Trichotomy for Dynamical Systems in Banach Spaces

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 of 𝜔-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.

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)∈ℝ²₊∶𝑡≥𝑡₀}. 𝐼 is the identity operator.

Definition 1. A mapping 𝜑∶𝑇×𝑋→𝑋 is called evolution semiflow on 𝑋 if the following properties are satisfied: 𝜑(𝑡,𝑡,𝑥)=𝑥,∀(𝑡,𝑥)∈ℝ₊×𝑋,es₁𝜑𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥=𝜑𝑡,𝑡₀,𝑥,∀(𝑡,𝑠),𝑠,𝑡₀∈𝑇,∀𝑥∈𝑋.es₂

Definition 2. A mapping Φ∶𝑇×𝑋→ℬ(𝑉) is called evolution cocycle over an evolution semiflow 𝜑 if it satisfies the following properties: Φ(𝑡,𝑡,𝑥)=𝐼,∀𝑡≥0,∀𝑥∈𝑋,ec₁Φ𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥Φ𝑠,𝑡₀,𝑥=Φ𝑡,𝑡₀,𝑥,∀(𝑡,𝑠),𝑠,𝑡₀∈𝑇,∀𝑥∈𝑋.ec₂

Definition 3. The mapping 𝐶∶𝑇×𝑌→𝑌 defined by 𝐶(𝑡,𝑠,𝑥,𝑣)=𝜑(𝑡,𝑠,𝑥),Φ(𝑡,𝑠,𝑥)𝑣,(1) where Φ is an evolution cocycle over an evolution semiflow 𝜑, is called skew-evolution semiflow on 𝑌.

Example 4. Let 𝑓∶ℝ₊→ℝ^∗₊ be a decreasing function with the property that there exists lim_𝑡→∞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 𝑑𝑥,𝑦=^∞_𝑛=112^𝑛𝑑_𝑛𝑥,𝑦1+𝑑_𝑛,𝑥,𝑦where𝑑_𝑛𝑥,𝑦=sup_{[]𝑡∈0,𝑛}𝑥(𝑡)−𝑦(𝑡).(2)
If 𝑥∈𝒞, then, for all 𝑡∈ℝ₊, we denote that 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠), 𝑥_𝑡∈𝒞. Let 𝑋 be the closure in 𝒞 of the set {𝑓_𝑡,𝑡∈ℝ₊}. It follows that (𝑋,𝑑) is a metric space, and the mapping 𝜑∶Δ×𝑋→𝑋, 𝜑(𝑡,𝑠,𝑥)=𝓍_𝑡−𝑠 is an evolution semiflow on 𝑋.
We consider that 𝑉=ℝ², with the norm ‖𝑣‖=|𝑣₁|+|𝑣₂|, 𝑣=(𝑣₁,𝑣₂)∈𝑉. If 𝑢∶ℝ₊→ℝ^∗₊, then the mapping Φ_𝑢∶Δ×𝑋→ℬ(𝑉) defined by Φ_𝑢()𝑢(𝑠)𝑡,𝑠,𝓍𝑣=𝑢(𝑡)𝑒^{−∫𝑡𝑠𝓍(𝜏−𝑠)𝑑𝜏}𝑣₁,𝑢(𝑡)𝑢(𝑠)𝑒^{∫𝑡𝑠𝓍(𝜏−𝑠)𝑑𝜏}𝑣₂(3) is an evolution cocycle over 𝜑, and 𝐶=(𝜑,Φ_𝑢) 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 semiflow 𝐶=(𝜑,Φ) and a parameter 𝜆∈ℝ. We define the mapping Φ_𝜆∶𝑇×𝑋⟶ℬ(𝑉),Φ_𝜆𝑡,𝑡₀,𝑥=𝑒^{−𝜆(𝑡−𝑡0)}Φ𝑡,𝑡₀,𝑥.(4)
We observe that 𝐶_𝜆=(𝜑,Φ_𝜆) is also a skew-evolution semiflow, called 𝜆-shifted skew-evolution semiflow on 𝑌. We will call Φ_𝜆 the 𝜆-shifted evolution cocycle.

Example 6. Let 𝑋 be the metric space defined in the first Example. We define the mapping 𝜑₀∶ℝ₊×𝑋→𝑋, 𝜑₀(𝑡,𝑥)=𝑥_𝑡, where 𝑥_𝑡(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: 𝜕𝑣(𝑡)𝜕𝜕𝑡𝑡,𝑦=𝑥²𝑣𝜕𝑦²(),𝑣𝑡,𝑦,𝑡>0,𝑦∈0,10,𝑦=𝑣₀𝑦,𝑦∈(0,1),𝜕𝑣()=𝜕𝑦𝑡,0𝜕𝑣()𝜕𝑦𝑡,1=0,𝑡>0.(5)
Let 𝑉=ℒ²(0,1) be a separable Hilbert space with the orthonormal basis {𝑒_𝑛}_𝑛∈ℕ, 𝑒₀=1, 𝑒_𝑛(𝑦)=√2cos𝑛𝜋𝑦, where 𝑦∈(0,1), 𝑛∈ℕ.
We denote that 𝐷(𝐴)={𝑣∈ℒ²(0,1),𝑣(0)=𝑣(1)=0}, and we define the operator (𝐴)𝑑𝐴∶𝐷⊂𝑉⟶𝑉,𝐴𝑣=²𝑣𝑑𝑦²,(6)
which generates a 𝒞₀-semigroup 𝑆, defined by 𝑆(𝑡)𝑣=^∞_𝑛=0𝑒^{−𝑛2𝜋2𝑡}𝑣,𝑒_𝑛𝑒_𝑛,(7) 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 𝑣̇(𝑡)=𝐴𝜑₀()𝑣(𝑡)𝑣(0)𝑡,𝑥,𝑡>0,=𝑣₀.(8)
The mapping Φ₀∶ℝ₊×𝑋⟶ℬ(𝑉),Φ₀(𝑡,𝑥)𝑣=𝑆^𝑡₀𝑥(𝑠)𝑑𝑠𝑣(9)
is a classic cocycle over the semiflow 𝜑₀, 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 𝜑(𝑡,𝑠,𝑥)=𝜑₀(𝑡−𝑠,𝑥),Φ(𝑡,𝑠,𝑥)=Φ₀(),∀()𝑡−𝑠,𝑥𝑡,𝑠,𝑥∈𝑇×𝑋,(10)
is a skew-evolution semiflow on 𝑌. Hence, the skew-evolution semiflows generalize the notion of skew-product semiflows.
More directly, if Φ(𝑡,𝑠,𝑥) is the solution of the Cauchy problem 𝑣^′(𝑡)𝜑()𝑣(𝑡)𝑣(𝑠)=𝐴𝑡,𝑠,𝑥,𝑡>𝑠,=𝑥,(11)
then 𝐶=(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 mapping 𝑆∶ℝ₊→ℬ(𝑉) is called semigroup of linear operators on 𝑉 if the following relations hold: 𝑆(0)=𝐼,sgr₁𝑆(𝑡)𝑆(𝑠)=𝑆(𝑡+𝑠),∀(𝑡,𝑠)∈ℝ²₊.sgr₂

Example 8. One can naturally associate to every semigroup of operators the mapping Φ_𝑆∶𝑇×𝑋→ℬ(𝑉), defined by Φ_𝑆(𝑡,𝑠,𝑥)=𝑆(𝑡−𝑠), which is an evolution cocycle on 𝑉 over evolution semiflows given, for example, by 𝜑(𝑡,𝑠,𝑥)=𝑥_𝑡−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 semiflow 𝐶=(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, the 𝜔-trichotomy. Some examples and connections with the classic concept of exponential trichotomy are also provided.

Let 𝐶∶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 𝑃(𝑥)²=𝑃(𝑥),∀𝑥∈𝑋(12)

is called projections family on 𝑉.

Definition 10. A projections family 𝑃∶𝑋→ℬ(𝑉) is said to be invariant relative to the skew-evolution semiflow 𝐶=(𝜑,Φ) if Φ()𝑃(𝑥)𝜑()Φ(),∀()𝑡,𝑠,𝑥=𝑃𝑡,𝑠,𝑥𝑡,𝑠,𝑥𝑡,𝑠,𝑥∈𝑇×𝑋.(13)
The splitting of the state space into three subspaces will be assured by the following.

Definition 11. Three projections families {𝑃_𝑘}_{𝑘∈{1,2,3}} are said to be compatible with a skew-evolution semiflow 𝐶=(𝜑,Φ) if
(c₁) each of 𝑃_𝑘, 𝑘∈{1,2,3} is invariant relative to 𝐶;
(c₂)  for all 𝑥∈𝑋, the projections verify the relations 𝑃₁(𝑥)+𝑃₂(𝑥)+𝑃₃(𝑥)=𝐼,𝑃_𝑖(𝑥)𝑃_𝑗(𝑥){}=0,∀𝑖,𝑗∈1,2,3𝑖≠𝑗.(14)
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 mapping 𝜓∶ℝ₊→ℝ^∗₊, and we define 𝐴_𝜓=𝜔∈ℝ,sup_𝑡≥0𝜓(𝑡)𝑒^−𝜔𝑡,𝐵<∞_𝜓=𝜔∈ℝ,inf_𝑡≥0𝜓(𝑡)𝑒^−𝜔𝑡>0.(15)

Remark 12. For every function 𝜓∶ℝ₊→ℝ^∗₊, the following statements hold:(i)𝐴_𝜓≠∅ and 𝐵_𝜓≠∅;(ii)𝜇∈𝐴_𝜓 implies the existence of a constant 𝑀∈ℝ₊ such that 𝜓(𝑡)≤𝑀𝑒^𝜇𝑡,∀𝑡≥0;(16)(iii)𝜆∈𝐵_ℎ implies that there exists a constant 𝑚∈ℝ₊ such that 𝜓(𝑡)≥𝑚𝑒^𝜆𝑡,∀𝑡≥0.(17)
Let us denote that 𝜔_𝜓=inf𝐴_𝜓,if𝐴_𝜓≠∅,∞,if𝐴_𝜓𝜔=∅,_𝜓=sup𝐵_𝜓,if𝐵_𝜓≠∅,−∞,if𝐵_𝜓=∅.(18)

Definition 13. A skew-evolution semiflow 𝐶=(𝜑,Φ) is called 𝜔-trichotomic if there exist three projections families {𝑃_𝑘}_{𝑘∈{1,2,3}} compatible with 𝐶 and some functions 𝜓,𝜁,𝜒∶ℝ₊→ℝ^∗₊ with the properties 𝜔_𝜓<0,𝜔_𝜁>0,𝜔_𝜒<∞,𝜔_𝜒<0,(19)
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. Let 𝑓∶ℝ₊→(0,∞) be a decreasing function with the property that there exists lim_𝑡→∞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 (𝑉),ΦΦ∶𝑇×𝑋⟶ℬ𝑡,𝑡₀𝑣=𝑒,𝑥_{−𝜆(𝑡−𝑡0)+∫𝑡𝑡0𝑥(𝜏−𝑡0)𝑑𝜏}𝑣₁,𝑒^{∫𝑡𝑡0𝑥(𝜏−𝑡0)𝑑𝜏}𝑣₂,𝑒^{−(𝑡−𝑡0)𝑥(0)+∫𝑡𝑡0𝑥(𝜏−𝑡0)𝑑𝜏}𝑣₃,(20) 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 Φ𝑡,𝑡₀,𝑥𝑃₁(𝑥)𝑣≤𝑒^{[−𝜆+𝑥(0)](𝑡−𝑠)}Φ𝑠,𝑡₀,𝑥𝑃₁(𝑥)𝑣,Φ𝑡,𝑡₀,𝑥𝑃₂(𝑥)𝑣≥𝑒^{𝑙(𝑡−𝑠)}Φ𝑠,𝑡₀,𝑥𝑃₂(𝑥)𝑣,Φ𝑡,𝑡₀,𝑥𝑃₃(𝑥)𝑣≤𝑒^{−𝑥(0)(𝑡−𝑠)}Φ𝑠,𝑡₀,𝑥𝑃₃(𝑥)𝑣,Φ𝑡,𝑡₀,𝑥𝑃₃(𝑥)𝑣≥𝑒^{−𝑥(0)(𝑡−𝑠)}Φ𝑠,𝑡₀,𝑥𝑃₃(𝑥)𝑣(21)
hold for all (𝑡,𝑠), (𝑠,𝑡₀)∈𝑇 and all (𝑥,𝑣)∈𝑌. The mappings 𝜓, 𝜁, 𝜒∶ℝ₊→ℝ^∗₊, defined by 𝜓(𝑢)=𝑒^{[−𝜆+𝑥(0)]𝑢},𝜁(𝑢)=𝑒^𝑙𝑢,𝜒(𝑢)=𝑒^{−𝑥(0)𝑢}(22)
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. For 𝑃₃=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, if 𝑃_𝑘 is a given projections family, we will denote that Φ_𝑘(𝑡,𝑠,𝑥)=Φ(𝑡,𝑠,𝑥)𝑃_𝑘(𝑥),(23)
for every (𝑡,𝑠),(𝑠,𝑡₀)∈𝑇 and 𝑥∈𝑋. We remark that the following relations hold:(i)Φ_𝑘(𝑡,𝑡,𝑥)=𝑃_𝑘(𝑥), ∀(𝑡,𝑥)∈ℝ₊×𝑋;(ii)Φ_𝑘(𝑡,𝑠,𝜑(𝑠,𝑡₀,𝑥))Φ_𝑘(𝑠,𝑡₀,𝑥)=Φ_𝑘(𝑡,𝑡₀,𝑥), ∀(𝑡,𝑠), (𝑠,𝑡₀)∈𝑇, ∀𝑥∈𝑋.

Proposition 16. A skew-evolution semiflow 𝐶=(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 {𝑃_𝑘}_{𝑘∈{1,2,3}} compatible with 𝐶 such that 𝑒^{𝜈1(𝑡−𝑡0)}Φ₁𝑡,𝑡₀,𝑥𝑣≤𝑁₁𝑃₁(𝑥)𝑣,et₁𝑒^{𝜈2(𝑡−𝑡0)}𝑃₂(𝑥)𝑣≤𝑁₂Φ₂𝑡,𝑡₀,𝑥𝑣,et₂𝑃₃(𝑥)𝑣≤𝑁₃𝑒^{𝜈3(𝑡−𝑡0)}Φ₃𝑡,𝑡₀,𝑥𝑣≤𝑁²₃𝑒^{2𝜈3(𝑡−𝑡0)}𝑃₃(𝑥)𝑣,et₃
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 𝜔_𝜓<0,𝜔_𝜁>0,𝜔_𝜒<∞,𝜔_𝜒<0,(24)
such that relations (i)–(iii) of Definition 13 are verified. For 𝜇∈𝐴_𝜓, there exists a constant 𝑀∈ℝ₊ such that 𝜓(𝑡)≤𝑀𝑒^𝜇𝑡, for all  𝑡≥0, and for 𝜆∈𝐵_𝜓 there exists a constant 𝑚∈ℝ₊ with the property 𝜓(𝑡)≥𝑚𝑒^𝜆𝑡, for all 𝑡≥0.
As 𝜔_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)≥𝑁₂𝑒^𝜈2𝑡, 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 1𝑁₃𝑒^−𝜈3𝑡≤𝜒(𝑡)≤𝑁₃𝑒^𝜈3𝑡,∀𝑡≥0.(25)
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 𝜓(𝑡)=𝑁₁𝑒^−𝜈1𝑡,𝜁(𝑡)=𝑁₂⁻¹𝑒^𝜈2𝑡,𝜒(𝑡)=𝑁₃𝑒^𝜈3𝑡.(26)
We obtain the relations 𝜔_𝜓<0,𝜔_𝜁>0,𝜔_𝜒<∞,𝜔_𝜒<0.(27)
Hence, relations (i)–(iii) of Definition 13 are verified, and, therefore, 𝐶 is 𝜔-trichotomic, which ends the proof.

Remark 17. Proposition 16 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 semiflow 𝐶=(𝜑,Φ) is 𝜔-trichotomic if and only if there exist three projections families {𝑃_𝑘}_{𝑘∈{1,2,3}} compatible with 𝐶 and(i)the evolution cocycle Φ₁ is exponentially stable;(ii)the evolution cocycle Φ₂ is exponentially instable;(iii)there exists a constant 𝜆>0 such that the 𝜆-shifted evolution cocycle Φ³_𝜆 is exponentially stable and the −𝜆-shifted evolution cocycle Φ³_−𝜆 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 Φ₃𝑡,𝑡₀,𝑥𝑣≤𝑁₃𝑒^{𝜈3(𝑡−𝑡0)}𝑃₃(𝑥)𝑣,∀𝑡,𝑡₀∈𝑇,∀(𝑥,𝑣)∈𝑌.(28)
Let us consider that 𝜆=2𝜈₃>0. We obtain successively Φ³_𝜆𝑡,𝑡₀,𝑥𝑣=𝑒^{−𝜆(𝑡−𝑡0)}Φ₃𝑡,𝑡₀,𝑥𝑣≤𝑁₃𝑒^{−𝜆(𝑡−𝑡0)}𝑒^{𝜈3(𝑡−𝑡0)}𝑃₃(𝑥)𝑣=𝑁₃𝑒^{−𝜈3(𝑡−𝑡0)}𝑃₃(𝑥)𝑣,(29)
for all (𝑡,𝑡₀)∈𝑇 and all (𝑥,𝑣)∈𝑌, which shows that Φ³_𝜆 is exponentially stable. Also, we have 𝑒^{−𝜈3(𝑡−𝑡0)}𝑃₃(𝑥)𝑣≤𝑁₃Φ₃𝑡,𝑡₀𝑣,∀,𝑥𝑡,𝑡₀∈𝑇,∀(𝑥,𝑣)∈𝑌.(30)
If we consider that 𝜆=2𝜈₃>0, we obtain, according also to Definition 2  ec₂, Φ³_−𝜆𝑠,𝑡₀,𝑥𝑣=𝑒^{𝜆(𝑠−𝑡0)}Φ₃𝑠,𝑡₀,𝑥𝑣≤𝑁₃𝑒^{𝜈3(𝑡−𝑠)}𝑒^{𝜆(𝑠−𝑡0)}Φ₃𝑡,𝑡₀,𝑥𝑣=𝑁₃𝑒^{−𝜈3(𝑡−𝑠)}Φ³_−𝜆𝑡,𝑡₀,𝑥𝑣,(31)
for all (𝑡,𝑠),(𝑠,𝑡₀)∈𝑇 and all (𝑥,𝑣)∈𝑌, which proves that Φ³_−𝜆 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 Φ³_𝜆 is exponentially stable, then there exists some constants 𝑁≥1 and 𝜈>0 such that Φ³_𝜆𝑡,𝑡₀,𝑥𝑣≤𝑁𝑒^{−𝜈(𝑡−𝑡0)}𝑃₃(𝑥)𝑣,∀𝑡≥𝑡₀≥0,∀(𝑥,𝑣)∈𝑌.(32)
Further, we obtain Φ₃𝑡,𝑡₀,𝑥𝑣=𝑒^{𝜆(𝑡−𝑡0)}Φ³_𝜆𝑡,𝑡₀,𝑥𝑣≤𝑁𝑒^{𝜆(𝑡−𝑡0)}𝑒^{−𝜈(𝑡−𝑠)}Φ³_𝜆𝑠,𝑡₀,𝑥𝑣=𝑁𝑒^{𝜆(𝑡−𝑠)}𝑒^{−𝜈(𝑡−𝑠)}Φ₃𝑠,𝑡₀,𝑥𝑣,(33)
for all (𝑡,𝑠),(𝑠,𝑡₀)∈𝑇 and all (𝑥,𝑣)∈𝑌. Denoting 𝜈₃=𝜆−𝜈,if𝜆>𝜈,1,if𝜆≤𝜈,(34)
the first relation in et₃ is obtained.
If there exists a constant 𝜆>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 Φ𝑠,𝑡₀,𝑥𝑣=𝑒^{−𝜆(𝑠−𝑡0)}Φ_−𝜆𝑠,𝑡₀,𝑥𝑣≤𝑁𝑒^{−𝜈(𝑡−𝑠)}𝑒^{−𝜆(𝑠−𝑡0)}Φ_−𝜆𝑡,𝑡₀,𝑥𝑣=𝑁𝑒^{−𝜈(𝑡−𝑠)}𝑒^{𝜆(𝑡−𝑠)}Φ𝑡,𝑡₀,𝑥𝑣,(35)
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 of 𝜔-trichotomy is given relative to the dual space 𝑉^∗ of the Banach space 𝑉. To this aim, let us consider three projections families {𝑃_𝑘}_{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 exists 𝑁≥1 such that ^𝑡_𝑡0Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗𝑝𝑑𝑠≤𝑁𝑃₁(𝑥)𝑣^∗𝑝,(36) for all 𝑝>1, all (𝑡,𝑡₀)∈𝑇, and all (𝑥,𝑣^∗)∈𝑋×𝑉^∗;(ii)there exists 𝑁≥1 such that ^𝑡_𝑡0Φ₂𝑠,𝑡₀,𝑥𝑣^𝑝𝑑𝑠^1/𝑝≤𝑁Φ₂𝑡,𝑡₀,𝑥𝑣,(37) for all 𝑝>1, all (𝑡,𝑡₀)∈𝑇, and all (𝑥,𝑣)∈𝑌;(iii)there exist 𝛼<0 and 𝑀≥1 such that ^∞_𝑡0𝑒^{𝛼(𝜏−𝑡0)}Φ₃𝜏,𝑡₀,𝑥𝑣𝑑𝜏≤𝑀𝑃₃(𝑥)𝑣,(38) for all 𝑡₀∈ℝ₊ and all (𝑥,𝑣)∈𝑌 and there exist 𝛽<0 and 𝑀≥1 such that ^𝑡_𝑡0𝑒^{𝛽(𝑡−𝜏)}Φ₃𝜏,𝑡₀,𝑥𝑣𝑑𝜏≤𝑀Φ₃𝑡,𝑡₀,𝑥𝑣,(39) 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 Φ𝑡,𝑡₀,𝑥𝑃₁(𝑥)𝑣≤𝜓𝑡−𝑡₀𝑃₁(𝑥)𝑣,∀𝑡,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌.(40)
An equivalent relation is obtained, if we consider Definition 2  ec₂, given by Φ₁𝑡,𝑡₀,𝑥𝑣=Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥Φ₁𝑠,𝑡₀𝑣(Φ,𝑥≤𝜓𝑡−𝑠)₁𝑠,𝑡₀,𝑥𝑣,(41)
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: ^𝑡_𝑡0Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗𝑝≤𝑑𝑠_𝑡𝑡0𝜓^𝑝(𝑡−𝑠)Φ₁𝑠,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗𝑝𝑑𝑠≤𝑁^𝑝𝑡_𝑡0𝑒^{−𝑝𝜈(𝑡−𝑠)}𝑃₁(𝑥)𝑣^∗𝑝≤𝑁𝑃𝑑𝑠₁(𝑥)𝑣^∗𝑝,(42)
for all (𝑡,𝑡₀)∈𝑇 and all (𝑥,𝑣^∗)∈𝑋×𝑉^∗, where we have denoted that 𝑁=𝑁^𝑝/𝜈𝑝.
(ii) According to Definition 13, there exists a mapping 𝜁∶ℝ₊→ℝ^∗₊, with the property 𝜔_𝜁<0, such that 𝜁𝑡−𝑡₀𝑃₂(𝑥)𝑣≤Φ₂𝑡,𝑡₀𝑣,∀,𝑥𝑡,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌.(43)
The property of function 𝜁 assures the existence of some constants 𝑁≥1 and 𝜈>0, such that, for all 𝑝>0, the following relations hold: ^𝑡_𝑡0Φ₂𝑠,𝑡₀,𝑥𝑣^𝑝𝑑𝑠^1/𝑝≤𝑁Φ₂𝑡,𝑡₀𝑣×,𝑥_𝑡𝑡0𝑒^{−𝜈𝑝(𝑡−𝑠)}𝑑𝑠^1/𝑝≤𝑁Φ₂𝑡,𝑡₀,𝑥𝑣,(44)
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 Theorem 18.
Sufficiency. (i) As the evolution cocycle Φ₁ has exponential growth, there exist 𝑀≥1 and 𝜌>0 such that Φ₁𝑡,𝑡₀,𝑥𝑣≤𝑀𝑒^{𝜌(𝑡−𝑠)}Φ₁𝑠,𝑡₀𝑣,∀(),,𝑥𝑡,𝑠𝑠,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌.(45)
Let 𝑡≥𝑡₀+1. We consider that ^𝑡0_𝑡⁺¹₀𝑒^{−𝜌(𝑠−𝑡0)}𝑑𝑠=¹₀𝑒^−𝜌𝜏𝑑𝜏=1−𝑒^−𝜌𝜌=𝑐>0.(46)
For every 𝑝>1 there exists 𝑞>1 such that 1/𝑝+1/𝑞=1. We obtain 𝑐𝑣^∗,Φ₁𝑡,𝑡₀𝑣≤,𝑥_𝑡0𝑡+10𝑒^{−𝜌(𝑠−𝑡0)}𝑣^∗,Φ₁𝑡,𝑡₀𝑣≤,𝑥𝑑𝑠_𝑡0𝑡+10𝑒^{−𝜌(𝑠−𝑡0)}Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗×Φ₁𝑠,𝑡₀𝑣‖𝑣‖,𝑥𝑑𝑠≤𝑀^𝑡0_𝑡⁺¹₀Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗≤𝑑𝑠_𝑡0𝑡+10Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗𝑝𝑑𝑠^1/𝑝×^𝑡0_𝑡⁺¹₀𝑀^𝑞𝑃₁(𝑥)𝑣^𝑞𝑑𝑠^1/𝑞≤𝑀‖𝑣‖^𝑡_𝑡0Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗𝑝𝑑𝑠^1/𝑝≤𝑀𝑁𝑃₁(𝑥)𝑣𝑃₁(𝑥)𝑣^∗.(47)
Hence, we have that Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀𝑣≤𝑀𝑁,𝑥𝑐𝑃₁(𝑥)𝑣,∀𝑡≥𝑡₀+1,∀(𝑥,𝑣)∈𝑌.(48)
For 𝑡∈[𝑡₀,𝑡₀+1] we obtain Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥𝑣≤𝑀𝑒^𝜌𝑃₁(𝑥)𝑣,∀(𝑥,𝑣)∈𝑌,(49)
which implies that Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥𝑣≤𝑀₁𝑃₁(𝑥)𝑣,∀𝑡,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌,(50) where we have denoted 𝑀₁=𝑀𝑒^𝜌+𝑁𝑐.(51)
Further, the following relations hold: 𝑣^∗,Φ₁𝑡,𝑡₀𝑣=𝑣,𝑥_∗,Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥Φ₁𝑠,𝑡₀𝑣=Φ,𝑥₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗,Φ₁𝑠,𝑡₀,𝑥𝑣≤𝑀₁𝑃₁(𝑥)𝑣Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗.(52)
By integrating on [𝑡₀,𝑡], it follows that 𝑡−𝑡₀𝑣^∗,Φ₁𝑡,𝑡₀,𝑥𝑣≤𝑀₁𝑃₁(𝑥)𝑣^𝑡_𝑡0Φ₁𝑡,𝑠,𝜑𝑠,𝑡₀,𝑥^∗𝑣^∗𝑑𝑠≤𝑀₁𝑁𝑃₁(𝑥)𝑣𝑃₁(𝑥)𝑣^∗,(53)
and further 𝑡−𝑡₀Φ₁𝑡,𝑡₀,𝑥𝑣≤𝑀₁𝑁𝑃₁(𝑥)𝑣,(54)
which implies that Φ₁𝑡,𝑡₀,𝑥𝑣≤𝜓𝑡−𝑡₀𝑃₁(𝑥)𝑣,∀𝑡,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌,(55) 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 𝐹(𝑡)=^𝑡_𝑡0Φ₂𝑠,𝑡₀,𝑥𝑣^𝑝𝑑𝑠.(56)
According to the fact that 𝐹(𝑡)≤𝑁^𝑝𝐹^′(𝑡), for all 𝑡≥0, we obtain, for all (𝑥,𝑣)∈𝑌, 𝐹𝑡₀+1𝑒^{(𝑡−𝑡0−1)/𝑁𝑝}≤𝐹(𝑡)≤𝑁^𝑝Φ₂𝑡,𝑡₀,𝑥𝑣^𝑝.(57)
As the evolution cocycle Φ₂ has exponential decay, there exist some constants 𝑀≥1 and 𝜌>0 such that Φ₂𝑠,𝑡₀,𝑥𝑣≤𝑀𝑒^{𝜌(𝑡−𝑠)}Φ₂𝑡,𝑡₀𝑣,∀(),,𝑥𝑡,𝑠𝑠,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌.(58) Let 𝑠∈[𝑡₀,𝑡₀+1). We obtain 𝑃₂(𝑥)𝑣^𝑝≤𝑀^𝑝𝑒^{𝑝𝜌(𝑡−𝑠)𝑡0}_𝑡⁺¹₀Φ₂𝑠,𝑡₀,𝑥𝑣^𝑝𝑑𝑠(59)
and further 𝑀^−𝑝𝑒^{−𝑝𝜌(𝑡−𝑠)}𝑃₂(𝑥)𝑣^𝑝≤𝐹𝑡₀+1.(60)
For 𝑡≥𝑡₀+1, we obtain 1𝑀^𝑝𝑒^𝜌𝑝𝑒^{−1/𝑁𝑝}𝑒^{(𝑡−𝑡0)/𝑁𝑝}‖𝑣‖^𝑝≤𝑁^𝑝Φ₂𝑡,𝑡₀,𝑥𝑣^𝑝,(61)
and, if 𝑡∈[𝑡₀,𝑡₀+1],𝑃₂(𝑥)𝑣≤𝑀𝑒^𝜌𝑒^{1/𝑝𝑁𝑝}𝑒^{(−1/𝑝𝑁𝑝)(𝑡−𝑡0)}Φ₂𝑡,𝑡₀,𝑥𝑣,(62)
for all (𝑥,𝑣)∈𝑌. Hence, 𝜁𝑡−𝑡₀𝑃₂(𝑥)𝑣≤Φ₂𝑡,𝑡₀𝑣,∀,𝑥𝑡,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌,(63) where we have considered 𝜁∶ℝ₊→ℝ^∗₊ defined by 𝜁(𝑢)=1𝑁𝑀𝑒^𝜌𝑒^{−1/𝑝𝑁𝑝}𝑒^{(1/𝑝𝑁𝑝)𝑢},(64)
with the property 𝜔_𝜁>0. Statement (𝑡₂) of Definition 13 is proved.
(iii) A similar proof for the 𝛼-shifted evolution cocycle Φ³_𝛼 in (i) and for the 𝛽-shifted evolution cocycle Φ³_𝛽 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 Φ³_𝛼𝑡,𝑡₀,𝑥𝑣≤𝜒₁𝑡−𝑡₀𝑃₃(𝑥)𝑣,∀𝑡,𝑡₀()𝜒∈𝑇,𝑥,𝑣∈𝑌,₂𝑡−𝑡₀𝑃₃(𝑥)𝑣≤Φ³_𝛽𝑡,𝑡₀𝑣,∀,𝑥𝑡,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌.(65)
If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent with Φ₃𝑡,𝑡₀,𝑥𝑣≤𝑒^{−𝛼(𝑡−𝑡0)}𝜒₁𝑡−𝑡₀𝑃₃(𝑥)𝑣,∀𝑡,𝑡₀()𝑒∈𝑇,𝑥,𝑣∈𝑌,_{−𝛽(𝑡−𝑡0)}𝜒₂𝑡−𝑡₀𝑃₃(𝑥)𝑣≤Φ₃𝑡,𝑡₀𝑣,∀,𝑥𝑡,𝑡₀∈𝑇,(𝑥,𝑣)∈𝑌.(66)
We define 𝜒,𝜒∶ℝ₊→ℝ^∗₊ by 𝜒(𝑢)=𝑒^−𝛼𝑢𝜒₁(𝑢) and 𝜒(𝑢)=𝑒^−𝛽𝑢𝜒₂(𝑢), which have the properties 𝜔𝜒<∞ 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.

Acknowledgments

This paper was conceived during the visit at the Institute of Mathematics of Bordeaux, France. The author wishes to express her profound and respectful gratitude to Professor Bernard Chevreau and Professor Mihail Megan. Also, the author gratefully acknowledges helpful comments and suggestions from the referees.