# Mixed-type Functional Differential Equations

Part of the Operator Theory: Advances and Applications book series (OT, volume 182)

## Abstract

In this chapter we study linear functional differential equations of the form
$$x'\left( t \right) = \int_{ - q}^p {d\eta \left( \theta \right)x\left( {t + \theta } \right) + h\left( t \right),}$$
(8.1)
where −q<0<p, x(t) ∈ ℂ M , and dη(θ) is an M×M matrix of finite (complex-valued) Lebesgue-Stieltjes measures on [−q, p]. Equation (8.1) is called of mixed type if the measure matrix dη(θ) is supported on both of the subintervals [0, p] and [−q, 0]. As an initial condition we assume x(t) to be known for t∈[−q, p]:
$$x\left( t \right) = \phi \left( t \right), - q \leqslant t \leqslant p.$$
The special case studied most has the form
$$x'\left( t \right) = \sum\limits_{j = 1}^N {A_j x\left( {t + r_j } \right) + h\left( t \right),}$$
(8.2)
where ∼r1,…,r N } is a subset of [−q, p] consisting of discrete shifts and A1,…,A N are complex M×M matrices. Here the measure matrix $$d\eta \left( \theta \right) = \sum\nolimits_{j = 1}^N {\delta \left( {\theta - r_j } \right)Aj}$$ is discrete. Equations (8.1) and (8.2) are called autonomous, because dη(θ) does not depend on t∈[−q, p].

## Keywords

Bounded Linear Operator Functional Differential Equation Vertical Strip Exponential Dichotomy Delay Equation
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