R E S E A R C H
Open Access
C
1
regularity of the stable subspaces with a
general nonuniform dichotomy
Jie Wang
Correspondence: wjhehehe@126. com
Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
Abstract
For nonautonomous linear difference equations, we establishC1 regularity of the stable subspaces under sufficientlyC1-parameterized perturbations. We consider the general case of nonuniform dichotomies, which corresponds to the existence of what we call nonuniform (μ,ν)-dichotomies.
Mathematics Subject Classification 2000: Primary 34D09; 34D10; 37D99.
Keywords:difference equations, parameter dependence, nonuniform dichotomies
1. Introduction
We consider nonautonomous linear difference equations
vm+1=Amvm+Bm(λ)vm (1:1)
in a Banach space, wherelis a parameter in some open subsetYof a Banach space (the parameter space), andl ® Bm(l) is of class C1 for eachm Î J =N. Assuming
that the unperturbed dynamics
vm+1=Amvm (1:2)
admits a very general nonuniform dichotomy (see Section 2 for the definition), and that
sup
m∈J,λ∈Y
Bm(λ) and sup m∈J,λ∈Y
Bm(λ)
are sufficiently small, we establish the optimal C1regularity of the stable subspaces on the parameterlfor Equation (1.1).
The classical notion of (uniform) exponential dichotomy, essentially introduced by Perron in [1], plays an important role in a large part of the theory of differential equa-tions and dynamical systems. We refer the reader to the books [2-5] for details and references. Inspired both in the classical notion of exponential dichotomy and in the notion of nonuni-formly hyperbolic trajectory introduced by Pesin in [6,7], Barreira and Valls [8-11] have introduced the notion of nonuniform exponential dichotomies and have developed the corresponding theory in a systematic way for the continuous and discrete dynamics during the last few years. See also the book [12] for details. As mentioned in [12], in finite-dimensional spaces essentially any linear differential equa-tion with nonzero Lyapunov exponents admits a nonuniform exponential dichotomy.
The works of Barreira and Valls can be regarded as a nice contribution to the nonuni-form hyperbolicity theory [13].
There are some works concerning the smooth dependence of the stable and unstable sub-spaces on the parameter. For example, in the case of continuous time, that is, for linear differential equations
v= [A(t) +B(t,λ)]v,
Johnson and Sell [14] considered exponential dichotomies in ℝ(in a finite dimen-sional space), and proved that for Ckperturbations, if the perturbation and its deriva-tives in lare bounded and equicontinuous in the parameter, then the projections are of class Ckinl. In the case of discrete time, Barreira and Valls established the optimal
C1 dependence of the stable and unstable subspaces on the parameter in [15] for the uniform exponential dichotomies and in [16] for the nonuniform exponential dichotomies.
In our study, we establish the optimalC1 dependence of the stable subspaces on the parameter for very general nonuniform dichotomies (which was first introduced by Bento and Silva in [17]) for (1.1). Such dichotomies include for example the classical notion of uniform exponential dichotomies, as well as the notions of nonuniform expo-nential dichotomies and nonuniform polynomial dichotomies. The proof in this article follows essentially the ideas in [16], with some essential difficulties because we consider the new dichotomies. We also note that we can establish the optimalC1 dependence of the unstable subspaces on the parameter using the similar discussion as in [16], and we omit the detail for short.
2. Setup
LetB(X)be the set of bounded linear operations in the Banach space X. Let (Am)mÎJ
be a sequence of invertible operators inB(X). For each m, nÎ J, we set
A(m,n) = ⎧ ⎨ ⎩
Am−1...An, ifm>n,
Id, ifm=n,
A−1
m ...A−n−11 , ifm<n.
In order to introduce the notion of nonuniform dichotomy, it is convenient to con-sider the notion of growth rate. We say that an increasing function μ:J®(0, +∞) is a growth rate if
μ(0) = 1 and lim
n→+∞μ(n) = +∞.
Given two growth ratesμ andν, we say that the sequence (Am)mÎJ (or the cocycle
A(m,n)) admits a nonuniform (μ,ν) dichotomy if there exist projectionsPm∈B(X)for
eachmÎJsuch that
A(m,n)Pn=PmA(m,n), m,n,∈J
and there exist constantsa, D> 0 andε> 0 such that A(m,n)Pn≤D
μ (m)
μ(n) −α
and
A(m,n)−1Qm≤D
μ (m)
μ(n) −α
νε(m), (2:2)
for eachm≥n, whereQm= Id -Pmis the complementary projection ofPm.
When μ(m) = ν(m) = er(m), we recover the notion of r-nonuniform exponential dichotomy, while we recover the notion of nonuniform polynomial dichotomy when μ(m) =ν(m) = 1 +m.
For example, if μandνare arbitrary growth rates and ε, a> 0, consider a sequence of linear operators An:ℝ2 ®ℝ2given by diagonal matrices
An=
an0
0 bn
,
where ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
am=
μ (m+ 1)
μ(m) −α
e ε
2logν(m+1)(cos(m+1)−1)−
ε
2logν(m)(cosm−1),
bm =
μ(m+ 1)
μ(m) α
e− ε
2logν(m+1)(cos(m+1)−1)+
ε
2logν(m)(cosm−1),
for anymÎ J. Then (Am)mÎJadmits a nonuniform (μ,ν) dichotomy with the
projec-tionsPm:ℝ2® ℝ2 defined byPm(x, y) = (x, 0), and we have
A(m,n)Pn≤
μ (m)
μ(n) −α
νε(n),
and
A(m,n)−1Qm≤
μ (m)
μ(n) −α
νε(m)
for eachm≥n.
In this article, for each nÎ J, we define the stable and unstable subspaces by
En=Pn(X) and Fn=Qn(X).
3. Main results
We establish the existence of stable subspaces EλnonJ for each l ÎY, such that the maps λ→Eλnare of class C1. As the same in [10], we look for each spaceEλnas a graph overEn. More precisely, we look for linear operatorsFn,l:En®Fnsuch that
Eλn= graph(IdEn+n,λ), n∈J, λ∈Y.
Given a constant < 1, letX be the space of familiesF = (Fn,l)nÎJ,lÎYof linear
operators Fn,l:En®Fnsuch that
:= sup n,λνε(|n|) : (n,λ)∈J×Y≤κ
and
for eachl,μÎY. EquippingX with the distance
−= sup n,λ−n,λνε(|n|) : (n,λ)∈J×Y,
it becomes a complete metric space. Given∈X and l Î Y, for each nÎ J, we consider the vector space
Eλn= graph(IdEn+n,λ) = ξ,n,λξ:ξ ∈En
.
Moreover, for eachm, nÎJ, we set
Aλ(m,n) =
⎧ ⎨ ⎩
Cm−1...Cn, ifm>n,
Id, ifm=n,
C−1
m ...C−n−11 , ifm<n,
whereCk=Ak+Bk(l) for eachkÎJ.
Now we state the main result of this article.
Theorem 3.1. Assume that the sequence(Am)mÎJadmits a nonuniform(μ,ν)
dichot-omy, and for eachm∈J,Bm:Y →B(X)are C1 functions satisfying
Bm(λ)≤δν−β(m+ 1) and Bm(λ)≤δν−β(m+ 1). (3:1)
Suppose further that
ϑ= ∞
n=1
μ
(n)
μ(n+ 1) −α
ν3ε−β(n+ 1)<∞. (3:2)
Then forδsufficiently small there exists a unique∈Xsuch that
Eλm=Aλ(m,n)Eλn (3:3)
for each m, n ÎJ. Moreover,
(1)for each nÎJ, m≥n and l ÎY we have
Aλ(m,n)Eλn≤D
μ (m)
μ(n) α
νε(n) (3:4)
for some constant D’> 0;
(2)The mapl↦Fn,lis of class C1for each n ÎJ. Proof. GivennÎJ and (ξ,h)ÎEn×Fn, the vector
(xm,ym) =Aλ(m,n)(ξ,η)∈Em×Fm
satisfies
xm=A(m,n)ξ+ m−1
l=n
PmA(m,l+ 1)Bl(λ)(xl,yl) (3:5)
and
ym=A(m,n)η+ m−1
l=n
for eachm≥n.
Due to the required invariance in (3.3), given(xn,yn)∈Eλnwe must have ym =Fm,l
xmfor eachm, and thus Equations (3.5)-(3.6) are equivalent to
xm=A(m,n)xn+ m−1
l=n
PmA(m,l+ 1)Bl(λ)(IdEl+l,λ)xl (3:7)
and
m,λxm=A(m,n)n,λxn+ m−1
l=n
QmA(m,l+ 1)Bl(λ)(IdEl+l,λ)xl (3:8)
for eachm≥n.
Now we introduce linear operators related to (3.7). Given∈X, nÎ Jandl ÎY, we consider the linear operatorsWn
m,λ=Wmn,,λ:En→Emdetermined recursively by the identities
Wmn,λ=PmA(m,n) + m−1
l=n
PmA(m,l+ 1)Bl(λ)(IdEl+l,λ)W
n
l,λ (3:9)
for m>n, settingWn
n,λ= IdEn. We note that forxn=ξ ÎEn, the sequence
xm=Wmn,λxn=Wmn,λξ (3:10)
is the solution of Equation (3.5) with yl=Fl,lxlfor each l ≥n. Equivalently, it is a
solution of Equation (3.7).
Using (3.10) we can rewrite (3.8) in the form
m,λWmn,λ=A(m,n)n,λ+
m−1
l=n
QmA(m,l+ 1)Bl(λ)(IdEl+l,λ)W
n
l,λ (3:11)
Lemma 3.2.Given δsufficiently small, for each∈Xandl Î Y, the following prop-erties are equivalent:
(1) (3.11)holds for every nÎ J and m≥n;
(2)for every nÎJ and m≥n we have
n,λ=− ∞
l=n
QnA(l+ 1,n)−1Bl(λ)(IdEl+l,λ)W
n
l,λ (3:12)
Proof of the lemma. We first show that the series in (3.12) are well defined. Using (2.2) and (3.1), we obtain
∞
l=n
QnA(l+ 1,n)−1Bl(λ)(IdEl+l,λ)W
n l,λνε(n)
≤δD(1 +κ) ∞
l=n
μ (l+ 1)
μ(n) −α
νε−β(l+ 1)Wn l,λνε(n)
≤2δD
∞
l=n
μ(l+ 1)
μ(n) −α
νε−β(l+ 1)Wn
l,λνε(n).
By (3.9), for each m≥nwe have Wmn,λ≤D
μ (m)
μ(n) −α
νε(n)
+δD(1 +κ)
m−1
l=n
μ(m)
μ(l+ 1) −α
νε−β(l+ 1)Wn l,λ.
(3:14)
Setting
ϒ= sup
m≥n
μ
(m)
μ(n) α
Wmn,λ
.
Then we have
ϒ≤Dνε(n) +δD(1 +κ)ϒ
m−1
l=n
μ
(l)
μ(l+ 1) −α
νε−β(l+ 1)
≤Dνε(n) + 2δDϑϒ.
(3:15)
Takingδsufficiently small such that 2δϑD< 1/2 (independently ofn) we obtain
ϒ≤2Dνε(n),
and therefore,
Wmn,λ≤2D
μ (m)
μ(n) −α
νε(n). (3:16)
Combined (3.13) and (3.16), we have ∞
l=n
QnA(l+ 1,n)−1Bl(λ)(IdEl+l,λ)W
n l,λνε(n)
≤4δD2
∞
l=n
μ
(l)μ(l+ 1)
μ2(n) −α
νε−β(l+ 1)ν2ε(n)
≤4δD2 ∞
l=n
ν3ε−β(l+ 1)
≤κ
(3:17)
provided that δsufficiently small.
Now we assume that identity (3.11) holds. It is equivalent to
n,λ=QnA(m,n)−1m,λWmn,λ
− m−1
l=n
QnA(l+ 1,n)−1Bl(λ)(IdEl+l,λ)W
n l,λ.
(3:18)
Using (3.16), for eachm≥nwe have
QnA(m,n)−1m,λWmn,λ≤2κD2 μ
(m)
μ(n) −2α
νε(n)
Conversely, let us assume that identity (3.12) holds. Then Proof of the lemma. By (3.17) the operatorAis well-defined and
A()≤κ
for δsufficiently small. Furthermore, writing
Setting
ϒl=
μ(l)
μ(n) α
Wl,λ−Wl,μ.
Then we have from the above inequality that
ϒm≤6δD2ϑνε(n)λ−μ+ 2δD m−1
l=n ϒl
μ
(l)
μ(l+ 1) −α
νε−β(l).
Setting ϒ= sup{ϒm:m≥n}, we obtain
ϒ≤6δD2ϑνε(n)λ−μ+ 2δDϑϒ.
Takingδsufficiently small such that 2δDϑ< 1/2, we obtain
ϒ≤12δD2ϑνε(n)λ−μ,
and therefore,
Wm,λ−Wm,μ≤12δD2ϑ
μ (m)
μ(n) −α
νε(n)λ−μ. (3:19)
Therefore, it follows form (3.19) that
bl≤6δDλ−μ
μ (l)
μ(n) −α
ν−β(l+ 1)νε(n)
+ 2δν−β(l+ 1)·12δD2ϑ μ
(l)
μ(n) −α
νε(n)λ−μ
=Kδλ−μ μ
(l)
μ(n) −α
ν−β(l+ 1)νε(n)
whereK= 6D+ 24δD2ϑ > 0. Therefore, we obtain
A()n,λ−A()n,μνε(n)
≤∞
l=n
QnA(l+ 1,n)−1blνε(n)
≤δKDλ−μ
∞
l=n
μ(l)μ(l+ 1)
μ2(n) −α
νε−β(l+ 1)ν2ε(n)
≤δKDλ−μ
∞
l=n
ν3ε−β(l+ 1)
≤δKDϑλ−μ
and provided that δis sufficiently small, we obtainClμ(A(F))≤∥l-μ∥. This shows that A(X)⊂X.
Now we noteF be the space of sequencesU= (Un,l)nÎN,lÎYof linear operatorsUn,l
: En®Fnindexed byYsuch thatl↦ Un,lis continuous for eachnÎ J, and
U= sup
(n,λ)∈J×Y
Equipping F with this norm, it becomes a complete metric space, For each (,U)∈X×F,nÎ J, andl ÎY, we also define linear operatorsB(F, U)n,lby
B(,U)n,λ=− ∞
l=n
QnA(l+ 1,n)−1Gl,λ, (3:21)
where
Gl,λ=Bl(λ)(Zl,λ+l,λZl,λ+Ul,λWln,λ) +Bl(λ)(IdEl+l,λ)Wl,λ (3:22)
and the linear operatorsZm,l=Zm,F,U,l:En®Emare determined recursively by the
identities
Zm,λ= m−1
l=n
PmA(m,l+ 1)Gl,λ (3:23)
form >n, settingZn,l= 0. One observe that by the continuity of the functionsFl,l
and Ul,lonlthe functionsl↦Wl,land l↦Zl,lare also continuous.
Lemma 3.4. For δ sufficiently small, the operator B is well defined, and
B(X×F)⊂F.
Proof of the lemma. By (3.16) and (3.20) we have Zm,λ≤
m−1
l=n
PmA(m,l+ 1)Bl(λ)1 +l,λZl,λ
+Bl(λ)Ul,λ+Bl(λ)1 +l,λWln,λ
≤2δD m−1
l=n
μ
(m)
μ(l+ 1) −α
νε−β(l+ 1)Zl,λ
+ 6δD2
m−1
l=n
μ
(m)
μ(l+ 1) −α
νε−β(l+ 1)μ(l) μ(n)
−α νε(n)
≤2δD m−1
l=n
μ
(m)
μ(l+ 1) −α
νε−β(l+ 1)Z
l,λ+ 6δD2ϑ μ
(m)
μ(n) −α
Settingϒm=
μ (m)
μ(n) α
Zm,λ, we obtain
ϒm≤2δD m−1
l=n
μ (l+ 1)
μ(n) α
νε−β(l+ 1)Z
l,λ+ 6δD2ϑ
≤2δDϑϒl+ 6δD2ϑ
and settingϒ= sup{ϒm:m≥n},
ϒ≤2δDϑϒ+ 6δD2ϑ.
Thus, takingδsufficiently small such that2δDϑ < 1
2, we have
ϒ≤12δD2ϑ,
and therefore
Zm,λ≤12δD2ϑ
μ (m)
μ(n) −α
Setting
it follows from (3.16) and (3.20) that
G≤D
provided that δis sufficiently small. This shows that Bis well defined for eachn, and that∥B(F,U)∥≤1. Therefore,B(X×F)⊂F.
Now we define another mapS:X×F→X×F by
S(,U) = (A(),B(,U)).
By Lemmas 3.3 and 3.4, it is clearly that the maps S is well defined and
S(X×F)⊂X×F.
Lemma 3.5.Forδsufficiently small, the operator S is a contraction.
By (3.16) we obtain
Thus, takingδsufficiently small so that2δDϑ < 1
2 we have
Using (3.32) in (3.30) we obtain
from (3.29) and the above inequality that for δ sufficiently small the operatorS is a contraction.
Now we proceed with the proof of Theorem 3.1. By Lemma 3.5 and its proof, there exists a unique pair(,U)∈X×F such that S(,U) = (,U)andis the unique sequence inX such thatA()n,λ=n,λfor eachnÎJ,l ÎY. Namely,is the unique solution of Equation (3.12) as well as Equation (3.11). Together with (3.9) this implies that ifξÎ En, then
m→Wmn,λξ,m,λ
Wmn,λξ
By (3.16)
(xm,ym)≤4D
μ (m)
μ(n) −α
νε(n)x n.
On the other hand,
(xn,yn)=xn+yn≥xn−yn=xn −n,λxn≥(1−κ)xn.
Thus
(xm,ym)≤
4D
1−κ μ
(m)
μ(n) −α
νε(n)xn,yn,
which implies that (3.4) holds with D= 4D 1−κ >0.
For the C1 regularity of the maps λ→n,λ we consider the pair (1,U1) = (0, 0)∈X×F. Clearly,
Un1,λ= d
dλ
1
n,λ
for eachnÎJ andl ÎY. We define a sequence(m,Um)∈X ×F by (m+1,Um+1) =S(m,Um) = (A(m),B(m,Um))
For a given m Î J, if λ→mn,λis of class C1 for eachn ÎJ, andUmn,λ=
d
dλmm,λfor every nÎJ andl ÎY, then the linear operatorsWm,l andZm,lsatisfyZm,λ= ddλWm,λ for m ≥ n and l Î Y. Therefore we can apply Leibniz’s rule to conclude that
λ→m+1
n,λ is of classC
1
for every nÎ J, with
Unm,+1λ =B(m,Um)n,λ
=− ∞
l=n ∂ ∂λ
QnA(l+ 1,n)−1Bl(λ)(Wl,λ+ml,λWl,λ)
= d
dλA( m)
n,λ
= d
dλ m+1
n,λ
(3:33)
for eachnÎJ andl ÎY.
Moveover, if(,U)is the unique fixed point for the contraction mapS. then the sequencemn,λ
m∈Nconverge uniformly ton,λand the sequence
Um n,λ
m∈Nconverge
uniformly toUn,λfor eachnÎ Jandl ÎY.
We know that if a sequence fmofC1 functions converges uniformly, and its
deriva-tives fm also converges uniformly, then the limit of fm is of classC1, and its derivative
is the limit of fm. Therefore, by (3.33) each functionλ→n,λis of classC1, and
d
dλn,λ=Un,λ
Acknowledgements
This study was supported by the National Natural Science Foundation of China (Grant No. 11171090), the Program for New Century Excellent Talents in University (Grant No. NCET-10-0325), China Postdoctoral Science Foundation funded project (Grant No. 20110491345) and the Fundamental Research Funds for the Central Universities.
Competing interests
The author declares that they have no competing interests.
Received: 13 December 2011 Accepted: 14 March 2012 Published: 14 March 2012
References
1. Perron, O: Die Stabilitätsfrage bei Differentialgleichungen. Math Z.32, 703–728 (1930). doi:10.1007/BF01194662 2. Coppel, W: Dichotomies in Stability Theory. In Lect Notes in Math, vol. 629,Springer (1978)
3. Dalec’kiĭ, Ju, Kreĭn, M: Stability of solutions of differential equations in Banach space. Translations of mathematical monographs. Am Math Soc.43, 1974
4. Massera, J, Schaffer, J: Linear Differential Equations and Function Spaces. In Pure Appl Math, vol. 21,Academic Press, New York/London (1966)
5. Mitropolsky, Yu, Samoilenko, A, Kulik, V: Dichotomies and Stability in Nonautonomous Linear Systems. In Stab Control Theory Methods Appl, vol. 14,Taylor & Francis (2003)
6. Pesin, Y: Families of invariant manifolds that corresponding to nonzero characteristic exponents. Izv Akad Nauk SSSR Ser Mat40, 1332–1379 (1976). (Russian) English Trans. Math. USSR-Izv. 10, 1261- 1305 (1976)
7. Pesin, Y: Characteristic Lyapunov exponents and smooth ergodic theory. Rus Math Surv.32, 55–114 (1977). doi:10.1070/ RM1977v032n04ABEH001639
8. Barreira, L, Valls, C: Robustness of nonuniform exponential dichotomies in Banach spaces. J Diff Equ.244, 2407–2447 (2008). doi:10.1016/j.jde.2008.02.028
9. Barreira, L, Valls, C: Parameter dependence of stable manifolds for difference equations. Nonlinearity.23, 341–367 (2010). doi:10.1088/0951-7715/23/2/007
10. Barreira, L, Valls, C: Optimal regularity of robustness for parameterized perturbations. Bull Sci Math.134, 767–785 (2010). doi:10.1016/j.bulsci.2009.12.003
11. Barreira, L, Valls, CL: Robust nonuniform dichotomies and parameter dependence. J Math Anal Appl.373, 690–708 (2011). doi:10.1016/j.jmaa.2010.08.026
12. Barreira, L, Valls, C: Stability of Nonautonomous Differential Equations. Lect Notes in Math 1926. Springer (2008) 13. Barreira, L, Pesin, Y: Nonuniform hyperbolicity. In Encycl Math Appl, vol. 115,Cambridge Univ. Press, Cambridge (2007) 14. Johnson, R, Sell, G: Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems. J Diff
Equ.41, 262–288 (1981). doi:10.1016/0022-0396(81)90062-0
15. Barreira, L, Valls, C: Smooth robustness of exponential dichotomies. Proc Am Math Soc.139, 999–1012 (2011). doi:10.1090/S0002-9939-2010-10531-2
16. Barreira, L, Valls, C: Smooth robustness of parameterized perturbations of exponential dichotomies. J Diff Equ.249, 2021–2043 (2010). doi:10.1016/j.jde.2010.06.011
17. Bento, AJG, Silva, C: Nonuniform (μ,ν)-dichotomies and local dynamics of difference equations. Nonlinear Anal.75, 78–90 (2012). doi:10.1016/j.na.2011.08.008
doi:10.1186/1687-1847-2012-31
Cite this article as:Wang:C1regularity of the stable subspaces with a general nonuniform dichotomy.Advances in Difference Equations20122012:31.
Submit your manuscript to a
journal and benefi t from:
7Convenient online submission 7Rigorous peer review
7Immediate publication on acceptance 7Open access: articles freely available online 7High visibility within the fi eld
7Retaining the copyright to your article
