Stability of linear differential equations with a distributed delay

Leonid Berezansky,Elena Braverman

doi:10.3934/cpaa.2011.10.1361

Abstract

We present some new stability results forthe scalar linear equation with a distributed delay$\dot{x}(t) + \sum_{k=1}^m \int_{h_k(t)}^t x(s)d_s R_k(t,s) =0, h_k(t)\leq t,$ su$p_{t\geq 0}(t-h_k(t))where the functions involved in the equation are not required to be continuous.The results are applied to integro-differential equations,equations with several concentrated delays and equations of a mixed type.

Highlights

It is usually believed that equations with a distributed delay provide a more realistic description for models of population dynamics and mathematical biology in general, see, for example, [1]
Volterra considered the logistic equation with a distributed delay in 1926 [2], before the Hutchinson’s equation was introduced in 1948 [3]
To the best of our knowledge the first systematic study of equations with a distributed delay can be found in the monograph of Myshkis [4], the results obtained by 1993 are summarized in the book of Kuang [1]

Summary

Introduction

It is usually believed that equations with a distributed delay provide a more realistic description for models of population dynamics and mathematical biology in general, see, for example, [1]. Equations with a distributed delay were studied even before relevant models with concentrated delays appeared. To the best of our knowledge the first systematic study of equations with a distributed delay can be found in the monograph of Myshkis [4], the results obtained by 1993 are summarized in the book of Kuang [1]. Equations with a distributed delay are intensively studied. Most of the obtained results are not relevant for non-autonomous models and do not involve equations with a concentrated delay as a special case. Let us notice that in the present paper we study delay equations in the most general framework from the following points of view.

LEONID BEREZANSKY AND ELENA BRAVERMAN

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