# Admissibility for Nonuniform(μ,ν) Contraction and Dichotomy

Authors: Yongxin Jiang; Fang-fang Liao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The relation between the notions of nonuniform asymptotic stability and admissibility is considered. Using appropriate Lyapunov norms, it is showed that if any of their associatedℒp spaces, with p∈(1,∞], is admissible for a given evolution process, then this process is a nonuniform (μ,ν) contraction and dichotomy. A collection of admissible Banach spaces for any given nonuniform (μ,ν) contraction
and dichotomy is provided.


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## Body

## 1. Introduction

The study of the admissibility property has a fairly long history, and it goes back to the pioneering work of Perron [1] in 1930. Perron concerned originally the existence of bounded solutions of the equation
(1.1)x'=A(t)x+f(t)
in ℝn for any bounded continuous perturbation f:ℝ0+→ℝn. This property can be used to deduce the stability or the conditional stability under sufficiently small perturbations of a given linear equation:
(1.2)x'=A(t)x.
His result served as a starting point for many works on the qualitative theory of the solutions of differential equations. Moreover, a simple consequence of one of the main results in that paper stated explicitly in [2, Theorem 1] is probably the first step in the literature concerning the study of the relation between admissibility and the notions of stability and conditional stability. We refer the reader to [2] for details. Relevant results concerning the extension of Perron's problem in the more general framework of the infinite-dimensional Banach spaces with bounded A(t) were obtained by Daleckij and Krein [3], Massera and Schäffer [4], and the work of Levitan and Zhikov [5] for certain cases of unbounded A(t).Over the last decades an increasing interest can be seen in the study of the asymptotic behavior of evolution equations in abstract spaces. In [6, 7], Latushkin et al. studied the dichotomy of linear skew-product semiflows defined on compact spaces. Using the so-called evolution semigroup, they expressed its dichotomy in terms of hyperbolicity of a family of weighted shift operators. In [8–10], Preda et al. considered related problems in the particular case of uniform exponential behavior. A large class of Schäffer spaces, which were introduced by Schäffer in [11] (see also [4]), acted as admissible spaces for the case of uniform exponential dichotomies. It is worth noting here the works by Huy [12–16] in the study of the existence of an exponential dichotomy for evolution equations.In the case of nonuniform exponential dichotomies, Preda and Megan [17] obtained related results also for the class of Schäffer spaces, but using a notion of dichotomy which is different from the original one motivated by ergodic theory and the nonuniform hyperbolicity theory, as detailed, for example, in [18, 19]. In the more recent work [20], the authors consider the same weaker notion of exponential dichotomy and obtain sharper relations between admissibility and stability for perturbations and solutions in C0. Important contributions in this aspect have been made by Barreira et al. [2, 18, 19, 21–25]. Particularly, in [22], Barreira and Valls showed an equivalence between the admissibility of their associated ℒp spaces (p∈(1,∞]) and the nonuniform exponential stability of certain evolution families by using appropriate adapted norms. They also establish a collection of admissible Banach spaces for any given nonuniform exponential dichotomy in [2]. Recently, Preda et al. [26] studied the connection between the (non)uniform exponential dichotomy of a non(uniform) exponentially bounded, strongly continuous evolution family and the admissibility of some function spaces, which extended those results established in [2, 22].In the present paper, inspired by Barreira and Valls [2, 22], we give a characterization of nonuniform asymptotic stability in terms of admissibility property. We consider a more general type of dichotomy which is called (μ,ν) dichotomy in [21], also proposed in [27]. In this dichotomy, not only the usual exponential behavior is replaced by an arbitrary, which may correspond, for example, to situations when the Lyapunov exponents are all infinity or are all zero, but also different growth rates for the uniform and nonuniform parts of the dichotomy are considered. It extended exponential dichotomy in various ways. In [21], it has also been showed that there is a large class of equations exhibiting this behavior. We emphasize that the characterization in our paper is a very general one; it includes as particular cases many interesting situations among them we can mention some results in previous references. To some extent, our results have a certain significance to study the theory of nonuniform hyperbolicity.


## 2. Admissibility for Nonuniform(μ,ν) Contractions

We first concentrate on the simpler case of admissibility for nonuniform(μ,ν) contractions, leaving the more elaborate case of admissibility for nonuniform (μ,ν) dichotomies for the second part of the paper. This allows us to present the results and their proofs without some accessory technicalities. After the introduction of some basic notions, using appropriate adapted Lyapunov norms, we show that the admissibility with respect to some space ℒp with p∈(1,∞] is sufficient for an evolution process to be a nonuniform (μ,ν) contraction.


### 2.1. Basic Notions

We say that an increasing functionμ:ℝ+→[1,+∞) is a growth rate if
(2.1)μ(0)=1,limt→+∞μ(t)=+∞.We say that a family of linear operatorsT(t,s), t≥s≥0 in a Banach space X is an evolution process if:(1)
T(t,t)=Id and T(t,τ)T(τ,s)=T(t,s),t,τ,s>0;(2)
(t,s,x)↦T(t,s)x is continuous for t≥s≥0 and x∈X.In this section, we also assume that(3)
there existω≥0, D>0 and two growth rates μ(t), ν(t) such that
(2.2)∥T(t,s)∥≤D(μ(t)μ(s))ωνε(s),t≥s≥0.We consider the new norms
(2.3)∥x∥t′=sup{∥T(σ,t)x∥(μ(σ)μ(t))-ω,σ≥t},x∈X,t∈ℝ0+.
These satisfy
(2.4)∥x∥≤∥x∥t′≤Dνε(t)∥x∥,x∈X,t∈ℝ0+.
Moreover, with respect to these norms the evolution process has the following bounded growth property.Proposition 2.1.
IfT is an evolution process, then
(2.5)∥T(t,s)x∥t′≤(μ(t)μ(s))ω∥x∥s′
for every t≥s≥0 and x∈X.Proof.
We have(2.6)∥T(t,s)x∥t′=sup{∥T(σ,t)x∥(μ(σ)μ(t))-ω,σ≥t}≤(μ(t)μ(s))ωsup{∥T(σ,t)x∥(μ(σ)μ(s))-ω,σ≥s}=(μ(t)μ(s))ω∥x∥s′
which yields the desired inequality.Definition 2.2.
We say that an evolution processT is a nonuniform (μ,ν) contraction in ℝ0+ if there exist some constants α, D>0, ε≥0 and two growth rates μ(t), ν(t) such that
(2.7)∥T(t,s)∥≤D(μ(t)μ(s))-ανε(s),t≥s≥0.
When ε=0, we say that  (1.2)  has a uniform (μ,ν) contraction or simply a (μ,ν) contraction.In the following, we introduce several Banach spaces that are used throughout the paper. We first set(2.8)Lp={f:ℝ0+→ℝLebesgue-measurable:∥f∥p<∞}
for each p∈[1,∞), and
(2.9)L∞={f:ℝ0+→ℝLebesgue-measurable:∥f∥∞<∞}
Respectively, with the norms
(2.10)∥f∥p=(∫0∞‍|f(t)|p)1/p,∥f∥∞=esssupt∈ℝ0+|f(t)|.
Then for each p∈[1,∞] the set ℒp of the equivalence classes [f] of functions g∈Lp such that g=f Lebesgue-almost everywhere is a Banach space (again with the norms in  (2.10)).For each Banach spaceE=ℒp, with p∈[1,∞], we set
(2.11)E(X)={f:ℝ0+→XBochner-measurable:t↦∥f(t)∥t′∈E}
using the norms ∥·∥t′ in  (2.3), and we endow E(X)=ℒp(X) with the norm
(2.12)∥f∥p′=∥F∥p,whereF(t)=∥f(t)∥t′.Repeating arguments in the proof of Theorem 3 in [22], we obtain the following statement.Lemma 2.3.
For eachp∈[1,∞] and E=ℒp, the set E(X) is a Banach space with the norm in  (2.12), and the convergence in E(X) implies the pointwise convergence Lebesgue-almost everywhere.Definition 2.4.
We say that a Banach spaceE is admissible for the evolution process T if for each f∈E(X) the function xf:ℝ0+→X defined by
(2.13)xf(t)=∫0∞‍T(t,τ)f(τ)dτ
is in ℒ∞ (see  (2.11)).By Lemma2.3 we know that ℒ∞ is a Banach space with the norm
(2.14)∥g∥∞′=esssupt∈ℝ0+∥g(t)∥t′.Lemma 2.5.
There existsK>0 such that
(2.15)∥xf∥∞′≤K∥f∥p′foreveryf∈E(X).Proof.
We define a linear operatorG:E(X)→ℒ∞(X) by Gf=xf. We use the closed graph theorem to show that G is bounded. For this, let us take a sequence (fn)n∈N⊂E(X) and f∈E(X) such that fn→f in E(X) when n→∞ and also h∈ℒ∞(X) such that Gfn→h in ℒ∞(X) when n→∞. We need to show that Gf=h Lebesgue-almost everywhere. For each t≥0 and n∈N we have
(2.16)∥(Gfn)(t)-(Gf)(t)∥t′=sup{∥∫0t‍T(σ,t)T(t,τ)(fn(τ)-f(τ))dτ∥(μ(σ)μ(t))-ω:σ≥t}=sup{∥∫0t‍T(σ,τ)(fn(τ)-f(τ))dτ∥(μ(σ)μ(t))-ω:σ≥t}≤sup{∫0t‍∥T(σ,τ)(fn(τ)-f(τ))∥(μ(σ)μ(t))-ωdτ:σ≥t}=sup{∫0t‍∥T(σ,τ)(fn(τ)-f(τ))∥(μ(σ)μ(τ))-ω(μ(t)μ(τ))ωdτ:σ≥t}=∫0t‍∥fn(τ)-f(τ)∥τ′(μ(t)μ(τ))ωdτ≤μ(t)ω∫0t‍∥fn(τ)-f(τ)∥τ′dτ.
According to Hölder's inequality, there exists α=α([0,t]) such that
(2.17)∥(Gfn)(t)-(Gf)(t)∥t′≤μ(t)ω∫0t‍∥fn(τ)-f(τ)∥τ′dτ≤μ(t)ωα∥(fn(τ)-f(τ))∥p′.
Therefore, for each t≥0, letting n→∞ we find that (Gfn)(t)→(Gf)(t). This shows that Gf=h Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that G is a bounded operator. This completes the proof of the lemma.


### 2.2. Criterion for Nonuniform(μ,ν) Contraction

Theorem 2.6.
If for somep∈(1,∞] the space E=ℒp is admissible for the evolution process T, then T is a nonuniform (μ,ν) contraction.Proof.
We follow arguments in [22]. Given x∈X and t0≥0, we define a function f:ℝ0+→X by
(2.18)f(t)={T(t,t0)x,t∈[t0,t0+1]0,t∈ℝ0+∖[t0,t0+1].
We note that
(2.19)∥f(t)∥t′≤∥T(t,t0)x∥t′χ[t0,t0+1](t).
Then, for each t∈[t0,t0+1] and x∈X, we have
(2.20)∥T(t,t0)x∥t′=sup{∥T(σ,t)T(t,t0)x∥(μ(σ)μ(t))-ω,σ≥t}≤(μ(t)μ(t0))ωsup{∥T(σ,t0)x∥(μ(σ)μ(t0))-ω,σ≥t0}=(μ(t)μ(t0))ω∥x∥t0′≤(μ(t0+1)μ(t0))ω∥x∥t0′.
Therefore,
(2.21)∥f(t)∥p′≤(μ(t0+1)μ(t0))ω∥x∥t0′∥χ[t0,t0+1](t)∥p=(μ(t0+1)μ(t0))ω∥x∥t0′
and in particular f∈E(X). On the other hand, according to  (2.13) and  (2.18), we have
(2.22)xf(t)=∫t0t0+1‍T(t,τ)T(τ,t0)xdτ=T(t,t0)x
for all t≥t0+1, which implies that
(2.23)∥T(t,t0)x∥t′=∥xf∥t′≤∥xf∥∞′.
By Lemma 2.5 and  (2.21)–(2.23), we obtain
(2.24)∥T(t,t0)x∥t′≤∥xf∥∞′≤K∥f∥p′≤K(μ(t0+1)μ(t0))ω∥x∥t0′
for all t≥t0+1, t0≥0, and x∈X. We claim that
(2.25)∥T(t,t0)∥′:=supx≠0∥T(t,t0)x∥t′∥x∥t0′≤L,L=(μ(t0+1)μ(t0))ωmax{K,1}
for all t≥t0. Indeed, for t≥t0+1 inequality  (2.25) follows from  (2.24), and for t∈[t0,t0+1] the inequality follows from  (2.20).
Now givenx∈X, t0≥0, and δ>0, we define a function g:ℝ0+→X by
(2.26)g(t)={T(t,t0)x,t∈[t0,t0+δ]0,t∈ℝ0+∖[t0,t0+δ].
It follows from  (2.25) that
(2.27)∥g(t)∥t′≤∥T(t,t0)x∥t′≤L∥x∥t0′
and thus,
(2.28)g∈E(X),∥g∥p′≤Lδ1/p∥x∥t0′.
On the other hand, writing y=T(t0+δ,t0)x,
(2.29)δ22∥y∥t0+δ′=∥∫t0t0+δ‍(τ-t0)ydτ∥t0+δ′=sup{∥T(σ,t0+δ)∫t0t0+δ‍(τ-t0)ydτ∥(μ(σ)μ(t0+δ))-ω:σ≥t0+δ}=sup{∥∫t0t0+δ‍(τ-t0)T(σ,t0)xdτ∥(μ(σ)μ(t0+δ))-ω:σ≥t0+δ}≤sup{∫t0t0+δ‍(τ-t0)∥T(σ,t0)x∥(μ(σ)μ(t0+δ))-ωdτ:σ≥t0+δ}=∫t0t0+δ‍(τ-t0)sup{∥T(σ,t0+δ)y∥(μ(σ)μ(t0+δ))-ω:σ≥t0+δ}dτ=∫t0t0+δ‍(τ-t0)∥y∥t0+δ′dτ=∫t0t0+δ‍(τ-t0)∥T(t0+δ,τ)T(τ,t0)x∥t0′dτ
Since
(2.30)xg(t)=∫0t‍T(t,τ)g(τ)dτ={0,t∈[0,t0](t,t0)T(t,t0)x,t∈[t0,t0+δ],δT(t,t0)x,t∈[t0+δ,∞),
it follows from Lemma 2.5,  (2.25), and  (2.28) that
(2.31)δ22∥T(t0+δ,t0)x∥t0+δ′≤L∫t0t0+δ‍(τ-t0)∥T(τ,t0)x∥τ′dτ=L∫t0t0+δ‍∥xg(τ)∥τ′dτ≤Lδ∥xg∥∞′≤KLδ∥g∥p′≤KL2δ(p+1)/p∥x∥t0′
for all t0≥0, δ>0, and x∈X; we thus obtain
(2.32)δ22∥T(t0+δ,t0)x∥t0+δ′≤KL2δ(p+1)/p∥x∥t0′
so
(2.33)∥T(t0+δ,t0)∥′≤2KL2δ(1-p)/p
for all t0≥0 and δ>0. Since (1-p)/p<0 for p∈(1,∞], there exists δ0>0 sufficiently large such that
(2.34)K0:=2KL2δ(1-p)/p<1.
Setting n=[(lnμ(t)-lnμ(t0))/δ0] for each t≥t0, we have
(2.35)T(t,t0)=T(t,t0+nδ0)T(t0+nδ0,t0).
By  (2.25) and  (2.33) we obtain
(2.36)∥T(t,t0)∥′≤L∥T(t0+nδ0,t0)∥′≤L∏i=0n-1‍∥T(t0+(i+1)δ0,t0+iδ0)∥′≤LK0n
for t≥t0. By  (2.34) and
(2.37)n=[lnμ(t)-lnμ(t0)δ0]≥lnμ(t)-lnμ(t0)δ0-1
this implies that
(2.38)∥T(t,t0)∥′≤d(μ(t)μ(t0))-α,
where
(2.39)d=LK0,α=-1δ0lnK0.
We note that d,α>0. Since
(2.40)∥T(t,t0)x∥t′≥∥T(t,t0)x∥,
and by  (2.4),
(2.41)∥x∥t0′=sup{∥T(σ,t0)x∥(μ(σ)μ(t0))-ω:σ≥t0}≤Dνε(t0)∥x∥.
It follows from  (2.38) that
(2.42)∥T(t,t0)∥=supx≠0∥T(t,t0)x∥∥x∥≤Dνε(t0)supx≠0∥T(t,t0)x∥t′∥x∥t0′≤dDνε(t0)(μ(t)μ(t0))-α
for any t≥t0. Therefore, the evolution process T is a nonuniform (μ,ν) contraction with α and D~=dD. This concludes the proof of Theorem 2.6.


### 2.3. Admissible Spaces for Nonuniform(μ,ν) Contractions

We consider the spaces(2.43)LDp={f:ℝ0+→ℝLebesgue-measurable:∥f∥p,D<∞}
for each p∈[1,∞), and
(2.44)LD∞={f:ℝ0+→ℝLebesgue-measurable:∥f∥∞,D<∞},
respectively, with the norms
(2.45)∥f∥p,D=(∫0∞‍|f(t)|p(Dνε(t)(μ(t)μ′(t))1/q)p)1/p,∥f∥∞,D=esssupt∈ℝ0+(|f(t)|Dνε(t)μ(t)μ′(t)).
In a similar manner to that in Lemma 2.3 these normed spaces induce Banach spaces ℒDp and ℒDp(X) for each p∈[1,∞], the last one with norm
(2.46)∥f∥p,D′=∥F∥p,D,whereF(t)=∥f(t)∥t′.Theorem 2.7.
If the evolution processT is a nonuniform (μ,ν) contraction, then for any p∈[1,∞] the space ℒDp is admissible for T.Proof.
We first takef∈ℒD∞. Then
(2.47)∥xf(t)∥t′=sup{∥∫0t‍T(σ,t)T(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}=sup{∥∫0t‍T(σ,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}≤sup{∫0t‍∥T(σ,τ)∥·∥f(τ)∥dτ:σ≥t}≤sup{∫0t‍Dντ(τ)(μ(σ)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}≤sup{∫0t‍Dντ(τ)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}≤∥f∥∞,D′∫0t‍(μ(t)μ(τ))-αμ′(τ)μ(τ)dτ≤∥f∥∞,D′1-μ(t)-αα≤1α∥f∥∞,D′.
Therefore,
(2.48)∥xf∥∞′=supt≥0∥xf(t)∥t′≤supt≥01α∥f∥∞,D′<∞
and ℒD∞ is admissible for T.
Now we takef∈ℒDp(X) for some p∈[1,∞). Using Hölder's inequality we obtain
(2.49)∥xf(t)∥t′=sup{∥∫0t‍T(σ,t)T(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}≤sup{∫0t‍Dντ(t)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}≤∥f∥p,D′(∫0t‍(μ(t)μ(τ))-αqμ′(τ)μ(τ)dτ)1/q≤∥f∥p,D′(1-μ(t)-αqαq)1/q≤1(αq)1/q∥f∥p,D′,
where 1/p+1/q=1. We conclude that ℒDp is also admissible for T.


## 2.1. Basic Notions

We say that an increasing functionμ:ℝ+→[1,+∞) is a growth rate if
(2.1)μ(0)=1,limt→+∞μ(t)=+∞.We say that a family of linear operatorsT(t,s), t≥s≥0 in a Banach space X is an evolution process if:(1)
T(t,t)=Id and T(t,τ)T(τ,s)=T(t,s),t,τ,s>0;(2)
(t,s,x)↦T(t,s)x is continuous for t≥s≥0 and x∈X.In this section, we also assume that(3)
there existω≥0, D>0 and two growth rates μ(t), ν(t) such that
(2.2)∥T(t,s)∥≤D(μ(t)μ(s))ωνε(s),t≥s≥0.We consider the new norms
(2.3)∥x∥t′=sup{∥T(σ,t)x∥(μ(σ)μ(t))-ω,σ≥t},x∈X,t∈ℝ0+.
These satisfy
(2.4)∥x∥≤∥x∥t′≤Dνε(t)∥x∥,x∈X,t∈ℝ0+.
Moreover, with respect to these norms the evolution process has the following bounded growth property.Proposition 2.1.
IfT is an evolution process, then
(2.5)∥T(t,s)x∥t′≤(μ(t)μ(s))ω∥x∥s′
for every t≥s≥0 and x∈X.Proof.
We have(2.6)∥T(t,s)x∥t′=sup{∥T(σ,t)x∥(μ(σ)μ(t))-ω,σ≥t}≤(μ(t)μ(s))ωsup{∥T(σ,t)x∥(μ(σ)μ(s))-ω,σ≥s}=(μ(t)μ(s))ω∥x∥s′
which yields the desired inequality.Definition 2.2.
We say that an evolution processT is a nonuniform (μ,ν) contraction in ℝ0+ if there exist some constants α, D>0, ε≥0 and two growth rates μ(t), ν(t) such that
(2.7)∥T(t,s)∥≤D(μ(t)μ(s))-ανε(s),t≥s≥0.
When ε=0, we say that  (1.2)  has a uniform (μ,ν) contraction or simply a (μ,ν) contraction.In the following, we introduce several Banach spaces that are used throughout the paper. We first set(2.8)Lp={f:ℝ0+→ℝLebesgue-measurable:∥f∥p<∞}
for each p∈[1,∞), and
(2.9)L∞={f:ℝ0+→ℝLebesgue-measurable:∥f∥∞<∞}
Respectively, with the norms
(2.10)∥f∥p=(∫0∞‍|f(t)|p)1/p,∥f∥∞=esssupt∈ℝ0+|f(t)|.
Then for each p∈[1,∞] the set ℒp of the equivalence classes [f] of functions g∈Lp such that g=f Lebesgue-almost everywhere is a Banach space (again with the norms in  (2.10)).For each Banach spaceE=ℒp, with p∈[1,∞], we set
(2.11)E(X)={f:ℝ0+→XBochner-measurable:t↦∥f(t)∥t′∈E}
using the norms ∥·∥t′ in  (2.3), and we endow E(X)=ℒp(X) with the norm
(2.12)∥f∥p′=∥F∥p,whereF(t)=∥f(t)∥t′.Repeating arguments in the proof of Theorem 3 in [22], we obtain the following statement.Lemma 2.3.
For eachp∈[1,∞] and E=ℒp, the set E(X) is a Banach space with the norm in  (2.12), and the convergence in E(X) implies the pointwise convergence Lebesgue-almost everywhere.Definition 2.4.
We say that a Banach spaceE is admissible for the evolution process T if for each f∈E(X) the function xf:ℝ0+→X defined by
(2.13)xf(t)=∫0∞‍T(t,τ)f(τ)dτ
is in ℒ∞ (see  (2.11)).By Lemma2.3 we know that ℒ∞ is a Banach space with the norm
(2.14)∥g∥∞′=esssupt∈ℝ0+∥g(t)∥t′.Lemma 2.5.
There existsK>0 such that
(2.15)∥xf∥∞′≤K∥f∥p′foreveryf∈E(X).Proof.
We define a linear operatorG:E(X)→ℒ∞(X) by Gf=xf. We use the closed graph theorem to show that G is bounded. For this, let us take a sequence (fn)n∈N⊂E(X) and f∈E(X) such that fn→f in E(X) when n→∞ and also h∈ℒ∞(X) such that Gfn→h in ℒ∞(X) when n→∞. We need to show that Gf=h Lebesgue-almost everywhere. For each t≥0 and n∈N we have
(2.16)∥(Gfn)(t)-(Gf)(t)∥t′=sup{∥∫0t‍T(σ,t)T(t,τ)(fn(τ)-f(τ))dτ∥(μ(σ)μ(t))-ω:σ≥t}=sup{∥∫0t‍T(σ,τ)(fn(τ)-f(τ))dτ∥(μ(σ)μ(t))-ω:σ≥t}≤sup{∫0t‍∥T(σ,τ)(fn(τ)-f(τ))∥(μ(σ)μ(t))-ωdτ:σ≥t}=sup{∫0t‍∥T(σ,τ)(fn(τ)-f(τ))∥(μ(σ)μ(τ))-ω(μ(t)μ(τ))ωdτ:σ≥t}=∫0t‍∥fn(τ)-f(τ)∥τ′(μ(t)μ(τ))ωdτ≤μ(t)ω∫0t‍∥fn(τ)-f(τ)∥τ′dτ.
According to Hölder's inequality, there exists α=α([0,t]) such that
(2.17)∥(Gfn)(t)-(Gf)(t)∥t′≤μ(t)ω∫0t‍∥fn(τ)-f(τ)∥τ′dτ≤μ(t)ωα∥(fn(τ)-f(τ))∥p′.
Therefore, for each t≥0, letting n→∞ we find that (Gfn)(t)→(Gf)(t). This shows that Gf=h Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that G is a bounded operator. This completes the proof of the lemma.


## 2.2. Criterion for Nonuniform(μ,ν) Contraction

Theorem 2.6.
If for somep∈(1,∞] the space E=ℒp is admissible for the evolution process T, then T is a nonuniform (μ,ν) contraction.Proof.
We follow arguments in [22]. Given x∈X and t0≥0, we define a function f:ℝ0+→X by
(2.18)f(t)={T(t,t0)x,t∈[t0,t0+1]0,t∈ℝ0+∖[t0,t0+1].
We note that
(2.19)∥f(t)∥t′≤∥T(t,t0)x∥t′χ[t0,t0+1](t).
Then, for each t∈[t0,t0+1] and x∈X, we have
(2.20)∥T(t,t0)x∥t′=sup{∥T(σ,t)T(t,t0)x∥(μ(σ)μ(t))-ω,σ≥t}≤(μ(t)μ(t0))ωsup{∥T(σ,t0)x∥(μ(σ)μ(t0))-ω,σ≥t0}=(μ(t)μ(t0))ω∥x∥t0′≤(μ(t0+1)μ(t0))ω∥x∥t0′.
Therefore,
(2.21)∥f(t)∥p′≤(μ(t0+1)μ(t0))ω∥x∥t0′∥χ[t0,t0+1](t)∥p=(μ(t0+1)μ(t0))ω∥x∥t0′
and in particular f∈E(X). On the other hand, according to  (2.13) and  (2.18), we have
(2.22)xf(t)=∫t0t0+1‍T(t,τ)T(τ,t0)xdτ=T(t,t0)x
for all t≥t0+1, which implies that
(2.23)∥T(t,t0)x∥t′=∥xf∥t′≤∥xf∥∞′.
By Lemma 2.5 and  (2.21)–(2.23), we obtain
(2.24)∥T(t,t0)x∥t′≤∥xf∥∞′≤K∥f∥p′≤K(μ(t0+1)μ(t0))ω∥x∥t0′
for all t≥t0+1, t0≥0, and x∈X. We claim that
(2.25)∥T(t,t0)∥′:=supx≠0∥T(t,t0)x∥t′∥x∥t0′≤L,L=(μ(t0+1)μ(t0))ωmax{K,1}
for all t≥t0. Indeed, for t≥t0+1 inequality  (2.25) follows from  (2.24), and for t∈[t0,t0+1] the inequality follows from  (2.20).
Now givenx∈X, t0≥0, and δ>0, we define a function g:ℝ0+→X by
(2.26)g(t)={T(t,t0)x,t∈[t0,t0+δ]0,t∈ℝ0+∖[t0,t0+δ].
It follows from  (2.25) that
(2.27)∥g(t)∥t′≤∥T(t,t0)x∥t′≤L∥x∥t0′
and thus,
(2.28)g∈E(X),∥g∥p′≤Lδ1/p∥x∥t0′.
On the other hand, writing y=T(t0+δ,t0)x,
(2.29)δ22∥y∥t0+δ′=∥∫t0t0+δ‍(τ-t0)ydτ∥t0+δ′=sup{∥T(σ,t0+δ)∫t0t0+δ‍(τ-t0)ydτ∥(μ(σ)μ(t0+δ))-ω:σ≥t0+δ}=sup{∥∫t0t0+δ‍(τ-t0)T(σ,t0)xdτ∥(μ(σ)μ(t0+δ))-ω:σ≥t0+δ}≤sup{∫t0t0+δ‍(τ-t0)∥T(σ,t0)x∥(μ(σ)μ(t0+δ))-ωdτ:σ≥t0+δ}=∫t0t0+δ‍(τ-t0)sup{∥T(σ,t0+δ)y∥(μ(σ)μ(t0+δ))-ω:σ≥t0+δ}dτ=∫t0t0+δ‍(τ-t0)∥y∥t0+δ′dτ=∫t0t0+δ‍(τ-t0)∥T(t0+δ,τ)T(τ,t0)x∥t0′dτ
Since
(2.30)xg(t)=∫0t‍T(t,τ)g(τ)dτ={0,t∈[0,t0](t,t0)T(t,t0)x,t∈[t0,t0+δ],δT(t,t0)x,t∈[t0+δ,∞),
it follows from Lemma 2.5,  (2.25), and  (2.28) that
(2.31)δ22∥T(t0+δ,t0)x∥t0+δ′≤L∫t0t0+δ‍(τ-t0)∥T(τ,t0)x∥τ′dτ=L∫t0t0+δ‍∥xg(τ)∥τ′dτ≤Lδ∥xg∥∞′≤KLδ∥g∥p′≤KL2δ(p+1)/p∥x∥t0′
for all t0≥0, δ>0, and x∈X; we thus obtain
(2.32)δ22∥T(t0+δ,t0)x∥t0+δ′≤KL2δ(p+1)/p∥x∥t0′
so
(2.33)∥T(t0+δ,t0)∥′≤2KL2δ(1-p)/p
for all t0≥0 and δ>0. Since (1-p)/p<0 for p∈(1,∞], there exists δ0>0 sufficiently large such that
(2.34)K0:=2KL2δ(1-p)/p<1.
Setting n=[(lnμ(t)-lnμ(t0))/δ0] for each t≥t0, we have
(2.35)T(t,t0)=T(t,t0+nδ0)T(t0+nδ0,t0).
By  (2.25) and  (2.33) we obtain
(2.36)∥T(t,t0)∥′≤L∥T(t0+nδ0,t0)∥′≤L∏i=0n-1‍∥T(t0+(i+1)δ0,t0+iδ0)∥′≤LK0n
for t≥t0. By  (2.34) and
(2.37)n=[lnμ(t)-lnμ(t0)δ0]≥lnμ(t)-lnμ(t0)δ0-1
this implies that
(2.38)∥T(t,t0)∥′≤d(μ(t)μ(t0))-α,
where
(2.39)d=LK0,α=-1δ0lnK0.
We note that d,α>0. Since
(2.40)∥T(t,t0)x∥t′≥∥T(t,t0)x∥,
and by  (2.4),
(2.41)∥x∥t0′=sup{∥T(σ,t0)x∥(μ(σ)μ(t0))-ω:σ≥t0}≤Dνε(t0)∥x∥.
It follows from  (2.38) that
(2.42)∥T(t,t0)∥=supx≠0∥T(t,t0)x∥∥x∥≤Dνε(t0)supx≠0∥T(t,t0)x∥t′∥x∥t0′≤dDνε(t0)(μ(t)μ(t0))-α
for any t≥t0. Therefore, the evolution process T is a nonuniform (μ,ν) contraction with α and D~=dD. This concludes the proof of Theorem 2.6.


## 2.3. Admissible Spaces for Nonuniform(μ,ν) Contractions

We consider the spaces(2.43)LDp={f:ℝ0+→ℝLebesgue-measurable:∥f∥p,D<∞}
for each p∈[1,∞), and
(2.44)LD∞={f:ℝ0+→ℝLebesgue-measurable:∥f∥∞,D<∞},
respectively, with the norms
(2.45)∥f∥p,D=(∫0∞‍|f(t)|p(Dνε(t)(μ(t)μ′(t))1/q)p)1/p,∥f∥∞,D=esssupt∈ℝ0+(|f(t)|Dνε(t)μ(t)μ′(t)).
In a similar manner to that in Lemma 2.3 these normed spaces induce Banach spaces ℒDp and ℒDp(X) for each p∈[1,∞], the last one with norm
(2.46)∥f∥p,D′=∥F∥p,D,whereF(t)=∥f(t)∥t′.Theorem 2.7.
If the evolution processT is a nonuniform (μ,ν) contraction, then for any p∈[1,∞] the space ℒDp is admissible for T.Proof.
We first takef∈ℒD∞. Then
(2.47)∥xf(t)∥t′=sup{∥∫0t‍T(σ,t)T(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}=sup{∥∫0t‍T(σ,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}≤sup{∫0t‍∥T(σ,τ)∥·∥f(τ)∥dτ:σ≥t}≤sup{∫0t‍Dντ(τ)(μ(σ)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}≤sup{∫0t‍Dντ(τ)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}≤∥f∥∞,D′∫0t‍(μ(t)μ(τ))-αμ′(τ)μ(τ)dτ≤∥f∥∞,D′1-μ(t)-αα≤1α∥f∥∞,D′.
Therefore,
(2.48)∥xf∥∞′=supt≥0∥xf(t)∥t′≤supt≥01α∥f∥∞,D′<∞
and ℒD∞ is admissible for T.
Now we takef∈ℒDp(X) for some p∈[1,∞). Using Hölder's inequality we obtain
(2.49)∥xf(t)∥t′=sup{∥∫0t‍T(σ,t)T(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}≤sup{∫0t‍Dντ(t)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}≤∥f∥p,D′(∫0t‍(μ(t)μ(τ))-αqμ′(τ)μ(τ)dτ)1/q≤∥f∥p,D′(1-μ(t)-αqαq)1/q≤1(αq)1/q∥f∥p,D′,
where 1/p+1/q=1. We conclude that ℒDp is also admissible for T.


## 3. Admissibility for Nonuniform(μ,ν) Dichotomies

We consider in this second part admissibility for nonuniform(μ,ν) dichotomies. It generalizes the usual notion of exponential dichotomy in several ways: besides introducing a nonuniform term, causing that any conditional stability may be nonuniform, we consider rates that may not be exponential as well as different rates in the uniform and nonuniform parts. After introducing some basic notions, we show that the admissibility with respect to some space ℒp with p∈(1,∞] is sufficient for an evolution process to be a nonuniform (μ,ν) dichotomy. When compared to the case of contractions, this creates substantial complications. We also provide a collection of admissible Banach spaces for any given nonuniform (μ,ν) dichotomy.


### 3.1. Basic Notions

We consider an evolution processT(t,s), t≥s≥0 satisfied 1, 2 in Section 2.We also consider a functionP:ℝ0+→B(X), where B(X) is the set of bounded linear operators in X, such that(1)
P(t)2=P(t), for every t≥0;(2)
(t,x)↦P(t)x is continuous in ℝ0+×X.We will refer toP as a projection function. Given an evolution process T, we say that a projection function P is compatible with T if: (1)
T(t,s)p(s)=P(t)T(t,s), for every t,τ,s>0;(2)
the map(3.1)T(t,σ)∣kerP(σ):kerP(σ)→kerP(t)is invertible for everyt≥s≥0.We also assume that(3)
there existD>0, ω≥0 and two growth rates μ(t), ν(t) such that
(3.2)∥T(t,s)P(s)∥≤D(μ(t)μ(s))ωνε(s),t≥s≥0.(3.3)∥T(t,s)Q(s)∥≤D(μ(s)μ(t))ωνε(s),s≥t≥0.We note that due to the invertibility assumption in condition (1.2), condition  (3.3) is simply a version of  (3.2) when time goes backwards.We always consider in the paper an evolution processT together with a projection function P which is compatible with T (and which satisfies  (3.2) and  (3.3)). We write
(3.4)U(t,s)=T(t,s)P(s),V(t,s)=T(t,s)Q(s),
where Q(t)=Id-P(t) for each t≥0. we consider the new norms
(3.5)∥x∥t′=sup{∥U(σ,t)x∥(μ(σ)μ(t))-ω,σ≥t}+sup{∥V(σ,t)x∥(μ(t)μ(σ))-ω,0≤σ≤t}.
for each x∈X and t∈ℝ0+, where V(σ,t) denotes the inverse of the map in  (3.1). We have
(3.6)∥x∥t′≥∥P(t)x∥+∥Q(t)x∥≥∥P(t)x+Q(t)x∥=∥x∥.
Moreover, by  (3.2) and  (3.3),
(3.7)∥x∥t′≤2Dνε(t)∥x∥,x∈X,t∈ℝ0+.Definition 3.1.
We say that an evolution processT is a nonuniform (μ,ν) dichotomy in ℝ+ if there exist a projection function P:ℝ0+→B(X) compatible with T, some constants α, D>0, ε≥0 and two growth rates μ(t), ν(t) such that
(3.8)∥T(t,s)P(s)∥≤D(μ(t)μ(s))-ανε(s),t≥s≥0.∥T(t,s)Q(s)∥≤D(μ(s)μ(t))-βνε(s),s≥t≥0,
When ε=0, we say that  (1.2)  has a uniform (μ,ν) dichotomy or simply a (μ,ν) dichotomy.In the following, we still consider several spacesLp, L∞, respectively, with the norms  (2.10), which induce Banach spaces ℒp for each p∈[1,∞]. We also set E(X) as in  (2.11) but using the norms ∥·∥t′ in  (3.5), we endow E(X)=ℒp(X) with the norm in  (2.12).We also obtain easily the same statement in Lemma2.3 for the set E(X) using the norms ∥·∥t′ in  (3.5).Definition 3.2.
We say that a Banach spaceE is admissible for the evolution process T if for each f∈E(X)(1)
the function(3.9)ℝ0+∋τ↦{V(t,τ)f(τ),τ≥t,0,0≤τ<tis inℒ1(X) for each t≥0;(2)
the functionxf:ℝ0+→X defined by
(3.10)xf(t)=∫0t‍U(t,τ)f(τ)dτ-∫t∞‍V(t,τ)f(τ)dτis inℒ∞(X).We note that since∥Q(t)x∥≤∥Q(t)x∥t′ for every x∈X, and t≥0, any function in ℒ1(X) is also integrable in R0+, and thus the first condition ensures that the function xf is well defined. By Lemma 2.3 we know that ℒ∞(X) is a Banach space with the norm
(3.11)∥g∥∞′=esssupt∈ℝ0+∥g(t)∥t′.Lemma 3.3.
If for somep∈[1,∞] the space E=ℒp is admissible for the evolution process T, then there exists K>0 such that
(3.12)∥xf∥∞′≤K∥f∥p′foreveryf∈E(X).Proof.
We follow arguments in [2]. For each t≥0, we define a map Ht:E(X)→ℒ1(X) by
(3.13)(Htf)(τ)={V(t,τ)f(τ),τ≥t,0,0≤τ<t.
Clearly, Ht is linear. We use the closed graph theorem to show that Ht is bounded. For this, let us take a sequence (fn)n∈N⊂E(X) and f∈E(X) such that fn→f in E(X) when n→∞, and also g∈ℒ1(X) such that Htfn→g in ℒ1(X) when n→∞. We need to show that Htf=g Lebesgue-almost everywhere. By Lemma 2.3, the sequence fn converges pointwise Lebesgue-almost everywhere. Therefore,
(3.14)(Htfn)(τ)=V(t,τ)fn(τ)→V(t,τ)f(τ)=(Htf)(τ)
when n→∞, for Lebesgue-almost every τ∈[t,+∞). On the other hand, since Htfn→g in ℒ1(X) when n→∞, we also have (Htfn)(τ)→g(τ) when n→∞, for Lebesgue-almost every τ∈[t,+∞). This shows that Htf=g Lebesgue-almost everywhere, and Ht is bounded for each t≥0.
We define a linear operatorG:E(X)→ℒ∞(X) by Gf=xf. We use again the closed graph theorem to show that G is bounded. For this, let us take a sequence (fn)n∈N⊂E(X) and f∈E(X) such that fn→f in E(X) when n→∞, and also h∈ℒ∞(X) such that Gfn→h in ℒ∞(X) when n→∞. We write
(3.15)(G1f)(t)=P(t)(Gf)(t)=∫0t‍U(t,τ)f(τ)dτ,(G2f)(t)=Q(t)(Gf)(t)=-∫t∞‍V(t,τ)f(τ)dτ.
Using the similar proof of Lemma2.5, for each t≥0 and n∈N we have
(3.16)∥(G1fn)(t)-(G1f)(t)∥t′=sup{∥∫0t‍U(σ,t)U(t,τ)(fn(τ)-f(τ))dτ∥(μ(σ)μ(t))-ω:σ≥t}≤μ(t)ω∫0t‍∥fn(τ)-f(τ)∥τ′dτ.
According to Hölder's inequality, there exists α=α([0,t]) such that
(3.17)∥(G1fn)(t)-(G1f)(t)∥t′≤μ(t)ωα∥(fn(τ)-f(τ))∥p′.
Furthermore, we have
(3.18)∥(G2fn)(t)-(G2f)(t)∥t′≤sup{∫t∞‍∥V(σ,t)V(t,τ)(fn(τ)-f(τ))∥(μ(t)μ(σ))-ωdτ:0≤σ≤t}=∫t∞‍sup{∥V(σ,t)Ht(fn-f)(τ)∥(μ(t)μ(σ))-ω:0≤σ≤t}dτ=∫t∞‍∥Ht(fn-f)(τ)∥t′dτ=∥Ht(fn-f)∥t1.
It follows from  (3.17) and  (3.18) that
(3.19)∥(Gfn)(t)-(Gf)(t)∥t′≤μ(t)ωα∥fn-f∥p′+∥Ht(fn-f)∥t1.
Therefore, for eacht≥0, letting n→∞ we find that (Gfn)(t)→(Gf)(t). This shows that Gf=h Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that G is a bounded operator. This completes the proof of the lemma.


### 3.2. Criterion for Nonuniform(μ,ν) Dichotomy

Theorem 3.4.
If for somep∈(1,∞] the space E=ℒp is admissible for the evolution process T, then T is a nonuniform (μ,ν) dichotomy.Proof.
We first consider the spacePt0=ImP(t0). Given x∈Pt0 and t0≥0, repeating argument of the proof in Theorem 2.6, except limiting T(t,t0) on Pt0, we obtain
(3.20)∥T(t,t0)∣Pt0∥′≤d(μ(t)μ(t0))-α,
where
(3.21)d=LK0,α=-1δ0lnK0.
We note that d,α>0. For each x∈Pt0, we have
(3.22)∥T(t,t0)x∥t′≥∥T(t,t0)x∥,
and by  (3.2),
(3.23)∥x∥t0′=sup{∥U(σ,t0)x∥(μ(σ)μ(t0))-ω:σ≥t0}≤Dνε(t0)∥x∥.
It follows from  (3.5) and  (3.20) that
(3.24)∥T(t,t0)∣Pt0∥=supx∈Pt0∖{0}∥T(t,t0)x∥∥x∥≤Dνε(t0)supx∈Pt0∖{0}∥T(t,t0)x∥t′∥x∥t0′≤dDνε(t0)(μ(t)μ(t0))-α
for any t≥t0.
Now we consider the spaceQt0=KerP(t0). Given x∈Qt0 and t0≥0, we define a function f:ℝ0+→X by
(3.25)f(t)={V(t,t0)x,t∈ℝ0+⋂‍[t0-1,t0]0,t∈ℝ0+∖[t0-1,t0].
Clearly, f(t)∈Qt for every t≥0. Moreover, for each t∈[0,t0-1] (this interval may be empty), we have
(3.26)xf(t)=-∫t0-1t0‍V(t,τ)V(τ,t0)xdτ=-∫t0-1t0‍V(t,t0)xdτ=-V(t,t0)x,
and it follows from Lemma 3.3 that
(3.27)∥V(t,t0)x∥t′=∥xf∥∞′≤K∥f∥p′
for t∈[0,t0-1].
On the other hand, for eacht∈[t0-1,t0], we have
(3.28)∥f(t)∥t′=sup{∥V(σ,t)f(t)∥(μ(t)μ(σ))-ω,0≤σ≤t}≤sup{∥V(σ,t)V(t,t0)x∥(μ(t)μ(σ))-ω,0≤σ≤t}≤(μ(t0)μ(t))ωsup{∥V(σ,t0)x∥(μ(t0)μ(σ))-ω,0≤σ≤t0}≤(μ(t0)μ(t0-1))ω∥x∥t0′.
So
(3.29)∥f∥p′=∫ℝ0+⋂‍[t0-1,t0]‍∥f(t)∥t′dt≤(μ(t0)μ(t0-1))ω∥x∥t0′
and in particular f∈E(X). We thus have
(3.30)∥V(t,t0)x∥t′≤K∥f∥p′≤K(μ(t0)μ(t0-1))ω∥x∥t0′
for every t∈[0,t0-1], and x∈Qt0. This implies that
(3.31)∥V(t,t0)∥′≔supx∈Qt0∖{0}∥V(t,t0)x∥t′∥x∥t0′≤L,L=(μ(t0)μ(t0-1))ωmax{K,1}
for all t≤t0. Indeed, for t∈[0,t0-1] inequality  (3.31) follows from  (3.30), and for t∈ℝ0+⋂‍[t0-1,t0] the inequality follows from  (3.28).
Now givenx∈Qt0, t0≥0, and δ>0, we define a function g:ℝ0+→X by
(3.32)g(t)={V(t,t0)x,t∈ℝ0+⋂‍[t0-δ,t0]0,t∈ℝ0+[t0-δ,t0].
It follows from  (3.31) that
(3.33)∥g(t)∥t′≤∥V(t,t0)x∥t′≤L∥x∥t0′,
and thus,
(3.34)g∈E(X),∥g∥p′≤Lδ1/p∥x∥t0′.
On the other hand, in a similar manner to that in  (2.29),
(3.35)δ22∥V(t0-δ,t0)x∥t0-δ′=∥∫t0-δt0‍(τ-t0)V(t0-δ,t0)xdτ∥t0-δ′≤∫t0-δt0‍(t0-τ)sup{∥V(t0-δ,t0)x∥t0-δ′:0≤σ≤t0-δ}dτ=∫t0-δt0‍(t0-τ)∥V(t0-δ,t0)x∥t0′dτ=∫t0-δt0‍(t0-τ)∥V(t0-δ,τ)V(τ,t0)x∥t0′dτ.
Since
(3.36)xg(t)=-∫t∞‍V(t,τ)g(τ)dτ={0,t∈[t0,∞)(t-t0)V(t,t0)x,t∈[t0-δ,t0].-δV(t,t0)x,t∈[0,t0-δ].
It follows from Lemma 3.3,  (3.31), and  (3.34) that
(3.37)δ22∥V(t0-δ,t0)x∥t0-δ′≤L∫t0-δt0‍(t0-τ)∥V(τ,t0)x∥τ′dτ=L∫t0-δt0‍∥xg(τ)∥τ′dτ≤Lδ∥xg∥∞′≤KLδ∥g∥p′≤KL2δ(p+1)/p∥x∥t0′
for all t0≥0, δ>0, and x∈Qt0; we thus obtain
(3.38)δ22∥V(t0-δ,t0)x∥t0-δ′≤KL2δ(p+1)/p∥x∥t0′
so
(3.39)∥V(t0-δ,t0)∥′≤2KL2δ(1-p)/p
for all t0≥0 and δ>0. Taking the same δ0 as before, and setting n=[(lnμ(t0)-lnμ(t))/δ0] for each t≤t0, we have
(3.40)V(t,t0)=V(t,t0-nδ0)V(t0-nδ0,t0).
By  (3.31) and  (3.39) we obtain
(3.41)∥V(t,t0)∥′≤L∥V(t0-nδ0,t0)∥′≤L∏i=0n-1‍∥V(t0-(i+1)δ0,t0-iδ0)∥′≤LK0n
for t≤t0, where K0:=2KL2δ0(1-p)/p<1. Since
(3.42)n=[lnμ(t0)-lnμ(t)δ0]≥lnμ(t0)-lnμ(t)δ0-1,
this implies that
(3.43)∥V(t,t0)∥′≤d(μ(t)μ(t0))α,
where
(3.44)d=LK0,α=-1δ0lnK0.
We note that d,α>0. By  (3.6)
(3.45)∥V(t,t0)x∥t′≥∥V(t,t0)x∥,
and by  (3.3),
(3.46)∥x∥t0′=sup{∥V(σ,t0)x∥(μ(t0)μ(σ))-ω:0≤σ≤t0}≤Dνε(t0)∥x∥
for x∈Qt0. It follows from  (3.43) that
(3.47)∥V(t,t0)∣Qt0∥=supx∈Qt0∖{0}∥V(t,t0)x∥∥x∥≤Dνε(t0)supx∈Qt0∖{0}∥V(t,t0)x∥t′∥x∥t0′≤dDνε(t0)(μ(t)μ(t0))α
for any t≤t0. To show that T is a nonuniform exponential dichotomy, we note that setting t=s in  (3.2) and  (3.3) yields
(3.48)∥P(s)∥≤Dνε(s),∥Q(s)∥≤Dνε(s)
for every s≥0. Together with  (3.24) and  (3.47) this implies that
(3.49)∥T(t,s)P(s)∥≤∥T(t,s)∣Ps∥∥P(s)∥≤dD2(μ(t)μ(s))-αν2ε(s),t≥s≥0.∥T(t,s)Q(s)∥≤∥T(t,s)∣Qs∥∥P(s)∥≤dD2(μ(s)μ(t))-αν2ε(s),s≥t≥0.
This shows that T is a nonuniform (μ,ν) dichotomy with α,α, dD2 and 2ε.


### 3.3. Admissible Spaces for a Nonuniform(μ,ν) Dichotomy

We consider the spaces(3.50)LDp={f:ℝ0+→ℝLebesgue-measurable:∥f∥p,D<∞}
for each p∈[1,∞), and
(3.51)LD∞={f:ℝ0+→ℝLebesgue-measurable:∥f∥∞,D<∞},
respectively, with the norms
(3.52)∥f∥p,D=(∫0∞‍|f(t)|p(Dνε(t)(μ(t)μ′(t))1/q)p)1/p,∥f∥∞,D=esssupt∈ℝ0+(|f(t)|Dνε(t)μ(t)μ′(t)).
In a similar manner to Lemma 2.3 these normed spaces induce Banach spaces ℒDp and ℒDp(X) for each p∈[1,∞], the last one with norm
(3.53)∥f∥p,D′=∥F∥p,D,whereF(t)=∥f(t)∥t′.Theorem 3.5.
If the evolution processT is a nonuniform (μ,ν) dichotomy, then for any p∈[1,∞] the space ℒDp is admissible for T.Proof.
We first takef∈LD∞. Then
(3.54)∥xf(t)∥t′=sup{∥∫0t‍U(σ,t)U(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}+sup{∥∫t∞‍V(σ,t)V(t,τ)f(τ)dτ∥(μ(t)μ(σ))-ω:0≤σ≤t}≤sup{∫0t‍∥U(σ,τ)∥·∥f(τ)∥(μ(σ)μ(t))-ωdτ:σ≥t}+sup{∫t∞‍∥V(σ,τ)∥·∥f(τ)∥(μ(t)μ(σ))-ωdτ:0≤σ≤t}≤sup{∫0t‍Dνε(τ)(μ(σ)μ(τ))-α∥f(τ)∥dτ:σ≥t}+sup{∫t∞‍Dνε(τ)(μ(σ)μ(τ))β∥f(τ)∥dτ:0≤σ≤t}≤sup{∫0t‍Dνε(τ)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}+sup{∫t∞‍Dνε(τ)(μ(t)μ(τ))β∥f(τ)∥τ′dτ:0≤σ≤t}≤∥f∥∞,D′∫0t‍(μ(t)μ(τ))-αμ′(τ)μ(τ)dτ+∥f∥∞,D′∫t∞‍(μ(t)μ(τ))βμ′(τ)μ(τ)dτ=1α∥f∥∞,D′(1-μ(t)-α)+1β∥f∥∞,D′≤(1α+1β)∥f∥∞,D′.
Therefore,
(3.55)∥xf∥∞′=supt≥0∥xf(t)∥t′≤supt≥0(1α+1β)∥f∥∞,D′<∞
and ℒD∞ is admissible for T.
Now we takef∈ℒDp(X) for some p∈[1,∞). Using Hölder's inequality we obtain
(3.56)∥xf(t)∥t′=sup{∥∫0t‍U(σ,t)U(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}+sup{∥∫t∞‍V(σ,t)V(t,τ)f(τ)dτ∥(μ(t)μ(σ))-ω:0≤σ≤t}≤sup{∫0t‍Dνε(τ)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}+sup{∫t∞‍Dνε(τ)(μ(t)μ(τ))β∥f(τ)∥τ′dτ:0≤σ≤t}≤∥f∥p,D′(∫0t‍(μ(t)μ(τ))-αqμ′(τ)μ(τ)dτ)1/q+∥f∥p,D′(∫t∞‍(μ(t)μ(τ))βqμ′(τ)μ(τ)dτ)1/q=∥f∥p,D′(1-μ(t)-αqαq)1/q+(1βq)1/q∥f∥p,D′≤(1(αq)1/q+1(βq)1/q)∥f∥p,D′,
where 1/p+1/q=1. We conclude that ℒDp is also admissible for T.


## 3.1. Basic Notions

We consider an evolution processT(t,s), t≥s≥0 satisfied 1, 2 in Section 2.We also consider a functionP:ℝ0+→B(X), where B(X) is the set of bounded linear operators in X, such that(1)
P(t)2=P(t), for every t≥0;(2)
(t,x)↦P(t)x is continuous in ℝ0+×X.We will refer toP as a projection function. Given an evolution process T, we say that a projection function P is compatible with T if: (1)
T(t,s)p(s)=P(t)T(t,s), for every t,τ,s>0;(2)
the map(3.1)T(t,σ)∣kerP(σ):kerP(σ)→kerP(t)is invertible for everyt≥s≥0.We also assume that(3)
there existD>0, ω≥0 and two growth rates μ(t), ν(t) such that
(3.2)∥T(t,s)P(s)∥≤D(μ(t)μ(s))ωνε(s),t≥s≥0.(3.3)∥T(t,s)Q(s)∥≤D(μ(s)μ(t))ωνε(s),s≥t≥0.We note that due to the invertibility assumption in condition (1.2), condition  (3.3) is simply a version of  (3.2) when time goes backwards.We always consider in the paper an evolution processT together with a projection function P which is compatible with T (and which satisfies  (3.2) and  (3.3)). We write
(3.4)U(t,s)=T(t,s)P(s),V(t,s)=T(t,s)Q(s),
where Q(t)=Id-P(t) for each t≥0. we consider the new norms
(3.5)∥x∥t′=sup{∥U(σ,t)x∥(μ(σ)μ(t))-ω,σ≥t}+sup{∥V(σ,t)x∥(μ(t)μ(σ))-ω,0≤σ≤t}.
for each x∈X and t∈ℝ0+, where V(σ,t) denotes the inverse of the map in  (3.1). We have
(3.6)∥x∥t′≥∥P(t)x∥+∥Q(t)x∥≥∥P(t)x+Q(t)x∥=∥x∥.
Moreover, by  (3.2) and  (3.3),
(3.7)∥x∥t′≤2Dνε(t)∥x∥,x∈X,t∈ℝ0+.Definition 3.1.
We say that an evolution processT is a nonuniform (μ,ν) dichotomy in ℝ+ if there exist a projection function P:ℝ0+→B(X) compatible with T, some constants α, D>0, ε≥0 and two growth rates μ(t), ν(t) such that
(3.8)∥T(t,s)P(s)∥≤D(μ(t)μ(s))-ανε(s),t≥s≥0.∥T(t,s)Q(s)∥≤D(μ(s)μ(t))-βνε(s),s≥t≥0,
When ε=0, we say that  (1.2)  has a uniform (μ,ν) dichotomy or simply a (μ,ν) dichotomy.In the following, we still consider several spacesLp, L∞, respectively, with the norms  (2.10), which induce Banach spaces ℒp for each p∈[1,∞]. We also set E(X) as in  (2.11) but using the norms ∥·∥t′ in  (3.5), we endow E(X)=ℒp(X) with the norm in  (2.12).We also obtain easily the same statement in Lemma2.3 for the set E(X) using the norms ∥·∥t′ in  (3.5).Definition 3.2.
We say that a Banach spaceE is admissible for the evolution process T if for each f∈E(X)(1)
the function(3.9)ℝ0+∋τ↦{V(t,τ)f(τ),τ≥t,0,0≤τ<tis inℒ1(X) for each t≥0;(2)
the functionxf:ℝ0+→X defined by
(3.10)xf(t)=∫0t‍U(t,τ)f(τ)dτ-∫t∞‍V(t,τ)f(τ)dτis inℒ∞(X).We note that since∥Q(t)x∥≤∥Q(t)x∥t′ for every x∈X, and t≥0, any function in ℒ1(X) is also integrable in R0+, and thus the first condition ensures that the function xf is well defined. By Lemma 2.3 we know that ℒ∞(X) is a Banach space with the norm
(3.11)∥g∥∞′=esssupt∈ℝ0+∥g(t)∥t′.Lemma 3.3.
If for somep∈[1,∞] the space E=ℒp is admissible for the evolution process T, then there exists K>0 such that
(3.12)∥xf∥∞′≤K∥f∥p′foreveryf∈E(X).Proof.
We follow arguments in [2]. For each t≥0, we define a map Ht:E(X)→ℒ1(X) by
(3.13)(Htf)(τ)={V(t,τ)f(τ),τ≥t,0,0≤τ<t.
Clearly, Ht is linear. We use the closed graph theorem to show that Ht is bounded. For this, let us take a sequence (fn)n∈N⊂E(X) and f∈E(X) such that fn→f in E(X) when n→∞, and also g∈ℒ1(X) such that Htfn→g in ℒ1(X) when n→∞. We need to show that Htf=g Lebesgue-almost everywhere. By Lemma 2.3, the sequence fn converges pointwise Lebesgue-almost everywhere. Therefore,
(3.14)(Htfn)(τ)=V(t,τ)fn(τ)→V(t,τ)f(τ)=(Htf)(τ)
when n→∞, for Lebesgue-almost every τ∈[t,+∞). On the other hand, since Htfn→g in ℒ1(X) when n→∞, we also have (Htfn)(τ)→g(τ) when n→∞, for Lebesgue-almost every τ∈[t,+∞). This shows that Htf=g Lebesgue-almost everywhere, and Ht is bounded for each t≥0.
We define a linear operatorG:E(X)→ℒ∞(X) by Gf=xf. We use again the closed graph theorem to show that G is bounded. For this, let us take a sequence (fn)n∈N⊂E(X) and f∈E(X) such that fn→f in E(X) when n→∞, and also h∈ℒ∞(X) such that Gfn→h in ℒ∞(X) when n→∞. We write
(3.15)(G1f)(t)=P(t)(Gf)(t)=∫0t‍U(t,τ)f(τ)dτ,(G2f)(t)=Q(t)(Gf)(t)=-∫t∞‍V(t,τ)f(τ)dτ.
Using the similar proof of Lemma2.5, for each t≥0 and n∈N we have
(3.16)∥(G1fn)(t)-(G1f)(t)∥t′=sup{∥∫0t‍U(σ,t)U(t,τ)(fn(τ)-f(τ))dτ∥(μ(σ)μ(t))-ω:σ≥t}≤μ(t)ω∫0t‍∥fn(τ)-f(τ)∥τ′dτ.
According to Hölder's inequality, there exists α=α([0,t]) such that
(3.17)∥(G1fn)(t)-(G1f)(t)∥t′≤μ(t)ωα∥(fn(τ)-f(τ))∥p′.
Furthermore, we have
(3.18)∥(G2fn)(t)-(G2f)(t)∥t′≤sup{∫t∞‍∥V(σ,t)V(t,τ)(fn(τ)-f(τ))∥(μ(t)μ(σ))-ωdτ:0≤σ≤t}=∫t∞‍sup{∥V(σ,t)Ht(fn-f)(τ)∥(μ(t)μ(σ))-ω:0≤σ≤t}dτ=∫t∞‍∥Ht(fn-f)(τ)∥t′dτ=∥Ht(fn-f)∥t1.
It follows from  (3.17) and  (3.18) that
(3.19)∥(Gfn)(t)-(Gf)(t)∥t′≤μ(t)ωα∥fn-f∥p′+∥Ht(fn-f)∥t1.
Therefore, for eacht≥0, letting n→∞ we find that (Gfn)(t)→(Gf)(t). This shows that Gf=h Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that G is a bounded operator. This completes the proof of the lemma.


## 3.2. Criterion for Nonuniform(μ,ν) Dichotomy

Theorem 3.4.
If for somep∈(1,∞] the space E=ℒp is admissible for the evolution process T, then T is a nonuniform (μ,ν) dichotomy.Proof.
We first consider the spacePt0=ImP(t0). Given x∈Pt0 and t0≥0, repeating argument of the proof in Theorem 2.6, except limiting T(t,t0) on Pt0, we obtain
(3.20)∥T(t,t0)∣Pt0∥′≤d(μ(t)μ(t0))-α,
where
(3.21)d=LK0,α=-1δ0lnK0.
We note that d,α>0. For each x∈Pt0, we have
(3.22)∥T(t,t0)x∥t′≥∥T(t,t0)x∥,
and by  (3.2),
(3.23)∥x∥t0′=sup{∥U(σ,t0)x∥(μ(σ)μ(t0))-ω:σ≥t0}≤Dνε(t0)∥x∥.
It follows from  (3.5) and  (3.20) that
(3.24)∥T(t,t0)∣Pt0∥=supx∈Pt0∖{0}∥T(t,t0)x∥∥x∥≤Dνε(t0)supx∈Pt0∖{0}∥T(t,t0)x∥t′∥x∥t0′≤dDνε(t0)(μ(t)μ(t0))-α
for any t≥t0.
Now we consider the spaceQt0=KerP(t0). Given x∈Qt0 and t0≥0, we define a function f:ℝ0+→X by
(3.25)f(t)={V(t,t0)x,t∈ℝ0+⋂‍[t0-1,t0]0,t∈ℝ0+∖[t0-1,t0].
Clearly, f(t)∈Qt for every t≥0. Moreover, for each t∈[0,t0-1] (this interval may be empty), we have
(3.26)xf(t)=-∫t0-1t0‍V(t,τ)V(τ,t0)xdτ=-∫t0-1t0‍V(t,t0)xdτ=-V(t,t0)x,
and it follows from Lemma 3.3 that
(3.27)∥V(t,t0)x∥t′=∥xf∥∞′≤K∥f∥p′
for t∈[0,t0-1].
On the other hand, for eacht∈[t0-1,t0], we have
(3.28)∥f(t)∥t′=sup{∥V(σ,t)f(t)∥(μ(t)μ(σ))-ω,0≤σ≤t}≤sup{∥V(σ,t)V(t,t0)x∥(μ(t)μ(σ))-ω,0≤σ≤t}≤(μ(t0)μ(t))ωsup{∥V(σ,t0)x∥(μ(t0)μ(σ))-ω,0≤σ≤t0}≤(μ(t0)μ(t0-1))ω∥x∥t0′.
So
(3.29)∥f∥p′=∫ℝ0+⋂‍[t0-1,t0]‍∥f(t)∥t′dt≤(μ(t0)μ(t0-1))ω∥x∥t0′
and in particular f∈E(X). We thus have
(3.30)∥V(t,t0)x∥t′≤K∥f∥p′≤K(μ(t0)μ(t0-1))ω∥x∥t0′
for every t∈[0,t0-1], and x∈Qt0. This implies that
(3.31)∥V(t,t0)∥′≔supx∈Qt0∖{0}∥V(t,t0)x∥t′∥x∥t0′≤L,L=(μ(t0)μ(t0-1))ωmax{K,1}
for all t≤t0. Indeed, for t∈[0,t0-1] inequality  (3.31) follows from  (3.30), and for t∈ℝ0+⋂‍[t0-1,t0] the inequality follows from  (3.28).
Now givenx∈Qt0, t0≥0, and δ>0, we define a function g:ℝ0+→X by
(3.32)g(t)={V(t,t0)x,t∈ℝ0+⋂‍[t0-δ,t0]0,t∈ℝ0+[t0-δ,t0].
It follows from  (3.31) that
(3.33)∥g(t)∥t′≤∥V(t,t0)x∥t′≤L∥x∥t0′,
and thus,
(3.34)g∈E(X),∥g∥p′≤Lδ1/p∥x∥t0′.
On the other hand, in a similar manner to that in  (2.29),
(3.35)δ22∥V(t0-δ,t0)x∥t0-δ′=∥∫t0-δt0‍(τ-t0)V(t0-δ,t0)xdτ∥t0-δ′≤∫t0-δt0‍(t0-τ)sup{∥V(t0-δ,t0)x∥t0-δ′:0≤σ≤t0-δ}dτ=∫t0-δt0‍(t0-τ)∥V(t0-δ,t0)x∥t0′dτ=∫t0-δt0‍(t0-τ)∥V(t0-δ,τ)V(τ,t0)x∥t0′dτ.
Since
(3.36)xg(t)=-∫t∞‍V(t,τ)g(τ)dτ={0,t∈[t0,∞)(t-t0)V(t,t0)x,t∈[t0-δ,t0].-δV(t,t0)x,t∈[0,t0-δ].
It follows from Lemma 3.3,  (3.31), and  (3.34) that
(3.37)δ22∥V(t0-δ,t0)x∥t0-δ′≤L∫t0-δt0‍(t0-τ)∥V(τ,t0)x∥τ′dτ=L∫t0-δt0‍∥xg(τ)∥τ′dτ≤Lδ∥xg∥∞′≤KLδ∥g∥p′≤KL2δ(p+1)/p∥x∥t0′
for all t0≥0, δ>0, and x∈Qt0; we thus obtain
(3.38)δ22∥V(t0-δ,t0)x∥t0-δ′≤KL2δ(p+1)/p∥x∥t0′
so
(3.39)∥V(t0-δ,t0)∥′≤2KL2δ(1-p)/p
for all t0≥0 and δ>0. Taking the same δ0 as before, and setting n=[(lnμ(t0)-lnμ(t))/δ0] for each t≤t0, we have
(3.40)V(t,t0)=V(t,t0-nδ0)V(t0-nδ0,t0).
By  (3.31) and  (3.39) we obtain
(3.41)∥V(t,t0)∥′≤L∥V(t0-nδ0,t0)∥′≤L∏i=0n-1‍∥V(t0-(i+1)δ0,t0-iδ0)∥′≤LK0n
for t≤t0, where K0:=2KL2δ0(1-p)/p<1. Since
(3.42)n=[lnμ(t0)-lnμ(t)δ0]≥lnμ(t0)-lnμ(t)δ0-1,
this implies that
(3.43)∥V(t,t0)∥′≤d(μ(t)μ(t0))α,
where
(3.44)d=LK0,α=-1δ0lnK0.
We note that d,α>0. By  (3.6)
(3.45)∥V(t,t0)x∥t′≥∥V(t,t0)x∥,
and by  (3.3),
(3.46)∥x∥t0′=sup{∥V(σ,t0)x∥(μ(t0)μ(σ))-ω:0≤σ≤t0}≤Dνε(t0)∥x∥
for x∈Qt0. It follows from  (3.43) that
(3.47)∥V(t,t0)∣Qt0∥=supx∈Qt0∖{0}∥V(t,t0)x∥∥x∥≤Dνε(t0)supx∈Qt0∖{0}∥V(t,t0)x∥t′∥x∥t0′≤dDνε(t0)(μ(t)μ(t0))α
for any t≤t0. To show that T is a nonuniform exponential dichotomy, we note that setting t=s in  (3.2) and  (3.3) yields
(3.48)∥P(s)∥≤Dνε(s),∥Q(s)∥≤Dνε(s)
for every s≥0. Together with  (3.24) and  (3.47) this implies that
(3.49)∥T(t,s)P(s)∥≤∥T(t,s)∣Ps∥∥P(s)∥≤dD2(μ(t)μ(s))-αν2ε(s),t≥s≥0.∥T(t,s)Q(s)∥≤∥T(t,s)∣Qs∥∥P(s)∥≤dD2(μ(s)μ(t))-αν2ε(s),s≥t≥0.
This shows that T is a nonuniform (μ,ν) dichotomy with α,α, dD2 and 2ε.


## 3.3. Admissible Spaces for a Nonuniform(μ,ν) Dichotomy

We consider the spaces(3.50)LDp={f:ℝ0+→ℝLebesgue-measurable:∥f∥p,D<∞}
for each p∈[1,∞), and
(3.51)LD∞={f:ℝ0+→ℝLebesgue-measurable:∥f∥∞,D<∞},
respectively, with the norms
(3.52)∥f∥p,D=(∫0∞‍|f(t)|p(Dνε(t)(μ(t)μ′(t))1/q)p)1/p,∥f∥∞,D=esssupt∈ℝ0+(|f(t)|Dνε(t)μ(t)μ′(t)).
In a similar manner to Lemma 2.3 these normed spaces induce Banach spaces ℒDp and ℒDp(X) for each p∈[1,∞], the last one with norm
(3.53)∥f∥p,D′=∥F∥p,D,whereF(t)=∥f(t)∥t′.Theorem 3.5.
If the evolution processT is a nonuniform (μ,ν) dichotomy, then for any p∈[1,∞] the space ℒDp is admissible for T.Proof.
We first takef∈LD∞. Then
(3.54)∥xf(t)∥t′=sup{∥∫0t‍U(σ,t)U(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}+sup{∥∫t∞‍V(σ,t)V(t,τ)f(τ)dτ∥(μ(t)μ(σ))-ω:0≤σ≤t}≤sup{∫0t‍∥U(σ,τ)∥·∥f(τ)∥(μ(σ)μ(t))-ωdτ:σ≥t}+sup{∫t∞‍∥V(σ,τ)∥·∥f(τ)∥(μ(t)μ(σ))-ωdτ:0≤σ≤t}≤sup{∫0t‍Dνε(τ)(μ(σ)μ(τ))-α∥f(τ)∥dτ:σ≥t}+sup{∫t∞‍Dνε(τ)(μ(σ)μ(τ))β∥f(τ)∥dτ:0≤σ≤t}≤sup{∫0t‍Dνε(τ)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}+sup{∫t∞‍Dνε(τ)(μ(t)μ(τ))β∥f(τ)∥τ′dτ:0≤σ≤t}≤∥f∥∞,D′∫0t‍(μ(t)μ(τ))-αμ′(τ)μ(τ)dτ+∥f∥∞,D′∫t∞‍(μ(t)μ(τ))βμ′(τ)μ(τ)dτ=1α∥f∥∞,D′(1-μ(t)-α)+1β∥f∥∞,D′≤(1α+1β)∥f∥∞,D′.
Therefore,
(3.55)∥xf∥∞′=supt≥0∥xf(t)∥t′≤supt≥0(1α+1β)∥f∥∞,D′<∞
and ℒD∞ is admissible for T.
Now we takef∈ℒDp(X) for some p∈[1,∞). Using Hölder's inequality we obtain
(3.56)∥xf(t)∥t′=sup{∥∫0t‍U(σ,t)U(t,τ)f(τ)dτ∥(μ(σ)μ(t))-ω:σ≥t}+sup{∥∫t∞‍V(σ,t)V(t,τ)f(τ)dτ∥(μ(t)μ(σ))-ω:0≤σ≤t}≤sup{∫0t‍Dνε(τ)(μ(t)μ(τ))-α∥f(τ)∥τ′dτ:σ≥t}+sup{∫t∞‍Dνε(τ)(μ(t)μ(τ))β∥f(τ)∥τ′dτ:0≤σ≤t}≤∥f∥p,D′(∫0t‍(μ(t)μ(τ))-αqμ′(τ)μ(τ)dτ)1/q+∥f∥p,D′(∫t∞‍(μ(t)μ(τ))βqμ′(τ)μ(τ)dτ)1/q=∥f∥p,D′(1-μ(t)-αqαq)1/q+(1βq)1/q∥f∥p,D′≤(1(αq)1/q+1(βq)1/q)∥f∥p,D′,
where 1/p+1/q=1. We conclude that ℒDp is also admissible for T.



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*Source: 741696-2012-12-23.xml*