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Shadowing, Hyers--Ulam stability and hyperbolicity for nonautonomous linear delay differential equations

Lucas Backes, Davor Dragicevic, Mihaly Pituk

Abstract

It is known that hyperbolic non\-autonomous linear delay differential equations in a finite dimensional space are Hyers--Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.

Shadowing, Hyers--Ulam stability and hyperbolicity for nonautonomous linear delay differential equations

Abstract

It is known that hyperbolic non\-autonomous linear delay differential equations in a finite dimensional space are Hyers--Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.
Paper Structure (3 sections, 5 theorems, 116 equations)

This paper contains 3 sections, 5 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of the main theorem

Key Result

Theorem 1

BDPS If Eq. LDE has an exponential dichotomy, then it is Hyers--Ulam stable and hence shadowable.

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • proof : Proof of Theorem \ref{['T2']}
  • Claim 1
  • ...and 27 more