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Rolewicz-type characterizations for the uniform

and nonuniform stability and instability of linear

skew-product semiflows

Petre Preda, Andreea Babaita, Raluca Muresan

Abstract

In this paper we give necessary and sufficient conditions for the uniformand nonuniform exponential stability of linear skew-product semiflows.We provide only sufficient conditions for the uniform and nonuniform in-stability of linear skew-product semiflows. We give several Rolewicz typetheorems and thus we generalize important results due to S. Rolewicz, J.Zabczyk and A. L. Sasu, B. Sasu. In order to obtain nonuniform stabilityand instability, the cocycle associated to the linear skew-product semi-flow is nonuniformly exponentially bounded. Also we use an alternativeLyapunov norm introduced by Barreira-Valls in [1].

AMS Subject Classifications (2010): 34D05, 34D09.Key words and phrases: linear skew-product semiflow, cocycle, uni-

form exponential stability, nonuniform exponential stability, uniform ex-ponential instability, nonuniform exponential instability, Rolewicz

1 Introduction

In recent years, the field of evolution equations in infinite-dimensional Ba-nach spaces has seen an important and significant progress. The asymt-potic behaviour of differential equations has been studied using differentgeneralizations: C0-semigroups that model autonomous differential equa-tions, evolution families that model nonautonomous differential equationsand finally linear skew-product semiflows that model nonlinear differentialequations. Referring to the later generalization, what is of utmost impor-tance is the fact that many classical problems relating to the asymptoticalbehaviour of differential equations have been solved in an unified approachusing linear skew-evolution semiflows (see, for example, Sacker and Sell[26], Chow and Leiva [3]-[6], Chicone and Latushkin [2] and Latushkin,Montgomery - Smith and Randolph [9], Megan, Sasu and Sasu [12]- [16],Pliss and Sell [22]). In this way the theory of linear skew-product semi-flows has seen great progress in recent years.

The linear skew-product semiflow arise as solution operators for vari-ational equations

d

dtu(t) = A((, t))u(t),

1

where is a semiflow on a locally compact metric space and A() anunbounded linear operator on X, for every .

Dynamical systems described by linear skew-product semiflows werefirst studied in the papers of Sacker and Sell [25], [26]. Later on, severalauthors extended the the studies in this domain: Chow and Leiva [3]-[6], Chicone, Latushkin, Montgomery-Smith, Randolph [2], [9], and manyothers.

One of the most important results of the stability theory is due to R.Datko, who proved in 1970 in [7] that a C0-semigroup T = {T (t)}t0defined on a complex Hilbert space H is uniformly exponentially stableif and only if the map ||T ()x|| is in L2(R+) for all x X. A. Pazygeneralizes the previous result for Lp(R+), p 1, and C0-semigroups inBanach spaces (see [20], [21]).

Later on R. Datko proves in 1972 in [8] that an evolution familyU = {U(t, s)}ts0 which is uniformly exponentially bounded is uniformlyexponentially stable if and only if there exists p 1 such that

sups0

s

||U(t, s)x||pdt 0for all u > 0 and N(, u) is nondecreasing for a fixed u. Let U ={U(t, s)}ts0 be a strongly continuous evolution family on a Banachspace X.

If for every x X there is (x) > 0 such that

sups0

s

N((x), ||U(t, s)x||)dt 0, for allt > 0, such that

0

N(||T (t)x||)dt 0.

In this paper we study the stability and instability of linear skew-product semiflows. We obtain characterizations for uniform and nonuni-form stability and sufficient conditions in the case of unstable behaviour.

2

We consider two types of cocycles associated with the linear skew- productsemiflows: cocycles that are uniformly exponentially bounded and thosethat have nonuniform exponential growth. In the first case we obtainuniform asymptotic behaviour and in the second case we have observed anonuniform one. As a result, the main results are split into two sections,each treating one type of behaviour. The nonuniform case was studiedusing some interesting Lyapunov norms introduced by L. Barreira and C.Valls in [1].

Thus we answer important questions concerning the asymptotic be-haviour of linear skew-product semiflows and obtain generalizations ofsome well-known results due to Datko ([7]), Pazy ([20]), Rolewicz ([24]),van Neerven ([18]), Megan, Sasu and Sasu ([17]), Zabczyk ([28]). Thetheory developed here is applicable for a large class of systems.

2 Notations and preliminaries

Let (X, || ||) be a Banach space and (, d) a metric space which form thetrivial Banach bundle = X . We denote by (B(X), || ||) the Banachspace of all linear operators from X into itself.

Definition 2.1. A map : R+ is called a semiflow on if ithas the following properties:

(i) (, 0) = for all ;(ii) (, t+ s) = ((, s), t) for all (t, s, ) R2+ ;

(iii) (, t) 7 (, t) is continuous on R+.Definition 2.2. Let be a semiflow. A strongly continuous cocycle overthe semiflow is a function : R+ B(X) that satisfies the followingconditions:

(i) (, 0) = I for all , where I is the identity operator on X;(ii) (, t+s) = ((, t), s)(, t) for all (t, s, ) R2+ (the cocycle

identity);

(iii) (, t) 7 (, t)x is continous for all and x X.If there exists M, > 0 such that

||(, t)|| Met, for all t R+ and ,then the cocycle is uniformly exponentially bounded.

If there exists M : R+ and > 0 such that||(, t)|| M()et, for all t R+ and ,

then the cocycle is nonuniformly exponentially bounded.The classical notion of uniform exponential behavior is very stringent

for the dynamics and it is of interest to look for more general types ofhyperbolic behavior. These generalizations can be much more typical.This is precisely what happens with the notion of nonuniform exponentialboundedness. For examples of nonuniform exponential boundedness see[1], [23] and the references therein.

3

Definition 2.3. The linear skew-product semiflow associated with theabove cocycle is the dynamical system pi = (, ) defined by pi : X R+ X ,

pi(x, , t) = ((, t)x, (, t)).

Remark 2.1. If pi = (, ) is a linear skew-product semiflow on =X , then for every R the pair pi = ( , ), where (, t) =et(, t), for all (, t) R+, is also a linear skew-product semiflowon = X .

Indeed, all the conditions of a cocycle over the semiflow are obviouslyverified by .

Example 2.1. Let be a locally compact metric space, a semiflow on and T = {T (t)}t0 a C0-semigroup on X. Then the pair piT = (T, ),where T (, t) = T (t), for all (, t) R+, is a linear skew-productsemiflow on = X called the linear skew-product semiflow generatedby the C0-semigroup T and the semiflow .

Example 2.2. Let = R+, (, t) = + t and U = {U(t, s)}ts0 be anevolution family on X. We define

(, t) = U( + t, t), for all (, t) R2+.Then pi = (, ) is a linear skew-product semiflow on = X calledthe linear skew-product semiflow generated by the evolution family U andthe semiflow .

Example 2.3. Let X be a Banach space, a locally compact metricspace, T = {T (t)}t0 a C0-semigroup and {U()} a bounded stronglycontinuous family of idempotent operators with the property that

U()T (t) = T (t)U(), (, t) R+,then the pair pi = (, ) defined by

(, t) = , (, t) = U()T (t), (, t) R+is a linear skew-product semiflow.

Example 2.4. Let be a compact metric space and : R+ bea semiflow on . Let A : B(X) be a continuous mapping, where Xis a Banach space and let (, t) be the solution of the linear differentialsystem

u(t) = A((, t))u(t),t 0.Then the pair pi = (, ) is a linear skew-product semiflow on = X.

These equations arise from the linearization of nonlinear equations (see[?] and the references therein).

Example 2.5. Let X be a Banach space and let C(R+,R) be the spaceof all continuous functions with the topology of uniform convergence oncompact subsets on R+. This space is metrizable with the metric

d(x, y) =

n=1

1

2ndn(x, y)

1 + dn(x, y),

4

where dn(x, y) = supt[0,n]

|x(t) y(t)|.On the Banach space X, we consider the nonautonomous differential

equationx(t) = a(t)x(t), t 0,

where a : R+ R+ is an uniformly continuous function such that thereexists = lim

ta(t) 0 such that

||(, t)|| Net, for all (, t) R+.If N is not a constant, but a function on , i.e. N : R+, then

pi = (, ) is said to be nonuniformly exponentially stable.In the following example we present two linear skew-product semiflows,

one being uniformly exponentially stable and the other being uniformlyexponentially instable.

Example 2.6. Let C(R+,R) be the space of all continuous functions onR and a, b : R R+ two functions with the following properties: a isincreasing, continuous and = lim

ta(t) < , b is decreasing such that

there exists = limt

b(t) > 0.

We consider as(t) = a(t + s), = {as : s R}, : R ,(, t)(s) = (t+ s). It can easily be seen that is a semi-flow on .

If > and : R+ B(X),

(, t)x = et+ t0 ()sx,

then pi = (, ) is a linear skew-product semiflow on X which isuniformly exponentially stable.

We also consider bs(t) = b(t + s), = {bs : s R}, : R+ B(X), (, t)(s) = (t+ s), which is a semiflow on .

If : R+ B(X),

(, t)x = e t0 ()dx,

then pi = (, ) is a linear skew-product semiflow on X that is uni-formly exponentially instable.

5

Definition 2.5. A linear skew-product semiflow pi = (, ) on = Xis said to be uniformly exponentially instable if there exist N, > 0 suchthat

||(, t)|| Net, for all (, t) R+.!!!Ex de cociclu care este unif exp instabil.In the case that N is a function on taking positive real values, then

pi = (, ) is said to be nonuniformly exponentially instable.!!!Ex de cociclu care nu este unif exp instabil.

3 Uniform stability and instability for lin-ear skew-evolution semiflows

In this section we shall give necessary and sufficient conditions for theuniform exponential stability of linear skew-product semiflows in Banachspaces. Also we present sufficient conditions for the uniform exponentialinstability of linear skew-product semiflows in Banach spaces.

3.1 Uniform stability for linear skew-evolution semi-flows

Firstly, we give a Pata-like theorem that is the key point in the proof ofour main results. V. pata proved this theorem in [19] for C0 semigroups,thus our result is more general.

Theorem 3.1. Let pi = (, ) be a linear skew-product semiflow on =X such that the cocycle is uniformly exponentially bounded.

The linear skew-product semiflow pi = (, ) on = X is uniformlyexponentially stable if and only if there are c (0, 1) and T > 0, for all and x X there is ,x (0, T ] such that

||(, ,x)x|| c||x||.Proof. Necessity. Assume that pi is uniformly exponentially stable. Thenthere are N, > 0 such that

||(, t)x|| Net||x||, (, t, x) R+ X.Let T > 0, T > 1

ln 1

N, c = NeT (0, 1), , x X and

,x = T . The above inequality now becomes

||(, ,x)x|| c||x||.Sufficiency. Let and x X, then there are T > 0, c (0, 1) and

,x such that||(, ,x)x|| c||x||.

We consider = (, ,x) and y = (, ,x), then there is ,y (0, T ] such that

||(, ,y)y|| c||y|| c2||x||.But by the cocycle identity it follows that

(, ,y)y = ((, ,x), ,y)(, ,x)x = (, ,x +

,y)x.

6

We denote t0 = 0, t1 = ,x, t2 = t1 + ,y and we obtain

||(, t1)x|| c||x||, ||(, t2)x|| c2||x||.By proceeding in a similar way, we obtain a sequence t0 < t1 < . . . 0, u, > 0,N(, ) is increasing, if is fixed, and N(, u) is increasing, if u is fixed.

The linear skew-product semiflow pi = (, ) on = X is uniformlyexponentially stable if and only if for all x X there is x > 0 such that

sup

0

N(x, ||(, t)x||)dt

Proof. Necessity. If pi is uniformly exponentially stable, then there aretwo constants N , > 0 such that

||(, t)x|| Net||x||, (, t, x) R+ X.Since the function N is increasing in the second variable, we have that

N(, ||(, t)x||) N(, Net||x||), > 0, (, t, x) R+ X.Also, since N is increasing and is continuous in both variables, then thefunctions N(x, ||(, t)x||) and , Net||x|| are measurable and there-fore integrable.

This implies that 0

N(x, ||(, t)x||)dt

0

N(, Net||x||)dt

and so

sup

0

N(x, ||(, t)x||)dt

0

N(, Net||x||)dt 0 thereare and x X such that for all (0, T ],

||(, )x|| > c||x||.It follows from this strict inequality that x 6= 0.

SinceN is increasing in the second variable, it follows thatN(, ||(, )x||) N(, c||x||). Since N is a monotone function and is continuous we canintegrate the last inequality on [0, T ]. We get

0

N(, ||(, )x||)d T

0

N(, ||(, )x||)d TN(, c||x||)

and so

sup

0

N(, ||(, )x||)d TN(, c||x||), T > 0, x X \ {0}.

But from the hypothesis of the theorem we have that

sup

0

N(, ||(, )x||)d 0, for all > 0, x X \ {0}.

??The next corollary also appears in [27].

8

Corollary 3.1. Let pi = (, ) be a linear skew-product semiflow on = X such that the cocycle is uniformly exponentially bounded.

The pair pi = (, ) is uniformly exponentially stable if and only if forall x X there is px > 0 such that

0

||(, )x||px 1 and T > 0, for all x X and there is ,x (0, T ]such that

||(, ,x)x|| c||x||.Proof. Necessity. We assume that pi is uniformly exponentially instable,then there are N, > 0 such that

||(, t)x|| Net||x||, (, t, x) R+ X.Let T > 0, T > 1

ln 1

N, c = NeT > 1 , x X, and ,x = T .

Then we have that(, ,x) c||x||.

Sufficiency. Let and x X, then there is ,x (0, T ] such that||(, ,x)x|| c||x||.

Now for (, ,x) and y = (, ,x)x there is (0, T ] suchthat

||((, ,x), )(, ,x)|| = ||(, + ,x)x|| c2||x||.We denote t0 = 0, t1 = ,x, t2 = t1 +

and we obtain

||(, ti)x|| ci||x||, i = 0, 1, 2.By proceeding in a similar manner, we find an increasing sequence

(tn)n, tn (0, nT ], such that||(, tn)x|| cn||x||, (, x) X,n N.

9

Since c > 1, then limn

tn = and so for t 0 there is n0 N suchthat tn0 t < tn0+1. Let = 1T ln c > 0, we have that

||x||et ||x||etn0+1 ||x||e(n0+1)T = ||x||cn0+1 ||(, tn0+1)x|| =

= ||((, t), tn0+1t)(, t)x|| Me(tn0+1tn0 )||(, t)x|| MeT ||(, t)x||.Therefore there are N = 1

MeTand = 1

Tln c such that

||(, t)x|| Net||x||, (, t, x) R+ X.

The next theorem is a Rolewicz type theorem.

Theorem 3.4. Let pi = (, ) be a linear skew-product semiflow on =X such that the cocycle is uniformly exponentially bounded andinjective in both variables. Let N : R+R+ R+ be a function such that

N(, 0) = 0, N(, u) > 0, u, > 0,N(, ) is increasing, if is fixed, and N(, u) is increasing, if u is fixed.

If for all x X there is x > 0 such that

sup

0

N(x,1

||(, )x|| )d 1 and T > 0 there arex X and such that for all (0, T ],

||(, )x|| < c||x||.The last inequality shows that x 6= 0 and is equivalent to

1

(, )x>

1

c||x|| .

Since N is increasing, it follows that

N(,1

(, )x) > N(,

1

c||x|| )

and by integrating this inequality on (0, T ] it follows that 0

N(,1

(, )x)d

T0

N(,1

(, )x)d > TN(,

1

c||x|| ), T > 0.

This shows that

0N(, 1

(,)x)d =, which is absurd.

10

Corollary 3.2. Let pi = (, ) be a linear skew-product semiflow on = X such that the cocycle is uniformly exponentially bounded andinjective in both variables.

The pair pi = (, ) is uniformly exponentially instable if and only iffor all x X \ {0} there is px > 0 such that

0

1

||(, )x||px d 0 such that

||(, t)x|| Net||x||, (, t, x) R+ X.It follows that

0

1

||(, )x||px d 1

(N ||x||)px

0

epxtdt =1

px(N ||x||)px

The following theorem is the a Pata type theorem.

Theorem 4.1. Let pi = (, ) be a linear skew-product semiflow on =X such that the cocycle is nonuniformly exponentially bounded.

There are two constants N, > 0 such that

||(, t)||(,t) Net||x||, (, t, x) R+ Xif and only if there are c (0, 1) and T > 0 for all and x X thereis ,x (0, T ] such that

||(, ,x)x||(,,x) c||x||.?As a remark, the inequality

||(, t)||(,t) Net||x||, (, t, x) R+ X,although looks like the one in the definition of the uniform exponentialstability, it shows nonuniform exponential stability, since it implies

||(, t)|| NM()et||x||, (, t, x) R+ X.Proof. Necessity. It is identical with the proof in the uniform case.

Sufficiency. Let and x X, by the hypotheses of the theoremit follows that there there is ,x (0, T ] such that

||(, ,x)x||(,,x) c||x||.Now we put (, ,x) instead of and (, ,x) instead of x in the

previous inequality. Therefore by the hypotheses there is 1,x (0, T ]such that

||((, ,x), 1,x)(, ,x)x||((,,x),1,x) c||(, ,x)x||(,,x),

which is equivalent to

||(, ,x + 1,x)x||(,,x+1,x) c2||x||.

By doing this repeatedly and by denoting s0 = 0, s1 = ,x, s2 =s1 +

1,x, so on, sn = sn1 +

n1,x , we have that sn sn1 (0, T ], so

(sn)n is a strictly increasing sequence of positive real numbers and we alsohave

||(, sn)x||(,sn) cn||x||. (2)We have two cases: (sn)n is not bounded and so converges to and

(sn)n is bounded and therefore convergent.In the first case we assume that (sn)n is unbounded and so lim

nsn =

. If t 0, then there is n N such that sn t < sn+1. In this case,using the cocycle identity, the nonuniform boundedness of the cocycle and the fact that

((, sn), t sn) = (, t),we have that

||(, t)x||(,t) = ||((, sn), t sn)(, sn)x||(,t)

12

e(tsn)||(, sn)x||(,sn) eT cn||x||.We put c = eT , then = 1

Tln c > 0. Also, since t < sn+1

(n + 1)T , we get et > e(n+1)T . Therefore the last inequality nowbecomes

||(, t)x||(,t) eT enT ||x|| == eT eT e(n+1)T ||x|| e(+)T et||x|| (, t, x) R+ X.

We have shown that there are N = e(+)T > 0 and = 1T

ln c > 0such that

||(, t)x||(,t) Net||x||, (, t, x) R+ X.In the second case (sn)n is bounded and therefore there is s > 0 such

that limn

= s. By Remark 4.1 inequality 2 becomes

||(, sn)x|| cn||x||, (, x) X.By computing the limit of the last inequality, when n , we have

that||(, s)x|| = 0, (, x) X,

since c (0, 1). This proves that(, t)x = ((, s), t s)(, s)x = 0, t s.

Now if t [0, s), then there is n N such that sn t < sn+1. Byrepeating the steps of the first case we have that

||(, t)x||(,t) eT cn||x||.By proceeding identically as in the first case we deduce that

||(, t)x||(,t) Net||x||, (, t, x) R+ X,

where N = e(+)T > 0 and = 1T

ln c > 0.

The following theorems represent our main results of this section andare Rolewicz type theorems. Also they are sufficient conditions for nonuni-form exponential stability and instability.



If for every x X there is x > 0 such that

sup

0

N(x, ||(, t)x||(,t))dt

Proof. It is similar to the uniform case.



If for every x X there is x > 0 such that

sup

0

N(x,1

||(, t)x||(,t) )dt

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