Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 741696, 23 pages
doi:10.1155/2012/741696
Research Article
Admissibility for Nonuniform μ, ν
Contraction and Dichotomy
Yongxin Jiang and Fang-fang Liao
Department of Mathematics, College of Science, Hohai University, Nanjing, Jiangsu 210098, China
Correspondence should be addressed to Yongxin Jiang, yxinjiang@163.com
Received 4 October 2012; Accepted 27 November 2012
Academic Editor: Juntao Sun
Copyright q 2012 Y. Jiang and F.-f. Liao. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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 associated
Lp 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.
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
x Atx ft
1.1
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:
x Atx.
1.2
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
2
Abstract and Applied Analysis
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 At were obtained by Daleckij and Krein 3, Massera and Schäffer 4, and
the work of Levitan and Zhikov 5 for certain cases of unbounded At.
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 socalled 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 nonuniform exponential dichotomy of a nonuniform
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.
Abstract and Applied Analysis
3
2.1. Basic Notions
We say that an increasing function μ : R → 1, ∞ is a growth rate if
μ0 1,
lim μt ∞.
2.1
t → ∞
We say that a family of linear operators T t, s, t ≥ s ≥ 0 in a Banach space X is an
evolution process if:
1 T t, t Id and T t, τT τ, s T t, s, t, τ, s > 0;
2 t, s, x → T t, sx is continuous for t ≥ s ≥ 0 and x ∈ X.
In this section, we also assume that
3 there exist ω ≥ 0, D > 0 and two growth rates μt, νt such that
μt
T t, s ≤ D
μs
We consider the new norms
xt
ω
νε s,
t ≥ s ≥ 0.
μσ −ω
sup T σ, tx
,σ ≥ t ,
μt
2.2
x ∈ X, t ∈ R0 .
2.3
These satisfy
x ≤ xt ≤ Dνε tx,
x ∈ X, t ∈ R0 .
2.4
Moreover, with respect to these norms the evolution process has the following bounded
growth property.
Proposition 2.1. If T is an evolution process, then
μt ω
xs
T t, sxt ≤
μs
2.5
for every t ≥ s ≥ 0 and x ∈ X.
Proof. We have
T t, sxt
μσ −ω
sup T σ, tx
,σ ≥ t
μt
μt ω
μσ −ω
sup T σ, tx
,σ ≥ s
≤
μs
μs
μt
μs
which yields the desired inequality.
ω
xs
2.6
4
Abstract and Applied Analysis
Definition 2.2. We say that an evolution process T is a nonuniform μ, ν contraction in R0 if
there exist some constants α, D > 0, ε ≥ 0 and two growth rates μt, νt such that
T t, s ≤ D
μt
μs
−α
νε s,
t ≥ s ≥ 0.
2.7
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
Lp f : R0 −→ R Lebesgue-measurable : f p < ∞
2.8
for each p ∈ 1, ∞, and
L∞ f : R0 −→ R Lebesgue-measurable : f ∞ < ∞
2.9
Respectively, with the norms
f p
∞
ftp
1/p
,
0
f ess supft.
∞
t∈R0
2.10
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 space E Lp , with p ∈ 1, ∞, we set
EX f : R0 −→ X Bochner-measurable : t −→ ftt ∈ E
2.11
using the norms · t in 2.3, and we endow EX Lp X with the norm
f F ,
p
p
where Ft ftt .
2.12
Repeating arguments in the proof of Theorem 3 in 22, we obtain the following
statement.
Lemma 2.3. For each p ∈ 1, ∞ and E Lp , the set EX is a Banach space with the norm in
2.12, and the convergence in EX implies the pointwise convergence Lebesgue-almost everywhere.
Definition 2.4. We say that a Banach space E is admissible for the evolution process T if for
each f ∈ EX the function xf : R0 → X defined by
xf t ∞
0
is in L∞ see 2.11.
T t, τfτdτ
2.13
Abstract and Applied Analysis
5
By Lemma 2.3 we know that L∞ is a Banach space with the norm
g ess supgt .
∞
t
t∈R0
2.14
Lemma 2.5. There exists K > 0 such that
xf ≤ K f p
∞
for every f ∈ EX.
2.15
Proof. We define a linear operator G : EX → 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 ⊂ EX and
f ∈ EX such that fn → f in EX 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
t
−ω
μσ
Gfn t − Gf t sup :σ≥t
T σ, tT t, τ fn τ − fτ dτ t
μt
0
t
−ω
μσ
:σ≥t
sup T σ, τ fn τ − fτ dτ μt
0
t
μσ −ω
T σ, τ fn τ − fτ dτ : σ ≥ t
≤ sup
μt
0
t
μσ −ω μt ω
T σ, τ fn τ − fτ
dτ : σ ≥ t
sup
μτ
μτ
0
t
0
fn τ − fτ
τ
≤ μtω
t
0
μt
μτ
ω
dτ
fn τ − fτ dτ.
τ
2.16
According to Hölder’s inequality, there exists α α0, t such that
Gfn t − Gf t ≤ μtω
t
t
0
fn τ − fτ dτ ≤ μtω α fn τ − fτ .
τ
p
2.17
Therefore, for each t ≥ 0, letting n → ∞ we find that Gfn t → Gft. 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 some p ∈ 1, ∞ the space E Lp is admissible for the evolution process T , then
T is a nonuniform μ, ν contraction.
6
Abstract and Applied Analysis
Proof. We follow arguments in 22. Given x ∈ X and t0 ≥ 0, we define a function f : R0 → X
by
T t, t0 x,
ft 0,
t ∈ t0 , t0 1
t ∈ R0 \ t0 , t0 1.
2.18
We note that
ft ≤ T t, t0 x χt ,t 1 t.
0 0
t
t
2.19
Then, for each t ∈ t0 , t0 1 and x ∈ X, we have
T t, t0 xt
μσ −ω
sup T σ, tT t, t0 x
,σ ≥ t
μt
μσ −ω
μt ω
sup T σ, t0 x
, σ ≥ t0
≤
μt0 μt0 ≤
μt
μt0 ω
2.20
xt0
μt0 1
μt0 ω
xt0 .
Therefore,
ft ≤
p
μt0 1
μt0 ω
xt0 χt0 ,t0 1 tp
μt0 1
μt0 ω
xt0
2.21
and in particular f ∈ EX. On the other hand, according to 2.13 and 2.18, we have
xf t t0 1
T t, τT τ, t0 xdτ T t, t0 x
2.22
t0
for all t ≥ t0 1, which implies that
T t, t0 xt xf t ≤ xf ∞ .
2.23
By Lemma 2.5 and 2.21–2.23, we obtain
T t, t0 xt
μt0 1 ω
≤ xf ∞ ≤ K f p ≤ K
xt0
μt0 2.24
Abstract and Applied Analysis
7
for all t ≥ t0 1, t0 ≥ 0, and x ∈ X. We claim that
T t, t0 : sup
T t, t0 xt
xt0
x
/0
≤ L,
L
μt0 1
μt0 ω
max{K, 1}
2.25
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 given x ∈ X, t0 ≥ 0, and δ > 0, we define a function g : R0 → X by
gt t ∈ t0 , t0 δ
t ∈ R0 \ t0 , t0 δ.
T t, t0 x,
0,
2.26
It follows from 2.25 that
gt ≤ T t, t0 x ≤ Lx
t
t0
t
2.27
and thus,
g ∈ EX,
g ≤ Lδ1/p x .
t0
p
2.28
On the other hand, writing y T t0 δ, t0 x,
t0 δ
δ2 y −
t
τ
ydτ
0
t
δ
0
t0
2
t0 δ
t0 δ
−ω
μσ
sup T σ, t0 δ
: σ ≥ t0 δ
τ − t0 ydτ μt0 δ
t0
t δ
−ω
0
μσ
: σ ≥ t0 δ
sup τ − t0 T σ, t0 xdτ μt0 δ
t0
−ω
μσ
dτ : σ ≥ t0 δ
≤ sup
τ − t0 T σ, t0 x
μt0 δ
t0
t0 δ
−ω
μσ
: σ ≥ t0 δ dτ
τ − t0 sup T σ, t0 δy
μt0 δ
t0
t δ
0
t0 δ
t0
t0 δ
t0
τ − t0 yt0 δ dτ
τ − t0 T t0 δ, τT τ, t0 xt0 dτ
2.29
8
Abstract and Applied Analysis
Since
⎧
⎪
⎪
⎨0,
xg t T t, τgτdτ t, t0 T t, t0 x,
⎪
⎪
0
⎩δT t, t x,
0
t
t ∈ 0, t0 t ∈ t0 , t0 δ,
t ∈ t0 δ, ∞,
2.30
it follows from Lemma 2.5, 2.25, and 2.28 that
δ2
T t0 δ, t0 xt0 δ ≤ L
2
L
t0 δ
t0
t0 δ
t0
τ − t0 T τ, t0 xτ dτ
xg τ dτ ≤ Lδxg τ
∞
2.31
≤ KLδg p ≤ KL2 δp1/p xt0
for all t0 ≥ 0, δ > 0, and x ∈ X; we thus obtain
δ2
T t0 δ, t0 xt0 δ ≤ KL2 δp1/p xt0
2
2.32
T t0 δ, t0 ≤ 2KL2 δ1−p/p
2.33
so
for all t0 ≥ 0 and δ > 0. Since 1 − p/p < 0 for p ∈ 1, ∞, there exists δ0 > 0 sufficiently large
such that
K0 : 2KL2 δ1−p/p < 1.
2.34
Setting n ln μt − ln μt0 /δ0 for each t ≥ t0 , we have
T t, t0 T t, t0 nδ0 T t0 nδ0 , t0 .
2.35
By 2.25 and 2.33 we obtain
T t, t0 ≤ LT t0 nδ0 , t0 ≤L
n−1
T t0 i 1δ0 , t0 iδ0 ≤ LK0n
2.36
i0
for t ≥ t0 . By 2.34 and
ln μt − ln μt0 ln μt − ln μt0 −1
≥
n
δ0
δ0
2.37
Abstract and Applied Analysis
9
this implies that
T t, t0 ≤ d
μt
μt0 −α
2.38
,
where
d
L
,
K0
α−
1
ln K0 .
δ0
2.39
We note that d, α > 0. Since
T t, t0 xt ≥ T t, t0 x,
2.40
and by 2.4,
xt0
μσ
sup T σ, t0 x
μt0 −ω
: σ ≥ t0
≤ Dνε t0 x.
2.41
It follows from 2.38 that
μt −α
T t, t0 xt
T t, t0 x
ε
≤
dDν
≤ Dνε t0 sup
t
0
μt0 x
xt0
x/
0
x/
0
T t, t0 sup
2.42
for any t ≥ t0 . Therefore, the evolution process T is a nonuniform μ, ν contraction with α
dD. This concludes the proof of Theorem 2.6.
and D
2.3. Admissible Spaces for Nonuniform μ, ν Contractions
We consider the spaces
p
LD f : R0 −→ R Lebesgue-measurable : f p,D < ∞
2.43
for each p ∈ 1, ∞, and
f L∞
f
:
R
−→
R
Lebesgue-measurable
:
<
∞
,
0
D
∞,D
2.44
respectively, with the norms
p 1/p
p
μt 1/q
ε
ft
Dν t ,
p,D
μ t
0
ε μt
f .
ess sup ft Dν t ∞,D
μ t
t∈R
f ∞
0
2.45
10
Abstract and Applied Analysis
p
In a similar manner to that in Lemma 2.3 these normed spaces induce Banach spaces LD and
p
LD X for each p ∈ 1, ∞, the last one with norm
f F ,
p,D
p,D
where Ft ftt .
2.46
Theorem 2.7. If the evolution process T is a nonuniform μ, ν contraction, then for any p ∈ 1, ∞
p
the space LD is admissible for T .
Proof. We first take f ∈ L∞
D . Then
t
μσ −ω
xf t sup :σ≥t
T σ, tT t, τfτdτ t
μt
0
t
μσ −ω
:σ≥t
sup T σ, τfτdτ μt
0
t
≤ sup
T σ, τ · fτ dτ : σ ≥ t
0
≤ sup
≤ sup
t
−α
−α
μσ
Dν τ
μτ
0
τ
t
μt
Dν τ
μτ
0
≤ f ∞,D
τ
t 0
μt
μτ
−α
fτ dτ : σ ≥ t
τ
fτ dτ : σ ≥ t
τ
2.47
μ τ
dτ
μτ
1 − μt−α 1 ≤ f ∞,D
≤ f ∞,D .
α
α
Therefore,
xf supxf t ≤ sup 1 f <∞
∞,D
∞
t
t≥0
t≥0 α
and L∞
D is admissible for T .
2.48
Abstract and Applied Analysis
11
p
Now we take f ∈ LD X for some p ∈ 1, ∞. Using Hölder’s inequality we obtain
t
μσ −ω
xf t sup :σ≥t
T σ, tT t, τfτdτ t
0
μt
≤ sup
t
≤ f p,D
≤ f p,D
μt
Dν t
μτ
0
t 0
−α
τ
μt
μτ
−αq
1 − μt−αq
αq
fτ dτ : σ ≥ t
τ
μ τ
dτ
μτ
1/q
1/q
2.49
1
1/q f p,D ,
αq
≤
p
where 1/p 1/q 1. We conclude that LD 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 process T t, s, t ≥ s ≥ 0 satisfied 1, 2 in Section 2.
We also consider a function P : R0 → BX, where BX is the set of bounded linear
operators in X, such that
1 P t2 P t, for every t ≥ 0;
2 t, x → P tx is continuous in R0 × X.
We will refer to P as a projection function. Given an evolution process T , we say that a
projection function P is compatible with T if:
1 T t, sps P tT t, s, for every t, τ, s > 0;
2 the map
T t, σ | ker P σ : ker P σ −→ ker P t
is invertible for every t ≥ s ≥ 0.
3.1
12
Abstract and Applied Analysis
We also assume that
3 there exist D > 0, ω ≥ 0 and two growth rates μt, νt such that
T t, sP s ≤ D
T t, sQs ≤ D
ω
μt
μs
νε s,
μs
μt
t ≥ s ≥ 0.
3.2
s ≥ t ≥ 0.
3.3
ω
νε s,
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 process T together with a projection
function P which is compatible with T and which satisfies 3.2 and 3.3. We write
Ut, s T t, sP s,
V t, s T t, sQs,
3.4
where Qt Id − P t for each t ≥ 0. we consider the new norms
xt
μσ −ω
sup Uσ, tx
,σ ≥ t
μt
μt −ω
,0 ≤ σ ≤ t .
sup V σ, tx
μσ
3.5
for each x ∈ X and t ∈ R0 , where V σ, t denotes the inverse of the map in 3.1. We have
xt ≥ P tx Qtx ≥ P tx Qtx x.
3.6
Moreover, by 3.2 and 3.3,
xt ≤ 2Dνε tx,
x ∈ X, t ∈ R0 .
3.7
Definition 3.1. We say that an evolution process T is a nonuniform μ, ν dichotomy in R if
there exist a projection function P : R0 → BX compatible with T , some constants α, D > 0,
ε ≥ 0 and two growth rates μt, νt such that
T t, sP s ≤ D
μt
μs
μs
T t, sQs ≤ D
μt
−α
−β
νε s,
t ≥ s ≥ 0.
3.8
νε s,
s ≥ t ≥ 0,
When ε 0, we say that 1.2 has a uniform μ, ν dichotomy or simply a μ, ν dichotomy.
Abstract and Applied Analysis
13
In the following, we still consider several spaces Lp , L∞ , respectively, with the norms
2.10, which induce Banach spaces Lp for each p ∈ 1, ∞. We also set EX as in 2.11 but
using the norms · t in 3.5, we endow EX Lp X with the norm in 2.12.
We also obtain easily the same statement in Lemma 2.3 for the set EX using the
norms · t in 3.5.
Definition 3.2. We say that a Banach space E is admissible for the evolution process T if for
each f ∈ EX
1 the function
R0
τ −→
V t, τfτ,
τ ≥ t,
0≤τ <t
0,
3.9
is in L1 X for each t ≥ 0;
2 the function xf : R0 → X defined by
xf t t
Ut, τfτdτ −
0
∞
V t, τfτdτ
3.10
t
is in L∞ X.
We note that since Qtx ≤ Qtxt for every x ∈ X, and t ≥ 0, any function in
L 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
1
g ess supgt .
∞
t
t∈R0
3.11
Lemma 3.3. If for some p ∈ 1, ∞ the space E Lp is admissible for the evolution process T , then
there exists K > 0 such that
xf ≤ K f p
∞
for every f ∈ EX.
3.12
Proof. We follow arguments in 2. For each t ≥ 0, we define a map Ht : EX → L1 X by
Ht f τ V t, τfτ, τ ≥ t,
0,
0 ≤ τ < t.
3.13
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 ⊂ EX and f ∈ EX such that fn → f in EX when n → ∞,
and also g ∈ L1 X such that Ht fn → g in L1 X when n → ∞. We need to show that
14
Abstract and Applied Analysis
Ht f g Lebesgue-almost everywhere. By Lemma 2.3, the sequence fn converges pointwise
Lebesgue-almost everywhere. Therefore,
Ht fn τ V t, τfn τ −→ V t, τfτ Ht f τ
3.14
when n → ∞, for Lebesgue-almost every τ ∈ t, ∞. On the other hand, since Ht fn → g
in L1 X when n → ∞, we also have Ht fn τ → gτ when n → ∞, for Lebesgue-almost
every τ ∈ t, ∞. This shows that Ht f g Lebesgue-almost everywhere, and Ht is bounded
for each t ≥ 0.
We define a linear operator G : EX → 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 ⊂ EX and
f ∈ EX such that fn → f in EX when n → ∞, and also h ∈ L∞ X such that Gfn → h
in L∞ X when n → ∞. We write
G1 f t P t Gf t t
Ut, τfτdτ,
0
G2 f t Qt Gf t −
∞
3.15
V t, τfτdτ.
t
Using the similar proof of Lemma 2.5, for each t ≥ 0 and n ∈ N we have
t
−ω
μσ
G1 fn t − G1 f t sup :σ≥t
Uσ, tUt, τ fn τ − fτ dτ t
μt
0
≤ μtω
t
0
fn τ − fτ dτ.
τ
3.16
According to Hölder’s inequality, there exists α α0, t such that
G1 fn t − G1 f t ≤ μtω α fn τ − fτ .
t
p
3.17
Furthermore, we have
μt −ω
V σ, tV t, τ fn τ − fτ dτ : 0 ≤ σ ≤ t
μσ
t
∞
μt −ω
sup V σ, tHt fn − f τ
: 0 ≤ σ ≤ t dτ
μσ
t
∞
Ht fn − f τ dτ Ht fn − f 1 .
t
t
G2 fn t − G2 f t ≤ sup
t
t
∞
3.18
Abstract and Applied Analysis
15
It follows from 3.17 and 3.18 that
Gfn t − Gf t ≤ μtω αfn − f Ht fn − f 1 .
p
t
t
3.19
Therefore, for each t ≥ 0, letting n → ∞ we find that Gfn t → Gft. 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 some p ∈ 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 space Pt0 Im P 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
T t, t0 | Pt0 ≤ d
μt
μt0 −α
,
3.20
where
d
L
,
K0
α−
1
ln K0 .
δ0
3.21
We note that d, α > 0. For each x ∈ Pt0 , we have
T t, t0 xt ≥ T t, t0 x,
3.22
and by 3.2,
μσ −ω
sup Uσ, t0 x
: σ ≥ t0 ≤ Dνε t0 x.
μt0 xt0
3.23
It follows from 3.5 and 3.20 that
T t, t0 | Pt0 T t, t0 x
x
\{0}
sup
x∈Pt0
≤ Dνε t0 sup
T t, t0 xt
x∈Pt0 \{0}
≤ dDνε t0 for any t ≥ t0 .
μt
μt0 xt0
−α
3.24
16
Abstract and Applied Analysis
Now we consider the space Qt0 Ker P t0 . Given x ∈ Qt0 and t0 ≥ 0, we define a
function f : R0 → X by
ft t0 − 1, t0 t ∈ R0 \ t0 − 1, t0 .
V t, t0 x, t ∈ R0
0,
3.25
Clearly, ft ∈ Qt for every t ≥ 0. Moreover, for each t ∈ 0, t0 − 1 this interval may be
empty, we have
t0
xf t −
t0 −1
t0
−
t0 −1
V t, τV τ, t0 xdτ
3.26
V t, t0 xdτ −V t, t0 x,
and it follows from Lemma 3.3 that
V t, t0 xt xf ∞ ≤ K f p
3.27
for t ∈ 0, t0 − 1.
On the other hand, for each t ∈ t0 − 1, t0 , we have
μt −ω
ft sup V σ, tft
,0 ≤ σ ≤ t
t
μσ
μt −ω
,0 ≤ σ ≤ t
≤ sup V σ, tV t, t0 x
μσ
μt0 −ω
μt0 ω
sup V σ, t0 x
, 0 ≤ σ ≤ t0
≤
μt
μσ
≤
μt0 μt0 − 1
ω
3.28
xt0 .
So
f p
R0
t0 −1,t0 ft dt ≤
t
μt0 μt0 − 1
ω
xt0
3.29
and in particular f ∈ EX. We thus have
V t, t0 xt
≤ K f p ≤ K
μt0 μt0 − 1
ω
xt0
3.30
Abstract and Applied Analysis
17
for every t ∈ 0, t0 − 1, and x ∈ Qt0 . This implies that
V t, t0 :
V t, t0 xt
sup
x∈Qt0 \{0}
≤ L,
xt0
L
μt0 μt0 − 1
ω
max{K, 1}
3.31
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 given x ∈ Qt0 , t0 ≥ 0, and δ > 0, we define a function g : R0 → X by
V t, t0 x,
gt 0,
t ∈ R0 t0 − δ, t0 t ∈ R0 t0 − δ, t0 .
3.32
It follows from 3.31 that
gt ≤ V t, t0 x ≤ Lx ,
t
t0
t
3.33
and thus,
g ∈ EX,
g ≤ Lδ1/p x .
t0
p
3.34
On the other hand, in a similar manner to that in 2.29,
t0
δ2
τ − t0 V t0 − δ, t0 xdτ V t0 − δ, t0 xt0 −δ t0 −δ
2
t0 −δ
≤
t0
t0 −δ
t0
t0 −δ
t0
t0 −δ
t0 − τ sup V t0 − δ, t0 xt0 −δ : 0 ≤ σ ≤ t0 − δ dτ
3.35
t0 − τV t0 − δ, t0 xt0 dτ
t0 − τV t0 − δ, τV τ, t0 xt0 dτ.
Since
xg t −
∞
t
⎧
⎪
t ∈ t0 , ∞
⎪
⎨0,
V t, τgτdτ t − t0 V t, t0 x, t ∈ t0 − δ, t0 .
⎪
⎪
⎩−δV t, t x,
t ∈ 0, t0 − δ.
0
3.36
18
Abstract and Applied Analysis
It follows from Lemma 3.3, 3.31, and 3.34 that
δ2
V t0 − δ, t0 xt0 −δ ≤ L
2
L
t0
t0 −δ
t0
t0 −δ
t0 − τV τ, t0 xτ dτ
xg τ dτ ≤ Lδxg τ
∞
3.37
≤ KLδg p ≤ KL2 δp1/p xt0
for all t0 ≥ 0, δ > 0, and x ∈ Qt0 ; we thus obtain
δ2
V t0 − δ, t0 xt0 −δ ≤ KL2 δp1/p xt0
2
3.38
V t0 − δ, t0 ≤ 2KL2 δ1−p/p
3.39
so
for all t0 ≥ 0 and δ > 0. Taking the same δ0 as before, and setting n ln μt0 − ln μt/δ0 for each t ≤ t0 , we have
V t, t0 V t, t0 − nδ0 V t0 − nδ0 , t0 .
3.40
By 3.31 and 3.39 we obtain
V t, t0 ≤ LV t0 − nδ0 , t0 ≤L
n−1
V t0 − i 1δ0 , t0 − iδ0 ≤ LK0n
3.41
i0
1−p/p
for t ≤ t0 , where K0 : 2KL2 δ0
< 1. Since
ln μt0 − ln μt
ln μt0 − ln μt
− 1,
≥
n
δ0
δ0
3.42
this implies that
μt
V t, t0 ≤ d
μt0 α
,
3.43
where
d
L
,
K0
α−
1
ln K0 .
δ0
3.44
Abstract and Applied Analysis
19
We note that d, α > 0. By 3.6
V t, t0 xt ≥ V t, t0 x,
3.45
and by 3.3,
μt0 −ω
sup V σ, t0 x
: 0 ≤ σ ≤ t0 ≤ Dνε t0 x
μσ
xt0
3.46
for x ∈ Qt0 . It follows from 3.43 that
V t, t0 | Qt0 V t, t0 x
x
\{0}
sup
x∈Qt0
≤ Dνε t0 sup
V t, t0 xt
x∈Qt0 \{0}
xt0
3.47
μt α
≤ dDνε t0 μt0 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
P s ≤ Dνε s,
Qs ≤ Dνε s
3.48
for every s ≥ 0. Together with 3.24 and 3.47 this implies that
T t, sP s ≤ T t, s | Ps P s
−α
2 μt
ν2ε s,
≤ dD
μs
T t, sQs ≤ T t, s | Qs P s
−α
2 μs
ν2ε s,
≤ dD
μt
t ≥ s ≥ 0.
3.49
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
p
LD f : R0 −→ R Lebesgue-measurable : f p,D < ∞
3.50
20
Abstract and Applied Analysis
for each p ∈ 1, ∞, and
L∞
D f : R0 −→ R Lebesgue-measurable : f ∞,D < ∞ ,
3.51
respectively, with the norms
p 1/p
p
μt 1/q
ε
ft
Dν t ,
p,D
μ t
0
ε μt
f .
ess sup ft Dν t ∞,D
μ t
t∈R
∞
f 3.52
0
p
p
In a similar manner to Lemma 2.3 these normed spaces induce Banach spaces LD and LD X
for each p ∈ 1, ∞, the last one with norm
f F ,
p,D
p,D
where Ft ftt .
3.53
Theorem 3.5. If the evolution process T is a nonuniform μ, ν dichotomy, then for any p ∈ 1, ∞
p
the space LD is admissible for T .
Proof. We first take f ∈ L∞
D . Then
t
μσ −ω
xf t sup Uσ, tUt, τfτdτ :σ≥t
t
μt
0
∞
μt −ω
V σ, tV t, τfτdτ :0≤σ≤t
sup μσ
t
μσ −ω
dτ : σ ≥ t
≤ sup
Uσ, τ · fτ
μt
0
∞
μt −ω
dτ : 0 ≤ σ ≤ t
sup
V σ, τ · fτ
μσ
t
t
μσ −α ε
fτ dτ : σ ≥ t
Dν τ
≤ sup
μτ
0
∞
μσ β ε
fτdτ : 0 ≤ σ ≤ t
sup
Dν τ
μτ
t
t
μt −α ε
≤ sup
fτ τ dτ : σ ≥ t
Dν τ
μτ
0
∞
μt β ε
fτ τ dτ : 0 ≤ σ ≤ t
Dν τ
sup
μτ
t
t
Abstract and Applied Analysis
≤ f ∞,D
21
t 0
μt
μτ
−α
μ τ
dτ f ∞,D
μτ
∞ t
μt
μτ
β
μ τ
dτ
μτ
1 1
f 1 − μt−α f ∞,D
∞,D
α
β
1 1 f .
≤
∞,D
α β
3.54
Therefore,
xf supxf t ≤ sup 1 1 f <∞
∞,D
∞
t
α β
t≥0
t≥0
3.55
and L∞
D is admissible for Tp.
Now we take f ∈ LD X for some p ∈ 1, ∞. Using Hölder’s inequality we obtain
t
μσ −ω
xf t sup Uσ, tUt, τfτdτ :σ≥t
t
μt
0
∞
μt −ω
V σ, tV t, τfτdτ :0≤σ≤t
sup μσ
t
≤ sup
t
sup
μt
Dν τ
μτ
0
∞
t
≤ f p,D
≤
μt
Dν τ
μτ
0
p,D
f p,D
μt
μτ
∞ t
−αq
μt
μτ
1 − μt−αq
αq
β
p
fτ dτ : 0 ≤ σ ≤ t
τ
μ τ
dτ
μτ
βq
3.56
1/q
μ τ
dτ
μτ
1/q
1
1
1/q 1/q
αq
βq
fτ dτ : σ ≥ t
τ
ε
t f −α
ε
1
βq
1/q
1/q
f p,D
f ,
p,D
where 1/p 1/q 1. We conclude that LD is also admissible for T .
22
Abstract and Applied Analysis
Acknowledgments
This work is partly supported by the National Natural Science Foundation of China under
Grant no. 11171090 and the Fundamental Research Funds for the Central Universities.
The authors would like to show their great thanks to Professor Jifeng Chu for his useful
suggestions and comments.
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