# Mixed-type Functional Differential Equations

Chapter

## Abstract

In this chapter we study linear functional differential equations of the form where − The special case studied most has the form where ∼

$$
x'\left( t \right) = \int_{ - q}^p {d\eta \left( \theta \right)x\left( {t + \theta } \right) + h\left( t \right),}
$$

(8.1)

*q*<0<*p*,*x*(*t*) ∈ ℂ^{ M }, and*d*η(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*d*η(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*]:$$
x\left( t \right) = \phi \left( t \right), - q \leqslant t \leqslant p.
$$

$$
x'\left( t \right) = \sum\limits_{j = 1}^N {A_j x\left( {t + r_j } \right) + h\left( t \right),}
$$

(8.2)

*r*_{1},…,*r*_{ N }} is a subset of [−*q, p*] consisting of discrete shifts and*A*_{1},…,*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*d*η(θ) does not depend on*t*∈[−*q, p*].## Keywords

Bounded Linear Operator Functional Differential Equation Vertical Strip Exponential Dichotomy Delay Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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