Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems
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September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
Dissipativity of some classes of equations 227
(1) there are positive numbers M and q < 1 such that
||U(k, A)U
−1
(m, A)|| ≤ Mq
k−m
(k ≥ m); (6.3)
(2) |F (k, u| ≤ C + D|u| (C ≥ 0, 0 ≤ D < (1 − q)M
−1
) for all u ∈ E
n
.
Then the equation (6.2) is dissipative.
Proof. Let ϕ(·, u, Φ) (Φ(k, u) := A(k)u + F (k, u)) be a solution of the equation
(6.2) passing through the point u ∈ E
n
for k = 0. According to the formula of
variation of constants (see, for example,
[
170
]
) we have
ϕ(k, u, Φ) = U (k, A)
u +
k−1
X
m=0
U
−1
(m + 1, A)F (m, ϕ(m, u, Φ))
,
and, consequently,
|ϕ(k, u, Φ)| ≤ M q
k
|u|
+
k−1
X
m=0
q
−m+1
(C + D|ϕ(m, u, Φ)|)
. (6.4)
We set u(k) := q
−k
|ϕ(k, u, Φ)| and, taking into account (6.4), we obtain
u(k) ≤ M |u| + CM q
k−1
X
m=0
q
−m
+ DM
k−1
X
m=0
u(m). (6.5)
Denote the right hand side of the equality (6.5) by v(k). Note, that
v(k + 1) −v(k) = q
−k
CM
q
+
DM
q
u(k) ≤
DM
q
v(k) +
CM
q
q
−k
,
and,hence,
v(k + 1) ≤
1 +
DM
q
v(k) +
CM
q
q
−k
.
From this inequality we obtain
v(k) ≤
1 +
DM
q
k−1
v(1) +
CM
q
1 − q
k−1
1 − q
.
Therefore,
|ϕ(k, u, Φ)| ≤ (q + DM )
k−1
qM|u|+
CM
q − 1
(q
k−1
−1), (6.6)
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
228 Global Attractors of Non-autonomous Dissipative Dynamical Systems
because v(1) = M |u|. From (6.6) follows that
lim
k→+∞
|ϕ(k, u, Φ)| ≤
CM
1 − q
.
The theorem is proved.
Corollary 6.1 Under the conditions of Theorem 6.5 the equation (6.2) has at
least one p-periodic solution.
Proof. This statement follows from Theorems 6.2 and 6.5.
6.2 Equations with impulse
Consider the following differential equation:
˙u = f(t, u) (f ∈ C(R × E
n
, E
n
)). (6.7)
Suppose that f(t + τ, u) = f(t, u) (t ∈ R, u ∈ E
n
) and f is regular, i.e. for the
equation
˙v = g(t, v) (6.8)
the conditions of existence, uniqueness and continuability on R
+
are fulfilled for
all g ∈ H(f) := {f
τ
: τ ∈ [0, τ[}. Denote by ϕ(·, v, g) a solution of (6.8) passing
through the point v for t = 0. Note, that
ϕ(t + τ, v, g) = ϕ(t, ϕ(τ, v, g), g
τ
)
for all t, τ ∈ R
+
, v ∈ E
n
and g ∈ H(f).
Let {t
k
} ⊂ R (t
0
= 0) and {s
k
} ⊂ E
n
be such that for certain p > 0 (p ∈
Z) t
k+p
− t
k
= τ and s
k+p
= s
k
for all k ∈ Z. It is well known
[
170
]
that
these conditions are necessary and sufficient for τ –periodicity of the distribution
+∞
P
k=−∞
s
k
δ
t
k
.
Consider a nonlinear τ–periodic equation with impulse
˙u = f(t, u) +
+∞
X
k=−∞
s
k
δ
t
k
= F. (6.9)
It is known (see, for example,
[
170
]
) that, under the conditions above, the equa-
tion (6.9) has a unique generalized solution (which is piecewise continuous) passing
through the point u when t = 0 for every u ∈ E
n
. This solution we denote by
ϕ(·, u, F). Thus, lim
t↓0
ϕ(t, u, F) = u. In addition, on every segment ]t
k
, t
k+1
[ the
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
Dissipativity of some classes of equations 229
equation (6.9) coincides with (6.7). Therefore, the equality
ϕ(t, u, F) = ϕ(t − t
k
, c
k
, f
t
k
) (6.10)
holds, where the sequence {c
k
} ⊂ E
n
satisfies the difference equation
c
k+1
= ϕ(t
k+1
−t
k
, c
k
, f
t
k
) + s
k
. (6.11)
The equation with impulse (6.9) is said to be dissipative if there exists r > 0
such that
lim sup
t→+∞
|ϕ(t, v,
e
F)| < r (6.12)
for all v ∈ E
n
and
e
F ∈ H(F) := {F
s
: s ∈ [0, τ[}, where F
s
is an s-translation of the
distribution F.
Remark 6.1 For periodic equations the condition of dissipativity (6.12) can be
simplified because (6.12) is equivalent to the following relation:
lim sup
t→+∞
|ϕ(t, u, F)| < r. (6.13)
Proof. In fact, (6.12) implies (6.13). Conversely. From the equality ϕ(t + s, u, F) =
ϕ(t, ϕ(s, u, F), F
s
) (if in the point s ∈ R the function ϕ(t, u, F) is discontinuous, then
ϕ(s, u, F) := ϕ(s + 0, u, F)) for all t, τ ∈ R
+
. Therefore, we have
ϕ(t, v,
e
F) = ϕ(t − τ + s, ϕ(τ − s, v, F
k
), F)
for all t ≥ τ −s, where s ∈ [0, τ [ and
e
F = F
k
.
Lemma 6.1 The equation with impulse (6.9) is dissipative if and only if the
difference equation (6.11) is dissipative.
Proof. Let (6.9) be dissipative. Then there exists r > 0 such that the condition
(6.13) is fulfilled. If u ∈ E
n
, then
ϕ(k, u, Φ) = ϕ(t
k
, u, F)
and, hence,
lim sup
k→+∞
|ϕ(k, u, Φ)| < r. (6.14)
Conversely. Let the equation (6.11) be dissipative. Then there is r > 0 such that
the condition (6.14) holds. Thus, for all u ∈ E
n
there exists k
0
(u) ∈ Z
+
such that
|ϕ(k, u, Φ)| < r for all k ≥ k
0
(u). Let t ≥ t
k
0
and t ∈ [t
k
, t
k+1
[ k ≥ k
0
). By (6.10),
ϕ(t, u, F) = ϕ(t − t
k
, c
k
, f
t
k
), where c
k
= ϕ(t
k
, u, F). We set
R
0
:= max{|ϕ(s, u, g)| : s ∈ [0, τ ], |u| ≤ r, g ∈ H(f)}
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
230 Global Attractors of Non-autonomous Dissipative Dynamical Systems
and note that |ϕ(t, u, F)| = |ϕ(t −t
k
, c
k
, f
t
k
)| ≤ R
0
for all t ≥ t
k
0
and, consequently,
lim sup
t→+∞
|ϕ(t, u, F)| ≤ R
0
. (6.16)
Moreover, the number R
0
does not depend on u ∈ E
n
in the inequality (6.16). The
lemma is proved.
Define a mapping P : E
n
→ E
n
by the equality
P (u) := ϕ(τ, u, F) = ϕ(t
p
, u, F) = ϕ(p, u, Φ). (6.17)
From the equality (6.17) and general properties of solutions of difference equations
follows the continuity of the mapping P .
Theorem 6.6 The equation with impulse (6.9) is dissipative if and only if the
cascade (E
n
, P ), where P is defined by the equality (6.17), is dissipative.
Proof. This affirmation directly follows from Lemma 6.1 and Theorem 6.1.
Corollary 6.2 If the equation (6.9) is dissipative, then there exists a nonempty
compact connected set I ⊂ E
n
possessing the following properties:
(1) P (I) = I, where P is a mapping of E
n
onto itself, defined by the formula (6.17);
(2) for an arbitrary ε > 0 there is δ(ε) > 0 such that ρ(u, I) < δ implies
ρ(P
k
(u), I) < ε for all k ∈ Z
+
;
(3) the equality lim
k→+∞
ρ(P
k
(u), I) = 0 holds for all u ∈ E
n
.
Proof. Corollary 6.2 follows from Theorems 6.6 and 4.9.
Corollary 6.3 If the equation (6.9) is dissipative, then:
(1) there is r > 0 and for all a > 0 there is k(a) ∈ Z
+
such that P
k
B[0, a] ⊆ B[0, r]
for all k ≥ k(a);
(2) there is u
0
∈ I such that P (u
0
) = u
0
and, consequently, ϕ(t, u
0
, F) is a τ -
periodic solution of (6.9).
Consider a quasi-linear equation with impulse
˙u = A(t)u + F (t, u) +
+∞
X
k=−∞
s
k
δ
t
k
. (6.18)
The following statement holds.
Theorem 6.7 Let A and F be τ-periodic in t ∈ R. The equation (6.18) is
dissipative, if the following conditions are fulfilled:
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
Dissipativity of some classes of equations 231
(1) there are positive numbers N and ν such that
||U(t, A)U
−1
(s, A)|| ≤ Ne
−ν(t−s)
,
where U(t, A) is a Cauchy matrix for the equation
˙u = A(t)u.
(2) there exists a sufficiently small positive number ε
0
such that |F (t, u)| ≤ C +
D|u| (u ∈ E
n
, C ≥ 0 and 0 ≤ D ≤ ε
0
).
Proof. Let f(t, u) := A(t)u + F (t, u). According to Lemma 6.1, to prove Theorem
6.7 it is sufficient to show that the difference equation (6.11) is dissipative. Let us
rewrite the equation (6.11) as follows:
c
k+1
=
e
A(k)c
k
+
e
F (k, c
k
), (6.19)
where
e
A(k) := U(t
k+1
−t
k
, A
t
k
) and
e
F (k, u) := ϕ(t
k+1
−t
k
, u, f
t
k
) + s
k
−
e
A(k)u.
Note, that under the conditions of Theorem 6.7
e
A(k) and
e
F (k, u) are p–periodic in
k ∈ Z.
We will show that the trivial solution of the homogeneous equation
c
k+1
=
e
A(k)c
k
(6.20)
is uniformly asymptotically stable. In virtue of the periodicity of
e
A(k) it is sufficient
to show that it is asymptotically stable. Note, that
U(k,
e
A) =
k−1
Y
m=0
U(t
m+1
−t
m
, A
t
m
) = U(t
k
−t
p−1
, A
t
p−1
),
therefore, ||U (k,
e
A)|| ≤ N e
−ν(t
k
−t
p−1
)
. From this fact follows the asymptotic sta-
bility of the trivial solution of (6.20).
We will show now that |
e
F (k, u)| ≤
e
C +
e
D(D)|u| for all u ∈ E
n
, where
e
C and
e
D(D) are some positive constants, moreover,
e
D(D) → 0 as D → 0. Indeed, since
ϕ(t, u, f
l
) is a solution of the equation
˙u = f
l
(t, u) = A
l
(t)u + F
l
(t, u),
then ϕ(t, u, f
l
) = U(t, A
l
)
u +
R
t
o
U
−1
(s, A
l
)F
l
(, ϕ(s, u, f
l
))ds
and, consequently,
e
F (k, u) =
Z
t
k+1
−t
k
0
U(t
k+1
−t
k
, A
t
k
)U
−1
(s, A
t
k
)F
t
k
(s, ϕ(s, u, f
t
k
))ds + s
k
. (6.21)
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
232 Global Attractors of Non-autonomous Dissipative Dynamical Systems
Using the same arguments as in Theorem 3.21, we can show that
|ϕ(t, u, f
l
)| ≤ N|u|e
(−ν+ND)t
+
CN
ν − DN
(6.22)
for all t ≥ 0, l ∈ R and u ∈ E
n
(0 ≤ D ≤ νN
−1
). From (6.21) and (6.22) we have
|
e
F (k, u)| ≤
t
k+1
−t
k
Z
0
Ne
−ν(t
k+1
−t
k
−s)
(C + D|ϕ(s, u, f
l
)|)ds ≤
Ne
−ν(t
k+1
−t
k
)
t
k+1
−t
k
Z
0
Ce
νs
+ De
νs
h
CN
ν − DN
+ N|u|e
−ν+ND)s
i
ds
= Ne
−ν(t
k+1
−t
k
)
h
C + D
CN
ν − DN
i
e
ν(t
k+1
−t
k
)
−1
ν
+
+N|u|D
e
ND(t
k+1
−t
k
)
−1
ND
=
NC
ν − DN
1 − e
−ν(t
k+1
−t
k
)
+
N|u|
e
(ND−ν)(t
k+1
−t
k
)
−e
−ν(t
k+1
−t
k
)
≤
e
C +
e
D(D)|u|,
where
e
C := NC(ν − ND)
−1
and
e
D(D) := N sup
0≤k≤p−1
h
e
(ND−ν)(t
k+1
−t
k
)
−e
−ν(t
k+1
−t
k
)
i
. (6.23)
From (6.23) follows that
e
D(D) → 0 as D → 0. Therefore, there is ε
0
> 0, such
that
e
D(ε
0
) < M
−1
(1 − q) (see Theorem 6.5). According to Theorem 6.5, the
difference equation (6.19) is dissipative and, hence, the equation with impulse (6.18)
is dissipative too. The theorem is proved.
6.3 Convergent periodic equations with impulse
Definition 6.2 Following
[
137
]
,
[
270
]
, the equation with impulse (6.9) we will
call convergent, if (6.9) has a unique τ–periodic solution ϕ(t, u, F) which is globally
uniformly asymptotically stable.
By analogy, it is possible to define the notion of convergence for the difference
equation (6.1). From this definition follows that convergent equations form that
part of dissipative equations which has the Levinson’s center consisting from a
single periodic trajectory (solution) of (6.9).
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
Dissipativity of some classes of equations 233
Lemma 6.2 The τ-periodic equation with impulse (6.9) is convergent if and only
if (6.11) is convergent.
Proof. Let (6.9) be convergent. Then there is a unique τ-periodic solution
ϕ(t, u
0
, F) of (6.9) which is globally uniformly asymptotically stable. Directly
from the corresponding definitions follows that ϕ(t, u
0
, Φ) (Φ(k, u) := ϕ(t
k+1
−
t
k
, u, f
t
k
) + s
k
) is a unique p-periodic solution of (6.11) and is globally uniformly
asymptotically stable.
Conversely. Let (6.11) be convergent and ϕ(k, u
0
, Φ) be its unique p-periodic
globally uniformly asymptotically stable solution. Obviously, ϕ(t, u
0
, F) is a unique
τ-periodic solution of (6.9). Let k ∈ Z and t ∈ [t
k
, t
k+1
[. According to (6.10),
|ϕ(t, u, F) − ϕ(t, u
0
, F)| = |ϕ(t − t
k
, c
k
, f
τ
k
) − ϕ(t − t
k
, c
0
k
, f
t
k
)|, (6.24)
where c
k
:= ϕ(t
k
, u, F) and c
0
k
:= ϕ(t
k
, u
0
, F). The mapping ϕ : [0, τ] ×K ×H(f) →
E
n
(ϕ : (s, u, g) → ϕ(s, u, g)) is uniformly continuous for every compact K ⊂ E
n
and, hence, for all ε > 0 there is δ(ε) > 0 such that
|ϕ(s, u
1
, g) −ϕ(s, u
2
, g)| < ε, (6.25)
for all s ∈ [0, τ], |u
1
−u
2
| < δ, u
1
, u
2
∈ K and g ∈ H(f ).
From (6.24) and (6.25) and the global uniform asymptotic stability of
ϕ(k, u
0
, Φ) = c
0
k
follows an analogous property for ϕ(t, u
0
, F). The lemma is proved.
Theorem 6.8 Let A(k) and F (k, u) be p–periodic in k ∈ Z and the following
conditions be fulfilled:
(1) there exist positive numbers M and q < 1 such that the inequality (6.3) holds;
(2) |F (k, u
1
) − F (k, u
2
)| ≤ L|u
1
− u
2
| (0 ≤ L < M
−1
(1 − q)) for all k ∈ Z and
u
1
, u
2
∈ E
n
.
Then the equation (6.2) is convergent.
Proof. Let ϕ(k, u, Φ) (Φ(k, u) := A(k)u + F (k, u)) be a solution of (6.2) passing
through the point u for k = 0. In virtue of the formula of variation of constants we
have
ϕ(k, u, Φ) = U (k, A)
u +
k−1
X
m=0
U
−1
(m + 1, A)F (m, ϕ(m, u, Φ))
.
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
234 Global Attractors of Non-autonomous Dissipative Dynamical Systems
Consequently,
ϕ(k, u
1
, Φ) − ϕ(k, u
2
, Φ) = U(k, A)
u
1
−u
2
+
k−1
X
m=1
U
−1
(m + 1, A)[F (m, ϕ(m, u, Φ)) − F (m, ϕ(m, u
2
, Φ))]
.
Thus,
|ϕ(k, u
1
, Φ) − ϕ(k, u
2
, Φ)| ≤ Mq
k
|u
1
−u
2
|
+Lq
−1
k−1
X
m=0
q
−m
|ϕ(k, u
1
, Φ) − ϕ(k, u
2
, Φ)|
. (6.26)
Let u(k) := |ϕ(k, u
1
, Φ) − ϕ(k, u
2
, Φ)|q
−k
. From (6.26) follows that
u(k) ≤ M
|u
1
−u
2
|+ Lq
−1
k−1
X
m=0
u(m)
. (6.27)
Denote by v(k) the right hand side of (6.27). The following inequality
v(k + 1) −v(k) = LMq
−1
u(k) ≤ LM q
−1
v(k). (6.28)
holds. From (6.28) we obtain
v(k) ≤ (1 + LMq
−1
)
k−1
v(1)
and, hence,
u(k) ≤ (1 + LM q
−1
)M|u
1
−u
2
|, (6.29)
since v(1) = M |u
1
−u
2
|. From (6.29) we have
|ϕ(k, u
1
, Φ) − Φ(k, u
2
, Φ)| ≤ (q + LM )
k−1
qM|u
1
−u
2
| (6.30)
for all u
1
, u
2
∈ E
n
and k ∈ Z.
Let us note that under the conditions of Theorem 6.8 we can apply Theorem
6.5 and, consequently, the equation (6.2) admits at least one p-periodic solution
ϕ(k, u
0
, Φ) (see also Corollary 6.1). From (6.30) follows the global asymptotic sta-
bility of the solution ϕ(k, u
0
, Φ) and, according to the periodicity of (6.2), the
solution ϕ(k, u
0
, Φ) will be globally uniformly asymptotically stable. The theorem
is proved.
Theorem 6.9 Let A(t) and F (t, u) be τ -periodic in t ∈ R. The equation (6.18)
is convergent, if the following conditions are fulfilled:
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
Dissipativity of some classes of equations 235
(1) there are positive numbers N and ν such that
||U(t, A)U
−1
(s, A)|| ≤ Ne
−ν(t−s)
(t ≥ s);
(2) there is a sufficiently small positive number ε
0
such that
|F (t, u
1
) − F (t, u
2
)| ≤ L|u
1
−u
2
|
(u
1
, u
2
∈ E
n
, t ∈ R and 0 ≤ L ≤ ε
0
).
Proof. According to Lemma 6.2 to prove the convergence of (6.18) it is sufficient
to establish that under the conditions of Theorem 6.9 the difference equation (6.11)
possesses the same property. As well as in the proof of Theorem 6.7, we will present
the equation (6.11) in the form (6.19). Under the conditions of Theorem 6.9 the
trivial solution of (6.20) is uniformly asymptotically stable. Further, (6.21) implies
that
|
e
F (k, u
1
) −
e
F (k, u
2
)| = |
t
k+1
−t
k
Z
0
U(t
k+1
−t
k
, A
t
k
)U
−1
(s, A
t
k
)×
[F
t
k
(s, ϕ(s, u
1
, f
t
k
) − F
t
k
(s, ϕ(s, u
2
, f
t
k
))]ds ≤ (6.31)
t
k+1
−t
k
Z
0
Ne
−ν(t
k+1
−t
k
−s)
L|ϕ(s, u
1
, f
t
k
−ϕ(s, u
2
, f
t
k
)|ds.
Reasoning in the same way that in Theorem 5.23 we obtain
|ϕ(t, u
1
, f
t
k
−ϕ(t, u
2
, f
t
k
)| ≤ N|u
1
−u
2
|e
−(ν−LN)t
(6.32)
for all u
1
, u
2
∈ E
n
and t ∈ R
+
, where f
t
k
(t, u) := A
t
k
(t)u + F
t
k
(t, u). From (6.31)
and (6.32) we have
|
e
F (k, u
1
) −
e
F (k, u
2
)| ≤ Ne
−ν(t
k+1
−t
k
)
L ×
t
k+1
−t
k
Z
0
e
LN
2
s
N|u
1
−u
2
|ds ≤
e
L(L)|u
1
−u
2
|,
where
e
L(L) := sup
0≤k≤p−1
h
e
(−ν+LN
2
)(t
k+1
−t
k
)
−e
−ν(t
k+1
−t
k
)
i
. (6.33)
From (6.33) follows that
e
L(L) → 0 for L → 0. Therefore, there exists ε
0
> 0 such
that
e
L(L) < M
−1
(1 − q) for 0 ≤ L ≤ ε
0
. By Theorem 6.8, the equation (6.20) is
convergent and to finish the proof of Theorem 6.9 it is sufficient to refer to Lemma
6.2.
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 6
236 Global Attractors of Non-autonomous Dissipative Dynamical Systems
Remark 6.2 a. A statement similar to Theorem 6.9 has also been proved in the
work
[
256
]
.
b. The problem of the convergence and dissipativity of equations with impulse
are studied in the works
[
256
]
,
[
115
]
,
[
116
]
,
[
256
]
.
6.4 Asymptotic stability of linear functional differential equations
Using some ideas and methods developed while studying dissipative dynamical
systems we may receive a series of conditions which are equivalent to the asymp-
totic stability of linear non-autonomous dynamical systems with infinite dimensional
phase space. As an application of these result we may receive the corresponding
results for linear functional differential equations.
Let (X, h, Y ) be a vectorial fibering with the fiber E (E is a Banach space)
and | · | : X → R be a norm on X, compatible with the distance of X, i.e. | · | is
continuous and |x| = ρ(x, θ
y
), where x ∈ X
y
, θ
y
is the null element of the space X
y
and ρ is a distance on X.
Recall that a dynamical system (X, S
+
, π) is called locally compact (locally
completely continuous) if for any x ∈ X there are δ
x
> 0 and l
x
> 0 such that
π
t
B(x, δ
x
) (t ≥ l
x
) is relatively compact.
Theorem 6.10 Let (X, S
+
, π) be locally compact and Y be compact. Then the
following conditions are equivalent:
1. lim
t→+∞
|xt| = 0 for all x ∈ X;
2. all the motions in (X, S
+
, π) are relatively compact and (X, S
+
, π) does not
admit nontrivial compact motions defined on S;
3. there are positive numbers N and ν such that |xt| ≤ Ne
−νt
|x| for all x ∈ X
and t ∈ S
+
.
Proof. From the equality lim
t→+∞
|xt| = 0 follows that Σ
+
x
is relatively compact
and ω
x
⊆ Θ := {θ
y
: y ∈ J
Y
, where θ
y
is the null element of X
y
}. Thus, the
dynamical system (X, S
+
, π) is pointwise dissipative and, according to Theorem
1.10, it is compactly dissipative. Denote by J
X
the Levinson’s center of the
dynamical system (X, S
+
, π). We will show that J
X
= Θ. Obviously, the set Θ
is compact and invariant (π
t
Θ = Θ for all t ∈ S
+
) and, consequently, Θ ⊆ J
X
.
From the last inclusion follows that h(J
X
) = J
Y
. Now let us show that J
X
= Θ. If
we suppose that it is not true, then J
X
\Θ 6= ∅ and, hence, there is x
0
∈ J
X
\Θ. Since
in the set J
X
all motions are continuable on S (see Theorem 1.6), then there exists
a continuous mapping ϕ : S → J
X
such that: ϕ(0) = x
0
and π
t
ϕ(s) = ϕ(t + s)
for all s ∈ S and t ∈ S
+
. On the other hand, by the linearity of the system
(X, S
+
, π), (Y, S
+
, σ), h
along with the point x
0
all the points λx
0
(λ ∈ R) be-