Lyapunov Function and Exponential Trichotomy on Time Scales

Abstract

We study the relation between Lyapunov function and exponential trichotomy for the linear equation 𝑥^Δ=𝐴(𝑡)𝑥 on time scales. Furthermore, as an application of these results, we give the roughness of exponential trichotomy on time scales.

1. Introduction

Exponential trichotomy is important for center manifolds theorems and bifurcation theorems. When people analyze the asymptotic behavior of dynamical systems, exponential trichotomy is a powerful tool. The conception of trichotomy was first introduced by Sacker and Sell [1]. They described SS-trichotomy for linear differential systems by linear skew-product flows. Furthermore, Elaydi and Hájek [2, 3] gave the notions of exponential trichotomy for differential systems and for nonlinear differential systems, respectively. These notions are stronger notions than SS-trichotomy. In 1991, Papaschinopoulos [4] discussed the exponential trichotomy for linear difference equations. And in 1999 Hong and his partners [5, 6] studied the relationship between exponential trichotomy and the ergodic solutions of linear differential and difference equations with ergodic perturbations. Recently, Barreira and Valls [7, 8] gave the conception of nonuniform exponential trichotomy. From their papers, we can see that the exponential trichotomy studied before is just a special case of the nonuniform exponential trichotomy. For more information about exponential trichotomy we refer the reader to papers [9–14].

Many phenomena in nature cannot be entirely described by discrete system, or by continuous system, such as insect population model, the large population in the summer, the number increases, a continuous function can be shown. And in the winter the insects freeze to death or all sleep, their number reduces to zero, until the eggs hatch in the next spring, the number increases again, this process is a jump process. Therefore, this population model is a discontinuous jump function, it cannot be expressed by a single differential equation, or by a single difference equations. Is it possible to use a unified framework to represent the above population model? In 1988, Hilger [15] first introduced the theory of time scales. From then on, there are numerous works on this area (see [16–23]). Time scales provide a method to unify and generalize theories of continuous and discrete dynamical systems.

In this paper, motivated by [8], we study the exponential trichotomy on time scales. We firstly introduce (𝜆,𝜇)-Lyapunov function on time scales. Then we study the relationship between exponential trichotomy and (𝜆,𝜇)-Lyapunov function on time scales. We obtain that the linear equation 𝑥^Δ=𝐴(𝑡)𝑥 admits exponential trichotomy, if it has two (𝜆,𝜇)-Lyapunov functions with some property; conversely, the linear equation has two (𝜆,𝜇)-Lyapunov functions, if it admits strict exponential trichotomy. At last, by using these results we investigate the roughness of exponential trichotomy on time scales. Above all, our paper gives a way to unify the analysis of continuous and discrete exponential trichotomy.

This paper is organized as follows. In Section 2, we review some useful notions and basic properties on time scales. Our main results will be stated and proved in Section 3. Finally, in Section 4, we study the roughness of exponential trichotomy on time scales.

2. Preliminaries on Time Scales

In order to make our paper independent, some preliminary definitions and theories on time scales are listed below.

Definition 2.1. Let 𝕋 be a time scale which is an arbitrary nonempty closed subset of the real numbers. The forward jump operator is defined by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡}, (2.1) while the backward jump operator is defined by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶s<𝑡}, (2.2) for every 𝑡∈𝕋. If 𝜎(𝑡)=𝑡, then 𝑡 is called right-dense. And if 𝜌(𝑡)=𝑡, then 𝑡 is called left-dense. Let 𝜇(𝑡)∶=𝜎(𝑡)−𝑡 be the graininess function.

For example, the set of real numbers ℝ is a time scale with 𝜎(𝑡)=𝑡 and 𝜇(𝑡)=0 for 𝑡∈𝕋=ℝ and the set of integers ℤ is a time scale with 𝜎(𝑡)=𝑡+1 and 𝜇(𝑡)=1 for 𝑡∈𝕋=ℤ.

Definition 2.2. A function 𝑓∶𝕋→ℝ^𝑛 is called rd-continuous if it is continuous at right-dense points in 𝕋 and left-sided limits exist at left dense points in 𝕋.

We denote the set of rd-continuous functions by 𝐶_rd(𝕋,ℝ^𝑛).

Definition 2.3. We say that a function 𝑓∶𝕋→ℝ^𝑛 is differentiable at 𝑡∈𝕋, if for any 𝜀>0, there exist 𝕋-neighborhood 𝑈 of 𝑡 and 𝑓^Δ(𝑡)∈ℝ^𝑛 such that for any 𝑠∈𝑈 one has 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤𝜀(𝜎(𝑡)−𝑠), (2.3) and 𝑓^Δ(𝑡) is called the derivative of 𝑓 at 𝑡.

Relate to the differential properties, if 𝑓 and 𝑔 is differentiable at 𝑡, then we have the following equalities:𝑓(𝜎(𝑡))=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡),𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡). (2.4)(2.5)

Definition 2.4. We say that a function 𝑝∶𝕋→ℝ is regressive if 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋. Furthermore, if 1+𝜇(𝑡)𝑝(𝑡)>0, 𝑝 is called positive regressive. An 𝑛×𝑛 matrix valued function 𝐴(𝑡) on time scale 𝕋 is called regressive if Id+𝜇(𝑡)𝐴(𝑡) is invertible for all 𝑡∈𝕋.

Let 𝑎,𝑏 be regressive. Define (𝑎⊕𝑏)(𝑡)=𝑎(𝑡)+𝑏(𝑡)+𝜇(𝑡)𝑎(𝑡)𝑏(𝑡) and (⊖𝑎)(𝑡)=−𝑎(𝑡)/(1+𝜇(𝑡)𝑎(𝑡)) for all 𝑡∈𝕋. Then the regressive set is a Abelian group and it is not hard to verify that the following properties hold:(1)𝑎⊖𝑎=0; (2)⊖(⊖𝑎)=𝑎; (3)𝑎⊖𝑏=(𝑎−𝑏)/(1+𝜇(𝑡)𝑏); (4)⊖(𝑎⊕𝑏)=(⊖𝑎)⊕(⊖𝑏), where 𝑎,𝑏 are regressive.

In order to make our paper intelligible, the exponential function on time scales which we will use in our paper is defined as following. For the more general conception of exponential function on time scales, please refer to [17].

Definition 2.5. Let 𝑝 be positive regressive. We define the exponential function by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠1𝜇(𝜏)ln1+𝜇(𝜏)𝑝(𝜏)Δ𝜏, (2.6) for all 𝑠,𝑡∈𝕋. Here the integral is always understood in Lebesgue's sense and if 𝑓∈𝐶_rd(𝕋,ℝ^𝑛), for any 𝑡∈𝕋 one has _𝑡^𝜎(𝑡)𝑓(𝑠)Δ𝑠=𝜇(𝑡)𝑓(𝑡). (2.7)

From the definition of exponential function we have that if 𝑎,𝑏 are positive regressive, then(1)𝑒₀(𝑡,𝑠)=1 and 𝑒_𝑎(𝑡,𝑡)=1;(2)𝑒_𝑎(𝑡,𝑠)=𝑒_⊖𝑎(𝑠,𝑡);(3)𝑒_𝑎(𝑡,𝑠)𝑒_𝑎(𝑠,𝑟)=𝑒_𝑎(𝑡,𝑟);(4)𝑒_𝑎(𝑡,𝑠)𝑒_𝑏(𝑡,𝑠)=𝑒_𝑎⊕𝑏(𝑡,𝑠);(5)[𝑒_𝑎(𝑐,𝑡)]^Δ_𝑡=−𝑎(𝑡)𝑒_𝑎(𝑐,𝜎(𝑡)), where [𝑒_𝑎(𝑐,𝑡)]^Δ_𝑡 stands for the delta derivative of 𝑒_𝑎(𝑐,𝑡) with respect to 𝑡.

Lemma 2.6 (L'Hôpital's Rule). Suppose that 𝑓 and 𝑔 are differentiable on 𝕋. Then if 𝑔(𝑡) satisfies 𝑔(𝑡)>0, 𝑔^Δ>0 and lim_𝑡→+∞𝑔(𝑡)=∞, then the existence of lim_𝑡→+∞𝑓^Δ(𝑡)/𝑔^Δ(𝑡) implies that lim_𝑡→+∞𝑓(𝑡)/𝑔(𝑡) exists and lim_𝑡→+∞𝑓(𝑡)/𝑔(𝑡)=lim_𝑡→+∞𝑓^Δ(𝑡)/𝑔^Δ(𝑡).

3. Exponential Trichotomy and Lyapunov Function on Time Scales

From now on, we always suppose that 𝕋 is a two-sides infinite time scale and the graininess function 𝜇(𝑡) is bounded, which means that there exists a 𝑀>0 such that 0≤𝜇(𝑡)≤𝑀. We consider the following linear equation𝑥^Δ=𝐴(𝑡)𝑥, (3.1) where 𝐴(𝑡) is an 𝑛×𝑛 matrix valued function on time scale 𝕋, satisfying that 𝐴(𝑡) is rd-continuous and regressive. To make it simple, in this paper we always require that 𝐴(𝑡) is regressive. If 𝐴(𝑡) is not regressive, from [20] we can know that the linear evolution operator 𝑇(𝑡,𝜏) associated to (3.1) is not invertible and exists only for 𝜏≤𝑡 which will make our paper complicated. Here, 𝑇(𝑡,𝜏) is called the linear evolution operator if 𝑇(𝑡,𝜏) satisfies the following conditions: (1)𝑇(𝜏,𝜏)=Id; (2)𝑇(𝑡,𝜏)𝑇(𝜏,𝑠)=𝑇(𝑡,𝑠); (3)the mapping (𝑡,𝜏)→𝑇(𝑡,𝜏)𝑥 is continuous for any fixed 𝑥∈ℝ^𝑛.

Equation (3.1) is said to admit an exponential trichotomy on time scale 𝕋 if there exist projections 𝑃(𝑡), 𝑄(𝑡), 𝑅(𝑡)∶ℝ^𝑛→ℝ^𝑛 for each 𝑡∈𝕋 such that 𝑃(𝑡)+𝑄(𝑡)+𝑅(𝑡)=Id,𝑇(𝑡,𝜏)𝑃(𝜏)=𝑃(𝑡)𝑇(𝑡,𝜏),𝑇(𝑡,𝜏)𝑄(𝜏)=𝑄(𝑡)𝑇(𝑡,𝜏),𝑇(𝑡,𝜏)𝑅(𝜏)=𝑅(𝑡)𝑇(𝑡,𝜏) (3.2) for 𝑡,𝜏∈𝕋 and there are some constants 𝑎>𝑏≥0 and 𝐷>1 such that for 𝑡≥𝜏,𝑡,𝜏∈𝕋‖𝑇(𝑡,𝜏)𝑃(𝜏)‖≤𝐷𝑒_𝑎(𝜏,𝑡),‖𝑇(𝑡,𝜏)𝑅(𝜏)‖≤𝐷𝑒_𝑏(𝑡,𝜏), (3.3) and for 𝑡≤𝜏,𝑡,𝜏∈𝕋‖𝑇(𝑡,𝜏)𝑄(𝜏)‖≤𝐷𝑒_𝑎(𝑡,𝜏),‖𝑇(𝑡,𝜏)𝑅(𝜏)‖≤𝐷𝑒_𝑏(𝜏,𝑡). (3.4) We say that (3.1) admits a strong exponential trichotomy on time scale 𝕋 if there exist 𝑃(𝑡), 𝑄(𝑡), 𝑅(𝑡)∶ℝ^𝑛→ℝ^𝑛 for each 𝑡∈𝕋 and constants 𝑎>𝑏≥0, 𝐷>1 satisfying (3.2), (3.3), (3.4), as well as a constant 𝑐 with 𝑐≥𝑎 and 𝑐>1 such that‖𝑇(𝑡,𝜏)𝑃(𝜏)‖≤𝐷𝑒_𝑐()‖𝑇()𝑄(𝜏𝜏,𝑡,𝑡≤𝜏,𝑡,𝜏∈𝕋,𝑡,𝜏)‖≤𝐷𝑒_𝑐(𝑡,𝜏),𝑡≥𝜏,𝑡,𝜏∈𝕋. (3.5) Obviously, if for any 𝑡∈𝕋 we have 𝜇(𝑡)≡0 or 𝜇(𝑡)≡1, then the notion of exponential trichotomy on time scale 𝕋 becomes the exponential trichotomy for differential equation or difference equation, respectively.

Next, we give the notion of Lyapunov function. Consider a function 𝑉∶𝕋×ℝ^𝑛→ℝ^𝑛. If the following two conditions are satisfied (H1)for each 𝜏∈𝕋 set 𝑉_𝜏∶=𝑉(𝜏,⋅) and 𝐶^𝑠𝑉_𝜏∶={0}∪𝑉_𝜏⁻¹(−∞,0),𝐶^𝑢𝑉_𝜏∶={0}∪𝑉_𝜏⁻¹(0,+∞). (3.6)let 𝑟_𝑠 and 𝑟_𝑢 be, respectively, the maximal dimensions of linear subspaces inside 𝐶^𝑠(𝑉_𝜏) and 𝐶^𝑢(𝑉_𝜏), then we have 𝑟_𝑠+𝑟_𝑢=𝑛; (H2)for every 𝑡≥𝜏, 𝑡,𝜏∈𝕋 and 𝑥∈ℝ^𝑛, we have 𝑇(𝜏,𝑡)𝐶^𝑠(𝑉_𝑡)⊂𝐶^𝑠(𝑉_𝜏) and 𝑇(𝑡,𝜏)𝐶^𝑢(𝑉_𝜏)⊂𝐶^𝑢(𝑉_𝑡),

then we say that the function 𝑉 is a Lyapunov function.

Let 𝐻^𝑠_𝜏∶=_𝑟∈𝕋𝑇(𝜏,𝑟)𝐶^𝑠𝑉_𝑟⊂𝐶^𝑠𝑉_𝜏,𝐻^𝑢_𝜏∶=_𝑟∈𝕋𝑇(𝜏,𝑟)𝐶^𝑢𝑉_𝑟⊂𝐶^𝑢𝑉_𝜏. (3.7) By the notion of Lyapunov function, we can see that there are subspaces 𝐷^𝑠_𝜏(𝑉) and 𝐷^𝑢_𝜏(𝑉) such that 𝐷^𝑠_𝜏(𝑉)⊕𝐷^𝑢_𝜏(𝑉)=ℝ^𝑛. The function 𝑉 is called a (𝜆,𝜇)-Lyapunov function if 𝑉 is a Lyapunov function and for 𝑡≥𝜏,𝑡,𝜏∈𝕋, 𝑉 satisfies the following conditions: (L1)|𝑉(𝑡,𝑥)|≤𝑁‖𝑥‖, (L2)for 𝑥∈𝐷^𝑠_𝜏(𝑉), 𝑉²(𝑡,𝑇(𝑡,𝜏)𝑥)≤𝑒_𝜆(𝑡,𝜏)𝑉²(𝜏,𝑥), (L3)for 𝑥∈𝐷^𝑢_𝜏(𝑉), 𝑉²(𝑡,𝑇(𝑡,𝜏)𝑥)≥𝑒_𝜇(𝑡,𝜏)𝑉²(𝜏,𝑥), (L4)for 𝑥∈𝐷^𝑠_𝜏(𝑉)⋃𝐷^𝑢_𝜏(𝑉), |𝑉(𝑡,𝑥)|≥‖𝑥‖/𝑁,

where 𝜆<𝜇 and 𝑁 is a constant with 𝑁>1.

Furthermore, let 𝑆(𝑡) for each 𝑡∈𝕋 be a symmetric invertible 𝑛×𝑛 matrix. Set 𝐻(𝑡,𝑥)∶=⟨𝑆(𝑡)𝑥,𝑥⟩. If 𝑉(𝑡,𝑥)∶=−sign𝐻(𝑡,𝑥)|𝐻(𝑡,𝑥)| is a Lyapunov function ((𝜆,𝜇)-Lyapunov function), then 𝑉(𝑡,𝑥) is called a quadric Lyapunov function ((𝜆,𝜇)-quadric Lyapunov function). Let ‖𝑆(𝑡)‖∶=sup_{𝑥∈ℝ𝑛}|𝐻(𝑡,𝑥)|/‖𝑥‖². Now, we state and prove our main results.

Theorem 3.1. Suppose that (3.1) has two (𝜆,𝜇)-quadratic Lyapunov functions, 𝑉(𝑡,𝑥) with (𝑟₁,𝑟₂) and 𝑊(𝑡,𝑥) with (𝑙₁,𝑙₂) satisfying 𝑟₁<𝑟₂<0<𝑙₁<𝑙₂. If the symmetric invertible 𝑛×𝑛 matrixes 𝑆(𝑡) and 𝑇(𝑡) for 𝑉(𝑡,𝑥) and 𝑊(𝑡,𝑥), respectively, satisfy sup_𝑡∈𝕋‖𝑆(𝑡)‖<∞ and sup_𝑡∈𝕋‖𝑇(𝑡)‖<∞, then (3.1) has an exponential trichotomy.

In order to prove Theorem 3.1, we need some lemmas.

Lemma 3.2. The subspaces 𝐷^𝑠_𝜏(𝑉), 𝐷^𝑢_𝜏(𝑉), 𝐷^𝑠_𝜏(𝑊), and 𝐷^𝑢_𝜏(𝑊) according to Lyapunov functions 𝑉 and 𝑊 have the following relations 𝐷^𝑠_𝜏(𝑉)⊂𝐷^𝑠_𝜏(𝑊),𝐷^𝑢_𝜏(𝑊)⊂𝐷^𝑢_𝜏(𝑉). (3.8)

Proof. By the knowledge of mathematical analysis, we can easily get that ln(1+𝑙𝑥)/𝑥 is a decreasing function for any 𝑙∈ℝ. Thus, when 0≤𝑥≤𝑀, we have ln(1+𝑙𝑀)/𝑀≤ln(1+𝑙𝑥)/𝑥≤𝑙. Now, we prove this lemma by contradiction. Suppose that there is 𝑥∈𝐷^𝑠_𝜏(𝑉)⧵𝐷^𝑠_𝜏(𝑊). Then there are 𝑦∈𝐷^𝑠_𝜏(𝑊) and 𝑧∈𝐷^𝑢_𝜏(𝑊)⧵{0} such that 𝑥=𝑦+𝑧. Since 𝑟₁<𝑙₁<𝑙₂, then by Lemma 2.6 we get lim_𝑡→+∞1𝑡ln|𝑉(𝑡,𝑇(𝑡,𝜏)𝑥)|≥lim_𝑡→+∞1𝑡‖𝑇()𝑥‖ln𝑡,𝜏𝑁≥lim_𝑡→+∞1𝑡ln‖𝑇(𝑡,𝜏)𝑧‖−𝑇(𝑡,𝜏)𝑦≥lim_𝑡→+∞1𝑡𝑉(()𝑧)ln𝑡,𝑇𝑡,𝜏𝑁−𝑁𝑉𝑡,𝑇(𝑡,𝜏)𝑦≥lim_𝑡→+∞1𝑡ln𝑒_𝑙2(𝑡,𝜏)𝑉(𝜏,𝑧)𝑁−𝑁𝑒_𝑙1(𝑡,𝜏)𝑉𝜏,𝑦≥lim_𝑡→+∞1𝑡ln𝑒_𝑙1(𝑡,𝜏)=lim_𝑡→+∞12𝑡^𝑡_𝜏1𝜇(𝑠)ln1+𝜇(𝑠)𝑙₁Δ𝑠≥lim_𝑡→+∞𝑡−𝜏2𝑡ln1+𝑙₁𝑀𝑀=ln1+𝑙₁𝑀2𝑀>0. (3.9) But for 𝑥∈𝐷^𝑠_𝜏(𝑉), one has lim_𝑡→+∞1𝑡ln|𝑉(𝑡,𝑇(𝑡,𝜏)𝑥)|≤lim_𝑡→+∞1𝑡ln𝑒_𝑟1(𝑉(≤𝑟𝑡,𝜏)|𝜏,𝑥)|¹2<0. (3.10) It is a contradiction. So we have 𝐷^𝑠_𝜏(𝑉)⊂𝐷^𝑠_𝜏(𝑊).
By the concept of Lyapunov function, we get that for 𝑡≤𝜏, 𝑉^′(𝜏,𝑥)≤𝑒_𝜆(𝜏,𝑡)𝑉^′(𝑡,𝑇(𝑡,𝜏)𝑥),𝑥∈𝐷^𝑠_𝜏,𝑉^′(𝜏,𝑥)≥𝑒_𝜇(𝜏,𝑡)𝑉^′(𝑡,𝑇(𝑡,𝜏)𝑥),𝑥∈𝐷^𝑢_𝜏, (3.11) where 𝑉′=𝑉 or 𝑊, 𝜆=𝑟₁ or 𝑙₁ and 𝜇=𝑟₂ or 𝑙₂. Using above inequalities and the similar way by which we get 𝐷^𝑠_𝜏(𝑉)⊂𝐷^𝑠_𝜏(𝑊), we can directly obtain that 𝐷^u_𝜏(𝑊)⊂𝐷^𝑢_𝜏(𝑉). Here we omit the detailed proofing. Thus, the proof is completed.

Obviously, from Lemma 3.2 we have (𝐷^𝑢_𝜏(𝑉)⋂𝐷^𝑠_𝜏(𝑊))⋂𝐷^𝑢_𝜏(𝑊)={0}, (𝐷^𝑢_𝜏(𝑉)∩𝐷^𝑠_𝜏(𝑊))⋂𝐷^𝑠_𝜏(𝑉)={0} and 𝐷^𝑢_𝜏(𝑊)⋂𝐷^𝑠_𝜏(𝑉)={0}. Furthermore, dim𝐷^𝑢_𝜏(𝑉)𝐷^𝑠_𝜏(𝑊)=dim𝐷^𝑢_𝜏(𝑉)+dim𝐷^𝑠_𝜏(𝑊)−dim𝐷^𝑢_𝜏(𝑉)+𝐷^𝑠_𝜏(𝑊)=𝑛−dim𝐷^𝑠_𝜏(𝑉)+𝑛−dim𝐷^𝑢_𝜏(𝑊)−dim𝐷^𝑢_𝜏(𝑉)+𝐷^𝑠_𝜏(𝑊). (3.12) Thus, dim𝐷^𝑢_𝜏(𝑉)𝐷^𝑠_𝜏(𝑊)+dim𝐷^𝑠_𝜏(𝑉)+dim𝐷^𝑢_𝜏(𝑊)≥2𝑛−dim𝐷^𝑢_𝜏(𝑉)+𝐷^𝑠_𝜏(𝑊)≥𝑛. (3.13) So we obtain 𝐷^𝑢_𝜏(𝑉)𝐷^𝑠_𝜏(𝑊)⊕𝐷^𝑠_𝜏(𝑉)⊕𝐷^𝑢_𝜏(𝑊)=ℝ^𝑛. (3.14) Then from the conditions in Theorem 3.1 we have the following results.

When 𝑡≥𝜏, for 𝑥∈𝐷^𝑠_𝜏(𝑉), we get‖𝑇(𝑡,𝜏)𝑥‖≤𝑁|𝑉(𝑡,𝑇(𝑡,𝜏)𝑥)|≤𝑁𝑒_𝑟1(𝑡,𝜏)|𝑉(𝜏,𝑥)|≤𝑁²𝑒_𝑟1(𝑡,𝜏)‖𝑥‖, (3.15) and for 𝑥∈𝐷^𝑢_𝜏(𝑉)⋂𝐷^𝑠_𝜏(𝑊), we get‖𝑇(𝑡,𝜏)𝑥‖≤𝑁|𝑉(𝑡,𝑇(𝑡,𝜏)𝑥)|≤𝑁𝑒_𝑙1(𝑡,𝜏)|𝑉(𝜏,𝑥)|≤𝑁²𝑒_𝑙1(𝑡,𝜏)‖𝑥‖. (3.16)

When 𝑡≤𝜏, for 𝑥∈𝐷^𝑢_𝜏(𝑊), we get‖𝑇(𝑡,𝜏)𝑥‖≤𝑁|𝑉(𝑡,𝑇(𝑡,𝜏)𝑥)|≤𝑁𝑒_𝑙2(𝑡,𝜏)|𝑉(𝜏,𝑥)|≤𝑁²𝑒_𝑙2(𝑡,𝜏)‖𝑥‖, (3.17) and for 𝑥∈𝐷^𝑢_𝜏(𝑉)⋂𝐷^𝑠_𝜏(𝑊), we get‖𝑇(𝑡,𝜏)𝑥‖≤𝑁|𝑉(𝑡,𝑇(𝑡,𝜏)𝑥)|≤𝑁𝑒_𝑟2(𝑡,𝜏)|𝑉(𝜏,𝑥)|≤𝑁²𝑒_𝑟2(𝑡,𝜏)‖𝑥‖. (3.18)

Let 𝑃_𝑉(𝜏)∶ℝ^𝑛⟶𝐷^𝑠_𝜏(𝑉),𝑄_𝑉(𝜏)∶ℝ^𝑛⟶𝐷^𝑢_𝜏(𝑉),𝑃_𝑊(𝜏)∶ℝ^𝑛⟶𝐷^𝑠_𝜏(𝑊),𝑄_𝑊(𝜏)∶ℝ^𝑛⟶𝐷^𝑢_𝜏(𝑊) (3.19) be projections. Set 𝑃(𝜏)∶=𝑃_𝑉(𝜏), 𝑄(𝜏)∶=𝑄_𝑊(𝜏) and 𝑅(𝜏)∶=𝑃_𝑊(𝜏)+𝑄_𝑉(𝜏). From Lemma 3.2, we have 𝑄_𝑉(𝜏)𝑃_𝑊(𝜏)=𝑃_𝑊(𝜏)𝑄_𝑉(𝜏)=0. Thus, by simply deducing, we get that 𝑅(𝜏) is also a projection. Set 𝐿₁(𝜏)=Id−𝑃(𝜏). Next, we will give the boundedness of 𝑃(𝜏), 𝑄(𝜏), and 𝑅(𝜏). By the conditions of Theorem 3.1, we suppose that there is a constant 𝐿>0 such that sup_𝑡∈𝕋‖𝑆(𝑡)‖≤𝐿 and sup_𝑡∈𝕋‖𝑇(𝑡)‖≤𝐿. We have the following lemma.

Lemma 3.3. ‖𝑃(𝑡)𝑥‖≤√2𝑁²‖𝑥‖,‖𝑄(𝑡)𝑥‖≤√2𝑁²‖𝑥‖,‖𝑅(𝑡)𝑥‖≤2√2𝑁²‖𝑥‖. (3.20)

Proof. Firstly, one has 𝑉((𝑡)𝑥)=⟨𝑆(𝑡)𝑃(𝑡)(𝑡)𝑥⟩≥‖𝑃(𝑡)𝑥‖𝑡,𝑃𝑥,𝑃²𝑁²,𝑉𝑡,𝐿₁(𝑡)𝑥=−𝑆(𝑡)𝐿₁(𝑡)𝑥,𝐿₁(𝑡)𝑥≥‖𝑄(𝑡)𝑥‖²𝑁². (3.21) Since 1𝑁²𝑃(𝑡)𝑁𝑥−²2𝑆(𝑡)𝑥²+1𝑁²𝐿₁(𝑡)𝑁𝑥−²2𝑆(𝑡)𝑥²=1𝑁²‖𝑃(𝑡)𝑥‖²+1𝑁²𝐿₁(𝑡)𝑥²+𝑁²2‖𝑆(𝑡)𝑥‖²−⟨𝑃(𝑡)𝑥,𝑆(𝑡)𝑥⟩+⟨𝐿₁(𝑡)(𝑡)≤⟨𝑆(𝑡)𝑃(𝑡)(𝑡)𝑥⟩−𝑆(𝑡)𝐿𝑥,𝑆𝑥⟩𝑥,𝑃₁(𝑡)𝑥,𝐿₁(𝑡)𝑥+𝑁²2‖𝑆(𝑡)𝑥‖²−⟨𝑃(𝑡)𝑥,𝑆(𝑡)𝑥⟩+𝐿₁(𝑡)(𝑡)𝑥=𝑁𝑥,𝑆₂2‖𝑆(𝑡)𝑥‖², (3.22) then we obtain ‖𝑃(𝑡)𝑥‖≤𝑃(𝑡)𝑁𝑥−²2𝑆(𝑡)𝑥+N²2‖𝑆(𝑡)𝑥‖≤𝑁²√2‖𝑆(𝑡)𝑥‖+𝑁²2‖𝑆(𝑡)𝑥‖≤√2𝑁²‖𝑆(𝑡)𝑥‖≤√2𝑁²𝐿‖𝑥‖. (3.23) Similarly, we get ‖𝑄(𝑡)𝑥‖≤√2𝑁²𝐿‖𝑥‖,‖𝑅(𝑡)𝑥‖≤2√2𝑁²𝐿‖𝑥‖. (3.24)

Proof of Theorem 3.1. Firstly, we need to show that there are constants 𝑟^′₁,𝑟^′₂<0 and 𝑙^′₁,𝑙^′₂>0 such that when 𝑡≥𝜏, one has 𝑒_𝑟1(𝑡,𝜏)≤𝑒_𝑟′1(𝑡,𝜏),𝑒_𝑙1(𝑡,𝜏)≤𝑒_𝑙′1(𝑡,𝜏), (3.25)(3.26) and when 𝑡≤𝜏, one has 𝑒_𝑙2(𝑡,𝜏)≤𝑒_𝑙′2(𝑡,𝜏),𝑒_𝑟2(𝑡,𝜏)≤𝑒_𝑟′2(𝑡,𝜏). (3.27) For inequality (3.25), by the definition of exponential function, we only need to find a constant 𝑟^′₁<0 such that 1+𝜇(𝑡)𝑟₁≤1+𝜇(𝑡)𝑟^′₁². (3.28) That means 𝑟₁≤2𝑟^′₁+𝜇(𝑡)𝑟₁^′2. (3.29) Thus, if 𝑟₁/2≤𝑟^′₁<0, (3.25) always holds. Similarly, we get that if 𝑙^′₁≥𝑙₁/2, 0<𝑙^′₂<(1+𝑀𝑙₂−1)/𝑀 and (−1+𝑀𝑟₂−1)/𝑀<𝑟^′₂<0, then (3.26), (3.27) always hold, respectively. Here, we notice that since 𝑟₂ is positive regressive, then 1+𝑀𝑟₂ is significative.
By (3.15)–(3.18) and Lemma 3.3, we get that when 𝑡≥𝜏, ‖𝑇(𝑡,𝜏)𝑃(𝜏)𝑥‖≤𝑇(𝑡,𝜏)|_𝐷^𝑠_𝜏(𝑉)𝑃(𝜏)𝑥‖≤√‖2𝑁²𝐿𝑒_𝑟1(𝑡,𝜏)‖𝑥‖≤√2𝑁²𝐿𝑒_𝑟′1(𝑥‖,‖𝑇()𝑅(𝜏)𝑥‖≤𝑇(𝑡,𝜏)‖𝑡,𝜏𝑡,𝜏)|_𝐷^𝑠_{𝜏⋂𝐷(𝑊)}^𝑢_𝜏(𝑉)𝑅(𝜏)𝑥‖√‖≤22𝑁²𝐿𝑒_𝑙1(𝑡,𝜏)‖𝑥‖≤2√2𝑁²𝐿𝑒_𝑙′1(𝑡,𝜏)‖𝑥‖, (3.30) and when 𝑡≤𝜏, ‖𝑇(𝑡,𝜏)𝑄(𝜏)𝑥‖≤𝑇(𝑡,𝜏)|_𝐷^𝑢_𝜏(𝑊)𝑄(𝜏)𝑥‖≤√‖2𝑁²𝐿𝑒_𝑙2(𝑡,𝜏)‖𝑥‖≤√2𝑁²𝐿𝑒_𝑙′2(𝑥‖,‖𝑇()𝑅(𝜏)𝑥‖≤𝑇(𝑡,𝜏)‖𝑡,𝜏𝑡,𝜏)|_𝐷^𝑠_{𝜏⋂𝐷(𝑊)}^𝑢_𝜏(𝑉)𝑅(𝜏)𝑥‖√‖≤22𝑁²𝐿𝑒_𝑟2(𝑡,𝜏)‖𝑥‖≤2√2𝑁²𝐿𝑒_𝑟′2(𝑡,𝜏)‖𝑥‖. (3.31) Set 𝐷∶=2√2𝑁²𝐿, 𝑎∶=min{|𝑟^′₁|,𝑙^′₂}, and 𝑏∶=max{|𝑟^′₂|,𝑙^′₁}. Obviously, we have 𝑎>𝑏 and the above inequalities imply that for 𝑡≥𝜏,𝑡,𝜏∈𝕋 one has ‖𝑇(𝑡,𝜏)𝑃(𝜏)‖≤𝐷𝑒_𝑎(𝜏,𝑡),‖𝑇(𝑡,𝜏)𝑅(𝜏)‖≤𝐷𝑒_𝑏(𝑡,𝜏), (3.32) and for 𝑡≤𝜏,𝑡,𝜏∈𝕋 one has ‖𝑇(𝑡,𝜏)𝑄(𝜏)‖≤𝐷𝑒_𝑎(𝑡,𝜏),‖𝑇(𝑡,𝜏)𝑅(𝜏)‖≤𝐷𝑒_𝑏(𝜏,𝑡). (3.33) Thus, from the definition of exponential trichotomy on time scales, we get that (3.1) has an exponential trichotomy on time scale 𝕋. This completes the proof of Theorem 3.1.

Next, we give an approximately converse result of Theorem 3.1.

Theorem 3.4. If (3.1) admits a strict exponential trichotomy, then for (3.1) there exist two (𝜆,𝜇)-Lyapunov functions, 𝑉′(𝑡,𝑥) with (⊖𝑟^′₁,⊖𝑟^′₂) and 𝑊′(𝑡,𝑥) with (𝑙^′₁,𝑙^′₂) satisfying ⊖𝑟^′₁<⊖𝑟^′₂<0<𝑙^′₁<𝑙^′₂.

Proof. Suppose that (3.1) admits a strict exponential trichotomy with projections 𝑃(𝑡), 𝑄(𝑡), 𝑅(𝑡), and constants 𝑐≥𝑎>𝑏≥0 satisfying (3.2)–(3.5). Set 𝑆(𝑡)∶=_𝑡^+∞(𝑇(𝜎(𝑣),𝑡)𝑃(𝑡))^∗𝑇(𝜎(𝑣),𝑡)𝑃(𝑡)𝑒_𝑐1⊕𝑏(𝜎(𝑣))−,𝑡Δ𝑣_𝑡−∞𝑇(𝜎(𝑣),𝑡)𝐿₁(𝑡)^∗𝑇(𝜎(𝑣),𝑡)𝐿₁(𝑡)𝑒_𝑐2⊕𝑏(𝜎(𝑣),𝑡)Δ𝑣, (3.34) where 𝐿₁(𝑡)=𝑄(𝑡)+𝑅(𝑡) and 𝑏<𝑐₂<𝑐₁<𝑎. Let 𝐻(𝑡,𝑥)∶=⟨𝑆(𝑡)𝑥,𝑥⟩.
When 𝑡≥𝜏,𝑡,𝜏∈𝕋, we get 𝐻(𝑡,𝑇(𝑡,𝜏)𝑥)=_𝑡^+∞‖𝑇(𝜎(𝑣),𝜏)𝑃(𝜏)𝑥‖²𝑒_𝑐1⊕𝑏(𝜎(𝑣))−,𝑡Δ𝑣_𝑡−∞𝑇(𝜎(𝑣),𝜏)𝐿₁(𝜏)𝑥²𝑒_𝑐2⊕𝑏(𝜎(𝑣))≤,𝑡Δ𝑣_𝜏+∞‖𝑇(𝜎(𝑣),𝜏)𝑃(𝜏)𝑥‖²𝑒_𝑐1⊕𝑏(𝜎(𝑣),𝜏)Δ𝑣𝑒_𝑐1⊕𝑏()−𝜏,𝑡_𝜏−∞𝑇(𝜎(𝑣),𝜏)𝐿₁(𝜏)𝑥²𝑒_𝑐2⊕𝑏(𝜎(𝑣),𝜏)Δ𝑣𝑒_𝑐2⊕𝑏(𝜏,𝑡)≤𝑒_𝑐2⊕𝑏(𝜏,𝑡)𝐻(𝜏,𝑥). (3.35) Set 𝑉(𝑡,𝑥)∶=−sign𝐻(𝑡,𝑥)|𝐻(𝑡,𝑥)|. Then if 𝑉(𝜏,𝑥)>0, that is, 𝑥∈𝐶^𝑢(𝑉_𝜏), we have 𝐻(𝜏,𝑥)<0. By (3.35) we obtain that 𝐻(𝑡,𝑇(𝑡,𝜏))<0. Then we get 𝑉(𝑡,𝑇(𝑡,𝜏)𝑥)>0, that is, 𝑇(𝑡,𝜏)𝑥∈𝐶^𝑢(𝑉_𝑡). That means 𝑇(𝑡,𝜏)𝐶^𝑢(𝑉_𝜏)⊂𝐶^𝑢(𝑉_𝑡). Similarly, we have 𝑇(𝜏,𝑡)𝐶^𝑠(𝑉_𝑡)⊂𝐶^𝑠(𝑉_𝜏). Thus, 𝑉(𝑡,𝑥) is a Lyapunov function. By (3.3), (3.4) and the characters of exponential function on time scales, one has |𝐻(𝑡,𝑥)|≤_𝑡^+∞‖𝑇(𝜎(𝑣),𝑡)𝑃(𝑡)𝑥‖²𝑒_𝑐1⊕𝑏(𝜎(𝑣))+,𝑡Δ𝑣_𝑡−∞𝑇(𝜎(𝑣),𝑡)𝐿₁(𝑡)𝑥²𝑒_𝑐2⊕𝑏(𝜎(𝑣))≤,𝑡Δ𝑣_𝑡+∞𝐷²𝑒_𝑎(𝑡,𝜎(𝑣))𝑒_𝑎(𝑡,𝜎(𝑣))𝑒_𝑐1⊕𝑏(𝜎(𝑣),𝑡)Δ𝑣‖𝑥‖²+^𝑡_−∞𝐷²𝑒_𝑎(𝜎(𝑣),𝑡)𝑒_𝑎(𝜎(𝑣),𝑡)𝑒_𝑐2⊕𝑏(𝜎(𝑣),𝑡)Δ𝑣‖𝑥‖²+^𝑡_−∞𝐷²𝑒_𝑏(𝑡,𝜎(𝑣))𝑒_𝑏(𝑡,𝜎(𝑣))𝑒_𝑐2⊕𝑏(𝜎(𝑣),𝑡)Δ𝑣‖𝑥‖²≤_𝑡^+∞𝐷²𝑒_𝑎⊖𝑏(𝑡,𝜎(𝑣))Δ𝑣‖𝑥‖²+^𝑡_−∞𝐷²𝑒_{𝑎⊕𝑎⊕𝑐2⊕𝑏}(𝜎(𝑣),𝑡)Δ𝑣‖𝑥‖²+^𝑡_−∞𝐷²𝑒_𝑐2⊖𝑏(𝜎(𝑣),𝑡)Δ𝑣‖𝑥‖²=_𝑡^+∞𝐷²−𝑒_𝑎⊖𝑏(𝑡,𝑣)^Δ_𝑣‖𝑥‖𝑎⊖𝑏Δ𝑣²+^𝑡_−∞𝐷²−𝑒_{⊖(𝑎⊕𝑎⊕𝑐2⊕𝑏)}(𝑡,𝑣)^Δ_𝑣⊖𝑎⊕𝑎⊕𝑐₂‖𝑥‖⊕𝑏Δ𝑣²+^𝑡_−∞𝐷²−𝑒_𝑏⊖𝑐2(𝑡,𝑣)^Δ_𝑣𝑏⊖𝑐₂Δ𝑣‖𝑥‖². (3.36) Since 𝜇(𝑡)≤𝑀, then 1=(𝑡)𝑏𝑎⊖𝑏1+𝜇≤𝑎−𝑏1+𝑀𝑏,𝑎−𝑏−1⊖𝑎⊕𝑎⊕𝑐₂=(𝑡)⊕𝑏1+𝜇𝑎⊕𝑎⊕𝑐₂⊕𝑏𝑎⊕𝑎⊕𝑐₂≤1⊕𝑏𝑎⊕𝑎⊕𝑐₂1⊕𝑏+𝑀≤2𝑎+𝑐₂+𝑏+𝑀,−1𝑏⊖𝑐₂=1+𝜇(𝑡)𝑐₂𝑐₂≤−𝑏1+𝑀𝑐₂𝑐₂.−𝑏 (3.37) Thus, |𝐻(≤𝐷𝑡,𝑥)|²(1+𝑀𝑏)𝑎−𝑏+𝐷²12𝑎+𝑐₂+𝑏+𝑀+𝐷²1+𝑀𝑐₂𝑐₂𝑥‖−𝑏‖². (3.38) Set 𝐾∶=𝐷²(1+𝑀𝑏)/(𝑎−𝑏)+𝐷²(1/(2𝑎+𝑐₂+𝑏)+𝑀)+𝐷²((1+𝑀𝑐₂)/(𝑐₂−𝑏)).
Let 𝐻^𝑠_𝜏(𝑉)∶=𝑃(𝜏)ℝ^𝑛 and 𝐻^𝑢_𝜏(𝑉)∶=𝐿₁(𝜏)ℝ^𝑛=𝑅(𝜏)ℝ^𝑛⊕𝑄(𝜏)ℝ^𝑛. For 𝑥∈𝐻^𝑠_𝜏(𝑉), one has 𝐻(𝑡,𝑥)=_𝑡^+∞‖𝑇(𝜎(𝑣),𝑡)𝑃(𝑡)𝑥‖²𝑒_c1⊕𝑏(𝜎(𝑣))=,𝑡Δ𝑣_𝑡+∞‖𝑇(𝜎(𝑣),𝑡)𝑥‖²𝑒_𝑐1⊕𝑏(𝜎(𝑣),𝑡)Δ𝑣. (3.39) Obviously, one has ‖𝑇(𝑡,𝜏)‖≤‖𝑇(𝑡,𝜏)𝑃(𝜏)‖+‖𝑇(𝑡,𝜏)𝑄(𝜏)‖+‖𝑇(𝑡,𝜏)𝑅(𝜏)‖. (3.40) Then by (3.3)–(3.5), we obtain that ‖𝑇(𝑡,𝜏)‖≤𝐷𝑒_𝑎(𝜏,𝑡)+𝐷𝑒_𝑏(𝑡,𝜏)+𝐷𝑒_𝑐()‖𝑇(𝑡,𝜏for𝑡≥𝜏,𝑡,𝜏)‖≤𝐷𝑒_𝑎(𝑡,𝜏)+𝐷𝑒_𝑏(𝜏,𝑡)+𝐷𝑒_𝑐(𝜏,𝑡)for𝑡≤𝜏. (3.41)(3.42) Thus, |𝐻(𝑡,𝑥)|≥_𝑡^+∞‖𝑥‖²‖𝑇(𝑡,𝜎(𝑣))‖²𝑒_𝑐1⊕𝑏(𝜎(𝑣))≥,𝑡Δ𝑣_𝑡+∞‖𝑥‖²𝐷𝑒_𝑎(𝑡,𝜎(𝑣))+𝐷𝑒_𝑏(𝜎(𝑣),𝑡)+𝐷𝑒_𝑐(𝜎(𝑣),𝑡)²𝑒_𝑐1⊕𝑏(𝜎(𝑣))≥‖𝑥‖,𝑡Δ𝑣₂9𝐷²_𝑡^+∞𝑒_𝑐1⊕𝑏(𝜎(𝑣),𝑡)𝑒_𝑐⊕𝑐((𝑣=‖𝑥‖𝑡,𝜎))Δ𝑣₂9𝐷²_𝑡^+∞−𝑒_{𝑐⊕𝑐⊖(𝑐1⊕𝑏)}(𝑡,𝑣)^Δ_𝑣𝑐⊕𝑐⊖𝑐₁⊕𝑏Δ𝑣. (3.43) Since 𝑐⊕𝑐⊖(𝑐₁⊕𝑏)=(2𝑐+𝜇(𝑠)𝑐²−(𝑐₁+𝑏+𝜇(𝑠)𝑐₁𝑏))/(1+𝜇(𝑠)(𝑐₁+𝑏+𝜇(𝑠)𝑏))≤2𝑐−𝑐₁−𝑏+𝑀𝑐²−𝑀𝑐₁𝑏, then we get |𝐻(≥‖𝑥‖𝑡,𝑥)|²9𝐷²2𝑐−𝑐₁−𝑏+𝑀𝑐²−𝑀𝑐₁𝑏. (3.44) Similarly, for 𝑥∈𝐻^𝑢_𝜏(𝑉), we get 𝐻(𝑡,𝑥)=−^𝑡_−∞‖𝑇(𝜎(𝑣),𝑡)𝑥‖²𝑒_𝑐2⊕𝑏(𝜎(𝑣))|𝐻(≥‖𝑥‖,𝑡Δ𝑣,𝑡,𝑥)|²9𝐷²𝑐₂+𝑏+2𝑐+𝑀2𝑐₂𝑐+2𝑏𝑐+𝑐²+𝑀²𝑏𝑐+𝑐₂𝑐²+𝑏𝑐²+𝑀³𝑏𝑐². (3.45)
Let 𝐿′∶=max{𝑐₂+𝑏+2𝑐+𝑀(2𝑐₂𝑐+2𝑏𝑐+𝑐²)+𝑀²(𝑏𝑐+𝑐₂𝑐²+𝑏𝑐²)+𝑀³𝑏𝑐²,2𝑐−𝑐₁−𝑏+𝑀𝑐²−𝑀𝑐₁𝑏}. Then when 𝑥∈𝐻^𝑠_𝜏(𝑉)⋃𝐻^𝑢_𝜏(𝑉), we get |𝐻(𝑡,𝑥)|≥‖𝑥‖²/9𝐷²𝐿′. Let 𝑁²∶=max{9𝐷²𝐿′,𝐾}. We get |𝐻(𝑡,𝑥)|≤𝐾≤𝑁²‖𝑥‖² for any 𝑡∈𝕋, 𝑥∈ℝ^𝑛, and when 𝑥∈𝐻^𝑠_𝜏(𝑉)⋃𝐻^𝑢_𝜏(𝑉), we get |𝐻(𝑡,𝑥)|≥‖𝑥‖²/𝑁².
Since 𝑐₂<𝑐₁, then we have 𝑐₂⊕𝑏<𝑐₁⊕𝑏 and there are constants 𝑟^′₁ and 𝑟^′₂ with 𝑐₂⊕𝑏≤𝑟^′₂<𝑟^′₁≤𝑐₁⊕𝑏. Thus, when 𝑥∈𝐻^𝑠_𝜏(𝑉), for any 𝑡≥𝜏 we get 𝐻(𝑡,𝑇(𝑡,𝜏)𝑥)=_𝑡^+∞‖𝑇(𝜎(𝑣),𝜏)𝑃(𝜏)𝑥‖²𝑒_𝑐1⊕𝑏(𝜎(𝑣),𝑡)Δ𝑣≤𝑒_𝑐1⊕𝑏(𝜏,𝑡)𝐻(𝜏,𝑥), (3.46) that is, 𝑉²(𝑡,𝑇(𝑡,𝜏)𝑥)≤𝑒_𝑐1⊕𝑏(𝜏,𝑡)𝑉²(𝜏,𝑥)≤𝑒_𝑟′1(𝜏,𝑡)𝑉²(𝜏,𝑥), (3.47) and when 𝑥∈𝐻^𝑢_𝜏(𝑉), for any 𝑡≥𝜏 we get 𝐻(𝑡,𝑇(𝑡,𝜏)𝑥)=−^𝑡_−∞𝑇(𝜎(𝑣),𝜏)𝐿₁(𝜏)𝑥²𝑒_𝑐2⊕𝑏(𝜎(𝑣),𝑡)Δ𝑣≤𝑒_𝑐2⊕𝑏(𝜏,𝑡)𝐻(𝜏,𝑥), (3.48) that is, 𝑉²(𝑡,𝑇(𝑡,𝜏)𝑥)≥𝑒_𝑐2⊕𝑏(𝜏,𝑡)𝑉²(𝜏,𝑥)≥𝑒_𝑟′2(𝜏,𝑡)𝑉²(𝜏,𝑥). (3.49) Therefore, 𝑉(𝑡,𝑥) is a (⊖𝑟^′₁,⊖𝑟^′₂)-Lyapunov function.
Set 𝑇(𝑡)∶=_𝑡^+∞𝑇(𝜎(𝑣),𝑡)𝐿₂(𝑡)^∗𝑇(𝜎(𝑣),𝑡)𝐿₂(𝑡)𝑒_𝑑1⊕𝑏((𝑣−𝑡,𝜎))Δ𝑣_𝑡−∞(𝑇(𝜎(𝑣),𝑡)𝑄(𝑡))^∗𝑇(𝜎(𝑣),𝑡)𝑄(𝑡)𝑒_𝑑2⊕𝑏(𝑡,𝜎(𝑣))Δ𝑣, (3.50) where 𝐿₂(𝑡)=𝑃(𝑡)+𝑅(𝑡) and 𝑏<𝑑₁<𝑑₂<𝑎. Let 𝐺(𝑡,𝑥)∶=⟨𝑇(𝑡)𝑥,𝑥⟩ and 𝑊(𝑡,x)∶=−sign𝐺(𝑡,𝑥)|𝐺(𝑡,𝑥)|. Then 𝐺(𝑡,𝑥)=_𝑡^+∞𝑇(𝜎(𝑣),𝑡)𝐿₂(𝑡)𝑥²𝑒_𝑑1⊕𝑏((𝑣−𝑡,𝜎))Δ𝑣_𝑡−∞‖(𝑇(𝜎(𝑣),𝑡)𝑄(𝑡))𝑥‖²𝑒_𝑑2⊕𝑏(𝑡,𝜎(𝑣))Δ𝑣. (3.51) Let 𝐻^𝑠_𝜏(𝑊)∶=𝐿₂(𝜏)ℝ^𝑛 and 𝐻^𝑢_𝜏(𝑊)∶=𝑄(𝜏)ℝ^𝑛. Similar to the consideration for 𝑉(𝑡,𝑥), we can see that 𝐺(𝑡,𝑥) is also a Lyapunov function satisfying 𝐺(𝑡,𝑇(𝑡,𝜏)𝑥)≤𝑒_𝑑2⊕𝑏()𝐺(),|𝐺(𝑡,𝜏𝜏,𝑥𝑡,𝑥)|≤𝐷²1+𝑀𝑎+1𝑎−𝑏2𝑎+𝑑₁+𝑏+𝑀+1+𝑀𝑏𝑑₁‖𝑥‖−𝑏²,|𝐺(≥‖𝑥‖𝑡,𝑥)|²9𝐷²𝐿^′′,for𝑥∈𝐻^𝑠_𝜏(𝑊)𝐻^𝑢_𝜏(𝑊), (3.52) where 𝐿^′′=max{𝑑₁+𝑏+2𝑐+𝑀(2𝑑₁𝑐+2𝑏𝑐+𝑐²)+𝑀²(𝑏𝑐+𝑑₁𝑐²+𝑏𝑐²)+𝑀³𝑏𝑐², 2𝑐−𝑑₂−𝑏+𝑀𝑐²−𝑀𝑑₂𝑏}.
Since 𝑑₁<𝑑₂, then we have 𝑑₁⊕𝑏<𝑑₂⊕𝑏 and there are constants 𝑙^′₁ and 𝑙^′₂ with 𝑑₁⊕𝑏≤𝑙^′₁<𝑙^′₂≤𝑑₂⊕𝑏. Thus, for any 𝑡≥𝜏, we obtain that when 𝑥∈𝐻^𝑠_𝜏(𝑊), 𝐺(𝑡,𝑇(𝑡,𝜏)𝑥)=_𝑡^+∞𝑇(𝜎(𝑣),𝜏)𝐿₂(𝜏)𝑥²𝑒_𝑑1⊕𝑏(𝑡,𝜎(𝑣))Δ𝑣≤𝑒_𝑑1⊕𝑏(𝑡,𝜏)𝐺(𝜏,𝑥), (3.53) that is, 𝑊²(𝑡,𝑇(𝑡,𝜏)𝑥)≤𝑒_𝑑1⊕𝑏(𝑡,𝜏)𝑊²(𝜏,𝑥)≤𝑒_𝑙′1(𝑡,𝜏)𝑊²(𝜏,𝑥), (3.54) and when 𝑥∈𝐻^𝑢_𝜏(𝑊), 𝐺(𝑡,𝑇(𝑡,𝜏)𝑥)=−^𝑡_−∞𝑇(𝜎(𝑣),𝜏)𝐿₁(𝜏)𝑥²𝑒_𝑑2⊕𝑏(𝑡,𝜎(𝑣))Δ𝑣≤𝑒_𝑑2⊕𝑏(𝑡,𝜏)𝐺(𝜏,𝑥), (3.55) that is, 𝑊²(𝑡,𝑇(𝑡,𝜏)𝑥)≥𝑒_𝑑2⊕𝑏(𝑡,𝜏)𝑊²(𝜏,𝑥)≥𝑒_𝑙′2(𝑡,𝜏)𝑉²(𝜏,𝑥). (3.56) Thus, 𝑊(𝑡,𝑥) is a (𝑙^′₁,𝑙^′₂)-Lyapunov function.
According to all the discussions above we get that when (3.1) has a strict exponential trichotomy, there exists two (𝜆,𝜇)-Lyapunov functions, 𝑉(𝑡,𝑥) with (⊖𝑟^′₁,⊖𝑟^′₂) and 𝑊(𝑡,𝑥) with (𝑙^′₁,𝑙^′₂) satisfying ⊖𝑟^′₁<⊖𝑟^′₂<0<𝑙^′₁<𝑙^′₂. This completes the proof.

4. Roughness of Exponential Trichotomy on Time Scales

The roughness of exponential dichotomy on time scales had been studied by Zhang and his cooperators in their paper [24] in 2010. In this section, we go further to study the roughness of exponential trichotomy on time scales, using the results which we get from Section 3. We have known that the notion of exponential trichotomy plays a central role when we study center manifolds, so it is important to understand how exponential trichotomy vary under perturbations. Here, we discuss the following linear perturbed equation𝑥^Δ=𝐴(𝑡)𝑥+𝐵(𝑡)𝑥, (4.1) where 𝐵(𝑡) is an 𝑛×𝑛 matrix valued function on time scale 𝕋 with ‖𝐵(𝑡)‖≤𝛿. For this linear perturbed (4.1) we get the following theorem.

Theorem 4.1. Suppose that (3.1) has a strict exponential trichotomy. If 𝛿>0 is sufficiently small, then the linear perturbed (4.1) also has an exponential trichotomy.

Proof. Since (3.1) has a strict exponential trichotomy, then by Theorem 3.4 there are two (𝜆,𝜇)-Lyapunov functions, 𝑉(𝑡,𝑥) with (𝑟₁,𝑟₂) and 𝑊(𝑡,𝑥) with (𝑙₁,𝑙₂) satisfying r₁<𝑟₂<0<𝑙₁<𝑙₂. Here, 𝑉(𝑡,𝑥) and 𝑊(𝑡,𝑥) are defined in Theorem 3.4. Let 𝑥(𝑡) be a solution of (3.1) satisfying 𝑥(𝜏)=𝑥. Differentiating on both sides of 𝐻(𝑡,𝑥(𝑡))=⟨𝑆(𝑡)𝑥(𝑡),𝑥(𝑡)⟩, one has 𝐻^Δ(𝑡,𝑥(𝑡))=⟨𝑆(𝑡)𝑥(𝑡),𝑥(𝑡)⟩^Δ=𝑆^Δ(𝑡)𝑥(𝑡),𝑥(𝑡)+𝑆(𝜎(𝑡))𝑥^Δ(𝑡),𝑥(𝑡)+𝑆(𝜎(𝑡))𝑥(𝜎(𝑡)),𝑥^Δ(𝑡=𝑆_Δ)(𝑡)+𝑆(𝜎(𝑡))𝐴(𝑡)+𝐴^∗(𝑡)𝑆(𝜎(𝑡(𝑡)𝐴))+𝜇^∗(𝑡)𝑆(𝜎(𝑡))𝐴(𝑡)𝑥(𝑡),𝑥(𝑡). (4.2) Let 𝑇(𝑡,𝜏) be the linear evolution operator related to (3.1). Then we get 𝑥(𝑡)=𝑇(𝑡,𝜏)𝑥. And we can see that 𝑃(𝑡)𝑥(𝑡)=𝑃(𝑡)𝑇(𝑡,𝜏)𝑥=𝑇(𝑡,𝜏)𝑃(𝜏)𝑥 is also a solution of (3.1). By (4.2) we get 𝐻^Δ(𝑡,𝑃(𝑡)𝑥(𝑡))=𝑆^Δ(𝑡)+𝑆(𝜎(𝑡))𝐴(𝑡)+𝐴^∗(𝑡)𝑆(𝜎(𝑡(𝑡)𝐴))+𝜇^∗(𝑡)𝑆(𝜎(𝑡))𝐴(𝑡)𝑃(𝑡)𝑥(𝑡),𝑃(𝑡)𝑥(𝑡). (4.3) At the same time, by (2.4), (2.7), and (3.46) one has 𝜇(𝑡)𝐻^Δ((𝑡)𝑥(𝑡(𝜎(𝑡)(𝜎(𝑡𝑥(𝜎(𝑡((𝑡)𝑥(𝑡((𝑡)𝑥(𝑡𝑒𝑡,𝑃))=𝐻,𝑃)))))−𝐻𝑡,𝑃))≤𝐻𝑡,𝑃))_𝑐1⊕𝑏((𝑡((𝑡)𝑥(𝑡((𝑡)𝑥(𝑡𝑡,𝜎))−𝐻𝑡,𝑃))=𝐻𝑡,𝑃))exp_𝑡^𝜎(𝑡)1𝜇(𝑠)ln1+⊖𝑐₁𝜇(𝑠𝑐⊕𝑏)Δ𝑠−1=⊖₁⊕𝑏𝜇(𝑡)𝐻(𝑡,𝑃(𝑡)𝑥(𝑡)). (4.4) Thus, we obtain that 𝐻^Δ(𝑡,𝑃(𝑡)𝑥(𝑡))≤⊖𝑐₁⊕𝑏𝐻(𝑡,𝑃(𝑡)𝑥(𝑡)). (4.5) Then by (4.3) and (4.5) we can see 𝑆^Δ(𝑡)+𝑆(𝜎(𝑡))𝐴(𝑡)+𝐴^∗(𝑡)𝑆(𝜎(𝑡))+𝜇(𝑡)𝐴^∗(𝑡)𝑆(𝜎(𝑡))𝐴(𝑡)≤⊖𝑐₁⊕𝑏𝑆(𝑡). (4.6) Now, Let 𝑦(𝑡) be a solution of equation 𝑥^Δ=𝐴(𝑡)𝑥+𝐵(𝑡)𝑥 satisfying 𝑦(𝜏)=𝑦. Since 𝑦(𝜎(𝑡))=𝑦(𝑡)+𝜇(𝑡)𝑦^Δ(𝑡)=𝑦(𝑡)+𝜇(𝑡)(𝐴(𝑡)+𝐵(𝑡))𝑦(𝑡), one has 𝐻^Δ𝑡,𝑦(𝑡)=𝑆(𝑡)𝑦(𝑡),𝑦(𝑡)^Δ=𝑆^Δ(𝑡)𝑦(𝑡),𝑦(𝑡)+𝑆(𝜎(𝑡))𝑦^Δ(𝑡),𝑦(𝑡)+𝑆(𝜎(𝑡))𝑦(𝜎(𝑡)),𝑦^Δ(𝑡=𝑆_Δ)(𝑡)𝑦(𝑡),𝑦(𝑡)+𝑆(𝜎(𝑡))𝐴(𝑡)𝑦(𝑡),𝑦(𝑡)+𝐴^∗(𝑡)𝑆(𝜎(𝑡𝑦(𝑡)(𝑡+𝜇(𝑡)𝐴)),𝑦)_∗(𝑡)𝑆(𝜎(𝑡𝐴(𝑡)𝑦(𝑡)(𝑡+𝑆(𝜎(𝑡𝐵(𝑡)𝑦(𝑡)(𝑡+𝜇(𝑡)𝑆(𝜎(𝑡𝐵(𝑡)𝑦(𝑡)(𝑡)𝑦(𝑡+𝑆(𝜎(𝑡𝑦(𝑡)(𝑡)𝑦(𝑡+𝜇(𝑡)𝑆(𝜎(𝑡𝐴(𝑡)𝑦(𝑡)(𝑡)𝑦(𝑡+𝜇(𝑡)𝑆(𝜎(𝑡𝐵(𝑡)𝑦(𝑡)(𝑡)𝑦(𝑡𝑐)),𝑦))),𝑦))),𝐴))),𝐵))),𝐵))),𝐵)≤⊖₁⊕𝑏𝐻𝑡,𝑦(𝑡)+‖𝑆(𝜎(𝑡))‖‖𝐵(𝑡)‖𝑦(𝑡)²+2𝜇(𝑡)𝐴(𝑡)‖𝑆(𝜎(𝑡))‖‖𝐵(𝑡)‖𝑦(𝑡)²+‖𝑆(𝜎(𝑡))‖‖𝐵(𝑡)‖𝑦(𝑡)²+𝜇(𝑡)‖𝑆(𝜎(𝑡))‖‖𝐵(𝑡)‖²𝑦(𝑡)². (4.7)
By the variation of constants formula, we have 𝑇(𝑡,𝑠)=Id+∫^𝑡_𝑠𝐴(𝑣)𝑇(𝑣,𝑠)Δ𝑣. Thus, by (2.7) we obtain 𝑇(𝜎(𝑡),𝑡)=𝐼+_𝑡^𝜎(𝑡)𝐴(𝑣)𝑇()(𝑡)𝐴(𝑡)𝑇()(𝑡)𝐴(𝑡).𝑣,𝑡Δ𝑣=Id+𝜇𝑡,𝑡=Id+𝜇 (4.8) Therefore, by Definition 2.5 and (2.7), (3.41), we obtain Id+𝜇(𝑡)𝐴(𝑡)≤𝐷𝑒_𝑎(𝑡,𝜎(𝑡))+𝐷𝑒_𝑏(𝜎(𝑡),𝑡)+𝐷𝑒_𝑐(𝜎(𝑡))(𝑡(𝑡)𝑏(𝑡)𝑐().,𝑡=𝐷1+𝜇)(⊖𝑎)+𝐷1+𝜇+𝐷1+𝜇≤3𝐷+𝐷𝑀𝑏+𝑐 (4.9) Thus, one has 𝜇(𝑡)𝐴(𝑡)≤Id+𝜇(𝑡)𝐴(𝑡)+‖Id‖≤3𝐷+𝐷𝑀(𝑏+𝑐)+1. (4.10) From Theorem 3.4, we know ‖𝑆(𝑡)‖=sup_{𝑥∈ℝ𝑛}𝐻(𝑡,𝑥)‖𝑥(𝑡)‖²≤𝑁²,𝑦(𝑡)≤𝑁𝑉𝑡,𝑦(𝑡). (4.11) Thus, 𝐻^Δ𝑡,𝑦(𝑡)≤⊖𝑐₁⊕𝑏𝐻𝑡,𝑦(𝑡)+2𝑁²𝛿+2(3𝐷+1+𝐷𝑀𝑏+𝐷𝑀𝑐)𝑁²𝛿+𝑀𝑁²𝛿²𝑁²𝐻𝑡,𝑦(𝑡). (4.12) Let 𝑀(𝛿)∶=[2𝑁²𝛿+2(3𝐷+1+𝐷𝑀𝑏+𝐷𝑀𝑐)𝑁²𝛿+𝑀𝑁²𝛿²]𝑁². It is not hard to see that 𝑀(𝛿)→0 when 𝛿→0. Set 𝐸(𝛿)∶=⊖(𝑐₁⊕𝑏)+𝑀(𝛿). Then when 𝛿 is sufficiently small, we have 𝐸(𝛿)<0 and (𝑡)𝐸(𝛿)(𝑡−𝑐1+𝜇=1+𝜇)¹⊕𝑏1+𝜇(𝑡)𝑐₁(𝛿=1⊕𝑏+𝑀)1+𝜇(𝑡)𝑐₁(𝑡)𝑀(𝛿)⊕𝑏+𝜇>0. (4.13) So 𝐸(𝛿) is positive regressive. Then for 𝐻(𝑡,𝑦(𝑡))>0, (4.12) can be written as 𝐻^Δ𝑡,𝑦≤𝐸(𝛿)𝐻𝑡,𝑦(𝑡). (4.14) Notice that 𝐻𝑡,𝑦(𝑡)𝑒_{⊖𝐸(𝛿)}(𝑡,𝜏)^Δ=𝐻^Δ𝑡,𝑦(𝑡)𝑒_{⊖𝐸(𝛿)}(𝜎(𝑡),𝜏)+⊖𝐸(𝛿)𝑒_{⊖𝐸(𝛿)}(𝑡,𝜏)𝐻𝑡,𝑦(𝑡)=𝐻^Δ𝑡,𝑦(𝑡)𝑒_{⊖𝐸(𝛿)}(𝜎(𝑡))(𝛿)𝑒,𝜏+⊖𝐸_{⊖𝐸(𝛿)}(𝑡,𝜎(𝑡))𝑒_{⊖𝐸(𝛿)}(𝜎(𝑡),𝜏)𝐻𝑡,𝑦(𝑡)=𝐻^Δ𝑡,𝑦(𝑡)𝑒_{⊖𝐸(𝛿)}(𝜎(𝑡))(𝛿)1,𝜏+⊖𝐸(𝛿)𝜇(𝑡)𝑒1+⊖𝐸^{⊖𝐸(𝛿)}(𝜎(𝑡))𝐻(𝑡=𝐻,𝜏𝑡,𝑦)_Δ𝑡,𝑦−𝐸(𝛿)𝐻𝑡,𝑦(𝑡)𝑒_{⊖𝐸(𝛿)}(𝜎(𝑡),𝜏)≤0. (4.15) Thus, we get 𝐻𝑡,𝑦(𝑡)≤𝑒_𝐸(𝛿)(𝑡,𝜏)𝐻𝜏,𝑦(𝜏). (4.16) Then we can see that if 𝑦(𝑡)∈𝐶^𝑠(𝑉_𝑡), we have 𝑉(𝑡,𝑦(𝑡))<0 and thus 𝐻(𝑡,𝑦(𝑡))>0. By (4.16) we have 𝐻(𝜏,𝑦(𝜏))>0 and thus 𝑉(𝜏,𝑦(𝜏))<0, that is, 𝑦(𝜏)∈𝐶^𝑠(𝑉_𝜏). Let 𝑈(𝑡,𝜏) be the linear evolution operator associated to (4.1). Then 𝑦(𝑡)=𝑈(𝑡,𝜏)𝑦(𝜏). Thus, we get 𝑈(𝜏,𝑡)𝐶^𝑠𝑉_𝑡⊂𝐶^𝑠𝑉_𝜏. (4.17) Let 𝐿₁(𝑡)=𝑄(𝑡)+𝑅(𝑡). We can see that 𝐿₁(𝑡)𝑥(𝑡) is also a solution of (3.1). By (4.2) one has 𝐻^Δ𝑡,𝐿₁(𝑡)𝑥(𝑡)=𝑆^Δ(𝑡)+𝑆(𝜎(𝑡))𝐴(𝑡)+𝐴^∗(𝑡)𝑆(𝜎(𝑡(𝑡)𝐴))+𝜇^∗(𝑡)𝑆(𝜎(𝑡))𝐴(𝑡)𝐿₁(𝑡)𝑥(𝑡),𝐿₁(𝑡)𝑥(𝑡). (4.18) Similar to the proof of (4.5), we have 𝐻^Δ(𝑡,𝐿₁(𝑡)𝑥(𝑡))≤⊖(𝑐₂⊕𝑏)𝐻(𝑡,𝐿₁(𝑡)𝑥(𝑡)). Then we get 𝑆^Δ(𝑡)+𝑆(𝜎(𝑡))𝐴(𝑡)+𝐴^∗(𝑡)𝑆(𝜎(𝑡))+𝜇(𝑡)𝐴^∗(𝑡)𝑆(𝜎(𝑡))𝐴(𝑡)≤⊖𝑐₂⊕𝑏𝑆(𝑡). (4.19) Following the process for getting (4.12), one has 𝐻^Δ𝑡,𝑦(𝑡)≤⊖𝑐₂⊕𝑏𝐻𝑡,𝑦(𝑡)+𝑀(𝛿)𝐻𝑡,𝑦(𝑡). (4.20) Let 𝐹(𝛿)∶=⊖(𝑐₂⊕𝑏)−𝑀(𝛿). When 𝛿 is sufficiently small, we obtain 𝐹(𝛿)<0 and (𝑡)𝐹(𝛿)(𝑡−𝑐1+𝜇=1+𝜇)²⊕𝑏1+𝜇(𝑡)𝑐₂(𝛿=1⊕𝑏−𝑀)(𝑡)(𝛿).1+𝜇)(𝑎⊕𝑏−𝑀 (4.21) So 𝐹(𝛿) is positive regressive. Then for 𝐻(𝑡,𝑦(𝑡))<0, (4.20) can be written as 𝐻^Δ𝑡,𝑦≤𝐹(𝛿)𝐻𝑡,𝑦(𝑡). (4.22) Following the similar process as (4.15), we get 𝐻𝑡,𝑦(𝑡)≤𝑒_𝐹(𝛿)(𝑡,𝜏)𝐻𝜏,y(𝜏). (4.23) Then if 𝑦(𝜏)∈𝐶^𝑢(𝑉_𝜏), we have 𝑉(𝜏,𝑦(𝜏))>0 and thus 𝐻(𝜏,𝑦(𝜏))<0. By (4.23) we have 𝐻(𝑡,𝑦(𝑡))<0 and thus 𝑉(𝑡,𝑦(𝑡))>0, that is, 𝑦(𝑡)∈𝐶^𝑢(𝑉_𝑡). So we get 𝑈(𝑡,𝜏)𝐶^𝑢𝑉_𝜏⊂𝐶^𝑢𝑉_𝑡. (4.24) By the notion of Lyapunov function and (4.17), (4.24) we can see that 𝑉(𝑡,𝑥) is a Lyapunov function of (4.1).
Let 𝐻_𝜏^′𝑢(𝑉)∶=_𝑟∈𝕋𝑈(𝜏,𝑟)𝐶^𝑢𝑉_𝑟⊂𝐶^𝑢𝑉_𝜏,𝐻_𝜏^′𝑠(𝑉)∶=_𝑟∈𝕋𝑈(𝜏,𝑟)𝐶^𝑠𝑉_𝑟⊂𝐶^𝑠𝑉_𝜏. (4.25) By the notion of Lyapunov function, we know that there exist two subspaces 𝐷_𝜏^′𝑢(𝑉) and 𝐷_𝜏^′𝑠(𝑉) such that 𝐷_𝜏^′𝑢(𝑉)⊂𝐻_𝜏^′𝑢(𝑉), 𝐷_𝜏^′𝑠(𝑉)⊂𝐻_𝜏^′𝑠(𝑉), and 𝐷_𝜏^′𝑢(𝑉)⊕𝐷_𝜏^′𝑠(𝑉)=ℝ^𝑛.
Now, we prove that 𝑉(𝑡,𝑥) is a (𝜆,𝜇)-Lyapunov function. Since 𝑐₁>𝑐₂, we have 𝑐₁⊕𝑏>𝑐₂⊕𝑏. By simple computation, we obtain ⊖(𝑐₁⊕𝑏)<⊖(𝑐₂⊕𝑏). Therefore, when 𝛿 is sufficiently small, we get 𝐸(𝛿)<𝐹(𝛿). Then there are constants 𝑟₁,𝑟₂ such that 𝐸(𝛿)≤𝑟₁<𝑟₂≤𝐹(𝛿). Thus, by inequality (4.16), for 𝑦(𝜏)∈𝐷_𝜏^′𝑠 one has 𝑉²𝑡,𝑦(𝑡)≤𝑒_𝐸(𝛿)(𝑡,𝜏)𝑉²𝜏,𝑦(𝜏)≤𝑒_𝑟1(𝑡,𝜏)𝑉²𝜏,𝑦(𝜏). (4.26) By inequality (4.23), for 𝑦(𝜏)∈𝐷_𝜏^′𝑢, we obtain 𝑉²𝑡,𝑦(𝑡)≥𝑒_𝐹(𝛿)(𝑡,𝜏)𝑉²𝜏,𝑦(𝜏)≥𝑒_𝑟2(𝑡,𝜏)𝑉²𝜏,𝑦(𝜏). (4.27) Then by (4.26) and (4.27) we can see that 𝑉(𝑡,𝑦) is a (𝑟₁,𝑟₂)-Lyapunov function of (4.1).
Next, we will prove that 𝑊(𝑡,𝑦) is also a (𝜆,𝜇)-Lyapunov function of (4.1). Similar to the consideration of 𝐻(𝑡,𝑦), for 𝐺(𝑡,𝑦) we have that if 𝑦∈𝐶^𝑠(𝑊_𝑡), then 𝑊(𝑡,𝑦(𝑡))<0, that is, 𝐺(𝑡,𝑦(𝑡))>0. Thus, we get 𝐺^Δ𝑡,𝑦(𝑡)≤𝑑₁⊕𝑏+𝑀(𝛿)𝐺𝑡,𝑦(𝑡). (4.28) Let 𝐸′(𝛿)∶=𝑑₁⊕𝑏+𝑀(𝛿). By the similar deducing as (4.16), one has 𝐺𝑡,𝑦(𝑡)≤𝑒_{𝐸′(𝛿)}(𝑡,𝜏)𝐺𝜏,𝑦(𝜏). (4.29) Thus, when 𝐺(𝑡,𝑦(𝑡))>0, that is, 𝑊(𝑡,𝑦(𝑡))<0, we have 𝐺(𝜏,𝑦(𝜏))>0, that is, 𝑊(𝜏,𝑦(𝜏))<0. That means 𝑈(𝜏,𝑡)𝐶^𝑠𝑊_𝑡⊂𝐶^𝑠𝑊_𝜏. (4.30)
When 𝑦∈𝐶^𝑢(𝑊_𝑡), we have 𝑊(𝑡,𝑦(𝑡))>0, that is, 𝐺(𝑡,𝑦(𝑡))<0. Then 𝐺^Δ𝑡,𝑦(𝑡)≤𝑑₂⊕𝑏−𝑀(𝛿)𝐺𝑡,𝑦(𝑡). (4.31) Let 𝐹′(𝛿)=𝑑₂⊕𝑏−𝑀(𝛿). By the similar deducing as (4.15), one has 𝐺𝑡,𝑦(𝑡)≤𝑒_{𝐹′(𝛿)}(𝑡,𝜏)𝐺𝜏,𝑦(𝜏). (4.32) Then when 𝐺(𝜏,𝑦(𝜏))<0, that is, 𝑊(𝜏,𝑦(𝜏))>0, we have 𝐺(𝑡,𝑦(𝑡))<0, that is, 𝑊(𝜏,𝑦(𝜏))>0. That means 𝑈(𝑡,𝜏)𝐶^𝑢𝑊_𝜏⊂𝐶^𝑢𝑊_𝑡. (4.33) By (4.30) and (4.33), we know that 𝑊(𝑡,𝑦) is a Lyapunov function of (4.1).
Since 𝑑₁<𝑑₂, we have 𝑑₁⊕𝑏<𝑑₂⊕𝑏. Then when 𝛿 is sufficiently small, we get 𝐸′(𝛿)<𝐹′(𝛿). Therefore, there are constants 𝑙₁,𝑙₂ such that 𝐸′(𝛿)≤𝑙₁<𝑙₂≤𝐹′(𝛿).
Let 𝐻_𝜏^′𝑢(𝑊)∶=_𝑟∈𝕋𝑈(𝜏,𝑟)𝐶^𝑢𝑊_𝑟⊂𝐶^𝑢𝑊_𝜏,𝐻_𝜏^′𝑠(𝑊)∶=_𝑟∈𝕋𝑈(𝜏,𝑟)𝐶^𝑠𝑊_𝑟⊂𝐶^𝑠𝑊_𝜏. (4.34) By the notion of Lyapunov function, we can know that there exist 𝐷_𝜏^′𝑢(𝑊) and 𝐷_𝜏^′𝑠(𝑊) such that 𝐷_𝜏^′𝑢(𝑊)⊂𝐻_𝜏^′𝑢(𝑊), 𝐷_𝜏^′𝑠(𝑊)⊂𝐻_𝜏^′𝑠(𝑊), and 𝐷_𝜏^′𝑢(𝑊)⊕𝐷_𝜏^′𝑠(𝑊)=ℝ^𝑛.
Furthermore, by (4.29) and (4.32), for any 𝑡≥𝜏, when 𝑦(𝜏)∈𝐷_𝜏^′𝑠(𝑊), one has 𝑊²𝑡,𝑦(𝑡)≤𝑒_{𝐸′(𝛿)}(𝑡,𝜏)𝑊²𝜏,𝑦(𝜏)≤𝑒_𝑙1(𝑡,𝜏)𝑊²𝜏,𝑦(𝜏), (4.35) and when 𝑦(𝜏)∈𝐷_𝜏^′𝑢(𝑊), one has 𝑊²𝑡,𝑦(𝑡)≥𝑒_{𝐹′(𝛿)}(𝑡,𝜏)𝑊²𝜏,𝑦(𝜏)≥𝑒_𝑙2(𝑡,𝜏)𝑊𝜏,𝑦(𝜏). (4.36) Thus, we can see that 𝑊(𝑡,𝑦) is a (𝑙₁,𝑙₂)-Lyapunov function of (4.1). Therefore, we know that (4.1) has two (𝜆,𝜇)-quadratic Lyapunov functions, 𝑉(𝑡,𝑥) with (𝑟₁,𝑟₂) and 𝑊(𝑡,𝑥) with (𝑙₁,𝑙₂) satisfying 𝑟₁<𝑟₂<0<𝑙₁<𝑙₂. Then by Theorem 3.1 we get that (4.1) has an exponential trichotomy. The proof is completed.

Acknowledgment

The author expresses her sincere thanks to Professors Yong Li and Zhenxin Liu for their encouragement and valuable suggestions.