Exponential dichotomy for delay linear
non-autonomous equations
Patricia H. Tacuri/ Miguel Frasson
ICMC - University of São Paulo
[email protected]/[email protected]
1. Introduction
2. Mathematical
Results
We are concerned with the study of the asymptotic behavior of
In our first theorem we show a correspondence between solutions
solutions to non-autonomous linear retarded functional differential
of the initial value problem for the nonautonomous inhomogeneous
equations, and an important tool to study it is the concept of expo-
retarded functional diferential equation (2) with solutions of the ab-
nential dichotomies, which has been studied with much emphasis
stract integral equations (AIE)
in the last fifty years by many authors ([1]–[7]). This concept was
Z
(1)
where xt(θ) = x(t + θ) and L(t) is a linear operator from the Ban
s
value prototype problem ẋ = 0 and C(t) : X → X ∗ is defined
by C(t)ϕ = B(t)ϕ + (f (t), 0) where B(t) : X → X ∗ is a family of
bounded operators given by B(t)ϕ = (L(t)ϕ, 0).
n
nach space X = C([−h, 0], C ) into C . In the spirit of the paper
of Latushkin et. al [4] if we want to write system (1) as an abstract
evolutionary system in a Banach space X,
u̇(t) = A(t)u(t),
(3)
where T0 is a C0-semigroup defined by the solution of the initial-
dimensional setting [7]. In this work we consider the IVP
xs = ϕ ∈ X
T0∗(t − τ )C(τ )U (τ, s)ϕdτ
U (t, s)ϕ = T0(t − s)ϕ +
introduced by Perron in his classical paper on stability in a finite-
ẋ(t) = L(t)xt, t ≥ s,
t
u(s) = ϕ ∈ D(A(s)),
Theorem 2.1. Let, for X, T0(t) and C(t) as described above, U (t, s)
denote the evolutionary system defined by the AIE (3). Then x(t)
defined by
t ≥ s,
then, by the general theory we have that
n
o
D(A(s)) = φ ∈ X : φ̇ ∈ X, φ̇(0) = L(s)φ ,
s, t ∈ R
x(s + θ) = ϕ(θ),
−h ≤ θ ≤ 0,
(4)
t≥s
(5)
x(t) = (U (t, s)ϕ)(0),
A(s)φ = φ̇.
satisfies (2), and conversely, if x is a solution of the RFDE (2) sat-
Thereafter, notice that the Equation (1) enters in the definition of de
isfying the initial condition (4), then for t ≥ s and −h ≤ θ ≤ 0, we
domain of A, this means that to change Equation (1) is to change
have
the domain of the infinitesimal generator. This would lead to techni(U (t, s)ϕ)(θ) =
cal complications if we want to relate solutions of various equations
to each other by means of a variation-of-constants formula. In order to solve this problem and principally to ensure the existence of
an evolutionary family of operators {U (t, s)}t≥s which gives a solution xt of (1), we use the (sun-star) ∗-framework developed by
Clement et. al [3] which gives a variation-of-constant formula that
is the core of a the good perturbation theory. This theory is a kind
of extrapolation theory for strongly continuous semigroups and we

x(t + θ),
unique bounded solution xt ∈ C([−h, 0], Cn) to the following integral
equation
(2)
we say that an evolution family {U (t, s)}t≥s have exponential dichotomy on R (with a constant α > 0) if there exists a projectionvalued function P : R → B(X), such that the function t 7→ P (t)ϕ
is continuous and bounded for each ϕ ∈ X, and for some constant
M = M (α) > 0, the following holds:
1. P (t)U (t, s) = U (t, s)P (s);
2. The restriction U (t, s)|Im Q(s) is invertible as an operator from
4. kU (t, s)Q(s)k ≤ M e−α(s−t), for t ≤ s.
T0∗(t − τ )C(τ )xτ dτ.
xt = T0(t − s)ϕ +
s
References
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Math. Ann. 277 (1987), no. 4, 709–725.
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.
t
Z
[1] Coppel, W. A. Dichotomies in stability theory. Lecture Notes in Mathematics,
has a unique solution for every bounded forcing function f . Where
3. kU (t, s)P (s)k ≤ M e−α(t−s), for s ≤ t;
t + θ ≥ s.
tial dichotomy if and only if for every f ∈ BC(R, Cn) there exists a
tem (1) has the property of exponential dichotomy if and only if the
Im Q(s) to Im Q(t);
(6)
Theorem 2.2. The evolutionary family {U (t, s)}t≥s has an exponen-
conditions. Using these tools we show that the solution of the sys-
ẋ(t) = L(t)xt + f (t), t ≥ s
t + θ ≤ s,
The next theorem is our main result.
can use it for evolutionary system with two parameters under some
corresponding inhomogeneus equation


ϕ(t − s + θ),