DOI: 10.1515/auom-2015-0013
An. Şt. Univ. Ovidius Constanţa
Vol. 23(1),2015, 199–212
Nonuniform exponential stability for evolution
families on the real line
Claudia Isabela Morariu and Petre Preda
Abstract
The purpose of the present paper is to investigate the problem of
nonuniform exponential stability of evolution families on the real line
using the input-output technique known in the literature as the
Perron method for the study of exponential stability. In this manuscript
we describe an evolution family on the real line and we present sufficient
conditions for the nonuniform exponential stability of an evolution family on the real line that does not have exponential growth.
1
Indroduction
One of the most important asymptotic properties of a differential system is
the exponential dichotomy, notion introduced by O. Perron in 1930 in [14].
J.L. Daleckij and M.G. Krein in [5], J.L. Massera and J.J. Schäffer in [11,
Chapter 8] have obtained dichotomy results for differential equations on R, for
the infinite dimensional case and W.A. Coppel in [4] and P. Hartman in [6]
for the finite dimensional case.
In 1974 M.G. Krein and J.L. Daleckij in [5, Theorem 4.1, p. 81] shows
that if A ∈ B(X) then σ(A) ∩ iR = ∅ if and only if the differential equation
ẋ(t) = Ax(t) + f (t) has an unique solution x ∈ C, for all f ∈ C, where C
represents the Banach space of the continuous and bounded functions on R
and σ(A) represent the spectrum of the operator A.
Key Words: Evolution family on R, nonuniform exponential stability, Perron condition.
2010 Mathematics Subject Classification: Primary 34D05; Secondary 34C35, 47B48.
Received: 27 April, 2014.
Revised: 15 June, 2014.
Accepted: 29 June, 2014.
199
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
200
An important contribution in the study of the asymptotic behaviour of the
dynamical systems described by the evolution families is represented by [3],
published in 1999 by C. Chicone and Y. Latushkin. Another important results
in the study of the evolution equations were obtained by B.M. Levitan and
V.V. Zhikov in [9] and A. Pazy in [13]. Some of the results were extended for
the evolution families with nonuniform exponential growth by L. Barreira and
C. Valls in [1] and [2].
In 1998 Y. Latushkin, T. Randolph and R. Schnaubelt in [8] study the
dichotomy on R for the evolution families with uniform exponential growth
through the assigned evolution semigroup. The dichotomy on R+ was studied
by N. Van Minh, F. Räbiger and R. Schnaubelt in [12] and N.T. Huy in [7].
Similar results for the dichotomy on the real line were obtained by A.L.
Sasu and B. Sasu in [15] and A.L. Sasu in [16]. All the above results are
obtained for t0 ∈ R+ . In [15] and [16] are considered systems described by
evolution families with exponential growth on the real line. It can also be
mentioned the results obtained by M. Marin and O. Florea in [10] as well as
K. Sharma and M. Marin in [17].
It is known that the exponential dichotomy is a generalization of the exponential stability, so it is expected that the above results in more stringent
conditions should imply the exponential stability on R.
The main purpose of this paper is to give a sufficient condition for the
nonuniform exponential stability of an evolution family without exponential
growth on the real line using the concept of Perron condition.
Section 2 is devoted to the preliminaries while Section 3 is dedicated to
the main results. First there are specified the following concepts: evolution
family on R, nonuniform exponentially stable evolution family and uniformly
stable evolution family. In Definition 3.1 we describe when we say that an
evolution family satisfies the Perron condition. The Theorem 3.3 will be used
in the demonstration of one of the most important result of this paper, namely
Theorem 3.4.
Further in Definition 3.5 it is specified when an evolution family satisfies
the (p, ∞)− Perron condition, where p > 1 and in Theorem 3.7 is presented
another important result related to the nonuniform exponential stability of an
evolution family. For the last result, considering p = 1, we obtain a characterization for the uniform exponential stability of an evolution family on the
real line.
2
Preliminaries
Let X be a Banach space and B(X) the Banach algebra of all linear and
bounded operators acting on X. We will denote by k · k the norm on X and
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
201
B(X) and ∆ = {(t, t0 ) ∈ R2 : t ≥ t0 }.
Definition 2.1. An application Φ : ∆ → B(X), Φ = {Φ(t, t0 )}t≥t0 , is called
an evolution family on R if it satisfies the following properties:
(i) Φ(t, t) = I, for all t ∈ R, where I is the identity operator on X;
(ii) Φ(t, τ )Φ(τ, t0 ) = Φ(t, t0 ), for all t ≥ τ ≥ t0 ;
(iii) The map Φ(·, t0 )x is continuous on [t0 , ∞), for all x ∈ X and
Φ(t, ·)x is continuous on (−∞, t], for all x ∈ X.
We mention the following function spaces:
C = {f : R → R : f is continuous and bounded},
C00 = {f ∈ C : lim f (t) = lim f (t) = 0},
t→−∞
t→∞
Z∞
Lp (X) = {f : R → X : f is measurable and
kf (t)kp dt < ∞}, where p ∈ [1, ∞)
−∞
and
L∞ (X) = {f : R → X : f is measurable and ess sup kf (t)k < ∞}.
t∈R
The norm on C and C00 is |||f ||| = sup kf (t)k.
t∈R
∞
p
The norms on L (X) and L (X) are denoted by
Z∞
kf kp =
p1
kf (t)k dt , respectively kf k∞ = ess sup kf (t)k.
p
−∞
t∈R
Let {Φ(t, t0 )}t≥t0 be an evolution family on R.
Definition 2.2. We say that the evolution family {Φ(t, t0 )}t≥t0 is nonuniform
exponentially stable if there exists N : R → R∗+ and ν > 0 such that
kΦ(t, t0 )xk ≤ N (t0 )e−ν(t−t0 ) kxk, for all t ≥ t0 and x ∈ X.
Definition 2.3. We say that the evolution family {Φ(t, t0 )}t≥t0 is uniformly
stable if there exists a constant N > 0 such that
kΦ(t, t0 )xk ≤ N kxk, for all t ≥ t0 and x ∈ X.
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
202
3
Main results
Definition 3.1. We say that the evolution family {Φ(t, t0 )}t≥t0 satisfies the
Perron condition if:
(i) For all f ∈ C00 it results that xf ∈ C, where xf (t) =
Rt
Φ(t, τ )f (τ )dτ ;
−∞
(ii) If there is w ∈ C such that w(t) = Φ(t, s)w(s), for all t ≥ s, it results
that w = 0.
Remark 3.2. If {Φ(t, t0 )}t≥t0 satisfies the Perron condition then
Zt
Φ(t, τ )f (τ )dτ, for all t ≥ s.
xf (t) = Φ(t, s)xf (s) +
s
Theorem 3.3. If {Φ(t, t0 )}t≥t0 satisfies the Perron condition then there is a
constant K > 0 such that
|||xf ||| ≤ K|||f |||, for all f ∈ C00 .
Proof. Let U : C00 → C, defined by Uf = xf . We notice that U is a linear
operator and we will prove that it is closed.
Let (fn )n∈N be a sequence in C00 , f ∈ C00 and g ∈ C such that
fn → f in C00 and Ufn → g in C.
We have that
Ufn (t) = xfn (t) = Φ(t, s)xfn (s) +
Zt
Φ(t, τ )fn (τ )dτ, for all t ≥ s
(3.1)
s
and
t
Z
Zt
Zt
Φ(t, τ )fn (τ )dτ − Φ(t, τ )f (τ )dτ ≤ kΦ(t, τ )(fn (τ ) − f (τ ))kdτ. (3.2)
s
s
s
Since the function τ 7→ Φ(t, τ )x : [s, t] → X is continuous, so it is bounded,
we have that there is Ms,t,x > 0 such that
kΦ(t, τ )xk ≤ Ms,t,x , for all t ∈ R and x ∈ X
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
203
and from the Uniform Boundedness Principle it results that there is Ms,t > 0
such that
kΦ(t, τ )xk ≤ Ms,t kxk, for all t ∈ R and x ∈ X.
From the relation (3.2) it follows that
Zt
Zt
kΦ(t, τ )(fn (τ ) − f (τ ))kdτ ≤ Ms,t
s
kfn (τ ) − f (τ )kdτ
s
≤ Ms,t (t − s)|||fn − f ||| −−−−→ 0.
n→∞
From the relation (3.1), for n → ∞, it results that
Zt
Φ(t, τ )f (τ )dτ, for all t ≥ s.
g(t) = Φ(t, s)g(s) +
s
We consider now w(t) = g(t) − xf (t) = Φ(t, s)w(s), for all t ≥ s, which
implies that w = 0 because w ∈ C. It follows that g = xf = Uf .
We obtain that U is a closed operator and by the Closed Graph Theorem
it is also bounded. Therefore there is K > 0 such that
|||xf ||| ≤ K|||f |||, for all f ∈ C00 .
Theorem 3.4. If {Φ(t, t0 )}t≥t0 satisfies the Perron condition then
{Φ(t, t0 )}t≥t0 is nonuniform exponentially stable.
Proof. Let x ∈ X, t0 ∈ R, δ > 0 and


0, t < t0


 4 (t − t0 ), t0 ≤ t < t0 + δ
δ
2
δ
δ
χt0 : R → R+ , χt0 (t) =
4
δ

4 − δ (t − t0 ), t0 + 2 ≤ t < t0 + δ




0, t ≥ t0 + δ.
It follows that χδt0 ∈ C00 and |||χδt0 ||| = 2.
Now we consider f : R → X, f (t) = χ1t0 (t)Φ(t, t0 )x. Thus f ∈ C00 and
|||f |||
≤
2 supt∈[t0 ,t0 +1] kΦ(t, t0 )kkxk
=
2M (t0 )kxk,
where
M (t0 ) = supt∈[t0 ,t0 +1] kΦ(t, t0 )k.
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
204
We obtain that
Zt
xf (t) =
Zt
Φ(t, τ )f (τ )dτ =
−∞
t0


0, t < t0
χ1t0 (τ )dτ Φ(t, t0 )x = (t − t0 )Φ(t, t0 )x, t ∈ [t0 , t0 + 1)


Φ(t, t0 )x, t ≥ t0 + 1.
But xf ∈ C and from Theorem 3.3 it results that there is K > 0 such that
kΦ(t, t0 )xk ≤ K|||f ||| ≤ 2KM (t0 )kxk, for all t ≥ t0 + 1 and x ∈ X.
For t ∈ [t0 , t0 + 1) we have that kΦ(t, t0 )xk ≤ M (t0 )kxk. Therefore
kΦ(t, t0 )xk ≤ L(t0 )kxk, for all t ≥ t0 and x ∈ X,
where L(t0 ) = M (t0 ) max{1, 2K}.
We set now x ∈ X, t0 ∈ R, δ > 0 and
g : R → X, g(t) = χδt0 (t)Φ(t, t0 )x.
It follows that g ∈ C00 and |||g||| ≤ 2L(t0 )kxk.
We obtain that
Zt
xg (t) =
Zt
Φ(t, τ )g(τ )dτ =
−∞
t0


0, t < t0
χδt0 (τ )dτ Φ(t, t0 )x = (t − t0 )Φ(t, t0 )x, t ∈ [t0 , t0 + δ)


δΦ(t, t0 )x, t ≥ t0 + δ.
But xg ∈ C and from Theorem 3.3 it results that there is K > 0 such that
δkΦ(t, t0 )xk ≤ K|||g||| ≤ 2KL(t0 )kxk, for all t ≥ t0 + δ, x ∈ X and δ > 0.
For δ = t − t0 it follows that
(t − t0 )kΦ(t, t0 )xk ≤ 2KL(t0 )kxk, for all t ≥ t0 and x ∈ X.
Let now x ∈ X, n ∈ N∗ , t0 ∈ R, δ > 0 and


0, t < t0

(t − t )Φ(t, t )x, t ∈ [t , t + δ]
0
0
0 0
y : R → X, y(t) =
δ(1 − nt + nt0 + nδ)Φ(t, t0 )x, t ∈ (t0 + δ, t0 + δ + n1 ]



0, t > t0 + δ + n1 .
It follows that y ∈ C00 and |||y||| ≤ 2KL(t0 )kxk.
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
205
We obtain that
Zt
xy (t) =
Zt
Φ(t, τ )y(τ )dτ =
−∞
Φ(t, τ )y(τ )dτ =
(t − t0 )2
Φ(t, t0 )x,
2!
t0
for all t ∈ [t0 , t0 + δ] and δ > 0.
But xy ∈ C and from Theorem 3.3 it results that there is K > 0 such that
(t − t0 )2
kΦ(t, t0 )xk ≤ 2K 2 L(t0 )kxk, for all t ≥ t0 and x ∈ X.
2!
Let x ∈ X, t0 ∈ R, n ∈ N∗ , δ > 0 and


0, t < t0



2


 (t − t0 ) Φ(t, t0 )x, t ∈ [t0 , t0 + δ]
2!
h : R → X, h(t) =
δ2


(1 − nt + nt0 + nδ)Φ(t, t0 )x, t ∈ (t0 + δ, t0 + δ + n1 ]



2!

0, t > t + δ + 1 .
0
n
It results that h ∈ C00 and |||h||| ≤ 2K 2 L(t0 )kxk.
We obtain that
Zt
xh (t) =
Zt
Φ(t, τ )h(τ )dτ =
−∞
Φ(t, τ )h(τ )dτ =
(t − t0 )3
Φ(t, t0 )x,
3!
t0
for all t ∈ [t0 , t0 + δ] and δ > 0.
But xh ∈ C and from Theorem 3.3 it follows that there is K > 0 such that
(t − t0 )3
kΦ(t, t0 )xk ≤ 2K 3 L(t0 )kxk, for all t ≥ t0 and x ∈ X.
3!
Inductively we obtain that
(t − t0 )n
kΦ(t, t0 )xk ≤ 2K n L(t0 )kxk, for all t ≥ t0 , x ∈ X and n ∈ N.
n!
Sharing with 2n K n it results that
(t − t0 )n
1
kΦ(t, t0 )xk ≤ 2 n L(t0 )kxk, for all t ≥ t0 and x ∈ X.
2n K n n!
2
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
206
We have that
∞
∞
X
X
(t − t0 )n
1
L(t0 )kxk, for all t ≥ t0 and x ∈ X,
kΦ(t,
t
)xk
≤
2
0
n
n
2 K n!
2n
n=0
n=0
or equivalently
e
(t−t0 )
2k
kΦ(t, t0 )xk ≤ 4L(t0 )kxk, for all t ≥ t0 and x ∈ X.
So
kΦ(t, t0 )xk ≤ 4L(t0 )e−
(t−t0 )
2k
kxk, for all t ≥ t0 and x ∈ X.
1
we will obtain that there exists
Denoting by N (t0 ) = 4L(t0 ) and ν =
2K
∗
N : R → R+ and ν > 0 such that
kΦ(t, t0 )xk ≤ N (t0 )e−ν(t−t0 ) kxk, for all t ≥ t0 and x ∈ X,
so {Φ(t, t0 )}t≥t0 is nonuniform exponentially stable.
Definition 3.5. We say that the evolution family {Φ(t, t0 )}t≥t0 satisfies the
(p, ∞)-Perron condition if:
(i) For all f ∈ Lp (X) it results that xf ∈ L∞ (X), where
Zt
xf (t) =
Φ(t, τ )f (τ )dτ ;
−∞
(ii) If there is w ∈ L∞ (X) such that w(t) = Φ(t, s)w(s), for all t ≥ s, it
results that w = 0.
Theorem 3.6. If {Φ(t, t0 )}t≥t0 satisfies the (p, ∞)-Perron condition there is
a constant K > 0 such that
kxf k∞ ≤ Kkf kp , for all f ∈ Lp (X).
Proof. Let U : Lp (X) → L∞ (X), defined by Uf = xf . We notice that U is a
linear operator and we will prove that it is closed.
Let (fn )n∈N be a sequence in Lp (X), f ∈ Lp (X) and g ∈ L∞ (X) such that
fn → f in Lp (X) and Ufn → g in L∞ (X).
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
207
Using the same technique as in Theorem 3.3 we obtain that
Zt
Φ(t, τ )f (τ )dτ, for all t ≥ s.
g(t) = Φ(t, s)g(s) +
s
Considering w(t) = g(t) − xf (t) = Φ(t, s)w(s), ∀ t ≥ s, it follows that
w = 0 because w ∈ L∞ (X). It results that g = xf a.e and thus g = xf = Uf
in L∞ (X).
We obtain that U is a closed operator and by the Closed Graph Theorem
it is also bounded. Therefore there is K > 0 such that
||xf ||∞ ≤ K||f ||p , for all f ∈ Lp (X).
Theorem 3.7. If {Φ(t, t0 )}t≥t0 satisfies the (p, ∞)-Perron condition, p > 1
then there exists N : R → R∗+ and ν > 0 such that
1− 1
p
kΦ(t, t0 )xk ≤ N (t0 )e−ν(t−t0 )
kxk, for all t ≥ t0 and x ∈ X.
Proof. Let x ∈ X, t0 ∈ R and
f : R → X, f (t) = ϕ[t0 ,t0 +1] (t)Φ(t, t0 )x,
where ϕ[t0 ,t0 +1] denotes the characteristic function of the interval [t0 , t0 + 1].
It results that
f ∈ Lp (X) and kf kp ≤ M (t0 )kxk, where M (t0 ) =
sup
kΦ(t, t0 )k.
t∈[t0 ,t0 +1]
We have that
Zt
xf (t) =
Φ(t, τ )f (τ )dτ =
−∞
Zt
=
t0


0, t < t0
ϕ[t0 ,t0 +1] (τ )dτ Φ(t, t0 )x = (t − t0 )Φ(t, t0 )x, t ∈ [t0 , t0 + 1)


Φ(t, t0 )x, t ≥ t0 + 1.
But xf ∈ L∞ (X) and from Theorem 3.4 it results that there is K > 0 such
that
kΦ(t, t0 )xk ≤ Kkf kp ≤ KM (t0 )kxk, for all t ≥ t0 + 1 and x ∈ X.
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
208
For t ∈ [t0 , t0 + 1) we have that kΦ(t, t0 )xk ≤ M (t0 )kxk. Therefore
kΦ(t, t0 )xk ≤ L(t0 )kxk, for all t ≥ t0 and x ∈ X,
where L(t0 ) = M (t0 ) max{1, K}.
Let now x ∈ X, t0 ∈ R, δ > 0 and
g : R → X, g(t) = ϕ[t0 ,t0 +δ] (t)Φ(t, t0 )x.
1
It results that g ∈ Lp (X) and kgkp ≤ δ p L(t0 )kxk.
We have that
Zt
xg (t) =
Φ(t, τ )g(τ )dτ =
−∞
Zt
=
t0


0, t < t0


(t
− t0 )Φ(t, t0 )x, t ∈ [t0 , t0 + δ)
ϕ[t0 ,t0 +δ] (τ )dτ Φ(t, t0 )x =


δΦ(t, t0 )x, t ≥ t0 + δ.
But xg ∈ L∞ (X) and from Theorem 3.4 it results that there is K > 0 such
that
1
δkΦ(t, t0 )xk ≤ Kkgkp ≤ KL(t0 )δ p kxk, for all t ≥ t0 + δ, x ∈ X and δ > 0.
If we put δ = t − t0 then we obtain that
1
(t − t0 )kΦ(t, t0 )xk ≤ KL(t0 )(t − t0 ) p kxk, for all t ≥ t0 and x ∈ X,
or equivalently
1
(t − t0 )1− p kΦ(t, t0 )xk ≤ KL(t0 )kxk, for all t ≥ t0 and x ∈ X.
Now we consider
1
h : R → X, h(t) = ϕ[t0 ,t0 +δ] (t)(t − t0 )1− p Φ(t, t0 )x.
1
It results that h ∈ Lp (X) and khkp ≤ KL(t0 )δ p kxk.
We have that
Zt
xh (t) =
Zt
Φ(t, τ )h(τ )dτ =
−∞
t0
1
(τ − t0 )1− p ϕ[t0 ,t0 +δ] (τ )dτ Φ(t, t0 )x.
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
209
1
(t − t0 )2− p
Φ(t, t0 )x.
2 − p1
But xh ∈ L∞ (X) and from Theorem 3.4 there is K > 0 such that
If t ≥ t0 + δ then xh (t) =
1
1
(t − t0 )2− p
kΦ(t, t0 )xk ≤ Kkhkp ≤ K 2 L(t0 )(t−t0 ) p kxk, for all t ≥ t0 and x ∈ X.
2!
Inductively we obtain that
1
(t − t0 )n(1− p )
kΦ(t, t0 )xk ≤ K n L(t0 )kxk.
n!
By sharing with 2n K n it results that
1
L(t0 )
(t − t0 )n(1− p )
kΦ(t, t0 )xk ≤
kxk.
n
n
2 K n!
2n
Thus
1
∞
∞
X
X
(t − t0 )n(1− p )
L(t0 )
kΦ(t,
t
)xk
≤
kxk,
0
n K n n!
2
2n
n=0
n=0
or equivalently
1
e 2K (t−t0 )
1− 1
p
kΦ(t, t0 )xk ≤ 2L(t0 )kxk.
We have that
1
1− 1
p
kΦ(t, t0 )xk ≤ 2L(t0 )e− 2K (t−t0 )
kxk, for all t ≥ t0 and x ∈ X.
Denoting by N (t0 ) = 2L(t0 ) and ν =
N : R → R∗+ and ν > 0 such that
kΦ(t, t0 )xk ≤ N (t0 )e−ν(t−t0 )
1− 1
p
1
2K
we will obtain that there exists
kxk, for all t ≥ t0 and x ∈ X.
Further we will analyze the case (1, ∞).
Theorem 3.8. {Φ(t, t0 )}t≥t0 satisfies the (1, ∞)-Perron condition if and only
if {Φ(t, t0 )}t≥t0 is uniformly stable.
Proof. Necessity. Let t0 ∈ R, δ > 0, x ∈ X such that Φ(t, t0 )x 6= 0, for all
t ≥ t0 and
Φ(t, t0 )x
,
f : R → X, f (t) = ϕ[t0 ,t0 +δ] (t)
kΦ(t, t0 )xk
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
210
where ϕ[t0 ,t0 +δ] denotes the characteristic function of the interval [t0 , t0 + δ].
It results that f ∈ L1 (X) and kf k1 = δ.
We have that
tZ
0 +δ
Zt
xf (t) =
Φ(t, τ )f (τ )dτ =
−∞
dτ
Φ(t, t0 )x, for all t ≥ t0 + δ.
kΦ(τ, t0 )xk
t0
But xf ∈ L∞ (X) and by Theorem 3.4 it results that there is K > 0 such
that
tZ
0 +δ
dτ
1
kΦ(t, t0 )xk ≤ K, for all x ∈ X and δ > 0.
δ
kΦ(τ, t0 )xk
t0
For δ → 0 we obtain that
kΦ(t, t0 )xk ≤ Kkxk, for all t ≥ t0 and x ∈ X.
Let now t0 ∈ R, x ∈ X and t1 > t0 such that Φ(t1 , t0 )x = 0. It implies
that
Φ(t, t0 )x = 0, for all t ≥ t1 .
Denoting by σ = inf{t ≥ t0 : Φ(t, t0 )x = 0} it follows that Φ(σ, t0 )x = 0,
or equivalently Φ(t, t0 )x 6= 0, for all t ∈ [t0 , σ). Therefore
kΦ(t, t0 )xk ≤ Kkxk, for all t ≥ t0 and x ∈ X.
Sufficiency. We consider f ∈ L1 (X) and xf (t) =
Rt
Φ(t, τ )f (τ )dτ. We have
−∞
that
Zt
kxf (t)k ≤
N kf (τ )kdτ ≤ N kf k1 < ∞, for all t ∈ R.
−∞
It result that xf ∈ L∞ (X).
Let now w ∈ L∞ (X) such that w(t) = Φ(t, s)w(s), for all t ≥ s. We also
set t ∈ R and s ∈ [t − 1, t]. It follows that
kw(t)k ≤ N kw(s)k, for all s ∈ [t − 1, t].
We obtain that
Zt
kw(t)k ≤ N
kw(s)kds ≤ N kwk∞ < ∞, for all t ≥ s,
t−1
thus kw(t)k = 0, for all t ∈ R, so w = 0.
In this way we obtain that the evolution family is uniformly stable.
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
211
Acknowledgment. This work was supported by the strategic grant
POSDRU/159/1.5/S/137750, Project ”Doctoral and Postdoctoral programs
support for increased competitiveness in Exact Sciences research” cofinanced
by the European Social Found within the Sectorial Operational Program Human Resources Development 2007-2013.
References
[1] Barreira, L., Valls, C., Admissibility for nonuniform exponential contractions, Journal of Differential Equations 249 (2010), 2889-2904.
[2] Barreira, L., Valls, C., Nonuniform exponential dichotomies and admissibility, Discrete and Continuous Dynamical Systems 30 (2011), 39-53.
[3] Chicone, C., Latushkin, Y., Evolution semigroups in dynamical systems
and differential equations, Mathematical Surveys and Monographs, vol.
70, Providence, RO: American Mathematical Society, 1999.
[4] Coppel, W.A., Dichotomies in Stability Theory, Lecture Notes in Math.,
vol. 629, Springer-Verlag, New York, 1978.
[5] Daleckij, J.L., Krein, M.G., Stability of Differential Equations in Banach
Space, Amer. Math. Soc., Providence, RI, 1974.
[6] Hartman,
P. Ordinary
York/London/Sidney, 1964.
Differential
Equations,
Wiley,
New
[7] Huy, N.T., Invariant manifolds of admissibile classes for semi-linear evolution equations, J. Differential Equations 246 (2009), 1820-1844.
[8] Latushkin, Y., Randolph, T., Schnaubelt, R., Exponential dichotomy and
mild solutions of non-autonomous equations in Banach spaces, J. Dynam.
Differential Equations 3 (1998), 489-510.
[9] Levitan, B.M., Zhikov, V.V., Almost Periodic Functions and Differential
Equations, Cambridge Univ. Press, Cambridge, 1982.
[10] Marin, M., Florea, O., On temporal behaviour of solutions in Thermoelasticity of porous micropolar bodies, An. Sti. Univ. Ovidius Constanţa,
Vol. 22, issue 1, (2014) 169-188.
[11] Massera, J.L., Schäffer, J.J., Linear Differential Equations and Function
Spaces, Academic Press, New York, 1966.
NONUNIFORM EXPONENTIAL STABILITY FOR EVOLUTION FAMILIES ON
THE REAL LINE
212
[12] Van Minh, N., Räbiger, F., Schnaubelt, R., Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations
on the half-line, Integral Equations Operator Theory 32 (1998), 332-353.
[13] Pazy, A., Semigroups of Linear Operators and Applications to Partial
Differential Equations, Springer-Verlag, New York, 1983.
[14] Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32
(1930), 703-728.
[15] Sasu, A.L., Sasu, B. Exponential dichotomy on the real line and admissibility of function spaces, Integr. equ. oper. theory 54 (2006), 113-130.
[16] Sasu, A.L., Integral equations on function spaces and dichotomy on the
real line, Integr. equ. oper. theory 58 (2007), 133-152.
[17] Sharma, K., Marin, M., Reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic
solids, An. Sti. Univ. Ovidius Constanţa, Vol. 22, issue 2, (2014) 151175.
Claudia Isabela MORARIU,
Department of Mathematics,
Faculty of Mathematics and Computer Science,
West University of Timişoara,
Vasile Pârvan Blvd. No. 4, 300223 Timişoara, Romania.
Email: claudia morariu@yahoo.com
Petre PREDA,
Department of Mathematics,
Faculty of Mathematics and Computer Science,
West University of Timişoara,
Vasile Pârvan Blvd. No. 4, 300223 Timişoara, Romania.
Email: preda@math.uvt.ro