R E S E A R C H
Open Access
Exponential dichotomy on time scales and
admissibility of the pair
(C
b
rd
(
T
+
,
X
),
L
p
(
T
+
,
X
))
Liu Yang
1, Jimin Zhang
1*, Xiaoyuan Chang
2and Zhifeng Liu
1*Correspondence:
[email protected] 1School of Mathematical Sciences,
Heilongjiang University, 74 Xuefu Street, Harbin, Heilongjiang 150080, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we study a relation between exponential dichotomy on time scales and admissibility of the pair (Cbrd(T+,X),Lp(T+,X)) for an evolution family on time scales. We establish a sufficient criterion for the existence of exponential dichotomy on time scales in terms of the admissibility of the pair (Cbrd(T+,X),Lp(T+,X)) for the evolution family. Conversely, with the help of exponential dichotomy on time scales, we give the admissibility of the pair (Cb
rd(T+,X),Lp(T+,X)) for an input-output equation on
time scales.
MSC: 34N05; 34D09
Keywords: time scales; exponential dichotomy; admissibility
1 Introduction
The concept of exponential dichotomies was first introduced by Perron in [] to study the conditional stability of the linear differential equations and the existence of bounded solutions of the nonlinear differential equations. Then Li [] established the correspond-ing analogous concept for the linear difference equations. The theory of exponential di-chotomies has been playing an important role in the study of the theory of differential equations and difference equations (see [–]). An interesting and challenging problem is to establish necessary and sufficient criteria for the existence of exponential dichotomies. Among the many methods and tools, the admissibility techniques or input-output meth-ods have been extensively applying to study the existence of exponential dichotomies for differential equations and difference equations [–].
It is well known that the theory of dynamic equations on time scales provides a unify-ing structure for the study of differential equations in the continuous case and difference equations in the discrete case and has tremendous potential for applications in mathe-matical models of real processes and phenomena [–]. In recent years, the theory of exponential dichotomies on time scales for the linear dynamic equations on time scales extends the idea of hyperbolicity from autonomous dynamic equations on time scales to explicitly nonautonomous ones and has progressed greatly [–]. In view of the im-portant role of the admissibility techniques or input-output methods in the study of the exponential dichotomy on differential equations and difference equations, it is natural for us to ask whether the admissibility techniques or input-output methods can be applied to deal with problems of exponential dichotomies on time scales for an evolution family on time scales.
Motivated by the results of admissibility and exponential dichotomy for differential equations and difference equations in [–], in this paper, we establish a relation between exponential dichotomy on time scales and admissibility of the pair (Cbrd(T+,X),Lp(T+,X))
for an evolution family on time scales. The paper is organized as follows. In the next section, we present some basic information concerning exponential dichotomies on time scales and admissibility for an evolution family. In Section , we construct an equivalent relation between exponential dichotomy on time scales and the admissibility of the pair (Cb
rd(T+,X),Lp(T+,X)) for the evolution family on time scales. Our result extends related
results known for differential equations and difference equations on the half-line to more general time scales.
2 Preliminaries and basic definitions
In this section, we first introduce the following concepts related to the notion of time scales, which can be found in [, , ]. A time scaleTis defined as a nonempty closed subset of the real numbers. Define the forward jump operatorσ :T→Tand the grain-iness functionμ(t) =σ(t) –tfor anyt∈T. In the following discussion, the time scaleT is assumed to be unbounded above and below. Let Crd(T,R) be the set of rd-continuous
functionsg:T→RandR+(T,R) :={g∈C
rd(T,R) : +μ(t)g(t) > ,t∈T}be the space of
positively regressive functions. We define the exponential function on time scales by
eϕ(t,s) =exp
t
s
ζμ(τ)
ϕ(τ)τ
withζh(z) = ⎧ ⎨ ⎩
z ifh= ,
Log( +hz)/h ifh= ,
for anyϕ∈R+(T,R) ands,t∈T, whereLogis the principal logarithm. Define
(ϕ⊕ψ)(t) :=ϕ(t) +ψ(t) +μ(t)ϕ(t)ψ(t),
ϕ:= – ϕ(t) +μ(t)ϕ(t),
(ωϕ)(t) := lim
hμ(t)
( +hϕ(t))ω–
h
for a givenω∈R+and for anyt∈T,ϕ,ψ∈R+(T,R). Let
T+=T∩[, +∞), κ=mint∈T+, [t,s]
T+= [t,s]∩T+, t,s∈T+, [ϕ]∗:=sup
t∈T+
ϕ(t), [ϕ]∗:= inf t∈T+
ϕ(t)
for any bounded functionϕ∈Crd(T+,R).
Let (X,·) be a Banach space andB(X) be the space of bounded linear operators defined onX. Now we give some definitions on time scales.
Definition . {U(t,s)}t≥s⊂B(X) is said to be an evolution family on a time scaleT+if
(i) U(t,t) = idfor everyt∈T+andU(t,τ)U(τ,s) =U(t,s)for anyt≥τ≥s≥κ; (ii) for eachs∈T+and anyx∈X,U(·,s)xis rd-continuous on[s,∞)T+for the first
Remark . In the general case, an evolution family{U(t,s)}t≥sis related to an evolution
operator of a linear dynamic equation on time scales.
Definition . An evolution family{U(t,s)}t,s∈T+ is said to be exponential growth on a time scaleT+if there exist positive constantsLandρsuch that
U(t,s)≤Leρ(t,s), t≥s,t,s∈T+. (.)
Definition . An evolution family {U(t,s)}t,s∈T+ is said to admit an exponential di-chotomy on a time scale T+ if there exist projections{P(t)}
t∈T+ such thatU(t,s)P(s) =
P(t)U(t,s) for anyt≥s≥κandU(t,s)|KerP(s):KerP(s)→KerP(t) is an isomorphism for
anyt≥s,t,s∈T+and there exist a constantK> andα∈C
rd(T+,R) with [α]∗> such
that
(i) U(t,s)x ≤Keα(t,s)xfor allx∈RangeP(s)and anyt≥s,t,s∈T+;
(ii) U(t,s)y ≥
Keα(t,s)yfor ally∈KerP(s)and anyt≥s,t,s∈T
+.
Remark . The exponential function on time scales can display different forms when we choose different time scales. For example, whenT=RorT=Z, we haveeα(t,s) =e–α(t–s)
oreα(t,s) = (/( +α))t–sifαis a constant. LetT=qN,q> , theneα(t,s) =
τ∈[s,t)[/
( + (q– )ατ)]. More examples for the exponential function on different time scales can be found in []. This shows that the exponential dichotomy on time scales is more general and unifies the notions of exponential dichotomies on the continuous and discrete case. On the other hand, we have
eα(t,s)≤eα(t–s), e–α(t–s)≤eα(t,s) (.)
for anyt≥sand any time scaleT(see (.) in []). We let
CbrdT+,X:=
u∈Crd
T+,X|u
∞:=sup
t∈T+
u(t)<∞
and
LpT+,X:=
ff :T+→Xis a Bochner measurable function with
fp:=
∞
κ
f(t)pt /p
<∞
forp> . It is not difficult to show that (Cb
rd(T+,X), · ∞) and (Lp(T+,X), · p) are both
Banach spaces (see []). We consider the integral equation on time scales
u(t) =U(t,s)u(s) +
t
s
U(t,τ)f(τ)τ, t≥s,t,s∈T+, (.)
wheref ∈Lp(T+,X) andu∈Cb
Definition . The pair (Cbrd(T+,X),Lp(T+,X)) is said to be admissible for an evolution
family{U(t,s)}t,s∈T+ if for everyf ∈Lp(T+,X) there exists a functionu∈Cb
rd(T+,X) such
that the pair (u,f) satisfies (.). We say thatLp(T+,X) is the input space and Cb
rd(T+,X) is
the output space.
We easily show that a pair (u,f) satisfies (.) if and only if (u,f) satisfies
u(t) =U(t,κ)u(κ) +
t
κ
U(t,τ)f(τ)τ, t≥κ,t∈T+. (.)
In fact, if (.) holds, then for eachs≥κ
u(s) =U(s,κ)u(κ) +
s
κ
U(s,τ)f(τ)τ
and
U(t,s)u(s) =U(t,s)U(s,κ)u(κ) +
s
κ
U(t,s)U(s,τ)f(τ)τ
=U(t,κ)u(κ) +
s
κ
U(t,τ)f(τ)τ
=u(t) –
t
s
U(t,τ)f(τ)τ
for anyt≥s,t∈T+.
3 Main result
In this section, we establish a relation between exponential dichotomy on time scales and admissibility of the pair (Cb
rd(T+,X),Lp(T+,X)) for an evolution family on time scales. Let
the linear subspaceEκ:={x∈X|U(·,κ)x∈Cbrd(T+,X)}. Now we state our main result.
Theorem . Assume that an evolution family U(t,s)t≥sadmits an exponential growth on a time scaleT+ with[u]∗<∞.Then the pair(Cbrd(T+,X),Lp(T+,X))is admissible for the evolution family U(t,s)t≥son the time scaleT+and Eκ is closed and complemented in X if
and only if U(t,s)t≥sadmits an exponential dichotomy on the time scaleT+.
The proof of Theorem . is nontrivial, we shall divide it into several steps and assume that the conditions in Theorem . are always satisfied. We first establish some auxiliary results. IfEκ is closed and complemented inX, then there is a closed linear subspace
Fκ such thatX=Eκ⊕Fκ. We define the linear subspace Cbrd,Fκ(T+,X) :={u∈Cbrd(T+,X) :
u(κ)∈Fκ}. Using similar arguments to that of Lemma . in [], we conclude that if the
pair (Cbrd(T+,X),Lp(T+,X)) is admissible, then for everyf∈Lp(T+,X) there exists a unique
functionu¯∈Cb,Fκ
rd (T+,X) such that the pair (u¯,f) satisfies (.). Therefore, we can define
the input-output operator J:Lp(T+,X)→Cb,Fκ
rd (T+,X) byJ(f) =u¯, where the pair (¯u,f)
satisfies (.).
Proof According to the closed graph theorem, we only need to prove thatJ is closed. We assume that{fn}n∈N⊂Lp(T+,X),f∈Lp(T+,X), andfn→f inLp(T+,X) asn→ ∞and
there exists a functionu¯∈Cb,Fκ
rd (T+,X) such thatu¯n=J(fn)→ ¯uin Cbrd,Fκ(T+,X) asn→ ∞.
It follows from (.) that
¯
un(t) =U(t,κ)u¯n(κ) + t
κ
U(t,τ)fn(τ)τ, t∈T+. (.)
On the other hand, by (.) and the Hölder inequality on time scales, we have
κtU(t,τ)fn(τ) –f(τ)
τ≤
t
κ
U(t,τ)fn(τ) –f(τ)τ
≤L t
κ
eρ(t,τ)fn(τ) –f(τ)τ
≤L
t
κ
eqρ(t,τ)τ
/q t
κ
fn(τ) –f(τ) p
τ
/p
=L
t
κ
/(qρ)e
(qρ)(τ,t)τ
/q
fn–fp
≤ + [(qρ)μ]∗
[qρ]∗ eρ(t,κ)fn–fp
for each t∈T+, where /q+ /p= . Thent
κU(t,τ)fn(τ)τ →
t
κU(t,τ)f(τ)τ since
fn→f inLp(T+,X) asn→ ∞. Combining with (.) gives
¯
u(t) =U(t,κ)¯u(κ) +
t
κ
U(t,τ)f(τ)τ
since u¯n→ ¯uin Cbrd,Fκ(T+,X) asn→ ∞. This implies thatJ(f) =u¯. The proof is
com-pleted.
For each givens∈T+, we let
Es:=
x∈X:sup
t≥s
U(t,s)x<∞
, Fs:=U(s,κ)Fκ. (.)
Lemma . If the pair (Cbrd(T+,X),Lp(T+,X)) is admissible for the evolution family U(t,s)t≥son the time scale T+, then the subspace Es is closed for every s∈T+ and there is a positive constant Kandα∈Crd(T+,R)with[α]∗> such that
U(t,s)x≤Keα(t,s)x (.)
for any x∈Esand t≥s.
Proof Let
γ := J
[pβ]∗
+ [(pβ)μ]∗
/p
whereJis defined in Lemma .. A direct calculation gives [γ β]∗> . For each given
Next we consider two different cases.
= eγ(t,κ) +μ(t)γ
–γ
t
s
eβ(τ,s)
U(τ,s)xτ+
eβ(t,s)
U(t,s)x
≥
for any t∈ (s,ds,x)T+, which implies that eγ(t,κ)t
s eβ(τ,s)
U(τ,s)xτ is nondecreasing on
(s,ds,x)T+. Then
eγ(t,κ)
t
s
eβ(τ,s)
U(τ,s)xτ≥eγ(ηs,κ) ηs
s
eβ(τ,s)
U(τ,s)xτ (.)
for anyt∈[ηs,ds,x)T+. By (.), we have
U(t,s)x≤Leρ(t,s)x ≤Leρ(ηs,s)x (.)
for anyt∈[s,ηs]T+. It follows from (.), (.), and (.) that
Leρ(ηs,s)x
≤
ηs
s
Leρ(ηs,s)x
τ≤
ηs
s
eβ(τ,s)
U(τ,s)xτ
≤eγ(t,ηs) t
s
eβ(τ,s)
U(τ,s)xτ≤
eγ(t,ηs)
γ
eβ(t,s)
U(t,s)x
for anyt∈[ηs,ds,x)T+. Then
U(t,s)x≤ L
γeρ(ηs,s)eγ(t,ηs)eβ(t,s)x
= L
γeρ(ηs,s)eγ(s,ηs)eβγ(t,s)x
= L
γeρ⊕γ(ηs,s)eα(t,s)x
for anyt∈[ηs,ds,x)T+. To obtain the conclusion, we need to show thatδ(s) :=eρ⊕γ(ηs,s) is bounded for anys∈T+. For the definition ofηs(see (.)), there are the following three
different cases:
Case . s+ ∈T+. We haveη
s=inf{t∈T+|t≥s+ }=s+ <s+ + [μ]∗.
Case . s+ /∈T+and(s,s+ ]∩T+=∅. Lett∗=max{t∈[s,s+ ]T+}. We have
σ(t∗) >t∗. In fact, ifσ(t∗) =t∗, thent∗is a right-dense point, which implies that there is a pointt∗∗>t∗andt∗∗∈[s,s+ ]T+. This is a contradiction. By the
definition oft∗, we getηs=σ(t∗)andηs≤s+ +σ(t∗) –t∗≤s+ + [μ]∗.
Case . (s,s+ ]∩T+=∅. We haveηs=σ(s) >sandηs≤s+σ(s) –s≤s+ + [μ]∗.
In view of the above discussion and (.), we have
δ(s)≤eρ(ηs,s)eγ(ηs,s)≤eρ(ηs–s)eγ(ηs–s)≤e(ρ+γ)(ηs–s)≤e(ρ+γ)(+[μ] ∗)
:=L
for anys∈T+. Then
for allt∈[ηs,ds,x)T+. Moreover, by (.), we get
U(t,s)x≤Leρ(t,s)x=Leρ(t,s)eα(t,s)eα(t,s)x
≤Leρ(t,s)eγ(t,s)eα(t,s)x ≤Leρ(ηs,s)eγ(ηs,s)eα(t,s)x
≤LLeα(t,s)x (.)
for allt∈[s,ηs]T+. It follows from (.) and (.) that
U(t,s)x≤Keα(t,s)x (.)
for allt∈[s,ds,x)T+, whereK=max{LL, (LL/γ)}.
The second case iss≤ds,x≤ηs. It follows from (.) and (.) that
U(t,s)x≤Leρ(t,s)x ≤Leρ(ηs,s)eγ(ηs,s)eα(t,s)x ≤Keα(t,s)x (.)
for allt∈[s,ds,x]T+.
Based on (.), (.), and the definition ofds,x, we conclude that (.) holds. Lets∈T+
and{xn}n∈N⊂Eswithxn→xasn→ ∞. Combining with (.) givesU(t,s)xn ≤Kxn
for anyn∈Nand anyt≥s. Thus, we getU(t,s)x ≤Kxfor anyt≥s. This implies
thatx∈EsandEsis closed. The proof is completed.
Lemma . If the pair (Cbrd(T+,X),Lp(T+,X)) is admissible for the evolution family U(t,s)t≥s on the time scaleT+,then the subspace Fsis closed for every s∈T+ and there is a positive constant Kandα∈Crd(T+,R)with[α]∗> such that
KU(t,s)y≥eα(t,s)y (.)
for any y∈Fsand t≥s.Moreover,U(t,s)|Fs :Fs→Ft is an isomorphism for any t≥s,
t,s∈T+.
Proof Letβ,γ be positive constants andαbe a rd-continuous function defined in (.). Fory∈Fκ\ {}, we haveU(t,κ)y= for anyt∈T+. In fact, if there is¯t∈T+such that
U(¯t,κ)y= , thenU(t,κ)y=U(t,t¯)U(¯t,κ)y= for anyt≥ ¯tandy∈Eκ. This means that
y∈Eκ∩Fκ andy= . This is a contradiction toy∈Fκ\ {}. For eacht∈T+, we choose
{τt
n}n∈N⊂T+such thatt<τt<τt<· · ·<τnt<· · · andτnt→ ∞asn→ ∞. We definefτnt : T+→Xby
fτnt(s) = –χ[t,τnt]T+eβ(s,κ)
U(s,κ)y
U(s,κ)y
anduτnt:T
+→Xby
uτnt(s) =
∞
s
χ[t,τt
n]T+(τ)eβ(τ,κ)
U(τ,κ)y τU(s,κ)y.
It follows that
fτntp≤
∞
t
e(pβ)(s,κ)s
/p
≤
+ [(pβ)μ]∗ [pβ]∗
/p
By (.) and (.), we have
We are now at the right position to establish Theorem ..
Proof of Theorem. (Sufficiency). IfU(t,s)t≥sadmits an exponential growth and an
ex-ponential dichotomy on the time scaleT+, then
x+y ≥ positive constantˆcsuch that
ˆ
It follows from (i) and (ii) in Definition . that
for anyt∈T+. Therefore, (id –P(κ))x= andx∈RangeP(κ). On the other hand, it is clear
thatRangeP(κ)⊂Eκ. This means thatEκ=RangeP(κ) is closed and complemented inX.
(Necessity). By Lemmas . and ., if the pair (Cb
rd(T+,X),Lp(T+,X)) is admissible for
the evolution familyU(t,s)t≥s on the time scaleT+, thenEsandFs(see (.)) are both
closed linear subspaces for everys∈T+. LetP(s) be the projection satisfyingP(s)(X) =E
s
for everys∈T+. To obtain the conclusions, we need to prove (I–P(s))(X) =F
s. Ifz∈Es∩Fs
for everys∈T+, then there iszˆ∈F
κ such thatU(s,κ)zˆ=z. ByU(t,κ)zˆ=U(t,s)U(s,κ)ˆz=
U(t,s)x∈Cb
rd(T+,X), we getzˆ∈Eκ∩Fκ={}andz=U(s,κ)ˆz= . Thus,Es∩Fs={}. For
anyz∈X, we havef(t) :=χ[s,ηs)T+U(t,s)z∈Lp(T+,X) and there existsu∈Cbrd.Fκ(T+,X) such that
u(t) =J(f) =U(t,s)u(s) +
t
s
U(t,τ)f(τ)τ
≥U(t,s)u(s) +
ηs
s
U(t,τ)f(τ)τ
≥U(t,s)u(s) +z
for any t≥ηs, whereηs can be found in (.). Then we getu(s) +z∈Es. This implies
together with the fact thatu(s)∈Fssinceu(κ)∈Fκthatz=u(s) +z–u(s)∈Es+Fs.
Com-bining withEs∩Fs={}givesX=Es⊕Fs. This means that (I–P(s))(X) =Fsis well defined.
Hence, we haveU(t,s)P(s) =P(t)U(t,s),RangeP(s) =EsandKerP(s) =F(s). It follows from
Lemma . and Lemma . thatU(t,s)t≥sadmits an exponential dichotomy on the time
scaleT+, whereK=max{K
,K}andβ,γ,αcan be found in (.).
Remark . Our result extends related results known for differential equations [] and difference equations [] on the half-line to more general time scales.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
Author details
1School of Mathematical Sciences, Heilongjiang University, 74 Xuefu Street, Harbin, Heilongjiang 150080, P.R. China.
2School of Applied Sciences, Harbin University of Science and Technology, 52 Xuefu Street, Harbin, Heilongjiang 150080,
P.R. China.
Acknowledgements
This research is supported by National Natural Science Foundation of China (No. 11201128 and No. 11426077), China Postdoctoral Science Foundation, Natural Science Foundation of Heilongjiang Province-A201414, Science and
Technology Innovation Team in Higher Education Institutions of Heilongjiang Province (No. 2014TD005), the Heilongjiang University Fund for Distinguished Young Scholars (JCL201203) and the Fund of Heilongjiang Education Committee (No. 12541127).
Received: 15 December 2014 Accepted: 9 February 2015
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