Lyapunov Functions for General Nonuniform Dichotomies

Lyapunov Functions for GeneralNonuniform Dichotomies

Luis Barreira, Jifeng Chu and Claudia Valls

Abstract. For nonautonomous linear equations x′ = A(t)x, we give a completecharacterization of the existence of exponential behavior in terms of Lyapunovfunctions. In particular, we obtain an inverse theorem giving explicitly Lyapunovfunctions for each exponential dichotomy. The main novelty of our work is thatwe consider a very general type of nonuniform exponential dichotomy. This in-cludes for example uniform exponential dichotomies, nonuniform exponential di-chotomies and polynomial dichotomies. We also consider the case of differentgrowth rates for the uniform and the nonuniform parts of the dichotomy. As anapplication of our work, we establish in a very direct manner the robustness ofnonuniform exponential dichotomies under sufficiently small linear perturbations.

Mathematics Subject Classification (2010). Primary: 34D09, 34D10.

Keywords. Lyapunov functions, nonuniform dichotomies, robustness.

1. Introduction

We consider nonautonomous linear equations

x′ = A(t)x, (1.1)

where A : R → Mp is a continuous function with values in the set Mp of p × pmatrices. Our main aim is to characterize the existence of nonuniform exponentialbehavior for the solutions of equation (1.1) in terms of Lyapunov functions. As anapplication, we provide a very direct proof of the robustness of the nonuniformexponential behavior, that is, of the persistence of the exponential behavior in theequation

x′ = [A(t) + B(t)]x (1.2)

Luis Barreira and Claudia Valls were partially supported by FCT through CAMGSD, Lisbon. Jifeng

Chu was supported by the National Natural Science Foundation of China (Grants No. 11171090,

No. 11271078 and No. 11271333), the Program for New Century Excellent Talents in University

(Grant No. NCET-10-0325), China Postdoctoral Science Foundation funded project (Grant No.

2012T50431).

Milan J. Math. Vol. 81 (2013) 153–169DOI 10.1007/s00032-013-0198-yPublished online March 19, 2013© 2013 Springer Basel Milan Journal of Mathematics

154 L. Barreira, J. Chu and C. Valls Vol.81 (2013)

for any sufficiently small linear perturbation B(t).

The main novelty of our work is that we consider a very general type of exponen-tial dichotomy. The classical notion of (uniform) exponential dichotomy, essentiallyintroduced by Perron in [24], plays an important role in the theory of differentialequations and dynamical systems. In particular, the existence of an exponential di-chotomy for equation (1.1) implies the existence of topological conjugacies and ofstable and unstable invariant manifolds for any sufficiently small nonlinear pertur-bation. We refer the reader to the books [9, 14, 15, 27] for details and references.On the other hand, it is also true that the existence of an exponential dichotomy isa stringent condition and it is important to look for more general notions, particu-larly in view of the applications. In particular, the notion of nonuniform exponentialdichotomy (see [5]) was inspired both on the classical notion of (uniform) exponen-tial dichotomy and on the notion of nonuniformly hyperbolic trajectory introducedby Pesin (see [2]). We emphasize that in comparison to the notion of (uniform)exponential dichotomy, this notion is a much weaker requirement. For example, infinite-dimensional spaces essentially any linear equation with nonzero Lyapunov ex-ponents admits a nonuniform exponential dichotomy.

In this paper we consider an even more general type of exponential dichotomyin which the usual exponential behavior is replaced by an arbitrary growth rate.This may correspond for example to situations when the Lyapunov exponents areall infinity or are all zero. As proposed in [7], we also consider the case of differentgrowth rates for the uniform and nonuniform parts of the dichotomy. We showedin [1] that there is a large class of equations exhibiting such behavior, and that itis possible to provide natural conditions for the existence of general dichotomies interms of appropriate Lyapunov exponents.

The importance of Lyapunov functions is well established, particularly in thestudy of the stability of trajectories both under linear and nonlinear perturbations.This study goes back to the seminal work of Lyapunov in his 1892 thesis, repub-lished most recently in [18]. Among the first accounts of the theory we mention thebooks by LaSalle and Lefschetz [16], Hahn [13], and Bhatia and Szego [8]. Accordingto Coppel in [11], the connection between Lyapunov functions and uniform expo-nential dichotomies was first considered by Maızel’ in [19]. We refer to the bookby Mitropolsky, Samoilenko and Kulik [21] for a detailed discussion and furtherreferences.

The robustness problem also has a long history. In particular, the problem wasdiscussed by Massera and Schaffer [20], Perron [24], Coppel [11] and in the caseof Banach spaces by Dalec′kiı and Kreın [12]. The continuous dependence of theprojections of the exponential dichotomies on the perturbation was obtained byPalmer [23] for ordinary differential equations. For more recent work we refer to[10, 17, 22, 25, 26] and the references therein. We also refer to [4, 6] for the study ofrobustness in the nonuniform setting.

Vol.81 (2013) Lyapunov Functions for General Nonuniform Dichotomies 155

2. The Case of Contractions

We first formulate our results in the particular case of nonuniform contractions.The statements in this section follow immediately from the results for nonuniformdichotomies in Sections 3 and 4.

Let A : R → Mp be a continuous function. Then each solution of equation (1.1)is global, and we denote the corresponding evolution operator by T (t, s). This is thelinear operator such that

T (t, s)x(s) = x(t), t, s ∈ R,

for any solution x(t) of equation (1.1). Clearly, T (t, t) = Id and

T (t, τ)T (τ, s) = T (t, s), t, τ, s ∈ R.

In order to introduce the notion of a nonuniform contraction, we shall say thata differentiable function μ : R → R

+ is a growth rate if μ′ > 0. Examples of growthrates are et, et

3,√t2 + 1 + t and log(1 + et). Given growth rates μ and ν, we say

that equation (1.1) admits a (μ, ν)-nonuniform contraction if there exist constantsα,D > 0 and ε ≥ 0 such that

‖T (t, s)‖ ≤ D

(μ(t)μ(s)

)−α

ν(|s|)ε, t ≥ s.

The following is an example of a nonuniform contraction.

Example 1. Consider the differential equation

x′ =( −1√

t2 + 1+ cos t− t sin t− 1

)x (2.1)

The evolution operator associated to equation (2.1) is

T (t, s) =(t +

√t2 + 1

s +√s2 + 1

)−1

e−t+s+t cos t−s cos s

=(t +

√t2 + 1

s +√s2 + 1

)−1

et(cos t−1)−s(cos s−1).

(2.2)

For t, s ≥ 0, it follows from (2.2) that

‖T (t, s)‖ ≤(t +

√t2 + 1

s +√s2 + 1

)−1

e2s

For t ≥ 0 and s ≤ 0, it follows from (2.2) that

‖T (t, s)‖ ≤(t +

√t2 + 1

s +√s2 + 1

)−1

.

Finally, for s ≤ t ≤ 0, it follows from (2.2) that

‖T (t, s)‖ ≤(t +

√t2 + 1

s +√s2 + 1

)−1

e2|t| ≤(t +

√t2 + 1

s +√s2 + 1

)−1

e2|s|.

This implies that the scalar equation (2.1) admits a (μ, ν)-nonuniform contractionwith α = D = 1, ε = 2, μ(t) =

√t2 + 1 + t and ν(t) = et.


More generally, if μ and ν are arbitrary growth rates, then the scalar equation

x′ =(−αμ′(t)

μ(t)+

εν ′(t)2ν(t)

(cos t− 1) − ε

2log ν(t) sin t

)x

admits a (μ, ν)-nonuniform contraction with D = 1.Before formulating any results, we would like to comment on the notion of non-

uniform contraction and its ubiquity (similar comments apply with simple changes tononuniform dichotomies). Considering for simplicity of the exposition the particularcase when μ(t) = ν(t) = et and only positive times, we note that if

lim supt→+∞

1t

log ‖x(t)‖ < 0

for any nonzero solution x = x(t) of equation (1.1), and all Lyapunov exponents ofthe equation x′ = A(t)x and its adjoint y′ = −A(t)∗y are finite, then (1.1) admits anonuniform exponential contraction (see for example [2, 5]). This means that thereexist constants α,D > 0 and ε ≥ 0 such that

‖T (t, s)‖ ≤ De−α(t−s)+εs, t ≥ s.

Of course, it may be that the constant ε vanishes, in which case one recovers theclassical notion of uniform exponential contraction. In a related direction, in thecontext of ergodic theory the constant ε can be made arbitrarily small for almostall trajectories. More precisely, if ϕt is a measure-preserving flow generated by anautonomous equation x′ = f(x) in a finite-dimensional space, then for almost all xthe linear variational equation y′ = Ax(t)y, with Ax(t) = dϕt(x)f , admits a nonuni-form exponential contraction with an arbitrarily small constant ε whenever thereare only negative Lyapunov exponents. On the other hand, it follows from work ofBarreira and Schmeling in [3] that for several classes of measure-preserving trans-formations, the set of points for which the constant ε cannot be made zero hasfull topological entropy and full Hausdorff dimension. Therefore, at least from thepoints of view of topological entropy and Hausdorff dimension it is unavoidable toconsider nonuniform exponential contractions (again, a similar comment applies tononuniform exponential dichotomies).

Now we formulate our results in the particular case of nonuniform contractions.Given matrices S(t) ∈ Mp for each t ∈ R, we consider the functions

H(t, x) = 〈S(t)x, x〉, (2.3)

andH(t, x) =

d

dhH(t + h, T (t + h, t)x

)∣∣h=0

, (2.4)

whenever the derivative is well defined.

Theorem 2.1. If equation (1.1) admits a (μ, ν)-nonuniform contraction, then thereexist symmetric positive-definite p× p matrices S(t) for t ∈ R such that:

1. S(t) is of class C1 in t and

lim supt→±∞

log ‖S(t)‖log ν(|t|) < ∞; (2.5)


2. for every t ∈ R,

S′(t) + S(t)A(t) + A(t)∗S(t) ≤ −μ′(t)μ(t)

Id; (2.6)

3. there exists K > 0 such that for every t ∈ R and x ∈ Rp,

H(t, x) ≤ −Kμ′(t)μ(t)

|H(t, x)|. (2.7)

For the following two results, we assume that there exist constants c > 0 andγ, L ≥ 0 such that

‖T (t, s)‖ ≤ Lν(t)γ for |t− s| ≤ c. (2.8)For example, if t �→ A(t) is bounded, then property (2.8) is automatically satisfied.

Theorem 2.2. If there exist symmetric positive-definite p× p matrices S(t) for t ∈ R

satisfying conditions 1–3 in Theorem 2.1 with K > 2γ, andμ(t)μ(s)

≥ ν(t)ν(s)

for every t ≥ s, (2.9)

then equation (1.1) admits a (μ, ν)-nonuniform contraction.

For example, for the growth rates μ(t) = eat and ν(t) = ebt, condition (2.9)holds provided that a ≥ b.

Theorem 2.3. Let A,B : R → Mp be continuous functions such that equation (1.1)admits a (μ, ν)-nonuniform contraction with growth rates μ and ν satisfying (2.9).If γ = 2ε, 4ε < α, and

‖B(t)‖ ≤ δμ′(t)μ(t)

ν(|t|)−2ε, t ∈ R

for some δ > 0 sufficiently small, then equation (1.1) admits a (μ, ν)-nonuniformcontraction.

We note that these three theorems are particular cases of Theorems 3.1, 3.2and 4.1.

3. Characterization of Nonuniform Dichotomies

We consider in this section the more general case of nonuniform dichotomies. Themain difficulty is caused by the existence of stable and unstable behavior alongdirections that may approach each other.

Let us first introduce the notion of nonuniform dichotomy. We say that equa-tion (1.1) admits a (μ, ν)-nonuniform dichotomy if there exist projections P (t) ∈ Mp

for t ∈ R satisfyingT (t, s)P (s) = P (t)T (t, s), t, s ∈ R,

and there exist constants α, β,D > 0 and ε ≥ 0 such that

‖T (t, s)P (s)‖ ≤ D

(μ(t)μ(s)

)−α

ν(|s|)ε, t ≥ s,


and

‖T (t, s)Q(s)‖ ≤ D

(μ(s)μ(t)

)−β

ν(|s|)ε, s ≥ t,

where Q(t) = Id − P (t) for each t ∈ R. Proceeding in a similar manner to that inSection 2, one can obtain examples of (μ, ν)-nonuniform dichotomies.

The following result gives a characterization of (μ, ν)-nonuniform dichotomiesin terms of the functions H and H in (2.3) and (2.4).

Theorem 3.1. If equation (1.1) admits a (μ, ν)-nonuniform dichotomy, then thereexist symmetric invertible p× p matrices S(t) for t ∈ R such that conditions 1–3 inTheorem 2.1 hold.

Proof. Since equation (1.1) admits a (μ, ν)-nonuniform dichotomy, for each t ∈ R

one can define subspaces

F st = P (t)(Rp) and F u

t = Q(t)(Rp).

We consider the matrices

S(t) =∫ ∞

tT (v, t)∗P (v)∗P (v)T (v, t)

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

−∫ t

−∞T (v, t)∗Q(v)∗Q(v)T (v, t)

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv,

(3.1)

for some positive constant < min{α, β}. Clearly, the matrix S(t) is symmetric foreach t. Now let H be the function defined by (2.3). Since H(t, x) > 0 for x ∈ F s

t \{0},and H(t, x) < 0 for x ∈ F u

t \ {0}, it follows easily from the identity F st ⊕ F u

t = Rp

that the matrix S(t) is invertible for each t. Indeed, by (3.1), the restrictions

S(t)|F st =

∫ ∞

tT (v, t)∗P (v)∗P (v)T (v, t)

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

and

S(t)|F ut = −

∫ t

−∞T (v, t)∗Q(v)∗Q(v)T (v, t)

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

are nondegenerate definite quadratic forms and hence are invertible. Moreover, withrespect to the decomposition R

p = F st ⊕ F u

t , one can write S(t) in the block form

(S(t)|F s

t 00 S(t)|F u

t

)

and hence it is also invertible.


Now we observe that

|H(t, x)|

≤∫ ∞

t‖T (v, t)P (t)x‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

+∫ t

−∞‖T (v, t)Q(t)x‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

≤ D2ν(|t|)2ε‖x‖2{∫ ∞

t

(μ(v)μ(t)

)−2� μ′(v)μ(v)

dv +∫ t

−∞

(μ(t)μ(v)

)−2� μ′(v)μ(v)

dv

}

=D2

2ν(|t|)2ε‖x‖2

{−∫ ∞

t

[(μ(v)μ(t)

)−2� ]′dv +

∫ t

−∞

[(μ(t)μ(v)

)−2� ]′dv

}

=D2

2ν(|t|)2ε‖x‖2

{2 − lim

v→−∞

(μ(t)μ(v)

)−2�}

≤ D2

ν(|t|)2ε‖x‖2,

because μ is strictly increasing.Since the matrix S(t) is symmetric, we obtain

‖S(t)‖ = supx �=0

|H(t, x)|‖x‖2 ≤ D2

ν(|t|)2ε, (3.2)

and hence (2.5) holds. Since

∂

∂tT (τ, t) = −T (τ, t)A(t) and

∂

∂tT (τ, t)∗ = −A(t)∗T (τ, t)∗,

the function t �→ S(t) is of class C1, and has derivative

S′(t) = −P (t)∗P (t)μ′(t)μ(t)

−∫ ∞

tA(t)∗T (v, t)∗P (v)∗P (v)T (v, t)

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

−∫ ∞

tT (v, t)∗P (v)∗P (v)T (v, t)A(t)

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(α− )μ′(t)μ(t)

∫ ∞

tT (v, t)∗P (v)∗P (v)T (v, t)

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv


−Q(t)∗Q(t)μ′(t)μ(t)

+∫ t

−∞A(t)∗T (v, t)∗Q(v)∗Q(v)T (v, t)

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

+∫ t

−∞T (v, t)∗Q(v)∗Q(v)T (v, t)A(t)

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞T (v, t)∗Q(v)∗Q(v)T (v, t)

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

= −(P (t)∗P (t) + Q(t)∗Q(t)

)μ′(t)μ(t)

−A(t)∗S(t) − S(t)A(t)

− 2(α− )μ′(t)μ(t)

∫ ∞

tT (v, t)∗P (v)∗P (v)T (v, t)

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞T (v, t)∗Q(v)∗Q(v)T (v, t)

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv.

Since < min{α, β}, we obtain

⟨(S′(t) + S(t)A(t) + A(t)∗S(t) +

(P (t)∗P (t) + Q(t)∗Q(t)

)μ′(t)μ(t)

)x, x

⟩

= −2(α− )μ′(t)μ(t)

∫ ∞

t‖T (v, t)P (t)x‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞‖T (v, t)Q(t)x‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv ≤ 0.

(3.3)

On the other hand, we have

2⟨(P (t)∗P (t) + Q(t)∗Q(t)

)x, x

⟩= 2‖P (t)x‖2 + 2‖Q(t)x‖2≥ (‖P (t)x‖ + ‖Q(t)x‖)2 ≥ ‖x‖2, (3.4)

and hence,

P (t)∗P (t) + Q(t)∗Q(t) ≥ 12Id.

Therefore, it follows from (3.3) that (2.6) holds with S(t) replaced by the matrix2S(t) for every t ∈ R.

Furthermore, if x(t) is a solution of equation (1.1), then

d

dtH(t, x(t)) = 〈S′(t)x(t), x(t)〉 + 〈S(t)x′(t), x(t)〉 + 〈S(t)x(t), x′(t)〉

=⟨(S′(t) + S(t)A(t) + A(t)∗S(t)

)x(t), x(t)

⟩.

(3.5)


It follows from (3.3) and (3.4) that

d

dtH(t, x(t))

≤ −⟨(P (t)∗P (t) + Q(t)∗Q(t)

)x(t), x(t)

⟩μ′(t)μ(t)

− 2(α− )μ′(t)μ(t)

∫ ∞

t‖T (v, t)P (t)x(t)‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞‖T (v, t)Q(t)x(t)‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

≤ −2(α− )μ′(t)μ(t)

∫ ∞

t‖T (v, t)P (t)x(t)‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞‖T (v, t)Q(t)x(t)‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv.

(3.6)

Now we note that

H(t, x(t)) =∫ ∞

t‖T (v, t)P (t)x(t)‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

−∫ t

−∞‖T (v, t)Q(t)x(t)‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv.

(3.7)

If H(t, x(t)) ≥ 0, then since < min{α, β}, it follows from (3.6) that

d

dtH(t, x(t))

≤ −2(α− )μ′(t)μ(t)

∫ ∞

t‖T (v, t)P (t)x(t)‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

+ 2(α− )μ′(t)μ(t)

∫ t

−∞‖T (v, t)Q(t)x(t)‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

≤ −2(α− )μ′(t)μ(t)

|H(t, x(t))|.

Similarly, if H(t, x(t)) ≤ 0, then

d

dtH(t, x(t))

≤ 2(β − )μ′(t)μ(t)

∫ ∞

t‖T (v, t)P (t)x(t)‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞‖T (v, t)Q(t)x(t)‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

≤ −2(β − )μ′(t)μ(t)

|H(t, x(t))|.

Taking K = min{α, β} − we obtain (2.7) and the proof is complete. �


The following result is a partial converse to Theorem 3.1. It gives a criterionfor the existence of a nonuniform dichotomy.

Theorem 3.2. Assume that there exist symmetric invertible p × p matrices S(t) fort ∈ R satisfying conditions 1–3 in Theorem 2.1 as well as conditions (2.8) and (2.9)with K > 2γ. Then equation (1.1) admits a (μ, ν)-nonuniform dichotomy.

Proof. We consider the sets

Esτ = {0} ∪ {

x ∈ Rp : H(t, T (t, τ)x) > 0 for every t ≥ τ

},

andEu

τ = {0} ∪ {x ∈ R

p : H(t, T (t, τ)x) < 0 for every t ≥ τ}.

Lemma 3.3. Let x ∈ Esτ ∪ Eu

τ . Then

|H(τ, x)| ≥ c

L2ν(|τ |)−2γ‖x‖2.

Proof of the lemma. Set x(t) = T (t, τ)x. It follows from (3.5) and condition 2 thatd

dtH(t, x(t)) ≤ −‖x(t)‖2. (3.8)

Using (3.8) and the definition of x(t), we obtain

H(τ + c, x(τ + c)) −H(τ, x) =∫ τ+c

τ

d

drH(r, x(r)) dr

= −∫ τ+c

τ‖T (r, τ)x‖2 dr

≤ −‖x‖2∫ τ+c

τ

dr

‖T (τ, r)‖2 .

It thus follows from (2.8) that

H(τ + c, x(τ + c)) −H(τ, x) ≤ −‖x‖2∫ τ+c

τ

1L2

ν(|τ |)−2γ dr

= − c

L2ν(|τ |)−2γ‖x‖2.

Therefore, for x ∈ Esτ , since H(τ + c, x(τ + c)) ≥ 0, we obtain

H(τ, x) ≥ H(τ, x) −H(τ + c, x(τ + c)) ≥ c

L2ν(|τ |)−2γ‖x‖2.

Similarly, by (3.8) we have

H(τ, x) −H(τ − c, x(τ − c)) ≤ −‖x‖2∫ τ

τ−c

dr

‖T (τ, r)‖2 .

Therefore, if x ∈ Euτ , since H(τ − c, x(τ − c)) ≤ 0, it follows from (2.8) that

|H(τ, x)| ≥ |H(τ, x)| − |H(τ − c, x(τ − c))|= H(τ − c, x(τ − c)) −H(τ, x)

≥ c

L2ν(|τ |)−2γ‖x‖2.

This completes the proof of the lemma. �


Lemma 3.4. Given continuous functions w, f : [τ, t] → R+ and K > 0, if

w(x) − w(τ) ≥ K

∫ x

τw(z)f(z) dz (3.9)

for every x ∈ [τ, t], then w(x) ≥ w(τ) exp(K

∫ xτ f(z) dz

)for every x ∈ [τ, t].

Proof of the lemma. The proof follows one of the usual proofs of Gronwall’s lemmasimply reversing inequalities. For each x ∈ [τ, t], multiplying (3.9) by f(x), we have

w(x)f(x)w(τ)/K +

∫ xτ w(z)f(z) dz

≥ Kf(x). (3.10)

Integrating in (3.10) from τ to x, we obtain

log(w(τ)/K +

∫ x

τw(z)f(z) dv

)− log

(w(τ)/K

) ≥ K

∫ x

τf(z) dz,

which implies that

w(τ) + K

∫ x

τw(z)f(z) dv ≥ w(τ) exp

(K

∫ x

τf(z) dz

).

Again by (3.9), we obtain

w(x) ≥ w(τ) + K

∫ x

τw(z)f(z) dv ≥ w(τ) exp

(K

∫ x

τf(z) dz

).

This yields the desired inequality. �Lemma 3.5. If x ∈ Es

τ , then

H(t, T (t, τ)x) ≤(μ(t)μ(τ)

)−K

H(τ, x), t ≥ τ, (3.11)

and if x ∈ Euτ , then

|H(t, T (t, τ)x)| ≥(μ(t)μ(τ)

)K

|H(τ, x)|, t ≥ τ. (3.12)

Proof of the lemma. Take x ∈ Euτ . For every t ≥ u ≥ τ , it follows from (2.7) that

H(t, T (t, τ)x) −H(u, T (u, τ)x) =∫ t

uH(v, T (v, τ)x) dv

≤ K

∫ t

u

μ′(v)μ(v)

H(v, T (v, τ)x) dv.

We thus obtain

|H(t, T (t, τ)x)| − |H(u, T (u, τ)x)| ≥ K

∫ t

u

μ′(v)μ(v)

|H(v, T (v, τ)x)| dv,and it follows from Lemma 3.4 that

|H(t, T (t, τ)x)| ≥ |H(u, T (u, τ)x)|exp(K

∫ t

u

μ′(v)μ(v)

dv

)

= |H(u, T (u, τ)x)|(μ(t)μ(u)

)K


In particular, taking u = τ we obtain (3.12).Now take x ∈ Es

τ . For every t ≥ u ≥ τ , we have

H(t, T (t, τ)x) −H(u, T (u, τ)x) =∫ t

uH(v, T (v, τ)x) dv

≤ −K

∫ t

u

μ′(v)μ(v)

H(v, T (v, τ)x) dv.

The function w : [τ, t] → R+ defined by

w(z) = H(t + τ − z, T (t + τ − z, τ)x)

satisfiesw(τ) = H(t, T (t, τ)x), w(t + τ − u) = H(u, T (u, τ)x).

Hence,

w(τ) − w(t + τ − u) ≤ −K

∫ t

u

μ′(v)μ(v)

w(t + τ − v) dv

= −K

∫ t+τ−u

τ

μ′(t + τ − z)μ(t + τ − z)

w(z) dz.(3.13)

Now we write α = t+ τ −u. Since u ∈ [τ, t], we have α ∈ [τ, t], and inequality (3.13)is equivalent to

w(α) − w(τ) ≥ K

∫ α

τ

μ′(t + τ − z)μ(t + τ − z)

w(z) dz, α ∈ [τ, t].

By Lemma 3.4, we obtain

w(α) ≥ w(τ)exp(K

∫ α

τ

μ′(t + τ − z)μ(t + τ − z)

dz

)= w(τ)

(μ(t)

μ(t + τ − α)

)K

Therefore,

H(t, T (t, τ)x) = w(τ) ≤(μ(t)μ(τ)

)−K

w(t) =(μ(t)μ(τ)

)−K

H(τ, x).

This establishes (3.11) and the proof of the lemma is complete. �

Now we observe that by condition 1, there exist constants d, δ > 0 such that‖S(t)‖ ≤ dν(|t|)δ for every t ∈ R, and hence

|H(t, x)| ≤ dν(|t|)δ‖x‖2. (3.14)

Lemma 3.6. For every t, τ ∈ R with t ≥ τ , we have

‖T (t, τ)|Esτ‖2 ≤

dL2

c

(μ(t)μ(τ)

)−K+2γ

ν(|τ |)δ+2γ ,

and

‖T (t, τ)−1|Eut ‖2 ≤

dL2

c

(μ(t)μ(τ)

)−K

ν(|t|)δ+2γ .


Proof of the lemma. Take x ∈ Esτ . By Lemmas 3.3 and 3.5 together with (2.8)

and (3.14), for every t ≥ τ we have

‖T (t, τ)x‖2 ≤ L2

cν(t)2γH(t, T (t, τ)x)

≤ L2

cν(t)2γ

(μ(t)μ(τ)

)−K

H(τ, x)

≤ L2

cν(t)2γ

(μ(t)μ(τ)

)−K

‖S(τ)‖ · ‖x‖2

≤ dL2

cν(t)2γ

(μ(t)μ(τ)

)−K

ν(|τ |)δ‖x‖2

≤ dL2

c

(ν(t)ν(τ)

)2γ( μ(t)μ(τ)

)−K

ν(|τ |)δ+2γ‖x‖2

≤ dL2

c

(μ(t)μ(τ)

)−K+2γ

ν(|τ |)δ+2γ‖x‖2.

Similarly, for every x ∈ Euτ and t ≥ τ , we have

‖T (t, τ)x‖2 ≥ 1dν(|t|)−δ|H(t, T (t, τ)x)|

≥ 1dν(|t|)−δ

(μ(t)μ(τ)

)K

|H(τ, x)|

≥ c

dL2ν(|t|)−δ

(μ(t)μ(τ)

)K

ν(τ)−2γ‖x‖2

=c

dL2ν(|t|)−(2γ+δ)

(μ(t)μ(τ)

)K(ν(|t|)ν(τ)

)2γ

‖x‖2

≥ c

dL2ν(|t|)−(2γ+δ)

(μ(t)μ(τ)

)K(ν(t)ν(τ)

)2γ

‖x‖2

≥ c

dL2

(μ(t)μ(τ)

)K

ν(|t|)−(2γ+δ)‖x‖2.

This completes the proof of the lemma. �

The proof of the following lemma is analogous to that of Lemma 5 in [6].

Lemma 3.7. For each t ∈ R, the sets Est , Eu

t are subspaces, and Est ⊕ Eu

t = Rp.

We denote by

P (t) : Rp → Est and Q(t) : Rp → Eu

t

the projections associated to the decomposition Est ⊕ Eu

t = Rp. Repeating the ar-

guments in the proof of Lemma 6 in [6] with a(t) = cν(|t|)−2γL2, we obtain thefollowing.


Lemma 3.8. We have

‖P (t)‖ = ‖Q(t)‖ ≤√

2L2

cν(|t|)2γ‖S(t)‖, t ∈ R.

To complete the proof of the theorem, we observe that

‖T (t, τ)P (τ)‖ ≤ ‖T (t, τ)|Esτ‖ · ‖P (τ)‖,

and‖T (t, τ)−1Q(t)‖ ≤ ‖T (t, τ)−1|Eu

t ‖ · ‖Q(t)‖.Therefore, it follows from Lemmas 3.6 and 3.8 that equation (1.1) admits a (μ, ν)-nonuniform dichotomy. �

4. Robustness of Nonuniform Dichotomies

We establish in this section the robustness of nonuniform dichotomies with the helpof Lyapunov functions. Roughly speaking, a nonuniform dichotomy for equation (1.1)is said to be robust if equation (1.2) still admits a nonuniform dichotomy for anysufficiently small perturbation B(t). We follow closely the arguments in the proof ofTheorem 7 in [6].

Theorem 4.1. Let A,B : R → Mp be continuous functions such that equation (1.1)admits a (μ, ν)-nonuniform dichotomy. If conditions (2.8) and (2.9) hold with γ = 2εand 4ε < min{α, β}, and

‖B(t)‖ ≤ δμ′(t)μ(t)

ν(|t|)−2ε, t ∈ R

for some δ > 0 sufficiently small, then equation (1.2) admits a (μ, ν)-nonuniformdichotomy.

Proof. Let U(t, s) be the evolution operator associated to equation (1.2). For everyt, s ∈ R with |t− s| ≤ c, we have

U(t, s) = T (t, s) +∫ t

sT (t, τ)B(τ)U(τ, s) dτ,

and hence,

‖U(t, s)‖ ≤ Lν(t)2ε +∫ t

sLν(t)2εδν(|τ |)−2ε‖U(τ, s)‖ dτ.

Now letU∗(t, s) = ‖U(t, s)‖ν(t)−2ε.

Then

U∗(t, s) ≤ L + δL

∫ t

sU∗(τ, s)

(ν(τ)ν(|τ |)

)2ε

dτ ≤ L + δL

∫ t

sU∗(τ, s) dτ.

Using Gronwall’s inequality, we obtain

U∗(t, s) ≤ L exp(∫ t

sδL dτ

)≤ L exp(δLc)


for every t, s ∈ R with |t− s| ≤ c. Therefore,

‖U(t, s)‖ ≤ L exp(δLc)ν(t)2ε,

which implies that condition (2.8) also holds for the perturbed equation (1.2).Now we consider the matrices S(t) in (3.1). Condition 1 is established in the

proof of Theorem 3.1. For condition 2, since we already know that (2.6) holds, it issufficient to show that

S(t)B(t) + B(t)∗S(t) ≤ ημ′(t)μ(t)

Id (4.1)

for some constant η < 1. By (3.2) we have

S(t)B(t) + B(t)∗S(t) ≤ 2‖S(t)‖ · ‖B(t)‖ ≤ 2δD2

· μ

′(t)μ(t)

, (4.2)

and taking δ sufficiently small, we find that (4.1) holds with some η < 1. For condi-tion 3 we note that if x(t) is any solution of equation (1.2), then

d

dtH(t, x(t)) = 〈S′(t)x(t), x(t)〉

+ 〈S(t)A(t)x(t), x(t)〉 + 〈S(t)B(t)x(t), x(t)〉+ 〈A(t)∗S(t)x(t), x(t)〉 + 〈B(t)∗S(t)x(t), x(t)〉.

(4.3)

By (3.3) and (3.4) we obtain

〈S′(t)x(t), x(t)〉 + 〈S(t)A(t)x(t), x(t)〉 + 〈A(t)∗S(t)x(t), x(t)〉

≤ −12· μ

′(t)μ(t)

‖x(t)‖2

− 2(α− )μ′(t)μ(t)

∫ ∞

t‖T (v, t)P (t)x(t)‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞‖T (v, t)Q(t)x(t)‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv

(4.4)

Furthermore, by (4.2) we have

〈S(t)B(t)x(t), x(t)〉 + 〈B(t)∗S(t)x(t), x(t)〉 ≤ 2δD2

· μ

′(t)μ(t)

‖x(t)‖2.

Therefore, it follows from (4.3) and (4.4) that if δ is so small that 2δD2/ < 1/2,then

d

dtH(t, x(t))

≤ −2(α− )μ′(t)μ(t)

∫ ∞

t‖T (v, t)P (t)x(t)‖2

(μ(v)μ(t)

)2(α−�) μ′(v)μ(v)

dv

− 2(β − )μ′(t)μ(t)

∫ t

−∞‖T (v, t)Q(t)x(t)‖2

(μ(t)μ(v)

)2(β−�) μ′(v)μ(v)

dv.

(4.5)


In view of (3.7), where x(t) is now a solution of equation (1.2), if H(t, x(t)) ≥ 0,then by (4.5) and in a similar manner to that in (3.6), since α− > 0 and β− > 0we obtain

d

dtH(t, x(t)) ≤ −2(α− )|H(t, x(t))|.

Analogously, if H(t, x(t)) ≤ 0, then

d

dtH(t, x(t)) ≤ −2(β − )|H(t, x(t))|.

Thus, taking K = min{α, β} − with sufficiently small, it follows from (4.5) that

d

dtH(t, x(t))|t=τ ≤ −K|H(τ, x(τ))|,

and condition 3 holds with K > 2γ. It thus follows from Theorem 3.2 that equa-tion (1.2) admits a (μ, ν)-nonuniform dichotomy. �

References

[1] L. Barreira, J. Chu and C. Valls, Robustness of nonuniform dichotomies with different

growth rates, Sao Paulo J. Math. Sci. 5 (2011), 1–29.[2] L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and

Its Applications 115, Cambridge University Press, 2007.[3] L. Barreira and J. Schmeling, Sets of “non-typical” points have full topological entropy

and full Hausdorff dimension, Israel J. Math. 116 (2000), 29–70.[4] L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach

spaces, J. Differential Equations 244 (2008), 2407–2447.[5] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect.

Notes. in Math. 1926, Springer, 2008.[6] L. Barreira and C. Valls, Robustness via Lyapunov functions, J. Differential Equations

246 (2009), 2891–2907.[7] A. Bento and C. Silva, Generalized nonuniform dichotomies and local stable manifolds,

preprint.[8] N. Bhatia and G. Szego, Stability Theory of Dynamical Systems, Grundlehren der math-

ematischen Wissenschaften 161, Springer, 1970.[9] C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differ-

ential Equations, Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999.[10] S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for

skew-product semiflow in Banach spaces, J. Differential Equations 120 (1995), 429–477.[11] W. Coppel, Dichotomies in Stability Theory, Lect. Notes in Math. 629, Springer, 1978.[12] Ju. Dalec′kiı and M. Kreın, Stability of Solutions of Differential Equations in Banach

Space, Translations of Mathematical Monographs 43, Amer. Math. Soc. 1974.[13] W. Hahn, Stability of Motion, Grundlehren der mathematischen Wissenschaften 138,

Springer, 1967.[14] J. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Mono-

graphs 25, Amer. Math. Soc., 1988.


[15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math.840, Springer, 1981.

[16] J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method, with Applications,Mathematics in Science and Engineering 4, Academic Press, 1961.

[17] X.-B. Lin, Exponential dichotomies and homoclinic orbits in functional-differential equa-

tions, J. Differential Equations 63 (1986), 227–254.[18] A. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, 1992.[19] A. Maızel’, On stability of solutions of systems of differential equations, Ural. Politehn.

Inst. Trudy 51 (1954), 20–50.[20] J. Massera and J. Schaffer, Linear differential equations and functional analysis. I, Ann.

of Math. (2) 67 (1958), 517–573.[21] Yu. Mitropolsky, A. Samoilenko and V. Kulik, Dichotomies and Stability in Nonau-

tonomous Linear Systems, Stability and Control: Theory, Methods and Applications14, Taylor & Francis, 2003.

[22] R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness,Nonlinear Anal. 31 (1998), 559–571.

[23] K. Palmer, Exponential dichotomies and transversal homoclinic points, J. DifferentialEquations 55 (1984), 225–256.

[24] O. Perron, Die Stabilitatsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703–728.[25] V. Pliss and G. Sell, Robustness of exponential dichotomies in infinite-dimensional dy-

namical systems, J. Dynam. Differential Equations 11 (1999), 471–513.[26] L. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl.

314 (2006), 436–454.[27] G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sci-

ences 143, Springer, 2002.

Luis BarreiraDepartamento de MatematicaInstituto Superior Tecnico1049-001 Lisboa, Portugale-mail: [email protected]

Jifeng ChuDepartment of MathematicsCollege of ScienceHohai UniversityNanjing 210098, Chinae-mail: [email protected]

Claudia VallsDepartamento de MatematicaInstituto Superior Tecnico1049-001 Lisboa, Portugale-mail: [email protected]

Received: May 30, 2012.

Documents

Lyapunov Functions for General Nonuniform Dichotomies