A Perron-type theorem for nonautonomous delay equations

Cent. Eur. J. Math. • 11(7) • 2013 • 1283-1295DOI: 10.2478/s11533-013-0244-6

Central European Journal of Mathematics

A Perron-type theorem for nonautonomousdelay equations

Research Article

Luis Barreira1∗, Claudia Valls1†

1 Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal

Received 29 December 2011; accepted 15 September 2012

Abstract: We show that if the Lyapunov exponents of a linear delay equation x′ = L(t)xt are limits, then the same happenswith the exponential growth rates of the solutions to the equation x′ = L(t)xt + f(t, xt ) for any sufficiently smallperturbation f .

MSC: 34D08, 34D10

Keywords: Lyapunov exponents • Nonautonomous delay equations© Versita Sp. z o.o.

1. Introduction

We show that the asymptotic exponential behavior of the solutions to a linear delay equation persists under sufficientlysmall perturbations. More precisely, under the assumption that all Lyapunov exponents are limits, we show that theasymptotic exponential behavior of the solutions to the linear equationx ′ = L(t)xt (1)

in a Banach space is reproduced exactly by the solutions to the nonlinear equationx ′ = L(t)xt + f(t, xt) (2)

∗ E-mail: [email protected]† E-mail: [email protected]

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A Perron-type theorem for nonautonomous delay equations

for any sufficiently small perturbation f (see (3) for the precise assumption), in the sense that for any solution x toequation (2) the limitλ = lim

t→+∞ 1t log ‖xt‖

exists, and coincides with a Lyapunov exponent to equation (1). The required smallness of the perturbation correspondsto assuming that limt→+∞

∫ t+1teδτ sup

x 6=0‖f(τ, x)‖‖x‖ dτ = 0, (3)

for some δ > 0, or simply that the particular solution x = x(t) satisfieslimt→+∞

∫ t+1teδτ ‖f(τ, xτ )‖‖xτ‖

dτ = 0,for some δ > 0. Condition (3) has the advantage that it can be verified without any a priori knowledge of the solutions.In the particular case of perturbations of a differential equation x ′ = Ax with constant coefficients (for which theLyapunov exponents are always limits), a related result can be found in Coppel’s book [4]. Earlier results were obtainedby Perron [9], Lettenmeyer [7], and Hartman and Wintner [6]. Corresponding results for perturbations of autonomousdelay equations x ′ = Lxt were obtained by Pituk [10, 11] (for values in Cn and finite delay) and Matsui, Matsunagaand Murakami [8] (for values in a Banach space and infinite delay). Related results for perturbations of autonomousdifference equations were first obtained by Coffman [3].2. Standing assumptions and exponential behavior

Given r > 0, let D = D([−r, 0], X ) be the Banach space of continuous functions φ : [−r, 0]→ X with values in a Banachspace X , with the norm‖φ‖ = sup{|φ(θ)| : −r ≤ θ ≤ 0},

where | · | is the norm in X . We consider linear operators L(t) : D → X such that (t, x) 7→ L(t)x is continuous, andequation (1), where x ′ denotes the right-hand derivative and where xt(θ) = x(t + θ), θ ∈ [−r, 0]. We assume that thereexists κ > 0 such that ∫ t+rt‖L(τ)‖dτ ≤ κ(1 + |t|)

for every t ∈ R. Then for each (s, φ) ∈ R×D there is a unique solution xt( · , s, φ), t ≥ s, to equation (1) with xs( · , s, φ) =φ, see for example [5]. We define the evolution operator T (t, s) : D → D for t ≥ s by T (t, s)φ = xt( · , s, φ). Clearly,T (t, t) = Id for t ≥ 0, and

T (t, s)T (s, r) = T (t, r)for t ≥ s ≥ r ≥ 0. It is convenient to extend the domain of T (t, s) to a space that contains some discontinuous functions.For this we write L(t) in the form

L(t)φ = ∫ 0−rdθ [η(t, θ)]φ(θ), (4)

for some linear operators η(t, θ) measurable in (t, θ) ∈ R× [−r, 0] and continuous from the left in θ. We also setmη(t) = Var η(t, · ),

where Var denotes the total variation in [−r, 0]. Note that ‖L(t)‖ = mη(t). Now let D̂ be the space of functionsφ : [−r, 0]→ X such that for each s ∈ [−r, 0] the limits

limθ→s−

φ(θ) and limθ→s+φ(θ)

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L. Barreira, C. Valls

exist and φ is right-continuous at s, i.e., limθ→s+φ(θ) = φ(s). We also consider the supremum norm in D̂. Each operatorL(t) can be extended to D̂ using the integral in (4), provided that the associated Riemann–Stieltjes sums take the value[η(t, b)−η(t, a)]φ(b−) for each subinterval [a, b] ⊂ [−r, 0] (so that points at which both φ and η(t, · ) have discontinuitiescause no problem).It can be shown that for each (s, φ) ∈ R× D̂ there is a unique solution vt( · , s, φ) ∈ D̂ for t ≥ s with vs( · , s, φ) = φ to theintegral equation obtained from equation (1), see [5] for a related discussion. We also consider the evolution operatorT̂ (t, s) : D̂→ D̂ defined for t ≥ s by

T̂ (t, s)φ = vt( · , s, φ).We note that T̂ (t, s)|D = T (t, s). For simplicity of the notation, from now on we shall denote T̂ (t, s) simply by T (t, s).We always assume that the evolution operator is invertible and that it is in block form, with each block correspondingto a Lyapunov exponent

λ(x) = lim supt→+∞

1t log‖T (t, 0)x‖,

for some x ∈ D̂. More precisely, we assume that there exist decompositionsD̂ = F1(t)⊕F2(t)⊕ · · ·⊕Fp(t) (5)

for t ≥ 0, and real numbers λ1 < . . . < λp such thatH1 for each t, s ≥ 0 and i = 1, . . . , p,

T (t, s)Fi(s) = Fi(t);H2 given ε > 0, there exists ρ > 0 such that

ρ−1e(λi−ε)(t−s)−εs‖x‖ ≤ ‖T (t, s)x‖ ≤ ρe(λi+ε)(t−s)+εs‖x‖ (6)for every t ≥ s ≥ 0, i = 1, . . . , p and x ∈ Fi(s);

H3 for each i = 1, . . . , p the projections Pi(t) associated to the decomposition (5) satisfylimt→+∞ 1

t log‖Pi(t)‖ = 0. (7)We note that for ordinary differential equations in finite-dimensional spaces, conditions H1–H3 are automatically satisfiedwhen the coefficients L(t) are constant or periodic (as a consequence of Floquet theory) and, more generally, when theequation is reducible (that is, kinematically similar to an equation with constant coefficients; this means that there existsa coordinate change transforming equation (1) into an equation with constant coefficients preserving the values λi of theLyapunov exponents and the subexponential behavior of the norms of the projections Pi(t) as required in (7)). In thegeneral nonautonomous case, further motivation for conditions H1–H3 comes from ergodic theory. Namely, consider aflow (φt)t∈R defined by an autonomous equation x ′ = f(x) preserving a finite measure µ. This means that µ(φt(A)) = µ(A)for any measurable set A and any t ∈ R. Then the trajectory φt(x) of µ-almost every point x gives rise to a linearvariational equation

v ′ = Ax (t)v, with Ax (t) = dφt (x)f,satisfying conditions H1–H3 (again up to a coordinate change that preserves the values of the Lyapunov exponents andthe subexponential behavior of the norms of the associated projections). In particular, it follows from (6) thatlimt→+∞ 1

t log{ infx∈Gi(s)‖T (t, s)x‖} = lim

t→+∞ 1t log{ sup

x∈Gi(s)‖T (t, s)x‖} = λi,

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whereGi(s) = {x ∈ Fi(s) : ‖x‖ = 1}.In other words, there exists a uniform behavior of the Lyapunov exponent in each set Gi(s). We refer the reader to thebooks [1, 2] for details and references (also for the infinite-dimensional setting).Given a number b ∈ R that is not a Lyapunov exponent λi, we consider the decompositions

D̂ = E(t)⊕F (t), (8)where

E(t) =⊕λi<b

Fi(t) and F (t) =⊕λi>b

Fi(t).are subspaces for each t ≥ 0. Let

P(t) = ∑λi<b

Pi(t) and Q(t) = ∑λi>b

Pi(t) (9)be the projections associated to the decomposition (8). Take also a < b < c such that the interval [a, c] contains noLyapunov exponent λi.Proposition 2.1.The following properties hold:1. E(0) = {x ∈ D̂ : λ(x) < b} and λ(x) > b for x ∈ F (0) \ {0};2. given ε > 0, there exists K = K (ε) > 1 such that

‖T (t, s)|E(s)‖ ≤ Kea(t−s)+εs, t ≥ s, (10)‖T (t, s)|F (s)‖ ≤ Kec(t−s)+εs, t ≤ s;

3. given ε > 0, there exists M = M(ε) > 0 such that

‖P(t)‖ ≤ Meεt and ‖Q(t)‖ ≤ Meεt (11)for every t ≥ 0.

Proof. The last property in the proposition is a simple consequence of condition H3 and (9). Moreover, properties 1.and 2. follow readily from the remaining assumptions. Indeed, by the second inequality in (6) we have‖T (t, s)|Fj (s)‖ ≤ ρe(λj+ε)(t−s)+εs ≤ ρe(λi+ε)(t−s)+εs

for t ≥ s and j ≤ i. Similarly, by the first inequality in (6) we have‖T (s, t)|Fj (t)‖ ≤ ρe(λj−ε)(s−t)+εs = ρeλj (s−t)+εt ≤ ρeλi+1(s−t)+εt

for t ≥ s and j ≥ i+ 1, that is,‖T (t, s)|Fj (s)‖ ≤ ρeλi+1(t−s)+εs

for t ≤ s and j ≥ i+ 1. This allows us to choose a and c as desired.In particular, taking d > λp it follows from (10) that given ε > 0, there exists N = N(ε) > 0 such that

‖T (t, s)‖ ≤ Ned(t−s)+εs, t ≥ s. (12)1286


3. A nonautonomous Perron-type theorem

In this section we consider the nonlinear equationxt = T (t, s)xs +∫ t

sT (t, τ)X0f(τ, xτ )dτ (13)

for some continuous function f : R×D→ D, where (X0u)(0) = u and (X0u)(θ) = 0 for θ < 0. The following is our mainresult.Theorem 3.1.Let x be a solution to equation (13) such that

‖f(t, xt)‖ ≤ γ(t)‖xt‖, t ≥ 0, (14)for some continuous function γ : R→ R satisfying

limt→+∞

∫ t+1teδτγ(τ)dτ = 0, (15)

for some δ > 0. Then one of the following alternatives hold:

• xt = 0 for all sufficiently large t;

• there exists i ∈ {1, . . . , p} such that

λi = limt→+∞ 1

t log ‖xt‖. (16)Proof. We first establish an auxiliary result. Take ε = δ/6.Lemma 3.2.We have

‖xt‖ ≤ N‖xs‖ed(t−s)+εs exp(N∫ t

seεuγ(u)du) (17)

for all t ≥ s. Hence, given r > 0, there exists C = C (r) > 0 such that

C−1e−ε(k+1)r‖x(k+1)r‖ ≤ ‖xt‖ ≤ Ceεkr‖xkr‖ (18)for all integers k ≥ s/r and all kr ≤ t ≤ (k + 1)r.Proof of the lemma. By (13), it follows from (12) and (14) that

‖xt‖ ≤ Ned(t−s)+εs‖xs‖+N∫ t

sed(t−τ)+ετγ(τ)‖xτ‖dτ,

and hence,e−d(t−s)‖xt‖ ≤ Neεs‖xs‖+N

∫ t

se−d(τ−s)+ετγ(τ)‖xτ‖dτ.

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By Gronwall’s lemma, this yields property (17). By (15), we haveS = sup

t≥0∫ t+rteετγ(τ)dτ <∞.

It follows from (17) that (18) holds with C = N exp(NS) max{1, edr}. �Now let a, b, c ∈ R be as in Section 2. We consider the norm‖x‖t = sup

σ≥te−a(σ−t)‖T (σ, t)P(t)x‖+ sup

σ≤te−c(σ−t)‖T (σ, t)Q(t)x‖ (19)

for each t ≥ 0 and x ∈ D̂. We have‖x‖t = ‖P(t)x‖t + ‖Q(t)x‖t , (20)

and one can easily verify that‖x‖ ≤ ‖x‖t ≤ 2KMe2εt‖x‖. (21)

Lemma 3.3.For t ≥ s we have

‖T (t, s)P(s)x‖t ≤ ea(t−s)‖P(s)x‖s, (22)‖T (t, s)Q(s)x‖t ≥ ec(t−s)‖Q(s)x‖s. (23)

Proof of the lemma. For t ≥ s we have‖T (t, s)P(s)x‖t = sup

σ≥t‖T (σ, t)T (t, s)P(s)x‖e−a(σ−t) = sup

σ≥t‖T (σ, s)P(s)x‖e−a(σ−s+s−t)

= ea(t−s) supσ≥t‖T (σ, s)P(s)x‖e−a(σ−s) ≤ ea(t−s) sup

σ≥s‖T (σ, s)P(s)x‖e−a(σ−s) = ea(t−s)‖P(s)x‖s.

Similarly, for t ≥ s we have‖T (t, s)Q(s)x‖t = sup

σ≤t‖T (σ, t)T (t, s)Q(s)x‖e−c(σ−t) = sup

σ≤t‖T (σ, s)Q(s)x‖e−c(σ−s+s−t)

= ec(t−s) supσ≤t‖T (σ, s)Q(s)x‖e−c(σ−s) ≥ ec(t−s) sup

σ≤s‖T (σ, s)Q(s)x‖e−c(σ−s) = ec(t−s)‖Q(s)x‖s.

This completes the proof of the lemma. �

Now let x be a solution to equation (13). Using the decomposition in (8), one can write xt = yt + zt , whereyt = P(t)xt and zt = Q(t)xt .

By (13), we haveyt = T (t, s)ys +∫ t

sT (t, τ)P(τ)X0f(τ, xτ )dτ, zt = T (t, s)zs +∫ t

sT (t, τ)Q(τ)X0f(τ, xτ )dτ.

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Lemma 3.4.One of the following alternatives holds: lim sup

t→+∞1t log ‖xt‖ < b, (24a)

limk→+∞ ‖zkr‖kr‖ykr‖kr

= 0; (24b)or lim inf

t→+∞ 1t log ‖xt‖ > b (25a)

limk→+∞ ‖ykr‖kr‖zkr‖kr

= 0. (25b)Proof of the lemma. For t ≥ kr we have

yt = T (t, kr)ykr +∫ t

krT (t, σ )P(σ )X0f(σ, xσ )dσ (26)

andzt = T (t, kr)zkr +∫ t

krT (t, σ )Q(σ )X0f(σ, xσ )dσ. (27)

By (21) and (23), it follows from (27) that for t ≥ kr,‖zt‖t ≥ ‖T (t, kr)zkr‖t − ∥∥∥∥∫ t

krT (t, τ)Q(τ)X0f(τ, xτ )dτ∥∥∥∥

t≥ ec(t−kr)‖zkr‖kr − 2KMe2εt∫ t

kr‖T (t, τ)Q(τ)X0f(τ, xτ )‖dτ,

and hence, by (11) and (12),‖zt‖t ≥ ec(t−kr)‖zkr‖kr − 2KM2Ne2εt∫ t

kred(t−τ)+2ετγ(τ)‖xτ‖dτ = ec(t−kr)‖zkr‖kr − 2KM2N

∫ t

kre(d+2ε)(t−τ)+4ετγ(τ)‖xτ‖dτ.

On the other hand, by Lemma 3.2, for t ≤ (k + 1)r we obtain‖zt‖t ≥ ec(t−kr)‖zkr‖kr − 2KM2NC‖xkr‖krγk max{1, e(d+2ε)r},

whereγk = ∫ (k+1)r

kre6ετγ(τ)dτ.

By (15), we haveγk → 0 when k →∞. (28)

By (20) and (21), we find that for kr ≤ t ≤ (k + 1)r,‖zt‖t ≥ ec(t−kr)‖zkr‖kr −D1γk(‖ykr‖kr + ‖zkr‖kr), (29)

for some constant D1 > 0. By (26) and (22), it follows from similar estimates that for t ≥ kr,‖yt‖t ≤ ea(t−kr)‖ykr‖kr + 2KM2e2εt∫ t

krea(t−τ)+ετγ(τ)‖xτ‖dτ

≤ ea(t−kr)‖ykr‖kr + 2KM2∫ t

kre(a+2ε)(t−τ)+3ετγ(τ)‖xτ‖dτ

≤ ea(t−kr)‖ykr‖kr + 2KM2C‖xkr‖krγk max{1, e(a+2ε)r}.(30)

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By (30), we obtain that for kr ≤ t ≤ (k + 1)r,‖yt‖t ≤ ea(t−kr)‖ykr‖kr +D2γk(‖ykr‖kr + ‖zkr‖kr), (31)

for some constant D2 > 0. Inequalities (29) and (31) yield that‖z(k+1)r‖(k+1)r ≥ α‖zkr‖kr −Dγk

(‖ykr‖kr + ‖zkr‖kr), (32)

and‖y(k+1)r‖(k+1)r ≤ β‖ykr‖kr +Dγk

(‖ykr‖kr + ‖zkr‖kr), (33)

for all integers k ≥ s/r, whereD = D1 +D2, α = ecr and β = ear . (34)

Now we claim that either‖zkr‖kr ≤ ‖ykr‖kr for all large k, (35)

or‖ykr‖kr < ‖zkr‖kr for all large k. (36)

We shall show that if (35) fails, then (36) holds. Let us assume that (35) does not hold. Then‖zkr‖kr > ‖ykr‖kr for infinitely many k. (37)

By (28), given δ > 0, there exists k1 such that γk < δ for k ≥ k1. By (32) and (33), we find that for infinitely manyintegers k ≥ k1,‖z(k+1)r‖(k+1)r ≥ (α −Dδ)‖zkr‖kr −Dδ‖ykr‖kr , (38)

and‖y(k+1)r‖(k+1)r ≤ (β +Dδ)‖ykr‖kr +Dδ‖zkr‖kr . (39)

By (37), there exists k2 ≥ k1 such that ‖yk2r‖k2r < ‖zk2r‖k2r . We show by induction on k that‖ykr‖kr < ‖zkr‖kr for all k ≥ k2. (40)

If ‖ykr‖kr < ‖zkr‖kr for some k ≥ k2, then, by (38) and (39),‖z(k+1)r‖(k+1)r ≥ (α − 2Dδ)‖zkr‖kr

and‖y(k+1)r‖(k+1)r ≤ (β + 2Dδ)‖zkr‖kr .Since β < α , we find that

‖y(k+1)r‖(k+1)r ≤ β + 2Dδα − 2Dδ ‖z(k+1)r‖(k+1)r < ‖z(k+1)r‖(k+1)r ,

provided that δ is sufficiently small. This shows that (40) holds. Thus, we showed that if (35) fails, then (36) holds. Asa consequence, we have the following two cases.1290


Case 1. Assume that (35) holds. We show that (24a) and (24b) hold. Given η > 0, take k0 so large that γk < η and‖zkr‖kr ≤ ‖ykr‖kr for all k ≥ k0. By (33), we find that for k ≥ k0,

‖y(k+1)r‖(k+1)r ≤ (β + 2Dη)‖ykr‖kr ,which implies that

‖ykr‖kr ≤ N1(β + 2Dη)k , k ≥ k0,where N1 = (β + 2Dη)−k0‖yk0r‖k0r . Together with (18), (20) and (21), this yields that for k ≥ k0 and kr ≤ t ≤ (k + 1)r,‖xt‖ ≤ Ceεkr‖xkr‖ ≤ Ceεkr‖xkr‖kr = Ceεkr

(‖ykr‖kr + ‖zkr‖kr) ≤ 2Ceεkr‖ykr‖kr ≤ 2CN1eεkr(β + 2Dη)k .

The last inequality implies that if β + 2Dη ≥ 1, then‖xt‖ ≤ 2CN1(β + 2Dη)t/reεkr , t ≥ k0r,

and that if β + 2Dη < 1, then‖xt‖ ≤ 2CN1(β + 2Dη)(t−r)/reεkr , t ≥ k0r.In both cases, we obtain lim sup

t→+∞1t log‖xt‖ ≤ log(β + 2Dη)

r + ε,

and since η > 0 is arbitrary, lim supt→∞

1t log‖xt‖ ≤ logβ

r + ε = a+ ε < b,

provided that ε is sufficiently small. This yields (24a). Now we establish (24b). We note that ‖ykr‖kr > 0 for all large k ,since otherwise (20) and (35) yield‖xkr‖kr = ‖ykr‖kr + ‖zkr‖kr ≤ 2‖ykr‖kr = 0

for infinitely many k , contradicting the hypothesis that ‖xt‖γ > 0 for all t ≥ s. Now we defineS = lim sup

k→+∞‖zkr‖kr‖ykr‖kr

.

By (35), we have 0 ≤ S ≤ 1. It follows from (35) and (33) that for all large k ,‖y(k+1)r‖(k+1)r ≤ (β + 2Dγk )‖ykr‖kr .

Together with (32), this yields that for all large k ,‖z(k+1)r‖(k+1)r‖y(k+1)r‖(k+1)r ≥

α −Dγkβ + 2Dγk · ‖zkr‖kr‖ykr‖kr

− Dγkβ + 2Dγk .

Taking limsup on both sides and using (28), we obtain S ≥ (α/β)S. Since α/β > 1, see (34), this implies that S = 0,and (24b) holds.Case 2. Now assume that (36) holds. We show that (25a) and (25b) hold. Given η such that 0 < η < α/(2D), take k0such that γk < η and ‖ykr‖kr < ‖zkr‖kr for all k ≥ k0. By (32), we find that for k ≥ k0,

‖z(k+1)r‖(k+1)r ≥ (α − 2Dη)‖zkr‖kr1291


and hence,‖zkr‖kr ≥ N2(α − 2Dη)k ,where N2 = (α−2Dη)−k0‖zk0r‖k0r > 0. Together with (18), (21) and (20), this yields that for k ≥ k0 and kr ≤ t ≤ (k+1)r,

‖xt‖ ≥ C−1e−ε(k+1)r‖x(k+1)r‖ ≥ C−1(2KM)−1e−3ε(k+1)r‖x(k+1)r‖(k+1)r ≥ C−1(2KM)−1e−3ε(k+1)r‖z(k+1)r‖(k+1)r≥ C−1(2KM)−1N2e−3ε(k+1)r(α − 2Dη)k+1.

Therefore, if α − 2Dη ≥ 1, then‖xt‖ ≥ C−1(2KM)−1e−3ε(k+1)rN2(α − 2Dη)t/r , t ≥ k0r,

and if α − 2Dη < 1, then‖xt‖ ≥ C−1(2KM)−1e−3ε(k+1)rN2(α − 2Dη)(t+r)/r , t ≥ k0r.

In both cases, we obtain lim inft→+∞ 1

t log‖xt‖ ≥ log(α − 2Dη)r − 3ε,

and letting η→ 0, lim inft→+∞ 1

t log‖xt‖ ≥ log αr − 3ε = c − 3ε > b,

provided that ε is sufficiently small. This yields (25a). Now we establish (25b). We defineR = lim sup

k→+∞‖ykr‖kr‖zkr‖kr

.

By (36), we have 0 ≤ R ≤ 1. It follows from (36) in (32) that for all large k ,‖z(k+1)r‖(k+1)r ≥ (α − 2Dγk )‖ykr‖kr .

Together with (33), this yields that for all large k ,‖y(k+1)r‖(k+1)r‖z(k+1)r‖(k+1)r ≤

β +Dγkα − 2Dγk · ‖ykr‖kr‖zkr‖kr

+ Dγkα − 2Dγk .

Taking limsup on both sides and using (28), we obtain R ≤ (β/α)R . Since β/α < 1, this implies that R = 0, and (25b)holds. �

We proceed with the proof of the theorem. Let x be a solution to equation (13) satisfying the hypotheses of the theorem.If xs = 0 for some s, then it follows from (17) that xt = 0 for all t ≥ s, and hence, the first alternative in the theoremholds. Now let us assume that xt 6= 0 for all t ≥ s. Take real numbers bj such thatλj < bj < λj+1 for 1 ≤ j < p.

Take also b0 < λ1 and bp > λp. Applying Lemma 3.4 to each number b = bj , we conclude that there exists j ∈ {1, . . . , p}such that lim supt→+∞

1t log‖xt‖ < bj and lim inf

t→+∞ 1t log‖xt‖ > bj−1.

Letting bj ↘ λj and bj−1 ↗ λj , we find that limt→+∞ 1

t log‖xt‖ = λj .

This completes the proof of the theorem.1292


Now we show that any solution x(t) to equation (13) satisfying the second alternative in Theorem 3.1 is essentiallyasymptotically tangent to the spaces Fi(t) with i as in (16). We consider the decompositionsD̂ = E(t)⊕F (t)⊕Fi(t),

whereE(t) =⊕

j<iFj (t) and F (t) =⊕

j>iFj (t)

for each t ≥ 0. Let also P(t), Q(t) and R(t) be the projections associated to this decomposition.Theorem 3.5.Let x be a solution to equation (13) such that condition (14) holds for some continuous function γ : R→ R satisfying (15),for some δ > 0. If (16) holds, then lim

k→+∞ e−2εkr ‖P(kr)xkr‖‖R(kr)xkr‖ = 0 (41)

and limk→+∞ e−2εkr ‖Q(kr)xkr‖

‖R(kr)xkr‖ = 0 (42)for every r, ε > 0.

Proof. We writext = yt + zt + wt ,

whereyt = P(t)xt , zt = Q(t)xt and wt = R(t)xt .

Take numbers a < b < c such that [a, c] ⊂ (λi−1, λi). Thenlimt→+∞ 1

t log‖xt‖ = λi > b

and it follows from Lemma 3.4 that limk→+∞ ‖ykr‖kr

‖zkr + wkr‖kr= 0. (43)

Now take numbers a′ < b′ < c′ such that [a′, c′] ⊂ (λi, λi+1). Note that a′ > a and c′ > c. Thenlimt→+∞ 1

t log‖xt‖ = λi < b′

and it follows from Lemma 3.4 that limk→+∞ ‖zkr‖′kr

‖ykr + wkr‖′kr= 0, (44)

with the norms ‖·‖′kr defined as in (19), with the numbers a and c replaced by a′ and c′. Given δ > 0, take η ∈ (0, 1)such that 2KMη/(1− η) < δ. By (44), for all sufficiently large k , we have‖zkr‖′kr ≤ η‖ykr + wkr‖′kr . (45)

Similarly, by (43), for all sufficiently large k ,‖ykr‖kr ≤ η‖zkr + wkr‖kr . (46)

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By (19), since a′ > a, we have‖ykr‖kr = sup

σ≥kre−a(σ−kr)‖T (σ, kr)P(kr)x‖ = sup

σ≥kre−(a−a′)(σ−kr)e−a′(σ−kr)‖T (σ, kr)P(kr)x‖

≥ supσ≥kr

e−a′(σ−kr)‖T (σ, kr)P(kr)x‖ = ‖ykr‖′kr . (47)Analogously, by (19), since c′ > c, we have

‖zkr‖kr = supσ≤kr

e−c(σ−kr)‖T (σ, kr)Q(kr)x‖ = supσ≤kr

e−(c−c′)(σ−kr)e−c′(σ−kr)‖T (σ, kr)Q(kr)x‖≤ sup

σ≤kre−c′(σ−kr)‖T (σ, kr)Q(kr)x‖ = ‖zkr‖′kr . (48)

Using (47) and (48), we deduce from (46) that‖ykr‖′kr ≤ η‖zkr + wkr‖kr ≤ η‖zkr‖′kr + η‖wkr‖kr . (49)

Therefore, it follows from (45) together with (49) that‖zkr‖′kr ≤ η2‖zkr‖′kr + η2‖wkr‖kr + η‖wkr‖′kr and ‖zkr‖′kr ≤ η(1− η2)−1(η‖wkr‖kr + ‖wkr‖′kr).

Hence, using (21), we obtain‖zkr‖ ≤ ‖zkr‖′kr ≤ 2KMη(1 + η)(1− η2)−1e2εkr‖wkr‖ ≤ δe2εkr‖wkr‖.

Since δ is arbitrary, we obtain limk→+∞ ‖zkr‖

e2εkr‖wkr‖ = 0,which establishes identity (42). Identity (41) can be established in a similar manner interchanging the roles of yand z.Now we describe two nontrivial consequences of Theorem 3.1. We first consider perturbations of linear equations havingonly negative Lyapunov exponents.Theorem 3.6.Let f : R× D̂→ D̂ be a continuous function such that

‖f(t, x)‖ ≤ γ(t)‖x‖, t ≥ 0, x ∈ D̂,

for some continuous function γ : R→ R satisfying (15) for some δ > 0. If λp < 0, then

limt→+∞ 1

t log ‖xt‖ < 0for all solutions x to equation (2).Now we consider the particular case of linear perturbations. Consider the (linear) equation

x ′ = L(t)xt +M(t)xt (50)for some linear operators M(t) varying continuously with t ≥ 0.Theorem 3.7.If ‖M(t)‖ ≤ γ(t) for some continuous function γ : R→ R satisfying (15) for some δ > 0, then the Lyapunov exponents ofequations (1) and (50) have the same values.

Theorem 3.7 can be seen as a criterion for the persistence of the (Lyapunov) spectrum of a linear delay equation.1294


Acknowledgements

We would like to thank the referees for the very careful reading of an earlier version of the manuscript and for manysuggestions.Research supported by Portuguese National Funds through FCT – Fundação para a Ciência e a Tecnologia within theproject PTDC/MAT/117106/2010 and by CAMGSD.

References

[1] Barreira L., Pesin Ya., Nonuniform Hyperbolicity, Encyclopedia Math. Appl., 115, Cambridge University Press,Cambridge, 2007[2] Barreira L., Valls C., Stability of Nonautonomous Differential Equations, Lecture Notes in Math., 1926, Springer,Berlin, 2008[3] Coffman C.V., Asymptotic behavior of solutions of ordinary difference equations, Trans. Amer. Math. Soc., 1964, 110,22–51[4] Coppel W.A., Stability and Asymptotic Behavior of Differential Equations, D.C. Heath, Boston, 1965[5] Hale J.K., Verduyn Lunel S.M., Introduction to Functional-Differential Equations, Appl. Math. Sci., 99, Springer,New York, 1993[6] Hartman P., Wintner A., Asymptotic integrations of linear differential equations, Amer. J. Math., 1955, 77, 45–86[7] Lettenmeyer F., Über das asymptotische Verhalten der Lösungen von Differentialgleichungen und Differentialgle-ichungssystemen, Sitzungsberichte der Königl. Bayerischen Akademie der Wissenschaften, München, 1929, 201–252[8] Matsui K., Matsunaga H., Murakami S., Perron type theorem for functional differential equations with infinite delayin a Banach space, Nonlinear Anal., 2008, 69(11), 3821–3837[9] Perron O., Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math.Z., 1929, 29(1), 129–160[10] Pituk M., A Perron type theorem for functional differential equations, J. Math. Anal. Appl., 2006, 316(1), 24–41[11] Pituk M., Asymptotic behavior and oscillation of functional differential equations, J. Math. Anal. Appl., 2006, 322(2),1140–1158

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A Perron-type theorem for nonautonomous delay equations