Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 409049 8 pageshttpdxdoiorg1011552013409049

Research ArticleGeneralized Exponential Trichotomies for Abstract EvolutionOperators on the Real Line

Nicolae Lupa1 and Mihail Megan23

1 Faculty of Economics and Business Administration West University of Timisoara Boulevard Pestalozzi 16300115 Timisoara Romania

2 Academy of Romanian Scientists Independentei 54 050094 Bucharest Romania3 Faculty of Mathematics and Computer Science West University of Timisoara Boulevard V Parvan 4300223 Timisoara Romania

Correspondence should be addressed to Mihail Megan meganmathuvtro

Received 22 May 2013 Accepted 30 August 2013

Academic Editor Messaoud Bounkhel

Copyright copy 2013 N Lupa and M Megan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper considers two trichotomy concepts in the context of abstract evolution operators The first one extends the notion ofexponential trichotomy in the sense of Elaydi-Hajek for differential equations to abstract evolution operators and it is a naturalextension of the generalized exponential dichotomy considered in the paper by Jiang (2006)The second concept is dual in a certainsense to the first oneWe prove that these types of exponential trichotomy imply the existence of generalized exponential dichotomyon both half-lines We emphasize that we do not assume the invertibility of the evolution operators on the whole space X (unlikethe case of evolution operators generated by differential equations)

1 Introduction

The notion of exponential dichotomy plays a central rolein the qualitative theory of differential equations dynamicalsystems and many other domains (see eg [1ndash4] and thereferences therein) While dichotomy assumes the existenceof two complementary projections trichotomy implies theexistence of three projections The exponential trichotomiesare split into two qualitative different classes depending onthe behavior of the evolution operators with respect to thestructural projections The first type introduced by Sackerand Sell [5] involves a continuous decomposition of thestate space into three closed subspaces (the stable subspacethe unstable subspace and the neutral subspace) (for moredetails about this type of trichotomy we refer the reader to[6ndash10]) The second one introduced by Elaydi and Hajek in[11] implies the existence of exponential dichotomy on bothhalf lines with structural projections 119875

+and 119875

minussuch that

119875+119875minus= 119875minus119875+= 119875minus As far as we know the case of exponential

trichotomy in the sense of Elaydi-Hajek has been studied

before only for reversible evolution operators in [12 13] andin particular for differential equations in [11 14 15]

This paper considers two trichotomy concepts in thesense of Elaydi-Hajek in the general case of abstract evolutionoperators The first one extends the exponential trichotomyin [11] to evolution operators which are not invertible onthe whole space 119883 and it is a natural extension of thegeneralized exponential dichotomy considered in [1] Thistype of exponential dichotomy was defined by Muldowneyin [4] for differential equations without assuming conditions(7) and (8) The second concept is dual in a certain senseto the first one and implies the existence of generalizedexponential dichotomy on both R

minusand R

+ with projection

valued functions 119875minusand 119875

+ respectively such that

119875+(119905) 119875minus(119905) = 119875

minus(119905) 119875+(119905) = 119875

+(119905) forall119905 isin R (1)

The aim of the paper is to extend some results from [11] tothe concepts of exponential trichotomymentioned aboveWenote that we do not need to assume neither the invertibility of

2 Journal of Function Spaces and Applications

the evolution operators on the whole space119883 nor a boundedgrowth condition on 119880(119905 119904)

2 Preliminary Notions

Let 119883 be a real or complex Banach space and let B(119883) bethe Banach algebra of all bounded linear operators on119883 Thenorms on 119883 and on B(119883) will be denoted by sdot Also weconsider Δ = (119905 119904) isin R2 119905 ge 119904 and the subsets

Δminus= (119905 119904) isin Δ 119905 le 0 Δ

+= (119905 119904) isin Δ 119904 ge 0 (2)

In this section we give some preliminary definitions

Definition 1 An operator valued function 119880 Δ rarr B(119883)

is said to be an evolution operator (on 119883) if the followingconditions are satisfied

(1198901) 119880(119905 119905) = 119868 (the identity operator on119883) for 119905 isin R

(1198902) 119880(119905 119904)119880(119904 119905

0) = 119880(119905 119905

0) for all 119905 ge 119904 ge 119905

0

If 119880(119905 119904) are defined for all 119905 119904 isin R and relation (1198902) holds

for all 119905 119904 1199050isin R then we say that 119880 is a reversible evolution

operator

The notion of evolution operator arises naturally from thetheory of well-posed evolution equations Roughly speakingwhen the Cauchy problem

(119905) = 119860 (119905) 119906 (119905) 119905 ge 119904

119906 (119904) = 119909(3)

is well-posedwith regularity subspaces119884119905 119905 isin R the operator

119880 (119905 119904) 119909 = 119906 (119905 119904 119909) for 119905 ge 119904 119909 isin 119884119904 (4)

where 119906(sdot 119904 119909) is the unique solution of (3) can be extendedby continuity to an evolution operator For more details onwell-posed nonautonomous Cauchy problems we refer thereader to Nagel and Nickel [16] and the references therein

Definition 2 An operator valued function 119875 R rarr B(119883) issaid to be a projection valued function if

1198752

(119905) = 119875 (119905) for every 119905 isin R (5)

Definition 3 Three projection valued functions 119875119894 R rarr

B(119883) 119894 isin 1 2 3 are called supplementary if

(1) 1198751(119905) + 119875

2(119905) + 119875

3(119905) = 119868 for 119905 isin R

(2) 119875119894(119905)119875119895(119905) = 0 for 119905 isin R and 119894 119895 isin 1 2 3 119894 = 119895

Definition 4 Given an evolution operator 119880 Δ rarr B(119883)we say that a projection valued function 119875 R rarr B(119883) isinvariant for 119880 if

119875 (119905) 119880 (119905 119904) = 119880 (119905 119904) 119875 (119904) forall (119905 119904) isin Δ (6)

This implies that the family R(119875(119905)) 119905 isin R of ranges of theprojections 119875(119905) is invariant in the sense that if 119909 isin R(119875(119904))

for some 119904 isin R then 119880(119905 119904)119909 isin R(119875(119905)) for all 119905 ge 119904 For

an evolution operator 119880 and a projection valued function 119875

invariant for119880 such that the restriction of the operator119880(119905 119904)

onR(119875(119904)) viewed as amap fromR(119875(119904)) intoR(119875(119905)) is anisomorphism for certain (119905 119904) isin Δ we denote by 119880

119875(119904 119905) the

inverse operator from the range of119875(119905) onto the range of119875(119904)

Definition 5 Let 119880 Δ rarr B(119883) be an evolution operatorand let119875 R rarr B(119883)be a projection valued functionwhichis invariant for 119880 We say that 119875 is

(i) compatible with 119880 if the restriction of 119880(119905 119904) onR(119875(119904)) intoR(119875(119905)) is an isomorphism for all (119905 119904) isinΔ

(ii) compatible on the left with 119880 if the restriction of119880(119905 119904) on R(119875(119904)) into R(119875(119905)) is an isomorphismfor all (119905 119904) isin Δ

minus

(iii) compatible on the right with 119880 if the restriction of119880(119905 119904) on R(119875(119904)) into R(119875(119905)) is an isomorphismfor all (119905 119904) isin Δ

+

We denote by U the set of all bounded and continuousfunctions 119906 R rarr (0infin) satisfying

int

119905

119904

119906 (120591) 119889120591 997888rarr infin as 119905 997888rarr infin for fixed 119904 isin R (7)

int

119905

119904

119906 (120591) 119889120591 997888rarr infin as 119904 997888rarr minusinfin for fixed 119905 isin R (8)

If there is a constant 119886 gt 0 such that 119906(119905) ge 119886 for all 119905 isin Rthen 119906 isin U In particular all the positive constants belong tothe setU However there exist functions different from thosementioned above which belong toU For example

119906 (119905) =1

1 + |119905| 119905 isin R (9)

(there exists no 119886 gt 0 such that 119906(119905) ge 119886 for all 119905 isin R)

3 Generalized Exponential Trichotomies

In this section we consider two trichotomy concepts in thesense of Elaydi-Hajek in the general case of abstract evolutionoperators

31 Generalized ℓ-Exponential Trichotomy The notion ofexponential trichotomy given below extends the exponentialtrichotomy in [11] to evolution operators which are notinvertible on the whole space 119883 and it is a natural extensionof the generalized exponential dichotomy considered in [1]

Definition 6 We say that an evolution operator 119880 hasa generalized ℓ-exponential trichotomy if there exist threesupplementary projection valued functions 119875

1 1198752 1198753and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(l1) 1198751is invariant for 119880

(l2) 1198752is compatible with 119880

Journal of Function Spaces and Applications 3

(l3) 1198753is compatible on the left with 119880

(l4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(l5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(l6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(l7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

If there exists a positive constant 119886 gt 0 such that 119906(119905) ge 119886 forall 119905 isin R then we say that 119880 has an ℓ-exponential trichotomy(in the sense of Elaydi-Hajek)

The notion of ℓ-exponential trichotomy was introducedin [11] for linear differential equations in finite-dimensionalspaces and in [13] for reversible evolution operators In thispaper we do not assume the invertibility of the evolutionoperator on the whole space119883 This degree of generalizationis motivated by possible applications to partial differentialequations in which the evolution operators are not invertible(see [17] and the references therein) Furthermore in [11] theauthors require thematrix119860(119905) to be bounded on the real lineIn the case of abstract evolution operators this corresponds tothe assumption that the evolution operators are exponentiallybounded (ie there exist 120596 gt 0 and 119872 ge 1 such that119880(119905 119904) le 119872119890

120596(119905minus119904) for all 119905 ge 119904) Unfortunately most ofthe evolution operators do not possess this property (see [18pp 12])

Remark 7 If 119880 has a generalized ℓ-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (10)

that is the projection valued functions 1198751 1198752 and 119875

3are

uniformly bounded

Remark 8 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line) (see [1]) This means that if anevolution operator has a generalized exponential dichotomy(on the real line) then it also has a generalized ℓ-exponentialtrichotomy

Remark 9 If 119880 has a generalized ℓ-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then condi-

tions (7) and (8) imply that

119880 (119905 119904) 119875119894(119904) 119909 997888rarr 0 as 119905 997888rarr +infin 119894 isin 1 3

119880119875119894

(119905 119904) 119875119894(119904) 119909 997888rarr 0 as 119905 997888rarr minusinfin 119894 isin 2 3

(11)

If the evolution operator119880 is generated by a well-possed evo-lution equation then the relations above lead to a geometricdescription of the extended state space R times 119883 of (3) Moreprecisely it splits into three invariant vector bundles (thestable bundle the unstable bundle and the center bundle)For more details about the geometric theory of discrete andcontinuous nonautonomous dynamical systems we refer thereader to [19] and the references therein

One can easily give an example of an evolution operatorwhich has a generalized ℓ-exponential trichotomy and doesnot have an ℓ-exponential trichotomy

Example 10 The evolution operator 119880 Δ rarr B(R3)

defined by

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890minusint

119905

119904sign(120591)119906(120591)119889120591

1199093)

(12)

has a generalized ℓ-exponential trichotomy with canonicalprojections for each 119906 isin U However it does not have anℓ-exponential trichotomy for the function 119906 considered inrelation (9)

Themain goal of the paper is to extend some results from[11] to the general case of abstract evolution operators Wenote that we do not need to assume neither the invertibility ofthe evolution operators on the whole space119883 nor a boundedgrowth condition on the evolution operators (unlike the caseof differential equations with bounded coefficients as in [11])

Theorem 11 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875

+ 119875minus R rarr B(119883) which

are invariant for 119880 and there exist a constant 119870 ge 1 and afunction 119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

minus(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

Proof Necessity We consider

119875+(119905) = 119875

1(119905) + 119875

3(119905) 119875

minus(119905) = 119875

1(119905) for 119905 isin R (13)

It is easy to see that the first two conditions hold and we havethat

119876+(119905) = 119875

2(119905) 119876

minus(119905) = 119875

2(119905) + 119875

3(119905) forall119905 isin R (14)

These imply that 119880(119905 119904)|R(119876

+(119904))

R(119876+(119904)) rarr R(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+with the inverse

119880119876+

(119904 119905) = 1198801198752

(119904 119905) (15)

4 Journal of Function Spaces and Applications

and 119880(119905 119904)|R(119876

minus(119904))

R(119876minus(119904)) rarr R(119876

minus(119905)) is an

isomorphism for all (119905 119904) isin Δminuswith

119880119876minus

(119904 119905) = 1198801198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (16)

Hence

119880119876minus

(119904 119905) 119876minus(119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (17)

for all (119905 119904) isin Δminus We first prove that 119880

119876minus

(119904 119905) consideredabove is correctly defined Indeed for every 119909 isin R(119876

minus(119905))

we have that

119880119876minus

(119904 119905) 119909

= 1198801198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) 1198801198752

(119904 119905) 1198752(119905) 119909 + 119875

3(119904) 1198801198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

+ 1198753(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

= 119876minus(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

(18)

which belongs toR(119876minus(119904)) Moreover a simple computation

shows that

119880 (119905 119904) 119880119876minus

(119904 119905) 119876minus(119905) = 119876

minus(119905)

119880119876minus

(119904 119905) 119880 (119905 119904) 119876minus(119904) = 119876

minus(119904)

for (119905 119904) isin Δminus

(19)

For (119905 119904) isin Δ+ we have

1003817100381710038171003817119880 (119905 119904) 119875+(119904)

1003817100381710038171003817 le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817

+1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817 le 2119870119890minusint

119905

119904119906(120591)119889120591

(20)

10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817=100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119870119890minusint

119905

119904119906(120591)119889120591

(21)

If (119905 119904) isin Δminus then we get

1003817100381710038171003817119880 (119905 119904) 119875minus(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(22)

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817le100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(23)

Sufficiency We set

1198751(119905) = 119875

minus(119905) 1198752(119905)

= 119876+(119905) 1198753(119905)

= 119875+(119905) minus 119875

minus(119905)

for 119905 isin R

(24)

First we observe that

119876+(119905) 119876minus(119905) = 119876

minus(119905) 119876+(119905) = 119876

+(119905) 119905 isin R (25)

119875+(119905) minus 119875

minus(119905) = 119875

+(119905) 119876minus(119905) = 119876

minus(119905) 119875+(119905) 119905 isin R (26)

The restriction 119880(119905 119904)|R(1198752(119904))

R(1198752(119904)) rarr R(119875

2(119905)) is an

isomorphism for all 119905 ge 119904 and by relation (25) we have

1198801198752

(119904 119905) 1198752(119905)

=

119880119876+

(119904 119905) 119876+(119905) if 119905 ge 119904 ge 0

119876+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119876+(119905) if 0 ge 119905 ge 119904

119876+(119904) 119880119876minus

(119904 0) 119876minus(0) 119880119876+

(0 119905) 119876+(119905) if 119905 ge 0 ge 119904

(27)

Also the restriction 119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905))

is an isomorphism for all (119905 119904) isin Δ with 0 ge 119905 ge 119904 and byrelation (26) we deduce that

1198801198753

(119904 119905) 1198753(119905) = 119875

+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119875+(119905) (28)

We now verify the inequalities in Definition 6 consideringthree cases

(1) For 119905 ge 119904 ge 0 we have

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119875minus(119904)

1003817100381710038171003817

le 119872minus119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119876minus(119904)

1003817100381710038171003817

le (1 +119872minus)119870119890minusint

119905

119904119906(120591)119889120591

(29)

(2) If 0 ge 119905 ge 119904 then we obtain

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119876+ (119905)

1003817100381710038171003817 le (1 +119872+)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le1003817100381710038171003817119875+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119875+ (119905)

1003817100381710038171003817 le 1198722

+119870119890minusint

119905

119904119906(120591)119889120591

(30)

Journal of Function Spaces and Applications 5

(3) For 119905 ge 0 ge 119904 using the evolution property (1198902) we

get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817

=1003817100381710038171003817119880 (119905 0) 119875

+(0) 119880 (0 119904) 119875

minus(119904)

1003817100381710038171003817

le1003817100381710038171003817119880 (119905 0) 119875

+(0)

1003817100381710038171003817

1003817100381710038171003817119880 (0 119904) 119875minus(119904)

1003817100381710038171003817

le 119870119890minusint

119905

0119906(120591)119889120591

119870119890minusint

0

119904119906(120591)119889120591

= 1198702

119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

times10038171003817100381710038171003817119880119876minus

(119904 0) 119876minus(0)

10038171003817100381710038171003817

10038171003817100381710038171003817119880119876+

(0 119905) 119876+(119905)

10038171003817100381710038171003817

le (1 +119872+)119870119890minusint

0

119904119906(120591)119889120591

119870119890minusint

119905

0119906(120591)119889120591

= (1 +119872+)1198702

119890minusint

119905

119904119906(120591)119889120591

(31)

These complete the proof of the theorem

Remark 12 Note that in comparison to Lemma 12 in [11] weassume that 119875

+(119905) is bounded for 119905 le 0 and 119875

minus(119905) is bounded

for 119905 ge 0 In our opinion these conditions must also beadded in the particular case of evolution operators generatedby differential equations with bounded coefficients This ismotivated by the fact that the proof of the above mentionedlemma seems to not be quite accurateMore precisely the firstcomputation in (ii) rArr (iii) holds only for 119905 ge 0 ge 119904 and notfor all 119905 ge 119904 Indeed using the notations from [11] and takingfor example the case 119905 ge 119904 ge 0 we have

10038171003817100381710038171003817119883 (119905) 119875

1119883minus1

(119904)10038171003817100381710038171003817=10038171003817100381710038171003817119883 (119905) 119875

+119875minus119883minus1

(119904)10038171003817100381710038171003817

le10038171003817100381710038171003817119883 (119905) 119875

+119883minus1

(119904)10038171003817100381710038171003817

10038171003817100381710038171003817119883 (119904) 119875

minus119883minus1

(119904)10038171003817100381710038171003817

le 119871119890minus120572(119905minus119904) 10038171003817100381710038171003817

119883 (119904) 119875minus119883minus1

(119904)10038171003817100381710038171003817

(32)

Thus in order to prove (iii) one must add the assumptionthat there exists a constant119872

minusge 1 such that

10038171003817100381710038171003817119883 (119905) 119875

minus119883minus1

(119905)10038171003817100381710038171003817le 119872minus forall119905 ge 0 (33)

In our case this is equivalent to

sup119905ge0

1003817100381710038171003817119875minus (119905)1003817100381710038171003817 le 119872

minuslt infin (34)

The projection valued function 119875minusis defined on the whole

real line Because the evolution operator 119880 has a generalizedexponential dichotomy with 119875

minusonly on the left half-line we

have that 119875minus(119905) is bounded for 119905 le 0 Still this does not give

any information about the boundedness for 119905 gt 0 (the samecomment applies to 119875

+)

Theorem 13 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875119876 R rarr B(119883) which areinvariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 and 119875(119905)119876(119905) = 119876(119905)119875(119905)for all 119905 isin R

(2) sup119905le0

119875(119905) le 1198721and sup

119905ge0119876(119905) le 119872

2

(3) the restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ with 119905 ge 0(4) the restriction of119880(119905 119904) onR(119876(119904)) intoR(119876(119905)) is an

isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119905 ge 0

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119904 le 0

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

Proof NecessityWe consider

119875 (119905) = 1198751(119905) + 119875

3(119905) 119876 (119905) = 119875

2(119905) + 119875

3(119905) (35)

for 119905 isin R It is easy to see that 119875(119905)119876(119905) = 119876(119905)119875(119905) = 1198753(119905)

Hence 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 On the other hand since119868minus119875(119905) = 119875

2(119905) we have that the restriction119880(119905 119904)

|R(119868minus119875(119904)) isan isomorphism from the range of 119868 minus 119875(119904) onto the range of119868 minus 119875(119905) for all (119905 119904) isin Δ with 119905 ge 0 and

119880119868minus119875

(119904 119905) = 1198801198752

(119904 119905) (36)

By a similar argument as in the proof ofTheorem 11 we obtainthat 119880(119905 119904)

|R(119876(119904)) is an isomorphism from the range of 119876(119904)

onto the range of 119876(119905) for all (119905 119904) isin Δminusand

119880119876(119904 119905) 119876 (119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (37)

For (119905 119904) isin Δ+ we have

119880 (119905 119904) 119875 (119904) le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 +1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(38)

and for (119905 119904) isin Δminus we get

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 le

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(39)

If (119905 119904) isin Δ with 119905 ge 0 then it follows that

1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))1003817100381710038171003817 =

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

(40)

6 Journal of Function Spaces and Applications

and if (119905 119904) isin Δ with 119904 le 0 then we have

119880 (119905 119904) (119868 minus 119876 (119904)) =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(41)

SufficiencyWe set

1198751(119905) = 119868 minus 119876 (119905)

1198752(119905) = 119868 minus 119875 (119905)

1198753(119905) = 119875 (119905) 119876 (119905)

(42)

for 119905 isin R We observe that 119868 minus 119875(119905) = 119876(119905)(119868 minus 119875(119905)) = (119868 minus

119875(119905))119876(119905) for all 119905 isin R Now it is easy to see that the restrictionof119880(119905 119904) onR(119875

2(119904)) intoR(119875

2(119905)) is an isomorphism for all

119905 ge 119904 and

1198801198752

(119904 119905) 1198752(119905)

=119880119868minus119875

(119904 119905) (119868 minus 119875 (119905)) for 119905 ge 119904 with 119905 ge 0

(119868 minus 119875 (119904)) 119880119876(119904 119905) 119876 (119905) (119868 minus 119875 (119905)) for 0 ge 119905 ge s

(43)

Also the restriction119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905)) is

an isomorphism for all (119905 119904) isin Δminusand

1198801198753

(119904 119905) 1198753(119905) = 119875 (119904) 119880

119876(119904 119905) 119876 (119905) 119875 (119905) (44)

Let (119905 119904) isin Δ If 119905 ge 119904 ge 0 then we have1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

= 119880 (119905 119904) 119875 (119904) (119868 minus 119876 (119904))

le 119880 (119905 119904) 119875 (119904) 119868 minus 119876 (119904)

le (1 +1198722)119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 le 119880 (119905 119904) 119875 (119904) 119876 (119904)

le 1198722119870119890minusint

119905

119904119906(120591)119889120591

(45)

If 0 ge 119905 ge 119904 then it follows1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119868 minus 119875 (119904)

times1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)

1003817100381710038171003817 119868 minus 119875 (119905)

le (1 +1198721)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le 119875 (119904)

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 119875 (119905)

le 1198722

1119870119890minusint

119905

119904119906(120591)119889120591

(46)

Finally if 119905 ge 0 ge 119904 then we get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904)) le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(47)

Therefore119880 has a generalized ℓ-exponential trichotomy

32 Generalized 119903-Exponential Trichotomy Nowwe considera concept of generalized exponential trichotomy which isdual in a certain sense to the one given in Definition 6

Definition 14 We say that an evolution operator 119880 hasa generalized 119903-exponential trichotomy if there exist threesupplementary projection valued functions119875

11198752 and119875

3and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(r1) 1198751is invariant for 119880

(r2) 1198752is compatible with 119880

(r3) 1198753is compatible on the right with 119880

(r4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(r7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

If there is a positive constant 119886 gt 0 such that 119906(119905) ge 119886 for all119905 isin R then we say that 119880 has an 119903-exponential trichotomy

The notion of 119903-exponential trichotomy was defined in[13] for reversible evolution operators

Remark 15 If 119880 has a generalized 119903-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (48)

Remark 16 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line)

Example 17 Theevolution operator119880 Δ rarr B(R3) definedby

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890int

119905

119904sign(120591)119906(120591)119889120591

1199093)

(49)

has a generalized 119903-exponential trichotomy with canonicalprojections for each 119906 isin U

Proceeding in a similarmanner to that inTheorems 11 and13 we get the following

Journal of Function Spaces and Applications 7

Theorem 18 Let 119880 Δ rarr B(119883) be an evolution operatorThe following statements are equivalent

(i) 119880 has a generalized 119903-exponential trichotomy(ii) There exist two projection valued functions 119875

+and 119875

minus

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

+(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

(iii) There exist two projection valued functions 119875 and 119876

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that

(1) 119875(119905)119876(119905) = 119876(119905)119875(119905) = 0 for all 119905 isin R(2) sup

119905le0119875(119905) le 119872

1and sup

119905ge0119876(119905) le 119872

2

(3) The restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of 119880(119905 119904) on R(119876(119904)) into R(119876(119905)) isan isomorphism for all (119905 119904) isin Δ with 119904 le 0

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119905 ge 0

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119904 le 0

Let us notice that the exponential trichotomy consideredin Definition 14 seems to be closer to exponential dichotomyon the real line than the other trichotomies since in thecase of linear differential equations it implies exponentialdichotomy on both half lines and (3) has no nontrivialbounded solution (see Lemma 1 in [20])

Remark 19 As a consequence of Lemma 22 from [11] thedifferential equation

(119905) = 119860 (119905) 119909 (119905) 119905 isin R (50)

where 119909(119905) isin C119899 and 119860(119905) is a bounded and continuous 119899 times 119899

matrix on the whole real line that is sup119905isinR119860(119905) lt infin has

an ℓ-exponential trichotomy if and only if its adjoint equation

119910 (119905) = minus119860lowast

(119905) 119910 (119905) 119905 isin R (51)

has an 119903-exponential trichotomy where 119860lowast(119905) is the adjoint

matrix of 119860(119905) When assuming that (50) is defined on a

Hilbert spaceH this result still holds In fact it remains validfor any reversible evolution operator on reflexive Banachspaces in a certain sense which we explain below

If 119880 R2 rarr B(119883) is a reversible evolution operator ona Banach space119883 then we put

119881 (119905 119904) = 119880(119904 119905)lowast

for 119905 s isin R (52)

Notice that 119881 R2 rarr B(119883) is also a reversible evolutionoperator on the dual space 119883

lowast of 119883 (see eg [21]) Let usdenote the elements of the dual space 119883

lowast by 119909lowast and denote

⟨119909 119909lowast⟩ = 119909

lowast(119909) If 119883lowastlowast is the dual space of 119883lowast then there

exists a continuous linear transformation 119869 119883 rarr 119883lowastlowast

defined by

⟨119909lowast

119869119909⟩ = ⟨119909 119909lowast

⟩ forall119909 isin 119883 and 119909lowast

isin 119883lowast

(53)

As a consequence of the Hahn-Banach theorem the operator119869 is an isometry (ie 119869119909 = 119909 for every 119909 isin 119883) and henceit is a one-to-one mapping

Proposition 20 A reversible evolution operator119880 on a reflex-ive Banach space119883 has a generalized ℓ-exponential trichotomy(generalized 119903-exponential trichotomy respectively) if andonly if its adjoint evolution operator 119881 has a generalized 119903-exponential trichotomy (generalized ℓ-exponential trichotomyrespectively)

Proof Necessity It is easy to see that if 119880 has a generalizedℓ-exponential trichotomy (a generalized 119903-exponential tri-chotomy) with projection valued functions 119875

+(119905) and 119875

minus(119905)

as in Theorem 11 (Theorem 18) then the adjoint evolutionoperator 119881 has a generalized 119903-exponential trichotomy (ageneralized ℓ-exponential trichotomy) with projections

119868 minus 119875lowast

+(119905) and 119868 minus 119875

lowast

minus(119905) (54)

Sufficiency We now assume that 119881 has a generalized ℓ-exponential trichotomy with projection valued functions 119875

+

and 119875minusas in Theorem 11 For each 119905 isin R we consider the

following operators

+(119905) 119883 997888rarr 119883

+(119905) 119909 = 119869

minus1

119875lowast

+(119905) 119869119909

minus(119905) 119883 997888rarr 119883

minus(119905) 119909 = 119869

minus1

119875lowast

minus(119905) 119869119909

(55)

We notice that +and

minusare projection valued functions on

119883 We first prove that they are invariant for 119880 We have

⟨+(119905) 119880 (119905 119904) 119909 119909

lowast

⟩

= ⟨119909lowast

119869+(119905) 119880 (119905 119904) 119909⟩ = ⟨119909

lowast

119875lowast

+(119905) 119869119880 (119905 119904) 119909⟩

= ⟨119875+(119905) 119909lowast

119869119880 (119905 119904) 119909⟩ = ⟨119880 (119905 119904) 119909 119875+(119905) 119909lowast

⟩

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

the evolution operators on the whole space119883 nor a boundedgrowth condition on 119880(119905 119904)

2 Preliminary Notions

Let 119883 be a real or complex Banach space and let B(119883) bethe Banach algebra of all bounded linear operators on119883 Thenorms on 119883 and on B(119883) will be denoted by sdot Also weconsider Δ = (119905 119904) isin R2 119905 ge 119904 and the subsets

Δminus= (119905 119904) isin Δ 119905 le 0 Δ

+= (119905 119904) isin Δ 119904 ge 0 (2)

In this section we give some preliminary definitions

Definition 1 An operator valued function 119880 Δ rarr B(119883)

is said to be an evolution operator (on 119883) if the followingconditions are satisfied

(1198901) 119880(119905 119905) = 119868 (the identity operator on119883) for 119905 isin R

(1198902) 119880(119905 119904)119880(119904 119905

0) = 119880(119905 119905

0) for all 119905 ge 119904 ge 119905

0

If 119880(119905 119904) are defined for all 119905 119904 isin R and relation (1198902) holds

for all 119905 119904 1199050isin R then we say that 119880 is a reversible evolution

operator

The notion of evolution operator arises naturally from thetheory of well-posed evolution equations Roughly speakingwhen the Cauchy problem

(119905) = 119860 (119905) 119906 (119905) 119905 ge 119904

119906 (119904) = 119909(3)

is well-posedwith regularity subspaces119884119905 119905 isin R the operator

119880 (119905 119904) 119909 = 119906 (119905 119904 119909) for 119905 ge 119904 119909 isin 119884119904 (4)

where 119906(sdot 119904 119909) is the unique solution of (3) can be extendedby continuity to an evolution operator For more details onwell-posed nonautonomous Cauchy problems we refer thereader to Nagel and Nickel [16] and the references therein

Definition 2 An operator valued function 119875 R rarr B(119883) issaid to be a projection valued function if

1198752

(119905) = 119875 (119905) for every 119905 isin R (5)

Definition 3 Three projection valued functions 119875119894 R rarr

B(119883) 119894 isin 1 2 3 are called supplementary if

(1) 1198751(119905) + 119875

2(119905) + 119875

3(119905) = 119868 for 119905 isin R

(2) 119875119894(119905)119875119895(119905) = 0 for 119905 isin R and 119894 119895 isin 1 2 3 119894 = 119895

Definition 4 Given an evolution operator 119880 Δ rarr B(119883)we say that a projection valued function 119875 R rarr B(119883) isinvariant for 119880 if

119875 (119905) 119880 (119905 119904) = 119880 (119905 119904) 119875 (119904) forall (119905 119904) isin Δ (6)

This implies that the family R(119875(119905)) 119905 isin R of ranges of theprojections 119875(119905) is invariant in the sense that if 119909 isin R(119875(119904))

for some 119904 isin R then 119880(119905 119904)119909 isin R(119875(119905)) for all 119905 ge 119904 For

an evolution operator 119880 and a projection valued function 119875

invariant for119880 such that the restriction of the operator119880(119905 119904)

onR(119875(119904)) viewed as amap fromR(119875(119904)) intoR(119875(119905)) is anisomorphism for certain (119905 119904) isin Δ we denote by 119880

119875(119904 119905) the

inverse operator from the range of119875(119905) onto the range of119875(119904)

Definition 5 Let 119880 Δ rarr B(119883) be an evolution operatorand let119875 R rarr B(119883)be a projection valued functionwhichis invariant for 119880 We say that 119875 is

(i) compatible with 119880 if the restriction of 119880(119905 119904) onR(119875(119904)) intoR(119875(119905)) is an isomorphism for all (119905 119904) isinΔ

(ii) compatible on the left with 119880 if the restriction of119880(119905 119904) on R(119875(119904)) into R(119875(119905)) is an isomorphismfor all (119905 119904) isin Δ

minus

(iii) compatible on the right with 119880 if the restriction of119880(119905 119904) on R(119875(119904)) into R(119875(119905)) is an isomorphismfor all (119905 119904) isin Δ

+

We denote by U the set of all bounded and continuousfunctions 119906 R rarr (0infin) satisfying

int

119905

119904

119906 (120591) 119889120591 997888rarr infin as 119905 997888rarr infin for fixed 119904 isin R (7)

int

119905

119904

119906 (120591) 119889120591 997888rarr infin as 119904 997888rarr minusinfin for fixed 119905 isin R (8)

If there is a constant 119886 gt 0 such that 119906(119905) ge 119886 for all 119905 isin Rthen 119906 isin U In particular all the positive constants belong tothe setU However there exist functions different from thosementioned above which belong toU For example

119906 (119905) =1

1 + |119905| 119905 isin R (9)

(there exists no 119886 gt 0 such that 119906(119905) ge 119886 for all 119905 isin R)

3 Generalized Exponential Trichotomies

In this section we consider two trichotomy concepts in thesense of Elaydi-Hajek in the general case of abstract evolutionoperators

31 Generalized ℓ-Exponential Trichotomy The notion ofexponential trichotomy given below extends the exponentialtrichotomy in [11] to evolution operators which are notinvertible on the whole space 119883 and it is a natural extensionof the generalized exponential dichotomy considered in [1]

Definition 6 We say that an evolution operator 119880 hasa generalized ℓ-exponential trichotomy if there exist threesupplementary projection valued functions 119875

1 1198752 1198753and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(l1) 1198751is invariant for 119880

(l2) 1198752is compatible with 119880

Journal of Function Spaces and Applications 3

(l3) 1198753is compatible on the left with 119880

(l4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(l5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(l6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(l7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

If there exists a positive constant 119886 gt 0 such that 119906(119905) ge 119886 forall 119905 isin R then we say that 119880 has an ℓ-exponential trichotomy(in the sense of Elaydi-Hajek)

The notion of ℓ-exponential trichotomy was introducedin [11] for linear differential equations in finite-dimensionalspaces and in [13] for reversible evolution operators In thispaper we do not assume the invertibility of the evolutionoperator on the whole space119883 This degree of generalizationis motivated by possible applications to partial differentialequations in which the evolution operators are not invertible(see [17] and the references therein) Furthermore in [11] theauthors require thematrix119860(119905) to be bounded on the real lineIn the case of abstract evolution operators this corresponds tothe assumption that the evolution operators are exponentiallybounded (ie there exist 120596 gt 0 and 119872 ge 1 such that119880(119905 119904) le 119872119890

120596(119905minus119904) for all 119905 ge 119904) Unfortunately most ofthe evolution operators do not possess this property (see [18pp 12])

Remark 7 If 119880 has a generalized ℓ-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (10)

that is the projection valued functions 1198751 1198752 and 119875

3are

uniformly bounded

Remark 8 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line) (see [1]) This means that if anevolution operator has a generalized exponential dichotomy(on the real line) then it also has a generalized ℓ-exponentialtrichotomy

Remark 9 If 119880 has a generalized ℓ-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then condi-

tions (7) and (8) imply that

119880 (119905 119904) 119875119894(119904) 119909 997888rarr 0 as 119905 997888rarr +infin 119894 isin 1 3

119880119875119894

(119905 119904) 119875119894(119904) 119909 997888rarr 0 as 119905 997888rarr minusinfin 119894 isin 2 3

(11)

If the evolution operator119880 is generated by a well-possed evo-lution equation then the relations above lead to a geometricdescription of the extended state space R times 119883 of (3) Moreprecisely it splits into three invariant vector bundles (thestable bundle the unstable bundle and the center bundle)For more details about the geometric theory of discrete andcontinuous nonautonomous dynamical systems we refer thereader to [19] and the references therein

One can easily give an example of an evolution operatorwhich has a generalized ℓ-exponential trichotomy and doesnot have an ℓ-exponential trichotomy

Example 10 The evolution operator 119880 Δ rarr B(R3)

defined by

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890minusint

119905

119904sign(120591)119906(120591)119889120591

1199093)

(12)

has a generalized ℓ-exponential trichotomy with canonicalprojections for each 119906 isin U However it does not have anℓ-exponential trichotomy for the function 119906 considered inrelation (9)

Themain goal of the paper is to extend some results from[11] to the general case of abstract evolution operators Wenote that we do not need to assume neither the invertibility ofthe evolution operators on the whole space119883 nor a boundedgrowth condition on the evolution operators (unlike the caseof differential equations with bounded coefficients as in [11])

Theorem 11 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875

+ 119875minus R rarr B(119883) which

are invariant for 119880 and there exist a constant 119870 ge 1 and afunction 119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

minus(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

Proof Necessity We consider

119875+(119905) = 119875

1(119905) + 119875

3(119905) 119875

minus(119905) = 119875

1(119905) for 119905 isin R (13)

It is easy to see that the first two conditions hold and we havethat

119876+(119905) = 119875

2(119905) 119876

minus(119905) = 119875

2(119905) + 119875

3(119905) forall119905 isin R (14)

These imply that 119880(119905 119904)|R(119876

+(119904))

R(119876+(119904)) rarr R(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+with the inverse

119880119876+

(119904 119905) = 1198801198752

(119904 119905) (15)

4 Journal of Function Spaces and Applications

and 119880(119905 119904)|R(119876

minus(119904))

R(119876minus(119904)) rarr R(119876

minus(119905)) is an

isomorphism for all (119905 119904) isin Δminuswith

119880119876minus

(119904 119905) = 1198801198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (16)

Hence

119880119876minus

(119904 119905) 119876minus(119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (17)

for all (119905 119904) isin Δminus We first prove that 119880

119876minus

(119904 119905) consideredabove is correctly defined Indeed for every 119909 isin R(119876

minus(119905))

we have that

119880119876minus

(119904 119905) 119909

= 1198801198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) 1198801198752

(119904 119905) 1198752(119905) 119909 + 119875

3(119904) 1198801198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

+ 1198753(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

= 119876minus(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

(18)

which belongs toR(119876minus(119904)) Moreover a simple computation

shows that

119880 (119905 119904) 119880119876minus

(119904 119905) 119876minus(119905) = 119876

minus(119905)

119880119876minus

(119904 119905) 119880 (119905 119904) 119876minus(119904) = 119876

minus(119904)

for (119905 119904) isin Δminus

(19)

For (119905 119904) isin Δ+ we have

1003817100381710038171003817119880 (119905 119904) 119875+(119904)

1003817100381710038171003817 le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817

+1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817 le 2119870119890minusint

119905

119904119906(120591)119889120591

(20)

10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817=100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119870119890minusint

119905

119904119906(120591)119889120591

(21)

If (119905 119904) isin Δminus then we get

1003817100381710038171003817119880 (119905 119904) 119875minus(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(22)

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817le100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(23)

Sufficiency We set

1198751(119905) = 119875

minus(119905) 1198752(119905)

= 119876+(119905) 1198753(119905)

= 119875+(119905) minus 119875

minus(119905)

for 119905 isin R

(24)

First we observe that

119876+(119905) 119876minus(119905) = 119876

minus(119905) 119876+(119905) = 119876

+(119905) 119905 isin R (25)

119875+(119905) minus 119875

minus(119905) = 119875

+(119905) 119876minus(119905) = 119876

minus(119905) 119875+(119905) 119905 isin R (26)

The restriction 119880(119905 119904)|R(1198752(119904))

R(1198752(119904)) rarr R(119875

2(119905)) is an

isomorphism for all 119905 ge 119904 and by relation (25) we have

1198801198752

(119904 119905) 1198752(119905)

=

119880119876+

(119904 119905) 119876+(119905) if 119905 ge 119904 ge 0

119876+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119876+(119905) if 0 ge 119905 ge 119904

119876+(119904) 119880119876minus

(119904 0) 119876minus(0) 119880119876+

(0 119905) 119876+(119905) if 119905 ge 0 ge 119904

(27)

Also the restriction 119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905))

is an isomorphism for all (119905 119904) isin Δ with 0 ge 119905 ge 119904 and byrelation (26) we deduce that

1198801198753

(119904 119905) 1198753(119905) = 119875

+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119875+(119905) (28)

We now verify the inequalities in Definition 6 consideringthree cases

(1) For 119905 ge 119904 ge 0 we have

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119875minus(119904)

1003817100381710038171003817

le 119872minus119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119876minus(119904)

1003817100381710038171003817

le (1 +119872minus)119870119890minusint

119905

119904119906(120591)119889120591

(29)

(2) If 0 ge 119905 ge 119904 then we obtain

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119876+ (119905)

1003817100381710038171003817 le (1 +119872+)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le1003817100381710038171003817119875+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119875+ (119905)

1003817100381710038171003817 le 1198722

+119870119890minusint

119905

119904119906(120591)119889120591

(30)

Journal of Function Spaces and Applications 5

(3) For 119905 ge 0 ge 119904 using the evolution property (1198902) we

get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817

=1003817100381710038171003817119880 (119905 0) 119875

+(0) 119880 (0 119904) 119875

minus(119904)

1003817100381710038171003817

le1003817100381710038171003817119880 (119905 0) 119875

+(0)

1003817100381710038171003817

1003817100381710038171003817119880 (0 119904) 119875minus(119904)

1003817100381710038171003817

le 119870119890minusint

119905

0119906(120591)119889120591

119870119890minusint

0

119904119906(120591)119889120591

= 1198702

119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

times10038171003817100381710038171003817119880119876minus

(119904 0) 119876minus(0)

10038171003817100381710038171003817

10038171003817100381710038171003817119880119876+

(0 119905) 119876+(119905)

10038171003817100381710038171003817

le (1 +119872+)119870119890minusint

0

119904119906(120591)119889120591

119870119890minusint

119905

0119906(120591)119889120591

= (1 +119872+)1198702

119890minusint

119905

119904119906(120591)119889120591

(31)

These complete the proof of the theorem

Remark 12 Note that in comparison to Lemma 12 in [11] weassume that 119875

+(119905) is bounded for 119905 le 0 and 119875

minus(119905) is bounded

for 119905 ge 0 In our opinion these conditions must also beadded in the particular case of evolution operators generatedby differential equations with bounded coefficients This ismotivated by the fact that the proof of the above mentionedlemma seems to not be quite accurateMore precisely the firstcomputation in (ii) rArr (iii) holds only for 119905 ge 0 ge 119904 and notfor all 119905 ge 119904 Indeed using the notations from [11] and takingfor example the case 119905 ge 119904 ge 0 we have

10038171003817100381710038171003817119883 (119905) 119875

1119883minus1

(119904)10038171003817100381710038171003817=10038171003817100381710038171003817119883 (119905) 119875

+119875minus119883minus1

(119904)10038171003817100381710038171003817

le10038171003817100381710038171003817119883 (119905) 119875

+119883minus1

(119904)10038171003817100381710038171003817

10038171003817100381710038171003817119883 (119904) 119875

minus119883minus1

(119904)10038171003817100381710038171003817

le 119871119890minus120572(119905minus119904) 10038171003817100381710038171003817

119883 (119904) 119875minus119883minus1

(119904)10038171003817100381710038171003817

(32)

Thus in order to prove (iii) one must add the assumptionthat there exists a constant119872

minusge 1 such that

10038171003817100381710038171003817119883 (119905) 119875

minus119883minus1

(119905)10038171003817100381710038171003817le 119872minus forall119905 ge 0 (33)

In our case this is equivalent to

sup119905ge0

1003817100381710038171003817119875minus (119905)1003817100381710038171003817 le 119872

minuslt infin (34)

The projection valued function 119875minusis defined on the whole

real line Because the evolution operator 119880 has a generalizedexponential dichotomy with 119875

minusonly on the left half-line we

have that 119875minus(119905) is bounded for 119905 le 0 Still this does not give

any information about the boundedness for 119905 gt 0 (the samecomment applies to 119875

+)

Theorem 13 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875119876 R rarr B(119883) which areinvariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 and 119875(119905)119876(119905) = 119876(119905)119875(119905)for all 119905 isin R

(2) sup119905le0

119875(119905) le 1198721and sup

119905ge0119876(119905) le 119872

2

(3) the restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ with 119905 ge 0(4) the restriction of119880(119905 119904) onR(119876(119904)) intoR(119876(119905)) is an

isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119905 ge 0

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119904 le 0

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

Proof NecessityWe consider

119875 (119905) = 1198751(119905) + 119875

3(119905) 119876 (119905) = 119875

2(119905) + 119875

3(119905) (35)

for 119905 isin R It is easy to see that 119875(119905)119876(119905) = 119876(119905)119875(119905) = 1198753(119905)

Hence 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 On the other hand since119868minus119875(119905) = 119875

2(119905) we have that the restriction119880(119905 119904)

|R(119868minus119875(119904)) isan isomorphism from the range of 119868 minus 119875(119904) onto the range of119868 minus 119875(119905) for all (119905 119904) isin Δ with 119905 ge 0 and

119880119868minus119875

(119904 119905) = 1198801198752

(119904 119905) (36)

By a similar argument as in the proof ofTheorem 11 we obtainthat 119880(119905 119904)

|R(119876(119904)) is an isomorphism from the range of 119876(119904)

onto the range of 119876(119905) for all (119905 119904) isin Δminusand

119880119876(119904 119905) 119876 (119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (37)

For (119905 119904) isin Δ+ we have

119880 (119905 119904) 119875 (119904) le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 +1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(38)

and for (119905 119904) isin Δminus we get

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 le

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(39)

If (119905 119904) isin Δ with 119905 ge 0 then it follows that

1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))1003817100381710038171003817 =

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

(40)

6 Journal of Function Spaces and Applications

and if (119905 119904) isin Δ with 119904 le 0 then we have

119880 (119905 119904) (119868 minus 119876 (119904)) =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(41)

SufficiencyWe set

1198751(119905) = 119868 minus 119876 (119905)

1198752(119905) = 119868 minus 119875 (119905)

1198753(119905) = 119875 (119905) 119876 (119905)

(42)

for 119905 isin R We observe that 119868 minus 119875(119905) = 119876(119905)(119868 minus 119875(119905)) = (119868 minus

119875(119905))119876(119905) for all 119905 isin R Now it is easy to see that the restrictionof119880(119905 119904) onR(119875

2(119904)) intoR(119875

2(119905)) is an isomorphism for all

119905 ge 119904 and

1198801198752

(119904 119905) 1198752(119905)

=119880119868minus119875

(119904 119905) (119868 minus 119875 (119905)) for 119905 ge 119904 with 119905 ge 0

(119868 minus 119875 (119904)) 119880119876(119904 119905) 119876 (119905) (119868 minus 119875 (119905)) for 0 ge 119905 ge s

(43)

Also the restriction119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905)) is

an isomorphism for all (119905 119904) isin Δminusand

1198801198753

(119904 119905) 1198753(119905) = 119875 (119904) 119880

119876(119904 119905) 119876 (119905) 119875 (119905) (44)

Let (119905 119904) isin Δ If 119905 ge 119904 ge 0 then we have1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

= 119880 (119905 119904) 119875 (119904) (119868 minus 119876 (119904))

le 119880 (119905 119904) 119875 (119904) 119868 minus 119876 (119904)

le (1 +1198722)119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 le 119880 (119905 119904) 119875 (119904) 119876 (119904)

le 1198722119870119890minusint

119905

119904119906(120591)119889120591

(45)

If 0 ge 119905 ge 119904 then it follows1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119868 minus 119875 (119904)

times1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)

1003817100381710038171003817 119868 minus 119875 (119905)

le (1 +1198721)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le 119875 (119904)

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 119875 (119905)

le 1198722

1119870119890minusint

119905

119904119906(120591)119889120591

(46)

Finally if 119905 ge 0 ge 119904 then we get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904)) le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(47)

Therefore119880 has a generalized ℓ-exponential trichotomy

32 Generalized 119903-Exponential Trichotomy Nowwe considera concept of generalized exponential trichotomy which isdual in a certain sense to the one given in Definition 6

Definition 14 We say that an evolution operator 119880 hasa generalized 119903-exponential trichotomy if there exist threesupplementary projection valued functions119875

11198752 and119875

3and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(r1) 1198751is invariant for 119880

(r2) 1198752is compatible with 119880

(r3) 1198753is compatible on the right with 119880

(r4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(r7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

If there is a positive constant 119886 gt 0 such that 119906(119905) ge 119886 for all119905 isin R then we say that 119880 has an 119903-exponential trichotomy

The notion of 119903-exponential trichotomy was defined in[13] for reversible evolution operators

Remark 15 If 119880 has a generalized 119903-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (48)

Remark 16 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line)

Example 17 Theevolution operator119880 Δ rarr B(R3) definedby

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890int

119905

119904sign(120591)119906(120591)119889120591

1199093)

(49)

has a generalized 119903-exponential trichotomy with canonicalprojections for each 119906 isin U

Proceeding in a similarmanner to that inTheorems 11 and13 we get the following

Journal of Function Spaces and Applications 7

Theorem 18 Let 119880 Δ rarr B(119883) be an evolution operatorThe following statements are equivalent

(i) 119880 has a generalized 119903-exponential trichotomy(ii) There exist two projection valued functions 119875

+and 119875

minus

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

+(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

(iii) There exist two projection valued functions 119875 and 119876

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that

(1) 119875(119905)119876(119905) = 119876(119905)119875(119905) = 0 for all 119905 isin R(2) sup

119905le0119875(119905) le 119872

1and sup

119905ge0119876(119905) le 119872

2

(3) The restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of 119880(119905 119904) on R(119876(119904)) into R(119876(119905)) isan isomorphism for all (119905 119904) isin Δ with 119904 le 0

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119905 ge 0

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119904 le 0

Let us notice that the exponential trichotomy consideredin Definition 14 seems to be closer to exponential dichotomyon the real line than the other trichotomies since in thecase of linear differential equations it implies exponentialdichotomy on both half lines and (3) has no nontrivialbounded solution (see Lemma 1 in [20])

Remark 19 As a consequence of Lemma 22 from [11] thedifferential equation

(119905) = 119860 (119905) 119909 (119905) 119905 isin R (50)

where 119909(119905) isin C119899 and 119860(119905) is a bounded and continuous 119899 times 119899

matrix on the whole real line that is sup119905isinR119860(119905) lt infin has

an ℓ-exponential trichotomy if and only if its adjoint equation

119910 (119905) = minus119860lowast

(119905) 119910 (119905) 119905 isin R (51)

has an 119903-exponential trichotomy where 119860lowast(119905) is the adjoint

matrix of 119860(119905) When assuming that (50) is defined on a

Hilbert spaceH this result still holds In fact it remains validfor any reversible evolution operator on reflexive Banachspaces in a certain sense which we explain below

If 119880 R2 rarr B(119883) is a reversible evolution operator ona Banach space119883 then we put

119881 (119905 119904) = 119880(119904 119905)lowast

for 119905 s isin R (52)

Notice that 119881 R2 rarr B(119883) is also a reversible evolutionoperator on the dual space 119883

lowast of 119883 (see eg [21]) Let usdenote the elements of the dual space 119883

lowast by 119909lowast and denote

⟨119909 119909lowast⟩ = 119909

lowast(119909) If 119883lowastlowast is the dual space of 119883lowast then there

exists a continuous linear transformation 119869 119883 rarr 119883lowastlowast

defined by

⟨119909lowast

119869119909⟩ = ⟨119909 119909lowast

⟩ forall119909 isin 119883 and 119909lowast

isin 119883lowast

(53)

As a consequence of the Hahn-Banach theorem the operator119869 is an isometry (ie 119869119909 = 119909 for every 119909 isin 119883) and henceit is a one-to-one mapping

Proposition 20 A reversible evolution operator119880 on a reflex-ive Banach space119883 has a generalized ℓ-exponential trichotomy(generalized 119903-exponential trichotomy respectively) if andonly if its adjoint evolution operator 119881 has a generalized 119903-exponential trichotomy (generalized ℓ-exponential trichotomyrespectively)

Proof Necessity It is easy to see that if 119880 has a generalizedℓ-exponential trichotomy (a generalized 119903-exponential tri-chotomy) with projection valued functions 119875

+(119905) and 119875

minus(119905)

as in Theorem 11 (Theorem 18) then the adjoint evolutionoperator 119881 has a generalized 119903-exponential trichotomy (ageneralized ℓ-exponential trichotomy) with projections

119868 minus 119875lowast

+(119905) and 119868 minus 119875

lowast

minus(119905) (54)

Sufficiency We now assume that 119881 has a generalized ℓ-exponential trichotomy with projection valued functions 119875

+

and 119875minusas in Theorem 11 For each 119905 isin R we consider the

following operators

+(119905) 119883 997888rarr 119883

+(119905) 119909 = 119869

minus1

119875lowast

+(119905) 119869119909

minus(119905) 119883 997888rarr 119883

minus(119905) 119909 = 119869

minus1

119875lowast

minus(119905) 119869119909

(55)

We notice that +and

minusare projection valued functions on

119883 We first prove that they are invariant for 119880 We have

⟨+(119905) 119880 (119905 119904) 119909 119909

lowast

⟩

= ⟨119909lowast

119869+(119905) 119880 (119905 119904) 119909⟩ = ⟨119909

lowast

119875lowast

+(119905) 119869119880 (119905 119904) 119909⟩

= ⟨119875+(119905) 119909lowast

119869119880 (119905 119904) 119909⟩ = ⟨119880 (119905 119904) 119909 119875+(119905) 119909lowast

⟩

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

(l3) 1198753is compatible on the left with 119880

(l4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(l5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(l6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(l7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

If there exists a positive constant 119886 gt 0 such that 119906(119905) ge 119886 forall 119905 isin R then we say that 119880 has an ℓ-exponential trichotomy(in the sense of Elaydi-Hajek)

The notion of ℓ-exponential trichotomy was introducedin [11] for linear differential equations in finite-dimensionalspaces and in [13] for reversible evolution operators In thispaper we do not assume the invertibility of the evolutionoperator on the whole space119883 This degree of generalizationis motivated by possible applications to partial differentialequations in which the evolution operators are not invertible(see [17] and the references therein) Furthermore in [11] theauthors require thematrix119860(119905) to be bounded on the real lineIn the case of abstract evolution operators this corresponds tothe assumption that the evolution operators are exponentiallybounded (ie there exist 120596 gt 0 and 119872 ge 1 such that119880(119905 119904) le 119872119890

120596(119905minus119904) for all 119905 ge 119904) Unfortunately most ofthe evolution operators do not possess this property (see [18pp 12])

Remark 7 If 119880 has a generalized ℓ-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (10)

that is the projection valued functions 1198751 1198752 and 119875

3are

uniformly bounded

Remark 8 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line) (see [1]) This means that if anevolution operator has a generalized exponential dichotomy(on the real line) then it also has a generalized ℓ-exponentialtrichotomy

Remark 9 If 119880 has a generalized ℓ-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then condi-

tions (7) and (8) imply that

119880 (119905 119904) 119875119894(119904) 119909 997888rarr 0 as 119905 997888rarr +infin 119894 isin 1 3

119880119875119894

(119905 119904) 119875119894(119904) 119909 997888rarr 0 as 119905 997888rarr minusinfin 119894 isin 2 3

(11)

If the evolution operator119880 is generated by a well-possed evo-lution equation then the relations above lead to a geometricdescription of the extended state space R times 119883 of (3) Moreprecisely it splits into three invariant vector bundles (thestable bundle the unstable bundle and the center bundle)For more details about the geometric theory of discrete andcontinuous nonautonomous dynamical systems we refer thereader to [19] and the references therein

One can easily give an example of an evolution operatorwhich has a generalized ℓ-exponential trichotomy and doesnot have an ℓ-exponential trichotomy

Example 10 The evolution operator 119880 Δ rarr B(R3)

defined by

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890minusint

119905

119904sign(120591)119906(120591)119889120591

1199093)

(12)

has a generalized ℓ-exponential trichotomy with canonicalprojections for each 119906 isin U However it does not have anℓ-exponential trichotomy for the function 119906 considered inrelation (9)

Themain goal of the paper is to extend some results from[11] to the general case of abstract evolution operators Wenote that we do not need to assume neither the invertibility ofthe evolution operators on the whole space119883 nor a boundedgrowth condition on the evolution operators (unlike the caseof differential equations with bounded coefficients as in [11])

Theorem 11 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875

+ 119875minus R rarr B(119883) which

are invariant for 119880 and there exist a constant 119870 ge 1 and afunction 119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

minus(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

Proof Necessity We consider

119875+(119905) = 119875

1(119905) + 119875

3(119905) 119875

minus(119905) = 119875

1(119905) for 119905 isin R (13)

It is easy to see that the first two conditions hold and we havethat

119876+(119905) = 119875

2(119905) 119876

minus(119905) = 119875

2(119905) + 119875

3(119905) forall119905 isin R (14)

These imply that 119880(119905 119904)|R(119876

+(119904))

R(119876+(119904)) rarr R(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+with the inverse

119880119876+

(119904 119905) = 1198801198752

(119904 119905) (15)

4 Journal of Function Spaces and Applications

and 119880(119905 119904)|R(119876

minus(119904))

R(119876minus(119904)) rarr R(119876

minus(119905)) is an

isomorphism for all (119905 119904) isin Δminuswith

119880119876minus

(119904 119905) = 1198801198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (16)

Hence

119880119876minus

(119904 119905) 119876minus(119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (17)

for all (119905 119904) isin Δminus We first prove that 119880

119876minus

(119904 119905) consideredabove is correctly defined Indeed for every 119909 isin R(119876

minus(119905))

we have that

119880119876minus

(119904 119905) 119909

= 1198801198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) 1198801198752

(119904 119905) 1198752(119905) 119909 + 119875

3(119904) 1198801198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

+ 1198753(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

= 119876minus(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

(18)

which belongs toR(119876minus(119904)) Moreover a simple computation

shows that

119880 (119905 119904) 119880119876minus

(119904 119905) 119876minus(119905) = 119876

minus(119905)

119880119876minus

(119904 119905) 119880 (119905 119904) 119876minus(119904) = 119876

minus(119904)

for (119905 119904) isin Δminus

(19)

For (119905 119904) isin Δ+ we have

1003817100381710038171003817119880 (119905 119904) 119875+(119904)

1003817100381710038171003817 le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817

+1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817 le 2119870119890minusint

119905

119904119906(120591)119889120591

(20)

10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817=100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119870119890minusint

119905

119904119906(120591)119889120591

(21)

If (119905 119904) isin Δminus then we get

1003817100381710038171003817119880 (119905 119904) 119875minus(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(22)

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817le100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(23)

Sufficiency We set

1198751(119905) = 119875

minus(119905) 1198752(119905)

= 119876+(119905) 1198753(119905)

= 119875+(119905) minus 119875

minus(119905)

for 119905 isin R

(24)

First we observe that

119876+(119905) 119876minus(119905) = 119876

minus(119905) 119876+(119905) = 119876

+(119905) 119905 isin R (25)

119875+(119905) minus 119875

minus(119905) = 119875

+(119905) 119876minus(119905) = 119876

minus(119905) 119875+(119905) 119905 isin R (26)

The restriction 119880(119905 119904)|R(1198752(119904))

R(1198752(119904)) rarr R(119875

2(119905)) is an

isomorphism for all 119905 ge 119904 and by relation (25) we have

1198801198752

(119904 119905) 1198752(119905)

=

119880119876+

(119904 119905) 119876+(119905) if 119905 ge 119904 ge 0

119876+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119876+(119905) if 0 ge 119905 ge 119904

119876+(119904) 119880119876minus

(119904 0) 119876minus(0) 119880119876+

(0 119905) 119876+(119905) if 119905 ge 0 ge 119904

(27)

Also the restriction 119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905))

is an isomorphism for all (119905 119904) isin Δ with 0 ge 119905 ge 119904 and byrelation (26) we deduce that

1198801198753

(119904 119905) 1198753(119905) = 119875

+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119875+(119905) (28)

We now verify the inequalities in Definition 6 consideringthree cases

(1) For 119905 ge 119904 ge 0 we have

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119875minus(119904)

1003817100381710038171003817

le 119872minus119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119876minus(119904)

1003817100381710038171003817

le (1 +119872minus)119870119890minusint

119905

119904119906(120591)119889120591

(29)

(2) If 0 ge 119905 ge 119904 then we obtain

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119876+ (119905)

1003817100381710038171003817 le (1 +119872+)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le1003817100381710038171003817119875+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119875+ (119905)

1003817100381710038171003817 le 1198722

+119870119890minusint

119905

119904119906(120591)119889120591

(30)

Journal of Function Spaces and Applications 5

(3) For 119905 ge 0 ge 119904 using the evolution property (1198902) we

get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817

=1003817100381710038171003817119880 (119905 0) 119875

+(0) 119880 (0 119904) 119875

minus(119904)

1003817100381710038171003817

le1003817100381710038171003817119880 (119905 0) 119875

+(0)

1003817100381710038171003817

1003817100381710038171003817119880 (0 119904) 119875minus(119904)

1003817100381710038171003817

le 119870119890minusint

119905

0119906(120591)119889120591

119870119890minusint

0

119904119906(120591)119889120591

= 1198702

119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

times10038171003817100381710038171003817119880119876minus

(119904 0) 119876minus(0)

10038171003817100381710038171003817

10038171003817100381710038171003817119880119876+

(0 119905) 119876+(119905)

10038171003817100381710038171003817

le (1 +119872+)119870119890minusint

0

119904119906(120591)119889120591

119870119890minusint

119905

0119906(120591)119889120591

= (1 +119872+)1198702

119890minusint

119905

119904119906(120591)119889120591

(31)

These complete the proof of the theorem

Remark 12 Note that in comparison to Lemma 12 in [11] weassume that 119875

+(119905) is bounded for 119905 le 0 and 119875

minus(119905) is bounded

for 119905 ge 0 In our opinion these conditions must also beadded in the particular case of evolution operators generatedby differential equations with bounded coefficients This ismotivated by the fact that the proof of the above mentionedlemma seems to not be quite accurateMore precisely the firstcomputation in (ii) rArr (iii) holds only for 119905 ge 0 ge 119904 and notfor all 119905 ge 119904 Indeed using the notations from [11] and takingfor example the case 119905 ge 119904 ge 0 we have

10038171003817100381710038171003817119883 (119905) 119875

1119883minus1

(119904)10038171003817100381710038171003817=10038171003817100381710038171003817119883 (119905) 119875

+119875minus119883minus1

(119904)10038171003817100381710038171003817

le10038171003817100381710038171003817119883 (119905) 119875

+119883minus1

(119904)10038171003817100381710038171003817

10038171003817100381710038171003817119883 (119904) 119875

minus119883minus1

(119904)10038171003817100381710038171003817

le 119871119890minus120572(119905minus119904) 10038171003817100381710038171003817

119883 (119904) 119875minus119883minus1

(119904)10038171003817100381710038171003817

(32)

Thus in order to prove (iii) one must add the assumptionthat there exists a constant119872

minusge 1 such that

10038171003817100381710038171003817119883 (119905) 119875

minus119883minus1

(119905)10038171003817100381710038171003817le 119872minus forall119905 ge 0 (33)

In our case this is equivalent to

sup119905ge0

1003817100381710038171003817119875minus (119905)1003817100381710038171003817 le 119872

minuslt infin (34)

The projection valued function 119875minusis defined on the whole

real line Because the evolution operator 119880 has a generalizedexponential dichotomy with 119875

minusonly on the left half-line we

have that 119875minus(119905) is bounded for 119905 le 0 Still this does not give

any information about the boundedness for 119905 gt 0 (the samecomment applies to 119875

+)

Theorem 13 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875119876 R rarr B(119883) which areinvariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 and 119875(119905)119876(119905) = 119876(119905)119875(119905)for all 119905 isin R

(2) sup119905le0

119875(119905) le 1198721and sup

119905ge0119876(119905) le 119872

2

(3) the restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ with 119905 ge 0(4) the restriction of119880(119905 119904) onR(119876(119904)) intoR(119876(119905)) is an

isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119905 ge 0

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119904 le 0

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

Proof NecessityWe consider

119875 (119905) = 1198751(119905) + 119875

3(119905) 119876 (119905) = 119875

2(119905) + 119875

3(119905) (35)

for 119905 isin R It is easy to see that 119875(119905)119876(119905) = 119876(119905)119875(119905) = 1198753(119905)

Hence 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 On the other hand since119868minus119875(119905) = 119875

2(119905) we have that the restriction119880(119905 119904)

|R(119868minus119875(119904)) isan isomorphism from the range of 119868 minus 119875(119904) onto the range of119868 minus 119875(119905) for all (119905 119904) isin Δ with 119905 ge 0 and

119880119868minus119875

(119904 119905) = 1198801198752

(119904 119905) (36)

By a similar argument as in the proof ofTheorem 11 we obtainthat 119880(119905 119904)

|R(119876(119904)) is an isomorphism from the range of 119876(119904)

onto the range of 119876(119905) for all (119905 119904) isin Δminusand

119880119876(119904 119905) 119876 (119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (37)

For (119905 119904) isin Δ+ we have

119880 (119905 119904) 119875 (119904) le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 +1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(38)

and for (119905 119904) isin Δminus we get

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 le

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(39)

If (119905 119904) isin Δ with 119905 ge 0 then it follows that

1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))1003817100381710038171003817 =

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

(40)

6 Journal of Function Spaces and Applications

and if (119905 119904) isin Δ with 119904 le 0 then we have

119880 (119905 119904) (119868 minus 119876 (119904)) =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(41)

SufficiencyWe set

1198751(119905) = 119868 minus 119876 (119905)

1198752(119905) = 119868 minus 119875 (119905)

1198753(119905) = 119875 (119905) 119876 (119905)

(42)

for 119905 isin R We observe that 119868 minus 119875(119905) = 119876(119905)(119868 minus 119875(119905)) = (119868 minus

119875(119905))119876(119905) for all 119905 isin R Now it is easy to see that the restrictionof119880(119905 119904) onR(119875

2(119904)) intoR(119875

2(119905)) is an isomorphism for all

119905 ge 119904 and

1198801198752

(119904 119905) 1198752(119905)

=119880119868minus119875

(119904 119905) (119868 minus 119875 (119905)) for 119905 ge 119904 with 119905 ge 0

(119868 minus 119875 (119904)) 119880119876(119904 119905) 119876 (119905) (119868 minus 119875 (119905)) for 0 ge 119905 ge s

(43)

Also the restriction119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905)) is

an isomorphism for all (119905 119904) isin Δminusand

1198801198753

(119904 119905) 1198753(119905) = 119875 (119904) 119880

119876(119904 119905) 119876 (119905) 119875 (119905) (44)

Let (119905 119904) isin Δ If 119905 ge 119904 ge 0 then we have1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

= 119880 (119905 119904) 119875 (119904) (119868 minus 119876 (119904))

le 119880 (119905 119904) 119875 (119904) 119868 minus 119876 (119904)

le (1 +1198722)119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 le 119880 (119905 119904) 119875 (119904) 119876 (119904)

le 1198722119870119890minusint

119905

119904119906(120591)119889120591

(45)

If 0 ge 119905 ge 119904 then it follows1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119868 minus 119875 (119904)

times1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)

1003817100381710038171003817 119868 minus 119875 (119905)

le (1 +1198721)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le 119875 (119904)

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 119875 (119905)

le 1198722

1119870119890minusint

119905

119904119906(120591)119889120591

(46)

Finally if 119905 ge 0 ge 119904 then we get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904)) le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(47)

Therefore119880 has a generalized ℓ-exponential trichotomy

32 Generalized 119903-Exponential Trichotomy Nowwe considera concept of generalized exponential trichotomy which isdual in a certain sense to the one given in Definition 6

Definition 14 We say that an evolution operator 119880 hasa generalized 119903-exponential trichotomy if there exist threesupplementary projection valued functions119875

11198752 and119875

3and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(r1) 1198751is invariant for 119880

(r2) 1198752is compatible with 119880

(r3) 1198753is compatible on the right with 119880

(r4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(r7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

If there is a positive constant 119886 gt 0 such that 119906(119905) ge 119886 for all119905 isin R then we say that 119880 has an 119903-exponential trichotomy

The notion of 119903-exponential trichotomy was defined in[13] for reversible evolution operators

Remark 15 If 119880 has a generalized 119903-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (48)

Remark 16 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line)

Example 17 Theevolution operator119880 Δ rarr B(R3) definedby

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890int

119905

119904sign(120591)119906(120591)119889120591

1199093)

(49)

has a generalized 119903-exponential trichotomy with canonicalprojections for each 119906 isin U

Proceeding in a similarmanner to that inTheorems 11 and13 we get the following

Journal of Function Spaces and Applications 7

Theorem 18 Let 119880 Δ rarr B(119883) be an evolution operatorThe following statements are equivalent

(i) 119880 has a generalized 119903-exponential trichotomy(ii) There exist two projection valued functions 119875

+and 119875

minus

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

+(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

(iii) There exist two projection valued functions 119875 and 119876

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that

(1) 119875(119905)119876(119905) = 119876(119905)119875(119905) = 0 for all 119905 isin R(2) sup

119905le0119875(119905) le 119872

1and sup

119905ge0119876(119905) le 119872

2

(3) The restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of 119880(119905 119904) on R(119876(119904)) into R(119876(119905)) isan isomorphism for all (119905 119904) isin Δ with 119904 le 0

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119905 ge 0

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119904 le 0

Let us notice that the exponential trichotomy consideredin Definition 14 seems to be closer to exponential dichotomyon the real line than the other trichotomies since in thecase of linear differential equations it implies exponentialdichotomy on both half lines and (3) has no nontrivialbounded solution (see Lemma 1 in [20])

Remark 19 As a consequence of Lemma 22 from [11] thedifferential equation

(119905) = 119860 (119905) 119909 (119905) 119905 isin R (50)

where 119909(119905) isin C119899 and 119860(119905) is a bounded and continuous 119899 times 119899

matrix on the whole real line that is sup119905isinR119860(119905) lt infin has

an ℓ-exponential trichotomy if and only if its adjoint equation

119910 (119905) = minus119860lowast

(119905) 119910 (119905) 119905 isin R (51)

has an 119903-exponential trichotomy where 119860lowast(119905) is the adjoint

matrix of 119860(119905) When assuming that (50) is defined on a

Hilbert spaceH this result still holds In fact it remains validfor any reversible evolution operator on reflexive Banachspaces in a certain sense which we explain below

If 119880 R2 rarr B(119883) is a reversible evolution operator ona Banach space119883 then we put

119881 (119905 119904) = 119880(119904 119905)lowast

for 119905 s isin R (52)

Notice that 119881 R2 rarr B(119883) is also a reversible evolutionoperator on the dual space 119883

lowast of 119883 (see eg [21]) Let usdenote the elements of the dual space 119883

lowast by 119909lowast and denote

⟨119909 119909lowast⟩ = 119909

lowast(119909) If 119883lowastlowast is the dual space of 119883lowast then there

exists a continuous linear transformation 119869 119883 rarr 119883lowastlowast

defined by

⟨119909lowast

119869119909⟩ = ⟨119909 119909lowast

⟩ forall119909 isin 119883 and 119909lowast

isin 119883lowast

(53)

As a consequence of the Hahn-Banach theorem the operator119869 is an isometry (ie 119869119909 = 119909 for every 119909 isin 119883) and henceit is a one-to-one mapping

Proposition 20 A reversible evolution operator119880 on a reflex-ive Banach space119883 has a generalized ℓ-exponential trichotomy(generalized 119903-exponential trichotomy respectively) if andonly if its adjoint evolution operator 119881 has a generalized 119903-exponential trichotomy (generalized ℓ-exponential trichotomyrespectively)

Proof Necessity It is easy to see that if 119880 has a generalizedℓ-exponential trichotomy (a generalized 119903-exponential tri-chotomy) with projection valued functions 119875

+(119905) and 119875

minus(119905)

as in Theorem 11 (Theorem 18) then the adjoint evolutionoperator 119881 has a generalized 119903-exponential trichotomy (ageneralized ℓ-exponential trichotomy) with projections

119868 minus 119875lowast

+(119905) and 119868 minus 119875

lowast

minus(119905) (54)

Sufficiency We now assume that 119881 has a generalized ℓ-exponential trichotomy with projection valued functions 119875

+

and 119875minusas in Theorem 11 For each 119905 isin R we consider the

following operators

+(119905) 119883 997888rarr 119883

+(119905) 119909 = 119869

minus1

119875lowast

+(119905) 119869119909

minus(119905) 119883 997888rarr 119883

minus(119905) 119909 = 119869

minus1

119875lowast

minus(119905) 119869119909

(55)

We notice that +and

minusare projection valued functions on

119883 We first prove that they are invariant for 119880 We have

⟨+(119905) 119880 (119905 119904) 119909 119909

lowast

⟩

= ⟨119909lowast

119869+(119905) 119880 (119905 119904) 119909⟩ = ⟨119909

lowast

119875lowast

+(119905) 119869119880 (119905 119904) 119909⟩

= ⟨119875+(119905) 119909lowast

119869119880 (119905 119904) 119909⟩ = ⟨119880 (119905 119904) 119909 119875+(119905) 119909lowast

⟩

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

and 119880(119905 119904)|R(119876

minus(119904))

R(119876minus(119904)) rarr R(119876

minus(119905)) is an

isomorphism for all (119905 119904) isin Δminuswith

119880119876minus

(119904 119905) = 1198801198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (16)

Hence

119880119876minus

(119904 119905) 119876minus(119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (17)

for all (119905 119904) isin Δminus We first prove that 119880

119876minus

(119904 119905) consideredabove is correctly defined Indeed for every 119909 isin R(119876

minus(119905))

we have that

119880119876minus

(119904 119905) 119909

= 1198801198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) 1198801198752

(119904 119905) 1198752(119905) 119909 + 119875

3(119904) 1198801198753

(119904 119905) 1198753(119905) 119909

= 1198752(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

+ 1198753(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

= 119876minus(119904) (119880

1198752

(119904 119905) 1198752(119905) 119909 + 119880

1198753

(119904 119905) 1198753(119905) 119909)

(18)

which belongs toR(119876minus(119904)) Moreover a simple computation

shows that

119880 (119905 119904) 119880119876minus

(119904 119905) 119876minus(119905) = 119876

minus(119905)

119880119876minus

(119904 119905) 119880 (119905 119904) 119876minus(119904) = 119876

minus(119904)

for (119905 119904) isin Δminus

(19)

For (119905 119904) isin Δ+ we have

1003817100381710038171003817119880 (119905 119904) 119875+(119904)

1003817100381710038171003817 le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817

+1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817 le 2119870119890minusint

119905

119904119906(120591)119889120591

(20)

10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817=100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119870119890minusint

119905

119904119906(120591)119889120591

(21)

If (119905 119904) isin Δminus then we get

1003817100381710038171003817119880 (119905 119904) 119875minus(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(22)

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817le100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(23)

Sufficiency We set

1198751(119905) = 119875

minus(119905) 1198752(119905)

= 119876+(119905) 1198753(119905)

= 119875+(119905) minus 119875

minus(119905)

for 119905 isin R

(24)

First we observe that

119876+(119905) 119876minus(119905) = 119876

minus(119905) 119876+(119905) = 119876

+(119905) 119905 isin R (25)

119875+(119905) minus 119875

minus(119905) = 119875

+(119905) 119876minus(119905) = 119876

minus(119905) 119875+(119905) 119905 isin R (26)

The restriction 119880(119905 119904)|R(1198752(119904))

R(1198752(119904)) rarr R(119875

2(119905)) is an

isomorphism for all 119905 ge 119904 and by relation (25) we have

1198801198752

(119904 119905) 1198752(119905)

=

119880119876+

(119904 119905) 119876+(119905) if 119905 ge 119904 ge 0

119876+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119876+(119905) if 0 ge 119905 ge 119904

119876+(119904) 119880119876minus

(119904 0) 119876minus(0) 119880119876+

(0 119905) 119876+(119905) if 119905 ge 0 ge 119904

(27)

Also the restriction 119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905))

is an isomorphism for all (119905 119904) isin Δ with 0 ge 119905 ge 119904 and byrelation (26) we deduce that

1198801198753

(119904 119905) 1198753(119905) = 119875

+(119904) 119880119876minus

(119904 119905) 119876minus(119905) 119875+(119905) (28)

We now verify the inequalities in Definition 6 consideringthree cases

(1) For 119905 ge 119904 ge 0 we have

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119875minus(119904)

1003817100381710038171003817

le 119872minus119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=10038171003817100381710038171003817119880119876+

(119904 119905) 119876+(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

+(119904) 119876minus(119904)

1003817100381710038171003817

le (1 +119872minus)119870119890minusint

119905

119904119906(120591)119889120591

(29)

(2) If 0 ge 119905 ge 119904 then we obtain

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119876+ (119905)

1003817100381710038171003817 le (1 +119872+)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le1003817100381710038171003817119875+ (119904)

1003817100381710038171003817

10038171003817100381710038171003817119880119876minus

(119904 119905) 119876minus(119905)

10038171003817100381710038171003817

times1003817100381710038171003817119875+ (119905)

1003817100381710038171003817 le 1198722

+119870119890minusint

119905

119904119906(120591)119889120591

(30)

Journal of Function Spaces and Applications 5

(3) For 119905 ge 0 ge 119904 using the evolution property (1198902) we

get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817

=1003817100381710038171003817119880 (119905 0) 119875

+(0) 119880 (0 119904) 119875

minus(119904)

1003817100381710038171003817

le1003817100381710038171003817119880 (119905 0) 119875

+(0)

1003817100381710038171003817

1003817100381710038171003817119880 (0 119904) 119875minus(119904)

1003817100381710038171003817

le 119870119890minusint

119905

0119906(120591)119889120591

119870119890minusint

0

119904119906(120591)119889120591

= 1198702

119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

times10038171003817100381710038171003817119880119876minus

(119904 0) 119876minus(0)

10038171003817100381710038171003817

10038171003817100381710038171003817119880119876+

(0 119905) 119876+(119905)

10038171003817100381710038171003817

le (1 +119872+)119870119890minusint

0

119904119906(120591)119889120591

119870119890minusint

119905

0119906(120591)119889120591

= (1 +119872+)1198702

119890minusint

119905

119904119906(120591)119889120591

(31)

These complete the proof of the theorem

Remark 12 Note that in comparison to Lemma 12 in [11] weassume that 119875

+(119905) is bounded for 119905 le 0 and 119875

minus(119905) is bounded

for 119905 ge 0 In our opinion these conditions must also beadded in the particular case of evolution operators generatedby differential equations with bounded coefficients This ismotivated by the fact that the proof of the above mentionedlemma seems to not be quite accurateMore precisely the firstcomputation in (ii) rArr (iii) holds only for 119905 ge 0 ge 119904 and notfor all 119905 ge 119904 Indeed using the notations from [11] and takingfor example the case 119905 ge 119904 ge 0 we have

10038171003817100381710038171003817119883 (119905) 119875

1119883minus1

(119904)10038171003817100381710038171003817=10038171003817100381710038171003817119883 (119905) 119875

+119875minus119883minus1

(119904)10038171003817100381710038171003817

le10038171003817100381710038171003817119883 (119905) 119875

+119883minus1

(119904)10038171003817100381710038171003817

10038171003817100381710038171003817119883 (119904) 119875

minus119883minus1

(119904)10038171003817100381710038171003817

le 119871119890minus120572(119905minus119904) 10038171003817100381710038171003817

119883 (119904) 119875minus119883minus1

(119904)10038171003817100381710038171003817

(32)

Thus in order to prove (iii) one must add the assumptionthat there exists a constant119872

minusge 1 such that

10038171003817100381710038171003817119883 (119905) 119875

minus119883minus1

(119905)10038171003817100381710038171003817le 119872minus forall119905 ge 0 (33)

In our case this is equivalent to

sup119905ge0

1003817100381710038171003817119875minus (119905)1003817100381710038171003817 le 119872

minuslt infin (34)

The projection valued function 119875minusis defined on the whole

real line Because the evolution operator 119880 has a generalizedexponential dichotomy with 119875

minusonly on the left half-line we

have that 119875minus(119905) is bounded for 119905 le 0 Still this does not give

any information about the boundedness for 119905 gt 0 (the samecomment applies to 119875

+)

Theorem 13 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875119876 R rarr B(119883) which areinvariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 and 119875(119905)119876(119905) = 119876(119905)119875(119905)for all 119905 isin R

(2) sup119905le0

119875(119905) le 1198721and sup

119905ge0119876(119905) le 119872

2

(3) the restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ with 119905 ge 0(4) the restriction of119880(119905 119904) onR(119876(119904)) intoR(119876(119905)) is an

isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119905 ge 0

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119904 le 0

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

Proof NecessityWe consider

119875 (119905) = 1198751(119905) + 119875

3(119905) 119876 (119905) = 119875

2(119905) + 119875

3(119905) (35)

for 119905 isin R It is easy to see that 119875(119905)119876(119905) = 119876(119905)119875(119905) = 1198753(119905)

Hence 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 On the other hand since119868minus119875(119905) = 119875

2(119905) we have that the restriction119880(119905 119904)

|R(119868minus119875(119904)) isan isomorphism from the range of 119868 minus 119875(119904) onto the range of119868 minus 119875(119905) for all (119905 119904) isin Δ with 119905 ge 0 and

119880119868minus119875

(119904 119905) = 1198801198752

(119904 119905) (36)

By a similar argument as in the proof ofTheorem 11 we obtainthat 119880(119905 119904)

|R(119876(119904)) is an isomorphism from the range of 119876(119904)

onto the range of 119876(119905) for all (119905 119904) isin Δminusand

119880119876(119904 119905) 119876 (119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (37)

For (119905 119904) isin Δ+ we have

119880 (119905 119904) 119875 (119904) le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 +1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(38)

and for (119905 119904) isin Δminus we get

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 le

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(39)

If (119905 119904) isin Δ with 119905 ge 0 then it follows that

1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))1003817100381710038171003817 =

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

(40)

6 Journal of Function Spaces and Applications

and if (119905 119904) isin Δ with 119904 le 0 then we have

119880 (119905 119904) (119868 minus 119876 (119904)) =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(41)

SufficiencyWe set

1198751(119905) = 119868 minus 119876 (119905)

1198752(119905) = 119868 minus 119875 (119905)

1198753(119905) = 119875 (119905) 119876 (119905)

(42)

for 119905 isin R We observe that 119868 minus 119875(119905) = 119876(119905)(119868 minus 119875(119905)) = (119868 minus

119875(119905))119876(119905) for all 119905 isin R Now it is easy to see that the restrictionof119880(119905 119904) onR(119875

2(119904)) intoR(119875

2(119905)) is an isomorphism for all

119905 ge 119904 and

1198801198752

(119904 119905) 1198752(119905)

=119880119868minus119875

(119904 119905) (119868 minus 119875 (119905)) for 119905 ge 119904 with 119905 ge 0

(119868 minus 119875 (119904)) 119880119876(119904 119905) 119876 (119905) (119868 minus 119875 (119905)) for 0 ge 119905 ge s

(43)

Also the restriction119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905)) is

an isomorphism for all (119905 119904) isin Δminusand

1198801198753

(119904 119905) 1198753(119905) = 119875 (119904) 119880

119876(119904 119905) 119876 (119905) 119875 (119905) (44)

Let (119905 119904) isin Δ If 119905 ge 119904 ge 0 then we have1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

= 119880 (119905 119904) 119875 (119904) (119868 minus 119876 (119904))

le 119880 (119905 119904) 119875 (119904) 119868 minus 119876 (119904)

le (1 +1198722)119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 le 119880 (119905 119904) 119875 (119904) 119876 (119904)

le 1198722119870119890minusint

119905

119904119906(120591)119889120591

(45)

If 0 ge 119905 ge 119904 then it follows1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119868 minus 119875 (119904)

times1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)

1003817100381710038171003817 119868 minus 119875 (119905)

le (1 +1198721)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le 119875 (119904)

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 119875 (119905)

le 1198722

1119870119890minusint

119905

119904119906(120591)119889120591

(46)

Finally if 119905 ge 0 ge 119904 then we get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904)) le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(47)

Therefore119880 has a generalized ℓ-exponential trichotomy

32 Generalized 119903-Exponential Trichotomy Nowwe considera concept of generalized exponential trichotomy which isdual in a certain sense to the one given in Definition 6

Definition 14 We say that an evolution operator 119880 hasa generalized 119903-exponential trichotomy if there exist threesupplementary projection valued functions119875

11198752 and119875

3and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(r1) 1198751is invariant for 119880

(r2) 1198752is compatible with 119880

(r3) 1198753is compatible on the right with 119880

(r4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(r7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

If there is a positive constant 119886 gt 0 such that 119906(119905) ge 119886 for all119905 isin R then we say that 119880 has an 119903-exponential trichotomy

The notion of 119903-exponential trichotomy was defined in[13] for reversible evolution operators

Remark 15 If 119880 has a generalized 119903-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (48)

Remark 16 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line)

Example 17 Theevolution operator119880 Δ rarr B(R3) definedby

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890int

119905

119904sign(120591)119906(120591)119889120591

1199093)

(49)

has a generalized 119903-exponential trichotomy with canonicalprojections for each 119906 isin U

Proceeding in a similarmanner to that inTheorems 11 and13 we get the following

Journal of Function Spaces and Applications 7

Theorem 18 Let 119880 Δ rarr B(119883) be an evolution operatorThe following statements are equivalent

(i) 119880 has a generalized 119903-exponential trichotomy(ii) There exist two projection valued functions 119875

+and 119875

minus

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

+(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

(iii) There exist two projection valued functions 119875 and 119876

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that

(1) 119875(119905)119876(119905) = 119876(119905)119875(119905) = 0 for all 119905 isin R(2) sup

119905le0119875(119905) le 119872

1and sup

119905ge0119876(119905) le 119872

2

(3) The restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of 119880(119905 119904) on R(119876(119904)) into R(119876(119905)) isan isomorphism for all (119905 119904) isin Δ with 119904 le 0

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119905 ge 0

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119904 le 0

Let us notice that the exponential trichotomy consideredin Definition 14 seems to be closer to exponential dichotomyon the real line than the other trichotomies since in thecase of linear differential equations it implies exponentialdichotomy on both half lines and (3) has no nontrivialbounded solution (see Lemma 1 in [20])

Remark 19 As a consequence of Lemma 22 from [11] thedifferential equation

(119905) = 119860 (119905) 119909 (119905) 119905 isin R (50)

where 119909(119905) isin C119899 and 119860(119905) is a bounded and continuous 119899 times 119899

matrix on the whole real line that is sup119905isinR119860(119905) lt infin has

an ℓ-exponential trichotomy if and only if its adjoint equation

119910 (119905) = minus119860lowast

(119905) 119910 (119905) 119905 isin R (51)

has an 119903-exponential trichotomy where 119860lowast(119905) is the adjoint

matrix of 119860(119905) When assuming that (50) is defined on a

Hilbert spaceH this result still holds In fact it remains validfor any reversible evolution operator on reflexive Banachspaces in a certain sense which we explain below

If 119880 R2 rarr B(119883) is a reversible evolution operator ona Banach space119883 then we put

119881 (119905 119904) = 119880(119904 119905)lowast

for 119905 s isin R (52)

Notice that 119881 R2 rarr B(119883) is also a reversible evolutionoperator on the dual space 119883

lowast of 119883 (see eg [21]) Let usdenote the elements of the dual space 119883

lowast by 119909lowast and denote

⟨119909 119909lowast⟩ = 119909

lowast(119909) If 119883lowastlowast is the dual space of 119883lowast then there

exists a continuous linear transformation 119869 119883 rarr 119883lowastlowast

defined by

⟨119909lowast

119869119909⟩ = ⟨119909 119909lowast

⟩ forall119909 isin 119883 and 119909lowast

isin 119883lowast

(53)

As a consequence of the Hahn-Banach theorem the operator119869 is an isometry (ie 119869119909 = 119909 for every 119909 isin 119883) and henceit is a one-to-one mapping

Proposition 20 A reversible evolution operator119880 on a reflex-ive Banach space119883 has a generalized ℓ-exponential trichotomy(generalized 119903-exponential trichotomy respectively) if andonly if its adjoint evolution operator 119881 has a generalized 119903-exponential trichotomy (generalized ℓ-exponential trichotomyrespectively)

Proof Necessity It is easy to see that if 119880 has a generalizedℓ-exponential trichotomy (a generalized 119903-exponential tri-chotomy) with projection valued functions 119875

+(119905) and 119875

minus(119905)

as in Theorem 11 (Theorem 18) then the adjoint evolutionoperator 119881 has a generalized 119903-exponential trichotomy (ageneralized ℓ-exponential trichotomy) with projections

119868 minus 119875lowast

+(119905) and 119868 minus 119875

lowast

minus(119905) (54)

Sufficiency We now assume that 119881 has a generalized ℓ-exponential trichotomy with projection valued functions 119875

+

and 119875minusas in Theorem 11 For each 119905 isin R we consider the

following operators

+(119905) 119883 997888rarr 119883

+(119905) 119909 = 119869

minus1

119875lowast

+(119905) 119869119909

minus(119905) 119883 997888rarr 119883

minus(119905) 119909 = 119869

minus1

119875lowast

minus(119905) 119869119909

(55)

We notice that +and

minusare projection valued functions on

119883 We first prove that they are invariant for 119880 We have

⟨+(119905) 119880 (119905 119904) 119909 119909

lowast

⟩

= ⟨119909lowast

119869+(119905) 119880 (119905 119904) 119909⟩ = ⟨119909

lowast

119875lowast

+(119905) 119869119880 (119905 119904) 119909⟩

= ⟨119875+(119905) 119909lowast

119869119880 (119905 119904) 119909⟩ = ⟨119880 (119905 119904) 119909 119875+(119905) 119909lowast

⟩

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

(3) For 119905 ge 0 ge 119904 using the evolution property (1198902) we

get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 =1003817100381710038171003817119880 (119905 119904) 119875

minus(119904)

1003817100381710038171003817

=1003817100381710038171003817119880 (119905 0) 119875

+(0) 119880 (0 119904) 119875

minus(119904)

1003817100381710038171003817

le1003817100381710038171003817119880 (119905 0) 119875

+(0)

1003817100381710038171003817

1003817100381710038171003817119880 (0 119904) 119875minus(119904)

1003817100381710038171003817

le 119870119890minusint

119905

0119906(120591)119889120591

119870119890minusint

0

119904119906(120591)119889120591

= 1198702

119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le1003817100381710038171003817119876+ (119904)

1003817100381710038171003817

times10038171003817100381710038171003817119880119876minus

(119904 0) 119876minus(0)

10038171003817100381710038171003817

10038171003817100381710038171003817119880119876+

(0 119905) 119876+(119905)

10038171003817100381710038171003817

le (1 +119872+)119870119890minusint

0

119904119906(120591)119889120591

119870119890minusint

119905

0119906(120591)119889120591

= (1 +119872+)1198702

119890minusint

119905

119904119906(120591)119889120591

(31)

These complete the proof of the theorem

Remark 12 Note that in comparison to Lemma 12 in [11] weassume that 119875

+(119905) is bounded for 119905 le 0 and 119875

minus(119905) is bounded

for 119905 ge 0 In our opinion these conditions must also beadded in the particular case of evolution operators generatedby differential equations with bounded coefficients This ismotivated by the fact that the proof of the above mentionedlemma seems to not be quite accurateMore precisely the firstcomputation in (ii) rArr (iii) holds only for 119905 ge 0 ge 119904 and notfor all 119905 ge 119904 Indeed using the notations from [11] and takingfor example the case 119905 ge 119904 ge 0 we have

10038171003817100381710038171003817119883 (119905) 119875

1119883minus1

(119904)10038171003817100381710038171003817=10038171003817100381710038171003817119883 (119905) 119875

+119875minus119883minus1

(119904)10038171003817100381710038171003817

le10038171003817100381710038171003817119883 (119905) 119875

+119883minus1

(119904)10038171003817100381710038171003817

10038171003817100381710038171003817119883 (119904) 119875

minus119883minus1

(119904)10038171003817100381710038171003817

le 119871119890minus120572(119905minus119904) 10038171003817100381710038171003817

119883 (119904) 119875minus119883minus1

(119904)10038171003817100381710038171003817

(32)

Thus in order to prove (iii) one must add the assumptionthat there exists a constant119872

minusge 1 such that

10038171003817100381710038171003817119883 (119905) 119875

minus119883minus1

(119905)10038171003817100381710038171003817le 119872minus forall119905 ge 0 (33)

In our case this is equivalent to

sup119905ge0

1003817100381710038171003817119875minus (119905)1003817100381710038171003817 le 119872

minuslt infin (34)

The projection valued function 119875minusis defined on the whole

real line Because the evolution operator 119880 has a generalizedexponential dichotomy with 119875

minusonly on the left half-line we

have that 119875minus(119905) is bounded for 119905 le 0 Still this does not give

any information about the boundedness for 119905 gt 0 (the samecomment applies to 119875

+)

Theorem 13 An evolution operator 119880 Δ rarr B(119883) has ageneralized ℓ-exponential trichotomy if and only if there existtwo projection valued functions 119875119876 R rarr B(119883) which areinvariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 and 119875(119905)119876(119905) = 119876(119905)119875(119905)for all 119905 isin R

(2) sup119905le0

119875(119905) le 1198721and sup

119905ge0119876(119905) le 119872

2

(3) the restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ with 119905 ge 0(4) the restriction of119880(119905 119904) onR(119876(119904)) intoR(119876(119905)) is an

isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119905 ge 0

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with

119904 le 0

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

Proof NecessityWe consider

119875 (119905) = 1198751(119905) + 119875

3(119905) 119876 (119905) = 119875

2(119905) + 119875

3(119905) (35)

for 119905 isin R It is easy to see that 119875(119905)119876(119905) = 119876(119905)119875(119905) = 1198753(119905)

Hence 119875(119905) + 119876(119905) minus 119875(119905)119876(119905) = 119868 On the other hand since119868minus119875(119905) = 119875

2(119905) we have that the restriction119880(119905 119904)

|R(119868minus119875(119904)) isan isomorphism from the range of 119868 minus 119875(119904) onto the range of119868 minus 119875(119905) for all (119905 119904) isin Δ with 119905 ge 0 and

119880119868minus119875

(119904 119905) = 1198801198752

(119904 119905) (36)

By a similar argument as in the proof ofTheorem 11 we obtainthat 119880(119905 119904)

|R(119876(119904)) is an isomorphism from the range of 119876(119904)

onto the range of 119876(119905) for all (119905 119904) isin Δminusand

119880119876(119904 119905) 119876 (119905) = 119880

1198752

(119904 119905) 1198752(119905) + 119880

1198753

(119904 119905) 1198753(119905) (37)

For (119905 119904) isin Δ+ we have

119880 (119905 119904) 119875 (119904) le1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 +1003817100381710038171003817119880 (119905 119904) 119875

3(119904)

1003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(38)

and for (119905 119904) isin Δminus we get

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 le

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817+100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817

le 2119870119890minusint

119905

119904119906(120591)119889120591

(39)

If (119905 119904) isin Δ with 119905 ge 0 then it follows that

1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))1003817100381710038171003817 =

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

(40)

6 Journal of Function Spaces and Applications

and if (119905 119904) isin Δ with 119904 le 0 then we have

119880 (119905 119904) (119868 minus 119876 (119904)) =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(41)

SufficiencyWe set

1198751(119905) = 119868 minus 119876 (119905)

1198752(119905) = 119868 minus 119875 (119905)

1198753(119905) = 119875 (119905) 119876 (119905)

(42)

for 119905 isin R We observe that 119868 minus 119875(119905) = 119876(119905)(119868 minus 119875(119905)) = (119868 minus

119875(119905))119876(119905) for all 119905 isin R Now it is easy to see that the restrictionof119880(119905 119904) onR(119875

2(119904)) intoR(119875

2(119905)) is an isomorphism for all

119905 ge 119904 and

1198801198752

(119904 119905) 1198752(119905)

=119880119868minus119875

(119904 119905) (119868 minus 119875 (119905)) for 119905 ge 119904 with 119905 ge 0

(119868 minus 119875 (119904)) 119880119876(119904 119905) 119876 (119905) (119868 minus 119875 (119905)) for 0 ge 119905 ge s

(43)

Also the restriction119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905)) is

an isomorphism for all (119905 119904) isin Δminusand

1198801198753

(119904 119905) 1198753(119905) = 119875 (119904) 119880

119876(119904 119905) 119876 (119905) 119875 (119905) (44)

Let (119905 119904) isin Δ If 119905 ge 119904 ge 0 then we have1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

= 119880 (119905 119904) 119875 (119904) (119868 minus 119876 (119904))

le 119880 (119905 119904) 119875 (119904) 119868 minus 119876 (119904)

le (1 +1198722)119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 le 119880 (119905 119904) 119875 (119904) 119876 (119904)

le 1198722119870119890minusint

119905

119904119906(120591)119889120591

(45)

If 0 ge 119905 ge 119904 then it follows1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119868 minus 119875 (119904)

times1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)

1003817100381710038171003817 119868 minus 119875 (119905)

le (1 +1198721)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le 119875 (119904)

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 119875 (119905)

le 1198722

1119870119890minusint

119905

119904119906(120591)119889120591

(46)

Finally if 119905 ge 0 ge 119904 then we get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904)) le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(47)

Therefore119880 has a generalized ℓ-exponential trichotomy

32 Generalized 119903-Exponential Trichotomy Nowwe considera concept of generalized exponential trichotomy which isdual in a certain sense to the one given in Definition 6

Definition 14 We say that an evolution operator 119880 hasa generalized 119903-exponential trichotomy if there exist threesupplementary projection valued functions119875

11198752 and119875

3and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(r1) 1198751is invariant for 119880

(r2) 1198752is compatible with 119880

(r3) 1198753is compatible on the right with 119880

(r4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(r7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

If there is a positive constant 119886 gt 0 such that 119906(119905) ge 119886 for all119905 isin R then we say that 119880 has an 119903-exponential trichotomy

The notion of 119903-exponential trichotomy was defined in[13] for reversible evolution operators

Remark 15 If 119880 has a generalized 119903-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (48)

Remark 16 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line)

Example 17 Theevolution operator119880 Δ rarr B(R3) definedby

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890int

119905

119904sign(120591)119906(120591)119889120591

1199093)

(49)

has a generalized 119903-exponential trichotomy with canonicalprojections for each 119906 isin U

Proceeding in a similarmanner to that inTheorems 11 and13 we get the following

Journal of Function Spaces and Applications 7

Theorem 18 Let 119880 Δ rarr B(119883) be an evolution operatorThe following statements are equivalent

(i) 119880 has a generalized 119903-exponential trichotomy(ii) There exist two projection valued functions 119875

+and 119875

minus

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

+(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

(iii) There exist two projection valued functions 119875 and 119876

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that

(1) 119875(119905)119876(119905) = 119876(119905)119875(119905) = 0 for all 119905 isin R(2) sup

119905le0119875(119905) le 119872

1and sup

119905ge0119876(119905) le 119872

2

(3) The restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of 119880(119905 119904) on R(119876(119904)) into R(119876(119905)) isan isomorphism for all (119905 119904) isin Δ with 119904 le 0

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119905 ge 0

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119904 le 0

Let us notice that the exponential trichotomy consideredin Definition 14 seems to be closer to exponential dichotomyon the real line than the other trichotomies since in thecase of linear differential equations it implies exponentialdichotomy on both half lines and (3) has no nontrivialbounded solution (see Lemma 1 in [20])

Remark 19 As a consequence of Lemma 22 from [11] thedifferential equation

(119905) = 119860 (119905) 119909 (119905) 119905 isin R (50)

where 119909(119905) isin C119899 and 119860(119905) is a bounded and continuous 119899 times 119899

matrix on the whole real line that is sup119905isinR119860(119905) lt infin has

an ℓ-exponential trichotomy if and only if its adjoint equation

119910 (119905) = minus119860lowast

(119905) 119910 (119905) 119905 isin R (51)

has an 119903-exponential trichotomy where 119860lowast(119905) is the adjoint

matrix of 119860(119905) When assuming that (50) is defined on a

Hilbert spaceH this result still holds In fact it remains validfor any reversible evolution operator on reflexive Banachspaces in a certain sense which we explain below

If 119880 R2 rarr B(119883) is a reversible evolution operator ona Banach space119883 then we put

119881 (119905 119904) = 119880(119904 119905)lowast

for 119905 s isin R (52)

Notice that 119881 R2 rarr B(119883) is also a reversible evolutionoperator on the dual space 119883

lowast of 119883 (see eg [21]) Let usdenote the elements of the dual space 119883

lowast by 119909lowast and denote

⟨119909 119909lowast⟩ = 119909

lowast(119909) If 119883lowastlowast is the dual space of 119883lowast then there

exists a continuous linear transformation 119869 119883 rarr 119883lowastlowast

defined by

⟨119909lowast

119869119909⟩ = ⟨119909 119909lowast

⟩ forall119909 isin 119883 and 119909lowast

isin 119883lowast

(53)

As a consequence of the Hahn-Banach theorem the operator119869 is an isometry (ie 119869119909 = 119909 for every 119909 isin 119883) and henceit is a one-to-one mapping

Proposition 20 A reversible evolution operator119880 on a reflex-ive Banach space119883 has a generalized ℓ-exponential trichotomy(generalized 119903-exponential trichotomy respectively) if andonly if its adjoint evolution operator 119881 has a generalized 119903-exponential trichotomy (generalized ℓ-exponential trichotomyrespectively)

Proof Necessity It is easy to see that if 119880 has a generalizedℓ-exponential trichotomy (a generalized 119903-exponential tri-chotomy) with projection valued functions 119875

+(119905) and 119875

minus(119905)

as in Theorem 11 (Theorem 18) then the adjoint evolutionoperator 119881 has a generalized 119903-exponential trichotomy (ageneralized ℓ-exponential trichotomy) with projections

119868 minus 119875lowast

+(119905) and 119868 minus 119875

lowast

minus(119905) (54)

Sufficiency We now assume that 119881 has a generalized ℓ-exponential trichotomy with projection valued functions 119875

+

and 119875minusas in Theorem 11 For each 119905 isin R we consider the

following operators

+(119905) 119883 997888rarr 119883

+(119905) 119909 = 119869

minus1

119875lowast

+(119905) 119869119909

minus(119905) 119883 997888rarr 119883

minus(119905) 119909 = 119869

minus1

119875lowast

minus(119905) 119869119909

(55)

We notice that +and

minusare projection valued functions on

119883 We first prove that they are invariant for 119880 We have

⟨+(119905) 119880 (119905 119904) 119909 119909

lowast

⟩

= ⟨119909lowast

119869+(119905) 119880 (119905 119904) 119909⟩ = ⟨119909

lowast

119875lowast

+(119905) 119869119880 (119905 119904) 119909⟩

= ⟨119875+(119905) 119909lowast

119869119880 (119905 119904) 119909⟩ = ⟨119880 (119905 119904) 119909 119875+(119905) 119909lowast

⟩

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces and Applications

and if (119905 119904) isin Δ with 119904 le 0 then we have

119880 (119905 119904) (119868 minus 119876 (119904)) =1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(41)

SufficiencyWe set

1198751(119905) = 119868 minus 119876 (119905)

1198752(119905) = 119868 minus 119875 (119905)

1198753(119905) = 119875 (119905) 119876 (119905)

(42)

for 119905 isin R We observe that 119868 minus 119875(119905) = 119876(119905)(119868 minus 119875(119905)) = (119868 minus

119875(119905))119876(119905) for all 119905 isin R Now it is easy to see that the restrictionof119880(119905 119904) onR(119875

2(119904)) intoR(119875

2(119905)) is an isomorphism for all

119905 ge 119904 and

1198801198752

(119904 119905) 1198752(119905)

=119880119868minus119875

(119904 119905) (119868 minus 119875 (119905)) for 119905 ge 119904 with 119905 ge 0

(119868 minus 119875 (119904)) 119880119876(119904 119905) 119876 (119905) (119868 minus 119875 (119905)) for 0 ge 119905 ge s

(43)

Also the restriction119880(119905 119904)|R(1198753(119904))

R(1198753(119904)) rarr R(119875

3(119905)) is

an isomorphism for all (119905 119904) isin Δminusand

1198801198753

(119904 119905) 1198753(119905) = 119875 (119904) 119880

119876(119904 119905) 119876 (119905) 119875 (119905) (44)

Let (119905 119904) isin Δ If 119905 ge 119904 ge 0 then we have1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

= 119880 (119905 119904) 119875 (119904) (119868 minus 119876 (119904))

le 119880 (119905 119904) 119875 (119904) 119868 minus 119876 (119904)

le (1 +1198722)119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817

le 119870119890minusint

119905

119904119906(120591)119889120591

1003817100381710038171003817119880 (119905 119904) 1198753(119904)

1003817100381710038171003817 le 119880 (119905 119904) 119875 (119904) 119876 (119904)

le 1198722119870119890minusint

119905

119904119906(120591)119889120591

(45)

If 0 ge 119905 ge 119904 then it follows1003817100381710038171003817119880 (119905 119904) 119875

1(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904))

le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817le 119868 minus 119875 (119904)

times1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)

1003817100381710038171003817 119868 minus 119875 (119905)

le (1 +1198721)2

119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198753

(119904 119905) 1198753(119905)

10038171003817100381710038171003817le 119875 (119904)

1003817100381710038171003817119880119876 (119904 119905) 119876 (119905)1003817100381710038171003817 119875 (119905)

le 1198722

1119870119890minusint

119905

119904119906(120591)119889120591

(46)

Finally if 119905 ge 0 ge 119904 then we get

1003817100381710038171003817119880 (119905 119904) 1198751(119904)

1003817100381710038171003817 = 119880 (119905 119904) (119868 minus 119876 (119904)) le 119870119890minusint

119905

119904119906(120591)119889120591

100381710038171003817100381710038171198801198752

(119904 119905) 1198752(119905)

10038171003817100381710038171003817=1003817100381710038171003817119880119868minus119875 (119904 119905) (119868 minus 119875 (119905))

1003817100381710038171003817 le 119870119890minusint

119905

119904119906(120591)119889120591

(47)

Therefore119880 has a generalized ℓ-exponential trichotomy

32 Generalized 119903-Exponential Trichotomy Nowwe considera concept of generalized exponential trichotomy which isdual in a certain sense to the one given in Definition 6

Definition 14 We say that an evolution operator 119880 hasa generalized 119903-exponential trichotomy if there exist threesupplementary projection valued functions119875

11198752 and119875

3and

there exist a constant 119870 ge 1 and a function 119906 isin U such thatthe following properties hold

(r1) 1198751is invariant for 119880

(r2) 1198752is compatible with 119880

(r3) 1198753is compatible on the right with 119880

(r4) 119880(119905 119904)119875

1(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r5) 1198801198752

(119904 119905)1198752(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

(r6) 119880(119905 119904)119875

3(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(r7) 1198801198753

(119904 119905)1198753(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

If there is a positive constant 119886 gt 0 such that 119906(119905) ge 119886 for all119905 isin R then we say that 119880 has an 119903-exponential trichotomy

The notion of 119903-exponential trichotomy was defined in[13] for reversible evolution operators

Remark 15 If 119880 has a generalized 119903-exponential trichotomywith projection valued functions 119875

1 1198752 and 119875

3 then we have

that

sup119905isinR

1003817100381710038171003817119875119894 (119905)1003817100381710038171003817 lt infin for 119894 isin 1 2 3 (48)

Remark 16 If in Definition 6 we consider 1198753(119905) = 0 for all

119905 isin R then we obtain the concept of generalized exponentialdichotomy (on the real line)

Example 17 Theevolution operator119880 Δ rarr B(R3) definedby

119880 (119905 119904) (1199091 1199092 1199093)

= (119890minusint

119905

119904119906(120591)119889120591

1199091 119890int

119905

119904119906(120591)119889120591

1199092 119890int

119905

119904sign(120591)119906(120591)119889120591

1199093)

(49)

has a generalized 119903-exponential trichotomy with canonicalprojections for each 119906 isin U

Proceeding in a similarmanner to that inTheorems 11 and13 we get the following

Journal of Function Spaces and Applications 7

Theorem 18 Let 119880 Δ rarr B(119883) be an evolution operatorThe following statements are equivalent

(i) 119880 has a generalized 119903-exponential trichotomy(ii) There exist two projection valued functions 119875

+and 119875

minus

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

+(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

(iii) There exist two projection valued functions 119875 and 119876

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that

(1) 119875(119905)119876(119905) = 119876(119905)119875(119905) = 0 for all 119905 isin R(2) sup

119905le0119875(119905) le 119872

1and sup

119905ge0119876(119905) le 119872

2

(3) The restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of 119880(119905 119904) on R(119876(119904)) into R(119876(119905)) isan isomorphism for all (119905 119904) isin Δ with 119904 le 0

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119905 ge 0

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119904 le 0

Let us notice that the exponential trichotomy consideredin Definition 14 seems to be closer to exponential dichotomyon the real line than the other trichotomies since in thecase of linear differential equations it implies exponentialdichotomy on both half lines and (3) has no nontrivialbounded solution (see Lemma 1 in [20])

Remark 19 As a consequence of Lemma 22 from [11] thedifferential equation

(119905) = 119860 (119905) 119909 (119905) 119905 isin R (50)

where 119909(119905) isin C119899 and 119860(119905) is a bounded and continuous 119899 times 119899

matrix on the whole real line that is sup119905isinR119860(119905) lt infin has

an ℓ-exponential trichotomy if and only if its adjoint equation

119910 (119905) = minus119860lowast

(119905) 119910 (119905) 119905 isin R (51)

has an 119903-exponential trichotomy where 119860lowast(119905) is the adjoint

matrix of 119860(119905) When assuming that (50) is defined on a

Hilbert spaceH this result still holds In fact it remains validfor any reversible evolution operator on reflexive Banachspaces in a certain sense which we explain below

If 119880 R2 rarr B(119883) is a reversible evolution operator ona Banach space119883 then we put

119881 (119905 119904) = 119880(119904 119905)lowast

for 119905 s isin R (52)

Notice that 119881 R2 rarr B(119883) is also a reversible evolutionoperator on the dual space 119883

lowast of 119883 (see eg [21]) Let usdenote the elements of the dual space 119883

lowast by 119909lowast and denote

⟨119909 119909lowast⟩ = 119909

lowast(119909) If 119883lowastlowast is the dual space of 119883lowast then there

exists a continuous linear transformation 119869 119883 rarr 119883lowastlowast

defined by

⟨119909lowast

119869119909⟩ = ⟨119909 119909lowast

⟩ forall119909 isin 119883 and 119909lowast

isin 119883lowast

(53)

As a consequence of the Hahn-Banach theorem the operator119869 is an isometry (ie 119869119909 = 119909 for every 119909 isin 119883) and henceit is a one-to-one mapping

Proposition 20 A reversible evolution operator119880 on a reflex-ive Banach space119883 has a generalized ℓ-exponential trichotomy(generalized 119903-exponential trichotomy respectively) if andonly if its adjoint evolution operator 119881 has a generalized 119903-exponential trichotomy (generalized ℓ-exponential trichotomyrespectively)

Proof Necessity It is easy to see that if 119880 has a generalizedℓ-exponential trichotomy (a generalized 119903-exponential tri-chotomy) with projection valued functions 119875

+(119905) and 119875

minus(119905)

as in Theorem 11 (Theorem 18) then the adjoint evolutionoperator 119881 has a generalized 119903-exponential trichotomy (ageneralized ℓ-exponential trichotomy) with projections

119868 minus 119875lowast

+(119905) and 119868 minus 119875

lowast

minus(119905) (54)

Sufficiency We now assume that 119881 has a generalized ℓ-exponential trichotomy with projection valued functions 119875

+

and 119875minusas in Theorem 11 For each 119905 isin R we consider the

following operators

+(119905) 119883 997888rarr 119883

+(119905) 119909 = 119869

minus1

119875lowast

+(119905) 119869119909

minus(119905) 119883 997888rarr 119883

minus(119905) 119909 = 119869

minus1

119875lowast

minus(119905) 119869119909

(55)

We notice that +and

minusare projection valued functions on

119883 We first prove that they are invariant for 119880 We have

⟨+(119905) 119880 (119905 119904) 119909 119909

lowast

⟩

= ⟨119909lowast

119869+(119905) 119880 (119905 119904) 119909⟩ = ⟨119909

lowast

119875lowast

+(119905) 119869119880 (119905 119904) 119909⟩

= ⟨119875+(119905) 119909lowast

119869119880 (119905 119904) 119909⟩ = ⟨119880 (119905 119904) 119909 119875+(119905) 119909lowast

⟩

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 7

Theorem 18 Let 119880 Δ rarr B(119883) be an evolution operatorThe following statements are equivalent

(i) 119880 has a generalized 119903-exponential trichotomy(ii) There exist two projection valued functions 119875

+and 119875

minus

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that the following properties hold

(1) 119875+(119905)119875minus(119905) = 119875

minus(119905)119875+(119905) = 119875

+(119905) for all 119905 isin R

(2) sup119905le0

119875+(119905) le 119872

+and sup

119905ge0119875minus(119905) le 119872

minus

(3) The restriction of119880(119905 119904) onR(119876+(119904)) intoR(119876

+(119905)) is

an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of119880(119905 119904) onR(119876minus(119904)) intoR(119876

minus(119905)) is

an isomorphism for all (119905 119904) isin Δminus

(5) 119880(119905 119904)119875+(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(6) 119880119876+

(119904 119905)119876+(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

+

(7) 119880(119905 119904)119875minus(119904) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

(8) 119880119876minus

(119904 119905)119876minus(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for all (119905 119904) isin Δ

minus

where 119876+(119905) = 119868 minus 119875

+(119905) and 119876

minus(119905) = 119868 minus 119875

minus(119905) 119905 isin R

(iii) There exist two projection valued functions 119875 and 119876

invariant for119880 and there exist a constant119870 ge 1 and a function119906 isin U such that

(1) 119875(119905)119876(119905) = 119876(119905)119875(119905) = 0 for all 119905 isin R(2) sup

119905le0119875(119905) le 119872

1and sup

119905ge0119876(119905) le 119872

2

(3) The restriction of119880(119905 119904) onR(119868minus119875(119904)) intoR(119868minus119875(119905))

is an isomorphism for all (119905 119904) isin Δ+

(4) The restriction of 119880(119905 119904) on R(119876(119904)) into R(119876(119905)) isan isomorphism for all (119905 119904) isin Δ with 119904 le 0

(5) 119880(119905 119904)119875(119904) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119905 ge 0

(6) 119880119868minus119875

(119904 119905)(119868 minus 119875(119905)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

+

(7) 119880(119905 119904)(119868 minus 119876(119904)) le 119870119890minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ

minus

(8) 119880119876(119904 119905)119876(119905) le 119870119890

minusint

119905

119904119906(120591)119889120591 for (119905 119904) isin Δ with 119904 le 0

Let us notice that the exponential trichotomy consideredin Definition 14 seems to be closer to exponential dichotomyon the real line than the other trichotomies since in thecase of linear differential equations it implies exponentialdichotomy on both half lines and (3) has no nontrivialbounded solution (see Lemma 1 in [20])

Remark 19 As a consequence of Lemma 22 from [11] thedifferential equation

(119905) = 119860 (119905) 119909 (119905) 119905 isin R (50)

where 119909(119905) isin C119899 and 119860(119905) is a bounded and continuous 119899 times 119899

matrix on the whole real line that is sup119905isinR119860(119905) lt infin has

an ℓ-exponential trichotomy if and only if its adjoint equation

119910 (119905) = minus119860lowast

(119905) 119910 (119905) 119905 isin R (51)

has an 119903-exponential trichotomy where 119860lowast(119905) is the adjoint

matrix of 119860(119905) When assuming that (50) is defined on a

Hilbert spaceH this result still holds In fact it remains validfor any reversible evolution operator on reflexive Banachspaces in a certain sense which we explain below

If 119880 R2 rarr B(119883) is a reversible evolution operator ona Banach space119883 then we put

119881 (119905 119904) = 119880(119904 119905)lowast

for 119905 s isin R (52)

Notice that 119881 R2 rarr B(119883) is also a reversible evolutionoperator on the dual space 119883

lowast of 119883 (see eg [21]) Let usdenote the elements of the dual space 119883

lowast by 119909lowast and denote

⟨119909 119909lowast⟩ = 119909

lowast(119909) If 119883lowastlowast is the dual space of 119883lowast then there

exists a continuous linear transformation 119869 119883 rarr 119883lowastlowast

defined by

⟨119909lowast

119869119909⟩ = ⟨119909 119909lowast

⟩ forall119909 isin 119883 and 119909lowast

isin 119883lowast

(53)

As a consequence of the Hahn-Banach theorem the operator119869 is an isometry (ie 119869119909 = 119909 for every 119909 isin 119883) and henceit is a one-to-one mapping

Proposition 20 A reversible evolution operator119880 on a reflex-ive Banach space119883 has a generalized ℓ-exponential trichotomy(generalized 119903-exponential trichotomy respectively) if andonly if its adjoint evolution operator 119881 has a generalized 119903-exponential trichotomy (generalized ℓ-exponential trichotomyrespectively)

Proof Necessity It is easy to see that if 119880 has a generalizedℓ-exponential trichotomy (a generalized 119903-exponential tri-chotomy) with projection valued functions 119875

+(119905) and 119875

minus(119905)

as in Theorem 11 (Theorem 18) then the adjoint evolutionoperator 119881 has a generalized 119903-exponential trichotomy (ageneralized ℓ-exponential trichotomy) with projections

119868 minus 119875lowast

+(119905) and 119868 minus 119875

lowast

minus(119905) (54)

Sufficiency We now assume that 119881 has a generalized ℓ-exponential trichotomy with projection valued functions 119875

+

and 119875minusas in Theorem 11 For each 119905 isin R we consider the

following operators

+(119905) 119883 997888rarr 119883

+(119905) 119909 = 119869

minus1

119875lowast

+(119905) 119869119909

minus(119905) 119883 997888rarr 119883

minus(119905) 119909 = 119869

minus1

119875lowast

minus(119905) 119869119909

(55)

We notice that +and

minusare projection valued functions on

119883 We first prove that they are invariant for 119880 We have

⟨+(119905) 119880 (119905 119904) 119909 119909

lowast

⟩

= ⟨119909lowast

119869+(119905) 119880 (119905 119904) 119909⟩ = ⟨119909

lowast

119875lowast

+(119905) 119869119880 (119905 119904) 119909⟩

= ⟨119875+(119905) 119909lowast

119869119880 (119905 119904) 119909⟩ = ⟨119880 (119905 119904) 119909 119875+(119905) 119909lowast

⟩

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces and Applications

= ⟨119909119880(119905 119904)lowast

119875+(119905) 119909lowast

⟩ = ⟨119909 119875+(119904) 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119875+(119904) 119880(119905 119904)

lowast

119909lowast

119869119909⟩ = ⟨119880(119905 119904)lowast

119909lowast

119875lowast

+(119904) 119869119909⟩

= ⟨119880(119905 119904)lowast

119909lowast

119869+(119904) 119909⟩ = ⟨

+(119904) 119909 119880(119905 119904)

lowast

119909lowast

⟩

= ⟨119880 (119905 119904) +(119904) 119909 119909

lowast

⟩

(56)

for all 119909 isin 119883 and 119909lowastisin 119883lowast and we similarly get that

⟨minus(119905) 119880 (119905 119904) 119909 119909

lowast

⟩ = ⟨119880 (119905 119904) minus(119904) 119909 119909

lowast

⟩

119909 isin 119883 119909lowast

isin 119883lowast

(57)

These imply

+(119905) 119880 (119905 119904) = 119880 (119905 119904)

+(119904)

minus(119905) 119880 (119905 119904) = 119880 (119905 119904)

minus(119904)

(58)

Hence +and

minusare both invariant for 119880 We can easily get

+(119905) minus(119905) =

minus(119905) +(119905) =

minus(119905) 119905 isin R (59)

It is not difficult to prove that 119880 has a generalized 119903-exponential trichotomywith projections 119868minus

+(119905) and 119868minus

minus(119905)

in Theorem 18 Similarly we can obtain that if the evolutionoperator 119881 has a generalized 119903-exponential trichotomy then119880 has a generalized ℓ-exponential trichotomy

Remark 21 The necessity from the previous proposition stillholds in a general Banach space At this point we donot knowwhether the sufficiency is also valid

Acknowledgment

The authors would like to thank the referees for carefullyreading their paper and for their helpful suggestions andcomments which improved the quality of the paper

References

[1] L Jiang ldquoGeneralized exponential dichotomy and global lin-earizationrdquo Journal of Mathematical Analysis and Applicationsvol 315 no 2 pp 474ndash490 2006

[2] N Lupa andMMegan ldquoExponential dichotomies of evolution-operators in Banach spacesrdquo Monatshefte fur Mathematik Inpress

[3] R O Mosincat C Preda and P Preda ldquoDichotomies withno invariant unstable manifolds for autonomous equationsrdquoJournal of Function Spaces and Applications vol 2012 ArticleID 527647 23 pages 2012

[4] J S Muldowney ldquoDichotomies and asymptotic behavior forlinear differential systemsrdquo Transactions of the American Math-ematical Society vol 283 pp 465ndash484 1984

[5] R J Sacker and G R Sell ldquoExistence of dichotomies andinvariant splittings for linear differential systems IIIrdquo Journalof Differential Equations vol 22 no 2 pp 497ndash522 1976

[6] M Megan and C Stoica ldquoOn uniform exponential trichotomyof evolution operators in Banach spacesrdquo Integral Equations andOperator Theory vol 60 no 4 pp 499ndash506 2008

[7] M Megan and C Stoica ldquoEquivalent definitions for uni-form exponential trichotomy of evolution operators in BanachspacesrdquoHot Topics in Operator Theory vol 9 pp 151ndash158 2008

[8] B Sasu and A L Sasu ldquoExponential trichotomy and p-admissibility for evolution families on the real linerdquoMathema-tische Zeitschrift vol 253 no 3 pp 515ndash536 2006

[9] B Sasu and A L Sasu ldquoNonlinear criteria for the existenceof the exponential trichotomy in infinite dimensional spacesrdquoNonlinear Analysis Theory Methods and Applications vol 74no 15 pp 5097ndash5110 2011

[10] J Zhang ldquoLyapunov function and exponential trichotomy ontime scalesrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 958381 22 pages 2011

[11] S Elaydi and O Hajek ldquoExponential trichotomy of differentialsystemsrdquo Journal ofMathematical Analysis andApplications vol129 no 2 pp 362ndash374 1988

[12] L H Popescu N Lupa and M Megan ldquoExponential tri-chotomy on Banach spacesrdquo in Proceedings of the Seminarof Mathematical Analysis and Applications in Control TheoryUniversity of Timisoara 2010

[13] L H Popescu and T Vesselenyi ldquoTrichotomy and topologicalequivalence for evolution familiesrdquo Bulletin of the BelgianMathematical Society vol 18 no 4 pp 679ndash694 2011

[14] S Elaydi and O Hajek ldquoExponential dichotomy of nonlinearsystems of ordinary differential equationsrdquo Differential IntegralEquations vol 3 pp 1201ndash1224 1990

[15] J Hong R Obaya and A S Gil ldquoExponential trichotomy and aclass of ergodic solutions of differential equations with ergodicperturbationsrdquo Applied Mathematics Letters vol 12 no 1 pp7ndash13 1999

[16] R Nagel and G Nickel ldquoWellposedness for nonautonomousabstract Cauchy problemsrdquo in Progress in Nonlinear DifferentialEquations andTheir Applications vol 50 pp 279ndash293 2002

[17] P Acquistapace ldquoEvolution operators and strong solutionsof abstract linear parabolic equationsrdquo Differential IntegralEquations vol 1 pp 433ndash457 1988

[18] WACoppelDichotomies in StabilityTheory vol 629 ofLectureNotes in Mathematics Springer Berlin Germany 1978

[19] C Potzsche ldquoNonautonomous bifurcation of bounded solu-tions II a shovel-bifurcation patternrdquo Discrete and ContinuousDynamical Systems vol 31 no 3 pp 941ndash973 2011

[20] K J Palmer ldquoExponential dichotomy and expansivityrdquo Annalidi Matematica Pura ed Applicata vol 185 supplement 5 ppS171ndashS185 2006

[21] H M Rodrigues and J G Ruas ldquoEvolution equationsdichotomies and the Fredholm alternative for bounded solu-tionsrdquo Journal of Differential Equations vol 119 no 2 pp 263ndash283 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of