DSpace at VNU: Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows

Introduction One of the most important results in the theory of stability for a strongly continuous semigroup of linear operators has been obtained by Datko [1] in 1970; it states that t

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Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows

Pham Viet Hai∗

Department of Mathematics, Viet Nam National University, Ha Noi, College of Science, Viet Nam

a r t i c l e i n f o

Article history:

Received 25 October 2009

Accepted 25 January 2010

MSC:

93D20

34D05

46E30

47D06

Keywords:

Exponential stability

Linear skew-evolution semiflows

Function spaces

a b s t r a c t

In this paper, we will consider the concept ‘‘linear skew-evolution semiflows’’ and extend theorems of R Datko, S Rolewicz, Zabczyk and J.M.A.M van Neerven for this case [15]

© 2010 Elsevier Ltd All rights reserved

1 Introduction

One of the most important results in the theory of stability for a strongly continuous semigroup of linear operators has been obtained by Datko [1] in 1970; it states that the semigroup(T(t))t≥ 0is uniformly exponentially stable if and only if,

for each x∈X , the map t→ kT (t)xk lies in L2(R+) Later, in [2] Pazy proved that the result remains true even if we replace

L2(R+)with L p(R+), where p∈ [1, ∞) In 1972, Datko generalized the results above as follows, [3

Theorem 1.1 An evolution family(U(t,s))t≥s≥ 0with exponential growth is uniformly exponentially stable if and only if there exists p∈ [1, ∞)such that:

sup

s≥ 0

s

kU (τ,s)xk pdτ < ∞

for all x∈X

The result provided byTheorem 1.1was extended to dichotomy by Preda and Megan [4] in 1985 The same result was generalized in 1986 by Rolewicz [5] in the following way

Theorem 1.2 Letφ :R+→R+be a continuous, non-decreasing function withφ(0) =0 andφ(t) >0 for each positive t, and

(U(t,s))t≥s≥ 0an evolution family, with exponential growth If

sup

s≥ 0

s

φ (kU(τ,s)xk)dτ < ∞, (x∈X)

∗Corresponding address: Faculty of Mathematics, Mechanics and Informatics, College of Science, Viet Nam National University, Ha Noi, 334, Nguyen Trai Road, Thanh Xuan Dist., Ha Noi, Viet Nam.

E-mail address:phamviethai86@gmail.com

0362-546X/$ – see front matter © 2010 Elsevier Ltd All rights reserved.

then(U(t,s))t≥s≥ 0is uniformly exponentially stable.

W Litman gave another proof ofTheorem 1.2for strongly continuous semigroup of linear operators, [6] In [7], the author generalize some results due to S Rolewicz, Z Zabczyk A unified treatment of the Datko–Pazy and Rolewicz theorem is presented by Neerven [8] In fact, Neerven presented in [8] a more general result

Theorem 1.3 Let C0-semigroup(T(t))t≥ 0on a Banach space X and E be a Banach function space over R+with lim t→∞ϕE(t) =

∞whereϕE(t X[ 0 ,t) E If for all x ∈ X , the map t → kT (t)xk belongs to E, then(T(t))t≥ 0is uniformly exponentially stable.

This method can be generalized for the study of uniform exponential stability of linear skew-product semiflows In [9 Megan, A.L Sasu, B Sasu proved a more general result than Neerven’s result The same direction is given in [10]

Theorem 1.4 The linear skew-product semiflowπ0 = (Φ0, σ0)is uniformly exponentially stable if and only if there are B ∈

B(N)and a sequence(t n)of positive real numbers with the following properties

(1)

Sup

n∈ N

|t n+ 1−t n| < ∞.

(2) The function:

ϕx,θ(.) :N→R+

ϕx,θ(n) = kΦ0(θ,t n)xk

belongs to B.

(3) There exists K:X→ (0, ∞)such that:

| ϕx,θ|B≤K(x)

for all(x, θ) ∈E.

In [11], the author proved:

Theorem 1.5 The linear skew-evolution semiflowπ = (Φ, σ )is uniformly exponentially stable if and only if there are B∈B(N)

and a constant L>0 such that:

1 The mapping:ϕ(x, θ,m, ) :N→R+

ϕ(x, θ,m,n) := kΦ(m+n,m, θ)xk

ϕ(x, θ,m, ) ∈B for all(x, θ,m) ∈U××N.

2 k ϕ(x, θ,m, )kB≤L for all(x, θ,m) ∈U××N.

Theorem 1.6 The linear skew-evolution semiflowπ = (Φ, σ )is uniformly exponentially stable if and only if there are N∈N

and a constant L such that:

∞

X

j= 0

N(kΦ(m+j,m, θ)xk) ≤L

for all(m, θ,x) ∈N××U.

By extending techniques in [11], this paper will study continuous characterizations for the uniform exponential stability of linear skew-evolution semiflows

2 Preliminaries

2.1 Linear skew-evolution semiflow

Let us recall basic notions of linear skew-evolution semiflows, X a Banach space,(,d)a metric space We denote byL(X)

be the Banach algebra of all bounded linear operators acting on X ; T= { (t,s) ∈R2+|t ≥s}and∆= { (m,n) ∈N2|m≥n}

Definition 2.1 A continuous mappingσ :T×is called a evolution semiflow onif:

(1) σ(t,t, θ) = θ

(2) σ(t,s, σ(s,r, θ)) = σ(t,r, θ), for all t≥s≥r≥0; θ ∈

Definition 2.2 A pairπ = (Φ, σ )is called a linear skew-evolution semiflow onE =X×ifσis an evolution semiflow

onandΦ:T×→L(X)satisfies the following conditions:

(1) Φ(t,t, θ) =I the identity operator on X , for all(t, θ) ∈R+×.

(2) Φ(t,r, θ) =Φ(t,s, σ(s,r, θ))Φ(s,r, θ)for all t≥s≥r≥0; θ ∈

(3) there are M, ω >0 such thatkΦ(t,s, θ)xk ≤ Meω(t−s)kxkfor all((t,s), θ,x) ∈T××X

Example 2.1 It is easy to see that C0-semigroups, evolution families and linear skew-product semiflows are particular cases

of linear skew-evolution semiflows

Example 2.2 Let{U (t,s)}t≥s≥ 0be an evolution family, with uniform exponential growth If there exists a bounded strongly continuous family of idempotent operators{P (θ)}θ∈ , with the property that

P(θ)U(t,s) =U(t,s)P(θ), t≥s≥0, θ ∈

then the pairπ = (Φ, σ )defined by

σ(t,s, θ) = θ, Φ(t,s, θ) =P(θ)U(t,s)

is a linear skew-evolution semiflow

Example 2.3 Letbe a compact metric space,σ an evolution semiflow on, X a Banach space and A :  → L(X)a continuous map IfΦ(t,t0, θ)x is the solution of the Cauchy problem

then the pairπ = (Φ, σ )is a linear skew-evolution semiflow Eq.(2.1)is the starting point of our paper

Definition 2.3 A linear skew-evolution semiflowπ = (Φ, σ )is said to be uniformly exponentially stable if there are K >0 andν >0 such that:

kΦ(t,s, θ)xk ≤ K e−ν(t−s)kxk

for all((t,s), θ,x) ∈T××X

Throughout this paper we shall denote:

U= {x ∈X: kxk =1}

2.2 Function spaces, sequence spaces

Let(Ω,Σ, µ)be a positiveσ-finite measure space ByMwe denote the linear space ofµ-measure functions f :Ω →C,

identifying the functions which are equal toµ-a.e

Definition 2.4 A Banach function norm is a function N :M→ [0; ∞]with the following properties:

(1) N(f) =0 if and only if f =0µ-a.e

(2) If|f | ≤ |g | µ-a.e then N(f) ≤N(g)

(3) N(af) = |a|N(f)for all a∈C and f ∈Mwith N(f) < ∞

(4) N(f+g) ≤N(f) +N(g)for all f,g∈M

Let B=B Nbe the set defined by:

B:= {f ∈M: kf kB:=N(f) < ∞}.

It is easy to see that(B; k.kB)is a normed linear space If B is complete then B is called Banach function space overΩ For more details about function spaces, we can see [12–14]

Definition 2.5 If(Ω,Σ, µ) = (R+,L,m)whereLis theσ-algebra of all Lebesgue measurable sets and m the Lebesgue

measure then

(1) For each Banach function space over R+we define:

F B:R+→R+∪ {∞}

F B(t) := X[ 0 ;t) B;X[ 0 ;t)∈B

∞;X[ 0 ;t)6∈B

whereXA denotes the characteristic function of A F B is called the fundamental function of the Banach space B.

(2) B(R+)is the set of all Banach function space:







lim

t→∞F B(t) = ∞

inf X[t;t+ 1 ) B>0.

Example 2.4 A trivial example of Banach function space over R+which belongsB(R+)is L p(R+,C)with p∈ [1, ∞).

Definition 2.6 If(Ω,Σ, µ) = (N,P(N), µc)whereµcis the countable measure then

(1) For each Banach function space over N (or Banach sequence space), B, we define:

F B:N∗→R+∪ {∞}

F B(n) = X{ 0 ; ;n− 1 } B;X{ 0 ; ;n− 1 }∈B

∞;X{ 0 ; ;n− 1 }6∈B

called the fundamental function of B.

(2) B(N)the set of all Banach sequence spaces B:

(lim

n→∞F B(n) = ∞

inf

n∈ N

X{n} B>0.

Example 2.5 If p∈ [1, ∞)then B=l pwith:

kskp=

∞

X

n= 0

|s (n)|p

!1

is a Banach sequence space which belongs toB(N)

Remark 2.6 If B is a Banach function space over R+which belongs toB(R+)then:

S B:=

(

(αn)n:

∞

X

n= 0

αnX[n,n+ 1 )∈B

)

with respect to the norm:

k (αn)nkS

B:=

∞

X

n= 0

αnX[n,n+ 1 )

B

is a Banach sequence space which belongs toB(N)

Definition 2.7 LetN be the set of all non-decreasing continuous functions:

N: [0; ∞ ) → [0; ∞ )

with properties:



N(0) =0

N(t) >0; t>0.

3 Main results

3.1 Discrete characterization

Lemma 3.1 If there are two constants p∈N∗and c∈ (0;1)(does not belong to(x, θ,m)) such that:

kΦ(p+m,m, θ)xk ≤ ckxk

for all(x, θ,m) ∈X××N,

then the linear skew-evolution semiflowπ = (Φ, σ)is uniformly exponentially stable.

Proof For each(t,s) ∈R+×R+:

If t∈ [0;1]:

kΦ(t+s,s, θ)xk ≤ Meωtkxk

≤Meωkxk

If t≥1:

t+s≥ [t ] + [s] ≥1+ [s] >s.

There exist k∈N and r∈ [0,p)such that:

[t] −1=kp+r

k> [t ] −1

p −1> t−2

p −1.

It is clear that:

kΦ(t+s,s, θ)xk ≤ Meω(t+s−[t]−[s] )kΦ(kp+r+1+ [s] ,s, θ)xk

≤Meω(t+s−[t]−[s] )MeωrkΦ(kp+1+ [s] ,s, θ)xk

≤M2eω(t+s−[t]−[s] )eωp c kkΦ(1+ [s] ,s, θ)xk

≤M3eω(t+s−[t]−[s] )eωp c keω( 1 +[s]−s)kxk

≤M3eωp+ 2 ωc t− 2 − 1

kxk

where M, ωinDefinition 2.2 So Lemma is proved 

Theorem 3.2. π = (Φ, σ )is uniformly exponentially stable if and only if there are d∈N;B∈B(N)and a constant L>0 such

that:

(1) The mapping:ϕd(x, θ,m, ) :N→R+

ϕd(x, θ,m,n) := kΦ(m+n+d,m, θ)xk

ϕd(x, θ,m, ) ∈B for all(x, θ,m) ∈U××N.

(2) k ϕd(x, θ,m, )kB≤L for all(x, θ,m) ∈U××N.

Proof Necessity:π = (Φ, σ)is uniformly exponentially stable There are K, ν >0 such that:

kΦ(m+n,m, θ)xk ≤ K e−νn

for all(m,n,x) ∈N×N×U.

∞

X

n= 0

kΦ(m+n,m, θ)xk ≤ K

∞

X

n= 0

e−νn

1−e− ν

So d=0;B:=l1andϕ(x, θ,m, ) ∈B.

Sufficiency: Let x∈U and B∈B(N)

(lim

n→∞F B(n) = ∞

c=inf

n∈N

X{n} B>0.

We see that

ϕd(x, θ,m,n)X{n}≤ ϕd(x, θ,m, )

L ≥ k ϕd(x, θ,m, )kB

≥ ϕd(x, θ,m,n X{n} B

≥ ϕd(x, θ,m,n)c

ϕd(x, θ,m,n) ≤ L

c.

For j∈ {0; ;n}andθ1:= σ (m+j+d,m, θ), we see that

kΦ(m+n+2d,m, θ)xk ≤ L

ckΦ(m+j+d,m, θ)xk

kΦ(m+n+2d,m, θ)xkX{ 0 ; ;n}≤ L

cϕd(x, θ,m, )

kΦ(m+n+2d,m, θ)xk F B(n+1) ≤ L

c| ϕd(x, θ,m, )|B

≤ L2 c

lim kΦ(m+n+2d,m, θ)xk =0.

There exists n0∈N such that:

kΦ(m+n0+2d,m, θ)xk ≤1

2.

ByLemma 3.1, Theorem is proved 

3.2 Continuous characterization

Theorem 3.3 The linear skew-evolution semiflowπ = (Φ, σ )is uniformly exponentially stable if and only if there are B ∈

B(R+)and L>0 such that:

(1) The mapping:

ψ(x, θ,m, ) :R+→R+

ψ(x, θ,m,t) := kΦ(m+t,m, θ)xk

ψ(x, θ,m, ) ∈B

for all(x, θ,m) ∈U××N.

(2) L≥ k ψ(x, θ,m, )kB

Proof Necessity: B:=L1(R+,C)and L:= Kν.

Indeed, we have:

0

kΦ(m+ τ,m, θ)xkdτ ≤ Z

∞

0

K e−ντdτ

ν =L.

Sufficiency

For each t∈R+, there exists n∈N such that t∈ [n ,n+1)

kΦ(m+n+1,m, θ)xk ≤ Meω(n+ 1 −t)kΦ(m+t,m, θ)xk

≤ MeωkΦ(m+t,m, θ)xk

ϕ1(x, θ,m,n) ≤Meωψ(x, θ,m,t)

∞

X

n= 0

ϕ1(x, θ,m,n)X[n,n+ 1 )≤Meωψ(x, θ,m, ).

So

k ϕ1(x, θ,m, )kS B =

∞

X

n= 0

ϕ1(x, θ,m,n)X[n,n+ 1 )

B

≤Meωk ψ(x, θ,m, )kB

≤MeωL

where M, ωinDefinition 2.2 By usingTheorem 3.2, Theorem is proved 

Corollary 3.4 The linear skew-evolution semiflowπ = (Φ, σ)is uniformly exponentially stable if and only if there exists p∈ [1, ∞)such that:

sup

θ∈ 

m∈ N

0

kΦ(τ +m,m, θ)xkpdτ < ∞

for all x∈X

Proof Necessity It is trivial.

Sufficiency It results byTheorem 3.3for B:=L p(R+,C) 

Theorem 3.5 The linear skew-evolution semiflowπ = (Φ, σ )is uniformly exponentially stable if and only if there are N∈N

and a constant L such that:

0

N(kΦ(τ +m,m, θ)xk)dτ ≤L

for all(m, θ,x) ∈N××U.

Proof The necessity is obvious for N(t) =t.

The sufficiency: M; ωinDefinition 2.2 We putγ (t) =N Meω t

∞

X

n= 0

γ (kΦ(n+m,m, θ)xk) =

∞

X

n= 1

γ (kΦ(n+m,m, θ)xk) + γ (1)

≤

∞

X

n= 0

γ (kΦ(n+1+m,m, θ)xk) +N



1

Meω



≤

∞

X

n= 0

Z n+ 1

n

N(kΦ(τ +m,m, θ)xk)dτ +N



1

Meω



≤L+N



1

Meω



By Theorem 3.4 in [11], Theorem is proved 

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