Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations
We obtain conditions for existence of piecewise continuous almost periodic solutions of a system of impulsive differential equations with exponentially dichotomous linear part. The robustness of exponential dichotomy and exponential contraction for linear systems with small perturbations of right-ha...
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Tkachenko, V.I. 2021-02-10T13:08:16Z 2021-02-10T13:08:16Z 2014 Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations / V.I. Tkachenko // Нелінійні коливання. — 2014. — Т. 17, № 4. — С. 546-557 — Бібліогр.: 14 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177107 517.9 We obtain conditions for existence of piecewise continuous almost periodic solutions of a system of impulsive differential equations with exponentially dichotomous linear part. The robustness of exponential dichotomy and exponential contraction for linear systems with small perturbations of right-hand sides and points of impulsive action are studied. Отримано умови iснування кусково-неперервних майже перiодичних розв’язкiв систем диференцiальних рiвнянь з iмпульсною дiєю та експоненцiально дихотомiчною лiнiйною частиною. Вивчено грубiсть експоненцiальної дихотомiї та експоненцiальної стiйкостi лiнiйних систем при малих збуреннях правих частин та точок iмпульсної дiї. en Інститут математики НАН України Нелінійні коливання Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations Експоненціальна дихотомія та існування майже періодичних розв’язків імпульсних диференціальних рівнянь Экспоненциальная дихотомия и существование почти периодических решений импульсных дифференциальных уравнений Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations |
| spellingShingle |
Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations Tkachenko, V.I. |
| title_short |
Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations |
| title_full |
Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations |
| title_fullStr |
Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations |
| title_full_unstemmed |
Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations |
| title_sort |
exponential dichotomy and existence of almost periodic solutions of impulsive differential equations |
| author |
Tkachenko, V.I. |
| author_facet |
Tkachenko, V.I. |
| publishDate |
2014 |
| language |
English |
| container_title |
Нелінійні коливання |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Експоненціальна дихотомія та існування майже періодичних розв’язків імпульсних диференціальних рівнянь Экспоненциальная дихотомия и существование почти периодических решений импульсных дифференциальных уравнений |
| description |
We obtain conditions for existence of piecewise continuous almost periodic solutions of a system of impulsive differential equations with exponentially dichotomous linear part. The robustness of exponential dichotomy and exponential contraction for linear systems with small perturbations of right-hand sides and points of impulsive action are studied.
Отримано умови iснування кусково-неперервних майже перiодичних розв’язкiв систем диференцiальних рiвнянь з iмпульсною дiєю та експоненцiально дихотомiчною лiнiйною частиною. Вивчено грубiсть експоненцiальної дихотомiї та експоненцiальної стiйкостi лiнiйних систем при малих збуреннях правих частин та точок iмпульсної дiї.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/177107 |
| citation_txt |
Exponential dichotomy and existence of almost periodic solutions of impulsive differential equations / V.I. Tkachenko // Нелінійні коливання. — 2014. — Т. 17, № 4. — С. 546-557 — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
AT tkachenkovi exponentialdichotomyandexistenceofalmostperiodicsolutionsofimpulsivedifferentialequations AT tkachenkovi eksponencíalʹnadihotomíâtaísnuvannâmaižeperíodičnihrozvâzkívímpulʹsnihdiferencíalʹnihrívnânʹ AT tkachenkovi éksponencialʹnaâdihotomiâisuŝestvovaniepočtiperiodičeskihrešeniiimpulʹsnyhdifferencialʹnyhuravnenii |
| first_indexed |
2025-11-27T07:42:29Z |
| last_indexed |
2025-11-27T07:42:29Z |
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1850806774901243904 |
| fulltext |
UDC 517.9
EXPONENTIAL DICHOTOMY AND EXISTENCE OF ALMOST PERIODIC
SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS
ЕКСПОНЕНЦIАЛЬНА ДИХОТОМIЯ ТА IСНУВАННЯ
МАЙЖЕ ПЕРIОДИЧНИХ РОЗВ’ЯЗКIВ
IМПУЛЬСНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
V. I. Tkachenko
Inst. Math. Nat. Acad. Sci. Ukraine
Ukraine, 01601, Kyiv 4, Tereshchenkivska str., 3
We obtain conditions for existence of piecewise continuous almost periodic solutions of a system of impulsi-
ve differential equations with exponentially dichotomous linear part. The robustness of exponential di-
chotomy and exponential contraction for linear systems with small perturbations of right-hand sides and
points of impulsive action are studied.
Отримано умови iснування кусково-неперервних майже перiодичних розв’язкiв систем диферен-
цiальних рiвнянь з iмпульсною дiєю та експоненцiально дихотомiчною лiнiйною частиною. Ви-
вчено грубiсть експоненцiальної дихотомiї та експоненцiальної стiйкостi лiнiйних систем при
малих збуреннях правих частин та точок iмпульсної дiї.
1. Introduction. We investigate the problem of existence of a piecewise continuous almost peri-
odic solution for the semilinear impulsive differential equation
du
dt
= A(t)u+ f(t, u), t 6= τj , (1)
∆u|t=τj = u(τj + 0)− u(τj) = Bju(τj) + gj(u(tj)), j ∈ Z, (2)
where u : R → Rn. We use the concept of discontinuous almost periodic functions in the sense
of [1, 2]. There are many works (see, e.g., [3 – 6] and references given there) devoted to a study
of almost periodic solutions of impulsive systems.
We assume that the corresponding linear homogeneous equation (if f ≡ 0, gj ≡ 0) has
an exponential dichotomy. Matrices (I + Bj) may degenerate, det(I + Bj) = 0, for some (or
all) j ∈ Z therefore, solutions of the system are not extendable to the negative semiaxis or
are ambiguously extendable. Defining exponential dichotomy we require that only solutions
of linear system from the unstable manifold can be unambiguously extended to the negative
semiaxis. This corresponds to the definition of exponential dichotomy for evolution equations
in an infinite dimensional Banach space [7 – 9].
Robustness is an impotent property of exponential dichotomy [8 – 10]. We mention arti-
cles [11 – 14] where the robustness of exponential dichotomy for impulsive systems by small
perturbations in the right-hand sides is proved. In this article we prove robustness of exponenti-
al dichotomy also by small perturbation of points of the impulsive action. We use change of time
in the system. Then approximation of the impulsive system by difference systems (see [7]) can
be used.
c© V. I. Tkachenko, 2014
546 ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
EXPONENTIAL DICHOTOMY AND EXISTENCE OF ALMOST PERIODIC SOLUTIONS . . . 547
2. Preliminaries and main results. Let X be an abstract Banach space and R and Z be the
sets of real and integer numbers, respectively.
We will consider the space PC(J,X), J ⊂ R, of all piecewise continuous functions x : J →
→ X such that
i) the set T = {τj ∈ J : τj+1 > τj , j ∈ Z} is the set of discontinuities of x;
ii) x(t) is left-continuous, x(tj−0) = x(tj), and there exists limt→tj+0 x(t) = x(tj+0) < ∞.
Definition 1. A strictly increasing sequence {τk} of real numbers has uniformly almost peri-
odic sequences of differences if for any ε > 0 there exists a relatively dense set of ε-almost periods
common for all the sequences {τ jk}, where τ jk = τk+j − τk, j ∈ Z.
Recall that an integer p is called an ε-almost period of a sequence {xk} if ‖xk+p − xk‖ < ε
for any k ∈ Z. A sequence {xk} is almost periodic if for any ε > 0 there exists a relatively dense
set of its ε-almost periods.
Definition 2. A function ϕ(t) ∈ PC(R, X) is said to be W -almost periodic if
i) the sequence {τk} of discontinuities of ϕ(t) has uniformly almost periodic sequences of
differences;
ii) for any ε > 0 there exists a positive number δ = δ(ε) such that if the points t′ and t′′ belong
to the same interval of continuity and |t′ − t′′| < δ then ‖ϕ(t′)− ϕ(t′′)‖ < ε;
iii) for any ε > 0 there exists a relatively dense set Γ of ε-almost periods such that if τ ∈ Γ,
then ‖ϕ(t+ τ)− ϕ(t)‖ < ε for all t ∈ R which satisfy the condition |t− τk| ≥ ε, k ∈ Z.
We consider the impulsive equation (1), (2) with the following assumptions:
(H1) the matrix-valued function A(t) is Bohr almost periodic,
(H2) the sequence of real numbers τk has uniformly almost periodic sequences of differences,
and there exists θ > 0 such that infk τ
1
k = θ > 0,
(H3) the sequence {Bj} of (n× n)-matrices is almost periodic,
(H4) we shell use the notation Uρ = {x ∈ Rn : ‖x‖ ≤ ρ}; the function f(t, u) : R× Rn →
→ Rn is continuous in u and is W -almost periodic in t uniformly with respect to u ∈ Uρ with
some ρ > 0,
(H5) the sequence {gj(u)} of continuous functions Uρ → Rn is almost periodic uniformly
with respect to u ∈ Uρ.
By Lemma 22 [9, p. 192] for a sequence {τj} with uniformly almost periodic sequences of
differences there exists the limit
lim
T→∞
i(t, t+ T )
T
= p
uniformly with respect to t ∈ R, where i(s, t) is the number of the points τk lying in the
interval (s, t).
The next lemma is proved in [9].
Lemma 1. Assume that the sequence of real numbers {τj} has uniformly almost periodic
sequences of differences, the sequence {Bj} is almost periodic and the function f(t) : R → Rn
is W -almost periodic. Then for any ε > 0 there exist a real number ν, 0 < ν < ε, and relatively
dense sets of real numbers Γ and integers Q such that the following relations hold:
‖f(t+ r)− f(t)‖ < ε, t ∈ R, |t− τj | > ε, j ∈ Z,
‖Bk+q −Bk‖ < ε, ‖τ qk − r‖ < ν,
for k ∈ Z, r ∈ Γ, q ∈ Q.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
548 V. I. TKACHENKO
Definition 3. A function x(t) : [t0, t1] → Rn is said to be a solution of the initial problem
u(t0) = u0 ∈ Rn for the equation (1), (2) on [t0, t1] if
(i) it is continuous in [t0, τk], (τk, τk+1], . . . , (tk+s, t1] with discontinuities of the first kind at
the moments t = τj ,
(ii) x(t) is continuously differentiable in each of the intervals (t0, τk), (τk, τk+1), . . . , (tk+s, t1)
and satisfies the equations (1) and (2) if t ∈ (t0, t1), t 6= τj and t = τj respectively,
(iii) the initial value condition u(t0) = u0 is fulfilled.
Together with equation (1), (2) we consider the linear homogeneous equation
du
dt
= A(t)u, t 6= τj , (3)
∆u|t=τj = u(τj + 0)− u(τj) = Bju(τj), j ∈ Z. (4)
Denote byX(t, s) the evolution operator of the linear equation without impulses (3). It satisfies
X(τ, τ) = I, X(t, s)X(s, τ) = X(t, τ), t, s, τ ∈ R.
We define an evolution operator for equation (3), (4) by
U(t, s) = X(t, s) if τk < s ≤ t ≤ τk+1,
and
U(t, s) = X(t, τk)(I +Bk)X(τk, τk−1) . . . (I +Bm)X(τm, s), (5)
if τm−1 < s ≤ τm < τm+1 < . . . < τk < t ≤ τk+1.
Definition 4. We say that system (3), (4) has an exponential dichotomy on R with exponent
β > 0 and bound M ≥ 1 if there exist projections P (t), t ∈ R, such that
(i) U(t, s)P (s) = P (t)U(t, s), t ≥ s;
(ii) U(t, s)|Im (P (s)) for t ≥ s is an isomorphism on Im (P (s)), then U(s, t) is defined as an
inverse map from Im (P (t)) to Im (P (s));
(iii) ‖U(t, s)(1− P (s))‖ ≤ Me−β(t−s), t ≥ s;
(iv) ‖U(t, s)P (s)‖ ≤ Meβ(t−s), t ≤ s.
Now we formulate our main result.
Theorem 1. Suppose that system (1), (2) satisfies assumptions (H1) – (H5), linear system (3),
(4) is exponentially dichotomous with constants β and M ≥ 1.
Assume that the functions f(t, u) and gj(u) satisfy the Lipschitz condition
‖f(t, u1)− f(t, u2)‖ ≤ L‖u1 − u2‖, ‖gj(u1)− gj(u2)‖ ≤ L‖u1 − u2‖, j ∈ Z,
with a positive constant L and are uniformly bounded in the region t ∈ R, u ∈ Uρ :
sup
(t,u)
‖f(t, u)‖ ≤ H < ∞, sup
u
‖gj(u)‖ ≤ H < ∞, j ∈ Z.
Then for a sufficiently smallL the system (1), (2) has a unique piecewise continuousW -almost
periodic solution.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
EXPONENTIAL DICHOTOMY AND EXISTENCE OF ALMOST PERIODIC SOLUTIONS . . . 549
3. Robustness of exponential dichotomy. If system (3), (4) has an exponential dichotomy on
R, then the nonhomogeneous equation
du
dt
= A(t)u+ f(t), t 6= τj , (6)
∆u|t=τj = u(τj + 0)− u(τj) = Bju(τj) + gj , j ∈ Z, (7)
has a unique solution bounded on R,
u0(t) =
∞∫
−∞
G(t, s)f(s)(x)ds+
∑
j∈Z
G(t, τj)gj , (8)
where
G(t, s) =
{
U(t, s)(I − P (s)), t ≥ s,
−U(t, s)P (s), t < s,
is a Green function such that
‖G(t, s)‖ ≤ Me−β|t−s|, t, s ∈ R. (9)
Analogously to [7, p. 250] it can be proved that a function u(t) is a bounded solution on the
semiaxis [t0,+∞) if and only if
u(t) = U(t, t0)(I − P (t0))u(t0) +
+∞∫
t0
G(t, s)f(s)ds+
∑
t0≤τj
G(t, τj)gj , t ≥ t0. (10)
A function u(t) is a bounded solution on the semiaxis (−∞, t0] if and only if
u(t) = U(t, t0)P (t0)u(t0) +
t0∫
−∞
G(t, s)f(s)ds+
∑
t0>τj
G(t, τj)gj , t ≤ t0. (11)
Lemma 2. Let the impulsive system (3), (4) be exponentially dichotomous with positive con-
stants β and M. Then there exists δ0 > 0 such that the perturbed systems
du
dt
= Ã(t)u, t 6= τ̃j , (12)
∆u|t=τ̃j = u(τ̃j + 0)− u(τ̃j) = B̃ju(τ̃j), j ∈ Z, (13)
with supj |τj − τ̃j | ≤ δ0, supj ‖Bj − B̃j‖ ≤ δ0, supt ‖A(t) − Ã(t)‖ ≤ δ0, are also exponentially
dichotomous with some constants β1 ≤ β and M1 ≥ M.
Proof. In system (3), (4), we introduce the change of time t = α(t′) such that τj = α(τ̃j),
j ∈ Z, and the functionα is continuously differentiable and monotone on each interval (τ̃j , τ̃j+1).
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
550 V. I. TKACHENKO
The function α can be chosen in a piecewise linear form,
t = ajt
′ + bj , aj =
τj+1 − τj
τ̃j+1 − τ̃j
, bj =
τj+1τ̃j − τj τ̃j+1
τ̃j+1 − τ̃j
if t′ ∈ (τ̃j , τ̃j+1).
The function α(t′) satisfies the conditions
|α(t′)− t′| ≤ δ0,
∣∣∣∣dα(t′)
dt′
− 1
∣∣∣∣ ≤ aδ0
with some positive constant a independent of j and δ0.
System (3), (4) in the new coordinates v(t′) = u(α(t′)) has the form
dv
dt′
= A1(t
′)v, t 6= τ̃j , (14)
∆v|t′=τ̃j = v(τ̃j + 0)− v(τ̃j) = Bjv(τ̃j), j ∈ Z, (15)
where A1(t
′) =
dα(t′)
dt′
A(α(t′)). System (14), (15) has evolution the operator U1(t
′, s′) =
= U(α(t′), α(s′)). If system (3), (4) has exponential dichotomy with a projection P (t) at point
t, then system (14), (15) has exponential dichotomy with the projection P1(t
′) = P (α(t′)) at
point t′. Indeed,
‖U1(t
′, s′)(1− P1(s
′))‖ = ‖U(α(t′), α(s′))(1− P (α(s′))‖ ≤
≤ Me−β(α(t
′)−α(s′)) ≤ M1e
−β(t′−s′), t′ ≥ s′,
where M1 = Me2δ0 . The inequality for an unstable manifold is proved analogously.
The linear systems (14), (15) and (12), (13) have the same points of impulsive actions τ̃j ,
j ∈ Z, and
‖A1(t
′)− Ã(t′)‖ ≤
∥∥∥∥dα(t′)
dt′
A(α(t′))−A(α(t′))
∥∥∥∥+ ‖A(α(t′))−A(t′)‖+
+ ‖A(t′)− Ã(t′)‖ ≤ δ0
(
a sup
t
‖A‖+ sup
t
∥∥∥∥dAdt
∥∥∥∥+ 1
)
.
Let Ũ(t′, s′) be an evolution operator for system (12), (13). To show that for a sufficiently
small δ0 system (12), (13) is exponentially dichotomous we use the following version of Theo-
rem 7.6.10 [7]:
Assume that the evolution operator U1(t
′, s′) has an exponential dichotomy on R and satis-
fies
sup
0≤t′−s′≤d
‖U1(t
′, s′)‖ < ∞ (16)
for some positive d. Then there exists η > 0 such that
‖Ũ(t′, s′)− U1(t
′, s′)‖ < η, whenever t− s ≤ d
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
EXPONENTIAL DICHOTOMY AND EXISTENCE OF ALMOST PERIODIC SOLUTIONS . . . 551
and the evolution operator Ũ(t′, s′) has an exponential dichotomy on R.
For proving this statement we set
tn = s′ + dn, Tn = U1(s
′ + d(n+ 1), s′ + dn), T̃n = Ũ(s′ + d(n+ 1), s′ + dn) for n ∈ Z.
If the evolution operator U1(t, s) has an exponential dichotomy, then {Tn} has a discrete dicho-
tomy in the sense of [7] (Definition 7.6.4).
By [7] (Theorem 7.6.7), there exists η > 0 such that {T̃n} with supn ‖Tn − T̃n‖ ≤ η has a
discrete dichotomy.
Now we are in the conditions of [7, p. 229, 230], Excersise 10 (see also a more general
statement [8], Theorem 4.1), that finishes the proof.
The exponentially dichotomous system (12), (13) has the Green function
G̃(t, s) =
{
Ũ(t, s)(I − P̃ (s)), t ≥ s,
−Ũ(t, s)P̃ (s), t < s,
such that
‖G̃(t, s)‖ ≤ M1e
−β1|t−s|, t, s ∈ R.
Lemma 3. The difference of the Green functions of the exponentially dichotomous linear
systems (12), (13) and (14), (15) satisfies
G̃(t, τ)−G1(t, τ) =
∞∫
−∞
G1(t, s)(Ã(s)−A1(s))G̃(s, τ)ds+
+
∑
j
G1(t, τ̃j)(B̃j −Bj)G̃(τ̃j , τ), t, τ ∈ R, (17)
where G1(t, τ) = G(α(t), α(τ)).
Proof. G̃(t, τ) satisfies the equation
du
dt
= A1(t)u+ (Ã(t)−A1(t))G̃(t, τ),
∆u|t=τ̃j = Bju+ (B̃j −Bj)G̃(τ̃j , τ).
By (10), we have, for t ≥ τ,
G̃(t, τ) = U1(t, τ)(I − P1(τ))G̃(τ, τ) +
+∞∫
τ
G1(t, s)(Ã(s)−A1(s))G̃(s, τ)ds+
+
∑
τ≤τ̃j
G1(t, τ̃j)(B̃j −Bj)G̃(τ̃j , τ). (18)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
552 V. I. TKACHENKO
Analogously, by (11), we have, for t < τ,
G̃(t, τ) = U1(t, τ)P1(τ)G̃(τ − 0, τ) +
τ∫
−∞
G1(t, s)(Ã(s)−A1(s))G̃(s, τ)ds+
+
∑
τ>τ̃j
G1(t, τ̃j)(B̃j −Bj)G̃(τ̃j , τ). (19)
Putting t = τ in (18), we get
P1(τ)G̃(τ, τ) =
+∞∫
τ
G1(τ, s)(Ã(s)−A1(s))G̃(s, τ)ds+
∑
τ≤τ̃j
G1(τ, τ̃j)(B̃j −Bj)G̃(τ̃j , τ).
Since G̃(τ, τ)− G̃(τ − 0, τ) = I, we have by (19), for t < τ,
G̃(t, τ) = U1(t, τ)
( +∞∫
τ
G1(τ, s)(Ã(s)−A1(s))G̃(s, τ)ds+
+
∑
τ≤τ̃j
G1(τ, τ̃j)(B̃j −Bj)G̃(τ̃j , τ)
)
− U1(t, τ)P1(τ)+
+
τ∫
−∞
G1(t, s)(Ã(s)−A1(s))G̃(s, τ)ds+
∑
τ>τ̃j
G1(t, τ̃j)(B̃j −Bj)G̃(τ̃j , τ) =
= G1(t, τ) +
∞∫
−∞
G1(t, s)(Ã(s)−A1(s))G̃(s, τ)ds+
∑
j
G1(t, τ̃j)(B̃j −Bj)G̃(τ̃j , τ).
The case t ≥ τ is considered analogously.
Lemma 3 is proved.
By (17), we obtain the estimate
‖G̃(t, τ)−G1(t, τ)‖ ≤ δ0M2e
−β2|t−τ |, t, τ ∈ R, (20)
with some β2 ≤ β1 and M2 ≥ 1.
Lemma 4. Let systems (3), (4) and (12), (13) satisfy assumptions of Lemma 2 with sufficiently
small δ0 > 0. Then the corresponding Green functions of these systems satisfy the inequality
‖G̃(t, τ)−G(t, τ)‖ ≤ δ0M̃2e
−β2|t−s|, (21)
for all t and s such that |t− τj | > δ0, |s− τj | > δ0 for all j ∈ Z.
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EXPONENTIAL DICHOTOMY AND EXISTENCE OF ALMOST PERIODIC SOLUTIONS . . . 553
Proof. We have
‖G(t, τ)− G̃(t, τ)‖ ≤ ‖G(t, τ)−G(α(t), α(τ))‖+ ‖G(α(t), α(τ))− G̃(t, τ)‖.
Let t > s (the case t < s is considered analogously). Then
‖G(t, s)−G(α(t), α(s))‖ = ‖U(t, s)P (s)− U(α(t), α(s))P (α(s))‖ ≤
≤ ‖U(t, s)P (s)− U(α(t), s)P (s)‖+
+ ‖U(α(t), s)P (s)− U(α(t), α(s))P (α(s))‖ ≤
≤ ‖U(t, s)P (s)− U(α(t), t)U(t, s)P (s)‖+
+ ‖U(α(t), s)P (s)− U(α(t), s)U(s, α(s))P (α(s))‖ ≤
≤ ‖I − U(α(t), t)‖‖U(t, s)P (s)‖+
+ ‖U(α(t), s)P (s)‖‖I − U(s, α(s))‖ ≤
≤ Me−γ(t−s)‖I − U(α(t), t)‖+Me−γ(α(t)−s)‖I − U(s, α(s)‖.
Here, for definiteness s ≥ α(s), t ≤ α(t). If t ∈ (τj + δ0, τj+1 − δ0), then α(t) ∈ (τj , τj+1).
Therefore, by continuity there exists a positive constant C1 independent of t such that ‖I −
−U(α(t), t)‖ ≤ C1δ0. As a result, we obtain ‖G(t, s) − G(α(t), α(s))‖ ≤ δ0M3e
−β|t−s| with
some positive constant M3 independent of t, s ∈ R and δ0. Now, taking into account (20) we
obtain (21).
Lemma 4 is proved.
Corollary 1. Assume that system (3), (4) satisfies conditions (H1) – (H3) and is exponentially
dichotomous with constants β and M. Then for any ε > 0, t, s ∈ R, |t − τj | > ε, |s − τj | > ε,
j ∈ Z, there exists a relatively dense set of ε-almost periods r such that
‖G(t+ r, s+ r)−G(t, s)‖ ≤ εC1 exp
(
−β
2
|t− s|
)
, (22)
where C1 is a positive constant independent on ε.
Proof. If u(t) = U(t, s)u0, u(s) = u0, is a solution of the impulsive equation (3), (4), then
u1(t) = U(t+ r, s+ r)u0 is a solution of the equation
du
dt
= A(t+ r)u, t 6= τj , (23)
∆u|t=τj+q = u(τj+q + 0)− u(τj+q) = Bj+qu(τj+q), j ∈ Z. (24)
By Lemma 1, there exists a positive integer q such that τj+q ∈ (s + r, t + r) if τj ∈ (s, t). Now
we apply Lemma 4.
4. Almost periodic solutions of linear inhomogeneous system. We prove existence of almost
periodic solutions in a linear inhomogeneous system.
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554 V. I. TKACHENKO
Lemma 5. Let a linear homogeneous system satisfy assumptions (H1) – (H3) and be exponenti-
ally dichotomous on the axis. If the function f(t) is W -almost periodic and the sequence {gj} is
almost periodic, then the linear inhomogeneous system (6), (7) has unique solution, which is
bounded on R and W -almost periodic.
Proof. The unique solution, bounded on R, of the system (6), (7) is defined by formula (8).
We show that it is W -almost periodic.
Take an ε-almost period r for the right-hand side of the equation. Then
u0(t+ r)− u0(t) =
+∞∫
−∞
G(t+ r, s)f(s)ds+
∑
j
G(t+ r, τj)gj −
+∞∫
−∞
G(t, s)f(s)ds−
−
∑
j
G(t, τj)gj =
+∞∫
−∞
(G(t+ r, s+ r)−G(t, s))f(s+ h)ds+
+
+∞∫
−∞
G(t, s)(f(s+ r)− f(s))ds+
∑
j
(G(t+ r, τj+q)−G(t, τj))gj+q+
+
∑
j
G(t, τj)(gj+q − gj).
We estimate the first integral,
∞∫
−∞
‖(G(t+ r, s+ r)−G(t, s))f(s+ r)‖ds≤
≤
∑
k∈Z
τk+1−ε∫
τk+ε
‖(G(t+ r, s+ r)−G(t, s))f(s+ r)‖ds+
+
∑
k∈Z
τk+ε∫
τk−ε
‖(G(t+ r, s+ r)−G(t, s))f(s+ r)‖ds ≤
≤
∞∫
−∞
εC1e
−β
2
|t−s|‖f(s)‖ds+
∑
k∈Z
τk+ε∫
τk−ε
‖G(t+ r, s+ r)f(s+ r)‖ds+
+
∑
k∈Z
τk+ε∫
τk−ε
‖G(t, s)f(s+ r)‖ds.
By Lemma 1, |τj+q − τj − r| < ε, therefore τj + r + ε > τj+q (we assume that r > 0 for
definiteness). The difference G(t, τj) − G(t + r, τj+q) is estimated as follows. Let t − τj ≥ ε.
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EXPONENTIAL DICHOTOMY AND EXISTENCE OF ALMOST PERIODIC SOLUTIONS . . . 555
Then
‖G(t, τj)−G(t+ r, τj+q)‖ = ‖U(t, τj)(I − P (τj))− U(t+ r, τj+q)(I − P (τj+q))‖ ≤
≤ ‖U(t, τj)(I − P (τj))− U(t, τj + ε)(I − P (τj + ε))‖+
+ ‖U(t, τj + ε)(I − P (τj + ε))− U(t+ r, τj + ε+ r)(I − P (τj + ε+ r))‖+
+ ‖U(t+ r, τj + ε+ r)(I − P (τj + ε+ r))− U(t+ r, τj+q)(I − P (τj+q))‖. (25)
The first and the third differences are small because of continuity of the function U(t, s) at
intervals between the points of impulses,
‖U(t, τj)(I − P (τj))− U(t, τj + ε)(I − P (τj + ε))‖ ≤
≤ ‖U(t, τj + ε)(I − P (τj + ε))(U(τj + ε, τj)− I)‖ ≤
≤ εC2e
−β(t−τj−ε),
‖U(t+ r, τj + ε+ r)(I − P (τj + ε+ r))− U(t+ r, τj+q)(I − P (τj+q))‖ =
= ‖U(t+ r, τj + ε+ r)(I − P (τj + ε+ r))(U(τj + ε+ r, τj+q)− I)‖ ≤
≤ εC2e
−γ(t−τj−ε).
The second difference in (25) is small because of (22).
5. Proof of Theorem 1. Denote by M the space of all W -almost periodic functions with
discontinuous at points of the same sequence {τj}. The norm in the space M is defined as
‖ϕ‖0 = supt∈R ‖ϕ(t)‖, ϕ ∈ M. We define an operator T on M as follows: if ϕ(t) ∈ M, then
(Tϕ)(t) =
∞∫
−∞
G(t, s)f(s, ϕ(s))ds+
∑
j
G(t, τj)g(ϕ(τj)).
First, we prove that T (Dh) ⊆ Dh for some h > 0 where
Dh = {ϕ ∈ M, ‖ϕ‖0 ≤ h}.
Indeed, if ‖ϕ‖0 ≤ h, then
‖Tϕ‖ ≤
∞∫
−∞
‖G(t, s)‖‖f(s, ϕ(s))‖ds+
∑
j
‖G(t, τj)‖‖g(ϕ(τj))‖ ≤
≤
∞∫
−∞
Me−β|t−s|Hds+
∑
j
Me−β|t−τj |H ≤
≤ 2MH
(
1
β
+
1
1− e−βθ
)
≤ h.
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556 V. I. TKACHENKO
By Lemma 37 [2, p. 214], if ϕ(t) is an W -almost periodic function and infi τ
1
i = θ > 0, then
{ϕ(τi)} is an almost periodic sequence. Using the method of finding common almost periods, it
is possible to show that the sequence {gi(ϕ(τi))} is almost periodic.
Let r be an ε-almost period of the function ϕ(t). Analogously to the proof of Lemma 5 we
show that for t ∈ R, |t− τj | > ε, j ∈ Z, the following inequality holds:
‖(Tϕ)(t+ r)− (Tϕ)(t)‖ =
∥∥∥∥∥∥
+∞∫
−∞
G(t+ r, s)f(s)ds+
∑
j
G(t+ r, τj)gj −
−
+∞∫
−∞
G(t, s)f(s)ds−
∑
j
G(t, τj)gj
∥∥∥∥∥∥ ≤ Γ(ε)ε,
where Γ(ε) is some bounded function of ε. Hence, we proved that T (Dh) ⊆ Dh.
If ϕ,ψ ∈ Dh, then
‖(Tϕ)(t)− (Tψ)(t)‖ ≤
∞∫
−∞
‖G(t, s)‖‖f(s, ϕ(s))− f(s, φ(s))‖ds+
+
∑
k
‖G(t, τk)‖‖gk(ϕ(τk))− gk(φ(τk))‖ ≤
≤ 2MH
(
1
β
+
1
1− e−βθ
)
‖ϕ− ψ‖0.
For sufficiently small N > 0, the operator T is a contraction in the domain Dh, and so there
exists a unique W -almost periodic solution of system (1), (2).
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Received 20.03.14
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