Page 87 - Textos de Matemática Vol. 36

Appendix A
A.1 Basic results for linear equations
In this section we recall some well-known facts about first order linear systems of differential equations with piecewise continuous, locally bounded coefficients and, in the case of constant coefficients, about the corresponding algebraic equations.
Throughout the whole section we assume that A,B:R→Cn×n and b:R→Cn
are piecewise continuous, locally bounded and that x0 ∈ Cn, t0 ∈ R.
First we cite without proof some properties of linear systems of the form
x˙ = A(t)x + b(t), x(t0) = x0,
and of the corresponding homogeneous system x˙ = A(t)x;
(A.1) (A.2)
(A.3)
these results can be found in many textbooks on ordinary differential equations. We agree that a solution of the differential equation (A.1) is a continuous
and piecewise differentiable function φ which satisfies φ˙(t)=A(t)φ(t)+b(t) for t∈I\E,
where E ⊂ I is a discrete set, which may be empty; similarly we define so- lutions for all other first order systems of differential equations with piecewise continuous, locally bounded coefficients.
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