# Trichotomy for Dynamical Systems in Banach Spaces

Authors: Codruţa Stoica
Journal: The Scientific World Journal
 (2013)

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## Abstract

We construct a framework for the study of dynamical systems that describe phenomena from physics and engineering in infinite dimensions and whose state evolution is set out by skew-evolution semiflows. Therefore, we introduce the concept ofω-trichotomy. Characterizations in a uniform setting are proved, using techniques from the
domain of nonautonomous evolution equations with unbounded coefficients, and connections with the classic notion of trichotomy are given. The statements are sustained by several examples.


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## Body

## 1. Introduction

The possibility of reducing the nonautonomous case in the study of associated evolution operators to the autonomous case of evolution semigroups on various Banach function spaces can be considered as an important way towards applications issued from the real world. Of great importance in the study the solution of differential equations is the approach by evolution families, as the techniques from the domain of non autonomous equations with unbounded coefficients in infinite dimensions can be extended in this direction.Appropriate for the study of evolution equations in infinite dimensions are also the skew-evolution semiflows, introduced by us in [1], as generalizations for evolution operators and skew-product semiflows, and whose applicability is pointed out, for example, by Bento and Silva in [2] and by Hai in [3, 4]. Other asymptotic properties for skew-evolution semiflows are defined and characterized in [5, 6].The techniques used in the investigation of exponential stability and exponential instability were generalized for the case of exponential dichotomy in [7, 8] and for the case of exponential trichotomy in [9, 10]. The main idea in the study of trichotomy, introduced for the finite dimensional case by Sacker and Sell in [11], as a natural generalization of the concept of dichotomy is to obtain, at any moment, a decomposition of the state space into three subspaces: a stable subspace, an instable one, and a third one, the central manifold. The aim of the present study is to emphasize the property of ω-trichotomy for skew-evolution semiflows in Banach spaces and to give several conditions in order to describe the behavior related to the third subspace.


## 2. Skew-Evolution Semiflows

Let(X,d) be a metric space, V a Banach space, and V* its topological dual. Let ℬ(V) be the space of all V-valued bounded operators defined on V. The norm of vectors on V and on V* and of operators on ℬ(V) is denoted by ∥·∥. Let us consider that Y=X×V and T={(t,t0)∈ℝ+2:t≥t0}. I is the identity operator.Definition 1.
A mappingφ:T×X→X is called evolution semiflow on X if the following properties are satisfied:
(es1)φ(t,t,x)=x,∀(t,x)∈ℝ+×X,(es2)φ(t,s,φ(s,t0,x))=φ(t,t0,x),∀(t,s),(s,t0)∈T,∀x∈X.Definition 2.
A mappingΦ:T×X→ℬ(V) is called evolution cocycle over an evolution semiflow φ if it satisfies the following properties:
(ec1)Φ(t,t,x)=I,∀t≥0,∀x∈X,(ec2)Φ(t,s,φ(s,t0,x))Φ(s,t0,x)=Φ(t,t0,x),∀(t,s),(s,t0)∈T,∀x∈X.Definition 3.
The mappingC:T×Y→Y defined by
(1)C(t,s,x,v)=(φ(t,s,x),Φ(t,s,x)v),
where Φ is an evolution cocycle over an evolution semiflow φ, is called skew-evolution semiflow on Y.Example 4.
Letf:ℝ+→ℝ+* be a decreasing function with the property that there exists limt→∞f(t)=a>0. We denote by 𝒞=𝒞(ℝ+,ℝ+) the set of all continuous functions x:ℝ+→ℝ+, endowed with the topology of uniform convergence on compact subsets of ℝ+, metrizable by means of the distance
(2)d(x,y)=∑n=1∞12ndn(x,y)1+dn(x,y),wheredn(x,y)=supt∈[0,n]|x(t)-y(t)|.
Ifx∈𝒞, then, for all t∈ℝ+, we denote that xt(s)=x(t+s), xt∈𝒞. Let X be the closure in 𝒞 of the set {ft,t∈ℝ+}. It follows that (X,d) is a metric space, and the mapping φ:Δ×X→X, φ(t,s,x)=𝓍t-s is an evolution semiflow on X.
We consider thatV=ℝ2, with the norm ∥v∥=|v1|+|v2|, v=(v1,v2)∈V. If u:ℝ+→ℝ+*, then the mapping Φu:Δ×X→ℬ(V) defined by
(3)Φu(t,s,𝓍)v=(u(s)u(t)e-∫st𝓍(τ-s)dτv1,u(t)u(s)e∫st𝓍(τ-s)dτv2)
is an evolution cocycle over φ, and C=(φ,Φu) is a skew-evolution semiflow.A particular class of skew-evolution semiflows is emphasized in the following.Remark 5.
Let us consider a skew-evolution semiflowC=(φ,Φ) and a parameter λ∈ℝ. We define the mapping
(4)Φλ:T×X⟶ℬ(V),Φλ(t,t0,x)=e-λ(t-t0)Φ(t,t0,x).
We observe thatCλ=(φ,Φλ) is also a skew-evolution semiflow, called λ-shifted skew-evolution semiflow on Y. We will call Φλ the λ-shifted evolution cocycle.Example 6.
LetX be the metric space defined in the first Example. We define the mapping φ0:ℝ+×X→X, φ0(t,x)=xt, where xt(τ)=x(t+τ), for all τ≥0, which is a classic semiflow on X. Let us consider for every x∈X the parabolic system with Neumann's boundary conditions as follows:
(5)∂v∂t(t,y)=x(t)∂2v∂y2(t,y),t>0,y∈(0,1),v(0,y)=v0(y),y∈(0,1),∂v∂y(t,0)=∂v∂y(t,1)=0,t>0.
LetV=ℒ2(0,1) be a separable Hilbert space with the orthonormal basis {en}n∈ℕ, e0=1, en(y)=2cosnπy, where y∈(0,1), n∈ℕ.
We denote thatD(A)={v∈ℒ2(0,1),v(0)=v(1)=0}, and we define the operator
(6)A:D(A)⊂V⟶V,Av=d2vdy2,
which generates a𝒞0-semigroup S, defined by
(7)S(t)v=∑n=0∞e-n2π2t〈v,en〉en,
where 〈·,·〉 denotes the scalar product in V. For every 𝓍∈X, let us define an operator A(x):D(A)⊂V→V, A(x)=x(0)A, which allows us to rewrite system (5) in V as
(8)v˙(t)=A(φ0(t,x))v(t),t>0,v(0)=v0.
The mapping(9)Φ0:ℝ+×X⟶ℬ(V),Φ0(t,x)v=S(∫0tx(s)ds)v
is a classic cocycle over the semiflowφ0, and C0=(φ0,Φ0) is a linear skew-product semiflow strongly continuous on Y. Also, for all v0∈D(A), we have obtained that v(t)=Φ(t,x)x0, t≥0, is a strongly solution of system (8). As C0=(φ0,Φ0) is a skew-product semiflow on Y, then the mapping C:T×Y→Y, C(t,s,x,v)=(φ(t,s,x),Φ(t,s,x)v), where
(10)φ(t,s,x)=φ0(t-s,x),Φ(t,s,x)=Φ0(t-s,x),∀(t,s,x)∈T×X,
is a skew-evolution semiflow onY. Hence, the skew-evolution semiflows generalize the notion of skew-product semiflows.
More directly, ifΦ(t,s,x) is the solution of the Cauchy problem
(11)v′(t)=A(φ(t,s,x))v(t),t>s,v(s)=x,
thenC=(φ,Φ) is a linear skew-evolution semiflow.
Let us recall the definition of a semigroup of linear operators, and let us give an example which shows that this is generating a skew-evolution semiflow.Definition 7.
A mappingS:ℝ+→ℬ(V) is called semigroup of linear operators on V if the following relations hold:
(sgr1)S(0)=I,(sgr2)S(t)S(s)=S(t+s),∀(t,s)∈ℝ+2.Example 8.
One can naturally associate to every semigroup of operators the mappingΦS:T×X→ℬ(V), defined by ΦS(t,s,x)=S(t-s), which is an evolution cocycle on V over evolution semiflows given, for example, by φ(t,s,x)=xt-s (see Example 4).Other examples of skew-evolution semiflows are given in [6]. The asymptotic properties, as well as their characterizations, are given by means of norms of the trajectories or orbits of v, given by t→Φ(t,t0,x)v, which are considered measurable. A particular case of skew-evolution semiflows is given by the following.Definition 9.
A skew-evolution semiflowC=(φ,Φ) is *-strongly measurable if, for every (t,t0,x,v*)∈T×X×V*, the mapping given by s↦∥Φ(t,s,φ(s,t0,x))*v*∥ is measurable on [t0,t].


## 3. On Trichotomy Issues

We intend to give a new approach for the property of trichotomy for skew-evolution semiflows, theω-trichotomy. Some examples and connections with the classic concept of exponential trichotomy are also provided.LetC:T×Y→Y, C(t,s,x,v)=(φ(t,s,x),Φ(t,s,x)v) be a skew-evolution semiflow on Y. We recall that a mapping P:X→ℬ(V) with the property
(12)P(x)2=P(x),∀x∈Xis calledprojections family on V.Definition 10.
A projections familyP:X→ℬ(V) is said to be invariant relative to the skew-evolution semiflow C=(φ,Φ) if
(13)Φ(t,s,x)P(x)=P(φ(t,s,x))Φ(t,s,x),0∀(t,s,x)∈T×X.
The splitting of the state space into three subspaces will be assured by the following.Definition 11.
Three projections families{Pk}k∈{1,2,3} are said to be compatible with a skew-evolution semiflow C=(φ,Φ) if
(
c
1
) each of Pk, k∈{1,2,3} is invariant relative to C;
(
c
2
)  for all x∈X, the projections verify the relations
(14)P1(x)+P2(x)+P3(x)=I,Pi(x)Pj(x)=0,0∀i,j∈{1,2,3}i≠j.
In what follows we introduce the elements which will allow us to introduce a new concept of trichotomy for skew-evolution semiflows. We consider a mappingψ:ℝ+→ℝ+*, and we define
(15)Aψ={ω∈ℝ,supt≥0ψ(t)e-ωt<∞},Bψ={ω∈ℝ,inft≥0ψ(t)e-ωt>0}.Remark 12.
For every functionψ:ℝ+→ℝ+*, the following statements hold:(i)
Aψ≠∅ and Bψ≠∅;(ii)
μ∈Aψ implies the existence of a constant M∈ℝ+ such that
(16)ψ(t)≤Meμt,∀t≥0;(iii)
λ∈Bh implies that there exists a constant m∈ℝ+ such that
(17)ψ(t)≥meλt,∀t≥0.
Let us denote that(18)ω-ψ={infAψ,ifAψ≠∅,∞,ifAψ=∅,ω_ψ={supBψ,ifBψ≠∅,-∞,ifBψ=∅.Definition 13.
A skew-evolution semiflowC=(φ,Φ) is called ω-trichotomic if there exist three projections families {Pk}k∈{1,2,3} compatible with C and some functions ψ,ζ,χ:ℝ+→ℝ+* with the properties
(19)ω-ψ<0,ω_ζ>0,ω¯χ<∞,ω_χ<0,
such that(t1)
∥Φ(t,t0,x)P1(x)v∥≤ψ(t-t0)∥P1(x)v∥;(t2)
ζ(t-t0)∥P2(x)v∥≤∥Φ(t,t0,x)P2(x)v∥;(t3)
∥Φ(t,t0,x)P3(x)v∥≤χ(t-t0)∥P3(x)v∥ and χ(t-t0)∥Φ(t,t0,x)P3(x)v∥≥∥P3(x)v∥,for all(t,t0)∈T and all (x,v)∈Y.Example 14.
Letf:ℝ+→(0,∞) be a decreasing function with the property that there exists limt→∞f(t)=l>0. Let us denote that λ>f(0). Let us consider the Banach space V=ℝ3 with the norm ∥(v1,v2,v3)∥=|v1|+|v2|+|v3|, v=(v1,v2,v3)∈V. The mapping
(20)Φ:T×X⟶ℬ(V),Φ(t,t0,x)v=(e-λ(t-t0)+∫t0tx(τ-t0)dτv1,e∫t0t‍x(τ-t0)dτv2,0e-(t-t0)x(0)+∫t0tx(τ-t0)dτv3),
where t≥t0≥0, (x,v)∈Y, is an evolution cocycle over the evolution semiflow given in Example 4. We define the projections families P1,P2,P3:X→ℬ(ℝ3) by P1(x)v=(v1,0,0), P2(x)v=(0,v2,0), P3(x)v=(0,0,v3), for all x∈X and all v=(v1,v2,v3)∈ℝ3. The following inequalities
(21)∥Φ(t,t0,x)P1(x)v∥≤e[-λ+x(0)](t-s)∥Φ(s,t0,x)P1(x)v∥,∥Φ(t,t0,x)P2(x)v∥≥el(t-s)∥Φ(s,t0,x)P2(x)v∥,∥Φ(t,t0,x)P3(x)v∥≤e-x(0)(t-s)∥Φ(s,t0,x)P3(x)v∥,∥Φ(t,t0,x)P3(x)v∥≥e-x(0)(t-s)∥Φ(s,t0,x)P3(x)v∥
hold for all(t,s), (s,t0)∈T and all (x,v)∈Y. The mappings ψ, ζ, χ:ℝ+→ℝ+*, defined by
(22)ψ(u)=e[-λ+x(0)]u,ζ(u)=elu,χ(u)=e-x(0)u
satisfy relations (19). For s=t0 and according to Definition 2, (et1) we obtain relations (i)–(iii) in Definition 13. Hence, C is ω-trichotomic.Remark 15.
ForP3=0, the property of ω-dichotomy is obtained. If we consider P2=P3=0, we obtain the property of ω-stability and for P1=P3=0, the property of ω-instability.
In what follows, ifPk is a given projections family, we will denote that
(23)Φk(t,s,x)=Φ(t,s,x)Pk(x),
for every(t,s),(s,t0)∈T and x∈X. We remark that the following relations hold:(i)
Φk(t,t,x)=Pk(x), ∀(t,x)∈ℝ+×X;(ii)
Φk(t,s,φ(s,t0,x))Φk(s,t0,x)=Φk(t,t0,x), ∀(t,s), (s,t0)∈T, ∀x∈X.Proposition 16.
A skew-evolution semiflowC=(φ,Φ) is ω-trichotomic if and only if there exist some constants N1, N2, N3≥1, ν1, ν2, ν3>0 and three projections families {Pk}k∈{1,2,3} compatible with C such that
(et1)eν1(t-t0)∥Φ1(t,t0,x)v∥≤N1∥P1(x)v∥,(et2)eν2(t-t0)∥P2(x)v∥≤N2∥Φ2(t,t0,x)v∥,(et3)∥P3(x)v∥≤N3eν3(t-t0)∥Φ3(t,t0,x)v∥≤N32e2ν3(t-t0)∥P3(x)v∥,
for all(t,t0)∈T and all (x,v)∈Y.Proof.
Necessity. Let C be ω-trichotomic. Then there exist three projections families compatible with C and three functions ψ,ζ,χ:ℝ+→ℝ+* with the properties
(24)ω-ψ<0,ω_ζ>0,ω-χ<∞,ω_χ<0,
such that relations (i)–(iii) of Definition13 are verified. For μ∈Aψ, there exists a constant M∈ℝ+ such that ψ(t)≤Meμt, for all  t≥0, and for λ∈Bψ there exists a constant m∈ℝ+ with the property ψ(t)≥meλt, for all t≥0.
Asω-ψ<0, there exist N1≥1 and ν1>0, such that ψ(t)≤N1e-ν1t, for all t≥0. Hence, relation (et1) is obtained. As ω_ζ>0, it follows that there exist N2≥1 and ν2>0, such that ζ(t)≥N2eν2t, for all t≥0, which implies (et2). From ω-χ<∞ and ω_χ<0, it follows that there exist N3≥1, ν3>0 with the property
(25)1N3e-ν3t≤χ(t)≤N3eν3t,∀t≥0.
Hence, relation(et3) is satisfied.
Sufficiency. Let us assume that there exist three projections families compatible with C and N1, N2, N3≥1, ν1, ν2, ν3>0 such that relations (et1)–(et3) hold. Let us define the mappings ψ, ζ, χ:ℝ+→ℝ+* by
(26)ψ(t)=N1e-ν1t,ζ(t)=N2-1eν2t,χ(t)=N3eν3t.
We obtain the relations(27)ω-ψ<0,ω_ζ>0,ω-χ<∞,ω_χ<0.
Hence, relations (i)–(iii) of Definition13 are verified, and, therefore, C is ω-trichotomic, which ends the proof.Remark 17.
Proposition16 is in fact the classic definition of exponential trichotomy. On the other hand, in Definition 13, the exponentials are not implied.


## 4. Main Results

We obtain a characterization for the property of trichotomy, by means of the shifted skew-evolution semiflow.Theorem 18.
A skew-evolution semiflowC=(φ,Φ) is ω-trichotomic if and only if there exist three projections families {Pk}k∈{1,2,3} compatible with C and(i)
the evolution cocycleΦ1 is exponentially stable;(ii)
the evolution cocycleΦ2 is exponentially instable;(iii)
there exists a constantλ>0 such that the λ-shifted evolution cocycle Φλ3 is exponentially stable and the -λ-shifted evolution cocycle Φ-λ3 is exponentially instable.Proof.
Necessity. Statements (i) and (ii) are obtained immediately from the necessity of Proposition 16. According to (et3), there exist N3≥1 and ν3>0 such that
(28)∥Φ3(t,t0,x)v∥≤N3eν3(t-t0)∥P3(x)v∥,0∀(t,t0)∈T,∀(x,v)∈Y.
Let us consider thatλ=2ν3>0. We obtain successively
(29)∥Φλ3(t,t0,x)v∥=e-λ(t-t0)∥Φ3(t,t0,x)v∥≤N3e-λ(t-t0)eν3(t-t0)∥P3(x)v∥=N3e-ν3(t-t0)∥P3(x)v∥,
for all(t,t0)∈T and all (x,v)∈Y, which shows that Φλ3 is exponentially stable. Also, we have
(30)e-ν3(t-t0)∥P3(x)v∥≤N3∥Φ3(t,t0,x)v∥,0∀(t,t0)∈T,∀(x,v)∈Y.
If we consider thatλ=2ν3>0, we obtain, according also to Definition 2  (ec2),
(31)∥Φ-λ3(s,t0,x)v∥=eλ(s-t0)∥Φ3(s,t0,x)v∥≤N3eν3(t-s)eλ(s-t0)∥Φ3(t,t0,x)v∥=N3e-ν3(t-s)∥Φ-λ3(t,t0,x)v∥,
for all(t,s),(s,t0)∈T and all (x,v)∈Y, which proves that Φ-λ3 is exponentially instable.
Sufficiency. Relations (i) and (ii) in Definition 13 are obtained from the sufficiency of Proposition 16. As there exists a constant λ>0 such that the skew-evolution semiflow Φλ3 is exponentially stable, then there exists some constants N≥1 and ν>0 such that
(32)∥Φλ3(t,t0,x)v∥≤Ne-ν(t-t0)∥P3(x)v∥,∀t≥t0≥0,∀(x,v)∈Y.
Further, we obtain(33)∥Φ3(t,t0,x)v∥=eλ(t-t0)∥Φλ3(t,t0,x)v∥≤Neλ(t-t0)e-ν(t-s)∥Φλ3(s,t0,x)v∥=Neλ(t-s)e-ν(t-s)∥Φ3(s,t0,x)v∥,
for all(t,s),(s,t0)∈T and all (x,v)∈Y. Denoting
(34)ν3={λ-ν,ifλ>ν,1,ifλ≤ν,
the first relation in(et3) is obtained.
If there exists a constantλ>0 such that the -λ-shifted skew-evolution semiflow C-λ is exponentially instable, then there exist some constants N≥1 and ν>0 such that
(35)∥Φ(s,t0,x)v∥=e-λ(s-t0)∥Φ-λ(s,t0,x)v∥≤Ne-ν(t-s)e-λ(s-t0)∥Φ-λ(t,t0,x)v∥=Ne-ν(t-s)eλ(t-s)∥Φ(t,t0,x)v∥,
for all(t,s),(s,t0)∈T and all (x,v)∈Y. If we consider ν3 defined as in (34), the second relation in (et3) is obtained. Hence, also (iii) in Definition 13 is true and C is ω-trichotomic.Another characterization for the property ofω-trichotomy is given relative to the dual space V* of the Banach space V. To this aim, let us consider three projections families {Pk}k∈{1,2,3} compatible with C such that the evolution cocycle Φ1 has exponential growth and the evolution cocycle Φ2 has exponential decay.Theorem 19.
A*-strongly measurable skew-evolution semiflow C=(φ,Φ) is ω-trichotomic if the and only if the following statements hold:(i)
there existsN~≥1 such that
(36)∫t0t∥Φ1(t,s,φ(s,t0,x))*v*∥pds≤N~∥P1(x)v*∥p,
for all p>1, all (t,t0)∈T, and all (x,v*)∈X×V*;(ii)
there existsN¯≥1 such that
(37)(∫t0t‍∥Φ2(s,t0,x)v∥pds)1/p≤N¯∥Φ2(t,t0,x)v∥,
for all p>1, all (t,t0)∈T, and all (x,v)∈Y;(iii)
there existα<0 and M~≥1 such that
(38)∫t0∞eα(τ-t0)∥Φ3(τ,t0,x)v∥dτ≤M~∥P3(x)v∥,
for all t0∈ℝ+ and all (x,v)∈Y and there exist β<0 and M-≥1 such that
(39)∫t0teβ(t-τ)∥Φ3(τ,t0,x)v∥dτ≤M¯∥Φ3(t,t0,x)v∥,
for all (t,t0)∈T and all (x,v)∈Y.Proof.
Necessity. (i) As C is ω-trichotomic, according to Definition 13, there exists a mapping ψ:ℝ+→ℝ+*, with the property ω¯ψ<0, such that
(40)∥Φ(t,t0,x)P1(x)v∥≤ψ(t-t0)∥P1(x)v∥,0∀(t,t0)∈T,(x,v)∈Y.
An equivalent relation is obtained, if we consider Definition2  (ec2), given by
(41)∥Φ1(t,t0,x)v∥=∥Φ1(t,s,φ(s,t0,x))Φ1(s,t0,x)v∥≤ψ(t-s)∥Φ1(s,t0,x)v∥,
for all(t,t0)∈T and all (x,v)∈Y. According to the properties of function ψ, there exist N≥1 and ν>0 such that the following relations hold:
(42)∫t0t∥Φ1(t,s,φ(s,t0,x))*v*∥pds≤∫t0tψp(t-s)∥Φ1(s,s,φ(s,t0,x))*v*∥pds≤Np∫t0t‍e-pν(t-s)∥P1(x)v*∥pds≤N~∥P1(x)v*∥p,
for all(t,t0)∈T and all (x,v*)∈X×V*, where we have denoted that N~=Np/νp.
(ii) According to Definition13, there exists a mapping ζ:ℝ+→ℝ+*, with the property ω_ζ<0, such that
(43)ζ(t-t0)∥P2(x)v∥≤∥Φ2(t,t0,x)v∥,0∀(t,t0)∈T,(x,v)∈Y.
The property of functionζ assures the existence of some constants N≥1 and ν>0, such that, for all p>0, the following relations hold:
(44)(∫t0t‍∥Φ2(s,t0,x)v∥pds)1/p≤N∥Φ2(t,t0,x)v∥×(∫t0t‍e-νp(t-s)ds)1/p≤N¯∥Φ2(t,t0,x)v∥,
for all(t,t0)∈T and all (x,v)∈Y, where we have denoted that N-=N/(νp)1/p.
(iii) Both relations are obtained by a similar proof as in (i) and (ii), according to  Theorem18.
Sufficiency. (i) As the evolution cocycle Φ1 has exponential growth, there exist M≥1 and ρ>0 such that
(45)∥Φ1(t,t0,x)v∥≤Meρ(t-s)∥Φ1(s,t0,x)v∥,0∀(t,s),(s,t0)∈T,(x,v)∈Y.
Lett≥t0+1. We consider that
(46)∫t0t0+1e-ρ(s-t0)ds=∫01e-ρτdτ=1-e-ρρ=c>0.
For everyp>1 there exists q>1 such that 1/p+1/q=1. We obtain
(47)c|〈v*,Φ1(t,t0,x)v〉|≤∫t0t0+1e-ρ(s-t0)|〈v*,Φ1(t,t0,x)v〉|ds≤∫t0t0+1e-ρ(s-t0)∥Φ1(t,s,φ(s,t0,x))*v*∥×∥Φ1(s,t0,x)v∥ds≤M∥v∥∫t0t0+1∥Φ1(t,s,φ(s,t0,x))*v*∥ds≤(∫t0t0+1∥Φ1(t,s,φ(s,t0,x))*v*∥pds)1/p×(∫t0t0+1Mq∥P1(x)v∥qds)1/q≤M∥v∥(∫t0t∥Φ1(t,s,φ(s,t0,x))*v*∥pds)1/p≤MN~∥P1(x)v∥∥P1(x)v*∥.
Hence, we have that(48)∥Φ1(t,s,φ(s,t0,x))v∥≤MN~c∥P1(x)v∥,0∀t≥t0+1,∀(x,v)∈Y.
Fort∈[t0,t0+1] we obtain
(49)∥Φ1(t,s,φ(s,t0,x))v∥≤Meρ∥P1(x)v∥,∀(x,v)∈Y,
which implies that(50)∥Φ1(t,s,φ(s,t0,x))v∥≤M1∥P1(x)v∥,0∀(t,t0)∈T,(x,v)∈Y,
where we have denoted
(51)M1=M(eρ+N~c).
Further, the following relations hold:(52)|〈v*,Φ1(t,t0,x)v〉|=|〈v*,Φ1(t,s,φ(s,t0,x))Φ1(s,t0,x)v〉|=|〈Φ1(t,s,φ(s,t0,x))*v*,Φ1(s,t0,x)v〉|≤M1∥P1(x)v∥∥Φ1(t,s,φ(s,t0,x))*v*∥.
By integrating on[t0,t], it follows that
(53)(t-t0)|〈v*,Φ1(t,t0,x)v〉|≤M1∥P1(x)v∥∫t0t∥Φ1(t,s,φ(s,t0,x))*v*∥ds≤M1N~∥P1(x)v∥∥P1(x)v*∥,
and further(54)(t-t0)∥Φ1(t,t0,x)v∥≤M1N~∥P1(x)v∥,
which implies that(55)∥Φ1(t,t0,x)v∥≤ψ(t-t0)∥P1(x)v∥,∀(t,t0)∈T,(x,v)∈Y,
where we have considered ψ(u)=M1(N~+1)/u, which satisfies ω-ψ<0. Statement (t1) of Definition 13 is hence obtained.
(ii) Let(t,t0)∈T. Let us denote that
(56)F(t)=∫t0t∥Φ2(s,t0,x)v∥pds.
According to the fact thatF(t)≤N¯pF′(t), for all t≥0, we obtain, for all (x,v)∈Y,
(57)F(t0+1)e(t-t0-1)/N¯p≤F(t)≤N¯p∥Φ2(t,t0,x)v∥p.
As the evolution cocycleΦ2 has exponential decay, there exist some constants M≥1 and ρ>0 such that
(58)∥Φ2(s,t0,x)v∥≤Meρ(t-s)∥Φ2(t,t0,x)v∥,0∀(t,s),(s,t0)∈T,(x,v)∈Y.
Let s∈[t0,t0+1). We obtain
(59)∥P2(x)v∥p≤Mpepρ(t-s)∫t0t0+1∥Φ2(s,t0,x)v∥pds
and further(60)M-pe-pρ(t-s)∥P2(x)v∥p≤F(t0+1).
Fort≥t0+1, we obtain
(61)1Mpeρpe-1/N¯pe(t-t0)/N¯p∥v∥p≤N¯p∥Φ2(t,t0,x)v∥p,
and, ift∈[t0,t0+1],(62)∥P2(x)v∥≤Meρe1/pN-pe(-1/pN-p)(t-t0)∥Φ2(t,t0,x)v∥,
for all(x,v)∈Y. Hence,
(63)ζ(t-t0)∥P2(x)v∥≤∥Φ2(t,t0,x)v∥,0∀(t,t0)∈T,(x,v)∈Y,
where we have considered ζ:ℝ+→ℝ+* defined by
(64)ζ(u)=1N¯Meρe-1/pN-pe(1/pN-p)u,
with the propertyω_ζ>0. Statement (t2) of Definition 13 is proved.
(iii) A similar proof for theα-shifted evolution cocycle Φα3 in (i) and for the β-shifted evolution cocycle Φβ3 in (ii) leads to the existence of two functions χ1,χ2:ℝ+→ℝ+*, with the properties ω¯χ1<0 and ω_χ2>0, such that
(65)∥Φα3(t,t0,x)v∥≤χ1(t-t0)∥P3(x)v∥,0∀(t,t0)∈T,(x,v)∈Y,χ2(t-t0)∥P3(x)v∥≤∥Φβ3(t,t0,x)v∥,∀(t,t0)∈T,(x,v)∈Y.
If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent with(66)∥Φ3(t,t0,x)v∥≤e-α(t-t0)χ1(t-t0)∥P3(x)v∥,00∀(t,t0)∈T,(x,v)∈Y,e-β(t-t0)χ2(t-t0)∥P3(x)v∥≤∥Φ3(t,t0,x)v∥,00∀(t,t0)∈T,(x,v)∈Y.
We defineχ~,χ¯:ℝ+→ℝ+* by χ~(u)=e-αuχ1(u) and χ¯(u)=e-βuχ2(u), which have the properties ω¯χ~<∞ and ω_χ¯<0. Hence, property (t3) of Definition 13 is obtained.Remark 20.
The property described by relation (36) is also called *-strong integral stability for a skew-evolution semiflow. Relation (36) is a characterization of Barbashin type in the strong topology, and relations (37), (38), and (39) are characterizations of Datko type for the asymptotic properties of skew-evolution semiflows involved in the definition of ω-trichotomy.



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*Source: 793813-2013-09-29.xml*