Revista Colombiana de Matemáticas Volumen 33 (1999), páginas 9–13

A remark on exponential dichotomies

Raúl Naulin^{∗}

Universidad de Oriente, Cumaná, Venezuela

Abstract. A proof of the existence of an exponential dichotomy for the linear system x′(t) = A(t)x(t) is given, based on the admissibility of the pair (B(∞), B_{A}(∞)), where B(∞) is the space of continuous functions on the semiaxis J = [0, ∞), values in ℂ^{n} and having a limit as t → ∞, and B_{A}(∞) is the space of bounded functions f on J such that A^{−1}f ∈ B(∞). Keywords and phrases. Exponential dichotomies, admissibility.

1991 Mathematics Subject Classification. Primary 34A30.

1. Introduction

In this paper we consider the system of differential equations

x′(t) = A(t)x(t) + f(t), t ∈ J := [0, ∞),  (1)

where A(t) is a continuous matrix function with complex entries. The function f(t) belongs to a functional space we will define in the course of the paper.

Definition 1. Let C and D be function spaces. We say that the pair (C, D) is admissible for equation (1) if for each f in the space D there exists a solution of (1) belonging to C.

∗Supported by Proyecto CI-5-025-00730/95

Admissible pairs are important in the theory of differential equations (see [1], [2]), as they define the dichotomic behavior of the linear system

x′(t) = A(t)x(t).  (2)

Definition 2. We say that equation (2) has an exponential dichotomy on J, if there exist a fundamental matrix Φ of (2), a projection matrix P (i.e., P P = P) and positive constants K, α such that

\begin{aligned}|Φ(t)P Φ^{−1}(s)| & ≤ Keα(s − t), t ≥ s ≥ 0, \\ |Φ(t)(I − P)Φ^{−1}(s)| & ≤ Ke^{α(t−s)}, s ≥ t ≥ 0. \end{aligned}  (3)

In this paper, we are concerned with the following classical result [1]: Theorem A. Equation (2) has an exponential dichotomy on J if for any bounded and continuous function f(t) on J, equation (1) has at least one bounded solution.

The aim of this paper is the characterization of exponential dichotomy by means of the admissibility of a pair of spaces of functions with limit at infinity.

2. Preliminaries

We will make use of the following spaces

B := {f : J → ℂ^{n} : f is bounded and continuous} ,
B(∞) := {f ∈ B : lim_{t→∞} f(t) exists} .

We call B(∞) the space of functions with limit at infinity. These spaces, endowed with the norm |f|_{∞} = sup{|f(t)| : t ∈ J}, become Banach spaces. Furthermore, if F : J → ℂ^{n×n} and F (t) is invertible for each t ∈ J, we define

B_{F} (∞) := {f ∈ B : F^{−1}f ∈ B(∞)} .

To this space we give the norm |f|_{F} = |F^{−1}f|_{∞}. Provided that F is bounded on J, also B_{F} (∞) is a Banach space. If equation (2) has an exponential dichotomy, then for any f ∈ B, equation (1) has the following bounded solution:

x_{f}(t) = ∫_{0}^{t} Φ(t)P Φ(s)f(s) ds − ∫_{t}^{∞} Φ(t)(I − P)Φ^{−1}(s)f(s) ds.

Let us introduce the following Green function:

G(t, s) = \begin{cases}Φ(t)P Φ(s), & t ≥ s, \\ −Φ(t)(I − P)Φ^{−1}(s), & s > t. \end{cases}  (4)

By means of this function we can write the solution x_{f} in the form:

x_{f}(t) = ∫_{J} G(t, s)f(s)ds.  (5)

If A(t) is a bounded function, we will use the following identity:

Φ(t)P Φ^{−1}(0) − I = ∫_{J} G(t, s)A(s)ds.  (6)

3. The main result

Theorem 1. If the function A(t) is bounded on J and the matrix A(t) is invertible for each t ∈ J, then the following assertions are equivalent: (A) The pair (B, B) is admissible. (B) The pair (B(∞), B_{A}(∞)) is admissible. (C) Equation (2) has an exponential dichotomy on J.

Proof. (A) ⇔ (C). This follows from Theorem A. We observe that this equivalence holds without the requirements of invertibility of the matrices A(t) or the boundedness of the function A(t). (A) ⇒ (B). Let f ∈ B_{A}(∞). Since f ∈ B_{A}, formula (5) makes sense. Therefore x_{f} defines a solution of (1) belonging to B(∞). We have to prove that lim_{t→∞} x_{f} exists. Using (6) we may write

x_{f}(t) = −A^{−1}(t)f(t) + Φ(t)P Φ^{−1}(0)A^{−1}(t)f(t) + I_{1}(t) + I_{2}(t),  (7)

where

I_{1}(t) := ∫_{0}^{t} G(t, s)A(s) [A^{−1}(s)f(s) − A^{−1}(t)f(t)] ds,
I_{2}(t) := ∫_{t}^{∞} G(t, s)A(s) [A^{−1}(s)f(s) − A^{−1}(t)f(t)] ds.

Taking into account (3), we can estimate I_{i}, i = 1, 2. We have

\begin{aligned}|I_{1}(t)| & ≤ ∫_{0}^{t} |Φ(t)P Φ^{−1}(s)||A(s)||A^{−1}(s)f(s) − A^{−1}(t)f(t)| ds \\ & ≤ ∫_{0}^{t/2} ⋯ + ∫_{t/2}^{t} ⋯ \\ & ≤ 2K|A|_{∞}α^{−1}e^{− \frac{α}{2} t}|A^{−1}f|_{∞} \\ & + α^{−1}K sup_{s∈[t/2, t]} |A^{−1}(s)f(s) − A^{−1}(t)f(t)| \end{aligned}  (8)

and

\begin{aligned}|I_{2}(t)| & ≤ |A|_{∞} ∫_{t}^{∞} e^{α(t−s)}|A^{−1}(s)f(s) − A^{−1}(t)f(t)|ds \\ & ≤ 2α^{−1}|A|_{∞}K sup_{s∈[t, ∞)} |A^{−1}(s)f(s) − A^{−1}(t)f(t)|. \end{aligned}  (9)

Since f ∈ B_{A}(∞), it is clear from (8) and (9) that lim_{t→∞} I_{i}(t) = 0. From (7) we obtain that lim_{t→∞} x_{f}(t) = −(A^{−1}f)(∞). Therefore, the function pair (B(∞), B_{A}(∞)) is admissible.

(B) ⇒ (A). Let S be the subspace of ℂ^{n} of values of initial conditions of solutions of equation (2) belonging to B(∞), and let U be a supplementary subspace of S. We have the direct sum ℂ^{n} = S ⨁ U. Then it is easy to prove that equation (1) has, for any f ∈ B_{A}(∞), a unique solution, which we denote by T (f), that belongs to B(∞) and is such that the initial condition satisfies T (f)(0) ∈ U. It is also easy to verify that this correspondence is linear. Thus, we define this way a linear map T : B_{A}(∞) → B(∞) such that T (f) satisfies (1) and T (f)(0) ∈ U. This map has a closed graph (the proof of this assertion is exactly the same as that of Proposition 3.4 in [1]). Therefore, it is bounded, i.e., there exists a constant M, such that

|T (f)| ≤ M|f|_{A}.  (10)

Let f ∈ B and for each n = 1, 2, ..., let θ_{n}(t) be a continuous function such that |θ_{n}|_{∞} = 1, θ_{n}(t) = 1 if t ∈ [0, n] and θ_{n}(t) = 0 if t ≥ n + 1. Let {f_{n}} be the sequence in B_{A} defined by

f_{n}(t) = θ_{n}(t)f(t).  (11)

For each function f_{n}, we consider the solution x_{n} = T (f_{n}) of the equation

x′(t) = A(t)x(t) + f_{n}(t).  (12)

According to (10), for any index n we have

|x_{n}|_{∞} ≤ M|f_{n}|_{A} ≤ M|f|_{A}.  (13)

From (12) and (13) we obtain that the sequences {x_{n}} and {x′_{n}} are bounded on any compact subinterval of J. From the Ascoli-Arzelá theorem, there exists then a subsequence {x_{n}^{1}} of {x_{n}} uniformly convergent on [0, 1] to a continuous function u_{1} on the interval [0, 1]. By the same argument, there exists a subsequence {x_{n}^{2}} of {x_{n}^{1}} converging uniformly on the interval [0, 2] to a continuous function u_{2} such that u_{1} = u_{2} on [0, 1]. Carrying out this process iteratively, we obtain, for any natural number N, a subsequence {x_{n}^{N}} of {x_{n}^{N−1}}, converging uniformly to a continuous function u_{N} on the interval [0, N], and such that u_{N} = u_{N−1} on [0, N −1]. Defining u(t) = u_{N}(t) if t ∈ [0, N], we obtain that the diagonal sequence {x_{n}^{n}} converges uniformly to u on each compact subinterval of J. From (13), we obtain that u ∈ B. From (11) and (12) it follows that u satisfies u′ = A(t)u + f. This means that u is a solution of (1) in the space B.

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References

[1] W. A. Coppel, Dichotomies in Stability Theory, Lectures Notes in Mathematics, 629, Springer-Verlag, Berlin, 1978.

[2] J. L. Massera, J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, 1966.

Departamento de Matemáticas

Universidad de Oriente Cumaná 6101 A-285, Venezuela