Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems
We examine the robustness of exponential dichotomies of boundary-value problems for general linear first-order one-dimensional hyperbolic systems. It is assumed that the boundary conditions guarantee an increase in the smoothness of solutions in a finite time interval, including the reflection bound...
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Kmit, I.Ya. Recke, L. Tkachenko, V.I. 2020-02-11T11:55:43Z 2020-02-11T11:55:43Z 2013 Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems / I.Ya. Kmit, L. Recke, V.I. Tkachenko // Український математичний журнал. — 2013. — Т. 65, № 2. — С. 236-251. — Бібліогр.: 21 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164979 517.9 We examine the robustness of exponential dichotomies of boundary-value problems for general linear first-order one-dimensional hyperbolic systems. It is assumed that the boundary conditions guarantee an increase in the smoothness of solutions in a finite time interval, including the reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations. Вивчається грубiсть експоненцiальної дихотомiї для крайових задач для загальних лiнiйних гiперболiчних систем першого порядку. Припускається, що крайовi умови забезпечують пiдвищення гладкостi розв’язкiв за скiнченний промiжок часу, що дозволяє також розглядати умови вiдбиття вiд межi областi. Показано, що дихотомiя зберiгається у просторi неперервних функцiй при малих збуреннях всiх коефiцiєнтiв диференцiальних рiвнянь. en Інститут математики НАН України Український математичний журнал Статті Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems Грубiсть експоненцiальних дихотомiй крайових задач для загальних гiперболiчних систем першого порядку Article published earlier |
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Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems |
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Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems Kmit, I.Ya. Recke, L. Tkachenko, V.I. Статті |
| title_short |
Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems |
| title_full |
Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems |
| title_fullStr |
Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems |
| title_full_unstemmed |
Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems |
| title_sort |
robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems |
| author |
Kmit, I.Ya. Recke, L. Tkachenko, V.I. |
| author_facet |
Kmit, I.Ya. Recke, L. Tkachenko, V.I. |
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Статті |
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Статті |
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2013 |
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English |
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Український математичний журнал |
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Інститут математики НАН України |
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Грубiсть експоненцiальних дихотомiй крайових задач для загальних гiперболiчних систем першого порядку |
| description |
We examine the robustness of exponential dichotomies of boundary-value problems for general linear first-order one-dimensional hyperbolic systems. It is assumed that the boundary conditions guarantee an increase in the smoothness of solutions in a finite time interval, including the reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.
Вивчається грубiсть експоненцiальної дихотомiї для крайових задач для загальних лiнiйних гiперболiчних систем першого порядку. Припускається, що крайовi умови забезпечують пiдвищення гладкостi розв’язкiв за скiнченний промiжок часу, що дозволяє також розглядати умови вiдбиття вiд межi областi. Показано, що дихотомiя зберiгається у просторi неперервних функцiй при малих збуреннях всiх коефiцiєнтiв диференцiальних рiвнянь.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/164979 |
| citation_txt |
Robustness of the exponential dichotomies of boundary-value problems for the general first-order hyperbolic systems / I.Ya. Kmit, L. Recke, V.I. Tkachenko // Український математичний журнал. — 2013. — Т. 65, № 2. — С. 236-251. — Бібліогр.: 21 назв. — англ. |
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2025-11-25T20:45:51Z |
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2025-11-25T20:45:51Z |
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| fulltext |
UDC 517.9
I. Ya. Kmit
(Inst. Appl. Problems Mech. and Math. Nat. Acad. Sci. Ukraine, Lviv and Humboldt Univ. Berlin, Germany),
L. Recke (Humboldt Univ. Berlin, Germany),
V. I. Tkachenko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE
PROBLEMS FOR GENERAL FIRST-ORDER HYPERBOLIC SYSTEMS*
ГРУБIСТЬ ЕКСПОНЕНЦIАЛЬНИХ ДИХОТОМIЙ КРАЙОВИХ ЗАДАЧ
ДЛЯ ЗАГАЛЬНИХ ГIПЕРБОЛIЧНИХ СИСТЕМ ПЕРШОГО ПОРЯДКУ
We examine the robustness of exponential dichotomies of boundary-value problems for general linear first-order one-
dimensional hyperbolic systems. It is assumed that the boundary conditions guarantee an increase in the smoothness of
solutions in a finite time interval, which includes reflection boundary conditions. We show that the dichotomy survives in
the space of continuous functions under small perturbations of all coefficients in the differential equations.
Вивчається грубiсть експоненцiальної дихотомiї для крайових задач для загальних лiнiйних гiперболiчних систем
першого порядку. Припускається, що крайовi умови забезпечують пiдвищення гладкостi розв’язкiв за скiнченний
промiжок часу, що дозволяє також розглядати умови вiдбиття вiд межi областi. Показано, що дихотомiя зберiгається
у просторi неперервних функцiй при малих збуреннях всiх коефiцiєнтiв диференцiальних рiвнянь.
1. Introduction and main results. The concept of exponential dichotomy plays a crucial role in
various aspects of the perturbation and the stability theory [3, 4, 17 – 19]. An important problem
here is robustness of the exponential dichotomy of a system, i.e., its stability with respect to small
perturbations in the system. This problem is extensively studied in the literature, e.g., in [6, 13, 14, 20]
for finite-dimensional case and in [2, 7, 15] for infinite-dimensional case. It should be noted that the
hyperbolic case (see, e.g., [16]) seems more complicated here in comparison to ODEs and parabolic
PDEs, mostly due to worse regularity properties of hyperbolic operators.
We address the issue of stability of exponential dichotomies for general linear one-dimensional
first-order hyperbolic systems(
∂t + a(x, t, ε)∂x + b(x, t, ε)
)
u = 0, x ∈ (0, 1), (1.1)
subjected to (nonlocal) boundary conditions
uj(0, t) =
n∑
k=m+1
pjk(t)uk(0, t) +
m∑
k=1
pjk(t)uk(1, t), 1 ≤ j ≤ m,
uj(1, t) =
m∑
k=1
qjk(t)uk(0, t) +
n∑
k=m+1
qjk(t)uk(1, t), m < j ≤ n.
(1.2)
Here u = (u1, . . . , un) is a vector of real-valued functions, a = diag(a1, . . . , an) and b = {bjk}nj,k=1
are matrices of real-valued functions, and 0 ≤ m ≤ n are fixed integers.
Set
*Partially supported by the Alexander von Humboldt Foundation (I. Ya. Kmit) and the DFG Research Center MATHEON
mathematics for key technologies (project D8) (I. Ya. Kmit and L. Recke).
c© I. YA. KMIT, L. RECKE, V. I. TKACHENKO, 2013
236 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 237
Π = {(x, t) : 0 < x < 1, −∞ < t <∞}.
Assume that there exists ε0 > 0 such that for all ε ∈ [0, ε0] and all (x, t) ∈ Π the following conditions
are fulfilled:
aj , bjk, pjk, qjk are continuously differentiable in x, t, ε for all j, k ≤ n, (1.3)
aj > 0 for all j ≤ m and aj < 0 for all j > m, (1.4)
inf
x,t
|aj | > 0 for all j ≤ n, (1.5)
sup
x,t
{|aj |, |∂xaj |, |∂taj |, |∂εaj |} <∞ for all j ≤ n, (1.6)
sup
x,t
{|pjk|, |qjk|, |bjk|, |∂εbjk|, |∂tbjk|} <∞ for all j, k ≤ n, (1.7)
for all 1 ≤ j 6= k ≤ n there exist βjk, γjk ∈ C1 ([0, 1]× R× [0, ε0))
such that bjk(x, t, 0) = βjk(x, t, ε) (ak(x, t, ε)− aj(x, t, 0))
and bjk(x, t, ε) = γjk(x, t, ε) (ak(x, t, ε)− aj(x, t, ε)),
(1.8)
and
sup
x,t
{
|∂xβjk|, |∂tβjk|, |∂xγjk|, |∂tγjk|
}
<∞ for all j 6= k. (1.9)
Given s ∈ R, set
Πs =
{
(x, t) : 0 < x < 1, s < t <∞
}
.
We subject the system (1.1), (1.2) by the initial conditions at time t = s:
u(x, s) = ϕ(x), x ∈ [0, 1], (1.10)
and consider the initial boundary-value problem (1.1), (1.2), (1.10) in Πs for arbitrarily fixed s ∈ R.
Now we intend to switch to a weak formulation of the latter using integration along characteristic
curves: For given j ≤ n, x ∈ [0, 1], t ∈ R, and ε ∈ [0, ε0] the j-th characteristic of (1.1) passing
through the point (x, t) ∈ Πs is defined as the solution ξ ∈ [0, 1] 7→ ωj(ξ;x, t, ε) ∈ R of the initial
value problem
∂ξωj(ξ;x, t, ε) =
1
aj(ξ, ωj(ξ;x, t, ε), ε)
, ωj(x;x, t, ε) = t. (1.11)
Write
cj(ξ, x, t, ε) = exp
ξ∫
x
(
bjj
aj
)
(η, ωj(η;x, t, ε), ε) dη, dj(ξ, x, t, ε) =
cj(ξ, x, t, ε)
aj(ξ, ωj(ξ;x, t, ε), ε)
.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
238 I. YA. KMIT, L. RECKE, V. I. TKACHENKO
Due to (1.5), the characteristic curve τ = ωj(ξ;x, t, ε) reaches the boundary of Πs in two points with
distinct ordinates. Let xj(x, t, ε) denote the abscissa of that point whose ordinate is smaller. Let us
introduce linear bounded operators R : C
(
Πs
)n 7→ C ([s,∞))n and Bε : C
(
Πs
)n 7→ C
(
Πs
)n
and
an affine bounded operator S : C
(
Πs
)n 7→ C
(
Πs
)n
by
(Ru)j(t) =
n∑
k=m+1
pjk(t)uk(0, t) +
m∑
k=1
pjk(t)uk(1, t), 1 ≤ j ≤ m,
(Ru)j(t) =
m∑
k=1
qjk(t)uk(0, t) +
n∑
k=m+1
qjk(t)uk(1, t), m < j ≤ n,
(1.12)
(Bεu)j(x, t) = cj(xj(x, t, ε), x, t, ε)uj (xj(x, t, ε), ωj(xj(x, t, ε);x, t, ε)) , (1.13)
and
(Su)j(x, t) =
(Ru)j(t) if t > s,
ϕj(x) if t = s.
(1.14)
By abuse of notation, we did not indicate the dependence of the above operators on s; in fact, in the
consideration below the value of s ∈ R will be arbitrarily fixed.
Straightforward calculations show that a C1-map u : [0, 1] × [0,∞) → Rn is a solution to (1.1),
(1.2), (1.10) if and only if it satisfies the following system of integral equations:
uj(x, t) = (BεSu)j(x, t)−
−
x∫
xj(x,t,ε)
dj(ξ, x, t, ε)
n∑
k=1
k 6=j
bjk(ξ, ωj(ξ;x, t, ε), ε)uk(ξ, ωj(ξ;x, t, ε))dξ, j ≤ n. (1.15)
Now, the notion of weak (continuous) solution in Πs can be naturally defined as follows.
Definition 1.1. A continuous function u is called a continuous solution to (1.1), (1.2), (1.10)
in Πs if it satisfies (1.15).
For given ε > 0, denote by U ε(t, s) : C([0, 1])n 7→ C([0, 1])n the evolution operator of the
system (1.1), (1.2) whose existence is given by Theorem 2.1, i.e., a bounded operator mapping the
values of solutions at time s into their values at time t and satisfying the properties U ε(s, s) = I and
U ε(t, s)U ε(s, τ) = U ε(t, τ) for all t ≥ s ≥ τ.
We examine robustness of exponential dichotomies for a range of boundary operators ensuring
that smoothness of solutions increases in finite time. With this aim we will assume that the sys-
tem (1.1), (1.2) has a smoothing property studied in [9, 10].
Definition 1.2. Let ε > 0. The evolution operator U ε(t, s) to the problem (1.1), (1.2) is called
smoothing if, for every s ∈ R, there exists t > s such that U ε(t, s)ϕ ∈ C1 ([0, 1])n for every
ϕ ∈ C ([0, 1])n .
In the following definition the range of an operator P will be denoted by ImP.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 239
Definition 1.3. Let ε > 0. We say that the system (1.1), (1.2) has an exponential dichotomy on
R with exponent β > 0 and bound M if there exist projections P ε(t), t ∈ R, such that
(i) U ε(t, s)P ε(s) = P ε(t)U ε(t, s), t ≥ s;
(ii) U ε(t, s)|Im(P ε(s)) for t ≥ s is an isomorphism on Im(P ε(s)), then U ε(s, t) is defined as an
inverse map from Im(P ε(t)) to Im(P ε(s));
(iii) ‖U ε(t, s)(1− P ε(s))‖ ≤Me−β(t−s), t ≥ s;
(iv) ‖U ε(t, s)P ε(s)‖ ≤Meβ(t−s), t ≤ s.
Here and below by ‖ · ‖ we denote the operator norm in L (C([0, 1])n) .
Now we formulate our main result.
Theorem 1.1. Suppose that the system (1.1), (1.2) with ε = 0 has an exponential dichotomy
and the corresponding evolution operator U0(t, s) is bounded:
sup
0≤t−s≤1
‖U0(t, s)‖ <∞. (1.16)
Moreover, assume that there is ε0 > 0 such that the following conditions are fulfilled: (1.3) – (1.9)
and
there exists k ∈ N such that (BεR)k = 0 for all ε ≤ ε0. (1.17)
Then there exists ε′ ≤ ε0 such that for all ε ≤ ε′ the system (1.1), (1.2) has an exponential dichotomy.
Remark 1.1. Note that the boundary conditions (1.2) together with the property (1.17) general-
ize boundary conditions appearing in models of chemical kinetics [1, 21].
2. Basic facts. The first fact follows from the results obtained in [8, 11] and entails, in particular,
the existence of an evolution operator.
Theorem 2.1. Under the conditions (1.3) – (1.9), for given ε > 0, s ∈ R, T > 0, and ϕ ∈
∈ C ([0, 1])n fulfilling the zero-order compatibility conditions
ϕj(0) = (Rϕ)j(s), 1 ≤ j ≤ m,
ϕj(1) = (Rϕ)j(s), m < j ≤ n,
(2.1)
the initial boundary-value problem (1.1), (1.2), (1.10) has a unique continuous solution in Πs and
this solution satisfies the apriori estimate
‖u‖C(Πs\Πs+T )
n ≤ C(T )‖ϕ‖C([0,1])n (2.2)
with a constant C(T ) > 0 depending on T, but not on s, ϕ, and ε ≤ ε0.
The second fact can be readily obtained by [10] (Theorem 2.7) and the argument used in its
proof. It states the smoothing property of the evolution operator as well as the fact that the time at
which the continuous solution to (1.1), (1.2), (1.10) reaches the C1-regularity does not exceed a fix
number d, whatsoever initial time s ∈ R.
Lemma 2.1. Under the conditions (1.3) – (1.9) and (1.17) the evolution operator is smoothing
and satisfies the following property:
there exists d > 0 such that for any s ∈ R and t as in Definition 1.2,
the inequality |t− s| ≤ d is true for all ε ≤ ε0.
(2.3)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
240 I. YA. KMIT, L. RECKE, V. I. TKACHENKO
The third fact is a variant of [5] (Theorem 7.6.10).
Theorem 2.2. Assume that the evolution operator U0(t, s) has an exponential dichotomy on
R and satisfies (1.16). Then there exists η > 0 such that for all ε > 0 with
‖U0(t, s)− U ε(t, s)‖ < η, whenever t− s = 2d
the evolution operator U ε(t, s) has an exponential dichotomy on R also.
Proof. Given s ∈ R and ε > 0, set
tn = s+ 2dn, T εn = U ε(t0 + 2d(n+ 1), t0 + 2dn) for n ∈ Z.
If the evolution operator U0(t, s) has an exponential dichotomy, then the sequence
{
T 0
n
}
has a
discrete dichotomy in the sense of [5] (Definition 7.6.4).
By [5] (Theorem 7.6.7), there exists η > 0 such that for all ε > 0 with
sup
n
‖T 0
n − T εn‖ ≤ η
{T εn} has a discrete dichotomy.
Now we are in the conditions of [5, p. 229, 230], Excersise 10 (see also a more general statement
[7], Theorem 4.1), what finishes the proof.
3. Proof of Theorem 1.1. Given s ∈ R and ε > 0, let us introduce linear bounded operators Dε,
F ε : C
(
Πs
)n → C
(
Πs
)n
by
(Dεw)j (x, t) = −
x∫
xj(x,t,ε)
dj(ξ, x, t, ε)
n∑
k=1
k 6=j
bjk(ξ, ωj(ξ;x, t, ε), ε)wk(ξ, ωj(ξ;x, t, ε)) dξ,
(F εf)j (x, t) =
x∫
xj(x,t,ε)
dj(ξ, x, t, ε)fj(ξ, ωj(ξ;x, t, ε))dξ.
Here again we dropped the dependence of Dε and F ε on s, as throughout the proof s ∈ R is
arbitrarily fixed. To simplify further notation, set
a(x, t) = a(x, t, 0), b(x, t) = b(x, t, 0), cj(x, t) = cj(x, t, 0), dj(x, t) = dj(x, t, 0),
aε(x, t) = a(x, t, ε), bε(x, t) = b(x, t, ε), βεjk(x, t) = βjk(x, t, ε),
ωj(ξ;x, t) = ωj(ξ;x, t, 0), xj(x, t) = xj(x, t, 0), D = D0, F = F 0.
(3.1)
Fix arbitrary values s ∈ R and ε ≤ ε0 and an arbitrary initial function ϕ ∈ C ([0, 1])n in
(1.10). Let u and v be the continuous solutions to the problem (1.1), (1.2), (1.10) with ε = 0 and ε,
respectively. By Lemma 2.1, the evolution operator U ε(t, s) is smoothing with the time of smoothing
not exceeding d. This means that starting at t = s + d the solutions u and v are continuously
differentiable and, therefore, satisfy the system (1.1) pointwise. Hence, the difference u − v fulfills
the equation
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 241
(∂t + a(x, t)∂x + b(x, t))(u− v) = (aε(x, t)− a(x, t)) ∂xv + (bε(x, t)− b(x, t)) v, (x, t) ∈ Πs+d,
(3.2)
and the boundary conditions
(uj − vj)(0, t) = (R(u− v))j (t), 1 ≤ j ≤ m, t ≥ s,
(uj − vj)(1, t) = (R(u− v))j (t), m < j ≤ n, t ≥ s,
(3.3)
or, the same, the operator equation
(u− v)
∣∣
Πs+d
= BR(u− v) +D(u− v) + F ((aε − a) ∂xv) + F ((bε − b) v) . (3.4)
A similar equation is true for u− v under the operator BR, what entails
(u− v)
∣∣
Πs+d
= (BR)2(u− v) + (I +BR)D(u− v)+
+(I +BR)F ((aε − a) ∂xv) + (I +BR)F ((bε − b) v) .
Doing this iteration, on the k-th step we meet the property (see (1.17))
(BR)k (u− v) ≡ 0 (3.5)
and, hence, get the formula
(u− v)
∣∣
Πs+d
=
k−1∑
i=0
(BR)iD(u− v) +
k−1∑
i=0
(BR)iF ((aε − a) ∂xv) +
k−1∑
i=0
(BR)iF ((bε − b) v).
In particular,
(u− v)(x, s+ 2d) =
k−1∑
i=0
[
(BR)iD(u− v)
]
(x, s+ 2d)+
+
k−1∑
i=0
[
(BR)iF ((aε − a) ∂xv)
]
(x, s+ 2d) +
k−1∑
i=0
[
(BR)iF ((bε − b) v)
]
(x, s+ 2d). (3.6)
Therefore, on the account of Theorem 2.2, we are done if we show that, given η > 0, there is ε′ ≤ ε0
such that
‖(u− v)(·, s+ 2d)‖C([0,1])n ≤ η‖ϕ‖C([0,1])n , (3.7)
the bound being uniform in s ∈ R, ε ≤ ε′, and ϕ ∈ C ([0, 1])n . To derive (3.7), we estimate each of
the three sums in (3.6) separately.
To obtain the desired estimate for the first sum in (3.6), we first derive the formula for D(u− v)
contributing into this summand. To this end, use the operator representation for u and v, namely,
u = BSu+Du, v = BεSv +Dεv,
where the functions u and v are restricted to Πs \ Πs+2d and the operators Bε, S, and Dε are
restricted to C
(
Πs \Πs+2d
)n
. Note that, as it follows from the definition, Bε, S, and Dε map
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
242 I. YA. KMIT, L. RECKE, V. I. TKACHENKO
C
(
Πs \Πs+2d
)n
into C
(
Πs \Πs+2d
)n
. Thus, for the difference we have
u− v = BS(u− v) + (B −Bε)Sv +D(u− v) + (D −Dε)v, (3.8)
hence
D(u− v) = DBS(u− v) +D (B −Bε)Sv +D2(u− v) +D(D −Dε)v. (3.9)
Our next objective is to rewrite the last equation with respect to the new variable
w = D(u− v). (3.10)
With this aim we substitute (3.8) into the first summand in the right-hand side of (3.9) and get
w = D(BS)2(u− v) +D(I +BS) (B −Bε)Sv +D(I +BS)w +D(I +BS)(D −Dε)v.
(3.11)
Continuing in this fashion (again substituting (3.8) into the first summand in the right-hand side of
(3.11)), on the r-th step we arrive at the formula
w = D(BS)r(u− v) +D
r−1∑
i=0
(BS)i (B −Bε)Sv+
+D
r−1∑
i=0
(BS)iw +D
r−1∑
i=0
(BS)i(D −Dε)v. (3.12)
Since (u − v)(·, s) ≡ 0 on [0, 1], there exists r0 ∈ N such that (BS)r0(u − v) = 0. Therefore, the
resulting equation for w restricted to Πs \Πs+2d can be written as
w = D
r0−1∑
i=0
(BS)i (B −Bε)Sv +D
r0−1∑
i=0
(BS)i(D −Dε)v +D
r0−1∑
i=0
(BS)iw. (3.13)
Our goal now is to show the existence of a function α : [0, 1] → R with α(ε) → 0 as ε → 0 for
which we have
‖w‖C(Πs\Πs+2d)
n ≤ α(ε)‖ϕ‖C([0,1])n , (3.14)
the estimate being uniform in s ∈ R and ϕ ∈ C([0, 1])n satisfying the zero-order compatibility
conditions (2.1). With this aim we first show that there is a function α̃(ε) meeting the same properties
as α(ε) such that the C
(
Πs \Πs+2d
)n
-norm of the first two summands in the right-hand side of
(3.13) is bounded from above by α̃(ε)‖ϕ‖C([0,1])n . Afterwords, we use the boundedness of the
operators B, S, D restricted to C
(
Πs \Πs+2d
)n
, then apply Gronwall’s inequality to (3.13), and
this way derive (3.14). To this end, observe that the integral operator D can be considered as Volterra
operator of the second kind. This follows from the fact that D can be equivalently defined by the
formula
(Dw)j (x, t) = −
t∫
tj(x,t)
d̃j(τ, x, t)
n∑
k=1
k 6=j
bjk(ω̃j(τ ;x, t), τ)wk(ω̃j(τ ;x, t), τ) dτ,
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ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 243
where τ ∈ R 7→ ω̃j(τ ;x, t) ∈ [0, 1] is the inverse form of the j-th characteristic of (1.1) passing
through the point (x, t) ∈ Π, tj(x, t) is a minimum value of τ at which the characteristic τ =
= ω̃j(τ ;x, t) reaches ∂Πs, and
d̃j(τ, x, t) = exp
τ∫
t
bjj(ω̃j(η;x, t), η) dη.
Thus, the estimate (3.14) will be proved as soon as we derive the upper bound α̃(ε)‖ϕ‖C([0,1])n
for the absolute value of the first two summands in (3.13). The idea behind the proof is a smoothing
property of the operators representing those summands. We prove this only for one summand in each
sum (when i = 0). For all other summands we apply similar argument.
Thus, to get the desired estimate for the summand D(D −Dε)v, it suffices to show that, given
j ≤ n, the function (DDεv)j (x, t) is continuously differentiable in ε and that the derivative is
bounded on Πs \ Πs+2d uniformly in s ∈ R and ε ≤ ε0. Indeed, following the techniques from [9],
fix a sequence vl ∈ C1
(
Π
)n
such that
vl → v in C
(
Πs \Πs+2d
)n
as l→∞. (3.15)
We are done if we prove that ∂ε
[(
DDεvl
)
j
(x, t)
]
converges in C
(
Πs \Πs+2d
)
as l→∞ and that
the limit function is bounded on Πs \Πs+2d uniformly in s ∈ R and ε ≤ ε0. Consider the following
expression for
(
DDεvl
)
j
(x, t):
(
DDεvl
)
j
(x, t) =
n∑
k=1
k 6=j
n∑
i=1
i6=k
x∫
xj(x,t)
ξ∫
xk(ξ,ωj(ξ;x,t),ε)
djki(ξ, η, x, t, ε)bjk(ξ, ωj(ξ;x, t))×
×vli(η, ωk(η; ξ, ωj(ξ;x, t), ε)) dηdξ (3.16)
with
djki(ξ, η, x, t, ε) = dj(ξ, x, t)dk(η, ξ, ωj(ξ;x, t), ε)bki(η, ωk(η; ξ, ωj(ξ;x, t), ε), ε).
Let xjk(x, t, ε) denote the x-coordinate of the point (if any) where the characteristics ωj(ξ;x, t) and
ωk(ξ; 0, s, ε) if k ≤ m and the characteristics ωj(ξ;x, t, ε) and ωk(ξ; 1, s, ε) if k > m intersect.
Hence, xjk(x, t, ε) satisfies the equation
ωj(xjk(x, t, ε);x, t) = ωk(xjk(x, t, ε); 0, s, ε) (3.17)
if k ≤ m, and the equation
ωj(xjk(x, t, ε);x, t) = ωk(xjk(x, t, ε); 1, s, ε) (3.18)
if k > m. Suppose for definiteness that j ≤ m and k > m (similar argument works for all other
j 6= k). Thus, if xjk(x, t, ε) exists for some (x, t, ε), then the integrals in (3.16) admit the decompo-
sition
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
244 I. YA. KMIT, L. RECKE, V. I. TKACHENKO
x∫
xj(x,t)
ξ∫
xk(ξ,ωj(ξ;x,t),ε)
dηdξ =
xjk(x,t,ε)∫
xj(x,t)
ξ∫
xk(ξ,ωj(ξ;x,t),ε)
dηdξ +
x∫
xjk(x,t,ε)
ξ∫
1
dηdξ, (3.19)
where the function xk(ξ, ωj(ξ;x, t), ε) in the right-hand side satisfies the equality
ωk(xk(ξ, ωj(ξ;x, t), ε); ξ, ωj(ξ;x, t), ε) = s. (3.20)
Now we intend to show that the derivatives ∂εxk(ξ, ωj(ξ;x, t), ε) and ∂εxjk(x, t, ε) exist. With this
aim we introduce a couple of useful formulas
∂xωj(ξ;x, t, ε) = − 1
aj(x, t, ε)
exp
x∫
ξ
(
∂taj
a2
j
)
(η, ωj(η;x, t, ε), ε)dη, (3.21)
∂tωj(ξ;x, t, ε) = exp
x∫
ξ
(
∂taj
a2
j
)
(η, ωj(η;x, t, ε), ε)dη, (3.22)
∂εωj(ξ;x, t, ε) = exp
x∫
ξ
(
∂taj
a2
j
)
(η, ωj(η;x, t, ε), ε)dη×
×
x∫
ξ
(
∂εaj
a2
j
)
(η, ωj(η;x, t, ε), ε)×
× exp
η∫
x
(
∂taj
a2
j
)
(η1, ωj(η1;x, t, ε), ε) dη1 dη. (3.23)
Then the existence of the derivatives ∂εxk(ξ, ωj(ξ;x, t), ε) and ∂εxjk(x, t, ε) follow from the equal-
ities (3.20) and (3.18), respectively. Furthermore, we derive the formulas
∂εxk(ξ, ωj(ξ;x, t), ε) = −ak(xk(ξ, ωj(ξ;x, t), ε), s)∂4ωk(xk(ξ, ωj(ξ;x, t), ε); ξ, ωj(ξ;x, t), ε)
(3.24)
and
∂εxjk(x, t, ε)
(
aεk − aj
ajaεk
)
(xjk(x, t, ε), ωj(xjk(x, t, ε);x, t)) = ∂4ωk(xjk(x, t, ε); 1, s, ε). (3.25)
Hence, on the account of the assumption (1.8), from the last equality we get
∂εxjk(x, t, ε)bjk(xjk(x, t, ε), ωj(xjk(x, t, ε);x, t)) =
=
(
βεjkaja
ε
k
)
(xjk(x, t, ε), ωj(xjk(x, t, ε);x, t))∂4ωk(xjk(x, t, ε); 1, s, ε). (3.26)
Now, using the regularity assumption (1.3), we are able to compute the derivative
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ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 245
∂ε
[
(DDεvl)j(x, t)
]
=
=
n∑
k=1
k 6=j
n∑
i=1
i 6=k
x∫
xj(x,t)
x∫
xk(ξ,ωj(ξ;x,t),ε)
∂ε
[
djki(ξ, η, x, t, ε)bjk(ξ, ωj(ξ;x, t))
]
×
×vli(η, ωk(η; ξ, ωj(ξ;x, t), ε)) dηdξ+
+
n∑
k=1
k 6=j
n∑
i=1
i6=k
x∫
xj(x,t)
x∫
xk(ξ,ωj(ξ;x,t),ε)
djki(ξ, η, x, t, ε)bjk(ξ, ωj(ξ;x, t))×
×∂εωk(η; ξ, ωj(ξ;x, t), ε)∂2v
l
i(η, ωk(η; ξ, ωj(ξ;x, t), ε)) dηdξ+
+
n∑
k=1
k 6=j
n∑
i=1
i 6=k
(
βεjkaja
ε
k
)
(xjk(x, t, ε), ωj(xjk(x, t, ε);x, t))∂4ωj(xjk(x, t, ε); 1, s, ε)×
×
ξ∫
xk(xjk(x,t,ε),ωj(xjk(x,t,ε);x,t),ε)
[
djki(ξ, η, x, t, ε)v
l
i(η, ωk(η; ξ, ωj(ξ;x, t), ε))
]
ξ=xjk(x,t,ε)
dη−
−
n∑
k=1
k 6=j
n∑
i=1
i 6=k
xjk(x,t,ε)∫
xj(x,t)
∂εxk(ξ, ωj(ξ;x, t), ε)bjk(ξ, ωj(ξ;x, t))×
×
[
djki(ξ, η, x, t, ε)v
l
i(η, ωk(η; ξ, ωj(ξ;x, t), ε))
]
η=xk(ξ,ωj(ξ;x,t),ε)
dξ−
−
n∑
k=1
k 6=j
n∑
i=1
i6=k
(
βεjkaja
ε
k
)
(xjk(x, t, ε), ωj(xjk(x, t, ε);x, t))∂4ωj(xjk(x, t, ε); 1, s, ε)×
×
ξ∫
1
[
djki(ξ, η, x, t, ε)v
l
i(η, ωk(η; ξ, ωj(ξ;x, t), ε))
]
ξ=xjk(x,t,ε)
dη, (3.27)
where ∂rg here and below denotes the derivative of g with respect to the r-th argument. Note that
xk(xjk(x, t, ε), ωj(xjk(x, t, ε);x, t), ε) = 1, hence the third and the fifth summands in the right-hand
side cancel out. The first and the fourth summands converge in C
(
Πs \Πs+2d
)
as l→∞. Our task
is therefore reduced to show the uniform convergence of all integrals in the second summand. For
this purpose we will transform the integrals as follows: Changing the order of integration and using
(1.3) and (1.8), we get (to simplify notation in the calculation below we drop the dependence of xj
on x and t)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
246 I. YA. KMIT, L. RECKE, V. I. TKACHENKO
x∫
xj
x∫
η
djki(ξ, η, x, t, ε)bjk(ξ, ωj(ξ;x, t))∂εωk(η; ξ, ωj(ξ;x, t), ε)×
×∂2v
l
i(η, ωk(η; ξ, ωj(ξ;x, t), ε)) dξdη =
=
x∫
xj
x∫
η
djki(ξ, η, x, t, ε)∂εωk(η; ξ, ωj(ξ;x, t), ε)bjk(ξ, ωj(ξ;x, t))×
×
[(
∂ξωk
)
(η; ξ, ωj(ξ;x, t), ε)
]−1(
∂ξv
l
i
)
(η, ωk(η; ξ, ωj(ξ;x, t), ε))dξdη =
=
x∫
xj
x∫
η
djki(ξ, η, x, t, ε)∂εωk(η; ξ, ωj(ξ;x, t), ε))
(
βεjkaja
ε
k
)
(ξ, ωj(ξ;x, t))×
×∂3ωk(η; ξ, ωj(ξ;x, t), ε)
(
∂ξv
l
i
)
(η, ωk(η; ξ, ωj(ξ;x, t), ε))dξdη =
=
x∫
xj
x∫
η
d̃jki(ξ, η, x, t, ε)
(
∂ξv
l
i
)
(η, ωk(η; ξ, ωj(ξ;x, t), ε))dξdη =
= −
x∫
xj
x∫
η
∂ξd̃jki(ξ, η, x, t, ε)v
l
i (η, ωk(η; ξ, ωj(ξ;x, t), ε)) dξdη+
+
x∫
xj
[
d̃jki(ξ, η, x, t, ε)v
l
i (η, ωk(η; ξ, ωj(ξ;x, t), ε))
]ξ=x
ξ=η
dη. (3.28)
Here
d̃jki(ξ, η, x, t, ε) = djki(ξ, η, x, t, ε)∂εωk(η; ξ, ωj(ξ;x, t), ε)×
×∂3ωk(η; ξ, ωj(ξ;x, t), ε)
(
βεjkaja
ε
k
)
(ξ, ωj(ξ;x, t)).
Now, the desired convergence follows from (3.15) and (3.21) – (3.23). The desired boundedness of
the limit function is a consequence of the assumptions (1.6), (1.7), and (1.9).
Returning to the formula (3.13), similar argument works also for the operators contributing into
the first sum: Again, for i = 0, on the account of the definition of the operators D and Bε, we have
to show that the ε-derivative of
(DBεvl)j(x, t) =
=
n∑
k=1
k 6=j
xj(x,t)∫
x
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))ck(xk(ξ, ωj(ξ;x, t), ε), ξ, ωj(ξ;x, t), ε)×
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ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 247
×vlk(xk(ξ, ωj(ξ;x, t), ε), ωk(xk(ξ, ωj(ξ;x, t), ε); ξ, ωj(ξ;x, t), ε)dξ (3.29)
converges uniformly on Πs \ Πs+2d and that the limit function is bounded uniformly in s ∈ R and
ε ≤ ε0. To show this, we differentiate (3.29) in ε, use (1.8), and integrate by parts. To be more
precise, fix arbitrary j ≤ m and k > m (similarly for all other j 6= k) and rewrite the k-th summand
in the right-hand side of (3.29) as (up to the sign)
xjk(x,t,ε)∫
xj(x,t)
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))ck(xk(ξ, ωj(ξ;x, t), ε), ξ, ωj(ξ;x, t), ε)×
×vlk(xk(ξ, ωj(ξ;x, t), ε), s)dξ+
+
x∫
xjk(x,t,ε)
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))ck(1, ξ, ωj(ξ;x, t), ε)×
×vlk(1, ωk(1; ξ, ωj(ξ;x, t), ε))dξ. (3.30)
Then the ε-derivative of this expression equals
x∫
xj(x,t)
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))∂εck(xk(ξ, ωj(ξ;x, t), ε), ξ, ωj(ξ;x, t), ε)×
×vlk(xk(ξ, ωj(ξ;x, t), ε), s)dξ+
+
xjk(x,t,ε)∫
xj(x,t)
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))ck(xk(ξ, ωj(ξ;x, t), ε), ξ, ωj(ξ;x, t), ε)×
×∂εxk(ξ, ωj(ξ;x, t), ε)∂1v
l
k(xk(ξ, ωj(ξ;x, t), ε), s)dξ+
+
x∫
xjk(x,t,ε)
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))ck(1, ξ, ωj(ξ;x, t), ε)×
×∂εωk(1; ξ, ωj(ξ;x, t), ε)∂2v
l
k(1, ωk(1; ξ, ωj(ξ;x, t), ε))dξ. (3.31)
For the first summand the desired convergence and the uniform boundedness of the limit function is
obvious. The last two summands are equal to
xjk(x,t,ε)∫
xj(x,t)
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))ck(xk(ξ, ωj(ξ;x, t), ε), ξ, ωj(ξ;x, t), ε)×
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
248 I. YA. KMIT, L. RECKE, V. I. TKACHENKO
×∂εxk(ξ, ωj(ξ;x, t), ε) [∂ξxk(ξ, ωj(ξ;x, t), ε)]
−1 ∂ξv
l
k(xk(ξ, ωj(ξ;x, t), ε), s)dξ+
+
x∫
xjk(x,t,ε)
dj(ξ, x, t)bjk(ξ, ωj(ξ;x, t))ck(1, ξ, ωj(ξ;x, t), ε)∂εωk(1; ξ, ωj(ξ;x, t), ε)×
× [∂ξωk(1; ξ, ωj(ξ;x, t), ε))]
−1 ∂ξv
l
k(1, ωk(1; ξ, ωj(ξ;x, t), ε))dξ. (3.32)
Next we use the formulas (3.20), (3.21), and (3.22) and calculate
∂ξxk(ξ, ωj(ξ;x, t), ε) = ak(xk(ξ, ωj(ξ;x, t), ε), s)
(
aεk − aj
ajaεk
)
(ξ, ωj(ξ;x, t))×
×∂3ωk(xk(ξ, ωj(ξ;x, t), ε); ξ, ωj(ξ;x, t), ε),
∂ξωk(1; ξ, ωj(ξ;x, t), ε) =
(
aεk − aj
ajaεk
)
(ξ, ωj(ξ;x, t))∂3ωk(1; ξ, ωj(ξ;x, t), ε).
Now, due to the assumptions (1.3) and (1.8), we are in a position to bring the expression (3.32) to a
desirable form
xjk(x,t,ε)∫
xj(x,t)
dj(ξ, x, t)ck(xk(ξ, ωj(ξ;x, t), ε), ξ, ωj(ξ;x, t), ε)∂εxk(ξ, ωj(ξ;x, t), ε)×
×ak(xk(ξ, ωj(ξ;x, t), ε), s)∂3ωk(xk(ξ, ωj(ξ;x, t), ε); ξ, ωj(ξ;x, t), ε)×
×
(
aja
ε
kβ
ε
jk
)
(ξ, ωj(ξ;x, t))∂ξv
l
k(xk(ξ, ωj(ξ;x, t), ε), s)dξ+
+
x∫
xjk(x,t,ε)
dj(ξ, x, t)ck(1, ξ, ωj(ξ;x, t), ε)∂εωk(1; ξ, ωj(ξ;x, t), ε)×
×∂3ωk(1; ξ, ωj(ξ;x, t), ε)
(
ajakβ
ε
jk
)
(ξ, ωj(ξ;x, t))∂ξv
l
k(1, ωk(1; ξ, ωj(ξ;x, t), ε))dξ =
= −
xjk(x,t,ε)∫
xj(x,t)
∂ξejk(ξ, x, t, ε)v
l
k(xk(ξ, ωj(ξ;x, t), ε), s)dξ+
+ejk(ξ, x, t, ε)v
l
k(xk(ξ, ωj(ξ;x, t), ε), s)
∣∣∣xjk(x,t,ε)
ξ=xj(x,t)
−
−
x∫
xjk(x,t,ε)
∂ξ ẽjk(ξ, x, t, ε)v
l
k(1, ωk(1; ξ, ωj(ξ;x, t), ε))dξ+
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ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 249
+ẽjk(ξ, x, t, ε)v
l
k(1, ωk(1; ξ, ωj(ξ;x, t), ε))
∣∣∣x
ξ=xjk(x,t,ε)
, (3.33)
where
ejk(ξ, x, t, ε) = dj(ξ, x, t)ck(xk(ξ, ωj(ξ;x, t), ε), ξ, ωj(ξ;x, t), ε)∂εxk(ξ, ωj(ξ;x, t), ε)×
×ak(xk(ξ, ωj(ξ;x, t), ε), s)∂3ωk(xk(ξ, ωj(ξ;x, t), ε); ξ, ωj(ξ;x, t), ε)
(
aja
ε
kβ
ε
jk
)
(ξ, ωj(ξ;x, t)),
ẽjk(ξ, x, t, ε) = dj(ξ, x, t)ck(1, ξ, ωj(ξ;x, t), ε)∂εωk(1; ξ, ωj(ξ;x, t), ε)×
× (aja
ε
kβjk) (ξ, ωj(ξ;x, t))∂3ωk(1; ξ, ωj(ξ;x, t), ε).
To finish with the first summand in (3.6) it remains, similarly to (3.28), apply the conditions (1.6),
(1.7), (1.9), (3.15), and the formulas (3.21) – (3.23).
The last two summands in (3.6) are treated by means of the assumptions (1.5), (1.6), (1.9) (entail-
ing, in particular, the uniform boundedness of the operators B and F restricted to C
(
Πs \Πs+2d
)
)
as well as by the smoothing apriori estimate
‖v‖C(Πs+d\Πs+2d)
n + ‖∂xv‖C(Πs+d\Πs+2d)
n ≤ C‖ϕ‖C([0,1])n , (3.34)
where the constant C > 0 depends on d but not on ε ≤ ε0 and s ∈ R. We are therefore reduced to
prove the estimate (3.34). To this end, we start with the operator representation of v in Πs+d \Πs+2d,
namely,
v = BεRv +Dεv.
After a number of iterations we derive the following formula suitable for our purposes:
v =
k−1∑
i=0
(BεR)i
(
DεBεR+ (Dε)2
)
v. (3.35)
The estimate (3.34) now readily follows from the smoothing property in x of the operators DεBε and
(Dε)2 and the apriori estimate (2.2) with 2d in place of T. Showing the smoothing property of the
operators DεBε and (Dε)2 in x, we follow a similar argument as in the proof above of the smoothing
property in ε. We illustrate this by example of the operator (Dε)2 (and similarly for DεBε): We take
into account that
[
(Dε)2 vl
]
j
(x, t) on Πs+d \ Πs+2d is given by the formula (3.16) where bjk is
replaced by bεjk; xj(x, t) ≡ 0 if j ≤ m; and xj(x, t) ≡ 1 if j > m. Below we therefore drop the
dependence of xj on x and t. Changing the order of integration, we have
∂x
[(
(Dε)2vl
)
j
(x, t)
]
=
=
n∑
k=1
k 6=j
n∑
i=1
i 6=k
x∫
xj
x∫
η
∂x
[
djki(ξ, η, x, t, ε)bjk(ξ, ωj(ξ;x, t, ε), ε)
]
vli(η, ωk(η; ξ, ωj(ξ;x, t, ε), ε))dξdη+
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
250 I. YA. KMIT, L. RECKE, V. I. TKACHENKO
+
n∑
k=1
k 6=j
n∑
i=1
i 6=k
x∫
xj
x∫
η
djki(ξ, η, x, t, ε)bjk(ξ, ωj(ξ;x, t, ε), ε)×
×∂3ωk(η; ξ, ωj(ξ;x, t, ε), ε)∂xωj(ξ;x, t, ε)∂2v
l
i(η, ωk(η; ξ, ωj(ξ;x, t, ε), ε))dξdη. (3.36)
Let us transform the second summand similarly to (3.28): For given k 6= j and i 6= k we have (using
the assumptions (1.3) and (1.8))
x∫
xj
x∫
η
djki(ξ, η, x, t, ε)bjk(ξ, ωj(ξ;x, t, ε), ε)×
×∂3ωk(η; ξ, ωj(ξ;x, t, ε), ε)∂xωj(ξ;x, t, ε)∂2v
l
i(η, ωk(η; ξ, ωj(ξ;x, t, ε), ε))dξdη =
=
x∫
xj
x∫
η
djki(ξ, η, x, t, ε)∂3ωk(η; ξ, ωj(ξ;x, t, ε), ε)∂xωj(ξ;x, t, ε)×
×bjk(ξ, ωj(ξ;x, t, ε), ε)
[(
∂ξωk
)
(η; ξ, ωj(ξ;x, t, ε), ε)
]−1
×
×
(
∂ξv
l
i
)
(η, ωk(η; ξ, ωj(ξ;x, t, ε), ε))dξdη =
=
x∫
xj
x∫
η
djki(ξ, η, x, t, ε)∂xωj(ξ;x, t, ε)
(
akajγjk
)
(ξ, ωj(ξ;x, t, ε), ε)×
×
(
∂ξv
l
i
)
(η, ωk(η; ξ, ωj(ξ;x, t, ε), ε))dξdη =
=
x∫
xj
x∫
η
d̃jki(ξ, η, x, t, ε)
(
∂ξv
l
i
)
(η, ωk(η; ξ, ωj(ξ;x, t, ε), ε))dξdη =
= −
x∫
xj
x∫
η
∂ξd̃jki(ξ, η, x, t, ε)v
l
i (η, ωk(η; ξ, ωj(ξ;x, t, ε), ε)) dξdη+
+
x∫
xj
[
d̃jki(ξ, η, x, t, ε)v
l
i (η, ωk(η; ξ, ωj(ξ;x, t, ε), ε))
]ξ=x
ξ=η
dη,
where
d̃jki(ξ, η, x, t) = djki(ξ, η, x, t, ε)∂xωj(ξ;x, t, ε) (akajγjk) (ξ, ωj(ξ;x, t, ε), ε).
Now in (3.36) we can pass to the limit as l→∞ and then to the right-hand side apply the apriori es-
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ROBUSTNESS OF EXPONENTIAL DICHOTOMIES OF BOUNDARY-VALUE PROBLEMS . . . 251
timate (2.2). Combining the resulting inequality with the formula (3.35) and the apriori estimate (2.2)
gives (3.34).
Theorem 1.1 is proved.
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Received 25.12.12
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