Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 793813, 8 pages http://dx.doi.org/10.1155/2013/793813
Research Article Trichotomy for Dynamical Systems in Banach Spaces Codruua Stoica Department of Mathematics and Computer Science, Aurel Vlaicu University of Arad, 2 Elena Dr˘agoi Str, 310330 Arad, Romania Correspondence should be addressed to Codrut¸a Stoica; [email protected] Received 25 July 2013; Accepted 13 August 2013 Academic Editors: G. Bonanno, F. Minh´os, and G.-Q. Xu Copyright © 2013 Codrut¸a Stoica. 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. 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,
𝜑 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) = 𝜑 (𝑡, 𝑡0 , 𝑥) ,
(es1 )
∀ (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,
(ec1 )
2
The Scientific World Journal Φ (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) Φ (𝑠, 𝑡0 , 𝑥) = Φ (𝑡, 𝑡0 , 𝑥) ,
∀ (𝑡, 𝑠) , (𝑠, 𝑡0 ) ∈ 𝑇, ∀𝑥 ∈ 𝑋.
(ec2 )
𝜕2 V 𝜕V (𝑡, 𝑦) = 𝑥 (𝑡) 2 (𝑡, 𝑦) , 𝜕𝑡 𝜕𝑦
Definition 3. The mapping 𝐶 : 𝑇 × 𝑌 → 𝑌 defined by 𝐶 (𝑡, 𝑠, 𝑥, V) = (𝜑 (𝑡, 𝑠, 𝑥) , Φ (𝑡, 𝑠, 𝑥) V) ,
Example 4. Let 𝑓 : 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
(5)
𝑡 > 0.
Let 𝑉 = L2 (0, 1) be a separable Hilbert space with the orthonormal basis {𝑒𝑛 }𝑛∈N , 𝑒0 = 1, 𝑒𝑛 (𝑦) = √2 cos 𝑛𝜋𝑦, 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 : 𝐷 (𝐴) ⊂ 𝑉 → 𝑉,
∞
If 𝑥 ∈ 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 that 𝑉 = 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 𝑢 (𝑠) − ∫𝑠𝑡 x(𝜏−𝑠)𝑑𝜏 𝑢 (𝑡) ∫𝑠𝑡 x(𝜏−𝑠)𝑑𝜏 𝑒 𝑒 V1 , V2 ) 𝑢 (𝑡) 𝑢 (𝑠) (3)
is an evolution cocycle over 𝜑, and 𝐶 = (𝜑, Φ𝑢 ) is a skewevolution 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 Φ𝜆 : 𝑇 × 𝑋 → B (𝑉) ,
𝑦 ∈ (0, 1) ,
𝜕V 𝜕V (𝑡, 0) = (𝑡, 1) = 0, 𝜕𝑦 𝜕𝑦
𝐴V =
𝑑2 V , 𝑑𝑦2
Φ𝜆 (𝑡, 𝑡0 , 𝑥) = 𝑒−𝜆(𝑡−𝑡0 ) Φ (𝑡, 𝑡0 , 𝑥) . (4)
We observe that 𝐶𝜆 = (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. Let 𝑋 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
2 2
𝑆 (𝑡) V = ∑ 𝑒−𝑛 𝜋 𝑡 ⟨V, 𝑒𝑛 ⟩ 𝑒𝑛 ,
(2)
𝑡∈[0,𝑛]
Φ𝑢 (𝑡, 𝑠, x) V = (
𝑡 > 0, 𝑦 ∈ (0, 1) ,
(6)
which generates a C0 -semigroup 𝑆, defined by
∞
𝑑 (𝑥, 𝑦) = ∑
where 𝑑𝑛 (𝑥, 𝑦) = sup 𝑥 (𝑡) − 𝑦 (𝑡) .
V (0, 𝑦) = V0 (𝑦) ,
(1)
where Φ is an evolution cocycle over an evolution semiflow 𝜑, is called skew-evolution semiflow on 𝑌.
1 𝑑𝑛 (𝑥, 𝑦) , 𝑛 1 + 𝑑 (𝑥, 𝑦) 2 𝑛 𝑛=1
the parabolic system with Neumann’s boundary conditions as follows:
(7)
𝑛=0
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) = 𝐴 (𝜑0 (𝑡, 𝑥)) V (𝑡) ,
𝑡 > 0,
(8)
V (0) = V0 . The mapping Φ0 : R+ × 𝑋 → B (𝑉) ,
𝑡
Φ0 (𝑡, 𝑥) V = 𝑆 (∫ 𝑥 (𝑠) 𝑑𝑠) V 0
(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, 𝑥) , ∀ (𝑡, 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) = 𝐴 (𝜑 (𝑡, 𝑠, 𝑥)) V (𝑡) , V (𝑠) = 𝑥,
𝑡 > 𝑠,
(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.
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Definition 7. A mapping 𝑆 : R+ → B(𝑉) is called semigroup of linear operators on 𝑉 if the following relations hold: 𝑆 (0) = 𝐼, 𝑆 (𝑡) 𝑆 (𝑠) = 𝑆 (𝑡 + 𝑠) ,
∀ (𝑡, 𝑠) ∈ R2+ .
(sgr1 )
In what follows we introduce the elements which will allow us to introduce a new concept of trichotomy for skewevolution semiflows. We consider a mapping 𝜓 : R+ → R∗+ , and we define
(sgr2 )
𝐴 𝜓 = {𝜔 ∈ R, sup 𝜓 (𝑡) 𝑒−𝜔𝑡 < ∞} ,
Example 8. One can naturally associate to every semigroup of operators the mapping Φ𝑆 : 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 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].
𝑡≥0
(15) −𝜔𝑡
𝐵𝜓 = {𝜔 ∈ R, inf 𝜓 (𝑡) 𝑒 𝑡≥0
> 0} .
Remark 12. For every function 𝜓 : R+ → R∗+ , the following statements hold: (i) 𝐴 𝜓 ≠ 0 and 𝐵𝜓 ≠ 0; (ii) 𝜇 ∈ 𝐴 𝜓 implies the existence of a constant 𝑀 ∈ R+ such that 𝜓 (𝑡) ≤ 𝑀𝑒𝜇𝑡 ,
∀𝑡 ≥ 0;
(16)
(iii) 𝜆 ∈ 𝐵ℎ implies that there exists a constant 𝑚 ∈ R+ such that 𝜓 (𝑡) ≥ 𝑚𝑒𝜆𝑡 ,
∀𝑡 ≥ 0.
(17)
Let us denote that inf 𝐴 𝜓 , if 𝐴 𝜓 ≠ 0, ∞, if 𝐴 𝜓 = 0,
3. On Trichotomy Issues
𝜔𝜓 = {
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
sup 𝐵𝜓 , if 𝐵𝜓 ≠ 0, 𝜔𝜓 = { −∞, if 𝐵𝜓 = 0.
𝑃(𝑥)2 = 𝑃 (𝑥) ,
∀𝑥 ∈ 𝑋
Definition 13. A skew-evolution semiflow 𝐶 = (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 𝜔𝜓 < 0,
(12)
𝜔𝜁 > 0,
𝜔𝜒 < ∞,
𝜔𝜒 < 0,
(19)
such that
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 (𝑡, 𝑠, 𝑥) 𝑃 (𝑥) = 𝑃 (𝜑 (𝑡, 𝑠, 𝑥)) Φ (𝑡, 𝑠, 𝑥) , ∀(𝑡, 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 {𝑃𝑘 }𝑘∈{1,2,3} are said to be compatible with a skew-evolution semiflow 𝐶 = (𝜑, Φ) if (c1 ) each of 𝑃𝑘 , 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,
(18)
𝑃𝑖 (𝑥) 𝑃𝑗 (𝑥) = 0, ∀𝑖, 𝑗 ∈ {1, 2, 3} 𝑖 ≠ 𝑗.
(14)
(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 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 Φ : 𝑇 × 𝑋 → B (𝑉) , Φ (𝑡, 𝑡0 , 𝑥) V 𝑡
−𝜆(𝑡−𝑡0 )+∫𝑡 𝑥(𝜏−𝑡0 )𝑑𝜏
= (𝑒
0
𝑡
−(𝑡−𝑡0 )𝑥(0)+∫𝑡 𝑥(𝜏−𝑡0 )𝑑𝜏
𝑒
0
𝑡
∫ 𝑥(𝜏−𝑡0 )𝑑𝜏
V1 , 𝑒 𝑡0
V3 ) ,
V2 ,
(20)
4
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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 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 , (21) 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 𝜓 (𝑢) = 𝑒[−𝜆+𝑥(0)]𝑢 ,
𝜁 (𝑢) = 𝑒𝑙𝑢 ,
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 𝜔instability. In what follows, if 𝑃𝑘 is a given projections family, we will denote that Φ𝑘 (𝑡, 𝑠, 𝑥) = Φ (𝑡, 𝑠, 𝑥) 𝑃𝑘 (𝑥) ,
(23)
for every (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and 𝑥 ∈ 𝑋. We remark that the following relations hold:
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 𝜔𝜓 < 0,
𝜔𝜁 > 0,
𝜔𝜒 < ∞,
𝜔𝜒 < 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) ≤ 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. As 𝜔𝜓 < 0, there exist 𝑁1 ≥ 1 and ]1 > 0, such that 𝜓(𝑡) ≤ 𝑁1 𝑒−]1 𝑡 , for all 𝑡 ≥ 0. Hence, relation (et1 ) is obtained. As 𝜔𝜁 > 0, it follows that there exist 𝑁2 ≥ 1 and ]2 > 0, such that 𝜁(𝑡) ≥ 𝑁2 𝑒]2 𝑡 , for all 𝑡 ≥ 0, which implies (et2 ). From 𝜔𝜒 < ∞ and 𝜔𝜒 < 0, it follows that there exist 𝑁3 ≥ 1, ]3 > 0 with the property 1 −]3 𝑡 𝑒 ≤ 𝜒 (𝑡) ≤ 𝑁3 𝑒]3 𝑡 , 𝑁3
∀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, ]1 , ]2 , ]3 > 0 such that relations (et1 )–(et3 ) hold. Let us define the mappings 𝜓, 𝜁, 𝜒 : R+ → R∗+ by 𝜓 (𝑡) = 𝑁1 𝑒−]1 𝑡 ,
𝜁 (𝑡) = 𝑁2−1 𝑒]2 𝑡 ,
𝜒 (𝑡) = 𝑁3 𝑒]3 𝑡 . (26)
We obtain the relations 𝜔𝜓 < 0,
𝜔𝜁 > 0,
𝜔𝜒 < ∞,
𝜔𝜒 < 0.
(27)
Hence, relations (i)–(iii) of Definition 13 are verified, and, therefore, 𝐶 is 𝜔-trichotomic, which ends the proof.
(i) Φ𝑘 (𝑡, 𝑡, 𝑥) = 𝑃𝑘 (𝑥), ∀(𝑡, 𝑥) ∈ R+ × 𝑋;
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.
(ii) Φ𝑘 (𝑡, 𝑠, 𝜑(𝑠, 𝑡0 , 𝑥))Φ𝑘 (𝑠, 𝑡0 , 𝑥) = Φ𝑘 (𝑡, 𝑡0 , 𝑥), ∀(𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇, ∀𝑥 ∈ 𝑋.
4. Main Results
Proposition 16. A skew-evolution semiflow 𝐶 = (𝜑, Φ) is 𝜔trichotomic if and only if there exist some constants 𝑁1 , 𝑁2 , 𝑁3 ≥ 1, ]1 , ]2 , ]3 > 0 and three projections families {𝑃𝑘 }𝑘∈{1,2,3} compatible with 𝐶 such that 𝑒]1 (𝑡−𝑡0 ) Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁1 𝑃1 (𝑥) V ,
(et1 )
𝑒]2 (𝑡−𝑡0 ) 𝑃2 (𝑥) V ≤ 𝑁2 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
(et2 )
] (𝑡−𝑡 ) 𝑃3 (𝑥) V ≤ 𝑁3 𝑒 3 0 Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁32 𝑒2]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V , for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌.
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 𝜔trichotomic if and only if there exist three projections families {𝑃𝑘 }𝑘∈{1,2,3} compatible with 𝐶 and (i) the evolution cocycle Φ1 is exponentially stable; (ii) the evolution cocycle Φ2 is exponentially instable;
(et3 )
(iii) there exists a constant 𝜆 > 0 such that the 𝜆-shifted evolution cocycle Φ3𝜆 is exponentially stable and the −𝜆shifted evolution cocycle Φ3−𝜆 is exponentially instable.
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Proof. Necessity. Statements (i) and (ii) are obtained immediately from the necessity of Proposition 16. According to (et3 ), there exist 𝑁3 ≥ 1 and ]3 > 0 such that ] (𝑡−𝑡 ) Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁3 𝑒 3 0 𝑃3 (𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, ∀ (𝑥, V) ∈ 𝑌.
(28)
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 ] > 0 such that −𝜆(𝑠−𝑡0 ) Φ−𝜆 (𝑠, 𝑡0 , 𝑥) V Φ (𝑠, 𝑡0 , 𝑥) V = 𝑒 −](𝑡−𝑠) −𝜆(𝑠−𝑡0 ) Φ−𝜆 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁𝑒 𝑒 = 𝑁𝑒−](𝑡−𝑠) 𝑒𝜆(𝑡−𝑠) Φ (𝑡, 𝑡0 , 𝑥) V ,
Let us consider that 𝜆 = 2]3 > 0. We obtain successively 3 Φ𝜆 (𝑡, 𝑡0 , 𝑥) V = 𝑒−𝜆(𝑡−𝑡0 ) Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁3 𝑒−𝜆(𝑡−𝑡0 ) 𝑒]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V = 𝑁3 𝑒−]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V ,
(35)
(29)
for all (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌. If we consider ]3 defined as in (34), the second relation in (et3 ) is obtained. Hence, also (iii) in Definition 13 is true and 𝐶 is 𝜔trichotomic.
(30)
Another characterization for the property of 𝜔trichotomy 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.
If we consider that 𝜆 = 2]3 > 0, we obtain, according also to Definition 2 (ec2 ),
Theorem 19. A ∗-strongly measurable skew-evolution semiflow 𝐶 = (𝜑, Φ) is 𝜔-trichotomic if the and only if the following statements hold:
for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌, which shows that Φ3𝜆 is exponentially stable. Also, we have 𝑒−]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V ≤ 𝑁3 Φ3 (𝑡, 𝑡0 , 𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, ∀ (𝑥, V) ∈ 𝑌.
3 Φ−𝜆 (𝑠, 𝑡0 , 𝑥) V = 𝑒𝜆(𝑠−𝑡0 ) Φ3 (𝑠, 𝑡0 , 𝑥) V ≤ 𝑁3 𝑒]3 (𝑡−𝑠) 𝑒𝜆(𝑠−𝑡0 ) Φ3 (𝑡, 𝑡0 , 𝑥) V = 𝑁3 𝑒−]3 (𝑡−𝑠) Φ3−𝜆 (𝑡, 𝑡0 , 𝑥) V ,
̃ ≥ 1 such that (i) there exists 𝑁 (31)
𝑡 ∗ 𝑝 ̃ 𝑃1 (𝑥) V∗ 𝑝 , ∫ Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠 ≤ 𝑁 𝑡
for all 𝑝 > 1, all (𝑡, 𝑡0 ) ∈ 𝑇, and all (𝑥, V∗ ) ∈ 𝑋 × 𝑉∗ ;
for all (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌, which proves that Φ3−𝜆 is exponentially instable.
(ii) there exists 𝑁 ≥ 1 such that
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 Φ3𝜆 is exponentially stable, then there exists some constants 𝑁 ≥ 1 and ] > 0 such that
𝑝 (∫ Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠) 𝑡
3 Φ𝜆 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁𝑒−](𝑡−𝑡0 ) 𝑃3 (𝑥) V , ∀𝑡 ≥ 𝑡0 ≥ 0, ∀(𝑥, V) ∈ 𝑌.
𝑡
0
the first relation in (et3 ) is obtained.
≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
(37)
̃ ≥ 1 such that (iii) there exist 𝛼 < 0 and 𝑀 ∞
(32)
̃ 𝑃3 (𝑥) V , ∫ 𝑒𝛼(𝜏−𝑡0 ) Φ3 (𝜏, 𝑡0 , 𝑥) V 𝑑𝜏 ≤ 𝑀 𝑡0
(38)
for all 𝑡0 ∈ R+ and all (𝑥, V) ∈ 𝑌 and there exist 𝛽 < 0 and 𝑀 ≥ 1 such that 𝑡
(33)
∫ 𝑒𝛽(𝑡−𝜏) Φ3 (𝜏, 𝑡0 , 𝑥) V 𝑑𝜏 ≤ 𝑀 Φ3 (𝑡, 𝑡0 , 𝑥) V , 𝑡0
(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 𝜔𝜓 < 0, such that
for all (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌. Denoting 𝜆 − ], if 𝜆 > ], ]3 = { 1, if 𝜆 ≤ ],
1/𝑝
for all 𝑝 > 1, all (𝑡, 𝑡0 ) ∈ 𝑇, and all (𝑥, V) ∈ 𝑌;
Further, we obtain 𝜆(𝑡−𝑡0 ) Φ3 (𝑡, 𝑡 , 𝑥) V Φ3 (𝑡, 𝑡0 , 𝑥) V = 𝑒 𝜆 0 ≤ 𝑁𝑒𝜆(𝑡−𝑡0 ) 𝑒−](𝑡−𝑠) Φ3𝜆 (𝑠, 𝑡0 , 𝑥) V = 𝑁𝑒𝜆(𝑡−𝑠) 𝑒−](𝑡−𝑠) Φ3 (𝑠, 𝑡0 , 𝑥) V ,
(36)
0
(34)
Φ (𝑡, 𝑡0 , 𝑥) 𝑃1 (𝑥) V ≤ 𝜓 (𝑡 − 𝑡0 ) 𝑃1 (𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
(40)
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An equivalent relation is obtained, if we consider Definition 2 (ec2 ), given by Φ1 (𝑡, 𝑡0 , 𝑥) V = Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) Φ1 (𝑠, 𝑡0 , 𝑥) V (41) ≤ 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 ] > 0 such that the following relations hold:
0
∗ 𝑝 ≤ ∫ 𝜓𝑝 (𝑡 − 𝑠) Φ1 (𝑠, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠 𝑡0
−𝑝](𝑡−𝑠)
𝑡0
(42)
∗ 𝑝
𝜁 (𝑡 − 𝑡0 ) 𝑃2 (𝑥) V ≤ Φ2 (𝑡, 𝑡0 , 𝑥) V ,
The property of function 𝜁 assures the existence of some constants 𝑁 ≥ 1 and ] > 0, such that, for all 𝑝 > 0, the following relations hold: 1/𝑝
𝑝 (∫ Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠) 𝑡
≤ (∫
𝑡0 +1
Sufficiency. (i) As the evolution cocycle Φ1 has exponential growth, there exist 𝑀 ≥ 1 and 𝜌 > 0 such that 𝜌(𝑡−𝑠) Φ1 (𝑠, 𝑡0 , 𝑥) V , Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑀𝑒 ∀ (𝑡, 𝑠) , (𝑠, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
𝑝 Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥))∗ V∗ 𝑑𝑠)
𝑡0 +1
𝑞 𝑀 𝑃1 (𝑥) V 𝑑𝑠) 𝑞
1/𝑞
𝑡 ∗ 𝑝 ≤ 𝑀 ‖V‖ (∫ Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠) 𝑡
1/𝑝
0
̃ 𝑃1 (𝑥) V 𝑃1 (𝑥) V∗ . ≤ 𝑀𝑁
̃ 𝑀𝑁 𝑃 (𝑥) V , Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V ≤ 𝑐 1
(48)
∀𝑡 ≥ 𝑡0 + 1, ∀(𝑥, V) ∈ 𝑌.
= 𝑐 > 0.
∀ (𝑥, V) ∈ 𝑌, (49)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
(50)
where we have denoted 𝑀1 = 𝑀 (𝑒𝜌 +
̃ 𝑁 ). 𝑐
(51)
Further, the following relations hold: (45)
Let 𝑡 ≥ 𝑡0 + 1. We consider that 1−𝑒 𝜌
(47)
1/𝑝
Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V ≤ 𝑀1 𝑃1 (𝑥) V ,
for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌, where we have denoted that 𝑁 = 𝑁/(]𝑝)1/𝑝 . (iii) Both relations are obtained by a similar proof as in (i) and (ii), according to Theorem 18.
𝑒−𝜌(𝑠−𝑡0 ) 𝑑𝑠 = ∫ 𝑒−𝜌𝜏 𝑑𝜏 =
Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥))∗ V∗ 𝑑𝑠
which implies that
≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
−𝜌
𝑡0 +1
(44)
𝑡0
0
𝑡0
𝜌 Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V ≤ 𝑀𝑒 𝑃1 (𝑥) V , 1/𝑝
𝑡
𝑡0
∗ 𝑒−𝜌(𝑠−𝑡0 ) Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗
For 𝑡 ∈ [𝑡0 , 𝑡0 + 1] we obtain
≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V × (∫ 𝑒−]𝑝(𝑡−𝑠) 𝑑𝑠)
1
𝑡0 +1
Hence, we have that (43)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
𝑡0 +1
≤∫
𝑡0
∗
for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V ) ∈ 𝑋 × 𝑉 , where we have ̃ = 𝑁𝑝 /]𝑝. denoted that 𝑁 (ii) According to Definition 13, there exists a mapping 𝜁 : R+ → R∗+ , with the property 𝜔𝜁 < 0, such that
∫
𝑡0
× (∫ ∗
0
𝑒−𝜌(𝑠−𝑡0 ) ⟨V∗ , Φ1 (𝑡, 𝑡0 , 𝑥) V⟩ 𝑑𝑠
𝑡0
̃ 𝑃1 (𝑥) V∗ 𝑝 , ≤𝑁
𝑡
𝑡0 +1
𝑡0
𝑃1 (𝑥) V 𝑑𝑠
≤𝑁 ∫ 𝑒
≤∫
≤ 𝑀 ‖V‖ ∫
𝑡
𝑝
𝑐 ⟨V∗ , Φ1 (𝑡, 𝑡0 , 𝑥) V⟩
× Φ1 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠
𝑡 ∗ 𝑝 ∫ Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠 𝑡
𝑡
For every 𝑝 > 1 there exists 𝑞 > 1 such that 1/𝑝 + 1/𝑞 = 1. We obtain
(46)
∗ ⟨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∗ .
(52)
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By integrating on [𝑡0 , 𝑡], it follows that (𝑡 − 𝑡0 ) ⟨V∗ , Φ1 (𝑡, 𝑡0 , 𝑥) V⟩ ≤ 𝑀1 𝑃1 (𝑥) V ∫
𝑡
𝑡0
𝜁 (𝑢) =
Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥))∗ V∗ 𝑑𝑠 (53)
̃ 𝑃1 (𝑥) V𝑃1 (𝑥) V∗ , ≤ 𝑀1 𝑁 and further ̃ 𝑃1 (𝑥) V , (𝑡 − 𝑡0 ) Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑀1 𝑁
(54)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
(55)
̃ + 1)/𝑢, which where we have considered 𝜓(𝑢) = 𝑀1 (𝑁 satisfies 𝜔𝜓 < 0. Statement (𝑡1 ) of Definition 13 is hence obtained. (ii) Let (𝑡, 𝑡0 ) ∈ 𝑇. Let us denote that 𝑡
𝑝 𝐹 (𝑡) = ∫ Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠. 𝑡
with the property 𝜔𝜁 > 0. Statement (𝑡2 ) of Definition 13 is proved. (iii) A similar proof for the 𝛼-shifted evolution cocycle Φ3𝛼 in (i) and for the 𝛽-shifted evolution cocycle Φ3𝛽 in (ii) leads to the existence of two functions 𝜒1 , 𝜒2 : R+ → R∗+ , with the properties 𝜔𝜒1 < 0 and 𝜔𝜒2 > 0, such that
(56)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌, 𝜒2 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V ≤ Φ3𝛽 (𝑡, 𝑡0 , 𝑥) V ,
𝑝
According to the fact that 𝐹(𝑡) ≤ 𝑁 𝐹 (𝑡), for all 𝑡 ≥ 0, we obtain, for all (𝑥, V) ∈ 𝑌,
If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent with −𝛼(𝑡−𝑡0 ) 𝜒1 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V , Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑒
𝜌(𝑡−𝑠) Φ2 (𝑡, 𝑡0 , 𝑥) V , Φ2 (𝑠, 𝑡0 , 𝑥) V ≤ 𝑀𝑒 ∀ (𝑡, 𝑠) , (𝑠, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
(58)
Let 𝑠 ∈ [𝑡0 , 𝑡0 + 1). We obtain 𝑡 +1
0 𝑝 𝑝 𝑝𝜌(𝑡−𝑠) ∫ 𝑃2 (𝑥) V ≤ 𝑀 𝑒 𝑡 0
𝑝 Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠
(59)
and further 𝑝 𝑀−𝑝 𝑒−𝑝𝜌(𝑡−𝑠) 𝑃2 (𝑥) V ≤ 𝐹 (𝑡0 + 1) . 1
𝑝 𝑝 𝑝 𝑝 𝑒−1/𝑁 𝑒(𝑡−𝑡0 )/𝑁 ‖V‖𝑝 ≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
(61)
and, if 𝑡 ∈ [𝑡0 , 𝑡0 + 1], 𝑝
𝜌 1/𝑝𝑁 (−1/𝑝𝑁 𝑒 𝑃2 (𝑥) V ≤ 𝑀𝑒 𝑒
𝑝
)(𝑡−𝑡0 )
Φ2 (𝑡, 𝑡0 , 𝑥) V ,
for all (𝑥, V) ∈ 𝑌. Hence, 𝜁 (𝑡 − 𝑡0 ) 𝑃2 (𝑥) V ≤ Φ2 (𝑡, 𝑡0 , 𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
𝑒−𝛽(𝑡−𝑡0 ) 𝜒2 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V ≤ Φ3 (𝑡, 𝑡0 , 𝑥) V ,
(66)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌. ̃ 𝜒 : R+ → R∗+ by 𝜒(𝑢) ̃ We define 𝜒, = 𝑒−𝛼𝑢 𝜒1 (𝑢) and 𝜒(𝑢) = 𝑒−𝛽𝑢 𝜒2 (𝑢), which have the properties 𝜔𝜒̃ < ∞ and 𝜔𝜒 < 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 𝜔trichotomy.
Acknowledgments (60)
For 𝑡 ≥ 𝑡0 + 1, we obtain 𝑀𝑝 𝑒𝜌𝑝
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
(57)
As the evolution cocycle Φ2 has exponential decay, there exist some constants 𝑀 ≥ 1 and 𝜌 > 0 such that
(65)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
0
𝑝 𝑝 𝑝 𝐹 (𝑡0 + 1) 𝑒(𝑡−𝑡0 −1)/𝑁 ≤ 𝐹 (𝑡) ≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V .
(64)
3 Φ𝛼 (𝑡, 𝑡0 , 𝑥) V ≤ 𝜒1 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V ,
which implies that Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝜓 (𝑡 − 𝑡0 ) 𝑃1 (𝑥) V ,
𝑝 𝑝 𝑁 𝑁 1 𝑒−1/𝑝 𝑒(1/𝑝 )𝑢 , 𝑁𝑀𝑒𝜌
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.
References (62)
(63)
[1] M. Megan and C. Stoica, “Exponential instability of skew-evolution semi ows in Banach spaces,” Studia Universitatis “BabesBolyai” Mathematica, vol. 53, no. 1, pp. 17–24, 2008. [2] A. J. G. Bento and C. M. Silva, “Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs,” http://arxiv .org/abs/1210.7740.
8 [3] P. V. Hai, “Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 12, pp. 4390–4396, 2010. [4] P. V. Hai, “Discrete and continuous versions of Barbashintype theorem of linear skew-evolution semiflows,” Applicable Analysis, vol. 90, no. 12, pp. 1897–1907, 2011. [5] M. Megan and C. Stoica, “Concepts of dichotomy for skewevolution semiflows in banach spaces,” Annals of the Academy of Romanian Scientists, vol. 2, no. 2, pp. 125–140, 2010. [6] C. Stoica and M. Megan, “On uniform exponential stability for skew-evolution semiflows on Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 3-4, pp. 1305–1313, 2010. [7] J. Appell, V. Lakshmikantham, N. V. Minh, and P. P. Zabreiko, “A general model of evolutionary processes. Exponential dichotomy-I,” Nonlinear Analysis, vol. 21, no. 3, pp. 207–218, 1993. [8] A. L. Sasu and B. Sasu, “Integral equations, dichotomy of evolution families on the half-line and applications,” Integral Equations and Operator Theory, vol. 66, no. 1, pp. 113–140, 2010. [9] S. Elaydi and O. Hajek, “Exponential trichotomy of differential systems,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 362–374, 1988. [10] M. Megan and C. Stoica, “On uniform exponential trichotomy of evolution operators in banach spaces,” Integral Equations and Operator Theory, vol. 60, no. 4, pp. 499–506, 2008. [11] R. J. Sacker and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems, III,” Journal of Differential Equations, vol. 22, no. 2, pp. 497–522, 1976.
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Research Article Trichotomy for Dynamical Systems in Banach Spaces Codruua Stoica Department of Mathematics and Computer Science, Aurel Vlaicu University of Arad, 2 Elena Dr˘agoi Str, 310330 Arad, Romania Correspondence should be addressed to Codrut¸a Stoica; [email protected] Received 25 July 2013; Accepted 13 August 2013 Academic Editors: G. Bonanno, F. Minh´os, and G.-Q. Xu Copyright © 2013 Codrut¸a Stoica. 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. 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,
𝜑 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) = 𝜑 (𝑡, 𝑡0 , 𝑥) ,
(es1 )
∀ (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,
(ec1 )
2
The Scientific World Journal Φ (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) Φ (𝑠, 𝑡0 , 𝑥) = Φ (𝑡, 𝑡0 , 𝑥) ,
∀ (𝑡, 𝑠) , (𝑠, 𝑡0 ) ∈ 𝑇, ∀𝑥 ∈ 𝑋.
(ec2 )
𝜕2 V 𝜕V (𝑡, 𝑦) = 𝑥 (𝑡) 2 (𝑡, 𝑦) , 𝜕𝑡 𝜕𝑦
Definition 3. The mapping 𝐶 : 𝑇 × 𝑌 → 𝑌 defined by 𝐶 (𝑡, 𝑠, 𝑥, V) = (𝜑 (𝑡, 𝑠, 𝑥) , Φ (𝑡, 𝑠, 𝑥) V) ,
Example 4. Let 𝑓 : 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
(5)
𝑡 > 0.
Let 𝑉 = L2 (0, 1) be a separable Hilbert space with the orthonormal basis {𝑒𝑛 }𝑛∈N , 𝑒0 = 1, 𝑒𝑛 (𝑦) = √2 cos 𝑛𝜋𝑦, 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 : 𝐷 (𝐴) ⊂ 𝑉 → 𝑉,
∞
If 𝑥 ∈ 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 that 𝑉 = 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 𝑢 (𝑠) − ∫𝑠𝑡 x(𝜏−𝑠)𝑑𝜏 𝑢 (𝑡) ∫𝑠𝑡 x(𝜏−𝑠)𝑑𝜏 𝑒 𝑒 V1 , V2 ) 𝑢 (𝑡) 𝑢 (𝑠) (3)
is an evolution cocycle over 𝜑, and 𝐶 = (𝜑, Φ𝑢 ) is a skewevolution 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 Φ𝜆 : 𝑇 × 𝑋 → B (𝑉) ,
𝑦 ∈ (0, 1) ,
𝜕V 𝜕V (𝑡, 0) = (𝑡, 1) = 0, 𝜕𝑦 𝜕𝑦
𝐴V =
𝑑2 V , 𝑑𝑦2
Φ𝜆 (𝑡, 𝑡0 , 𝑥) = 𝑒−𝜆(𝑡−𝑡0 ) Φ (𝑡, 𝑡0 , 𝑥) . (4)
We observe that 𝐶𝜆 = (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. Let 𝑋 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
2 2
𝑆 (𝑡) V = ∑ 𝑒−𝑛 𝜋 𝑡 ⟨V, 𝑒𝑛 ⟩ 𝑒𝑛 ,
(2)
𝑡∈[0,𝑛]
Φ𝑢 (𝑡, 𝑠, x) V = (
𝑡 > 0, 𝑦 ∈ (0, 1) ,
(6)
which generates a C0 -semigroup 𝑆, defined by
∞
𝑑 (𝑥, 𝑦) = ∑
where 𝑑𝑛 (𝑥, 𝑦) = sup 𝑥 (𝑡) − 𝑦 (𝑡) .
V (0, 𝑦) = V0 (𝑦) ,
(1)
where Φ is an evolution cocycle over an evolution semiflow 𝜑, is called skew-evolution semiflow on 𝑌.
1 𝑑𝑛 (𝑥, 𝑦) , 𝑛 1 + 𝑑 (𝑥, 𝑦) 2 𝑛 𝑛=1
the parabolic system with Neumann’s boundary conditions as follows:
(7)
𝑛=0
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) = 𝐴 (𝜑0 (𝑡, 𝑥)) V (𝑡) ,
𝑡 > 0,
(8)
V (0) = V0 . The mapping Φ0 : R+ × 𝑋 → B (𝑉) ,
𝑡
Φ0 (𝑡, 𝑥) V = 𝑆 (∫ 𝑥 (𝑠) 𝑑𝑠) V 0
(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, 𝑥) , ∀ (𝑡, 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) = 𝐴 (𝜑 (𝑡, 𝑠, 𝑥)) V (𝑡) , V (𝑠) = 𝑥,
𝑡 > 𝑠,
(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.
The Scientific World Journal
3
Definition 7. A mapping 𝑆 : R+ → B(𝑉) is called semigroup of linear operators on 𝑉 if the following relations hold: 𝑆 (0) = 𝐼, 𝑆 (𝑡) 𝑆 (𝑠) = 𝑆 (𝑡 + 𝑠) ,
∀ (𝑡, 𝑠) ∈ R2+ .
(sgr1 )
In what follows we introduce the elements which will allow us to introduce a new concept of trichotomy for skewevolution semiflows. We consider a mapping 𝜓 : R+ → R∗+ , and we define
(sgr2 )
𝐴 𝜓 = {𝜔 ∈ R, sup 𝜓 (𝑡) 𝑒−𝜔𝑡 < ∞} ,
Example 8. One can naturally associate to every semigroup of operators the mapping Φ𝑆 : 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 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].
𝑡≥0
(15) −𝜔𝑡
𝐵𝜓 = {𝜔 ∈ R, inf 𝜓 (𝑡) 𝑒 𝑡≥0
> 0} .
Remark 12. For every function 𝜓 : R+ → R∗+ , the following statements hold: (i) 𝐴 𝜓 ≠ 0 and 𝐵𝜓 ≠ 0; (ii) 𝜇 ∈ 𝐴 𝜓 implies the existence of a constant 𝑀 ∈ R+ such that 𝜓 (𝑡) ≤ 𝑀𝑒𝜇𝑡 ,
∀𝑡 ≥ 0;
(16)
(iii) 𝜆 ∈ 𝐵ℎ implies that there exists a constant 𝑚 ∈ R+ such that 𝜓 (𝑡) ≥ 𝑚𝑒𝜆𝑡 ,
∀𝑡 ≥ 0.
(17)
Let us denote that inf 𝐴 𝜓 , if 𝐴 𝜓 ≠ 0, ∞, if 𝐴 𝜓 = 0,
3. On Trichotomy Issues
𝜔𝜓 = {
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
sup 𝐵𝜓 , if 𝐵𝜓 ≠ 0, 𝜔𝜓 = { −∞, if 𝐵𝜓 = 0.
𝑃(𝑥)2 = 𝑃 (𝑥) ,
∀𝑥 ∈ 𝑋
Definition 13. A skew-evolution semiflow 𝐶 = (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 𝜔𝜓 < 0,
(12)
𝜔𝜁 > 0,
𝜔𝜒 < ∞,
𝜔𝜒 < 0,
(19)
such that
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 (𝑡, 𝑠, 𝑥) 𝑃 (𝑥) = 𝑃 (𝜑 (𝑡, 𝑠, 𝑥)) Φ (𝑡, 𝑠, 𝑥) , ∀(𝑡, 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 {𝑃𝑘 }𝑘∈{1,2,3} are said to be compatible with a skew-evolution semiflow 𝐶 = (𝜑, Φ) if (c1 ) each of 𝑃𝑘 , 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,
(18)
𝑃𝑖 (𝑥) 𝑃𝑗 (𝑥) = 0, ∀𝑖, 𝑗 ∈ {1, 2, 3} 𝑖 ≠ 𝑗.
(14)
(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 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 Φ : 𝑇 × 𝑋 → B (𝑉) , Φ (𝑡, 𝑡0 , 𝑥) V 𝑡
−𝜆(𝑡−𝑡0 )+∫𝑡 𝑥(𝜏−𝑡0 )𝑑𝜏
= (𝑒
0
𝑡
−(𝑡−𝑡0 )𝑥(0)+∫𝑡 𝑥(𝜏−𝑡0 )𝑑𝜏
𝑒
0
𝑡
∫ 𝑥(𝜏−𝑡0 )𝑑𝜏
V1 , 𝑒 𝑡0
V3 ) ,
V2 ,
(20)
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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 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 , (21) 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 𝜓 (𝑢) = 𝑒[−𝜆+𝑥(0)]𝑢 ,
𝜁 (𝑢) = 𝑒𝑙𝑢 ,
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 𝜔instability. In what follows, if 𝑃𝑘 is a given projections family, we will denote that Φ𝑘 (𝑡, 𝑠, 𝑥) = Φ (𝑡, 𝑠, 𝑥) 𝑃𝑘 (𝑥) ,
(23)
for every (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and 𝑥 ∈ 𝑋. We remark that the following relations hold:
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 𝜔𝜓 < 0,
𝜔𝜁 > 0,
𝜔𝜒 < ∞,
𝜔𝜒 < 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) ≤ 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. As 𝜔𝜓 < 0, there exist 𝑁1 ≥ 1 and ]1 > 0, such that 𝜓(𝑡) ≤ 𝑁1 𝑒−]1 𝑡 , for all 𝑡 ≥ 0. Hence, relation (et1 ) is obtained. As 𝜔𝜁 > 0, it follows that there exist 𝑁2 ≥ 1 and ]2 > 0, such that 𝜁(𝑡) ≥ 𝑁2 𝑒]2 𝑡 , for all 𝑡 ≥ 0, which implies (et2 ). From 𝜔𝜒 < ∞ and 𝜔𝜒 < 0, it follows that there exist 𝑁3 ≥ 1, ]3 > 0 with the property 1 −]3 𝑡 𝑒 ≤ 𝜒 (𝑡) ≤ 𝑁3 𝑒]3 𝑡 , 𝑁3
∀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, ]1 , ]2 , ]3 > 0 such that relations (et1 )–(et3 ) hold. Let us define the mappings 𝜓, 𝜁, 𝜒 : R+ → R∗+ by 𝜓 (𝑡) = 𝑁1 𝑒−]1 𝑡 ,
𝜁 (𝑡) = 𝑁2−1 𝑒]2 𝑡 ,
𝜒 (𝑡) = 𝑁3 𝑒]3 𝑡 . (26)
We obtain the relations 𝜔𝜓 < 0,
𝜔𝜁 > 0,
𝜔𝜒 < ∞,
𝜔𝜒 < 0.
(27)
Hence, relations (i)–(iii) of Definition 13 are verified, and, therefore, 𝐶 is 𝜔-trichotomic, which ends the proof.
(i) Φ𝑘 (𝑡, 𝑡, 𝑥) = 𝑃𝑘 (𝑥), ∀(𝑡, 𝑥) ∈ R+ × 𝑋;
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.
(ii) Φ𝑘 (𝑡, 𝑠, 𝜑(𝑠, 𝑡0 , 𝑥))Φ𝑘 (𝑠, 𝑡0 , 𝑥) = Φ𝑘 (𝑡, 𝑡0 , 𝑥), ∀(𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇, ∀𝑥 ∈ 𝑋.
4. Main Results
Proposition 16. A skew-evolution semiflow 𝐶 = (𝜑, Φ) is 𝜔trichotomic if and only if there exist some constants 𝑁1 , 𝑁2 , 𝑁3 ≥ 1, ]1 , ]2 , ]3 > 0 and three projections families {𝑃𝑘 }𝑘∈{1,2,3} compatible with 𝐶 such that 𝑒]1 (𝑡−𝑡0 ) Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁1 𝑃1 (𝑥) V ,
(et1 )
𝑒]2 (𝑡−𝑡0 ) 𝑃2 (𝑥) V ≤ 𝑁2 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
(et2 )
] (𝑡−𝑡 ) 𝑃3 (𝑥) V ≤ 𝑁3 𝑒 3 0 Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁32 𝑒2]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V , for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌.
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 𝜔trichotomic if and only if there exist three projections families {𝑃𝑘 }𝑘∈{1,2,3} compatible with 𝐶 and (i) the evolution cocycle Φ1 is exponentially stable; (ii) the evolution cocycle Φ2 is exponentially instable;
(et3 )
(iii) there exists a constant 𝜆 > 0 such that the 𝜆-shifted evolution cocycle Φ3𝜆 is exponentially stable and the −𝜆shifted evolution cocycle Φ3−𝜆 is exponentially instable.
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Proof. Necessity. Statements (i) and (ii) are obtained immediately from the necessity of Proposition 16. According to (et3 ), there exist 𝑁3 ≥ 1 and ]3 > 0 such that ] (𝑡−𝑡 ) Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁3 𝑒 3 0 𝑃3 (𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, ∀ (𝑥, V) ∈ 𝑌.
(28)
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 ] > 0 such that −𝜆(𝑠−𝑡0 ) Φ−𝜆 (𝑠, 𝑡0 , 𝑥) V Φ (𝑠, 𝑡0 , 𝑥) V = 𝑒 −](𝑡−𝑠) −𝜆(𝑠−𝑡0 ) Φ−𝜆 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁𝑒 𝑒 = 𝑁𝑒−](𝑡−𝑠) 𝑒𝜆(𝑡−𝑠) Φ (𝑡, 𝑡0 , 𝑥) V ,
Let us consider that 𝜆 = 2]3 > 0. We obtain successively 3 Φ𝜆 (𝑡, 𝑡0 , 𝑥) V = 𝑒−𝜆(𝑡−𝑡0 ) Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁3 𝑒−𝜆(𝑡−𝑡0 ) 𝑒]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V = 𝑁3 𝑒−]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V ,
(35)
(29)
for all (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌. If we consider ]3 defined as in (34), the second relation in (et3 ) is obtained. Hence, also (iii) in Definition 13 is true and 𝐶 is 𝜔trichotomic.
(30)
Another characterization for the property of 𝜔trichotomy 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.
If we consider that 𝜆 = 2]3 > 0, we obtain, according also to Definition 2 (ec2 ),
Theorem 19. A ∗-strongly measurable skew-evolution semiflow 𝐶 = (𝜑, Φ) is 𝜔-trichotomic if the and only if the following statements hold:
for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌, which shows that Φ3𝜆 is exponentially stable. Also, we have 𝑒−]3 (𝑡−𝑡0 ) 𝑃3 (𝑥) V ≤ 𝑁3 Φ3 (𝑡, 𝑡0 , 𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, ∀ (𝑥, V) ∈ 𝑌.
3 Φ−𝜆 (𝑠, 𝑡0 , 𝑥) V = 𝑒𝜆(𝑠−𝑡0 ) Φ3 (𝑠, 𝑡0 , 𝑥) V ≤ 𝑁3 𝑒]3 (𝑡−𝑠) 𝑒𝜆(𝑠−𝑡0 ) Φ3 (𝑡, 𝑡0 , 𝑥) V = 𝑁3 𝑒−]3 (𝑡−𝑠) Φ3−𝜆 (𝑡, 𝑡0 , 𝑥) V ,
̃ ≥ 1 such that (i) there exists 𝑁 (31)
𝑡 ∗ 𝑝 ̃ 𝑃1 (𝑥) V∗ 𝑝 , ∫ Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠 ≤ 𝑁 𝑡
for all 𝑝 > 1, all (𝑡, 𝑡0 ) ∈ 𝑇, and all (𝑥, V∗ ) ∈ 𝑋 × 𝑉∗ ;
for all (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌, which proves that Φ3−𝜆 is exponentially instable.
(ii) there exists 𝑁 ≥ 1 such that
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 Φ3𝜆 is exponentially stable, then there exists some constants 𝑁 ≥ 1 and ] > 0 such that
𝑝 (∫ Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠) 𝑡
3 Φ𝜆 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑁𝑒−](𝑡−𝑡0 ) 𝑃3 (𝑥) V , ∀𝑡 ≥ 𝑡0 ≥ 0, ∀(𝑥, V) ∈ 𝑌.
𝑡
0
the first relation in (et3 ) is obtained.
≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
(37)
̃ ≥ 1 such that (iii) there exist 𝛼 < 0 and 𝑀 ∞
(32)
̃ 𝑃3 (𝑥) V , ∫ 𝑒𝛼(𝜏−𝑡0 ) Φ3 (𝜏, 𝑡0 , 𝑥) V 𝑑𝜏 ≤ 𝑀 𝑡0
(38)
for all 𝑡0 ∈ R+ and all (𝑥, V) ∈ 𝑌 and there exist 𝛽 < 0 and 𝑀 ≥ 1 such that 𝑡
(33)
∫ 𝑒𝛽(𝑡−𝜏) Φ3 (𝜏, 𝑡0 , 𝑥) V 𝑑𝜏 ≤ 𝑀 Φ3 (𝑡, 𝑡0 , 𝑥) V , 𝑡0
(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 𝜔𝜓 < 0, such that
for all (𝑡, 𝑠), (𝑠, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌. Denoting 𝜆 − ], if 𝜆 > ], ]3 = { 1, if 𝜆 ≤ ],
1/𝑝
for all 𝑝 > 1, all (𝑡, 𝑡0 ) ∈ 𝑇, and all (𝑥, V) ∈ 𝑌;
Further, we obtain 𝜆(𝑡−𝑡0 ) Φ3 (𝑡, 𝑡 , 𝑥) V Φ3 (𝑡, 𝑡0 , 𝑥) V = 𝑒 𝜆 0 ≤ 𝑁𝑒𝜆(𝑡−𝑡0 ) 𝑒−](𝑡−𝑠) Φ3𝜆 (𝑠, 𝑡0 , 𝑥) V = 𝑁𝑒𝜆(𝑡−𝑠) 𝑒−](𝑡−𝑠) Φ3 (𝑠, 𝑡0 , 𝑥) V ,
(36)
0
(34)
Φ (𝑡, 𝑡0 , 𝑥) 𝑃1 (𝑥) V ≤ 𝜓 (𝑡 − 𝑡0 ) 𝑃1 (𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
(40)
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An equivalent relation is obtained, if we consider Definition 2 (ec2 ), given by Φ1 (𝑡, 𝑡0 , 𝑥) V = Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) Φ1 (𝑠, 𝑡0 , 𝑥) V (41) ≤ 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 ] > 0 such that the following relations hold:
0
∗ 𝑝 ≤ ∫ 𝜓𝑝 (𝑡 − 𝑠) Φ1 (𝑠, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠 𝑡0
−𝑝](𝑡−𝑠)
𝑡0
(42)
∗ 𝑝
𝜁 (𝑡 − 𝑡0 ) 𝑃2 (𝑥) V ≤ Φ2 (𝑡, 𝑡0 , 𝑥) V ,
The property of function 𝜁 assures the existence of some constants 𝑁 ≥ 1 and ] > 0, such that, for all 𝑝 > 0, the following relations hold: 1/𝑝
𝑝 (∫ Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠) 𝑡
≤ (∫
𝑡0 +1
Sufficiency. (i) As the evolution cocycle Φ1 has exponential growth, there exist 𝑀 ≥ 1 and 𝜌 > 0 such that 𝜌(𝑡−𝑠) Φ1 (𝑠, 𝑡0 , 𝑥) V , Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑀𝑒 ∀ (𝑡, 𝑠) , (𝑠, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
𝑝 Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥))∗ V∗ 𝑑𝑠)
𝑡0 +1
𝑞 𝑀 𝑃1 (𝑥) V 𝑑𝑠) 𝑞
1/𝑞
𝑡 ∗ 𝑝 ≤ 𝑀 ‖V‖ (∫ Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠) 𝑡
1/𝑝
0
̃ 𝑃1 (𝑥) V 𝑃1 (𝑥) V∗ . ≤ 𝑀𝑁
̃ 𝑀𝑁 𝑃 (𝑥) V , Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V ≤ 𝑐 1
(48)
∀𝑡 ≥ 𝑡0 + 1, ∀(𝑥, V) ∈ 𝑌.
= 𝑐 > 0.
∀ (𝑥, V) ∈ 𝑌, (49)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
(50)
where we have denoted 𝑀1 = 𝑀 (𝑒𝜌 +
̃ 𝑁 ). 𝑐
(51)
Further, the following relations hold: (45)
Let 𝑡 ≥ 𝑡0 + 1. We consider that 1−𝑒 𝜌
(47)
1/𝑝
Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V ≤ 𝑀1 𝑃1 (𝑥) V ,
for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V) ∈ 𝑌, where we have denoted that 𝑁 = 𝑁/(]𝑝)1/𝑝 . (iii) Both relations are obtained by a similar proof as in (i) and (ii), according to Theorem 18.
𝑒−𝜌(𝑠−𝑡0 ) 𝑑𝑠 = ∫ 𝑒−𝜌𝜏 𝑑𝜏 =
Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥))∗ V∗ 𝑑𝑠
which implies that
≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
−𝜌
𝑡0 +1
(44)
𝑡0
0
𝑡0
𝜌 Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V ≤ 𝑀𝑒 𝑃1 (𝑥) V , 1/𝑝
𝑡
𝑡0
∗ 𝑒−𝜌(𝑠−𝑡0 ) Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗
For 𝑡 ∈ [𝑡0 , 𝑡0 + 1] we obtain
≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V × (∫ 𝑒−]𝑝(𝑡−𝑠) 𝑑𝑠)
1
𝑡0 +1
Hence, we have that (43)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
𝑡0 +1
≤∫
𝑡0
∗
for all (𝑡, 𝑡0 ) ∈ 𝑇 and all (𝑥, V ) ∈ 𝑋 × 𝑉 , where we have ̃ = 𝑁𝑝 /]𝑝. denoted that 𝑁 (ii) According to Definition 13, there exists a mapping 𝜁 : R+ → R∗+ , with the property 𝜔𝜁 < 0, such that
∫
𝑡0
× (∫ ∗
0
𝑒−𝜌(𝑠−𝑡0 ) ⟨V∗ , Φ1 (𝑡, 𝑡0 , 𝑥) V⟩ 𝑑𝑠
𝑡0
̃ 𝑃1 (𝑥) V∗ 𝑝 , ≤𝑁
𝑡
𝑡0 +1
𝑡0
𝑃1 (𝑥) V 𝑑𝑠
≤𝑁 ∫ 𝑒
≤∫
≤ 𝑀 ‖V‖ ∫
𝑡
𝑝
𝑐 ⟨V∗ , Φ1 (𝑡, 𝑡0 , 𝑥) V⟩
× Φ1 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠
𝑡 ∗ 𝑝 ∫ Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥)) V∗ 𝑑𝑠 𝑡
𝑡
For every 𝑝 > 1 there exists 𝑞 > 1 such that 1/𝑝 + 1/𝑞 = 1. We obtain
(46)
∗ ⟨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∗ .
(52)
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7 where we have considered 𝜁 : R+ → R∗+ defined by
By integrating on [𝑡0 , 𝑡], it follows that (𝑡 − 𝑡0 ) ⟨V∗ , Φ1 (𝑡, 𝑡0 , 𝑥) V⟩ ≤ 𝑀1 𝑃1 (𝑥) V ∫
𝑡
𝑡0
𝜁 (𝑢) =
Φ1 (𝑡, 𝑠, 𝜑 (𝑠, 𝑡0 , 𝑥))∗ V∗ 𝑑𝑠 (53)
̃ 𝑃1 (𝑥) V𝑃1 (𝑥) V∗ , ≤ 𝑀1 𝑁 and further ̃ 𝑃1 (𝑥) V , (𝑡 − 𝑡0 ) Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑀1 𝑁
(54)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
(55)
̃ + 1)/𝑢, which where we have considered 𝜓(𝑢) = 𝑀1 (𝑁 satisfies 𝜔𝜓 < 0. Statement (𝑡1 ) of Definition 13 is hence obtained. (ii) Let (𝑡, 𝑡0 ) ∈ 𝑇. Let us denote that 𝑡
𝑝 𝐹 (𝑡) = ∫ Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠. 𝑡
with the property 𝜔𝜁 > 0. Statement (𝑡2 ) of Definition 13 is proved. (iii) A similar proof for the 𝛼-shifted evolution cocycle Φ3𝛼 in (i) and for the 𝛽-shifted evolution cocycle Φ3𝛽 in (ii) leads to the existence of two functions 𝜒1 , 𝜒2 : R+ → R∗+ , with the properties 𝜔𝜒1 < 0 and 𝜔𝜒2 > 0, such that
(56)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌, 𝜒2 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V ≤ Φ3𝛽 (𝑡, 𝑡0 , 𝑥) V ,
𝑝
According to the fact that 𝐹(𝑡) ≤ 𝑁 𝐹 (𝑡), for all 𝑡 ≥ 0, we obtain, for all (𝑥, V) ∈ 𝑌,
If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent with −𝛼(𝑡−𝑡0 ) 𝜒1 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V , Φ3 (𝑡, 𝑡0 , 𝑥) V ≤ 𝑒
𝜌(𝑡−𝑠) Φ2 (𝑡, 𝑡0 , 𝑥) V , Φ2 (𝑠, 𝑡0 , 𝑥) V ≤ 𝑀𝑒 ∀ (𝑡, 𝑠) , (𝑠, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
(58)
Let 𝑠 ∈ [𝑡0 , 𝑡0 + 1). We obtain 𝑡 +1
0 𝑝 𝑝 𝑝𝜌(𝑡−𝑠) ∫ 𝑃2 (𝑥) V ≤ 𝑀 𝑒 𝑡 0
𝑝 Φ2 (𝑠, 𝑡0 , 𝑥) V 𝑑𝑠
(59)
and further 𝑝 𝑀−𝑝 𝑒−𝑝𝜌(𝑡−𝑠) 𝑃2 (𝑥) V ≤ 𝐹 (𝑡0 + 1) . 1
𝑝 𝑝 𝑝 𝑝 𝑒−1/𝑁 𝑒(𝑡−𝑡0 )/𝑁 ‖V‖𝑝 ≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V ,
(61)
and, if 𝑡 ∈ [𝑡0 , 𝑡0 + 1], 𝑝
𝜌 1/𝑝𝑁 (−1/𝑝𝑁 𝑒 𝑃2 (𝑥) V ≤ 𝑀𝑒 𝑒
𝑝
)(𝑡−𝑡0 )
Φ2 (𝑡, 𝑡0 , 𝑥) V ,
for all (𝑥, V) ∈ 𝑌. Hence, 𝜁 (𝑡 − 𝑡0 ) 𝑃2 (𝑥) V ≤ Φ2 (𝑡, 𝑡0 , 𝑥) V , ∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
𝑒−𝛽(𝑡−𝑡0 ) 𝜒2 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V ≤ Φ3 (𝑡, 𝑡0 , 𝑥) V ,
(66)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌. ̃ 𝜒 : R+ → R∗+ by 𝜒(𝑢) ̃ We define 𝜒, = 𝑒−𝛼𝑢 𝜒1 (𝑢) and 𝜒(𝑢) = 𝑒−𝛽𝑢 𝜒2 (𝑢), which have the properties 𝜔𝜒̃ < ∞ and 𝜔𝜒 < 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 𝜔trichotomy.
Acknowledgments (60)
For 𝑡 ≥ 𝑡0 + 1, we obtain 𝑀𝑝 𝑒𝜌𝑝
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌,
(57)
As the evolution cocycle Φ2 has exponential decay, there exist some constants 𝑀 ≥ 1 and 𝜌 > 0 such that
(65)
∀ (𝑡, 𝑡0 ) ∈ 𝑇, (𝑥, V) ∈ 𝑌.
0
𝑝 𝑝 𝑝 𝐹 (𝑡0 + 1) 𝑒(𝑡−𝑡0 −1)/𝑁 ≤ 𝐹 (𝑡) ≤ 𝑁 Φ2 (𝑡, 𝑡0 , 𝑥) V .
(64)
3 Φ𝛼 (𝑡, 𝑡0 , 𝑥) V ≤ 𝜒1 (𝑡 − 𝑡0 ) 𝑃3 (𝑥) V ,
which implies that Φ1 (𝑡, 𝑡0 , 𝑥) V ≤ 𝜓 (𝑡 − 𝑡0 ) 𝑃1 (𝑥) V ,
𝑝 𝑝 𝑁 𝑁 1 𝑒−1/𝑝 𝑒(1/𝑝 )𝑢 , 𝑁𝑀𝑒𝜌
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.
References (62)
(63)
[1] M. Megan and C. Stoica, “Exponential instability of skew-evolution semi ows in Banach spaces,” Studia Universitatis “BabesBolyai” Mathematica, vol. 53, no. 1, pp. 17–24, 2008. [2] A. J. G. Bento and C. M. Silva, “Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs,” http://arxiv .org/abs/1210.7740.
8 [3] P. V. Hai, “Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 12, pp. 4390–4396, 2010. [4] P. V. Hai, “Discrete and continuous versions of Barbashintype theorem of linear skew-evolution semiflows,” Applicable Analysis, vol. 90, no. 12, pp. 1897–1907, 2011. [5] M. Megan and C. Stoica, “Concepts of dichotomy for skewevolution semiflows in banach spaces,” Annals of the Academy of Romanian Scientists, vol. 2, no. 2, pp. 125–140, 2010. [6] C. Stoica and M. Megan, “On uniform exponential stability for skew-evolution semiflows on Banach spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 3-4, pp. 1305–1313, 2010. [7] J. Appell, V. Lakshmikantham, N. V. Minh, and P. P. Zabreiko, “A general model of evolutionary processes. Exponential dichotomy-I,” Nonlinear Analysis, vol. 21, no. 3, pp. 207–218, 1993. [8] A. L. Sasu and B. Sasu, “Integral equations, dichotomy of evolution families on the half-line and applications,” Integral Equations and Operator Theory, vol. 66, no. 1, pp. 113–140, 2010. [9] S. Elaydi and O. Hajek, “Exponential trichotomy of differential systems,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 362–374, 1988. [10] M. Megan and C. Stoica, “On uniform exponential trichotomy of evolution operators in banach spaces,” Integral Equations and Operator Theory, vol. 60, no. 4, pp. 499–506, 2008. [11] R. J. Sacker and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems, III,” Journal of Differential Equations, vol. 22, no. 2, pp. 497–522, 1976.
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