On (ℎ,𝑘)-Dichotomies for Nonautonomous Linear Difference Equations in Banach Spaces

Abstract

This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial). In the particular case of strongly invariant dichotomy projections, we present characterizations of these sequences and connections with other dichotomy concepts existent in the literature. Some illustrative examples clarify the implications between these concepts.

1. Introduction

In the last few years, the theory of difference equations has witnessed an impressive development. The notion of (uniform) exponential dichotomy 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 theory of differential equations and dynamical systems.

The notion of dichotomy for differential equations has gained prominence since the appearance of two fundamental monographs of Massera and Schäffer [3] and Daleckii and Krein [4]. These were followed by the important book of Coppel [5] who synthesized and improved the results that existed in the literature up to 1978. Early works in the counterpart results of difference equations appeared in the paper of Coffman and Schäffer [6] and later, in 1981, when Henry included discrete dichotomies in his book [7]. This was followed by the classical monographs due to Agarwal [8] where ordinary and exponential dichotomy properties of difference equations are studied and various applications are provided. Significant work was reported by Pötzsche in [9].

One important and useful concept of dichotomy in the study of difference equations is the so-called (uniform) (ℎ,𝑘)-dichotomy concept introduced by Pinto in [10]. Since then, this concept has been extensively studied and applied; see, for example, Naulin and Pinto [11], Megan [12], Fenner [13], and Lin [14] where several examples are presented.

A new notion called nonuniform (𝜇,𝜈)-dichotomy is proposed by Bento and Silva in [15] for the continuous case and in [16, 17] for discrete time settings, for invertible systems, with growth rates given by increasing functions (sequences) which go to infinity. In the last years, this subject (resp., nonuniform exponential dichotomy and nonuniform polynomial dichotomy) became one of the subjects of large interest, significant results being obtained (see, e.g., [18–24]).

On the other hand, we can consider the case of noninvertible systems. From this point of view the paper of Aulbach and Kalkbrenner [25] is of interest, where the notion of exponential forward splitting is introduced, motivated by the fact that there are equations whose backward solutions are not guaranteed to exist. Indeed, this study is of interest in applications; see, for example, random dynamical systems, generated by random parabolic equations, which are not invertible (for more details, see Zhou et al. [26]).

Motivated by the above studies, in this paper, we consider two concepts of nonuniform (ℎ,𝑘)-dichotomies for difference equations in Banach spaces. These concepts use two concepts of sequences of projections: invariant and strongly invariant for the respective difference systems. Thus, we offer characterizations for these concepts and present connections between them. It is worth to mention the fact that if a sequence of projections is invariant for a reversible system then it is also strongly invariant for that system, and in this way the concepts merge. In this way, our study is related to the case of noninvertible systems.

2. Preliminaries

Let 𝑋 be a Banach space and ℬ(𝑋) the Banach space of all bounded linear operators on 𝑋. The norms on 𝑋 and on ℬ(𝑋) will be denoted by ‖⋅‖. The identity operator on 𝑋 is denoted by 𝐼. If 𝑇∈ℬ(𝑋), then we will denote by Ker𝑇 the kernel of 𝑇; that is, Ker𝑇={𝑥∈𝑋with𝑇𝑥=0},Range𝑇={𝑇𝑥;with𝑥∈𝑋}, (1)(2) respectively. We also denote by Δ the set of all pairs of all natural numbers (𝑚,𝑛) with 𝑚≥𝑛; that is, Δ={(𝑚,𝑛)∈ℕ²with𝑚≥𝑛}.

We consider the linear difference system 𝑥_𝑛+1=𝐴_𝑛𝑥_𝑛, (𝔄) where 𝐴∶ℕ→ℬ(𝑋) is a given sequence. For (𝑚,𝑛)∈Δ, we define 𝐴^𝑛_𝑚=𝐴_𝑚−1⋅…⋅𝐴_𝑛,if𝑚>𝑛,𝐼,if𝑚=𝑛. (3) It is obvious that 𝐴^𝑛_𝑚𝐴^𝑝_𝑛=𝐴^𝑝_𝑚,∀(𝑚,𝑛),𝑛,𝑝∈Δ, (4) and every solution of (𝔄) satisfies 𝑥_𝑚=𝐴^𝑛_𝑚𝑥_𝑛∀(𝑚,𝑛)∈Δ. (5) If for every 𝑛∈ℕ the operator 𝐴_𝑛 is invertible, then the system (𝔄) is called reversible.

Definition 1. A sequence 𝑃∶ℕ→ℬ(𝑋) is called a sequence of projections if 𝑃²_𝑛=𝑃_𝑛,forevery𝑛∈ℕ. (6)

Remark 2. If 𝑃 is a sequence of projections, then 𝑄=𝐼−𝑃 is also a sequence of projections (which is called the complementary sequence of projections of 𝑃) with Ker𝑄_𝑛=Range𝑃_𝑛 and Range𝑄_𝑛=Ker𝑃_𝑛 for every 𝑛∈ℕ.

3. (ℎ,𝑘)-Dichotomy with Invariant Projections

Definition 3. A projection sequence 𝑃 is called invariant for the system (𝔄) if 𝐴_𝑛𝑃_𝑛=𝑃_𝑛+1𝐴_𝑛,∀𝑛∈ℕ. (7)

Remark 4. If 𝑃 is invariant for (𝔄), then its complementary 𝑄 is also invariant for (𝔄). Furthermore, as a consequence of (7), we have 𝐴^𝑛_𝑚𝑃_𝑛=𝑃_𝑚𝐴^𝑛_𝑚,𝐴^𝑛_𝑚𝑄_𝑛=𝑄_𝑚𝐴^𝑛_𝑚 (8) for all (𝑚,𝑛)∈Δ.

Remark 5. If 𝑃 is invariant for (𝔄), then(i)𝐴_𝑛(Ker𝑃_𝑛)⊂Ker𝑃_𝑛+1 for all 𝑛∈ℕ.(ii)𝐴^𝑛_𝑚(Ker𝑃_𝑛)⊂Ker𝑃_𝑚 and 𝐴^𝑛_𝑚(Range𝑃_𝑛)⊂Range𝑃_𝑚 for all (𝑚,𝑛)∈Δ.

Definition 6. An nondecreasing sequence 𝜑∶ℕ→[1,+∞) is said to be a growth rate sequence if 𝜑₀=1 and lim_𝑛→+∞𝜑_𝑛=+∞.
Let ℎ,𝑘∶ℕ→[1,+∞) be two growth rate sequences.

Definition 7. We say that the linear difference system (𝔄) admits a strong (ℎ,𝑘)-dichotomy if there exist a sequence of projections 𝑃 invariant for (𝔄) and constants 𝑁≥1,𝛼>0, and 𝛽≥0 such that ℎ^𝛼_𝑚𝐴^𝑛_𝑚𝑃_𝑛𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑛𝑃_𝑛𝑥ℎ^𝛼_𝑚𝑄_𝑛𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝐴^𝑛_𝑚𝑄_𝑛𝑥 (9)(10) for all (𝑚,𝑛,𝑥)∈Δ×𝑋.
As particular cases of (ℎ,𝑘)-dichotomy, we have(1)If ℎ_𝑛=𝑘_𝑛=𝑒^𝑛, then we say that (𝔄) admits a strong exponential dichotomy.(2)If ℎ_𝑛=𝑒^𝑛 and 𝛽=0, then we say that system (𝔄) admits a strong uniform exponential dichotomy.(3)If ℎ_𝑛=𝑘_𝑛=𝑛+1, then we say that system (𝔄) admits a strong polynomial dichotomy.(4)If ℎ_𝑛=𝑛+1 and 𝛽=0, then we say that system (𝔄) admits a strong uniform polynomial dichotomy.

Remark 8. If the linear difference system (𝔄) admits a strong (ℎ,𝑘)-dichotomy, then for all (𝑚,𝑛)∈Δ the following assertion holds: 𝐴^𝑛_𝑚𝑄_𝑛𝑥=0⟹𝑄_𝑛𝑥=0. (11)

Definition 9. We say that the linear difference system (𝔄) admits an (ℎ,𝑘) dichotomy if there exist a sequence of projections 𝑃 invariant for (𝔄) and constants 𝑁≥1,𝛼>0, and 𝛽≥0 such that ℎ^𝛼_𝑚𝐴^𝑛_𝑚𝑃_𝑛≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑛ℎ^𝛼_𝑚≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝐴^𝑛_𝑚𝑄_𝑛 (12)(13) for all (𝑚,𝑛,𝑥)∈Δ×𝑋.
If the system (𝔄) admits a (ℎ,𝑘)-dichotomy and we take into account the particular cases from above, we obtain the notions of exponential dichotomy, uniform exponential dichotomy, polynomial dichotomy, and uniform polynomial dichotomy, respectively.
From the previous definition, we have that 𝑃_𝑛≤𝑁𝑘^𝛽_𝑛,∀𝑛∈ℕ. (14)

Definition 10. If 𝑘∶ℕ→[1,+∞) is a given growth rate sequence and 𝑃 is a sequence of projections that verifies (14), then we say that 𝑃 is 𝑘-bounded.
As particular cases of k-boundedness, we have the following.(i)If there exist 𝑀≥1 and 𝛾≥0 such that ‖𝑃(𝑛)‖≤𝑀𝑒^𝛾𝑛 for all 𝑛∈ℕ, then we say that 𝑃 is exponentially bounded. (ii)If there exist 𝑀≥1 and 𝛾≥0 such that ‖𝑃(𝑛)‖≤𝑀(𝑛+1)^𝛾 for all 𝑛∈ℕ, then we say that 𝑃 is polynomially bounded. (iii)If there exist 𝑀≥1 such that ‖𝑃(𝑛)‖≤𝑀 for all 𝑛∈ℕ, then we say that 𝑃 is bounded.

Remark 11. It is worth to mention that the concept that we name “strong (ℎ,𝑘)-dichotomy,” in the exponential case with bounded sequences of projections, is in fact the notion of exponential forward splitting considered in [25, Definition 2.1]. The notion of strong (uniform) exponential dichotomy has been also considered in [27]. The name of “strong dichotomy” derives from the fact that it implies the other dichotomy concept in this paper (under the 𝑘-boundedness assumption of the sequence of the projections). It is important to state that the “strong dichotomy” from this paper differs from the “strong dichotomy” defined in [18].

Remark 12. Through the following two examples we will show that for a difference system (𝔄) the concepts of strong (ℎ,𝑘)-dichotomy and (ℎ,𝑘)-dichotomy are distinct, neither of them implying the other one.

Example 13. Consider 𝑋=𝑙¹ the Banach space of all real valued sequences 𝑥=(𝑥_𝑛)_𝑛≥0 endowed with the norm ‖𝑥‖₁=∑^+∞_𝑗=0|𝑥_𝑗|. Fix two growth rate sequences ℎ,𝑘∶ℕ→[1,+∞). Define, for every 𝑛∈ℕ, 𝑃_𝑛∶𝑙¹→𝑙¹ by 𝑃_𝑛𝑥=(𝑦^𝑛_𝑗)_𝑗≥0 given by 𝑦^𝑛_𝑗=𝑥_𝑗+𝑘^𝑘𝑛_𝑛⋅𝑥_𝑗+1,if𝑗iseven,0,if𝑗isodd. (15) One can see that 𝑃_𝑛∈ℬ(𝑙¹) for all 𝑛∈ℕ, 𝑃 is a sequence of projections on 𝑙¹ and if we pick 𝑥∈𝑙¹ satisfying 𝑥_2𝑗+1=1/(𝑗+1)² and 𝑥_2𝑗=0, for all 𝑗∈ℕ, a simple computation gives us that 𝑃_𝑛𝑥₁=𝑘^𝑘𝑛_𝑛‖𝑥‖₁, (16) from where we deduce that 𝑃_𝑛≥𝑘^𝑘𝑛_𝑛,∀𝑛∈ℕ. (17) In addition, we have the following properties: 𝑃_𝑚𝑃_𝑛=𝑃_𝑛,𝑃_𝑛𝑃_𝑚=𝑃_𝑚,𝑄_𝑚𝑥₁≥𝑄_𝑛𝑥₁ (18) for all (𝑚,𝑛,𝑥)∈Δ×𝑙¹.
Consider the linear difference system 𝑥_𝑛+1=ℎ_𝑛ℎ_𝑛+1𝑃_𝑛𝑥_𝑛+ℎ_𝑛+1ℎ_𝑛𝑄_𝑛+1𝑥_𝑛, (19) where 𝑄∶ℕ→ℬ(𝑙¹) is the complementary sequence of projections of 𝑃.
Some computations show us that 𝐴^𝑛_𝑚 is given by 𝐴^𝑛_𝑚=ℎ_𝑛ℎ_𝑚𝑃_𝑛+ℎ_𝑚ℎ_𝑛𝑄_𝑚 (20) for all (𝑚,𝑛)∈Δ.
One can check that the sequence of projections is invariant for the system (𝔄). From the fact that ℎ_𝑚𝐴^𝑛_𝑚𝑃_𝑛𝑥₁=ℎ_𝑛𝑃_𝑛𝑥₁≤ℎ_𝑛𝑘_𝑛𝑃_𝑛𝑥₁,ℎ_𝑛𝑘_𝑚𝐴^𝑛_𝑚𝑄_𝑛𝑥₁=𝑘_𝑚ℎ_𝑚𝑄_𝑚𝑥₁≥ℎ_𝑚𝑄_𝑛𝑥₁ (21) for all (𝑚,𝑛,𝑥)∈Δ×𝑙¹, we have that the system (𝔄) admits a strong (ℎ,𝑘)-dichotomy.
Indeed, assume that the system (𝔄) admits a (ℎ,𝑘)-dichotomy. Then, by (14) and (17), we get the inequalities 𝑘^𝑘𝑛_𝑛≤𝑃_𝑛≤𝑁𝑘^𝛽_𝑛∀𝑛∈ℕ. (22) Because the sequence 𝑘 is a growth rate, choose 𝑛_𝛽 such that 𝑘_𝑛≥𝛽+1∀𝑛≥𝑛_𝛽. (23) Hence, the preceding inequality implies that 𝑘_𝑛≤𝑘^𝑘𝑛_𝑛^−𝛽≤𝑁∀𝑛≥𝑛_𝛽, (24) which, by making 𝑛→∞, contradicts the fact that 𝑘 is a growth rate. Thus, we can see that the system (𝔄) fails to admit a (ℎ,𝑘)-dichotomy.

Example 14. On 𝑋=ℝ³ consider the sequence 𝐴∶ℕ→ℬ(ℝ³) defined, for 𝑛∈ℕ and 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³, by 𝐴_𝑛ℎ𝑥=^𝑛ℎ_𝑛+1𝑥₁,ℎ_𝑛+1ℎ_𝑛𝑥₂,0. (25) By considering the linear difference system (𝔄) associated to the above-defined sequence of bounded linear operators, we obtain that for all (𝑚,𝑛,𝑥)∈Δ×ℝ³ we have 𝐴^𝑛_𝑚ℎ𝑥=^𝑛ℎ_𝑚𝑥₁,ℎ_𝑚ℎ_𝑛𝑥₂,0. (26) We define the sequence of projections 𝑃∶ℕ→ℬ(ℝ³) by 𝑃_𝑛𝑥₁,𝑥₂,𝑥₃=𝑥₁,0,0 (27) for all 𝑛∈ℕ and (𝑥₁,𝑥₂,𝑥₃)∈ℝ³. One can see that 𝑃 is invariant for the system (𝔄) and ℎ_𝑚𝐴^𝑛_𝑚𝑃_𝑛=ℎ_𝑛≤ℎ_𝑛𝑘_𝑚 (28) for all (𝑚,𝑛)∈Δ. Moreover, for (𝑚,𝑛)∈Δ, if we pick 𝑥=(0,1,0), then we get that ℎ_𝑛𝐴^𝑛_𝑚𝑄_𝑛𝑥=ℎ_𝑚‖𝑥‖, (29) from where we obtain that ℎ_𝑚≤ℎ_𝑛𝐴^𝑛_𝑚𝑄_𝑛≤ℎ_𝑛𝑘_𝑚𝐴^𝑛_𝑚𝑄_𝑛 (30) for all (𝑚,𝑛)∈Δ.
By (28) and (30), we conclude that the system (𝔄) admits a (ℎ,𝑘)-dichotomy.
Because 𝑄₀(0,0,1)≠(0,0,0) and 𝐴⁰₁(0,0,1)=(0,0,0), by Remark 8, it follows that the system (𝔄) does not admit a strong (ℎ,𝑘)-dichotomy.

4. (ℎ,𝑘)-Dichotomy with Strongly Invariant Projections

Definition 15. A sequence of projections 𝑃 is called strongly invariant for the system (𝔄) if 𝑃 is invariant for (𝔄) and for all 𝑛∈ℕ the restriction of 𝐴_𝑛 to Ker𝑃_𝑛 is an isomorphism from Ker𝑃_𝑛 to Ker𝑃_𝑛+1.

Remark 16. 𝑃 is strongly invariant for the system (𝔄) if and only if for all (𝑚,𝑛)∈Δ the restriction of 𝐴^𝑛_𝑚 to Ker𝑃_𝑛 is an isomorphism from Ker𝑃_𝑛 to Ker𝑃_𝑚.

Remark 17. In the exponential setting, under the hypotheses that 𝑃 is a strongly invariant for a system (𝔄) and it is bounded, it can be easily checked that the concept of strong exponential dichotomy defined in our paper is in fact the concept of exponential splitting defined in [25, Definition 4.1] or the concept of exponential dichotomy from [28, Definition 1.1].

Remark 18. If the sequence of projections 𝑃 is invariant for the reversible system (𝔄), then it is also strongly invariant for (𝔄).
Indeed, if 𝑃 is invariant for the reversible system (𝔄), then for every 𝑦∈Ker𝑃_𝑛+1 we have 𝑦=𝐴_𝑛(𝐴_𝑛)⁻¹𝑦=𝐴_𝑛𝑥 with 𝑥=(𝐴_𝑛)⁻¹𝑦∈Ker𝑃_𝑛.
There are linear difference systems which admit (ℎ,𝑘)-dichotomy with respect to projection sequences which are invariant but are not strongly invariant for the system. This phenomenon is illustrated by the following example.

Example 19. On 𝑋=ℝ³, with ‖𝑥‖=|𝑥₁|+|𝑥₂|+|𝑥₃|, we consider the system generated by the sequence 𝐴_𝑛 defined by 𝐴_𝑛𝑥₁,𝑥₂,𝑥₃=𝑥₁𝑒,𝑒𝑥₂,0 (31) for all 𝑛∈ℕ and 𝑥=(𝑥₁,𝑥₂,𝑥₃)∈ℝ³. It is immediate to see that 𝐴^𝑛_𝑚=𝑒^𝑛−𝑚𝑥₁,𝑒^𝑚−𝑛𝑥₂,0,𝑚≠𝑛,𝐼,𝑚=𝑛. (32) Define the sequence of projections 𝑃∶ℕ→ℬ(𝑋) by 𝑃_𝑛𝑥₁,𝑥₂,𝑥₃=𝑥₁,0,0∀𝑛∈ℕ,𝑥₁,𝑥₂,𝑥₃∈ℝ³. (33) We have that the sequence 𝑃 is invariant for the system (𝔄). Moreover, one can see that 𝐴^𝑛_𝑚𝑃_𝑛𝑥=𝑒^𝑛−𝑚𝑃_𝑛𝑥, (34) and hence 𝐴^𝑛_𝑚𝑃_𝑛=𝑒^{−(𝑚−𝑛)}𝑃_𝑛∀(𝑚,𝑛)∈Δ. (35) Moreover, for (𝑚,𝑛)∈Δ, 𝑚≠𝑛, and 𝑥₀=(0,1,0)∈ℝ³, we have that 𝐴^𝑛_𝑚𝑄_𝑛𝑥₀=𝑒^𝑚−𝑛𝑄_𝑛𝑥₀. (36) On the other hand, it is easy to check that 𝐴^𝑛_𝑛𝑄_𝑛𝑥₀=𝑒^𝑛−𝑛𝑄_𝑛𝑥₀; (37) hence 𝐴^𝑛_𝑚𝑄_𝑛≥𝑒^𝑚−𝑛𝑄_𝑛=𝑒^𝑚−𝑛∀(𝑚,𝑛)∈Δ. (38) This shows us that we have a system (𝔄) which is (uniformly) exponentially dichotomic with the invariant projection sequence 𝑃. But 𝑃 is not strongly invariant because, for example, 𝐴₁𝑄_𝑛(0,0,1)=(0,0,0).

Remark 20. Our second concept of (ℎ,𝑘)-dichotomy introduced in this paper (see Definition 9) can be viewed in the light of the general dichotomies presented in [17], by making the choice 𝑎(𝑚,𝑛)=(ℎ(𝑛)/ℎ(𝑚))^𝛼⋅𝑘(𝑛)^𝛽 and 𝑏(𝑚,𝑛)=(ℎ(𝑛)/ℎ(𝑚))^𝛼⋅𝑘(𝑚)^𝛽 in the case of strongly invariant sequences of projections and by observing that the inequality (𝐷2) from [17], namely, ‖(𝐴^𝑛_𝑚∣_Ker𝑃𝑛)⁻¹𝑄_𝑚‖≤𝑏(𝑚,𝑛), implies the corresponding instability inequality from our dichotomy concept by observing that 1≤𝐴^𝑛_𝑚∣_Ker𝑃𝑛𝐴^𝑛_𝑚∣_Ker𝑃𝑛⁻¹𝑄_𝑚≤𝐴^𝑛_𝑚∣_Ker𝑃𝑛⋅𝑏(𝑚,𝑛). (39) Moreover, under the assumption of 𝑘-boundedness of the sequence of the projections and strong invariance, the notion of strong (ℎ,𝑘)-dichotomy from our paper is a particular case of the general dichotomies from [17], by making the same choice of the sequences 𝑎(𝑚,𝑛) and 𝑏(𝑚,𝑛).
The above-mentioned concept is useful in input-output techniques, and we mention the work of B. Sasu and A. L. Sasu [29], where, in the uniform exponential setting, from a sufficient condition of admissibility of pairs of function spaces to an associated control system of (𝔄), they deduce the property of uniform exponential dichotomy and the boundedness of the sequence of projections.
Even in the particular case of exponential dichotomy, the implication that gives (39) from the preceding remark fails to hold, showing that there exist reversible systems that verify Definition 9 and do not satisfy the dichotomic behavior pointed out by (39), even in the case of bounded sequences of projections.

Remark 21. If the sequence of projections 𝑃 is strongly invariant for the system (𝔄), then (i)for every 𝑛∈ℕ there is an isomorphism 𝐵_𝑛 from the Ker𝑃_𝑛+1 to Ker𝑃_𝑛 such that 𝐴_𝑛𝐵_𝑛𝑄_𝑛+1=𝑄_𝑛and𝐵_𝑛𝐴_𝑛𝑄_𝑛=𝑄_𝑛;(ii)for all (𝑚,𝑛)∈Δ there is an isomorphism 𝐵^𝑛_𝑚 from Ker𝑃_𝑚 to Ker𝑃_𝑛 with 𝐴^𝑛_𝑚𝐵^𝑛_𝑚𝑄_𝑚=𝑄_𝑚,𝐵^𝑛_𝑚𝐴^𝑛_𝑚𝑄_𝑛=𝑄_𝑛 (40)(41)for all (𝑚,𝑛)∈Δ.
In particular case when (𝔄) is invertible and 𝑃 is invariant for (𝔄), then 𝐵_𝑛=(𝐴_𝑛)⁻¹ for every 𝑛∈ℕ and 𝐵^𝑛_𝑚=(𝐴^𝑛_𝑚)⁻¹ for all (𝑚,𝑛)∈Δ.

Proposition 22. Let ℎ and 𝑘 be growth rate sequences and 𝑃 a sequence of projections that is strongly invariant for a system (𝔄). Then the system (𝔄) admits a strong (ℎ,𝑘)-dichotomy if and only if there exist 𝑁≥1, 𝛼>0, and 𝛽≥0 such that ℎ^𝛼_𝑚𝐴^𝑛_𝑚𝑃_𝑛𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑛𝑃_𝑛𝑥,ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝑄_𝑚𝑥 (42)(43) for all (𝑚,𝑛,𝑥)∈Δ×𝑋, where 𝐵^𝑛_𝑚 are the images of the function 𝐵 defined in Remark 21.

Proof. We have to prove the equivalence of (10) and (43). Let (𝑚,𝑛,𝑥)∈Δ×𝑋. For the necessity, by (40), we have ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝐴^𝑛_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝑥=𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝑄_𝑚𝑥. (44) For the sufficiency, by (41), we have ℎ^𝛼_𝑚𝑄_𝑛𝑥=ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝐴^𝑛_𝑚𝑄_𝑛𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝑄_𝑚𝐴^𝑛_𝑚𝑄_𝑛𝑥=𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝐴^𝑛_𝑚𝑄_𝑛𝑥 (45) which concludes our proof.

Proposition 23. Let ℎ and 𝑘 be growth rate sequences and 𝑃 a sequence of projections that is strongly invariant for a system (𝔄). If there exist 𝑁≥1, 𝛼>0, and 𝛽≥0 such that the system (𝔄) has the properties ℎ^𝛼_𝑚𝐴^𝑛_𝑚𝑃_𝑛≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑛,ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚 (46)(47) for all (𝑚,𝑛)∈Δ, where 𝐵^𝑛_𝑚 are the images of the function 𝐵 defined in Remark 21. Then, (𝔄) admits a (ℎ,𝑘)-dichotomy.

Proof. We have to show that (47) implies (13). Let (𝑚,𝑛)∈Δ. The conclusion follows from ℎ^𝛼_𝑚≤ℎ^𝛼_𝑚𝑄_𝑛=ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝐴^𝑛_𝑚𝑄_𝑛≤ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝐴^𝑛_𝑚𝑄_𝑛≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝐴^𝑛_𝑚𝑄_𝑛. (48)

The converse of the preceding proposition is not generally valid, as shown in the following example.

Example 24. On 𝑋=𝑙¹, consider (𝑘_𝑛) an arbitrary growth rate, ℎ_𝑛=𝑒^𝑛 for 𝑛∈ℕ, 𝛼=1 and 𝛽=0. Define the sequence 𝐴∶ℕ→ℬ(𝑙¹) by 𝐴_𝑛𝑥=(𝑦_𝑗(𝑛))_𝑗≥0, where 𝑦_3𝑗+1(𝑛)=1𝑒⋅𝑥_3𝑗+1𝑦_3𝑗+2(𝑛)=𝑒⋅𝑥_3𝑗+2𝑦_3𝑗(𝑛)=𝑒^{𝑒𝑛−𝑒𝑛+1}⋅𝑥_3𝑗𝑗∈ℕ. (49) Then, we have that for (𝑚,𝑛,𝑥)∈Δ×𝑙¹, 𝐴^𝑛_𝑚𝑥=(𝑦_𝑗(𝑚,𝑛))_𝑗≥0 with 𝑦_3𝑗+1(𝑚,𝑛)=𝑒^𝑛−𝑚⋅𝑥_3𝑗+1𝑦_3𝑗+2(𝑚,𝑛)=𝑒^𝑚−𝑛⋅𝑥_3𝑗+2𝑦_3𝑗(𝑚,𝑛)=𝑒^{𝑒𝑛−𝑒𝑚}⋅𝑥_3𝑗𝑗∈ℕ. (50) Define, for every 𝑛∈ℕ, the projection 𝑃_𝑛∶𝑙¹→𝑙¹ by 𝑃_𝑛𝑥=(𝑎_𝑗)_𝑗≥0 where 𝑎_3𝑗+1=𝑥_3𝑗+1𝑎_3𝑗+2𝑎=0_3𝑗=0𝑗∈ℕ. (51) It is easy to check that the sequence of projections 𝑃 is strongly invariant for the system (𝔄), and for (𝑚,𝑛,𝑥)∈Δ×𝑙¹, 𝐵^𝑛_𝑚𝑄_𝑚𝑥=(𝑧_𝑗(𝑚,𝑛))_𝑗≥0 is given by 𝑧_3𝑗+1()𝑧𝑚,𝑛=0_3𝑗+2(𝑚,𝑛)=𝑒^𝑛−𝑚⋅𝑥_3𝑗+2𝑧_3𝑗(𝑚,𝑛)=𝑒^{𝑒𝑚−𝑒𝑛}⋅𝑥_3𝑗𝑗∈ℕ. (52) A simple computation shows us that 𝑒^𝑚𝐴^𝑛_𝑚𝑃_𝑛=𝑒^𝑛 (53) for all (𝑚,𝑛)∈Δ and by choosing 𝑥∈𝑙¹ with 𝑥_3𝑗+1=𝑥_3𝑗=0 and 𝑥_3𝑗+2=1/(𝑗+1)², it follows that 𝑒^𝑚≤𝐴^𝑛_𝑚𝑄_𝑛𝑒^𝑛. (54) Hence, the system (𝔄) admits a (ℎ,𝑘)-dichotomy.
Assume now that there exist 𝛼>0, 𝑁≥1, and 𝛽≥0 with ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚 (55) for all (𝑚,𝑛)∈Δ. In our case, the preceding inequality becomes 𝑒^𝛼𝑚𝐵^𝑛_𝑚𝑄_𝑚≤𝑁𝑒^𝛼𝑛 (56) for all (𝑚,𝑛)∈Δ.
Consider that 𝑛=0, 𝑚∈ℕ, and 𝑥∈𝑙¹ given by 𝑥_3𝑗+1=𝑥_3𝑗+2𝑥=0_3𝑗=1𝑗+1²𝑗∈ℕ. (57) It follows that 𝐵⁰_𝑚𝑄_𝑚𝑥₁=𝑒^𝑒𝑚−1⋅𝜋√6=𝑒^𝑒𝑚−1‖𝑥‖₁, (58) from where we get that 𝐵⁰_𝑚𝑄_𝑚≥𝑒^𝑒𝑚−1. (59) Combining the above inequality with (56), we get that 𝑒^{𝑒𝑚+𝛼𝑚−1}≤𝑁 (60) for all 𝑚∈ℕ, which is a contradiction.
Moreover, the converse of the implication from Proposition 23 does not hold not even in the finite dimensional case, as we can see from the following example.

Example 25. Consider that ℎ_𝑛=𝑘_𝑛=𝑒^𝑛, 𝑛∈ℕ. Consider, on 𝑋=ℝ³, the sequence 𝐴∶ℕ→ℬ(ℝ³) is defined by 𝐴_𝑛𝑥₁,𝑥₂,𝑥₃=𝑥₁𝑒,𝑒𝑥₂,𝑥₃𝑒². (61) A simple computation shows us that 𝐴^𝑛_𝑚𝑥₁,𝑥₂,𝑥₃=𝑒^𝑛−𝑚𝑥₁,𝑒^𝑚−𝑛𝑥₂,𝑒^{𝑛2−𝑚2}𝑥₃ (62) for all (𝑚,𝑛)∈Δ and (𝑥₁,𝑥₂,𝑥₃)∈ℝ³. Define the (constant) sequence of projections 𝑃∶ℕ→ℬ(𝑋) by 𝑃_𝑛𝑥₁,𝑥₂,𝑥₃=𝑥₁,0,0∀𝑛∈ℕ,𝑥₁,𝑥₂,𝑥₃∈ℝ³. (63) We have that 𝑃 is strongly invariant to the system (𝔄) with 𝐴^𝑛_𝑚∣_Ker𝑃𝑛⁻¹0,𝑥₂,𝑥₃=0,𝑒^𝑛−𝑚𝑥₂,𝑒^{𝑚2−𝑛2}𝑥₃ (64) for all (𝑚,𝑛)∈Δ and for all (0,𝑥₂,𝑥₃)∈Ker𝑃_𝑛. A straightforward computation shows us that 𝐴^𝑛_𝑚𝑃_𝑛=𝑒^{−(𝑚−𝑛)},𝐴^𝑛_𝑚𝑄_𝑛≥𝑒^𝑚−𝑛 (65) for all (𝑚,𝑛)∈Δ; hence, (𝔄) satisfies the conditions from Definition 9 (in the exponential case).
Assume by a contradiction that there exist 𝑁≥1, 𝛼>0, and 𝛽≥0 such that 𝐴^𝑛_𝑚𝑃_𝑛≤𝑁𝑒^𝛽𝑛𝑒^{−𝛼(𝑚−𝑛)}𝐴^𝑛_𝑚∣⁻¹_Ker𝑃𝑛≤𝑁𝑒^𝛽𝑚𝑒^{−𝛼(𝑚−𝑛)} (66)(67) for all (𝑚,𝑛)∈Δ. Choose 𝑛=0. From the fact that 𝐴⁰_𝑚∣_Ker𝑃0⁻¹=𝑒^𝑚2∀𝑚∈ℕ. (68) From (67) and (68), we get the contradiction 𝑒^𝑚2≤𝑁𝑒^{(𝛽−𝛼)𝑚}∀𝑚∈ℕ. (69)

Proposition 26. Let ℎ and 𝑘 be growth rate sequences and 𝑃 a sequence of projections that is strongly invariant for a system (𝔄). If there exist 𝑁≥1, 𝛼>0, and 𝛽≥0 such that the system (𝔄) verifies (46) and (47), then (𝔄) admits a strong (ℎ,𝑘)-dichotomy.

Proof. Let (𝑚,𝑛,𝑥)∈Δ×𝑋. It follows that ℎ^𝛼_𝑚𝐴^𝑛_𝑚𝑃_𝑛𝑥≤ℎ^𝛼_𝑚𝐴^𝑛_𝑚𝑃_𝑛𝑃_𝑛𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑛𝑃_𝑛𝑥,ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝑥≤ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝑄_𝑚𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝑄_𝑚𝑥 (70) which, by Proposition 22, concludes the proof.

Theorem 27. Let ℎ and 𝑘 be growth rates and 𝑃 a sequence of projections that is strongly invariant for a system (𝔄). Then, the system (𝔄) verifies (46) and (47) if and only if (𝔄) admits a strong (ℎ,𝑘)-dichotomy and 𝑃 is 𝑘-bounded.

Proof. The necessity follows from Proposition 26 and (14).
For the sufficiency, let 𝑀≥1 and 𝛾≥0 such that 𝑃_𝑛≤𝑀𝑘^𝛾_𝑛,∀𝑛∈ℕ. (71) Let (𝑚,𝑛,𝑥)∈Δ×𝑋. From ℎ^𝛼_𝑚𝐴^𝑛_𝑚𝑃_𝑛𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑛𝑃_𝑛𝑥≤2𝑀𝑁ℎ^𝛼_𝑛𝑘_𝑛^𝛽+𝛾‖𝑥‖,ℎ^𝛼_𝑚𝐵^𝑛_𝑚𝑄_𝑚𝑥≤𝑁ℎ^𝛼_𝑛𝑘^𝛽_𝑚𝑄_𝑚𝑥≤2𝑀𝑁ℎ^𝛼_𝑛𝑘_𝑚^𝛽+𝛾‖𝑥‖, (72) the conclusion follows.

Corollary 28. Let 𝑃 be a sequence of projections that is strongly invariant for a system (𝔄). (i)If ℎ_𝑛=𝑘_𝑛=𝑒^𝑛, then the system (𝔄) verifies (46) and (47) if and only if it is exponentially dichotomic and 𝑃 is exponentially bounded. (ii)If ℎ_𝑛=𝑘_𝑛=𝑛+1, then the system (𝔄) verifies (46) and (47) if and only if it is polynomially dichotomic and 𝑃 is polynomially bounded. (iii)If ℎ_𝑛=𝑒^𝑛 (resp., ℎ_𝑛=𝑛+1) and 𝛽=0, then the system (𝔄) verifies (46) and (47) if and only if it is uniformly exponentially (polynomially) dichotomic and 𝑃 is bounded.

Acknowledgments

The authors would like to thank the referees for carefully reading the paper and for their valuable suggestions and comments, which led to the current improved form of the paper. Mihai Gabriel Babuţia is supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, Project no. PN-II-RU-TE-2011-3-0103.