Volterra and Functional Differential Equations

ON WEIGHTED MEASURES AND A NEUTRAL FUNCTIONL DIFFERENTIAL EQUATLON

Olof J. Staffans

Heisinki University of Technology Institute of Mathematics Otaniemi, Finland

DOI: 10.1201/9781003420026-11

1. INTRODUCTION

We study the linear, autonomous, neutral system of functional differential equations

$$\frac{\text{d}}{\text{dt}}\text{(μ*x(t)+f(t))=v*x(t)+g(t),}\text{t}\epsilon {ℝ}^{+}$$ (1.1)

$$\text{x(t})=\varphi (\text{t)},\text{t}\in {ℝ}^{-}$$ (1.2)

Here ℝ⁺=[0,∞),R^- =[-∞,0],μ and v are matrix-valued measures on ℝ⁺, finite with respect to a weight function, and f, g and ϕ are continuous and satisfy certain growth candicions as $\text{f}+\text{k}=$. We give conditions which imply that solutions of $\left(1,1\right),\left(1,2\right)$ can be decomposed into components with different exponential growth rate. Smilar results have earlier ...