Admissibility and general dichotomies for evolution families

Admissibility and general dichotomies for evolution families

Davor Dragiˇcevi´c

, Nevena Jurˇcevi´c Peˇcek

and Nicolae Lupa

1Department of Mathematics, University of Rijeka, Radmile Matejˇci´c 2, Rijeka, 51000, Croatia

2Department of Mathematics, Politehnica University of Timis,oara, Piat,a Victoriei 2, Timis,oara, 300006, Romania

Received 9 March 2020, appeared 21 October 2020 Communicated by Christian Pötzsche

Abstract. For an arbitrary noninvertible evolution family on the half-line and for ρ: [0,∞) → [0,∞) in a large class of rate functions, we consider the notion of a ρ- dichotomy with respect to a family of norms and characterize it in terms of two ad- missibility conditions. In particular, our results are applicable to exponential as well as polynomial dichotomies with respect to a family of norms. As a nontrivial application of our work, we establish the robustness of general nonuniform dichotomies.

Keywords: admissibility, dichotomies with growth rates, robustness.

2020 Mathematics Subject Classification: 34D09.

1 Introduction

Among many methods used to study the asymptotic behavior of nonautonomous systems, one of the most famous is the so-called admissibility method. This line of research in the context of differential equations has a long history that goes back to the pioneering work of Perron [26]. The main idea of Perron’s work was to characterize the asymptotic properties of the linear differential equation

x˙(t) =A(t)x(t), t∈_J, in terms of the (unique) solvability inO(_J,X)of the equation

˙

x(t) = A(t)x(t) + f(t), t ∈_J,

for each test function f ∈ I(_J,X), where J ∈ {[0,∞),R}. Here X is a Banach space, while I(_J,X) – the input-space and O(_J,X) – the output space are suitably constructed function spaces. The milestones of this theory were grounded in the sixtieth in the remarkable works of Massera and Schäffer [18–20] and respectively in the seventies in the outstanding monographs of Coppel [10] and Daleck˘ıi and Kre˘ın [11].

BCorresponding author. Email: ddragicevic@math.uniri.hr

Since then various authors obtained valuable contributions to this line of the research.

For the results dealing with characterizations of uniform exponential behavior in terms of appropriate admissibility properties, we refer to the works of Huy [15], Latushkin, Randolph and Schnaubelt [16], Van Minh, Räbiger and Schnaubelt [22], Van Minh and Huy [23], Preda, Pogan and Preda [28,29] as well as Sasu and Sasu [31–35]. For contributions dealing with various concepts of nonuniform exponential behavior, we refer to [4,5,17,21,27,30,36] and references therein. For a detailed description of this line of the research, we refer to [6].

We point out that all the above works deal withexponential behavior. Although this type of behavior has a somewhat privileged role due to its connections with the hyperbolic smooth dynamics, it is certainly not the only type of behavior that appears in the qualitative study of nonautonomous differential equations. To the best of our knowledge, the study of di- chotomies with not necessarily exponential rates of expansion and contraction was initiated by Muldowney [24] and Naulin and Pinto [25]. More recently, in the context of nonuniform asymptotic behavior such dichotomies have been studied by Barreira and Valls [1,3] and Bento and Silva [8,9]. A special emphasis was devoted to the so-called polynomial dichotomies [2,7].

A complete characterization of polynomial dichotomies in terms of admissibility for evolution families was obtained by Dragiˇcevi´c [12] (see also [13] for related results in the case of discrete time) by building on the work of Hai [14], who considered polynomial stability.

The main objective of the present paper is to obtain similar results to that in [12] but for a much wider class of dichotomies. More precisely, for a large class of rate functions ρ: [0,∞) → [0,∞), we introduce the notion of a ρ-dichotomy with respect to a family of norms. We then obtain a complete characterization of this concept in terms of appropriate admissibility conditions. We point out that our results are new even in the particular case of uniformρ-dichotomies. Indeed, although the proofs use the somewhat standard techniques, the major task accomplished in the present paper was to formulate appropriate admissibility conditions for the general dichotomies we study. In addition, the obtained results are new even for the class of polynomial dichotomies since in comparison to [12], we do not require that our evolution family exhibits polynomial bounded growth property. Consequently, we need to impose two admissibility conditions (rather than just one as in [12]) to characterize polynomial dichotomies. We stress that in the present paper we also use different admissibility spaces from those in [12].

The paper is organized as follows. In Section2 we introduce the class of dichotomies we study as well as input and output spaces we are going to use. In Section3, we show that the existence ofρ-dichotomies yields two types of admissibility properties. Then, in Section4we obtain a converse result by showing that those admissibility properties imply the existence of aρ-dichotomy. Finally, in Section 5 we apply those results to establish the robustness of ρ-dichotomies.

2 Preliminaries

2.1 Generalized dichotomies

Let X = (X,k·k) be an arbitrary Banach space and let B(X) be the Banach algebra of all bounded linear operators onX. A family T = {T(t,s)}_t≥s≥0 of operators in B(X)is said to be anevolution familyonXif the following properties hold:

• T(t,t) =_{Id, for}t≥_0;

• T(t,s)T(s,τ) =T(t,τ), fort ≥s≥τ≥0;

• for all s ≥ 0 and x ∈ X the mapping t 7→ T(t,s)x is continuous on [s,∞) and the mappingt7→ T(s,t)x is continuous on[0,s].

In this paper we always assume thatT = {T(t,s)}_t≥s≥0 is an evolution family on X and letρ: [0,∞)→[0,∞)be a strictly increasing function of classC¹such that

ρ(0) =0 and lim

t→_∞ρ(t) =_∞.

In particular, observe that ρ is bijective. Furthermore, assume that {k·k_t}_t≥0 is a family of norms onXsuch that:

• there existC>0 andε≥0 with

kxk ≤ kxk_t ≤Ce^ερ(t)kxk, forx ∈Xandt≥0; (2.1)

• the mappingt 7→ kxk_t is continuous for eachx ∈X.

We say that the evolution family T admits a ρ-dichotomy with respect to the family of normsk·k_t,t≥0, if there exists a family{P(t)}_t≥0 of projections onXsatisfying

T(t,s)P(s) =P(t)T(t,s), fort≥s ≥0, (2.2) such that

T(t,s)|_KerP(s): KerP(s)→KerP(t)is invertible for all t≥s ≥0, (2.3) and there exist constantsλ,D>0 such that:

• forx ∈Xandt≥ s≥0,

kT(t,s)P(s)xk_t≤ De^{−λ(ρ(t)−ρ(s))}kxk_s; (2.4)

• forx ∈Xand 0≤t≤ s,

kT(t,s)(Id−P(s))xk_t ≤ De^{−λ(ρ(s)−ρ(t))}kxk_s, (2.5) where

T(t,s):=

T(s,t)|_KerP(t) −1

: KerP(s)→KerP(t), for 0≤t≤ s.

In the following we recall the concept ofρ-nonuniform exponential dichotomy for evolution families (see [1,3]) and establish its connection with the notion ofρ-dichotomy with respect to a family of norms. An evolution familyT is said to admit aρ-nonuniform exponential dichotomy if there exists a family {P(t)}_t≥0of projections on X satisfying (2.2) and (2.3), and there exist constantsλ,D>0 andε≥0 such that

kT(t,s)P(s)k ≤De^{−λ(ρ(t)−ρ(s))+ερ(s)}, fort≥ s≥0, (2.6) and

kT(t,s)(_Id−P(s))k ≤De^{−λ(ρ(s)−ρ(t))+ερ(s)}, for 0≤ t≤s. (2.7)

The concept ofρ-nonuniform exponential dichotomy includes as a special case the usual exponential behaviorwhenρ(t) = t. Also, forρ(t) =ln(t+1)we obtain the concept ofnonuni- form polynomial dichotomyintroduced independently by Barreira and Valls [2] and Bento and Silva [7], and more general for ρ(t) = Rt

0 µ(t)dt, where µ : [0,∞) → (0,∞) is a continuous function such that limt→_∞Rt

0µ(t)dt= ∞, we obtain the nonuniform version of thegeneralized dichotomyin the sense of Muldowney [24].

Proposition 2.1. The following statements are equivalent:

1. T admits aρ-nonuniform exponential dichotomy;

2. T admits a ρ-dichotomy with respect to a family of normsk · k_t, t ≥ ₀ such that t 7→ kxk_t is continuous for each x∈X.

Proof. Assume that T admits a ρ-nonuniform exponential dichotomy. For t ≥ 0 and x ∈ X, set

kxk_t =sup

τ≥t

e^{λ(ρ(τ)−ρ(t))}kT(τ,t)P(t)xk+ sup

τ∈[0,t]

e^{λ(ρ(t)−ρ(τ))}kT(τ,t)(Id−P(t))xk.

A simple computation shows that (2.1) holds for C = 2D. Moreover, by repeating the argu- ments in the proof of [6, Proposition 5.6], one can show thatt 7→ kxk_t is continuous for each x∈ X. Furthermore, fort ≥s≥0 andx ∈Xwe have

kT(t,s)P(s)xk_t =sup

τ≥t

e^{λ(ρ(τ)−ρ(t))}kT(τ,s)P(s)xk

=_sup

τ≥t

e^{−λ(ρ(t)−ρ(s))}e^{λ(ρ(τ)−ρ(s))}kT(_τ,s)P(s)xk

≤e^{−λ(ρ(t)−ρ(s))}sup

τ≥s

e^{λ(ρ(τ)−ρ(s))}kT(τ,s)P(s)xk

≤e^{−λ(ρ(t)−ρ(s))}kxk_s,

and thus (2.4) holds. Similarly, one can prove (2.5). Hence, the evolution familyT admits a ρ-dichotomy with respect to the family of norms k · k_t,t≥0, defined above.

Conversely, assume that T admits a ρ-dichotomy with respect to a family of normsk · k_t onXsatisfying (2.1) for someC>0 andε≥0. Fort ≥s≥0 andx ∈Xwe have

kT(t,s)P(s)xk ≤ kT(t,s)P(s)xk_t

≤De^{−λ(ρ(t)−ρ(s))}kxk_s

≤DCe^ερ(s)e^{−λ(ρ(t)−ρ(s))}kxk,

and thus (2.6) holds. Similarly, one can show (2.7). Therefore, the evolution familyT admits aρ-nonuniform exponential dichotomy.

2.2 Admissible spaces

LetY₁ be the space of all Bochner measurable functionsx: [0,∞)→X such that kxk₁:=

Z _∞

0

kx(t)k_tdt<_∞,

identifying functions that are equal Lebesque-almost everywhere. It is easy to show that (Y₁,k·k₁) is a Banach space (see [4, Theorem 1]). Moreover, consider the space Y_∞ of all continuous functions x: [0,∞)→ Xsuch that

kxk_∞ :=sup

t≥0

kx(t)k_t <_∞.

One can easily prove that (Y_∞,k·k_∞)is a Banach space. For a closed subspace Z ⊂ X, Y_∞^Z is the space of allx∈Y_∞such thatx(0)∈Z. Obviously,Y_∞^Zis a closed subspace ofY_∞, therefore it is also a Banach space.

We consider another Banach function space(Y_∞⁰ ,k·k⁰_∞), which consists of all Bochner mea- surable functions x: [0,∞)→Xsuch that

kxk⁰_∞ :=ess sup

t≥0

kx(t)k_t <_∞,

where ess sup is taken with respect to the Lebesgue measure on[0,∞).

3 From dichotomy to admissibility

In this section we show that the existence of aρ-dichotomy with respect to a family of norms for an evolution family T = {T(t,s)}_t≥s≥0 yields the admissibility of the pairs Y_∞^Z,Y₁

, Y_∞^Z,Y_∞⁰

for a certain closed subspace Z⊂X.

Proposition 3.1. Assume that the evolution family T admits aρ-dichotomy with respect to a family of normsk·k_t, t ≥0, and set Z= KerP(0). Then, for each y∈Y₁there exists a unique x ∈Y_∞^Zsuch that

x(t) =T(t,s)x(s) +

Z _t

s T(t,τ)y(τ)dτ, for t≥s ≥0. (3.1) Proof. Take an arbitraryy∈Y1. For t≥0, set

x(t) =

Z _t

0 T(t,s)P(s)y(s)ds−

Z _∞

t T(t,s)(Id−P(s))y(s)ds.

It follows from (2.4) and (2.5) that kx(t)k_t ≤

Z _t

0

kT(t,s)P(s)y(s)k_tds+

Z _∞

t

kT(t,s)(Id−P(s))y(s)k_tds

≤D Z _t

0 e^{−λ(ρ(t)−ρ(s))}ky(s)k_sds+D Z _∞

t e^{−λ(ρ(s)−ρ(t))}ky(s)k_sds

≤D Z _t

0

ky(s)k_sds+D Z _∞

t

ky(s)k_sds=Dkyk₁,

for every t ≥ 0, and thus x ∈ Y_∞. On the other hand, it is easy to check that x(0) ∈ Z.

Therefore, x∈Y_∞^Z. Moreover, fort ≥s≥0 we have x(t)−T(t,s)x(s) =

Z _t

0 T(t,τ)P(τ)y(τ)dτ−T(t,s)

Z _s

0 T(s,τ)P(τ)y(τ)dτ

−

Z _∞

t T(t,τ)(Id−P(τ))y(τ)dτ +T(t,s)

Z _∞

s T(s,τ)(Id−P(τ))y(τ)dτ

=

Z _t

s T(t,τ)P(τ)y(τ)dτ+

Z _t

s T(t,τ)(Id−P(τ))y(τ)dτ

=

Z _t

s T(t,τ)y(τ)dτ,

and therefore we conclude that (3.1) holds. In order to establish the uniqueness, it is sufficient to consider the case wheny=0. Let x∈Y_∞^Z such that

x(t) =T(t,s)x(s), fort ≥s≥0.

Then, from (2.5) we have

kx(0)k₀ =k(Id−P(0))x(0)k₀ =kT(0,t)(Id−P(t))x(t)k₀

≤De^−λρ(t)kx(t)k_t

≤De^−λρ(t)kxk_∞,

for everyt ≥ 0. Passing to the limit when t → ∞, we conclude that x(0) =0, which implies thatx =0.

Proposition 3.2. Assume that the evolution familyT admits a ρ-dichotomy with respect to a family of normsk·k_t, t≥0, and set Z=KerP(0). Then, for each y∈Y_∞⁰ there exists a unique x∈Y_∞^Z such that

x(t) =T(t,s)x(s) +

Z _t

s ρ⁰(τ)T(t,τ)y(τ)dτ, for t ≥s≥0. (3.2) Proof. Takey∈Y_∞⁰ . Fort ≥0, set

x(t) =

Z _t

0 ρ⁰(s)T(t,s)P(s)y(s)ds−

Z _∞

t ρ⁰(s)T(t,s)(Id−P(s))y(s)ds.

It follows from (2.4) and (2.5) that kx(t)k_t≤

Z _t

0 ρ⁰(s)kT(t,s)P(s)y(s)k_tds+

Z _∞

t ρ⁰(s)kT(t,s)(Id−P(s))y(s)k_tds

≤ D Z _t

0 ρ⁰(s)e^{−λ(ρ(t)−ρ(s))}ky(s)k_sds+D Z _∞

t ρ⁰(s)e^{−λ(ρ(s)−ρ(t))}ky(s)k_sds

≤ Dkyk⁰_{∞ Z t}

0 ρ⁰(s)e^{−λ(ρ(t)−ρ(s))}ds+

Z _∞

t ρ⁰(s)e^{−λ(ρ(s)−ρ(t))}ds

≤ ^2D

λ kyk⁰_∞, for every t≥0.

Sincex(₀)∈ Z, we conclude that x ∈ Y_∞^Z. A simple computation shows that (3.2) holds. The uniqueness part can be established as in the proof of Proposition3.1.

4 From admissibility to dichotomy

The aim of this section is to prove that the admissibility of the pairs Y_∞^Z,Y₁

, Y_∞^Z,Y_∞⁰ for a closed subspaceZ ⊂ X yields the existence of aρ-dichotomy with respect to the family of norms{k·k_t}_t≥0. More precisely, our goal is to establish the following result.

Theorem 4.1. Assume that there exists a closed subspace Z⊂ X such that:

(i) for each y∈Y₁ there exists a unique x∈Y_∞^Z satisfying(3.1);

(ii) for each y ∈Y_∞⁰ there exists a unique x∈Y_∞^Z satisfying(3.2).

Then, the evolution familyT admits aρ-dichotomy with respect to the family of normsk·k_t, t≥0.

Proof. Let

T_Z :D(T_Z)⊂Y_∞^Z →Y₁, T_Zx =y, where

D(T_Z) =ⁿx∈Y_∞^Z : there existsy∈Y₁satisfying (3.1)o . Furthermore, let

T_Z⁰ :D(T_Z⁰)⊂Y_∞^Z →Y_∞⁰ , T_Z⁰x=y, where

D(T_Z⁰) =ⁿx∈Y_∞^Z: there exists y∈Y_∞⁰ satisfying (3.2) o

.

Lemma 4.2. The operators T_Z: D(T_Z)→Y₁, T_Z⁰ : D(T_Z⁰)→Y_∞⁰ are well-defined, linear and closed.

Proof of the lemma. Assume thatx∈Y_∞^Z andy₁,y2 ∈Y₁ such that x(t) =T(t,τ)x(τ) +

Z _t

τ

T(t,s)y_i(s)ds, fort ≥τ≥0 andi∈ {1, 2}. Hence,

Z _t

τ

T(t,s)(y₁(s)−y₂(s))ds=0, fort> τ≥0.

Dividing by t−τand letting t−τ→0, it follows from the Lebesgue differentiation theorem that

y₁(t) =y₂(t) for almost everyt≥_0.

We conclude thaty₁=y₂inY₁. Thus,T_Z is well-defined and, by definition it is linear.

We now show thatT_Zis closed. Let(x_n)_n∈Nbe a sequence inD(T_Z)converging tox ∈Y_∞^Z such thatyn=TZxnconverges toy∈Y₁. Then, fort ≥τ≥0 we have that

x(t)−T(t,τ)x(τ) = lim

n→_∞(x_n(t)−T(t,τ)x_n(τ)) = lim

n→_∞ Z _t

τ

T(t,s)y_n(s)ds.

On the other hand, we have

Z _t

τ

T(t,s)yn(s)ds−

Z _t

τ

T(t,s)y(s)ds

≤ M Z _t

τ

kyn(s)−y(s)kds

≤ M Z _t

τ

kyn(_s)−y(_s)k_sds

≤ Mky_n−yk_1,

where M = M(t,τ) = sup{kT(t,s)k : s ∈ [τ,t]}is finite by the Banach–Steinhaus theorem.

Sincey_n→yinY₁, we conclude that

nlim→_∞ Z _t

τ

T(t,s)yn(s)ds=

Z _t

τ

T(t,s)y(s)ds,

and therefore (3.1) holds. We conclude that x∈ D(T_Z)andT_Zx= y. Therefore,T_Zis a closed linear operator. Similarly, one can show that T_Z⁰ is well-defined, linear and closed.

By the assumption in Theorem 4.1, the linear operators T_Z, T_Z⁰ are bijective, and by pre- vious lemma and the Closed Graph Theorem they have bounded inverse GZ: Y₁ → Y_∞^Z and G⁰_Z: Y_∞⁰ →Y_∞^Z, respectively.

Forτ≥0, set S(τ) =

v∈ X: sup

t≥τ

kT(t,τ)vk_t <_∞

and U(τ) =T(τ, 0)Z.

Clearly,S(τ)andU(τ)are subspaces ofX for eachτ≥0.

Lemma 4.3. Forτ≥0, we have that

X =S(τ)⊕U(τ). (4.1)

Proof of the lemma. Letτ≥0 and takev∈ X. Set

g(s) =χ_[τ,τ+1](s)T(s,τ)v, s ≥0.

Clearly, g ∈ Y₁. Since TZ is invertible, there exists h ∈ D(TZ) ⊂ Y_∞^Z such that TZh = g. It follows from (3.1) that

h(t) =T(t,τ)(h(τ) +v) fort ≥τ+1.

Sinceh∈Y_∞, we conclude thath(τ) +v∈S(τ). Similarly, it follows from (3.1) that h(τ) =T(_{τ, 0})h(₀)_.

Sinceh(0)∈ Z, we have thath(τ)∈U(τ)and thus

v= (h(τ) +v) + (−h(τ))∈S(τ) +U(τ). We have proved thatX=S(τ) +U(τ).

Take nowv ∈S(τ)∩U(τ). Then, there existsz ∈Z such thatv=T(τ, 0)z. We consider a functionh: [0,∞)→X, defined by

h(t) =T(t, 0)z fort≥0.

Clearly, h ∈ Y_∞^Z. Since h(t) = T(t,s)h(s)for all t ≥ s ≥ 0, it follows that T_Zh = 0 and thus h = 0. We conclude that v = h(τ) = 0, and hence S(τ)∩U(τ) = {0}. This completes the proof of the lemma.

Let P(τ): X → S(τ)and Q(τ): X →U(τ)be the projections associated with the decom- position (4.1), withP(τ) +Q(τ) =Id. Observe that (2.2) holds. Indeed, observe that

T(t,τ)S(τ)⊂ S(t) and T(t,τ)U(τ)⊂U(t), fort ≥τ≥0.

Hence, we have that for everyx ∈Xandt≥τ≥0,

P(t)T(t,τ)x =P(t)T(t,τ)P(τ)x+P(t)T(t,τ)Q(τ)x=T(t,τ)P(τ)x.

We conclude that (2.2) holds.

Lemma 4.4. For t≥τ≥0, the restriction T(t,τ)|_U(τ): U(τ)→U(t)is invertible.

Proof of the lemma. Let t ≥ τ ≥ 0 and take x ∈ U(t). Then, there exists z ∈ Z such that x = T(t, 0)z. Since T(τ, 0)z ∈ U(τ) and x = T(t,τ)T(τ, 0)z, we conclude that T(t,τ)|_U(τ) is surjective.

Let now x ∈ U(τ) such thatT(t,τ)x = 0. Take z ∈ Z such that x = T(τ, 0)z. We define u: [0,∞)→ X byu(s) = T(s, 0)z, s ≥ 0. Since u(s) = 0 for s ≥ t, we have that u ∈ Y_∞^Z and T_Zu=0. Consequently,u=0 andx= u(τ) =0. This proves that T(t,τ)|_U(τ) is also injective.

The proof of the lemma is completed.

Lemma 4.5. There exists M>0such that

kP(τ)vk_τ ≤ Mkvk_τ, for all v∈X and τ≥0. (4.2) Proof of the lemma. Take v ∈ X and τ ≥ 0 . Moreover, given h > 0, we define a function g_h: [0,∞)→X by

g_h(t) = ¹

hχ_[τ,τ+h](t)T(t,τ)v.

Clearly, g_h ∈Y₁ and thus there existsx_h ∈ D(T_Z)such thatT_Zx_h =g_h. We have kP(τ)vk_τ =kx_h(τ) +vk_τ ≤ kx_h(τ)k_τ+kvk_τ ≤ kG_Zg_hk_∞+kvk_τ. Moreover,

kG_Zg_hk_∞ ≤ kG_Zk · kg_hk₁=kG_Zk¹ h

Z _τ+h τ

kT(t,τ)vk_tdt.

Lettingh→0, we obtain

kP(τ)vk_τ ≤(1+kG_Zk)kvk_τ, and we conclude that (4.2) holds forM =₁+kG_Zk_.

Lemma 4.6. There exist constantsλ,D>0such that

kT(t,τ)vk_t ≤ De^{−λ(ρ(t)−ρ(τ))}kvk_τ, for t ≥τ≥0and v∈S(τ). (4.3) Proof of the lemma. Fixτ≥0 and letv∈ S(τ). We consider the function

u: [_0,∞)→ X, u(_t) =χ_[τ,∞)(_t)_T(_t,τ)_v.

Moreover, for any fixedh >0, we define two functionsϕ_h: [0,∞)→_Randg_h: [0,∞)→Xby

ϕ_h(t) =











0, 0≤t≤τ,

1

h(t−τ)_, τ≤t≤ τ+h, 1, t≥ τ+h, and

g_h(t) = ¹

hχ_[τ,τ+h](t)T(t,τ)v, t≥0.

It is easy to show thatg_h∈Y₁, ϕ_hu∈ D(T_Z)andT_Z(ϕ_hu) =g_h. We have sup

t≥τ+h

ku(t)k_t = sup

t≥τ+h

kϕ_h(t)u(t)k_t ≤ kϕ_huk_∞= kGZg_hk_∞

≤ kG_Zk · kg_hk₁

=kG_Zk¹ h

Z _τ+h

τ

ku(s)k_sds.

Hence, lettingh→0 we obtain the inequality

ku(t)k_t≤ kG_Zk · kvk_τ, for every t≥τ.

Thus,

kT(t,τ)vk_t≤ kG_Zk · kvk_τ, for every t≥τ. (4.4) Let us take t ≥ τand v ∈ S(τ)_{such that} T(t,τ)v 6= _{0, thus}T(s,τ)v 6= _{0 for all}s ∈ [_τ,t]_. Let us considerx,y: [0,∞)→Xdefined by

y(s) =χ_[τ,t](s) ^T(s,τ)v

kT(s,τ)vk_s^{, s}≥0, and

x(s) =











0, 0≤s≤τ,

Rs

τ ρ⁰(r)_k^T(s,τ)v

T(r,τ)vk_r dr, τ<s≤t, Rt

τρ⁰(r)_kT^T₍⁽_r,τ^s,τ₎⁾_v^v_k

r dr, s>t.

Note thaty ∈Y_∞⁰ andkyk⁰_∞ =1. Furthermore, sincev∈S(τ)we get that kx(s)k_s≤

Z _t

τ

ρ⁰(r)

kT(r,τ)vk_r^drkT(s,τ)vk_s≤a_t,τ,v sup

r≥τ

kT(r,τ)vk_r <_∞, for alls≥τ, where

a_t,τ,v=

Z _t

τ

ρ⁰(r)

kT(r,τ)vk_r ^dr<_∞,

and thusx∈Y_∞^Z. It is straightforward to show thatT_Z⁰x= y. Consequently, kxk_∞ =kG_Z⁰ yk_∞ ≤ kG_Z⁰k · kyk⁰_∞ =kG_Z⁰k_.

Therefore,

kG_Z⁰k ≥ kxk_∞ ≥ kx(t)k_t= kT(t,τ)vk_t

Z _t

τ

ρ⁰(r)

kT(_r,τ)_vk_r^{dr. (4.5)} From (4.4) it follows that

1

kT(r,τ)vk_r ≥ ¹

kGZk · kvk_τ^{, for allr}∈ [τ,t], and thus, from (4.5) we get

kG⁰_Zk · kG_Zk · kvk_τ ≥ kT(t,τ)vk_t(ρ(t)−ρ(τ)), for t≥τandv∈S(τ). Consequently,

(t−τ)T

ρ⁻¹(t),ρ⁻¹(τ)v

ρ⁻¹(t)≤ kG⁰_Zk · kG_Zk · kvk_ρ−1(τ),

fort ≥τ andv∈ S ρ⁻¹(τ). Let N₀ ∈ _N^∗ such that N₀ > ekG⁰_Zk · kG_Zk, and lett ≥ τ+N₀. Then,

N0

T

ρ⁻¹(t),ρ⁻¹(τ)v

ρ⁻¹(t)≤(t−τ) T

ρ⁻¹(t),ρ⁻¹(τ)v ρ⁻¹(t)

≤ kG⁰_Zk · kG_Zk · kvk_ρ−1(τ),

which implies that there existsN₀∈_N^∗ such that

kT(ρ⁻¹(t),ρ⁻¹(τ))vk_ρ−1(t) ≤ ¹

ekvk_ρ−1(τ), (4.6) for t ≥ τwith t−τ ≥ N₀ andv ∈ S ρ⁻¹(τ). Take an arbitraryt ≥ τwith t−τ ≥ N₀ and writet−τin the form

t−τ=kN₀+r, k=k(t,τ)∈_N^∗ and r=r(t,τ)∈[0,N₀). Observing that

T

ρ⁻¹(t),ρ⁻¹(τ)

=T

ρ⁻¹(t),ρ⁻¹(τ+kN₀)

k−1

∏

T

ρ⁻¹(τ+ (k−j)N₀),ρ⁻¹(τ+ (k−j−1)N₀), it follows from (4.4) and (4.6) that

kT

ρ⁻¹(_t)_,ρ⁻¹(τ)_vk_ρ−1(t) ≤ kGZke^−kkvk_ρ−1(τ)

≤ekG_Zke^{−N10(t−τ)}kvk_ρ−1(τ),

and thus (4.3) holds withλ=1/N₀andD= ekG_Zk. The proof of the lemma is completed.

Lemma 4.7. There existλ,D>₀such that

kT(t,τ)vk_t ≤ De^{−λ(ρ(τ)−ρ(t))}kvk_τ, for0≤t ≤τand v∈U(τ). (4.7) Proof of the lemma. Takeτ>0 andz∈ Z. We define a functionu: [0,∞)→ Xby

u(t) =T(t, 0)z, fort ≥0.

For sufficiently smallh>0, we defineψ_h: [0,∞)→_R,

ψ_h(t) =











1, 0≤t ≤τ−h,

−^t−_h^τ, τ−h≤t ≤τ, 0, t ≥τ.

Finally, we consider

g_h: [0,∞)→X, g_h =−¹

hχ_[τ−h,τ]u.

It is easy to check that g_h ∈Y₁,ψ_hu∈ D(T_Z)andT_Z(ψ_hu) =g_h. Hence, sup

t∈[0,τ−h]

ku(t)k_t = _sup

t∈[0,τ−h]

kψ_h(t)u(t)k_t≤ kψ_huk_∞ =kG_Zg_hk_∞

≤ kG_Zk · kg_hk₁

=kG_Zk · ¹ h

Z _τ

τ−h

ku(s)k_sds.

Lettingh→0, we get

ku(t)k_t ≤ kG_Zk · ku(τ)k_{τ, for 0}≤ t≤_τ,

which implies

kT(t, 0)zk_t ≤ kG_Zk · kT(τ, 0)zk_τ, forz ∈Zand 0≤t≤τ. (4.8) Take nowz∈Z\ {0}and 0≤t ≤τ. We definex,y: [0,∞)→Xby

y(s) =

(−_k^T(s,0)z

T(s,0)zk_s, 0≤s ≤τ, 0, s >τ, and

x(s) = (Rτ

s ρ⁰(r)_kT^T₍⁽_r,0^s,0₎⁾_z^z_k

r dr, 0≤s≤τ,

0, s>τ.

Observe thaty ∈ Y_∞⁰ andkyk⁰_∞ = 1. Moreover, x ∈ Y_∞^Z and it is easy to check that T_Z⁰x = y.

Hence,

kxk_∞ =kG⁰_Zyk_∞≤ kG⁰_Zk. Consequently, for each 0≤s ≤τwe have

kG⁰_Zk ≥ kT(s, 0)zk_s

Z _τ

s ρ⁰(r) ¹

kT(r, 0)zk_r^dr.

Lettingτ→∞, we conclude that kG⁰_Zk ≥ kT(s, 0)zk_s

Z _∞

s ρ⁰(r) ¹

kT(r, 0)zk_r^{dr fors} ≥0 andz ∈Z\ {0}. (4.9) Take now 0≤ t≤τandz∈ Z\ {0}. It follows from (4.8) and (4.9) that

1

kT(ρ⁻¹(t), 0)zk_ρ−1(t)

≥ ¹ kG⁰_Zk

Z _∞

ρ⁻¹(t)ρ⁰(r) ¹

kT(r, 0)zk_r^dr

≥ ¹ kG⁰_Zk

Z _ρ⁻¹(τ)

ρ⁻¹(t) ρ⁰(r) ¹

kT(r, 0)zk_r^dr

≥ ¹ kG⁰_Zk

Z _ρ⁻¹(τ)

ρ⁻¹(t) ρ⁰(r) ¹

kG_Zk · kT(ρ⁻¹(τ), 0)zk_ρ−1(τ)

dr

= ^τ−t

kG⁰_Zk · kG_Zk· ¹

kT(ρ⁻¹(τ)_{, 0})zk_ρ−1(τ)

and thus

(τ−t)kT(ρ⁻¹(t), 0)zk_ρ−1(t)≤ kG_Zk · kG⁰_Zk · kT(ρ⁻¹(τ), 0)zk_ρ−1(τ). We conclude that there existsN0∈ _N^∗ such that

kT(ρ⁻¹(t), 0)zk_ρ−1(t)≤ ¹

ekT(ρ⁻¹(τ), 0)zk_ρ−1(τ), forz ∈Zand 0≤t≤τsuch thatτ−t≥ N₀. Hence,

kT(ρ⁻¹(t),ρ⁻¹(τ))vk_ρ−1(t) ≤ ¹

ekvk_ρ−1(τ),

for v ∈ U(ρ⁻¹(τ)) and 0 ≤ t ≤ τ such that τ−t ≥ N0. By arguing as in the proof of Lemma4.6, we find that there existλ,D>0 such that

kT(ρ⁻¹(t),ρ⁻¹(τ))vk_ρ−1(t) ≤De^{−λ(τ−t)}kvk_ρ−1(τ),

forv ∈U(ρ⁻¹(τ))_{and 0}≤t ≤τ, which readily implies the conclusion of the lemma.

In order to complete the proof of the theorem, it is sufficient to observe that (4.2), (4.3) and (4.7) imply that (2.4) and (2.5) hold.

Remark 4.8. It is worth observing that in order to deduce the existence of a ρ-dichotomy we imposed two admissibility conditions. In the following two examples we will illustrate that this was necessary.

Example 4.9. We consider an evolution familyT = {T(t,s)}_t≥s≥0given by T(t,s) =Id, t≥s ≥0.

Furthermore, take Z = {0} and let k·k_t = k·k for t ≥ 0. Then for each y ∈ Y₁, the unique x∈Y_Z satisfying (3.1) is given by

x(t) =

Z _t

0

T(t,s)y(s)ds=

Z _t

0

y(s)ds, t≥_0.

Thus, the first assumption of Theorem4.1is fulfilled. On the other hand,T obviously doesn’t admit aρ-dichotomy with respect to the family of norms k·k_t,t≥0.

The following example is a simple modification of [12, Example 1].

Example 4.10. LetX=_Rwith the standard Euclidean norm|·|. Furthermore, letk·k_t=|·|for t≥0 and takeZ={0}. Furthermore, letρ(t) =ln(1+t)fort≥0. We consider the sequence (A_n)_n∈Nof operators onX(which can of course be identified with numbers) given by

An =

(n ifn=2^l for somel∈_N, 0 otherwise.

Furthermore, fort ≥s≥0 we define T(t,s) =

(A_btc−1· · ·A_bsc, btc ≥ bsc+1, 1, btc=bsc.

Then, T = {T(t,s)}_t≥s≥0 is an evolution family. By arguing as in [12, Example 1], it is easy to check that the second assumption of Theorem 4.1 is satisfied and T doesn’t admit a ρ- dichotomy with respect to the family of normsk·k_t,t ≥0.

5 Robustness of generalized dichotomies

In this section we apply our main results to prove that the concept ofρ-dichotomy with respect to a family{k·k_t}_t≥0of norms on Xpersist under sufficiently small linear perturbations. As a consequence, we establish the robustness property ofρ-nonuniform exponential dichotomy.

Theorem 5.1. Assume that the evolution family{T(t,s)}_t≥s≥0admits aρ-dichotomy with respect to a family{k·k_t}_t≥0of norms on X satisfying

kxk ≤ kxk_t≤Ce^ερ(t)kxk, for x∈ X and t≥0,

for some C > 0 and ε ≥ 0, such that the mapping t 7→ kxk_t is continuous for each x ∈ X. If B:[0,∞)→ B(X)is a strongly continuous operator-valued function such that

kB(t)k ≤δe^{−(ε+a)ρ(t)}ρ⁰(t)_, t≥_{0, (5.1)}

for some a > 0 and sufficiently small δ > 0, then the perturbed evolution family {U(t,s)}_t≥s≥0 satisfying

U(t,s) =T(t,s) +

Z _t

s T(t,τ)B(τ)U(τ,s)dτ, t ≥s≥0, (5.2) admits aρ-dichotomy with respect to the family of normsk·k_t, t≥0.

Proof. Since {T(t,s)}_{t≥s≥0 admits a} ρ-dichotomy with respect to the family of norms k·k_t, t ≥ 0, it follows from Proposition 3.1 and Proposition3.2 that there exists a closed subspace Z⊂ Xsuch that the operators

T_Z: D(T_Z)⊂Y_∞^Z →Y₁ and T_Z⁰ : D(T_Z⁰)⊂Y_∞^Z →Y_∞⁰ ,

defined in the proof of Theorem4.1, are invertible and closed. We consider the graph norms:

kxk_TZ :=kxk_∞+kTZxk₁, x∈ D(TZ), and

kxk_T0

Z :=kxk_∞+kT_Z⁰xk⁰_∞, x∈ D(T_Z⁰). SinceTZ, T_Z⁰ are closed, it follows that(D(TZ),k · k_TZ), D(T_Z⁰),k · k_T⁰

Z

are Banach spaces.

Furthermore,

T_Z:(D(T_Z),k · k_TZ)→(Y₁,k · k₁) and

T_Z⁰ :

D(T_Z⁰),k · k_T⁰

Z

→ Y_∞⁰ ,k · k⁰_∞

are bounded linear operators, denoted simply byT_Z andT_Z⁰, respectively.

We consider the linear operatorsD:D(T_Z)→Y₁, D⁰ :D(T_Z⁰)→Y_∞⁰ defined by (Dx)(t) = B(t)x(t)and(D⁰x)(t) = ¹

ρ⁰(t)^B(t)x(t), fort ≥0.

One can easy check that these operators are well-defined. Furthermore, for each x ∈ D(T_Z) we have

kDxk₁ =

Z _∞

0

kB(t)x(t)k_tdt

≤C Z _∞

0 e^ερ(t)kB(t)x(t)kdt

≤δC Z _∞

0

e^−aρ(t)ρ⁰(t)kx(t)kdt

≤ ^δC a kxk_∞, and thus

kDxk₁≤ ^δC

a kxk_TZ, x∈ D(T_Z). (5.3) On the other hand, forx ∈ D(T_Z⁰)we get

k(D⁰x)(t)k_t = ¹

ρ⁰(t)kB(t)x(t)k_t

≤ ¹ ρ⁰(t)^Ce

ερ(t)kB(t)x(t)k

≤δCe^−aρ(t)kx(t)k

≤δCkxk_T⁰

Z,

for all t≥0, hence

kD⁰xk⁰_∞ ≤δCkxk_T⁰

Z, x∈ D(T_Z⁰). (5.4)

We define now the linear operators

U_Z :D(U_Z)→Y₁, U_Zx =y,

whereD(U_Z)is the set of all functionsx∈Y_∞^Z such that there existsy ∈Y₁ satisfying x(t) =U(t,s)x(s) +

Z _t

s U(t,τ)y(τ)dτ, fort≥s ≥0, and respectively,

U_Z⁰ :D(U_Z⁰ )→Y_∞⁰ , U_Z⁰ x=y,

whereD(U_Z⁰)is the set of all functionsx∈Y_∞^Z such that there existsy ∈Y_∞⁰ satisfying x(t) =U(t,s)x(s) +

Z _t

s ρ⁰(τ)U(t,τ)y(τ)dτ, fort≥s ≥0.

Lemma 5.2. We have:

D(T_Z) =D(U_Z) and T_Z =U_Z+D, (5.5) and respectively,

D(T_Z⁰) =D(U_Z⁰) and T_Z⁰ =U⁰_Z+D⁰. (5.6) Proof of the lemma. Takex∈ D(U_Z), that isx ∈Y_∞^Zsuch that there existsy∈Y₁withU_Zx=y.

Then, fort≥s ≥0 we have x(t) =U(t,s)x(s) +

Z _t

s U(t,τ)y(τ)dτ

= T(t,s)x(s) +

Z _t

s T(t,τ)B(τ)U(τ,s)x(s)dτ+

Z _t

s T(t,τ)y(τ)dτ +

Z _t

s

Z _t

τ

T(t,r)B(r)U(r,τ)y(τ)dr dτ

= T(t,s)x(s) +

Z _t

s T(t,r)y(r)dr+

Z _t

s T(t,r)B(r)U(r,s)x(s)dr +

Z _t

s

Z _r

s T(t,r)B(r)U(r,τ)y(τ)dτdr

= T(t,s)x(s) +

Z _t

s T(t,r) (y(r) +B(r)x(r))dr, thusx ∈ D(T_Z)and

T_Zx= y+Dx= (U_Z+D)x.

Reversing the arguments, we conclude that (5.5) holds. Similarly, one can prove (5.6).

Now, we continue the proof of the theorem. From (5.5) and (5.3) we have k(U_Z−T_Z)xk₁ =kDxk₁≤ ^δC

a kxk_TZ, for all x∈ D(T_Z) =D(U_Z),

which implies that U_Z : D(U_Z) → Y₁ is bounded. Since T_Z is invertible, we obtain that U_Z is also invertible for sufficiently small δ > 0. Similarly, one can show that U_Z⁰ is invertible for sufficiently small δ >0. By Theorem4.1 we conclude that the perturbed evolution family {U(t,s)}_{t≥s≥0 admits a}ρ-dichotomy with respect to the family of normsk · k_t,t ≥_0.