On Dichotomous Behavior of Variational Difference Equations and Applications

Abstract

We give new and very general characterizations for uniform exponential dichotomy of variational difference equations in terms of the admissibility of pairs of sequence spaces over ℕ with respect to an associated control system. We establish in the variational case the connections between the admissibility of certain pairs of sequence spaces over ℕ and the admissibility of the corresponding pairs of sequence spaces over ℤ. We apply our results to the study of the existence of exponential dichotomy of linear skew-product flows.

1. Introduction

In recent years, a number of papers have added important contributions to the existing literature on the relation between exponential dichotomy of systems and solvability properties of associated difference or differential equations, or the so-called admissibility properties with sequence or function spaces (see [1–10]). In the case of classical differential equations, the literature on this subject is rich and the main techniques are presented in the valuable contributions of Coffman and Schäffer [11], Coppel [12], Daleckii and Kreĭn [13], and Massera and Schäffer [14]. In the last few years, the methods have been improved and extended for general cases like those of evolution families or variational equations. The recent development in the theory of linear skew-product flows led to important generalizations of the classical results (see [4, 6, 7, 15–17]). A significant achievement was obtained in [15], where Chow and Leiva deduced the structure of the stable and unstable subspaces for an exponentially dichotomic linear skew-product flow. Various discrete-time characterizations for uniform exponential dichotomy of linear skew-product flows were obtained in [4, 6, 7]. In [4] Chow and Leiva introduced and characterized the concept of pointwise discrete dichotomy for a skew-product sequence over 𝑋×Θ, with 𝑋 a Banach space and Θ a compact Haussdorff space.

Let 𝑋 be a Banach space, let Θ be a metric space, and let 𝜋=(Φ,𝜎) be a linear skew-product flow on ℰ=𝑋×Θ, that is, 𝜎∶Θ×ℝ→Θ, 𝜎(𝜃,0)=𝜃 and 𝜎(𝜃,𝑡+𝑠)=𝜎(𝜎(𝜃,𝑡),𝑠), for all (𝜃,𝑡,𝑠)∈Θ×ℝ², and the mapping Φ∶Θ×ℝ₊→ℒ(𝑋) satisfies the following conditions: Φ(𝜃,0)=𝐼, (the identity operator on 𝑋), Φ(𝜃,𝑡+𝑠)=Φ(𝜎(𝜃,𝑡),𝑠)Φ(𝜃,𝑡), for all (𝜃,𝑡,𝑠)∈Θ×ℝ²₊ and there are 𝑀≥1,𝜔>0 such that ||Φ(𝜃,𝑡)||≤𝑀𝑒^𝜔𝑡, for all (𝜃,𝑡)∈Θ×ℝ₊. We associate with 𝜋 the variational discrete-time system 𝑥(𝜃)(𝑛+1)=Φ(𝜎(𝜃,𝑛),1)𝑥(𝜃)(𝑛),∀(𝜃,𝑛)∈Θ×ℕ.( (𝐴_𝜋)) Using the consequences given by the pointwise behavior, Chow and Leiva established in [4] an important result concerning the exponential dichotomy of linear skew-product flows.

Theorem 1.1. Let 𝜋=(Φ,𝜎) be a linear skew-product flow on ℰ=𝑋×Θ. Then 𝜋 is uniformly exponentially dichotomic if and only if the system (𝐴_𝜋) is uniformly exponentially dichotomic.A direct proof of the above theorem was presented in [6], without requiring continuity properties.

The impressive development of difference equations in the past few years (see, e.g., [1–4, 6–10, 15–25] and the references therein) led to important contributions at the study of the qualitative behavior of solutions of variational equations via discrete-time techniques. If {𝐴(𝜃)}_𝜃∈Θ⊂ℒ(𝑋), then one considers the linear system of variational difference equations 𝑥(𝜃)(𝑛+1)=𝐴(𝜎(𝜃,𝑛))𝑥(𝜃)(𝑛),(𝜃,𝑛)∈Θ×ℕ,( (𝐴)) and associates to (𝐴) the control system: 𝛾(𝜃)(𝑚+1)=𝐴(𝜎(𝜃,𝑚))𝛾(𝜃)(𝑚)+𝑠(𝑚+1),(𝜃,𝑚)∈Θ×ℤ.( (𝑆_𝐴)) We denote by (𝑄_𝐴) the restriction of (𝑆_𝐴) to ℕ. One of the main results in [6] states that a linear skew-product flow is uniformly exponentially dichotomic if and only if the pair (𝑐₀(ℕ,𝑋),𝑐₀₀(ℕ,𝑋)) is admissible with respect to the system (𝑄_𝐴) and the space 𝑋 may be decomposed at every moment into a sum of stable fiber and unstable fiber, that is, X=𝒮₀(𝜃)+𝒰₀(𝜃), for all 𝜃∈Θ. Here, 𝑐₀(ℕ,𝑋) denotes the space of all sequences 𝑠∶ℕ→𝑋 with lim_𝑛→∞𝑠(𝑛)=0 and 𝑐₀₀(ℕ,𝑋)={𝑠∈𝑐₀(ℕ,𝑋)∶𝑠(0)=0}.

In [7], we introduced distinct concepts of admissibility considering as the input space Δ(ℤ,𝑋)-the space of all sequences of finite support and we gave a unified treatment considering both cases when the output space is ℓ^𝑝(ℤ,𝑋) or 𝑐₀(ℤ,𝑋)∶={𝑠∶ℤ→𝑋∶lim_𝑘→±∞𝑠(𝑘)=0}. The approach given in [7] relies on the unique solvability of the discrete-time system (𝑆_𝐴) between certain sequence spaces. The main results in [7] expressed the uniform exponential dichotomy of a discrete linear skew-product flow 𝜋 in terms of the uniform 𝑞-admissibility (𝑞∈(1,∞]) of one of the pairs (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) and (𝑐₀(ℤ,𝑋),Δ(ℤ,𝑋)), with respect to the system (𝑆_𝐴).

The natural question arises whether in the characterization of the exponential dichotomy the unique solvability of the associated discrete control system can be dropped. Another question is which are the connections between the solvability of the system (𝑆_𝐴) and the solvability of the system (𝑄_𝐴). Using constructive techniques, we will provide complete answers for both questions and we will obtain new and optimal characterizations for uniform exponential dichotomy of variational difference equations. We denote by Δ₀(ℕ,𝑋) the space of sequences 𝑠∶ℕ→𝑋 with finite support and 𝑠(0)=0. The admissibility concepts introduced in the present paper are new and simplify the solvability conditions in [6]. Specifically, the input space considered is the smallest possible input space, the solvability study is reduced to the behavior on the half-line and the boundedness condition is imposed only on the unstable fiber. Moreover we extend the applicability area to the general context of discrete variational systems.

Another purpose of this paper is to provide a complete study concerning the relation between the discrete admissibility on the whole line and the corresponding discrete admissibility on the half-line. Our main strategy will be to provide an almost exhaustive analysis, providing the connections between admissibility of concrete pairs of sequence spaces defined on ℕ and the uniform exponential dichotomy of a general system of discrete variational equations, emphasizing the appropriate techniques for each case considered therein. Finally, applying the main results, we will deduce new characterizations for uniform exponential dichotomy of linear skew-product flows, in terms of the uniform 𝑞-admissibility of the pairs (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) and (𝑐₀(ℕ,𝑋),Δ₀(ℕ,𝑋)).

2. Uniform Exponential Dichotomy of Variational Difference Equations

Let 𝑋 be a real or complex Banach space, let (Θ,𝑑) be a metric space and let ℰ=𝑋×Θ. The norm on 𝑋 and on ℒ(𝑋)-the Banach algebra of all bounded linear operators on 𝑋 will be denoted by ||⋅||. Notations
Let ℤ denote the set of the integers, let ℕ be the set of the nonnegative integers (𝑚∈ℤ,𝑚≥0) and let ℤ₋ be the set of the nonpositive integers (𝑚∈ℤ,𝑚≤0). For every 𝐴⊂ℤ, let 𝜒_𝐴 denote the characteristic function of 𝐴. If 𝐽∈{ℤ,ℕ,ℤ₋} let Δ(𝐽,𝑋) be the linear space of all sequences 𝑠∶𝐽→𝑋 with the property that the set {𝑘∈𝐽∶𝑠(𝑘)≠0} is finite and let Δ₀(ℕ,𝑋)∶={𝑠∈Δ(ℕ,𝑋)∶𝑠(0)=0}.

If 𝑝∈[1,∞) let ℓ^𝑝(𝐽,𝑋) denote the set of all sequences 𝑠∶𝐽→𝑋 with the property that ∑_𝑗∈𝐽‖𝑠(𝑗)‖^𝑝<∞, which is a Banach space with respect to the norm ‖𝑠‖_𝑝∶=(_𝑗∈𝐽‖𝑠(𝑗)‖^𝑝)^1/𝑝. (2.1) Let ℓ^∞(𝐽,𝑋) be the set of all bounded sequences 𝑠∶𝐽→𝑋, which is a Banach space with respect to the norm ‖𝑠‖_∞∶=sup_𝑗∈𝐽||𝑠(𝑗)||. If 𝜔∈{−∞,∞}, then 𝑐₀(𝐽,𝑋)-the set of all sequences 𝑠∶𝐽→𝑋 with lim_𝑗→𝜔𝑠(𝑗)=0, is a closed linear subspace of ℓ^∞(𝐽,𝑋). For 𝑝∈[1,∞], we set ℓ^𝑝₀(ℕ,𝑋)∶={𝑠∈ℓ^𝑝(ℕ,𝑋)∶𝑠(0)=0}. Similarly, 𝑐₀₀(ℕ,𝑋)∶={𝑠∈𝑐₀(ℕ,𝑋)∶𝑠(0)=0}.

Definition 2.1. A mapping 𝜎∶Θ×ℤ→Θ is called discrete flow on Θ if 𝜎(𝜃,0)=𝜃 and 𝜎(𝜃,𝑚+𝑛)=𝜎(𝜎(𝜃,𝑚),𝑛), for all (𝜃,𝑚,𝑛)∈Θ×ℤ².Let {𝐴(𝜃)}_𝜃∈Θ⊂ℒ(𝑋). We consider the linear system of variational difference equations (𝐴). The discrete cocycle associated with the system (𝐴) is Φ∶Θ×ℕ→ℒ(𝑋),Φ(𝜃,𝑛)=𝐴(𝜎(𝜃,𝑛−1))⋯𝐴(𝜃),𝑛∈ℕ^∗,𝐼,𝑛=0, (2.2) where 𝐼 is the identity operator on 𝑋. It is easy to see that Φ(𝜃,𝑚+𝑛)=Φ(𝜎(𝜃,𝑛),𝑚)Φ(𝜃,𝑛), for all (𝜃,𝑚,𝑛)∈Θ×ℕ².

Definition 2.2. The system (𝐴) is said to be uniformly exponentially dichotomic if there exist a family of projections {𝑃(𝜃)}_𝜃∈Θ⊂ℒ(𝑋) and two constants 𝐾≥1 and 𝜈>0 such that the following properties hold:(i)Φ(𝜃,𝑛)𝑃(𝜃)=𝑃(𝜎(𝜃,𝑛))Φ(𝜃,𝑛), for all (𝜃,𝑛)∈Θ×ℕ;(ii)‖Φ(𝜃,𝑛)𝑥‖≤𝐾𝑒^−𝜈𝑛‖𝑥‖, for all 𝑥∈Im𝑃(𝜃) and all (𝜃,𝑛)∈Θ×ℕ;(iii)‖Φ(𝜃,𝑛)𝑦‖≥(1/𝐾)𝑒^𝜈𝑛‖𝑦‖, for all 𝑦∈Ker𝑃(𝜃) and all (𝜃,𝑛)∈Θ×ℕ;(iv)the restriction Φ(𝜃,𝑛)_|∶Ker𝑃(𝜃)→Ker𝑃(𝜎(𝜃,𝑛)) is an isomorphism, for all (𝜃,𝑛)∈Θ×ℕ.

For every 𝜃∈Θ we denote by ℱ(𝜃) the linear space of all sequences 𝜑∶ℤ₋→𝑋 with the property 𝜑(𝑚)=𝐴(𝜎(𝜃,𝑚−1))𝜑(𝑚−1), for all 𝑚∈ℤ₋.

Let 𝑝∈[1,∞]. For every 𝜃∈Θ we consider the stable space 𝒮_𝑝(𝜃)={𝑥∈𝑋∶Φ(𝜃,⋅)𝑥∈ℓ^𝑝(ℕ,𝑋)}, (2.3) and the unstable space 𝒰_𝑝(𝜃)={𝑥∈𝑋∶thereexists𝜑∈ℱ(𝜃)∩ℓ^𝑝(ℤ₋,𝑋)with𝜑(0)=𝑥}. (2.4) We note that 𝒮_𝑝(𝜃) and 𝒰_𝑝(𝜃) are linear subspaces, for every 𝜃∈Θ.Remark 2.3. If the system (𝐴) is uniformly exponentially dichotomic and sup_𝜃∈Θ‖𝐴(𝜃)‖<∞, (2.5) then the family of projections given by Definition 2.2 is uniquely determined and sup_𝜃∈Θ‖𝑃(𝜃)‖<∞. (2.6) Moreover, for every 𝑝∈[1,∞], we have that Im𝑃(𝜃)=𝒮_𝑝(𝜃) and Ker𝑃(𝜃)=𝒰_𝑝(𝜃), for all 𝜃∈Θ (see [7, Proposition 2.1]).We associate to the system (𝐴) the input-output control system (𝑆_𝐴).

Definition 2.4. Let 𝑝,𝑞∈[1,∞]. The pair (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) is said to be uniformly 𝑞-admissible for the system (𝑆_𝐴) if the following assertions hold: (i)for every 𝑠∈Δ(ℤ,𝑋) there is a unique 𝛾_𝑠∶Θ→ℓ^𝑝(ℤ,𝑋) solution of the system (𝑆_𝐴) corresponding to the input sequence 𝑠;(ii)there is 𝜆>0 such that ‖𝛾_𝑠(𝜃)‖_𝑝≤𝜆‖𝑠‖_𝑞, for all (𝜃,𝑠)∈Θ×Δ(ℤ,𝑋).

For the proof of the next result we refer to [7, Theorem 3.6].

Theorem 2.5. Let 𝑝,𝑞∈[1,∞] be such that (𝑝,𝑞)≠(∞,1). The following assertions hold:(i)if the pair (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴), then the system (𝐴) is uniformly exponentially dichotomic;(ii) if 𝑝≥𝑞 and sup_𝜃∈Θ‖𝐴(𝜃)‖<∞, then the system (𝐴) is uniformly exponentially dichotomic if and only if the pair (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴).

We consider the input-output system 𝛼(𝜃)(𝑛+1)=𝐴(𝜎(𝜃,𝑛))𝛼(𝜃)(𝑛)+𝑢(𝑛+1),(𝜃,𝑛)∈Θ×ℕ.( (𝑄_𝐴))

The main question arises whether in the characterization of the exponential dichotomy the unique solvability of the associated discrete equation can be dropped. Another question is which are the connections between the solvability of the system (𝑄_𝐴) and the solvability of the system (𝑆_𝐴). In what follows we will give complete answers for both questions, our study focusing on these central purposes.

Definition 2.6. Let 𝑝,𝑞∈[1,∞]. The pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) is said to be uniformly 𝑞-admissible for the system (𝑄_𝐴) if there is 𝜆>0 such that the following assertions hold:(i) for every 𝑢∈Δ₀(ℕ,𝑋) there is 𝛼∶Θ→ℓ^𝑝(ℕ,𝑋) solution of the system (𝑄_𝐴) corresponding to 𝑢;(ii)if 𝑢∈Δ₀(ℕ,𝑋) and 𝛼∶Θ→ℓ^𝑝(ℕ,𝑋) is a solution of (𝑄_𝐴) corresponding to 𝑢 with the property that 𝛼(𝜃)(0)∈𝒰_𝑝(𝜃), for every 𝜃∈Θ, then ‖𝛼(𝜃)‖_𝑝≤𝜆‖𝑢‖_𝑞,∀𝜃∈Θ. (2.7)

Lemma 2.7. Let 𝑝,𝑞∈[1,∞]. If the pair (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴) then 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋, for all 𝜃∈Θ.

Proof. Let 𝑥∈𝑋. We consider the sequence 𝑠∶ℤ→𝑋,𝑠(𝑘)=𝜒_{0}(𝑘)𝑥. From hypothesis, there is 𝛾_𝑠 solution of the system (𝑆_𝐴) corresponding to 𝑠.
Let 𝜃∈Θ. From 𝛾_𝑠(𝜃)(𝑛)=Φ(𝜃,𝑛)𝛾_𝑠(𝜃)(0) for all 𝑛∈ℕ and since 𝛾_𝑠(𝜃)∈ℓ^𝑝(ℤ,𝑋) we deduce that 𝛾_𝑠(𝜃)(0)∈𝒮_𝑝(𝜃). In addition, considering the sequence 𝜑_𝜃∶ℤ₋→𝑋,𝜑_𝜃(𝑘)=𝜒_{0}(𝑘)𝑥−𝛾_𝑠(𝜃)(𝑘) we have that 𝜑_𝜃∈ℱ(𝜃) and 𝜑_𝜃∈ℓ^𝑝(ℤ₋,𝑋). This shows that 𝑥−𝛾_𝑠(𝜃)(0)=𝜑_𝜃(0)∈𝒰_𝑝(𝜃). It follows that 𝑥=𝛾_𝑠(𝜃)(0)+(𝑥−𝛾_𝑠(𝜃)(0))∈𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃). Since 𝜃∈Θ was arbitrary, the proof is complete.

The first main result of this section is as follows.

Theorem 2.8. Let 𝑝,𝑞∈[1,∞]. The following assertions are equivalent:(i)the pair (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴);(ii)the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴) and 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋, for all 𝜃∈Θ.

Proof. (i)⇒(ii) Let 𝜆>0 be given by Definition 2.4. Let 𝑢∈Δ₀(ℕ,𝑋). Then the sequence 𝑠_𝑢∶ℤ→𝑋,𝑠_𝑢(𝑘)=𝜒_ℕ(𝑘)𝑢(𝑘) belongs to Δ(ℤ,𝑋). From the uniform 𝑞-admissibility of the pair (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) it follows that there is a unique 𝛾_𝑢∶Θ→ℓ^𝑝(ℤ,𝑋) solution of (𝑆_𝐴) corresponding to 𝑠_𝑢. Moreover ‖𝛾_𝑢(𝜃)‖_𝑝≤𝜆‖𝑠_𝑢‖_𝑞,∀𝜃∈Θ. (2.8) Taking 𝛼_𝑢∶Θ→ℓ^𝑝(ℕ,𝑋),𝛼_𝑢(𝜃)(𝑛)=𝛾_𝑢(𝜃)(𝑛), (2.9) we have that 𝛼_𝑢 is a solution of (𝑄_𝐴) corresponding to 𝑢.
Let 𝛼∶Θ→ℓ^𝑝(ℕ,𝑋) be a solution of (𝑄_𝐴) corresponding to 𝑢 with the property that 𝛼(𝜃)(0)∈𝒰_𝑝(𝜃), for all 𝜃∈Θ. Then for every 𝜃∈Θ, there is 𝜑_𝜃∈ℓ^𝑝(ℤ₋,𝑋) with 𝜑_𝜃(0)=𝛼(𝜃)(0) and 𝜑_𝜃(𝑚)=𝐴(𝜎(𝜃,𝑚−1))𝜑_𝜃(𝑚−1),∀𝑚∈ℤ₋. (2.10) Considering the sequence 𝛾_𝜃∶ℤ→𝑋,𝛾_𝜃𝜑(𝑚)=𝛼(𝜃)(𝑚),𝑚∈ℕ,_𝜃(𝑚),𝑚∈ℤ⧵N, (2.11) we obtain that 𝛾_𝜃∈ℓ^𝑝(ℤ,𝑋) and 𝛾_𝜃(𝑚+1)=𝐴(𝜎(𝜃,𝑚))𝛾_𝜃(𝑚)+𝑠_𝑢(𝑚+1),∀𝑚∈ℤ. (2.12) This implies that 𝛾∶Θ→ℓ^𝑝(ℤ,𝑋),𝛾(𝜃)=𝛾_𝜃 (2.13) is solution of the system (𝑆_𝐴) corresponding to 𝑠_𝑢. From the uniqueness, we deduce that 𝛾=𝛾_𝑢. Then, using (2.8) we have that ‖𝛼(𝜃)‖_𝑝≤‖𝛾_𝜃‖_𝑝=‖𝛾_𝑢(𝜃)‖_𝑝≤𝜆‖𝑠_𝑢‖_𝑞=𝜆‖𝑢‖_𝑞,∀𝜃∈Θ. (2.14) Taking into account that 𝜆 does not depend on 𝑢 or 𝜃 we obtain that the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴).
In addition, from Lemma 2.7 we have that 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋, for all 𝜃∈Θ.
(ii)⇒(i) Let 𝜆>0 be given by Definition 2.6.
Let 𝑠∈Δ(ℤ,𝑋). Then, there is ℎ∈ℤ₋ such that 𝑠(𝑗)=0, for all 𝑗≤ℎ.
Consider the sequence 𝑢∶ℕ→𝑋,𝑢(𝑛)=𝑠(𝑛+ℎ). Then 𝑢∈Δ₀(ℕ,𝑋). From the uniform 𝑞-admissibility of the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) there is 𝛼∶Θ→ℓ^𝑝(ℕ,𝑋) solution of the system (𝑄_𝐴) corresponding to 𝑢.
Let 𝜃∈Θ. Since 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋 there is 𝑥^𝜃_𝑠∈𝒮_𝑝(𝜃) and 𝑥^𝜃_𝑢∈𝒰_𝑝(𝜃) such that 𝛼(𝜃)(0)=𝑥^𝜃_𝑠+𝑥^𝜃_𝑢. Let
𝛿_𝜃∶ℕ→𝑋,𝛿_𝜃(𝑛)=𝛼(𝜃)(𝑛)−Φ(𝜃,𝑛)𝑥^𝜃_𝑠. (2.15) Then 𝛿_𝜃∈ℓ^𝑝(ℕ,𝑋),𝛿_𝜃(0)=𝑥^𝜃_𝑢 and 𝛿_𝜃(𝑛+1)=𝐴(𝜎(𝜃,𝑛))𝛿_𝜃(𝑛)+𝑢(𝑛+1),∀𝑛∈ℕ. (2.16) From 𝑥^𝜃_𝑢∈𝒰_𝑝(𝜃) it follows that there is 𝜑_𝜃∈ℓ^𝑝(ℤ₋,𝑋) with 𝜑_𝜃(0)=𝑥^𝜃_𝑢 and 𝜑_𝜃(𝑚)=𝐴(𝜎(𝜃,𝑚−1))𝜑_𝜃(𝑚−1),∀𝑚∈ℤ₋. (2.17) Let 𝑣_𝜃∶ℤ→𝑋,𝑣_𝜃𝛿(𝑘)=^𝜃𝜑(𝑘),𝑘∈ℕ,_𝜃(𝑘),𝑘∈ℤ⧵ℕ. (2.18) Then 𝑣_𝜃∈ℓ^𝑝(ℤ,𝑋) and from (2.16) and (2.17) we obtain that 𝑣_𝜃(𝑚+1)=𝐴(𝜎(𝜃,𝑚))𝑣_𝜃(𝑚)+𝑠(𝑚+ℎ+1),∀𝑚∈ℤ. (2.19) We define 𝛾∶Θ→ℓ^𝑝(ℤ,𝑋),𝛾(𝜃)(𝑚)=𝑣_{𝜎(𝜃,ℎ)}(𝑚−ℎ). (2.20) Then, using (2.19) we have that 𝑣_{𝜎(𝜃,ℎ)}(𝑚+1)=𝐴(𝜎(𝜃,𝑚+ℎ))𝑣_{𝜎(𝜃,ℎ)}(𝑚)+𝑠(𝑚+ℎ+1),∀𝑚∈ℤ,∀𝜃∈Θ, (2.21) which implies that 𝛾(𝜃)(𝑘+1)=𝐴(𝜎(𝜃,𝑘))𝛾(𝜃)(𝑘)+𝑠(𝑘+1),∀𝑘∈ℤ,∀𝜃∈Θ. (2.22) This shows that 𝛾 is a solution of the system (𝑆_𝐴) corresponding to 𝑠.
Let 𝛾∶Θ→ℓ^𝑝(ℤ,𝑋) be a solution of (𝑆_𝐴) corresponding to 𝑠 and let 𝛽=𝛾−𝛾. Then 𝛽(𝜃)(𝑚+1)=𝐴(𝜎(𝜃,𝑚))𝛽(𝜃)(𝑚),∀(𝑚,𝜃)∈ℤ×Θ. (2.23) Let 𝑚∈ℤ. Considering 𝛼∶Θ→ℓ^𝑝(ℕ,𝑋),𝛼(𝜃)(𝑛)=𝛽(𝜎(𝜃,−𝑚))(𝑛+𝑚), (2.24) we have that 𝛼(𝜃)(𝑛+1)=𝐴(𝜎(𝜃,𝑛))𝛼(𝜃)(𝑛), for all (𝑛,𝜃)∈ℕ×Θ, so 𝛼 is a solution of the system (𝑄_𝐴) corresponding to 𝑤=0. From (2.23) it follows that 𝛼(𝜃)(0)∈𝒰_𝑝(𝜃), for all 𝜃∈Θ. Then, from the uniform 𝑞-admissibility of the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) we obtain that 𝛼=0. In particular 𝛽(𝜃)(𝑚)=𝛼(𝜎(𝜃,𝑚))(0)=0, for all 𝜃∈Θ. Since 𝑚∈ℤ was arbitrary it follows that 𝛽(𝜃)=0, for all 𝜃∈Θ, so 𝛾 was uniquely determined.
It remains to verify that condition (ii) in Definition 2.4 is fulfilled. Let 𝑠∈Δ(ℤ,𝑋) and let 𝛾∈ℓ^𝑝(ℤ,𝑋) be the solution of (𝑆_𝐴) corresponding to the input 𝑠. Let ℎ∈ℤ₋ be such that 𝑠(𝑗)=0, for all 𝑗≤ℎ. We consider the sequence 𝑢_ℎ∶ℕ→𝑋,𝑢_ℎ(𝑛)=𝑠(𝑛+ℎ) and let
𝛼_ℎ∶Θ→ℓ^𝑝(ℕ,𝑋),𝛼_ℎ(𝜃)(𝑛)=𝛾(𝜎(𝜃,−ℎ))(𝑛+ℎ). (2.25) Then we have that 𝛼_ℎ is the solution of the system (𝑄_𝐴) corresponding to 𝑢_ℎ. From 𝛾(𝜃)(𝑚)=𝐴(𝜎(𝜃,𝑚−1))𝛾(𝜃)(𝑚−1),∀𝑚≤ℎ,∀𝜃∈Θ, (2.26) we deduce that 𝛾(𝜎(𝜃,−ℎ))(𝑘+ℎ)=𝐴(𝜎(𝜃,𝑘−1))𝛾(𝜎(𝜃,−ℎ))(𝑘+ℎ−1),∀𝑘∈ℤ₋,∀𝜃∈Θ. (2.27)
Let 𝜃∈Θ. Denoting by 𝜑_𝜃∶ℤ₋→𝑋,𝜑_𝜃(𝑘)=𝛾(𝜎(𝜃,−ℎ))(𝑘+ℎ) we deduce that 𝜑_𝜃(𝑘)=𝐴(𝜎(𝜃,𝑘−1))𝜑_𝜃(𝑘−1),∀𝑘∈ℤ₋. (2.28) Moreover, since 𝛾(𝜃)∈ℓ^𝑝(ℤ,𝑋), for all 𝜃∈Θ, it follows that 𝜑_𝜃∈ℓ^𝑝(ℤ₋,𝑋).
This implies that 𝛼_ℎ(𝜃)(0)=𝜑_𝜃(0)∈𝒰_𝑝(𝜃), for all 𝜃∈Θ. From the uniform 𝑞-admissibility of the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) we obtain that ‖𝛼_ℎ(𝜃)‖_𝑝≤𝜆‖𝑢_ℎ‖_𝑞,∀𝜃∈Θ. (2.29)
Since ||𝑢_ℎ||_𝑞=||𝑠||_𝑞, from (2.29) we deduce that (^∞_𝑗=0‖𝛼_ℎ(𝜃)(𝑗)‖^𝑝)^1/𝑝≤𝜆‖𝑠‖_𝑞,∀𝜃∈Θ, (2.30) which is equivalent with (^∞_𝑗=0‖𝛾(𝜎(𝜃,−ℎ))(𝑗+ℎ)‖^𝑝)^1/𝑝≤𝜆‖𝑠‖_𝑞,∀𝜃∈Θ, (2.31) and, respectively, with (^∞_𝑘=ℎ‖𝛾(𝜃)(𝑘)‖^𝑝)^1/𝑝≤𝜆‖𝑠‖_𝑞,∀𝜃∈Θ. (2.32)
Since relation (2.32) holds for all ℎ∈ℤ₋ with the property that 𝑠(𝑗)=0, for all 𝑗≤ℎ, for ℎ→−∞ in (2.32) we obtain that ||𝛾(𝜃)||_𝑝≤𝜆||𝑠||_𝑞, for all 𝜃∈Θ. Taking into account that 𝜆 does not depend on 𝜃 or 𝑠, we conclude that the pair (ℓ^𝑝(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴) and the proof is complete.

The second main result of this section is as follows.

Theorem 2.9. Let 𝑝,𝑞∈[1,∞] be such that (𝑝,𝑞)≠(∞,1). The following assertions hold: (i)if the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴) and 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋, for all 𝜃∈Θ, then the system (𝐴) is uniformly exponentially dichotomic;(ii)if 𝑝≥𝑞 and sup_𝜃∈Θ‖𝐴(𝜃)‖<∞, then the system (𝐴) is uniformly exponentially dichotomic if and only if the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴) and 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋, for all 𝜃∈Θ.

Proof. This follows from Theorems 2.5 and 2.8.

Remark 2.10. Naturally, the question arises whether, generally, the uniform 1-admissibility of the pair (ℓ^∞(ℕ,𝑋),Δ₀(ℕ,𝑋)) for the system (𝑄_𝐴) and the property that 𝒮_∞(𝜃)+𝒰_∞(𝜃)=𝑋, for all 𝜃∈Θ, implies the uniform exponential dichotomy of the system (𝐴). The answer is negative, as the following example shows.

Example 2.11. Let 𝑊 be a Banach space and let 𝑋=𝑊×𝑊. On 𝑋 we consider the norm ||(𝑤₁,𝑤₂)||=||𝑤₁||_𝑊+‖𝑤₂‖_𝑊. Let 1𝑓∶ℤ→(0,∞),𝑓(𝑘)=𝑘+1,𝑘∈ℕ,1−𝑘,𝑘∈ℤ⧵ℕ. (2.33) Let Θ=ℤ and let 𝜎∶Θ×ℤ→Θ,𝜎(𝜃,𝑚)=𝜃+𝑚. For every 𝜃∈Θ we consider the operator 𝐴(𝜃)∶𝑋→𝑋,𝐴(𝜃)(𝑤₁,𝑤₂)=(𝑓(𝜃+1)𝑤𝑓(𝜃)¹,𝑓(𝜃)𝑤𝑓(𝜃+1)²). (2.34) The discrete cocycle associated with the system 𝑥(𝜃)(𝑛+1)=𝐴(𝜎(𝜃,𝑛))𝑥(𝜃)(𝑛),∀(𝜃,𝑛)∈Θ×ℕ,( (𝐴′)) is given by Φ(𝜃,𝑛)∶𝑋→𝑋,Φ(𝜃,𝑛)(𝑤₁,𝑤₂)=(𝑓(𝜃+𝑛)𝑤𝑓(𝜃)¹,𝑓(𝜃)𝑤𝑓(𝜃+𝑛)²) (2.35) for all (𝜃,𝑛)∈Θ×ℕ.

Step 1. We prove that 𝒮_∞(𝜃)+𝒰_∞(𝜃)=𝑋, for all 𝜃∈Θ.
Let 𝜃∈Θ. Using (2.35) it is easy to see that 𝒮_∞(𝜃)=𝑊×{0}. We prove that 𝒰_∞(𝜃)={0}×𝑊. Indeed, if 𝑥=(0,𝑤)∈{0}×𝑊, by defining 𝜑∶ℤ₋→𝑋,𝜑(𝑚)=(𝑓(𝜃)/𝑓(𝜃+𝑚))𝑥, we observe that 𝜑∈ℱ(𝜃) and 𝜑(0)=𝑥. This shows that 𝑥∈𝒰_∞(𝜃), so {0}×𝑊⊂𝒰_∞(𝜃).
Conversely, let 𝑥∈𝒰_∞(𝜃). Then there is 𝜑∈ℱ(𝜃)∩ℓ^∞(ℤ₋,𝑋) with 𝜑(0)=𝑥. If 𝜑=(𝜑₁,𝜑₂), in particular, we have that
𝜑₁(0)=𝑓(𝜃)𝜑𝑓(𝜃+𝑘)¹(𝑘),∀𝑘∈ℤ₋. (2.36) Since 𝜑∈ℓ^∞(ℤ₋,𝑋) we have that 𝜑₁∈ℓ^∞(ℤ₋,𝑊). Then, from (2.36) we obtain that ‖𝜑₁(0)‖≤𝑓(𝜃)𝑓(𝜃+𝑘)‖𝜑₁‖_∞,∀𝑘∈ℤ₋. (2.37) As 𝑘→−∞ in (2.37) it follows that 𝜑₁(0)=0. Then 𝑥=(0,𝜑₂(0))∈{0}×𝑊. This shows that 𝒰_∞(𝜃)={0}×𝑊. So, 𝒮_∞(𝜃)+𝒰_∞(𝜃)=𝑋, for all 𝜃∈Θ.

Step 2. We prove that the pair (ℓ^∞(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 1-admissible for the system (𝑄_𝐴). Let 𝑢∈Δ₀(ℕ,𝑋) and let 𝜃∈Θ. If 𝑢=(𝑢₁,𝑢₂), then we define 𝛼_𝜃∶ℕ→𝑋,𝛼_𝜃(𝑛)=(^𝑛_𝑘=0𝑓(𝜃+𝑛)𝑢𝑓(𝜃+𝑘)¹(𝑘),−^∞_𝑘=𝑛+1𝑓(𝜃+𝑘)𝑢𝑓(𝜃+𝑛)²(𝑘)). (2.38) Since 𝑢∈Δ₀(ℕ,𝑋) there is 𝑙∈ℕ^∗ such that 𝑢(𝑘)=0, for all 𝑘≥𝑙. Then 𝛼_𝜃(𝑛)=(𝑓(𝜃+𝑛)^𝑙_𝑘=0𝑢₁(𝑘)𝑓(𝜃+𝑘),0),∀𝑛≥𝑙. (2.39) From (2.39) it follows that 𝛼_𝜃∈𝑐₀(ℕ,𝑋), so, in particular, 𝛼_𝜃∈ℓ^∞(ℕ,𝑋). We define 𝛼∶Θ→ℓ^∞(ℕ,𝑋),𝛼(𝜃)=𝛼_𝜃 and an easy computation shows that 𝛼 is a solution of the system (𝑄_𝐴) corresponding to 𝑢.
Let 𝛼=(𝛼₁,𝛼₂)∶Θ→ℓ^∞(ℕ,𝑋) be a solution of (𝑄_𝐴) corresponding to 𝑢 with the property that 𝛼(𝜃)(0)∈𝒰_∞(𝜃), for all 𝜃∈Θ.
Let 𝜃∈Θ. From 𝛼(𝜃)(0)∈𝒰_∞(𝜃) and Step 1 it follows that 𝛼₁(𝜃)(0)=0. This implies that 𝛼₁(𝜃)(1)=𝑢₁(1), so
‖𝛼₁(𝜃)(2)‖≤²_𝑘=1‖𝑢₁(𝑘)‖. (2.40) Inductively, we obtain that ‖𝛼₁(𝜃)(𝑛)‖≤^𝑛_𝑘=1‖𝑢₁(𝑘)‖,∀𝑛∈ℕ (2.41) so ‖𝛼₁(𝜃)‖_∞≤‖𝑢₁‖₁. (2.42) Since 𝑢(𝑘)=0, for all 𝑘≥𝑙, we have that 𝛼₂(𝜃)(𝑘)=𝑓(𝜃+𝑙−1)𝛼𝑓(𝜃+𝑘)²(𝜃)(𝑙−1),∀𝑘≥𝑙−1. (2.43) Taking into account that 𝛼(𝜃)∈ℓ^∞(ℕ,𝑋), using relation (2.43) we deduce that ‖𝛼₂(𝜃)(𝑙−1)‖≤𝑓(𝜃+𝑘)‖𝛼𝑓(𝜃+𝑙−1)²(𝜃)‖_∞,∀𝑘≥𝑙−1. (2.44) For 𝑘→∞ in (2.44) we obtain that 𝛼₂(𝜃)(𝑙−1)=0. Then, from (2.43) it follows that 𝛼₂(𝜃)(𝑘)=0, for all 𝑘≥𝑙−1. Moreover, if 𝑙≥2, from 0=𝛼₂(𝜃)(𝑙−1)=𝑓(𝜃+𝑙−2)𝛼𝑓(𝜃+𝑙−1)²(𝜃)(𝑙−2)+𝑢₂(𝑙−1), (2.45) we deduce that ‖𝛼₂(𝜃)(𝑙−2)‖≤‖𝑢₂(𝑙−1)‖. Inductively, it follows that ‖𝛼₂(𝜃)(𝑗)‖≤^𝑙−1_𝑘=𝑗+1‖𝑢₂(𝑘)‖,∀𝑗∈{0,…,𝑙−2}. (2.46) In particular, this implies that ‖𝛼₂(𝜃)‖_∞=sup_{𝑗∈{0,…,𝑙−2}}‖𝛼₂(𝜃)(𝑗)‖≤‖𝑢₂‖₁. (2.47) From relations (2.42) and (2.47) we obtain that ‖𝛼(𝜃)‖_∞=‖𝛼₁(𝜃)‖_∞+‖𝛼₂(𝜃)‖_∞≤‖𝑢₁‖₁+‖𝑢₂‖₁=‖𝑢‖₁, for all 𝜃∈Θ. This shows that the pair (ℓ^∞(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 1-admissible for the system (𝑄_𝐴).

Step 3. We prove that the system (𝐴) is not uniformly exponentially dichotomic.
Supposing that the system (𝐴) is uniformly exponentially dichotomic, there exists a family of projections {𝑃(𝜃)}_𝜃∈Θ and two constants 𝐾,𝜈>0 given by Definition 2.2. Then ‖Φ(𝜃,𝑛)𝑥‖≤𝐾𝑒^−𝜈𝑛‖𝑥‖,∀𝑥∈Im𝑃(𝜃),∀(𝜃,𝑛)∈Θ×ℕ. (2.48) According to Remark 2.3 and Step 1 we have that Im𝑃(𝜃)=𝒮_∞(𝜃)=𝑊×{0}, for all 𝜃∈Θ. Then (2.48) yields 𝑓(𝜃+𝑛)/𝑓(𝜃)≤Ke^−𝜈𝑛, for all (𝜃,𝑛)∈Θ×ℕ. In particular, for 𝜃=0, from the above inequality we obtain that 1/(𝑛+1)≤Ke^−𝜈𝑛, for all 𝑛∈ℕ, which is absurd. This shows that the system (𝐴) is not uniformly exponentially dichotomic.

Definition 2.12. Let 𝑞∈[1,∞). The pair (𝑐₀(ℤ,𝑋),Δ(ℤ,𝑋)) is said to be uniformly 𝑞-𝑎𝑑𝑚𝑖𝑠𝑠𝑖𝑏𝑙𝑒 for the system (𝑆_𝐴) if the following assertions hold:(i)for every 𝑠∈Δ(ℤ,𝑋) there is a unique 𝛾_𝑠∶Θ→𝑐₀(ℤ,𝑋) solution of the system (𝑆_𝐴) corresponding to 𝑠;(ii)there is 𝜆>0 such that ‖𝛾_𝑠(𝜃)‖_∞≤𝜆‖𝑠‖_𝑞, for all (𝜃,𝑠)∈Θ×Δ(ℤ,𝑋).

For the proof of next theorem we refer to [7, Theorem 3.7].

Theorem 2.13. Let 𝑞∈(1,∞). The following assertions hold:(i)if the pair (𝑐₀(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴), then the system (𝐴) is uniformly exponentially dichotomic;(ii)if sup_𝜃∈Θ‖𝐴(𝜃)‖<∞, then the system (𝐴) is uniformly exponentially dichotomic if and only if the pair (𝑐₀(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴).

For every 𝜃∈Θ we consider the subspaces 𝒮₀(𝜃)={𝑥∈𝑋∶Φ(𝜃,⋅)𝑥∈𝑐₀𝒰(ℕ,𝑋)},₀(𝜃)={𝑥∈𝑋∶thereis𝜑∈ℱ(𝜃)∩𝑐₀(ℤ₋,𝑋)with𝜑(0)=𝑥}. (2.49)

Definition 2.14. Let 𝑞∈[1,∞). The pair (𝑐₀(ℕ,𝑋),Δ₀(ℕ,𝑋)) is said to be uniformly 𝑞-𝑎𝑑𝑚𝑖𝑠𝑠𝑖𝑏𝑙𝑒 for the system (𝑄_𝐴) if there is 𝜆>0 such that the following assertions hold:(i)for every 𝑢∈Δ₀(ℕ,𝑋) there is 𝛼∶Θ→𝑐₀(ℕ,𝑋) solution of the system (𝑄_𝐴) corresponding to 𝑢;(ii)if 𝑢∈Δ₀(ℕ,𝑋) and 𝛼∶Θ→𝑐₀(ℕ,𝑋) is a solution of (𝑄_𝐴) corresponding to 𝑢 with the property that 𝛼(𝜃)(0)∈𝒰₀(𝜃) for every 𝜃∈Θ, then ‖𝛼(𝜃)‖_∞≤𝜆‖𝑢‖_𝑞 for all 𝜃∈Θ.

Working with the subspaces 𝒮₀(𝜃),𝒰₀(𝜃) and using similar arguments with those in Lemma 2.7 and Theorem 2.8 we obtain the following.

Theorem 2.15. Let 𝑞∈[1,∞). The following assertions are equivalent: (i)the pair (𝑐₀(ℤ,𝑋),Δ(ℤ,𝑋)) is uniformly 𝑞-admissible for the system (𝑆_𝐴);(ii) the pair (𝑐₀(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴) and 𝒮₀(𝜃)+𝒰₀(𝜃)=𝑋, for all 𝜃∈Θ.

As a consequence of Theorems 2.13 and 2.15 we deduce the following result.

Theorem 2.16. Let 𝑞∈(1,∞). The following assertions hold:(i)if the pair (𝑐₀(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴) and 𝒮₀(𝜃)+𝒰₀(𝜃)=𝑋, for all 𝜃∈Θ, then the system (𝐴) is uniformly exponentially dichotomic;(ii)if sup_𝜃∈Θ‖𝐴(𝜃)‖<∞, then the system (𝐴) is uniformly exponentially dichotomic if and only if the pair (𝑐₀(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴) and 𝒮₀(𝜃)+𝒰₀(𝜃)=𝑋, for all 𝜃∈Θ.

Remark 2.17. It is easy to see that the system presented in Example 2.11 has the property that 𝒮₀(𝜃)+𝒰₀(𝜃)=𝑋, for all 𝜃∈Θ. Using similar arguments with those from Example 2.11, we deduce that the pair (𝑐₀(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 1-admissible for the system (𝑄_𝐴). But, for all that, the system (𝐴) is not uniformly exponentially dichotomic.

Remark 2.18. As an immediate consequence of Theorem 2.16 we obtain the main result of the paper [6]. The techniques involved in the proofs from [6] are different from those presented above.

3. Applications for Uniform Exponential Dichotomy of Linear Skew-Product Flows

In this section we apply the results obtained in the previous section in order to deduce characterizations for uniform exponential dichotomy of linear skew-product flows.

Let 𝑋 be a real or complex Banach space, let (Θ,𝑑) be a metric space and let ℰ=𝑋×Θ. A mapping 𝜎∶Θ×ℝ→Θ is called a flow on Θ if 𝜎(𝜃,0)=𝜃 and 𝜎(𝜃,𝑡+𝑠)=𝜎(𝜎(𝜃,𝑡),𝑠), for all (𝜃,𝑡,𝑠)∈Θ×ℝ².

Definition 3.1. A pair 𝜋=(Φ,𝜎) is called a linear skew-product flow if 𝜎 is a flow on Θ and the mapping Φ∶Θ×ℝ₊→ℒ(𝑋) satisfies the following conditions:(i)Φ(𝜃,0)=𝐼, (the identity operator), for all 𝜃∈Θ;(ii)Φ(𝜃,𝑡+𝑠)=Φ(𝜎(𝜃,𝑡),𝑠)Φ(𝜃,𝑡), for all (𝜃,𝑡,𝑠)∈Θ×ℝ²₊;(iii)there are 𝑀≥1 and 𝜔>0 such that ||Φ(𝜃,𝑡)||≤𝑀𝑒^𝜔𝑡, for all (𝜃,𝑡)∈Θ×ℝ₊.

Definition 3.2. A linear skew-product flow 𝜋=(Φ,𝜎) is said to be uniformly exponentially dichotomic if there are a family of projections {𝑃(𝜃)}_𝜃∈Θ and two constants 𝐾,𝜈>0 such that the following properties hold:(i)Φ(𝜃,𝑡)𝑃(𝜃)=𝑃(𝜎(𝜃,𝑡))Φ(𝜃,𝑡), for all (𝜃,𝑡)∈Θ×ℝ₊;(ii)||Φ(𝜃,𝑡)𝑥||≤Ke^−𝜈𝑡||𝑥||, for all (𝜃,𝑡)∈Θ×ℝ₊ and 𝑥∈Im𝑃(𝜃);(iii)||Φ(𝜃,𝑡)𝑥||≥(1/𝐾)𝑒^𝜈𝑡||𝑥||, for all (𝜃,𝑡)∈Θ×ℝ₊ and 𝑥∈Ker𝑃(𝜃);(iv)for every (𝜃,𝑡)∈Θ×ℝ₊, the operator Φ(𝜃,𝑡)_|∶Ker𝑃(𝜃)→Ker𝑃(𝜎(𝜃,𝑡)) is invertible.

Remark 3.3. Let 𝜋=(Φ,𝜎) be a linear skew-product flow on ℰ=𝑋×Θ. We associate with 𝜋 the variational discrete-time system (𝐴_𝜋). The discrete cocycle associated with the system (𝐴_𝜋) is {Φ(𝜃,𝑛)}_{(𝜃,𝑛)∈Θ×ℕ}.If 𝜋=(Φ,𝜎) is a linear skew-product flow on ℰ=𝑋×Θ, we consider the system 𝛼(𝜃)(𝑛+1)=Φ(𝜎(𝜃,𝑛),1)𝛼(𝜃)(𝑛)+𝑢(𝑛+1),∀(𝜃,𝑛)∈Θ×ℕ.( (𝑄_𝐴𝜋))

As consequences of the main results from the previous section we deduce the following.

Theorem 3.4. Let 𝜋=(Φ,𝜎) be a linear skew-product flow on ℰ=𝑋×Θ and let 𝑝,𝑞∈[1,∞] be such that (𝑝,𝑞)≠(∞,1). The following assertions hold:(i) if the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴𝜋) and 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋, for all 𝜃∈Θ, then 𝜋 is uniformly exponentially dichotomic;(ii)if 𝑝≥𝑞, then 𝜋 is uniformly exponentially dichotomic if and only if the pair (ℓ^𝑝(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴𝜋) and 𝒮_𝑝(𝜃)+𝒰_𝑝(𝜃)=𝑋, for all 𝜃∈Θ.

Proof. This follows from Theorems 2.9 and 1.1.

Theorem 3.5. Let 𝜋=(Φ,𝜎) be a linear skew-product flow on ℰ=𝑋×Θ and let 𝑞∈(1,∞). Then 𝜋 is uniformly exponentially dichotomic if and only if the pair (𝑐₀(ℕ,𝑋),Δ₀(ℕ,𝑋)) is uniformly 𝑞-admissible for the system (𝑄_𝐴𝜋) and 𝒮₀(𝜃)+𝒰₀(𝜃)=𝑋, for all 𝜃∈Θ.

Proof. This follows from Theorems 2.16 and 1.1.

4. Conclusions

In order to study the existence of exponential dichotomy of a variational difference equation (𝐴), we associate to (𝐴) the linear control system: 𝛾(𝜃)(𝑚+1)=𝐴(𝜎(𝜃,𝑚))𝛾(𝜃)(𝑚)+𝑠(𝑚+1),(𝜃,𝑚)∈Θ×𝐽 (4.1) which is denoted by (𝑆_𝐴) if 𝐽=ℤ and by (𝑄_𝐴) if 𝐽=ℕ. The uniform exponential dichotomy of (𝐴) may be expressed either in terms of the unique solvability of the system (𝑆_𝐴) or using the solvability of the system (𝑄_𝐴) in certain hypotheses, provided (in both cases) that the norm of the solution of the control system satisfies a boundedness condition with respect to the norm of the input sequence. In the first case the smaller input space is Δ(ℤ,𝑋), while in the second case the smaller input space may be considered Δ₀(ℕ,𝑋). When the associated control system is on the half-line, then the uniqueness of solution may be dropped. In this case, the space 𝑋 should be the sum between the stable and the unstable space at every point 𝜃∈Θ, but the boundedness condition may hold only for output sequences starting in the unstable space. Our study is explicitly done for the case when the output space is an ℓ^𝑝(𝐽,𝑋)-space as well as when this a 𝑐₀(𝐽,𝑋)-space, emphasizing the particular properties of each case. In base of Theorem 1.1 the main results are applicable not only to the general case of variational difference equations but also to the class of linear skew-product flows in infinite-dimensional spaces, without requiring measurability or continuity properties.

Acknowledgments

The author would like to thank the referee for thoroughly reading of the paper and for helpful suggestions and comments. The work is supported by the PN II Exploratory Research Grant ID 1081.