# On Weak Exponential Expansiveness of Evolution Families in Banach Spaces

Authors: Tian Yue; Xiao-qiu Song; Dong-qing Li
Journal: The Scientific World Journal
 (2013)

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## Abstract

The aim of this paper is to give several characterizations for the property of weak exponential expansiveness for evolution families in Banach spaces. Variants for weak exponential expansiveness of some well-known results in stability theory (Datko (1973), Rolewicz (1986), Ichikawa (1984), and Megan et al. (2003)) are obtained.


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## Body

## 1. Introduction

In recent years, the exponential stability theory of one parameter semigroups of operators and evolution families has witnessed significant development. A number of long-standing open problems have been solved, and the theory seems to have obtained a certain degree of maturity. One of the most important results of the stability theory is due to Datko, who proved in 1970 in [1] that a strongly continuous semigroup of operators {T(t)}t≥0 is uniformly exponentially stable if and only if for each vector x from the Banach space X the function t→∥T(t)x∥ lies in L2(ℝ+). Later, Pazy generalizes the result in [2] for Lp(ℝ+), p≥1. In 1973, Dakto [3] generalized the results above and proved that an evolutionary process 𝒰={U(t,s)}t≥s≥0 with uniform exponential growth is uniformly exponentially stable if and only if there exists an exponent p≥1 such that sups≥0∫s∞‍∥U(t,s)x∥pdt<∞, for each x∈X. This result was improved by Rolewicz in 1986 (see [4]). In [5, 6], the authors generalized the results above in the case of C0-semigroups and evolutionary process, respectively, and presented a unified treatment in terms of Banach function spaces.In the last few years, new concepts of exponential expansiveness and in particular, of exponential instability, have been introduced and characterized (see[7–14]). The cases of uniform exponential instability have been considered in [8] for evolution families and in [10] for linear skew-product flows.In the present paper, we introduce the concept of weak exponential expansiveness for evolution families which is an extension of classical concept of exponential expansiveness. Our main objective is to give some characterizations for weak exponential expansiveness properties of evolution families in Banach spaces, and variants for weak exponential expansiveness of some well-known results in stability theory (Datko [3], Rolewicz [4], Ichikawa [15], and Megan et al. [8]) are obtained.


## 2. Preliminaries

LetX be a real or complex Banach space. The norm on X and on the space B(X) of all bounded linear operators on X will be denoted by ∥·∥.Definition 1.
A family𝒰={U(t,s)}t≥s≥0 of bounded linear operators is called an evolution family if the following conditions are satisfied:(i)
U(t,t)=I, the identity operator on X, for all t≥0;(ii)
U(t,r)U(r,s)=U(t,s) for all t≥r≥s≥0;(iii)
there existM≥1 and ω>0 such that ∥U(t,s)x0∥≤Meω(t-s)∥x0∥ for all t≥s≥0 and x0∈X;(iv)
for everyx0∈X and every t0≥0, the mapping r↦∥U(r,t0)x0∥ is continuous on [t0,∞);(v)
for everyt≥s≥0, the operator U(t,s) is injective.Definition 2.
An evolution family𝒰={U(t,s)}t≥s≥0 is called uniformly expansive if there exists a constant N>0 such that
(1)∥U(t,t0)x0∥≥N∥U(r,t0)x0∥,
for all t≥r≥t0≥0 and x0∈X.Definition 3.
An evolution family𝒰={U(t,s)}t≥s≥0 is said to be uniformly exponentially expansive if there are N,v>0 such that
(2)∥U(t,t0)x0∥≥Nev(t-r)∥U(r,t0)x0∥,
for all t≥r≥t0≥0 and x0∈X.Remark 4.
It is obvious that an evolution family𝒰 is uniformly exponentially expansive if and only if there are N,v>0 such that
(3)∥U(t,t0)x0∥≥Nev(t-t0)∥x0∥,
for all t≥t0≥0 and x0∈X.Remark 5.
If the evolution family𝒰 is uniformly exponentially expansive, then it is uniformly expansive. The converse is not necessarily valid. To show this, we consider the following example.Example 6.
Letf:ℝ+→ℝ+ be a monotone increasing and bounded continuous function, X=ℝ+. The evolution family 𝒰 defined by
(4)U(t,t0)x0=ef(t)-f(t0)x0,
for all t≥t0≥0 and x0∈X.Proof.
As a first step, we prove that𝒰 is uniformly expansive. The evolution family 𝒰 satisfies the inequality
(5)∥U(t,t0)x0∥=ef(t)-f(t0)|x0|≥ef(r)-f(t0)|x0|=N∥U(r,t0)x0∥,
for all t≥r≥t0≥0 and x0∈X, where N=1. Hence, 𝒰 is uniformly expansive.
As a second step, we prove that𝒰 is not uniformly exponentially expansive. If we suppose that 𝒰 is uniformly exponentially expansive, then, by Remark 4, there exist some constants N,v>0 such that
(6)ef(t)-f(t0)≥Nev(t-t0),
for all t≥t0≥0.
In particular, fort0=0, we obtain ef(t)-f(0)≥Nevt, which is absurd for t→∞. Hence, 𝒰 is not uniformly exponentially expansive.Definition 7.
An evolution family𝒰={U(t,s)}t≥s≥0 is called weakly exponentially expansive if there are N,v>0 such that for all x0∈X there exists t0≥0 with
(7)∥U(t,t0)x0∥≥Nev(t-r)∥U(r,t0)x0∥,
for all t≥r≥t0.Remark 8.
If the evolution family𝒰 is uniformly exponentially expansive, then it is weakly exponentially expansive.The following example shows that the converse is not valid.Example 9.
LetX=ℝ2 with the Euclidean norm. Consider the evolution family generated by the matrix U(t,t0)=P(t,t0)Q(t0), where
(8)P(t,t0)=(et-t0sinte-(t-t0)cost-et-t0coste-(t-t0)sint),Q(t0)=(cost0sint0sint0-cost0).Proof.
We divide the proof into two steps.
As a first step, we prove that𝒰 is weakly exponentially expansive. For every x0∈ℝ2, there exist ρ≥0 and t0∈[0,2π) such that x0=(ρcost0,ρsint0)T.
It is easy to see that(9)U(t,t0)x0=P(t,t0)Q(t0)x0=P(t,t0)(ρ,0)T=(ρet-t0sint,-ρet-t0cost)T,
and hence
(10)∥U(t,t0)x0∥=ρet-t0≥Net-r∥U(r,t0)x0∥,
for all t≥r≥t0 with N=1, which shows that 𝒰 is weakly exponentially expansive.
As a second step, we prove that𝒰 is not uniformly exponentially expansive. If we assume that 𝒰 is uniformly exponentially expansive, then there exist some constants N,v>0 such that
(11)∥U(t,t0)y0∥≥Nev(t-r)∥U(r,t0)y0∥,
for all t≥r≥t0≥0 and y0∈X.
In particular, fory0=(sint0,-cost0)T, we obtain
(12)U(t,t0)y0=P(t,t0)(0,1)T=(e-(t-t0)cost,e-(t-t0)sint)T,
and hence
(13)∥U(t,t0)y0∥=e-(t-t0)=e-(t-r)∥U(r,t0)y0∥,
which shows that 𝒰 is not uniformly exponentially expansive.


## 3. The Main Results

Theorem 10.
The following assertions are equivalent:(i)
𝒰 is weakly exponentially expansive;(ii)
there areδ>0 and c>1 such that for every x0∈X there exist t0≥0 and h0∈(0,δ] with
(14)∥U(h0+r,t0)x0∥≥c∥U(r,t0)x0∥,for allr≥t0;(iii)
there existδ>0 and c>1 such that for each x0∈X there exists t0≥0 with the property that for every r≥t0 there is h0∈(0,δ] with
(15)∥U(h0+r,t0)x0∥≥c∥U(r,t0)x0∥.Proof.
(i)⇒(ii) If 𝒰 is weakly exponentially expansive, then by Definition 7, there exist N,v>0 such that for all x0∈X there exists t0≥0 with the property
(16)∥U(t,t0)x0∥≥Nev(t-r)∥U(r,t0)x0∥,
for all t≥r≥t0. Let δ>0 satisfy that Nevδ>1. Then, for h0=δ, we have
(17)∥U(r+h0,t0)x0∥≥Nevδ∥U(r,t0)x0∥=c∥U(r,t0)x0∥,
for all r≥t0.
(ii)⇒(iii) It is obvious.
(iii)⇒(i) We define N=1/Meωδ and v=lnc/δ, where δ>0 and c>1 are given by (iii).
From (iii), it results that for eachx0∈X, there exists t0≥0 with the property that for every r≥t0 there is h0∈(0,δ] such that
(18)∥U(h0+r,t0)x0∥≥c∥U(r,t0)x0∥.
Letr≥t0, and we have that there is h1∈(0,δ] with
(19)∥U(h1+h0+r,t0)x0∥≥c∥U(h0+r,t0)x0∥≥c2∥U(r,t0)x0∥.
By induction, we have that
(20)∥U(rn+r,t0)x0∥≥cn∥U(r,t0)x0∥,∀n∈ℕ,
where
(21)rn={0,n=0,∑i=0n-1hi,n∈ℕ*,hi∈(0,δ].
It is easy to see that (rn) is unbounded. In fact, if (rn) is bounded, then there exists r*∈ℝ with rn→r* (n→∞). From the relation (20) and c>1, it follows that
(22)∥U(r+r*,t0)x0∥≥limn→∞cn∥U(r,t0)x0∥⟶∞,
which is a contradiction because {U(t,s)}t≥s≥0⊆B(X).
So, (rn) is unbounded, and then for t≥r, there is n∈ℕ such that
(23)rn≤t-r≤rn+1≤(n+1)δ.
Then,(24)∥U(rn+1+r,t0)x0∥≤Meω(r+rn+1-t)∥U(t,t0)x0∥≤Meω(rn+1-rn)∥U(t,t0)x0∥≤Meωδ∥U(t,t0)x0∥,
and hence
(25)∥U(t,t0)x0∥≥1Meωδ∥U(r+rn+1,t0)x0∥≥1Meωδcn+1∥U(r,t0)x0∥=1Meωδev(n+1)δ∥U(r,t0)x0∥≥Nev(t-r)∥U(r,t0)x0∥.Remark 11.
Theorem10 can be considered a generalization of some results from uniform exponential instability proved in [8].An important set in what follows isℱ1, the set of all nondecreasing functions F:ℝ+→ℝ+ with the properties:(f1)
F(tr)≤F(t)F(r), for all (t,r)∈ℝ+2;(f2)
F(t)>0, for every t>0.Theorem 12.
An evolution family𝒰 is weakly exponentially expansive if and only if there are F∈ℱ1 and K>0 such that for every x0∈X∖{0} there is t0≥0 with
(26)∫r∞F(1∥U(τ,t0)x0∥)dτ≤KF(1∥U(r,t0)x0∥),
for all r≥t0.Proof.
Necessity. If 𝒰 is weakly exponentially expansive, then by Definition 7, there are N,v>0 such that for all x0∈X∖{0} there exists t0≥0 with
(27)∫r∞1∥U(τ,t0)x0∥dτ≤∫r∞‍1Nev(τ-r)∥U(r,t0)x0∥dτ=1Nv∥U(r,t0)x0∥,
for all r≥t0.
Thus, the inequality (26) is satisfied for F(t)=t and K=1/Nv.
Sufficiency. We assume for a contradiction that for all δ>0 and c>1 there exists x0∈X such that for every t0≥0 there is r≥t0 with
(28)∥U(t+r,t0)x0∥<c∥U(r,t0)x0∥,
for all t∈(0,δ].
In particular, forδ=KF(3) and c=3, the inequality (28) implies
(29)F(3)∫r∞‍F(1∥U(t,t0)x0∥)dt≥∫r∞‍F(3∥U(t,t0)x0∥)dt=∫0∞‍F(3∥U(τ+r,t0)x0∥)dτ≥∫0δ‍F(3∥U(τ+r,t0)x0∥)dτ>∫0δ‍F(1∥U(r,t0)x0∥)dτ=δF(1∥U(r,t0)x0∥),
which contradicts the inequality (26). This contradiction proves that 𝒰 is weakly exponentially expansive.It makes sense to consider also the setℱ2 all non-decreasing functions F:ℝ+→ℝ+ with the properties:(g1)
F(tr)≥F(t)F(r), for all (t,r)∈ℝ+2;(g2)
F(t)>0, for every t>0.Theorem 13.
An evolution family𝒰 is weakly exponentially expansive if and only if there are F∈ℱ2 and K>0 such that for every x0∈X∖{0} there is t0≥0 with the relation (26).Proof.
Necessity. This is a simple verification for F(t)=t.
Sufficiency. It is similar to the proof of Theorem 12. Indeed, from (28) for c∈(1,+∞) and δ=2K/F(1/c), we have
(30)∫r∞‍F(1∥U(t,t0)x0∥)dt=∫0∞‍F(1∥U(τ+r,t0)x0∥)dτ≥∫0δ‍F(1∥U(τ+r,t0)x0∥)dτ≥∫0δ‍F(1c∥U(r,t0)x0∥)dτ≥δF(1c)F(1∥U(r,t0)x0∥)>KF(1∥U(r,t0)x0∥),
which contradicts the inequality (26).Remark 14.
The preceding theorems are variants for the case of weak exponential expansiveness property of a well-known theorem due to Rolewicz [4].Corollary 15.
An evolution family𝒰 is weakly exponentially expansive if and only if there are p>0 and K>0 such that for all x0∈X∖{0} there exists t0≥0 with
(31)∫r∞‍1∥U(t,t0)x0∥pdt≤K1∥U(r,t0)x0∥p,
for all r≥t0.Proof.
It is immediate from Theorem12 for F(t)=tp.Remark 16.
Corollary15 is the version of a well-known theorem due to Datko [3], for the case of weak exponential expansiveness of evolution families.In the following corollary, we give a discrete version of Theorems12 and 13.Corollary 17.
An evolution family𝒰 is weakly exponentially expansive if and only if there are F∈ℱ1∪ℱ2 and K>0 such that for all x0∈X∖{0} there exists t0≥0 with
(32)∑n=0∞‍F(1∥U(r+n,t0)x0∥)≤KF(1∥U(r,t0)x0∥),
for all r≥t0.Proof.
Necessity. This is a simple verification for F(t)=t.
Sufficiency. By Definition 1, we know that ∥U(r+τ,r+n)∥≤Meω, τ∈[n,n+1] and τ↦∥U(r+τ,r+n)∥ is continuous on [n,n+1] for all n∈ℕ, so there exist Ln>0 such that
(33)Ln=minτ∈[n,n+1]∥U(r+τ,r+n)∥,n∈ℕ.
LetL=infnLn. We suppose that F∈ℱ1, and from (32), it results that
(34)∫r∞‍F(1∥U(t,t0)x0∥)dt=∑n=0∞‍∫nn+1‍F(1∥U(r+τ,r+n)U(r+n,t0)x0∥)dτ≤∑n=0∞‍∫nn+1‍F(1L∥U(r+n,t0)x0∥)dτ≤F(1L)∑n=0∞‍F(1∥U(r+n,t0)x0∥)≤KF(1L)F(1∥U(r,t0)x0∥),
for all r≥t0.
IfF∈ℱ2, in a similar way we have
(35)F(L)∫r∞‍F(1∥U(t,t0)x0∥)dt≤∫0∞‍F(L∥U(r+τ,t0)x0∥)dτ=∑n=0∞‍∫nn+1‍F(L∥U(r+τ,r+n)U(r+n,t0)x0∥)dτ≤∑n=0∞‍∫nn+1‍F(1∥U(r+n,t0)x0∥)dτ≤KF(1∥U(r,t0)x0∥),
for all r≥t0.
Applying Theorems12 and 13, we conclude that 𝒰 is weakly exponentially expansive.Corollary 18.
An evolution family𝒰 is weakly exponentially expansive if and only if there is K>0 such that for all x0∈X∖{0} there exists t0≥0 with
(36)∑n=0∞‍1∥U(r+n,t0)x0∥≤K1∥U(r,t0)x0∥,
for all r≥t0.Another characterization of the weak exponential expansiveness is given by the following.Theorem 19.
An evolution family𝒰 is weakly exponentially expansive if and only if there are K,α>0 such that for every x0∈X∖{0} there is t0≥0 with
(37)1t-r∫rt‍eα(τ-r)1∥U(τ,t0)x0∥dτ≤K1∥U(r,t0)x0∥,
for all t>r≥t0.Proof.
Necessity. If 𝒰 is weakly exponentially expansive then by Definition 7, there are N,v>0 such that for all x0∈X∖{0} there is t0>0 with the property that for α=v/2 we have
(38)1t-r∫rt‍eα(τ-r)1∥U(τ,t0)x0∥dτ≤1N(t-r)∫rt‍e-α(τ-r)dτ1∥U(r,t0)x0∥=1-e-α(t-r)Nα(t-r)1∥U(r,t0)x0∥≤K1∥U(r,t0)x0∥,
for all t>r≥t0, where K=(1/N)supλ>0((1-e-λ)/λ)<∞.
Sufficiency. Let δ>0 be such that eαδ>1+3αδK, where K and α are given by (37). We suppose that 𝒰 is not weakly exponentially expansive. Then, by Theorem 10, for c=3, there exists x0∈X such that for all t0≥0 and all u∈(0,δ] there is r≥t0 with
(39)∥U(u+r,t0)x0∥<3∥U(r,t0)x0∥.
Then, for t=r+δ, we have
(40)1t-r∫rt‍eα(τ-r)1∥U(τ,t0)x0∥dτ=1δ∫0δ‍eαu1∥U(u+r,t0)x0∥du≥13δ∫0δ‍eαudu1∥U(r,t0)x0∥=eαδ-13αδ1∥U(r,t0)x0∥>K1∥U(r,t0)x0∥,
which contradicts the inequality (37), the proof is completed.



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