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Exponential Ordering for Neutral Functional Differential Equations With Non-Autonomous Linear D-Operator

Rafael Obaya, Víctor M. Villarragut

Abstract

We study neutral functional differential equations with stable linear non-autonomous $D$-operator. The operator of convolution $\hat{D}$ transforms $BU$ into $BU$. We show that, if $D$ is stable, then $\hat{D}$ is invertible and, besides, $\hat{D}$ and $\hat{D}^{-1}$ are uniformly continuous for the compact-open topology on bounded sets. We introduce a new transformed exponential order and, under convenient assumptions, we deduce the 1-covering property of minimal sets. These conclusions are applied to describe the amount of material in a class of compartmental systems extensively studied in the literature.

Exponential Ordering for Neutral Functional Differential Equations With Non-Autonomous Linear D-Operator

Abstract

We study neutral functional differential equations with stable linear non-autonomous -operator. The operator of convolution transforms into . We show that, if is stable, then is invertible and, besides, and are uniformly continuous for the compact-open topology on bounded sets. We introduce a new transformed exponential order and, under convenient assumptions, we deduce the 1-covering property of minimal sets. These conclusions are applied to describe the amount of material in a class of compartmental systems extensively studied in the literature.
Paper Structure (6 sections, 28 theorems, 97 equations)

This paper contains 6 sections, 28 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Non-Autonomous stable linear $D$-operators
  4. Transformed exponential order and structure of omega-limit sets
  5. Neutral compartmental systems
  6. Some specific neutral compartmental systems

Key Result

Lemma 3.1

For each $\omega\in\Omega$, there exists an $m\times m$ matrix $\mu(\omega)=[\mu_{ij}(\omega)]_{ij}$ of real Borel regular measures with finite total variation such that

Theorems & Definitions (50)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • Theorem 3.6
  • ...and 40 more