Punjab University
Journal of Mathematics (ISSN 1016-2526)
Vol.47(1)(2015) pp.00.00
Asymptotic Behavior of Linear Evolution Difference System
Akbar Zada
Department of Mathematics,
University of Peshawar, Peshawar, Pakistan
Email: [email protected], [email protected]
Tongxing Li
Qingdao Technological University, Feixian, Shandong 273400, P. R. China
Email: [email protected]
Muhammad Arif
Department of Mathematics,
University of Peshawar, Peshawar, Pakistan
Email: [email protected]
Received: 12 March, 2015 / Accepted: 04 May, 2015 / Published online: 15 May, 2015
Abstract. In this article we give some characterizations of exponential
stability for a periodic discrete evolution family of bounded linear operators acting on a Banach space in terms of discrete evolution semigroups,
acting on a special space of almost periodic sequences. As a result, a
spectral mapping theorem is stated.
AMS (MOS) Subject Classification Codes: Primary 35B35.
Key Words: Discrete evolution semigroups, Exponential stability, Periodic sequences,
Almost periodic sequences.
1. I NTRODUCTION
In the recent times Y. Wang et al. in [7] has proved that the discrete system λu+1 =
Au λu is uniformly exponentially stable if and only if the unique solution of the initial
value problem
(
λu+1 = Au λu + z(u + 1), u ∈ Z+ ,
(Au , 0)
λ0 = 0,
is bounded for any natural number u and any almost d-periodic sequence z(u) with z(0) =
0. Here, Au is a sequence of bounded linear operators on Banach space X. It is well
known, see e.g. [1, 4, 6, 8] that if the initial value problem
dλ
= Aλ(t) + eiβt γ, t ≥ 0, λ(0) = 0,
dt
1
2
Akbar Zada, Tongxing Li and Muhammad Arif
has a bounded solution on R+ for every β ∈ R and any γ ∈ X then the homogenous
system dλ
dt = Aλ(t), is uniformly exponentially stable.
In case of discrete semigroups recently Zada et al. [10] proved that the system λu+1 =
D(1)λu is uniformly exponentially stable if and only if for each d-periodic bounded sequence f (u) with f (0) = 0 the solution of the initial value problem
(
λu+1 = D(1)λu + eiβ(u+1) f (u + 1),
(D(1), µ, 0)
λ(0) = 0
is bounded, where D(1) is the algebraic generator of the discrete semigroup T (u), u ∈ Z+ .
In this article we extended the result of last quoted paper to space of almost periodic
sequences denoted by AP 1 (Z+ , X), for such spaces we recommend [5].
2. N OTATIONS AND P RELIMINARIES
We denote by k · k the norms of operators and vectors. Denote by R+ the set of real
numbers and by Z+ the set of all non-negative integers.
Let B(Z+ , X) be the space of X-valued bounded sequences with the supremum norm,
and Pd (Z+ , X) be the space of d-periodic (with d ≥ 2) sequences z(n). Then Pd (Z+ , X)
is a closed subspace of B(Z+ , X).
Throughout this paper, A ∈ B(X), σ(A) denotes the spectrum of A, and r(A) :=
sup{|λ| : λ ∈ σ(A)} denotes the spectral radius of A. It is well known that r(A) =
1
lim kAn k n . The resolvent set of A is defined as ρ(A) := C\σ(A), i.e., the set of all
n→∞
λ ∈ C for which A − λI is an invertible operator in B(X).
We give some results in the framework of general Banach space and spaces of sequences
as defined above.
Recall that A is power bounded if there exists a positive constant M such that kAn k ≤
M for all n ∈ Z+ .
The family Ω := {ξ(u, v) : u, v ∈ Z+ , u ≥ v} of bounded linear operators is called
d-periodic discrete evolution family, for a fixed integer d ∈ {2, 3, . . .}, if it satisfies the
following properties:
• ξ(u, u) = I, ∀ u ∈ Z+ .
• ξ(u, v)ξ(v, r) = ξ(u, r), ∀ u ≥ v ≥ r, u, v, r ∈ Z+ .
• ξ(u + d, v + d) = ξ(u, v), ∀ u ≥ v, u, v ∈ Z+ .
It is well known that any d-periodic evolution family Ω is exponentially bounded, that
is, there exist ρ ∈ R and Mρ ≥ 0 such that
kξ(u, v)k ≤ Mρ eρ(u−v) , ∀ u ≥ v ∈ Z+ .
(2. 1)
When family Ω is exponentially bounded its growth bound, ρ0 (Ω), is the infimum of all
ρ ∈ R for which there exists Mρ ≥ 1 such that the relation ( 2. 1 ) is fulfilled. It is known
that
ρ0 (Ω)
=
=
lnkξ(u, 0)k
u
(2. 2)
1
ln(r(ξ(d, 0))).
d
(2. 3)
lim
u7→∞
Asymptotic Behavior of Linear Evolution Difference System
3
In fact
ρ0 (Ω) :
=
=
=
=
=
ln kξ(u, 0)k
u
ln kξ(ud, 0)k
lim
u7→∞
ud
1
1
lim ln kξ u (d, 0)k u
d u7→∞
1
1
ln lim kξ u (d, 0)k u
d u7→∞
1
ln(r(ξ(d, 0))).
d
lim
u7→∞
A family Ω is uniformly exponentially stable if ρ0 (Ω) is negative, or equivalently, there
exists M ≥ 1 and ρ ≥ 0 such that kξ(u, v)k ≤ M e−ρ(u−v) , for all u ≥ v ∈ Z+ .
The following lemma is a consequence of (2.1).
Lemma 2.1. [9] The discrete evolution family Ω is uniformly exponentially stable if and
only if r(ξ(d, 0)) < 1.
The map ξ(d, 0) is also called the Poincare map or monodromy operator of the evolution
family Ω.
Proposition 2.2. Let Ω = {ξ(u, v) : u, v ≥ 0} be a d-periodic discrete evolution family
acting on the Banach space X. The following four statements are equivalent:
(1) ξ(u, v) is uniformly exponentially stable.
(2) There exists two positive constants M and w such that
kξ(u, v)k ≤ M e−w(u−v) , ∀ u, v ≥ 0.
(3) The spectral radius of ξ(u, 0) is less than one; i.e.,
1
r(ξ(u, 0)) = sup{|λ| : λ ∈ σ(ξ(u, 0))} = lim kξ(u)k k k < 1.
k→∞
(4) For each µ ∈ R, one has
sup k
u≥1
u
X
e−iµk ξ(u, k)k k = M (µ) < ∞.
k=1
The proof of the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) is obvious from the definitions. The
implication of (4) ⇒ (1) can be found in Lemma (1) of [3].
We recall the following result from [7] which is very helpful in the proof of our main
result.
Theorem 2.3. [7] Let Ω := {ξ(u, v) : u, v ∈ Z+ , u ≥ v} be a discrete evolution family
on X. If the sequence
u
X
eiθk ξ(u, k)z(k)
ζu =
k=0
is bounded for each real number θ and each d-periodic sequence z(u) ∈ W, then Ω is
uniformly exponentially stable.
4
Akbar Zada, Tongxing Li and Muhammad Arif
3. D ISCRETE E VOLUTION S EMIGROUP
Here we consider a space of X-valued sequences and define a discrete evolution semigroup acting on it. For this purpose, we need the following spaces:
B(Z, X) which is the space of all X-valued bounded and uniformly convergent sequences
defined on Z, endowed with the norm kf k∞ = supu∈Z kf (u)k.
P d (Z, X) which is the subspace of B(Z, X) consisting of all sequences F such that F(u +
d) = F(u) f or all u ∈ Z.
AP 1 (Z, X) which is the space of all X-valued sequences defined on Z representable in
k=∞
P iµk u
the form f(u) =
e
ck (f) f or all u ∈ Z, where µk ∈ Z, ck (f) ∈ X and
k=−∞
kfk1 =
k=∞
X
kck (f)k < ∞.
k=−∞
For almost periodic sequences, see [2, 5].
For an arbitrary u ≥ 0, we denote by Iˆu the set ofTall X-valued sequences defined on
Z such that there exists a sequence F in P d (Z, X) AP 1 (Z, X) with F(u) = 0, f =
F|{u,u+1,...} and f(v) = 0 for all v < u. Set Iˆ = {eiµ ⊗ f : µ ∈ R and f ∈ ∪u≥0 Iˆu } and let
e
ˆ Consider the space E(Z,
ˆ which is a closed subspace
E(Z, X) = span(I).
X) = span(I)
of B(Z, X) endowed with sup norm. The discrete evolution semigroup T¸ = {T¸(u)}u≥0
e
associated to a q-periodic discrete family Ω = {ξ(u, v)}u,v≥0 on E(Z,
X) is defined as:
½
¡
¢
ξ(u, u − v)ef(u − v), if u ≥ v,
T¸(v)ef (u) =
(3. 4)
0,
if u < v,
e
for ef ∈ E(Z,
X).
e
Proposition 3.1. The space E(Z,
X) is invariant under the discrete evolution semigroup
T¸, defined in (3.4).
Proof. Let ef(u) = eiµu f(u), with µ ∈ R and f ∈ ∪u≥0 Iˆu . Then there exists r ≥ 0 and a
sequence F(u) ∈ P d (Z, X) ∩ AP 1 (Z, X) such that F(r) = 0, f(u) = F(u) f or u ≥ r
and f(u) = 0 for u < r. Thus, for each u ≥ 0 and v ∈ Z, we have
½ iµ(u−v)
¡
¢
e
ξ(u − v)F(u − v), if u ≥ v + r,
e
T¸(v)f (u) =
(3. 5)
0,
if 0 ≤ u < v + r.
The sequence G(u) = e−iµ(u−v) ξ(u−v)F(u−v) is d-periodic and belongs to AP 1 (Z, X).
Moreover
kG(.)k1 ≤ kξ(u − v)kk
k=∞
X
eiµk (u−v) ck (F)k ≤ M eνu kF(.)k1 < ∞
k=−∞
for some M ≥ 1 and w ∈ R. Thus T¸(u)ef ∈ Iˆ.
e
As an operator from E(Z,
X) to B(Z, X), T¸(u) is linear. When ef = αe
g + βe
h ∈
e
e
E(Z, X), with e
g, h ∈ Iˆ and α, β are complex scalars, one has T¸(u)f = αT¸(u)e
g + β T¸(u)e
h.
e
But T¸(u)e
g, T¸(u)e
h ∈ Iˆ and therefore T¸(u)ef belongs to E(Z, X). Thus E(Z,
X) is invariant
under the discrete evolution semigroup T¸.
Asymptotic Behavior of Linear Evolution Difference System
5
4. R ESULTS
Let G = T¸(1) − I, where T¸(1) is called the algebraic generator of the discrete evolution
semigroup T¸. Thus, for discrete semigroups, the Taylor formula of order one is:
T¸(u)fe − fe =
u−1
X
T¸(k)Gfe,
for all u ∈ Z+ with u ≥ 1,
(4. 6)
k=0
for all fe ∈ X.
e
X). The following two statements are equivalent:
Lemma 4.1. Let ef, ye ∈ E(Z,
• Ge
y = −ef.
u
P
• ye(u) =
ξ(u, k)ef(k) for all n ∈ Z+ .
k=0
Proof. (1 ⇒ 2): Using the Taylor formula (4.6), we have
T¸(u)e
y − ye =
u−1
X
T¸(k)Ge
y=−
k=0
u−1
X
T¸(k)fe.
k=0
Hence, for every u ∈ Z+ ,
ye(u) =
(T¸(u)e
y )(u) +
u−1
X
(T¸(k)fe)(u)
k=0
=
ξ(u, 0)e
y (0) +
u−1
X
ξ(u, u − k)fe(u − k)
k=0
=
u
X
ξ(u, k)fe(k).
k=0
(2 ⇒ 1): For the converse implication as G = T¸(1) − I, thus
Ge
y (u) = (T¸(1) − I)e
y (u)
= T¸(1)e
y (u) − ye(u)
= ξ(u, u − 1)e
y (u − 1) − ye(u)
= ξ(u, u − 1)
u−1
X
ξ(u − 1, k)fe(k) − ye(u)
k=0
=
u−1
X
ξ(u, k)fe(k) −
k=0
u
X
ξ(u, k)fe(k)
k=0
= −fe(u)
The proof is complete.
In the next theorem we give our main result.
Theorem 4.2. Let U be a q-periodic discrete evolution family acting on a Banach space
e
X and let T¸ be its associated discrete evolution semigroup on E(Z,
X). Denote by G
the operator T¸(1) − I, where T¸(1) is the algebraic generator of T¸ . The following are
equivalent.
(1) Ω is uniformly exponentially stable.
(2) T¸ is uniformly exponentially stable.
6
Akbar Zada, Tongxing Li and Muhammad Arif
(3) G is invertible.
u
P
e
e
(4) For each ef ∈ E(Z,
X), the series
ξ(u − k, 0)ef(k) belongs to E(Z,
X).
k=0
u
P
d
(5) For each f ∈ P (Z, X), the series
ξ(u − k, 0)f(k) is bounded on Z+ .
k=0
Proof. (1) ⇒ (2). Let N and v be two positive constants such that kξ(u, v)k ≤ N e−ν(u−v)
e
for all u ≥ v. Then, for all u ≥ 0 and any ef belonging to E(Z,
X), one has
kT¸(v)efkE(Z,X)
e
=
sup kξ(u − v, 0)ef(u − v)k
u≥v
≤
N e−ν(u−v) sup kef(u − v)k
u≥v
=
Ne
−ν(u−v)
kefkE(Z,X)
.
e
(2) ⇒ (3). It is well known that the evolution semigroup T¸ is uniformly exponentially
stable if and only if r(T¸(1)) < 1. It means that 1 is not an eigenvalue of T¸(1) i.e. 1 ∈
ρ(T¸(1)) and so G = T¸(1) − I is invertible.
e
e
(3) ⇒ (4). As G is invertible. Thus for every ef ∈ E(Z,
X) there exists ye ∈ E(Z,
X) such
n
P
that Ge
y = −ef. Thus by Lemma 4.1 we get ye(u) =
ξ(u − k, 0)ef(k) and by Lemma 3.1
u
P
k=0
e
ξ(u − k, 0)ef(k) belongs to E(Z,
X).
k=0
(4) ⇒ (5). The series
u
P
e
U(u − k, 0)ef(k) is bounded because it belongs to E(Z,
X)
k=0
which is a subset of B(Z, X).
(5) ⇒ (1). It can be seen as a direct consequence of Theorem 2.3.
In terms of initial value problems, the result contained in Theorem 4.2 may be read as
follows.
e
Corollary 4.3. Ω is uniformly exponentially stable if and only if for each ef ∈ E(Z,
X),
the solution of the problem
λu+1 = A(u)λu + ef(u + 1), u ∈ Z+
λ0 = 0,
is bounded on Z+ .
5. A PPLICATIONS
An immediate consequence of Theorem 4.2 is the spectral mapping theorem for the
e
discrete evolution semigroup T¸ on E(Z,
X).
Theorem 5.1. Let Ω be a d-periodic discrete evolution family acting on X and let T¸ be its
e
associated discrete evolution semigroup on E(Z,
X). Denote by G the operator T¸(1) − I,
where T¸(1) is the algebraic generator of T¸. Then
σ(G) = {z ∈ T : Re(z) ≤ r(G)}.
Proof. It is well-known that ρ(G) ⊇ {z ∈ T : Re(z) > r(G)}. To establish the converse
inclusion, let α ∈ ρ(G) and µ ∈ T with Re(µ) ≥ Re(α). We prove that µ ∈ ρ(G). Consider the discrete evolution family Uα (u, v) = e−α(u−v) U(u, v), where u ≥ v ≥ 0, whose
associated discrete evolution semigroup is T¸α (u) = e−αu T¸(n). Obviously, αI − G is the
Asymptotic Behavior of Linear Evolution Difference System
infinitesimal generator of T¸α . Because αI − G is invertible and applying Theorem 4.2,
C¸α (and then T¸µ ) is uniformly exponentially stable. Therefore, by applying again Theorem
4.2, µ ∈ ρ(G).
Acknowledgments: The authors express their sincere gratitude to the
Editor and referees for the careful reading of the original manuscript and
useful comments that helped to improve the presentation of the results.
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