Nonuniform Exponential Dichotomies in Terms of Lyapunov Functions for Noninvertible Linear Discrete-Time Systems

Abstract

The aim of this paper is to give characterizations in terms of Lyapunov functions for nonuniform exponential dichotomies of nonautonomous and noninvertible discrete-time systems.

1. Introduction

The notion of (uniform) exponential dichotomy essentially introduced by Perron in [1] for differential equations and by Li in [2] for difference equations plays a central role in a substantial part of the asymptotic behaviors theory of dynamical systems.

In some situations, particularly in the nonautonomous settings, the concept of uniform exponential dichotomy is too restrictive and it is important to consider more general behaviors.

One of the main reasons for weakening the assumption of uniform exponential dichotomy is that from the point of view of ergodic theory almost all variational equations in a finite-dimensional space admit a nonuniform exponential dichotomy. On the other hand it is important to treat the case of noninvertible systems because of their interest in applications (e.g., random dynamical systems, generated by random parabolic equations, are not invertible).

The importance of Lyapunov functions is well established in the study of linear and nonlinear systems in both continuous and discrete-time. Thus, after the seminal work of Lyapunov republished most recently in [3] relevant results using Lyapunov's direct method (or second method) are presented in the books due to LaSalle, Lefschetz [4], Hahn [5], Halanay, Wexler [6], and Maliso and Mazenc [7].

This paper considers the general notion of nonuniform exponential dichotomy for nonautonomous linear discrete-time systems in Banach spaces. The main objective is to give characterizations of nonuniform exponential dichotomy in terms of Lyapunov functions for the general case of noninvertible linear discrete-time systems, as a particular case is the concept of (nonuniform) exponential dichotomy for the discrete-time linear systems which are invertible in the unstable directions. This approach can be found in the works of Barreira and Valls (see [8, 9]), and for the uniform approach we can mention the paper of Papaschinopoulos (see [10]).

In the nonuniform exponential dichotomies of linear discrete-time systems presented in this paper we consider two types of projection sequences: invariant and strongly invariant, which are distinct even in the finite-dimensional case.

We remark that we consider linear discrete-time systems having the right hand side not necessarily invertible and the dichotomy concepts studied in this paper use the evolution operators in forward time. In the definition of nonuniform exponential dichotomy we assume the existence of invariant projections sequence. At a first view the existence of such sequence is a strong hypothesis. This impediment can be eliminated using the notion of admissibility (see [11]).

The main theme of our paper is the relation between the notion of nonuniform exponential dichotomy with invariant projection sequences and the notion of Lyapunov functions. The case of exponential dichotomy with strongly invariant projection sequences was considered by Barreira and Valls (see [8, 9]).

2. Definitions, Notations, and Preliminary Results

We first fix the notions and introduce the basic concepts underlying this paper. By ℕ we denote the positive integers and ℝ₊ denotes the set of positive real numbers. 𝑋 is a real or complex Banach space and ℬ(𝑋) is the Banach algebra of all bounded linear operators on 𝑋. The norm on 𝑋 and ℬ(𝑋) will be denoted by ‖⋅‖. We denote by 𝐼 the identity operator on 𝑋.

If 𝐴∈ℬ(𝑋) then we will denote by Ker𝐴 and by Im𝐴, respectively, the kernel and range of 𝐴; that is, Ker𝐴={𝑥∈𝑋∶𝐴𝑥=0},(1) respectively, Im𝐴={𝐴𝑥∶𝑥∈𝑋}.(2) We also denote by Δ the set of all pairs of natural numbers (𝑚,𝑛) with 𝑚≥𝑛.

Throughout this paper, we consider the linear discrete-time systems of the form 𝑥_𝑛+1=𝐴_𝑛𝑥_𝑛,𝑛∈ℕ,(𝔄) where (𝐴_𝑛) is a sequence in ℬ(𝑋). If for every 𝑛∈ℕ the operator (𝐴_𝑛) is invertible, then the linear discrete-time system (𝔄) is called reversible. Every solution 𝑥=(𝑥_𝑛) of the system (𝔄) is given by 𝑥_𝑚=𝒜(𝑚,𝑛)𝑥_𝑛,(3) for all (𝑚,𝑛)∈Δ, where 𝒜∶Δ→ℬ(𝑋) is defined by 𝒜()=𝐴𝑚,𝑛^𝑚−1⋯𝐴_𝑛if𝑚>𝑛𝐼if𝑚=𝑛.(4) The map 𝒜 is called the discrete evolution operator associated to the system (𝔄).

Remark 1. The discrete evolution operator 𝒜(𝑚,𝑛) satisfies the propagator property; that is, 𝒜(𝑚,𝑛)𝒜(𝑘,𝑛)=𝒜(𝑚,𝑛),(5) for all (𝑚,𝑘), (𝑘,𝑛)∈Δ.

Definition 2. A sequence (𝑃_𝑛) in ℬ(𝑋) is called a projections sequence on 𝑋, if 𝑃²_𝑛=𝑃_𝑛,forevery𝑛∈ℕ.(6)

Remark 3. If (𝑃_𝑛) is a projections sequence on 𝑋 the sequence (𝑄_𝑛) defined by 𝑄_𝑛=𝐼−𝑃_𝑛,∀𝑛∈ℕ(7) is a projections sequence on 𝑋, which is called the complementary projections of (𝑃_𝑛). We remark that 𝑃_𝑛𝑄_𝑛=𝑄_𝑛𝑃_𝑛=0, Ker𝑃_𝑛=Im𝑄_𝑛, and Ker𝑄_𝑛=Im𝑃_𝑛.

Definition 4. A projection sequence (𝑃_𝑛) is called invariant for the system (𝔄) if 𝐴_𝑛𝑃_𝑛=𝑃_𝑛+1𝐴_𝑛,(8) for all 𝑛∈ℕ.

Remark 5. In the particular case when (𝔄) is autonomous, that is, 𝐴_𝑛=𝐴∈ℬ(𝑋) and 𝑃_𝑛=𝑃∈ℬ(𝑋) for all 𝑛∈ℕ, then (𝑃_𝑛) is invariant for (𝔄) if and only if 𝐴𝑃=𝑃𝐴.

Remark 6. The relation (8) from Definition 4 is also true for the complementary projection (𝑄_𝑛) and, as a consequence of (8), we have that 𝒜(𝑚,𝑛)𝑃_𝑛=𝑃_𝑚𝒜(𝑚,𝑛),(9) respectively, 𝒜(𝑚,𝑛)𝑄_𝑛=𝑄_𝑚𝒜(𝑚,𝑛),(10) for all (𝑚,𝑛)∈Δ.

Remark 7. If (𝑃_𝑛) is a projections sequence invariant for the reversible system (𝔄) then 𝒜(𝑚,𝑛) is invertible for all (𝑚,𝑛)∈ℕ² and 𝒜(𝑚,𝑛)⁻¹𝑃_𝑚=𝑃_𝑛𝒜(𝑚,𝑛)⁻¹,𝒜(𝑚,𝑛)⁻¹𝑄_𝑚=𝑄_𝑛𝒜(𝑚,𝑛)⁻¹,(11) for all (𝑚,𝑛)∈Δ.

Definition 8. Let (𝑃_𝑛) be a projections sequence which is invariant for the system (𝔄). We say that (𝑃_𝑛) is strongly invariant for (𝔄) if for every (𝑚,𝑛)∈Δ the linear operator 𝒜(𝑚,𝑛) is an isomorphism from Ker𝑃_𝑛 to Ker𝑃_𝑚.

A characterization of strongly invariant projections sequence is given by the following.

Proposition 9. Let (𝑃_𝑛) be a projections sequence which is invariant for the system (𝔄). Suppose that for all (𝑚,𝑛)∈Δ the evolution operator 𝒜(𝑚,𝑛) is injective on Ker𝑃_𝑛. Then (𝑃_𝑛) is strongly invariant for (𝔄) if and only if Ker𝑃_𝑚⊂Im𝒜(𝑚,𝑛)(12) for all (𝑚,𝑛)∈Δ.

Proof.
Necessity. If (𝑃_𝑛) is strongly invariant for (𝔄) and 𝑦∈Ker𝑃_𝑚 then there is 𝑥∈Ker𝑃_𝑛 with 𝑦=𝒜(𝑚,𝑛)𝑥 and hence 𝑦∈Im𝒜(𝑚,𝑛).
Sufficiency. We will prove that for every 𝑦∈Ker𝑃_𝑚 there exists 𝑥∈Ker𝑃_𝑛 with 𝑦=𝒜(𝑚,𝑛)𝑥.
Let 𝑦∈Ker𝑃_𝑚. Then 𝑦∈Im𝑄_𝑚 and hence 𝑦=𝑄_𝑚𝑦. Moreover, from the hypothesis there is 𝑥₀∈𝑋 such that 𝑦=𝒜(𝑚,𝑛)𝑥₀. Then 𝑦=𝑄_𝑚𝑦=𝑄_𝑚𝒜(𝑚,𝑛)𝑥₀=𝒜(𝑚,𝑛)𝑄_𝑛𝑥₀=𝒜(𝑚,𝑛)𝑥,(13) where 𝑥=𝑄_𝑛𝑥₀∈Ker𝑃_𝑛.

Corollary 10. If the projections sequence (𝑃_𝑛) is invariant for the reversible system (𝔄) then it is also strongly invariant for (𝔄).

An example of invariant projections sequence which is not strongly invariant is given by the following.

Example 11. Let 𝑋=ℝ³ with the norm 𝑥₁,𝑥₂,𝑥₃=𝑥₁+𝑥₂+𝑥₃(14) and let (𝔄) be the discrete-time system defined by the sequence 𝐴_𝑛𝑥₁,𝑥₂,𝑥₃=𝑥₁𝑒,𝑒𝑥₂,𝑒𝑥₃𝑥if𝑛≥1₁,0,𝑥₃if𝑛=0.(15) It is easy to verify that the sequence (𝑃_𝑛) defined by 𝑃_𝑛𝑥₁,𝑥₂,𝑥₃=𝑥₁,𝑥₂𝑥,0if𝑛=0₁+𝑥₂𝑒^−2𝑛,0,0if𝑛≥1(16) is a projections sequence which is invariant for the system (𝔄). Moreover, the evolution operator associated to the system (𝔄) is given by 𝒜(𝑚,𝑛)𝑥₁,𝑥₂,𝑥₃=𝑥₁𝑒^𝑛−𝑚,𝑥₂𝑒^𝑚−𝑛,𝑥₃𝑒^𝑚−𝑛𝑥if𝑚≥𝑛≥1or𝑚=𝑛=0₁𝑒^−𝑚,0,𝑥₃𝑒^𝑚if𝑚>𝑛=0(17) for all (𝑚,𝑛)∈Δ. We can see that the evolution operator is injective on Ker𝑃_𝑛. The sequence (𝑃_𝑛) is not strongly invariant for (𝔄) because 𝑦=(−1/𝑒²,1,0)∈Ker𝑃₁ and 𝑦∉Im𝒜(1,0).

Remark 12. If the projections sequence (𝑃_𝑛) is strongly invariant for the system (𝔄) then there exist ℬ∶Δ→ℬ(𝑋) such that for all (𝑚,𝑛)∈Δ the evolution operator ℬ(𝑚,𝑛) is an isomorphism from Ker𝑃_𝑚 to Ker𝑃_𝑛.

Proposition 13. The function ℬ∶Δ→ℬ(𝑋) has the following properties:(b₁)𝒜(𝑚,𝑛)ℬ(𝑚,𝑛)𝑄_𝑚=𝑄_𝑚, (b₂)ℬ(𝑚,𝑛)𝒜(𝑚,𝑛)𝑄_𝑛=𝑄_𝑛, (b₃)ℬ(𝑚,𝑛)𝑄_𝑚=𝑄_𝑛ℬ(𝑚,𝑛)𝑄_𝑚, (b₄)𝑄_𝑚=ℬ(𝑚,𝑚)𝑄_𝑚=𝑄_𝑚ℬ(𝑚,𝑚)𝑄_𝑚
for all (𝑚,𝑛)∈Δ.

Proof. The properties (b₁) and (b₂) are immediate.
(b₃) We observe that for every (𝑚,𝑛,𝑥)∈Δ×𝑋 we have that 𝑄_𝑚𝑥∈Im𝑄_𝑚=Ker𝑃_𝑚(18) and hence ℬ(𝑚,𝑛)𝑄_𝑚𝑥∈Ker𝑃_𝑛=Im𝑄_𝑚,(19) which implies (b₃).
(b₄) follows immediately from (b₁) and (b₃).

Remark 14. If the projections sequence (𝑃_𝑛) is invariant for the reversible system (𝔄) then ℬ(𝑚,𝑛)=𝒜(𝑚,𝑛)⁻¹,(20) for all (𝑚,𝑛)∈Δ.

3. Nonuniform Exponential Dichotomies

In this section we investigate some dichotomy concepts of linear discrete-time systems (𝔄) with respect to a projections sequence (𝑃_𝑛) invariant for (𝔄).

Definition 15. We say that system (𝔄) admits a nonuniform exponential dichotomy (n.e.d.) with respect to the projections sequence (𝑃_𝑛), if there exist a constant 𝛼>0 and a function 𝐷∶ℕ→[1,∞) such that the following properties hold: 𝑒^{𝛼(𝑚−𝑛)}‖𝒜()𝑥‖(𝑛𝑥‖,𝑒𝑚,𝑛≤𝐷)‖(21)_{𝛼(𝑚−𝑛)}𝑦≤𝐷(𝑚)𝒜(𝑚,𝑛)𝑦,(22) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑛.

As particular cases of nonuniform exponential dichotomy, we have the following. (1)If 𝐷(𝑛)=𝑐𝑒^𝜀𝑛 with 𝑐≥1 and 𝜀≥0 then we say that system (𝔄) admits an exponential dichotomy (e.d.).(2)If 𝐷(𝑛)=𝐷≥1 for all 𝑛∈ℕ then we say that system (𝔄) is uniformly exponentially dichotomic (u.e.d.).

Remark 16. If 𝐷(𝑛)=𝐷≥1 for all 𝑛∈ℕ and (𝑃_𝑛) is bounded (i.e., there exist 𝑀≥1 such that ‖𝑃_𝑛‖≤𝑀) then it can be easily checked that the concept of nonuniform exponential dichotomy considered in our paper is in fact a particular case of exponential forward splitting considered in [12], with (𝑚,𝑛)∈Δ.

Remark 17. If the system (𝔄) is uniformly exponentially dichotomic and (𝑃_𝑛) is bounded then the sequence 𝐴_𝑛 is uniformly bounded with respect to 𝑛. (i.e., sup_𝑛∈ℕ‖𝐴_𝑛‖<∞). On the other hand, sup_𝑛∈ℕ‖𝐴_𝑛‖ may be finite and system (𝔄) admits a nonuniform behavior. See [13] for various examples for nonuniform exponential contractions and [14, 15] for nonuniform dichotomies. For the differential equations case, see [16].

Remark 18. The connection between the dichotomy concepts considered in this paper can be synthesized as (u.e.d.) ⇒ (e.d.) ⇒ (n.e.d.). The following examples show that the converse implications are not valid.

Example 19. Let (𝔄) be the linear discrete-time system and (𝑃_𝑛) the projections sequence considered in Example 11. By a simple computation we can see that 𝑒^𝑚−𝑛‖𝒜(𝑚,𝑛)𝑥‖≤𝑒^2𝑛‖𝑥‖,(23) respectively, 𝑒^𝑚−𝑛𝑦≤𝑒^2𝑚𝒜(𝑚,𝑛)𝑦(24) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑛. Hence, for 𝛼=1 and 𝐷(𝑛)=𝑒^2𝑛 the system (𝔄) is nonuniform exponentially dichotomic. Obviously, the nonuniform part cannot be removed.

Example 20. Let 𝑋=𝑙^∞(ℝ) be the Banach space of bounded real-valued sequences, endowed with the norm ‖𝑥‖=sup_𝑛≥0𝑥_𝑛,for𝑥=𝑥₀,𝑥₁,…,𝑥_𝑛,…∈𝑋.(25)
Let (𝑃_𝑛) be a sequence in ℬ(𝑋) defined by 𝑃_𝑛𝑥₀,𝑥₁,𝑥₂=𝑥,…₀+2^𝑛2−1𝑥₁,0,𝑥₂+2^𝑛2−1𝑥₃,0,….(26) It is a simple verification to see that (𝑃_𝑛) is a projections sequence with the complementary 𝑄_𝑛𝑥₀,𝑥₁,𝑥₂,…=1−2^𝑛2𝑥₁,𝑥₁,1−2^𝑛2𝑥₃,𝑥₃,….(27)
We consider the linear discrete-time system (𝔄) defined by the sequence (𝐴_𝑛) given by 𝐴_𝑛=2^{𝑎𝑛−𝑎𝑛+1}𝑃_𝑛+2^{𝑎𝑛+1−𝑎𝑛}𝑄_𝑛+1,(28) where 𝑎_𝑛=2𝑘3if𝑛=2𝑘2𝑘+1if𝑛=2𝑘+1,(29) for all 𝑛∈ℕ and all 𝑥∈𝑋.
We have that the evolution operator associated to (𝔄) is 𝒜(𝑚,𝑛)=2^{𝑎𝑛−𝑎𝑚}𝑃_𝑛+2^{𝑎𝑚−𝑎𝑛}𝑄_𝑛,(30) for all (𝑚,𝑛)∈Δ.
We observe that for all (𝑚,𝑛)∈Δ we obtain 𝑎_𝑛−𝑎_𝑚≤𝑛−𝑚3+2𝑛3(31) hence 𝑎_𝑚−𝑎_𝑛≥𝑚−𝑛3−2𝑛3.(32) We can see that (𝑃_𝑛) is strongly invariant for the system (𝔄) and ‖𝒜(𝑚,𝑛)𝑥‖≤2^2𝑛/32^{−(1/3)(𝑚−𝑛)}‖𝑥‖(33) and, respectively, ℬ(𝑚,𝑛)𝑦≤2^(2/3)𝑚2^{−(1/3)(𝑚−𝑛)}𝑦(34) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑛. Finally, we observe that system (𝔄) is exponentially dichotomic.
On the other side, if we suppose that the system (𝔄) admits a uniform approach, taking into account (21) from Definition 15 with 𝐷(𝑛)=𝐷 and (33) for 𝑚=𝑛+1 and 𝑛=2𝑘+1, we have that 2^(4𝑘+1)/3‖𝑥‖≤𝐷‖𝑥‖(35) for all 𝑥∈Im𝑃_𝑛, which shows that nonuniform part cannot be removed.

Example 21. Consider, on 𝑋=ℝ², the sequence (𝐴_𝑛) in ℬ(ℝ²) given by 𝐴_𝑛𝑥₁,𝑥₂=𝑒𝑎_𝑛𝑥₁,𝑥₂𝑒(36) for all (𝑛,𝑥₁,𝑥₂)∈ℕ×ℝ², where 𝑎_𝑛=𝑒^{𝑛(1+2𝑛)}𝑒if𝑛=2𝑘_{−(𝑛+1)(1+2𝑛+1)}if𝑛=2𝑘+1.(37) Then for (𝑃_𝑛) in ℬ(ℝ²) defined by 𝑃_𝑛𝑥₁,𝑥₂=𝑥₁,0(38) for all (𝑛,𝑥₁,𝑥₂)∈ℕ×ℝ², we have that for 𝛼=1 and 𝐷(𝑛)=𝑒^{𝑛(1+2𝑛)} the system (𝔄) is nonuniform exponentially dichotomic. Obviously, system (𝔄) is neither exponentially dichotomic nor uniformly exponentially dichotomic.

Remark 22. The system (𝔄) is nonuniform exponentially dichotomic with respect to the projections sequence (𝑃_𝑛) invariant for (𝔄) if and only if there exist a constant 𝑟∈(0,1) and a function 𝐷∶ℕ→[1,∞) such that ‖𝒜(𝑚,𝑛)𝑥‖≤𝐷(𝑛)𝑟^𝑚−𝑛‖𝑥‖,𝑦≤𝐷(𝑚)𝑟^𝑚−𝑛𝒜(𝑚,𝑛)𝑦,(39) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑛.
A characterization of nonuniform exponential dichotomy of reversible systems is given by the following.

Proposition 23. The reversible system (𝔄) is nonuniform exponentially dichotomic if and only if there exist a projections sequence (𝑃_𝑛) invariant for (𝔄), a function 𝐷∶ℕ→[1,∞), and a constant 𝛼>0 such that ‖𝒜(𝑚,𝑛)𝑥‖≤𝐷(𝑛)𝑒^{−𝛼(𝑚−𝑛)}‖𝑥‖,𝒜(40)₋₁(𝑚,𝑛)𝑦≤𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝑦(41) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑚.

Proof. It is sufficient to prove the equivalence between (22) and (41).
Necessity. If (22) holds then for all (𝑚,𝑛,𝑦)∈Δ×Ker𝑃_𝑚 we have 𝒜(𝑚,𝑛)⁻¹𝑦=𝒜(𝑚,𝑛)⁻¹𝑄_𝑚𝑦=𝑄_𝑛𝒜(𝑚,𝑛)⁻¹𝑄_𝑚𝑦≤𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝒜(𝑚,𝑛)𝑄_𝑛𝒜(𝑚,𝑛)⁻¹𝑄_𝑚𝑦=𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝑄_𝑚𝑦=𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝑦.(42)
Sufficiency. From (41) it results that for all (𝑚,𝑛,𝑦)∈Δ×Ker𝑃_𝑚 we have 𝑦=𝑄_𝑛𝑦=𝒜(𝑚,𝑛)⁻¹𝑄_𝑚𝒜(𝑚,𝑛)𝑄_𝑛𝑦≤𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝑄_𝑚𝒜(𝑚,𝑛)𝑄_𝑛𝑦=𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝒜(𝑚,𝑛)𝑄_𝑛𝑦.(43)

A characterization of nonuniform exponential dichotomy property with respect to strongly invariant projection sequences is given by the following.

Theorem 24. Let (𝑃_𝑛) be a projections sequence which is strongly invariant for the system (𝔄). Then (𝔄) is nonuniform exponentially dichotomic with respect to (𝑃_𝑛) if and only if there exist a function 𝐷∶ℕ→[1,∞) and a constant 𝛼>0 such that ‖𝒜(𝑚,𝑛)𝑥‖≤𝐷(𝑛)𝑒^{−𝛼(𝑚−𝑛)}‖𝑥‖,ℬ()𝑦(𝑚)𝑒(44)𝑚,𝑛≤𝐷^{−𝛼(𝑚−𝑛)}𝑦(45) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑚.

Proof. It is sufficient to prove the equivalence between (22) and (45).
Necessity. We observe that from (22) and the properties (b₁) and (b₃) from Proposition 13, we obtain 𝒜()𝑦=ℬ()𝑄𝑚,𝑛𝑚,𝑛_𝑚𝑦=^(b3)𝑄_𝑛ℬ(𝑚,𝑛)𝑄_𝑚𝑦≤𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝒜(𝑚,𝑛)𝑄_𝑛ℬ(𝑚,𝑛)𝑄_𝑚𝑦=𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝑄_𝑚𝒜(𝑚,𝑛)ℬ(𝑚,𝑛)𝑄_𝑚𝑦_b1=𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝑄_𝑚𝑦=𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝑦,(46) for all (𝑚,𝑛,𝑦)∈Δ×Ker𝑃_𝑚.
Sufficiency. By (45), using the property (b₂) from Proposition 13 we obtain 𝑦=𝑄_𝑛𝑦=^(b2)ℬ(𝑚,𝑛)𝒜(𝑚,𝑛)𝑄_𝑛𝑦=ℬ(𝑚,𝑛)𝑄_𝑚𝒜(𝑚,𝑛)𝑄_𝑛𝑦≤𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝒜(𝑚,𝑛)𝑄_𝑛𝑦=𝐷(𝑚)𝑒^{−𝛼(𝑚−𝑛)}𝒜(𝑚,𝑛)𝑦(47) for all (𝑚,𝑛,𝑦)∈Δ×Ker𝑃_𝑛.

4. Lyapunov Functions and Nonuniform Exponential Dichotomies

Let (𝔄) be a linear discrete-time system on a Banach space 𝑋 and (𝑃_𝑛) a projections sequence which is invariant for (𝔄).

Definition 25. We say that 𝐿∶Δ×𝑋→ℝ₊ is a Lyapunov function for the system (𝔄) with respect to projections sequence (𝑃_𝑛) if there exists a constant 𝑎∈(1,+∞) such that the following properties hold: 𝐿(𝑚,𝑛,𝑥)−𝑎𝐿(𝑚+1,𝑛,𝑥)≥‖𝒜(𝑚,𝑛)𝑥‖(48) for all (𝑚,𝑛,𝑥)∈Δ×Im𝑃_𝑛𝐿𝑚+1,𝑛,𝑦−𝑎𝐿𝑚,𝑛,𝑦≥𝒜(𝑚+1,𝑛)𝑦(49) for all (𝑚,𝑛,𝑦)∈Δ×Ker𝑃_𝑛, 𝐿𝑛,𝑛,𝑦≥𝑦(50) for all (𝑛,𝑦)∈ℕ×Ker𝑃_𝑛.

Example 26. Let (𝔄) be the linear discrete-time system and (𝑃_𝑛) the projections sequence considered in Example 11. Let 𝐿(𝑚,𝑛,𝑥)=2^𝑛−𝑚𝑃_𝑛𝑥+2^−𝑚𝑒^{𝑚𝑚−𝑛}_𝑘=𝑛2^𝑘𝑄_𝑘𝑥,(51) for all (𝑚,𝑛,𝑥)∈Δ×𝑋. By a simple computation for 𝑎=2𝑒/5 we can see that 𝐿(𝑚,𝑛,𝑥)−2𝑒5𝐿(𝑚+1,𝑛,𝑥)=2−2𝑒5‖𝒜()𝑥‖≥‖𝒜()𝑥‖𝑚,𝑛𝑚,𝑛(52) for all (𝑚,𝑛,𝑥)∈Δ×Im𝑃_𝑛 and 𝐿𝑚+1,𝑛,𝑦−2𝑒5𝐿𝑚,𝑛,𝑦≥𝑒^{𝑚−𝑛+1}𝑄_𝑚+1𝑦=𝒜(𝑚+1,𝑛)𝑦(53) for all (𝑚,𝑛,𝑦)∈Δ×Ker𝑃_𝑛.
Moreover 𝐿𝑛,𝑛,𝑦=𝑄_𝑛𝑦=𝑦(54) for all (𝑛,𝑦)∈ℕ×Ker𝑃_𝑛.

The main result of this paper is as follows.

Theorem 27. The linear discrete-time system (𝔄) is nonuniform exponentially dichotomic with respect to the projections sequence (𝑃_𝑛) invariant for (𝔄) if and only if there exists a nondecreasing sequence 𝛽∶ℕ→[1,∞) such that 𝐿(𝑚,𝑛,𝑥)+𝐿𝑚,𝑛,𝑦≤𝛽(𝑛)‖𝑥‖+𝛽(𝑚)𝒜(𝑚,𝑛)𝑦(55) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑛.

Proof.
Necessity. Suppose that (𝔄) is nonuniform exponentially dichotomic with respect to the projections sequence (𝑃_𝑛). We define 𝐿∶Δ×𝑋→ℝ₊ by 𝐿(𝑚,𝑛,𝑥)=^∞_𝑘=𝑚𝑑^𝑘−𝑚𝒜(𝑘,𝑛)𝑃_𝑛𝑥+^𝑚_𝑘=𝑛𝑑^𝑚−𝑘𝒜(𝑘,𝑛)𝑄_𝑛𝑥,(56) where 𝑑∈(1,1/𝑟) and 𝑟 is given by Remark 22. First, we observe that 𝐿(𝑚,𝑛,𝑥)=^∞_𝑘=𝑚𝑑^𝑘−𝑚𝒜(𝑘,𝑛)𝑃_𝑛𝑥+^𝑚_𝑘=𝑛𝑑^𝑚−𝑘𝒜(𝑘,𝑛)𝑄_𝑛𝑥≤^∞_𝑘=𝑚𝑑^𝑘−𝑚𝐷(𝑛)𝑟^𝑘−𝑛𝑃_𝑛𝑥+^𝑚_𝑘=𝑛𝑑^𝑚−𝑘𝑟^𝑚−𝑘𝐷(𝑚)𝒜(𝑚,𝑛)𝑄_𝑛𝑥≤𝐷(𝑛)𝑃1−𝑑𝑟^𝑛𝑥+𝐷(𝑚)𝒜()𝑄1−𝑑𝑟𝑚,𝑛_𝑛𝑥=𝛽(𝑛)𝑃_𝑛𝑥+𝛽(𝑚)𝒜(𝑚,𝑛)𝑄_𝑛𝑥,(57) where 𝛽(𝑛)=𝐷(𝑛)/(1−𝑑𝑟), and thus (55) is verified.
On the other hand, for 𝑥∈Im𝑃_𝑛 we have that 𝐿(𝑚,𝑛,𝑥)=^∞_𝑘=𝑚𝑑^𝑘−𝑚‖𝒜(𝑘,𝑛)𝑥‖=𝑑⁰‖𝒜(𝑚,𝑛)𝑥‖+𝑑¹‖𝒜(𝑚+1,𝑛)𝑥‖+⋯=𝑎^𝑛_𝑚+𝑑𝑎^𝑛_𝑚+1+𝑑²𝑎^𝑛_𝑚+2+⋯,(58) where 𝑎^𝑛_𝑚=‖𝒜(𝑚,𝑛)𝑥‖. Moreover 𝐿()=𝑚+1,𝑛,𝑥_{∞𝑘=𝑚+1}𝑑^{𝑘−𝑚−1}‖𝒜(𝑘,𝑛)𝑥‖=𝑑⁰‖𝒜(𝑚+1,𝑛)𝑥‖+𝑑¹‖𝒜(𝑚+2,𝑛)𝑥‖+⋯=𝑎^𝑛_𝑚+1+𝑑𝑎^𝑛_𝑚+2+𝑑²𝑎^𝑛_𝑚+3+⋯.(59) From (58) and (59) we have that 𝐿(𝑚,𝑛,𝑥)=𝑎^𝑛_𝑚+𝑑𝐿(𝑚+1,𝑛,𝑥).(60) Hence 𝐿(𝑚,𝑛,𝑥)−𝑎𝐿(𝑚+1,𝑛,𝑥)≥‖𝒜(𝑚,𝑛)𝑥‖,(61) for all (𝑚,𝑛,𝑥)∈Δ×Im𝑃_𝑛.
In the same manner we can see that for 𝑦∈Ker𝑃_𝑛 we have that 𝐿𝑚+1,𝑛,𝑦=^𝑚+1_𝑘=𝑛𝑑^𝑚−𝑘𝒜()𝑦+𝒜()𝑦.𝑘,𝑛=𝑑𝐿𝑚,𝑛,𝑦𝑚+1,𝑛(62) Hence 𝐿𝑚+1,𝑛,𝑦−𝑎𝐿𝑚,𝑛,𝑦≥𝒜(𝑚+1,𝑛)𝑦.(63)
Sufficiency. According to (48) for every (𝑚,𝑛,𝑥)∈Δ×Im𝑃_𝑛 we have that 𝐿()()≥‖𝒜()𝑥‖𝐿()()≥‖𝒜()𝑥‖⋮𝑛,𝑛,𝑥−𝑎𝐿𝑛+1,𝑛,𝑥𝑛,𝑛𝑛+1,𝑛,𝑥−𝑎𝐿𝑛+2,𝑛,𝑥𝑛+1,𝑛(64) which implies 𝐿(𝑛,𝑛,𝑥)≥^∞_𝑗=𝑛𝑎^𝑗−𝑛𝒜𝑗,𝑛𝑥=^∞_𝑘=0𝑎^𝑘‖𝒜(𝑛+𝑘,𝑛)𝑥‖.(65) From (65) we have that 𝑎^𝑚−𝑛‖𝒜(𝑚,𝑛)𝑥‖≤𝐿(𝑛,𝑛,𝑥)≤𝛽(𝑛)‖𝑥‖.(66) Hence ‖𝒜()𝑥‖≤1𝑚,𝑛𝑎^𝑚−𝑛𝛽(𝑛)‖𝑥‖(67) for all (𝑚,𝑛,𝑥)∈Δ×Im𝑃_𝑛.
In a completely analog way, from (49) and (50) for 𝑦∈Ker𝑃_𝑛 we have that 𝐿≥𝒜()𝑦𝐿≥𝒜()𝑦⋮𝐿≥𝒜()𝑦𝐿≥𝑦𝑚,𝑛,𝑦−𝑎𝐿𝑚−1,𝑛,𝑦𝑚,𝑛𝑚−1,𝑛,𝑦−𝑎𝐿𝑚−2,𝑛,𝑦𝑚−1,𝑛𝑛+1,𝑛,𝑦−𝑎𝐿𝑛,𝑛,𝑦𝑛+1,𝑛𝑛,𝑛,𝑦(68) which implies 𝐿𝑚,𝑛,𝑦≥^𝑚_𝑗=𝑛𝑎^𝑚−𝑗𝒜𝑗,𝑛𝑦≥𝑎^𝑚−𝑛𝒜(𝑛,𝑛)𝑦=𝑎^𝑚−𝑛𝑦.(69) By (69) we have 𝑎^𝑚−𝑛𝑦≤𝐿𝑚,𝑛,𝑦≤𝛽(𝑚)𝒜(𝑚,𝑛)𝑦.(70) Hence 𝑦≤𝛽(𝑚)𝑎^𝑚−𝑛𝒜(𝑚,𝑛)𝑦(71) for all (𝑚,𝑛,𝑦)∈Δ×Ker𝑃_𝑛. From (67) and (71) we obtain that system (𝔄) is nonuniform exponentially dichotomic. Thus, the proof is completed.

Corollary 28. The linear discrete-time system (𝔄) is exponentially dichotomic with respect to the projections sequence (𝑃_𝑛) if and only if there exist some constants 𝐾, 𝑝≥1 and a Lyapunov function 𝐿 with respect to (𝑃_𝑛) such that 𝐿(𝑚,𝑛,𝑥)+𝐿𝑚,𝑛,𝑦≤𝐾𝑝^𝑛‖𝑥‖+𝑝^𝑚𝒜(𝑚,𝑛)𝑦(72) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑛.

Corollary 29. The linear discrete-time system (𝔄) is uniformly exponentially dichotomic with respect to the projections sequence (𝑃_𝑛) if and only if there exist a Lyapunov function with respect to (𝑃_𝑛) and a constant 𝐾≥1 such that 𝐿(𝑚,𝑛,𝑥)+𝐿𝑚,𝑛,𝑦≤𝐾‖𝑥‖+𝒜(𝑚,𝑛)𝑦(73) for all (𝑚,𝑛)∈Δ and all (𝑥,𝑦)∈Im𝑃_𝑛×Ker𝑃_𝑛.