution (16).
In this case,
lim
t→+∞
h(t)
(1 − δ)α −
1
lim
2 t→+∞
=
β′
1 − δ − 12 αβ
α′ α′ α 1
1
=
lim lim 2 = 0.
=
2(1 − δ) t→+∞ αβ
2(1 − δ) t→+∞ α β
1 β′
2 β
=
√ ′
β 2
α′ 2
( αβ
) + ( αβ
)
Consequently, σ
e2 (t) = o(1), by Lemma 1. Further, we have
lim
t→+∞
h(t)
(1 − δ)α −
1 β′
2 β
σ
e2 (t) = 0.
Then σ
e1 (t) = o(1). This implies that Theorem 9 is valid. Moreover,
(
zi (t) = o e
δ
∫t
t0
α dτ )
(i = 1, 2), δ ∈ (0, 1).
On Asymptotic Stability of Solutions. . .
91
Then
β′
α
(
)
1
αβ
α2
lim
R(t) = lim
+
= 0,
2
t→+∞ 1 + ( β )2
t→+∞ α
(α
β) +1
α
′
( α
1
α2
lim
I(t) = lim
t→+∞ α
t→+∞ α +
β
β′
αβ
′
β
α
−
α
β
+
)
β
α
= 0.
Thus
λ′i (t)
= o(α) (i = 1, 2).
λi (t)
Therefore, just as in Theorem 6,
(
yi (t) = o e
δ
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
Theorem 10. Let the condition (9) and the following conditions
α(+∞) = −∞,
β(+∞) = ∞,
α′
= O(1),
α
α
→ ∞,
β
β′
= O(1)
β2
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the following replacement:
z1 (t)λ1 (t) = ξ1 (t),
z2 (t)λ2 (t) = ξ2 (t).
Then the system (10) takes the form

(
λ′ (t) )
λ′1 (t)


− 1
ξ1 (t)+
ξ1′ (t) = λ1 (t) +


λ2 (t) − λ1 (t) λ1 (t)





λ′2 (t)
λ1 (t)


+
ξ2 (t),

λ (t) − λ (t) λ (t)
2
1
2

λ′1 (t)
λ2 (t)
′


ξ
(t)
=
−
ξ1 (t)+

2

λ2 (t) − λ1 (t) λ1 (t)



(


λ′2 (t)
λ′ (t) )


+ λ2 (t) −
− 2
ξ2 (t).
λ2 (t) − λ1 (t) λ2 (t)
(17)
(18)
In accordance with Theorem 1, we write an auxiliary system of differential
equations:

)
(
1 β′

′

−
R(t)
σ1 (t) + h(t)σ2 (t),
σ
(t)
=
α
−
 1
2 β
(
)

1 β′

σ2′ (t) = h(t) + α −
− R(t) σ2 (t).
2 β
92
Tatiyana Barinova and Alexander Kostin
Consider a particular solution with the initial conditions σi (t0 ) = 0 (i =
1, 2):

) ∫t
)
∫τ (
∫t (
′
′

−
α− 21 ββ −R(t) dτ1
α− 21 ββ −R(t) dτ



h(τ )e τ0
dτ,
σ
e2 (t) = et0



t
) ∫0t
)
∫t (
∫τ (
′
′
1 β

α−
α− 21 ββ −R(t) dτ1
−R(t)
dτ
−

2 β



σ
e (t) = et0
h(τ )e
σ2 (τ )e τ0
dτ.

 1
t0
In this case,
β′
α
(
)
1
β2
α
lim
R(t) = lim
+
= 0.
2
t→+∞ β
t→+∞ β(1 + ( β )2 )
(α
β) +1
α
′
Then
lim
t→+∞
√ ′
′
( ββ2 )2 + ( βα2 )2
α−
h(t)
1
=
lim
=
1 β′
1
2 t→+∞ α
− R(t)
β − 2 β 2 − β R(t)
√
α′ ( β ′ β )2 ( α ′ )2
1
1
=
lim
+
=
lim
= 0.
2 t→+∞
β2 α
αβ
2 t→+∞ αβ
1 β′
2 β
Therefore, σ
e2 (t) = o(1), by Lemma 1. Next,
lim
t→+∞
α−
h(t)
σ
e2 (t) = 0.
− R(t)
1 β′
2 β
Then σ
e1 (t) = o(1). This implies that Theorem 10 is valid. To obtain
the estimate of solutions yi (t) (i = 1, 2), we make in the system (18) the
following replacement:
δ
ξi (t) = e
∫t
t0
α dτ
ηi (t) (i = 1, 2), δ ∈ (0, 1).
Then system (18) takes the form

)
(
λ′ (t)
λ′1 (t)


− 1
− δα η1 (t)+
η1′ (t) = λ1 (t) +


λ2 (t) − λ1 (t) λ1 (t)



′


λ1 (t)
λ2 (t)


η2 (t),
+

λ2 (t) − λ1 (t) λ2 (t)

λ′1 (t)
λ2 (t)
′


η
(t)
=
−
η1 (t)+

2

λ2 (t) − λ1 (t) λ1 (t)



(
)


λ′2 (t)
λ′ (t)


+ λ2 (t) −
− 2
− δα η2 (t).
λ2 (t) − λ1 (t) λ2 (t)
(19)
(20)
On Asymptotic Stability of Solutions. . .
93
In accordance with Theorem 1, we write an auxiliary system of differential
equations:

(
)
1 β′

′

− R(t) σ1 (t) + h(t)σ2 (t),
σ1 (t) = (1 − δ)α −
2 β
(21)
(
)
′

σ ′ (t) = h(t) + (1 − δ)α − 1 β − R(t) σ (t).

2
2
2 β
Let us consider a particular solution with the initial conditions σi (t0 ) = 0
(i = 1, 2):

)
∫t (
′

(1−δ)α− 12 ββ −R(t) dτ


t


×
σ
e2 (t) = e 0




)

∫τ (
′


(1−δ)α− 21 ββ −R(t) dτ1
−

∫t



× h(τ )e τ0
dτ,

t0
(22)
(
)
∫t
′


(1−δ)α− 21 ββ −R(t) dτ



σ
e1 (t) = et0
×





)

∫τ (
′


−
(1−δ)α− 21 ββ −R(t) dτ1
∫t


τ0

× h(τ )e
dτ.
σ2 (τ )e

t0
It is not difficult to see that the replacement (19) does not affect the stability.
At the same time,
(
δ
∫t
α dτ )
t0
ξi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, according to (17),
∫t
(
t0
zi (t) = o e
λ′ (t)
(δα− λ1 (t) ) dτ )
1
(i = 1, 2), δ ∈ (0, 1).
Further,
β′
α
(
1
β2
α
lim
R(t) = lim
+ α 3
t→+∞ α
t→+∞ α(1 + ( β )2 )
(
)
+
β
α
′
)
α
β
β′
= 0,
α
(
)
1
β2
α
lim
I(t) = lim
− α 2
= 0.
β
α
t→+∞ α
t→+∞ α( + )
(β) + 1
β
α
′
Consequently,
λ′i (t)
= o(α) (i = 1, 2)
λi (t)
and
(
zi (t) = o e
δ
∫t
t0
α dτ )
(i = 1, 2), δ ∈ (0, 1).
94
Tatiyana Barinova and Alexander Kostin
Then
{
y1 (t) = z1 (t) + z2 (t),
y2 (t) = λ1 (t)z1 (t) + λ2 (t)z2 (t)
(
δ
=⇒ yi (t) = o e
∫t
{
=⇒
α dτ )
y1 (t) = z1 (t) + z2 (t),
y2 (t) = ξ1 (t) + ξ2 (t)
=⇒
(i = 1, 2), δ ∈ (0, 1).
t0
Theorem 11. Let the condition (9) and the following conditions
α(+∞) = −∞,
′
α
= o(1),
α
β(+∞) ∈ R \ {0},
β′
= O(1)
β2
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replacement (17). We get the system (18). In the system (18) we make the replacement (19). We obtain
the system (20), an auxiliary system of differential equations (21) and its
particular solution (22).
In this case,
β′
α
(
)
1
β2
α
lim
R(t) = lim
+
= 0.
2
t→+∞ β
t→+∞ β(1 + ( β )2 )
(α
β) +1
α
′
Then
√ ′
′
( ββ2 )2 + ( βα2 )2
1
h(t)
=
lim
=
1 β′
1 β′
1
2 t→+∞ (1 − δ) α
(1 − δ)α − 2 β − R(t)
β − 2 β 2 − β R(t)
√
α′ ( β ′ β )2 ( α ′ )2
1
1
=
lim
+
=
lim
= 0.
2(1 − δ) t→+∞
β2 α
αβ
2(1 − δ) t→+∞ αβ
lim
t→+∞
Therefore, σ
e2 (t) = o(1), by Lemma 1. Next,
h(t)
σ
e2 (t) = 0.
′
t→+∞ (1 − δ)α − 1 β − R(t)
2 β
lim
Then σ
e1 (t) = o(1). This implies that Theorem 11 is valid. Thus
(
ξi (t) = o e
δ
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
Then, according to (17),
(
∫t
zi (t) = o et0
λ′ (t)
(δα− λ1 (t) ) dτ )
1
(i = 1, 2), δ ∈ (0, 1).
On Asymptotic Stability of Solutions. . .
95
Further,
β′
α
(
1
β2
α
lim
R(t) = lim
+ α 3
t→+∞ α
t→+∞ α(1 + ( β )2 )
(β) +
α
′
)
α
β
= 0,
β′
α
(
)
1
β2
α
I(t) = lim
lim
−
= 0.
α
β
2
t→+∞ α
t→+∞ α( α + )
(β) + 1
β
α
′
Thus
λ′i (t)
= o(α) (i = 1, 2)
λi (t)
and
(
∫t
δ
α dτ )
t0
zi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, just as in Theorem 10,
(
yi (t) = o e
δ
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
Theorem 12. Let the condition (9) and the following conditions
α(+∞) = −∞,
α′
= o(β),
α
β = o(1),
β′
= O(1)
β2
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replacement (17). We get the system (18). In the system (18) we make the replacement (19). We obtain
the system (20), an auxiliary system of differential equations (21) and its
particular solution (22).
In this case,
β′
)
(
1
αβ
β2
lim
R(t) = lim
+ α 2
= 0.
β
t→+∞ β
t→+∞ 1 + ( )2
(β) + 1
α
α′
Then
√ ′
′
( ββ2 )2 + ( βα2 )2
1
h(t)
=
=
lim
1 β′
1
1 β′
2 t→+∞ (1 − δ) α
(1 − δ)α − 2 β − R(t)
β − 2 β 2 − β R(t)
√
α′ ( β ′ β )2 ( α ′ )2
1
1
=
lim
+
=
lim
= 0.
2(1 − δ) t→+∞
β2 α
αβ
2(1 − δ) t→+∞ αβ
lim
t→+∞
Therefore, σ
e2 (t) = o(1), by Lemma 1. Next,
lim
t→+∞
h(t)
σ
e2 (t) = 0.
′
(1 − δ)α − 21 ββ − R(t)
96
Tatiyana Barinova and Alexander Kostin
Then σ
e1 (t) = o(1). This implies that Theorem 12 is valid. Thus
(
∫t
δ
α dτ )
t0
ξi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, according to (17),
∫t
(
t0
zi (t) = o e
λ′ (t)
(δα− λ1 (t) ) dτ )
(i = 1, 2), δ ∈ (0, 1).
1
Then
β′
(
1
αβ
β2
lim
R(t) = lim
+ α 3
t→+∞ α
t→+∞ α + β
(
)
+
β
β
α
α′
)
= 0,
α
β
β′
(
)
1
αβ
β2
lim
I(t) = lim
− α 2
= 0.
α
2
t→+∞ α
t→+∞ ( ) + 1
(β) + 1
β
α′
Hence
λ′i (t)
= o(α) (i = 1, 2)
λi (t)
and
(
∫t
δ
α dτ )
t0
zi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, just as in Theorem 10,
(
yi (t) = o e
δ
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
Theorem 13. Let the condition (9) and thr following conditions
α(+∞) = −∞,
β(+∞) = ∞,
α′
= O(1),
α
α
→ const ̸= 0,
β
β′
= o(1)
β2
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replacement (17). We get the system (18). In the system (18) we make the replacement (19). We obtain
the system (20), an auxiliary system of differential equations (21) and its
particular solution (22).
In this case,
β′
α
(
1
β2
α
R(t) = lim
+ α 3
lim
t→+∞ β( α + β )
t→+∞ α
(β) +
β
α
′
)
α
β
= 0.
On Asymptotic Stability of Solutions. . .
97
Then
h(t)
h(t)
= lim
=
′
′
β
1
t→+∞ α(1 − δ − 1 β − 1 R(t))
t→+∞ (1 − δ)α −
2 β − R(t)
2 αβ
α
v
u(
β′ β
α′
)2 (
)2
u
1
β2 α
αβ
t
=
lim
+
=
′
′
β
β
2 t→+∞
1 − δ − 12 ββ2 α
1 − δ − 12 ββ2 α
α′ 1
lim =
= 0.
2(1 − δ) t→+∞ αβ
lim
Therefore, σ
e2 (t) = o(1), by Lemma 1. Further, we have
lim
t→+∞
h(t)
σ
e2 (t) = 0.
′
(1 − δ)α − 21 ββ − R(t)
Then σ
e1 (t) = o(1). This implies that Theorem 13 is valid. Thus
(
∫t
δ
α dτ )
t0
ξi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, according to (17),
(
∫t
zi (t) = o et0
λ′ (t)
(δα− λ1 (t) ) dτ )
1
(i = 1, 2), δ ∈ (0, 1).
As is shown above,
lim
t→+∞
Then
Thus
1
R(t) = 0.
α
β′
α
)
(
1
β2
α
= 0.
lim
I(t) = lim
−
2
t→+∞ α
t→+∞ α( α + β )
(α
β) +1
β
α
′
λ′i (t)
= o(α) (i = 1, 2)
λi (t)
and
(
∫t
δ
α dτ )
t0
zi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, just as in Theorem 10,
(
δ
yi (t) = o e
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
Theorem 14. Let the condition (9) and the following conditions
α = o(1),
′
α
= O(1),
α
β(+∞) = ∞,
β′
= o(α)
β
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
98
Tatiyana Barinova and Alexander Kostin
Proof. In the system (10) we make the replacement (17). We get the system (18). In the system (18) we make the replacement (19). We obtain
the system (20), an auxiliary system of differential equations (21) and its
particular solution (22).
In this case,
β′
α
(
)
1
αβ
α
lim
+
R(t) = lim
= 0.
2
t→+∞ α
t→+∞ β( α + β )
(α
β) +1
β
α
′
Then
lim
t→+∞
√ ′
β 2
α′ 2
( αβ
) + ( αβ
)
h(t)
1
=
lim
1 β′
2 t→+∞ (1 − δ −
(1 − δ)α − 2 β − R(t)
1 β′
2 αβ
−
1
α
R(t))
= 0.
Therefore, σ
e2 (t) = o(1), by Lemma 1. Further, we have
lim
t→+∞
h(t)
σ
e2 (t) = 0.
′
(1 − δ)α − 12 ββ − R(t)
Then σ
e1 (t) = o(1). This implies that Theorem 14 is valid. Moreover,
(
∫t
δ
α dτ )
t0
ξi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, according to (17),
∫t
(
t0
zi (t) = o e
λ′ (t)
(δα− λ1 (t) ) dτ )
1
(i = 1, 2), δ ∈ (0, 1).
As is shown above,
lim
t→+∞
1
R(t) = 0.
α
Then
( α
1
α
lim
I(t) = lim
−
t→+∞ α
t→+∞ α α + β
β
β′
αβ
′
Thus
α
β
+
)
β
α
= 0.
λ′i (t)
= o(α) (i = 1, 2)
λi (t)
and
(
∫t
δ
α dτ )
t0
zi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, just as in Theorem 10,
(
yi (t) = o e
δ
∫t
t0
α dτ )
(i = 1, 2), δ ∈ (0, 1).
On Asymptotic Stability of Solutions. . .
99
Theorem 15. Let the condition (9) and the following conditions
α(+∞) ∈ R− ,
β(+∞) = ∞,
′
β′
= o(1)
β
α
= O(1),
α
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replacement (17). We get the system (18). In the system (18) we make the replacement (19). We obtain
the system (20), an auxiliary system of differential equations (21) and its
particular solution (22).
In this case,
β′
α
(
)
1
β
α
lim
R(t) = lim
+
= 0.
2
t→+∞ α
t→+∞ β( α + β )
α(( α
β ) + 1)
β
α
′
Then
√
′
′
β 2
α 2
( αβ
) + ( αβ
)
1
h(t)
=
= 0.
lim
lim
′
′
β
β
t→+∞ (1 − δ)α − 1
2 t→+∞ (1 − δ − 21 αβ − α1 R(t))
2 β − R(t)
Therefore, σ
e2 (t) = o(1), by Lemma 1. Further, we have
lim
t→+∞
h(t)
σ
e2 (t) = 0.
′
(1 − δ)α − 21 ββ − R(t)
Then σ
e1 (t) = o(1). This implies that Theorem 15 is valid. Hence
(
δ
∫t
α dτ )
t0
ξi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, according to (17),
(
∫t
zi (t) = o et0
λ′ (t)
(δα− λ1 (t) ) dτ )
(i = 1, 2), δ ∈ (0, 1).
1
As is shown above,
lim
t→+∞
Then
1
R(t) = 0.
α
′
Thus
β′
α
)
(
1
β
α
lim
−
= 0.
I(t) = lim
β
t→+∞ α
t→+∞ α( α + β )
α( α
β
α
β + α)
λ′i (t)
= o(α) (i = 1, 2)
λi (t)
and
(
zi (t) = o e
δ
∫t
t0
α dτ )
(i = 1, 2), δ ∈ (0, 1).
100
Tatiyana Barinova and Alexander Kostin
Then, just as in Theorem 10,
(
δ
∫t
α dτ )
t0
yi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Theorem 16. Let the condition (9) and the following conditions
α
→ 0,
α(+∞) = −∞, β(+∞) = ∞,
β
α′
β′
=
O(1),
= O(1)
α2
β
be fulfilled. Then the trivial solution of equation (1) is asymptotically stable.
Proof. In the system (10) we make the replacement (17). We get the system (18). In the system (18) we make the replacement (19). We obtain
the system (20), an auxiliary system of differential equations (21) and its
particular solution (22).
In this case,
β′
α
(
)
1
β
α2
lim
R(t) = lim
+
= 0.
2
t→+∞ α
t→+∞ 1 + ( β )2
α(( α
β ) + 1)
α
′
Then
√ ′
β 2
α′ 2
( αβ
) + ( αβ
)
h(t)
1
lim
=
lim
= 0.
′
′
β
β
1
1
1
t→+∞ (1 − δ)α −
2 t→+∞ (1 − δ − 2 αβ − α R(t))
2 β − R(t)
Consequently, σ
e2 (t) = o(1), by Lemma 1. Further, we have
h(t)
σ
e2 (t) = 0.
′
t→+∞ (1 − δ)α − 1 β − R(t)
2 β
lim
Then σ
e1 (t) = o(1). This implies that Theorem 16 is valid. Moreover,
(
ξi (t) = o e
δ
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
Then, according to (17),
∫t
(
t0
zi (t) = o e
λ′ (t)
(δα− λ1 (t) ) dτ )
(i = 1, 2), δ ∈ (0, 1).
1
As is shown above,
lim
t→+∞
Then
Thus
1
R(t) = 0.
α
( α
1
α2
lim
I(t) = lim
t→+∞ α
t→+∞ α +
β
β′
β
′
β
α
−
β
α( α
β + α)
λ′i (t)
= o(α) (i = 1, 2)
λi (t)
)
= 0.
On Asymptotic Stability of Solutions. . .
and
(
∫t
δ
101
α dτ )
t0
zi (t) = o e
(i = 1, 2), δ ∈ (0, 1).
Then, just as in Theorem 10,
(
δ
yi (t) = o e
∫t
α dτ )
t0
(i = 1, 2), δ ∈ (0, 1).
2.3. The case of purely imaginary roots λi (t) (i = 1, 2). Let us analyze
equation (6):
y ′′ + p(t)y = 0
√
√
where p(t) ∈ CI2 , p(t) > 0 in I. Then λ1 (t) = −i p, λ2 (t) = i p, p = p(t).
Theorem 17. Let the conditions
β(+∞) = +∞,
β′
= o(1),
β2
′
( ββ2 )′
β′
β
= o(1)
be fulfilled. Then the trivial solution of equation (6) is asymptotically stable.
Proof. In this case the system (10) takes the form

)
( 1 β′
1 β′

′

− iβ z1 (t) +
z2 (t),
z1 (t) = −
2 β
2 β
( 1 β′
)
 ′
1 β′

z2 (t) =
z1 (t) + −
+ iβ z2 (t).
2 β
2 β
In this system we make the following replacement:
δ
zi (t) = e
∫t
t0
(− 12
β′
β
) dτ
φi (t) (i = 1, 2), δ ∈ (0, 1).
As a result, we obtain the following system:

( 1 − δ β′
)
1 β′

′

φ
(t)
=
−
−
iβ
φ1 (t) +
φ2 (t),
 1
2 β
2 β
( 1 − δ β′
)

1 β′

φ′2 (t) =
φ1 (t) + −
+ iβ φ2 (t).
2 β
2 β
Then in the system (23) we make the following replacement:
(
) (
)(
)
φ1 (t)
1
0
η1 (t)
=
,
φ2 (t)
r(t) 1
η2 (t)
(23)
where ηi (t) are the new unknown functions (i = 1, 2). Then the system (23)
takes the form

)
( 1 − δ β′
1 β′
1 β′


+
r(t) − iβ η1 (t) +
η2 (t),
η1′ (t) = −



2 β
2 β
2 β



)
( 1 β′
1 β′ 2
(24)
+ 2iβr(t) −
r (t) − r′ (t) η1 (t)+
η2′ (t) =

2 β
2 β



( 1 − δ β′
)

1 β′


+ −
−
r(t) + iβ η2 (t).

2 β
2 β
102
Tatiyana Barinova and Alexander Kostin
Suppose
1 β′
+ 2iβr(t) = 0.
2 β
Then
r(t) =
1 β′
i = o(1).
4 β2
Then the system (24) takes the form

)
( 1 − δ β′
1 β′ β′
1 β′


+
i − iβ η1 (t) +
η2 (t),
η1′ (t) = −


2

2 β
8 β β
2 β



( 1 ( β ′ )′
1 β ′ ( β ′ )2 )
η2′ (t) = −
i
+
η1 (t)+

4 β2
8 β β2



( 1 − δ β′
)

1 β′ β′


+ −
−
i
+
iβ
η2 (t).

2 β
8 β β2
(25)
In accordance with Theorem 1, for the system (25) we write an auxiliary
system of differential equations:

1 − δ β′
1 β ′ ′


σ
(t)
=
−
σ
(t)
+
σ2 (t),
1

 1
2 β
2 β
√
(26)
(( β ′ )′ )2 ( β ′ )2 ( β ′ )4 1 − δ β ′

1

′

4
+
−
σ
(t).
σ2 (t) =
2
8
β2
β
β2
2 β
We denote
√
1
g(t) =
8
(( β ′ )′ )2 ( β ′ )2 ( β ′ )4
+
.
4
β2
β
β2
Consider a particular solution of the system (26) with the initial conditions
σi (t0 ) = 0 (i = 1, 2):

∫τ
∫t
∫t
β′
β′

− (− 1−δ
(− 1−δ
2
β ) dτ
2
β ) dτ1


τ
t

dτ,
g(τ )e 0
σ
e2 (t) = e 0



∫t

(− 1−δ

2

t0


σ
e (t) = e

 1
β′
β
) dτ
t0
∫t
∫τ
− (−
1 β ′ σ2 (τ )e τ0
e
2 β
1−δ β ′
2
β
) dτ1
dτ.
t0
In this case,
lim
t→+∞
g(t)
β′
− 1−δ
2 β
v
u ( β′ ′ )
u ( β 2 ) 2 ( β ′ )4
1
=−
lim t4 β ′
+
=
4(1 − δ) t→+∞
β2
β
( β2′ )′ 1
β lim ′ = 0.
=−
1 − δ t→+∞ ββ
On Asymptotic Stability of Solutions. . .
103
Therefore, σ
e2 (t) = o(1), by Lemma 1. Further, we have
′
1 β
2 |β |
lim
′
t→+∞ − 1−δ β
2 β
σ
e2 (t) = 0.
Then σ
e1 (t) = o(1). This implies that Theorem 17 is valid. Moreover,
ηi (t) = o(1) (i = 1, 2). Then φi (t) = o(1) (i = 1, 2). Then we have obtained
(
δ
zi (t) = o e
∫t
t0
(− 12
β′
β
) dτ )
(i = 1, 2), δ ∈ (0, 1).
Then
{
y1 (t) = z1 (t) + z2 (t),
y2 (t) = −iβz1 (t) + iβz2 (t)
∫t
(
=⇒
(− 21 δ
β′
β
(
y1 (t) = o e
δ
∫t
t0
(− 21
β′
β
) dτ )
,

y (t) = −iβz (t) + iβz (t)
2
1
2
+ ββ2 ) dτ )
=⇒ |y2 (t)| = o e

t

( δ ∫ (− 12


|y1 (t)| = o e t0
=⇒
t

( ∫ δ(− 12


|y2 (t)| = o et0
t0



=⇒
′
=⇒
β′
β
) dτ )
β′
β
+o(1)) dτ )
,
δ ∈ (0, 1).
,
Remark 1. The condition
′
( ββ2 )′
β′
β
= o(1)
is satisfied if there exists the corresponding limit.
Proof.
′
lim
t→+∞
( ββ2 )′
β′
β
[
]
= we use the inverse de L’Hospital’s rule =
=−
1
β′
lim 2
= 0.
2 t→+∞ β ln β
Remark 2. The conditions of I. T. Kiguradze’s Theorem 4 are equivalent
to those of Theorem 17. But, in addition to Theorem 17, we have obtained
the estimate of solutions of equation (6).
104
Tatiyana Barinova and Alexander Kostin
Proof.
2ββ ′
β′
= 2 lim 2 = 0,
3
t→+∞ β
t→+∞ β
lim p′ p− 2 = lim (β 2 )′ β −3 = lim
3
t→+∞
t→+∞
lim (ln p)−1
∫t
t→+∞
[
]
′ − 3 ′
(p p 2 ) dτ = we use de L’Hospital’s rule =
a
|(p′ p− 2 )′ |
3
= lim
t→+∞
p′
p
= lim
t→+∞
|(2ββ ′ β −3 )′ |
2ββ ′
β2
′
= lim
t→+∞
|( ββ2 )′ |
β′
β
= 0.
Conclusion
In the present paper we have revealed the sufficient conditions for asymptotic stability, as well as the estimate of solutions of the homogeneous linear
non-autonomous second order differential equation in terms of the behavior
of roots of the characteristic equation in the case of complex roots. The
results of the work allow one to proceed both to investigating equations
of higher order and to considering the problems on a simple stability and
instability.
References
1. L. Cesari, Un nuovo criterio di stabilità per le soluzioni delle equazioni differenziali
lineari. Ann. Scuola Norm. Super. Pisa (2) 9 (1940), 163–186.
2. K. P. Persidski, On characteristic numbers of a linear system of differential equations. (Russian) Izv. Akad. Nauk Kaz. SSR, Ser. Matem. i Mekh 42 (1947), 5–47.
3. N. Ya. Lyashchenko, On asymptotic stability of solutions of a system of differential
equations. (Russian) Doklady Akad. Nauk SSSR (N.S.) 96 (1954), 237–239.
4. J. K. Hale and A. P. Stokes, Conditions for the stability of nonautonomous differential equations. J. Math. Anal. Appl. 3 (1961), 50–69.
5. A. V. Kostin, Stability and asymptotics of quasilinear nonautonomous differential
systems. (Russian) OGU, Odessa, 1984.
6. I. T. Kiguradze, On the asymptotic properties of solutions of the equation u′′ +
a(t)un = 0. (Russian) Soobshch. Akad. Nauk Gruzin. SSR 30 (1963), 129–136.
(Received 30.01.2014)
Authors’ address:
Odessa I. I. Mechnikov National University, 2 Dvoryanska St., Odessa
65082, Ukraine.
E-mail: Tani112358@mail.ru; kostin− a@ukr.net