Linear autonomous equations

Abstract

In this chapter, we consider a special but important class of linear equations; namely, linear autonomous equations. A linear autonomous RFDE has the form

$$ \dot x(t) = L({x_t}) $$

((1))

where L is a continuous linear function mapping C into ℝⁿ. This hypothesis implies there exists an n × n matrix η(θ) -r ≤ θ ≤ 0, whose elements are of bounded variation such that

$$ \matrix{ {L(\phi ) = \int_{ - r}^0 {[d\eta (\theta )]\phi (\theta )} ,\,} & {\phi {\rm{in}}C.} \cr } $$

((2))