Nonuniform Exponential Trichotomy for Linear Discrete-Time Systems in Banach Spaces

Abstract

The aim of this paper is to give several characterizations for nonuniform exponential trichotomy properties of linear difference equations in Banach spaces. Well-known results for exponential stability and exponential dichotomy are extended to the case of nonuniform exponential trichotomy.

1. Introduction

In the mathematical literature of the last decades, the asymptotic properties of solutions of evolution equations in finite or infinite dimensional space have proved to be research area of large intensity. There were defined and developed concepts of the asymptotic behaviors, as stability, expansivity, dichotomy, and trichotomy (see [1–25] and the references therein), based on the fact that the dynamical systems which describe processes from economics, physical sciences, or engineering are extremely complex and the identification of the proper mathematical model is difficult.

As a natural generalization of exponential dichotomy, exponential trichotomy is one of the most complex asymptotic properties of dynamical systems arising from the central manifold theory. When people analyze the asymptotic behavior of dynamical systems, exponential trichotomy is a powerful tool. Starting from the idea that the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold, it is obvious that the concept of exponential dichotomy describes a rather idealistic situation when the solution is either exponentially stable on the stable subspaces or exponentially unstable on the unstable subspaces (see [4, 7, 8, 16]). Thus, with motivation from the properties arising in bifurcation theory, a new asymptotic concept called exponential trichotomy which reflects a deeper analysis of the behavior of solutions of dynamical systems is introduced. Under this case the main idea in the study of the asymptotic behavior is to obtain, at any moment, a decomposition of the state space in three subspaces: a stable one, an instable one, and a third one, the central manifold.

The conception of trichotomy firstly arose in the works of Sacker and Sell [19] in 1976. They described trichotomy for linear differential systems by linear skew-product flows. Later, Elaydi and Hájek [6, 7] gave the notions of exponential trichotomy for differential systems and for nonlinear differential systems, respectively. The case of difference equations received a special attention in the paper of Elaydi and Janglajew [8] where the authors deduced the first input-output criteria for exponential trichotomy, on one hand, and they introduced the first nonlinear discrete concepts of exponential trichotomy, on the other hand. Despite the increasing interest on this topic, most of papers were devoted to problems regarding the robustness of the exponential trichotomy (see [1, 9, 10, 12, 15, 25]) and only in the past few years the existence criteria started to be obtained (see [2, 13, 18, 20–22]). For instance, in [9, 10] Hong and his partners studied the relationship between exponential trichotomy and the ergodic solutions of linear differential and difference equations with ergodic perturbations. In [20], the connections between the existence of exponential trichotomy of variational difference equations and the solvability of the associated variational control system were established. And in [22] B. Sasu and A.L. Sasu obtained some nonlinear conditions for the existence of the exponential trichotomy of skew-product flows in infinite dimensional spaces.

In this paper, we introduce the concept of nonuniform exponential trichotomy for linear difference equations which is an extension of classical concept of uniform exponential trichotomy. Our main objective is to give some characterizations for nonuniform exponential trichotomy properties of linear difference equations in Banach spaces, and variants for nonuniform exponential trichotomy of some well-known results in uniform exponential stability theory (Datko [5], Przyłuski and Rolewicz [17]) and exponential dichotomy theory (Popa et al. [16]) are obtained.

2. Preliminaries

Let 𝑋 be a real or complex Banach space. The norm on 𝑋 and on ℬ(𝑋) the Banach algebra of all bounded linear operators acting on 𝑋 will be denoted by ||⋅||. We denote Δ={(𝑚,𝑛)∈ℕ²,𝑚≥𝑛} and 𝑇={(𝑚,𝑝,𝑛)∈ℕ³,𝑚≥𝑝≥𝑛}. Let ℱ be the set of all nondecreasing functions 𝐹∶ℝ₊→ℝ₊ with the properties 𝐹(0)=0 and 𝐹(𝑡)>0 for every 𝑡>0. Let 𝐼 be the identity operator on 𝑋.

In the present paper we consider linear discrete-time system of difference equations 𝑥_𝑛+1=𝐴(𝑛)𝑥_𝑛,𝑛∈ℕ, (1) where 𝐴∶ℕ→ℬ(𝑋) is a sequence in ℬ(𝑋). Then every solution 𝑥={𝑥_𝑛} of the system (1) is given by 𝑥_𝑚=𝑈(𝑚,𝑛)𝑥_𝑛 (2) for all (𝑚,𝑛)∈Δ, where the mapping 𝑈∶Δ→ℬ(𝑋) is defined by 𝑈()𝐴()(𝑛)𝑚,𝑛∶=𝐼,𝑚=𝑛𝑚−1⋯𝐴,𝑚>𝑛. (3) It is easy to see that 𝑈(𝑚,𝑝)𝑈(𝑝,𝑛)=𝑈(𝑚,𝑛), for all (𝑚,𝑝,𝑛)∈𝑇.

For the particular case when (1) is autonomous, that is, 𝐴(𝑛)=𝐴∈ℬ(𝑋) for all 𝑛∈ℕ, then 𝑈(𝑚,𝑛)=𝐴^𝑚−𝑛 for all (𝑚,𝑛)∈Δ.

Definition 1. An application 𝑃∶ℕ→ℬ(𝑋) is said to be a projection family on 𝑋 if 𝑃²(𝑛)=𝑃(𝑛), (4) for all 𝑛∈ℕ.

Definition 2. Three projection families 𝑃₁,𝑃₂,𝑃₃∶ℕ→ℬ(𝑋) are said to be compatible with the system (1), if
(c1)  𝑃₁(𝑛)+𝑃₂(𝑛)+𝑃₃(𝑛)=𝐼, for all 𝑛∈ℕ,
(c2)  𝑃_𝑖(𝑛)𝑃_𝑗(𝑛)=0, for all 𝑛∈ℕ, for all 𝑖,𝑗∈{1,2,3},𝑖≠𝑗,
(c3)  ‖𝑃_𝑖(𝑛)𝑥+𝑃_𝑗(𝑛)𝑥‖²=‖𝑃_𝑖(𝑛)𝑥‖²+‖𝑃_𝑗(𝑛)𝑥‖², for all 𝑛∈ℕ, all 𝑥∈𝑋, and all 𝑖,𝑗∈{1,2,3}, 𝑖≠𝑗,
(c4)  𝐴(𝑛)𝑃_𝑖(𝑛)=𝑃_𝑖(𝑛+1)𝐴(𝑛), for all 𝑛∈ℕ and all 𝑖∈{1,2,3}.

Remark 3. If 𝑃₁,𝑃₂,𝑃₃∶ℕ→ℬ(𝑋) are projection families compatible with the system (1) then 𝑈(𝑚,𝑛)𝑃_𝑖(𝑛)=𝑃_𝑖(𝑚)𝑈(𝑚,𝑛) (5) for all 𝑖∈{1,2,3} and all (𝑚,𝑛)∈Δ.

Definition 4. The linear discrete-time system (1) is said to be uniformly exponentially trichotomic if there exist three projection families {𝑃_𝑖}_{𝑖∈{1,2,3}} compatible with the system (1), constants 𝑣₁,𝑣₂,𝑣₃, and 𝑣₄ with the property 𝑣₁≤𝑣₂<0<𝑣₃≤𝑣₄, and 𝑁≥1 such that (uet₁)  ‖𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥‖≤𝑁‖𝑃₁(𝑛)𝑥‖𝑒^{𝑣1(𝑚−𝑛)},  (uet₂)  ‖𝑃₂(𝑛)𝑥‖≤𝑁‖𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥‖𝑒^{−𝑣4(𝑚−𝑛)},  (uet₃)  ‖𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥‖≤𝑁‖𝑃₃(𝑛)𝑥‖𝑒^{𝑣3(𝑚−𝑛)},  (uet₄)  ‖𝑃₃(𝑛)𝑥‖≤𝑁‖𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥‖𝑒^{−𝑣2(𝑚−𝑛)}, for all (𝑚,𝑛,𝑥)∈Δ×𝑋.

Example 5. Let 𝑋=ℝ³ with the Euclidean norm. The application A∶ℕ→ℬ(ℝ³) is defined by 𝐴(𝑛)𝑥₁,𝑥₂,𝑥₃=𝑎𝑒^𝜆𝑛𝑥₁,𝑥₂𝑎,𝑥₃𝑒,∀𝑛,𝑥₁,𝑥₂,𝑥₃∈ℕ×ℝ³, (6) where 𝑎∈(0,𝑒⁻³) and 𝜆_𝑛=(𝑛+2)cosln(𝑛+2)−(𝑛+1)cosln(𝑛+1).
Let us consider the projection families 𝑃₁,𝑃₂,𝑃₃∶ℕ→ℬ(ℝ³) defined by 𝑃₁(𝑛)𝑥₁,𝑥₂,𝑥₃=𝑥₁,𝑃,0,0₂(𝑛)𝑥₁,𝑥₂,𝑥₃=0,𝑥₂,𝑃,0₃(𝑛)𝑥₁,𝑥₂,𝑥₃=0,0,𝑥₃ (7) for all 𝑛∈ℕ and all 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈𝑋.
A simple calculus shows that 𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥=𝑎^𝑚−𝑛𝑒^𝜇𝑚𝑛𝑥₁,𝑈()𝑃,0,0𝑚,𝑛₂(𝑛)𝑥=0,𝑎^{−(𝑚−𝑛)}𝑥₂,𝑈()𝑃,0𝑚,𝑛₃(𝑛)𝑥=0,0,𝑒^{−(𝑚−𝑛)}𝑥₃, (8) where 𝜇_𝑚𝑛=(𝑚+1)cosln(𝑚+1)−(𝑛+1)cosln(𝑛+1).
If we denote (𝑥)=𝑥cosln𝑥, then by Lagrange’s mean value theorem it results that for every 𝑛∈ℕ there exists 𝜏_𝑛∈(𝑛+1,𝑛+2) such that 𝜆_𝑛=𝜑(𝑛+2)−𝜑(𝑛+1)=cosln𝜏_𝑛−sinln𝜏_𝑛 (9) and hence |𝜆_𝑛|≤√2≤2 for every 𝑛∈ℕ.
This implies that sup_𝑛∈ℕ‖𝐴(𝑛)‖≤𝑎𝑒²+1𝑎+1𝑒,𝜇_𝑚𝑛≤2(𝑚−𝑛). (10) In the following we obtain the relations 𝑒^𝑚−𝑛𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥≤𝑒^{𝜀(𝑚−𝑛)+𝜇𝑚𝑛}𝑎^𝑚−𝑛𝑥₁≤𝑥₁=𝑃₁(𝑛)𝑥,𝑒^𝑚−𝑛𝑃₂(𝑛)𝑥≤𝑒^{3(𝑚−𝑛)}𝑥₂≤𝑎^{−(𝑚−𝑛)}𝑥₂=𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥,𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥=𝑒^{−(𝑚−𝑛)}𝑥₃≤𝑒^{(𝑚−𝑛)}𝑥₃=𝑃₃(𝑛)𝑥𝑒^{(𝑚−𝑛)},𝑃₃(𝑛)𝑥=𝑥₃≤𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥𝑒^{(𝑚−𝑛)}, (11) for all (𝑚,𝑛,𝑥)∈Δ×ℝ³, where 𝜀=ln(𝑎𝑒²)⁻¹>ln𝑒=1. Thus Definition 4 is satisfied for 𝑁=−𝑣₁=−𝑣₂=𝑣₃=𝑣₄=1; hence, the system (1) is uniformly exponentially trichotomic.

Definition 6. The linear discrete-time system (1) is said to be nonuniformly exponentially trichotomic if there exist a nondecreasing sequence of real numbers 𝑁∶ℕ→ℝ^∗₊, constants 𝑣₁, 𝑣₂, 𝑣₃, and 𝑣₄ with the property 𝑣₁≤𝑣₂<0<𝑣₃≤𝑣₄, and three projection families {𝑃_𝑖}_{𝑖∈{1,2,3}} compatible with the system (1) such that (net₁)  ‖𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥‖≤𝑁(𝑛)‖𝑃₁(𝑛)𝑥‖𝑒^{𝑣1(𝑚−𝑛)},  (net₂)  ‖𝑃₂(𝑛)𝑥‖≤𝑁(𝑚)‖𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥‖𝑒^{−𝑣4(𝑚−𝑛)},  (net₃)  ‖𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥‖≤𝑁(𝑛)‖𝑃₃(𝑛)𝑥‖𝑒^{𝑣3(𝑚−𝑛)},  (net₄)  ‖𝑃₃(𝑛)𝑥‖≤𝑁(𝑚)‖𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥‖𝑒^{−𝑣2(𝑚−𝑛)}, for all (𝑚,𝑛,𝑥)∈Δ×𝑋.

Remark 7. Consider the following.(i)For 𝑃₃=0 in Definition 6 we obtain the property of nonuniform exponential dichotomy.(ii)For 𝑃₂=𝑃₃=0, the property of nonuniform exponential stability is obtained. It follows that a nonuniformly exponentially stable linear discrete-time system is nonuniformly exponentially dichotomic and, further, nonuniformly exponentially trichotomic.(iii)For 𝑃₁=𝑃₃=0, we obtain the property of nonuniform exponential expansivity. Also it is easy to see that the property of nonuniform exponential expansivity implies the nonuniform exponential dichotomy and, further, the nonuniform exponential trichotomy.

Remark 8. The linear discrete-time system (1) is nonuniformly exponentially trichotomic if and only if there exist a nondecreasing sequence of real numbers 𝑁∶ℕ→ℝ^∗₊, constants 𝑣₁, 𝑣₂, 𝑣₃, and 𝑣₄ with the property 𝑣₁≤𝑣₂<0<𝑣₃≤𝑣₄, and three projection families {𝑃_𝑖}_{𝑖∈{1,2,3}} compatible with the system (1) such that (a) 𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥≤𝑁𝑝𝑈𝑝,𝑛𝑃₁(𝑛)𝑥𝑒^{𝑣1(𝑚−𝑝)}, (12) (b) 𝑈𝑝,𝑛𝑃₂(𝑛)𝑥≤𝑁(𝑚)𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥𝑒^{−𝑣4(𝑚−𝑝)}, (13) (c) 𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥≤𝑁𝑝𝑈𝑝,𝑛𝑃₃(𝑛)𝑥𝑒^{𝑣3(𝑚−𝑝)}, (14) (d) 𝑈𝑝,𝑛𝑃₃(𝑛)𝑥≤𝑁(𝑚)𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥𝑒^{−𝑣2(𝑚−𝑝)}, (15)for all (𝑚,𝑝,𝑛,𝑥)∈𝑇×𝑋.

Remark 9. It is obvious that if the system (1) is uniformly exponentially trichotomic then it is nonuniformly exponentially trichotomic. But the converse statement is not necessarily valid. This fact is illustrated by the following example.

Example 10. Let X=ℝ³ with the Euclidean norm. The application 𝐴∶ℕ→ℬ(ℝ³) is defined by 𝐴(𝑛)𝑥₁,𝑥₂,𝑥₃=𝑎𝑥₁,𝑥₂,𝑥₃,∀𝑛,𝑥₁,𝑥₂,𝑥₃∈ℕ×ℝ³, (16) where 𝑎∈(0,𝑒⁻¹]. We consider the projection families 𝑃₁,𝑃₂,𝑃₃∶ℕ→ℬ(ℝ³), 𝑃₁(𝑛)𝑥₁,𝑥₂,𝑥₃=𝑥₁,𝑃,0,0₂(𝑛)𝑥₁,𝑥₂,𝑥₃=0,𝑥₂,𝑃,0₃(𝑛)𝑥₁,𝑥₂,𝑥₃=0,0,𝑥₃ (17) for all 𝑛∈ℕ and all 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈𝑋, compatible with the system (1).
Then 𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥=𝑎^𝑚−𝑛𝑥₁,𝑈()𝑃,0,0𝑚,𝑛₂(𝑛)𝑥=0,𝑥₂,𝑈()𝑃,0𝑚,𝑛₃(𝑛)𝑥=0,0,𝑥₃. (18) Further we have that 𝑒^𝑚−𝑛𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥=(𝑒𝑎)^𝑚−𝑛𝑥₁≤𝑥₁≤𝑒^𝑛𝑥₁=𝑁(𝑛)𝑃₁(𝑛)𝑥,𝑒^𝑚−𝑛𝑃₂(𝑛)𝑥=𝑒^𝑚−𝑛𝑥₂≤𝑒^𝑚𝑥₂=𝑁(𝑚)𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥,1𝑁(𝑚)𝑒^{−(𝑚−𝑛)}𝑃₃(𝑛)𝑥=𝑒^−𝑚⋅𝑒^{−(𝑚−𝑛)}𝑥₃≤𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥=𝑥₃≤𝑒^𝑛⋅𝑒^𝑚−𝑛𝑃₃(𝑛)𝑥 (19) for all (𝑚,𝑛,𝑥)∈Δ×ℝ³. Thus Definition 6 is satisfied for 𝑁(𝑛)=𝑒^𝑛 and 𝑣₁=𝑣₂=−𝑣₃=−𝑣₄=−1; hence, the system (1) is nonuniformly exponentially trichotomic.
On the other hand, if we suppose that system (1) is uniformly exponentially trichotomic, then there exist two constants 𝑁≥1 and 𝑣₄>0 such that 𝑥₂=𝑃₂(𝑛)𝑥≤𝑁𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥𝑒^{−𝑣4(𝑚−𝑛)}=𝑁𝑥₂𝑒^{−𝑣4(𝑚−𝑛)} (20) for all (𝑚,𝑛,𝑥)∈Δ×ℝ³. This implies that 𝑒^{𝑣4(𝑚−𝑛)}≤𝑁 for all (𝑚,𝑛)∈Δ, which is false.

3. The Main Results

Theorem 11. The linear discrete-time system (1) is nonuniformly exponentially trichotomic if and only if there exist a function 𝐹∈ℱand three projection families {𝑃_𝑖}_{𝑖∈{1,2,3}} compatible with the system (1) such that the following relations hold:
(i) There exist a constant 𝑑₁>0 and a sequence of positive real numbers {𝜉_𝑛}_𝑛∈ℕ such that ^𝑚_𝑘=𝑛𝐹𝑒^{𝑑1(𝑘−𝑛)}𝑈(𝑘,𝑛)𝑃₁(𝑛)𝑥≤𝐹𝜉_𝑛𝑃₁(𝑛)𝑥, (21)
(ii) There exist a constant 𝑑₂>0 and a sequence of positive real numbers {𝜂_𝑛}_𝑛∈ℕ such that ^𝑚_𝑘=𝑛𝐹𝑒^{−𝑑2(𝑘−𝑛)}𝑈(𝑘,𝑛)𝑃₂(𝑛)𝑥≤𝐹𝜂_𝑚𝑒^{−𝑑2(𝑚−𝑛)}𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥, (22)
(iii) There exist a constant 𝑑₃>0 and a sequence of positive real numbers {𝛼_𝑛}_𝑛∈ℕ such that ^𝑚_𝑘=𝑝𝐹𝑒^{−𝑑3(𝑘−𝑛)}𝑈(𝑘,𝑛)𝑃₃(𝑛)𝑥≤𝐹𝛼_𝑛𝑒^{−𝑑3(𝑝−𝑛)}𝑈𝑝,𝑛𝑃₃(𝑛)𝑥, (23)
(iv) There exist a constant 𝑑₄>0 and a sequence of positive real numbers {𝛽_𝑛}_𝑛∈ℕ such that ^𝑚_𝑘=𝑝𝐹𝑒^{𝑑4(𝑘−𝑛)}𝑈(𝑘,𝑛)𝑃₃(𝑛)𝑥≤𝐹𝛽_𝑚𝑒^{𝑑4(𝑚−𝑛)}𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥 (24) for all (𝑚,𝑝,𝑛,𝑥)∈𝑇×𝑋.

Proof. (a)⇔(i) Necessity. We consider 𝐹(𝑡)=𝑡, 𝑡≥0. As system (1) is nonuniformly exponentially trichotomic, Remark 8 assures the existence of a constant 𝑣₁<0, a sequence of real numbers 𝑁∶ℕ→ℝ^∗₊, and a projection family 𝑃₁ such that (a) holds. We obtain for 𝑑₁=−(𝑣₁/2)>0 and according to (a)^𝑚_𝑘=𝑛𝑒^{𝑑1(𝑘−𝑛)}𝑈(𝑘,𝑛)𝑃₁(𝑛)𝑥≤𝑁(𝑛)^𝑚_𝑘=𝑛𝑒^{𝑑1(𝑘−𝑛)}𝑒^{𝑣1(𝑘−𝑛)}𝑈(𝑛,𝑛)𝑃₁(𝑛)𝑥=𝑁(𝑛)𝑃₁(𝑛)𝑥^𝑚_𝑘=𝑛𝑒^{(𝑣1/2)(𝑘−𝑛)}≤𝜉_𝑛𝑃₁(𝑛)𝑥,∀(𝑚,𝑛,𝑥)∈Δ×𝑋, (25) where we have denoted 𝜉_𝑛=(1/(1−𝑒^−𝑑1))𝑁(𝑛), 𝑛∈ℕ.
Sufficiency. According to the hypothesis, if we consider 𝑘=𝑚 then 𝐹𝑒^{𝑑1(𝑚−𝑛)}𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥≤𝐹𝜉_𝑛𝑃₁(𝑛)𝑥 (26) for all (𝑚,𝑛,𝑥)∈Δ×𝑋. By means of the properties of function 𝐹, we have 𝑈(𝑚,𝑛)𝑃₁(𝑛)𝑥≤𝜉_𝑛𝑃₁(𝑛)𝑥𝑒^{−𝑑1(𝑚−𝑛)}. (27) Thus relation (a) is obtained.
Similarly the other equivalences can also be proved, that is, (b)⇔(ii), (c)⇔(iii), and (d)⇔(iv).

Remark 12. The preceding theorem is an extension for the case of nonuniform exponential trichotomy of a result due to Popa et al. in [16]. Additionally, it is variant for the case of nonuniform exponential trichotomy property of a well-known theorem due to Przyłuski and Rolewicz [17] for exponential stability. If we consider 𝐹(𝑡)=𝑡^𝑝, 𝑡≥0, 𝑝>0, then Theorem 11 can be considered a version for the case of nonuniform exponential trichotomy of some results due to Datko [5].
It is well known that the exponential dichotomy involves two commuting projection families. In order to emphasize the natural extension of the nonuniform exponential trichotomy relative to the property of dichotomy, we will present a characterization by means of two commuting projection families, introduced by the following.

Definition 13. Two projection families 𝑅₁, 𝑅₂∶ℕ→ℬ(𝑋) are said to be compatible with the system (1), if (r1) 𝑅_𝑖(𝑛)𝑅_𝑗(𝑛)=0, for all 𝑛∈ℕ, for all 𝑖,𝑗∈{1,2}, 𝑖≠𝑗, (r2) ‖𝑅₁(𝑛)𝑥+𝑅₂(𝑛)𝑥‖²=‖𝑅₁(𝑛)𝑥‖²+‖𝑅₂(𝑛)𝑥‖², for all 𝑛∈ℕ, all 𝑥∈𝑋, (r3) ‖[𝐼−𝑅₁(𝑛)]‖²=‖[𝐼−𝑅₁(𝑛)−𝑅₂(𝑛)]𝑥‖²+‖𝑅₂(𝑛)𝑥‖², for all 𝑛∈ℕ, all 𝑥∈𝑋, (r4) ‖[𝐼−𝑅₂(𝑛)]‖²=‖[𝐼−𝑅₁(𝑛)−𝑅₂(𝑛)]𝑥‖²+‖𝑅₁(𝑛)𝑥‖², for all 𝑛∈ℕ, all 𝑥∈𝑋, (r5) 𝐴(𝑛)𝑅_𝑖(𝑛)=𝑅_𝑖(𝑛+1)𝐴(𝑛), for all 𝑛∈ℕ and all 𝑖∈{1,2}.

Theorem 14. The linear discrete-time system (1) is nonuniformly exponentially trichotomic if and only if there exist a nondecreasing sequence of real numbers 𝑁₁∶ℕ→ℝ^∗₊, constants 𝜇>𝑣>0, and two projection families {𝑅_𝑖}_𝑖∈{1,2} compatible with the system (1) such that (𝑎^′) 𝑈(𝑚,𝑛)𝑅₁(𝑛)𝑥≤𝑁₁𝑝𝑈𝑝,𝑛𝑅₁(𝑛)𝑥𝑒^{−𝜇(𝑚−𝑝)}, (28) (𝑏^′) 𝑈𝑝,𝑛𝑅₂(𝑛)𝑥≤𝑁₁(𝑚)𝑈(𝑚,𝑛)𝑅₂(𝑛)𝑥𝑒^{−𝑣(𝑚−𝑝)}, (29) (𝑐^′) 𝑈(𝑚,𝑛)𝐼−𝑅₂(𝑛)𝑥≤𝑁₁𝑝𝑈𝑝,𝑛𝐼−𝑅₂(𝑛)𝑥𝑒^{𝑣(𝑚−𝑝)}, (30) (𝑑^′) 𝑈𝑝,𝑛𝐼−𝑅₁(𝑛)𝑥≤𝑁₁(𝑚)𝑈(𝑚,𝑛)𝐼−𝑅₁(𝑛)𝑥𝑒^{𝜇(𝑚−𝑝)}, (31)for all (𝑚,𝑝,𝑛,𝑥)∈𝑇×𝑋.

Proof. Necessity. As the system (1) is nonuniformly exponentially trichotomic, according to Remark 8 there exist three projection families {𝑃_𝑖}_{𝑖∈{1,2,3}} compatible with the system (1), constants 𝑣₁≤𝑣₂<0<𝑣₃≤𝑣₄, and a nondecreasing sequence of real numbers 𝑁∶ℕ→ℝ^∗₊ such that relations (a)–(d) hold.
If we denote 𝑅₁=𝑃₁ and 𝑅₂=𝑃₂ then conditions (r1)–(r5) of Definition 13 result from (c1)–(c4) of Definition 2. Thus, the projection families 𝑅₁ and 𝑅₂ are compatible with the system (1).
Let us define 𝜇=−𝑣₁=−𝑣₂>0, 𝑣=𝑣₃=𝑣₄>0, and 𝑁₁(𝑛)=𝑁(𝑛), 𝑛∈ℕ. It is obvious that relations (𝑎^′) and (𝑏^′) hold. Using conditions (c3), (c4), (b), and (d) we obtain 𝑈(𝑝,𝑛)[𝐼−𝑅₁(𝑛)]𝑥²=𝑈𝑝,𝑛𝑃₂(𝑛)+𝑃₃(𝑛)𝑥²=𝑈𝑝,𝑛𝑃₂(𝑛)𝑥²+𝑈𝑝,𝑛𝑃₃(𝑛)𝑥²≤𝑁(𝑚)²𝑈(𝑚,𝑛)𝑃₂(𝑛)𝑥²𝑒^{−2𝑣4(𝑚−𝑝)}+𝑁(𝑚)²𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥²𝑒^{−2𝑣2(𝑚−𝑝)}≤𝑁(𝑚)²𝑈(𝑚,𝑛)𝑃₂(𝑛)+𝑃₃(𝑛)𝑥²𝑒^{−2𝑣2(𝑚−𝑝)}=𝑁₁(𝑚)²𝑈(𝑚,𝑛)[𝐼−𝑅₁(𝑛)]𝑥²𝑒^{2𝜇(𝑚−𝑝)} (32) for all (𝑚,𝑝,𝑛,𝑥)∈𝑇×𝑋. Hence, relation (𝑑^′) holds.
The proof of (𝑐^′) is similar.
Sufficiency. We consider the projection families 𝑃₁=𝑅₁,𝑃₂=𝑅₂,𝑃₃=𝐼−𝑅₁−𝑅₂=𝐼−𝑅₁𝐼−𝑅₂. (33) These are compatible with the system (1).
Let us define 𝑣₁=𝑣₂=−𝜇<0, 𝑣₃=𝑣₄=𝑣>0, and 𝑁(𝑛)=𝑁₁(𝑛), 𝑛∈ℕ. It is clear that (𝑎^′)⇔(a) and (𝑏^′)⇔(b). By condition (𝑐^′) we obtain 𝑈(𝑚,𝑛)𝑃₃(𝑛)𝑥=𝑈(𝑚,𝑛)𝐼−𝑅₁(𝑛)𝐼−𝑅₂(𝑛)𝑥≤𝑁₁𝑝𝑈𝑝,𝑛𝐼−𝑅₁(𝑛)𝐼−𝑅₂(𝑛)𝑥𝑒^{𝑣(𝑚−𝑝)}=𝑁𝑝𝑈𝑝,𝑛𝑃₃(𝑛)𝑥𝑒^{𝑣3(𝑚−𝑝)} (34) for all (𝑚,𝑝,𝑛,𝑥)∈𝑇×𝑋, and hence (c) holds.
Similarly, we can prove that (𝑑^′) implies relation (d). Finally, by Remark 8 we conclude that the system (1) is nonuniformly exponentially trichotomic.

Acknowledgments

The authors would like to express their sincere thanks to the reviewer for the valuable suggestions and comments which have improved the quality of the paper. This work was supported by “the Fundamental Research Funds for the Central Universities” (no. 2013XK03).