Admissibility for Nonuniform (𝜇,𝜈) Contraction and Dichotomy

Abstract

The relation between the notions of nonuniform asymptotic stability and admissibility is considered. Using appropriate Lyapunov norms, it is showed that if any of their associated ℒ^𝑝 spaces, with 𝑝∈(1,∞], is admissible for a given evolution process, then this process is a nonuniform (𝜇,𝜈) contraction and dichotomy. A collection of admissible Banach spaces for any given nonuniform (𝜇,𝜈) contraction and dichotomy is provided.

1. Introduction

The study of the admissibility property has a fairly long history, and it goes back to the pioneering work of Perron [1] in 1930. Perron concerned originally the existence of bounded solutions of the equation 𝑥^′=𝐴(𝑡)𝑥+𝑓(𝑡) (1.1) in ℝ_𝑛 for any bounded continuous perturbation 𝑓∶ℝ⁺₀→ℝ_𝑛. This property can be used to deduce the stability or the conditional stability under sufficiently small perturbations of a given linear equation: 𝑥^′=𝐴(𝑡)𝑥. (1.2) His result served as a starting point for many works on the qualitative theory of the solutions of differential equations. Moreover, a simple consequence of one of the main results in that paper stated explicitly in [2, Theorem 1] is probably the first step in the literature concerning the study of the relation between admissibility and the notions of stability and conditional stability. We refer the reader to [2] for details. Relevant results concerning the extension of Perron's problem in the more general framework of the infinite-dimensional Banach spaces with bounded 𝐴(𝑡) were obtained by Daleckij and Krein [3], Massera and Schäffer [4], and the work of Levitan and Zhikov [5] for certain cases of unbounded 𝐴(𝑡).

Over the last decades an increasing interest can be seen in the study of the asymptotic behavior of evolution equations in abstract spaces. In [6, 7], Latushkin et al. studied the dichotomy of linear skew-product semiflows defined on compact spaces. Using the so-called evolution semigroup, they expressed its dichotomy in terms of hyperbolicity of a family of weighted shift operators. In [8–10], Preda et al. considered related problems in the particular case of uniform exponential behavior. A large class of Schäffer spaces, which were introduced by Schäffer in [11] (see also [4]), acted as admissible spaces for the case of uniform exponential dichotomies. It is worth noting here the works by Huy [12–16] in the study of the existence of an exponential dichotomy for evolution equations.

In the case of nonuniform exponential dichotomies, Preda and Megan [17] obtained related results also for the class of Schäffer spaces, but using a notion of dichotomy which is different from the original one motivated by ergodic theory and the nonuniform hyperbolicity theory, as detailed, for example, in [18, 19]. In the more recent work [20], the authors consider the same weaker notion of exponential dichotomy and obtain sharper relations between admissibility and stability for perturbations and solutions in 𝐶₀. Important contributions in this aspect have been made by Barreira et al. [2, 18, 19, 21–25]. Particularly, in [22], Barreira and Valls showed an equivalence between the admissibility of their associated ℒ^𝑝 spaces (𝑝∈(1,∞]) and the nonuniform exponential stability of certain evolution families by using appropriate adapted norms. They also establish a collection of admissible Banach spaces for any given nonuniform exponential dichotomy in [2]. Recently, Preda et al. [26] studied the connection between the (non)uniform exponential dichotomy of a non(uniform) exponentially bounded, strongly continuous evolution family and the admissibility of some function spaces, which extended those results established in [2, 22].

In the present paper, inspired by Barreira and Valls [2, 22], we give a characterization of nonuniform asymptotic stability in terms of admissibility property. We consider a more general type of dichotomy which is called (𝜇,𝜈) dichotomy in [21], also proposed in [27]. In this dichotomy, not only the usual exponential behavior is replaced by an arbitrary, which may correspond, for example, to situations when the Lyapunov exponents are all infinity or are all zero, but also different growth rates for the uniform and nonuniform parts of the dichotomy are considered. It extended exponential dichotomy in various ways. In [21], it has also been showed that there is a large class of equations exhibiting this behavior. We emphasize that the characterization in our paper is a very general one; it includes as particular cases many interesting situations among them we can mention some results in previous references. To some extent, our results have a certain significance to study the theory of nonuniform hyperbolicity.

2. Admissibility for Nonuniform (𝜇,𝜈) Contractions

We first concentrate on the simpler case of admissibility for nonuniform (𝜇,𝜈) contractions, leaving the more elaborate case of admissibility for nonuniform (𝜇,𝜈) dichotomies for the second part of the paper. This allows us to present the results and their proofs without some accessory technicalities. After the introduction of some basic notions, using appropriate adapted Lyapunov norms, we show that the admissibility with respect to some space ℒ^𝑝 with 𝑝∈(1,∞] is sufficient for an evolution process to be a nonuniform (𝜇,𝜈) contraction.

2.1. Basic Notions

We say that an increasing function 𝜇∶ℝ⁺→[1,+∞) is a growth rate if 𝜇(0)=1,lim_𝑡→+∞𝜇(𝑡)=+∞. (2.1)

We say that a family of linear operators 𝑇(𝑡,𝑠), 𝑡≥𝑠≥0 in a Banach space 𝑋 is an evolution process if:(1)𝑇(𝑡,𝑡)=𝐼𝑑 and 𝑇(𝑡,𝜏)𝑇(𝜏,𝑠)=𝑇(𝑡,𝑠),𝑡,𝜏,𝑠>0;(2)(𝑡,𝑠,𝑥)↦𝑇(𝑡,𝑠)𝑥 is continuous for 𝑡≥𝑠≥0 and 𝑥∈𝑋. In this section, we also assume that(3) there exist 𝜔≥0, 𝐷>0 and two growth rates 𝜇(𝑡), 𝜈(𝑡) such that ‖𝑇(𝜇(𝑡)𝑡,𝑠)‖≤𝐷𝜇(𝑠)^𝜔𝜈^𝜀(𝑠),𝑡≥𝑠≥0. (2.2)We consider the new norms ‖𝑥‖^′_𝑡𝑇()𝑥‖𝜇(𝜎)=sup‖𝜎,𝑡𝜇(𝑡)^−𝜔,𝜎≥𝑡,𝑥∈𝑋,𝑡∈ℝ⁺₀. (2.3) These satisfy ‖𝑥‖≤‖𝑥‖^′_𝑡≤𝐷𝜈^𝜀(𝑡)‖𝑥‖,𝑥∈𝑋,𝑡∈ℝ⁺₀. (2.4) Moreover, with respect to these norms the evolution process has the following bounded growth property.

Proposition 2.1. If 𝑇 is an evolution process, then ‖𝑇(𝑡,𝑠)𝑥‖^′_𝑡≤𝜇(𝑡)𝜇(𝑠)^𝜔‖𝑥‖^′_𝑠 (2.5) for every 𝑡≥𝑠≥0 and 𝑥∈𝑋.

Proof. We have ‖𝑇(𝑡,𝑠)𝑥‖^′_𝑡𝑇()𝑥‖𝜇(𝜎)=sup‖𝜎,𝑡𝜇(𝑡)^−𝜔≤𝜇(𝑡),𝜎≥𝑡𝜇(𝑠)^𝜔𝑇()𝑥‖𝜇(𝜎)sup‖𝜎,𝑡𝜇(𝑠)^−𝜔=𝜇(𝑡),𝜎≥𝑠𝜇(𝑠)^𝜔‖𝑥‖^′_𝑠 (2.6) which yields the desired inequality.

Definition 2.2. We say that an evolution process 𝑇 is a nonuniform (𝜇,𝜈) contraction in ℝ⁺₀ if there exist some constants 𝛼, 𝐷>0, 𝜀≥0 and two growth rates 𝜇(𝑡), 𝜈(𝑡) such that ‖𝑇(𝜇(𝑡)𝑡,𝑠)‖≤𝐷𝜇(𝑠)^−𝛼𝜈^𝜀(𝑠),𝑡≥𝑠≥0. (2.7) When 𝜀=0, we say that (1.2) has a uniform (𝜇,𝜈) contraction or simply a (𝜇,𝜈) contraction.

In the following, we introduce several Banach spaces that are used throughout the paper. We first set 𝐿^𝑝=𝑓∶ℝ⁺₀→ℝLebesgue-measurable∶𝑓_𝑝<∞ (2.8) for each 𝑝∈[1,∞), and 𝐿^∞=𝑓∶ℝ⁺₀→ℝLebesgue-measurable∶𝑓_∞<∞ (2.9) Respectively, with the norms 𝑓_𝑝=^∞₀𝑓(𝑡)^𝑝1/𝑝,𝑓_∞=esssup_𝑡∈ℝ+0𝑓(𝑡). (2.10) Then for each 𝑝∈[1,∞] the set ℒ^𝑝 of the equivalence classes [𝑓] of functions 𝑔∈𝐿^𝑝 such that 𝑔=𝑓 Lebesgue-almost everywhere is a Banach space (again with the norms in (2.10)).

For each Banach space 𝐸=ℒ^𝑝, with 𝑝∈[1,∞], we set 𝐸(𝑋)=𝑓∶ℝ⁺₀→𝑋Bochner-measurable∶𝑡↦𝑓(𝑡)^′_𝑡∈𝐸 (2.11) using the norms ‖⋅‖^′_𝑡 in (2.3), and we endow 𝐸(𝑋)=ℒ^𝑝(𝑋) with the norm 𝑓^′_𝑝=‖𝐹‖_𝑝,where𝐹(𝑡)=𝑓(𝑡)^′_𝑡. (2.12)

Repeating arguments in the proof of Theorem 3 in [22], we obtain the following statement.

Lemma 2.3. For each 𝑝∈[1,∞] and 𝐸=ℒ^𝑝, the set 𝐸(𝑋) is a Banach space with the norm in (2.12), and the convergence in 𝐸(𝑋) implies the pointwise convergence Lebesgue-almost everywhere.

Definition 2.4. We say that a Banach space 𝐸 is admissible for the evolution process 𝑇 if for each 𝑓∈𝐸(𝑋) the function 𝑥_𝑓∶ℝ⁺₀→𝑋 defined by 𝑥_𝑓(𝑡)=^∞₀𝑇(𝑡,𝜏)𝑓(𝜏)𝑑𝜏 (2.13) is in ℒ^∞ (see (2.11)).

By Lemma 2.3 we know that ℒ^∞ is a Banach space with the norm 𝑔^′_∞=esssup_𝑡∈ℝ+0𝑔(𝑡)^′_𝑡. (2.14)

Lemma 2.5. There exists 𝐾>0 such that 𝑥_𝑓^′_∞≤𝐾𝑓^′_𝑝forevery𝑓∈𝐸(𝑋). (2.15)

Proof. We define a linear operator 𝐺∶𝐸(𝑋)→ℒ^∞(𝑋) by 𝐺𝑓=𝑥_𝑓. We use the closed graph theorem to show that 𝐺 is bounded. For this, let us take a sequence (𝑓_𝑛)_𝑛∈N⊂𝐸(𝑋) and 𝑓∈𝐸(𝑋) such that 𝑓_𝑛→𝑓 in 𝐸(𝑋) when 𝑛→∞ and also ℎ∈ℒ^∞(𝑋) such that 𝐺𝑓_𝑛→ℎ in ℒ^∞(𝑋) when 𝑛→∞. We need to show that 𝐺𝑓=ℎ Lebesgue-almost everywhere. For each 𝑡≥0 and 𝑛∈𝑁 we have 𝐺𝑓_𝑛(𝑡)−𝐺𝑓(𝑡)^′_𝑡=sup^t₀𝑇(𝜎,𝑡)𝑇(𝑡,𝜏)𝑓_𝑛(𝜏)(𝜏)𝜇(𝜎)−𝑓𝑑𝜏𝜇(𝑡)^−𝜔∶𝜎≥𝑡=sup^𝑡₀𝑇(𝜎,𝜏)𝑓_𝑛(𝜏)(𝜏)𝜇(𝜎)−𝑓𝑑𝜏𝜇(𝑡)^−𝜔∶𝜎≥𝑡≤sup^𝑡₀𝑇(𝜎,𝜏)𝑓_𝑛(𝜏)(𝜏)𝜇(𝜎)−𝑓𝜇(𝑡)^−𝜔𝑑𝜏∶𝜎≥𝑡=sup^𝑡₀𝑇(𝜎,𝜏)𝑓_𝑛(𝜏)(𝜏)𝜇(𝜎)−𝑓𝜇(𝜏)^−𝜔𝜇(𝑡)𝜇(𝜏)^𝜔=𝑑𝜏∶𝜎≥𝑡_𝑡0𝑓_𝑛(𝜏)−𝑓(𝜏)^′_𝜏𝜇(𝑡)𝜇(𝜏)^𝜔(𝑡)𝑑𝜏≤𝜇^𝜔𝑡₀𝑓_𝑛(𝜏)−𝑓(𝜏)^′_𝜏𝑑𝜏. (2.16) According to Hölder's inequality, there exists 𝛼=𝛼([0,𝑡]) such that 𝐺𝑓_𝑛(𝑡)−𝐺𝑓(𝑡)^′_𝑡≤𝜇(𝑡)^𝜔𝑡₀𝑓_𝑛(𝜏)−𝑓(𝜏)^′_𝜏𝑑𝜏≤𝜇(𝑡)^𝜔𝛼𝑓_𝑛(𝜏)−𝑓(𝜏)^′_𝑝. (2.17) Therefore, for each 𝑡≥0, letting 𝑛→∞ we find that (𝐺𝑓_𝑛)(𝑡)→(𝐺𝑓)(𝑡). This shows that 𝐺𝑓=ℎ Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that 𝐺 is a bounded operator. This completes the proof of the lemma.

2.2. Criterion for Nonuniform (𝜇,𝜈) Contraction

Theorem 2.6. If for some 𝑝∈(1,∞] the space 𝐸=ℒ^𝑝 is admissible for the evolution process 𝑇, then 𝑇 is a nonuniform (𝜇,𝜈) contraction.

Proof. We follow arguments in [22]. Given 𝑥∈𝑋 and 𝑡₀≥0, we define a function 𝑓∶ℝ⁺₀→𝑋 by 𝑓(𝑡)=𝑇𝑡,𝑡₀𝑥,𝑡∈𝑡₀,𝑡₀+10,𝑡∈ℝ⁺₀⧵𝑡₀,𝑡₀+1. (2.18) We note that 𝑓(𝑡)^′_𝑡≤𝑇𝑡,𝑡₀𝑥^′_𝑡𝜒_{[𝑡0,𝑡0+1]}(𝑡). (2.19) Then, for each 𝑡∈[𝑡₀,𝑡₀+1] and 𝑥∈𝑋, we have 𝑇𝑡,𝑡₀𝑥^′_𝑡=sup𝑇(𝜎,𝑡)𝑇𝑡,𝑡₀𝑥𝜇(𝜎)𝜇(𝑡)^−𝜔≤𝜇(𝑡),𝜎≥𝑡𝜇𝑡₀^𝜔sup𝑇𝜎,𝑡₀𝑥𝜇(𝜎)𝜇𝑡₀^−𝜔,𝜎≥𝑡₀=𝜇(𝑡)𝜇𝑡₀^𝜔‖𝑥‖^′_𝑡0≤𝜇𝑡₀+1𝜇𝑡₀^𝜔‖𝑥‖^′_𝑡0. (2.20) Therefore, 𝑓(𝑡)^′_𝑝≤𝜇𝑡₀+1𝜇𝑡₀^𝜔‖𝑥‖^′_𝑡0𝜒_{[𝑡0,𝑡0+1]}(𝑡)_𝑝=𝜇𝑡₀+1𝜇𝑡₀^𝜔‖𝑥‖^′_𝑡0 (2.21) and in particular 𝑓∈𝐸(𝑋). On the other hand, according to (2.13) and (2.18), we have 𝑥_𝑓(𝑡)=^𝑡0_𝑡⁺¹₀𝑇(𝑡,𝜏)𝑇𝜏,𝑡₀𝑥𝑑𝜏=𝑇𝑡,𝑡₀𝑥 (2.22) for all 𝑡≥𝑡₀+1, which implies that 𝑇(𝑡,𝑡₀)𝑥^′_𝑡=𝑥_𝑓^′_𝑡≤𝑥_𝑓^′_∞. (2.23) By Lemma 2.5 and (2.21)–(2.23), we obtain 𝑇𝑡,𝑡₀𝑥^′_𝑡≤𝑥_𝑓^′_∞≤𝐾𝑓^′_𝑝𝜇𝑡≤𝐾⁰+1𝜇𝑡₀^𝜔‖𝑥‖^′_𝑡0 (2.24) for all 𝑡≥𝑡₀+1, 𝑡₀≥0, and 𝑥∈𝑋. We claim that 𝑇𝑡,𝑡₀^′∶=sup_𝑥≠0𝑇𝑡,𝑡₀𝑥^′_𝑡‖𝑥‖^′_𝑡0𝜇𝑡≤𝐿,𝐿=⁰+1𝜇𝑡₀^𝜔max{𝐾,1} (2.25) for all 𝑡≥𝑡₀. Indeed, for 𝑡≥𝑡₀+1 inequality (2.25) follows from (2.24), and for 𝑡∈[𝑡₀,𝑡₀+1] the inequality follows from (2.20).
Now given 𝑥∈𝑋, 𝑡₀≥0, and 𝛿>0, we define a function 𝑔∶ℝ⁺₀→𝑋 by 𝑔(𝑡)=𝑇𝑡,𝑡₀𝑥,𝑡∈𝑡₀,𝑡₀+𝛿0,𝑡∈ℝ⁺₀⧵𝑡₀,𝑡₀+𝛿. (2.26) It follows from (2.25) that 𝑔(𝑡)^′_𝑡≤𝑇𝑡,𝑡₀𝑥^′_𝑡≤𝐿‖𝑥‖^′_𝑡0 (2.27) and thus, 𝑔∈𝐸(𝑋),𝑔^′_𝑝≤𝐿𝛿^1/𝑝‖𝑥‖^′_𝑡0. (2.28) On the other hand, writing 𝑦=𝑇(𝑡₀+𝛿,𝑡₀)𝑥, 𝛿²2𝑦^′_𝑡0+𝛿=^𝑡0_𝑡^+𝛿₀𝜏−𝑡₀𝑦𝑑𝜏^′_𝑡0+𝛿=sup𝑇𝜎,𝑡₀+𝛿^𝑡0_𝑡^+𝛿₀𝜏−𝑡₀𝜇(𝜎)𝑦𝑑𝜏𝜇𝑡₀+𝛿^−𝜔∶𝜎≥𝑡₀+𝛿=sup^𝑡0_𝑡^+𝛿₀𝜏−𝑡₀𝑇𝜎,𝑡₀𝜇(𝜎)𝑥𝑑𝜏𝜇𝑡₀+𝛿^−𝜔∶𝜎≥𝑡₀+𝛿≤sup^𝑡0_𝑡^+𝛿₀𝜏−𝑡₀𝑇𝜎,𝑡₀𝑥𝜇(𝜎)𝜇𝑡₀+𝛿^−𝜔𝑑𝜏∶𝜎≥𝑡₀=+𝛿_{𝑡0𝑡+𝛿0}𝜏−𝑡₀sup𝑇𝜎,𝑡₀𝑦𝜇(𝜎)+𝛿𝜇𝑡₀+𝛿^−𝜔∶𝜎≥𝑡₀=+𝛿𝑑𝜏_{𝑡0𝑡+𝛿0}𝜏−𝑡₀𝑦^′_𝑡0+𝛿=𝑑𝜏_{𝑡0𝑡+𝛿0}𝜏−𝑡₀𝑇𝑡₀+𝛿,𝜏𝑇𝜏,𝑡₀𝑥^′_𝑡0𝑑𝜏 (2.29) Since 𝑥_𝑔(𝑡)=^𝑡₀𝑇(𝑡,𝜏)𝑔(𝜏)𝑑𝜏=0,𝑡∈0,𝑡₀𝑡,𝑡₀𝑇𝑡,𝑡₀𝑥,𝑡∈𝑡₀,𝑡₀+𝛿,𝛿𝑇𝑡,𝑡₀𝑥,𝑡∈𝑡₀+𝛿,∞, (2.30) it follows from Lemma 2.5, (2.25), and (2.28) that 𝛿²2𝑇𝑡₀+𝛿,𝑡₀𝑥^′_𝑡0+𝛿≤𝐿^𝑡0_𝑡^+𝛿₀𝜏−𝑡₀𝑇𝜏,𝑡₀𝑥^′_𝜏𝑑𝜏=𝐿^𝑡0_𝑡^+𝛿₀𝑥_𝑔(𝜏)^′_𝜏𝑑𝜏≤𝐿𝛿𝑥_𝑔^′_∞≤𝐾𝐿𝛿𝑔^′_𝑝≤𝐾𝐿²𝛿^{(𝑝+1)/𝑝}‖𝑥‖^′_𝑡0 (2.31) for all 𝑡₀≥0, 𝛿>0, and 𝑥∈𝑋; we thus obtain 𝛿²2𝑇𝑡₀+𝛿,𝑡₀𝑥^′_𝑡0+𝛿≤𝐾𝐿²𝛿^{(𝑝+1)/𝑝}‖𝑥‖^′_𝑡0 (2.32) so 𝑇𝑡₀+𝛿,𝑡₀^′≤2𝐾𝐿²𝛿^{(1−𝑝)/𝑝} (2.33) for all 𝑡₀≥0 and 𝛿>0. Since (1−𝑝)/𝑝<0 for 𝑝∈(1,∞], there exists 𝛿₀>0 sufficiently large such that 𝐾₀∶=2𝐾𝐿²𝛿^{(1−𝑝)/𝑝}<1. (2.34) Setting 𝑛=[(ln𝜇(𝑡)−ln𝜇(𝑡₀))/𝛿₀] for each 𝑡≥𝑡₀, we have 𝑇𝑡,𝑡₀=𝑇𝑡,𝑡₀+𝑛𝛿₀𝑇𝑡₀+𝑛𝛿₀,𝑡₀. (2.35) By (2.25) and (2.33) we obtain 𝑇𝑡,𝑡₀^′≤𝐿𝑇𝑡₀+𝑛𝛿₀,𝑡₀^′≤𝐿^𝑛−1_𝑖=0𝑇𝑡₀+(𝑖+1)𝛿₀,𝑡₀+𝑖𝛿₀^′≤𝐿𝐾^𝑛₀ (2.36) for 𝑡≥𝑡₀. By (2.34) and (𝑡)𝑡𝑛=ln𝜇−ln𝜇₀𝛿₀≥ln𝜇(𝑡)−ln𝜇𝑡₀𝛿₀−1 (2.37) this implies that 𝑇(𝑡,𝑡₀)^′𝜇(𝑡)≤𝑑𝜇𝑡₀^−𝛼, (2.38) where 𝐿𝑑=𝐾₀1,𝛼=−𝛿₀ln𝐾₀. (2.39) We note that 𝑑,𝛼>0. Since 𝑇(𝑡,𝑡₀)𝑥^′_𝑡≥𝑇𝑡,𝑡₀𝑥, (2.40) and by (2.4), ‖𝑥‖^′_𝑡0=sup𝑇𝜎,𝑡₀𝑥𝜇(𝜎)𝜇𝑡₀^−𝜔∶𝜎≥𝑡₀≤𝐷𝜈^𝜀𝑡₀‖𝑥‖. (2.41) It follows from (2.38) that 𝑇𝑡,𝑡₀=sup_𝑥≠0𝑇𝑡,𝑡₀𝑥‖𝑥‖≤𝐷𝜈^𝜀𝑡₀sup_𝑥≠0𝑇𝑡,𝑡₀𝑥^′_𝑡‖𝑥‖^′_𝑡0≤𝑑𝐷𝜈^𝜀𝑡₀𝜇(𝑡)𝜇𝑡₀^−𝛼 (2.42) for any 𝑡≥𝑡₀. Therefore, the evolution process 𝑇 is a nonuniform (𝜇,𝜈) contraction with 𝛼 and 𝐷=𝑑𝐷. This concludes the proof of Theorem 2.6.

2.3. Admissible Spaces for Nonuniform (𝜇,𝜈) Contractions

We consider the spaces 𝐿^𝑝_𝐷=𝑓∶ℝ⁺₀→ℝLebesgue-measurable∶𝑓_𝑝,𝐷<∞ (2.43) for each 𝑝∈[1,∞), and 𝐿^∞_𝐷=𝑓∶ℝ⁺₀→ℝLebesgue-measurable∶𝑓_∞,𝐷<∞, (2.44) respectively, with the norms 𝑓_𝑝,𝐷=^∞₀𝑓(𝑡)^𝑝𝐷𝜈^𝜀(𝑡)𝜇(𝑡)𝜇^′(𝑡)^{1/𝑞𝑝1/𝑝},𝑓_∞,𝐷=esssup_𝑡∈ℝ+0𝑓(𝑡)𝐷𝜈^𝜀(𝑡)𝜇(𝑡)𝜇^′(𝑡). (2.45) In a similar manner to that in Lemma 2.3 these normed spaces induce Banach spaces ℒ^𝑝_𝐷 and ℒ^𝑝_𝐷(𝑋) for each 𝑝∈[1,∞], the last one with norm 𝑓^′_𝑝,𝐷=‖𝐹‖_𝑝,𝐷,where𝐹(𝑡)=𝑓(𝑡)^′_𝑡. (2.46)

Theorem 2.7. If the evolution process 𝑇 is a nonuniform (𝜇,𝜈) contraction, then for any 𝑝∈[1,∞] the space ℒ^𝑝_𝐷 is admissible for 𝑇.

Proof. We first take 𝑓∈ℒ^∞_𝐷. Then 𝑥_𝑓(𝑡)^′_𝑡=sup^𝑡₀𝑇()𝑇()𝑓(𝜏)𝜇(𝜎)𝜎,𝑡𝑡,𝜏𝑑𝜏𝜇(𝑡)^−𝜔∶𝜎≥𝑡=sup^𝑡₀𝑇()𝑓(𝜏)𝜇(𝜎)𝜎,𝜏𝑑𝜏𝜇(𝑡)^−𝜔∶𝜎≥𝑡≤sup^𝑡₀‖𝑇(𝜎,𝜏)‖⋅𝑓(𝜏)𝑑𝜏∶𝜎≥𝑡≤sup^𝑡₀𝐷𝜈^𝜏(𝜏𝜇(𝜎))𝜇(𝜏)^−𝛼𝑓(𝜏)^′_𝜏𝑑𝜏∶𝜎≥𝑡≤sup^𝑡₀𝐷𝜈^𝜏(𝜏)𝜇(𝑡)𝜇(𝜏)^−𝛼𝑓(𝜏)^′_𝜏≤𝑓𝑑𝜏∶𝜎≥𝑡_{′∞,𝐷𝑡0}𝜇(𝑡)𝜇(𝜏)^−𝛼𝜇^′(𝜏)𝜇(𝜏)≤𝑓𝑑𝜏_′∞,𝐷1−𝜇(𝑡)^−𝛼𝛼≤1𝛼𝑓^′_∞,𝐷. (2.47) Therefore, 𝑥_𝑓^′_∞=sup_𝑡≥0𝑥_𝑓(𝑡)^′_𝑡≤sup_𝑡≥01𝛼𝑓^′_∞,𝐷<∞ (2.48) and ℒ^∞_𝐷 is admissible for 𝑇.
Now we take 𝑓∈ℒ^𝑝_𝐷(𝑋) for some 𝑝∈[1,∞). Using Hölder's inequality we obtain 𝑥_𝑓(𝑡)^′_𝑡=sup^𝑡₀𝑇()𝑇()𝑓(𝜏)𝜇(𝜎)𝜎,𝑡𝑡,𝜏𝑑𝜏𝜇(𝑡)^−𝜔∶𝜎≥𝑡≤sup^𝑡₀𝐷𝜈^𝜏(𝑡)𝜇(𝑡)𝜇(𝜏)^−𝛼𝑓(𝜏)^′_𝜏≤𝑓𝑑𝜏∶𝜎≥𝑡_{′𝑝,𝐷𝑡0}𝜇(𝑡)𝜇(𝜏)^−𝛼𝑞𝜇^′(𝜏)𝜇(𝜏)𝑑𝜏^1/𝑞≤𝑓^′_𝑝,𝐷1−𝜇(𝑡)^−𝛼𝑞𝛼𝑞^1/𝑞≤1𝛼𝑞^1/𝑞𝑓^′_𝑝,𝐷, (2.49) where 1/𝑝+1/𝑞=1. We conclude that ℒ^𝑝_𝐷 is also admissible for 𝑇.

3. Admissibility for Nonuniform (𝜇,𝜈) Dichotomies

We consider in this second part admissibility for nonuniform (𝜇,𝜈) dichotomies. It generalizes the usual notion of exponential dichotomy in several ways: besides introducing a nonuniform term, causing that any conditional stability may be nonuniform, we consider rates that may not be exponential as well as different rates in the uniform and nonuniform parts. After introducing some basic notions, we show that the admissibility with respect to some space ℒ^𝑝 with 𝑝∈(1,∞] is sufficient for an evolution process to be a nonuniform (𝜇,𝜈) dichotomy. When compared to the case of contractions, this creates substantial complications. We also provide a collection of admissible Banach spaces for any given nonuniform (𝜇,𝜈) dichotomy.

3.1. Basic Notions

We consider an evolution process 𝑇(𝑡,𝑠), 𝑡≥𝑠≥0 satisfied 1, 2 in Section 2.

We also consider a function 𝑃∶ℝ⁺₀→𝐵(𝑋), where 𝐵(𝑋) is the set of bounded linear operators in 𝑋, such that(1)𝑃(𝑡)²=𝑃(𝑡), for every 𝑡≥0;(2)(𝑡,𝑥)↦𝑃(𝑡)𝑥 is continuous in ℝ⁺₀×𝑋.

We will refer to 𝑃 as a projection function. Given an evolution process 𝑇, we say that a projection function 𝑃 is compatible with 𝑇 if: (1)𝑇(𝑡,𝑠)𝑝(𝑠)=𝑃(𝑡)𝑇(𝑡,𝑠), for every 𝑡,𝜏,𝑠>0;(2) the map 𝑇(𝑡,𝜎)∣ker𝑃(𝜎)∶ker𝑃(𝜎)→ker𝑃(𝑡) (3.1) is invertible for every 𝑡≥𝑠≥0. We also assume that(3) there exist 𝐷>0, 𝜔≥0 and two growth rates 𝜇(𝑡), 𝜈(𝑡) such that ‖𝑇()𝑃(𝑠𝜇(𝑡)𝑡,𝑠)‖≤𝐷𝜇(𝑠)^𝜔𝜈^𝜀(𝑠),𝑡≥𝑠≥0.‖𝑇()𝑄(𝑠𝜇(𝑠)𝑡,𝑠)‖≤𝐷𝜇(𝑡)^𝜔𝜈^𝜀(𝑠),𝑠≥𝑡≥0. (3.2)(3.3)

We note that due to the invertibility assumption in condition (1.2), condition (3.3) is simply a version of (3.2) when time goes backwards.

We always consider in the paper an evolution process 𝑇 together with a projection function 𝑃 which is compatible with 𝑇 (and which satisfies (3.2) and (3.3)). We write 𝑈(𝑡,𝑠)=𝑇(𝑡,𝑠)𝑃(𝑠),𝑉(𝑡,𝑠)=𝑇(𝑡,𝑠)𝑄(𝑠), (3.4) where 𝑄(𝑡)=𝐼𝑑−𝑃(𝑡) for each 𝑡≥0. we consider the new norms ‖𝑥‖^′_𝑡𝑈()𝑥‖𝜇(𝜎)=sup‖𝜎,𝑡𝜇(𝑡)^−𝜔𝑉()𝑥‖𝜇(𝑡),𝜎≥𝑡+sup‖𝜎,𝑡𝜇(𝜎)^−𝜔,0≤𝜎≤𝑡. (3.5) for each 𝑥∈𝑋 and 𝑡∈ℝ⁺₀, where 𝑉(𝜎,𝑡) denotes the inverse of the map in (3.1). We have ‖𝑥‖^′_𝑡≥‖𝑃(𝑡)𝑥‖+‖𝑄(𝑡)𝑥‖≥‖𝑃(𝑡)𝑥+𝑄(𝑡)𝑥‖=‖𝑥‖. (3.6) Moreover, by (3.2) and (3.3), ‖𝑥‖^′_𝑡≤2𝐷𝜈^𝜀(𝑡)‖𝑥‖,𝑥∈𝑋,𝑡∈ℝ⁺₀. (3.7)

Definition 3.1. We say that an evolution process 𝑇 is a nonuniform (𝜇,𝜈) dichotomy in ℝ⁺ if there exist a projection function 𝑃∶ℝ⁺₀→𝐵(𝑋) compatible with 𝑇, some constants 𝛼, 𝐷>0, 𝜀≥0 and two growth rates 𝜇(𝑡), 𝜈(𝑡) such that ‖𝑇()𝑃(𝑠𝜇(𝑡)𝑡,𝑠)‖≤𝐷𝜇(𝑠)^−𝛼𝜈^𝜀(𝑠)‖𝑇()𝑄(𝑠𝜇(𝑠),𝑡≥𝑠≥0.𝑡,𝑠)‖≤𝐷𝜇(𝑡)^−𝛽𝜈^𝜀(𝑠),𝑠≥𝑡≥0, (3.8) When 𝜀=0, we say that (1.2) has a uniform (𝜇,𝜈) dichotomy or simply a (𝜇,𝜈) dichotomy.

In the following, we still consider several spaces 𝐿^𝑝, 𝐿^∞, respectively, with the norms (2.10), which induce Banach spaces ℒ^𝑝 for each 𝑝∈[1,∞]. We also set 𝐸(𝑋) as in (2.11) but using the norms ‖⋅‖^′_𝑡 in (3.5), we endow 𝐸(𝑋)=ℒ^𝑝(𝑋) with the norm in (2.12).

We also obtain easily the same statement in Lemma 2.3 for the set 𝐸(𝑋) using the norms ‖⋅‖^′_𝑡 in (3.5).

Definition 3.2. We say that a Banach space 𝐸 is admissible for the evolution process 𝑇 if for each 𝑓∈𝐸(𝑋)(1) the function ℝ⁺₀𝑉()𝑓(𝜏)∋𝜏↦𝑡,𝜏,𝜏≥𝑡,0,0≤𝜏<𝑡 (3.9) is in ℒ¹(𝑋) for each 𝑡≥0;(2) the function 𝑥_𝑓∶ℝ⁺₀→𝑋 defined by 𝑥_𝑓(𝑡)=^𝑡₀𝑈(𝑡,𝜏)𝑓(𝜏)𝑑𝜏−^∞_𝑡𝑉(𝑡,𝜏)𝑓(𝜏)𝑑𝜏 (3.10) is in ℒ^∞(𝑋).

We note that since ‖𝑄(𝑡)𝑥‖≤‖𝑄(𝑡)𝑥‖^′_𝑡 for every 𝑥∈𝑋, and 𝑡≥0, any function in ℒ¹(𝑋) is also integrable in 𝑅⁺₀, and thus the first condition ensures that the function 𝑥_𝑓 is well defined. By Lemma 2.3 we know that ℒ^∞(𝑋) is a Banach space with the norm 𝑔^′_∞=esssup_𝑡∈ℝ+0𝑔(𝑡)^′_𝑡. (3.11)

Lemma 3.3. If for some 𝑝∈[1,∞] the space 𝐸=ℒ^𝑝 is admissible for the evolution process 𝑇, then there exists 𝐾>0 such that 𝑥_𝑓^′_∞≤𝐾𝑓^′_𝑝forevery𝑓∈𝐸(𝑋). (3.12)

Proof. We follow arguments in [2]. For each 𝑡≥0, we define a map 𝐻_𝑡∶𝐸(𝑋)→ℒ¹(𝑋) by 𝐻_𝑡𝑓𝜏)=𝑉()𝑓(𝜏)(𝑡,𝜏,𝜏≥𝑡,0,0≤𝜏<𝑡. (3.13) Clearly, 𝐻_𝑡 is linear. We use the closed graph theorem to show that 𝐻_𝑡 is bounded. For this, let us take a sequence (𝑓_𝑛)_𝑛∈𝑁⊂𝐸(𝑋) and 𝑓∈𝐸(𝑋) such that 𝑓_𝑛→𝑓 in 𝐸(𝑋) when 𝑛→∞, and also 𝑔∈ℒ¹(𝑋) such that 𝐻_𝑡𝑓_𝑛→𝑔 in ℒ¹(𝑋) when 𝑛→∞. We need to show that 𝐻_𝑡𝑓=𝑔 Lebesgue-almost everywhere. By Lemma 2.3, the sequence 𝑓_𝑛 converges pointwise Lebesgue-almost everywhere. Therefore, 𝐻_𝑡𝑓_𝑛(𝜏)=𝑉(𝑡,𝜏)𝑓_𝑛(𝜏)→𝑉(𝑡,𝜏)𝑓(𝜏)=𝐻_𝑡𝑓(𝜏) (3.14) when 𝑛→∞, for Lebesgue-almost every 𝜏∈[𝑡,+∞). On the other hand, since 𝐻_𝑡𝑓_𝑛→𝑔 in ℒ¹(𝑋) when 𝑛→∞, we also have (𝐻_𝑡𝑓_𝑛)(𝜏)→𝑔(𝜏) when 𝑛→∞, for Lebesgue-almost every 𝜏∈[𝑡,+∞). This shows that 𝐻_𝑡𝑓=𝑔 Lebesgue-almost everywhere, and 𝐻_𝑡 is bounded for each 𝑡≥0.
We define a linear operator 𝐺∶𝐸(𝑋)→ℒ^∞(𝑋) by 𝐺𝑓=𝑥_𝑓. We use again the closed graph theorem to show that 𝐺 is bounded. For this, let us take a sequence (𝑓_𝑛)_𝑛∈𝑁⊂𝐸(𝑋) and 𝑓∈𝐸(𝑋) such that 𝑓_𝑛→𝑓 in 𝐸(𝑋) when 𝑛→∞, and also ℎ∈ℒ^∞(𝑋) such that 𝐺𝑓_𝑛→ℎ in ℒ^∞(𝑋) when 𝑛→∞. We write 𝐺₁𝑓(𝑡)=𝑃(𝑡)𝐺𝑓(𝑡)=^𝑡₀𝑈()𝑓(𝜏)𝐺𝑡,𝜏𝑑𝜏,₂𝑓(𝑡)=𝑄(𝑡)𝐺𝑓(𝑡)=−^∞_𝑡𝑉(𝑡,𝜏)𝑓(𝜏)𝑑𝜏. (3.15)
Using the similar proof of Lemma 2.5, for each 𝑡≥0 and 𝑛∈𝑁 we have 𝐺₁𝑓_𝑛(𝑡)−𝐺₁𝑓(𝑡)^′_𝑡=sup^𝑡₀𝑈(𝜎,𝑡)𝑈(𝑡,𝜏)𝑓_𝑛(𝜏)(𝜏)𝜇(𝜎)−𝑓𝑑𝜏𝜇(𝑡)^−𝜔(𝑡)∶𝜎≥𝑡≤𝜇^𝜔𝑡₀𝑓_𝑛(𝜏)−𝑓(𝜏)^′_𝜏𝑑𝜏. (3.16) According to Hölder's inequality, there exists 𝛼=𝛼([0,𝑡]) such that 𝐺₁𝑓_𝑛(𝑡)−𝐺₁𝑓(𝑡)^′_𝑡≤𝜇(𝑡)^𝜔𝛼𝑓_𝑛(𝜏)−𝑓(𝜏)^′_𝑝. (3.17) Furthermore, we have 𝐺₂𝑓_𝑛(𝑡)−𝐺₂𝑓(𝑡)^′_𝑡≤sup^∞_𝑡𝑉(𝜎,𝑡)𝑉(𝑡,𝜏)𝑓_𝑛(𝜏)(𝜏)𝜇(𝑡)−𝑓𝜇(𝜎)^−𝜔𝑑𝜏∶0≤𝜎≤𝑡=^∞_𝑡sup𝑉(𝜎,𝑡)𝐻_𝑡𝑓_𝑛𝜏𝜇(𝑡)−𝑓()𝜇(𝜎)^−𝜔=∶0≤𝜎≤𝑡𝑑𝜏_∞𝑡𝐻_𝑡𝑓_𝑛−𝑓(𝜏)^′_𝑡𝑑𝜏=𝐻_𝑡𝑓_𝑛−𝑓¹_𝑡. (3.18)
It follows from (3.17) and (3.18) that 𝐺𝑓_𝑛(𝑡)−𝐺𝑓(𝑡)^′_𝑡≤𝜇(𝑡)^𝜔𝛼𝑓_𝑛−𝑓^′_𝑝+𝐻_𝑡𝑓_𝑛−𝑓¹_𝑡. (3.19)
Therefore, for each 𝑡≥0, letting 𝑛→∞ we find that (𝐺𝑓_𝑛)(𝑡)→(𝐺𝑓)(𝑡). This shows that 𝐺𝑓=ℎ Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that 𝐺 is a bounded operator. This completes the proof of the lemma.

3.2. Criterion for Nonuniform (𝜇,𝜈) Dichotomy

Theorem 3.4. If for some 𝑝∈(1,∞] the space 𝐸=ℒ^𝑝 is admissible for the evolution process 𝑇, then 𝑇 is a nonuniform (𝜇,𝜈) dichotomy.

Proof. We first consider the space 𝑃_𝑡0=Im𝑃(𝑡₀). Given 𝑥∈𝑃_𝑡0 and 𝑡₀≥0, repeating argument of the proof in Theorem 2.6, except limiting 𝑇(𝑡,𝑡₀) on 𝑃_𝑡0, we obtain 𝑇𝑡,𝑡₀∣𝑃_𝑡0^′𝜇(𝑡)≤𝑑𝜇𝑡₀^−𝛼, (3.20) where 𝐿𝑑=𝐾₀1,𝛼=−𝛿₀ln𝐾₀. (3.21) We note that 𝑑,𝛼>0. For each 𝑥∈𝑃_𝑡0, we have 𝑇𝑡,𝑡₀𝑥^′_𝑡≥𝑇𝑡,𝑡₀𝑥, (3.22) and by (3.2), ‖𝑥‖^′_𝑡0=sup𝑈𝜎,𝑡₀𝑥𝜇(𝜎)𝜇𝑡₀^−𝜔∶𝜎≥𝑡₀≤𝐷𝜈^𝜀𝑡₀‖𝑥‖. (3.23) It follows from (3.5) and (3.20) that 𝑇𝑡,𝑡₀∣𝑃_𝑡0=sup_{𝑥∈𝑃𝑡0⧵{0}}𝑇𝑡,𝑡₀𝑥‖𝑥‖≤𝐷𝜈^𝜀𝑡₀sup_{𝑥∈𝑃𝑡0⧵{0}}𝑇𝑡,𝑡₀𝑥^′_𝑡‖𝑥‖^′_𝑡0≤𝑑𝐷𝜈^𝜀𝑡₀𝜇(𝑡)𝜇𝑡₀^−𝛼 (3.24) for any 𝑡≥𝑡₀.
Now we consider the space 𝑄_𝑡0=Ker𝑃(𝑡₀). Given 𝑥∈𝑄_𝑡0 and 𝑡₀≥0, we define a function 𝑓∶ℝ⁺₀→𝑋 by 𝑓(𝑡)=𝑉𝑡,𝑡₀𝑥,𝑡∈ℝ⁺₀⋂𝑡₀−1,𝑡₀0,𝑡∈ℝ⁺₀⧵𝑡₀−1,𝑡₀. (3.25) Clearly, 𝑓(𝑡)∈𝑄_𝑡 for every 𝑡≥0. Moreover, for each 𝑡∈[0,𝑡₀−1] (this interval may be empty), we have 𝑥_𝑓(𝑡)=−^𝑡0_𝑡0−1𝑉(𝑡,𝜏)𝑉𝜏,𝑡₀𝑥𝑑𝜏=−^𝑡0_𝑡0−1𝑉𝑡,𝑡₀𝑥𝑑𝜏=−𝑉𝑡,𝑡₀𝑥, (3.26) and it follows from Lemma 3.3 that 𝑉𝑡,𝑡₀𝑥^′_𝑡=𝑥_𝑓^′_∞≤𝐾𝑓^′_𝑝 (3.27) for 𝑡∈[0,𝑡₀−1].
On the other hand, for each 𝑡∈[𝑡₀−1,𝑡₀], we have 𝑓(𝑡)^′_𝑡𝑉()𝑓(𝑡𝜇(𝑡)=sup𝜎,𝑡)𝜇(𝜎)^−𝜔𝑉()𝑉,0≤𝜎≤𝑡≤sup𝜎,𝑡𝑡,𝑡₀𝑥𝜇(𝑡)𝜇(𝜎)^−𝜔≤𝜇𝑡,0≤𝜎≤𝑡₀𝜇(𝑡)^𝜔sup𝑉𝜎,𝑡₀𝑥𝜇𝑡₀𝜇(𝜎)^−𝜔,0≤𝜎≤𝑡₀≤𝜇𝑡₀𝜇𝑡₀−1^𝜔‖𝑥‖^′_𝑡0. (3.28) So 𝑓^′_𝑝=_{ℝ+0⋂[𝑡0−1,𝑡0]}𝑓(𝑡)^′_𝑡𝜇𝑡𝑑𝑡≤⁰𝜇𝑡₀−1^𝜔‖𝑥‖^′_𝑡0 (3.29) and in particular 𝑓∈𝐸(𝑋). We thus have 𝑉(𝑡,𝑡₀)𝑥^′_𝑡≤𝐾𝑓^′_𝑝𝜇𝑡≤𝐾⁰𝜇𝑡₀−1^𝜔‖𝑥‖^′_𝑡0 (3.30) for every 𝑡∈[0,𝑡₀−1], and 𝑥∈𝑄_𝑡0. This implies that 𝑉𝑡,𝑡₀^′∶=sup_{𝑥∈𝑄𝑡0⧵{0}}𝑉𝑡,𝑡₀𝑥^′_𝑡‖𝑥‖^′_𝑡0𝜇𝑡≤𝐿,𝐿=⁰𝜇𝑡₀−1^𝜔max{𝐾,1} (3.31) for all 𝑡≤𝑡₀. Indeed, for 𝑡∈[0,𝑡₀−1] inequality (3.31) follows from (3.30), and for 𝑡∈ℝ⁺₀⋂[𝑡₀−1,𝑡₀] the inequality follows from (3.28).
Now given 𝑥∈𝑄_𝑡0, 𝑡₀≥0, and 𝛿>0, we define a function 𝑔∶ℝ⁺₀→𝑋 by 𝑔(𝑡)=𝑉𝑡,𝑡₀𝑥,𝑡∈ℝ⁺₀⋂𝑡₀−𝛿,𝑡₀0,𝑡∈ℝ⁺₀𝑡₀−𝛿,𝑡₀. (3.32) It follows from (3.31) that 𝑔(𝑡)^′_𝑡≤𝑉𝑡,𝑡₀𝑥^′_𝑡≤𝐿‖𝑥‖^′_𝑡0, (3.33) and thus, 𝑔∈𝐸(𝑋),𝑔^′_𝑝≤𝐿𝛿^1/𝑝‖𝑥‖^′_𝑡0. (3.34) On the other hand, in a similar manner to that in (2.29), 𝛿²2𝑉𝑡₀−𝛿,𝑡₀𝑥^′_𝑡0−𝛿=^𝑡0_𝑡0−𝛿𝜏−𝑡₀𝑉𝑡₀−𝛿,𝑡₀𝑥𝑑𝜏^′_𝑡0−𝛿≤^𝑡0_𝑡0−𝛿𝑡₀−𝜏sup𝑉𝑡₀−𝛿,𝑡₀𝑥^′_𝑡0−𝛿∶0≤𝜎≤𝑡₀=−𝛿𝑑𝜏_{𝑡0𝑡0−𝛿}𝑡₀−𝜏𝑉𝑡₀−𝛿,𝑡₀𝑥^′_𝑡0=𝑑𝜏_{𝑡0𝑡0−𝛿}𝑡₀−𝜏𝑉𝑡₀−𝛿,𝜏𝑉𝜏,𝑡₀𝑥^′_𝑡0𝑑𝜏. (3.35) Since 𝑥_𝑔(𝑡)=−^∞_𝑡𝑉()𝑔(𝜏)𝑡𝑡,𝜏𝑑𝜏=0,𝑡∈₀,∞𝑡−𝑡₀𝑉𝑡,𝑡₀𝑥,𝑡∈𝑡₀−𝛿,𝑡₀.−𝛿𝑉𝑡,𝑡₀𝑥,𝑡∈0,𝑡₀−𝛿. (3.36) It follows from Lemma 3.3, (3.31), and (3.34) that 𝛿²2𝑉𝑡₀−𝛿,𝑡₀𝑥^′_𝑡0−𝛿≤𝐿^𝑡0_𝑡0−𝛿𝑡₀−𝜏𝑉𝜏,𝑡₀𝑥^′_𝜏𝑑𝜏=𝐿^𝑡0_𝑡0−𝛿𝑥_𝑔(𝜏)^′_𝜏𝑑𝜏≤𝐿𝛿𝑥_𝑔^′_∞≤𝐾𝐿𝛿𝑔^′_𝑝≤𝐾𝐿²𝛿^{(𝑝+1)/𝑝}‖𝑥‖^′_𝑡0 (3.37) for all 𝑡₀≥0, 𝛿>0, and 𝑥∈𝑄_𝑡0; we thus obtain 𝛿²2𝑉𝑡₀−𝛿,𝑡₀𝑥^′_𝑡0−𝛿≤𝐾𝐿²𝛿^{(𝑝+1)/𝑝}‖𝑥‖^′_𝑡0 (3.38) so 𝑉𝑡₀−𝛿,𝑡₀^′≤2𝐾𝐿²𝛿^{(1−𝑝)/𝑝} (3.39) for all 𝑡₀≥0 and 𝛿>0. Taking the same 𝛿₀ as before, and setting 𝑛=[(ln𝜇(𝑡₀)−ln𝜇(𝑡))/𝛿₀] for each 𝑡≤𝑡₀, we have 𝑉𝑡,𝑡₀=𝑉𝑡,𝑡₀−𝑛𝛿₀𝑉𝑡₀−𝑛𝛿₀,𝑡₀. (3.40) By (3.31) and (3.39) we obtain 𝑉𝑡,𝑡₀^′≤𝐿𝑉𝑡₀−𝑛𝛿₀,𝑡₀^′≤𝐿^𝑛−1_𝑖=0𝑉𝑡₀−(𝑖+1)𝛿₀,𝑡₀−𝑖𝛿₀^′≤𝐿𝐾^𝑛₀ (3.41) for 𝑡≤𝑡₀, where 𝐾₀∶=2𝐾𝐿²𝛿₀^{(1−𝑝)/𝑝}<1. Since 𝑡𝑛=ln𝜇₀−ln𝜇(𝑡)𝛿₀≥ln𝜇𝑡₀−ln𝜇(𝑡)𝛿₀−1, (3.42) this implies that 𝑉𝑡,𝑡₀^′𝜇(𝑡)≤𝑑𝜇𝑡₀^𝛼, (3.43) where 𝐿𝑑=𝐾₀1,𝛼=−𝛿₀ln𝐾₀. (3.44) We note that 𝑑,𝛼>0. By (3.6) 𝑉𝑡,𝑡₀𝑥^′_𝑡≥𝑉𝑡,𝑡₀𝑥, (3.45) and by (3.3), ‖𝑥‖^′_𝑡0=sup𝑉𝜎,𝑡₀𝑥𝜇𝑡₀𝜇(𝜎)^−𝜔∶0≤𝜎≤𝑡₀≤𝐷𝜈^𝜀𝑡₀‖𝑥‖ (3.46) for 𝑥∈𝑄_𝑡0. It follows from (3.43) that 𝑉𝑡,𝑡₀∣𝑄_𝑡0=sup_{𝑥∈𝑄𝑡0⧵{0}}𝑉𝑡,𝑡₀𝑥‖𝑥‖≤𝐷𝜈^𝜀𝑡₀sup_{𝑥∈𝑄𝑡0⧵{0}}𝑉𝑡,𝑡₀𝑥^′_𝑡‖𝑥‖^′_𝑡0≤𝑑𝐷𝜈^𝜀𝑡₀𝜇(𝑡)𝜇𝑡₀^𝛼 (3.47) for any 𝑡≤𝑡₀. To show that 𝑇 is a nonuniform exponential dichotomy, we note that setting 𝑡=𝑠 in (3.2) and (3.3) yields ‖𝑃(𝑠)‖≤𝐷𝜈^𝜀(𝑠),‖𝑄(𝑠)‖≤𝐷𝜈^𝜀(𝑠) (3.48) for every 𝑠≥0. Together with (3.24) and (3.47) this implies that ‖𝑇(𝑡,𝑠)𝑃(𝑠)‖≤𝑇(𝑡,𝑠)∣𝑃_𝑠‖𝑃(𝑠)‖≤𝑑𝐷²𝜇(𝑡)𝜇(𝑠)^−𝛼𝜈^2𝜀(𝑠)‖𝑇()𝑄(𝑠≤𝑇(),𝑡≥𝑠≥0.𝑡,𝑠)‖𝑡,𝑠∣𝑄_𝑠‖𝑃(𝑠)‖≤𝑑𝐷²𝜇(𝑠)𝜇(𝑡)^−𝛼𝜈^2𝜀(𝑠),𝑠≥𝑡≥0. (3.49) This shows that 𝑇 is a nonuniform (𝜇,𝜈) dichotomy with 𝛼,𝛼, 𝑑𝐷² and 2𝜀.

3.3. Admissible Spaces for a Nonuniform (𝜇,𝜈) Dichotomy

We consider the spaces 𝐿^𝑝_𝐷=𝑓∶ℝ⁺₀→ℝLebesgue-measurable∶𝑓_𝑝,𝐷<∞ (3.50) for each 𝑝∈[1,∞), and 𝐿^∞_𝐷=𝑓∶ℝ⁺₀→ℝLebesgue-measurable∶𝑓_∞,𝐷<∞, (3.51) respectively, with the norms 𝑓_𝑝,𝐷=^∞₀𝑓(𝑡)^𝑝𝐷𝜈^𝜀(𝑡)𝜇(𝑡)𝜇^′(𝑡)^{1/𝑞𝑝1/𝑝},𝑓_∞,𝐷=esssup_𝑡∈ℝ+0𝑓(𝑡)𝐷𝜈^𝜀(𝑡)𝜇(𝑡)𝜇^′(𝑡). (3.52) In a similar manner to Lemma 2.3 these normed spaces induce Banach spaces ℒ^𝑝_𝐷 and ℒ^𝑝_𝐷(𝑋) for each 𝑝∈[1,∞], the last one with norm 𝑓^′_𝑝,𝐷=‖𝐹‖_𝑝,𝐷,where𝐹(𝑡)=𝑓(𝑡)^′_𝑡. (3.53)

Theorem 3.5. If the evolution process 𝑇 is a nonuniform (𝜇,𝜈) dichotomy, then for any 𝑝∈[1,∞] the space ℒ^𝑝_𝐷 is admissible for 𝑇.

Proof. We first take 𝑓∈𝐿^∞_𝐷. Then 𝑥_𝑓(𝑡)^′_𝑡=sup^𝑡₀𝑈()𝑈()𝑓(𝜏)𝜇(𝜎)𝜎,𝑡𝑡,𝜏𝑑𝜏𝜇(𝑡)^−𝜔∶𝜎≥𝑡+sup^∞_𝑡𝑉()𝑉()𝑓(𝜏)𝜇(𝑡)𝜎,𝑡𝑡,𝜏𝑑𝜏𝜇(𝜎)^−𝜔∶0≤𝜎≤𝑡≤sup^𝑡₀‖𝑈(⋅𝑓(𝜏𝜇(𝜎)𝜎,𝜏)‖)𝜇(𝑡)^−𝜔𝑑𝜏∶𝜎≥𝑡+sup^∞_𝑡‖𝑉(⋅𝑓(𝜏𝜇(𝑡)𝜎,𝜏)‖)𝜇(𝜎)^−𝜔𝑑𝜏∶0≤𝜎≤𝑡≤sup^𝑡₀𝐷𝜈^𝜀(𝜏)𝜇(𝜎)𝜇(𝜏)^−𝛼𝑓(𝜏)𝑑𝜏∶𝜎≥𝑡+sup^∞_𝑡𝐷𝜈^𝜀(𝜏)𝜇(𝜎)𝜇(𝜏)^𝛽𝑓(𝜏)𝑑𝜏∶0≤𝜎≤𝑡≤sup^𝑡₀𝐷𝜈^𝜀(𝜏)𝜇(𝑡)𝜇(𝜏)^−𝛼𝑓(𝜏)^′_𝜏𝑑𝜏∶𝜎≥𝑡+sup^∞_𝑡𝐷𝜈^𝜀(𝜏)𝜇(𝑡)𝜇(𝜏)^𝛽𝑓(𝜏)^′_𝜏≤𝑓𝑑𝜏∶0≤𝜎≤𝑡_{′∞,𝐷𝑡0}𝜇(𝑡)𝜇(𝜏)^−𝛼𝜇^′(𝜏)𝜇(𝜏)𝑑𝜏+𝑓^′_∞,𝐷^∞_𝑡𝜇(𝑡)𝜇(𝜏)^𝛽𝜇^′(𝜏)𝜇(𝜏)=1𝑑𝜏𝛼𝑓^′_∞,𝐷1−𝜇(𝑡)^−𝛼+1𝛽𝑓^′_∞,𝐷≤1𝛼+1𝛽𝑓^′_∞,𝐷. (3.54) Therefore, 𝑥_𝑓^′_∞=sup_𝑡≥0𝑥_𝑓(𝑡)^′_𝑡≤sup_𝑡≥01𝛼+1𝛽𝑓^′_∞,𝐷<∞ (3.55) and ℒ^∞_𝐷 is admissible for 𝑇.
Now we take 𝑓∈ℒ^𝑝_𝐷(𝑋) for some 𝑝∈[1,∞). Using Hölder's inequality we obtain 𝑥_𝑓(𝑡)^′_𝑡=sup^𝑡₀𝑈()𝑈()𝑓(𝜏)𝜇(𝜎)𝜎,𝑡𝑡,𝜏𝑑𝜏𝜇(𝑡)^−𝜔∶𝜎≥𝑡+sup^∞_𝑡𝑉()𝑉()𝑓(𝜏)𝜇(𝑡)𝜎,𝑡𝑡,𝜏𝑑𝜏𝜇(𝜎)^−𝜔∶0≤𝜎≤𝑡≤sup^𝑡₀𝐷𝜈^𝜀(𝜏)𝜇(𝑡)𝜇(𝜏)^−𝛼𝑓(𝜏)^′_𝜏𝑑𝜏∶𝜎≥𝑡+sup^∞_𝑡𝐷𝜈^𝜀(𝜏)𝜇(𝑡)𝜇(𝜏)^𝛽𝑓(𝜏)^′_𝜏≤𝑓𝑑𝜏∶0≤𝜎≤𝑡_{′𝑝,𝐷𝑡0}𝜇(𝑡)𝜇(𝜏)^−𝛼𝑞𝜇^′(𝜏)𝜇(𝜏)𝑑𝜏^1/𝑞+𝑓^′_𝑝,𝐷^∞_𝑡𝜇(𝑡)𝜇(𝜏)^𝛽𝑞𝜇^′(𝜏)𝜇(𝜏)𝑑𝜏^1/𝑞=𝑓^′_𝑝,𝐷1−𝜇(𝑡)^−𝛼𝑞𝛼𝑞^1/𝑞+1𝛽𝑞^1/𝑞𝑓^′_𝑝,𝐷≤1𝛼𝑞^1/𝑞+1𝛽𝑞^1/𝑞𝑓^′_𝑝,𝐷, (3.56) where 1/𝑝+1/𝑞=1. We conclude that ℒ^𝑝_𝐷 is also admissible for 𝑇.

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China under Grant no. 11171090 and the Fundamental Research Funds for the Central Universities. The authors would like to show their great thanks to Professor Jifeng Chu for his useful suggestions and comments.