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On: 29 October 2014, At: 16:12

Publisher: Taylor & Francis

Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Applicable Analysis: An International Journal

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Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows

Pham Viet Hai a

a

Department of Mathematics , College of Science, Vietnam National University , Hanoi, Vietnam

Published online: 23 Feb 2011

To cite this article: Pham Viet Hai (2011) Discrete and continuous versions of Barbashin-type

theorem of linear skew-evolution semiflows, Applicable Analysis: An International Journal, 90:12, 1897-1907, DOI: 10.1080/00036811.2010.534728

To link to this article: http://dx.doi.org/10.1080/00036811.2010.534728

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Vol 90, No 12, December 2011, 1897–1907

Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows

Pham Viet Hai*

Department of Mathematics, College of Science, Vietnam National University, Hanoi, Vietnam Communicated by R.P Gilbert (Received 12 July 2010; final version received 19 October 2010) This article is concerned with a well-known theorem of Barbashin which states that an evolution family fU(t, s)}ts0, or simple U, is uniformly exponentially stable if and only if U satisfies the integral condition supt0 Rt

0Uðt, Þd 5 1 In fact, the author formulated the above result for non-autonomous differential equations in the frame work of finite-dimensional spaces The aim of this article is to give discrete and continuous versions of Barbashin-type theorem for the case linear skew-evolution semiflows Giving up disadvantages in Barbashin’s proof, we shall extend this problem, based on the recent methods Thus we obtain necessary and sufficient conditions for uniform exponential stability, generalizing a classical stability theorem due to Barbashin

Keywords: exponential stability; Barbashin theorem; linear skew-evolution semiflows

AMS Subject Classifications: 34D05; 34D20

1 Introduction

The understanding of the asymptotic behaviour of evolution equations is one of the most important problems of modern mathematical analysis There are many ways to study the problem: input-output criterions, discrete-time methods, the Dato– Pazy–Rolewicz theorem, and so on The results of this area have become increasingly important We shall abbreviate research works, which appear in recent times For more details about these results, we can see references

The earliest study on the input-output method or the so-called admissibility may

be [1], which is concerned with the problem of conditional stability of a system

x0¼A(t)x and its connection with the existence of bounded solutions of the equation

x0¼A(t)x þ f(t) After the seminal researches of Perron, there have been a great number of works devoted to this problem, such as [1–9] For the case of discrete-time systems analogous results were first obtained in 1934 by Ta Li [5] A nice proof for

*Email: phamviethai86@gmail.com

ISSN 0003–6811 print/ISSN 1563–504X online

ß 2011 Taylor & Francis

http://dx.doi.org/10.1080/00036811.2010.534728

Ta Li’s result is presented in [6] Perron’s ideas have been successfully extended by Massera, Scha¨ffer and by Daleckii, Krein, respectively, in infinite-dimensional spaces, [7,8] Especially, Latushkin and Schnaubelt established the relation between the exponential dichotomy of a strongly continuous cocycle over a flow and the dichotomy of the associated discrete cocycle, employing an evolution semigroup technique, [4] The authors extended some important theorems in the field

of evolution families, proving that the uniform exponential dichotomy of a linear skew-product semiflow is equivalent to the hyperbolicity of its evolution semigroup

on C0(€, X) This result can be interpreted as a generalization of a dichotomy theorem due to Minh, Ra¨biger and Schnaubelt, [2] A significant step has been made

by Henry in [10] The author characterized the dichotomy of a sequence of bounded linear operators (Tn)n2Z in terms of the existence and uniqueness of bounded solutions for xnþ1¼Tnxnþfn, for every bounded sequence ( fn)n2Z Moreover, the author showed the relation between the discrete dichotomy and the exponential dichotomy for an evolution family

Another approach was given in [11] where, Novo and Obaya constructed continuous separation of state spaces on the compact positively invariant subset M under assumptions that skew-product semiflows are eventually strongly monotone, which has consanguineous relations with the exponential dichotomy of linear skew-product semiflows

Recently, a great number of articles about the Dato–Pazy–Rolewicz theorem were published, [10,12–23] This theorem was the starting point for outstanding results concerning the exponential stability A new idea has been presented by Preda, Pogan and Preda, [12] The authors characterized the uniform exponential stability

of evolution families in terms of the existence of some functionals on sequence (function) spaces In fact, these functionals are generalizations of the integrationR1

0

or seriesP1

j¼0 This interesting idea provides us a way to attack Barbashin’s theorem For more details about this result, we refer to [24]

This article is orgainized as follows In Section 2, for the reader’s convenience, we recall some concepts and results on linear skew-evolution semiflows Section 3 is devoted to the proof of main results First, the discrete version of Barbashin’s theorem is proved And then we prove Barbashin’s theorem by using the discrete version Thus, we obtain a connection between the discrete version and the continuous version

2 Notations and preliminaries

Let X be a Banach space, L(X) the Banach algebra of all bounded linear operators acting on X, (€, d ) a metric space The norm on X, L(X) will be denoted by kk and

T:¼ fðt, sÞ 2 R2þ, t  s  0g

Definition 2.1 The mapping  : T  € ! € is called an evolution semiflow on € if: (1) (t, t, ) ¼  for all , t

(2) (t, s, (s, r, )) ¼  (t, r, ) for all t  s  r  0,  2 €

(3)  is continuous

Given an evolution semiflow, the linear skew-evolution semiflow can be defined

as follows [18,19,25]

Definition 2.2 A pair  ¼ (, ) is called a linear skew-evolution semiflow on

E:¼ X  € if  is an evolution semiflow on € and  : T  € ! L(X) has the following properties:

(1) (t, t, ) ¼ I, the identity operator on X for all (t, ) 2 Rþ €

(2) (t, r, ) ¼ (t, s, (s, r, ))(s, r, ) for all t  s  r  0 and  2 €

(3) There are M, !40 such that k(t þ s, s, )xk  Me!tkxk for all

ðt, s, , xÞ 2 R2þ € X

Remark

(1) The mapping  given Definition 2.2 is called the cocycle associated to the linear skew-evolution semiflow 

(2) In what follows, we shall denote by M, ! the constants defined in Definition 2.2

Example 2.3

(1) One can easily check that C0-semigroups, evolution families and linear skew-product semiflows are particular cases of linear skew-evolution semiflows (2) If  ¼ (, ) is a linear skew-evolution semiflow on E then for every  2 R the pair ¼(, ), where (t, s, ) ¼ e(ts)(t, s, ), is also a linear skew-evolution semiflow on E

(3) Let € be a compact metric space,  an evolution semiflow on € and A :

€! L(X) a continuous mapping If (t, t0, )x is the solution of the Cauchy problem

u0ðtÞ ¼ Aððt, t0, ÞÞuðtÞ, t  t0, then the pair  ¼ (, ) is a linear skew-evolution semiflow

We denote by Cs(€, L(X)) the space of all strongly continuous bounded mapping

H: € ! L(X), which is a Banach space with respect to the norm

H

k k:¼ sup

 2 €

HðÞ

 

:

P 2 Cs(€, L(X)), there is an unique linear skew-evolution semiflow P¼(P, P) on

Esuch that

Pðt, s, Þx ¼ ðt, s, Þx þ

Zt

s

ðt, , ð, s, ÞÞPðð, s, ÞÞPð, s, Þx d ð1Þ

for t  s 0 and (, x) 2 €  X

Proof First, we shall show that for every  2 € and t  s  0, the integral equation (1) has a solution which is a bounded linear operator on X Therefore,

we define the sequence

0ðt, s, Þx ¼ ðt, s, Þx,

nþ1ðt, s, Þx ¼

Zt

ðt, , ð, s, ÞÞPðð, s, ÞÞnð, s, Þx d:

8

<

:

One can easily check that

nðt, s, Þ

 Me!ðtsÞðM Pk kðt  sÞÞn

It makes sense to define

Pðt, s, Þ :¼X1

n¼0

nðt, s, Þ,

for every , t  s So P(t, t, ) ¼ I, for all , t Using (2), we have for every , t  s,

P(t, s, ) 2 L(X) and

Pðt, s, Þ

   Með!þM P k kÞðtsÞ: Moreover,

Pðt, s, Þx ¼ ðt, s, Þx þX1

n¼1

nðt, s, Þx

¼ðt, s, Þx þX1

n¼1

Zt

s

ðt, , ð, s, ÞÞPðð, s, ÞÞn1ð, s, Þx d

¼ðt, s, Þx þ

Zt

s

ðt, , ð, s, ÞÞPðð, s, ÞÞPð, s, Þx d:

It is easy to show that P verifies the cocycle identity Finally, we prove the uniqueness Suppose that 0

Pis a cocycle which verifies the conditions of Theorem 2.4 Then we have

Pðt, s, Þx  0Pðt, s, Þx

Zt

s

M Pk ke!ðtÞPð, s, Þx  0Pð, s, Þxd: From Gronwall’s lemma, it follows that P¼0

(1) If P 2 Cs(€, L(X)),  is an evolution semiflow and (T(t))t0is a C0-semigroup, there is a unique linear skew-evolution semiflow P¼(P, P) on E such that

Pðt, s, Þx ¼ Tðt  sÞx þ

Zt

s

Tðt  ÞPðð, s, ÞÞPð, s, Þx d

for t  s 0 and (, x) 2 €  X

(2) If P 2 Cs(€, L(X)),  is an evolution semiflow and (U(t, s))ts0is an evolution family, there is a unique linear skew-evolution semiflow P¼(P, P) on E such that

Pðt, s, Þx ¼ Uðt, sÞx þ

Zt

s

Uðt, ÞPðð, s, ÞÞPð, s, Þx d

for t  s 0 and (, x) 2 €  X

Definition 2.6 The linear skew-evolution semiflow  ¼ (, ) is said to be uniformly exponentially stable if there are K40 and 40 such that

ðt þ s, s, Þx

 Ketk k,x for all ðt, s, , xÞ 2 R2þ € X

A condition for the uniform exponential stability of linear skew-evolution semiflows is given by the following lemma

LEMMA 2.7 If there are two constants p and c 2(0, 1) such that

ð p þ m, m, Þ

 c, for all(, m) 2 €  N, then the linear skew-evolution semiflow  ¼ (, ) is uniformly exponentially stable

3 Main results

As said above, to investigate the problem of Barbashin, we use the method introduced in [12,20] That is, we shall generalize the seriesP1

j¼0to obtain a class of functionals We next use the discrete-time method to find the discrete version of Barbashin’s theorem and then convert the result to the continuous version Throughout this section, we shall denote Sþ

(Mþ

) the set of all positive sequences (functions) and s1s2if s1(j)  s2(j), for every j 2 N or Rþ N is the set of all non-decreasing functions b : Rþ! Rþwith the property b(t)40, for all t40 We need the following notion

Definition 3.1 H(N) is the set of all functions F : Sþ

![0, 1] with the property (1) F(s1)  F(s2) provided that s1s2

(2)

lim

 4 0

F Xf0, ,ng

 ¼ 1:

Remark

(1) If F1, F22 H(N) then F1þ 22 H (2) If F1F2, F12 H(N) and F2satisfies the condition (1) of Definition 3.1 then

F22 H(N)

Example 3.2 Let F1, F2: Sþ![0, 1] be maps defined by

F1ðsÞ:¼X1

j¼0

sð j Þ,

F2ðsÞ:¼Y1

j¼0

ð1 þ sð j ÞÞ:

One can check that F1, F22 H(N).

Definition 3.3 H(Rþ) is the set of all functionals G : Mþ![0, 1] with the property (1) G(s1)  G(s2) provided that s1s2

(2)

lim

 4 0

G X½0,n

 

 ¼ 1:

Remark

(1) If G1, G22 H(Rþ), then G1þ 22 H(Rþ

(2) If G1G2, G12 H(Rþ) and G2 satisfies the condition (1) of Definition 3.3, then G22 H(Rþ)

Example 3.4 The mapping

Gð f Þ:¼

Z1

0

f ðÞd

belongs to H(Rþ)

We start with the discrete-time version of Barbashin’s theorem

THEOREM 3.5 The linear skew-evolution semiflow  is uniformly exponentially stable

if and only if there exist F 2 H(N), K40, b 2 N and a non-decreasing sequence (tn)  Rþsuch that

sup

n, m, 2 N,  2 €

F ’ð bð, m, n, :ÞÞ  K,

where

’bð, m, n, j Þ ¼ b ðm þ tn, m þ tj, ðm þ tj, m, ÞÞ

, for j 2 f0, , ng,

0, for j =2 f0, , ng:



Proof Necessity Let F 2 H(N) be the functional defined by FðsÞ ¼P1

n¼0sðnÞ, tj¼j and b(t) ¼ t Definition 2.6 guarantees that there are K, 40 such that k(t, s, )k 

Ke(ts) It follows that the uniform boundedness is well-defined from inequalities

F ’ð bð, m, n, :ÞÞ ¼Xn

j¼0

ðm þ n, m þ j, ðm þ j, m, ÞÞ

Xn j¼0

Keðnj Þ¼Xn

j¼0

KejX1

j¼0

Kej5 1:

Sufficiency To prove the converse, we divide the proof into two steps

Step 1 Let us first prove the uniform boundedness of (m þ n, m, ) This means,

we need to show that there is L satisfying the inequality

sup ðm þ tn, m, Þ

  L 5 1:

From the condition (2) of Definition 3.1, there is k 2 N* satisfying the condition

F X f0, , kg

 K bð1Þ for every  2 Rþ We consider two cases as follows If n  k then

ðm þ tn, m, Þ

 Me!tk:

If n  k then taking j 2 {0, , k} randomly, one can easily check

ðm þ tn, m, Þ

 ¼ ðm þ t n, m þ tj, ðm þ tj, m, ÞÞðm þ tj, m, Þ

ðm þ tn, m þ tj, ðm þ tj, m, ÞÞðm þ tj, m, Þ

Me!tkðm þ tn, m þ tj, ðm þ tj, m, ÞÞ,

’bð, m, n, :Þ  b ðm þ tn, m, Þ

Me!t k

Xf0, ,kg,

K F b ðm þ tn, m, Þ

Me!t k

Xf0, ,kg

b ðm þ tn, m, Þ

Me!t k

K bð1Þ, which implies

ðm þ tn, m, Þ

 Me!tk: The uniform boundedness of (m þ tn, m, ) is proved since we only take L  Me!tk Step 2 We prove that the conditions of Lemma 2.7 works

Using the condition (2) of Definition 3.1 again, we get the natural number r such that

F Xf0, , rg

 K

bð1 2LÞ: For m 2 N and j 2 {0, , r}, it is clear that

ðm þ tr, m, Þ

  ¼ ðm þ t r, m þ tj, ðm þ tj, m, ÞÞðm þ tj, m, Þ

ðm þ tr, m þ tj, ðm þ tj, m, ÞÞðm þ tj, m, Þ

Lðm þ tr, m þ tj, ðm þ tj, m, ÞÞ,

’bð, m, r, :Þ  b ðm þ tr, m, Þ

L

Xf0, , rg,

K F b ðm þ tr, m, Þ

L

Xf0, , rg

b ðm þ tr, m, Þ

L

K

bð1Þ:

This is enough to show that

ðm þ tr, m, Þ

  1

2: Applying Lemma 2.7, we obtain the uniform exponential stability of  g

equivalent:

(1) U is uniformly exponentially stable

(2) There is b 2 N such that supn 2 N Pn

j¼0bð Uðn, j Þ Þ5 1:

(3) There is b 2 N such that supn, m 2 N Pn

j¼0bð Uðn þ m, j þ mÞ Þ5 1:

Proof

(1) ¼4(2)

It is a simple exercise, for b(t) ¼ t

(2) ¼4(3)

Indeed, for n, m 2 N, we have the inequality

Xn j¼0

bð Uðn þ m, j þ mÞ 

Þ  Xnþm

j¼0

bð Uðn þ m, j Þ 

Þ,

this yields supn, m 2 N Pn

j¼0bð Uðn þ m, j þ mÞ Þ5 1

(3) ¼4(1)

Putting (t, s, ) ¼ U(t, s) and (t, s, ) ¼ , for every  and t  s, one can easily check that  ¼ (, ) is a linear skew-evolution semiflow and

’bð, m, n, j Þ ¼ b Uðm þ n, m þ j Þ

, for j 2 f0, , ng,

0, for j =2 f0, , ng,



Thus we have

X1 j¼0

’bð, m, n, j Þ ¼Xn

j¼0

bUðm þ n, m þ j Þ

¼Xmþn j¼m

bUðm þ n, j Þ

Xmþn j¼0

bUðm þ n, j Þ

:

It follows supm, n, P1

j¼0’bð, m, n, j Þ 5 1

Applying Theorem 3.5 for FðsÞ ¼ P1

n¼0sðnÞ, the proof is complete g

COROLLARY 3.7 An evolution family U is uniformly exponentially stable if and only if

sup

n, m, 2 N

Yn j¼0

1 þ Uðn þ m, j þ mÞ 

5 1:

Proof The necessity can be followed from the inequality

Yn

j¼0

1 þ Uðn þ m, j þ mÞ 

Yn j¼0

ekUðnþm,jþmÞk ¼ ePnj¼0kUðnþm, jþmÞk: From Theorem 3.5, we can follow the sufficiency, using FðsÞ ¼Q1

ð1 þ sð j ÞÞ g

Now, we give a characterization of the uniform exponential stability of linear skew-evolution semiflows, which generalizes the well-known theorem of Barbashin

THEOREM 3.8 The linear skew-evolution semiflow  is uniformly exponentially stable

if and only if there exist G 2 H(Rþ), b 2 N , K40 such that

sup

m, n, 

G ð bð, m, n, :ÞÞ  K,

where

bð, m, n, Þ ¼ b ðm þ n, m þ , ðm þ , m, ÞÞ

, for  2 ½0, n,

0, for  =2 ½0, n:



Proof Necessity Let G: Mþ

![0, 1] be the mapping defined by

Gð f Þ ¼R1

0 f ðÞd and b(t) ¼ t From Definition 2.6, there are K, 40 such that k(t, s, )k  Ke(ts) Hence we get inequalities

Gð bð, m, n, :ÞÞ ¼

Zn

0

ðm þ n, m þ , ðm þ , m, ÞÞ



Zn

0

KeðnÞd ¼

Zn

0

Ked 

Z1

0

Ked 5 1:

Sufficiency For t 2 Rþ, we put ðtÞ :¼ b t

Me !

  , then 2 N Also for s 2 Sþ let fs :

Rþ! Rþbe the mapping given by fs() ¼ s([]) and FG: Sþ![0, 1] the functional defined by FG(s) :¼ G( fs) Using the fact that G 2 H(Rþ), one can easily verify that

FG2 H(N) and by observing that

ðm þ n, m þ ½, ðm þ ½, m, ÞÞ

ðm þ n, m þ , ðm þ , m, ÞÞ

 ðm þ , m þ ½, ðm þ ½, m, ÞÞ 

Me!ð½Þðm þ n, m þ , ðm þ , m, ÞÞ

Me!ðm þ n, m þ , ðm þ , m, ÞÞ for every  2 [0, n] It follows that

bð, m, n, Þ  b ðm þ n, m þ ½, ðm þ ½, m, ÞÞ

Me!

¼ððm þ n, m þ ½, ðm þ ½, m, ÞÞ

Þ ¼ f’ ð, m, n, :ÞðÞ, which implies that

bð, m, n, :Þ  f’ð, m, n, :ÞðÞ,

K G ð bð, m, n, :ÞÞ G f ’ ð, m, n, :ÞðÞ

¼FGð’ð, m, n, :ÞÞ:

This proves the uniform boundedness of FG(’(, m, n, )) Applying Theorem 3.5 we obtain the uniform exponential stability of  g

equivalent

(1) U is uniformly exponentially stable