# ON WEIGHTED MEASURES AND A NEUTRAL FUNCTIONL DIFFERENTIAL EQUATLON

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

$$\frac{\text{d}}{\text{dt}}\text{(\mu *x(t)+f(t))=v*x(t)+g(t),}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{t}\epsilon {\mathbb{R}}^{+}$$ (1.1)

$$\text{x(t})=\varphi (\text{t)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{t}\in {\mathbb{R}}^{-}$$ (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 ...

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