Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case
The double singular perturbation for boundary-value problem for nonlinear system of ordinary differential equations is considered. For a formal asymptotic solution, constructed by the method of boundary functions and generalized inverse matrices and projectors, we prove an asymptotic property of th...
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2018
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irk-123456789-1771792021-02-12T01:26:23Z Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case Karandzhulov, L.I. The double singular perturbation for boundary-value problem for nonlinear system of ordinary differential equations is considered. For a formal asymptotic solution, constructed by the method of boundary functions and generalized inverse matrices and projectors, we prove an asymptotic property of the formal series. Розглянуто подвiйно сингулярне збурення граничної задачi для нелiнiйної системи звичайних диференцiальних рiвнянь. Для формального асимптотичного розв’язку, який побудовано за методом граничних функцiй та узагальнених обернених матриць i проекторiв, доведено асимптотичну властивiсть формального ряду. 2018 Article Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case / L.I. Karandzhulov // Нелінійні коливання. — 2018. — Т. 21, № 1. — С. 36-53. — Бібліогр.: 14 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177179 517.9 en Нелінійні коливання Інститут математики НАН України |
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The double singular perturbation for boundary-value problem for nonlinear system of ordinary differential equations is considered. For a formal asymptotic solution, constructed by the method of boundary
functions and generalized inverse matrices and projectors, we prove an asymptotic property of the formal
series. |
| format |
Article |
| author |
Karandzhulov, L.I. |
| spellingShingle |
Karandzhulov, L.I. Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case Нелінійні коливання |
| author_facet |
Karandzhulov, L.I. |
| author_sort |
Karandzhulov, L.I. |
| title |
Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case |
| title_short |
Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case |
| title_full |
Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case |
| title_fullStr |
Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case |
| title_full_unstemmed |
Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case |
| title_sort |
asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case |
| publisher |
Інститут математики НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/177179 |
| citation_txt |
Asymptotic behavior of solutions for differential equations with double singularity in conditionally stable case / L.I. Karandzhulov // Нелінійні коливання. — 2018. — Т. 21, № 1. — С. 36-53. — Бібліогр.: 14 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT karandzhulovli asymptoticbehaviorofsolutionsfordifferentialequationswithdoublesingularityinconditionallystablecase |
| first_indexed |
2025-07-15T15:12:43Z |
| last_indexed |
2025-07-15T15:12:43Z |
| _version_ |
1837726290706169856 |
| fulltext |
UDC 517.9
ASYMPTOTIC BEHAVIOR OF SOLUTIONS
FOR DIFFERENTIAL EQUATIONS WITH DOUBLE SINGULARITY
IN CONDITIONALLY STABLE CASE
АСИМПТОТИЧНА ПОВЕДIНКА РОЗВ’ЯЗКIВ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ З ПОДВIЙНОЮ СИНГУЛЯРНIСТЮ
В УМОВНО СТIЙКОМУ ВИПАДКУ
L. I. Karandzhulov
Techn. Univ., Sofia, Bulgaria
e-mail: likar@tu-sofia.bg
The double singular perturbation for boundary-value problem for nonlinear system of ordinary differenti-
al equations is considered. For a formal asymptotic solution, constructed by the method of boundary
functions and generalized inverse matrices and projectors, we prove an asymptotic property of the formal
series.
Розглянуто подвiйно сингулярне збурення граничної задачi для нелiнiйної системи звичайних
диференцiальних рiвнянь. Для формального асимптотичного розв’язку, який побудовано за ме-
тодом граничних функцiй та узагальнених обернених матриць i проекторiв, доведено асимп-
тотичну властивiсть формального ряду.
1. Introduction. In the paper we investigate the asymptotic behavior of the formal solution of
the boundary-value problem (BVP)
ε
dx
dt
= Ax+ εF (t, x, ε, f(t, ε)) + ϕ(t), t ∈ [a, b], 0 < ε � 1, (1)
lx(·) = h, h ∈ Rn, (2)
where ε is a small positive parameter. The BVP (1), (2) will be considered under the following
conditions:
(C1) The (n×n)-matrix A with constant elements has p eigenvalues with negative real part,
and the remaining (n − p) eigenvalues have positive real part, i.e., λj ∈ σ(A), Reλj < 0,
j = 1, p, and Reλj > 0, j = p+ 1, n.
(C2) The vector-function ϕ(t) is an n-dimensional vector-function of the class C∞([a, b]).
(C3) The function F (t, x, ε, f(t, ε)) is an n-dimensional vector-function, having arbitrary
order continuous partial derivatives with respect to all arguments in the domain G = [a, b] ×
×Dx×[0, ε]×Df ,whereDx ∈ Rn is some neighborhood of the solution x0(t) of the degenerate
system Ax0(t) +ϕ(t) = 0, Df ∈ Rp is a bounded and closed domain, 0 < ε � 1. The function
f(t, ε) is smooth of arbitrary order with respect to all arguments in the domainG1 = [a, b]×(0, ε]
and its values belongs to Df .
(C4) l is a linear n-dimensional bounded vector functional, l = col
(
l1, . . . , ln
)
, l ∈
∈ (C[a, b] → Rn, Rn) .
c© L. I. Karandzhulov, 2018
36 ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 37
We assume that the function f from (1) contains singular elements (for example, f =
= f(exp(−t/ε), sin(t/ε))). Thus the system (1) has a double singularity. On the one hand, the
small parameter ε appears before the derivative, and, on the other hand, it brings in a singularity
of the function f.
The condition (C1) shows that we consider system (1) in a conditionally stable case [13, 14].
We seek to determine existence and uniqueness of a n-dimensional asymptotic solution
x(t, ε) of the BVP (1), (2) such that x(·, ε) ∈ C1([a, b]), x(t, ·) ∈ C((0, ε0]) and limε→0 x(t, ε) =
= x0(t).
The construction of the asymptotic solution of problem (1), (2) is based on the boundary
functions method (see, for example, [13, 14]). The initial research for a Cauchy problem with
double singularity is carried out in [9] in the case Reλi < 0 ∀i, λi ∈ σ(A). The BVP in this case
are analyzed in the papers [7, 11].
Primary research of the problem (1), (2) with conditions (C1) – (C4) is conducted in [5]
and [6]. In the papers [5, 6] the formal asymptotic solutions of the BVP (1), (2) have been
constructed in various cases. In [6] we considered the case which uses generalized inverse matri-
ces and projectors [1, 3, 10].
2. Preliminary results and problem formulation. In the paper [6] a formall asymptotic soluti-
on of the BVP (1), (2) was obtained after introducing a second parameter µ and studyind the
BVP with two parameters ε ∈ [0, ε] and µ ∈ (0, ε], 0 < ε � 1,
εż = Az + εF (t, z, ε, f(t, µ)) + ϕ(t), t ∈ [a, b],
lz(·) = h, h ∈ Rn.
(3)
The solution of the BVP (3) be found in a unique formal expression of the form
z(t, ε, µ) =
∞∑
k=0
(zk(t, µ) + Πk(τ, µ) +Qk(ν, µ)) εk, τ =
t− a
ε
, ν =
t− b
ε
. (4)
After the determination of zk(t, µ), Πk(τ, µ), and Qk(ν, µ), a solution of (1), (2) takes the
form
x(t, ε) =
∞∑
k=0
(zk(t, ε) + Πk(τ, ε) +Qk(ν, µ)) εk.
Using the condition (C1) it is easy to obtain functions zk(t, ε), which are elements of the
regular series. Elements of the singular series Πk(τ, ε), τ =
t− a
ε
and Qk(ν, µ), ν =
t− b
ε
,
were obtained by solving sequential linear differential systems. We will indicate some of the
operations made in the article [6] and necessary for the present work.
We substitute series (4) into system (3) and we represent the function F (t, z, ε, f(t, µ)) in
the form
F
(
t,
∞∑
k=0
(zk(t, µ) + Πk(τ, µ) +Qk(ν, µ)) εk, ε, f(t, µ)
)
= F (t, ε, µ) + ΠF (τ, ε, µ) +QF (ν, ε, µ),
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
38 L. I. KARANDZHULOV
where
F (t, ε, µ) = F
(
t,
∞∑
k=0
zk(t, µ)εk, ε, f(t, µ)
)
,
ΠF (τ, ε, µ) = F
(
a+ ετ,
∞∑
k=0
(zk(a+ ετ, µ) + Πk(τ, µ)) εk, ε, f(a+ ετ, µ)
)
−
− F
(
a+ ετ,
∞∑
k=0
zk(a+ ετ, µ)εk, ε, f(a+ ετ, µ)
)
, (5)
QF (ν, ε, µ) = F
(
b+ εν,
∞∑
k=0
(zk(b+ εν, µ) +Qk(ν, µ)) εk, ε, f(b+ εν, µ)
)
−
− F
(
b+ εν,
∞∑
k=0
zk(b+ εν, µ)εk, ε, f(b+ εν, µ)
)
.
We decompose the functions F (t, ε, µ), ΠF (τ, ε, µ), QF (ν, ε, µ) in Taylor series in a neighbor-
hood of the points (t, z0(t), 0, f), (a, z0(a) + Π0(τ), 0, f), (b, z0(b) + Q0(ν), 0, f), respectively.
We get
F (t, ε, µ) =
∞∑
k=0
F k(t, µ)εk,
where
F k(t, µ) =
F (t, z0(t), 0, f(t, µ)), k = 0,
Fz(t, z0(t), 0, f(t, µ))zk(t, µ)+
+gk (z0(t), . . . , zk−1(t, µ), f(t, µ)) , k = 1, 2, . . . .
(6)
In (6) the functions gk contain derivative up to the (k−1)th order of the functionF (t, z, ε, f(t, µ))
with respect to z and ε, calculated in the point (t, z0(t), 0, f)
ΠF (τ, ε, µ) =
∞∑
k=0
ΠFk(τ, µ)εk,
where
ΠFk(τ, µ) =
F (a, z0(a, µ) + Π0(τ), 0, f(a, µ))−
−F (a, z0(a, µ), 0, f(a, µ)), k = 0,
Fz(a, z0(a) + Π0(τ), 0, f(a, µ))Πk(τ, µ)+
+Gk (τ,Π0(τ), . . . ,Πk−1(τ, µ), f(a, µ)) , k = 1, 2, . . . .
(7)
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 39
The function Gk contain derivatives up to the (k − 1)th order of the function F (t, z, ε, f(t, µ))
with respect to t, z and ε in the point (a, z0(a) + Π0(τ), 0, f(a, µ)), and the derivatives up to the
(k − 1)th order of the function zk(t, µ) with respect to t in the point (a, µ),
QF (ν, ε, µ) =
∞∑
k=0
QFk(ν, µ)εk,
where
QFk(ν, µ) =
F (b, z0(b) +Q0(ν), 0, f(b, µ))−
−F (b, z0(b), 0, f(b, µ)), k = 0,
Fz(b, z0(b) +Q0(ν), 0, f(b, µ))Qk(ν, µ)+
+Rk (ν,Q0(ν), . . . , Qk−1(ν, µ), f(b, µ)) , k = 1, 2, . . . .
(8)
The functions Rk contain derivatives up to the (k − 1)th order of the function F (t, z, ε, f(t, µ))
with respect to t, z and ε in the point (b, z0(b) +Q0(ν), 0, f(b, µ)), and the derivatives up to the
(k − 1)th order of the function zk(t, µ) with respect to t in the point (b, µ).
A similar approach is used for the BVP (1), (2), which does not contain the function f(t, ε),
in the articles [4, 8]. In these articles and in the article [6] Lemma 1 is essential. The problem
connected with condition (C1) for a differential equation in a Banach space is discussed, for
example, in [2].
Lemma 1. Let the matrixA satisfy the condition (C1), P be a spectral projection on the left half
plane of the matrix A, and functions g(τ) ∈ C(0,+∞), g(ν) ∈ C(−∞, 0) satisfy the inequalities
‖g(τ)‖ ≤ C∗ exp(−α∗τ), C∗ > 0, α∗ > 0, τ ≥ 0,
‖g(ν)‖ ≤ C
∗
exp(α∗ν), C
∗
> 0, α∗ > 0, ν ≤ 0.
Then the systems
dx
dτ
= Ax + g(τ), τ ∈ [0,+∞) and
dy
dν
= Ay + g(ν), ν ∈ (−∞, 0], have
particular solutions (Lτg)(τ) and (Lνg)(ν), respectively, in the forms
(Lτg)(τ) =
+∞∫
0
K(τ, s)g(s)ds and (Lνg)(ν) =
0∫
−∞
K(ν, s)g(s) ds,
satisfying the inequalities
‖(Lτg)(τ)‖ ≤ C exp(−γτ), τ ≥ 0; ‖(Lνg)(ν)‖ ≤ C exp(γν), ν ≤ 0,
where C, C, γ, γ are certain positive constants, and
K(τ, s) =
X(τ)PX−1(s), 0 ≤ s ≤ τ < +∞,
−X(τ)(I − P )X−1(s), 0 ≤ τ ≤ s < +∞,
K(ν, s) =
−X(ν)(I − P )X−1(s), −∞ < ν ≤ s ≤ 0,
X(ν)PX−1(s), −∞ < s ≤ ν < 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
40 L. I. KARANDZHULOV
Let the linear system
dx
dt
= Ax have a fundamental solution matrix X(t) = exp(At),
X(0) = En andB be an (n×n) nonsingular constant matrix such thatB−1AB = diag (A+, A−),
where A+ is an (p× p)-matrix having eigenvalues with negative real parts, Reλi < 0, i = 1, p,
and A− is ((n − p) × (n − p))-matrix having eigenvalues with positive real parts, Reλi > 0,
i = p+ 1, n.
The system
dx
dt
= Ax has stable manifold S+ in the form S+ : x = Hx, where H =
= B21B
−1
11 is an ((n − p) × p)-matrix, and unstable manifold S− in the form S− : x = Hx,
where H = B12B
−1
22 is an (p × (n − p))-matrix. The cells Bij , i, j = 1, 2, are elements of the
block representation of the matrix B =
(
B11 B12
B21 B22
)
.
Let Xp(τ) = X(τ)
(
Ep
H
)
be a (n× p)-matrix, Xn−p(ν) = X(ν)
(
H
En−p
)
be a (n× (n−
−p))-matrix.
Let us introduce the following notations:
D1(ε) = lXp(·) = lXp
(
(·)− a
ε
)
is a (n× p)-matrix;
D2(ε) = lXn−p(·) = lXn−p
(
(·)− b
ε
)
is a (n× (n− p))-matrix;
D(ε) = (D1(ε)D2(ε)) is a (n× n)-matrix.
Let D(ε) = D0 +O (εq exp(−α/ε)) , D0 be a (n×n)-matrix, with constant elements. In this
case rangD0 = r < n and D−1
0 does not exist. But according to [1, 3, 10] we can use unite
Mur – Penrous pseudoinverse matrix of D0, which is written as D+
0 . More details in connection
to pseudoinverse matrices can be found in the cited literature. Let PD0 and PD∗
0
be (n × n)-
matrices (orthogonal projections) projecting Rn onto N(D0) = kerD0 and onto N(D∗0) =
= kerD∗0, respectively, i.e., PD0 : Rn → kerD0, PD∗
0
: Rn → kerD∗0, D
∗
0 = DT
0 , P
2
D0
= PD0 ,
P 2
D∗
0
= PD∗
0
. Having in mind that rangD0 = r < n, then rangPD0 = rangPD∗
0
= n −
−r = q. There exists q linearly independent columns in the (n × n)-matrix PD0 and q linearly
independent rows in the (n × n)-matrix PD∗
0
. By PD0q we denote the (n × q)-matrix consisting
of q arbitrary linearly independent columns of PD0 , and by PD∗
0q
we denote the (q × n)-matrix
consisting of q arbitrary linearly independent rows of PD∗
0
.
Theorem 1 [6]. Suppose that the following conditions are satisfied:
(H1) (C1) – (C4);
(H2) the matrix D(ε) has the representation D(ε) = D0 +O (εq exp(−α/ε)) , q ∈ N, α > 0,
and rankD0 = r < n;
(H3) PD∗
0
h0 = 0, where h0 = h+ l
(
A−1ϕ(·)
)
;
(H4) the nonlinear equation PD∗
0
h1(ε, µ, ξ0) = 0 for all 0 < ε ≤ ε, 0 < µ ≤ ε has a unique
bounded solution with respect to ξ0 = ψ0(ε, µ) ∈ Rq, where
h1(ε, µ, ξ0) = −lz1(·, µ)− l(LτΠF0)(·, µ, ξ0)− l(LνQF0)(·, µ, ξ0)
at ΠF0(τ, µ) = F (a, z0(a, µ) + Π0(τ), 0, f(a, µ)) − F (a, z0(a, µ), 0, f(a, µ)) and QF0(ν, µ) =
= F (b, z0(b) +Q0(ν), 0, f(b, µ))− F (b, z0(b), 0, f(b, µ)).
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 41
Then the principal part of the expansion (4) has the forms
z0(t, µ) = −A−1ϕ(t),
Π0(τ) = Xp(τ)[PD0q ]pψ0(ε, µ) +Xp(τ)[D+
0 h0]p, (9)
Q0(ν) = Xn−p(ν)[PD0q ]n−pψ0(ε, µ) +Xn−p(ν)
[
D+
0 h0
]
n−p ,
respectively.
We introduce the following notations:
hk(ε, µ, ξk−1) = D1(ε, µ)ξk−1 + Sk−1(ε, µ), ξk−1 ∈ Rq, k ≥ 2,
D1(ε, µ) = −l
(
Fz
(
a, z0(a) + Π0
(
(·)− a
ε
)
, 0, f(a, µ)
)
Xp(·)[D0q]p +
+ Fz
(
b, z0(b) +Q0
(
(·)− b
ε
)
, 0, f(b, µ)
)
Xn−p(·)[D0q]n−p
)
,
Sk(ε, µ) = −lzk+1(·, µ)− l
(
Fz (a, z0(a) + Π0(·), 0, f(a, µ)) Φk(·, ε, µ)+
+Fz(b, z0(b) +Q0(·), 0, f(b, µ))Φk(·, ε, µ)
)
−
− l (Gk(·,Π0(·), . . . ,Πk−1(·, µ), f(a, µ))+
+Rk(·, Q0(·), . . . , Qk−1(·, µ), f(b, µ))) , k ≥ 1, (10)
Φk(τ, ε, µ) = Xp(τ)
[
D+
0 hk
]
p
+ (LτΠF k)(τ, µ),
Φk(ν, ε, µ) = Xn−p(ν)
[
D+
0 hk
]
n−p + (LνQF k)(ν, µ),
hk = hk(ε, µ, ξk−1) = hk(ε, µ, ψk−1(ε, µ)) = hk(ε, µ),
D1(ε, µ) = PD0qD1(ε, µ),
Sk(ε, µ) = −PD∗
0q
Sk(ε, µ), k ≥ 1.
It should be noted that in (10) D1(ε, µ) is an (n× q)-matrix, Sk(ε, µ) is an n-vector, D1(ε, µ)
is a (q × q)-matrix and Sk(ε, µ) is a q-vector. Besides, the functions Gk contain derivative up to
the (k − 1)th order of the function a F (t, z, ε, f(t, µ)) with respect to t, z, and ε, in the point
(a, z0(a)+Π0(τ), 0, f(a, µ)), and derivative up to the (k−1)th order of the function zk(t, µ) with
respect to t in the point (a, µ). The functions Rk contain derivatives up to the (k− 1)th order of
the function F (t, z, ε, f(t, µ)) with respect to t, z, and ε, in the point (b, z0(b)+Q0(ν), 0, f(b, µ)),
and derivatives up to the (k−1)th order of the function zk(t, µ) with respect t in the point (b, µ).
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
42 L. I. KARANDZHULOV
Theorem 2 [6]. Let the conditions (H1) – (H4) of the Theorem 1 and the condition
(H5) rankD1(ε, µ) 6= 0 ∀0 < ε ≤ ε, 0 < µ ≤ ε
be fulfilled. Then the coefficients zk(t, µ), Πk(τ, µ), and Qk(ν, µ) for k ≥ 1 of the series (4) have
the forms
zk(t, µ) = −
(
A−1
)k+1 dkϕ(t)
dtk
−
1∑
i=k
(A−1)i
di−1
dti−1
F k−i(t, µ),
Πk(τ, µ) = −Xp(τ)[PD0q ]pD
−1
1 (ε, µ)Sk(ε, µ) + Φk(τ, ε, µ),
Qk(ν, µ) = −Xn−p(ν)[PD0q ]n−pD
−1
1 (ε, µ)Sk(ε, µ) + Φk(ν, ε, µ).
Theorem 3 [6]. Let the conditions (H1) – (H5) of the Theorem 2 be fulfilled. Then
1) the functions zk(t, µ), k ≥ 0, are bounded, i. e., ∃Mk > 0: ‖zk(t, µ)‖ ≤ Mk ∀t ∈ [a, b],
µ ∈ (0, ε], k ≥ 0;
2) the boundary functions Πk(τ, µ) and Qk(ν, µ), k ≥ 0, decrease exponentially at τ → ∞
and ν → −∞ respectively, µ ∈ (0, ε].
In the next section we will show that the obtained formal series (4) is asymptotic.
2. Main results. In BVP (3) we make a change of variables,
u(t, ε, µ) = z(t, ε, µ)− Zn(t, ε, µ), (11)
where Zn(t, ε, µ) =
∑n
k=0 [zk(t, µ) + Πk(τ, µ) +Qk(ν, µ)] εk is the nth partial sum of the series
(4). Keeping in mind the expressions (9) in Theorem 1, the notations (10) and zk,Πk, Qk, k ≥ 1,
in Theorem 2 we obtain that the new variable u satisfy the BVP
εu̇ = Au+Hn(t, u, ε, µ), lu(·, ε, µ) = 0, (12)
where
Hn(t, u, ε, µ) = εF (t, u+ Zn, ε, f(t, µ)) + L(t, τ, ν, ε, µ). (13)
The expression L(t, τ, ν, ε, µ) have the form
L(t, τ, ν, ε, µ) = ∆1 + ∆2 + ∆3, (14)
where
∆1 = A
n∑
k=1
zk(t, µ)εk −
n∑
k=0
d
dt
zk(t, µ)εk+1,
∆2 = A
n∑
k=0
Πk(τ, µ)εk −
n∑
k=0
d
dτ
Πk(τ, µ)εk, (15)
∆3 = A
n∑
k=0
Qk(ν, µ)εk −
n∑
k=0
d
dν
Qk(ν, µ)εk.
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 43
For the function Hn(t, u, ε, µ) from (13) we will prove three lemmas.
Lemma 2. Let u = 0 in (13). Then the function Hn(t, 0, ε, µ) with C > 0, t ∈ [a, b], ε ∈
∈ (0, ε1], µ ∈ (0, ε1], 0 < ε1 < ε, satisfies the inequality ‖Hn(t, 0, ε, µ)‖ ≤ Cεn+1.
Proof. By means of equalities (5) – (8) for the representation of Hn(t, 0, ε, µ) we obtain
Hn(t, 0, ε, µ) = εF (t, Zn, ε, f) + L(t, τ, ν, ε, µ) =
= εF
(
t,
n∑
k=0
[zk + Πk +Qk] ε
k, ε, f
)
+ L(t, τ, ν, ε, µ) =
= ∆0 + ∆1 + ∆2 + ∆3, (16)
where ∆0 = εF + εΠF + εQF, and ∆i, i = 1, 2, 3 it, are given by (15). For a representation of
the sums ∆i we use (5) for ∆0, Theorem 2 for ∆1. For representation ∆2 and ∆3 we consider
receiving the boundary functions Πk, Qk by linear BVPs, from which result of the boundary
function in Theorem 2,
dΠk(τ)
dτ
= AΠk(τ) + ΠF k(τ, µ), τ ∈
[
0,
b− a
ε
]
, µ ∈ (0, ε],
dQk(ν)
dν
= AQk(ν) +QF k(ν, µ), ν ∈
[
a− b
ε
, 0
]
, µ ∈ (0, ε],
l
(
Πk
(
(·)− a
ε
, µ
)
+Qk
(
(·)− b
ε
, µ
))
=
{
h− l(z0(·, µ)), k = 0,
−l(zk(·, µ)), k = 1, 2, . . . ,
ΠF k(τ, µ) =
{
0, k = 0,
ΠFk−1(τ, µ), k = 1, 2, . . . ,
QF k(ν, µ) =
{
0, k = 0,
QFk−1(τ, µ), k = 1, 2, . . . .
Thus we get
∆0 = εF + εΠF + εQF =
n∑
k=0
F kε
k+1 +
n∑
k=0
ΠFkε
k+1 +
n∑
k=0
QFkε
k+1,
∆1 = A
n∑
k=1
zk(t, µ)εk −
n∑
k=0
d
dt
zk(t, µ)εk+1 = −
n∑
k=1
F k−1(t, µ)εk − d
dt
znε
n+1,
∆2 = A
n∑
k=0
Πkε
k −
n∑
k=0
d
dτ
Πkε
k = −
n∑
k=1
ΠF k−1(τ, µ)εk,
∆3 = A
n∑
k=0
Qkε
k −
n∑
k=0
d
dν
Qkε
k = . . . = −
n∑
k=1
ΠQk−1(ν, µ)εk.
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44 L. I. KARANDZHULOV
We substitute ∆i, i = 0, 1, 2, 3, in (16) and obtain
Hn(t, 0, ε, µ) = (−Azn(t, µ) + ΠFn(τ, µ) +QFn(ν, µ)) εn+1.
Let the following inequalities be fulfilled:
‖A‖ ≤ C0, C0 > 0, ‖zn(t, µ)‖ ≤ Mn, Mn > 0.
On the other hand Theorem 2 shows that by the equalities (14) and (15) we have
‖ΠFn(τ, µ)‖ ≤ Cn exp(−αnτ), Cn > 0, αn > 0, τ ≥ 0,
‖QFn(ν, µ)‖ ≤ Cn exp(αnν), Cn > 0, αn > 0, ν ≤ 0.
Then
‖Hn(t, 0, ε, µ)‖ = ‖ −Azn(t, µ) + ΠFn(τ, µ) +QFn(ν, µ)‖εn+1 ≤
≤ [‖A‖‖zn(t, µ)‖+ ‖ΠFn(τ, µ)‖+ ‖QFn(ν, µ)‖] εn+1 ≤
≤
[
C0Mn + Cn exp(−αnτ) + Cn exp(αnν)
]
εn+1.
Keeping in mind that at t ∈ [a, b], τ =
t− a
ε
≥ 0, ν =
t− b
ε
≤ 0, for the upper bound, we
finally obtain
‖Hn(t, 0, ε, µ)‖ ≤
[
C0Mn + Cn.1 + Cn.1
]
εn+1 = Cεn+1,
where C = C0Mn + Cn + Cn.
Let us also estimate the function Hn(t, u, ε, µ).
Lemma 3. There exists a constant C∗ > 0 such that ‖Hn(t, u, ε, µ)‖ ≤ C∗ε for ‖u‖ ≤ 2R,
R > 0.
Proof. From (13) we get
Hn(t, u, ε, µ) = εF (t, u+ Zn, ε, f(t, µ)) + L(t, τ, ν, ε, µ).
Then
Hn(t, u, ε, µ) = εF (t, u+ Zn, ε, f(t, µ))− εF (t, Zn, ε, f(t, µ))+
+ εF (t, Zn, ε, f(t, µ)) + L(t, τ, ν, ε, µ) =
= ε [F (t, u+ Zn, ε, f(t, µ))− F (t, Zn, ε, f(t, µ))] +
+ εF (t, Zn, ε, f(t, µ)) + L(t, τ, ν, ε, µ) =
= ε
1∫
0
Fx(t, Zn + θu, ε, f(t, µ))dθu+Hn(t, 0, ε, µ).
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 45
According to the condition (C3) and x = Zn + θu the function Fx is continuous with respect to
all arguments in the domain G, i. e., there exists a constant C̃ > 0 such that ‖Fx‖ ≤ C̃. Using
the last lemma we obtain the estimate
‖Hn(t, u, ε, µ)‖ ≤ ε
1∫
0
‖Fx(t, Zn + θu, ε, f(t, µ))‖dθ‖u‖+ ‖Hn(t, 0, ε, µ)‖ ≤
≤ εC̃‖u‖+ Cεn+1 ≤ 2RC̃ε+ Cεn+1 ≤ C∗ε,
where 2RC̃ ≤ C∗ < 2RC̃ +
˜̃
C, 0 <
˜̃
C � 1.
Lemma 4. Let, in the some neighborhood of the degenerate solution ‖z0‖ < δ,we have ‖z‖ ≤
≤ ρ < δ and t ∈ [a, b], ε ∈ [0, ε1], µ ∈ (0, ε1]. Then there is a positive constant K1 such that in
case of ‖u‖ ≤ δ and ‖u‖ ≤ δ, where 0 < δ < δ and δ+ρ < δ, the function Hn(t, u, ε, µ) satisfies
the inequality
‖∆Hn‖ = ‖Hn(t, u, ε, µ)−Hn(t, u, ε, µ)‖ ≤ K1ε‖u− u‖.
Proof. From (13) we get
∆Hn = ε
[
F (t, u+ zn, ε, f(t, µ))− F (t, u+ zn, ε, f(t, µ))
]
.
The estimate of the last difference is realized analogously to Lemma 3. Consequently,
‖∆Hn‖ ≤ K1ε‖u− u‖.
Let B be a (n × n)-matrix satisfying the condition B−1AB = diag (A+ A−) , where
Reλi(A+) < 0, j = 1, p, Reλi(A−) > 0, j = p+ 1, n.
In the system (12) we make a change of variables,
u(t, ε, µ) = B
(
η(t, ε, µ)
δ(t, ε, µ)
)
,
where η(t, ε, µ) = (η1, . . . , ηp)
T , δ(t, ε, µ) = (δ1, . . . , δn−p)
T . Then (8) take the form
εη̇(t, ε, µ) = A+η +
[
H̃n(t, η, δ, ε, µ)
]
p
,
(17)
εδ̇(t, ε, µ) = A−δ +
[
H̃n(t, η, δ, ε, µ)
]
n−p
,
lB
(
η(t, ε, µ)
δ(t, ε, µ)
)
= 0, (18)
where
H̃n(t, η, δ, ε, µ) = B−1Hn(t, η, δ, ε, µ) =
= B−1εF
(
t, B
(
η
δ
)
+ Zn, ε, f(t, µ)
)
+B−1L(t, τ, ν, ε, µ). (19)
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
46 L. I. KARANDZHULOV
The notations
[
H̃n
]
p
and
[
H̃n
]
n−p
are used to denote the first p and the later n− p elements of
the vector function H̃n, respectively.
According to Lemmas 2 and 4 it is possible to assert that for the equality (19) we have the
inequalities
‖H̃n(t, η, δ, ε, µ)‖ ≤ C̃1ε
n+1, C̃1 > 0,
‖∆H̃n‖ = ‖H̃n(t, η, δ, ε, µ)− H̃n(t, η, δ, ε, µ) ≤ ˜̃
C1ε
(
‖η − η‖+ ‖δ − δ‖
)
,
˜̃
C1 > 0.
(20)
The system of differential equations (17) transforms to a system of integral equations,
η(t, ε, µ) = W (t, s, ε, µ)η(a, ε, µ) +
t∫
a
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
p
ds,
δ(t, ε, µ) = W (t, s, ε, µ)δ(b, ε, µ) +
b∫
t
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
n−p
ds,
(21)
where η(a, ε, µ) and δ(b, ε, µ) are arbitrary vectors depending on ε and µ. In (21) the fundamental
matrices W (t, s, ε, µ) and W (t, s, ε, µ) are solutions of the systems
ε
dη
dt
= A+η, W (s, s, ε, µ) = Ep and ε
dδ
dt
= A−δ, W (s, s, ε, µ) = En−p
respectively. If σ > 0 and K0 > 0, then the inequalities
‖W (t, s, ε, µ)‖ ≤ K0 exp
(
−σ t− s
ε
)
, a ≤ s ≤ t ≤ b,
‖W (t, s, ε, µ)‖ ≤ K0 exp
(
−σ s− t
ε
)
, a ≤ t ≤ s ≤ b,
(22)
are fulfilled. In consequence we will consider the iterative process
η0(a, ε, µ) = 0, δ0(b, ε, µ) = 0,
ηi(t, ε, µ) = W (t, s, ε, µ)η(a, ε, µ) +
t∫
a
W (t, s, ε, µ)
1
ε
[
H̃n
(
t, ηi−1, δi−1, ε, µ
)]
p
ds, (23)
δi(t, ε, µ) = W (t, s, ε, µ)δ(b, ε, µ) +
b∫
t
W (t, s, ε, µ)
1
ε
[
H̃n
(
t, ηi−1, δi−1, ε, µ
)]
n−p
ds.
Our further aim is to show that the integral system (21) has a uniquel and continuous soluti-
on. Therefore we introduce a Banach space M [12] which consists of the all continuous n-
dimensional functions
y(t, ε, µ) = (η1(t, ε, µ), . . . , ηp(t, ε, µ), δ1(t, ε, µ), . . . , δn−p(t, ε, µ))T
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 47
in the domain G2 = [a, b] × (0, ε] × (0, ε] = {(t, ε, µ)|t ∈ [a, b], ε ∈ (0, ε], µ ∈ (0, ε]} with the
norm
‖y(t, ε, µ)‖ =
p∑
i=1
max
G2
|ηi(t, ε, µ)|+
n−p∑
i=1
max
G2
|δi(t, ε, µ)|
and the distance between the elements
‖y2(t, ε, µ)− y1(t, ε, µ) =
p∑
i=1
max
G2
∣∣η2
i (t, ε, µ)− η1
i (t, ε, µ)
∣∣+
n−p∑
i=1
max
G2
∣∣δ2
i (t, ε, µ)− δ1
i (t, ε, µ)
∣∣ .
The right-hand side of (21) we consider as an action of the operator L(·) on the vector
function
y(·, ε, µ) =
(
η(·, ε, µ)
δ(·, ε, µ)
)
,
L(y) =
[
L1(y)
L2(y)
]
=
W (t, s, ε, µ)η(a, ε, µ) +
t∫
a
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
p
ds
W (t, s, ε, µ)δ(b, ε, µ) +
b∫
t
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
n−p
ds
.
(24)
We will prove that the operator L(y) is a contractive operator.
Lemma 5. Let the conditions be fulfilled:
1) ˜̃C1 < σ and C∗ < σ,
2) 0 < K0 <
1
4 ,
3) 0 < R <
2K0C
∗
σ(1− 2K0)
,
4) 0 < ε ≤ ε <
σ
2K0C∗
(1− 2K0)R,
5) ‖η(a, ε, µ)‖ ≤ R, ‖δ(b, ε, µ)‖ ≤ R.
Then the operator L(y) is a contraction.
Proof. Step 1. Primary we show that the operator L(y) maps the space M into itself,
‖L(y)‖ ≤ max
G2
∥∥∥∥∥∥W (t, s, ε, µ)η(a, ε, µ) +
t∫
a
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
p
ds
∥∥∥∥∥∥+
+ max
G2
∥∥∥∥∥∥W (t, s, ε, µ)δ(b, ε, µ) +
b∫
t
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
n−p
ds
∥∥∥∥∥∥ ≤
≤ ∆1 + ∆2, (25)
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48 L. I. KARANDZHULOV
where
∆1 = max
G2
[
‖W (t, s, ε, µ)‖ ‖η(a, ε, µ)‖+ ‖W (t, s, ε, µ)‖ ‖δ(b, ε, µ)‖
]
,
∆2 = max
G2
t∫
a
‖W (t, s, ε, µ)‖1
ε
‖[H̃n(s, η, δ, ε, µ)]p‖ds
+
+ max
G2
b∫
t
‖W (t, s, ε, µ)‖1
ε
‖[H̃n(s, η, δ, ε, µ)]n−p‖ds
.
Keeping in mind (22), the condition 4, 5 of the present lemma and Lemma 3 for estimates
of ∆1 and ∆2 we have
∆1 ≤ max
G2
[
K0 exp
(
−σ t− s
ε
)
R+K0 exp(−σs− t
ε
)R
]
≤ K0R(1 + 1) = 2K0R,
∆2 ≤ max
G2
t∫
a
K0 exp
(
−σ t− s
ε
)
1
ε
C∗εds+ max
G2
b∫
t
K0 exp
(
−σ s− t
ε
)
1
ε
C∗ε ds =
= K0C
∗max
G2
t∫
a
exp
(
−σ t− s
ε
)
ds+
b∫
t
exp
(
−σ s− t
ε
)
ds
=
= K0C
∗max
G2
ε
σ
[
1− exp
(
−σ t− a
ε
)
− exp
(
−σ b− t
ε
)
+ 1
]
≤
≤ K0C
∗ ε
σ
2
[
1− exp
(
−σb− a
ε
)]
≤ 2K0C
∗ ε
σ
.
We substitute the estimates for ∆1, ∆2 in (25) and get
‖L(y)‖ ≤ ∆1 + ∆2 ≤ 2K0R+ 2K0C
∗ ε
σ
.
Keeping in mind the condition 4, we get ‖L(y)‖ ≤ R, that is, the operator L(y) maps the
space M into itself.
Step 2. We will estimate the difference L(y2) − L(y1) (see (24)). According to (22), second
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 49
inequality in (20) and condition 4 we obtain
‖L(y2)− L(y1)‖ =
∥∥∥∥[ L1(y2)− L1(y1)
L2(y2)− L2(y1)
]∥∥∥∥ =
=
∥∥∥∥∥∥∥∥∥∥∥∥∥∥
t∫
a
W (t, s, ε, µ)
1
ε
[
H̃n
(
η2, δ2, s, ε, µ
)
− H̃n
(
η1, δ1, s, ε, µ
)]
p
ds
b∫
t
W (t, s, ε, µ)
1
ε
[
H̃n
(
η2, δ2, s, ε, µ
)
− H̃n
(
η1, δ1, s, ε, µ
)]
n−p
ds
∥∥∥∥∥∥∥∥∥∥∥∥∥∥
≤
≤
t∫
a
K0 exp
(
−σ t− s
ε
)
1
ε
˜̃
C1ε
(∥∥η2 − η1
∥∥+
∥∥δ2 − δ1
∥∥) ds+
+
b∫
t
K0 exp
(
−σ s− t
ε
)
1
ε
˜̃
C1ε
(∥∥η2 − η1
∥∥+
∥∥δ2 − δ1
∥∥) ds ≤
≤ 2K0
˜̃
C1
σ
ε
(
1− exp
(
−σ b− a
ε
))
‖y2 − y1‖ ≤
2K0
˜̃
C1
σ
ε‖y2 − y1‖ ≤
≤ 2K0
˜̃
C1
σ
ε‖y2 − y1‖ <
2K0
˜̃
C1
σ
σ
2K0C∗
(1− 2K0)R‖y2 − y1‖ =
=
˜̃
C1
C∗
(1− 2K0)R︸ ︷︷ ︸
Θ
‖y2 − y1‖ = Θ‖y2 − y1‖.
The conditions 1 – 3 of the lemma show that 0 < Θ < 1. Consequently, the operator L(y) is
a contraction.
We introduce the following notations:
D(ε) = l
([
B11W (·, a, ε) B12W (·, b, ε)
B21W (·, a, ε) B22W (·, b, ε)
])
— (n× n)-matrix, (26)
g(ε, µ) = −l
B
(·)∫
a
W (·, s, ε) 1
ε
[
H̃n(s, η, δ, ε, µ)
]
p
ds
b∫
(·)
W (·, s, ε)1
ε
[
H̃n(s, η, δ, ε, µ)
]
n−p
ds
— (n× 1)-vector. (27)
Let D(ε) = D0 +O
(
εs exp
(
−α
ε
))
, where D0 is a constant (n× n)-matrix.
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
50 L. I. KARANDZHULOV
Theorem 4. Let the conditions (H1) – (H5) (see Theorems 1 and 2), the conditions of Lemma 5,
and the conditions rankD0 = n, K3b1‖B‖ ≤ 2, where ‖D−1
0 ‖ ≤ K3, K3 > 0, ‖l(ψ)‖ ≤ b1‖ψ‖,
b1 > 0, be satisfied. Then there exist constants ε∗ > 0, C̃∗ > 0 such that the problem (1), (2) has
a unique solution x(t, ε) and it satisfies the inequality
‖x(t, ε)−Xn(t, ε)‖ ≤ C̃∗εn+1 (28)
for t ∈ [a, b] and ε ∈ (0, ε∗] .
Proof. To prove that (1), (2) has the only solution satisfying (28) means to prove that the
boundary problem (3) has a unique solution satisfying
||z(t, ε, µ)− Zn(t, ε, µ)|| ≤ C̃∗εn+1
for t ∈ [a, b], ε ∈ (0, ε∗] and µ ∈ (0, ε∗] .
Therefore (3) we make the following replacement (11) and obtain the boundary problem
(12). To prove the theorem it is sufficient to show that (12) has a unique solution such that
‖u(t, ε, µ)‖ ≤ C̃∗εn+1. For system (12) we make a change of the variable
u(t, ε, µ) = B
(
η(t, ε, µ)
δ(t, ε, µ)
)
,
where η(t, ε, µ) = (η1, . . . , ηp)
T , δ(t, ε, µ) = (δ1, . . . , δn−p)
T , and consider the equivalent integral
equation (21).
Lemma 5 shows that system (21) has only a solution which does not go out of the area
Ω = {(t, η, δ, ε, µ)|t ∈ [a, b], ‖η‖ ≤ R, ‖δ‖ ≤ R, ε ∈ (0, ε], µ ∈ (0, ε]}
and depends on arbitrary constant vectors η(a, ε, µ) and δ(b, ε, µ). The determination of the
vectors η(a, ε, µ) and δ(b, ε, µ) is performed using the algebraic system (18),
lu(·, ε, µ) = lB
(
η(·, ε, µ)
δ(·, ε, µ)
)
= 0.
We substitute η and δ of (21) in the last equation, and according to the notations of (26),
(27) we find that η(a, ε, µ) and δ(b, ε, µ) satisfy the algebraic system
D(ε)
η(a, ε, µ)
δ(b, ε, µ)
= g(ε, µ).
After dropping exponentially small elements in D(ε), we get the system
D0
η(a, ε, µ)
δ(b, ε, µ)
= g(ε, µ). (29)
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ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR DIFFERENTIAL EQUATIONS . . . 51
Under the condition of the theorem, the system (29) has a solution η(a, ε, µ)
δ(b, ε, µ)
= D
−1
0 g(ε, µ). (30)
An estimate of (30), according to the conditions of the theorem and the proof of a Lemma 5,
has the form∥∥∥∥[ η(a, ε, µ)
δ(b, ε, µ)
]∥∥∥∥ ≤ ∥∥∥D−1
0
∥∥∥ ‖g(ε, µ)‖ ≤ K3‖g(ε, µ)‖ = K3b1 ‖B‖∆2 ≤ 2R.
In the latter inequality we used the estimate for ∆2 (see step 1 of the Lemma 5) and the condi-
tion K3b1‖B‖ ≤ 2. Thus for integral equations (21) we find the representation
η(t, ε, µ) = W (t, s, ε, µ)
[
D
−1
0 g(ε, µ)
]
p
+
t∫
a
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
p
ds,
δ(t, ε, µ) = W (t, s, ε, µ)
[
D
−1
0 g(ε, µ)
]
n−p
+
b∫
t
W (t, s, ε, µ)
1
ε
[
H̃n(t, η, δ, ε, µ)
]
n−p
ds.
(31)
For the integral equations (31) we apply the iteration process (23). For the first approximation
of (23), keeping in mind Lemma 3 and that u = (η1, . . . , ηp, δ1, . . . , δn−p) we find
‖u1 − u0‖ ≤
p∑
i=1
max
∣∣η1
i − η0
i
∣∣+
n−p∑
i=1
max
∣∣δ1
i − δ0
i
∣∣ =
= max
G2
∥∥η1 − η0
∥∥+ max
G2
∥∥δ1 − δ0
∣∣ ≤
≤ max
G2
[
‖W (t, a, ε, µ)‖‖η(a, ε, µ)‖+ ‖W (t, b, ε, µ)‖‖δ(b, ε, µ)‖
]
+
+ max
G2
{ t∫
a
∥∥∥∥W (t, a, ε, µ)
1
ε
∥∥∥∥∥∥∥∥[H̃n(s, 0, 0, ε, µ)
]
p
∥∥∥∥ ds+
+
b∫
t
∥∥W (t, b, ε, µ)
∥∥ 1
ε
∥∥∥∥[H̃n(s, 0, 0, ε, µ)
]
n−p
∥∥∥∥ ds
}
= ∆1 + ∆
0
2,
where
∆
0
2 = max
G2
t∫
a
∥∥∥∥W (t, s, ε, µ)
1
ε
∥∥∥∥∥∥∥∥[H̃n(s, 0, 0, ε, µ)
]
p
∥∥∥∥ ds+
ISSN 1562-3076. Нелiнiйнi коливання, 2018, т . 21, N◦ 1
52 L. I. KARANDZHULOV
+
b∫
t
∥∥W (t, s, ε, µ)
∥∥ 1
ε
∥∥∥∥[H̃n(s, 0, 0, ε, µ)
]
n−p
∥∥∥∥ ds
.
It is possible to prove that ∆1 ≤ K1εn+1, K1 > 0, ∆
0
2 ≤ K2εn+1, K2 > 0. Consequently,
‖u1 − u0‖ ≤ Kεn+1 or ‖u1 − u0‖ ≤
α
2
, α = 2Kεn+1.
With the help of the second inequalities in (20) we obtain
∥∥uj − uj−1
∥∥ ≤ 1
2
∥∥uj−1 − uj−2
∥∥ .
Therefore we have
‖u1 − u0‖ ≤
α
2
, ‖u2 − u1‖ ≤
α
22
, . . . , ‖uj − uj−1‖ ≤
α
2j
.
Then, in the domain G2,
‖uj(t, ε, εµ)‖ ≤
j∑
k=1
‖uk(t, ε, µ)− uk−1(t, ε, µ)‖ ≤
≤
(
1 +
1
2
+
1
22
+ . . .+
1
2j−1
+ . . .
)
α
2
≤ Kεn+1.
Let limj→∞ uj(t, ε, εµ) = u(t, ε, εµ). Then (22) becomes an identify. Then there exist constants
C∗ > 0 and ε∗ > 0, ε∗ ≤ ε such that, in the region,
Ω1 = {(t, η, δ, ε, µ)|t ∈ [a, b], ‖η‖ ≤ R, ‖δ‖ ≤ R, ε ∈ (0, ε∗] , µ ∈ (0, ε∗]}
we have
‖u(t, ε, µ)‖ =
∥∥∥∥[ η(a, ε, µ)
δ(b, ε, µ)
]∥∥∥∥ ≤ C̃∗εn+1.
Theorem 4 shows that the obtained formal series (4) is asymptotic and limε→0 x(t, ε) = x0(t).
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Received 20.10.17
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