doi:10.1155/2010/347670
Research Article
Pairs of Function Spaces and Exponential
Dichotomy on the Real Line
Adina Luminit¸a Sasu
Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timis¸oara, C. Coposu Boulevard. no. 4, 300223 Timis¸oara, Romania
Correspondence should be addressed to Adina Luminit¸a Sasu,[email protected]
Received 15 January 2010; Accepted 21 January 2010
Academic Editor: Gaston Mandata N’Guerekata
Copyrightq2010 Adina Luminit¸a Sasu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We provide a complete diagram of the relation between the admissibility of pairs of Banach function spaces and the exponential dichotomy of evolution families on the real line. We prove that ifW ∈ HRandV ∈ TRare two Banach function spaces with the property that either
W∈ WRorV ∈ VR, then the admissibility of the pairWR, X, VR, Ximplies the existence of the exponential dichotomy. We study when the converse implication holds and show that the hypotheses on the underlying function spaces cannot be dropped and that the obtained results are the most general in this topic. Finally, our results are applied to the study of exponential dichotomy ofC0-semigroups.
1. Introduction
In the study of the asymptotic behavior of evolution equations the input-output conditions are very efficient tools, with wide applicability area, and give a nice connection between control theory and the qualitative theory of differential equations see 1–16 and the reference therein. Starting with the pioneering work of Perron see 8 these methods were developed and improved in remarkable books see1,4,6. A new and interesting perspective on this framework was proposed in5, where the authors presented a complete study of stability, expansiveness, and dichotomy of evolution families on the half-line in terms of input-output methods. This paper was the starting point for an entire collection of studies dedicated to the input-output techniques and their applications to the qualitative theory of differential and difference equations.
on the half-linesee5,9,10and the case of evolution families on the real linesee11–
16, which require a distinct analysis for each case. For instance, when one determines sufficient conditions for the existence of exponential dichotomy on the half-line, an important hypothesis is that the initial stable subspace is closed and complemented see, e.g., 5, Theorem 4.3 or 9, Theorem 3.3. This assumption may be dropped when we study the exponential dichotomy on the real line see, e.g., 11, Theorem 5.1 or 16, Theorem 5.3. These facts implicitly generate the differences between the admissibility concepts used on the real line compared with those used on the half-line and also interesting technical approaches in each case.
The aim of the present paper is to provide new and very general conditions for the existence of exponential dichotomy on the real line. We consider the problem of finding connections between the solvability of an integral equation and the existence of exponential dichotomy of evolution families on the real line. The main purpose is to obtain a complete diagram and a classification of the classes of function spaces that may be used in the study of exponential dichotomy via admissibility.
For the beginning we will present the previous results in this topic and the main objectives will be clearly specified in the context of the actual state of knowledge. We denote byTRthe class of all Banach sequence spaces Bwhich are invariant under translations, contain the continuous functions with compact support, satisfy an integral property and if
B\L1R,R/∅, then there is a continuous functionϕ ∈ B\L1R,R. We consider HR
the subclass of TR satisfying the ideal property. We associate two subclasses of HR:
WR—the class of all Banach function spaces with unbounded fundamental function and
VR—the class of all Banach function spaces which contain at least a nonintegrable function. A pair of function spaces WR, X, VR, X is called admissible for an evolution family
U {Ut, s}t≥s on the Banach spaceXif for every test function in the input spaceVR, X there exists a unique solution function in the output spaceWR, Xfor the associated integral equation given by the variation of constants formulaseeDefinition 3.5below.
For the first time, we have proposed in 11 a sufficient condition for exponential dichotomy, using certain Banach function spaces which are invariant under translations and we obtained the following theorem.
Theorem 1.1. IfV ∈ VRand the pairCbR, X, VR, Xis admissible for an evolution family
U{Ut, s}t≥s, thenUis exponentially dichotomic.
Our study has been continued and extended in 16, both for uniform dichotomy and exponential dichotomy. According to the proof of Theorem 4.8 in16we may give the following sufficient condition for uniform dichotomy.
Theorem 1.2. IfW ∈ HR, V ∈ TR, and the pairWR, X, VR, X is admissible for an evolution familyU{Ut, s}t≥s, thenUis uniformly dichotomic.
From the proof of Theorem 5.3iin16we deduce the following sufficient condition for exponential dichotomy.
Theorem 1.3. If W ∈ WR,V ∈ TR, and the pair WR, X, VR, Xis admissible for an evolution familyU{Ut, s}t≥s, thenUis exponentially dichotomic.
if so, which is the most general class where the input space should belong to. The aim of the present paper is to answer this question and to provide a complete study of the exponential dichotomy on the real line via integral admissibility. The answer to the above question will establish clearly how should one modify the hypotheses ofTheorem 1.2such that the admissibility of the pairWR, X, VR, Ximplies the existence of the exponential dichotomy.
We will prove that ifW ∈ HR and V ∈ VR, then the admissibility of the pair
WR, X, VR, X is a sufficient condition for exponential dichotomy. Consequently, we will deduce a complete diagram of the study of exponential dichotomy on the real line in terms of the admissibility of function spacesseeTheorem 3.11. Specifically, ifW ∈ HR
and V ∈ TR are two Banach function spaces with the property that either W ∈ WR
or V ∈ VR, then the admissibility of the pair WR, X, VR, X implies the existence of the exponential dichotomy. Also, in certain conditions, we deduce that the exponential dichotomy of an evolution familyU{Ut, s}t≥sis equivalent with the admissibility of the pairWR, X, VR, X.
By an example we motivate our techniques and show that the hypotheses from our main results cannot be removed. Precisely, if W ∈ HR and V ∈ TR are such that
W /∈ WRandV /∈ VR, then we prove that the admissibility of the pairWR, X, VR, X does not imply the exponential dichotomy. Moreover, we show that the obtained results and their consequences are the most general in this topic.
Finally, our results are applied at the study of the exponential dichotomy of C0
-semigroups. Using function spaces which are invariant under translations, we obtain a classification of the classes of input and output spaces which may be used in the study of exponential dichotomy of semigroups in terms of input-output techniques with respect to associated integral equations.
2. Preliminaries: Banach Function Spaces
In this section, for the sake of clarity, we present some definitions and notations and we introduce the main classes of function spaces that will be used in our study. LetMRbe the linear space of all Lebesgue measurable functionsu :R → R, identifying the functions equal almost everywhere.
Definition 2.1. A linear subspaceBofMRis callednormed function spaceif there is a mapping
| · |B:B → R such that
i|u|B0 if and only ifu0 a.e.;
ii|αu|B|α||u|B, for allα, u∈R×B;
iii|u v|B≤ |u|B |v|B, for allu, v∈B;
ivifu, v∈Band|u| ≤ |v|a.e. then|u|B≤ |v|B;
vifu∈B, then|u| ∈B.
IfB,| · |Bis complete, thenBis calledBanach function space.
Notations 1. LetCcR,Rdenote the linear space of all continuous functionsv:R → Rwith compact support. Throughout this paper, we denote byTRthe class of all Banach function spaces B, which are invariant under translations,CcR,R ⊂ B, and satisfy the following conditions:
ifor everyt > sthere isαt, s>0 such thatts|uτ|dτ ≤αt, s|u|B, for allu∈B;
iiifB\L1R,R/∅then there is a continuous functionϕ∈B\L1R,R.
For examples of Banach function spaces from the classTRwe refer to11.
LetHRbe the class of all Banach function spacesB∈ TRwith the property that if
|u| ≤ |v|a.e. andv∈B, thenu∈B.
For everyA ⊂ Rwe denote byχA the characteristic function of the setA. Then, if
B∈ HR, we have thatχa,b∈B, for everya, b∈Rwitha < b.
Definition 2.3. LetB ∈ HR. The mapping FB : 0,∞ → R, FBt |χ0,t|B is calledthe
fundamental functionof the spaceB.
For the proof of the next proposition we refer to16, Proposition 2.8.
Proposition 2.4. LetB ∈ HRandν > 0. Ifu :R → R is a function, which belongs toBand
with the property thatqu:R → R , qut
t 1
t usdsbelongs toB, then the functions
fu, gu:R−→R , fut
t
−∞e
−νt−susds, g
ut
∞
t
e−νs−tusds 2.1
belong toB.
Example 2.5. Let M1R,R be the linear space of all u ∈ MR with the property that
supt∈Rtt 1|us|ds <∞. With respect to the norm
u M1 :sup
t∈R t 1
t
|us|ds, 2.2
this is a Banach function space which belongs toHR.
Lemma 2.6. IfB∈ TR, thenB⊂M1R,R.
Proof. Letα >0 be such that10|uτ|dτ≤α|u|B, for allu∈B. Then, we have that
t 1
t
|vτ|dτ
1
0
Notations 2. In what follows we denote by
From the above inequality we deduce thatvnis fundamental in the Banach spaceB, so there isw ∈Bsuch thatvn → winB. According to16, Lemma 2.4there is a subsequencevkn
such thatvkn → wa.e. This implies thatv≡wa.e., sovwinB. Thusv∈Band the proof
is complete.
Notation 2. Let X be a real or complex Banach space. For everyB ∈ TR we denote by
BR, Xthe linear space of all Bochner measurable functionsv :R → Xwith the property that the mappingNv:R → R , Nvt vt lies inB. With respect to the norm v BR,X:
|Nv|B,BR, Xis a Banach space.
3. Exponential Dichotomy for Evolution Families on the Real Line
LetXbe a real or complex Banach space. The norm onXand onBX, the Banach algebra of all bounded linear operators onX, will be denoted by · . Denote byIdthe identity operator onX. First, we remind some basic definitions.
Definition 3.1. A familyU{Ut, s}t≥sof bounded linear operators onXis called anevolution
familyif the following properties hold:
iUt0, t0 IdandUt, sUs, t0 Ut, t0, for allt≥s≥t0;
iifor everyx∈Xand everyt0∈Rthe mappingt→Ut, t0xis continuous ont0,∞
and the mappings→Ut0, sxis continuous on−∞, t0;
Definition 3.2. An evolution familyU{Ut, s}t≥sis said to beuniformly dichotomicif there are a family of projections{Pt}t∈Rand a constantK≥1 such that
iUt, t0Pt0 PtUt, t0, for allt≥t0;
iithe restrictionUt, t0|: kerPt0 → kerPtis an isomorphism, for allt≥t0;
iii Ut, t0x ≤K x , for allx∈ImPt0and allt≥t0;
iv Ut, t0y ≥1/K y , for ally∈kerPt0and allt≥t0.
Definition 3.3. An evolution familyU {Ut, s}t≥s is said to be exponentially dichotomicif
there exist a family of projections{Pt}t∈Rand two constantsK≥1 andν >0 such that
iUt, t0Pt0 PtUt, t0, for allt≥t0;
iithe restrictionUt, t0|: kerPt0 → kerPtis an isomorphism, for allt≥t0;
iii Ut, t0x ≤Ke−νt−t0 x , for allx∈ImPt0and allt≥t0;
iv Ut, t0y ≥1/Keνt−t0 y , for ally∈kerPt0and allt≥t0.
Remark 3.4. It is obvious that if an evolution family is exponentially dichotomic, then it is
uniformly dichotomic.
One of the most efficient tool in the study of the dichotomic behavior of an evolution family is represented by the so-called input-output techniques. The input-output method considered in this paper is the admissibility of a pair of function spaces. Indeed, letW, V be two Banach function spaces such thatW∈ HRandV ∈ TR.
Definition 3.5. The pair WR, X, VR, X is said to be admissible for U if for every v ∈
VR, Xthere exists a uniquef∈WR, Xsuch that the pairf, vsatisfies the equation
ft Ut, sfs
t
s
Ut, τvτdτ, ∀t≥s. EU
Remark 3.6. If the pairWR, X, VR, Xis admissible forU, then it makes sense to define
the operatorQ : VR, X → WR, X, Qv f, wheref ∈WR, Xis such that the pair
f, vsatisfiesEU. ThenQis a bounded linear operatorsee16, Proposition 4.4.
LetU{Ut, s}t≥sbe an evolution family onXandW ∈ HR. For everyt∈R, we considerthe stable subspaceXstas the space of allx∈Xwith the property that the function
δx:R−→X, δxτ
⎧ ⎨ ⎩
Uτ, tx, τ≥t,
0, τ < t 3.1
belongs toWR, Xand we definethe unstable subspaceXutas the space of allx∈Xwith the property that there is a functionϕx ∈WR, Xsuch thatϕxt xandϕxτ Uτ, sϕxs, for alls≤τ ≤t.
Theorem 3.7. LetU{Ut, s}t≥sbe an evolution family onXand letW, V be two Banach function spaces withW ∈ HRandV ∈ TR. If the pairWR, X, VR, Xis admissible for the evolution familyU, thenUis uniformly dichotomic with respect to the family of projections{Pt}t∈R, where
ImPt Xst, kerPt Xut, ∀t∈R. 3.2
Taking into account the results obtained in 11,16, an interesting open question is whether in the study of exponential dichotomy, the output space may belong to the general classHR. To answer this question, the first purpose of this paper is to prove the following theorem.
Theorem 3.8. LetU {Ut, s}t≥sbe an evolution family on the Banach spaceX and let W, V be
two Banach function spaces with W ∈ HRand V ∈ VR. If the pair WR, X, VR, X is
admissible forU, thenUis uniformly exponentially dichotomic.
The proof will be constructive and therefore, we will present several intermediate results.
Theorem 3.9. LetU {Ut, s}t≥sbe an evolution family on the Banach spaceX and letW, V be
two Banach function spaces with W ∈ HRand V ∈ VR. If the pair WR, X, VR, X is
admissible forU, then there areK, ν >0such that
Ut, t0x ≤Ke−νt−t0 x , ∀t≥t0, ∀x∈Xst0. 3.3
Proof. According toTheorem 3.7andDefinition 3.2iiiwe have that there isλ >0 such that
Lett0∈Rand letx∈Xst0. We consider the functions
v:R → X, vt ψt−t0Ut, t0x, 3.8
f :R → X, ft
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
t
t0
ψτ−t0dτUt, t0x, t≥t0,
0, t < t0.
3.9
Since v ∈ CcR, X it follows that v ∈ VR, X. Setting q
r
0ψτdτ we observe that
ft qUt, t0x, for allt ≥ t0 r. Sincex ∈Xst0we deduce thatf ∈WR, X. A simple
computation shows that the pairf, vsatisfiesEU, sof Qv. This implies that
fWR,X ≤ Q v VR,X. 3.10
According to relation3.4we observe that
vt ψt−t0 Ut, t0x ≤λ x ψt0t, ∀t∈R, 3.11
and using the invariance under translations of the space V we deduce that v VR,X ≤
λ x |ψ|V. Fromψt≤ϕt, for allt∈R, we have that|ψ|V ≤ |ϕ|V. Thus we obtain that
v VR,X≤λ x ϕV. 3.12
From Ut0 r 1, t0x ≤λ Uτ, t0x λ/q fτ , for allτ ∈t0 r, t0 r 1, we have
that Ut0 r 1, t0x χt0 r,t0 r 1τ≤λ/q fτ , for allτ∈R, which implies that
FW1 Ut0 r 1, t0x ≤
λ
qfWR,X. 3.13
Settinglr 1, from relations3.10–3.13it follows that
qFW1 Ut0 l, t0x ≤λ2 Q ϕV x . 3.14
Using relations3.7and3.14we deduce that Ut0 l, t0x ≤1/e x .Taking into account
thatldoes not depend ont0orx, we have that
Ut0 l, t0x ≤
1
e x , ∀t0∈R, ∀x∈Xst0. 3.15
Letν1/landKλe. Lett≥t0andx∈Xst0. Then, there arej∈Nands∈0, lsuch that
tt0 jl s. Using relations3.4and3.15we obtain that Ut, t0x ≤λ Ut0 jl, t0x ≤
Theorem 3.10. LetU {Ut, s}t≥s be an evolution family on the Banach spaceX and letW, V
Proof. Let M ≥ 1 and ω > 0 be given by Definition 3.1. According to Theorem 3.7 and
Definition 3.2ivwe have that there isλ >0 such that
Using relation3.17we have that
vt ψt−t0 Ut, t0x ≤ψt−t0λ Ut0 r, t0x , ∀t∈R 3.24
which implies that
v VR,X≤ψVλ Ut0 r, t0x ≤ϕVλ Ut0 r, t0x . 3.25
Sincexϕxt0 Ut0, tϕxt, for allt∈t0−1, t0, we deduce that
x χt0−1,t0t≤Me
ωϕ
xtχt0−1,t0t≤
Meω
q ft, ∀t∈R. 3.26
This shows that
q x FW1≤MeωfWR,X. 3.27
From relations3.19–3.27it follows that Ut0 r, t0x ≥e x . Taking into account thatr
does not depend ont0orx, we have that
Ut0 r, t0x ≥e x , ∀t0∈R, ∀x∈Xut0. 3.28
We setν1/randKλe. Lett≥t0andx∈Xut0. Then, there arej ∈Nands∈0, rsuch
thattt0 jr s. Using relations3.17and3.28we obtain that Ut, t0x ≥1/λ Ut0
jr, t0x ≥1/λej x ≥1/Keνt−t0 x , which completes the proof.
Now, we may give the proof ofTheorem 3.8.
Proof ofTheorem 3.8. This immediately follows from Theorems3.7,3.9, and3.10.
Now, we may give the main result of the paper, which establishes a complete diagram concerning the study of exponential dichotomy on the real line in terms of integral admissibility.
Theorem 3.11. LetU{Ut, s}t≥sbe an evolution family on the Banach spaceXand letW, V be
two Banach function spaces withW ∈ HRandV ∈ TR. IfW ∈ WRorV ∈ VR, then the
following assertions hold:
iif the pair WR, X, VR, X is admissible for U, then U is uniformly exponentially
dichotomic;
iiifV ⊂ W and one of the spacesV, W belongs to the classOR, thenUis exponentially dichotomic if and only if the pairWR, X, VR, Xis admissible forU.
Proof. iThis follows from Theorems1.3and3.8.
iiNecessity.Suppose thatUis exponentially dichotomic with respect to the family of
Letv∈VR, X. We consider the function
An easy computation shows that the pair f, v satisfies EU. The uniqueness of
f is immediate see, e.g., 16, the Necessity part of Theorem 5.3. In conclusion, the pair
WR, X, VR, Xis admissible for the evolution familyU.
The natural question arises whether the hypotheses from Theorem 3.11 can be dropped and also if the conditions given by this theorem are the most general in this topic. The answers are given by the following example.
Example 3.12. Let W ∈ HR and V ∈ TR be such thatW /∈ WRand V /∈ VR. Then
everyt≥swe consider the operator
We have thatfis correctly defined and an easy computation shows that the pairf, vsatisfies
that there exists limt→ ∞f1t 0. Using similar arguments with those in3.34we deduce
From relation3.38we have that
g2t e−τ−tg2τ, ∀τ ≥t. 3.40
Integrating in relation3.40onτ, τ 1and usingLemma 2.6, we deduce that
g2tτ 1
τ
e−ξ−tg2ξdξ≤e−τ−tgM1R,X, ∀τ≥t. 3.41
Forτ → ∞in3.41it follows thatg2t 0. Sincet∈Rwas arbitrary we have thatg0, so
ff. Thus, the pairWR, X, VR, Xis admissible for the evolution familyU.
Suppose thatUis exponentially dichotomic with respect to the family of projections
{Pt}t∈Rand the constantsK, ν >0. According to13, Proposition 3.1we have that ImPt
{x∈X : supτ≥t Uτ, tx <∞}, which implies that ImPt R× {0}, for allt∈R. Then, we obtain that
ϕsϕtx1
≤Ke−νt−s|x1|, ∀t≥s, ∀x1∈R 3.42
or equivalently
ϕt
ϕs ≤Ke−νt−s, ∀t≥s 3.43
which is absurd. In conclusion, the pairWR, X, VR, Xis admissible forU, but, for all that, the evolution familyUis not exponentially dichotomic.
4. Applications to the Case of
C
0-SemigroupsIn this section, by applying the central theorems from the Section 3 we deduce several consequences of the main results for the study of exponential dichotomy ofC0-semigroups.
LetXbe a real or complex Banach space.
Definition 4.1. A family T {Tt}t≥0 of bounded linear operators onX is said to be a C0
-semigroupif the following properties are satisfied:
iT0 Id, the identity operator onX;
iiTt s TtTs, for allt, s≥0;
Definition 4.2. AC0-semigroupT{Tt}t≥0is said to beexponentially dichotomicif there exist
a projectionP ∈ BXand two constantsK≥1 andν >0 such that
iP Tt TtP, for allt≥0;
ii Ttx ≤Ke−νt x , for allx∈ImPand allt≥0;
iii Ttx ≥1/Keνt x , for allx∈kerPand allt≥0;
ivthe restrictionTt|: kerP → kerPis an isomorphism, for everyt≥0.
Remark 4.3. iIfT{Tt}t≥0 is aC0-semigroup, we can associate toTan evolution family
UT {UTt, s}t≥s, byUTt, s Tt−s, for everyt≥s.
ii A C0-semigroup T {Tt}t≥0 is exponentially dichotomic if and only if
the associated evolution family UT {UTt, s}t≥s is exponentially dichotomic see 12, Proposition 4.4.
LetW, V be two Banach function spaces such thatW∈ HRandV ∈ TR.
Definition 4.4. The pairWR, X, VR, Xis said to beadmissiblefor theC0-semigroupT
{Tt}t≥0if for everyv∈VR, Xthere is a uniquef∈WR, Xsuch that
ft Tt−sfs
t
s
Tt−τvτdτ, ∀t≥s. ET
Theorem 4.5. LetT {Tt}t≥0 be aC0-semigroup on the Banach space X and letW, V be two
Banach function spaces withW ∈ HRand V ∈ TR. IfW ∈ WRor V ∈ VR, then the
following assertions hold:
iif the pair WR, X, VR, X is admissible for T, then T is uniformly exponentially
dichotomic;
iiifV ⊂ W and one of the spacesV, W belongs to the classOR, thenTis exponentially dichotomic if and only if the pairWR, X, VR, Xis admissible forT.
Proof. This follows fromTheorem 3.11andRemark 4.3ii.
Remark 4.6. According to the example given in the previous section we deduce that the
hypothesisW ∈ WRorV ∈ VRcannot be removed. Moreover,Theorem 4.5provides a complete answer concerning the study of exponential dichotomy of semigroups using input-output techniques with respect to the associated integral equation.
Acknowledgment
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